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https://arxiv.org/abs/1703.02629 | Online Learning Without Prior Information | The vast majority of optimization and online learning algorithms today require some prior information about the data (often in the form of bounds on gradients or on the optimal parameter value). When this information is not available, these algorithms require laborious manual tuning of various hyperparameters, motivating the search for algorithms that can adapt to the data with no prior information. We describe a frontier of new lower bounds on the performance of such algorithms, reflecting a tradeoff between a term that depends on the optimal parameter value and a term that depends on the gradients' rate of growth. Further, we construct a family of algorithms whose performance matches any desired point on this frontier, which no previous algorithm reaches. | \section{Problem Definition and Prior Work}
Data streams, large datasets, and adversarial environments require online optimization algorithms, which continually adapt model parameters to the data. At iteration $t$, these algorithms pick a point $w_t\in W$, are presented with a loss function $\ell_t:W\to \mathbb{R}$, and suffer loss $\ell_t(w_t)$. The algorithm's performance is measured by \emph{regret}, which is defined as the loss relative to some comparison point $u$:
\begin{align*}
R_T(u)&= \sum_{t=1}^T \ell_t(w_t)-\ell_t(u)
\end{align*}
When $W$ is a convex set and the $\ell_t$ are guaranteed to be convex, the regret can be minimized using only information about the gradients of $\ell_t$ at $w_t$, leading to simple and efficient algorithms.
All online convex optimization algorithms require either a bound $B$ on the diameter of $W$ or a bound $L_{\max}$ on the gradients of $\ell_t$, or suffer a penalty that is exponential in gradients' rate of growth when no information is given \citep{cutkosky2016online}. When $B$ is known but $L_{\max}$ is unknown, there are algorithms that can obtain regret $O(BL_{\max}\sqrt{T})$ \citep{duchi10adagrad,mcmahan2010adaptive}. Conversely, when $B$ is infinite (e.g. $W$ is an entire vector space) but $L_{\max}$ is known, there are algorithms that obtain $O(\|u\|L_{\max}\sqrt{T\log(\|u\|T)})$ or $O(\|u\|L_{\max}\sqrt{T}\log(\|u\|T))$ regret \citep{mcmahan2012no,orabona2013dimension, mcmahan2013minimax,orabona2014simultaneous, orabona2016coin}. The situation does not improve when both $B$ and $L_{\max}$ are known. In this case it is impossible to do better than $O(BL_{\max} \sqrt{T})$, so knowing just one of these parameters is essentially as good as knowing both \citep{abernethy2008optimal}. In the case where no prior information is given, it was recently proved by \citet{cutkosky2016online} that the regret must contain an additional exponential penalty $\exp(\max_t\sqrt{L_t/L_{t-1}})$,
where $L_t$ is the maximum gradient observed by iteration $t$.
The case in which we have no bound on either $B$ or $L_{\max}$ is common in practice. A standard pragmatic approach to this lack of information is to simply make a guess for these parameters and then apply an algorithm that uses the guess as input, but this approach is theoretically unsound in online learning, and rather laborious and inelegant in general. We explore lower bounds and algorithms that adapt to the unknown quantities in a principled way in this paper.
Where no information is given, we prove that there is a frontier of matching lower and upper bounds on $R_T(u)$ that trades-off a $\|u\|L_{\max}\sqrt{T}\log(\|u\|T)$ term with a $\exp(\max_tL_t/L_{t-1})$ term along two dimensions, which we parametrize by $k$ and $\gamma$.\footnote{The square root is missing from the exponential term because we improved the lower bound given in \citet{cutkosky2016online} (see Section \ref{sec:lowerbound}).} Along the first dimension, the exponential penalty is reduced to $\exp((L_t/L_{t-1})/k^2)$ for any $k>0$ at the expense of rescaling the regret's $\sqrt{T}$ term to $k\|u\|L_{\max}\sqrt{T}\log(\|u\|T)$. Along the second dimension, the logarithm's power in the $\sqrt{T}$ term is reduced to $\|u\|L_{\max}\sqrt{T}\log^\gamma(\|u\|T)$ for any $\gamma\in (1/2,1]$ at the expense of increasing the exponential penalty to $\exp((L_t/L_{t-1})^{1/(2\gamma-1)})$. We prove the lower bounds by constructing a specific adversarial loss sequence, and we prove the upper bounds by providing a family of algorithms whose regret matches the lower bound frontier for any $k$ and $\gamma$.
\section{Notation and Setup}\label{sec:background}
Before proceeding further, we provide a few definitions that will be useful throughout this paper. A set $W$ is a \emph{convex set} if $W$ is a subset of some real vector space and $tx+(1-t)y\in W$ for all $x,y\in W$ and $t\in[0,1]$. Throughout this paper we will assume that $W$ is closed. A function $f$ is a \emph{convex function} if $f(tx+(1-t)y)\ge tf(x)+(1-t)f(y)$ for all $x,y$ and $t\in[0,1]$. If $f:V\to \mathbb{R}$ for some vector space $V$, then a vector $g\in V^*$ is a \emph{subgradient} of $f$ at $x$, denoted $g\in \partial f(x)$, if $f(y)\ge f(x)+g\cdot (y-x)$ for all $y$. Here we use the dot product to indicate application of linear functionals in the dual space since this should cause no confusion. A \emph{norm} $\|\cdot\|$ is a function such that $\|x\|=0$ if and only if $x=0$, $\|cx\|=|c|\|x\|$ for any scalar $c$, and $\|x+y\|\le \|x\|+\|y\|$ for all $x$ and $y$. The \emph{dual norm} is a norm $\|\cdot\|_\star$ defined by $\|x\|_\star = \sup_{\|y\|=1}x\cdot y$. As a special case, when $\|x\|=\sqrt{x\cdot x}$ (the $L_2$ norm), then $\|\cdot\|_\star=\|\cdot\|$.
Online convex optimization problems can be reduced to online \emph{linear} optimization problems in which the loss functions are constrained to be linear functions. The reduction follows by replacing the loss function $\ell_t(w)$ with the linear function $g_t\cdot w$, where $g_t$ is a subgradient of $\ell_t$ at $w_t$. Then, by definition, $g_t\cdot w_t-g_t\cdot u\ge \ell_t(w_t)-\ell_t(u)$. Therefore the regret of our algorithm with respect to the linear loss functions $g_t\cdot w$ is an upper-bound on the regret with respect to the real loss functions $\ell_t$. Because of this reduction, many online convex optimization algorithms (including ours) are \emph{first order} algorithms, meaning they access the loss functions only through their subgradients. For the rest of this paper we will therefore assume that the losses are linear, $\ell_t(w)=g_t\cdot w$.
We will focus all of our lower bounds in Section \ref{sec:lowerbound} and algorithms in Section \ref{sec:algorithms} on the case in which the domain $W$ is an entire Hilbert space, so that $W$ has infinite diameter and no boundary. This case is very common in practical optimization optimization problems encountered in machine learning, in which any constraints are often only implicitly enforced via regularization. Our objective is to design lower bounds and algorithms such that $R_T(u)$ depends on $\|u\|$, $T$, and $L_{\max}$ without prior knowledge of these parameters.
In the following sections we use a compressed-sum notation where subscripts with colons indicate summations: $\sum_{t=1}^T g_t=g_{1:T}$, $ \sum_{t=1}^T \|g_t\|^2=\|g\|^2_{1:T}$, $\sum_{t=1}^T g_tw_t=(gw)_{1:T}$ and similarly for other indexed sums. Proofs are in the appendix when they do not immediately follow the result.
\section{A Frontier of Lower Bounds}\label{sec:lowerbound}
In this section we give our frontier of lower bounds for online optimization without prior information. First we describe our adversarial loss sequence and lower bound frontier along the $k$ dimension, and then we extend the argument to obtain the full two dimensional frontier parametrized by both $k$ and $\gamma$.
\subsection{Trade-offs in the multiplicative constant $k$}
Given an algorithm, we establish a lower bound on its performance by constructing an adversarial sequence of subgradients $g_t\in \mathbb{R}$. This sequence sets $g_t=-1$ for $T-1$ iterations, where $T$ is chosen adversarially but can be made arbitrarily large, then sets $g_T=O(k\sqrt{T})$. Perhaps surprisingly, we prove that this simple strategy forces the algorithm to experience regret that is exponential in $\sqrt{T}/k$. We then express $\sqrt{T}/k$ as a constant multiple of $\tfrac{1}{k^2}L_t/L_{t-1}$, where $L_t=\max_{t'\le t}|g_t|$, capturing the algorithm's sensitivity to the big jump in the gradients between $T-1$ and $T$ in the adversarial sequence.
The cost that an algorithm pays when faced with the adversarial sequence is stated formally in the following Theorem.
\begin{Theorem}\label{thm:lowerboundkonly}
For any $k>0$, $T_0>0$, and any online optimization algorithm picking $w_t\in \mathbb{R}$, there exists a $T>T_0$, a $u\in \mathbb{R}$, and a fixed sequence $g_t\in \mathbb{R}$ on which the regret is:
\begin{align*}
R_T(u)&=\sum_{t=1}^T g_tw_t-g_tu\\
&\ge k\|u\|L_{\max} \log(T\|u\|+1)\sqrt{T} + \frac{L_{\max}}{T-1}\exp\left(\frac{\sqrt{T-1}}{8k}\right)\\
&\ge k\|u\|L_{\max} \log(T\|u\|+1)\sqrt{T} + \max_{t\le T}L_{\max}\frac{L_{t-1}^2}{\|g\|^2_{1:t-1}}\exp\left[\frac{1}{2}\left(\frac{L_t/L_{t-1}}{288 k^2}\right)\right]
\end{align*}
where $L_t=\max_{t'\le t} \|g_{t'}\|$, and $L_{\max}=L_T=\max_{t\le T}\|g_t\|$.
\end{Theorem}
The first inequality in this bound demonstrates that it is impossible to guarantee sublinear regret without prior information while maintaining $O(L_{\max}\|u\|\log(\|u\|))$ dependence on $L_{\max}$ and $\|u\|$,\footnote{it \emph{is} possible to guarantee sublinear regret in exchange for $O(L_{\max}\|u\|^2)$ dependence, see \citet{orabona2016scale}} but the second inequality provides hope that if the loss sequence is limited to small jumps in $L_t$, then we might be able to obtain sublinear regret. Specifically, from the first inequality, observe that in order to bring the exponential term to lower than $O(T)$, the value of $k$ needs to be at least $\Omega(\sqrt{T}/\log(T))$, which causes the non-exponential term to become $O(T)$. However, the second inequality emphasizes that our high regret is the result of a large jump in the value of $L_t$, so that we might expect to do better if there are no such large jumps. Our upper bounds are given in the form of algorithms that guarantee regret matching the second inequality of this lower bound for any $k$, showing that we can indeed do well without prior information so long as $L_t$ does not increase too quickly.
\subsection{Trade-offs in the Logarithmic exponent $\gamma$}
To extend the frontier to the $\gamma$ dimension, we modify our adversarial sequence by setting $g_T = O(\gamma k^{1/\gamma} T^{1-1/2\gamma})$ instead of $O(k\sqrt{T})$. This results in a penalty that is exponential in $(\sqrt{T}/k)^{1/\gamma}$, which we express as a multiple of $(L_t/\gamma k^2L_{t-1})^{1/(2\gamma-1)}$. Since $\gamma\in(1/2,1]$, we are getting a larger exponential penalty even though the adversarial subgradients have decreased in size, illustrating that decreasing the logarithmic factor is very expensive.
The full frontier is stated formally in the following Theorem.
\begin{restatable}{Theorem}{lowerbound}\label{thm:lowerbound}
For any $\gamma\in(1/2,1]$, $k>0$, $T_0>0$, and any online optimization algorithm picking $w_t\in \mathbb{R}$, there exists a $T>T_0$, a $u\in \mathbb{R}$, and a sequence $g_1,\dots, g_T\in \mathbb{R}$ with $\|g_t\|\le \max(1,18\gamma(4k)^{1/\gamma}(t-1)^{1-1/2\gamma})$ on which the regret is:\footnote{The same result holds with in expectation for randomized algorithms with a deterministic sequence $g_t$.}
\begin{align*}
R_T(u)&=\sum_{t=1}^T g_tw_t-g_tu\\
&\ge k\|u\|L_{\max} \log^\gamma(T\|u\|+1)\sqrt{T} + \frac{L_{\max}}{T-1}\exp\left(\frac{(T-1)^{1/2\gamma}}{2(4k)^{1/\gamma}}\right)\\
&\ge k\|u\|L_{\max} \log^\gamma(T\|u\|+1)\sqrt{T} + \max_{t\le T}L_{\max}\frac{L_{t-1}^2}{\|g\|^2_{1:t-1}}\exp\left[\frac{1}{2}\left(\frac{L_t/L_{t-1}}{288\gamma k^2}\right)^{1/(2\gamma-1)}\right]
\end{align*}
where $L_t=\max_{t'\le t} \|g_{t'}\|$ and $L_{\max}=L_T=\max_{t\le T}\|g_t\|$.
\end{restatable}
Again, the first inequality tells us that adversarial sequences can always deny the algorithm sublinear regret and the second inequality says that so long as $L_t$ grows slowly, we can still hope for sublinear regret. This time, however, the second inequality appears to blow up when $\gamma\to 1/2$. In this case, $L_{\max}=O(k^2)$ regardless of $T$ and so the value of $L_t/L_{t-1}$ is never very large, keeping the exponent in the second inequality less than 1 so that the singularity in the exponent does not send the bound to infinity.
This singularity at $\gamma=1/2$ tells us that the adversary does not need to be ``very adversarial'' in order to force us to experience exponential regret.
To gain some more intuition for what happens at $\gamma=1/2$, consider a model in which the adversary must commit ahead of time to some $L_{\max}$ (which corresponds to picking $k$), unknown to the optimization algorithm, such that $\|g_t\|\le L_{\max}$ for all $t$. When a bound $L_{\text{bound}}\ge L_{\max}$ is known to the algorithm ahead of time, then it is possible to achieve $O(\|u\| L_{\text{bound}}\sqrt{T\log(\|u\|T)})$ regret (e.g. see \citet{orabona2016coin}). However, note that when $\gamma=1/2$, committing to an appropriate $L_{\max}$ would not prevent an adversary from using the sequence of Theorem \ref{thm:lowerbound}. Therefore, Theorem \ref{thm:lowerbound} tells us that algorithms which achieve $O(\|u\| L_{\text{bound}}\sqrt{T\log(\|u\|T)})$ regret are inherently very fragile because if the bound is incorrect (which happens for large enough $k$), then the adversary can force the algorithm to suffer $L_{\max}\exp(O(T/L_{\max}))$ regret for arbitrarily large $T$.
Continuing with the model in which the adversary must commit to some unknown $L_{\max}$ ahead of time, suppose we are satisfied with $O(\|u\|L_{\max}\sqrt{T}\log^\gamma(\|u\|T))$ regret for some $\gamma>1/2$. In this case, after some (admittedly possibly very large) number of iterations, the exponential term in the second inequality no longer grows with $T$, and the adversarial strategy of Theorem \ref{thm:lowerbound} is not available because this strategy requires a choice of $L_{\max}$ that depends on $T$. Therefore an algorithm that guarantees regret matching the second inequality for some $k$ and $\gamma$ will obtain an asymptotic dependence on $T$ that is only $\log^\gamma(T)\sqrt{T}$.
These lower bounds show that there is a fundamental frontier of tradeoffs the between parameters $\gamma$ and $k$ and the exponential penalty. Now we proceed to derive algorithms that match any point on the frontier without prior information.
\section{Regret Analysis without Information}\label{sec:analysis}
In this section we provide the tools used to derive algorithms whose regret matches the lower bounds in the previous section.
Our algorithms make use of the Follow-the-Regularized-Leader (FTRL) framework, which is an elegant and intuitive way to design online learning algorithms (see \citet{shalev2011online,mcmahan2014survey} for detailed discussions). After seeing the $t^{th}$ loss of the online learning game, an FTRL algorithm chooses a function $\psi_t$ (called a \emph{regularizer}), and picks $w_{t+1}$ according to:
\begin{align*}
w_{t+1} &= \mathop{\text{argmin}}_{w\in W} \psi_t(w) +\sum_{t'=1}^t\ell_{t'}(w)
\end{align*}
Careful choice of regularizers is obviously crucial to the success of such an algorithm, and in the following we provide simple conditions on $\psi$ sufficient for FTRL to achieve optimal regret without prior information. Our analysis generalizes many previous works for online learning with unconstrained $W$ (e.g. \citet{orabona2013dimension, orabona2014simultaneous,cutkosky2016online}) in which regret bounds were proved via arduous ad-hoc constructions. Further, our techniques improve the regret bound in the algorithm that does not require prior information of \citet{cutkosky2016online}. We note that an alternative set of conditions on regularizers was given in \citet{orabona2016coin} via an elegant reduction to coin-betting algorithms, but this prior analysis requires a known bound on $L_{\max}$.
Our regularizers $\psi_t$ take the form $\psi_t(w) = \frac{k}{a_t\eta_t}\psi(a_tw)$ for some fixed function $\psi$ and numbers $a_t$ and $\eta_t$. The value $k$ specifies the corresponding tradeoff parameter in the lower-bound frontier, while the function $\psi$ specifies the value of $\gamma$. The values for $a_t$ and $\eta_t$ do not depend on $k$ or $\psi$, but are carefully chosen functions of the observed gradients $g_1,\dots,g_t$ that guarantee the desired asymptotics in the regret bound.
\subsection{Generalizing Strong Convexity}
Prior analyses of FTRL often make use of strongly-convex regularizers to simplify regret analysis, but it turns out that strongly-convex regularizers cannot match our lower bounds. Fortunately, there is a simple generalization of strong-convexity that will suffice for our purposes. This generalized notion is very similar to a dual version of the ``local smoothness" condition used in \citet{orabona2013dimension}. We define this generalization of strong-convexity below.
\begin{Definition}
Let $W$ be a convex space and
let $\sigma:W^2\to \mathbb{R}$ by an arbitrary function. We say a convex function $f:W\to \mathbb{R}$ is $\sigma(\cdot,\cdot)$-strongly convex with respect to a norm $\|\cdot\|$ if for all $x,y\in W$ and $g\in \partial f(x)$ we have
\[
f(y)\ge f(x) + g\cdot(y-x) + \frac{\sigma(x,y)}{2}\|x-y\|^2
\]
As a special case (and by abuse of notation), for any function $\sigma:W\to \mathbb{R}$ we define $\sigma(w,z)=\min(\sigma(w),\sigma(z))$ and define $\sigma(\cdot)$-strong convexity accordingly.
\end{Definition}
We'll usually just write $\sigma$-strongly convex instead of $\sigma(\cdot,\cdot)$-strongly convex since our definition is purely a generalization of the standard one. We will also primarily make use of the special case $\sigma(w,z)=\min(\sigma(w),\sigma(z))$.
\subsection{Adaptive regularizers}
Now we present a few definitions that will allow us to easily construct sequences of regularizers that achieve regret bounds without information. Intuitively, we require that our regularizers $\psi_t$ grow super-linearly in order to ensure that $\psi_t(w)+g_{1:t}w$ always has a minimal value. However, we do not want $\psi_t$ to grow quadratically because this will result in $O(\|u\|^2)$ regret. The formal requirements on the shape of $\psi_t$ are presented in the following definition:
\begin{Definition}
Let $W$ be a closed convex subset of a vector space such that $0\in W$. Any differentiable function $\psi:W\to \mathbb{R}$ that satisfies the following conditions:
\begin{enumerate}
\item $\psi(0)=0$.
\item $\psi(x)$ is $\sigma$-strongly-convex with respect to some norm $\|\cdot\|$ for some $\sigma:W\to \mathbb{R}$ such that $\|x\|\ge \|y\|$ implies $\sigma(x)\le \sigma(y)$.
\item For any $C$, there exists a $B$ such that $\psi(x)\sigma(x)\ge C$ for all $\|x\|\ge B$.
\end{enumerate}
is called a \emph{$(\sigma,\|\cdot\|)$-adaptive regularizer}. We also define the useful auxiliary function $h(w) = \psi(w)\sigma(w)$ and by mild abuse of notation, we define $h^{-1}(x) = \max_{h(w)\le x}\|w\|$.
\end{Definition}
We will use adaptive regularizers as building blocks for our FTRL regularizers $\psi_t$, so it is important to have examples of such functions. We will provide some tools for finding adaptive regularizers in Section \ref{sec:algorithms}, but to keep an example in mind for now, we remark that $\psi(w)=(\|w\|+1)\log(\|w\|+1)-\|w\|$ is a $\left(\frac{1}{\|\cdot\|+1},\|\cdot\|\right)$-adaptive regularizer where $\|\cdot\|$ is the $L_2$ norm.
The following definition specifies the sequences $\eta_t$ and $a_t$ which we use to turn an adaptive regularizer into the regularizers used for our FTRL algorithms:
\begin{Definition}\label{dfn:regularizers}
Let $\|\cdot\|$ be a norm and $\|\cdot\|_\star$ be the dual norm ($\|x\|_\star = \sup_{\|y\|=1}x\cdot y$). Let $g_1,\dots,g_T$ be a sequence of subgradients and set $L_t=\max_{t'\le t} \|g_t\|_\star$.
Define the sequences $\frac{1}{\eta_t}$ and $a_t$ recursively by:
\begin{align*}
\frac{1}{\eta_0^2}&=0\\
\frac{1}{\eta_{t}^2} &= \max\left(\frac{1}{\eta_{t-1}^2}+2\|g_t\|_\star^2,L_t\|g_{1:t}\|_\star\right)\\
a_1&=\frac{1}{(L_1\eta_1)^2}\\
a_t&=\max\left(a_{t-1},\frac{1}{(L_t\eta_t)^2}\right)
\end{align*}
Suppose $\psi$ is a $(\sigma,\|\cdot\|)$-adaptive regularizer and $k>0$. Define
\begin{align*}
\psi_t(w)&=\frac{k}{\eta_t a_t}\psi(a_t w)\\
w_{t+1} &= \mathop{\text{argmin}}_{w\in W} \psi_t(w) + g_{1:t}\cdot w\\
\end{align*}
\end{Definition}
Now without further ado, we give our regret bound for FTRL using these regularizers.
\begin{restatable}{Theorem}{parameterfreeregret}\label{thm:parameterfreeregret}
Suppose $\psi$ is a $(\sigma,\|\cdot\|)$-adaptive regularizer and $g_1,\dots,g_T$ is some arbitrary sequence of subgradients. Let $k\ge 1$, and let $\psi_t$ be defined as in Definition \ref{dfn:regularizers}.
Set
\begin{align*}
\sigma_{\text{min}} &= \inf_{\|w\|\le h^{-1}\left(10/k^2\right)} k\sigma(w)\\
D &= \max_{t} \frac{L_{t-1}^2}{(\|g\|_\star^2)_{1:t-1}}h^{-1}\left(\frac{5L_t}{k^2L_{t-1}}\right)\\
Q_T&=2\frac{\|g\|_{1:T}}{L_{\max}}
\end{align*}
Then FTRL with regularizers $\psi_t$ achieves regret
\begin{align*}
R_T(u) & \le \frac{k}{Q_T\eta_T}\psi(Q_Tu) +\frac{45L_{\max}}{\sigma_{\text{min}}}+2L_{\max} D\\
&\le kL_{\max}\frac{\psi(2uT)}{\sqrt{2T}} +\frac{45L_{\max}}{\sigma_{\text{min}}}+2L_{\max} D
\end{align*}
\end{restatable}
This bound consists of three terms, the first of which will correspond to the $\sqrt{T}$ term in our lower bounds and the last of which will correspond to the exponential penalty. The middle term is a constant independent of $u$ and $T$. To unpack a specific instantiation of this bound, consider the example adaptive regularizer $\psi(w)=(\|w\|+1)\log(\|w\|+1)-\|w\|$. For this choice of $\psi$, we have $\psi(2uT)/\sqrt{2T}=O(\|u\|\sqrt{T}\log(T\|u\|+1))$ so that the first term in the regret bound matches the $\sqrt{T}$ term in our lower bound with $\gamma=1$. Roughly speaking, $h(w)\approx \log(w)$, so that $h^{-1}(x)\approx \exp(x)$ and the quantity $D=\max_t\frac{L_{t-1}^2}{(\|g\|_\star^2)_{1:t-1}}h^{-1}\left(\frac{5L_t}{k^2L_{t-1}}\right)$ matches the exponential penalty in our lower bound. In the following section we formalize this argument and exhibit a family of adaptive regularizers that enable us to design algorithms whose regret matches any desired point on the lower bound frontier.
\section{Optimal Algorithms}\label{sec:algorithms}
In this section we construct specific adaptive regularizers in order to obtain optimal algorithms using our regret upper bound of Theorem \ref{thm:parameterfreeregret}. The results in the previous section hold for arbitrary norms, but from this point on we will focus on the $L_2$ norm.
Our regret upper bound expresses regret in terms of the function $h^{-1}$. Inspection of the bound shows that if $h^{-1}(x)$ is exponential in $x^{1/(2\gamma-1)}$, and $\psi(w)=O(\|w\|\log^\gamma(\|w\|+1))$, then our upper bound will match (the second inequality in) our lower bound frontier. The following Collary formalizes this observation.
\begin{Corollary}\label{thm:orderbound}
If $\psi$ is an $(\sigma,\|\cdot\|)$-adaptive regularizer such that
\begin{align*}
\psi(x)\sigma(x) &\ge \Omega(\gamma\log^{2\gamma-1}(\|x\|))\\
\psi(x)&\le O(\|x\|\log^\gamma(\|x\|+1))
\end{align*}
then for any $k\ge 1$, FTRL with regularizers $\psi_t(w) = \frac{k}{a_t\eta_t}\psi(a_tw)$ yields regret
\begin{align*}
R_T(u) &\le O
\left[
kL_{\max}\sqrt{T}\|u\|\log^\gamma(T\|u\|+1) +
\max_t\frac{L_{\max} L_{t-1}^2}{\|g\|^2_{1:t-1}}\exp
\left[
O\left(\left(
\frac{L_t}{k^2\gamma L_{t-1}}
\right)^{1/(2\gamma-1)}\right)
\right]
\right]
\end{align*}
We call regularizers that satisfy these conditions $\gamma$-optimal.
\end{Corollary}
With this Corollary in hand, to match our lower bound frontier we need only construct a $\gamma$-optimal adaptive regularizer for all $\gamma\in(1/2,1]$. Constructing adaptive regularizers is made much simpler with Proposition \ref{thm:onedimconditions} below. This proposition allows us to design adaptive regularizers in high dimensional spaces by finding simple one-dimensional functions. It can be viewed as taking the place of arguments in prior work \citep{mcmahan2014unconstrained,orabona2016coin,cutkosky2016online} that reduce high dimensional problems to one-dimensional problems by identifying a ``worst-case" direction for each subgradient $g_t$.
\begin{restatable}{Proposition}{onedimconditions}\label{thm:onedimconditions}
Let $\|\cdot\|$ be the $L_2$ norm $(\|w\|=\|w\|_2=\sqrt{w\cdot w})$. Let $\phi$ be a three-times differentiable function from the non-negative reals to the reals that satisfies
\begin{enumerate}
\item $\phi(0)=0$.
\item $\phi'(x)\ge 0$.
\item $\phi''(x)\ge 0$.
\item $\phi'''(x)\le 0$.
\item $\lim_{x\to\infty} \phi(x)\phi''(x)=\infty$.
\end{enumerate}
Then $\psi(w)=\phi(\|w\|)$ is a $(\phi''(\|\cdot\|),\|\cdot\|)$-adaptive regularizer.
\end{restatable}
Now we are finally ready to derive our first optimal regularizer:
\begin{Proposition}\label{thm:rexregularizer}
Let $\|\cdot\|$ be the $L_2$ norm. Let
$\phi(x) = (x+1)\log(x+1)-x$. Then $\psi(w)=\phi(\|w\|)$ is a $1$-optimal, $(\phi''(\|\cdot\|),\|\cdot\|)$-adaptive regularizer.
\end{Proposition}
\begin{proof}
We can use Proposition \ref{thm:onedimconditions} to prove this with a few simple calculations:
\begin{align*}
\phi(0)&=0\\
\phi'(x) &= \log(x+1)\\
\phi''(x) &= \frac{1}{x+1}\\
\phi'''(x) &= -\frac{1}{(x+1)^2}\\
\phi(x)\phi''(x) &= (\log(x+1)-\frac{x}{x+1})
\end{align*}
Now the conclusion of the Proposition is immediate from Proposition \ref{thm:onedimconditions} and inspection of the above equations.
\end{proof}
A simple application of Corollary \ref{thm:orderbound} shows that FTRL with regularizers $\psi_t(w)=\frac{k}{\eta_t}((\|w\|+1)\log(\|w\|+1)-\|w\|)$ matches our lower bound with $\gamma=1$ for any desired $k$.
In fact, the result of Proposition \ref{thm:rexregularizer} is a more general phenomenon:
\begin{Proposition}\label{thm:matchthebound}
Let $\|\cdot\|$ be the $L_2$ norm. Given $\gamma\in(1/2,1]$, set $\phi(x) = \int_0^x \log^\gamma(z+1)\ dz$. Then $\psi(w)=\phi(\|w\|)$ is a $\gamma$-optimal, $(\phi''(\|\cdot\|),\|\cdot\|)$-adaptive regularizer.
\end{Proposition}
\begin{proof}
\begin{align*}
\phi(0) &= 0\\
\phi'(x) &= \log^\gamma(x+1)\\
\phi''(x) &= \gamma\frac{\log^{\gamma-1}(x+1)}{x+1}\\
\phi'''(x) &= \gamma(\gamma-1)\frac{\log^{\gamma-2}(x+1)}{(x+1)^2}-\gamma\frac{\log^{\gamma-1}(x+1)}{(x+1)^2}
\end{align*}
Since $\gamma\le 1$, $\phi'''(x)\le 0$ and so $\phi$ satisfies the first four conditions of Proposition \ref{thm:onedimconditions}. It remains to characterize $\phi(x)$ and $\phi(x)\phi''(x)$, which we do by finding lower and upper bounds on $\phi(x)$:
For a lower bound, we have
\begin{align*}
\frac{1}{2}\frac{d}{dx}x\log^\gamma(x+1)&= \frac{1}{2}\left(\log^\gamma(x+1)+\gamma\frac{x}{x+1}\log^{\gamma-1}(x+1)\right)\\
&\le \log^\gamma(x+1)
\end{align*}
where the inequality follows since $\frac{x}{x+1}\le \log(x+1)$, which can be verified by differentiating both sides.
Therefore $\phi(x) \ge \frac{1}{2}x\log^\gamma(x+1)$. This lower-bound implies
\begin{align*}
\phi(x)\phi''(x) &\ge \frac{1}{2}\gamma\frac{x}{x+1} \log^{2\gamma-1}(x+1)
\end{align*}
which gives us the last condition in Proposition \ref{thm:onedimconditions}, as well as the first condition for $\gamma$-optimality.
Similarly, we have
\begin{align*}
\frac{d}{dx}x\log^\gamma(x+1)&=\left(\log^\gamma(x+1)+\gamma\frac{x}{x+1}\log^{\gamma-1}(x+1)\right)\\
&\ge \log^\gamma(x+1)
\end{align*}
This implies $\phi(x)\le x\log(x+1)$ which gives us the second condition for $\gamma$-optimality.
\end{proof}
Thus, by applying Theorem \ref{thm:parameterfreeregret} to the regularizers of Proposition \ref{thm:matchthebound}, we have a family of algorithms that matches our family of lower-bounds up to constants. The updates for these regularizers are extremely simple:
\begin{align*}
w_{t+1} =
-\frac{g_{1:t}}{a_t\|g_{1:t}\|}\left[\exp\left((\eta_t\|g_{1:t}\|/k)^{1/\gamma}\right)-1\right]
\end{align*}
The guarantees of Theorem \ref{thm:parameterfreeregret} do not make any assumptions on how $k$ is chosen, so that we could choose $k$ using prior knowledge if it is available. For example, if a bound on $L_t/L_{t-1}$ is known, we can set $k\ge \sqrt{\max_t L_t/L_{t-1}}$. This reduces the exponentiated quantity $\max_t L_t/k^2L_{t-1}$ to a constant, leaving a regret of $O(\|u\|\log(T\|u\|+1)L_{\max}\sqrt{T\max_t L_t/L_{t-1}})$. This bound holds without requiring a bound on $L_{\max}$. Thus our algorithms open up an intermediary realm in which we have no bounds on $\|u\|$ or $L_{\max}$, and yet we can leverage some other information to avoid the exponential penalty.
\section{\textsc{FreeRex}}\label{sec:freerex}
Now we explicitly describe an algorithm, along with a fully worked-out regret bound. The norm $\|\cdot\|$ used in the following is the $L_2$ norm ($\|w\|=\sqrt{w\cdot w}$), and our algorithm uses the adaptive regularizer $\psi(w)=(\|w\|+1)\log(\|w\|+1)-\|w\|$. Similar calculations could be performed for arbitrary $\gamma$ using the regularizers of Proposition \ref{thm:matchthebound}, but we focus on the $\gamma=1$ because it allows for simpler and tighter analysis through our closed-form expression for $\psi$. Since we do not require any information about the losses, we call our algorithm \textsc{FreeRex}\ for Information-\textbf{free} \textbf{R}egret via \textbf{ex}ponential updates.
\begin{algorithm}
\caption{\textsc{FreeRex}}
\label{alg:freerex}
\begin{algorithmic}
\STATE {\bfseries Input:} $k$.
\STATE {\bfseries Initialize:} $\frac{1}{\eta_0^2}\gets 0$, $a_0\gets 0$, $w_1\gets 0$, $L_0\gets 0$, $\psi(w)=(\|w\|+1)\log(\|w\|+1)-\|w\|$.
\FOR{$t=1$ {\bfseries to} $T$}
\STATE Play $w_t$, receive subgradient $g_t\in \partial \ell_t(w_t)$.
\STATE $L_t\gets \max(L_{t-1},\|g_t\|)$.
\STATE $\frac{1}{\eta_t^2}\gets\max\left(\frac{1}{\eta_{t-1}^2}+2\|g_t\|^2, L_t \|g_{1:t}\|\right)$.
\STATE $a_t\gets\max(a_{t-1},1/(L_t\eta_t)^2)$.
\STATE //Set $w_{t+1}$ using FTRL update
\STATE $w_{t+1} \gets -\frac{g_{1:t}}{a_t\|g_{1:t}\|}\left[\exp\left(\frac{\eta_t\|g_{1:t}\|}{k}\right)-1\right]$ // $=\mathop{\text{argmin}}_w\left[\frac{k\psi(a_tw)}{a_t\eta_t}+g_{1:t}w\right]$
\ENDFOR
\end{algorithmic}
\end{algorithm}
\begin{Theorem}\label{thm:freerexregret}
The regret of \textsc{FreeRex}\ (Algorithm \ref{alg:freerex}) is bounded by
\begin{align*}
R_T(u) &\le k\|u\|\sqrt{2\|g\|^2_{1:T}+L_{\max}\max_{t\le T}\|g_{1:t}\|} \log\left(\frac{2\|g\|_{1:T}}{L_{\max}}\|u\|+1\right)+\frac{45L_{\max}}{k}\exp(10/k^2+1)\\
&\quad\quad+ 2L_{\max}\max_t \frac{L_{t-1}^2}{\|g\|^2_{1:t-1}}\left[\exp\left(\frac{5L_t}{k^2L_{t-1}}+1\right)-1\right]
\end{align*}
\end{Theorem}
\begin{proof}
Define $\phi(x)=(x+1)\log(x+1)-x$. Then $\psi(w)=(\|w\|+1)\log(\|w\|+1)-\|w\|$ is a $(\phi''(\|\cdot\|),\|\cdot\|)$-adaptive regularizer by Proposition \ref{thm:rexregularizer}. Therefore we can immediately apply Theorem \ref{thm:parameterfreeregret} to obtain
\begin{align*}
R_T(u) &\le \frac{k}{Q_T\eta_T}\psi(Q_Tu)+\frac{45L_{\max}}{\phi''_{\text{min}}}+ 2L_{\max} D
\end{align*}
where we've defined $\phi''_{\text{min}}=\inf_{\|w\|\le h^{-1}(10/k^2)}k\phi''(\|w\|)$.
We can compute (for non-negative $x$):
\begin{align*}
\phi(x) &\le (x+1)\log(x+1)\\
\phi''(x) &=\frac{1}{x+1}\\
h(w)&=\phi(\|w\|)\phi''(\|w\|)=\left(\log(\|w\|+1)-\frac{\|w\|}{\|w\|+1}\right)\\
&\ge \log(\|w\|+1)-1
\end{align*}
From Proposition \ref{thm:etarates} (part 2) we have $\frac{1}{\eta_T}\le \sqrt{2\|g\|^2_{1:T} + L_{\max}\max_{t\le T}\|g_{1:t}\|}$. We also have $(\|w\|+1)\log(\|w|+1)-\|w\|=\|w\|\log(\|w\|+1)+\log(\|w\|+1)-\|w\|\le \|w\|\log(\|w\|+1)$, so we are left with
\begin{align*}
R_T(u) &\le \frac{k}{\eta_T} \|u\|\log(Q_T\|u\|+1)+\sup_{\|w\|\le h^{-1}(\frac{10}{k^2})}\frac{45(\|w\|+1)}{k} + 2L_{\max} D\\
&=k\sqrt{2\|g\|^2_{1:T}+L_{\max}\max_{t\le T}\|g_{1:t}\|}\|u\|\log(a_T\|u\|)+1)+\frac{45L_{\max}}{k}\left[h^{-1}\left(\frac{10}{k^2}\right)+1\right] \\
&\quad+ 2L_{\max} D
\end{align*}
Now it remains to bound $h^{-1}(10/k^2)$ and $D$. From our expression for $h$, we have
\begin{align*}
h^{-1}(x/k^2) &\le \exp\left[\frac{x}{k^2}+1\right]-1
\end{align*}
Therefore we have
\begin{align*}
h^{-1}(10/k^2)&\le \exp(10/k^2+1)-1\\
D &= 2\max_t \frac{L_{t-1}^2}{(\|g\|_\star^2)_{1:t-1}}h^{-1}\left(\frac{5L_t}{k^2L_{t-1}}\right)\\
&\le 2\max_t \frac{L_{t-1}^2}{(\|g\|_\star^2)_{1:t-1}}\left[\exp\left(\frac{5L_t}{k^2L_{t-1}}+1\right)-1\right]
\end{align*}
Substituting the value $Q_T=2\frac{\|g\|_{1:T}}{L_{\max}}$, we conclude
\begin{align*}
R_T(u) &\le k\sqrt{2\|g\|^2_{1:T}+L_{\max}\max_{t\le T}\|g_{1:t}\|} \|u\|\log\left(\frac{2\|g\|_{1:T}}{L_{\max}}\|u\|+1\right) \\
&\quad\quad+\frac{45L_{\max}}{k}\exp(10/k^2+1)+ 2L_{\max} D
\end{align*}
From which the result follows by substituting in our expression for $D$.
\end{proof}
As a specific example, for $k=\sqrt{5}$ we numerically evaluate the bound to get
\begin{align*}
R_T(u) &\le \|u\|\sqrt{10\|g\|^2_{1:T}+5L_{\max}\max_{t\le T}\|g_{1:t}\|} \log\left(\frac{2\|g\|_{1:T}}{L_{\max}}\|u\|+1\right)+405L_{\max}\\
&\quad\quad+ 2L_{\max}\max_t \frac{L_{t-1}^2}{\|g\|^2_{1:t-1}}\left[\exp\left(\frac{L_t}{L_{t-1}}+1\right)-1\right]
\end{align*}
\section{Conclusions}
We have presented a frontier of lower bounds on the worst-case regret of any online convex optimization algorithm without prior information. This frontier demonstrates a fundamental trade-off at work between $kuL_{\max}\log^\gamma(Tu+1)$ and $\exp\left[\left(\max_t \frac{L_t}{\gamma k^2L_{t-1}}\right)^{\frac{1}{2\gamma-1}}\right]$ terms. We also present some easy-to-use theorems that allow us to construct algorithms that match our lower bound for any chosen $k$ and $\gamma$. Note that by virtue of not requiring prior information, our algorithms are nearly hyperparameter-free. They only require the essentially unavoidable trade-off parameters $k$ and $\gamma$. Since our analysis does not make assumptions about the loss functions or comparison point $u$, the parameters $k$ and $\gamma$ can be freely chosen by the user. Unlike other algorithms that require $\|u\|$ or $L_{\max}$, there are no unknown constraints on these parameters.
Our results also open a new perspective on optimization algorithms by casting using prior information as a tool to avoid the exponential penalty. Previous algorithms that require bounds on the diameter of $W$ or $L_{\max}$ can be viewed as addressing this issue. We show that it also possible to avoid the exponential penalty by using a known bound on $\max_t L_t/L_{t-1}$, leading to a regret of $\tilde O(\|u\|L_{\max}\sqrt{T\max_t L_t/L_{t-1}})$.
Although we answer some important questions, there is still much to do in online learning without prior information. For example, it is possible to obtain $O(\|u\|^2L_{\max}\sqrt{T})$ regret without prior information \citep{orabona2016scale}, so it should be possible to extend our lower-bound frontier beyond $\|u\|\log(\|u\|)$. Further, it would be valuable to further characterize the conditions for which the adversary can guarantee regret that is exponential in $T$. We showed that one such condition is that there must be a large jump in the value of $L_t$, but there may very well be others. Fully characterizing these conditions should allow us design algorithms that smoothly interpolate between ``nice'' environments that do not satisfy the conditions and fully adversarial ones that do.
Finally, while our analysis allows for the use of arbitrary norms, we focus our examples on the $L_2$ norm. It may be interesting to design adaptive regularizers with respect to a more diverse set of norms, or to extend our theory to encompass time-changing norms.
\small
| {
"timestamp": "2017-06-07T02:02:43",
"yymm": "1703",
"arxiv_id": "1703.02629",
"language": "en",
"url": "https://arxiv.org/abs/1703.02629",
"abstract": "The vast majority of optimization and online learning algorithms today require some prior information about the data (often in the form of bounds on gradients or on the optimal parameter value). When this information is not available, these algorithms require laborious manual tuning of various hyperparameters, motivating the search for algorithms that can adapt to the data with no prior information. We describe a frontier of new lower bounds on the performance of such algorithms, reflecting a tradeoff between a term that depends on the optimal parameter value and a term that depends on the gradients' rate of growth. Further, we construct a family of algorithms whose performance matches any desired point on this frontier, which no previous algorithm reaches.",
"subjects": "Machine Learning (cs.LG); Machine Learning (stat.ML)",
"title": "Online Learning Without Prior Information",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808759252645,
"lm_q2_score": 0.7217432122827968,
"lm_q1q2_score": 0.7075110683296941
} |
https://arxiv.org/abs/1706.04540 | On Error Detection in Asymmetric Channels | We study the error detection problem in $ q $-ary asymmetric channels wherein every input symbol $ x_i $ is mapped to an output symbol $ y_i $ satisfying $ y_i \geq x_i $. A general setting is assumed where the noise vectors are (potentially) restricted in: 1) the amplitude, $ y_i - x_i \leq a $, 2) the Hamming weight, $ \sum_{i=1}^n 1_{\{y_i \neq x_i\}} \leq h $, and 3) the total weight, $ \sum_{i=1}^n (y_i - x_i) \leq t $. Optimal codes detecting these types of errors are described for certain sets of parameters $ a, h, t $, both in the standard and in the cyclic ($ \operatorname{mod}\, q $) version of the problem. It is also demonstrated that these codes are optimal in the large alphabet limit for every $ a, h, t $ and every block-length $ n $. | \section{Introduction}
\label{sec:intro}
\IEEEPARstart{P}{hysical} properties of certain optical and flash memory devices
that are used for information storage and transmission are such that the received
symbol (voltage, number of photons, etc.) can never be larger than the corresponding
transmitted symbol.%
\footnote{For convenience, we assume that the received symbol cannot be
\emph{smaller} than the transmitted one, which is of course equivalent.}
For this reason, the resulting communication models are usually referred to as
\emph{asymmetric} channels.
In the present paper we study the problem of error \emph{detection} in such channels.
In particular, we analyze the effect of the \emph{amplitude}, the \emph{Hamming weight},
and the \emph{total weight} of noise vectors on the size of optimal error-detecting codes.
This approach unifies and generalizes several known error models from the literature.
Our main results are proofs of optimality of a family of error-detecting codes for
some classes of channel parameters, and a proof that the same family is optimal in the
limit of large alphabets for all channel parameters and all block-lengths.
The paper is structured as follows.
In Section \ref{sec:model} we give a description of the type of asymmetric channels
we have in mind and the problem that will be analyzed in the sequel, as well as a
brief overview of the relevant literature.
Sections \ref{sec:infinite}, \ref{sec:finite}, \ref{sec:cyclic} contain our main
results concerning optimal error-detecting codes for asymmetric channels with infinite,
finite, and cyclic alphabets, respectively.
A brief conclusion and some pointers for further work are stated in Section \ref{sec:conclusion}.
\vspace{-1mm}
\section{Model description and problem formulation}
\label{sec:model}
Let $ \mathbb{A} $ denote the channel alphabet, which we shall take to be
either $ \mathbb{Z} $---the set of all integers---or a subset of $ \mathbb{Z} $
of the form $ \{ 0, 1, \ldots, q - 1 \} $.
For any input vector $ {\bf x} = (x_1, \ldots, x_n) \in \mathbb{A}^n $, \pagebreak
the channel outputs a vector $ {\bf y} = (y_1, \ldots, y_n) \in \mathbb{A}^n $
satisfying the following conditions:
\begin{itemize}
\item[1)]
$ 0 \leq y_i - x_i \leq a $,
\item[2)]
$ \sum_{i=1}^n \mathbbm{1}_{\{y_i \neq x_i\}} \leq h $,
\item[3)]
$ \sum_{i=1}^n (y_i - x_i) \leq t $.
\end{itemize}
Hence, we consider asymmetric channels with additional constraints imposed on:
\begin{inparaenum}
\item[1)]
the amplitude of the noise at each coordinate (so-called limited-magnitude errors),
\item[2)]
the number of symbols hit by noise, i.e., the Hamming weight of the error
vector $ {\bf y} - {\bf x} $, and
\item[3)]
the total weight of the error vector.
\end{inparaenum}
The error vectors satisfying the above constraints will be referred to as
$ (a, h, t) $-asymmetric errors.
Note that the situations where only some of the above three constraints are
imposed on the noise are special cases of our setting.
Namely, by taking $ a = t $ the constraint 1) is effectively excluded (i.e.,
becomes redundant), as it is when $ a = q - 1 $ in the finite alphabet case.
Similarly, one can exclude constraint 2) by setting $ h = n $, and
constraint 3) by setting $ t = a h $.
We shall refer to the $ (q-1, n, t) $-asymmetric errors (when only the constraint
3) is in effect) as the $ (\cdot, \cdot, t) $-asymmetric errors, and similarly
for the other cases.
Hence, the `$ \cdot $' indicates that the corresponding constraint is either
redundant, or is not being considered at all.
\textit{\textbf{Convention:}
To avoid discussing trivial cases, as well as to simplify the exposition, we
shall assume hereafter that $ q, n, a, h, t $ are positive integers satisfying
$ a \leq q - 1 $ (in the finite alphabet case), $ a \leq t \leq a h $, and $ h \leq n $.}
\myqed
\begin{figure
\centering
\subfloat[$ (2, 1, \cdot) $-asymmetric errors.]
{
\centering
\includegraphics[width=0.38\columnwidth]{out_a2_h1}
\label{fig:region_a2_h1}
}
\hspace{5ex}
\subfloat[$ (\cdot, \cdot, 2) $-asymmetric errors.]
{
\centering
\includegraphics[width=0.38\columnwidth]{out_t2}
\label{fig:region_t2}
}\\
\subfloat[$ (2, \cdot, 3) $-asymmetric errors.]
{
\centering
\includegraphics[width=0.38\columnwidth]{out_a2_t3}
\label{fig:region_a2_t3}
}
\hspace{5.7ex}
\subfloat[$ (2, \cdot, \cdot) $-asymmetric errors.]
{
\centering
\includegraphics[width=0.38\columnwidth]{out_a2}
\label{fig:region_a2}
}
\caption{The region $ \out{0} $---the set of channel outputs that can be produced by the
input vector $ \bf 0 $ (depicted as black dot)---for various channel parameters and
block-length $ n = 2 $.}%
\label{fig:out0}
\end{figure}%
For $ {\bf x} \in \mathbb{A}^n $, denote by $ \out{x} $ the set of all channel
outputs $ {\bf y} \in \mathbb{A}^n $ that can be produced by the input $ {\bf x} $
and the noise vectors satisfying the constraints 1)--3) (see Figure \ref{fig:out0}).
The dependence of $ \out{x} $ on the parameters $ a, h, t $ is suppressed
for notational simplicity; this should not cause any confusion.
A code $ {\mathcal C} \subseteq \mathbb{A}^n $ is said to detect certain error
patterns if no codeword $ {\bf x} $ can produce another codeword $ {\bf y} \neq {\bf x} $
at the channel output, when any of those error patterns occur.
In symbols, if $ {\bf x} \in {\mathcal C} $ and $ {\bf y} \in \out{x} $,
$ {\bf y} \neq {\bf x} $, then $ {\bf y} \notin {\mathcal C} $.
This ensures that the received vector is either the transmitted codeword, or
not a codeword at all, meaning that the receiver can unambiguously determine
whether an error has happened during transmission.
Our object of study in this paper are codes detecting $ (a, h, t) $-asymmetric
errors;%
\footnote{Codes detecting asymmetric errors can also detect so-called unidirectional
errors---errors which are either positive or negative within a codeword, but the sign
is not known in advance; see, e.g., \cite{bose+lin}.}
they will be referred to as $ (a, h, t) $-asymmetric-error-detecting codes, or
$ (a, h, t) $-AED codes for short.
\vspace{-1mm}
\subsection*{Related work}
The work most closely related to ours, in which optimal $ (\cdot, \cdot, t) $-AED
codes over $ q $-ary alphabets were determined, is \cite{borden}.
In fact, we show that the same family of codes that was studied there remains
optimal in the case of $ (a, h, t) $-asymmetric errors in some instances.
We also extend and generalize the results of \cite{borden} to infinite alphabets.
The mentioned construction from \cite{borden} was also used in \cite{ahlswede1}
for the $ (a, \cdot, \cdot) $ case, and a very similar construction (also for the
$ (a, \cdot, \cdot) $ case, but in a quite different setting) appears in \cite{kovacevic+stojakovic+tan}.
Systematic $ (a, \cdot, \cdot) $-AED codes (as well as $ (a, h, \cdot) $-AED codes) were studied in \cite{elarief}.
Some other works on the error detection problem in asymmetric channels should
also be mentioned, e.g., \cite{albassam+bose, bose+elmougy+tallini, bose+lin,
bose+pradhan, bose+rao, naydenova+klove}.
All of these works are focused on special cases of the model introduced above---%
$ (\cdot, \cdot, t) $, $ (a, h, \cdot) $, binary alphabet, etc.
In \cite{tallini+bose}, a generalization of the $ (\cdot, \cdot, t) $ error model
is studied where both positive and negative errors are allowed.
Finally, for a study of the error \emph{correction} problem in asymmetric channels
we refer the reader to \cite{klove}; see also \cite{cassuto, klove+bose+elarief,
kloveetal, schwartz, yari+klove+bose} for some of the more recent works.
\section{Asymmetric channels with infinite alphabet}
\label{sec:infinite}
We first consider the case when the channel alphabet is the set of all integers
$ \mathbb{Z} $.
Though this is clearly not practically motivated, there are several reasons why
these results are relevant for the study of communication over asymmetric channels.
First, this will provide geometric intuition about the problem and allow us to
``visualize'' AED codes.
Second, the codes described below will be shown optimal for all channel parameters
in the infinite alphabet case, suggesting that the corresponding codes in the finite
alphabet case are nearly optimal, at least in some asymptotic regimes.
In fact, as we already mentioned, they will be proven optimal in some special
instances of the finite alphabet case as well.
Third, infinite alphabet can be seen as a limiting case and an approximation of
a finite alphabet; this is relevant in situations where the alphabet size $ q $ is
large compared to the ``error radius'' $ t $.
Before stating the results we need a few definitions.
To quantify what it means for a code in $ \mathbb{Z}^n $ to be optimal,
we define the density of $ {\mathcal C} \subseteq \mathbb{Z}^n $ as follows:
\begin{equation}
\label{eq:density}
\mu({\mathcal C}) \defeq \lim_{k \to \infty}
\frac{ \left| {\mathcal C} \cap \{-k, \ldots, k\}^n \right| }{ (2k+1)^{n} } .
\end{equation}
This parameter represents the infinite-space analog of the cardinality of
codes in finite spaces.
In case the limit in \eqref{eq:density} does not exist, one can naturally define
the upper ($ \overline{\mu}({\mathcal C}) $) and the lower
($ \underline{\mu}({\mathcal C}) $) density by replacing $ \lim $ with
$ \limsup $ and $ \liminf $, respectively.
We say that $ {\mathcal C} $ is an optimal $ (a, h, t) $-AED code in $ \mathbb{Z}^n $
if no $ (a, h, t) $-AED code in $ \mathbb{Z}^n $ has upper density larger than
$ \overline{\mu}({\mathcal C}) $.
A code $ {\mathcal C} \subseteq \mathbb{Z}^n $ is said to be linear if it is a sublattice
of $ \mathbb{Z}^n $, i.e., if $ ({\mathcal C},+) $ is a subgroup of $ (\mathbb{Z}^n,+) $.
For a linear code we have
$ \mu({\mathcal C}) = \frac{1}{|\mathbb{Z}^n / {\mathcal C}|} $, where
$ \mathbb{Z}^n / {\mathcal C} $ is the quotient group of the lattice $ \mathcal C $.
For $ S \subset \mathbb{Z}^n $, we say that $ (S, {\mathcal C}) $ is a
\emph{packing} in $ \mathbb{Z}^n $ if the translates $ {\bf x} + S $ and $ {\bf y} + S $
are disjoint for any two distinct codewords $ {\bf x}, {\bf y} \in {\mathcal C} $
(here $ {\bf x} + S = \{{\bf x} + {\bf s} : {\bf s} \in S \} $).
If $ (S, {\mathcal C}) $ is a packing and $ {\bf 0} \in S $, then each of these translates
contains exactly one codeword, and so we must have $ \overline{\mu}({\mathcal C}) \leq \frac{1}{|S|} $.
The following claim gives an upper bound on the density of $ (a, h, t) $-AED
codes in $ \mathbb{Z}^n $.
\begin{theorem}
\label{thm:density}
Let $ {\mathcal C} $ be an $ (a, h, t) $-AED code in $ \mathbb{Z}^n $.
Then $ \overline{\mu}({\mathcal C}) \leq \frac{1}{t+1} $.
\end{theorem}
\begin{IEEEproof}
Let $ {\bf e}_i $ be the unit vector having a $ 1 $ at the $ i $'th coordinate
and $ 0 $'s elsewhere.
Observe the following vectors:
\begin{equation}
\label{eq:list}
\begin{aligned}
& {\bf 0}, \quad {\bf e}_1, \quad 2 \cdot {\bf e}_1, \quad \ldots, \quad a \cdot {\bf e}_1, \\
& a \cdot {\bf e}_1 + {\bf e}_2, \quad \ldots, \quad a \cdot {\bf e}_1 + a \cdot {\bf e}_2, \\
& \;\; \vdots \\
& a \cdot {\bf e}_1 + a \cdot {\bf e}_2 + \cdots + a \cdot {\bf e}_{k-1} + \alpha_k \cdot {\bf e}_k
\end{aligned}
\end{equation}
where the list extends until one of the constraints on the noise is violated.
In other words, the last vector on the list is of the form
$ \sum_{i=1}^{h} \alpha_i \cdot {\bf e}_i $, where, for some $ k \in \{1, \ldots, h\} $,
$ \alpha_i = a $ for all $ i < k $, $ \alpha_i = 0 $ for all $ i > k $, and
$ \sum_{i=1}^k \alpha_i = t $.
Denote the set of all vectors on the resulting list by $ S $, and note that $ |S| = t + 1 $.
An important observation about this set is that, for any two vectors $ {\bf f}, {\bf g} \in S $,
where $ \bf f $ precedes $ \bf g $ on the list \eqref{eq:list}, the vector $ {\bf g} - {\bf f} $
satisfies all the noise constraints:
$ {\bf g} - {\bf f} = \sum_{i=1}^{h} \gamma_i \cdot {\bf e}_i $ with $ 0 \leq \gamma_i \leq a $
and $ \sum_{i=1}^{h} \gamma_i \leq t $.
This means that, for any two such vectors $ {\bf f}, {\bf g} $ we must have
$ {\bf g} - {\bf f} \in \out{0} $.\\
Now, let $ {\mathcal C} \subseteq \mathbb{Z}^n $ be an $ (a, h, t) $--AED code,
$ |{\mathcal C}| \geq 2 $.
We claim that $ (S, {\mathcal C}) $ is a packing in $ \mathbb{Z}^n $.
Suppose that this is not the case, i.e., that $ {\bf x} + {\bf f} = {\bf y} + {\bf g} $
for two distinct codewords $ {\bf x}, {\bf y} \in {\mathcal C} $ and two distinct
vectors $ {\bf f}, {\bf g} \in S $.
Without loss of generality, we can assume that $ \bf f $ precedes $ \bf g $ on the
list \eqref{eq:list}.
We then have $ {\bf g} - {\bf f} \in \out{0} $ and so
$ {\bf x} = {\bf y} + {\bf g} - {\bf f} \in \out{y} $, which means that the code
$ \mathcal C $ is not $ (a, h, t) $-AED, a contradiction.
Therefore, any $ (a, h, t) $-AED code defines a packing of the set $ S $ in
$ \mathbb{Z}^n $, and so its density cannot exceed $ \frac{1}{|S|} = \frac{1}{t+1} $.
\end{IEEEproof}
We next give an explicit construction of \emph{linear} codes achieving the upper
bound just derived.
A particular such code is depicted in Figure \ref{fig:code}.
\begin{figure
\centering
\includegraphics[width=0.96\columnwidth]{code_a2_h1b}
\caption{The code $ {\mathcal C}(2;2) \subseteq \mathbb{Z}^2 $ and an illustration of
its $ (2,1,\cdot) $-AED property. Codewords are depicted as black dots.}
\label{fig:code}
\end{figure}%
\begin{theorem}
\label{thm:optimalcodeZ}
The code
\begin{equation}
\label{eq:code}
{\mathcal C}(n;t) \defeq \left\{ {\bf x} \in \mathbb{Z}^n : \sum_{i=1}^n x_i \equiv 0 \mod (t + 1) \right\}
\end{equation}
is an optimal $ (a, h, t) $-AED code in $ \mathbb{Z}^n $.
\end{theorem}
\pagebreak
\begin{IEEEproof}
Due to our assumptions about the noise, each of the allowed error vectors from
$ \out{0} $ can change the sum of the codeword symbols, $ \sum_{i=1}^n x_i $, by
at most $ t $.
Therefore, no codeword of $ {\mathcal C}(n;t) $, other than the one that was
transmitted, can be produced at the output of the channel, proving that this
code is indeed $ (a, h, t) $-AED.
To demonstrate its optimality, observe that the quotient group of the lattice
$ {\mathcal C}(n;t) $ is $ \mathbb{Z}^n / {\mathcal C}(n;t) \cong \mathbb{Z}_{t+1} $.
The density of $ {\mathcal C}(n;t) $ is therefore
$ \mu\left({\mathcal C}(n;t)\right) = \frac{1}{|\mathbb{Z}_{t+1}|} = \frac{1}{t+1} $,
which is by Theorem \ref{thm:density} the largest possible value.
\end{IEEEproof}
The code $ {\mathcal C}(n;t) $ can also be written in the form
$ {\mathcal C}(n;t) = \big\{ \boldsymbol{\xi} \cdot {\bf G}(n;t) : \boldsymbol{\xi} \in \mathbb{Z}^n \big\} $,
where $ {\bf G}(n;t) $ is an $ n \times n $ generator matrix:
\begin{equation}
{\bf G}(n;t) =
\begin{pmatrix}
t+1 & 0 & 0 & \cdots & 0 \\
-1 & 1 & 0 & \cdots & 0 \\
-1 & 0 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
-1 & 0 & 0 & \cdots & 1
\end{pmatrix}.
\end{equation}
\begin{remark}
Note that the codes $ {\mathcal C}(n;t) $---optimal $ (a, h, t) $-AED codes in
$ \mathbb{Z}^n $---do not depend on the parameters $ a, h $, a somewhat counter-%
intuitive fact.
Hence, detecting $ (\cdot, \cdot, t) $-asymmetric errors incurs no loss in code
efficiency compared to the case of detecting more restrictive $ (a, h, t) $-%
asymmetric errors.
The corresponding statement for error-\emph{correcting} codes is in general false.
Note that the density of $ {\mathcal C}(n;t) $ is independent of the block-length
$ n $ as well.%
\footnote{In the finite alphabet case, it is known that the number of redundant
symbols of an optimal $ (\cdot, \cdot, t) $-AED code depends only on $ t $ and not on
the block-length $ n $, see for example \cite{borden, bose+elmougy+tallini, bose+lin}.}
\myqed
\end{remark}
\section{Asymmetric channels with finite alphabet}
\label{sec:finite}
The main idea in constructing codes over finite alphabets is simple: take an
$ (a, h, t) $-AED code in $ \mathbb{Z}^n $ and restrict it to the hypercube
$ \{0, 1, \ldots, q-1\}^n $.
This will clearly yield an $ (a, h, t) $-AED code with alphabet
$ [q] \defeq \{0, 1, \ldots, q-1\} $.
We say that $ {\mathcal C} $ is an optimal $ (a, h, t) $-AED code in $ [q]^n $
if it has the largest cardinality among all $ (a, h, t) $-AED codes in $ [q]^n $.
Since the code $ {\mathcal C}(n;t) $ from \eqref{eq:code} is optimal in $ \mathbb{Z}^n $,
it is natural to take it, or any of its translations, as the basis for construction.
In other words, we consider codes in $ [q]^n $ of the form
$ \left( {\bf z} + {\mathcal C}(n;t) \right) \cap [q]^n $,
for an arbitrary vector $ {\bf z} \in \mathbb{Z}^n $.
The resulting family of codes can be written as
\begin{equation}
\label{eq:codeq}
{\mathcal C}_q^{(j)}(n;t) \defeq \left\{ {\bf x} \in [q]^n : \sum_{i=1}^n x_i \equiv j \mod (t + 1) \right\} ,
\end{equation}
where $ j \in \{0, 1, \ldots, t\} $.
The cardinality of $ {\mathcal C}_q^{(j)}(n;t) $ is maximized (with respect to $ j $)
for $ j = j^* \defeq \lfloor \frac{(q-1)n}{2} \rfloor \mymod (t+1) $, see \cite{borden}.
The codes $ {\mathcal C}_q^{(j^*)}(n;t) $ are known \cite{borden} to be optimal
$ (\cdot,\cdot,t) $-AED codes for every $ q, n, t $.
We prove below their optimality in some other cases as well.
The proof method from \cite{borden}, however, does not seem to be applicable to
these cases due to a different shape of the regions $ \out{x} $.
\begin{theorem}
\label{thm:densityq}
Let $ {\mathcal C} $ be a $ (\cdot,h,t) $-AED code in $ [q]^n $.
Then $ |{\mathcal C}| \leq q^{n-1} \lceil \frac{q}{t+1} \rceil $.
\end{theorem}
\begin{IEEEproof}
Partition the space $ [q]^n $ into $ q^{n-1} $ ``lines'', each containing $ q $
points whose coordinates $ 2,\ldots,n $ are fixed and the first coordinate
varies through $ [q] $.
If $ {\bf x} = (x_1, x_2, \ldots, x_n) $ is a codeword of a $ (\cdot,h,t) $-AED code,
then $ (x_1 + k, x_2, \ldots, x_n) $ cannot be a codeword for any $ 1 \leq k \leq t $,
because $ (x_1 + k, x_2, \ldots, x_n) \in \out{x} $.
This implies that each of the mentioned lines contains at most $ \lceil \frac{q}{t+1} \rceil $
codewords, and therefore $ |{\mathcal C}| \leq q^{n-1} \lceil \frac{q}{t+1} \rceil $.
\end{IEEEproof}
\begin{theorem}
\label{thm:optimalcodeq}
Suppose that $ t + 1 $ divides $ q $.
Then the codes $ {\mathcal C}_q^{(j)}(n;t) $ are optimal $ (\cdot,h,t) $-AED codes in $ [q]^n $.
Their cardinality is $ \big| {\mathcal C}_q^{(j)}(n;t) \big| = \frac{q^n}{t+1} $,\;
$ \forall j \in \{0, 1, \ldots, t\} $.
\end{theorem}
\begin{IEEEproof}
For every choice of the values $ x_1, \ldots, x_{n-1} \in [q] $ there are
$ \frac{q}{t+1} $ possible values of $ x_n \in [q] $ satisfying the
congruence $ \sum_{i=1}^n x_i \equiv j \mymod (t+1) $, for any fixed $ j $.
Therefore, $ |{\mathcal C}_q^{(j)}(n;t)| = q^{n-1} \cdot \frac{q}{t+1} $ for
all $ j \in \{0, 1, \ldots, t\} $, which is by Theorem \ref{thm:densityq}
the largest possible value.
\end{IEEEproof}
The following statement refers to codes detecting a single ($ h = 1 $)
asymmetric error, with no bound on the amplitude of the error, other than the
implicit one $ a \leq q - 1 $.
It is a special case of Theorem \ref{thm:optimalcodeq} but we state it
separately nonetheless because the corresponding problem for error\linebreak
correction has been studied in some detail in the literature \cite{klove+bose+elarief, kloveetal, zhang+ge}.
\begin{corollary}
The codes $ {\mathcal C}_q^{(j)}(n;q-1) $ are optimal $ (\cdot,1,\cdot) $-AED
codes in $ [q]^n $.
Their cardinality is $ \big| {\mathcal C}_q^{(j)}(n; q-1) \big| = q^{n-1} $,
$ \forall j \in \{0, 1, \ldots, q-1\} $.
\end{corollary}
\begin{IEEEproof}
Take $ t = q - 1 $ in Theorem \ref{thm:optimalcodeq} and notice that
$ (\cdot, 1, q - 1) $-asymmetric errors are in fact $ (\cdot, 1, \cdot) $-%
asymmetric errors.
\end{IEEEproof}
It should be noted that the codes $ {\mathcal C}_q^{(j)}(n;t) $ are not
optimal for general $ (a, h, t) $-asymmetric errors---counterexamples can
be constructed for small values of these parameters (see \cite{ahlswede1}
for a counterexample for the $ (a, \cdot, \cdot) $ case).
\section{Asymmetric channels with cyclic alphabet}
\label{sec:cyclic}
In this section we discuss briefly the cyclic version of the asymmetric channel
(as in, e.g., \cite{klove+bose+elarief, yari+klove+bose}).
Our motivating example is the so-called noisy typewriter channel wherein each
transmitted symbol $ x_i $ is received as either $ x_i $, or $ x_i + 1 \mymod q $.
Let $ +_q $ denote addition modulo $ q $.
The cyclic asymmetric channel we have in mind is defined as follows:
for any input vector $ {\bf x} \in [q]^n $ the channel outputs $ {\bf y} = {\bf x} +_q {\bf f} \in [q]^n $,
where $ {\bf f} \in [q]^n $ is an arbitrary noise vector satisfying the constraints
1)--3) described in Section \ref{sec:model}.
Hence, the model is the same as before, the only difference being that the sum
of the input vector and the noise vector is now taken $ \operatorname{mod}\, q $;
in other words, we now allow the errors to ``wrap around''.
To distinguish between cyclic and non-cyclic cases, we shall refer to the errors
just described as $ (a, h, t)^\circ $-asymmetric errors, and similarly for the
corresponding codes.
The code space in this setting can be represented as the torus $ \mathbb{Z}_q^n $
in which there are no ``boundary effects'' that are present in the non-cyclic case.
This enables one to derive a simple upper bound on the cardinality of optimal
codes by using a method identical to the one used for the infinite alphabet
case.
\begin{theorem}
\label{thm:densitymodq}
Let $ {\mathcal C} $ be an $ (a, h, t)^\circ $-AED code in $ [q]^n $.
Then $ |{\mathcal C}| \leq \frac{q^n}{t+1} $.
\end{theorem}
\begin{IEEEproof}
Analogous to the proof of Theorem \ref{thm:density}.
\end{IEEEproof}
We next identify a class of parameters for which the above bound is tight.
(As we shall point out in Section \ref{sec:conclusion}, it cannot be tight in
general.)
\begin{theorem}
\label{thm:optimalcodemodq}
Suppose that $ t + 1 $ divides $ q $.
Then the codes $ {\mathcal C}_q^{(j)}(n;t) $ are optimal $ (a, h, t)^\circ $-AED codes in $ [q]^n $.
\end{theorem}
\begin{IEEEproof}
We have shown in Theorem \ref{thm:optimalcodeq} that
$ \big| {\mathcal C}_q^{(j)}(n;t) \big| = \frac{q^n}{t+1} $ when $ t + 1 $ divides
$ q $, which is by Theorem \ref{thm:densitymodq} the maximum possible cardinality
of an $ (a, h, t)^\circ $-AED code in $ [q]^n $.
It is left to prove that $ {\mathcal C}_q^{(j)}(n;t) $ are indeed $ (a, h, t)^\circ $-AED.
We prove this fact below for $ j = 0 $; the statement for an arbitrary $ j $ is an
easy consequence.
The key observation is that, when $ t + 1 $ divides $ q $,
$ \big\{ {\bf x} \in \mathbb{Z}^n :
\exists {\bf x}' \in {\mathcal C}_q^{(0)}(n;t) \; \text{s.t.} \; {\bf x} \equiv {\bf x}' \mymod q \big\} =
{\mathcal C}(n;t) $, which follows from the definition of the codes $ {\mathcal C}(n;t) $
and $ {\mathcal C}_q^{(0)}(n;t) $.
In other words, the code $ {\mathcal C}(n;t) $ is a periodic extension to
$ \mathbb{Z}^n $ of the code $ {\mathcal C}_q^{(0)}(n;t) $.
With this interpretation in mind it is easy to see that the statement that $ {\mathcal C}_q^{(0)}(n;t) $
is $ (a, h, t)^\circ $-AED is equivalent to the statement that $ {\mathcal C}(n;t) $
is $ (a, h, t) $-AED, which we already know is true.
\end{IEEEproof}
\begin{corollary}
The codes $ {\mathcal C}_q^{(j)}(n;q-1) $ are optimal $ (\cdot,1,\cdot)^\circ $-AED
codes in $ [q]^n $.
\hfill \IEEEQED
\end{corollary}
\section{Concluding remarks and further work}
\label{sec:conclusion}
The problem we have addressed in this letter is that of finding optimal error-%
detecting codes for asymmetric channels with various constraints on the noise.
The solution has been obtained in the infinite alphabet case for all parameters,
but for finite alphabets the general question is still open.
The task of settling it for every $ q, n, a, h, t $ may turn out to be too
difficult and it is instructive to focus on \emph{asymptotic} optimality instead.
For example, observe the regime where $ q, a, h, t $ are fixed and $ n \to \infty $.
Note that $ \big\{ {\mathcal C}_q^{(j)}(n;t) : j \in \{0, 1, \ldots, t\} \big\} $
is a partition of $ [q]^n $, meaning that the codes in this set are mutually disjoint
and their union is all of $ [q]^n $.
It is not difficult to argue that the members of this partition are of ``approximately
the same cardinality'' for large $ n $, and consequently
$ \big| {\mathcal C}_q^{(j)}(n;t) \big| \sim \frac{q^n}{t+1} $ for any
$ j \in \{0, 1, \ldots, t\} $
(here $ a_n \sim b_n $ is a shorthand for $ \lim_{n \to \infty} \frac{a_n}{b_n} = 1 $).
Whether this family of codes is asymptotically optimal is an interesting
question which we summarize below.
In light of Theorem \ref{thm:optimalcodeZ} one may conjecture that the answer
is positive.
{\it Problem:} Fix $ q, a, h, t $, and let $ D_{q}(n;a,h,t) $ denote the size
of an optimal $ (a, h, t) $-AED code in $ [q]^n $.
Is it true that $ D_{q}(n;a,h,t) \sim \frac{q^n}{t+1} $ as $ n \to \infty $?
\hfill \myqed
As for the cyclic case, we note that the cardinality of optimal codes cannot
scale as $ \frac{q^n}{t+1} $ in general.
For example, a code is $ (\cdot, h, \cdot)^\circ $-AED if and only if its
minimum Hamming distance is $ > h $, and it is known that such codes cannot
have size $ \sim c q^n $ for $ h \geq 2 $ (this follows from the sphere packing
bound in the $ q $-ary Hamming space).
\vfill
\section*{Acknowledgment}
The author would like to thank Vincent Y. F. Tan for reading a preliminary version
of this work and for several helpful discussions on the subject matter.
\vfill
| {
"timestamp": "2017-09-12T02:04:55",
"yymm": "1706",
"arxiv_id": "1706.04540",
"language": "en",
"url": "https://arxiv.org/abs/1706.04540",
"abstract": "We study the error detection problem in $ q $-ary asymmetric channels wherein every input symbol $ x_i $ is mapped to an output symbol $ y_i $ satisfying $ y_i \\geq x_i $. A general setting is assumed where the noise vectors are (potentially) restricted in: 1) the amplitude, $ y_i - x_i \\leq a $, 2) the Hamming weight, $ \\sum_{i=1}^n 1_{\\{y_i \\neq x_i\\}} \\leq h $, and 3) the total weight, $ \\sum_{i=1}^n (y_i - x_i) \\leq t $. Optimal codes detecting these types of errors are described for certain sets of parameters $ a, h, t $, both in the standard and in the cyclic ($ \\operatorname{mod}\\, q $) version of the problem. It is also demonstrated that these codes are optimal in the large alphabet limit for every $ a, h, t $ and every block-length $ n $.",
"subjects": "Information Theory (cs.IT); Discrete Mathematics (cs.DM)",
"title": "On Error Detection in Asymmetric Channels",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808759252645,
"lm_q2_score": 0.7217432122827968,
"lm_q1q2_score": 0.7075110683296941
} |
https://arxiv.org/abs/1602.01307 | On an explicit lower bound for the star discrepancy in three dimensions | Following a result of D.~Bylik and M.T.~Lacey from 2008 it is known that there exists an absolute constant $\eta>0$ such that the (unnormalized) $L^{\infty}$-norm of the three-dimensional discrepancy function, i.e, the (unnormalized) star discrepancy $D^{\ast}_N$, is bounded from below by $D_{N}^{\ast}\geq c (\log N)^{1+\eta}$, for all $N\in\mathbb{N}$ sufficiently large, where $c>0$ is some constant independent of $N$. This paper builds upon their methods to verify that the above result holds with $\eta<1/(32+4\sqrt{41})\approx 0.017357\ldots$ | \section{Introduction and statement of the result}
Suppose we are given a set $\mathcal{P}_N$ consisting of $N$ points in the $d$-dimensional unit cube. We intend to investigate how \emph{well} this set is distributed in $[0,1)^{d}$. To this end we introduce the \emph{discrepancy function}
\begin{equation*}
D_N(x):=N \lambda_d([0,x))-\#(\mathcal{P}_N\cap[0,x)),\qquad x\in[0,1)^{d},
\end{equation*}
i.e. the difference between the expected and actual number of points of $\mathcal{P}_N$ in $[0,x)$ if we assume uniform distribution. Here, $\lambda_d$ denotes the $d$-dimensional Lebesgue measure and we abbreviated $[0,x)=[0,x_1)\times[0,x_2)\times\cdots\times[0,x_d)$ for $x=(x_1,x_2,\ldots,x_d)$. Furthermore, we refer to its $L^{\infty}$-norm
\begin{equation*}
D^{\ast}_N:=\sup_{x\in[0,1)^{d}}|D_N(x)|
\end{equation*}
as \emph{star discrepancy}. Notice that, in other literature, this entity often appears as a normalized version, i.e. $D^{\ast}_N/N$.
Over time an extensive theory has evolved around the magnitude of $D^{\ast}_N$ in terms of $N$ for arbitrary as well as for specific point sets. See, for instance, the books \cite{DicDig10,MatGeo99,KuiUni74}, just to name a few. Finding the exact order of growth seems to be an intriguingly difficult question and has not yet been solved for dimensions three or higher. In this paper we focus on a lower bound for the star discrepancy of arbitrary sets of $N$ points in the three-dimensional case based on the work of D.~Bilyk and M.T.~Lacey \cite{BilOnt083}. More precisely, we show
\begin{theorem}
\label{thm:main}
For all $N$-point sets in $[0,1)^{3}$, with $N$ sufficiently large, the star discrepancy satisfies
\begin{equation*}
D^{\ast}_N\geq C(\log N)^{1+\eta}, \quad\text{for all } \eta<1/(32+4\sqrt{41})\es0.017357\ldots
\end{equation*}
\end{theorem}
To the author's best knowledge this is the first quantitative result with respect to $\eta$.
It is worth mentioning that the basic inherent ideas reach back to K.F. Roth's seminal paper \cite{RotOni54}, in which he showed
\begin{theorem}[Roth, 1954]
We have $D^{\ast}_N\geq \|D_N\|_2\geq c_d \left(\log N\right)^{(d-1)/2}$ for all $d\geq2$.
\end{theorem}
Although this bound is now known not to be sharp for $D^{\ast}_N$ (see Schmidt's theorem below) it was his approach using the system of Haar functions and Haar decompositions which struck a chord at that time and lead to a completely new methodology for proving discrepancy bounds. For a comprehensive survey see \cite{BilOnr14}, for instance.
It took as much as 18 years until a better estimate for $D^{\ast}_N$ in the two-dimensional case was discovered by W.M. Schmidt, see \cite{SchIrr72}:
\begin{theorem}[Schmidt, 1972]
\label{thm:schmidt}
For $d=2$ we have $ D^{\ast}_N\geq C \log N$.
\end{theorem}
This bound is even known to be sharp. Later, in 1981, G.~Hal{\'a}sz managed to give a proof of Schmidt's result by refining Roth's approach via introducing special auxiliary functions, namely Riesz products, and using duality, see \cite{HalOnr81}. Both, Roth's and Hal{\'a}sz' proof are also to be found in \cite{MatGeo99}. Unfortunately, Hal{\'a}sz' methods are not directly applicable to higher dimensions, due to a shortfall of certain orthogonality properties.
This shortfall leads us to yet another main ingredient of the proof of Bilyk and Lacey as well as of this paper. In \cite{BecAtw89} J.~Beck laid the groundwork for combining Hal{\'a}sz' approach to graph theory and probability theory in three dimensions. He thereby gave the first improvement to Roth's bound by a double logarithmic factor in this case. In fact, he proved the following theorem.
\begin{theorem}[Beck, 1989]
\label{thm:beck}
For all $N$-point sets in $[0,1)^{3}$ and for all $\varepsilon>0$ we have
\begin{equation*}
D^{\ast}_N\geq C_{\varepsilon} \log N \cdot(\log\log N)^{1/8-\varepsilon}
\end{equation*}
\end{theorem}
For the sake of completeness one needs to add that an analogue of Theorem~\ref{thm:main} for arbitrary dimension $d\geq4$ was proven by Bilyk and Lacey together with A.~Vagharshakyan in \cite{BilOnt08d}. Within their paper they showed that the exponent of the logarithm in Roth's theorem can be increased to $(d-1)/2+\eta_d$ with an (unspecified) $\eta_d>0$. Due to the transition to higher dimensions and to simplification reasons several arguments were refined and the overall strategy was slightly changed in comparison to the three-dimensional case. Apart from the increasing combinatorial effort this is one of the main reasons why the same line of reasoning as in the proof of Theorem~\ref{thm:main} would not (yet) work in higher dimensions. This might be an interesting subject to be investigated in the future.
The author would also like to mention that a new proof for the lower bound of the star discrepancy of the first $N$ points of a sequence in the unit interval has recently been discovered by G. Larcher, see \cite{LarOnt15}, and has been slightly improved upon in \cite{LarAni16}, which transfers to two-dimensional point sets by a result from \cite{KuiUni74}.
The second section is dedicated to briefly describe the main ideas of Hal{\'a}sz' proof of Theorem~\ref{thm:schmidt} as well as to explain why his strategy cannot be directly extended to higher dimensions. This serves as an incentive to present the result of Bilyk and Lacey, i.e. Theorem~\ref{thm:main} without the specific bound for $\eta$, in Section~\ref{sec:bylik}, as they incorporate these ideas and provide the tools to fill the aforementioned gaps. We focus on one of these tools, the so-called \emph{Littlewood-Paley inequalities}, in Section~\ref{sec:lp} since they play an integral role in our proof. Finally, in Section~\ref{sec:main}, we carefully estimate the $L^1$-norm of a certain auxiliary function $\Psi^{\neg}$ which already appeared in \cite{BilOnt083}. This, in turn, contributes the crucial bound for $\eta$ and thus completes the proof of Theorem~\ref{thm:main}.
\section{Hal{\'a}sz' proof of Theorem~\ref{thm:schmidt}}
\label{sec:halasz}
The essential idea behind this proof is to choose an auxiliary function $\Phi$ in such a way that it is complicated enough to recapture the overall structure of $D_N$ well, while, on the other hand, it remains relatively easy to handle. More precisely, one constructs $\Phi$ such that $\|\Phi\|_1\leq2$ and $\langle\Phi,D_N\rangle\geq c\log N$ for some $c>0$ since then, by duality,
\begin{equation*}
2D^{\ast}_N=2\|D_N\|_{\infty}\geq\langle\Phi,D_N\rangle\geq c\log N.
\end{equation*}
This behaviour can be achieved by using sums of signed Haar functions.
\begin{definition}
\label{def:haarfunction}
Let $\mathcal{D}$ denote the class of dyadic intervals, i.e.
\begin{equation*}
\mathcal{D}=\{[a2^{-k},(a+1)2^{-k}):~k\in\mathbb{N} \text{ and }0\leq a<2^{k}\}.
\end{equation*}
Furthermore, we subdivide each $J\in\mathcal{D}$ into a left and a right half, $J_{l}$ and $J_{r}$, respectively, and define the \emph{one-dimensional Haar function} as $h_J=-\mathds{1}_{J_l}+\mathds{1}_{J_r}$. In higher dimensions $d\geq2$ we take a dyadic rectangle $R=J_1\times J_2\times\cdots\times J_d\in\mathcal{D}^{d}$ and $x=(x_1,x_2,\ldots,x_d)\in[0,1)^{d}$ and set
\begin{equation*}
h_{R}(x)=h_{J_1}(x_1)h_{J_2}(x_2)\cdots h_{J_d}(x_d).
\end{equation*}
\end{definition}
One of the main advantages of working in this function system is that products of Haar functions again yield Haar functions in some cases. This is indicated in the following lemma, see \cite{BilOnt083}.
\begin{lemma}[Product rule]
\label{lemma:productrule}
Let $R_1, R_2,\ldots,R_k\in\mathcal{D}^{d}$ with non-empty intersection. If, additionally, the $t$-th coordinates of all rectangles are mutually different for all $1\leq t\leq d$, then
\begin{equation*}
h_{R_1}h_{R_2}\cdots h_{R_k}=\sigma h_{S}, \quad\text{where }S=R_1\cap\cdots \cap R_k\text{ and }\sigma\in\{-1,+1\}.
\end{equation*}
\end{lemma}
Let us now set
\begin{equation}
\label{eqn:halaszrfunction}
f_k=\sum_{R=J_1\times J_2\in\mathcal{D}^2,|R|=2^{-n}, |J_1|=2^{-k}}\varepsilon_{R}h_{R},\qquad 0\leq k\leq n,
\end{equation}
for a specific choice of signs $\varepsilon_{R}$ which we do not want to specify here, and where $n$ is chosen such that $2^{n-2}\leq N<2^{n-1}$. Subsequently, we define $\Phi$ as the Riesz product
\begin{equation*}
\Phi=\prod_{k=0}^{n}(1+\gamma f_k)-1 = \gamma \sum_{k=0}^{n}f_k+\Phi_{>n},\quad \gamma\in(0,1),
\end{equation*}
where $\Phi_{>n}$ contains all sums of products of functions of the form (\ref{eqn:halaszrfunction}). The key observation is that in dimension $d=2$ two or more \emph{hyperbolic} dyadic rectangles (i.e. they share the same volume) cannot coincide in any of their coordinates and, thus, their product is a Haar function again as a result of the product rule. The upper bound on the norm $\|\Phi\|_1$ can now be easily obtained with the help of Lemma~\ref{lemma:productrule} and the lower bound for $\langle D_n,\Phi\rangle$ follows from a special choice of coefficients $\varepsilon_R$ and a standard argument involving the product rule again (see, e.g.,~\cite{MatGeo99}).
Observe that the \emph{key observation} from above deprives us of the possibility to repeat this proof verbatim for $d\geq3$. Indeed, already in dimension 3 the length of one coordinate of a hyperbolic rectangle does not fully specify the lengths of the other two, and, hence, \emph{coincidences} may occur.
\section{An outline of the strategy behind the existence result}
\label{sec:bylik}
In order to make the machinery of Hal{\'a}sz work in dimension $d=3$ and in order to improve upon Beck's result, Theorem~\ref{thm:beck}, Bilyk and Lacey had to modify the auxiliary function on the one hand, and used more involved analytical tools adjusted to it, on the other. Also, they had to make up for the shortfall of the product rule in certain cases, as stated in the above paragraph. Since this is the part on which this paper emphasizes, this is dealt with in the next section. We shall now turn to the construction of our auxiliary function.
\begin{definition}
\label{def:rademacher}
For $n\in\mathbb{N}$ let
\begin{equation*}
\mathds{H}_n=\left\{ \vec{r}=(r_1,r_2,r_3)\in\mathbb{N}^3:~|\vec{r}|:=r_1+r_2+r_3=n \right\},
\end{equation*}
where the letter ``$\mathds{H}$'' is used to resemble the term \emph{hyperbolic}. Two or more hyperbolic vectors have a \emph{coincidence} if their entries agree in one coordinate and are said to be \emph{strongly distinct} in the other case. Furthermore, for $\vec{r}\in\mathds{H}_n$ we define the set $\mathcal{D}^{3}_{\vec{r}}=\{J_1\times J_2\times J_3\in\mathcal{D}^3:|J_t|=2^{-r_t}\}$ and, subsequently, call the function
\begin{equation*}
f_{\vec{r}}=\sum_{R\in\mathcal{D}_{\vec{r}}^3}\alpha(R)h_R,\qquad\alpha(R)\in\{-1,1\}
\end{equation*}
an $\mathsf{r}$-\emph{function} with parameter $\vec{r}\in\mathds{H}_n$. Naturally, they generalize (\ref{eqn:halaszrfunction}).
\end{definition}
\begin{remark}
\label{rem:rademacher}
These functions have mean zero and $f_{\vec{r}}^2=\mathds{1}_{[0,1)^3}$. Moreover, the product $f_{\vec{r}} f_{\vec{s}}$ gives an $\mathsf{r}$-function if $\vec{r},\vec{s}\in\mathds{H}_n$ are strongly distinct, as a consequence of the product rule. Also, products of two or more $\mathsf{r}$-functions have mean zero if the maximum of the entries of the underlying vectors is unique in some coordinate.
\end{remark}
For the rest of this paper we write $A\lesssim B$ if there exists an absolute constant $c$ independent of $N$ such that $A\leq c B$. Correspondingly, $A\approx B$ indicates equality up to a multiplicative constant. Furthermore, we fix $n\approx\log N$ as in Section~\ref{sec:halasz} and set
\begin{equation*}
q=n^{\varepsilon},
\quad
\rho=q^{1/2}n^{-1},
\quad
\tilde{\rho}=aq^{b}n^{-1}=aq^{b-1/2}\rho,
\qquad
a,b,\varepsilon>0.
\end{equation*}
As a matter of fact, we work with $q$ as if it was an integer, since its fractional part is of negligible size. Moreover, the proof of (\ref{eqn:psil1})
dictates $b<1/4$. Here, too, we shall continue our calculations with $b=1/4$ ,wich is compensated for by using a strict inequality sign for $\varepsilon$ in Theorem~\ref{thm:main}. Additionally, we partition the set $\{1,2,\ldots,n\}$ into $q$ equal parts $I_1,\ldots,I_q$ , $I_v=\{(v-1)n/q+1,(v-1)n/q+2,\ldots,vn/q\}$, and group hyperbolic vectors into collections $\mathds{A}_v$, $1\leq v \leq q$, according to their first coordinate:
\begin{equation*}
\mathds{A}_v:=\left\{\vec{r}=(r_1,r_2,r_3)\in\mathds{H}_n:~r_1\in I_v\right\}.
\end{equation*}
The Riesz product we intend to consider is now defined as
\begin{equation*}
\Psi=\prod_{v=1}^{q}(1+\tilde{\rho} F_v)=1+\Psi^{\textsf{sd}}+\Psi^{\neg},\qquad F_{v}=\sum_{\vec{r}\in\mathds{A}_{v}}f_{\vec{r}},
\end{equation*}
where $\Psi^{\textsf{sd}}$ comprises the sums of products of strongly distinct collections of $\mathsf{r}$-functions and $\Psi^{\neg}$ contains the rest.
The main ingredient of the proof of Theorem~\ref{thm:main} is the lemma below.
\begin{lemma}
\label{lemma:bilykmainlemma}
One has the following estimates:
\begin{align}
\label{eqn:psil1}
\|\Psi \|_1 &\lesssim 1, \\
\label{eqn:psinsdl1}
\|\Psi^{\neg}\|_1&\ls1, \\
\label{eqn:psisdl1}
\|\Psi^{\textsf{sd}}\|_1&\ls1,
\end{align}
where we require $b<1/4$, and $\varepsilon<\min\{1/3,1/(1+12b)\}$ for (\ref{eqn:psil1}) and $\varepsilon<(8-\sqrt{41})/23$ for (\ref{eqn:psinsdl1}) and (\ref{eqn:psisdl1}), respectively.
\end{lemma}
The plain proof of this lemma without the bounds for $\varepsilon$ requires Littlewood-Paley theory, properties of exponential Orlicz classes as well as conditional expectation arguments and can be found in \cite{BilOnt083}. For a detailed derivation of the bound for $\varepsilon$ for (\ref{eqn:psil1}) the reader is encouraged to study the author's PhD-thesis \cite{PucDis17}. The proof of (\ref{eqn:psinsdl1}) is dealt with in Section~\ref{sec:main}.
Let us remark that by choosing $\Psi^{\textsf{sd}}$ as our auxiliary function the product rule (Lemma~\ref{lemma:productrule}) is applicable and, hence, similar arguments as those used in Section~\ref{sec:halasz} (see \cite{BilOnt083}) for relatively moderate values of $\varepsilon$ (see \cite{PucDis17}) lead to the estimate
\begin{equation*}
\langle D_N,\Psi^{\textsf{sd}}\rangle \gtrsim a q^{b}n \approx (\log N)^{1+\varepsilon/4}.
\end{equation*}
Thus, considering (\ref{eqn:psisdl1}) we obtain our main result by Hölder's inequality. Notice that the value of $\varepsilon$ directly determines that of $\eta$ in Theorem~\ref{thm:main}. Therefore, it is essential to meticulously keep trace of $\varepsilon$ while proving (\ref{eqn:psinsdl1}).
\section{A brief note on the Littlewood-Paley inequalites}
\label{sec:lp}
Roth's proof heavily relies on Parseval's identity and orthogonality which, of course, are distinctive features of $L^{2}$. To apply similar methods in other functions spaces -- above all, in $L^{p}$ spaces with $1<p<\infty$ -- we require a powerful tool from harmonic analysis, the so-called \emph{Littlewood-Paley inequalities}. Since most of the proofs from this paper extensively make use of these inequalities, this section is dedicated to provide a brief introduction to this topic tailored to our requirements. More information can be found in~\cite{BurSha87,SteHar93,WanSha91}, for instance.
Let us consider the case $d=1$ first. For suitable functions $f$ defined on the unit interval the \emph{dyadic square function} is given by
\begin{equation*}
\mathcal{S} f=\Bigg[| \mathds{E} f|^2+\sum_{k=0}^{\infty}\Bigg( \sum_{J\in\mathcal{D},|J|=2^{-k}} \frac{\langle f, h_{J}\rangle}{|J|}h_{J}\Bigg)^{2} \Bigg]^{1/2}.
\end{equation*}
If we choose $f$ to be of the form $f=\sum_{J\in\mathcal{D}}\alpha(J)h_{J}$ this simplifies to
\begin{equation*}
\mathcal{S} f=\Bigg[ \sum_{k=0}^{\infty}\Bigg( \sum_{J\in\mathcal{D}, |J|=2^{-k}} \alpha(J)h_{J} \Bigg)^{2} \Bigg]^{1/2}=\Bigg[ \sum_{J\in\mathcal{D}}\alpha^{2}(J)\mathds{1}_{J} \Bigg]^{1/2}.
\end{equation*}
Observe that Parseval's identity may be reformulated as $\|f \|_{2}=\| \mathcal{S} f\|_2$. Hence, the Littlewood-Paley inequalities as stated in the proposition below (cf. \cite{WanSha91}) can be seen as its extension to other $L^{p}$ spaces.
\begin{proposition}[Littlewood-Paley inequalites]
For all $1<p<\infty$ there exist positive constants $A_p\geq 1+1/\sqrt{p-1}$ and $B_p\lesssim\sqrt{p}$ for $p\geq2$ such that
\begin{equation*}
A_p\|\mathcal{S} f\|_{p}\leq \|f\|_{p}\leq B_p\|\mathcal{S} f\|_P.
\end{equation*}
\end{proposition}
The key observation is given by the fact that there is a version of the Littlewood-Paley inequalities (with exactly the same constants $A_p$ and $B_p$) which is valid for Hilbert space-valued functions, where the integrals involved are understood as Bochner integrals. This version allows us to apply the Littlewood-Paley inequality in, say, the first coordinate while keeping the other coordinates fixed in the sense of vector-valued coefficients. For full details of this discussion and for an illustrative example referring to Roth's proof the reader is once again advised to consult \cite{BilOnr14,BilOnt083}.
\section{The study of coincidences of hyperbolic vectors}
\label{sec:main}
The structure of coincidences within collections of hyperbolic vectors can probably be best explained by two-colored graphs. These are triples $G=(V(G),E_2,E_3)$, where $V(G)\subseteq\{1,2,\ldots,q\}=:[q]$ denotes the set of vertices and the symmetric subsets of $V(G)\times V(G)\setminus\{(k,k):k\in V(G)\}$, $E_2$ and $E_3$, are the edge sets of color 2 and 3, respectively. Additionally, we say that $Q\subseteq V(G)$ is a \emph{clique of color} $j$ iff it is subject to
\begin{equation*}
\forall v,w\in Q,~v\neq w:~(v,w)\in E_j
\end{equation*}
and $Q$ is maximal with this property. Here, maximality is understood in the sense that if $\tilde{Q}\supseteq Q$ is subject to the above condition, then $\tilde{Q}=Q$. Notice that edges serve to indicate that two vectors have a coincidence and its color states the coordinate. Hence, vertices from one clique of, say, color 2 shall correspond to a collection of hyperbolic vectors which have a coincidence in the second coordinate.
\begin{definition}
\label{def:admissible}
A two-colored graph $G$ is called \emph{admissible} if the following four conditions are fulfilled:
\begin{enumerate}[(i)]
\item Each $E_j$ decomposes into a union of cliques,
\item If $Q_2$ and $Q_3$ are cliques of color 2 and 3, respectively, then $|Q_2\cap Q_3|\in\{0,1\}$.
\item Every vertex is contained in at least one clique.
\item Cliques of the same color are disjoint.
\end{enumerate}
Moreover, we subdivide the class of admissible \emph{connected} (a.c.) graphs on a given vertex set $V$ further into $\mathcal{T}(V)$ and $\mathcal{C}(V)$. Here, $\mathcal{T}(V)$ comprises all a.c.\,graphs $G$ defined on $V$ such that either
\begin{enumerate}[(i)]
\item $G$ is a tree or
\item if $G$ contains a cycle then this cycle is composed of edges of one color only,
\end{enumerate}
and $\mathcal{C}(V)$ contains the rest. That is, graphs in $\mathcal{C}(V)$ contain cycles composed of edges of both colors. We shall refer to such cycles as \emph{bicolored}.
\end{definition}
Observe that if we regard the individual cliques as vertices themselves, the elements of $\mathcal{T}(V)$ admit of a tree representation. This is why we refer to them as \emph{generalized trees} in all that follows.
A bound for the number of admissible graphs on a given vertex set is given in the lemma below.
\begin{lemma}
\label{lemma:numberofgraphs}
Let $V\subseteq[q]$. The number of admissible graphs on $V$ is bounded by $c|V|^{2|V|}$, $c>0$. For generalized tree graphs this number reduces to $2^{|V|}|V|^{|V|-2}$.
\end{lemma}
\begin{proof}
The first bound is derived in \cite[p. 144]{BilOnr14} and the estimate for generalized trees is better known as \emph{Cayley's formula} without the additional factor $2^{|V|}$ which arises from choosing one of two colors for each edge. Since elements of $\mathcal{T}(V)$ can deviate from actual trees in a prescribed manner only (see item (iv) of Definition~\ref{def:admissible}) this estimate continues to hold for generalized trees. Cayley's formula was initially shown by C.W. Borchardt. Four more recent proofs can be found in the book~\cite{AigPro14}, for instance.
\end{proof}
The connection to our problem can now be drawn via the functions
$$
\sprod(\mathds{X}(G))=\sum_{(\vec{r}_{1},\vec{r}_{2},\ldots,\vec{r}_{{|V|}})\in\mathds{X}(G)}f_{\vec{r}_{1}}\cdots f_{\vec{r}_{|V|}},
$$
where $G$ is an admissible graph, and
\begin{equation*}
\mathds{X}(G):=\Big\{(\vec{r}_{1},\vec{r}_{2},\ldots,\vec{r}_{{|V|}})\in\prod_{v\in V}\mathds{A}_v:~(v_1,v_2)\in E_j \Rightarrow r_{v_1}^{(j)}=r_{v_2}^{(j)}\Big\}.
\end{equation*}
The norms of these functions for a.c.\,graphs can be estimated as follows.
\begin{lemma}
\label{lemma:beckgain}
Let $G$ be an a.c.\,graph with vertex set $V$, $|V|\geq2$, comprising exactly $t$ disjoint bicolored cycles. For all $\varepsilon<1/3$ and all $1\leq l\leq q$ we have
\begin{align*}
\tilde{\rho}^{|V|}\|\sprod(\mathds{X}(G))\|_{lq^{1/2}}
&\lesssim
\min\Big\{
l^{\frac{3}{2}}qn^{-\frac{1}{2}},
l^{\frac{|V|}{2}}n^{1-\frac{|V|}{2}}
\Big\}
l^{-\frac{t}{2}} q^{\frac{t}{4}}n^{-\frac{t}{2}}\\
&=: M_{|V|,l} l^{-\frac{t}{2}} q^{\frac{t}{4}}n^{-\frac{t}{2}}
\end{align*}
\end{lemma}
\begin{proof}
In \cite{BilOnt083} Bilyk and Lacey derive an algorithm for estimating the above norm. In short, they repeatedly apply the Littlewood-Paley inequality and/or the triangle inequality to successively specify all hyperbolic vectors. As $G$ is connected, some vertices might have one or all of its coordinates fully determined even earlier.
To provide a clearer picture of their argument let us consider the graph $G_0$ on three vertices associated to the first picture in Figure~\ref{fig:graphs}.
\begin{figure}[H]
\begin{equation*}
\begin{matrix}
\vec{r}&&\vec{s}&&\vec{t} \\ \hline
r_1 && s_1 && t_1 \\
r_2&=&s_2 &\neq& t_2 \\
r_3 &\neq&s_3&=&t_3
\end{matrix}
\qquad \qquad
\begin{matrix}
\vec{r}&&\vec{s}&&\vec{t} \\ \hline
r_1 && s_1 && \mu\\
r_2&=&s_2 &\neq&\nu \\
r_3 &\neq&n-\mu-\nu&=& n-\mu-\nu
\end{matrix}
\end{equation*}
\caption{Hyperbolic vectors associated to the graphs $G_0$ (left) and $\tilde{G}_0$ (right).}
\label{fig:graphs}
\end{figure}
W.l.o.g. assume $t_1\in I_1$. One application of the Littlewood-Paley inequality in the first coordinate yields
\begin{multline*}
\| \sprod(\mathds{X}(G_0)) \|_{lq^{1/2}}=\Bigg\| \sum_{(\vec{r},\vec{s},\vec{t})\in\mathds{X}(G_0)}f_{\vec{r}}f_{\vec{s}}f_{\vec{t}}\Bigg\|_{lq^{1/2}}\\
\lesssim l^{\frac{1}{2}}q^{\frac{1}{4}} \Bigg\| \Bigg[ \sum_{\mu\in I_1} \Bigg| \sum_{\substack{(\vec{r},\vec{s},\vec{t})\in\mathds{X}(G_0) \\ \vec{t}=(\mu, t_2,t_3)}} f_{\vec{r}}f_{\vec{s}}f_{\vec{t}} \Bigg|^{2} \Bigg]^{\frac{1}{2}} \Bigg\|_{lq^{1/2}}.
\end{multline*}
Subsequently, we fix $t_2$ with the help of the triangle inequality
\begin{equation*}
\| \sprod(\mathds{X}(G_0)) \|_{lq^{1/2}} \lesssim l^{\frac{1}{2}}q^{\frac{1}{4}} \sum_{\nu=1}^{n} \Bigg\| \Bigg[ \sum_{\mu\in I_1} \Bigg| \sum_{\substack{(\vec{r},\vec{s},\vec{t})\in\mathds{X}(G_0) \\ \vec{t}=(\mu, \nu,t_3)}} f_{\vec{r}}f_{\vec{s}}f_{\vec{t}} \Bigg|^{2} \Bigg]^{\frac{1}{2}} \Bigg\|_{lq^{1/2}}.
\end{equation*}
Notice that $\vec{t}$ is already fully specified, since its coordinates add up to $n$. Consequently, we can pull $f_{\vec{t}}$ out of the sum, where it simplifies to 1. Taking the supremum w.r.t. $\mu$ and $\nu$ then finally yields
\begin{equation*}
\| \sprod(\mathds{X}(G_0)) \|_{lq^{1/2}} \lesssim l^{\frac{1}{2}} q^{-\frac{1}{4}}n^{\frac{3}{2}} \sup_{\mu,\nu}\|\sprod(\mathds{X}(\tilde{G}_0)) \|_{lq^{1/2}},
\end{equation*}
where we used $|I_1|=n/q$. The vectors from $\mathds{X}(\tilde{G}_0)$ are depicted in the right picture of Fig.~\ref{fig:graphs}. Observe that we only need to carry out the first of the above steps, i.e. the Littlewood-Paley inequality, in order to completely determine $\vec{s}$. We continue in this direction until we have considered every vertex as then the expression in modulus equals to 1.
Our approach works in the following way. We make direct use of the discussion above and subsequently distinguish between a.c.\,graphs either belonging to $\mathcal{T}$ or to $\mathcal{C}$. In the first case we apply both the Littlewood-Paley and the triangle inequaliy at a cost of $l^{1/2}q^{-1/4}n^{3/2}$ once, thus fully specifying one vertex and simultaneously fixing one coordinate of an adjacent vector. For this vertex, in turn, we only need to apply the Littlewood-Paley inequality. In doing so we save an entire power of $n$ in each step.
This gives the second entry from the minimum from the claim. The first entry is a revised version of the case $|V|=2$ from \cite{BilOnt083}, which (by \cite{PucDis17}) is valid for $\varepsilon<1/2$. If, additionally, $G$ contains a cycle with edges of different color, i.e. $G\in\mathcal{C}$, then, due to the hyperbolic assumption, there is at least one vertex which is fully specified by the other vertices of the cycle. Consequently, we gain a factor of $l^{1/2}q^{-1/4}n^{1/2}$ for each of the $t$ bicolored cycles. For further details the reader is referred to \cite{PucDis17}.
\end{proof}
Now, our strategy becomes more visible. While a.c.\,graphs with bicolored cycles are hard to handle combinatorically speaking, they yield much better estimates in terms of Lemma~\ref{lemma:beckgain} compared to graphs from $\mathcal{T}$. As it turns out, generalized trees account for the lion share in our estimates. To see this, we adhere to the approach of Bilyk and Lacey once again and, additionally, keep trace of $\varepsilon$ to find that
\begin{equation}
\label{eqn:sumadm}
\|\Psi^{\neg}\|_1\lesssim\sum_{v=2}^{q} \sum_{G \text{ admissible}, |V(G)|=v}\tilde{\rho}^{v}\|\sprod(\mathds{X}(G))\|_{q^{1/2}}\qquad\text{for all }\varepsilon<1/4.
\end{equation}
One can easily check that the summands for $v=2,3$ are bounded by an absolute constant if $\varepsilon<1/6$.
In what follows we abbreviate ${[q] \choose v}=\{V\subseteq [q]:|V|=v\}$ as well as $\mathcal{V}(V,l)=\{\mathbf{V}=(V_1,V_2,\ldots,V_l):V_j\neq\emptyset\text{ and } \mathbf{V} \text{ is a partition of } V\}$. Observe that the cardinality of the above set is given by the \emph{Stirling number of the second kind}, which is known to satisfy $ \#\mathcal{V}(l,V)\lesssim {|V| \choose l} l^{|V|-l}$, see \cite{RenOns69}.
Let us continue with the remaining sum in (\ref{eqn:sumadm}). We decompose each admissible graph $G$ into a union of its a.c.\,components. Due to a reduction lemma from \cite{BilOnt083} we know that for any two such subgraphs $G=G_1\cup G_2$ we have $\sprod(\mathds{X}(G))=\sprod(\mathds{X}(G_1))\sprod(\mathds{X}(G_2))$. Subseqently, we may apply Hölder's inequality, which yields
\begin{align}
\|\Psi^{\neg}\|_1
&\!\lesssim\!\!
\sum_{v=4}^{q} \!\sum_{V\in{[ q] \choose v }}\! \sum_{l=1}^{v/2} \sum_{\substack{(V_1,\ldots, V_l)\in\mathcal{V}(V,l)}} \sum_{\substack{G=G_1\cup\cdots\cup G_l \\ G_j \text{ is a.c.\,on }V_j }} \prod_{j=1}^{l} \tilde{\rho}^{|V_j|}\left\| \sprod(\mathds{X}(G_j))\right\|_{lq^{1/2}}\nonumber\\
\label{eqn:streescycle}
&=:
\sum_{v=4}^{q} \sum_{V\in{[ q] \choose v }} \sum_{l=1}^{v/2} \sum_{\substack{(V_1,\ldots, V_l) \in\mathcal{V}(V,l)}}
\left( \Sigma_{\text{tree}}+\Sigma_{\text{cycle}}\right),
\end{align}
where
\begin{align*}
\Sigma_{\text{tree}}
&=
\sum_{\substack{G=G_1\cup\cdots\cup G_l \\ G_j \in \mathcal{T}(V_j)}} \prod_{j=1}^{l}\tilde{\rho}^{|V_j|}\left\| \sprod(\mathds{X}(G_j))\right\|_{lq^{1/2}}\quad\text{and}\\
\Sigma_{\text{cycle}}
&=
\sum_{\substack{G=G_1\cup\cdots\cup G_l \\ G_j \text{ a.c.\,on }V_j \text{ and } \exists j_0:T(G_{j_0})\geq1}} \prod_{j=1}^{l} \tilde{\rho}^{|V_j|}\left\| \sprod(\mathds{X}(G_j))\right\|_{lq^{1/2}}.
\end{align*}
with $T(G_j)=\max\{\tau:G_j \text{ contains } \tau \text{ disjoint bicolored cycles}\}$.
Before we proceed with the estimation we shall give one more technical lemma.
\begin{lemma}
\label{lemma:lagrange}
Let $l$, $k$, and $v$ be integers with $1\leq k\leq l\leq v/2$. Furthermore, consider $v_1,v_2,\ldots,v_l\in\mathbb{N}$ with $v_j\geq2$, $1\leq j\leq l$, and $v_1+v_2+\cdots+v_l=v$. Then
\begin{equation*}
\left(\prod_{j=1}^{k} v_j^{ v_j-2}\right)\cdot\left(\prod_{j=k+1}^{l}v_j^{2v_j} \right)\lesssim\left( \frac{v}{k}\right)^{v-2k},
\end{equation*}
and if $k=0$, i.e. the first product vanishes, we obtain $(v/l)^{2v}$ as an upper bound.
\end{lemma}
\begin{proof}
We confine ourselves to the case where $k\geq1$, since the other case follows the same spirit. Let us consider the Lagrangian
\begin{equation*}
\mathcal{L}(v_1,\ldots,v_l;\lambda) =\sum_{j=1}^{k}(v_j-2)\log v_j+2\sum_{j=k+1}^{l}v_{j}\log v_j-\lambda(v_1+\cdots+v_l-v).
\end{equation*}
Simple algebraic manipulations lead to the solution
\begin{gather}
v_j = \frac{2}{w}, \quad1\leq j\leq k, \qquad \qquad v_j=e^{\lambda/2-1},\quad k<j\leq l,\nonumber\\
\label{eqn:lagrangesolutions}
\lambda= \frac{2}{3}\left(2+\log\left(\frac{wv-2k}{e^{1-\lambda} w(l-k)}\right)\right) = \frac{2}{3}\left(2+\log\left(\frac{ve^{1-\lambda}-ke^w}{e^{2-2\lambda}(l-k)}\right)\right),
\end{gather}
where $w=W(2e^{1-\lambda})$ with $W$ denoting the \emph{Lambert $W$ function}. Observe that $0<w\leq1$ since $v_j\geq2$, and, consequently,
\begin{equation*}
v_j=\frac{2}{w}\leq \frac{2}{w}e^{1-w}=\frac{2e}{w e^w}=e^{\lambda},\quad 1\leq j\leq k.
\end{equation*}
Furthermore, if $l-k>\sqrt{kv}$ we immediately get $v_j=e^{\lambda/2-1}\leq \frac{v}{l-k}<\sqrt{v/k}$ for $\quad k<j\leq l
$ by (\ref{eqn:lagrangesolutions}). On the other hand, if $l-k\leq\sqrt{kv}$ we can solve the last expression in (\ref{eqn:lagrangesolutions}) for $\lambda$, giving
\begin{equation*}
\lambda=2 \log\left(\frac{l-k+\sqrt{(l-k)^2+4e^{w+1}kv}}{2ke^{w}}\right),
\end{equation*}
and, thus, $e^{\lambda/2}\lesssim\sqrt{v/k}$ can be obtained without difficulty. Consequently, the left-hand side from the claim can be estimated by
\begin{equation*}
e^{\lambda(v_1+\cdots+v_k-2k)} e^{2(\frac{\lambda}{2}-1)(v_{k+1}+\cdots+v_l)}
\lesssim
e^{\lambda (v_1+\cdots +v_l)-2\lambda k }
\!=\!
e^{\lambda(v-2k)}
\!\lesssim\!
\left(\frac{v}{k}\right)^{v-2k}\!.
\end{equation*}
\end{proof}
Within the subsequent paragraphs we show
\begin{equation}
\label{eqn:stree}
\sum_{v=4}^{q}\sum_{V\in{[q]\choose v}}\sum_{l=1}^{v/2}\sum_{(V_1,\ldots,V_l)\in \mathcal{V}(V,l)}\Sigma_{\text{tree}} \ls1\qquad\text{for all }\varepsilon<(8-\sqrt{41})/23.
\end{equation}
Indeed, for all $\varepsilon<1/3$ we have by Lemmas~\ref{lemma:numberofgraphs}, \ref{lemma:beckgain}, and \ref{lemma:lagrange}
\begin{equation}
\label{eqn:streem}
\Sigma_{\text{tree}}
\lesssim
\prod_{j=1}^{l}M_{|V_j|,l} |V_j|^{|V_j|-2}\lesssim l^{-v+2l}v^{v-2l}\prod_{j=1}^{l}M_{|V_j|,l}.
\end{equation}
Let us choose $\alpha_{\tau}\in(0,1/2)$ arbitrarily for now and consider the sum over $l$. For the first $\alpha_{\tau} v$ summands we choose the second entry of the minimum $M_{|V_j|,l}$ and the first entry for all the others. Since $|V_1|+\cdots+|V_l|=v$ this yields
\begin{align*}
\sum_{l=1}^{v/2}\sum_{(V_1,\ldots,V_l)\in \mathcal{V}(V,l)}\!\!\!\!\!\Sigma_{\text{tree}}
&\lesssim
\sum_{l=1}^{\alpha_{\tau} v}\! {v \choose l}\! l^{\frac{v}{2}+l}v^{v-2l}n^{-\frac{v}{2}+l}\!+\!\!\sum_{l=\alpha_{\tau} v+1}^{v/2}\!\!\! {v \choose l} l^{\frac{5}{2}l}v^{v-2l}q^{l}n^{-\frac{l}{2}}\\
&=:\Sigma_1^{\tau} +\Sigma_2^{\tau}.
\end{align*}
By Stirling's formula we immediately obtain
\begin{equation}
\label{eqn:sti}
\Sigma_1^{\tau}\lesssim \sum_{l=1}^{\alpha_{\tau} v} l^{\frac{v}{2}-\frac{1}{2}}v^{v-l}n^{-\frac{v}{2}+l}
\leq v^{\frac{3}{2}-\frac{1}{2}}n^{-\frac{v}{2}}\sum_{l=1}^{\alpha_{\tau} v}(v^{-1}n)^{l}
\lesssim v^{v(\frac{3}{2}-\alpha_{\tau})-\frac{1}{2}}n^{-v(\frac{1}{2}-\alpha_{\tau})}.
\end{equation}
For the estimation of $\Sigma_2^{\tau}$ we observe that
\begin{equation*}
{v\choose l\!+\!\alpha_{\tau} v\!+\!1}\!\leq\! {v\!-\!\alpha_{\tau} v\!-\!1\choose l}(l+\alpha_{\tau} v +1)^{-\alpha_{\tau} v-1}v^{\alpha_{\tau} v+1}\! \lesssim\! {v\!-\!\alpha_{\tau} v\!-\!1\choose l}
\end{equation*}
and, consequently,
\begin{align}
\Sigma_2^{\tau}
&\lesssim
v^{v(1+\frac{\alpha_{\tau}}{2})-\frac{1}{2}}q^{\alpha_{\tau} v+1}n^{-\frac{\alpha_{\tau}}{2}v-\frac{1}{2}}\sum_{l=0}^{\frac{v}{2}-\alpha_{\tau} v-1}{v-\alpha_{\tau} v-1\choose l}(v^{\frac{1}{2}}qn^{-\frac{1}{2}})^{l}\nonumber\\
\label{eqn:stii}
&\lesssim v^{v(1+\frac{\alpha_{\tau}}{2})+\frac{1}{2}}q^{\alpha_{\tau} v+1} n^{-\frac{\alpha_{\tau} }{2}v-\frac{1}{2}},
\end{align}
since $(1+v^{1/2}qn^{-1/2})^{v-\alpha_{\tau} v-1}\lesssim \exp(v^{3/2}q n^{-1/2})\ls1$ for $\varepsilon<1/5$.
In the same spirit we may now derive (\ref{eqn:stree}):
\begin{align*}
& \sum_{v=4}^{q}\sum_{V\in{[q]\choose v}}(\Sigma_1^{\tau}+\Sigma_2^{\tau})\\
&\lesssim\!\sum_{v=0}^{q-4}\!{q\!-\!4\choose v} \!\!\left( \!\!q^{v(\!\frac{3}{2}-\alpha_{\tau}\!)+\frac{11}{2}-4\alpha_{\tau}} \! n^{-v(\!\frac{1}{2}-\alpha_{\tau}\!)-2+4\alpha_{\tau}}\!\! +\! q^{v(\!1+\frac{3\alpha_{\tau}}{2}\!)+\frac{11}{2}+6\alpha_{\tau}}n^{-\frac{\alpha_{\tau}}{2}v-\frac{1}{2}-2\alpha_{\tau}} \!\!\right)\\
&\lesssim
q^{\frac{11}{2}-4\alpha_{\tau}}n^{-2+4\alpha_{\tau}}e^{q^{5/2-\alpha_{\tau}}n^{-1/2+\alpha_{\tau}}}+q^{\frac{11}{2}+6\alpha_{\tau}}n^{-\frac{1}{2}-2\alpha_{\tau}}e^{q^{2+3\alpha_{\tau}/2}n^{-\alpha_{\tau}/2}}.
\end{align*}
The latter expression is bounded by a constant if
\begin{equation*}
\varepsilon<\varepsilon^{\tau}(\alpha_{\tau}):=\min\left\{\frac{4-8\alpha_{\tau}}{11-8\alpha_{\tau}},\frac{1-2\alpha_{\tau}}{5-2\alpha_{\tau}},\frac{1+4\alpha_{\tau}}{11+12\alpha_{\tau}},\frac{\alpha_{\tau}}{4+3\alpha_{\tau}} \right\}.
\end{equation*}
It is fairly easy to see that the optimal $\alpha_{\tau}\in(0,1/2)$ is $\alpha_{\tau}^{\text{opt}}=(\sqrt{41}-5)/4$, for which we have $\varepsilon^{\tau}(\alpha_{\tau}^{\text{opt}})=(8-\sqrt{41})/23$, and (\ref{eqn:stree}) follows.
What is left to show is that the part of (\ref{eqn:streescycle}) comprising $\Sigma_{\text{tree}}$ outweighs the part with $\Sigma_{\text{cycle}}$. To this end we notice that a bicolored cycle can only occur if it contains at least four vertices and we may thus estimate
\begin{align*}
\Sigma_{\text{cycle}}
&\lesssim
\sum_{t=1}^{v/4}\sum_{\substack{t1,\ldots,t_l\geq0 \\ t_1+\cdots+t_l=t}} \sum_{\substack{G_1 \text{ a.c.\,on } V_1\\T(G_1)=t_1}} \cdots \sum_{\substack{G_l \text{ a.c.\,on } V_l\\T(G_l)=t_l}}\prod_{j=1}^{l} \tilde{\rho}^{|V_j|}\|\sprod(\mathds{X}(G_j))\|_{lq^{1/2}}\\
&\lesssim (S_{<l}+S_{\geq l})\prod_{j=1}^{l}M_{|V_j|,l},
\end{align*}
where we used Lemma~\ref{lemma:beckgain} with $\varepsilon<1/3$ and where we set
\begin{align*}
S_{<l}
&=
\sum_{t=1}^{l-1}l^{-\frac{t}{2}}q^{\frac{t}{4}}n^{-\frac{t}{2}} \sum_{\substack{t_1,t_2,\ldots,t_l\geq0 \\ t_1+t_2+\cdots+t_l=t}} \sum_{\substack{G_1 \text{ a.c.\,on } V_1 \\ T(G_1)=t_1}}\cdots\sum_{\substack{G_l \text{ a.c.\,on } V_l \\ T(G_l)=t_l}}1, \text{ and}\\
S_{\geq l}
&=
\sum_{t=l}^{v/4}l^{-\frac{t}{2}}q^{\frac{t}{4}}n^{-\frac{t}{2}} \sum_{\substack{t_1,t_2,\ldots,t_l\geq0 \\ t_1+t_2+\cdots+t_l=t}} \sum_{\substack{G_1 \text{ a.c.\,on } V_1 \\ T(G_1)=t_1}}\cdots\sum_{\substack{G_l \text{ a.c.\,on } V_l \\ T(G_l)=t_l}}1.
\end{align*}
Observe that, for fixed $t$, at least $l-t$ of the subgraphs occuring in $S_{<l}$ do not contain a bicolored cycle. Hence, we may apply Lemma~\ref{lemma:lagrange} with $k=l-t$, and together with Stirling's formula this yields
\begin{align}
S_{<l}
&\lesssim
\sum_{t=1}^{l-1}l^{-\frac{t}{2}}q^{\frac{t}{4}}n^{-\frac{t}{2}}{t+l-1 \choose l-1}\left(\frac{v}{l-t}\right)^{v-2(l-t)}\nonumber\\
\label{eqn:ssl}
& \lesssim
l^{-\frac{1}{2}}v^{v-2l}\sum_{t=1}^{l-1}(l-t)^{-v+2(l-t)}\left(l^{-\frac{1}{2}}v^2q^{\frac{1}{4}}n^{-\frac{1}{2}}\right)^{t}.
\end{align}
Moreover, as $\varepsilon\leq1/15$, each summand from the latter expression is bounded by $H(t)=(l-t)^{-v+2(l-t)}(l^{-1/2}v^{-21/4})^t$. Obviously, $H(1)\leq l^{-v+2l}$, which is the corresponding part of $\Sigma_{\text{tree}}$, see (\ref{eqn:streem}). Furthermore, the only critical point of $H$ is given by $t_0$ with $l-t_0=v/(2W(z_0))$, $z_0=e/2\cdot l^{1/4}v^{29/8}$. Since $z_0\geq e\cdot2^{54/4}$ we have $W(z_0)>79/20$. On the other hand, for each $\kappa>0$
there exists a constant $c>0$ such that $W(z_0)\leq z_0^{\kappa}+c$. By choosing $\kappa=1/1711$, for instance, we see that $H(t_0)$ is bounded by $H(l-1)$ from above.
Thus, it remains to investigate $H(l-1)$. For $l\geq\alpha_{\tau}^{\text{opt}} v+1$ this number is bounded by $H(1)$, which has already been dealt with. For $l\leq\alpha_{\tau}^{\text{opt}}$ the summands themselves are not well comparable to parts of $\Sigma_{\text{tree}}$ individually. However, in average, that is, considering everything down to the sum over $l$, they are bounded by (\ref{eqn:sti}). Indeed, for all $\varepsilon<4/35$ we have
\begin{multline*}
\sum_{l=1}^{\alpha_{\tau}^{\text{opt}} v}\!\!\sum_{(V_1,\ldots,V_l)\in\mathcal{V}(V,l)} \prod_{j=1}^{l}\!M_{|V_j|,l}S_{<l}
\lesssim
\sum_{l=1}^{\alpha_{\tau}^{\text{opt}} v}\!\!\!{v\choose l} l^{v-l}l^{\frac{v}{2}}n^{-\frac{v}{2}+l}l^{-\frac{1}{2}}v^{v-2l}H(l-1)\\
\lesssim
v^{v(\frac{5}{2}-\frac{35}{4}\alpha_{\tau}^{\text{opt}})+\frac{19}{4}}n^{-v(\frac{1}{2}-\alpha_{\tau}^{\text{opt}})}
\lesssim
v^{v(\frac{3}{2}-\alpha_{\tau}^{\text{opt}})-\frac{1}{2}}n^{-v(\frac{1}{2}-\alpha_{\tau}^{\text{opt}})}.
\end{multline*}
For the study of $S_{\geq l}$ we proceed similarly to (\ref{eqn:ssl}). We exploit Lemma~\ref{lemma:lagrange}, but now for $k=0$, and obtain
\begin{equation*}
S_{\geq l} \lesssim l^{-2v-l+\frac{1}{2}} v^{2v+l-1}\sum_{t=l}^{v/4}\left( l^{-\frac{1}{2}} q^{\frac{1}{4}} n^{-\frac{1}{2}} \right)^{t}
\lesssim l^{-2v-\frac{3}{2} l+\frac{1}{2}} v^{2v+l-1} q^{\frac{l}{4}}n^{-\frac{l}{2}}.
\end{equation*}
The rest follows more or less the same strategy as proving (\ref{eqn:stree}). I.e., we split up the sum over $l$ at $\alpha_{\tau}^{\text{opt}} v$, estimate accordingly as we did for $\Sigma_1^{\tau}$ and $\Sigma_2^{\tau}$ and find that their upper bounds (\ref{eqn:sti}) and (\ref{eqn:stii}) dominate the resulting expressions. For full details, see \cite{PucDis17}, again. This finishes the proof of Theorem~\ref{thm:main}.
\section{Acknowledgements}
The author is extremely grateful to his supervisor, Gerhard Larcher, for his valuable consultation, for proof reading and for his general assistance and encouragement during the writing of this paper.
| {
"timestamp": "2016-10-06T02:03:58",
"yymm": "1602",
"arxiv_id": "1602.01307",
"language": "en",
"url": "https://arxiv.org/abs/1602.01307",
"abstract": "Following a result of D.~Bylik and M.T.~Lacey from 2008 it is known that there exists an absolute constant $\\eta>0$ such that the (unnormalized) $L^{\\infty}$-norm of the three-dimensional discrepancy function, i.e, the (unnormalized) star discrepancy $D^{\\ast}_N$, is bounded from below by $D_{N}^{\\ast}\\geq c (\\log N)^{1+\\eta}$, for all $N\\in\\mathbb{N}$ sufficiently large, where $c>0$ is some constant independent of $N$. This paper builds upon their methods to verify that the above result holds with $\\eta<1/(32+4\\sqrt{41})\\approx 0.017357\\ldots$",
"subjects": "Number Theory (math.NT)",
"title": "On an explicit lower bound for the star discrepancy in three dimensions",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808736209154,
"lm_q2_score": 0.7217432122827968,
"lm_q1q2_score": 0.7075110666665458
} |
https://arxiv.org/abs/1707.04043 | Quasi-steady state reduction for the Michaelis-Menten reaction-diffusion system | The Michaelis-Menten mechanism is probably the best known model for an enzyme-catalyzed reaction. For spatially homogeneous concentrations, QSS reductions are well known, but this is not the case when chemical species are allowed to diffuse. We will discuss QSS reductions for both the irreversible and reversible Michaelis-Menten reaction in the latter case, given small initial enzyme concentration and slow diffusion. Our work is based on a heuristic method to obtain an ordinary differential equation which admits reduction by Tikhonov-Fenichel theory. We will not give convergence proofs but we provide numerical results that support the accuracy of the reductions. | \section{Introduction}
The Michaelis-Menten mechanism \cite{michaelismenten} is probably the best known model for an enzyme-catalyzed reaction. In this reaction network, a substrate $S$ and an enzyme $E$ combine to form a complex $C$, which degrades back to substrate and enzyme, or to product P and enzyme. In the reversible setting there is also a back reaction combining $E$ and $P$ to complex. The reaction scheme thus reads
\[
E+S \xrightleftharpoons[k_{-1}]{k_{1}} C \xrightleftharpoons[k_{-2}]{k_{2}} E+P.
\]
In the irreversible case one assumes that product and enzyme cannot combine to form complex, i.e. one has $C\xrightarrow{k_{2}}E+P$. Typically, no complex or product are assumed present initially.
Assuming mass action kinetics and spatially homogeneous concentrations, the evolution of the concentrations $(s,e,c,p)$ of $S,E,C,P$ can be described by a system of four ordinary differential equations, from which by stoichiometry one obtains a two-dimensional system (first discussed from a mathematical perspective by Briggs and Haldane \cite{briggshaldane}). Employing the familiar quasi-steady state (QSS) assumption for complex, based on small initial concentration of enzyme, further reduces the system to dimension one.\\
For reaction systems, quasi-steady state (QSS) assumptions frequently lead to singular perturbation problems for which the classical theories of Tikhonov \cite{tikh} and Fenichel \cite{fenichel} are applicable. (Moreover, one should note results by Hoppensteadt \cite{Hoppensteadt} on unbounded time intervals; see also \cite{lws}.) \\
For spatially inhomogeneous concentrations in a reaction vessel, thus for reaction-diffusion systems, Tikhonov's and Fenichel's theory is not applicable since their fundamental results are limited to finite dimensional systems. Therefore, reaction-diffusion systems are much more difficult to analyze, and only partial results are known. As for the Michaelis-Menten reaction with diffusion and small initial enzyme concentration, Britton \cite{britton} and Yannacopoulos et al. \cite{Yannacopoulos} derived QSS reductions with the additional assumptions of immobile complex and enzyme. Kalachev et al. \cite{kkkpz} used asymptotic expansions with respect to a small parameter to obtain results about the behavior of the solutions under different time scales for diffusion, with the diffusion time scale different from the time scale for the slow reaction part. (As \cite{kkkpz} indicates, even finding candidates for reduced reaction-diffusion systems may be a nontrivial task.) Starting from different assumptions about the reaction mechanism (viz., smallness of certain rate constants), Bothe and Pierre \cite{BothePierre1} as well as Bisi et al. \cite{bisi} discussed reductions for a related system, including convergence proofs. \\
In the present paper we will discuss QSS reductions for both the irreversible and the reversible Michaelis-Menten reaction with diffusion, under the conditions of small initial enzyme concentration and slow diffusion. Our work is based on a heuristic method described in \cite{laxgoeke}, which utilizes a spatial discretization to obtain an ordinary differential equation which admits reduction by Tikhonov-Fenichel theory. In many relevant cases, the reduced ODE system can, in turn, be identified as the spatial discretization of another partial differential equation system. This resulting PDE is a candidate for the reduced system and, as pointed out in \cite{laxgoeke}, it is the only possible candidate. In the present paper we will not discuss convergence issues, which seem to be quite technically involved, but we provide numerical simulations that support the accuracy of the reduction. \\
The plan of the paper is as follows. In Section 2 we will briefly recall the most important aspects of the spatially homogeneous system and moreover note some general features of the inhomogeneous case. \\
In Section 3 the ``classical'' QSS assumption is discussed, i.e. we assume small initial enzyme concentration and slow diffusion. We first review some relevant results from the literature, and give an informal description of the reduction procedure from \cite{laxgoeke}. Following a (degenerate) scaling similar to the one in Heineken, Tsuchiya and Aris \cite{hta} we derive a reduction via the approach in \cite{laxgoeke}; to the authors' knowledge, the form of the reduced PDE system has not been known in the literature to date. \\ The reduction is consistent with the spatially homogeneous case, thus setting the diffusion constants equal to zero yields the usual Michaelis-Menten equation. The degenerate scaling seems unavoidable in the PDE case (while one can circumvent it for the ODE), thus we need to go beyond the classical singular perturbation reduction due to Tikhonov and Fenichel. The scaling requires a consistency condition which is intuitively likely to hold in general; we can justify it mathematically in the case when enzyme and complex diffuse at the same rate. In Section 4 we present numerical simulations which exhibit very good agreement with the reduced system. \\
In the Appendix, employing the heuristic method from \cite{laxgoeke}, we carry out the necessary computations for the reductions and also determine suitable initial values for the reduced system.
\section{Preliminaries}
\subsection{The spatially homogeneous setting}
We recall some facts about the Michaelis-Menten reaction with homogeneously distributed concentrations. The evolution of the concentrations $(s,e,c,p)$ of $S,E,C,P$ is governed by the four-dimensional ordinary differential equation
\begin{align*}
&\dot s=-k_1es+k_{-1}c\\
&\dot e=-k_1es+(k_{-1}+k_2)c-k_{-2}ep\\
&\dot c=k_1es-(k_{-1}+k_2)c+k_{-2}ep\\
&\dot p=k_2c-k_{-2}ep.
\end{align*}
This system admits the (stoichiometric) first integrals $\Psi_1(s,e,c,p)=e+c$ and $\Psi_2(s,e,c,p)=s+c+p$. Therefore a two-dimensional system remains:
\begin{align}
&\dot s=-k_1e_0s+(k_1s+k_{-1})c\label{mm1}\\
&\dot c=k_1e_0s-(k_1s+k_{-1}+k_2)c+k_{-2}(e_0-c)(s_0-s-c)\label{mm2},
\end{align}
where $s_0$ and $e_0$ are the initial concentrations of $S$ and $E$, and initially no product $P$ or complex $C$ are present. The system is called irreversible whenever $k_{-2}=0$, and reversible otherwise. The most common quasi-steady state assumption is that the initial enzyme concentration is small, one considers $e_0=\varepsilon e_0^*$ in the asymptotic limit $\varepsilon\to 0$, for the irreversible system.\\
Heineken, Tsuchiya and Aris \cite{hta} were the first to discuss the Michaelis-Menten system from the perspective of singular perturbations, and Segel and Slemrod \cite{ss} were the first to directly prove a rigorous convergence result for the unbounded time interval: Writing \eqref{mm1}--\eqref{mm2} in the slow time scale $\tau=\varepsilon t$
\begin{align}
&s'=-k_1se_0^*+\varepsilon^{-1}(k_1s+k_{-1})c\label{mm1slow}\\
&c'=k_1se_0^*-\varepsilon^{-1}(k_1s+k_{-1}+k_2)c\label{mm2slow},
\end{align}
the solutions of \eqref{mm1slow}--\eqref{mm2slow} converge for all $t_0>0$ uniformly on $[t_0,\infty)$ to the solutions of
\begin{equation}\label{red}
s'= - \frac{k_1k_2se_0^*}{k_1s+k_{-1}+k_2}
\end{equation}
on the asymptotic slow manifold $\mathcal V=\{(s,0),\ s\geq0\}$ as $\varepsilon\to0$. (Below we will sometimes change between time scales without mentioning this explicitly.) \\
Both the approach by Heineken et al. \cite{hta} and the proof by Segel and Slemrod \cite{ss} use appropriate scalings of the variables, in particular they introduce $z:=c/e_0$. It is possible to avoid such a scaling, which becomes degenerate as $e_0\to 0$, in the spatially homogeneous case (see e.g. \cite{gw}) but as it turns out we will need to utilize such a degenerate scaling to obtain a reduction when concentrations are not homogeneously distributed in the reaction vessel.\\
We will also discuss the reversible Michaelis-Menten system, which appears less frequently in the literature; in part this may be due to the unwieldy expression for the QSS reduction; see Miller and Alberty \cite{MiAl}. The singular perturbation reduction (see \cite{nw11} and \cite{gwz2}) of \eqref{mm1slow}--\eqref{mm2slow} for $k_{-2}>0$ and $e_0=\varepsilon e_0^*$ leads to
\begin{equation}\label{redrev}
s'= - \frac{(k_1k_2s+k_{-1}k_{-2}(s-s_0))e_0^*}{k_1s+k_{-2}(s_0-s)+k_{-1}+k_2}
\end{equation}
on the asymptotic slow manifold $\mathcal V=\{(s,0),\ s\geq0\}$ as $\varepsilon\to0$. (Here, uniform convergence again holds on $[t_0,\infty)$; see \cite{lws}). Both the QSS and the singular perturbation reductions agree up to first order in the small parameter; see \cite{gwz2}.
\subsection{The spatially inhomogeneous setting}
When the concentrations are inhomogeneously distributed and diffusion is present then the system is described by a reaction-diffusion equation. Thus, let $\Omega$ be a bounded region with a smooth boundary and let $\delta_s,\delta_e,\delta_c,\delta_p\geq0$ denote the diffusion constants. The governing equations are
\begin{align}
\partial_{t} s&= \delta_s \Delta s-k_1se+k_{-1} c, &\text{in } (0,\infty)\times \Omega \label{mm1diff}\\
\partial_{t} e&= \delta_e\Delta e -k_1se+ (k_{-1}+k_2) c-k_{-2}ep, &\text{in } (0,\infty)\times \Omega \label{mm2diff} \\
\partial_{t} c&= \delta_c\Delta c +k_1se-(k_{-1}+k_2) c+k_{-2}ep, &\text{in } (0,\infty)\times \Omega \label{mm3diff} \\
\partial_{t} p&= \delta_p\Delta p+k_2 c -k_{-2}ep, &\text{in } (0,\infty)\times \Omega \label{mm4diff}
\end{align}
with continuous initial values
\[
s(0,x)=s_0(x),\quad e(0,x)=e_0(x),\quad c(0,x)=c_0(x),\quad p(0,x)=p_0(x),\quad \text{in } \Omega
\]
and one has Neumann boundary conditions
\[
\frac{\partial s}{\partial \nu}=\frac{\partial e}{\partial \nu}=\frac{\partial c}{\partial \nu}=\frac{\partial p}{\partial \nu}=0,\quad \text{in } (0,\infty)\times \partial\Omega
\]
with $\frac{\partial }{\partial \nu}$ denoting the outer normal derivative.
We collect a few general properties.
\begin{remark}\label{remark1}
\begin{itemize}
\item From Smith \cite{Smith}, Ch.~7, Thm.~3.1 and Cor.~3.2--3.3 one sees that all the solution entries remain nonnegative for all $t>0$ whenever they are nonnegative at $t=0$. Moreover, Bothe and Rolland \cite{BotheRolland1} (see in particular Remark 1) have shown that there exists a classical solution of {class $C^{\infty}$} whenever one has initial values of class $W^{s,p}(\Omega;\mathbb{R}^4_+)$ for $p>1$, $s>0$.
\item When $\delta_e=\delta_c$ then
\[
\partial_t(e+c)=\delta_e\Delta(e+c)
\]
and as a consequence of the strong maximum principle (see Smith \cite{Smith} Theorem 2.2) $e+c$ is uniformly bounded by ${\rm max}(e_0+c_0)$ for all $t\geq 0$.\\
Furthermore, in the case that $\delta_s=\delta_e=\delta_c=\delta_p$ one gets
\[
\partial_t(s+e+2c+p)=\delta_e\Delta(s+e+2c+p),
\]
whence $s+e+2c+p$ is bounded by ${\rm max}(s_0+e_0+2c_0+p_0)$ for all $t\geq 0$; in particular nonnegativity implies that every component is bounded.
\item The stoichiometric first integrals of the spatially homogeneous setting survive as conservation laws
\[\frac{1}{\abs{\Omega}}\int_{\Omega}e(0,x)+c(0,x)\: dx=\frac{1}{\abs{\Omega}}\int_{\Omega}e_0(x)+c_0(x)\: dx\]
resp.
\[\frac{1}{\abs{\Omega}}\int_{\Omega}s(0,x)+c(0,x)+p(0,x)\: dx=\frac{1}{\abs{\Omega}}\int_{\Omega}s_0(x)+c_0(x)+p_0(x)\: dx,\]
but a reduction of dimension (i.e., elimination of certain variables) is no longer possible.
\item In the irreversible case one may consider only the first three equations as their right-hand sides do not depend on $p$.
\item Results regarding the long time behavior of solutions of the reversible Michaelis-Menten reaction can be found in Elia\v{s} \cite{elias}.
\end{itemize}
\end{remark}
\section{Reduction given slow diffusion and small initial enzyme concentration}
\subsection{Review of results in the literature}
As noted above, there exists no counterpart to Tikhonov's and Fenichel's theorems for infinite dimensional systems, hence the reduction of reaction-diffusion equations is not possible in a similarly direct manner.\\
Regarding the reduction of the Michaelis-Menten reaction with diffusion, one sometimes finds the one-dimensional equation \eqref{red} augmented by a diffusion term for substrate, with no further argument given. This ad-hoc method is problematic, since it amounts to ignoring diffusion in the reduction step.
The appropriate approach is to start with the full system \eqref{mm1diff}--\eqref{mm4diff} and consider possible reductions in the limiting case of small initial concentration for enzyme, with slow diffusion. This will be the vantage point in the present paper.\\
With regard to such an approach, the authors are aware only of three papers for the irreversible system (i.e. \eqref{mm1diff}--\eqref{mm3diff} with $k_{-2}=0$). Yannacopoulos et al. \cite{Yannacopoulos} assumed $P$ and $C$ to be immobile (i.e. $\delta_e=\delta_c=0$; see their equation (71)) and gave a second order approximation for the case of a one dimensional domain (see in particular equation (80) which in lowest order reduces to the Michaelis-Menten equation for substrate, augmented by diffusion). Britton \cite{britton}, Ch.~8 gave the first order approximation
\begin{equation}\label{immobile}
\partial_{\tau} s= \delta_s \Delta s- \frac{k_1k_2s{(e_0+c_0)}}{k_1s+k_{-1}+k_2}
\end{equation}
which is in agreement with the lowest order terms given in \cite{Yannacopoulos}. He made no assumptions on diffusion constants for enzyme or complex, and instead started with system \eqref{mm1}--\eqref{mm2}, augmented by diffusive terms for $s$ and $c$. This is problematic because the elimination of $e$ via stoichiometry is no longer possible when diffusion is present. Therefore Britton's approach is limited to the case considered by Yannacopoulos at al. \cite{Yannacopoulos}. \\
Kalachev et al. \cite{kkkpz} started from \eqref{mm1diff}--\eqref{mm3diff} and considered up to three time scales, with the slow reaction part of order $\varepsilon$ (the total initial mass of enzyme divided by the total initial mass of substrate), a fast reaction part, and diffusion of order $\delta$, deriving asymptotic expansions for the solutions and reductions in different time regimes. They did not discuss the case that slow reaction and diffusion are in the same time scale (i.e., $\delta=\varepsilon$) which we will consider. (In \cite{kkkpz}, Remark 1.2 further work was announced for this case, but apparently this has not been published yet.)
\subsection{Informal review of the reduction heuristics}\label{heuristic}
We will employ a heuristic method to construct a candidate for a reduced system that was introduced in \cite{laxgoeke}. In contrast to the convergence property for the ODE after discretization (which is a consequence of Tikhonov's and Fenichel's theorems) we will not prove convergence here; generally this seems a very hard task (see Section \ref{conclrem}). However, as remarked in \cite{laxgoeke}, Proposition 4.3, the reduced PDE determined by the heuristics represents the only possible reduction of the reaction-diffusion system as $\varepsilon\to 0$. \\
Briefly the heuristics can be described as follows: By spatial discretization of a reaction-diffusion system which depends on a small parameter $\varepsilon$, one obtains a system of ordinary differential equations depending on $\varepsilon$. If the ODE system admits a Tikhonov-Fenichel reduction and the reduced ODE is the spatial discretization of another partial differential equation system, then we will call the latter {\em the reduced PDE of the reaction-diffusion system}. (The conditions stated above are frequently satisfied; see e.g.\cite{Lax}.) The following results are in part based on the second author's doctoral thesis \cite{Lax}. {Detailed computations will be presented in the Appendix.
\subsection{The irreversible case}\label{mainresultirrev}
In order to determine the reduced PDE systems, we need some preparations.
We consider first the irreversible reaction-diffusion system \eqref{mm1diff}--\eqref{mm3diff}.
Defining total enzyme concentration $y:=e+c$, we get
\begin{align}
\partial_{t} s&= \delta_s \Delta s-k_1s(y-c)+k_{-1} c, &\text{in } (0,\infty)\times \Omega \label{mm1diffy}\\
\partial_{t} c&= \delta_c\Delta c +k_1s(y-c)-(k_{-1}+k_2) c, &\text{in } (0,\infty)\times \Omega\label{mm2diffy}\\
\partial_{t} y&= \delta_c\Delta c + \delta_e(\Delta y -\Delta c), &\text{in } (0,\infty)\times \Omega\label{mm3diffy}
\end{align}
with initial values $s(0,x)=s_0(x)$, $c(0,x)=c_0(x)$, $y(0,x)=e_0(x)+c_0(x)$. Our basic assumptions are:
\begin{itemize}
\item Diffusion is slow, and therefore we introduce the scaling
\[
\delta_z=\varepsilon \delta_z^*\text{ for }z=s,e,c.
\]
\item Total enzyme concentration is small for all $t\geq 0$, and therefore we set
\[
y=\varepsilon y^*{\text{ and } c=\varepsilon c^*},\text{ and also }e_0=\varepsilon e_0^*,\quad c_0=\varepsilon c_0^*.
\]
\end{itemize}
Incorporating these assumptions we have
\begin{align*}
\partial_{t} s&= \varepsilon\delta_s^* \Delta s+\varepsilon(k_1s+k_{-1}) c^*- \varepsilon k_1s y^*, &\text{in } (0,\infty)\times \Omega \\
\partial_{t} c^*&= \varepsilon\delta_c^* \Delta c-(k_1s+k_{-1}+k_2) c^* + k_1s y^*, &\text{in } (0,\infty)\times \Omega\\
\partial_{t} y^*&= \varepsilon\delta_e^*\Delta y^*+\varepsilon(\delta_c^*-\delta_e^*) \Delta c^*, &\text{in } (0,\infty)\times \Omega
\end{align*}
with initial values \[s(0,x)=s_0(x),\quad c^*(0,x)=c_0^*(x),\quad y^*(0,x)=y^*_0(x):=e_0^*(x)+c_0^*(x).\] In slow time $\tau=\varepsilon t$ one now finds
\begin{align}
\partial_{\tau} s&= \delta_s^* \Delta s+(k_1s+k_{-1}) c^*- k_1s y^*, &\text{in } (0,\infty)\times \Omega \label{mm1diffskal}\\
\partial_{\tau} c^*&= \delta_c^* \Delta c^*-\varepsilon^{-1}(k_1s+k_{-1}+k_2) c^* + \varepsilon^{-1} k_1s y^*, &\text{in } (0,\infty)\times \Omega\label{mm2diffskal}\\
\partial_{\tau} y^*&= \delta_e^*\Delta y^*+\delta \Delta c^*, &\text{in } (0,\infty)\times \Omega\label{mm3diffskal}
\end{align}
with the abbreviation
\begin{equation}\label{delteq}
\delta:=\delta_c^*-\delta_e^*.
\end{equation}
We will discuss two different cases: If the diffusion constants $\delta_e^*$ and $\delta_c^*$ are close in the sense that $\delta=\varepsilon \delta^*$, then equation \eqref{mm3diffskal} reads
\begin{equation}
\partial_{\tau} y^*= \delta_e^*\Delta y^*+\varepsilon\delta^* \Delta c
\end{equation}
and the reduced system for $\varepsilon\to 0$ is again a reaction-diffusion system (with a rational reaction term). Otherwise, the reduced system becomes highly nonlinear.
\begin{remark}
The argument is based on the critical assumption that the ``degenerate'' scalings $c^*=\varepsilon^{-1} c$ and $y^*=\varepsilon^{-1} y$ hold for all $t\geq 0$; to state it more precisely, one needs a uniform bound (with respect to $\varepsilon$) for $c^*$ and $y^*$.
In the special case $\delta_c^*=\delta_e^*$ (e.g. if the molecules of enzyme and complex are of the same size; see Keener and Sneyd \cite{ksI}, Subsection 2.2.2), Remark \ref{remark1} implies that $c^*$ and $y^*$ are uniformly bounded by $e_0^*+c_0^*$. We are not able to extend this property to the case $\delta_c^*\neq\delta_e^*$, but we will verify in the Appendix that the corresponding uniform boundedness property holds for the ODEs obtained via discretization. Furthermore, numerical results indicate that degenerate scaling poses no problem for the Michaelis-Menten system (see Section \ref{numeric}).
\end{remark}
\subsubsection{Irreversible case with $\delta_c^*-\delta_e^*=\mathcal O(\varepsilon)$}\label{mainresultirrevsub1}
In this case the reduced PDE (as defined in subsection \ref{heuristic}) is given by
\begin{align}
\partial_{\tau} s&= \delta_s^* \Delta s-\frac{k_1k_2y^*s}{k_1s+k_{-1}+k_2}, &\text{in } (0,T)\times \Omega\label{mm1diffred1} \\
\partial_{\tau} y^*&= \delta_e^*\Delta y^*, &\text{in } (0,T)\times \Omega \label{mm3diffred1}
\end{align}
on the asymptotic slow manifold \[\mathcal V=\left\{(s,c^*,y^*)\in\mathbb{R}^3_+,\ c^*=\frac{k_1s y^*}{k_1s+k_{-1}+k_2}\right\}.\]
Appropriate initial values on $\mathcal V$ are given by
$(\tilde s_0,\tilde y_0^*)=\left(s_0,y_0^*\right)$.
This assertion is a direct consequence of Proposition \ref{irrevdiscred} in the Appendix.\\
Total enzyme concentration in the reduced equation is subject only to diffusion, and there remains a reaction-diffusion equation for substrate, with the reaction part similar to the usual Michaelis-Menten term. It is worth looking at some special cases: When $\delta_e^*=\delta_c^*=0$, $y^*=y_0^*$ is constant in time and we have
\begin{equation*}
\partial_{\tau} s= \delta_s \Delta s- \frac{k_1k_2s_0y_0^*}{k_1s+k_{-1}+k_2}
\end{equation*}
as in Yannacopoulos et al. \cite{Yannacopoulos}, Equation (80) and in Britton \cite{britton}, Ch.~8. Moreover, setting all diffusion constants to zero (and assuming $c_0^*=0$ as well as constant $y_0^*$) leads to the usual spatially homogeneous reduction as given in \eqref{red}.\\
As far as the authors know, this reduced system has not appeared in the literature so far. The numerical simulations in Section \ref{numeric} indicate convergence.
\subsubsection{Irreversible case with $\delta_c^*-\delta_e^*=\mathcal O(1)$}\label{mainresultirrevsub2}
In this case the reduction is given by
\begin{align}
\partial_{\tau} s&= \delta_s^* \Delta s-\frac{k_1k_2y^*s}{k_1s+k_{-1}+k_2}, &\text{in } (0,T)\times \Omega\label{mm1diffred} \\
\partial_{\tau} y^*&= \delta_e^*\Delta y^*+\delta \Delta\left(\frac{k_1y^*s}{k_1s+k_{-1}+k_2}\right), &\text{in } (0,T)\times \Omega \label{mm3diffred}
\end{align}
on the asymptotic slow manifold \[\mathcal V=\left\{(s,c^*,y^*)\in\mathbb{R}^3_+,\ c^*=\frac{k_1s y^*}{k_1s+k_{-1}+k_2}\right\}.\]
The appropriate initial values are as before (also following from Proposition \ref{irrevdiscred}).\\
This case may be said to correspond to the one mentioned but not treated in Kalachev et al. \cite{kkkpz}; there seems to be no discussion of this in the literature. Note that now the equations for $s$ and $y^*$ are fully coupled; this is a more complex situation than before. Again, numerical simulations (Section \ref{numeric}) are in good agreement with the reduction.
\subsection{The reversible case}\label{mainresultrev}
We will determine a reduced system for the reversible Michaelis-Menten reaction with diffusion, i.e.,
\begin{align*}
\partial_{\tau} s&= \delta_s^* \Delta s+(k_1s+k_{-1}) c^*- k_1s y^*, &\text{in } (0,T)\times \Omega \\
\partial_{\tau} c^*&= \delta_c^* \Delta c^*-\varepsilon^{-1}[(k_1s+k_{-1}+k_{-2}p+k_2) c^* + (k_1s+k_{-2}p) y^*], &\text{in } (0,T)\times \Omega\\
\partial_{\tau} y^*&= \delta_e^*\Delta y^*+\delta \Delta c^*, &\text{in } (0,T)\times \Omega\\
\partial_{\tau} p&= \delta_p^* \Delta p+(k_{-2}p+k_{2}) c^*- k_{-2}py^*, &\text{in } (0,T)\times \Omega
\end{align*}
Here we choose the same scaling as in \eqref{mm1diffskal}--\eqref{mm3diffskal} and additionally we let $\delta_p=\varepsilon \delta_p^*$. If $\delta_c^*-\delta_e^*=\mathcal O(\varepsilon)$ then we get
\begin{align}
\partial_{\tau} s&= \delta_s^* \Delta s-\frac{(k_1k_2s-k_{-1}k_{-2}p)y^*}{k_1s+k_{-2}p+k_{-1}+k_2}, &\text{in } (0,T)\times \Omega\label{mm1diffred3} \\
\partial_{\tau} y^*&= \delta_e^*\Delta y^*, &\text{in } (0,T)\times \Omega \label{mm3diffred3}\\
\partial_{\tau} p&= \delta_p^*\Delta p+\frac{(k_1k_2s-k_{-1}k_{-2}p)y^*}{k_1s+k_{-2}p+k_{-1}+k_2}, &\text{in } (0,T)\times \Omega \label{mm4diffred3}
\end{align}
on the asymptotic slow manifold \[\mathcal V=\left\{(s,c^*,y^*,p)\in\mathbb{R}^4_+,\ c^*=\frac{(k_1s+k_{-2}p) y^*}{k_1s+k_{-2}p+k_{-1}+k_2}\right\}.\] Note that \eqref{mm3diffred3} is uncoupled from the remaining system.
In case $\delta_c^*-\delta_e^*=\mathcal O(1)$ we get
\begin{align}
\partial_{\tau} s&= \delta_s^* \Delta s-\frac{(k_1k_2s-k_{-1}k_{-2}p)y^*}{k_1s+k_{-2}p+k_{-1}+k_2}, &\text{in } (0,T)\times \Omega\label{mm1diffred4} \\
\partial_{\tau} y^*&= \delta_e^*\Delta y^*+\delta\Delta\left(\frac{(k_1s+k_{-2}p)y^*}{k_1s+k_{-2}p+k_{-1}+k_2}\right), &\text{in } (0,T)\times \Omega \label{mm3diffred4}\\
\partial_{\tau} p&= \delta_p^*\Delta p+\frac{(k_1k_2s-k_{-1}k_{-2}p)y^*}{k_1s+k_{-2}p+k_{-1}+k_2}, &\text{in } (0,T)\times \Omega \label{mm4diffred4}
\end{align}
on the same asymptotic slow manifold,
but here one has a fully coupled system for $s$, $y^*$ and $p$.\\
The proofs follow from Proposition \ref{revdiscred}.
In both settings, appropriate initial values are given by
$(\tilde s_0,\tilde y_0^*,\tilde p_0)=\left(s_0,y_0^*,p_0\right)$.\\
Again, setting all diffusion constants to zero (and assuming $c_0^*=p_0=0$ as well as constant $y_0^*$ and $s_0$) leads to $s(\tau,x)+p(\tau,x)=s_0$ and thus to the usual reduction as given in \eqref{redrev}.
\section{Numerical simulations}\label{numeric}
In the following we will provide numerical results that are in good agreement with the reduction given above.
The solutions have been obtained using MATLAB's \texttt{pdepe} function. This function solves an initial-boundary value problem for spatially one-dimensional systems of parabolic and elliptic partial differential equations in the self-adjoint form
$$
C(x,t,u,\partial_x u)\partial_t u = x^{-m} \partial_x (x^m F(x,t,u,\partial_x u)) + S(x,t,u,\partial_x u).
$$
In our case, $m=0$ and $C$ is the identity matrix.
Furthermore, in the case of system \eqref{mm1diffskal}--\eqref{mm3diffskal}, using the unknown $u = ( s, c^*, y^*)^T$, the flux $F$ and the source $S$ become
$$
F = \begin{pmatrix} \delta_s\partial_x s \\ \delta_c^*\partial_x c^* \\ \delta_e^*\partial_x y^* +\delta\partial_x c^* \end{pmatrix}, \quad
S = \begin{pmatrix} (k_1s +k_{-1})c^* -k_1s y^* \\ \varepsilon^{-1}(k_1s +k_{-1}+k_2) c^* + \varepsilon^{-1} k_1s y^* \\ 0 \end{pmatrix}.
$$
The reduced system \eqref{mm1diffred}--\eqref{mm3diffred}, using the unknown $u = ( s, y^*)^T$, the flux $F$ and the source $S$ become
$$
F = \begin{pmatrix} \delta_s\partial_x s \\ \frac{\delta k_1y(k_{-1}+k_2)}{(k_1s+k_{-1}+k_2)^2}\partial_xs+(\frac{\delta k_1s}{k_1s+k_{-1}}+\delta_e^*)\partial_x y^* \end{pmatrix}, \quad
S = \begin{pmatrix} \frac{k_1 k_2 s y^*}{k_1 s+k_{-1}+k_2} \\ 0 \end{pmatrix}.
$$
As boundary conditions, we use homogeneous Neumann boundary conditions, i.e., for each unknown we set the spatial derivative equal to zero at the boundary.
The \texttt{pdepe} function uses a self-adjoint finite difference semi-discretization in space, and solves the obtained system ordinary differential equations by the implicit, adaptive multistep solver \texttt{ode15s}. In all our experiments we have set the tolerances to values below the accuracy we intend to observe (absolute tolerance $10^{-14}$, relative tolerance $10^{-10}$). We have used 100 equidistant grid cells.
Figure \ref{fig:InitialCondition} shows the initial condition we have used; a step function in $s$, a smooth cosine profile for $c$, and a cosine profile with an additional Gaussian bump for $y$. We have set $\delta_s=\delta_e=k_1=k_{-1}=k_2=1$ and $\delta_c=2$ (so $\delta=1$; see case \ref{mainresultirrevsub2}).
Figure \ref{fig:sol10} shows the solutions at time $T=0.005$ for $\varepsilon=1.0$. Already, one can see that the concentration $s$ is described well by the reduced system, whereas we see a discrepancy in $y$. For $\varepsilon=0.0001$, shown in Figure \ref{fig:sol00001}, to the eye there is no difference between the solutions of the original and the reduced systems. In Figure \ref{fig:convergence} we investigate the convergence of the solution of the full system to the solution of the reduced system. The error is measured in the $L^\infty$ norm in all three solution components. As $\varepsilon\to 0$, we observe rather clean first-order convergence in double-logarithmic plot. Finally, we also set $\delta_c=1$ (so $\delta=0$; see case \ref{mainresultirrevsub1}) and measure in Figure \ref{fig:convergenceglsc} and Figure \ref{fig:convergencegly} again the error. This confirms what the theory has predicted.
\begin{figure}
\centering\includegraphics[width=0.8\linewidth]{initial.pdf}
\caption{Initial condition for $s$, $c^*$ and $y^*$.}
\label{fig:InitialCondition}
\end{figure}
\begin{figure}
\centering\includegraphics[width=0.8\linewidth]{solungle1.pdf}
\caption{Solutions $s$ and $y^*$ at time $T=0.005$. Comparison between Michaelis-Menten and reduced system for $\varepsilon=1.0$.}
\label{fig:sol10}
\end{figure}
\begin{figure}
\centering\includegraphics[width=0.8\linewidth]{solungle00001.pdf}
\caption{Solutions $s$ and $y^*$ at time $T=0.005$. Comparison between Michaelis-Menten and reduced system for $\varepsilon=0.0001$.}
\label{fig:sol00001}
\end{figure}
\begin{figure}
\centering\includegraphics[width=0.8\linewidth]{convergenceungl.pdf}
\caption{Convergence of the full solution to the reduced solution as $\varepsilon\to 0$. Error measured in the $L^\infty$ norm.}
\label{fig:convergence}
\end{figure}
\begin{figure}
\centering\includegraphics[width=0.8\linewidth]{convergenceglsc.pdf}
\caption{Convergence of the full solution to the reduced solution for equal diffusion constants as $\varepsilon\to 0$. Error measured in the $L^\infty$ norm.}
\label{fig:convergenceglsc}
\end{figure}
\begin{figure}
\centering\includegraphics[width=0.8\linewidth]{convergencegly.pdf}
\caption{Convergence of the full solution to the reduced solution for equal diffusion constants as $\varepsilon\to 0$. Error measured in the $L^\infty$ norm.}
\label{fig:convergencegly}
\end{figure}
\section{Concluding remarks}\label{conclrem}
\begin{itemize}
\item As already noted, we do not discuss convergence results. But it is easy to see that the uniform bound for $y^*$ implies that $c$ converges uniformly to 0 as $\varepsilon\to0$. Moreover, up to taking a subsequence, $c^*:=\varepsilon^{-1}c$ and $y^*$ converge $\text{weakly}^*$ in $C^{0}$ and weakly in $L^{p}$ for all $1<p<\infty$. This may be a starting point for a convergence proof.
\item As already mentioned, this above reductions can be obtained only after a degenerate scaling of certain variables; then a Tikhonov-Fenichel reduction is applicable. (The corresponding scaling by Heineken et al. \cite{hta} in the ODE case is convenient, but not necessary.) This may also be the underlying reason why the approach by Yannacopoulos et al. \cite{Yannacopoulos} was not directly applicable to the given setting. The scaled quantities $y^*$ and $c^*$ can be seen as first order approximations of $y$ and $s$ of the solution of \eqref{mm1diffy}--\eqref{mm3diffy} (with respect to the assumptions regarding slow diffusion and small total initial enzyme concentration) where the zero order terms are equal to zero. The effect of degenerate scalings in general is investigated in a forthcoming paper \cite{lw2}.
\item A reduction similar to the one above was already given in the dissertation \cite{Lax}, but it was based on writing the system in $(s,e,c,p)$ and scaling both $e=\varepsilon e^*$ and $c=\varepsilon c^*$. The reduced system is equivalent to the reduced system given here. We chose to change the variables to $(s,c,y,p)$ in order to emphasize the resemblance to the non-diffusive case which is otherwise lost.
\item It is also possible to only scale $y$ instead of both $c$ and $y$ (and still obtain that $c$ will be of order $\varepsilon$). But there are some disadvantages: The computation of the reduced system gets more involved as the results of \cite{laxgoeke} cannot be used directly. Moreover, we only get a zero order approximation to the slow manifold, given by $c=0$.
\item Different QSS assumptions are also being discussed in the literature. Various choices of small rate constants can be found in \cite{laxgoeke,Lax}; for example the assumptions of slow product formation ($k_2=\varepsilon k_2^*$) and slow diffusion ($\delta_z=\varepsilon \delta_z^*\text{ for }z=s,e,c,p$) as well as only slow product formation are discussed.\\
Moreover, the assumption of slow complex formation ($k_1=\varepsilon k_1^*$ and $k_{-2}=\varepsilon k_{-2}^*$) and slow diffusion can be discussed by employing the method developed in \cite{laxgoeke}. A reduced system is given by
\begin{align*}
\partial_{\tau} s&= \delta_s \Delta s-\frac{k_1k_2}{k_{-1}+k_2}se+\frac{k_{-1}k_{-2}}{k_{-1}+k_2} ep\\
\partial_{\tau} e&= \delta_e\Delta e \\
\partial_{\tau} p&= \delta_p\Delta p+\frac{k_1k_2}{k_{-1}+k_2}se-\frac{k_{-1}k_{-2}}{k_{-1}+k_2} ep
\end{align*}
on the slow manifold defined by $c=0$. This corresponds to the convergence results of Bothe and Pierre \cite{BothePierre1} and Bisi et al. \cite{bisi} for a related system which is defined by the reaction $A_1+A_2 \rightleftharpoons A_3 \rightleftharpoons A_4+A_5$. (Note that the latter reaction is easier to analyze, due to the structure of the conservation laws; see Elia\v{s} \cite{elias}). In all cases, the numerical results are in good agreement with the reduction.
\item By analogous methods one can derive a reduction given the assumption of small total initial enzyme concentration, but with fast diffusive terms. Scaling again $y=\varepsilon y^*$ and $c=\varepsilon c^*$ and using results of \cite{Lax} one obtains the classical reduction: the fast diffusion yields a homogenization of the concentrations, enzyme and complex are in QSS and the reduced dynamics of the substrate are described by \eqref{red} (again, the reduction is in good agreement with numerical results). We omit details here.
\end{itemize}
\section{Acknowledgement}
The second-named author was supported by the DFG Research Training Group ``Experimental and Constructive Algebra'' (GRK 1632).
| {
"timestamp": "2017-07-14T02:04:36",
"yymm": "1707",
"arxiv_id": "1707.04043",
"language": "en",
"url": "https://arxiv.org/abs/1707.04043",
"abstract": "The Michaelis-Menten mechanism is probably the best known model for an enzyme-catalyzed reaction. For spatially homogeneous concentrations, QSS reductions are well known, but this is not the case when chemical species are allowed to diffuse. We will discuss QSS reductions for both the irreversible and reversible Michaelis-Menten reaction in the latter case, given small initial enzyme concentration and slow diffusion. Our work is based on a heuristic method to obtain an ordinary differential equation which admits reduction by Tikhonov-Fenichel theory. We will not give convergence proofs but we provide numerical results that support the accuracy of the reductions.",
"subjects": "Analysis of PDEs (math.AP)",
"title": "Quasi-steady state reduction for the Michaelis-Menten reaction-diffusion system",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808736209154,
"lm_q2_score": 0.7217432122827968,
"lm_q1q2_score": 0.7075110666665458
} |
https://arxiv.org/abs/0902.4762 | A combinatorial analysis of interacting diffusions | We consider a particular class of n-dimensional homogeneous diffusions all of which have an identity diffusion matrix and a drift function that is piecewise constant and scale invariant. Abstract stochastic calculus immediately gives us general results about existence and uniqueness in law and invariant probability distributions when they exist. These invariant distributions are probability measures on the $n$-dimensional space and can be extremely resistant to a more detailed understanding. To have a better analysis, we construct a polyhedra such that the inward normal at its surface is given by the drift function and show that the finer structures of the invariant probability measure is intertwined with the geometry of the polyhedra. We show that several natural interacting Brownian particle models can thus be analyzed by studying the combinatorial fan generated by the drift function, particularly when these are simplicial. This is the case when the polyhedra is a polytope that is invariant under a Coxeter group action, which leads to an explicit description of the invariant measures in terms of iid Exponential random variables. Another class of examples is furnished by interactions indexed by weighted graphs all of which generate simplicial polytopes with $n !$ faces. We show that the proportion of volume contained in each component simplex corresponds to a probability distribution on the group of permutations, some of which have surprising connections with the classical urn models. | \section{Introduction}
Consider the following two examples of stochastic processes. The first one, called the bang-bang process is classical and is particularly important in stochastic control theory. See the articles by Karatzas and Shreve \cite{karshbang}, Shreve \cite{shrevebang}, or the chapter by Warnecke \cite{warbang}. This is a one dimensional diffusion which solves the following SDE with a single real parameter $\alpha$:
\[
dX_t = -\alpha\;\text{sign}(X_t)dt + dW_t.
\]
It can be easily shown that the process is recurrent if and only if $\alpha$ is positive. In that case, the process has a unique reversible stationary distribution under which $\abs{X_t}$ is distributed as Exponential with rate $2\alpha$.
The other example is the class of Brownian motions with rank-based interactions. This is a family of $n$ one dimensional diffusions which is parametrized by a single vector $\delta$ in $\mathbb{R}^n$. These diffusions have an identity diffusion matrix and a drift that depends on the order in which the coordinates can be arranged in increasing values. If we think of each diffusion as recording the position of a particle moving on a line, then at any instant of time the particle with the $i$th smallest position gets an instantaneous drift $\delta_i$. The formal SDE for the diffusion can be described by
\begin{equation}\label{ranksde}
d X_t (i) = \sum_{j=1}^n \delta_j 1\left\{\;X_t(i)= X_{t}[j]\;\right\} dt + dW_t(i), \quad i =1,2,\ldots,n,
\end{equation}
where $X_{t}[1] \le X_{t}[2] \le \ldots \le X_{t}[n]$ are the coordinates arranged in increasing order. The $W_t(i)$'s are asumed to be independent Brownian motions for some suitable underlying filtration.
The rank-based interacting Brownian motions or closely related models have appeared in several veins of the literature. Extensive reviews can be found in the articles by Pal and Pitman \cite{palpitman} and Chatterjee and Pal \cite{chatpal}. Some of the recent work include the articles by Jourdain and Malrieu \cite{joumal} and Banner, Fernholz, and Karatzas \cite{atlasmodel}. Also see the related discrete time models by Ruzmaikina and Aizenman \cite{ruzaizenman}, and L-P. Arguin \cite{arguinPD}.
The recurrence, transience, and other long term properties of the rank-based Brownian motions can be precisely determined. The following result is from \cite{palpitman}.
\begin{thm}[Theorem 4 in \cite{palpitman}]\label{theoremN}
For $1 \le k \le n$ let
\begin{equation}
\label{alphak}
\alpha_k:= \sum_{i = 1}^k ( \delta_i - \bar{\delta} ), \quad \bar{\delta}:=\frac{1}{n}\sum_{i=1}^n\delta_i.
\end{equation}
For each fixed initial distribution of the $n$ particle system with
drifts $\delta_i$, the collection of laws of
$X_t[n] - X_t{[1]}$ for $t \ge 0$ is tight if
and only if
\begin{equation}
\label{conal}
\alpha _k >0 \mbox{ for all } 1 \le k \le N-1,
\end{equation}
in which case the following result holds:
The distribution of the spacings system $(X_t{[j+1]} - X_t{[j]},\;1\le j\le n-1)$ at time $t$ converges in total
variation norm as $t$ tends to infinity to a unique stationary distribution for
the spacings system, which is that of independent Exponential
variables $Y_k$ with rates $2\alpha_k$, $1 \le k \le n-1$.
Moreover, the spacings system is reversible at equilibrium.
\end{thm}
The independence of the spacings under the invariant distribution is somewhat puzzling since, due to the interaction, there is no independence between the spacing processes. The proof, which does not shed light on this phenomenon, invokes Williams's results on reflected Brownian motions \cite{williams87r}.
There are some similarities between the two examples. Both involve drift functions that are piecewise constant; in fact the drift is a single vector under the action of a group (sign flips for the former and permutations for the latter). Moreover, in both cases the invariant distribution involves independent Exponentials which provide a friendly description of an otherwise abstract probability measure.
We provide alternate proofs of these results as an application of the following general theory. Notice that the drift function in both the examples is the negative of the gradient (in the sense of distributions) of a positively homogenous (homogenous of degree one) function in $\mathbb{R}^n$. For the bang-bang process, this function is $\alpha\abs{x}$, while for the rank-based processes it is given by $-\sum_{i=1}^n \delta_i x[i]$. Any continuous positively homogenous function which is nonnegative for all $x$ can be viewed as the Minkowski (or, the gauge) function of a certain body containing the origin. Since we consider piecewise constant drifts, these bodies are $n$ dimensional polyhedras. The corresponding stochastic process is shown to be recurrent when the polyhedra is a bounded polytopes, which in turn happens when $k$ is strictly positive for all $x\neq 0$. When this is the case, the invariant distribution for the diffusion can be obtained from the uniform distribution on the polytope.
Let us now focus on the uniform distribution on an $n$-dimensional polytopes. If the polytope is a unit simplex $\{x: x_i \ge0,\; \sum_i x_i \le 1\}$, the uniform distribution can be effectively generated by dividing iid Exponential random variables by their total sum. Now suppose a polytope can be triangulated, i.e., decomposed as a union of $K$ simplices with disjoint interiors. It is a well-known theorem that such a triangulation is possible for every polytope. Every component simplex, say $S_i$, in the triangulation is a linear map $A_i$ of the unit simplex. Thus, an algorithm to generate a point uniformly from the polytope would be to generate a point $X$ uniformly from the unit simplex, choose $I$ between $\{ 1,2,\ldots,K\}$ with probability
\begin{equation}\label{probi}
P\left( I=i \right)=\frac{\text{Vol}(S_i)}{\sum_{j=1}^K \text{Vol}(S_j)}, \quad i=1,2,\ldots,K,
\end{equation}
and let $Y=A_I X$. Such an $Y$ is clearly uniformly chosen from the polytope.
A particularly explicit triangulation exists when the polytope is simplicial. That is, each of its extremal face is an $(n-1)$ dimensional simplex. One can then simply connect the origin (which is in the interior) with each of these faces to generate a nice triangulation. How does one check if a polytope is simplicial ? We demonstrate a simple condition when the symmetry group of the polytope is Coxeter, i.e., generated purely by reflections. Coxeter groups find applications in several ares of mathematics. The finite Coxeter groups include symmetries of regular polytopes and the Weyl groups of simple Lie algebras. They are usually defined formally as a set of generators and relations among them. However, we consider them in their original geometric form as treated by H. S. M. Coxeter in the classics \cite{cox1} and \cite{cox2}. The definitions and properties of irreducible group actions and Coxeter groups have been described in Subsection \ref{groupcase}. This is a particularly nice case, when not only the polytope is simplicial, but it is also regular, and hence the random variable $I$ is uniformly distributed among $\{1,2,\ldots,K\}$. To connect the dots with independent Exponentials, we simply need to describe the maps $A_i$'s. In fact, due to regularity, determining $A_1$ is enough, since the other maps are merely orbits under the group action. This is exactly the case for the bang-bang or the rank-based processes. The regularity corresponds to exchangeability among the particles, i.e., the condition that if the initial distribution is exchangeable among the coordinates, then so is the distribution at every other point of time.
The second case we consider is not regular and does not involve any groups. The interaction is parametrized by all graphs with $n$ vertices and possible edge-weights. In this case, the maps $A_i$'s are simple and explicit. However, the probabilities in \eqref{probi} are not. In fact, these probabilities correspond to probabilities of various orderings of particles, increasingly arranged, under the invariant distribution. Hence, these polytopes induce probability distributions on permutations of $n$ labels. We take up a few examples and show surprising connections with existing probability models on permutations.
\medskip
In the regular case, one of the results we prove is the following.
\begin{prop}\label{gpdiff1}
Consider the SDE
\begin{equation}\label{sdegenform}
d X_t = b(X_t) dt + dW_t, \quad \text{where}\; W_t=(W_t(1), W_t(2),\ldots, W_t(n))
\end{equation}
is an $n$-dimensional Brownian motion. Assume that the drift function $b:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is piecewise constant and satisfies
\begin{equation}\label{bcond}
b(\alpha x)=b(x), \quad \forall\;\alpha >0, \quad \text{and},\quad b(Ax)=Ab(x), \quad \forall\; A\in G,
\end{equation}
where $G$ is a finite irreducible group of orthogonal matrices.
Then the following conclusions hold.
\begin{enumerate}
\item Let $k(x)=-\iprod{x,b(x)}$. A sufficient condition for $X$ to be recurrent is that, for some non-zero vector $\lambda \in \mathbb{R}^n$, we have
\[
k(x)=\max_{A\in G} \iprod{A\lambda, x}.
\]
In that case $X$ has a unique, reversible invariant distribution $\mu$ on $\mathbb{R}^n$. The marginal law of $X_t$ converges in total variation to $\mu$ as $t$ tends to infinity.
\item If $G$ is Coxeter, there is a set of $n$ many linearly independent vectors $\{\eta_1, \eta_2, \ldots, \eta_n \}$ and $n$ many constants $\{\alpha_1, \alpha_2,\ldots,\alpha_n\}$ such that under $\mu$, the random variables
\[
Y_i= \alpha_i\iprod{A\eta_i, x}, \quad \text{if}\quad k(x)=\iprod{A\lambda,x}, \quad i=1,2,\ldots,n,
\]
are iid exponential random variables with rate two.
\item Additionally, if the stabilizer subgroup of $\lambda$ in $G$ is trivial, then the vectors $\{\eta_1, \eta_2, \ldots, \eta_n \}$ are determined as the generators of the conic hull of the set of vectors $\{\lambda - A\lambda, \; A\in G \}$. That is to say, every vector in the set $\{\lambda - A\lambda, \; A\in G \}$ can be represented as a linear combination of the subset $\{\eta_1, \eta_2, \ldots, \eta_n \}$ with nonnegative coefficients.
The constants $\alpha_1,\ldots,\alpha_n$ are the unique positive coefficients of $\eta_1,\ldots,\eta_n$ in the expansion
\begin{equation*}
\lambda= \sum_{i=1}^n \alpha_i \eta_i.
\end{equation*}
\end{enumerate}
\end{prop}
Note that, we really do not need to know the details of the group structure to apply the previous result, except for the information that $G$ is Coxeter. As we show in the examples, necessary and sufficient conditions can be obtained if we have a better knowledge of the group structure. Finally, let us mention that a list of Coxeter groups up to isomorphisms is available and can be found in any standard textbook, say \cite{reflectiongroups}.
In Subsection \ref{examples}, we describe several families of interacting diffusions that can be analyzed by the previous theorem. They all appear as solutions to stochastic differential equations of the type \eqref{sdegenform} with a piecewise constant drift function satisfying conditions \eqref{bcond}, but involving different families of orthogonal groups.
When the group is the group of permutation matrices, we get back rank based interactions. Using Proposition \ref{gpdiff1}, we provide an alternative proof (not involving reflected Brownian motions) of the Pal-Pitman result.
The second class of examples are called \textit{sign-rank based interactions}. Here the drift vector not only changes when the coordinate values get permuted, but also, when when the signs of coordinates change. The relevant group is the one which generated by all the permutation matrices and all the diagonal matrices whose diagonal elements are either plus or minus one. In one dimension, this boils down to the simple Bang-bang process.
The third class of examples are similar to sign-rank based processes, but with more constraints. Here, too, the drift vector changes when we permute coordinates. It also changes when we change signs of coordinates, but only when done in pairs. The group behind the curtain is generated by permutation matrices and diagonal matrices whose diagonal elements are $\pm 1$ with the additional constraint that only even number of $-1$'s are allowed.
Readers acquainted with the theory of Coxeter groups will recognize that the previous three examples correspond to the three well-known families of Coxeter groups, denoted by $\mathcal{A}_n$, $\mathcal{B}_n$, and $\mathcal{D}_n$ for each $n\in \mathbb{N}$. In each case Proposition \ref{gpdiff1} allows us to formulate a simple sufficient condition for checking the existence of a unique invariant probability distribution and provides a complete description of the distribution in terms of independent Exponentials.
\medskip
In the case of interaction through graphs we consider the following class of interacting diffusions. Let $\mathcal{G}$ be a graph on $n$ vertices where the vetices are labeled by $\{1,2,\ldots,n\}$. The edge between $i$ and $j$ have an associated edge weight $\beta_{ij}$, which is zero if there is no edge between the two vertices. Consider the SDE on $\mathbb{R}^n$ given by
\[
dX_t(i) = \sum_{j=1}^n\beta_{ij}\text{sign}\left( X_t(j) - X_t(i) \right)dt + dW_t(i),\quad i=1,2,\ldots,n,
\]
where, $W$ is again an $n$-dimensional Brownian motion.
When all the edge weights are nonnegative, the model can be described by saying that the Brownian motions, which are indexed by the vertices of the graphs, get attracted towards one another. The constants $\beta_{ij}$ measure the strength of their attraction.
Unless the graph is the complete graph with constant edge-weights, the interaction is not regular. However, if we define
\[
\bar{X_t} = \frac{1}{n}\sum_{i=1}^n X_t(i),
\]
the centered vector $(X(1)-\bar{X}, X(2) - \bar{X}, \ldots, X(n) - \bar{X})$ have an invariant distribution whenever the graph is connected and the edge-weights are nonnegative.
For this class of interactions our main focus of investigation is the law of random permutation that takes indices of coordinates to their ranks under the invariant distribution. This is not uniform by virtue of not being regular. As an interesting example, we consider the case when we assume that each particle has a mass $m_i$, $i=1,2,\ldots,n$, and that $\beta_{ij}=m_im_j$ in the sense that the strength of the mutual attraction is proportional to the product of their masses. Due to this \textit{gravitational} intuition, under the invariant distribution, we should expect heavier particles to stay at the middle of the pile, while the lighter ones should be at the edge (the Sun being at the center, and Pluto at the far end).
In general, this is very difficult to prove. However, in one particular case, this becomes apparent. For any $\alpha > 0$, consider $n$ particles with the interaction described in the previous paragraph, where the mass of the first particle is $\alpha$ and the rest of the masses are $1$. Let $\sigma(1)$ denote the rank of the first particle under the invariant distribution of the centered vector $(X(1)-\bar{X}, X(2) - \bar{X}, \ldots, X(n) - \bar{X})$. A surprising connection with Polya's urn scheme emerges. We prove the following.
\begin{prop}\label{massbeta1} Consider a Polya's urn scheme which initially has $2\alpha$ red balls and $2\alpha$ black balls. At every step one picks ups a ball at random, returns the ball to the urn and adds an extra ball of the same color. Then, the distribution of $\sigma(1)-1$ is the same as the number of red balls picked when we run the urn scheme described above for $n-1$ steps.
In particular, the sequence of random variables $\sigma(1)/n$ converges weakly to the Beta$(2\alpha,2\alpha)$ distribution as $n$ tends to infinity.
\end{prop}
\subsection{Outline of the paper}
In the next section we describe the set-up of the paper and prove general results about recurrence of interacting diffusions and their invariant distributions when they exist. Section \ref{cones} describes the combinatorics involved in the invariant distributions. Subsection \ref{groupcase} describes the connection with Coxeter groups followed by several examples in Subsection \ref{examples}. The following subsection \ref{graphcase} proves results about interactions parametrized by graphs.
\section{Diffusions with piecewise constant drift}
Consider a sequence of $n$-dimensional cones $C_1, C_2, \ldots, C_r$ whose interiors are disjoint and the closure of their union is the entire space $\mathbb{R}^n$. Ignoring their mutual intersections (a set of measure zero) they can be thought of as a partition of $\mathbb{R}^n$. Let $b:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a function that is constant over each $C_i$. In this section we prove some general results about the class of diffusions which satisfy the following stochastic differential equation:
\begin{equation}\label{sde1}
d X_t = b(X_t) dt + dW_t,
\end{equation}
where $W_t=(W_t(1), W_t(2),\ldots, W_t(n))$ is an $n$-dimensional Brownian motion.
The existence and uniqueness in law is immediate by an application of Girsanov's theorem. Define the function $k(x)= -\iprod{x,b(x)}$. Since $b$ is constant over cones, it follows that $k$ is a positively homogeneous function. For example, $k$ could be the Minkowski functional (i.e., the gauge function) of a convex body containing the origin. If $k$ is continuous, by virtue of being piecewise linear, it follows easily that $b$ is the negative of the gradient of $k$ in the sense of distributions. In that case the SDE in \eqref{sde1} is an example of the Langevin equation. The following proposition is well-known about Langevin SDE.
\begin{prop}\label{invariantdist}
Consider the stochastic differential equation \eqref{sde1}. Let $k:\mathbb{R}^n \rightarrow \mathbb{R}$ be a continuous function such that $b$ represents $-\nabla k$ in the sense of distribution.
Assume that $\exp(-2k(x))$ is integrable. Then the probability distribution given by the un-normalized density $\exp(-2k(x))d x$ provides a reversible, invariant probability distribution $\mu$ for the process in \eqref{sde1}.
\end{prop}
\begin{proof} Let $\mu$ be the measure on $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n)$ defined by $\mu(d x)= e^{-2k(x)}d x$. Consider the Sobolev space, $H^{1,2}$, of all measurable functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$, such that $f$ and all of its partial derivatives $\partial f/\partial x_i$, $i=1,2,\ldots,n$ (in the sense of distributions) are in $\mathbb{L}^2(\mu)$. Then we can define the following symmetric bilinear form, on the domain $H^{1,2}$, given by
\[
\mathcal{E}(f,g)= \int_{\mathbb{R}^n} \langle \nabla f,\nabla g \rangle e^{-2k(x)}d x.
\]
Since $e^{-2k(x)}$ is never zero, it follows that $H^{1,2}$ is a Hilbert space. Thus, $\mathcal{E}$ is closed, since it is defined everywhere on the Hilbert space $H^{1,2}$. It is also known to be $\mathcal{E}$ is Markovian (see, e.g., \cite{dirichlet}, example 1.2.1.). Thus, it is clear that this is a Dirichlet form in $\mathbb{L}^2(\mathbb{R}^n,\mu)$.
By Theorem 1.3.1 in \cite{dirichlet}, we claim the existence of a unique non-positive definite self-adjoint operator $\mathcal{L}'$ on $H^{1,2}$ such that
\[
\mathcal{E}(f,g) = \langle \sqrt{-\mathcal{L}'}f, \sqrt{-\mathcal{L}'}g \rangle_{\mu}, \quad \forall \; f,g \in \; H^{1,2}.
\]
Here $\iprod{\cdot}_{\mu}$ refers to the usual inner product in $\mathbb{L}^2(\mu)$. Or, in other words, (Corollary 1.3.1 of \cite{dirichlet}) there is a unique self-adjoint operator $\mathcal{L}'$ on a domain $\mathcal{D}(\mathcal{L}')\subseteq H^{1,2}$ such that
\begin{equation}\label{uniqueop}
\mathcal{E}(f,g) = \iprod{-\mathcal{L}'f, g}_{\mu},\quad \forall\; f \in \mathcal{D}(\mathcal{L}'),\;\forall\; g \in H^{1,2}.
\end{equation}
We now show that \eqref{uniqueop} is satisfied by a multiple of the generator of the Markov process in \eqref{sde1}. The generator, $\mathcal{L}$, is given by
\[
\mathcal{L}f = \langle b, \nabla f \rangle + \frac{1}{2}\Delta f.
\]
By our assumption $b=-\nabla k$ takes finitely many values. Thus, we can define $\mathcal{L}$ on the domain
\[
H^{2,2} = \left\{f \in \mathbb{L}^2(\mu)\;\Big\vert\; \frac{\partial f}{\partial x_i}\in \mathbb{L}^2(\mu), \text{and}\; \frac{\partial^2 f}{\partial x_i^2} \in \mathbb{L}^2(\mu)\;\forall\; i=1,2\ldots,n.\; \right\}.
\]
It is clear that the domain of $\mathcal{L}$ above is a subset of $H^{1,2}$.
We claim that $2\mathcal{L}$ satisfies \eqref{uniqueop}. In that direction, consider any $f\in H^{2,2}$ and any $g\in H^{1,2}$, we have
\begin{equation}\label{equalgen}
\begin{split}
\int_{\mathbb{R}^n}\mathcal{L}(f)ge^{-2k(x)}d x &= \int_{\mathbb{R}^n} \left( \langle b, \nabla f \rangle + \frac{1}{2}\Delta f \right)g e^{-2k(x)}d x\\
&= \int_{\mathbb{R}^n} \langle b, \nabla f \rangle g e^{-2k(x)}d x + \int_{\mathbb{R}^n}\frac{1}{2}\Delta f g e^{-2k(x)}d x\\
&= -\int_{\mathbb{R}^n} \langle \nabla k, \nabla f \rangle g e^{-2k(x)}d x - \frac{1}{2}\int_{\mathbb{R}^n}\langle \nabla f , \nabla(g e^{-2k(x)}) \rangled x \\
&= -\frac{1}{2}\int_{\mathbb{R}^n}\langle \nabla f, \nabla g\rangle e^{-2k(x)}d x =-\frac{1}{2}\mathcal{E}(f,g).
\end{split}
\end{equation}
Note that the boundary terms are zero in the integration by parts above since both $\partial f/\partial x_i$ and $g$ are in $\mathbb{L}^2(\mu)$, and thus
\[
\frac{\partial f}{\partial x_i}g e^{-2k(x)}\Big\vert^{\infty}_{-\infty}=0, \quad \forall i=1,2,\ldots,n.
\]
We can rewrite \eqref{equalgen} as
\[
\mathcal{E}(f,g) = \iprod{-2\mathcal{L}f,g},\quad \forall\; f\in H^{2,2},\; \forall\; g \in H^{1,2}
\]
which, compared with \eqref{uniqueop}, proves that $2\mathcal{L}$ to be the unique operator associated with the Dirichlet form $\mathcal{E}$. Further, from self-djointness of $\mathcal{L}$, we infer
\begin{equation}\label{reverse}
\langle \mathcal{L}f, g \rangle_{\mu} = \iprod{f,\mathcal{L}g}_{\mu}, \quad \forall\; f,g \in H^{2,2},
\end{equation}
where $\iprod{\cdot,\cdot}_{\mu}$ refers to the usual $\mathbb{L}^2$ inner product. We can take $g\equiv 1$ to get that $\mu$ is an invariant measure for the process $X_t$. This proves the claim.
\end{proof}
When is the function $\exp(-2k(x))$ integrable ? This question is critical to both the recurrence property of the diffusion process as well as the existence of a unique long term stationary distribution. Its answer, however, is geometric in nature. It is intuitive that $k$ needs to be nonnegative. Notice that if $k$ is nonnegative, by virtue of being positively homogeneous, it is the gauge function (Minkowski functional) of a set containing the origin. That is to say, if we define the \textit{unit ball} and the \textit{surface} given by $k$ respectively as
\begin{equation}\label{ballandsurface}
\begin{split}
C&=\{x\;:\; k(x) \le 1\},\quad\mcal{S}=\{ x\;: \; k(x)=1\},
\end{split}
\end{equation}
it is not difficult to see that $k$ satisfies the relation $k(x)=\inf\left\{ \alpha > 0:\; x\in \alpha C \right\}$. We have the following definition.
\begin{defn}\label{perfect}
A continuous, nonnegative, positively homogeneous function $k:\mathbb{R}^n \rightarrow \mathbb{R}\cup\{\infty\}$ is said to be \emph{irreducible} if it satisfies $k(x)=0$ if and only if $x=0$.
\end{defn}
\begin{lemma}\label{firstproperties}
If $k$ is continuous and irreducible either $k(x) \ge 0$, $\forall\; x\in \mathbb{R}^n$, or $k(x)\le 0$, $\forall\; x\in \mathbb{R}^n$. Moreover, if $k(x) > 0$, $\forall\; x \in \mathbb{R}^n, x\neq 0$, then $\vol{C} < \infty$.
\end{lemma}
\begin{proof}
Suppose that there are points $x_0$ and $x_1$ such that $k(x_0) > 0$ and $k(x_1) < 0$. We can choose a continuous curve $\gamma_t,\; t \in [0,1]$, in $\mathbb{R}^n$ such that $\gamma_0=x_1$ and $\gamma_1=x_2$ and $0 \notin \gamma[0,1]$. Since $k$ is continuous, by the intermediate value theorem, there exists a $t^* \in (0,1)$ such that $k(\gamma_{t^*}) = 0$ but $\gamma_{t^*}\neq 0$. But this is impossible if $k$ is irreducible, and we have proved the first assertion of the lemma.
For the second assertion, we need to show that $C$ is bounded. Suppose, on the contrary, we can find a sequence $\{x_n\} \subseteq C$ such that $\lim_{n\rightarrow \infty}\norm{x_n}=\infty$. One can assume that $\norm{x_n} \ge 1$, for all $n \in \mathbb{N}$. Then the points $y_n = x_n / \norm{x_n}$ satisfy
\[
k(y_n) = k(x_n) / \norm{x_n} \le 1,\quad \forall\; n \in \mathbb{N}.
\]
Thus, $y_n \in C$, for all $n=1,2,\ldots$. However, there exists a subsequence of $y_n$, say $\{y_{n_m}\}$ such that $\lim_{m\rightarrow \infty} y_{n_m}=z$, for some $z$ with $\norm{z}=1$. Hence, by continuity of $k$, we infer
\[
k(z) = \lim_{m \rightarrow \infty} k(y_{n_m}) = \lim_{m \rightarrow \infty} \frac{k(x_{n_m})}{\norm{x_{n_m}}} = 0.
\]
The final equality is due to the fact that $0 \le k(x_n) \le 1$ for all $n$ and $\lim \norm{x_n} =\infty$. Since $z\neq 0$, this contradicts our assumption that $k$ is irreducible. Hence we are done.
\end{proof}
We now show that the process in \eqref{sde1} is Harris recurrent if $k$ is nonnegative and irreducible. It then follows (see \cite{durrettstoccalculus}, Section 7.5) that it has a unique invariant measure $\mu$ described above in Proposition \ref{invariantdist}. Moreover, if $P_t(x)$ is the marginal distribution of $X_t$ when $X_0=x$, then
$\lim_{t\rightarrow \infty}\norm{P_t(x) - \mu}_{\text{TV}} =0$. Here $\norm{\cdot}_{\text{TV}}$ refers to the total variation norm on measures.
The following claim settles the argument.
\begin{lem}\label{homogrecurrent} Consider the notations and assumptions in Proposition \ref{invariantdist}. Suppose that the function $k$ is a nonnegative, irreducible, positively homogeneous function. Then the process $X_t$ is recurrent.
\end{lem}
\begin{proof}
We will use Corollary 7.5.4 in \cite{durrettstoccalculus}. We need to consider the quantity
\[
d(x) = n + 2\iprod{x,b(x)},\quad x \in \mathbb{R}^n.
\]
By our definition we have $\iprod{x,b(x)}=-k(x)$. Thus, $d(x) = n - 2k(x)$.
Now, since $k$ is non-negative and irreducible, it is growing to infinity \emph{uniformly} in all directions radially outward from zero. The way to see this is to note
\[
k(x) = \norm{x}k(x/\norm{x})\;\ge\; \norm{x} \inf_{\norm{y}=1}k(y)= c_1 \norm{x}
\]
The constant $c_1=\inf_{\norm{y}=1}k(y)$ is positive since $k$ is a strictly positive continuous function on the compact set $\{ y:\norm{y}=1\}$.
Now, if we fix an $\epsilon > 0$, there exists $R > 0$ such that $d(x) < -\epsilon$ for all $x$ with $\norm{x} > R$. Let $T_R$ be the first hitting time of the compact set $B_R=\{x\in \mathbb{R}^n:\; \norm{x}\le R\}$. By Corollary 7.5.4 in \cite{durrettstoccalculus}, we immediately obtain $E^{x}(T_R) \le{\norm{x}^2}/{\epsilon}$. Thus $B_R$ gets visited infinitely often and hence the process $X_t$ is recurrent.
\end{proof}
\comment{
\section{A polar decomposition formula}
Throughout we consider the Euclidean space of dimension $n$. Consider a measurable, positively homogeneous function $k:\mathbb{R}^n \rightarrow \mathbb{R}^+ \cup \{\infty\}$, where $\mathbb{R}^+$ denotes the subset of nonnegative real numbers. Define the \textit{unit ball} and the \textit{surface} given by $k$ respectively as
\begin{equation}\label{ballandsurface}
\begin{split}
C&=\{x\;:\; k(x) \le 1\},\quad\mcal{S}=\{ x\;: \; k(x)=1\}.
\end{split}
\end{equation}
Since $k$ is continuous, both $C$ and $\mcal{S}$ are Borel measurable. The following definition will be of much use later in the text.
\begin{defn}\label{perfect}
A continuous, nonnegative, positively homogeneous function $k$ to be \emph{irreducible} if it satisfies $k(x)=0$ if and only if $x=0$.
\end{defn}
It is not difficult to see that if $k$ is irreducible then volume of the unit ball $C$ is finite. In fact, there is a one-to-one correspondence between compact bodies containing the origin and such irreducible functions $k$ which can be interpreted as their Minkowski functionals. That is to say, $k(x) = \inf\left\{ r\ge 0: x \in rC \right\}$.
Thus for any nonnegative, irreducible $k$, consider the surface defined in \eqref{ballandsurface} and define the surface projection function $\Theta:\mathbb{R}^n \rightarrow \mcal{S}$ by
\begin{equation}\label{whatistheta}
\Theta(x) = \frac{x}{k(x)},\quad x \in \mathbb{R}^n \backslash \{0\}, \;\text{and}\; \Theta(0) = 0.
\end{equation}
We take the following two measurable spaces
\[
M_1 = \Big(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n)\Big)\quad\text{and}\quad M_2 = \Big( \mathbb{R}^+ \times \mcal{S},\;\; \mathcal{B}(\mathbb{R}^+)\otimes \mathcal{B}(\mcal{S})\Big).
\]
One can construct a measurable map $T:M_1 \rightarrow M_2$ given by $T(x) = \left( k(x), \Theta(x)\right)$, $\forall\; x\in \mathbb{R}^n$. It clearly follows from the definition that $T$ is a one-to-one map.
\begin{prop}\label{changeofvariable} For any nonnegative, irreducible, positively homogeneous function $k:\mathbb{R}^n \rightarrow \mathbb{R}^+ \cup \{\infty\}$, and any integrable $f:\mathbb{R}^n \rightarrow \mathbb{R}$, we have
\begin{equation}\label{cngvble}
\int_{\{x:k(x) < \infty\}} f(x)d x = n \vol{C} \int_{0}^{\infty} r^{n-1} \int_{\mcal{S}}f(r\cdot z)d \mu(z)\; d r,
\end{equation}
where $dx$ refers to the $n$-dimensional Lebesgue measure and $\mu$ is the \textit{cone measure} on $\mcal{S}$ defined as
\begin{equation}\label{conemsr}
\mu(E) = \frac{1}{\vol{C}}\vol{T^{-1}([0,1]\times E)}, \quad \forall \; E \in \mathcal{B}(\mcal{S}).
\end{equation}
\end{prop}
The proof is straightforward and follows first by restricting $f$ among the indicators of product sets $\{k(x) \le b\}\cap\{\Theta(x) \in E\}$ and then making a change of variable by defining $y=x/b$. Applying standard approximation arguments then completes the proof for all integrable functions. This is a very special example of a general measure theoretic factorization theorem as stated in Lemma 4 of \cite{burglar}. In particular, here the \textit{factor space} is $\mathbb{R}$ and the group is the positive real numbers under multiplication.
An immediate corollary of the previous polar decomposition formula is the following proposition.
\begin{prop}\label{changeofvble}
Consider a nonnegative, irreducible, positively homogeneous function $k$ on $\mathbb{R}^n$. Let $Q$ be a probability measure on $\mathbb{R}^n$ with density $q$ which is a function of $k(x)$. That is, for some non-negative measurable function $h:\mathbb{R} \rightarrow \mathbb{R}^+$ we have
\[
Q(A) = \int_A q(x)d x = \int_A h(k(x)) d x, \qquad \forall \; A \in \mathcal{B}(\mathbb{R}^n).
\]
Then, under $Q$, the random vector $\Theta$ of \eqref{whatistheta} is independent of $k$. The marginal law of $\Theta$ is the cone measure on $\mcal{S}$, while irrespective of the functional form of $k$, the law of the random variable $k(X)$ is given by the density proportional to $nr^{n-1}h(r)$.
\end{prop}
\begin{proof}
Consider a measurable subset $A$ on $\mcal{S}$. By Proposition \ref{changeofvariable} we get
\[
\begin{split}
Q\left( \Theta \in A, k \le R \right) &= n \text{vol}(C) \int_{0}^{R} r^{n-1} \int_{\mcal{S}} q(r.z)1_{\{z \in A\}} d \mu( z) d r\\
& = n \text{vol}(C) \int_{0}^{R} r^{n-1} h(r) \int_{\mcal{S}} 1_{\{z \in A\}} d \mu(z) d r\\
&= \mu(A) \cdot n \text{vol}(C) \int_{0}^{R} r^{n-1} h(r) d r.
\end{split}
\]
Thus, under $Q$, the random variables $k$ and $\Theta$ are independent and the law of $\Theta$ is the cone measure $\mu$, and the density of $k$ is proportional to $nr^{n-1}h(r)$.
\end{proof}
A particularly important case arises when we consider the density on $\mathbb{R}^n$ given by normalizing $\exp(-k(x))$ in which case the random variable $k(X)$ has law Gamma($n$).
\medskip
A similar situation arises when instead of $\mathbb{R}^n$ we consider the $n$-dimensional space of complex numbers $\mathbb{C}^n$. We use the same notation $k$ to denote an irreducible homogeneous function from $\mathbb{C}^n$ to $\mathbb{C} \cup \infty$, by which we mean that for all $z=(z_1, z_2, \ldots, z_n)\in \mathbb{C}^n$ and for all non-zero $\alpha \in \mathbb{C}$, we have
\[
k\left( \alpha z \right) = \alpha k(z), \quad \text{and}\quad k(z)\neq 0, \quad \text{for all} \; z\neq 0.
\]
Let us also reserve the same notation for the unit ball and the surface
\[
C= \left\{ z\in \mathbb{C}^n:\; \abs{k(z)}\le 1 \right\}, \quad \mcal{S} = \left\{z\in \mathbb{C}^n:\; \abs{k(z)}=1 \right\}.
\]
We use Lemma 4 of \cite{burglar} where the {factor space} is $\mathbb{C}$ and the group is the set of nonzero complex numbers under multiplication. This leads us to the following counterpart of Proposition \ref{changeofvble} for complex variables.
\begin{prop}\label{changeofvblec}
Consider any irreducible homogeneous function $k:\mathbb{C}^n \rightarrow \mathbb{C}\cup\{\infty\}$. Then, under the Lebesgue measure, the variable $k(z)$ and the vector $z/k(z)$ are independent.
Consider now any probability measure $Q$ on $\mathbb{C}^n$ whose Radon-Nikod\'ym derivative with respect to the Lebesgue measure on $\mathbb{C}^n$ is $h(\mathfrak{Re}(k(z)))$ for some nonnegative measurable function $h:\mathbb{R} \rightarrow \mathbb{R}^+$.
\end{prop}
\begin{proof}
As before, we first consider $f$ to be the indicator of the sets
\[
\left\{ \abs{k(z)} \le r \right\}\cap \left\{ \arg(k(z))\ge \theta \right\}\cap \left\{ \frac{z}{k(z)} \in E \right\},
\]
where $r$ is positive, $\theta$ is a real number, and $E$ is some Borel subset of $\mathbb{C}$. In this case, it follows that
\[
\int_{\{z: k(z) < \infty\}} f(z) dz = \int_{\mathbb{C}^n}1\left\{ \abs{k(z)} \le r \right\}1\left\{ \arg(k(z))\ge \theta \right\}1\left\{ \frac{z}{k(z)} \in E \right\}dz.
\]
We make a change of variable by defining $y=zr^{-1}e^{-i\theta}$, from which it follows that $\abs{k(y)}=\abs{k(z)}/r$, $\arg{k(y)}=\arg{k(z)}-\theta$, and $z/k(z)=y/k(y)$. Thus
\[
\begin{split}
\int_{z: k(z) < \infty} f(z) dz &= r^{2n}\int_{\mathbb{C}^n} 1\left\{ \abs{k(y)} \le 1 \right\}1\left\{ \arg(k(y))\ge 0 \right\}1\left\{ \frac{y}{k(y)} \in E \right\}dy\\
&= r^{2n}\vol{D}\mu(A).
\end{split}
\]
\end{proof}
}
The integrability of the function $\exp(-2k(x))$ requires precisely the same condition as in the last lemma.
\begin{lemma}\label{homisinteg}
Suppose that $k:\mathbb{R}^n \rightarrow \mathbb{R}$ is a nonnegative, irreducible, continuous, positively homogeneous function. Then the $\exp(-2k(x))$ is an integrable function.
\end{lemma}
To prove the previous lemma we need the following \textit{polar decomposition} formula. For any nonnegative, irreducible $k$, define the surface projection function $\Theta:\mathbb{R}^n \rightarrow \mcal{S}$ by
\begin{equation}\label{whatistheta}
\Theta(x) = \frac{x}{k(x)},\quad x \in \mathbb{R}^n \backslash \{0\}, \;\text{and}\; \Theta(0) = 0.
\end{equation}
Clearly the range of $\Theta$ is the surface $\mcal{S}$ defined in \eqref{ballandsurface}.
We take the following two measurable spaces
\[
M_1 = \Big(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n)\Big)\quad\text{and}\quad M_2 = \Big( \mathbb{R}^+ \times \mcal{S},\;\; \mathcal{B}(\mathbb{R}^+)\otimes \mathcal{B}(\mcal{S})\Big).
\]
One can construct a measurable map $T:M_1 \rightarrow M_2$ given by $T(x) = \left( k(x), \Theta(x)\right)$, $\forall\; x\in \mathbb{R}^n$. It clearly follows from the definition that $T$ is a one-to-one map. We prove the following slightly general result for the sake of completeness.
\begin{lem}\label{changeofvariable} For any nonnegative, irreducible, positively homogeneous function $k:\mathbb{R}^n \rightarrow \mathbb{R}^+ \cup \{\infty\}$, and any integrable $f:\mathbb{R}^n \rightarrow \mathbb{R}$, we have
\begin{equation}\label{cngvble}
\int_{\{x:k(x) < \infty\}} f(x)d x = n \vol{C} \int_{0}^{\infty} r^{n-1} \int_{\mcal{S}}f(r\cdot z)d \mu(z)\; d r,
\end{equation}
where $dx$ refers to the $n$-dimensional Lebesgue measure and $\mu$ is the \textit{cone measure} on $\mcal{S}$ defined as
\begin{equation}\label{conemsr}
\mu(E) = \frac{1}{\vol{C}}\vol{T^{-1}([0,1]\times E)}, \quad \forall \; E \in \mathcal{B}(\mcal{S}).
\end{equation}
\end{lem}
\begin{proof}
We first prove \eqref{cngvble} for functions $f$ equal to indicators of sets $A= T^{-1}\left([0,b]\times E\right)$, where $E \in \mathcal{B}(\mcal{S})$ and $b \ge 0$. We have
\begin{eqnarray*}
\vol{A} = \int 1_{A}d x &=& \int 1{\{k(x) \le b\}}1{\{\Theta(x) \in E\}}d x.
\end{eqnarray*}
If we make let $y=b^{-1}x$, then, by positive homogeneity of $k$, one can write the last equation as
\[
\begin{split}
\vol{A} &= b^n \int 1{\{k(y) \le 1 \}}1{\{\Theta(y) \in E\}}d y = b^n \vol{T^{-1}([0,1]\times E)} \\ &= b^n \vol{C}\cdot \mu(E) = \vol{C} n \int_{0}^{b} r^{n-1} \int_E d \mu \; d r,
\end{split}
\]
which proves \eqref{cngvble} for this particular case. The rest of the argument follows from standard measure theoretic approximation results.
\end{proof}
\begin{proof}[Proof of Lemma \ref{homisinteg}] Since $2k$ is another nonnegative, irreducible, positively homogeneous function, it suffices to show that $\exp(-k(x))$ is integrable.
By Lemma \ref{firstproperties}, the set $C=\{x\in \mathbb{R}^n:\; k(x)\le 1\}$ has a finite volume, and hence the cone measure on $\mcal{S}=\partial C$ is well-defined. From the change of variable formula in Lemma \ref{changeofvariable} we then obtain
\[
\begin{split}
\int_{\mathbb{R}^n}e^{-k(x)}d x &= n\vol{C} \int_{0}^{\infty}r^{n-1}e^{-r}d r\\
&= {n}\vol{C} \int_{0}^{\infty}s^{n-1}e^{-s}d s= \vol{C} {n}\Gamma\left({n}\right)\\
&=\vol{C}\Gamma\left({n}+1\right) < \infty.
\end{split}
\]
\end{proof}
Note that, under the probability measure with the unnormalized density $e^{-k(x)}$, the random variable $k(X)$ is always distributed as Gamma($n$) irrespective of $k$ and independently of the vector $\Theta(X)$. Similarly, under the uniform measure on $C$, the random variable $k(X)$ is always distributed as Beta$(n,1)$ independently of $\Theta(X)$. Under both these measures, $\Theta(X)$ has the same law. This provides a link between the two probability measures which is important in their understanding.
\bigskip
What can be recovered when $k$ is not irreducible ? In general little, except when $k$ is the gauge function of a lower dimensional polytope. The following proposition generalizes Proposition \ref{invariantdist}.
\begin{prop}\label{additivesplit}
Let $k:\mathbb{R}^n \rightarrow \mathbb{R}$ be a continuous function whose derivative in the sense of distribution is represented by a bounded function. Suppose there exists a subspace $H \subseteq \mathbb{R}^n$ such that if $y \in H$ and $z \in H^{\perp}$, then
\[
k(y + z) = k_1(y) + k_2(z).
\]
Additionally, assume that $k_1$ is a nonnegative, irreducible, positive homogeneous function on $H$. Consider the solution to \eqref{sde1} when $b(x)$ represents $-\nabla{k(x)}$, where $\nabla$ is in the sense of distributions. Assume that a solution to \eqref{sde1} with this drift exists. Let $A:\mathbb{R}^n \rightarrow H$ be a projection matrix onto the subspace $H$. Then
\begin{enumerate}
\item the process $Y_t=AX_t$ has a unique reversible stationary probability distribution $\mu$.
\item Suppose $k'$ is any other function defined as
\[
k'(x) = k_1(Ax) + k_2'(x-Ax),\quad \forall\; x\in \mathbb{R}^n,
\]
for some non-negative function $k_2'$ such that $\exp(-2k')$ is integrable. Then $\mu$ is the law of the random vector $Y=AX$, where $X$ is a random vector with density proportional to $\exp(-2k')$.
\end{enumerate}
\end{prop}
\begin{proof} Define $k'_2:\mathbb{R}^n\rightarrow \mathbb{R}$ to be
\begin{equation}\label{whatisk2prime}
k_2'(x) = \sup_{1\le j\le n-d}\vert \iprod{x,\delta_j}\vert,
\end{equation}
where $\delta_i,\;i=1,\ldots,n-d$ is an orthogonal basis of $H^{\perp}$. Modify the function $k$ by defining
\[
k'(x) = k_1(y) + k'_2(z),\quad \forall\; y\in H,\; z\in H^{\perp},\; x=y+z.
\]
\medskip
\noindent $\bullet$ We claim that $k'$ is a non-negative, irreducible, positively homogeneous function on $\mathbb{R}^n$.
Let us verify the condition in definition \ref{perfect}. Since $k_1$ and $k_2'$ are both positive, for $x=y+z$, $y\in H$, $z\in H^{\perp}$, we get
\[
\{h'(x)=0\}\; \Leftrightarrow\; \{k'(x)=0\}\;\Leftrightarrow\;\{ k_1(y)=0\}\; \text{and} \; \{k_2'(z)=0\}.
\]
But, by assumption for $k_1$ and by construction for $k_2'$, we get that $k'(x)=0$ if and only if $y=0$ and $z=0$, that is, if and only if $x=0$. This proves our claim.
Consider the solutions of the following two stochastic differential equations
\begin{eqnarray}
d X_t &=& -\nabla k(X'_t)d t + d W_t, \label{orgeqn}\\
d X'_t &=& -\nabla k'(X'_t)d t + d W_t, \label{alteqn}
\end{eqnarray}
where the drifts are function representing the derivatives in the sense of distributions. That $X_t'$ exists (in the weak sense) is clear.
\medskip
\noindent $\bullet$ We claim that $X'$ has a unique reversible invariant distribution $\mu'$ with unnormalized density $\exp(-2k'(x))d x$. From Lemma \ref{homisinteg}, we deduce that $\exp(-2k')$ is integrable. Hence we can suitably normalize and get a probability measure whose density is proportional to $\exp(-2k'(x))$. Now, we can apply Proposition \ref{invariantdist} to claim that the unique reversible invariant distribution of the process $X'$ exists and is given by the unnormalized density $\exp(-2k'(x))$.
Now, consider a projection matrix $A$ with range space $H$. Thus, $A^{\perp}=I-A$ is a projection onto $H^{\perp}$. Now, by assumption, the function $k$ splits additively as
\[
\begin{split}
k(x) &= k_1(Ax) + k_2(A^{\perp}x)
\end{split}
\]
Taking gradient on both sides, one obtains
\[
\begin{split}
\Rightarrow \; \nabla k(x) &= A'\nabla k_1(Ax) + \left(A^{\perp}\right)' \nabla k_2(A^{\perp}x).
\end{split}
\]
Here $A'$ and $(A^{\perp})'$ refers to taking adjoints. But $A$ being a projection is self-adjoint and satisfies $A^2=A$. Thus, it follows that
\begin{equation}\label{newgrad}
A\nabla k(x) = A^2\nabla k_1(Ax) + A(I-A)\nabla k_2(A^{\perp}x)= A\nabla k_1(Ax).
\end{equation}
Exactly in the same way we also have
\begin{equation}\label{newgrad2}
A\nabla k'(x) = AA'\nabla k_1(Ax) + A(A^{\perp})'\nabla k'_2(A^{\perp}x)= A k_1(Ax).
\end{equation}
Now, consider the processes $Y=AX$ and $Y'=AX'$ which clearly satisfy the following differential equations
\begin{equation}
\begin{split}
d Y_t &= -A\nabla k(X_t)d t + Ad W_t,\quad \text{and} \\
d Y'_t &= -A\nabla k'(X'_t)d t + Ad W_t.
\end{split}
\end{equation}
Using equations \eqref{newgrad}, and \eqref{newgrad2} we can rewrite the above equations as
\[
\begin{split}
d Y_t &= -A\nabla k_1(Y) + Ad W_t,\quad \text{and}\\
d Y'_t &= -A\nabla k_1(Y') + Ad W_t.
\end{split}
\]
Clearly the laws of the process $Y$ and $Y'$ are identical due to the uniqueness in law of the weak solutions of their stochastic differential equation. To get past the arbitrariness of the hyperplane $H$, one can simply observe that if dimension$(H)=d \le n$, there exists a $(d\times n)$ matrix $D$ which is a bijection between $H$ and $\mathbb{R}^d$. The laws of $DY$ and $DY'$ are identical by standard theory of SDE. Now one simply inverts $D$ onto $H$ to obtain our conclusion.
Now, since $A$ is a linear map, the process $Y'$ has a unique invariant distribution $\nu$ induced by the invariant distribution of $X'$, and given by
\begin{equation}\label{whatisnu}
\nu(B) = \mu'(Ax\in B),\qquad \forall \; B \in \mathcal{B}(H),
\end{equation}
where $\mu'$ is the unique invariant distribution for the process $X'$. Thus, by the equality in law of the processes $Y$ and $Y'$ it follows that the process $Y_t=AX_t$ has a unique invariant distribution given by $\nu$ according to the recipe above.
It is easy to give an intrinsic description of $\nu$ from \eqref{whatisnu}. If $X'$ is distributed according to $\mu'$, then for any $B \in \mathcal{B}(H)$, we have
\[
\begin{split}
\nu(B)=\mu'\left(AX'\in B\right)&= \int_{Ax\in B}\exp\left(-2k_1(Ax) - 2k_2'(x-Ax)\right)d x\\
& =\int_{y\in B}\exp\left(-2k_1(y)\right)d y \int_{H^{\perp}}\exp\left(-2k_2'(z)\right)d z.
\end{split}
\]
Clearly, $\nu$ has a density proportional to $\exp(-2k_1(y))$ with respect to the Lebesgue measure restricted to the hyperplane $H$.
\end{proof}
\section{Simplicial cones and Exponential distributions}\label{cones}
We have seen in the last section that the invariant probability distributions for the SDEs described in \eqref{sde1} have unnormalized densities with respect to the Lebesgue measure given by $\exp(-2k(x))$, where $k$ is a nonnegative, irreducible, positively homogeneous function. Even with such an explicit description of the density function it can be very hard to compute any means, variances, or one-dimensional marginal distributions. Our objective in this section is to link them to the combinatorial structures of the unit ball generated by the positively homogeneous function $2k$, particularly when these are star-shaped or convex polytopes. In the special case of simplicial polytopes this allows us to furnish a complete description of the invariant measure in terms of independent Exponential random variables.
The main geometric idea is the following. Consider, as before, a drift function $b$ which is constant over cones $C_1, C_2, \ldots, C_r$. We assume that $b$ is the negative gradient in the sense of distributions of the function $k(x)=-\iprod{x,b(x)}$. Thus, $2k(x)$ is a linear function inside each cone $C_i$.
Suppose now that $C_i$ is simplicial. Simplicial cones are those that can be transformed to the positive quadrant by applying a non-singular linear transformation. That is to say, for each $C_i$ there exists $n$ linearly independent vectors in $\mathbb{R}^n$, denoted by $\beta_1, \beta_2, \ldots, \beta_n$, such that $C_i=\left\{ x\in \mathbb{R}^n: \; x=\sum_i a_i \beta_i, \; a_i \ge 0\; \forall\; i \right\}$. It is now not hard to see that in that case the probability measure given by $\exp(-2k(x))$ restricted to the cone $C_i$ must be a linear transformation of independent Exponential distributions. The difficulty in the execution of the previous argument is to identify from the function $b$ that a simplicial polytope is lurking behind the scenes and to compute the necessary linear transformations. In the following two subsections we consider two general classes of examples where the argument can be fully carried out. One, the regular case, is where each $C_i$ can be mapped to any other by a group of orthogonal transformations. This leads us to a connection with finite irreducible Coxeter groups. The other, which is not regular, deals with graphs and the combinatorics hidden in their structure.
\subsection{The regular case: groups of orthogonal transformations}\label{groupcase}
Consider a finite dimensional Euclidean vector space $V$. A linear transformation from $V$ to itself is called orthogonal if the corresponding matrix is orthogonal. In that case the determinant of the transformation is $\pm 1$. An important class of orthogonal matrices is given by reflections. A reflection along a unit vector $u\in V$ corresponds to the matrix $I - 2uu'$ (all vectors are columns and $u'$ denotes the transpose of $u$). Geometrically it produces the mirror image of any vector with respect to the hyperplane orthogonal to $u$. A reflection group is a group of matrices, each element of which is a reflection matrix.
Let $G$ be a group of orthogonal matrices. $G$ is called irreducible if there is no non-trivial subspace $W$ of $V$ which is stable under the action of $G$, i.e., $\rho(s)W \subseteq W$, for all $s\in G$.
As a recurring example, considering the symmetric group of permutations on $n$ elements. It has a natural representation as permutation matrices on $\mathbb{R}^n$. This is not irreducible since the one-dimensional subspace $W_1$ spanned by the vector of all ones remains invariant under the action of the group. However, the action restricted to $W_1$ and $W_2=W_1^\perp$ is irreducible.
For a finite group of orthogonal linear transformations $G$ we now define what is known as a \textit{fundamental region}. Please see Chapter 3 in \cite{reflectiongroups} for more details.
\begin{fundamental} A subset $F\subseteq V$ is known as a fundamental region for a group of orthogonal transformations $G$ if
\begin{enumerate}
\item $F$ is open,
\item $F\cap AF=\emptyset$ if $A \neq I$, where $I$ is the identity matrix and $A \in G$,
\item $V= \cup_{A\in G} \overline{AF}$, where $\overline{B}$ denotes the topological closure of a subset $B$.
\end{enumerate}
\end{fundamental}
For the representation of the symmetric group as permutation matrices such a region is provided by the cone $\left\{ x\in \mathbb{R}^n: \; x_1 < x_2 < \ldots < x_n \right\}$. In general, fundamental regions are not unique.
Henceforth we will work with $V = \mathbb{R}^n$ even though proper subspaces of $\mathbb{R}^n$ provide another rich class of examples.
The groups we will be interested in, which includes the permutations as a special case, are generated purely by reflections. When irreducible, these groups are known as Coxeter groups and we review their basic structure below.
A reflection along a vector $r$ is uniquely characterized by the fact that it keeps every vector orthogonal to $r$ unchanged and flips the sign of every multiple of $r$. A Coxeter group is a finite irreducible group of orthogonal transformations generated by finitely many reflections. A frequent class of examples are the Dihedral groups which are the symmetry groups of regular polygons. Although, these groups contain both rotations (think of a unit square being rotated by $\pi/2$) and reflections (the square getting reflected along the mid-axis), one can show that these groups can be generated purely by the reflection elements (reflecting the square twice along different axes amounts to a rotation).
Closely associated with the Coxeter groups is the idea of \textit{root systems}. If $G$ contains a reflection along $r$, then both $r$ and $-r$ are known as \textit{roots} of $G$. Let $\Delta$ be the set of all the roots of $G$, usually referred to as the \textit{root system} of the Coxeter group. Recall the definition of a fundamental region. We are going to construct a fundamental region for $G$ which is a simplicial cone.
Let $G$ be a Coxeter group acting on $\mathbb{R}^n$. Thus, in particular, it is irreducible.
Choose any vector $u\in \mathbb{R}^n$ such that $\iprod{u,r}\neq 0$ for any root $r$ of $G$. Then the root system is partitioned into two subsets
\[
\Delta^+_u =\{r\in \Delta:\; \iprod{r,u} > 0 \}, \qquad \Delta^-_u= \{ r\in \Delta:\; \iprod{r,u} < 0\}.
\]
Theorem 4.1.7 in \cite{reflectiongroups} (and several lemmas preceding it) proves the following result.
\begin{thm}
There is a unique collection of $n$ many vectors $\Pi_u$ in $\Delta_u^+$ such that every vector in $\Delta_u^+$ can be written as a linear combination of elements in $\Pi_u$ with non-negative coefficients. This collection, known as the $u$-base or fundamental roots, and denoted by $\Pi_u$, is linearly independent and forms a basis of $\mathbb{R}^n$.
\end{thm}
A $u$-base provides a fundamental region for a Coxeter group $G$, by defining
\begin{equation}\label{fundacox}
F_u := \left\{ x\in \mathbb{R}^n:\; \iprod{x,r} >0 \quad \forall\; r\in \Pi_u \right\}.
\end{equation}
That this is indeed a fundamental region has been proved in Theorem 4.2.4 in \cite{reflectiongroups}. That this is a simplicial cone follows since the vectors in $\Pi_u$ are linearly independent.
\medskip
Let us now return to the framework in Proposition \ref{invariantdist}. We start with a drift function $b:\mathbb{R}^n \rightarrow \mathbb{R}$ that is scale invariant, i.e., $b(\alpha x)= b(x)$ for all $\alpha > 0$. We would like to analyze the probability measure given by normalizing $\exp(-2k(x))$ where $k(x)=-\iprod{x,b(x)}$. Suppose the drift function $b$ takes finitely many values on $\mathbb{R}^n$ and satisfies the property that $b(Ax)=Ab(x)$ for all $A\in G$ for some Coxeter group $G$. Can we claim that there is a finite sequence of non-overlapping simplicial cones whose union is the whole space and such that $b$ takes a constant value inside each cone ?
The answer is no in general. However, there is a simple sufficient condition which indeed guarantees an affirmative answer to the question. Let, as before, $k(x)=-\iprod{x,b(x)}$. If $k$ is non-negative, then clearly, by positive homogeneity, $k$ is the Minkowski functional of a star-shaped body containing the origin. The question in the previous paragraph is equivalent to asking whether this star-shaped body can be triangulated as a disjoint union of simplices each of which contains the origin as a extreme point. We are going to show below that the answer to the question is yes, if $k(x)=\max_{A\in G}\iprod{\lambda,Ax}$ for some vector $\lambda$. Seen through a geometric lens, this is equivalent to the statement that the star-shaped body generated by $k$ is a convex polytope which then turns out to be simplicial.
\begin{lemma}\label{constructk}
Let $G$ be a finite irreducible group of orthogonal matrices on $\mathbb{R}^n$. For any $\lambda \in \mathbb{R}^n$, $\lambda \neq 0$, the function
\begin{equation}\label{whatisk}
k(x)= \max_{A\in G}\iprod{\lambda,Ax}, \quad x\in \mathbb{R}^n,
\end{equation}
is a nonnegative positively homogeneous function on $\mathbb{R}^n$, which is irreducible. Moreover, $k$ is invariant under the action of the group. That is $k(x) = k\left(Ax\right)$, for all $A\in G$, and for all $x\in \mathbb{R}^n$.
\end{lemma}
\begin{proof}
It is trivial to see that $k$ is positively homogeneous. To show that it is nonnegative we use the fact (see \cite{diaconisbook}) that for any non-trivial irreducible group $G$ of orthogonal matrices, the sum $\sum_{A \in G} A $ is the zero matrix. Since $G$ is irreducible, it follows that
\[
\sum_{A\in G} \iprod{\lambda,Ax} = \iprod{\lambda,\sum_A Ax} = 0.
\]
But this implies that the maximum must be non-negative. Thus $k$ is nonnegative.
To prove that $k$ must be strictly positive for all nonzero vectors, note that for the previous argument, $k(x)=0$ for some $x$ would imply that
\[
\iprod{\lambda, Ax}=0, \quad \forall\; A\in G.
\]
Define the subspace
\begin{equation}\label{whatisvl}
V_{\lambda}= \left\{ x\in \mathbb{R}^n:\; \iprod{\lambda, Ax}=0 \quad \forall\; A\in G \right\}.
\end{equation}
Next, note that $V_{\lambda}$ is stable under the action of $G$. To see this take any $B\in G$ and any $x\in V_{\lambda}$, then clearly $Bx \in V_{\lambda}$. Thus $AV_{\lambda} \subseteq V_{\lambda}$ for all $A\in G$. But, since $G$ is irreducible, $V_{\lambda}$ must be either zero or the entire space. If $V_{\lambda}$ is the entire subspace, then by putting $A$ to be the identity in the definition \eqref{whatisvl}, we get $\iprod{\lambda,x}=0 \;\forall\; x\in V$. This shows that $\lambda$ must be zero, which we have ruled out in our assumption.
The invariance of $k$ under the action of $G$ is clear by the homomorphism property.\end{proof}
The last lemma proves that $k$ is the gauge function of a convex polytope containing the origin. The following lemma shows that the number of extremal faces of the polytope is given by the size of the orbit of $\lambda$.
\begin{lemma}\label{maximumwhere}
Consider $k$ as in Lemma \ref{constructk}. Given any $\lambda$, there exists $x\neq 0$ such that $k(x)=\iprod{\lambda,x}$. Moreover, $k\left( Ax \right) = \iprod{A\lambda, Ax}$ for all $A\in G$.
\end{lemma}
\begin{proof}
To prove the first assertion, suppose that for all $x$, we have
\[
\iprod{\lambda,x} < k(x)= \max_{A\in G} \iprod{A\lambda,x}.
\]
Then, for any such $x$, for any $B\in G$, we also have
\[
\begin{split}
\iprod{B\lambda, x} &= \iprod{\lambda,B' x} < k\left( B'x\right)=\max_{A\in G}\iprod{A\lambda, B'x}=\max_{A\in G}\iprod{BA\lambda,x} = k(x).
\end{split}
\]
But that would imply $\max_{A\in G}\iprod{A\lambda, x} < k(x)$ which is clearly a contradiction.
For the second assertion, consider $x,\lambda$ such that $k(x)=\iprod{x,\lambda}$. Now
\[
k\left( Bx \right) = \max_{A\in G}\iprod{A\lambda,Bx}= \max_{A\in G}\iprod{B'A\lambda,x}=\max_{A\in G}\iprod{(B^{-1}A)\lambda,x}.
\]
The right hand side is maximized when $B^{-1}A=I$ which proves the lemma.
\end{proof}
We now show that for Coxeter groups that the polytope generated by $k$ is simplicial. That is to say, all its extremal facets are simplices.
\begin{lemma}\label{maxsimp}
Consider any $n$-dimensional irreducible group of orthogonal matrices $G$. Let $\lambda \in \mathbb{R}^n$ be such that $A\lambda \neq \lambda$ for all $\lambda \neq e$. In other words, $\lambda$ has no non-trivial stabilizer.
Consider the region $\mathcal{C}= \left\{ x\in \mathbb{R}^n:\; k(x)=\iprod{\lambda,x} \right\}$. Then the interior of $\mathcal{C}$, given by
$\mathcal{C}_0 = \left\{ \iprod{\lambda,x} > \iprod{A\lambda,x}, \; \forall \; A \in G \right\}$, provides a fundamental region for the group. Additionally, if $G$ is a Coxeter group, $\mathcal{C}$ is an $n$-dimensional closed simplicial cone.
\end{lemma}
\begin{proof}
Note that $\mathcal{C}$ is the region $\left\{x\in \mathbb{R}^n:\; \iprod{\lambda, x} \ge \iprod{A\lambda, x}\; \forall A\in G \right\}$. We first show that $\mathcal{C}$ is a $n$-dimensional convex cone. Label the non-identity elements of the group $G$ by $A_1,A_2,\ldots,A_N$ where $N+1=\lvert G\rvert$.
Consider the matrix $Q$ defined by
\begin{equation}\label{livectors}
Q = \left[\begin{array}{c}
\lambda - A_1\lambda\\
\lambda - A_2\lambda\\
\vdots \\
\lambda - A_N\lambda\\
\end{array}\right],
\end{equation}
where all vectors are row vectors. This matrix $Q$ when applied to vectors of $\mathcal{C}$ produces nonnegative entries. The dimension of $Q$ is $N \times n$. We first show that the rank of $Q$ is $n$. Note that, trivially the rank cannot be more than $n$. We show that the dimension of the kernel is zero which proves that the rank must be exactly $n$.
Let $\mathcal{K}$ denote the kernel, $\left\{ x:\; Qx=0 \right\}$. Then we claim that $\mathcal{K}$ is invariant under the action of the group. This is because, $x\in \mathcal{K}$ iff $\iprod{\lambda,x}=\iprod{A\lambda,x}$ for all $A\in G$. But, for any $B,A\in G$, we also have
\[
\begin{split}
\iprod{A\lambda, Bx}&= \iprod{B^{-1}A\lambda, x}=\iprod{\lambda,x},\quad \text{since}\; x\in \mathcal{K},\\
&=\iprod{B^{-1}\lambda, x}= \iprod{B'\lambda,x}= \iprod{\lambda, Bx}.
\end{split}
\]
Thus $Bx\in \mathcal{K}$. This proves that $\mathcal{K}$ stable under the action of the group. But since the representation is irreducible, this implies that $\mathcal{K}$ is either zero or the full space. But, it is easy to see that if $\mathcal{K}$ is the full space, then $\lambda$ must be zero. This proves that $\mathcal{C}$ is $n$-dimensional. That it is a convex cone is obvious.
Since the dimension of $\mathcal{C}$ is $n$ and the stabilizer of $\lambda$ is the identity, the interior of the cone is given by $\mathcal{C}_0$. We now show that $\mathcal{C}_0$ is a fundamental region for $G$ by verifying the definition. $\mathcal{C}_0$ is open by definition. For any $A\in G $, $A\neq I$, note that $A\mathcal{C}_0$ is the following set
\[
\begin{split}
\left\{Ax,\; x\in \mathcal{C}_0\right\}&= \left\{y:\;\iprod{\lambda, A^{-1}y} > \iprod{B\lambda, A^{-1}y},\; \forall B\in G \right\}\\
&= \left\{y: \iprod{A\lambda, y} > \iprod{B\lambda,y},\; \forall B \in G \right\}.
\end{split}
\]
Thus $x\in \mathcal{C}_0 \cap A\mathcal{C}_0$ implies $\iprod{\lambda,x} > \iprod{A\lambda, x} > \iprod{\lambda, x}$ which is impossible. Thus the intersection must be empty. It is also trivial to see that $\cup_{A\in G}A \mathcal{C}= \mathbb{R}^n$. This shows that $\mathcal{C}_0$ is a fundamental region.
For Coxeter groups we now show that $\mathcal{C}_0$ is the same region as $F_{\lambda}$ defined in \eqref{fundacox}. Notice first that if $A$ is a reflection along a vector $r$ for some $A\in G$, then
\begin{equation}\label{bothfunda}
\iprod{\lambda - A\lambda , x}= 2\frac{\iprod{r,\lambda}\iprod{r,x}}{\norm{r}^2}.
\end{equation}
Now, suppose $x\in \mathcal{C}_0$. Then $\iprod{\lambda - A\lambda, x} > 0$ for all non-identity $A\in G$, in particular, for all $A$ which corresponds to reflections along the roots. Thus, for any root $r\in \Delta_{\lambda}^+$, from the above equality we get that $\iprod{r,x} > 0$. From the definition of $F_{\lambda}$, it is now obvious that $x\in F_{\lambda}$. Thus we have shown that $\mathcal{C}_0 \subseteq F_{\lambda}$.
For the reverse equality, note that if $\mathcal{C}_0$ is a proper subset of $F_{\lambda}$, then for every $A\in G$, the set $A\mathcal{C}_0$ is a proper subset of $AF_{\lambda}$. But, each $AF_{\lambda}$ is disjoint and the union of the closures of $A\mathcal{C}_0$ is the entire $\mathbb{R}^n$. This is clearly impossible. Thus, we have shown that equality holds among the two fundamental regions $\mathcal{C}_0$ and $F_{\lambda}$. Since $F_{\lambda}$ is a simplicial cone, so is $\mathcal{C}_0$. Thus $\mathcal{C}$ is a closed simplicial cone.
\end{proof}
The connection between simplicial cones and Exponential distributions is made precise in the next lemma.
\begin{lemma}\label{simpexp}
Consider a sequence of simplicial cones $C_1, C_2, \ldots, C_r$ which are open, disjoint, and the closure of their union is the whole space.
Let $k$ be a nonnegative, irreducible, positively homogeneous function such that $k$ is linear on each $C_i$. That is, $k(x)= \iprod{\lambda_i,x}$, {for all} $x\in C_i$, for some sequence of vectors $\lambda_1,\lambda_2,\ldots,\lambda_r$ which may not be all distinct.
Let $X$ be a random variable whose density with respect to the Lebesgue measure on $\mathbb{R}^n$ is proportional to $e^{-2k(x)}$. Let $B_i$, $i=1,2,\ldots,r$, be any set of invertible matrices such that for each $i$, the matrix $B_i$ maps the cone $C_i$ onto the $n$-dimensional quadrant. Also, let $\alpha_1(i),\ldots,\alpha_n(i)$ be the coefficients in the unique representation $\lambda_i'= \alpha'(i) B_i$. Then, the random vector $Y=(Y_1,Y_2,\ldots,Y_n)$, where
\[
Y_j:= \alpha_j(i)\iprod{(B_i)_{j*}, X}, \quad \text{if}\; X\in C_i,
\]
is a vector of iid Exponential$(2)$ random variables. Here $(B_i)_{j*}$ denotes the $j$th row of the matrix $B_i$.
\end{lemma}
\begin{proof}
Since the cones $C_i$'s are simplicial, the existence of the matrices $B_i$'s follows from the definition. Moreover, it follows from the definition of $\alpha(i)$ that if $x\in C_i$ and $z=B_ix$, then $k(x)=\iprod{\lambda_i,x}=\sum_j \alpha_j(i) z_j$ irrespective of $i$. Let $B_i^*$ be the matrix given by
\[
\left(B_i^*\right)_{j*}= \alpha_j(i)(B_i)_{j*}, \quad j=1,2,\ldots,d.
\]
Since the transformations are piecewise linear, it follows that
\[
Y:= \sum_j B^*_j X 1\left( X\in C_j \right)
\]
has a density proportional to $\exp\{ -2\sum_j y_j\}$ over the quadrant $\{y:\; y_1\ge 0, y_2\ge 0, \ldots, y_d\ge 0 \}$. This immediately identifies itself as the joint density of iid exponentials with rate two. This proves part (2).
\end{proof}
\begin{prop}\label{mainthmstatic}
Let $G$ denote a Coxeter group acting on $\mathbb{R}^n$. For $\lambda\in \mathbb{R}^n$, $\lambda\neq 0$, let $k(x)=\max_{A\in G}\iprod{A\lambda, x}$.
Let $\nu$ be the probability measure with unnormalized density $\exp\{-2k(x)\}$ on $\mathbb{R}^n$. Then the following statements hold true.
\begin{enumerate}
\item When the stabilizer of $\lambda$ is trivial, the conic hull of the finite set $\{\lambda - A\lambda,\; A\in G \}$ contains $n$ linearly independent generating vectors $\{\eta_1,\eta_2,\ldots,\eta_n\}$. That is, every other vector in the set can be expressed as a linear combination of the generators with nonnegative coefficients.
\item Let $X$ denote a random variable with distribution $\nu$. Also let $\alpha_i$ denote the unique positive coefficient of $\eta_i$ in the expansion
\begin{equation*}\label{whatisalpha}
\lambda= \sum_{i=1}^n \alpha_i \eta_i.
\end{equation*}
For $i=1,2,\ldots,n$, define the change of variable
\begin{equation*}\label{howtochange}
Y_i= \alpha_i\iprod{A\eta_i,X}, \quad \text{when}\quad k(X)=\iprod{A\lambda, X}.
\end{equation*}
Then, the vector $Y=(Y_1,Y_2,\ldots,Y_n)$ are iid Exponential$(2)$ random variables.
\end{enumerate}
\end{prop}
\begin{proof}
To prove the first assertion, we use Lemma \ref{maxsimp}. Assume first that the stabilizer of $\lambda$ is trivial. Then, Lemma \ref{maxsimp} tells us that the cone
\begin{equation}\label{pfcone}
\mathcal{C}=\{x:\; \iprod{\lambda-A\lambda,x} \ge 0 \}
\end{equation}
is an $n$-dimensional simplicial cone. Hence there exists exactly $n$ many linearly independent generators among the set $\{\lambda - A\lambda, \; A\in G \}$ such that every other vector is a linear combination with nonnegative coefficients. The rest of the result follows directly from Lemma \ref{simpexp}. Notice that the coefficients of $\lambda$ in the expansion $\lambda= \sum_{i=1}^n \alpha_i \eta_i$ are positive by Farkas lemma. This is because, for any non-zero $y\in \mathcal{C}$, the inner product $\iprod{\lambda,y}= k(y) > 0$ by irreducibility and nonnegativity of $k$.
\medskip
When $\lambda$ has a nontrivial stabilizer, the cone in \eqref{pfcone} is a union of several simplicial cones. The simplest way to see this is to take a sequence $\lambda_l$ which have no nontrivial stabilizers and which converges to $\lambda$. The component cones are then given by the limits of the sequence of simplicial cones generated by them. In any case Lemma \ref{simpexp} still holds, however, the vectors $\{\eta_1, \ldots, \eta_n\}$ have to be determined by the limiting procedure.
\end{proof}
The proof of Proposition \ref{gpdiff1} in the Introduction now follows easily.
\subsection{Examples}\label{examples}
Let us consider some examples of consequences of Proposition \ref{gpdiff1}.
\medskip
\noindent\textbf{Example 1: Rank based interactions.} Brownian motions with rank based interactions have been considered in equation \eqref{ranksde}. Clearly the drift function $b$ is constant over finitely many cones determined by the permutation generated by the ordered coordinates. Let $x[1] \le x[2] \le \ldots x[n]$ denote the coordinates of an $n$ dimensional vector arranged in increasing order. It is easy to see that
\[
k(x)= \sum_{i=1}^n \delta_i x{[i]},
\]
is a positively homogeneous function which is not irreducible since it takes a constant value over the linear span of the vector of all ones. However, if we let $H$ be the subspace orthogonal to $1$, then $k$ splits additively as
\[
k(x) = -\sum_{i=1}^n \delta_i \left( x{[i]} - \bar{x}\right) + n\bar{\delta}\bar{x}.
\]
Let $k_1:H\rightarrow \mathbb{R}$ denote the restriction of $k$ to $H$, then it is clear that $k_1(x)=k_1(A_{\sigma}x)$ for any permutation matrix $A_{\sigma}$. Now, the group of permutation matrices acting on $H$ is well known to be irreducible and generated by reflections along $e_{i+1}-e_i$ for $i=1,2,\ldots,n-1$. This is just a restatement of the fact that every permutation can be written as a product of transposes. Thus, it is a Coxeter group, often denoted by $\mathcal{A}_{n-1}$. Thus the conclusions of Proposition \ref{constructk} applies and $k_1$ is irreducible if
\begin{equation}\label{whenkrank}
k_1(x)= \max_{A_{\sigma}}\iprod{-A_{\sigma}\delta,x}.
\end{equation}
This condition is equivalent to the condition $\delta_1 \ge \delta_2 \ge \ldots \ge \delta_n$.
Hence, from Proposition \ref{additivesplit} it also follows that the projection $P_HX_t$ of the diffusion on to $H$ has an invariant distribution whose density with respect to Lebesgue measure on $H$ is proportional to $\exp(-2k_1(x))$.
We now apply Proposition \ref{mainthmstatic}. One can see that the set of vectors $-\delta + A_{\sigma}\delta$, as $\sigma$ ranges over permutations, contains positive multiples of vectors $e_{i+1} - e_i$, since they correspond to the transposition of $i$ and $i+1$. These $n-1$ linearly independent vectors are the conic extremes of the set. Thus, by Proposition \ref{mainthmstatic}, the spacings $X[i+1]- X[i]$ are independent Exponential random variables under the invariant distribution. The correct rates can be easily verified.
In this example it is easy to see the shortfall of the sufficient condition \eqref{whenkrank}. The drift function is constant over a fundamental region $F=\{x\in H:\; x_1 < x_2 < \ldots < x_n \}$ which is clearly simplicial. Now $k(x)$ is irreducible if and only if the unit ball generated by $k$ is compact. By symmetry, we can restrict our attention to $F$. Since $F$ is simplicial we can apply a suitable linear transformation to map it to the positive quadrant. Thus, it can be easily verified that if $\bar{\delta}$ denoted the average of the coordinates of $\delta$, the intersection of the unit ball with $F$ is compact if and only~if
\[
\alpha_k:=\sum_{i=1}^k\left( \delta_i - \bar{\delta} \right) >0, \quad \forall\; i=1,2,\ldots, n-1.
\]
This is precisely the condition derived by Pal \& Pitman in \cite{palpitman} using the theory of reflected Brownian motions and is weaker than the sufficient condition that the coordinates of $\delta$ decreases with increasing values of the coordinates of $x$.
\medskip
\noindent\textbf{Example 2: Sign-rank based interactions.}
An example of interactions similar to rank-based can be generated by allowing both the rank and signs coordinates to determine the drift. As before, we start with the $n$-dimensional SDE:
\begin{equation}\label{gensgnrank}
d X_t = b(X_t) dt + d W_t,
\end{equation}
where $W_t$ is an $n$-dimensional Brownian motion. Suppose that the drift function takes finitely many values, is scale invariant, and $b(Ax)=Ab(x)$ whenever $A$ is either a permutation matrix or a diagonal matrix with each diagonal entry being plus or minus one. Thus, not only that the values of the drift get permuted whenever the coordinates get permuted, but also the sign of the drift changes with the sign of the corresponding coordinate.
The group generated by the collection of permutation matrices and the diagonal matrices of sign flips is a Coxeter group denoted by $\mathcal{B}_n$. Please see pages 66--71 of \cite{reflectiongroups} for more details. We can safely apply Proposition \ref{gpdiff1}. Thus, the $n$ dimensional process under such a sign-rank based interaction is recurrent if there is a vector $\lambda \in \mathbb{R}^n$, $\lambda \neq 0$, such that
\[
k(x)=-\iprod{x, b(x)} = \max_{A\in \mathcal{B}_n}\iprod{A\lambda, x}.
\]
If we restrict the above condition to the cone $\{ x: 0 < x_1 < x_{2} < \ldots < x_n \}$ we see that the vector of drifts $b(x)=-\lambda$ where $\lambda$ satisfis that $0 \le \lambda_1 \le \lambda_2 \le \ldots \le \lambda_n $.
When this does hold true, $X$ has an unique long term stationary distribution. To find the decomposition of this probability distribution in terms of independent Exponentials we consider a $\delta$ all of whose coordinates are non-zero and distinct. That is, it has a trivial stabilizer subgroup in $\mathcal{B}_n$. Consider the conic hull generated by the set of vectors $\{ \lambda - A\lambda, \; A\in \mathcal{B}_n \}$. As in the case of rank-based interactions one can see that the generators of the conic hull are positive multiples of the vectors $e_1$ and $\{e_{i+1} - e_i , \; i=1,2,\ldots n-1\}$.
Now we apply the final conclusion of Prop \ref{gpdiff1}. To get the vector of Exponentials under the invariant distribution, note that
\[
k(x) = \sum_{i=1}^n \lambda_i \abs{x}[i],
\]
where $\abs{x}[1] \le \abs{x}[2] \le \ldots \abs{x}[n]$ are the ordered values of the absolute values of the coordinates $(\abs{x_1}, \abs{x_2}, \ldots, \abs{x_n})$.
Thus, from Proposition \ref{gpdiff1} it follows that the random vector $(\abs{X}[1], \abs{X}[i+1] - \abs{X}[i], \; i=1,2,\ldots, n-1)$ are distributed as independent Exponentials.
To compute the rates of these Exponentially distributed random variables, one needs to compute the coefficient of $\lambda$ with respect to the basis vector $e_1$ and $\{e_{i+1} - e_i , \; i=1,2,\ldots n-1\}$. A simple computation leads us to the conclusion that the corresponding vector of rates of these Exponentials are given by
\[
\left(2\sum_{s=j}^n \lambda_s, \quad j=1,2,\ldots, n\right).
\]
\medskip
\noindent{\textbf{Example 3: Constrained sign-rank based interactions.}}
This is a interesting class of constrained sign-rank based interactions where not all sign changes of coordinates affect the drift vector. Consider again the stochastic differential equation \eqref{gensgnrank}. Suppose that the drift function $b$ takes finitely many values, is scale invariant, and $b(Ax)=Ab(x)$ for all permutation matrices $A$ and all diagonal matrices with each diagonal entry being positive or negative one \textit{with the constraint} that there are even number of negative ones. Thus, the sign of the drift vector changes when either the ordering of coordinates change or when pairs of coordinates have flipped their signs.
The groups generated by the permutation matrices and the diagonal matrices with even number of sign flips is clearly a subgroup of the $\mathcal{B}_n$. They form, in fact, a family of Coxeter groups, usually denoted by $\mathcal{D}_n$ where $n$ denotes the dimension of the underlying space. We again refer the reader to pages 66--71 of \cite{reflectiongroups} for more details.
We apply Proposition \ref{gpdiff1}. Thus, the $n$ dimensional process under such a constrained sign-rank based interaction is recurrent if there is a vector $\lambda \in \mathbb{R}^n$, $\lambda \neq 0$, such that
\[
k(x)=-\iprod{x, b(x)} = \max_{A\in \mathcal{D}_n}\iprod{A\lambda, x}.
\]
The above condition is more difficult to analyze than the previous examples. One can show using known results about the fundamental root systems of $\mathcal{D}_n$ (page 71 in \cite{reflectiongroups}) that the drift is determined by the fact that over the cone
\begin{equation}\label{concons}
\left\{ x: 0 < x_1 + x_2, \quad x_1 < x_{2} < \ldots < x_n \right\},
\end{equation}
the drift is a constant $b(x)=-\lambda$ where $\lambda$ satisfis that $0 \le \lambda_1 + \lambda_2$ and $ \lambda_1 \le \lambda_2 \le \ldots \le \lambda_n $. This cone is actually a fundamental region for the group $\mathcal{D}_n$. Thus, the drift vector is now determined over entire $\mathbb{R}^n$ by the property $b(Ax)=Ab(x)$ for all $A\in \mathcal{D}_n$.
Under this condition the process has a long term stationary distribution. To find what functions turn out to be independent Exponentials, we need to understand, for a given $x$, what unique $A\in \mathcal{D}_n$ will produce $k(x)=\iprod{\lambda, Ax}$. Clearly, this will happen if $A$ is chosen such that $Ax$ belongs to the cone \eqref{concons}. There are two cases to consider. One, when the number of coordinates of $x$ that are negative is even. In this case, one simply flips the signs of these coordinates, and then ranks the absolute values to get a vector in \eqref{concons}. Both these actions are permissible since they correspond to multiplication by matrices in $\mathcal{D}_n$. The other case is when $x$ has odd number of negative coordinates. First, one has to flip the sign of all the negative coordinates except the least negative one and then rank all the coordinates. In this ordering, either the absolute value of the second least negative coordinate is less than the least positive coordinate in which case, the resulting vector is in \eqref{concons}. Or, it is not, in which case we need to compare the least negative coordinate with the least positive coordinate. Their sum is either positive or negative, and we make the appropriate (zero or two) sign flips to get the right transformation. Let $H(x)$ be the resulting vector produced by the above procedure.
Under this stationary distribution, the vector of random vector $H(X_1, X_2, \ldots, X_n)$ is distributed as $n$ independent Exponentials. Furthermore, as in the case of sign-rank interactions, one can work out the linear algebra to compute that the corresponding vector of rates of these Exponentials are given by
\[
\left(\sum_{s=1}^n \lambda_s,\quad -\lambda_1+ \sum_{s=2}^n \lambda_s, \quad \quad 2\sum_{s=j}^n \lambda_s, \quad j=3,4,\ldots, n\right).
\]
\subsection{An example of irregular interaction}\label{graphcase}
In this subsection we consider an example of interacting Brownian motions with drift functions that are still piecewise constant on cones, but are not consistent with any group action.
Consider a graph $\mathcal{G}$ on $n$ vertices where the vetices are labeled by $\{1,2,\ldots,n\}$. The edge between $i$ and $j$ have an associated edge weight $\beta_{ij}$, which is zero if there is no edge between the two vertices.
Consider the SDE on $\mathbb{R}^n$ given by
\begin{equation}\label{sdegraph}
dX_t(i) = \sum_{j=1}^n\beta_{ij}\text{sign}\left( X_t(j) - X_t(i) \right)dt + dW_t(i),
\end{equation}
where, as before, $W=(W(1), W(2), \ldots, W(n))$ is an $n$-dimensional Brownian motion.
When all the edge weights are nonnegative, the model can be described by saying that the Brownian motions, which are indexed by the vertices of the graphs, get attracted towards one another. The constants $\beta_{ij}$ measure the strength of their attraction.
The appropriately defined drift function $b(x)$ is piecewise constant on the family of cones $C_{\pi}:=\{ x: x_{\pi(1)}\le x_{\pi(2)}\le \ldots \le x_{\pi(n)} \}$, where $\pi$ ranges over all permutations of $n$ labels. However, it might not satisfy the condition that $b(A_{\pi}x)=A_{\pi}b(x)$ where $A_{\pi}$ is the permutation matrix corresponding to $\pi$.
Note that the drift function $b$ is the negative of the gradient of the positively homogeneous function
\[
k(x)=\sum_{i<j}\beta_{ij}\abs{x_i-x_j}, \qquad \forall \; x\in \mathbb{R}^n.
\]
It can be easily verified (and intuitive) that if $\beta_{ij}$'s are nonnegative and $\mathcal{G}$ is connected, the function $k$ is irreducible when restricted to the subspace $H$ orthogonal to the vector of all ones (which we denote by $1$). Then the conclusions of Proposition~\ref{additivesplit} applies. In particular, if we define
\[
k'(x) = k(x) + \abs{\iprod{x,1}}
\]
then $k'$ is integrable and both the probability measures induced by functions $\exp(-2k)$ and $\exp(-2k')$ on $H$ must be the same.
Assume $\beta_{ij}\ge 0$ and $\mathcal{G}$ is connected. For convenience absorb the factor of two in $\exp(-2k)$ in the definition of $\beta$. Let $\mu_n$ be the probability measure whose unnormalized density is given by
\begin{equation}\label{invgraph}
\exp\left(-\sum_{i<j} \beta_{ij} \abs{x_i - x_j} - \abs{\sum_i x_i}\right).
\end{equation}
What properties of the probability measure can we explicitly describe ? Clearly, any deep inspection of such a general family is extremely difficult. We will improve our chances if we restrict the edge weights to the following class. Consider $n$ positive constants $m_1, m_2, \ldots, m_n$. Let $\beta_{ij}=m_im_j$ for all pairs $i,j$. In particular the graph is complete. One can think of $m_i$ as the \textit{mass} of the $i$th particle, and hence the strength of attraction between particles $i$ and $j$ is proportional to the product of their masses. In fact, the case when all the $m_i$'s are equal to one have been dealt with in \cite{chatpal} where they were named the one-dimensional gravity model.
Now suppose $X = (X_1,\ldots,X_n)$ follows the p.d.f.
\[
C_n \exp\left(-\sum_{i<j} m_im_j \abs{x_i - x_j} - \abs{\sum_i x_i}\right),
\]
where $C_n$ is the normalizing constant. Let $M = \sum_{i=1}^n m_i$ be the total mass of the system. For each $i$, let $Y_i = X_{(i)}$ and $\Pi(i)$ be the (random) index $j$ such that $Y_i = X_j$. The joint p.m.f.\ of $(\Pi, Y)$ at a point $(\pi,y)$ is
\[
C_n \exp\biggl(-\sum_{i<j} m_{\pi_i} m_{\pi_j} (y_j-y_i) - \abs{\sum_i y_i}\biggr),
\]
where $\pi=(\pi_1,\ldots,\pi_n)$ is a permutation of $\{1,\ldots,n\}$ and $y_1 < \cdots < y_n\in \mathbb{R}$.
Now let $\Delta_i = Y_{i+1}-Y_i$, $i=1,\ldots,n-1$. For each $i$, and each $\pi\in S_n$, let
\[
F_i(\pi) = \frac{\sum_{j=1}^i m_{\pi_j}}{M}.
\]
Then $M^{-1}\sum_{j=i+1}^n m_{\pi_j} = 1- F_i(\pi)$, and $F_n(\pi) \equiv 1$. The joint density of $(\Pi, \Delta, Y_1)$ at a point $(\pi, \delta, y_1)$ is
\begin{align*}
&C_n \exp \left(- \sum_{i<j} m_{\pi_i} m_{\pi_j} \biggl(\sum_{k=i}^{j-1} \delta_k \biggr) - \abs{\sum_{i=2}^n\sum_{j=1}^{i-1} \delta_j + n y_1}\right)\\
&= C_n \exp\left(-M^2\sum_{i=1}^{n-1} F_i(\pi)(1-F_i(\pi)) \delta_i - \abs{\sum_{i=2}^n\sum_{j=1}^{i-1} \delta_j + n y_1}\right).
\end{align*}
Now we can easily integrate out $Y_1$ to get the joint density of $(\Pi, \Delta)$:
\[
C \exp\biggl(-M^2\sum_{i=1}^{n-1} F_i(\pi)(1-F_i(\pi)) \delta_i\biggr),
\]
where $C$ is now a different constant. Thus, conditional on $\Pi = \pi$, $Y_1,\ldots,Y_n$ are independent, with $Y_i \sim \mathrm{Exp}(M^2F_i(\pi)(1-F_i(\pi)))$. It is easy to see from this observation that the marginal p.m.f.\ of $\Pi$ must be
\begin{equation}\label{pmfpi}
C(m) \prod_{i=1}^{n-1} \frac{1}{F_i(\pi)(1-F_i(\pi))},
\end{equation}
where $\pi = (\pi_1,\ldots,\pi_n)$ is any permutation of $\{1,\ldots,n\}$ and $C(m)$ is the normalizing constant that depends on the values of $m_1,\ldots, m_n$.
If $m_1 = m_2 =\cdots = m_n$, then this is the uniform distribution on $S_n$. Otherwise, it is a non-uniform distribution on the set of permutations. Thus the cost we pay for losing the regularity of group actions is that the spacings between the order statistics are only conditionally independent Exponentials, conditioned on the random permutation generated by the ranks.
It is very difficult to see what sort of distributions on the space of permutations the probability mass function \eqref{pmfpi} induces. Clearly the p.m.f.\ is large when $F_i(\pi)$ is close to zero or one for most values of $i$. The intuition from gravity predicts that heavier particles should be close and should avoid being too high or too low in rank.
We now show this to be true in a particularly simple case when there is a single distinguished particle. Suppose that $m_1=\alpha$ and $m_2=m_3=\ldots=m_n=1$. We consider the joint distribution as before
\[
\frac{d\mu}{dx} = C_n \exp\left(-\sum_{i < j}m_i m_j\vert x_i - x_j\vert + \abs{ \sum x_i } \right).
\]
We are interested in the derived joint distribution of the ranks of each particle given by the general expression in the previous section.
Let us compute the distribution of the rank of the first particle which is distinguished from the others due to a different mass.
\begin{equation}\label{beta}
\begin{split}
P(\text{rank of the first particle} = j)&= P(\Pi(j)=1)\\
&= \sum_{\sigma: \sigma(j)=1} C \prod_{i=1}^{n-1}\frac{1}{F_i(\sigma)(1-F_i(\sigma))},
\end{split}
\end{equation}
where $C$ is a constant depending on $n$ and $\alpha$. In the following text, we will freely use $C$ as the normalizing constant keeping in mind that the constants might be different from one another although they only depend on $n$ and $\alpha$.
Now, there are $(n-1)!$ many permutations $\sigma$ such that $\sigma(j)=1$. For any of them
\[
\begin{split}
F_i(\sigma) =
\begin{cases}
\frac{i}{\alpha + n-1} & \text{if}\; i < j,\\
\frac{\alpha + i-1}{\alpha + n-1} & \text{otherwise}.
\end{cases}
\end{split}
\]
And thus we can rewrite \eqref{beta} as
\begin{equation}\label{finitebeta}
\begin{split}
P(\Pi(j)=1) &= C \frac{1}{\prod_{i=1}^{j-1}\frac{i}{\alpha + n - 1}\left(1- \frac{i}{\alpha+ n -1} \right)\prod_{i=j}^{n-1} \frac{\alpha + i - 1}{\alpha + n -1}\left( 1 - \frac{\alpha + i - 1}{\alpha + n - 1} \right) }\\
&= C \frac{1}{(j-1)! (n-j)! \prod_{i=1}^{j-1}(\alpha + n - i -1) \prod_{i=j}^{n-1}(\alpha + i -1)}\\
&= C \comb{n-1}{j-1}\frac{\alpha(\alpha+1)\ldots(\alpha+n-j-1)\; \alpha(\alpha+1)\ldots(\alpha + j-2) }{\alpha(\alpha+1)\ldots(\alpha + n -2)\;\alpha(\alpha+1)\ldots(\alpha + n - 2)}\\
&= C \comb{n-1}{j-1}\alpha(\alpha+1)\ldots(\alpha+n-j-1)\; \alpha(\alpha+1)\ldots(\alpha + j-2).
\end{split}
\end{equation}
We can immediately infer from the previous expression the following fact:
\[
\begin{split}
\frac{P(\text{rank of the first particle}=j+1)}{P(\text{rank of the first particle}=j)} & = \frac{P(\Pi(j+1)=1)}{P(\Pi(j)=1)}\\
& = \frac{(n-j)(\alpha + j -1)}{j(\alpha + n - j -1)}.
\end{split}
\]
In other words
\begin{eqnarray*}
P(\text{rank of the first particle}=j+1) &>& P(\text{rank of the first particle}=j)\\
\text{iff}\quad (n-j)(\alpha + j -1) & >& j(\alpha + n - j -1),\\
\text{iff}\quad n(\alpha - 1) + (n - \alpha + 1)j - j^2 &>& (\alpha + n -1)j - j^2,\\
\text{iff}\quad 2(\alpha - 1)j &<& n(\alpha - 1).
\end{eqnarray*}
Thus, if $\alpha$ is more than $1$, the probability of the rank being $j$ increases till $j=\lceil n/2\rceil$, and then strictly decreases. Clearly, the most likely position for the heavier particle is going to be the median. On the other hand, if $\alpha < 1$, just the opposite happens, and we are likely to see the lighter particle either at the top or trailing behind.
The probability computed in \eqref{finitebeta}, although seemingly unfriendly, is a very familiar object. Consider a Polya's urn scheme which has $\alpha$ red balls and $\alpha$ black balls. We play a game where at each step we pick a ball at random and replace it in the urn with a ball of the same color. It is well known (see Feller \cite{feller}) that if we play this game for $n-1$ steps the probability we pick exactly $j-1$ red balls is given by
\[
\comb{n-1}{j-1}\frac{\alpha(\alpha+1)\ldots(\alpha+n-j-1)\; \alpha(\alpha+1)\ldots(\alpha + j-2)}{2\alpha (2\alpha +1)(2\alpha +2)\ldots (2\alpha + n-2)}.
\]
If we compare the previous expression with \eqref{finitebeta}, the differences are merely in the expression of the normalizing constants. Thus, if $\sigma(1)=\Pi^{-1}(1)$ is the rank of the first particle, it is clear that $\sigma(1)-1$ has the same distribution as the number of red balls picked in a Polya's urn scheme run for $n-1$ steps.
\begin{prop}\label{massbeta} For any $\alpha > 0$, consider the SDE \eqref{sdegraph} with a distribution of mass such that the mass of the first particle being $\alpha$ and the rest being of mass $1$.
Then the sequence of random variables $\sigma(1)/n$ converges weakly to the Beta$(2\alpha, 2\alpha)$.
\end{prop}
\begin{proof}
The proof follows from known results about Polya's urn. The factor of $2$ is due to the fact that we had earlier absorbed the $2$ in \eqref{invgraph}.
\end{proof}
\bibliographystyle{amsalpha}
| {
"timestamp": "2009-02-27T07:24:53",
"yymm": "0902",
"arxiv_id": "0902.4762",
"language": "en",
"url": "https://arxiv.org/abs/0902.4762",
"abstract": "We consider a particular class of n-dimensional homogeneous diffusions all of which have an identity diffusion matrix and a drift function that is piecewise constant and scale invariant. Abstract stochastic calculus immediately gives us general results about existence and uniqueness in law and invariant probability distributions when they exist. These invariant distributions are probability measures on the $n$-dimensional space and can be extremely resistant to a more detailed understanding. To have a better analysis, we construct a polyhedra such that the inward normal at its surface is given by the drift function and show that the finer structures of the invariant probability measure is intertwined with the geometry of the polyhedra. We show that several natural interacting Brownian particle models can thus be analyzed by studying the combinatorial fan generated by the drift function, particularly when these are simplicial. This is the case when the polyhedra is a polytope that is invariant under a Coxeter group action, which leads to an explicit description of the invariant measures in terms of iid Exponential random variables. Another class of examples is furnished by interactions indexed by weighted graphs all of which generate simplicial polytopes with $n !$ faces. We show that the proportion of volume contained in each component simplex corresponds to a probability distribution on the group of permutations, some of which have surprising connections with the classical urn models.",
"subjects": "Probability (math.PR); Combinatorics (math.CO)",
"title": "A combinatorial analysis of interacting diffusions",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808718926533,
"lm_q2_score": 0.7217432122827968,
"lm_q1q2_score": 0.7075110654191844
} |
https://arxiv.org/abs/1901.07189 | Imaging point sources in heterogeneous environments | Imaging point sources in heterogeneous environments from boundary or far-field measurements has been extensively studied in the past. In most existing results, the environment, represented by the refractive index function in the model equation, is assumed known in the imaging process. In this work, we investigate the impact of environment uncertainty on the reconstruction of point sources inside it. Following the techniques developed by El Badia and El Hajj (C. R. Acad. Sci. Paris, Ser. I, 350 (2012), 1031-1035), we derive stability of reconstructing point sources in heterogeneous media with respect to measurement error as well as smooth changes in the environment, that is, the refractive index. Numerical simulations with synthetic data are presented to further explore the derived stability properties. | \section{Introduction}
\label{SEC:Intro}
Recovering radiative sources inside heterogeneous media from boundary or far-field measurements has applications in many branches of science and technology~\cite{AmBaFl-SIAM02,Baltes-Book78,BaLiTr-JDE10,BaBeBeLe-IP05,CaEw-IPBVP75,Chin-ACM10,ElVa-IP09,GaOsTa-JIIPP13,HoCl-MB97,Ikehata-IP99,ImYa-IP98,IsLu-IPI18,JiLiYa-JDE17,Larsen-JQSRT73,MaTs-IP13,Nicaise-SIAM00,PuYa-IP96,Sezginer-IP87,Siewart-JQSRT93,Yamamoto-IP95}. Extensive mathematical and computational studies of such inverse source problems have been performed in the past decades; see, for instance, ~\cite{AnBuEr-Book97,Isakov-Book90,Isakov-Book06} and references therein for recent reviews on the subject.
In this work, we are interested in a source recovery problem where the source to be reconstructed is the superposition of point sources~\cite{BaBeBeLe-IP05,CaEw-IPBVP75,ElBadia-IP05,ElHa-JIIPP02,ElEl-CRASP12,ElNa-IP11,FaHaEs-IPI13,GaZhCoWa-BOE10,KaLe-IP04,KoYa-IP02,LiTa-CiCP09,MaTs-IP13,OhInOh-IP11,Vessella-IP92}. Unlike general source functions, point sources are efficiently characterized by their locations and strengths, a fact that significantly reduces the dimension of the parameter space of the inverse problems. This dimension reduction often enables one to obtain uniqueness in the inverse problem with minimum amount of observed data and provides the possibility of utilizing efficient reconstruction algorithms, for instance these based on compressive sensing~\cite{ChMoPa-IP12,FaStYa-SIAM10}, in the source recovery process.
To formulate our problem, let $\Omega\subset \mathbb R} \newcommand{\bbS}{\mathbb S^d$ ($d\ge 3$) be a simply connected domain with $\cC^2$ boundary $\partial\Omega$. Let $u(\bx)$ be the solution to the following boundary value problem to the Helmholtz equation:
\begin{equation}\label{EQ:Helmholtz}
\begin{array}{rcll}
\Delta u + k^2 \big(1+n(\bx)\big) u &=& q(\bx), &\mbox{in}\ \ \Omega\\
u &=& f(\bx), & \mbox{on}\ \partial\Omega
\end{array}
\end{equation}
where the real-valued function $n(\bx)$ is the refractive index, $q(\bx)$ and $f(\bx)$ are internal and boundary source terms respectively. We assume that $n(\bx)$ has a compact support in $\Omega$, that is, $\operatorname{supp}(n) \subset\subset \Omega$, and that $1+n(\bx)>0$, $\forall \bx\in\Omega$. We assume that $0$ is not an eigenvalue of the operator $\Delta + k^2(1+n)$ with homogeneous Dirichlet boundary condition such that the problem~\eqref{EQ:Helmholtz} admits a unique solution for given source functions $q$ and $f$.
We assume that the internal source function $q(\bx)$ is a superposition of $m$ point sources located at $\{\bx_j\}_{j=1}^m$ with strengths $\{\lambda_j\}_{j=1}^m$, that is,
\begin{equation}\label{EQ:Source}
q(x) = \sum_{j=1}^m \lambda_j \delta(\bx-\bx_j).
\end{equation}
The strengths $\{\lambda_j\}$ are all assumed to be real-valued so that there is no physical absorption occurring at the point sources.
The Helmholtz equation~\eqref{EQ:Helmholtz} can be viewed as a simplified frequency-domain model for either electromagnetic or ultrasound wave propagation, depending on the value of the parameters, mainly the wavenumber $k$, in the equation. The mathematical derivations in the rest of the paper implicitly assume that the wavenumber $k$ is real-valued and $k>0$. We believe, however, the same types of calculations can be carried out for the zero-frequency case ($k=0$) as those studied in ~\cite{BaBeBeLe-IP05,ElBadia-IP05} or in~\cite{WaLiJi-MP05} with an extra absorption term.
Let us also mention that since linearizing inverse coefficient problems often results in inverse source problems, the point source reconstruction problem we study in this paper is closely related to the problem of reconstructing small volume inclusions in background media~\cite{AmKa-IP03,AmMoVo-ESAIM03} and the problem of imaging small scatterers in complex media~\cite{BoPaTs-IP03,BoPaTs-IP05,ChMoPa-IP16,DeMaGr-JASA05}. The main difference is that in our derivation below, we can utilize explicitly the fact that point sources are singular. The secondary sources created by small scatterers or inclusions, however, do not carry the same level of singularity of the point sources.
We are interested in the problem of reconstructing the point sources, i.e. their locations and strengths, from Cauchy data $(f, g)$ where the boundary measurement $g(\bx)$ is given as
\begin{equation}\label{EQ:Helmholtz Data}
g(\bx)=\left( {\dfrac{\partial u}{\partial \nu}}\right){\bigg|}_{\partial\Omega} \equiv {\boldsymbol \nu}\cdot\nabla u|_{\partial\Omega},
\end{equation}
${\boldsymbol \nu}(\bx)$ being the unit outer normal vector of the domain boundary at $\bx\in\partial\Omega$.
To our best knowledge, in all the previous work on point source recovery, the environment, that is, the refractive index $n(\bx)$ in our formulation, in which point sources (or point source like localized objects), are to be sought is assumed to be known exactly, with the only exception in ~\cite{BuLaTaTs-AFR09} where the authors tried to reconstruct point sources in homogeneous media with unknown impenetrable obstacles. In other words, in the environment under which the data are collected is the same as the environment that data are back-propagated to reconstruct the point sources. Moreover, the mathematical works where stability of source reconstructions are derived with respect to noise in measured data are all done under the assumption that the environment is homogeneous (so that one could have access to the explicit form of the associated Green's function).
In the rest of the paper, we prove a stability result, following the techniques developed by El Badia and El Hajj (\emph{C. R. Acad. Sci. Paris, Ser. I, 350 (2012), 1031-1035})~\cite{ElEl-CRASP12}, on the recovering of point sources in smooth \emph{inhomogeneous} environment, that is, when the refractive index $n(\bx)$ varies smoothly in space. We also prove the stability of point source reconstruction with respect to smooth changes in the environment: if we perform reconstructions in a medium that is only slightly, in appropriate sense, different than the medium from which we collected the data, then the reconstructions are only slightly different from the reconstructions in the exact medium that generated the data.
\section{Main results}
\label{SEC:Results}
We now present the main result of this short paper: (i) the stability of reconstructing point sources in an heterogeneous medium; and (ii) the stability of the reconstructions with respect to smooth changes of the medium.
\subsection{Stability in heterogeneous media}
\label{SUBSEC:Stab Hetero Media}
We first consider the case where the underlying medium is known but heterogeneous, that is, we have a known but spatially varying refractive index $n(\bx)$. Under this circumstance, we can uniquely reconstruct the source $q(\bx)$ from a single pair of Cauchy data $(f, g)$ with a given H\"older type of stability. This is a generalization of the stability results of El Badia and El Hajj established in~\cite{ElEl-CRASP12,ElNa-IP11}.
We make the following general assumptions on the setup of the problem.
\noindent{\bf (A).} The domain $\Omega$ has a $\cC^2$ boundary $\partial\Omega$. The refractive index $n(\bx)$ is real-valued and smooth, with support $\operatorname{supp}(n)\subset\Omega$. The point sources are well separated in the sense that $\displaystyle\min_{i\neq j}|\bx_i-\bx_j|\ge \mathfrak{c}>0$ for some $\mathfrak{c}$. The point sources are sufficiently far away from the boundary of the domain such that ${\rm dist}(\bx_j,\partial\Omega)\ge \widetilde\mathfrak{c}>0$, $\forall j$, for some $\widetilde\mathfrak{c}$. The strengths of the point sources satisfy $0<\underline{\lambda} \le \lambda_j\le \overline{\lambda} <+\infty$, $\forall j$, for some $\underline{\lambda}$ and $ \overline{\lambda}$. The illumination boundary source $f$ is the restriction of a $\cC^\infty$ function to $\partial\Omega$.
It will be clear that the smoothness assumptions on the refractive index $n(\bx)$ and the boundary source $f$ are not completely necessary. In fact, being $\cC^3$ is sufficient for all the results to hold. It should also be noted that the assumptions on the point sources imply that $\bx_i\neq \bx_j$ whenever $i\neq j$, a fact that is implicitly used later when we study uniqueness of reconstructions.
\begin{lemma}\label{LEM:Uniqueness Known Media}
Let
\begin{equation}\label{EQ:Source12}
q_\ell = \sum_{j=1}^{m_\ell} \lambda_{\ell, j} \delta(\bx - \bx_{\ell, j}), \quad \ell=1,2
\end{equation}
be two sets of point sources satisfying the assumptions in~{\bf (A)}, $u_\ell$ ($\ell=1, 2$) the corresponding solutions to the Helmholtz equation~\eqref{EQ:Helmholtz}. Then $(f_1, g_1)=(f_2, g_2)$ implies that $m_1 = m_2$ $(:= m)$ and
\[
\quad \bx_{1,j} = \bx_{2,\pi(j)},\quad \lambda_{1,j} = \lambda_{2, \pi(j)},\ \ 1\le j\le m
\]
for some permutation $\pi \in {\rm Sym(m)}$.
\end{lemma}
\begin{proof}
This result follows from the unique continuation principle for Cauchy problems of elliptic equations. Let $D_\varepsilon^{\ell, j}:=\cB(\bx_{\ell, j}, \varepsilon)$ be the disk centered at $\bx_{\ell, j}$ with radius $\varepsilon > 0$ small enough such that $D_\varepsilon^{\ell, j}\subset \Omega$. We define $w = u_1 - u_2$ and verify that $w$ solves
\begin{equation}
\begin{array}{cl}
\Delta w + k^2 (1+n) w = 0, & \mbox{in}\ \Omega_{\varepsilon} := \Omega\backslash \bigcup_{\ell, j} D_\varepsilon^{\ell, j}\\
\\
w = 0, \quad \partial_{\nu} w = 0, & \mbox{on}\ \partial\Omega
\end{array}
\end{equation}
Since $\partial\Omega \subset \partial \Omega_{\varepsilon}$, we then conclude from the unique continuation principle~\cite[Theorem 3.3.1]{Isakov-Book06} that $w(\bx) = 0$, $\forall \bx\in\Omega_{\varepsilon}$ with any $\varepsilon > 0$. If we take $\varepsilon\to 0$, this implies that $w = 0$ except at the locations $\bx_{\ell, j}$. Therefore $w$ must be a finite linear combination of point sources and their derivatives. This is impossible. Therefore $q_1 = q_2$ up to a possible permutation $\pi$, that is, renumbering of the point sources.
\end{proof}
We now study the stability of the reconstruction. Following~\cite{ElEl-CRASP12,ElNa-IP11}, we look at the stability issue for an algebraic reconstruction technique that is based on the projection of the point sources into planes in $\mathbb R} \newcommand{\bbS}{\mathbb S^3$. Due to the fact that the medium is heterogeneous, we need to find good ways to do the projection. In the next two lemmas, we introduce our method of projection onto surfaces determined by the solutions of the Helmholtz equation (which are controlled by the medium).
\begin{lemma}\label{LEM:FRAME}
Under the assumptions in~{\bf(A)} on the refractive index and $\Omega$, there exists a complex-valued function $\phi(\bx)$ and a constant $\mu>0$ such that $\phi(\bx)$ solves
\begin{equation}\label{EQ:Schrodinger}
\Delta \phi + k^2(1 + n) \phi = 0, \ \ \mbox{in}\ \ \Omega
\end{equation}
$|\nabla \phi(\bx)|\neq 0$, $\forall \bx\in\Omega$, and
\[
\sup_{\bx\in\Omega}(|\phi(\bx)| + |\nabla \phi(\bx)|) < \mu \inf_{\bx\in\Omega} |\phi(\bx)|.
\]
\end{lemma}
\begin{proof}
With the regularity of $n(\bx)$ assumed in~{\bf(A)}, we can take $\phi$ as the well-known complex geometrical optics (CGO) solution to~\eqref{EQ:Schrodinger}; see for instance~\cite{NaUhWa-IP13}. More precisely, let $\zeta = \eta + \mathfrak{i}\xi$ with $\eta\in\mathbb R} \newcommand{\bbS}{\mathbb S^3$ and $\xi \in \mathbb R} \newcommand{\bbS}{\mathbb S^3$ given vectors such that $\eta \cdot \xi = 0$ and $\zeta\cdot\zeta=k^2$ (i.e. $|\eta|^2-|\xi|^2=k^2$). It is shown in~\cite{NaUhWa-IP13} that~\eqref{EQ:Schrodinger} has a solution of the form
\begin{equation*}
\phi(\bx) = e^{i\zeta\cdot \bx} (1 + r(\bx)),\;\text{ with }\; \|r\|_{\cH^3(\Omega)} \le \mathfrak{c} \frac{k^2}{|\zeta|} \|n\|_{\cH^3(\Omega)}
\end{equation*}
when $|\zeta|$ is sufficiently large. Then by the Sobolev embedding theorem~\cite{AdFo-Book03}, $r\in \cC^{1,1/2}(\Omega)$. If we choose $|\zeta|$ large such that $\|r\|_{\cC^{1,1/2}(\Omega)} < 1/2$, then we will have $|\nabla \phi|\neq 0$ for all $\bx\in \Omega$ and $\phi\in \cC^{1,1/2}(\Omega)$. Therefore we can find a constant $\mu > 0$ such that
\begin{equation}
\sup_{\bx\in\Omega}(|\phi(\bx)| + |\nabla \phi(\bx)|) < \mu \inf_{\bx\in\Omega} |\phi(\bx)|.
\end{equation}
This completes the proof.
\end{proof}
\begin{definition}\label{DEF:PROJ}
The function $\phi$ introduced in Lemma~\ref{LEM:FRAME} defines a local frame $(\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3)$ on $\Omega$, where $\mathbf{e}_1 = \frac{\nabla \phi}{|\nabla \phi|}\in \mathbb C} \newcommand{\bbE}{\mathbb E^3$, and $[\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3]$ forms an unitary matrix, which also determines a local coordinate change $\mathcal{U}: \mathbb{R}^3\rightarrow \mathbb{C}^3$, denoted by $\mathcal{U}(\bx) = (\mathfrak{u}_1(\bx), \mathfrak{u}_2(\bx), \mathfrak{u}_3(\bx))$, from the Cartesian coordinate to the local frame. We define a projection $\mathcal S} \newcommand{\cT}{\mathcal T:\mathbb C} \newcommand{\bbE}{\mathbb E^3\rightarrow \mathbb{C}$ as:
\begin{equation*}
\mathcal S} \newcommand{\cT}{\mathcal T(\bx) = \overline{(\mathfrak{u}_2 + \mathfrak{i} \mathfrak{u}_3)}.
\end{equation*}
This projection defines a pseudo distance function $\text{\rm dist}_{\mathcal S} \newcommand{\cT}{\mathcal T}(\bx, \by) = |\mathcal S} \newcommand{\cT}{\mathcal T(\bx) - \mathcal S} \newcommand{\cT}{\mathcal T(\by)|$ and
the diameter of $\Omega$ under the projection $\mathcal S} \newcommand{\cT}{\mathcal T$ is denoted by ${\rm diam}_{\mathcal S} \newcommand{\cT}{\mathcal T}(\Omega) := \displaystyle\sup_{\bx, \by\in\Omega} {\rm dist}_{\mathcal S} \newcommand{\cT}{\mathcal T}(\bx, \by)$.
\end{definition}
The project $\mathcal S} \newcommand{\cT}{\mathcal T$ we defined here is clearly not unique in the sense that one can rotate the coordinates to use $(\mathfrak{u}_3,\mathfrak{u}_1)$ or $(\mathfrak{u}_1,\mathfrak{u}_2)$ to replace $\mathfrak{u}_2$ and $\mathfrak{u}_3$. Moreover, since $\phi$ can be chosen differently using the complex vector $\zeta$ (which controls the boundary condition needed), we can construct the $\mathcal S} \newcommand{\cT}{\mathcal T$ that we need by selection a specific vector.
The projection we just introduced allows us to construct the following test functions.
\begin{lemma}\label{LEM:Test Func}
Let $\{ \bx_j \}_{j=1}^m \subset \Omega$ be arbitrary distinct points. Then under assumption~{\bf(A)}, there exists a function $\psi(\bx)$ solving
\begin{equation}\label{EQ:Test Func}
\Delta \psi + k^2(1+n) \psi = 0, \ \ \mbox{in}\ \ \Omega
\end{equation}
such that
\begin{equation}\label{EQ:Test Func Zero}
\psi(\bx_j) = 0, \qquad 1\le j\le m.
\end{equation}
\end{lemma}
\begin{proof}
We construct the function $\psi$ as follows:
\begin{equation}\label{EQ:Test Func Form}
\psi(\bx) = \phi(\bx) \prod_{j=1}^m \big(\mathcal S} \newcommand{\cT}{\mathcal T(\bx)-\mathcal S} \newcommand{\cT}{\mathcal T(\bx_j)\big).
\end{equation}
Then clearly $\psi(\bx_j) = 0$, $j=1, 2, \cdots, m$. It is straightforward to verify that $\Delta \mathcal S} \newcommand{\cT}{\mathcal T = 0$ and $ \nabla \phi\cdot \nabla \mathcal S} \newcommand{\cT}{\mathcal T = 0$, which allow us to check that $\psi$ solves ~\eqref{EQ:Test Func}.
\end{proof}
We are now ready to prove the stability of the reconstruction. Our reconstruction scheme follows a two-step process. In the first step, the locations of the point sources are probed by a projection method. In the second step, we use the reconstructed locations $\{\bx_j\}_{j=1}^m$ to reconstruct the strengths of the point sources $\{\lambda_j\}_{j=1}^m$.
\begin{theorem}\label{THM:Stab Data}
Let $q_1$ and $q_2$ be two sources of the form~\eqref{EQ:Source12} with $m:=m_1=m_2$ that are reconstructed from the Cauchy data $(f_1, g_1)$ and $(f_2, g_2)$ respectively. Let $\mathcal{S}$ be the projection in Definition~\ref{DEF:PROJ}. Let $\sigma := \displaystyle\min_{\ell, i\neq j}|\mathcal S} \newcommand{\cT}{\mathcal T(\bx_{\ell,i}) - \mathcal S} \newcommand{\cT}{\mathcal T(\bx_{\ell ,j})|$ and assume that $\sigma>0$. Then, under the assumptions in {\bf(A)}, there exists a permutation $\pi\in{\rm Sym(m)}$ acting on $\{1, 2, \cdots, m\}$ and a constant $\mathfrak{c}_1$ depending on $\Omega$, $\phi$, $\mathcal S} \newcommand{\cT}{\mathcal T$, and $m$, such that
\begin{multline}\label{EQ:Stab Loc Data}
\rho_{\bx}:=\max_{1\le j\le m} |\mathcal{S}(\bx_{1,j})-\mathcal{S}(\bx_{2,\pi(j)})| \\
\le \mathfrak{c}_1 \left( \frac{\sqrt{|\partial\Omega|}({\rm diam}_{\mathcal S} \newcommand{\cT}{\mathcal T}(\Omega))^{2m-1} }{\underline{\lambda} \sigma^{m-1}}\left( \|g_1 - g_2\|_{L^2(\partial\Omega)} + \|f_1 - f_2\|_{L^2(\partial\Omega)} \right)\right)^{\frac{1}{m}}
\end{multline}
where $|\partial\Omega|$ is the surface measure of $\partial\Omega$. Assume further that $\rho_{\bx}< \sigma$, and let $\widetilde\rho_x:=\displaystyle \max_{1\le j\le m}|\mathcal U} \newcommand{\cV}{\mathcal V(x_{1,j}) -\mathcal U} \newcommand{\cV}{\mathcal V(\bx_{2,\pi(j)})|$, then there exists constants $\mathfrak{c}_2$ and $\mathfrak{c}_3$, again depending on $\Omega$, $\phi$, $\mathcal S} \newcommand{\cT}{\mathcal T$, and $m$, such that
\begin{multline}\label{EQ:Stab Strength Data}
\rho_\lambda:=\max_{1\le j\le m}|\lambda_{1,j} - \lambda_{2,\pi(j)}| \\
\le \mathfrak{c}_2 \overline{\lambda} \widetilde{\rho}_\bx + \mathfrak{c}_3 \sqrt{|\partial\Omega|}({\rm diam}_{\mathcal S} \newcommand{\cT}{\mathcal T}(\Omega))^{2m-2} \big(\|g_1 - g_2\|_{L^2(\partial\Omega)} + \|f_1 - f_2\|_{L^2(\partial\Omega)}\big).
\end{multline}
\end{theorem}
\begin{proof}
Let $u_\ell$ $(\ell=1, 2)$ be the solution to the Helmholtz equation~\eqref{EQ:Helmholtz} with source $q_\ell$. We define $w:=u_1-u_2$. Then $w$ solves the Helmholtz equation~\eqref{EQ:Helmholtz} with boundary data $(w, \partial_\nu w):=(f_1-f_2, g_1-g_2)$. Let $1\le j' \le m$ be an integer. From Lemma~\ref{LEM:Test Func}, we find a function, $\phi$ being defined in Lemma~\ref{LEM:FRAME},
\begin{equation}\label{EQ:Psi j'}
\psi_{j'}(\bx)=\phi(\bx)\prod_{i=1}^m(\mathcal S} \newcommand{\cT}{\mathcal T(\bx) - \mathcal S} \newcommand{\cT}{\mathcal T(\bx_{1,i})) \prod_{j=1, j\neq j'}^m (\mathcal S} \newcommand{\cT}{\mathcal T(\bx) - \mathcal S} \newcommand{\cT}{\mathcal T(\bx_{2,j}))
\end{equation}
that solves the equation
\begin{equation}\label{EQ:Psi-t}
\Delta \psi_{j'} + k^2(1 + n) \psi_{j'} = 0, \ \ \mbox{in}\ \ \Omega
\end{equation}
and satisfies
\begin{equation}
\psi_{j'}(\bx_{\ell,j}) = 0,\ \ \forall (\ell, j)\neq (2, j') .
\end{equation}
Multiplying the equation for $w$ by $\psi_{j'}$ and the equation for $\psi_{j'}$ by $w$, taking the difference of the results, and applying Green's identity, we have, with $\mathfrak{s}(\bx)$ the Lebesgue measure on $\partial\Omega$,
\begin{equation*}
\begin{aligned}
&\left| \int_{\partial\Omega} \big((f_1-f_2) \partial_{\nu} \psi_{j'} - \psi_{j'} (g_1 - g_2)\big)d\mathfrak{s} \right| =\left| \lambda_{2,j'} \psi_{j'}(\bx_{2,j'})\right|\\
=& \left|\lambda_{2,j'} \phi(\bx) \prod_{i=1}^m(\mathcal S} \newcommand{\cT}{\mathcal T(\bx_{2,j'}) - \mathcal S} \newcommand{\cT}{\mathcal T(\bx_{1,i})) \prod_{j=1, j\neq j'}^m (\mathcal S} \newcommand{\cT}{\mathcal T(\bx_{2,j'}) - \mathcal S} \newcommand{\cT}{\mathcal T(\bx_{2,j})) \right|
\ge \theta \underline{\lambda} \rho_{j'}^m \sigma^{m-1}.
\end{aligned}
\end{equation*}
where $\theta := \inf_{\bx\in\Omega}|\phi(\bx)|$, $\rho_{j'} := \displaystyle\min_{1\le j\le m} |\mathcal S} \newcommand{\cT}{\mathcal T(\bx_{2,j'})-\mathcal S} \newcommand{\cT}{\mathcal T(\bx_{1,j})|$, and we have used $|\lambda_{2,j'}|\ge \underline{\lambda}$ by the assumptions in {\bf(A)}.
Meanwhile, by the Cauchy-Schwartz inequality, we have
\begin{multline*}
\left| \int_{\partial\Omega} \big((f_1-f_2) \partial_{\nu} \psi_{j'} - \psi_{j'} (g_1 - g_2)\big)d\mathfrak{s} \right| \\
\le \|g_1 - g_2\|_{L^2(\partial\Omega)} \|\psi_{j'}\|_{L^2(\partial\Omega)} + \|f_1 -f_2\|_{L^2(\partial\Omega)} \|\partial_{\nu} \psi_{j'}\|_{L^2(\partial\Omega)},
\end{multline*}
where we can estimate
\begin{equation*}
\begin{aligned}
\|\psi_{j'}\|_{L^2(\partial\Omega)} &\le \vartheta \sqrt{|\partial\Omega|} ({\rm diam}_{\mathcal S} \newcommand{\cT}{\mathcal T}(\Omega))^{2m-1},\\ \|\partial_{\nu}\psi_{j'}\|_{L^2(\partial\Omega)} &\le \vartheta \sqrt{|\partial\Omega|}({\rm diam}_{\mathcal S} \newcommand{\cT}{\mathcal T}(\Omega))^{2m-1} + \mathfrak{c}\theta \sqrt{|\partial\Omega|}({\rm diam}_{\mathcal S} \newcommand{\cT}{\mathcal T}(\Omega))^{2m-2},
\end{aligned}
\end{equation*}
with $\vartheta := \|\phi\|_{W^{1,\infty}(\Omega)}$ and $\mathfrak{c} =\mathfrak{c}(\Omega, \mathcal S} \newcommand{\cT}{\mathcal T, m)$ a bounded constant. Using the fact that $\vartheta < \mu \theta$, given in Lemma~\ref{LEM:FRAME}, we conclude from the above calculations that
\begin{equation}
\underline{\lambda} \rho_{j'}^m \sigma^{m-1} \le \widetilde\mathfrak{c} \sqrt{|\partial\Omega|}({\rm diam}_{\mathcal S} \newcommand{\cT}{\mathcal T}(\Omega))^{2m-1} \big(\|g_1 - g_2\|_{L^2(\partial\Omega)} + \|f_1 - f_2\|_{L^2(\partial\Omega)}\big)
\end{equation}
with $\widetilde\mathfrak{c}$ a bounded constant that depends on $\Omega$, $\phi$, $\mathcal S} \newcommand{\cT}{\mathcal T$, and $m$.
Since $j'$ is taken arbitrarily, we conclude that
\begin{equation}
\max_{1\le j' \le m} \rho_{j'} \le \left( \frac{\widetilde\mathfrak{c} \sqrt{|\partial\Omega|}({\rm diam}_{\mathcal S} \newcommand{\cT}{\mathcal T}(\Omega))^{2m-1} }{\underline{\lambda} \sigma^{m-1}}\left( \|g_1 - g_2\|_{L^2(\partial\Omega)} + \|f_1 - f_2\|_{L^2(\partial\Omega)} \right)\right)^{1/m}.
\end{equation}
By the symmetry in our calculations between the two groups of point sources, we see that we could replace the left hand side of the above inequality with the Hausdorff distance between the two groups of projected points $\{\mathcal S} \newcommand{\cT}{\mathcal T(\bx_{1,j})\}_{j=1}^m$ and $\{\mathcal S} \newcommand{\cT}{\mathcal T(\bx_{2,j})\}_{j=1}^m$:
\[
\max\{ \max_{1\le j'\le m} \displaystyle\min_{1\le j\le m} |\mathcal S} \newcommand{\cT}{\mathcal T(\bx_{2,j'})-\mathcal S} \newcommand{\cT}{\mathcal T(\bx_{1,j})|, \max_{1\le j'\le m} \displaystyle\min_{1\le j\le m} |\mathcal S} \newcommand{\cT}{\mathcal T(\bx_{1,j'})-\mathcal S} \newcommand{\cT}{\mathcal T(\bx_{2,j})|\}.
\]
The stability result~\eqref{EQ:Stab Loc Data} then follows from this fact and the Hall theorem~\cite{Cameron-Book94,ElEl-CRASP12}, which states that there exists a permutation $\pi$ acting on $\{1,2,\cdots,m\}$, that is a renumbering of the points, such that the Hausdorff distance can be realized by $\rho_\bx$.
The next step is to establish the stability for the strengths of point sources. For an integer $1\le j'\le m$, we introduce the function
\begin{equation*}
\varphi_{j'}(\bx)=\phi(\bx)\prod_{j=1, j\neq j'}^m(\mathcal S} \newcommand{\cT}{\mathcal T(\bx) - \mathcal S} \newcommand{\cT}{\mathcal T(\bx_{1,j})) (\mathcal S} \newcommand{\cT}{\mathcal T(\bx) - \mathcal S} \newcommand{\cT}{\mathcal T(\bx_{2,\pi(j)})).
\end{equation*}
Then $\varphi_{j'}$ solves the equation
\begin{equation*}
\Delta \varphi_{j'} + k^2(1 + n) \varphi_{j'} = 0, \ \ \mbox{in}\ \ \Omega
\end{equation*}
and satisfies
\begin{equation*}
\varphi_{j'}(\bx_{\ell,j}) = 0,\ \ \forall (\ell, j) \notin \{(1, j'), (2, \pi(j'))\}.
\end{equation*}
Following the same procedure as before, we multiply the equation for $w$ by $\varphi_{j'}$ and the equation for $\varphi_{j'}$ by $w$, take the difference of the results, and apply Green's identity to obtain,
\begin{equation*}
\left| \lambda_{2,\pi(j')} \varphi_{j'}(\bx_{2,\pi(j')}) - \lambda_{1,j'} \varphi_{j'}(\bx_{1,j'})\right|= \left| \int_{\partial\Omega} \left( (f_1 - f_2)\partial_{\nu}\varphi_{j'} - \varphi_{j'} (g_1 - g_2) \right) d\mathfrak{s} \right|.
\end{equation*}
By the Cauchy-Schwartz inequality, we have
\begin{multline*}
\left| \int_{\partial\Omega} \big((f_1-f_2) \partial_{\nu} \varphi_{j'} - \varphi_{j'} (g_1 - g_2)\big)d\mathfrak{s} \right| \\
\le \|g_1 - g_2\|_{L^2(\partial\Omega)} \|\varphi_{j'}\|_{L^2(\partial\Omega)} + \|f_1 -f_2\|_{L^2(\partial\Omega)} \|\partial_{\nu} \varphi_{j'}\|_{L^2(\partial\Omega)},
\end{multline*}
where
\begin{equation*}
\begin{aligned}
\|\varphi_{j'}\|_{L^2(\partial\Omega)} &\le \vartheta \sqrt{|\partial\Omega|} ({\rm diam}_{\mathcal S} \newcommand{\cT}{\mathcal T}(\Omega))^{2m-2},\\ \|\partial_{\nu}\varphi_{j'}\|_{L^2(\partial\Omega)} &\le \vartheta \sqrt{|\partial\Omega|}({\rm diam}_{\mathcal S} \newcommand{\cT}{\mathcal T}(\Omega))^{2m-2} + \mathfrak{c}\theta \sqrt{|\partial\Omega|}({\rm diam}_{\mathcal S} \newcommand{\cT}{\mathcal T}(\Omega))^{2m-3},
\end{aligned}
\end{equation*}
with $\vartheta$ and $\theta$ defined as before, and $\mathfrak{c} =\mathfrak{c}(\Omega, \mathcal S} \newcommand{\cT}{\mathcal T, m)$ a bounded constant. We therefore have
\begin{multline}\label{EQ:Stab Strength A}
\left| \lambda_{2,\pi(j')} \varphi_{j'}(\bx_{2,\pi(j')}) - \lambda_{1,j'} \varphi_{j'}(\bx_{1,j'})\right| \\
\le \widetilde\mathfrak{c} \sqrt{|\partial\Omega|}({\rm diam}_{\mathcal S} \newcommand{\cT}{\mathcal T}(\Omega))^{2m-1} \big(\|g_1 - g_2\|_{L^2(\partial\Omega)} + \|f_1 - f_2\|_{L^2(\partial\Omega)}\big).
\end{multline}
We now verify, using the assumption that $\sigma > \bar \rho_\bx$, that,
\begin{multline*}
|\varphi_{j'}(\bx_{1,j'})|=|\phi(\bx_{1,j'})\prod_{j=1, j\neq j'}^m(\mathcal S} \newcommand{\cT}{\mathcal T(\bx_{1,j'}) - \mathcal S} \newcommand{\cT}{\mathcal T(\bx_{1,j})) (\mathcal S} \newcommand{\cT}{\mathcal T(\bx_{1,j'}) - \mathcal S} \newcommand{\cT}{\mathcal T(\bx_{2,\pi(j)}))|\\ \ge \theta \sigma^{m-1} \prod_{j=1, j\neq j'}^m \big| \left|\mathcal S} \newcommand{\cT}{\mathcal T(\bx_{1,j'}) -\mathcal S} \newcommand{\cT}{\mathcal T(\bx_{1,j})\right| - \left| \mathcal S} \newcommand{\cT}{\mathcal T(\bx_{2,\pi(j)}) -\mathcal S} \newcommand{\cT}{\mathcal T(\bx_{1,j})\right| \big|
\ge \sigma^{m-1} |\sigma - \rho_\bx|^{m-1}.
\end{multline*}
This allows us to conclude that, for some constants $\mathfrak{c}'$ and $\mathfrak{c}''$ (for instance, one could take $\mathfrak{c}' = \theta^{-1} \sigma^{1-m}(\sigma -\rho_{\bx})^{1-m}$), we have
\begin{equation}\label{EQ:Stab Strength B}
\begin{aligned}
\max_{1\le j'\le m} |\lambda_{2,\pi(j')}-\lambda_{1,j'}|&\le \mathfrak{c}' \max_{1\le j'\le m}|\lambda_{2,\pi(j')}\varphi_{j'}(\bx_{1,j'})-\lambda_{1,j'}\varphi_{j'}(\bx_{1,j'})| \\
&\le \mathfrak{c}' \max_{1\le j'\le m}\lambda_{2,\pi(j')}|\varphi_{j'}(\bx_{1,j'})-\varphi_{j'}(\bx_{2,\pi(j')})| \\
&\quad + \mathfrak{c}' \max_{1\le j'\le m}|\lambda_{2,\pi(j')}\varphi_{j'}(\bx_{2,\pi(j')})-\lambda_{1,j'}\varphi_{j'}(\bx_{1,j'})|\\ &\le \mathfrak{c}'' \overline{\lambda}\widetilde{\rho}_\bx + \mathfrak{c}' \max_{1\le j'\le m}|\lambda_{2,\pi(j')}\varphi_{j'}(\bx_{2,\pi(j')})-\lambda_{1,j'}\varphi_{j'}(\bx_{1,j'})|.
\end{aligned}
\end{equation}
The stability bound~\eqref{EQ:Stab Strength Data} then follows from ~\eqref{EQ:Stab Strength A} and~\eqref{EQ:Stab Strength B}.
\end{proof}
The stability of reconstructing the locations of the point sources is H\"older type with exponent $\frac{1}{m}$, $m$ being the number of point sources included. The stability deteriorates quickly when $m$ increases. Therefore, we could only hope to reconstruct stably a very small number of point sources in practice.
The conditional stability of the reconstructing the strengths of the point sources contains two parts. The second part is from the Cauchy data and is Lipschitz type. The constant in front of it, however, depends on $m$. When ${\rm diam}_{\mathcal S} \newcommand{\cT}{\mathcal T}(\Omega)$ is large, this constant blows up quickly with $m$, another indication that one can not hope to stably reconstruct a large number of point sources. The first part comes from error in the determination of the locations of the point sources. If the locations are reconstructed perfectly, this term disappear. If, on the other hand, there is a large error in the reconstructing of the locations, the error in the reconstruction of the strengths is also large.
\subsection{Stability with respect to media changes}
\label{SUBSEC:Uncertainty}
Here we study the stability of the reconstruction of point sources with respect to smooth media changes. We assume that the measured data are collected with a medium $n_1$ that we do not know exactly. We then reconstruct the point sources pretending that the medium in which the data were collected is $n_2$. We show that the reconstructions in $n_2$ is not too different from the reconstructions in $n_1$ if $n_2$ is not too different from $n_1$, in appropriate sense.
\begin{theorem}\label{THM:Stab Medium}
Let $q_1$ and $q_2$ be two sources of the form~\eqref{EQ:Source12} $(m:=m_1=m_2)$, reconstructed for two media with refractive index $n_1$ and $n_2$ respectively, using Cauchy data $(f, g)$. Under the assumptions in~{\bf(A)} for $(q_\ell, n_\ell)$ ($\ell=1,2$), there exists a permutation $\pi$ acts on $\{1,2\dots,m\}$ such that
\begin{equation}\label{EQ:Stab Medium}
\max_{1\le j \le m} \left| \mathcal S} \newcommand{\cT}{\mathcal T(\bx_{1,j}) - \mathcal S} \newcommand{\cT}{\mathcal T(\bx_{2,\pi(j)}) \right| \le \mathfrak{c} \left[ \frac{\mu k^2 }{\underline{\lambda} \sigma^{m-1}} \|n_1 - n_2\|_{L^{2}(\Omega)} \right]^{\frac{1}{m}},
\end{equation}
$\sigma = \min_{i\neq j}|\mathcal S} \newcommand{\cT}{\mathcal T(\bx_{2,i}) - \mathcal S} \newcommand{\cT}{\mathcal T(\bx_{2,j})|$, $\mu$ is from Lemma~\ref{LEM:FRAME}, and $\mathfrak{c}=\mathfrak{c}(\Omega,m,\overline{\lambda}, g)$ is a bounded constant.
\end{theorem}
\begin{proof}
Let $u_\ell$ $(\ell=1, 2)$ be the solution to the Helmholtz equation~\eqref{EQ:Helmholtz} with the source and refractive index pair $(n_\ell, q_\ell)$. We define $w:=u_1-u_2$, $\delta n=n_1-n_2$, and $\delta q=q_1-q_2$. Then $w$ solves
\begin{equation*}\label{EQ:Helmholtz Diff Media}
\Delta w + k^2 (1+n_1) w = \delta q(\bx) -k^2 \delta n(\bx) u_2(\bx),\ \ \mbox{in}\ \ \Omega
\end{equation*}
with boundary data $(w, \partial_\nu w):=(0, 0)$.
Let $\phi$ be the function defined in Lemma~\ref{LEM:FRAME} for the medium with refractive index $n_1$. Let $1\le j' \le m$ be an integer. We use the function
\begin{equation*}
\psi_{j'}(\bx)=\phi(\bx)\prod_{i=1}^m(\mathcal S} \newcommand{\cT}{\mathcal T(\bx) - \mathcal S} \newcommand{\cT}{\mathcal T(\bx_{1,i})) \prod_{j=1, j\neq j'}^m (\mathcal S} \newcommand{\cT}{\mathcal T(\bx) - \mathcal S} \newcommand{\cT}{\mathcal T(\bx_{2,j})) .
\end{equation*}
This function now solves the equation
\begin{equation*}
\Delta \psi_{j'} + k^2(1 + n_1) \psi_{j'} = 0, \ \ \mbox{in}\ \ \Omega
\end{equation*}
and satisfies
\begin{equation*}
\psi_{j'}(\bx_{\ell,j}) = 0,\ \ \forall (\ell, j)\neq (2, j') .
\end{equation*}
Multiplying the equation for $w$ by $\psi_{j'}$ and the equation for $\psi_{j'}$ by $w$, taking the difference of the results, and applying Green's identity, we have
\begin{equation}
k^2\int_{\Omega} (n_2 - n_1)(\bx) \psi_{j'}(\bx) u_2(\bx) d\bx = \lambda_{2,j'} \psi_{j'}(\bx_{2,j'})
\end{equation}
By the Cauchy-Schwartz inequality, we have
\begin{equation*}
k^2 \left| \int_{\Omega} (n_2 - n_1)(\bx) \psi_{j'}(\bx) u_2(\bx) d\bx \right| \le k^2 \|n_2 - n_1\|_{L^{2}(K)} \|\psi_{j'}\|_{L^{\infty}(\Omega)} \|u_2\|_{L^2(K)},
\end{equation*}
where $K := \operatorname{supp}(n_2 - n_1)\subset \subset \Omega$, $\|\psi_{j'} \|_{L^{\infty}(\Omega)} \le \|\phi\|_{L^{\infty}(\Omega)} ({\rm diam}_{\mathcal S} \newcommand{\cT}{\mathcal T}(\Omega))^{2m-1}$.
Let $G$ be the fundamental solution for $\Delta + k^2(1+n_2)$ with homogeneous Neumann boundary condition. We have the following representation for $u_2$:
\begin{equation}\label{EQ:U2}
u_2(\bx) = \sum_{j=1}^{m} \lambda_{2,j} G(\bx, \bx_{2,j}) - \int_{\partial\Omega} G(\bx, \by) g(\by) d\mathfrak{s}
\end{equation}
This allows us to conclude that $\|u_2\|_{L^2(K)}\le \mathfrak{c}(1+m \overline{\lambda})$ for some constant $\mathfrak{c} = \mathfrak{c}(\Omega, K, g)$.
The rest of the proof is identical to that of Theorem~\ref{THM:Stab Data}. We first verify that
\[
\left| \lambda_{2,j} \psi_{j'}(\bx_{2,j'})\right| \ge \theta \underline{\lambda} \rho_{j'}^{m} \sigma^{m-1}
\]
where $\rho_{j'}: = \displaystyle\min_{1\le j\le m} {\rm dist}_{\mathcal S} \newcommand{\cT}{\mathcal T}(\bx_{2,j'}, \bx_{1,j})$ is defined the same way as before.
Therefore, we have obtained
\begin{equation*}
\max_{1\le j' \le m} \rho_{j'} \le \mathfrak{c} \left[ \frac{\mu k^2 }{\underline{\lambda} \sigma^{m-1}} \|n_1 - n_2\|_{L^2(\Omega)} \right]^{\frac{1}{m}},
\end{equation*}
The stability bounded~\eqref{EQ:Stab Medium} then follows from symmetry argument and the Hall theorem~\cite{Cameron-Book94,ElEl-CRASP12}.
\end{proof}
This result shows that the reconstruction of the locations of the point sources, up to a permutation, is relatively robust against changes in the underlying medium. However, the stability again deteriorates fast when the number of point sources increases.
Note that the above result is based on the assumption that we know exactly the number of point sources inside the medium. We do not have a general uniqueness result that allows us to determine the number of point sources from the measurement in this case. However, in some special cases, we can hope to reconstruct uniquely the point sources in a unknown medium that is has sufficiently simple structures, utilizing the fact that point sources are more singular compared to media variations.
\paragraph{Media with localized perturbations.} Let us consider the case where the medium $n_2$ is the homogeneous medium $n_1$ with a finite number of additional localized anomalies. That is, the refractive index $n_2$ is of the form:
\[
n_2(\bx) = n_1+\sum_{k=1}^K \tau_k \chi_{\Omega_k},
\]
where $\Omega_k$ is the support of the $k$-th anomaly and $\tau_k$ is its strength. We assume that the point sources are away from the local anomalies of the medium, that is, ${\rm dist}(\bx_j, \Omega_k)\ge \mathfrak{c}>0$ $\forall j,k$, for some $\mathfrak{c}$. Then we could follow the same proof in Lemma~\ref{LEM:Uniqueness Known Media} to show that $w$ must vanish outside the support of $\delta n=n_2-n_1$. This allows us to uniquely (up to a permutation as before) determine the point sources that do NOT live in the support of the local anomalies of the medium. This provides a uniqueness argument for the numerical point source reconstructions in~\cite{BuLaTaTs-AFR09}.
\section{Numerical experiments}
\label{SEC:Num}
We now perform some numerical simulations in the context of the theoretical study in the previous section. We have two specific aims in mind: (i) when the underlying medium is known but heterogeneous, we want to see how well we can reconstruct point sources inside the medium; and (ii) when the underlying medium is not known, we want to see how the reconstruction of point sources are affected by the reconstruction of the medium.
We therefore intend to reconstruct the medium as well as the point sources inside the medium. We assume that we have access to multiple Cauchy data sets. We use a two-step reconstruction process. In the first step, we form \emph{differential} data sets to eliminate the effect of the point sources and focus only on the medium. In the second step, we use a \emph{single} data set to reconstruct the point sources, so that our simulations are in consistent with the theory developed in the previous section.
\paragraph{Media reconstruction.} Let $u$ be the solution to the Helmholtz equation~\eqref{EQ:Helmholtz}, and $\widetilde u$ be the solution of the same equation but with boundary source $f(\bx)+h(\bx)$. Then we check that $v:=\widetilde u-u$ solves the Helmholtz equation
\begin{equation}\label{EQ:Helmholtz Diff Dif}
\begin{array}{rcll}
\Delta v + k^2\big(1+n(\bx)\big) v &=& 0,& \mbox{in}\ \ \Omega\\
v & = & h(\bx), & \mbox{on}\ \ \partial\Omega
\end{array}
\end{equation}
By changing the probe source $h(\bx)$, we could obtain data determined by the Dirichlet-to-Neumann operator
\begin{equation}\label{EQ:Data Diff}
\Lambda_n: h(\bx) \mapsto g(\bx)=\partial_\nu v_{|\partial\Omega}: =\partial_\nu \widetilde u_{|\partial\Omega}-\partial_\nu u_{|\partial\Omega} .
\end{equation}
These data allow us to reconstruct the refractive index $n(\bx)$ since that is the only unknown quantity in~\eqref{EQ:Helmholtz Diff Dif}. This inverse problem has been studied extensively; see for instance~\cite{Isakov-Book06,NaUhWa-IP13} and references therein.
We perform the reconstruction by reformulate the inverse problem as a minimization problem. Let us assume that we have data generated from $J$ different probe sources $\{h_j\}_{j=1}^J$. We reconstruct $n(\bx)$ by minimizing the following mismatch functional:
\begin{equation}
\Phi(n):= \dfrac{1}{2}\sum_{j=1}^J \int_{\partial\Omega} (\Lambda_n h_j - g_j^*)^2 d\mathfrak{s}(\bx) + \frac{\beta}{2} \int_{\Omega} |\nabla n|^2 d\bx
\end{equation}
where $g^*_j$ is the measured differential data corresponding to the probe source $h_j$ and the parameter $\beta$ is the strength of the regularization term.
We solve this minimization problem with a quasi-Newton method~\cite{NoWr-Book06,ReBaHi-SIAM06} where we use the adjoint state method to calculate the gradient of the objective functional with respect to the refractive index. Let $w_j$ $(1\le j\le J)$ be the solution to adjoint equation
\begin{equation}\label{EQ:Helmholtz Adj}
\Delta w_j + k^2(1+n) w_j = 0,\ \ \mbox{in}\ \ \Omega, \qquad \ w_j=-(\Lambda_n h_j - g_j^*),\ \ \mbox{on}\ \ \partial\Omega
\end{equation}
We can then show that the Fr\'echet derivative of $\Phi$ with respect to $n$ in direction $\delta n$ is given as
\begin{equation}
\Phi'(n)[\delta n] =k^2 \sum_{j=1}^J \int_\Omega w_j v_j \delta n(\bx) d\bx-\beta\Big[\int_{\Omega} (\Delta n) \delta n(\bx) d\bx-\int_{\partial\Omega}\partial_\nu n \delta n(\bx) d\mathfrak{s}(\bx)\Big].
\end{equation}
In the minimization process, we solve the forward and adjoint Helmholtz problems~\eqref{EQ:Helmholtz Diff Dif} and~\eqref{EQ:Helmholtz Adj} with a standard $P_1$ finite element solver.
\paragraph{Source reconstruction.} Once the refractive index $n(\bx)$ is reconstructed, we can reconstruct the unknown point sources, encoded in $q(\bx)$ in the Helmholtz equation~\eqref{EQ:Helmholtz}, from observed boundary data. We do this again with a minimization strategy. More precisely, we minimize the functional
\begin{equation}\label{EQ:Obj Src}
\Psi(\bx_1, \cdots, \bx_m, \lambda_1, \cdots, \lambda_m):= \frac{1}{2}\int_{\partial\Omega} (\bn\cdot\nabla u - g^{\ast})^2 d\mathfrak{s}(\bx)
\end{equation}
over the locations and strengths of the point sources.
To avoid dealing with the singularity of the solution $u$ due to the point sources, we explicitly factorize out the singular part of $u$ as follows. Let $G(\bx;\by)$ be the fundamental solution of the homogeneous Helmholtz operator in the whole space, that is,
\[
\Delta G + k^2 G = -\delta(\bx-\by), \ \ \mbox{in}\ \ \mathbb R} \newcommand{\bbS}{\mathbb S^d.
\]
We represent the solution of~\eqref{EQ:Helmholtz} through the integral equation
\begin{multline}
u(\bx)=-\sum_{j=1}^m \lambda_j G(\bx_j; \bx)+k^2\int_\Omega n(\by)u(\by)G(\by;\bx)d\by \\
+\int_{\partial\Omega} G(\by;\bx) \bn\cdot \nabla u(\by) d\by-\int_{\partial\Omega} f(\by) \bn\cdot\nabla G(\by;\bx) d\by.
\end{multline}
Let $\widehat u(\bx):=u(\bx)+\sum_{j=1}^m \lambda_j G(\bx_j; \bx)$, then $\widehat u$ solves the integral equation
\begin{equation}\label{EQ:U Nonsingular}
\widehat u(\bx)=k^2\int_\Omega n(\by)\widehat u(\by)G(\by;\bx)d\by +\int_{\partial\Omega} G(\by;\bx) \bn\cdot \nabla \widehat u(\by) d\by+Q(\bx),
\end{equation}
where the source term
\begin{multline*}
Q(\bx)=-k^2\sum_{j=1}^m \lambda_j\Big[ \int_\Omega n(\by)G(\bx_j;\by)G(\by;\bx)d\by +\int_{\partial\Omega} G(\by;\bx) \bn\cdot \nabla G(\bx_j;\by) d\by\Big] \\ -\int_{\partial\Omega} f(\by) \bn\cdot\nabla G(\by;\bx) d\by.
\end{multline*}
To find the solution $u$, we solve for $\widehat u$ using~\eqref{EQ:U Nonsingular} and then form $u=\widehat u- \sum_{j=1}^m \lambda_j G(\bx_j; \bx)$.
To evaluate the gradient of the objective function, we introduce the adjoint problem
\begin{equation}
\Delta w + k^2(1+n) w = 0,\ \ \mbox{in}\ \ \Omega, \qquad \ w=\bn\cdot\nabla u-g^*,\ \ \mbox{on}\ \ \partial\Omega .
\end{equation}
We can then show that the gradient of $\Psi$ with respect to a parameter the strength $\lambda_k$ and location $\bx_k$ are given respectively as
\begin{equation}
\dfrac{d\Psi}{d\lambda_k} =w(\bx_j),\qquad\mbox{and},\qquad \nabla_{\bx_k} \Psi=\lambda_j \nabla_\bx w(\bx_j).
\end{equation}
The numerical simulations we present below are all done in a two-dimensional domain for simplicity. The best way to make this consistent with the theory in the previous section, which are constructed in dimension three, is to view the the simulations as simplifications of three-dimensional ones for which the refractive index and the illumination sources are invariant in the $z$-direction. We set the domain $\Omega = [0, 1]^2$ and the wave number $k = 8$ in our experiments. We collect $J=6$ \emph{differential} data sets generated from sources $f$ and $\{h_j\}_{j=1}^{J}$ to reconstruct the refractive index. To avoid the inverse crime, the synthetic measurements are generated on a fine grid while the inversion is fulfilled on another coarse grid. Moreover, we pollute our synthetic data with multiplicative random noise by perform the operation: $ g^*_j\to g^*_j(1 + \tau \mathcal U} \newcommand{\cV}{\mathcal V([-1,1]))$ with $\mathcal U} \newcommand{\cV}{\mathcal V[-1, 1]$ the uniformly distributed random variable in $[-1, 1]$ and $\tau$ the level of noise that we will specify later. The algorithms are implemented in the $\verb|MATLAB|$ software with the source codes deposited at $\verb|github|$~\footnote{The github repository for our source codes is at \href{https://github.com/lowrank/ips}{https://github.com/lowrank/ips}.}.
We performed simulations on several different media. Here we present results on two typical ones that have refractive indices respectively.
\begin{equation}\label{EQ:Med 1}
n(\bx) = 0.5 + 0.5\displaystyle\sum_{k=1}^2\cos \left(\dfrac{\pi |\bx - \by_k|}{2R}\right)\chi_{D_R(\by_k)}
\end{equation}
where $\by_1 = (0.25, 0.25)$, $\by_2 = (0.75, 0.75)$, $R=0.25$ and $\chi_{D_R(\by_k)}$ is the characteristic function of the disk of radius $R$ centered at $\by_k$, and
\begin{equation}\label{EQ:Med 2}
n(\bx)=0.4\chi_{Rec} + 0.2 \chi_{D_{0.2}(\by_3)},
\end{equation}
where $\chi_{Rec}$ is the characteristic function of the rectangle $Rec=(0.5\ 0.75)\times(0.25\ 0.75)$; see Figure~\ref{FIG:Unknown Media 1} and Figure~\ref{FIG:Unknown Media 2} respectively for the plots of these refractive indices.
\begin{figure}[htb!]
\begin{center}
\includegraphics[scale=0.5]{d1}
\includegraphics[scale=0.5]{d5}
\caption{Locations of the true (crosses: $\times$) and reconstructed (circles: $\circ$) point sources in medium~\eqref{EQ:Med 1} in Experiment 1. Shown are results with data contain $1\%$ (left) and $5\%$ (right) random noise respectively.}
\end{center}
\label{FIG:Known Med 1}
\end{figure}
\paragraph{Experiment 1 [Recovery in Known Environments].} In the first set of numerical experiments, we perform reconstructions of point sources in heterogeneous media with \emph{known} refractive indices. In Figure~\ref{FIG:Known Med 1} and Figure~\ref{FIG:Known Med 2}, we show reconstructions of the locations of the point sources in the media~\eqref{EQ:Med 1} and~\eqref{EQ:Med 2} respectively. The true and reconstructed strengths are summarized in the first two rows of Table~\ref{TAB:Intensities}.
\begin{figure}[htb!]
\begin{center}
\includegraphics[scale=0.5]{s1}
\includegraphics[scale=0.5]{s5}
\caption{Same as Figure~\ref{FIG:Known Med 1} but for the medium with refractive index~\eqref{EQ:Med 2}.}
\end{center}
\label{FIG:Known Med 2}
\end{figure}
\begin{table}[hbt!]
\centering
\small \caption{True and reconstructed intensities of the point sources, $(\lambda_1, \lambda_2, \lambda_3, \lambda_4)$, in different numerical experiments.}
\begin{center}
\begin{tabular}{lccc}
\hline
Experiment & True value & \multicolumn{2}{c}{Reconstructions with noisy data} \\
& & 1\% noise & 5\% noise \\
\hline
1: medium~\eqref{EQ:Med 1}& (0.89,0.73,0.71,0.52) & (0.91,0.73,0.68,0.54) & (0.92,0.65,0.76,0.46) \\
1: medium~\eqref{EQ:Med 2}& (0.89,0.73,0.71,0.52) & (0.88,0.72,0.71,0.52) & (0.88,0.72,0.69,0.54)\\
2 & (0.89,0.73,0.71,0.52) & (0.91,0.76,0.65,0.58) & (0.87,0.83,0.56,0.64) \\
3 & (0.89,0.73,0.71,0.52) & (0.90,0.81,0.65,0.56) & (0.89,0.85,0.61,0.60)\\
\hline
\end{tabular}
\end{center}
\label{TAB:Intensities}
\end{table}
The simulations show that one can indeed reconstruct point sources, both their locations and their intensities, inside heterogeneous media when the media are not unreasonably complex. We want to emphasize here that in our theoretical analysis as well as numerical simulations, both the true sources and the sources to be reconstructed are explicitly assumed to be point sources. In other words, we explicitly search for the locations and strengths of the point sources, instead of reconstructing spatially distributed sources hoping that the result will give us point sources. In general, we observe from our extensive numerical simulations that when the refractive index is exactly known, the reconstructions are quite stable when the number of point sources is small. However, the reconstructions become too sensitive to algorithmic parameters when the number of point sources gets large.
We also want to emphasize that it is important to impose the constraints on the separability of the point sources in the numerical simulations. In other words, we have to explicitly ensure that the point sources to be reconstructed are far away from each other. Even in this case, the reconstructions are sensitive to the initial guess of the locations of the point sources. The objective function that we minimize to reconstruct the point sources can not differentiate between the true point sources and the equivalent class of re-labeled point sources. Therefore, the minimization algorithm could be easily fooled to jump between different intermediate configurations if the point sources are not well-separated. To further illustrate on this issue, we plot in Figure~\ref{FIG:Sentitivity Obj} the (normalized) objective functional $\Psi$ defined in~\eqref{EQ:Obj Src} as a function of the location of a single point source (the intensity of the source being assumed known). The true point source is located at $(0.443, 0.298)$. While it is clear from the plot that the objective function is convex with respect to the location of the point source when $k=5$, this is not true anymore when $k=8$ and $k=12$. In the case of $k=8$, two local minmizers emerge at $y\approx 0.85$. More local minimizers emerge when $k=12$. These plots show that even in the case of a single point source, when the initial guess is far from the true position, the minimization algorithm could return wrong reconstructions. We can not visualize this phenomenon in the case of more than one point source. However, one can easily imagine that the situation would be far worse in that scenario.
\begin{figure}[htb!]
\begin{center}
\includegraphics[width=0.30\textwidth]{Obj-Landscape-k-5.png}\,
\includegraphics[width=0.33\textwidth]{Obj-Landscape-k-8.png}\,
\includegraphics[width=0.315\textwidth]{Obj-Landscape-k-12.png}
\caption{Normalized objective function $\Psi$, defined in ~\eqref{EQ:Obj Src}, as a function of the location of a single point source. The true location is at $(0.443, 0.298)$. Shown from left to right are the three cases of $k=5$, $k=8$ and $k=12$ respectively.}
\end{center}
\label{FIG:Sentitivity Obj}
\end{figure}
\paragraph{Experiment 2 [Recovery in an Unknown Environment].} In the second set of simulations, we reconstruct point sources in medium~\eqref{EQ:Med 1} assuming that both the medium and the point sources are not known. The reconstructions of the medium and the locations of the sources are shown in Figure~\ref{FIG:Unknown Media 1} and the reconstructed intensities of the point sources are summarized in the third row of Table~\ref{TAB:Intensities}.
\begin{figure}[!htb]
\begin{center}
\includegraphics[height=0.18\textwidth]{true_d_4.png}
\includegraphics[height=0.18\textwidth]{rec_d_4.png}
\includegraphics[height=0.18\textwidth]{err_d_4.png}
\includegraphics[height=0.18\textwidth]{diag_d_4.png}\\
\includegraphics[scale=0.5]{../Figures/d1u}
\includegraphics[scale=0.5]{../Figures/d5u}
\end{center}
\caption{Simultaneous reconstruction of the point sources and the refractive index $n(\bx)$ in~\eqref{EQ:Med 1}. Top row: from left to right are the true $n(\bx)$, $n(\bx)$ reconstructed with data containing $1\%$ random noise, the difference between the true and the reconstructed $n(\bx)$, and the cross section of $n(\bx)$ along the diagonal (red and blue lines are for the true and the reconstruction respectively). Bottom row: true (crosses: $\times$) and reconstructed (circles: $\circ$) locations of the point sources using data with $1\%$ (left) and $5\%$ (right) random noise. }
\label{FIG:Unknown Media 1}
\end{figure}
Let us emphasize again that the reconstruction here is done in two steps. In the first step, we reconstruct the refractive index using multiple differential data sets. In the second step, we fix the refractive index, which is the reconstructed one, and reconstruct the point sources from \emph{one} Cauchy data set. If we compare the reconstructions in Figure~\ref{FIG:Unknown Media 1} with those in Figure~\ref{FIG:Known Med 1} (which are reconstructed under the true medium $n$ in~\eqref{EQ:Med 1}), and the reconstructions of intensities in the third row of Table.~\ref{TAB:Intensities} with those in the first row of the same table, we see that the reconstructions of the point sources are different but are of similar quality. That is, smooth changes in the refractive index introduces relatively small error in the reconstruction of point sources. This confirms our stability result in Theorem~\ref{THM:Stab Medium}.
\paragraph{Experiment 3 [Recovery in an Unknown Environment].} We repeat here the simulations in Experiment 2 for the medium~\eqref{EQ:Med 2}. The reconstructions of the medium and the locations of the sources are shown in Figure~\ref{FIG:Unknown Media 2} and the reconstructed intensities of the point sources are summarized in the last row of Table~\ref{TAB:Intensities}.
\begin{figure}[!htb]
\begin{center}
\includegraphics[height=0.18\textwidth]{true_s_4.png}
\includegraphics[height=0.18\textwidth]{rec_s_4.png}
\includegraphics[height=0.18\textwidth]{err_s_4.png}
\includegraphics[height=0.18\textwidth]{diag_s_4.png}\\
\includegraphics[scale=0.5]{s1u}
\includegraphics[scale=0.5]{s5u}
\end{center}
\caption{Simultaneous reconstruction of the point sources and the refractive index $n(\bx)$ in~\eqref{EQ:Med 2}. Top row: from left to right are the true $n(\bx)$, $n(\bx)$ reconstructed with data containing $1\%$ random noise, the difference between the true and the reconstructed $n(\bx)$, and the cross section of $n(\bx)$ along the diagonal (red and blue lines are for the true and the reconstruction respectively). Bottom row: true (crosses: $\times$) and reconstructed (circles: $\circ$) locations of the point sources using data with $1\%$ (left) and $5\%$ (right) random noise.}
\label{FIG:Unknown Media 2}
\end{figure}
The numerical results in this experiment again confirms the stability result in Theorem~\ref{THM:Stab Medium}. This can be seen by comparing the reconstructions in Figure~\ref{FIG:Unknown Media 2} with those in Figure~\ref{FIG:Known Med 2}, and the reconstructions of intensities in the fourth row of Table.~\ref{TAB:Intensities} with those in the second row of the same table.
The results in Experiment 2 and Experiment 3 demonstrate that smooth uncertainty in the medium produces relatively small errors in the reconstructions of the point sources. In other words, if we collect data from a medium that we know only approximately, we can simply perform reconstructions using our best known approximation to the medium. The results are not very different from those obtained using the true medium.
\section{Concluding remarks}
\label{SEC:Concl}
In this short paper, we studied, both theoretically and numerically, the reconstruction of point sources in heterogeneous media from boundary Cauchy data.
Our first result is derived when the underlying medium is known. This is on the stability of the location and intensity reconstructions with respect to noise in the Cauchy data. This is a generalization of the results of El Badia and El Hajj in~\cite{ElEl-CRASP12} for the same reconstructions but in homogeneous media. Our numerical simulations confirm the theoretical predictions. More precisely, when only a very small number of point sources are to be reconstructed, numerical experiments suggest that they can be relatively stably recovered when the medium is known.
The motivation for our second result is to see how stable an imaging result, which could be the imaging of a point source as in our case, or a point scatter~\cite{AmKa-IP03,AmMoVo-ESAIM03}, or a reflector~\cite{FoGaPaSo-Book07}, or an extended target~\cite{BaCaLiRe-IP07,BaRe-SIAM08,Garnier-SIAM05,Zhao-SIAM04}, is with respect to uncertainties in the medium properties. This is an important problem to be addressed since in most applications, the underlying media are either assumed known or have to be reconstructed as well. In either case, targets are imaged with medium properties that are not the true medium properties. We established a stability result on the reconstruction of point sources with respect to smooth changes of the medium. This result says that if the medium is known up to a small (smooth) error, one can hope that the reconstructions are close to the true reconstructions. Numerical experiments show that even in the complicated case of simultaneous reconstructions of the refractive index and the point sources, the location of the sources can often be reconstructed in a robust way, indicating that the error caused by the uncertainty in the medium property, i.e. the refractive index, is relatively small. More quantitative characterization of the uncertainty in the reconstructions needs to be performed, for instance, following the ideas presented in~\cite{ReVa-Prep18} in the context of photoacoustic imaging.
\section*{Acknowledgments}
We would like to thank Professor Abdellatif El Badia for useful discussion on algebraic methods for reconstructing point sources. This work is partially supported by the National Science Foundation through grant DMS-1620473.
{\small
| {
"timestamp": "2019-07-01T02:02:56",
"yymm": "1901",
"arxiv_id": "1901.07189",
"language": "en",
"url": "https://arxiv.org/abs/1901.07189",
"abstract": "Imaging point sources in heterogeneous environments from boundary or far-field measurements has been extensively studied in the past. In most existing results, the environment, represented by the refractive index function in the model equation, is assumed known in the imaging process. In this work, we investigate the impact of environment uncertainty on the reconstruction of point sources inside it. Following the techniques developed by El Badia and El Hajj (C. R. Acad. Sci. Paris, Ser. I, 350 (2012), 1031-1035), we derive stability of reconstructing point sources in heterogeneous media with respect to measurement error as well as smooth changes in the environment, that is, the refractive index. Numerical simulations with synthetic data are presented to further explore the derived stability properties.",
"subjects": "Analysis of PDEs (math.AP); Numerical Analysis (math.NA)",
"title": "Imaging point sources in heterogeneous environments",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808713165662,
"lm_q2_score": 0.7217432122827968,
"lm_q1q2_score": 0.7075110650033974
} |
https://arxiv.org/abs/1506.01534 | Roots of Dehn twists about multicurves | A \textit{multicurve} $\C$ on a closed orientable surface is defined to be a finite collection of disjoint non-isotopic essential simple closed curves. The Dehn twist $t_{\C}$ about $\C$ is the product of the Dehn twists about the individual curves. In this paper, we give necessary and sufficient conditions for the existence of a root of such a Dehn twist, that is, a homeomorphism $h$ such that $h^n = t_{\C}$. We give combinatorial data that corresponds to such roots, and use it to determine upper bounds for $n$. Finally, we classify all such roots up to conjugacy for surfaces of genus 3 and 4. | \section{Introduction}
For $g \geq 0$, let $S_g$ be the closed, orientable surface of genus $g$, and let $\text{Mod}(S_g)$ denote the mapping class group of $S_g$. By a \textit{multicurve} $\operatorname{{\mathcal{C}}}$ in $S_g$, we mean a finite collection of disjoint non-isotopic essential simple closed curves in $S_g$. Let $t_c$ denote the left-handed Dehn twist about an essential simple closed curve $c$ on $S_g$. Since the Dehn twists about any two curves in $\operatorname{{\mathcal{C}}}$ commute, we will define the \textit{left-handed Dehn twist about} $\operatorname{{\mathcal{C}}}$ to be
$$
t_{\operatorname{{\mathcal{C}}}} := \prod_{c\in \operatorname{{\mathcal{C}}}} t_c
$$
A \textit{root of $t_{\operatorname{{\mathcal{C}}}}$ of degree $n$} is an element $h \in \text{Mod}(S_g)$ such that $h^n = t_{\operatorname{{\mathcal{C}}}}$.
When $\operatorname{{\mathcal{C}}}$ comprises a single nonseparating curve, D. Margalit and S. Schleimer \cite{MS} showed the existence of roots of $t_{\operatorname{{\mathcal{C}}}}$ of degree $2g - 1$ in $\operatorname{Mod}(S_g)$, for $g \geq 2$. This motivated \cite{MK1}, in which D. McCullough and the first author derived necessary and sufficient conditions for the existence of a root of degree $n$. As immediate applications of the main theorem in the paper, they showed that roots of even degree cannot exist and that $n \leq 2g -1$. When $\operatorname{{\mathcal{C}}}$ consists of a single separating curve, the first author derived conditions \cite{KR1} for the existence of a root of $t_{\operatorname{{\mathcal{C}}}}$. A stable quadratic upper bound on $n$, and complete classifications of roots for $S_2$ and $S_3$, were derived as corollaries
to the main result. In this paper, we shall derive conditions for the existence of a root of $t_{\operatorname{{\mathcal{C}}}}$ when $|\operatorname{{\mathcal{C}}}| \geq 2$, and since there are no such multicurves in $S_1$ or $S_0$, we shall assume henceforth that $g \geq 2$.
In general, a root $h$ of $t_{\operatorname{{\mathcal{C}}}}$ may permute some curves in $\operatorname{{\mathcal{C}}}$, while preserving other curves. So we define an \textit{$(r,k)$-permuting root} of $t_{\operatorname{{\mathcal{C}}}}$ to be one that induces a partition of $\operatorname{{\mathcal{C}}}$ into $r$ singletons and $k$ other subsets of size greater than one. The theory for $(r,0)$-permuting roots, as we will see, can be obtained by generalizing the theories developed in \cite{MK1} and \cite{KR1}, which involved the analysis of the fixed point data of finite cyclic actions.
The theory that we intend to develop for $(r,k)$-permuting roots when $k>0$ can be motivated by the following example. Consider the multicurve $\operatorname{{\mathcal{C}}}$ in $S_5$ shown in Figure \ref{fig:perm_s5}.
\begin{figure}[h!]
\labellist
\small
\pinlabel $2\pi/5$ at 30 200
\endlabellist
\centering
\includegraphics[width=35 ex]{Perm_S5}
\caption{Nonseparating multicurve of size 5 in $S_5$}\label{fig:perm_s5}
\end{figure}
\noindent It is apparent that the rotation of $S_5$ by $2\pi/5$ composed with $t_c$ for some fixed $c\in \operatorname{{\mathcal{C}}}$ is a 5$^{th}$ root of $t_{\operatorname{{\mathcal{C}}}}$ in $\operatorname{Mod}(S_5)$. This is a simple example of a $(0,1)$-permuting root, which is obtained by removing invariant disks around pairs of points in two distinct orbits of the $2\pi/5$ rotation of $S_0$, and then attaching five 1-handles with full twists. This example indicates that a classification of such roots would require the examination of the orbit information of finite cyclic actions, in addition to their fixed point data. This is a significant departure from existing theories developed in~\cite{MK1} and~\cite{KR1}.
Any subset of a multicurve will be called a \textit{submulticurve}. A multicurve $\operatorname{{\mathcal{C}}}$ in $S_g$ is said to be \textit{pseudo-nonseparating} if $\operatorname{{\mathcal{C}}}$ separates $S_g$, but no proper submulticurve of $\operatorname{{\mathcal{C}}}$ separates $S_g$. A multicurve that contains no pseudo-nonseparating submulticurves will be called a \textit{nonseparating multicurve}, while a multicurve which is a disjoint union of pseudo-nonseparating multicurves will be called a \textit{separating multicurve}. A multicurve that is neither separating nor nonseparating will be called a \textit{mixed multicurve}. In Figure~\ref{fig:s5mixed} below, the collection of curves $\operatorname{{\mathcal{C}}} = \{c_1,c_2,c_3,c_4\}$ is a mixed multicurve, while the subcollections $\{c_2,c_3\}$, $\{c_1,c_2,c_3\}$ and $\{c_2,c_4\}$ form pseudo-nonseparating, separating and nonseparating multicurves, respectively.
\begin{figure}[h]
\labellist
\small
\pinlabel $c_1$ at 205 120
\pinlabel $c_2$ at 565 160
\pinlabel $c_3$ at 565 85
\pinlabel $c_4$ at 1053 85
\endlabellist
\centering
\includegraphics[width= 75 ex]{s5}
\caption{The surface $S_5$ with a mixed multicurve}\label{fig:s5mixed}
\end{figure}
We start by generalizing the notion of a nestled $(n,\ell)$-action from~\cite{KR1} to a \textit{permuting $(n,r,k)$-action}. These are $C_n$-actions on $S_g$ that have $r$ distinguished fixed points, and $k$ distinguished non-trivial orbits. In Section~\ref{sec: actions_data_triples}, we introduce the notion of a \textit{permuting data set}, which is a generalization of a data set from \cite{KR1}. We use Thurston's orbifold theory \cite[Chapter 13]{T1} in Theorem~\ref{thm:action_triple_correspondence} to establish a correspondence between permuting $(n,r,k)$-actions on $S_g$ and permuting data sets of genus $g$ and degree $n$. In other words, permuting data sets algebraically encode these permuting actions and contain all the relevant orbit and fixed-point information required to classify the roots that will be constructed from these actions.
Let $S_g(\operatorname{{\mathcal{C}}})$ denote the surface obtained from $S_g$ by deleting an annular neighbourhood of $\operatorname{{\mathcal{C}}}$ and capping. In Section~\ref{sec:nonsepmulcurves}, we prove that conjugacy classes of roots of Dehn twists about nonseparating multicurves correspond to a special subclass of permuting actions on $S_g(\operatorname{{\mathcal{C}}})$. We use this to obtain the following bounds for the degree of such a root
\begin{coro}
Let $\operatorname{{\mathcal{C}}}$ be a nonseparating multicurve in $S_g$ of size $m$, and let $h$ be an $(r,k)$-permuting root of $t_{\operatorname{{\mathcal{C}}}}$ of degree $n$.
\begin{enumerate}[(i)]
\item If $r\geq 0$, then
$$
n\leq \begin{cases}
4(g-m) + 2 &: g-m \geq 1 \\
g &: g = m.
\end{cases}
$$
Furthermore, if $g=m$, then this upper bound is realizable.
\item If $r\geq 1$, then $n$ is odd.
\item If $r=1$, then $n\leq 2(g-m)+1$.
\item If $r\geq 2$, then $\displaystyle n \leq \frac{g-m+r-1}{r-1}$.
\end{enumerate}
\end{coro}
If $\operatorname{{\mathcal{C}}}$ is a separating or a mixed multicurve, $S_g(\operatorname{{\mathcal{C}}})$ will have multiple connected components. In order to classify roots of $t_{\operatorname{{\mathcal{C}}}}$ in this case, we will require multiple actions on the various components of $S_g(\operatorname{{\mathcal{C}}})$ which can be put together and extended to a root of $t_{\operatorname{{\mathcal{C}}}}$ on $S_g$. In Section~\ref{sec:sepmulcurves}, we will show that this extension will require compatibility of orbits across components of $S_g(\operatorname{{\mathcal{C}}})$ in addition to compatibility of fixed points as in \cite{KR1}. As an immediate consequence to this theorem, we obtain quadratic bounds for the degree of the root in terms of the genera of the individual components. Furthermore, we obtain a quadratic stable upper bound for the degree of the root as in \cite[Theorem 8.14]{KR1}.
In Sections \ref{sec:classify_genus3} and \ref{sec:classify_genus4}, we use our theory to obtain a complete classification of roots of $t_{\operatorname{{\mathcal{C}}}}$ on surfaces of genus 3 and 4 respectively. We conclude by proving that a root of $t_{\operatorname{{\mathcal{C}}}}$ cannot lie in the Torelli group of $S_g$, and also indicate how our results can be extended to classify roots of finite products of powers of commuting Dehn twists.
integrated
\section{Roots and their induced partitions}
In this section we shall introduce some preliminary notions, which will be used in later sections.
\begin{notation}\label{defn:sgc}
Let $\operatorname{{\mathcal{C}}}$ be a multicurve in $S_g$, and let $N$ be a closed annular neighbourhood of $\operatorname{{\mathcal{C}}}$.
\begin{enumerate}[(i)]
\item We denote the surface $\overline{S_g\setminus N}$ by $\widehat{S_g(\operatorname{{\mathcal{C}}})}$.
\item The closed orientable surface obtained from $\widehat{S_g(\operatorname{{\mathcal{C}}})}$ by capping off its boundary components is denoted by $S_g(\operatorname{{\mathcal{C}}})$.
\item If $\operatorname{{\mathcal{C}}}$ is a nonseparating multicurve, then $S_g(\operatorname{{\mathcal{C}}})$ is a connected surface whose genus we denote by $g_{\operatorname{{\mathcal{C}}}}$.
\end{enumerate}
\end{notation}
\begin{notation}\label{defn:pseudo_multicurve}
Recall that a multicurve $\operatorname{{\mathcal{C}}}$ in $S_g$ is said to be \emph{pseudo-nonseparating} if $S_g(\operatorname{{\mathcal{C}}})$ is disconnected, but $S_g(\operatorname{{\mathcal{C}}}')$ is not disconnected for any proper submulticurve $\operatorname{{\mathcal{C}}}'\subset \operatorname{{\mathcal{C}}}$.
\begin{enumerate}[(i)]
\item If $|\operatorname{{\mathcal{C}}}| = k$, we write $\operatorname{{\mathcal{C}}}^{(k)}$ for such a multicurve. Note that $\operatorname{{\mathcal{C}}}^{(1)}$ is a single separating curve.
\item A disjoint union of $m$ copies of $\operatorname{{\mathcal{C}}}^{(k)}$ is denoted by $\operatorname{{\mathcal{C}}}^{(k)}(m)$.
\item For integers $g \geq 0$ and $m \geq 1$, we define $\S_g(m)$ to be the disjoint union of $m$ copies $\{S_g^1, S_g^2, \ldots, S_g^m\}$ of $S_g$ isometrically imbedded in $\mathbb{R}^3$. In particular, $\S_g(1) \cong S_g$, and hence we shall write $S_g$ for $\S_g(1)$.
\item Given two surfaces $S_{g_1}$ and $\S_{g_2}(m)$ and a fixed $k\in \mathbb{N}$, we construct a new surface $S_g$ with $g=(g_1+mg_2 + (k-1)m)$, containing a multicurve of type $\operatorname{{\mathcal{C}}}^{(k)}(m)$, in the following manner. We remove $km$ disks $\{D_{i,j}^1 : 1\leq j\leq k, 1\leq i\leq m\}$ on $S_{g_1}$ and $k$ disks $\{D_{i,j}^2 : 1\leq j\leq k\}$ on each $S_{g_2}^i$. Now connect $\partial D_{i,j}^1$ to $\partial D_{i,j}^2$ along a 1-handle $A_{i,j}$, and choose the unique curve (upto isotopy) $c_{i,j}$ on $A_{i,j}$. Let $\operatorname{{\mathcal{C}}} = \{c_{i,j}\}$, then note that $\operatorname{{\mathcal{C}}} = \operatorname{{\mathcal{C}}}^{(k)}(m)$, so we write
$$
S_{g_1}\#_{\operatorname{{\mathcal{C}}}} \S_{g_2}(m)
$$
for the new surface $S_g$.
\item Similarly, given surfaces $\{S_{g_1}, \S_{g_2,1}(m_1), \ldots, \S_{g_2,s}(m_s)\}$ and non-negative integers $\{k_1,k_2,\ldots, k_s\}$, we construct a new surface $S_g$ with $g= g_1 + \sum_{i=1}^s m_i(g_{2,i} + k_i-1)$, containing a multicurve of type $\operatorname{{\mathcal{C}}} = \sqcup_{i=1}^s \operatorname{{\mathcal{C}}}^{(k_i)}(m_i)$. Let
$$
S_{g_i'} := S_{g_1}\#_{\operatorname{{\mathcal{C}}}^{(k_i)}(m_i)} \S_{g_2,i}(m_i)
$$
and $\operatorname{{\mathcal{C}}}_i := \operatorname{{\mathcal{C}}}\setminus \operatorname{{\mathcal{C}}}^{(k_i)}(m_i)$, we now define
$$
\overline{\textbf{\Large \#}}_{i=1}^s \left( S_{g_1}{\#_{\operatorname{{\mathcal{C}}}^{(k_i)}(m_i)}} \S_{g_2,i}(m_i) \right) := \bigcup_{i=1}^s \widehat{S_{g_i'}(\operatorname{{\mathcal{C}}}_i)}.
$$
If $s = 2$, we simply write $S_g = \S_{g_{2,1}}(m_1)\#_{\operatorname{{\mathcal{C}}}^{(k_1)}(m_1)} S_{g_1}\#_{\operatorname{{\mathcal{C}}}^{(k_2)}(m_2)} \S_{g_{2,2}}(m_2)$.
\end{enumerate}
\end{notation}
\noindent In Figure~\ref{fig:sg_sgm} below, we give an example of a such a surface $S_{22}$ with a multicurve $\operatorname{{\mathcal{C}}} = \operatorname{{\mathcal{C}}}^{(2)}(2) \sqcup \operatorname{{\mathcal{C}}}^{(1)}(3)$ constructed as in Notation~\ref{defn:pseudo_multicurve} from the surfaces $S_5, \S_3(2)$, and $\S_3(3)$.
\begin{figure}[h]
\labellist
\tiny
\pinlabel $\operatorname{{\mathcal{C}}}^{(2)}(2)$ at 70 95
\pinlabel $\operatorname{{\mathcal{C}}}^{(1)}(3)$ at 620 35
\endlabellist
\centering
\includegraphics[width= 70 ex]{sg-sgm}
\caption{The surface $S_{22} = \S_3(2) \#_{\operatorname{{\mathcal{C}}}_1} S_5 \#_{\operatorname{{\mathcal{C}}}_2} \S_3(3),$ where $\operatorname{{\mathcal{C}}}_1 = \operatorname{{\mathcal{C}}}^{(2)}(2)$ and $\operatorname{{\mathcal{C}}}_2 = \operatorname{{\mathcal{C}}}^{(1)}(3)$.}
\label{fig:sg_sgm}
\end{figure}
\begin{rem}\label{rem:root_isotopy}
Let $\operatorname{{\mathcal{C}}} = \{c_1,c_2,\ldots, c_m\}$, and suppose that $h$ is a root of $t_{\operatorname{{\mathcal{C}}}}$ of degree $n$ in $\operatorname{Mod}(S_g)$.
Then, we claim that, up to isotopy, $h(\operatorname{{\mathcal{C}}}) = \operatorname{{\mathcal{C}}}$.
Suppose first that $h(c_i) \nsim c_1$ for all $i$. Then there exists a neighbourhood $N$ of $c_1$ such that
$$
t_{h(c_i)} \mid_N = \text{id}_N \text{ and } t_{c_i}\mid_N = \text{id}_N \quad\forall i\neq 1
$$
However, since $h^n = t_{\operatorname{{\mathcal{C}}}}$, it follows that
$$
t_{\operatorname{{\mathcal{C}}}} = ht_{\operatorname{{\mathcal{C}}}}h^{-1} = ht_{c_1}h^{-1}ht_{c_2}h^{-1}\ldots ht_{c_m}h^{-1} = t_{h(c_1)}t_{h(c_2)}\ldots t_{h(c_m)}
$$
Hence
$$
t_{c_1}\mid_N = t_{\operatorname{{\mathcal{C}}}}\mid_N = t_{h(c_1)}t_{h(c_2)}\ldots t_{h(c_m)}\mid_N = \text{id}_N
$$
which is a contradiction.
So we assume without loss of generality that $h(c_1) \sim c_1$. Then we choose a
neighbourhood $N'$ of $c_1$ disjoint from $c_i$ for $i\neq 1$, and choose an isotopy $\varphi_t$ such that
$$
\varphi_t\mid_{S_g\setminus N'} = \text{id}_{S_g\setminus N'}
$$
and $\varphi_0 = h$ and $\varphi_1(c_1) = c_1$. Replacing $h$ by $\varphi_1$, we may assume that $h(c_1) = c_1$. Now note that $h(c_i) \nsim c_1$ for all $i\neq 1$, which allows us to proceed
by induction on $|\operatorname{{\mathcal{C}}}|$ to conclude that, up to isotopy, $h(\operatorname{{\mathcal{C}}}) = \operatorname{{\mathcal{C}}}$.
\end{rem}
\begin{defn}\label{defn:rkpartition}
Let $\operatorname{{\mathcal{C}}}$ be a multicurve of size $m$ in $S_g$. Then for integers $r,k \geq 0$, an \textit{$(r,k)$-partition}
of $\operatorname{{\mathcal{C}}}$ is a partition $\operatorname{{\mathbb P}}_{r,k}(\operatorname{{\mathcal{C}}}) = \{\operatorname{{\mathcal{C}}}_1',\ldots,\operatorname{{\mathcal{C}}}_r',\operatorname{{\mathcal{C}}}_1,\ldots,\operatorname{{\mathcal{C}}}_k\}$ of the set $\operatorname{{\mathcal{C}}}$ into subsets such that for all $i$,
\begin{enumerate}[(i)]
\item $|\operatorname{{\mathcal{C}}}_i'| =1$, $|\operatorname{{\mathcal{C}}}_i| > 1$, and
\item $\operatorname{{\mathcal{C}}}_i$ comprises only separating or only nonseparating curves.
\end{enumerate}
\end{defn}
\begin{defn}
Let $\operatorname{{\mathcal{C}}}$ be a multicurve in $S_g$. Then for integers $r,k \geq 0$, a root $h$ of $t_{\operatorname{{\mathcal{C}}}}$ of is said to be \textit{$(r,k)$-permuting} if it induces an $(r,k)$-partition of $\operatorname{{\mathcal{C}}}$.
\end{defn}
\begin{notation}\label{notation:rkpartition}
Let $\operatorname{{\mathcal{C}}}$ be a multicurve of size $m$ in $S_g$ and consider an $(r,k)$-partition $\operatorname{{\mathbb P}}_{r,k}(\operatorname{{\mathcal{C}}}) = \{\operatorname{{\mathcal{C}}}_1',\ldots,\operatorname{{\mathcal{C}}}_r',\operatorname{{\mathcal{C}}}_1,\ldots, \operatorname{{\mathcal{C}}}_k\}$ as in Definition~\ref{defn:rkpartition}.
\begin{enumerate}[(i)]
\item We shall denote the multiset $\llbrace |\operatorname{{\mathcal{C}}}_1|,|\operatorname{{\mathcal{C}}}_2|, \ldots,|\operatorname{{\mathcal{C}}}_k|\rrbrace$ by $S(\operatorname{{\mathbb P}}_{r,k})$. (From here on, we shall denote a multiset using $\llbrace\,\,\rrbrace$).
\item If $\operatorname{{\mathbb P}}_{r,k}(\operatorname{{\mathcal{C}}})$ is induced by an $(r,k)$-permuting root $h$ of $t_{\operatorname{{\mathcal{C}}}}$, then we shall denote it by $\operatorname{{\mathcal{C}}}_{r,k}(h)$.
\end{enumerate}
\end{notation}
\section{Permuting Actions and Permuting Data Sets}\label{sec: actions_data_triples}
In this section we shall introduce permuting $(n,r,k)$-actions, which are generalizations of the nestled $(n,\ell)$-actions from~\cite{KR1}. We shall also introduce the notion of a permuting $(n,r,k)$-data set, which is an abstract tuple involving non-negative integers that would algebraically encode a permuting $(n,r,k)$-action.
\begin{defn}\label{defn:permuting_actions}
For integers $n\geq 1 \text{ and }r,k\geq 0$, an orientation-preserving $C_n$-action $t$ on $S_g$ is called a \emph{permuting $(n,r,k)$-action} if
\begin{enumerate}[(i)]
\item there is a set $\operatorname{{\mathbb P}}(t) \subset S_g$ of $r$ distinguished fixed points of $t$, and
\item there is a set $\o(t) \subset S_g$ of $k$ distinguished non-trivial orbits of $t$.
\end{enumerate}
\end{defn}
\begin{notation}\label{defn:orbit_distribution}
Let $t$ be a permuting $(n,r,k)$-action on $S_g$.
\begin{enumerate}[(i)]
\item Fix a point $P \in S_g$, and consider $t_{\ast}: T_P(S_g) \to T_{t(P)}(S_g)$. By the Nielsen realisation theorem~\cite{K1}, we may change $t$ by isotopy in $\operatorname{Mod}(S_g)$ so that $t_{\ast}$ is an isometry. Hence, $t_{\ast}$ induces a local rotation by an angle, which we shall denote by $\theta_P(t)$. Note that if $P \in \operatorname{{\mathbb P}}(t)$, then $\theta_P(t) = 2\pi a/n,$ where $\gcd(a,n) = 1$.
\item Fix an orbit $\o=\{P_1,\ldots,P_s\} \in \o(t)$. If $s<n$, then $s\mid n$, and there exists a cone point in the quotient orbifold of degree $n/s$. Each $P_i$ has stabilizer generated by $t^s$ and the rotation induced by $t^s$ around each $P_i$ must be the same, since its action at one point is conjugate by a power of $t$ to its action at each other point in the orbit. So the rotation angle is of the form $2\pi c^{-1}/(n/s)\pmod{2\pi}$, where $(c,n/s) = 1$ and $c^{-1}$ denotes the inverse of $c\pmod{n/s}$. We now associate to this orbit a pair $p(\o)$ as follows:
$$
p(\o) := \begin{cases}
(c, n/s) &\text{if } s<n \\
(0,1) &\text{if } s=n.
\end{cases}
$$
\item For any orbit $\o \in \o(t)$, if $p(\o) = (a,b)$, then we define
$$
\theta_{\o}(t) :=
\begin{cases}
2\pi a^{-1}/b &\text{if } a\neq 0 \\
0 &\text{if }a=0.
\end{cases}
$$
\end{enumerate}
\end{notation}
\begin{defn}\label{defn: orbit_distribution}
Consider a permuting $(n,r,k)$-action $t$ on $S_g$ with $\operatorname{{\mathbb P}}(t) = \{P_1,\ldots,P_r\}$ and $\o(t) = \{\o_1,\o_2,\ldots, \o_k\}$.
\begin{enumerate}[(i)]
\item We write
$$
S(t) = \llbrace |\o_1|, |\o_2|, \ldots, |\o_k|\rrbrace.
$$
\item For each $p \in \{p(\o_i) : 1\leq i \leq k\}$, define
$$
m_p = |\{j : p(\o_j) = x\}|.
$$
We define the \emph{orbit distribution} of $t$ to be the set
$$
\o_t = \{(p,m_p) : \in \{p(\o_i): 1\leq i\leq k\}\}.
$$
\end{enumerate}
\end{defn}
\begin{defn}
\label{defn:eq_perm_actions}
Let $t_1$ and $t_2$ be two permuting $(n,r,k)$-actions on $S_g$ with $\operatorname{{\mathbb P}}(t_s) = \{P_{s,1},P_{s,2},\ldots,P_{s,r}\}$ and $\o(t_s) = \{\o_{s,1},\o_{s,2},\ldots, \o_{s,k}\}$ for $s=1,2$. We say $t_1$ is \emph{equivalent} to $t_2$ if $\o_{t_1} = \o_{t_2}$ and there is an orientation-preserving homeomorphism $\phi \in \operatorname{Mod}(S_g)$ such that
\begin{enumerate}[(i)]
\item $\phi(P_{1,i}) = P_{2,i}$ for $1 \leq i \leq r$,
\item for each $1\leq j\leq k$, if $\o_{s,j} = \{Q_{j,1}^s, Q_{j,2}^s, \ldots, Q_{j,m_{s_j}}^s\}$, then $m_{1_j} = m_{2_j}$ and $\phi(Q_{j,i}^1) = Q_{j,i}^2$ for all $1\leq i\leq m_{1_j}$, and
\item $\phi t_1 \phi^{-1}$ is isotopic to $t_2$ relative to $\operatorname{{\mathbb P}}(t_2)\sqcup(\cup_{j=1}^k \o_{2,j})$.
\end{enumerate}
The equivalence class of a permuting $(n,r,k)$-action is denoted by $\l t\r$.
\end{defn}
We now introduce the notion of an \textit{$(n,r)$-data set}, which encodes the signature of the quotient orbifold of a permuting $(n,r,k)$-action and the turning angles around its distinguished fixed points. Furthermore, the $(n,r)$-data set will be combined with the orbit distribution of the action to form a pair, which we will call a \textit{permuting $(n,r,k)$-data set}.
\begin{defn}\label{defn:data_sets}
Given $n\geq 1$ and $r\geq 0$, an \textit{$(n,r)$-data set} is a tuple
$$
\operatorname{{\mathcal D}} = (n,g_0, (a_1,a_2,\ldots, a_r); (c_1,n_1), (c_2,n_2),\ldots, (c_s,n_s))
$$
where $n\geq 1$ and $ g_0 \geq 0$ are integers, each $a_i$ is a residue class modulo $n$, and each $c_i$ is a residue class modulo $n_i$ such that:
\begin{enumerate}[(i)]
\item each $n_i\mid n$,
\item $gcd(a_i,n) = gcd(c_i,n_i) = 1$, and
\item $\displaystyle \sum_{i=1}^r a_i + \sum_{j=1}^s \frac{n}{n_i}c_i \equiv 0\pmod{n}$.
\end{enumerate}
The number $g$ determined by the equation
\begin{equation}\label{eqn:riemann_hurwitz}
\frac{2-2g}{n} = 2-2g_0 + r\left(\frac{1}{n}-1 \right) + \sum_{j=1}^s \left(\frac{1}{n_j} - 1 \right)
\end{equation}
is called the \emph{genus} of the data set.
\end{defn}
\begin{defn} Fix an $(n,r)$-data set $\operatorname{{\mathcal D}}$ of genus $g$ as above.
\begin{enumerate}[(i)]
\item For each $(a,b) \in \{(0,1),(c_1,n_1), \ldots, (c_s,n_s)\}$,
we write
$$
\theta((a,b)) := \begin{cases}
0 & \text{if }a=0, \\
2\pi a^{-1}/b &\text{if } a\neq 0.
\end{cases}
$$
\item For each $p \in \{(0,1),(c_1,n_1), \ldots, (c_s,n_s)\}$, choose a non-negative integer $m_p$. Then the set $\o_{\operatorname{{\mathcal D}}} = \{(p,m_p) : m_p > 0\}$
is called an \emph{orbit distribution} of $\operatorname{{\mathcal D}}$.
\item Given an orbit distribution $\o_{\operatorname{{\mathcal D}}}$ associated with an $(n,r)$-data set $\operatorname{{\mathcal D}}$, the pair $(\operatorname{{\mathcal D}}, \o_{\operatorname{{\mathcal D}}})$ is called a \emph{permuting $(n,r,k)$-data set} of genus $g$, where $k=\sum_p m_p$.
\end{enumerate}
\end{defn}
\begin{defn} Let
\begin{equation*}
\begin{split}
\operatorname{{\mathcal D}} &= (n,g_0, (a_1,a_2,\ldots, a_r); (c_1,n_1), (c_2,n_2),\ldots, (c_s,n_s)) \\
\text{ and } \operatorname{{\mathcal D}}' &= (n,g_0', (a_1',a_2',\ldots, a_r'); (c_1',n_1'), (c_2',n_2'),\ldots, (c_s',n_s'))
\end{split}
\end{equation*}
be two $(n,r)$-data sets as in Definition \ref{defn:data_sets}.
\begin{enumerate}[(i)]
\item $\operatorname{{\mathcal D}}$ and $\operatorname{{\mathcal D}}'$ are said to be \emph{equivalent} if
$$
\llbrace a_1,a_2,\ldots, a_r\rrbrace = \llbrace a_1', a_2', \ldots, a_r'\rrbrace, \text{ and }
$$
$$
\llbrace (c_1,n_1),\ldots, (c_s,n_s)\rrbrace = \llbrace (c_1',n_1'), \ldots (c_s',n_s')\rrbrace.
$$
\item Two permuting $(n,r,k)$-data sets $(\operatorname{{\mathcal D}}, \o_{\operatorname{{\mathcal D}}})$ and $(\operatorname{{\mathcal D}}', \o_{\operatorname{{\mathcal D}}'})$ are said to be \emph{equivalent} if $\operatorname{{\mathcal D}}$ and $\operatorname{{\mathcal D}}'$ are equivalent as above, and $\o_{\operatorname{{\mathcal D}}} = \o_{\operatorname{{\mathcal D}}'}$.
\end{enumerate}
\end{defn}
\noindent Note that equivalent data sets have the same genus.
\begin{thm}\label{thm:action_triple_correspondence}
Given $n\geq 1$ and $g\geq 0$, equivalence classes of permuting $(n,r,k)$-data sets of genus $g$ correspond to equivalence classes of permuting $(n,r,k)$-actions on $S_g$.
\end{thm}
\begin{proof}
Let $t$ be a permuting $(n,r,k)$-action on $S_g$ with quotient orbifold $\O$ whose underlying surface has genus $g_0$. If $t$ is a free action, then $\O = S_{g_0}$, and we simply write $\operatorname{{\mathcal D}} = (n,g_0;)$ and $\o_{\operatorname{{\mathcal D}}} = \{(0,1),k)\}$. If $t$ is not free, let $p_j$ be the image in $\O$ of the $P_j$, for $1 \leq j\leq r$, and let $q_1,q_2,\ldots, q_s$ be the other possible cone points of $\O$ as in Figure \ref{fig:orb}.
\begin{figure}[h]
\label{fig:orb}
\labellist
\small
\pinlabel $p_1$ at 810 50
\pinlabel $p_2$ at 785 91
\pinlabel $q_1$ at 686 103
\pinlabel $q_2$ at 645 75
\endlabellist
\centering
\includegraphics[width = 75 ex]{orbifold}
\caption{The quotient orbifold $\O$.}
\end{figure}
Let $\alpha_i$ be the generator of the orbifold fundamental group $\pi_1^{orb}(\O)$ that goes around the point $p_i, 1\leq i\leq r$ and let $\gamma_j$ be the generators going around $q_j,1\leq j\leq s$. Let $x_p$ and $y_p,1\leq p\leq g_0$ be the standard generators of the ``surface part'' of $\O$, chosen to give the following presentation of $\pi_1^{orb}(\O)$:
\begin{gather*}
\pi_1^{orb}(\O) = \langle \alpha_1,\alpha_2,\ldots, \alpha_r, \gamma_1,\gamma_2,\ldots, \gamma_s,x_1,y_1,x_2,y_2,\ldots x_{g_0},y_{g_0} \lvert \\ \alpha_1^n = \alpha_2^n = \ldots = \gamma_1^{n_1} = \gamma_2^{n_2} = \ldots = \gamma_s^{n_s} = 1, \alpha_1\ldots \alpha_r\gamma_1\ldots \gamma_s = \prod_{p=1}^{g_0}[x_p,y_p]\rangle
\end{gather*}
From orbifold covering space theory \cite{T1}, we have the following exact sequence
$$
1 \rightarrow \pi_1(S_g) \rightarrow \pi_1^{orb}(\O) \xrightarrow{\rho} C_n\rightarrow 1,
$$
where $C_n = \langle t \rangle$.
The homomorphism $\rho$ is obtained by lifting path representatives of elements of $\pi_1^{orb}(\O)$. Since these do not pass through the cone points, the lifts are uniquely determined.
For $1\leq i\leq s$, the preimage of $q_i$ consists of $n/n_i$ points cyclically permuted by $t$. As in Notation~\ref{defn:orbit_distribution}, the rotation angle at each point is of the form $2\pi c_i^{-1}/n_i$ where $c_i$ is a residue class modulo $n_i$ and $gcd(c_i,n_i) = 1$. Lifting the $\gamma_i$, we have that $\rho(\gamma_i) = h^{(n/n_i)c_i}$. Similarly, lifting the $\alpha_i$ gives $\rho(\alpha_i)=t^{a_i}$ where $gcd(a_i,n)=1$. Finally, we have
$$
\rho(\prod_{p=1}^{g_0}[x_p,y_p]) = 1,
$$
since $C_n$ is abelian, so
$$
1 = \rho(\alpha_1\ldots \alpha_r\gamma_1\ldots \gamma_s) = t^{a_1+\ldots + a_r+(n/n_1)c_1 + \ldots + (n/n_s)c_s}
$$
giving
$$
\sum_{i=1}^r a_i + \sum_{j=1}^s \frac{n}{n_j}c_j \equiv 0 \pmod{n}.
$$
The fact that the data set $\operatorname{{\mathcal D}}$ has genus $g$ follows easily from the multiplicativity of the orbifold Euler characteristic for the orbifold covering $S_g \to \O$:
$$
\frac{2-2g}{n} = 2-2g_0 + r\left(\frac{1}{n}-1 \right) + \sum_{j=1}^s \left(\frac{1}{n_j} - 1 \right).
$$
Thus, $h$ gives a $(n,r)$-data set
$$
\operatorname{{\mathcal D}} = (n,g_0, (a_1,a_2,\ldots, a_r);(c_1,n_1),(c_2,n_2),\ldots, (c_s,n_s))
$$
of genus $g$, and hence $(\operatorname{{\mathcal D}}, \o_t)$ forms a permuting $(n,r,k)$-data set.
Consider another permuting $(n,r,k)$-action $t'$ in the equivalence class of
$t$ with a distinguished fixed point set $\operatorname{{\mathbb P}}(t') = \{P_1',P_2',\ldots,P_r'\}$. Then by definition there exists an orientation-preserving homeomorphism $\phi \in \operatorname{Mod}(S_g)$
such that $\phi(P_j) = P_j'$ for all $j$ and $\phi t\phi^{-1}$ is isotopic to $t'$ relative to $\operatorname{{\mathbb P}}(t')$. Therefore, $\theta_{P_j}(t) = \theta_{P_j'}(t')$, for $1\leq j\leq r$, and since $\o_t = \o_{t'}$, the two actions will produce the same permuting $(n,r,k)$-data sets.
Conversely, given a permuting $(n,r,k)$-data set $(\operatorname{{\mathcal D}}, \o_{\operatorname{{\mathcal D}}})$, we construct the orbifold $\O$ and a representation $\rho : \pi_1^{orb}(\O) \to C_n$. Any finite subgroup of $\pi_1^{orb}(\O)$ is conjugate to one of the cyclic subgroups generated by $\alpha_j$ or a $\gamma_i$, so condition (iv) in the definition of the data set ensures that the kernel of $\rho$ is torsion-free. Therefore, the orbifold covering $S \to \O$ corresponding to the kernel is a manifold, and calculation of the Euler characteristic shows that $S = S_g$. Thus we obtain a $C_n$-action $t$ on $S_g$ with $r$ distinguished fixed points $\operatorname{{\mathbb P}}(t)$. We now construct $\o_t$ from $\o_{\operatorname{{\mathcal D}}}$ in the following manner. For each pair $(p,m_p) \in \o_{\operatorname{{\mathcal D}}}$, write $p = (a,b)$. If $a=0$, then choose $m_p$ orbits of size $n$ (if $t$ is a free action, this choice is trivial, but otherwise, such an orbit would always exist in a small neighbourhood around any fixed point of $t$). If $a\neq 0$, then there exists a cone point in $\O$ of degree $b$, so there exists an orbit of $t$ of size $n/b$ in $S_g$. Once again, by considering a small neighbourhood of this orbit, we may choose $m_p$ distinct orbits $\{\o_p^1, \o_p^2, \ldots, \o_p^{m_p}\}$ and set
$$
\o(t) := \bigsqcup_{(p,m_p) \in \o_{\operatorname{{\mathcal D}}}} \{\o_p^1, \o_p^2, \ldots, \o_p^{m_p}\},
$$
which in turn gives $\o_t = \o_{\operatorname{{\mathcal D}}}$.
It remains to show that the resulting action on $S_g$ is determined upto our equivalence in $\operatorname{Mod}(S_g)$. Suppose that two permuting $(n,r,k)$-actions $t$ and $t'$ have the same permuting $(n,r,k)$-data set $(\operatorname{{\mathcal D}},\o_{\operatorname{{\mathcal D}}})$. $\operatorname{{\mathcal D}}$ encodes the fixed point data of the periodic transformation $t$, so by a result of J. Nielsen~\cite{N1} (or by a subsequent result of A. Edmonds~\cite[Theorem 1.3]{AE}), $t$ and $t'$ have to be conjugate by an orientation-preserving homeomorphism $\phi$. Let $\O'$ be the quotient orbifold of the action $t'$, and $\rho': \pi_1^{orb}(\O') \to C_n$ be the induced representations. Then $\phi$ induces a map $\varphi_{\#} : \pi_1^{orb}(\O) \to \pi_1^{orb}(\O')$ such that $\rho'\circ \varphi_{\#} = \rho$ as in~\cite[Theorem 2.1]{MK1}. If $\gamma$ is a loop around a cone point in $\O$, then $\rho(\gamma)$ is a loop around a cone point in $\O'$, and these cone points are associated to the same pair in $\operatorname{{\mathcal D}}$ since $\rho'(\varphi_{\#}(\gamma)) = \rho(\gamma)$. Hence, $\phi$ maps $\operatorname{{\mathbb P}}(t)$ to $\operatorname{{\mathbb P}}(t')$ and $\o(t)$ to $\o(t')$ as in Definition \ref{defn:eq_perm_actions}. Furthermore, $\o_t = \o_{\operatorname{{\mathcal D}}} = \o_{t'}$ by construction, and hence the permuting data set determines $t$ upto equivalence.
\end{proof}
\section{Nonseparating multicurves}
\label{sec:nonsepmulcurves}
Recall that a multicurve $\operatorname{{\mathcal{C}}}$ is said to be nonseparating if it does not contain any pseudo-nonseparating submulticurves. In this section, we establish that a root of $t_{\operatorname{{\mathcal{C}}}}$ corresponds to a special kind of permuting action on the connected surface $S_g(\operatorname{{\mathcal{C}}})$.
\begin{defn}\label{defn:orbit_equiv_sep}
Let $t_i$ be a permuting $(n_i,r_i,k_i)$-action on $S_{g_i}$ for $i =1,2$. Two orbits $\o_i \in \o(t_i)$ are said to be \textit{equivalent} (in symbols, $\o_1 \sim \o_2$) if
\begin{enumerate}[(i)]
\item $|\o_1| = |\o_2|$, and
\item if $|\o_1| < n := \operatorname{lcm}(n_1,n_2)$, then we further require that $$\theta_{\o_1}(t_1) + \theta_{\o_2}(t_2) \equiv 2\pi/n \pmod{2\pi}.$$
\end{enumerate}
\end{defn}
\noindent In this section, we will only need the case when $t_1 = t_2$, but we will need the general case in Section~\ref{sec:sepmulcurves}.
\begin{defn}
\label{def:nesnl2m}
Let $\operatorname{{\mathcal{C}}}$ be a nonseparating multicurve in $S_g$. A permuting $(n,2r,2k)$-action $t$ on $S_g(\operatorname{{\mathcal{C}}})$ is said to be \textit{nonseparating with respect to $\operatorname{{\mathcal{C}}}$} if there exists a $(r,k)$-partition $\operatorname{{\mathbb P}}_{r,k}(\operatorname{{\mathcal{C}}})$ of $\operatorname{{\mathcal{C}}}$ such that
\begin{enumerate}[(i)]
\item there exists $r$ mutually disjoint pairs $\{P_i,P_i'\}$ of distinguished fixed points in $\operatorname{{\mathbb P}}(t)$ such that $\theta_{P_i}(t) + \theta_{P_i'}(t) = 2\pi/n$ modulo $2\pi$, for $1 \leq i \leq r$, and
\item there exists $k$ mutually disjoint pairs $\{\o_i,\o_i'\}$ of distinguished nontrivial orbits in $\o_t$ such that $\o_i\sim \o_i'$, for $1 \leq i \leq k$, and \item $S(\operatorname{{\mathbb P}}_{r,k}) = S(t)$.
\end{enumerate}
\end{defn}
\begin{thm}
\label{thm:actions-nonseproots}
Let $\operatorname{{\mathcal{C}}}$ be a nonseparating multicurve in $S_g$. Then for $n \geq 1$, equivalence classes of permuting $(n,2r,2k)$-actions on $S_g(\operatorname{{\mathcal{C}}})$ that are nonseparating with respect to $\operatorname{{\mathcal{C}}}$ correspond to the conjugacy classes in $\operatorname{Mod}(S_g)$ of $(r,k)$-permuting roots of $t_{\operatorname{{\mathcal{C}}}}$ of degree $n$.
\end{thm}
\begin{proof}
First, we shall prove that a conjugacy class of an $(r,k)$-permuting root $h$ of $t_{\operatorname{{\mathcal{C}}}}$ of degree $n$ yields an equivalence class of a permuting $(n,2r,2k)$-action that is nonseparating with respect to $\operatorname{{\mathcal{C}}}$. We assume without loss of generality that $r,k > 0$, with the implicit understanding that, when either of them is zero, the corresponding arguments may be disregarded.
Let $\operatorname{{\mathcal{C}}}_{r,k}(h) = \{\operatorname{{\mathcal{C}}}_1', \operatorname{{\mathcal{C}}}_2',\ldots, \operatorname{{\mathcal{C}}}_r', \operatorname{{\mathcal{C}}}_1,\operatorname{{\mathcal{C}}}_2,\ldots, \operatorname{{\mathcal{C}}}_k\}$ be the $(r,k)$-partition of $\operatorname{{\mathcal{C}}}$ associated with $h$. Choose a closed tubular neighborhood $N$ of $\operatorname{{\mathcal{C}}}$, and consider $S_g(\operatorname{{\mathcal{C}}})$ as in Definition \ref{defn:sgc}. By isotopy, we may assume that $t_{\operatorname{{\mathcal{C}}}}(\operatorname{{\mathcal{C}}})=\operatorname{{\mathcal{C}}}$, $t_{\operatorname{{\mathcal{C}}}}(N)=N$, and $t_{\operatorname{{\mathcal{C}}}}\vert_{\widehat{S_g(\operatorname{{\mathcal{C}}})}}=\text{id}_{\widehat{S_g(\operatorname{{\mathcal{C}}})}}$. Suppose that $h$ is a root of $t_{\operatorname{{\mathcal{C}}}}$ of degree $n$, then by Remark \ref{rem:root_isotopy}, we may assume that $h$ preserves $\mathcal{C}$ and takes $N$ to $N$.
By the Nielsen-Kerckhoff theorem~\cite{K1}, $\hat{t} := h\vert_{\widehat{S_g(\operatorname{{\mathcal{C}}})}}$ is isotopic to a homeomorphism whose $n^{th}$ power is $\text{id}_{\widehat{S_g(\operatorname{{\mathcal{C}}})}}$. So we may change $h$ by isotopy so that $\hat{t}^n = \text{id}_{\widehat{S_g(\operatorname{{\mathcal{C}}})}}$. We fill in the $2m$ boundary circles of $\widehat{S_g(\operatorname{{\mathcal{C}}})}$ with disks and extend $\hat{t}$ to a homeomorphism $t$ on $S_g(\operatorname{{\mathcal{C}}})$ by coning. Thus $t$ defines a effective $C_n$-action on $S_g(\operatorname{{\mathcal{C}}})$, where $C_n=\langle t\;\vert\;t^n=1\rangle$.
The $C_n$-action $t$ fixes the centers $P_i$ and $P_i'$ of the $2r$ disks $D_i$ and $D_i'$, $1 \leq i \leq r$, of $\overline{S_g\setminus \widehat{S_g(\operatorname{{\mathcal{C}}})}}$ whose boundaries are the components of $\partial N$ which are fixed by $t$. The orientation of $S_g$ determines one for $S_g(\operatorname{{\mathcal{C}}})$, so we may speak of directed angles of rotation about the centres of these disks. Since $h^n = t_{\operatorname{{\mathcal{C}}}}$, it follows from \cite[Theorem 2.1]{MK1} that
$$
\theta_{P_i}(t) + \theta_{P_i'}(t) = 2\pi/n \pmod{2\pi},
$$
as illustrated in Figure~\ref{fig:twist} below.
\begin{figure}[h]
\labellist
\tiny
\pinlabel $P_i$ at 75 80
\pinlabel $P_i'$ at 270 115
\pinlabel $\theta_{P_i}(t)$ at 85 140
\pinlabel $\theta_{P_i'}(t)$ at 295 52
\endlabellist
\centering
\includegraphics[width = 50 ex]{pq}
\caption{Angle compatibility at each pair $\{P_i,P_i'\} \subset \operatorname{{\mathbb P}}(t)$.}
\label{fig:twist}
\end{figure}
The remaining disks occuring in $\overline{S_g \setminus \widehat{S_g(\operatorname{{\mathcal{C}}})}}$ form $k$ pairs of orbits of sizes $m_1, m_2,\ldots, m_k$ where $S(\operatorname{{\mathcal{C}}}_{r,k}(h)) = \llbrace m_1,m_2,\ldots, m_k\rrbrace$. For $1 \leq j \leq k$, we denote the centres of these pairs of disks by $Q_{i,j}$ and $Q_{i,j}'$, and the orbits of these centres by $\o_j$ and $\o_j'$. Thus $t$ is a permuting $(n,2r,2k)$-action with $\operatorname{{\mathbb P}}(t) = \{P_1,P_1',\ldots,P_r,P_r'\}$ and $\o(t) = \{\o_1,\o_1',\ldots,\o_k,\o_k'\}$.
It remains to show that $\o_i \sim \o_i'$ for each $i$. By construction, $|\o_i|=|\o_i'| = m_i$, and if $m_i =n$, then $\o_i\sim \o_i'$ holds trivially. If not, then we need to show that the angle compatibility condition from Definition \ref{def:orbit_equiv} holds. Write $\o_i= \{Q_{i,1},Q_{i,2},\ldots, Q_{i,m_i}\}$, $\o_i' = \{Q_{i,1}',Q_{i,2}',\ldots, Q_{i,m_i}'\}$ and note that $h^{m_i}$ is an $(n/m_i)^{\text{th}}$ root of $t_{\operatorname{{\mathcal{C}}}}$ such that $h^{m_i}(c_{i,1}) = c_{i,1}$. Hence,
$$
\theta_{Q_{i,1}}(t^{m_i}) + \theta_{Q_{i,1}'}(t^{m_i}) = 2\pi/(n/m_i) \pmod{2\pi},
$$
which implies that
$$
m_i\theta_{\o_i}(t) + m_i\theta_{\o_i'}(t) \equiv 2\pi/(n/m_i) \pmod{2\pi}.
$$
Since $t^n$ is a restriction of a single Dehn twist, it follows that
$$
\theta_{\o_i}(t) + \theta_{\o_i'}(t) \equiv 2\pi/n \pmod{2\pi}.
$$
Hence $\o_i \sim \o_i'$, and we obtain a permuting $(n,2r,2k)$-action $t$ on $S_g(\operatorname{{\mathcal{C}}})$ that is nonseparating with respect to $\operatorname{{\mathcal{C}}}$.
Now suppose $h_1, h_2\in \operatorname{Mod}(S_g)$ are two roots of $t_{\operatorname{{\mathcal{C}}}}$ that are conjugate in $\operatorname{Mod}(S_g)$ via $\Phi \in \operatorname{Mod}(S_g)$, and let $t_s$ denote the finite order homeomorphisms on $S_g(\operatorname{{\mathcal{C}}})$ induced by $h_s$, for $s = 1,2$. Then $t_{\operatorname{{\mathcal{C}}}} = \phi t_{\operatorname{{\mathcal{C}}}}\Phi^{-1} = t_{\Phi(\operatorname{{\mathcal{C}}})}$, so we may assume upto isotopy that $\Phi(\operatorname{{\mathcal{C}}}) = \operatorname{{\mathcal{C}}}$ (as in Remark \ref{rem:root_isotopy}) and that $\Phi(N) = N$. We extend $\Phi\mid_{\widehat{S_g(\operatorname{{\mathcal{C}}})}}$ to an element $\phi \in \operatorname{Mod}(S_g(\operatorname{{\mathcal{C}}}))$ by coning. Now, $\phi$ maps $\operatorname{{\mathbb P}}(t_1)$ to $\operatorname{{\mathbb P}}(t_2)$, and $\o(t_1)$ to $\o(t_2)$ bijectively as in Definition~\ref{defn:eq_perm_actions}. Since the $h_s$ and $\Phi$ all preserve $N$, $\phi t_1\phi^{-1}$ is isotopic to $t_2$ preserving $\operatorname{{\mathbb P}}(t_2)$ and $\o(t_2)$. Furthermore, for each $\o \in \o(t_1)$, $p(\phi(\o)) = p(\o)$. Hence, $\o_{t_1} = \o_{t_2}$ and so $t_1$ and $t_2$ will be equivalent permuting $(n,2r,2k)$-actions.
Conversely, given a permuting $(n,2r,2k)$-action $t$ on $S_g(\operatorname{{\mathcal{C}}})$ that is nonseparating with respect to $\operatorname{{\mathcal{C}}}$, we can reverse the argument to produce the $(r,k)$-permuting root $h$. Let $\operatorname{{\mathbb P}}(t) = \{P_1, P_1', \ldots, P_r, P_r'\}$ and $\o(t) = \{\o_1,\o_1', \ldots, \o_k,\o_k'\}$. For $1\leq i\leq r$, we remove disks $D_i$ and $D_i'$ invariant under the action of $t$ around the $P_i$ and $P_i'$ and attaching $r$ annuli to obtain the surface $S_g(\operatorname{{\mathcal{C}}}\setminus \operatorname{{\mathcal{C}}}')$. The condition on the angles $\{\theta_{P_i}(t), \theta_{P_i'}(t)\}$ ensures that the rotation angles work correctly to allow an extension of $t$ to obtain an $h_0$ with $h_0^n = t_{\operatorname{{\mathcal{C}}}'}$ in $\operatorname{Mod}(S_g(\operatorname{{\mathcal{C}}}\setminus \operatorname{{\mathcal{C}}}'))$, where $\operatorname{{\mathcal{C}}}'= \cup_{i=1}^r \operatorname{{\mathcal{C}}}_i'$.
Now write $\o_1 = \{Q_{1,1}, Q_{1,2}, \ldots, Q_{1,m_1}\}$ and consider disks $D_{1,i}$ around $Q_{1,i}$ such that $t(D_{1,i}) = t(D_{1,i+1})$. Similarly, write $\o_1'=\{Q_{1,1}',Q_{1,2}',\ldots, Q_{1,m_1}'\}$ and consider disks $D_{1,i}'$ as earlier. Then we attach $m_1$ annuli connecting $\partial D_{1,i}$ to $\partial D_{1,i}'$. Each such annulus contains a nonseparating curve $c_{1,i}$, which is unique unto isotopy. Repeating this process for $1\leq i\leq m_1$, we obtain the surface $S_g(\operatorname{{\mathcal{C}}}\setminus (\operatorname{{\mathcal{C}}}'\cup\operatorname{{\mathcal{C}}}_1))$. Since $t(D_{1,i}) = D_{1,i+1}$, we may extend the homeomorphism $h_0$ to a homeomorphism $\widetilde{h_0} \in \operatorname{Mod}(S_g(\operatorname{{\mathcal{C}}}'\cup \operatorname{{\mathcal{C}}}_1))$, which cyclically permutes the $c_{1,i}$. If $|\o_1| = |\o_1'| = n$, then define $h_1 := \widetilde{h_0}t_{c_{1,1}}$. Otherwise, since $\o_1 \sim \o_1'$, the difference in the turning angles around $Q_{1,i}$ and $Q_{1,i}'$ is $2\pi/n$. Let $\widetilde{h_1}$ be the $(1/n)^{th}$-twist around $c_{1,1}$. Now $h_1 := \widetilde{h_0}\widetilde{h_1}$ is an $(r,1)$-permuting root of $t_{\operatorname{{\mathcal{C}}}' \cup \operatorname{{\mathcal{C}}}_1}$ of degree $n$ in $\operatorname{Mod}(S_g(\operatorname{{\mathcal{C}}} \setminus (\operatorname{{\mathcal{C}}}' \cup \operatorname{{\mathcal{C}}}_1)))$. We now repeat this process inductively to obtain an $(r,k)$-permuting root $h:= h_k\in \operatorname{Mod}(S_g)$ of $t_{\operatorname{{\mathcal{C}}}}$ of degree $n$.
It remains to show that the resulting root $h$ of $t_{\operatorname{{\mathcal{C}}}}$ is determined up to conjugacy. Suppose $t_1$ and $t_2$ are two equivalent $(n,2r,2k)$-actions on $S_g(\operatorname{{\mathcal{C}}})$ that are nonseparating with respect to $\operatorname{{\mathcal{C}}}$ with $\operatorname{{\mathbb P}}(t_s) = \{P_{s,1},P_{s,2},\ldots, P_{s,r}\} \text{ and }\o(t_s) = \{\o_{s,1},\o_{s,2},\ldots, \o_{s,k}\}$, for $s=1,2$. Let $\phi \in \operatorname{Mod}(S_g(\operatorname{{\mathcal{C}}}))$ be an orientation-preserving homeomorphism satisfying the conditions in Definition \ref{defn:eq_perm_actions}. Then repeating the argument from \cite[Theorem 2.1]{MK1}, $\phi$ extends to a homeomorphism $\Phi_0 \in \operatorname{Mod}(S_g(\operatorname{{\mathcal{C}}}\setminus \operatorname{{\mathcal{C}}}'))$ such that $\Phi_0 h_{1,0} \Phi_0^{-1} = h_{2,0}$, where $h_{s,0}$ is the root of $t_{\operatorname{{\mathcal{C}}}'}$ obtained from $t_s$, for $s=1,2$, as above. Furthermore, since $\phi$ maps $\o_{1,i}$ to $\o_{2,i}$ as in Definition \ref{defn:eq_perm_actions}, we may once again extend $\Phi_0$ to a homeomorphism $\Phi \in \operatorname{Mod}(S_g)$ satisfying $\Phi h_1 \Phi^{-1} = h_2$, where $h_s$ is the root of $t_{\operatorname{{\mathcal{C}}}}$ obtained from $t_s$, for $s=1,2$.
\end{proof}
\begin{defn}
\label{def:nonsep-dataset}
A permuting $(n,2r,2k)$-data set $(\operatorname{{\mathcal D}},\o_{\operatorname{{\mathcal D}}})$ is called \emph{nonseparating} if
\begin{enumerate}[(i)]
\item
$\displaystyle \operatorname{{\mathcal D}} = (n,g_0,(a_1,a_1',\ldots,a_r,a_r');(c_1,n_1),\ldots,(c_s,n_s)),$
where \\ $a_i + a_i' \equiv a_ia_i' \bmod n,$ for $1 \leq i \leq r$
\item For each $(p,m_p) \in \o_{\operatorname{{\mathcal D}}}$, there exists $(p',m_p') \in \o_{\operatorname{{\mathcal D}}}$ such that
$m_p = m_p'$ and
$$
\theta(p) + \theta(p') = \begin{cases}
0 &: p = p' = (0,1) \\
2\pi/n \pmod{2\pi} &: \text{otherwise.}
\end{cases}
$$
\end{enumerate}
\end{defn}
\noindent We now deduce the following theorem from Theorems \ref{thm:action_triple_correspondence} and \ref{thm:actions-nonseproots}.
\begin{thm}
Let $\operatorname{{\mathcal{C}}}$ be a nonseparating multicurve in $S_g$. For $n\geq 1$, equivalence classes of nonseparating permuting $(n,2r,2k)$-data sets of genus $g_{\operatorname{{\mathcal{C}}}}$ correspond to equivalence classes of permuting $(n,2r,2k)$-actions on $S_g(\operatorname{{\mathcal{C}}})$ that are nonseparating with respect to $\operatorname{{\mathcal{C}}}$.
\end{thm}
\begin{cor}\label{cor:nonsepdsets-nonseproots}
Let $\operatorname{{\mathcal{C}}}$ be a nonseparating multicurve in $S_g$. For $n\geq 1$, equivalence classes of nonseparating permuting $(n,2r,2k)$-data sets of genus $g_{\operatorname{{\mathcal{C}}}}$ correspond to conjugacy classes in $\operatorname{Mod}(S_g)$ of $(r,k)$-permuting roots of $t_{\operatorname{{\mathcal{C}}}}$ of degree $n$.
\end{cor}
If $\operatorname{{\mathcal{C}}}$ is a nonseparating multicurve of size $m$, then $g_{\operatorname{{\mathcal{C}}}} = g-m$. Hence we obtain the following restrictions on the degree of a root of $t_{\operatorname{{\mathcal{C}}}}$.
\begin{cor}
\label{cor:bound_nonsep}
Let $\operatorname{{\mathcal{C}}}$ be a nonseparating multicurve in $S_g$ of size $m$, and let $h$ be an $(r,k)$-permuting root of $t_{\operatorname{{\mathcal{C}}}}$ of degree $n$.
\begin{enumerate}[(i)]
\item If $r\geq 0$, then
$$
n\leq \begin{cases}
4(g-m) + 2, &\text{if } g-m \geq 1 \\
g, &\text{if } g = m.
\end{cases}
$$
Furthermore, if $g=m$, then this upper bound is realizable.
\item If $r\geq 1$, then $n$ is odd.
\item If $r=1$, then $n\leq 2(g-m)+1$.
\item If $r\geq 2$, then $\displaystyle n \leq \frac{g-m+r-1}{r-1}$.
\end{enumerate}
\end{cor}
\begin{proof}
If $r\geq 0$ and $g-m\geq 1$, then by a result of Wiman \cite[Theorem 6]{H1}, the highest order of a cyclic action on $S_g(\operatorname{{\mathcal{C}}})$ is $4(g-m) + 2$. If $r\geq 0$ and $g=m$, then consider the permuting $(n,2r,2k)$-action $t$ on $S_0$ that is nonseparating with respect to $\operatorname{{\mathcal{C}}}$ guaranteed by Theorem \ref{thm:actions-nonseproots}. Since $t \in \operatorname{Mod}(S_0)$ is an element of order $n$, it must be a rotation by $2\pi/n$ radians. Since the two fixed points of this action are not compatible in the sense of Definition \ref{def:nesnl2m}, $r=0$ and every non-trivial orbit has size $n$. Hence, $m = nk$ and so $n\mid m$, and in particular, $n\leq m=g$. Furthermore, if $m = g$, let $t$ be the rotatiuon of the sphere by $2\pi/n$, and $c \in \operatorname{{\mathcal{C}}}$ be any curve. Then $h := tt_c$ is a root of $t_{\operatorname{{\mathcal{C}}}}$ of degree $m$ and (i) follows.
(ii) and (iii) follow from \cite[Corollary 2.2]{MK1}. If $r\geq 2$, then consider the corresponding permuting $(n,2r,2k)$-data set from Theorem \ref{thm:action_triple_correspondence}. Note that the genus $(g-m)$ of the permuting data set is given by
$$\frac{2-2(g-m)}{n} = 2-2g_0 + 2r\left(\frac{1}{n}-1 \right) + \sum_{j=1}^s \left(\frac{1}{n_j} - 1 \right).$$ Since $(1-1/n_j) \geq 0$ and $g_0 \geq 0,$ we have $(g-m) - 1 +r \geq n(-1+r),$ from which (iv) follows.
\end{proof}
\section{Separating Multicurves}
\label{sec:sepmulcurves}
A separating multicurve $\operatorname{{\mathcal{C}}}$ in $S_g$ is the disjoint union of finitely many pseudo-nonseparating curves. In this case $S_g(\operatorname{{\mathcal{C}}})$ is disconnected, so we will require multiple finite order actions on the individual components of $S_g(\operatorname{{\mathcal{C}}})$ to come together to form a root of $t_{\operatorname{{\mathcal{C}}}}$ on $S_g$. To improve the exposition, we begin with the case $\operatorname{{\mathcal{C}}} = \operatorname{{\mathcal{C}}}^{(m)}$ so that $S_g(\operatorname{{\mathcal{C}}})$ has two components.
\begin{defn}\label{defn:mpair_permact}
Equivalence classes $\l t_i\r$ of permuting $(n_i,r_i,k_i)$-actions on $S_{g_i}$, for $i = 1,2$, are said to form an \textit{(r,k)-compatible pair $(\l t_1 \r, \l t_2 \r)$ of degree $n$} for integers $r,k \geq 0$, if
\begin{enumerate}[(i)]
\item $n = \operatorname{lcm}(n_1,n_2)$, and
\item there exists $\{P_{i,1}, P_{i,2},\ldots, P_{i,r}\} \subset \operatorname{{\mathbb P}}(t_i)$ such that for $1\leq j\leq r$,
$$
\displaystyle \theta_{P_{1,j}}(t_1) + \theta_{P_{2,j}}(t_2) = \frac{2\pi}{n} \pmod{2\pi} \text{, and }
$$
\item there exists $\{\o_{i,1},\o_{i,2},\ldots, \o_{i,k}\} \subset \o(t_i)$ for $i=1,2$ such that $\o_{1,j} \sim \o_{2,j}$ as in Definition~\ref{defn:orbit_equiv_sep}, for $1\leq j\leq k$.
\end{enumerate}
If
$$
\alpha := \sum_{j=1}^k |\o_{1,j}| = \sum_{j=1}^k |\o_{2,j}|
$$
then the number $g := g_1+g_2+r+\alpha - 1$ is called the \emph{genus} of the pair $(\l t_1\r, \l t_2\r)$. Note that two actions are $(1,0)$-compatible if they are compatible as nestled actions in the sense of \cite[Definition 3.2]{KR1}.
\end{defn}
\begin{notation}
Let $(\l t_1\r, \l t_2\r)$ be an $(r,k)$-compatible pair of degree $n$ as in Definition \ref{defn:mpair_permact}. We write $\operatorname{{\mathbb P}}(t_1,t_2):= \{P_{1,1},P_{1,2},\ldots, P_{1,r}\}$ and $\o(t_1,t_2):= \{\o_{1,1},\o_{1,2},\ldots, \o_{1,k}\}$. We define $\operatorname{{\mathbb P}}(t_2,t_1)$ and $\o(t_2,t_1)$ similarly.
\end{notation}
\begin{lemma}\label{lem:powers_of_twists}
Let $\operatorname{{\mathcal{C}}}=\{c_1,c_2,\ldots, c_m\}$ be a multicurve on $S_g$, and $N_i$ be annular neighbourhood of $c_i$. Write $N = \sqcup_{i=1}^m N_i$, and suppose $t\in \operatorname{Mod}(S_g)$ is such that $t\lvert_{S_g\setminus N} = \text{id}_{S_g\setminus N}$, then $\exists d_1,d_2,\ldots, d_m\in \mathbb{N}\cup\{0\}$ such that $t = t_{c_1}^{d_1}\ldots t_{c_m}^{d_m}$.
\end{lemma}
\begin{proof}
Since $t\lvert_{S_g\setminus N} = \text{id}_{S_g\setminus N}$, it follows that $t$ fixes $\partial N$. The lemma now follows from the fact \cite[Lemma 4.1.A, Chapter 12]{DS} that $$\operatorname{Mod}(N\text{fix}(\partial N)) \cong \oplus_{i=1}^m \operatorname{Mod}(N_i\text{fix}(\partial N_i)) = \oplus_{i=1}^m \langle t_{c_i}\rangle.$$
\end{proof}
\begin{thm}\label{thm:homologous_root_action}
Suppose that $S_g = S_{g_1}\#_{\operatorname{{\mathcal{C}}}} S_{g_2}$, where $\operatorname{{\mathcal{C}}} = {\operatorname{{\mathcal{C}}}}^{(m)}$. Then $(r,k)$ -permuting roots of $t_{\operatorname{{\mathcal{C}}}}$ of degree $n$ correspond to the $(r,k)$-compatible pairs $(\l t_1\r, \l t_2\r)$ of equivalence classes of permuting $(n_i,r_i,k_i)$-actions on the $S_{g_i}$, of degree $n$.
\end{thm}
\begin{proof}
As before, we assume $m>1$ and first show that every $(r,k)$-permuting root $h$ of $t_{\operatorname{{\mathcal{C}}}}$ of degree $n$ yields a compatible pair $(\l t_1 \r, \l t_2 \r)$ of degree $n$. Consider the $(r,k)$-partition $\operatorname{{\mathcal{C}}}_{r,k}(h) = \{\operatorname{{\mathcal{C}}}_1',\ldots \operatorname{{\mathcal{C}}}_r',\operatorname{{\mathcal{C}}}_1,\ldots, \operatorname{{\mathcal{C}}}_k\}$ of $\operatorname{{\mathcal{C}}}$ induced by $h$. Let $\widehat{S_{g_s}}$ for $s=1,2$ denote the two components of $\widehat{S_g(\operatorname{{\mathcal{C}}})}$. Let $N$ be a closed annular neighborhood of $\operatorname{{\mathcal{C}}}$. By isotopy, we may assume that
$t_{\operatorname{{\mathcal{C}}}}(\operatorname{{\mathcal{C}}}) = \operatorname{{\mathcal{C}}}$, $t_{\operatorname{{\mathcal{C}}}}(N) = N$ , and $t_{\operatorname{{\mathcal{C}}}}\vert_{\widehat{S_{g_s}}} = id_{\widehat{S_{g_s}}}$.
Putting $\widehat{t_s} = h\vert_{\widehat{S_{g_s}}}$, we may assume up to isotopy that
$\widehat{t_s}^n\vert_{\widehat{S_{g_s}}} = \text{id}_{\widehat{S_{g_s}}}$ for $s=1,2$.
Let $n_s$ be the smallest positive integer such that $\widehat{t_s}^{n_s}=\text{id}_{\widehat{S_{g_s}}}$ for $s=1,2$, and let $q=\operatorname{lcm}(n_1,n_2)$. Then $t:= h^q$ satisfies the hypotheses of Lemma \ref{lem:powers_of_twists}. Hence, $\exists d_c \in \mathbb{N}\cup\{0\}$ such that
$$
h^q = \prod_{c\in \operatorname{{\mathcal{C}}}} t_c^{d_c}
$$
Since $h^n\lvert_{\widehat{S_{g_1}}} = \text{id}_{\widehat{S_{g_1}}}$ it follows that $n_1\mid n$, and similarly $n_2\mid n$. Hence, $q\mid n$ and so
$$
\prod_{c\in \operatorname{{\mathcal{C}}}}t_c = t_{\operatorname{{\mathcal{C}}}} = (h^q)^{n/q} = \prod_{c\in \operatorname{{\mathcal{C}}}}t_c^{nd_c/q}
$$
Fix $c\in \operatorname{{\mathcal{C}}}$ and restrict the functions on both sides of this equation to a closed annular neighbourhood of $c$ disjoint from other curves in $\operatorname{{\mathcal{C}}}$. As in Remark \ref{rem:root_isotopy}, we see that $nd_c/q = 1$ and hence $n = q = \operatorname{lcm}(n_1,n_2)$.
We fill in $\partial \widehat{S_{g_s}}$ with disks to obtain the closed oriented surfaces $S_{g_s}$ for $s=1,2$. We then extend $\widehat{t_s}$ to the $S_{g_s}$ by coning. Thus $t_s$ defines an effective $C_{n_s}$-action on the $S_{g_s}$, where $n_s \mid n$ for $s =1,2$, and $n = \operatorname{lcm}(n_1,n_2)$.
When $r > 0$, the homeomorphism $t_s$ fixes the center points $\{P_{s,1},P_{s,2},\ldots, P_{s,r}\}$ of $r$ disks in $\overline{S_{g_s}\setminus \widehat{S_{g_s}}}$ for $s =1,2$. Hence we may write $\operatorname{{\mathbb P}}(t_1,t_2) = \{P_{1,1},P_{1,2},\ldots, P_{1,r}\}$ and $\operatorname{{\mathbb P}}(t_2,t_1) = \{P_{2,1},P_{2,2},\ldots, P_{2,r}\}$. For $1\leq j\leq r$, the proof of \cite[Theorem 3.4]{KR1} implies that the corresponding turning angles around $P_{1,j}$ and $P_{2,j}$ must be compatible in the sense of condition (ii) of Definition \ref{defn:mpair_permact}.
When $k > 0$, let $\operatorname{{\mathcal{C}}}_i = \{c_{i,1},c_{1,2},\ldots, c_{i,m_i}\}$, for $1 \leq i \leq k$. Associated with each curve $c_{i,j} \in \operatorname{{\mathcal{C}}}_i$, is a disk
$D_{i,j}^s$ in each $\overline{S_{g_s}\setminus \widehat{S_{g_s}}}$ for $s = 1,2$. The centers $Q_{i,j}^s$ of the $m_i$ disks $D_{i,j}^s$ for $1 \leq j \leq m_i$ form an orbit $\o_{s,i}$ in $S_{g_s}$ for $s = 1,2$. Thus we obtain a collection $\o_{t_s} = \{\o_{s,1}, \ldots , \o_{s,k}\}$ of $k$ distinguished nontrivial orbits on $S_{g_s}$ for $s = i,j$. It remains to show that $\o_{1,i} \sim \o_{2,i}$, for $1\leq i\leq k$, but the argument for this is similar to that of Theorem~\ref{thm:actions-nonseproots}. Hence, the pair $(\l t_1\r, \l t_2\r)$ forms an $(r,k)$-compatible pair of degree $n$.
The argument for the converse, and the fact that the resulting root is determined upto conjugacy, is analogous to that of Theorem~\ref{thm:actions-nonseproots}.
\end{proof}
\begin{defn}
\label{defn:doubpermdspair}
Permuting data sets $\widetilde{\operatorname{{\mathcal D}}_i} := (\operatorname{{\mathcal D}}_i,\o_{\operatorname{{\mathcal D}}_i})$ for $i=1,2$, where
$$\operatorname{{\mathcal D}}_i = (n_i, \widetilde{g_i}, (a_{i,1},\ldots, a_{i,r_i}); ((c_{i,1},y_{i,1}) ,\ldots, (c_{i,s_i},y_{i,s_i})),$$ are said to be \textit{$(r,k)$-compatible of degree $n$} for integers $r \geq 0$ and $k \geq 0$, if
\begin{enumerate}[(i)]
\item $n = \operatorname{lcm}(n_1,n_2)$,
\item $r \leq r_i$, for $i=1,2$, and for $1 \leq j \leq r$,
$$a_{1,j}+a_{2,j} \equiv a_{1,j}a_{2,j} \pmod n,$$
\item there exists $S_i \subseteq \o_{\operatorname{{\mathcal D}}_i}$ such that
\begin{enumerate}
\item
$$
k = \displaystyle \sum_{(p,m_p) \in S_1} m_p = \sum_{(q,m_q) \in S_2} m_q,
$$
\item for every $(p,m_p) \in S_1$, there exists $(q,m_q) \in S_2$ such that
$$
\theta(p)+\theta(q) \equiv
\begin{cases}
0 &: \text{ if } p = q = (0,1) \\
2\pi/n \pmod{2\pi} &: \text{ otherwise}
\end{cases}
$$
\end{enumerate}
\end{enumerate}
If $$\alpha := \sum_{((a,b),m_p)\in S_1} n_1/b = \sum_{((c,d),m_q) \in S_2} n_2/d$$ and
if $g_i$ is the genus of $\operatorname{{\mathcal D}}_i$, then the number $g = g_1+g_2+r+\alpha-1$ is called \textit{genus} of the pair
$(\widetilde{\operatorname{{\mathcal D}}_1},\widetilde{\operatorname{{\mathcal D}}_2})$.
\end{defn}
\noindent The next theorem and its corollary follow directly from Theorems~\ref{thm:action_triple_correspondence} and \ref{thm:homologous_root_action}.
\begin{thm}
\label{thm:mcomp_per_pairs-mcompdspairs}
Suppose that $S_g = S_{g_1}\#_{\operatorname{{\mathcal{C}}}} S_{g_2}$, where $\operatorname{{\mathcal{C}}} = {\operatorname{{\mathcal{C}}}}^{(m)}$. Then $(r,k)$-compatible pairs $(\l t_1\r, \l t_2\r)$ of equivalence classes of permuting $(n_i,r_i,k_i)$-actions on the $S_{g_i}$ correspond to $(r,k)$-compatible pairs $(\widetilde{\operatorname{{\mathcal D}}_1}, \widetilde{\operatorname{{\mathcal D}}_2})$ of data sets of genus $g$, where $\widetilde{D_i}$ is a permuting $(n_i,r_i,k_i)$-data set of genus $g_i$.
\end{thm}
\begin{cor}
Suppose that $S_g = S_{g_1}\#_{\operatorname{{\mathcal{C}}}} S_{g_2}$, where $\operatorname{{\mathcal{C}}} = {\operatorname{{\mathcal{C}}}}^{(m)}$. Then conjugacy classes of $(r,k)$-permuting roots of $t_{\operatorname{{\mathcal{C}}}}$ of degree $n$ correspond to $(r,k)$-compatible pairs $(\widetilde{\operatorname{{\mathcal D}}_1},\widetilde{\operatorname{{\mathcal D}}_2})$ of degree $n$ and genus $g$, where $\widetilde{D_i}$ is a permuting $(n_i,r_i,k_i)$ data set of genus $g_i$.
\end{cor}
We now consider the case of a $(0,k')$-permuting root $h$ of $t_{\operatorname{{\mathcal{C}}}}$ where $\operatorname{{\mathcal{C}}} = \operatorname{{\mathcal{C}}}^{(k)}(m)$. Here, the restriction of $h$ to $S_g(\operatorname{{\mathcal{C}}})$ induces a non-trivial permutation of the components of $S_g(\operatorname{{\mathcal{C}}})$. Since $h$ is a homeomorphism, it maps one component to another of the same genus, thus inducing an action on each subsurface of the form $\S_g(m) \subset S_g(\operatorname{{\mathcal{C}}})$. Therefore, we generalize the notion of a permuting $(n,r,k)$-action to encompass the action on $\S_g(m)$ induced by $h$.
\begin{defn} \label{defn:permuting_action_sgm}
Fix integers $g \geq 0$ and $m \geq 1$.
\begin{enumerate}[(i)]
\item A homeomorphism $\sigma_m : \S_g(m) \to \S_g(m)$ is said to be \textit{essential} if for all $i$, $\sigma_m\vert_{S_g^i} (S_g^i)= S_g^{i+1}$.
\noindent An essential homeomorphism $\sigma_m$ on $\S_g(m)$ can be viewed simply as the $m$-cycle $(1,2,\ldots,m)$ permuting its $m$ components $S_g^i$.
\item An orientation-preserving $C_n$-action $t$ on $\S_g(m)$ is said to be a \emph{permuting $(n,r,k)$-action} if $m \mid n$ and there exists an essential homeomorphism $\sigma_m : \S_g(m) \to \S_g(m)$ such that $t = \sigma_m \circ \widetilde{t}$, where $\widetilde{t}$ is a permuting $(n,r,k)$-action on each $S_g^i$.
\end{enumerate}
\end{defn}
\begin{rem}\label{rem: permuting_action_sgm}
Suppose that $\widetilde{t} \in \operatorname{Mod}(S_g)$ defined a permuting $(n,r,k)$-action and $t = \sigma_m\circ \widetilde{t}$, then $t_i := t^m\lvert_{S_g^i} \in \operatorname{Mod}(S_g^i)$ defines a permuting $(n/m,\widetilde{r},\widetilde{k})$-action on $S_g^i$. Furthermore, all the $t_i$ are conjugate to each other via $\sigma_m$.
Conversely, if $t'\in \operatorname{Mod}(S_g)$ is a permuting $(n/m, \widetilde{r},\widetilde{k})$-action on $S_g$ which has an $m^{th}$ root $\widetilde{t} \in \operatorname{Mod}(S_g)$, then the map $t := \sigma_m\circ\widetilde{t}$ defines a permuting $(n,r,k)$-action on $\S_g(m)$. Thus, a permuting $(n,r,k)$-action on $\S_g(m)$ corresponds to a permuting $(n/m,\widetilde{r},\widetilde{k})$-action on $S_g$ which has an $m^{th}$ root in $\operatorname{Mod}(S_g)$.
\end{rem}
\begin{defn}
Let $t_1$ and $t_2$ be two permuting $(n,r,k)$-actions on $\S_g(m)$.
Then we say $t_1$ is \emph{equivalent} to $t_2$ if ${t_1}^m\vert_{S_g^i}$ and ${t_2}^m\vert_{S_g^i}$ are equivalent as permuting $(n/m,\widetilde{r},\widetilde{k})$-actions on $S_g^i$ in the sense of Definition~\ref{defn:eq_perm_actions}.
\end{defn}
\begin{defn}
\label{defn:uv_comp_pair}
Let $u,v\geq 0$ be fixed integers, $t_1$ be a permuting $(n_1,r_1,k_1)$-action on $S_{g_1}$ and $t_2$ be a permuting $(n_2,r_2,k_2)$-action on $\S_{g_2}(m)$ such that $t_2 = \sigma_m \circ \widetilde{t_2}$ as in Remark \ref{rem: permuting_action_sgm}. Then the equivalence classes $(\l t_1\r, \l t_2\r)$ are said to form a \textit{(u,v)-compatible pair} if
\begin{enumerate}[(i)]
\item for $1 \leq i \leq m$, $(\l {t_1}^m \r, \l t_2^m\vert_{S_{g_2}^i} \r)$ is an $(u,v)$-compatible pair of degree $n/m$, where $n=\operatorname{lcm}(n_1,n_2)$,
\item for $1\leq i\leq m, 1\leq j \leq u,$ there mutually disjoint pairs $\{P_{i,j}^1,P_{i,j}^2\}$, where $P_{i,j}^1 \in \operatorname{{\mathbb P}}(t_1^m)$ and $P_{i,j}^2 \in \operatorname{{\mathbb P}}(t_2^m\vert_{S_{g_2}^i})$ such that
$$
\theta_{P_{i,j}^1}(t_1) + \theta_{P_{i,j}^2}(\widetilde{t_2}) = 2\pi/n \pmod{2\pi}, \text{ and }
$$
\item for $1\leq i\leq m, 1\leq j \leq v,$ mutually disjoint pairs $\{\o_{i,j}^1, \o_{i,j}^2\}$, where $\o_{i,j}^1 \in \o(t_1)$ and $\o_{i,j}^2 \in \o(\widetilde{t_2})$, such that
$\o_{i,j}^1 \sim \o_{i,j}^2$, as in Definition~\ref{defn:orbit_equiv_sep}.
\end{enumerate}
The number $n = \operatorname{lcm}(n_1,n_2)$ is called the \textit{degree} of the pair, and the number $g = g_1 + m(g_2+k-1)$ is called the \textit{genus} of the pair.
\end{defn}
\begin{notation}
Let $(\l t_1\r, \l t_2\r)$ be a $(u,v)$-compatible pair of degree $n$ as in Definition \ref{defn:uv_comp_pair}. We write
\begin{enumerate}[(i)]
\item $\operatorname{{\mathbb P}}(t_1, t_2) = \{P_{i,j}^1 : 1\leq i\leq m, 1\leq j\leq u\}$ and $\operatorname{{\mathbb P}}(t_2,t_1) = \{P_{i,j}^2 : 1\leq i\leq m, 1\leq j\leq u\}$.
\item Similarly, we define $\o(t_1,t_2) = \{\o_{i,j}^1 : 1\leq i\leq m, 1\leq j\leq v\}$ and $\o(t_1,t_2) = \{\o_{i,j}^2 : 1\leq i\leq m, 1\leq j\leq v\}$.
\end{enumerate}
\end{notation}
\begin{lemma}\label{lem:single_orbit_actions}
Suppose that $\operatorname{{\mathcal{C}}} = \operatorname{{\mathcal{C}}}^{(k)}(m)$ be a multicurve in $S_g$. Then conjugacy classes of $(0,k')$-permuting roots of $t_{\operatorname{{\mathcal{C}}}}$ of degree $n$ correspond to $(u,v)$-compatible pairs of degree $n$ and genus $g$, where $k'=u+v$.
\end{lemma}
\begin{proof}
Let $h\in\operatorname{Mod}(S_g)$ be a $(0,k')$-permuting root of degree $n$ on $S_g$. Then as in Theorem \ref{thm:homologous_root_action}, we obtain an effective $C_{n_1}$-action $t_1$ on $S_{g_1}$ and an effective $C_{n_2}$-action on $S_{g_2}(m)$, where $n=\operatorname{lcm}(n_1,n_2)$. Furthermore, $h$ restricts to an essential homeomorphism $\sigma_m : \S_g(m) \to \S_g(m)$ such that $\sigma_m\circ t_2 = t_2\circ \sigma_m$. Hence the maps
$$
t_{2,i}:= \sigma_m^{-1}t\lvert_{S_{g_2}^i} : S_{g_2}^i \to S_{g_2}^i
$$
are conjugate to each other when considered as elements of $\operatorname{Mod}(S_g)$, and so $t_2$ is a permuting $(n_2,r_2,k_2)$-action on $S_{g_2}(m)$ as in Definition \ref{defn:permuting_action_sgm}. Since $h^m$ is a root of $t_{\operatorname{{\mathcal{C}}}}$ of degree $(n/m)$ that preserves each $\operatorname{{\mathcal{C}}}^{(k)}_i$, we may restrict $h^m$ and cone to obtain maps
$$
h_i \in \operatorname{Mod}(\Sigma_i) \text{, where } \Sigma_i := S_{g_1}\#_{\operatorname{{\mathcal{C}}}_i^{(k)}}S_{g_2}^i,
$$
which are pairwise conjugate to each other via $h$. Thus, it follows from Theorem \ref{thm:homologous_root_action} that there exist integers $u,v\geq 0$ such that $(\l t_1^m\r, \l t_2^m\lvert_{S_{g_2}^i}\r)$ forms a $(u,v)$-compatible pair of degree $n/m$, for $1\leq i\leq m$.
By condition (ii) of Definition \ref{defn:mpair_permact}, there exists mutually disjoint pairs $\{P_{i,j}^1, P_{i,j}^2\}$, where $P_{i,j}^1 \in \operatorname{{\mathbb P}}(t_1^m)$ and $P_{i,j}^2 \in \operatorname{{\mathbb P}}(t_2^m\vert_{S_{g_2}^i})$, such that
$$
\theta_{P_{i,j}^1}(t_1^m) + \theta_{P_{i,j}^2}(\widetilde{t_2}^m) = \frac{2\pi}{(n/m)} \pmod{2\pi}.
$$
Once again, since $h^m$ is a root of $t_{\operatorname{{\mathcal{C}}}}$ it follows that
$$
\theta_{P_{i,j}^1}(t_1) + \theta_{P_{i,j}^2}(\widetilde{t_2}) = 2\pi/n \pmod{2\pi}.
$$
Similarly, one obtains condition (iii) of Definition \ref{defn:uv_comp_pair} as well. Note that every fixed point of $h^m$, and every orbit of $h^m$, induces one orbit of $h$, and thus $k'=u+v$. The converse is a just a matter of reversing this argument.
\end{proof}
We now consider the case of an $(r,k)$-permuting root of $t_{\operatorname{{\mathcal{C}}}}$ where $r,k>0$. We begin by writing $S_g$ as a connected sum of subsurfaces $S_{g_i}$ across those pseudo-nonseparating submulticurves which are preserved by $h$. The restriction of $h$ to each $S_{g_i}$ is then a $(0,k_i)$-permuting root of the Dehn twist about the submulticurve $\operatorname{{\mathcal{C}}}\cap S_{g_i}$. This allows us to apply Theorem \ref{lem:single_orbit_actions} to obtain finite order actions on $S_{g_i}$ such that pairs of actions on adjacent subsurfaces are compatible (in the sense of Theorem \ref{thm:homologous_root_action}).
\begin{notation}\label{not:rk_permuting_root}
Let $\operatorname{{\mathcal{C}}}$ be a separating multicurve in $S_g$, and let $h$ be an $(r,k)$-permuting root of $t_{\operatorname{{\mathcal{C}}}}$.
\begin{enumerate}[(i)]
\item We shall denote the set of all pseudo-nonseparating submulticurves of $\operatorname{{\mathcal{C}}}$ that are preserved by $h$ by $\operatorname{Fix}_h(\operatorname{{\mathcal{C}}}) = \{\operatorname{{\mathcal E}}_1,\ldots,\operatorname{{\mathcal E}}_{m(h)}\}$.
\item Writing
$$
S_g = \overline{\textbf{\Large \#}}_{i=1}^{m(h)} (S_{g_i} \#_{\operatorname{{\mathcal E}}_i} S_{g_{i+1}})
$$
and $D_i := \operatorname{{\mathcal{C}}} \cap S_{g_i}$, we have that for each $i$, $S_{g_i} \cap \operatorname{{\mathcal{C}}}_{r,k}(h)$ is a $(0,k_i)$-partition of $D_i$, which has the form
$$
S_{g_i} \cap \operatorname{{\mathcal{C}}}_{r,k}(h) = \{ \operatorname{{\mathcal{C}}}^{(k_{i,j})}_{i,j}(m_{i,j})\,:\, 1 \leq j \leq k_i\}.$$
\item For $1 \leq i \leq r+1$, we write
$$
S_{g_i} = \overline{\textbf{\Large \#}}_{j=1}^{k_i} (S_{g_{i,1}} \#_{D_{i,j}} \S_{g_{i,2,j}}(m_{i,j})),
$$
where $D_{i,j} = \operatorname{{\mathcal{C}}}_{i,j}^{(k_{i,j})}(m_{i,j})$ and $g_i = g_{i,1} + \sum_{j=1}^{k_i} m_{i,j} (g_{i,2,j} + k_{i,j} - 1)$.
\item We will denote $\displaystyle \min_{1 \leq i \leq m(h)} g_i+g_{i+1}$ by $g(\operatorname{{\mathcal{C}}})$.
\end{enumerate}
\end{notation}
\noindent In Figure~\ref{fig:split-surface} below, we have the surface $S_{24}$ with the multicurve $$\operatorname{{\mathcal{C}}} = \operatorname{{\mathcal{C}}}^{(2)}(2) \sqcup \operatorname{{\mathcal{C}}}^{(1)}(2) \sqcup \operatorname{{\mathcal{C}}}^{(3)} \sqcup \operatorname{{\mathcal{C}}}^{(1)} \sqcup \operatorname{{\mathcal{C}}}^{(1)}(3).$$ According to Notation~\ref{not:rk_permuting_root},
$$S_{24} = \left(\S_3(2) \#_{\operatorname{{\mathcal{C}}}^{(2)}(2)} S_2 \#_{\operatorname{{\mathcal{C}}}^{(1)}(2)} \S_1(2)\right) \#_{\operatorname{{\mathcal{C}}}^{(3)}} S_2 \#_{\operatorname{{\mathcal{C}}}^{(1)}} \left(S_1 \#_{\operatorname{{\mathcal{C}}}^{(1)}(3)}\S_3(3)\right).$$
\begin{figure}[h]
\labellist
\tiny
\pinlabel $\operatorname{{\mathcal{C}}}^{(2)}(2)$ at 70 95
\pinlabel $\operatorname{{\mathcal{C}}}^{(1)}(2)$ at 225 145
\pinlabel $\operatorname{{\mathcal{C}}}^{(3)}$ at 333 40
\pinlabel $\operatorname{{\mathcal{C}}}^{(1)}$ at 545 145
\pinlabel $\operatorname{{\mathcal{C}}}^{(1)}(3)$ at 620 40
\endlabellist
\centering
\includegraphics[width = 70 ex]{split-surface1}
\caption{$S_{24}$ with a separating multicurve $\operatorname{{\mathcal{C}}}$.}
\label{fig:split-surface}
\end{figure}
\begin{defn}
\label{defn:separating_actions_rk}
Let $t_1$ be a permuting $(n_1,0,k_1)$-action on $S_{g_1}$, and let $t_{2,j}$ be a permuting $(n_{2,j},0,k_{2,j})$-action on $\S_{g_{2,j}}(m_j)$, for $1 \leq j \leq s$. Then $(\l t_1\r ,\l t_{2,1}\r ,\ldots,\l t_{2,s}\r )$ forms a \textit{(s+1)-compatible tuple of degree n} if
\begin{enumerate}[(i)]
\item for each $1\leq j\leq s$, $(\l t_1\r ,\l t_{2,j}\r )$ forms an $(u_j,v_j)$-compatible pair of degree $n$, for some $u_j, v_j \geq 0$ such that $k_{2,j} = u_j+v_j$.
\item For each $i\neq j$,
$$
\o(t_1,t_{2,i})\cap\o(t_1,t_{2,j}) = \emptyset = \operatorname{{\mathbb P}}(t_1,t_{2,i})\cap\operatorname{{\mathbb P}}(t_1,t_{2,j}).
$$
\end{enumerate}
The number $g = g_1+ \sum_{j=1}^s m_j(g_{2,j}+ k_{2,j} -1)$ is called the \textit{genus} of the $(s+1)$-tuple.
\end{defn}
\begin{defn}\label{defn:separating_multituples}
Fix $m,n \in \mathbb{N}$, and for $1 \leq i \leq m+1$, let
$$\bar{t_i} = (\l t_{i,1}\r ,\l t_{i,2,1}\r ,\ldots,\l t_{i,2,s_i}\r )$$ be a $(s_i+1)$-compatible tuple as in Definition \ref{defn:separating_actions_rk}. Then the tuple
$$
( \overline{t_1} , \overline{t_2}, \ldots, \overline{t_{m+1}} )
$$
is said to form an \textit{(m+1)-compatible multituple of degree n} if for each $1\leq i\leq (m+1)$,
\begin{enumerate}[(i)]
\item the pair $(\l t_{i,1}\r ,\l t_{i+1,1}\r )$ forms an $(r_{i,1},k_{i,1})$-compatible pair of degree $n$, and
\item
$
\o(t_{i,1},t_{i+1,1})\cap \left( \sqcup_{j=1}^{s_i} \o(t_{i,1},t_{i,2,j})\right) = \emptyset = \operatorname{{\mathbb P}}(t_{i,1},t_{i+1,1})\cap \left( \sqcup_{j=1}^{s_i} \operatorname{{\mathbb P}}(t_{i,1},t_{i,2,j})\right).
$
\end{enumerate}
If $g(\overline{t_i})$ denotes the genus of $\overline{t_i}$, and
$$
\alpha_i := \sum_{\o \in\o(t_{1,i},t_{1,i+1})} |\o|
$$
then the number
$$
g = \sum_{i=1}^{m+1} g(\overline{t_i}) + \sum_{i=1}^m (r_{1,i} + k_{1,i}\alpha_i-1)
$$
is called the \textit{genus} of the multituple.
\end{defn}
\noindent The following theorem follows from Lemmas~\ref{thm:homologous_root_action} and \ref{lem:single_orbit_actions}.
\begin{thm}\label{thm: compatible_multituple}
Let $\operatorname{{\mathcal{C}}}$ be separating multicurve in $S_g$. Then the conjugacy class of a root $h$ of $t_{\operatorname{{\mathcal{C}}}}$ of degree $n$ with $|\operatorname{Fix}_h(\operatorname{{\mathcal{C}}})| = m$ corresponds to an $(m+1)$-compatible multituple of degree $n$ and genus $g$.
\end{thm}
\noindent The following corollary, which gives an upper bound for the degree of a root of $t_{\operatorname{{\mathcal{C}}}}$, follows from \cite[Theorem 8.6]{KR1} and Theorem~\ref{thm: compatible_multituple}.
\begin{cor}
\label{cor:bound_sep}
Let $\operatorname{{\mathcal{C}}}$ be a separating multicurve in $S_g$ and $h$ be a root of $t_{\operatorname{{\mathcal{C}}}}$ of degree $n$. Then $n \leq 4{g(\operatorname{{\mathcal{C}}})}^2 + 2g(\operatorname{{\mathcal{C}}})$.
\end{cor}
\noindent The following corollary, which gives a stable upper bound for the degree of a root of $t_{\operatorname{{\mathcal{C}}}}$, follows from \cite[Theorem 8.14]{KR1} and Theorem~\ref{thm: compatible_multituple}.
\begin{cor}
\label{cor:stablebound_sep}
Let $\operatorname{{\mathcal{C}}}$ be a separating multicurve in $S_g$ and $h$ be a root of $t_{\operatorname{{\mathcal{C}}}}$ of degree $n$. Then $n \leq \frac{16}{5}g^2 + 12g + \frac{45}{4}$,
if $g(\operatorname{{\mathcal{C}}}) \geq 10$.
\end{cor}
The primary purpose behind developing the notion of data sets was for algebraically describing actions that are involved in the construction of roots. Owing to complexity of notation, we shall abstain from developing separate notation involving data sets for encoding compatible tuples or multituples. We will see later in Sections \ref{sec:classify_genus3} and \ref{sec:classify_genus4} that one can easily circumvent the need for complex notation by a judicious use of Remark \ref{rem: permuting_action_sgm}.
Recall that a multicurve $\operatorname{{\mathcal{C}}}$ is said to be mixed if it is neither separating nor nonseparating. For the sake of brevity, we do not develop a separate theory to describe roots of Dehn twists about mixed multicurves as such a theory would merely be a combination of the separating and nonseparating cases. However, we will classify such roots on $S_3$ and $S_4$, thereby indicating how such a theory would follow from the ideas developed in Sections \ref{sec:nonsepmulcurves} and \ref{sec:sepmulcurves}.
\section{Classification of roots for $S_3$}
\label{sec:classify_genus3}
In this, and the subsequent section, we classify roots of Dehn twists about multicurves on surfaces of genus 3 and 4. Once again, we break up the classification into nonseparating, separating and mixed multicurves.
When classifying an $(m+1)$-compatible multituples $(\overline{t_1},\overline{t_2},\ldots, \overline{t_{m+1}})$ that corresponds to a root, Condition (i) of Definition \ref{defn:separating_multituples} and Condition (ii) of Definition \ref{defn:separating_actions_rk} help in eliminating data sets that do not lead to roots. For the sake of brevity, we only list those data sets that do lead to roots. Furthermore, in each case, a careful examination of the data set $\operatorname{{\mathcal D}}$ also gives $\o_{\operatorname{{\mathcal D}}}$, and so we only display the former.
Finally, when $\overline{t_i}$ is a permuting $(n,r,k)$-action on $\S_g(m)$, we use Remark \ref{rem: permuting_action_sgm} and replace $\overline{t_i}$ by the corresponding action on $S_g$ which has an $m^{th}$ root $\widetilde{t_i}$, whose equivalence class can be encoded by a data set $D_i$. Therefore an $(m+1)$-compatible multituple is described by a tuple $(D_1,D_2,\ldots, D_{m+1})$ of data sets, which will be listed in a table. While enumerating the curves in a multicurve, we use the convention that separating curves are denoted with the letter $c$, while nonseparating curves are denoted with the letter $d$. \\
\noindent \textbf{$\operatorname{{\mathcal{C}}}$ is a nonseparating multicurve.}
\begin{table}[H]
\begin{tabular}{|l|c|}
\hline
Degree & $D_1$ \\
\hline
3 & $(3,0;(1,3),(1,3))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \{d_1,d_2,d_3\}$ and $(r,k)=(0,1)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|}
\hline
Degree & $D_1$ \\
\hline
2 & $(2,1;)$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \{d_1,d_2\}$ and $(r,k)=(0,1)$}
\end{table}
\noindent{\textbf{$\operatorname{{\mathcal{C}}}$ is a separating multicurve.}}
\begin{table}[H]
\begin{tabular}{|l|c|c|}
\hline
Degree & $D_1$ & $D_2$ \\
\hline
3 & $(3,0;(1,3),(1,3))$ & $(1,1;)$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \{c_1,c_2,c_3\}, S_3 = S_0\#_{\operatorname{{\mathcal{C}}}} S_1(3)$ and $(r,k)=(0,1)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|c|}
\hline
Degree & $D_1$ & $D_2$ & $D_3$ \\
\hline
2 & $(1,1,1;)$ & $(2,0,1;(1,2))$ & $(1,1;)$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}}=\{c_1,c_2,c_3\}, S_3=S_0 \#_{c_1}S_1\#_{\{c_2,c_3\}}S_1(2)$ and $(r,k)=(1,1)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|c|c|}
\hline
Degree & $D_1$ & $D_2$ & $D_3$ \\
\hline
3 & $(3,0,2;(2,3),(2,3))$ & $(3,0,(2,2);(2,3))$ & $(3,0,2;(2,3),(2,3))$ \\
6 & $(6,0,5;(1,2),(2,3))$ & $(3,0,(1,1);(1,3))$ & $(6,0,5;(1,2),(2,3))$ \\
6 & $(6,0,1;(1,2),(1,3))$ & $(1,1,(1,1);)$ & $(6,0,1;(1,2),(1,3))$ \\
6 & $(3,0,2;(2,3),(2,3))$ & $(2,0,(1,1);(1,2),(1,2))$ & $(3,0,2;(2,3),(2,3))$ \\
6 & $(2,0,1;(1,2),(1,2),(1,2))$ & $(3,0,(2,2);(2,3))$ & $(2,0,1;(1,2),(1,2),(1,2))$ \\
12 & $(3,0,1;(1,3),(1,3))$ & $(4,0,(3,3);(1,2))$ & $(3,0,1;(1,3),(1,3))$ \\
12 & $(4,0,3;(1,2),(3,4))$ & $(3,0,(1,1);(1,3))$ & $(4,0,3;(1,2),(3,4))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \{c_1,c_2\}, S_3=S_1\#_{c_1}S_1\#_{c_2}S_1$ and $(r,k)=(2,0)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|}
\hline
Degree & $D_1$ & $D_2$ \\
\hline
2 & $(2,1;)$ & $(1,1;)$ \\
12 & $(4,0;(1,4),(1,2),(1,4))$ & $(6,0;(5,6),(1,2),(2,3))$ \\
12 & $(4,0;(3,4),(1,2),(3,4))$ & $(6,0;(5,6),(1,2),(2,3))$ \\
12 & $(6,0;(5,6),(1,2),(2,3))$ & $(4,0;(1,4),(1,2),(1,4))$ \\
12 & $(6,0;(5,6),(1,2),(2,3))$ & $(4,0;(3,4),(1,2),(3,4))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \{c_1,c_2\}, S_3 = S_1\#_{\operatorname{{\mathcal{C}}}}S_1(2)$ and $(r,k)=(0,1)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|c|}
\hline
Degree & $D_1$ & $D_2$ \\
\hline
2 & $(2,0,(1,1);(1,2),(1,2),(1,2))$ & $(1,0,(1,1);)$ \\
3 & $(3,0,(1,1);(1,3))$ & $(1,0,(1,1);)$ \\
3 & $(3,0,(2,2);(2,3))$ & $(3,0,(2,2);(2,3))$ \\
4 & $(4,0,(1,1);(1,2))$ & $(1,0,(1,1);)$ \\
4 & $(4,0,(3,3);(1,2))$ & $(2,0,(1,1);(1,2),(1,2),(1,2))$ \\
6 & $(3,0,(2,2);(2,3))$ & $(2,0,(1,1);(1,2),(1,2),(1,2))$ \\
12 & $(4,0,(3,3);(1,2))$ & $(3,0,(1,1);(1,3))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \operatorname{{\mathcal{C}}}^{(2)}$, $S_3 = S_1 \#_{\operatorname{{\mathcal{C}}}} S_1$, and $(r,k)=(2,0)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|}
\hline
Degree & $D_1$ & $D_2$ \\
\hline
6 & $(2,0;(1,2),(1,2),(1,2),(1,2))$ & $(6,0;(1,6),(1,2),(1,3))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \operatorname{{\mathcal{C}}}^{(2)},S_3 = S_1\#_{\operatorname{{\mathcal{C}}}} S_1$ and $(r,k)=(0,1)$}
\end{table}
\noindent \textbf{$\operatorname{{\mathcal{C}}}$ is a mixed multicurve.}
\begin{table}[H]
\begin{tabular}{|l|c|c|}
\hline
Degree & $D_1$ & $D_2$ \\
\hline
2 & $(3,0;(1,3),(1,3))$ & $(1,1;)$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}}=\{c_1,c_2,c_3,d_1,d_2,d_3\}, S_3=S_0\#_{\{c_1,c_2,c_3\}}\# S_1(3)$ where $d_i\in \widehat{S_1^i}$ and $(r,k)=(0,2)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|c|}
\hline
Degree & $D_1$ & $D_2$ & $D_3$ \\
\hline
2 & $(1,1,1;)$ & $(2,0,1;(1,2))$ & $(1,1;)$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \{c_1,c_2,c_3,d_1,d_2\}$, $S_3=S_1\#_{c_1}S_0\#_{\{c_2,c_3\}}S_1(2)$ where $d_i \in \widehat{S_1^i}$ for $i=1,2$ and $(r,k)=(1,2)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|}
\hline
Degree & $D_1$ & $D_2$ \\
\hline
2 & $(2,1;)$ & $(2,0;)$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}}=\{c_1,c_2,d_1,d_2\}, S_3=S_1\#_{\{c_1,c_2\}}S_1(2)$ and $d_i \in \widehat{S_1^i}$ for $i=1,2$ and $(r,k)=(0,2)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|}
\hline
Degree & $D_1$ & $D_2$ \\
\hline
2 & $(2,0,1;(1,2),(1,2),(1,2))$ & $(1,1,1;)$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}}=\{c_1,d_1,d_2\}, S_3=S_1\#_{c_1}S_2$ and $\{d_1,d_2\} \subset \widehat{S_2}$ and $(r,k)=(1,1)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|c|}
\hline
Degree & $D_1$ & $D_2$ \\
\hline
3 & $(3,0,2;(2,3),(2,3))$ & $(3,0,(2,2,2);)$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \{c_1,d_1\}$ mixed and $S_3 = S_1\#_{c_1}S_2$ where $d_1 \in \widehat{S_2}$ and $(r,k)=(2,0)$}
\end{table}
\section{Classification of roots for $S_4$}
\label{sec:classify_genus4}
\noindent\textbf{$\operatorname{{\mathcal{C}}}$ is a nonseparating multicurve.}
\begin{table}[H]
\begin{tabular}{|l|c|}
\hline
Degree & $D_1$ \\
\hline
4 & $(4,0;(1,4),(1,4))$ \\
2 & $(2,1;(1,2),(1,2))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}}= \{d_1,d_2,d_3,d_4\}$ and $(r,k)=(0,1)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|}
\hline
Degree & $D_1$ \\
\hline
3 & $(3,1;)$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}}= \{d_1,d_2,d_3\}$ and $(r,k)=(0,1)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|}
\hline
Degree & $D_1$ \\
\hline
2 & $(2,1;(1,2),(1,2))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \{d_1,d_2\}$ and $(r,k)=(0,1)$}
\end{table}
\noindent\textbf{$\operatorname{{\mathcal{C}}}$ is a separating multicurve.}
\begin{table}[H]
\begin{tabular}{|l|c|c|}
\hline
Degree & $D_1$ & $D_2$ \\
\hline
4 & $(4,0;(1,4),(1,4))$ & $(1,1;)$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \{c_1,c_2,c_3,c_4\}, S_4 = S_0\#_{\operatorname{{\mathcal{C}}}} S_1(4)$ and $(r,k)=(0,1)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|c|}
\hline
Degree & $D_1$ & $D_2$ & $D_3$ \\
\hline
2 & $(1,1;)$ & $(2,0;)$ & $(1,1;)$ \\
6 & $(6,0;(1,6),(1,2),(1,3))$ & $(2,0;)$ & $(6,0;(1,6),(1,2),(1,3))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \{c_1,c_2,c_3,c_4\},S_4 = S_1(2)\#_{\{c_1,c_2\}}S_0 \#_{\{c_3,c_4\}}S_1(2)$ and $(r,k)=(0,2)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|}
\hline
Degree & $D_1$ & $D_2$ \\
\hline
3 & $(3,1;)$ & $(1,1;)$ \\
6 & $(6,0; (1,6),(1,2),(1,3))$ & $(1,1;)$\\
6 & $(6,0; (5,6),(1,2),(2,3))$ & $(1,1;)$ \\
6 & $(3,1;)$ & $(6,0; (1,6),(1,2),(1,3))$ \\
6 & $(3,1;)$ & $(6,0; (5,6),(1,2),(2,3))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}}=\{c_1,c_2,c_3\}, S_4 = S_1\#_{\operatorname{{\mathcal{C}}}} S_1(3)$ and $(r,k)=(0,1)$.}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|c|}
\hline
Degree & $D_1$ & $D_2$ & $D_3$ \\
\hline
4 & $(1,1,1;)$ & $(4,0,1;(1,2),(1,4))$ & $(1,1;)$ \\
6 & $(1,1,1;)$ & $(6,0,1;(1,2),(1,3))$ & $(1,1;)$ \\
6 & $(3,0,2;(2,3),(2,3))$ & $(2,0,1;(1,2),(1,2),(1,2))$ & $(6,0;(1,6),(1,2),(1,3))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \{c_1,c_2,c_3\}, S_4 = S_1 \#_{c_1}S_1\#_{\{c_2,c_3\}}S_1(2)$ and $(r,k)=(1,1)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|c|c|}
\hline
Degree & D1 & D2 & D3 & D4 \\
\hline
3 & $(3,0,2;(2,3),(2,3))$ & $(3,0,2;(2,3),(2,3))$ & $(3,0,2;(2,3),(2,3))$ & $(3,0,2;(2,3),(2,3))$ \\
6 & $(6,0,5;(1,2),(2,3))$ & $(3,0,1;(1,3),(1,3))$ & $(6,0,5;(1,2),(2,3))$ & $(3,0,1;(1,3),(1,3))$ \\
12 & $(3,0,1;(1,3),(1,3))$ & $(4,0,3;(1,2),(3,4))$ & $(3,0,1;(1,3),(1,3))$ & $(4,0,3;(1,2),(3,4))$ \\
12 & $(4,0,3;(1,2),(3,4))$ & $(3,0,1;(1,3),(1,3))$ & $(4,0,3;(1,2),(3,4))$ & $(3,0,1;(1,3),(1,3))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \{c_1,c_2,c_3\}, S_4 = S_1\#_{c_1}S_1\#_{c_2}S_1\#_{c_3}S_1$ and $(r,k)=(3,0)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|}
\hline
Degree & $D_1$ & $D_2$ \\
\hline
6 & $(2,0;(1,2),(1,2),(1,2),(1,2),(1,2),(1,2))$ & $(6,0;(1,6),(1,2),(1,3))$ \\
6 & $(2,1;(1,2),(1,2))$ & $(6,0;(1,6),(1,2),(1,3))$ \\
6 & $(6,0;(5,6),(1,3),(5,6))$ & $(1,1;)$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}}= \{c_1,c_2\}, S_4 = S_2 \#_{\operatorname{{\mathcal{C}}}}S_1(2)$ and $(r,k)=(0,1)$.}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|c|}
\hline
Degree & $D_1$ & $D_2$ & $D_3$ \\
\hline
6 & $(6,0,5;(1,2),(2,3))$ & $(3,0,1;(1,3),(1,3))$ & $(6,0,5;(1,3),(5,6))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \{c_1,c_2\}, S_4 = S_1\#_{c_1}S_1\#_{c_2}S_2$ and $(r,k)=(2,0)$.}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|}
\hline
Degree & $D_1$ & $D_2$\\
\hline
3 & $(3,0;(1,3),(1,3),(1,3))$ & $(3,0;(1,3),(1,3),(1,3))$ \\
3 & $(3,0;(1,3),(1,3),(1,3))$ & $(3,0;(2,3),(2,3),(2,3))$ \\
3 & $(3,0;(2,3),(2,3),(2,3))$ & $(3,0;(2,3),(2,3),(2,3))$ \\
3 & $(3,0;(2,3),(2,3),(2,3))$ & $(3,0;(1,3),(1,3),(1,3))$ \\
6 & $(3,0;(1,3),(1,3),(1,3))$ & $(6,0;(1,6),(1,2),(1,3))$ \\
6 & $(3,0;(1,3),(1,3),(1,3))$ & $(6,0;(5,6),(1,2),(2,3))$ \\
6 & $(3,0;(2,3),(2,3),(2,3))$ & $(6,0;(1,6),(1,2),(1,3))$ \\
6 & $(3,0;(2,3),(2,3),(2,3))$ & $(6,0;(5,6),(1,2),(2,3))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}}=\operatorname{{\mathcal{C}}}^{(3)}, S_4= S_1\#_{\operatorname{{\mathcal{C}}}}S_1$ and $(r,k)=(0,1)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|}
\hline
Degree & $D_1$ & $D_2$\\
\hline
4 & $(2,0,1;(1,2),(1,2),(1,2))$ & $(4,0,3;(1,2),(3,4))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}}=\operatorname{{\mathcal{C}}}^{(3)}, S_4 = S_1\#_{\operatorname{{\mathcal{C}}}}S_1$ and $(r,k)=(1,1)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|}
\hline
Degree & $D_1$ & $D_2$ \\
\hline
3 & $(3,0,(2,2,2);)$ & $(3,0,(2,2,2);)$ \\
6 & $(3,0,(2,2,2);)$ & $(2,0,(1,1,1);(1,2))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \operatorname{{\mathcal{C}}}^{(3)}, S_4 = S_1\#_{\operatorname{{\mathcal{C}}}} S_1$ and $(r,k)=(3,0)$.}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|c|}
\hline
Degree & $D_1$ & $D_2$ & $D_3$ \\
\hline
4 & $(4,0;(1,4),(1,2),(1,4))$ & $(2,0;(1,2),(1,2))$ & $(4,0;(1,4),(1,2),(1,4))$ \\
4 & $(4,0;(1,4),(1,2),(1,4))$ & $(2,0;(1,2),(1,2))$ & $(4,0;(3,4),(1,2),(3,4))$ \\
4 & $(4,0;(3,4),(1,2),(3,4))$ & $(2,0;(1,2),(1,2))$ & $(4,0;(3,4),(1,2),(3,4))$ \\
6 & $(6,0;(1,6),(1,2),(1,3))$ & $(2,0;(1,2),(1,2))$ & $(6,0;(1,6),(1,2),(1,3))$ \\
6 & $(6,0;(1,6),(1,2),(1,3))$ & $(2,0;(1,2),(1,2))$ & $(6,0;(5,6),(1,2),(2,3))$ \\
6 & $(6,0;(5,6),(1,2),(2,3))$ & $(2,0;(1,2),(1,2))$ & $(6,0;(5,6),(1,2),(2,3))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \operatorname{{\mathcal{C}}}^{(2)}\sqcup \operatorname{{\mathcal{C}}}^{(2)}, S_4 = S_1\#_{\operatorname{{\mathcal{C}}}^{(2)}} S_0 \#_{\operatorname{{\mathcal{C}}}^{(2)}} S_1$ and $(r,k)=(0,2)$.}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|}
\hline
Degree & $D_1$ & $D_2$ \\
\hline
2 & $(2,0;(1,2),(1,2))$ & $(1,1;)$ \\
6 & $(2,0;(1,2),(1,2))$ & $(6,0;(1,6),(1,2),(1,3))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \operatorname{{\mathcal{C}}}^{(2)}\sqcup \operatorname{{\mathcal{C}}}^{(2)}, S_4 = S_0\#_{\operatorname{{\mathcal{C}}}} S_1(2)$ and $(r,k)=(0,2)$.}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|}
\hline
Degree & $D_1$ & $D_2$ \\
\hline
4 & $(2,0;(1,2),(1,2),(1,2),(1,2),(1,2),(1,2))$ & $(4,0;(1,4),(1,2),(1,4))$ \\
4 & $(2,0;(1,2),(1,2),(1,2),(1,2),(1,2),(1,2))$ & $(4,0;(3,4),(1,2),(3,4))$ \\
4 & $(2,1;(1,2),(1,2))$ & $(4,0;(1,4),(1,2),(1,4))$ \\
4 & $(2,1;(1,2),(1,2))$ & $(4,0;(3,4),(1,2),(3,4))$ \\
4 & $(4,0;(1,2),(1,2),(3,4))$ & $(2,0;(1,2),(1,2),(1,2),(1,2))$ \\
4 & $(4,0;(1,2),(1,2),(3,4))$ & $(2,1;)$ \\
4 & $(4,0;(1,2),(1,2),(2,4))$ & $(2,0;(1,2),(1,2),(1,2),(1,2))$ \\
4 & $(4,0;(1,2),(1,2),(2,4))$ & $(2,1;)$ \\
6 & $(2,0;(1,2),(1,2),(1,2),(1,2),(1,2),(1,2))$ & $(6,0;(1,6),(1,2),(1,3))$ \\
6 & $(2,1;(1,2),(1,2))$ & $(6,0;(1,6),(1,2),(1,3))$ \\
6 & $(6,0;(5,6),(1,3),(5,6))$ & $(2,0;(1,2),(1,2),(1,2),(1,2))$ \\
6 & $(6,0;(5,6),(1,3),(5,6))$ & $(2,1;)$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \operatorname{{\mathcal{C}}}^{(2)}, S_4 = S_2\#_{\operatorname{{\mathcal{C}}}} S_1$ and $(r,k)=(0,1)$.}
\end{table}
\begin{table} [H]
\begin{tabular}{|l|l|l|}
\hline
Degree & $D_1$ & $D_2$ \\
\hline
2 & $(1,2,(1,1);)$ & $(2,0,(1,1);(1,2),(1,2))$ \\
2 & $(2,0,(1,1);(1,2),(1,2),(1,2),(1,2))$ & $(1,1,(1,1);)$ \\
2 & $(2,1,(1,1);)$ & $(1,1,(1,1);)$ \\
3 & $(1,2,(1,1);)$ & $(3,0,(1,1);(1,3))$ \\
3 & $(3,0,(1,1);(2,3),(2,3))$ & $(1,1,(1,1);)$ \\
3 & $(3,0,(2,2);(1,3),(1,3))$ & $(3,0,(2,2);(1,3),(1,3))$ \\
4 & $(1,2,(1,1);)$ & $(4,0,(1,1);(1,2))$ \\
5 & $(5,0,(1,1);(3,5))$ & $(1,1,(1,1);)$ \\
6 & $(6,0,(1,1);(2,3))$ & $(1,1,(1,1);)$ \\
6 & $(2,0,(1,1);(1,2),(1,2),(1,2),(1,2))$ & $(3,0,(2,2);(2,3))$ \\
6 & $(2,0,(1,1);(1,2),(1,2),(1,2),(1,2))$ & $(3,0,(2,2);(2,3))$ \\
6 & $(2,1,(1,1);)$ & $(3,0,(2,2);(2,3))$ \\
6 & $(3,0,(2,2);(1,3),(1,3))$ & $(2,0,(1,1);(1,2),(1,2))$ \\
6 & $(6,0,(5,5);(1,3)$ & $(3,0,(1,1);(1,3))$ \\
12 & $(3,0,(1,1);(2,3),(2,3))$ & $(4,0,(3,3);(1,2))$ \\
12 & $(6,0,(5,5);(1,3))$ & $(4,0,(1,1);(1,2))$ \\
15 & $(5,0,(3,3);(4,5))$ & $(3,0,(2,2);(2,3))$ \\
20 & $(5,0,(4,4);(2,5))$ & $(4,0,(1,1);(1,2))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \operatorname{{\mathcal{C}}}^{(2)}, S_4 = S_2 \#_{\operatorname{{\mathcal{C}}}} S_1$ and $(r,k)=(2,0)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|c|}
\hline
Degree & $D_1$ & $D_2$ & $D_3$\\
\hline
6 & $(3,0,2;(2,3),(2,3))$ & $(2,0,1;(1,2),(1,2),(1,2))$ & $(6,0;(1,6),(1,2),(1,3))$ \\
6 & $(1,1,1;)$ & $(6,0,1;(1,2),(1,3))$ & $(2,0;(1,2),(1,2),(1,2),(1,2))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \operatorname{{\mathcal{C}}}^{(2)}\sqcup\{c_1\}, S_4 = S_1\#_{c_1} S_1\#_{\operatorname{{\mathcal{C}}}^{(2)}} S_1$ and $(r,k)=(1,1)$.}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|c|c|}
\hline
Degree & $D_1$ & $D_2$ & $D_3$ \\
\hline
3 & $(3,0,2;(2,3),(2,3))$ & $(3,0,(2,2,2);)$ & $(3,0,(2,2);(2,3))$ \\
12 & $(4,0,3;(1,2),(3,4))$ & $(3,0,(1,1,1);)$ & $(4,0,(3,3);(1,2))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \{c_1\}\sqcup \operatorname{{\mathcal{C}}}^{(2)}, S_4 = S_1 \#_{c_1} S_1 \#_{\operatorname{{\mathcal{C}}}^{(2)}} S_1$ and $(r,k)=(3,0)$.}
\end{table}
\noindent\textbf{$\operatorname{{\mathcal{C}}}$ is a mixed multicurve.}
\begin{table}[H]
\begin{tabular}{|l|c|c|}
\hline
Degree & $D_1$ & $D_2$ \\
\hline
4 & $(4,0;(1,4),(1,4)$ & $(1,0;)$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \{c_1,\ldots, c_4, d_1,\ldots, d_4\}$, $S_4 = S_0\#_{\{c_1,\ldots, c_4\}} S_1(4)$ with $d_i \in \widehat{S_1^i}$ and $(r,k)=(0,2)$.}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|c|}
\hline
Degree & $D_1$ & $D_2$ & $D_3$ \\
\hline
2 & $(1,0;)$ & $(2,0;(1,2),(1,2))$ & $(1,0;)$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \{c_1,\ldots, c_4, d_1,\ldots, d_4\}$, $S_4 = S_1(2)\#_{\{c_1,c_2\}}S_0\#_{\{c_3,c_4\}}S_1(2)$ with $d_i \in \widehat{S_1^i}$ and $(r,k)=(0,4)$.}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|c|}
\hline
Degree & $D_1$ & $D_2$ & $D_3$ \\
\hline
2 & $(1,1;)$ & $(2,0;(1,2),(1,2))$ & $(1,0;)$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}}=\{c_1,\ldots, c_4,d_1,d_2\}, S_4 = S_1(2)\#_{\{c_1,c_2\}}S_0\#_{\{c_3,c_4\}}S_1(2)$ with $d_i \in \widehat{S_1^i}$ and $(r,k)=(0,3)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|c|c|}
\hline
Degree & $D_1$ & $D_2$ & $D_3$ & $D_4$ \\
\hline
2 & $(1,0;)$ & $(1,1,1;)$ & $(2,0,1;(1,2))$ & $(1,1,1;)$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}}=\{c_1,\ldots, c_4, d_1,d_2\}, S_4 = S_1(2)\#_{\{c_1,c_2\}}\left(S_1\#_{c_3}S_0\#_{c_4}S_1\right)$ with $d_i \in \widehat{S_1^i}$ and $(r,k)=(2,2)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|}
\hline
Degree & $D_1$ & $D_2$\\
\hline
3 & $(3,1;)$ & $(1,0;)$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}}=\{c_1,c_2,c_3,d_1,d_2,d_3\}, S_4 = S_1\#_{\{c_1,c_2,c_3\}}S_1(3)$ with $d_i \in \widehat{S_1^i}$ and $(r,k)=(0,2)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|c|}
\hline
Degree & $D_1$ & $D_2$ & $D_3$ \\
\hline
2 & $(2,0,1;(1,2),(1,2),(1,2))$ & $(1,1,1;)$ & $(4,0;(1,4),(1,4))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}}=\{c_1,c_2,c_3,d_1,d_2\}, S_4=S_1\#_{c_1}S_1\#_{\{c_2,c_3\}}S_1(2)$ with $d_i \in \widehat{S_1^i}$ and $(r,k)=(1,2)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|}
\hline
Degree & $D_1$ & $D_2$\\
\hline
2 & $(2,0;(1,2),(1,2),(1,2),(1,2),(1,2),(1,2))$ & $(4,0;(1,4),(1,4))$ \\
2 & $(2,1;(1,2),(1,2))$ & $(4,0;(1,4),(1,4))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}}=\{c_1,c_2,d_1,d_2\}, S_4 = S_2\#_{\{c_1,c_2\}}S_1(2)$ with $d_i \in \widehat{S_1^i}$ and $(r,k)=(0,2)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|}
\hline
Degree & $D_1$ & $D_2$\\
\hline
2 & $(2,0;(1,2),(1,2))$ & $(4,0;(1,4),(1,2),(1,4))$ \\
2 & $(2,0;(1,2),(1,2))$ & $(4,0;(3,4),(1,2),(3,4))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}}=\{c_1,c_2,d_1,d_2\}, S_4=S_2\#_{\{c_1,c_2\}}S_1(2)$ with $d_i \in \widehat{S_2}$ and $(r,k)=(0,2)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|}
\hline
Degree & $D_1$ & $D_2$\\
\hline
2 & $(1,1,1;)$ & $(2,0,1;(1,2),(1,2),(1,2))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}}=\{c_1,d_1,d_2\}, S_4 = S_1\#_{c_1}S_3$ with $\{d_1,d_2\}\subset \widehat{S_3}$ and $(r,k)=(1,1)$}
\end{table}
\begin{table}[H]
\begin{tabular}{|l|c|c|}
\hline
Degree & $D_1$ & $D_2$\\
\hline
2 & $(2,0;(1,2),(1,2))$ & $(2,0;(1,2),(1,2),(1,2),(1,2))$ \\
\hline
\end{tabular}
\caption{$\operatorname{{\mathcal{C}}} = \operatorname{{\mathcal{C}}}^{(2)}\sqcup\{d_1,d_2\},S_4 = S_2\#_{\operatorname{{\mathcal{C}}}^{(2)}}S_1$ with $\{d_1,d_2\} \subset \widehat{S_2}$ and $(r,k)=(0,2)$.}
\end{table}
\section{Concluding remarks}
\subsection{Roots and the Torelli Group}
Let $\Psi : \operatorname{Mod}(S_g) \to \operatorname{Sp}(2g,\mathbb{Z})$ be the symplectic representation of $\operatorname{Mod}(S_g)$ arising out of its action on $H_1(S_g,\mathbb{Z})$, and let $\operatorname{{\mathcal I}}(S_g)$ denote the kernel of $\Psi$, the Torelli group of $S_g$. If $\operatorname{{\mathcal{C}}}$ contains nonseparating curves, then $\Psi(t_{\operatorname{{\mathcal{C}}}})$ is a product of commuting elementary matrices, and thus a root of $t_{\operatorname{{\mathcal{C}}}}$ induces a root of such a matrix in $Sp(2g,\mathbb{Z})$. However, if every curve in $\operatorname{{\mathcal{C}}}$ is separating, then $\Psi(h)$ is a root of unity in $Sp(2g,\mathbb{Z})$. Using the theory we have developed for multicurves, we shall now give a succinct proof of the fact that such a root cannot lie in $\operatorname{{\mathcal I}}(S_g)$.
\begin{thm}\label{thm:no_roots_Tor}
Let $h$ be the root of the Dehn twist $t_{\operatorname{{\mathcal{C}}}}$ about a multicurve $\operatorname{{\mathcal{C}}}$ in $S_g$. Then $h \notin \operatorname{{\mathcal I}}(S_g)$.
\end{thm}
\begin{proof}
Since $\operatorname{{\mathcal I}}(S_g)$ is normal in $\operatorname{Mod}(S_g)$, it suffices to prove that the conjugacy class of $h$ intersects $\operatorname{{\mathcal I}}(S_g)$ non-trivially. If $c$ is an essential simple closed curve in $S_g$, it follows from \cite[Proposition 6.3, Example 6.5.2]{MF} that $t_c \in \operatorname{{\mathcal I}}(S_g)$ if and only if $c$ is a separating curve. Consequently, it suffices to assume that every curve in $\operatorname{{\mathcal{C}}}$ is a separating curve.
In that case, let $\operatorname{Fix}_h(\operatorname{{\mathcal{C}}}) = \{\operatorname{{\mathcal E}}_1,\ldots,\operatorname{{\mathcal E}}_m\}$, where $m = |\operatorname{Fix}_h(\operatorname{{\mathcal{C}}})|$, and write
$$
S_g = \overline{\textbf{\Large \#}}_{i=1}^{m(h)} (S_{g_i} \#_{\operatorname{{\mathcal E}}_i} S_{g_{i+1}}),
$$
as in Notation~\ref{not:rk_permuting_root}. By Theorem~\ref{thm: compatible_multituple}, the conjugacy class of $h$ corresponds to an $(m+1)$-compatible multituple $( \overline{t_1} , \overline{t_2}, \ldots, \overline{t_{m+1}} )$, where
$\bar{t_i} = (\l t_{i,1}\r ,\l t_{i,2,1}\r ,\ldots,\l t_{i,2,s_i}\r )$ is a $(s_i+1)$-compatible tuple of genus $g_i$.
Choose $1\leq i\leq m$ such that at least one component of $\overline{t_i}$ is non-trivial, so we write
$$
S_{g_i} = \overline{\textbf{\Large \#}}_{j=1}^{k_i} (S_{g_{i,1}} \#_{D_{i,j}} \S_{g_{i,2,j}}(m_{1,j})),
$$
as in Notation \ref{not:rk_permuting_root}. If $g_{i,1} \geq 1$, and $t_{i,1}$ is a non-trivial finite order element in $\operatorname{Mod}(S_{g_{i,1}})$, then $t_{i,1} \notin \operatorname{{\mathcal I}}(S_{g_{i,1}})$ by \cite[Theorem 6.8]{MF}. Since $t_{i,1}$ is obtained from a restriction of $h$ by coning, it follows that $h\notin \operatorname{{\mathcal I}}(S_g)$. If $t_{i,1}$ is trivial or $g_{i,1}=0$, then it follows $k_i \geq 1$ and that $t_{i,2,j}$ is non-trivial for some $1\leq j\leq k_i$. Since each curve in $\operatorname{{\mathcal{C}}}$ is separating, $g_{i,2,j} \geq 1$, and so the surface $\S_{g_{i,2,j}}(m_{i,j})$ contributes $2m_{i,j}g_{i,2,j}$ generators to the standard geometric symplectic basis of $H_1(S_g,\mathbb{Z})$. Now $t_{i,2,j}$ (and consequently $h$) cyclically permutes the components of $\S_{g_{i,2,j}}(m_{i,j})$, and thus $h\notin \operatorname{{\mathcal I}}(S_g)$.
\end{proof}
\subsection{Roots of finite product of powers}
The theory developed in this paper for classifying roots up to conjugacy for finite products of commuting Dehn twists can be naturally generalised to one that classifies roots of finite products of powers of commuting twists.
Currently, the compatibility condition requires that pairs of distinguished orbits (or fixed points) of permuting actions should have associated angles that add up to $2\pi/n \pmod{2\pi}$. When $c$ is a single nonseparating curve, the roots of $t_c^{\ell}$ for $1 \leq \ell < n$, were classified in \cite{KR2} by using a variant of this condition, which required that the angles associated with compatible fixed points add up to $2\pi\ell/n \pmod{2\pi}$. This notion of compatibility of fixed points can be generalized to orbits, and this will eventually lead to the classification of roots of homeomorphisms of the form $\prod_{i=1}^m t_{c_i}^{\ell_i}$, where $\{c_1,c_2,\ldots,c_m\}$ is a multicurve. It is also apparent that such roots would not lie in $\operatorname{{\mathcal I}}(S_g)$ for the same reasons as above.
\section{Acknowledgements}
The authors would like to thank Dan Margalit and Darryl McCullough for helpful suggestions and examples that made the theory more comprehensive.
\bibliographystyle{plain}
| {
"timestamp": "2015-06-05T02:09:22",
"yymm": "1506",
"arxiv_id": "1506.01534",
"language": "en",
"url": "https://arxiv.org/abs/1506.01534",
"abstract": "A \\textit{multicurve} $\\C$ on a closed orientable surface is defined to be a finite collection of disjoint non-isotopic essential simple closed curves. The Dehn twist $t_{\\C}$ about $\\C$ is the product of the Dehn twists about the individual curves. In this paper, we give necessary and sufficient conditions for the existence of a root of such a Dehn twist, that is, a homeomorphism $h$ such that $h^n = t_{\\C}$. We give combinatorial data that corresponds to such roots, and use it to determine upper bounds for $n$. Finally, we classify all such roots up to conjugacy for surfaces of genus 3 and 4.",
"subjects": "Geometric Topology (math.GT)",
"title": "Roots of Dehn twists about multicurves",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808707404788,
"lm_q2_score": 0.7217432122827968,
"lm_q1q2_score": 0.7075110645876103
} |
https://arxiv.org/abs/1804.10929 | Diagonal asymptotics for symmetric rational functions via ACSV | We consider asymptotics of power series coefficients of rational functions of the form $1/Q$ where $Q$ is a symmetric multilinear polynomial. We review a number of such cases from the literature, chiefly concerned either with positivity of coefficients or diagonal asymptotics. We then analyze coefficient asymptotics using ACSV (Analytic Combinatorics in Several Variables) methods. While ACSV sometimes requires considerable overhead and geometric computation, in the case of symmetric multilinear rational functions there are some reductions that streamline the analysis. Our results include diagonal asymptotics across entire classes of functions, for example the general 3-variable case and the Gillis-Reznick-Zeilberger (GRZ) case, where the denominator in terms of elementary symmetric functions is $1 - e_1 + c e_d$ in any number $d$ of variables. The ACSV analysis also explains a discontinuous drop in exponential growth rate for the GRZ class at the parameter value $c = (d-1)^{d-1}$, previously observed for $d=4$ only by separately computing diagonal recurrences for critical and noncritical values of $c$. | \section{Introduction}
We study the power series coefficients of rational functions
of the form $F(x_1, \dots , x_d) = 1/Q(x_1, \ldots , x_d)$
where $Q$ is a symmetric multilinear function with $Q({\bf 0}) \neq 0$.
Let
$$ F({\bf x}) = \frac{1}{Q({\bf x})} = \sum_{{\bf r}\in\mathbb{Z}^d} a_{\bf r} {\bf x}^{\bf r}, $$
converging in some polydisk ${\cal D} \subset \mathbb{C}^d$. Often one focuses
on the diagonal coefficients $\delta_n := a_{n, \ldots , n}$, whose
univariate generating function $\textsf{diag}_F (z) := \sum_n \delta_n z^n$
satisfies a linear differential equation with polynomial coefficients,
but may be transcendental. A number of questions are natural, including
nonnegativity (are all coefficients nonnegative), eventual nonnegativity
(all but finitely many coefficients nonnegative), diagonal extraction
(computing $\textsf{diag}_F$ from $Q$), diagonal asymptotics, multivariate
asymptotics and phase transitions in the asymptotics of $\{ a_{\bf r} \}$.
The positivity (nonnegativity) question is the most classical, dating back
at least to Szeg{\H o}'s work in~\cite{szego-pos33}. The techniques, some
of which are indicated in the next section, used in the literature are diverse
and include integral methods and special functions, positivity preserving
operators, combinatorial identities, computer algebra such as cylindrical
algebraic decomposition, or determinantal methods.
Contrasting to these methods are analytic combinatorial several-variable
methods (ACSV) as developed in~\cite{PW-book}. These are typically
asymptotic, rather than exact, and therefore less useful for proving
classical positivity statements, though they can be used to disprove them.
Their chief advantages are their broad applicability and, increasingly,
the level to which they have been automated. Our aim in this paper is
to apply ACSV methods to a number of previously studied families of
rational coefficient sequences, thereby extending what is known as well
as illuminating the relative advantages of each method.
\subsection{Previously studied instances}
Let ${\cal M}_d$ denote the class of symmetric functions of $d$ variables that are
multilinear (degree~1 in each variable). This class of generating functions
$F({\bf x}) := 1/Q({\bf x})$ where $Q \in {\cal M}_d$ includes a great number of previously
studied cases, some of which we now review. Here and in the following,
we use $d$ for the
number of variables and boldface ${\bf x}, {\bf y}, {\bf z}$, etc., for vectors of
length $d$ of integer, real or complex numbers. When $d$ is small we use
$x, y, z, w$ for $x_1, x_2, x_3, x_4$. Let $e_k = e_{k,d}$ denote the $k^{th}$
elementary symmetric function of $d$ variables, the sum of all distinct $k$ element
products from the set of $d$ variables. An equivalent description of
the class ${\cal M}_d$ is that it contains all linear combinations of
$\{ e_{k,d} : 0 \leq k \leq d \}$.
The Askey-Gasper rational function is
\begin{equation} \label{eq:askey-gasper}
A(x,y,z) := \frac{1}{1 - x - y - z + 4xyz},
\end{equation}
which, in the previous notation, is $A({\bf x}) = F({\bf x})$ when $d=3$
and $Q = 1 - e_1 + 4 e_3$.
Gillis, Reznick and Zeilberger \cite{zb-pos-el83} deduce positivity of
$A$ from positivity of a $4$-variate extension due to
Koornwinder~\cite{koornwinder-pos78}, for which they give a short
elementary proof using a positivity preserving operation. Gillis,
Reznick and Zeilberger also provide an elementary proof of the
stronger result by Askey and Gasper \cite{askey-pos77} that
$A^\beta$ is positive for $\beta \geq ( \sqrt{17} - 3 ) / 2 \approx 0.56$,
by deriving a recurrence relation for the coefficients that makes positivity
apparent.
Specific functions in ${\cal M}_4$ that have shown up in the literature include
the Szeg{\H o} rational function
\begin{equation} \label{eq:szego}
S(x,y,z,w) := \frac{1}{e_3(1-x,1-y,1-z,1-w)}
\end{equation}
as well as the Lewy-Askey function
\begin{equation} \label{eq:lewy}
L(x,y,z,w) := \frac{1}{e_2(1-x,1-y,1-z,1-w)},
\end{equation}
which is a rescaled version of $1/Q({\bf x})$ with $d=4$ and
$Q = 1 - e_1 + \frac23 e_2$.
Szeg{\H o} \cite{szego-pos33} proved that \eqref{eq:szego} is positive.
In fact, he showed that $e_{d - 1, d}^{- \beta} (1 - {\bf x})$ is nonnegative
if $\beta \geq 1 / 2$. His proof relates the power series coefficients to
integrals of products of Bessel functions and, among other ingredients,
employs the Gegenbauer--Sonine addition theorem. Scott and
Sokal~\cite{ss-pos13} establish a vast and powerful generalization of
this result by showing that, if $T_G$ is the spanning-tree polynomial
of a connected series-parallel graph, then $T_G^{- \beta} (1 - {\bf x})$
is nonnegative if $\beta \geq 1 / 2$. In the simplest non-trivial case,
if $G$ is a $d$-cycle, then $T_G = e_{d - 1, d}$, thus recovering
Szeg{\H o}'s result. Relaxing the condition on $\beta$, Scott and
Sokal further extend their results to spanning-tree polynomials of
general connected graphs. They do so by realizing that Kirchhoff's
matrix-tree theorem implies that these polynomials can be expressed
as determinants, and by proving that determinants of this kind are
nonnegative. As another consequence of this determinantal nonnegativity,
Scott and Sokal conclude that~\eqref{eq:lewy} is nonnegative,
thus answering a question originating with Lewy~\cite{askey-pos72}
(with positivity replaced by nonnegativity).
Kauers and Zeilberger~\cite{kz-pos08} show that positivity of the
Lewy-Askey rational function~\eqref{eq:lewy} would follow from
positivity of the four variable function
\begin{equation} \label{eq:kz}
K(x,y,z,w) := \frac{1}{1 - e_1 + 2 e_3 + 4 e_4}.
\end{equation}
However, the conjectured positivity (or even nonnegativity)
of~\eqref{eq:kz} remains open.
As noted above, $e_{d - 1, d}^{- \beta} (1 - {\bf x})$ is nonnegative if $\beta
\geq 1 / 2$. The asymptotics of $e_{k,d}^{-\beta}(1 - {\bf x})$ are computed
in~\cite{BP-cones} for $(k,d)=(2,3)$. In the cone $2(rs + rt + st) >
r^2 + s^2 + t^2$, the coefficient $a_{r,s,t}$ is asymptotically positive
when $\beta > 1/2 = (d-k)/2$ and not when $\beta < 1/2$.
A conjecture of Scott and Sokal that remains open in both directions
is that, for general $k$ and $d$, the condition $\beta \geq (d-k)/2$
is necessary and sufficient for nonnegativity of the coefficients of
$e_{k,d}^{-\beta}(1 - {\bf x})$.
Gillis, Reznick and Zeilberger~\cite{zb-pos-el83} consider the family
\begin{equation} \label{eq:GRZ}
F_{c,d} (x_1 , \ldots , x_d) := \frac{1}{1 - e_1 + c \, e_d}
\end{equation}
of rational functions, where $c$ is a real parameter.
When $c < 0$, the coefficients are trivially positive, therefore it is usual
to assume $c > 0$. Gillis, Reznick and Zeilberger show that $F_{c,3}$ has
nonnegative coefficients if $c \leq 4$ (and this condition is shown to be
necessary in \cite{straub-pos}), but they conjecture that the threshold
for $d \geq 4$ has a different form, namely that $F_{c,d}$ has
nonnegative coefficients if and only if $c \leq d!$.
It is claimed in~\cite{zb-pos-el83}, but the proof is omitted
due to its length, that nonnegativity of $F_{d!, d}$ is
implied by nonnegativity of the diagonal power series coefficients.
In the cases $d = 4, 5, 6$, Kauers~\cite{kauers-pos07} proved
nonnegativity of these diagonal coefficients by applying cylindrical
algebraic decomposition (CAD) to the respective recurrences. On the
other hand, it is suggested in~\cite{sz-pos} that the diagonal
coefficients are eventually positive if $c < (d - 1)^{d - 1}$.
\subsection{Previous questions and results on diagonals}
The diagonal generating function $\textsf{diag}_F$ and the sequence
$\delta_n := a_{n, \ldots , n}$ it generates have received
special attention. One reason is that the question of multivariate
asymptotics in the diagonal direction is simply stated, whereas the
question of asymptotics in all possible directions requires discussion
of different possible phase regimes, a notion of uniformity over
directions, degeneracies when the coordinates are not of comparable
magnitudes, and so forth. Another reason is that there are effective
methods for determining $\textsf{diag}_F$ from $Q$, transferring the problem
to the familiar univariate realm.
We briefly recall the theory of diagonal extraction. A $d$-variate power
series $F$ is said to be D-finite if the formal derivatives
$\{ \partial_{\bf r} F : {\bf r} \in (\mathbb{Z}^+)^d \}$ form a finite
dimensional vector space over $\mathbb{C} [{\bf x}]$. In one variable, this is
equivalent to $F$ satisfying a linear differential equation with
polynomial coefficients,
$$\sum_{i=0}^k q_i (z) \frac{d^i}{dz^i} F = 0, \quad q_i \in \mathbb{C}[z] .$$
\begin{prop}[D-finite closure under diagonals~\cite{lipshitz-diagonal}]
\label{pr:lipshitz}
Let $F({\bf x})$ be a D-finite power series. Then $\textsf{diag} (z) := \sum_n
\delta_n z^n$ is D-finite, where $\delta_n := a_{n, \ldots , n}$.
\end{prop}
When $F$ is a rational function and $d=2$, it was known that $\textsf{diag}$
is algebraic (and thus D-finite) at least by the late
1960's~\cite{Furstenberg1967,hautus-klarner-diagonal},
and in special cases by P{\'o}lya in the 1920's~\cite{Polya1921}.
In the rational function $F(x,y) = P(x,y)/Q(x,y)$ one substitutes
$y=1/x$ and computes a residue integral to extract the constant coefficient.
The basis for Lipshitz' proof was the realization that the complex
integration can be viewed as purely formal. With the advent of
computer algebra this formal D-module computation was automated,
with an early package in Macaulay and more widely used modern
implementations in Magma, Mathematica and Maple. Due to advances
in software and processor speed, these computations are often completable
on functions arising in applications. Christol~\cite{Christol1984} was the first to show
that diagonals of \emph{rational} functions are D-finite.
The following relationship between D-finiteness of a univariate function
and the existence of a polynomial recursion satisfied by its coefficient
sequence is the result of translating a formal differential equation
into a relation among the coefficients.
\begin{prop} \label{pr:P-recursion}
The series $f(z) = \sum_{n \geq 0} a_n z^n$ is D-finite if and only
if it is polynomially recursive, meaning that there is a $k > 0$ and
there are polynomials $p_0, \ldots, p_k$, not all zero, such that for
all but finitely many $n$,
$$\sum_{i=0}^k p_i (n) f(n+i) = 0 \, .$$
\end{prop}
Let $f$ be a D-finite power series in one variable. If $f$ has positive
finite radius of convergence and integer coefficients, then it is a
so-called {\em G-function} and has well behaved asymptotics according
to following result.
\begin{prop}[Asymptotics of G-Function Coefficients] \label{pr:G}
Suppose $f$ is D-finite with finite radius of convergence and integer
coefficients annihilated by a minimal order linear differential operator
$\mathcal{L}$ with polynomial coefficients. Then $\mathcal{L}$ has only
regular singular points in the Frobenius sense. Consequently,
the coefficients $\{ a_n \}$ are given asymptotically by a formula
\begin{equation} \label{eq:regular}
a_n \sim \sum_\alpha C_\alpha n^{\beta_\alpha} \rho_\alpha^{-n}
(\log n)^{k_\alpha}
\end{equation}
where the sum is over quadruples $(C_\alpha, b_\alpha, \rho_\alpha, k_\alpha)$
as $\alpha$ ranges over a finite set $A$ with the following properties.
The base $\rho_\alpha$ is an algebraic number, a root of the leading
polynomial coefficient of $\mathcal{L}$. The $\beta_\alpha$ are
rational and for each value of $\rho_\alpha$ can be determined as roots
of an explicit polynomial constructed from $\rho_\alpha$ and $\mathcal{L}$.
The log powers $k_\alpha$ are nonnegative integers, zero unless for fixed
$\rho_\alpha$ there exist two values of $\beta_\alpha$ differing by an
integer (including multiplicities in the construction of $\beta_\alpha$).
The $C_\alpha$ are not in general closed form analytic expressions, but
may be determined rigorously to any desired accuracy.
\end{prop}
\begin{proof}
The discussion in~\cite[page~37]{melczer-phd}
gives references to several published results that together establish
this proposition; see also Flajolet and
Sedgewick~\cite[Section VII. 9]{flajolet-sedgewick-anacomb}.
Determination of all rational and algebraic numbers
other than $C_\alpha$ is known to be effective.
\end{proof}
Because there are computational methods for the study of diagonals,
it is of interest to reduce positivity questions to those involving
only diagonals. For the Gillis-Reznick-Zeilberger class $F_{c,d}$,
such a result is conjectured.
\begin{conj}[\cite{zb-pos-el83}] \label{conj:diagonal}
For $d \geq 4$, the following three statements are equivalent.
\begin{enumerate}[(i)]
\item $c \leq d!$
\item The diagonal coefficients of $F_{c,d}$ are nonnegative
\item All coefficients of $F_{c,d}$ are nonnegative
\end{enumerate}
\end{conj}
To be precise, $(iii) \Rightarrow (ii) \Rightarrow (i)$ is trivial
(look at $\delta_1$); nonnegativity of all coefficients of $F_{c,d}$
holds for some interval $c \in [0 , c_{\max}]$, therefore the conjecture
comes down to nonnegativity of $F_{d! , d}$. A proof for $(ii)
\Rightarrow (iii)$ in the case $c = d!$ is claimed in~\cite{zb-pos-el83}
but omitted from the paper due to length.
This question is generalized in~\cite{sz-pos} to all of ${\cal M}_d$.
\begin{question}[\protect{\cite[Question 1.1 and following]{sz-pos}}]
For $Q \in {\cal M}_d$ and $F = 1/Q$, under what conditions does
nonnegativity of the coefficients of $\textsf{diag}_F$ imply nonnegativity
of all coefficients of $F$?
\end{question}
More specifically, with nonnegativity in place of positivity, the authors
of that paper wonder whether positivity of $F$ is equivalent to positivity
of $\textsf{diag}_F$ together with positivity of $F(x_1, \ldots , x_{d-1} , 0)$.
They prove that this is true for $d=2$ and, with additional evidence,
conjecture this to be true for $d=3$ as well. Combined
with~\cite[Conjecture~1]{straub-pos} and \cite[Conjecture 3.3]{sz-pos},
we obtain the following explicit predictions on the diagonal coefficients.
\begin{conj} \label{conj:M3}
Let $F = 1/Q$ where $Q = 1 - e_1 + a e_2 + b e_3$, which is, up to rescaling,
the general element of ${\cal M}_3$.
Then $\textsf{diag}_F$ is nonnegative if and only if
\begin{equation}
b \leq \begin{cases}
6(1-a) & a \leq a_0 \\ 2-3a+2(1-a)^{3/2} & a_0 \leq a \leq 1 \\
-a^3 & a \geq 1, \end{cases}
\end{equation}
where $a_0 \approx -1.81$ is characterized by $6(1-a_0) = 2-3a_0+2(1-a_0)^{3/2}$.
\end{conj}
\subsection{Present results}
In the present work we use ACSV to answer asymptotic versions of
these questions. Aside from computing special cases, the main
new results are (1) simplification for diagonals with symmetric denominators
via the Grace-Walsh-Szeg{\H o} Theorem (Lemma~\ref{lem:GWS} below);
(2) an easy further simplification for the Gillis-Reznick-Zeilberger
class (Lemma~\ref{lem:Fcd} below); and (3) a topological computation
to explain the drop in magnitude of coefficients at critical parameter values
(Theorem~\ref{th:lacuna} below).
The first special case we look at is the diagonal of the general
element of ${\cal M}_3$, corresponding to Conjecture~\ref{conj:M3}.
\begin{theorem} \label{th:diag M3}
Let $Q = 1 - e_1 + a e_2 + b e_3$, let $F = 1/Q = \sum_{\bf r} a_{\bf r} {\bf z}^{{\bf r}}$
and let $\delta_n = a_{n, \dots , n}$ be the diagonal coefficients of $F$.
Then $\delta_n$ is eventually positive when
\begin{equation}
b < \begin{cases}
-9a & a \leq -3 \\ 2 - 3a + 2(1-a)^{3/2} & -3 \leq a \leq 1 \\
- a^3 & a \geq 1
\end{cases}
\label{eq:M3cases}
\end{equation}
while, when the inequality is reversed, $\delta_n$ attains an infinite
number of positive and negative values.
\end{theorem}
Theorem~\ref{th:diag M3} is obtained by examining asymptotic regimes,
captured in the following result.
\begin{theorem} \label{th:asm M3}
Let $Q , F,$ and $\delta_n$ be as in Theorem~\ref{th:diag M3}.
Assuming that $b$ is not equal to the piecewise function in
Equation~\eqref{eq:M3cases},
\begin{equation}
\delta_n = \sum_{x \in E}\left(\frac{x^{-3n}}{n} \cdot
\left| \frac{1-2ax-bx^2}{1-ax} \right| \cdot
\frac{1}{2\sqrt{3}(1-2x+ax^2)} \right)
\left(1+O\left(\frac{1}{n}\right)\right), \label{eq:asm M3}
\end{equation}
where $E$ consists of the minimal modulus roots of the polynomial
$Q(x,x,x)=1 - 3x + 3a x^2 + b x^3$.
\end{theorem}
The situation for eventual positivity on the diagonal when equality holds
in Equation~\eqref{eq:M3cases} is more delicate. When $a < -3$
it follows from seeing that there are two diagonal minimal points, $(r,r,r)$
and $(-r,-r,-r)$, with a greater constant at the positive point.
When $-3 < a < 1$, it follows from a dominant positive real cone point.
When $a = -3$ a quadratically degenerate smooth point at $(-1/3,-1/3,-1/3)$
may be shown via rigorous numerical diagonal extraction to dominate
the cone point at $(1/3,1/3,1/3)$, leading to alternation.
When $a = 1$, $a_{\bf r} \equiv 1$. Finally, when $a > 1$, there are
three smooth points on the unit circle, with nonnegativity conjectured
because the positive real point is degenerate and should dominate.
Our second set of results concern the diagonal of the
general element of the GRZ rational function $F_{c,d}$.
Let
\begin{equation} \label{eq:c_*}
c_* = c_* (d) := (d-1)^{d-1} \, .
\end{equation}
The following corresponds to Conjecture~\ref{conj:diagonal}.
\begin{theorem} \label{th:asym pos}
Let $d \geq 4$. Then the diagonal coefficients of $F_{c,d}$ are
eventually positive when $c < c_*$ and contain an infinite number
of positive and negative values when $c > c_*$. When $c<c_*$,
there is a conical neighborhood ${\mathcal N}$ of the diagonal
such that $a_{\bf r} > 0$ for all but finitely many ${\bf r} \in {\mathcal N}$.
\end{theorem}
Again, the result is obtained through an explicit asymptotic analysis.
\begin{theorem} \label{th:asm Fcd}
Let $\delta_n$ be the diagonal coefficients of $F_{c,d}$. Then when
$c \neq c_*$,
\[ \delta_n = \sum_{x \in E}\left(\frac{x^{-dn}}{n^{(d-1)/2}} \cdot
\left(\frac{2\pi(1-(d-1)r)}{r^{(d-1)/2}}\right)^{(d-1)/2} \cdot
\frac{1}{d^{1/2}(1-(d-1)r)}\right) \left(1+O\left(\frac{1}{n}
\right)\right), \]
where $E$ consists of the minimal modulus roots of the polynomial
$1/F_{c,d}(x,\dots,x)=1 - dx + c x^d$.
\end{theorem}
These theorems are proven in Section~\ref{sec:Fcd}, using ACSV smooth
point methods summarized in Section~\ref{sec:ACSV}, however the case
$c = c_*$ for the GRZ rational function requires the more delicate
results of Section~\ref{sec:lacuna}.
\subsection{Exponential drop and further results}
In the GRZ family, for even values of $d \geq 4$ the exponential growth
rate of the coefficients drops at the special value $c = (d-1)^{d-1}$.
This special value, and the corresponding drop in exponential growth, may
be identified for each fixed $d$ from the differential equation annihilating
the diagonal. For example, when $d=4$ an annihilating differential equation
for the diagonal of $F_{c,4}$ is computed by D-module integration in the Mathematica package
of Koutschan~\cite{Koutschan2010b}
producing the annihilating operator
$\mathcal{L}$, of order~3 and maximum coefficient degree~8, such that
$\mathcal{L} \textsf{diag}_{F_{c,4}} = 0$:
\begin{equation}
\label{eq:Fc4}
\begin{split}
\mathcal{L} &= z^2(c^4z^4+4c^3z^3+6c^2z^2+4cz-256z+1)(3cz-1)^2
\partial_z^3 \\
& + 3z(3cz-1)(6c^5z^5+15c^4z^4+8c^3z^3-6c^2z^2-384cz^2-6cz+384z-1)
\partial_z^2 \\
& + (cz+1)(63c^5z^5-3c^4z^4-66c^3z^3+18c^2z^2+720cz^2+19cz-816z+1)
\partial_z \\
& + 9c^6z^5-3c^5z^4-6c^4z^3+18c^3z^2-360c^2z^2+13c^2z-384cz+c-24.
\end{split}
\end{equation}
When $c = 27$, all coefficients in~\eqref{eq:Fc4} acquire
enough zeros at $z = 1/81$ that the quantity $(81 z - 1)^4$
may be factored out of the entire operator, leaving the following
operator of order~3 and maximum degree~4:
\begin{equation}
\begin{split}
{\mathcal L}_{27} := &z^2 (81 z^2 + 14 z + 1) \; \partial_z^3
+ 3 z (162 z^2 + 21 z + 1) \; \partial_z^2 \\
&+ (21 z + 1) (27 z + 1) \partial_z + 3 (27 z + 1).
\end{split}
\end{equation}
Asymptotics for $\delta_n$ may be extracted via the methodology
described in Proposition~\ref{pr:G}. In the special case $d=4, c=27$,
the recursion may be found on the OEIS (entry A125143) and
identifies $\{ \delta_n \}$ as the
{\em Almkvist--Zudilin numbers}\footnote{That these are the diagonals
of the rational function $F_{27, 4}$ was observed in \cite{s-apery},
where it is further conjectured that the coefficients of $F_{27,4}$
satisfy very strong congruences.}
from~\cite[sequence (4.12)($\, \delta$)]{asz-clausen}. The known
asymptotic formula implies that $|\delta_n|^{1/n} \to 9$. However,
as $c \neq 27$ approaches 27 from either side, we have
$$\lim_{c \to 27} \lim_{n \to \infty} |\delta_n|^{1/n} = 81;$$
in other words, the growth rate at $c=27$ drops suddenly from~81 to~9.
The occurrence of a phase change at $(d-1)^{d-1}$ for all $d$
and drop in exponential rate for even $d \geq 4$ had not
previously been proved. The special role of the case $c=(d-1)^{d-1}$ was
observed in~\cite[Example~4.4]{sz-pos} and claimed to agree with intuition
from hypergeometric functions.
We verify this, first by identifying the singularity from an ACSV point
of view and then by checking that this singularity indeed produces the
observed dimension drop.
\begin{theorem}[exponential growth approaching criticality] \label{th:d-1}
For all $d \geq 2$,
$$ \lim_{c \to c_*} \limsup_{n \to \infty} |\delta_n|^{1/(dn)} = d-1 \, .$$
\end{theorem}
\begin{theorem}[dimension drop at criticality] \label{th:d-1 critical}
When $c = c_*$ and $d\geq4$ is even,
$$\limsup_{n \to \infty} |\delta_n|^{1/(dn)} < d-1 \, .$$
\end{theorem}
Theorem~\ref{th:d-1 critical} is proved in Section~\ref{sec:lacuna}.
\section{ACSV} \label{sec:ACSV}
In this section we describe the basic setup for ACSV and state some
existing results. Definitions for the topological and geometric
quantities used below can be found in Pemantle and Wilson~\cite{PW-book}.
Throughout this section let $F({\bf z}) = P({\bf z})/Q({\bf z})
= \sum_{\bf r} a_{\bf r} {\bf z}^{\bf r}$ denote a rational series in $d$ variables,
with $P$ and $Q$ co-prime polynomials.
Assume that $F$ has a (finite) positive radius of convergence; that is,
$Q({\bf 0}) \neq 0$ and $P/Q$ is not a polynomial. Let ${\mathcal V} :=
\{{\bf z} \in \mathbb{C}^d : Q({\bf z}) = 0\}$ denote the singular variety for $F$ and let
${\cal M} = (\mathbb{C}^*)^d \setminus {\mathcal V}$ where $\mathbb{C}^* = \mathbb{C} \setminus \{ 0 \}$.
Coefficients $a_{\bf r}$ are extracted via the multivariate Cauchy formula
\begin{equation} \label{eq:cauchy}
a_{\bf r} = \frac{1}{(2 \pi i)^d} \int_{{\bf T}} {\bf z}^{-{\bf r}} F({\bf z}) \frac{d{\bf z}}{{\bf z}},
\end{equation}
where $d{\bf z} / {\bf z}$ denotes the holomorphic logarithmic volume form
$(dz_1 / z_1) \wedge \cdots \wedge (dz_d/z_d)$ and ${\bf T}$ denotes a
small torus (a product of sufficiently small circles about
the origin in each coordinate, so that the product of the corresponding
disks is disjoint from ${\mathcal V}$). The fundamental insight of ACSV is
that the integral depends only on the homology class of ${\bf T}$ in
$H_d({\cal M})$. Therefore, one tries to replace ${\bf T}$ by some homologous
chain ${\cal C}$ over which the integral is easier, typically via some
combination of residue reductions and saddle point estimates.
A {\em direction} of asymptotics is an element $\hat{\rr} \in (\mathbb{RP}^d)^+$;
that is, a projective vector in the positive orthant. If ${\bf r} \in (\mathbb{R}^d)^+$
we write $\hat{\rr}$ to denote the representative ${\bf r}/|{\bf r}|$ of the
projective equivalence class containing
${\bf r}$, where $|{\bf r}|=|{\bf r}|_1 := r_1+\cdots+r_d$.
Given a Whitney stratification of ${\mathcal V}$ into smooth manifolds, the
{\em critical
set} $\textsf{crit}(\hat{\rr})$ for a direction $\hat{\rr}$ is the set of ${\bf z} \in {\mathcal V}$
such that $\hat{\rr}$ is orthogonal to the tangent space of the stratum of
${\bf z}$ in ${\mathcal V}$. If ${\bf z}$ is a smooth point of ${\mathcal V}$ and $Q$ is
square-free, this means $\hat{\rr}$ should be parallel to the logarithmic gradient
$(z_1 \partial Q / \partial z_1, \ldots , z_d \partial Q / \partial z_d)$.
A {\em minimal} point for direction $\hat{\rr}$ is a point ${\bf z} \in \textsf{crit} (\hat{\rr})$
such that the open polydisk ${\cal D} ({\bf z}) := \{ {\bf w} : |w_j| < |z_j| \,
\forall 1 \leq j \leq d \}$ does not intersect ${\mathcal V}$. The minimal
point ${\bf z}$ is called {\em strictly minimal} if the closed polydisk
$\overline{{\cal D} ({\bf z})}$ intersects ${\mathcal V}$ only at ${\bf z}$.
For any $\ensuremath{\boldsymbol \beta} \in \mathbb{R}^d$,
let ${\bf T} (\ensuremath{\boldsymbol \beta}) = \{ {\bf w} : |w_j| = \exp (\beta_j) \,\, \forall \, 1 \leq
j \leq d \}$ denote the torus of points with log modulus vector $\ensuremath{\boldsymbol \beta}$.
The \emph{amoeba} of $Q({\bf z})$ is the image of ${\mathcal V}$ under the map
$\text{Relog}({\bf z}) = (\log|z_1|,\dots,\log|z_d|)$, while the
\emph{height} of a point ${\bf z}$ is $h_{\hat{\rr}}({\bf z}) = -\hat{\rr} \cdot \text{Relog}({\bf z})$.
Except in Section~\ref{sec:lacuna},
all ACSV computations are based on the following result.
\begin{theorem}[smooth point formula] \label{th:smooth}
Fix $F = P/Q = \sum_{\bf r} a_{{\bf r}} {\bf z}^{\bf r}$ and vector ${\bf r} \in (\mathbb{R}^d)^+$
in direction $\hat{\rr}$. Assume there exists $\ensuremath{\boldsymbol \beta} \in \mathbb{R}^d$ such that
the following two hypotheses hold.
\begin{enumerate}[{\bf 1}]
\item {\bf Finite critical points on the torus.} \label{i:smooth}
The set $E : = {\bf T}(\ensuremath{\boldsymbol \beta}) \cap \textsf{crit}(\hat{\rr})$ is finite, nonempty
and contains only minimal smooth points.
\item {\bf Quadratic nondegeneracy}.
At each ${\bf z} \in E$ fix $k=k({\bf z})$ such $\partial Q / \partial z_k ({\bf z}) \neq 0$
and let $z_k = g(z_1,\dots,\hat{z_k},\dots,z_d)$ be a smooth local
parametrization of $z_k$ on ${\mathcal V}$ as a function of $\{ z_j : j \neq k \}$.
We assume that the Hessian determinant ${\mathcal H}_{k({\bf z})}$ of second partial
derivatives of $g\left(w_1e^{i\theta_1},\dots,w_de^{i\theta_d}\right)$
with respect to the $\theta_j$ at the origin is non-zero for each
${\bf z} \in E$.
\end{enumerate}
Then there exists a closed neighborhood ${\mathcal N}$ of $\hat{\rr}$ in $(\mathbb{R}^d)^+$ on which all
the above hypotheses hold and, for any ${\bf r}$ with $\hat{\rr}$
in this neighborhood,
\begin{equation} \label{eq:smooth}
a_{\bf r} = ( 2 \pi)^{(1-d)/2} \sum_{{\bf z} \in E}
\det {\mathcal H}_{k({\bf z})}^{-1/2}
\frac{P({\bf z})}{z_k (\partial Q / \partial z_k)({\bf z})} r_k^{(1-d)/2} {\bf z}^{-{\bf r}}
+ O \left ( r_k^{-d/2} {\bf z}^{-{\bf r}} \right ) \, .
\end{equation}
\end{theorem}
\begin{unremark}
A number of other formulae for $a_{\bf r}$ are equivalent to this one
and hold under the same hypotheses.
An explicit formula for ${\mathcal H}_k$ in terms of partial derivatives
of $Q$ is given in~\cite[Theorem 54]{melczer-phd}. The following
coordinate-free formula for the constants involved in terms of the
complexified Gaussian curvature ${\mathcal K}$ at a smooth point ${\bf z} \in {\mathcal V}$
is given in~\cite[(9.5.2)]{PW-book} as
\begin{equation}
\label{eq:curv}
a_{\bf r} = (2 \pi)^{(1-d)/2} \left [ \sum_{{\bf z} \in E}
{\mathcal K} ({\bf z})^{-1/2} \; |{\nabla}_{\log} Q({\bf z})|^{-1} P({\bf z}) \;
|{\bf r}|^{(1-d)/2} \, {\bf z}^{-{\bf r}} \right ]
+ O \left ( |{\bf r}|^{-d/2} |{\bf z}|^{-{\bf r}} \right )
\end{equation}
\end{unremark}
\begin{proof}
Assume first that $\log |{\bf w}|$ is
the unique minimizer of ${\bf r} \cdot {\bf x}$ on the boundary of the log domain
of convergence (this being a component of the complement of the amoeba).
Under no assumptions on $E$ or ${\mathcal K}$, Theorem~9.3.2 of~\cite{PW-book}
writes the multivariate Cauchy integral~\ref{eq:cauchy} as the integral
of a residue form $\omega$ over an intersection cycle, ${\cal C}$. Taking into
account that $E$ is finite, and assuming an extra hypothesis that ${\bf r}$ is
a {\em proper direction} (see~\cite[Definition~2.3]{BP-cones}),
Theorem~9.4.2 of~\cite{PW-book}
identifies ${\cal C}$ as a sum of quasi-local cycles near the points of $E$.
For each such ${\bf z}$, if $\partial Q / \partial z_k$ and $\det {\mathcal H}_k$
do not vanish, Theorem~9.2.7 of~\cite{PW-book} identifies the integral
as the corresponding summand in~\eqref{eq:smooth}.
Nonvanishing of ${\mathcal H}_k$ is equivalent to nonvanishing of ${\mathcal K}$,
leading to the coordinate-free formula~\eqref{eq:curv}, which may be
found in~\cite[Theorem~9.3.7]{PW-book}. This proves the theorem under
an extra hypothesis on the amoeba boundary.
To remove the properness hypothesis, consider the intersection cycle ${\cal C}$
obtained from expanding the torus ${\bf T}(\ensuremath{\boldsymbol \beta} - \epsilon{\bf r})$ inside the domain of convergence
of $F$ to a torus ${\bf T}(\ensuremath{\boldsymbol \beta} + \epsilon{\bf r})$.
The construction in~\cite[Section~A4]{PW-book}
gives a compact $(d-1)$-chain representing a relative cycle in
$H_{d-1} ({\mathcal V}^{c+\varepsilon} , {\mathcal V}^{c-\varepsilon})$; that is, a chain of
maximum height $c+\varepsilon$ with maximum boundary height $c-\varepsilon$.
Applying the downward gradient flow of $h_{\hat{\rr}}$ on ${\mathcal V}$ for arbitrarily small time,
we arrive again at a chain satisfying the conclusions
of~\cite[Theorem~9.4.2]{PW-book}. Because the deformed chain
has nonvanishing boundary, one must add a term for the chain swept
out by the deformation applied to this boundary, but the elements of this
chain have height at most $c - \varepsilon$ so the resulting integral will
be within the error term above.
\end{proof}
\begin{corollary} \label{cor:asym pos}
Assume the hypotheses of Theorem~\ref{th:smooth}, and fix a vector
$\mathbf{v}$ in direction $\hat{\rr}$.
\begin{enumerate}[(i)]
\item If $E = \{ {\bf z} \}$ for some ${\bf z}$ in the positive real orthant
in $\mathbb{C}^d$ and the leading constant of Equation~\eqref{eq:smooth} is positive,
then there exists a neighbourhood of $\hat{\rr}$ such that all but finitely
many coefficients $\{ a_{\bf r} : \hat{\rr} \in {\mathcal N} \}$ are positive. \label{item:1}
\item If $E = \{ {\bf z} \}$ for some ${\bf z}$ such that
${\bf z}^{\bf v} := \prod_{j=1}^d z_j^{v_j}$ is positive real and the leading
constant of Equation~\eqref{eq:smooth} is positive,
then all but finitely many coefficients $a_{n\bf v}$ are positive.
\label{item:2}
\item If $E$ does not contain a point ${\bf z}$ with ${\bf z}^{\bf v}$
positive real and the sum in Equation~\eqref{eq:smooth} is not
identically zero, then infinitely many coefficients $a_{n\bf v}$
are positive and infinitely many $a_{n\bf v}$ are negative. \label{item:3}
\end{enumerate}
\end{corollary}
\begin{unremark}
When $E$ contains a point in the positive real orthant but it is not
a singleton, the corollary does not provide information as to eventual
positivity.
\end{unremark}
\begin{proof}
Conclusions~\eqref{item:1} and~\eqref{item:2}
follow immediately from~\eqref{eq:smooth} because the sum is a single
positive term.
For conclusion~\eqref{item:3}, grouping the elements of $E$ by
conjugate pairs we note that up to scaling by ${\bf z}^{n \bf v}n^{d/2}$
the asymptotic leading term of $a_{n\bf v}$ has the form
\[ l_n = \sum_{i=1}^{|E|}a_i \cos(2\pi \theta_i n + \beta_i), \]
where each $\theta_i,a_i,\beta_i$ is real, and $\theta_i \in (0,1)$.
If $r_n$ is any sequence satisfying a linear recurrence
relation with constant coefficients, and $r_n=O(1/n)$,
then Bell and Gerhold~\cite[Section 3]{BellGerhold2007} show that
$l_n > r_n$ infinitely often. Since the modulus of the error term
in Equation~\eqref{eq:smooth} can be bounded by a linear recurrence
sequence with growth $O(1/n)$, we see that $a_{n\bf v}$ is positive
infinitely often. Repeating the argument with $-l_n$ shows that
$a_{n\bf v}$ is negative infinitely often.
\end{proof}
Any computer algebra system can compute the set of smooth critical
points in $\textsf{crit}(\hat{\rr})$ by
solving the $d-1$ equations $({\nabla}_{\log}Q) ({\bf z}) \parallel \hat{\rr}$
together with the equation $Q({\bf z}) = 0$, where
${\nabla}_{\log}Q = \left(z_1 \partial Q/\partial z_1, \dots, z_d \partial Q/\partial z_d\right)$.
Identifying which points
in $\textsf{crit}$ are minimal is more difficult, although still
effective~\cite{MelczerSalvy2016}. For our cases, we can use results
about symmetric functions to help with the computations.
For any polynomial $Q$ in $d$ variables, let $\delta^Q$ denote the
codiagonal: the univariate polynomial defined by $\delta^Q (x)
= Q(x, \ldots , x)$.
\begin{lemma}[polynomials in ${\cal M}_d$ have diagonal minimal points] \label{lem:GWS}
Let $F = 1/Q$ with $Q \in {\cal M}_d$. Let $x$ be a zero of $\delta^Q$ of
minimal modulus. Then ${\bf x} := (x, \ldots , x)$ is a minimal point
for $F$ in $\textsf{crit} (1, \ldots , 1)$.
\end{lemma}
This follows directly from the classical Grace-Walsh-Szegő Theorem,
a modern proof of which is contained in the following.
\begin{proof}
Let $\alpha_1 , \ldots , \alpha_k$ be the roots of $\delta^Q$,
where $k \leq d$ is the common degree of $Q$ and $\delta^Q$ and
$|\alpha_1|$ is minimal among $\{ |\alpha_j| : j \leq k \}$.
For any $\varepsilon > 0$, the polynomial
$$M({\bf x}) := \prod_{j=1}^k (x_j - \alpha_j)$$
has no zeros in the polydisk ${\cal D}$ centered at the origin whose
radii are $\alpha_1 - \varepsilon$. The symmetrization of $M$
(see~\cite{borcea-branden-LYPS2})
is defined to be the multilinear symmetric function $m$ such that
$m(x,\dots,x) = M(x,\dots,x)$. In our case $M(x,\dots,x)= \delta^Q(x)$, and
it immediately follows that $m=Q$. By the
Borcea-Br{\"a}nd{\'e}n symmetrization lemma
(see~\cite[Theorem~2.1]{borcea-branden-LYPS2}), the polynomial
$Q$ has no zeros in the polydisk ${\cal D}$. We conclude that the
zero ${\bf x}$ of $Q$ is a minimal point of $F$.
\end{proof}
\section{Symmetric multilinear functions of three variables} \label{sec:M3}
In this section we determine the diagonal asymptotics for general
$Q = 1 - e_1 + a e_2 + b e_3 \in {\cal M}_3$. Taking the coefficient of $e_1$
to be~1 loses no generality because of the rescaling $x_j \to \lambda x_j$
which preserves ${\cal M}_d$ and affects coefficient asymptotics in a trivial way.
In order to use Theorem~\ref{th:smooth}, we begin by identifying
minimal points. Lemma~\ref{lem:GWS} dictates that our search should
be on the diagonal.
To that end, let $\delta^Q(x) = Q(x,x,x) = 1 - 3x + 3a x^2 + b x^3$.
The discriminant of $\delta^Q$ is a positive real multiple of
$p(a,b) := 4a^3 - 3a^2 + 6ab + b^2 - 4b = (a-1 + 3(b-1))^2 - 4 (b-1)^3$, and
the zero set of $\delta^Q$ is obtained from that of the cubic $4 b^3 = - a^2$
by centering at $(1,-1)$ and shearing via $(a,b) \mapsto (a+3b , b)$.
The discriminant $p(a,b)$ vanishes along the red curve (solid and dashed)
in Figure~\ref{fig:1}. Let $r_1 (a)$ and $r_2 (a)$ denote respectively
the upper and lower branches of the solution to $p(a,b) = 0$.
\begin{figure}
\centering
\begin{tikzpicture}[x=0.8cm,y=0.08cm]
\draw[-,color=black] (-5,0) -- (4,0);
\foreach \x in {-4,-3,-2,-1,1,2,3}\draw[shift={(\x,0)},color=black] (0pt,2pt) -- (0pt,-2pt) node[below] {\footnotesize $\x$};
\draw[-,color=black] (0,-30) -- (0,45);
\foreach \y in {-30,-20,-10,10,20,30,40}\draw[shift={(0,\y)},color=black] (2pt,0pt) -- (-2pt,0pt) node[left] {\footnotesize $\y$};
\clip(-5,-30) rectangle (4,45);
\draw[line width=1pt,color=red,smooth,samples=100,domain=-5:1,opacity=0.5, dashed] plot(\x,{-3*(\x)+2-2*sqrt(-(\x)^(3)+3*(\x)^(2)-3*(\x)+1)});
\draw[line width=1pt,color=red,smooth,samples=100,domain=-5:-3,opacity=0.5, dashed] plot(\x,{-3*(\x)+2+2*sqrt(-(\x)^(3)+3*(\x)^(2)-3*(\x)+1)});
\draw[line width=1pt,color=red,smooth,samples=100,domain=-3:1,opacity=1] plot(\x,{-3*(\x)+2+2*sqrt(-(\x)^(3)+3*(\x)^(2)-3*(\x)+1)});
\draw[line width=1pt,color=Green,smooth,samples=100,domain=-5:1,opacity=0.4, dashed] plot(\x,{-(\x)^3});
\draw[line width=1pt,color=Green,smooth,samples=100,domain=1:4,opacity=1] plot(\x,{-(\x)^3});
\draw[line width=1pt,color=blue,smooth,samples=100,domain=-5:-3,opacity=1] plot(\x,{-9*(\x)});
\draw[line width=1pt,color=blue,smooth,samples=100,domain=-3:4,opacity=0.4, dashed] plot(\x,{-9*(\x)});
\end{tikzpicture}
\caption{The three regimes defined by Proposition~\ref{pr:1},
made up of the curves {\color{blue}$b=-9a$}, {\color{red}$p(a,b)=0$},
and {\color{Green}$b=-a^3$}. Dashed lines represent the curves where they
do not determine positivity of coefficients; note smoothness in the
transitions between regimes.}
\label{fig:1}
\end{figure}
\begin{lemma}
Let $p$ be a minimal modulus root of $\delta^Q$. Then any critical point of $1/Q$
on the torus $T(p,p,p)$ has the form $(q,q,q)$ where $\delta^Q(q)=0$.
\end{lemma}
\begin{proof}
Gr{\"o}bner basis computations show nondiagonal critical points to be
permutations of $\left(\frac{1}{a},\frac{1}{a},\frac{a(1-a)}{a^2+b}\right)$,
occurring when $b = a^2 (a-2)$. When $a \leq 1$, the only time the positive
root of $\delta^Q(x)$ has modulus $1/|a|$ is the trivial case $(a,b)=(1,-1)$.
When $b = a^2(a-2)$ and $a>1$, the modulus of the product of the roots
of $\delta^Q(x)$ equals $\frac{1}{a^2(a-2)}$ and the minimal
roots of $\delta^Q(x)$ are a pair of complex conjugates. If this pair has
modulus $1/a$, then the real root of $\delta^Q(x)$ is $\pm \frac{1}{a^4(a-2)}$,
but $\delta^Q\left(\pm \frac{1}{a^4(a-2)}\right) \neq 0$ for $a>1$.
\end{proof}
Determining asymptotics is thus a matter of determining the minimal
modulus roots of $\delta^Q(x)$. The following may be proved by comparing
moduli of roots, separating cases according to the sign of $p(a,b)$.
\begin{prop} \label{pr:1}
The function $\delta^Q$ has a minimal positive real zero if and only if
$$b \leq \begin{cases}
-9a & a \leq -3 \\ r_1(a) & -3 \leq a \leq 1 \\ - a^3 & a \geq 1
\end{cases}
$$
This corresponds to the set of points lying on and below the solid curve
in Figure~\ref{fig:1}.
\end{prop}
\begin{proof}[Proof of Theorems~\ref{th:diag M3} and~\ref{th:asm M3}:]
Suppose $b$ is greater than the piecewise expression in the
proposition; then $\delta^Q$ has no minimal positive zero,
so the product of the three coordinates of the minimal points
determined above do not lie in the positive orthant.
By part~\eqref{item:3} of Corollary~\ref{cor:asym pos}, the
diagonal coefficients are not eventually positive.
Asymptotics of $\delta_n$ are determined by Theorem~\ref{th:smooth},
and when $b$ is less than the piecewise expression it can be verified
that the dominant term is positive.
\end{proof}
\section{The Gillis-Reznick-Zeilberger classes} \label{sec:Fcd}
Throughout this section, let $F = F_{c,d} = 1/Q_{c,d} = 1 / (1 - e_1 + c e_d)$
and recall that $c_* = (d-1)^{d-1}$.
Lemma~\ref{lem:GWS} implies that for $Q \in {\cal M}_d$, in the diagonal
direction, one may find diagonal minimal points. For $F_{c,d}$,
things are even simpler: all critical points for diagonal asymptotics
are diagonal points.
\begin{lemma} \label{lem:Fcd}
Let $F_{c,d} = 1/Q_{c,d}$.
If ${\bf z} \in \textsf{crit} (1, \ldots , 1)$ then $z_i = z_j$ for all $1 \leq i, j \leq d$.
\end{lemma}
\begin{proof} From $Q = Q_{c,d} = 1 - e_1 + c e_d$ we see that
$({\nabla}_{\log} Q)_j = -z_j - c e_d$ and hence that $({\nabla}_{\log} Q)_i
= ({\nabla}_{\log} Q)_j$ if and only if $z_i = z_j$.
\end{proof}
\begin{prop}[Smoothness of $F_{c,d}$ for $c \neq c_*$] \label{pr:d-1}
Let $F_{c,d} = 1/Q_{c,d}$. If $c \neq c_* $
then ${\mathcal V}$ is smooth. If $c = c_*$ then ${\mathcal V}$ fails to be smooth
at the single point ${\bf z}_* = (1/(d-1), \ldots , 1/(d-1))$. When $c = c_*$,
the singularity at ${\bf z}_*$ has tangent cone $e_2$.
\end{prop}
\begin{proof}
Checking smoothness of ${\mathcal V}$ we observe that for $d$ fixed and
$c$ and $x_1, \ldots , x_d$ variable, vanishing of the gradient
of $Q_{c,d}$ with respect to the $x$ variables implies $x_j = c e_d$
for all $j$. This common value, $x$, cannot be zero, hence $x_j \equiv x$
and $c = x^{1-d}$. Vanishing of $Q_{c,d}$ then implies vanishing of $1 - dx + x$,
hence $x = 1/(d-1)$ and $c = c_*$. This proves the first
two statements. Setting $c = c_*$ and $x_j = 1/(d-1) + y_j$ centers
$Q_{c_*,d}$ at the singularity and produces a leading term of
$(d-1) e_2 ({\bf y})$, proving the third statement.
\end{proof}
\subsection{Proof of Theorems~\protect{\ref{th:asym pos}}
and~\protect{\ref{th:asm Fcd}} in the case $c < c_*$}
When $c \leq 0$, the denominator of $F_{c,d}$ is one minus the sum
of positive monomials, which leaves no doubt as to positivity.
Assume, therefore, that $0 < c < c_*$.
Apply Lemma~\ref{lem:GWS} to see that if $x$ is a
minimum modulus zero of $\delta^Q := Q_{c,d} (x, \ldots ,x)$
then $(x, \ldots , x)$ is a minimal point for $F_{c,d}$ in the
diagonal direction. Apply Lemma~\ref{lem:Fcd} to conclude that
the set $E$ in Theorem~\ref{th:smooth} of minimal critical points
on ${\bf T} (|x|, \ldots , |x|)$ consists only of points $(y, \ldots , y)$
such that $y$ is a root of $\delta^Q$. By part~\eqref{item:1} of
Corollary~\ref{cor:asym pos}, it suffices to check that
$\delta^Q = 1 - dx + c x^d$ has a unique minimal modulus root
$\rho$ and that $\rho \in \mathbb{R}^+$. Thus, the conclusion follows
from the following proposition.
\begin{prop} \label{pr:c < c_*}
For $c \in (0,c_*)$, the polynomial $\delta^Q = 1 - dx + c x^d$ has
a root $\rho \in \left [ \frac{1}{d} , \frac{1}{d-1} \right]$
which is the unique root of $\delta^Q$ of modulus less than $1/(d-1)$.
\end{prop}
\begin{proof}
Checking signs we find that $\delta^Q (1/d) =
c d^{-d} > 0$ while $\delta^Q (1/(d-1)) = - (d-1)^{-1} + c (d-1)^{-d}
< - (d-1)^{-1} + c_* (d-1)^{-d} = 0$, therefore there is at
least one root, call it $\rho$, of $\delta^Q$ in the interval
$[1/d, 1/(d-1)]$. On the other hand, when $|z| = 1/(d-1)$, we see
that $|d z| \geq |1 + c z^d|$ and therefore, by applying Rouch{\'e}'s
theorem to the functions $-dz$ and $1 + c z^d$, we see that
$\delta^Q$ has as many zeros on $|z| < 1/(d-1)$ as does $-dz$:
precisely one root, $\rho$.
\end{proof}
\subsection{Proof of Theorems~\protect{\ref{th:asym pos}}
and~\protect{\ref{th:asm Fcd}} in the case $c > c_*$}
Again, by Lemmas~\ref{lem:GWS} and~\ref{lem:Fcd}, we may apply
part~\eqref{item:3} of Corollary~\ref{cor:asym pos} to the set $E$ of points
$(y, \ldots , y)$ for all minimal modulus roots $y$ of $\delta^Q$.
The result then reduces to the following proposition.
\begin{prop}
For $c > c_*$, the set of minimal modulus roots of the polynomial
$\delta^Q = 1 - dx + c x^d$ contains no point whose $d^{th}$ power
is real and positive.
\end{prop}
\begin{proof}
First, if $z^d$ is real then the imaginary part
of $\delta^Q (z)$ is equal to the imaginary part of $-dz$, hence
any root $z$ of $\delta^Q$ with $z^d$ real is itself real.
Next we check that $\delta^Q$ has no positive real roots.
Differentiating $\delta^Q (x)$ with respect to $x$ gives
the increasing function $d (-1 + cx^{d-1})$ with a
unique zero at $c^{-1/(d-1)}$. This gives the location of the
minimum of $\delta^Q$ on $\mathbb{R}^+$, where the function value is
$1 - d c^{-1/(d-1)} + c^{1 - d/(d-1)} = 1 - (d-1) / c^{1/(d-1)}$
which is positive because $c > (d-1)^{d-1}$.
If $d$ is even, $\delta^Q$ clearly has no negative real roots,
hence no real roots at all, finishing the proof in this case.
If $d$ is odd $\delta^Q$ will have a negative real root $u$, however
because $d$ is odd, the product of the coordinates of
$(u, \ldots , u)$ is $u^d < 0$.
\end{proof}
We conjecture that the roots of minimal modulus when $c>c_*$ are
always a complex conjugate pair, however this determination does not
affect our positivity results.
\subsection{Proof of Theorem~\protect{\ref{th:d-1}}}
When $c < c_*$ we have seen that there is a single real minimal point
$(\rho_c, \ldots , \rho_c)$ in the diagonal direction and that
$\rho_c \uparrow 1/(d-1)$ as $c \uparrow c_*^-$.
The limit from below in Theorem~\ref{th:d-1} then follows
directly from Theorem~\ref{th:asm Fcd}.
For the limit from above,
it suffices to show that in the diagonal direction,
for $c$ sufficiently close to $c_*$ and greater, $E$ consists
of a single diagonal complex conjugate pair $(\zeta_c, \ldots , \zeta_c)$
and $(\overline{\zeta_c}, \ldots , \overline{\zeta_c})$, and that
$\overline{\zeta_c} \to 1/(d-1)$ as $c_* \downarrow c$.
First, we check that at $c = c_*$ the unique minimum modulus
root of $\delta^Q$ is the doubled root at $1/(d-1)$. For $c = c_*$,
the first and third terms of $\delta^Q = 1 - dz + c_* z^d$ have modulus~1
and $1/(d-1)$ when $|z|=1/(d-1)$, respectively, summing to the modulus of
the middle term; therefore if $\delta^Q(z) = 0$ and $|z| = 1/(d-1)$ then
the third term is positive real. But then the second term must be positive
real too, hence the unique solution of modulus at most $1/(d-1)$ is
$z = 1/(d-1)$. A quick computation shows the multiplicity to be precisely~2.
We know that for $c > c_*$ there are no real roots. Therefore, as $c$
increases from $c_*$, the minimum modulus doubled root splits into two
conjugate roots, which, in a neighborhood of $c_*$, are still the only
minimum modulus roots.
\section{Lacuna computations} \label{sec:lacuna}
Theorem~\ref{th:lacuna} is the subject of forthcoming work~\cite{BMP-lacuna}.
Theorem~\ref{th:d-1 critical} follows immediately,
with the specifications: $d \geq 4$ and even, $c = c_*$, $k=1$,
$P = 1$, $Q = Q_{c,d}$, ${\bf z}_* = (1/d, \ldots , 1/d)$,
$\hat{\rr} = (1, \ldots , 1)$, $B$ is the component of the
complement of the amoeba of $Q$ containing $(a, \ldots , a)$
for $a < - \log d$, ${\bf x}_* = (-\log d, \ldots , - \log d)$,
${\bf y}_* = {\bf 0}$ and ${\mathcal N}$ taken to be the diagonal.
Proposition~\ref{pr:d-1} guarantees the correct shape for
the tangent cone to $Q$ at ${\bf z}_*$.
\begin{theorem} \label{th:lacuna}
Suppose $F = P/Q^k$ with $P$ a holomorphic function and $Q$ a real
Laurent polynomial. Fix $\hat{\rr} \in \mathbb{RP}^d$, let $B$ be a component
of the complement of the amoeba of $Q$, let $\sum_{\bf r} a_{\bf r} {\bf z}^{\bf r}$
be the Laurent expansion for $F$ convergent for ${\bf z} = \exp ({\bf x} + i {\bf y})$
and ${\bf x} \in B$. Let ${\bf x}_* \in \partial B$ be a maximizing point
for ${\bf r} \cdot {\bf x}$ on $\partial B$. Assume that ${\mathcal V}$ has a
unique singularity ${\bf z}_* = \exp ({\bf x}_* + i {\bf y}_*)$, and that
the tangent cone of $Q$ at ${\bf z}$ transforms by a real linear map
to $z_d^2 - \sum_{j=1}^{d-1} z_j^2$. Let ${\mathcal N}$ be any closed cone
such that ${\bf x}_*$ maximizes ${\bf r} \cdot {\bf x}$ for all ${\bf r} \in {\mathcal N}$.
If $d > 2k$ is even then there is an $\varepsilon > 0$ and a chain
$\Gamma$ contained in the set ${\mathcal V}_{\-\varepsilon} := \{ {\bf z} \in {\mathcal V} :
|{\bf z}^{-{\bf r}}| \leq \exp ( - {\bf r} \cdot {\bf x}_* - \varepsilon |{\bf r}|)$ such that
\begin{equation} \label{eq:int chain}
a_{\bf r} = \int_\Gamma {\bf z}^{-{\bf r}} \frac{P}{Q^k} \frac{d{\bf z}}{{\bf z}} \, .
\end{equation}
In other words, the chain of integration can be slipped below
the height of the singular point.
\end{theorem}
\begin{proof}[Sketch of proof:]
Expand the torus ${\bf T}$ of integration to ${\bf z}_*$ and just beyond.
The integral~\eqref{eq:cauchy} turns into a residue integral
over an intersection cycle swept out by the expanding torus;
see, e.g.~\cite[Appendix~A.4]{PW-book}. For small perturbations
$Q_\varepsilon$ of $Q$, the residue cycle is the union of a sphere surrounding ${\bf z}_*$
and a hyperboloid intersecting the sphere. As $Q_\varepsilon \to Q$, this
cycle may be deformed so that the sphere shrinks to a point while the
hyperboloid's neck also constricts to a point. The hyperboloid may then
be folded back on itself so that in a neighborhood of ${\bf z}_*$, the
chain vanishes, leaving a chain $\Gamma$ supported below the height
of ${\bf z}_*$.
\end{proof}
| {
"timestamp": "2018-05-01T02:09:24",
"yymm": "1804",
"arxiv_id": "1804.10929",
"language": "en",
"url": "https://arxiv.org/abs/1804.10929",
"abstract": "We consider asymptotics of power series coefficients of rational functions of the form $1/Q$ where $Q$ is a symmetric multilinear polynomial. We review a number of such cases from the literature, chiefly concerned either with positivity of coefficients or diagonal asymptotics. We then analyze coefficient asymptotics using ACSV (Analytic Combinatorics in Several Variables) methods. While ACSV sometimes requires considerable overhead and geometric computation, in the case of symmetric multilinear rational functions there are some reductions that streamline the analysis. Our results include diagonal asymptotics across entire classes of functions, for example the general 3-variable case and the Gillis-Reznick-Zeilberger (GRZ) case, where the denominator in terms of elementary symmetric functions is $1 - e_1 + c e_d$ in any number $d$ of variables. The ACSV analysis also explains a discontinuous drop in exponential growth rate for the GRZ class at the parameter value $c = (d-1)^{d-1}$, previously observed for $d=4$ only by separately computing diagonal recurrences for critical and noncritical values of $c$.",
"subjects": "Combinatorics (math.CO); Symbolic Computation (cs.SC); Number Theory (math.NT)",
"title": "Diagonal asymptotics for symmetric rational functions via ACSV",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808701643912,
"lm_q2_score": 0.7217432122827968,
"lm_q1q2_score": 0.707511064171823
} |
https://arxiv.org/abs/2205.12546 | Some equivalence relation between persistent homology and morphological dynamics | In Mathematical Morphology (MM), connected filters based on dynamics are used to filter the extrema of an image. Similarly, persistence is a concept coming from Persistent Homology (PH) and Morse Theory (MT) that represents the stability of the extrema of a Morse function. Since these two concepts seem to be closely related, in this paper we examine their relationship, and we prove that they are equal on n-D Morse functions, n $\ge$ 1. More exactly, pairing a minimum with a 1-saddle by dynamics or pairing the same 1-saddle with a minimum by persistence leads exactly to the same pairing, assuming that the critical values of the studied Morse function are unique. This result is a step further to show how much topological data analysis and mathematical morphology are related, paving the way for a more in-depth study of the relations between these two research fields. | \section{Introduction}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Grimaud2.png}
\caption{Low sensibility of dynamics to noise (extracted from~\cite{grimaud1992new}).}
\label{fig.grimaud.2}
\end{figure}
In \emph{Mathematical Morphology}~\cite{najman2013mathematical,serra1986introduction,serra2012mathematical}, \emph{dynamics}~\cite{grimaud1991geodesie,grimaud1992new,vachier1995extraction}, defined in terms of continuous paths and optimization problems, represents a very powerful tool to measure the significance of extrema in a gray-level image (see Figure~\ref{fig.grimaud.2}). Thanks to dynamics, we can efficiently select markers of objects in an image. These markers (that do not depend on the size or on the shape of objects) help to select relevant components in an image; hence, this process is a way to filter objects depending on their contrast, whatever the scale of the objects, and is often combined with the watershed~\cite{najman1996geodesic,vincent1991watersheds} for image segmentation. This contrasts with convolution filters often used in digital signal processing or morphological filters~\cite{najman2013mathematical,serra1986introduction,serra2012mathematical} where geometrical properties do matter.
\medskip
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Grimaud3.png}
\caption{The dynamics of a minimum of a given function can be computed thanks to a flooding algorithm (extracted from~\cite{grimaud1992new}).}
\label{fig.grimaud.3}
\end{figure}
Note that there exists an interesting relation between flooding algorithms and the computation of dynamics (see Figure~\ref{fig.grimaud.3}). Indeed, when we flood the topographical view of a function, at a given level, two basins merge, and the dynamics of the highest minima of the two basins is the difference between the current level of water and the altitude of this highest minima.
\medskip
Similarly, in \emph{Persistent Homology}~\cite{edelsbrunner2008persistent,edelsbrunner2000topological} well-known in \emph{Computational Topology}~\cite{edelsbrunner2010computational}, we can find the same paradigm: topological features whose~\emph{persistence} is high are \textquote{true} when the ones whose persistence is low are considered as sampling artifacts, whatever their scale. An example of application of persistence is the filtering of \emph{Morse-Smale complexes}~\cite{edelsbrunner2003hierarchical,edelsbrunner2003morse,gunther2012efficient} used in \emph{Morse Theory}~\cite{milnor1963morse,forman2002user} where pairs of extrema of low persistence are canceled for simplification purpose. This way, we obtain simplified topological representations of \emph{Morse functions}. A discrete counterpart of Morse theory, known as \emph{Discrete Morse Theory} can be found in~\cite{forman1995discrete,jollenbeck2009minimal,forman2002user,forman1998morse}.
\medskip
As detailed in~\cite{dey2007stability}, pairing by persistence of critical values can be extended in a more general setting to pairing by \emph{interval persistence} of critical points. The result is that it is possible to perform function matching based on their critical points, and then to pair all critical points of a given function (see Figure~2 in~\cite{dey2007stability}) where persistent homology does not succeed. However, due to the modification of the definition introduced in~\cite{dey2007stability}, this matching is not applicable when we consider usual threshold sets.
\medskip
In this paper, we prove that the relation between Mathematical Morphology and Persistent Homology is strong in the sense that pairing (of minima) by dynamics and pairing $1$-saddles by persistence is equivalent (and then dynamics and persistence of the corresponding pair are equal) in $n$-D ($n\geq 1$), when we work with Morse functions. For $n=1$, the proof is much simpler (with some extra condition on the limits of the domain), but contains the essence of the proof for $n\geq 1$, which is more technical. In order to ease the reading, we provide the complete proofs for both cases, first for the 1D case and then for the $n$-D case. This paper is the extension of \cite{boutry2019equivalence} (which contains the 1D case) and \cite{boutry2021equivalence} (which generalizes \cite{boutry2019equivalence} to the $n$-D case, $n\geq 1$).
\medskip
The plan of the paper is the following: Section~\ref{sec.background} recalls the mathematical background needed in this paper, Section~\ref{sec.sketches} provides sketches of the equivalence of pairing by dynamics and by persistence in 1D and in $n$-D, Section~\ref{sec.equivalence1D} contains the complete proof of the 1D equivalence, while Section~\ref{sec.equivalence} contains the complete proof of the $n$-D equivalence. In Section~\ref{sec:perspectives}, we discuss several research directions opened by the results of this paper. Section~\ref{sec.conclusion} concludes the paper.
\section{Mathematical pre-requisites}
\label{sec.background}
We call \emph{path} from $\mathbf{x}\xspace$ to $\mathbf{x}\xspace'$ both in $\mathbb{R}\xspace^n$ a continuous mapping from $[0,1]$ to $\mathbb{R}\xspace^n$. Let $\Pi_1$, $\Pi_2$ be two paths satisfying $\Pi_1(1) = \Pi_2(0)$, then we denote by $\Pi_1 <> \Pi_2$ the \emph{join} between these two paths. For any two points $\mathbf{x}\xspace^1,\mathbf{x}\xspace^2 \in \mathbb{R}\xspace^n$, we denote by $[\mathbf{x}\xspace^1,\mathbf{x}\xspace^2]$ the path:
$$\lambda \in [0,1] \rightarrow (1-\lambda) . \mathbf{x}\xspace^1 + \lambda . \mathbf{x}\xspace^2.$$
\medskip
Also, we work with $\mathbb{R}\xspace^n$ supplied with the Euclidean norm: $$\|.\|_2 : \mathbf{x}\xspace \rightarrow \|\mathbf{x}\xspace\|_2 = \sqrt{\sum_{i = 1}^n \mathbf{x}\xspace_i^2}.$$
\medskip
In the sequel, we use \emph{lower threshold sets} coming from cross-section topology~\cite{meyer1989skeletons,bertrand1996topological,beucher1992morphological} of a function $f$ defined for some real value $\lambda \in \mathbb{R}\xspace$ by:
$$[f < \lambda] = \left\{x \in \mathbb{R}\xspace^n \ \Big| \ f(x) < \lambda \right\},$$
and
$$[f \leq \lambda] = \left\{x \in \mathbb{R}\xspace^n \ \Big| \ f(x) \leq \lambda \right\}.$$
\subsection{Morse functions}
We call \emph{Morse functions} the real functions in $\ensuremath{\mathcal{C}^{\infty}(\Reals^n)\xspace}$ whose Hessian is not degenerated at \emph{critical values}, that is, where their gradient vanishes. A strong property of Morse functions is that their critical values are isolated. In particular, we call $\mathfrak{D}$-Morse\xspace functions the Morse functions which tend to $\pm\infty$ when the $2$-norm of their argument tends to $+\infty$. Note that this last property will only be used to treat the 1D case in this paper.
\begin{Lem}[Morse Lemma~\cite{audin2014morse}]
\label{lemma.morse}
Let $f : \ensuremath{\mathcal{C}^{\infty}(\Reals^n)\xspace} \rightarrow \mathbb{R}\xspace$ be a Morse function. When $x^*\xspace \in \mathbb{R}\xspace^n$ is a critical point of $f$, then there exists some neighborhood $V$ of $x^*\xspace$ and some diffeomorphism $\varphi : V \rightarrow \mathbb{R}\xspace^n$ such that $f$ is equal to a second order polynomial function of $\mathbf{x}\xspace = (x_1,\dots,x_n)$ on $V$: $\forall \; \mathbf{x}\xspace \in V$, $$f \circ \varphi^{-1} (\mathbf{x}\xspace) = f(x^*\xspace) - x_1^2 - x_2^2 - \dots - x_k^2 + x_{k+1}^2 + \dots + x_n^2.$$
\end{Lem}
We call \emph{$k$-saddle} of a Morse function a point $x \in \mathbb{R}\xspace^n$ such that the Hessian matrix has exactly $k$ strictly negative eigenvalues (and then $(n-k)$ strictly positive eigenvalues); in this case, $k$ is sometimes called the \emph{index} of $f$ at $x$. We say that a Morse function has \emph{unique critical values} when for any two different critical values $x_1,x_2 \in \mathbb{R}\xspace^n$ of $f$, we have $f(x_1) \neq f(x_2)$. (See Appendix \ref{app:ambiguities} for a discussion \nico{about} this hypothesis.)
\subsection{Pairing by dynamics (1D)}
\label{ssec.dyn}
\begin{figure}
\centering
\includegraphics[width=0.6\linewidth]{pairingbydynamics.pdf}
\caption{Example of pairing by dynamics: the abscissa $\mathbf{x_{\mathrm{min}}}\xspace$ of the red point is paired by dynamics relatively to $f$ with the abscissa $\mathbf{x}_{\max}\xspace$ of the green point on its left because the \textquote{effort} needed to reach a point of lower height than $f(\mathbf{x_{\mathrm{min}}}\xspace)$ (like the two black points) following the graph of $f$ is minimal on the left.
}
\label{fig.pairingbydynamics}
\end{figure}
Let $f : \mathbb{R}\xspace \rightarrow \mathbb{R}\xspace$ be a $\mathfrak{D}$-Morse\xspace function with unique critical values. For $\mathbf{x_{\mathrm{min}}}\xspace \in \mathbb{R}\xspace$ a local minimum of $f$, if there exists at least one abscissa $\mathbf{x_{\mathrm{min}}}\xspace' \in \mathbb{R}\xspace$ of $f$ such that $f(\mathbf{x_{\mathrm{min}}}\xspace') < f(\mathbf{x_{\mathrm{min}}}\xspace)$, then we define the \emph{dynamics}~\cite{grimaud1992new} of $\mathbf{x_{\mathrm{min}}}\xspace$ by:
$$\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) := \min_{\gamma \in C} \max_{s \in [0,1]} f(\gamma(s)) - f(\mathbf{x_{\mathrm{min}}}\xspace),$$
\noindent
where $C$ is the set of paths $\gamma : [0,1] \rightarrow \mathbb{R}\xspace$ verifying $\gamma(0) := \mathbf{x_{\mathrm{min}}}\xspace$ and verifying that there exists some $s \in ]0,1]$ such that $f(\gamma(s)) < f(\mathbf{x_{\mathrm{min}}}\xspace)$.
\medskip
Let us now define $\gamma^*$ as a path of $C$ verifying:
$$\max_{s \in [0,1]} f(\gamma^*(s)) - f(\mathbf{x_{\mathrm{min}}}\xspace) = \min_{\gamma \in C} \max_{s \in [0,1]} f(\gamma(s)) - f(\mathbf{x_{\mathrm{min}}}\xspace),$$
then we say that this path is \emph{optimal}. The real value $\mathbf{x}_{\max}\xspace$ \emph{paired by dynamics} to $\mathbf{x_{\mathrm{min}}}\xspace$ (relatively to $f$) is the local maximum of $f$ characterized by:
$$\mathbf{x}_{\max}\xspace := \gamma^*(s^*),$$
with $f(\gamma^*(s^*)) = \max_{s \in [0,1]} f(\gamma^*(s))$. We obtain then:
$$f(\mathbf{x}_{\max}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace) = \mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace).$$
Note that the local maximum $\mathbf{x}_{\max}\xspace$ of $f$ does not depend on the path $\gamma^*$ (see Figure~\ref{fig.pairingbydynamics}), and its value is unique (by hypothesis on $f$), which shows that in some way $\mathbf{x}_{\max}\xspace$ and $\mathbf{x_{\mathrm{min}}}\xspace$ are \textquote{naturally} paired by dynamics.
\subsection{Pairing by persistence (1D)}
From now on, we denote by $\overline{\mathbb{R}\xspace}\xspace := \{+\infty,-\infty\} \cup \mathbb{R}\xspace$ the complete real line, and by $\mathrm{cl}_{\realscomplete}\xspace(A)$ the closure in $\overline{\mathbb{R}\xspace}\xspace$ of the set $A \subseteq \mathbb{R}\xspace$.
\medskip
\begin{algorithm}[h]
\caption{Pairing by persistence of $\mathbf{x}_{\max}\xspace$.}
\label{algo.pairingbypersistence}
$\mathbf{x_{\mathrm{min}}}\xspace := \emptyset$\;
$[\mathbf{x}_{\max}\xspace^-,\mathbf{x}_{\max}\xspace^+] := \mathrm{cl}_{\realscomplete}\xspace (\mathcal{CC}\xspace([f \leq f(\mathbf{x}_{\max}\xspace)],\mathbf{x}_{\max}\xspace))$\;
\If{$\mathbf{x}_{\max}\xspace^- > -\infty \; \| \; \mathbf{x}_{\max}\xspace^+ < +\infty$}{
$\mathbf{x_{\mathrm{min}}}\xspace^- := \mathrm{rep}\xspace([\mathbf{x}_{\max}\xspace^-,\mathbf{x}_{\max}\xspace],f)$\;
$\mathbf{x_{\mathrm{min}}}\xspace^+ := \mathrm{rep}\xspace([\mathbf{x}_{\max}\xspace,\mathbf{x}_{\max}\xspace^+],f)$\;
\If{$\mathbf{x}_{\max}\xspace^- > -\infty \; \& \& \; \mathbf{x}_{\max}\xspace^+ < +\infty$}{
$\mathbf{x_{\mathrm{min}}}\xspace := {\arg \max}_{x \in \{\mathbf{x_{\mathrm{min}}}\xspace^-,\mathbf{x_{\mathrm{min}}}\xspace^+\}} f(x)$\;
}
\If{$\mathbf{x}_{\max}\xspace^- > -\infty \; \&\& \; \mathbf{x}_{\max}\xspace^+ = +\infty$}{
$\mathbf{x_{\mathrm{min}}}\xspace := \mathbf{x_{\mathrm{min}}}\xspace^-$\;
}
\If{$\mathbf{x}_{\max}\xspace^- = -\infty \; \&\& \; \mathbf{x}_{\max}\xspace^+ < +\infty$}{
$\mathbf{x_{\mathrm{min}}}\xspace := \mathbf{x_{\mathrm{min}}}\xspace^+$\;
}
}
\return $\mathbf{x_{\mathrm{min}}}\xspace$\;
\end{algorithm}
\begin{figure}
\centering
\includegraphics[width=0.6\linewidth]{persistencebis.pdf}
\caption{Example of pairing by persistence: the abscissa $\mathbf{x}_{\max}\xspace$ of the local maximum in red is paired by persistence relatively to $f$ with the abscissa $\mathbf{x_{\mathrm{min}}}\xspace$ of the local minimum in green, since its image by $f$ is greater than the image by $f$ of the abscissa $\mathbf{x_{\mathrm{min}}}\xspace^2\xspace$ of the local minimum drawn in pink.
}
\label{fig.persistence}
\end{figure}
Let $f : \mathbb{R}\xspace \rightarrow \mathbb{R}\xspace$ be a $\mathfrak{D}$-Morse\xspace function with unique critical values, and let $\mathbf{x}_{\max}\xspace$ be a local maximum of $f$. Let us recall the 1D procedure~\cite{edelsbrunner2008persistent} which pairs (relatively to $f$) local maxima to local minima (see Algorithm~\ref{algo.pairingbypersistence}). Roughly speaking, the representatives $\mathbf{x_{\mathrm{min}}}\xspace^-$ and $\mathbf{x_{\mathrm{min}}}\xspace^+$ are the abscissas where connected components of respectively $$[f \leq (f(\mathbf{x_{\mathrm{min}}}\xspace^-)]\text { and }[f \leq (f(\mathbf{x_{\mathrm{min}}}\xspace^+)]$$ \textquote{emer\-ge} (see Figure~\ref{fig.persistence}), when $\mathbf{x}_{\max}\xspace$ is the abscissa where two connected components of $[f < f(\mathbf{x}_{\max}\xspace)]$ \textquote{mer\-ge} into a single component of $[f \leq f(\mathbf{x}_{\max}\xspace)]$. Pairing by persistence associates then $\mathbf{x}_{\max}\xspace$ to the value $\mathbf{x_{\mathrm{min}}}\xspace$ belonging to $\{\mathbf{x_{\mathrm{min}}}\xspace^-,\mathbf{x_{\mathrm{min}}}\xspace^+\}$ which maximizes $f(\mathbf{x_{\mathrm{min}}}\xspace)$. The \emph{persistence} of $\mathbf{x}_{\max}\xspace$ relatively to $f$ is then equal to $\mathrm{Per}\xspace(\mathbf{x}_{\max}\xspace) := f(\mathbf{x}_{\max}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$.
\subsection{Pairing by dynamics ($n$-D)}
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{dynpair.png}
\caption{Pairing by dynamics on a Morse function: the red and blue paths are both in $(D_{\xmin})\xspace$ but only the blue one reaches a point $\mathbf{x_{<}}\xspace$ whose height is lower than $f(\mathbf{x_{\mathrm{min}}}\xspace)$ with a minimal effort.}
\label{fig.dynpair}
\end{figure}
From now on, $f : \mathbb{R}\xspace^n \rightarrow \mathbb{R}\xspace$ is a Morse function with unique critical values.
\medskip
Let $\mathbf{x_{\mathrm{min}}}\xspace$ be a local minimum of $f$. Then we call \emph{set of descending paths starting from $\mathbf{x_{\mathrm{min}}}\xspace$} (shortly $(D_{\xmin})\xspace$) the set of paths going from $\mathbf{x_{\mathrm{min}}}\xspace$ to some element $\mathbf{x_{<}}\xspace \in \mathbb{R}\xspace^n$ satisfying $f(\mathbf{x_{<}}\xspace) < f(\mathbf{x_{\mathrm{min}}}\xspace)$.
\medskip
The \emph{effort} of a path $\Pi : [0,1] \rightarrow \mathbb{R}\xspace^n$ (relatively to $f$) is equal to:
$$\mathrm{Effort}\xspace(\Pi) := \max_{\ell \in [0,1], \ell' \in [0,1]} (f(\Pi(\ell)) - f(\Pi(\ell'))).$$
\medskip
A local minimum $\mathbf{x_{\mathrm{min}}}\xspace$ of $f$ is said to be \emph{matchable} if there exists some $\mathbf{x_{<}}\xspace \in \mathbb{R}\xspace^n$ such that $f(\mathbf{x_{<}}\xspace) < f(\mathbf{x_{\mathrm{min}}}\xspace)$. We call \emph{dynamics} of a matchable local minimum $\mathbf{x_{\mathrm{min}}}\xspace$ of $f$ the value:
$$\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) = \min_{\Pi \in (D_{\xmin})\xspace} \max_{\ell \in [0,1]} \left(f(\Pi(\ell)) - f(\mathbf{x_{\mathrm{min}}}\xspace) \right),$$
and we say that $\mathbf{x_{\mathrm{min}}}\xspace$ is \emph{paired by dynamics} (see Figure~\ref{fig.dynpair}) with some $1$-saddle $\mathbf{x_{\mathrm{sad}}}\xspace \in \mathbb{R}\xspace^n$ of $f$ when:
$$\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) = f(\mathbf{x_{\mathrm{sad}}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace).$$
\medskip
An \emph{optimal} path $\Pi^{\mathrm{opt}}\xspace$ is an element of $(D_{\xmin})\xspace$ whose effort is equal to $\min_{\Pi \in (D_{\xmin})\xspace}(\mathrm{Effort}\xspace(\Pi))$. Note that for any local minimum $\mathbf{x_{\mathrm{min}}}\xspace$ of $f$, there always exists some optimal path $\Pi^{\mathrm{opt}}\xspace$ such that: $$\mathrm{Effort}\xspace(\Pi^{\mathrm{opt}}\xspace) = \mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace).$$
\medskip
Thanks to the uniqueness of critical values of $f$, there exists only one critical point of $f$ which can be paired with $\mathbf{x_{\mathrm{min}}}\xspace$ by dynamics.
\medskip
Dynamics are always positive, and the dynamics of an absolute minimum of $f$ is set at $+\infty$ (by convention).
\subsection{Pairing by persistence ($n$-D)}
\begin{figure}
\centering
\includegraphics[width=0.45\linewidth]{perpair.png}~
\includegraphics[width=0.45\linewidth]{merge.png}
\caption{Pairing by persistence on a Morse function: we compute the plane whose height is reaching $f(\mathbf{x_{\mathrm{sad}}}\xspace)$ (see the left side), which allows us to compute $C^{\mathrm{sad}}\xspace$, to deduce the components $C^I_i\xspace$ whose closure contains $\mathbf{x_{\mathrm{sad}}}\xspace$, and to decide which representative is paired with $\mathbf{x_{\mathrm{sad}}}\xspace$ by persistence by choosing the one whose height is the greatest. We can also observe (see the right side) the \emph{merge phase} where the two components merge and where the component whose representative is paired with $\mathbf{x_{\mathrm{sad}}}\xspace$ dies.}
\label{fig.perpair}
\end{figure}
Let us denote by $\mathrm{clo}\xspace$ the closure operator, which adds to a subset of $\mathbb{R}\xspace^n$ all its accumulation points, and by $\mathcal{CC}\xspace(X)$ the connected components of a subset $X$ of $\mathbb{R}\xspace^n$. We also define the \emph{representative} of a subset $X$ of $\mathbb{R}\xspace^n$ relatively to a Morse function $f$ the point which minimizes $f$ on $X$:
$$\mathrm{rep}\xspace(X) = {\arg \min}_{\mathbf{x}\xspace \in X} f(\mathbf{x}\xspace).$$
\medskip
\begin{Def}
\label{def.persistence}
Let $f$ be some Morse function with unique critical values, and let $\mathbf{x_{\mathrm{sad}}}\xspace$ be the abscissa of some $1$-saddle point of $f$. Now we define the following expressions. First, $$C^{\mathrm{sad}}\xspace = \mathcal{CC}\xspace([f \leq f(\mathbf{x_{\mathrm{sad}}}\xspace)],\mathbf{x_{\mathrm{sad}}}\xspace)$$ denotes the component of the set $[f \leq f(\mathbf{x_{\mathrm{sad}}}\xspace)]$ which contains $\mathbf{x_{\mathrm{sad}}}\xspace$. Second, we denote by:
$$\{C^I_i\xspace\}_{i \in I} = \mathcal{CC}\xspace([f < f(\mathbf{x_{\mathrm{sad}}}\xspace)])$$
the connected components of the open set $[f < f(\mathbf{x_{\mathrm{sad}}}\xspace)]$. Third, we define
$$\{C_i^{\mathrm{sad}}\xspace\}_{i \in I^{\mathrm{sad}}\xspace} = \left\{ C^I_i\xspace \ |\ \mathbf{x_{\mathrm{sad}}}\xspace \in \mathrm{clo}\xspace(C^I_i\xspace) \right\}$$
the subset of components $C^I_i\xspace$ whose closure contains $\mathbf{x_{\mathrm{sad}}}\xspace$. Fourth, for each $i \in I^{\mathrm{sad}}\xspace$, we denote $$\mathrm{rep}\xspace_i = {\arg \min}_{x \in C_i^{\mathrm{sad}}\xspace} f(x)$$ the representative of $C_i^{\mathrm{sad}}\xspace$. Fifth, we define the abscissa
$$\mathbf{x_{\mathrm{min}}}\xspace = \mathrm{rep}\xspace_{i_{\mathrm{paired}}\xspace}$$
with
$$i_{\mathrm{paired}}\xspace = {\arg \max}_{i \in I^{\mathrm{sad}}\xspace} f(\mathrm{rep}\xspace_i),$$
thus $\mathbf{x_{\mathrm{min}}}\xspace$ is the representative of the component $C_i^{\mathrm{sad}}\xspace$ of minimal depth. In this context, we say that $\mathbf{x_{\mathrm{sad}}}\xspace$ is \emph{paired by persistence} to $\mathbf{x_{\mathrm{min}}}\xspace$. Then, the \emph{persistence} of $\mathbf{x_{\mathrm{sad}}}\xspace$ is equal to:
$$\mathrm{Per}\xspace(\mathbf{x_{\mathrm{sad}}}\xspace) = f(\mathbf{x_{\mathrm{sad}}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace).$$
\end{Def}
\section{Sketches of the proofs (1D vs. $n$-D)}
\label{sec.sketches}
\subsection{Pairing by dynamics implies pairing by persistence}
\begin{table*}[!htbp]
\centering
\caption{Sketches of the 1D/$n$-D proofs that pairing by dynamics implies pairing by persistence.}
\begin{tabular}{l|c|l}
\toprule
\multicolumn{3}{c}{Hypotheses:}\\
\toprule
\toprule
$f$ is a $\mathfrak{D}$-Morse\xspace function & & $f$ is a Morse function\\
\midrule
\multicolumn{3}{c}{$f$ has unique critical values}\\
\midrule
\multicolumn{3}{c}{$\mathbf{x_{\mathrm{min}}}\xspace$ is a local minimum of $f$}\\
\midrule
\multicolumn{3}{c}{$\mathbf{x_{\mathrm{min}}}\xspace$ and $\mathbf{x}_{\mathrm{max/sad}}\xspace$ are paired by \textbf{dynamics}}\\
\midrule
$\mathbf{x}_{\max}\xspace > \mathbf{x_{\mathrm{min}}}\xspace$ & \hspace{1em} & $\mathbf{x_{\mathrm{min}}}\xspace \neq \mathbf{x_{\mathrm{sad}}}\xspace$\\
\toprule
\toprule
\multicolumn{3}{c}{Notations}\\
\toprule
\toprule
$[\xmax^-\xspace,\xmax^+\xspace] = \mathrm{cl}_{\realscomplete}\xspace (\mathcal{CC}\xspace([f \leq f(\mathbf{x}_{\max}\xspace)],\mathbf{x}_{\max}\xspace))$ & & $C^{\mathrm{sad}}\xspace = \mathcal{CC}\xspace([f \leq f(\mathbf{x_{\mathrm{sad}}}\xspace)],\mathbf{x_{\mathrm{sad}}}\xspace)$\\
\midrule
& & $\{C^I_i\xspace\}_{i \in I} = \mathcal{CC}\xspace([f < f(\mathbf{x_{\mathrm{sad}}}\xspace)])$\\
\midrule
& & $\{C_i^{\mathrm{sad}}\xspace\}_{i \in I^{\mathrm{sad}}\xspace} = \left\{ C^I_i\xspace \ |\ \mathbf{x_{\mathrm{sad}}}\xspace \in \mathrm{clo}\xspace(C^I_i\xspace) \right\}$\\
\bottomrule
\bottomrule
\multicolumn{3}{c}{Step 1:}\\
\midrule
& & $\exists\ i \in I$ s.t. $\mathbf{x_{\mathrm{min}}}\xspace \in C^I_i\xspace$\\
$\mathbf{x_{\mathrm{min}}}\xspace$ represents $[\xmax^-\xspace,\mathbf{x}_{\max}\xspace]$ & & with $\mathbf{x_{\mathrm{min}}}\xspace$ representing $C^I_i\xspace$\\
\midrule
\multicolumn{3}{c}{(otherwise $\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) < f(\mathbf{x}_{\mathrm{max/sad}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$ which leads to a contradiction)}\\
\midrule
& & $C^I_i\xspace$ belongs to $\{C_i^{\mathrm{sad}}\xspace\}_{i \in I^{\mathrm{sad}}\xspace}$\\
& & then $\mathbf{x_{\mathrm{min}}}\xspace$ represents some $C_{\imin}^{\mathrm{sad}}\xspace$\\
\bottomrule
\bottomrule
\multicolumn{3}{c}{Step 2:}\\
\toprule
\toprule
$f(\mathrm{rep}\xspace([\mathbf{x}_{\max}\xspace,\xmax^+\xspace],f)) < f(\mathbf{x_{\mathrm{min}}}\xspace)$ & & $\forall \ i \neq i_{\mathrm{min}}\xspace$, $f(\mathrm{rep}\xspace(C_i^{\mathrm{sad}}\xspace,f) < f(\mathbf{x_{\mathrm{min}}}\xspace)$\\
\midrule
\multicolumn{3}{c}{(otherwise $\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) > f(\mathbf{x}_{\mathrm{max/sad}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$ which leads to a contradiction)}\\
\bottomrule
\bottomrule
\multicolumn{3}{c}{Step 3:}\\
\toprule
\toprule
\multicolumn{3}{c}{$\mathbf{x_{\mathrm{min}}}\xspace$ and $\mathbf{x}_{\mathrm{max/sad}}\xspace$ are paired by persistence}\\
\bottomrule
\end{tabular}
\label{table.dyn.to.per}
\end{table*}
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{sketch1bis.pdf}
\caption{Pairing by dynamics implies pairing by persistence in 1D: when $\mathbf{x_{\mathrm{min}}}\xspace$ (in black) is paired with $\mathbf{x}_{\max}\xspace$ (in purple) by dynamics, we observe easily that $\mathbf{x_{\mathrm{min}}}\xspace$ is the representative of the basin where it lies. Furthermore, the optimal path descending lower than $f(\mathbf{x_{\mathrm{min}}}\xspace)$ goes on the right side and goes through $\mathbf{x}_{\max}\xspace$ (since we look for a minimal effort and $f(\mathbf{x}_{\max}\xspace^2\xspace)$ is greater than $f(\mathbf{x}_{\max}\xspace)$). This implies that the right basin contains a representative lower than $f(\mathbf{x_{\mathrm{min}}}\xspace)$. Since $\mathcal{CC}\xspace([f \leq f(\mathbf{x}_{\max}\xspace)],\mathbf{x}_{\max}\xspace)$ is made of the two described basins, we obtain easily that $\mathbf{x}_{\max}\xspace$ is paired with $\mathbf{x_{\mathrm{min}}}\xspace$ by persistence.}
\label{fig:sketch1D-dyn-per}
\end{figure}
Let us start from the 1D case (see Figure~\ref{fig:sketch1D-dyn-per}). We assume (see Table~\ref{table.dyn.to.per}) that we have some Morse function $f$ defined on the real line and that the critical values are unique, that is, for two different extrema $x_1,x_2$ of $f$, we have $f(x_1) \neq f(x_2)$. Furthermore, we assume that the abscissas $\{\mathbf{x_{\mathrm{min}}}\xspace,\mathbf{x}_{\max}\xspace\}$ with $\mathbf{x}_{\max}\xspace > \mathbf{x_{\mathrm{min}}}\xspace$ are paired by dynamics, that is, starting from $\mathbf{x_{\mathrm{min}}}\xspace$ and following the graph of $f$, the lower effort to reach a lower value is on the right side. Using these properties, we want to show that $\mathbf{x}_{\max}\xspace$ and $\mathbf{x_{\mathrm{min}}}\xspace$ are paired by persistence.
\medskip
\textbf{\underline{1D proof:}} Let us proceed in three steps. First, we want to show that $\mathbf{x_{\mathrm{min}}}\xspace$ is the representative of the basin $[\xmax^-\xspace,\mathbf{x}_{\max}\xspace]$ of level $f(\mathbf{x}_{\max}\xspace)$ containing it. This is easily proven by contradiction: if $\mathbf{x_{\mathrm{min}}}\xspace$ is not the representative of this basin, there exists some $x^*\xspace$ in it where $f(x^*\xspace) < f(\mathbf{x_{\mathrm{min}}}\xspace)$, and then the dynamics of $\mathbf{x_{\mathrm{min}}}\xspace$ is lower than $f(\mathbf{x}_{\max}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$, which is impossible by hypothesis.
\medskip
Now that we know that $\mathbf{x_{\mathrm{min}}}\xspace$ represents the basin $[\xmax^-\xspace,\mathbf{x}_{\max}\xspace]$, we can show that $f(\mathbf{x_{\mathrm{min}}}\xspace)$ is greater than the image by $f$ of the representative of $[\mathbf{x}_{\max}\xspace,\xmax^+\xspace]$ corresponding also to the lower threshold set $[f \leq f(\mathbf{x}_{\max}\xspace)]$. By assuming the contrary, we would imply that any descending path starting from $\mathbf{x_{\mathrm{min}}}\xspace$ would go outside the component $[\xmax^-\xspace,\xmax^+\xspace] = \mathcal{CC}\xspace([f \leq f(\mathbf{x}_{\max}\xspace),\mathbf{x}_{\max}\xspace])$, which means that we would obtain a dynamics of $\mathbf{x_{\mathrm{min}}}\xspace$ greater than $f(\mathbf{x}_{\max}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$, which is impossible.
\medskip
Since we have obtained that $\mathbf{x_{\mathrm{min}}}\xspace$ is the representative of the highest basin starting for the extrema $\mathbf{x}_{\max}\xspace$, we can conclude easily that $\mathbf{x}_{\max}\xspace$ is paired with $\mathbf{x_{\mathrm{min}}}\xspace$ by persistence.
\medskip
\textbf{\underline{$n$-D proof:}} The proof in $n$-D, $n \geq 2$, is very similar, except that we have more complex notations. Indeed, we study $1$-saddles instead of maxima; the path between the two points is not \textquote{unique} anymore; and we do not have anymore a natural order between two abscissas.
\medskip
We cannot define $\xmax^-\xspace$ and $\xmax^+\xspace$, but instead we can define the closed connected component $\mathcal{CC}\xspace([f \leq f(\mathbf{x}_{\max}\xspace)],\mathbf{x}_{\max}\xspace)$ containing $\mathbf{x}_{\max}\xspace$. Also, we cannot define $]\xmax^-\xspace,\mathbf{x}_{\max}\xspace[$ or $]\mathbf{x}_{\max}\xspace,\xmax^+\xspace[$ but instead we can define the connected components $C^I_i\xspace$ which are components of $[f < f(\mathbf{x_{\mathrm{sad}}}\xspace)]$, and the components $C_i^{\mathrm{sad}}\xspace$ of $[f < f(\mathbf{x_{\mathrm{sad}}}\xspace)]$ with the additional property that their closure contains $\mathbf{x_{\mathrm{sad}}}\xspace$. Last point, we do not need anymore the condition that the studied function tends to infinity when the norm of the abscissa tends to infinity, but the consequence is that the proof is a little more complex.
\medskip
After having introduced these notations, we can follow the same three steps as before. We first prove that $\mathbf{x_{\mathrm{min}}}\xspace$, paired to $\mathbf{x_{\mathrm{sad}}}\xspace$ by dynamics, is the representative of some $C^I_i\xspace$ (otherwise we would obtain that the dynamics of $\mathbf{x_{\mathrm{min}}}\xspace$ is lower than $f(\mathbf{x_{\mathrm{sad}}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$ since we can reach a point on the graph of $f$ which is lower than $f(\mathbf{x_{\mathrm{min}}}\xspace)$). Then, the proof that this $C^I_i\xspace$ is in fact one of the $C_i^{\mathrm{sad}}\xspace$ follows from the fact that otherwise, any descending path of $\mathbf{x_{\mathrm{min}}}\xspace$ must go out of $C^I_i\xspace$ to reach a lower value than $f(\mathbf{x_{\mathrm{min}}}\xspace)$, and then the dynamics of $\mathbf{x_{\mathrm{min}}}\xspace$ would be greater than $f(\mathbf{x_{\mathrm{sad}}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$.
\medskip
Now that we know that $\mathbf{x_{\mathrm{min}}}\xspace$ belongs to some $C_i^{\mathrm{sad}}\xspace$, we can use the property that there exists exactly two basins in the component $\mathcal{CC}\xspace([f \leq f(\mathbf{x_{\mathrm{sad}}}\xspace)],\mathbf{x_{\mathrm{sad}}}\xspace)$ (since we work with a Morse function). By assuming that $\mathbf{x_{\mathrm{min}}}\xspace$ is not the highest representative among the open components $C_i^{\mathrm{sad}}\xspace$, we obtain one more time that any path starting from $\mathbf{x_{\mathrm{min}}}\xspace$ must go outside $$\mathcal{CC}\xspace([f \leq f(\mathbf{x_{\mathrm{sad}}}\xspace)],\mathbf{x_{\mathrm{sad}}}\xspace)$$ to descend lower than $f(\mathbf{x_{\mathrm{min}}}\xspace)$, which would lead to a greater dynamics than $f(\mathbf{x_{\mathrm{sad}}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$. Thus, $\mathbf{x_{\mathrm{min}}}\xspace$ is the highest representative among the ones of the components $\{C_i^{\mathrm{sad}}\xspace\}_i$.
\medskip
We conclude one more time that $\mathbf{x_{\mathrm{sad}}}\xspace$ is paired to $\mathbf{x_{\mathrm{min}}}\xspace$ by persistence when $\mathbf{x_{\mathrm{min}}}\xspace$ is paired to $\mathbf{x_{\mathrm{sad}}}\xspace$ by dynamics.
\subsection{Pairing by persistence implies pairing by dynamics}
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{sketch2bis.pdf}
\caption{Pairing by persistence implies pairing by dynamics in 1D: starting from the local maximum $\mathbf{x}_{\max}\xspace$ (in black), we define the component $[\xmax^-\xspace,\xmax^+\xspace]$ of the lower threshold set of $f$ which contains $\mathbf{x}_{\max}\xspace$. By definition of pairing by persistence, we know that the representative of the component $[\xmax^-\xspace,\mathbf{x}_{\max}\xspace]$ is $\mathbf{x_{\mathrm{min}}}\xspace$ drawn in purple (since $\mathbf{x_{\mathrm{min}}}\xspace < \mathbf{x}_{\max}\xspace$) and we call $\nico{x^{\forall}_{\mathrm{min}}\xspace}$ (drawn in red) the representative of the component $[\mathbf{x}_{\max}\xspace,\xmax^+\xspace]$. From these facts, we deduce easily that $\mathbf{x_{\mathrm{min}}}\xspace$ is matchable since $f(\nico{x^{\forall}_{\mathrm{min}}\xspace}) < f(\mathbf{x_{\mathrm{min}}}\xspace)$. We also deduce that there exists a descending path from $\mathbf{x_{\mathrm{min}}}\xspace$ to $\mathbf{x}_{\max}\xspace$ to $\nico{x^{\forall}_{\mathrm{min}}\xspace}$ which lies inside $[\xmax^-\xspace,\xmax^+\xspace]$ and then its associated effort is equal to $f(\mathbf{x}_{\max}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$, which means that the dynamics of $\mathbf{x_{\mathrm{min}}}\xspace$ is lower than or equal to this same value. Additionally, we can show that every optimal path connects $\mathbf{x_{\mathrm{min}}}\xspace$ to $\mathbf{x}_{\max}\xspace$ and thus the dynamics of $\mathbf{x_{\mathrm{min}}}\xspace$ is greater than or equal to $f(\mathbf{x}_{\max}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$. It is then easy to conclude that the dynamics of $\mathbf{x_{\mathrm{min}}}\xspace$ is equal to $f(\mathbf{x}_{\max}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$, and then by \nico{uniqueness} of the critical values, $\mathbf{x_{\mathrm{min}}}\xspace$ is paired with $\mathbf{x}_{\max}\xspace$ by dynamics.}
\label{fig:sketch1D-per-dyn}
\end{figure}
\begin{table*}[!htbp]
\centering
\caption{Sketches of the 1D/$n$-D proofs that pairing by persistence implies pairing by dynamics.}
\begin{tabular}{l|c|l}
\toprule
\multicolumn{3}{c}{Hypotheses:}\\
\toprule
\toprule
$f$ is a $\mathfrak{D}$-Morse\xspace function & & $f$ is a Morse function\\
\midrule
\multicolumn{3}{c}{$f$ has unique critical values}\\
\midrule
\multicolumn{3}{c}{$\mathbf{x}_{\mathrm{max/sad}}\xspace$ is a local maximum/$1$-saddle of $f$}\\
\midrule
\multicolumn{3}{c}{$\mathbf{x}_{\mathrm{max/sad}}\xspace$ and $\mathbf{x_{\mathrm{min}}}\xspace$ are paired by \textbf{persistence}}\\
\midrule
$\mathbf{x}_{\max}\xspace > \mathbf{x_{\mathrm{min}}}\xspace$ & \hspace{1em} & $\mathbf{x_{\mathrm{min}}}\xspace \neq \mathbf{x_{\mathrm{sad}}}\xspace$\\
\toprule
\toprule
\multicolumn{3}{c}{Notations:}\\
\toprule
\toprule
$[\xmax^-\xspace,\xmax^+\xspace] = \mathcal{CC}\xspace([f \leq f(\mathbf{x}_{\max}\xspace)],\mathbf{x}_{\max}\xspace)$ & & $C^{\mathrm{sad}}\xspace = \mathcal{CC}\xspace([f \leq f(\mathbf{x_{\mathrm{sad}}}\xspace)],\mathbf{x_{\mathrm{sad}}}\xspace)$\\
\midrule
$\nico{x^{\forall}_{\mathrm{min}}\xspace} = \mathrm{rep}\xspace([\xmax^-\xspace,\xmax^+\xspace],f)$ & & $\{C^I_i\xspace\}_{i \in I} = \mathcal{CC}\xspace([f < f(\mathbf{x_{\mathrm{sad}}}\xspace)])$\\
\midrule
& & $\{C_i^{\mathrm{sad}}\xspace\}_{i \in I^{\mathrm{sad}}\xspace} = \left\{ C^I_i\xspace \ |\ \mathbf{x_{\mathrm{sad}}}\xspace \in \mathrm{clo}\xspace(C^I_i\xspace) \right\}$\\
\midrule
& & $i_{\mathrm{min}}\xspace \in I^{\mathrm{sad}}\xspace$ s.t. $\mathbf{x_{\mathrm{min}}}\xspace$ represents $C_{\imin}^{\mathrm{sad}}\xspace$\\
\bottomrule
\bottomrule
\multicolumn{3}{c}{Step 1:}\\
\toprule
\toprule
$\gamma := [\mathbf{x_{\mathrm{min}}}\xspace,\nico{x^{\forall}_{\mathrm{min}}\xspace}]$ & & $\mathrm{Card}\xspace(I^{\mathrm{sad}}\xspace) > 1$\\
with $f(\nico{x^{\forall}_{\mathrm{min}}\xspace}) < f(\mathbf{x_{\mathrm{min}}}\xspace)$ & & $\Rightarrow$ $\exists \ \mathbf{i_{<}}\xspace \in I^{\mathrm{sad}}\xspace$, $\exists \ \mathbf{x_{<}}\xspace \in C_{\ILOWER}^{\mathrm{sad}}\xspace$,\\
& & \hspace{0.4cm} s.t. $f(\mathbf{x_{<}}\xspace) < f(\mathbf{x_{\mathrm{min}}}\xspace)$\\
\midrule
\multicolumn{3}{c}{$\mathbf{x_{\mathrm{min}}}\xspace$ is matchable}\\
\bottomrule
\bottomrule
\multicolumn{3}{c}{Step 2:}\\
\toprule
\toprule
$\gamma$ is a descending path & & $\exists\ \gamma_1$ from $\mathbf{x_{\mathrm{min}}}\xspace$ to $\mathbf{x_{\mathrm{sad}}}\xspace$ in $C_{\imin}^{\mathrm{sad}}\xspace$\\
& & $\forall\ i \in I^{\mathrm{sad}}\xspace \setminus\{i_{\mathrm{min}}\xspace\}, \exists\ \gamma_2$ from $\mathbf{x_{\mathrm{sad}}}\xspace$ to $\mathbf{x_{<}}\xspace$\\
& & $\Rightarrow \gamma := \gamma_1 <>\gamma_2$ is a descending path\\
\midrule
\multicolumn{3}{c}{the dynamics of $\gamma$ is equal to $f(\mathbf{x}_{\max}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$}\\
\multicolumn{3}{c}{$\Rightarrow \mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) \leq f(\mathbf{x}_{\mathrm{max/sad}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$}\\
\bottomrule
\bottomrule
\multicolumn{3}{c}{Step 3:}\\
\toprule
\toprule
If $\mathbf{x_{\mathrm{min}}}\xspace$ is paired by dynamics with $x^*\xspace$ & & $\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) < f(\mathbf{x_{\mathrm{sad}}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$ $\textbf{(H)}\xspace$\\
Then $x^*\xspace > \mathbf{x_{\mathrm{min}}}\xspace$ & & $ \Rightarrow \exists$ a descending $\gamma$ from $\mathbf{x_{\mathrm{min}}}\xspace$ in $C_{\imin}^{\mathrm{sad}}\xspace$\\
$\mathbf{x_{<}}\xspace := \inf\{x > \mathbf{x_{\mathrm{min}}}\xspace \; ; \; f(x) < f(\mathbf{x_{\mathrm{min}}}\xspace)\}$ & & $\Rightarrow \mathbf{x_{\mathrm{min}}}\xspace$ does not represent $C_{\imin}^{\mathrm{sad}}\xspace$\\
$\mathbf{x_{<}}\xspace > \mathbf{x}_{\max}\xspace$ & & $\Rightarrow \textbf{(H)}\xspace$ is false\\
$\gamma$ optimal path $\Rightarrow$ $\{\mathbf{x_{\mathrm{min}}}\xspace, \mathbf{x}_{\max}\xspace, \mathbf{x_{<}}\xspace\} \in \gamma$ & & \\
\midrule
\multicolumn{3}{c}{$\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) \geq f(\mathbf{x}_{\mathrm{max/sad}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$}\\
\bottomrule
\bottomrule
\multicolumn{3}{c}{Step 4:}\\
\toprule
\toprule
\multicolumn{3}{c}{$\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) = f(\mathbf{x}_{\mathrm{max/sad}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$}\\
\midrule
\multicolumn{3}{c}{$\mathbf{x}_{\mathrm{max/sad}}\xspace$ and $\mathbf{x_{\mathrm{min}}}\xspace$ are paired by dynamics}\\
\bottomrule
\end{tabular}
\label{table.sketch.per.dyn}
\end{table*}
We assume as usual that $f$ is a Morse function (see Table~\ref{table.sketch.per.dyn}), that its critical values are unique. Let us prove that when some maximum of $f$ in the 1D case (or some $1$-saddle of $f$ in the $n$-D case) is paired by persistence to some minimum of this same function $f$, then this minimum is paired with this maximum (resp. this $1$-saddle) by dynamics.
\medskip
\textbf{\underline{1D proof:}} Let us start with the 1D case (see Figure~\ref{fig:sketch1D-per-dyn}). By considering that some maximum $\mathbf{x}_{\max}\xspace$ is paired with some minimum $\mathbf{x_{\mathrm{min}}}\xspace$ by persistence (with $\mathbf{x_{\mathrm{min}}}\xspace < \mathbf{x}_{\max}\xspace$), we obtain at the same time several properties (by definition of the pairing by persistence):
\begin{itemize}
\item we can draw the threshold set $[f \leq f(\mathbf{x}_{\max}\xspace)]$ at level $f(\mathbf{x}_{\max}\xspace)$,
\item we know that it draws a connected component $$\mathcal{CC}\xspace([f \leq f(\mathbf{x}_{\max}\xspace)],\mathbf{x}_{\max}\xspace)$$ containing $\mathbf{x}_{\max}\xspace$ that we can define as $[\xmax^-\xspace,\xmax^+\xspace]$ with $\xmax^-\xspace < \mathbf{x}_{\max}\xspace < \xmax^+\xspace$,
\item we know then that $\mathbf{x_{\mathrm{min}}}\xspace$ is the representative of $[\xmax^-\xspace,\mathbf{x}_{\max}\xspace]$ and we can define some $\nico{x^{\forall}_{\mathrm{min}}\xspace}$ as being the representative of $[\mathbf{x}_{\max}\xspace,\xmax^+\xspace]$, with $f(\mathbf{x_{\mathrm{min}}}\xspace) > f(\nico{x^{\forall}_{\mathrm{min}}\xspace})$.
\end{itemize}
Now let us prove that $\mathbf{x_{\mathrm{min}}}\xspace$ is paired by dynamics to $\mathbf{x}_{\max}\xspace$ in four steps. First, we know that there exists some path $\gamma : [0,1] \rightarrow [\mathbf{x_{\mathrm{min}}}\xspace,\nico{x^{\forall}_{\mathrm{min}}\xspace}] : \lambda \rightarrow (1-\lambda) \mathbf{x_{\mathrm{min}}}\xspace + \lambda \nico{x^{\forall}_{\mathrm{min}}\xspace}]$ joining $\mathbf{x_{\mathrm{min}}}\xspace$ to $\nico{x^{\forall}_{\mathrm{min}}\xspace}$ with $f(\nico{x^{\forall}_{\mathrm{min}}\xspace}) < f(\mathbf{x_{\mathrm{min}}}\xspace)$, then $\mathbf{x_{\mathrm{min}}}\xspace$ is matchable.
\medskip
Then, the second step is straightforward: since $\gamma$ reaches some $\nico{x^{\forall}_{\mathrm{min}}\xspace}$ with an altitude lower than the one of $\mathbf{x_{\mathrm{min}}}\xspace$, it is a descending path. Furthermore, the effort associated to $\gamma$ is equal to $f(\mathbf{x}_{\max}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$, since we have to reach $(\mathbf{x}_{\max}\xspace,f(\mathbf{x}_{\max}\xspace))$ when we start from $(\mathbf{x_{\mathrm{min}}}\xspace,f(\mathbf{x_{\mathrm{min}}}\xspace))$ to be able to go down to $$(\nico{x^{\forall}_{\mathrm{min}}\xspace},f(\nico{x^{\forall}_{\mathrm{min}}\xspace})).$$ Then the optimal effort associated to $\mathbf{x_{\mathrm{min}}}\xspace$, that is the dynamics of $\mathbf{x_{\mathrm{min}}}\xspace$, is lower than or equal to $f(\mathbf{x}_{\max}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$.
\medskip
Now, for the third step, we assume that $\mathbf{x_{\mathrm{min}}}\xspace$ is paired with some $x^*\xspace < \mathbf{x_{\mathrm{min}}}\xspace$, which is clearly impossible: otherwise dynamics of $\mathbf{x_{\mathrm{min}}}\xspace$ would be greater than $f(\mathbf{x}_{\max}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$ (we would need to go outside the connected component $[\xmax^-\xspace,\xmax^+\xspace]$ to reach some altitude lower than $f(\mathbf{x_{\mathrm{min}}}\xspace)$). Then $\mathbf{x_{\mathrm{min}}}\xspace$ is paired with some maximum $x^*\xspace$ greater than $\mathbf{x_{\mathrm{min}}}\xspace$. Now, we define $\mathbf{x_{<}}\xspace$ as the \textquote{first} abscissa of altitude lower than $f(\mathbf{x_{\mathrm{min}}}\xspace)$ on the right side of $\mathbf{x_{\mathrm{min}}}\xspace$; obviously this abscissa is greater than $\mathbf{x}_{\max}\xspace$ since $\mathbf{x_{\mathrm{min}}}\xspace$ is the representative of the basin $[\xmax^-\xspace,\mathbf{x}_{\max}\xspace]$. Since any optimal descending path starting from $\mathbf{x_{\mathrm{min}}}\xspace$ goes through the abscissas $\mathbf{x_{\mathrm{min}}}\xspace$, $\mathbf{x}_{\max}\xspace$ and then $\mathbf{x_{<}}\xspace$, its associated effort is greater than or equal to $f(\mathbf{x}_{\max}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$.
\medskip
The fourth step combines the previous properties and leads to the conclusion that the dynamics of $\mathbf{x_{\mathrm{min}}}\xspace$ is equal to $f(\mathbf{x}_{\max}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$, which means that the maxima associated to $\mathbf{x_{\mathrm{min}}}\xspace$ by dynamics is $\mathbf{x}_{\max}\xspace$ (by \nico{uniqueness} of the critical values).
\medskip
\textbf{\underline{$n$-D proof:}} The main steps of the $n$-D proof are very similar to the 1D case. However, the notations are very different, due to the fact that the number of path from one point to another in $\mathbb{R}\xspace^n$ is infinite (and there is no \textquote{left} nor \textquote{right}). Starting from the $1$-saddle $\mathbf{x_{\mathrm{sad}}}\xspace$ paired by persistence to $\mathbf{x_{\mathrm{min}}}\xspace$, we have to use the following notations:
\begin{itemize}
\item we define the closed component $C^{\mathrm{sad}}\xspace = \mathcal{CC}\xspace([f \leq f(\mathbf{x_{\mathrm{sad}}}\xspace)],\mathbf{x_{\mathrm{sad}}}\xspace)$,
\item we define also the open components $\{C_{\imin}\xspace\}_i$ of $[f < f(\mathbf{x_{\mathrm{sad}}}\xspace)]$, whose subset $\{C_i^{\mathrm{sad}}\xspace\}_i$ corresponds to these components whose closure contains $\mathbf{x_{\mathrm{sad}}}\xspace$,
\item we call $i_{\mathrm{min}}\xspace$ the index of the component $C_{\imin}^{\mathrm{sad}}\xspace$ that $\mathbf{x_{\mathrm{min}}}\xspace$ represents.
\end{itemize}
\medskip
The first step consists of recalling that the number of components of $C_i^{\mathrm{sad}}\xspace$ is equal to two, then greater than one, and thus there exists some index $\mathbf{i_{<}}\xspace$ and some abscissa $\mathbf{x_{<}}\xspace \in C_{\ILOWER}^{\mathrm{sad}}\xspace$ such that $f(\mathbf{x_{<}}\xspace) < f(\mathbf{x_{\mathrm{sad}}}\xspace)$ (since pairing by persistence associates $\mathbf{x_{\mathrm{sad}}}\xspace$ to the local minimum of the highest altitude). Thus, $\mathbf{x_{\mathrm{min}}}\xspace$ is matchable.
\medskip
As a second step, we construct a path $\gamma_1$ from $\mathbf{x_{\mathrm{min}}}\xspace$ to $\mathbf{x_{\mathrm{sad}}}\xspace$ in $C_{\imin}^{\mathrm{sad}}\xspace$ and another path $\gamma_2$ from $\mathbf{x_{\mathrm{sad}}}\xspace$ to $\mathbf{x_{<}}\xspace$ in the component $C_i^{\mathrm{sad}}\xspace$ containing it, from which we deduce a descending path $\gamma := \gamma_1 <> \gamma_2$ associated to $\mathbf{x_{\mathrm{min}}}\xspace$. Thus, the effort associated to $\gamma$ is lower than or equal to $f(\mathbf{x_{\mathrm{sad}}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$ (since this path has not yet been shown to be optimal).
\medskip
The third step uses a proof by contradiction. We assume that the dynamics of $\mathbf{x_{\mathrm{min}}}\xspace$ is lower than $f(\mathbf{x_{\mathrm{sad}}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$; we call this hypothesis $\mathrm{HYP}\xspace$. Then, $\mathrm{HYP}\xspace$ implies that there exists a descending path inside the component $C_{\imin}^{\mathrm{sad}}\xspace$, which implies that $\mathbf{x_{\mathrm{min}}}\xspace$ does not represent $C_{\imin}^{\mathrm{sad}}\xspace$, which is impossible (it contradicts the hypotheses). Then, the dynamics of $\mathbf{x_{\mathrm{min}}}\xspace$ is greater than or equal to $f(\mathbf{x_{\mathrm{sad}}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$.
\medskip
As for the 1D case, the fourth steps concludes: since the dynamics of $\mathbf{x_{\mathrm{min}}}\xspace$ is equal to $f(\mathbf{x_{\mathrm{sad}}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$ thanks to the combination of the previous steps, the only possible local maximum paired by dynamics to $\mathbf{x_{\mathrm{min}}}\xspace$ is $\mathbf{x_{\mathrm{sad}}}\xspace$.
\section{Pairings by dynamics and by persistence are equivalent in 1D}
\label{sec.equivalence1D}
In this section, we prove that under some constraints, pairings by dynamics and by persistence are equivalent in the 1D case.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{proofDYNPER.pdf}
\caption{A $\mathfrak{D}$-Morse\xspace function where the local extrema $\mathbf{x_{\mathrm{min}}}\xspace$ and $\mathbf{x}_{\max}\xspace$ are paired by dynamics.}
\label{fig.proofDYNPER}
\end{figure}
\begin{Proposition}
Let $f : \mathbb{R}\xspace \rightarrow \mathbb{R}\xspace$ be a $\mathfrak{D}$-Morse\xspace function with unique critical values. Now, let us assume that a local minimum $\mathbf{x_{\mathrm{min}}}\xspace \in \mathbb{R}\xspace$ of $f$ is paired with a local maximum $\mathbf{x}_{\max}\xspace$ of $f$ by dynamics. We assume \nico{without loss of generality} that $\mathbf{x_{\mathrm{min}}}\xspace < \mathbf{x}_{\max}\xspace$ (the reasoning is the same for the opposite assumption). Also, we denote by $(\mathbf{x}_{\max}\xspace^-,\mathbf{x}_{\max}\xspace^+) \in \overline{\mathbb{R}\xspace}\xspace^2$ the two values verifying: $$[\mathbf{x}_{\max}\xspace^-,\mathbf{x}_{\max}\xspace^+] = \mathrm{cl}_{\realscomplete}\xspace (\mathcal{CC}\xspace([f \leq f(\mathbf{x}_{\max}\xspace)],\mathbf{x}_{\max}\xspace)).$$
Then the following properties are true:
\begin{itemize}
\setlength\itemsep{1em}
\item[(P1)] $\mathbf{x_{\mathrm{min}}}\xspace = \mathrm{rep}\xspace([\mathbf{x}_{\max}\xspace^-,\mathbf{x}_{\max}\xspace],f)$,
\item[(P2)] When $\mathbf{x}_{\max}\xspace^+$ is finite, $\mathbf{x_{\mathrm{min}}}\xspace^2\xspace := \mathrm{rep}\xspace([\mathbf{x}_{\max}\xspace,\mathbf{x}_{\max}\xspace^+],f)$ \nico{satisfies} $f(\mathbf{x_{\mathrm{min}}}\xspace^2\xspace) < f(\mathbf{x_{\mathrm{min}}}\xspace)$,
\item[(P3)] $\mathbf{x}_{\max}\xspace$ and $\mathbf{x_{\mathrm{min}}}\xspace$ are paired by persistence.
\end{itemize}
\label{proposition.DYNPER}
\end{Proposition}
\textbf{Proof:}\xspace Figure~\ref{fig.proofDYNPER} depicts an example of $\mathfrak{D}$-Morse\xspace function where $\mathbf{x_{\mathrm{min}}}\xspace$ and $\mathbf{x}_{\max}\xspace$ are paired by dynamics.
\medskip
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{absurde1.pdf}
\caption{Proof of $(P1)$.}
\label{fig:absurde1}
\end{figure}
Let us prove $(P1)$; we proceed by \textit{reductio ad absurdum}. When $\mathbf{x_{\mathrm{min}}}\xspace$ is not the lowest local minimum of $f$ on the interval $[\mathbf{x}_{\max}\xspace^-,\mathbf{x}_{\max}\xspace]$, then there exists another local minimum $x^* \in [\mathbf{x}_{\max}\xspace^-,\mathbf{x}_{\max}\xspace]$ of $f$ (see Figure~\ref{fig:absurde1}) which \nico{satisfies} $f(x^*) < f(\mathbf{x_{\mathrm{min}}}\xspace)$ ($x^*$
and $\mathbf{x_{\mathrm{min}}}\xspace$ being distinct local extrema of $f$, their images by $f$ are not equal). Then, because the path joining $\mathbf{x_{\mathrm{min}}}\xspace$ and $x^*$ belongs to $C$ (defined in Subsection~\ref{ssec.dyn}), we have: $$\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) \leq \max\{f(x) - f(\mathbf{x_{\mathrm{min}}}\xspace) \; ; \; x \in \mathrm{iv}(x^*,\mathbf{x_{\mathrm{min}}}\xspace)\}.$$
Let us call $x^{**} := \arg \max_{x \in [\mathrm{iv}(\mathbf{x_{\mathrm{min}}}\xspace,x^*)]} f(x)$, we can deduce that $f(x^{**}) < f(\mathbf{x}_{\max}\xspace)$ since $x^{**} \in \mathrm{iv}(x^*,\mathbf{x_{\mathrm{min}}}\xspace) \subseteq ]\mathbf{x}_{\max}\xspace^-, \mathbf{x}_{\max}\xspace[$. In this way, $$\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) \leq f(x^{**}) - f(\mathbf{x_{\mathrm{min}}}\xspace),$$ which is lower than $f(\mathbf{x}_{\max}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$; this is a contradiction since $\mathbf{x_{\mathrm{min}}}\xspace$ and $\mathbf{x}_{\max}\xspace$ are paired by dynamics. $(P1)$ is then proved.
\medskip
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{absurde2.pdf}
\caption{Proof of $(P2)$ in the case where $\mathbf{x}_{\max}\xspace^+$ is finite.}
\label{fig:absurde2}
\end{figure}
Now let us prove $(P2)$. Let us assume that $\mathbf{x}_{\max}\xspace^+$ is finite and let $\mathbf{x_{\mathrm{min}}}\xspace^2\xspace$ be the representative of $[\mathbf{x}_{\max}\xspace,\mathbf{x}_{\max}\xspace^+]$ relatively to $f$. Let us assume that $f(\mathbf{x_{\mathrm{min}}}\xspace^2\xspace) > f(\mathbf{x_{\mathrm{min}}}\xspace)$. Note that we cannot have equality of $f(\mathbf{x_{\mathrm{min}}}\xspace^2\xspace)$ and $f(\mathbf{x_{\mathrm{min}}}\xspace)$, since $\mathbf{x_{\mathrm{min}}}\xspace$ and $\mathbf{x_{\mathrm{min}}}\xspace^2\xspace$ are both local extrema of $f$. Then we obtain Figure~\ref{fig:absurde2}. Since with $x \in [\mathbf{x}_{\max}\xspace,\mathbf{x}_{\max}\xspace^+]$, we have $f(x) \geq f(\mathbf{x_{\mathrm{min}}}\xspace^2\xspace) > f(\mathbf{x_{\mathrm{min}}}\xspace)$, and because $\mathbf{x_{\mathrm{min}}}\xspace$ is paired with $\mathbf{x}_{\max}\xspace$ by dynamics with $\mathbf{x_{\mathrm{min}}}\xspace < \mathbf{x}_{\max}\xspace$, then there exists a value $x$ on the right of $\mathbf{x}_{\max}\xspace$ where $f(x)$ is lower than $f(\mathbf{x_{\mathrm{min}}}\xspace)$. In other words, there exists: $$x^< := \inf\{ x \in [\mathbf{x}_{\max}\xspace,+\infty[ \; ; \; f(x) < f(\mathbf{x_{\mathrm{min}}}\xspace)\}$$ such that for some arbitrarily small value $\varepsilon > 0$, $f(x^< + \varepsilon) < f(\mathbf{x_{\mathrm{min}}}\xspace)$. Since $x^< > \mathbf{x}_{\max}\xspace^+$, any path $\gamma$ joining $\mathbf{x_{\mathrm{min}}}\xspace$ to $x^<$ goes through a local maximum $\mathbf{x}_{\max}\xspace^2\xspace$ defined by
$$\mathbf{x}_{\max}\xspace^2\xspace := \arg \max_{x \in [\mathbf{x}_{\max}\xspace^+,x^<]} f(x)$$
which \nico{satisfies} $f(\mathbf{x}_{\max}\xspace^2\xspace) > f(\mathbf{x}_{\max}\xspace^+)$. Then the dynamics of $\mathbf{x_{\mathrm{min}}}\xspace$ is greater than or equal to $f(\mathbf{x}_{\max}\xspace^2\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$ which is greater than $f(\mathbf{x}_{\max}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$. We obtain a contradiction. Then we have $f(\mathbf{x_{\mathrm{min}}}\xspace^2\xspace) < f(\mathbf{x_{\mathrm{min}}}\xspace)$. The proof of $(P2)$ is done.
\medskip
Thanks to $(P1)$ and $(P2)$, we obtain directly $(P3)$ by applying Algorithm~\ref{algo.pairingbypersistence}. \qed
\begin{Proposition}
Let $f : \mathbb{R}\xspace \rightarrow \mathbb{R}\xspace$ be a $\mathfrak{D}$-Morse\xspace function with unique critical values. Now, let us assume that a local minimum $\mathbf{x_{\mathrm{min}}}\xspace \in \mathbb{R}\xspace$ of $f$ is paired with a local maximum $\mathbf{x}_{\max}\xspace$ of $f$ by persistence. We assume \nico{without loss of generality} that $\mathbf{x_{\mathrm{min}}}\xspace < \mathbf{x}_{\max}\xspace$ (the reasoning is the same for the opposite assumption). Then, $\mathbf{x}_{\max}\xspace$ and $\mathbf{x_{\mathrm{min}}}\xspace$ are paired by dynamics.
\label{proposition.PERDYN}
\end{Proposition}
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{proofPERDYNbis.pdf}
\caption{A $\mathfrak{D}$-Morse\xspace function $f : \mathbb{R}\xspace \rightarrow \mathbb{R}\xspace$ where the local extrema $\mathbf{x_{\mathrm{min}}}\xspace$ and $\mathbf{x}_{\max}\xspace$ are paired by persistence relatively to $f$.}
\label{fig.proofPERDYN}
\end{figure}
\textbf{Proof:}\xspace We denote by $(\mathbf{x}_{\max}\xspace^-,\mathbf{x}_{\max}\xspace^+) \in \overline{\mathbb{R}\xspace}\xspace^2$ the two values verifying: $$[\mathbf{x}_{\max}\xspace^-,\mathbf{x}_{\max}\xspace^+] = \mathrm{cl}_{\realscomplete}\xspace(\mathcal{CC}\xspace([f \leq f(\mathbf{x}_{\max}\xspace)],\mathbf{x}_{\max}\xspace)).$$
Since $\mathbf{x_{\mathrm{min}}}\xspace$ is paired by persistence to $\mathbf{x}_{\max}\xspace$ with $\mathbf{x_{\mathrm{min}}}\xspace < \mathbf{x}_{\max}\xspace$ (see Figure~\ref{fig.proofPERDYN}), then:
$$\mathbf{x_{\mathrm{min}}}\xspace = \mathrm{rep}\xspace([\mathbf{x}_{\max}\xspace^-,\mathbf{x}_{\max}\xspace],f) \in \mathbb{R}\xspace,$$
and, by Algorithm~\ref{algo.pairingbypersistence}, we know that $\mathbf{x}_{\max}\xspace^- > - \infty$ (then $\mathbf{x}_{\max}\xspace^-$ is finite).
\medskip
When $\mathbf{x}_{\max}\xspace^+ < +\infty$ (Case $1$), the representative $\nico{x^{\forall}_{\mathrm{min}}\xspace}$ of $[\mathbf{x}_{\max}\xspace,\mathbf{x}_{\max}\xspace^+]$ relatively to $f$ is exists in $]\mathbf{x}_{\max}\xspace,\mathbf{x}_{\max}\xspace^+[$ and is unique, and its image by $f$ is lower than $f(\mathbf{x_{\mathrm{min}}}\xspace)$. When $\mathbf{x}_{\max}\xspace^+ = +\infty$ (Case $2$), $\lim_{x \rightarrow +\infty} f(x) = -\infty$, and then there exists one more time an abscissa $\nico{x^{\forall}_{\mathrm{min}}\xspace} \in \mathbb{R}\xspace$ whose image by $f$ is lower than $f(\mathbf{x_{\mathrm{min}}}\xspace)$. So, in both cases, there exists a (finite) value $\nico{x^{\forall}_{\mathrm{min}}\xspace} \in ]\mathbf{x}_{\max}\xspace,\mathbf{x}_{\max}\xspace^+[$ verifying $f(\nico{x^{\forall}_{\mathrm{min}}\xspace}) < f(\mathbf{x_{\mathrm{min}}}\xspace)$. This way, we know that $\mathbf{x_{\mathrm{min}}}\xspace$ is paired with some abscissa in $\mathbb{R}\xspace$ by dynamics.
\medskip
In Case $1$, we know that the path defined as:
$$\gamma : \lambda \in [0,1] \rightarrow \gamma(\lambda) := (1-\lambda) \mathbf{x_{\mathrm{min}}}\xspace + \lambda \nico{x^{\forall}_{\mathrm{min}}\xspace}$$
belongs to the set of paths $C$ defining the dynamics of $\mathbf{x_{\mathrm{min}}}\xspace$ (see Subsection~\ref{ssec.dyn}). Then, $$\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) \leq \max\{f(x) - f(\mathbf{x_{\mathrm{min}}}\xspace) \; ; \; x \in \gamma([0,1])\},$$
which is lower than or equal to $f(\mathbf{x}_{\max}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$ since $f$ is maximal at $\mathbf{x}_{\max}\xspace$ on $[\mathbf{x}_{\max}\xspace^-,\mathbf{x}_{\max}\xspace^+]$. Then we have the following property: $$\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) \leq f(\mathbf{x}_{\max}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace).$$ In Case $2$, since $f(x)$ is lower than $f(\mathbf{x}_{\max}\xspace)$ for $x \in ]\mathbf{x}_{\max}\xspace,+\infty[$, then one more time we get $\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) \leq f(\mathbf{x}_{\max}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$. Let us call this property $(P)$.
\medskip
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{absurde3.pdf}
\caption{The proof that it is impossible to obtain a local maximum $x^* < \mathbf{x_{\mathrm{min}}}\xspace$ paired with $\mathbf{x_{\mathrm{min}}}\xspace$ by dynamics when $\mathbf{x_{\mathrm{min}}}\xspace$ is paired with $\mathbf{x}_{\max}\xspace > \mathbf{x_{\mathrm{min}}}\xspace$ by persistence.}
\label{fig:absurde3}
\end{figure}
Even if we know that there exists some local maximum of $f$ which is paired with $\mathbf{x_{\mathrm{min}}}\xspace$ by dynamics, we do not know whether the abscissa of this local maximum is lower than or greater than $\mathbf{x_{\mathrm{min}}}\xspace$. Then, let us assume that there exists a local maximum $x^* < \mathbf{x_{\mathrm{min}}}\xspace$ (lower case) which is associated to $\mathbf{x_{\mathrm{min}}}\xspace$ by dynamics. We denote this property $(H)$ and we depict it in Figure~\ref{fig:absurde3}. Since $f(x)$ is greater than or equal to $f(\mathbf{x_{\mathrm{min}}}\xspace)$ for $x \in [\mathbf{x}_{\max}\xspace^-,\mathbf{x_{\mathrm{min}}}\xspace]$, $(H)$ implies that $x^* < \mathbf{x}_{\max}\xspace^-$. Then, we can observe that the local maximum $x^1$ of $f$ of maximal abscissa in $[x^*,\mathbf{x}_{\max}\xspace^-]$ \nico{satisfies} $f(x^1) > f(\mathbf{x}_{\max}\xspace)$, which implies that $\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) \geq f(x^1) - f(\mathbf{x_{\mathrm{min}}}\xspace) > f(\mathbf{x}_{\max}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$ (since we go through $x^1$ to reach $x^*$), which contradicts $(P)$. $(H)$ is then false. In other words, we are in the upper case: the local maximum paired by dynamics to $\mathbf{x_{\mathrm{min}}}\xspace$ belongs to $]\mathbf{x_{\mathrm{min}}}\xspace, +\infty[$, let us call this property $(P')$.
\medskip
Now let us define: $$x^< := \inf\{x > \mathbf{x_{\mathrm{min}}}\xspace \; ; \; f(x) < f(\mathbf{x_{\mathrm{min}}}\xspace)\},$$ (see again Figure~\ref{fig.proofPERDYN}) and let us remark that $x^< > \mathbf{x}_{\max}\xspace$ (because $\mathbf{x_{\mathrm{min}}}\xspace$ is the representative of $f$ on $[\mathbf{x}_{\max}\xspace^-,\mathbf{x}_{\max}\xspace]$). Since we know by $(P')$ that a local maximum $x > \mathbf{x_{\mathrm{min}}}\xspace$ of $f$ is paired by dynamics with $\mathbf{x_{\mathrm{min}}}\xspace$, then the image of every optimal path belonging to $C$ contains $\{x^<\}$, and then contains $[\mathbf{x_{\mathrm{min}}}\xspace,x^<]$. Indeed, an optimal path in $C$ whose image would not contain $\{x^<\}$ would then contain an abscissa $x < \mathbf{x}_{\max}\xspace^-$ and then we would obtain $\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) > f(\mathbf{x}_{\max}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$, which would contradict $(P)$.
\medskip
Now, the maximal value of $f$ on $[\mathbf{x_{\mathrm{min}}}\xspace,x^<]$ is equal to $f(\mathbf{x}_{\max}\xspace)$, then $\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) = f(\mathbf{x}_{\max}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$. The only local maximum of $f$ whose value is $f(\mathbf{x}_{\max}\xspace)$ is $\mathbf{x}_{\max}\xspace$, then $\mathbf{x}_{\max}\xspace$ is paired with $\mathbf{x_{\mathrm{min}}}\xspace$ by dynamics relatively to $f$. \qed
\begin{Th}
Let $f : \mathbb{R}\xspace \rightarrow \mathbb{R}\xspace$ be a $\mathfrak{D}$-Morse\xspace function with a finite number of local extrema and unique critical values. A local minimum $\mathbf{x_{\mathrm{min}}}\xspace \in \mathbb{R}\xspace$ of $f$ is paired by dynamics to a local maximum $\mathbf{x}_{\max}\xspace \in \mathbb{R}\xspace$ of $f$ iff $\mathbf{x}_{\max}\xspace$ is paired by persistence to $\mathbf{x_{\mathrm{min}}}\xspace$. In other words, pairings by dynamics and by persistence lead to the same result. Furthermore, we obtain $\mathrm{Per}\xspace(\mathbf{x}_{\max}\xspace) = \mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace)$.
\label{th.equivalent}
\end{Th}
\textbf{Proof:}\xspace This theorem results from Propositions~\ref{proposition.DYNPER} and~\ref{proposition.PERDYN}. \qed
\medskip
Note that pairing by persistence has been proved to be \emph{symmetric} in~\cite{cohen2009extending} for Morse functions defined on manifolds: the pairing is the same for a Morse function and its negative.
\section{The $n$-D equivalence}
\label{sec.equivalence}
Let us make two important remarks that will help us in the sequel.
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{optsad.png}
\caption{Every optimal descending path goes through a $1$-saddle. Observe the path in blue coming from the left side and decreasing when following the topographical view of the Morse function $f$. The effort of this path to reach the minimum of $f$ is minimal thanks to the fact that it goes through the saddle point at the middle of the image.}
\label{fig.optsad}
\end{figure}
\begin{Lem}
\label{lemma.saddle}
Let $f : \mathbb{R}\xspace^n \rightarrow \mathbb{R}\xspace$ be a Morse function and let $\mathbf{x_{\mathrm{min}}}\xspace$ be a local minimum of $f$. Then for any optimal path $\Pi^{\mathrm{opt}}\xspace$ in $(D_{\xmin})\xspace$, there exists some $\ell^*\xspace \in ]0,1[$ such that it is a maximum of $f \circ \Pi^{\mathrm{opt}}\xspace$ and at the same time $\Pi^{\mathrm{opt}}\xspace(\ell^*\xspace)$ is the abscissa of a $1$-saddle point of $f$.
\end{Lem}
\textbf{Proof}\xspace: This proof is depicted in Figure~\ref{fig.optsad}. Let us proceed by counterposition, and let us prove that when a path $\Pi$ in $(D_{\xmin})\xspace$ does not go through a $1$-saddle of $f$, it cannot be optimal.
\medskip
Let $\Pi$ be a path in $(D_{\xmin})\xspace$. Let us define $\ell^*\xspace \in [0,1]$ as one of the positions where the mapping $f \circ \Pi$ is maximal:
$$\ell^*\xspace \in {\arg \max}_{\ell \in [0,1]} f( \Pi (\ell)),$$
and $x^*\xspace = \Pi(\ell^*\xspace)$. Let us prove that we can find another path $\Pi'$ in $(D_{\xmin})\xspace$ whose effort is lower than the one of $\Pi$.
\medskip
\begin{figure}
\centering
\includegraphics[width=0.45\linewidth]{gradnotzero.png}~\includegraphics[width=0.3\linewidth]{2-saddle.png}
\caption{How to compute descending paths of lower efforts. The initial path going through $x^*\xspace$ (the little gray ball) is in red, the new path of lower effort is in green (the non-zero gradient case is on the left side, the zero-gradient case is on the right side).}
\label{fig.lower}
\end{figure}
At $x^*\xspace$, $f$ can satisfy three possibilities:
\begin{itemize}
\item When we have $\nabla f (x^*\xspace) \neq 0$ (see the left side of Figure~\ref{fig.lower}), then locally $f$ is a plane of slope $\|\nabla f (x^*\xspace)\|$, and then we can easily find some path $\Pi'$ in $(D_{\xmin})\xspace$ with a lower effort than $\mathrm{Effort}\xspace(\Pi)$. More precisely, let us fix some arbitrary small value $\varepsilon > 0$ and draw the closed topological ball $\bar{B}(\xstar,\varepsilon)\xspace$, we can define three points:
\begin{align*}
\ell_{min}\xspace & = \min\{\ell \ |\ \Pi(\ell) \in \bar{B}(\xstar,\varepsilon)\xspace\},\\
\ell_{max}\xspace & = \max\{\ell \ |\ \Pi(\ell) \in \bar{B}(\xstar,\varepsilon)\xspace\},\\
x_B & = x^*\xspace - \varepsilon . \frac{\nabla f (x^*\xspace)}{\|\nabla f (x^*\xspace)\|}.
\end{align*}
Thanks to these points, we can define a new path $\Pi'$:
\scalebox{0.85}{\parbox{\linewidth}{%
$$\Pi|_{[0,\ell_{min}\xspace]} <> [\Pi(\ell_{min}\xspace),x_B] <> [x_B,\Pi(\ell_{max}\xspace)] <> \Pi|_{[\ell_{max}\xspace,1]}.$$
}}
By doing this procedure at every point in $[0,1]$ where $f \circ \Pi$ reaches its maximal value, we obtain a new path whose effort is lower than the one of $\Pi$.
\item When we have $\nabla f (x^*\xspace) = 0$, then we are at a critical point of $f$. It cannot be a $0$-saddle, that is, a local minimum, due to the existence of the descending path going through $x^*\xspace$. It cannot be a $1$-saddle neither (by hypothesis). It is then a $k$-saddle point with $k \in [2,n]$ (see the right side of Figure~\ref{fig.lower}). Using Lemma~\ref{lemma.morse}, $f$ is locally equal to a second order polynomial function (up to a change of coordinates $\varphi$ s.t. $\varphi(x^*\xspace) = \mathbf{0}\xspace$): $$ \ f \circ \varphi^{-1} (\mathbf{x}\xspace) = f(x^*\xspace) - x_1^2 - x_2^2 - \dots - x_k^2 + x_{k+1}^2 + \dots + x_n^2.$$
Now, let us define for some arbitrary small value $\varepsilon > 0$:
\begin{align*}
\ell_{min}\xspace & = \min\{\ell \ |\ \Pi(\ell) \in \bar{B}(\veczero,\varepsilon)\xspace\},\\
\ell_{max}\xspace & = \max\{\ell \ |\ \Pi(\ell) \in \bar{B}(\veczero,\varepsilon)\xspace\},\\
\end{align*}
and
\scalebox{0.85}{\parbox{\linewidth}{%
$$\mathfrak{B}\xspace = \left\{\mathbf{x}\xspace \ \Big| \ \sum_{i \in [1,k]} x_i^2 \leq \varepsilon^2 \ \textbf{ and } \forall j \in [k+1,n], x_j = 0\right\} \setminus \{\mathbf{0}\xspace\}.$$
}}
This last set is connected since it is equal to a $k$-manifold (with $k \geq 2$) minus a point. Let us assume \nico{without loss of generality} that $\Pi(\ell_{min}\xspace)$ and $\Pi(\ell_{max}\xspace)$ belong to $\mathfrak{B}\xspace$ (otherwise we can consider their orthogonal projections on the hyperplane of lower dimension containing $\mathfrak{B}\xspace$ but the reasoning is the same). Thus, there exists some path $\Pi_\mathfrak{B}\xspace$ joining $\Pi(\ell_{min}\xspace)$ to $\Pi(\ell_{max}\xspace)$ in $\mathfrak{B}\xspace$, from which we can deduce the path $\Pi' = \Pi|_{[0,\ell_{min}\xspace]} <> \Pi_\mathfrak{B}\xspace <> \Pi|_{[\ell_{max}\xspace,1]}$ whose effort is lower than the one of $\Pi$ since its image is inside $[f < f(x^*\xspace)]$.
\end{itemize}
Since we have seen that, in any possible case, $\Pi$ is not optimal, it concludes the proof.\qed
\begin{figure}
\centering
\includegraphics[width=0.4\linewidth]{saddle.png}
\caption{A $1$-saddle point leads to two open connected components. At a $1$-saddle point whose abscissa is $\mathbf{x_{\mathrm{sad}}}\xspace$ (at the center of the image), the component $[f \leq f(\mathbf{x_{\mathrm{sad}}}\xspace)]$ is locally the merge of the closure of two connected components (in orange) of $[f < f(\mathbf{x_{\mathrm{sad}}}\xspace)]$ when $f$ is a Morse function.}
\label{fig.saddle}
\end{figure}
\begin{Proposition}
\label{proposition.2}
Let $f$ be a Morse function from $\mathbb{R}\xspace^n$ to $\mathbb{R}\xspace$ with $n \geq 1$. When $x^*\xspace$ is a critical point of index $1$, then there exists $\varepsilon > 0$ such that:
$$\mathrm{Card}\xspace\left(\mathcal{CC}\xspace(B(x^*\xspace,\varepsilon) \cap [f < f(x^*\xspace)])\right) = 2,$$
where $\mathrm{Card}\xspace$ is the \emph{cardinality operator}.
\end{Proposition}
\textbf{Proof}\xspace: The case $n = 1$ is obvious, let us then treat the case $n \geq 2$ (see Figure~\ref{fig.saddle}). Thanks to Lemma~\ref{lemma.morse} and thanks to the fact that $\mathbf{x_{\mathrm{sad}}}\xspace$ is the abscissa of a $1$-saddle, we can say that (up to a change of coordinates and in a small neighborhood around $\mathbf{x_{\mathrm{sad}}}\xspace$) for any $\mathbf{x}\xspace$:
$$f(x) = f(\mathbf{x_{\mathrm{sad}}}\xspace) + \mathbf{x}\xspace^T . \MATRICEHESSENNESADDLE . \mathbf{x}\xspace,$$
where $\mathbb{I}_{n-1}\xspace$ is the identity matrix of dimension $(n-1) \times (n-1)$. In other words, around $\mathbf{x_{\mathrm{sad}}}\xspace$, we obtain that:
$$[f < f(\mathbf{x_{\mathrm{sad}}}\xspace)] = \left\{\mathbf{x}\xspace \ \Big| \ - x_1^2 + \sum_{i = 2}^n x_i^2 < 0 \right\} = C_+\xspace \cup C_-\xspace,$$
with:
\begin{align*}
C_+\xspace & = \left\{\mathbf{x}\xspace \ \Big| \ x_1 > \sqrt{\sum_{i = 2}^n x_i^2} \right\},\\
\text{and}\\
C_-\xspace & = \left\{\mathbf{x}\xspace \ \Big| \ x_1 < - \sqrt{\sum_{i = 2}^n x_i^2} \right\},
\end{align*}
where $C_+\xspace$ and $C_-\xspace$ are two open connected components of $\mathbb{R}\xspace^n$. Indeed, for any pair $(M,M')$ of $C_+\xspace$, we have $x_1^M\xspace > \sqrt{\sum_{i = 2}^n (x_i^M)\xspace^2}$ and $x_1^{M'}\xspace > \sqrt{\sum_{i = 2}^n \XIMPRIME^2}$, from which we define $N = (x_1^M\xspace,0,\dots,0)^T \in C_+\xspace$ and $N' = (x_1^{M'}\xspace,0,\dots,0)^T \in C_+\xspace$ from which we deduce the path $[M,N] <> [N,N'] <> [N',M']$ joining $M$ to $M'$ in $C_+\xspace$. The reasoning with $C_-\xspace$ is the same. Since $C_+\xspace$ and $C_-\xspace$ are two connected (separated) disjoint sets, the proof is done.\qed
\subsection{Pairing by persistence implies pairing by dynamics in $n$-D}
\begin{Th}
Let $f$ be a Morse function from $\mathbb{R}\xspace^n$ to $\mathbb{R}\xspace$. We assume that the $1$-saddle point of $f$ whose abscissa is $\mathbf{x_{\mathrm{sad}}}\xspace$ is paired by persistence to a local minimum $\mathbf{x_{\mathrm{min}}}\xspace$ of $f$. Then, $\mathbf{x_{\mathrm{min}}}\xspace$ is paired by dynamics to $\mathbf{x_{\mathrm{sad}}}\xspace$.
\end{Th}
\textbf{Proof}\xspace: Let us assume that $\mathbf{x_{\mathrm{sad}}}\xspace$ is paired by persistence to $\mathbf{x_{\mathrm{min}}}\xspace$, then we have the hypotheses described in Definition~\ref{def.persistence}. Let us denote by $C^{\mathrm{min}}\xspace$ the connected component in $\{C_i\}_{i \in I^{\mathrm{sad}}\xspace}$ satisfying that $\mathbf{x_{\mathrm{min}}}\xspace = \mathrm{rep}\xspace(C_{i_{\mathrm{min}}\xspace})$.
\medskip
Since $\mathbf{x_{\mathrm{sad}}}\xspace$ is the abscissa of a $1$-saddle, by Proposition~\ref{proposition.2}, we know that $\mathrm{Card}\xspace(I^{\mathrm{sad}}\xspace) = 2$, then there exists: $\mathbf{x_{<}}\xspace = \mathrm{rep}\xspace(C^<\xspace)$ with $C^<\xspace$ the component $C_i$ with $i \in I \setminus \{i_{\mathrm{min}}\xspace\}$, then $\mathbf{x_{\mathrm{min}}}\xspace$ is matchable. Let us assume that the dynamics of $\mathbf{x_{\mathrm{min}}}\xspace$ satisfies:
$$\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) < f(\mathbf{x_{\mathrm{sad}}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace). \ \ \ (\mathrm{HYP}\xspace)$$
This means that there exists a path $\Pi_<\xspace$ in $(D_{\xmin})\xspace$ such that:
$$\max_{\ell \in [0,1]} f(\Pi_<\xspace(\ell)) - f(\mathbf{x_{\mathrm{min}}}\xspace) < f(\mathbf{x_{\mathrm{sad}}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace),$$
that is, for any $\ell \in [0,1]$, $f(\Pi_<\xspace(\ell)) < f(\mathbf{x_{\mathrm{sad}}}\xspace)$, and then by continuity in space of $\Pi_<\xspace$, the image of $[0,1]$ by $\Pi_<\xspace$ is in $C^{\mathrm{min}}\xspace$. Because $\Pi_<\xspace$ belongs to $(D_{\xmin})\xspace$, there exists then some $\mathbf{x_{<}}\xspace \in C^{\mathrm{min}}\xspace$ satisfying $f(\mathbf{x_{<}}\xspace) < f(\mathbf{x_{\mathrm{min}}}\xspace)$. We obtain a contradiction, $(\mathrm{HYP}\xspace)$ is then false. Then, we have $\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) \geq f(\mathbf{x_{\mathrm{sad}}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace).$
\medskip
Because for any $i \in I^{\mathrm{sad}}\xspace$, $\mathbf{x_{\mathrm{sad}}}\xspace$ is an accumulation point of $C_i$ in $\mathbb{R}\xspace^n$, there exist a path $\Pi_m$ from $\mathbf{x_{\mathrm{min}}}\xspace$ to $\mathbf{x_{\mathrm{sad}}}\xspace$ such that:
\begin{align*}
\forall \ell \in [0,1], & \Pi_m(\ell) \in C^{\mathrm{sad}}\xspace,\\
\forall \ell \in [0,1[, & \Pi_m(\ell) \in C^{\mathrm{min}}\xspace.
\end{align*}
In the same way, there exists a path $\Pi_M$ from $\mathbf{x_{<}}\xspace$ to $\mathbf{x_{\mathrm{sad}}}\xspace$ such that:
\begin{align*}
\forall \ell \in [0,1], & \Pi_M(\ell) \in C^{\mathrm{sad}}\xspace,\\
\forall \ell \in [0,1[, & \Pi_M(\ell) \in C^<\xspace.
\end{align*}
We can then build a path $\Pi$ which is the concatenation of $\Pi_m$ and $\ell \rightarrow \Pi_M(1-\ell)$, which goes from $\mathbf{x_{\mathrm{min}}}\xspace$ to $\mathbf{x_{<}}\xspace$ and goes through $\mathbf{x_{\mathrm{sad}}}\xspace$. Since this path stays inside $C^{\mathrm{sad}}\xspace$, we know that $\mathrm{Effort}\xspace(\Pi) \leq f(\mathbf{x_{\mathrm{sad}}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$, and then $\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) \leq f(\mathbf{x_{\mathrm{sad}}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$.
\medskip
By grouping the two inequalities, we obtain that $\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) = f(\mathbf{x_{\mathrm{sad}}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$, and then by uniqueness of the critical values of $f$, $\mathbf{x_{\mathrm{min}}}\xspace$ is then paired by dynamics to $\mathbf{x_{\mathrm{sad}}}\xspace$. \qed
\subsection{Pairing by dynamics implies pairing by persistence in $n$-D}
\begin{Th}
Let $f$ be a Morse function from $\mathbb{R}\xspace^n$ to $\mathbb{R}\xspace$. We assume that the local minimum $\mathbf{x_{\mathrm{min}}}\xspace$ of $f$ is paired by dynamics to a $1$-saddle of $f$ of abscissa $\mathbf{x_{\mathrm{sad}}}\xspace$. Then, $\mathbf{x_{\mathrm{sad}}}\xspace$ is paired by persistence to $\mathbf{x_{\mathrm{min}}}\xspace$.
\end{Th}
\textbf{Proof}\xspace: Let us assume that $\mathbf{x_{\mathrm{min}}}\xspace$ is paired to $\mathbf{x_{\mathrm{sad}}}\xspace$ by dynamics. Let us recall the usual framework relative to persistence:
\begin{align*}
C^{\mathrm{sad}}\xspace &= \mathcal{CC}\xspace([f \leq f(\mathbf{x_{\mathrm{sad}}}\xspace)],\mathbf{x_{\mathrm{sad}}}\xspace),\\
\{C^I_i\xspace\}_{i \in I} &= \mathcal{CC}\xspace([f < f(\mathbf{x_{\mathrm{sad}}}\xspace)]),\\
\{C_i^{\mathrm{sad}}\xspace\}_{i \in I^{\mathrm{sad}}\xspace} &= \left\{ C^I_i\xspace | \mathbf{x_{\mathrm{sad}}}\xspace \in \mathrm{clo}\xspace(C^I_i\xspace) \right\},\\
\forall i \in I^{\mathrm{sad}}\xspace, \ \mathrm{rep}\xspace_i &= {\arg \min}_{x \in C_i^{\mathrm{sad}}\xspace} f(x).
\end{align*}
By Definition~\ref{def.persistence}, $\mathbf{x_{\mathrm{sad}}}\xspace$ is paired to the representative $\mathrm{rep}\xspace_i$ of $C_i^{\mathrm{sad}}\xspace$ which maximizes $f(\mathrm{rep}\xspace_i)$.
\medskip
\begin{enumerate}
\item Let us show that there exists some index $i_{\mathrm{min}}\xspace$ such that $\mathbf{x_{\mathrm{min}}}\xspace$ is the representative of a component $C_{\imin}^{\mathrm{sad}}\xspace$ of $\{C_i^{\mathrm{sad}}\xspace\}_{i \in I^{\mathrm{sad}}\xspace}$.
\begin{enumerate}
\item First, $\mathbf{x_{\mathrm{min}}}\xspace$ is paired by dynamics with $\mathbf{x_{\mathrm{sad}}}\xspace$ and $\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace)$ is greater than zero, then $f(\mathbf{x_{\mathrm{sad}}}\xspace) > f(\mathbf{x_{\mathrm{min}}}\xspace)$, then $\mathbf{x_{\mathrm{min}}}\xspace$ belongs to $[f < f(\mathbf{x_{\mathrm{sad}}}\xspace)]$, then there exists some $i_{\mathrm{min}}\xspace \in I$ such that $\mathbf{x_{\mathrm{min}}}\xspace \in C_{\imin}\xspace$ (see Equation $(2)$ above).
\item Now, if we assume that $\mathbf{x_{\mathrm{min}}}\xspace$ is not the representative of $C_{\imin}\xspace$, there exists then some $\mathbf{x_{<}}\xspace$ in $C_{\imin}\xspace$ satisfying that $f(\mathbf{x_{<}}\xspace) < f(\mathbf{x_{\mathrm{min}}}\xspace)$, and then there exists some $\Pi$ in $(D_{\xmin})\xspace$ whose image is contained in $C_{\imin}\xspace$. In other words,
$$\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) \leq \mathrm{Effort}\xspace(\Pi) < f(\mathbf{x_{\mathrm{sad}}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace),$$
which contradicts the hypothesis that $\mathbf{x_{\mathrm{min}}}\xspace$ is paired with $\mathbf{x_{\mathrm{sad}}}\xspace$ by dynamics.
\item Let us show that $i_{\mathrm{min}}\xspace$ belongs to $I^{\mathrm{sad}}\xspace$, that is, $\mathbf{x_{\mathrm{sad}}}\xspace \in \mathrm{clo}\xspace(C_{\imin}\xspace)$. Let us assume that:
$$\mathbf{x_{\mathrm{sad}}}\xspace \not \in \mathrm{clo}\xspace(C_{\imin}\xspace). \ \ \ (\mathrm{HYP2}\xspace)$$
Every path in $(D_{\xmin})\xspace$ goes outside of $C_{\imin}\xspace$ to reach some point whose image by $f$ is lower than $f(\mathbf{x_{\mathrm{min}}}\xspace)$ since $\mathbf{x_{\mathrm{min}}}\xspace$ has been proven to be the representative of $C_{\imin}\xspace$. Then this path intersects the boundary $\partial\xspace$ of $C_{\imin}\xspace$. Since by $(\mathrm{HYP2}\xspace)$, $\mathbf{x_{\mathrm{sad}}}\xspace$ does not belong to the boundary $\partial\xspace$ of $C_{\imin}\xspace$, any optimal path $\Pi^*\xspace$ in $(D_{\xmin})\xspace$ goes through one $1$-saddle $\mathbf{x_{\mathrm{sad}}}\xspace_2 = {\arg \max}_{\ell \in [0,1]} f(\Pi^*\xspace(\ell))$ (by Lemma~\ref{lemma.saddle}) different from $\mathbf{x_{\mathrm{sad}}}\xspace$ and \nico{satisfying} then $f(\mathbf{x_{\mathrm{sad}}}\xspace_2) > f(\mathbf{x_{\mathrm{sad}}}\xspace)$. Thus, $\mathrm{dyn}\xspace(\mathbf{x_{\mathrm{min}}}\xspace) > f(\mathbf{x_{\mathrm{sad}}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$, which contradicts the hypothesis that $\mathbf{x_{\mathrm{min}}}\xspace$ is paired with $\mathbf{x_{\mathrm{sad}}}\xspace$ by dynamics. Then, we have:
$$\mathbf{x_{\mathrm{sad}}}\xspace \in \mathrm{clo}\xspace(C_{\imin}\xspace).$$
\end{enumerate}
\item Now let us show that $f(\mathbf{x_{\mathrm{min}}}\xspace) > f(\mathrm{rep}\xspace(C_i^{\mathrm{sad}}\xspace))$ for any $i \in I^{\mathrm{sad}}\xspace \setminus \{i_{\mathrm{min}}\xspace\}$. For this aim, we prove that there exists some $i \in I^{\mathrm{sad}}\xspace$ such that $f(\mathrm{rep}\xspace(C_i^{\mathrm{sad}}\xspace)) < f(\mathbf{x_{\mathrm{min}}}\xspace)$ and we conclude with Proposition~\ref{proposition.2}. Let us assume that the representative $r$ of each component $C_i^{\mathrm{sad}}\xspace$ except $C^{\mathrm{min}}\xspace$ satisfies $f(r) > f(\mathbf{x_{\mathrm{min}}}\xspace)$, then any path $\Pi$ of $(D_{\xmin})\xspace$ has to go outside $C^{\mathrm{sad}}\xspace$ to reach some point whose image by $f$ is lower than $f(\mathbf{x_{\mathrm{min}}}\xspace)$. We obtain the same situation as before (see $(1.c)$), and then we obtain that the effort of $\Pi$ is greater than $f(\mathbf{x_{\mathrm{sad}}}\xspace) - f(\mathbf{x_{\mathrm{min}}}\xspace)$, which leads to a contradiction with the hypothesis that $\mathbf{x_{\mathrm{min}}}\xspace$ is paired with $\mathbf{x_{\mathrm{sad}}}\xspace$ by dynamics. We have then that there exists $i \in I^{\mathrm{sad}}\xspace$ such that $f(\mathrm{rep}\xspace(C_i^{\mathrm{sad}}\xspace)) < f(\mathbf{x_{\mathrm{min}}}\xspace)$. Thanks to Proposition~\ref{proposition.2}, we know then that $\mathbf{x_{\mathrm{min}}}\xspace$ is the representative of the components of $[f < f(\mathbf{x_{\mathrm{sad}}}\xspace)]$ whose image by $f$ is the greatest.
\item It follows that $\mathbf{x_{\mathrm{sad}}}\xspace$ is paired with $\mathbf{x_{\mathrm{min}}}\xspace$ by persistence.
\end{enumerate}
\section{Perspectives: a research program linking Topological Data Analysis and MM}
\label{sec:perspectives}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{TDAanalysisPipeline.png}
\caption{An example of segmentation of a microscopy image of cells and their nuclei~\cite{lukasczyk2020localized} with the topological data analysis framework. The very same example can be seen as an application of the morphological data analysis framework (see text).}
\label{fig.TDApipeline}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=.7\linewidth]{saliencyCells.png}
\caption{Saliency map corresponding to Fig.\ref{fig.TDApipeline}. In this image, the contours that are the more persistent are darker than the others (see text for details.)}
\label{fig:saliencyCells}
\end{figure*}
This paper is a step towards exploring the possible interactions between Topological Data Analysis (TDA) and MM. In this section, we detail some ideas for a research program linking these two fields.
As a very first example, let us look at
Fig.\ref{fig.TDApipeline}, which provides an illustration of an image analysis pipe\-line originally performed in the context of topological data analysis using the library \nico{called \emph{Topology Toolkit}~\cite{tierny2017topology,masood2019overview} (shortly TTK).}
In the original publication \cite{lukasczyk2020localized}, the steps are the following
\begin{enumerate}
\item The original data (microscopy image of cells and their nuclei) are simplified with a small threshold of persistence (Fig.~\ref{fig.TDApipeline}.a)
\item The Morse-Smale complex leads to an oversegmentation ((Fig.~\ref{fig.TDApipeline}.b)
\item The persistence curve (Fig.~\ref{fig.TDApipeline}.c) is the number of persistent pairs as a function of their persistence. The vertical dashed line on the left corresponds to the level of simplification of Fig.~\ref{fig.TDApipeline}.a and b. The vertical dashed line on the right corresponds to the level of simplification of Fig.~\ref{fig.TDApipeline}.e and f.
\item The diagram of persistence (Fig.~\ref{fig.TDApipeline}.d)
\item The image is simplified (Fig.~\ref{fig.TDApipeline}.e) with a threshold corresponding to the vertical dashed line on the right of Fig.~\ref{fig.TDApipeline}.c.
\item The Morse-Smale complex separatrices of Fig.~\ref{fig.TDApipeline}.e provides 1 maximum per nuclei, while the nuclei are the maxima of the same image (Fig.~\ref{fig.TDApipeline}.f).
\end{enumerate}
Thanks to the result of this paper and some previous work, we can translate this process in mathematical morphology. The filtering by persistence belong to a class of morphological filters called {\em connected filters} \cite{salembier2009connected}, with a criterion named dynamics. The Morse-Smale complexe is replaced by the watershed \cite{vcomic2005morse,vcomic2016computing}. The persistence curve is called a granulometric curve \cite{matheron1972random}. Hence, from a morphological perspective, the same example can be done using Higra \cite{perret2019higra}, a (morphological) library that computes the various steps, and this leads to the following description.
\begin{enumerate}
\item A connected filter with a small dynamics threshold is first applied on the original data (Fig.~\ref{fig.TDApipeline}.a)
\item The watershed of Fig.~\ref{fig.TDApipeline}.a is oversegmented (see Fig.~\ref{fig.TDApipeline}.b)
\item The granulometric curve (Fig.~\ref{fig.TDApipeline}.c) provides the number of maximum as a function of the dynamics
\item A connected filter of Fig.~\ref{fig.TDApipeline}.a with a dynamics thres\-hold corresponding to the vertical dashed line on the right of Fig.~\ref{fig.TDApipeline}.c leads to Fig.~\ref{fig.TDApipeline}.e.
\item The watershed of Fig.~\ref{fig.TDApipeline}.e gives one region per cell, while the nuclei are the maxima of the same image (Fig.~\ref{fig.TDApipeline}.f).
\end{enumerate}
It is worthwhile to explore the differences between the two approaches. In mathematical morphology, there is no persistence diagram. On the other hand, there exist saliency maps \cite{najman1996geodesic,najman2011equivalence,cousty2018hierarchical}. Intuitively, a saliency map can be obtained by filtering the original image/data for all values of the criterion (here, dynamics), and stacking (summing) the watersheds of all the filtered images. A contour that is persistent is present many times in the stack, and has a high value in the resulting saliency map. Fig.~\ref{fig:saliencyCells} shows the saliency map of the original data of Fig.~\ref{fig.TDApipeline} for the dynamics criterion.
In TDA, only a few criteria other than dynamics have been studied \cite{carr2004simplifying} but MM has many more, see \cite{Higra::criteria} for a few of them. There exist also several ways to simplify using non-increasing criteria \cite{salembier1998antiextensive,urbach2007connected,xu2015connected}.
The links between Morse-Smale Complex and watershed \cite{vcomic2005morse,vcomic2016computing} need to be explored, specifically in the context of Discrete Morse Theory~\cite{forman1998morse}. We envision doing such a study based on watershed cuts \cite{cousty2009watershed}, see also \cite{cousty2014collapses} that highlights some links between the watershed and topology.
Many other comparisons should be done. To mention one of those, the {\em contour tree} \cite{freeman1967searching} from TDA is closely related to the {\em tree of shapes} \cite{caselles2009geometric} from MM. Comparing those trees and the algorithms for computing them from TDA \cite{carr2003computing,gueunet2019task} and from MM \cite{geraud2013quasi,crozet2014first,carlinet2018tree} would be rewarding. In particular, the morphological algorithms for computing the tree of shapes, which are quasi-linear whatever the dimension of the space, are based on the ones for computing the tree of upper or lower level sets, called the {\em component trees} \cite{carlinet2014comparative}, and seem more efficient than the ones from TDA.
\section{Conclusion}
\label{sec.conclusion}
In this paper, we have proved that persistence and dynamics lead to the same pairings in $n$-D, $n \geq 1$, which implies that they are equal whatever the dimension. Concerning the future works, we propose to investigate the relationship between persistence and dynamics in the discrete case~\cite{forman1998morse} (that is, on complexes). We will also check under which conditions pairings by persistence and by dynamics are equivalent for functions that are not Morse. Furthermore, we will examine if the fast algorithms used in MM like watershed cuts, Betti numbers computations or attribute-based filtering are applicable to PH. Conversely, we will study if some PH concepts can be seen as the generalization of some MM concepts (for example, dynamics seems to be a particular case of persistence).
More generally, we believe that exploring the links and differences between TDA and MM would benefit to the two communities.
\section*{Acknowledgements}
The authors would like to thank Julien Tierny for many interesting discussions and for providing \nico{us} Fig. \ref{fig.TDApipeline}.
| {
"timestamp": "2022-05-26T02:11:03",
"yymm": "2205",
"arxiv_id": "2205.12546",
"language": "en",
"url": "https://arxiv.org/abs/2205.12546",
"abstract": "In Mathematical Morphology (MM), connected filters based on dynamics are used to filter the extrema of an image. Similarly, persistence is a concept coming from Persistent Homology (PH) and Morse Theory (MT) that represents the stability of the extrema of a Morse function. Since these two concepts seem to be closely related, in this paper we examine their relationship, and we prove that they are equal on n-D Morse functions, n $\\ge$ 1. More exactly, pairing a minimum with a 1-saddle by dynamics or pairing the same 1-saddle with a minimum by persistence leads exactly to the same pairing, assuming that the critical values of the studied Morse function are unique. This result is a step further to show how much topological data analysis and mathematical morphology are related, paving the way for a more in-depth study of the relations between these two research fields.",
"subjects": "Computer Vision and Pattern Recognition (cs.CV); Image and Video Processing (eess.IV); Signal Processing (eess.SP); Algebraic Topology (math.AT); Differential Geometry (math.DG)",
"title": "Some equivalence relation between persistent homology and morphological dynamics",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808690122163,
"lm_q2_score": 0.7217432122827968,
"lm_q1q2_score": 0.7075110633402486
} |
https://arxiv.org/abs/1611.09443 | Star bodies with completely symmetric sections | We say that a star body $K$ is completely symmetric if it has centroid at the origin and its symmetry group $G$ forces any ellipsoid whose symmetry group contains $G$, to be a ball. In this short note, we prove that if all central sections of a star body $L$ are completely symmetric, then $L$ has to be a ball. A special case of our result states that if all sections of $L$ are origin symmetric and 1-symmetric, then $L$ has to be a Euclidean ball. This answers a question from \cite{R2}. Our result is a consequence of a general theorem that we establish, stating that if the restrictions in almost all equators of a real function $f$ defined on the sphere, are isotropic functions, then $f$ is constant a.e. In the last section of this note, applications, improvements and related open problems are discussed and two additional open questions from \cite{R} and \cite{R2} are answered.} | \section{Introduction}
\hspace*{1.5em}In what follows, we fix an orthonormal basis in the $n$-dimensional Euclidean space $\mathbb{R}^n$ and we denote by $\langle x,y\rangle$ the scalar product of two vectors $x,\ y$, with respect to this orthonormal basis. We denote by $S^{n-1}$ the unit sphere in $\mathbb{R}^n$, i.e. the set $\{x\in\mathbb{R}^n:|x|=1\}$, where $|x|=\sqrt{\langle x,x\rangle}$ is the length of $x$. We denote by $x^{\perp}$ the subspace which is orthogonal to $x$, i.e. $x^{\perp}=\{u\in\mathbb{R}^n: \langle x,u\rangle=0\}$. Every set of the form $S^{n-1}\cap x^{\perp}$ will be called an ``equator of $S^{n-1}$''. Denote, also, by $dx$ (or $dy$, $du$) the volume element on $S^{n-1}$ or on equators of $S^{n-1}$. The set of all isometries in $\mathbb{R}^n$ will be denoted by $ISO(n)$ and the set of all linear isometries (i.e. the orthogonal group) will be denoted by $O(n)$. Let $H$ be linear space. We denote by $SO(H)$, the set of all rotations in $H$. As usual, we set $SO(n):=SO(\mathbb{R}^n)$. A convex body in $\mathbb{R}^n$ will be a convex compact set with non-empty interior. The term ``star body'' will always refer to a set which is star-shaped with respect to the origin and has non-empty interior. A star body $K$ will be called origin symmetric (or symmetric) if $K=-K$ and $K$ will be called centrally symmetric if $K$ is a translation of $-K$. Finally, we say that $K$ is \emph{1-symmetric} if its symmetry group contains the symmetry group of a cube in $\mathbb{R}^n$.
Several important results concerning characterizations of Euclidean spaces by properties of their subspaces or quotients
have appeared in the past; we refer to \cite{Gr}, \cite{M} and \cite{Sch2}. The starting point of this manuscript is the study of the following related problem.
\begin{question}\cite{R2}\label{question-main}
Let $K$ be a convex body in $\mathbb{R}^n$, $n\geq 3$. If all orthogonal projections of $K$ are 1-symmetric, is it true that $K$ must be a Euclidean ball?
\end{question}
We will show that the answer to this question is affirmative.
It turns out that we are able to prove much more, as we provide with a rather general sufficient condition for the structure of symmetries of projections or sections of $K$ that forces $K$ to be a ball.
First we will need some definitions. Recall that the symmetry group of a set (function on $S^{n-1}$) is the group of isometries $Sym(K):=\{T\in ISO(n):TK=K\}$ (resp. $Sym(f):=\{T\in O(n):f(Tx)=f(x)\ , \ \textnormal{ for all } x\in S^{n-1}\}$).
\begin{definition}\label{def-main}
A subgroup $G$ of $ISO(n)$ will be called \emph{complete}, if every ellipsoid whose symmetry group contains $G$ is a ball.
\end{definition}
Famous examples of complete groups of isometries (see also Remarks \ref{rem-0} and \ref{rem-0'} below) are dihedral groups (in $\mathbb{R}^2$), symmetry groups of platonic solids (in $\mathbb{R}^3$), the symmetry group of the regular $n$-simplex and the symmetry group of the $n$-cube.
\begin{definition}\label{def-comp-sym-2}
A subset of $\mathbb{R}^n$ or a function $f:S^{n-1}\to\mathbb{R}$ is called \emph{completely symmetric} if its centroid is at the origin and its symmetry group is complete.
\end{definition}
We have the following result:
\begin{theorem}\label{cor-main}
Let $f:S^{n-1}\to\mathbb{R}$ be a continuous function whose restriction to every equator is completely symmetric. Then, $f$ is constant.
\end{theorem}
The following corollaries will follow immediately from Theorem \ref{cor-main}.
\begin{corollary}\label{cor-1}
Let $K$ be a star body with continuous radial function in $\mathbb{R}^n$, $n\geq 3$, whose central sections (i.e. intersections with subspaces) are all completely symmetric. Then, $K$ is an origin symmetric Euclidean ball.
\end{corollary}
For instance, if all central sections of a star body $K$ with continuous boundary are origin symmetric and 1-symmetric, then $K$ has to be an origin symmetric ball.
\begin{corollary}\label{cor-2}
Let $K$ be a convex body in $\mathbb{R}^n$, $n\geq 3$, whose orthogonal projections are all completely symmetric. Then, $K$ is an origin symmetric Euclidean ball.
\end{corollary}
\begin{corollary}\label{cor-3}
Let $K$ be a convex body in $\mathbb{R}^n$, $n\geq 3$, whose central sections are all 1-symmetric. Then, $K$ is a Euclidean ball.
\end{corollary}
\begin{corollary}\label{cor-4}
Let $K$ be a convex body in $\mathbb{R}^n$, $n\geq 3$, whose orthogonal projections are all 1-symmetric. Then, $K$ is a Euclidean ball.
\end{corollary}
We call a measurable function $f:S^{n-1}\to\mathbb{R}$ \emph{ isotropic} if the signed measure $fdx$ is isotropic. Recall that a signed Borrel measure $\mu$ on $S^{n-1}$ is called isotropic if its center of mass is at the origin and the map
$$S^{n-1}\ni x\mapsto\int_{S^{n-1}}\langle x,y\rangle^2 d\mu(y)\in\mathbb{R}$$
is constant.
Theorem \ref{cor-main} will follow easily from the following general result:
\begin{theorem}\label{thm-main}
Let $f:S^{n-1}\to\mathbb{R}$ be a measurable, bounded a.e. and even function, $n\geq 3$. If for almost every $u\in S^{n-1}$ the restriction $f|_{S^{n-1}\cap u^{\perp}}$ of $f$ to $S^{n-1}\cap u^{\perp}$ is isotropic (i.e. the restriction of $f$ to almost every equator is isotropic), then $f$ is almost everywhere equal to a constant.
\end{theorem}
Proofs of the aforementioned results will be given in Section 3. In Section 4, we establish some related results (namely, we prove a stability version of Theorem \ref{thm-main} and we answer two questions from \cite{R} and \cite{R2}) and we pose some open problems. Let us comment on the method of the proof of Theorem \ref{thm-main}: The novelty consists of the fact that we avoid using involved geometric and topological arguments, that often appear in results of Geometric Tomography (i.e. problems of determination of sets by data of their projections or sections). Instead, we make use of some integral geometric formulae together with an isoperimetric inequality and its equality cases.
\begin{remark}\label{rem-0}
In the plane, there is a very simple description of completeness. A subgroup $G$ of $ISO(2)$ is complete if and only if it contains an operator which can be written as the composition of a rotation $T$, different than $\pm Id$, and a translation.
\end{remark}
To see this, assume for simplicity that $G\subseteq O(2)$ and assume that $G$ contains a rotation different than $\pm Id$. The reader can check that if $E$ is an origin symmetric ellipse with $TE=E$, then $E$ must be a disk. Conversely, assume that $G$ does not contain a rotation different than $\pm Id$. Since the composition of two reflections is a rotation, $G$ contains at most three elements: $Id$, $-Id$ and a reflection $R$. Then, for any origin symmetric ellipse $E$, with $RE=E$, $E$ is invariant under the action of $G$, which proves our claim.
\begin{corollary}\label{cor-3-D}
Let $f:S^2\to\mathbb{R}$ be a continuous fuction. Assume that for every $u\in S^2$, there exists $T_u\in SO(u^{\perp})\setminus\{\pm Id\}$, such that $f(T_ux)=f(x)$, for all $x\in u^{\perp}$. Then, $f$ is constant.
\end{corollary}
\noindent Proof. Immediate from Theorem \ref{cor-main} and the previous remark. $\square$
\section{Background}
In this section, we mention some basic facts about convex and star bodies that will be needed subsequently. We refer to \cite{G} and \cite{Sc} for more details and references.
Let $K$ be a star body in $\mathbb{R}^n$, that contains the origin in its interior. Its radial function $\rho_K:S^{n-1}\to \mathbb{R}_+$ of $K$ is defined by
$$\rho_K(u)=\sup\{tu:tu\in K\}\ .$$
It is true that $\rho_{K\cap u^{\perp}}=\rho_K|_{S^{n-1}\cap u^{\perp}}$,
for all $u\in S^{n-1}$. The star body $K$ is called isotropic if its centroid is at the origin and
\begin{equation}\label{eq-isotropicity-1st}
\int_K\langle x,y\rangle^2dy=|K|^{(n+2)/n}L_K^2\ ,\qquad\textnormal{for all }x\in S^{n-1}\ ,
\end{equation}
where $L_K$ is a constant that depends only on $K$ and it is called ``the isotropic constant of K'' (see \cite{MP}). If $K$ is origin symmetric (or more generally if its centroid is at the origin), then it follows directly by integration in polar coordinates that $K$ is isotropic if and only if the function $\rho^{n+2}_K$ is isotropic. Moreover, one can easily check (see again \cite{MP}) that an equivalent statement to (\ref{eq-isotropicity-1st}) is the following:
\begin{equation}\label{eq-isotropicity-2nd}
\int_Kx_ix_jdx=\delta_{ij}|K|^{(n+2)/n}L_K^2\ ,\qquad i,j\in\{1,\dots,n\}\ .
\end{equation}
Let $L$ be convex a convex body in $\mathbb{R}^n$. Its support function $h_L:\mathbb{R}^n\to \mathbb{R}$ is given by
$$h_L(x)=\max\{\langle x,y\rangle:y\in L\}\ .$$
Note that for $u\in S^{n-1}$, we have $h_{L|u^{\perp}}|_{S^{n-1}}=h_L|_{S^{n-1}\cap u^{\perp}}$, where $L|u^{\perp}$ denotes the orthogonal projection of $L$ onto the subspace $u^{\perp}$.
The function $h_L$ is clearly convex and positively homogeneous. Conversely, every convex and positively homogeneous function is the support function of a unique convex body.
The surface area measure of $L$ (viewed as a measure on $S^{n-1}$) is defined by $$S_L(\Omega)=\mathcal{H}^{n-1}\Big(\big\{x\in \textnormal{bd}L:\exists u\in \Omega, \textnormal{ such that }\langle x,u\rangle=h_K(u)\big\}\Big)\ ,\ \ \Omega\textnormal{ is a Borel subset of }S^{n-1}\ ,$$
where $\textnormal{bd}L$ is the boundary of $L$, $\mathcal{H}^{n-1}$ is the $(n-1)$-dimensional Haussdorff measure.
It holds that
\begin{equation}\label{eq-total-curvature}
\int_{S^{n-1}}dS_L(x)=\partial L\ ,
\end{equation}
where $\partial L$ denotes the surface area of $L$.
If $S_L$ is absolutely continuous with respect to the Lebesgue measure, its density is denoted by $f_L$ and it is usually called ``the curvature function of $L$''.
Assume, now that the convex body $L$ contains the origin in its interior. Its polar body $L^\circ$ is defined by:
$$L^{\circ}=\{x\in\mathbb{R}^n:\langle x,y\rangle\leq 1, \textnormal{ for all }y\in L\}\ .$$
Then, $L^{\circ}$ is also a convex body that contains the origin in its interior. Moreover, $(L^{\circ})^{\circ}=L$ and
\begin{equation}\label{eq-polarity}
(L\cap u^{\perp})^{\circ}=L^{\circ}|u^{\perp}\qquad\textnormal{and}\qquad (L|u^{\perp})^{\circ}=L^{\circ}\cap u^{\perp}\ ,
\end{equation}
for all $u\in S^{n-1}$.
\section{Proofs}
\hspace*{1.5em}We start this section with the proof of our main result, Theorem \ref{thm-main}. All constants that appear will be positive constants that depend only on the dimension $n$.
Let $K$ be a symmetric star body in $\mathbb{R}^n$. The centroid body $\Gamma K$ of $K$ is the symmetric convex body whose support function is defined as follows:
$$h_{\Gamma K}(u)=\int_K|\langle x,u\rangle|dx\ .$$
Define also the quantity
\begin{equation}\label{eq-B(K)}
B(K):=\int_K\dots\int_K\textnormal{det}(x_1,\dots,x_n)^2dx_1\dots dx_n\ .
\end{equation}
We will need the following fact:
\begin{lemma}\label{lemma-centroid-density}
The surface area measure of $\Gamma K$ is absolutely continuous and its density $f_{\Gamma K}$ is given by:
$$f_{\Gamma K}(u)=c_1B(K\cap u^{\perp})\ ,\qquad u\in S^{n-1}\ .$$
\end{lemma}
\noindent Proof.
Note that by integrating in polar coordinates, the support function of the centroid body of $K$ can be written as:
\begin{equation}\label{eq-centroid-projection-body}
h_{\Gamma K}(u)=\frac{1}{n+1}\int_{S^{n-1}}|\langle x,u\rangle|\rho_K^{n+1}(x)dx\ ,\qquad u\in S^{n-1}\ .
\end{equation}
Lemma \ref{lemma-centroid-density} is actually a reformulation of a result of Weil about the curvature function of the projection body of a convex body with absolutely continuous surface area measure. Recall the definition of the projection body $\Pi L$ of a convex body $L$:$$h_{\Pi L}(u)=\frac{1}{2}\int_{S^{n-1}}|\langle x,u\rangle| dS_L(x)\ ,\qquad u\in S^{n-1}\ . $$As Weil showed (see \cite{W}), if $f=f_L=dS_L/dx$, then for all $u\in S^{n-1}$, it holds:
\begin{equation}\label{eq-Weil}
f_{\Pi L}(u)=c_1'\int_{S^{n-1}\cap u^{\perp}}\dots\int_{S^{n-1}\cap u^{\perp}}\textnormal{det}(x_1,\dots,x_{n-1})^2f(x_1)\dots f(x_{n-1})dx_1\dots dx_{n-1}
\end{equation}
By the Minkowski Existence and Uniqueness Theorem (see again \cite{Sc}), there exists a unique (up to translation) convex body $C(K)$ with surface area measure given by
$$dS_{C(K)}=\frac{2}{n+1}\rho_K^{n+1}dx\ .$$
It is then clear that $\Gamma K=\Pi C(K)$, so by (\ref{eq-centroid-projection-body}) and (\ref{eq-Weil}), for $u\in S^{n-1}$, we get:
\begin{eqnarray*}
f_{\Gamma K}(u)&=&f_{\Pi C(K)}(u)\\
&=&c_1''\int_{S^{n-1}\cap u^{\perp}}\dots\int_{S^{n-1}\cap u^{\perp}}\textnormal{det}(x_1,\dots,x_{n-1})^2\rho_K^{n+1}(x_1)\dots \rho_K^{n+1}(x_{n-1})dx_1\dots dx_{n-1}\\
&=&c_1B(K\cap u^{\perp})\ ,
\end{eqnarray*}
where the last equality follows again by integration in polar coordinates.
$\square$\\
\\
The following easy fact (see e.g. \cite{MP}) will be also needed.
\begin{lemma}\label{lemma-legendre-isotropic}
If $K$ is isotropic, then $$\int_K|x|^2dx=c_2B(K)^{1/n}\ .$$
\end{lemma}
\noindent Proof.
It follows immediately by (\ref{eq-isotropicity-1st}) and (\ref{eq-isotropicity-2nd}) that
$$\int_K |x|^2dx=n|K|^{(n+2)/n}L_K^2\ .$$
On the other hand, expanding the determinant in the definition of $B(K)$, one easily sees:
$$B(K)=c_2'(L_K^2)^n
\ ,$$
as required. $\square$\\
\\
Proof of Theorem \ref{thm-main}: Let $c\in\mathbb{R}$. It is clear by our assumption and by the obvious fact that $c$ is isotropic in every equator of $S^{n-1}$ that $f+c$ is bounded a.e., even and isotropic in almost every equator of $S^{n-1}$. Since $f$ is bounded, we may choose $c$ to be so large that $f+c$ is positive a.e. Therefore, by replacing $f$ by $f+c$ if necessary, we may assume that $f$ is positive a.e.. Consequently, we may view the function $f^{1/(n+1)}$ as the radial function $\rho_K$ of an origin-symmetric star body $K$. Then, by our assumption $\rho_K^{n+1}$ is isotropic in almost every equator, thus $K\cap u^{\perp}$ is isotropic, for almost every $u\in S^{n-1}$. Integrating in polar coordinates twice, one has:
$$\int_K|x|dx=\frac{1}{n+1}\int_{S^{n-1}}\rho_K^{n+1}dx=c_3\int_{S^{n-1}}\int_{S^{n-1}\cap u^{\perp}}\rho_K^{n+1}dxdu=c_4\int_{S^{n-1}}\int_{K\cap u^{\perp}}|x|^2dxdu\ .$$
Using the assumption that $K\cap u^{\perp}$ is isotropic, together with Lemma \ref{lemma-legendre-isotropic}, we get:
$$\int_K|x|dx=c_5\int_{S^{n-1}}B(K\cap u^{\perp})^{1/(n-1)}du\ ,$$
so using H\"older's inequality we obtain:
\begin{equation}\label{eq-Holder}
\int_K|x|dx\leq c_6\Big(\int_{S^{n-1}}B(K\cap u^{\perp})du\Big)^{1/(n-1)}\ .
\end{equation}
Next, using Lemma \ref{lemma-centroid-density} and (\ref{eq-total-curvature}), (\ref{eq-Holder}) immediately becomes:
\begin{equation}\label{eq-after-Holder}
\int_K|x|dx\leq c_7\Big(\int_{S^{n-1}}f_{\Gamma K}du\Big)^{1/(n-1)}=c_8\partial \big(\Gamma K\big)^{1/(n-1)}\ ,
\end{equation}
where $\partial L$ denotes the surface area of a convex body $L$. Recall the particular case of the Aleksandrov-Fenchel inequality (see e.g. \cite{Sc}):
\begin{equation}\label{eq-A-F}
\big(\partial L\big)^{1/(n-1)}\leq c_0W(L):=\int_{S^{n-1}}h_Ldx\ ,
\end{equation}with equality if and only if $L$ is a ball (note that the quantity $W(L)$ is proportional to the mean width of $L$). Applying this to (\ref{eq-after-Holder}), we deduce:
\begin{equation}\label{eq-last}
\int_K|x|dx\leq c_9W(\Gamma K)
=c_{10}\int_K|x|dx\ .
\end{equation}
Note that equality holds in both (\ref{eq-Holder}) and (\ref{eq-A-F}) (and therefore in (\ref{eq-last})) if $K$ is a ball. Thus, $c_{10}=1$. On the other hand, if $K$ is not a ball, it is well known (see again \cite{Sc} or \cite{G}) that (since $K$ is centrally symmetric), $\Gamma K$ is not a ball, so the inequality (\ref{eq-A-F}) is strict. This is a contradiction and our result follows. $\square$\\
\\
\begin{remark}\label{rem-l1}
It follows from the proof of Theorem \ref{thm-main} that if we restrict ourselves to non-negative functions (or more generally bounded from below a.e.), the assertion of the Theorem remains true if we assume $f$ to be just integrable instead of a.e. bounded.
\end{remark}
\begin{remark}
The evenness assumption in Theorem \ref{thm-main} cannot be dropped. For instance, it is well known that in any dimension $n\geq 2$, there exist convex bodies of constant width that are not Euclidean balls, i.e. convex bodies $K$ such that $h_K+h_{-K}$ is constant but $K$ is not a ball. Clearly, the support function $h_K$ of a body of constant width $K$ is isotropic in every equator, but $h_K$ is not constant if $K$ is not a ball.
\end{remark}
Before we proceed to the proof of Theorem \ref{cor-main}, we will need the following lemma:
\begin{lemma}\label{lemma-comp-sym-implies-isotropy}
Let $f:S^{n-1}\to \mathbb{R}_+$ be an integrable function. If $f$ is completely symmetric, then $f$ is isotropic.
\end{lemma}
\noindent Proof. Let $G$ be the symmetry group of $f$. Since the centroid operator intertwines isometries, it follows immediately that $G\subseteq O(n)$. Furthermore, notice that $$\int_{S^{n-1}}f(x)\langle x,u\rangle^2dx\leq \int_{S^{n-1}}f(x)|x|^2dx=\int_{S^{n-1}}f(x)dx<\infty\ ,$$for all $u\in S^{n-1}$.
Define the ellipsoid $E$, whose support function is given by:
$$h_E(u)^2=\int_{S^{n-1}}f(x)\langle x,u\rangle^2 dx\ ,\qquad u\in S^{n-1}\ .$$
Then, for any $T\in G$ and for any $u\in S^{n-1}$, we have:
\begin{eqnarray*}
h_E(u)^2&=&\int_{S^{n-1}}f(x)\langle x,u\rangle^2 dx\\
&=& \int_{S^{n-1}}f(Tx)\langle x,u\rangle^2 dx\\
&=&\int_{S^{n-1}}f(x)\langle T^{-1}x,u\rangle^2 dx\\
&=& \int_{S^{n-1}}f(x)\langle x,Tu\rangle^2 dx=h_E(Tu)^2\ .
\end{eqnarray*}
This shows that the symmetry group of $h_E$ and hence the symmetry group of $E$ contains $G$. Since $G$ is complete, it follows that $E$ is a Euclidean ball, thus $h_E$ is constant on $S^{n-1}$. This is equivalent to the fact that $f$ is isotropic. $\square$
\begin{remark}\label{rem-0'}
In view of the proof of Lemma \ref{lemma-comp-sym-implies-isotropy}, define the $O(n)$-symmetry group of a set $K$ (resp. a function $f$ on $S^{n-1}$) to be the subgroup of $O(n)$: $Sym(K)\cap O(n)$ (resp. $Sym(f)\cap O(n)$).
The following is an equivalent definition of complete symmetry: A real function defined on the sphere or a subset of $\mathbb{R}^n$ is called completely symmetric, if its $O(n)$-symmetry group is complete.
\end{remark}
To see this, notice that since the centroid operator (for sets or functions) intertwines isometries, it follows that the centroid $x$ of all sets or functions, whose $O(n)$-symmetry group is complete, has to be the origin. Indeed, if $x$ is not the origin and $G$ is the $O(n)$-symmetry group of a set or a function on the sphere, then for all $T\in G$, we must have $Tx=x$. This shows that the line $\mathbb{R}x$ is invariant under the action of $G$. Then, for any origin-symmetric ellipsoid $E$ one of whose major axis' is $\mathbb{R}x$ and its intersection with the hyperplane $x^{\perp}$ is an $(n-1)$-dimensional ball, $Sym(E)\supseteq G$, but is not necessarily a ball. Thus, the centroid of a set or function, whose $O(n)$-symmetry group is complete, is always the origin. But then, as in the proof of Lemma \ref{lemma-comp-sym-implies-isotropy}, it follows that its symmetry group is contained in $O(n)$.\\
\\
Proof of Theorem \ref{cor-main}: As in the proof of Theorem \ref{thm-main}, we may assume that $f$ is non-negative. Then, for every $p> 0$, the function $f^p$ is completely symmetric in every equator of $S^{n-1}$, thus by Lemma \ref{lemma-comp-sym-implies-isotropy}, $f(x)^p$ is isotropic in every equator of $S^{n-1}$. But then, it is immediate to check that the function $f(-x)^p$ is also isotropic in every equator of $S^{n-1}$. It follows that the function $F_p(x):=(1/2)(f(x)^p+f(-x)^p)$ is isotropic in every equator of $S^{n-1}$. Since $F_p$ is also even for all $p\geq 0$ and since $F_p$ is continuous, it follows by Theorem \ref{thm-main} that $F_p\equiv d_p$, where $d_p\geq 0$ is a constant which depends only on $p$. In particular, $f(x)+f(-x)\equiv 2d_1$. Furthermore, we get $\max\{f(x),f(-x)\}\equiv d_{\infty}:=\lim_{p\to\infty}(d_p)^{1/p}$. Thus, for every $x\in S^{n-1}$, we have $f(x)=d_{\infty}$ or $f(x)=2d_1-d_{\infty}$. This together with the continuity of $f$ prove our claim. $\square$
\\
\\
Proof of Corollaries \ref{cor-1} and \ref{cor-2}: Let $K$ be a star body with continuous radial function whose sections are completely symmetric (resp. a convex body whose projections are completely symmetric). Then, $\rho_K$ (resp. $h_K$) is continuous and completely symmetric in every equator of $S^{n-1}$. This shows that $\rho_K$ (resp. $h_K$) is constant thus $K$ is a ball. $\square$
\\
\\
Proof of Corollaries \ref{cor-3} and \ref{cor-4}: It is clear that an 1-symmetric convex body is centrally symmetric and if its center of symmetry is the origin, then it is also completely symmetric. Let $K$ be a convex body with 1-symmetric sections (resp. projections).
It is known (see \cite[Corollaries 7.1.3 and 3.1.5]{G}) that if a convex body $K$ has centrally symmetric central sections (resp. projections), then it is itself centrally symmetric. Therefore, there is a vector $x\in \mathbb{R}^n$, such that all sections (resp. projections) of $K+x$ are completely symmetric, which by Corollary \ref{cor-1} (resp. \ref{cor-2}) shows that $K+x$ is an origin symmetric ball, hence $K$ is a ball, as claimed. $\square$
\section{Further remarks and open problems}
$\S$ 1. As we have seen, the method for proving Theorem \ref{thm-main} (and therefore all its applications described here) is remarkably quick. Another advantage is that a modification of this method implies an improvement of Theorem \ref{thm-main} for positive functions, namely a stability result. To demonstrate this, let us recall that $B(K)$ (defined by (\ref{eq-B(K)})) is invariant under volume-preserving linear transformations. This, easily implies that if $K$ is a compact set with centroid at the origin, then $$\int_K|x|^2dx\geq c_2B(K)^{1/n}\ ,$$with equality if and only if $K$ is isotropic ($c_2$ is the constant from Lemma (\ref{lemma-legendre-isotropic}). It is, therefore, reasonable to define $K$ to be ``$\varepsilon$-isotropic'' (for some $\varepsilon\geq 0$), if $K$ has centroid at the origin and $\int_K|x|^2dx\leq (1+\varepsilon)c_2B(K)^{1/n}$. Similarly, for a non-negative function $f:S^{n-1}\to\mathbb{R}$, we say that $f$ is $\varepsilon$-isotropic, if
\begin{equation}\label{eq-1+epsilon-isotropic}
\int_{S^{n-1}}fdx\leq (1+\varepsilon)\overline{c}\bigg(\int_{S^{n-1}}\dots\int_{S^{n-1}}\textnormal{det}(x_1,\dots,x_n)^2f(x_1)\dots f(x_n)dx_1\dots dx_n\bigg)^{1/n} \ ,
\end{equation}
where the constant $\overline{c}$ is chosen so that there is equality in (\ref{eq-1+epsilon-isotropic}) when $f\equiv 1$ and $\varepsilon=0$. Note that a set or a function is isotropic if and only if it is 0-isotropic. Integration in polar coordinates implies immediately that a symmetric star body (or more generally a star body with centroid at the origin) $K$ is $\varepsilon$-isotropic if and only if the function $\rho_K^{n+2}$ is $\varepsilon$-isotropic. Now, let $f=\rho_K^{n+1}:S^{n-1}\to (0,\infty)$ ($K$ is as always a symmetric star body) be such that $f|_{S^{n-1}\cap u^{\perp}}$ is $\varepsilon$-isotropic, for all $u\in U$, where $U$ is a measurable subset of $S^{n-1}$, whose complement has measure less than $\delta>0$. Following the steps of the proof of Theorem \ref{thm-main}, we arrive at:
\begin{eqnarray*}
c_0W(\Gamma K)\leq(1+\varepsilon)\Big(\partial(\Gamma K)\Big)^{1/(n-1)}+\overline{c}'\delta\esssup\rho_K^{n+1} \leq \big(\partial(\Gamma K)\big)^{1/(n-1)}+\overline{c}''(\varepsilon+\delta)\esssup\rho_K^{n+1}\ ,
\end{eqnarray*}
where $\overline{c}'$ and $\overline{c}''$ are positive constants that depend only on $n$. On the other hand, since trivially $\Gamma K$ contains the centroid body of the ball of radius $\essinf \rho_K$ and is contained in the centroid body of the ball of radius $\esssup\rho_K$ (recall that the centroid body of an origin symmetric ball is a ball), it follows from \cite[(7.124)]{Sc} that $$c_0W(\Gamma K)-(\partial (\Gamma K))^{1/(n-1)}\geq Cd_H(\Gamma K, \overline{B})\ ,$$
where $d_H$ is the Haussdorff metric, $\overline{B}$ is the ball centered at the origin with the same mean width as $\Gamma K$ and $C$ is a constant that depends only on $n$, $\esssup \rho_K^{n+1}$ and $\essinf\rho_K^{n+1}$. Since $(n+1)h_{\Gamma K}(u)=\int_{S^{n-1}}f(x)|\langle x,u\rangle|dx$ and $W(\overline{B})=W(\Gamma K)$ is proportional to $\int_{S^{n-1}}fdx$, we conclude the following:
\begin{theorem}\label{thm-stability}
There exists a function $C:(0,\infty)^2\to (0,\infty)$, with the following property: Let $\varepsilon,\delta>0$ and let $f:S^{n-1}\to\mathbb{R}$ be an even, measurable, strictly positive a.e. and bounded a.e. function, such that $f|_{S^{n-1}\cap u^{\perp}}$ is $\varepsilon$-isotropic, for all $u\in U$, for some measurable set $U\subseteq S^{n-1}$ with $|S^{n-1}\setminus U|<\delta$. Then,
$$\sup_{u\in S^{n-1}}\bigg|\int_{S^{n-1}}f(x)|\langle x,u\rangle |dx-k\int_{S^{n-1}}f(x)dx\bigg|\leq C(\essinf f,\esssup f)\cdot (\varepsilon+\delta)\ ,$$
where $k:=\int_{S^{n-1}}|x_1 |dx$.
\end{theorem}
To see that Theorem \ref{thm-stability} is actually stronger than Theorem \ref{thm-main}, recall the fact (see again \cite{Sc} or \cite{G}) that the cosine transform: $f\mapsto \int_{S^{n-1}}f(x)|\langle x,u\rangle |dx$ is injective in the class of even functions.
\\
\\
$\S$ 2. Another possible strengthening of Theorem \ref{thm-main} would be a local version of it. More precisely, it would be extremely interesting if the answer to the following question was affirmative:
\begin{question}\label{question-open-1}
Assume that for a measurable subset $U$ of $S^{n-1}$ and for an even bounded measurable function $f:S^{n-1}\to\mathbb{R}$, $f|_{S^{n-1}\cap u^{\perp}}$ is isotropic, for all $u\in U$. Is it true that $f$ is a.e. equal to a constant on the set $\bigcup_{u\in U} (S^{n-1}\cap u^{\perp})$?
\end{question}
Question \ref{question-open-1} is closely related to \cite[Problem 10]{R2}. We believe that such a result, if true, would have various applications in Geometric Tomography. To illustrate the usefulness of our conjectured local version of Theorem \ref{thm-main}, let us mention that together with Lemma \ref{lemma-old} below and an inductive argument, this would provide a quick alternative proof of a result due to the second named author \cite{R}: Let $f,\ g$ be two continuous functions on $S^{n-1}$, $n\geq 3$. If for each 2-dimensional subspace $H$, there exists $T_H\in SO(H)$, such that $f(x)=g(T_Hx)$, for all $x\in S^{n-1}\cap H$, then $f(x)=g(x)$, for all $x\in S^{n-1}$ or $f(x)=g(-x)$, for all $x\in S^{n-1}$.
\begin{lemma} \label{lemma-old}\cite{R}
Let $f,\ g$ be two continuous functions on $S^2$, such that for every $u\in S^2$, $f(x)=g(x)$, for all $x\in u^{\perp}$ or $f(x)=g(-x)$, for all $x\in u^{\perp}$. Then, $f(x)=g(x)$, for all $x\in S^{2}$ or $f(x)=g(-x)$, for all $x\in S^{2}$.
\end{lemma}
\begin{remark}\label{rem-1}
Our methods allow us to drop the continuity assumption in Lemma \ref{lemma-old} (a fact which was conjectured in \cite{R}). More precisely, we can show the following: If $f$, $g$ are $L^1$ functions on $S^{n-1}$, such that for almost every $u\in S^{n-1}$, $f(x)=g(x)$, for almost every $x\in S^{n-1}\cap u^{\perp}$ or $f(x)=g(-x)$, for almost every $x\in S^{n-1}\cap u^{\perp}$, then $f(x)=g(x)$, a.e. in $S^{n-1}$ or $f(x)=g(-x)$, a.e. in $S^{n-1}$. The proof is an immediate consequence of a well know integral-geometric formula due to Bussemann (see also \cite{S-W} for generalizations and references): For any non-negative functions $F_1,\dots,F_{n-1}$ on the sphere $S^{n-1}$, we have:
\begin{eqnarray*}&&\int_{S^{n-1}}F_1dx\dots\int_{S^{n-1}}F_{n-1}dx\\
&=&c\int_{S^{n-1}}\int_{S^{n-1}\cap u^{\perp}}\dots\int_{S^{n-1}\cap u^{\perp}}|det(x_1,\dots,x_{n-1})|F(x_1)\dots F(x_n)dx_1\dots dx_{n-1}du\ ,
\end{eqnarray*}
as long as both parts exist. Here as always, $c>0$ is a constant that depends only on the dimension. Indeed, setting $F_1(x):=(f(x)-g(x))^2$, $F_2(x)=\dots =F_{n-1}(x)=(f(x)-g(-x))^2$, our assumption gives $$\int_{S^{n-1}}(f(x)-g(x))^2dx\bigg(\int_{S^{n-1}}(f(x)-g(-x))^2dx\bigg)^{n-2}=0\ ,$$which shows that $f(x)=g(x)$ a.e or $f(x)=g(-x)$ a.e. as claimed.
\end{remark}
\textit{}\\
$\S$ 3. Recall that a subset of $\mathbb{R}^n$ is called \emph{unconditional} if it is symmetric with respect to the hyperplanes $e_1^{\perp},\dots,e_{n}^{\perp}$, for some orthonormal basis $\{e_1,\dots,e_n\}$ of $\mathbb{R}^n$.
In view of Question \ref{question-main}, observe that ellipsoids and bodies of revolution have the following property: Every section (projection) is unconditional. One might ask if these are the only examples of convex bodies with this property. The answer to this question is negative as the following shows.
\begin{proposition}\label{prop-counterexample}
There exists an origin symmetric convex body in $\mathbb{R}^n$, $n\geq 3$, which is not an ellipsoid or a body of revolution and whose sections (projections) are all unconditional.
\end{proposition}
\noindent Proof. As (\ref{eq-polarity}) shows, it suffices to find a counterexample for projections, since its polar would give a counterexample for sections. It is well known that the Minkowski sum of two ellipsoids is not always an ellipsoid. It follows that the sum of an ellipsoid and a ball is not always an ellipsoid (otherwise, we would be able to apply linear transformations to show that the sum of any two ellipsoids is an ellipsoid). It follows by continuity (since the family of ellipsoids which are not bodies of revolution is dense to the family of all ellipsoids, in the sense of the Haussdorff metric) that we can find an ellipsoid $E$ which is not a body of revolution and a ball $B$ such that $E+B$ is not an ellipsoid. On the other hand, $E+B$ is clearly not a body of revolution but all its projections are unconditional. $\square$
\\
It should be remarked that Proposition \ref{prop-counterexample} gives a negative answer to Problem 13 from \cite{R2}.
Nevertheless, the following questions remain open:
\begin{question}\label{question-open-2}
Is it true that if all projections of a convex body $K$ in $\mathbb{R}^n$, $n\geq 3$, are translations of completely symmetric sets, then $K$ is a ball?
\end{question}
\begin{remark}\label{rem-2}
It should be noted that if all projections of $K$ are translations of completely symmetric sets, then the restriction of the (even) function $h_K(x)+h_K(-x)=h_K(x)+h_{-K}(x)$ on every equator of $S^{n-1}$ is completely symmetric. Thus, Theorem \ref{thm-main} immediately implies that $h_K+h_{-K}$ is constant everywhere on $S^{n-1}$, i.e. $K$ is a convex body of constant width.
\end{remark}
\begin{question}\label{question-open-3}
Is it true that if all projections in $\mathbb{R}^n$, $n\geq 3$, of a convex body $K$ are linear (or more generally affine) images of completely symmetric sets (e.g. all projections are linear images of 1-symmetric sets), then $K$ is an origin-symmetric ellipsoid?
\end{question}
Variants about sections of star bodies (or equivalently restrictions of functions) are also interesting. In this connection, see again \cite{Gr} or \cite{M}.
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"timestamp": "2016-11-30T02:01:40",
"yymm": "1611",
"arxiv_id": "1611.09443",
"language": "en",
"url": "https://arxiv.org/abs/1611.09443",
"abstract": "We say that a star body $K$ is completely symmetric if it has centroid at the origin and its symmetry group $G$ forces any ellipsoid whose symmetry group contains $G$, to be a ball. In this short note, we prove that if all central sections of a star body $L$ are completely symmetric, then $L$ has to be a ball. A special case of our result states that if all sections of $L$ are origin symmetric and 1-symmetric, then $L$ has to be a Euclidean ball. This answers a question from \\cite{R2}. Our result is a consequence of a general theorem that we establish, stating that if the restrictions in almost all equators of a real function $f$ defined on the sphere, are isotropic functions, then $f$ is constant a.e. In the last section of this note, applications, improvements and related open problems are discussed and two additional open questions from \\cite{R} and \\cite{R2} are answered.}",
"subjects": "Metric Geometry (math.MG)",
"title": "Star bodies with completely symmetric sections",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808678600414,
"lm_q2_score": 0.7217432122827968,
"lm_q1q2_score": 0.7075110625086741
} |
https://arxiv.org/abs/2002.02339 | Robust $k$-means Clustering for Distributions with Two Moments | We consider the robust algorithms for the $k$-means clustering problem where a quantizer is constructed based on $N$ independent observations. Our main results are median of means based non-asymptotic excess distortion bounds that hold under the two bounded moments assumption in a general separable Hilbert space. In particular, our results extend the renowned asymptotic result of Pollard, 1981 who showed that the existence of two moments is sufficient for strong consistency of an empirically optimal quantizer in $\mathbb{R}^d$. In a special case of clustering in $\mathbb{R}^d$, under two bounded moments, we prove matching (up to constant factors) non-asymptotic upper and lower bounds on the excess distortion, which depend on the probability mass of the lightest cluster of an optimal quantizer. Our bounds have the sub-Gaussian form, and the proofs are based on the versions of uniform bounds for robust mean estimators. | \section{Introduction}
Statistical (sample-based) $k$-means clustering is the classical form of quantization for probability measures. In this framework, given a distribution $P$ defined on a normed space $(E, \norm{\cdot})$ and an integer $k \ge 1$, one wants to find $A^* \subset E$ such that the \emph{distortion}
\[
D(A) = \E\min\limits_{a \in A} \norm{X - a}^2 \quad\text{is minimized among all}\quad A \subset E,\; |A| = k.
\]
It is well known that if $(E, \norm{\cdot})$ is ${\mathbb R}^{d}$ with the Euclidean norm and if $\E \norm{X}^2 < \infty$ then this \emph{optimal quantizer} $A^*$ exists (see e.g., Theorem~1 in \citep{linder2002learning}), although it is not necessarily unique for $k \ge 2$. The value of the optimal distortion can be written as $D(A^*)$. In the statistical setup, the access to $P$ is achieved via $N$ independent observations $X_1, \ldots, X_N$ sampled according to $P$. Consider again the case of ${\mathbb R}^d$ and the Euclidean norm. The following renowned result due to \cite{pollard1981strong} states strong consistency of (any) \emph{empirically optimal quantizer}, which is defined by
\begin{equation}
\label{empquantizer}
\hat{A} \in \argmin\limits_{A \subset {\mathbb R}^d, |A| = k} \frac{1}{N} \sum\limits_{i = 1}^N \min\limits_{a \in A} \norm{X_i - a}^2.
\end{equation}
\begin{Theorem}[Strong consistency of $k$-means \citep{pollard1981strong}]\label{thm_pollard}
For any distribution $P$ such that $\E \norm{X}^2 < \infty$ and any integer $k \ge 1$, it holds that
\[
D(\hat{A}) - D(A^*) \overset{a.s.}{\to} 0,\quad \text{as}\quad N \to \infty.
\]
\end{Theorem}
This consistency result was extended to the case where the space $(E, \norm{\cdot})$ is a general separable Hilbert space \citep*{biau2008performance, levrard2015nonasymptotic}. Clearly, the consistency alone does not provide any information on how many training samples are needed
to ensure that the excess distortion is below a certain level. Moreover, it does not allow the underlying distribution to be different for each sample size $N$.
Over the last three decades a lot of efforts have been made in order to prove non-asymptotic results for the \emph{excess distortion} $D(\hat{A}) - D(A^*)$ where the space is ${\mathbb R}^d$ or a general separable Hilbert space. We refer to various bounds in \citep*{bartlett1998minimax, linder2002learning, biau2008performance, graf2007foundations, maurer2010k, narayanan2010sample, levrard2013fast, levrard2015nonasymptotic, fefferman2016testing, maurer2016vector}
and the references therein. However, almost all the results were provided under the strong assumption that the domain is bounded. That is, it is usually assumed that $\norm{X} \le T$ almost surely where $X$ is distributed according to $P$ and $T > 0$ is a constant. This simple setup allows one to use the tools of Empirical Process Theory in order to prove results of the form (see, e.g., Theorem~2.1 by \cite*{biau2008performance}, where the space $(E, d)$ is assumed to be a separable Hilbert space)
\begin{equation}
\label{boundedcase}
D(\hat{A}) - D(A^*) \lesssim T^2 \sqrt{\frac{k^2 + \log \frac{1}{\delta}}{N}},
\end{equation}
holding, with probability at least $1 - \delta$, for $\delta \in (0,1)$, where the notation $\lesssim$ suppresses absolute multiplicative constants. The question of general unbounded distributions is more challenging and has been studied less. The case where the vectors $X_i$ have well behaved exponential moments was analyzed in \citep{cadre2012holder}. Some results under less restrictive assumptions include: the uniform deviation bounds in \citep*{telgarsky2013moment, bachem2017uniform}; a \emph{sub-Gaussian} distortion bound in \citep*{brownlees2015empirical} for the so-called $k$-medians problem; and the results in \citep*{brecheteau2018robust} for trimmed quantizers. We discuss some of these results in more detail in what follows. However, we emphasize that in our particular setup the results we are aware of require the existence of at least \emph{four moments} (that is, $\E \norm{X}^4 < \infty$) compared to the minimal assumption under which the problem makes sense~--- $\E \norm{X}^2 < \infty$~--- which we are aiming for in this paper (this assumption is required to define the distortion $D(A^*)$). The question whether non-asymptotic results of the form \eqref{boundedcase} are possible under the minimal assumption $\E \norm{X}^2 < \infty$ (as in \citep{pollard1981strong}) appeared naturally in several papers (see, e.g., \citep{levrard2013fast}) but has not yet been addressed.
Our motivating example is the sub-Gaussian mean estimator introduced in \citep{lugosi2019sub}. Consider the situation where $E = {\mathbb R}^d$ with the Euclidean norm, set $\mu = \E X$, and assume that the covariance matrix $\Sigma = \E(X - \mu)(X - \mu)^{\top}$ exists. If $k = 1$, we obviously have that the optimal quantizer $A^*$ is actually the mean $\mu$.
In this case, our problem boils down to the estimation of the mean of a random vector. It was shown by \citeauthor{lugosi2019sub} that there is an estimator (denoted by $\hat{a}$) such that, with probability at least $1 - \delta$,
\begin{equation}
\label{lugosimendelson}
\E \norm{X - \hat{a}}^2 - \E \norm{X - \mu}^2
\lesssim \frac{\E \norm{X - \mu}^2 + \lambda_{\max}(\Sigma) \log \frac{1}{\delta} }{N},
\end{equation}
where $\lambda_{\max}(\Sigma)$ is the largest eigenvalue of the covariance matrix $\Sigma$, the expectation is taken only with respect to $X$, and $\hat{a} = \hat{a}(X_1, \ldots, X_N)$ is random.
It is known that this bound is valid for the sample mean in the case where the underlying distribution is multivariate Gaussian. The bound \eqref{lugosimendelson} has some remarkable properties:
\begin{itemize}
\item The dependence on $N$ is $O\left(\frac{1}{N}\right)$.
\item It only requires the existence of two moments, that is $\E \norm{X}^2 < \infty$. We note that $\lambda_{\max}(\Sigma) \le \Tr(\Sigma) = \E \norm{X - \mu}^2$.
\item It has the logarithmic dependence on the confidence, which is $\log \frac{1}{\delta}$ and corresponds to the situation where $\norm*{\mu - \hat{a}}$ has \emph{sub-Gaussian tails} (see, e.g., \citep{Vershynin2016HDP} for various equivalent definitions of sub-Gaussian distributions).
\item Finally, even in the favorable bounded case where $\norm{X - \mu} \le T$ almost surely, the bound \eqref{lugosimendelson} does not scale as $T^2$ (compare it with the typical $k$-means bound \eqref{boundedcase}) but as $\E \norm{X - \mu}^2$ which can be much smaller than $T^2$.
\end{itemize}
Therefore, extending the original question of whether the non-asymptotic excess distortion bounds are possible under $\E \norm{X}^2 < \infty$, it is natural to ask if one can prove a result of the form \eqref{lugosimendelson} for $k \ge 2$. Unfortunately, a fully general picture is much more subtle. In particular, even in the favorable bounded case for $k \ge 2$, lower bounds of order $\Omega\left(\frac{1}{\sqrt{N}}\right)$ are known (see, e.g., \citep{antos2005improved}) making the simple bound \eqref{boundedcase} sharp with respect to $N$.
Further, if $k = 1$ we observe that the right-hand side of \eqref{lugosimendelson} converges to zero as $N$ goes to infinity even if the underlying distribution $P$ depends on the sample size $N$ (see e.g., Example \ref{pminexample} below). Our only condition is that $\E \norm{X - \mu}^2 = \Tr(\Sigma)$ does not grow too fast as $N$ goes to infinity. Example~\ref{pminexample} below shows that the same is not true for general $k \ge 2$.
Risk bounds having the sub-Gaussian form for heavy-tailed distributions have attracted a lot of attention recently. Among these advances are (almost) optimal results on mean estimation in various norms \citep{minsker2018uniform, lugosi2019near} and in classification \citep*{lecue2018robust}, covariance estimation \citep{mendelson2018robust} and robust regression \citep{minsker2019excess, lugosi2019regularization, lecue2017robust}. All the technical results in this area are based on different versions of the so-called \emph{median of means estimator}, which was first introduced and analyzed by \cite{nemirovsky1983problem} and independently in \citep*{alon1999space}. For the sake of completeness, let us recall this basic result.
Assume that $Y_1, \ldots, Y_N$ are independent random variables with the same mean $\mu$ and the same variance $\sigma^2$. Fix the confidence level $\delta$ and assume that $\ell = \left\lceil 8 \log \frac{1}{\delta}\right\rceil$ is such that $N = m \ell$, where $m$ is integer. Split the set $\{1, \ldots, N\}$ into $\ell$ blocks $I_1, \ldots, I_{\ell}$ of equal size such that $I_j = \{1 + m(j - 1), \ldots, mj\}$. Denote the \emph{median of means} (MOM for short) estimator by
\[
\hat{\mu} = \mathrm{Median}\left(\frac{\ell}{N}\sum\limits_{i \in I_1}Y_i, \ldots, \frac{\ell}{N} \sum\limits_{i \in I_\ell}Y_i\right).
\]
For this estimator we have the following sub-Gaussian behaviour. It holds, with probability at least $1 - \delta$, that
\[
\abs*{\hat{\mu} - \mu} \le \sigma\sqrt{\frac{32 \log \frac{1}{\delta}}{N}}.
\]
Returning to the question of $k$-means clustering and the inequalities of the form \eqref{lugosimendelson} for general $k \ge 2$, the following simple example inspired by \citeauthor*{bachem2017uniform} highlights some of the obstacles we have to handle.
\begin{Example}
\label{pminexample}
Let $N$ be the sample size. Consider the real line ${\mathbb R}$, $k = 2$ and the distribution $P$ supported on $\{0, \sqrt{N}\}$ such that $P(\{0\}) = 1 - \frac{1}{N}$ and $P(\{\sqrt{N}\}) = \frac{1}{N}$. In this case we have $D^*(A) = 0$.
One may easily see that, with constant probability, the value $\sqrt{N}$ is not among the observations $X_1, \ldots, X_N$. That obviously forces $\hat{A} = \{0\}$ and
\[
D(\hat{A}) - D(A^*) = \E X^2 = 1,
\]
which is not converging to zero as $N$ goes to infinity.
\end{Example}
In Section~\ref{sec:lowerbound} we significantly extend this construction. Of course, Example~\ref{pminexample} does not contradict the strong consistency result of Theorem~\ref{thm_pollard}. Although $\E X^2 = 1$, the distribution $P = P(N)$ changes with $N$, which is, of course, not allowed in Theorem~\ref{thm_pollard}. However, in the statistical learning theory literature the underlying distribution $P$ is usually allowed to be different for each value of $N$ as in Example \ref{pminexample}. This provides an additional motivation for our study. Our general bounds provide consistency even for some sequences of distributions changing with the sample size $N$.
\subsection*{On Voronoi cells and clustering}
From now on we assume that $(E, \norm{\cdot})$ is a separable Hilbert space with the inner product denoted by $\langle\cdot, \cdot \rangle$.
Any quantizer $A = \{a_1, \ldots, a_k\}$ induces a partition of \(E\) into the so-called \emph{Voronoi cells}, which for \(a \in A\) consists of the points that have \(a\) as the closest point from \(A\). To avoid the uncertainty at the boundaries, we assume that the elements of each quantizer \( A = \{a_1, \dots, a_k \} \) are ordered, and define the cells for $j = 1, \ldots, k$,
\begin{align*}
V_{j}(A) = \bigl\{x \in E \;:\; & \norm{x - a_j} < \norm{x - a_{j'}}, \; j' = 1, \dots, j - 1, \\
& \norm{x - a_j} \le \norm{x - a_{j'}}, \; j' = j + 1, \dots, k \bigr\}.
\end{align*}
This way, we ensure that the cells are non-intersecting, and each of them is an intersection of \( k - 1\) open or closed half-spaces.
Slightly abusing the notation, we sometimes write $V_{a_j}$ instead of $V_{j}$.
We recall some properties of an \emph{optimal} quantizer under the assumption $\E \norm{X}^2 < \infty$.
\begin{enumerate}
\item\label{prop:existence} For any distribution $P$ with $\E \norm{X}^2 < \infty$ and any $k$ there exists an optimal $k$-elements quantizer $A^*$ (see \citep[Corollary~3.1]{fischer2010quantization}) Note that an optimal quantizer is not necessarily unique.
\item\label{zero_intesec} For any optimal $A^*$ and $i \neq j$,
\[
P\left(\norm{X - a_j} = \norm{X - a_i}\right) = 0,
\]
which means that the measure of intersection of any two cells is zero, thus it does not matter to which cell the boundary points are assigned (see \citep{graf2007foundations}, Theorem~4.2.)
\item The \emph{centroid condition} \citep{graf2007foundations}: for $j = 1, \ldots, k$,
\begin{align}
&\E \norm{X - a_j}^2 \Ind[X \in V_{j}] = \inf\limits_{a \in E} \E \norm{X - a}^2 \Ind[X \in V_{j}], \nonumber \\
&a_j = \frac{\E X \Ind[X \in V_{j}]}{P(V_j)}, \;\; \text{whenever}\;\; P(V_j) > 0.
\end{align}
\item Once the support of $P$ consists of at least $k$ elements, there is a well-defined real number $M = M(P, k)$ such that for any optimal $A^*$,
\begin{equation}
\label{upperm}
\norm{a} \le M \quad \text{for all}\quad a \in A^*.
\end{equation}
We refer to the original proof of \citeauthor{pollard1981strong} or to Lemma~5.1 in \citep{fischer2010quantization}.
\item\label{prop:p_min} Due to Theorem~4.1 in \citep{graf2007foundations} provided that the support of $P$ consists of at least $k$ elements, there exists $p_{\min} > 0$ such that for any optimal $A^*$,
\begin{equation}
\label{pmin}
\min\limits_{j}P(V_{j}(A^*)) \ge p_{\min}.
\end{equation}
\end{enumerate}
Observe that the same conclusions work if we replace $P$ by its empirical counterpart $P_N$.
In particular, a version of centroid condition is also valid for the empirically optimal quantizer defined by \eqref{empquantizer}.
However, it is not true for a MOM based estimator in general.
\subsection*{Structure of the paper}
\begin{itemize}
\item Section~\ref{sec:warmup} is devoted to a high probability excess distortion bound that holds in the case where a good guess on the localization radius of the optimal quantizer $A^*$ is available. The result generalizes naturally several known bounds for the empirically optimal quantizer in separable Hilbert spaces.
\item Section~\ref{sec:pminbound} contains our main results. We show that there is a consistent median-of-means based estimator that gives the sub-Gaussian performance under our minimal moment assumption provided that a good guess on $p_{\min}$ is given and $p_{\min} N \to \infty$. We also prove a lower bound showing that our dependence on $p_{\min}$ and $N$ is sharp up to constant factors in the special case of ${\mathbb R}^d$.
\item Finally, Section~\ref{sec:Generalcase} is devoted to the generalization of our main results. We show that it is possible to prove a slightly weaker bound using the procedure that does not require the knowledge of the parameters of $P$.
\item Section~\ref{Discussions} is devoted to some final remarks.
\end{itemize}
\subsection*{Notation}
For $a, b \in {\mathbb R}$, we set $a \wedge b = \min\{a, b\}, a \lor b = \max\{a, b\}$ and for two real valued functions $f, g$, we write $f \lesssim g$ iff there is an absolute constant $c > 0$ such that $f \le c g$.
We set $f \simeq g$ if $f \lesssim g$ and $g \lesssim f$.
Given a probability measure $P$ let $P^{\otimes N}$ denote the measure which is $N$-times product of $P$. For the sake of simplicity, we always assume that $\log x$ is equal to $\log x\: \lor\: 1$. The indicator of an event $A$ is denoted by $\Ind[A]$. We also use the standard $O(\cdot)$, $\Omega(\cdot)$, $\Theta(\cdot)$ notation as well as $\mathrm{KL}(P, Q)$ and $\mathrm{TV}(P, Q)$ for Kullback--Leibler divergence and Total Variation distance between two measures $P$ and $Q$ (see, e.g., \citep{boucheron2013concentration}). The support of a measure $P$ is denoted by $\supp(P)$. For a normed space $(E, \norm{\cdot})$ let $B_{R}$ denote the closed ball of radius $R$ centered at the origin. To avoid the measurability issues we use the standard convention for the supremum of stochastic processes (see Paragraph~2 in \citep{Talagrand2014}). Given the sample $X_{1}, \ldots, X_{N}$ sampled i.i.d.\ from $P$ and a function $f \colon E \to {\mathbb R}$ we denote $P_N f = P_N f(X) = \frac{1}{N} \sum\limits_{i = 1}^N f(X_i)$. In general, the symbol $P_N$ denotes the empirical measure.
We are interested in $L_2(P_N)$ space, and the corresponding covering number of a functional class $\mathcal{G}$ is denoted by $\mathcal{N}_{2}(\mathcal{G}, x, P_N)$, where $x$ is the corresponding radius (see e.g., \citep{Vershynin2016HDP} for more details on covering numbers).
\section{Simple Case: Known Magnitude of an Optimal Quantizer}
\label{sec:warmup}
In this section we provide our simplest result which can serve as a good illustration of the underlying techniques. In Sections~\ref{sec:pminbound} and~\ref{sec:Generalcase} we are focusing on sharpening our basic bound as well as weakening some of the assumptions.
We first show a simple bound which holds in the situations where a good guess on $M$ is available (recall the property~\eqref{upperm}). The result of Theorem~\ref{thm:simpleupperm} below can be seen as a strengthening of Theorem~11 in \citep{brownlees2015empirical}.
\begin{Remark}
\label{importantremark}
It is important to note that the boundedness of the vectors in the finite set $A^*$ has nothing in common with the boundedness of the observations $X_1, \ldots, X_N$. The latter can still be unbounded and the distribution $P$ can be heavy-tailed.
\end{Remark}
We proceed with the main result of this section.
\begin{Theorem}
\label{thm:simpleupperm}
Fix $\delta \in (0, 1)$.
Let some $M$ satisfying \eqref{upperm} be known. There is an estimator $\hat{A}_{\delta, M}$ that depends on $M$ and $\delta$ such that, with probability at least $1 - \delta$,
\[
D(\hat{A}_{\delta, M}) - D(A^*) \lesssim M \left(M + \sqrt{\E \norm{X}^{2}}\right) \left(\frac{k}{\sqrt{N}} + \sqrt{\frac{\log \frac{1}{\delta}}{N}}\right).
\]
\end{Theorem}
Let us now define the estimator that we use in Theorem~\ref{thm:simpleupperm}.
Notice that minimizing \(\E \min_{a \in A} \norm{a - X}^{2} \) with respect to $A$ is equivalent to minimizing \(\E l_{A}(X)\), where
\[
l_{A}(x) = \min_{a \in A} -2 \langle x, a\rangle + \norm{a}^{2} \, .
\]
Fix $1 \leq \ell \leq N$ and assume without loss of generality that $m = N / \ell$ is integer. Split the set $\{1, \ldots, N\}$ into $\ell$ blocks $I_1, \ldots, I_{\ell}$ of equal size such that $I_j = \{1 + (j - 1) m, \ldots, j m\}$. For any real-valued function $f$ and random variables $X_1, \ldots, X_N$ define
\begin{equation}\label{def:mom}
\MOM(f) = \mathrm{Median}\left(\frac{\ell}{N} \sum\limits_{i \in I_1} f(X_i), \ldots, \frac{\ell}{N} \sum\limits_{i \in I_\ell} f(X_i)\right).
\end{equation}
Slightly abusing the notation we also set
\begin{equation}\label{def:Ak}
\mathcal{A}^k = \left\{A \subset E \;:\; |A| \le k\right\} \quad \text{and} \quad
\mathcal{A}^k_M = \left\{A \in \mathcal{A}^k \;:\; \max\limits_{a \in A} \norm{a} \le M\right\}.
\end{equation}
\begin{framed}
\textbf{The estimator of Theorem~\ref{thm:simpleupperm}}. Define
\[
\hat{A}_{\delta, M} = \arg\min_{A \in \mathcal{A}_{M}^k} \MOM(l_{A}),
\]
with the number of blocks \(\ell = 8 \left\lceil \log \tfrac{2}{\delta}\right\rceil + 1\). If there are many minimizers, we may choose any of them.
\end{framed}
The proof of Theorem~\ref{thm:simpleupperm} relies on the uniform bound for the median of means estimator. However, instead of restricting our attention to the medians only, we consider the \emph{quantiles of means} (QOM). That is, for a given level \( \alpha \in (0, 1)\),
\[
\QOM_{\alpha}(f) = \mathrm{Quant}_{\alpha}\left( \frac{\ell}{N} \sum\limits_{i \in I_1} f(X_i), \ldots, \frac{\ell}{N} \sum\limits_{i \in I_\ell} f(X_i) \right),
\]
where \( \mathrm{Quant}_{\alpha}(x_1, \dots, x_\ell) = x^{(\lceil \alpha \ell \rceil)} \), for \( x^{(1)}, \dots, x^{(\ell)} \) being a non-decreasing rearrangement of the original sequence. For the sake of simplicity, we always assume that \( \ell \alpha \) is non-integer, such that the quantile is uniquely defined, and, in particular
\( \mathrm{Quant}_{\alpha}(x_1, \dots, x_\ell) = - \mathrm{Quant}_{1 - \alpha}(-x_1, \dots, -x_\ell)\). It is usually enough to assume that $\ell$ is not even which can be always achieved by adding at most one extra block. Obviously, $\QOM_{\frac{1}{2}}$ corresponds to the median of means.
\begin{Lemma}
\label{lem:momuniform}
Fix $\alpha \in (0, 1)$ and consider a separable class $\mathcal{F}$ of square integrable real-valued functions. Suppose, we have $\ell$ blocks and \( \ell \alpha \) is a non-integer. It holds that, with probability at least \(1 - e^{-\alpha^{2} \ell / 2}\),
\begin{equation}
\label{absvalueequation}
\sup\limits_{f \in \mathcal{F}} (\E f - \QOM_{\alpha}(f))
\le \frac{16}{\alpha} \E \sup\limits_{f \in \mathcal{F}} \left(\frac{1}{N} \sum_{i = 1}^{N} \epsilon_i f(X_i)\right) + \sqrt{\frac{8}{\alpha} \sup\limits_{f \in \mathcal{F}} \Var(f(X))\ \frac{\ell}{N}},
\end{equation}
as well as, with probability at least \( 1 - e^{-(1-\alpha)^2 \ell / 2} \),
\begin{align}
\label{absvalueequation_second}
&\sup\limits_{f \in \mathcal{F}} (\QOM_{\alpha}(f) - \E f) \nonumber
\\
&\quad\quad \le \frac{16}{1 - \alpha} \E \sup\limits_{f \in \mathcal{F}} \left(\frac{1}{N} \sum_{i = 1}^{N} \epsilon_i f(X_i)\right) + \sqrt{\frac{8}{1 - \alpha} \sup\limits_{f \in \mathcal{F}} \Var(f(X))\ \frac{\ell}{N}},
\end{align}
where \( \epsilon_{1}, \dots, \epsilon_{N} \) are i.i.d.\ Rademacher signs.
\end{Lemma}
\begin{Remark}
In the case where \( \alpha \) is fixed, we can take \( \ell \simeq \log \tfrac{1}{\delta} \), so that with probability at least \( 1 - \delta \),
\[
\sup\limits_{f \in \mathcal{F}} \abs{\E f - \QOM_{\alpha}(f)}
\lesssim \; \E \sup\limits_{f \in \mathcal{F}}\left(\frac{1}{N} \sum_{i = 1}^{N} \epsilon_i f(X_i)\right) + \sqrt{\sup\limits_{f \in \mathcal{F}} \Var(f(X))\ \frac{\log\tfrac{1}{\delta}}{N}},
\]
where the first term represents the expectation of the empirical process, whereas the second term corresponds to a tail with the sub-Gaussian behavior.
Compare this inequality with Talagrand's inequality for empirical processes, where the assumption \(\sup\limits_{f \in \mathcal{F}} \abs{f(X)} \le C\) almost surely is needed (see Chapter~12 in \cite{boucheron2013concentration}).
\end{Remark}
As noticed by \cite{minsker2018uniform} (see equation~(2.7)) an inequality similar to \eqref{absvalueequation} of Lemma~\ref{lem:momuniform} for $\alpha = \frac{1}{2}$ follows from the proof of Theorem~2 in \citep*{lecue2018robust}. However, to the best of our knowledge, Lemma~\ref{lem:momuniform} in this form is not presented explicitly in the literature. We provide its proof in the appendix for the sake of completeness.
\begin{proof}[Proof of Theorem~\ref{thm:simpleupperm}]
\textbf{Step 1.}
First, we provide the high probability part of the analysis. Observe that
\begin{align*}
D(\hat{A}_{\delta, M}) - D(A^*) &= \E l_{\hat{A}_{\delta, M}} - \E l_{A^*}
\\
&\le \E l_{\hat{A}_{\delta, M}} - \MOM(l_{\hat{A}_{\delta, M}}) + \MOM(l_{A^*}) - \E l_{A^*}
\\
&\le 2 \sup\limits_{A \in \mathcal{A}_M^k} \abs*{\E l_{A} - \MOM(l_{A})},
\end{align*}
where we used $\MOM(l_{A^*}) \ge \MOM(l_{\hat{A}_{\delta, M}})$ since $A^* \in \mathcal{A}_M^k$.
We have by Lemma~\ref{lem:momuniform} that, with probability at least \( 1 - \delta \),
\begin{align*}
\sup_{A \in \mathcal{A}_{M}^k} \abs{\E l_{A}(X) - \MOM(l_{A})}
&\lesssim \E \sup_{A \in \mathcal{A}_{M}^k} \frac{1}{N} \sum_{i = 1}^{N} \epsilon_i l_{A}(X_i) + \sqrt{\sup_{A \in \mathcal{A}_{M}^k} \Var(l_{A}(X)) \frac{\log \frac{1}{\delta}}{N}},
\end{align*}
where \( \epsilon_1, \dots, \epsilon_N \) are independent Rademacher signs.
Here, we have for each \( A \in \mathcal{A}_{M}^k \),
\begin{align*}
l_{A}(x)^2 & = \left(\norm{x - a_x}^{2} - \norm{x}^{2}\right)^2 = (\norm{x - a_x} - \norm{x})^2 (\norm{x - a_x} + \norm{x})^2 \\
& \leq \norm{a_x}^2 (2 \norm{x} + \norm{a_x})^2,
\end{align*}
where \( a_x \in \mathrm{Arg}\min_{a \in A} \norm{x - a} \). Then, since \( \norm{a_x} \leq M \) for any \( x \), we have
\[
\Var(l_{A}(X)) \leq \E l_{A}(X)^2 \lesssim M^2 \left(M^2 + \E \norm{X}^{2}\right) \, .
\]
\textbf{Step 2.}
Note that $\hat{A}_{\delta, M}$ can consist of less than $k$ points. However, in this case we can always add the copies of some of them and identify $\hat{A}_{\delta, M}$ with $(a_1, \dots, a_k)$.
This does not change $l_{\hat{A}_{\delta, M}}$ and preserves the Voronoi partition of the space since the cells corresponding to the newly added points are empty. Finally, we estimate
\begin{equation}
\label{supofproc}
\E \sup_{A \in \mathcal{A}_{M}^k} \frac{1}{N} \sum_{i = 1}^{N} \epsilon_i l_{A}(X_i).
\end{equation}
Consider the set $\mathcal{F}_{\mathcal{A}} = \left\{f_{A} \;:\; A \in \mathcal{A}_{M}^k\right\}$ of ${\mathbb R}^k$-valued functions such that for any $A = \{a_1, \ldots, a_k\},\ A \in \mathcal{A}_{M}^k$ we set
\begin{equation}
\label{fax}
f_{A}(x) = \left(-2\langle x,a_1 \rangle + \norm{a_1}^2, \ldots, -2\langle x,a_k \rangle + \norm{a_k}^2\right).
\end{equation}
For $\mathbf{c} \in {\mathbb R}^k$ let $\phi(\mathbf{c}) = \min\limits_{i \leq k} c_i$. We obviously have $l_{A}(X) = \phi(f_{A}(X))$. Following the analysis of Section~3.2 in \citep{maurer2016vector} we have for any two $A$ and $B$ in $\mathcal{A}_M^k$,
\[
\abs*{\phi(f_{A}(X_i)) - \phi(f_{B}(X_i))} \le \norm*{f_A(X_i) - f_B(X_i)}_2.
\]
This allows us to use the $\ell_2$-contraction to upper bound \eqref{supofproc} with the quantity scaling linearly in $k$. To do so, we observe that Maurer's vector contraction inequality (Theorem~3 in \citep{maurer2016vector}) implies
\[
\E \sup_{A \in \mathcal{A}_{M}^k} \frac{1}{N} \sum_{i = 1}^{N} \epsilon_i l_{A}(X_i)
\le \frac{\sqrt{2}}{N} \left(2 \E \sup\limits_{A \in \mathcal{A}_{M}^k} \sum_{i, j = 1}^{N, k}\epsilon_{i, j}\langle X_i, a_j\rangle + \E\sup\limits_{A \in \mathcal{A}_{M}^k} \sum_{i, j = 1}^{N, k} \epsilon_{i, j} \norm{a_j}^2\right),
\]
where $\epsilon_{i, j}$, $i = 1, \ldots, N$, $j = 1, \ldots, k$, are independent Rademacher signs, and where $A = \{a_1, \ldots, a_k\}$. We further have by Khintchine's inequality,
\begin{align*}
\E \sup\limits_{A \in \mathcal{A}_{M}^k} \sum_{i, j = 1}^{N, k} \epsilon_{i, j} \langle X_i, a_j\rangle &\le \sum\limits_{j = 1}^{k} \E \sup\limits_{A \in \mathcal{A}_{M}^k} \left\langle \sum \limits_{i = 1}^{N} \epsilon_{i,j} X_i, a_j\right\rangle \le \sum\limits_{j = 1}^{k} \E \sup\limits_{A \in \mathcal{A}_{M}^k} \norm*{\sum\limits_{i = 1}^{k} \epsilon_{i,j} X_i} \norm{a_j}
\\
&\le k M \max\limits_{j \leq k} \E \norm*{\sum\limits_{i = 1}^{N} \epsilon_{i,j} X_i} \le k M \max\limits_{j \leq k} \sqrt{\E \norm*{\sum\limits_{i = 1}^{N} \epsilon_{i,j} X_i}^2}
\\
&\le k M \sqrt{\sum\limits_{i = 1}^{N} \E \norm{X_i}^2},
\end{align*}
and also
\[
\E \sup\limits_{A \in \mathcal{A}_{M}^k} \sum_{i, j = 1}^{N, k} \epsilon_{i, j} \norm{a_j}^2
\le \sum\limits_{j = 1}^{k} \E \sup\limits_{A \in \mathcal{A}_{M}^k} \abs*{\sum_{i = 1}^{N}\epsilon_{i, j}} \norm{a_j}^2
\lesssim k M^2 \sqrt{N}.
\]
Finally, taking the expectation with respect to $X_1, \ldots, X_N$ and using Jensen's inequality we obtain an analog of~\eqref{boundedcase}. That is,
\begin{equation}
\label{lineark}
\E \sup_{A \in \mathcal{A}_{M}^k} \frac{1}{N} \sum_{i = 1}^{N} \epsilon_i l_{A}(X_i) \lesssim \frac{k M \left(\sqrt{\E \norm{X}^2} + M\right)}{\sqrt{N}}.
\end{equation}
Combining the above bounds we prove the claim.
\end{proof}
It is by now well known that in our setup in the bounded case (e.g., when $\norm{X} \le T$ almost surely) the right dependence of the excess distortion on the number of clusters is $\sqrt{k}$ up to logarithmic factors \citep{fefferman2016testing, narayanan2010sample}. It is natural to ask if the same dependence is possible in our Theorem~\ref{thm:simpleupperm}. First, observe that in the unbounded case, there are some complications. In particular, our parameter $M = M(P, k)$ can also depend on $k$. This means that the overall dependence of the excess distortion on $k$ can be more complicated. Nevertheless, in the next section we show, among other things, that these improvements are possible and, in particular, the $k$-term is replaced by the $\sqrt{k}$-term.
\section{Towards Better Bounds Based on \texorpdfstring{\(p_{\min}\)}{pmin}}
\label{sec:pminbound}
This section is devoted to our main results. We prove almost optimal non-asymptotic bounds for $k$-means. Recall that if $\E \norm{X}^2 < \infty$ we have for any optimal quantizer
\[
p_{\min} = \min_{a \in A^*} P(V_{a}) > 0,
\]
unless the support of \(P\) has less than \(k\) points. Notice that \( p_{\min} \) controls the magnitude of the largest vector in \(A^{*}\). Indeed, using the centroid condition, Jensen's inequality, and the Cauchy--Schwarz inequality we have for any $a \in A^*$,
\begin{equation}
\label{eq:boundsona}
\norm{a} = \norm*{\E[X \vert \; X \in V_{a}]}
= \frac{\norm*{\E X \Ind[X \in V_a]}}{P(V_{a})}
\leq \frac{\E^{1/2} \norm{X}^{2}}{\sqrt{P(V_{a})}}
\le \frac{\E^{1/2} \norm{X}^{2}}{\sqrt{p_{\min}}} .
\end{equation}
This suggests that the mass of the lightest cluster of an optimal quantizer should affect the quality of any empirical quantizer.
Let us return to Example~\ref{pminexample}. In this case we have $k = 2$, $M \le \sqrt{N}$, $p_{\min} = \frac{1}{N}$, $\E \norm{X}^2 \le 1$ and the bound \eqref{eq:boundsona} is tight. However, the bound of Theorem~\ref{thm:simpleupperm} is not tight anymore as it scales as $O(\sqrt{N})$. Indeed, Theorem~\ref{thm:simpleupperm} implies the bound
$O\left(\frac{k \left(M^2 + M \sqrt{\E \norm{X}^2}\right)}{\sqrt{N}}\right)$ which is $O\left(\frac{k \E \norm{X}^2}{p_{\min} \sqrt{N}}\right)$ whenever \eqref{eq:boundsona} is tight.
The challenging part is to get the optimal dependence on $p_{\min}$ and $N$ in the excess distortion bound. In what follows, we show that the dependence $\Theta \left(\frac{1}{\sqrt{N p_{\min}}}\right)$ is achievable with respect to these parameters. The result of this form guarantees the consistency for sequences of distributions depending on $N$ as long as \(N p_{\min} \to \infty\) and the second moment is uniformly bounded. This extends the original asymptotic result of \cite{pollard1981strong} to the case where the distribution is allowed to change with $N$.
Suppose that we know the value of \(p_{\min} > 0\) for at least one optimal quantizer.
Denote for short, \(P_N(V) = \frac{1}{N} \sum_{i = 1}^{N} \Ind[X_i \in V]\). Naturally, we want to find a solution \(\hat{A}\) such that the corresponding Voronoi cells are of measure at least \(p_{\min} > 0\) which translates into \(P_N(V_j) \ge p_{\min}/2\) due to concentration, provided that $N$ is large enough.
It implies that each cell corresponding to $\hat{A}$ contains enough sample points, which corresponds to the so-called \emph{constrained} $k$-means clustering.
In ${\mathbb R}^d$ the algorithmic side of constrained clustering is well studied in the context of optimal transport and has numerous practical applications, see \citep*{ng2000note, cuturi2014fast, genevay2019differentiable} and references therein.
We have additional motivation to introduce $p_{\min}$ since this quantity appears naturally in the condition implying the so-called \emph{fast rates} of the excess distortion in the bounded case \citep{levrard2015nonasymptotic}.
Finally, recalling Example~\ref{pminexample} we know that in any reasonable clustering problem $p_{\min} \gg \frac{1}{N}$ which means that the optimal solution $A^*$ has enough observations in each cell. At the same time, we do not have such a natural preliminary guess on $M$.
As before, the number of blocks depends solely on the desired confidence level.
Our main result is the following theorem.
\begin{Theorem}\label{thm:pmin}
Fix $\delta \in (0, 1)$. Suppose, \(\min_{a \in A^{*}} P(V_a) \geq p_{\min} \) for some optimal quantizer \(A^{*} \).
There is an estimator $\hat{A}_{\delta, p_{\min}}$ that depends on $p_{\min}$ and $\delta$ such that, with probability at least \( 1 - \delta\),
\[
D(\hat{A}_{\delta, p_{\min}}) - D(A^*) \lesssim \E \norm{X - \mu}^{2} \left((\log N)^2 \sqrt{\frac{k}{N p_{\min}}} + \sqrt{\frac{\log \frac{1}{\delta}}{N p_{\min}}}\right) \, .
\]
\end{Theorem}
Let us now present our estimator.
\begin{framed}
\textbf{The estimator of Theorem~\ref{thm:pmin}}. We set
\begin{equation}\label{pmin_estimator}
\hat{A}_{\delta, p_{\min}} = \argmin_{\substack{A \in \mathcal{A}^k \\ \min\limits_{a \in A} P_N(V_a) \geq p_{\min} / 2}} \MOM(l_{A}),
\end{equation}
with the number of blocks $\ell = 12 \left\lceil \log \tfrac{6}{\delta}\right\rceil + 1$.
\end{framed}
The idea behind this estimator is quite natural: we guarantee the robustness by using the MOM principle and by restricting our attention only to the cells containing enough points. As already mentioned, this is essentially a robust version of the constrained $k$-means quantizer introduced in \cite{ng2000note}.
We introduce several technical results that together lead us to the proof of Theorem~\ref{thm:pmin} at the end of this section.
Since the estimator we consider is translation invariant, we can assume that \( \E X = 0 \) in the proof without loss of generality. As previously, our main tool is the concentration of MOM for a suitably chosen subset of \(\left\{l_{A} \;:\; A \in \mathcal{A}^k\right\}\). We show that the restriction \( P_{N}(V_j) \geq p_{\min} / 2 \) in \eqref{pmin_estimator} implies a convenient bound for the vectors in the resulting empirical quantizer. Let us define the following class of quantizers:
\begin{equation}\label{def:A_Mm}
\mathcal{A}_{M,m}^k = \Bigl\{A \in \mathcal{A}^k \;:\; \min_{a \in A} \norm{a} \le m,\; \max_{a \in A} \norm{a} \le M\Bigr\}, \quad 0 < m \le M.
\end{equation}
The following lemma says that with high probability all the solutions corresponding to $\hat{A}_{\delta, p_{\min}}$ are bounded which is, of course, natural in view of the proof of Theorem~\ref{thm:simpleupperm}. However, the key technical observation is that we also need to control the smallest norm by saying that there is at least one center in $\hat{A}_{\delta, p_{\min}}$ which is relatively close to the actual expectation. In order to show this we do not have to use any uniform results that hold simultaneously for the entire class $\mathcal{A}^k$. Therefore, we have the following property.
\begin{Lemma}
\label{lem:optimizer_Mm}
With probability at least $1 - e^{- \ell / 12}$, it holds that simultaneously for all $A \in \mathcal{A}^k$ such that $\MOM(l_A) \le 0$,
\[
\min_{a \in A} \norm{a} \le m = 4 \sqrt{2 \E \norm{X}^2}.
\]
Moreover, with probability at least $1 - \left(e^{- \ell / 12} + e^{- N p_{\min} / 12}\right)$, it holds that
\[
\hat{A}_{\delta, p_{\min}} \in \mathcal{A}^k_{M, m}
\quad\text{for } m \text{ defined above and } M = 10 \sqrt{\frac{\E \norm{X}^2}{p_{\min}}}.
\]
\end{Lemma}
\begin{Remark}\label{rem:simple_bound}
Note that the first statement of the above lemma gives us a prior bound on the excess distortion $D(\hat{A}_{\delta, p_{\min}}) - D(A^*)$. Indeed, since $\ell \ge 12 \log \tfrac{1}{\delta}$, with probability at least $1 - \delta$, we have $\min_{a \in \hat{A}_{\delta, p_{\min}}} \norm{a} \le m$, thus
\begin{equation}\label{eq:simple_bound}
D(\hat{A}_{\delta, p_{\min}}) - D(A^*) \le D(\hat{A}_{\delta, p_{\min}}) = \E \min_{a \in \hat{A}_{\delta, p_{\min}}} \norm{a - X}^2 \le \E (m + \norm{X})^2 \lesssim \E \norm{X}^2.
\end{equation}
\end{Remark}
Before going to the proof of Lemma~\ref{lem:optimizer_Mm}, let us state the following trivial result on empirical quantiles. We postpone its proof to the appendix.
\begin{Lemma}
\label{lem:quant_ratio}
Let $\xi_1, \dots, \xi_\ell$ be i.i.d.\ random values such that $\E \xi < \infty$ and $\xi \ge 0$ almost surely.
Then for any $0 < \alpha < 1$ we have
\[
\mathbb{P}\left(\mathrm{Quant}_{1 - \alpha}(\xi_1, \dots, \xi_\ell) \ge \frac{2}{\alpha} \E \xi\right) \le \exp\left(- \frac{\alpha \ell}{6}\right).
\]
\end{Lemma}
\begin{proof}[Proof of Lemma~\ref{lem:optimizer_Mm}]
\textbf{Step 1.}
First, let us prove the bound on the minimal norm.
Consider $A \in \mathcal{A}^k$ such that $\min_{a \in A} \norm{a} \ge m$. Then for any $x \in B_{m / 2}$ (recall that $B_{m/2}$ is a ball of radius $m/2$ centered at the origin) it holds that $\min_{a \in A} \norm{a - x} \ge \frac{m}{2}$, thus for all $x \in E$,
\begin{align*}
l_A(x) = \min_{a \in A} \norm{a - x}^2 - \norm{x}^2
&\ge \frac{m^2}{4} \Ind\left[\norm{x} \le \frac{m}{2}\right] - \norm{x}^2
\\
&= \frac{m^2}{4} - \left(\frac{m^2}{4} \Ind\left[\norm{x} > \frac{m}{2}\right] + \norm{x}^2\right),
\end{align*}
and hence
\[
\MOM(l_A) \ge \frac{m^2}{4} - \MOM\left(\frac{m^2}{4} \Ind\left[\norm{X} > \frac{m}{2}\right] + \norm{X}^2\right).
\]
According to Lemma~\ref{lem:quant_ratio}, with probability at least $1 - e^{- \ell / 12}$, it holds that
\begin{align*}
\MOM\left(\frac{m^2}{4} \Ind\left[\norm{X} > \frac{m}{2}\right] + \norm{X}^2\right)
&\le 4 \E \left(\frac{m^2}{4} \Ind\left[\norm{X} > \frac{m}{2}\right] + \norm{X}^2\right) \\
&= 4 \left(\frac{m}{2}\right)^{2} \mathbb{P}\left(\norm{X} > \frac{m}{2} \right) + 4 \E \norm{X}^{2} \\
&< 8 \E \norm{X}^2,
\end{align*}
where in the last inequality we apply strict Chebyshev's inequality (assuming \(\E \norm{X}^2 > 0\)). Thus, simultaneously for all $A \in \mathcal{A}^k$ satisfying $\min_{a \in A} \norm{a} \ge m$ we have
\[
\MOM(l_A) > \frac{m^2}{4} - 8 \E \norm{X}^2 = 0.
\]
In particular, since $\{0\}$ is one of the potential candidates for $\hat{A}_{\delta, p_{\min}}$ and $\MOM(l_{\{0\}}) = 0$, we have
\[
\min_{a \in \hat{A}_{\delta, p_{\min}}} \norm{a} < m.
\]
\textbf{Step 2.}
Now consider $A \in \mathcal{A}^k$ such that there is $b \in A$ with $\norm{b} \le m$.
It is easy to see that for any $a \in A$, $x \in V_a$ implies $\norm{a - x} \le \norm{b - x}$, thus
\begin{equation}\label{eq:norm_x}
\norm{x} \ge \frac{\norm{a} - \norm{b}}{2}.
\end{equation}
Assume $\norm{a} > M$ for some $a \in A$, then $\norm{x} > \frac{M - m}{2}$ for any $x \in V_a$. Hence,
\[
P_N(V_a) \le P_N\left(\norm{X} > (M - m)/2\right).
\]
At the same time, by strict Chebyshev's inequality we have (once $\E \norm{X}^2 > 0$) that
\[
P\left(\norm{X} > (M - m)/2\right) < \frac{4 \E \norm{X}^2}{(M - m)^2} \le \frac{p_{\min}}{4}.
\]
Now Chernoff's bound for Bernoulli random variables (p.~48, \cite{boucheron2013concentration}) yields that, with probability at least \(1 - e^{- N p_{\min} / 12}\),
\[
P_N\left(\norm{X} > (M - m)/2\right)
\le P\left(\norm{X} > (M - m)/2\right) + \frac{p_{\min}}{4}
< \frac{p_{\min}}{2}.
\]
This implies \(P_N(V_a) < \frac{p_{\min}}{2}\),
what means that none of such $A$ can be chosen by our estimator.
By the union bound, we finally get that $\hat{A}_{\delta, p_{\min}} \in \mathcal{A}^k_{M, m}$, with probability at least \(1 - \left(e^{- \ell / 12} + e^{- N p_{\min} / 12}\right)\).
\end{proof}
The next step is to provide a uniform concentration of MOM over a class of quantizers \(\mathcal{A}^k_{M, m}\). First, we estimate the $L_2$-diameter and covering numbers of the functional class corresponding to $\mathcal{A}_{M, m}^k$:
\begin{equation}\label{def:F_Mm}
\mathcal{F}_{M, m}^k = \left\{l_A \;:\; A \in \mathcal{A}_{M, m}^k\right\}.
\end{equation}
\begin{Lemma}
\label{lem:A_radius}
For any distribution $P$ and any set $A \in \mathcal{A}^k_{M, m}$ it holds that
\begin{equation}\label{eq:A_radius}
\sum_{a \in A} \norm{a}^2 P(V_a) \le 2 m^2 + 8 \E \norm{X}^2,
\end{equation}
and
\begin{equation}\label{eq:var_Mm}
\E l_A^2(X) \le 4 M^2 \left(m^2 + 6 \E \norm{X}^2\right).
\end{equation}
\end{Lemma}
\begin{proof}
Fix $A \in \mathcal{A}^k$ and let $b \in A$ be such that $\norm{b} \le m$.
Then for any $a \in A$ and $x \in V_a$ it holds from~\eqref{eq:norm_x} that \(\norm{a} \le \norm{b} + 2 \norm{x} \le m + 2 \norm{x}\).
Therefore,
\[
\sum_{a \in A} P(V_a) \norm{a}^2 = \E \sum_{a \in A} \Ind[X \in V_a] \norm{a}^2
\le \E \left(m + 2 \norm{X}\right)^2 \le 2 m^2 + 8 \E \norm{X}^2.
\]
Further, we easily have using \eqref{eq:A_radius}
\begin{align*}
\E l_A^2(X)
&\le \sum_{a \in A} P(V_a) \E\left[\left(\norm{a}^2 + 2 \norm{a} \cdot \norm{X}\right)^2 \middle| X \in V_a\right] \\
& \le 2 M^2 \left(\sum_{a \in A} P(V_a) \norm{a}^2 + 4 \E \norm{X}^2\right) \\
& \le 4 M^2 \left(m^2 + 6 \E \norm{X}^2\right).
\end{align*}
\end{proof}
The next technical lemma is one of our main contributions which can be of independent interest. It states the upper bounds on $\log \mathcal{N}_{2}\left(\mathcal{F}_{M, m}^k, t, P_N\right)$ for general separable Hilbert spaces as well as for ${\mathbb R}^d$. The question on sharp bounds on covering numbers for the classes of functions indexed by $\mathcal{A}^k$ appeared naturally in the analysis of $k$-means clustering in the uniformly bounded case. The way to do it is to estimate the so-called fat-shattering dimension \citep*{narayanan2010sample, fefferman2016testing} or to decompose the covering numbers as a product of $k$ covering numbers of some simpler classes indexed by $\mathcal{A}$ as in \citep*{brownlees2015empirical, foster2019ell_}. Furthermore, in the special case of ${\mathbb R}^d$, the analysis can be done via the computation of Pollard's pseudodimension \citep{bachem2017uniform}.
Unfortunately, it seems that these approaches are better tuned to the analysis of uniformly bounded distributions or to the finite dimensional case. Our approach is based on direct computations of these covering numbers via the Johnson--Lindenstrauss lemma \citep*{johnson1984extensions}, and in ${\mathbb R}^d$ our analysis removes the unnecessary logarithmic factors appearing in some previous works in the bounded case.
\begin{Lemma}
\label{lem:cov_number}
For any \(0 < t < \diam_{2}\left(\mathcal{F}_{M, m}^k, P_N\right)\) it holds that
\begin{equation}
\label{eq:cov_number}
\log \mathcal{N}_{2}\left(\mathcal{F}_{M, m}^k, t, P_N\right)
\lesssim \frac{k M^2 \left(m^2 + P_N \norm{X}^2\right) \log N}{t^2} \log \frac{M \left(m + \sqrt{P_N \norm{X}^2}\right)}{t}.
\end{equation}
Moreover, if $E = {\mathbb R}^d$, then
\begin{equation}
\label{eq:cov_number_Rd}
\log \mathcal{N}_{2}\left(\mathcal{F}_{M, m}^k, t, P_N\right)
\lesssim k d \log \frac{M \left(m + \sqrt{P_N \norm{X}^2}\right)}{t}.
\end{equation}
\end{Lemma}
The proof of this fact is deferred to the appendix. With this result in mind we are ready to show the following uniform bound.
\begin{Lemma}
\label{lem:concentration_Mm}
Fix the number of blocks $\ell$ and assume that $\ell$ divides $N$.
Then for any fixed $\alpha \in (0, 1)$ with \( \ell \alpha \) being non-integer, with probability at least \(1 - e^{- \alpha^2 \ell / 2}\),
\begin{equation}
\label{eq:concentration_Mm}
\sup_{A \in \mathcal{A}_{M, m}^k} \left(\E l_{A}(X) - \QOM_{\alpha}(l_{A})\right)
\lesssim M \left(m + \sqrt{\E \norm{X}^2}\right) \left(\frac{(\log N)^2}{\alpha} \sqrt{\frac{k}{N}} + \sqrt{\frac{\ell}{\alpha N}}\right),
\end{equation}
as well as, with probability at least \(1 - e^{- (1 - \alpha)^2 \ell / 2}\),
\begin{align}
\label{eq:concentration_Mm_uppertail}
&\sup_{A \in \mathcal{A}_{M, m}^k} \left(\QOM_{\alpha}(l_{A}) - \E l_{A}(X) \right)
\\
&\quad\quad\lesssim M \left(m + \sqrt{\E \norm{X}^2}\right) \left(\frac{(\log N)^2}{1 - \alpha} \sqrt{\frac{k}{N}} + \sqrt{\frac{\ell}{(1 - \alpha) N}}\right) \nonumber.
\end{align}
\end{Lemma}
\begin{proof}
By Lemma~\ref{lem:momuniform}, with probability at least \(1 - e^{- \alpha^2 \ell / 2}\), it holds that
\begin{align*}
&\sup_{A \in \mathcal{A}_{M, m}^k} \left(\E l_{A}(X) - \QOM_{\alpha}(l_{A})\right)
\\
&\quad\quad\lesssim \E \sup_{A \in \mathcal{A}_{M, m}^k} \left(\frac{1}{\alpha N} \sum_{i = 1}^{N} \epsilon_i l_A(X_i)\right) + \sqrt{\sup_{A \in \mathcal{A}_{M, m}^k} \Var(l_A(X)) \frac{\ell}{\alpha N}}.
\end{align*}
Now we are going to bound the first term of the right-hand side for a fixed sample $X_1, \dots, X_N$ using Dudley's chaining argument.
It follows from~\eqref{eq:var_Mm} applied to the empirical distribution $P_N$ that
\[
\diam_2\left(\mathcal{F}_{M, m}^k, P_N\right) \le 10 M \sigma_N, \quad\text{where}\quad \sigma_N = m + \sqrt{P_N \norm{X}^2},
\]
thus, Dudley's chaining argument (e.g., Lemma~A.3 in \citep{srebro2010smoothness}) together with Lemma~\ref{lem:cov_number} ensure the following bound on the Rademacher averages of $l_A$ for any $\beta > 0$,
\begin{align*}
\E_{\epsilon} \sup_{A \in \mathcal{A}_{M, m}^k} \left(\frac{1}{N} \sum_{i = 1}^{N} \epsilon_i l_A(X_i)\right)
&\lesssim \beta + \frac{1}{\sqrt{N}} \int_\beta^{\diam_2\left(\mathcal{F}_{M, m}^k, P_N\right)} \sqrt{\log\mathcal{N}_2 \left(\mathcal{F}_{M, m}^k, t, P_N\right)} d t \\
&\lesssim \beta + \frac{1}{\sqrt{N}} \int_\beta^{10 M \sigma_N} \frac{M \sigma_N}{t} \sqrt{k \log N \log \frac{M \sigma_N}{t}} d t \\
&\lesssim \beta + M \sigma_N \sqrt{\frac{k \log N}{N}} \log^{3/2}\left(\frac{M \sigma_N}{\beta}\right)
\end{align*}
Further, choosing \(\beta = M \sigma_N \sqrt{\frac{k}{N}}\) we have
\[
\E_{\epsilon} \sup_{A \in \mathcal{A}_{M, m}^k} \left(\frac{1}{N} \sum_{i = 1}^{N} \epsilon_i l_A(X_i)\right)
\lesssim M \sigma_N \sqrt{\frac{k}{N} \log N \log^{3}\frac{N}{k}}
\le M \sigma_N (\log N)^2 \sqrt{\frac{k}{N}}.
\]
Finally,~\eqref{eq:var_Mm} implies
\[
\Var(l_A(X)) \le \E l_A^2(X) \lesssim M^2 \left(m^2 + \E \norm{X}^2\right),
\]
thus we conclude that, with probability at least \(1 - e^{- \alpha^2 \ell / 2}\),
\begin{align*}
\sup_{A \in \mathcal{A}_{M, m}^k} \left(\E l_{A}(X) - \QOM_{\alpha}(l_{A})\right)
& \lesssim \frac{\E M \sigma_N (\log N)^2}{\alpha} \sqrt{\frac{k}{N}} + \sqrt{\sup_{A \in \mathcal{A}_{M, m}^k} \Var(l_A(X)) \frac{\ell}{\alpha N}} \\
& \lesssim M \left(m + \sqrt{\E \norm{X}^2}\right) \left(\frac{(\log N)^2}{\alpha} \sqrt{\frac{k}{N}} + \sqrt{\frac{\ell}{\alpha N}}\right).
\end{align*}
Inequality \eqref{eq:concentration_Mm_uppertail} can be similarly derived from \eqref{absvalueequation_second}.
\end{proof}
We are now ready to prove Theorem~\ref{thm:pmin}.
\begin{proof}[Proof of Theorem~\ref{thm:pmin}]
In order to finish the proof we need to combine several results.
Let us fix some optimal quantizer \(A^*\), satisfying \(P(V_a) \ge p_{\min}\) for all \(a \in A^*\).
We derive the bound on the union of the events below:
\begin{itemize}
\item by Chernoff's and the union bounds, it holds with probability at least \(1 - k e^{-N p_{\min} / 8}\) that for any \(a \in A^*\)
\[
P_N(V_a) \ge \frac{P(V_a)}{2} \ge \frac{p_{\min}}{2},
\]
hence \(A^*\) is in the set of possible solutions on this event;
\item by Lemma~\ref{lem:optimizer_Mm}, with probability at least \( 1 - e^{- \ell / 12} - e^{- N p_{\min} / 12}\), we have \(\hat{A}_{\delta, p_{\min}} \in \mathcal{A}^k_{M, m}\) with
\(M = 10 \sqrt{\frac{\E \norm{X}^{2}}{p_{\min}}}\) and \(m = 4 \sqrt{2 \E \norm{X}^{2}}\).
In addition, a similar property can be derived for \(A^*\). Indeed, if \(\min_{a \in A^*} \norm{a} > m\), then
\begin{align*}
\E l_{A^*}(X) &\ge \frac{m^2}{4} P\left(\norm{X} \le \frac{m}{2}\right) - \E \norm{X}^2
\\
&\ge 8 \E \norm{X}^{2} \left(1 - \frac{1}{8}\right) - \E \norm{X}^{2}
> 0 = \E l_{\{0\}}(X).
\end{align*}
This contradicts the optimality of \(A^*\).
Now assume \(\min_{a \in A^*} \norm{a} \le m\), but \(\norm{a} > M\) for some \(a \in A^*\). Arguing as in the proof of Lemma~\ref{lem:optimizer_Mm}, we obtain
\[
P(V_a) \le P\left(\norm{X} > \frac{M - m}{2}\right) < \frac{4 \E \norm{X}^2}{(M - m)^2} \le \frac{p_{\min}}{4}.
\]
This contradicts the lower bound \(P(V_a) \ge p_{\min}\);
\item by Lemma~\ref{lem:concentration_Mm}, taking $M$ and $m$ as above,
we have that, with probability at least \(1 - 2 e^{-\ell/8}\),
\[
\sup_{A \in A_{M, m}} \abs{\E l_{A} - \MOM(l_{A})}
\lesssim \E \norm{X}^2 \left((\log N)^2 \sqrt{\frac{k}{N p_{\min}}} + \sqrt{\frac{\ell}{N p_{\min}}}\right) \, .
\]
\end{itemize}
All three assertions take place with probability at least \( 1 - 3 e^{-\ell / 12} - (k + 1) e^{-N p_{\min} / 12}\). Suppose for a moment that \( N p_{\min} \geq 12 \log \tfrac{2 (k + 1)}{\delta} \). Then, additionally, due to the choice \(\ell = 12 \left\lceil \log \tfrac{6}{\delta} \right\rceil + 1 \), we have that the total probability is at least \( 1 - \delta \).
Since we know that on this event \(\hat{A}_{\delta, p_{\min}}, A^* \in A_{M, m} \) and \( \MOM\left(l_{\hat{A}_{\delta, p_{\min}}}\right) \leq \MOM(l_{A^*}) \), we have
\begin{align*}
D(\hat{A}_{\delta, p_{\min}}) - D(A^*) & = \E l_{\hat{A}_{\delta, p_{\min}}} - \E l_{A^{*}}\\
& \le \E l_{\hat{A}_{\delta, p_{\min}}} - \MOM\left(l_{\hat{A}_{\delta, p_{\min}}}\right) - \E l_{A^*} + \MOM(l_{A^*})\\
& \lesssim \E \norm{X}^2 \left((\log N)^2 \sqrt{\frac{k}{N p_{\min}}} + \sqrt{\frac{\log \frac{1}{\delta}}{N p_{\min}}}\right) \, .
\end{align*}
Finally, consider the case \( N p_{\min} < 12 \log \tfrac{2 (k + 1)}{\delta} \). According to~\eqref{eq:simple_bound} one has, with probability at least $1 - \delta$,
\[
D(\hat{A}_{\delta, p_{\min}}) - D(A^*) \lesssim \E \norm{X}^2
\lesssim \E \norm{X}^2 \left((\log N)^2 \sqrt{\frac{k}{N p_{\min}}} + \sqrt{\frac{\log \frac{1}{\delta}}{N p_{\min}}}\right) \, .
\]
The claim follows.
\end{proof}
Finally, we present an analog of Theorem~\ref{thm:pmin} in ${\mathbb R}^d$. We are able to completely remove the $(\log N)$-factor by making $d$ appear in the bound. First, we need the following simple result.
\begin{Corollary}
\label{lem:concentration_Mm_Rd}
For any $\alpha \in (0, 1)$ with \(\ell \alpha\) being non-integer, we have, with probability at least \(1 - e^{- \alpha^2 \ell / 2}\),
\[
\sup_{A \in \mathcal{A}_{M, m}^k} \left(\E l_{A}(X) - \QOM_{\alpha}(l_{A})\right)
\lesssim M \left(m + \sqrt{\E \norm{X}^2}\right) \left(\frac{1}{\alpha} \sqrt{\frac{k d}{N}} + \sqrt{\frac{\ell}{\alpha N}}\right).
\]
as well as, with probability at least \(1 - e^{- (1 - \alpha)^2 \ell / 2}\),
\[
\sup_{A \in \mathcal{A}_{M, m}^k} \left(\QOM_{\alpha}(l_{A}) - \E l_{A}(X) \right)
\lesssim M \left(m + \sqrt{\E \norm{X}^2}\right) \left(\frac{1}{1 - \alpha} \sqrt{\frac{k d}{N}} + \sqrt{\frac{\ell}{(1 - \alpha) N}}\right).
\]
\end{Corollary}
\begin{proof}
Using the Dudley integral argument again we have
\begin{align*}
\E_{\epsilon} \sup_{A \in \mathcal{A}_{M, m}^k} \left(\frac{1}{N} \sum_{i = 1}^{N} \epsilon_i l_A(X_i)\right)
&\lesssim \frac{1}{\sqrt{N}} \int_0^{\diam_2\left(\mathcal{F}_{M, m}^k, P_N\right)} \sqrt{\log\mathcal{N}_2 \left(\mathcal{F}_{M, m}^k, t, P_N\right)} d t \\
&\lesssim \frac{1}{\sqrt{N}} \int_0^{10 M \sigma_N} \sqrt{k d \log\left(\frac{M \sigma_N}{t}\right)} d t \\
&\lesssim M \sigma_N \sqrt{\frac{k d}{N}}.
\end{align*}
The rest of the proof is exactly the same as for Lemma~\ref{lem:concentration_Mm}.
\end{proof}
With this result in mind, we can immediately prove our second main result.
\begin{Theorem}
\label{thm:pmin_Rd}
Consider the case of ${\mathbb R}^d$ with the Euclidean distance. Fix \(\delta \in (0, 1)\). Suppose, \(\min_{a \in A^{*}} P(V_a) \geq p_{\min} \) for some optimal quantizer \(A^{*} \).
The same estimator $\hat{A}_{\delta, p_{\min}}$ satisfies, with probability at least \( 1 - \delta\),
\[
D(\hat{A}_{\delta, p_{\min}}) - D(A^*)
\lesssim \E \norm{X - \mu}^{2} \sqrt{\frac{k d + \log \frac{1}{\delta}}{N p_{\min}}} \, .
\]
Moreover, if \(N p_{\min} \gtrsim d\log N + \log \frac{1}{\delta}\), then
\[
D(\hat{A}_{\delta, p_{\min}}) - D(A^*)
\lesssim \sqrt{\Tr(\Sigma) \left(\lambda_{\max}(\Sigma) + p_{\min} \Tr(\Sigma)\right)} \sqrt{\frac{k d + \log \frac{1}{\delta}}{N p_{\min}}} \, ,
\]
where $\Sigma$ is the covariance matrix of $X$ and $\lambda_{\max}(\Sigma)$ is its largest eigenvalue.
\end{Theorem}
\begin{proof}
The proof of the first statement repeats the same lines of the proof of Theorem~\ref{thm:pmin}. The only difference is that the bound of Lemma~\ref{lem:concentration_Mm} is replaced by the bound of Corollary~\ref{lem:concentration_Mm_Rd}.
We proceed with the proof of the second inequality. In what follows, we only emphasize the differences with the proof of Theorem~\ref{thm:pmin}.
Our idea is to use the Vapnik--Chervonenkis type argument to provide a slightly sharper upper bound on $M$.
Recall that according to Lemma~\ref{lem:optimizer_Mm}, with probability at least \( 1 - e^{-\ell/12}\), it holds that
\[
\min_{a \in \hat{A}_{\delta, p_{\min}}} \norm{a} \le m = 4 \sqrt{2 \E \norm{X}^{2}}
\]
and, with total probability $1 - \left(e^{- \ell / 12} + e^{- N p_{\min} / 12}\right)$, we have
\[
\max_{a \in \hat{A}_{\delta, p_{\min}}} \norm{a} \le 10 \sqrt{\frac{\E \norm{X}^2}{p_{\min}}}.
\]
Now, we are going to show that, with probability $1 - \left(e^{- \ell / 12} + 4 e^{- N p_{\min} / 100}\right)$, \(\hat{A}_{\delta, p_{\min}} \in \mathcal{A}^k_{M, m}\), where \(M = 2 m \lor 8 \sqrt{\frac{\lambda_{\max}(\Sigma)}{p_{\min}}}\), provided that $N p_{\min} \gtrsim d \log N$.
Consider any $A \in \mathcal{A}^k$ such that there is $b \in A$ with $\norm{b} \le m$. Notice that $x \in V_a$ for $a \in A$ implies
\begin{equation}
\label{eq:condition}
2 \langle x, a - b \rangle \ge \norm{a}^2 - \norm{b}^2.
\end{equation}
Using \(\E\left(\langle X, a - b \rangle\right)^2 \le \lambda_{\max}(\Sigma) \norm{a - b}^2\) and assuming that $\norm{a} > M$, we have by Chebyshev's inequality
\begin{equation}
\label{eq:pva}
P\left(2 \langle X, a - b \rangle \ge \norm{a}^2 - \norm{b}^2\right)
\le \frac{4 \lambda_{\max}(\Sigma) \norm{a - b}^2}{(\norm{a}^2 - \norm{b}^2)^2} < \frac{16 \lambda_{\max}(\Sigma)}{M^2} \le \frac{p_{\min}}{4},
\end{equation}
where we additionally used
\[
\norm{a}^2 - \norm{b}^2 = \left(\norm{a} - \norm{b}\right) \left(\norm{a} + \norm{b}\right) > \frac{M \norm{a - b}}{2}.
\]
Observe that \eqref{eq:condition} implies
\[
P_N(V_a) \le P_N\left(2 \langle X, a - b \rangle \ge \norm{a}^2 - \norm{b}^2\right).
\]
Applying Theorem~5.1 in \citep*{boucheron2005theory} to the class induced by half-spaces \( {\mathcal F} = \left\{ \Ind[\langle X, u \rangle \ge t] \;:\; u \in {\mathbb R}^{d}, t \in {\mathbb R}\right\}\), we get that simultaneously for all \(u \in {\mathbb R}^{d} \) and \( t \in {\mathbb R}\), with probability at least \(1 - 4 e^{- N p_{\min} / 100}\),
\begin{align*}
&P_N(\langle X, u \rangle \ge t) - P(\langle X, u \rangle \ge t)
\\
&\quad\quad\le 2 \sqrt{P(\langle X, u \rangle \ge t) \frac{\log\mathbb{S}_{2N}({\mathcal F}) + N p_{\min} / 100}{N}} + 4 \frac{\log\mathbb{S}_{2N}({\mathcal F}) + N p_{\min} / 100}{N},
\end{align*}
where \( \mathbb{S}_{N}({\mathcal F}) \) denotes the shatter-coefficient of the class \( {\mathcal F} \) (see the definition in \cite{boucheron2005theory}). By the \citeauthor{Vapnik74} lemma and since the VC-dimension of \( {\mathcal F} \) is known to be $d + 1$, we have \( \log \mathbb{S}_{2N}({\mathcal F}) \lesssim d \log N\). Due to \eqref{eq:pva} and provided that \( N p_{\min} \ge 100 \log \mathbb{S}_{2N}({\mathcal F})\), with probability at least \( 1 - 4 e^{- N p_{\min} / 100}\), we have
\[
P_N\left(2 \langle X, a - b \rangle \ge \norm{a}^2 - \norm{b}^2\right) < \frac{p_{\min}}{2}.
\]
Therefore, we obtain a contradiction with the definition of $\hat{A}_{\delta, p_{\min}}$ and prove that, with probability at least \(1 - \left(e^{- \ell / 12} + 4 e^{- N p_{\min} / 100}\right)\), we have the inclusion \(\hat{A}_{\delta, p_{\min}} \in \mathcal{A}^k_{M, m}\). Similarly, \(A^* \in \mathcal{A}^k_{M, m}\).
Since \(N \gtrsim \frac{1}{p_{\min}} \ge k\), we have \( N p_{\min} \gtrsim d \log N + \log \frac{1}{\delta} \gtrsim \log \frac{k}{\delta}\).
Then the remainder of the proof repeats the lines of the proof of Theorem~\ref{thm:pmin} in the regime \(N p_{\min} \gtrsim \log \tfrac {k}{\delta}\) with Lemma~\ref{lem:concentration_Mm} replaced by Corollary~\ref{lem:concentration_Mm_Rd}.
The claim follows.
\end{proof}
Finally, we compare the results obtained in this section with the results from the previous section, were we study quantizers with known magnitude \( M \). First, we notice that the leading term \(M^2 + M \sqrt{\E \norm{X}^2}\) in Theorem~\ref{thm:simpleupperm} is replaced in~\eqref{eq:concentration_Mm} by a much better term \(M m + M \sqrt{\E \norm{X}^2}\). Indeed, in view of Lemma~\ref{lem:optimizer_Mm}, we are interested in the regime \(m \lesssim \sqrt{\E \norm{X}^{2}}\) and \( M \lesssim \sqrt{{\E \norm{X}^{2}}/{p_{\min}}} \). Therefore, the new bound allows us to obtain a better dependence on $p_{\min}$ since we may otherwise get an additional factor $\frac{1}{\sqrt{p_{\min}}}$.
Second, similar results can be obtained in the case where $M$ is known instead of $p_{\min}$, i.e., in the setting of Section~\ref{sec:warmup}. Namely, applying Lemma~\ref{lem:concentration_Mm} and the first part of Lemma~\ref{lem:optimizer_Mm} to the estimator of Theorem~\ref{thm:simpleupperm} we conclude that (see also the details of the proof of Theorem~\ref{thm:general_rate} below), with probability at least \(1 - \delta\), it holds that
\[
D(\hat{A}_{\delta, M}) - D(A^*) \lesssim M \sqrt{\E \norm{X}^{2}} \left((\log N)^2 \sqrt{\frac{k}{N}} + \sqrt{\frac{\log \frac{1}{\delta}}{N}}\right) \, .
\]
In the case of ${\mathbb R}^d$, we obtain respectively that, with probability at least \( 1 - \delta\), it holds that
\[
D(\hat{A}_{\delta, M}) - D(A^*) \lesssim M \sqrt{\E \norm{X}^{2}} \left(\sqrt{\frac{k d}{N}} + \sqrt{\frac{\log \frac{1}{\delta}}{N}}\right) \, .
\]
\subsection{A lower bound with \texorpdfstring{\(p_{\min}\)}{pmin}}
\label{sec:lowerbound}
Here we study the question of the optimality of Theorem~\ref{thm:pmin} and Theorem~\ref{thm:pmin_Rd}. The lower bounds for the excess distortion appeared first in \citep{bartlett1998minimax} for the bounded case ($\norm{X} \le 1$ almost surely), where they showed a lower bound of order \( \Omega\left(\sqrt{\frac{k^{1 - 4 / d}}{N}}\right) \). Furthermore, \cite{linder2002learning} recovers this bound for constant \( d \) and \( k \geq 3 \), while \cite{antos2005improved} shows the same bound for \( k = 2 \) but only for empirically optimal quantizers.
Below, we focus on how the mass of the lightest cluster affects the excess distortion in the unbounded case. We extend the construction of \cite{linder2002learning} to derive a bound that confirms that the dependence on \( p_{\min} \), $N$, and $\E \norm{X - \mu}^2$ in Theorem~\ref{thm:pmin_Rd} is sharp in some cases. The optimal dependence on the remaining parameters remains open.
Fix \( k = 4 \) and \( d = 1 \). Consider a class of probability measures on \( {\mathbb R}\),
\[
\mathcal{P}(p_{\min}, \sigma) = \left\{ P \;:\; \E X^{2} \leq \sigma^{2} \ \text{and}\; \exists A^* \in \argmin D(A, P) \ \text{s.t.}\ \min_{a \in A^*} P(V_{a}) \geq p_{\min} \right\},
\]
i.e., the probability measures that have an optimal quantizer based on \(k \) points such that the probability of \( X \) falling into each Voronoi cell under \( P \) is at least \( p_{\min} \). Theorem~\ref{thm:pmin_Rd} implies that for \( k = 4 \) and \( d = 1 \) there is an estimator \( \hat{A}_N \) based on the i.i.d.\ sample \( X_1, \dots, X_N \), such that for any \( P \in \mathcal{P}(p_{\min}, \sigma) \), we have with probability at least (say) $0.99$,
\[
D(\hat{A}_N, P) - D(A^{*}, P) \lesssim \sigma^2 \sqrt{\frac{1}{N p_{\min}}},
\]
where the probability of the event is measured with respect to the joint distribution \( \mathbb{P} = P^{\otimes N} \).
The following result shows that when $d$, $k$, and $\delta$ are constants, the result is sharp up to a constant factor.
\begin{Theorem}\label{pmin_lowerbound}
Under the notation introduced above let \( \sigma > 0 \), \( p_{\min} \leq {1}/{10}\). Let also \(N p_{\min} > 1/8\). Then, for any empirically designed quantizer \( \hat{A}_N \) there is a distribution \( P \in \mathcal{P}(p_{\min}, \sigma) \), such that, with probability at least $\frac{1}{4}$,
\[
D(\hat{A}_N, P) - D(A^{*}, P) \geq \frac{\sigma^{2}}{60}\sqrt{\frac{1}{N p_{\min}}}.
\]
\end{Theorem}
Let us first present a heuristic argument showing the validity of Theorem~\ref{pmin_lowerbound}. Slightly abusing the notation for \( p \in (0, 1/2) \), \(\delta \in (-1/2, 1/2) \) consider a distribution \( P_{p, \delta} \) supported on five points
\begin{align*}
P_{p, \delta}(X = - \tfrac{1}{2} p^{-1/2}) &= P_{p, \delta}(X = - p^{-1/2}) = \frac{p(1 - \delta)}{4},
\qquad
P_{p, \delta}(X = 0)= 1 - p, \\
P_{p, \delta}(X = \tfrac{1}{2} p^{-1/2}) &= P_{p, \delta}(X = p^{-1/2}) = \frac{p(1 + \delta)}{4}.
\end{align*}
We have \( \E X^{2} = 5/8 \). Obviously, we can rescale these values, so it is enough to consider the case \( \sigma^{2} = 5/8\). It is easy to see that for \( \delta > 0 \) the optimal quantizer is \( A^{*} = (0, \tfrac{1}{2} p^{-1/2}, p^{-1/2}, -\tfrac{3}{4} p^{-1/2}) \) with the distortion
\[
D(A^*, P_{p, \delta}) = \frac{p(1 - \delta)}{2} \left( \frac{p^{-1/2}}{4}\right)^{2} = \frac{1 - \delta}{32} .
\]
For \( \delta = \frac{1}{\sqrt{Np}} \), the number of points on the negative side is greater, with constant probability (see p. 27 in \citep{linder2002learning}). In such a case, the empirically optimal quantizer must be \( \hat{A} = (0, - p^{-1/2}, -\tfrac{1}{2} p^{-1/2}, (\frac{1}{2} + a)p^{-1/2}) \), where \( a \) is some value between \( 0\) and \( \tfrac{1}{2} \). Thus, the distortion of any empirically optimal quantizer is at least
\[
D(\hat{A}, P_{p, \delta}) \geq \frac{p(1 + \delta)}{2} \left(\frac{p^{-1/2}}{4}\right)^{2} = \frac{1 + \delta}{32},
\]
which implies
\[
D(\hat{A}, P_{p, \delta}) - D(A^{*}, P_{p, \delta}) \geq \frac{\delta}{16} = \frac{1}{16} \frac{1}{\sqrt{Np}} \,.
\]
However, this only touches the empirically optimal quantizer. The proof of the lower bound relies on a standard reduction to hypothesis testing.
\begin{proof}[Proof of Theorem~\ref{pmin_lowerbound}]
As pointed out above, we can fix \( \sigma^{2} = 5/8 \) without loss of generality. Set \( p = 4 p_{\min} \leq 2/5 \). Then we have that \( P_{p, \delta}, P_{p, -\delta} \in \mathcal{P}( p_{\min}, 5/8) \). Denote, \( P_1 = P_{p, \delta} \) and \( P_2 = P_{p, -\delta} \). The Kullback-Leibler divergence between the two satisfies for $\delta \le \frac{1}{2}$,
\[
\mathrm{KL}(P_1, P_2) = p \delta \log\frac{1 + \delta}{1 - \delta} \leq 4 p \delta^{2} .
\]
Using Pinsker's inequality and the additivity of the KL-divergence for product measures (see \citep{boucheron2013concentration}) we have
\[
\mathrm{TV}\left(P_1^{\otimes N}, P_{2}^{\otimes N}\right) \leq \sqrt{\frac{N}{2} \mathrm{KL}(P_1, P_2)}
\leq \sqrt{2 N p \delta^{2}} = 1/2,
\]
where we choose \( \delta = 1 / \sqrt{8 N p} \) (the condition \( N p_{\min} > 1/8 \) ensures that \( \delta < 1/2\)). Below, we only consider the distributions \( P \in \{P_1, P_2\} \) instead of the whole class \( \mathcal{P}\left(p_{\min}, \sqrt{{5}/{8}}\right) \). Consider an empirical quantizer denoted as \( \hat{A}_{N} = \hat{A}_N(X_1, \dots, X_N)\) that takes only the values \( \{A_1, A_2\} \), where
\[
A_1 = (-\tfrac{3}{4} p^{-1/2}, 0, \tfrac{1}{2} p^{-1/2}, p^{-1/2} ) \quad \text{and} \quad A_2 = ( -p^{-1/2}, -\tfrac{1}{2} p^{-1/2}, 0, \tfrac{3}{4} p^{-1/2} ).
\]
Let \( \Omega_{1} \subset {\mathbb R}^{N} \) be the set such that \( \hat{A}_N = A_1 \) on it, and \( \hat{A}_N = A_2 \) outside of $\Omega_{1}$. Since \( \mathrm{TV}\left(P_1^{\otimes N}, P_2^{\otimes N}\right) \leq 1/2 \), we have
\begin{align*}
\max_{j = 1,2} P_j^{\otimes N}\left\{\hat{A}_N \neq A_j\right\} &= \max\left(1 - P_1^{\otimes N}(\Omega_1), P_2^{\otimes N}(\Omega_1)\right)
\\
&\geq \max\left(1/2 - P_2^{\otimes N}(\Omega_1), P_2^{\otimes N}(\Omega_1)\right) \geq 1/4 .
\end{align*}
Notice that under \( \mathbb{P}_{j} \) the event \( \hat{A}_N \neq A_j \) corresponds to the distortion
\[
D(A_1, P_2) = D(A_2, P_1) = \frac{p (1 + \delta)}{2} \left(\frac{p^{-1/2}}{4}\right)^{2} = \frac{1 + \delta}{32},
\]
whereas the minimal distortion is
\[
D(A_j, P_j) = \frac{p(1 - \delta)}{2} \left(\frac{p^{-1/2}}{4}\right)^{2} = \frac{1 - \delta}{32} \, .
\]
Since \( \delta = 1 / \sqrt{8 N p} \), \( p = 4 p_{\min} \), and \( \sigma^2 = 5/8\), with probability at least \(1/4\), we have
\[
D(\hat{A}_N, P_j) - D(A_j, P_j) = \frac{\delta}{16} = \frac{1}{16 \sqrt{8 N p}} > \frac{\sigma^{2}}{60 \sqrt{N p_{\min}}} \, .
\]
It remains to show why only \( \hat{A}_N \in \{A_1, A_2\} \) matters. For an arbitrary \( \hat{A}_N \), the corresponding Voronoi cells could be one of the following:
\begin{enumerate}
\item \( \bigl\{\{ -p^{-1/2}, -\tfrac{1}{2}p^{-1/2}\}, \{0\}, \{\tfrac{1}{2}p^{-1/2}\}, \{p^{-1/2}\}\bigr\}, \)
\item \( \bigl\{\{ -p^{-1/2} \}, \{-\tfrac{1}{2}p^{-1/2}, 0\}, \{\tfrac{1}{2}p^{-1/2}\}, \{p^{-1/2}\}\bigr\}, \)
\item \( \bigl\{\{ -p^{-1/2} \}, \{-\tfrac{1}{2}p^{-1/2}\}, \{0, \tfrac{1}{2}p^{-1/2}\}, \{p^{-1/2}\}\bigr\}, \)
\item \( \bigl\{\{ -p^{-1/2}\}, \{-\tfrac{1}{2}p^{-1/2}\}, \{0\}, \{\tfrac{1}{2}p^{-1/2}, p^{-1/2}\}\bigr\} \).
\end{enumerate}
Denote by \( \tilde{A}_{N} \) an empirical quantizer such that it equals to \(A_1\) in the cases~1. and~2., and equals to \(A_2\) in the cases~3. and~4. Let us show case by case, that the distortion of \( \tilde{A}_N \) is smaller under either measure.
\begin{enumerate}
\item This case is trivial: by the centroid condition under both measures the optimal center for the cluster \( \{ -p^{-1/2}, -\tfrac{1}{2}p^{-1/2} \} \) is \( -\tfrac{3}{4} p^{-1/2} \), which corresponds to \( A_1 \).
\item It is easy to calculate that the minimal distortion of a cluster on two points \( a, b \) with probabilities \(q, r\), respectively, is \( (a - b)^{2} \tfrac{qr}{q + r} \). Therefore, using only the distortion on \( \{-\tfrac{1}{2}p^{-1/2}, 0\} \),
\[
D(\hat{A}_N, P_1) \geq \frac{1}{4p} \frac{p(1 - \delta) (1 - p)}{p(1 - \delta) + 4(1 - p)} > \frac{1 - \delta}{32} = D(A_1, P_1),
\]
where the second inequality follows from \( p \leq 2/5\). Using additionally \( \delta < 1\), we have
as well
\[
D(\hat{A}_N, P_2) \geq \frac{1}{4p} \frac{p(1 + \delta) (1 - p)}{p(1 + \delta) + 4(1 - p)} >
\frac{1 + \delta}{32} = D(A_1, P_2) \, .
\]
\end{enumerate}
Due to the symmetry, case~3. is similar to case~2. and case~4. is similar to case~1. We conclude that we always have \( D(\hat{A}_N, P_j) \geq D(\tilde{A}_N, P_j) \) for both \(j = 1, 2 \).
\end{proof}
\section{Unknown Parameters of Distributions}
\label{sec:Generalcase}
In this section we show that a convergence rate similar to one in Theorem~\ref{thm:simpleupperm} and Theorem~\ref{thm:pmin} holds without any prior knowledge on $M$ or $p_{\min}$. Our motivation is that in practice we may not have any information about the underlying distribution $P$. We show that even in this case the sub-Gaussian excess distortion bounds are possible. However, as a result, our bounds become more sensitive to some specific properties of $P$. The following theorem is the main result of this section.
\begin{Theorem}
\label{thm:general_rate}
Fix $\delta \in (0, 1)$.
There is an estimator $\hat{A}_{\delta}$ depending on $\delta$ such that, with probability at least $1 - \delta$,
\[
D(\hat{A}_\delta) - D(A^*) \lesssim R\; \sqrt{\E \norm{X - \mu}^2} \left((\log N)^2 \sqrt{\frac{k}{N}} + \sqrt{\frac{\log\frac{1}{\delta}}{N}}\right),
\]
where $R$ is such that
\begin{equation}\label{eq:R}
\E \norm{X - \mu}^2 \Ind[\norm{X - \mu} > R] \le \frac{\Delta}{64},
\end{equation}
and
\begin{equation}\label{def:Delta}
\Delta = \inf_{A \in \mathcal{A}^{k-1}} D(A) - \inf_{A \in \mathcal{A}^k} D(A).
\end{equation}
\end{Theorem}
\begin{Remark}
Observe that both $R$ and $\Delta$ played an import role in the original proof of the strong consistency by \citeauthor{pollard1981strong}.
\end{Remark}
Let us first define our estimator. As before, in this section we use the notation~\eqref{def:Ak}.
\begin{framed}
\textbf{The estimator of Theorem~\ref{thm:general_rate}.} We set
\[
\hat{A}_{\delta} = \argmin_{A \in \mathcal{A}^k} \MOM(l_{A}),
\]
with the number of blocks $\ell = 32 \left\lceil \log \tfrac{4}{\delta}\right\rceil + 1$.
\end{framed}
As before, our estimator $\hat{A}_{\delta}$ is an analog of an empirically optimal quantizer \eqref{empquantizer} with the only difference that instead of the sample mean we minimize the MOM criterion.
Note that the estimator is translation invariant, so we can once again assume that \(\E X = 0\) without loss of generality.
\begin{proof}[Proof of Theorem~\ref{thm:general_rate}]
The proof is based on the following simple observation: if for $A \in \mathcal{A}^k$ there is $a \in A$ such that \(\norm{a} \gg R\), then by considering \(A^{\prime} = A \setminus \{a\}\) we obtain that \(A' \in \mathcal{A}^{k-1}\) and \(D(A') - D(A) \ll \Delta\), thus $A$ cannot be a (nearly) optimal empirical solution.
Namely, we are going to compare $\hat{A}_\delta$ with $\hat{A}_\delta \cap B_M$ for some $M \gtrsim R$ and show that with high probability either $\E l_{\hat{A}_\delta \cap B_M}$ is close to $\E l_{\hat{A}_\delta}$ (for small $N$) or \(\hat{A}_\delta \subset B_M\) (for large $N$), where $B_M$ is the ball of radius $M$ centred at the origin.
Moreover, \(\min_{a \in \hat{A}_\delta} \norm{a} \lesssim \sqrt{\E \norm{X}^2}\) with high probability,
thus in both cases we can apply Lemma~\ref{lem:concentration_Mm} to obtain the convergence rate of the form
\[
D(\hat{A}_\delta) - D(A^*)
\lesssim M \sqrt{\E \norm{X}^2} \left((\log N)^2 \sqrt{\frac{k}{N}} + \sqrt{\frac{\log\frac{1}{\delta}}{N}}\right).
\]
First, according to the first part of Lemma~\ref{lem:optimizer_Mm}, with probability at least \(1 - e^{- \ell / 12} \ge 1 - \delta / 4\),
\[
\min_{a \in \hat{A}_\delta} \norm{a} \le m = 4 \sqrt{2 \E \norm{X}^2}.
\]
Let us define $M = m + 2 (R \lor m)$. Note that
\[
\E \norm{X}^2 \le R^2 + \E \norm{X}^2 \Ind[\norm{X} > R] \le R^2 + \frac{\Delta}{64} \le R^2 + \frac{\E \norm{X}^2}{64},
\]
thus $R \ge 0.99 \sqrt{\E \norm{X}^2}$, which implies $M \simeq R$.
Now fix $A \in \mathcal{A}^k$ such that \(\min_{a \in A} \norm{a} \le m\). Then by~\eqref{eq:norm_x} for any $a \in A$ (if it exists) such that \(\norm{a} > M\) and any $x \in V_a$, one has \(\norm{x} > \frac{M - m}{2} = R \lor m\), thus \(l_A \equiv l_{A \cap B_M}\) on $B_{R \lor m}$. Moreover, recall that for all $x \in E$,
\[
\min_{a \in A} \norm{a - x} \le m + \norm{x}, \quad
\min_{a \in A \cap B_M} \norm{a - x} \le m + \norm{x},
\]
and thus
\begin{align*}
l_{A \cap B_M}(x) - l_A(x)
&= \min_{a \in A \cap B_M} \norm{a - x}^2 - \min_{a \in A} \norm{a - x}^2
\\
&\le (m + \norm{x})^2 \Ind[\norm{x} > R \lor m]
\le 4 \norm{x}^2 \Ind[\norm{x} > R].
\end{align*}
To simplify the notation by writing $\MOM(f(X))$ and $\QOM_{\alpha}(f(X))$ we mean $\MOM(f)$ and $\QOM_{\alpha}(f)$ respectively. Therefore, we have
\begin{align*}
\MOM(l_A) &\ge \MOM\left(l_{A \cap B_M}(X) - 4 \norm{X}^2 \Ind[\norm{X} > R]\right) \\
&\ge \QOM_{1/4}(l_{A \cap B_M}) - 4 \QOM_{3/4}\left(\norm{X}^2 \Ind[\norm{X} > R]\right).
\end{align*}
The last term can be bounded by Lemma~\ref{lem:quant_ratio}: with probability at least \(1 - \delta / 4\),
\[
\QOM_{3/4} \left(\norm{X}^2 \Ind[\norm{X} > R]\right)
\le 8 \E \norm{X}^2 \Ind[\norm{X} > R]
\le \frac{\Delta}{8},
\]
thus
\begin{equation}\label{eq:QOM_AM_bound}
\QOM_{1/4}(l_{A \cap B_M}) \le \MOM(l_A) + \frac{\Delta}{2}.
\end{equation}
Further, we can assume without loss of generality\ that $A^*$ belongs to $\mathcal{A}^k_{M, m}$. Indeed, \(\min_{a \in A^*} \norm{a} \le m\) according to the proof of Theorem~\ref{thm:pmin}, and if \(A^* \not\subset B_M\), then \(|A^* \cap B_M| < k\) and hence
\[
\E l_{A^*} + \Delta \le \E l_{A^* \cap B_M}
\le \E l_{A^*} + 4 \E \norm{X}^2 \Ind[\norm{X} > R]
\le \E l_{A^*} + \frac{\Delta}{16},
\]
which is possible only if $\Delta = 0$. But in this case \(\norm{X} \le R \le M\) almost surely and \(|\supp(P)| \le k - 1\), thus one can choose \(A^* = \supp(P) \in \mathcal{A}^k_{M, m}\).
Lemma~\ref{lem:concentration_Mm} ensures that, with probability at least $1 - \delta / 2$, for any $\alpha \in \left\{\frac{1}{4}, \frac{1}{2}\right\}$ it holds that
\begin{equation}\label{eq:l_concentration}
\sup_{A \in \mathcal{A}_{M, m}^k} \abs{\E l_{A}(X) - \QOM_{\alpha}(l_{A})}
\le C R \sqrt{\E \norm{X}^2} \left((\log N)^2 \sqrt{\frac{k}{N}} + \sqrt{\frac{\log \frac{1}{\delta}}{N}}\right),
\end{equation}
where $C > 0$ is an absolute constant.
Finally, we get the following lines of inequalities, which hold with probability at least $1 - \delta$,
\begin{align*}
\E l_{\hat{A}_\delta \cap B_M}
&\le \QOM_{1/4}(l_{\hat{A}_\delta \cap B_M}) + C R \sqrt{\E \norm{X}^2} \left((\log N)^2 \sqrt{\frac{k}{N}} + \sqrt{\frac{\log \frac{1}{\delta}}{N}}\right) \\
&\le \MOM(l_{\hat{A}_\delta}) + \frac{\Delta}{2} + C R \sqrt{\E \norm{X}^2} \left((\log N)^2 \sqrt{\frac{k}{N}} + \sqrt{\frac{\log \frac{1}{\delta}}{N}}\right) \\
&\le \MOM(l_{A^*}) + \frac{\Delta}{2} + C R \sqrt{\E \norm{X}^2} \left((\log N)^2 \sqrt{\frac{k}{N}} + \sqrt{\frac{\log \frac{1}{\delta}}{N}}\right) \\
&\le \E l_{A^*} + \frac{\Delta}{2} + 2 C R \sqrt{\E \norm{X}^2} \left((\log N)^2 \sqrt{\frac{k}{N}} + \sqrt{\frac{\log \frac{1}{\delta}}{N}}\right).
\end{align*}
Now there are two possible cases.
If
\[
C R \sqrt{\E \norm{X}^2} \left((\log N)^2 \sqrt{\frac{k}{N}} + \sqrt{\frac{\log \frac{1}{\delta}}{N}}\right) \ge \frac{\Delta}{4},
\]
then
\[
\E l_{\hat{A}_\delta} \le \E l_{\hat{A}_\delta \cap B_M}
\le \E l_{A^*} + 4 C R \sqrt{\E \norm{X}^2} \left((\log N)^2 \sqrt{\frac{k}{N}} + \sqrt{\frac{\log \frac{1}{\delta}}{N}}\right).
\]
Otherwise, we have
\[
C R \sqrt{\E \norm{X}^2} \left((\log N)^2 \sqrt{\frac{k}{N}} + \sqrt{\frac{\log \frac{1}{\delta}}{N}}\right) < \frac{\Delta}{4},
\]
then \(\hat{A}_\delta \subset B_M\): indeed, \(\E l_{\hat{A}_\delta \cap B_M} < \E l_{A^*} + \Delta\), now assume \(|\hat{A}_\delta \cap B_M| < k\), then
\[
\E l_{\hat{A}_\delta \cap B_M}
\ge \inf_{A \in \mathcal{A}^{k-1}} \E l_A
= \E l_{A^*} + \Delta,
\]
and we obtain a contradiction.
Thus, \(\hat{A}_\delta \in \mathcal{A}^k_{M, m}\), and~\eqref{eq:l_concentration} again yields
\begin{align*}
D(\hat{A}_\delta) - D(A^*) & = \E l_{\hat{A}_\delta} - \E l_{A^*} \\
& \le \E l_{\hat{A}_\delta} - \MOM(l_{\hat{A}_\delta}) - \E l_{A^*} + \MOM(l_{A^*}) \\
& \lesssim R \sqrt{\E \norm{X}^2} \left((\log N)^2 \sqrt{\frac{k}{N}} + \sqrt{\frac{\log \frac{1}{\delta}}{N}}\right) \, .
\end{align*}
\end{proof}
We conclude this section by comparing Theorem~\ref{thm:general_rate} to Theorem~2.2 presented in \citep*{biau2008performance}. The form of the latter result is somewhat similar to our excess distortion bound. However, the proof of Theorem~2.2 contains an inaccuracy which, to the best of our understanding, cannot be immediately fixed. The problem in the proof comes from the application of Corollary~2.1 in \citep{biau2008performance} which requires that the centres belong to the set $\mathcal{A}_{M}^k$ (which is called $\mathcal{F}_M^k$ there) and also that the observations $X_1, \ldots, X_N$ are in a bounded domain with probability one. The last fact does not hold for the unbounded distributions considered there (recall our Remark~\ref{importantremark}). Fortunately, with additional technical efforts and by replacing the empirically optimal quantizer with our MOM minimizer, we achieve the result even stronger in a manner than one claimed in Theorem~2.2 by \citeauthor{biau2008performance}
\section{Discussion}
\label{Discussions}
Finally, we discuss several previous results related to clustering for heavy-tailed distributions as well as directions for future work.
The results of \cite*{brownlees2015empirical} are only presented for $k$-medians (where the distortion is defined as $D(A) = \E \min\limits_{a \in A} \norm{X - a}$ instead of $D(A) = \E \min\limits_{a \in A} \norm{X - a}^2$). However, we believe that their techniques, at least if applied straightforwardly, would require $\E \norm{X}^4 < \infty$. Similarly, \citeauthor{brecheteau2018robust} require $\E \norm{X}^2 < \infty $, but the targeted quantizer is different from ours.
Our Theorem~\ref{thm:simpleupperm} only requires $\E \norm{X}^2 < \infty$ and is valid for any separable Hilbert space, whereas Theorem~11 in \citep{brownlees2015empirical} depends explicitly on the dimension and has a worse dependence on the $\log \frac{1}{\delta}$-term.
The uniform bounds in \citep{telgarsky2013moment, bachem2017uniform} provide uniform convergence bounds under $\E \norm{X}^4 < \infty$ in ${\mathbb R}^d$ that cannot be immediately converted into the excess distortion bounds similar to ours.
Since these uniform bounds are tuned to the analysis of empirically optimal quantizers, they obviously have a suboptimal dependence on the confidence parameter $\delta$.
A natural course of further research is to introduce some favorable assumptions on the distribution $P$ leading to the so-called \emph{fast rates} for the excess distortion. These are the excess distortion bounds scaling as $O\left(\frac{1}{N}\right)$ instead of $O\left(\frac{1}{\sqrt{N}}\right)$ which, of course, cannot be obtained for free \citep{antos2005improved}. By now, these assumptions and their analysis are well-understood in the bounded case (see \citep{levrard2015nonasymptotic} and references therein).
Another interesting direction is to sharpen our bounds and make our robust algorithms more practical. As already mentioned, we believe that making some assumptions on $p_{\min}$ and thus restricting the sizes of clusters is somewhat more natural than assuming that $M$ is known in advance. Finally, it is natural to further extend our main results to the situation where the adversarial corruption of the observations is allowed. We refer to \citep{minsker2018uniform} for the related techniques.
\subsection*{Acknowledgements}
We would like to thank Olivier Bachem for stimulating discussions, G\'{a}bor Lugosi for a valuable feedback and Marco Cuturi and Nikita Puchkin for providing several important references. We are also thankful to the three anonymous referees for their useful comments and suggestions.
The work of Alexey Kroshnin was conducted within the framework of the HSE University Basic Research Program.
Results of Section 4 have been obtained under support of the RSF grant No. 19-71-30020.
| {
"timestamp": "2020-11-04T02:24:42",
"yymm": "2002",
"arxiv_id": "2002.02339",
"language": "en",
"url": "https://arxiv.org/abs/2002.02339",
"abstract": "We consider the robust algorithms for the $k$-means clustering problem where a quantizer is constructed based on $N$ independent observations. Our main results are median of means based non-asymptotic excess distortion bounds that hold under the two bounded moments assumption in a general separable Hilbert space. In particular, our results extend the renowned asymptotic result of Pollard, 1981 who showed that the existence of two moments is sufficient for strong consistency of an empirically optimal quantizer in $\\mathbb{R}^d$. In a special case of clustering in $\\mathbb{R}^d$, under two bounded moments, we prove matching (up to constant factors) non-asymptotic upper and lower bounds on the excess distortion, which depend on the probability mass of the lightest cluster of an optimal quantizer. Our bounds have the sub-Gaussian form, and the proofs are based on the versions of uniform bounds for robust mean estimators.",
"subjects": "Statistics Theory (math.ST); Machine Learning (cs.LG)",
"title": "Robust $k$-means Clustering for Distributions with Two Moments",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808753491773,
"lm_q2_score": 0.7217432062975979,
"lm_q1q2_score": 0.7075110620467311
} |
https://arxiv.org/abs/0912.3568 | A Continuum Version of the Kunz-Souillard Approach to Localization in One Dimension | We consider continuum one-dimensional Schrödinger operators with potentials that are given by a sum of a suitable background potential and an Anderson-type potential whose single-site distribution has a continuous and compactly supported density. We prove exponential decay of the expectation of the finite volume correlators, uniform in any compact energy region, and deduce from this dynamical and spectral localization. The proofs implement a continuum analog of the method Kunz and Souillard developed in 1980 to study discrete one-dimensional Schrödinger operators with potentials of the form background plus random. | \section{Introduction}
In their work \cite{ks}, Kunz and Souillard introduced the first
method which allowed one to give a rigorous proof of localization for
the one-dimensional discrete Anderson model; see also \cite{cfks}
for a presentation of the method. Since then, other, more
versatile, methods have been developed which allowed one to establish
Anderson localization also for multi-dimensional and continuum
Anderson models. However, the Kunz-Souillard method has several
important features which still deserve interest. First, it
directly establishes a strong form of dynamical localization which
was proven with other methods only much later and with more effort.
Second, and most importantly, among the available methods which
establish one-dimensional localization, it is the only one which
shows localization at all energies, arbitrary disorder, and
without requiring ergodicity. It completely avoids the use of
Lyapunov exponents and provides an extremely direct path to
dynamical localization.
The virtues of the Kunz-Souillard methods were demonstrated in the
work \cite{desiso}, which proves localization for one-dimensional
discrete Anderson models with arbitrary bounded background
potential, and in \cite{simon}, which applies the method to
decaying random potentials.
It is our goal here to extend the Kunz-Souillard method to
one-dimensional {\it continuum} Anderson-type models and provide a
localization proof which allows for a rather general class of
bounded, not necessarily periodic, background potentials.
We mention one earlier work of Royer \cite{r} which carried over
many of the features of the Kunz-Souillard method to a continuum
model. There the Brownian motion generated random potential from
the work \cite{gmp} by Goldsheid, Molchanov and Pastur is considered. The
argument in \cite{r} does not carry over to Anderson-type models
and does not include deterministic background potentials.
Specifically, we consider the random Schr\"odinger operator
\begin{equation} \label{model}
H_{\omega} = -\frac{d^2}{dx^2} + W_0(x) + V_{\omega}(x)
\end{equation}
in $L^2({\mathbb R})$, where the random potential is given by
\begin{equation} \label{randompot}
V_{\omega}(x) = \sum_{n\in {\mathbb Z}} \omega_n f(x-n).
\end{equation}
For the single-site potential $f$, the random
coupling constants $\omega_n$, and the background potential $W_0$,
we fix the following assumptions:
The coupling constants $ \omega = (\omega_n)_{n\in{\mathbb Z}}$ are i.i.d.\
real random variables, whose distribution has a continuous and
compactly supported density $r$.
For the single-site potential $f$, we assume that
\begin{equation} \label{singlesite1}
c \chi_I(x) \le f(x) \le C \chi_{[-1,0]}
\end{equation}
for constants $0< c \le C < \infty$ and a non-trivial subinterval
$I$ of $[-1,0]$. We also assume that there exists an interval
$[a,b] \subset [-1,0]$ such that
\begin{equation} \label{singlesite2}
f(x)>0 \:\mbox{for a.e.}\: x\in [a,b], \quad f(x)=0 \:\mbox{for}\: x\in [-1,0]\setminus [a,b].
\end{equation}
The background potential $W_0$ is bounded and such that
\begin{equation} \label{relcomp}
\{ W_0(\cdot-n)|_{[-1,0]}: n\in {\mathbb Z}\} \: \mbox{is relatively compact in}\: L^{\infty}(-1,0).
\end{equation}
This includes 1-periodic potentials $W_0$, but, much more
generally, allows for many situations where $H_{\omega}$ is not
ergodic with respect to 1-shifts, examples being
(i) almost periodic potentials $W_0$, that is, $\{W_0(\cdot-\tau):\tau \in {\mathbb R}\}$ is relatively compact in $L^{\infty}({\mathbb R})$,
(ii) any $W_0$ that is uniformly continuous on ${\mathbb R}$, in which case \eqref{relcomp} follows from equicontinuity and the Arzela-Ascoli theorem.
Let $\chi_x = \chi_{[x-1,x]}$ and, for $x,y \in {\mathbb Z}$ and $E_{max} >0$, define
\begin{align}
\label{correlators} \rho(x,y;E_{max}) & := {\mathbb E} \Big( \sup\{
\|\chi_x g(H_{\omega}) \chi_y\|: \, g:{\mathbb R}\to {\mathbb C} \:\mbox{Borel
measurable}, \\
\nonumber & \qquad |g|\le 1, \: \mbox{supp}\,g \subset
[-E_{max},E_{max}]\} \Big).
\end{align}
Our main result is
\begin{theorem} \label{main}
For every $E_{max} >0$, there exist $C<\infty$ and $\eta>0$ such that for all $x,y \in {\mathbb Z}$,
\begin{equation} \label{dynloc}
\rho(x,y;E_{max}) \le Ce^{-\eta|x-y|}.
\end{equation}
\end{theorem}
Let us discuss some immediate consequences of this result. First, Theorem~\ref{main} implies a strong form of dynamical localization. Indeed, choosing $g_t(E) = e^{-itE} \chi_{[-E_{max},E_{max}]}(E)$, we find that
$$
{\mathbb E} \Big( \sup_{t \in {\mathbb R}} \|\chi_x e^{-i t H_{\omega}} \chi_{[-E_{max},E_{max}]}(H_{\omega}) \chi_y\| \Big) \le Ce^{-\eta|x-y|}.
$$
Dynamical localization implies spectral localization. Explicitly, we can deduce the following result.
\begin{coro} \label{specloc}
For almost every $\omega$, $H_\omega$ has pure point spectrum with exponentially decaying eigenfunctions.
\end{coro}
We refer the reader to \cite{aenss} for an explicit derivation of this consequence and other interesting forms and signatures of localization from a statement like Theorem~\ref{main}. Note that spectral localization throughout the spectrum is obtained by taking a countable intersection over an unbounded sequence of values of $E_{max}$.
Spectral localization and a weaker form of dynamical localization with periodic background potential can be established by different methods for more general single-site distributions. The most general result, which merely assumes non-trivial bounded support, can be found in \cite{dss}.
It has also been shown recently in \cite{hss} how the fractional moment method can be used to show the exponentially decaying dynamical localization bound \eqref{dynloc} for the model \eqref{model}, \eqref{randompot}. This work requires periodicity of the background potential and the existence of a bounded and compactly supported density for the single-site distribution.
Given these results for the case of periodic background and the limitation of the methods used in their proofs to this particular choice of background, we see that the method we present here should be regarded as the primary tool to perturb a given Schr\"odinger operator by the addition of a (preferably small) random potential to generate pure point spectrum. In fact, a question of this kind posed by T.\ Colding and P.\ Deift triggered our work. Thus, as a sample application, we state the following corollary, which answers the specific question we were faced with.
\begin{coro} \label{ColdingDeift}
Given $W_0 : {\mathbb R} \to {\mathbb R}$ with $C^1$-norm $\|W_0\|_{\infty} + \|W_0'\|_{\infty} < \infty$, there exists a $V : {\mathbb R} \to {\mathbb R}$
with arbitrarily small $C^1$-norm such that $-d^2/dx^2 + W_0 +V$ has pure point spectrum.
\end{coro}
This is an immediate consequence of Theorem~\ref{main} and Corollary~\ref{specloc}, since $W_0$ is uniformly continuous and we may choose $V$ of the form (\ref{randompot}) with single-site potential $f$ of small $C^1$-norm and $\omega_n$ supported in $[0,1]$.
Apart from the ability to handle quite general background potentials, the flexibility of the Kunz-Souillard method manifests itself in the discrete case in another way: it allows one to prove powerful results for random decaying potentials; see Simon \cite{simon} and the follow-up paper \cite{dss2}. We have not been able to establish a continuum analogue of Simon's work using our method. The obstacle to doing this is in making Proposition~\ref{t1normthm} in Section~\ref{s.T122strongbound} quantitative. We regard it as an interesting open problem to establish such a quantitative estimate. \footnote{Incidentally, it was realized sixteen years after Simon's 1982 work that there is a different approach to random decaying potentials which is applicable in more general situations and which does extend easily to the continuum \cite{kls}. Another approach to continuum random decaying potentials was given in \cite{ku}.}
The remainder of the paper is organized as follows. Section~\ref{s.reduction} describes the overall strategy of the proof of Theorem~\ref{main}, adapting the strategy of \cite{ks} and culminating in Section~\ref{ss.endofproof}. In the remaining sections we prove various norm bounds on integral operators which are used in this argument, with the crucial contraction property being established in Section~\ref{s.T122strongbound}. A technical result on the large coupling limit of Pr\"ufer amplitudes, used in Section~\ref{s.T122strongbound}, is proven in Appendix~\ref{s.appA}, while Appendix~\ref{s.appB} summarizes a number of standard ODE facts.
\vspace{.3cm}
\noindent {\bf Acknowledgements:} We would like to thank B.~Simon for useful discussions at an early stage of this work, as well as T.~Colding and P.~Deift for posing a question which motivated us to finish it.
\section{Reduction to Integral Operator Bounds} \label{s.reduction}
\subsection{Finite Volume Correlators} \label{ss:fvcorrelators}
Consider finite-volume restrictions of $H_\omega$, that is, for $L
\in {\mathbb Z}_+$, we denote by $H_\omega^L$ the restriction of $H_\omega$
to the interval $[-L,L]$ with Dirichlet boundary conditions. For fixed $L$ we will use the abbreviation $\omega = (\omega_{-L+1}, \ldots, \omega_L)$, as these are the only coupling constants which $H_{\omega}^L$ depends on.
Moreover, for $x,y \in {\mathbb Z}$ and $E_{max} >0$, we introduce the {\it finite volume correlators}
$$
\rho_L(x,y;E_{max}) := {\mathbb E} \Big( \sup\{ \|\chi_x g(H_{\omega}^L)
\chi_y\|: \, |g|\le 1, \: \mbox{supp}\,g \subset
[-E_{max},E_{max}]\} \Big).
$$
\begin{lemma}
We have
$$
\rho(x,y;E_{max}) \le \liminf_{L \to \infty} \rho_L(x,y;E_{max}).
$$
\end{lemma}
\begin{proof}
See \cite[Eq.~(2.28)]{aenss} and its discussion there.
\end{proof}
We will often suppress the dependence of these quantities on
$E_{max}$ in what follows. Moreover, we can without loss of
generality restrict our attention to the case $x = 1$ and $y = n
\in {\mathbb Z}_+$. Thus, we aim to estimate $\rho(1,n)$ by means of
finding estimates for $\rho_L(1,n)$ that are uniform in $L$.
Explicitly, our goal is to show the following:
\begin{prop}\label{l.finitevolkeylemma}
There exist $C<\infty$ and $\eta>0$ such that, for all $n,L \in
{\mathbb Z}_+$ with $n \le L$, we have
\begin{equation} \label{dynloc2}
\rho_L(1,n) \le Ce^{-\eta n}.
\end{equation}
\end{prop}
In order to estimate $\rho_L(1,n)$, we consider the eigenfunction
expansion of $H_\omega^L$. Thus, for $L
\in {\mathbb Z}_+$ and $\omega = (\omega_{-L+1}, \ldots, \omega_L)$, we denote the (simple) eigenvalues of $H_\omega^L$ by
$\{ E_k : k \ge 1 \}$ and the associated normalized eigenvectors
by $\{ v_k : k \ge 1\}$. Here we leave the dependence of these
quantities on $\omega$ and $L$ implicit.
The proof of Proposition~\ref{l.finitevolkeylemma} starts with the
following observation:
\begin{align}
\label{e.rholest} \rho_L(1,n) & \le {\mathbb E} \left( \sum_{|E_k| \le
E_{max}} \|\chi_n v_k \| \cdot \|\chi_1 v_k\| \right) \\
\nonumber & = \sum_{k = 1}^\infty {\mathbb E} \left( \chi_{\{ \omega :
|E_k(\omega)| \le E_{max} \} } \|\chi_n v_k \| \cdot \|\chi_1 v_k\| \right) \\
\nonumber & = \sum_{k = 1}^\infty \int_{{\mathbb R}^{2L}} \chi_{\{ \omega :
|E_k(\omega)| \le E_{max} \} } \|\chi_n v_k \| \cdot \|\chi_1
v_k\| \prod_{j = -L+1}^L r(\omega_j) \, d\omega_j
\end{align}
In the next subsection we will rewrite the integral in the last line of
\eqref{e.rholest} by introducing a change of variables based on
the Pr\"ufer phase of the eigenfunctions $v_k$ at the integer
sites in $[-L,L]$. The Jacobian of this change of variables will
be computed in Subsection~\ref{ss.jacobian}. This will then lead
to a formula for $\rho_L(1,n)$ involving integral operators, which
will be made explicit in Subsection~\ref{ss.intop}.
\subsection{Change of Variables} \label{sec:changevar}
In this subsection we introduce the change of variables according
to which we will rewrite the integrals appearing in the last line
of \eqref{e.rholest}.
To introduce Pr\"ufer variables, let $u_{-L}(\cdot,\omega,E)$ be the solution of $-u''+(W_0+V_{\omega})u=Eu$ satisfying $u(-L)=0$ and $u'(-L)=1$. The corresponding Pr\"ufer phase $\varphi_{-L}(\cdot,\omega,E)$ and amplitude $R_{-L}(\cdot,\omega,E)$ are defined by
$$
u_{-L} = R_{-L} \sin \varphi_{-L}, \quad u_{-L}' = R_{-L} \cos \varphi_{-L}
$$
normalized so that $\varphi_{-L}(-L)=0$ and $\varphi_{-L}(\cdot,\omega,E)$ continuous to get uniqueness of the phase. The Pr\"ufer phase satisfies the first order equation (cf.\ Lemma~\ref{lem:xder})
\begin{equation} \label{e.phaseDE}
\varphi_{-L}' = 1 - (1+W_0+V_{\omega}-E) \sin^2 \varphi_{-L}.
\end{equation}
Fix an $M>0$ such that
\begin{equation}\label{f.Mdef}
\mathrm{supp} \, r \subseteq [-M,M].
\end{equation}
Thus
\begin{equation} \label{e.shift1}
\|\varphi'\|_\infty \le 2 + \|W_0\|_{\infty} + M \|f\|_\infty +E_{max} < \infty.
\end{equation}
Choose $N \in {\mathbb Z}_+$ such that
\begin{equation} \label{e.shift2}
2 + \| W_0\|_\infty
+ M \|f\|_\infty + E_{max} < N \pi.
\end{equation}
Hence the change of the Pr\"ufer phase over any interval of length
one is uniformly bounded in absolute value by $N
\pi$.\footnote{Actually, we need $N \pi$ to bound the
\emph{growth} of the Pr\"ufer phase over a unit interval, but it
follows from the differential equation that it can never decrease
by more than $\pi$.}
With the circle
$$
{\mathbb T}_N := {\mathbb R} / (2 \pi N {\mathbb Z})
$$
and
$$
\Omega :=\left\{ (\omega,k) \in [-M,M]^{2L} \times {\mathbb Z}_+ \,:\; E_k(\omega) \in [-E_{max},E_{max}] \right\}
$$
we can now define our change of variables
\begin{eqnarray*}
\mathcal{C} \quad : \quad \Omega \quad
& \longrightarrow & {\mathbb T}_N^{2L-1} \times
[-E_{max} , E_{max}] \times \{ 0 , \ldots , 2N-1 \} \\
(\omega, k) & \mapsto & (\theta_{-L+1}
, \ldots , \theta_{L-1} , E , j )
\end{eqnarray*}
as follows:
\begin{itemize}
\item For $i = -L + 1, \ldots, L-1$, $\theta_i \in {\mathbb T}_N$ is chosen so that
$$
\varphi_{-L}(i,\omega,E_k(\omega)) \equiv \theta_i \mod 2\pi N.
$$
\item $E \in [-E_{max} , E_{max}]$ is given by
$$
E = E_k(\omega).
$$
\item Finally, $j \in \{ 0 , \ldots , 2N-1 \}$ is defined so that
$$
k \equiv j \mod 2N.
$$
\end{itemize}
\begin{lemma}\label{l.cis11}
The change of variables $\mathcal{C}$ is one-to-one.
\end{lemma}
\begin{proof}
The key point is that the Pr\"ufer phase of the $k$-th
eigenfunction of $H_\omega$ runs from $0$ (at $-L$) to $k \pi$ (at
$L$), i.e.\ $\varphi_{-L}(L,\omega,E_k(\omega)) =k\pi$. Since we are taking phases modulo $2N\pi$, we need to ensure
that no ambiguities are generated. The desired uniqueness follows
from our choice of $N$, which can be seen as follows.
Suppose that $(\theta_{-L+1}, \ldots, \theta_{L-1}, E, j)$ belongs to the range of $\mathcal{C}$ and that $\mathcal{C}(\omega,k) = (\theta_{-L+1}, \ldots, \theta_{L-1}, E, j)$, $(\omega,k) \in \Omega$. Let $\theta_{-L} :=0$, $\theta_{L} :=j\pi$. Thus, by definition of $\mathcal{C}$, for $i=-L+1,\ldots, L$,
\begin{equation} \label{e.Cinv}
\varphi_{-L}(i,\omega,E) \equiv \theta_i \mod 2\pi N.
\end{equation}
This determines $\omega_i$, $i=-L+1,\ldots,L$, uniquely, as is seen iteratively in $i$: For fixed $i$, $\varphi_{-L}(i-1,\omega,E)$ is determined by $\omega_{-L+1}$, \ldots, $\omega_{i-1}$ and, by \eqref{e.shift1} and \eqref{e.shift2},
$$
|\varphi_{-L}(i,\omega,E) - \varphi_{-L}(i-1,\omega,E)| < N\pi.
$$
We also know from Appendix~\ref{s.appB} (cf.\ Lemma~\ref{lem:lamderphi}) that $\varphi_{-L}(i,\omega,E)$ is strictly decreasing in $\omega_i$. Therefore, given $\omega_{-L+1}, \ldots, \omega_{i-1}$, there can be at most one $\omega_i \in [-M,M]$ satisfying \eqref{e.Cinv}.
Finally, with the unique values of $\omega_{-L+1}$, \ldots, $\omega_L$ reconstructed, $k \in {\mathbb Z}_+$ is uniquely determined by $\varphi_{-L}(L,\omega,E)=k\pi$.
\end{proof}
Now we carry out the change of variables in \eqref{e.rholest} and consider the
resulting integral.
Similarly to the definition of $u_{-L}$, let $u_L(\cdot,\omega,E)$ be the solution of $-u''+(W_0+V_{\omega})u=Eu$ determined by $u_L(L)=0$, $u'_L(L)=1$, with corresponding Pr\"ufer variables $\varphi_L$ and $R_L$, where $\varphi_L(L)=0$.
For $g\in L^{\infty}([-1,0])$ and $E, \lambda , \theta , \eta \in {\mathbb R}$, let $u_0(\cdot , \theta,
\lambda, g, E)$ be the unique solution of
\begin{equation}\label{geve}
-u'' + gu + \lambda fu = Eu
\end{equation}
with $u_0'(0) = \cos \theta$, $u_0(0) = \sin \theta$ and let
$u_{-1}(\cdot , \eta, \lambda, g, E)$ be the unique solution of
\eqref{geve} with $u_{-1}'(-1) = \cos \eta$, $u_{-1}(-1) = \sin
\eta$.
Let $\varphi_0(\cdot , \theta , \lambda , g, E)$ and $R_0(\cdot ,
\theta , \lambda , g, E)$ be the Pr\"ufer phase and amplitude,
respectively, for $u_0(\cdot , \theta , \lambda , g, E)$.
Similarly, let
$\varphi_{-1}(\cdot , \eta , \lambda , g, E)$ and
$R_{-1}(\cdot , \eta , \lambda , g, E)$ be the Pr\"ufer variables for $u_{-1}(\cdot , \eta,
\lambda, g, E)$.
For the next definition, in order to make use of \eqref{e.shift1} and \eqref{e.shift2}, we assume $\|g\|_{\infty} \le \|W_0\|_{\infty}$ and $|E|\le E_{max}$.
Note that $\varphi_0(\cdot , \theta+\pi , \lambda , g, E) = \varphi_0(\cdot , \theta , \lambda , g, E) + \pi$ and $\varphi_{-1}(\cdot , \eta+\pi , \lambda , g, E) = \varphi_{-1}(\cdot , \eta , \lambda , g, E)+\pi$. Thus, in particular, $\varphi_0(-1 , \cdot , \lambda , g, E)$ and $\varphi_{-1}(0 , \cdot , \lambda , g, E)$ induce well-defined mappings from ${\mathbb T}_N$ to ${\mathbb T}_N$, which is how we use them below.
If $\alpha, \beta \in {\mathbb T}_N$ are such that there exists a coupling
constant $\lambda \in [-M,M]$ with $\varphi_0(-1,\alpha,\lambda,g,E) = \beta$
(or, equivalently, $\varphi_{-1}(0,\beta,\lambda,g,E) = \alpha$), we
define $\lambda(\beta,\alpha,g,E) = \lambda$. Note that by \eqref{e.shift1} and \eqref{e.shift2}, using the same argument as in the proof of Lemma~\ref{l.cis11}, this $\lambda$ is uniquely determined if it exists.
Finally, write $f_i := f(\cdot -i)$ and $g_i := W_0(\cdot-i)$.
\begin{lemma}\label{l.aftercov}
We have
$$
\rho_L(1,n) \le \int_{-E_{max}}^{E_{max}} \rho_L(1,n,E) \, dE,
$$
where, for $E \in [-E_{max}, E_{max}]$, we write
\begin{align} \label{e.fixenergy}
\rho_L(1,n,E) := \sum_{j=0}^{2N-1} \int_{{\mathbb T}_N^{2L-1}} &
r(\lambda(\theta_{L-1}, j\pi, g_L, E)) \cdot r(\lambda(0,
\theta_{-L+1}, g_{-L+1}, E)) \\
& \left( \prod_{i - -L+2}^{L-1} r(\lambda(\theta_{i-1}, \theta_i,
g_i, E)) \right) \left( \int_{n-1}^n u_{L}^2 \right)^{1/2} \left(
\int_0^1 u_{L}^2 \right)^{1/2} \nonumber \\
& \frac{R_{-L}^2(-L+1) \cdots R_{-L}^2(0) \cdot R_{L}^2(1) \cdots
R_{L}^2(L-1)}{\int f_{-L+1} u_{-L}^2 \cdots \int f_{0} u_{-L}^2
\cdot \int f_{1} u_{L}^2 \cdots \int f_{L} u_{L}^2} \nonumber \\
& d \theta_{-L+1} \cdots d \theta_{L-1} \nonumber
\end{align}
and interpret $r(\lambda(\cdots))$ as zero if $\lambda(\cdots)$
does not exist. Here the argument $\omega$ in the functions $u_{\pm L} = u_{\pm L}(\cdot,\omega,E)$ and $R_{\pm L} = R_{\pm L}(\cdot,\omega,E)$ is the one uniquely determined via Lemma~\ref{l.cis11} by $\theta_{-L+1}$, \ldots, $\theta_{L}$, $E$ and $j$.
\end{lemma}
\begin{proof}
For $k \in {\mathbb Z}_+$, let
$$
\Omega_k := \{ \omega \in [-M,M]^{2L} : |E_k(\omega)| \le E_{max} \}
$$
and write
$$
A_k := \int_{{\mathbb R}^{2L}} \chi_{\Omega_k} \|\chi_n v_k\| \|\chi_1 v_k\| \prod_{i=-L+1}^L r(\omega_i)\,d\omega_i.
$$
On $\Omega_k$ we change variables by the map
\begin{eqnarray*}
\mathcal{C}_k \quad : \quad \Omega_k & \longrightarrow &
{\mathbb T}_N^{2L-1} \times
[-E_{max} , E_{max}] \\
\omega & \mapsto & (\theta_{-L+1} ,
\ldots , \theta_{L-1} , E_k(\omega)).
\end{eqnarray*}
Let $J_k = \partial \mathcal{C}_k/\partial \omega$ be its Jacobian. Pick $j\in \{0,\ldots,2N-1\}$ so that $j\equiv k \mod 2N$. Noting that, in terms of the new variables on $\mathcal{C}_k(\Omega_k)$,
\[ v_k = \frac{u_L(\cdot,\omega(\theta_{-L+1},\ldots,\theta_L,E,j),E)}{\|u_L(\cdot,\omega(\theta_{-L+1},\ldots,\theta_L,E,j),E)\|},\]
we get
\begin{eqnarray*}
A_k & = & \int_{-E_{max}}^{E_{max}} \int_{{\mathbb T}_N^{2L-1}} \chi_{\mathcal{C}_k(\Omega_k)} |\det J_k|^{-1} r(\lambda(0,\theta_{-L+1},g_{-L+1},E)) \\
& & \left( \prod_{i=-L+2}^{L-1} r(\lambda(\theta_{i-1}, \theta_i, g_i, E)) \right) r(\lambda(\theta_{L-1},j\pi, g_L,E)) \\
& & \frac{\left(\int_{n-1}^n u_L^2\right)^{1/2} \left(\int_0^1 u_L^2 \right)^{1/2}}{\int_{-L}^L u_L^2} \,d\theta_{-L+1} \ldots d\theta_{L-1}\,dE.
\end{eqnarray*}
The Jacobian determinant is calculated in Lemma~\ref{l.jacobidet} of the next subsection. Inserting the result yields
\begin{eqnarray*}
A_k & = & \int_{-E_{max}}^{E_{max}} \int_{{\mathbb T}_N^{2L-1}} \chi_{\mathcal{C}_k(\Omega_k)} r(\lambda(\theta_{L-1},j\pi, g_L,E)) r(\lambda(0,\theta_{-L+1},g_{-L+1},E)) \\
& & \left( \prod_{i=-L+2}^{L-1} r(\lambda(\theta_{i-1}, \theta_i, g_i, E)) \right) \left(\int_{n-1}^n u_L^2\right)^{1/2} \left(\int_0^1 u_L^2 \right)^{1/2} \\
& & \frac{R_{-L}^2(-L+1) \ldots R_{-L}^2(0) R_L^2(1) \ldots R_L^2(L-1)}{\int f_{-L+1} u_{-L}^2 \ldots \int f_0 u_{-L}^2 \int f_1 u_L^2 \ldots \int f_L u_L^2} \, d\theta_{-L+1} \ldots d\theta_{L-1}\,dE.
\end{eqnarray*}
Let $k_1 \not= k_2$ and $j_1$, $j_2 \in \{0,\ldots,2N-1\}$ with $j_{\ell} \equiv k_{\ell} \mod 2N$. Then Lemma~\ref{l.cis11} says that
$$
\left( \mathcal{C}_{k_1}(\Omega_{k_1}) \times \{j_1\} \right) \cap \left( \mathcal{C}_{k_2}(\Omega_{k_2}) \times \{j_2\} \right) = \emptyset.
$$
Thus it follows that $\sum_k A_k \le \int_{-E_{max}}^{E_{max}} \rho_L(1,n,E)$, with $\rho_L(1,n,E)$ defined in \eqref{e.fixenergy}. But by \eqref{e.rholest} we have $\rho_L(1,n) \le \sum_k A_k$, which completes the proof.
\end{proof}
\subsection{Calculation of the Jacobian}\label{ss.jacobian}
It will turn out that the Jacobians arising above, up to constant row and column multipliers, have the simple structure considered in the next lemma.
\begin{lemma}\label{l.detformula}
We have
$$
\det \begin{pmatrix} a_1 & a_1 & a_1 & \cdots & a_1 & a_1
\\ b_2 & a_2 & a_2 & \cdots & a_2 & a_2 \\
b_3 & b_3 & a_3 & \cdots & a_3 & a_3 \\
\vdots & \vdots & \vdots & & \vdots & \vdots\\
b_n & b_n & b_n & \cdots & b_n & a_n \end{pmatrix} = a_1 (a_2 -
b_2)(a_3 - b_3) \cdots (a_n - b_n).
$$
\end{lemma}
\begin{proof}
Observe that
\begin{align*}
\det \begin{pmatrix} a_1 & a_1 & a_1 & \cdots & a_1 & a_1
\\ b_2 & a_2 & a_2 & \cdots & a_2 & a_2 \\
b_3 & b_3 & a_3 & \cdots & a_3 & a_3 \\
\vdots & \vdots & \vdots & & \vdots & \vdots\\
b_n & b_n & b_n & \cdots & b_n & a_n \end{pmatrix} & = \det
\begin{pmatrix} a_1 & a_1 & a_1 & \cdots & a_1 & a_1
\\ b_2 - a_2 & 0 & 0 & \cdots & 0 & 0 \\
b_3 & b_3 & a_3 & \cdots & a_3 & a_3 \\
\vdots & \vdots & \vdots & & \vdots & \vdots\\
b_n & b_n & b_n & \cdots & b_n & a_n \end{pmatrix} \\
& = (a_2 - b_2) \det \begin{pmatrix} a_1 & a_1 & a_1 & \cdots &
a_1 & a_1
\\ b_3 & a_3 & a_3 & \cdots & a_3 & a_3 \\
b_4 & b_4 & a_4 & \cdots & a_4 & a_4 \\
\vdots & \vdots & \vdots & & \vdots & \vdots\\
b_n & b_n & b_n & \cdots & b_n & a_n \end{pmatrix}
\end{align*}
and then obtain the result by iteration (or induction).
\end{proof}
\begin{lemma}\label{l.jacobidet}
With the conventions for the arguments of $u_{\pm L}$ and $R_{\pm L}$ made in Lemma~\ref{l.aftercov} we have
$$
\det J_k = \frac{\int f_{-L+1} u_{-L}^2 \cdots \int f_{0} u_{-L}^2
\cdot \int f_{1} u_{L}^2 \cdots \int f_{L} u_{L}^2}{R_{-L}^2(-L+1)
\cdots R_{-L}^2(0) \cdot R_{L}^2(1) \cdots R_{L}^2(L-1)} \left(
\int_{-L}^L u_L^2 \right)^{-1}.
$$
\end{lemma}
\begin{proof}
With a slight adjustment of the notation introduced above, we have for $i = -L+1 , \ldots ,
0$,
\begin{equation} \label{e.isonleft}
\theta_i \equiv \varphi_{-L}(i,0,(\omega_{-L+1} , \ldots ,
\omega_i),E_k(\omega)) \mod 2\pi N,
\end{equation}
and for $i = 1 , \ldots , L-1$,
\begin{equation} \label{e.isonright}
\theta_i \equiv \varphi_{L}(i, j\pi, (\omega_{i+1} , \ldots ,
\omega_L),E_k(\omega)) \mod 2\pi N.
\end{equation}
The notational adjustment made here consists in stressing that $\varphi_{-L}(i)$ depends explicitly on $\omega_n$ only for $n=-L+1,\ldots,i$ (we only need to know the potential on $[-L,i]$ to calculate it), while it depends on all $\omega_n$ implicitly through $E_k(\omega)$. Similar reasoning applies to $\varphi_L(i)$ in \eqref{e.isonright}.
Thus, using \eqref{e.isonright} for $1 \le i \le L-1$ and $n \le i$, we have by Corollary~\ref{cor:Ederphi} and the Feynman-Hellmann formula
\begin{align*}
\frac{\partial \theta_i}{\partial \omega_n} & = \frac{\partial
\varphi_L(i)}{\partial E} \cdot \frac{\partial E_k}{\partial
\omega_n} \\
& = - \frac{1}{R_L^2(i)} \int_i^L u_L^2 \cdot \int f_n v_k^2 \\
& = - \frac{1}{R_L^2(i)} \frac{\int_i^L u_L^2}{\int_{-L}^L u_L^2}
\cdot \int f_n u_L^2;
\end{align*}
while for $1 \le i \le L-1$ and $n > i$, we have
\begin{align*}
\frac{\partial \theta_i}{\partial \omega_n} & = \frac{\partial
\varphi_L(i)}{\partial \omega_n} + \frac{\partial
\varphi_L(i)}{\partial E} \cdot \frac{\partial E_k}{\partial
\omega_n} \\
& = \frac{1}{R_L^2(i)} \int f_n u_L^2 - \frac{1}{R_L^2(i)} \int_i^L u_L^2 \cdot \int f_n v_k^2 \\
& = \frac{1}{R_L^2(i)} \int f_n u_L^2 - \frac{1}{R_L^2(i)}
\frac{\int_i^L u_L^2}{\int_{-L}^L u_L^2} \cdot \int f_n u_L^2 \\
& = \frac{1}{R_L^2(i)} \frac{\int_{-L}^i u_L^2}{\int_{-L}^L u_L^2}
\cdot \int f_n u_L^2.
\end{align*}
Analogous formulae, based on \eqref{e.isonleft}, hold in the case $ -L+1 \le i \le 0$. In this case we get for $n>i$ that
$$
\frac{\partial \theta_i}{\partial \omega_n} = \frac{1}{R^2_{-L}(i)} \frac{\int_{-L}^i u_{-L}^2}{\int_{-L}^L u_{-L}^2} \int f_n u_{-L}^2
$$
and, for $n\le i$,
$$
\frac{\partial \theta_i}{\partial \omega_n} = - \frac{1}{R_{-L}^2(i)} \frac{\int_i^L u_{-L}^2}{\int_{-L}^L u_{-L}^2} \int f_n u_{-L}^2.
$$
Also, writing $E$ as the first of the new variables, the first row of the Jacobian has entries $\partial E/\partial \omega_n = \int f_n v_k^2 = \int f_n u_L^2 / \int_{-L}^L u_L^2$.
Using these formulae and factoring out common factors in rows and
columns, we find that
$$
\det J_k = \frac{\int f_{-L+1} u_L^2 \cdots \int f_L
u_L^2}{R_{-L}^2(-L+1) \cdots R_{-L}^2(0) \cdot R_L^2(1) \cdots
R_L^2(L-1)} \left( \int_{-L}^L u_L^2 \right)^{-2L} \cdot \det A
$$
where
$$
A = \begin{pmatrix} 1 & 1 & 1 & \cdots & 1 & 1 & 1
\\ -I_{-L+1 , L}^- & I_{-L,-L+1}^- & I_{-L,-L+1}^- & \cdots & \cdots & I_{-L,-L+1}^- & I_{-L,-L+1}^- \\
-I_{-L+2 , L}^- & -I_{-L+2 , L}^- & I_{-L , -L+2}^- & \cdots & \cdots & I_{-L , -L+2}^- & I_{-L , -L+2}^- \\
\vdots & \vdots & \vdots & & \vdots & \vdots & \vdots \\
-I_{0 , L}^- & \cdots & -I_{0 , L}^- & I_{-L , 0}^- & I_{-L ,
0}^- & \cdots & I_{-L , 0}^- \\
-I_{1 , L}^+ & \cdots & -I_{1 , L}^+ & -I_{1 ,
L}^+ & I_{-L , 1}^+ & \cdots & I_{-L , 1}^+ \\
\vdots & \vdots & \vdots & & \vdots & \vdots & \vdots \\
-I_{L-1 , L}^+ & -I_{L-1 , L}^+ & -I_{L-1 , L}^+ & \cdots &
-I_{L-1 , L}^+ & -I_{L-1 , L}^+ & I_{-L , L-1}^+
\end{pmatrix}
$$
and
$$
I_{m,l}^\pm := \int_m^l u_{\pm L}^2.
$$
Applying Lemma~\ref{l.detformula}, we obtain
\begin{eqnarray*}
\det A & = & (I_{-L,-L+1}^- + I_{-L+1 , L}^-) \cdots (I_{-L ,0}^-
+ I_{0 , L}^-) \\ & & \cdot (I_{-L , 1}^+ + I_{1 , L}^+) \cdots (I_{-L ,
L-1}^+ + I_{L-1 , L}^+) \\
& = & \left( \int_{-L}^L u_{-L}^2 \right)^{L} \cdot \left(
\int_{-L}^L u_L^2 \right)^{L-1}.
\end{eqnarray*}
Plugging this into the formula for $\det J_k$ obtained above, the
result follows.
\end{proof}
\subsection{The Integral Operator Formula}\label{ss.intop}
The expression in Lemma~\ref{l.aftercov} may be written in a more
succinct form once we have introduced a number of quantities.
If, for given $g \in L^\infty(-1,0)$ with $\|g\|_{\infty} \le \|W_0\|_{\infty}$, $E \in [-E_{max} , E_{max}]$ and $\beta, \alpha \in {\mathbb T}_N$, $\lambda(\beta,\alpha, g,E)$ as defined in Section~\ref{sec:changevar} exists, we
write
\begin{eqnarray} \label{upmdef}
u_+(\cdot, \beta, \alpha, g,E) & := & u_0(\cdot , \alpha, \lambda(\beta,
\alpha, g,E), g,E), \\ u_-(\cdot, \beta, \alpha, g,E) & := &
u_{-1}(\cdot , \beta, \lambda(\beta, \alpha, g,E), g,E) \nonumber
\end{eqnarray}
and
\begin{eqnarray} \label{Rpmdef}
R_+(\cdot, \beta, \alpha, g,E) & := & R_0(\cdot , \alpha, \lambda(\beta,
\alpha, g,E), g,E), \\ R_-(\cdot, \beta, \alpha, g,E) & := &
R_{-1}(\cdot , \beta, \lambda(\beta, \alpha, g,E), g,E). \nonumber
\end{eqnarray}
For later use, note that
\begin{align}\label{upmrel}
u_+^2 (\cdot, \beta , \alpha,g,E) & = R_+^2 (-1,\beta,\alpha,g,E) \cdot
u_-^2(\cdot,\beta,\alpha,g,E) \\
\nonumber & = R_-^{-2} (0,\beta,\alpha,g,E) \cdot
u_-^2(\cdot,\beta,\alpha,g,E).
\end{align}
We introduce the following integral operators defined on functions $F$ on ${\mathbb T}_N$:
\begin{align*}
(T_0(g,E) F)(\beta) & = \int \frac{R_+^2(-1,\beta,\alpha,g,E)}{\int
f u_+^2(\cdot,\beta,\alpha,g,E)} r(\lambda(\beta,\alpha,g,E))
F(\alpha) \, d\alpha, \\
(\tilde T_0(g,E) F)(\alpha) & = \int
\frac{R_-^2(0,\beta,\alpha,g,E)}{\int f
u_-^2(\cdot,\beta,\alpha,g,E)} r(\lambda(\beta,\alpha,g,E))
F(\beta) \, d\beta, \\
(T_1(g,E) F)(\beta) & = \int \frac{R_+(-1,\beta,\alpha,g,E)}{\int f
u_+^2(\cdot,\beta,\alpha,g,E)} r(\lambda(\beta,\alpha,g,E))
F(\alpha) \, d\alpha,
\end{align*}
and, for $j = 0, \ldots, 2N-1$, the functions
\begin{align*}
(\Psi_j(g,E))(\theta) & = \frac{R_+^2(-1,\theta,j\pi,g,E)}{\int f
u_+^2(\cdot,\theta,j\pi,g,E)} r(\lambda(\theta,j\pi,g,E)), \\
(\Phi(g,E))(\theta) & = \frac{R_-^2(0,0,\theta,g,E)}{\int
f u_-^2(\cdot,0,\theta,g,E)} r(\lambda(0,\theta,g,E)).
\end{align*}
Now we are finally in a position to state the integral operator
formula, which bounds $\rho_L(1,n,E)$ from above.
\begin{lemma}
There exists a constant $C = C(E_{max})$ such that for every $E
\in [-E_{max} , E_{max}]$, we have
\begin{eqnarray} \label{e.intopformula}
\lefteqn{\rho_L(1,n,E)} \\ & \le & C \sum_{j = 0}^{2N-1} \Big\langle \tilde T_0(g_0,E)
\cdots \tilde T_0(g_{-L+2},E) \Phi(g_{-L+1},E) , \nonumber \\
& & T_1(g_1,E) \cdots
T_1(g_n,E) T_0(g_{n+1},E) \cdots T_0(g_{L-1},E) \Psi_j(g_L,E) \Big\rangle. \nonumber
\end{eqnarray}
Here $\langle \cdot, \cdot \rangle$ denotes the inner product on $L^2({\mathbb T}_N)$.
\end{lemma}
\begin{proof}
It follows from the a priori bounds in Lemma~\ref{lem:solest} that there exists a constant $C=C(E_{max})$ such that
$$
\int_{n-1}^n u_L^2 \le C R_L^2(n), \quad \int_0^1 u_L^2 \le C R_L^2(0)
$$
uniformly in $E\in [-E_{max},E_{max}]$. After using these bounds in the integrand on the right hand side of \eqref{e.fixenergy} we rearrange the terms in the integrand as
\begin{eqnarray*}
\lefteqn{ R_L(n) R_L(0) \frac{R_{-L}^2(-L+1) \cdots R_{-L}^2(0) R_L^2(1) \cdots R_L^2(L-1)}{ \int f_{-L+1} u_{-L}^2 \cdots \int f_0 u_{-L}^2 \int f_1 u_L^2 \cdots \int f_L u_L^2}} \\
& = & \left( \prod_{i=-L+1}^0 \frac{R_{-L}^2(i)}{\int f_i u_{-L}^2} \right) \left( \prod_{i=1}^n \frac{R_L(i-1)R_L(i)}{\int f_i u_L^2} \right) \left( \prod_{i=n+1}^L \frac{R_L^2(i-1)}{\int f_i u_L^2} \right).\nonumber
\end{eqnarray*}
Taking into account the scaling properties of Pr\"ufer amplitudes, we get the following relations between $u_{\pm L}$, $R_{\pm L}$ and $u_{\pm}$, $R_{\pm}$:
$$
\frac{R_{-L}^2(i)}{\int f_i u_{-L}^2} = \frac{R_-^2(0,\theta_{i-1},\theta_i,g_i,E)}{\int f u_-^2(\cdot,\theta_{i-1}, \theta_i,g_i,E)},
$$
$$
\frac{R_L^2(i-1)}{\int f_i u_L^2} = \frac{R_+^2(-1, \theta_{i-1}, \theta_i, g_i,E)}{\int f u_+^2(\cdot, \theta_{i-1}, \theta_i, g_i, E)},
$$
and
\begin{eqnarray*}
\frac{R_L(i-1) R_L(i)}{\int f_i u_L^2} & = & \frac{R_L^2(i-1)}{\int f_i u_L^2} \cdot \frac{R_L(i)}{R_L(i-1)} \\
& = & \frac{R_+^2(-1, \theta_{i-1}, \theta_i, g_i, E)}{\int f u_+^2 (\cdot, \theta_{i-1}, \theta_i, g_i, E)} \cdot \frac{1}{R_+(-1,\theta_{i-1}, \theta_i, g_i, E)} \\
& = & \frac{R_+(-1, \theta_{i-1}, \theta_i, g_i, E)}{\int f u_+^2(\cdot, \theta_{i-1}, \theta_i, g_i, E)}.
\end{eqnarray*}
Plugging all this into \eqref{e.fixenergy} and using the definitions of the integral operators $T_0$, $\tilde{T}_0$, $T_1$ as well as the functions $\Psi_j$ and $\Phi$, we obtain \eqref{e.intopformula}.
\end{proof}
\subsection{Proof of Proposition~\ref{l.finitevolkeylemma} and Theorem~\ref{main}} \label{ss.endofproof}
We are now in a position to describe how our main result Theorem~\ref{main} follows from norm bounds for the operators $T_1$, $T_0$ and $\tilde{T}_0$, which we will establish in the remaining sections of this paper.
As was explained in Section~\ref{ss:fvcorrelators}, it suffices to prove Proposition~\ref{l.finitevolkeylemma}. By Lemma~\ref{l.aftercov} it suffices to establish a bound
\begin{equation} \label{e.rho1nEbound}
\rho_L(1,n,E) \le Ce^{-\eta n}
\end{equation}
with constants $C<\infty$ and $\eta>0$ which are uniform in $E\in [-E_{max},E_{max}]$. For this we will use the integral formula \eqref{e.intopformula}.
Denote the norm of a linear operator $T$ from $L^p({\mathbb T}_N)$ to
$L^q({\mathbb T}_N)$ by $\| T \|_{p,q}$.
By Lemmas~\ref{l.T011bound} and \ref{l.T012bound} we have $\|T_0(g,E)\|_{1,1}=1$ as well as $\|T_0(g,E)\|_{1,2} \le C$ and $\|\Psi_j(g,E)\|_1 \le C$ uniformly in $E\in [-E_{max},E_{max}]$, $\|g\|_{\infty} \le \|W\|_{\infty}$ and $j=0,\ldots,2N-1$. Thus
$$
\| T_0(g_{n+1},E) \cdots T_0(g_{L-1},E) \Psi_j(g_L,E)\|_2 \le C
$$
uniformly in $E\in [-E_{max},E_{max}]$, $L\in {\mathbb N}$ and $j=0,\ldots,2N-1$. Similarly, also uniformly,
$$
\|\tilde{T}_0(g_0,E) \cdots \tilde{T}(g_{-L+2},E) \Phi(g_{-L+1},E)\|_2 \le C.
$$
This yields, by \eqref{e.intopformula} and Cauchy-Schwarz, that there is $C=C(E_{max},W_0,N)$ such that
$$
\rho_L(1,n,E) \le C \prod_{i=1}^n \|T_1(g_i,E)\|_{2,2}.
$$
In Section~\ref{s.T122strongbound} we will show that $\|T_1(g,E)\|_{2,2} < 1$ for every $g\in L^{\infty}(-1,0)$ and $E\in {\mathbb R}$. Finally, we establish in Section~\ref{s.T122continuity} that $\|T_1(g,E)\|_{2,2} = \|T_1(g-E,0)\|_{2,2}$ is continuous in $(g,E) \in L^{\infty}(-1,0) \times {\mathbb R}$. By assumption \eqref{relcomp} it is guaranteed that $\{g_i:i\in {\mathbb Z}\}$ is relatively compact in $L^{\infty}(-1,0)$ and thus $\{g_i:i\in {\mathbb Z}\} \times [-E_{max},E_{max}]$ is relatively compact in $L^{\infty}(-1,0)\times {\mathbb R}$. Thus
$$
\|T_1(g_i,E)\|_{2,2} \le \gamma < 1
$$
uniformly in $i$ and $E\in [-E_{max},E_{max}]$. This proves \eqref{e.rho1nEbound} with $\eta = \ln(1/\gamma)$.
\section{Elementary Results for the Integral Operators}
In this section we consider the integral operators introduced in Section~\ref{ss.intop} and establish several elementary results for them.
We begin with $T_0$ and $\tilde T_0$.
\begin{lemma} \label{l.T011bound}
We have
$$
\|T_0(g,E)\|_{1,1} = \|\tilde T_0(g,E)\|_{1,1} = 1.
$$
\end{lemma}
\begin{proof}
As $g$, $E$ are fixed here, we will suppress them in this proof.
Suppose $F \in L^1({\mathbb T}_N)$. Then,
\begin{eqnarray} \label{e.t0bound}
\|T_0 F\|_1 & \le & \int_{{\mathbb T}_N} \int_{{\mathbb T}_N} \frac{R_+^2(-1,\beta,\alpha)}{\int f
u_+^2(\cdot,\beta,\alpha)} r(\lambda(\beta,\alpha)) |F(\alpha)|
\, d\alpha \, d\beta \\
& = & \int_{{\mathbb T}_N} \int_{{\mathbb T}_N} \frac{R_+^2(-1,\beta,\alpha)}{\int f
u_+^2(\cdot,\beta,\alpha)} r(\lambda(\beta,\alpha))
\, d\beta \, |F(\alpha)| \, d\alpha. \nonumber
\end{eqnarray}
For fixed $\alpha$, $\lambda(\beta,\alpha)$ is strictly increasing in $\beta$ and the inverse function satisfies, see Lemma~\ref{lem:lamderphi}(b),
\begin{equation} \label{e.betalambda}
\frac{\partial \beta}{\partial \lambda} = - R_+^{-2}(-1,\beta,\alpha) \int_0^{-1} f u_+^2(\cdot,\beta,\alpha) =R_+^{-2}(-1,\beta,\alpha) \int_{-1}^0 f u_+^2(\cdot,\beta,\alpha).
\end{equation}
Thus we can change variables and find that the right hand side of \eqref{e.t0bound} is equal to
$$
\int_{{\mathbb T}_N} \int_{{\mathbb R}} r(\lambda) \, d\lambda \, |F(\alpha)| \, d\alpha = \|r\|_1 \|F\|_1 = \|F\|_1,
$$
where we also used the fact
that $r$ is the density of a probability distribution. This shows that $\|T_0\|_{1,1} \le 1$. Since the first
step in \eqref{e.t0bound} becomes an identity when $F \ge 0$, we get
$\|T_0\|_{1,1} = 1$.
Using
\begin{equation} \label{e.alphalambda}
\frac{d\alpha}{d\lambda} = - R_-^{-2}(0,\beta,\alpha) \int_{-1}^0 f
u_-^2(\cdot,\beta,\alpha).
\end{equation}
instead of \eqref{e.betalambda}, the proof of $\|\tilde T_0\|_{1,1} = 1$
is completely analogous.
\end{proof}
\begin{lemma} \label{l.T012bound}
We have
$$
\|T_0(g,E)\|_{1,2} \le C <\infty
$$
and
$$
\|\tilde T_0(g,E)\|_{1,2} \le C <\infty
$$
uniformly in $E\in [-E_{max},E_{max}]$ and $\|g\|_{\infty} \le \|W_0\|_{\infty}$.
Denoting by $\|\cdot\|_1$ the $L^1$-norm on ${\mathbb T}_N$, we also have
$$
\|\Psi_j(g,E)\|_1 \le C<\infty
$$
and
$$
\|\Phi(g,E)\|_1 \le C<\infty
$$
uniformly in $E\in [-E_{max},E_{max}]$, $\|g\|_{\infty} \le \|W_0\|_{\infty}$ and $j=0,\ldots, 2N-1$.
\end{lemma}
\begin{proof}
Lemmas~\ref{lem:solest} and \ref{lem:L2b} provide bounds $C_1<\infty$ and $C_2>0$ such that
\begin{equation} \label{auxbound1}
R_+^2(-1,\beta,\alpha) = R_0^2(-1,\alpha,\lambda(\beta,\alpha)) \le C_1 \end{equation}
and
\begin{equation} \label{auxbound2}
\int f u_+^2(\cdot,\beta,\alpha) = \int f u_0^2(\cdot,\alpha,\lambda(\beta,\alpha)) \ge C_2
\end{equation}
uniformly in $E\in [-E_{max},E_{max}]$, $\|g\|_{\infty} \le \|W_0\|_{\infty}$ and $\alpha, \beta$ such that $\lambda(\beta,\alpha) \in \mbox{supp}\,r$. In \eqref{auxbound2} we have also exploited the assumption \eqref{singlesite1} on the single site potential~$f$.
Thus we get for $F\in L^1({\mathbb T}_N)$ that
\begin{eqnarray*}
\|T_0 F\|_2^2 & = & \int_{{\mathbb T}_N} \left| \int_{{\mathbb T}_N} \frac{R_+^2(-1,\beta,\alpha)}{\int f u_+^2(\cdot,\beta,\alpha)} r(\lambda(\beta,\alpha)) F(\alpha)\,d\alpha \right|^2\,d\beta \\
& \le & 2\pi N (C_1/C_2)^2 \|r\|_{\infty}^2 \|F\|_1^2,
\end{eqnarray*}
resulting in the required norm bound for $T_0$. The bound for $\tilde{T}_0$ is found similarly.
From (\ref{auxbound1}) and (\ref{auxbound2}) we also get the bounds for $\Psi_j$ and $\Phi$, first in the $L^{\infty}$-norm and then in the $L^1$-norm since ${\mathbb T}_N$ has finite volume.
\end{proof}
Let us now turn to $T_1$. For $\alpha,\beta \in {\mathbb T}_N$, we write
\begin{equation}\label{t1kerdef}
T_1(\beta,\alpha) = \begin{cases} \frac{R_+(-1,\beta,\alpha)
r(\lambda(\beta,\alpha))}{\int f(x) u_+^2(x,\beta,\alpha) \, dx} &
\text{ if } \lambda(\beta,\alpha) \text{ exists,} \\
0 & \text{ otherwise}
\end{cases}
\end{equation}
for its integral kernel.
\begin{prop} \label{proposition1}
We have
\begin{align}
\label{t1kerper} & T_1(\beta+ \pi , \alpha + \pi) = T_1(\beta ,
\alpha) \text{ for all
} \beta,\alpha \in {\mathbb T}_N, \\
\label{t1kercont} & T_1(\cdot,\cdot) \text{ is continuous on }
{\mathbb T}_N^2.
\end{align}
\end{prop}
\begin{proof}
Note that $\varphi(x,\alpha+\pi,\lambda) =
\varphi(x,\alpha,\lambda) + \pi$ for every $x$. This follows from
$u_0(x,\alpha+\pi,\lambda) = - u_0(x,\alpha,\lambda)$ for every
$x$, along with the initial condition $\phi(0,\alpha,\lambda) =
\alpha$.
To derive \eqref{t1kerper} from this, consider a pair
$(\beta,\alpha)$ such that $\lambda(\beta,\alpha)$ exists. Then
$\lambda(\beta+\pi,\alpha+\pi)$ exists, too, and is equal to
$\lambda(\beta,\alpha)$. Moreover, it then follows readily from
the definition that $T_1(\beta+\pi,\alpha+\pi) =
T_1(\beta,\alpha)$.
To show \eqref{t1kercont}, we will need the following:
\begin{equation}\label{dopen}
D := \{ (\beta,\alpha) \in {\mathbb T}_N^2 : \lambda(\beta,\alpha) \text{
exists} \} \text{ is open and $\lambda (\cdot,\cdot)$ is
continuous on } D.
\end{equation}
To see this, fix some $(\beta,\alpha) \in D$ and $\varepsilon >
0$. Keep $\beta$ initially fixed and increase $\alpha$ a bit.
Clearly, there will still be a corresponding $\lambda$ that sends
$\beta$ to $\alpha + \delta_1$, $\delta_1 > 0$. Choose $\delta_1$
small enough so that $\lambda(\beta,\alpha + \delta_1) \le
\lambda(\beta,\alpha) + \frac{\varepsilon}{2}$. Similarly, keeping
$\alpha + \delta_1$ fixed and decreasing $\beta$ a bit, we find
$\delta_2
> 0$ so that $\lambda(\beta - \delta_2 , \alpha + \delta_1) \le
\lambda (\beta,\alpha) + \varepsilon$. Similarly, we can choose
$\delta_3,\delta_4
> 0$ with $\lambda(\beta + \delta_4,\alpha - \delta_3) \ge
\lambda(\beta, \alpha) - \varepsilon$. It then follows, again by
the monotonicity properties, that the set $\{ (\beta + \delta ,
\alpha + \tilde \delta) : - \delta_2 \le \delta \le \delta_4 , \,
- \delta_3 \le \tilde \delta \le \delta_1 \}$ is contained in $D$
and $\lambda$ restricted to this set takes values in the interval
$[\lambda(\beta, \alpha) - \varepsilon , \lambda(\beta, \alpha) +
\varepsilon]$. The assertion \eqref{dopen} follows.
With the closed subset $A := \lambda^{-1} (\mathrm{supp} \, r)$ of
$D$, we can rewrite $T_1(\beta,\alpha)$ as
$$
T_1(\beta,\alpha) = \begin{cases} \frac{R_+(-1,\beta,\alpha)
r(\lambda(\beta,\alpha))}{\int f(x) u_+^2(x,\beta,\alpha) \, dx} &
\text{ if } (\beta,\alpha) \in D, \\
0 & \text{ if } (\beta,\alpha) \in {\mathbb T}_N^2 \setminus A.
\end{cases}
$$
Notice that this is well-defined. Since $\{ D , {\mathbb T}_N^2 \setminus
A\}$ is an open cover of ${\mathbb T}_N^2$, it suffices to check continuity
for each of these two open sets. Continuity on ${\mathbb T}_N^2 \setminus A$
is obvious. Continuity on $D$ follows by \eqref{dopen}, the
continuity of $\lambda$ in $(\beta,\alpha)$, and (via \eqref{upmdef} and \eqref{Rpmdef}) the joint continuity of
$R_0(-1,\alpha,\lambda)$ and $\int f u_0^2(\cdot,\alpha,\lambda)$ in
$(\alpha,\lambda)$. The latter is a consequence of the a priori bound provided in Lemma~\ref{lem:contdep}. This concludes the proof of \eqref{t1kercont}.
\end{proof}
\begin{lemma} \label{l.T122elembound}
We have
$$
\|T_1(g,E)\|_{2,2} \le 1.
$$
\end{lemma}
\begin{proof}
Let
$$
K_1(\beta,\alpha) = \frac{r(\lambda(\beta,\alpha))}{\int f
u_+^2(\cdot,\beta,\alpha)}
$$
and
$$
K_2(\beta,\alpha) = \frac{R_+^2(-1,\beta,\alpha)
r(\lambda(\beta,\alpha))}{\int f u_+^2(\cdot,\beta,\alpha)}
$$
if $\lambda(\beta,\alpha)$ exists and $K_1(\beta,\alpha) =
K_2(\beta,\alpha) = 0$ otherwise.
Thus, using \eqref{upmrel} and the change of variables \eqref{e.alphalambda},
\begin{align}
\label{k1one} \int_{{\mathbb T}_N} K_1(\beta,\alpha) \, d\alpha & = \int_{{\mathbb T}_N}
\frac{r(\lambda(\beta,\alpha))}{\int f u_+^2(\cdot,\beta,\alpha)} \, d\alpha\\
\nonumber & = \int_{{\mathbb T}_N} \frac{R_-^2(0,\beta,\alpha) \cdot
r(\lambda(\beta,\alpha))}{\int f
u_-^2(\cdot,\beta,\alpha)} \, d\alpha \\
\nonumber & = \int_{{\mathbb T}_N} \frac{R_{-1}^2(0,\beta,\lambda) \cdot
r(\lambda)}{\int f u_{-1}^2(\cdot,\beta,\lambda)} \left|
\frac{d\alpha}{d\lambda} \right|
d\lambda \\
\nonumber & = \int_{{\mathbb T}_N} r(\lambda) \, d\lambda \\
\nonumber & = 1.
\end{align}
Similarly, using \eqref{e.betalambda},
\begin{align}
\label{k2one} \int_{{\mathbb T}_N} K_2(\beta,\alpha) \, d\beta & = \int_{{\mathbb T}_N}
\frac{R_+^2(-1,\beta,\alpha)
r(\lambda(\beta,\alpha))}{\int f u_+^2(\cdot,\beta,\alpha)} \, d\beta \\
\nonumber & = \int_{{\mathbb T}_N} \frac{R_0^2(-1,\alpha,\lambda) \cdot
r(\lambda)}{\int f u_0^2(\cdot,\alpha,\lambda)} \left|
\frac{d\beta}{d\lambda} \right|
d\lambda \\
\nonumber & = \int_{{\mathbb T}_N} r(\lambda) \, d\lambda \\
\nonumber & = 1.
\end{align}
We have $T_1(\beta,\alpha) = \sqrt{K_1(\beta,\alpha)}
\sqrt{K_2(\beta,\alpha)}$, so that the Schur Test, e.g.\ \cite{Weidmann}, immediately
gives $\|T_1\|_{2,2} \le 1$.
\end{proof}
\section{The Operator $T_1$ has $\|\cdot\|_{2,2}$-Norm Less Than One} \label{s.T122strongbound}
The purpose of this section is to establish to following strengthening of Lemma~\ref{l.T122elembound}, which is the key technical result of our work.
\begin{prop}\label{t1normthm}
We have $\|T_1(g,E)\|_{2,2} < 1$.
\end{prop}
We will suppress the $(g,E)$-dependence in our notation for the remainder of this section.
We have already seen that $T_1$ is a bounded operator on $L^2({\mathbb T}_N)$. Moreover, \eqref{t1kerper} suggests that we decompose $T_1$ as a direct sum of integral operators on $L^2(0,\pi)$. Let us implement this:
\begin{lemma}\label{intdeclem}
{\rm(a)} Suppose $h$ is continuous on $(\pi n, \pi (n+1))$ for $n = 0,1,\ldots,2N-1$, $j \in \{ 0 , 1 , \ldots , 2N-1 \}$, and $x \in (0,\pi)$, and let
$$
(Uh)_j(x) = \frac{1}{\sqrt{2N}} \sum_{n = 0}^{2N-1} e^{\frac{-i \pi j n}{N}} h(x + \pi n).
$$
Then $U$ extends to a unitary operator
$$
U : L^2({\mathbb T}_N) \to \bigoplus_{j=0}^{2N-1} L^2(0,\pi).
$$
{\rm (b)} We have
$$
U T_1 U^{-1} = \bigoplus_{j=0}^{2N-1} L_j,
$$
where $L_j$ is the integral operator in $L^2(0,\pi)$ with kernel
$$
L_j(\beta , \alpha) = \sum_{n = 0}^{2N-1} T_1(\beta , \alpha + n \pi) e^{\frac{i \pi j n}{N}}.
$$
{\rm (c)} We have $\|T_1\| = \|L_0\|$, with both norms being the operator norm in the respective $L^2$ space.
\end{lemma}
\begin{proof}
(a) Suppose $h$ is continuous on $(\pi n, \pi (n+1))$ for $n = 0,1,\ldots,2N-1$. Then,
\begin{align*}
\| Uh \|^2 & = \sum_{j = 0}^{2N-1} \int_0^\pi \left| \frac{1}{\sqrt{2N}} \sum_{n = 0}^{2N-1} e^{\frac{-i \pi j n}{N}} h(x + \pi n) \right|^2 \, dx \\
& = \int_0^\pi \sum_{j = 0}^{2N-1} \left| \sum_{n = 0}^{2N-1} \frac{e^{\frac{-i \pi j n}{N}}}{\sqrt{2N}}\, h(x + \pi n) \right|^2 \, dx \\
& = \int_0^\pi \sum_{j = 0}^{2N-1} |h(x + \pi j)|^2 \, dx \\
& = \int_0^{2N\pi} |h(x)|^2 \, dx \\
& = \|h\|^2.
\end{align*}
Here, all steps save the third follow by simple rewriting and the third step follows from the Parseval identity for ${\mathbb C}^{2N}$.
For a continuous $g = (g_j) \in \bigoplus_{j=0}^{2N-1} L^2(0,\pi)$, we define
$$
h(x + \pi n) = \frac{1}{\sqrt{2N}} \sum_{j = 0}^{2N-1} e^{\frac{i \pi j n}{N}} g_j(x),
$$
where $x \in (0,\pi)$ and $n \in \{ 0 , 1 , \ldots 2N-1 \}$ and note that $g = U h$. Since $h$ is continuous on $(\pi n, \pi (n+1))$ for $n = 0,1,\ldots,2N-1$, we may conclude that $U$ is a densely defined isometry with dense range, and hence $U$ extends to a unitary operator from $L^2({\mathbb T}_N)$ onto $\bigoplus_{j=0}^{2N-1} L^2(0,\pi)$.
(b) We have
\begin{align*}
\Big( \bigoplus_{k=0}^{2N-1} L_k U h \Big)_j (\beta) & = \int_0^\pi L_j(\beta,\alpha) (Uh)_j(\alpha) \, d\alpha \\
& = \int_0^\pi \sum_{n = 0}^{2N-1} T_1(\beta , \alpha + n \pi) e^{\frac{i \pi j n}{N}} \frac{1}{\sqrt{2N}} \sum_{m = 0}^{2N-1} e^{\frac{-i \pi j m}{N}} h(\alpha + \pi m) \, d\alpha \\
& = \frac{1}{\sqrt{2N}} \sum_{n = 0}^{2N-1} \; \sum_{m = 0}^{2N-1} e^{\frac{-i \pi j (m-n)}{N}} \int_0^\pi T_1(\beta , \alpha + n \pi) h(\alpha + \pi m) \, d\alpha
\end{align*}
and
\begin{align*}
\Big( U T_1 h \Big)_j (\beta) & = \Big( U \int_0^{2N\pi} T_1(\cdot , \alpha) h (\alpha) \, d\alpha \Big)_j (\beta) \\
& = \frac{1}{\sqrt{2N}} \sum_{n = 0}^{2N-1} e^{\frac{-i \pi j n}{N}} \int_0^{2N\pi} T_1(\beta + \pi n , \alpha) h (\alpha) \, d\alpha \\
& = \frac{1}{\sqrt{2N}} \sum_{n = 0}^{2N-1} e^{\frac{-i \pi j n}{N}} \sum_{m=0}^{2N-1} \int_0^{\pi} T_1(\beta + \pi n , \alpha + \pi m) h (\alpha + \pi m) \, d\alpha \\
& = \frac{1}{\sqrt{2N}} \sum_{n = 0}^{2N-1} \; \sum_{m=0}^{2N-1} e^{\frac{-i \pi j n}{N}} \int_0^{\pi} T_1(\beta , \alpha + \pi(m-n)) h (\alpha + \pi m) \, d\alpha \\
& = \frac{1}{\sqrt{2N}} \sum_{\tilde n = 0}^{2N-1} \; \sum_{m=0}^{2N-1} e^{\frac{-i \pi j (m - \tilde n)}{N}} \int_0^{\pi} T_1(\beta , \alpha + \pi \tilde n) h (\alpha + \pi m) \, d\alpha,
\end{align*}
from which the asserted identity follows.
(c) From the decomposition established above, we get
$\|T_1\| = \max_{0 \le j \le 2N-1} \| L_j \|$. As $T_1(\beta,\alpha) \ge 0$, we have $|L_j(\beta,\alpha)|
\le L_0(\beta,\alpha)$ and therefore $\| L_j \| \le \| L_0 \|$ for every $j$. This yields the claim.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{t1normthm}.]
By Lemma~\ref{intdeclem}.(c), it suffices to show $\|L_0\| < 1$. By
Lemma~\ref{l.T122elembound} and Lemma~\ref{intdeclem}.(c), $\|L_0\| \le 1$.
Suppose that $\|L_0\| = 1$. By
compactness, there exists $f \not= 0$ such that $\|L_0 f\| =
\|f\|$ (choose $f$ as an eigenvector to the eigenvalue $1 = \|L_0\|
= \| |L_0| \|$ of $| L_0 |$ and use $\|L_0 f\| = \| |L_0| f \|$).
The operator $L_0$ has a positive kernel and we may therefore
assume that $f \ge 0$.
Consider the $\pi$-periodic extension $\tilde f$ of $f$ to ${\mathbb T}_N$. Then,
\begin{align*}
(L_0 f)(\beta) & = \int_0^\pi \sum_{n = 0}^{2N-1} T_1(\beta , \alpha + n \pi) f(\alpha) \, d\alpha\\
& = \sum_{n = 0}^{2N-1} \int_0^\pi T_1(\beta , \alpha + n \pi) \tilde f(\alpha + n \pi) \, d\alpha\\
& = \int_{{\mathbb T}_N} T_1(\beta,\alpha) \tilde f(\alpha) \,
d\alpha.
\end{align*}
By \eqref{k1one}, \eqref{k2one} along with $K_1(\beta + \pi,\alpha
+ \pi) = K_1(\beta,\alpha)$ and $K_2(\beta + \pi,\alpha + \pi) =
K_2(\beta,\alpha)$, we find
\begin{align*}
\| L_0 f \|^2 & = \int_0^\pi | (L_0 f)(\beta) |^2 \, d\beta \\
& = \int_0^\pi \left| \int_{{\mathbb T}_N} T_1(\beta,\alpha) \tilde f(\alpha) \,
d\alpha \right|^2 \, d\beta \\
& = \int_0^\pi \left| \int_{{\mathbb T}_N}
\sqrt{K_1(\beta,\alpha)} \sqrt{K_2(\beta,\alpha)} \tilde f(\alpha)
\, d\alpha \right|^2 \, d\beta \\
& \le \int_0^\pi \left( \int_{{\mathbb T}_N} K_1(\beta,\alpha) \, d\alpha \;
\int_{{\mathbb T}_N} K_2(\beta,\alpha) |\tilde f(\alpha)|^2 \, d\alpha \right)
\, d\beta \\
& = \int_0^\pi \int_{{\mathbb T}_N} K_2(\beta,\alpha) |\tilde f(\alpha)|^2 \, d\alpha
\, d\beta \\
& = \int_0^\pi \sum_{n=0}^{2N-1} \int_0^\pi K_2(\beta,\alpha - \pi n) |\tilde f(\alpha - \pi n)|^2 \,
d\alpha \, d\beta \\
& = \int_0^\pi \sum_{n=0}^{2N-1} \int_0^\pi K_2(\beta + \pi n,\alpha) |\tilde f(\alpha)|^2 \,
d\alpha \, d\beta \\
& = \int_0^\pi \left( \sum_{n=0}^{2N-1} \int_0^\pi K_2(\beta + \pi n,\alpha) \, d\beta \right) |\tilde f(\alpha)|^2 \,
d\alpha \\
& = \int_0^\pi \left( \int_{{\mathbb T}_N} K_2(\beta,\alpha) \, d\beta \right) |\tilde f(\alpha)|^2 \,
d\alpha \\
& = \int_0^\pi |\tilde f(\alpha)|^2 \, d\alpha \\
& = \int_0^\pi |f(\alpha)|^2 \, d\alpha \\
& = \| f \|^2 \\
& = \| L_0 f \|^2.
\end{align*}
Thus, we have equality in the application of the Cauchy-Schwarz
inequality, which implies that for almost every $\beta \in
(0,\pi)$, the functions $\sqrt{K_1(\beta,\cdot)}$ and
$\sqrt{K_2(\beta,\cdot)} \tilde f(\cdot)$ are linearly dependent
in $L^2({\mathbb T}_N)$. Since they are both non-negative and non-zero, we
see that for $\beta \in (0,\pi) \setminus N$, $\mathrm{Leb}(N) =
0$, there is $C_\beta > 0$ such that
$$
C_\beta K_1(\beta, \cdot) = K_2(\beta,\cdot) \tilde f(\cdot)^2.
$$
Fix $\beta \in [0,\pi) \setminus N$ and let
$$
M_\beta := \{ \alpha : \lambda(\beta,\alpha) \in \mathrm{supp}\, r
\}.
$$
Then, for almost every $\alpha \in M_\beta$, we have $C_\beta =
R_+^2(-1,\beta,\alpha)\tilde f(\alpha)^2$, or
\begin{equation}\label{14}
\tilde f(\alpha)^2 = C_\beta
R_{-1}^2(0,\beta,\lambda(\beta,\alpha)).
\end{equation}
Let $[A,B]$ be a non-trivial interval that is contained in the
support of $r$ (recall that $r$ is continuous). If $c_\beta$ and $d_\beta$ are the unique phases
determined by $\lambda(\beta,c_\beta) = B$ and
$\lambda(\beta,d_\beta) = A$ (i.e., $c_\beta =
\varphi_{-1}(0,\beta,B)$ and $d_\beta = \varphi_{-1}(0,\beta,A)$),
then $c_\beta < d_\beta$ and $[c_\beta,d_\beta] \subset M_\beta$.
Moreover, $c_\beta$ and $d_\beta$ are strictly increasing and
continuous in $\beta$ (cf.~Lemma~\ref{lem:thetader}) and we have $[c_{\beta + \pi},d_{\beta +
\pi}] = [c_\beta + \pi , d_\beta + \pi]$.
It follows that
$$
I := \bigcup_{\beta \in [0,\pi) \setminus N} ( c_\beta , d_\beta )
$$
is an open interval of length greater than $\pi$. For fixed $\beta
\in (0,\pi) \setminus N$, $\tilde f^2$ is real-analytic on
$(c_\beta , d_\beta)$ by \eqref{14}. This uses that (i) $\alpha(\lambda)$ is analytic in $\lambda$ with $\alpha'(\lambda)<0$ (and thus its inverse function $\lambda(\beta,\alpha)$ is analytic in $\alpha$) and (ii) $R_{-1}^2(0,\beta,\lambda)$ is analytic in $\lambda$. Property (ii) follows from part (a) of Lemma~\ref{lem:lamderphi}. This also implies the analyticity of the right hand side of \eqref{e.alphalambda} in $\lambda$ which in turn gives (i).
We conclude
that $\tilde f^2$ is analytic on $I$ and, due to
$\pi$-periodicity, on all of ${\mathbb R}$.
Now, we again fix a $\beta \in [0,\pi) \setminus N$ and conclude
by analytic continuation that \eqref{14} holds for all $\alpha$
for which $\lambda(\beta,\alpha)$ exists. Therefore,
$R_{-1}^2(0,\beta,\lambda(\beta,\alpha))$ is bounded in $\alpha$
(as this holds for the $\pi$-periodic analytic function on the LHS
of \eqref{14}). But $\lambda(\beta,\alpha)$ takes on arbitrary
real values as $\alpha$ varies and hence
\begin{equation}\label{prampbdd}
\sup_{\lambda \in {\mathbb R}} R_{-1}(0,\beta,\lambda) < \infty,
\end{equation}
which is impossible by Proposition~\ref{proposition2}. This
contradiction completes the proof of $\|T_1\| = \| L_0 \| < 1$.
\end{proof}
\section{The Dependence of $T_1$ on the Background} \label{s.T122continuity}
In this section we study the map $(g,E) \mapsto T_1(g,E)$. Note that the energy $E$ can be absorbed in $g$, that is, $T_1(g,E) = T_1(g-E,0)$. For this reason, we will consider without loss of generality the case $E = 0$. Consequently, in this section, $E$ is dropped from the notation and assumed to be zero. For example, we write $T_1(g)$ for $T_1(g,0)$ and $\lambda(\beta,\alpha,g)$ for $\lambda(\beta,\alpha,g,0)$.
Write
$$
D(g) = \{ (\beta,\alpha) \in {\mathbb R}^2 : \lambda(\beta,\alpha,g) \text{
exists} \}
$$
and
$$
A(g) = \lambda(\cdot,\cdot,g)^{-1}([-M,M]) \subset D(g),
$$
where, as in Section~\ref{s.reduction}, supp$\,r \subset [-M,M]$.
\begin{lemma}
Suppose $g_n \to g$ in $L^\infty(-1,0)$. Then, we have
\begin{equation}\label{dincl}
D(g) \subseteq \liminf_{n \to \infty} D(g_n)
\end{equation}
and
\begin{equation}\label{aincl}
{\mathbb R}^2 \setminus A(g) \subseteq \liminf_{n \to \infty} {\mathbb R}^2
\setminus A(g_n).
\end{equation}
\end{lemma}
\begin{proof}
Let $(\beta,\alpha) \in D(g)$ so that $\bar \lambda :=
\lambda(\beta,\alpha,g)$ exists. Fix some $\varepsilon > 0$. Then,
by monotonicity,
$$
\varphi_0(-1,\alpha,\bar \lambda - \varepsilon, g) < \beta <
\varphi_0(-1,\alpha,\bar \lambda + \varepsilon, g).
$$
It follows from $L^1_\mathrm{loc}$-continuity of solutions in $g$, specifically the bound provided in Lemma~\ref{lem:contdep}, that for
$n$ sufficiently large,
$$
\varphi_0(-1,\alpha,\bar \lambda - \varepsilon, g_n) < \beta <
\varphi_0(-1,\alpha,\bar \lambda + \varepsilon, g_n).
$$
Thus, for such values of $n$, there is $\bar \lambda_n \in (\bar
\lambda - \varepsilon , \bar \lambda + \varepsilon)$ with
$\varphi_0(-1,\alpha,\bar \lambda_n,g_n) = \beta$. In particular,
$\lambda(\beta,\alpha,g_n)$ exists (and is given by $\bar
\lambda_n$). This proves \eqref{dincl}. For later use, we note
that the proof also shows $\lambda(\beta,\alpha,g_n) \to
\lambda(\beta,\alpha,g)$.
Now consider $(\beta,\alpha) \in {\mathbb R}^2 \setminus A(g)$. That is,
either $\lambda(\beta,\alpha,g)$ does not exist or it does exist
but lies outside the interval $[-M,M]$. Suppose there is a sequence
$n_k \to \infty$ such that $\lambda(\beta,\alpha,g_{n_k})$ exists
and belongs to $[a,b]$ for every $k$. This means that
$\varphi_{-1}(0,\beta,\lambda(\beta,\alpha),g_{n_k}) = \alpha$. By
monotonicity, this gives
$$
\varphi_{-1}(0,\beta,b,g_{n_k}) \le \alpha \quad \text{and} \quad
\varphi_{-1}(0,\beta,a,g_{n_k}) \ge \alpha.
$$
Taking $k \to \infty$, we find
$$
\varphi_{-1}(0,\beta,b,g) \le \alpha \quad \text{and} \quad
\varphi_{-1}(0,\beta,a,g) \ge \alpha.
$$
This means, however, that there exists $\lambda \in [-M,M]$ such
that $\varphi_{-1}(0,\beta,\lambda,g) = \alpha$, which is a
contradiction. This proves \eqref{aincl}.
\end{proof}
\begin{lemma}\label{contkerlem}
Suppose $g_n \to g$ in $L^\infty(-1,0)$. Then,
$$
\lim_{n \to \infty} T_1(\beta,\alpha,g_n) = T_1(\beta,\alpha,g).
$$
for every $(\beta,\alpha) \in {\mathbb R}^2$.
\end{lemma}
\begin{proof}
We first consider the case $(\beta,\alpha) \in {\mathbb R}^2 \setminus
A(g)$. As seen above, this implies $(\beta,\alpha) \in {\mathbb R}^2
\setminus A(g_n)$ for $n \ge N_1$. Consequently,
$T_1(\beta,\alpha,g) = T_1(\beta,\alpha,g_n) = 0$ for $n \ge N_1$,
which trivially implies convergence.
If $(\beta,\alpha) \in D(g)$, we know that $(\beta,\alpha) \in
D(g_n)$ for $n \ge N_2$. Then, using continuous dependence of solutions on the potential again, it is readily
seen that
$$
T_1(\beta,\alpha,g_n) =
\frac{R_0(-1,\alpha,\lambda(\beta,\alpha,g_n),g_n)
r(\lambda(\beta,\alpha,g_n))}{\int f(x)
u_0^2(x,\alpha,\lambda(\beta,\alpha,g_n),g_n) \, dx} \to
T_1(\beta,\alpha,g).
$$
Here we also used that $\lambda(\beta,\alpha,g_n) \to
\lambda(\beta,\alpha,g)$, which was proven above.
\end{proof}
\begin{prop}
The real-valued map $g \mapsto \|T_1(g)\|_{2,2}$ is continuous on the ball of radius $\|W_0\|_{\infty}$ in $L^{\infty}(-1,0)$.
\end{prop}
\begin{proof}
We have to show that for $g,g_n \in L^\infty(-1,0)$, $\|g\|_{\infty}, \|g_n\|_{\infty} \le \|W_0\|_{\infty}$, $n\ge 1$,
with $\|g_n - g\|_\infty \to 0$, we have
$$
\lim_{n \to \infty} \|T_1(g_n)\|_{2,2} =
\|T_1(g)\|_{2,2}.
$$
By Lemma~\ref{intdeclem}.(c), it suffices to show that
\begin{equation}\label{t2toshow}
\lim_{n \to \infty} \| L_0(g_n) \|_{2,2} = \| L_0(g) \|_{2,2}.
\end{equation}
Recall that
$$
L_0 (\beta,\alpha,\cdot) = \sum_{n = 0}^{2N-1} T_1(\beta,\alpha + \pi n,\cdot);
$$
compare Lemma~\ref{intdeclem}.(b). Using (\ref{t1kerdef}) and the a priori bounds Lemma~\ref{lem:solest} and Lemma~\ref{lem:L2b}, this
implies that $L_0 (\beta,\alpha,\cdot)$ is uniformly bounded,
uniformly for $\{g\} \cup \{ g_n \}_{n \ge 1}$. By Lemma~\ref{contkerlem}, the
functions $L_0 (\cdot,\cdot,g_n)$ converge pointwise to $L_0
(\cdot,\cdot,g)$. Consequently,
\begin{align*}
\| L_0(g_n) - L_0(g) \|^2_{2,2} & \le \|
L_0(g_n) - L_0(g) \|^2_\mathrm{HS} \\
& = \int_0^\pi \int_0^\pi \left| L_0(\beta,\alpha,g_n) -
L_0(\beta,\alpha,g) \right|^2 \, d\alpha \, d\beta \\
& \to 0
\end{align*}
by dominated convergence. This proves \eqref{t2toshow} and hence
the theorem.
\end{proof}
\begin{appendix}
\section{Large Coupling Limit of the Pr\"ufer Amplitude} \label{s.appA}
Here we establish a technical fact which was used in the proof of Proposition~\ref{t1normthm}.
\begin{prop} \label{proposition2}
It holds that
\begin{equation}\label{rlaminf}
\lim_{\lambda \to \infty} R_{-1}(0,\beta,\lambda) = \infty.
\end{equation}
\end{prop}
\begin{proof}
For $[a,b]$ from \eqref{singlesite2} let $\theta \in [0,\pi)$ be such
that $\theta = \varphi_{-1}(a,\beta,\lambda,g) \mod \pi$ and
denote by $\varphi(x,\lambda) := \varphi_a(x,\theta,\lambda,g)$
and $R(x,\lambda) := R_a(x,\theta,\lambda,g)$ the Pr\"ufer phase
and amplitude for the solution of $-u''+gu+\lambda fu=0$ with
$u'(a)=\cos\theta$, $u(a)=\sin\theta$. It suffices to show that
\begin{equation} \label{prop2I}
\lim_{\lambda\to \infty} R(b,\lambda) = \infty.
\end{equation}
This follows as supp$\,f \cap ([-1,a)\cup (b,0]) =\emptyset$ and
therefore, by Lemma~\ref{lem:solest}, $R_{-1}(a,\alpha,\lambda,g) \approx 1$ and
$R_{-1}(0,\alpha,\lambda,g) \approx R_{-1}(b,\alpha,\lambda,g)
\approx R(b,\lambda)$.
Note that $\varphi$ and $R$ satisfy the Pr\"ufer differential
equations
\begin{equation} \label{prop2II}
\varphi' = 1-(1+g+\lambda f)\sin^2 \varphi
\end{equation}
and
\begin{equation} \label{prop2III}
(\ln R)' = \frac{1}{2} (1+g+\lambda f) \sin 2\varphi.
\end{equation}
We will first show that there exists $\lambda_0 \in {\mathbb R}$ and $\eta
\in (0,\pi)$ such that
\begin{equation} \label{prop2IV}
\varphi(x,\lambda) < \eta \quad \mbox{for all } \lambda \ge
\lambda_0 \mbox{ and all } x\in [a,b].
\end{equation}
By \eqref{prop2II} and $\theta\ge 0$ we know that
$\varphi(b,\lambda)>0$ for all $\lambda$. Sturm comparison Lemma~\ref{lem:sturm} or, more directly, Lemma~\ref{lem:thetader} shows that in proving \eqref{prop2IV} it suffices to assume that $\theta
\in [\pi/2, \pi)$. Choose $\eta \in (\theta, \pi)$ with $\sin^2
\eta = \frac{1}{2} \sin^2\theta$ and let
$$
M_{\lambda} := \Big\{ x\in[a,b]: 1+g(x)+\lambda f(x) \ge
\frac{1}{\sin^2 \eta} \Big\}.
$$
It follows from \eqref{singlesite2} that $|M_{\lambda}| \to b-a$ as
$\lambda\to\infty$. To show \eqref{prop2IV} for the given choice
of $\eta$, we assume, by way of contradiction, that there are
arbitrarily large $\lambda>0$ for which the set $\{x\in [a,b]:
\varphi(x,\lambda)\ge \eta\}$ is non-empty and thus has a minimum
$b_{\lambda}$ with $\varphi(b_{\lambda},\lambda)= \eta$. Also, let
$a_{\lambda} := \max \{x\in [a,b_{\lambda}]: \varphi(x,\lambda) =
\theta\}$. Thus $\varphi(x,\lambda) \in [\theta,\eta]$ for all
$x\in [a_{\lambda}, b_{\lambda}]$.
By \eqref{prop2II} we have $\varphi'(x) \le 1+\|g\|_{\infty}$ for
all $x\in [a_{\lambda}, b_{\lambda}]$ and $\varphi'(x)\le 1 -
\sin^2 \phi(x)/\sin^2 \eta \le 0$ for $x\in M_{\lambda} \cap
[a_{\lambda}, b_{\lambda}]$. Thus
\begin{eqnarray*}
\eta - \theta & = & \varphi(b_{\lambda},\lambda) - \varphi(a_{\lambda},\lambda) = \int_{a_{\lambda}}^{b_{\lambda}} \varphi'(x,\lambda)\,dx \\
& \le & \int_{[a_{\lambda},b_{\lambda}] \setminus M_{\lambda}}
(1+\|g\|_{\infty})\,dx \le (1+\|g\|_{\infty}) (b-a-|M_{\lambda}|).
\end{eqnarray*}
Choosing a sufficiently large $\lambda > 0$, we can make the
right-hand side arbitrarily small and hence we obtain the
contradiction $\eta-\theta \le 0$, proving \eqref{prop2IV}.
Next, consider the set
$$
N_{\lambda} := \Big\{ x\in[a,b]:\, \varphi(x,\lambda)\ge
\frac{\pi}{4}\Big\}.
$$
As $f\ge 0$, we see by Lemma~\ref{lem:lamderphi}(b) that $N_{\lambda}$ is
decreasing for increasing $\lambda$. We will show that
\begin{equation} \label{prop2VI}
\lim_{\lambda\to\infty} |N_{\lambda}| = 0
\end{equation}
and
\begin{equation} \label{prop2VII}
\sup_{\lambda>0} \lambda \int_{N_{\lambda}} f(x)\,dx < \infty.
\end{equation}
By \eqref{prop2II} we have for all $\lambda$ that
\begin{equation} \label{prop2VIII}
\pi > |\varphi(b,\lambda)-\varphi(a,\lambda)| = \left|b-a-\int_a^b
(1+g(x)+\lambda f(x)) \sin^2 \varphi(x,\lambda)\,dx \right|.
\end{equation}
For $\lambda\ge \lambda_0$ from \eqref{prop2IV} we can bound
$$
\int_a^b (1+g(x)+\lambda f(x))\sin^2\varphi(x,\lambda)\,dx \ge
|b-a|(1-\|g\|_{\infty}) +\lambda \min\{\sin^2\eta,1/2\}
\int_{N_{\lambda}} f(x)\,dx.
$$
By \eqref{prop2VIII} it follows that $\lambda \int_{N_{\lambda}}
f(x)\,dx$ is bounded in $\lambda$, proving \eqref{prop2VII}. This
implies \eqref{prop2VI} as $|N_\lambda| \to |N|$, where $N=
\bigcap_{\lambda} N_{\lambda}$ and $f>0$ almost everywhere on $N$.
We will now use \eqref{prop2III} to prove \eqref{prop2I}. We
estimate
\begin{align} \label{prop2IX}
\int_a^b (1+g(x)+\lambda f(x))\sin 2\varphi(x,\lambda)\,dx \ge &
-(1+\|g\|_{\infty})(b-a) \\
& +\lambda \int_a^b f(x) \sin 2\varphi(x,\lambda)\,dx. \nonumber
\end{align}
By \eqref{prop2VII},
\begin{equation} \label{prop2X}
\lambda \int_a^b f(x) \sin 2\varphi(x,\lambda)\,dx \ge -\lambda
\int_{N_{\lambda}} f(x)\,dx \ge -C
\end{equation}
uniformly in $\lambda$. Moreover, Sturm comparison (Lemma~\ref{lem:sturm}) with the
solution $u(x)=\exp(\sqrt{K}x)$ of $-u''+Ku=0$, where
$K=\lambda\|f\|_{\infty} +\|g\|_{\infty}$, gives for
$\theta\not=0$ and $\lambda$ sufficiently large (such that
$\theta>1/\sqrt{K}$) that $\varphi(x,\lambda)\ge 1/\sqrt{K}$. This
shows, using the definition of $N_{\lambda}$ and \eqref{prop2VI},
\begin{equation} \label{prop2XI}
\lambda \int_{[a,b]\setminus N_{\lambda}} f(x) \sin
2\varphi(x,\lambda) \gtrsim \sqrt{\lambda} \int_{[a,b]\setminus
N_{\lambda}} f(x)\,dx \gtrsim \sqrt{\lambda}.
\end{equation}
Finally, \eqref{prop2IX}, \eqref{prop2X} and \eqref{prop2XI} yield
$$
\ln R(b,\lambda) = \int_a^b (1+g(x)+\lambda f(x))\sin
2\varphi(x,\lambda)\,dx \to \infty \quad \mbox{as
$\lambda\to\infty$}.
$$
Thus we have shown \eqref{prop2I} for $\theta\not=0$. The case
$\theta=0$ is slightly different. Comparing with the solution of
$-u''+Ku=0$, $u(a)=0$, in this case gives, for suitable $C_1>0$
and $C_2>0$,
$$
\varphi(x,\lambda) \ge \frac{C_1}{\lambda} \quad \mbox{if } |x-a|
\ge \frac{C_2}{\sqrt{\lambda}}.
$$
This suffices to get the bound \eqref{prop2XI} and thus lets the
above argument go through.
\end{proof}
\section{Pr\"ufer Variables and A Priori Bounds} \label{s.appB}
This appendix contains a brief summary of standard facts on Pr\"ufer variables and a priori bounds on solutions which have been used throughout the paper. Proofs can for example be found in the appendix of \cite{hss} (for Lemma~\ref{lem:solest}, Lemma~\ref{lem:L2b}, Lemma~\ref{lem:xder}, Lemma~\ref{lem:lamderphi}(b) and Corollary~\ref{cor:Ederphi}) and the appendix of \cite{poisson} (Lemma~\ref{lem:thetader}). Lemma~\ref{lem:lamderphi}(a) and Lemma~\ref{lem:sturm} are special cases of Theorems~2.1 and 13.1 in \cite{weidmann2}. Lemma~\ref{lem:contdep} is proven as Lemma~A.2 in \cite{dss} for the special case that $u_1$ and $u_2$ satisfy the same initial condition at $y$. The proof given there extends easily to give the result we need here.
\begin{lemma} \label{lem:solest}
For every $q\in L^1_{\rm loc}({\mathbb R})$, every solution $u$ of $-u''+qu=0$, and all $x,y \in {\mathbb R}$ one has
\begin{eqnarray*}
\lefteqn{(|u(y)|^2 +|u'(y)|^2) \exp \left(- \int_{\min(x,y)}^{\max(x,y)} |1+q(t)|\, dt \right)} \nonumber \\ & \leq & |u(x)| ^2 + |u'(x)|^2 \leq \left( |u(y)|^2 + |u'(y)|^2 \right)
\exp \left( \int_{\min(x,y)}^{\max(x,y)} |1+q(t)|\, dt \right).
\end{eqnarray*}
\end{lemma}
\begin{lemma} \label{lem:contdep}
For $i=1,2$, let $q_i \in L^1_{\rm loc}({\mathbb R})$ and let $u_i$ be solutions of $-u_i''+q_iu_i=0$. Then for any $x\in {\mathbb R}$,
\begin{eqnarray*}
\lefteqn{\left( |u_1(x)-u_2(x)|^2 + |u_1'(x)-u_2'(x)|^2 \right)^{1/2}} \nonumber \\
& \le & \left( |u_1(y)-u_2(y)|^2 + |u_1'(y)-u_2'(y)|^2 \right)^{1/2} \exp\Big\{ \int_{\min(x,y)}^{\max(x,y)} (|q_2(t)|+1)\,dt\Big\} \nonumber \\
& & \mbox{} + \left( |u_1(y)|^2 + |u_1'(y)|^2 \right) \exp\Big\{ \int_{\min(x,y)}^{\max(x,y)} (|q_1(t)|+|q_2(t)|+2)\,dt\Big\} \nonumber \\ & & \mbox{} \times \int_{\min(x,y)}^{\max(x,y)} |q_1(t)-q_2(t)|\,dt.
\end{eqnarray*}
\end{lemma}
\begin{lemma} \label{lem:L2b}
For any positive real numbers $\ell$ and $M$ there exists $C>0$ such that
\begin{equation} \label{eq:intbd2}
\int_c^{c+\ell} |u(t)|^2 dt \geq C \left( |u(c)|^2 + | u'(c)|^2 \right)
\end{equation}
for every $c\in {\mathbb R}$, every $L^1_{\rm loc}$-function $q$ with $\int_c^{c+\ell} |q(t)|\,dt \le M$, and any solution $u$ of $-u''+qu=0$ on $[c,c+\ell]$.
\end{lemma}
Our remaining results relate to Pr\"ufer variables.
In general, for any real potential $q\in L^1_{\rm loc}( \mathbb{R})$ and
real parameters $c$ and $\theta$ let $u_c$ be
the solution of
\[
-u'' + q u = 0
\]
with $u_c(c)=\sin \theta$, $u_c'(c)=\cos \theta$. By regarding this
solution and its derivative in polar coordinates, we define the
Pr\"ufer amplitude $R_c(x)$ and the Pr\"ufer phase
$\phi_c(x)$ by writing
\begin{equation} \label{eq:pruferdef2}
u_c(x)= R_c(x) \sin \phi_c(x)
\quad \mbox{and} \quad u_c'(x) = R_c(x) \cos
\phi_c(x).
\end{equation}
For uniqueness of the Pr\"ufer phase we declare
$\phi_c(c)=\theta$ and require continuity of $\phi_c$ in
$x$. In what follows the initial phase $\theta$ will be fixed and we thus leave the dependence of $u_c$, $R_c$ and $\phi_c$ on $\theta$ implicit in our notation.
In the new variables $R$ and $\phi$ the second order equation $-u''+qu=0$ becomes a system of two first order equations, where the equation for $\phi$ is not coupled with $R$:
\begin{lemma} \label{lem:xder} For fixed $c$ and $\theta$, one has
that
\begin{equation} \label{eq:xderR}
(\ln R^2_c(x))' \, = \, \left( 1 + q(x) \right) \, \sin \left( 2 \,
\phi_c(x) \right),
\end{equation}
and
\begin{equation} \label{eq:xderphi}
\phi_c'(x) \, = \, 1 \, -
\, \left( 1 + q(x)\right) \, \sin^2 \left( \phi_c(x) \right).
\end{equation}
\end{lemma}
When considering $\phi_c(x)$ at fixed $x$ as a function of the initial phase $\theta$ one can show
\begin{lemma} \label{lem:thetader}
For fixed $c$ and $x$, one has
\begin{equation} \label{eq:thetader}
\frac{\partial}{\partial \theta} \phi_c (x,\theta) = \frac{1}{R_c^2(x,\theta)}.
\end{equation}
\end{lemma}
Next we provide some results about the dependence of solutions on a coupling constant at a potential.
\begin{lemma} \label{lem:lamderphi} Let $W$ and $V$ be real valued
functions in $L^1_{\rm loc}( \mathbb{R})$. For real parameters $c$, $\theta$
and $\lambda$, let $u_c(\cdot,\lambda)$ be the solution of
\[
-u'' + Wu + \lambda V u = 0,
\]
normalized so that $u_c(c,\lambda) = \sin \theta$ and $u_c'(c,\lambda) = \cos
\theta$. Denote the Pr\"ufer variables of $u_c(\cdot,\lambda)$ and $u_c'(\cdot,\lambda)$ by $\phi_c(x,\lambda)$ and $R_c(x,\lambda)$.
{\rm (a)} For fixed $x$, $u_c(x,\lambda)$ and $u_c'(x,\lambda)$ (and thus also $\phi_c(x,\lambda)$ and $R_c(x,\lambda)$) are entire functions of $\lambda$.
{\rm (b)} One has that
\begin{equation} \label{eq:lamder}
\frac{\partial}{\partial \lambda} \phi_c(x,\lambda) \, = \,
-R_c^{-2}(x,\lambda) \, \int_c^x V(t) \, u_c^2(t,\lambda) \, dt.
\end{equation}
\end{lemma}
As a special case one finds the energy derivative of the Pr\"ufer
phase.
\begin{coro} \label{cor:Ederphi}
Let $u$ be the solution of $-u'' + Wu = Eu$
normalized so that $u(c) = \sin \theta$ and $u'(c) = \cos
\theta$, and let $\phi_c(x,E)$ and $R_c(x,E)$ be the corresponding Pr\"ufer variables. Then
\begin{equation} \label{eq:Ederphi}
\frac{ \partial}{ \partial E} \phi_c(x,E) \, = \,
R_c^{-2}(x,E) \, \int_c^x u^2(t) \, dt.
\end{equation}
\end{coro}
Finally, we state a version of Sturm's comparison theorem.
\begin{lemma} \label{lem:sturm}
For $i=1,2$, let $u_i$ be the solution of $-u_i''+q_iu_i=0$ with $u_i(c)=\sin \theta_i$ and $u_i'(c)=\cos \theta_i$. Define the Pr\"ufer phases $\phi_i(x)$ to $(u_i,u_i')$ as in (\ref{eq:pruferdef2}).
Suppose that $q_1(t)\ge q_2(t)$ for all $t\in [c,x]$ and $\theta_2 \ge \theta_1$, then $\phi_2(x) \ge \phi_1(x)$.
\end{lemma}
\end{appendix}
| {
"timestamp": "2009-12-18T02:28:50",
"yymm": "0912",
"arxiv_id": "0912.3568",
"language": "en",
"url": "https://arxiv.org/abs/0912.3568",
"abstract": "We consider continuum one-dimensional Schrödinger operators with potentials that are given by a sum of a suitable background potential and an Anderson-type potential whose single-site distribution has a continuous and compactly supported density. We prove exponential decay of the expectation of the finite volume correlators, uniform in any compact energy region, and deduce from this dynamical and spectral localization. The proofs implement a continuum analog of the method Kunz and Souillard developed in 1980 to study discrete one-dimensional Schrödinger operators with potentials of the form background plus random.",
"subjects": "Mathematical Physics (math-ph)",
"title": "A Continuum Version of the Kunz-Souillard Approach to Localization in One Dimension",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.98028087477309,
"lm_q2_score": 0.7217432062975979,
"lm_q1q2_score": 0.707511061630944
} |
https://arxiv.org/abs/0810.1348 | Double Schubert polynomials for the classical groups | For each infinite series of the classical Lie groups of type B,C or D, we introduce a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the appropriate flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. They are also positive in a certain sense, and when indexed by maximal Grassmannian elements, or by the longest element in a finite Weyl group, these polynomials can be expressed in terms of the factorial analogues of Schur's Q- or P-functions defined earlier by Ivanov. | \section{Introduction}
A classical result of Borel states that
the (rational) cohomology ring of
the flag variety of any
simple complex Lie group $G$
is isomorphic, as a graded ring, to the {\it coinvariant\/} algebra
of the corresponding Weyl group $W$,
i.e. to the quotient of a polynomial
ring modulo the ideal generated by
the $W$-invariant polynomials
of positive degree.
The Schubert classes form a
distinguished additive basis of the cohomology ring
indexed by the elements in the Weyl group.
Bernstein-Gelfand-Gelfand \cite{BGG} (see also Demazure \cite{D})
showed that if one starts with a polynomial that represents the
cohomology class of
highest codimension (the Schubert
class of a point), one obtains all the other Schubert
classes by applying a succession of
{\it divided difference\/} operators
corresponding to simple roots. This construction depends on
the choice of a polynomial representative for the ``top'' cohomology class.
For $SL(n,\mathbb{C})$, Lascoux and Sch\"utzenberger \cite{LS} considered one particular choice, which yielded polynomials - the Schubert polynomials -
with particularly good combinatorial and geometric
properties.
It is a natural problem to extend
the construction in \cite{LS} to the other
classical Lie groups.
To this end, Fomin and Kirillov
\cite{FK} listed up five properties that characterize
the Schubert polynomials in type $\mathrm{A},$
but they showed that it is impossible to
construct a theory of ``Schubert polynomials''
in type $\mathrm{B}$ satisfying the same properties.
For type $\mathrm{B}_{n}$, they constructed several families
of polynomials which satisfy all but one of these properties.
There is another approach to this
problem due to Billey and Haiman
\cite{BH}. Consider one of the series of Lie types
$\mathrm{B}_{n},\mathrm{C}_{n}$, and $\mathrm{D}_{n}$
and denote by $G_{n},T_{n},$ and $B_{n}$
the corresponding classical Lie group,
a maximal torus, and a Borel subgroup
containing the maximal torus. The associated flag variety is $\mathcal{F}_{n}=G_{n}/B_{n}$, and the cohomology Schubert classes $\sigma_{w}^{(n)}$ are labeled by elements $w$ in the Weyl group
$W_{n}$ of $G_n$. There is a natural embedding of groups $G_{n}\hookrightarrow G_{n+1}$, and this induces an embedding
of the flag varieties
$\mathcal{F}_{n}\hookrightarrow \mathcal{F}_{n+1}$
and reverse maps in cohomology $H^*(\mathcal{F}_{n+1},\mathbb{Q}) \to H^{*}(\mathcal{F}_n,\mathbb{Q})$, compatible with the Schubert classes.
For each element $w$ in the infinite Weyl group $W_\infty = \bigcup_{n \ge 1} W_n$, there is
a {\it stable\/} Schubert class
$\sigma_{w}^{(\infty)}=\underleftarrow{\lim} \,
\sigma_{w}^{(n)}$
in the inverse system
$\underleftarrow{\lim} \,H^{*}(\mathcal{F}_{n},\mathbb{Q})$
of the cohomology rings.
A priori, the class $\sigma_{w}^{(\infty)}$
is represented by a homogeneous element in the ring of
power series $\mathbb{Q}[[z_1, z_2, \dots ]]$, but
Billey and Haiman showed in \cite{BH} that
it is represented by
a {\em unique} element $\mathfrak{S}_{w}$
in the subring\footnote{
The elements $\{z_{1},z_{2},\ldots;p_{1}(z),p_{3}(z),p_{5}(z),\ldots\}$ are algebraically independent, so
the ring (\ref{eq:BHring}) can also be
regarded simply as the polynomial
ring in $z_{i}$ and $p_{k}$ ($k=1,3,5,\ldots)$. }
\begin{equation}
\mathbb{Q}[z_1,z_2, \ldots;
p_{1}(z),p_{3}(z),p_{5}(z),\ldots],\label{eq:BHring}
\end{equation}
where $p_{k}(z)=\sum_{i=1}^{\infty}z_{i}^{k}$
denotes the power-sum
symmetric function.
Note that
the images of the even power-sums $p_{2i}(z)$ in the limit of the coinvariant rings
vanish for each of the types $\mathrm{B,C}$ and $\mathrm{D}$.
The elements $\{\mathfrak{S}_{w}\}$ are obtained as
the {\em unique} solutions of a system of equations involving infinitely many
divided difference operators.
These polynomials will satisfy the main combinatorial properties of the type $\mathrm{A}$ Schubert polynomials, if interpreted appropriately.
In particular, the polynomial
$\mathfrak{S}_{w}$
is stable, i.e.
it represents the Schubert classes
$\sigma_{w}^{(n)}$
in $H^*(\mathcal{F}_n,\mathbb{Q})$ simultaneously for all positive integers $n.$
The flag varieties admit an action of
the maximal torus $T_{n}$, and the inclusion $\mathcal{F}_n \hookrightarrow \mathcal{F}_{n+1}$ is equivariant
with respect to $T_{n}\hookrightarrow T_{n+1}
$. Therefore one can define an {\em equivariant} version
of the stable Schubert classes, and one can ask whether
we can `lift' the polynomials of Billey and Haiman
to the equivariant setting.
These will be the ``double Schubert polynomials", which we will define and study next.
The terminology comes from type $\mathrm{A}$, where Lascoux and Sch\"utzenberger defined a double version of their Schubert polynomials in \cite{LS} (see also \cite{MacSchubert}).
As shown by Fulton in \cite{F:deg}, the type $\mathrm{A}$ double Schubert polynomials can also be constructed as polynomials which represent the cohomology classes of some degeneracy loci. More recently, two related constructions connecting double Schubert polynomials to equivariant cohomology of flag manifolds, using either Thom polynomials or Gr\"obner degenerations were obtained independently by Feh\'er and Rim\'anyi \cite{FR} and by Knutson and Miller \cite{KM}. The degeneracy locus construction was extended to other types by Fulton \cite{F}, Pragacz-Ratajski \cite{PR} and Kresch-Tamvakis \cite{KT}. The resulting polynomials are expressed in terms of Chern classes canonically associated to the geometric situation at hand.
Their construction depends again on the choice of a
polynomial to represent the ``top class" - the diagonal class
in the cohomology of a flag bundle.
Unfortunately
different choices lead to polynomials having some desirable combinatorial properties - but not all. In particular, the polynomials in \cite{KT,F} do not satisfy the
stability property.
In this paper we will work in the equivariant cohomology of flag varieties.
As in \cite{BH}, there is a unique family of stable polynomials,
which is the unique solution of a system of divided difference operators. In our study we will make full use of {\it localization\/} techniques in equivariant cohomology. In the process, we will reprove, and put on a more solid geometric foundation, the results from \cite{BH}.
\subsection{Infinite hyperoctahedral groups}\label{ssec:infinite_weyl} To fix notations, let $W_\infty$ be the infinite hyperoctahedral group, i.e.~the Weyl group of type $\mathrm{C}_\infty$ (or $\mathrm{B}_\infty$). It is generated by elements $s_0,s_1, \ldots$ subject to the braid relations described in (\ref{eq:Wrel}) below. For each nonnegative integer $n$, the subgroup $W_n$ of $W_\infty$ generated by $s_0, \ldots , s_{n-1}$ is the Weyl group of type $\mathrm{C}_n$. $W_\infty$ contains a distinguished subgroup $W_\infty'$ of index $2$ - the Weyl group of $\mathrm{D}_\infty$ - which is generated by $s_{\hat{1}},s_{1}, s_2, \ldots$, where $s_{\hat{1}} = s_0s_1s_0$. The corresponding finite subgroup $W_n'=W_\infty' \cap W_n$ is the type $\mathrm{D}_n$ Weyl group. To be able to make statements which are uniform across all classical types, we use $\pmb{W}_{\infty}$ to denote $W_{\infty}$ when we consider types $\mathrm{C}$ or $\mathrm{B}$ and
$W_{\infty}'$ for type $\mathrm{D}$; similar notation is used for $\pmb{W}_{n}\subset \pmb{W}_{\infty}$. Finally,
set $\pmb{I}_{\infty}$ to be the indexing set $\{0,1,2,\ldots\}$
for types $\mathrm{B,C}$ and $\{\hat{1},1,2,\ldots\}$ for type $\mathrm{D}.$
\subsection{Stable Schubert classes}\label{ssec:SSch} To each element $w \in \pmb{W}_n$ there corresponds a torus-fixed point $e_{w}$ in the flag variety $\mathcal{F}_{n}=G_{n}/B_{n}$ (see \S \ref{ssec:NotationFlag} below).
The {\em Schubert variety}
$X_{w}$ is defined to be the closure of the {\em Schubert cell}
$B^{-}_{n}e_{w}$ in $\mathcal{F}_{n},$
where $B^{-}_{n}$ is the Borel subgroup opposite
to $B_{n}$. The fundamental class of $X_w$ determines a {\em Schubert class}
$\sigma_{w}^{(n)}$ in the
equivariant cohomology ring\begin{footnote}{
Unless otherwise stated, from now on we work over cohomology with coefficients over $\mathbb{Z}$.}\end{footnote} $H_{T_{n}}^{2 \ell(w)}(\mathcal{F}_{n})$, where $\ell(w)$ is the length of $w$. The classes $\{\sigma_{w}^{(n)}\}_{w\in \pmb{W}_{n}}$
form an $H_{T_{n}}^{*}(pt)$-basis
of the equivariant cohomology ring
$H_{T_{n}}^{*}(\mathcal{F}_{n}).$
Note that $H_{T_{n}}^{*}(pt)$ is canonically
identified with the polynomial ring
$\mathbb{Z}[t]^{(n)}:=\mathbb{Z}[t_{1},\ldots,t_{n}]$ generated by
a standard basis $\{t_{1},\ldots,t_{n}\}$ of the character group of $T_{n}.$
Since the torus $T_n$ acts trivially on $e_v$, the inclusion map $\iota_v:e_v \to \mathcal{F}_n$ is equivariant, and induces the {\em localization map} $\iota_{v}^{*}: H_{T_{n}}^{*}(\mathcal{F}_{n})
\rightarrow H_{T_{n}}^{*}(e_{v})$.
It is well-known (cf. e.g. \cite{A}) that
the product map
$$
\iota^{*}=(\iota_{v}^{*})_{v}: H_{T_{n}}^{*}(\mathcal{F}_{n})
\longrightarrow
\prod_{v\in \pmb{W}_{n}}
H_{T_{n}}^{*}(e_{v})
=\prod_{v\in \pmb{W}_{n}}
\mathbb{Z}[t]^{(n)}$$
is injective, so we will often
identify $\sigma_{w}^{(n)}$ with an
element in $\prod_{v\in \pmb{W}_{n}}
\mathbb{Z}[t]^{(n)}$ via $\iota^{*}.$
It turns out that the localizations of Schubert classes stabilize, in the sense that for $v,w \in \pmb{W}_n \subset \pmb{W}_m$, the polynomial $\iota_v^*(\sigma_w^{(m)})$ in $H^*_{T_{m}}(e_v)=\mathbb{Z}[t]^{(m)}$ remains constant as $m$ varies.
Therefore, we can pass to the limit to define the {\em stable (equivariant) Schubert class} $\sigma_{w}^{(\infty)
$
in $\prod_{v\in \pmb{W}_{\infty}}\mathbb{Z}[t]$,
where $\mathbb{Z}[t]$ is the polynomial ring
in the variables $t_i$ ($i \ge 1$).
Denote by $H_{\infty}$ the $\mathbb{Z}[t]$-submodule of $\prod_{v\in \pmb{W}_{\infty}}\mathbb{Z}[t]$ spanned by the stable Schubert classes. We will show that $H_{\infty}$ is actually a subalgebra of $\prod_{v\in \pmb{W}_{\infty}}\mathbb{Z}[t]$ which has a $\mathbb{Z}[t]$-basis consisting of stable Schubert classes $\{\sigma_{w}^{(\infty)}\}.$
A crucial part in the theory of Schubert polynomials of classical types is played by the $P$ and $Q$ Schur functions \cite{Schur}. These are symmetric functions $P_\lambda(x)$ and $Q_\lambda(x)$ in a new set of variables $x=(x_1, x_2, \ldots)$, and are indexed by strict partitions $\lambda$ (see \S \ref{sec:Schur} below for details). The $P$ or $Q$ Schur function corresponding to $\lambda$ with one part of length $i$ is denoted respectively by $P_i(x)$ and $Q_i(x)$.
Define $\Gamma=\mathbb{Z}[Q_{1}(x),Q_{2}(x),\ldots]$ and
$\Gamma'=\mathbb{Z}[P_{1}(x),P_{2}(x),\ldots]$. Note that $\Gamma$ and $\Gamma'$ are not polynomial rings, since $Q_i(x)$ respectively $P_i(x)$ are not algebraically independent (see \S \ref{sec:Schur} for the relations among them), but they have canonical $\mathbb{Z}$-bases consisting of the $Q$-Schur functions $Q_\lambda(x)$ (respectively $P$-Schur functions $P_\lambda(x)$).
We define next the $\mathbb{Z}[t]$-algebras of {\em Schubert polynomials} $$R_{\infty}=\mathbb{Z}[t]\otimes_{\mathbb{Z}}\Gamma\otimes_{\mathbb{Z}}\mathbb{Z}[z],\quad
R'_{\infty}=\mathbb{Z}[t]\otimes_{\mathbb{Z}}\Gamma'\otimes_{\mathbb{Z}}\mathbb{Z}[z],
$$
where $\mathbb{Z}[z]=\mathbb{Z}[z_{1},z_{2},\ldots]$ is the polynomial
ring in $z=(z_{1},z_{2},\ldots).$ We will justify the terminology in the next paragraph.
Again, in order to state results uniformly in all types, we use the bold letter $\pmb{R}_{\infty}$
to denote $R_{\infty}$ for type $\mathrm{C}$ and $R'_{\infty}$
for types $\mathrm{B}$ and $\mathrm{D}$.
There exists a homomorphism
$$
\Phi=(\Phi_{v})_{v\in \pmb{W}_{\infty}} : \pmb{R}_{\infty}\longrightarrow
\prod_{v\in \pmb{W}_{\infty}}\mathbb{Z}[t]
$$
of graded $\mathbb{Z}[t]$-algebras,
which we call {\em universal localization map}. Its precise (algebraic) definition is given in \S \ref{sec:UnivLoc} and it has a natural geometrical interpretation explained in \S \ref{sec:geometry}. One of the main results in this paper is that $\Phi$ is an isomorphism
from $\pmb{R}_{\infty}$ onto $H_{\infty}$
(cf. Theorem \ref{thm:isom} below). While injectivity is easily proved algebraically, surjectivity is more subtle. It uses the {\em transition equations}, which are recursive formulas which allow writing any stable Schubert class $\sigma_w^{(\infty)}$ in terms of
the (Lagrangian or Orthogonal) Grassmannian Schubert classes. Once reduced to the Grassmannian case, earlier results of the first and third author \cite{Ik,IN}, which show that the classes in question are represented by Ivanov's {\em factorial $Q$ (or $P$)} Schur functions \cite{Iv}, finish the proof of surjectivity.
By pulling back - via $\Phi$ - the stable Schubert classes,
we introduce polynomials $\mathfrak{S}_{w}
=\mathfrak{S}_{w}(z,t;x)$ in $\pmb{R}_{\infty}$, which are uniquely determined by
the "localization equations"
\begin{equation}
\Phi_{v}\left(
\mathfrak{S}_{w}(z,t;x)\right)
=\iota_{v}^{*}(\sigma_{w}^{(\infty)}),
\label{eq:Phisigma}
\end{equation}
where
$\iota_{v}^{*}(\sigma_{w}^{(\infty)})
\in \mathbb{Z}[t]$ is the stable limit
of $\iota_{v}^{*}(\sigma_{w}^{(n)})$.
\subsection{Divided difference operators}
Alternatively, $\{\mathfrak{S}_{w}(z,t;x)\}$ can be characterized
in a purely algebraic manner by using the divided difference operators.
There are two families of operators $\partial_i, \delta_i$ ($i \in \pmb{I}_\infty$) on $\pmb{R}_\infty$, such that operators from one family commute with those from the other (see \S \ref{ssec:divdiff}
for the definition). Then:
\begin{thm
\label{existC}
There exists a unique family of elements $\mathfrak{\mathfrak S}_{w}= \mathfrak S_w(z,t;x) $ in $\pmb{R}_\infty$, where $w\in \pmb{W}_{\infty}$,
satisfying the equations
\begin{equation}\label{E:dd} \partial_{i}\mathfrak{S}_{w}=\begin{cases}
\mathfrak{S}_{ws_{i}}\quad\mbox{if}\quad \ell(ws_{i})<\ell(w)\\
0 \quad \mbox{otherwise}
\end{cases},\quad
\delta_{i}\mathfrak{S}_{w}=\begin{cases}
\mathfrak{S}_{s_{i}w}\quad\mbox{if}\quad \ell(s_{i}w)<\ell(w)\\
0 \quad \mbox{otherwise}
\end{cases},
\end{equation}
for all $i\in \pmb{I}_{\infty}$,
and such that $\mathfrak{S}_{w}$ has no
constant term except for
$\mathfrak{S}_{e}=1$.
\end{thm}
The operators $\partial_{i},\delta_{i}$
are the limits of the same operators on the equivariant
cohomology $H^*_{T_n} (\mathcal{F}_n)$, since the latter are compatible with the projections $H^*_{T_{n+1}}(\mathcal{F}_{n+1}) \to H^*_{T_n}(\mathcal{F}_n)$. In this context, the operator $\partial_i$ is an equivariant generalization the operator defined in \cite{BGG,D}, and it can shown that it is induced by the right action of the Weyl group on the equivariant cohomology (cf. \cite{KK,Kn}). The operator $\delta_i$ exists only in equivariant cohomology, and it was used in \cite
{Kn,Ty} to study equivariant Schubert classes. It turns out that it corresponds to a left Weyl group action on $H^*_{T_n} (\mathcal{F}_n)$.
\subsection{Billey and Haiman's polynomials and a change of variables}
The polynomials from Theorem \ref{existC} are lifts of Billey-Haiman polynomials from the non-equivariant
to equivariant cohomology.
Concretely, if we forget the torus action,
then
$$
\mathfrak{S}_{w}(z,0;x)=\mathfrak{S}_{w}^{BH}(z;x)
$$
where $\mathfrak{S}_{w}^{BH}(z;x)$ denotes
the Billey-Haiman's polynomial. This follows immediately from the Theorem \ref{existC}, since $\{\mathfrak{S}_{w}^{BH}(z;x)\}$
are the unique elements in
$\Gamma\otimes_{\mathbb{Z}}\mathbb{Z}[z]$ (for type $\mathrm{C}$)
or $\Gamma'\otimes_{\mathbb{Z}}\mathbb{Z}[z]$ (for type $\mathrm{B,D}$)
which satisfy the equation involving the right divided difference operators
$\partial_{i}.$
The variables $z_i$ in $\mathfrak{S}_w(z,t;x)$ correspond geometrically to the limits of Chern classes of the tautological line bundles, while the variables $t_i$ to the equivariant parameters. To understand why the variables $x_i$ are needed - both algebraically and geometrically - we comment next on the ``change of variables'' which relates the Billey-Haiman polynomials to those defined by Fomin and Kirrilov in \cite[Thm. 7.3]{FK}. The general formula in our situation - along with its geometrical explanation - will be given in section \ref{sec:geometry} below. The relation between
$x$ and $z$ is given by
$$
\prod_{i=1}^{\infty}\frac{1+x_{i}u}{1-x_{i}u}
=\prod_{i=1}^{\infty}(1-z_{i}u),
$$
or equivalently
$p_{2i}(z)=0$ and $-2p_{2i+1}(x)=p_{2i+1}(z).$ In type $\mathrm{C}$, it is known that $p_{2i}(z)$ generates the (limit of the) ideal of relations in cohomology, therefore such a variable change eliminates the ambiguity of representatives
coming from $p_{2i}(z)=0.$ Note that the change of variables can be also
expressed as $(-1)^{i}e_{i}(z)=Q_{i}(x)=2P_{i}(x)\;(i\geq 1).$
It follows that after extending the scalars from $\mathbb{Z}$ to $\mathbb{Q}$, the ring $\Gamma\otimes_{\mathbb{Z}}\mathbb{Z}[z]$
(or $\Gamma'\otimes_{\mathbb{Z}}\mathbb{Z}[z]$) is identified with
the ring (\ref{eq:BHring}).
Since both $\Gamma$ and $\Gamma'$ have
distinguished $\mathbb{Z}$-bases,
the polynomials $\mathfrak{S}^{BH}_{w}(z;x)$ will expand uniquely as a combination of $Q$-Schur (or $P$-Schur) functions with coefficients in $\mathbb{Q}[z]$.
In type $\mathrm{C}$, the change of variables corresponds to making $Q_i(x)=c_i(\mathcal{S}^*)$ - where $c_i(\mathcal{S}^*)$ is the limit of the Chern classes of the duals of the tautological subbundles of the Lagrangian Grassmannians, regarded as elements in $\underleftarrow{\lim} H^*(\mathcal{F}_n)$. This is the identification which was used by Pragacz (see e.g. \cite[pag. 32]{FP}) to study the cohomology of the Lagrangian Grassmannian.
\subsection{Combinatorial properties of the double Schubert polynomials} We state next
the combinatorial properties of the double Schubert polynomials $\mathfrak S_w(z,t;x)$:
\begin{itemize}
\item(Basis) $\mathfrak S_w(z,t;x)$ form a $\mathbb{Z}[t]$-basis of $\pmb{R}_\infty$;
\item(Symmetry) $\mathfrak S_w(z,t;x) = \mathfrak S_{w^{-1}}(-t,-z;x)$;
\item(Positivity) The double Schubert polynomial $\mathfrak S_w(z,t;x)$ can be uniquely written as \[ \mathfrak S_w(z,t;x) = \sum_{\lambda} f_\lambda(z,t) F_\lambda(x) \/, \] where the sum is over strict partitions $\lambda=(\lambda_1, \dots , \lambda_r)$ such that $\lambda_1+ \dots +\lambda_r \le \ell(w)$, $f_\lambda(z,t)$ is a homogeneous polynomial in $\mathbb{N}[z,-t]$,
and $F_\lambda(x)$ is the $Q$-Schur function $Q_\lambda(x)$ in type $\mathrm{C}$, respectively the $P$-Schur function $P_\lambda(x)$ in types $\mathrm{B,D}$.
For a precise combinatorial formula for the
coefficients $f_\lambda(z,t)$ see Cor. \ref{cor:typeAexpand} and Lem. \ref{lem:S00} below.
\end{itemize}
The basis property implies that we can define the structure constants
$c_{uv}^{w}(t)\in \mathbb{Z}[t]$ by the
expansion
$$
\mathfrak{S}_{u}(z,t;x)
\mathfrak{S}_{v}(z,t;x)
=\sum_{w \in \pmb{W}_{\infty}}c_{uv}^{w}(t)\mathfrak{S}_{w}(z,t;x).
$$ These coincide with the structure constants in equivariant cohomology of $\mathcal{F}_n$, written in a stable form. The same phenomenon happens in \cite{LS,BH}.
\subsection{Grassmannian Schubert classes}
To each strict partition $\lambda$ one can associate a {\em Grassmannian element} $w_\lambda \in \pmb{W}_\infty$.
Geometrically these arise as the elements
in $\pmb{W}_{\infty}$ which index the pull-backs of the Schubert classes from the appropriate Lagrangian or Orthogonal Grassmannian, via the natural projection from the flag variety. For the Lagrangian Grassmannian,
the first author \cite{Ik} identified the equivariant Schubert
classes with the factorial analogues
of Schur $Q$-function defined by Ivanov \cite{Iv}.
This result was extended
to the maximal isotropic
Grassmannians of orthogonal types $\mathrm{B}$ and $\mathrm{D}$
by Ikeda and Naruse \cite{IN}.
See \S \ref{sec:Schur} for the definition
of Ivanov's functions
$Q_{\lambda}(x|t),P_{\lambda}(x|t).$
We only mention here that if
all $t_{i}=0$ they coincide with the ordinary $Q$ or $P$
functions;
in that case, these results recover Pragacz's results from \cite{Pr} (see also \cite{Jo}).
We will show in Theorem \ref{PhiFacQ}
that the polynomial $\mathfrak S_{w_\lambda}(z,t;x)$ coincides with $Q_\lambda(x|t)$
or $P_\lambda(x|t)$, depending on the type at hand.
In particular, the double Schubert polynomials
for the Grassmannian elements
are Pfaffians - this is a {\em Giambelli formula} in this case.
\subsection{Longest element formulas}
Next we present the combinatorial formula for the double Schubert polynomial indexed by $w_0^{(n)}$, the longest element in $\pmb{W}_n$ (regarded as a subgroup of $\pmb{W}_\infty$). This formula has a particular significance since this is the top class mentioned in the first section.
We denote by $\mathfrak{B}_{w},\mathfrak{C}_{w},
\mathfrak{D}_{w}$ the double Schubert polynomial
$\mathfrak{S}_{w}$ for types $\mathrm{B,C},$ and $\mathrm{D}$ respectively.
Note that $\mathfrak{B}_{w}=2^{-s(w)}
\mathfrak{C}_{w},$ where $s(w)$ is the number of
signs changed by $w$ (cf. \S \ref{sec:signed.permut} below).
\begin{thm}[Top classes]\label{thm:Top} The double Schubert polynomial
associated with the longest element $w_{0}^{(n)}$ in $\pmb{W}_{n}$
is equal to:
\begin{enumerate}
\item
$\mathfrak{C}_{w_{0}^{(n)}}(z,t;x) =
Q_{\rho_{n}+\rho_{n-1}}
(x|t_{1},-z_{1},t_{2},-z_{2},\ldots,t_{n-1},-z_{n-1}),\label{longC}$
\item $\mathfrak{D}_{w_{0}^{(n)}}(z,t;x) =P_{2\rho_{n-1}}
(x|t_{1},-z_{1},t_{2},-z_{2},\ldots,t_{n-1},-z_{n-1}),\label{longD}$
\end{enumerate}
where $\rho_{k}=(k,k-1,\ldots,1).$
\end{thm}
\subsection{Comparison with degeneracy loci formulas}
One motivation for the present paper was to give a
geometric interpretation
to the factorial Schur $Q$-function by means of degeneracy loci formulas.
In type $\mathrm{A}$,
this problem was treated by the second author in \cite{Mi},
where the Kempf-Laksov formula for degeneracy loci
is identified with the Jacobi-Trudi type
formula for the factorial (ordinary) Schur function. To this end,
we will reprove a multi-Pfaffian expression for $\sigma_{w_\lambda}$ (see \S \ref{sec:Kazarian} below) obtained by Kazarian \cite{Ka} while studying Lagrangian degeneracy loci.
\subsection{Organization}
Section \ref{sec:EqSch} is devoted to some general
facts about the equivariant cohomology of the
flag variety.
In section \ref{sec:Classical} we fix notation concerning
root systems and Weyl groups, while in section \ref{sec:Schur} we give the definitions and some
properties of
$Q$- and $P$-Schur functions, and of their factorial analogues.
The stable (equivariant) Schubert classes
$\{\sigma_{w}^{(\infty)}\}$
and the ring $H_{\infty}$ spanned by these classes are introduced in section \ref{sec:StableSchubert}.
In section \ref{sec:UnivLoc} we define the ring of Schubert polynomials $\pmb{R}_{\infty}$
and establish the isomorphism
$\Phi: \pmb{R}_{\infty}\rightarrow H_{\infty}.$
In the course of the proof, we recall
the previous results on isotropic Grassmannians
(Theorem \ref{PhiFacQ}).
In section \ref{sec:WactsR} we define the left and right action of the infinite Weyl group on ring
$\pmb{R}_{\infty},$ and then use them to define the divided
difference operators.
We also discuss the compatibility
of the actions on both $\pmb{R}_{\infty}$ and $H_{\infty}$
under the isomorphism $\Phi.$
We will prove the existence and
uniqueness theorem for the double Schubert polynomials
in section \ref{sec:DSP}, along with some basic combinatorial properties
of them. The formula for the Schubert polynomials indexed by the longest Weyl group element is proved in
section \ref{sec:Long}.
Finally, in section \ref{sec:geometry} we give an alternative geometric
construction of our universal localization map $\Phi$, and
in section \ref{sec:Kazarian}, we prove the formula for $Q_{\lambda}(x|t)$
in terms of a multi-Pfaffian.
\subsection{Note}
After the present work was completed
we were informed that
A. Kirillov \cite{K} had introduced
double Schubert polynomials
of type $\mathrm{B}$ (and $\mathrm{C}$) in 1994
by using Yang-Baxter operators (cf. \cite{FK}), independently to us,
although no connection with (equivariant) cohomology had been established.
His approach is quite different from ours, nevertheless
the polynomials are the same, after a suitable identification
of variables. Details will be given elsewhere.
\bigskip
This is the full paper version of
`extended abstract' \cite{fpsac} for the FPSAC 2008
conference held in Vi\~na del Mar, Chile, June 2008.
Some results in this paper
were announced without proof in \cite{rims}.
\subsection{Acknowledgements}
We would like to thank S.~Billey and H.~Tamvakis for stimulating conversations
that motivated this work, and to S. Kumar, K.~Kuribayashi, M.~Mimura, M.~Nakagawa, T. Ohmoto, N.~Yagita and M. Yasuo for helpful comments. This work was facilitated by the "Workshop on Contemporary Schubert Calculus and Schubert Geometry" organized at Banff in March 2007. We are grateful to the organizers J. Carrell and F. Sottile for inviting all the authors there.
\section{Equivariant Schubert classes of the flag variety}
\label{sec:EqSch}
\setcounter{equation}{0}
In this section we will recall some basic facts about the equivariant cohomology of the flag variety $\mathcal{F}=G/B$.
The main references are \cite{A} and \cite{KK} (see also \cite{Ku}).
\subsection{Schubert varieties and equivariant cohomology}\label{ssec:NotationFlag}
Let $G$ be a complex connected semisimple Lie group,
$T$ a maximal torus, $W=N_{G}(T)/T$ its Weyl group,
and $B$ a Borel subgroup such that $T\subset B.$ The flag variety is the variety $\mathcal{F}=G/B$ of translates of the Borel subgroup $G$, and it admits a $T$-action, induced by the left $G$-action. Each Weyl group element determines a $T$-fixed point $e_w$ in the flag variety (by taking a representative of $w$), and these are all the torus-fixed points. Let $B^{-}$ denote the opposite Borel subgroup. The {\em Schubert variety} $X_w$ is the closure of $B^- e_w$ in the flag variety; it has codimension $\ell(w)$ - the length of $w$ in the Weyl group $W$.
In general, if $X$ is a topological space with a left $T$-action, the equivariant cohomology of $X$ is the ordinary cohomology of a ``mixed space'' $(X)_T$, whose definition (see e.g.
\cite{GKM} and references therein) we recall. Let $ET
\longrightarrow BT$ be the universal $T-$bundle. The $T-$action on
$X$ induces an action on the product $ET \times X$ by $t\cdot
(e,x)=(et^{-1},tx)$. The quotient space $(X)_T=(ET \times X)/T$ is
the ``homotopic quotient'' of $X$ and the ($T-$)equivariant
cohomology of $X$ is by definition
\[ H^{i}_T(X)=H^i(X_T). \] In particular, the equivariant
cohomology of a point, denoted by $\mathcal{S}$, is equal to the
ordinary cohomology of the classifying space $BT$. If $\chi$ is a
character in $\hat{T}=Hom(T, \mathbb{C}^*)$ it determines a line
bundle $L_\chi: ET\times_T \mathbb{C}_\chi \longrightarrow BT$ where
$\mathbb{C}_\chi$ is the $1-$dimensional $T-$module determined by $\chi$.
It turns out that the morphism $\hat{T} \longrightarrow
H^2_T(pt)$ taking the character $\chi$ to the first Chern
class $c_1(L_\chi)$ extends to an isomorphism from the symmetric
algebra of $\hat{T}$ to $H^*_T(pt)$. Therefore, if one chooses a basis $t_1, \ldots , t_n$ for $\hat{T}$, then $\mathcal{S}$ is the polynomial ring $\mathbb{Z}[t_1, \ldots , t_n]$.
Returning to the situation when $X=\mathcal{F}$, note that $X_{w}$ is a $T$-stable,
therefore its fundamental class determines
the {\em (equivariant) Schubert class} $\sigma_{w}=[X_{w}]_T$ in
$H_{T}^{2\ell(w)}(\mathcal{F}).$
It is well-known that the Schubert classes form an $H_{T}^{*}(pt)$-basis of $H_{T}^{*}(\mathcal{F}).$
\subsection{Localization map}
Denote by $\mathcal{F}^T= \{e_v| v \in W\}$ the set of $T$-fixed points in $\mathcal{F}$; the inclusion
$\iota: \mathcal{F}^{T}\hookrightarrow \mathcal{F}$
is $T$-equivariant and
induces
a homomorphism
$\iota^{*}: H_{T}^{*}(\mathcal{F})\longrightarrow
H_{T}^{*}(\mathcal{F}^{T})=\prod_{v\in W}H_{T}^{*}(e_{v}).$
We identify each $H_{T}^{*}(e_{v})$ with $\mathcal{S}$ and for $\eta \in H^*_T (\mathcal{F})$ we denote its localization in $H_{T}^{*}(e_{v})$ by $\eta|_v$.
Let $R^{+}$ denote the set of positive roots corresponding
to $B$ and set $R^{-}=-R^{+},
R=R^{+}\cup R^{-}.$
Each root $\alpha$ in $R$ can be
regarded as a linear
form in $\mathcal{S}.$ Let $s_{\alpha}$ denote the reflection
corresponding to the root $\alpha.$ Remarkably, the localization map $\iota^*$ in injective,
and the elements $\eta=(\eta|_{v})_{v}$ in $\prod_{v\in W}\mathcal{S}$ in the image of $\iota^*$
are characterized by the {\em GKM conditions} (see e.g. \cite{A}):
$$
\eta|_{v}-\eta|_{s_{\alpha}v}\;\mbox{is a multiple
of}\;\alpha
$$
for all $v$ in $W$ and $\alpha\in R^{+},$ where $s_\alpha \in W$ is the reflection associated to $\alpha$.
\subsection{Schubert classes}
We recall a characterization of the Schubert class $\sigma_{w}.$
Let $\leq$ denote the Bruhat-Chevalley ordering
on $W$; then $e_{v}\in X_{w}$ if and only if
$w\leq v.$
\begin{prop}\cite{A},\cite{KK} \label{xi}
The Schubert class $\sigma_{w}$ is characterized by the
following conditions:
\begin{enumerate}
\item $\sigma_w|_{v}$ vanishes unless $w\leq v,$
\item If $w\leq v$ then $\sigma_{w}|_{v}$ is homogeneous of degree $\ell(w)$,
\item $\sigma_w|_{w}=\prod_{\alpha\in R^{+}\cap wR^{-}}\alpha.$
\end{enumerate}
\end{prop}
\begin{prop}\label{prop:basis}
Any cohomology class $\eta$ in $H_{T}^{*}(\mathcal{F})$
can be written uniquely as an
an $H_{T}^{*}(pt)$-linear
combination of $\sigma_{w}$ using only
those $w$ such that $w\geq u$ for some
$u$ with $\eta|_{u}\ne 0.$
\end{prop}
\begin{proof}
The corresponding fact
for the Grassmann variety
is proved in \cite{KnTa}.
The same proof works for the general flag variety also.
\end{proof}
\subsection{Actions of Weyl group}\label{ssec:WeylAction}
There are two actions of the Weyl group on the equivariant cohomology ring $H_{T}^{*}(\mathcal{F})$, which are used to define corresponding divided-difference operators.
In this section we will follow the approach presented in \cite{Kn}. Identify $\eta \in H_{T}^{*}(\mathcal{F})$ with the sequence of polynomials $(\eta|_v)_{v \in W}$ arising from the localization map. For $w \in W$ define
\[ (w^{R}\eta)|_v = \eta|_{vw} \quad (w^{L}\eta)|_v = w \cdot (\eta|_{w^{-1}v}) \/.\]
It is proved in \cite{Kn} that these are well defined actions on $H^*_{T_n}(\mathcal{F}_n),$ and that $w^{R}$ is $H^*_{T}(pt)$-linear, while $w^{L}$ it is not (precisely because it acts on the polynomials' coefficients).
\subsection{Divided difference operators}\label{ssec:divdiff}
For each simple root $\alpha_{i}$, we define
the {\it divided difference operators\/}
$\partial_{i}$ and $\delta_{i}$
on
$H_{T}^{*}(\mathcal{F})$
by
$$
(\partial_{i}\eta)|_{v}
=\frac{\eta|_{v}-(s_{i}^{R}\eta)|_{v}}{-v(\alpha_{i})},\quad
(\delta_{i}\eta)_{v}=\frac{\eta|_{v}-(s_{i}^{L}\eta)|_{v}}{\alpha_{i}}\quad (v\in W).
$$
These rational functions are proved to be actually polynomials. They satisfy the GKM conditions, and thus
give elements in $H_{T}^{*}(\mathcal{F})$
(see \cite{Kn}).
We call $\partial_{i}$'s (resp. $\delta_{i}$'s) {\it right\/}
(resp. {\it left\/}) divided difference operators.
The operator $\partial_{i}$ was introduced
in \cite{KK}.
On the ordinary cohomology, analogous operators
to $\partial_{i}$'s are introduced
independently by Bernstein et al. \cite{BGG}
and Demazure \cite{D}.
The left divided difference operators
$\delta_{i}$ was studied by Knutson in \cite{Kn}
(see also \cite{Ty}). Note that $\partial_{i}$ is $H_{T}^{*}(pt)$-linear
whereas $\delta_{i}$ is not. The next proposition was stated \cite[Prop.2]{Kn} (see also \cite{KK,Ty}).
\begin{prop
\label{prop:propertiesdiv}
\begin{enumerate}
\item \label{welldef} Operators $\partial_{i}$ and $\delta_{i}$
are well-defined on the ring $H_{T}^{*}(\mathcal{F})$;
\item The left and right divided difference
operators commute with each other;
\item \label{eq:partial_sigma} We have
\begin{equation}
\partial_{i}\sigma_{w}=\begin{cases}
\sigma_{ws_{i}} &\mbox{if}\;\ell(ws_{i})=\ell(w)-1\\
0 &\mbox{if}\;\ell(ws_{i})=\ell(w)+1
\end{cases},\quad
\delta_{i}\sigma_{w}=\begin{cases}
\sigma_{s_{i}w} &\mbox{if}\;\ell(s_{i}w)=\ell(w)-1\\
0 &\mbox{if}\;\ell(s_{i}w)=\ell(w)+1
\end{cases}.\label{eq:divdiffsigma}
\end{equation}
\end{enumerate}
\end{prop}
\begin{proof} We only prove (\ref{eq:divdiffsigma})
for $\delta_{i}$ here, as the rest is proved in \cite{Kn}.
By Prop. \ref{xi}
$(\delta_{i}{\sigma}_{w})|_{v}$ is nonzero
only for $\{v|\;v\geq w\;\mbox{or}\; s_{i}v\geq w\}.$
This implies that
the element $\delta_{i}{\sigma}_{w}$
is a $H_{T}^{*}(pt)$-linear combination of
$\{{\sigma}_{v}\,|\,v\geq w\;\mbox{or}\; s_{i}v\geq w\}$
by Prop. \ref{prop:basis}.
Moreover $\sigma_{v}$ appearing in the linear combination
have degree at most $\ell(w)-1.$
Thus if $\ell(s_{i}w)=\ell(w)+1$ then
$\delta_{i}{\sigma}_{w}$ must vanish.
If $\ell(s_{i}w)=\ell(w)-1$
the only possible
term is a multiple of ${\sigma}_{s_{i}w}.$
In this case we calculate
$$
\left(\delta_{i}{\sigma}_{w}\right)|_{s_{i}w}
=\frac{{\sigma}_{w}|_{s_{i}w}
-s_{i}({\sigma}_{w}|_{w})}{\alpha_{i}}
=-\frac{s_{i}({\sigma}_{w}|_{w})}{\alpha_{i}},
$$
where we used
${\sigma}_{w}|_{s_{i}w}=0$
since $s_{i}w<w.$
Here we recall the following well-known fact that
$$w>s_{i}w\Longrightarrow
s_{i}(R^{+}\cap w R^{-})=
(R^{+}\cap s_{i}w R^{-})\sqcup \{-\alpha_{i}\}.$$
So we have $$s_{i}({\sigma}_{w}|_{w})
=\prod_{\beta\in s_{i}(R^{+}\cap w R^{-})}\beta
=(-\alpha_{i})\prod_{\beta\in (R^{+}\cap s_{i}w R^{-})}\beta
=(-\alpha_{i})\cdot{\sigma}_{s_{i}w}|_{s_{i}w}.$$
By the characterization (Prop. \ref{xi}),
we have $\delta_{i}{\sigma}_{w}=\sigma_{s_{i}w}.$
\end{proof}
\section{Classical groups}
\label{sec:Classical}
\setcounter{equation}{0}
In this section, we fix the notations
for the root systems, Weyl groups,
for the classical groups
used throughout the paper.
\subsection{Root systems}\label{ssection:root_systems} Let $G_n$ be the classical Lie group of one of the types $\mathrm{B}_n,\mathrm{C}_n$ or $\mathrm{D}_n$, i.e. the symplectic group $Sp(2n,\mathbb{C})$ in type $\mathrm{C}_n$, the odd orthogonal group $SO(2n+1,\mathbb{C})$
in types $\mathrm{B}_n$ and $SO(2n,\mathbb{C})$ in type $\mathrm{D}_n$.
Correspondingly we have the set $R_{n}$ of
roots, and the set of simple roots. These are subsets of
the character group
$\hat{T}_{n}=\bigoplus_{i=1}^{n}\mathbb{Z} t_{i}$
of $T_{n},$ the maximal torus of $G_{n}.$
The positive roots $R_{n}^{+}$
(set $R_{n}^{-}:=-R_{n}^{+}$ the negative roots)
are given by
\begin{eqnarray*}
\mbox{Type}\; \mathrm{B}_{n}:&
R_{n}^{+}&=\{t_{i}\;|\; 1\leq i\leq n\}
\cup\{t_{j}\pm t_{i}\;|\;1\leq i<j\leq n\},\\
\mbox{Type}\; \mathrm{C}_{n}:&
R_{n}^{+}&=\{2t_{i}\;|\; 1\leq i\leq n\}
\cup\{t_{j}\pm t_{i}\;|\;1\leq i<j\leq n\},\\
\mbox{Type}\; \mathrm{D}_{n}:&
R_{n}^{+}&=\{t_{j}\pm t_{i}\;|\;1\leq i<j\leq n\}.
\end{eqnarray*}
The following are the simple roots:
\begin{eqnarray*}
\mbox{Type}\; \mathrm{B}_{n}:&
\alpha_{0}&=t_{1},
\quad \alpha_{i}=t_{i+1}-t_{i}\quad(1\leq i\leq n-1) ,\\
\mbox{Type}\; \mathrm{C}_{n}:&
\alpha_{0}&=2t_{1},
\quad
\alpha_{i}=t_{i+1}-t_{i}\quad(1\leq i\leq n-1),\\
\mbox{Type}\; \mathrm{D}_{n}:&
\alpha_{\hat{1}}&=t_{1}+t_{2},
\quad \alpha_{i}=t_{i+1}-t_{i}\quad(1\leq i\leq n-1).
\end{eqnarray*}
We introduce a symmetric
bilinear form $(\cdot,\cdot)$ on
$\hat{T}_{n}\otimes_{\mathbb{Z}}\mathbb{Q}$
by $(t_{i},t_{j})=\delta_{i,j}.$
The simple {\it coroots\/} $\alpha_{i}^{\vee}$
are defined to be $\alpha_{i}^{\vee}=2\alpha_{i}/(\alpha_{i},\alpha_{i}).$
Let $\omega_{i}$ denote the {\it fundamental weights}, i.e. those elements in $\hat{T}_{n}\otimes_{\mathbb{Z}}\mathbb{Q}$ such that $(\omega_{i},\alpha_{i}^{\vee})=\delta_{i,j}$.
They are explicitly given as follows:
\begin{eqnarray*}
\mbox{Type}\; \mathrm{B}_{n}:&
\omega_{0}&={\textstyle\frac{1}{2}}\left(t_{1}+t_{2}+\cdots+t_{n}\right),
\quad \omega_{i}=t_{i+1}+\cdots+t_{n}\quad(1\leq i\leq n-1),\\
\mbox{Type}\; \mathrm{C}_{n}:&
\omega_{i}&=t_{i+1}+\cdots+t_{n}\quad(0\leq i\leq n-1),\\
\mbox{Type}\; \mathrm{D}_{n}:&
\omega_{\hat{1}}&={\textstyle\frac{1}{2}}\left(t_{1}+t_{2}+\cdots+t_{n}\right),\quad
\omega_{1}={\textstyle\frac{1}{2}}\left(-t_1+t_2+\cdots+t_n\right),\\
&\omega_{i}&=t_{i+1}+\cdots+t_{n}\quad(2\leq i\leq n-1).
\end{eqnarray*}
\subsection{Weyl groups}\label{ssec:Weyl}
Set $I_{\infty}=\{0,1,2,\ldots\}$ and $I_{\infty}'
=\{\hat{1},1,2,\ldots\}.$
We define the Coxeter
group $(W_{\infty},I_{\infty})$
(resp. $(W_{\infty}',I_{\infty})$)
of infinite rank, and its finite {\it parabolic\/} subgroup
$W_{n}$
(resp. $W'_{n}$)
by the following Coxeter graphs:
\noindent
$\mathrm{C}_{n}\subset \mathrm{C}_{\infty}$ ($\mathrm{B}_{n}\subset \mathrm{B}_{\infty}$)
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\put(4,16.5){\line(1,0){12}}
\put(4,18.5){\line(1,0){12}}
\multiput(15,15)(15,0){4}{
\put(0,0){$\circ$}
\put(4,2.4){\line(1,0){12}}}
\put(75,15){$\circ$}
\put(120,10)
{\put(0,5){$\circ$}
\put(4,6.5){\line(1,0){12}}
\put(4,8.5){\line(1,0){12}}
\multiput(15,5)(15,0){5}{
\put(0,0){$\circ$}
\put(4,2.4){\line(1,0){12}}}
\put(90,5){$\circ$}}
\put(0,8){\tiny{$s_0$}}
\put(15,8){\tiny{$s_1$}}
\put(30,8){\tiny{$s_2$}}
\put(72,8){\tiny{$s_{n-1}$}}
\put(120,8){\tiny{$s_0$}}
\put(135,8){\tiny{$s_1$}}
\put(150,8){\tiny{$s_2$}}
\put(190,8){\tiny{$s_{n-1}$}}
\put(210,8){\tiny{$s_{n}$}}
\put(170,8){\tiny{$\cdots$}}
\put(50,8){\tiny{$\cdots$}}
\put(95,13){$\hookrightarrow$}
\put(214.5,17.5){\line(1,0){10}}
\put(226,15){$\cdots$}
\end{picture}
\end{center}
$\mathrm{D}_{n}\subset \mathrm{D}_{\infty}$
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\put(0,20){\tiny{$s_{\hat{1}}$}}
\put(0,0){\tiny{$s_1$}}
\put(15,8){\tiny{$s_2$}}
\put(30,8){\tiny{$s_3$}}
\put(72,8){\tiny{$s_{n-1}$}}
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More explicitly,
the group $W_{\infty}$ (resp. $W_{\infty}'$ ) is
generated by the simple reflections
$s_{i}\,(i\in I_{\infty})$ (resp. $s_{i}\,(i\in I_{\infty}')$)
subject to the relations:
\begin{equation}
\begin{cases}s_{i}^{2}=e\;(i\in I_{\infty})\\
s_{0}s_{1}s_{0}s_{1}=s_{1}s_{0}s_{1}s_{0}\\
s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1}
\; (i\in I_{\infty}\setminus \{0\})\\
s_{i}s_{j}=s_{j}s_{i}\;(|i-j|\geq 2)
\end{cases},\quad
\begin{cases}
s_{i}^{2}=e\;( i\in I_{\infty}')\\
s_{\hat{1}}s_{2}s_{\hat{1}}=s_{2}s_{\hat{1}}s_{2}\\
s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1}
\; (i\in I_{\infty}'\setminus \{\hat{1}\})\\
s_{\hat{1}}s_{i}=s_{i}s_{\hat{1}}\;(i\in I_{\infty}',\;i\ne 2)\\
s_{i}s_{j}=s_{j}s_{i}\;(\;i,j\in I_{\infty}'\setminus \{\hat{1}\},
\,|i-j|\geq 2)
\end{cases}.\label{eq:Wrel}
\end{equation}
For general facts on Coxeter groups, we refer to
\cite{BjBr}.
Let $\leq $ denote the {\it Bruhat-Chevalley order\/}
on $W_{\infty}$ or $W_{\infty}'.$ The {\it length\/} $\ell(w)$ of $w\in W_{\infty}$ (resp. $w\in W'_{\infty}$) is defined to be the least number $k$ of simple reflections
in any reduced expression of $w\in W_{\infty}$.
The subgroups $W_{n}
\subset W_{\infty}$,
$W_{n}'\subset W_{\infty}'$ are
the Weyl groups of the following types:
$$
\mbox{Type}\; B_{n},C_{n}:
W_{n}=\langle s_{0},s_{1},s_{2},\ldots,s_{n-1}
\rangle,\quad
\mbox{Type}\; D_{n}:
W_{n}'=\langle s_{\hat{1}},s_{1},s_{2},\ldots,s_{n-1}\rangle.
$$
It is known that the inclusion $W_{n}\subset W_{\infty}$
(resp. $W_{n}'\subset W_{\infty}'$) preserves the length and the Bruhat-Chevalley order, while $W_{\infty}'\subset W_{\infty},$
(resp. $W_{n}'\subset W_{n}$) is not (using terminology from \cite{BjBr} this says that $W_n$ is a {\em parabolic subgroup} of $W_\infty$, while $W'_\infty$ is not). From now on, whenever possible, we will employ the notation explained in \S \ref{ssec:infinite_weyl}, and use bold fonts $\pmb{W}_\infty$ respectively $\pmb{W}_n$ to make uniform statements.
\subsection{Signed permutations}\label{sec:signed.permut}
The group $W_\infty$
is identified with the set of all permutations $w$ of the set $\{1,2,\ldots\}\cup \{\bar{1},\bar{2},\ldots\}$ such that $w(i)\ne i$ for only finitely many $i$, and $\overline{w(i)}=w(\bar{i})$ for all $i$. These can also be considered as
signed (or barred) permutation of $\{1,2,\ldots\}$; we often use one-line notation
$w=(w(1),w(2),\ldots)$ to denote an element $w\in W_{\infty}$. The simple reflections are
identified with the transpositions
$s_{0}=(1,\bar{1})$ and $s_{i}=(i+1,i)(\overline{i},\overline{i+1})$ for $i\geq 1.$
The subgroup $W_n\subset W_{\infty}$
is described as
$$
W_{n}=\{w\in W_{\infty}\;|\; w(i)=i\;\mbox{for} \; i>n\}.
$$
In one-line notation,
we often denote
an element $w\in W_{n}\subset W_{\infty}$
by the finite sequence
$(w(1),\ldots,w(n)).$
The group $W_{\infty}'$, as a (signed) permutation group,
can be realized
as the subgroup of $W_{\infty}$
consisting of elements in $W_\infty$ with even number of sign changes.
The simple reflection $s_{\hat{1}}$ is identified with $s_{0}s_{1}s_{0}
\in W_{\infty}$, so as a permutation
$s_{\hat{1}}=(1,\bar{2})(2,\bar{1}).
\subsection{Grassmannian elements}
An element $w\in W_{\infty}$ is
a {\it Grassmannian element\/}
if $$w(1)<w(2)<\cdots<w(i)<\cdots$$ in the ordering
$ \cdots<\bar{3}<\bar{2}<\bar{1}<1<2<3<\cdots
.$
Let $W_{\infty}^{0}$ denote the set of all Grassmannian elements in $W_{\infty}.$
For $w\in W_{\infty}^{0}$, let $r$ be the number such that
\begin{equation} w(1)<\cdots<w(r)<1\quad\mbox{and}
\quad \bar{1}<w(r+1)<w(r+2)<\cdots.
\label{eq:GrassPerm}
\end{equation}
Then we define the $r$-tuple of positive integers
$\lambda=(\lambda_{1},\ldots,\lambda_{r})$ by
$ \lambda_{i}=\overline{w(i)}$
for $1\leq i\leq r.$
This is a {\it strict partition\/}
i.e. a partition with distinct parts:
$\lambda_{1}>\cdots>\lambda_{r}>0.$
Let $\mathcal{SP}$ denote the set of all
strict partitions.
The correspondence gives a bijection
$
W_{\infty}^{0}\longrightarrow
\mathcal{SP}.
$
We denote by $w_{\lambda}\in W_{\infty}^{0}$
the Grassmannian element corresponding
to $\lambda\in \mathcal{SP}$; then
$\ell(w_{\lambda})=|\lambda|=\sum_{i}\lambda_{i}$.
Note that this bijection preserves the partial order
when $\mathcal{SP}$ is considered to be a partially ordered set given by
the inclusion $\lambda\subset \mu$
of strict partitions.
We denote by $W_{\infty}^{\hat{1}}$
the set of all
Grassmannian elements contained in $W_{\infty}'.$
For $w\in W_{\infty}^{\hat{1}},$
the number $r$ in (\ref{eq:GrassPerm}) is always even.
Define the strict partition
$\lambda'=(\lambda'_{1}>\cdots>\lambda'_{r}\geq 0)$ by setting $ \lambda'_{i}=\overline{w(i)}-1$
for $1\leq i\leq r.$
Note that $\lambda'_{r}$ can be zero
this time.
This correspondence gives also a
bijection $
W_{\infty}^{\hat{1}}
\longrightarrow
\mathcal{SP}.$
We denote by $w'_{\lambda}\in W_{\infty}^{\hat{1}}$
the element corresponding
to $\lambda\in \mathcal{SP}.$
As before, $\ell(w'_{\lambda})=|\lambda|$ where
$\ell(w)$ denotes the length of $w$ in $W_\infty'$.
\bigskip
{\bf Example.} Let $\lambda=(4,2,1).$
Then the corresponding Grassmannian elements are given by
$w_{\lambda}=\bar{4}\bar{2}\bar{1}3
=s_{0}s_{1}s_{0}s_{3}s_{2}s_{1}s_{0}$
and $w'_{\lambda}=\bar{5}\bar{3}\bar{2}\bar{1}4
=s_{\hat{1}}s_{2}s_{1}s_{4}s_{3}s_{2}s_{\hat{1}}.$
\bigskip
The group $W_{\infty}$ (resp. $W_{\infty}'$)
has a parabolic subgroup
generated by $s_{i}\;(i\in I_{\infty}\setminus \{0\})$
(resp. $s_{i}\,(i\in I'_{\infty}\setminus \{\hat{1}\}).$
We denote these subgroups by $S_{\infty}
=\langle s_{1},s_{2},\ldots\rangle$
since it is isomorphic to the infinite Weyl group of type $\mathrm{A}.$
The product map
$$
W_{\infty}^{0}\times S_{\infty}
\longrightarrow W_{\infty}
\quad(\mbox{resp.}\;
W_{\infty}^{\hat{1}}\times S_{\infty}
\longrightarrow W'_{\infty}),
$$
given by $(u,w)\mapsto uw$
is a bijection satisfying $\ell(uw)=\ell(u)+\ell(w)$
(cf. \cite[Prop. 2.4.4]{BjBr}).
As a consequence, $w_{\lambda}$ (resp. $w_{\lambda}'$) is the unique element
of minimal length
in the left coset $w_{\lambda}S_{\infty}$
(resp. $w_{\lambda}'S_{\infty}$).
\section{Schur's $Q$-functions and its factorial
analogues}\label{sec:Schur}
\setcounter{equation}{0}
\subsection{Schur's $Q$-functions}
Our main reference for symmetric functions is
\cite{Mac}.
Let $x=(x_{1},x_{2},\ldots)$
be infinitely many
indeterminates.
Define $Q_i(x)$ as the coefficient of $u^i$ in
the generating function
$$f(u)=
\prod_{i=1}^{\infty}
\frac{1+x_{i}u}{1-x_{i}u}
=\sum_{k\geq 0}Q_{k}(x)u^{k}.
$$
Note that $Q_{0}=1.$
Define $\Gamma$ to be $\mathbb{Z}[Q_1(x),Q_2(x),\ldots].$
The identity
$f(u)f(-u)=1$ yields
\begin{equation}\label{eq:quadraticQ}
Q_{i}(x)^{2}+2\sum_{j=1}^{i}(-1)^{j}Q_{i+j}(x)Q_{i-j}(x)=0
\quad \mbox{for}\quad i\geq 1.
\end{equation}
It is known that the ideal of relations among the functions $Q_k(x)$ is
generated by the previous relations.
For $i\geq j\geq 0,$ define elements
$$
Q_{i,j}(x):=Q_{i}(x)Q_{j}(x)+2\sum_{k=1}^{j}(-1)^{k}
Q_{i+k}(x)Q_{j-k}(x).
$$
Note that $Q_{i,0}(x)=Q_{i}(x)$
and $Q_{i,i}(x)\,(i\geq 1)$ is identically zero.
For $\lambda$ a strict partition
we write $\lambda=(\lambda_{1}>\lambda_{2}>\cdots
>\lambda_{r}\geq 0)$ with $r$ even.
Then the corresponding
Schur's $Q$-function
$Q_{\lambda}=Q_{\lambda}(x)$ is defined by
$$
Q_{\lambda}(x)=\mathrm{Pf}(Q_{\lambda_{i},\lambda_{j}}(x))_{1\leq i<j\leq r},
$$
where $\mathrm{Pf}$ denotes the Pfaffian.
It is known then that the functions
$Q_\lambda(x)$ for $\lambda\in \mathcal{SP}$ form
a $\mathbb{Z}$-basis of $\Gamma.$
The {\em $P$-Schur function} is defined to be $P_{\lambda}(x)=2^{-\ell(\lambda)}Q_{\lambda}(x)$ where $\ell(\lambda)$ is the number of non-zero parts in $\lambda.$
The next lemma shows that the $Q$-Schur function is {\em supersymmetric\/.}
\begin{lem}\label{lem:super}
Each element
$\varphi(x)$ in $\Gamma$
satisfies
$$
\varphi(t,-t,x_{1},x_{2},\ldots)
=\varphi(x_{1},x_{2},\ldots)
$$
where $t$ is an indeterminate.
\end{lem}
\begin{proof}
It suffices to show this for the ring generators
$Q_{i}(x).$ This follows immediately from the generating function.
\end{proof}
\subsection{Factorial $Q$ and $P$-Schur functions}\label{ssec:FacSchur} In this section we recall the definition and some properties of the factorial $Q$-Schur and $P$-Schur functions defined by V.N. Ivanov in \cite{Iv}. Fix $n \ge 1$ an integer, $\lambda$ a strict partition of length $r \le n$ and $a=(a_i)_{i \ge 1}$ an infinite sequence. By $(x|a)^k$ we denote the product $(x-a_1)\cdots (x-a_k)$. According to \cite[Def. 2.10]{Iv} the factorial $P$-Schur function $P_\lambda^{(n)}(x_1, \ldots x_n|a)$ is defined by: \[ P_\lambda^{(n)}(x_1, \ldots x_n|a) = \frac{1}{(n-r)!} \sum_{w \in S_n} w\cdot \bigl( \prod_{i=1}^r (x_i|a)^{\lambda_i} \prod_{i \le r, i < j \le n} \frac{x_i+x_j}{x_i-x_j} \bigr) \/, \] where $w$ acts on variables $x_i$. If $a_1=0$ this function is stable, i.e. $P_\lambda^{(n+1)}(x_1, \ldots x_n,0|a) = P_\lambda^{(n)}(x_1, \ldots x_n|a)$, therefore there is a well defined limit denoted $P_\lambda(x|a)$. It was proved in \cite[Prop. 8]{IN} that if $a_1 \neq 0$, $P_\lambda^{(n)}(x_1, \ldots x_n|a)$ is stable modulo $2$, i.e. $P_\lambda^{(n+2)}(x_1, \ldots x_n,0,0|a)=P_\lambda^{(n)}(x_1, \ldots x_n|a)$; so in this case there is a well-defined even and odd limit. From now on we will denote by $P_\lambda(x|a)$ the {\em even} limit of these functions. Define also the factorial $Q$-Schur function $Q_\lambda(x|a)$ to be \[ Q_\lambda(x|a) = 2^{\ell(\lambda)}P_\lambda(x|0,a) \/,\] where $\ell(\lambda)$ is the number of non-zero parts of $\lambda$. As explained in \cite{IN}, the situation $a_1 \neq 0$ is needed to study type $\mathrm{D}$; in types $\mathrm{B,C}$, the case $a_1=0$ will suffice. For simplicity, we will also denote $P_\lambda^{(n)}(x_1, \ldots x_n|a)$ by $P_\lambda(x_1, \ldots x_n|a)$.
Let now $t=(t_{1},t_{2},\ldots)$ be indeterminates. Define:
$$
t_{\lambda}=
(t_{\lambda_{1}},\ldots,
t_{\lambda_{r}},0,0,\ldots),
\quad
t_{\lambda}'=\begin{cases}
(t_{\lambda_{1}+1},\ldots,
t_{\lambda_{r}+1},0,0,\ldots) &\mbox{if}\;r\;\mbox{is even}\\
(t_{\lambda_{1}+1},\ldots,
t_{\lambda_{r}+1},t_{1},0,\ldots)&\mbox{if}\;r\;\mbox{is odd}
\end{cases}.
$$
Let also $w_{\lambda}\in W_{\infty}^{0}$ (resp. $w_{\lambda}'\in W_{\infty}^{\hat{1}}$ ) be the Grassmann element corresponding
to $\lambda\in\mathcal{SP}.$
We associate to $\lambda$ its
{\it shifted Young diagram} $Y_\lambda$
as the set of {\it boxes}
with coordinates $(i,j)$
such that $1\leq i\leq r$ and $i\leq j\leq i+\lambda_i-1.$
We set $\lambda_j=0$ for $j>r$ by convention.
Define
$$
H_{\lambda}(t)=\prod_{(i,j)\in Y_{\lambda}}
(t_{\overline{w_{\lambda}(i)}}
+t_{\overline{w_{\lambda}(j)}}),\quad
H_{\lambda}'(t)=\prod_{(i,j)\in Y_{\lambda}}
(t_{\overline{w_{\lambda}'(i)}}
+t_{\overline{w_{\lambda}'(j+1)}}).
$$
{\bf Example.} Let $\lambda=(3,1).$
Then $w_{\lambda}=\bar{3}\bar{1}2,$
$w_{\lambda}'=\bar{4}\bar{2}13,$ and
$$H_{\lambda}(t)=4t_{1}t_{3}(t_{3}+t_{1})(t_{3}-t_{2}),\quad
H_{\lambda}'(t)=(t_{4}+t_{2})(t_{4}-t_{1})(t_{4}-t_{3})(t_{2}-t_{1}).$$
\setlength{\unitlength}{0.6mm}
\begin{center}
\begin{picture}(200,45)
\put(-25,30){Type $\mathrm{C}$}
\put(5,35){\line(1,0){60}}
\put(5,25){\line(1,0){60}}
\put(25,15){\line(1,0){20}}
\put(5,25){\line(0,1){10}}
\put(25,15){\line(0,1){20}}
\put(45,15){\line(0,1){20}}
\put(65,25){\line(0,1){10}}
\put(11,28){\small{$2t_{3}$}}
\put(31,18){\small{$2t_{1}$}}
\put(27,28){\small{$t_{3}\!+\!t_{1}$}}
\put(47,28){\small{$t_{3}\!-\!t_{2}$}}
\put(75,30){Type $D$}
\put(105,35){\line(1,0){60}}
\put(105,25){\line(1,0){60}}
\put(125,15){\line(1,0){20}}
\put(105,25){\line(0,1){10}}
\put(125,15){\line(0,1){20}}
\put(145,15){\line(0,1){20}}
\put(165,25){\line(0,1){10}}
%
\put(107,28){\small{$t_{4}\!+\!t_{2}$}}
\put(127,18){\small{$t_{2}\!-\!t_{1}$}}
\put(127,28){\small{$t_{4}\!-\!t_{1}$}}
\put(147,28){\small{$t_{4}\!-\!t_{3}$}}
\put(0,28){\small{$3$}}
\put(0,18){\small{$1$}}
\put(13,38){\small{$3$}}
\put(33,38){\small{$1$}}
\put(53,38){\small{$\bar{2}$}}
%
\put(100,28){\small{$4$}}
\put(100,18){\small{$2$}}
\put(113,38){\small{${2}$}}
\put(133,38){\small{$\bar{1}$}}
\put(153,38){\small{$\bar{3}$}}
%
\put(5,5){$w_{\lambda}=s_{0}s_{2}s_{1}s_{0}=\bar{3}\bar{1}2$}
\put(105,5){$w_{\lambda}'=s_{1}s_{3}s_{2}s_{\hat{1}}=\bar{4}\bar{2}13$}
\end{picture}
\end{center}
\begin{prop} (\cite{Iv})
For any strict partition $\lambda$,
the factorial $Q$-Schur function $Q_{\lambda}(x|t)$
(resp. $P_{\lambda}(x|t)$)
satisfies the following properties:
\begin{enumerate}
\item $Q_{\lambda}(x|t)$ (resp. $P_{\lambda}(x|t)$) is
homogeneous of degree $|\lambda|=\sum_{i=1}^{r}\lambda_{i},$
\item $Q_{\lambda}(x|t)=Q_{\lambda}(x)+\mbox{lower order terms in}\;x$\\
(resp. $P_{\lambda}(x|t)=P_{\lambda}(x)+\mbox{lower order terms in}\;x$),
\item $Q_{\lambda}(t_{\mu}|t)=0$ (resp. $P_{\lambda}(t_{\mu}'|t)=0$) unless $\lambda\subset \mu,$
\item $Q_{\lambda}(t_{\lambda}|t)=
H_{\lambda}(t)$
(resp. $P_{\lambda}(t'_{\lambda}|t)=
H'_{\lambda}(t)$).
\end{enumerate}
Moreover $Q_{\lambda}(x|t)$
(resp. $P_{\lambda}(x|t)$)
($\lambda\in \mathcal{SP}$)
form a $\mathbb{Z}[t]$-basis of $\mathbb{Z}[t]\otimes_{\mathbb{Z}}
\Gamma$ (resp.
$\mathbb{Z}[t]\otimes_{\mathbb{Z}}
\Gamma'$).
\end{prop}
\begin{proof} In the case $t_1=0$ this was proved in \cite[Thm. 5.6]{Iv}. If $t_1 \neq 0$, the identity (3) follows from definition (cf. \cite[Prop.~9]{IN}), while (4) follows from a standard computation.\end{proof}
\begin{remark}{\rm The statement in the previous proposition can be strengthened by showing that the properties (1)-(4) characterize the factorial $Q-$Schur (respectively $P$-Schur) functions. For $t_1=0$ this was shown in \cite[Thm.~5.6]{Iv}. A similar proof can be given for $t_1 \neq 0$, but it also follows from Thm. \ref{thm:isom} below. The characterization statement will not be used in this paper.} \end{remark}
\begin{remark}{\rm The function
$Q_{\lambda}(x|t)$ belongs actually to
$\Gamma\otimes_{\mathbb{Z}}\mathbb{Z}[t_{1},t_{2},\ldots,t_{\lambda_{1}-1}]$ and $P_{\lambda}(x|t)$ to
$\Gamma'\otimes_{\mathbb{Z}}\mathbb{Z}[t_{1},t_{2},\ldots,t_{\lambda_{1}}]$. For example we have
$$
Q_{i}(x|t)=\sum_{j=0}^{i-1}(-1)^{j}
e_{j}(t_{1},\ldots,t_{i-1})Q_{i-j}(x),\quad
P_{i}(x|t)=\sum_{j=0}^{i-1}(-1)^{j}
e_{j}(t_{1},\ldots,t_{i})P_{i-j}(x).
$$}
\end{remark}
\begin{remark}
{\rm
An alternative formula for $Q_\lambda(x|t)$, in terms of a {\em multi-Pfaffian}, will be given below in \S \ref{sec:Kazarian}}.
\end{remark}
The following proposition will only be used within the proof of the formula for the Schubert polynomial for the longest element in each type, presented in \S \ref{sec:Long} below.
\begin{prop} [\cite{Iv}]\label{prop:Pf}
Let $\lambda=(\lambda_{1}>\cdots>\lambda_{r}\geq 0)$ a strict partition
with $r$ even. Then
$$
Q_{\lambda}(x|t)=\mathrm{Pf}\left(
Q_{\lambda_{i},\lambda_{j}}(x|t)
\right)_{1\leq i<j\leq r},\quad
P_{\lambda}(x|t)=\mathrm{Pf}\left(
P_{\lambda_{i},\lambda_{j}}(x|t)
\right)_{1\leq i<j\leq r}.
$$
\end{prop}
\begin{proof} Again, for $t_1=0$, this was proved in \cite[Thm.3.2]{Iv}, using the approach described in \cite[III.8 Ex.13]{Mac}. The same approach works in general, but for completeness we briefly sketch an argument. Lemma \ref{injective} below shows that there is an injective universal localization map $\Phi: \mathbb{Z}[z] \otimes \mathbb{Z}[t] \otimes \Gamma' \to \prod_{w \in W_\infty'} \mathbb{Z}[t]$. The image of $P_\lambda(x|t)$ is completely determined by the images at Grassmannian Weyl group elements $w_\mu'$ and it is given by $P_\lambda(t'_\mu|t)$. But by the results from \cite[\S 10]{IN} we have that $P_\lambda(t'_\mu|t)= \mathrm{Pf}( P_{\lambda_i, \lambda_j}(t'_\mu|t))_{1 \le i< j \le r}$. The result follows by injectivity of $\Phi$.\end{proof}
We record here the following formula used later. The proof is by a standard computation (see e.g. the proof of \cite[Thm. 8.4]{Iv}).
\begin{lem}
We have
\begin{equation}
P_{k,1}(x|t)=
P_{k}(x|t)P_{1}(x|t)
-P_{k+1}(x|t)-(t_{k+1}+t_{1})P_{k}(x|t)\quad
\mbox{for}\quad k\geq 1.\label{eq:P2row}
\end{equation}
\end{lem}
\subsection{Factorization formulae}
In this section we present several
{\em factorization} formulas for the factorial $P$ and $Q$-Schur functions,
which will be used later in \S \ref{sec:Long}. To this end, we first consider the case of ordinary factorial Schur functions.
\subsubsection{Factorial Schur polynomials}
Let $\lambda=(\lambda_{1},\ldots,\lambda_{n})$
be a partition.
Define the {\it factorial
Schur polynomial\/} by
$$s_{\lambda}(x_{1},\ldots,x_{n}|t)
=\frac{\det((x_{j}|t)^{\lambda_{i}+n-i})_{1\leq i,j\leq n}}{\prod_{1\leq i<j\leq n}(x_{i}-x_{j})} \/, $$
where $(x|t)^{k}$ denotes $\prod_{i=1}^{k}(x-t_{i})$.
It turns out that $s_{\lambda}(x|t)$ is an element
in
$\mathbb{Z}[x_{1},\ldots,x_{n}]^{S_{n}}
\otimes \mathbb{Z}[t_{1},\ldots,t_{\lambda_{1}+n-1}].$
For some basic properties of these polynomials,
the reader can consult \cite{MS}.
The following formula will be used to prove
Lem. \ref{lem:piDelta} in \S \ref{sec:Long}.
\begin{lem}\label{lem:A-long} We have
$
s_{\rho_{n-1}}(t_{1},\ldots,t_{n}|t_{1},-z_{1},t_{2},-z_{2},
\ldots,
t_{n-1},-z_{n-1})=\prod_{1\leq i<j\leq n}(t_{j}+z_{i}).
$
\end{lem}
\begin{proof} When variables $z_i,t_i$ are specialized as in this Lemma,
the numerator is
an anti-diagonal lower triangular matrix. The entry on the $i$-th row
on the anti-diagonal is given by
$\prod_{j=1}^{i-1}(t_{i}-t_{j})(t_{i}+z_{j}).$
The Lemma follows immediately from this.
\end{proof}
Next formula is a version of Lem. \ref{lem:A-long}
which will be used in the proof of Lem. \ref{lem:piDeltaD}.
\begin{lem}\label{lem:A-longOdd}
If $n$ is odd
then we have
$$
s_{\rho_{n-1}+1^{n-1}}(t_{2},\ldots,t_{n}|t_{1},-z_{1},\ldots,
t_{n-1},-z_{n-1})
=\prod_{j=2}^{n}(t_{j}-t_{1})
\prod_{1\leq i<j\leq n}(t_{j}+z_{i}).
$$
\end{lem}
\begin{proof}
Similar to the proof of Lem. \ref{lem:A-long}.
\end{proof}
\subsubsection{$P$ and $Q$-Schur functions}
We need the following factorization formula.
\begin{lem}\label{lem:factorization} Let $\lambda=(\lambda_{1},\ldots,\lambda_{n})$ be a partition.
Then
we have
\begin{eqnarray*}
Q_{\rho_{n}+\lambda}(x_{1},\ldots,x_{n}|t)
&=&\prod_{i=1}^{n}2x_{i}\prod_{1\leq i< j\leq n}
(x_{i}+x_{j})\times
s_{\lambda}(x_{1},\ldots,x_{n}|t),
\end{eqnarray*}
\end{lem}
\begin{proof}
By their very definition
$$Q_{\rho_{n}+\lambda}(x_{1},\ldots,x_{n}|t)
=2^{n}\sum_{w\in S_{n}}w\left[\prod_{i=1}^{n}
x_{i}(x_{i}|t)^{\lambda_{i}+n-i}
\prod_{1\leq i<j\leq n}\frac{x_{i}+x_{j}}{x_{i}-x_{j}}
\right],
$$
where $w$ acts as permutation of the variables
$x_{1},\ldots,x_{n}.$
Since the polynomial
$\prod_{i=1}^{n}x_{i}
\prod_{1\leq i<j\leq n}(x_{i}+x_{j})$ in the parenthesis
is symmetric in $x$, the last expression factorizes into
$$
2^{n}\prod_{i=1}^{n}x_{i}
\prod_{1\leq i<j\leq n}(x_{i}+x_{j})\times
\sum_{w\in S_{n}}w\left[\prod_{i=1}^{n}
(x_{i}|t)^{\lambda_{i}+n-i}
\prod_{1\leq i<j\leq n}({x_{i}-x_{j}})^{-1}
\right]$$
Then by the definition of $s_{\lambda}(x_{1},\ldots,x_{n}|t)$
we have the lemma.
\end{proof}
The following two lemmas are proved in the same way:
\begin{lem}\label{lem:factorizationD} Assume $n$ is even.
Let
$\lambda=(\lambda_{1},\ldots,\lambda_{n})$
be a partition.
Then we have
$$
P_{\rho_{n-1}+\lambda}
(x_{1},\ldots,x_{n}|t)
=
\displaystyle{
\prod_{1\leq i<j\leq n}}(x_{i}+x_{j})
\cdot s_{\lambda}(x_{1},\ldots,x_{n}|t).
$$
\end{lem}
\begin{lem}\label{lem:factorizationDodd}
Assume $n$ is odd. Let
$\lambda=(\lambda_{1},\ldots,\lambda_{n-1})$
be a partition.
Then we have
$$
P_{\rho_{n-1}+\lambda}(x_{1},\ldots,x_{n-1}|t)
\prod_{1\leq i<j\leq n-1}(x_{i}+x_{j})
\times s_{\lambda+1^{n-1}}
(x_{1},\ldots,x_{n-1}|t).
$$
\end{lem}
\section{Stable Schubert classes}\label{sec:StableSchubert}
\setcounter{equation}{0}
The aim of this section is to introduce
{\it stable\/} Schubert classes
indexed by the Weyl group of
infinite rank $\pmb{W}_{\infty}$. Recall that the embeddings of Dynkin diagrams shown in \S \ref{ssec:Weyl} induce embeddings $i:\pmb{W}_n \to \pmb{W}_{n+1}$; then $\pmb{W}_\infty = \bigcup_{n \ge 1} \pmb{W}_n$.
\subsection{Stable Schubert classes}\label{ssec:SSC}
Let us denote by $\sigma_{w}^{(n)}$ the equivariant Schubert
class on $\mathcal{F}_{n}$ labeled by $w\in \pmb{W}_{n}.$
\begin{prop}\label{prop:StabSch}
The localization of Schubert classes is stable, i.e
$$\sigma_{i(w)}^{(n+1)}|_{i(v)}=\sigma_w^{(n)}|_v
\quad\mbox{for all}\quad
w,v\in \pmb{W}_{n}.$$
\end{prop}
\begin{proof}
First we claim that
$\sigma_{i(w)}^{(n+1)}|_{i(v)}\in \mathbb{Z}[t]^{(n)}$
for any $w,v\in \pmb{W}_{n}.$
Let $w_{0}^{(n)}$ be
the longest element in $\pmb{W}_{n}$.
By Prop. \ref{xi}, we have for $v\in \pmb{W}_{n}$
$$
\sigma_{i(w_{0}^{(n)})}^{(n+1)}|_{i(v)}
=\begin{cases}
\prod_{\beta\in R^{+}_{n}}\beta
&\mbox{if} \;v= w_{0}^{(n)}
\\
0 &\mbox{if} \;v\ne w_{0}^{(n)}
\end{cases}.
$$
In particular these polynomials belong to
$\mathbb{Z}[t]^{(n)}.$
For arbitrary $w\in \pmb{W}_{n}$,
any reduced expression of $i(w)$ contains only simple reflections $s_0, \dots , s_{n-1}$ (in type $D$, $s_0$ is replaced by $s_{\hat{1}}$).
Hence we obtain
the Schubert
class $\sigma_{i(w)}^{(n+1)}$
by applying the divided difference
operators $\partial_{0},\ldots,\partial_{n-1}$
(in type $D$, $\partial_0$ is replaced by $\partial_{\hat{1}}$)
successively to the class $\sigma_{i(w_{0}^{(n)})}^{(n+1)}.$
In this process only the variables $t_{1},\ldots,t_{n}$
are involved to compute
$\sigma_{i(w)}^{(n+1)}|_{i(v)}\;(v\in \pmb{W}_{n}).$
Hence the claim is proved.
For $w\in \pmb{W}_{n},$
we consider the element $\eta_{w}$
in $\prod_{v\in \pmb{W}_{n}}\mathbb{Z}[t]^{(n)}$
given by
$\eta_{w}|_{v}=\sigma_{i(w)}^{(n+1)}|_{i(v)}\;(v\in \pmb{W}_{n}).$
We will show
the element $\eta_{w}$
satisfies the conditions in Prop. \ref{xi} that
characterize $\sigma_{w}^{(n)}.$
In fact, the vanishing condition
holds since $i(w)\leq i(v)$
if and only if $w\leq v.$
Homogeneity and the degree condition is
satisfied because $\ell(i(w))=\ell(w).$
The normalization follows from the fact
$R_{n}^{+}\cap w R^{-}_{n}=
R_{n+1}^{+}\cap i(w) R^{-}_{n+1}.$
Thus we have $\eta_{w}=\sigma_{w}^{(n)}$
and the proposition is proved.
\end{proof}
Fix $w$ be in $\pmb{W}_{\infty}.$
Then, by the previous proposition, for any $v\in \pmb{W}_{\infty}$, and for any sufficiently
large $n$ such that $w,v\in \pmb{W}_{n}$,
the polynomial
$\sigma_{w}^{(n)}|_{v}$ does not depend on
the choice of $n.$
Thus we can introduce
a unique element
$\sigma_{w}^{(\infty)}
=(\sigma_{w}^{(\infty)}|_{v})_{v}$ in $\prod_{v\in \pmb{W}_{\infty}}\mathbb{Z}[t]$ such that
$$
\sigma_{w}^{(\infty)}|_{v}
=\sigma_{w}^{(n)}|_{v}$$
for all sufficiently large $n.$
We call this element the {\it stable\/} Schubert class.
\begin{Def} Let $H_{\infty}$ be the $\mathbb{Z}[t]$-submodule of $\prod_{v\in \pmb{W}_{\infty}}\mathbb{Z}[t]$ spanned by the stable Schubert classes $\sigma_{w}^{(\infty)},w\in \pmb{W}_{\infty}$, where the $\mathbb{Z}[t]$-module structure is given by the diagonal multiplication.
\end{Def}
We will show later in Cor. \ref{cor:Hsubalg} that
$H_{\infty}$ is actually a $\mathbb{Z}[t]$-subalgebra
in the product ring $\prod_{v\in \pmb{W}_{\infty}}\mathbb{Z}[t].$
The properties of the (finite-dimensional) Schubert classes extend immediately to the stable case. For example, the classes $\sigma_{w}^{(\infty)}\;(w\in W_{\infty})$ are
linearly independent over $\mathbb{Z}[t]$
(Prop. \ref{prop:basis}), and they satisfy the properties from Prop. \ref{xi}. To state the latter, define $R^{+}=\bigcup_{n\geq 1}R_{n}^{+}$, regarded
as a subset of $\mathbb{Z}[t].$ Then:
\begin{prop}\label{StableClass}
The stable Schubert class satisfies the following:
\begin{enumerate}
\item[(1)](Homogeneity) $\sigma_w^{(\infty)}|_v$ is homogeneous of degree $\ell(w)$ for each $v\geq w,$
\item[(2)](Normalization) \label{char:normal}$\sigma_w^{(\infty)}|_w=\prod_{\beta\in R^{+}\cap w(R^{-})}\beta,$
\item[(3)](Vanishing) $\sigma_w^{(\infty)}|_{v}$ vanishes unless $v\geq w.$
\end{enumerate}
\end{prop}
It is natural to
consider the following stable version of the GKM conditions
in the ring $\prod_{v\in\pmb{W}_{\infty}}\mathbb{Z}[t]$:
$$\eta|_{v}-\eta|_{s_{\alpha}v}\;\;\mbox{is a multiple
of}\;\alpha\;\mbox{for all}\;
\alpha\in R^{+},\; v\in \pmb{W}_{\infty}
$$
Then the stable Schubert class $\sigma_{w}^{(\infty)}$
is the unique element in
$\prod_{v\in W_{\infty}}\mathbb{Z}[t]$ that
satisfies the GKM conditions and the
three conditions Prop. \ref{StableClass}. It follows that all the elements
from $H_\infty$ satisfy the GKM conditions. In particular, the proofs from \cite{Kn} can be retraced, and one
can define the left and right actions of
$\pmb{W}_{\infty}$ on $H_{\infty}$
by the same formulas as in \S \ref{ssec:WeylAction}
but for $i\in \pmb{I}_{\infty}.$
Using these actions, we define also the divided difference operators
$\partial_{i},\delta_{i}$ on $H_{\infty}$
(see \S \ref{ssec:divdiff}). The next result follows again from the finite dimensional case (Prop. \ref{prop:propertiesdiv}).
\begin{prop}\label{prop:divdiff}
We have
$$
\partial_{i}\sigma_{w}^{(\infty)}=\begin{cases}
\sigma_{ws_{i}}^{(\infty)}& \ell(ws_{i})=\ell(w)-1\\
0 &\ell(ws_{i})=\ell(w)+1
\end{cases},\quad
\delta_{i}\sigma_{w}^{(\infty)}=\begin{cases}
\sigma_{s_{i}w}^{(\infty)}& \ell(s_{i}w)=\ell(w)-1\\
0 &\ell(s_{i}w)=\ell(w)+1
\end{cases}.
$$
\end{prop}
\subsection{Inverse limit of cohomology groups}\label{ssec:invlim}
Let $H_{n}$ denote the image of
the localization map
$$
\iota_{n}^{*}: H_{T_{n}}^{*}(\mathcal{F}_{n})
\longrightarrow H_{T_{n}}^{*}(\mathcal{F}_{n}^{T_{n}})
=\prod_{v\in \pmb{W}_{n}}\mathbb{Z}[t]^{(n)}.
$$
By the stability property for the localization of Schubert classes, the natural projections $H_\infty \to H_n \simeq H^*_{T_n} (\mathcal{F}_n)$ are compatible with the homomorphisms $H^*_{T_{n+1}}(\mathcal{F}_{n+1}) \to H^*_{T_n}(\mathcal{F}_n)$ induced by the equivariant embeddings $\mathcal{F}_n \to \mathcal{F}_{n+1}$. Therefore there is a $\mathbb{Z}[t]$-module homomorphism \[j: H_{\infty}\hookrightarrow \underleftarrow{\lim}\, H_{T_{n}}^{*}(\mathcal{F}_{n})\/.\] The injectivity of localization maps in the finite-dimensional setting implies that $j$ is injective as well.
\section{Universal localization map}\label{sec:UnivLoc}
In this section, we introduce a $\mathbb{Z}[t]$-algebra $\pmb{R}_\infty$
and establish an explicit isomorphism from
$\pmb{R}_{\infty}$ onto
$H_\infty,$ the $\mathbb{Z}[t]$-module spanned by the stable Schubert
classes. This isomorphism will be used
in the proof of the existence of
the double Schubert polynomials from \S \ref{sec:DSP}.
\subsection{The ring $\pmb{R}_{\infty}$ and the universal localization map}\label{ssec:UnivLoc}
Set $\mathbb{Z}[z]=\mathbb{Z}[z_{1},z_{2},z_{3},\ldots]$ and define the following rings:
$$R_{\infty}:=\mathbb{Z}[t]
\otimes_{\mathbb{Z}}\mathbb{Z}[z]
\otimes_{\mathbb{Z}} \Gamma,\quad \textrm{ and } \quad
R_{\infty}':=\mathbb{Z}[t]
\otimes_{\mathbb{Z}}\mathbb{Z}[z]
\otimes_{\mathbb{Z}} \Gamma' \/.
$$
As usual, we will use $\pmb{R}_{\infty}$ to denote $R_\infty$ for type $\mathrm{C}$ and
$R_{\infty}'$ for types $\mathrm{B}$ and $\mathrm{D}$.
We introduce next the most important algebraic tool of the paper.
Let $v$ be in $W_{\infty}.$
Set $t_{v}=(t_{v,1},t_{v,2},\ldots)$ to be
$$t_{v,i}=\begin{cases}t_{\overline{v(i)}}&
\mbox{if}\;v(i)\;\mbox{is negative}\\
0 & \mbox{otherwise}
\end{cases},
$$
where we set $t_{\overline{i}}$ to be $-t_{i}.$
Define a homomorphism of $\mathbb{Z}[t]$-algebras
$$
\Phi_{v}: R_{\infty}'\longrightarrow \mathbb{Z}[t]
\quad
\left(x\mapsto
t_{v},\quad
z_{i}\mapsto t_{v(i)}\right).$$
Note that since $v(i)=i$ for all
sufficiently large $i$, the substitution $x\mapsto t_{v}$
to $P_\lambda(x)$ gives a {\em polynomial} $P_{\lambda}(t_{v})$ in $\mathbb{Z}[t]$ (rather than a formal power series). Since $R_{\infty}$ is a subalgebra of $R_{\infty}'$, the restriction map
$\Phi_v:R_{\infty}\longrightarrow \mathbb{Z}[t]$ sends $Q_{\lambda}(x)$ to $Q_{\lambda}(t_{v}).$
\begin{Def} Define the "universal localization map"
to be the homomorphism of $\mathbb{Z}[t]$-algebras given
by
$$\Phi:
\pmb{R}_{\infty}\longrightarrow
\prod_{v\in \pmb{W}_{\infty}}\mathbb{Z}[t],\quad
f\mapsto (\Phi_{v}(f))_{v\in \pmb{W}_{\infty}}.
$$
\end{Def}
\begin{remark}{\rm A geometric interpretation of the map $\Phi$, in terms of the usual localization map, will be given later in \S \ref{sec:geometry}.}\end{remark}
The main result of this section is:
\begin{thm}\label{thm:isom}
The map
$\Phi$ is an isomorphism
of graded $\mathbb{Z}[t]$ algebras
from $\pmb{R}_\infty$ onto its image. Moreover, the image of $\Phi$ is equal to $H_\infty$.
\end{thm}
\begin{cor}\label{cor:Hsubalg}
$H_{\infty}$ is a $\mathbb{Z}[t]$-subalgebra in $\prod_{v\in \pmb{W}_{\infty}}\mathbb{Z}[t].$
\end{cor}
The proof of the theorem will be given in several lemmata and propositions, and it occupies the remaining part of section 6. The more involved part is to show surjectivity, which relies on the analysis of the "transition equations" implied by the equivariant Chevalley rule, and on study of factorial $P$ and $Q$- Schur functions. The proof of injectivity is rather short, and we present it next.
\begin{lem} \label{injective}
The map $\Phi$ is injective.
\end{lem}
\begin{proof}
We first consider type $\mathrm{B}$ case.
Write $f\in R_\infty'$ as
$f=\sum_{\lambda}c_\lambda(t,z) P_\lambda(x).$
Suppose $\Phi(f)=0.$
There are $m,n$ such that
$$c_\lambda\in \mathbb{Z}[t_1,\ldots,t_m,z_1,\ldots,z_n]$$
for all $\lambda$ such that $c_\lambda\ne 0.$
Define $v\in W_\infty$ by
$v(i)=m+i\;(1\leq i\leq n),$
$ v(n+i)=i\;(1\leq i\leq m),$
$v(m+n+i)=\overline{m+n+i}\;(1\leq i\leq N),$
and $
v(i)=i\;(i>N),$
where $N\geq m+n+1.$
Then we have
$\Phi_v(f)=\sum_\lambda c_\lambda(t_1,\ldots,t_m;t_{m+1},\ldots,t_{m+n})
P_\lambda(t_{m+n+1},t_{m+n+2},\ldots,t_{m+n+N})=0.$
Since this holds for all sufficiently large $N,$
we have $$\sum_\lambda c_\lambda(t_1,\ldots,t_m;t_{m+1},\ldots,t_{m+n})
P_\lambda(t_{m+n+1},t_{m+n+2},\ldots)=0.$$
Since $P_\lambda(t_{m+n+1},t_{m+n+2},\ldots)$ are linearly
independent over $\mathbb{Z}$
(see \cite{Mac}, III, (8.9)),
we have $$c_\lambda(t_1,\ldots,t_m;t_{m+1},\ldots,t_{m+n})=0$$
for all $\lambda.$
This implies $c_\lambda(t_1,\ldots,t_m;z_{1},\ldots,z_{n})=0$
for all $\lambda.$
Since $R_{\infty}\subset R_{\infty}',$
type $\mathrm{C}$ case follows immediately.
Type $\mathrm{D}$ case is proved by a minor modification.
Take $N$ to be always even, and consider
the sufficiently large {\it even} $N.$
\end{proof}
\subsection{Factorial $Q$-(and $P$-)functions
and Grassmannian Schubert classes}
Recall that there is a natural bijection between
$W_{\infty}^{0},W_{\infty}^{\hat{1}}$ and the set of strict partitions $\mathcal{SP}$.
The next result was proved by Ikeda in \cite{Ik} for type
$\mathrm{C}$, and Ikeda-Naruse in \cite{IN} for types $\mathrm{B,D}$.
\begin{thm}[\cite{Ik},\cite{IN}] \label{PhiFacQ}
Let $\lambda\in \mathcal{SP}$ and $w_{\lambda}
\in W_{\infty}^{0}$ and $w'_{\lambda}
\in W_{\infty}^{\hat{1}}$ be the corresponding Grassmannian
elements.
Then we have
\begin{enumerate}
\item $\Phi\left(Q_{\lambda}(x|t)\right)=\sigma_{w_{\lambda}}^{(\infty)}\;
\mbox{for type}\;C,$
\item $\Phi\left(P_{\lambda}(x|0,t)\right)
=\sigma_{w_{\lambda}}^{(\infty)}\mbox{for type}\;B,$
\item
$\Phi\left(P_{\lambda}(x|t)\right)
=\sigma_{w'_{\lambda}}^{(\infty)}\;
\mbox{for type}\;D.
$
\end{enumerate}
\end{thm}
\begin{proof} We consider first the type $\mathrm{C}$ case.
The map on $W_{\infty}$
given by $v\mapsto \sigma_{w_{\lambda}}^{(\infty)}|_{v}$
is constant on each left coset of $W_{\infty,0}\cong
S_{\infty}$
and it is determined by the values at the Grassmannian elements.
Let $v\in W_{\infty}$ and $w_{\mu}$ be the minimal length representative
of the coset $vS_{\infty}$ corresponding to
a strict partition $\mu.$
Then $t_{v}$ defined in \S \ref{ssec:UnivLoc}
is a permutation of $t_{\mu}.$
Since $Q_{\lambda}(x|t)$ is
symmetric with respect to $x$
we have
$\Phi_{v}(Q_{\lambda}(x|t))=Q_{\lambda}(t_{v}|t)
=Q_{\lambda}(t_{\mu}|t).
$
In \cite{Ik}, it was shown that
$Q_{\lambda}(t_{\mu}|t)=\sigma_{w_{\mu}}^{(\infty)}|_{w_{\mu}},$
which is equal to $\sigma_{w_{\mu}}^{(\infty)}|_v.$
This completes the proof in this case.
Proofs of the other cases are the same
with appropriate identification
of the functions and strict partitions.
\end{proof}
\subsection{Equivariant Chevalley formula}
The {\em
Chevalley formula} is a rule to multiply a Schubert class with a divisor class.
To state it we need some notation.
For a positive root $\alpha\in R^{+}$ and
a simple reflection $s_{i}$,
set
$$
c_{\alpha,s_{i}}=(\omega_i,\alpha^{\vee}),\quad
\alpha^{\vee}=2\alpha/(\alpha,\alpha),
$$
where $\omega_{i}$
is the $i$-th fundamental weight
of one of the classical types $\mathrm{A_{n}-D_n}$ for sufficiently large $n$.
The number $c_{\alpha,s_{i}}$ - called {\it Chevalley multiplicity} - does not depend on the choice of $n.$
\begin{prop}[cf. \cite{KK}]\label{prop:codim1}
For any $w\in \pmb{W}_{\infty}$, the Chevalley multiplicity $\sigma_{s_{i}}^{(\infty)}|_{w}$ is given
by $\omega_{i}-w(\omega_{i})$, where
$\omega_{i}$ is the fundamental weight
for a classical type $\mathrm{A_n-D_n}$ such that $n\geq i.$
\end{prop}
\begin{lem}[Equivariant Chevalley formula]
\label{lem:eqCh}
$$
\sigma_{s_{i}}^{(\infty)}
\sigma_{w}^{(\infty)}
=
\displaystyle\sum_{\alpha\in R^{+},\;\ell(w s_\alpha)=\ell(w)+1}
c_{\alpha,s_{i}}\,\sigma_{w s_\alpha}^{(\infty)} + \sigma_{s_{i}}^{(\infty)}|_{w} \cdot \sigma_{w}^{(\infty)}. $$
\end{lem}
\begin{proof}
The non-equivariant case is due to Chevalley \cite{C}, but the stable version of this formula was given in \cite{B93}.
An easy argument using localization shows that the only difference in the equivariant case is the appearance of the
equivariant term
$\sigma_{s_{i}}^{(\infty)}|_{w} \cdot \sigma_{w}^{(\infty)}.$
\end{proof}
\begin{remark}{\rm
There are only finitely many nonzero terms in the sum
in the right hand side.}
\end{remark}
\begin{lem} \label{lem:z}
The elements $\Phi(z_{i}) \in H_\infty$
are expressed in terms of Schubert classes as follows:
\noindent Type $\mathrm{B}$:
$\Phi(z_1)=\sigma_{s_1}^{(\infty)}-2\sigma_{s_{0}}^{(\infty)}+t_{1},\;
\Phi(z_i)=\sigma_{s_i}^{(\infty)}-\sigma_{s_{i-1}}^{(\infty)}+t_{i}
\; (i\geq 2),$
\noindent Type $\mathrm{C}$:
$\Phi({z}_i)=\sigma_{s_i}^{(\infty)}-\sigma_{s_{i-1}}^{(\infty)}+t_{i}
\; (i\geq 1),$
\noindent Type $\mathrm{D}$:
$\Phi({z}_1)=\sigma_{s_{1}}^{(\infty)}-\sigma_{s_{\hat{1}}} ^{(\infty)} +t_{1},\;
\Phi({z}_2)=\sigma_{s_{2}} ^{(\infty)}-\sigma_{s_{1}} ^{(\infty)}
-\sigma_{s_{\hat{1}}} ^{(\infty)} +t_{2}$, \quad and \\
$\Phi({z}_i)=\sigma_{s_{i} }^{(\infty)}-\sigma_{s_{i-1}} ^{(\infty)} +t_{i}\;(i\geq 3).
$
\end{lem}
\begin{proof}
This follows by localizing both sides of the formulas, and then using Prop. \ref{prop:codim1}.
\end{proof}
\begin{lem}\label{lem:PhisubH} We have $\mathrm{Im}(\Phi)\subset H_\infty.$
\end{lem}
\begin{proof}
The ring $R_{\infty}$ has a $\mathbb{Z}[t]$-basis
$z^{\alpha}Q_{\lambda}(x|t)$ where $z^{\alpha}$
are monomials in $\mathbb{Z}[z]$ and $\lambda$
are strict partitions.
Since $\Phi$ is $\mathbb{Z}[t]$-linear, it is enough to show that
$\Phi\left(z^{\alpha}Q_{\lambda}(x|t)\right)$
belongs to $H_{\infty}.$
We use induction on degree $d$ of the monomial $z^{\alpha}.$
The case $d=0$ holds by Thm. \ref{PhiFacQ}.
Let $d\geq 1$ and assume that
$\Phi(z^{\alpha}Q_{\lambda}(x|t))$ lies in $H_{\infty}$
for any monomial $z^{\alpha}$ of degree less than $d.$
Note that, by Lem. \ref{lem:z}, we have
$\Phi(z_{i})\in H_{\infty}.$
Choose any index $i$ such
that $z^{\alpha}=z_{i}\cdot z^{\beta}.$
By induction hypothesis $\Phi\left(z^{\beta}Q_{\lambda}(x|t)
\right)$ is an element in $H_{\infty},$
i.e., a linear combination
of $\sigma_{w}^{(\infty)}$'s with coefficients
in $\mathbb{Z}[t].$ Lem. \ref{lem:z} together with
equivariant Chevalley formula imply
that $\Phi(z_{i})\sigma_{w}^{(\infty)}$
belongs to $H_{\infty}.$
It follows that $z^{\alpha}Q_{\lambda}(x|t)$
belongs to $H_{\infty}.$
\end{proof}
\subsection{Transition equations}\label{ssec:trans} To finish the proof of surjectivity of $\Phi$, we need certain recursive relations for the Schubert classes - the {\it transition equations} - implied by the (equivariant) Chevalley formula. The arguments in this subsection are very similar to those given by S. Billey in \cite{B93}.
Let $t_{ij}$ denote the reflection with respect to the root $t_{j}-t_{i}$,
$s_{ij}$ the reflection with respect to $t_{i}+t_{j}$ and $s_{ii}$ the reflection
with respect to $t_{i}$ or $2t_{i}$ (depending on type).
From now on we regard $\mathbb{Z}[z]$
as subalgebra of $H_{\infty}$ via $\Phi$
and we identify $z_i$ with its image $\Phi(z_{i})$ in $H_{\infty}$
(cf. Lem \ref{lem:PhisubH}).
\begin{prop}[Transition equations]
The Schubert classes $\sigma_{w}$ of types
$\mathrm{B,C}$ and $\mathrm{D}$ satisfy the following recursion formula:
\begin{equation}
\sigma_w ^{(\infty)} =({z}_r-v(t_r))\;\sigma_v ^{(\infty)}
+
\sum_{1\leq i<r} \sigma^*_{v t_{ir}}+
\sum_{i\neq r}\sigma^*_{v s_{ir}}+
\chi\sigma^*_{v s_{rr}},\label{Transition}
\end{equation}
where $r$ is the last descent of $w$,
$s$ is the largest index such that $w(s)<w(r)$,
$v=wt_{rs}$,
$\chi=2,1,0$ according to the types $\mathrm{B,C,D}$,
and for each
$\sigma_{vt}^*=0$ unless $\ell(vt)
=\ell(v)+1=\ell(w)$ for $v,t\in \pmb{W}_{\infty}$
in which case $\sigma_{vt}^{*}=\sigma_{vt}^{(\infty)}.$
\end{prop}
\begin{proof}
The same as in \cite[Thm.4]{B93}
using the equivariant Chevalley formula (Lemma \ref{lem:eqCh}).
\end{proof}
\begin{remark}{\rm
The precise recursive nature of the equation (\ref{Transition}) will be explained
in the proof of the next Proposition below.}
\end{remark}
\begin{prop} \label{TransExp}
If $w\in \pmb{W}_{n}$ then
the Schubert class $\sigma_{w}^{(\infty)}$
is expressed as a $\mathbb{Z}[z,t]$-linear
combination of the Schubert classes of maximal Grassmannian type.
More precisely we have
\begin{equation}
\sigma_{w}^{(\infty)}=\sum_{\lambda}
g_{w,\lambda}({z},t)\sigma_{\lambda}^{(\infty)},\label{expansion}
\end{equation}
for some polynomials
$g_{w,\lambda}({z},t)$ in variables $t_i$ and $z_i$, and the sum is over strict partitions $\lambda$ such that $|\lambda|\leq n.$
\end{prop}
\begin{proof}
We will show that the recursion (\ref{Transition})
terminates in a finite number of steps
to get the desired expression.
Following \cite{B93}, we define a partial ordering on the elements of $\pmb{W}_{\infty}.$
Given $w$ in $\pmb{W}_{\infty}$,
let $LD(w)$ be the position of the last descent.
Define a partial ordering on the elements
of $\pmb{W}_{\infty}$ by $w<_{LD}u$
if $LD(u)<LD(w)$ or if $LD(u)=LD(w)$ and
$u(LD(u))<w(LD(w))$.
In \cite{B93} it was shown that each element
appearing on the right hand side of
(\ref{Transition}) is less than $w$ under this ordering.
Moreover it was proved in \cite[Thm.4]{B93} that
recursive applications of
(\ref{Transition}) give only terms which correspond to the elements in
$\pmb{W}_{n+r}$ where $r$ is the last descent of $w.$
Therefore we obtain the expansion (\ref{expansion}).
\end{proof}
\subsection{Proof of Theorem \ref{thm:isom}}
\begin{proof} By Lem. \ref{lem:PhisubH} we know
$\mathrm{Im}(\Phi)\subset H_{\infty}.$
Clearly $\Phi$ preserves the degree.
So it remains to show
$H_{\infty}\subset\mathrm{Im}(\Phi).$
In order to show this, it suffices to
$\sigma_{w}^{(\infty)}\in \mathrm{Im}(\Phi).$
In fact we have
\begin{equation}
\Phi\left(\sum_{\lambda}
g_{w,\lambda}(z,t)Q_{\lambda}(x|t)
\right)=\sigma_{w}^{(\infty)}.
\label{eq:DefSch}
\end{equation}
since
$\Phi$ is $\mathbb{Z}[z,t]$-linear.
\end{proof}
\section{Weyl group actions and divided difference operators on $\pmb{R}_{\infty}$}\label{sec:WactsR}
We define two commuting actions of $\pmb{W}_{\infty}$ on the ring $\pmb{R}_{\infty}.$
It is shown that the Weyl group actions
are compatible with the action on $H_{\infty}$
via $\Phi.$
\subsection{Weyl group actions on $R_{\infty}$}
We start from type $\mathrm{C}$. We make $W_{\infty}$ act as ring automorphisms
on $R_{\infty}$
by letting
$s_{i}^{z}$ interchange $z_{i}$ and $z_{i+1},$
for $i>0,$ and letting
$s_{0}^{z}$ replace $z_{1}$ and $-z_{1},$
and also
$$
s_{0}^{z}Q_{i}(x)=Q_{i}(x)+2\sum_{j=1}^{i}z_{1}^{j}Q_{i-j}(x).
$$
The operator $s_{0}^{z}$ was introduced in
\cite{BH}.
Let $\omega: R_{\infty}\rightarrow R_{\infty}$ be an involutive ring automorphism
defined by
$$
\omega(z_{i})=-t_{i},\quad
\omega(t_{i})=-z_{i},\quad
\omega(Q_{k}(x))=Q_{k}(x).
$$
Define the operators $s_{i}^{t}$ on $R_{\infty}$
by $s_{i}^{t}=\omega s_{i}^{z}\omega$
for $i\in I_{\infty}.$
More explicitly, $s_{i}^{t}$ interchange $t_{i}$ and
$t_{i+1}$, for $i>0$, and $s_{0}^{t}$
replace $t_{1}$ and $-t_{1}$ and also
$$
s_{0}^{t}Q_{i}(x)=Q_{i}(x)+2\sum_{j=1}^{i}(-t_{1})^{j}Q_{i-j}(x).
$$
\begin{lem}\label{lem:s0super} The action of operators
$s_{0}^{z},s_{0}^{t}$ on any $\varphi(x)\in \Gamma$
are written as
$$s_{0}^{z}\varphi(x_{1},x_{2},\ldots)
=\varphi(z_{1},x_{1},x_{2},\ldots),\quad
s_{0}^{t}\varphi(x_{1},x_{2},\ldots)
=\varphi(-t_{1},x_{1},x_{2},\ldots).$$
\end{lem}
Note that the right hand side
of both the formulas above belong to
$R_{\infty}.$
\begin{proof}
We show this for the generators $\varphi(x)=Q_{k}(x)$
of $\Gamma.$
By the definition of $s_{0}^{z}$ we have
$$
\sum_{k=0}^{\infty}
s_{0}^{z}
Q_{k}(x)\cdot u^{k}
=
\left(\prod_{i=1}^{\infty}
\frac{1+x_{i}u}{1-x_{i}u}
\right)\frac{1+z_{1}u}{1-z_{1}u}
=\sum_{k=0}^{\infty} Q_{k}(z_{1},x_{1},x_{2},\ldots)u^{k}.
$$
Thus we have the result for $s_{0}^{z}Q_{k}(x)$
for $k\geq 1.$
The proof for $s_{0}^{t}$ is similar.
\end{proof}
\begin{prop}
\begin{enumerate}
\item The operators
$s_{i}^{z}\;(i\geq 0)$ give an action of $W_{\infty}$ on $R_{\infty},$
\item The operators
$s_{i}^{t}\;(i\geq 0)$ give an action of $W_{\infty}$ on $R_{\infty},$
\item The two actions of $W_{\infty}$ commute with
each other.
\end{enumerate}
\end{prop}
\begin{proof}
We show that $s_{i}^{z}$ satisfy the Coxeter relations
for $W_{\infty}.$
The calculation for $s_{i}^{t}$ is the same.
We first show that $(s_{0}^{z})^{2}=1.$
For $f(z)\in \mathbb{Z}[z]$, $(s_{0}^{z})^{2}f(z)=f(z)$ is obvious.
We have for $\varphi(x)\in \Gamma,$
$$
(s_{0}^{z})^{2}
(\varphi(x))=
s_{0}^{z}\varphi(z_{1},x_{1},x_{2},\ldots)
=\varphi(z_{1},-z_{1},x_{1},x_{2},\ldots)
=\varphi(x_{1},x_{2},\ldots),
$$
where we used the super-symmetry
(Lemma \ref{lem:super})
at the last equality.
The verification of the remaining relations and the commutativity are left for the reader.
\end{proof}
In type $\mathrm{B}$, the action of $W_\infty$ on $R_\infty'$ is obtained by extending in the canonical way the action from $R_{\infty}.$ Finally, we consider the type $\mathrm{D}$ case.
In this case, the action is given by restriction the action of $W_{\infty}$ on $R'_\infty$ to the subgroup $W_{\infty}'.$
Namely, if we
set $s_{\hat{1}}^{z}=s_{0}^{z}s_{1}^{z}s_{0}^{z}$
and $s_{\hat{1}}^{t}=s_{0}^{t}s_{1}^{t}s_{0}^{t}$, then
we have the corresponding
formulas for $s_{\hat{1}}^{t}$ and $s_{\hat{1}}^{t}$ (in type D):
$$
s_{\hat{1}}^{z}\varphi(x_{1},x_{2},\ldots)
=\varphi(z_{1},z_{2},x_{1},x_{2},\ldots),\quad
s_{\hat{1}}^{t}\varphi(x_{1},x_{2},\ldots)
=\varphi(-t_{1},-t_{2},x_{1},x_{2},\ldots).
$$
\subsection{Divided difference operators}\label{ssec:divdiffgeom}
The divided difference operators
on $\pmb{R}_{\infty}$ are defined by
$$
\partial_{i}f=\frac{f-s_{i}^{z}f}{\omega(\alpha_{i})},\quad
\delta_{i}f=\frac{f-s_{i}^{t}f}{\alpha_{i}},
$$
where $s_{i}$ and $\alpha_{i}\;(i\in \pmb{I}_{\infty})$ are the
simple reflections and the corresponding simple roots.
Clearly we have
$\delta_{i}=\omega\partial_{i}\omega \quad (i\in \pmb{I}_{\infty}).$
\subsection{Weyl group action and
commutativity with divided difference operators}
\begin{prop}\label{prop:comm}
We have
$
(1)\;s_{i}^{L}\Phi=\Phi s_{i}^{t},\;
(2)\; s_{i}^{R}\Phi=\Phi s_{i}^{z}.
$
\end{prop}
\begin{proof} We will only prove this for type C; the other types can be treated similarly.
We first show $(1).$
This is equivalent to
$s_{i}\left(\Phi_{s_{i}v}(f)\right)=\Phi_{v}(s_{i}^{t}f)$
for all $f\in R_{\infty}.$
If $f\in \mathbb{Z}[z,t]$ the computation is
straightforward and we omit the proof.
Suppose $f=\varphi(x)\in \Gamma$.
We will only show
$s_{0}\left(\Phi_{s_{0}v}(f)\right)=\Phi_{v}(s_{0}^{t}f)$
since the case $i\geq 1$ is straightforward.
By Lem. \ref{lem:s0super}, the right hand side of this
equation is written as
\begin{equation}
\varphi(-t_1,x_1,x_2,\ldots)|_{x_j=t_{v,j}}.
\label{eq:rhs-t}
\end{equation}
Let $k$ be the (unique) index such that $v(k)=
1$ or $\overline{1}.$
Then the string $t_{s_0v}$ differs
from $t_v$ only in $k$-th position.
If $v(k)=\overline{1}$,
then
$t_{v,k}=t_1,\, t_{s_0v,k}=0$
and $t_{v,j}=t_{s_0v,j}$ for $j\ne k.$
In this case (\ref{eq:rhs-t}) is
$$
\varphi(-t_1,t_{v,1},\ldots,t_{v,k-1},t_1,t_{v,k+1},\ldots).
$$
This polynomial is equal to
$\varphi(t_{v,1},\ldots,t_{v,k-1},t_{v,k+1},\ldots)$
because $\varphi(x)$ is supersymmetric.
It is straightforward to see that
$s_{0}\Phi_{s_{0}v}(\varphi(x))$
is equal to $\varphi(t_{v,1},\ldots,t_{v,k-1},t_{v,k+1},\ldots).$ The case for $v(k)={1}$ is easier, so
we left it to the reader.
Next we show (2), i.e. $
\Phi_{vs_{i}}(f)=\Phi_{v}(s_{i}^{z}f)
$ for all $f\in R_{\infty}.$
Again, the case $f\in \mathbb{Z}[z,t]$ is straightforward,
so we we omit the proof of it.
We show $\Phi_{vs_{0}}\left(\varphi(x)\right)
=\Phi_{v}(s_{0}^{z}\varphi(x))$
for $\varphi(x)\in \Gamma.$
The right hand side is
\begin{equation}
\varphi(z_{1},x_{1},x_{2},\ldots)|_{z_{1}=v(t_{1}),\,
x_{j}=t_{v,j}},
\label{eq:rhs}
\end{equation}
where $t_{v,j}=t_{\overline{v(j)}}$ if
$v(j)$ is negative and otherwise $t_{v,j}$ is zero.
If $v(1)=-k$ is negative, the above function (\ref{eq:rhs}) is
$$
\varphi(-t_{k},t_{k},t_{v,2},t_{v,3},\ldots).
$$
This is equal to
$\varphi(0,0,t_{v,2},t_{v,3},\ldots)$
because $\varphi$ is supersymmetric.
Then also this is equal to $\varphi(0,0,t_{v,2},t_{v,3},\ldots)
=\varphi(0,t_{v,2},t_{v,3},\ldots)$ by stability
property.
Now since $\overline{v(1)}$ is positive we have
$t_{vs_{0}}=(0,t_{v,2},t_{v,3},\ldots).$
Therefore the polynomial (\ref{eq:rhs})
coincides with $\Phi_{vs_{0}}(\varphi(x)).$
If $v(1)$ is positive, then
$t_{v}=(0,t_{v,2},t_{v,3},\ldots)$ and
$t_{vs_{0}}=(t_{v(1)},t_{v,2},t_{v,3},\ldots).$
Hence the substitution
$x\mapsto t_{vs_{0}}$ to
the function $\varphi(x_{1},x_{2},\ldots)$
gives rise to the polynomial (\ref{eq:rhs}).
Next we show $
\Phi_{vs_{i}}(\varphi(x))=\Phi_{v}(s_{i}^{z}\varphi(x))
$
for $i\geq 1.$
First recall that $s_{i}^{z}\varphi(x)=\varphi(x).$
In this case
$t_{vs_{i}}$ is obtained from
$t_{v}$ by exchanging $t_{v,i}$ and $t_{v,i+1}.$
So $\varphi(t_{vs_{i}})=\varphi(t_{v}).$
This completes the proof.
\end{proof}
Using the above proposition, the next
result follows:
\begin{prop}\label{prop:PhiCommD} The localization map
$\Phi:\pmb{R}_{\infty}\rightarrow H_{\infty}$
commutes with the divided difference operators
both on $\pmb{R}_{\infty}$ and $H_{\infty},$ i.e.,
$$\Phi \,\partial_{i}=\partial_{i}\,\Phi,\quad
\Phi\,\delta_{i}=\delta_{i}\,\Phi$$
\end{prop}
\begin{proof}
Let $f\in R_{\infty}.$
Applying $\Phi$ on the both hand sides of
equation
$
{\omega(\alpha_{i})}\cdot
\partial_{i}f={f-s_{i}^{z}f}
$ we have
$\Phi(-\omega(\alpha_{i}))\cdot \Phi(\partial_{i}f)
=\Phi(f)-s_{i}^{R}\Phi(f)$,
where we used Prop. \ref{prop:comm} and linearity.
Localizing at $v$ we obtain
$v(\alpha_{i})\cdot \Phi_{v}(\partial_{i}f)
=\Phi_{v}(f)-\Phi_{vs_{i}}(f).$
Note that we used the definition of $s_{i}^{R}$ and
$\Phi_{v}(\omega(\alpha_{i}))=-v(\alpha_{i}).$
The proof for the statement regarding $\delta_{i}$
is similar, using $\Phi(\alpha_{i})=\alpha_{i}.$
\end{proof}
\subsection{Proof of the existence and uniqueness Theorem \ref{existC}}
\begin{proof} (Uniqueness)
Let
$\{\mathfrak{S}_{w}\}$ and $\{\mathfrak{S}_{w}'\}$
be two families
both satisfying the defining conditions
of the double Schubert polynomials.
By induction on the length of $w$,
we see
$\partial_{i}(\mathfrak{S}_{w}-\mathfrak{S}'_{w})=
\delta_{i}(\mathfrak{S}_{w}-\mathfrak{S}'_{w})=0$
for all $i\in \pmb{I}_{\infty}.$
This implies that
the difference $\mathfrak{S}_{w}-\mathfrak{S}'_{w}$ is
invariant for both left and right actions of $\pmb{W}_{\infty}.$
It is easy to see that the only such invariants
in $\pmb{R}_{\infty}$ are the constants.
So $\mathfrak{S}_{w}-\mathfrak{S}'_{w}=0$
by the constant term condition.
(Existence) Define $\mathfrak{S}_{w}(z,t;x)
=\Phi^{-1}(\sigma_{w}^{(\infty)}).$
By Prop. \ref{prop:PhiCommD}
and Prop. \ref{prop:divdiff},
$\mathfrak{S}_{w}(z,t;x)$
satisfies the defining equations for the double
Schubert polynomials.
The conditions on the constant term are
satisfied since $\sigma_{w}^{(\infty)}$
is homogeneous of degree $\ell(w)$ (Prop. \ref{StableClass})
and we have $\mathfrak{S}_{e}=1.$
\end{proof}
\begin{remark}\label{rem:TransDSP}
{\rm By construction, $\mathfrak{S}_{w}(z,t;x)$
satisfies the transition equation (\ref{Transition})
with $\sigma_{w}^{(\infty)}$ replaced by $\mathfrak{S}_{w}(z,t;x)$.
This equation provides an effective way to calculate
the double Schubert polynomials.}
\end{remark}
\begin{remark}\label{rem:A}{\rm The ring
$\mathbb{Z}[z]\otimes_{\mathbb{Z}}\mathbb{Z}[t]$ is
stable under the actions of the divided
difference operators $\partial_{i},\delta_{i}\,(i\geq 1)$ of type $\mathrm{A}$, and the type $\mathrm{A}$ double Schubert polynomials $\mathfrak{S}_{w}^{A}(z,t)$,
$w\in S_{\infty}$ form the unique family of solutions of the system of equations
involving only $\partial_{i},\delta_{i}$ for $i\geq 1$, and which satisfy the constant term conditions.}\end{remark}
\subsection{Projection to the cohomology of flag manifolds}\label{ssec:projection}
We close this section with a brief discussion
of the projection from $\pmb{R}_{\infty}$ onto
$H_{T_{n}}^{*}(\mathcal{F}_{n}).$
For $f\in \mathbb{Z}[t]$, we denote by $f^{(n)}\in \mathbb{Z}[t]^{(n)}$
the polynomial given
by setting $t_{i}=0$ for $i>n$ in $f.$
Let $\mathrm{pr}_{n}:
H_{\infty}\rightarrow H_{n}$ be the projection
given by
$(f_{v})_{v\in \pmb{W}_{\infty}}\mapsto
(f_{v}^{(n)})_{v\in \pmb{W}_{n}}.$
Consider the following composition of maps
$$
\pi_{n}: \pmb{R}_{\infty}\overset{\Phi}{\longrightarrow} H_{\infty}
\overset{\mathrm{pr}_{n}}{\longrightarrow} H_{n}
\cong
H_{T_{n}}^{*}
(\mathcal{F}_{n}).
$$
Explicitly, we have
$
\pi_{n}(f)|_{v}=\Phi_{v}(f)^{(n)}\;
(f\in \pmb{R}_{\infty},\; v\in \pmb{W}_{n}).
$
We will give an alternative geometric description
for $\pi_{n}$ in Section \ref{sec:geometry}.
\begin{prop}\label{prop:piCommD}
We have
$\pi_{n}(\mathfrak{S}_{w})=\sigma_{w}^{(n)}$
for $w\in \pmb{W}_{n}$ and $\pi_{n}(\frak{S}_{w})=0$
for $w\notin\pmb{W}_{n}.$
Moreover $\pi_{n}$ commutes with divided difference operators
$$
\partial_{i}^{(n)}\circ\pi_{n}=\pi_{n}\circ\partial_{i},\quad
\delta_{i}^{(n)}\circ\pi_{n}=\pi_{n}\circ\delta_{i}
\quad
(i\in \pmb{I}_{\infty}),
$$
where $\partial_{i}^{(n)},\delta_{i}^{(n)}$ are divided difference
operators on $H_{T_{n}}^{*}(\mathcal{F}_{n}).$
\end{prop}
\begin{proof}
The first statement follows from the construction
of $\sigma_{w}^{(\infty)}$
and the vanishing property (Prop. \ref{StableClass}).
The second statement follows from Prop. \ref{prop:PhiCommD}
and the commutativity $\partial_{i}^{(n)}\circ\mathrm{pr}_{n}=
\mathrm{pr}_{n}\circ\partial_{i},\,
\delta_{i}^{(n)}\circ\mathrm{pr}_{n}=
\mathrm{pr}_{n}\circ\delta_{i}$
which is obvious from the construction of
$\partial_{i},\delta_{i}.$
\end{proof}
\begin{cor} There exists an injective homomorphism of $\mathbb{Z}[t]$-algebras $\pi_\infty: \pmb{R}_\infty \to \underleftarrow{\lim} H^*_{T_n}(\mathcal{F}_n)$.\end{cor}
\begin{proof} The proof follows from the above construction and \S \ref{ssec:invlim}.\end{proof}
\section{Double Schubert polynomials}\label{sec:DSP}
\subsection{Basic properties} Recall that the {\em double Schubert polynomial} $\mathfrak{S}_w(z,t;x)$ is equal to the inverse image of the stable Schubert class $\sigma_w^{(\infty)}$ under the algebra isomorphism $\Phi:\pmb{R}_\infty \to H_\infty$. In the next two sections we will study the algebraic properties of these polynomials.
\begin{thm}\label{T:properties}
The double Schubert
polynomials satisfy the following:
\begin{enumerate}
\item (Basis)
The double Schubert polynomials $\{\mathfrak{S}_{w}\}_{w
\in \pmb{W}_{\infty}}$
form a $\mathbb{Z}[t]$-basis of
$\pmb{R}_{\infty}.$
\item (Relation to Billey-Haiman's polynomials) For all $w\in \pmb{W}_{\infty}$ we have
\begin{equation}
\mathfrak{S}_{w}(z,0;x)=\mathfrak{S}_{w}(z;x),
\label{eq:relBH}
\end{equation}
where $\mathfrak{S}_{w}(z;x)$ denotes
Billey-Haiman's polynomial.
\item (Symmetry) We have
$\mathfrak{S}_{w}(-t,-z;x)=\mathfrak{S}_{w^{-1}}(z,t;x).$
\end{enumerate}
\end{thm}
\begin{proof}
Property (1) holds because the stable Schubert classes $\sigma_w^{(\infty)}$ form a $\mathbb{Z}[t]$-basis for $H_\infty$ (cf. \S\ref{ssec:SSch}).
Property (2) holds because $\mathfrak{S}_{w}(z,0;x)\in \mathbb{Z}[z]\otimes \Gamma'$
satisfies the defining conditions for
Billey-Haiman's polynomials
involving the right divided difference operators $\partial_{i}.$
Then by the uniqueness of Billey-Haiman's polynomials,
we have the results.
For (3), set $\mathfrak{X}_{w}=\omega(\mathfrak{S}_{w^{-1}}).$
Then by the relation $\delta_{i}=\omega\partial_{i}\omega$
we can show that
$\{\mathfrak{X}_{w}\}$ satisfies the defining
conditions of the double Schubert polynomials.
So the uniqueness of the double Schubert polynomials
implies $\mathfrak{X}_{w}=\mathfrak{S}_{w}.$
Then we have
$\omega(\mathfrak{S}_{w})
=\omega(\mathfrak{X}_{w})=
\omega(\omega\mathfrak{S}_{w^{-1}})=
\mathfrak{S}_{w^{-1}}.$
\end{proof}
\begin{remark}{\rm
For type $D$ we have
$s_{0}^{z}s_{0}^{t}\mathfrak{D}_{w}=\mathfrak{D}_{\hat{w}}$
where $\hat{w}$ is the image of $w$ under the involution
of $W_{\infty}'$ given by interchanging
$s_{1}$ and $s_{\hat{1}}.$
This is shown by the uniqueness of
solution as in the proof of the symmetry property.
See \cite[Cor. 4.10]{BH} for the corresponding
fact for the Billey-Haiman polynomials.}
\end{remark}
\subsection{Relation to type $\mathrm{A}$ double Schubert polynomials}
Let $\mathfrak{S}_{w}^{A}(z,t)$ denote the
type $\mathrm{A}$ double Schubert polynomials.
Recall that $\pmb{W}_{\infty}$
has a parabolic subgroup
generated by $s_{i}\;(i\geq 1)$ which is
isomorphic to
$S_{\infty}.$
\begin{lem}\label{lem:typeA}
Let $w\in \pmb{W}_{\infty}.$
If $w\in S_{\infty}$ then
$\mathfrak{S}_{w}(z,t;0)=\mathfrak{S}_{w}^{A}(z,t)
$ otherwise we have
$\mathfrak{S}_{w}(z,t;0)=0.$
\end{lem}
\begin{proof} The polynomials
$\{\mathfrak{S}_{w}(z,t;0)\}$, $w\in S_{\infty},$
in $\mathbb{Z}[t]\otimes_{\mathbb{Z}}\mathbb{Z}[z]\subset \pmb{R}_{\infty}$ satisfy the defining divided difference
equations for the double Schubert polynomials of type $\mathrm{A}$
(see Remark \ref{rem:A}). This proves the first statement.
Suppose $w\not\in S_{\infty}.$
In order to show
$\mathfrak{S}_{w}(z,t;0)=0$,
we use the universal localization map
$\Phi^{A}: \mathbb{Z}[z]\otimes \mathbb{Z}[t] \to \prod_{v \in S_\infty}\mathbb{Z}[t]$ of type $\mathrm{A}$, which is defined in the obvious manner.
A similar proof to Lem. \ref{injective}
shows that the map
$\Phi^{A}$ is injective.
For any $v\in S_{\infty}$ we have
$\Phi_{v}(\mathfrak{S}_{w}(z,t;0))
=\Phi_{v}(\mathfrak{S}_{w}(z,t;x))
$, which is equal to $\sigma_{w}^{(\infty)}|_{v}$
by construction of $\mathfrak{S}_{w}(z,t;x).$
Since $v\not\geq w,$ we have
$\sigma_{w}^{(\infty)}|_{v}=0.$
This implies
that the image of $\mathfrak{S}_{w}$
under the universal localization map
$\Phi^{A}$ is zero, thus $\mathfrak{S}_{w}(z,t;0)=0.$
\end{proof}
\subsection{Divided difference operators and the double Schubert polynomials}
We collect here some properties concerning
actions of the divided difference operators
on the double Schubert polynomials.
These will be used in the next section.
\begin{prop}\label{prop:divdifSch}
Let $w=s_{i_{1}}\cdots s_{i_{r}}$ be a reduced expression
of $w\in \pmb{W}_{\infty}.$
Then the operators
$$\partial_{w}=\partial_{i_{1}}\cdots \partial_{i_{r}},
\quad
\delta_{w}=\delta_{i_{1}}\cdots \delta_{i_{r}}$$
do not depend on the reduced expressions and are
well-defined for $w\in \pmb{W}_{\infty}.$
Moreover we have
\begin{eqnarray}
\partial_{w}\mathfrak{S}_{u}&=&\begin{cases}
\mathfrak{S}_{uw^{-1}} & \mbox{if}\;\ell(uw^{-1})=\ell(u)-\ell(w)\\
0 &\mbox{otherwise}
\end{cases},\label{eq:partialSch}\\
\delta_{w}\mathfrak{S}_{u}&=&\begin{cases}
\mathfrak{S}_{wu}& \mbox{if}\;\ell(wu)=\ell(u)-\ell(w)
\\
0&\mbox{otherwise}
\end{cases}.
\end{eqnarray}
\end{prop}
\begin{proof}
Since $\{\mathfrak{S}_{u}\}$ is a
$\mathbb{Z}[t]$-basis of $\pmb{R}_{\infty}$,
the equation (\ref{eq:partialSch}) uniquely determine
a $\mathbb{Z}[t]$-linear operator, which we denote
by $\varphi_{w}.$
One can prove
$\partial_{i_{1}}\cdots \partial_{i_{r}}=\varphi_{w}$
by induction on the length of $w.$
The proof for $\delta_{i}$ is similar.
\end{proof}
\begin{remark}{\rm
The argument here is based on the
existence of $\{\mathfrak{S}_{w}\}$, but one can also prove it in the classical way - using braid relations (cf. e.g. \cite{BGG}) -
by a direct calculation.}
\end{remark}
The next result will be used in the next section
(Prop. \ref{prop:InterP}).
\begin{lem}\label{Phie}
We have $\Phi_{e}(\partial_{u}\mathfrak{S}_{w})=\delta_{u,w}.$
\end{lem}
\begin{proof}
First note that
$\Phi_{e}(\mathfrak{S}_{w})=\sigma_{w}^{(\infty)}|_{e}
=\delta_{w,e}.$
If $\ell(wu^{-1})=\ell(w)-\ell(u)$ is satisfied
then
by Prop. \ref{prop:divdifSch}, we have
$\Phi_{e}(\partial_{u}\mathfrak{S}_{w})
=\Phi_{e}(\mathfrak{S}_{wu^{-1}})=\delta_{w,u}.$
Otherwise we have $\partial_{u}\mathfrak{S}_{w}=0$
again by Prop. \ref{prop:divdifSch}.
\end{proof}
\subsection{Interpolation formulae and their applications}
In this section we obtain an explicit combinatorial formula for the double Schubert polynomials, based on the explicit formulas for the single Schubert polynomials from \cite{BH}. The main tool for doing this is the interpolation formula, presented next.
\begin{prop}[Interpolation formula]\label{prop:InterP} For any $f\in \pmb{R}_{\infty}$, we have
$$
f=\sum_{w\in \pmb{W}_{\infty}}\Phi_{e}(\partial_{w}(f))\mathfrak{S}_{w}(z,t;x).
$$
\end{prop}
\begin{proof}
Since the double Schubert polynomials
$\{\mathfrak{S}_{w}\}$
form a $\mathbb{Z}[t]$-basis
of the ring $\pmb{R}_{\infty}$, we write
$
f=\sum_{w\in \pmb{W}_{\infty}}c_{w}\mathfrak{S}_{w},\;
c_{w}(t)\in \mathbb{Z}[t].
$
As
$\partial_{w}$ is $\mathbb{Z}[t]$-linear
we obtain by using Lemma \ref{Phie}
$$
\Phi_{e}(\partial_{w}f)=\sum_{u\in \pmb{W}_{\infty}}c_{u}(t)
\Phi_{e}(\partial_{w}\mathfrak{S}_{u})
=\sum_{u\in \pmb{W}_{\infty}}c_{u}(t)
\delta_{w,u}=c_{w}(t).
$$
\end{proof}
\begin{remark}\label{rem:y}{\rm
Let $y=(y_{1},y_{2},\ldots)$ be formal parameters.
On the extended ring $\mathbb{Z}[y]\otimes \pmb{R}_{\infty}$,
we can introduce the Weyl group actions, divided difference operators, and the localization map
in the trivial way such that they are $\mathbb{Z}[y]$-linear.
Since the elements $\mathfrak{S}_{w} \,(w\in \pmb{W}_{\infty})$ clearly
form a $\mathbb{Z}[y]\otimes \mathbb{Z}[t]$-basis of $\mathbb{Z}[y]\otimes \pmb{R}_{\infty},$ the interpolation formula holds also
for any $f\in \mathbb{Z}[y]\otimes \pmb{R}_{\infty}.$}
\end{remark}
\begin{prop}\label{prop:LS-BH}
Let $y=(y_{1},y_{2},\ldots)$ be formal parameters. Then
$$
\mathfrak{S}_{w}(z,t;x)=\sum_{u,v}\mathfrak{S}_{u}^{A}(y,t)
\mathfrak{S}_{v}(z,y;x)
$$
summed over all $
u\in S_{\infty},\,v\in \pmb{W}_{\infty}$ such that
$w=uv,\,
\ell(u)+\ell(v)=\ell(w).$
\end{prop}
\begin{proof}
By the interpolation formula (see Remark \ref{rem:y}),
we have $$\mathfrak{S}_{w}(z,y;x)=\sum_{v}\Phi_{e}(\partial_{v}\mathfrak{S}_{w}(z,y;x))\mathfrak{S}_{v}(z,t;x).$$
By Prop. \ref{prop:divdifSch}, we see
$\partial_{v}\mathfrak{S}_{w}(z,y;x)
$ is equal to $\mathfrak{S}_{wv^{-1}}(z,y;x)$
if $\ell(wv^{-1})=\ell(w)-\ell(v)$,
and zero otherwise. Suppose $\ell(wv^{-1})=\ell(w)-\ell(v)$,
then
$\Phi_{e}\left(\mathfrak{S}_{wv^{-1}}(z,y;x)\right)=\mathfrak{S}_{wv^{-1}}(t,y;0)$ by the definition of $\Phi_{e}.$
By Lemma \ref{lem:typeA} this is
$\mathfrak{S}_{wv^{-1}}^{A}(t,y)$
if $wv^{-1}=u\in S_{\infty}$ and zero otherwise.
Then interchanging $t$ and $y$ we have the Proposition.
\end{proof}
Making $y=0$ in the previous proposition, and using that $\mathfrak{S}_u^A(y,t) = \mathfrak{S}_{u^{-1}}^A(-t,-y)$ (cf. Theorem \ref{T:properties} (3) above, for type $\mathrm{A}$ double Schubert polynomials) we obtain:
\begin{cor}\label{cor:typeAexpand} Let $\mathfrak{S}_{w}^{A}(z)$ denote the
(single) Schubert polynomial of type $\mathrm{A}.$
We have
$$
\mathfrak{S}_{w}(z,t;x)=\sum_{u,v}
\mathfrak{S}_{u^{-1}}^{A}(-t)\mathfrak{S}_{v}(z;x)
$$
summed over all $u\in S_{\infty}, v\in \pmb{W}_{\infty}$
such that $w=uv$ and $\ell(w)=\ell(u)+\ell(v).$
\end{cor}
There is an explicit combinatorial expression
for the Billey-Haiman polynomials $\mathfrak{S}_{w}(z;x)$
in terms of Schur $Q$-functions and type $\mathrm{A}$ (single)
Schubert polynomials (cf. Thms. 3 and 4 in \cite{BH}).
This, together with the above corollary
implies also an explicit formula in our case.
Moreover, the formula for
$\mathfrak{S}_{w}(z;x)$ is {\em positive},
and therefore this yields a positivity property for the double Schubert polynomials (see Thm. \ref{thm:positivity} below).
We will give an alternative proof for this
positivity result, independent of the results from {\em loc.cit.}
\subsection{Positivity property} To prove the positivity of the double Schubert polynomials, we begin with the following lemma (compare with Thms. 3 and 4 in \cite{BH}):
\begin{lem}\label{lem:S00}
We have
$
\mathfrak{S}_{w}(z;x)=
\sum_{u,v}\mathfrak{S}_{u}^{A}(z)\mathfrak{S}_{v}(0,0;x)
$
summed over all $u\in \pmb{W}_{\infty}, v\in S_{\infty}$
such that $w=uv$ and $\ell(w)=\ell(u)+\ell(v).$
\end{lem}
\begin{remark}{\rm
The function $\mathfrak{S}_{v}(0,0;x)$
is the Stanley's symmetric function
involved in the combinatorial expression for $\mathfrak{S}_{w}(z;x)$
from \cite{BH}. This follows from comparing the present
lemma and the Billey-Haiman's formulas 4.6 and 4.8.}
\end{remark}
\begin{proof}
By (\ref{eq:relBH}) and symmetry
property we have
$
\mathfrak{S}_{w}(z;x)=
\mathfrak{S}_{w}(z,0;x)
=\mathfrak{S}_{w^{-1}}(0,-z;x).$
Applying Prop. \ref{prop:LS-BH} with $y=0$
we can rewrite this as follows:
$$
\sum_{w^{-1}=u^{-1}v^{-1}}\mathfrak{S}_{u^{-1}}^{A}
(0,-z)\mathfrak{S}_{v^{-1}}(0,0;x)
=\sum_{w=vu}\mathfrak{S}_{u}^{A}
(z)\mathfrak{S}_{v}(0,0;x),$$
where the sum is over
$v\in \pmb{W}_{\infty}, u\in S_{\infty}$ such that
$w^{-1}=u^{-1}v^{-1},$ and $\ell(w^{-1})=\ell(u^{-1})+\ell(v^{-1}).$ The last equality
follows from symmetry property.
\end{proof}
We are finally ready to prove the positivity property of $\mathfrak{S}_{w}(z,t;x)$. Expand $\mathfrak{S}_{w}(z,t;x)$ as $$
\mathfrak{S}_{w}(z,t;x)=\sum_{\lambda
\in \mathcal{SP}}
f_{w,\lambda}(z,t)
F_{\lambda}(x),
$$ where
$F_{\lambda}(x)=Q_{\lambda}(x)$ for type
$\mathrm{C}$ and $P_{\lambda}(x)$ for type $\mathrm{D}.$
\begin{thm}[Positivity of double Schubert polynomials]\label{thm:positivity} For any $w \in W_n$, the coefficient $f_{w,\lambda}(z,t)$ is a polynomial in
$ \mathbb{N}[-t_{1},\ldots,-t_{n-1},
z_{1},\ldots,z_{n-1}]$.
\end{thm}
\begin{proof} The proof follows from the expression on Corollary \ref{cor:typeAexpand}, Lemma \ref{lem:Stanley} below, combined with Lemma \ref{lem:S00} and the fact that
$\mathfrak{S}_{u}^A(z)\in \mathbb{N}[z]$.
\end{proof}
\begin{lem}\label{lem:Stanley}
$\mathfrak{S}_{v}(0,0;x)$
is a linear combination of
Schur's $Q$- (respectively $P$-) Schur functions
with nonnegative integral coefficients. \end{lem}
\begin{proof} This follows from the transition equations
in \S \ref{ssec:trans}
(see also Remark \ref{rem:TransDSP}).
In fact, the functions
$\mathfrak{S}_{w}(0,0;x)$ satisfy the transition equations
specialized at $z=t=0$ with the Grassmannian
Schubert classes identified with the Schur's $Q$ or $P$-
functions.
In fact, the
recursive formula for $F_{w}(x)=\mathfrak{S}_{w}(0,0;x)$ is positive, in the sense that the right hand side of the
equation is a certain non-negative integral
linear combination of the functions $\{F_{w}(x)\}.$
This implies that
$F_{w}(x)=\mathfrak{S}_{w}(0,0;x)$ can be expressed as
a linear combination of Schur's $Q$ (or $P$)
functions with coefficients in non-negative integers.
\end{proof}
\section{Formula for the longest element}\label{sec:Long}
\setcounter{equation}{0}
In this section, we give explicit formula
for the double Schubert polynomials
associated with the longest element
$w_{0}^{(n)}$ in $W_{n}$ (and $W_{n}'$).
We note that our proof
of Theorem \ref{existC} is independent of this section.
\subsection{Removable boxes}
We start this section with some combinatorial
properties of factorial $Q$ and $P$-Schur functions.
The goal is to prove Prop. \ref{prop:deltaQ}, which shows how the divided difference operators act on the aforementioned functions.
See \S \ref{ssec:FacSchur} to recall the convention
for the shifted Young diagram $Y_{\lambda}.$
\begin{Def}
A box $x\in Y_\lambda$ is removable
if
$Y_\lambda-\{x\}$ is again a
shifted Young diagram of a strict partition.
Explicitly,
$x=(i,j)$ is removable if
$j=\lambda_i+i-1$ and $\lambda_{i+1}\leq \lambda_i-2.$
\end{Def}
To each box $x=(i,j)$ in $Y_\lambda$ we
define its {\it content} $c(x)\in I_\infty$,
$c'(x)\in I_\infty'$
by $c(x)=j-i,$
and
$c'(x)=j-i+1$ if $i\ne j,$
$c'(i,i)=\hat{1} $ if $i$ is odd, and $c'(i,i)=1$ if $i$ is even.
Let $i\in I_\infty$ (resp. $i\in I_\infty'$).
We call $\lambda$ $i$-{\it removable\/}
if there is a removable box $x$ in $Y_\lambda$
such that $c(x)=i$ (resp $c'(x)=i)$.
Note that there is at most one
such $x$ for each $i\in I_\infty$
(resp. $i\in I_\infty').$
We say $\lambda$ is $i$-{\it unremovable\/}
if it is not $i$-{\it removable}.
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\put(38.5,47){\small{$3$}}
\put(48.5,47){\small{$4$}}
\put(48.5,37){\small{$3$}}
\put(5,5){$0$ or $3$\,\mbox{-removable}}
\end{picture}
\begin{picture}(100,70)
\put(5,60){Type $D$}
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\put(48.5,37){\small{$4$}}
\put(5,5){$1$ or $4$\,\mbox{-removable}}
\end{picture}
\end{center}
The following facts are well-known (see e.g. \S 7 in \cite{IN}).
\begin{lem} \label{lem:s_iGrass}
Let $w_\lambda\in W_\infty^0$ (resp. $w'_\lambda\in W_\infty^{\hat{1}}$) denote the Grassmannian
element corresponding to $\lambda\in \mathcal{SP}.$
For $i\in I_\infty$ (resp. $i\in I_\infty'$),
a strict partition $\lambda$ is $i$-removable if and only if
$\ell(s_iw_\lambda)=\ell(w_\lambda)-1$
(resp. $\ell(s_iw_\lambda')=\ell(w_\lambda')-1$).
If $\lambda$ is $i$-removable
then $s_iw_\lambda$ (resp. $s_iw_\lambda'$) is also a
Grassmannian
element and the corresponding
strict partition is the one obtained from $\lambda$
by removing a (unique) box of content $i$.
\end{lem}
\begin{prop} \label{prop:deltaQ} Let $\lambda$ be a strict
partiton and $i\in I_\infty$ (resp. $i\in I_\infty'$).
\begin{enumerate}
\item \label{remove} If $\lambda$ is $i$-removable,
then $\delta_{i}Q_{\lambda}(x|t)=Q_{\lambda'}(x|t),$
(resp. $\delta_{i}P_{\lambda}(x|t)=P_{\lambda'}(x|t)$)
where $\lambda'$ is the strict partition obtained by removing
the (unique) box of content $i$ from $\lambda,$
\item \label{unremov} If $\lambda$ is $i$-unremovable, then
$\delta_{i}Q_{\lambda}(x|t)=0$
(resp. $\delta_{i}P_{\lambda}(x|t)=0$),
that is to say
$s_{i}^{t}Q_{\lambda}(x|t)=Q_{\lambda}(x|t)$
(resp. $s_{i}^{t}P_{\lambda}(x|t)=P_{\lambda}(x|t)$).
\end{enumerate}
\end{prop}
\begin{proof} This follows from Lemma \ref{lem:s_iGrass} and from the fact that $\mathfrak{C}_{w_\lambda}=Q_\lambda(x|t)$ and $\mathfrak{D}_{w_\lambda'}=P_\lambda(x|t)$, hence we can apply the divided difference equations from Theorem \ref{existC}.
\end{proof}
\subsection{Type $\mathrm{C}_{n}$ case}\label{ssec:LongC}
For $\lambda\in \mathcal{SP}$ we define
$$K_{\lambda}=
K_{\lambda}(z,t;x)
=Q_{\lambda}(x|t_{1},-z_{1},t_{2},-z_{2},\ldots,
t_{n},-z_{n},\ldots).
$$
We need the following two lemmata to
prove Thm. \ref{thm:Top}.
\begin{lem}\label{lem:deltaK} Set $\Lambda_{n}=\rho_{n}+\rho_{n-1}.$
We have
$
\delta_{n-1}\cdots\delta_{1}\delta_{0}
\delta_{1}\cdots\delta_{n-1}
K_{\Lambda_{n}}=K_{\Lambda_{n-1}}.
$
\end{lem}
\begin{lem}\label{lem:piDelta} We have
$
\pi_{n}(K_{\Lambda_{n}})
=\sigma_{w_{0}^{(n)}}^{(n)}$, where $\pi_n:R_\infty \to H^*_{T_n} (\mathcal{F}_n)$ is the projection defined in \S \ref{ssec:projection}.
\end{lem}
\subsubsection{Proof of Theorem \ref{thm:Top} for type $\mathrm{C}$}
\label{ssec:PfLongC}
\begin{proof}
Let $w_{0}^{(n)}$ in $W_{n}$ be the longest
element in $W_{n}.$
We need to show that
\begin{equation}
\mathfrak{C}_{w_{0}^{(n)}}(z,t;x)=K_{\Lambda_{n}}
(z,t;x).\label{LongElmFormula}
\end{equation}
Let $w\in W_{\infty}.$ Choose any
$n$ such that $w\in W_{n}$
and set
$$
F_{w}:=\delta_{ww_{0}^{(n)}}K_{\Lambda_{n}}.
$$ Since $\ell(w w_0^{(n)}) + 2n-1 = \ell(w w_0^{(n+1)})$ and $ww_0^{(n)} s_n \cdots s_1s_0s_1 \cdots s_n = ww_0^{(n+1)}$, it follows that $\delta_{ww_0^{(n)}} \cdot \delta_n \cdots \delta_{1}\delta_0 \delta_{1}\cdots \delta_n = \delta_{ww_0^{(n+1)}}$. Then
Lem. \ref{lem:deltaK} yields
$\delta_{ww_{0}^{(n+1)}}K_{\Lambda_{n+1}}
=\delta_{ww_{0}^{(n)}}K_{\Lambda_{n}}$
for any $w\in W_{n}$, so $F_{w}$ is
independent of the choice of $n.$
In order to prove the theorem
it is enough to prove $F_{w}=\mathcal{C}_{w}$
for all $w\in W_{\infty}.$
By definition of $F_{w}$
and basic properties of divided differences
we can show that
\begin{equation}
\delta_{i}F_{w}=\begin{cases}
F_{s_{i}w}& \ell(s_{i}w)=\ell(w)-1\\
0& \mbox{otherwise
\end{cases}.\label{eq:divF}
\end{equation}
Now we claim that
$\pi_{n}(F_{w})=\sigma_{w}^{(n)}$
(for any $n$ such that $w\in W_{n}$).
In fact,
by commutativity of $\pi_{n}$ and
divided difference operators (Prop. \ref{prop:piCommD}), we have
$$
\pi_{n}(F_{w})=\delta_{ww_{0}^{(n)}}\pi_{n}(K_{\Lambda_{n}})
=\delta_{ww_{0}^{(n)}}\sigma_{w_{0}^{(n)}}^{(n)}
=\sigma_{w}^{(n)}.
$$
In the second equality we used
Lem. \ref{lem:piDelta},
and the last equality
is a consequence of (\ref{eq:divF}). Thus the claim is proved.
Since the claim holds for any
sufficiently large $n$,
we have $\Phi(F_{w})=\sigma_{w}^{(\infty)}$
(cf. Prop. \ref{prop:StabSch}).
\end{proof}
\subsubsection{Proof of Lemma \ref{lem:deltaK}}
\begin{proof} The
Lemma follows from the successive use of the following
equations (see the example below):
\begin{enumerate}
\item \label{deltaQ1}
$
\delta_{i}K_{\Lambda_{n}-1^{n-i-1}}
=K_{\Lambda_{n}-1^{n-i}}\quad (0\leq i\leq n-1),
$
\item \label{deltaQ2}
$
\delta_{i}K_{\Lambda_{n}-1^{n}-0^{n-i}1^{i-1}}
=K_{\Lambda_{n}-1^{n}-0^{n-i-1}1^{i}}\quad (1\leq i\leq n-1).
$
\end{enumerate}
We first prove (\ref{deltaQ1}).
For the case $i=0$,
we can apply Prop. \ref{prop:deltaQ} (\ref{remove})
directly to get the equation.
Suppose $1\leq i\leq n-1.$
Before applying $\delta_{i}$
to $K_{\Lambda_{n}-1^{n-i-1}}$
we switch the parameters
at $(2i-1)$-th and $2i$-th positions to get
$$
K_{\Lambda_{n}-1^{n-i-1}}
=Q_{\Lambda_{n}-1^{n-i-1}}
(x|t_{1},-z_{1},\ldots,-z_{i},t_{i},t_{i+1},-z_{i+1},\ldots,t_{n},-z_{n}).
$$
This is valid in view of Prop. \ref{prop:deltaQ} (\ref{unremov}) and
the fact that
$\Lambda_{n}-1^{n-i-1}$ is $(2i-1)$-unremovable.
In the right hand side,
the parameters $t_{i}$ and $t_{i+1}$
are on $2i$-th and $(2i+1)$-th
positions. Thus the operator $\delta_{i}$ on
this function is equal to the $2i$-th divided difference
operator ``$\delta_{2i}$'' with respect to
the sequence of the rearranged parameters
$(t_{1},-z_{1},\ldots,-z_{i},t_{i},t_{i+1},-z_{i+1},\ldots,t_{n},-z_{n})$
(see example below).
Thus we have by Prop. \ref{prop:deltaQ} (\ref{remove})
$$
\delta_{i}K_{\Lambda_{n}-1^{n-i-1}}
=Q_{\Lambda_{n}-1^{n-i}}(x|t_{1},-z_{1},\ldots,-z_{i},t_{i},t_{i+1},-z_{i+1},\ldots,t_{n},-z_{n}),
$$
namely we remove the box of content $2i$
from $\Lambda_{n}-1^{n-i-1}.$
Then again by Prop. \ref{prop:deltaQ} (\ref{unremov}), the last
function is equal to
$K_{\Lambda_{n}-1^{n-i}}$;
here we notice $\Lambda_{n}-1^{n-i}$ is $(2i-1)$-unremovable.
Next we prove (\ref{deltaQ2}).
In this case, by Prop. \ref{prop:deltaQ} (\ref{unremov}),
we can switch $2i$-th and $(2i+1)$-th
parameters to get
$$
K_{\Lambda_{n}-1^{n}-0^{n-i}1^{i-1}}
=Q_{\Lambda_{n}-1^{n}-0^{n-i}1^{i-1}}(x|t_{1},-z_{1},\ldots,-z_{i-1},t_{i},t_{i+1},-z_{i},\ldots,
t_{n},-z_{n}).
$$
Here we used the fact that
$\Lambda_{n}-1^{n}-0^{n-i}1^{i-1}$
is $2i$-unremovable.
Now we apply $\delta_i$ to the function.
The operator $\delta_i$
is now ``$\delta_{2i-1}$''
with respect to
the sequence of the rearranged parameters
$(t_{1},-z_{1},\ldots,-z_{i-1},t_{i},t_{i+1},-z_{i},\ldots,
t_{n},-z_{n}).
$
By applying Prop.
\ref{prop:deltaQ} (\ref{remove}),
we have
$$
\delta_i
K_{\Lambda_{n}-1^{n}-0^{n-i}1^{i-1}}
=Q_{\Lambda_{n}-1^{n}-0^{n-i-1}1^{i}}
(x|t_{1},-z_{1},\ldots,-z_{i-1},t_{i},t_{i+1},-z_{i},\ldots,
t_{n},-z_{n}).
$$
The last expression is equal to
$K_{\Lambda_{n}-1^{n}-0^{n-i-1}1^{i}}$ since
$\Lambda_{n}-1^{n}-0^{n-i-1}1^{i}$ is $2i$-unremovable.
\end{proof}
{\bf Examples.}
Here we illustrate the process to
show $\delta_{2}\delta_{1}\delta_{0}\delta_{1}\delta_{2}
K_{\Lambda_{3}}=K_{\Lambda_{2}}$
(case $n=3$ in Lem. \ref{lem:deltaK}).
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%
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\put(120,43){\small{$\delta_{1}$}}
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\put(19,5){\small{$K_{5,3,1}$}}
\put(89,5){\small{$K_{4,3,1}$}}
\put(149,5){\small{$K_{4,2,1}$}}
\put(209,5){\small{$K_{4,2}$}}
\put(269,5){\small{$K_{4,1}$}}
\put(319,5){\small{$K_{3,1}$}}
\end{picture}
\end{center}
We pick up the first arrow:
$\delta_{2}K_{5,3,1}=K_{4,3,1}$
(equation (\ref{deltaQ1}) in
Lem. \ref{lem:deltaK} for $n=3,\,i=2$).
As is indicated in the proof, we divide
this equality into the following
four steps:
\begin{eqnarray*}
K_{5,3,1}=Q_{5,3,1}(x|t_{1},-z_{1},\underline{t_{2},-z_{2}},t_{3},-z_{3})
\underset{(a)}{=}
Q_{5,3,1}(x|t_{1},-z_{1},-z_{2},t_{2},t_{3},-z_{3})\\
\overset{\delta_{2}}{\longrightarrow}
Q_{4,3,1}(x|t_{1},-z_{1},\underline{-z_{2},t_{2}},t_{3},-z_{3})
\underset{(b)}{=}Q_{4,3,1}(x|t_{1},-z_{1},t_{2},-z_{2},t_{3},-z_{3})=K_{4,3,1}.
\end{eqnarray*}
In the equality $(a)$ we used the fact that
$\Lambda_{3}=(5,3,1)$ is $3$-unremovable,
so the underlined pair of variables can be exchanged
(by Prop. \ref{prop:deltaQ}, (2)).
Then we apply $\delta_{2}$ to this function.
Note that the variables $t_{2},t_{3}$ are in the
$4$-th and $5$-th positions in the parameters of
the function. So if we rename the parameters as
$
f=Q_{5,3,1}(x|t_{1},-z_{1},-z_{2},t_{2},t_{3},-z_{3})
=Q_{5,3,1}(x|u_{1},u_{2},u_{3},u_{4},u_{5},u_{6}),$
then $\delta_{2}$ is ``$\delta_{4}$''
with respect to the parameter sequence $(u_{i})_{i}.$
Namely we have
$$
\delta_{2}f=\frac{f-s_{2}^{t}f}{t_{3}-t_{2}}
=\frac{f-s_{4}^{u}f}{u_{5}-u_{4}},
$$
where $s_{4}^{u}$ exchanges $u_{4}$ and $u_{5}.$
Since $\Lambda_{3}=(5,3,1)$ is $4$-removable,
we see from Prop. \ref{prop:deltaQ}, (1) that
$\delta_{2}=$``$\delta_{4}$'' removes the box of content $4$
from $(5,3,1)$ to obtain the shape $(4,3,1).$
Then finally, in the equality $(b)$, we exchange the variables
$-z_{2},t_{2}$ again using Prop. \ref{prop:deltaQ}, (2).
This is valid since $(4,3,1)$ is $3$-unremovable.
Thus we obtained $K_{4,3,1}.$
\subsubsection{Proof of Lemma \ref{lem:piDelta}}
\begin{proof}
We calculate $\Phi_{v}(K_{\Lambda_{n}})$ for $v\in W_{n}.$
Recall that the map $\Phi_{v}: R_{\infty}\rightarrow
\mathbb{Z}[t]$ is the $\mathbb{Z}[t]$-algebra homomorphism
given by $x_{i}\mapsto t_{v,i}$
and $z_{i}\mapsto t_{v(i)}.$
So we have
$$
\Phi_{v}(K_{\Lambda_{n}})
=Q_{\Lambda_{n}}
(t_{v,1},\ldots,t_{v,n}|t_{1},-{t_{v(1)}},
\ldots, t_{n},-{t_{v(n)}}).
$$
Note that $t_{v,i}=0$ for $i>n$ since $v$ is an element
in $W_{n}.$
From the factorization formula \ref{lem:factorization},
this is equal to
$$\prod_{1\leq i\leq n}2t_{v,i}
\prod_{1\leq i<j\leq n}(t_{v,i}+t_{v,j})
\times s_{\rho_{n-1}}
(t_{v,1},\ldots,t_{v,n}|t_{1},-{t_{v(1)}},
\ldots, t_{n},-{t_{v(n)}}).
$$
The presence of the
factor $\prod_{i}2t_{v,i}$
implies that
$\Phi_{v}(K_{\Lambda_{n}})$ vanishes unless
$v(1),\ldots,v(n)$ are all negative.
So from now on we assume
$v=(\overline{\sigma(1)},\ldots,\overline{\sigma(n)})$
for some permutation $\sigma \in S_{n}.$
Then we have $t_{v,i}=t_{\sigma(i)}$
and $t_{v(i)}=-t_{\sigma(i)}$ so the last factor
of factorial Schur polynomial becomes
$$s_{\rho_{n-1}}
(t_{\sigma(1)},\ldots,t_{\sigma(n)}|t_{1},{t_{\sigma(1)}},
\ldots, t_{n},{t_{\sigma(n)}}).$$
This is equal to
$s_{\rho_{n-1}}
(t_{1},\ldots,t_{n}|t_{1},{t_{\sigma(1)}},
\ldots, t_{n},{t_{\sigma(n)}})$ because
$s_{\rho_{n-1}}$ is symmetric in ther first set of variables.
From Lem. \ref{lem:A-long} we know that this polynomial factors into
$\prod_{1\leq i<j\leq n}
(t_{j}-t_{\sigma(i)}).$
This is zero except for the case $\sigma=\mathrm{id},$
namely $v=w_{0}^{(n)}.$
If $\sigma=\mathrm{id}$ then
$\Phi_{v}(K_{\Lambda_{n}})$ becomes
$\prod_{1\leq i\leq n}2t_{i}
\prod_{1\leq i<j\leq n}(t_{i}+t_{j})
\prod_{1\leq i<j\leq n}
(t_{j}-t_{i})=\sigma_{w_{0}^{(n)}}^{(n)}|_{w_{0}^{(n)}}.$
\end{proof}
\subsection{Type $\mathrm{D}_{n}$ case}
Set
$K'_{\lambda}(z,-t;x)=
P_{\lambda}
(x|t_{1},-z_{1},\ldots,t_{n-1},-z_{n-1},\ldots).
$
Our goal in this section is
$$
\mathfrak{D}_{w_{0}^{(n)}}
=K'_{2\rho_{n-1}}
(z,t;x).
$$
We use the same strategy
as in \S \ref{ssec:LongC} to
prove this.
Actually the proof in \S \ref{ssec:PfLongC}
works also in this case
using the following two lemmata,
which will be proved below.
\begin{lem}\label{lem:DeltaKD} We have
$\delta_{n-1}\cdots\delta_{2}\delta_{\hat{1}}\delta_{1}\delta_{2}
\cdots\delta_{n-1}K_{2\rho_{n-1}}'
=K_{2\rho_{n-2}}'.$
\end{lem}
\begin{lem}\label{lem:piDeltaD} We have
$\pi_{n}(K'_{2\rho_{n-1}})=\sigma_{w^{(n)}_{0}}^{(n)}.$
\end{lem}
\subsubsection{A technical lemma}
We need the following technical lemma which is used
in the proof of Lem. \ref{lem:DeltaKD}.
Throughout the section,
$(u_{1},u_{2},u_{3},\ldots)$
denote any sequence of variables independent of $t_{1},t_{2}.$
\begin{lem}\label{lem:hat1}
Let $\lambda=(\lambda_{1},\ldots,\lambda_{r})$
be a strict partition such that $r$ is odd and $\lambda_{r}\geq 3.$
Set $\tilde{t}=(u_{1},t_{1},t_{2},u_{2},u_{3},\ldots).$
Then
$
\delta_{\hat{1}}P_{\lambda_{1},\ldots,\lambda_{r},1}(x|\tilde{t})
=P_{\lambda_{1},\ldots,\lambda_{r}}(x|\tilde{t}).
$
\end{lem}
\begin{sublem}\label{lem:deltahat1} Suppose
$\lambda$ is $1, 2$ and $\hat{1}$-unremovable.
Then we have
$
\delta_{\hat{1}}
P_{\lambda}(x|\tilde{t})=0.\label{eq:deltahat1}
$
\end{sublem}
\begin{proof}
Since
$\lambda$
is $1,2$-unremovable,
we can rearrange the first three parameters
by using Prop. \ref{prop:deltaQ}, so we have
$P_{\lambda}(x|\tilde{t})=P_{\lambda}(x|t_{1},t_{2},u_{1},u_{2},u_{3},\ldots).$
Because $\lambda$ is also $\hat{1}$-unremovable,
it follows that $\delta_{\hat{1}}P_{\lambda}(t_{1},t_{2},u_{1},u_{4},\ldots)=0$ from Prop. \ref{prop:deltaQ}.
\end{proof}
\begin{sublem}[Special case of Lem. \ref{lem:hat1} for $r=1$]\label{lem:Pk1}
We have $
\delta_{\hat{1}}P_{k,1}(x|\tilde{t})=P_{k}(x|\tilde{t})$
for $k\geq 3.$
\end{sublem}
\begin{proof}
Substituting $\tilde{t}$ for $t$ into (\ref{eq:P2row})
we have
$$P_{k,1}(x|\tilde{t})=
P_{k}(x|\tilde{t})P_{1}(x|\tilde{t})
-P_{k+1}(x|\tilde{t})-(u_{k-1}+u_{1})P_{k}(x|\tilde{t}).$$
By the explicit formula $P_{1}(x|\tilde{t})=P_{1}(x),$
we have
$\delta_{\hat{1}}P_{1}(x|\tilde{t})=1.$
We also have $\delta_{\hat{1}}P_{k}(x|\tilde{t})=\delta_{\hat{1}}P_{k+1}(x|\tilde{t})=0$
by Sublemma \ref{lem:deltahat1}.
Then we use the Leibnitz rule
$\delta_{\hat{1}}(fg)=\delta_{\hat{1}}(f)g+(s_{\hat{1}}f)\delta_{\hat{1}}(g)$
to get $\delta_{\hat{1}}P_{k,1}(x|\tilde{t})=P_{k}(x|\tilde{t}).$
\end{proof}
\begin{proof}[Proof of Lem. \ref{lem:hat1}.]
From the definition of the Pfaffian
it follows that
$$
P_{\lambda_{1},\ldots,\lambda_{r},1}(x|\tilde{t})
=\sum_{j=1}^{r}(-1)^{r-j}P_{\lambda_{j},1}(x|\tilde{t})P_{\lambda_{1},\ldots,
\widehat{\lambda_{j}},\ldots,\lambda_{r}}(x|\tilde{t}).
$$
Then the Leibnitz rule
combined with Sublemma
\ref{eq:deltahat1} and Sublemma \ref{lem:Pk1}
implies
$$
\delta_{\hat{1}}
P_{\lambda_{1},\ldots,\lambda_{r},1}(x|\tilde{t})
=\sum_{j=1}^{r}(-1)^{r-j}P_{\lambda_{j}}(x|\tilde{t})P_{\lambda_{1},\ldots,
\widehat{\lambda_{j}},\ldots,\lambda_{r}}(x|\tilde{t})
=P_{\lambda_{1},\ldots,\lambda_{r}}(x|\tilde{t}),
$$
where in the last equality we used the expansion
formula of Pfaffian again. \end{proof}
\subsubsection{Proof of Lem. \ref{lem:DeltaKD}}
\begin{proof}
Consider the case when $n$ is even.
By applying the same method of calculation as in
type $\mathrm{C}$ case, we have
$$
\delta_{1}\delta_{2}\cdots\delta_{n-1}K'_{2\rho_{n-1}}
=P_{\rho_{n-1}+\rho_{n-2}}(x|-z_{1},t_{1},t_{2},-z_{2},t_{3},-z_{3},\ldots).
$$
The problem here is that $\rho_{n-1}+\rho_{n-2}$ is not
$1$-unremovable when $n$ is even.
So we can not rewrite the function
as $K_{\rho_{n-1}+\rho_{n-2}}'.$
Nevertheless, by using Lem. \ref{lem:hat1},
we can show
$$
\delta_{\hat{1}}P_{\rho_{n-1}+\rho_{n-2}}(x|-z_{1},t_{1},t_{2},-z_{2},t_{3},-z_{3},\ldots)
=K'_{\rho_{n-1}+\rho_{n-2}-0^{n-2}1}.
$$
The rest of calculation is similar to type $\mathrm{C}$ case.
If $n$ is odd, we can show this equation
using only Prop. \ref{prop:deltaQ}
as in type $\mathrm{C}$ case.
\end{proof}
\subsubsection{Proof of Lem. \ref{lem:piDeltaD}}
\begin{proof}
Similar to the proof of Lem. \ref{lem:piDelta} using
Lem. \ref{lem:A-long}, \ref{lem:A-longOdd},
\ref{lem:factorizationD}, and
\ref{lem:factorizationDodd}.
We calculate $\Phi_{v}(K'_{2\rho_{n-1}})$ for $v\in W'_{n}.$
We have
$$
\Phi_{v}(K'_{2\rho_{n-1}})
=P_{2\rho_{n-1}}
(t_{v,1},\ldots,t_{v,n}|t_{1},-{t_{v(1)}},
\ldots, t_{n},-{t_{v(n)}}).
$$
Note that $t_{v,i}=0$ for $i>n$ since $v$ is an element
in $W'_{n}.$
Assume now that $n$ is even.
From the factorization formula (Lem. \ref{lem:factorizationD}),
this is equal to
$
\prod_{1\leq i<j\leq n}(t_{v,i}+t_{v,j})
\times s_{\rho_{n-1}}
(t_{v,1},\ldots,t_{v,n}|t_{1},-{t_{v(1)}},
\ldots, t_{n},-{t_{v(n)}}).
$$
The factorial Schur polynomial
factorizes further into linear terms by Lem. \ref{lem:A-long},
and we finally obtain $$
\Phi_{v}(K'_{2\rho_{n-1}})=
\prod_{1\leq i<j\leq n}(t_{v,i}+t_{v,j})
\prod_{1\leq i<j\leq n}
(t_{j}+t_{v(i)}).
$$
We set
$v=(\overline{\sigma(1)},\ldots,\overline{\sigma(n)})$
for some $\sigma \in S_{n}$ since otherwise
$t_{v,i}=t_{v,j}=0$ for some $i,j$ with $i\ne j$
and then $\prod_{1\leq i<j\leq n}
(t_{v,i}+t_{v,j})$ vanishes.
Then
the factor $\prod_{1\leq i<j\leq n}
(t_{j}+t_{v(i)})$ is $\prod_{1\leq i<j\leq n}
(t_{j}-t_{\sigma(i)}).$
This is zero except for the case $\sigma=\mathrm{id},$
namely $v=w_{0}^{(n)}.$
If $w=w_{0}^{(n)}$, we have
$\Phi_{v}(K'_{\rho_{n-1}})=
\prod_{1\leq i<j\leq n}(t_{i}+t_{j})
\prod_{1\leq i<j\leq n}
(t_{j}-t_{i})=\sigma_{w_{0}^{(n)}}^{(n)}|_{w_{0}^{(n)}}.$
Next we consider the case when $n$ is odd.
Note that the longest element $w_{0}^{(n)}$ in this case
is $1\bar{2}\bar{3}\cdots\bar{n}.$
Let $s(v)$ denote the number of
nonzero entries in $t_{v,1},\ldots,t_{v,n}.$
Then we have
$s(v)\leq n-1$ since $v\in W_{n}'$ and $n$ is odd.
We use the following identity:
\begin{eqnarray}
&&P_{2\rho_{n-1}}(x_{1},\ldots,x_{n-1}|t_{1},-z_{1},\ldots,t_{n-1},-z_{n-1})\nonumber\\
&=&\prod_{1\leq i<j\leq n-1}(x_{i}+x_{j})
\times s_{\rho_{n-1}+1^{n-1}}(x_{1},\ldots,x_{n-1}|t_{1},-z_{1},\ldots,t_{n-1},-z_{n-1}).\label{eq:facD}
\end{eqnarray}
If $s(v)<n-1$,
then $s(v)\leq n-3$ because $v\in W_{n}'.$
This means that there are at least $3$
zeros in $t_{v,1},\ldots,t_{v,n}.$
Because there is the factor
$\prod_{1\leq i<j\leq n-1}(x_{i}+x_{j})$
in (\ref{eq:facD}) we have $\Phi_{v}(K'_{2\rho_{n-1}})=0.$
So we suppose $s(v)=n-1.$
By a calculation using the definition of the factorial
Schur polynomial, we see that
$ s_{\rho_{n-1}+1^{n-1}}(x_{1},\ldots,x_{n-1}|t_{1},-z_{1},\ldots,
t_{n-1},-z_{n-1})$
is divisible by the factor $\prod_{i=1}^{n-1}(t_{1}-x_{i}).$
By this fact we may assume
$t_{v,1},\ldots,t_{v,n}$ is a permutation of
$0,t_{2},t_{3},\ldots,t_{n}$ since
otherwise $\Phi_{v}(K'_{2\rho_{n-1}})$ is zero.
Thus under the assumption, we have
$$
\Phi_{v}(K'_{2\rho_{n-1}})
=\prod_{2\leq i<j\leq n}(t_{i}+t_{j})
s_{\rho_{n-1}+1^{n-1}}(t_{2},\ldots,t_{n}|t_{1},-t_{v(1)},t_{2},-t_{v(2)},\ldots,
t_{n-1},-t_{v(n-1)}).
$$
By Lem. \ref{lem:A-longOdd}
this factorizes into
$
\prod_{2\leq i<j\leq n}(t_{i}+t_{j})
\prod_{j=2}^{n}(t_{j}-t_{1})
\prod_{1\leq i<j\leq n}(t_{j}+t_{v(i)}).
$
Now our assumption is that
the negative elements in
$\{v(1),\ldots,v(n)\}$ are exactly $\{2,3,\ldots,n\}.$
Among these elements, only
$w_{0}^{(n)}$ gives a non-zero
polynomial, which
is shown to be
$\prod_{1\leq i<j\leq n}(t_{i}+t_{j})(t_{j}-t_{i})
=\sigma_{w_{0}^{(n)}}^{(n)}|_{w_{0}^{(n)}}.$
\end{proof}
\section{Geometric construction
of the universal localization map}\label{sec:geometry}
\setcounter{equation}{0}
In this section, we construct the morphism of $\mathbb{Z}[t]$-algebras $\tilde{\pi}_{\infty}:\pmb{R}_\infty \longrightarrow \underleftarrow{\lim}\, H^*_{T_n}(\mathcal{F}_n)$
from a geometric point of view.
We start this section by describing
the embedding $\mathcal{F}_{n}\hookrightarrow \mathcal{F}_{n+1}$
explicitly, and calculate the localization
of the Chern roots of tautological bundles.
Then we introduce some particular cohomology classes
$\beta_{i}$ in $H_{T_{n}}^{*}(\mathcal{F}_{n}) $, by using the geometry of isotropic flag varieties. These classes satisfy the relations of the $Q$-Schur functions $Q_{i}(x)$, and mapping $Q_i(x)$ to $\beta_i$ ultimately leads to the homomorphism
$\tilde{\pi}_{\infty}.$ In particular, this provides an explanation on why
the Schur $Q$-functions enter into our
theory (cf. Prop. \ref{prop:pi}).
The final goal is to establish the connection
of $\tilde{\pi}_{\infty}$ and the universal
localization map $\Phi$ (Thm. \ref{thm:piQpiQ}).
The arguments in the preceding sections are
logically independent from this section.
However, we believe that the results in this section provide
the reader with some
insight into the underlying geometric idea
of the algebraic construction.
\subsection{Flag varieties of isotropic flags}\label{ssection:flagv} The groups $G_n$ are the group of automorphisms preserving a non-degenerate, bilinear form $\langle \cdot, \cdot \rangle$ on a complex vector space $V_n$. The pair $(V_n, \langle \cdot , \cdot \rangle)$ is the following:
\begin{enumerate}
\item In type $\mathrm{C}_n$, $V_n = \mathbb{C}^{2n}$; fix $\pmb{e}_n^{*}, \dots, \pmb{e}_1^{*}, \pmb{e}_1, \dots ,\pmb{e}_n$ a basis for $V_n$. Then $\langle\cdot , \cdot \rangle$ is the skew-symmetric form given by $\langle\pmb{e}_i, \pmb{e}_j^*\rangle = \delta_{i,j}$ (the ordering of the basis elements will be important later, when we will embed $G_n$ into $G_{n+1}$).
\item In types $\mathrm{B}_n$ and $\mathrm{D}_n$, $V_n$ is an odd, respectively even-dimensional complex vector space. Let $\pmb{e}_n^{*}, \dots , \pmb{e}_1^{*}, \pmb{e}_0, \pmb{e}_1, \dots , \pmb{e}_n$ respectively $\pmb{e}_n^{*}, \dots , \pmb{e}_1^{*}, \pmb{e}_1, \dots , \pmb{e}_n$ be a basis of $V_n$. Then $\langle \cdot , \cdot \rangle$ is the symmetric form such that $\langle\pmb{e}_i,\pmb{e}_j^*\rangle=\delta_{i,j}$.
\end{enumerate}
A subspace $V$ of $V_n$ will be called {\em isotropic} if $\langle \pmb{u},\pmb{v}\rangle = 0 $ for any $\pmb{u},\pmb{v} \in V$. Then $\mathcal{F}_n$ is the variety consisting of complete isotropic flags with respect to the appropriate bilinear form. For example, in type $\mathrm{C}_n$, $\mathcal{F}_n$ consists of nested sequence of vector spaces
\[ F_1 \subset F_2 \subset \dots \subset F_n \subset V_n = \mathbb{C}^{2n} \/, \] such that each $F_i$ is isotropic and $\dim F_i =i$. Note that the maximal dimension of an isotropic subspace of $\mathbb{C}^{2n}$ is $n$; but the flag above can be completed to a full flag of $\mathbb{C}^{2n}$ by taking $V_{n+i}= V_{n-i}^\perp$, using the non-degeneracy of the form $\langle \cdot , \cdot \rangle$. A similar description can be given in types $\mathrm{B}_n$ and $\mathrm{D}_n$, with the added condition that, in type $\mathrm{D}_n$, \[\dim F_n \cap \langle \pmb{e}_n^{*}, \dots , \pmb{e}_1^*\rangle \equiv 0 \mod 2 \/; \] in this case we say that all $F_n$ are {\em in the same family} (cf. \cite[pag.68]{FP}).
The flag variety $\mathcal{F}_{n}$ carries
a transitive left action of the group $G_n,$
and can be identified with the homogeneous space $G_n/B_n$, where $B_{n}$ is the Borel subgroup consisting of
upper triangular matrices in $G_{n}$.
Let $T_{n}$ be the maximal torus
in $G_{n}$ consisting of
diagonal matrices in $G_{n}.$
Let $t = {\mathrm{diag}}(\xi_n^{-1}, \dots ,\xi_1^{-1},\xi_1, \dots , \xi_n)$ be a torus element in types $\mathrm{C_n,D_n}$, and $t= {\mathrm{diag}}(\xi_n^{-1}, \dots ,\xi_1^{-1},1,\xi_1, \dots , \xi_n)$ in type $\mathrm{B}_n$. We denote by $t_{i}$ the character
of $T_{n}$ defined by $t\mapsto \xi_{i}^{-1}\;(t\in T_{n}).$
Then the weight of $\mathbb{C} \,\pmb{e}_i$ is $-t_i$ and that of $\mathbb{C}\, \pmb{e}_i^*$ is $t_i$.
We identify $t_i \in H^2_{T_n}(pt)$ with $c_1^T(\mathbb{C}\pmb{e}_i^*)$, where $\mathbb{C}\pmb{e}_i^*$ is the (trivial, but not equivariantly trivial) line bundle over $pt$
with fibre $\mathbb{C}\pmb{e}_i^*$. For $v \in W_n$, the corresponding $T_{n}$-fixed point ${e}_v$ is \[{e}_v: \langle \pmb{e}_{v(n)}^{*} \rangle \subset \langle \pmb{e}_{v(n)}^{*}, \pmb{e}_{v(n-1)}^{*}\rangle \subset \dots \subset \langle \pmb{e}_{v(n)}^{*}, \pmb{e}_{v(n-1)}^{*}, \dots , \pmb{e}_{v(1)}^{*}\rangle \subset V_n \/. \]
\subsection{Equivariant embeddings of flag varieties}\label{ssec:embeddings}
There is a natural embedding $G_n \hookrightarrow G_{n+1}$, given explicitly by
$$
g\rightarrow
\left(\begin{array}{c|c|c}
1 & & \\
\hline
& g & \\
\hline
& & 1
\end{array}\right).
$$
This corresponds to the embedding of Dynkin diagrams in each type.
This also induces embeddings $B_{n}\hookrightarrow B_{n+1},$ $T_{n}\hookrightarrow T_{n+1},$ and ultimately $\varphi_n
:\mathcal{F}_{n}\hookrightarrow \mathcal{F}_{n+1}.$
The embedding $\varphi_n$ sends
the complete isotropic flag $F_1 \subset \cdots \subset F_n$ of $V_n$ to the complete isotropic flag of $V_{n+1}=
\mathbb{C}\, \pmb{e}_{n+1}^{*} \oplus V_{n}\oplus \mathbb{C}\, \pmb{e}_{n+1}$:
\[ \mathbb{C}\, \pmb{e}_{n+1}^* \subset \mathbb{C}\, \pmb{e}_{n+1}^* \oplus F_1 \subset \cdots \subset \mathbb{C}\, \pmb{e}_{n+1}^* \oplus F_n \/. \]
Cleary $\varphi_n$ is
equivariant with respect to the embedding
$T_{n}\hookrightarrow T_{n+1}.$
\subsection{Localization of Chern classes of tautological bundles}
Consider the flag of tautological (isotropic) vector bundles
$$
0=\mathcal{V}_{n+1}\subset \mathcal{V}_{n}\subset
\cdots
\subset \mathcal{V}_{1}\subset \mathcal{E},
\quad \mathrm{rank}\,\mathcal{V}_{i}=n-i+1,
$$
where $\mathcal{E}$ is the trivial bundle
with fiber $V_{n}$ and
$\mathcal{V}_{i}$ is defined to be the vector
subbundle of $\mathcal{E}$ whose
fiber over the point $F_{\bullet}= F_1 \subset \dots \subset F_n$ in $\mathcal{F}_{n}$
is $F_{n-i+1}.$
Let $z_{i}=c_{1}^{T}(\mathcal{V}_{i}/\mathcal{V}_{i+1})$
\begin{footnote}{The bundle $\mathcal{V}_i/\mathcal{V}_{i+1}$ is in fact negative; for example, in type $\mathrm{C}$, if $n=1$, $\mathcal{F}_1=\mathbb{P}^1$ and $\mathcal{V}_1 = \mathcal{O}(-1)$. The reason for choosing positive sign for $z_i$ is to be consistent with the conventions used by Billey-Haiman in \cite{BH}.}\end{footnote}denote the equivariant Chern class of the line bundle
$\mathcal{V}_{i}/\mathcal{V}_{i+1}$.
\begin{prop}\label{prop:loc} Let $v \in W_n$. Then the localization map $\iota_v^*: H^*_{T_n} (\mathcal{F}_n) \to H^*_{T_n} ({e}_v)$ satisfies $\iota_v^*(z_i) = t_{v(i)}$. \end{prop}
\begin{proof} The pull-back of the line bundle $\mathcal{V}_{i}/\mathcal{V}_{i+1}$ via $\iota_v^*$ is the line bundle over ${e}_v$ with fibre $\mathbb{C}\pmb{e}_{v(i)}^{*}$, which has (equivariant) first Chern class $t_{v(i)}$. \end{proof}
\subsection{The cohomology class $\beta_{i}$}
In this section we introduce the cohomology classes
$\beta_{i}$, which will later be identified to $Q$-Schur functions $Q_i(x)$.
The torus action on $V_{n}$ induces a $T_{n}$-equivariant splitting $\mathcal{E}=\oplus_{i=1}^{n}\mathcal{L}_{i}\oplus\mathcal{L}_{i}^{*}$ ($\mathcal{E}=\oplus_{i=1}^{n}\mathcal{L}_{i}\oplus\mathcal{L}_{i}^{*}\oplus\mathcal{L}_{0}$ for type $\mathrm{B}_{n}$)
where $\mathcal{L}_{i}$ (resp. $\mathcal{L}_{i}^{*}$) is the trivial line bundle over $\mathcal{F}_n$
with fiber $\mathbb{C}\pmb{e}_{i}$ (resp. $\mathbb{C}\pmb{e}_{i}^{*}$). Recall from \S \ref{ssection:flagv} that $T_n$ acts on $\mathcal{L}_i^*$ by weight $t_i$ and that $t_i = c_1^T(\mathcal{L}_i^*)$.
Let $\mathcal{F}_n$ be the flag variety of type $\mathrm{C}_{n}$ or $\mathrm{D}_{n}$ and
set $\mathcal{V}=\mathcal{V}_{1}$.
We have the following
exact sequence of $T_{n}$-equivariant
vector bundles:
\begin{equation}
0\longrightarrow \mathcal{V}\longrightarrow
\mathcal{E}
\longrightarrow
\mathcal{V}^{*}\longrightarrow 0.\label{tauto}
\end{equation}
where $\mathcal{V}^{*}$ denotes the dual bundle
of $\mathcal{V}$ in $\mathcal{E}$ with respect to
the bilinear form.
Let $\mathcal{L}=\oplus_{i=1}^{n}\mathcal{L}_{i}$
and $\mathcal{L}^{*}=\oplus_{i=1}^{n}\mathcal{L}_{i}^{*}.$
Since $\mathcal{E}=\mathcal{L}\oplus
\mathcal{L}^{*},$
we have $c^{T}(\mathcal{E})=c^{T}(\mathcal{L})c^{T}(\mathcal{L}^{*}).$ Define
the class $\beta_{i}\in H_{T_{n}}^{*}(\mathcal{F}_{n})$ by
$$
\beta_{i}=c_{i}^{T}(\mathcal{V}^{*}-\mathcal{L}),
$$
where $c_{i}^{T}(\mathcal{A}-\mathcal{B})$
is the term of degree $i$ in the formal expansion
of $c^{T}(\mathcal{A})/c^{T}(\mathcal{B}).$
Using the relation $c^{T}(\mathcal{L})c^{T}(\mathcal{L}^{*})
=c^{T}(\mathcal{V})c^{T}(\mathcal{V}^{*})$,
we also have the expression:
$$
\beta_{i}=c_{i}^{T}(\mathcal{L}^{*}-\mathcal{V}).
$$
In terms of the Chern classes $z_{i},t_{i}$,
the class $\beta_{i}$ has
the following two equivalent expressions:
\begin{equation}
\sum_{i=0}^{\infty}\beta_{i}u^{i}=\prod_{i=1}^{n}\frac{1-z_{i}u}{1-t_{i}u}=
\prod_{i=1}^{n}\frac{1+t_{i}u}{1+z_{i}u}.\label{eq:twoexpr}
\end{equation}
\begin{lem}\label{lem:beta}
The classes $\beta_{i}$ satisfy
the same relations as the $Q$-Schur functions
of $Q_{i}(x)$, i.e.
$$
\beta_{i}^{2}+2\sum_{j=1}^{i}(-1)^{j}\beta_{i+j}\beta_{i-j}=0\quad
\mbox{for}\; i\geq 1.
$$
\end{lem}
\begin{proof}
We have the following two expressions:
$$
\sum_{i=0}^{\infty}\beta_{i}u^{i}
=
\prod_{i=1}^{n}\frac{1-z_{i}u}{1-t_{i}u},\quad
\sum_{j=0}^{\infty}(-1)^{j}\beta_{j}u^{j}=
\prod_{i=1}^{n}\frac{1-t_{i}u}{1-z_{i}u}.
$$
The lemma follows from multiplying both sides, and then extracting
the degree $2i$ parts.
\end{proof}
Minor modifications need to be done if $\mathcal{F}_n$ is the flag variety of type $\mathrm{B}_{n}$. In this case the tautological sequence of isotropic flag subbundles consists of $
0=\mathcal{V}_{n+1}\subset \mathcal{V}_{n}\subset
\cdots
\subset \mathcal{V}_{1}\subset \mathcal{E}= \mathbb{C}^{2n+1} \times \mathcal{F}_n$, but the dual bundle $\mathcal{V}_1^*$ of $\mathcal{V}_1$ is not isomorphic to $\mathcal{E}/\mathcal{V}_1$, which has rank $n+1$.
However, the line bundle $\mathcal{V}_1^\perp/\mathcal{V}_1$ is equivariantly isomorphic to $\bigwedge^{2n+1} \mathcal{E}$ - cf. \cite[pag.75]{FP} - so $c_1^T(\mathcal{V}_1^\perp/\mathcal{V}_1)=0$; here $\mathcal{V}_1^\perp$ denotes the bundle whose fibre over $V_1 \subset \cdots \subset V_n$ is the subspace of vectors in $\mathbb{C}^{2n+1}$ perpendicular to those in $V_n$ with respect to the non-degenerate form $\langle \cdot , \cdot \rangle$.
It follows that the bundle $\mathcal{E}/\mathcal{V}_{1}$ has (equivariant) total Chern class $(1-z_1u) \cdots (1-z_nu)$, which is the same as the total Chern class of $\mathcal{V}_{1}^*.$ Similarly, the total Chern class of $\mathcal{E}/\mathcal{L}$
with $\mathcal{L}=\oplus_{i=1}^{n}\mathcal{L}_{i}$ is $(1+t_1u)\cdots (1+t_nu)$ and equals $c^T(\mathcal{L}^*)$. So the definition of $\beta_i$ and the proofs of its properties remain unchanged.
Recall that in \S \ref{ssec:projection} we introduced $\pi_{n}:\pmb{R}_{\infty}
\longrightarrow H_{T_{n}}^{*}(\mathcal{F}_{n})$
by using the universal localization map $\Phi$. The following is the key fact used in the proof of the main
result of this section.
\begin{lem}\label{lem:key} We have
$
\pi_{n}(Q_{i}(x))=\beta_{i}.
$
\end{lem}
\begin{proof}
It is enough to show that
$\iota_{v}^{*}(\beta_{i})=Q_{i}(t_{v})
$ for $v\in \pmb{W}_{n}.$
By Prop. \ref{prop:loc} and the definition of
$\beta_{i}$, we have
$$
\iota_{v}^{*}\left(
\sum_{i=0}^{\infty}\beta_{i}u^{i}
\right)
=\iota_{v}^{*}\left(\prod_{i=1}^{n}\frac{1-z_{i}u}{1-t_{i}u}
\right)
=\prod_{i=1}^{n}\frac{1-t_{v(i)}u}{1-t_{i}u}.$$
If $v(i)$ is positive, the factors $1-t_{v(i)}u$ cancel out and
the last expression
becomes
$$
\prod_{v(i)\;{\small \mbox{negative}}}\frac{1-t_{v(i)}u}{1+t_{v(i)}u}
=\sum_{i=0}^{\infty}
Q_{i}(t_{v})u^{i}$$
where the last equality follows
from the definition of $Q_{i}(x)$ and
that of $t_{v}.$
\end{proof}
\subsection{Homomorphism $\tilde{\pi}_{n}$}
We consider $\mathcal{F}_{n}$ of one of the types
$\mathrm{B}_{n},\mathrm{C}_{n},$ and $\mathrm{D}_{n}.$
We will define next the projection homomorphism
from $\pmb{R}_{\infty}$ to $H_{T_{n}}^{*}(\mathcal{F}_{n})$, which will be used to construct the geometric analogue $\tilde{\pi}_n$ of $\pi_n$.
Note that $R_{\infty}$ is a proper subalgebra of $\pmb{R}_{\infty}$ in types $\mathrm{B}$ and $\mathrm{D}.$
We regard $H_{T_{n}}^{*}(\mathcal{F}_{n})$
as $\mathbb{Z}[t]$-module via the natural projection
$\mathbb{Z}[t]\rightarrow \mathbb{Z}[t_{1},\ldots,t_{n}].$
\begin{prop}\label{prop:pi}
There exists a homomorphism of graded
$\mathbb{Z}[t]$-algebras
$\tilde{\pi}_{n}:R_{\infty}\rightarrow H_{T_{n}}^{*}(\mathcal{F}_{n})$
such that
$$
\tilde{\pi}_{n}(Q_{i}(x))=\beta_{i}\quad
(i\geq 1)
\quad\mbox{,}\quad
\tilde{\pi}_{n}(z_{i})=z_{i}\quad
(1\leq i\leq n)\quad\mbox{and}\quad
\tilde{\pi}_{n}(z_{i})=0\quad(i>n).$$
\end{prop}
\begin{proof} This follows from the fact that $R_\infty$ is generated as a $\mathbb{Z}[t]$-algebra by $Q_{i}(x),z_{i}\;(i\geq 1)$, and that the ideal of relations among $Q_i(x)$ is generated by those in (\ref{eq:quadraticQ}) (see \cite{Mac}, III \S 8). Since the elements $\beta_i$ satisfy also those relations by Lemma \ref{lem:beta}, the result follows.\end{proof}
\subsection{Types $\mathrm{B}$ and $\mathrm{D}$} In this section, we extend $\tilde{\pi}_{n}$
from $R_{\infty}'$ to $H_{T_{n}}^{*}(\mathcal{F}_{n})$. The key to that is the identity $P_i(x) = \frac{1}{2} Q_i(x)$.
\begin{prop} Let
$\mathcal{F}_{n}$ be the flag variety of type $\mathrm{B}_{n}$ or $\mathrm{D}_{n}.$
Then there is an (integral) cohomology class
$\gamma_{i}$ such that
$2\gamma_{i}=\beta_{i}.$
Moreover, the classes $\gamma_i$ satisfy the following quadratic relations:
$$
\gamma_{i}^{2}+2\sum_{j=1}^{i-1}(-1)^{j}
\gamma_{i+j}\gamma_{i-j}
+(-1)^{i}\gamma_{2i}=0 \quad(i>0).
$$
\end{prop}
\begin{proof} Define $\gamma_i=\frac{1}{2} \beta_i$. Then, as in the proof of Lemma \ref{lem:key}, the localization $\iota^*_v(\gamma_i) = \frac{1}{2} Q_i(t_v) = P_i(t_v)$ which is a polynomial with integer coefficients.
The quadratic relations follow immediately from Lem. \ref{lem:beta}.
\end{proof}
The proposition implies immediately the following:
\begin{prop}\label{prop:piBD}
Let $\mathcal{F}_{n}$ be the flag variety of type $\mathrm{B}_{n}$ or $\mathrm{D}_{n}.$
There exists a homomorphism of graded
$\mathbb{Z}[t]$-algebras
$\tilde{\pi}_{n}:R_{\infty}'\rightarrow H_{T_{n}}^{*}(\mathcal{F}_{n})$
such that
$$
\tilde{\pi}_{n}(P_{i}(x))=\gamma_{i}\quad
(i\geq 1)
\quad\mbox{and}\quad
\tilde{\pi}_{n}(z_{i})=z_{i}\quad
(1\leq i\leq n)\quad\mbox{and}\quad
\tilde{\pi}_{n}(z_{i})=0\quad(i>n).$$
\end{prop}
\begin{remark} {\rm It is easy to see (cf, \cite[\S 6.2]{FP}) that the morphism $\tilde{\pi}_n:R_\infty \to H^*_{T_n}(\mathcal{F}_n)$ is surjective in type $\mathrm{C}$, and also in types $\mathrm{B,D}$, but with coefficients over $\mathbb{Z}[1/2]$. But in fact, using that $\Phi:R_\infty' \to H_\infty$ is an isomorphism, one can show that surjectivity holds over $\mathbb{Z}$ as well.}\end{remark}
\subsection{The geometric interpretation of the universal localization map $\Phi$}
From Prop. \ref{prop:pi} and Prop. \ref{prop:piBD},
we have $\mathbb{Z}[t]$-algebra homomorphism
$
\tilde{\pi}_{n}: \pmb{R}_{\infty}\longrightarrow
H_{T_{n}}^{*}(\mathcal{F}_{n})
$
for all types $\mathrm{B,C,D}.$
Since $\tilde{\pi}_n$ is compatible with maps $\varphi_{n}^{*}:H^*_{T_n}(\mathcal{F}_{n+1}) \to H^*_{T_n}(\mathcal{F}_n)$ induced by embeddings $\mathcal{F}_n \to \mathcal{F}_{n+1}$
there is an induced homomorphism $$\tilde{\pi}_{\infty}:\pmb{R}_\infty \longrightarrow \underleftarrow{\lim} H^*_{T_n} (\mathcal{F}_n).$$
Recall from \S \ref{ssec:projection} that we have the natural embedding
$\pi_\infty: R_{\infty}\hookrightarrow
\underleftarrow{\lim} H^*_{T_n} (\mathcal{F}_n)$, defined via the localization map $\Phi$. Then:
\begin{thm}\label{thm:piQpiQ}
We have that $\tilde{\pi}_{\infty}=\pi_\infty$.
\end{thm}
\begin{proof}
It is enough to show that $\tilde{\pi}_{n}=\pi_{n}$. To do that, we
compare both maps on the generators
of $\pmb{R}_{\infty}.$
We know that $\tilde{\pi}_{n}(Q_{i}(x))={\pi}_{n}(Q_{i}(x))
=\beta_{i}$
by Lem. \ref{lem:key} and this implies that
$\tilde{\pi}_{n}(P_{i}(x))={\pi}_{n}(P_{i}(x))$
for types $\mathrm{B}_{n}$ and $\mathrm{D}_{n}.$
It remains to show
$\pi_{n}(z_{i})=\tilde{\pi}_{n}(z_{i}).$
In this case, for $v\in \pmb{W}_{n}$,
\[
\iota_{v}^{*}\pi_{n}(z_{i})
=\Phi_{v}(z_{i})^{(n)}=t_{v(i)}^{(n)}
=\iota_{v}^{*}\tilde{\pi}_{n}(z_{i})\/.
\]
This completes the proof.
\end{proof}
\subsection{Integrality of Fulton's classes $c_{i}$}
We take to opportunity to briefly discuss, in the present setting, an integrality property of some
cohomology classes considered by Fulton in \cite{F}, in relation to degeneracy loci in classical types. This property was proved before, in a more general setting, by Edidin and Graham \cite{EG}, by using the geometry of quadric bundles.
Let $\mathcal{F}_{n}$ be the flag variety of type
$\mathrm{B}_{n}$ or $\mathrm{D}_{n}.$ Recall that $c_i$ is the equivariant cohomology class in $H_{T_{n}}^{*}(\mathcal{F}_{n},\mathbb{Z}[{\textstyle\frac{1}{2}}])$ defined by:
$$
c_{i}={\textstyle\frac{1}{2}}
\left(
e_{i}(-z_{1},\ldots,-z_{n})+e_{i}(t_{1},\ldots,t_{n})
\right)\quad(0\leq i\leq n).
$$
\begin{prop}
We have
$
c_{i}=
\sum_{j=1}^{i-\epsilon}(-1)^{j}e_{j}(t_{1},\ldots,t_{n})\gamma_{i-j}\;(0\leq i\leq n),
$
where $\epsilon=0$ if $i$ is even and $\epsilon=1$
if $i$ is odd.
In particular, $c_{i}$ are classes defined over $\mathbb{Z}$.
\end{prop}
\begin{proof}
Using the definition of $\beta_{i}$
and $
\sum_{i=0}^{n}2c_{i}u^{i}
=\prod_{i=1}^{n}(1-z_{i}u)
+\prod_{i=1}^{n}(1+t_{i}u)
$
we have
$$\sum_{i=0}^{n}2c_{i}u^{i}
=\prod_{i=1}^{n}(1+t_{i}u)+
\sum_{i=0}^{\infty}\beta_{i}u^{i}\prod_{i=1}^{n}(1-t_{i}u).
$$
Comparing both hand side of degree $i$ and using $2\gamma_{i}=\beta_{i}$
we have the equation.
\end{proof}
\section{Kazarian's formula for Lagrangian Schubert classes}
\label{sec:Kazarian}
\setcounter{equation}{0}
In this section, we give a brief discussion
of a ``multi-Schur Pfaffian'' expression for the Schubert classes
of the Lagrangian Grassmannian.
This formula appeared in a preprint of Kazarian, regarding a
degeneracy loci formula for the Lagrangian
vector bundles \cite{Ka}.
\subsection{Multi-Schur Pfaffian}
We recall the definition of the multi-Schur Pfaffian from \cite{Ka}.
Let $\lambda=(\lambda_{1}>\cdots>\lambda_{r}\geq 0)$
be any strict partition with $r$ even.
Consider an $r$-tuple of infinite sequences
$c^{(i)}=\{c^{(i)}_k\}_{k=0}^\infty \;(i=1,\ldots,r)$,
where each $c^{(i)}_k$ is an element in
a commutative ring with unit.
For $a\geq b\geq 0$, we set
$$
c_{a,b}^{(i),(j)}
:=c_{a}^{(i)}
c_{b}^{(j)}
+2\sum_{k=1}^{b}(-1)^{k}
c_{a+k}^{(i)}
c_{b-k}^{(j)}.
$$
Assume that the matrix $(c_{\lambda_i,\lambda_j}^{(i),(j)})_{i,j}$
is skew-symmetric, i.e.
$c_{\lambda_i,\lambda_j}^{(i),(j)}
=-c_{\lambda_j,\lambda_i}^{(j),(i)}$ for
$1\leq i,j\leq r.$
Then we consider its Pfaffian
$$\mathrm{Pf}_\lambda(c^{(1)},\ldots,c^{(r)})
=
\mathrm{Pf}\left(
c_{\lambda_i,\lambda_j}^{(i),(j)}
\right)_{1\leq i<j\leq r},
$$
called {\it multi-Schur Pfaffian}.
\subsection{Factorial Schur functions
as a multi-Schur Pfaffian}
We introduce the following
versions of factorial $Q$-Schur functions $Q_k(x|t)$:
$$
\sum_{k=0}^\infty Q_k^{(l)}(x|t)u^k=
\sum_{i=1}^\infty
\frac{1+x_iu}{1-x_iu}
\prod_{j=1}^{l-1}(1-t_ju).
$$
Note that, by definition,
$Q_k^{(k)}(x|t)=Q_k(x|t)$ and
$Q_k^{(1)}(x|t)=Q_k(x).$
\begin{prop} \label{prop:multiPf} Let $\lambda=(\lambda_{1}>\cdots>\lambda_{r}\geq 0)$
be any strict partition with $r$ even.
Set $c^{(i)}_k=Q_{k}^{(\lambda_i)}(x|t)$
for $i=1,\ldots,r.$
Then the matrix $(c_{\lambda_i,\lambda_j}^{(i),(j)})_{i,j}$
is skew-symmetric
and
we have
$$
\mathrm{Pf}_\lambda(c^{(1)},\ldots,c^{(r)})
=Q_\lambda(x|t).
$$
\end{prop}
\begin{proof}
In view of the Pfaffian formula for $Q_{\lambda}(x|t)$ (Prop. \ref{prop:Pf}),
it suffices to show the following identity:
\begin{equation}
Q_{k,l}(x|t)
=Q_{k}^{(k)}(x|t)Q_{l}^{(l)}(x|t)
+2\sum_{i=1}^{l}(-1)^{i}Q_{k+i}^{(k)}(x|t)Q_{l-i}^{(l)}(x|t).\label{Fkl}
\end{equation}
By induction we can show that for $k\geq 0$
\begin{eqnarray*}
Q_{j}^{(j+k)}(x|t)&=&\sum_{i=0}^{j}(-1)^{i}
e_{i}(t_{j+k-1},t_{j+k-2},\ldots,t_{j-i+1})Q_{j-i}(x|t),\\
Q_{j}^{(j-k)}(x|t)&=&\sum_{i=0}^{k}
h_{i}(t_{j-k},t_{j-k+1},\ldots,t_{j-i})Q_{j-i}(x|t).
\end{eqnarray*}
Substituting these
expressions into (\ref{Fkl}), we get
a quadratic expression
in $Q_i(x|t)$'s.
The obtained expression
coincides with a formula for
$Q_{k,l}(x|t)$ proved in \cite[Prop.7.1]{Ik}.
\end{proof}
\subsection{Schubert classes in the Lagrangian Grassmannian
as multi-Pfaffians} We use the notations from \S \ref{sec:geometry}.
The next formula expresses the equivariant Schubert class $\sigma_{w_{\lambda}}^{(n)}$ in a flag variety of type $\mathrm{C}$ in terms of a multi-Pfaffian. Recall that this is also the equivariant Schubert class for the Schubert variety indexed by $\lambda$ in the Lagrangian Grassmannian, so this is a "Giambelli formula" in this case. Another such expression, in terms of ordinary Pfaffians, was proved by the first author in \cite{Ik}.
\begin{prop}[cf. \cite{Ka}, Thm. 1.1] Set $\mathcal{U}_{k}
=\oplus_{j=k}^n \mathcal{L}_i.$
Then $
\sigma_{w_{\lambda}}^{(n)}
=\mathrm{Pf}_{\lambda}(c^{T}(\mathcal{E}-\mathcal{V}-\mathcal{U}_{\lambda_{1}}),\ldots,
c^{T}(\mathcal{E}-\mathcal{V}-\mathcal{U}_{\lambda_{r}})).$
\end{prop}
\begin{proof} By Thm. \ref{PhiFacQ}, we know
$\pi_{n}(Q_{\lambda}(x|t))=\sigma_{w_{\lambda}}^{(n)}.$
On the other hand the formula of Prop. \ref{prop:multiPf}
writes $Q_{\lambda}(x|t)$ as a multi-Pfaffian.
So it is enough to show that:
$$
c^{T}_{i}(\mathcal{E}-\mathcal{V}-\mathcal{U}_{k})=
\pi_{n}(Q_{i}^{(k)}(x|t)).
$$
We have
$$
c^{T}(\mathcal{E}-\mathcal{V}-\mathcal{U}_{k})
=\frac{\prod_{i=1}^n(1-t_i^2u^{2})}{
\prod_{j=1}^{n}(1+z_{i}u)\prod_{j=k}^n(1-t_ju)}
=\prod_{i=1}^n
\frac{1+t_iu}{1+z_{i}u}
\prod_{j=1}^{k-1}(1-t_ju).
$$
The first factor of the
right hand side is the generating
function for $\beta_{i}=\pi_{n}(Q_{i}(x))
(i\geq 0).$
So the last expression is
$$
\sum_{i=0}^{\infty}\pi_{n}(Q_{i}(x))u^{i}
\prod_{j=1}^{k-1}(1-t_ju)
=\sum_{i=0}^{\infty}\pi_{n}(Q_{i}^{(k)}(x|t))u^{i}.
$$
Hence the proposition is proved.
\end{proof}
\section{Type $\mathrm{C}$ double Schubert polynomials for $w \in W_3$}
\tiny
\renewcommand{\arraystretch}{1.2}
$\begin{array}{|c|l|}
\hline
123 & 1 \\
\hline
\123 & Q_1 \\
\hline
213 & Q_1+(z_1-t_1) \\
\hline
\213 & Q_2+Q_1(-t_1) \\
\hline
2\13 & Q_2+Q_1z_1 \\
\hline
\bar{2}\13 & Q_{21} \\
\hline
1\23 & Q_3+Q_2(z_1-t_1)+Q_1(-z_1t_1) \\
\hline
\bar{1}\23 & Q_{31}+Q_{21}(z_1-t_1) \\
\hline
132 & Q_{1}+(z_1+z_2-t_1-t_2) \\
\hline
\132 & 2Q_{2}+Q_{1}(z_1+z_2-t_1-t_2) \\
\hline
312 & Q_2+Q_1(z_1-t_1-t_2)+(z_1-t_1)(z_1-t_2) \\
\hline
\312 & Q_3+Q_2(-t_1-t_2)+Q_1 t_1t_2 \\
\hline
3\12 & Q_{3}+Q_{21}+Q_{2}(2z_1-t_1-t_2)+Q_{1}(z_1)(z_1-t_1-t_2) \\
\hline
\bar{3}\12 & Q_{31}+Q_{21}(-t_1-t_2) \\
\hline
1\32 & Q_{4}+Q_{3}(z_1-t_1-t_2)+Q_{2}(t_1t_2-z_1(t_1+t_2))+Q_1z_1t_1t_2 \\
\hline
\bar{1}\32 & Q_{41}+Q_{31}(z_1-t_1-t_2)+Q_{21}(t_1t_2-z_1(t_1+t_2)) \\
\hline
231 & Q_2+Q_{1}(z_1+z_2-t_1)+(z_1-t_1)(z_2-t_1) \\
\hline
\231 & Q_{3}+Q_{21}+Q_{2}(z_1+z_2-2t_1)+Q_{1}(-t_1)(z_1+z_2-t_1) \\
\hline
321 & Q_{3}+Q_{21}+Q_{2}(2z_1+z_2-2t_1-t_2)+Q_{1}(z_1+z_2-t_1)(z_1-t_1-t_2)+(z_1-t_1)(z_1-t_2)(z_2-t_1) \\
\hline
\321 & Q_{4}+Q_{31}+Q_{3}(z_1+z_2-2t_1-t_2)+Q_{21}(-t_1-t_2)+\\ & Q_{2}(t_1t_2-(t_1+t_2)(z_1+z_2-t_1)+Q_{1}t_1t_2(z_1+z_2-t_1) \\
\hline
3\21 & Q_{31}+Q_{3}(z_1-t_1)+Q_{21}(z_1-t_1)+Q_{2}(z_1-t_1)^2+Q_1 z_1(-t_1)(z_1-t_1) \\
\hline
\bar{3}\21 & Q_{32}+Q_{31}(-t_1)+Q_{21} t_1^2 \\
\hline
2\31 & Q_{41}+Q_{4}(z_1-t_1)+Q_{31}(z_1-t_1-t_2)+Q_{3}(z_1-t_1)(z_1-t_1-t_2)+Q_{21}(t_1t_2-z_1(t_1+t_2))+\\ &
Q_{2}(z_1-t_1)(t_1t_2-z_1(t_1+t_2))+Q_{1}(z_1-t_1)z_1t_1t_2 \\
\hline
\bar{2}\31 & Q_{42}+Q_{32}(z_1-t_1-t_2)+Q_{41}(-t_1)+Q_{31}(-t_1)(z_1-t_1-t_2)+Q_{21}t_1^2(z_1-t_2) \\
\hline
23\bar{1} & Q_{3}+Q_{2}(z_1+z_2)+Q_{1} z_1z_2 \\
\hline
\23\bar{1} & Q_{31}+Q_{21}(z_1+z_2) \\
\hline
32\bar{1} & Q_{4}+Q_{31}+Q_{3}(2z_1+z_2-t_1-t_2)+Q_{21}(z_1+z_2)+\\ & Q_{2}((z_1+z_2)(z_1-t_1-t_2)+z_1z_2)+Q_{1}z_1z_2(z_1-t_1-t_2) \\
\hline
\32\bar{1} & Q_{32}+Q_{41}+Q_{31}(z_1+z_2-t_1-t_2)+Q_{21}(z_1+z_2)(-t_1-t_2) \\
\hline
3\bar{2}\bar{1} & Q_{32}+Q_{31}z_1+Q_{21}z_1^2 \\
\hline
\bar{3}\bar{2}\bar{1} & Q_{321} \\
\hline
2\bar{3}\bar{1} & Q_{42}+Q_{32}(z_1-t_1-t_2)+Q_{41}z_1+Q_{31} z_1(z_1-t_1-t_2)+Q_{21}z_1^2(-t_1-t_2) \\
\hline
\bar{2}\bar{3}\bar{1} & Q_{421}+Q_{321}(z_1-t_1-t_2) \\
\hline
13\bar{2} & Q_{4}+Q_{3}(z_1+z_2-t_1)+Q_{2}(z_1z_2-t_1(z_1+z_2))+Q_1(-t_1)z_1z_2 \\
\hline
\13\bar{2} & Q_{41}+Q_{31}(z_1+z_2-t_1)+Q_{21}(z_1z_2-t_1(z_1+z_2)) \\
\hline
31\bar{2} & Q_{41}+Q_{4}(z_1-t_1)+Q_{31}(z_1+z_2-t_1)+Q_3(z_1-t_1)(z_1+z_2-t_1)+Q_{21}(z_1z_2-t_1(z_1+z_2))+\\ & Q_{2}(z_1-t_1)(z_1z_2-t_1(z_1+z_2))+Q_{1}(z_1-t_1)z_1z_2(-t_1) \\
\hline
\31\bar{2} & Q_{42}+Q_{32}(z_1+z_2-t_1)+Q_{41}(-t_1)+Q_{31}(z_1+z_2-t_1)(-t_1)+Q_{21}t_1^2(z_1+z_2) \\
\hline
3\bar{1}\bar{2} & Q_{42}+Q_{41}z_1+Q_{32}(z_1+z_2-t_1)+Q_{31}z_1(z_1+z_2-t_1)+Q_{21}z_1^2(z_2-t_1) \\
\hline
\bar{3}\bar{1}\bar{2} & Q_{421}+Q_{321}(z_1+z_2-t_1) \\
\hline
\end{array}$
$\begin{array}{|c|l|}
\hline
1\bar{3}\bar{2} & Q_{43}+Q_{42}(z_1-t_1)+Q_{32}(z_1^2+t_1^2-z_1t_1)+Q_{41}(-z_1t_1)+Q_{31}z_1(-t_1)(z_1-t_1)+Q_{21}(z_1^2t_1^2) \\
\hline
\bar{1}\bar{3}\bar{2} & Q_{431}+Q_{421}(z_1-t_1)+Q_{321}(z_1^2-z_1t_1+t_1^2) \\
\hline
12\bar{3} & Q_{5}+Q_4(z_1+z_2-t_1-t_2)+Q_3(z_1z_2+t_1t_2-(z_1+z_2)(t_1+t_2))+\\ & Q_2(z_1z_2(-t_1-t_2)+t_1t_2(z_1+z_2) )+Q_1 z_1z_2t_1t_2 \\
\hline
\12\bar{3} & Q_{51}+Q_{41}(z_1+z_2-t_1-t_2)+Q_{31}(z_1z_2+t_1t_2-(z_1+z_2)(t_1+t_2))+Q_{21}(z_1z_2(-t_1-t_2)+t_1t_2(z_1+z_2)) \\
\hline
21\bar{3} & Q_{51}+Q_{5}(z_1-t_1)+Q_{41}(z_1+z_2-t_1-t_2)+Q_{4}(z_1-t_1)(z_1+z_2-t_1-t_2)+ \\ & Q_{31}(z_1z_2+t_1t_2-(z_1+z_2)(t_1+t_2)) +Q_{3}(z_1-t_1)(z_1z_2+t_1t_2-(z_1+z_2)(t_1+t_2))+ \\ & Q_{21}(z_1z_2(-t_1-t_2)+t_1t_2(z_1+z_2)) +Q_2(z_1-t_1)(z_1z_2(-t_1-t_2)+t_1t_2(z_1+z_2))+Q_1 z_1z_2t_1t_2(z_1-t_1) \\
\hline
\21\bar{3} & Q_{52}+Q_{42}(z_1+z_2-t_1-t_2)+Q_{32}(z_1z_2+t_1t_2-(z_1+z_2)(t_1+t_2))+Q_{51}(-t_1)+\\ & Q_{41}(-t_1)(z_1+z_2-t_1-t_2)+Q_{31}(-t_1)(z_1z_2+t_1t_2-(z_1+z_2)(t_1+t_2)) \\
& +Q_{21}(t_1)^2(z_1z_2-(z_1+z_2)t_2)) \\
\hline
2\bar{1}\bar{3} & Q_{52}+Q_{42}(z_1+z_2-t_1-t_2)+Q_{32}(z_1z_2+t_1t_2-(z_1+z_2)(t_1+t_2))+Q_{51}z_1+Q_{41}z_1(z_1+z_2-t_1-t_2)+\\ & Q_{31}z_1(z_1z_2+t_1t_2-(z_1+z_2)(t_1+t_2)) +Q_{21} z_1^2(t_1t_2-z_2(t_1+t_2)) \\
\hline
\bar{2}\bar{1}\bar{3} & Q_{521}+Q_{421}(z_1+z_2-t_1-t_2)+Q_{321}(z_1z_2+t_1t_2-(z_1+z_2)(t_1+t_2)) \\
\hline
1\bar{2}\bar{3} & Q_{53}+Q_{52}(z_1-t_1)+Q_{51}(-z_1t_1)+Q_{43}(z_1+z_2-t_1-t_2)+Q_{42}(z_1-t_1)(z_1+z_2-t_1-t_2)+\\ & Q_{41}(-z_1t_1)(z_1+z_2-t_1-t_2) +Q_{32}(z_1(z_1-t_1)(z_2-t_1-t_2)+t_1^2(z_2-t_2))+\\ & Q_{31}(-z_1t_1)(z_1(z_2-t_1-t_2)-t_1(z_2-t_2))+Q_{21}z_1^2t_1^2(z_2-t_2) \\
\hline
\bar{1}\bar{2}\bar{3} & Q_{531}+Q_{431}(z_1+z_2-t_1-t_2)+Q_{521}(z_1-t_1)+
Q_{421}(z_1-t_1)(z_1+z_2-t_1-t_2)+\\ & Q_{321}((z_1^2-z_1t_1+t_1^2)(z_2-t_2)+z_1t_1(t_1-z_1)) \\
\hline
\end{array}
$
\normalsize
\section{Double Schubert polynomials in type $\mathrm{D}$ for $w \in W'_3$}
\tiny
\renewcommand{\arraystretch}{1.2}
$\begin{array}{|c|l|}
\hline
123 & 1 \\
\hline
213 & P_1+(z_1-t_1) \\
\hline
\bar{2}\13 & P_1 \\
\hline
\bar{1}\23 & P_2+P_1(z_1-t_1) \\
\hline
132 & 2P_1+(z_1+z_2-t_1-t_2) \\
\hline
312 & P_2+P_1(2z_1-t_1-t_2)+(z_1-t_1)(z_1-t_2) \\
\hline
\bar{3}\12 & P_2+P_1(-t_1-t_2) \\
\hline
\bar{1}\32 & P_3+P_2(z_1-t_1-t_2)+P_1(t_1t_2-z_1t_1-z_1t_2) \\
\hline
231 & P_2+P_1(z_1+z_2-2t_1)+(z_1-t_1)(z_2-t_1) \\
\hline
321 & P_{3}+P_{21}+P_2(2z_1+z_2-2t_1-t_2)+P_1(z_1^2+2z_1z_2+t_1^2+2t_1t_2-3t_1z_1-t_1z_2-t_2z_1-t_2z_2)+\\ & (z_1-t_1)(z_1-t_2)(z_2-t_1) \\
\hline
\bar{3}\21 & P_{21}+P_2(-t_1)+P_1t_1^2 \\
\hline
\bar{2}\31 & P_{31}+P_{21}(z_1-t_1-t_2)+P_3(-t_1)+P_2(-t_1)(z_1-t_1-t_2)+P_1 t_1^2(z_1-t_2) \\
\hline
\23\bar{1} & P_2+P_1(z_1+z_2) \\
\hline
3\bar{2}\bar{1} & P_{21}+P_{2}z_1+P_{1}z_1^2 \\
\hline
\32\bar{1} & P_3+P_{21}+P_2(z_1+z_2-t_1-t_2)+P_1(z_1+z_2)(-t_1-t_2) \\
\hline
2\bar{3}\bar{1} & P_{31}+P_{3}z_1+P_{21}(z_1-t_1-t_2)+P_2 z_1(z_1-t_1-t_2)+P_1 z_1^2(-t_1-t_2) \\
\hline
\13\bar{2} & P_3+P_2(z_1+z_2-t_1)+P_1(z_1z_2-t_1(z_1+z_2)) \\
\hline
3\bar{1}\bar{2} & P_{31}+P_{21}(z_1+z_2-t_1)+P_3 z_1+P_2z_1(z_1+z_2-t_1)+P_1 z_1^2(z_2-t_1) \\
\hline
\31\bar{2} & P_{31}+P_{21}(z_1+z_2-t_1)+P_3(-t_1)+P_2(-t_1)(z_1+z_2-t_1)+P_1(z_1+z_2)t_1^2 \\
\hline
1\bar{3}\bar{2} & P_{32}+P_{31}(z_1-t_1)+P_3(-z_1t_1)+P_{21}(z_1^2-z_1t_1+t_1^2)+P_2(-z_1t_1)(z_1-t_1)+P_1z_1^2t_1^2 \\
\hline
\12\bar{3} & P_4+P_3(z_1+z_2-t_1-t_2)+P_2(z_1z_2+t_1t_2-(z_1+z_2)(t_1+t_2))+P_1(z_1z_2(-t_1-t_2)+t_1t_2(z_1+z_2)) \\
\hline
2\bar{1}\bar{3} & P_{41}+P_{4}z_1+P_{31}(z_1+z_2-t_1-t_2)+P_3(z_1)(z_1+z_2-t_1-t_2)+P_{21}(z_1z_2+t_1t_2-(z_1+z_2)(t_1+t_2)) \\
& +P_2z_1(z_1z_2+t_1t_2-(z_1+z_2)(t_1+t_2))+P_1z_1^2(t_1t_2-z_2t_1-z_2t_2) \\
\hline
\21\bar{3} & P_{41}+P_{4}(-t_1)+P_{31}(z_1+z_2-t_1-t_2)+P_{3}(-t_1)(z_1+z_2-t_1-t_2)+P_{21}(z_1z_2+t_1t_2-(z_1+z_2)(t_1+t_2)) \\
& +P_2(-t_1)(z_1z_2+t_1t_2-(z_1+z_2)(t_1+t_2))+P_1t_1^2(z_1z_2-z_1t_2-z_2t_2) \\
\hline
1\bar{2}\bar{3} & P_{42}+P_{32}(z_1+z_2-t_1-t_2)+P_{41}(z_1-t_1)+P_{31}(z_1-t_1)(z_1+z_2-t_1-t_2)+ \\ & P_{21}(z_1^2z_2-t_1^2t_2+z_1t_1t_2-z_1z_2t_1+z_1^2(-t_1-t_2)+t_1^2(z_1+z_2)) +P_4(-z_1t_1)+\\ & P_3(-z_1t_1)(z_1+z_2-t_1-t_2)+P_2(-z_1t_1)(-z_1t_1-z_2t_1-z_1t_2+z_1z_2+t_1t_2)+P_1(z_1^2t_1^2)(z_2-t_2) \\
\hline
\end{array}
$
\nocite{*}
| {
"timestamp": "2008-10-09T21:59:53",
"yymm": "0810",
"arxiv_id": "0810.1348",
"language": "en",
"url": "https://arxiv.org/abs/0810.1348",
"abstract": "For each infinite series of the classical Lie groups of type B,C or D, we introduce a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the appropriate flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. They are also positive in a certain sense, and when indexed by maximal Grassmannian elements, or by the longest element in a finite Weyl group, these polynomials can be expressed in terms of the factorial analogues of Schur's Q- or P-functions defined earlier by Ivanov.",
"subjects": "Combinatorics (math.CO); Algebraic Geometry (math.AG)",
"title": "Double Schubert polynomials for the classical groups",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808741970027,
"lm_q2_score": 0.7217432062975979,
"lm_q1q2_score": 0.7075110612151569
} |
https://arxiv.org/abs/2209.06551 | On the complex valued metric-like spaces | The main purpose of this paper is to study complex valued metric-like spaces as an extension of metric-like spaces, complex valued partial metric spaces, partial metric spaces, complex valued metric spaces and metric spaces. In this article, the concepts such as quasi-equal points, completely separate points, convergence of a sequence, Cauchy sequence, cluster points and complex diameter of a set are defined in a complex valued metric-like space. | \section{Introduction and preliminaries}
Distance is an important and fundamental notion in mathematics and there exist many generalizations of this concept in the literature (see \cite{De}). One of such generalizations is the partial metric which was introduced by Matthews (see \cite{Mat}). It differs from a metric in that points are allowed to have non-zero "self-distances" (i.e., $d(x,x) \geq 0$), and the triangle inequality is modified to account for positive self-distances. O'Neill \cite{O} extended Matthews definition to partial metrics with "negative distances". Before describing the material of this paper, let us recall some definitions and set the notations which we use in what follows.
\begin{definition} A mapping $\mathfrak{p} : X \times X \rightarrow \mathbb{R}^{+}$, where $X$ is a non-empty set, is said to be a
partial metric on $X$ if for any $x, y, z \in X$, the following four conditions hold true:\\
(i) $x = y$ if and only if $\mathfrak{p}(x, x) = \mathfrak{p}(y, y) = \mathfrak{p}(x, y)$;\\
(ii) $\mathfrak{p}(x, x) \leq \mathfrak{p}(x, y)$;\\
(iii) $\mathfrak{p}(x, y) = \mathfrak{p}(y, x)$;\\
(iv) $\mathfrak{p}(x, z) \leq \mathfrak{p}(x, y) + \mathfrak{p}(y, z) - \mathfrak{p}(y, y)$.
\end{definition}
The pair $(X, \mathfrak{p})$ is then called a \emph{partial metric space.} A sequence $\{x_n\}$ in a partial metric space $(X, \mathfrak{p})$ converges to a point $x_0 \in X$ if $\lim_{n \rightarrow \infty}\mathfrak{p}(x_n, x_0) = \mathfrak{p}(x_0, x_0)$. A sequence $\{x_n\}$ of elements of $X$ is called \emph{Cauchy} if the limit $\lim_{m, n \rightarrow \infty} \mathfrak{p}(x_n, x_m)$ exists and is finite. The partial metric space $(X, \mathfrak{p})$ is called complete if for each Cauchy sequence $\{x_n\}$, there is some $x \in X$ such that
$$ \lim_{n \rightarrow \infty}\mathfrak{p}(x_n, x) = \mathfrak{p}(x, x) = \lim_{m, n \rightarrow \infty}\mathfrak{p}(x_n, x_m).$$
An example of a partial metric space is the pair $(\mathbb{R}^{+}, \mathfrak{p})$, where $\mathfrak{p}(x, y) = \max\{x, y\}$ for all $x, y \in \mathbb{R}^{+}$. For more material about the partial metric spaces, see, e.g. \cite{A, B, K} and references therein.
In 2012, A. Amini-Harandi \cite{AH} introduced a new extension of the concept of partial metric space, called a \emph{metric-like space}. After that, the concept of $b$-metric-like space which generalizes the notions of partial metric space, metric-like space, and $b$-metric space was introduced by Alghamdi et al. in \cite{Al}. Recently, Zidan and Mostefaoui \cite{Z} introduced the double controlled quasi metric-like spaces and studied some topological properties of this space. Here, we state the concept of a metric-like space.
\begin{definition} A mapping $\mathfrak{D}:X \times X \rightarrow \mathbb{R}^{+}$, where $X$ is a non-empty set, is said to be a metric-like on $X$ if for any $x, y, z \in X$, the following three conditions hold true:\\
(i) $\mathfrak{D}(x, y) = 0 \Rightarrow x = y$;\\
(ii) $\mathfrak{D}(x, y) = \mathfrak{D}(y, x)$;\\
(iii) $\mathfrak{D}(x, y) \leq \mathfrak{D}(x, z) + \mathfrak{D}(z, y)$.
\end{definition}
The pair $(X, \mathfrak{D})$ is called a metric-like space. A metric-like on $X$ satisfies all of the conditions of a metric except that $\mathfrak{D}(x, x)$ may be positive for some $x \in X$. The study of partial metric spaces has wide area of application, especially in computer sciences, see, e.g. \cite{B, Ma, R} and references therein. That is why working on this topic can be very useful in practice. Since metric-likes are generalizations of partial metrics, knowing them can therefore provide us more applicable fields. In fact, this is our motivation to study the metric-like spaces. Each metric-like $\mathfrak{D}$ on $X$ generates a topology $\tau_\mathfrak{D}$ on $X$ whose base is the family of open balls. An open ball in a metric-like space $(X, \mathfrak{D})$, with center $x$ and radius $r > 0$, is the set
$$B(x, r) = \{y \in X : |\mathfrak{D}(x, y) - \mathfrak{D}(x, x)| < r \}.$$
It is clear that a sequence $\{x_n\}$ in the metric-like space $(X, \mathfrak{D})$ converges to a point $x \in X$ if and only if $\lim_{n \rightarrow \infty}\mathfrak{D}(x_n, x) = \mathfrak{D}(x, x)$. A sequence $\{x_n\}$ of elements of a metric-like space $(X, \mathfrak{D})$ is called Cauchy if the limit $\lim_{n, m \rightarrow \infty}\mathfrak{D}(x_n, x_m)$ exists and is finite. The metric-like space $(X, \mathfrak{D})$ is called complete if for each Cauchy sequence $\{x_n\}$, there is some $x_0 \in X$ such that
$$\lim_{n \rightarrow \infty}\mathfrak{D}(x_n, x_0) = \mathfrak{D}(x_0, x_0) = \lim_{n, m \rightarrow \infty}\mathfrak{D}(x_n, x_m).$$
For more details on this topic, see, e.g. \cite{AH, Ha}. Note that every partial metric space is a metric-like space. But, the converse is not true in general. For example, let $X = \mathbb{R}$, and let $\mathfrak{D}(x, y) = \max\{|x - 5|, |y - 5|\}$ for all $x, y \in \mathbb{R}$. Then $(X, \mathfrak{D})$ is a metric-like space, but since $\mathfrak{D}(0, 0) \nleq \mathfrak{D}(1, 2)$, then $(X, \mathfrak{D})$ is not a partial metric space. We now state another extension of the notion of distance that allows distance to be a complex value. Azam et al., \cite{Az} introduced the concept of a complex valued metric space and obtained sufficient conditions for the existence of common fixed points of a pair of mappings satisfying contractive type conditions. In that article, they consider a partial order $\precsim$ on the set of complex numbers $\mathbb{C}$ and then introduce a complex valued metric. The partial order $\precsim$ is as follows:
\begin{align*}
z_1 \precsim z_2 \Leftrightarrow \ Re(z_1) \leq Re(z_2), \ Im(z_1) \leq Im(z_2).
\end{align*}
Let $X$ be a non-empty set. Suppose that the mapping $d :X \times X \rightarrow \mathbb{C}$ satisfies the following conditions:\\
(i) $0 \precsim d(x, y)$ for all $x, y \in X$;\\
(ii) $d(x,y) = 0 \Leftrightarrow x = y$;\\
(iii) $d(x,y) = d(y,x)$ for all $x, y \in X$;\\
(iv) $d(x,y) \precsim d(x,z) + d(z, y)$ for all $x, y, z \in X$.\\
Then $d$ is called a complex valued metric on $X$ , and $(X , d)$ is called a complex valued metric space. Combining the two concepts complex valued metric spaces and metric-like spaces, we get complex valued metric-like spaces. Also, we introduce the notion of a complex valued partial metric space. As will be seen, the notion of complex valued metric-like space is a generalization of the notions of metric-like space, complex valued metric space, partial metric space, complex valued partial metric space and metric space. Therefore, it is interesting to investigate this general notion.
In this article, we focus on the structure of complex valued metric-like spaces and study some topological properties
of this space. For instance, we introduce some concepts such as quasi-equal points, completely separate points, convergence of a sequence, Cauchy sequence, cluster point, limit point, complex absolute value and complex diameter of a subset of a complex valued metric-like space.
Additionally, we present several results about complex valued metric-like spaces.
\section{Results and proofs}
Let $\mathbb{C}$ be the set of complex numbers and $z_1, z_2 \in \mathbb{C}$. Following \cite{Az}, we define a partial order $\precsim$ on $\mathbb{C}$ as follows:
\begin{align*}
z_1 \precsim z_2 \Leftrightarrow Re(z_1) \leq Re(z_2), \ Im(z_1) \leq Im(z_2).
\end{align*}
Hence, $z_1 \precsim z_2$ if one of the following conditions is satisfied:\\
(i) $Re(z_1) = Re(z_2)$ and $Im(z_1) < Im(z_2)$;\\
(ii) $Re(z_1) < Re(z_2)$ and $\ Im(z_1) = Im(z_2)$;\\
(iii) $Re(z_1) < Re(z_2)$ and $\ Im(z_1) < Im(z_2)$;\\
(iv) $Re(z_1) = Re(z_2)$ and $\ Im(z_1) = Im(z_2)$.\\
We write $z_1 \precnsim z_2$ if $z_1 \neq z_2$ and one of (i), (ii), and (iii) is satisfied. Also, we write $z_1 \prec z_2$ if only (iii) is satisfied. Note that
\begin{align*}
& 0 \precsim z_1 \precnsim z_2 \Rightarrow |z_1| < |z_2|, \\ & z_1 \precsim z_2 \prec z_3 \Rightarrow z_1 \prec z_3.
\end{align*}
Also, $z_2 \succsim z_1$ (resp. $z_2 \succ z_1$) means that $z_1 \precsim z_2$ (resp. $z_1 \prec z_2$). Throughout the paper, the set $\{z \in \mathbb{C} \ | \ z \succsim 0 \}$ is denoted by $\mathbb{C} \succsim 0$, i.e. $\mathbb{C} \succsim 0 = \{z \in \mathbb{C} \ | \ z \succsim 0 \}$. A complex number $z$ is called positive if $0 \prec z$.
\begin{definition}
Let $X$ be a non-empty set. A mapping $d: X \times X \rightarrow \mathbb{C}\succsim 0$ is called a
complex valued metric-like on $X$ if for any $x, y, z\in X$, the following conditions hold:\\
(D1) $d(x, y)=0 \Rightarrow x=y$;\\
(D2) $d(x, y)=d(y, x)$;\\
(D3)$d(x, y)\precsim d(x, z)+d(z,y).$
\end{definition}
The pair $(X, d)$ is then called a complex valued metric-like space. Indeed, a complex valued metric-like on $X$ satisfies all of the conditions of complex valued metric except that may be $ 0 \precnsim d(x, x) $ for some $x \in X$. For convenience, we write (CVML) for "complex valued metric-like".
In the following, we provide an example of a CVML space.
\begin{example}
Let $ X=\mathbb{C}$ . A mapping
$d: \mathbb{C}\times \mathbb{C} \rightarrow\mathbb{C}$ defined by
$d\left(z_{1}, z_{2}\right)=e^{i \theta}\left(| z_{1} |+\left|z_{2}\right|\right)$, where $0 \leqslant \theta \leqslant \frac{\pi}{2}$
is a CVML on $\mathbb{C}$.
\end{example}
In the following, we introduce the complex absolute value of $z \in \mathbb{C}$ which is denoted by $|\cdot|_{c}$.
$$|z|_{c}=\left|\operatorname{Re(z)}\right|+i\left|Im(z)\right| $$ for any $z \in \mathbb{C}$.
Clearly, $0\precsim |z|_c$ for all $z \in \mathbb{C}$.
\begin{definition}
Let $(X, d)$ be a CVML space and $A\subseteq X$. An open ball with center $x_0 \in X$ and radius $0 \prec r \in \mathbb{C}$ is the set
\begin{align*}
N(x_0, r)=\left\{y \in X :|d(x_0, y)-d(x_0, x_0)|_{c} \prec r \right\}.
\end{align*}
\end{definition}
\begin{definition} Let $X$ be a non-empty set. A mapping $p: X \times X\rightarrow\mathbb{C} \succsim 0$ is said to be a
complex valued partial metric on $X$ if for all $x, y, z \in X,$ the following conditions hold:
\begin{itemize}
\item $x=y \Leftrightarrow p(x, x) = p(y, y) = p(x, y)$;
\item $\max \{p(x, x), p(y, y)\} \precsim p(x, y)$;
\item $ p(x, y)=p(y, x)$;
\item $p(x, z)\precsim p(x, y)+p(y, z)-p(y, y)$
\end{itemize}
The pair $(X, p)$ is called a complex valued partial metric space.
\end{definition}
\begin{definition}
Let $(X, d)$ be a CVML space, let $\{x_{n}\}_{n\geq 1}$ be a sequence of $X$ and let $x_{0} \in X$.
We say that
\begin{itemize}
\item The sequence $\{x_{n}\}_{n \geqslant 1}$ converges to $x_{0}$ if for every $0 \prec r \in \mathbb{C} $ there exists $ n_{0} \in \mathbb{N}$ such that $x_n \in N(x_{0}, r )$ for all $n>n_0$, i.e.
$|d(x_{n}, x_{0})-d(x_{0}, x_{0})|_{c} \prec r$ for all $n>n_0$.
We denote this by $\lim_{n \rightarrow +\infty} x_{n}=x_{0},$ or $x_{n} \rightarrow x_{0}$
as $n \rightarrow +\infty$.
\item The sequence $\{x_{n}\}_{n \geqslant 1}$ is called Cauchy if $\lim_{m,n \rightarrow +\infty} d(x_{n}, x_{m})$ exists and is finite.
It means that sequence $\{x_{n}\}_{n \geqslant 1}$ is Cauchy if and only if
$\lim_{m,n \rightarrow +\infty} d(x_{n}, x_{m})=z_0$ for some $z_0\in \mathbb{C}$.
\item The complex valued metric-like space $(X, d)$ is complete if every Cauchy sequence of $X$ is convergent. On the other hand, the complex valued metric-like space $(X, d)$ is complete if for each Cauchy sequence $\{x_n\}$, there is some $x \in X$ such that \begin{align*} \lim_{n \rightarrow +\infty} d(x_{n}, x) = d(x, x) = \lim_{m, n \rightarrow +\infty}d(x_n, x_m)\end{align*}
\end{itemize}
\end{definition}
\begin{remark} Note that in CVML spaces the limit of a convergent sequence is not necessarily unique and this means that topology of these spaces is not necessarily a Hausdorff topology. For instance, suppose that $X = \mathbb{C}$ and $d(x, y) = i \max\{|x|, |y|\}$ for each $x, y \in X$. Putting $x_n = \frac{i}{n}$, we have $\lim_{n \rightarrow \infty}d(\frac{i}{n}, i) = \lim_{n \rightarrow \infty} i \max\{|\frac{i}{n}|, |i|\} = i = d(i, i)$. It means that the sequence $\{\frac{i}{n}\}$ converses to $i$, i.e. $\frac{i}{n} \rightarrow i$. Moreover, we have $\lim_{n \rightarrow \infty}d(\frac{i}{n}, 2i) = \lim_{n \rightarrow \infty} i \max\{|\frac{i}{n}|, |2i| \} = 2i = d(2i, 2i)$, and consequently, $\frac{i}{n} \rightarrow 2i$ as well. This demonstrates that the sequence $\{\frac{i}{n}\}$ converges to two different points.
\end{remark}
The above example leads us to the next definition.
\begin{definition} (Quasi-equal points) Let $(X, d)$ be a CVML space. The points $x, y \in X$ are called \emph{quasi-equal points} if there exists a sequence $\{x_n\}$ of $X$ converging to both $x$ and $y$, i.e. $x_n \rightarrow x$ and $x_n \rightarrow y$.
\end{definition}
From the previous remark, one can easily deduce that if $X = \mathbb{C}$ and $d(x, y ) = i \max\{|x|, |y|\}$ for each $x, y \in \mathbb{C}$, then $i$ and $2i$ are two quasi-equal points.
\begin{definition} (Completely separate points) Let $(X, d)$ be a CVML space. The points $x, y$ of $X$ are called \emph{completely separate points} if the following condition holds true:
$$d(x, x) + d(y,y) \prec d(x, y) $$
\end{definition}
\begin{theorem}
Let $(X, d)$ be a CVML space. Then there are no convergent sequences to two completely separate points.
\end{theorem}
\begin{proof} Suppose that $x, y$ are two completely separate points. To obtain a contradiction, let $\{x_n\}$ be a sequence of $X$ converging to both $x, y$. Put $r_1 = \frac{1}{3}r$, where $r = d(x, y) - d(x, x) - d(y, y) \succ 0$. Since $x_n \rightarrow x$ and $x_n \rightarrow y$, for $\varepsilon = r_1$, there exist two positive integers $N_1$ and $N_2$ such that $$\max\{|d(x_N, x) - d(x, x)|, |d(x_N, y) - d(y, y)|\} \prec r_1$$
for all $n \geq N = \max\{N_1, N_2\}$. Therefore,
\begin{align*}
d(x, y) = |d(x, y)|_{c} & \precsim |d(x, x_N) + d(x_N, y)|_{c} \\ & = |d(x, x_N) + d(x_N, y) - d(x, x) + d(x, x) - d(y, y) + d(y, y)|_{c} \\ & \precsim |d(x, x_N) - d(x, x)|_{c} + |d(x_N, y) - d(y, y)|_{c} + d(x, x) + d(y, y) \\ & \prec r_1 + r_1 + d(x, x) + d(y,y) \\ & = 2 r_1 + d(x,x) + d(y,y) \\ & = \frac{2}{3}\Big[d(x, y) - d(x, x) - d(y, y)\Big] + d(x, x) + d(y, y) \\ & = \frac{2}{3}d(x, y) + \frac{1}{3}d(x, x) + \frac{1}{3}d(y, y).
\end{align*}
So we obtain that $d(x, y) \prec d(x, x) + d(y, y)$, a contradiction. This contradiction proves our claim.
\end{proof}
Immediate conclusion from the above theorem demonstrates that completely separate points are not quasi-equal.
\begin{definition} (Cluster points) Let $(X, d)$ be a CVML space and let $A$ be a subset of $X$. A point $x_0 \in X$ is said to be a cluster point of $A$ if for every positive complex number $r$ there exists an element $a \in A$ such that $|d(a, x_0) - d(x_0, x_0)|_{c} \prec r$.
\end{definition}
As usual, the set of all \emph{cluster points} of $A$ is called the closure of $A$ and is denoted by $\overline{A}$. Note that $x_0 \in \overline{A}$ if and only if $N(x_0, r) \cap A \neq \phi$ for all $0 \prec r \in \mathbb{C}$. Indeed, we have
\begin{align*}
\overline{A} = \Big\{x_0 \in X : N(x_0, r) \cap A \neq \phi \ \ for \ all \ 0 \prec r \in \mathbb{C}\Big\}
\end{align*}
It is clear that if $(X, d)$ is a CVML space and $A$ is a subset of $X$, then $A \subseteq \overline{A}$. In the following, we establish a theorem to present a necessary and sufficient condition for cluster points in the CVML spaces. First, we prove a theorem about the convergence of sequences in CVML spaces.
\begin{theorem
\label{1}
Let $(X, d)$ be a CVML space and let $\left\{x_{n}\right\}$ be a sequence of $X$. Then $\left\{x_{n}\right\}$ converges to $x$
if and only if $\lim_{n \rightarrow+\infty}\Big|d\left(x_{n}, x\right)-d(x, x)\Big| =0$.
\end{theorem}
\begin{proof}
Suppose that $\left\{x_{n}\right\}$ converges to $x$. Let $\varepsilon = \frac{\alpha}{\sqrt{2}}+i \frac{\alpha}{\sqrt{2}}=\frac{\alpha}{\sqrt{2}}(1+i)$, where
$\alpha$ is a given positive real number. Since $\left\{x_{n}\right\}$
converges to $x$, there exists a positive integer $N$
such that
$\left|d\left(x_{n}, x\right)-d(x, x)\right|_{c} \prec \varepsilon $ for all $n>N$.
Therefore,
$$\Big| | d\left(x_{n}, x\right)- d(x, x) |_{c}\Big| < |\varepsilon| = \alpha$$
for all $n>N$,
which means that
$\left|d\left(x_{n}, x\right)-d(x, x)\right| < \alpha$ for all $n > N$.
So
$$ \lim_{n \rightarrow+\infty} \left|d\left(x_{n}, x\right)-d(x, x)\right|=0.$$
Conversely, suppose that $\lim_{n \rightarrow+\infty} |d\left(x_{n}, x\right)-d(x, x) | = 0$. Therefore, for each positive real number $\alpha$, there exists a positive integer $N$ such that $\left|d(x_{n}, x)-d(x, x)\right| < \alpha$ for all $n > N$.
It is clear that for any $ 0 \prec \varepsilon \in \mathbb{C}$, there exists a positive real number $\alpha$ such that
$ \alpha+ i\alpha=(1+i) \alpha \prec \varepsilon.$
Since $\lim_{n \rightarrow+\infty} \left|d\left(x_{n}, x\right)-d(x, x)\right|=0,$ there
exists a positive integer $n_{0} \in \mathbb{N}$ such that
$\left|d\left(x_{n}, x\right)-d(x, x)\right|<\alpha$ for all $ n \geq n_{0}$.
Hence, for all $n \geq n_{0}$, we have
\begin{eqnarray*} \left| d\left(x_{n}, x\right)- d(x, x)\right|_{c} \precsim\left(1+i \right) | d\left(x_{n}, x\right)-d\left(x , x\right)| \prec (1+i) \alpha \prec \varepsilon \end{eqnarray*}
Indeed, we get that for all $0 \prec \varepsilon \in \mathbb{C}$ there exists a positive integer $n_0$ such that $|d \left(x_{n}, x\right)-\left.d(x, n)\right|_{c} \prec \varepsilon$ for all $n \geq n_0$, and
this means that
$
\lim_{n \rightarrow+\infty}x_{n} = x $, as desired.
\end{proof}
\begin{theorem} \label{2*}Let $(X, d)$ be a CVML space and let $A$ be a subset of $X$. Then $x_0 \in \overline{\mathcal{A}}$ if and only if there is a sequence $\{a_n\} \subseteq \mathcal{A}$ converging to $x_0$.
\end{theorem}
\begin{proof} Suppose that $x_0 \in \overline{A}$. So for each $r_n = \frac{1+i}{n}$ ($n \in \mathbb{N}$), there is an element $a_n \in \mathcal{A}$ such that $|d(a_n, x_0) - d(x_0, x_0)|_{c} \prec \frac{1+i}{n}$. It implies that $\lim_{n \rightarrow +\infty}|d(a_n, x_0) - d(x_0, x_0)|= 0$, and it follows from Theorem \ref{1} that the sequence $\{a_n\} \subseteq A$ converges to $x_0$. Conversely, assume that $\{a_n\}$ is a sequence of $A$ converging to $x_0$. We must to show that $x_0 \in \overline{A}$. Let $r$ be an arbitrary positive complex number. Therefore, there exists a positive number $N$ such that $a_n \in N(x_0, r)$ for all $n \geq N$. It means that $A \cap N(x_0, r) \neq \phi$. Since we are assuming that $r$ is an arbitrary positive complex number, it is deduced that $x_0 \in \overline{A}$, as desired.
\end{proof}
\begin{definition} (Limit points) Let $(X, d)$ be a CVML space and let $A$ be a subset of $X$. A point $x_0 \in X$ is said to be a limit point of $A$ if $N(x_0,r) \cap (A - \{x_0\}) \neq \phi$ for all positive complex numbers $r$.
\end{definition}
As usual, the set of all \emph{limit points} of $A$ is denoted by $A'$. A subset $A$ of $X$ is called a closed set, whenever each limit point of $A$ belongs to itself, i.e. $A' \subseteq A.$ One can easily prove the following theorem.
\begin{theorem} Let $(X, d)$ be a CVML space and let $A$ be a subset of $X$. Then $\overline{A} = A \cup A'$.
\end{theorem}
\begin{definition} Let $(X, d)$ be a CVML space and let $A$ be a subset of $X$. The complex diameter of $A$ is defined as follows:
$$ diam_c(A) : = \sup \Big\{|d(x, y) - d(x, x)|_{c}, |d(x, y) - d(y, y)|_{c} : x, y \in A \Big\}.$$
If $|diam_c(A)| < \infty$, then the subset $A \subseteq X$ is said to be bounded.
\end{definition}
\begin{example} Let $X = \mathbb{C}$, $d(x, y) = (1+i)(|z_1| + |z_2|)$, and let $A = \{z \in \mathbb{C} : 3 < |z| < 5\}$. In this case, we have
\begin{align*}
diam_c(A) & = \sup\Big\{|d(z_1, z_2) - d(z_1, z_1)|_{c}, |d(z_1, z_2) - d(z_2,z_2)|_{c} : z_1, z_2 \in A \Big\} \\ & = \sup\Big\{\big||z_1| - |z_2|\big| + i \big||z_1| - |z_2|\big| : z_1, z_2 \in A \Big\} \\ & = \sup\Big\{\big||z_1| - |z_2|\big|(1+i) : z_1, z_2 \in A \Big\} \\ & = 2(1+i).
\end{align*}
\end{example}
\vspace{.25cm}
\bibliographystyle{amsplain}
| {
"timestamp": "2022-09-15T02:12:05",
"yymm": "2209",
"arxiv_id": "2209.06551",
"language": "en",
"url": "https://arxiv.org/abs/2209.06551",
"abstract": "The main purpose of this paper is to study complex valued metric-like spaces as an extension of metric-like spaces, complex valued partial metric spaces, partial metric spaces, complex valued metric spaces and metric spaces. In this article, the concepts such as quasi-equal points, completely separate points, convergence of a sequence, Cauchy sequence, cluster points and complex diameter of a set are defined in a complex valued metric-like space.",
"subjects": "General Topology (math.GN)",
"title": "On the complex valued metric-like spaces",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808736209154,
"lm_q2_score": 0.7217432062975979,
"lm_q1q2_score": 0.7075110607993699
} |
https://arxiv.org/abs/0804.0387 | Projective spectrum in Banach algebras | For a tuple $A=(A_0, A_1, ..., A_n)$ of elements in a unital Banach algebra ${\mathcal B}$, its {\em projective spectrum} $p(A)$ is defined to be the collection of $z=[z_0, z_1, ..., z_n]\in \pn$ such that $A(z)=z_0A_0+z_1A_1+... +z_nA_n$ is not invertible in ${\mathcal B}$. The pre-image of $p(A)$ in ${\cc}^{n+1}$ is denoted by $P(A)$. When ${\mathcal B}$ is the $k\times k$ matrix algebra $M_k(\cc)$, the projective spectrum is a projective hypersurface. In infinite dimensional cases, projective spectrums can be very complicated, but also have some properties similar to that of hypersurfaces. When $A$ is commutative, $P(A)$ is a union of hyperplanes. When ${\mathcal B}$ is reflexive or is a $C^*$-algebra, the {\em projective resolvent set} $P^c(A):=\cc^{n+1}\setminus P(A)$ is shown to be a disjoint union of domains of holomorphy. Later part of this paper studies Maurer-Cartan type ${\mathcal B}$-valued 1-form $A^{-1}(z)dA(z)$ on $P^c(A)$. As a consequence, we show that if ${\mathcal B}$ is a $C^*$-algebra with a trace $\phi$, then $\phi(A^{-1}(z)dA(z))$ is a nontrivial element in the de Rham cohomology space $H^1_d(P^c(A), \cc)$. | \section{Introduction}
The classical spectrum of an element $A$ in a unital Banach algebra ${\mathcal B}$ is defined through the invertibility of
$A-\lambda I$. If $A=(A_0,\ A_1,\ ...,\ A_n)$ is a commutative tuple of elements in ${\mathcal B}$, then classical notions of joint spectrum
are defined through the invertibility of $(A_0-\lambda_0I,\ A_1-\lambda_1I,\ ...,\ A_n-\lambda_nI)$ in various senses (H\"{o}rmander [H\"{o}] Ch3, and Taylor [Ta]).
In all these cases, the identity
$I$ serves as a base against which the invertibilities of other elements are measured. The idea of projective spectrum, which we will define and
study, is to set $I$ free, and consider the invertibility of $z_0A_0+z_1A_1$, or more generally,
$A(z):=z_0A_0+z_1A_1+\cdots +z_nA_n.$
This is a measurement of how the elements behave against each other. Unlike classical notions of joint spectrums,
projective spectrum is defined for all tuples, not just commutative ones. This paper is organized as follows.\\
Section 1. {\em Preparation}. Here we define the projective spectrum and prove its non-triviality.
Section 2. {\em Projective spectrum and hypersurfaces}. When the Banach algebra is the matrix algebra $M_k({\mathbb C})$, projective spectrums are degree $k$
projective hypersurfaces. A comparision
between general projective spectrums and hypersurfaces is made in Section 2. We will see that when $A$ is a commutative tuple, its projective
spectrum is a union of hyperplanes. The main results are regarding the complement of projective spectrum (which we call projective
resolvent set). We show that when the Banach algebra is of certain type, for instance $C^*$, the complement is made of domains
of holomorphy.
Section 3. {\em ${\mathcal B}$-valued 1-form $A^{-1}(z)dA(z)$ and the de Rham cohomology space $H^1_d(P^c(A),\ {\mathbb C})$}.
This section makes a study on the topology of projective resolvent set. Since the tuple $A$ in general is of infinite dimensional nature, its
projective resolvent can be very complicated. Nonetheless, with the aid of Maurer-Cartan type form $A^{-1}(z)dA(z)$ and
central linear functionals on ${\mathcal B}$, we manage to peek into the de Rham cohomology space of projective resolvent sets.
Section 4. {\em The case when when $A$ is commutative}. A few observations and remarks are made here
concerning an Arnold and Briskorn's theorem in Hyperplane Arrangements.\\
{\bf Acknowlegement:} This paper was benefited from conversations with many of the authors colleagues at SUNY at Albany,
to whom the author is deeply indebted. In particular, the author would like to thank Michael Range, Mark Steinberger,
Michael Stessin, and Alex Tchernev for references and valuable discussions.
\section{Preparation}
We let $z=(z_0,\ z_1,\ ...,\ z_n)$ denote a general point in ${{\mathbb C}}^{n+1}$. The group ${{\mathbb C}}^{\times}$ of nonzero complex
numbers acts on ${{\mathbb C}}^{n+1}$ by scalar multiplications. The $n$ dimensional projective space ${\mathbb P}^n$ is the quotient
$({{\mathbb C}}^{n+1}\setminus \{0\})/{{\mathbb C}}^{\times}$. With topology induced from this quotient, ${\mathbb P}^n$ is a compact complex manifold.
The fibres of this quotient map are the integral curves of the {\em Euler vector field}
$\theta =\sum_{j=0}^{n}z_j\frac{\partial}{\partial z_j}$.
For a subset $S\subset {\mathbb C}^{n+1}$ invariant under ${{\mathbb C}}^{\times}$, $(S\setminus \{0\})/{{\mathbb C}}^{\times}$ will be denoted by $S^T$.
$[z_0,\ z_1,\ ...,\ z_n]$ denotes the homogenous coordinate of a general point in ${\mathbb P}^n$. On the open subset $U_0=\{z_0\neq 0\}\subset {\mathbb P}^n$,
$[z_0,\ z_1,\ ...,\ z_n]=[1,\ z_1/z_0,\ z_2/z_0,\ ...,\ z_n/z_0]$. The tuple $(z_1/z_0,\ z_2/z_0,\ ...,\ z_n/z_0)$ is the affine
coordinate for $U_0$, and is denoted simply by $(\xi_1,\ \xi_2,\ ...,\ \xi_n)$.
Throughout this paper, ${\mathcal B}$ is a Banach algebra with identity $I$. As usual, the set of bounded linear functionals on ${\mathcal B}$ is denoted by ${\mathcal B}^*$.
An element $\phi\in {\mathcal B}^*$ is said to be {\em central} if $\phi(XY)=\phi(YX)$ for all
$X,\ Y\in {\mathcal B}$. The set of central linear functionals on ${\mathcal B}$ shall be denoted by ${\mathcal B}_c^*$.
It is easy to see that ${\mathcal B}_c^*$ is a closed subspace of ${\mathcal B}^*$. A bounded linear functional $\phi$ on
${\mathcal B}$ is said to be {\em multiplicative} if $\phi(XY)=\phi(X)\phi(Y)$. Clearly, multiplicative linear
functionals are central. When ${\mathcal B}$ is commutative, the collection of multiplicative linear
functionals is called the {\em maximal ideal space} of ${\mathcal B}$.
If ${\mathcal B}$ is a $C^*$-algebra, then a positive central linear functional is called a {\em trace}. Of course, not every
$C^*$-algebra possesses a trace.
Unless stated otherwise, $A=(A_0,\ A_1,\ ...,\ A_n)$ always stands for an $(n+1)$-tuple of general elements in
${\mathcal B}$. A tuple $A$ is said to be commutative if $A_iA_j=A_jA_i$, $\forall 0\leq i,\ j\leq n$.
In this paper, the ${\mathcal B}$-valued linear function $A(z)=z_0A_0+z_1A_1+\cdots z_nA_n$ is a primary associate of a tuple $A$. Without loss of generality, we assume
the elements $A_0,\ A_1,\ ...,\ A_n$ are linearly independent, hence the range of $A(z)$ is an $n+1$ dimensional subspace of
of ${\mathcal B}$, which we denote by $E_A$. A subalgebra ${\mathcal A}$ of ${\mathcal B}$ is said to be inversion-closed if
for every invertible element $a\in {\mathcal A}$, $a^{-1}$ is also in ${\mathcal A}$. For a tuple $A=(A_0,\ A_1,\ ...,\ A_n)$,
we let ${\mathcal B}_A$ denote the smallest inversion-closed Banach sub-algebra of ${\mathcal B}$ that
contains $A_0,\ A_1,\ ...,\ A_n.$ Clearly, $A(z)$ is invertible in ${\mathcal B}$ if and only if it is invertible in
${\mathcal B}_A$. Moreover, when $A$ is a commutative tuple, ${\mathcal B}_A$ is a commutative Banach algebra. In this case,
the maximal ideal space shall be denoted by $M_A$.\\
{\bf Definition.} For a tuple $A$, we let
\[P(A)=\{z\in {{\mathbb C}}^{n+1}:\ A(z)\ \text{is not invertible}.\}\]
The {\em projective spectrum} $p(A)$ of $A$ is $P(A)^T$, e.g.
\[p(A)=\{z=[z_0,\ z_1,\ ...,\ z_n]\in {\mathbb P}^n:\ A(z)\ \text{is not invertible}.\}\]
For simplicity, we also refer to $P(A)$ as projective spectrum. The {\em projective resolvent sets} refer to
their complements $p^c(A)={\mathbb P}^n \setminus p(A)$ and $P^c(A)={{\mathbb C}}^{n+1}\setminus P(A)$.\\
We let ${\mathcal B}^{-1}$ be the set of invertible
elements in ${\mathcal B}$. It is easy to see that the linear isomorphism $A(\cdot):\ {\mathbb C}^{n+1}\longrightarrow E_A$ is a homeomorphism from
$P^c(A)$ to $E_A\cap {\mathcal B}^{-1}$, as well as from $p^c(A)$ to $(E_A\cap {\mathcal B}^{-1})/{\mathbb C}^{\times}$.
Since $E_A\cap {\mathcal B}^{-1}$ is open in $E_A$, $P^c(A)$ is open, hence $p(A)$ is a
compact subset of ${\mathbb P}^n$. In some cases $p(A)$ can be equal to the entire space ${\mathbb P}^n$, for example, when $A$ is a tuple of compact
operators on an infinite dimensional Hilbert space. But one can always consider the slightly bigger tuple $(I,\ A_0,\ A_1,\ ...,\ A_n)$
which clearly has a more interesting projective spectrum.
So without loss of generality, we assume throughout of the paper that $p(A)$ is a proper subset of ${\mathbb P}^n$, or equivalently
$P(A)\neq {\mathbb C}^{n+1}$.
First, we establish the nontriviality of $p(A)$. Idea of proof is from [Ya].
\begin{prop}
For any tuple $A$, $p(A)$ is a nontrivial compact subset of ${\mathbb P}^n$.
\end{prop}
\begin{pf}
It only remains to show that $p(A)$ nontrivial, or equivalently,
$P(A)$ contains elements other than the origin $0$.
One first checks that on $P^c(A)$,
\begin{align*}
&A^{-1}(z_0,\ z_1,\ ...,\ z_n)-A^{-1}(z'_0,\ z_1,\ ...,\ z_n)\\
&=A^{-1}(z)\left(I-(z_0A_0+z_1A_1+\cdots z_nA_n)(z'_0A_0+z_1A_1+\cdots z_nA_n)^{-1}\right)\\
&=A^{-1}(z)\big(I-((z_0-z'_0)A_0+(z'_0A_0+z_1A_1+\cdots z_nA_n))(z'_0A_0+z_1A_1+\cdots z_nA_n)^{-1}\big)\\
&=-(z_0-z'_0)A^{-1}(z)A_0(z'_0A_0+z_1A_1+\cdots z_nA_n)^{-1}.
\end{align*}
This shows that $A^{-1}(z)$ is analytic in $z_0$, and likewise in all other variables. Moreover, the calculations show that
\begin{equation*}
\frac{\partial}{\partial z_j}A^{-1}(z)=-A^{-1}(z)A_jA^{-1}(z),\ \ \ 0\leq j\leq n. \tag{1.1}
\end{equation*}
By Hartogs extension theorem, the origin $0$ cannot be an isolated singularity of $A^{-1}(z)$, and hence $p(A)$ is nontrivial.
\end{pf}
Proposition 1.1 is an interesting fact, since the elements $A_0,\ A_1,\ ...,\ A_n$ may have nothing to do with each other.
Let us look at a few examples.\\
{\bf Example 1}. When ${\mathcal B}$ is the matrix algebra $M_k({\mathbb C})$, $A=(A_0,\ A_1,\ ...,\ A_n)$ is a tuple of $k\times k$ matrices.
Then $A(z)$ is invertible if and only if $detA(z)\neq 0$. Since $detA(z)$ is homogenous of degree $k$,
\[p(A)=\{z=[z_0,\ ,z_1,\ ...,\ z_n]\in {\mathbb P}^n:\ detA(z)=0\}\]
is a projective hypersurface of degree $k$. And $p^c(A)$ in this case is a {\em hypersurface complement}.\\
{\hfill \rule{.25cm}{.25cm} \medbreak}
{\bf Example 2}. Let $A_0$ be any element in ${\mathcal B}$, and $A_1=-I$. Then for the tuple $A=(A_0,\ A_1)$,
$A(z)=z_0A_0-z_1I$. Clearly, if $[z_0,\ z_1]$ is in $p(A)$
then $z_0\neq 0$, and $p(A)$ under the affine coordinate $z_1/z_0$ is the classical spectrum $\sigma(A_0)$. So Proposition 1.1 in fact implies the
nonemptiness of the classical spectrum.\\
{\hfill \rule{.25cm}{.25cm} \medbreak}
{\bf Example 3}. Now consider $L^2({\mathbb T},\ m)$, where $m$ is the normalized Lebesgue measure on the unit circle ${\mathbb T}$.
$\{w^n:\ \ n\in {\mathbb Z},\ |w|=1\}$ is an orthonormal basis for $L^2({\mathbb T},\ m)$. Let $\theta$ be an irrational number and
set $\lambda=exp(2\pi \sqrt{-1}\theta)$. Consider the two unitaries defined by
\[A_0f(w)=wf(w),\ \ \ A_1f(w)=f(\lambda w),\ \ f\in L^2({\mathbb T},\ m),\]
and let ${\mathcal B}$ be the $C^*$-algebra generated by $A_0$ and $A_1$.
Clearly, $A(z)$ is invertible if and only if $z_0A_0A^*_1+z_1I$ is invertible. So by Example 2,
$p(A)=-\sigma(A_0A_1^*)$. One checks that $A_1^*f(w)=f({\bar \lambda}w)$, hence
\[A_0A_1^*w^n=w({\bar \lambda}w)^n={\bar \lambda}^nw^{n+1}.\]
So $A_0A_1^*$ is a unitary bilateral weighted shift. Since $A_1A_0=\lambda A_0A_1$,
\[A_1^*A_0A_1^*A_1=A_1^*A_0={\bar \lambda}A_0A_1^*,\]
hence $\sigma(A_0A_1^*)$ is invariant under multiplication by ${\bar \lambda}$ which implies $\sigma(A_0A_1^*)={\mathbb T}$. Therefore
\[p(A)={\mathbb T},\ \ \ \text{and}\ \ \ P(A)=\{(z_0,\ z_1)\in {\mathbb C}^2:\ |z_0|=|z_1|\}.\]
In this case, $P^c(A)$ consists of two connected components:
\[\Omega_0=\{ |z_0|>|z_1|\},\ \ \text{and}\ \ \Omega_1=\{ |z_0|<|z_1|\}.\]
The $C^*$ algebra generated by $A_0$ and $A_1$ is the {\em irrational rotation algebra} often denoted by
${\mathcal A}_{\theta}$. We will come back to this algebra in Section 3.\\
{\hfill \rule{.25cm}{.25cm} \medbreak}
\section{Projective spectrum and projective hypersurface}
As we have seen in Example 1, for ${\mathcal B}=M_k({\mathbb C})$, a projective spectrum is a projective hypersurface. Naturally, things could become very
different for other Banach algebras. For instance, in Example 3 the projective resolvent sets are disjoint unions of two connected
components, while a hypersurface complement is always connected. Nevertheless, as it turns out, projective spectrum resembles hypersurface
in many other ways.
In ${\mathbb P}^n$ for $n\geq 2$, a {\em line} is the quotient of a two dimensioal subspace of ${\mathbb C}^{n+1}$ (removing the origin) over ${\mathbb C}^{\times}$.
By virtue of the Fundamental Theorem of Algebra, a hypersurface in ${\mathbb P}^n$ intersects with every line. This fact holds true
for $p(A)$.
\begin{corr}
For $n\geq 2$, every line in ${\mathbb P}^n$ intersects with $p(A)$.
\end{corr}
\begin{pf}
It is equivalent to show that every two dimensional subspace in ${\mathbb C}^{n+1}$ intersects with $P(A)$ nontrivially. In fact,
for any two linearly independent vectors $\lambda=(\lambda_0,\ \lambda_1,\ ...,\ \lambda_n)$
and $\eta=(\eta_0,\ \eta_1,\ ...,\ \eta_n)$ in ${\mathbb C}^{n+1}$, let
\[A'=\sum_{j=0}^n \lambda_jA_j,\ \ A''=\sum_{j=0}^n\eta_jA_j.\]
By Proposition 1.1 for the case $A=(A',\ A'')$, there exists scalars $a$ and $b$, not both zero, such that
$aA'+bA''$ is not invertible, and hence $a\lambda+b\eta \in P(A)$.
\end{pf}
In the case $A$ is a commutative tuple, the projective spectrum can be explicitly calculated.
\begin{prop}
If $A$ is a commutative tuple, then $P(A)$ is a union of hyperplanes.
\end{prop}
\begin{pf}
As remarked in Section 1, ${\mathcal B}_A$ in this case is commutative, and $P(A)$ is unchanged when considered in
${\mathcal B}_A$. Let $M_A$ denote the maximal ideal space of ${\mathcal B}_A$. Then by Gelfand theorem $A(z)$ is not
invertible in ${\mathcal B}_A$ if and only if there exists $\phi\in M_A$ such that
\begin{equation*}
\phi(A(z))=\sum_{j=0}^{n}z_j\phi(A_j)=0.\tag{2.1}
\end{equation*}
For simplicity, we let $H_\phi=\{z\in {\mathbb C}^{n+1}:\ \sum_{j=0}^{n}z_j\phi(A_j)=0\}.$
If $\phi$ is such that $\phi(A_j)=0$ for all $j$, then $P(A)={\mathbb C}^{n+1}$, which is clearly a union
(uncountable) of hyperplanes. Otherwise $H_\phi$ is a hyperplane, and one sees that
\begin{equation*}
P(A)=\cup_{\phi\in M_A}H_{\phi}.\tag{2.2}
\end{equation*}
\end{pf}
In the case when $M_A$ is a finite set, for instance when ${\mathcal B}=M_k({\mathbb C})$, $P(A)$ is a union of a finite number
of hyperplanes, e.g., $P(A)$ is a {\em central hyperplane arrangement}. In this case the topology of $P^c(A)$ is a primary topic in
Hyperplane Arrangement (cf. Orlik and Terao [OT]). It is worth mentioning that every central hyperplane arrangement can be represented as the projective spectrum of
a tuple of diagonal matrices. For example, for the {\em braid arrangement} in ${\mathbb C}^3$ defined by equation $(z_0-z_1)(z_1-z_2)(z_2-z_0)=0$,
one can let
\[
A_0=\begin{pmatrix}
1 & 0 & 0\\
0 & -1 & 0\\
0 & 0 & 0 \\
\end{pmatrix},\ \ \
A_1=\begin{pmatrix}
-1 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 1 \\
\end{pmatrix},\ \ \
A_2=\begin{pmatrix}
0 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & -1 \\
\end{pmatrix},\]
and verifies easily that $P(A)=\{z\in {\mathbb C}^3:\ (z_0-z_1)(z_1-z_2)(z_2-z_0)=0\}$.\\
{\bf Example 4}. Now consider the disk algebra ${\mathcal B}=A({\bf D})$ and let $A_j=w^j,\ 0\leq j\leq n$. Then
$A$ is a commutative tuple, and
\[A(z)=\sum_{j=0}^n z_jw^j\]
is a degree $n$ polynomial in $w$ (when $z_n\neq 0$). In this case, $A(z)$ is invertible in ${\mathcal B}$ if and only if it has no zero in the
closed unit disk ${\overline {\bf D}}$. Here, the maximal ideal space of ${\mathcal B}$ is equal to ${\overline {\bf D}}$ (cf. Douglas [Do]), and a point
$w\in {\overline {\bf D}} $ acts on ${\mathcal B}$ by point evaluation
\[\phi_w(f):=f(w),\ \ f\in {\mathcal B}.\]
Then by Proposition 2.2.
\[P(A)=\bigcup_{|w|\leq 1}H_{\phi_{w}}.\]
{\hfill \rule{.25cm}{.25cm} \medbreak}
When ${\mathcal B}=M_k({\mathbb C})$, $P(A)$ is a hypersurface in ${\mathbb C}^{n+1}$ defined by the polynomial equation $detA(z)=0$. So its complement is clearly a domain of holomorphy, because
$1/detA(z)$ is holomorphic on $P^c(A)$ and cannot be extended analytically to a neighborhood of any point in $P(A)$. On another account,
one direct consequence of Proposition 2.2 is that when $A$ is commutative, each path connected component of $P^c(A)$ is a domain of
holomorphy. To see this, we let $U$ be a connected component of $P^c(A)$, and $\lambda$ be any point in $\partial U$.
Since $P(A)$ is a union of hyperplanes, $\lambda$ is in one of these hyperplanes, say $\{\sum_{j=0}^na_jz_j=0\}$. So
$(\sum_{j=0}^na_jz_j)^{-1}$ is holomorphic on $U$ and does not have a holomorphic extension to any neighborhood of $\lambda$.
On a general projective resolvent set $P^c(A)$, $A^{-1}(z)$ is holomorphic and cannot be extended to any greater region.
So it is natural to ask whether $P^c(A)$
is necessarily a domain of holomorphy, or a disjoint union of domains of holomorphy when it is not path connected. Of course,
the difference here is that $A^{-1}(z)$ is a ${\mathcal B}$-valued function.
Here, we show that for some interesting types of Banach algebra ${\mathcal B}$, the answer is positive.
\begin{thm}
If ${\mathcal B}$ is reflexive as a Banach space, i.e. ${\mathcal B}={\mathcal B}^{**}$, then every connected component of $P^c(A)$
is a domain of holomorphy.
\end{thm}
\begin{pf}
We let $U$ be a connected component of $P^c(A)$, and $\lambda$ be a point in $\partial U$. We will show by contracdiction that there exists
a $\phi\in {\mathcal B}^*$ such that $\phi(A^{-1}(z))$ does not extend holomorphicly to any neighborhood of $\lambda$.
Suppose on the contrary for every $\phi\in {\mathcal B}^*$, $\phi(A^{-1}(z))$ extends holomorphicly to a neighborhood of $\lambda$. Then one
observes that the function
\[F(\phi,\ z):=\phi(A^{-1}(z)),\ \ \phi\in {\mathcal B}^{*},\ z\in U,\]
is a bounded linear functional on ${\mathcal B}^*$ for every fixed $z$, and has, for every fixed $\phi$, a holomorphic continuation to
a neighborhood of $\lambda$.
Let $z^m,\ m\geq 0,$ be a sequence in $U$ that converges to $\lambda$, and consider the sequence $F_m(\phi):=F(\phi,\ z^m)$.
To take care of the case that $\partial U$ may intersect itself at $\lambda$, we assume that for every open neighborhood $V$
of $\lambda$, $z^m$ stay in the same connected component of $V\cap U$ when $m$ is sufficiently large.
Then $F_m\in {\mathcal B}^{**},\ \forall m,$ and for every fixed $\phi$
\[sup\{|F_m(\phi)|:\ m\geq 0\}<\infty.\]
The Uniform Boundedness Principle then implies that the limit
\[F_{\infty}(\phi):=\lim_{m\to \infty}F_m(\phi),\ \ \phi\in {\mathcal B}^*\]
is in ${\mathcal B}^{**}$. Since ${\mathcal B}={\mathcal B}^{**}$, there exists a $B\in {\mathcal B}$ such that
\begin{equation*}
F_{\infty}(\phi)=\phi(B),\ \ \ \forall \phi\in {\mathcal B}^*.\tag{2.3}
\end{equation*}
Moreover, for a fixed $C\in {\mathcal B}$ and any $\phi\in {\mathcal B}^*$, the functional $\phi_C$ defined by
\[\phi_C(X):=\phi(XC),\ \ X\in {\mathcal B}\]
is clearly in ${\mathcal B}^*$, so it follows from (2.3)
that
\begin{align*}
\lim_{m\to \infty}\phi(A^{-1}(z^m)C)&=\lim_{m\to \infty}F(\phi_c,\ z^m)\\
&=\lim_{m\to \infty}F_m(\phi_c)\\
&=F_{\infty}(\phi_c)\\
&=\phi(BC), \ \ \ \forall \phi\in {\mathcal B}^*.
\end{align*}
Letting $C=A(\lambda)$, we have
\begin{align*}
\phi(BA(\lambda))&=\lim_{m\to \infty}\phi(A^{-1}(z^m)A(\lambda))\\
&=\lim_{m\to \infty}\phi\left((A^{-1}(z^m)(A(z^m)+A(\lambda)-A(z^m)))\right)\\
&=\phi(I)+\sum_{j=0}^n\lim_{m\to \infty}(\lambda_j-z^m_j)\phi(A^{-1}(z^m)A_j)\\
&=\phi(I)+\sum_{j=0}^n\lim_{m\to \infty}(\lambda_j-z^m_j)\phi_{A_j}(A^{-1}(z^m)).
\end{align*}
Since $\phi_{A_j}(A^{-1}(z))$ extends analytically to a neighborhood of $\lambda$, $\phi_{A_j}(A^{-1}(z^m))$ is bounded, and it follows that
\[\phi(BA(\lambda))=\phi(I).\ \ \ \forall \phi\in {\mathcal B}^*. \]
Similarly, we can also show $\phi(A(\lambda)B)=\phi(I),\ \forall \phi\in {\mathcal B}^*$. These imply that
$BA(\lambda)=A(\lambda)B=I$, e.g, $A(\lambda)$ is invertible, which is a contradiction.
\end{pf}
The proof of Theorem 2.3 can be modified to work for other Banach algebras. For example, if ${\mathcal H}$ is a reflexive Banach space,
and ${\mathcal B}$ is a Banach sub-algebra of $B({\mathcal H})$---the set of bounded linear operators on ${\mathcal H}$, then for every
$x\in {\mathcal H}$ and $f\in {\mathcal H}^*$ , \[\phi_{x,f}(C)=f(Cx),\ \ C\in {\mathcal B}\] defines a bounded linear functional on
${\mathcal B}$. If we let $F_m(x,\ f):=f(A^{-1}(z^m)x)$ and apply the Uniform Boundedness Principle, then
\[F_{\infty}(x,\ f):=\lim_{m\to \infty}F_m(x,\ f),\ \ \ \ x\in {\mathcal H}, f\in {\mathcal H}^*\]
is a bounded bilinear form on ${\mathcal H}\times {\mathcal H}^*$. In particular, if we fix $x$ then $F_{\infty}(x,\ \cdot)$
is in ${\mathcal H}^{**}$. Now since ${\mathcal H}$ is reflexive, there is a unique $B(x)\in {\mathcal H}$ such that
\[F_{\infty}(x,\ f)=f(B(x)),\ \forall x\in {\mathcal H},\ f\in {\mathcal H}^*,\]
and it is not hard to see that $B$ is a linear and bounded.
Similar to the ending part of the proof of Theorem 2.3, we have
\[f(BA(\lambda)x)=f(x)=f(A(\lambda)Bx),\ \ \forall x\in {\mathcal H},\ f\in {\mathcal H}^*,\]
which means $A(\lambda)$ is invertible with inverse $B$. However, in general this $B$ may not be in ${\mathcal B}$.
But things can be pulled together in the case when ${\mathcal B}$ is a $C^*$-algebra. In this case, ${\mathcal B}$ can be
identified (up to a isometrically $*$-isomorphism) with a $C^*$-subalgebra of $B({\mathcal H})$ (cf. Davidson [Da]), where ${\mathcal H}$ is a Hilbert
space. And an element in ${\mathcal B}$ is invertible if and only if it is invertible in $B({\mathcal H})$ (cf. Douglas [Do]). We therefore have the following
\begin{thm}
If ${\mathcal B}$ is a unital $C^*$-algebra, then every connected component of $P^c(A)$ is a domain of holomorphy.\\
\end{thm}
{\bf Question A}. Is the statement in Theorem 2.4 true for any unital Banach algebra?\\
\section{${\mathcal B}$-valued 1-form $A^{-1}(z)dA(z)$ and the de Rham cohomology space $H^1_d(P^c(A),\ {\mathbb C})$}
If $S$ is a hypersurface defined by $\{q(z)=0\}$, where $q$ is an irreducible homogenous polynomial of degree $k>0$,
then the complements $S^c={\mathbb C}^{n+1}\setminus S$ and $(S^T)^c={\mathbb P}^n\setminus S^T$ are both connected. Moreover,
the singular homology group $H_1(S^c,\ {\mathbb Z})={\mathbb Z}$, and $H_1((S^T)^c,\ {\mathbb Z})={\mathbb Z}/k{\mathbb Z}$ (cf. Dimca [Di], Ch4), which indicates that
neither $S^c$ nor $(S^T)^c$ is simply connected. But, as indicated in Example 3, projective resolvent sets $P^c(A)$ and
$p^c(A)$ may not be connected. Furthermore, connected components of $p^c(A)$ may also be simply connected.
However, connected components of $P^c(A)$ behave somewhat differently. In this section we will have a peek on the
de Rham cohomology space $H^1_d(P^c(A),\ {\mathbb C})$. The Maurer-Cartan type ${\mathcal B}$-valued 1-form
$\omega_A(z):=A^{-1}(z)dA(z)$ and central linear functionals on ${\mathcal B}$ are important tools in our study.
Here $d=\sum_{j=0}^n\frac{\partial}{\partial z_j}dz_j.$
First of all, from a operator-theoretic point of view, $\omega_A(z)$ is a faithful associate of the tuple $A$ because it determines
$A$ up to a certain equivalence. Let $A$ and $B$ be two tuples with the same projective spectrum $S\subset {\mathbb C}^{n+1}$. Here, the two 1-forms $\omega_A(z)$ and
$\omega_B(z)$ are said to be {\em similar} if there is an invertible element $V\in {\mathcal B}$ such that
\[V^{-1}\omega_A(z)V=\omega_B(z),\ \ \ \forall z\in S^c.\]
\begin{prop}
Let $A$ and $B$ be two tuples with the same projective spectrum $S$. Then the two 1-forms $\omega_A(z)$ and
$\omega_B(z)$ are similar if and only if there are invertible element $U,\ V\in {\mathcal B}$ such that
$UA_jV=B_j$ for every $0\leq j\leq n$.
\end{prop}
\begin{pf}
For the sufficiency, one easily checks that $UA_jV=B_j$ for each $0\leq j\leq n$ implies $UA(z)V=B(z)$. Then on the projective resolvent set
$S^c$,
\begin{align*}
B^{-1}(z)dB(z)&=V^{-1}A^{-1}(z)U^{-1}dUA(z)V\\
&=V^{-1}A^{-1}(z)U^{-1}UdA(z)V\\
&=V^{-1}A^{-1}(z)dA(z)V.
\end{align*}
For the necessity, one checks that $V^{-1}\omega_A(z)V=\omega_B(z)$ implies
\[\sum_{j=0}^nV^{-1}A^{-1}(z)A_jVdz_j=\sum_{j=0}^nB^{-1}(z)B_jdz_j,\]
and hence $V^{-1}A^{-1}(z)A_jV=B^{-1}(z)B_j$ for each $j$, or equivalently,
\begin{equation*}
A_jV=A(z)VB^{-1}(z)B_j,\ \ \ \forall z\in S^c. \tag{3.1}
\end{equation*}
So for any fixed $w\in S^c$, one has
\[\sum_{j=0}^nw_jA_jV=A(z)VB^{-1}(z)\sum_{j=0}^nw_jB_j,\]
which implies that
\[A(w)VB^{-1}(w)=A(z)VB^{-1}(z),\ \ \ \forall z\in S^c.\]
So $A(z)VB^{-1}(z)$ is an invertible constant, for which we denote by $U^{-1}$. Then by (3.1), $UA_jV=B_j$ for every $0\leq j\leq n$.
\end{pf}
One observes that for a $\phi\in {\mathcal B}^*$, $\phi(\omega_A(z))=\sum_{j=0}^n\phi(A^{-1}(z)A_j)dz_j$ is a holomorphic 1-form
on $P^c(A)$.
\begin{thm}
Consider a bounded linear functional $\phi$ on ${\mathcal B}$.
(a) If $\phi$ is central, then $\phi(\omega_A(z))$ is a closed 1-form on $P^c(A)$.
(b) If $\phi(I)\neq 0$, then there is no holomorphic function $f(z)$ on $P^c(A)$ such that
$df(z)=\phi(\omega_A(z))$.
\end{thm}
\begin{pf}
(a) First of all, Maurer-Cartan type form $\omega$ has the property $d\omega=-\omega\wedge \omega$. This fact for $\omega_A(z)$
also follows easily from (1.1). By (1.1), for each $j$,
\[\frac{\partial}{\partial z_j}A^{-1}(z)=-A^{-1}(z)A_jA^{-1}(z),\]
hence
\begin{align*}
d\omega_A(z)&=\sum_{j=0}^ndA^{-1}(z)A_jdz_j\\
&=\sum_{i,j=1}^n\frac{\partial A^{-1}(z)}{\partial z_i}A_jdz_i\wedge dz_j\\
&=\sum_{i,j=0}^n-A^{-1}(z)A_iA^{-1}(z)A_jdz_i\wedge dz_j\\
&=\sum_{i<j}-(A^{-1}(z)A_iA^{-1}(z)A_j -A^{-1}(z)A_jA^{-1}(z)A_i)dz_i\wedge dz_j \tag{3.2}\\
&=-\omega_A(z)\wedge \omega_A(z). \tag{3.3}
\end{align*}
If $\phi$ is central, then by (3.2)
\begin{align*}
d\phi(\omega_A(z))&=\phi(d\omega_A(z))\\
&=\sum_{i<j}-\phi(A^{-1}(z)A_iA^{-1}(z)A_j -A^{-1}(z)A_jA^{-1}(z)A_i)dz_i\wedge dz_j\\
&=0,
\end{align*}
hence $\phi(\omega_A(z))$ is closed.
(b) If there exists an $f$ holomorphic on $P^c(A)$ such that
$df(z)=\phi(\omega_A(z))$, then
\[\frac{\partial f}{\partial z_j}(z)=\phi(A^{-1}(z)A_j),\ \ \forall j.\]
So for any $t\in {\mathbb C}^{\times }$,
\begin{equation*}
\frac{\partial f}{\partial z_j}(tz)=\phi(A^{-1}(tz)A_j)=t^{-1}\phi(A^{-1}(z)A_j),\ \ \forall j.\tag{3.4}
\end{equation*}
It follows that
\begin{align*}
d(f(tz))&=\sum_{j}t\frac{\partial f}{\partial z_j}(tz)dz_j\\
&=tt^{-1}\sum_{j}\phi(A^{-1}(z)A_j)dz_j\\
&=\phi(\omega_A(z))\\
&=df(z),
\end{align*}
hence $f(tz)-f(z)$ is a constant depending on $t$, say $c(t)$. To figure out $c(t)$, one computes using (3.4) that
\begin{align*}
c'(t)&=\frac{df(tz)}{dt}\\
&=\sum_{j}z_j\frac{\partial f}{\partial z_j}(tz)\\
&=t^{-1}\sum_{j}z_j\phi(A^{-1}(z)A_j)\\
&=t^{-1}\phi(I).
\end{align*}
Since $c(1)=0$, $c(t)=\phi(I)logt$, hence
\begin{equation*}
f(tz)-f(z)=\phi(I)logt.\tag{3.5}
\end{equation*}
Since $f$ is holomorphic on $P^c(A)$, and $tz\in P^c(A)$ for $t\in {\mathbb C}^{\times}$, $f(tz)$ is holomorphic in $t$, and hence
$\phi(I)logt$ is holomorphic on ${\mathbb C}^{\times}$, which is possible only if $\phi(I)=0$.
\end{pf}
If $P^c(A)$ is not path connected, Theorem 3.2 can be stated for every connected component of $P^c(A)$.
For a domain $U\in {\mathbb C}^{n+1}$ of holomorphy ( or equivalently, a {\em Stein domain}), its de Rham cohomology $H^*_d(U,\ {\mathbb C})$
can be calculated by holomorphic forms (cf. Range [Ra]). To be precise, if $\Omega^r(U)$ is the collection of
holomorphic $r$-forms on $U$, then
\[H^r_d(U,\ {\mathbb C})\simeq \{f\in \Omega^r(U):\ df=0\}/d\Omega^{r-1}(U),\ \ \ for\ r\geq 0.\]
This observation, combined with Theorem 2.4 and Theorem 3.2, leads to the following
\begin{thm}
Let ${\mathcal B}$ be a $C^*$-algebra with a trace $\phi$, and $U$ be a connected component of $P^c(A)$.
Then $\phi(\omega_A(z))|_U$ is a nontrivial element in the de Rham
cohomology space $H^1_d(U,\ {\mathbb C})$. In particlar, $U$ is not simply connected.
\end{thm}
\begin{pf}
By Theorem 2.4, $U$ is a domain of holomorphy. Hence
the de Rham cohomology space $H^1_d(U,\ {\mathbb C})$ can be calculated by holomorphic forms.
Now since $\phi$ is a trace, $\phi $ is central with $\phi(I)>0$. Theorem 3.2 then concludes that $\phi(\omega_A(z))|_U$ is a nontrivial element in $H^1_d(U,\ {\mathbb C})$.\\
\end{pf}
{\bf Example 5}. Let $A$ be a tuple of $k\times k$ matrices, and let $Tr$ be the ordinary trace on square matrices.
$P^c(A)$ is a hypersurface complement in this case.
It is a classical fact that for a one variable square matrix-valued differentiable function $M(t)$
\[Tr(M^{-1}(t)\frac{d}{dt}M(t))=\frac{d}{dt}\text{log(det}M(t)),\]
hence we have
\[Tr(\omega_A(z))=d\text{logdet}A(z),\ \ \ z\in P^c(A).\]
One sees that $\text{logdet}A(z)$ is not a global holomorphic function on $P^c(A)$, hence $Tr(\omega_A(z))$ is
closed but not exact.\\
{\hfill \rule{.25cm}{.25cm} \medbreak}
In order to have more traces, one can let ${\mathcal B}={\mathcal B}_A$.\\
{\bf Example 6}. Consider a tuple $A$ of $3\times 3$ matrices, where
\[
A_0=\begin{pmatrix}
1 & 0 & 0\\
0 & 0 & -1\\
0 & 1 & 0 \\
\end{pmatrix},\ \ \
A_1=\begin{pmatrix}
1 & 0 & 0\\
0 & \sqrt{-1} & 0\\
0 & 0 & -\sqrt{-1} \\
\end{pmatrix},\ \ \
A_2=\begin{pmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1 \\
\end{pmatrix}.\]
Then
\[A(z)=
\begin{pmatrix}
z_0+z_1+z_2 & 0 & 0\\
0 & \sqrt{-1}z_1+z_2 & -z_0\\
0 & z_0 & -\sqrt{-1}z_1+z_2 \\
\end{pmatrix},\]
and $detA(z)=(z_0+z_1+z_2)(z_0^2+z_1^2+z_2^2)$.
Hence \[P(A)=\{(z_0,\ z_1,\ z_2)\in {{\mathbb C}}^3:\ (z_0+z_1+z_2)(z_0^2+z_1^2+z_2^2)=0\}.\]
In this case, one verifies that ${\mathcal B}_A={\mathbb C}\oplus M_{2\times 2}({\mathbb C})$.
Let $\phi_1$ and $\phi_2$ be the linear functionals on ${\mathbb C}\oplus M_{2\times 2}({\mathbb C})$ defined by
\[\phi_1
\begin{pmatrix}
a & 0\\
0 & T\\
\end{pmatrix}=a,\ \ \
\phi_2
\begin{pmatrix}
a & 0\\
0 & T\\
\end{pmatrix}=Tr(T),\]
where $T\in M_{2\times 2}({\mathbb C})$. Then $\phi_1$ and $\phi_2$ are both traces, hence by Example 5
\[\phi_1(\omega_A(z))=\frac{dz_0+dz_1+dz_2}{z_0+z_1+z_2},\ \ \
\phi_2(\omega_A(z))=\frac{2z_0dz_0+2z_1dz_1+2z_2dz_2}{z_0^2+z_1^2+z_2^2}\]
are nontrivial 1-forms in $H^1_d(P^c(A),\ {\mathbb C})$. \\
{\hfill \rule{.25cm}{.25cm} \medbreak}
{\bf Example 7}. Now we continue with Example 3. We have remarked that is this case ${\mathcal B}$ is the irrational
rotation algebra ${\mathcal A}_{\theta}$. It is not hard to see that
monomials $A_0^kA_1^l,\ \ k,\ l\in {\mathbb Z},$ span a dense subspace of ${\mathcal A}_{\theta}$.
${\mathcal A}_{\theta}$ possesses a unique faithful unital trace $\phi$ defined by $\phi(I)=1$ and
$\phi(A_0^kA_1^l)=0$ for $k$ and $l$ not both zero.
We now compute $\phi(\omega_A(z))$, where $A=(A_0,\ A_1)$. We showed in Example 3 that in this case $P^c(A)$ has two
connected components:
\[\Omega_0=\{ |z_0|>|z_1|\},\ \ \text{and}\ \ \Omega_1=\{ |z_0|<|z_1|\}.\]
On $\Omega_0$,
\begin{align*}
A^{-1}(z)dA(z)&=(z_0A_0+z_1A_1)^{-1}(A_0dz_0+A_1dz_1)\\
&=z_0^{-1}(A_0+\frac{z_1}{z_0}A_1)^{-1}A_0(dz_0+A_0^{-1}A_1dz_1)\\
&=z_0^{-1}(I+\frac{z_1}{z_0}A_0^{-1}A_1)^{-1}(dz_0+A_0^{-1}A_1dz_1)\\
&=\left(I-\frac{z_1}{z_0}A_0^{-1}A_1+(\frac{z_1}{z_0}A_0^{-1}A_1)^2-\cdots \right)(\frac{dz_0}{z_0}+A_0^{-1}A_1\frac{dz_1}{z_0}).
\end{align*}
Hence $\phi(\omega_A(z))=\frac{dz_0}{z_0}$. Likewise, on $\Omega_1$, $\phi(\omega_A(z))=\frac{dz_1}{z_1}$.
As a matter of fact, in this case it is not hard to compute directly that
\[H^1_d(U_0,\ {\mathbb C})={\mathbb C} \frac{dz_0}{z_0},\ \ \ H^1_d(U_1,\ {\mathbb C})={\mathbb C} \frac{dz_1}{z_1}.\]\\
{\hfill \rule{.25cm}{.25cm} \medbreak}
We conclude this section with a remark on the value of $\phi(I)$. The case $\phi(I)\neq 0$ is important for Theorem 3.2(b), and
as we will see, the case $\phi(I)=0$ is also meaningful.
For $\phi\in {\mathcal B}^*$,
one easily sees that $\phi(\omega_A(z))$ is homogenous of degree $0$. So it makes one wonder for what type of $\phi$,
$\phi(\omega_A(z))$ defines a 1-form on the projective resolvent set $p^c(A)\subset {\mathbb P}^n$.
For a holomorphic 1-form $\eta=\sum_{j=0}^{n}f_j(z)dz_j$, its {\em contraction} with the Euler
field $\theta=\sum_{j=0}^{n}z_j\frac{\partial}{\partial z_j}$ is
\[\Delta(\eta):=\sum_{j=0}^{n}z_jf_j(z).\]
By Griffiths [Gr], a locally holomorphic 1-form $\eta=\sum_{j=0}^{n}f_j(z)dz_j$ on ${\mathbb C}^{n+1}$ comes from
a 1-form on ${\mathbb P}^n$ if and only if it is homogenous of degree $0$ and the contraction $\Delta (\eta)=0$.
It is easy to see that the Maurer-Cartan form $\omega_A(z)$ is homogenous of degree $0$.
But it itself is not a 1-form on $p^c(A)$. In fact, one checks easily that
\begin{align*}
\Delta(\omega_A(z))&=\Delta(\sum_{j=0}^nA^{-1}(z)A_jdz_j)\\
&=\sum_{j=0}^nA^{-1}(z)z_jA_j\\
&=I.
\end{align*}
So if $\phi$ is a linear functional on ${\mathcal B}$ such that $\phi(I)=0$, then
\begin{align*}
\Delta(\phi(\omega_A(z)))&=\phi(\Delta(\omega_A(z)))\\
&=\phi(I)\\
&=0,
\end{align*}
hence $\phi(\omega_A(z))$ defines a global holomorphic 1-form on $p^c(A)$. Furthermore, using arguments similar to that in
the proof of Theorem 3.2(a), one can easily check that when, in addition, $\phi$ is central, $\phi(\omega_A(z))$ is also closed
on $p^c(A)$.
\section{the case when $A$ is commutative}
When $A$ is a commutative tuple, ${\mathcal B}_A$ is an commutative Banach sub-algebra of ${\mathcal B}$.
One observes that in this case, $({\mathcal B}_A)^*_c={\mathcal B}_A^*$,
and every $\phi\in M_A$ has the property $\phi(I)=1$. Then, as remarked after Example 4,
every connected component of $P^c(A)$ is a domain of holomorphy, and the next corollary is another consequence of Theorem 3.2.
\begin{corr}
If $A$ is a commutative tuple, then for every $\phi\in M_A$, $\phi(\omega_A(z))$ is a nontrivial element in $H^1_d(P^c(A),\ {\mathbb C})$
\end{corr}
As stated in Proposition 2.2 that in this case $P(A)$ is a (possibly uncountable) union of hyperplanes. This section recalls
a theorem in Hyperplane Arrangements by Arnold and Briskorn, and discusses its possible analogue in the setting here.
First, if $\phi$ is a multiplicative linear functional on ${\mathcal B}_A$, then
\begin{align*}
\phi(\omega_A(z))&=\phi(A^{-1}(z))d\phi(A(z))\\
&=\frac{d\phi(A(z))}{\phi(A(z))}\\
&=\frac{d\sum_{j=0}^n z_j\phi(A_j)}{\sum_{j=0}^n z_j\phi(A_j)}\tag{3.6}
\end{align*}
Here one recalls that $\sum_{j=0}^n z_j\phi(A_j)$ is the defining function for the hyperplane $H_{\phi}$.
In the case $A$ is a tuple of generic commutative $k\times k$ matrices, the maximum ideal space $M_A$ consists of $k$ elements, say, $\phi_1,\ ...,\ \phi_k$, and hence
\[P(A)=\cup_{j=0}^k H_{\phi_j}\]
is a central arrangement. By a well-known result conjectured by Arnold [Ar] and proved by Briskorn [Br],
the cohomology algebra $H^*_d(P^c(A),\ {\mathbb C})$ is generated by $1$ and the 1-forms
\[\frac{d\sum_{j=0}^n z_j\phi(A_j)}{\sum_{j=0}^n z_j\phi(A_j)},\ \ \ \phi\in M_A.\]
Here we make two observations.\\
{\bf 1}. First, let $\wedge ({\mathcal B}_A^*)$ be the exterior algebra on ${\mathcal B}_A^*$.
By Theorem 3.2, $\omega_A(z)$ induces a homomorphism
$\omega_A(z)^*:\wedge ({\mathcal B}_A^*)\longrightarrow H^*_d(P^c(A),\ {\mathbb C})$ defined by $\omega_A(z)^*(1)=1$, and
\[\omega_A(z)^*(\phi_1\wedge \phi_2\wedge \cdots \wedge \phi_q)=
\phi_1(\omega_A(z))\wedge \phi_2(\omega_A(z))\wedge \cdots \wedge \phi_q(\omega_A(z)),\ \ q\geq 1.\]
Corollary 4.1 and the remarks above lead to the following simple consequence of the Arnold and Briskorn's theorem.
\begin{corr}
For ${\mathcal B}=M_k({\mathbb C})$ and any tuple $A$ of commutative matrices in $M_k({\mathbb C})$,
$\omega_A(z)^*:\ \wedge ({\mathcal B}_A^*) \longrightarrow H^*(P^c(A),\ {\mathbb C})$ is surjective.
\end{corr}
Corollary 4.1 indicates that for ${\mathcal B}=M_k({\mathbb C})$, topological information of $P^c(A)$ is all encoded in $\omega_A(z)$.
It will be interesting to see if there are other Banach algebras with this property.\\
{\bf 2}. For a general commutative tuple $A$, $P^c(A)$ may not be connected, hence $\omega_A(z)^*$ may not be surjective. But it makes sense to ask
whether there is a similar fact for every connected component $U$ of $P^c(A)$, e.g., whether the map
$\omega_A(z)^*:\ \wedge ({\mathcal B}_A^*) \longrightarrow H^*(U,\ {\mathbb C})$ is surjective. But since in general both
${\mathcal B}_A^*$ and $H^*(U,\ {\mathbb C})$ can be infinite dimensional, the nature of this problem is somewhat hard to see.
A related particular case is when the maximal ideal space $M_A$ is path connected. For any $\phi_0,\ \phi_1\in M_A$, let $\phi_t,\ t\in [0,\ 1]$ be a
continuous path in $M_A$. Taking any fixed cycle $\gamma \in H_1(U, {\mathbb Z})$ and using (4.1), we have
\[\frac{1}{2\pi \sqrt{-1}}\int_{\gamma}\phi_t(\omega_A(z))=\frac{1}{2\pi \sqrt{-1}}\int_{\gamma}dlog(\sum_{j=0}^n z_j\phi_t(A_j)),\]
which is an integer-valued continuous function in $t$, hence is a constant. This means that
$\omega_A(z)^*$ restricted to $M_A$ is a constant in $H^1_d(U,\ {\mathbb C})$.\\
{\bf Example 8.} Now we take another look at Example 4. Note that in this case ${\mathcal B}_A=A({\bf D})$. Then,
\[\omega_A(z)=\frac{\sum_{j=0}^nw^jdz_j}{\sum_{j=0}^nz_jw^j}.\]
Now $P^c(A)$ is connected (as we will see in a minute), and $M_A$ can be identified with $\overline{D}$ through evaluation(cf. Douglas [Do]),
so as remarked above $\omega_A(z)^*$ restricted to $M_A$
is a constant in $H^1_d(P^c(A),\ {\mathbb C})$.
For simplicity, we pick $\phi\in M_A$ to be the evaluation at $w=0$. Then
\[\omega_A(z)^*(\phi)=\phi(\omega_A(z))=\frac{dz_0}{z_0}.\]
On the other hand, the topology of $P^c(A)$ is not hard to determine directly. First of all, it is easy to see that
$P^c(A)\subset \{z_0\neq 0\}$. Hence $P^c(A)={{\mathbb C}}^{\times}\times p^c(A)$. Under the affine coordinate
$\xi_j=z_j/z_0,\ 1\leq j\leq n$,
\[p^c(A)=\{\xi=(\xi_1,\ ...,\ \xi_n)\in {{\mathbb C}}^n:\ 1+\sum_{j=1}^n \xi_jw^j\neq 0\ \text{on}\ {\overline{\bf D}}\}.\]
Now consider the maps $H_t$ on $p^c(A)$ defined by
\[H_t(\xi)=(t\xi_1,\ t^2\xi_2,\ ...,\ t^n\xi_n),\]
where $t\in [0,\ 1]$. Since $1+\sum_{j=1}^n \xi_jw^j\neq 0$ for every $w\in {\overline{\bf D}}$,
$1+\sum_{j=1}^n \xi_j(tw)^j$ doesn't vanish on ${\overline{\bf D}}$ as well. This shows that $H_t$ maps $p^c(A)$ into $p^c(A)$,
and it furnishes a retraction of $p^c(A)$ to the origin $0$.
Since $P^c(A)={{\mathbb C}}^{\times}\times p^c(A)$, $H^*_d(P^c(A),\ {\mathbb C})$ is isomorphic to $H^*_d({{\mathbb C}}^{\times},\ {\mathbb C})$.
In particular, $H^1_d(P^c(A),\ {\mathbb C})={\mathbb C}\frac{dz_0}{z_0}$, and $H^q_d(P^c(A),\ {\mathbb C})=0$ for $q\geq 2$. So Corollary 4.2
holds for the disk algebra $A({\bf D})$ and the tuple $(1,\ w,\ w^2,\ ...,\ w^n)$. It is not clear whether Corollary 4.2
holds for other tuples in $A({\bf D})$. \\
{\hfill \rule{.25cm}{.25cm} \medbreak}
It is worth mentioning that for every $\phi$ in the dual of $A({\bf D})$ such that
$\phi(1)=0$, $\phi(\omega_A(z))$ is a closed form on $p^c(A)$ (by the remarks at the end of section 3).
Using the affine coordinate $\xi$,
$\phi(\omega_A(z))$ is equal to
\[\phi^*(\xi):=\phi\left(\frac{\sum_{j=1}^n w^jd\xi_j}{1+\sum_{j=1}^n \xi_jw^j}\right)=d\phi\left(log(1+\sum_{j=1}^n \xi_jw^j)\right).\]
Now since in this case $p^c(A)$ is contractible to a point, $\phi^*(\xi)$ is exact, i.e. $\phi\left(log(1+\sum_{j=1}^n \xi_jw^j)\right)$ is
holomorphic on $p^c(A)$.
We end this paper with the following\\
{\bf Question B}. For the disk algebra $A({\bf D})$, is the map
\[\omega_A(z)^*:\ \wedge (A^*({\bf D})) \longrightarrow H^*_d(U,\ {\mathbb C})\] surjective
for every tuple $A$ of functions in $A({\bf D})$ and any connected component $U$ of $P^c(A)$?
\vspace{1cm}
| {
"timestamp": "2008-04-02T18:00:25",
"yymm": "0804",
"arxiv_id": "0804.0387",
"language": "en",
"url": "https://arxiv.org/abs/0804.0387",
"abstract": "For a tuple $A=(A_0, A_1, ..., A_n)$ of elements in a unital Banach algebra ${\\mathcal B}$, its {\\em projective spectrum} $p(A)$ is defined to be the collection of $z=[z_0, z_1, ..., z_n]\\in \\pn$ such that $A(z)=z_0A_0+z_1A_1+... +z_nA_n$ is not invertible in ${\\mathcal B}$. The pre-image of $p(A)$ in ${\\cc}^{n+1}$ is denoted by $P(A)$. When ${\\mathcal B}$ is the $k\\times k$ matrix algebra $M_k(\\cc)$, the projective spectrum is a projective hypersurface. In infinite dimensional cases, projective spectrums can be very complicated, but also have some properties similar to that of hypersurfaces. When $A$ is commutative, $P(A)$ is a union of hyperplanes. When ${\\mathcal B}$ is reflexive or is a $C^*$-algebra, the {\\em projective resolvent set} $P^c(A):=\\cc^{n+1}\\setminus P(A)$ is shown to be a disjoint union of domains of holomorphy. Later part of this paper studies Maurer-Cartan type ${\\mathcal B}$-valued 1-form $A^{-1}(z)dA(z)$ on $P^c(A)$. As a consequence, we show that if ${\\mathcal B}$ is a $C^*$-algebra with a trace $\\phi$, then $\\phi(A^{-1}(z)dA(z))$ is a nontrivial element in the de Rham cohomology space $H^1_d(P^c(A), \\cc)$.",
"subjects": "Functional Analysis (math.FA)",
"title": "Projective spectrum in Banach algebras",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808736209154,
"lm_q2_score": 0.7217432062975979,
"lm_q1q2_score": 0.7075110607993699
} |
https://arxiv.org/abs/1111.0245 | Partially Ordinal Sums and $P$-partitions | We present a method of computing the generating function $f_P(\x)$ of $P$-partitions of a poset $P$. The idea is to introduce two kinds of transformations on posets and compute $f_P(\x)$ by recursively applying these transformations. As an application, we consider the partially ordinal sum $P_n$ of $n$ copies of a given poset, which generalizes both the direct sum and the ordinal sum. We show that the sequence $\{f_{P_n}(\x)\}_{n\ge 1}$ satisfies a finite system of recurrence relations with respect to $n$. We illustrate the method by several examples, including a kind of 3-rowed posets and the multi-cube posets. | \section{Introduction}
A $P$-partition is an order-reversing map from a partially ordered set (poset) to
non-negative integers (see \cite[Ch. IV]{Stanley-97}). Denote the set of $P$-partitions of a poset $P$ by ${\rm Par}(P)$. The (multivariate) generating function of $P$-partitions of $P$ is
given by
\[
f_P({\mathbf x}) = \sum_{\sigma \in {\rm Par}(P)} \prod_{a \in P} x_a^{\sigma(a)},
\]
where ${\mathbf x}$ is the variable vector $(x_a)_{a\in P}$.
Stanley showed that $f_P({\mathbf x})$ can be expressed as a sum over linear extensions of $P$ \cite[Theorem 4.5.4]{Stanley-97}. Andrews, Paule and Riese \cite{Andre01-a, Andre01-c,
Andre01-e, Andre04, Andre04-a,Andre07b}
computed $f_{P_n}({\mathbf x})$ for several sequences $\{P_n\}_{n \ge 1}$ of posets by developing MacMahon's
partition analysis. Corteel, Savage et. al. \cite{ Davis-05a, Davis-06}
presented five guidelines for deriving recurrence relations of $f_{P_n}({\mathbf x})$.
D'Souza \cite{Souza}
provided a Maple package {\tt GFPartitions} which generates
recurrence relations of $f_{P_n}({\mathbf x})$ once the decomposition of the posets $P_n$ is given manually.
Ekhad and Zeilberger \cite{Ekhad-07} considered the umbral operator
on ``grafting'' of posets. The corresponding Maple package {\tt
RotaStanley} can generate the recurrence relations automatically.
Our main objective is to find an efficient method of computing $f_P({\mathbf x})$.
For this purpose, we introduce two kinds of transformations on posets in Section \ref{sec-transform}, which are the \emph{deletion}
and the \emph{partially linear extension}. We find that there exist simple relations between the generating function of $P$-partitions of a poset and those of its transformations. Thus the generating function $f_P({\mathbf x})$ can be computed by recursively applying these transformations.
We then consider the posets $P_n$ composed of $n$ copies of a given poset $P$ in Section \ref{sec-gfun}. More precisely, we introduce the \emph{partially ordinal sum} $\oplus_R$ of posets, and denote by $P_n$ the sum $P \oplus_R P \oplus_R \dots \oplus_R P$, where $P$ occurs $n$ times. By applying the above two transformations, we find that the sequence $\{f_{P_n}({\mathbf x})\}_{n \ge 1}$ satisfies a system of recurrence relations with respect to $n$.
Finally, we provide some examples in Section \ref{sec-examples}, including a kind of $3$-rowed posets which can not be dealt with by the packages {\tt GFPartitions} and {\tt RotaStanley}.
Before our further discussion, let us recall the Hasse diagram of a poset $(P,\leq)$. For $x,y\in P$, we say $y$
covers $x$, denoted by $x \lessdot y$, if $x < y$ and if no
element $z\in P$ satisfies $x<z<y$. The Hasse diagram of $P$ is a graphical representation of $P$, in which every element of
$P$ is represented by a vertex and two vertices are joined by a line
with vertex $x$ drawn below vertex $y$ if $x \lessdot y$.
To coincide with
the descriptions used by Andrews, Paule and Riese \cite{Andre01-a},
we rotate the Hasse diagram by $90$ degree clockwise so that smaller elements lie to the left. For
example, the diamond poset $D=\{1,2,3,4\}$ with cover relations $\{1 \lessdot 2 \lessdot 4, 1 \lessdot 3 \lessdot 4\}$ can be represented by Figure \ref{Fig-diamond}.
\begin{figure}[ht]
\centering
\begin{picture}(90,70)(0,0)
\multiput(15,35)(30,-30){2}{\line(1,1){30}}
\multiput(15,35)(30,30){2}{\line(1,-1){30}}
\multiput(15,35)(60,0){2}{\circle*{4}}
\multiput(45,5)(0,60){2}{\circle*{4}}
\put(7,31){\tiny $1$}\put(38,0){\tiny $2$}
\put(38,67){\tiny $3$}\put(78,31){\tiny $4$}
\end{picture}
\caption{The Hasse diagram of the Diamond poset $D$. }\label{Fig-diamond}
\end{figure}
\section{Two Transformations on Posets}
\label{sec-transform}
In this section, we introduce the deletions and the partially linear extensions. On one hand, they reduce a poset to simpler ones. On the other hand, the generating function of $P$-partitions of a poset can be expressed in terms of those of its transformations.
This enables us to compute $f_P({\mathbf x})$ by recursively transforming the poset $P$.
\subsection{The Deletions}
The first transformation we consider is removing an element from a poset. We only concern on the {\it removable} elements, which cover at
most one element and are covered by at most one element.
Let $P$ be a poset and $b\in P$ be a removable element of $P$. The deletion of $P$ with respect to $b$ is the transformation from $P$ to the induced sub-poset $P \setminus \{b\}$ of $P$ by deleting $b$ from $P$. The following theorem shows the relation between $f_P({\mathbf x})$ and $f_{P \setminus \{b\}}({\mathbf x})$.
\begin{theo}\label{theo-absorb}
Let $P$ be a poset and $b \in P$ be a removable element of $P$. Then
\begin{equation}\label{eq-absorb}
f_P({\mathbf x}) = \frac{g({\mathbf x}) - h({\mathbf x})}{1-x_b},
\end{equation}
where
\[
g({\mathbf x}) = \begin{cases}
f_{P \setminus \{b\}}({\mathbf x})|_{x_c=x_b x_c}, & \mbox{if $\exists\, c \in P$ such that $b \lessdot c$}, \\[5pt]
f_{P \setminus \{b\}}({\mathbf x}), & \mbox{otherwise,}
\end{cases}
\]
and
\[
h({\mathbf x}) = \begin{cases}
x_b f_{P \setminus \{b\}}({\mathbf x})|_{x_a=x_a x_b}, & \mbox{if $\exists\, a \in P$ such that $a \lessdot b$}, \\[5pt]
0, & \mbox{otherwise.}
\end{cases}
\]
\end{theo}
\noindent {\it Proof.} We only give the proof for the case when there exist $a$ and $c$ such that $a \lessdot b
\lessdot c$. The other three cases can be proved in a similar way.
By the definition of $P$-partitions, we have
\begin{eqnarray*}
f_P({\mathbf x})
&=& \sum_{\sigma \in Par(P \setminus \{b\})} \left( \sum_{m=\sigma(c)}^{\sigma(a)} x_b^m \prod_{u \in P \setminus \{b\}} x_u^{\sigma(u)}
\right) \\
&=& \sum_{\sigma \in Par(P \setminus \{b\})} \left( \frac{x_b^{\sigma(c)} - x_b^{\sigma(a)+1}}{1-x_b} \prod_{u \in P \setminus \{b\}} x_u^{\sigma(u)}
\right) \\
&=& \frac{1}{1-x_b} \sum_{\sigma \in Par(P \setminus \{b\})} (x_b
x_c)^{\sigma(c)} \prod_{u \in P \setminus \{b,c\}}
x_u^{\sigma(u)}\\
&& \quad - \frac{x_b}{1-x_b} \sum_{\sigma \in Par(P \setminus \{b\})}
(x_a x_b)^{\sigma(a)} \prod_{u \in P \setminus \{a,b\}}
x_u^{\sigma(u)},
\end{eqnarray*}
as desired. \hfill \rule{4pt}{7pt}
Now we give an example to illustrate the usage of Theorem~\ref{theo-absorb}.
\begin{exam}\label{exam-delete-gfun}
Let us consider the Diamond poset $D$ as shown in Figure~\ref{Fig-diamond}. We see that $2$ is a removable element of $D$ and $D \setminus \{2\}$ is a chain $C$ of length $3$ with
\[
f_C(x_1,x_2,x_3) = \frac{1}{(1-x_1)(1-x_1x_2)(1-x_1x_2x_3)}.
\]
Invoking equation \eqref{eq-absorb}, we derive that
\begin{align*}\label{eq-exam-absorb}
f_{D}({\mathbf x}) &= \frac{1}{1-x_2}
\left(f_C(x_1,x_3,x_2x_4)-x_2 f_C(x_1x_2,x_3,x_4)\right)\\[5pt]
&= \frac{1-x_1^2x_2x_3}{(1-x_1)(1-x_1x_2)(1-x_1x_3)(1-x_1x_2x_3)(1-x_1x_2x_3x_4)}.
\end{align*}
\end{exam}
\subsection{The Partially Linear Extensions}
The second transformation we consider is partially ordering the elements of an anti-chain of a poset.
Let $A$ be an anti-chain of a poset $P$ and let $M$ be a non-empty subset of $A$.
The partially linear extension (PLE in short) of $P$ with respect to the pair $(M,A)$ is the transformation from the poset $P$ to the poset $P(M,A)$ by gluing the elements of $M$ together and
setting the glued element cover the elements of $A\setminus M$.
More precisely, $P(M,A)$ is the poset
defined on $P \setminus M \cup \{M\}$ and partially
ordered by $x \le y$ if and only if
\begin{itemize}
\item[(a)] $x,y \in P\setminus M$ and $x \le_P y$, or
\item[(b)] $x,y \in P\setminus M$ and there exist $x' \in A$ and $y' \in M$
such that $x \le_P x', y' \le_P y$, or
\item[(c)] $x=M$ and there exists $y' \in M$ such that $y' \le_P y$,
or
\item[(d)] $y=M$ and there exists $x' \in A$ such that $x \le_P x'$,
or
\item[(e)] $x=y=M$.
\end{itemize}
\begin{exam} \label{exam-PLE}
Let $P=\{1,2,3,4,5\}$ be the poset as shown in Figure \ref{Fig-PLE0}.
\begin{figure}[ht]
\centering
\begin{picture}(70,70)(0,0)
\multiput(5,35)(30,-30){2}{\line(1,1){30}}
\multiput(5,35)(30,30){2}{\line(1,-1){30}}
\put(5,35){\line(1,0){60}}
\multiput(5,35)(30,0){3}{\circle*{4}}
\multiput(35,5)(0,60){2}{\circle*{4}}
\put(0,30){\tiny $1$}\put(37,0){\tiny $2$}
\put(37,28){\tiny $3$}
\put(37,67){\tiny $4$}\put(67,30){\tiny $5$}
\end{picture}
\caption{The Hasse diagram of the poset $P$. }\label{Fig-PLE0}
\end{figure}
Then the posets
$P(\{2\},\{2,3,4\})$ and $P(\{2,3 \},\{2,3,4\})$ are given as in Figure~\ref{Fig-ple}.
\end{exam}
\begin{figure}[ht]
\centering
\begin{picture}(350,50)(0,0)
\put(0,23){\small$P(\{2\},\{2,3,4\})=$}
\multiput(95,25)(40,0){2}{\circle*{4}}
\multiput(115,5)(0,40){2}{\circle*{4}}
\multiput(115,5)(-20,20){2}{\line(1,1){20}}
\multiput(115,5)(20,20){2}{\line(-1,1){20}}
\put(155,25){\circle*{4}}
\put(135,25){\line(1,0){20}}
\put(89,18){\tiny$1$}
\put(117,0){\tiny$3$}
\put(117,47){\tiny$4$}
\put(134,17){\tiny$\{2\}$}
\put(156,18){\tiny$5$}
\put(180,23){\small$P(\{2,3\},\{2,3,4\})=$}
\put(285,25){\line(1,0){60}}
\multiput(285,25)(20,0){4}{\circle*{4}}
\put(281,17){\tiny$1$}
\put(299,17){\tiny$4$}
\put(317,17){\tiny$\{2,3\}$}
\put(347,17){\tiny$5$}
\end{picture}
\caption{Two partially linear extensions of $P$. \label{Fig-ple}}
\end{figure}
As a generalization of Theorem~2.3 in \cite{GHX-07}, the generating
function of $P$-partitions of a poset can be expressed by those
of its PLE's.
\begin{theo}\label{theo-PLE}
Let $P$ be a poset and $A$ be an anti-chain of $P$. Then
\begin{equation}\label{eq-PLE}
f_P({\mathbf x}) = \sum_{\emptyset \not= M \subseteq A} (-1)^{|M|-1}
f_{P(M,A)}({\mathbf x}) |_{x_M = \prod_{a \in M}x_a}.
\end{equation}
\end{theo}
\noindent {\it Proof.} For each $a \in A$, we define
\[
S_a = \{\sigma \in {\rm Par}(P) \colon \sigma(a) \le \sigma(x), \forall\,
x \in A \}.
\]
Since ${\rm Par}(P) = \cup_{a \in A} S_a$, by the
inclusion-exclusion principle we derive that
\[
f_P({\mathbf x}) = \sum_{\emptyset \not= M \subseteq A} (-1)^{|M|-1}
\sum_{\sigma \in S_M} \prod_{u \in P} x_u^{\sigma{(u)}},
\]
where $S_M = \bigcap_{a \in M} S_a$.
Given a $P$-partition $\sigma \in S_M$, we denote $m=\min\{\sigma(a)
\colon a \in A\}$. Then by the definition of $S_M$ we have
$\sigma(x)=m$ for any $x \in M$. Let
\[
\sigma'(u) = \begin{cases} \sigma(u), & u \in P \setminus M, \\
m, & u=M. \end{cases}
\]
One sees that $\sigma'$ is a $P$-partition of $P(M,A)$.
Conversely, let $\sigma'$ be a $P$-partition of $P(M,A)$. By
defining
\[
\sigma(u) = \begin{cases} \sigma'(u), & u \in P \setminus M, \\
\sigma'(M), & u \in M, \end{cases}
\]
we obtain a $P$-partition of $P$ in $S_M$. We thus set up a one-to-one
corresponding between the $P$-partitions of $P$ in $S_M$ and the
$P$-partitions of $P(M,A)$. Therefore,
\begin{eqnarray*}
f_P({\mathbf x}) &=& \sum_{\emptyset \not= M \subseteq A} (-1)^{|M|-1}
\sum_{\sigma' \in {\rm Par}(P(M,A))} \prod_{u \in M} x_u^{\sigma'(M)}
\cdot \prod_{u \in P \setminus M} x_u^{\sigma'{(u)}} \\
&=& \sum_{\emptyset \not= M \subseteq A} (-1)^{|M|-1} f_{P(M,A)}({\mathbf x}) |_{x_M = \prod_{a \in M}x_a}.
\end{eqnarray*}
This completes the proof. \hfill \rule{4pt}{7pt}
The Hasse diagrams provide a simple graphical representation for Equation~\eqref{eq-PLE}.
\begin{exam}
Let $P=\{1,2,3,4\}$ be the poset as shown in Figure \ref{Fig-exam-PLE}.
\begin{figure}[htb]
\centering
\begin{picture}(60,40)(0,0)
\multiput(5,5)(0,30){2}{\line(1,0){30}}
\multiput(5,5)(30,0){2}{\circle*{4}}
\multiput(5,35)(30,0){2}{\circle*{4}}
\put(5,5){\line(1,1){30}}
\put(35,5){\line(-1,1){30}}
\put(0,0){\tiny$1$}
\put(0,37){\tiny$2$}
\put(37,0){\tiny$3$}
\put(37,37){\tiny$4$}
\end{picture}
\caption{The Hasse diagram of the poset $P$.}\label{Fig-exam-PLE}
\end{figure}
Taking the anti-chain $\{1,2\}$ into account, we find that the PLE's are shown as in Figure \ref{Fig-exam-PLE-gfun}, from which we read out
\begin{multline*}
f_P({\mathbf x}) = f_{P(\{1\},\{1,2\})}(x_2,x_1,x_3,x_4) + f_{P(\{2\},\{1,2\})}(x_1,x_2,x_3,x_4) \\
-f_{P(\{1,2\},\{1,2\})}(x_1x_2, x_3, x_4).
\end{multline*}
\begin{figure}[ht]
\centering
\begin{picture}(310,60)(0,0)
\multiput(5,15)(0,30){2}{\line(1,0){30}}
\multiput(5,15)(30,0){2}{\circle*{4}}
\multiput(5,45)(30,0){2}{\circle*{4}}
\put(5,15){\line(1,1){30}}
\put(35,15){\line(-1,1){30}}
\put(0,6){\tiny$1$}
\put(0,49){\tiny$2$}
\put(36,6){\tiny$3$}
\put(36,49){\tiny$4$}
\put(55,27){$\longrightarrow$}
\multiput(85,30)(30,0){2}{\circle*{4}}
\put(85,30){\line(1,0){30}}
\put(115,30){\line(1,1){25}}
\put(115,30){\line(1,-1){25}}
\multiput(140,5)(0,50){2}{\circle*{4}}
\put(82,19){\tiny$2$}
\put(105,19){\tiny$\{1\}$}
\put(144,1){\tiny$3$}
\put(144,52){\tiny$4$}
\put(150,27){$+$}
\multiput(175,30)(30,0){2}{\circle*{4}}
\put(175,30){\line(1,0){30}}
\put(205,30){\line(1,1){25}}
\put(205,30){\line(1,-1){25}}
\multiput(230,5)(0,50){2}{\circle*{4}}
\put(172,19){\tiny$1$}
\put(195,19){\tiny$\{2\}$}
\put(234,1){\tiny$3$}
\put(234,52){\tiny$4$}
\put(240,27){$-$}
\put(265,30){\circle*{4}}
\put(265,30){\line(1,1){25}}
\put(265,30){\line(1,-1){25}}
\multiput(290,5)(0,50){2}{\circle*{4}}
\put(272,28){\tiny$\{1,2\}$}
\put(296,3){\tiny$3$}
\put(296,53){\tiny$4$}
\end{picture}
\caption{A graphical representation of Equation~\eqref{eq-PLE}.}\label{Fig-exam-PLE-gfun}
\end{figure}
\end{exam}
\subsection{Computing $f_P({\mathbf x})$ via two Transformations}
We shall show that the deletion and the partially linear extension are powerful enough for computing $f_P({\mathbf x})$ for any poset $P$.
\begin{theo}\label{theo-transformations}
Any poset can be reduced to the empty poset by applying the deletion and the partially linear extension finite times.
\end{theo}
\noindent {\it Proof.} Let $ac(P)$ denote the number of distinct anti-chains of a poset $P$. Since each element of $P$ forms an anti-chain, the deletion reduces $ac(P)$ by at least one.
Suppose that $A$ is an anti-chain of $P$ with cardinality at least two and $M$ is a non-empty subset of $A$. If $M \not= A$, there exist $x \in M$ and $y \in A \setminus M$. Then $\{x,y\}$ is an anti-chain of $P$ but is not an anti-chain of $P(M,A)$. If $M = A$, the cardinality of $P(M,A)$ is strictly less than that of $P$. Thus in either case, we have $ac(P(M,A)) \le ac(P)-1$.
Now iteratively apply the deletion whenever there is a removable element and apply the partially linear extension whenever there is an anti-chain with cardinality at least two. Since $ac(P)$ is a finite number, the procedure eventually stops. The final poset contains no removable element and no anti-chain with cardinality at least two. The only poset satisfying this property is the empty poset. \hfill \rule{4pt}{7pt}
\section{Partially Ordinal Sums}\label{sec-gfun}
In this section, we consider a kind of posets composed of small blocks.
Let $P,Q$ be two posets and $R$ be a subset of the Cartesian product
$P\times Q$. The \emph{partially ordinal sum} (or \emph{$R$-plus}, for short) of $P$ and $Q$ with respect to $R$
is the poset $P \oplus_R Q$ defined on the disjoint union of $P$ and
$Q$ and partially ordered by $x \leq y$ in $P \oplus_R Q$ if and
only if
\begin{itemize}
\item[(a)] $x,y \in P$ and $x \leq_P y$, or
\item[(b)] $x,y\in Q$ and $x \leq_Q y$, or
\item[(c)] $x \in P, y \in Q$ and there exists $(x',y') \in R$ such that $x \le_P x'$ and
$y' \le_Q y$.
\end{itemize}
As special cases, $R$-plus reduces to the direct sum if $R=\emptyset$ and to the ordinal sum if $R=P \times Q$, respectively.
\begin{exam}\label{exam-R-plus}
Figure~\ref{R-plus} gives the $R$-plus of $P$ and $Q$, where $P=\{1,2\}$ is an anti-chain, $Q=\{3,4\}$ is a chain, and $R=\{(1,4), (2,3), (2,4)\}$.
\end{exam}
\begin{figure}[ht]
\centering
\begin{picture}(300,50)(0,0)
\put(0,23){\small$P=$}
\multiput(30,5)(0,40){2}{\circle*{4}}
\put(32,0){\tiny$1$}
\put(32,47){\tiny$2$}
\put(55,23){\small$Q=$}
\put(85,25){\line(1,0){40}}
\multiput(85,25)(40,0){2}{\circle*{4}}
\put(80,18){\tiny$3$}
\put(126,18){\tiny$4$}
\put(145,23){\small$P \oplus_R Q =$}
\multiput(210,5)(0,40){2}{\circle*{4}}
\multiput(250,25)(40,0){2}{\circle*{4}}
\put(210,5){\line(4,1){80}}
\put(210,45){\line(2,-1){40}}
\put(250,25){\line(1,0){40}}
\put(212,0){\tiny$1$}
\put(212,47){\tiny$2$}
\put(251,28){\tiny$3$}
\put(291,28){\tiny$4$}
\end{picture}
\caption{The partially ordinal sum of $P$ and $Q$ w.r.t.
$R$. \label{R-plus}}
\end{figure}
It is easy to check that the partially ordinal sum is associative up to isomorphic. Therefore we can naturally extend the definition of partially ordinal sum of two posets to several posets.
In particular, we denote by $P_R^n$ the partially ordinal sum $P \oplus_R P \oplus_R \cdots \oplus_R P$ of
$n$ copies of $P$ w.r.t. $R$.
The sequence $\{f_{P_R^n}({\mathbf x})\}_{n \ge 1}$ satisfies a kind of recurrence relation which is given as follows. We say a sequence $\{f_n({\mathbf x})\}_{n\ge1}$ of functions is \emph{substituted recursive} if there are finitely many sequences
\[
\{f_n^{(0)}({\mathbf x})\}_{n \ge 1},\ \{f_n^{(1)}({\mathbf x})\}_{n \ge 1},\ \ldots,\ \{f_n^{(I)}({\mathbf x})\}_{n \ge 1}
\]
such that
$f_n^{(0)}({\mathbf x}) = f_n({\mathbf x})$ and for $i=0,1,\ldots,I$,
\begin{equation} \label{eq-subrec}
f_n^{(i)}({\mathbf x}) = \sum _{j=0}^I \sum _{k=0}^{K} r_{i,j,k}({\mathbf x}) f_{n-1}^{(j)}({\mathbf y}^{(j,k)}),
\end{equation}
where $r_{i,j,k}({\mathbf x})$ are rational functions and each component of the variable vector ${\mathbf y}^{(j,k)}$ is a monomial in $x_1,\ldots,x_n$.
For example, suppose that
\[
f_n(x_1,\ldots,x_n) = \frac{1}{1-x_1} g_{n-1}(x_2, \ldots,x_n),
\]
and
\[
g_n(x_1,\ldots,x_n) = \frac{g_{n-1}(x_1x_2, x_3, \ldots,x_n)}{1-x_1x_2} - \frac{g_{n-1}(x_1, x_2x_3, x_4, \ldots, x_n)}{1-x_1x_3}.
\]
Then both $\{f_n({\mathbf x})\}_{n \ge 1}$ and $\{g_n({\mathbf x})\}_{n \ge 1}$ are substituted recursive.
To compute $f_{P_R^n}({\mathbf x})$, we consider the more general posets
\begin{equation}\label{eq-Xn}
X_n= A \oplus_{R_1} P_{R}^n \oplus_{R_2} B,
\end{equation}
where $A,B$ are posets and $R_1 \subseteq A \times P, R_2 \subseteq P \times B$.
\begin{theo}
\label{theo-subrec}
Let $X_n$ be given as in \eqref{eq-Xn}.
Then the sequence $\{f_{X_n}({\mathbf x})\}_{n \ge 1}$ of generating functions of $P$-partitions of
$X_n$ is substituted recursive.
\end{theo}
\noindent {\it Proof.}
Let ${\cal S}$ denote the set of all pairs $(C,R')$ such that $C$ is a chain, $R' \subseteq C\times P$, and none of the elements of $C$ in $C \oplus_{R'} P$ is removable. Since each element of $C$ is not removable, it must be covered by a certain element of $P$. Moreover, two distinct elements of $C$ can not be covered by the same element in $P$. This implies that the cardinality of $C$ is less than that of $P$. Therefore, ${\cal S}$ is a finite set.
Now we apply the deletion and the partially linear extension to the elements of $A \oplus_{R_1} P$ in $X_n$ whenever possible. As shown in the proof of Theorem~\ref{theo-transformations}, we eventually arrive at posets of the form $C \oplus_{R'} P_R^{n-1} \oplus_{R_2} B$ with $(C,R') \in {\cal S}$. Moreover, we have
\[
f_{X_n}({\mathbf x}) = \sum_{(C,R') \in \cal S} r(C,R',{\mathbf x}) f_{C \oplus_{R'} P_R^{n-1} \oplus_{R_2} B}({\mathbf y}^{(C,R')}),
\]
where $r(C,R',{\mathbf x})$ are rational functions of ${\mathbf x}$ depending on $C$ and $R'$ and each component of the variable vector ${\mathbf y}^{(C,R')}$ is a monomial in ${\mathbf x}$. By a similar discussion, for each $(C,R') \in {\cal S}$ we have
\[
f_{C \oplus_{R'} P_R^{n} \oplus_{R_2} B}({\mathbf x}) = \sum_{(C',R'') \in \cal S} r'(C,R', C',R'',{\mathbf x}) f_{C' \oplus_{R''} P_R^{n-1} \oplus_{R_2} B}({\mathbf y}^{(C,R',C',R'')}),
\]
where $r'(C,R',C',R'',{\mathbf x})$ are rational functions of ${\mathbf x}$ that depend on $C,R',C'$ and $R''$. This completes the proof. \hfill \rule{4pt}{7pt}
Note that the proof of Theorem \ref{theo-subrec} provides an algorithm for generating substituted
recurrence relations of $f_{X_n}({\mathbf x})$. The corresponding {\tt Maple} package is available at
{\tt http://www.combinatorics.net.cn/homepage/hou/}.
\section{Some Examples}\label{sec-examples}
In this section, we present some examples to illustrate our approach to the computation of $f_{X_n}({\mathbf x})$.
We begin with an introductory example, i.e., the $3$-rowed plane partition introduced by Souza \cite{Souza}. Then we provide some more examples, including the zigzag posets, the 2-rowed posets with double diagonals and the multi-cube posets.
In these examples, we consider the $q$-generating function $f_P(q)$ obtained from $f_P({\mathbf x})$ by setting all variables $x_i$ equal the indeterminant $q$. For brevity, we omit some variables equalling $q$ and write $f(x_1,x_2,\ldots,x_k)$ instead of $f(x_1,x_2,\ldots,x_k,q,q,\ldots,q)$. We also adopt the standard notation
\begin{eqnarray*}
(a;q)_n &:=& (1-a)(1-aq)\cdots(1-aq^{n-1}),\\
(a;q)_\infty &:=& \prod_{n=0}^\infty (1-aq^n).
\end{eqnarray*}
\subsection{An Introductory Example}
D'Souza\cite{Souza} introduced the $3$-rowed plane partition whose corresponding poset $P_n$ is given by Figure \ref{Fig-exam-3-rowed}.
He failed to find out recurrence relations of the generating function $f_{P_n}({\mathbf x})$.
Our approach gives substituted recurrence relations of $f_{P_n}({\mathbf x})$.
\begin{figure}[ht]
\centering
\begin{picture}(170,85)(0,0)
\multiput(10,40)(30,0){3}{\circle*{4}}
\multiput(40,10)(30,0){3}{\circle*{4}}
\multiput(40,70)(30,0){3}{\circle*{4}}
\multiput(10,40)(30,0){3}{\line(1,1){30}}
\multiput(10,40)(30,0){3}{\line(1,-1){30}}
\multiput(40,10)(0,60){2}{\line(1,0){70}}
\multiput(105,40)(6,0){3}{\circle*{2}}
\put(130,40){\circle*{4}}
\put(130,40){\line(1,1){30}}
\put(130,40){\line(1,-1){30}}
\multiput(160,10)(0,60){2}{\circle*{4}}
\put(160,10){\line(-1,0){10}}
\put(160,70){\line(-1,0){10}}
\put(2,37){\tiny$1$}
\put(39,0){\tiny$2$}
\put(39,74){\tiny$3$}
\put(32,37){\tiny$4$}
\put(69,0){\tiny$5$}
\put(69,74){\tiny$6$}
\put(62,37){\tiny$7$}
\put(97,0){\tiny$8$}
\put(99,74){\tiny$9$}
\put(135,38){\tiny$3n-2$}
\put(148,2){\tiny$3n-1$}
\put(158,74){\tiny$3n$}
\end{picture}
\caption{The graphical representation of the $3$-rowed poset $P_n$.}\label{Fig-exam-3-rowed}
\end{figure}
It is easy to see that $P_n = P_R^n$, where
$P$ is the poset on $\{1,2,3\}$ with $1 \lessdot 2$ and $1 \lessdot 3$, and $R=\{(2,2),(3,3)\}$.
By deleting the removable elements $2$ and $3$ of
$P$, we reduce the poset $P_n$ to $
Q_{n-1} = {\bf 1} \oplus_{R'} P_R^{n-1}$,
where ${\bf 1}$ is the poset with only one element and $R'=\{(1,2),(1,3)\} \subset {\bf 1} \times P$. See Figure~\ref{Fig-exam-3-rowed-absorb} for a demonstration.
According to Theorem \ref{theo-absorb}, we derive that
\begin{multline}
f_{P_n}(x_1,x_2,x_3) = \frac{1}{(1-x_2)(1-x_3)}\\[7pt] \times \big(
f_{Q_{n-1}}(x_1,q,qx_2,qx_3) -x_2 f_{Q_{n-1}}(x_1x_2,q,q,qx_3)\\
-x_3 f_{Q_{n-1}}(x_1 x_3,q,qx_2,q) + x_2x_3f_{Q_{n-1}}(x_1x_2 x_3,q,q,q) \big).
\label{eq-3-rowed-subrec1}
\end{multline}
\begin{figure}[ht]
\centering
\begin{picture}(270,50)(0,0)
\multiput(5,25)(20,0){3}{\circle*{2}}
\put(25,5){\circle{3}}
\multiput(45,5)(20,0){2}{\circle*{2}}
\multiput(25,45)(20,0){3}{\circle*{2}}
\multiput(5,25)(20,0){3}{\line(1,1){20}}
\multiput(5,25)(20,0){3}{\line(1,-1){20}}
\multiput(25,5)(0,40){2}{\line(1,0){47}}
\multiput(60,25)(6,0){3}{\circle*{1}}
\put(80,22){$\rightarrow$}
\multiput(105,25)(20,0){3}{\circle*{2}}
\multiput(145,5)(20,0){2}{\circle*{2}}
\put(125,45){\circle{3}}
\multiput(145,45)(20,0){2}{\circle*{2}}
\multiput(105,25)(20,0){3}{\line(1,1){20}}
\multiput(125,25)(20,0){2}{\line(1,-1){20}}
\put(105,25){\line(2,-1){40}}
\put(145,5){\line(1,0){27}}
\put(125,45){\line(1,0){47}}
\multiput(160,25)(6,0){3}{\circle*{1}}
\put(180,22){$\rightarrow$}
\multiput(205,25)(20,0){3}{\circle*{2}}
\multiput(245,5)(20,0){2}{\circle*{2}}
\multiput(245,45)(20,0){2}{\circle*{2}}
\multiput(225,25)(20,0){2}{\line(1,1){20}}
\multiput(225,25)(20,0){2}{\line(1,-1){20}}
\put(205,25){\line(2,-1){40}}
\put(205,25){\line(2,1){40}}
\put(245,5){\line(1,0){27}}
\put(245,45){\line(1,0){27}}
\multiput(260,25)(6,0){3}{\circle*{1}}
\end{picture}
\caption{The transformation from $P_n$ to $Q_{n-1}$.}\label{Fig-exam-3-rowed-absorb}
\end{figure}
Now let us consider $Q_n={\bf 1} \oplus_{R'} P_R^n$. It is readily to see that the unique element of ${\bf 1}$ and the minimal element of $P$ are not comparable. Thus we can apply PLE to the anti-chain consisting of these two elements, as shown in Figure~\ref{Fig-exam-3-rowed-PLE}.
\begin{figure}[ht]
\centering
\begin{picture}(320,50)(0,0)
\multiput(5,25)(20,0){2}{\circle{3}}
\put(45,25){\circle*{2}}
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\multiput(45,45)(20,0){2}{\circle*{2}}
\multiput(25,25)(20,0){2}{\line(1,1){20}}
\multiput(25,25)(20,0){2}{\line(1,-1){20}}
\multiput(45,5)(0,40){2}{\line(1,0){27}}
\put(5,25){\line(2,-1){40}}
\put(5,25){\line(2,1){40}}
\multiput(60,25)(4,0){3}{\circle*{1}}
\put(0,20){\tiny$a$}
\put(20,20){\tiny$b$}
\put(74,22){$=$}
\multiput(95,25)(20,0){2}{\circle{3}}
\put(135,25){\circle*{2}}
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\multiput(115,25)(20,0){2}{\line(1,-1){20}}
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\put(95,25){\line(1,0){20}}
\multiput(150,25)(4,0){3}{\circle*{1}}
\put(90,17){\tiny$b$}
\put(107,17){\tiny$\{a\}$}
\put(160,23){ \tiny$+$}
\multiput(175,25)(20,0){2}{\circle{3}}
\put(215,25){\circle*{2}}
\multiput(215,5)(20,0){2}{\circle*{2}}
\multiput(215,45)(20,0){2}{\circle*{2}}
\multiput(195,25)(20,0){2}{\line(1,1){20}}
\multiput(195,25)(20,0){2}{\line(1,-1){20}}
\multiput(215,5)(0,40){2}{\line(1,0){27}}
\put(175,25){\line(1,0){20}}
\multiput(230,25)(4,0){3}{\circle*{1}}
\put(170,17){\tiny$a$}
\put(187,17){\tiny$\{b\}$}
\put(240,23){ \tiny$-$}
\put(270,25){\circle{3}} \put(290,25){\circle*{2}}
\multiput(290,45)(20,0){2}{\circle*{2}}
\multiput(290,5)(20,0){2}{\circle*{2}}
\multiput(270,25)(20,0){2}{\line(1,1){20}}
\multiput(270,25)(20,0){2}{\line(1,-1){20}}
\multiput(290,5)(0,40){2}{\line(1,0){27}}
\multiput(305,25)(4,0){3}{\circle*{1}}
\put(252,17){\tiny$\{a,b\}$}
\end{picture}
\caption{The PLE transformation of $Q_n$.}\label{Fig-exam-3-rowed-PLE}
\end{figure}
After further deletions, all the posets generated by the PLE transformation reduce to $Q_{n-1}$. We thus obtain the recurrence relation
\begin{multline}
f_{Q_n}(x_1,x_2,x_3,x_4) = \frac{1-x_1x_2}{(1-x_1)(1-x_2)(1-x_3)(1-x_4)} \\[7pt]
\times \big( f_{Q_{n-1}}(x_1 x_2,q, qx_3,qx_4) -x_3f_{Q_{n-1}}(x_1x_2 x_3,q,q,qx_4) \\[7pt]
-x_4f_{Q_{n-1}}(x_1 x_2 x_4, q,qx_3,q)+x_3x_4f_{Q_{n-1}}(x_1x_2 x_3x_4,q,q,q) \big).
\label{eq-3-rowed-subrec2}
\end{multline}
By the recurrence relations \eqref{eq-3-rowed-subrec1} and \eqref{eq-3-rowed-subrec2} and the initial condition
\[
f_{Q_1}(x_1,x_2,x_3,x_4) = \frac{1-x_1^2x_2^2x_3x_4}{(1-x_1)(1-x_2)(1-x_1x_2x_3)(1-x_1x_2x_4)(1-x_1x_2x_3x_4)},
\]
we can compute $f_{P_n}(q)$ recursively.
\subsection{More Examples}
In this subsection, we give three more examples: the zigzag posets,
the $2$-rowed posets with double diagonals and the multi-cube posets.
\begin{exam}
Let $P=\{1,2\}$ be a chain with $2 \lessdot 1$ and let $R=\{(2,1)\}$.
The zig-zag poset of length $n$ is given by $Z_n = P_R^n$. We have
\[
f_{Z_n}(x_1,x_2) = \frac{f_{Z_{n-1}}(qx_2,q)}{(1-x_1)(1-x_2)}
-\frac{x_1 f_{Z_{n-1}}(qx_1x_2,q)}{(1-x_1)(1-x_1x_2)}.
\]
The initial condition is given by
\[
f_{Z_1}(x_1,x_2) = \frac{1}{(1-x_2)(1-x_1x_2)}.
\]
\end{exam}
Note that the $P$-partitions of $Z_n$ is exactly the up-down sequences defined by Carlitz \cite{Carlitz1973}.
\begin{exam}\label{exam-2-rowed-doublediagonal}
Let $P_n$ be the $2$-rowed poset with double diagonals depicted in Figure \ref{Fig-exam-2-rowed-doublediagonal}.
Then we have
\begin{equation}\label{eq-2-rowed}
f_{P_n}(q) = \frac{(-q^2;q^2)_{n-1}}{(q;q)_{2n}}.
\end{equation}
\end{exam}
\begin{figure}[ht]
\centering
\begin{picture}(225,60)(0,0)
\put(10,30){\circle*{4}}
\put(10,30){\line(1,-1){20}}
\put(10,30){\line(1,1){20}}
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\multiput(30,50)(40,0){3}{\circle*{4}}
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\multiput(30,10)(40,0){2}{\line(1,1){40}}
\multiput(70,10)(40,0){2}{\line(-1,1){40}}
\multiput(125,30)(6,0){3}{\circle*{2}}
\multiput(150,10)(40,0){2}{\circle*{4}}
\multiput(150,50)(40,0){2}{\circle*{4}}
\multiput(143,10)(0,40){2}{\line(1,0){47}}
\put(150,10){\line(1,1){40}}
\put(190,10){\line(-1,1){40}}
\put(190,10){\line(1,1){20}}
\put(210,30){\circle*{4}}
\put(210,30){\line(-1,1){20}}
\put(0,28){\tiny$1$}
\put(27,1){\tiny$2$}
\put(27,53){\tiny$3$}
\put(67,1){\tiny$4$}
\put(67,53){\tiny$5$}
\put(107,1){\tiny$6$}
\put(107,53){\tiny$7$}
\put(140,2){\tiny$2n-4$}
\put(140,53){\tiny$2n-3$}
\put(180,2){\tiny$2n-2$}
\put(180,53){\tiny$2n-1$}
\put(215,28){\tiny$2n$}
\end{picture}
\caption{The $2$-rowed poset with double diagonals.}\label{Fig-exam-2-rowed-doublediagonal}
\end{figure}
Davis, Souza, Lee and Savage \cite{Davis-06} used the
``digraph method'' to derive formulae \eqref{eq-2-rowed}. Using the inclusion-exclusion
principle, Gao, Hou and Xin \cite{GHX-07} obtained the same generating function. Our approach leads to a recurrence relation as follows
\begin{eqnarray*}
&&f_{P_n}(x_1,x_2,x_3,\ldots,x_{2n})\\[5pt]
=&&\frac{1-x_1^2x_2x_3}{(1-x_1)(1-x_1x_2)(1-x_1x_3)}
f_{P_{n-1}}(x_1x_2x_3,x_4,x_5,\ldots,x_{2n}).
\end{eqnarray*}
Note further that $P_2$ is the Diamond poset
$D$ given in Figure~\ref{Fig-diamond}. By iterating the recurrence relation and substituting all $x_i$ with $q$, we arrive at \eqref{eq-2-rowed}.
\begin{exam}
Let $D=\{1,2,3,4\}$ be the Diamond poset shown as in Figure~\ref{Fig-diamond} and
\[
R=\{(1,1), (2,2), (3,3), (4,4) \},
\]
The $n$-th multi-cube poset $C_n$ is defined by $C_n = D_R^n$, as shown in Figure \ref{Fig-exam-Cubics}.
Using the substituted recurrence relations, we compute $f_{C_n}(q)$ for $n \le 6$.
For example,
\[
f_{C_6}(q) = \frac{q^{192}+2q^{190}+\cdots+40660110q^{96}+\cdots+2q^2+1}{(q;q)_{24}}.
\]
\end{exam}
\begin{figure}[ht]
\centering
\begin{picture}(195,80)(0,0)
\multiput(30,10)(60,0){3}{\circle*{4}}
\multiput(50,30)(60,0){3}{\circle*{4}}
\multiput(10,50)(60,0){3}{\circle*{4}}
\multiput(30,70)(60,0){3}{\circle*{4}}
\multiput(30,10)(20,20){2}{\line(1,0){140}}
\multiput(10,50)(20,20){2}{\line(1,0){140}}
\multiput(30,10)(60,0){3}{\line(1,1){20}}
\multiput(10,50)(60,0){3}{\line(1,1){20}}
\multiput(30,10)(60,0){3}{\line(-1,2){20}}
\multiput(50,30)(60,0){3}{\line(-1,2){20}}
\multiput(180,50)(6,0){3}{\circle*{2}}
\put(3,42){\tiny$1$}
\put(32,2){\tiny$2$}
\put(23,73){\tiny$3$}
\put(51,22){\tiny$4$}
\put(63,42){\tiny$5$}
\put(92,2){\tiny$6$}
\put(83,73){\tiny$7$}
\put(111,22){\tiny$8$}
\put(123,42){\tiny$9$}
\put(150,2){\tiny$10$}
\put(143,73){\tiny$11$}
\put(171,22){\tiny$12$}
\end{picture}
\caption{The graphical representation of multi-cube posets.}\label{Fig-exam-Cubics}
\end{figure}
MacMahon \cite[Section $7$]{MacMahon-12} first considered the $P$-partitions of the poset $C_2$.
Under the framework of partition analysis, Andrews, Paule and Rises \cite{Andre01-c} computed the
$f_{C_2}(q)$ using the {\tt Omega} package. When $n \ge 3$, the computation of $f_{C_n}(q)$ seems not feasible by using {\tt Omega}.
\vskip 15pt
\noindent\textbf{Acknowledgments.}
This work was partially done during the second author's visiting at
RISC-Linz. He would like to thank Peter Paule for his invitation.
This work was supported by the PCSIRT project of the Ministry of
Education and the National Science Foundation of China.
| {
"timestamp": "2012-07-03T02:03:54",
"yymm": "1111",
"arxiv_id": "1111.0245",
"language": "en",
"url": "https://arxiv.org/abs/1111.0245",
"abstract": "We present a method of computing the generating function $f_P(\\x)$ of $P$-partitions of a poset $P$. The idea is to introduce two kinds of transformations on posets and compute $f_P(\\x)$ by recursively applying these transformations. As an application, we consider the partially ordinal sum $P_n$ of $n$ copies of a given poset, which generalizes both the direct sum and the ordinal sum. We show that the sequence $\\{f_{P_n}(\\x)\\}_{n\\ge 1}$ satisfies a finite system of recurrence relations with respect to $n$. We illustrate the method by several examples, including a kind of 3-rowed posets and the multi-cube posets.",
"subjects": "Combinatorics (math.CO); Number Theory (math.NT)",
"title": "Partially Ordinal Sums and $P$-partitions",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808724687407,
"lm_q2_score": 0.7217432062975979,
"lm_q1q2_score": 0.7075110599677956
} |
https://arxiv.org/abs/0908.4356 | Limit laws of transient excited random walks on integers | We consider excited random walks (ERWs) on integers with a bounded number of i.i.d. cookies per site without the non-negativity assumption on the drifts induced by the cookies. Kosygina and Zerner [KZ08] have shown that when the total expected drift per site, delta, is larger than 1 then ERW is transient to the right and, moreover, for delta>4 under the averaged measure it obeys the Central Limit Theorem. We show that when delta in (2,4] the limiting behavior of an appropriately centered and scaled excited random walk under the averaged measure is described by a strictly stable law with parameter delta/2. Our method also extends the results obtained by Basdevant and Singh [BS08b] for delta in (1,2] under the non-negativity assumption to the setting which allows both positive and negative cookies. | \section{Introduction and main results}
Excited random walk (ERW) on $\mathbb{Z}^d$ was introduced by Benjamini and
Wilson in \cite{BW03}. They proposed to modify the nearest neighbor
simple symmetric random walk by giving it a positive drift
(``excitation'') in the first coordinate direction upon reaching a
previously unvisited site. If the site had been visited before, then the
walk made unbiased jumps to one of its nearest neighbor sites. See
\cite{Ko03}, \cite{BR07}, \cite{HH08} and references therein for
further results about this particular model.
Zerner (\cite{Ze05}, \cite{Ze06}) generalized excited random walks by
allowing to modify the transition probabilities at each site not just
once but any number of times and, moreover, choosing them according to
some probability distribution. He obtained the criteria for recurrence
and transience and the law of large numbers for i.i.d.\ environments
on $\mathbb{Z}^d$ and strips and also for general stationary ergodic
environments on $\mathbb{Z}$. It turned out that this generalized model had
interesting behavior even for $d=1$, and this case was further studied
in \cite{MPV06}, \cite{BS08a}, \cite{BS08b}.
Results obtained in all these works rely on the assumption that
projections of all possible drifts on some fixed direction are
non-negative. In fact, the branching processes framework introduced in
\cite{KKS75} for random walks in random environment ($d=1$) and
employed in \cite{BS08a}, \cite{BS08b} for excited random walks,
does not depend on the positivity
assumption, and it seems natural to use this approach for extending the
analysis to environments which allow both positive and negative
drifts. This was done in \cite{KZ08}, where the authors discussed
recurrence and transience, laws of large numbers, positive speed, and
the averaged central limit theorem for multi-excited random walks on
$\mathbb{Z}$ in i.i.d.\ environments with bounded number of ``excitations''
per site. We postpone further discussion of known results for $d=1$
and turn to a precise description of the model considered in this
paper.
Given an arbitrary positive integer $M$ let
\begin{align*}
\Omega_M:=\big\{((\omega_z(i))_{i\in\mathbb{N}})_{z\in\mathbb{Z}}
\mid \, \omega_z(i)\in[0,1],\ &\text{for }
i\in\{1,2,\dots,M\}\\
\text{\ and}\
\omega_z(i)=1/2,\ &\text{for } i>M,\ z\in\mathbb{Z}\big\}.
\end{align*}
An element of $\Omega_M$ is called a cookie environment. For each
$z\in\mathbb{Z}$, the sequence $\{\omega_z(i)\}_{i\in\mathbb{N}}$ can be thought of
as a pile of cookies at site $z$, and $\omega_z(i)$ is referred to as
``the $i$-th cookie at $z$''. The number $\omega_z(i)$ is equal to the
transition probability from $z$ to $z+1$ of a nearest-neighbor random
walk upon the $i$-th visit to $z$. If $\omega_z(i)>1/2$ (resp.
$\omega_z(i)<1/2$) the corresponding cookie will be called positive
(resp.\ negative), $\omega_z(i)=1/2$ will correspond to a ``placebo''
cookie or, equivalently, the absence of an effective $i$-th cookie at
site $z$.
Let $\mathbb{P}$ be a probability measure on $\Omega_M$, which satisfies the
following two conditions:
\begin{itemize}
\item [(A1)] Independence: the sequence
$(\omega_z(\cdot))_{z\in\mathbb{Z}}$ is i.i.d.\ under $\mathbb{P}$.
\item [(A2)] Non-degeneracy:
\[ \mathbb{E}\left[\prod_{i=1}^M
\omega_0(i)\right]>0\ \text{ and }\ \mathbb{E}\left[\prod_{i=1}^M
(1-\omega_0(i))\right]>0.\]
\end{itemize}
Notice that we do not make any independence assumptions on the cookies
at the same site.
It will be convenient to define our ERW model using a
coin-toss construction. Let $(\Sigma,\mathcal{F})$ be some
measurable space equipped with a family of probability measures
$P_{x,\omega},\ x\in\mathbb{Z},\ \omega\in\Omega_M$, such that for each
choice of $x\in\mathbb{Z}$ and $\omega\in\Omega_M$ we have $\pm 1$-valued
random variables $B_{i}^{(z)},\ z\in\mathbb{Z},\ i\ge 1,$ which are
independent under $P_{x,\omega}$ with distribution given by
\begin{equation}\label{B}
P_{x,\omega}(B_{i}^{(z)}=1)=\omega_z(i)\quad\mbox{ and }\quad
P_{x,\omega}(B_{i}^{(z)}=-1)=1-\omega_z(i).
\end{equation}
Let $X_0$ be a random variable on $(\Sigma,\mathcal{F},P_{x,\omega})$ such that
$P_{x,\omega}(X_0=x)=1$. Then an ERW starting at
$x\in\mathbb{Z}$ in the environment $\omega$, $X:=\{X_n\}_{n\ge 0}$, can be defined on
the
probability space $(\Sigma,\mathcal{F},P_{x,\omega})$ by the relation
\begin{eqnarray}
X_{n+1}&:=&X_n+B_{\#\{r\le n\mid X_r=X_n\}}^{(X_n)},\quad n\ge 0.\label{x1}
\end{eqnarray}
Informally speaking, upon each visit to a site the walker eats a
cookie and makes one step to the right or to the left with
probabilities prescribed by this cookie. Since $\omega_z(i)=1/2$ for
all $i>M$, the walker will make unbiased steps from $z$ starting from
the $(M+1)$-th visit to $z$.
Events $\{B_{i}^{(z)}=1\}$, $i\in\mathbb{N}$, $z\in\mathbb{Z}$, will be referred to
as ``successes'' and events $\{B_{i}^{(z)}=-1\}$ will be called
``failures''.
The consumption of a cookie $\omega_z(i)$ induces a drift of size
$2\omega_z(i)-1$ with respect to $P_{x,\omega}$. Summing up over all
cookies at one site and taking the expectation with respect to $\mathbb{P}$
gives the parameter
\begin{equation}
\label{D}
\delta\ :=\ \mathbb{E}\Bigg[\sum_{i\ge 1}(2\omega_0(i)-1)\Bigg]\
=\ \mathbb{E}\left[\sum_{i=1}^M(2\omega_0(i)-1)\right],
\end{equation}
which we call the \textit{average total drift per site}. It
plays a key role in the classification of the asymptotic behavior of
the walk.
We notice that there is an obvious symmetry between positive
and negative cookies: if the environment $(\omega_z)_{z \in \mathbb{Z}}$ is
replaced by $(\omega^\prime_z)_{z \in \mathbb{Z}} $ where $\omega^\prime_z
(i) \ = \ 1- \omega_z (i)$, for all $i\in\mathbb{N},\ z\in\mathbb{Z}$, then
$X':=\{X_n^\prime\}_{n \ge 0}$, the ERW corresponding to the new
environment, satisfies
\begin{equation}
X'\overset{\mathcal{D}}{=} -X,
\end{equation}
where $\overset{\mathcal{D}}{=}$ denotes the equality in distribution.
Thus, it is sufficient to consider, say, only
non-negative $\delta$ (this, of course, allows both negative and
positive cookies), and we shall always assume this to be the
case.
Define the \textit{averaged} measure $P_x$ by setting $P_x(\ \cdot\
)=\mathbb{E}\left(P_{x,\omega}(\ \cdot\ )\right)$. Below we summarize known
results about this model.
\begin{theorem}[\cite{KZ08}] Assume (A1) and (A2).
\begin{itemize}
\item [(i)] If $\delta \in[0,1]$ then $X$ is recurrent, i.e.\ for
$\mathbb{P}$-a.a.\ $\omega$ it returns $P_{0,\omega}$-a.s.\ infinitely
many times to its starting point. If $\delta >1$ then $X$ is
transient to the right, i.e.\ for $\mathbb{P}$-a.a.\
$\omega$, $X_n\to\infty$ as $n\to\infty$ $P_{0,\omega}$-a.s..
\item [(ii)] There is a deterministic $v\in[0,1]$ such that $X$
satisfies for $\mathbb{P}$-a.a.\ $\omega$ the strong law of large numbers,
\begin{equation}\label{speed}
\lim_{n\to\infty}
\frac{X_n}{n}=v\quad P_{0,\omega}\text{-a.s..}
\end{equation}
Moreover, $v=0$ for $\delta\in[0,2]$ and $v>0$ for $\delta >2$.
\item [(iii)] If $\delta>4$ then the sequence
\[B_t^n:=\frac{X_{[tn]}-
[tn] v}{\sqrt{n}},\quad \text{ $t\ge 0$}
\]
converges weakly under $P_0$ to a non-degenerate Brownian
motion with respect to the Skorohod topology on the space of
c\'adl\'ag functions.
\end{itemize}
\end{theorem}
This theorem does not discuss the rate of growth of the ERW when it is
transient but has zero linear speed ($1<\delta\le 2$). It also
leaves open the question about fluctuations when $\delta\le 4$.
The rate of growth of the transient cookie walk with zero linear speed
was studied in \cite{BS08b} for the case of deterministic spatially
homogeneous non-negative cookie environments. For further discussion
we need some notation for the limiting stable distributions
that appear below. Given $\alpha\in (0,2]$ and $b>0$, denote by
$Z_{\alpha,b}$ a random variable (on some probability space) whose
characteristic function is determined by the relation
\begin{equation}
\label{sl}
\log Ee^{iuZ_{\alpha,b}}=
\begin{cases}
-b|u|^\alpha\left(1-i\frac{u}{|u|}\tan
\left(\frac{\pi\alpha}{2}\right)\right),
&\text{if } \alpha\ne
1;\\
-b|u|\left(1+\frac{2i}{\pi}\frac{u}{|u|}\log|u|\right),&\text{if }
\alpha=1.
\end{cases}
\end{equation}
Observe that $Z_{2,b}$ is a centered normal random variable
with variance $2b$. The weak convergence with respect to $P_0$ will
be denoted by $\Rightarrow$.
\begin{theorem}[\cite{BS08b}] \label{BS} Let $\omega_z(i)=p_i\in[1/2,1)$,
$i\in\mathbb{N}$ for all $z\in\mathbb{Z}$, where $p_i=1/2$ for $i>M$,
and $\delta$ be as in (\ref{D}), that is
$\delta=\sum_{i=1}^M(2p_i-1)$.
\begin{itemize}
\item [(i)] If $\delta\in(1,2)$ then there is a positive constant $b$
such that as
$n\to\infty$ \[X_n/n^{\delta/2}\Rightarrow(Z_{\delta/2,b})^{-\delta/2}.\]
\item [(ii)] If $\delta=2$ then $(X_n\log n)/n$ converges in
probability to some constant $c>0$.
\end{itemize}
The above results also hold if $X_n$ is replaced by $\sup_{i\le n}
X_i$ or $\inf_{i\ge n}X_i$.
\end{theorem}
The proof of Theorem~\ref{BS} used the non-negativity of cookies,
though this assumption does not seem to be essential for most parts of
the proof. It is certainly possible that the approach presented in
\cite{BS08b} could yield the same results without the non-negativity
assumption.
The functional central limit theorem for ERWs with $\delta\in(0,1)$ in
stationary ergodic non-negative cookie environments was obtained in
\cite{Do08}. The limiting process is shown to be Brownian motion
perturbed at extrema (see, for example, \cite{Da99}, \cite{CD99}).
The main results of this paper deal with the case when
$\delta\in(2,4]$, though they apply also to
$\delta\in(1,2]$. Moreover, our approach provides an alternative proof
of Theorem~\ref{BS} for general cookie environments that satisfy
conditions (A1) and (A2).
We establish the following theorem.
\begin{theorem}
\label{mainth}
Let $T_n=\inf\{j\ge 0\,|\,X_j=n\}$ and $v$ be the speed of the ERW (see
(\ref{speed})). The following statements hold under the averaged
measure $P_0$.
\begin{itemize}
\item [(i)] If $\delta\in(2,4)$ then there is a constant $b>0$ such
that as $n\to\infty$
\begin{align}
&\frac{T_n-v^{-1}n}{n^{2/\delta}}\Rightarrow Z_{\delta/2,b},
\label{itime}\quad\text{and}
\\&\frac{X_n-vn}{n^{2/\delta}}\Rightarrow
-v^{1+2/\delta}Z_{\delta/2,b}.\label{ix}
\end{align}
\item [(ii)] If $\delta=4$ then there is a constant $b>0$ such that
as $n\to\infty$
\begin{align}
&\frac{T_n-v^{-1}n}{\sqrt{n\log n}}\Rightarrow Z_{2,b}
,\label{iitime}\quad\text{and}
\\&\frac{X_n-vn}{\sqrt{n\log n}}\Rightarrow -v^{3/2}Z_{2,b}.\label{iix}
\end{align}
\end{itemize}
Moreover, (\ref{ix}) and (\ref{iix}) hold if $X_n$ is replaced by
$\sup_{i\le n} X_i$ or $\inf_{i\ge n}X_i$.
\end{theorem}
The paper is organized as follows. In Section~\ref{bp} we recall the
branching processes framework and formulate two statements
(Theorem~\ref{X} and Theorem~\ref{progeny}), from which we later infer
Theorem~\ref{mainth}. Section~\ref{soft} explains the idea of the
proof of Theorem~\ref{progeny} and studies properties of the
approximating diffusion process. In Section~\ref{4} we determine
sufficient conditions for the validity of
Theorem~\ref{progeny}. Section~\ref{5} contains the main technical
lemma (Lemma~\ref{main}). It is followed by three sections, where we
use the results of Section~\ref{5} to verify the sufficient conditions
of Section~\ref{4} and prove Theorem~\ref{X}. The proof of
Theorem~\ref{mainth} is given in Section~\ref{1from2}. The Appendix
contains proofs of several technical results.
\section{Reduction to branching processes}\label{bp}
From now on we assume that $\delta>1$, i.e.\ $X$ is transient to the
right. Suppose that the random walk $\{X_n\}_{n\ge
0}$ starts at $0$. At first, we recall the framework used in
\cite{BS08a}, \cite{BS08b}, and \cite{KZ08}. The main ideas go back
at least to \cite{K73} and \cite{KKS75}.
For $n\in\mathbb{N}$ and $k\le n$ define
\[D_{n,k}=\sum_{j=0}^{T_n-1}\mathbbm{1}_{\{X_j=k,\ X_{j+1}=k-1\}},\]
the number of jumps from $k$ to $k-1$ before time $T_n$.
Then
\begin{equation}\label{Tn}
T_n=n+2\sum_{k\le n}D_{n,k}=n+2\sum_{0\le k\le n}D_{n,k}+2\sum_{k<0}D_{n,k}.
\end{equation}
The last sum is bounded above by the total time spent by $X_n$ below $0$.
Since $X_n$ is transient to the right, this sum is $P_0$-a.s.\ finite,
and, therefore, for any $\alpha>0$
\begin{equation}\label{irrel}
\lim_{n\to\infty}\frac{\sum_{k<0}D_{n,k}}{n^\alpha}=0
\quad\text{$P_0$-a.s..}
\end{equation}
This will allow us to conclude that fluctuations of $T_n$ are
determined by those of $\sum_{0\le k\le n}D_{n,k}$, once we have shown that
the latter are of order $n^{2/\delta}$.
We now consider the ``reversed'' process
$\left(D_{n,n},D_{n,n-1}\dots,D_{n,0}\right)$. Obviously, $D_{n,n}=0$
for every $n\in\mathbb{N}$. Moreover, given
$D_{n,n},D_{n,n-1},\dots,D_{n,k+1}$, we can write
\begin{align*}
D_{n,k}=&\sum_{j=1}^{D_{n,k+1}+1}(\#\ \text{of jumps from $k$ to
$k-1$ between the $(j-1)$-th}\\&\text{
and $j$-th jump from $k$ to $k+1$ before time $T_n$}),\ k=0,1,\dots,n-1.
\end{align*}
Here we used the observation that the number of jumps from $k$ to $k+1$
before time $T_n$ is equal to $D_{n,k+1}+1$ for all $k \le n-1$. The
expression ``between the $0$-th and the $1$-st jump'' above should be
understood as ``prior to the $1$-st jump''.
Fix an $\omega\in\Omega_M$ and denote by $F^{(k)}_m$ the number of
``failures'' in the sequence $B^{(k)}$ (see (\ref{B}) with $z$
replaced by $k$) before the $m$-th ``success''. Then, given
$D_{n,k+1}$,
\[D_{n,k}=F^{(k)}_{D_{n,k+1}+1}.\]
Since the sequences $B^{(k)}$, $k\in\mathbb{Z}$, are i.i.d.\ under $P_0$,
we have that $F^{(k)}_m\overset{\mathcal{D}}{=}F^{(n-k-1)}_m$ and can
conclude that the distribution of
$\left(D_{n,n},D_{n,n-1}\dots,D_{n,0}\right)$ coincides with
that of $(V_0,V_1,\dots,V_n)$, where $V=\{V_k\}_{k\ge 0}$
is a Markov chain defined by
\[V_0=0,\quad V_{k+1}=F^{(k)}_{V_k+1},\quad k\ge 0.\]
For $x\ge 0$ we shall denote by $[x]$ the integer part of $x$ and by
$P_x^V$ the measure associated to the process $V$, which starts with
$[x]$ individuals in the $0$-th generation. Observe that $V$ is a
branching process with the following properties:
(i) $V$ has exactly $1$ immigrant in each generation (the
immigration occurs before the reproduction) and, therefore,
does not get absorbed at $0$.
(ii) The number of offspring of the $m$-th individual in
generation $k$ is given by the number of failures
between the $(m-1)$-th and $m$-th success in the sequence
$B^{(k)}$. In particular, if $V_{k}\ge M$ then the offspring
distribution of each individual after the $M$-th one is
$\mathrm{Geom(1/2)}$ (i.e.\ geometric on $\{0\}\cup\mathbb{N}$ with parameter
$1/2$).
Therefore (here and throughout taking any sum from $k$ to $\ell$ for
$k>\ell$ to be zero) we can write
\begin{equation}
\label{defV}
V_{k+1}=\sum_{m=1}^{M\wedge (V_k+1)}
\zeta^{(k)}_m+\sum_{m=1}^{V_k-M+1}\xi^{(k)}_m,\quad k\ge 0,
\end{equation}
where $\{\xi^{(k)}_m;\,k\ge 0,\,m\ge 1\}$ are i.i.d.\ $\mathrm{Geom(1/2)}$
random variables, vectors $\left(\zeta^{(k)}_1, \zeta^{(k)}_2, \cdots
\zeta^{(k)}_M\right)$, $k\ge 0$, are i.i.d.\ under $P^V_x$ and
independent of $\{\xi^{(k)}_m;\,k\ge 0,\,m\ge 1\}$. For each $k\ge 0$
the random variables $\{\zeta^{(k)}_m\}_{m=1}^M$ are neither
independent nor identically distributed, but, given that for some
$j<M$ \[\sum_{m=1}^j\zeta^{(k)}_m\ge M,\] that is all cookies at site $k$ have
been eaten before the $j$-th jump from $k$ to $(k+1)$, we are left with
$\{\zeta^{(k)}_m\}_{m=j+1}^M$ that are independent Geom(1/2) random variables.
Define
\begin{equation}
\label{ext}
\sigma^V_0=\inf\{j>0\,|\, V_j=0\},
\quad S^V=\sum_{j=0}^{\sigma^V_0-1}V_j.
\end{equation}
Detailed information about the tail behavior of $\sigma^V_0$ and $S^V$
will enable us to use the renewal structure and characterize the tails
of $\sum_{0\le k\le n}D_{n,k}$, and, therefore, of $T_n$ as
$n\to\infty$. We shall show in Section~\ref{1from2} that the following
two statements imply Theorem~\ref{mainth}.
\begin{theorem}\label{X}
Let $\delta>1$. Then
\begin{equation}
\label{ub}
\lim _{n \rightarrow \infty} n^\delta P_0^V(\sigma^V_0>n) = C_1\in(0,\infty).
\end{equation}
\end{theorem}
\begin{theorem}
\label{progeny}
Let $\delta>1$. Then
\begin{equation}
\label{ptails}
\lim_{n\to\infty}n^{\delta/2}P_0^V\left(S^V>n\right)=C_2 \in (0, \infty).
\end{equation}
\end{theorem}
\begin{remark}{\em In fact, a weaker result than (\ref{ub}) is
sufficient for our purpose: there is a constant $B$ such that $n
^\delta P_0^V (\sigma^V_0>n)\leq B $ for all $n\in\mathbb{N}$ (see
condition (A) in Lemma~\ref{4.1}). We also would like to point
out that the limits in (\ref{ub}) and (\ref{ptails}) exist for
every starting point $x\in\mathbb{N}\cup \{0\}$ with $C_1$ and $C_2$
depending on $x$. The proofs simply repeat those for $x=0$.
For the model described in Theorem~\ref{BS}, the convergence
(\ref{ub}) starting from $x\in\mathbb{N}$ is shown in \cite[Proposition
3.1]{BS08b} (for $\delta\in(1,2)$), and (\ref{ptails}) for
$\delta\in(1,2]$ is the content of \cite[Proposition
4.1]{BS08b}. Theorem~\ref{X} can also be derived from the construction
in \cite{KZ08} (see Lemma 17) and \cite{FYK90} (case $\theta<0$). We
use a different approach and obtain both results directly without
using the Laplace transform and Tauberian theorems. }
\end{remark}
We close this section by introducing some additional
notation. For $x\ge 0$ we set
\begin{align}
\tau^V_x&=\inf\{j> 0\,|\, V_j\ge x\};\label{tau}\\
\sigma^V_x&=\inf\{j> 0\,|\,V_j\le x\}.\label{sig}
\end{align}
We shall drop the superscript whenever there is no possibility of
confusion.
Since the random walk $X$ is transient to the right,
$P^V_y(\sigma^V_0<\infty)=1$ for every $y\ge 0$. This implies that
$P^V_y(\sigma^V_x<\infty)=1$ for every $x\in[0,y)$.
Let us remark that when we later deal with a continuous process on
$[0,\infty)$ we shall simply use the first hitting time of $x$ to
record the entrance time in $[x,\infty)$ (or $[0,x]$), given that the
process starts outside of the mentioned interval. We hope that
denoting the hitting time of $x$ for such processes also by $\tau_x$
will not result in ambiguity.
\section{The approximating diffusion process and its
properties}\label{soft}
The bottom-line of our approach is that the main features of
branching process $V$ killed upon reaching $0$ are reasonably well described by
a simple diffusion process.
The parameters of such diffusion processes can be easily computed at the
heuristic level. For $V_k\ge M$, (\ref{defV}) implies that
\begin{equation}\label{diffV}
V_{k+1}-V_k=\sum_{m=1}^M \zeta^{(k)}_m-M+1+\sum_{m=1}^{V_k-M+1}(\xi_m^{(k)}-1).
\end{equation}
By conditioning on the number of successes in the first $M$ tosses it is easy to
compute (see Lemma 3.3 in \cite{BS08a} or Lemma 17 in \cite{KZ08} for details)
that for all $x\ge 0$
\begin{equation}\label{drift}
E^V_x\left(\sum_{m=1}^M \zeta^{(k)}_m-M+1\right)=1-\delta.
\end{equation}
The term $\sum_{m=1}^M \zeta^{(k)}_m-M+1$ is independent of
$\sum_{m=1}^{V_k-M+1}(\xi_m^{(k)}-1)$. When $V_k$ is large, the latter is
approximately normal with mean $0$ and variance essentially equal to $2V_k$.
Therefore, the relevant diffusion should be given by the following
stochastic differential equation:
\begin{equation}\label{sde}
dY_t=(1-\delta)\,dt+\sqrt{2 Y_t}\,dB_t,\quad Y_0=y>0, \quad t\in[0,\tau_0^Y],
\end{equation}
where for $x\ge 0$ we set
\begin{equation}
\label{hit}
\tau_x^Y=\inf\{t\ge 0\,|\,Y_t=x\}.
\end{equation}
Observe that $\tau_0^Y<\infty$ a.s., since $2Y_t$ is a squared Bessel
process of negative dimension $2(1-\delta)$ (for a proof, set $a=0$
and let $b\to\infty$ in part (ii) of Lemma~\ref{elem} below).
The above heuristics are justified by the next lemma.
\begin{lemma} \label{weakcon}
Let $Y=\{Y_t\}_{t\ge 0}$ be the solution of (\ref{sde}).
Fix an arbitrary $\epsilon>0$. For $y\in(\epsilon,\infty )$ let $V_0=[ny]$,
and define
\begin{equation*}
Y^{\epsilon,n}_t = \frac{V_{[nt]\wedge \sigma^V_{\epsilon n}}}{n},\quad
t\in[0,\infty),
\end{equation*}
where $\sigma^V_x$ is given by (\ref{sig}).
Then the sequence of processes $Y^{\epsilon,n}=\{Y^{\epsilon,n}_t\}_{t\ge 0}$
converges in distribution as $n\to\infty$ with respect to the Skorokhod topology
on the space of c\`adl\`ag functions to the stopped diffusion
$Y^\epsilon=\{Y_{t\wedge\tau^Y_\epsilon}\}_{t\ge 0}$, $Y_0=y$.
\end{lemma}
\begin{proof}
We simply apply the (much more general) results of \cite{EK86}. We
first note that our convergence result considers the processes up to
the first entry into $(- \infty ,\epsilon ]$ for $\epsilon > 0$
fixed. So we can choose to modify the rules of evolution for $V$
when $V_k \leq \epsilon n $: we consider the process $(V^{n,\epsilon}_k)_{k
\geq 0}$ where, with the existing notation,
\begin{equation}
V^{n,\epsilon}_0=[ny],\quad V^{n,\epsilon}_{k+1}=\sum_{m=1}^{M}
\zeta^{(k)}_m+\sum_{m=1}^{V^{n,\epsilon}_k \vee
(\epsilon n)-M+1}\xi^{(k)}_m,\quad k\ge 0.
\end{equation}
Then (given the regularity of points for the limit process) it will suffice to
show the convergence of processes
\begin{equation*}
\tilde{Y}^{\epsilon,n}_t = \frac{V^{n,\epsilon}_{[nt]}}{n},\quad t\in[0,\infty),
\end{equation*}
to the solution of the stochastic integral equation
\begin{equation}
dY_t=(1-\delta)\,dt+\sqrt{2 (Y_t \vee \epsilon)}\,dB_t,\quad Y_0=y>0, \quad
t\in[0, \infty ).
\end{equation}
We can now apply Theorem 4.1 of Chapter 7 of \cite{EK86} with $X_n(t) =
\tilde{Y}^{\epsilon,n}_t $. The needed
uniqueness of the martingale problem corresponding to operator
\begin{equation}
Gf \ = \ (x \vee \epsilon ) f^{\prime \prime} + (1- \delta ) f^\prime
\end{equation}
follows from \cite{EK86}, Chapter 5, Section 3 (Theorems 3.6 and 3.7 imply the
distributional uniqueness for solutions of the corresponding stochastic integral
equation, and Proposition 3.1 shows that this implies the uniqueness for
the martingale problem).
\end{proof}
We shall see in a moment that this diffusion has the desired behavior
of the extinction time and of the total area under the path before the
extinction (see Lemma~\ref{difext} and Lemma~\ref{C} below).
Unfortunately, these properties in conjunction with
Lemma~\ref{weakcon} do not automatically imply Theorem~\ref{X} and
Theorem~\ref{progeny}, and work needs to be done to ``transfer'' these
results to the corresponding quantities of the process $V$.
Nevertheless, Lemma~\ref{weakcon} is very helpful when $V$ stays large
as we shall see later.
\medskip
In the rest of this section we state and prove several facts about
$Y$. When we need to specify that the process $Y$ starts at $y$ at
time $0$ we shall write $Y^y$. Again, whenever there is no ambiguity
about which process is being considered we shall drop the superscript
in $\tau^Y_x$ defined in (\ref{hit}).
\begin{lemma}
\label{elem}
Fix $y>0$.
\begin{itemize}
\item [(i)] (Scaling) Let $\widetilde{Y}=\{\widetilde{Y}_t\}_{t\ge 0}$, where
$\widetilde{Y}_t=\dfrac{Y^y_{ty}}{y}$.
Then $\widetilde{Y}\overset{\mathcal{D}}{=}Y^1$.
\item [(ii)] (Hitting probabilities) Let $0\le a<x<b$. Then
\[P_y^{Y}(\tau_a<\tau_b)=\frac{b^\delta-y^\delta}{b^\delta-a^\delta}.\]
\end{itemize}
\end{lemma}
\begin{proof}
Part (i) can be easily checked by It\^o's formula applied to $\widetilde{Y}_t$
or seen from scaling properties of the generator. The proof of part (ii) is
standard once we notice that the process $(Y^y_t)^{\delta}$ stopped upon
reaching the boundary of $[a,b]$ is a martingale. We omit the details.
\end{proof}
\begin{lemma}\label{difext}
Let $Y$ be the diffusion process defined by (\ref{sde}). Then
\[\lim_{x\to\infty}x^\delta P_1^Y(\tau_0>x)=C_3\in (0,\infty).\]
\end{lemma}
\begin{proof}
For every $\epsilon>0$ and for all $x>1/\epsilon$ we have by
Lemma~\ref{elem}
\begin{align*}
x^\delta P^Y_1(\tau_0>x)&\ge x^\delta P^Y_1(\tau_0>x\,|\,\tau_{\epsilon
x}<\tau_0) P^Y_1(\tau_{\epsilon x}<\tau_0) \\& \ge x^\delta
P^Y_{\epsilon x}(\tau_0>x)\left(\epsilon x\right)^{-\delta}=
\epsilon^{-\delta} P^Y_1(\tau_0>\epsilon^{-1})>0.
\end{align*}
This implies that for each $\epsilon>0$
\[\liminf_{x\to\infty}x^\delta P^Y_1(\tau_0>x)\ge \epsilon^{-\delta}
P^Y_1(\tau_0>\epsilon^{-1})>0.
\]
Taking the $\limsup_{\epsilon\to 0}$ in the right-hand side we get
\[\liminf_{x\to\infty}x^\delta P^Y_1(\tau_0>x)\ge \limsup_{\epsilon\to
0} \epsilon^{-\delta} P^Y_1(\tau_0>\epsilon^{-1})=\limsup_{x\to
\infty}x^\delta P^Y_1(\tau_0>x).\] This would immediately imply the
existence of a finite non-zero limit if we could show that
\[\limsup_{x\to \infty}x^\delta
P^Y_1(\tau_0>x)<\infty.\]
This is the content of the next lemma.
\begin{lemma}
\label{difup}
$\displaystyle\limsup_{x\to \infty}x^\delta P^Y_1(\tau_0>x)<\infty.$
\end{lemma}
The proof is very similar to the proof of the discrete version (see
(A) in Lemma~\ref{4.1} and its proof in Section~\ref{px}) and, thus,
is omitted.
\end{proof}
The final result of this section can be viewed as the ``continuous
counterpart'' of Theorem~\ref{progeny}.
It concerns the area under the path of $Y$.
\begin{lemma} \label{C} Let $Y$ be the diffusion process defined by (\ref{sde}).
Then
\[\lim_{y\to\infty}y^\delta P_1^Y\left(\int_0^{\tau_0}Y_t\,dt>y^2\right)
=C_4\in(0,\infty).\]
\end{lemma}
\begin{proof}
The proof uses scaling and follows the same steps as the proof of
Lemma~\ref{difext}. For every $\epsilon>0$ and $y>1/\epsilon$ we
have
\begin{align*}
y^\delta P_1^Y&\left(\int_0^{\tau_0}Y_t\,dt>y^2\right)\ge y^\delta
P_1^Y\left(\int_0^{\tau_0}Y_t\,dt>y^2\,\Big|\,\tau_{\epsilon
y}<\tau_0\right) P_1^Y(\tau_{\epsilon y}<\tau_0)\\ \ge\,&y^\delta
P_{\epsilon y}^Y\left(\int_0^{\tau_0}Y_t\,dt>y^2\right)(\epsilon
y)^{-\delta} =\epsilon^{-\delta}P_{\epsilon
y}^Y\left(\int_0^{\tau_0/(\epsilon y)} Y_{\epsilon ys}\,
ds>\frac{y}{\epsilon}\right) \\=\,&\epsilon^{-\delta}P_{\epsilon
y}^Y\left(\int_0^{\tau_0/(\epsilon y)} \frac{Y_{\epsilon
ys}}{\epsilon y}\, ds>\epsilon^{-2}\right)
=\epsilon^{-\delta} P_1^Y\left(\int_0^{\tau_0}
Y_s\,ds>\epsilon^{-2}\right)>0.
\end{align*}
This calculation, in fact, just shows that \[y^\delta
P_1^Y\left(\int_0^{\tau_0}Y_t\,dt>y^2\right)\] is a non-decreasing
positive function of $y$. Therefore, we only need to prove that it is
bounded as $y\to\infty$. But for $y>1$
\begin{align*}
&y^\delta P_1^Y\left(\int_0^{\tau_0}Y_t\,dt>y^2\right)\\&=
P_1^Y\left(\int_0^{\tau_0}Y_t\,dt>y^2\,\Big|\,
\tau_y<\tau_0\right)+
y^\delta P_1^Y\left(\int_0^{\tau_0}Y_t\,dt>y^2,\,\tau_y>\tau_0\right)
\\[2mm]&\le 1+y^\delta P_1^Y\left(\tau_0>y,\,
\tau_y>\tau_0\right)\le 1+y^\delta P_1^Y\left(\tau_0>y\right).
\end{align*}
An application of Lemma~\ref{difup} finishes the proof.
\end{proof}
\section{Conditions which imply Theorem~\ref{progeny}}\label{4}
We have shown that the diffusion process $Y$ has the desired asymptotic behavior
of the area under the path up to the exit time $\tau^Y_0$.
In this section we give sufficient conditions under which we can
``transfer'' this result to the process $V$ and obtain Theorem~\ref{progeny}.
\begin{lemma} \label{4.1}
Suppose that
\begin{itemize}
\item[(A)] There is a constant $B$ such that $ n ^\delta
P^V_0(\sigma _0 > n)\le B$ for all $n\in\mathbb{N}$;
\item[(B)] For every $\epsilon>0$ \[\lim_{n\to\infty} P_{\epsilon n}^V
\left( \sum_{i=0} ^{\sigma_0-1} V_i>n^2\right) = P_1^Y\left(
\int_0^{\tau_0} Y_t\,dt>\epsilon^{-2}\right);\]
\item[(C)] $\displaystyle\lim_{n\to\infty}n^\delta P_0^V\left(\tau_n<\sigma_0
\right)=C_5$.
\end{itemize}
Then \[\lim_{n\to\infty}n^\delta P_0^V\left(\sum_{i=0} ^{\sigma_0-1}
V_i>n^2\right)=C_4C_5,\] where $C_4$ is the constant from Lemma~\ref{C}.
\end{lemma}
\begin{proof}
Fix an $\epsilon\in(0,1)$ and split the path-space of $V$ into two parts,
the event $H_{n,\epsilon}:=\left\{\tau_{\epsilon n
}^V<\sigma_0^V\right\}$ and its complement, $H_{n,\epsilon}^c$.
First, consider the behavior of the total progeny on the event
$H_{n,\epsilon}^c$. On $H_{n,\epsilon}^c$, the process $V$ stays
below $\epsilon n$ until the time $\sigma_0$. Estimating each
$V_i$ from above by $\epsilon n$ and using (A) we get for all
$n\in\mathbb{N}$
\[n^\delta P_0^V \left(\sum_{i=0} ^{\sigma_0-1} V_i>n^2,\
H_{n,\epsilon}^c\right) \le n^\delta P_0^V(\sigma_0>n/\epsilon)\le
(2\epsilon)^\delta B. \]
Therefore, for all $n\in\mathbb{N}$
\begin{align*}
0\le
n^\delta P_0^V \left(\sum_{i=0} ^{\sigma_0-1} V_i>n^2\right)-n^\delta P_0^V
\left(\sum_{i=0} ^{\sigma_0-1} V_i>n^2,\
H_{n,\epsilon}\right)\le(2\epsilon)^\delta B.
\end{align*}
Hence, we only need to deal with the total progeny on the event
$H_{n,\epsilon}$. The rough idea is that, on $H_{n,\epsilon}$, it
is not unnatural for the total progeny to be of order $n^2$. This means that the
decay of the probability that the total progeny is over $n^2$
comes from the decay of the
probability of $H_{n,\epsilon}$, which is essentially given by
condition (C). This would suffice if we could let $\epsilon=1$ but
we need $\epsilon$ to be small, thus, some scaling is necessary to
proceed with the argument, and this brings into play condition
(B) and the result of Lemma~\ref{C}.
To get a lower bound on $F_n:=n^\delta P_0^V \left(\sum_{i=0} ^{\sigma_0-1}
V_i>n^2,\ H_{n,\epsilon}\right)$ we use monotonicity of $V$
with respect to the initial number of particles, conditions
(B) and (C), and Lemma~\ref{C}:
\begin{multline*}
\lim_{\epsilon\to 0}\liminf_{n\to\infty} F_n
\ge \lim_{\epsilon\to 0}\lim_{n\to\infty}n^\delta
P_0^V(H_{n,\epsilon})P_{\epsilon n}^V
\left(\sum_{i=0} ^{\sigma_0-1} V_i>n^2\right)\\
=C_5\lim_{\epsilon\to 0}\epsilon^{-\delta}P_1^Y\left( \int_0^{\tau_0}
Y_t\,dt>\epsilon^{-2}\right)=C_4C_5.
\end{multline*}
For an upper bound on $F_n$ we shall need two more parameters,
$K\in(1,1/\epsilon)$ and $R>1$. At the end, after taking the limits as
$n\to\infty$ and then $\epsilon\to 0$ we shall let $K\to\infty$ and
$R\to 1$.
\begin{multline*}
n^{-\delta} F_n=
P_0^V \left(\sum_{i=0}^{\tau_{ \epsilon n }-1}V_i+
\sum_{i=\tau_{\epsilon n }}
^{\sigma_0-1}V_i>n^2,\ H_{n,\epsilon}\right)\\
\le P_0^V \left(\sum_{i=\tau_{ \epsilon n }}
^{\sigma_0-1}V_i>n^2(1-K\epsilon),\ \sum_{i=0}^{\tau_{\epsilon n
}-1}V_i\le K\epsilon n^2,\ H_{n,\epsilon}\right)\\+P_0^V
\left(\sum_{i=0}^{\tau_{ \epsilon n
}-1}V_i>K\epsilon n^2,\ H_{n,\epsilon}\right)
\end{multline*}
We bound the first term on the right-hand side by the following sum:
\begin{multline*}
P_0^V
\left(\sum_{i=\tau_{\epsilon n }} ^{\sigma_0-1}V_i>n^2(1-K\epsilon),\
V_{\tau_{\epsilon n }}\le R\epsilon n,\
H_{n,\epsilon}\right)\\+P_0^V
\left(\sum_{i=\tau_{\epsilon n }} ^{\sigma_0-1}V_i>n^2(1-K\epsilon),\
V_{\tau_{\epsilon n }}>R\epsilon n,\
H_{n,\epsilon}\right).
\end{multline*}
Estimating these terms in an obvious way and putting everything back
together we get
\begin{align*}
n^{-\delta} F_n&\le P_{R\epsilon n}^V \left(\sum_{i=0}
^{\sigma_0-1}V_i>n^2(1-K\epsilon)\right)P_0^V(H_{n,\epsilon})
\\&+P_0^V \left(V_{\tau_{\epsilon n }}>R\epsilon n,\
H_{n,\epsilon}\right)+P_0^V \left(\sum_{i=0}^{\tau_{\epsilon n
}-1}V_i>K\epsilon n^2,\ H_{n,\epsilon}\right)\\&=(I)+(II)+(III).
\end{align*}
It only remains to multiply everything by $n^\delta$ and consider the
upper limits.
Term $n^\delta(I)$ gives the upper bound $C_4C_5$ in the same way as
we got a lower bound by sending $n\to\infty$, $\epsilon\to 0$, and
then $R\to 1$ and using easily verified continuity properties of the
relevant distributions. Parameter $K$ disappears when we let $\epsilon\to 0$.
Term $(II)$ is exponentially small in $n$ for fixed $\epsilon$ and $R$
(see Lemma~\ref{overshoot} below), thus $n^\delta (II)$ goes to zero as
$n\to\infty$.
Finally, since $V_i\le \epsilon n$ for all $i<\tau_{\epsilon n}$, we get
\begin{align*}
n^\delta P_0^V \left(\sum_{i=0}^{\tau_{\epsilon n }-1}V_i>K\epsilon n^2,\
H_{n,\epsilon}\right)&\le n^\delta P_0^V\left(\tau_{\epsilon n
}>Kn, H_{n,\epsilon}\right)\\ \le n^\delta
P_0^V\left(\sigma_0>Kn, H_{n,\epsilon}\right) &\le n^\delta
P_0^V(\sigma_0>Kn)\le \frac{2^\delta B}{K^\delta}. \qedhere
\end{align*}
\end{proof}
\section{Main tools}\label{5}
The main result of this section is Lemma~\ref{main}, which is a
discrete analog of Lemma~\ref{elem} (ii).
We start with two technical lemmas. The first one will be used many times
throughout the paper.
\begin{lemma}
\label{overshoot}
There are constants $c_1,c_2>0$ and $N\in\mathbb{N}$ such that for
every $x\ge N$ and $y\ge 0$,
\begin{align}
\sup_{0\le z<x}P_z^V\left(V_{\tau_x}>x+y\,|\,\tau_x<\sigma_0\right)&\le
c_1( e^{-c_2y^2/x} + e^{-c_2y});\label{over} \\ \sup_{x<z< 4x}
P_z^V(V_{\sigma_x\wedge \tau_{4x}}<x-y)&\le c_1
e^{-c_2y^2/x}\label{under}.
\end{align}
\end{lemma}
\noindent This statement is a consequence of the fact that the offspring
distribution of $V$ is essentially geometric. The proof is
given in the Appendix.
\begin{lemma}\label{scale}
Fix $a \in (1,2]$. Consider the process $V$ with $|V_0
-a^n| \le a^{2n/3}$ and let $\gamma=\inf
\{k\ge 0\,|\, V_k \notin (a^{n-1},a^{n+1} ) \}$. Then for all
sufficiently large $n$
\begin{align*}
(&i)\quad P^V\Big(\mathrm{dist}(V_\gamma ,(a^{n-1},a^{n+1})) \ge
a^{2(n-1)/3}\Big) \le \exp(-a^{n/4});\\(i&i)\quad
\left|P^V(V_\gamma\leq a^{n-1}) - \frac{a^\delta}{a^\delta +1}\right| \leq
a^{-n/4}.
\end{align*}
\end{lemma}
\noindent Part (i) is an immediate consequence of Lemma~\ref{overshoot}.
The proof of part (ii) is basic but technical and is given in the Appendix.
\begin{lemma}[Main lemma]\label{main}
For each $a \in (1,2]$ there is an $\ell_0\in\mathbb{N}$ such that if
$\ell,m,u,x\in\mathbb{N}$ satisfy $\ell_0\le\ell <m< u$ and $|x-a^m|\le
a^{2m/3}$ then
\[\frac{h_a^-(m)-1}{h_a^-(u)-1}\le
P_x^V(\sigma_{a^\ell}>\tau_{a^u})\le
\frac{h_a^+(m)-1}{h_a^+(u)-1},\] where
\[h_a^{\pm}(i)=\prod_{r=\ell+1}^i\left(a^\delta\mp a^{-\lambda
r}\right),\quad i>\ell,\] and $\lambda$ is some small positive number not
depending on $\ell$.
\end{lemma}
\begin{remark}\label{rml}
{\em It is to be noted that for fixed $\ell$ there are $K_1(\ell)$
and $K_2(\ell)$ such that
\[K_1(\ell) \le \frac{h^{\pm}_a(i)}{a^{(i-l)\delta}} \leq
K_2(\ell)\quad\text{for all } i >\ell\] and $K_i(\ell)\to 1$ as
$\ell \rightarrow \infty $, $i=1,2$.}
\end{remark}
\begin{proof}[Proof of Lemma \ref{main}] We will show the upper bound
by comparing the process $V$ with another process $\widetilde{V}$,
whose exit probabilities can be estimated by further reduction to an
exit problem for a birth-and-death-like Markov chain.
For $i\in\mathbb{N}$ set $x_i=[a^i+a^{2i/3}]$. By monotonicity, it is
enough to prove the upper bound when the starting point $x$ is equal
to $x_m$. Thus, we set $V_0=x_m$. The comparison will be done in two
steps.
{\em Step 1.} We shall construct a sequence of stopping times $\gamma_i$, $i\ge
0$, and a comparison process $\widetilde{V}=(\widetilde{V}_k)_{k\ge 0}$ with
$x_\ell$ as an absorbing point so that $\widetilde{V}_k\ge V_k$ for all $k$
before the absorption. Let $\gamma_0=0$,
\[\gamma_1=\inf\{k>0\,|\,V_k\not\in(a^{m-1},a^{m+1})\},\quad
\widetilde{V}_k=V_k\ \text{ for }\ k=0,1,\dots,\gamma_1-1,\] and at time
$\gamma_1$ add to $V_{\gamma_1}$ the necessary number of particles to get
\begin{equation*}
\widetilde{V}_{\gamma_1}=
\begin{cases}
x_{m-1},&\text{if } V_{\gamma_1}\le a^{m-1};\\
x_{m+j},&\text{if }
x_{m+j-1}<V_{\gamma_1}\le x_{m+j},\ j\in\mathbb{N}.
\end{cases}
\end{equation*}
Clearly, $\widetilde{V}_{\gamma_1}\ge V_{\gamma_1}$. By construction,
$\widetilde{V}_{\gamma_1}=x_n$ for some $n\ge m-1,\ n\ne m$. If
$\widetilde{V}_{\gamma_1}=x_\ell$, then we stop the
process.
Assume that we have already defined stopping times $\gamma_r$,
$r=0,1,\dots,i$, and the process $\widetilde{V}_k$ for all $k\le
\gamma_i$ so that $\widetilde{V}_{\gamma_i}=x_n$ for some $n>
\ell$. We define $\widetilde{V}_k$ for $k>\gamma_i$ by applying to it
the same branching mechanism as for $V$, namely, (\ref{defV}) with $V$
replaced by $\widetilde{V}$, $k\ge \gamma_i$. Denote by
$\gamma_{i+1}$ the first time after $\gamma_i$ when $\widetilde{V}$
exits the interval $(a^{n-1},a^{n+1})$. At time $\gamma_{i+1}$, if the
process exited through the lower end of the interval then we set
$\widetilde{V}_{\gamma_{i+1}}=x_{n-1}$, if the process exited the
through the upper end we add to $\widetilde{V}$ the minimal number of
particles needed to get $\widetilde{V}_{\gamma_{i+1}}=x_s$ for some
$s>n$. If $\widetilde{V}_{\gamma_{i+1}}=x_\ell$, then we stop the
process. Thus, we obtain a sequence of stopping times $\gamma_i$,
$i\ge 0$, and the desired dominating process $\widetilde{V}$ absorbed
at $x_\ell$ such that $\widetilde{V}_{\gamma_i}\in
\{x_\ell,x_{\ell+1},\dots\}$, $i\ge 0$.
{\em Step 2.} Define a Markov chain $R=(R_j)_{j\ge 0}$ on
$\{\ell,\ell+1,\dots\}$ by setting \[R_j=n\quad \text{if} \quad
\widetilde{V}_{\gamma_j}=x_n,\quad j\ge 0.\] The state $\ell$ is
absorbing. Let $\sigma^R_\ell=\inf\{j\ge 0\,|\, R_j=\ell\}$ and
$\tau^R_u=\inf\{j\ge 0\,|\, R_j\ge u\}$. By
construction, \[P^V_{x_m}(\sigma^V_{a^\ell}>\tau^V_{a^u})\le
P^{\widetilde{V}}_{x_m}(\sigma^{\widetilde{V}}_{x_\ell}>\tau^{\widetilde{V}}_{x_u})=
P^R_m(\sigma^R_\ell>\tau^R_u).\] We shall show that
$(h^+_a(R_j))_{j\ge 0}$ is a supermartingale with respect to the
natural filtration. (We set $h^+_a(\ell)=1$.) The optional stopping
theorem and monotonicity of function $h^+_a$ will immediately imply
the upper bound in the statement of the lemma.
For $i >\ell$ we have
\begin{multline*}
E^R_i\left(h_{a}^{+} (R_{1})\right) = h_{a}^{+} (i-1) P^R_i (R_1 =
i-1) \\+ h_{a}^{+} (i+1) P^R_i (R_1 = i+1) +
\sum_{n =i+2}^\infty h_{a}^{+} (n) P^R_i (R_1=n)
\end{multline*}
By the definition of $h^+_a$ this is less or equal than
\begin{multline}\label{h+}
h_{a}^{+}(i) \bigg[(a^{\delta} - a^{-\lambda i})^{-1}P^R_i (R_1 = i-1)
\\+(a^{\delta} - a^{-\lambda(i+1)}) P^R_i (R_1 =
i+1) + \sum_{n = i+2}^\infty
a^{\delta(n-i)} P^R_i (R_1 = n)\bigg].
\end{multline}
By Lemma~\ref{scale} and Lemma~\ref{overshoot} we have that for
all $i>\ell$, where $\ell$ is chosen sufficiently large,
\begin{align*}
&P^R_i (R_1 = i-1)=\frac{a^\delta}{a^\delta+1}+O(a^{-i/4}),\\
&P^R_i (R_1 = i+1)= \frac{1}{a^\delta+1}+O(a^{-i/4}),
\ \text{and }\\
&P^R_i (R_1 \ge n)\le P^R_{n-2}(R_1 \ge n)=O(\exp(-a^{n/4}))\
\text{for all }n\ge i+2.
\end{align*}
Substituting this into (\ref{h+}) and performing elementary
computations we obtain
\begin{equation*}
E^R_i\left(h_{a}^{+} (R_{1})\right) \le h_{a}^{+}(i)
\bigg[1-\frac{a^{-\lambda
i}}{a^\delta+1}\,\left(a^{-\lambda}-a^{-\delta}\right)
+ O\big(a^{-2\lambda i}\big)\bigg]
\le h^+_a(i),
\end{equation*}
provided that $\lambda\le 1/8$ and $\ell$ (therefore $i$) is
sufficiently large.
For the lower bound we argue in a similar manner, except that now we
choose $x_m=[a^m-a^{2m/3}]+1$, assume that $V_0=x_m$, and construct a
comparison process $\widetilde{V}=(\widetilde{V}_k)_{k\ge 0}$ absorbed
at $x_\ell$ so that $\widetilde{V}_0=V_0$ and $\widetilde{V}_k\le V_k$
for all $k$ before the absorption.
More precisely, we let $\gamma_0=0$ and assume that we have already
defined stopping times $\gamma_r$, $r=0,1,\dots,i$, and the process
$\widetilde{V}_k$ for all $k\le \gamma_i$ so that
$\widetilde{V}_{\gamma_i}=x_n$ for some $n\ne\ell$. We define
$\widetilde{V}_k$ for $k>\gamma_i$ by (\ref{defV}) with $V$ replaced
by $\widetilde{V}$, $k\ge \gamma_i$. Denote by $\gamma_{i+1}$ the
first time after $\gamma_i$ when $\widetilde{V}$ exits the interval
$(a^{n-1},a^{n+1})$. At time $\gamma_{i+1}$, if the process exited
through the upper end of the interval we set
$\widetilde{V}_{\gamma_{i+1}}=x_{n+1}$, if the process exited
through the lower end we reduce the number of particles by
removing the minimal number of particles to ensure that
$\widetilde{V}_{\gamma_{i+1}}=x_s$ for some $s<n$. If
$\widetilde{V}_{\gamma_{i+1}}\le x_\ell$, then we stop the process and redefine
$\widetilde{V}_{\gamma_{i+1}}$ to be $x_\ell$. This
procedure allows us to obtain a sequence of stopping times $\gamma_i$,
$i\ge 0$, and the desired comparison process $\widetilde{V}$
absorbed at $x_\ell$ such that $V_{\gamma_i}\in \{x_\ell,x_{\ell+1},\dots\}$,
$i\ge 0$.
Next, just as in the proof of the upper bound, we construct a Markov
chain $R=(R_j)_{j\ge 0}$ and show that $(h^-_a(R_j))_{j\ge 0}$ is a
submartingale (with $h^-_a(\ell)$ defined to be $1$). The optional
stopping theorem and monotonicity of function $h^-_a$ imply the lower
bound.
\end{proof}
\begin{corollary} \label{boundtau} For each non-negative integer $x$ there
exists a constant $C_6 = C_6(x)$ such that for every $n\in\mathbb{N}$
\begin{equation}\label{bt1}
n^\delta P^V_x( \tau_n < \sigma _0) \leq C_6.
\end{equation}
Moreover, for each $\epsilon>0$ there is a constant $c_3=c_3(\epsilon)$ such
that for all $n\in\mathbb{N}$
\begin{equation}\label{bt2}
P^V_n(\sigma_0 > \tau_{c_3n}) < \epsilon.
\end{equation}
\end{corollary}
\begin{remark}
{\em In fact, (\ref{bt1}) will be substantially improved by
Lemma \ref{triv}.}
\end{remark}
\begin{proof}
We choose arbitrarily $a \in (1,2]$ and an $\ell\ge \ell_0$ as in
Lemma \ref{main} but also such that $a^\ell > x $. We note that it is
sufficient to prove the statement for
$n$ of the form $[a^u]$. We define stopping times $\beta_i$,
$i\in\mathbb{N}$, by
\begin{align*}
\beta _1&= \inf \{k> 0\,|\, V_k\ge a^{\ell+1}\}; \\
\beta _{i+1}& = \inf \{k > \beta_i\,: \,V_k \ge a^{\ell+1} \mbox{
and } \exists
s\in(\beta_i,k)\,|\, V_s \le a^\ell\}.
\end{align*}
Lemma \ref{overshoot} and the monotonicity of $V$ with respect to its
starting point imply that
\begin{multline*}
P^V_x\left(\exists r \in [\beta_i, \beta_{i+1})\,|\, V_r \ge a^u
\,|\,\beta_i < \sigma_0\right) \\
\le \frac{h_a^+(\ell+1)-1}{h_a^+(u)-1} + \sum_{k=\ell+1} ^\infty
P^V_x (V_{\beta_i} \ge a^k + a^{2k/3}\,|\,\beta_i < \sigma_0)\,
\frac{h_a^+(k+1)-1}{h_a^+(u)-1} \\=
\frac{h_a^+(\ell+1)-1}{h_a^+(u)-1} \left(1 + \sum_{k=\ell+1} ^\infty
P^V_x (V_{\beta_i} \ge a^k + a^{2k/3}\,|\,\beta_i < \sigma_0)\,
\frac{h_a^+(k+1)-1}{h_a^+(\ell+1)-1} \right)\\\le
\frac{2(h_a^+(\ell+1)-1)}{h_a^+(u)-1}
\end{multline*}
supposing, as we may, that $\ell$ was fixed sufficiently large. Thus,
\[
a^{u \delta }P^V_x(\sigma_0 > \tau_{a^u}) \le 2a^{u \delta }\,
\frac{(h_a^+(\ell+1)-1 )}{h_a^+(u)-1}\sum_{i=1}^\infty P^V_x(\beta_i <
\sigma_0) .
\]
The bound (\ref{bt1}) now follows from noting that $P^V_x(\beta_i < \sigma_0)$
decays
geometrically fast to zero (with a rate which may depend on $\ell$ but
does not depend on $u$) and that $a^{u \delta }(h_a^+(u)-1)^{-1}$ is
bounded in $u$ (see Remark~\ref{rml}).
To prove (\ref{bt2}) we notice that by Lemma~\ref{main} and (\ref{bt1}) for all
$n>a^\ell$
\begin{align*}
P^V_n(\sigma_0 > \tau_{c_3n})&\le P^V_n(\sigma_0 >
\tau_{c_3n},\,\sigma_{a^\ell}>\tau_{c_3 n})+ P^V_n(\sigma_0 >
\tau_{c_3n},\,\sigma_{a^\ell}<\tau_{c_3 n})\\&\le
P^V_n(\sigma_{a^\ell}>\tau_{c_3
n})+\frac{C_6(a^\ell)}{(c_3n)^{\delta}}=O(c_3^{-\delta}).
\end{align*}
The constant $c_3$ can be chosen large enough to get (\ref{bt2}) for all
$n\in\mathbb{N}$.
\end{proof}
\section{Proof of (A)}\label{px}
\begin{proposition}\label{reentry}
There is a constant $c_4>0$ such that for all $k,\,x\in\mathbb{N}$ and $y\ge 0$
\begin{equation}
\label{ind}
P^V_y \left( \sum_{r=1}^{\sigma_0}\mathbbm{1}_{\{V_r \in [x,2x)\}} >
2xk\right) \le P^V_y(\rho_0<\sigma_0)(1- c_4)^k.
\end{equation}
where $\rho_0=\inf\{j\ge 0\,|\, V_j\in[x,2x)\}$.
\end{proposition}
\begin{proof}
First, observe that there is a constant $c>0$ such that for all
$x\in\mathbb{N}$
\begin{equation*}
(i) \quad P^V_{2x}(
\sigma_{x/2} < x) > c;
\qquad(ii)\quad P^V_{x/2}( \sigma_0 < \tau_x) > c.
\end{equation*}
The inequality $(i)$ is an immediate consequence of
Lemma~\ref{weakcon}. To prove the second inequality, we fix
$x_0\in\mathbb{N}$ and let $x>2x_0+1$. Then by Corollary~\ref{boundtau}
\begin{align*}
P^V_{x/2}( \sigma_0 < \tau_x)&= P^V_{x/2}( \sigma_0 < \tau_x\,|\,
\sigma_{x_0}<\tau_x)P^V_{x/2}( \sigma_{x_0} < \tau_x)\\&\ge
(1-C_6(x_0)x^{-\delta}) P^V_{x/2}( \sigma_{x_0} < \tau_x).
\end{align*}
Choosing $x_0$ large enough
and applying Lemma~\ref{main} to the last term in the right-hand side
we obtain $(ii)$ for all sufficiently large $x$. Adjusting the constant
$c$ if necessary we can extend $(ii)$ to all $x\in\mathbb{N}$.
Next, we show that $(i)$ and $(ii)$ imply (\ref{ind}) with $c_4=c^2$.
Denote by $\rho_0\ge 0$ the first entrance time of $V$ in $[x,2x)$ and
set \[\rho_j=\inf\{r\ge \rho_{j-1}+2x\,|\, V_r\in [x,2x)\},\quad
j\ge 1.\] Notice that for each $j\ge 1$, the time spent by $V$
in $[x,2x)$ during the time interval $[\rho_{j-1},\rho_j)$
is at most $2x$. If $V$ spends more than $2xk$ units of time in
$[x,2x)$ before time $\sigma_0$ then $\rho_k<\sigma^V_0$. Thus,
\begin{multline}\label{st}
P^V_y\left( \sum_{r=1}^{\sigma_0}\mathbbm{1}_{\{V_r \in [x,2x)\}} >
2xk\right) \le P^V_y(\rho_k<\sigma_0)
\\= P^V_y(\rho_k<\sigma_0\,|
\,\,\rho_{k-1}<\sigma_0)P^V_y(\rho_{k-1}<\sigma_0).
\end{multline}
Using the strong Markov property, monotonicity with respect to the
starting point, and inequalities $(i)$ and $(ii)$ we get
\begin{align*}
P^V_y&(\rho_k<\sigma_0\,| \,\,\rho_{k-1}<\sigma_0)\le \max_{x\le
z<2x} P^V_z(\rho_1<\sigma_0)\\&\le \max_{x\le z<2x}
\left(P^V_z(\rho_1<\sigma_0,\,
\sigma_{x/2}<x)+P^V_z(\rho_1<\sigma_0,\, \sigma_{x/2}\ge
x)\right)\\&\le \max_{x\le
z<2x}\left(P^V_z(\rho_1<\sigma_0\,|\,
\sigma_{x/2}<x)P^V_z(\sigma_{x/2}<x)+1-P^V_z(\sigma_{x/2}<
x)\right) \\&\le \max_{x\le z<2x}
\left(1-P^V_z(\sigma_{x/2}<x)(1-P^V_z(\rho_1<\sigma_0\,|\,
\sigma_{x/2}<x))\right)
\\&\le 1-P^V_{2x}(\sigma_{x/2}<x)(1-P^V_{x/2}(\rho_0<\sigma_0))
\\&\le 1-P^V_{2x}(\sigma_{x/2}<x)P^V_{x/2}(\tau_x>\sigma_0)
\le 1-c^2.
\end{align*}
Substituting this in (\ref{st}) and iterating in $k$ gives (\ref{ind}).
\end{proof}
\begin{proposition} \label{NEW} For every $h>0$
\[
\lim_{\epsilon\to 0}\limsup_{n \to \infty} n ^\delta P^V_0 \left( \sum
_{i=1}^{\sigma_0}
\mathbbm{1}_{\{V_i < \epsilon n\}} > n h\right)=0.
\]
\end{proposition}
\begin{proof}
Fix $\epsilon\in(0,1)$ and let $k=k(n,\epsilon)$ be
the smallest integer such that $2^k \ge \epsilon n$. Define
intervals $I_i = [2^{k-i}, 2^{k-i+1})$, $i\in\mathbb{N}$, and events
\[A_i=\left\{
\sum_{r=1}^{\sigma_0} \mathbbm{1}_{\{V_r \in I_i\}} > \frac{h2^{i-1}}
{\epsilon i(i+1)}\,|I_i|\right\}.
\]
Intervals $I_i$ and events $A_i$ depend on $n$ but this is not reflected
in our notation. Since $2^{k-1}<\epsilon n$ and
$\sum_{i=1}^k(i(i+1))^{-1}< 1$, we have
\[\left\{ \sum _{i=1}^{\sigma_0} \mathbbm{1}_{\{V_i < \epsilon n\}} > n h\right\}\subset
\bigcup_{i=1}^kA_i,\] and, therefore,
\[
P_0^V\left(\sum _{i=1}^{\sigma_0} \mathbbm{1}_{\{V_i < \epsilon n\}} > n h\right) \le
\sum_{i= 1}^k P(A_i).
\]
Using (\ref{bt1}) and (\ref{ind}) we get
\begin{align*}
n^\delta P_0^V\left(\sum _{i=1}^{\sigma_0} \mathbbm{1}_{\{V_i < \epsilon n\}}
> n h\right)&\le
n^\delta\sum_{i=1}^k \left(1-c_4\right)
^{\left[\frac{h2^{i-2}}{\epsilon i(i+1)}
\right]}P^V_0(\tau_{2^{k-i}}<\sigma_0)
\\&\le C_6(0)\,\epsilon^{-\delta}
\sum_{i\ge 1} \left(1-c_4\right) ^{\left[\frac{h2^{i-2}}{\epsilon i(i+1)}
\right]}2^{i \delta},
\end{align*}
and this quantity vanishes as $\epsilon\to 0 $.
\end{proof}
\begin{proof}[Proof of (A)]
To obtain (A) we apply (\ref{bt1}) and Proposition~\ref{NEW} to the right-hand
side of the following inequality:
\begin{align*}
n^\delta P^V_0 (\sigma_0 >n) &\leq n^\delta P^V_0 (\tau_{\epsilon n} < \sigma
_0) +n^\delta P^V_0 (\sigma_0 >n, \tau_{\epsilon n} > \sigma _0 )\\&\le
\epsilon^{-\delta}(\epsilon n)^{\delta}P^V_0 (\tau_{\epsilon n} < \sigma
_0)+n^{\delta}P^V_0\left( \sum _{i=1}^{\sigma_0}
\mathbbm{1}_{\{V_i < \epsilon n\}} > n\right). \qedhere
\end{align*}
\end{proof}
\section{Proof of (B)}
We shall need the following fact.
\begin{proposition}\label{T}
For each $\epsilon> 0$, there is a constant $C_7=C_7(\epsilon)>0$ such
that
\[
P^V_n(\sigma_0 > C_7n) < \epsilon\quad\text{for all $n\in\mathbb{N}$}.
\]
\end{proposition}
\begin{proof}
We have
\[P^V_n(\sigma_0 > C_7n)\le P^V_n(\sigma_0 >
\tau_{c_3n})+P^V_n(\sigma_0 > C_7n,\,\sigma_0 <\tau_{c_3n}).\] Using
(\ref{bt2}) we can choose $c_3>1$ so that $P^V_n (\sigma_0 > \tau_{c_3n}
) <\epsilon / 2$ for all $n\in\mathbb{N}$. Thus, we only need to estimate
the last term. Notice that it is bounded above by the probability that
the occupation time of the interval $(0,c_3n)$ up to the moment
$\sigma_0$ exceeds $C_7n$. The latter can be estimated by Markov
inequality:
\[P^V_n\left( \sum_{r=1}^{\sigma_0} \mathbbm{1}_{\{V_r < c_3n\}}>C_7n
\right)\le (C_7n)^{-1}E^V_n\left( \sum_{r=1}^{\sigma_0} \mathbbm{1}_{\{V_r <
c_3n\}} \right).\] We claim that the last expectation does not
exceed $4nc_3/c_4$ and so we can take $C_7>8c_3/(\epsilon c_4)$.
Indeed, let $m$ be the smallest positive integer such that $2^m\ge
c_3n$. Then writing the expectation of our non-negative integer-valued
random variable as the sum of the probabilities of its tails and using
(\ref{ind}) to estimate the tails we get
\begin{equation*}
E^V_n\left( \sum_{r=1}^{\sigma^V_0} \mathbbm{1}_{\{V_r < c_3n\}} \right) \le
\sum_{j=1}^mE^V_n\left( \sum_{r=1}^{\sigma^V_0}
\mathbbm{1}_{\{V_r\in[2^{j-1},2^j)\}} \right)
\le \sum_{j=1}^m\frac{2^j}{c_4}
\le \frac{2^{m+1}}{c_4}\le\frac{4nc_3}{c_4}. \qedhere
\end{equation*}
\end{proof}
\begin{proof}[Proof of (B)] For every $\alpha\in(0,\epsilon)$ and
$\beta\in(0,1)$ we have
\begin{multline}\label{Bee}
P^V_{\epsilon n}\left(\sum_{j=0}^{\sigma_{\alpha n}-1}
V_j>n^2\right)\le
P^V_{\epsilon n}\left(\sum_{j=0}^{\sigma_0-1} V_j>n^2\right)\\
\le P^V_{\epsilon n}\left(\sum_{j=0}^{\sigma_{\alpha n}-1}
V_j>(1-\beta)n^2\right)+P^V_{\epsilon
n}\left(\sum_{j=\sigma_{\alpha n}}^{\sigma_0-1} V_j>\beta
n^2\right).
\end{multline}
By Lemma~\ref{weakcon} for every $R>0$
\begin{equation}\label{Be}
\lim_{n\to\infty} P^V_{\epsilon
n}\left(\sum_{j=0}^{\sigma_{\alpha n}-1} V_j>Rn^2\right)=
P^Y_\epsilon\left (\int_0^{\sigma_\alpha}
Y_s\,ds>R\right),
\end{equation}
since, as is easily verified, under law $ P^Y_\epsilon $ the law of $
\int_0^{\sigma_\alpha} Y_s\,ds $ has no atoms. Next, we notice that
for all $x,\,\beta>0$
\begin{multline}\label{Bef}
P^Y_\epsilon\left
(\int_0^{\sigma_0} Y_s\,ds>(1+\beta)x\right) \ - \ P^Y_\epsilon\left
(\int_{\sigma_\alpha}^{\sigma_0} Y_s\,ds>\beta x\right)\\\le
P^Y_\epsilon\left
(\int_0^{\sigma_\alpha} Y_s\,ds>x\right)\le
P^Y_\epsilon\left
(\int_0^{\sigma_0} Y_s\,ds>x\right).
\end{multline}
By the strong Markov property and scaling,
\begin{align}\label{Beg}\nonumber
P^Y_\epsilon\left
(\int_{\sigma_\alpha}^{\sigma_0} Y_s\,ds> \beta x\right)&=P^Y_\alpha
\left
(\int_0^{\sigma_0} Y_s\,ds>\beta x\right)\\&=
P^Y_1\left
(\int_0^{\sigma_0} Y_s\,ds> \beta x\alpha^{-2}\right)\to
0\ \ \text{as } \alpha\to 0.
\end{align}
Letting $n\to\infty$, then $\alpha\to 0$, and finally $\beta\to 0$ we
obtain from (\ref{Bee})--(\ref{Beg})
that \[\liminf_{n\to\infty}P^V_{\epsilon
n}\left(\sum_{j=0}^{\sigma_0-1} V_j>n^2\right)\ge P^Y_1\left
(\int_0^{\sigma_0} Y_s\,ds>\epsilon^{-2}\right).\]
To get the matching upper bound it is enough to show that for every $\nu>0$ and
$\beta\in(0,1)$ there is an $\alpha\in(0,\epsilon)$ such that for all
sufficiently large $n$
\begin{equation}
\label{Beh}
P^V_{\epsilon n}
\left(\sum_{j=\sigma_{\alpha n}}^{\sigma_0-1} V_j>\beta n^2\right)
<2\nu.
\end{equation}
The left-hand side of (\ref{Beh}) does not exceed
\[P^V_{\alpha n} (\tau_{c_3n\alpha}<\sigma_0)+P^V_{\alpha n}
\left(\sum_{j=0}^{\sigma_0-1} V_j>\beta
n^2,\,\tau_{c_3n\alpha}>\sigma_0\right).\] Given $\nu$, define
$c_3(\nu)$ and $C_7(\nu)$ as in (\ref{bt2}) and Proposition~\ref{T}
respectively. Let $\alpha<\sqrt{\beta/(c_3 C_7)}$. By (\ref{bt2}) the
first term above is less than $\nu$ for all $n>1/\alpha$. On the set
$\{\tau_{c_3n\alpha}>\sigma_0\}$ the process $V$ is below $c_3n\alpha$
and, thus, the second term is bounded above by $P^V_{\alpha
n}(\sigma_0>(\beta n)/(c_3 \alpha))$. By Proposition~\ref{T} and our
choice of $\alpha$ the latter probability does not exceed $\nu$.
Using relations (\ref{Bee})--(\ref{Beg}) and, again, the absence of
atoms for the distribution of $\int_0^{\sigma_x} Y_s\,ds$ under
$P^Y_1$ for each $x\ge 0$, we get the desired upper bound.
\end{proof}
\section{Proofs of (C) and Theorem \ref{X}}
First we prove (C) of Lemma~\ref{4.1}. Then using the approach of
Lemma~\ref{4.1} we show the convergence claimed in Theorem \ref{X}.
The next lemma includes (C) as a special case ($k=0$, $C_5=f(0)$).
\begin{lemma}\label{triv}
There is a function $f:\,\mathbb{N}\cup\{0\}\to(0,\infty)$,
such that \[\lim_{n\to\infty} n^\delta
P_k^V(\sigma_0>\tau_n)=f(k)\quad \text{for each integer } k\ge 0.\]
\end{lemma}
We shall need the following proposition.
\begin{proposition}
\label{s}
For each $a\in(1,2]$ and $k\ge 0$
\begin{equation}
\label{ser}
\sum_{j=1}^\infty \Big|a^\delta
P_k^V(\sigma_0>\tau_{a^j}\,|\, \sigma_0>\tau_{a^{j-1}})-1\Big|<\infty.
\end{equation}
\end{proposition}
\begin{proof}
By the monotonicity in the initial number of particles we get a
lower bound: for all sufficiently large $j$ \[P_k^V(\sigma_0>\tau_{a^j}\,|\,
\sigma_0>\tau_{a^{j-1}})\ge P_{a^{j-1}}^V(\sigma_0>\tau_{a^j}).\] For
an upper bound we need to take into account the possibility of a
large ``overshoot''. Let $x=a^{j-1}+a^{\frac{2(j-1)}{3}}$, then
\begin{align*}
P_k^V&(\sigma_0>\tau_{a^j}\,|\, \sigma_0>\tau_{a^{j-1}})\le
P_x^V(\sigma_0>\tau_{a^j})\\
&+P_k^V(\sigma_0>\tau_{a^j}\,|\, \sigma_0>\tau_{a^{j-1}},
V_{\tau_{a^{j-1}}}>x)P_k^V(V_{\tau_{a^{j-1}}}>x\,|\,
\sigma_0>\tau_{a^{j-1}})\\&\le
P_x^V(\sigma_0>\tau_{a^j})+P_k^V(V_{\tau_{a^{j-1}}}>x\,|\,
\sigma_0>\tau_{a^{j-1}}).
\end{align*}
The last probability decays faster than any power of $a^{-j}$ as
$j\to\infty$ by Lemma~\ref{overshoot}. Therefore, it is enough to
show the convergence of the series
\begin{equation*}
\sum_{j=1}^\infty \Big|a^\delta
P_{x_{j-1}}^V(\sigma_0>\tau_{a^j})-1\Big|,
\end{equation*}
where $x_j=a^j+\epsilon_j$
and $0\le \epsilon_j\le a^{\frac{2j}{3}}$. We have for all sufficiently large
$j$ (with $\ell$ chosen appropriately for $a$ as in Lemma \ref{main}),
\begin{align*}
\Big| P_{x_{j-1}}^V&(\sigma_0>\tau_{a^j})-a^{-\delta}\Big|\\&\le\Big|
P_{x_{j-1}}^V(\sigma_{a^\ell}>\tau_{a^j})-a^{-\delta}\Big|+
P_{x_{j-1}}^V(\sigma_0>\tau_{a^j}>\sigma_{a^\ell})\\&\le\Big|
P_{x_{j-1}}^V(\sigma_{a^\ell}>\tau_{a^j})-a^{-\delta}\Big|+
P_{x_{j-1}}^V(\sigma_0>\tau_{a^j}\,|\,\tau_{a^j}>\sigma_{a^\ell})\\&\le\Big|
P_{x_{j-1}}^V(\sigma_{a^\ell}>\tau_{a^j})-a^{-\delta}\Big|+
P_{a^\ell}^V(\sigma_0>\tau_{a^j}).
\end{align*}
By Lemma \ref{main} \[\sum_{j=1}^\infty\Big|
P_{x_{j-1}}^V(\sigma_{a^\ell}>\tau_{a^j})-a^{-\delta}\Big|<\infty,\]
and to complete the proof of (\ref{ser}) we invoke the bound
provided by (\ref{bt1}) for $x = [a^\ell]$.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{triv}]
Fix an arbitrary non-negative integer $k$ and $a\in(1,2]$. For each
$n>a$ there is an $m\in\mathbb{N}$ such that $a^m\le n<a^{m+1}$. We
have
\[a^{m\delta}P_k^V(\sigma_0>\tau_{a^{m+1}})\le n^\delta
P_k^V(\sigma_0>\tau_n)\le a^{(m+1)\delta}P_k^V(\sigma_0>\tau_{a^m}).\]
If we can show that
\begin{equation}
\label{gak}
\lim_{m\to\infty}
a^{m\delta}P_k^V(\sigma_0>\tau_{a^m})=g(a,k)>0,
\end{equation}
for some $g(a,k)$, then
\begin{align*}
0<a^{-\delta}g(a,k)&\le
\liminf_{n\to\infty}n^\delta P_k^V(\sigma_0>\tau_n)\\&\le
\limsup_{n\to\infty} n^\delta P_k^V(\sigma_0>\tau_n)\le a^\delta
g(a,k).
\end{align*}
This implies
\[1\le\frac{\limsup_{n\to\infty} n^\delta P_k^V(\sigma_0>\tau_n)}
{\liminf_{n\to\infty}n^\delta P_k^V(\sigma_0>\tau_n)}\le a^{2\delta},\]
and we obtain the claimed result by letting $a$ go to $1$.
To show (\ref{gak}) we set $\ell=\min\{j\in\mathbb{N}\,|\, a^j>k\}$ and notice that for
$m>\ell$
\begin{align*}
a^{m\delta}P_k^V(\sigma_0>\tau_{a^m})&=a^\delta
P_k^V(\sigma_0>\tau_{a^m}\,|\,\sigma_0>\tau_{a^{m-1}})\times
a^{(m-1)\delta}
P_k^V(\sigma_0>\tau_{a^{m-1}})\\=\dots&=a^{\ell
\delta}P_k^V(\sigma_0>\tau_{a^\ell})\times
\prod_{j=\ell+1}^m a^\delta
P_k^V(\sigma_0>\tau_{a^j}\,|\,\sigma_0>\tau_{a^{j-1}}).
\end{align*}
Since all terms in the last product are strictly positive and $a^{\ell
\delta}P_k^V(\sigma_0>\tau_{a^\ell})$ does not depend on $m$, the
convergence (\ref{gak}) follows from (\ref{ser}).
\end{proof}
\begin{proof}[Proof of Theorem~\ref{X}]
We will show that $\lim\limits_{n \rightarrow \infty } n^\delta
P^V_0 (\sigma _0 > n) = C_3C_5$, where $C_3$ and $C_5$ are the same as in
Lemma~\ref{difext} and condition (C).
We begin with a lower bound. Fix positive $\epsilon$ and $\beta \ll
\epsilon$. We have
\[
P^V _0(\sigma _0 > n) \geq P^V _0(\tau _{n
\epsilon} < \sigma _ 0) P^V_{\epsilon n}(\sigma_0 > n )\ge
P^V _0(\tau _{\epsilon n} < \sigma _ 0) P^V_{\epsilon n}(\sigma_{\beta
n} > n ).
\]
By (C), Lemma~\ref{weakcon}, and scaling (Lemma \ref{elem} (i))
\[
\liminf_{n \rightarrow \infty } n ^\delta P^V _0(\sigma _0 > n)\ge
C_5\,\epsilon^{-\delta} P^Y_{1}\left(\tau^Y_{\beta/\epsilon}
>\epsilon^{-1}\right).
\]
Letting $\beta\to 0$ and then $\epsilon\to 0$ we obtain via
Lemma~\ref{difext} that \[\liminf_{n \rightarrow
\infty } n ^\delta P^V _0(\sigma _0 > n) \geq C_3 C_5 .\]
The upper bound is slightly more complicated. First, notice that
\[n^\delta P^V_0(\sigma_0>n,\,\tau_{\epsilon n} >\sigma_0)\le n^\delta
P^V_0\left(\sum_{i=1}^{\sigma_0}\mathbbm{1}_{\{V_i <\epsilon n\}}>n\right).\]
By Proposition~\ref{NEW} the right-hand side becomes negligible as
$n\to\infty$ and then $\epsilon \to 0$. Thus, it is enough to estimate $n^\delta
P^V_0(\sigma_0>n,\,\tau_{\epsilon n} <\sigma_0)$.
Let $R\in(1,3/2)$. Then
\begin{align*}
n ^\delta P^V _0(\sigma _0 > n,\tau_{\epsilon n}<\sigma_0)&\leq
n ^\delta P^V _0(\sigma_0>\tau_{\epsilon n} >(R-1)n)\\ &+
n ^\delta P^V _0(V_{\tau_{\epsilon n }} > R \epsilon n,\tau_{\epsilon
n}<\sigma_0 )\\&+
n ^\delta P^V _0( \sigma_0 - \tau_{\epsilon n} > (2-R)n,
V_{\tau_{\epsilon n }}\leq R \epsilon n ).
\end{align*}
By Proposition~\ref{NEW} the first term on the right-hand side
vanishes for every fixed $R>1$ when we let $n\to \infty$ and then
$\epsilon\to 0$. By Lemma~\ref{overshoot} the $\limsup_{n\to\infty}$
of the second term is zero. Thus it will be sufficient to bound the
last term. For $\beta\ll\epsilon$ let us define $\sigma^{\epsilon
n}_{\beta n} $ to be the the first time after $\tau_{\epsilon n} $
that $V$ falls below $\beta n$. Then the last term is bounded above
by
\begin{multline*}
n ^\delta \left(P^V _0( \sigma_0 - \sigma^{\epsilon n}_{\beta n}
> (R-1)n)+
P^V _0(\sigma^{\epsilon n}_{\beta n} - \tau_{\epsilon n} > (3-2R)n,
V_{\tau_{\epsilon n }}\leq R \epsilon n ) \right) \\\le \left(P^V _{\beta n}(
\sigma_0> (R-1)n)+ P^V _{R\epsilon n}(\sigma_{\beta n} > (3-2R)n)\right)n
^\delta P^V_0(\tau_{\epsilon n}<\sigma_0).
\end{multline*}
Taking $\limsup_{n\to\infty}$ and then letting $\beta\to 0$ we obtain
(by Proposition~\ref{T}, Lemma~\ref{weakcon}, and (C)) the following
upper bound for $\limsup_{n \rightarrow \infty }n ^\delta P^V _0(
\sigma_0 - \tau_{\epsilon n} > (2-R)n, V_{\tau_{\epsilon n }}\leq R
\epsilon n )$,
\[C_5\epsilon^{-\delta}P_{R\epsilon}^Y(\tau_0>(3-2R))=
C_5\epsilon^{-\delta}P_1^Y(\tau_0>\epsilon^{-1}(3-2R)/R).\] As $\epsilon\to 0$
and then $R\to 1$, the latter expression converges by Lemma~\ref{difext} to
$C_5C_3$. This completes the proof.
\end{proof}
\section{Proof of Theorem~\ref{mainth}}\label{1from2}
Let $\delta>2$. By (\ref{Tn}) and (\ref{irrel})), it is enough to show
that as $n\to\infty$
\begin{equation}
\label{ed}\frac{2\sum_{k=0}^n D_{n,k}-(v^{-1}-1)n}{n^{2/\delta}}\
\overset{\mathcal{D}}{=}\ \frac{2\sum_{j=0}^n
V_j-(v^{-1}-1)n}{n^{2/\delta}}
\end{equation}
converges in distribution to $Z_{\delta/2,b}$ for some $b>0$. Define
the consecutive times when $V_j=0$,\[\sigma_{0,0}=0,\quad
\sigma_{0,i}=\inf\{j>\sigma_{0,i-1}\,|\,V_j=0\},\quad i\in\mathbb{N},\] the
total progeny of $V$ over each lifetime,
$S_i=\sum_{j=\sigma_{0,i-1}}^{\sigma_{0,i}-1}V_j,\ i\in\mathbb{N}$, and the
number of renewals up to time $n$, $N_n=\max\{i\ge
0\,|\,\sigma_{0,i}\le n\}$. Then
$(\sigma_{0,i}-\sigma_{0,i-1},S_i)_{i\ge 1}$ are i.i.d.\ under
$P^V_0$. Moreover,
$\sigma_{0,i}-\sigma_{0,i-1}\overset{\mathcal{D}}{=}\sigma^V_0$ and
$S_i\overset{\mathcal{D}}{=}S^V$, $i\in\mathbb{N}$. By Theorem~\ref{progeny} the
distribution of $S^V$ is in the domain of attraction of the law of
$Z_{\delta/2,\tilde{b}}$ for some $\tilde{b}>0$ (see, for example \cite{Dur96},
Chapter 2, Theorem~7.7.) Since by Theorem~\ref{X} (in fact, the upper
bound (A) is sufficient) the second moment of $\sigma^V_0$ is finite,
it follows from standard renewal theory (see, for example,
\cite{Gu88}, Theorems~II.5.1 and II.5.2) that
\[\frac{N_n}{n}\overset{\mathrm{a.s.}}{\to}\lambda:=
\left(E^V_0\sigma_0\right)^{-1},\] and for each $\epsilon>0$ there is
$c_5>0$ such that $P^V_0\left(|N_n-\lambda
n|>c_5\sqrt{n}\right)<\epsilon$ for all sufficiently large
$n$. Using the fact that $T_n/n\to v^{-1}$ a.s.\ as $n\to\infty$ and
relations (\ref{Tn}) and (\ref{ed}) we get that
$E^V_0S_i=(v^{-1}-1)/(2\lambda)$.
\textit{Proof of part (i).} Let $\delta\in(2,4)$. We have
\begin{equation*}
\frac{\sum_{j=0}^nV_j-(v^{-1}-1)n/2}{n^{2/\delta}}
=\frac{\sum_{i=1}^{N_n}(S_i-E^V_0S_i)}{n^{2/\delta}}+
E^V_0S_1\,\frac{N_n-\lambda n}{n^{2/\delta}}+\frac{\sum_{j=\sigma_{0,N_n}}^n
V_j}{n^{2/\delta}}.
\end{equation*}
By Theorem I.3.2, \cite{Gu88}, the first term converges in
distribution to $Z_{\delta/2,\tilde{b}}$. The second term converges to
zero in probability by the above mentioned facts from renewal
theory. The last term is bounded above by $S_{N_{n+1}}/n^{2/\delta}$,
which converges to zero in probability. This finishes the proof of
(\ref{itime}), which immediately gives (\ref{ix}) with $X_n$ replaced
by $\sup_{i\le n}X_i$, since $\{\sup_{i\le n} X_i< m\}=\{T_m>n\}$.
Next we show (\ref{ix}) with $X_n$ replaced by $\inf_{i\ge n}X_i$. The proof is
the same as in, for example, \cite{BS08b}, p.\,849. We observe that for all
$m,n,p\in\mathbb{N}$
\[\{\sup_{i\le n}X_i<m\}\subset\{\inf_{i\ge n}X_i<m\}\subset \{\sup_{i\le
n}X_i<m+p\}\cup\{\inf_{i\ge T_{m+p}}X_i<m\}.\]
The following lemma completes the proof of part (i).
\begin{lemma}
\label{inf}
\[\lim_{k\to\infty}\sup_{n\ge 1}P_0\left(\inf_{i\ge
T_n}X_i<n-k\right)=0.\]
\end{lemma}
\noindent We postpone the proof of this lemma until the end of the section.
\textit{Proof of part (ii).} Let $\delta=4$. Theorem~\ref{progeny}
implies that the distribution of $S^V$ is in the domain of attraction
of the normal distribution (\cite{Fe71}, XVII.5). Norming constants
are easily computed to be (see \cite{Fe71}, XVII.5, (5.23) with $C=1$)
$\sqrt{C_2n\log n}$. The constant $b$, which appears in the statement
is equal to $C_2/2$. Relations (\ref{iitime}) and (\ref{iix}) follow
in the same way as for part (i). \qed
\begin{proof}[Proof of Lemma~\ref{inf}]
Let $P_{n,k}:=P_0\left(\inf_{i\ge T_n}X_i<n-k\right).$
\textit{Step 1.} The supremum over $n\ge 1$ can be reduced to the
maximum over $n\in\{1,2,\dots,k\}$: $\sup_{n\ge
1}P_{n,k}=\max_{1\le n\le k}P_{n,k}.$
Indeed, consider $P_{k,k}$ and $P_{m+k,k}$ for $m\ge 1$. The
corresponding events $\{\inf_{i\ge T_k}X_i<0\}$ and $\{\inf_{i\ge
T_{k+m}}X_i<m\}$ depend on the behavior of the process only at
times when $X_i$ is in $[0,\infty)$ and $[m,\infty)$ respectively.
But at times $T_0$ and $T_m$ the walk is at $0$ and $m$
respectively, and the distributions of the environments starting from
the current point to the right of it are the same under $P_0$. We
conclude that $P_{k,k}=P_{m+k,k}$. This is essentially the content
of Lemma~10 from \cite{Ze05}. The proof does not use the positivity of
cookies so it can be applied here.
\textit{Step 2.} We list four elementary properties of
$\{P_{n,k}\}$, $n,k\ge 1$.
\begin{itemize}
\item [(a)] $P_{n,k}\ge P_{n,k+m}$ for all $1\le n\le k$ and $m\ge 0$;
\item [(b)] $P_{n,k+m}$ converges to $0$ as $m\to\infty$ for each
$k\ge n\ge 1$;
\item [(c)] $P_{n,k}\ge P_{n+m,k+m}$ for all $n\le k$ and $m\ge 0$;
\item [(d)] $P_{n+m,k+m}$ converges to $0$ as $m\to\infty$ for each
$k\ge n\ge 1$.
\end{itemize}
Inequality (a) is obvious. Part (b) follows from the transience of
$X$. Namely, $\inf_{i\ge T_n} X_i>-\infty$ a.s.\ but $n-(k+m)\to-\infty$ as
$m\to\infty$. Inequality (c) is also obvious: since $T_n<T_{m+n}$ we
have
\begin{equation*}
\{\inf_{i\ge
T_n}X_i<n-k\}
\supset \{\inf_{i\ge
T_{n+m}}X_i<n-k\}=\{\inf_{i\ge
T_{n+m}}X_i<(n+m)-(k+m)\}.
\end{equation*}
The convergence in (d) again follows from the
transience: $X_i\to\infty$ as $i\to\infty$ a.s.\ implies that
$\inf_{i\ge T_{m+n}}X_i\to\infty$ as $m\to\infty$ a.s.\ but $(k+m)-(n+m)$
stays constant.
\textit{Step 3.} Take any $\epsilon>0$ and using (d) choose an $m$ so
that $P_{m,m}<\epsilon$. Properties (a) and (c) imply that
$P_{n,n+i}<\epsilon$ for all $i\ge 0$ and $n\ge m$. Using (b),
for $n=1,2,\dots,m-1$ choose $k_n$ so that $P_{n,k_n}<\epsilon$. Let
$K=\max_{1\le n\le m} k_n$ (naturally, we set $k_m=m$). Then
$P_{n,k}<\epsilon$ for all $n\le k$ and $k\ge K$ that is $\max_{1\le
n\le k} P_{n,k}<\epsilon$ for all $k\ge K$.
\end{proof}
\begin{remark}\label{alt}
{\em Theorem~\ref{X} and Theorem~\ref{progeny} imply Theorem~\ref{BS} for
general cookie environments satisfying conditions (A1) and (A2). The proof is
the same as in Section 6 of \cite{BS08b}) and uses Lemma~\ref{inf}. }
\end{remark}
| {
"timestamp": "2010-05-25T02:01:17",
"yymm": "0908",
"arxiv_id": "0908.4356",
"language": "en",
"url": "https://arxiv.org/abs/0908.4356",
"abstract": "We consider excited random walks (ERWs) on integers with a bounded number of i.i.d. cookies per site without the non-negativity assumption on the drifts induced by the cookies. Kosygina and Zerner [KZ08] have shown that when the total expected drift per site, delta, is larger than 1 then ERW is transient to the right and, moreover, for delta>4 under the averaged measure it obeys the Central Limit Theorem. We show that when delta in (2,4] the limiting behavior of an appropriately centered and scaled excited random walk under the averaged measure is described by a strictly stable law with parameter delta/2. Our method also extends the results obtained by Basdevant and Singh [BS08b] for delta in (1,2] under the non-negativity assumption to the setting which allows both positive and negative cookies.",
"subjects": "Probability (math.PR)",
"title": "Limit laws of transient excited random walks on integers",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808724687407,
"lm_q2_score": 0.7217432062975978,
"lm_q1q2_score": 0.7075110599677955
} |
https://arxiv.org/abs/2001.11385 | Theta surfaces | A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincaré showed that theta surfaces are precisely the surfaces of double translation, i.e. obtained as the Minkowski sum of two space curves in two different ways. These curves are parametrized by abelian integrals, so they are usually not algebraic. This paper offers a new view on this classical topic through the lens of computation. We present practical tools for passing between quartic curves and their theta surfaces, and we develop the numerical algebraic geometry of degenerations of theta functions. | \section{Introduction}
Our first example of a theta surface is {\em Scherk's minimal surface}, given by the equation
\begin{equation}
\label{eq:scherk}
{\rm sin}(X)\, \,-\,\, {\rm sin}(Y) \cdot {\rm exp}(Z)\,\,=\,\, 0.
\end{equation}
\begin{figure}[ht]
\centering
\includegraphics[scale = 0.45]{ScherkMinimalWhite.png}
\vspace{-0.2in}
\caption{Scherk's minimal surface.}
\label{polytope}
\end{figure}
This surface arises from the following quartic
curve in the complex projective plane~$\mathbb{P}^2$:
\begin{equation}
\label{eq:quartic1}
xy(x^2+y^2+z^2) \,\,=\,\,0.
\end{equation}
We use $X,Y,Z$ as affine coordinates for $\mathbb{R}^3$
and $x,y,z$ as homogeneous coordinates for $\mathbb{P}^2$.
Scherk's minimal surface is obtained as the Minkowski sum of two parametric space curves:
\begin{equation}
\label{eq:scherkpara1}
\begin{matrix}
\! (X,Y,Z) \,\,& \, = &
\bigl( {\rm arctan}(s), 0, {\rm log}(s) - {\rm log}(s^2{+}1)/2 \bigr) \\ & & +\,\,
\bigl( 0, -{\rm arctan}(t),- {\rm log}(t) + {\rm log}(t^2{+}1)/2 \bigr) .
\end{matrix}
\end{equation}
The derivation of (\ref{eq:scherk}) from (\ref{eq:quartic1})
via (\ref{eq:scherkpara1}) is given in Example \ref{ex:scherkalgo1}.
This computation is originally due to Richard~Kummer \cite[p.~52]{Kum} whose
1894 dissertation also displays a plaster model.
Following the classical literature (cf.~\cite{Little83,Sch}), a {\em surface of double translation} equals
\begin{equation*}
\mathcal{S} \quad = \quad \mathcal{C}_1 + \mathcal{C}_2 \,\, = \,\, \mathcal{C}_3 + \mathcal{C}_4 ,
\end{equation*}
where $\mathcal{C}_1,\mathcal{C}_2,\mathcal{C}_3,\mathcal{C}_4$ are curves in $\mathbb{R}^3$,
and the two decompositions are distinct.
We note that Scherk's minimal surface
(\ref{eq:scherk}) is a surface of double translation. A first representation
$ \mathcal{S} = \mathcal{C}_1 + \mathcal{C}_2$ was given in (\ref{eq:scherkpara1}).
A second representation $ \mathcal{S} = \mathcal{C}_3 + \mathcal{C}_4$ equals
\begin{equation}
\label{eq:scherkpara2}
\begin{matrix}
X & = & {\rm arctan}(u)\,\,\,+\,\,\,{\rm arctan} (v) & = &
{\rm arctan} \bigl(\frac{u+v}{1-uv} \bigr), \medskip \\
Y & = & {\rm arctan}(5u)\,\,+\,\,{\rm arctan} (5v) & = & \,
{\rm arctan} \bigl(\frac{5u+5v}{1-25uv} \bigr), \smallskip \\
Z & = & \frac{1}{2} {\rm log}\bigl( \frac{1+(5u)^2}{5(1+u^2)} \bigr) \, + \,
\frac{1}{2} {\rm log}\bigl( \frac{1+(5v)^2}{5(1+v^2)} \bigr).
& &
\end{matrix}
\end{equation}
It is instructive to verify that both parametrizations (\ref{eq:scherkpara1}) and
(\ref{eq:scherkpara2}) satisfy the equation~(\ref{eq:scherk}).
A remarkable theorem
due to Sophus Lie \cite{LieGes2}, refined by Henri Poincar\'e in \cite{Poin95},
states that these are precisely the surfaces derived from plane quartic curves by
the integrals appearing in Abel's Theorem.
The implicit equation of such a surface is an analytic object introduced by
Bernhard Riemann. Namely, if the quartic is smooth then this is Riemann's
{\em theta function} $\theta$.
Modern algebraic geometers view the surface
$\{ \theta = 0\}$ as the {\em theta divisor} in the Jacobian.
We replace that abelian threefold by
its universal cover $\mathbb{C} ^3$ and we focus on the subset $\mathbb{R}^3$ of real points.
Our object of study is
the real analytic surface
$\{ \theta = 0\}$ in $\mathbb{R}^3$.
The present article is organized as follows.
Section \ref{sec2} derives the parametrization of theta surfaces by abelian integrals.
We review Abel's Theorem~\ref{thm:abel},
Riemann's Theorem~\ref{thm:riemann} and
Lie's Theorem \ref{thm:lie}, all from the perspective developed by
Lie's successors in \cite{Eies08,Eies09,Kum,Sch,Wie}.
In Section \ref{sec3} we present a symbolic algorithm for computing theta
surfaces. Here the input is a reducible quartic curve whose abelian integrals
can be evaluated in closed form in a computer algebra system. As an illustration we show
how (\ref{eq:scherkpara1}) and (\ref{eq:scherk}) are derived from (\ref{eq:quartic1}).
In Section~\ref{sec5}
we discuss degenerations of curves and their Jacobians via tropical geometry.
This leads to the formula in Theorem \ref{thm:degtheta} for the implicit equation
of a degenerate theta surface.
Based on the combinatorics of Voronoi cells and Delaunay polytopes, this
offers a present-day explanation for formulas, such as (\ref{eq:scherk}),
that were found well over a century ago.
Theta surfaces are usually transcendental, but they can be algebraic in special~situations. Algebraic theta surfaces were classified by John Eiesland in~\cite{Eies08}.
Examples include the cubic surface mentioned by
Shiing-Shen Chern in \cite[p.~2]{Chern}, the quintic surface
shown by John Little in \cite[Example 4.3]{Little83} and the quadric surface arising from the union of four concurrent lines in Example~\ref{Quadric}.
In Section \ref{sec6} we revisit Eiesland's census of quartics with algebraic theta surfaces. We
present derivations and connections to sigma functions \cite{BEL,naka}.
In Section \ref{sec4} we study theta surfaces via numerical computation.
Building on
state-of-the-art methods for evaluating abelian integrals and theta functions,
we develop a numerical algorithm whose input is a
smooth quartic curve in~$\mathbb{P}^2$ and whose output is its
theta surface.
Our article proposes the name {\em theta surface}
for what Sophie Lie called a {\em surface of double translation}.
We return to historical sources in Section \ref{sec7}, by offering
a retrospective on the remarkable work done in Leipzig in the late
$19$th century by Lie's circle \cite{Lie,Sch}.
Our presentation here serves to connect differential geometry and algebraic geometry,
reconciling the $19$th and $20$th centuries, with a view towards applied mathematics in
the $21$st century.
On that note, there is a natural connection
to integrable systems and mathematical physics. The three-phase solutions \cite{DFS} to
the KP equation are closely related
to theta surfaces. Double translation surfaces also
represent invariants in the study of 4-wave interactions in~\cite{BalFer}.
\section{Parametrization by Abelian Integrals}
\label{sec2}
We begin with the parametric representation of theta surfaces.
Our point of departure is a real algebraic curve $\mathcal{Q}$
of degree four in the projective plane $\mathbb{P}^2$.
This is the zero set of a ternary quartic
that is unique up to scaling.
We thus identify the curve with its polynomial
\begin{equation*}
\label{eq:quartic2}
\qquad
Q(x,y,z) \quad =\,\,
\sum_{i+j+k = 4} \! \!c_{ijk} x^i y^j z^k, \qquad \qquad
c_{ijk} \in \mathbb{R}.
\end{equation*}
This quartic can be reducible, as in (\ref{eq:quartic1}), but we assume that it is squarefree.
The symbol $\mathcal{Q}$ refers to the complex curve, and we write
$\mathcal{Q}_\mathbb{R}$ for its subset of real points.
If $\mathcal{Q}$ is nonsingular then it is a non-hyperelliptic Riemann surface of genus $3$,
canonically embedded into $\mathbb{P}^2$.
The holomorphic differentials on $\mathcal{Q}$ comprise a $3$-dimensional vector space $H^0(\mathcal{Q}, \Omega^1_\mathcal{Q})$. Assuming that $z$ does not divide $Q$,
we choose a basis $\{\omega_1,\omega_2,\omega_3\}$ for this space as follows.
Consider the dehomogenization $q(x,y) = Q(x,y,1)$, set
$q_x = \partial q / \partial x$ and $q_y = \partial q / \partial y$, and fix
\begin{equation}
\label{eq:omegabasis1}
\omega_1 \, = \,\frac{x}{q_y} dx \, , \quad
\omega_2 \, = \,\frac{y}{q_y} dx \, , \quad
\omega_3 \, = \,\frac{1}{q_y} dx.
\end{equation}
As the quartic $q(x,y)$ is defined over $\mathbb{R}$, so is the basis.
To be precise, all coefficients appearing in the $\omega_i$ are real numbers.
Since $\,d q = q_x dx + q_y dy = 0 \,$ holds on $\mathcal{Q}$, we can also~write
\begin{equation}
\label{eq:omegabasis2}
\omega_1 \, = \,-\frac{x}{q_x} dy \, , \quad
\omega_2 \, = \,-\frac{y}{q_x} dy \, , \quad
\omega_3 \, = \,-\frac{1}{q_x} dy.
\end{equation}
\begin{remark}
\label{remk:canonicalemb}
If $p=(x:y:1)$ is any point on $\mathcal{Q}$, then
$\,(\omega_1(p):\omega_2(p):\omega_3(p)) \,=\, p$.
This reflects the fact that $\mathcal{Q}$ is the canonical embedding of
the abstract curve underlying $\mathcal{Q}$.
\end{remark}
Consider a path on the Riemann surface $\mathcal{Q}$ with end points $p$ and $r$.
We can integrate the form $\omega_i$ along that path and obtain
a complex number $\int_{p}^{r} \omega_i$.
If the path is real, from $p$ to $r$ on a single connected component of
the real curve
$\mathcal{Q}_\mathbb{R}$, then $\int_{p}^{r} \omega_i$ is a real number.
This number is computed in practise by regarding $y = y(x)$
as a function of $x$, defined by $q(x,y) = 0$, and by
computing the definite integral from the $x$-coordinate of $p$ to the $x$-coordinate of~$r$.
Alternatively, after replacing (\ref{eq:omegabasis1}) with (\ref{eq:omegabasis2}),
we can regard $x = x(y)$ as an implicitly defined function of $y$, and take
the definite integral from the $y$-coordinate of $p$ to that of~$r$.
We fix a line $\mathcal{L}(0)$ in $\mathbb{P}^2$ that intersects $\mathcal{Q}$ in four
distinct nonsingular points $p_{1}(0)$, $p_{2}(0),p_{3}(0)$ and $p_{4}(0)$. For $p_j \in \mathcal{Q}$ sufficiently close to $p_j(0)$,
we obtain an analytic curve
\begin{equation}
\label{eq:omega}
\begin{matrix}
\Omega_j (p_j) \,\,= \,\,(\Omega_{1j},\Omega_{2j},\Omega_{3j}) (p_j) \,\, = \,\,
\bigl(\, \int_{p_j(0)}^{p_j} \omega_1\, ,\, \int_{p_j(0)}^{p_j} \omega_2 \,,\, \int_{p_j(0)}^{p_j} \omega_3 \, \bigr).
\end{matrix}
\end{equation}
\begin{remark}
\label{rmk:omegatangents}
By the Fundamental Theorem of Calculus, the derivatives of these curves are
\begin{equation*}
\label{eq:omegaderivative}
\dot{\Omega}_j(p_j) \,\,\,= \,\,\,(\omega_1(p_j),\omega_2(p_j),\omega_3(p_j))\,\,\, = \,\,\, p_j .
\end{equation*}
Here, the derivative is performed with
respect to the local parameter of the point $p_j$. The second equality is Remark \ref{remk:canonicalemb}. In words,
the tangent direction to $\Omega_j$ at $p_j$ is $p_j$ itself.
\end{remark}
The following theorem paraphrases a basic result from the theory of Riemann surfaces:
\begin{theorem}[Abel's Theorem] \label{thm:abel}
Suppose that the points $p_1,p_2,p_3,p_4$ are collinear. Then
\begin{equation*}
\Omega_1(p_1)\,+\,\Omega_2(p_2)\,+\,\Omega_3(p_3)\,+\,\Omega_4(p_4) \,\,\,=\,\,\, 0.
\end{equation*}
\end{theorem}
Now, let us fix analytic coordinates $s,t$ on the curve, centered at $p_1(0)$ and $p_2(0)$ respectively. Then,
for every choice of $s,t$ in a small neighborhood of $0$, we obtain two nearby points $p_1(s),p_2(t)$.
The corresponding theta surface $\mathcal{S}$ is the image of the parametrization
\begin{equation}
\label{eq:stmap}
(s,t) \quad \mapsto \quad \Omega_1(p_1(s)) \,+ \, \Omega_2(p_2(t)) .
\end{equation}
The image is a complex analytic surface $\mathcal{S}$ in $\mathbb{C} ^3$.
However, if the points $p_1(0)$ and $p_2(0)$ are real
then we take $s$ and $t$ in a small neighborhood of $0$ in $\mathbb{R}$, and
$\mathcal{S}$ is a real surface in~$\mathbb{R}^3$.
This surface is analytic and only defined locally, since $s$ and $t$ are local parameters.
Shifting gears, let us now consider an arbitrary analytic
surface $\mathcal{T}$ in $\mathbb{C}^3$. We say that $\mathcal{T}$ is
a \emph{translation surface}
if there are two smooth analytic curves $\mathcal{C}_1,\mathcal{C}_2\subset \mathbb{C}^3$ such that
\[ \mathcal{T} \,\,=\,\, \mathcal{C}_1 \,+\, \mathcal{C}_2
\,\,=\,\,\bigl\{\, p_1 + p_2 \,\,|\,\, p_1\in \mathcal{C}_1, p_2\in \mathcal{C}_2 \,\bigr\} .\]
In words, $\mathcal{T}$ is the {\em Minkowski sum} of
the two {\em generating curves} $\mathcal{C}_1$ and $ \mathcal{C}_2$.
We require the parametrization to be injective,
i.e.~for each point $x\in \mathcal{T}$ there are unique points
$p_1\in \mathcal{C}_1,p_2\in \mathcal{C}_2$ such that $x=p_1+p_2$.
Note that,
if $\alpha_1,\alpha_2\colon \Delta \to \mathbb{C}^3$ are local parametrizations of
the generating curves $\mathcal{C}_1,\mathcal{C}_2$, then the translation surface $S$ has the parametrization
\[ \mathcal{T} \,\,= \,\,\{ \alpha_1(s) + \alpha_2(t) \} \,\,=\,\,
\left\{ \begin{pmatrix} \alpha_{11}(s)+\alpha_{21}(t) \\ \alpha_{12}(s)+\alpha_{22}(t) \\ \alpha_{13}(s) + \alpha_{23}(t) \end{pmatrix} \right\} .\]
\begin{definition}[Double translation surfaces] A translation surface $\mathcal{T}\subset \mathbb{C}^3$ as above is a \emph{double translation surface} if there exists other smooth analytic curves $\mathcal{C}_3,\mathcal{C}_4 \subset \mathbb{C}^3$ such that
\begin{equation} \label{eq:doubletranslation}
\mathcal{T} \,\,=\,\, \mathcal{C}_1\,+\,\mathcal{C}_2 \,\,= \,\,\mathcal{C}_3\,+\,\mathcal{C}_4.
\end{equation}
\end{definition}
Returning to the setting of algebraic geometry, let $\mathcal{S}$
be the theta surface derived as above from a quartic curve $\mathcal{Q}$
in the plane $\mathbb{P}^2$.
This formula (\ref{eq:stmap}) shows that $\mathcal{S}$ is a translation surface.
In fact, the theta surface $\mathcal{S}$ is a double translation surface.
This is a consequence of Abel's Theorem. To see this, consider the line
$\mathcal{L}$ that is spanned by the points $p_1(s)$ and $p_2(t)$ in $\mathbb{P}^2$.
This line intersects the quartic curve $\mathcal{Q}$ in two other
points $p_3(s,t)$ and $p_4(s,t)$, and these points are close to $p_3(0)$ and $p_4(0)$ respectively.
Theorem \ref{thm:abel} says that
\begin{equation}
\label{eq:stmapabel}
\Omega_1(p_1(s))\,+\,\Omega_2(p_2(t))\,\, =\,\, -\Omega_3(p_3(s,t))\, -\, \Omega_4(p_4(s,t)).
\end{equation}
The points $p_3(s,t)$ and $p_4(s,t)$ span the same line $\mathcal{L}$, so they determine $p_1(s)$
and $p_2(t)$ as the residual intersection points of the curve $\mathcal{Q}$ with $\mathcal{L}$. This means that the points $p_3(s,t)$ and $p_4(s,t)$ can move freely in neighborhoods of $p_3(0)$ and $p_4(0)$ on the curve
$\mathcal{Q}$. If $u,v$ are analytic coordinates on $\mathcal{Q}$,
centered in $p_3(0)$ and $p_4(0)$ respectively, then \eqref{eq:stmapabel} shows that
\begin{equation*}
\label{eq:uvmaps}
(u,v) \quad \mapsto \quad -\Omega_3(p_3(u))-\Omega_4(p_4(v))
\end{equation*}
is another parametrization of the surface $\mathcal{S}$.
Hence (\ref{eq:doubletranslation}) holds, with generating curves
\begin{equation}\label{eq:analyticcurves}
\mathcal{C}_1 = \Omega_1(p_1(s)), \quad \mathcal{C}_2 =\Omega_2(p_2(t)), \qquad \mathcal{C}_3 = -\Omega_3(p_3(u)), \quad
\mathcal{C}_4 = -\Omega_4(p_4(v)).
\end{equation}
In particular, we see that the two translation structures on
the
theta
surface
$\mathcal{S}$
are distinct. Indeed, Remark \ref{rmk:omegatangents} tells us that the tangent lines to these curves at $0$ correspond to the four points $p_1(0),p_2(0),p_3(0),p_4(0)\in \mathbb{P}^2$, and these are distinct by construction.
\begin{remark}\label{rmk:quarticfromcurves}
Remark \ref{rmk:omegatangents} shows that
the tangent directions to the analytic curves in \eqref{eq:analyticcurves}~are
\begin{align*}
\dot{\mathcal{C}}_1(s) &= \phantom{-} \dot{\Omega}_1(p_1(s)) = p_1(s),
& \dot{\mathcal{C}}_2(t) &= \phantom{-} \dot{\Omega}_2(p_2(t))=p_2(t),\\
\dot{\mathcal{C}}_3(u) &= -\dot{\Omega}_3(p_3(u)) = p_3(u), &
\dot{\mathcal{C}}_4(v) &= -\dot{\Omega}_4(p_4(v)) = p_4(v) .
\end{align*}
Here $p_1(s),p_2(t),p_3(u),p_4(v)$ are regarded as points in the projective
plane $\mathbb{P}^2$, indicating tangent directions in $\mathbb{C} ^3$, so the sign does not matter.
Hence, as in Remark \ref{remk:canonicalemb},
the analytic arcs $\dot{\mathcal{C}}_1,\dot{\mathcal{C}}_2,\dot{\mathcal{C}}_3,\dot{\mathcal{C}}_4$ lie on the quartic $\mathcal{Q}$. In particular, if we are given a theta surface $\mathcal{S}$ and one generating curve $\mathcal{C}_i$, but not the quartic $\mathcal{Q}$, then we can recover $\mathcal{Q}$ as a quartic in $\mathbb{P}^2$ that contains the analytic arc $\dot{\mathcal{C}}_i$.
This fact will be used in Section \ref{sec7}, together with a result of Lie, to give a differential-geometric solution
to Torelli's problem for genus three curves.
\end{remark}
\smallskip
Now assume that $\mathcal{Q}$ is a smooth quartic curve. Then we can find an implicit equation for our
surface via the \emph{Riemann theta function}. Recall \cite{Fay,Mum1} that this is the holomorphic function
\begin{equation}
\label{eq:theta}
\theta(\mathbf{x},B) \,\,= \,\,
\sum_{n\in \mathbb{Z}^3}\mathbf{e}\left( -\frac{1}{2}n^tBn \,+\,i\, n^t\mathbf{x} \right)
\,\, =\,\,
\sum_{n\in \mathbb{Z}^3}\exp\left(-\pi n^tBn\right)\cdot \operatorname{cos}(2\pi n^t\mathbf{x}),
\end{equation}
where $\mathbf{e}(t):=\exp(2\pi t)$, $\mathbf{x}=(X,Y,Z)\in \mathbb{C}^3$, and
$B \in \mathbb{C}^{3\times 3}$ is a symmetric matrix with positive-definite real part.
The second sum shows that $\theta$ takes real values if $B$ and $\mathbf{x}$ are~real.
This definition is slightly different from the usual ones, where $B$ is taken either with
positive definite imaginary part or with negative definite real part. We choose this version
of the Riemann theta function in order to highlight the real numbers. Namely,
the \emph{theta divisor}
\begin{equation}
\label{eq:thetadivisor}
\Theta_B \,\,:=\,\, \{ \,\mathbf{x} \,\,|\,\, \theta(\mathbf{x},B) = 0\, \}\, \,\,\subset \,\, \,\mathbb{C}^3.
\end{equation}
restricts to a real analytic surface when both $\mathbf{x}$ and $B$ are real. In general, we will show that the theta divisor coincides with the theta surface above. To this end, we choose a symplectic basis $\alpha_1,\beta_1,\alpha_2,\beta_2,\alpha_3,\beta_3$ for $H_1(\mathcal{Q},\mathbb{Z})$.
The intersection product on $\mathcal{Q}$ is given~by
\begin{equation*}\label{symplecticBasis}
(\alpha_j \cdot \alpha_k) =0, \qquad (\beta_j \cdot \beta_k )=0, \quad {\rm and} \quad (\alpha_j \cdot \beta_k) = \delta_{jk}.
\end{equation*}
The {\em period matrix} for the Riemann surface $\mathcal{Q}$ with respect to this
basis equals
\begin{align}\label{periodMatrix}
\Pi \,\,= \,\,\left( \Pi_{\alpha} \,|\, \Pi_{\beta} \right) \,\,\in\,\, \mathbb{C}^{3\times 6},
\end{align}
with entries $(\Pi_{\alpha})_{jk} = \int_{\alpha_k} \omega_j$ and $(\Pi_{\beta})_{jk} = \int_{\beta_k}\omega_j$. As a consequence of Riemann's relations \cite[Theorem 2.1]{Mum1}, the matrix $\Pi_{\alpha}$ is invertible,
and the corresponding \emph{Riemann matrix} is
\begin{equation}
\label{eq:riemannmatrix}
B \,\,=\,\, -i \cdot \Pi_{\alpha}^{-1}\Pi_{\beta} .
\end{equation}
Riemann's relations also show that the matrix $B$ is symmetric with positive-definite real part, so we can consider the theta divisor $\,\Theta_B \subset \mathbb{C}^3\,$ as in
(\ref{eq:thetadivisor}).
A fundamental theorem of Riemann implies that this coincides with our theta surface, up to a change of coordinates.
\begin{theorem}[Riemann's Theorem] \label{thm:riemann}
The theta surface $\mathcal{S}$ and the theta divisor $\Theta_B$ coincide up to an affine change
of coordinates on $\mathbb{C}^3$. More precisely, there is a vector $c\in \mathbb{C}^3$ such~that
\begin{equation}
\mathcal{S} \,\,= \,\,\Pi_{\alpha} \cdot \Theta_B + c,
\end{equation}
where the equality is meant on all points where the parametrized surface $\mathcal{S}$ is defined.
\end{theorem}
\begin{proof}
We outline how to obtain this result from the usual statement of Riemann's Theorem.
Let $(\eta_1,\eta_2,\eta_3)$ be the basis of $H^0(\mathcal{Q},\Omega^1_{\mathcal{Q}})$
obtained from $(\omega_1,\omega_2,\omega_3)$ by
coordinate change with the matrix $\Pi_{\alpha}^{-1}$. Then $\int_{\alpha_j}\eta_k = \delta_{kj}$
and $\int_{\beta_j}\eta_k = i \cdot B_{kj}$. Fix a point $r \in \mathcal{Q}$,
paths from $r$ to $p_1(0)$ and from $r$ to $p_2(0)$, and
local coordinates $s,t$ on $\mathcal{Q}$ around $p_1(0)$ and $p_2(0)$.
For $s,t$ small, we consider paths from $r$ to $p_1(s)$ and from $r$ to $p_2(t)$. This gives an analytic~map
\begin{equation*} \label{eq:Omega1Omega2}
(s,t) \,\,\,\mapsto \,\,\,\begin{pmatrix} \int_{r}^{p_1(s)} \eta_1 \\ \int_{r}^{p_1(s)} \eta_2 \\ \int_{r}^{p_1(s)} \eta_3 \end{pmatrix}
\, +\, \begin{pmatrix} \int_{r}^{p_2(t)} \eta_1 \\ \int_{r}^{p_2(t)} \eta_2 \\ \int_{r}^{p_2(t)} \eta_3 \end{pmatrix}.
\end{equation*}
The familiar form of Riemann's theorem \cite[Theorem 3.1]{Mum1} shows that
there is a constant $\kappa \in \mathbb{C}^3$ such that the image of this
map coincides with $\Theta_{B}-\kappa$. Now, it is enough to write
\begin{equation*}
\int_{r}^{p_1(s)} \!\! \! \eta_j \,\,=\,\, \int_r^{p_1(0)}\!\!\! \eta_j \,\,+\,\, \int_{p_1(0)}^{p_1(s)}\!\!\!\eta_j \qquad
{\rm and} \qquad \int_{r}^{p_2(t)}\!\!\! \eta_j \,\,=\,\, \int_r^{p_2(0)}\!\!\! \eta_j\,\, +\,\, \int_{p_2(0)}^{p_2(t)}\!\!\! \eta_j,
\end{equation*}
and then change the coordinates from the basis $(\hspace{-0.5mm}\eta_1,\hspace{-0.5mm}\eta_2,\hspace{-0.5mm}\eta_3\hspace{-0.5mm})$ back to the basis $(\hspace{-0.5mm}\omega_1,\hspace{-0.5mm}\omega_2,\hspace{-0.5mm}\omega_3\hspace{-0.5mm})$.
\end{proof}
Our discussion shows that each theta surface is also a surface of double translation, and the Riemann theta function provides an implicit equation when the quartic $\mathcal{Q}$ is smooth. A fundamental result of Lie
states that all \emph{nondegenerate} surfaces of double translation whose two parametrizations are \emph{distinct} arise in this way.
Here {nondegenerate} and distinct are technical conditions.
The precise definition, phrased in modern language, can be found in~\cite[Definition 2.2]{Little83}. In particular, the nondegeneracy hypothesis assures that none of the generating curves
can be a line. This rules out special surfaces such as cylinders or planes.
We recall briefly Lie's construction. Let
$\mathcal{S}=\mathcal{C}_1+\mathcal{C}_2=\mathcal{C}_3+\mathcal{C}_4$ be any surface of double translation in $\mathbb{C}^3$. Then we can identify the tangent lines to the curves $\mathcal{C}_j$ with points in $\mathbb{P}^2$, and taking all these tangent lines we obtain analytic arcs
$\dot{\mathcal{C}}_1,\dot{\mathcal{C}}_2,\dot{\mathcal{C}}_3,\dot{\mathcal{C}}_4 \subset \mathbb{P}^2$. If $\mathcal{S}$ is a theta surface then, by Remark \ref{rmk:quarticfromcurves}, all these arcs lie on a common quartic curve $\mathcal{Q}$. Lie proved that this property holds for all
nondegenerate surfaces of double translation~$\mathcal{S}$.
\begin{theorem}[Lie's Theorem] \label{thm:lie}
The arcs $\,\dot{\mathcal{C}}_1,\dot{\mathcal{C}}_2,\dot{\mathcal{C}}_3,\dot{\mathcal{C}}_4\,$ lie on a common
reduced quartic curve $\mathcal{Q}$
in the projective plane $\mathbb{P}^2$, and $\mathcal{S} $
coincides with the theta surface associated to~$\mathcal{Q}$.
\end{theorem}
Lie's original proof \cite{Lie} involves a complicated system of differential equations
satisfied by the parametrizations of the curves $\mathcal{C}_j$. These differential equations force the arcs
$\dot{\mathcal{C}}_j$ to lie on a quartic. A much simpler proof was subsequently given by Darboux \cite{Darb}.
A modern exposition is found in Little's paper \cite{Little83}, together with generalizations to higher dimensions.
\section{Symbolic Computations for Special Quartics}
\label{sec3}
There is a fundamental dichotomy in the study of theta surfaces, depending on the
nature of the underlying quartic in $\mathbb{P}^2$. If it is a smooth quartic
then Riemann's Theorem \ref{thm:riemann} furnishes the defining equation
of the theta surface.
This is the case to be studied numerically in Section \ref{sec4}.
At the other end of
the spectrum are the singular quartics considered in the classical literature.
Here methods from computer algebra can be used to compute the theta surface. This is our
topic in the current section. Our focus lies on evaluating the
abelian integrals in~\eqref{eq:omega} by exact symbolic computations,
as opposed to numerical evaluations of the integrals.
We begin by explaining these methods for our running example from the Introduction.
\begin{example}[Scherk's minimal surface] \label{ex:scherkalgo1}
We here derive \eqref{eq:scherkpara1} from \eqref{eq:quartic1}. This serves as a first
illustration for Algorithm \ref{alg:thetaSurface2} below.
We start with the quartic $q(x,y)=xy(x^2+y^2+1)$. This is the
dehomogenization of \eqref{eq:quartic1} with respect to $z$. Using the
partial derivatives $q_y(x,y)=x^3+3xy^2+x$ and $q_x(x,y)=3x^2y + y^3 + y$,
we compute the differential forms in
(\ref{eq:omegabasis1}) and (\ref{eq:omegabasis2}).
We choose to evaluate the integrals in (\ref{eq:omega})
over the lines $y=0$ and $x=0$. These will give us the two summands in
the parametrization (\ref{eq:stmap}). The first summand
is obtained by setting $x=s$ and $y=0$, so that $q_y(s,0)=s(s^2+1)$,
and by computing the antiderivatives of the resulting specialized forms $\omega_j(s,0)$
for $j=1,2,3$. The three coordinates of $\Omega_1(p_1(s))$ are
$$ \begin{matrix}
\Omega_{11}(p_1(s)) & =
&\int\frac{s}{s(s^2+1)}d s &=& \arctan(s), \smallskip \\
\Omega_{21}(p_1(s)) & =
& \int \frac{0}{s(s^2+1)}d s & =& 0, \smallskip \\
\Omega_{31}(p_1(s)) & =
& \int \frac{1}{s(s^2+1)} d s & =& \log(s) - \frac{1}{2}\log(s^2+1).
\end{matrix}
$$
The second summand is obtained by integrating over the line $x=0$,
with parameter $y=t$, so that $q_x(0,t)=t(t^2+1)$ in (\ref{eq:omegabasis2}).
The three coordinates of $\Omega_2(p_2(t))$ are found to be
$$ \begin{matrix}
\Omega_{12}(p_2(t)) & =
& - \int \frac{0}{t(t^2+1)}d t & = & 0, \smallskip \\
\Omega_{22}(p_2(t)) & =
& - \int\frac{t}{t(t^2+1)}d t & = & -\arctan(t), \smallskip \\
\Omega_{32}(p_2(t)) & =
&- \int\frac{1}{t(t^2+1)}d t & = & -\log (t)+\frac{1}{2}\log(t^2+1).
\end{matrix}
$$
By adding these integrals, we obtain the parametrization of the corresponding theta surface:
\begin{equation}
\label{eq:scherkparara}
X \,=\, \arctan(s)\,,\quad
Y\,=\, -\arctan(t)\,,\quad
Z\,=\, \log \Big(\frac{s}{\sqrt{s^2+1}}\Big) -\log \Big(\frac{t}{\sqrt{t^2+1}}\Big).
\end{equation}
This is Scherk's minimal surface (\ref{eq:scherkpara1}). Trigonometry now yields
the implicit equation in (\ref{eq:scherk}).
\end{example}
\smallskip
The following algorithm summarizes the steps we have performed in Example \ref {ex:scherkalgo1}.
Starting from the quartic curve in $\mathbb{P}^2$, we compute the two generating curves of
the associated theta surface in $\mathbb{C} ^3$. This is done by evaluating the
integrals in (\ref{eq:omega}) as explicitly as possible.
\medskip
\begin{algorithm}[H]\label{alg:thetaSurface2}
\KwIn{ A quartic equation $q(x,y)$ describing a reduced
plane quartic curve.}
\KwOut{The parametrization (\ref{eq:stmap}) of the theta surface $\mathcal{S}$ in affine $3$-space.}
\KwSty{Step 1:} Specify two points $p_1$ and $p_2$ on the quartic. \\
\KwSty{Step 2:} Fix local parameters $(x_1,y_1)$ and $(x_2,y_2)$
around $p_1$ and $p_2$ respectively. \\
\KwSty{Step 3:} Write $y_j$ as an algebraic function in $x_j$ on its branch. \\
\KwSty{Step 4:} Compute the partial derivative $q_y$ on the two branches. \\
\KwSty{Step 5:} Substitute $x_1=s$ and $x_2=t$ into the differential forms
$\omega_1,\omega_2,\omega_3$ in (\ref{eq:omegabasis1}). \\
\KwSty{Step 6:} By integrating these differential forms, compute the vectors
\begin{equation*}
\begin{matrix}
\Omega_1(p_1(s)) & = & (\int \frac{s}{q_y(s,y_1(s))} ds , \int \frac{y_1(s)}{q_y(s,y_1(s))} ds , \int \frac{1}{q_y(s,y_1(s))} ds), \\
\Omega_2(p_2(t)) & = &
(\int \frac{t}{q_y(t,y_2(t))} dt,\int \frac{y_2(t)}{q_y(t,y_2(t))} dt,\int \frac{1}{q_y(t,y_2(t))} dt).
\end{matrix}
\end{equation*}
\hspace{-1.8mm}
\KwSty{Step 7:}
Output the sum $\Omega_1(p_1(s))+\Omega_2(p_2(t))$ of the generating curves as in~(\ref{eq:stmap}).
\caption{Computing the parametrized theta surface from its plane quartic}
\end{algorithm}
\medskip
One important part of Algorithm \ref{alg:thetaSurface2}
is Step 3, where $y_j$ is
represented as an algebraic function in $x_j$.
This function has algebraic degree at most four,
so it can be written in radicals. After all, the steps above are meant as a symbolic algorithm.
However, we found the representation in radicals to be
infeasible for practical computations unless the quartic is very special.
Even more crucial is the computation of the indefinite integrals in Step 6. This can be done explicitly whenever the quartic is reducible and all the components are
rational: in that special case, the holomorphic differentials of (\ref{eq:omegabasis1}) restrict to a differential $\frac{f(t)}{g(t)}dt$ on $\mathbb{P}^1$ where $f(t),g(t)$ are polynomials, and any such expression can be integrated symbolically.
In what follows we focus on instances where the quartic $q(x,y)$ is reducible.
A reducible plane quartic is one of the followings: four straight lines,
a conic and two straight lines, two conics, or a cubic and a line. In the sequel, we show the computations of
such theta surfaces using Algorithm \ref{alg:thetaSurface2}.
The first three cases were worked out by Richard Kummer~\cite{Kum} in his thesis, and the latter case was
studied by Georg Wiegner~\cite{Wie}. We here present three further examples of non-algebraic theta surfaces. The algebraic ones will be discussed in Section~\ref{sec6}.
Consider the first three of the four cases above. Then $q(x,y)$ factors into two conics.
The two conics meet in four points in $\mathbb{P}^2$. We consider the pencil of conics
through these~points.
\begin{remark} \label{rmk:onsameconic}
A result due to Lie states that
the theta surface $\mathcal{S}$
for a product of two conics only depends on their pencil, provided
$p_1$ and $p_2$ lie on the same conic (cf.~\cite[Section 3]{Kum}).
Hence $\mathcal{S}$ has infinitely many distinct
representations $\mathcal{C}_1+\mathcal{C}_2$ as a translation surface.
We shall return to this topic in Theorem \ref{thm:lietetra},
where it is shown how to compute these representations.
\end{remark}
For instance, for Scherk's surface in Example~\ref{ex:scherkalgo1},
the four points in $\mathbb{P}^2$ are $(i:0:1),(-i:0:1),(0:i:1),(0:-i:1)$,
and the pencil is generated by $xy$ and $x^2+y^2+z^2$.
We can replace these two quadrics by any other pair in the pencil
and obtain two generating curves.
We now examine another case which is similar. It will lead to the theta surface in
(\ref{eq:fourterms}).
\begin{example} \label{eq:amoebaofline}
Fix the four points $(1:0:0),(0:1:0),(0:0:1),(1:1:1)$ in $\mathbb{P}^2$. Their pencil of conics
is generated by $y(x-z)$ and $(x-y)z$. We multiply the first conic with a general member of the pencil to get
the quartic $Q=y(x-z) \cdot ((1+\lambda )xy-xz-\lambda yz)$. Here $\lambda $ is a parameter,
which we included in order to illustrate Remark \ref{rmk:onsameconic}.
Dehomogenizing $Q$, we get $q=y(x-1)((1+\lambda)xy-x-\lambda y)$. We compute
$\omega_1,\omega_2,\omega_3$ from the partial derivatives
\begin{align*}
&q_x\,=\,y((1+\lambda)xy-x-\lambda y)+y(x-1)((1+\lambda)y-1)),& \\
&q_y\,=\,(x-1)((1+\lambda)xy-x-\lambda y)+y(x-1)((1+\lambda)x-\lambda).&
\end{align*}
We integrate over the two lines given by $y(x-1)=0$. On the first line
$\{y=0\}$, we have $q_y=-x(x-1)$. The indefinite forms of the three abelian integrals
in $\Omega_1(p_1(s))$ are
$$ \begin{matrix} \int\frac{s}{-s(s-1)}d s \,=\, \log \left(\frac{1}{s-1}\right)\,,\quad
\int \frac{0}{-s(s-1)}d s\,=\, 0 \,,\quad
\int \frac{1}{-s(s-1)} d s\,=\, \log \bigl(\frac{s}{s-1} \bigr).
\end{matrix}
$$
For the line $\{x-1=0\}$ we transform the differentials by
passing from \eqref{eq:omegabasis1} to \eqref{eq:omegabasis2}. We~find
$$ \begin{matrix}
\int \frac{1}{-t(t-1)}d t = \log \bigl( \frac{t}{t-1} \bigr) \,,\quad
\int \frac{t}{-t(t-1)}d t= \log\left(\frac{1}{t-1}\right) \, , \quad
\int \frac{1}{-t(t-1)}d t = \log \bigl( \frac{t}{t-1} \bigr).
\end{matrix}
$$
We conclude that the resulting theta surface has the parametric representation
\begin{equation}\label{thetaSurfex1}
\begin{matrix}
&X & = & \log \left(\frac{1}{s-1}\right)\,+\,\log \bigl( \frac{t}{t-1} \bigr),\smallskip \\
& Y & = & \qquad 0 \,\,\,\, + \,\,\,\log\left(\frac{1}{t-1}\right), \smallskip \\
&Z & = & \log \bigl(\frac{s}{s-1} \bigr) \,+\,\log \bigl( \frac{t}{t-1} \bigr) .
\end{matrix}
\end{equation}
\end{example}
\begin{remark} \label{rmk:implicitization}
The output of Algorithm \ref{alg:thetaSurface2} looks
like (\ref{eq:scherkparara}) or (\ref{thetaSurfex1}). It gives the
theta surface $\mathcal{S}$ in parametric form. Whenever the quartic is \emph{rational nodal}, meaning that
all the components are rational and with at most nodes as singularities,
as in Figure \ref{fig:nodalquartics},
the expressions
found for $X$, $Y$ and $Z$ are $\mathbb{C} $-linear combinations
of logarithms of linear functions in $s$ and $t$. Indeed, since the singularities are nodal,
the differentials of \eqref{eq:omegabasis1} restrict to each rational component $\Gamma\cong \mathbb{P}^1$
as meromorphic differentials with at most simple poles \cite[Chapter 2]{ACG}. Each such differential can be written as a
sum of terms of the form $\frac{1}{(t-a)}$, which integrate to $\log(t-a)$.
From a representation as a $\mathbb{C}$-linear combination of logarithms we find an implicit equation for
$\mathcal{S}$ by familiar elimination techniques
from symbolic computation, such as {\em resultants} or {\em Gr\"obner bases}.
Namely, we choose constants $\alpha, \beta, \gamma \in \mathbb{C} $ such that
${\rm exp}(\alpha X)$, ${\rm exp}(\beta Y)$ and ${\rm exp}(\gamma Z)$
are written as rational functions in $s$ and $t$, and we then eliminate
$s$ and $t$ to obtain a trivariate polynomial $\Psi(u,v,w)$ such that
$\mathcal{S}$ is defined locally~by
\begin{equation}
\label{eq:Psieqn}
\Psi \bigl( \,{\rm exp}(\alpha X), \,{\rm exp}(\beta Y),\,{\rm exp}(\gamma Z)\, \bigr) \,\,\, = \,\,\, 0 .
\end{equation}
For the output (\ref{eq:scherkparara}), we take $\alpha = \beta = i$, $\,\gamma = 1$ and
$\,\Psi = uv^2w-u^2v-uw+v $. With these choices, (\ref{eq:Psieqn})
is precisely the implicit
equation (\ref{eq:scherk}) of Scherk's surface, times a constant.
For Example~\ref{eq:amoebaofline}, the implicit equation is
especially nice. The output (\ref{thetaSurfex1}) suggests the choices
$\alpha=\beta=\gamma=1$ and $\,\Psi = u+v-w+1$, and hence the theta surface is given by
\begin{equation} \label{eq:fourterms} {\rm exp}(X) \,+\, {\rm exp}(Y) \,-\, {\rm exp}(Z) \,\,= \,\,-1 .
\end{equation}
An explanation for the occurrence of such exponential sums is offered
in Section~\ref{sec5}.
\end{remark}
We present two more illustrations of our
methodology for special quartics. In each case we carry out both
Algorithm \ref{alg:thetaSurface2}
and the subsequent implicitization step as in Remark~\ref{rmk:implicitization}.
\begin{example}
Consider the quartic $q=xy(1-x^2+y^2)$. This corresponds to the pencil of conics
through $(0:i:1),(0:-i:1),(1:0:1),(-1:0:1)$.
For the abelian integrals, we compute the partial derivatives $q_x=y(y^2-x^2+1)-2x^2y$ and $q_y=x(y^2-x^2+1)+2xy^2$.
We first integrate the differential forms in (\ref{eq:omegabasis2}) over the line $y=0$, with local parameter $x=s$.
On this component, $q_x=x(1-x^2)$. The indefinite integrals are found to be
$$ \begin{matrix}
\! \int \!\! \frac{s}{-s(s^2-1)}d s = \frac{1}{2}\log (\frac{{s{+}1}}{{s{-}1}})), \quad \quad
\int \!\! \frac{0}{-s(s^2-1)}d s = 0, \quad \quad
\int \!\! \frac{1}{-s(s^2-1)} d s = \log(s) - \!\frac{1}{2} \log (s^2{-}1).
\end{matrix}
$$
We next integrate over the line $x=0$, using
(\ref{eq:omegabasis1}) with $q_y=y(y^2+1)$.
The abelian integrals~are
$$ \begin{matrix}
\int\frac{0}{-t(1+t^2)}d t \,=\, 0 \, , \quad
\int \frac{t}{-t(1+t^2)}d t \,=\, -\arctan (t) \, , \quad
\int \frac{1}{-t(1+t^2)} d t \,=\, -\log(t) \,+\, \frac{1}{2} \log (1+t^2).
\end{matrix}
$$
We conclude that the output of Step 7 in Algorithm \ref{alg:thetaSurface2} equals
\begin{align*
&X \,\,=\,\, \frac{1}{2} \log (s+1) - \frac{1}{2} \log (s-1) , \nonumber\\
& Y \,\,=\,\, -\arctan (t) \, = \, -\frac{1}{2i} \log (t-i) + \frac{1}{2i} \log(t+i), \\
&Z\,\,=\,\, \log(s)- \frac{1}{2} \log (s^2-1)-\log(t) + \frac{1}{2} \log (1+t^2) \nonumber.
\end{align*}
From this we find that the implicit equation of the theta surface equals
$$
-2 \,{\rm exp}(Z)\,{\rm exp}(X)\,{\rm sin}(Y) \, - \, {\rm exp}(2X) \,- \,1 =0.
$$
\end{example}
Finding the two pairs of generating curves on a theta surface is particularly pleasant
when the underlying quartic is the union of four lines in $\mathbb{P}^2$ that
are defined over~$ \mathbb{Q}$.
\begin{example} \label{ex:fournicelines}
Consider the quartic $q = (y+x-1)(y-x-1)(y+x+1)(y-x+1)$.
For the first pair of generating curves,
we compute the abelian integrals over the first two lines:
$$ \begin{matrix}
\int \! \frac{1}{8(s - 1)}d s \,=\, \frac{1}{8}\log(s-1) ,&
\int \! - \frac{1}{8s}d s \,=\, -\frac{1}{8}\log(s) ,&
\int \! \frac{1}{8s(s-1)} d s \,=\, \frac{1}{8} \log (s -1)- \frac{1}{8} \log (s), \\
\int \! \frac{1}{8(t + 1)}d t \,=\, \frac{1}{8} \log(t + 1) ,&
\int \! \frac{1}{8t}d t= \frac{1}{8} \log(t) ,&
\int \! \frac{1}{8t(t+1)} d t \,=\, - \frac{1}{8}\log(t + 1) + \frac{1}{8} \log(t).
\end{matrix}
$$
Hence, the coordinates of the theta surface are given in terms of the parameters $s$ and $t$ by
\begin{align*}\label{thetaSurfaceEx3}
& 8X \,\,=\,\, \log(s - 1)+\log(t + 1), \nonumber\\
& 8Y\,\,=\,\, -\log(s)\,+\,\log(t), \\
&8Z\,\,=\,\, \log(s - 1) - \log(s)-\log(t + 1) + \log(t) \nonumber.
\end{align*}
Choosing $\alpha=\beta=\gamma=8$ in Remark \ref{rmk:implicitization},
and hence $\,u={\rm exp} (8X),\,v={\rm exp} (8Y)$ and $ w={\rm exp} (8Z)$, the
implicit equation (\ref{eq:Psieqn}) of the theta surface is represented by the polynomial
$$ \Psi(u,v,w) \,\, = \,\, u^2 v w^2-2 u^2 v w-u v^2 w+u v w^2+u^2 v-4 u v w-v^2 w+u v-u w-2 v w -w .$$
The same method works fairly automatically for any arrangement of four lines in the plane.
\end{example}
\section{Degenerations of Theta Functions}
\label{sec5}
We saw in Theorem \ref{thm:riemann}
that the equation of the theta surface
associated with a smooth plane quartic is a
Riemann theta function. However, all
theta surfaces seen in the classical literature were computed
from quartics that are singular. In this section we explain
how singularities induce degenerate theta functions.
These are finite sums of exponentials, given combinatorially by the
cells in the associated Delaunay subdivision of~$\mathbb{Z}^3$.
This furnishes a conceptual explanation of the
equations defining theta surfaces like (\ref{eq:scherk})~or~(\ref{eq:fourterms}).
Our approach is from the point of tropical geometry,
which mirrors the theory of toroidal
degenerations~\cite{GruHu}.
In what follows we focus on the \emph{rational nodal quartic curves}.
These are quartic curves whose irreducible components are rational.
The special properties of such curves were already discussed briefly in Remark \ref{rmk:implicitization}.
Rational nodal quartics
appear in five types: an irreducible quartic with three nodes, a nodal cubic together with a line, two smooth conics, a smooth conic with two lines, and
an arrangement of four lines.
\begin{figure}[]
\begin{center}$
\begin{array}{lllll}
\includegraphics[width=20mm]{nodalrationalquartic.pdf}&
\includegraphics[width=20mm]{nodalcubicline.pdf}&
\includegraphics[width=20mm]{twoconics.pdf}&\,\,
\includegraphics[width=12mm]{conictwolines.pdf}&\,\,
\includegraphics[width=20mm]{fourlines.pdf}
\end{array}$
\end{center}
\caption{The five types of rational nodal quartics.
\label{fig:nodalquartics}}
\end{figure}
To each of these curves we associate its dual graph $\Gamma = (V,E)$.
This has a vertex for each irreducible component and one edge for each intersection point between two components. A node on an irreducible component counts as an intersection of that component with itself.
\begin{figure}[h]
\begin{center}$
\begin{array}{lllll}
\includegraphics[width=20mm]{graph_nodalrationalquartic.pdf}&
\includegraphics[width=20mm]{graph_nodalcubicline.pdf}&
\includegraphics[width=20mm]{graph_twoconics.pdf}&
\includegraphics[width=20mm]{graph_conictwolines.pdf}&
\includegraphics[width=20mm]{graph_fourlines.pdf}
\end{array}$
\end{center}
\caption{The dual graphs corresponding to the five types of rational nodal quartics.
\label{fig:graphs}}
\end{figure}
The dual graphs play a key role in tropical geometry,
namely in the tropicalization of curves and their Jacobians.
We follow the combinatorial construction in \cite[Section 5]{BBC}.
Fix one of the graphs $\Gamma$ in Figure~\ref{fig:graphs}
with an orientation for each edge. The first homology group
\[ H_1(\Gamma,\mathbb{Z}) \,\,=\,\,
\ker \left( \partial\colon \mathbb{Z}^{E} \to \mathbb{Z}^{V} \right) \]
is free abelian of rank $3$.
Let us fix a $\mathbb{Z}$-basis $\gamma_1,\gamma_2,\gamma_3$ for
$H_1(\Gamma,\mathbb{Z})$. Each $\gamma_j$ is a directed
cycle in $\Gamma$. We encode our basis in a $3 \times |E|$ matrix $\Omega$.
The entries in the $j$-th row of $\Omega$ are the coordinates
of $\gamma_j$ with respect to the standard basis of $\mathbb{Z}^E$.
Our five matrices $\Omega$~are
$$ \!
\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\! ,\,
\begin{bmatrix} 1 & \!\! \!-1 & 0 & 0 \\ 0 & 1 & \!\! -1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\! , \,
\begin{bmatrix} 1 & \!\! \! -1 & 0 & 0 \\ 0 & 1 & \!\! -1 & 0 \\ 0 & 0 & 1 & \!\! -1 \end{bmatrix} \! , \,
\begin{bmatrix} 1 & 0 & 1 & 0 & 0 \\ \! -1 & 1 & 0 & \!\! -1 & 0 \\ 0 & \!\!\! -1 & 0 & 0 & \!\! -1 \end{bmatrix}\! , \,
\begin{bmatrix} 1 & \!\! \!-1 & 0 & 1 & 0 & 0 \\ \! -1 & 0 & 1 & 0 & \!\! -1 & 0 \\ 0 & 1
&\! \!\! -1 & 0 & 0 & 1 \end{bmatrix}\!. $$
We define the \emph{Riemann matrix} of $\Gamma$ to be
the positive definite symmetric $3 \times 3$ matrix
\begin{equation}
\label{eq:periodtropical}
B \,\, := \,\,\Omega \cdot \Omega^T.
\end{equation}
If we change the orientations and cycle bases then the matrix
$B$ transforms under the action of ${\rm GL}(3,\mathbb{Z})$ by
conjugation. The Riemann matrices of our five graphs are
the matrices in the fourth column in Figure~\ref{fig:vallentin}.
The label \ {\tt Form} \ refers to the quadratic form
represented by~$B$.
\begin{figure}
\includegraphics[scale=0.83]{val49.pdf}
\caption{The Voronoi polytopes corresponding to the
five types of rational nodal quartics, in reverse order to
Figure \ref{fig:graphs}. This diagram is taken
from the dissertation of Frank Vallentin \cite[Section 4.3, page 49]{Val}.
The first column gives the number $d$ of edges in the dual graph.
\label{fig:vallentin} }
\end{figure}
We now explain how a Riemann matrix $B$ of
the dual graph as in \eqref{eq:periodtropical}
induces a degeneration to a singular curve with dual graph
as in Figure~\ref{fig:graphs}. Let $B_0$ be a fixed
(real or complex) symmetric $3 \times 3$ matrix with positive definite real part.
Consider the one-parameter family of classical Riemann matrices
\begin{equation}
\label{eq:limitfamily}
B_t \,\,:=\,\, tB+B_0 \qquad \text{for } \,\, t\geq 0.
\end{equation}
We note that the real part of $B_t$ is always positive definite.
If the chosen matrix $B_0$ does not belong to the hyperelliptic locus, then
the set of positive real numbers $t$ such that $B_t$ lies on the hyperelliptic locus is discrete. Thus, almost all Riemann matrices $B_t$ correspond to non-hyperelliptic curves of
genus three, and hence to smooth quartic curves in the plane.
We consider the Riemann theta functions for these curves. More precisely, we
evaluate $\theta( \,\bullet\,, B_t)$ at ${\bf x}$ translated
by the vector $i\cdot tB\mathbf{a}$, where $\mathbf{a} = (a_1,a_2,a_3)^T \in \mathbb{R}^3$. This gives
$$ \begin{matrix}
\theta \bigl( \mathbf{x}-i\cdot tB\mathbf{a},B_t \bigr) & = & \sum_{n\in \mathbb{Z}^3} \mathbf{e}\left(-\frac{1}{2}n^T(tB+B_0) n + i\cdot n^T\left(\mathbf{x}-i\cdot tB\mathbf{a}\right) \right) \medskip \\
& = & \sum_{n\in \mathbb{Z}^3}\mathbf{e} \left( - \frac{1}{2}\left(n^T Bn-2n^TB\mathbf{a} \right)t - \frac{1}{2}n^TB_0 n + i\cdot n^T\mathbf{x}\right) \medskip \\
& = & \sum_{n\in \mathbb{Z}^3} \mathbf{e}\left(-\frac{1}{2}(n^T Bn-2n^TB\mathbf{a})t \right)\cdot \mathbf{e}\left( - \frac{1}{2}n^TB_0n+i\cdot n^T\mathbf{x} \right) \label{eq:thetalimit}.
\end{matrix}
$$
As $t\to +\infty$,
the term $\mathbf{e}\left(-\frac{1}{2}(n^T Bn-2n^TB\mathbf{a})t \right)$ converges
if and only if $\,n^TBn-2n^TB\mathbf{a}\geq 0$.
For each $n \in \mathbb{Z}^3$, this condition is a linear inequality in ${\bf a}$, which can be
rewritten as follows:
\begin{equation}
\label{eq:voronoicell}
\mathbf{a}^T B \mathbf{a} \,\,\,\leq \,\,(\mathbf{a}-n)^TB(\mathbf{a}-n).
\end{equation}
Hence, in order for the theta function above to converge to a degenerate theta
function, we must choose $\mathbf{a}$ in such a way that \eqref{eq:voronoicell} is satisfied for every $n\in \mathbb{Z}^3$. The positive definite quadratic form given by $B$
defines a metric on $\mathbb{R}^3$. The condition \eqref{eq:voronoicell}
says that, among all lattice points $n$ in $\mathbb{Z}^3$,
the origin is a closest one to $\mathbf{a}$ in this metric.
This means that $\mathbf{a}$ is contained in the {\em Voronoi cell}
with respect to the lattice $\mathbb{Z}^3$ and the metric defined by~$B$.
The Voronoi cell is a $3$-dimensional polytope. It belongs to the
class of {\em unimodular zonotopes}.
The third column of Figure \ref{fig:vallentin}
shows this for each of the five types of nodal
quartics in Figure \ref{fig:nodalquartics}. The translates of the Voronoi cell
by vectors in $\mathbb{Z}^3$
define a tiling of $\mathbb{R}^3$. Their boundaries
form an infinite $2$-dimensional polyhedral complex.
See \cite[Figure 15]{BBC} for the case of four lines, on the right
in Figures \ref{fig:nodalquartics} and \ref{fig:graphs}.
This surface is the {\em tropical theta divisor} in $\mathbb{R}^3$.
It can be viewed as a polyhedral model for our
degenerate theta surface. For details on these objects
see \cite{BBC,Chan} and references therein.
We encourage our readers to~spot Figure \ref{fig:graphs} in \cite[Figures 1 and 8]{Chan} and to
examine the tropical Torelli map in \cite[Theorem 6.2]{Chan}.
We now assume that $\mathbf{a}$ is a vertex of the Voronoi cell.
We write $\mathcal{D}_{{\bf a},B}$ for the set of all vectors
$n \in \mathbb{Z}^3$ for which equality holds in (\ref{eq:voronoicell}).
This set is finite for each Riemann matrix $B$ derived from a graph in
Figure~\ref{fig:graphs}. We note the following behavior for the summands above:
\begin{equation}
\label{eq:summandsabove}
\,\mathbf{e}\biggl(-\frac{1}{2}(n^T Bn-2n^TB\mathbf{a})t \biggr)
\,\,\, \xrightarrow{t \rightarrow +\infty} \,\,\,
\begin{cases}\, \,1 & {\rm if} \,\, n \in \mathcal{D}_{{\bf a},B},
\\ \,\,0 & {\rm if}\,\, n \in \mathbb{Z}^3 \backslash \mathcal{D}_{{\bf a},B}. \end{cases}
\end{equation}
From this we shall infer the following theorem, which is our main result in this section.
\begin{theorem}\label{thm:degtheta}
Fix a vertex ${\bf a}$ of the Voronoi cell for the
degeneration (\ref{eq:limitfamily}). The associated theta function
is the following finite sum over all vertices of the Delaunay polytope dual to~${\bf a}$:
\begin{equation}
\label{eq:finitethetasum}
\sum_{n\in \mathcal{D}_{{\bf a},B}}
\mathbf{e}\left( - \,\frac{1}{2}n^TB_0n\,+\,i\cdot n^T\mathbf{x} \right) .
\end{equation}
The number of summands in (\ref{eq:finitethetasum}) equals $\,8$ for a rational quartic,
$6$ for a nodal cubic plus line, $4$ or $6$ for two conics,
$4$ or $5$ for a conic plus two lines, and $4$ for four lines.
This recovers the equations for theta surfaces given by Eiesland in
\cite[eqns.(5),(6)]{Eies08} and \cite[eqn.(5)]{Eies09}.
\end{theorem}
\begin{proof}
The subdivision of $\mathbb{R}^3$ dual to the Voronoi decomposition
is the Delaunay subdivision; see \cite[Section 5]{BBC} or \cite[Section 4.2]{Chan}.
Its cells are dual to those of the Voronoi decomposition. In particular, each
vertex ${\bf a}$ of the Voronoi polytope corresponds to a $3$-dimensional
{\em Delaunay polytope}.
The vertices of the Delaunay polytope are the elements of the set $\mathcal{D}_{{\bf a},B}$.
For instance, consider the case of four lines, which is
listed last in Figures \ref{fig:nodalquartics}, \ref{fig:graphs} and
first in Figure \ref{fig:vallentin}. The Voronoi cell is the
{\em permutohedron}, also known as the truncated octahedron.
Each of its $24$ vertices ${\bf a}$ is dual to a tetrahedron
in the Delaunay subdivision. See \cite[Example 5.5]{BBC}
and the left diagram in \cite[Figure 15]{BBC}.
Hence all Delaunay polytopes are tetrahedra, i.e.~$|\mathcal{D}_{{\bf a},B}| = 4$.
This explains the four terms in
(\ref{eq:fourterms}) or~\cite[eqns.~(6)]{Eies08}.
The four other Delaunay subdivisions are obtained by
moving the matrix $B$ to a lower-dimensional stratum in the
tropical moduli space, proceeding downwards in \cite[Figure 8]{Chan}.
The resulting Delaunay polytopes are obtained by fusing the tetrahedra in
\cite[Figure 15]{BBC}. Hence all Delaunay polytopes
${\rm conv}(\mathcal{D}_{{\bf a},B})$
are naturally triangulated into unit tetrahedra.
We now present a list, up to symmetry, of all vertices ${\bf a}$
of the Voronoi polytopes. Our five special Riemann matrices $B$ here appear in
the order given in Figure \ref{fig:vallentin}. In each case, we display the
Delaunay set $\mathcal{D}_{{\bf a},B}$. This is the support
of the degenerate theta function in (\ref{eq:finitethetasum}):
$$
\begin{matrix}
\text{\em type of curve} & \text{\# \em orbit} & {\bf a}^T & \mathcal{D}_{{\bf a},B} \medskip \\
\text{$4$ lines} & 24 &
\bigl(\frac{3}{4},\frac{1}{2},\frac{1}{4} \bigr) & \{ (000),(100),(110),(111) \} \medskip \\
\text{conic + $2$ lines} & 8 &
\bigl(\frac{3}{4},\frac{1}{2},\frac{1}{4} \bigr) & \{ (000),(100),(110),(111) \} \smallskip \\
\text{conic + $2$ lines} & 10 &
\bigl(\frac{5}{8},\frac{1}{4},\frac{5}{8}\bigr) & \{ (000),(001),(100),(101), (111)\} \medskip \\
\text{$2$ conics} & 8 &
\bigl(\frac{3}{4},\frac{1}{2},\frac{1}{4} \bigr) & \{ (000),(100),(110),(111) \} \smallskip \\
\text{$2$ conics} & 6 &
\bigl(\frac{1}{2},1,\frac{1}{2}\bigr) & \{
(000),(010),(011),(110),(111),(121) \} \medskip \\
\text{cubic+line} & 12 &
\bigl(\frac{2}{3},\frac{1}{3},\frac{1}{2} \bigr) &
\{ (000),(001),(100),(101),(110),(111)\}
\medskip \\
\text{rational quartic} & 8 &
\bigl(\frac{1}{2},\frac{1}{2},\frac{1}{2} \bigr) &
\hspace{-10mm} \{ (000),(001),(010),(011)
\medskip \\
& &
& \quad
\hspace{9mm} (100),(101),(110), (111)
\} \medskip \\
\end{matrix}
$$
The column {\em \# orbit} gives the cardinality of
the symmetry class of the vertex ${\bf a}$ of the Voronoi cell.
We see that the Delaunay polytope
$\,{\rm conv}(\mathcal{D}_{{\bf a},B})\,$ is either a
tetrahedron, an Egyptian pyramid, an octahedron, or a cube.
Given the type in Figure~\ref{fig:nodalquartics},
for a suitable choice of Riemann matrix $B$
and Voronoi vertex ${\bf a}$, we recover precisely the
tetrahedron in \cite[eqn.~(6)]{Eies08},
the octahedron in \cite[eqn.~(6)]{Eies08},
and the cube in \cite[eqn.~(5)]{Eies09}.
Eiesland's coefficients $A,B,C,\ldots$ for these theta surfaces
are determined by the fixed symmetric matrix $B_0$
that define the degeneration~(\ref{eq:limitfamily}).
In conclusion, equation (\ref{eq:summandsabove}) implies that
$\,\theta \bigl( \mathbf{x}-i\cdot tB\mathbf{a},B_t \bigr) \,$
converges to the function given by the
finite sum in (\ref{eq:finitethetasum}),
with $\mathcal{D}_{{\bf a},B}$ as derived above.
\end{proof}
In the table above, the same four-element set $\mathcal{D}_{{\bf a},B}$
occurs for 4 lines, for conic + 2 lines and for 2 conics. This is explained by
Remark \ref{rmk:onsameconic} because all three choices
are possible for the basis of a fixed pencil of conics.
Our running example belongs to this tetrahedron case.
\begin{example}[Scherk's minimal surface] \label{ex:vierzwei}
We here use theta functions to recover the surface in Figure~\ref{polytope}.
The rational nodal quartic (\ref{eq:quartic1}) consists of a smooth conic and two lines.
The second row in Figure \ref{fig:vallentin} shows that the corresponding tropical period matrix is
$ \,B= \begin{tiny} \begin{pmatrix} \,\,2 & \! -1 & \,0 \,\\ -1 & 3 & \!\! -1 \\ \,\, 0 \,& \!\! -1 & 2 \,\end{pmatrix}\end{tiny} $.
We consider the degenerate theta function of Theorem \ref{thm:degtheta} with the data
$$ \begin{small}
B_0 \,=\, i\cdot \begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 &-1 \end{pmatrix},
\quad \mathbf{a} \,=\, \begin{pmatrix}
3/4 \\ 1/2 \\ 1/4 \end{pmatrix}, \quad \mathbf{x}
= \frac{1}{2\pi}\begin{pmatrix} 2X \\ -X+Y-iZ \\ -2Y \end{pmatrix} .
\end{small} $$
Then $\,\mathcal{D}_{\mathbf{a},B} \,=\, \{ (0,0,0),\, (1,0,0),\, (1,1,0),\, (1,1,1) \}$,
and the four-term sum in (\ref{eq:finitethetasum}) equals
\begin{equation}
\label{eq:scherkExp}
\begin{matrix} & 1 - {\rm exp}(2iX)+ {\rm exp}(iX+iY+Z) - {\rm exp}(iX-iY+Z) \\ = &
- 2i \cdot {\rm exp}(2i X) \cdot \bigl( \,
\sin(X) \,-\, \sin(Y)\exp(Z)\, \bigr).
\end{matrix}
\end{equation}
In the parentheses on the right we see
expression (\ref{eq:scherk}) from the beginning of this paper.
\end{example}
\begin{example}[Irreducible quartic]
This class of theta surfaces was studied by Eiesland in \cite{Eies09}.
His ``unicursal quartic'' is the rational quartic with three nodes
on the left in Figure \ref{fig:nodalquartics}. Here,
$B$ is the identity matrix and the Voronoi cell is the cube
with vertices $\left(\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2}\right)$.
Fixing the vertex $\mathbf{a}^T = \left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$,
the Delaunay polytope is the cube with vertex set $\{0,1\}^3$.
We choose an arbitrary real symmetric $3 \times 3$ matrix $B_0$, and we abbreviate
\begin{equation}
\label{eq:Ann} \begin{matrix} A_{n} \,\,: =\,\, \mathbf{e} \bigl(-\frac{1}{2}n^t B_0 n \bigr)
\qquad {\rm for} \,\, \,n \in \{0,1\}^3. \end{matrix}
\end{equation}
Note that $A_{(000)} = 1$.
Writing $\mathbf{x}= -i \cdot(X,Y,Z)$, the degenerate theta function in
(\ref{eq:finitethetasum}) equals
$$ \begin{matrix}
\lim_{t\to +\infty}\theta\left( \mathbf{x}-i\cdot tB\mathbf{a},B_t \right)
\,\,=\,\,\, 1
\,+\,A_{(100)}\,\mathbf{e}( X)
\,+\,A_{(010)}\,\mathbf{e}( Y)
\,+\,A_{(001)}\,\mathbf{e}( Z) \qquad \qquad \\ \qquad \qquad
+\,A_{(011)}\,\mathbf{e}( Y{+}Z)
+A_{(101)}\,\mathbf{e}( X{+}Z)
+A_{(110)}\,\mathbf{e}( X{+}Y)
+A_{(111)}\,\mathbf{e}( X{+}Y{+}Z).
\end{matrix}
$$
This is precisely the theta surface derived
in the theorem in \cite[page 176]{Eies09}. Here we have
\begin{equation*}
A_{(100)}A_{(010)}A_{(001)}A_{(111)} \,\,=\,\, A_{(000)} A_{(011)}A_{(101)}A_{(110)}.
\end{equation*}
This follows directly from (\ref{eq:Ann}), and it matches
Eiesland's identity in \cite[eqn.~(6)]{Eies09}.
\end{example}
\section{Algebraic Theta Surfaces}
\label{sec6}
Theta surfaces are usually transcendental. But, in some special cases, it can happen that~they are algebraic.
These cases were classified by Eiesland \cite{Eies08}. Our aim is to present his result.
\begin{example} \label{ex:TSdegree4}
We begin by showing that the following quartic surface is a theta surface:
\begin{equation} \label{eq:TSdegree4}
Y^4\,-\,4XY^2\,-\,4X^2\,+\,8Z\,\,=\,\,0.
\end{equation}
The underlying quartic curve consists of a cuspidal cubic together with its cuspidal tangent:
$$q\,=\,(y^2-x^3)y.$$
We evaluate the abelian integrals over the line $y=0$ and the cubic,
parametrized respectively $x=s$ and $x=t$. It turns out that all
required antiderivatives are algebraic functions, namely
$$ \begin{matrix}
X & = & \frac{1}{2}\int\frac{s}{s^3}d s \,-\,\int\frac{t}{t^3}d t & = & - \frac{1}{2s} +\frac{1}{t} ,\medskip \\
Y & = & \frac{1}{2}\int\frac{s^{3/2}}{s^3}d s \,-\,\int\frac{0}{t^3}d t & = & - \frac{1}{\sqrt{s}} \, , \medskip \\
Z & = & \frac{1}{2}\int\frac{1}{s^3}ds \,-\,\int\frac{1}{t^3}d t & = & -\frac{1}{4s^2}+\frac{1}{2t^2}.
\end{matrix} $$
This is a parametrization of the rational surface (\ref{eq:TSdegree4}), which is
singular along a line at~infinity.
\end{example}
\begin{example}\label{Quadric}
Another one is the quadric whose equation $3XY-Z=0$. It arises from the quartic
which is the union of the four concurrent lines $x=-y$, $x=y$, $2x=-y$, $2x=y$.
\end{example}
Eiesland \cite{Eies08} identified all scenarios where the abelian integrals are essentially rational.
\begin{theorem}[Eiesland] \label{thm:eiesland}
Every algebraic theta surface is rational and has degree $2, 3, 4, 5$~or~$6$.
The underlying quartic has rational components and none of its singularities are nodes.
\end{theorem}
In Example \ref{ex:TSdegree4} we already saw our first algebraic theta surface. In the next
examples we present further surfaces, of degrees $5, 4, 6, 3$, in this order. In each case,
the quartic curve satisfies the condition in the second sentence of Theorem~\ref{thm:eiesland}.
The section will conclude with a discussion on degenerations of abelian functions.
This will establish the link to Section~\ref{sec5}.
\begin{figure}[!htb]
\begin{minipage}{0.50\textwidth}
\centering
\includegraphics[width=0.95\linewidth]{CardioidSurfaceNoBox}\\
\end{minipage} \hfill
\begin{minipage}{0.50\textwidth}
\centering
\includegraphics[width=.6\linewidth]{CardioidSurfaceOld}\\
\end{minipage}
\vspace{-0.35in}
\caption{The cardioid surface in Example \ref{ex:cardiodsurface}.
On the right is Eiesland's plaster model~\cite{Eies08}.
\label{fig:CardioidSurface}}
\end{figure}
\begin{example}\label{ex:curvelittle}
The following toric curve a is rational quartic with a triple point:
$$ Q \,\,=\,\, x^4 - yz^3 .$$
Following \cite[Example 4.3]{Little83},
the abelian integrals on this curve evaluate to
$$ \begin{matrix}
X &= &\int sds +\int{t}d t & = & \frac{1}{2}(s^2+t^2), \smallskip \\
Y & = & \int s^4d s +\int t^4d t & = & \frac{1}{5}(s^5+t^5),\smallskip \\
Z & = & \int d s +\int d t & = & s+t.
\end{matrix} $$
Elimination of the parameters $s$ and $t$ reveals the equation for this quintic theta surface:
\begin{equation} \label{eq:TSdegree5}
Z^5-20X^2Z+20Y\,\,= \,\,0.
\end{equation}
Little \cite{Little83} generalizes this and other theta surfaces to higher dimensions.
He presents a geometric derivation of theta divisors in $\mathbb{C} ^g$ from
(degenerate) canonical curves of genus $g$.
\end{example}
Our next example shows how transcendental theta surfaces can degenerate to algebraic surfaces.
This kind of analysis plays an important role in Eiesland's proof of Theorem \ref{thm:eiesland}.
\begin{example} \label{ex:TSdegree3}
We consider an irreducible rational quartic with two singular points, namely
one ordinary cusp and one tacnode. The following curve
has these properties for general $\lambda$:
$$q\,\,=\,\,x^3y+\lambda x^4+(1-\lambda) x^2y -y^2 \qquad {\rm with} \quad
q_y\,\,=\,\,x^3 + (1-\lambda)x^2 -2y . $$
Following Eiesland's derivation in \cite[Section III]{Eies08}, we choose
the rational parametrization
$$ \begin{matrix}
x\,\,=\,\,\frac{w^2\,+\,(\lambda+1)w}{w\,+\,1}
\quad {\rm and} \quad
y\,\,= \,\,\frac{(w^2\,+\,(\lambda+1)w)^2}{w\,+\,1}
\end{matrix}
$$
The differential form $dx$ can be written in terms of the curve parameter $w$ as follows:
$$ \begin{matrix} dx \,\, = \,\, \bigl( 1 + \frac{\lambda}{(w+1)^2} \bigr) dw .
\end{matrix} $$
We substitute these expressions into \eqref{eq:omegabasis1},
and we find the three antiderivatives
$$ \begin{matrix}
\frac{\log(w+\lambda+1)}{\lambda+1} - \frac{\log(w)}{\lambda + 1} \, , \quad -w \, , \quad
\frac{(\lambda - 1)\log(w+\lambda+1 )}{(\lambda+1)^3}-\frac{(\lambda-1)\log(w )}{(\lambda+1)^3} -
\frac{(\lambda - 1)w - \lambda-1}{(\lambda+1)^2w^2+(\lambda+1)^3w}. \end{matrix}
$$
The two generating curves are found by setting $w=s$ and $w=t$. Hence our theta surface is
$$ \begin{matrix}
X & = & \frac{\log(s+\lambda+1)}{\lambda+1} - \frac{\log (s)}{\lambda + 1}
\,\,+\,\,\frac{\log(t+\lambda+1)}{\lambda+1} - \frac{\log(t)}{\lambda + 1} , \smallskip \\
Y & = & -\,s\,- t,\qquad \qquad \smallskip \\
Z & = & \frac{(\lambda - 1)\log(s+\lambda+1 )}{(\lambda+1)^3}-\frac{(\lambda-1)\log(s )}{(\lambda+1)^3} -
\frac{(\lambda - 1)s - \lambda-1}{(\lambda+1)^2s^2+(\lambda+1)^3s} \smallskip \\ &
&\quad \quad +\frac{(\lambda - 1)\log(t+\lambda+1 )}{(\lambda+1)^3}-\frac{(\lambda-1)\log(t )}{(\lambda+1)^3} - \frac{(\lambda - 1)t - \lambda-1}{(\lambda+1)^2t^2+(\lambda+1)^3t}.
\end{matrix} $$
From this parametrization we infer that the theta surface is transcendental for $\lambda \not= -1$.
Now let $\lambda=-1$. Then $q=x^3y-x^4+2yx^2-y^2$
and the tacnode is now a {\em node-cusp}.
This is a singular point obtained by merging a node and a cusp.
Using the calculus identity $ \,
{\rm lim}_{\lambda \rightarrow -1} \bigl\{\frac{\log(s+\lambda+1)}{\lambda+1}
- \frac{\log(s)}{\lambda + 1} \bigr\} =
\frac{d}{ds} {\rm log}(s) =
\frac{1}{s}$, the parametrization of our surface becomes
$$
\begin{matrix} X\,=\,\frac{1}{s }+\frac{1}{t },\quad \quad Y\,=\,-s-t, \quad \quad
Z\,=\,\frac{3s+2}{6s^3}+\frac{3t+2}{6t^3}. \end{matrix}
$$
Eliminating $s$ and $t$, we obtain the implicit equation.
$$ 2X^3Y+3X^2Y+6X^2-6YZ+6X\,\,=\,\,0. $$
Hence, the theta surface is a rational quartic for $\lambda=-1$,
and it is transcendental for $\lambda \not=-1$.
\end{example}
The maximum degree of any algebraic theta surface is six. The next example attains~this.
\begin{example} \label{ex:ats6}
Following Eiesland \cite[p.~381--383 and VI on p.~386]{Eies08},
we consider the quartic
$$ q \,\,= \,\, (y-x^2)^2 \,+\, 2xy(y-x^2) \,+\, y^3 . $$
The unique singular point, at the origin, is a {\em tacnode cusp}. The theta surface is given by
$$ \begin{matrix}
X & = & \int \! \frac{s+1}{s^4} ds \,+\, \int \! \frac{t+1}{t^4} dt & = &
-\frac{1}{2s^2} - \frac{1}{3s^3} \,- \,\frac{1}{2t^2} - \frac{1}{3t^3} ,
\smallskip \\
Y & = & \int \! \frac{1}{s^2}ds \, +\,\int \! \frac{1}{t^2}dt , & = &
-\frac{1}{s}\,-\,\frac{1}{t},
\smallskip \\
Z & = & \int \! \frac{2s+1}{s^6} ds \,+\,\int \! \frac{2t+1}{t^6} & = &
-\frac{1}{5s^5} - \frac{1}{2s^4} \,-\,\frac{1}{5t^5} - \frac{1}{2t^4}.
\\
\end{matrix}
$$
The implicit equation is found to be
$$
4 Y^6-24 Y^5-60 X Y^3+45 Y^4+180 X Y^2-180 X^2+180 Y Z-180 Z \,\,\,=\,\,\, 0.
$$
This sextic looks different from that in \cite[VI on p.~386]{Eies08}
because of a coordinate change.
\end{example}
We conclude our panorama of algebraic theta surfaces with two classical cubic surfaces.
\begin{example}[Cardioid Surface] \label{ex:cardiodsurface}
We first consider the cardioid $q=(x^2+y^2-2x)^2-4(x^2+y^2)$. Note that $q_y=4x^2y + 4y^3 -8xy -8y$. We choose a rational parametrization as follows:
$$ \begin{matrix}
x\,\,=\,\,\frac{4(1-w^2)}{(w^2+1)^2}\,,
\quad \hbox{hence} \quad
dx=\frac{8w(w^2 - 3)}{(w^2+1)^3}dw \,,
\quad \hbox{and} \quad
y\,\,= \,\,\frac{8w}{(w^2+1)^2}.
\end{matrix}
$$
We substitute this into the differential forms in (\ref{eq:omegabasis1}), and
we compute the three antiderivatives:
$$
\begin{matrix}
\int \frac{w^2-1}{2(1+w^2)^2}dw = -\frac{w}{2(w^2 + 1)} \, , \quad
\int \frac{-w}{(1+w^2)^2}dw = \frac{1}{2(w^2 + 1)} \, , \quad
\int \frac{-1}{8}dw = -\frac{w}{8}.
\end{matrix}
$$
Similar to the computations in the previous examples, we obtain the theta surface as follows:
\begin{equation*}
\label{eq:cardiodsurface}
\begin{matrix} X\,=\,-\frac{s}{2(s^2 + 1)}-\frac{t}{2(t^2 + 1)},\quad \quad Y\,=\,\frac{1}{2(s^2 + 1)}+\frac{1}{2(t^2 + 1)}, \quad \quad Z\,=\,-\frac{s}{8}-\frac{t}{8}, \medskip \\ 8X^2Z + 8Y^2Z - 4YZ - X \,\,=\,\,0.
\end{matrix}
\end{equation*}
Two pictures of the cardioid surface, from the 19th and 21st century,
are shown in Figure~\ref{fig:CardioidSurface}.
\end{example}
\begin{figure}[h!]
\vspace{-0.08in}
\begin{minipage}{0.50\textwidth}
\centering
\includegraphics[width=.9\linewidth]{DeltoidSurfaceNoBox}\\
\end{minipage}\hfill
\begin{minipage}{0.50\textwidth}
\centering
\includegraphics[width=.6\linewidth]{DeltoidSurfaceOld}\\
\end{minipage}
\vspace{-0.3in}
\caption{The Deltoid Surface. On the right ist Eiesland's plaster model~\cite{Eies08}.
\label{fig:DeltoidSurface}}
\vspace{-0.04in}
\end{figure}
\begin{example}[Deltoid Surface] \label{ex:deltoidsurface}
We consider the deltoid curve with parametrization
$$ \begin{matrix}
x\,\,=\,\,\frac{4}{(w+1)^2},
\quad {\rm and} \quad
y\,\,= \,\,\frac{4}{w^2+6w+9}.
\end{matrix}
$$
This defines the quartic $\,q=y^2+x^2-2xy+x^2y^2-2x^2y-2xy^2$.
The abelian integrals give
\begin{equation*}
\begin{matrix}
&X & = & \int \frac{1}{(s+1)^2}ds+\int \frac{1}{(t+1)^2}dt& = & -\frac{1}{(s + 1)}-\frac{1}{(t + 1)},
\smallskip \\
& Y & = & \int \frac{1}{(3+s)^2}ds+\int \frac{1}{(3+t)^2}dt & = &- \frac{1}{s+3}-\frac{1}{(t+3)},
\smallskip \\
&Z & = & \int \frac{1}{4}ds+\int \frac{1}{4}dt &= &\frac{s}{4}+\frac{t}{4}.
\end{matrix}
\end{equation*}
Elimination yields the following cubic equation. This surface is exhibited in
Figure~\ref{fig:DeltoidSurface}:
\begin{equation*}
4XYZ + 4XY + 2XZ - 2YZ + 3X - Y\,\,=\,\,0.
\end{equation*}
This equation can be transformed into the cubic $\,XYZ = a_1 X+a_2 Y+a_3 Z\,$ given by Chern \cite[p.~2]{Chern}
via a linear change of coordinates, similar to that in \cite[page 376, equation~(11)]{Eies08}.
We saw the algebraic theta surfaces of lowest degree three
in Figures \ref{fig:CardioidSurface} and \ref{fig:DeltoidSurface},
in views that take us back to the 19th century.
We will learn more about the plaster models
in Section~\ref{sec7}.
\end{example}
In Section \ref{sec5} we derived special transcendental theta surfaces
by degenerations from Riemann's theta function. An example
was the formula for Scherk's surface in (\ref{eq:scherkExp}).
This raises the question whether our algebraic theta surfaces
can be obtained in a similar manner.
While the answer is affirmative, the details are complicated
and we can only offer a glimpse. The role of the theta function is now played
by a variant called the {\em sigma function} \cite[\S 5]{naka}.
We illustrate this for the singular quartic of Example \ref{ex:curvelittle}, given by the affine equation
\begin{equation}
\label{eq:xzsingquar}
x^4 - z^3 \,\,=\,\, 0. \qquad \qquad
\end{equation}
This belongs to the family of {\em $(3,4)$-curves} \cite[eqn (5.1)]{BEL}.
These plane curves are defined~by
\begin{equation}
\label{eq:34curve}
\qquad
x^4 - z^3 \,+\, \lambda_1 x^2z + \lambda_2 x^2
+ \lambda_3 xz +\lambda_4 x + \lambda_5 z + \lambda_6 \,\,=\,\, 0 .
\end{equation}
The binomial (\ref{eq:xzsingquar})
is the most degenerate instance where all six coefficients $\lambda_i$ are zero.
Such curves, and the more general $(n,s)$ curves, were introduced in the
theory of integrable systems by Buchstaber, Enolski and Leykin \cite{BEL}
and studied further by Nakayashiki \cite{naka}. They
considered the sigma function associated to (\ref{eq:34curve}).
This is an
abelian function which generalizes Klein's classical sigma function for hyperelliptic curves.
The sigma function for a $(3,4)$-curve is a multigraded power series in $X,Y,Z$
whose coefficients are polynomials in $\lambda_1,\lambda_2,\ldots,\lambda_6$.
By \cite[Example 4.5]{BEL},
the term of lowest degree in the sigma function equals
$$\sigma_{3,4} \,\,\,=\,\,\,\,Z^5\, -\, 5X^2Z \,+\, 4 Y . $$
This is precisely the quintic in (\ref{eq:TSdegree5}),
after the coordinate scaling $(X,Y,Z) \mapsto (2X,5Y,Z)$.
Thus, our algebraic theta surface arises from the sigma function
by setting $\lambda_1 = \cdots = \lambda_6 = 0$.
Similarly, the theta surface in
Example \ref{ex:ats6} is closely related to the sextic $\sigma_{2,7}$ in
\cite[Example 4.5]{BEL}.
The polynomials $\sigma_{n,s}$ are known as {\em Schur-Weierstrass polynomials}.
These play a fundamental role for rational analogs of abelian functions, and hence in the
design of special solutions to the KP equation. For details we refer to
\cite{BEL,naka} and the references therein. Even the
genus $3$ case offers opportunities for further research. It would be interesting to
revisit this topic from the perspectives of
theta surfaces and tropical geometry, as in Section~\ref{sec5}.
\section{A Numerical Approach for Smooth Quartics}
\label{sec4}
In this section we assume that the given quartic curve $\mathcal{Q}$ is nonsingular in $\mathbb{P}^2$.
Then $\mathcal{Q}$ is a compact Riemann surface of genus $3$.
Riemann's Theorem \ref{thm:riemann} shows that its theta divisor coincides
with its theta surface, up to an affine transformation.
We here validate that result computationally
using current tools from numerical algebraic geometry.
We sample points on
the theta surface using Algorithm~\ref{alg:thetaSurface1} below, and we then check that Riemann's theta function vanishes at these points using the Julia package \texttt{Theta.jl}
by Agostini and Chua~\cite{Julia}.
\medskip
\begin{algorithm}[H]\label{alg:thetaSurface1}
\KwIn{ The inhomogeneous equation $q(x,y)$ of a smooth plane quartic}
\KwOut{A point on the corresponding theta surface in $\mathbb{R}^3$ or $\mathbb{C} ^3$}
\KwSty{Step 1:} Specify two points $p_1$ and $p_2$ on the quartic. \\
\KwSty{Step 2:} Take two other points $p_1'$ and $p_2'$ nearby $p_1$ and $p_2$ respectively. \\
\KwSty{Step 3:} Compute the following triples of integrals numerically:
\begin{equation*} \begin{matrix}
c_1 & = & (\int_{p_1}^{p_1'}\omega_1 , \int_{p_1}^{p_1'}\omega_2 , \int_{p_1}^{p_1'}\omega_3), &
c_2 & = &
(\int_{p_2}^{p_2'}\omega_1 , \int_{p_2}^{p_2'}\omega_2 , \int_{p_2}^{p_2'}\omega_3).
\end{matrix}
\end{equation*}
\\
\KwSty{Step 4:} Output the sum $c_1+c_2$.
\caption{Sampling from a theta surface given its plane quartic }
\end{algorithm}
\medskip
This algorithm is similar to Algorithm \ref{alg:thetaSurface2}.
However, the difference is that computation is now done by numerical evaluation.
Indeed, when the polynomial $q(x,y)$ defines a smooth quartic, it is impractical to work with
an algebraic formula for $y$ in terms of $x$, so we employ numerical methods even for
Steps 1 and 2 above. Of course, when such an expression is available, it can be used
to strengthen the numerical computations, as we will see later.
The central point of Algorithm \ref{alg:thetaSurface1} is computing the abelian integrals in Step 3.
Such integrals appear throughout mathematics, from algebraic geometry to number theory and integrable systems, and there is extensive work in evaluating them numerically. Notable
implementations are the library \texttt{abelfunctions} in SageMath~\cite{SwiDec},
the package \texttt{algcurves} in Maple~\cite{DecHoe}, and the MATLAB code presented in \cite{MATLAB}.
The software we used for our experiments is the package \verb|RiemannSurfaces| in SageMath
due to Bruin, Sijsling and Zotine~\cite{BruSijZot}.
The underlying algorithm views a plane algebraic curve $\mathcal{Q}$ as a ramified
cover $\mathcal{Q}\to\mathbb{P}^1$~of the Riemann sphere $\mathbb{P}^1$ via the projection $(x,y)\mapsto x$. The package lifts paths from $\mathbb{P}^1$ to paths on the Riemann surface $\mathcal{Q}$ and integrates the abelian differentials in \eqref{eq:omegabasis1} along these paths via certified homotopy continuation.
In order to carry this out, it is essential to avoid the ramification points of the projection to $\mathbb{P}^1$.
This is done by computing the Voronoi decomposition of the Riemann sphere
$\mathbb{P}^1$ given by the branch points
of $\mathcal{Q}\to \mathbb{P}^1$. The integration paths are obtained from edges of the
Voronoi cells. Avoidance of the ramification points is also
a feature in the other packages such as \texttt{algcurves}
and \texttt{abelfunctions}.
It is important to note that avoiding the ramification points conflicts with our
desire to create real theta surfaces and to
work with Riemann matrices and theta equations over~$\mathbb{R}$.
The cycle basis that is desirable for revealing the real structure, as in \cite{Sil},
forces us to compute integrals near ramification points.
We had to tweak the method in \cite{BruSijZot} to make this~work.
\smallskip
In what follows we present a case study
that illustrates Algorithm~\ref{alg:thetaSurface1}.
Our instance is the {\em Trott curve}.
This is a smooth plane quartic $\mathcal{Q}$, defined by the inhomogeneous polynomial
\begin{equation}
\label{eq:trottpolynomial}
q(x,y) \,\,=\,\, 144(x^4+y^4)-225(x^2+y^2)+350x^2y^2+81 .
\end{equation}
The Trott curve is a widely known example of a real quartic whose $28$ bitangent lines are all real
and touch at real points.
It is also a \emph{M-curve}, meaning that the real locus $\mathcal{Q}_{\mathbb{R}}$ has four connected components, which is the highest possible number for a real genus 3 curve.
\begin{figure}[h]
\centering
\includegraphics[width=0.4\linewidth]{TrottCurve.pdf} \\
\caption{The Trott curve, together with some branching points and some homology cycles.}
\label{fig:trott}
\end{figure}
Given any M-curve $\mathcal{Q}$, there exists a symplectic basis
for the homology group $H^1(\mathcal{Q},\mathbb{Z})$
whose associated period matrix
$\Pi$ in (\ref{periodMatrix}) respects the real structure (cf.~\cite{Sil}). With such a choice of basis, the
Riemann matrix $B$ in (\ref{eq:riemannmatrix}) is real, and the theta function
defines a surface in real $3$-space $\mathbb{R}^3$.
According to Theorem \ref{thm:riemann}, this surface is precisely
our theta surface.
We now verify this numerically. The first step consists in identifying a real period matrix for $\mathcal{Q}$.
To do so, we follow Silhol \cite{Sil}. First we observe that the Trott curve is highly symmetric:
its automorphism group is the dihedral group $D_4$, generated by the automorphisms
\begin{equation}\label{eq:autTrott}
(x,y) \mapsto (x,-y) ,\qquad (x,y)\mapsto(-x,y) ,\qquad (x,y)\mapsto (-x,-y), \qquad (x,y)\mapsto (y,x) .
\end{equation}
We choose a symplectic basis of $H^1(\mathcal{Q},\mathbb{Z})$ as follows: the cycles
$\alpha_1,\alpha_2,\alpha_3$ are as indicated in Figure \ref{fig:trott}, where we take
$\alpha_1,\alpha_3$ with clockwise orientation and $\alpha_2$ with
counterclockwise orientation. Furthermore, referring again to Figure \ref{fig:trott}, we take $\beta_1,\beta_3$ to be the two cycles lying over the interval $[s_1,s_2]$ and intersecting $\alpha_1,\alpha_3$ respectively. Instead, the path $\beta_2$ is the one lying over the interval $(-\infty,-s_4]\cup[s_4,+\infty)$. Then, with these choices,
there exist real numbers $a_j$ and purely imaginary numbers $b_i$ such that
the two $3 \times 3$ blocks in \eqref{periodMatrix} satisfy
\begin{equation}\label{eq:periodtrott}
\Pi_{\alpha} \,=\, \begin{pmatrix} 0 & -a_1 & 0 \,\\ -a_1 & 0 & a_1 \, \\ \,a_2 & -a_2 & a_2 \,\end{pmatrix}, \qquad
\Pi_{\beta} \,= \, \begin{pmatrix}\, b_1 & 2b_1 &\, b_1 \\ b_1 & 0 & -b_1 \\ \, b_2 & 0 & b_2 \end{pmatrix}.
\end{equation}
The scalars $a_j$ are real because the paths $\alpha_1,\alpha_2,\alpha_3$
and the differentials $\omega_1,\omega_2,\omega_3$ are real.
The scalars $b_i$ are purely imaginary because, by construction, the paths
$\beta_1,\beta_2, \beta_3$ are anti-invariant with respect to complex conjugation
on $\mathcal{Q}$. The symmetries in the matrices $\Pi_\alpha$ and $\Pi_\beta$ reflect the action
of the automorphism group of \eqref{eq:autTrott}. Since $\Pi_{\alpha}$ is real and $\Pi_{\beta}$ is purely imaginary,
we conclude that the Riemann matrix $\,B=-i\cdot \Pi_{\alpha}^{-1}\cdot \Pi_\beta\,$ has
all its entries real.
At this point it should be straightforward to compute all the above explicitly. However, as we see from Figure \ref{fig:trott}, the paths that we have chosen pass through the ramification points of the projection $\mathcal{Q}\to \mathbb{P}^1$ to the $x$-axis. Hence, we cannot use the existing routines, such as \verb|RiemannSurfaces| or \verb|abelfunctions| straight away. To solve this problem we mix the numerical strategy together with the symbolic one in Algorithm \ref{alg:thetaSurface2}.
Indeed, since the Trott curve is highly symmetric, we can compute its points symbolically in terms of radicals: more precisely, for any $x\in\mathbb{C}$, the four corresponding points $(x,y)$ on the Trott curve are given by
\begin{equation}\label{eq:trottsolution}
y \,\,=\,\, \pm {\frac{\sqrt{225-350x^2\pm \sqrt{39556 x^4 - 27900 x^2 + 3969}}}{12\sqrt{2}}} .
\end{equation}
With this formula, we represent the abelian integrals on the Trott curve as symbolic integrals in the single variable $x$,
and we then compute these
numerically. We have implemented this strategy in Maple
and we found the parameters in the period matrices of \eqref{eq:periodtrott} to be
\begin{align*}
a_1&\,=\,-0.02498252478, & a_2&\,=\,0.03154914935,\\
b_1&\,=\,\phantom{-} 0.01384015941i, & b_2&\,=\,0.02348847438i.
\end{align*}
Consequently, the Riemann matrix for the Trott with our choice of homology basis equals
\begin{equation*}
B\,\,=\,\,
\begin{pmatrix}0.926246 & 0.553994 & 0.372252 \\
0.553994 & 1.10799 & 0.553994 \\
0.372252 & 0.553994 & 0.926246
\end{pmatrix}.
\end{equation*}
The next step is to compute some points in the theta surface $\mathcal{S}$. This can be done via Algorithm \ref{alg:thetaSurface1} using the methods of \verb|RiemannSurfaces|, as long as we integrate away from the ramification points of the projection to the $x$ axis. Otherwise, we can employ again the symbolic representation of \eqref{eq:trottsolution}, together with numerical integration.
We proceed as follows: first, we can choose the points $p_1,p_2$ in Algorithm \ref{alg:thetaSurface1} in such a way that the line passing through them is bitangent to $\mathcal{Q}$ and parallel to the $x$ axis. Then we compute the integrals in Algorithm \ref{alg:thetaSurface1} for points in small neighborhoods of $p_1,p_2$.
Further details on such computations can be found in
Section \ref{sec7}, following the definition of an envelope in~\eqref{eq:stmaptangent}.
The rows below are five points on $\mathcal{S}$ we obtained with Maple:
\[
10^{-5} \cdot
\begin{pmatrix}
2.58293 & \: -4.55191 & \: -6.38839 \\
2.53203 & \: -4.46200 & \: -6.26220 \\
2.48111 & \: -4.37204 & \: -6.13596 \\
2.43015 & \: -4.28204 & \: -6.00964 \\
2.37916 & \: -4.19200 & \: -5.88327
\end{pmatrix}.
\]
The last step is to check that these points are zeroes of the theta function $\theta(\mathbf{x},B)$, up to an affine transformations. To do so, we employ the Julia package \texttt{Theta.jl} that is described in \cite{Julia}. This
package is the latest software for computing with theta functions.
It is especially optimized for the case of small genus and,
more importantly, for repeatedly evaluating a theta function $\theta(\mathbf{x},B)$ at multiple points $\mathbf{x}$
for the same fixed Riemann matrix~$B$. This allows for a fast evaluation of the theta function which is very helpful for our problems.
In our situation, we now have a sample of points $\mathbf{x}_1,\dots,\mathbf{x}_N$ on the theta surface.
These are given numerically.
We consider the transformed points $\Pi^{-1}_{\alpha}\mathbf{x}_1,\dots,\Pi^{-1}_{\alpha}\mathbf{x}_N$.
According to the proof of Riemann's Theorem \ref{thm:riemann}, there exists a
vector $\kappa\in \mathbb{C}^3$ such that the translated theta function
$\theta(\textbf{x}+\kappa,B)$ vanishes on
$\Pi^{-1}_{\alpha}\mathbf{x}_1,\dots,\Pi^{-1}_{\alpha}\mathbf{x}_N$.
According to the full version of Riemann's Theorem \cite[Appendix to \S 3]{Mum1},
which incorporates theta characteristics,
the vector $\kappa$ can be assumed to have the form
$\kappa = \frac{1}{2}\left( iB\varepsilon + \delta\right)$, where
$\varepsilon,\delta \in \{0,1\}^3$. In particular, there are only $64$
possible choices for $\kappa$. We can check explicitly all of the $64$ possibilities.
In our experiments, we computed $N=10404$ points on the surface $\mathcal{S}$, and we evaluated
\[ m(\varepsilon,\delta) := \max_{i=1,\dots,N} |\theta(\mathbf{x}_i+\kappa,B)| \]
for each of the 64 possible choices of $(\varepsilon,\delta)$. This was computed by \texttt{Theta.jl}
on a standard laptop in approximately 9.6 minutes. We found that
\[ m\left( \varepsilon_0, \delta_0 \right) \approx 6\cdot 10^{-12}, \qquad \text{ for } \qquad \varepsilon_0 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix},\,\, \delta_0 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}. \]
For all the other choices of the pair $\varepsilon,\delta$, we determined that $10^{-3} \leq m(\varepsilon,\delta) \leq 2$.
This computation amount to a numerical verification of Riemann's Theorem \ref{thm:riemann}.
We have
\[ \mathcal{S} = \Pi_{\alpha}\cdot (\Theta_B - \kappa_0), \qquad \text{ for }\,\, \kappa_0 = \frac{1}{2}\left( iB\varepsilon_0 + \delta_0\right). \]
To conclude, this gives also a real analytic equation for $\mathcal{S}$. Indeed, for any $\kappa=\frac{1}{2}\left( iB\varepsilon + \delta\right)$,
the translated theta divisor $\Theta_B-\kappa$ is cut out by the theta function with characteristic
\begin{align*}
\theta[\varepsilon,\delta](\mathbf{x},B) & = \sum_{n\in \mathbb{Z}^3}\mathbf{e}\left( -\frac{1}{2}\left(n+\frac{\varepsilon}{2}\right)^t B \left( n + \frac{\varepsilon}{2} \right) + i\left( n+\frac{\varepsilon}{2} \right)^t\left( \mathbf{x}+\frac{\delta}{2} \right) \right) \\
& = \sum_{n\in \mathbb{Z}^3}\exp \left(-\pi \left(n+\frac{\varepsilon}{2}\right)^t B \left( n + \frac{\varepsilon}{2} \right) \right)\cdot \cos \left( 2\pi \left( n+\frac{\varepsilon}{2} \right)^t\left( \mathbf{x}+\frac{\delta}{2} \right) \right) .
\end{align*}
This is a real analytic function since the matrix $B$ is real.
\section{Sophus Lie in Leipzig}
\label{sec7}
Felix Klein held the professorship for geometry at the University of Leipzig until 1886
when he moved to G\"ottingen. In the same year, Sophus Lie was appointed
to be Klein's successor and he moved from
Christiania (Oslo) to Leipzig. Lie also became one of the three directors of the Mathematical Seminar,
an institution that Felix Klein had founded with the aim of strengthening the connection between education and research. In his first years at Leipzig,
Lie was busy with completing his major work {\em Theory of Transformation Groups}
with the assistance of Friedrich Engel. It was released in three volumes in 1888, 1890 and 1893.
Thereafter, the subject of double translation surfaces moved back in the focus of his teaching and research,
and it caught the attention of the mathematical community for the first time.
In what follows we discuss notable historical developments, we revisit
Lie's pre-Leipzig work on these surfaces, and we show
how it relates to our discussion in the previous sections.
In 1892 Lie published an article explaining how theta surfaces can be
parametrized by abelian integrals \cite[p.~481]{LieGes2}.
He invited two of his Leipzig students, Richard Kummer and Georg Wiegner, to rework the classification
he had given in 1882 by means of abelian integrals.
The work of Kummer and Wiegner was published in their doctoral theses \cite{Kum,Wie}. Under the supervision of Lie's assistant Georg Scheffers, the two students also constructed a series of twelve plaster models
that visualize the diverse shapes exhibited by theta surfaces. It is surprising that the models were
commissioned by Lie, who, unlike his predecessor Klein, had
not been known for an engagement in popularizing mathematics in this manner.
The collection of mathematical models at the University of Leipzig was initiated by Felix Klein in 1880.
At the end of the 19th century, the collection included around $350$ models and drawings. During the
20th century, many models were lost or broken. A project for cataloging and restoring the collection
was initiated by Silvia Sch\"oneburg in 2014. A catalogue describing all
$240$ remaining models is expected to be published in 2021.
There are some very rare models in the collection, among them
nine of the surfaces created by Lie's students. These plaster models and their mathematics are the topic
of the third author's diploma thesis~\cite{Struwe}, submitted to Leipzig University in 2020.
It was her find of the models by Kummer and Wiegner that brought us together for our project on theta surfaces.
In 1895 Poincar{\'e} presented his proof of the relation between double translation surfaces and Abel's Theorem \cite{Poin1895,Poin95}. This led to the idea that these surfaces can
be seen as theta divisors of Jacobians \cite[p.2]{Chern}.
Darboux \cite{Darb} and Scheffers \cite{Sch} also published variants of the proof.
Eiesland \cite{Eies08,Eies09} completed the classification initiated by Kummer and Wiegner. He also constructed plaster models for some of his surfaces (cf.~Figures \ref{fig:CardioidSurface}
and \ref{fig:DeltoidSurface}). Eiesland's plaster models of theta surfaces were donated to
the collection at John Hopkins University.
Later on, our theme found its way into modern research.
Shiing-Shen Chern \cite{Chern}
characterized theta surfaces in terms of the \emph{web geometry}
that was developed by Blaschke and Bol in the 1930's.
In 1983, John Little \cite{Little83} studied the theory for curves of genus $g \geq 3$, and he proposed a solution of the \emph{Schottky problem} of recognizing Jacobians in terms of {\em translation manifolds}.
More precisely, he showed that a principally polarized abelian variety
of dimension $g$ is
the Jacobian of a non-hyperelliptic curve if and only if its theta divisor can be
written locally as a Minkowski sum of $g-1$ analytic curves.
Little's article \cite{Little92} connects this point of view to the integrable systems approach
(cf.~\cite{DFS}) to the Schottky problem.
\smallskip
The doctoral theses of Kummer and Wiegner built on Lie's earlier results.
One of these is the recovery of generating curves for a theta surface $\mathcal{S}$ from the equation of
$\mathcal{S}$ by means of differential geometry. This was helpful for
constructing real surfaces in situations when the abelian integrals delivered complex values.
We state Lie's result using a slight modification of the set-up in Section \ref{sec2}.
Fix a quartic $\mathcal{Q}\subset \mathbb{P}^2$ and let $\mathcal{L}(0)$ be a line that is \emph{tangent} to $\mathcal{Q}$ at a smooth point $p_1(0)$ and intersects $\mathcal{Q}$ in other two points $p_3(0),p_4(0)$. Using the points $p_1(z)$ near $p_1(0)$, where
$z$ is a local coordinate, we obtain a parametrization as in \eqref{eq:stmap}:
\begin{equation}\label{eq:stmaptangent}
(s,t)\,\,\mapsto \,\,\,\Omega_1(p_1(s)) \,+\,\Omega_1(p_1(t)).
\end{equation}
The surface $\mathcal{S}$ is the Minkowski sum $\mathcal{S}=\mathcal{C}+\mathcal{C}$,
where $\mathcal{C}$ is the curve $z\mapsto \Omega_1(p_1(z))$.
The scaled curve $2\cdot \mathcal{C}$ lies in $\mathcal{S}$. This curve
was called an \emph{envelope} by Lie.
In classical differential geometry, an {\em asymptotic curve} on a surface $\mathcal{S}$ is
a curve whose tangent direction at each point has normal curvature zero on $\mathcal{S}$.
This means that the tangent direction at each point is isotropic with respect to the
second fundamental form of~$\mathcal{S}$.
\begin{theorem}[Lie] \label{thm:lieasymptotic}
Let $\mathcal{S} = \mathcal{C}+\mathcal{C}$ be the theta surface given
by the parametrization~(\ref{eq:stmaptangent}). Then the envelope
$\,2\cdot \mathcal{C}\,$ is an asymptotic curve of the surface $\mathcal{S}$.
\end{theorem}
This result appears in
\cite[p.~211]{LieGes2}, albeit in a different formulation
that emphasizes minimal surfaces.
Our version in Theorem \ref{thm:lieasymptotic} was presented by Kummer in \cite[p.~15]{Kum}.
\smallskip
Lie's reconstruction is remarkable in that it solves \emph{Torelli's problem} for genus $3$ curves.
Indeed, Lie found his result several decades before Torelli \cite{Tor}
proved his famous theorem in algebraic geometry. Torelli's problem asks to recover an algebraic curve
$\mathcal{C}$ from its Jacobian $J(\mathcal{C})$ together with the theta divisor
$\Theta \subset J(\mathcal{C})$.
In our situation, once we recover the envelope $\mathcal{C}$ as the asymptotic curve of
the surface $\mathcal{S}$, we can reconstruct the quartic curve $\mathcal{Q}$
as in Remark \ref{rmk:quarticfromcurves}.
Note that this reconstruction technique also works
for singular quartics.
Algebraic geometers will notice a connection between Lie's approach and Andreotti's
geometric proof \cite{Andr} of Torelli's theorem. Indeed, the second fundamental
form of $\mathcal{S}$ is the differential of the Gauss map
$\, \mathcal{S} \rightarrow {(\mathbb{P}^2)}^*$.
This map associates to each point of $\mathcal{S}$ its tangent space in $\mathbb{C}^3$.
Andreotti observed that the Gauss map of the theta divisor
$\Theta \subset J(\mathcal{C})$ is branched
precisely over the curve in ${(\mathbb{P}^2)}^*$
dual to $\mathcal{C} \subset \mathbb{P}^2$.
Hence $\mathcal{C}$ can be recovered thanks to the biduality theorem.
It would be interesting to further study
Lie's differential-geometric approach to the Torelli problem via
the Gauss map. One natural question is whether
Theorem~\ref{thm:lieasymptotic} extends to curves of higher genus
and how this relates to Andreotti's method.
\begin{example}
To illustrate Lie's result, we determine an envelope for Scherk's minimal surface
directly from the equation \eqref{eq:scherk}.
After computing the second fundamental form, we see that a curve $(X(t),Y(t),Z(t))$ in
the surface is asymptotic if and only if it satisfies
\[ \frac{\dot{X}(t)^2}{\sin^2(X(t))}\, =\, \frac{\dot{Y}(t)^2}{\sin^2(Y(t))}\, , \quad \text{ or equivalently } \quad \frac{\dot{X}(t)}{\sin(X(t))} \,=\, \pm\frac{\dot{Y}(t)}{\sin(Y(t))}. \]
The solutions to this differential equation are given by the following two families of curves:
\[ \frac{\tan\left( X/2 \right)}{\tan\left( Y/2 \right)} \,=\, c
\qquad {\rm and} \qquad \tan\left( \frac{X}{2} \right)\cdot \tan\left( \frac{Y}{2} \right) \,=\, c
\qquad {\rm for} \quad c\in \mathbb{R}\setminus \{0\}. \]
Consider a curve from the first family. Setting $X=2\arctan(t)$, we can parametrize it as
\[ \left( 2\arctan(t)\,,\,\, 2\arctan\left( \frac{t}{c}\right)\,,\,\,
\log \left( \frac{c^2+t^2}{c(t^2+1)} \right) \right). \]
The last expression comes from the fact that $Z=\log\left( \frac{\sin(X)}{\sin(Y)} \right)$ holds
on Scherk's surface.
Now, setting $c=\frac{1}{5}$ and using Theorem \ref{thm:lieasymptotic}, we obtain the generating curve
$\mathcal{C}$ given by
$$ \left( \arctan(t), \arctan\left( 5t\right),\frac{1}{2}\log \left( \frac{1+(5t)^2}{5(t^2+1)} \right) \right). $$
The resulting representation $\mathcal{S} = \mathcal{C} + \mathcal{C}$ is
precisely the one we presented in equation~\eqref{eq:scherkpara2}.
\end{example}
Already in 1869, Lie studied the parametrization of {\em tetrahedral theta surfaces}
\begin{equation}
\label{eq:TTS} \mathcal{S} \,\,= \,\, \bigl\{\, \alpha \cdot \exp(X)
\,+\,\beta \cdot \exp(Y)\,+ \, \gamma \cdot \exp(Z)\,=\,\delta \,\bigr\}.
\end{equation}
Here $\alpha,\beta,\gamma,\delta$ are nonzero constants. These surfaces play a prominent role
in Theorem \ref{thm:degtheta}. The adjective ``tetrahedral'' refers to the fact that the
Delaunay polytope is a tetrahedron.
Example \ref{ex:vierzwei} shows
that Scherk's surface is tetrahedral, after a coordinate change over~$\mathbb{C} $.
Lie proved that tetrahedral theta surfaces admit infinitely many representations
$\mathcal{S} = \mathcal{C}_1 + \mathcal{C}_2$.
This was already mentioned in Remark \ref{rmk:onsameconic}.
We present Lie's method for identifying these infinitely many pairs of generating curves.
A key tool is the \emph{logarithmic transformation}
\begin{equation*}
\label{eq:logT}
X=\log(U),\quad Y=\log(V),\quad Z=\log(W) .
\end{equation*}
This transforms the surface $\mathcal{S}\subset \mathbb{C}^3$ into the
plane $\mathcal{P}\subset (\mathbb{C}^*)^3$ defined by the equation
\begin{equation} \label{eq:log(TTS)}
\mathcal{P} \,\,=\,\, \bigl\{\, \alpha \cdot U\,+\,\beta \cdot V\,\,+\,\,\gamma \cdot W\, = \, \delta \, \bigr\}.
\end{equation}
The generating curves in $\mathcal{S}=\mathcal{C}_1+\mathcal{C}_2$
correspond to curves $\mathcal{D}_1,\mathcal{D}_2$ such that $\mathcal{P} = \mathcal{D}_1\cdot \mathcal{D}_2$. Here $\mathcal{D}_1\cdot \mathcal{D}_2$ denotes the {\em Hadamard product} of the two curves,
i.e.~the set obtained from the coordinatewise product of all points in $\mathcal{D}_1$ with all points in $\mathcal{D}_2$. Lie studied this alternative formulation and found infinitely many pair of \emph{lines} $\mathcal{D}_1,\mathcal{D}_2\subset (\mathbb{C}^*)^3$ such that $\mathcal{P}=\mathcal{D}_1\cdot \mathcal{D}_2$.
We shall state Lie's result more precisely.
The action of the group of translations on $\mathbb{C}^3$ corresponds under the logarithmic transformation to the action of the torus $(\mathbb{C}^*)^3$ on itself.
Thus, we are free to rescale the coordinates $U,V,W$.
In particular, we can assume that our plane $\mathcal{P}$ and the desired
lines $\mathcal{D}_1$ and $\mathcal{D}_2$ contain the point $\mathbf{1}=(1,1,1)$.
With this, the identity
$\mathcal{P}=\mathcal{D}_1\cdot \mathcal{D}_2$ implies $\mathcal{D}_1,\mathcal{D}_2 \subset \mathcal{P}$. On the theta surface side, this corresponds to translating the surface and the curves until all of them pass through the origin $\mathbf{0}=(0,0,0)$.
We next consider the closure
of the plane $\mathcal{P}$ and the lines
$\mathcal{D}_1,\mathcal{D}_2$ inside the projective space $\mathbb{P}^3$ with coordinates $U,V,W,T$.
The arrangement of coordinate planes
$\{ UVWT= 0 \}$ in $\mathbb{P}^3$ intersects
our plane $\mathcal{P}$ in four lines $\mathcal{H}_1,\mathcal{H}_2,
\mathcal{H}_3,\mathcal{H}_4$.
Here now is the promised result.
\begin{theorem}[Lie]\label{thm:lietetra}
Let $\mathcal{D}_1,\mathcal{D}_2$ be lines through $\mathbf{1}=(1:1:1:1)$ in $\mathcal{P}$.
Then~$\mathcal{P}=\mathcal{D}_1\cdot \mathcal{D}_2$ if and only if the six lines $\,
\mathcal{D}_1,\mathcal{D}_2,\,
\mathcal{H}_1,\mathcal{H}_2,\mathcal{H}_3,\mathcal{H}_4\,$ are tangent to a common conic in $\,\mathcal{P}$.
\end{theorem}
This result is featured in
\cite[p.~526]{LieGes2}. We here present a self-contained proof.
\begin{proof}
We identify $\mathcal{P}$ with the affine plane with coordinates $s$ and $t$ by setting
\begin{equation}
\label{eq:UVWT}
U \,=\, 1 + a s + b t \,,\,\, \, V \,=\, 1 + c s + d t \,,\,\, \, W \,\,=\,\, 1+ e s+ f t \quad {\rm and} \quad T = 1.
\end{equation}
The origin $(s,t) = (0,0)$ corresponds to the distinguished point ${\bf 1}$. The two lines of interest are
$\mathcal{D}_1 = \{s = 0\}$ and $\mathcal{D}_2 = \{t = 0\}$.
The four coordinate lines are
$\mathcal{H}_1 = \{U= 0\}$,
$\mathcal{H}_2 = \{V = 0\}$,
$\mathcal{H}_3 = \{W = 0\}$,
and $\mathcal{H}_4$ is the line at infinity in the $(s,t)$-plane.
Thus, the six scalars $a,b,c,d,e,f$ in (\ref{eq:UVWT}) specify the inclusions
$\,\mathcal{D}_1, \mathcal{D}_2 \subset \mathcal{P} \subset \mathbb{P}^3$.
With these conventions, $\, \mathcal{D}_1,\mathcal{D}_2,\,
\mathcal{H}_1,\mathcal{H}_2,\mathcal{H}_3,\mathcal{H}_4\,$ are tangent to a common conic
in $\mathcal{P}$ if and only if
\begin{equation}
\label{eq:det=0} {\rm det} \begin{pmatrix}
a & b & ab \\
c & d & cd \\
e & f & ef \end{pmatrix} \,\, = \,\, 0 .
\end{equation}
The Hadamard product $\,\mathcal{D}_1 \cdot \mathcal{D}_2\,$ is a surface in $\mathbb{P}^3$.
It has the parametric representation
\begin{equation}
\label{eq:UVWTmult}
\tilde U \,=\, (1+as)(1+bt) \,,\,\,\,
\tilde V \,=\, (1+cs)(1+dt) \,,\,\,\,
\tilde W \,=\, (1+es)(1+ft) \,,\,\,\,
\tilde T = 1 .
\end{equation}
This can be rewritten as
$$ \tilde U \,=\, U \,+\, ab \cdot st \, ,\,\quad
\tilde V \,=\, V \,+\, cd \cdot st \, ,\,\quad \tilde W \,=\, W \,+\, ef \cdot st \, ,\,\quad
\tilde T \,= \,T . $$
Hence the surface $\,\mathcal{D}_1 \cdot \mathcal{D}_2\,$
equals the plane $\mathcal{P}$ in $\mathbb{P}^3$
if and only if the point $(ab:cd:ef:0)$ lies in $\mathcal{P}$.
This happens if and only if the condition (\ref{eq:det=0}) holds.
Now the proof is complete.
\end{proof}
Given the tetrahedral theta surface (\ref{eq:TTS}),
we can now construct a one-dimensional family of pairs $\mathcal{C}_1,\mathcal{C}_2$
of generating curves. The corresponding line pairs
$\mathcal{D}_1,\mathcal{D}_2$ in the plane (\ref{eq:log(TTS)})
are found as follows. We consider the one-dimensional family of
conics that are tangent to $\mathcal{H}_1,\mathcal{H}_2,\mathcal{H}_3,\mathcal{H}_4$.
Each such conic has tangent lines
pass through $\mathbf{1}$. These are $\mathcal{D}_1$ and $\mathcal{D}_2$.
In the algebraic formulation above,
the geometric constraints can be solved as follows.
The given theta surface (\ref{eq:TTS})
is specified by any solution to $\alpha + \beta+\gamma = \delta$.
The desired one-dimensional family is
the solution set to five equations in the six unknowns
$a,b,c,d,e,f$. In order for the planes in
(\ref{eq:UVWT}) and (\ref{eq:log(TTS)}) to agree, we need
$\, \alpha a + \beta c + \gamma e \,=\, \alpha b + \beta d + \gamma f \,=\, 0 $.
To get unique parameters for our lines, we may also fix
$a$ and $f $ in $\mathbb{C} $. Finally, the
quadratic equation (\ref{eq:det=0}) must be satisfied.
These five constraints define a curve in $\mathbb{C} ^6$
whose points are the solutions $\,(\mathcal{D}_1,\mathcal{D}_2)\,$
to $\,\mathcal{D}_1 \cdot \mathcal{D}_2 = \mathcal{P}\,$
and hence the solutions $\,(\mathcal{C}_1,\mathcal{C}_2)\,$
to $\,\mathcal{C}_1 + \mathcal{C}_2 = \mathcal{S}$.
We demonstrate this algorithm for computing generating curves of (\ref{eq:TTS})
in an example.
\begin{example} \label{ex:werevisit}
We revisit Example \ref{eq:amoebaofline} and the corresponding tetrahedral theta surface given
\eqref{eq:fourterms}. After replacing $Z$ with $Z+\log(3)$, the resulting surface
passes through $\mathbf{0}$. We have
$$ \begin{matrix}
\mathcal{S} & = & \,\,\{ \,\exp(X)\,+\,\exp(Y)\,-\,3\exp(Z)\,+\,1\,\,=\,\,0 \,\} & \subset \,\, \mathbb{C} ^3, \smallskip \\
\mathcal{P} & = & \{ \,U \, + \,V \, - \, 3 W \, + \, T \,\, = \,\,0 \, \} & \subset \,\, \mathbb{P}^3.
\end{matrix}
$$
To find a valid parametrization of $\mathcal{P}$ as in (\ref{eq:UVWTmult}),
we consider the equations in $a,b,c,d$, $e,f$ described above.
We fix $a=1 , f =0$ and we leave $b$ unspecified.
The remaining parameters are determined as $\, c = 1$, $\,d = -b$, $\, e = 2/3$,
by requiring
(\ref{eq:det=0}) and that (\ref{eq:UVWTmult}) lies on $\mathcal{P}$.
We conclude that the plane $\mathcal{P}$ has the parametrizations
$$
\tilde U \,=\, (1+s)\cdot (1+bt)\,, \,
\tilde V \,=\, (1+s)\cdot(1-bt)\,, \,
\tilde W \,=\, \left(1+\frac{2}{3}s \right)\cdot1, \,
\hbox{for all}\,\, \, \quad b \in \mathbb{C} \backslash \{0\}.
$$
Our tetrahedral theta surface $\mathcal{S}$ has
the one-dimensional family of parametrizations:
$$ X \,=\, {\rm log}(1+s) \,+\, {\rm log}(1+bt)\,, \quad
Y \,=\, {\rm log}(1+s) \,+\, {\rm log}(1-bt)\,, \quad
Z \,=\, {\rm log} \left( 1 + \frac{2}{3} s \right) .$$
This example is admittedly quite special, but the method
works for all tetrahedral theta surfaces, i.e.~whenever the quartic curve $\mathcal{Q}$ is
among the last three types in Figures \ref{fig:nodalquartics}
and \ref{fig:graphs}.
\end{example}
We conclude this article by returning to the twelve plaster models
of theta surfaces constructed by the doctoral students of Sophus Lie at Leipzig
in 1892. In Figure \ref{fig:KummerWiegner} we display
one model due to Richard Kummer \cite{Kum}
and one model due to Georg Wiegner \cite{Wie}.
\begin{figure}[h!]
\begin{minipage}{0.50\textwidth}
\centering
\includegraphics[width=.99\linewidth]{kummerpicreduced.jpg} \\
\end{minipage}\hfill
\begin{minipage}{0.50\textwidth}
\centering
\includegraphics[width=.98\linewidth]{wiegnerpicreduced.jpg} \\
\end{minipage}
\caption{Plaster models of theta surfaces
constructed in the 1890s by Lie's students
Kummer (left) and Wiegner (right).
These models are still in the collection at Universit\"at Leipzig.
\label{fig:KummerWiegner}
}
\vspace{-0.02in}
\end{figure}
The model on the left in Figure \ref{fig:KummerWiegner} shows
the tetrahedral theta surface
$$ \mathcal{S} \quad = \quad \bigl\{
\, 10^{-X} + 10^{-Y} + 10^{-Z} \, = \, 1 \,\bigr\}. $$
Kummer derives this surface in \cite[Section III.6]{Kum}
from a pencil of conics like in Example~\ref{eq:amoebaofline}.
In \cite[p.~32]{Kum} he applies a particular
transformation to the surface, which seems to be advantageous for the practical construction of a plaster model.
The one-dimensional family of Minkowski decompositions $\,\mathcal{S} = \mathcal{C}_1 + \mathcal{C}_2\,$
into curves can be found using our algorithm for Theorem \ref{thm:lietetra}. The model on the right in Figure \ref{fig:KummerWiegner} shows
another theta surface, namely
$$ \mathcal{S} \quad = \quad \biggl\{\,
{\rm tan}(Z) \, + \, \frac{2X}{X^2+2Y} \,\, = \,\,0 \,\biggr\}. $$
Wiegner derives this equation in \cite[Section IV.11]{Wie} from a quartic $\mathcal{Q}$
that decomposes into a cubic curve and one of its flex lines.
In \cite[Section II.4]{Wie}, Wiegner rederives the Weierstrass normal form,
and he fixes the flex line to be the line at infinity.
For the surface $\mathcal{S}$ he starts with
the rational cubic $\,q = y^2 - x^2(x-1)$, and he ends up
on \cite[p. 65]{Wie} with the equation seen above. The surface $\mathcal{S}$ is shown in \cite[Figure II, Tafel A]{Wie}.
In his appendix \cite[p.~82]{Wie}, Wiegner offers a delightful
description of how one actually builds a plaster model in practice.
\smallskip
This final section connects the 19th century with
the 21st century, and differential geometry with
algebraic geometry. Theta surfaces are beautiful objects,
not just for 3D printing, but they offer new vistas
on the moduli space of genus $3$ curves.
The explicit degenerations in Sections \ref{sec5}
and \ref{sec6}, and the tools from
numerical algebraic geometry in Section~\ref{sec4},
should be useful for many applications,
such as three-phase solutions of the KP equation~\cite{DFS}.
\medskip
\textbf{Acknowledgments:} We are thankful to the referees for their helpful comments and remarks. We are grateful to John Little for informing us of the quadric double translation surfaces of Example \ref{Quadric} and to Evgeny Ferapontov for mentioning his paper \cite{BalFer} to us.
\bibliographystyle{spmpsci}
| {
"timestamp": "2020-06-09T02:11:45",
"yymm": "2001",
"arxiv_id": "2001.11385",
"language": "en",
"url": "https://arxiv.org/abs/2001.11385",
"abstract": "A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincaré showed that theta surfaces are precisely the surfaces of double translation, i.e. obtained as the Minkowski sum of two space curves in two different ways. These curves are parametrized by abelian integrals, so they are usually not algebraic. This paper offers a new view on this classical topic through the lens of computation. We present practical tools for passing between quartic curves and their theta surfaces, and we develop the numerical algebraic geometry of degenerations of theta functions.",
"subjects": "Algebraic Geometry (math.AG); Complex Variables (math.CV); History and Overview (math.HO)",
"title": "Theta surfaces",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808718926534,
"lm_q2_score": 0.7217432062975979,
"lm_q1q2_score": 0.7075110595520085
} |
https://arxiv.org/abs/2004.06908 | Hessian quotient equations on exterior domains | It is well-known that a celebrated Jörgens-Calabi-Pogorelov theorem for Monge-Ampère equations states that any classical (viscosity) convex solution of $\det(D^2u)=1$ in $\mathbb{R}^n$ must be a quadratic polynomial. Therefore, it is an interesting topic to study the existence and uniqueness theorem of such fully nonlinear partial differential equations' Dirichlet problems on exterior domains with suitable asymptotic conditions at infinity. As a continuation of the works of Caffarelli-Li for Monge-Ampère equation and of Bao-Li-Li for $k$-Hessian equations, this paper is devoted to the solvability of the exterior Dirichlet problem of Hessian quotient equations $\sigma_k(\lambda(D^2u))/\sigma_l(\lambda(D^2u))=1$ for any $1\leq l<k\leq n$ in all dimensions $n\geq 2$. By introducing the concept of generalized symmetric subsolutions and then using the Perron's method, we establish the existence theorem for viscosity solutions, with prescribed asymptotic behavior which is close to some quadratic polynomial at infinity. | \section{Introduction}\label{sec:intro}
In this paper, we study the solvability of the exterior Dirichlet problem for Hessian quotient equations
\begin{equation}\label{eq:pro}
\left\{
\begin{array}{ll}
S_{k,l}(D^2u):=\frac{\sigma_k\left(\lambda\left(D^2u\right)\right)}{\sigma_l\left(\lambda\left(D^2u\right)\right)}=1,\quad& in \quad\mathbb{R}^n\setminus\overline{D},\\
u=\varphi, & on\quad\partial D,
\end{array}
\right.
\end{equation}
where $D$ is a bounded open set in $\mathbb{R}^n$, $n\geq2$, $1\leq l<k\leq n$, and $\lambda\left(D^2u\right)$ denotes the eigenvalues $\lambda=(\lambda_1,\cdots,\lambda_n)$ of the Hessian matrix of $u$. Here
$$\sigma_k(\lambda):=\sum_{1\leq i_1<\cdots<i_k\leq n}\lambda_{i_1}\cdots\lambda_{i_k}$$ is the $k$-th elementary symmetric function of $n$ variations, $k=1,\cdots,n$. This paper is a continuation to the work by Bao, Li and Li \cite{Bao-Li-Li-2014}, where the $k$-Hessian equations was considered
As a class of nonlinear elliptic second-order equations of the form $F(D^2u)=1$, \eqref{eq:pro} includes several typical cases. In particular, if $l=0$, then it is the classical Poisson equation $\Delta u=1$ when $k=1$, here we set $\sigma_0(\lambda)\equiv1$; when $2\leq k\leq n-1$, we have $k$-Hessian equations $\sigma_k\left(\lambda\left(D^2u\right)\right)=1$; and when $k=n$, it is the well-known Monge-Amp\`ere equation $\det(D^2u)=1$. If $l=1$ and $k=n=3$, then it happens to be the special Lagrangian equation $\det(D^2u)=\Delta u$ in $\mathbb{R}^3$ which originates from the study on the Lagrangian geometry \cite{Harvey-Lawson-1982}. If $l=n-1$ and $k=n$, then it is equivalent to the inverse harmonic Hessian equation $\frac{1}{\lambda_1(D^2u)}+\cdots+\frac{1}{\lambda_n(D^2u)}=1$.
The traditional (interior) Dirichlet problems for these equations have been extensively studied by many mathematicians since 1950s; one can see \cite{Aleksandrov-1958, Nirenberg-1953,Calabi-1958,Cheng-Yau-1977,Caffarelli-1984,Caffarelli-1985,Trudinger-1995, Trudinger-1997,Trudinger-2008, Urbas-1988,Urbas-1990,Wang-2009} and the references therein. Focusing on the topics of exterior Dirichlet problem, there has been many significant progresses in recent years. We would like to mention the work of Caffarelli and Li \cite{Caffarelli-Li-2003} on Monge-Amp\`ere equation, which is an extension of celebrated J\"{o}rgens-Calabi-Pogorelov theorem (\cite{Jorgens-1954, Calabi-1958, Pogorelov-1972, Cheng-Yau-1986, Jost-2001,Caffarelli-1995}) stating that any classical (viscosity) convex solution of $\det(D^2u)=1$ in $\mathbb{R}^n$ must be a quadratic polynomial. They showed that any convex viscosity solution of $\det(D^2u)=1$ in exterior domains of $\mathbb{R}^n, n\geq 3$ satisfies
\begin{equation}
\label{eq:C-Li}
\limsup_{|x|\to\infty}\left(|x|^{n-2}\left|u(x)-\left(\frac12x^TAx+b\cdot x+c\right)\right|\right)<\infty,
\end{equation}
for some symmetric positive definite matrix $A$ with $\det A=1$. In terms of prescribed asymptotic behavior \eqref{eq:C-Li}, they also established the existence and uniqueness theorem to the exterior Dirichlet problem. In $\mathbb{R}^2$, the similar problem was studied in \cite{Philippe-1992,Ferrer-1999, Ferrer-2000,Bao-Li-2012}. Very recently, Li and Lu \cite{Li-Lu-2018} completed the characterization of the existence and nonexistence of solutions with \eqref{eq:C-Li}.
For $k$-Hessian equations and Hessian quotient equations \eqref{eq:pro}, unlike the Monge-Amp\`ere case, they are not invariant under affine transformations. When $A$ in the prescribed asymptotic condition is a diagonal matrix, the corresponding existence theory in an exterior domain was investigated in \cite{Dai-Bao-2011, Bao-Li-2013}. As for the general positive definite matrix $A$, using a famliy of generlized symmetric subsolutions, Bao, Li and Li \cite{Bao-Li-Li-2014} successfully generalized the existence results of Monge-Amp\`ere equation \cite{Caffarelli-Li-2003} to the exterior Dirichlet problem of $k$-Hessian equations. Lately, Li, Li and Yuan \cite{Li-Li-Yuan-2019} studied the asymptotics \eqref{eq:C-Li} for convex solutions of $2$-Hessian equations and of inverse harmonic equations in exterior domains. We would like to point out that the Liouville type result for global solutions to \eqref{eq:pro} with $k=n$ and $l<n$ has been discussed by Bao et al. \cite{Bao-2003}.
As a continuation of \cite{Bao-Li-Li-2014}, in this paper, we shall establish the existence theorem of the exterior Dirichlet problem for Hessian quotient equations \eqref{eq:pro}, with some prescribed asymptotic behavior at infinity. We follow the notations and definitions in \cite{Caffarelli-1985,Trudinger-1995}.
First, we say that a function $u\in C^2(\mathbb{R}^n\setminus\overline{D})$ is admissible (or $k$-convex) if $\lambda(D^2u)\in\overline{\Gamma}_k$ in $\mathbb{R}^n\setminus\overline{D}$, where $\Gamma_k$ is the connected component of $\{\lambda\in\mathbb{R}^n|\sigma_k(\lambda)>0\}$ containing
$$\Gamma^{+}=\{\lambda\in\mathbb{R}^n~|~\lambda_i>0, i=1,\cdots,n\}.$$ Moreover,
$$\Gamma_k=\{\lambda\in\mathbb{R}^n~|~\sigma_j(\lambda)>0, 1\leq j\leq k\}.$$
Then we use $\mathrm{USC}(\Omega)$ and $\mathrm{LSC}(\Omega)$ to respectively denote the set of upper and lower semicontinuous real valued functions on $\Omega\subset\mathbb{R}^n$. The definition of viscosity solution of Hessian quotient equations is given by following.
\begin{Def}
A function $u\in \mathrm{USC}(\mathbb{R}^n\setminus\overline{D})$ is said to be a viscosity subsolution (supersolution) of equation \eqref{eq:pro} (or say that $u$ satisfies $S_{k,l}(D^2u)\geq(\leq)1$ in the viscosity sense), if for any $k$-convex function $\psi\in C^2(\mathbb{R}^n\setminus\overline{D})$ and point $\bar{x}\in \mathbb{R}^n\setminus\overline{D}$ satisfying $$\psi(\bar{x})=u(\bar{x})\quad\mathrm{and}\quad \psi\geq(\leq)u,$$ we have $$S_{k,l}(D^2\psi(\bar{x}))\geq(\leq)1.$$
A function $u\in C^0(\mathbb{R}^n\setminus\overline{D})$ is said to be a viscosity solution of \eqref{eq:pro} if it is both a viscosity subsolution and supersolution of \eqref{eq:pro}.
\end{Def}
\begin{Def}
Let $\varphi\in C^0(\partial D)$. A function $u \in \mathrm{USC}(\mathbb{R}^n\setminus D)$ $(u \in \mathrm{LSC}(\mathbb{R}^n\setminus D))$ is said to be a viscosity subsolution (supersolution) of problem \eqref{eq:pro}, if $u$ is a viscosity subsolution (supersolution) satisfying equation \eqref{eq:pro} in $\mathbb{R}^n\setminus\overline{D}$ and $u\leq (\geq)\varphi$ on $\partial D$. A function $u\in C^0(\mathbb{R}^n\setminus D)$ is said to be a viscosity solution of \eqref{eq:pro} if it is both a subsolution and a supersolution.
\end{Def}
We introduce the concept of generalized symmetric solutions to \eqref{eq:pro} in the following sense.
\begin{Def}
For a diagonal matrix $A=\mathrm{diag}(a_1,a_2,\cdots,a_n)$, we call $u$ a G-Sym (generalized symmetric) function with respect to $A$ if it is a function of $s=\frac12x^TAx=\frac12\sum_{i=1}^na_ix_i^2$, that is, $$u(x)=u(s):=u(\frac12x^TAx).$$ If $u$ is a solution of Hessian quotient equation \eqref{eq:pro} and is also a G-Sym function with respect to $A$, we say that $u$ is a G-Sym solution of \eqref{eq:pro}.
\end{Def}
Let
\begin{align*}\mathcal{A}_{k,l}:=\Big\{A~|~A ~\mbox{is a real $n\times n$ symmetric }&\mbox{positive definite matrix,}~ \\&\mbox{with }~\sigma_k(\lambda(A))=\sigma_l(\lambda(A))~\Big\}.
\end{align*} Note that $c^*(k,l)I\in\mathcal{A}_{k,l}$, where
\begin{equation}\label{c*}
c^*(k,l):=\left(C_n^l/C_n^k\right)^{\frac{1}{k-l}},
\end{equation} and
$$C_n^k=\frac{n!}{(n-k)!k!},\quad\,C_n^l=\frac{n!}{(n-l)!l!}$$ are two binomial coefficients. In order to avoid the abuse of symbol $\lambda$, we use $a:=(a_1,\cdots,a_n)$ to denote the eigenvalues $\lambda(A)$ in what follows. Clearly, for a diagonal matrix $A=\mathrm{diag}(a_1,\cdots,a_n)\in\mathcal{A}_{k,l}$ and any real constant $\mu$, it is obvious that
\begin{equation}\label{eq:linear-s}
w(s)=s+\mu, \quad\mbox{where}~ s=\frac12x^TAx=\frac12\sum_{i=1}^{n}a_{i}x_{i}^{2}
\end{equation}
is a G-Sym solution of \eqref{eq:pro} and satisfies $w''(s)\equiv0$. Except \eqref{eq:linear-s}, for looking for more G-Sym solutions of \eqref{eq:pro}, we have the following rigidity result.
\begin{Pro}\label{pro:general-solution}
Let $A=\mathrm{diag}(a_1,a_2,\cdots,a_n)\in\mathcal{A}_{k,l}$, $0\leq l<k\leq n$, and $0<\alpha<\beta<\infty$. Then there exists an $w\in C^2(\alpha,\beta)$ with $w''\not\equiv 0$ in $(\alpha,\beta)$, such that $w(x)=w(\frac12\sum_{i=1}^na_ix_i^2)$ is a G-Sym solution of equation \eqref{eq:pro} in $\{x\in\mathbb{R}^n|\alpha<\frac12\sum_{i=1}^na_ix_i^2<\beta\}$, if and only if \begin{equation}\label{rigid}
l=0~\mbox{and}~ k=n,\quad\mbox{or}\quad a_1=a_2=\cdots=a_n=c^*(k,l),
\end{equation}
where $c^*(k,l)=\left(C_n^l/C_n^k\right)^{\frac{1}{k-l}}$ is definded by \eqref{c*}.
\end{Pro}
This means that for $A\in\mathcal{A}_{k,l}$, \eqref{eq:pro} in general has no G-Sym solution unless the two cases in \eqref{rigid} hold. In order to use Perron's method to establish the existence theorem for problem \eqref{eq:pro}, it suffices to obtain enough subsolutions
with appropriate properties. We construct such subsolutions which are G-Sym functions with respect to $A$. This is the main new ingredient.
\begin{Rk}We remark that Proposition \ref{pro:general-solution} is an extension of Proposition 1.4 in \cite{Bao-Li-Li-2014} for $k$-Hessian equations. In particular, the Monge-Amp\`ere equation $\det(D^2u)=1$ has following radially symmetric solutions (see \cite{Caffarelli-Li-2003, Dai-Bao-2011}):
$$\omega_n(\frac12|x|^2)=\int_1^{\frac{|x|^2}{2}}\left(1+\alpha t^{-\frac{n}{2}}\right)^{\frac{1}{n}}\,dt,\quad \alpha>0,$$
which of course is a family of G-Sym solutions for each $A\in\mathcal{A}_{n,0}$ due to the invariance of affine transformations. It is because of this, that the radially symmetric solutions play an important role in the solvability of the exterior Dirichlet problems studied by Caffarelli-Li \cite{Caffarelli-Li-2003}. However, for $k$-Hessian equations and Hessian quotient equations, we are only allowed to assume that A is diagonal, but we cannot further assume that $A = c^{*}I$. This is the reason why we introduce the G-Sym functions.
\end{Rk}
In order to state our results precisely, we introduce some notations. For any fixed $t$-tuple $\{i_1,\cdots,i_t\}$, $1\leq t\leq n-k$, we set
$$\sigma_{k;i_1\cdots i_t}(a)=\sigma_k(a)|_{a_{i_1}= a_{i_2}=\cdots=a_{i_t}=0}.$$
We further define
\begin{equation}\label{eq:H-h}
H_k=H_k(\lambda(A)):=\max_{1\leq i\leq n}\frac{\sigma_{k-1;i}(a)a_i}{\sigma_k(a)},\ h_l=h_l(\lambda(A)):=\min_{1\leq i\leq n}\frac{\sigma_{l-1;i}(a)a_i}{\sigma_l(a)}.
\end{equation}
Set $h_0\equiv 0$ for completeness. By Perron's method, our first principal result is as follows.
\begin{Thm}\label{thm:main-1}
Let $D$ be a smooth, bounded, strictly convex open subset of $\mathbb{R}^n$, $n\geq 3$ and let $\varphi\in C^2(\partial D)$. For any $A\in\mathcal{A}_{k,l}$ with $1\leq l<k\leq n$ and $b\in\mathbb{R}^n$, if
\begin{itemize}
\item [(i)] $k-l\geq2$, or
\item [(ii)] $k-l=1$ and $H_k-h_l<\frac12$,
\end{itemize}
then there exists some constant $c_*$ depending only on $n,b,A,D$ and $\|\varphi\|_{C^2(\partial D)}$, such that for every $c>c_*$ there exists a unique viscosity solution $u\in C^0(\mathbb{R}^n\setminus D)$ of \eqref{eq:pro} and
\begin{equation}\label{eq:asym-1}
\limsup_{|x|\to\infty}\left(|x|^{\frac{k-l}{H_k-h_l}-2}\Big{|}u(x)-(\frac12x^TAx+b\cdot x+c)\Big{|}\right)<\infty.
\end{equation}
Especially, when $k-l\geq2$, \eqref{eq:asym-1} can be written as
\begin{equation}\label{eq:asym-2}
\limsup_{|x|\to\infty}\left(|x|^{\theta(n-2)}\Big{|}u(x)-(\frac12x^TAx+b\cdot x+c)\Big{|}\right)<\infty.
\end{equation}
where $\theta\in (\frac{k-l-2}{n-2},1]$ is a constant depending only on $n,k,l$ and $A$.
\end{Thm}
\begin{Rk}
When $l=0$, the asymptotics \eqref{eq:asym-2} is consistent in the well-known results. For instance, when $k=n$, for the Monge-Amp\`ere equation, see Caffarelli-Li \cite{Caffarelli-Li-2003}; when $2\leq k\leq n-1$, the Hessian equations, see Bao-Li-Li \cite{Bao-Li-Li-2014}. On the other hand, when $A=c^*(k,l)I$, then $H_k-h_l=\frac{k-l}{n}$, see \cite{Bao-Li-2013,Dai-Bao-2011}. We would like to point out that Theorem \ref{thm:main-1} is the main result of \cite{Zhao-2017}. Another independent proof is also given in \cite{Li-Li-2017}.
\end{Rk}
When $k=n$ and $l=n-1$, for the inverse harmonic equations in exterior domains, by using a different technique of constructing subsolutions, we have the following existence and uniqueness theorem to the exterior problem
\begin{equation}\label{eq:pro-i}
\left\{
\begin{array}{ll}
S_{n,n-1}(D^2u)=1,& in \quad\mathbb{R}^n\setminus\overline{D},\\
u=\varphi, & on\quad\partial D.
\end{array}
\right.
\end{equation}
\begin{Thm}\label{thm:P-L}
Let $D$ be a smooth, bounded, strictly convex open subset in $\mathbb{R}^n$ and let $\varphi\in C^2(\partial D)$. Then for any $A\in\mathcal{A}_{n,n-1}$, $b\in\mathbb{R}^n$,
(i) when $n\geq 3$, for each $\gamma<0$, there exists some constant $c_*$ depending only on $n,\gamma,A,b,D$ and $\|\varphi\|_{C^2(\partial D)}$, such that for every $c>c_*$, there exists a unique viscosity solution $u\in C^0(\mathbb{R}^n\setminus D)$ of \eqref{eq:pro-i} fulfilling
\begin{equation*}
\limsup_{|x|\to\infty}\left(|x|^{-\gamma}\left|u(x)-(\frac12x^TAx+b\cdot x+c)\right|\right)<\infty;
\end{equation*}
(ii) when $n=2$, there exists some constant $\alpha_*$ depending only on $A,b,D$ and $\|\varphi\|_{C^2(\partial D)}$, such that for every $\alpha>\alpha_*$, there exists a unique local convex solution $u\in C^\infty(\mathbb{R}^2\setminus \overline{D})\cap C^0(\mathbb{R}^2\setminus D)$ of \eqref{eq:pro-i} fulfilling
\begin{equation*
O(|x|^{-2})\leq u(x)-V(x)\leq M(\alpha)+O(|x|^{-2}),\quad\text{ as }|x|\to\infty,
\end{equation*}
where
\begin{gather*}
V(x)=\frac{1}{2}x^TAx+b\cdot x+\alpha\ln\sqrt{x^T(A-I)x}+c(\alpha),
\end{gather*}
and $M(\alpha), c(\alpha)$ are functions of $\alpha$.
\end{Thm}
\begin{Rk}
Here the restriction $H_{n}-h_{n-1}<\frac12$ is not required. The reason why we do not establish the existence for other cases $k-l=1$ and $H_k-h_l\geq\frac12$ is purely technical, and more discussions can be refered to the Appendix.
\end{Rk}
The remainder of this paper is organized as follows. As explained in \cite{Bao-Li-Li-2014}, in order to prove Theorems \ref{thm:main-1} and \ref{thm:P-L}, we only need to show that the assertion holds when $A$ is a diagonal matrix and $b=0$ by making use of an orthogonal transformation and by subtracting a linear function from $u$; but we cannot further assume that $A=c^*(k,l)I$ because the Hessian quotient equations are not invariant under affine transformations, like the Monge-Amp\`ere equation \cite{Caffarelli-Li-2003}.
In the next section, we first demonstrate that the Hessian quotient equations \eqref{eq:pro} with $1\leq l<k\leq n$ do not have G-Sym solutions with respect to $A$ for every $A\in\mathcal{A}_{k,l}$ unless $A=c^*(k,l)I$, see Proposition \ref{pro:general-solution}. Based on this rigidity property and to apply the Perron's method, the heart of our proof is to construct appropriate subsolutions which are G-Sym functions with respect to $A\in\mathcal{A}_{k,l}$ and verify certain asymptotic behaviors at infinity. Proposition \ref{pro:omega-alpha} provides us a family of G-Sym $k$-convex subsolutions of \eqref{eq:pro} with respect to $A$ for every $A\in\mathcal{A}_{k,l}$, no matter whether $H_k-h_l<\frac{k-l}{2}$ or not. Then by using Perron's method, we prove Theorem \ref{thm:main-1} and Theorem \ref{thm:P-L} in Section \ref{sec:3} and Section \ref{sec:4}, respectively. At last, in the Appendix,
we give several examples with $H_k-h_l=\frac12$ in the phase space of eigenvalues $\lambda(A)$ and explain more why we are not able to apply the Perron's method to build the existence result of viscosity solutions for problem \eqref{eq:pro} when $k-l=1$ and $H_k-h_l\geq \frac12$ with $k<n$. To solve it, new technique is needed.
\section{G-Sym solutions and G-Sym subsolutions}\label{sec:2}
This section is mainly concerned with the G-Sym solutions and subsolutions of equation \eqref{eq:pro} with respect to $A\in\mathcal{A}_{k,l}$, $0\leq l<k\leq n$. First, we show that for a diagonal matrix $A\in\mathcal{A}_{k,l}$, in general, there is no G-Sym solution, analogous to that in \cite{Bao-Li-Li-2014} using similar techniques. Then we utilize the $H_k$ and $h_l$ defined in \eqref{eq:H-h} to construct a family of G-Sym $k$-convex subsolutions in $\mathbb{R}^n\setminus\{0\}$. This will be important to prove Theorem \ref{thm:main-1} in the next section.
We start with recalling some elementary properties of $\sigma_k(a)$. Suppose $1\leq l\leq k\leq n$ and $a=(a_1,\cdots, a_n)$ with $a_i>0$, $i=1,\cdots,n$. Then
\begin{gather}
\sigma_k(a)=\sigma_{k;i}(a)+a_i\sigma_{k-1;i}(a), \quad\forall~i,\label{eq:sigma-k}\\
\sum_{i=1}^na_i\sigma_{k-1;i}(a)=k\sigma_{k}(a),\label{eq:k-sigma}\\
\sigma_l(a)\sigma_{k-1;i}(a)\geq \sigma_{l-1;i}(a)\sigma_k(a),\quad\forall~i. \label{eq:sigma-l-k}
\end{gather}
\subsection{G-Sym solutions and the proof of Proposition \ref{pro:general-solution}}
We here make use of the idea of Bao-Li-Li \cite{Bao-Li-Li-2014} to prove Proposition \ref{pro:general-solution}. The following lemma is also needed.
\begin{Lem}[Lemma 1.3 in \cite{Bao-Li-Li-2014}]\label{lem:D^2-omega}
For any $A=\mathrm{diag}(a_1,\cdots,a_n)$, if $w\in C^2(\mathbb{R}^n)$ is a G-Sym function with respect to $A$, then, with $a:=(a_1,\cdots,a_n)$,
\begin{equation}\label{eq:D^2-omega}
\sigma_k(\lambda(D^2w))=\sigma_k(a)(w')^k+w''(w')^{k-1}\sum_{i=1}^n\sigma_{k-1;i}(a)(a_ix_i)^2.
\end{equation}
\end{Lem}
\begin{proof}[Proof of Proposition \ref{pro:general-solution}]
When $l=0$ and $1\leq k\leq n$, for $k$-Hessian equations case, Proposition \ref{pro:general-solution} has been proved in \cite{Bao-Li-Li-2014}.
We here consider the Hessian quotient cases with $1\leq l<k\leq n$.
Given a fixed $s\in(\alpha,\beta)$ such that $w'(s)\neq 0$ and $w''(s)\neq 0$, take some integer $1\leq i\leq n$ and let $$x=(0,\cdots,0,\sqrt{2s/a_i},0,\cdots,0).$$
From \eqref{eq:D^2-omega}, we see that the G-Sym solution $w(s)$ of \eqref{eq:pro} satisfies the following ordinary equation
\begin{equation*
\sigma_k(a)(w')^k+2sw''(w')^{k-1}\sigma_{k-1;i}(a)a_i=\sigma_l(a)(w')^l+2sw''(w')^{l-1}\sigma_{l-1;i}(a)a_i.
\end{equation*}
We rewrite it as
$$\frac{(w')^l-(w')^k}{2sw''}=(w')^{k-1}\frac{\sigma_{k-1;i}(a)a_i}{\sigma_k(a)}-(w')^{l-1}\frac{\sigma_{l-1;i}(a)a_i}{\sigma_l(a)}.$$
Notice that the left hand side of the above equality depends only on $s$ and is independent of $i$, so we have, for any $i\neq j$,
\begin{align*}
&(w')^{k-1}\frac{\sigma_{k-1;i}(a)a_i}{\sigma_k(a)}-(w')^{l-1}\frac{\sigma_{l-1;i}(a)a_i}{\sigma_l(a)}\\
={}&(w')^{k-1}\frac{\sigma_{k-1;j}(a)a_j}{\sigma_k(a)}-(w')^{l-1}\frac{\sigma_{l-1;j}(a)a_j}{\sigma_l(a)}.
\end{align*}
That is,
$$
(w')^{k-1}\frac{\sigma_{k-1;i}(a)a_i-\sigma_{k-1;j}(a)a_j}{\sigma_k(a)}=(w')^{l-1}\frac{\sigma_{l-1;i}(a)a_i-\sigma_{l-1;j}(a)a_j}{\sigma_l(a)}.
$$
By using \eqref{eq:sigma-k} and $\sigma_k(a)=\sigma_l(a)$, we have
\begin{equation}\label{eq:w-i-j}
(w')^{k-1}[(a_i-a_j)\sigma_{k-1;ij}(a)]=(w')^{l-1}[(a_i-a_j)\sigma_{l-1;ij}(a)].
\end{equation}
If $k=n$,
then $\sigma_{k-1;ij}(a)=0$ for any $i\neq j$. Due to $\sigma_{l-1;ij}(a)>0$, we immediately get $a_i=a_j$ from \eqref{eq:w-i-j}. Thus, by arbitrariness of $i$ and $j$, we see $a_1=a_2=\cdots=a_n=c^*(n,l)$.
Now, we consider the case $1\leq l<k\leq n-1$. Assume for contradiction that $a_i\neq a_j$ for some $i\neq j$. From \eqref{eq:w-i-j}, it follows that
$$(w')^{k-1}\sigma_{k-1;ij}(a)=(w')^{l-1}\sigma_{l-1;ij}(a).$$
So that,
\begin{equation}\label{eq:i-j-C-s}
\frac{\sigma_{k-1;ij}(a)}{\sigma_{l-1;ij}(a)}=(w'(s))^{l-k}.
\end{equation}
Since the left side of \eqref{eq:i-j-C-s} is independent of $s$, we deduce that $w'(s)$ is a constant, which leads to $w''(s)\equiv 0$, a contradiction. Consequently, $a_1=a_2=\cdots=a_n=c^*(k,l)$.
\end{proof}
\subsection{G-Sym subsolutions}
By Proposition \ref{pro:general-solution}, it is only possible to find some G-Sym smooth $k$-convex subsolutions of equation \eqref{eq:pro} for any $A\in\mathcal{A}_{k,l}$. As before, we let $A=\mathrm{diag}(a_1,\cdots,a_n)\in\mathcal{A}_{k,l}$ and denote $\lambda(A)=(a_1,\cdots, a_n):=a$. We have $a_i>0$ $(i=1,\cdots,n)$ and $\sigma_k(a)=\sigma_l(a)$. Recall \eqref{eq:H-h},
$$H_k=\max_{1\leq i\leq n}\frac{\sigma_{k-1;i}(a)a_i}{\sigma_k(a)}, \quad h_l=\min_{1\leq i\leq n}\frac{\sigma_{l-1;i}(a)a_i}{\sigma_l(a)},$$
which indicates from \eqref{eq:sigma-k} and \eqref{eq:k-sigma} that, for all $0\leq l<k\leq n$,
$$\frac{k}{n}\leq H_k\leq1,\quad 0\leq h_l\leq\frac{l}{n},$$
and
\begin{equation}\label{eq:H-h-k-l}
\frac{k-l}{n}\leq H_k-h_l\leq1.
\end{equation}
In the following, we set
$$\mathcal{H}_{k,l}:=\frac{k-l}{2(H_k-h_l)},$$ for simplicity.
Now, using Lemma \ref{lem:D^2-omega} and recalling $s=\frac12x^TAx=\frac12\sum_{i=1}^na_ix_i^2$, we study the following ordinary equation
\begin{equation}\label{eq:ordinary-eq}
\left\{
\begin{array}{ll}
(w')^k+2w''(w')^{k-1}H_ks-(w')^l-2w''(w')^{l-1}h_ls=0, & s>0,\\
w'(s)>0, w''(s)<0.
\end{array}
\right.
\end{equation}
By setting $v(s)=w'(s)$, then
$$v^k+v'v^{k-1}H_k2s-v^l-v'v^{l-1}h_l2s=0.$$
Multiplying it by $\mathcal{H}_{k,l}v^{\frac{h_lk-H_kl}{H_k-h_l}}s^{\mathcal{H}_{k,l}-1}$ on both sides, we can rewrite it as
\begin{align*}
&\mathcal{H}_{k,l}v^{2\mathcal{H}_{k,l}H_k}s^{\mathcal{H}_{k,l}-1}+2\mathcal{H}_{k,l}H_kv'v^{2\mathcal{H}_{k,l}H_k-1}s^{\mathcal{H}_{k,l}}\\
=&\mathcal{H}_{k,l}v^{2\mathcal{H}_{k,l}h_l}s^{\mathcal{H}_{k,l}-1}+2\mathcal{H}_{k,l}hv'v^{2\mathcal{H}_{k,l}h_l-1}s^{\mathcal{H}_{k,l}}.
\end{align*}
Then, integrating it on both sides with respect to $s$, we see that
$$
v^{2\mathcal{H}_{k,l}H_k}s^{\mathcal{H}_{k,l}}=v^{2\mathcal{H}_{k,l}h_l}s^{\mathcal{H}_{k,l}}+\alpha,
$$
that is,
\begin{equation}\label{eq:v-k-l-H-h-s}
(v^{k-l}-1)v^{2\mathcal{H}_{k,l}h_l}=\alpha s^{-\mathcal{H}_{k,l}},
\end{equation}
where $\alpha$ is an arbitrary constant. Given $\alpha>0$, after differentiating \eqref{eq:v-k-l-H-h-s}, it is easy to find that there exists a solution $v_\alpha(s)>1$, such that
\begin{equation}\label{eq:d-v-alpha-s}
\frac{v'_\alpha}{v_\alpha}\left(H_kv_\alpha^{2\mathcal{H}_{k,l}H_k}-h_lv_\alpha^{2\mathcal{H}_{k,l}h_l}\right)=-\frac{\alpha}{2s^{1+\mathcal{H}_{k,l}}}.
\end{equation}
This indicates that $v'_\alpha(s)<0$. Therefore, for any $\alpha>0$, the ordinary equation \eqref{eq:ordinary-eq} has a family of solutions
\begin{align}
\omega_\alpha(s)&=\beta+\int_{\bar{s}}^sv_\alpha(t)\,dt\notag\\
&=\beta+s-\bar{s}+\int_{\bar{s}}^s\left(v_\alpha(t)-1\right)\,dt, \label{eq:ordinary-solution}
\end{align}
where $\beta\in\mathbb{R}$ and $\bar{s}>0$.
Next, let us characterize the asymptotic behavior of the solutions of \eqref{eq:ordinary-eq}. For $\alpha>0$, letting $s\to\infty$, we deduce by \eqref{eq:v-k-l-H-h-s} that
$$\lim_{s\to\infty}v_\alpha(s)=1,$$ and
\begin{equation}\label{eq:O-v-s}
v_\alpha(s)-1=O(s^{-\mathcal{H}_{k,l}}), \quad\mathrm{as}\ s\to\infty.
\end{equation}
We divide below into four cases.
\textbf{Case 1.} If $H_k-h_l<\frac{k-l}{2}$, it follows from \eqref{eq:ordinary-solution} and \eqref{eq:O-v-s} that
\begin{equation}\label{eq:omega-s-1}
\omega_\alpha(s)=s+\mu_1(\alpha)+O(s^{1-\mathcal{H}_{k,l}}),\quad s\to\infty,
\end{equation}
where $$\mu_1(\alpha)=\beta-\bar{s}+\int_{\bar{s}}^\infty(v_\alpha(t)-1)\,dt<\infty.$$ Moreover, in view of \eqref{eq:v-k-l-H-h-s},
\begin{equation*
\frac{\partial v_\alpha(s)}{\partial\alpha}=\frac{s^{-\mathcal{H}_{k,l}}}{2\mathcal{H}_{k,l}\left(H_kv_\alpha^{2\mathcal{H}_{k,l}H_k-1}-h_lv_\alpha^{2\mathcal{H}_{k,l}h_l-1}\right)}>0,
\end{equation*}
and we can see that $\mu_1(\alpha)$ increases with respect to $\alpha$ and $\lim_{\alpha\to\infty}\mu_1(\alpha)=\infty$. In particular, when $n\geq 3$ and $k-l\geq2$, we have by \eqref{eq:H-h-k-l} that $H_k-h_l<\frac{k-l}{2}$ holds for any $A\in\mathcal{A}_{k,l}$ and
\begin{equation*}
\frac{k-l}{2}-1<\mathcal{H}_{k,l}-1\leq\frac{n-2}{2}.
\end{equation*}
Thus, in this case, \eqref{eq:omega-s-1} can be written as
\begin{equation}\label{eq:omega-s-2}
\omega_\alpha(s)=s+\mu_1(\alpha)+O(s^{\frac{\theta(2-n)}{2}}),\quad\theta\in(\frac{k-l-2}{n-2},1].
\end{equation}
\textbf{Case 2.} If $n=k=2$ and $l=0$, then using \eqref{eq:H-h-k-l} we have $H_2-h_0=1$. From \eqref{eq:v-k-l-H-h-s} we immediately obtain $v_\alpha=\sqrt{1+\alpha/s}$ and \eqref{eq:ordinary-solution} becomes
\begin{align}
\omega_\alpha(s)&=\int_{\bar{s}}^s\sqrt{1+\alpha/s}\,ds+\beta\notag\\
&=\left[\sqrt{t}\sqrt{\alpha+t}+\alpha\ln\left(\sqrt{t}+\sqrt{\alpha+t}\right)\right]\Big{|}_{\bar{s}}^s+\beta\notag\\
&=s+\frac{\alpha}{2}\ln s+\mu_2(\alpha)+O(s^{-1}),\label{eq:omega-ln-2}
\end{align}
where $$\mu_2(\alpha)=\beta-\bar{s}-\frac{\alpha}{2}\ln\bar{s}+\int_{\bar{s}}^\infty\left(v_\alpha-1-\frac{\alpha}{2s}\right)\,dt<\infty.$$
\textbf{Case 3.} If $k-l=1$ and $H_k-h_l=\frac{k-l}{2}=\frac12$. By \eqref{eq:v-k-l-H-h-s} we obtain $(v_\alpha-1)v_\alpha^{2h_l}=\alpha/s$ and $v_\alpha-1=O(s^{-1})$. Since
$$\lim_{s\to\infty}\frac{v_\alpha-1-\alpha/s}{s^{-2}}=-2h_l\alpha^2<\infty.$$
which implies that $v_\alpha-1-\alpha/s=O(s^{-2}), s\to\infty$. We thereby derive from \eqref{eq:ordinary-solution} that
\begin{align}
\omega_\alpha(s) &=\beta+s-\bar{s}+\int_{\bar{s}}^s \alpha/s\,dt+\int_{\bar{s}}^s(v_\alpha-1-\alpha/s)\,dt\notag\\
&=s+\alpha\ln s+\mu_3(\alpha)+O(s^{-1}),\label{eq:omega-ln}
\end{align}
where $$\mu_3(\alpha)=\beta-\bar{s}-\alpha\ln\bar{s}+\int_{\bar{s}}^\infty(v_\alpha-1-\alpha/s)\,dt<\infty.$$
\textbf{Case 4.} If $k-l=1$ and $H_k-h_l>\frac{k-l}{2}=\frac12$, then \eqref{eq:v-k-l-H-h-s} reduces to $(v-1)v^{\frac{h_l}{H_k-h_l}}=\alpha s^{\frac{-1}{2(H_k-h_l)}}$. We easily verify that
$$\lim_{s\to\infty}\frac{v_\alpha-1-\alpha s^{\frac{-1}{2(H_k-h_l)}}}{1/s}=0,$$
and $$\lim_{s\to\infty}\frac{v_\alpha-1-\alpha s^{\frac{-1}{2(H_k-h_l)}}}{1/s^2}=\infty.$$
Hence, $v_\alpha-1-\alpha s^{\frac{-1}{2(H_k-h_l)}}=O(s^{\theta-1})$, $\theta\in (-1,0)$. Using \eqref{eq:ordinary-solution} gives
\begin{align}
\omega_\alpha(s)&=\beta+s-\bar{s}+\int_{\bar{s}}^s \alpha s^{\frac{-1}{2(H_k-h_l)}}\,dt+\int_{\bar{s}}^s(v_\alpha-1-\alpha s^{\frac{-1}{2(H_k-h_l)}})\,dt\notag\\
&=s+\frac{\alpha}{1-\frac{1}{2(H_k-h_l)}}s^{1-\frac{1}{2(H_k-h_l)}}+\mu_4(\alpha)+O(s^\theta),\label{eq:omega-power}
\end{align}
where $$\mu_4(\alpha)=\beta-\bar{s}-\frac{\alpha}{1-\frac{1}{2(H_k-h_l)}}\bar{s}^{1-\frac{1}{2(H_k-h_l)}}+\int_{\bar{s}}^\infty(v_\alpha-1-\alpha/s)\,dt<\infty.$$
Now, we are in a position to state explicitly that a family of functions $\omega_\alpha(s)$ given by \eqref{eq:ordinary-solution} are G-Sym subsolutions of equation \eqref{eq:pro} in $\mathbb{R}^n\setminus\{0\}$. Precisely, we have
\begin{Pro}\label{pro:omega-alpha}
For $n\geq2$, $0\leq l<k\leq n$ and $\alpha>0$, let $A\in\mathcal{A}_{k,l}$ be diagonal and $\omega_\alpha(x)=\omega_\alpha(\frac12x^TAx)$ be given by \eqref{eq:ordinary-solution}. Then $\omega_\alpha$ is a smooth $k$-convex subsolution of equation \eqref{eq:pro} in $\mathbb{R}^n\setminus\{0\}$. Furthermore,
(i) Assume $n\geq 3$. If $k-l=1$ and $H_k-h_l<\frac12$, then
\begin{equation}\label{eq:omega-alpha-1}
\omega_\alpha(x)=\frac12x^TAx+\mu_1(\alpha)+O(|x|^{2-2\mathcal{H}_{k,l}}), \quad |x|\to\infty;
\end{equation}
if $k-l\geq2$, then
\begin{equation}\label{eq:omega-alpha-2}
\omega_\alpha(x)=\frac12x^TAx+\mu_1(\alpha)+O(|x|^{\frac{\theta(2-n)}{2}}), \quad |x|\to\infty,
\end{equation}
where $$\theta\in\Big(\frac{k-l-2}{n-2},1\Big].$$
(ii) If $n=k=2$ and $l=0$, then
\begin{equation}\label{eq:omega-alpha-3}
\omega_\alpha(x)=\frac12x^TAx+\frac{\alpha}{2}\ln \left(\frac12x^TAx\right)+\mu_2(\alpha)+O(|x|^{-2}), \quad |x|\to\infty.
\end{equation}
(iii) If $k-l=1$ and $H_k-h_l=\frac12$, then
\begin{equation}\label{eq:omega-alpha-4}
\omega_\alpha(x)=\frac12x^TAx+\alpha\ln \left(\frac12x^TAx\right)+\mu_3(\alpha)+O(|x|^{-2}), \quad |x|\to\infty.
\end{equation}
(iv) If $k-l=1$ and $H_k-h_l>\frac12$, then, as $|x|\to\infty$,
\begin{equation}\label{eq:omega-alpha-5}
\omega_\alpha(x)=\frac12x^TAx+\frac{\alpha}{1-\frac{1}{2(H_k-h_l)}}\left(\frac12x^TAx\right)^{1-\frac{1}{2(H_k-h_l)}}+\mu_4(\alpha)+O(|x|^{2\theta}),
\end{equation}
where $\theta\in(-1,0)$.
\end{Pro}
\begin{Rk}
(1) We would like to remark that Li and Li \cite{Li-Li-2017} also obtain the assertion (i), by using variable $r=\sqrt{x^TAx}$. For the case $l=0$, $2\leq k\leq n$, the G-Sym subsolutions of $k$-Hessian equations \eqref{eq:pro} given by Proposition \ref{pro:omega-alpha} have been clarified by Bao, Li and Li in \cite{Bao-Li-Li-2014}.
(2) In particular, if we take $A=c^*(k,l)I$, then
$$\frac{\sigma_{k-1;i}(a)a_i}{\sigma_k(a)}=\frac{k}{n}, \quad\frac{\sigma_{l-1;i}(a)a_i}{\sigma_l(a)}=\frac{l}{n}, \quad\forall ~i=1,2,\cdots, n,$$
and
$$H_k-h_l=\frac{k-l}{n}.$$
If $n=k=2$ and $l=0$, then $c^*(2,0)=1$, $H_2=1$. By \eqref{eq:omega-ln-2} and \eqref{eq:omega-alpha-3}, we obtain its radial symmetric solution of the Monge-Amp\`ere equation $\det(D^2u)=1$ in dimension two
\begin{equation*
\omega_\alpha(x)=\frac12\left(|x|\sqrt{2\alpha+|x|^2}\right)+\alpha\ln\left(|x|^2+\sqrt{2\alpha+|x|^2}\right)+C,
\end{equation*}
satisfying $$\omega_\alpha(x)=\frac12|x|^2+\frac{\alpha}{2}\ln(\frac12|x|^2)+\mu_2(\alpha)+O(|x|^{-2}), \quad x\to\infty,$$
where $C$ is a constant depending on $\alpha$, see \cite{Wang-Bao-2013, Bao-Li-2012}. If $n\geq 3$, $2\leq k\leq n$ and $l=0$, then \eqref{eq:ordinary-solution} with $H_k=\frac{k}{n}$ and $h_l=0$ gives the radial symmetric solutions of $k$-Hessian equations \eqref{eq:pro} in $\mathbb{R}^n\setminus B_1$:
$$\omega_\alpha(x)=\int_1^{\frac{c^*(k,0)}{2}|x|^2}\left(1+\alpha t^{-\frac{n}{2}}\right)^{\frac{1}{k}}\,dt.$$
If $n\geq 3$ and $k=1$, then $c^*(1,0)=\frac{1}{n}$ and we obtain the radial symmetric solutions in $\mathbb{R}^n\setminus B_1$ for Poisson equation $\Delta u=1$,
\begin{align*}
\omega_\alpha(x)&=\int_1^{\frac{1}{2n}|x|^2}\left(1+\alpha t^{-\frac{n}{2}}\right)\,dt\\
&=\frac{1}{2n}|x|^2-\frac{\alpha}{n(n-2)}|x|^{2-n}-1+\frac{2\alpha}{n-2},
\end{align*}
which implies that the radial solution of Poisson equation in $\mathbb{R}^n\setminus\{0\}$ can only be a summation of Laplace's fundamental solution $\frac{\alpha}{n(2-n)}|x|^{2-n}$ and the solution $\frac{1}{2n}|x|^2+c$ in the form \eqref{eq:linear-s}.
\end{Rk}
We conclude this section by giving the proof of Proposition \ref{pro:omega-alpha}.
\begin{proof}[Proof of Proposition \ref{pro:omega-alpha}]
Clearly, \eqref{eq:omega-alpha-1}-\eqref{eq:omega-alpha-5} directly follow from \eqref{eq:omega-s-1}, \eqref{eq:omega-s-2}, \eqref{eq:omega-ln-2}, \eqref{eq:omega-ln} and \eqref{eq:omega-power}. We denote $\lambda(A)=(a_1,\cdots,a_n):=a$ throughout the proof. Since $v_\alpha'(s)=\omega_\alpha''(s)<0$, for any $1\leq m\leq k$,
\begin{align*}
\sigma_m(\lambda(D^2\omega_\alpha))&=\sigma_m(a)v_\alpha^m+v'_\alpha v_\alpha^{m-1}\sum_{i=1}^n\sigma_{m-1;i}(a)(a_ix_i)^2\\
&=\sigma_m(a)v_\alpha^{m-1}\left(v_\alpha+v'_\alpha\sum_{i=1}^n\frac{\sigma_{m-1;i}(a)}{\sigma_m(a)}(a_ix_i)^2\right)\\
&\geq \sigma_m(a)v_\alpha^{m-1}(v_\alpha+v'_\alpha H_m2s).
\end{align*}
From \eqref{eq:sigma-l-k} and \eqref{eq:A-H-h} , we know that $$H_m=\frac{\sigma_{m-1;n}(a)a_n}{\sigma_m(a)}\leq \frac{\sigma_{k-1;n}(a)a_n}{\sigma_k(a)}=H_k\quad \mathrm{for}\quad 1\le m\le k,$$ provided $a_1\le a_2\le\cdots\le a_n$. Hence,
$$\sigma_m(\lambda(D^2\omega_\alpha))\ge \sigma_m(a)v_\alpha^{m-1}(v_\alpha+v'_\alpha H_k2s).$$
In view of \eqref{eq:v-k-l-H-h-s} and \eqref{eq:d-v-alpha-s},
\begin{align*}
\frac{v'_\alpha}{v_\alpha}\frac{\alpha}{s^{\mathcal{H}_{k,l}}}H_k&=\frac{v'_\alpha}{v_\alpha}\left(H_kv^{2\mathcal{H}_{k,l}H_k}-H_kv^{2\mathcal{H}_{k,l}h_l}\right)\\
&\geq\frac{v'_\alpha}{v_\alpha}\left(H_kv^{2\mathcal{H}_{k,l}H_k}-h_lv^{2\mathcal{H}_{k,l}h_l}\right) =-\frac{\alpha}{2s^{\mathcal{H}_{k,l}+1}}.
\end{align*}
It shows that $v_\alpha+v'_\alpha H_k2s\geq 0$ and so $$\sigma_m(\lambda(D^2\omega_\alpha))>0, \quad x\in\mathbb{R}^n\setminus\{0\},$$ for any $1\leq m\leq k$.
On the other hand, by the definition of $H_k,h_l$ and the fact that $\omega''_\alpha(s)<0$,
\begin{align*}
&\sigma_k(\lambda(D^2\omega_\alpha))-\sigma_l(\lambda(D^2\omega_\alpha))\\
={}&\sigma_k(a)(\omega'_\alpha)^k+\omega''_\alpha(\omega'_\alpha)^{k-1}\sum_{i=1}^n\sigma_{k-1;i}(a)(a_ix_i)^2\\
&\quad-\sigma_l(a)(\omega'_\alpha)^l-\omega''_\alpha(\omega'_\alpha)^{l-1}\sum_{i=1}^n\sigma_{l-1;i}(a)(a_ix_i)^2\\
\geq{} & \sigma_k(a)\left((\omega'_\alpha)^k+2\omega''_\alpha(\omega'_\alpha)^{k-1}H_ks-(\omega'_\alpha)^l-2\omega''_\alpha(\omega'_\alpha)^{l-1}h_ls\right)=0.
\end{align*}
Consequently, $\omega_\alpha$ is a smooth $k$-convex subsolution of \eqref{eq:pro} in $\mathbb{R}^n\setminus\{0\}$.
\end{proof}
\section{Proof of Theorem \ref{thm:main-1}}\label{sec:3}
To prove Theorem \ref{thm:main-1}, we need apply an adapted Perron's method and comparison principle for general equation $f(\lambda(D^2(u)))=1$ to Hessian quotient equations \eqref{eq:pro}, see \cite{Li-Bao-2014, Bao-Li-Li-2014} and the references therein. For convenience, we present them as follows.
\begin{Lem}\label{lem:perron-m}
Let $\Omega$ be a domain in $\mathbb{R}^n$. Assume that there exist $\underline{u}, \bar{u}\in C^0(\overline{\Omega})$ respectively to be viscosity subsolution and supersolution of \eqref{eq:pro} such that $\underline{u}\leq \bar{u}$, and $\underline{u}=\varphi$ on $\partial \Omega$. Then
$$u(x):=\sup\{v(x)|\underline{u}\leq v\leq\bar{u}\, \mathrm{in}\, \Omega\,\mathrm{and}\, v\, \mbox{is a subsolution of \eqref{eq:pro}}, \text{with } v=\varphi\,\mathrm{on}\, \partial\Omega\}$$
is the unique viscosity solution of problem \eqref{eq:pro}.
\end{Lem}
\begin{Lem}\label{lem:compar}
Let $\Omega$ be a domain in $\mathbb{R}^n$. If $u\in\mathrm{USC}(\overline{\Omega})$, $v\in\mathrm{LSC}(\overline{\Omega})$ are respectively viscosity subsolution and supersolution of \eqref{eq:pro} in $\Omega$ and $u\leq v$ on $\partial\Omega$, then $u\leq v$ in $\Omega$.
\end{Lem}
Besides, we also need the following lemma which has proved in \cite{Caffarelli-Li-2003,Bao-Li-Li-2014}.
\begin{Lem}\label{lem:w-xi}
Let $D$ be a bounded strictly convex domain of $\mathbb{R}^n, n\geq2$, $\partial D\in C^2$, $\varphi\in C^2(\partial D)$ and let $A$ be an invertible and symmetric matrix. There exists some constant $C$, depending only on $n, \|\varphi\|_{C^2(\partial D)}$, the upper bound of $A$, the diameter and the convexity of $D$, and the $C^2$ norm of $\partial D$, such that for every $\xi\in\partial D$, there exists $\bar{x}(\xi)\in\mathbb{R}^n$ satisfying $$|\bar{x}(\xi)|\leq C\quad\text{and}\quad w_\xi<\varphi\quad\text{on}\quad \partial D\setminus\{\xi\},$$ where $$w_\xi(x)=\varphi(\xi)+\frac12\left((x-\bar{x}(\xi))^TA(x-\bar{x}(\xi))-(\xi-\bar{x}(\xi))^TA(\xi-\bar{x}(\xi))\right),\ x\in\mathbb{R}^n.$$
\end{Lem}
We now start to prove Theorem \ref{thm:main-1}, provided $A=\text{diag}(a_1,a_2,\cdots,a_n)$ and $b=0$. Actually, using Lemmas \ref{lem:perron-m}-\ref{lem:w-xi}, the proof is similar to that of \cite{Bao-Li-Li-2014} for $k$-Hessian equations. For the reader's convenience, we here present it as follows.
\begin{proof}[Proof of Theorem \ref{thm:main-1}]
For $s>0$, let $$E(s):=\left\{x\in\mathbb{R}^n~|~\frac12x^TAx<s\right\}.$$ Fix $\bar{s}>0$ such that $\overline{D}\subset E(\bar{s})$. Then recalling \eqref{eq:ordinary-solution}, for $\alpha>0,\beta\in\mathbb{R}$, $$\omega_\alpha(x)=\beta+\int_{\bar{s}}^{\frac12x^TAx}v_\alpha(t)\,dt.$$
By Proposition \ref{pro:omega-alpha}, we have that if $k-l=1$ and $H_k-h_l<\frac12$ or $k-l\geq2$, then $\omega_\alpha$ is a smooth $k$-convex subsolution of \eqref{eq:pro} in $\mathbb{R}^n\setminus\{0\}$ and satisfies $$\omega_\alpha(x)=\frac12x^TAx+\mu_1(\alpha)+O(|x|^{2-2\mathcal{H}_{k,l}}), \quad |x|\to\infty.$$ Also, the function $\mu_1(\alpha)$ is increasing and satisfies
\begin{equation}\label{eq:mu-alpha}
\lim_{\alpha\to\infty}\mu_1(\alpha)=\infty.
\end{equation}
Set
\begin{gather*}
\beta:=\min\{w_{\xi}(x)~|~\xi\in\partial D, x\in \overline{E(\bar{s})}\setminus D\},\\
\hat{b}:=\max\{w_{\xi}(x)~|~\xi\in\partial D, x\in \overline{E(\bar{s})}\setminus D\}.
\end{gather*}
where $w_\xi(x)$ is given by Lemma \ref{lem:w-xi}. Clearly, there holds that
\begin{equation}\label{eq:omega-beta}
\omega_\alpha\leq\beta,\quad\mathrm{in}\ E(\bar{s})\setminus\overline{D}, \forall \alpha>0.
\end{equation}
We will fix the value of $c_*$ in the proof. First we require that $c_*>\hat{b}$. It follows that $$\mu_1(0)=\beta-\bar{s}<\beta\leq \hat{b}<c_*.$$
Thus, in view of \eqref{eq:mu-alpha}, for every $c>c_*$, There exists a unique $\alpha(c)$ such that
\begin{equation}\label{eq:mu-alpha-c}
\mu_1(\alpha(c))=c.
\end{equation}
Set $$\underline{w}(x)=\max\{w_{\xi}(x)~|~\xi \in \partial D\}.$$
It is clear from Lemma \ref{lem:w-xi} that $\underline{w}$ is a locally Lipschitz function in $\mathbb{R}^n\setminus D$, and $\underline{w}=\varphi$ on $\partial D$. Since $w_\xi$ is a smooth convex solution of \eqref{eq:pro}, $\underline{w}$ is a viscosity subsolution of equation \eqref{eq:pro} in $\mathbb{R}^n\setminus\overline{D}$. We fix a number $\hat{s}>\bar{s}$, and then choose another number $\hat{\alpha}>0$ such that $$\min_{\partial E(\hat{s})}\omega_{\hat{\alpha}}>\max_{\partial E(\hat{s})}\underline{w}.$$ We require that $c_*$ also satisfies $c_*\geq \mu_1(\hat{\alpha})$ and fix now the value of $c_*$.
For $c\geq c_*$, we have $\alpha(c)=\mu_1^{-1}(c)\geq\mu_1^{-1}(c_*)\geq\hat{\alpha}$, and thereby
\begin{equation}\label{eq:omega-c-w}
\omega_{\alpha(c)}\geq \omega_{\hat{\alpha}}>\underline{w},\quad \mathrm{on}\ \partial E(\hat{s}).
\end{equation}
By \eqref{eq:omega-beta}, we have
\begin{equation}\label{eq:omega-beta-under-w}
\omega_{\alpha(c)}\leq \beta\leq \underline{w}, \quad \mathrm{in}\ E(\bar{s})\setminus\overline{D}.
\end{equation}
Now we define for $c>c_*$,
\begin{equation*}
\underline{u}(x)=\left\{
\begin{array}{ll}
\max\{\omega_{\alpha(c)}(x),\underline{w}(x)\}, & x\in E(\hat{s})\setminus D,\\
\omega_{\alpha(c)}(x),& x\in \mathbb{R}^n\setminus E(\hat{s}).
\end{array}
\right.
\end{equation*}
We know from \eqref{eq:omega-beta-under-w} that $$\underline{u}=\underline{w},\quad\mathrm{in}\ E(\bar{s})\setminus\overline{D},$$ and in particular
$$\underline{u}=\underline{w}=\varphi,\quad\mathrm{on}\ \partial D.$$
It follows from \eqref{eq:omega-c-w} that $\underline{u}=\omega_{\alpha(c)}$ in a neighborhood of $\partial E(\hat{s})$. Therefore $\underline{u}$ is locally Lipschitz in $\mathbb{R}^n\setminus D$. Since both $\omega_{\alpha(c)}$ and $\underline{w}$ are viscosity subsolutions of \eqref{eq:pro} in $\mathbb{R}^n\setminus\overline{D}$, so is $\underline{u}$.
For $c>c_*$, define $$\bar{u}(x):=\frac12x^TAx+c,$$ which is a smooth convex solution of \eqref{eq:pro}. By \eqref{eq:omega-beta-under-w}, we have
$$\omega_{\alpha(c)}\leq\beta\leq\hat{b}<c^*<\bar{u}, \quad\mathrm{on}\ \partial{D}.$$
Also, by \eqref{eq:omega-alpha-1} and \eqref{eq:omega-alpha-2} with \eqref{eq:mu-alpha-c},
$$\lim_{|x|\to\infty}\left(\omega_{\alpha(c)}-\bar{u}\right)=0.$$
Hence, applying Lemma \ref{lem:compar}, we deduce
\begin{equation}\label{eq:omega-c-bar-u}
\omega_{\alpha(c)}\leq \bar{u}, \quad\mathrm{on}\ \mathbb{R}^n\setminus D.
\end{equation}
From \eqref{eq:omega-c-w} and the above, one has, for $c>c^*$,
$$w_\xi\leq\bar{u}, \quad\mathrm{on}\ \partial(E(\hat{s})\setminus D),\forall\xi\in\partial D.$$
Using Lemma \ref{lem:compar} again, we obtain
$$w_\xi\leq\bar{u},\quad\mathrm{in}\ E(\hat{s})\setminus\overline{D},\forall\xi\in\partial D.$$
Therefore, $$\underline{w}\leq\bar{u},\quad\mathrm{in}\ E(\hat{s})\setminus\overline{D}.$$
Combining with \eqref{eq:omega-c-bar-u} and the above, we have
$$\underline{u}\leq\bar{u},\quad\mathrm{in}\ \mathbb{R}^n\setminus D. $$
For any $c>c^*$, let $\mathcal{S}_\alpha$ denote the set of $v\in \mathrm{USC}(\mathbb{R}^n\setminus D)$ which is the viscosity subsolution of \eqref{eq:pro} in $\mathbb{R}^n\setminus\overline{D}$ satisfying
\begin{equation*
v=\varphi\quad\mathrm{on}\ \partial D
\end{equation*}
and
\begin{equation*
\underline{u}\leq v\leq \bar{u}\quad\mathrm{in} \ \mathbb{R}^n\setminus D.
\end{equation*}
Apparently, $\underline{u}\in\mathcal{S}_\alpha$. Let
$$u(x):=\sup\{v(x)~|~v\in\mathcal{S}_\alpha\}, x\in \mathbb{R}^n\setminus D.$$
We have
\begin{equation*}
u(x)\geq\underline{u}=\omega_{\alpha(c)}(x)=\frac12x^TAx+c+O(|x|^{2-\frac{k-l}{H_k-h_l}}),\quad\mathrm{as}\quad x\to\infty,
\end{equation*}
especially, if $k-l\geq 2$, $$u(x)\geq\underline{u}=\frac12x^TAx+c+O(|x|^{\theta(2-n)}), \quad\mathrm{as}\quad x\to\infty$$
where $\theta\in(\frac{k-l-2}{n-2},1],$
and $$u(x)\leq\bar{u}(x)=\frac12x^TAx+c.$$
This indicates \eqref{eq:asym-1} and \eqref{eq:asym-2}.
Next, we prove that $u$ satisfies the boundary condition. It is obvious that $$\liminf_{x\to\xi}u(x)\geq\lim_{x\to\xi}\underline u(x)=\varphi(\xi), \quad\forall\xi\in\partial D.$$
So we only need to prove that $$\limsup_{x\to\xi}u(x)\leq\varphi(\xi),\quad\forall\xi\in\partial D.$$
Let $\omega^{+}\in C^2(\overline{E(\bar{s})\setminus D})$ be defined by
\begin{equation*
\left\{
\begin{array}{ll}
\Delta \omega^{+}=0,& in \ E(\bar{s})\setminus\overline{D}\\
\omega^{+}=\varphi, & on\ \partial D\\
\omega^{+}=\max_{\partial E(\bar{s})}\bar{u}=\bar{s}+c, & on\ \partial E(\bar{s}).
\end{array}
\right.
\end{equation*}
It is easy to see that a viscosity subsolution $v$ of \eqref{eq:pro} satisfies $\Delta v>0$ in viscosity sense. Therefore, for every $v\in\mathcal{S}_\alpha$, by $v\leq \omega^{+}$ on $\partial(E(\bar{s})\setminus D)$, we have $$v\leq\omega^{+}\quad\mathrm{in}\ E(\bar{s})\setminus\overline{D}.$$ It follows that $$u\leq \omega^{+}\quad\mathrm{in}\ E(\bar{s})\setminus\overline{D},$$ and then $$\limsup_{x\to\xi}u(x)\leq\lim_{x\to\xi}\omega^{+}(x)=\varphi(\xi),\quad\forall\xi\in\partial D.$$
Finally, applying the Perron's method Lemma \ref{lem:perron-m}, we find that $u\in C^0(\mathbb{R}^n\setminus D)$ is a viscosity solution of \eqref{eq:pro}. This completes our proof.
\end{proof}
We remark that Theorem \ref{thm:main-1} is also proved by Li and Li \cite{Li-Li-2017} by introducing a combersome notation $m_{k,l}$, which makes their proof looks rather long and difficult to read. Namely,
$$m_{k,l}(\lambda(A)):=\frac{k-l}{\bar{\xi}_k(\lambda(A))-\underline{\xi}_l(\lambda(A))}.$$ Denote $\lambda(A):=(a_1,\cdots,a_n)$. Then $\bar{\xi}_k$ and $\underline{\xi}_k$ respectively are
$$\bar{\xi}_k(a):=\sup_{\mathbb{R}^n\setminus\{0\}} \frac{\sum_{i=1}^n\sigma_{k-1;i}(a)a_i^2x_i^2}{\sigma_k(a)\sum_{i=1}^na_ix_i^2}$$ and $$\underline{\xi}_k(a):=\inf_{\mathbb{R}^n\setminus\{0\}} \frac{\sum_{i=1}^n\sigma_{k-1};i(a)a_i^2x_i^2}{\sigma_k(a)\sum_{i=1}^na_ix_i^2}.$$
In fact, these two quantities $\bar{\xi}_k$ and $\underline{\xi}_l$ are, respectively, equivalent to our $H_k$ and $h_l$ defined in \eqref{eq:H-h}. Indeed, it is clear that $H_k\geq \bar{\xi}_k$. On the other hand,
$$\frac{\sum_{i=1}^n\sigma_{k-1};i(a)a_i^2x_i^2}{\sigma_k(a)\sum_{i=1}^na_ix_i^2}=\frac{\sigma_{k-1;i_0}(a)a_{i_0}}{\sigma_k(a)}$$
if we choose $x=(0,\cdots,0,1,0,\cdots 0)$ whose the $i$-th component is $1$ and the others all are $0$. Thus, taking $\frac{\sigma_{k-1;i_0}(a)a_{i_0}}{\sigma_k(a)}=H_k(a)$ implies $H_k\leq\bar{\xi}_k$. Hence, $\bar{\xi}_k=H_k$. Likewise, we can verify $\underline{\xi}_l=h_l$.
\section{Proof of Theorem \ref{thm:P-L}}\label{sec:4}
In this section, we continue applying the Perron's method to prove Theorem \ref{thm:P-L}. The key is by making the Legendre transform to use a family of smooth convex supersolutions of Poisson equation, rather than that given by Proposition \ref{pro:omega-alpha}, to construct new subsolutions of \eqref{eq:pro-i} possessing suitable asymptotic property at infinity. The following lemma is needed and its proof can also be found in \cite{Dai-2012}.
\begin{Lem}\label{lem:Dai-s}
Let $D$ be a smooth, bounded, strictly convex open set in $\mathbb{R}^n$ and let $\varphi\in C^2(\overline{D})$ be $k$-convex. Assume that $D'\subset\subset D$ is an open subset and $V$ is a locally bounded function in $D$. Then there exists a $k$-convex function $u\in C^2(\overline{D})$ satisfying
\begin{equation}\label{eq:Dai-S-K-L}
\left\{
\begin{array}{ll}
S_{k,l}(D^2u)\geq 1, & x\in D,\\
u=\varphi, & x\in\partial D,\\
u<V, & x\in D'.
\end{array}
\right.
\end{equation}
\end{Lem}
\begin{proof}
From \cite{Trudinger-1995}, we let $v\in C^2(\overline{D})$ be a $k$-convex solution of problem
\begin{equation*}
\left\{
\begin{array}{ll}
S_{k,l}(D^2v)=1, & x\in D,\\
v=0, & x\in\partial D.
\end{array}
\right.
\end{equation*}
Since $\Delta v>0$, by the strong maximun principle, one has $v\leq -v_0$ on $\overline{D'}$ for some positive constant $v_0$. Set $$u(x)=\varphi(x)+\alpha v(x),\quad x\in D$$ where $\alpha>0$ is a constant to be determined later. Then $u\in C^2(\overline{D})$, $u=\varphi$ on $\partial D$ and $$u=\varphi+\alpha v\leq \sup_{D'}\varphi-\alpha v_0<\inf_{D'}V\leq V\quad \mathrm{in}\ D',$$
if $\alpha$ is large enough. Next we claim that $u$ is $k$-convex and is a solution of $$S_{k,l}(D^2u)\geq 1\quad \mathrm{in} \ D,$$ as long as $\alpha>1$.
Since $[S_{k,l}(S)]^{\frac{1}{k-l}}$ is a concave function of the elements of the symmetric matrix $S$ whenever $\lambda(S)\in \overline{\Gamma_k}$, we obtain
$$\left[S_{k,l}\left(\frac12(D^2\varphi+\alpha D^2v)\right)\right]^{\frac{1}{k-l}}\geq \frac12\left[S_{k,l}(D^2\varphi)\right]^{\frac{1}{k-l}}+\frac12\left[S_{k,l}(\alpha D^2 v)\right]^{\frac{1}{k-l}},$$
which indicates
\begin{align*}
\left[S_{k,l}(D^2\varphi+\alpha D^2v )\right]^{\frac{1}{k-l}}&\geq [S_{k,l}(D^2\varphi)]^{\frac{1}{k-l}}+[S_{k,l}(\alpha D^2 v)]^{\frac{1}{k-l}}\\
&\geq [S_{k,l}(\alpha D^2 v)]^{\frac{1}{k-l}}.
\end{align*}
Namely, $S_{k,l}(D^2u)\geq \alpha^{k-l}S_{k,l}(D^2 v)=\alpha^{k-l}$. Thus, if $\alpha>1$, then $S_{k,l}(D^2u)\ge1$ and our claim is true.
As argued above, we deduce that $u=\varphi+\alpha v$ is the solution of problem \eqref{eq:Dai-S-K-L} when $\alpha>1$ is sufficiently large.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:P-L}]
(i) Suppose $n\geq 3$. We split our proof into three steps.
\textbf{Step\, 1.} Construct a family of smooth convex subsolutions of $$S_{n,n-1}(D^2u)=1$$
in some exterior domain.
Given $A\in\mathcal{A}_{n,n-1}$, it is easy to check that $A^{-1}\in \mathcal{A}_{1,0}$. For $\gamma<0$, let
\begin{equation}\label{eq:u-y-poisson}
\bar{u}_{\alpha,\gamma,c}(y)=\left\{
\begin{array}{ll}
\frac12y^TA^{-1}y-c+\alpha|y|^{\gamma}, & 2-n\leq\gamma<0,\\\\
\frac12y^TA^{-1}y-c-\alpha|y|^{\gamma}, & \gamma\leq 2-n,
\end{array}
\right.
\end{equation}
where $c\in\mathbb{R}$ and $\alpha>0$.
Obviously, $\Delta\bar{u}_{\alpha,\gamma,c}\leq 1$ in $\mathbb{R}^n\setminus\{0\}$. Moreover, after a direct computation, for some $0<\delta<\min\{\lambda(A^{-1})\}:=\Lambda\leq \frac{1}{n}$, we have
$$D^2\bar{u}_{\alpha,\gamma,c}>\delta I,\quad\mbox{ if}~~ |y|>K,$$ where
\begin{equation*}
K:=\left\{
\begin{array}{ll}
{\big(\frac{-\alpha\gamma}{\Lambda-\delta}\big)}^{\frac{1}{2-\gamma}}, &\mathrm{if} \ 2-n\leq\gamma<0,\\\\
{\big[\frac{\alpha\gamma(\gamma-1)}{\Lambda-\delta}\big]^{\frac{1}{2-\gamma}}}, &\mathrm{if} \ \gamma\leq 2-n.
\end{array}
\right.
\end{equation*}
We then extend $\bar{u}_{\alpha,\gamma,c}(y)$ smoothly from $\mathbb{R}^n\setminus B_{K+\epsilon}$ ($\epsilon>0$) to $\mathbb{R}^n$ (still denoted by $\bar{u}_{\alpha,\gamma,c}$) such that $$D^2\bar{u}_{\alpha,\gamma,c}>\delta I\quad
\mathrm{in}\ \mathbb{R}^n.$$
We define the coordinate transformation
\begin{equation}\label{eq:y-x-D-L}
y\mapsto x=D\bar{u}_{\alpha,\gamma,c}(y)=A^{-1}y+O(|y|^{\gamma-1}).
\end{equation}
Since the Jacobian $\det D_yx=\det D^2 \bar{u}_{\alpha,\gamma,c}\neq 0$ and
\begin{align*}
|D(\bar{u}_{\alpha,\gamma,c}(y)-\bar{u}_{\alpha,\gamma,c}(y'))|&=\left|\int_0^1D^2\bar{u}_{\alpha,\gamma,c}(y'+t(y-y'))(y-y')\,dt\right|\\
&\geq\delta|y-y'|,
\end{align*}
for all $y,y'\in\mathbb{R}^n$. Therefore, the map \eqref{eq:y-x-D-L} is bijective.
For any $c\in\mathbb{R}$ and $\alpha>0$, we make a Legendre transform
\begin{equation}\label{eq:legendre-t}
u_{\alpha,\gamma,c}(x)=x\cdot y(x)-\bar{u}_{\alpha,\gamma,c}(y(x)).
\end{equation}
It is known that $y=Du_{\alpha,\gamma,c}(x)$ and $D^2\bar{u}_{\alpha,\gamma,c}=(D^2u_{\alpha,\gamma,c})^{-1}$. Setting $\tilde{D}:=D\bar{u}_{\alpha,\gamma,c}(B_{K+\epsilon})$, we thus obtain that
$$S_{n,n-1}(D^2u_{\alpha,\gamma,c})\geq1,\quad 0<D^2u_{c,\alpha}< \delta^{-1}I, \quad\mathrm{in}\ \mathbb{R}^n\setminus \tilde{D}.$$
Namely, we find that $u_{\alpha,\gamma,c}(x)$ are a family of smooth convex subsolutions of \eqref{eq:pro-i} in $\mathbb{R}^n\setminus \tilde{D}$. Furthermore, by virtue of \eqref{eq:legendre-t} with \eqref{eq:u-y-poisson} and \eqref{eq:y-x-D-L}, we derive that $y=Ax+O(|x|^{\gamma-1})$ and
\begin{equation}\label{eq:u-c-alpha-inf}
u_{\alpha,\gamma,c}(x)=\frac12x^TAx+c+O(|x|^{\gamma}),\quad |x|\to\infty.
\end{equation}
\textbf{Step\, 2.} Construct a viscosity subsolution and a viscosity supersolution to \eqref{eq:pro-i}.
For small $\alpha>0$ and $\epsilon>0$, we can assume that $\tilde{D}\subset D$. Fix now such $\alpha$ and choose a $R>0$ such that $D\subset D\bar{u}_{\alpha,\gamma,c}(B_R):=\hat{D}$. Set
\begin{gather*}
\underline{w}(x)=\max\{w_{\xi}(x)~|~\xi \in \partial D\}.
\end{gather*}
where $w_{\xi}$ is given by Lemma \ref{lem:w-xi}. In view of \eqref{eq:u-y-poisson} and \eqref{eq:legendre-t}, $$\lim_{c\to+\infty} u_{\alpha,\gamma,c}(x)=+\infty,\quad x\in \partial \hat{D}.$$ So we can choose a $c_*$, depending on $\alpha, \gamma, A, D,\varphi$, such that for every $c>c_*$,
\begin{equation}\label{eq:O-u-c-alpha}
\min_{\partial \hat{D}}u_{\alpha,\gamma,c}(x)>\max_{\partial \hat{D}}\underline{w}.
\end{equation}
Then from Lemma \ref{lem:Dai-s} there is a $n$-convex function $\tilde{u}$ fulfilling
\begin{equation}\label{eq:Dai-s}
\left\{
\begin{array}{ll}
S_{n,n-1}(D^2\tilde{u})\geq 1, & x\in \hat{D},\\
\tilde{u}=u_{\alpha,\gamma,c}, & x\in\partial \hat{D},\\
\tilde{u}<\underline{w}, & x\in D',
\end{array}
\right.
\end{equation}
where $D\subset D'\subset\subset \hat{D}$.
Now for every $c>c_*$, define
\begin{equation*}
\underline{u}(x)=\left\{
\begin{array}{ll}
\max\{\underline{w}(x),\tilde{u}\}, & x\in \hat{D}\setminus D,\\
u_{\alpha,\gamma,c}(x),& x\in \mathbb{R}^n\setminus\hat{D}.
\end{array}
\right.
\end{equation*}
By \eqref{eq:Dai-s}, $$\underline{u}=\underline{w},\quad\mathrm{in}\ D'\setminus D,$$ and in particular
$$\underline{u}=\underline{w}=\varphi,\quad\mathrm{on}\ \partial D.$$
Note from \eqref{eq:O-u-c-alpha} that $\underline{u}=u_{\alpha,\gamma,c}(x)$ in a neighborhood of $\partial \hat{D}$. Therefore $\underline{u}$ is locally Lipschitz in $\mathbb{R}^n\setminus D$. Since both $\tilde{u}$ and $\underline{w}$ are viscosity subsolutions of \eqref{eq:pro-i}, so is $\underline{u}$.
Next, for $c>c_*$, we define $$\bar{u}(x):=\frac12x^TAx+c.$$
Let $\max_{\partial D}\varphi \leq c_*<c\leq\bar{u}$, that is, $\underline{u}\leq \bar{u}$ on $\partial D$. By \eqref{eq:u-c-alpha-inf}, $$\lim_{|x|\to\infty}(\bar{u}-u_{\alpha,\gamma,c})=0.$$
Applying Lemma \ref{lem:compar}, we thus have for $c>c_*$, $$\underline{u}\leq \bar{u}\quad\mathrm{in}\ \mathbb{R}^n\setminus\overline{D}.$$
\textbf{Step\, 3.} Apply the Perron's method, Lemma \ref{lem:perron-m}, to obtain the existence of viscosity solution of \eqref{eq:pro-i}. This part is the same as the proof of Theorem \ref{thm:main-1}, we thus omit that.
(ii) Suppose $n=2$. Notice that $\det(A-I)=1$ for $A\in\mathcal{A}_{2,1}$, which implies $A-I\in \mathcal{A}_{2,0}$. We consider the Monge-Amp\`ere equation:
\begin{equation}\label{eq:pro-M-w}
\left\{
\begin{array}{ll}
\det(D^2w)=1,& in \quad\mathbb{R}^2\setminus\overline{D},\\
w=\varphi-\frac12|x|^2, & on\quad\partial D.
\end{array}
\right.
\end{equation}
From Theorem 1.1 in \cite{Bao-Li-2012}, there exists some constant $\alpha^*$ depending only on $A-I,b,D$ and $\|\varphi\|_{C^2(\partial D)}$ such that for every $\alpha>\alpha^*$, there exists a unique local convex solution $w\in C^\infty(\mathbb{R}^2\setminus \overline{D})\cap C^0(\mathbb{R}^2\setminus D)$ of \eqref{eq:pro-M-w} that satisfies
$$O(|x|^{-2})\leq w(x)-V'(x)\leq M(\alpha)+O(|x|^{-2}),\quad |x|\to\infty,$$
where
$$V'(x)=\frac12x^T(A-I)x+b\cdot x+\alpha\ln\sqrt{x^T(A-I)x}+c(\alpha),$$
and $M(\alpha),c(\alpha)$ are functions of $\alpha$.
Now, let $u(x)=w(x)+\frac12|x|^2$. After a direct computation, we find $u(x)$ is the local convex solution of \eqref{eq:pro-i} when $n=2$. Moreover,
$$O(|x|^{-2})\leq u(x)-V(x)\leq M(\alpha)+O(|x|^{-2}),\quad |x|\to\infty,$$
where
$$V(x)=V'(x)+\frac12|x|^2=\frac12x^TAx+b\cdot x+\alpha\ln\sqrt{x^T(A-I)x}+c(\alpha).$$
Thus, our proof is finished.
\end{proof}
\section*{Appendix: Examples for $H_k-h_{k-1}\geq \frac12$}
To study the existence of solutions for exterior Dirichlet problem \eqref{eq:pro} by the Perron's method, the key is to find enough subsolutions of Hessian quotient equations outside a bounded domain of $\mathbb{R}^n$. From the proof of Theorem \ref{thm:main-1}, we can see that when $k-l=1$, the assumption that $H_k-h_l<\frac12$ plays a core role in characterizing the asymptotic behavior at infinity of subsolutions. When $H_k-h_{k-1}\geq \frac12$, even though we have the corresponding subsolution with specific asymptotic behavior, see Proposition \ref{pro:omega-alpha}, we still can not use the current Perron's method to build the existence of Hessian quotient equations \eqref{eq:pro} for $A\in\mathcal{A}_{k,k-1}$.
Namely, when $k-l=1$ and $H_k-h_l<\frac{k-l}{2}$, since $\frac12x^TAx+c$ is a smooth solution (so is a supersolution) while
\begin{equation*
\omega_\alpha(x)=\frac12x^TAx+\mu_1(\alpha)+O(|x|^{2-\frac{k-l}{H_k-h_l}}), \quad |x|\to\infty.
\end{equation*}
is a $k$-convex subsolution, see Proposition \ref{pro:omega-alpha}, we can choose a suitable constant $\alpha>0$ such that $\mu_1(\alpha)=c$ and function $\frac12x^TAx+c$ can control $\omega_\alpha(x)$ at infinity, which makes Perron's method available. However, when $H_k-h_{k-1}=\frac12$ or $H_k-h_{k-1}>\frac12$, although we can construct some subsolutions $\omega_\alpha$ of \eqref{eq:pro}, respectively, satisfying
\begin{equation}\label{eq:app-omega-alpha-4}
\omega_\alpha(x)=\frac12x^TAx+\alpha\ln \left(\frac12x^TAx\right)+\mu_3(\alpha)+O(|x|^{-2}),
\end{equation}
and
\begin{equation}\label{eq:app-omega-alpha-5}
\omega_\alpha(x)=\frac12x^TAx+\frac{\alpha}{1-\frac{1}{2(H_k-h_l)}}\left(\frac12x^TAx\right)^{1-\frac{1}{2(H_k-h_l)}}+\mu_4(\alpha)+O(|x|^{2\theta}),
\end{equation}
where $\theta\in(-1,0)$,
apparently, the special solution $\frac12x^TAx+c$ is unable to control subsolution \eqref{eq:app-omega-alpha-4} and \eqref{eq:app-omega-alpha-5} at infinity, whatever $\alpha$ we choose. This is why we do not currently build the existence theorey for problem \eqref{eq:pro} with general diagonal matrix of $\mathcal{A}_{k,k-1}$ and $H_k-h_{k-1}\geq\frac12$.
For a given $A\in \mathcal{A}_{k,l}$, let $a:=\lambda(A)=(\lambda_1,\lambda_2,\cdots,\lambda_n)$ and $0<\lambda_1\leq\lambda_2\leq\cdots\leq\lambda_n$ denoting its positive eigenvalues. By the definition of \eqref{eq:H-h}, we easily calculate that, for $1\leq l<k\leq n$,
\begin{equation}\label{eq:A-H-h}
H_k=\frac{\sigma_{k-1;n}(a)\lambda_n}{\sigma_k(a)},\quad h_l=\frac{\sigma_{l-1;1}(a)\lambda_1}{\sigma_l(a)}.
\end{equation}
\begin{Ex}
When $n=k=2$ and $l=1$, we have, if $A\in \mathcal{A}_{2,1}$,
\begin{gather*}
\lambda_1\lambda_2=\lambda_1+\lambda_2, \quad H_2=1, \quad h_1=\frac{1}{\lambda_2}.
\end{gather*}
This implies $\lambda_2\geq 2$ and thus $H_2-h_1\geq \frac12$. Obviously, $H_2-h_1=\frac12$ if and only if $\lambda_1=\lambda_2=c^*(2,1)=2$, provided $n=2$.
\end{Ex}
\begin{Ex}
When $n\geq 3$, it is known that $H_k-h_{k-1}=\frac{1}{n}<\frac12$ if $A=c^*(k,k-1)I$.
\end{Ex}
\begin{Ex}
Let $A_1=\mathrm{diag}(2,4,4)$. By computations, we have
$$\sigma_3(A_1)=32=\sigma_2(A_1),$$
so $A_{1}\in\mathcal{A}_{3,2}$. Then by \eqref{eq:A-H-h}, we obtain
$$H_3(A_1)=1, h_2(A_1)=\frac12.$$
Thus, $H_3-h_2=\frac12$ for $A_1=\mathrm{diag}(2,4,4)$.
\end{Ex}
\begin{Ex}
Let $A_2=\mathrm{diag}(\frac{5}{3},5,5)$. We have
$$\sigma_3(A_2)=\frac{125}{3}=\sigma_2(A_2),\ \mbox{and}\ H_3(A_2)=1, h_2(A_2)=\frac{2}{5}. $$
This implies $H_3-h_2=\frac35>\frac12$ for $A_2=\mathrm{diag}(\frac{5}{3},5,5)$.
\end{Ex}
| {
"timestamp": "2020-05-08T02:08:29",
"yymm": "2004",
"arxiv_id": "2004.06908",
"language": "en",
"url": "https://arxiv.org/abs/2004.06908",
"abstract": "It is well-known that a celebrated Jörgens-Calabi-Pogorelov theorem for Monge-Ampère equations states that any classical (viscosity) convex solution of $\\det(D^2u)=1$ in $\\mathbb{R}^n$ must be a quadratic polynomial. Therefore, it is an interesting topic to study the existence and uniqueness theorem of such fully nonlinear partial differential equations' Dirichlet problems on exterior domains with suitable asymptotic conditions at infinity. As a continuation of the works of Caffarelli-Li for Monge-Ampère equation and of Bao-Li-Li for $k$-Hessian equations, this paper is devoted to the solvability of the exterior Dirichlet problem of Hessian quotient equations $\\sigma_k(\\lambda(D^2u))/\\sigma_l(\\lambda(D^2u))=1$ for any $1\\leq l<k\\leq n$ in all dimensions $n\\geq 2$. By introducing the concept of generalized symmetric subsolutions and then using the Perron's method, we establish the existence theorem for viscosity solutions, with prescribed asymptotic behavior which is close to some quadratic polynomial at infinity.",
"subjects": "Analysis of PDEs (math.AP)",
"title": "Hessian quotient equations on exterior domains",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.980280871316566,
"lm_q2_score": 0.7217432062975979,
"lm_q1q2_score": 0.7075110591362214
} |
https://arxiv.org/abs/1508.01610 | Sturm bounds for Siegel modular forms of degree 2 and odd weights | We correct the proof of the theorem in the previous paper presented by the first named author, which concerns Sturm bounds for Siegel modular forms of degree $2$ and of even weights modulo a prime number dividing $2\cdot 3$. We give also Sturm bounds for them of odd weights for any prime numbers, and we prove their sharpness. The results cover the case where Fourier coefficients are algebraic numbers. | \section{Introduction}
Sturm \cite{sturm1987congruence} studied how many Fourier coefficients we need, when we want to prove that an elliptic modular form vanishes modulo a prime ideal.
Its number is so called ``Sturm bound''. We shall explain it more precisely. For a modular form $f$, let $\Lambda$ be the index
set of the Fourier expansion of $f$. An explicitly given finite subset $S$ of $\Lambda$ is said to be a
\textit{Sturm bound} if vanishing modulo a prime ideal of Fourier coefficients of $f$ at $S$ implies vanishing modulo the prime ideal of all Fourier coefficients of $f$.
Poor-Yuen \cite{poor2015paramodular} studied initially Sturm bounds for Siegel
modular forms of degree $2$ for any prime number $p$. After their study, in \cite{choi2013sturm}, Choi, Choie and the
first named author gave other type bounds with simple descriptions for
them in the case of $p\ge 5$. Moreover, the first named author \cite{kikuta2015remark} attempted to supplement the case of $p\mid 2\cdot 3$. However, there are some gaps in the proof (of Theorem~2.1 in subsection 3.1, \cite{kikuta2015remark}). It seems that its method can only give more larger bounds. Richter-Raum \cite{richter2015sturm} gave some bounds for any $p$ in the case of general degree and any weight. However, their bounds seem not to be sharp except the case of $p\ge 5$ and even weight in degree 2 case. An improvement of their bounds depends on the case of degree $2$.
In this paper, we correct the proof of Theorem 2.1 in \cite{kikuta2015remark} by a new method. Namely we give the sharp Sturm bounds for Siegel modular forms of degree $2$ and even weight in the case of $p=2$, $3$. Moreover we give also the sharp bounds for them of odd weights modulo any prime number $p$. It should be remarked that, their sharpness become important to confirm congruences between two modular forms by numerical experiments, as the weights grow larger. Finally, we remark also that our results cover the case where Fourier coefficients are algebraic numbers.
\section{Statement of the results}
In order to state our results, we fix notation.
For a positive integer $n$,
we define the Siegel modular group $\Gamma_{n}$ of degree $n$ by
\begin{equation*}
\Gamma_{n} = \left\{\gamma \in \mathrm{GL}_{2n}(\mathbb{Z})
\bigm | \trn \gamma J_{n} \gamma = J_{n}
\right\},
\end{equation*}
where $J_{n} = \mat{0_{n}}{-1_{n}}{1_{n}}{0_{n}}$ and $0_{n}$ (resp. $1_{n}$)
is the zero matrix (resp. the identify matrix) of size $n$.
For a positive integer $N$, we define the principal congruence subgroup $\Gamma^{(n)}(N)$ of
level $N$ by
\begin{equation*}
\Gamma^{(n)}(N) = \left\{
\mat{a}{b}{c}{d} \in \Gamma_{n}\bigm | \begin{array}{l} a \equiv d \equiv 1_{n} \mod{N}\\ b\equiv c \equiv 0_n \mod{N} \end{array}
\right\}.
\end{equation*}
Here $a, b, c, d$ are $n \times n$ matrices.
A subgroup $\Gamma \subset \Gamma_{n}$ is said to be a congruence subgroup if
there exists a positive integer $N$ such that $\Gamma^{(n)}(N) \subset \Gamma
\subset \Gamma_{n}$.
For a congruence subgroup $\Gamma$, we say $\Gamma$ is of level $N$ if
$N = \min \left\{m \in \mathbb{Z}_{\ge 1}\bigm | \Gamma^{(n)}(m) \subset \Gamma
\right\}$.
We define the Siegel upper half space $\mathbb{H}_{n}$ of degree $n$ by
\begin{equation*}
\mathbb{H}_{n}=\left\{x + i y \bigm |
x \in \mathrm{Sym}_{n}(\mathbb{R}), \ y \in \mathrm{Sym}_{n}(\mathbb{R}), \ y \text{ is positive definite}
\right\},
\end{equation*}
where $\mathrm{Sym}_{n}(\mathbb{R})$ is a space of $n \times n$ symmetric matrices with
entries in $\mathbb{R}$.
For a congruence subgroup $\Gamma$ and $k \in \mathbb{Z}_{\ge 0}$,
a $\mathbb{C}$-valued holomorphic function $f$ on
$\mathbb{H}_{n}$ is said to be a (holomorphic) Siegel modular form of degree $n$,
of weight $k$ and of level $\Gamma$
if $f((a Z + b)(c Z + d)^{-1}) = \det \left(cZ + d\right)^{k} f(Z)$
for all $\mat{a}{b}{c}{d} \in \Gamma$.
If $n = 1$, we add the cusp condition.
We denote by $M_{k}(\Gamma)$ the space of Siegel modular forms of weight $k$
and of level $\Gamma$.
Any $f$ in $M_k(\Gamma )$ has a Fourier expansion of the form
\[
f(Z)=\sum_{0\le T\in \frac{1}{N}\Lambda _n}a_f(T)q^T,\quad q^T:=e^{2\pi i\text{tr}(TZ)},
\quad Z\in\mathbb{H}_2,
\]
where $T$ runs over all positive semi-definite elements of $\frac{1}{N}\Lambda _n$, $N$ is the level of $\Gamma $ and
\begin{align*}
\Lambda_n&:=\{ T=(t_{ij})\in\mathrm{Sym}_n(\mathbb{Q})\;|\; t_{ii},\;2t_{ij}\in\mathbb{Z}\; \}.
\end{align*}
For simplicity, we write $T=(m,r,n)$ for $T=\begin{pmatrix}m & r/2 \\ r/2 & n \end{pmatrix}\in \frac{1}{N}\Lambda _2$ and also $a_f(m,r,n)$ for $a_f(T)$.
Let $R$ be a subring of $\mathbb{C}$ and $M_{k}(\Gamma )_{R}\subset M_{k}(\Gamma )$ the $R$-module of all modular forms whose Fourier coefficients lie in $R$.
Let $f_1$, $f_2$ be two formal power series of the forms $f_i=\sum_{0\le T\in \frac{1}{N}\Lambda _n}a_{f_i}(T)q^T$ with $a_i\in R$. For an ideal $I$ of $R$, we write
\begin{equation*}
f_1 \equiv f_2 \mod{I},
\end{equation*}
if and only if $a_{f_1}(T) \equiv a_{f_2}(T) \mod{I}$ for all $T \in \frac{1}{N}\Lambda_{n}$
with $T \ge 0$.
If $I = (r)$ is a principal ideal, we simply denote $f_{1} \equiv f_{2} \mod{r}$.
Let $K$ be an algebraic number field and ${\mathcal O}={\mathcal O}_K$ the ring of integers in $K$. For a prime ideal $\frak{p}$ in ${\mathcal O}$, we denote by ${\mathcal O}_{\frak{p}}$ the localization of ${\mathcal O}$ at $\frak{p}$. Under these notation, we have
\begin{Thm}
\label{thm:1}
\noindent
Let $k$ be a non-negative integer, $\frak{p}$ an any prime ideal and $f \in
M_{k}(\Gamma _2)_{{\mathcal O}_{\frak{p}}}$.
We put
\begin{equation*}
b_{k} =
\begin{cases}
\left[ \frac{k}{10} \right] & \text{ if } k \text{ is even},\\
\left[ \frac{k - 5}{10} \right] & \text{ if } k \text{ is odd}.
\end{cases}
\end{equation*}
Here $[\cdot]$ is the Gauss symbol.
For $\nu \in \mathbb{Z}_{\ge 1}$,
assume that $a_f(m, r, n) \equiv 0 \bmod{\frak{p}^{\nu}}$ for all $m,\ r, \ n\in \mathbb{Z}$ with
\begin{align*}
0\le m, n\le b_{k},
\end{align*}
and $4mn - r^{2} \ge 0$, then we have $f \equiv 0 \bmod{\frak{p}^{\nu}}$.
\end{Thm}
\begin{Rem}
\begin{enumerate}
\item If $k$ is even and $\mathfrak{p} \nmid 2\cdot 3$, then the statement
of the theorem was essentially proved by Choi, Choie and the first named author
\cite{choi2013sturm}.
\item As mentioned in Introduction, in the case where $\frak{p}\mid 2\cdot 3$ and $k$ is even, the first named author stated the same property in \cite{kikuta2015remark}. However, the proof has some gaps and its method can give only more larger bounds. We give a new proof in subsection \ref{subsec:5.2}.
\item We note that $M_{k}(\Gamma_{2}) = \{0\}$ if $k$ is odd and $k < 35$.
\item Other type bounds also were given in \cite{kikuta2012note}.
\end{enumerate}
\end{Rem}
By the result of \cite{choi2013sturm} and a similar argument to them, we can prove the following.
\begin{Cor}
\label{cor:1}
Let $\Gamma \subset \Gamma_{2}$ be a congruence subgroup with level $N$, $k \in \mathbb{Z}_{\ge 0}$
and $f \in M_{k}(\Gamma)_{{\mathcal{O}}_{\mathfrak{p}}}$.
We put $i = [\Gamma_{2} : \Gamma]$.
For $\nu \in \mathbb{Z}_{\ge 1}$, assume that $a_{f}(m, r, n) \equiv 0
\bmod{\mathfrak{p}^{\nu}}$ for all $m, \ r, \ n
\in \frac{1}{N}\mathbb{Z}$ with
\begin{equation*}
0 \le m, n \le b_{ki}.
\end{equation*}
and $4mn - r^{2} \ge 0$, then we have $f \equiv 0 \bmod{\mathfrak{p}^{\nu}}$.
\end{Cor}
In the case of level $1$ (i.e., $N=1$), our bounds are sharp. More precisely, the following theorem holds.
\begin{Thm}
\label{thm:2}
Let $k \in \mathbb{Z}_{\ge 0}$ and $p$ be a prime number. We assume $M_{k}(\Gamma_{2}) \ne 0$.
Then there exists $f \in M_{k}(\Gamma_{2})_{\mathbb{Z}_{(p)}}$ with $f \not \equiv 0 \mod{p}$ such that
\begin{equation*}
a_{f}(m, r, n) = 0,\quad \text{ for all } m, \ n \le
b_{k} - 1.
\end{equation*}
\end{Thm}
\section{Notation}
\label{sec:3}
For a prime number $p$ and a $\mathbb{Z}_{(p)}$-module $M$, we put
\begin{equation*}
\widetilde{M} = M \otimes_{\mathbb{Z}_{(p)}}\mathbb{F}_{p}.
\end{equation*}
For an element $x \in M$, we denote by $\widetilde{x}$ the image of $x$ in $\widetilde{M}$.
For a $\mathbb{Z}_{(p)}$-linear map $\varphi : M \rightarrow N$, we denote by
$\widetilde{\varphi}$ the induced map from $\widetilde{M}$ to $\widetilde{N}$
by $\varphi$.
For $n \in \mathbb{Z}_{\ge 1}$, let $\Gamma$ be a congruence subgroup of $\Gamma_{n}$.
We denote $\widetilde{M}_{k}(\Gamma)_{\mathbb{Z}_{(p)}}$ by $\widetilde{M_{k}(\Gamma)_{\mathbb{Z}_{(p)}}}$.
For a commutative ring $R$ and an $R$-module $M$, we denote by
$\mathrm{Sym}^{2}(M) \subset M \otimes_{R} M$ the $R$-module generated by elements
$m \otimes m$ for $m \in M$.
Let $R$ be a $\mathbb{Z}_{(2)}$-algebra and $M$ an $R$-module.
We define an $R$-module $\wedge^{2}(M)$ by
\begin{math}
\wedge^{2}(M) = \left\{x \in M \bigm | x^{\iota} = - x\right\}.
\end{math}
Here $\iota$ is defined by $\iota(m \otimes n) = n \otimes m$ for
$m, \ n \in M$.
Let $q_{1}, q_{12}, q_{2}$ be variables and
\begin{math}
S = \left\{q_{1}^{m}q_{12}^{r}q_{2}^{n}\bigm |
m, n \in \mathbb{Z}_{\ge 0}, r \in \mathbb{Z}
\right\}
\end{math}
be a set of Laurent monomials.
We define an order of $S$ so that
$q_{1}^{m}q_{12}^{r}q_{2}^{n} \le q_{1}^{m'}q_{12}^{r'}q_{2}^{n'}$
if and only if one of the following conditions holds.
\begin{enumerate}
\item $m < m'$.
\item $m = m'$ and $n < n'$.
\item $m = m'$ and $n = n'$ and $r \le r'$.
\end{enumerate}
Let $K$ be a field and
\begin{math}
f = \sum_{m, r, n}
a_{f}(m, r, n)q_{1}^{m}q_{12}^{r}q_{2}^{n} \in
K[q_{12}, q_{12}^{-1}]\prsrs{q_{1}, q_{2}}
\end{math}
a formal power series. If $f \ne 0$,
let $q_{1}^{m_{0}}q_{12}^{r_{0}}q_{2}^{n_{0}}$ be the minimum
monomial which appears in $f$, that is the minimum monomial
of the set
\begin{math}
\left\{q_{1}^{m}q_{12}^{r}q_{2}^{n}\bigm |
a_{f}(m, r, n) \ne 0
\right\}.
\end{math}
We define the leading term $\ldt(f)$ of $f$ by
$a_{f}(m_{0}, r_{0}, n_{0})q_{1}^{m_{0}}q_{12}^{r_{0}}q_{2}^{n_{0}}$.
We also define the leading term of an element of
$K\prsrs{q_{1}, q_{2}} \setminus
\left\{0\right\}$
by the inclusion $K\prsrs{q_{1}, q_{2}} \subset
K[q_{12}, q_{12}^{-1}]\prsrs{q_{1}, q_{2}}$.
We regard $M_{k}(\Gamma_{2})$ as a subspace of $\mathbb{C}[q_{12}, q_{12}^{-1}]
\prsrs{q_{1}, q_{2}}$ by $\sum_{T = (m, r, n) \in \Lambda_{2}}a_{f}(m, r, n)
q^{T} \mapsto \sum_{m, r, n}a_{f}(m, r, n)q_{1}^{m}q_{12}^{r}q_{2}^{n}$.
For $f \in M_{k}(\Gamma_{2})$, we denote by $\ldt(f)$ the leading term of
the Fourier expansion of $f$.
For a field $K$, we regard $K\prsrs{q} \otimes_{K} K\prsrs{q}$ as a
subspace of $K\prsrs{q_{1}, q_{2}}$ by $q \otimes 1 \mapsto q_{1}$ and
$1 \otimes q \mapsto q_{2}$.
For a subring $R$ of $\mathbb{C}$ and a subset $S$ of $\mathbb{C}\prsrs{q_{1}, q_{2}}$, we
put
\begin{equation*}
S_{R}=\left\{f = \sum_{m, n}a_{f}(m, n)q_{1}^{m}q_{2}^{n}\in S\bigm |
a_{f}(m, n) \in R\right\}.
\end{equation*}
\section{Witt operators}
For the proof of the main results, we use Witt operators.
In this section, we define Witt operators and introduce basic properties of
them.
\subsection{Elliptic modular forms}
Since images of Witt operators can be written by elliptic modular forms,
we introduce some notation for elliptic modular forms.
For $k \in 2 \mathbb{Z}$ with $k \ge 4$, we denote by $e_{k} \in M_{k}(\Gamma_{1})$
the Eisenstein series of degree $1$ and weight $k$.
We normalize $e_{k}$ so that the constant term is equal to $1$.
We define Eisenstein series $e_{2}$ of degree $1$ and weight $2$ by
\begin{equation*}
e_{2}(q) = 1 - 24 \sum_{n = 1}^{\infty}\sigma_{1}(n)q^{n},
\end{equation*}
where $\sigma_{1}(n)$ is the sum of all positive divisors of $n$.
As is well known, $e_{2}$ satisfies the following identify:
\begin{equation*}
\tau^{-2}e_{2}(-\tau^{-1}) = \frac{12}{2\pi i\tau} + e_{2}(\tau).
\end{equation*}
We put $\Delta = 2^{-6} \cdot 3^{-3} (e_{4}^3 - e_{6}^{2})$. Then $\Delta$
is the Ramanujan's delta function.
For $k \ge 2$,
we define $N_{k}(\Gamma_{1})$ as the space of $\mathbb{C}$-valued holomorphic functions $f$
on $\mathbb{H}_{1}$ that satisfies the following three conditions:
\begin{enumerate}
\item $f(\tau + 1) = f(\tau)$.
\item There exists $g \in M_{k - 2}(\Gamma_{1})$ such that
\begin{equation*}
\tau^{-k}f(-\tau^{-1}) = \frac{1}{2 \pi i \tau}g(\tau) + f(\tau) \quad
\text{ for } \tau \in \mathbb{H}_{1}.
\end{equation*}
\item $f$ is holomorphic at the cusp $i \infty$.
\end{enumerate}
Since $f - e_{2}g/12 \in M_{k}(\Gamma_{1})$ for above $f$, we have the following
lemma.
\begin{Lem}
\label{lem:5}
\begin{equation*}
N_{k}(\Gamma_{1}) = M_{k}(\Gamma_{1}) \oplus e_{2}M_{k - 2}(\Gamma_{1}).
\end{equation*}
\end{Lem}
For $M = M_{k}(\Gamma_{1})$ or $N_{k}(\Gamma_{1})$, we regard
$M$ as a subspace of $\mathbb{C}\prsrs{q}$ via the Fourier expansion.
For $k = 2, 4, 6, 12$, we define elements of
$\mathrm{Sym}^{2}\left(N_{k}(\Gamma_{1})\right)_{\mathbb{Z}}$ as follows:
\begin{equation}
\label{eq:5}
x_{k} = e_{k} \otimes e_{k} ,\text{ for } k = 2, 4, 6, \qquad
x_{12} = \Delta \otimes \Delta, \qquad
y_{12} = e_{4}^{3} \otimes \Delta + \Delta \otimes e_{4}^{3}.
\end{equation}
We define $\alpha_{36} \in \wedge^{2}(M_{36}(\Gamma_{1}))_{\mathbb{Z}}$ by
\begin{equation*}
\alpha_{36} = x_{12}^{2}(\Delta \otimes e_{4}^{3} - e_{4}^{3} \otimes \Delta).
\end{equation*}
\subsection{Definition of Witt operators}
For $k \in \mathbb{Z}_{\ge 0}$ and $f \in M_{k}(\Gamma_{2})$, we consider the following
Taylor expansion
\begin{equation*}
f(Z)
= W(f)(\tau_{1}, \tau_{2}) + 2W'(f)(\tau_{1}, \tau_{2})
\left(2\pi i \tau_{12}\right) + W''(f)(\tau_{1}, \tau_{2})
\left(2 \pi i\tau_{12}\right)^{2} + O(\tau_{12}^{3}),
\end{equation*}
where $Z = \mat{\tau_{1}}{\tau_{12}}{\tau_{12}}{\tau_{2}} \in \mathbb{H}_{2}$.
We put $q_{1} = \mathbf{e}(\tau_{1}), q_{2} = \mathbf{e}(\tau_{2})$ and
$q_{12} = \mathbf{e}(\tau_{12})$.
By definition,
the following properties hold (see \cite[\S 9]{van2008siegel}).
\begin{enumerate}
\item $W'(f) = 0$ if $k$ is even and $W(f) = W''(f) = 0$ if $k$ is odd.
\item $W(f) \in \mathrm{Sym}^{2}(M_{k}(\Gamma_{1}))$ if $k$ is even
and $W'(f) \in \wedge^{2}(M_{k + 1}(\Gamma_{1}))$ if $k$ is odd.
Here we identify $q_{1}$ with $q\otimes 1$ and $q_{2}$ with $1 \otimes q$.
\item For $f \in M_{k}(\Gamma_{2})$ and $g \in
M_{l}(\Gamma_{2})$, we have
\begin{equation*}
W(fg) = W(f) W(g), \quad W'(fg) = W'(f) W(g) + W(f) W'(g).
\end{equation*}
Assume $k$ and $l$ are both even. Then we have
\begin{equation}
\label{eq:8}
W''(fg) = W''(f)W(g) + W(f)W''(g).
\end{equation}
\item For $f = \sum_{m, r, n}a_{f}(m, r, n)q_{1}^{m}q_{12}^{r}q_{2}^{n} \in
M_{k}(\Gamma_{2})$, we have
\begin{gather*}
W(f) = \sum_{m, r, n}a_{f}(m, r, n)q_{1}^{m}q_{2}^{n}, \quad
W'(f) = \frac{1}{2}\sum_{m, r, n}r a_{f}(m, r, n)q_{1}^{m}q_{2}^{n},\\
W''(f) = \frac{1}{2}\sum_{m, r, n}r^{2}a_{f}(m, r, n)q_{1}^{m}q_{2}^{n}.
\end{gather*}
\end{enumerate}
Let $k$ be even and $f \in M_{k}(\Gamma_{2})$. Then we have
\begin{gather*}
\tau_{1}^{-k-2}W''(f)(\tau_{1}^{-1}, \tau_{2}) = - \frac{1}{2\pi i}\theta_{2}
W(f)(\tau_{1}, \tau_{2}) + W''(f)(\tau_{1}, \tau_{2}),\\
W''(f)(\tau_{1}, \tau_{2}) = W''(f)(\tau_{2}, \tau_{1}).
\end{gather*}
Here $\theta_{2} = \frac{1}{2\pi i}\frac{d}{d\tau_{12}}$.
Therefore by Lemma \ref{lem:5}, we have the following lemma.
\begin{Lem}
Let $k \in 2\mathbb{Z}_{\ge 0}$ and $f \in M_{k}(\Gamma_{2})$. Then we have
$W''(f) \in \mathrm{Sym}^{2}(N_{k + 2}(\Gamma_{1}))$.
\end{Lem}
Let $R$ be a subring of $\mathbb{C}$.
If $k$ is even and $f \in M_{k}(\Gamma_{2})_{R}$, then we have
\begin{equation*}
W''(f) = \sum_{
\begin{subarray}{l}
m, r, n\\
r > 0
\end{subarray}}r^{2}a_{f}(m, r, n)q_{1}^{m}q_{12}^{r}q_{2}^{n},
\end{equation*}
since $a_{f}(m, -r, n) = a_{f}(m, r, n)$.
Thus we have $W''(f) \in \mathrm{Sym}^{2}(M_{k}(\Gamma_{1}))_{R}$.
By a similar reason, we have $W'(f) \in M_{k + 1}(\Gamma_{2})_{R}$ for
$f \in M_{k}(\Gamma_{2})_{R}$ with odd $k$.
For $k \in \mathbb{Z}_{\ge 0}$,
we define $R$-linear maps induced by $W, W'$ and $W''$ as follows.
\begin{align*}
&W_{R, 2k} : M_{2k}(\Gamma_{2})_{R} \rightarrow \mathrm{Sym}^{2}(M_{2k}(\Gamma_{1}))_{R},\\
&W'_{R, 2k - 1} : M_{2k - 1}(\Gamma_{2})_{R} \rightarrow
\wedge^{2}(M_{2k}(\Gamma_{1}))_{R},\\
&W''_{R, 2k}: M_{2k}(\Gamma_{2})_{R} \rightarrow
\mathrm{Sym}^{2}(N_{2 k + 2}(\Gamma_{1}))_{R}.
\end{align*}
\subsection{Igusa's generators and their images}
Let $X_{4}, \ X_{6}, \ X_{10}, \ X_{12}$ and $X_{35}$ be generators of
$\bigoplus_{k \in \mathbb{Z}}M_{k}(\Gamma_{2})$ given by Igusa \cite{igusa1962siegel},
\cite{igusa1964siegel}.
Here $X_{4}$ and $X_{6}$ are Siegel-Eisenstein series of weight 4 and 6
respectively. And $X_{10}, X_{12}$ and $X_{35}$ are cusp forms of weight
$10, 12$ and $35$ respectively.
We normalize these modular forms so that
\begin{equation*}
\ldt(X_{4}) = \ldt(X_{6}) = 1, \quad
\ldt(X_{10}) = \ldt(X_{12}) = q_{1}q_{12}^{-1}q_{2}^{2}, \quad
\ldt(X_{35}) = q_{1}^{2}q_{12}^{-1}q_{2}^{3}.
\end{equation*}
Here we note that $a_{X_{35}}(1, r, n) = 0$ for all $n, r \in \mathbb{Z}$,
because a weak Jacobi form of index $1$ and weight $35$ does not exist.
We also introduce $Y_{12} \in M_{12}(\Gamma_{2})_{\mathbb{Z}}$ and
$X_{k} \in M_{k}(\Gamma_{2})_{\mathbb{Z}}$ for
$k = 16,\allowbreak 18,\allowbreak 24,\allowbreak 28,\allowbreak
30,\allowbreak 36,\allowbreak 40,\allowbreak 42$
and $48$. Then by Igusa \cite{igusa1979ring},
\begin{equation*}
\left\{X_{k} \bigm | k = 4, 6, 10,
12, 16, 18, 24, 28, 30, 36, 40, 42,
48\right\} \cup \left\{Y_{12}\right\}
\end{equation*}
is a minimal set of generators of $\bigoplus_{k \in 2\mathbb{Z}} M_{k}(\Gamma_{2})_{\mathbb{Z}}$
as a $\mathbb{Z}$-algebra and we have $M_{k}(\Gamma_{2})_{\mathbb{Z}} = X_{35}M_{k-35}(\Gamma_{2})_{\mathbb{Z}}$
for odd $k$.
Igusa \cite{igusa1979ring} computed $W(X_{4}), \cdots, W(X_{48})$ and
$W(Y_{12})$, we introduce some of them.
\begin{align}
\label{eq:6}
\nonumber
&W(X_{4}) = x_{4},\quad
W(X_{6}) = x_{6}, \quad
W(X_{10}) = 0, \\
& W(X_{12}) = 2^{2}\cdot 3 x_{12},\quad W(Y_{12}) = y_{12}, \quad
W(X_{16})=x_4\cdot x_{12}
\end{align}
and
\begin{equation}
\label{eq:2}
W(X_{12i}) = d_{i}x_{12}^{i}, \text{ for } i = 1,2,3, 4.
\end{equation}
Here $x_{4}, x_{6}, x_{12}$ and $y_{12}$ are defined by \eqref{eq:5},
and $d_{i}$ is defined by $12/\mathrm{gcd}(12, i)$.
Images of $W'$ and $W''$ for some of the generators are given as follows.
\begin{Lem}
\label{lem:8}
We have
\begin{equation*}
W'(X_{35}) = \alpha_{36}.
\end{equation*}
and
\begin{equation*}
W''(X_{10}) = x_{12}, \quad W''(X_{12i}) = x_{2}x_{12}^{i}, \text{ for }
i = 1, 2, 3, 4.
\end{equation*}
\end{Lem}
\begin{proof}
By $\ldt(X_{35}) = q_{1}^{2}q_{12}^{-1}q_{2}^{3}$ and
$\wedge^{2}(M_{k}(\Gamma_{1})) = (\Delta \otimes e_{4}^{3} - e_{4}^{3} \otimes \Delta)
\mathrm{Sym}^{2}(M_{k - 12}(\Gamma_{1}))$, we see that $W'(X_{35})$ is a constant
multiple of $\alpha_{36}$. Since $a_{X_{35}}(2, r, 3) = 0$ if $r
\neq \pm 1$, we have $W'(X_{35}) = \alpha_{36}$.
Igusa computed $W''(X_{10})$ and $W''(X_{12})$ (see \cite[Lemma 12]
{igusa1979ring}). Note that our notation is different from his notation.
We denote his $W'$ by $W''$.
By this result, we can compute $W''(X_{12i})$ for $i = 2, 3, 4$.
\end{proof}
\subsection{Kernel of Witt operator modulo a prime}
Let $p$ be a prime number and $k$ even.
We consider the kernel of the Witt operator modulo $p$:
\begin{equation*}
\widetilde{W}_{\mathbb{Z}_{(p)}, k} : \widetilde{M}_{k}(\Gamma_{2})_{\mathbb{Z}_{(p)}} \rightarrow
\mathrm{Sym}^{2}(M_{k}(\Gamma_{2}))_{\mathbb{Z}_{(p)}} \otimes_{\mathbb{Z}_{(p)}}\mathbb{F}_{p}.
\end{equation*}
First we consider the case when $p \ge 5$. This case is easier.
\begin{Lem}
\label{lem:1}
Let $p$ be a prime number with $p \ge 5$.
Then we have
\begin{equation*}
\bigoplus_{k \in 2\mathbb{Z}_{\ge 0}}
\mathrm{Sym}^{2}\left(M_{k}(\Gamma_{1})\right)_{\mathbb{Z}_{(p)}} =
\mathbb{Z}_{(p)}[x_{4}, \ x_{6}, \ x_{12}].
\end{equation*}
\end{Lem}
\begin{proof}
It is easy to see that
$\mathrm{Sym}^{2}\left(M_{k}(\Gamma_{1})_{\mathbb{Z}_{(p)}}\right) = \mathrm{Sym}^{2}\left(M_{k}(\Gamma_{1})
\right)_{\mathbb{Z}_{(p)}}$ (see the remark after Theorem 5.12 of \cite{poor2015paramodular}).
Since $p \ge 5$,
we have $\bigoplus_{k \in 2\mathbb{Z}_{\ge 0}}M_{k}(\Gamma_{1})_{\mathbb{Z}_{(p)}}
= \mathbb{Z}_{(p)}[e_{4}, \ e_{6}]$ (see \cite{serre1973formes}).
We note that
$\bigoplus_{k \in 2\mathbb{Z}_{\ge 0}}\mathrm{Sym}^{2}\left(M_{k}(\Gamma_{1})_{\mathbb{Z}_{(p)}}\right)$
is generated by $x_{4}, \ x_{6}$ and $e_{4}^{3}\otimes e_{6}^{2} +
e_{6}^{2} \otimes e_{4}^{3}$ as an algebra over $\mathbb{Z}_{(p)}$.
Then the assertion of the lemma follows from the equation
\begin{equation*}
2^{12} \cdot 3^{6} x_{12} = x_{4}^{3} + x_{6}^{2} -
\left(e_{4}^{3}\otimes e_{6}^{2} + e_{6}^{2} \otimes e_{4}^{3} \right).
\end{equation*}
\end{proof}
The following is a key lemma for the proof of Theorem \ref{thm:1}
for $\mathfrak{p} \nmid 2 \cdot 3$. This lemma was also used in
\cite{choi2013sturm}.
\begin{Lem}
\label{lem:2}
Let $p \ge 5$ be a prime number and $k \in 2\mathbb{Z}_{\ge 0}$.
Then we have
\begin{equation*}
\ker\left(\widetilde{W}_{\zkp, k}\right) = \widetilde{X}_{10}
\widetilde{M}_{k - 10}(\Gamma_{2})_{\zkp}.
\end{equation*}
\end{Lem}
\begin{proof}
This lemma seems well-known. But for the sake of completeness,
we give a proof.
The inclusion $\widetilde{X}_{10}\widetilde{M}_{k - 10}(\Gamma_{2})_{\zkp} \subset
\ker\left(\widetilde{W}_{\zkp, k}\right)$ is obvious, because $W(X_{10}) = 0$.
Take $f \in M_{k}(\Gamma_{2})_{\zkp}$ with $W_{\zkp, k}(f) \equiv 0 \bmod{p}$.
By \eqref{eq:6} and Lemma \ref{lem:1}, $W_{\zkp, k}$ is surjective.
Take $g \in M_{k}(\Gamma_{2})_{\zkp}$ so that $W_{\zkp, k}(f) = pW_{\zkp, k}(g)$.
Then by \cite[Corollary 4.2]{nagaoka2000note}, there exists
$h \in M_{k - 10}(\Gamma_{2})_{\zkp}$ such that
$f - p g= X_{10} h$. This completes the proof.
\end{proof}
\begin{Rem}
Since $W(X_{12}) = 12 x_{12}$ and $M_{2}(\Gamma_{2}) = \left\{0\right\}$,
the assertion of the lemma does not hold if $p = 2, 3$.
\end{Rem}
Next we consider the case where $p = 2, 3$.
We recall the structure of the ring $\bigoplus_{k \in 2\mathbb{Z}}\widetilde{M}_{k}(\Gamma_{2})_{\zkp}$.
\begin{Thm}[Nagaoka \cite{nagaoka2005note}, Theorem 2]
\label{thm:4.7}
Let $p = 2, 3$. For $f\in \widetilde{M}_{k}(\Gamma_{2})_{\zkp}$, there exists a unique
polynomial $Q \in \mathbb{F}_{p}[x, y, z]$ such that
\begin{equation*}
\widetilde{f} = Q(\widetilde{X}_{10}, \widetilde{Y}_{12}, \widetilde{X}_{16}).
\end{equation*}
\end{Thm}
The above $Q$ for Igusa's generators are given as follows.
\begin{Lem}[Nagaoka \cite{nagaoka2005note}, proof of Lemma 1, Lemma 2]
\label{lem:10}
\begin{enumerate}
\item Suppose $p = 2$, then we have
\begin{align*}
& X_{4} \equiv X_{6} \equiv 1 \mod{p}, & & X_{12} \equiv X_{10} \mod{p}, \\
& X_{18} \equiv X_{16} \mod{p}, & & X_{24} \equiv X_{10}X_{16} \mod{p}, \\
& X_{28} \equiv X_{30} \equiv X_{16}^{2} \mod{p}, & & X_{36} \equiv X_{10} X_{16}^{2} \mod{p}, \\
& X_{40} \equiv X_{42} \equiv X_{16}^{3} \mod{p}, & & X_{48} \equiv X_{16}^{4} + X_{10}X_{16}^{3} + X_{10}^{4}Y_{12} \mod{p}, \\ & X_{35}^2\equiv X_{10}^2Y_{12}^2X_{16}^2+X_{10}^6 \mod{p}.
\end{align*}
\item Suppose $p = 3$, then we have
\begin{align*}
& X_{4} \equiv X_{6} \equiv 1 \mod{p}, & & X_{12} \equiv X_{10} \mod{p}, \\
& X_{18} \equiv X_{16} \mod{p}, & & X_{24} \equiv X_{10}X_{16} \mod{p}, \\
& X_{28} \equiv X_{30} \equiv X_{16}^{2} \mod{p}, & & X_{36} \equiv X_{16}^{3} + 2X_{10}^{3} Y_{12} + X_{10}X_{16}^{2} \mod{p}, \\
& X_{40} \equiv X_{16}^{3} + 2X_{10}^{3} Y_{12} \mod{p}, & & X_{42} \equiv X_{16}^{3} + X_{10}^{3}Y_{12} \mod{p}, \\
& X_{48} \equiv X_{10}X_{16}^{3} + 2X_{10}^{4}Y_{12} \mod{p}, & &
\end{align*}
and
\begin{multline*}
X_{35}^2\equiv 2X_{10}X_{16}^4+X_{10}Y_{12}^2X_{16}^3\\
+2X_{10}^2X_{16}^3+X_{10}^2Y_{12}^2X_{16}^2+ 2X_{10}^3Y_{12}X_{16}^2\\
+2X_{10}^4Y_{12}^3+X_{10}^4X_{16}^2+2X_{10}^7 \mod{p}.
\end{multline*}
\end{enumerate}
\end{Lem}
For later use, we prove the following lemma.
\begin{Lem}
\label{lem:11}
Let $p = 2, 3$ and $k \in 2\mathbb{Z}_{\ge 0}$ with $12 \nmid k$.
Then we have
\begin{equation*}
\widetilde{M}_{k}(\Gamma_{2})_{\zkp} \subset \widetilde{M}_{k + 2}(\Gamma_{2})_{\zkp}.
\end{equation*}
\end{Lem}
\begin{proof}
Take $f \in M_{k}(\Gamma_{2})_{\zkp}$. We show that there exists
$g \in M_{k + 2}(\Gamma_{2})_{\zkp}$ such that $f \equiv g \mod{p}$.
We may assume $f$ is an isobaric monomial
of Igusa's generators of even weights, that is $X_{4}, \cdots, X_{48}$ and $Y_{12}$.
If $f = X_{k}$ with $12 \nmid k$, then by Lemma \ref{lem:10},
we have $\widetilde{f} \in \widetilde{M}_{k + 2}(\Gamma_{2})_{\zkp}$.
In fact, we have $X_{18} \equiv X_{4}X_{16} \mod{p}$,
$X_{42}\equiv X_{16}X_{28} \bmod{2}$,
$X_{40} \equiv X_{42} + X_{10}^{3}Y_{12} \mod{3}$
and $X_{42} \equiv X_{16}X_{28} + X_{10}^{2}X_{12}Y_{12} \mod{3}$.
If $f$ is an isobaric monomial of weight $k$, then $f$ contains
some $X_{k}$ with $12 \nmid k$.
Therefore we have the assertion of the lemma.
\end{proof}
Let $f \in M_{k}(\Gamma_{2})_{\zkp}$ with $p = 2, 3$.
As we remarked before, $W(f) \equiv 0 \mod{p}$ does not imply
the existence of $g\in M_{k - 10}(\Gamma_{2})_{\zkp}$ such that
$f \equiv X_{10} g \mod{p}$.
Instead of Lemma \ref{lem:2}, we have the following proposition.
\begin{Prop}
\label{prop:1}
Let $p = 2, 3$ and $k \in 2\mathbb{Z}_{\ge 0}$.
\begin{enumerate}
\item Suppose $12 \nmid k$. Then we have
\begin{equation*}
\ker\left(\widetilde{W}_{\zkp, k}\right) = \widetilde{X}_{10} \widetilde{M}_{k - 10}(\Gamma_{2})_{\zkp}.
\end{equation*}
\item Suppose $k = 12 n$ with $n \in \mathbb{Z}$ and $p = 2$.
For $0 \le i \le n$ with $4 \nmid i$,
we put $i = 4s + t$ with $t \in \left\{1, 2, 3\right\}$ and
$m_{i} = X_{12t}X_{48}^{s}Y_{12}^{n - i}$.
Then we have
\begin{equation*}
\ker\left(\widetilde{W}_{\zkp, k}\right)
= \bigoplus_{
\begin{subarray}{c}
0 \le i \le n\\
4 \nmid i
\end{subarray}}
\mathbb{F}_{p}\widetilde{m}_{i} \oplus \widetilde{X}_{10} \widetilde{M}_{k - 10}(\Gamma_{2})_{\zkp}.
\end{equation*}
\item Suppose $k = 12 n$ with $n \in \mathbb{Z}$ and $p = 3$.
For $0 \le i \le n$ with $3 \nmid i$,
we put $i = 3s + t$ with $t \in \left\{1, 2\right\}$.
and $m_{i} =X_{12t}X_{36}^{s}Y_{12}^{n - i}$.
Then we have
\begin{equation*}
\ker\left(\widetilde{W}_{\zkp, k}\right)
= \bigoplus_{
\begin{subarray}{c}
0 \le i \le n\\
3 \nmid i
\end{subarray}}
\mathbb{F}_{p}\widetilde{m}_{i} \oplus \widetilde{X}_{10} \widetilde{M}_{k - 10}(\Gamma_{2})_{\zkp}.
\end{equation*}
\end{enumerate}
Moreover, if $f \in M_{k}(\Gamma_{2})_{\zkp}$ with $12 \mid k$ and
\begin{equation*}
W(f) \equiv W''(f) \equiv 0 \mod{p},
\end{equation*}
then there exists $g \in M_{k - 20}(\Gamma_{2})_{\zkp}$ such that
$f \equiv X_{10}^{2} g \mod{p}$.
\end{Prop}
\begin{proof}
Suppose $12 \nmid k$. Then by \cite[Lemma 13]{igusa1979ring},
$W_{\mathbb{Z}, k}$ is surjective. Therefore, $W_{\zkp, k}$ is surjective.
We can prove
\begin{math}
\ker\left(\widetilde{W}_{\zkp, k}\right) = \widetilde{X}_{10} \widetilde{M}_{k - 10}(\Gamma_{2})_{\zkp}
\end{math}
by a similar argument to the proof of Lemma \ref{lem:2}.
Next assume $k = 12n$ with $n \in \mathbb{Z}$.
For simplicity, we assume $p = 2$. We can prove the case when $p = 3$
in a similar way.
Take $f \in M_{k}(\Gamma_{2})_{\zkp}$ with $W(f) \equiv 0 \mod{p}$.
Put $d_{i} = 12/\mathrm{gcd}(12, i)$.
By \cite[Lemma 13]{igusa1979ring}, there exist
$a_{i, j}, b_{i}, c_{i} \in \zkp$ such that
\begin{equation*}
W(f) = \sum_{0 \le i\le j < n}a_{i, j}x_{4}^{3(n - j)}x_{12}^{i} y_{12}^{j
- i}
+ \sum_{
\begin{subarray}{c}
0 \le i \le n\\
4 \mid i
\end{subarray}}
b_{i}x_{12}^{i}y_{12}^{n - i}
+ \sum_{
\begin{subarray}{c}
0 \le i \le n\\
4 \nmid i
\end{subarray}}
c_{i}d_{i}x_{12}^{i}y_{12}^{n - i}.
\end{equation*}
By $x_{4} \equiv 1 \mod{p}$ and $W(f) \equiv 0 \mod{p}$, we have
$a_{i, j} \equiv b_{i} \equiv 0 \mod{p}$ for all $i, j$.
Here we note that $\widetilde{x}_{12}$ and $\widetilde{y}_{12}$ are algebraically independent
over $\mathbb{F}_{p}$. This is because $\ldt(x_{12}^{i}y_{12}^{j}) =
q_{1}^{i}q_{2}^{i + j}$.
By \cite[Lemma 13]{igusa1979ring}, there exists $f' \in M_{k}(\Gamma_{2})_{\zkp}$
such that
\begin{equation*}
W(f') = \sum_{0 \le i\le j < n}\frac{a_{i, j}}{p}
x_{4}^{3(n - j)}x_{12}^{i} y_{12}^{j - i}
+ \sum_{
\begin{subarray}{c}
0 \le i \le n\\
4 \mid i
\end{subarray}}
\frac{b_{i}}{p}x_{12}^{i}y_{12}^{n - i}.
\end{equation*}
By \eqref{eq:2},
there exists $u_{i} \in \mathbb{Z}_{(p)}^{\times}$ such that
$W(m_{i}) = u_{i}d_{i} x_{12}^{i}y_{12}^{n - i}$.
Therefore, there exist $a_{i} \in \mathbb{Z}_{(p)}$ such that
$W(f - pf' - \sum_{
\begin{subarray}{l}
0 \le i \le n\\
4 \nmid i
\end{subarray}
}a_{i}m_{i}
) = 0$. By \cite[Corollary 4.2]{nagaoka2000note}, there exists $g \in M_{k - 10}(\Gamma_{2})_{\zkp}$
such that
\begin{math}
\widetilde{f} = \sum_{i}\widetilde{a}_{i}\widetilde{m}_{i} + \widetilde{X}_{10}\widetilde{g}.
\end{math}
Thus we have
\begin{equation}
\label{eq:7}
\ker\left(\widetilde{W}_{\zkp, k}\right)
= \sum_{
\begin{subarray}{c}
0 \le i \le n\\
4 \nmid i
\end{subarray}}
\mathbb{F}_{p}\widetilde{m}_{i} + \widetilde{X}_{10} \widetilde{M}_{k - 10}(\Gamma_{2})_{\zkp}.
\end{equation}
We show that the sum \eqref{eq:7} is a direct sum.
Let $a_{i}\in \mathbb{Z}_{(p)}$ for $0 \le i \le n$ with $4 \nmid i$
and $g \in M_{k-10}(\Gamma_{2})_{\zkp}$. We put $f = \sum_{i}a_{i}m_{i} +
X_{10}g$.
By \eqref{eq:8}, we have
\begin{equation}
\label{eq:9}
W''(m_{i}) \equiv W''(X_{12t})W(X_{48}^{s}Y_{12}^{n-i}) \equiv
x_{12}^{i}y_{12}^{n-i} \mod{p}.
\end{equation}
Here we use $W(X_{12t}) \equiv 0 \mod{p}$ for $t = 1, 2, 3$
and $x_{2} \equiv 1 \mod{p}$.
By Igusa's computation, images of 14 generators $X_{4}, \cdots, X_{48}$
by $W$
can be written as $\mathbb{Z}$-coefficient polynomials of $x_{4}, x_{6}, x_{12}$ and $y_{12}$.
By Lemma \ref{lem:8}, we have $W''(X_{10}) = x_{12}$.
Thus there exist $\alpha_{a, b, c, d} \in \mathbb{Z}_{(p)}$ such that
\begin{equation*}
W''(X_{10}g)
= x_{12}W(g)
= \sum_{a, b, c, d}\alpha_{a, b, c, d} x_{4}^{a}x_{6}^{b}x_{12}^{c}y_{12}^{d},
\end{equation*}
where summation index runs over $\left\{(a, b, c, d) \in \mathbb{Z}_{\ge 0}^{4}
\bigm |
4a + 6b + 12 c + 12 d = k + 2\right\}$.
We assume $\widetilde{W}''_{\zkp, k}(\widetilde{f}) = \widetilde{W}''_{\zkp, k}
(\sum_{i}\widetilde{a}_{i}\widetilde{m}_{i} + \widetilde{X}_{10}\widetilde{g}) = 0$.
Then by \eqref{eq:9} and $x_{4} \equiv x_{6} \equiv 1 \mod{p}$, we have
\begin{equation*}
\sum_{i}\widetilde{a}_{i}\widetilde{x}_{12}^{i}\widetilde{y}_{12}^{n - i} +
\sum_{a, b, c, d}\widetilde{\alpha}_{a, b, c, d}\widetilde{x}_{12}^{c}\widetilde{y}_{12}^{d} = 0.
\end{equation*}
Since $4a + 6b = 0$ or $4a + 6b \ge 4$, the isobaric degree of
$\widetilde{x}_{12}^{c}\widetilde{y}_{12}^{d}$ is not equal to $k$. Therefore we have
$\widetilde{a}_{i} = 0$ for all $i$. This shows that the sum \eqref{eq:7} is a direct
sum. This also shows that if $f \in M_{k}(\Gamma_{2})_{\zkp}$ with $12 \mid k$
satisfies $W(f) \equiv W''(f) \equiv 0 \mod{p}$, then there exists
$h \in M_{k - 10}(\Gamma_{2})_{\zkp}$ such that $f \equiv X_{10}h\mod{p}$. By
$W''(f) \equiv 0 \mod{p}$, we have $W(h) \equiv 0 \mod{p}$.
Since $12 \nmid k - 10$, there exists $h' \in M_{k - 20}(\Gamma_{2})_{\zkp}$ such
that $h \equiv X_{10}h' \mod{p}$.
Therefore we have $f \equiv X_{10}^{2}h' \mod{p}$.
This completes the proof.
\end{proof}
\begin{Cor}
\label{cor:3}
Let $p = 2, 3$ and $f \in M_{k}(\Gamma_{2})_{\zkp}$ with $12 \mid k$.
If $W(f) \equiv 0 \mod{p}$, then there exists $g \in M_{k - 8}(\Gamma_{2})_{\zkp}$
such that $f \equiv X_{10}g \mod{p}$.
\end{Cor}
\begin{proof}
By Lemma \ref{lem:10}, the statement for $f = m_{i}$ is true for all $i$.
Then by Proposition \ref{prop:1}, we have
$f \equiv X_{10}(g + h) \mod{p}$ with $g \in M_{k - 8}(\Gamma_{2})_{\zkp}$ and
$h \in M_{k - 10}(\Gamma_{2})_{\zkp}$.
By Lemma \ref{lem:11}, we have our assertion.
\end{proof}
\section{Proof of the main results}
In this section, we give proofs of Theorem \ref{thm:1},
Corollary \ref{cor:1} and Theorem \ref{thm:2}.
We have
\begin{math}
\widetilde{M}_{k}(\Gamma_{2})_{\mathcal{O}_{\mathfrak{p}}}
=
\widetilde{M}_{k}(\Gamma_{2})_{\mathbb{Z}_{(p)}}\otimes_{\mathbb{F}_{p}}
\mathcal{O}_{\mathfrak{p}}/\mathfrak{p}.
\end{math}
Therefore Theorem \ref{thm:1} is reduced to the case of
$\mathcal{O}_{\mathfrak{p}} = \mathbb{Z}_{(p)}$, where $p$ is a prime number.
We also note that the statement of Theorem \ref{thm:1} for $\nu \ge 2$
is reduced to the case of $\nu = 1$ by repeatedly using the result.
This method was used in \cite{rasmussen2009higher}.
As we remarked before, the statement of Theorem \ref{thm:1}
was proved in \cite{choi2013sturm} if $k$ is even and $p \ge 5$.
Thus in this section, we assume
$k \equiv 0 \mod{2},\ p = 2, 3$
or $k \equiv 1 \mod{2}$.
First, we introduce the following notation, which is similar to
mod $p$ diagonal vanishing order defined by
Richter and Raum \cite{richter2015sturm}.
Let $\widetilde{f}$ be a $\mathbb{F}_{p}$-coefficients formal power series as follows;
\begin{equation*}
\widetilde{f} = \sum_{
\begin{subarray}{c}
m, r, n \in \mathbb{Q}\\
m, n, 4mn - r^{2}\ge 0
\end{subarray}}
\widetilde{a}_{f}(m, r, n)q_{1}^{m}q_{12}^{r}q_{2}^{n}
\in \bigcup_{N \in \mathbb{Z}_{\ge 1}}\mathbb{F}_{p}[q_{12}^{1/N}, q_{12}^{-1/N}]
\prsrs{q_{1}^{1/N}, q_{2}^{1/N}}.
\end{equation*}
We define $v_{p}(\widetilde{f})$ by
\begin{equation*}
v_{p}(\widetilde{f}) = \sup
\left\{
A \in \mathbb{R}\bigm |
\begin{array}{l}
\widetilde{a}_{f}(m, r, n) = 0, \\
\text{for all } m, r, n \in \mathbb{Q} \text{ with }
0 \le m, n < A
\end{array}
\right\}.
\end{equation*}
By definition, we have
\begin{equation}
\label{eq:11}
v_{p}(\widetilde{f} \widetilde{g}) \ge \max\left\{
v_{p}(\widetilde{f}), v_{p}(\widetilde{g})
\right\},
\end{equation}
for $\widetilde{f}, \widetilde{g} \in \bigcup_{N \in \mathbb{Z}_{\ge 1}}\mathbb{F}_{p}[q_{12}^{1/N}, q_{12}^{-1/N}]
\prsrs{q_{1}^{1/N}, q_{2}^{1/N}}$.
We note that
$v_{p}(\widetilde{f}) > A$ is equivalent to
$\widetilde{a}_{f}(m, r, n) = 0$ for all $m, n \le A$,
where $A \in \mathbb{R}$.
For the proof of Theorem \ref{thm:1}, we introduce the following three lemmas.
\begin{Lem}
\label{lem:3}
Let $p$ be a prime number and
$\widetilde{f} \in \widetilde{M}_{k}(\Gamma_{2})_{\zkp}$ with $k \in \mathbb{Z}_{\ge 0}$.
Then we have $v_{p}(\widetilde{X}_{10} \widetilde{f}) = v_{p}(\widetilde{f}) + 1$ and $v_p(\widetilde{X}_{35}\widetilde{f})\ge v_p(\widetilde{f})+2$.
\end{Lem}
\begin{proof}
We regard $\widetilde{X}_{10}$ and $\widetilde{X}_{35}$ as images
in the ring of formal power series
$\mathbb{F}_{p}(q_{12})\prsrs{q_{1}, q_{2}}$.
By the Borcherds product of $X_{10}$ (cf. \cite{gritsenko1997siegel}),
we have $\widetilde{X}_{10} = q_{1} q_{2} u$ where $u$ is a unit in
$\mathbb{F}_{p}(q_{12})\prsrs{q_{1}, q_{2}}$.
Similarly, we have $\widetilde{X}_{35} = q_{1}^2 q_{2}^2 (q_1-q_2)v$ for
some unit $v$ in $\mathbb{F}_{p}(q_{12})\prsrs{q_{1}, q_{2}}$ (cf. \cite{gritsenko1996igusa}).
The assertion of the lemma follows from these facts.
\end{proof}
\begin{Rem}
It is not easy to give an upper bound for $v_{p}(\widetilde{X}_{35}\widetilde{f}) - v_{p}(\widetilde{f})$
because of the factor $q_{1} - q_{2}$ in the Borcherds product of
$X_{35}$.
\end{Rem}
\begin{Lem}
\label{lem:12}
Let $p$ be a prime number and
\begin{equation*}
f = \sum_{m, n \ge 0}a_{f}(m, n)q_{1}^{m}q_{2}^{n}
\in \left( M_{k}(\Gamma_{1}) \otimes M_{k}(\Gamma_{1})
\right)_{\zkp}.
\end{equation*}
If $a_{f}(m, n)\equiv 0 \mod{p}$ for all $m, n \le [k/12]$,
then $f \equiv 0 \mod{p}$.
In particular, for $g \in M_{k}(\Gamma_{2})_{\zkp}$,
we have $W(g) \equiv 0 \mod{p}$ if $v_{p}(\widetilde{g})
> [k/12]$ and
$W'(g) \equiv 0 \mod{p}$ if $v_{p}(\widetilde{g}) > [(k + 1)/12]$.
\end{Lem}
\begin{proof}
By the original Sturm's theorem \cite{sturm1987congruence}, the map
\begin{equation*}
\widetilde{M}_{k}(\Gamma_{1})_{\zkp} \hookrightarrow \mathbb{F}_{p}\prsrs{q}/(q^{[k/12] + 1})
\end{equation*}
is injective.
Therefore we have the following injective map
\begin{multline*}
\mathrm{Sym}^{2}(M_{k}(\Gamma_{1}))_{\zkp}\otimes_{\zkp}\mathbb{F}_{p} =
\mathrm{Sym}^{2}(\widetilde{M}_{k}(\Gamma_{1})_{\zkp})\\
\hookrightarrow
F_{p}\prsrs{q}/(q^{[k/12] + 1})\otimes_{\mathbb{F}_{p}}
F_{p}\prsrs{q}/(q^{[k/12] + 1}).
\end{multline*}
Here we note that
$\mathrm{Sym}^{2}(M_{k}(\Gamma_{1})_{\zkp}) = \mathrm{Sym}^{2}(M_{k}(\Gamma_{1}))_{\zkp}$,
as we remarked in the proof of Lemma \ref{lem:1}.
Since the image of $\widetilde{f}$ by this map vanishes,
we have $\widetilde{f} = 0$.
\end{proof}
\begin{Lem}
\label{lem:4}
We define $f_{k} \in M_{k}(\Gamma_{2})_{\mathbb{Z}}$ for $k = 35, 39, 41, 43$ and $47$ as
follows.
\begin{gather*}
f_{35} = X_{35}, \quad f_{39} = X_{4} X_{35}, \quad f_{41} = X_{6}X_{35},
\quad
f_{43} = X_{4}^{2}X_{35}, \quad f_{47} = X_{12} X_{35}.
\end{gather*}
Then $\ldt(f_{k}) = q_{1}^{2}q_{12}^{-1}q_{2}^{3}$
for $k = 35, 39, 41, 43$
and $\ldt(f_{47}) = q_{1}^{3}q_{12}^{-2}q_{2}^{4}$.
\end{Lem}
\begin{proof}
This follows from $\ldt(X_{4}) = \ldt(X_{6}) = 1$,
$\ldt(X_{12}) = q_{1}q_{12}^{-1}q_{2}$ and
$\ldt(X_{35}) = q_{1}^{2}q_{12}^{-1}q_{2}^{3}$.
\end{proof}
\subsection{Proof of Theorem ~\ref{thm:1} for $p = 2, 3$ and even $k$}
\label{subsec:5.2}
Let $p = 2, 3$, $k \in 2\mathbb{Z}_{\ge 0}$ and $f \in M_{k}(\Gamma_{2})_{\zkp}$.
We assume
\begin{equation}
\label{eq:1}
v_{p}(\widetilde{f}) > b_{k},
\end{equation}
where $b_{k}$ is given in Theorem \ref{thm:1}.
We prove the statement of Theorem \ref{thm:1} by the induction on $k$.
First, we assume $k < 10$.
Then the statement is true because $M_{k}(\Gamma_{2})$ for $k = 4, 6, 8$ is one-dimensional
and $\ldt(X_{4}) = \ldt(X_{6}) = \ldt(X_{4}^{2}) = 1$.
Next, we assume $k \ge 10$ and the statement is true if the weight is
strictly less than $k$.
By \eqref{eq:1} and Lemma \ref{lem:12}, we have $W(f) \equiv 0 \mod{p}$.
If $12 \nmid k$, then by
Proposition \ref{prop:1}, there exists
$g \in M_{k - 10}(\Gamma_{2})_{\zkp}$ such that $f \equiv X_{10} g \mod{p}$.
By \eqref{eq:1} and Lemma \ref{lem:3}, we have
$v_{p}(\widetilde{g}) > b_{k - 10}$.
By the induction hypothesis, we have $g \equiv 0 \mod{p}$.
Thus we have the assertion of Theorem \ref{thm:1} in this case.
Next we assume $12 \mid k$. Then by Corollary \ref{cor:3}, there exists $g \in M_{k - 8}(\Gamma_{2})_{\zkp}$
such that $f \equiv X_{10}g \mod{p}$.
Since $b_{k - 10} \ge [(k - 8) / 12]$ for $k \ge 10$,
we have $W(g) \equiv 0 \mod{p}$ by
\eqref{eq:1}, Lemma \ref{lem:3} and Lemma \ref{lem:12}.
Therefore $W''(f) \equiv x_{12}W(g) \equiv 0 \mod{p}$.
By Proposition \ref{prop:1}, there exists $h \in M_{k - 20}(\Gamma_{2})_{\zkp}$ such that
$f \equiv X_{10}^{2}h \mod{p}$.
Since $v_{p}(\widetilde{h}) > b_{k - 20}$,
we have $h \equiv 0 \mod{p}$ by the induction hypothesis.
Thus we have $f\equiv 0 \mod{p}$. This completes the proof.\qed
\subsection{Proof of Theorem~\ref{thm:1} for the case $p \nmid 2\cdot 3$
and odd $k$}
Let $p$ be a prime number with $p \ge 5$ and $f \in M_{k}(\Gamma_{2})_{\mathbb{Z}_{(p)}}$
with $k$ odd.
We assume
\begin{equation}
v_{p}(\widetilde{f}) > b_{k}.
\label{eq:3}
\end{equation}
We prove the theorem by the induction on $k$.
Note that $M_{k}(\Gamma_{2}) = \left\{0\right\}$ if $k$ is odd and $k < 35$ or $k
= 37$.
First assume that $ 0 \le k - 35 < 10$ with $k \ne 37$.
Then $M_{k}(\Gamma_{2})$ is one-dimensional and spanned by $f_{k}$ given in Lemma
\ref{lem:4}.
By Lemma \ref{lem:4}, the assertion of the theorem holds if $k - 35 < 10$.
Next we assume $k -35 \ge 10$ and the assertion of the theorem holds if
the weight is strictly less than $k$.
By Igusa \cite{igusa1979ring}, there exists $g \in M_{k - 35}(\Gamma_{2})_{\zkp}$
such that $f = X_{35} g$.
By Lemma \ref{lem:8}, we have
\begin{align}
W'(f) = W'(X_{35}) W(g) = \alpha_{36} W(g). \label{eq:4}
\end{align}
By $[(k + 1)/12] \le b_{k}$ and Lemma \ref{lem:12}, we have $W'(f)
\equiv 0 \bmod{p}$.
Therefore,
we have $W(g) \equiv 0 \bmod{p}$ by \eqref{eq:4}. Then by Lemma \ref{lem:2}, there exists
$g' \in M_{k- 45}(\Gamma_{2})_{\mathbb{Z}_{(p)}}$ such that
$g \equiv X_{10} g' \bmod{p}$.
We put $f' = X_{35}g'$. Then we have $f \equiv X_{10}f' \bmod{p}$.
By \eqref{eq:3} and Lemma \ref{lem:3}, we have
$v_{p}(\widetilde{f}') > b_{k - 10}$.
By the induction hypothesis, we have $f' \equiv 0 \bmod{p}$.
Thus $f \equiv 0 \bmod{p}$. This completes the proof. \qed
\subsection{Proof of Theorem~\ref{thm:1} for $p = 2, 3$ and odd $k$}
In this subsection, we assume $p = 2, 3$ and $k$ is odd.
Since the case when $k = 48 + 35 = 83$ is special in our proof, we prove the
following two lemmas.
\begin{Lem}
\label{lem:6}
Let $\widetilde{f} \in \widetilde{M}_{48}(\Gamma_{2})_{\zkp}$ with $\widetilde{f} \ne 0$ and
$\ldt(\widetilde{f}) = \alpha q_{1}^{a}q_{12}^{b}q_{2}^{c}$ be the leading term of
$\widetilde{f}$. Here $\alpha \in \mathbb{F}_{p}^{\times}$.
Assume $\widetilde{W}_{\zkp, 48}(\widetilde{f}) = 0$.
Then we have $a \le 4$ and $c \le 4$.
\end{Lem}
\begin{proof}
By Proposition \ref{prop:1}, we have
\begin{equation*}
\ker(\widetilde{W}_{\zkp, 48}) = \bigoplus_{i}\mathbb{F}_{p}\widetilde{m}_{i}
\oplus \widetilde{X}_{10} \widetilde{M}_{38}(\Gamma_{2})_{\zkp}.
\end{equation*}
Here $i = 1, 2, 3$ if $p = 2$ and $i = 1, 2, 4$ if $p = 3$.
For $\widetilde{g} \in \widetilde{M}_{48}(\Gamma_{2})_{\zkp}$, let $Q_{g}=
\sum_{a, b,
c}\gamma_{a, b, c}x^{a}y^{b}z^{c}$ be
a $\mathbb{F}_{p}$-coefficients polynomial
such that $\widetilde{g} = Q_{g}(\widetilde{X}_{10}, \widetilde{Y}_{12}, \widetilde{X}_{16})$ as in
Theorem \ref{thm:4.7}.
Since
\begin{equation}
\label{eq:10}
\ldt\left(\widetilde{X}_{10}^{a}\widetilde{Y}_{12}^{b}\widetilde{X}_{16}^{c}\right) = q_{1}^{a + c}
q_{12}^{-a}q_{2}^{a + b + c},
\end{equation}
there exists a unique monomial
$\widetilde{X}_{10}^{a_{0}}\widetilde{Y}_{12}^{b_{0}}\widetilde{X}_{16}^{c_{0}}$ with $\gamma_{a_{0}, b_{0}, c_{0}} \ne 0$
such that
$\ldt(\widetilde{g}) = \ldt(\gamma_{a_0, b_0, c_0}\widetilde{X}_{10}^{a_{0}}\widetilde{Y}_{12}^{b_{0}}\widetilde{X}_{16}^{c_{0}})$.
We put $\phi(\widetilde{g}) = \widetilde{X}_{10}^{a_{0}} \widetilde{Y}_{12}^{b_{0}}\widetilde{X}_{16}^{c_{0}}$.
We define a set $S'$ by
\begin{gather*}
\left\{
1,
\widetilde{X}_{16},
\widetilde{Y}_{12},
\widetilde{X}_{10},
\widetilde{X}_{16}^{2},
\widetilde{Y}_{12} \widetilde{X}_{16},
\widetilde{Y}_{12}^{2},
\widetilde{X}_{10} \widetilde{X}_{16},
\widetilde{X}_{10} \widetilde{Y}_{12},
\widetilde{X}_{10}^{2},\right.\\
\left.
\widetilde{X}_{10} \widetilde{X}_{16}^{2},
\widetilde{X}_{10} \widetilde{Y}_{12} \widetilde{X}_{16},
\widetilde{X}_{10} \widetilde{Y}_{12}^{2},
\widetilde{X}_{10}^{2} \widetilde{X}_{16},
\widetilde{X}_{10}^{2} \widetilde{Y}_{12},
\widetilde{X}_{10}^{3}
\right\}.
\end{gather*}
Then $S'$ forms a basis of $\widetilde{M}_{38}(\Gamma_{2})_{\zkp}$.
This follows from $\dim_{\mathbb{F}_{p}}(\widetilde{M}_{38}(\Gamma_{2})_{\zkp}) =
\dim_{\mathbb{C}}M_{38}(\Gamma_{2}) = 16$ and Lemma \ref{lem:10}.
We put $S = \left\{\widetilde{X}_{10} a \bigm | a \in S'\right\}$.
We define a set $T$ by
\begin{equation*}
T =
\begin{cases}
\left\{\widetilde{m}_{1}, \widetilde{m}_{2}, \widetilde{m}_{3}\right\} & \text{ if } p = 2,\\
\left\{\widetilde{m}_{1}, \widetilde{m}_{2}, \widetilde{m}_{4}\right\} & \text{ if } p = 3.
\end{cases}
\end{equation*}
Then $S \cup T$ forms a basis of $\ker(\widetilde{W}_{\zkp, 48})$.
We have $\phi(s) = s$ except when $p = 3$ and $s = m_{4}$ for $s \in S\cup T$.
If $p = 3$, we have $\phi(\widetilde{m}_{4}) = \widetilde{X}_{10}\widetilde{Y}_{12}\widetilde{X}_{16}^{2}$.
Thus we see that
$\phi$ on $S \cup T$ is injective.
Therefore if $\widetilde{f} \in \ker(\widetilde{W}_{\zkp, 48})$ with $\widetilde{f} \ne 0$, then
there exists a unique $s \in S \cup T$ such that
$\ldt(\widetilde{f}) = \alpha \ldt(s)$ with $\alpha \ne 0$.
Note that degrees of
monomials $\left\{\phi(s) \bigm | s \in S\cup T\right\}$
are less than or equal to $4$.
Then by \eqref{eq:10}, we have the assertion of the lemma.
\end{proof}
\begin{Lem}
\label{lem:7}
Let $k = 83$, $\widetilde{f} = \widetilde{X}_{35}
\widetilde{g}\in \widetilde{M}_{k}(\Gamma_{2})_{\zkp}$ with
$g \in \widetilde{M}_{k - 35}(\Gamma_{2})_{\zkp}$ and
$\widetilde{W}_{\zkp, k - 35}(\widetilde{g}) = 0$.
Assume $v_{p}(\widetilde{f}) > b_{k} = 7$. Then we have
$\widetilde{f} = 0$.
\end{Lem}
\begin{proof}
Assume $\widetilde{f} \ne 0$.
We put $\ldt(\widetilde{g}) = \alpha q_{1}^{a}q_{12}^{b}q_{2}^{c}$, where
$\alpha \in \mathbb{F}_{p}^{\times}$.
Then by Lemma \ref{lem:6}, we have $a, c \le 4$.
Since $\ldt(\widetilde{X}_{35}) = q_{1}^{2}q_{12}^{-1}q_{2}^{3}$, we have
$\ldt(\widetilde{f}) = \alpha q_{1}^{a + 2}q_{12}^{b-1}q_{2}^{c + 3}$.
By $a + 2 \le 6$ and $c + 3 \le 7$, we have $v_{p}(\widetilde{f}) \le 7$.
\end{proof}
Let $k$ be odd and $f \in M_{k}(\Gamma_{2})_{\zkp}$. Assume that
\begin{align}
\label{k4}
v_p(\widetilde{f}) > b_{k}.
\end{align}
If $k < 45$, then the assertion follows from Lemma \ref{lem:4}.
Hence we suppose that $k \ge 45$. To apply an induction on $k$, suppose that the assertion is true for any weight strictly smaller than $k$.
We take $g\in M_{k-35}(\Gamma _2)_{\mathbb{Z}_{(p)}}$ such that $f=gX_{35}$.
By \eqref{k4}, \eqref{eq:4} and Lemma \ref{lem:12}, we have
$W(g) \equiv 0 \mod{p}$.
Now we separate into four cases:\\
(1) If $k \not \equiv 11 \mod{12}$, then there exists $g'\in M_{k-45}(\Gamma _2)_{\mathbb{Z}_{(p)}}$ such that
$g \equiv X_{10}g' \mod{p}$, by Proposition \ref{prop:1}. Then $f=X_{35}g=X_{35}X_{10}g'$. If we put $f':=X_{10}g'\in M_{k-10}(\Gamma _2)_{\mathbb{Z}_{(p)}}$, then
\[
b_{k} < v_p(\widetilde{f})=v_{p}(\widetilde{X}_{10}\widetilde{f} ')=1+v_p(\widetilde{f} ').
\]
This implies $v_p(\widetilde{f} ')> b_{k - 10}$. By the induction hypothesis, we get $f'\equiv 0$ mod $p$.
Therefore we have $f\equiv 0 \mod{p}$.
\noindent
(2) If $k \equiv 11 \mod{12}$ and $k \equiv 1, 5, 7, 9 \mod{10}$, then we have
$b_{k}= b_{k - 8} + 1$. By Corollary \ref{cor:3}, there exists
$g' \in M_{k - 43}(\Gamma_{2})_{\zkp}$ such that $g \equiv X_{10}g' \mod{p}$.
Put $f' = X_{35}g' \in M_{k-8}(\Gamma_{2})_{\zkp}$.
Then we have $v_{p}(\widetilde{f}') = v_{p}(\widetilde{f}) - 1 > b_{k - 8}$. By the induction
hypothesis, we have $f' \equiv 0 \mod{p}$.
Therefore we have $f \equiv 0 \mod{p}$.
\noindent
(3) If $k \equiv 11 \mod{12}$, $k \equiv 3 \mod{10}$ and $k < 115$,
then we have $k = 83$ because $k \ge 45$.
Then by Lemma \ref{lem:7}, we have $f \equiv 0 \mod{p}$.
\noindent
(4) Finally, we assume $k \equiv 11 \mod{12}$ and $k \ge 115$.
To prove this case, we start with proving the following lemma.
\begin{Lem}
\label{lem:9}
Let $f = X_{35}g \in M_{k}(\Gamma_{2})_{\zkp}$ with
$W(g) \equiv 0 \mod{p}$.
Assume $k \equiv 11 \mod{12}$, $k \ge 115$ and \eqref{k4}.
Then we have $W''(g) \equiv 0 \mod{p}$.
\end{Lem}
\begin{proof}
We show the statement only for $p=2$. The case $p=3$ also can be proved by a
similar argument.
By Corollary \ref{cor:3},
there exists $g'\in M_{k-43}(\Gamma _2)_{\mathbb{Z}_{(p)}}$ such that
$g\equiv X_{10}g' \mod{p}$.
Then, it follows from Lemma \ref{lem:10} that
\[fX_{35}\equiv X_{10}X_{35}^2g'
\equiv g'X_{10}^3(Y_{12}^2X_{16}^2+X_{10}^4) \mod{p}. \]
By Lemma \ref{lem:3} and the assumption (\ref{k4}), we have
\begin{align*}
b_{k}+2 < v_p(\widetilde{f})+2\le
v_p(\widetilde{f} \widetilde{X}_{35}) =v_{p}(\widetilde{g}' \widetilde{X}_{10}\widetilde{X}_{35}^2)=
3+v_p(\widetilde{g}'(\widetilde{Y}_{12}^2\widetilde{X}_{16}^2+\widetilde{X}_{10}^4)).
\end{align*}
This implies that
\begin{equation*}
v_p(\widetilde{g}'(\widetilde{Y}_{12}^2\widetilde{X}_{16}^2+\widetilde{X}_{10}^4))> [(k-15)/10].
\end{equation*}
On the other hand, we have
\begin{align*}
W(g'(Y_{12}^2X_{16}^2+X_{10}^4))=W(g'Y_{12}^2X_{16}^2)
\equiv W(g') \cdot y_{12}^2 \cdot x_{12}^2 \mod{p},
\end{align*}
where we used (\ref{eq:6}) and the fact $x_{4}\equiv 1$ mod $p$. By this
congruence, $W(\widetilde{g}'(\widetilde{Y}_{12}^2\widetilde{X}_{16}^2+\widetilde{X}_{10}^4))$ can be
regarded as of weight $k-43+48=k+5$. By $k\ge 115$, we have
\[v_p(\widetilde{g}'(\widetilde{Y}_{12}^2\widetilde{X}_{16}^2+\widetilde{X}_{10}^4))> [(k-15)/10] \ge [(k+5)/12]. \]
Applying Lemma \ref{lem:12}, we have
\[W(g'(Y_{12}^2X_{16}^2+X_{10}^4))\equiv W(g') \cdot y_{12}^2 \cdot x_{12}^2\equiv 0 \mod{p}. \]
This implies that
\[W''(g)\equiv W(g')\cdot x_{12} \equiv 0 \mod{p}. \]
This completes the proof of the lemma.
\end{proof}
We shall return to proof of the case (4).
Since $W(g)\equiv 0 \mod{p}$ and $W''(g) \equiv 0 \mod{p}$,
there exists $h\in M_{k-55}(\Gamma _2)_{\mathbb{Z}_{(p)}}$ such that
$g\equiv X_{10}^2h \mod{p}$ by Proposition \ref{prop:1}.
Note that $f\equiv X_{10}^2X_{35}h \mod{p}$. If we put $f':=X_{35}h\in M_{k-20}(\Gamma _2)_{\mathbb{Z}_{(p)}}$, then
\[v_{p}(\widetilde{f})=v_p(\widetilde{X}_{10}^2\widetilde{f}')=2+v_p (\widetilde{f}')> b_{k}. \]
This means that \[v_p(\widetilde{f}')> b_{k-20}. \]
By the induction hypothesis, we get $f'\equiv 0 \mod{p}$.
Therefore we have $f \equiv 0 \mod{p}$.
This completes the proof.
\qed
\subsection{Proof of Corollary \ref{cor:1}}
As explained in the beginning of this section, we may assume
$\mathcal{O}_{\mathfrak{p}} = \zkp$, where $p$ is a prime number.
Let $\Gamma \subset \Gamma_{2}$ be a congruence subgroup of level $N$
and $f \in M_{k}(\Gamma)_{\zkp}$.
By the proof of \cite[Theorem 1.3]{choi2013sturm}, there exists
$g \in M_{k (i - 1)}(\Gamma)_{\zkp}$ such that
\begin{equation*}
f g \in M_{ki}(\Gamma_{2})_{\zkp}, \text{ and } g \not \equiv 0 \mod{p}.
\end{equation*}
Here $i = [\Gamma_{2}: \Gamma]$.
We assume $v_{p}(\widetilde{f}) > b_{ki}$.
Then by \eqref{eq:11}, we have
\begin{equation*}
v_{p}(\widetilde{f} \widetilde{g}) \ge v_{p}(\widetilde{f}) > b_{ki}.
\end{equation*}
By Theorem \ref{thm:1}, we have $\widetilde{f} \widetilde{g} = 0$.
Since $\widetilde{g} \ne 0$, we have $\widetilde{f} = 0$, i.e., $f \equiv 0 \mod{p}$.
This completes the proof.
\qed
\subsection{Proof of the sharpness}
We prove Theorem \ref{thm:2}.
If $k$ is even, then we can show the assertion of the theorem
by a similar argument to \cite{choi2013sturm}.
Let $k$ be odd and $f_{k}$ for $k = 35, 39, 41, 43$ and $47$
be modular forms given in Lemma \ref{lem:4}.
Then by Lemma \ref{lem:4}, we have
$\ldt(f_{k}X_{10}^{i}) = q_{1}^{2 + i}q_{2}^{-1 - i}q_{3}^{3 + i}$
for $k = 35, 39, 41, 43$ and
$\ldt(f_{47}X_{10}^{i})=q_{1}^{3 + i}q_{2}^{-2 - i}q_{3}^{4 + i}$.
Thus we have the assertion of the theorem.\qed
\section*{Acknowledgment}
The first named author is supported by JSPS Grant-in-Aid for Young Scientists (B) 26800026.
The second named author is partially supported by JSPS Kakenhi 23224001.
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
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\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
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| {
"timestamp": "2015-08-10T02:04:51",
"yymm": "1508",
"arxiv_id": "1508.01610",
"language": "en",
"url": "https://arxiv.org/abs/1508.01610",
"abstract": "We correct the proof of the theorem in the previous paper presented by the first named author, which concerns Sturm bounds for Siegel modular forms of degree $2$ and of even weights modulo a prime number dividing $2\\cdot 3$. We give also Sturm bounds for them of odd weights for any prime numbers, and we prove their sharpness. The results cover the case where Fourier coefficients are algebraic numbers.",
"subjects": "Number Theory (math.NT)",
"title": "Sturm bounds for Siegel modular forms of degree 2 and odd weights",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808707404786,
"lm_q2_score": 0.7217432062975979,
"lm_q1q2_score": 0.7075110587204342
} |
https://arxiv.org/abs/1712.00708 | Study of the Sparse Superposition Codes and the Generalized Approximate Message Passing Decoder for the Communication over Binary Symmetric and Z Channels | In this project, the behavior of Generalized Approximate Message-Passing Decoder for BSC and Z Channel is studied using i.i.d matrices for constructing the codewords. The performance of GAMP in AWGN Channel is already evaluated in the previous scientific work of Jean Barbier, therefore, this project mainly focuses on the performance of GAMP decoder for BSC and Z Channel. We evaluate the performance of the GAMP decoder for sparse superposition codes at various settings and compare the performance of decoder for different channels and parameters. | \section{Introduction}
\subsection{Sparse Superposition Codes}
We can define sparse superposition code as a signal that consists of multiple sections and each section is a vector with length \(B\) where \(B-1\) of the values in a section are \(0\)s. For this project, we used \(B\) values where \(B\) is an integer power of 2. For instance, in case of \(B=4\), the alphabet has symbols \(1000, 0100, 0010, 0001\) and it can be said that the number of possible letters in the alphabet equals to \(B\). If we want to send a message \(\mathbf{x}\) with \(L\) sections, the total length of the message becomes \(N=BL\) and the sparsity of this vector increases when \(B\) increases. To send the sparse superposition codes through the channel, we multiply the message \(\mathbf{x}\) by a matrix \(\mathbf{A}\) and obtain a vector \(\mathbf{z}\) which is \(\mathbf{z}=\mathbf{A}\mathbf{x}\) and send \(\mathbf{z}\) through the noisy channel. \(\mathbf{A}\) is a transform matrix with size \(MN\) and the length of the obtained \(\mathbf{z}\) vector is equal to \(M\). For this project, we used a random gaussian matrix with 0 mean and variance \(\bigtriangleup\) as our \(\mathbf{A}\) matrix. However, we need to fix the power \(P\) of \(\mathbf{z}\) to one, therefore we created a Gaussian matrix with 0 mean and variance 1 and divided it by \(\sqrt{L}\) which becomes our \(\mathbf{A}\) matrix.
To determine the \(M\) values which allow this system to communicate through the channel we need to calculate our communication rate \(R\) with respect to \(M\). Number of informative bits in the message \(\mathbf{x}\) can be defined as \(K=L\log_2{B}\) and if we define a parameter called \(\alpha \) which is equal to \(\alpha=M/N\), the communication rate becomes \(R=K/M=L\log_2{B}/(N\alpha)=\log_2{B}/(B\alpha)\). Therefore, we can adjust the communication rate \(R\) by changing the number of rows in matrix \(\mathbf{A}\).
After sending the codeword \(\mathbf{z}\) through the AWGN channel we get an output \(\mathbf{y}\) which is \(\mathbf{y=z+N}\), where \(\mathbf{N}\) is the noise of the channel. However, BSC and Z channels are binary channels and only 0s and 1s can be sent through those channels and \(\mathbf{z}\) is not in a binary form. Therefore, instead of sending \(\mathbf{z}\), we send \(\text{sign} \mathbf{(z)}\) information through the channel. In binary symmetric channel, input bits 0 and 1 have a flip probability of \(\epsilon\) whereas in Z channel, only 0s have a flip probability of \(\epsilon\) and 1s are always transmitted correctly.
Our main ambition after transmission is to decode the message \(\mathbf{x}\) correctly, with the help of the output \(\mathbf{y}\), transform matrix \(\mathbf{A}\) and the signal-to-noise ratio (SNR) or \(\epsilon\) of the channel. To calculate the error obtained after the transmission, we use two measures which are mean square error(MSE or \(\widetilde{E}\)) and section error rate (SER). We can compute MSE and SER as:
\begin{equation}
\widetilde{E} =\frac{1}{L}\sum_{i=1}^{N}(x_{i}-\hat{x_{i}})^2
\end{equation}
\begin{equation}
SER=\frac{1}{L}\sum_{l=1}^{L}\mathbb{1}(x_{l}\ne\hat{x_{l}})
\end{equation}
By using SER and MSE values, it is possible to observe the behaviour of the decoder for different values of \(B\), \(R\), channel type and \(\epsilon\) or signal-to-noise ratio (SNR).
\subsection{Generalized Approximate Message Passing
Algorithm}
To decode the output vector \(\mathbf{y}\) to estimate the input \(\mathbf{x}\), we had to choose an algorithm to use. For this goal, after fruitful discussions with Jean Barbier in the beginning of the semester, we decided to use Generalized Approximate Message Passing (GAMP) algorithm instead of Belief Propagation (BP) since the factor graph associated to our estimation problem is densely connected (BP is used when the factor graph sparse and it is computationally inefficient to use in case of densely connected graph).
Generalized Approximate Message Passing gives us an approximation, but it is known as an efficient method for estimating unknown vector in a linear mixing problem. Let us consider our case as an example: There is an unknown vector, \( \mathbf{x} \), which is a sparse superposition vector with the entries of every section are zero except one component. Firstly, this vector is multiplied with a transform matrix \( \mathbf{A} \), which is a random gaussian matrix in our case, therefore we get a codeword \( \mathbf{z} \), namely \(\mathbf{z}=\mathbf{A}\mathbf{x} \). After this operation, \( \mathbf{z} \) generates an output vector \( \mathbf{y} \) with respect to a conditional probability distribution \( p_{Y |Z} (y |z ) \). The problem here is to estimate the input vector \( \mathbf{x} \) from the output vector \( \mathbf{y} \) and transform matrix \( \mathbf{A} \). Optimal estimation of \( \mathbf{x} \) is hard since the components of it are coupled in \( \mathbf{z} \). To achieve this goal, we use the GAMP algortihm, which is computationally simple, in Rangan's paper[2].
Given a matrix \( \mathbf{A}\in \mathbb{R}^{mxn} \), system inputs and outputs \( \mathbf{q} \) and \( \mathbf{y} \) and scalar estimation functions \(g_{in}(.)\) and \(g_{out}(.)\) generate a sequence of estimates \(\hat{ \mathbf{x}} \)(t), \( \hat{\mathbf{z}} \)(t), for \(t=0,1,...\) through the following recursions:
1) Initialization: Set \(t = 0\) and set \(\hat{x}_{j}(t) \) and \(\tau^{x}_{j}(t)\) to some initial values.
2) Output linear step: For each \(i\), compute:
\begin{equation}
\tau^{p}_{i}(t)=\sum\limits_{i} |a_{ij}|^{2} \tau^{x}_{j}(t)
\end{equation}
\begin{equation}
\hat{p}_{i}(t)=\sum\limits_{j} a_{ij}\hat{x}_{j}(t)-\tau^{p}_{i}(t) \hat{s}_{i}(t-1)
\end{equation}
\begin{equation}
\hat{z}_{i}(t)=\sum\limits_{j} a_{ij}\hat{x}_{j}(t)
\end{equation}
where initially, we take \( \hat{s}(-1) = 0. \)
3)Output nonlinear step: For each \( i \),
\begin{equation}
\hat{s}_{i}(t)=g_{out}(t,\hat{p}_{i}(t),y_{i}, \tau^{p}_{i}(t))
\end{equation}
\begin{equation}
\tau^{s}_{i}(t)=-\frac{\partial}{\partial \hat{p}}g_{out}(t,\hat{p}_{i}(t),y_{i}, \tau^{p}_{i}(t))
\end{equation}
4)Input linear step: For each \( j \),
\begin{equation}
\tau^{r}_{j}(t)=\left[\sum\limits_{i} |a_{ij}|^{2} \tau^{s}_{i}(t)\right]^{-1}
\end{equation}
\begin{equation}
\hat{r}_{j}(t)=\hat{x}_{j}(t)+\tau^{r}_{j}(t)\sum\limits_{i} a_{ij}\hat{s}_{i}(t)
\end{equation}
5) Input nonlinear step: For each \( j \),
\begin{equation}
\hat{x}_{j}(t+1)=g_{in}(t,\hat{r}_{j}(t),q_{j}, \tau^{r}_{j}(t))
\end{equation}
\begin{equation}
\tau^{x}_{j}(t+1)=\tau^{r}_{j}(t)\frac{\partial}{\partial \hat{r}}g_{in}(t,\hat{r}_{j}(t),q_{j}, \tau^{r}_{j}(t))
\end{equation}
Then increment \( t=t+1 \) and return to step 2 until a sufficient number of iterations have been performed.
In our case, we do not use \( \mathbf{q} \), since it is used to derive the input \( \mathbf{x} \) in Rangan's paper, whereas we directly have input vector \( \mathbf{x} \) in the beginning.
To adapt GAMP algorithm to our problem, we had to derive \(g_{in}\) and \(g_{out}\) mathematically using the problem parameters. Then, our decoder would be ready for the simulations.
\section{Mathematical Calculations for GAMP}
\subsection{Calculation of \(g_{in}\)}
To find \(\mathbf{\hat{x}}\) in the fifth step of GAMP, we need to compute \(g_{in}(\hat{\mathbf{r}},q,\boldsymbol{\tau}^{r})\), equivalently \(\mathbb{E}(\mathbf{x}|\hat{\mathbf{r}},q,\boldsymbol{\tau}^{r})\).
\begin{equation}
\mathbb{E}(\mathbf{x}|\hat{\mathbf{r}},q,\boldsymbol{\tau}^{r})=\frac {\int\mathrm{d}\mathbf{x} p_{0}(\mathbf{x}) \mathcal{N}(\mathbf{x}|\mathbf{r},\boldsymbol{\tau})\mathbf{x}} {\int\mathrm{d}\mathbf{x} p_{0}(\mathbf{x}) \mathcal{N}(\mathbf{x}|\mathbf{r},\boldsymbol{\tau}) }
\end{equation}
where
\begin{equation}
p_{0}(\mathbf{x})=\frac{1}{B}\sum_{i=1}^{B} \delta_{x_{i},1} \prod_{k\neq i}^{B} \delta_{x_{k},0}
\quad \quad \quad and \quad \quad \quad
\mathcal{N}(\mathbf{x}|\mathbf{r},\boldsymbol{\tau})= \frac{1}{\sqrt{(2\pi)^{B}|\boldsymbol{\Sigma}|}} e^{-\frac{1}{2} (\mathbf{x}-\mathbf{r})^{T}\boldsymbol{\Sigma}^{-1} (\mathbf{x}-\mathbf{r})}
\end{equation}
where
\begin{equation}
\Sigma=
\begin{pmatrix}
\diagentry{{\tau}_{1}}\\
&\diagentry{{\tau}_{2}}\\
&&\diagentry{\xddots}\\
&&&\diagentry{{\tau}_{B}}\\
\end{pmatrix}
\end{equation}
In the equations, \(\mathbf{r}\) and \(\boldsymbol{\tau}\) are B-dimensional vectors in which \(\mathbf{r}\) gives the mean and \(\boldsymbol{\tau}\) values constitute the covariance matrix of the gaussian pdf.
The results of the \(\mathbb{E}(\mathbf{x}|\hat{\mathbf{r}},q,\boldsymbol{\tau}^{r})\) are computed analytically in Matlab using the information above. After this computation, we need to compute \(\boldsymbol\tau^{x}\). For this, we must find
\(\boldsymbol\tau^{r}\frac{\partial}{\partial \boldsymbol{\hat{r}}}g_{in}(\boldsymbol{\hat{r}},q, \boldsymbol\tau^{r})\), equivalently \(var(\mathbf{x}|\hat{\mathbf{r}},q,\boldsymbol{\tau}^{r})\).
\begin{equation}
var(\mathbf{x}|\hat{\mathbf{r}},q,\boldsymbol{\tau}^{r})=\mathbb{E}(\mathbf{x}^{2}|\hat{\mathbf{r}},q,\boldsymbol{\tau}^{r})-\mathbb{E}(\mathbf{x}|\hat{\mathbf{r}},q,\boldsymbol{\tau}^{r})^{2}
\end{equation}
since, \(\mathbf{x}\) consists of 0s and 1s, \(\mathbb{E}(\mathbf{x}^{2}|\hat{\mathbf{r}},q,\boldsymbol{\tau}^{r})=\mathbb{E}(\mathbf{x}|\hat{\mathbf{r}},q,\boldsymbol{\tau}^{r})\). Therefore, we can conclude that:
\begin{equation}
\boldsymbol\tau^{x}=\mathbf{\hat{x}}-\mathbf{\hat{x}}^{2}
\end{equation}
where the multiplication \(\mathbf{\hat{x}}^{2}=\mathbf{\hat{x}}\mathbf{\hat{x}}\) is element-wise.
\subsection {Calculation of \(g_{out}\)}
\subsubsection{\(\mathbf{\hat{s}}\) and \(\boldsymbol{\tau}^s\) for AWGN Channel}
For the AWGN channel, we defined \(\mathbf{\hat{s}}\) as:
\begin{equation}
\mathbf{\hat{s}}=g_{out}(\mathbf{\hat{p}},\mathbf{y},\boldsymbol{\tau}^p)=\frac{(\mathbf{y}-\mathbf{\hat{p}})}{(\boldsymbol{\tau}^p+\tau^w)}
\end{equation}
where we define \(\tau^w\) as \(1/snr\). In the GAMP algorithm we defined \(\boldsymbol{\tau}^s\) as
\begin{equation}
\boldsymbol{\tau}^s=-\frac{\partial}{\partial \mathbf{\hat p}}(g_{out}(\mathbf{\hat{p}},\mathbf{y},\boldsymbol{\tau}^p))=-\frac{\partial}{\partial \mathbf{\hat p}}\frac{(\mathbf{y}-\mathbf{\hat{p}})}{(\boldsymbol{\tau}^p+\tau^w)}=\frac{1}{(\boldsymbol{\tau}^p+\tau^w)}
\end{equation}
\subsubsection{Computing \(\mathbf{\hat{s}}\) and \(\boldsymbol{\tau}^s\) for BSC}
While discussing GAMP algorithm, we said that \(\mathbf{\hat{s}}(t)=g_{out}(t,\mathbf{\hat{p}}(t),\mathbf{y}, \boldsymbol{\tau}^{p}(t))\) and for sum-product GAMP, we define \(g_{out}(\mathbf{\hat{p}},\mathbf{y},\boldsymbol{\tau}^p)\) as:
\begin{equation}
g_{out}(\mathbf{\hat{p}},\mathbf{y},\boldsymbol{\tau}^{p})=(\mathbf{\hat{z}}^0-\mathbf{\hat{p}})/\boldsymbol{\tau}^p
\end{equation}
where \(\mathbf{\hat{z}}^0:=\mathbb{E}(\mathbf{z}|\mathbf{\hat{p}},\mathbf{y},\boldsymbol{\tau}^p)\), \(Y\sim P_{Y|Z}\) and \(Z\sim \mathcal{N}(\mathbf{\hat{p}},\boldsymbol{\tau}^p)\). For BSC, we can define \(P_{Y|Z}\) as:
\begin{equation}
P_{Y|Z}=(1-\epsilon)\delta(\mathbf{y}-\pi(\mathbf{z}))+\epsilon\delta(\mathbf{y}+\pi(\mathbf{z}))
\end{equation}
where \(\epsilon\) is the flip probability and \(\pi(\mathbf{z})=sign(\mathbf{z})\). We can compute \(\mathbf{\hat{z}}^0\) such that:
\begin{equation}
\mathbf{\hat{z}}^0=\frac{1}{\mathcal{Z}}\int_{-\infty}^{\infty}\frac{\mathbf{z}e^{-(\mathbf{z}-\mathbf{\hat{p}})^2/2\boldsymbol{\tau}^p}}{\sqrt{2\pi \boldsymbol{\tau}^p}}((1-\epsilon)\delta(\mathbf{y}-\pi(\mathbf{z}))+\epsilon\delta(\mathbf{y}+\pi(\mathbf{z})))d\mathbf{z}
\end{equation}
\begin{multline}
\mathbf{\hat{z}}^0=\frac{1}{\mathcal{Z}}\int_{-\infty}^{0}\frac{\mathbf{z}e^{-(\mathbf{z}-\mathbf{\hat{p}})^2/2\boldsymbol{\tau}^p}}{\sqrt{2\pi \boldsymbol{\tau}^p}}((1-\epsilon)\delta(\mathbf{y}+1)+\epsilon\delta(\mathbf{y}-1))d\mathbf{z}+\\ \frac{1}{\mathcal{Z}}\int_{0}^{\infty}\frac{\mathbf{z}e^{-(\mathbf{z}-\mathbf{\hat{p}})^2/2\boldsymbol{\tau}^p}}{\sqrt{2\pi \boldsymbol{\tau}^p}}((1-\epsilon)\delta(\mathbf{y}-1)+\epsilon\delta(\mathbf{y}+1))d\mathbf{z}
\end{multline}
We can write \(\mathcal{Z}\) as:
\begin{multline}
\mathcal{Z}=\int_{-\infty}^{0}\frac{e^{-(\mathbf{z}-\mathbf{\hat{p}})^2/2\boldsymbol{\tau}^p}}{\sqrt{2\pi \boldsymbol{\tau}^p}}((1-\epsilon)\delta(\mathbf{y}+1)+\epsilon\delta(\mathbf{y}-1))d\mathbf{z}+\\ \int_{0}^{\infty}\frac{e^{-(\mathbf{z}-\mathbf{\hat{p}})^2/2\boldsymbol{\tau}^p}}{\sqrt{2\pi \boldsymbol{\tau}^p}}((1-\epsilon)\delta(\mathbf{y}-1)+\epsilon\delta(\mathbf{y}+1))d\mathbf{z}
\end{multline}
If we calculate the integrals we obtain \(\mathbf{\hat{z}}^0\) as:
\begin{multline}
\mathbf{\hat{z}}^0=\frac{1}{\mathcal{Z}}(\frac{-e^{-\mathbf{\hat{p}}^2/2\boldsymbol{\tau}^p}\sqrt{\boldsymbol{\tau}^p}}{\sqrt{2\pi}}+\mathbf{\hat{p}}\frac{erfc(\mathbf{\hat{p}}/\sqrt{2\boldsymbol{\tau}^p})}{2})((1-\epsilon)\delta(\mathbf{y}+1)+\epsilon\delta(\mathbf{y}-1))+\\ \frac{1}{\mathcal{Z}}(\frac{e^{-\mathbf{\hat{p}}^2/2\boldsymbol{\tau}^p}\sqrt{\boldsymbol{\tau}^p}}{\sqrt{2\pi}}+\mathbf{\hat{p}}\frac{1+erf(\mathbf{\hat{p}}/\sqrt{2\boldsymbol{\tau}^p})}{2})((1-\epsilon)\delta(\mathbf{y}-1)+\epsilon\delta(\mathbf{y}+1))
\end{multline}
where \(\mathcal{Z}\) is:
\begin{multline}
\mathcal{Z}=\frac{1}{2}erfc(\mathbf{\hat{p}}/\sqrt{2\boldsymbol{\tau}^p})((1-\epsilon)\delta(\mathbf{y}+1)+\epsilon\delta(\mathbf{y}-1))+\\ \frac{1}{2}(1+erf(\mathbf{\hat{p}}/\sqrt{2\boldsymbol{\tau}^p}))((1-\epsilon)\delta(\mathbf{y}-1)+\epsilon\delta(\mathbf{y}+1))
\end{multline}
We use \(\mathbf{\hat{z}}^0\) to compute \(\mathbf{\hat{s}}\) which is
\begin{equation}
\mathbf{\hat{s}}=\frac{\mathbf{\hat{z}}^0-\mathbf{\hat{p}}}{\boldsymbol{\tau}^p}
\end{equation}
After finding an expression for \(\mathbf{\hat{s}}\), we want to compute \(\boldsymbol{\tau}^s\) for BSC. To compute \(\boldsymbol{\tau}^{s}\), we need to find \(var(\mathbf{z}|\mathbf{\hat{p}},\mathbf{y},\boldsymbol{\tau}^p)\) which we can compute by:
\begin{equation}
var(\mathbf{z}|\mathbf{\hat{p}},\mathbf{y},\boldsymbol{\tau}^p)=\mathbb{E}(\mathbf{z}^2|\mathbf{\hat{p}},\mathbf{y},\boldsymbol{\tau}^p)-\mathbb{E}(\mathbf{z}|\mathbf{\hat{p}},\mathbf{y},\boldsymbol{\tau}^p)^2
\end{equation}
We know \(\mathbb{E}(\mathbf{z}|\mathbf{\hat{p}},\mathbf{y},\boldsymbol{\tau}^p)=\mathbf{\hat{z}}^0\) from the previous derivations, therefore we need to compute \(\mathbb{E}(\mathbf{z}^2)\) to compute \(var(\mathbf{z}|\mathbf{\hat{p}},\mathbf{y},\boldsymbol{\tau}^p)\):
\begin{multline}
\mathbb{E}(\mathbf{z}^2)=\frac{1}{\mathcal{Z}}\int_{-\infty}^{0}\frac{\mathbf{z}^2e^{-(\mathbf{z}-\mathbf{\hat{p}})^2/2\boldsymbol{\tau}^p}}{\sqrt{2\pi \boldsymbol{\tau}^p}}((1-\epsilon)\delta(\mathbf{y}+1)+\epsilon\delta(\mathbf{y}-1))d\mathbf{z}+\\ \frac{1}{\mathcal{Z}}\int_{0}^{\infty}\frac{\mathbf{z}^2e^{-(\mathbf{z}-\mathbf{\hat{p}})^2/2\boldsymbol{\tau}^p}}{\sqrt{2\pi \boldsymbol{\tau}^p}}((1-\epsilon)\delta(\mathbf{y}-1)+\epsilon\delta(\mathbf{y}+1))d\mathbf{z}
\end{multline}
After calculating the integrals, we obtain \(\mathbb{E}(\mathbf{z}^2)\) as
\begin{multline}
\mathbb{E}(\mathbf{z}^2)=\frac{1}{\mathcal{Z}}(\frac{-e^{-\mathbf{\hat{p}}^2/2\boldsymbol{\tau}^p}\mathbf{\hat{p}}\sqrt{\boldsymbol{\tau}^p}}{\sqrt{2\pi}}+(\mathbf{\hat{p}}^2+\boldsymbol{\tau}^p)\frac{erfc(\mathbf{\hat{p}}/\sqrt{2\boldsymbol{\tau}^p})}{2})((1-\epsilon)\delta(\mathbf{y}+1)+\epsilon\delta(\mathbf{y}-1))+\\ \frac{1}{\mathcal{Z}}(\frac{e^{-\mathbf{\hat{p}}^2/2\boldsymbol{\tau}^p}\mathbf{\hat{p}}\sqrt{\boldsymbol{\tau}^p}}{\sqrt{2\pi}}+(\mathbf{\hat{p}}^2+\boldsymbol{\tau}^p)\frac{1+erf(\mathbf{\hat{p}}/\sqrt{2\boldsymbol{\tau}^p})}{2})((1-\epsilon)\delta(\mathbf{y}-1)+\epsilon\delta(\mathbf{y}+1))
\end{multline}
Now we can compute \(var(\mathbf{z}|\mathbf{\hat{p}},\mathbf{y},\boldsymbol{\tau}^p)\) using the equation \(var(\mathbf{z}|\mathbf{\hat{p}},\mathbf{y},\boldsymbol{\tau}^p)=\mathbb{E}(\mathbf{z}^2)-(\mathbf{\hat{z}}^0)^2\). With the knowledge of \(var(\mathbf{z}|\mathbf{\hat{p}},\mathbf{y},\boldsymbol{\tau}^p)\), we can find an expression for \(\boldsymbol{\tau}^s\) which is:
\begin{equation}
\boldsymbol{\tau}^s=(\boldsymbol{\tau}^p-var(\mathbf{z}|\mathbf{\hat{p}},\mathbf{y},\boldsymbol{\tau}^p))/(\boldsymbol{\tau}^p)^2
\end{equation}
\subsubsection{Computing \(\mathbf{\hat{s}}\) and \(\boldsymbol{\tau}^s\) for Z Channel}
After finding \(\mathbf{\hat{s}}\) and \(\boldsymbol{\tau}^s\) for BSC, we can compute them for Z channel by replacing \(P_{Y|Z}\) by
\begin{equation}
P_{Y|Z}=\delta(\pi(\mathbf{z})-1)\delta(\mathbf{y}-\pi(\mathbf{z}))+\delta(\pi(\mathbf{z})+1)((1-\epsilon)\delta(\mathbf{y}-\pi(\mathbf{z}))+\epsilon\delta(\mathbf{y}+\pi(\mathbf{z})))
\end{equation}
where \(\epsilon\) is the flip probability of input -1. Now we replace the \(P_{Y|Z}\) term inside \(\mathbf{\hat{z}}^0\), \(\mathcal{Z}\) and \(\mathbb{E}(\mathbf{z})^2\) to find them for Z channel.
\begin{multline}
\mathbf{\hat{z}}^0=\frac{1}{\mathcal{Z}}(\frac{-e^{-\mathbf{\hat{p}}^2/2\boldsymbol{\tau}^p}\sqrt{\boldsymbol{\tau}^p}}{\sqrt{2\pi}}+\mathbf{\hat{p}}\frac{erfc(\mathbf{\hat{p}}/\sqrt{2\boldsymbol{\tau}^p})}{2})((1-\epsilon)\delta(\mathbf{y}+1))+\\ \frac{1}{\mathcal{Z}}(\frac{e^{-\mathbf{\hat{p}}^2/2\boldsymbol{\tau}^p}\sqrt{\boldsymbol{\tau}^p}}{\sqrt{2\pi}}+\mathbf{\hat{p}}\frac{1+erf(\mathbf{\hat{p}}/\sqrt{2\boldsymbol{\tau}^p})}{2})(\delta(\mathbf{y}-1)+\epsilon\delta(\mathbf{y}+1))
\end{multline}
\begin{multline}
\mathcal{Z}=\frac{1}{2}erfc(\mathbf{\hat{p}}/\sqrt{2\boldsymbol{\tau}^p})((1-\epsilon)\delta(\mathbf{y}+1))+\\ \frac{1}{2}(1+erf(\mathbf{\hat{p}}/\sqrt{2\boldsymbol{\tau}^p}))(\delta(\mathbf{y}-1)+\epsilon\delta(\mathbf{y}+1))
\end{multline}
\begin{multline}
\mathbb{E}(\mathbf{z}^2)=\frac{1}{\mathcal{Z}}(\frac{-e^{-\mathbf{\hat{p}}^2/2\boldsymbol{\tau}^p}\mathbf{\hat{p}}\sqrt{\boldsymbol{\tau}^p}}{\sqrt{2\pi}}+(\mathbf{\hat{p}}^2+\boldsymbol{\tau}^p)\frac{erfc(\mathbf{\hat{p}}/\sqrt{2\boldsymbol{\tau}^p})}{2})((1-\epsilon)\delta(\mathbf{y}+1))+\\ \frac{1}{\mathcal{Z}}(\frac{e^{-\mathbf{\hat{p}}^2/2\boldsymbol{\tau}^p}\mathbf{\hat{p}}\sqrt{\boldsymbol{\tau}^p}}{\sqrt{2\pi}}+(\mathbf{\hat{p}}^2+\boldsymbol{\tau}^p)\frac{1+erf(\mathbf{\hat{p}}/\sqrt{2\boldsymbol{\tau}^p})}{2})(\delta(\mathbf{y}-1)+\epsilon\delta(\mathbf{y}+1))
\end{multline}
Now we can find \(\mathbf{\hat{s}}\) and \(\boldsymbol{\tau}^s\) using the \(\mathbf{\hat{z}^0}\), \(\mathcal{Z}\) and \(\mathbb{E}(\mathbf{z})^2\) expressions for Z channel by using the formulas \(\mathbf{\hat{s}}=\frac{\mathbf{\hat{z}}^0-\mathbf{\hat{p}}}{\boldsymbol{\tau}^p}\) and \(\boldsymbol{\tau}^s=(\boldsymbol{\tau}^p-var(\mathbf{z}|\mathbf{\hat{p}},\mathbf{y},\boldsymbol{\tau}^p))/(\boldsymbol{\tau}^p)^2\) where \(var(\mathbf{z}|\mathbf{\hat{p}},\mathbf{y},\boldsymbol{\tau}^p)=\mathbb{E}(\mathbf{z}^2)-(\mathbf{\hat{z}}^0)^2\).
\section{State Evolution Function}
In order to evaluate the performance of our decoder, we need an objective measure to compare it. Thus, BSC and Z Channel are modeled as AWGN Channels with an effective noise variance \(\Sigma(E)^{2}\). Then, we iteratively compute the mean square error (MSE) \(\widetilde{E}^{(t)}=\mathbb{E}_{\mathbf{x,y}}[\frac{1}{L}\sum_{n=1}^{N}(\hat{x}^{(t)}_{n}-x_{n})^2]\) of the GAMP estimate (\(\mathbf{\hat{x}}^{(t)}\) in our case) for every iteration.
To do it, firstly, we calculated the effective noise variance \(\Sigma(E)^{2}\) by the relation:
\begin{equation}
\Sigma(E)^{2}=\frac{R}{\mathbb{E}_{p|E}[\mathcal{F}(p|E)] }
\end{equation}
where the expectation \(\mathbb{E}_{p|E}\) is with respect to \(\mathcal{N}(p|0,1-E)\) and
\begin{equation}
\mathcal{F}(p|E)=\int \mathrm{d}y f(y|p,E) (\partial_{m} \ln f(y|m,E))^{2}_{m=p}
\end{equation}
is the Fisher information of \(p\) associated with the distribution below:
\begin{equation}
f(y|p,E)=\int\mathrm{d}u P_{out}(y|u)\mathcal{N}(u|p,E)
\end{equation}
The calculation of the Fisher information is given for BSC below:
\begin{equation}
\begin{multlined}
f(y|p,E)=\int_{-\infty}^{0} \mathrm{d}u [(1-\epsilon)\delta(y+1)+\epsilon\delta(y-1)]\frac{e^{\frac{-(u-p)^{2}}{2E}}}{\sqrt {2\pi E}}\\ \\
+\int_{0}^{\infty} \mathrm{d}u [(1-\epsilon)\delta(y-1)+\epsilon\delta(y+1)]\frac{e^{\frac{-(u-p)^{2}}{2E}}}{\sqrt {2\pi E}}
\end{multlined}
\end{equation}
By change of variables, we write \(t=\frac{u-p}{\sqrt{E}}\), which then gives us:
\begin{equation}
\begin{multlined}
f(y|p,E)=\int_{-\infty}^{\frac{-p}{\sqrt{E}}} \mathrm{d}t [(1-\epsilon)\delta(y+1)+\epsilon\delta(y-1)]\frac{e^{\frac{-t^{2}}{2}}}{\sqrt {2\pi}}\\ \\
+\int_{\frac{-p}{\sqrt{E}}}^{\infty} \mathrm{d}t [(1-\epsilon)\delta(y-1)+\epsilon\delta(y+1)]\frac{e^{\frac{-t^{2}}{2}}}{\sqrt {2\pi}}
\end{multlined}
\end{equation}
As one can see, these integrals are in the form of \(Q\) functions, therefore we finally declare \(f(y|p,E)\) as:
\begin{equation}
\begin{multlined}
f(y|p,E)= (1-\epsilon)[\delta(y-1)\mathbb{Q}(\textstyle \frac{-p}{\sqrt{E}})+\delta(y+1)(1-\mathbb{Q}(\frac{-p}{\sqrt{E}}))]\\
+ \epsilon[\delta(y+1)\mathbb{Q}(\textstyle\frac{-p}{\sqrt{E}})+\delta(y-1)(1-\mathbb{Q}(\frac{-p}{\sqrt{E}}))]
\end{multlined}
\end{equation}
Now, we can calculate \(\mathcal{F}(p|E)\), since we have
\(f(y|p,E)\). From the definition of \(\mathcal{F}(p|E)\), it is equivalent to:
\begin{equation}
\mathcal{F}(p|E)=\int \mathrm{d}y \frac {(\partial_{p} f(y|p,E))^{2}} {f(y|p,E)}
\end{equation}
Let us say \(Q=\mathbb{Q}(\textstyle\frac{-p}{\sqrt{E}}) \)
for the \(Q\) functions (\(Q'\) to their derivatives with respect to \(p\)) which take place in \(f(y|p,E)\). We do this change for the ease of understanding of the calculation for the following steps:
\begin{equation}
\mathcal{F}(p|E)=\int \mathrm{d}y \frac {[(1-\epsilon)(\delta(y-1)Q'-\delta(y+1)Q')+\epsilon(\delta(y+1)Q'-\delta(y-1)Q')]^{2}} {(1-\epsilon)[\delta(y-1)Q+\delta(y+1)(1-Q)]+\epsilon[\delta(y+1)Q+\delta(y-1)(1-Q)]}
\end{equation}
Since \(\mathbf{y}\) is the output vector from the binary symmetric channel, it only consists of \(-1,1\)'s. Due to this fact, integration can be done by first making \(y=-1\), then \(y=1\) and finally summing these two results. Combining these facts, we finally get:
\begin{equation}
\mathcal{F}(p|E)=\frac{(Q'-2\epsilon Q')^{2}}{(Q+\epsilon-2\epsilon Q)(1-Q-\epsilon+2\epsilon Q)}
\end{equation}
where
\begin{equation}
Q=\int_{\frac{-p}{\sqrt{E}}}^{\infty} \mathrm{d}t \frac
{e^{\frac{-t^{2}}{2}}}{\sqrt{2\pi}} \qquad
and \qquad
Q'=\frac{e^{\frac{-p^{2}}{2E}}}{\sqrt{2\pi E}}
\end{equation}
As a result, Fisher information \(\mathcal{F}(p|E)\) calculation is completed. It can also be computed for the Z Channel in the same way just by changing \(P_{out}(y|u)\) from \( (1-\epsilon)(\delta(y-\pi(u))+\epsilon\delta(y+\pi(u))\) to \( \delta(\pi(u)-1)\delta(y-\pi(u))+\delta(\pi(u)+1)[(1-\epsilon)(\delta(y-\pi(u))+\epsilon\delta(y+\pi(u))\) where \(\pi(u)=sign(u)\).
After Fisher information calculation step, \(\mathbb{E}_{p|E}[\mathcal{F}(p|E)]\) can be found by taking the expectation of \(\mathcal{F}(p|E)\) with respect to \(\mathcal{N}(p|0,1-E)\). This is done by Monte Carlo integration using Matlab. Using this result and the formula (12), effective noise variance is calculated which makes us ready to continue for the second part to complete the state evolution function.
At the previous part, we computed \(\Sigma(E)^{2}\) for BSC and Z channel and we can model these channels as an effective AWGN channel with noise \(\mathcal{N}(0,\Sigma(E)^{2}/log_{2}(B))\). Then we create a denoiser \(f_{i}(\Sigma(E))\) which estimates the value of the \(i\)-th component of a section. We can define \(f_{i}(\Sigma(E))\) as:
\begin{equation}
f_{i}(\Sigma(E))=\frac{\Sigma_{\mathbf{x}}e^{-\frac{\|x-(s+z\Sigma(E)/\sqrt{log_{2}(B)}\|_{2}^{2}}{2\Sigma(E)^{2}/log_{2}(B)}}p_{0}(\mathbf{x})x_{i}}{\Sigma_{\mathbf{x}}e^{-\frac{\|x-(s+z\Sigma(E)/\sqrt{log_{2}(B)}\|_{2}^{2}}{2\Sigma(E)^{2}/log_{2}(B)}}p_{0}(\mathbf{x})}
\end{equation}
where \(s_{i}\) is the correct value of the \(i\)-th component of a section and \(z\) is a random variable with distribution \(\mathcal{N}(0,1)\). We can simplify that equation such that:
\begin{equation}
f_{i}(\Sigma(E))=[1+\Sigma_{k\ne i}^{B}e^{(s_{k}-s_{i})log_{2}(B)/\Sigma(E)^2+(z_{k}-z_{i})\sqrt{log_{2}(B)}/\Sigma(E)}]^{-1}
\end{equation}
Therefore MSE estimated from state evolution becomes:
\begin{equation}
T_{u}(E)=\mathbb{E}_{s,z}[\Sigma_{i=1}^B(f_{i}(\Sigma(E))-s_{i})^2]
\end{equation}
We can simplify that equation further, because we know that only one of the \(s_{i}\) values in a section is 1 while the other \((B-1)\) of them are 0s. Lets assume that \(s_{1}=1\) while other \(s_{i}=0\) \(\forall i\in\{2,...,B\}\). We know that \(f_{i}\) for \(\forall i\in\{2,...,B\}\) has an identical distribution because \(s_{i}=0\) and \(z_{i}\) is a random variable with distribution \(\mathcal{N}(0,1)\). Therefore we can compute the distribution of \(s_{2}\) and assign that value for \(\forall i\in\{3,...,B\}\).
\begin{equation}
f_{1}(\Sigma(E))=[1+e^{-log_{2}(B)/\Sigma(E)^2}\Sigma_{k\ge 2}^{B}e^{(z_{k}-z_{1})\sqrt{log_{2}(B)}/\Sigma(E)}]^ {-1}
\end{equation}
\begin{equation}
f_{2}(\Sigma(E))=[1+e^{log_{2}(B)/\Sigma(E)^2+(z_{1}-z_{2})\sqrt{log_{2}(B)}/\Sigma(E)}+\Sigma_{k\ge 3}^{B}e^{(z_{k}-z_{2})\sqrt{log_{2}(B)}/\Sigma(E)}]^ {-1}
\end{equation}
Therefore \(T_{u}(E)\) becomes,
\begin{equation}
T_{u}(E)=\int \mathcal{D} z (f_{1}(\Sigma(E))-1)^2+(B-1)f_{2}(\Sigma(E))^2
\end{equation}
To compute this integral, we can use Monte Carlo integration because \(z\) is a random variable with a distribution \(\mathcal{N}(0,1)\). Therefore, we can give z multiple random values with distribution \(\mathcal{N}(0,1)\) and take the mean of the results to obtain \(T_{u}(E)\).
Section error rate is another way to evaluate the performance of the GAMP decoder and we can also estimate the value of SER using state evolution. We obtained the equations for \(f_{1}(\Sigma(E))\) for \(s_{1}=1\) and \(f_{i}(\Sigma(E))\) for \(s_{i}=0\), \(\forall i\in\{2,...,B\}\). We know that \(f_{i}(\Sigma(E))\) estimates \(\hat{s_{i}}\) values, therefore we can estimate SER by using:
\begin{equation}
SER=\int \mathcal{D} z \mathbb{1}(\exists i\in\{2,...,B\} : f_{i}(\Sigma(E))>f_{1}(\Sigma(E)))
\end{equation}
We can compute this integral using Monte Carlo integration the way we computed \(T_{u}(E)\).
\section{Potential Function for AWGN Channel}
Another way to determine the performance of our decoder is to evaluate its potential function. For AWGN channel we can define the potential function as:
\begin{multline}
\Phi_{B}(E)=-\frac{\log_2(B)}{2R}\left(log(1/snr+E)+\frac{1-E}{1/snr+E}\right)+\\ \int \mathcal{D}\bar{z}log\left(e^{\frac{1}{2\Sigma(E)^2}+\frac{z_1}{\Sigma(E)}}+\Sigma_{i=2}^{B}e^{-\frac{1}{2\Sigma(E)^2}+\frac{z_1}{\Sigma(E)}}\right)
\end{multline}
where \(E=\frac{1}{L}\sum_{i}^{N}(x_{i}-\hat{x_{i}})^2\), \(\Sigma(E):=\sqrt{(\frac{1}{snr}+E)R/log_{2}B}\) and \(\mathcal{D}\bar{z}=\Pi_{i=1}^B\mathcal{D}z_{i}\) where \(\mathcal{D}z_{i}\) has the distribution of \(\mathcal{N}(0,1)\). To take the integral we can use Monte Carlo integration.
The information we gain from the potential function is the behavior of \(E\) for different \(R\),\(B\) and \(snr\). To observe the behaviour of \(E\), we need to search for the local maxima of the potential function by decreasing \(E\). By looking at the \(E\) value at the local maxima, we can observe the converging \(E\) value of the decoder.
What we can say about the shape of the potential function graph is that, if the potential function only has one local maxima, then the corresponding \(E\) is very small. However, if the potential function stays constant if we decrease \(E\) and starts to increase for smaller \(E\), we are at the belief propagation threshold and a small decrease in \(R\) may create a local maxima at that point, which may increase \(E\) by a big amount. Another scenario is to have two global maximas for different \(E\) values and the one with the higher error dominates the other one. We call that rate as the optimal threshold, which is the highest rate with a bad performance under optimal decoding.
\begin{figure}[t!]
\centering
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{PotFun.png}
\captionof{figure}{MSE vs Potential Function for B=2 and snr=30}
\label{fig:pot1}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{MSEPotFun.png}
\captionof{figure}{Iteartion vs MSE for B=2 and snr=30}
\label{fig:pot2}
\end{minipage}
\end{figure}
To observe the behaviour of the potential function, we selected our \(B=2\), \(snr=30\) and \(R=1.5, 1.68, 1.775\). From Figure 1, we can see that \(R=1.775\) is the optimal threshold because there are 2 global maximas at the potential function. Also, we can say that \(R=1.68\) is at the belief propagation threshold, because the potential function stays constant for a while then increases when \(E\) decreases. \(R=1.5\) is below the belief propagation threshold, because it only has one global maxima.
In Figure 1, when \(R=1.775\), the local maxima appears when \(E=0.1834\) while in Figure 2 \(E\) converges to \(0.1811\). So, we can estimate the converging \(E\) of the decoder by looking at the \(E\) value of the local maxima in the potential function when \(R\) is greater or equal to the optimal threshold. In Figure 1, when \(R=1.68\), the local maxima appears when \(E=0.1509\) while in Figure 2 \(E\) converges to \(0.1314\). Therefore, we can use \(E\) of the local maxima at the potential function to determine the \(E\) of the decoder. However, it is not as precise as when \(R\) is greater or equal to the optimal threshold, because at BP threshold region, small changes on \(R\) changes the \(E\) of the local maxima of the potential function and \(E\) obtained at the decoder by a huge amount. In Figure 1, when \(R=1.5\), the local maxima appears when \(E=0.0098\) while in Figure 2 \(E\) converges to \(0.0015\). We can still use the information about \(E\) of the local maxima of the potential function, but as \(E\) decreases below \(10^{-2}\), potential function graph stays constant, therefore we need to have extremely high number of z values in our Monte Carlo integration to determine the exact location of the local maxima.
\section{Experimental Results and Discussion}
We now present a number of numerical experiments testing the performance and behavior of the GAMP decoder in different practical scenarios with finite size signals. Communication rate \(R\), section length \(B\), flip probability \(\epsilon\), signal-to-noise ratio \(snr\) and the channel used in communication are varied in these experiments. To evaluate the results, we plotted many observations.
\begin{figure}[h!]
\centering
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{GaussB=2.png}
\captionof{figure}{Iteration vs SER in AWGN \newline(\(snr=15\) , \(B=2\))}
\label{fig:pot1}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{GaussB=4.png}
\captionof{figure}{Iteration vs SER in AWGN \newline( \(snr=15\) , \(B=4\))}
\label{fig:pot2}
\end{minipage}
\end{figure}
First of all, iteration vs symbol error rate (SER) plots for AWGN Channel where \(B=2, 4\) and \(SNR=15\) can be seen in Figures 3, 4. In the figures, 'o' denotes the decoder's SER and '*' denotes the state evolution's SER. The results of the decoder is followed by the state evolution function as expected. Therefore, the decoder works in AWGN Channel. It is also possible to confirm from the figures that when the communication rate increases it becomes harder to decode the signal.
\begin{figure}[b!]
\centering
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{BSC_SER_B=2.png}
\captionof{figure}{Iteration vs SER in BSC (\(\epsilon=0.1\) , \(B=2\))}
\label{fig:pot1}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{BSC_SER_B=4.png}
\captionof{figure}{Iteration vs SER in BSC ( \(\epsilon=0.1\) , \(B=4\))}
\label{fig:pot2}
\end{minipage}
\end{figure}
\begin{figure}[t!]
\centering
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{BSC_SER_B=8.png}
\captionof{figure}{Iteration vs SER in BSC (\(\epsilon=0.1\) , \(B=8\))}
\label{fig:pot1}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{BSC_SER_B=16.png}
\captionof{figure}{Iteration vs SER in BSC (\(\epsilon=0.1\) , \(B=16\))}
\label{fig:pot2}
\end{minipage}
\end{figure}
Secondly, iteration vs symbol error rate (SER) plots for BSC where \(B=2, 4, 8, 16\) and \(\epsilon=0.1\) can be seen in Figures 5, 6, 7 and 8. As seen, the results of the decoder and the state evolution function are coherent which proves that the decoder works well.
Also, we should say that there are some differences like the one in Figure 8 where the communication rate equals \(R=0.35\), and this is stemmed from the fact that decoder is working near the transition region, and because we are working with finite size signals, those kind of variations can be observed from simulation to simulation.
\begin{figure}[h!]
\centering
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{Z_SER_B=2.png}
\captionof{figure}{Iteration vs SER in Z Channel \newline(\(\epsilon=0.1\) , \(B=2\))}
\label{fig:pot1}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{Z_SER_B=4.png}
\captionof{figure}{Iteration vs SER in Z Channel \newline(\(\epsilon=0.1\) , \(B=4\))}
\label{fig:pot2}
\end{minipage}
\end{figure}
\begin{figure}[h!]
\centering
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{Z_SER_B=8.png}
\captionof{figure}{Iteration vs SER in Z Channel \newline(\(\epsilon=0.1\) , \(B=8\))}
\label{fig:pot1}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{Z_SER_B=16.png}
\captionof{figure}{Iteration vs SER in Z Channel \newline(\(\epsilon=0.1\) , \(B=16\))}
\label{fig:pot2}
\end{minipage}
\end{figure}
In third scenario, iteration vs symbol error rate (SER) plots for Z Channel where \(B=2, 4, 8, 16\) and \(\epsilon=0.1\) can be seen in Figures 9, 10, 11 and 12. The correctness of the results is again confirmed by the state evolution function. From the figures, it is obvious that it becomes easier to decode the original signal if the communication occurs in lower rates.
In Figures 13, 15 and 17, we can see the convergence rates(\(R_{u}\)) of our decoder(the rate where SER becomes 0) for BSC for \(\epsilon=(0.01,0.05,0.1)\). In Figures 14, 16 and 18, we can see the convergence rates of our decoder for Z channel for \(\epsilon=(0.01,0.05,0.1)\). In each plot we show the convergence region for \(B=2, 4, 8, 16\). By comparing the figures, we can see that \(R_{u}\) increases when \(\epsilon\) decreases if we keep all other parameters constant. Also at the convergence region, we can see that the SER estimated from state evolution decreases sharply, while the SER of the decoder decreases more slowly. This effect happens because our signal is a finite size signal and if we increase L, the decrease of the SER around the convergence region becomes sharper.
\begin{figure}[h!]
\centering
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{BSC_SER001.png}
\captionof{figure}{Rate vs SER in BSC (\(\epsilon=0.01\))}
\label{fig:pot1}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{Z_SER001.png}
\captionof{figure}{Rate vs SER in Z Channel (\(\epsilon=0.01\))}
\label{fig:pot2}
\end{minipage}
\end{figure}
\begin{figure}[h!]
\centering
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{BSC_SER005.png}
\captionof{figure}{Rate vs SER in BSC (\(\epsilon=0.05\))}
\label{fig:pot1}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{Z_SER005.png}
\captionof{figure}{Rate vs SER in Z Channel (\(\epsilon=0.05\))}
\label{fig:pot2}
\end{minipage}
\end{figure}
\begin{figure}[h!]
\centering
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{BSC_SER01.png}
\captionof{figure}{Rate vs SER in BSC (\(\epsilon=0.1\))}
\label{fig:pot1}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{Z_SER01.png}
\captionof{figure}{Rate vs SER in Z Channel (\(\epsilon=0.1\))}
\label{fig:pot2}
\end{minipage}
\end{figure}
For BSC and Z Channel, it is observed that when the same value of the communication rate \(R\) is used for different section length \(B's\), higher \(B\) gives lower section error rate. Besides, we can say that, when \(B\) increases \(R_{u}\) also increases if we keep all other parameters same.
We took the plots of the \(R_{u}^B(B)\) where \(B=2, 4, 8, 16, 32, 64\) and \(\epsilon=0.01, 0.05, 0.1\) for BSC and Z Channel. The plots can be seen in Figures 19, 20, 21, 22, 23, 24. To determine the correct \(R_{u}^B(B)\) values, we created an algorithm which has initial \(R_{min}\) and \(R_{max}\) values and we compute SER for \(R=(R_{min}+R_{max})/2\) and we repeat it for a huge number of times. If the decoder has \(SER=0\) for more than half of the runs, we set \(R_{min}=R\) and otherwise we set \(R_{max}=R\) and re-run the algorithm again. If \(R_{min}-R_{max}<T\), we stop the algorithm and select \(R_{u}^B(B)=(R_{min}+R_{max})/2\) for a given value of B and a threshold T.
\begin{figure}[h!]
\centering
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{RuBSC0-01.png}
\captionof{figure}{\(B\) vs \(R_{u}\) in BSC (\(\epsilon=0.01\))}
\label{fig:pot1}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{RuZ0-01.png}
\captionof{figure}{\(B\) vs \(R_{u}\) in Z Channel (\(\epsilon=0.01\))}
\label{fig:pot2}
\end{minipage}
\end{figure}
\begin{figure}[h!]
\centering
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{RuBSC0-05.png}
\captionof{figure}{\(B\) vs \(R_{u}\) in BSC (\(\epsilon=0.05\))}
\label{fig:pot1}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{RuZ0-05.png}
\captionof{figure}{\(B\) vs \(R_{u}\) in Z Channel (\(\epsilon=0.05\))}
\label{fig:pot2}
\end{minipage}
\end{figure}
\begin{figure}[h!]
\centering
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{RuBSC0-1.png}
\captionof{figure}{\(B\) vs \(R_{u}\) in BSC (\(\epsilon=0.1\))}
\label{fig:pot1}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{RuZ0-1.png}
\captionof{figure}{\(B\) vs \(R_{u}\) in Z Channel (\(\epsilon=0.1\))}
\label{fig:pot2}
\end{minipage}
\end{figure}
From the plots, it can be said that as \(B\) increases, \(R_{u}^B(B)\) increases. We know that, \(R_{u}^\infty\) can be expressed as a function of \(\epsilon\) and the type of channel. For BSC, \( R_{u}^\infty=(\pi ln(2))^{-1}(1-2\epsilon)^2\) and for \(\epsilon=0.01, 0.05, 0.1\), \(R_{u}^\infty=(0.441, 0.372, 0.294)\) respectively. For Z Channel \(R_{u}^\infty=(\pi ln(2)(1+\epsilon))^{-1}(1-\epsilon)\) and for \(\epsilon=0.01, 0.05, 0.1\), \(R_{u}^\infty=(0.45, 0.415, 0.375)\) respectively. If we look at the \(R_{u}^B(B)\) values for BSC and Z Channel in Figures 19, 20, 21, 22, 23, 24 we can see that \(R_{u}^B(B)\) becomes greater than \(R_{u}^\infty\) for \(B= 4, 8, 16, 32, 64\). The reasoning for that is unknown to us as we thought \(R_{u}^B(B)\) as an increasing function with respect to \(B\), when we looked at our figures. However, that might not be the case and they might be a limit where \(R_{u}^B(B)\) becomes a decreasing function and converges at \(R_{u}^\infty\).
We can compare the performances of BSC and Z Channel for GAMP decoder by looking at the \(R_{u}\) values. We know that for the same \(\epsilon\) value Z channel has higher convergence rates, because we only flip -1 values in Z channel while flip -1 and 1 values for BSC, which leads to higher SER in BSC for same \(\epsilon\) and \(R\). However, we can compare the performance of channels when their probability of error is same. For \(\epsilon=0.05\) in BSC and \(\epsilon=0.1\) in Z channel, the probability of error is 0.05 and we can see the convergence rates at Figures 21 and 24 respectively. From the figures, we can see that for the same probability of error, \(R_{u}^B(B)\) is higher for Z Channel than BSC for all B values. This is expected because our system performs better at asymmetric channels than the symmetric channels, because we can correct the errors in a superior way if the channel is asymmetric. If we compare \(R_{u}^\infty\) for those two cases, when \(\epsilon=0.05\) for BSC \(R_{u}^\infty=0.372\) while \(\epsilon=0.1\) for Z Channel \(R_{u}^\infty=0.375\) which proves the point that our decoder performs better at asymmetric channels than symmetric channels for the same probability of error.
\section{Conclusion and Future Work}
In this research, we presented an empirical study using GAMP algorithm to estimate the sparse superposition signal that is communicated through different channels at various parameter settings. The tested channels are AWGN, BSC and Z with the the changing parameters as section length \(B\), communication rate \(R\) and flip probability \(\epsilon\) or signal-to-noise ratio \(snr\). As a result, the implemented decoder works well as this can be proven by the behavior of the state evolution function.
Also, we can say that the figures of BSC and Z Channels are plotted with number of sections \(L=1000\) since bigger \(Ls\) lead to excessive computation time in Matlab using our laptops. Figures of AWGN Channels are plotted with \(L=4000\) since the computations are easier for AWGN. In all cases, the number of samples are \(10\) and Monte Carlo integrations are done with the \(MC size=10^{5}\). Only in \(B\) vs \(R_{u}\) of BSC and Z Channel, figures are plotted using the super-computer due to the very high values of \(B\) like \(32\) and \(64\).
In our project, we cannot achieve the Shannon capacity for the channels, therefore this study can be seen as sub-optimal. Spatial coupling, in conjunction with message-passing decoding, allows us to achieve Shannon capacity on memoryless channels (Donoho et al. 2013). Therefore, spatial coupling scheme can be integrated with our algorithm as a future work.
\section{Acknowledgement}
We could be able to complete this research with the helps and supervision of Jean Barbier and Mohamad Dia.
\newpage
\section{References}
\begin{enumerate}
\item Barbier, Jean. "Statistical physics and approximate message-passing algorithms for sparse linear estimation problems in signal processing and coding theory." arXiv preprint arXiv:1511.01650 (2015).
\item Rangan, Sundeep. "Generalized approximate message passing for estimation with random linear mixing." Information Theory Proceedings (ISIT), 2011 IEEE International Symposium on. IEEE, 2011.
\item Barbier, Jean, Mohamad Dia, and Nicolas Macris. "Proof of Threshold Saturation for Spatially Coupled Sparse Superposition Codes." arXiv preprint arXiv:1603.01817 (2016).
\item Barbier, Jean, Mohamad Dia, and Nicolas Macris. "Threshold Saturation of Spatially Coupled Sparse Superposition Codes for All Memoryless Channels." arXiv preprint arXiv:1603.04591
\item Barbier, Jean, and Florent Krzakala. "Approximate message-passing decoder and capacity-achieving sparse superposition codes." arXiv preprint arXiv:1503.08040 (2015).
\item Barbier, Jean, Christophe Schülke, and Florent Krzakala. "Approximate message-passing with spatially coupled structured operators, with applications to compressed sensing and sparse superposition codes." Journal of Statistical Mechanics: Theory and Experiment 2015.5 (2015): P05013.
\item Barbier, Jean, and Florent Krzakala. "Replica analysis and approximate message passing decoder for superposition codes." Information Theory (ISIT), 2014 IEEE International Symposium on. IEEE, 2014.
\item Donoho, David L., Adel Javanmard, and Alessandro Montanari. "Information-theoretically optimal compressed sensing via spatial coupling and approximate message passing." Information Theory, IEEE Transactions on 59.11 (2013): 7434-7464.
\end{enumerate}
\end{document}
| {
"timestamp": "2017-12-05T02:08:57",
"yymm": "1712",
"arxiv_id": "1712.00708",
"language": "en",
"url": "https://arxiv.org/abs/1712.00708",
"abstract": "In this project, the behavior of Generalized Approximate Message-Passing Decoder for BSC and Z Channel is studied using i.i.d matrices for constructing the codewords. The performance of GAMP in AWGN Channel is already evaluated in the previous scientific work of Jean Barbier, therefore, this project mainly focuses on the performance of GAMP decoder for BSC and Z Channel. We evaluate the performance of the GAMP decoder for sparse superposition codes at various settings and compare the performance of decoder for different channels and parameters.",
"subjects": "Information Theory (cs.IT)",
"title": "Study of the Sparse Superposition Codes and the Generalized Approximate Message Passing Decoder for the Communication over Binary Symmetric and Z Channels",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808707404786,
"lm_q2_score": 0.7217432062975978,
"lm_q1q2_score": 0.7075110587204341
} |
https://arxiv.org/abs/0902.0884 | Translated Poisson approximation to equilibrium distributions of Markov population processes | The paper is concerned with the equilibrium distributions of continuous-time density dependent Markov processes on the integers. These distributions are known typically to be approximately normal, and the approximation error, as measured in Kolmogorov distance, is of the smallest order that is compatible with their having integer support. Here, an approximation in the much stronger total variation norm is established, without any loss in the asymptotic order of accuracy; the approximating distribution is a translated Poisson distribution having the same variance and (almost) the same mean. Our arguments are based on the Stein-Chen method and Dynkin's formula. | \section{Introduction}
Density dependent Markov population processes, in which the transition rates depend
on the density of individuals in the population, have proved widely useful as models in the social and life
sciences: see, for example, the monograph of Kurtz~(1981), in which approximations in terms of
diffusions are extensively discussed, in the limit as the typical population size~$n$ tends to
infinity. Here, we are interested in the behavior at equilibrium.
Our starting point is the paper of Barbour~(1980), in which conditions are given
for the existence of an equilibrium distribution concentrated close to the
deterministic equilibrium, together with a bound of
order $O(1/\sqrt{n})$ on the Kolmogorov distance between the equilibrium distribution and a suitable
normal distribution.
We now show that this normal approximation can be substantially strengthened. Using a delicate argument based
on the Stein--Chen method, we are able to establish an approximation in total variation in terms of a
translated Poisson distribution. What is more, our error bounds with respect to this much stronger metric,
and under weaker assumptions than those previously considered, are still of ideal order
$O(1/\sqrt{n})$.
The first step in the argument is to establish the existence of an equilibrium distribution under
suitable conditions, and to show that it is appropriately concentrated around the `deterministic'
equilibrium, defined to be the stationary point of an associated system of differential equations
which describe the average drift of the process in the limit as $n\to\infty$; this is
accomplished in Section~\ref{equilibrium}. The closeness of
this distribution to our approximation is then established in Section~\ref{main}, by showing
that Dynkin's formula, applied in equilibrium, yields an equation not far removed from the Stein equation
for a centred Poisson distribution, enabling ideas related to Stein's method to be
brought into play. An important element in obtaining an approximation in total variation is to
show {\it a priori\/} that the equilibrium distribution is sufficiently smooth, in the sense
that translating it by a single unit changes the distribution only by order $O(1/\sqrt{n})$ in total
variation: see, for example, R\"ollin~(2005). The corresponding argument is to be found in
Section~\ref{one-shift}. We illustrate the results by applying them to a birth, death
and immigration process, with births occurring in groups.
\subsection{Basic approach}\label{prelims}
We start by defining our density dependent sequence of Markov processes. For each $n\in\mathbb N$,
let $Z_{n}(t)$, $t\ge0$, be an
irreducible continuous time pure jump Markov process taking values in $\mathbb Z$, with transition
rates given by
$$
i\ \to\ i+j \quad \mbox{ at rate }\quad n\lambda_j\Big(\frac{i}{n}\Big),\qquad i \in {\mathbb Z},\
j\in \mathbb Z\setminus \{0\},
$$
where the $\lambda_j(\cdot)$ are prescribed functions on $\mathbb R$; we set
$$
z_{n}(t)\ :=\ n^{-1}Z_{n}(t), \quad t \ge0.
$$
We then define an `average growth rate' of the process $z_{n}$ at $z \in n^{-1}{\mathbb Z}$ by
$$
F(z)\ :=\ \sum_{j\in \mathbb Z\setminus \{0\}}j{{\lambda}_{j}}(z),
$$
and a `quadratic variation' function by $n^{-1}{\sigma}^2(z)$,
where
$$
{\sigma}^2(z)\ =\ \sum_{j\in \mathbb Z\setminus \{0\}} j^2 {{\lambda}_{j}}(z),
$$
assumed to be finite for all~$z\in{\mathbb R}$.
\ignore{
For the rate at which the process leaves the state $z$, we write $n\lambda(z)$, where
$$
\lambda(z):=\sum_{j\in \mathbb Z\setminus \{0\}} \lambda_j(z).
$$
}
The `law of large numbers' approximation shows that, for large~$n$, the time dependent
development of the process~$z_n$ runs close to the solution of the differential equation system
$\dot z = F(z)$, with the same initial condition, and that there is a
approximately diffusive behaviour on a scale~$n^{-1/2}$ about this path (Kurtz~1970, 71). If~$F$ has a
single zero at a point~$c$, and is such that~$c$ is globally attracting for the
differential equation system, then~$Z_n$ has an
equilibrium distribution~$\Pi_n$ that is approximately normal,
and puts mass on a scale~$n^{1/2}$ around~$nc$ (Barbour~1980).
The corresponding asymptotic variance is given by $n^{1/2}v_c$ with $v_c := \frac{\sigma^2(c)}{-2F'(c)}$,
provided that $F'(c) < 0$, and the error of the approximation in Kolmogorov distance
is of ideal order~$O(n^{-1/2})$ if only finitely many of the functions~$\lambda_j$ are non-zero.
In this paper, we strengthen this result, by proving an accurate approximation to the
equilibrium distribution using another distribution on the integers. Under
assumptions similar to those needed for the previous normal approximation, we prove
that the distance in total variation between the centred
equilibrium distribution $\Pi_{n}-\lfloor nc \rfloor$ and the centred Poisson distribution
$$
\widehat{\rm Po}(nv_c)\ :=\ {\rm Po}(nv_c)*\delta_{-\lfloor nv_c\rfloor}
$$
is of order $O(n^{-1/2})$: here and subsequently, $\delta_r$ denotes the point mass on~$r$,
and~$*$ denotes convolution. If infinitely many of the~$\lambda_j$ are allowed to be non-zero,
but satisfy the analogue of a $(2+\alpha)$'th moment condition, for some $0 < \alpha \le 1$,
we prove that the error is of order $O(n^{-\alpha/2})$.
The proof of our approximation runs as follows.
The infinitesimal generator ${\mathcal A}_n$ of~$Z_{n}$, acting on a function $h$, is given by
$$
({\mathcal A}_n h)(i)\ :=\
\sum_{j\in \mathbb Z\setminus \{0\}}n{{\lambda}_{j}}\Big(\frac{i}{n}\Big)\big[h(i+j)-h(i)\big],
\quad i \in {\mathbb Z}.
$$
In equilibrium, under appropriate assumptions on~$h$, Dynkin's formula implies that
\begin{equation}\label{DF}
{\mathbb E}({\mathcal A}_n h)(Z_{n})=0.
\end{equation}
The following lemma, whose proof we omit, expresses ${\mathcal A}_n h$ in an alternative form.
\begin{lemma}\label{generator} Suppose that $\sum_{j\in\bZ\setminus\{0\}} j^2\lambda_j(z) < \infty$ for all~$z\in{\mathbb R}$.
Then, for any function $h\colon \mathbb Z\to \mathbb R$ with bounded differences, we have
\begin{equation}\label{basic}
({\mathcal A}_nh)(i)
\ =\ \frac{n}{2}{\sigma}^2\Big(\frac{i}{n}\Big)\bigtriangledown{g_h}(i)
+nF\Big(\frac{i}{n}\Big)g_h(i) + E_n(g,i),
\end{equation}
where $\bigtriangledown{f}(i):=f(i)-f(i-1)$ and $g_h(i) := \bigtriangledown{h}(i+1)$
and, for any $i\in{\mathbb Z}$,
\begin{eqnarray}
\lefteqn{E_n(g,i)}\nonumber\\
&:=& -\frac{n}{2}F\Big(\frac{i}{n}\Big)\bigtriangledown{g_h}(i)
+\sum_{j\geq 2}a_j(g,i) n\lambda_j\Big(\frac{i}{n}\Big)
-\sum_{j\geq 2}b_j(g,i) n{{\lambda}_{-j}}\Big(\frac{i}{n}\Big),
\label{En-def}
\end{eqnarray}
with
\begin{eqnarray}
2a_j(g,i) &:=& -j(j-1)\bigtriangledown g(i)
+ 2\sum_{k=1}^{j-1} {k \bigtriangledown g(i+j-k)} \label{aj-bnd-1}\\
&=& 2\sum_{k=2}^j \binom{k}{2} \bigtriangledown^2{g_h}(i+j-k+1);\label{aj-bnd-2}\\
2b_j(g,i) &:=& j(j-1)\bigtriangledown g(i)
- { 2\sum_{k=1}^{j-1} k \bigtriangledown g(i-j+k)} \nonumber \\
&=& 2\sum_{k=2}^j \binom{k}{2} \bigtriangledown^2{g_h}(i-j+k). \nonumber
\end{eqnarray}
\end{lemma}
\medskip
Writing \Ref{DF} using the result of Lemma~\ref{generator} leads to the required
approximation, as follows.
In equilibrium, $Z_n/n$ is close to~$c$, as is shown in the next section, and so the main
part of~\Ref{basic} is close to
$$
-F'(c)\left\{\frac{n\sigma^2(c)}{-2F'(c)}\bigtriangledown{g_h}(i) - (i-nc)g_h(i)\right\},
$$
because $F(c)=0$.
Here, the term in braces is very close to the Stein operator for the centred Poisson
distribution~${\widehat P}(nv_c)$ with $v_c = \frac{\sigma^2(c)}{-2F'(c)}$, applied to the
function~$g_h$: see R\"ollin~(2005). Indeed, for any~$v > 0$ and $B \subset {\bf Z}_v$,
where ${\bf Z}_v := \{l \in \mathbb Z,\,l \ge -\lfloor v\rfloor\}$, one can write
\begin{equation}\label{SEt}
1 \hspace*{-1mm} {\rm l}_{B}(l)-\widehat{{\rm Po}}(v)\{B\}\ =\ v\;\bigtriangledown{g}(l+1)-l{g}(l)
+\langle v \rangle{g}(l) , \quad l \in {\bf Z}_v,
\end{equation}
for a function~$g = g_{v,B}$ satisfying
\begin{equation}\label{iSEt}
\sup_{l\geq -\lfloor v \rfloor}|{g}(l+1)|\ \le\ \min\Bigl\{1,\frac{1}{\sqrt{v}}\Bigr\}; \qquad
\sup_{l\geq -\lfloor v \rfloor}|\bigtriangledown{g}(l+1)| \ \le\ \frac{1}{v};\qquad
g(l) \ =\ 0, \quad l \le -\lfloor v \rfloor,
\end{equation}
where $\langle x \rangle := x -\lfloor x \rfloor$ denotes the fractional part of~$x$;
note also, from~\Ref{SEt} and~\Ref{iSEt}, that
\begin{equation}\label{iSEt-2}
\sup_l |l{g}(l)| \le 3.
\end{equation}
Replacing~$l$ in~\Ref{SEt} by an integer valued random variable~$W$ then shows that, for
any $B \subset {\bf Z}_v$,
\begin{eqnarray}
\lefteqn{|{\mathbb P}[W \in B] - \widehat{{\rm Po}}(v)\{B\}|}\nonumber\\
&&\ \le\ \sup_{g\in{\mathcal G}_v}
|{\mathbb E}\{v\bigtriangledown{g}(W+1)-W{g}(W) +\langle v \rangle{g}(W)\}|
+ {\mathbb P}[W < -\lfloor v\rfloor], \label{SEt-2}
\end{eqnarray}
where~${\mathcal G}_v$ denotes the set of functions $g\colon\,{\mathbb Z}\to{\mathbb R}$ satisfying \Ref{iSEt}
and~\Ref{iSEt-2}.
Hence, replacing~$W$ by~$Z_n$ and $v$ by~$nv_c$ in~\Ref{SEt-2}, and comparing the expectation
with~\Ref{DF} expressed using Lemma~\ref{generator}, the required approximation in total
variation can be deduced; for this part of the argument,
we need in particular to show that, in equilibrium,
\begin{equation}\label{delta-2-bnd}
|{\mathbb E}\{\bigtriangledown{g}(Z_n+1)-\bigtriangledown{g}(Z_n)\}|
\ =\ |{\mathbb E}\{\bigtriangledown^2{g}(Z_n+1)\}| \ =\ O(n^{-3/2}),
\end{equation}
and also that ${\mathbb E} |E_n(g,Z_n)|
= O(n^{-\alpha/2})$ for any $g\in{\mathcal G}_{nv_c}$. The bound~\Ref{delta-2-bnd} follows from Corollary~\ref{lema2} in
Section~\ref{one-shift}, and the latter estimate, which also uses~\Ref{delta-2-bnd}, is the substance of
Section~\ref{main}.
\subsection{Assumptions}
We make the following assumptions on the functions ${\lambda}_{j}$.
The first ensures that the deterministic differential equations have a unique
equilibrium, which is sufficiently strongly attracting.
\\[0.7ex]
{\bf \small A1:} There exists a unique $c$ satisfying $F(c)=0$; furthermore, $F'(c) < 0$
and, for any $\eta>0$, $\mu_{\eta}:= \inf_{|z-c|\geq {\eta}}|F(z)|>0$.
\\[0.7ex]
The next assumption controls the global behaviour of the transition functions~$\lambda_j$.
\\[0.4ex]
{\bf \small A2:}
(a) For each $j\in {\mathbb Z}\setminus \{0,\}$, there exists $c_j\ge0$ such that
\begin{equation} \label{lamdaj}
\lambda_j(z)\leq c_j(1+|z-c|), \qquad z\in \mathbb R,
\end{equation}
where the~$c_j$ are such that, for some $0 < \alpha \le 1$,
$$
\sum_{j \in \mathbb Z \setminus \{0\}}|j|^{2+\alpha}c_{j}<\infty.
$$
\qquad (b) For some $\lambda^0>0$ ,
$$
{\lambda}_{1}(z)\ \geq\ 2 \lambda^0,\qquad z\in \mathbb R.
$$
The moment condition on the~$c_j$ in Assumption~A2\,(a) plays the same r\^ole as
the analogous moment condition in the Lyapounov central limit theorem. Under
this assumption, the ideal rate of convergence in the usual central limit
approximation is the rate~$O(n^{-\alpha/2})$ that we establish for our total
variation approximation.
Assumption A2\,(b) is important for establishing the smoothness of the equilibrium
distribution~$\Pi_n$. If, for instance, all jump sizes were multiples of~$2$, the
approximation that we are concerned with would not be accurate in total variation.
\\[0.7ex]
We also require some assumptions concerning the local properties of the functions~$\lambda_j$
near~$c$.\\[0.4ex]
{\bf \small A3:}
(a) There exist $\varepsilon>0$ and $0 < \delta \le 1$ and a set $J \subset
{\mathbb Z}\setminus \{0\}$ such that
\begin{eqnarray*}
\inf_{|z-c|\leq \delta}\lambda_j(z)\ \geq\ \varepsilon \lambda_j(c)\ > \ 0,\ \ \ j\in J;\\
\lambda_j(z)\Eq0 \quad \mbox{for all}\quad |z-c|\ \leq\ \delta,\ \ \ j\notin J.
\end{eqnarray*}
\qquad (b)\ For each $ j\in J$, ${\lambda}_{j}$ is of class $C^2$ on~$|z-c| \le \delta$.
\\[0.4ex]
Assumptions A2\,(a) and A3 imply in particular that the series
$\sum_{j \in \mathbb Z \setminus \{0\}}j\lambda_j(z)$ and
$\sum_{j \in \mathbb Z \setminus \{0\}}j^2\lambda_j(z)$ are uniformly convergent
on~$|z-c| \le \delta$, and that their sums,
$F$ and ${\sigma}^2$ respectively, are continuous there. They also imply that
$$
\sum_{j\in \mathbb Z\setminus \{0\}}|j|n \lambda_j(i/n)
\ =\ O(|i|),\ \ \ |i|\to \infty,
$$
so that the process $Z_n$ is a.s.\ non-explosive, in view of Hamza and Klebaner~(1995, Corollary~2.1).
\\[0.7ex]
The remaining assumptions control the derivatives of the functions~$\lambda_j$ near~$c$.\\[0.4ex]
{\bf \small A4:} For $\delta$ as in A2,
$$
L_1 \ :=\
\sup_{j\in J}\frac{\|{{\lambda}_j^{\prime}}\|_{\delta}}{\lambda_j(c)}
\ <\ \infty,
$$
where $\|f\|_{\delta}:=\sup_{|z-c|\leq \delta}|f(z)|$. \\[0.7ex]
This assumption implies in particular, in view of Assumptions A2--A3, that the series
$\sum_{j \in \mathbb Z \setminus \{0\}}j\lambda^{\prime}_j(z)$ and
$\sum_{j \in \mathbb Z \setminus \{0\}}j^2\lambda^{\prime}_j(z)$ are uniformly
convergent on $|z-c|\leq \delta$,
that their sums are
$F^{\prime}$ and~$(\sigma^2)^{\prime}$ respectively, and that
$F$ and $\sigma^2$ are of class $C^1$ on $|z-c|\leq \delta$. \\[0.7ex]
{\bf \small A5:} For $\delta$ as in A2,
$$
L_2\ :=\
\sup_{j\in J} \frac{\|{{\lambda }_j^{\prime \prime}}\|_{\delta}}{|j|\lambda_j(c)}
\ <\ \infty.
$$
This assumption implies, in view of A2--A3, that the series
$\sum_{j \in \mathbb Z \setminus \{0\}}j\lambda^{\prime \prime}_j(z)$
is uniformly convergent on $|z-c|\leq \delta$, its sum is
$F^{\prime \prime}$,
and~$F$ is of class $C^2$ on $|z-c|\leq \delta.$ \\[0.7ex]
\ignore{
In the example, {\bf \small Assumption 1} is satisfied if $d>\sum_{j\geq 1}jb_j,$ with
$c=a/(d-\sum_{j\geq 1}jb_j).$ {\bf \small Assumption 2} is satisfied with $\lambda^0=a/2$ and
$c_1=\max\{b_1, a+b_1c\},$ and, for instance, for $\delta=c/2$ and $\varepsilon=1/2,$ for $c_j=b_j
\max\{1,c\},$ with $j\geq 2,$ and $c_{-1}=d \max\{1,c\}.$ For any $\delta>0,$ we have
$\mu_{\delta}=\delta(d-\sum_{j\geq 1}jb_j).$ {\bf \small Assumption 3} is satisfied
if $\sum_{j\geq 1}|j|^{2+\alpha}b_j< \infty,$ in which case the other assumptions follow immediately.
}
Our arguments make frequent use of the following theorem, which is a restatement in our
setting of Hamza and Klebaner~(1995, Theorem~3.2), and justifies~\Ref{DF}.
\begin{theorem}\label{HK95}
Suppose that $Z_{n}$ is non-explosive. Let $h$ be a function satisfying
\begin{equation}\label{HK-1}
(|{\mathcal A}_n|h)(i)\ :=\
\sum_{j\in \mathbb Z\setminus \{0\}}{{\lambda}_{j}}\Big(\frac{i}{n}\Big)|h(i+j)-h(i)|
\ \leq\ c_{n,h}(1\vee |h(i)|),\;\;\;|i|\rightarrow \infty,
\end{equation}
for some $c_{n,h} < \infty$. Then,
if $h(Z_n(0))$ is integrable, so is $h(Z_n(t))$ for any $t\ge0$; moreover,
$$
h(Z_n(t))-h(Z_n(0))-\int_0^t({\mathcal A}_nh)(Z_n(s)))ds
$$
is a martingale, and {\it Dynkin's formula} holds:
\begin{equation}\label{HK-2}
{\mathbb E}[h(Z_{n}(t))-h(Z_n(0))]\ =\ \int_0^t{\mathbb E}({\mathcal A}_n h)(Z_{n}(s))ds.
\end{equation}
\end{theorem}
\setcounter{equation}{0}
\section{Existence of the equilibrium distribution}\label{equilibrium}
In this section, we prove that~$Z_n$ has an equilibrium distribution which is suitably
concentrated in the neighbourhood of~$nc$.
\begin{theorem}\label{exubj} Under Assumptions A1--A4, for all~$n$ large enough, $Z_n$ has an
equilibrium distribution $\Pi_n$, and
\begin{equation} \label{2ineq}
\begin{split}
&{\mathbb E}_{\Pi_n}\{|z_{n}-c| \cdot 1 \hspace*{-1mm} {\rm l}(|z_{n}-c| > \delta)\}\ =\ O(n^{-1})\\
&{\mathbb E}_{\Pi_n}\{(z_{n}-c)^2 \cdot 1 \hspace*{-1mm} {\rm l}(|z_{n}-c|\le \delta)\}\ =\ O(n^{-1}),
\end{split}
\end{equation}
for $\delta$ as in Assumption~A3: here, as before, $z_n := n^{-1}Z_n$.
\end{theorem}
\begin{proof} The argument is based on suitable choices of Lyapounov functions.
Consider the twice continuously differentiable function $V\colon \mathbb R \to \mathbb R_+$ defined by
$V(z):=|z-c|^{2+\alpha},$ for the $\alpha$ in Assumption~A2\,(a).
Since $V(c)=0$ and $V(z)>0$ for any $z\neq c$, and because
\begin{equation}\label{FVdash}
F(z)V^{\prime}(z) \ =\ -|F(z)|(2+\alpha)|z-c|^{1+\alpha}\ <\ 0\ \ \ \mbox{for any}\ z\neq c,
\end{equation}
while $F(c)V^{\prime}(c)=0$, we conclude that $V$ is a Lyapounov function
guaranteeing the asymptotic stability of the constant solution $c$ of the equation
$\dot{x}=F(x)$. We now use it to show the existence of~$\Pi_n$.
\begin{lemma}
\label{lem-3.2}
Under the assumptions of Theorem~\ref{exubj}, the function
$h_V(i):=V\big(\frac i n\big)=\big|\frac i n -c\big|^{2+\alpha}$ fulfils the
conditions of Theorem~\ref{HK95} with respect to the initial distribution
$\delta_l$, the point mass at~$l$, for any $l\in \mathbb Z$.
\end{lemma}
\begin{proof}
Checking~\Ref{HK-1}, we use Taylor approximation and Assumption~A2\,(a) to give
\begin{eqnarray}
(|{\mathcal A}_n|\, {h_V})(i)
& \leq& (2+\alpha)|z-c|^{1+\alpha}\sum_{j\in \mathbb Z \setminus \{0\}}|j|\ c_j(1+|z-c|) \nonumber\\
&& \qquad\mbox{}+\frac{(2+\alpha)(1+\alpha)|z-c|^{\alpha}}{2n}
\sum_{j\in \mathbb Z \setminus \{0\}} j^2c_j(1+|z-c|) \label{|an|} \\
&& \qquad\mbox{} +\frac{(2+\alpha)(1+\alpha)}{2n^{1+\alpha}}
\sum_{j\in \mathbb Z \setminus \{0\}}|j|^{2+\alpha}c_j(1+|z-c|),
\end{eqnarray}
where we write $z := i/n$. For $|z-c|<\delta\le 1$, the estimate in~\Ref{|an|} is
uniformly bounded by
$$
C_{1n}\ :=\ 2(2+\alpha)\Big\{\sum_j |j| c_j+\frac{(1+\alpha)}{2n}\sum_j j^2
c_j+\frac{(1+\alpha)}{2n^{1+\alpha}}\sum_j |j|^{2+\alpha} c_j\Big\} \ <\ \infty,
$$
because of Assumption~A2\,(a); for $|z-c|\geq \delta$, we have the bound
$$
(|{\mathcal A}_n|\,{h_V})(i)\ \leq\ C_{1n}|z-c|^{2+\alpha} \ =\ C_{1n}\, {h_V}(i),
$$
as required.
\end{proof}
The above lemma allows us to apply Dynkin's formula to the function $h_V$. Using Taylor
approximation as for~\eqref{|an|}, but now noting that the first order term
$$
\sum_{j\in \mathbb Z\setminus \{0\}}\lambda_j(z) j V^{\prime}(z) \ =\ F(z)V'(z)
$$
can be evaluated using~\Ref{FVdash}, it follows that
\begin{equation} \label{pti2}
({\mathcal A}_n \,h_V)(i)
\ \leq\ -|F(z)|(2+\alpha)|z-c|^{1+\alpha} + n^{-1}C_2 \ \le\ n^{-1}C_2
\end{equation}
on $|z-c|\le\delta$, for
$$
C_2 \ =\ (2+\alpha) (1+\alpha)\Bigl\{\sum_j j^2 c_j+ \sum_j |j|^{2+\alpha} c_j\Bigr\}
\ <\ \infty,
$$
where, once again, $z:= i/n$.
On $|z-c|> \delta$ and under Assumption A2\,(a), we have
\begin{eqnarray} \label{celneg}
({\mathcal A}_n \,h_V)(i)
&\le& -|F(z)|(2+\alpha)|z-c|^{1+\alpha}\ \nonumber\\
&&\quad\quad \Big[1- \frac{(1+\alpha)}{2n|F(z)|\cdot |z-c|}\sum_{j\in \mathbb Z \setminus \{0\}}j^2c_j(1+|z-c|)\nonumber\\
&&\quad\mbox{}\quad\quad -\frac{(1+\alpha)}{2n^{1+\alpha}|F(z)|\cdot |z-c|^{1+\alpha}}
\sum_{j\in \mathbb Z \setminus \{0\}}|j|^{2+\alpha}c_j(1+|z-c|)\Big] \nonumber\\
&\le& -\frac {\mu_{\delta}(2+\alpha)} 2 |z-c|^{1+\alpha}\
\leq\ - \mu_{\delta} |z-c|^{1+\alpha},
\end{eqnarray}
as long as $n$ is large enough that $n\delta \ge 1$ and
$$
\frac{(1+\delta)(1+\alpha)}{n\delta\mu_{\delta}}\sum_{j\in \mathbb Z \setminus \{0\}}|j|^{2+\alpha}c_j
\ <\ \frac 1 2.
$$
Dynkin's formula~\Ref{HK-2} then implies, for such $n$, that
\begin{eqnarray*}
0&\leq &{\mathbb E}_{i} h_V(Z_{n}(t))
\ =\ V(z) + \int_0^t{\mathbb E}_{i} ({\mathcal A}_n \,h_V)(Z_{n}(s))\,ds\\
&\leq & V(z) + \int_0^t\frac{C_2}{n}{\mathbb P}_{i}(|n^{-1}Z_{n}(s)-c|< \delta)\, ds\\
&&\mbox{}\quad -\mu_{\delta} \int_0^t{\mathbb E}_{i}\{|n^{-1}Z_{n}(s)-c|^{1+\alpha} \cdot
1 \hspace*{-1mm} {\rm l}{(|n^{-1}Z_{n}(s)-c|\geq \delta)}\}\,ds,
\end{eqnarray*}
for any $t>0$ and $i \in \mathbb Z$, where ${\mathbb P}_{i}$ and~${\mathbb E}_{i}$ denote
probability and expectation conditional on $Z_n(0) = i$. It now follows, for any $y\geq \delta$, that
\begin{eqnarray} \label{totpti2}
\lefteqn{\frac {\mu_{\delta}\, y^{1+\alpha}} {t} \int_0^t{\mathbb P}_{i}(|{n^{-1}Z_{n}(s)}-c|\geq y)\,ds}\nonumber\\
&&\ \le\ \frac {\mu_{\delta}}{t}
\int_0^t{\mathbb E}_{i}\{|{n^{-1}Z_{n}(s)}-c|^{1+\alpha} \cdot 1 \hspace*{-1mm} {\rm l}{(|{n^{-1}Z_{n}(s)}-c|\geq y)}\}\,ds\nonumber\\
&&\ \le\ \frac 1 t V({z})+\frac{C_2}{nt} \int_0^t {\mathbb P}_{i}(|{n^{-1}Z_{n}(s)}-c|< \delta)\,ds,
\end{eqnarray}
and, by letting $t\to \infty$, it follows that
$$
\limsup_{t\to \infty}\frac 1 t\int_0^t{\mathbb P}_{i}(|{n^{-1}Z_{n}(s)}-c|\geq y)\,ds
\ \leq\ \frac{C_2}{n\mu_\delta\, y^{1+\alpha}}.
$$
This implies that a limiting equilibrium distribution $\Pi_n$ for $Z_n$ exists,
see for instance Ethier and Kurtz (1986, Theorem~9.3, Chapter~4),
and that, writing $z_n := n^{-1}Z_n$, we have
$$
{\mathbb P}_{\Pi_n}(|z_{n}-c|\geq y)\ \leq\
\frac{C_2}{n\mu_{\delta}\, y^{1+\alpha}},
$$ for any $y\geq \delta$. Furthermore,
\begin{eqnarray*}
{\mathbb E}_{\Pi_n}\{|z_{n}-c| \cdot 1 \hspace*{-1mm} {\rm l}(|z_{n}-c|\geq \delta)\} &=&
\int_{\delta}^\infty{\mathbb P}_{\Pi_n}(|z_{n}-c|\geq y)\,dy \\
&\leq& \int_{\delta}^\infty \frac{C_2}{n\mu_{\delta}\, y^{1+\alpha}}\,dy \ =\ O(n^{-1}),
\end{eqnarray*}
proving the first inequality in~\eqref{2ineq}.\\
For the second inequality in~\eqref{2ineq}, we define a function $\tilde{V}\colon \mathbb R \to
\mathbb R$, which is of class $C^2(\mathbb R)$, is bounded and has uniformly
bounded first and second derivatives on $\mathbb R$, fulfils the conditions of Theorem~\ref{HK95},
and satisfies $F(z)\tilde{V}^{\prime}(z)=-(z-c)^2$ on $|z-c|\leq \delta$.
In view of the latter property, we begin by letting $v\colon [c-\delta,c+\delta]\to \mathbb R_+$ be
the function defined by
$$
v(z)\ :=\ \int_c^z \frac{-(x-c)^2}{F(x)}\,dx,
$$
with $v(c)=0$. Note that $v$ is well defined, since $F^{\prime}(x)<0$ on a small
enough neighborhood of~$c$, by Assumptions A1 and~A4, and that $v(z)>0$ for
any $z\neq c$. Furthermore, in view of Assumptions A1 and~A4,
$$
v^{\prime }(z) \ =\ -\frac{(z-c)^2}{F(z)} \ \ {\rm and}\ \ v^{\prime
\prime}(z) \ =\ \frac{(z-c)^2F^{\prime}(z)-2(z-c)F(z)}{F^2(z)}
$$
exist and are continuous on $|z-c|\leq \delta$,
since $|F(z)|>0$ for $z\neq c$, $F(z)\sim F^{\prime}(c)(z-c)$ for $z\to c$, and $F^{\prime}$ is
continuous. In particular, we have
\begin{equation}\label{Rk-1}
v^{\prime }(c) \ =\ \lim_{z\to c}v^{\prime}(z)=0 \ \ {\rm and}\ \
v^{\prime \prime}(c) \ =\ \lim_{z\to c}v^{\prime \prime}(z) \ =\ -\frac 1{F^{\prime}(c)}\ >\ 0.
\end{equation}
Now define the function $\tilde{V}$ to be identical with~$v$ on $|z-c| \le \delta$, and
continued
in $z \le c-\delta$ and in $z \ge c+\delta$ in such a way that the function is still~$C_2$,
and takes the same fixed value everywhere
on $|z-c| \ge 2\delta$.
\ignore{
as follows:
$$
\tilde{V}(z)\ :=\
\begin{cases}
v(c-\delta)-\delta+|z-c| + [v^{\prime}(c-\delta)+1]\sin(z-c+\delta)\\
\hspace{1.5in}\mbox{}+\frac 1 2 v^{\prime \prime}(c-\delta)\sin^2(z-c+\delta), &{\rm if}\ z<c-\delta; \\
v(z), &{\rm if}\ |z-c|\leq \delta; \\
v(c+\delta)-\delta+|z-c|+[v^{\prime}(c+\delta)-1]\sin(z-c-\delta)\\
\hspace{1.5in}\mbox{}+\frac 1 2 v^{\prime \prime}(c+\delta)\sin^2(z-c-\delta), &{\rm if}\ z>c+\delta.
\end{cases}
$$
Note that the function $\tilde{V}$ is of class $C^2(\mathbb R)$, and that
$$
|\tilde{V}^{\prime}(z)|\leq C_3 \ \ {\rm and}\ \ |\tilde{V}^{\prime \prime}(z)|\leq C_3,
$$
for any $z \in \mathbb R$, where
$$
C_3:=\max\{2+|v^{\prime}(c-\delta)|+3|v^{\prime \prime}(c-\delta)|,\sup_{|z-c|\leq \delta}(|v^{\prime }(z)|+|v^{\prime
\prime}(z)|),2+|v^{\prime}(c+\delta)|+3|v^{\prime \prime}(c+\delta)|\}.$$
}
Let
\[
C_3 \ :=\ \max\{\sup_{z\in \mathbb R} {\tilde V}(z),\,\sup_{z\in \mathbb R} |{\tilde V}'(z)|,\,
\sup_{z\in \mathbb R} |{\tilde V}''(z)|\}.
\]
\begin{lemma}\label{lem-3.3}
Under the assumptions of Theorem~\ref{exubj}, the function
$\tilde{h}_V(i):=\tilde{V}\big(\frac i n\big)$ fulfils the
conditions of Theorem~\ref{HK95} with respect to the initial distribution $\Pi_n$.
\end{lemma}
\begin{proof}
Since $\tilde{h}_V(i)$ is bounded, it follows that $\mathbb E_{\Pi_n}|\tilde{h}_V(Z_n)|<\infty$.
$|{\mathcal A}_n|\,\tilde{h}_V$ is also bounded, since, for $|n^{-1}i-c| \le 4\delta$,
by Assumption A2\,(a),
\[
(|{\mathcal A}_n|\,\tilde{h}_V)(i)
\ \le\ C_3 \sum_{j\in \mathbb Z\setminus \{0\}} c_j(1+4\delta),
\]
while, for $|n^{-1}i-c| > 4\delta$,
\begin{eqnarray*}
\lefteqn{(|{\mathcal A}_n|\,\tilde{h}_V)(i)
\ \le\ C_3 \sum_{j\colon\,|j+i-nc| \le 2n\delta} c_j(1+|n^{-1}i-c|)}\\
&&\ \le\ C_3 \Bigl\{\sum_{j\in \mathbb Z\setminus \{0\}} jc_j\Bigr\}
\frac{1+|n^{-1}i-c|}{|i-nc|-2n\delta}
\ \le\ C_3 \Bigl\{\sum_{j\in \mathbb Z\setminus \{0\}} jc_j\Bigr\}
\frac{1+4\delta}{2n\delta}.
\end{eqnarray*}
\ignore{
Taylor expansion then yields
\begin{eqnarray*} \
(|{\mathcal A}_n|\,\tilde{h}_V)(i)
&\leq& C_3\Big(\sum_{j\in \mathbb Z\setminus \{0\}}|j|c_j+\frac{1}{2n}\sum_{j\in \mathbb Z\setminus
\{0\}}j^2c_j\Big)(1+|n^{-1}i-c|)\\
&=& O(1\vee |\tilde{h}_V(i)|),
\end{eqnarray*}
by Assumption A2\,(a).
}
\end{proof}
We now apply Dynkin's formula to $\tilde h_{V}$, obtaining
$$
0 \ =\ \mathbb E_{\Pi_n}\{({\mathcal A}_n\,\tilde h_{V})(Z_n)\}
\ \le\ \mathbb E_{\Pi_n}\Big\{F(z_n)\tilde{V}^{\prime}(z_n)+
\sum_{j\in \mathbb Z \setminus \{0\}}\lambda_j(z_n)\frac{j^2}{2n}\, C_3\Big\}.
$$
Hence it follows that
\begin{eqnarray*}
\lefteqn{\mathbb E_{\Pi_n}\{-F(z_n)\tilde{V}^{\prime}(z_n)\cdot1 \hspace*{-1mm} {\rm l}(|z_n-c|\le\delta)\}}\\
&\le& \mathbb E_{\Pi_n}\Big\{F(z_n)\tilde{V}^{\prime}(z_n)\cdot1 \hspace*{-1mm} {\rm l}(|z_n-c| > \delta)
+\sum_{j\in \mathbb Z \setminus \{0\}}\lambda_j(z_n)\frac{j^2}{2n}\, C_3\Big\},
\end{eqnarray*}
whence we obtain
{\allowdisplaybreaks
\begin{eqnarray*}
\lefteqn{ \mathbb E_{\Pi_n}\{(z_n-c)^2\cdot 1 \hspace*{-1mm} {\rm l}(|z_n-c|\le\delta)\}}\\
&\leq& \mathbb E_{\Pi_n}\{|F(z_n)\tilde{V}^{\prime}(z_n)|\cdot1 \hspace*{-1mm} {\rm l}(|z_n-c|> \delta)\}
+ C_3 \,\mathbb E_{\Pi_n}\Big\{\sum_{j\in \mathbb Z \setminus \{0\}}\lambda_j(z_n)\frac{j^2}{2n}\Big\}\\
&\leq& C_3 \sum_{j\in \mathbb Z\setminus \{0\}}\Big(2|j|+\frac{j^2}{n}\Big)c_j\
\mathbb E_{\Pi_n}\{|z_n-c|\cdot 1 \hspace*{-1mm} {\rm l}(|z_n-c| > \delta)\}
+ \frac {C_3} {2n} \sup_{|z-c|\le\delta} \sigma^2(z).
\end{eqnarray*}}
Using the first inequality in~\eqref{2ineq} and Assumptions A2 and~A3, we conclude that
$$
{\mathbb E}_{\Pi_n}\{(z_{n}-c)^2 \cdot 1 \hspace*{-1mm} {\rm l}(|z_{n}-c|\le \delta)\} \ =\ O(n^{-1}),
$$
proving the second inequality in~\eqref{2ineq}.
\end{proof}
\begin{corollary} \label{prop1}
Under Assumptions A1--A4,
$$
{\mathbb E}_{\Pi_n}\{|z_{n}-c|\} \ =\ O(n^{-1/2}).
$$
\end{corollary}
\begin{proof}
Using H\"older's inequality, we obtain
\begin{eqnarray*}
\lefteqn{{\mathbb E}\{|z_{n}-c|\}}\\
&=&{\mathbb E}_{\Pi_n}\{|z_{n}-c| \cdot 1 \hspace*{-1mm} {\rm l}(|z_{n}-c| > \delta)\}+
{\mathbb E}_{\Pi_n}\{|z_{n}-c|\cdot 1 \hspace*{-1mm} {\rm l}(|z_{n}-c| \le \delta)\}\\
&\leq& {\mathbb E}\{|z_{n}-c|\cdot 1 \hspace*{-1mm} {\rm l}(|z_{n}-c| > \delta)\}+
\sqrt{{\mathbb E}_{\Pi_n}\{(z_{n}-c)^2\cdot 1 \hspace*{-1mm} {\rm l}(|z_{n}-c| \le \delta)\}}.
\end{eqnarray*}
The corollary now follows from Theorem~\ref{exubj}.
\end{proof}
\begin{corollary} \label{prob-n1}
Under Assumptions A1--A4, for any $0 < \delta' \le \delta$,
$$
{\mathbb P}_{\Pi_n}[|z_{n}-c|> \delta'] \ =\ O(n^{-1}).
$$
\end{corollary}
\begin{proof}
It follows from Chebyshev's inequality and Theorem~\ref{exubj} that
$$
{\mathbb P}_{\Pi_n}[|z_{n}-c|I[|z_n-c| \le \delta] > \delta'/2] \ \le\
4{\mathbb E}_{\Pi_n}\{|z_{n}-c|^2I[|z_n-c| \le \delta]\}/(\delta')^2 \ =\ O(n^{-1}),
$$
and that
$$
{\mathbb P}_{\Pi_n}[|z_{n}-c| > \delta] \ \le\ {\mathbb E}_{\Pi_n}\{|z_{n}-c|I[|z_n-c| > \delta]\}/\delta
\ =\ O(n^{-1}),
$$
from which the corollary follows.
\end{proof}
\section{The distance between $\Pi_n$ and its unit translation}\label{one-shift}
\setcounter{equation}{0}
\ignore{
If the equilibrium distribution $\Pi_n,$ suitably translated, is indeed $O(1/\sqrt{n})$ close to a
Poisson distribution with parameter $n\rho,$ say, then its unit translate is correspondingly close to
${\rm Po}(n\rho)$ translated by 1. Then, since the total-variation distance between ${\rm Po}(n\rho)$
and ${\rm Po}(n\rho)+1$ is of order $O(1/\sqrt{n}),$ the same has to be true of the total-variation
distance between $\Pi_n$ and its unit translate, where the \emph{distance in total variation} between
two probability measures $P$ and $Q$ on ${\mathbb Z}$
is defined as:
$$d_{TV}\{P,Q\}:=\sup_{A \subset \mathbb Z}|P(A)-Q(A)|.$$ However, as illustrated in Barbour and
Xia~(1999), Barbour and Cekanavicius~(2002) and R\"ollin~(2005), it is extremely useful to be able
to establish this latter fact in advance, in order to prove the translated Poisson approximation
heorem, using Stein's method, in the same way that proving a concentration inequality is a useful
prerequisite for deriving Berry-Essen approximations in the central limit theorem, see
Chen and Shao~(2003). In this section, we establish such a bound.
}
A key step in the argument leading to our approximation is to establish that the equilibrium
distribution~$\Pi_n$ of~$Z_n$ is sufficiently
smooth. In order to do so, we first need to prove an auxiliary result, showing that,
if the process~$Z_n$ starts near enough to~$nc$,
then it remains close to~$nc$ with high probability over any finite time interval.
This is the substance of the following lemma.
\begin{lemma}\label{lema3}
Under Assumptions A1--A4, for any $0 < \eta \le \delta$, there exists a constant $K_{U,\eta}<\infty$ such that
$$
\mathbb P[\sup_{t\in [0,U]}|Z_n(t)-nc|> n\eta \mid Z_n(0) = i]\ \leq\ n^{-1}K_{U,\eta},
$$
uniformly in $|i - nc| \le n\eta e^{-K_1U}/2$, where $K_1 := \|F'\|_{\delta}$.
\end{lemma}
\begin{proof}
It follows directly from Assumption A2\,(a) that~$h$ defined by $h(j)=j$ satisfies
condition~\Ref{HK-1}. Fix $Z_n(0) = i$, and define
\begin{equation}\label{tau-def}
\tau_\eta \ :=\ \inf\{t\ge0\colon\,|Z_n(t)-nc| > n\eta\}.
\end{equation}
Then it follows from Theorem~\ref{HK95} that
$$
{\mathcal M}_n(t)\ :=\ Z_n(t\wedge\tau_\eta) - i - \int_0^{t\wedge\tau_\eta} nF(z_n(s))\,ds
$$
is a martingale with expectation 0, and with expected quadratic variation no larger than
\begin{equation}\label{QV-2}
nt \sum_{j \in {\mathbb Z}\setminus\{0\}}j^2 c_j(1 + \eta)
\end{equation}
at time~$t$ (see Hamza and Klebaner~(1995, Corollary 3)); here, as earlier,
$z_n := n^{-1}Z_n$. Hence we have
\begin{equation*}
|z_n(t\wedge\tau_\eta)-c|\ \leq\ \frac 1 n \left\{\sup_{s\in [0,U]}|{\mathcal M}_n(s)|+|i-nc|\right\}
+\int_0^{t\wedge\tau_\eta}|F(z_n(s))|\,ds,
\end{equation*}
for any $0\leq t \leq U$, and also, from Assumptions A1--A4, we have
\begin{equation*}
|F(z)|
\ =\ |F(z)-F(c)|\ \leq\ \sup_{|y-c|\leq \delta}|F^{\prime}(y)|\, |z-c|.
\end{equation*}
Hence it follows that
\begin{equation*}
\int_0^{t\wedge\tau_\eta}|F(z_n(s))|\,ds\ \leq \ K_1 \int_0^{t\wedge\tau_\eta}|z_n(s)-c|\,ds.
\end{equation*}
Gronwall's inequality now implies that
\begin{equation*}
|z_n({t\wedge\tau_\eta})-c|\ \leq\
n^{-1}\left\{\sup_{s\in [0,U]}|{\mathcal M}_n(s)|+|i-nc|\right\}e^{K_1 t},
\end{equation*}
for any $0\leq t \leq U$, and so, for $|i - nc| \le n\eta e^{-K_1U}/2$,
\begin{equation}\label{wedge-bnd}
\sup_{t\in [0,U]}|z_n(t\wedge\tau_\eta)-c|
\ \le\ \eta/2 + n^{-1}\sup_{s\in [0,U]}|{\mathcal M}_n(s)|e^{K_1 U}.
\end{equation}
We have thus shown that
\begin{equation}\label{zc}
\mathbb P[\sup_{t\in [0,U]}|z_n(t)-c|> \eta \mid Z_n(0)=i\} \ \le\
\mathbb P[\sup_{s\in [0,U]}|{\mathcal M}_n(s)| > ne^{-K_1 U}\eta/2 \mid Z_n(0)=i].
\end{equation}
But by Kolomogorov's inequality, from~\Ref{QV-2}, we have
\begin{equation}\label{kolm}
\mathbb P[\sup_{s\in [0,U]}|{\mathcal M}_n(s)| > ne^{-K_1 U}\eta/2 \mid Z_n(0)=i]
\ \le\ 4 n^{-1}\eta^{-2}e^{2K_1U} U \sum_{j \in {\mathbb Z}\setminus\{0\}}j^2 c_j(1 + \eta),
\end{equation}
completing the proof.
\end{proof}
We can now prove the main theorem of this section.
\begin{theorem} \label{thdtvpi}
Under Assumptions A1--A4, there exists a constant $K>0$ such that
$$
d_{TV}\{\Pi_{n},\Pi_{n}*\delta_1\}\leq K n^{-1/2},
$$
where $\Pi_{n}*\delta_1$ denotes the equilibrium distribution $\Pi_{n}$ of $Z_n$, translated by 1.
\end{theorem}
\begin{proof}
Because we have little {\it a priori\/} information
about $\Pi_n$, we fix any~$U>0$, and use the stationarity of~$\Pi_n$ to give the inequality
\begin{eqnarray}
\lefteqn{d_{TV}\{\Pi_{n},\Pi_{n}*\delta_1\}}\nonumber\\
&\leq& \sum_{i \in {\mathbb Z}} \Pi_{n}(i)\, d_{TV}\{{\mathcal L}(Z_{n}(U) \mid
Z_{n}(0)=i),{\mathcal L}(Z_{n}(U)+1 \mid Z_{n}(0)=i)\}, \label{disttv}
\end{eqnarray}
By Corollary~\ref{prob-n1}, we thus have, for any $\delta' \le \delta$,
\begin{equation}\label{bound-1}
d_{TV}\{\Pi_{n},\Pi_{n}*\delta_1\} \ \le\ D_{1n}(\delta') + O(n^{-1}),
\end{equation}
where
$$
D_{1n}(\delta') \ :=\ \sum_{i\colon\,|i-nc| \le \delta'}\Pi_{n}(i)\, d_{TV}\{{\mathcal L}(Z_{n}(U) \mid
Z_{n}(0)=i),{\mathcal L}(Z_{n}(U)+1 \mid Z_{n}(0)=i)\}.
$$
This alters our problem to one of finding a bound of similar form, but now
involving the transition probabilities of the
chain~$Z_n$ over a finite time~$U$, and started in a fixed state~$i$ which is relatively
close to~$nc$.
We now use the fact that the upward jumps of length~$1$ occur at least as fast as a Poisson
process of rate~$\lambda^0$, something that will be used to derive the smoothness that we require.
We realize the chain~$Z_n$ with $Z_n(0) = i$ in the form $N_n + X_n$, for the bivariate chain
$(N_n,X_n)$ having transition rates
$$
\begin{array}{ll}
(l,m) \rightarrow (l+1,m) & \mbox{at rate}\ n\lambda^0 \\
(l,m) \rightarrow (l,m+1)
& \mbox{at rate}\ n\Bigl[{{\lambda}_{1}}\Bigl(\frac{l+m}{n}\Bigr)-\lambda^0\Bigr]\\
(l,m) \rightarrow (l,m+j) & \mbox{at rate}\ n{{\lambda}_{j}}\Bigl(\frac{l+m}{n}\Bigr),
\ \mbox{for any}\; j\in \mathbb Z,\, j\neq 0,1,
\end{array}
$$
and starting at $(0,i)$. This allows us to deduce that
\begin{eqnarray}
\lefteqn{d_{TV}\{{\mathcal L}(Z_{n}(U) \mid Z_{n}(0)=i),{\mathcal L}(Z_{n}(U)+1 \mid Z_{n}(0)=i)\}}\nonumber\\
&=& \frac{1}{2}\sum_{k \in {\mathbb Z}}|{\mathbb P}(Z_{n}(U)=k \mid Z_{n}(0)=i)
-{\mathbb P}(Z_{n}(U)=k-1 \mid Z_{n}(0)=i)| \nonumber\\
&=& \frac{1}{2}\sum_{k \in {\mathbb Z}}\left|\sum_{l\geq 0}{\mathbb P}(N_{n}(U)=l)
{\mathbb P}(X_{n}(U)=k-l \mid N_{n}(U)=l, X_{n}(0)=i) \right. \nonumber\\
&&\ \mbox{}\left. - \sum_{l\geq 1}{\mathbb P}(N_{n}(U)=l-1)
{\mathbb P}(X_{n}(U)=k-l \mid N_{n}(U)=l-1, X_{n}(0)=i) \right| \nonumber\\
&\le& \frac{1}{2}\sum_{k \in {\mathbb Z}}\sum_{l \geq 0}
|{\mathbb P}(N_{n}(U)=l)-{\mathbb P}(N_{n}(U)=l-1)|f^{U}_{l,i}(k-l) \nonumber\\
&& \ \mbox{}+\frac{1}{2}\sum_{k \in {\mathbb Z}}\sum_{l \geq 1}{\mathbb P}(N_{n}(U)=l-1)
|f^{U}_{l,i}(k-l)-f^U_{l-1,i}(k-l)|, \label{Poisson-part}
\end{eqnarray}
where
\begin{equation}\label{dens1}
f^{U}_{l,i}(m):={\mathbb P}(X_{n}(U)=m \mid N_{n}(U)=l, X_{n}(0)=i).
\end{equation}
Since, from Barbour, Holst and Janson~(1992, Theorem~1.C),
\begin{equation}\label{BHJ}
\sum_{l \geq 0}|{\mathbb P}(N_{n}(U)=l)-{\mathbb P}(N_{n}(U)=l-1)|
\ \leq\ \frac{1}{\sqrt{n\lambda^0U}} \ =\ O\Big(\frac{1}{\sqrt{n}}\Big),
\end{equation}
the first term in~\Ref{Poisson-part} is bounded by $1/\{\sqrt{n\lambda^0U}\}$,
yielding a contribution of the same size to~$D_{1n}(\delta')$ in~\Ref{bound-1}, and
it remains only to control the differences between the conditional probabilities
$f^{U}_{l,i}(m)$ and~$f^{U}_{l-1,i}(m)$.
To make the comparison between $f^{U}_{l,i}(m)$ and~$f^{U}_{l-1,i}(m)$, we first
condition on the whole Poisson paths of~$N_n$ leading to the events $\{N_{n}(U)=l\}$
and $\{N_{n}(U)=l-1\}$, respectively, chosen to be suitably matched; we write
\begin{eqnarray}
f^{U}_{l,i}(m) &=& \frac 1{U^l} \int_{[0,U]^l}ds_1\,\ldots\,ds_{l-1}\,ds^* \nonumber\\
&&\hspace{0.21in} {\mathbb P}(X_{n}(U)=m \mid N_n[0,U]=\nu^l(\cdot\;;s_1,\ldots,s_{l-1}, s^*), X_{n}(0)=i);\nonumber\\
f^U_{l-1,i}(m) &=& \frac 1 {U^l}\int_{[0,U]^{l}}ds_1...ds_{l-1}ds^* \nonumber \\
&& \quad {\mathbb P}(X_{n}(U)=m \mid N_n[0,U]=\nu^{l-1}(\cdot\;;s_1,\ldots,s_{l-1}), X_{n}(0)=i) ,
\label{dens2}
\end{eqnarray}
where
$$
\nu^r(u;t_1, \ldots, t_{r})\ :=\ \sum_{i=1}^{r} 1 \hspace*{-1mm} {\rm l}_{[0,u]}(t_i),
$$
and $Y[0,u]$ is used to denote $(Y(s),\,0\le s\le u)$. Fixing $s_1,s_2,\ldots,s_{l-1}$,
let $\bP_{i,s^*}$ denote the distribution of~$X_n$ conditional on
$N_n[0,U] = \nu^l(\cdot\;;s_1,\ldots,s_{l-1}, s^*)$ and $X_{n}(0)=i$, and let~${\mathbb P}_i$
denote that conditional on
$N_n[0,U] = \nu^{l-1}(\cdot\;;s_1,\ldots,s_{l-1})$ and $X_{n}(0)=i$; let
$\rho_{s^*}(u,x)$ denote the Radon--Nikodym derivative $d\bP_{i,s^*}/d{\mathbb P}_i$ evaulated at the
path $x[0,u]$. Then
$$
\bP_{i,s^*}[X_n(U)=m] \ =\ \int_{\{x[0,U]\colon x(U)=m\}} \rho_{s^*}(U,x)\,d{\mathbb P}_i(x[0,U]),
$$
and hence
\begin{equation}\label{pm-diff}
\bP_{i,s^*}[X_n(U)=m] - {\mathbb P}_i[X_n(U)=m] \ =\ \int 1 \hspace*{-1mm} {\rm l}_{\{m\}}(x(U))\{\rho_{s^*}(U,x)-1\}\,d{\mathbb P}_i(x[0,U]).
\end{equation}
Thus
\begin{eqnarray}
\lefteqn{\sum_{m\in{\mathbb Z}}|f^U_{l,i}(m) - f^U_{l-1,i}(m)|}\nonumber\\
&\le& \frac 1{U^l} \int_{[0,U]^l}ds_1\,\ldots\,ds_{l-1}\,ds^* \sum_{m\in{\mathbb Z}}
{\mathbb E}_i\left\{1 \hspace*{-1mm} {\rm l}_{\{m\}}(X_n(U))|\rho_{s^*}(U,X_n)-1|\right\} \nonumber\\
&\le& \frac 2{U^l} \int_{[0,U]^l}ds_1\,\ldots\,ds_{l-1}\,ds^*\,
{\mathbb E}_i\left\{[1 - \rho_{s^*}(U,X_n)]_+\right\}. \label{RN-bound}
\end{eqnarray}
To evaluate the expectation, note that $\rho_{s^*}(u,X_n)$, $u\ge0$, is a ${\mathbb P}_i$-martingale
with expectation~$1$.
Now, if the path $x[0,U]$ has~$r$ jumps at times $t_1 < \cdots < t_r$, writing
$$
y(v) \ :=\ x(v) + \nu^{l-1}(v\;;s_1,\ldots,s_{l-1}),\quad y_k \ :=\ y(t_k),\quad
j_k \ :=\ y_k - y_{k-1},
$$
we have
$$
\rho_{s^*}(u,x) \ =\
\begin{cases}
1
&\mbox{if}\quad u < s^*; \\
\exp\left( -n\int_{s^*}^u \{{\hat \l}(y({v})+n^{-1}) - {\hat \l}(y({v}))\}\,{dv}\right)\\
\qquad\prod_{\{k\colon s^* \le t_k \le u\}}
\left\{{\hat \l}_{j_k}(y_{k-1} + n^{-1}) / {\hat \l}_{j_k}(y_{k-1})\right\}
&\mbox{if}\quad u \ge s^*,
\end{cases}
$$
where ${\hat \l}_j(\cdot) = \lambda_j(\cdot)$ if $j\ne1$ and ${\hat \l}_1(\cdot) = \lambda_1(\cdot) - \lambda^0$,
and where ${\hat \l}(\cdot) := \sum_{j\in\bZ\setminus\{0\}} {\hat \l}_j(\cdot)$.
Thus, in particular, $\rho_{s^*}(u,x)$ is
absolutely continuous except for jumps at the times $t_k$. Then also, from
Assumptions A3\,(a) and~A4,
$$
\left|\frac{\lambda_j(y+n^{-1})}{\lambda_j(y)} - 1 \right| \ \le\ \frac{\|\lambda'_j\|_\delta}{n\varepsilon\lambda_j(c)}
\ \le\ |j|L_1/\{n\varepsilon\},
$$
uniformly in $|y-c| \le \delta$, for each $j\in J$. Hence it follows that, if we define
the stopping times
\begin{eqnarray}
\tau_\delta &:=& \inf\{u\ge0: |X_n(u) + \nu^{l-1}(u\;;s_1,\ldots,s_{l-1}) - nc| > n\delta\};\nonumber\\
\phi &:=& \inf\{u\ge0: \rho_{s^*}(u,X_n) \ge 2\}, \label{stopping}
\end{eqnarray}
then the expected quadratic variation of the martingale~$\rho_{s^*}(u,X_n)$ up to the time
$\min\{U,\tau_\delta,\phi\}$ is at most
\begin{equation}\label{QV}
4U\sum_{j \in {\mathbb Z}\setminus\{0\}} \left(\frac{|j|L_1}{n\varepsilon}\right)^2\,nc_j(1+\delta)
\ =:\ n^{-1}K(\delta,\varepsilon) U,
\end{equation}
where $K(\delta,\varepsilon) < \infty$ by Assumption~A2\,(a).
Clearly, from~\Ref{QV} and from Kolmogorov's inequality,
$$
{\mathbb P}_i[\phi < \min\{U,\tau_\delta\}] \ \le\ K(\delta,\varepsilon)U/n.
$$
Hence, again from~\Ref{QV},
$$
{\mathbb E}_i\left\{[1 - \rho_{s^*}(U,X_n)]_+\right\} \ \le\ n^{-1/2}\sqrt{K(\delta,\varepsilon) U} + n^{-1}K(\delta,\varepsilon)U
+ {\mathbb P}_i[\tau_\delta < U].
$$
Substituting this into~\Ref{RN-bound}, it follows that
\begin{eqnarray*}
\lefteqn{\sum_{l \geq 1}{\mathbb P}(N_{n}(U)=l-1)\sum_{m\in{\mathbb Z}}|f^U_{l,i}(m) - f^U_{l-1,i}(m)|}\\
&& \ \le\ 2\Bigl\{n^{-1/2}\sqrt{K(\delta,\varepsilon) U} + n^{-1}K(\delta,\varepsilon)U \\
&&\qquad\qquad\mbox{} + {\mathbb P}[\sup_{0 \le u \le U}|Z_n(u) - nc| > n\delta \mid Z_n(0) = i]\Bigr\}.
\end{eqnarray*}
But now, for all~$i$ such that $|i - nc| \le n\delta' = n\delta e^{-K_1U}/2$, the latter probability
is of order $O(n^{-1})$, by Lemma~\ref{lema3}, and hence the final term in~\Ref{Poisson-part}
is also of order $O(n^{-1/2})$, as required.
\end{proof}
\ignore{
We now need to compare ${\mathcal L}(Z_{n}(U) \mid Z_{n}(0)=z)$ with ${\mathcal L}(Z_{n}(U)+1 \mid
Z_{n}(0)=z)$ for any given and fixed $U,$ where $Z_n$ is the Markov process with generator ${\mathcal
A}_n.$ As in R\"ollin~(2005), a good way of doing this is to find a random variable $N$ embedded in
$Z_n(U)$ for which ${\mathcal L}(N)$ and ${\mathcal L}(N+1)$ are close enough, and to exploit this. Here,
we use a Poisson process with jumps of size $+1$ at rate $n\lambda^0$ to provide our $N.$ We thus need to
split $Z_n$ into a sum of two processes, one of which is this Poisson process. The construction of
the appropriate bivariate process $(X_n, N_n)$ is done at {\small \bf Step 2}.
Using the newly defined processes, we are able to bound $d_{TV}\{\Pi_{n},\Pi_{n}*\delta_1\}$
by the sum of two terms, see {\small \bf Step 3}. The first one of these we can further bound by the
total-variation distance between a Poisson distribution and its unit translation. This distance, by
Barbour, Holst and Janson~(1992, Theorem~1.C), can be bounded by $O(1/\sqrt{n}),$
see {\small \bf Step 4}.
The last step, {\small \bf Step 5}, is concerned with finding a bound of the same size on the second
term in the sum from {\small \bf Step 3}. Here, the essence of the argument is to show that the
conditional distribution of the paths of $X_n$ on $[0,U],$ given the path of the Poisson
process $\{N_n(t), \ 0\leq t\leq U\},$ changes only little in total variation if an extra jump
is added to $N_n:$ that is to say, ${\mathcal L}(X_{n}(U) \mid N_n(t)=n(t), 0\leq t\leq U; X_{n}(0)=z)$
and ${\mathcal L}(X_{n}(U) \mid N_n(t)=n(t)+1 \hspace*{-1mm} {\rm l}_{[s^*,\infty)}(t), 0\leq t\leq U; X_{n}(0)=z),$
where $s^*$ denotes the time of the extra jump, are $O(1/\sqrt{n})$ close. The detailed argument
requires several sub-steps, that we shall explain in due time.\\
}
As a consequence of this theorem, we have the following corollary.
\begin{corollary} \label{lema2} Under Assumptions A1--A4, for any bounded function~$f$,
$$
{\mathbb E}_{\Pi_n}\{\bigtriangledown f(Z_n)\} \ =\ O\Big(\frac{1}{\sqrt{n}}\,\|f\|\Big).
$$
\end{corollary}
\begin{proof} Immediate, because
$$
|{\mathbb E}_{\Pi_n}\{\bigtriangledown f(Z_n)\}| \ \le\ 2\|f\|\,d_{TV}(\Pi_n,\Pi_n*\delta_1).
$$
\end{proof}
\setcounter{equation}{0}
\section{Translated Poisson approximation to the equilibrium distribution}
\label{main}
We are now able to prove our main theorem. The centred equilibrium distribution of~$Z_n$
is $\widehat{\Pi}_n:=\Pi_n*\delta_{-\lfloor nc \rfloor}$, and we approximate it by a centred
Poisson distribution with similar variance.
\begin{theorem}\label{main-thm}
Under Assumptions A1--A5,
$$
d_{TV}(\widehat{\rm Po}(nv_c),\widehat{\Pi}_n) \ =\ O(n^{-\alpha/2}),
$$
where $v_c := \sigma^2(c)/\{-2F'(c)\}$.
\end{theorem}
\ignore{
For the example in the Preliminaries Section, we have $c=\frac{a}{d-\sum_{j\geq 1}jb_j},$
$-2F^{\prime}(c)=2(d-\sum_{j\geq 1}jb_j)$ and $\sigma^2(c)=a+\frac{a}{d-\sum_{j\geq 1}jb_j}(d+\sum_{j\geq 1}j^2b_j),$ so that the shifted equilibrium distribution $$\Pi_n-\Big\lfloor \frac{na}{d-\sum_{j\geq 1}jb_j}\Big\rfloor$$
can be approximated in total variation by the shifted Poisson distribution
$$\widehat{\rm Po}\Big(\frac{na(2d+\sum_{j\geq 1}j(j-1)b_j)}{2(d-\sum_{j\geq 1}jb_j)^2}\Big),$$
to order $O(1/\sqrt{n}).$
Note that if $b_j=0,$ for any $j\geq 1,$ than the process becomes a simple death with immigration process, whose equilibrium distribution is precisely the Poisson distribution
${\rm Po}\big(\frac{na}{d}\big)={\rm Po}(nc).$ In this case, the approximation is in fact exact.
}
\begin{proof}
We follow the recipe outlined in Section~\ref{prelims}.
From~\Ref{SEt-2}, we principally need to show that
$$
\sup_{g\in{\mathcal G}_{v}}
|{\mathbb E}\{{v}\;\bigtriangledown{g}(W+1)-W{g}(W) +\langle v \rangle{g}(W)\}| \ =\ O(n^{-\alpha/2}),
$$
for $W := Z_n - \lfloor nc \rfloor$, $v := nv_c$ and~${\mathbb E} := {\mathbb E}_{\Pi_n}$.
So, for any $g\in{\mathcal G}_{nv_c}$, write ${\tilde g}(i) := g(i - \lfloor nc \rfloor)$, and set
$$
h \ :=\ h_{n,g}(i)\ :=\
\begin{cases}
0, &\mbox{if}\quad i \le \lfloor nc \rfloor - \lfloor {nv_c} \rfloor;\\
\sum_{l = \lfloor nc \rfloor - \lfloor {nv_c} \rfloor}^{i - 1}{\tilde g}(l)
&\mbox{if}\quad i > \lfloor nc \rfloor - \lfloor {nv_c} \rfloor.
\end{cases}
$$
Note that, for $j\ge1$, by Assumption A2\,(a),
\begin{eqnarray*}
n\lambda_j(i/n)|h(i+j)-h(i)| &\le& njc_j\|{\tilde g}\| + c_j|i-\lfloor nc \rfloor|
\sum_{k=1}^j |g(i+j-k-\lfloor nc \rfloor)|\\
&\le& njc_j\|g\| + jc_j \sup_l|lg(l)| + c_j\sum_{k=1}^j |j-k|\|g\|,
\end{eqnarray*}
and that a similar bound, with $|j|$ replacing~$j$, is valid for $j\le -1$.
From the definition of~${\mathcal G}_{nv_c}$ in \Ref{SEt} and~\Ref{iSEt} and from Assumption~A2\,(a),
it thus follows
that $(|{\mathcal A}_n|h_{n,g})$ is a bounded function, and hence that
the function~$h_{n,g}$ satisfies condition~\Ref{HK-1}; furthermore, since
$|h_{n,g}(i)| \le |i - \lfloor nc \rfloor + \lfloor {nv_c} \rfloor|$, in view of~\Ref{iSEt},
$h_{n,g}$ is integrable with respect to~$\Pi_n$, because of Theorem~\ref{exubj}.
Hence it satisfies the conditions of Theorem~\ref{HK95}, from which we deduce, as
in~\Ref{DF}, that
$$
{\mathbb E}_{\Pi_n}({\mathcal A}_n h_{n,g})(Z_n) \ =\ 0.
$$
Applying Lemma~\ref{generator}, since~$h_{n,g}$ has bounded differences in view of~\Ref{iSEt},
it follows that
\begin{eqnarray}
0 &=& {\mathbb E}_{\Pi_n}\left\{\frac{n}{2}{\sigma}^2\Big(\frac{Z_n}{n}\Big)\bigtriangledown{{\tilde g}}(Z_n)
+nF\Big(\frac{Z_n}{n}\Big){\tilde g}(Z_n) + E_n({\tilde g},Z_n)\right\}\nonumber \\
&=& -F'(c){\mathbb E}_{\Pi_n}\left\{ nv_c\bigtriangledown{{\tilde g}}(Z_n) -
(Z_n-\lfloor nc\rfloor){\tilde g}(Z_n)
+\langle nv_c \rangle{{\tilde g}}(Z_n)\right\}\nonumber \\
&&\quad\mbox{} + {\mathbb E}_{\Pi_n}\{ E'_n({\tilde g},Z_n) + E_n({\tilde g},Z_n)\}, \label{DE-1}
\end{eqnarray}
where $E_n$ is as defined in~\Ref{En-def}, and
\begin{eqnarray*}
E'_n(g,i) &:=& \frac{n}{2}({\sigma}^2(i/n) - \sigma^2(c))\bigtriangledown{g}(i)\\
&&\quad\mbox{} + \{n(F(i/n) - F(c)) - F'(c)(i-\lfloor nc\rfloor)\}g(i)
+ F'(c) \langle nv_c \rangle{g}(i).
\end{eqnarray*}
The terms involving $E'_n({\tilde g},i)$ can be bounded, using~\Ref{iSEt}, as follows. First,
using Assumptions A2\,(a) and~A4,
\begin{eqnarray}
\lefteqn{ \frac{n}{2}|{\sigma}^2(i/n) - \sigma^2(c)|\,|\bigtriangledown{{\tilde g}}(i)|}\nonumber\\
&&\ \le\ \frac{1}{2nv_c}\|(\sigma^2)^{\prime}\|_{\delta} |i-nc|I[|i-nc| \le n\delta] \nonumber\\
&&\mbox{}\qquad + \frac{1}{2v_c}\Bigl(\sum_{j\in\bZ\setminus\{0\}} j^2c_j(1 + |i/n-c|) + \sigma^2(c)\Bigr)I[|i-nc| > n\delta];
\label{e-dash-1}
\end{eqnarray}
and then, under Assumptions A2\,(a) and~A5,
\begin{eqnarray}
\lefteqn{|n(F(i/n) - F(c)) - F'(c)(i-\lfloor nc\rfloor) + F'(c) \langle nv_c \rangle|
\,|{\tilde g}(i)|}\nonumber\\
&&\ =\ n|F(i/n) - F(c) - (i/n-c)F'(c)|\,|{\tilde g}(i)| \nonumber\\
&&\ \le\ \Bigl(\frac n2 (i/n-c)^2 I[|i/n-c| \le \delta] \sup_{|z-c|\le\delta}|F''(z)| \nonumber\\
&&\mbox{}\qquad + n\Bigl\{(1+|i/n-c|)\sum_{j\in\bZ\setminus\{0\}} |j|c_j + F'(c)|i/n-c|\Bigr\}I[|i-nc| > \delta]\Bigr)\frac1{\sqrt{nv_c}}.
\label{e-dash-2}
\end{eqnarray}
The contribution to~\Ref{DE-1} from ${\mathbb E}_{\Pi_n}\{ E'_n({\tilde g},Z_n)\}$ is thus of order
\begin{eqnarray}
\lefteqn{{\mathbb E}_{\Pi_n}\{|z_n-c| + (1 + |z_n-c|)I[|z_n-c| > \delta] + |z_n-c|^2 I[|z_n-c| \le \delta]\}}\nonumber\\
&&\ =\ O(n^{-1/2}), \hspace{3.5in}\label{Order-1}
\end{eqnarray}
by Theorem~\ref{exubj} and Corollaries \ref{prop1} and~\ref{prob-n1}.
The first term in $E_n({\tilde g},i)$ is also bounded in similar fashion:
from Assumptions A1, A2\,(a) and~A4,
\begin{eqnarray}
\lefteqn{\frac{n}{2}|F(i/n)|\,|\bigtriangledown{{\tilde g}}(i)|} \nonumber\\
&&\ \le\ \frac1{2nv_c} \{\|F'\|_\delta|i-nc| + \sum_{j\in\bZ\setminus\{0\}} c_j|j|(1 + |i-nc|)I[|i-nc| > \delta]\}.
\label{e-1}
\end{eqnarray}
giving a contribution to ${\mathbb E}_{\Pi_n}\{ E_n({\tilde g},Z_n)\}$ of the same order.
The remaining terms, involving $\bigtriangledown^2{{\tilde g}}$, need to be treated more
carefully.
We examine the first of them in detail, with the treatment of the second being
entirely similar. First, if either $|i/n-c| > \delta$ or $j > \sqrt n$, it is enough to use
the expression in~\Ref{aj-bnd-1} to give
\begin{eqnarray}
|a_j({\tilde g},i)|
&\le& j(j-1)\|\bigtriangledown{{\tilde g}}\| \ \le\ j(j-1)/(nv_c). \label{aj-part1}
\end{eqnarray}
For $|i/n-c| > \delta$, by Assumption~A2\,(a), this yields the estimate
\begin{eqnarray}
\lefteqn{\left|\sum_{j\ge2}a_j({\tilde g},i) n\lambda_j(i/n)\right|I[|i-nc| > \delta]}\nonumber\\
&\le& \sum_{j\ge2} \frac{j(j-1)c_j}{v_c}(1 + |i/n-c|)I[|i-nc| > \delta], \label{e-2}
\end{eqnarray}
with corresponding contribution to ${\mathbb E}_{\Pi_n}\{ E_n({\tilde g},Z_n)\}$ being of order $O(n^{-1})$,
by Theorem~\ref{exubj} and Corollary~\ref{prob-n1}.
Then, for $j > \sqrt n$ and $|i/n-c| \le \delta$, \Ref{aj-part1} yields
\begin{eqnarray}
\lefteqn{\left|\sum_{j>\sqrt n} a_j({\tilde g},i) n\lambda_j(i/n)\right|}\nonumber\\
&\le& \sum_{j>\sqrt n} \frac{j(j-1)c_j}{v_c}(1 + \delta)
\ \le\ \sum_{j\ge1}j^{2+\alpha}c_j n^{-\alpha/2}(1+\delta)/v_c,
\label{e-3}
\end{eqnarray}
making a contribution of order $O(n^{-\alpha/2})$ to ${\mathbb E}_{\Pi_n}\{ E_n({\tilde g},Z_n)\}$,
again using Assumption~A2\,(a).
In the remaining case, in which $j \le \sqrt n$ and $|i/n-c| \le \delta$, we use
\Ref{aj-bnd-2}, observing first that
\begin{eqnarray}
\lefteqn{n\bigtriangledown^2{{\tilde g}}(i+j-k+1) \lambda_j(i/n)} \nonumber\\
&&=\ n\bigtriangledown^2{{\tilde g}}(i+j-k+1) \lambda_j(c) +
n\bigtriangledown^2{{\tilde g}}(i+j-k+1) (\lambda_j(i/n) - \lambda_j(c)), \label{e-4}
\end{eqnarray}
the latter expression being bounded by
\begin{eqnarray}
|n\bigtriangledown^2{{\tilde g}}(i+j-k+1) (\lambda_j(i/n) - \lambda_j(c))|
&\le& \frac2{v_c}\|\lambda_j'\|_\delta \,|i/n-c|. \label{e-5}
\end{eqnarray}
The corresponding contribution to ${\mathbb E}_{\Pi_n}\{ E_n({\tilde g},Z_n)\}$ is thus at most
\begin{eqnarray}
&&\sum_{j=2}^{\lfloor\sqrt n\rfloor} (j^3/6)
\{\lambda_j(c) n\sup_l|{\mathbb E}_{\Pi_n}\bigtriangledown^2{{\tilde g}}(Z_n+l)|
+ 2 v_c^{-1}\|\lambda_j'\|_\delta \,{\mathbb E}_{\Pi_n}|z_n-c| \}\nonumber\\
&&\quad\ \le\ n^{(1-\alpha)/2} \sum_{j\ge2} j^{2+\alpha}c_j
\{n \sup_l|{\mathbb E}_{\Pi_n}\bigtriangledown^2{{\tilde g}}(Z_n+l)|
+ L_1 2 v_c^{-1}{\mathbb E}_{\Pi_n}|z_n-c| \}\nonumber\\
&&\quad\ =\ {n^{(1-\alpha)/2}\; O\bigl(n\cdot n^{-3/2}+n^{-1/2} \bigr) \ =\ O(n^{-\alpha/2})} , \label{e-6}
\end{eqnarray}
where we have used Assumptions A2\,(a) and~A4, and then Corollaries \ref{prop1} and~\ref{lema2},
and finally~\Ref{iSEt}.
Combining the bounds, and substituting them into~\Ref{DE-1}, it follows that
$$
|{\mathbb E}_{\Pi_n}\left\{ nv_c\bigtriangledown{g}(Z_n-\lfloor nc \rfloor) -
(Z_n-\lfloor nc\rfloor)g(Z_n-\lfloor nc \rfloor)
+\langle nv_c \rangle{g}(Z_n-\lfloor nc \rfloor)\right\}|
\ =\ O(n^{-\alpha/2}),
$$
uniformly in $g\in{\mathcal G}_{nv_c}$. Again from Corollary~\ref{lema2}, we also have
$$
|nv_c {\mathbb E}_{\Pi_n}\left\{ \bigtriangledown{g}(Z_n-\lfloor nc \rfloor) - \bigtriangledown{g}(Z_n-\lfloor nc \rfloor+1)
\right\}| \ =\ O(n^{-1/2}),
$$
for any $g\in{\mathcal G}_{nv_c}$. It thus follows from~\Ref{SEt-2} that
$$
d_{TV}(\widehat{\rm Po}(nv_c),\widehat{\Pi}_n) \ =\
O\left(n^{-\alpha/2} + {\mathbb P}_{\Pi_n}[Z_n - nc < -\lfloor nv_c\rfloor]\right),
$$
and the latter probability is of order~$O(n^{-1})$ by Corollary~\ref{prob-n1}.
This completes the proof.
\end{proof}
\medskip
\noindent{\bf Example.}\ Consider an immigration birth and death process~$Z$, with births occurring in groups
of more than one individual at a time. The process has transition rates as in Section~\ref{prelims},
with
$$
\lambda_{-1}(z):=dz, \ \ \ \lambda_{1}(z):=a+bq_1z \ \ \mbox{and} \ \ \lambda_j(z):=bq_jz, \ j\geq 2,
$$
while $\lambda_j(z):=0$, $j<-1$. Here, $b$ denotes the rate at which birth events occur,
and $a>0$ represents the immigration rate.
The quantity~$q_j$ denotes the probability that~$j$ offspring are born at a birth event, so
that $\sum_{j\ge1} q_j = 1$; we write $m_r := \sum_{j\ge1} j^rq_j$ for the $r$'th moment of
this distribution.
Then
$$
F(z)=a+z(bm_1-d), \quad\mbox{and}\quad \sigma^2(z)=a+z(bm_2+d).
$$
Assumption~A1 is satisfied if $d>bm_1$, with
$c=a/(d-bm_1)$ and $F'(c) = -(d-bm_1)$. Assumption~A2\,(a) is satisfied with
$c_j = bq_j \max\{1,c\}$, $j\ge2$, $c_1=\max\{bq_1, a+bq_1c\}$, and $c_{-1}=d \max\{1,c\}$,
provided that $m_{2+\alpha} < \infty$ for some $0 < \alpha \le 1$; for Assumption~A2\,(b), simply
take $\lambda^0=a/2$. The other assumptions are immediate.
The quantity $v_c$ appearing in Theorem~\ref{main-thm} then comes out to be
\[
v_c \ :=\ \frac{a(2d+b(m_2-m_1))}{2(d-bm_1)^2}\,,
\]
and the approximation to the equilibrium distribution of $Z_n - \lfloor nc \rfloor$
is the centred Poisson distribution $\widehat{\rm Po}(nv_c)$, accurate in total
variation to order $O\bigl(n^{-\alpha/2}\bigr)$.
Note that, if $b=0$, then the process becomes a simple immigration death
process, whose equilibrium distribution is precisely the Poisson distribution
${\rm Po}\big(na/d\big)={\rm Po}(nc)$. In this special case, the approximation is in fact exact.
\ignore{
\section*{Acknowledgment}
The author would like to thank her Ph.D. advisor A.~D.~Barbour for suggesting the topic
and for many helpful discussions.
}
| {
"timestamp": "2009-02-05T12:23:06",
"yymm": "0902",
"arxiv_id": "0902.0884",
"language": "en",
"url": "https://arxiv.org/abs/0902.0884",
"abstract": "The paper is concerned with the equilibrium distributions of continuous-time density dependent Markov processes on the integers. These distributions are known typically to be approximately normal, and the approximation error, as measured in Kolmogorov distance, is of the smallest order that is compatible with their having integer support. Here, an approximation in the much stronger total variation norm is established, without any loss in the asymptotic order of accuracy; the approximating distribution is a translated Poisson distribution having the same variance and (almost) the same mean. Our arguments are based on the Stein-Chen method and Dynkin's formula.",
"subjects": "Probability (math.PR)",
"title": "Translated Poisson approximation to equilibrium distributions of Markov population processes",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808695883037,
"lm_q2_score": 0.7217432062975979,
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https://arxiv.org/abs/1901.04219 | Ultraspherical moments on a set of disjoint intervals | Moment evaluations are important for the study of non-classical orthogonal polynomial systems for which explicit representations are not known. In this paper we compute, in terms of the hypergeometric function, the moments associated with a generalized ultraspherical weight on a collection of intervals with two symmetric gaps. These moments, parametrized by the endpoints of the gaps, are identified as a one parameter deformation between the full range ultraspherical moments and the half range ultraspherical moments. | \section{Introduction}
\noindent
The full range Ultraspherical moments are defined by the following integral over the interval $[-1,1]$
\begin{equation} \label{intclassic}
\mu_n := \int_{-1}^1 x^n(1-x^2)^{\mu-1/2} \; dx \qquad n = 0,1,2,3,.. \qquad Re(\mu) > -1/2.
\end{equation}
The moments have the following representation in terms of the Gamma function
\begin{equation} \label{intclassiceval}
\begin{split}
\mu_{2n} &= \frac{\Gamma(n+1/2)\Gamma(\mu+1/2)}{\Gamma(n+\mu+1)} \\
\mu_{2n+1} &= 0
\end{split}
\end{equation}
The half range Ultraspherical moments are defined over the interval $[0,1]$ as follows
\begin{equation} \label{halfeval}
\mu^H_{n}:= \int_0^1 x^n(1-x^2)^{\mu-1/2} \; dx \qquad n = 0,1,2,3,.. \qquad Re(\mu) > -1/2
\end{equation}
and have the following representation in terms of the Gamma function
\begin{equation} \label{halfeval2}
\mu^H_{n} = \frac{1}{2} \frac{\Gamma(n/2+1/2)\Gamma(\mu+1/2)}{\Gamma(n/2+\mu+1)}
\end{equation}
In this paper we construct a generalization of the classical ultraspherical moment integral (\ref{intclassic}) in terms of an integral over a set of disjoint intervals with two gaps of the same length. The particular configuration of disjoint intervals that we consider can be parametrized by one-parameter and we will show that the moments defined in this paper form a one-parameter deformation between the classical, full range, ultraspherical moments on the interval $[-1,1]$, and the half-range ultrapsherical moments on the interval $[0,1]$.
\vskip 0.2cm
\noindent
The construction we use follows an analagous construction to that used in \cite{Griffin-09} and \cite{Griffin-11} where one gap was considered, and is founded on the relationship between a collection of disjoint intervals and a diophantine polynomial equation known as a generalized Pell equation. The generalized ultraspherical moments form a one parameter generalization of the classical case and we show that the moments can be explicitly evaluated in terms of the hypergoemteric function and this parameter.
\vskip 0.2cm
\noindent
Explicit evaluation of moments such as these is important in general for further study of the associated orthogonal polynomials and related quantities, where one does not have an explicit representation for the polynomials or the recurrence coefficients in terms of special functions. See for example, \cite{Huybrechs} and \cite{Orel-Perne} where they make use of the half-range Chebyshev moments in a numerical study applied to the efficient computation of Fourier approximations. See also \cite{MRR} where computations performed with half range Legendre moments are applied to radiative transfer in spherical geometry.
\vskip 0.1cm
\noindent
This paper is organized as follows. In section $2$ we give a brief summary of the construction of the 1-gap case, studied in \cite{Griffin-09} and \cite{Griffin-11}, together with the explicit representation of the moments. Before moving to the two gap case we evaluate the odd moments again using a different method which is based on the polynomial mapping technique first outlined in \cite{GVA}. This method results in a different representation which involves the classical hypergeometric function. We finish section $2$ by describing in detail the construction of the two gap case. In section $3$ we compute the moment integral in the two gap case explicitly in terms of the classical hypergeometric function, making use of the revised method in section $2$. Finally we relate the limiting case, as the gaps become as large as possible, to the half range ultraspherical moments on the interval $[0,1]$.
\noindent
\section{Construction of the moment integral}
\noindent
\subsection{Revisiting the one gap case}
In this section we briefly review the construction of the moment integral for the case of one gap and use this opportunity to express the odd moments from \cite{Griffin-11} in a more convenient form for the purpose of this paper.
\vskip 0.2cm
\noindent
The Chebyshev polynomials of the first and second kind, which are orthogonal on the set $[-1,1]$, are well known to have parametric representations in terms of trigonometric functions. These representations derive mainly from the Polynomial Pell equation which is
\begin{equation} \label{Pell}
P_n^2(x) - (x^2-1)Q_n^2(x) = 1.
\end{equation}
If $P$ and $Q$ are polynomials then it can be shown that the only solutions to the above equation are the Chebyshev polynomials of the first and second kind, $T_n$ and $U_n$ respectively. See \cite{Nathanson} and associated references for more details on the connection between the Pell equation and the Chebyshev polynomials. Akhiezer's treatise on elliptic functions, \cite{Akhiezer}, see also \cite{Akhiezer-2}, introduced a generalization of the Chebyshev polynomials that were associated with the following modified Pell equation
\begin{equation}
(x-\alpha)P_n^2(x) - (x^2-1)(x-\beta)Q_n^2(x) = (x-\gamma_n).
\end{equation}
Akhiezer showed that the only polynomial solutions of this equation are polynomials orthogonal on the interval with one gap :
\begin{equation}
E := [-1,\alpha] \cup [\beta,1] \qquad -1 < \alpha \leq \beta < 1
\end{equation}
with respect to the weight function
\begin{equation}
w_E(x) = \sqrt{\frac{x-\alpha}{(1-x^2)(x-\beta)}}.
\end{equation}
Such polynomials have an analogous parametric representation in terms of elliptic functions. In \cite{ChLa} the technique of Akhiezer was generalized in order to obtain an explicit representation for generalized polynomials orthogonal on an interval with any number of gaps in terms of abelian functions.
\vskip 0.1cm
\noindent
In \cite{GVA}, a theory of orthogonal polynomials on several intervals with a general weight was developed. Under this theory the gaps could not be arbitrarily chosen, instead they had to be chosen in such a way that the several intervals were each a preimage of $[-1,1]$ under a polynomial mapping of some degree. For example, the set
\begin{equation}
E_2 = [-1,-b] \cup [b,1] \qquad b \in [0,1)
\end{equation}
can be mapped onto the set $[-1,1]$ using a polynomial of degree $2$. So, generalized Chebyshev polynomials on the set $E_2$ can be studied from the point of view of Akhiezer or from the point of view of the theory of polynomial mappings. In \cite{Griffin-09}, and later in \cite{Griffin-11}, a further connection between the Pell equation and the set $E_2$ was explored. Specifically, the Pell equation on $E_2$
\begin{equation} \label{PellE2}
(x+b)P_n^2(x) - (x^2-1)(x-b)Q_n^2(x) = (x- (-1)^{n+1}b),
\end{equation}
was normalized to obtain
\begin{equation}
\frac{x+b}{(x- (-1)^{n+1}b)}P_n^2(x) - \frac{(x^2-1)(x-b)}{(x- (-1)^{n+1}b)}Q_n^2(x) = 1.
\end{equation}
Under this normalization the following quantity
\begin{equation}
\sqrt{\frac{x+b}{(x- (-1)^{n+1}b)}}P_n(x) \qquad x \in E_2
\end{equation}
is a function that is somewhat analogous to the Chebyshev polynomial of degree $n$ in the sense that it tends to $T_n(x)$ as $b \to 0$. In particular for $n=1$ we have
\begin{equation}
\sqrt{\frac{x+b}{x-b}}P_1(x) = \sqrt{\frac{x^2-b^2}{1-b^2}}\frac{x}{|x|} \qquad x \in E_2
\end{equation}
In \cite{Griffin-11} this function was denoted as follows
\begin{equation}
\cos \phi_{E_2} := \sqrt{\frac{x^2-b^2}{1-b^2}}\frac{x}{|x|} \qquad x \in E_2
\end{equation}
to draw the analogy with the classical Chebyshev case where
\begin{equation}
\cos \theta = x \qquad x \in [-1,1].
\end{equation}
In \cite{Griffin-09} and \cite{Griffin-11} the following moment integrals over the set $E_2$ were computed where $n=0,1,2,...$ :
\begin{equation} \label{1-gap-even}
\int_{E_2} (\cos \phi_{E_2})^{2n}(1-\cos^2 \phi_{E_2})^\mu \; w_{E_2}(x) \; dx = \frac{\Gamma(\frac{1}{2})\Gamma(\mu + \frac{1}{2})}{\Gamma(\mu+1)}\frac{\left(\frac{1}{2} \right)_n}{(\mu+ 1)_n} \qquad Re(\mu) > -\frac{1}{2}
\end{equation}
\begin{equation}
\begin{split} \label{1-gap-odd}
& \int_{E_2} (\cos \phi_{E_2})^{2n+1}(1-\cos^2 \phi_{E_2})^\mu w_{E_2}(x) \; dx \\
& \qquad \qquad \qquad =\frac{\Gamma(\frac{1}{2})}{\Gamma(n+1+\mu)} \sum_{k=1}^{\infty}(-1)^{k+1}\frac{\Gamma(n+1-\frac{k}{2})\Gamma(\mu+\frac{k}{2})}{\Gamma(1-\frac{k}{2})\Gamma(\frac{1}{2}+\frac{k}{2})}b^k \qquad Re(\mu) > -\frac{1}{2}
\end{split}
\end{equation}
In (\ref{1-gap-even}) we are using the standard notation
\begin{equation} \label{shiftfac}
(a)_n := a(a+1)...(a+n-1) \qquad (a)_0 := 1
\end{equation}
and in (\ref{1-gap-odd}) the term
\begin{equation*}
\frac{\Gamma(n+1-\frac{k}{2})}{\Gamma(1-\frac{k}{2})}
\end{equation*}
is to be understood as a polynomial and is therefore well defined for even values of $k$.
Note that as $b \to 0$ the equations (\ref{1-gap-even}) and (\ref{1-gap-odd}) revert to
\begin{equation}
\int_{-1}^1x^{2n}(1-x^2)^{\mu-1/2}\; dx = \frac{\Gamma(\frac{1}{2})\Gamma(\mu + \frac{1}{2})}{\Gamma(\mu+1)}\frac{\left(\frac{1}{2} \right)_n}{(\mu+ 1)_n} \qquad Re(\mu) > -\frac{1}{2}
\end{equation}
and
\begin{equation}
\int_{-1}^1x^{2n+1}(1-x^2)^{\mu-1/2}\; dx = 0 \qquad Re(\mu) > -\frac{1}{2}
\end{equation}
which are the classical ultraspherical moments, this time with the $n$-dependance written in terms of the shifted factorial (\ref{shiftfac}).
\vskip 0.2cm
\noindent
Before proceeding to the two gap case we now compute the odd moments in (\ref{1-gap-odd}) using the method of polynomial mappings to arrive at a representation of the odd moments in terms of the hypergeometric function. We will make use of the following facts that can be easily verified :
\begin{itemize}
\item For $n=2$, up to a change in sign, the solutions of (\ref{PellE2}) are
\begin{equation} \label{P2Q2}
P_2(x) = \frac{2}{1-b^2}x^2 - \frac{1+b^2}{1-b^2} \qquad \textrm{and} \qquad Q_2(x) = \frac{2}{1-b^2}x + \frac{2b}{1-b^2}
\end{equation}
\item
\begin{equation}
\{ x \in {\bf R} \; | \; P_2(x) \in [-1,1] \} = E_2.
\end{equation}
\item
\begin{equation}
\cos^2{\phi_{E_2}} = \frac{P_2+1}{2} \qquad \textrm{specifically} \qquad
\cos \phi_{E_2} = \left\{
\begin{array}{cc}
-\sqrt{\frac{P_2+1}{2}} & x \in [-1,-b] \\
\sqrt{\frac{P_2+1}{2}} & x \in [b,1]
\end{array}
\right.
\end{equation}
\item
\begin{equation}
w_{E_2}(x) = \frac{|Q_2(x)|}{\sqrt{1-P_2^2(x)}}
\end{equation}
\end{itemize}
Making use of the points $3$ and $4$ above, the left hand side of (\ref{1-gap-odd}) and be written as
\begin{equation}
\begin{split}
& -\int_{-1}^{-b} \left(\frac{P_2(x)+1}{2} \right)^{n+1/2} \left(\frac{1-P_2(x)}{2} \right)^\mu \frac{|Q_2(x)|}{\sqrt{1-P_2^2(x)}} dx \\
& \qquad + \int_{b}^{1} \left(\frac{P_2(x)+1}{2} \right)^{n+1/2} \left(\frac{1-P_2(x)}{2} \right)^\mu \frac{|Q_2(x)|}{\sqrt{1-P_2^2(x)}} dx
\end{split}
\end{equation}
Making the substitution $ z = P_2(x) $ in the above integrals we obtain
\begin{equation}
\int_{-1}^1\left(\frac{z+1}{2} \right)^{n+\frac{1}{2}}\left(\frac{1-z}{2} \right)^{\mu}\left[-\frac{Q_2(P_{1,2}^{-1}(z))}{P_2^{\prime}(P_{1,2}^{-1}(z))}+\frac{Q_2(P_{2,2}^{-1}(z))}{P_2^{\prime}(P_{2,2}^{-1}(z))} \right] \frac{dz}{\sqrt{1-z^2}}
\end{equation}
Here the function $P_{1,2}^{-1}$ is defined to be the inverse of the function $P_2$ where $P_2$ is restricted to the domain $[-1,-b]$. Similarly, the function $P_{2,2}^{-1}$ is defined to be the inverse of the function $P_2$ where $P_2$ is restricted to the domain $[b,1]$. To simplify the terms in the square brackets we form the following partial fractions decomposition
\begin{equation}
\frac{Q_2(s)}{P_2(s) - z} = \frac{w_1(z)}{s - P_{1,2}^{-1}(z)} + \frac{w_2(z)}{s - P_{2,2}^{-1}(z)} \qquad z \in (-1,1)
\end{equation}
It follows that
\begin{equation}
w_i(z) = \frac{Q_2(P_{i,2}^{-1}(z))}{P_2^{\prime}(P_{i,2}^{-1}(z))} \qquad i=1,2 \qquad \textrm{and} \qquad w_1(z) + w_2(z) = 1.
\end{equation}
Therefore we rewrite the integral as
\begin{equation} \label{int1}
\int_{-1}^1\left(\frac{z+1}{2} \right)^{n+\frac{1}{2}}\left(\frac{1-z}{2} \right)^{\mu}\left[1-2\frac{Q_2(P_{1,2}^{-1}(z))}{P_2^{\prime}(P_{1,2}^{-1}(z))} \right] \frac{dz}{\sqrt{1-z^2}}
\end{equation}
We compute the following quantity directly from (\ref{P2Q2})
\begin{equation}
P_{1,2}^{-1}(x) = - \frac{1}{\sqrt{2}} \sqrt{(1+b^2)+x(1-b^2)}
\end{equation}
from which we find that
\begin{equation}
1-2\frac{Q_2(P_{1,2}^{-1}(z))}{P_2^{\prime}(P_{1,2}^{-1}(z))} = \frac{b\sqrt{2}}{\sqrt{1+b^2}}\frac{1}{\sqrt{1-\frac{b^2-1}{b^2+1}z}}
\end{equation}
Therefore (\ref{int1}) becomes
\begin{equation}
\frac{b\sqrt{2}}{\sqrt{1+b^2}}\frac{1}{2^{n+\frac{1}{2}+\mu}} \int_{-1}^1 (z+1)^{n}(1-z)^{\mu-\frac{1}{2}}\left(1 - \frac{b^2-1}{b^2+1}z \right)^{-\frac{1}{2}} dz
\end{equation}
which through the substitution $ u = \frac{1+z}{2} $ can be written as
\begin{equation}
\int_0^1 u^{n}(1-u)^{\mu-\frac{1}{2}}\left(1-\frac{(b^2-1)}{b^2}u\right)^{-\frac{1}{2}} du = \frac{\Gamma(n+1)\Gamma(\mu+\frac{1}{2})}{\Gamma(\mu+n+\frac{3}{2})}\;_2F_1\left(\frac{1}{2},n+1;\mu+n+\frac{3}{2}; \frac{b^2-1}{b^2}\right)
\end{equation}
Finally, using Pfaff's transformation \cite{AAR} page 68, on the hypergeometric function above we obtain
\begin{equation} \label{1-gap-odd-rep2}
\begin{split}
& \int_{E_2} (\cos \phi_{E_2})^{2n+1}(1-\cos^2 \phi_{E_2})^\mu w_{E_2}(x) \; dx \\
& \qquad \qquad \qquad = b \frac{\Gamma(n+1)\Gamma(\mu+\frac{1}{2})}{\Gamma(\mu+n+\frac{3}{2})}\;_2F_1\left(\frac{1}{2},\mu+\frac{1}{2};\mu+n+\frac{3}{2}; 1-b^2\right)
\end{split}
\end{equation}
as an alternative expression for the odd moments. Figure $1$ shows an example of the equivalence of the representations (\ref{1-gap-odd}) and (\ref{1-gap-odd-rep2}) in the case where $\mu=0$. In the graph, curve $A$ shows an example of the representation (\ref{1-gap-odd}) for the first odd moment with $\mu =0$, and curve $B$ shows the same example but using the representation (\ref{1-gap-odd-rep2}). We can see that the curves coincide for $b \in [0,1)$.
\begin{figure}[h]
\begin{center}
\center{\includegraphics[width=10cm,height=9cm,angle=0]{hypergeom1.pdf}}
\end{center}
\caption{Plot of the first odd moment as a function of $b$ in the case where $\mu = 0$.}
\label{3fns}
\end{figure}
\subsection{Two gap case}
For the remainder of this paper, we carry out a similar study to that described in the previous subsection, this time working with a set with two symmetric gaps. Specifically we work on the following set
\begin{equation}
E_4 := [-1,-b] \cup [-\sqrt{1-b^2},\sqrt{1-b^2}] \cup [b,1] \qquad b \in \left[\frac{1}{\sqrt{2}}, 1\right)
\end{equation}
This particular set is of interest as it can be thought of as the pre-image of $[-1,1]$ under the following polynomial mapping of degree $4$ (see figure $2$) :
\begin{equation} \label{map}
P_4(x) = \frac{2x^4-2x^2+b^2(1-b^2)}{b^2(1-b^2)}
\end{equation}
\begin{figure}[h]
\begin{center}
\center{\includegraphics[width=8cm,height=6cm,angle=0]{Fig2.pdf}}
\end{center}
\caption{Graph of $P_4$ with $b=0.8$. The gaps in the set $E_4$ lie inside both pairs of vertical lines.}
\end{figure}
\vskip 0.2cm
\noindent
The following lemma confirms this fact.
\begin{lemma} \label{preimage}
Let $ b \in \left[\frac{1}{\sqrt{2}}, 1\right)$. The function $P_4$ defined in equation (\ref{map}) satisfies the following properties:
\begin{enumerate}
\item $P_4(\pm 1) = P_4(0) = 1$
\item $P_4(\pm b) = P_4(\pm \sqrt{1-b^2}) = -1$
\item $P_4$ is decreasing on the intervals $(-1,-b)$, $(0, \sqrt{1-b^2})$
\item $P_4$ is increasing on the intervals $[\sqrt{1-b^2}, 0]$, $[b,1]$
\end{enumerate}
\end{lemma}
\begin{proof}
Items $1$ and $2$ follow directly from substitution of the points into the expression for $P_4$. The derivative of $P_4$, denoted as $P_4^{\prime}$ from now on, changes sign at $x = 0, \pm \frac{1}{\sqrt{2}} $. Since the intervals $(-1,-b)$ and $(0, \sqrt{1-b^2})$ do not contain any of these points for $b$ in the stated range it follows that $P_4$ is monotonic on these intervals. Combining this with $1$ and $2$ we find that $P_4$ is decreasing. $4$ follows from $3$ by symmetry.
\end{proof}
\noindent
We now show how the polynmomial $P_4$ appears as part of a solution to a generalized Pell equation. Consider the normalized general set with two gaps
\[ [-1,\alpha_1] \cup [\beta_1, \alpha_2] \cup [\beta_2,1] \qquad -1 < \alpha_1 < \beta_1 < \alpha_2 < \beta_2 < 1 \]
The generalized Pell equation for this set is
\[ (x-\alpha_1)(x-\alpha_2)P_n^2(x) - (x^2-1)(x-\beta_1)(x-\beta_2)Q_n^2(x) = (x-\gamma_1(n))(x-\gamma_2(n) \]
where $P_n$ is a polynomial of degree $n$ and $Q_n$ is a polynomial of degree $n-1$. As such, the Pell equation associated with the set $E_4$ is
\begin{equation} \label{PE4}
(x-\sqrt{1-b^2})(x+b)P_n^2(x) -(x^2-1)(x-b)(x+\sqrt{1-b^2})Q_n^2(x) = (x-\gamma_1(n))(x-\gamma_2(n))
\end{equation}
For the formulation of our integral, the cases $n=2$ and $n=4$ will be of particular interest. The following two lemmas deal with these cases individually.
\begin{lemma} \label{PellP4}
The polynomial $P_4$ appearing in (\ref{map}) together with the polynomial
\[ Q_4(x) = \frac{2x}{b^2(b^2-1)}\left( x^2 + (b- \sqrt{1-b^2})x - b\sqrt{1-b^2} \right) \]
form a solution of (\ref{PE4}).
\end{lemma}
\begin{proof}
Let
\begin{equation} \label{gamma4}
\gamma_1(4) = -b \qquad \gamma_2(4) = \sqrt{1-b^2}.
\end{equation}
Then we can write $Q_4$ as follows :
\[ Q_4(x) = \frac{2x}{b^2(b^2-1)} (x-\gamma_1(4))(x-\gamma_2(4)). \]
As such, the Pell equation simplifies to
\begin{equation}
P_4^2(x) - (x^2-1)(x^2-b^2)(x^2-(1-b^2))\left(\frac{2x}{b^2(b^2-1)}\right)^2 = 1
\end{equation}
Substituting (\ref{map}) into the above equation it can be easily verified that the left hand side cancels to $1$.
\end{proof}
\noindent
\begin{lemma}
The polynomials
\[ P_2(x) = \frac{(x-b)(x+\sqrt{1-b^2})}{b\sqrt{1-b^2}} \qquad \textrm{and} \qquad Q_2(x) = \frac{x}{b\sqrt{1-b^2}} \]
form a solution of (\ref{PE4}).
\end{lemma}
\begin{proof}
Let
\begin{equation}
\gamma_1(2) = -\sqrt{1-b^2} \qquad \gamma_2(2) = b.
\end{equation}
With these gamma points and $P_2$ and $Q_2$ as stated above, the Pell equation simplifies to
\begin{equation}
\frac{(x^2-(1-b^2))(x^2-b^2)}{b^2(1-b^2)} - \frac{x^2(x^2-1)}{b^2(1-b^2)} = 1
\end{equation}
The left hand side can be readily seen to cancel to $1$.
\end{proof}
\noindent
{\it Construction of the integral}
\vskip 0.2cm
\noindent
In analogy to the previous work on the set $E_2$ we define the quantity
\begin{equation}
\begin{split}
\cos \phi_{E_4}(x) & := \sqrt{\frac{(x-\alpha_1)(x-\alpha_2)}{(x-\gamma_1(2))(x-\gamma_2(2))}}P_2(x) \\
& = \sqrt{\frac{(x-\sqrt{1-b^2})(x+b)}{(x+\sqrt{1-b^2})(x-b)}}P_2(x) \\
& = \sqrt{\frac{(x-\sqrt{1-b^2})(x+b)}{(x+\sqrt{1-b^2})(x-b)}}\left(\frac{(x-b)(x+\sqrt{1-b^2})}{b\sqrt{1-b^2}}\right) \qquad x \in E_4
\end{split}
\end{equation}
\begin{figure}[t]
\begin{center}
\center{\includegraphics[width=8cm,height=6cm,angle=0]{Fig3.pdf}}
\end{center}
\caption{Plot of $\cos \phi_{E4}$ for two values of $b$}
\end{figure}
whose graph is shown in figure $3$. Furthermore we define the following weight function
\begin{equation}
w_{E_4}(x) = \sqrt{\frac{(x-\sqrt{1-b^2})(x+b)}{(1-x^2)(x-b)(x+\sqrt{1-b^2})}}
\end{equation}
which is the Akhiezer weight for the set $E_4$.
In terms of these quantities, we construct the following moment integral
\begin{equation} \label{def-int}
\int_{E_4} \left( \cos \phi_{E_4} \right)^{n}\left(1-\cos^2 \phi_{E_4} \right)^{\mu} w_{E_4} \; dx \qquad n = 0, 1, 2,... \;\; \textrm{and} \;\; Re(\mu) > -\frac{1}{2}
\end{equation}
where the dependence on $x$ has been suprressed for convenience of notation. Note that as $b \to \frac{1}{\sqrt{2}}$ the integral above becomes
\begin{equation}
\int_{-1}^1 \left( T_2(x)\right)^n \left(1-(T_2(x))^2 \right)^{\mu} \frac{dx}{\sqrt{1-x^2}}.
\end{equation}
\vskip 0.2cm
\noindent
It is the purpose of this paper to study the integral (\ref{def-int}). In the next section we will show, using the theory of polynomial mappings \cite{GVA}, that the even moments are the classical ultraspherical moments and are independent of the parameter $b$.
\vskip 0.2cm
\noindent
\begin{equation} \label{int-even}
\begin{split}
\int_{E_4} \left( \cos \phi_{E_4} \right)^{2n}\left(1-\cos^2 \phi_{E_4} \right)^{\mu} w_{E_4} \; dx = \frac{\Gamma(\frac{1}{2})\Gamma(\mu+\frac{1}{2})}{\Gamma(\mu+1)}\frac{\left(\frac{1}{2} \right)_n}{(\mu+1)_n} \\
\qquad \qquad \qquad n = 0, 1, 2,... \;\; \textrm{and} \;\; Re(\mu) > -\frac{1}{2}
\end{split}
\end{equation}
Note that the right hand side of the above equation is equal to $\mu_{2n}$ given in (\ref{intclassiceval}).
We will also show that the odd moments can be computed explicitly in terms of the classical hypergeometric function using the same method that was used in section $2.1$. Specifically we will show that
\vskip 0.2cm
\noindent
\begin{equation} \label{int-odd}
\begin{split}
& \int_{E_4} \left( \cos \phi_{E_4} \right)^{2n+1}\left(1-\cos^2 \phi_{E_4} \right)^{\mu} w_{E_4} \; dx = \\
& \qquad \qquad (1-2b\sqrt{1-b^2})\frac{(1)_n}{(\mu+\frac{1}{2})_{n+1}}\;_2F_1\left(\frac{1}{2}, \mu+1; \mu + n + \frac{3}{2} ; 4b^2(1-b^2) \right) \\
& \qquad \qquad \qquad n = 0, 1, 2,... \;\; \textrm{and} \;\; Re(\mu) > -\frac{1}{2}
\end{split}
\end{equation}
\noindent
\section{Main Results}
\begin{theorem} \label{even-moments}
For $ n = 0, 1, 2,...$, $Re(\mu) > -\frac{1}{2}$ and $ b \in \left( \frac{1}{\sqrt{2}},1 \right) $
\begin{equation}
\int_{E_4} \left( \cos \phi_{E_4} \right)^{2n}\left(1-\cos^2 \phi_{E_4} \right)^{\mu} w_{E_4} \; dx = \frac{\Gamma(\frac{1}{2})\Gamma(\mu+\frac{1}{2})}{\Gamma(\mu+1)}\frac{\left(\frac{1}{2} \right)_n}{(\mu+1)_n}
\end{equation}
\end{theorem}
\begin{proof}
The proof is based on a substitution involving the mapping function $P_4$ given in (\ref{map}). This substitution follows the general method of polynomial mappings outlined in \cite{GVA}. It can be easily verified that
\begin{equation}\label{evenP4}
\cos^2 \phi_{E_4} = \frac{P_4(x)+1}{2}
\end{equation}
In terms of $P_4$ the integral becomes
\begin{equation}
\int_{E_4} \left(\frac{P_4(x)+1}{2} \right)^n \left(\frac{1-P_4(x)}{2} \right)^\mu w_{E_4} dx
\end{equation}
We know from lemma {\bf \ref{PellP4}} that $P_4$ satisfies the Pell equation,
\begin{equation}
P_4^2(x) - \frac{(x^2-1)(x-b)(x+\sqrt{1-b^2})}{(x+b)(x-\sqrt{1-b^2})}Q_4^2(x) = 1.
\end{equation}
We can rearrange the above equation to obtain
\begin{equation}
w_{E_4}(x) = \frac{|Q_4(x)|}{\sqrt{1-P_4^2(x)}}
\end{equation}
Therefore the integral can be written as
\begin{equation}
\int_{E_4} \left(\frac{P_4(x)+1}{2} \right)^n \left(\frac{1-P_4(x)}{2} \right)^\mu \frac{|Q_4(x)|}{\sqrt{1-P_4^2(x)}} dx
\end{equation}
From lemma {\bf \ref{preimage}} we know that the set $E_4$ is the pre-image of $[-1,1]$ under the mapping $P_4$. Therefore we can write
\begin{equation}
E_4 = \cup_{i=1}^4 P^{-1}_{i,4}([-1,1])
\end{equation}
where $P_{1,4}$ is the function $P_4$ restricted to $[-1,-b]$. Similarly $P_{2,4}$ is restricted to $[-\sqrt{1-b^2},0]$, $P_{3,4}$ is restricted to $[0,\sqrt{1-b^2}]$ and $P_{4,4}$ is restricted to $[b,1]$. Using this notation the integral can now be written as a sum of four separate integrals
\begin{equation}
\sum_{i=1}^4 \int_{P_{i,4}^{-1}([-1,1])} \left(\frac{P_4(x)+1}{2} \right)^n \left(\frac{1-P_4(x)}{2} \right)^\mu \frac{|Q_4(x)|}{\sqrt{1-P_4^2(x)}} dx
\end{equation}
where for each integral the function $P_4$ is monotonic. In each of these intervals we make the substitution $z = P_4(x)$ to obtain
\begin{equation}
\sum_{i=1}^4 \int_{-1}^1 \left(\frac{z+1}{2} \right)^n \left(\frac{1-z}{2} \right)^\mu \frac{Q_4(P_{i,4}^{-1}(z))}{P_4^{\prime}(P_{i,4}^{-1}(z))}\frac{dz}{\sqrt{1-z^2}}
\end{equation}
where the absolute value signs have been removed owing to the orientation of the set $[-1,1]$. We move the summation sign inside the integral to obtain
\begin{equation}
\int_{-1}^1 \left(\frac{z+1}{2} \right)^n \left(\frac{1-z}{2} \right)^\mu \left( \sum_{i=1}^4 \frac{Q_4(P_{i,4}^{-1}(z))}{P_4^{\prime}(P_{i,4}^{-1}(z))}\right) \frac{dz}{\sqrt{1-z^2}}
\end{equation}
We now show that the sum appearing inside the integral is equal to $1$. We start with the partial fraction decomposition of the following rational function
\begin{equation}
\frac{Q_4(s)}{P_4(s) - z} = \sum_{i=1}^4 \frac{w_i(z)}{s - P_{i,4}^{-1}(z)} \qquad z \in (-1,1)
\end{equation}
remembering that $Q_4$ from the Pell equation is a polynomial of degree $3$ and that for $ z \in (-1,1) $ the polynomial $P_4(s) -z$ has four distinct roots. From the partial fractions decomposition it follows that
\begin{equation}
w_i(z) = \frac{Q_4(P_{i,4}^{-1}(z))}{P_4^{\prime}(P_{i,4}^{-1}(z))}.
\end{equation}
Since the leading coefficients of $Q_4$ and $P_4$ are the same we must have
\begin{equation} \label{sum}
\sum_{i=1}^4 w_i(z) = 1.
\end{equation}
So the integral becomes
\begin{equation}
\int_{-1}^1 \left(\frac{z+1}{2} \right)^n \left(\frac{1-z}{2} \right)^\mu \frac{dz}{\sqrt{1-z^2}}
\end{equation}
which can be recognized as a beta integral and easily evaluates to
\begin{equation}
\frac{\Gamma(\frac{1}{2})\Gamma(\mu+\frac{1}{2})}{\Gamma(\mu+1)}\frac{\left(\frac{1}{2} \right)_n}{(\mu+1)_n}
\end{equation}
\end{proof}
\noindent
\begin{theorem} \label{odd-moments}
For $ n = 0, 1, 2,...$, $Re(\mu) > -\frac{1}{2}$ and $ b \in \left( \frac{1}{\sqrt{2}},1 \right) $
\begin{equation}
\begin{split}
& \int_{E_4} \left( \cos \phi_{E_4} \right)^{2n+1}\left(1-\cos^2 \phi_{E_4} \right)^{\mu} w_{E_4} \; dx = \\
& \qquad \qquad (1-2b\sqrt{1-b^2})\frac{(1)_n}{(\mu+\frac{1}{2})_{n+1}}\;_2F_1\left(\frac{1}{2}, \mu+1; \mu + n + \frac{3}{2} ; 4b^2(1-b^2) \right)
\end{split}
\end{equation}
\end{theorem}
\begin{proof}
We first write the integral in terms of the polynomial $P_4(x)$ as we did in the previous proof. However, since the power of $ \cos \phi_{E_4}$ is now odd we cannot make use of (\ref{evenP4}). Instead we make use of the following :
\begin{equation}
\cos \phi_{E_4} = \left\{
\begin{array}{cc}
\sqrt{\frac{P_4+1}{2}} & x \in [-1,-b] \cup [b,1] \\
-\sqrt{\frac{P_4+1}{2}} & x \in [-\sqrt{1-b^2}, \sqrt{1-b^2} ]
\end{array}
\right.
\end{equation}
It therefore follows that
\begin{equation}
\begin{split}
& \int_{E_4} \left( \cos \phi_{E_4} \right)^{2n+1}\left(1-\cos^2 \phi_{E_4} \right)^{\mu} w_{E_4} \; dx = \\
& \qquad \int_{-1}^{-b} \left(\frac{P_4(x)+1}{2} \right)^{n+\frac{1}{2}} \left(\frac{1-P_4(x)}{2} \right)^{\mu} w_{E_4} \; dx \\
& \qquad - \int_{-\sqrt{1-b^2}}^{0} \left(\frac{P_4(x)+1}{2} \right)^{n+\frac{1}{2}} \left(\frac{1-P_4(x)}{2} \right)^{\mu} w_{E_4} \; dx \\
& \qquad - \int^{\sqrt{1-b^2}}_{0} \left(\frac{P_4(x)+1}{2} \right)^{n+\frac{1}{2}} \left(\frac{1-P_4(x)}{2} \right)^{\mu} w_{E_4} \; dx \\
& \qquad + \int_{b}^{1} \left(\frac{P_4(x)+1}{2} \right)^{n+\frac{1}{2}} \left(\frac{1-P_4(x)}{2} \right)^{\mu} w_{E_4} \; dx
\end{split}
\end{equation}
In each of the four integrals appearing on the right hand side we make the change of variables $z = P_4(x)$ to obtain
\begin{equation}
\int_{-1}^1\left(\frac{z+1}{2} \right)^{n+\frac{1}{2}}\left(\frac{1-z}{2} \right)^{\mu}\left[\frac{Q_4(P_{1,4}^{-1}(z))}{P_4^{\prime}(P_{1,4}^{-1}(z))}-\frac{Q_4(P_{2,4}^{-1}(z))}{P_4^{\prime}(P_{2,4}^{-1}(z))}-\frac{Q_4(P_{3,4}^{-1}(z))}{P_4^{\prime}(P_{3,4}^{-1}(z))}+\frac{Q_4(P_{4,4}^{-1}(z))}{P_4^{\prime}(P_{4,4}^{-1}(z))} \right] \frac{dz}{\sqrt{1-z^2}}
\end{equation}
which in light of (\ref{sum}) can be written as
\begin{equation} \label{int2}
\int_{-1}^1\left(\frac{z+1}{2} \right)^{n+\frac{1}{2}}\left(\frac{1-z}{2} \right)^{\mu}\left[1-2\left(\frac{Q_4(P_{2,4}^{-1}(z))}{P_4^{\prime}(P_{2,4}^{-1}(z))}+\frac{Q_4(P_{3,4}^{-1}(z))}{P_4^{\prime}(P_{3,4}^{-1}(z))}\right) \right] \frac{dz}{\sqrt{1-z^2}}
\end{equation}
We now compute the quantity
\begin{equation}
\frac{Q_4(P_{2,4}^{-1}(z))}{P_4^{\prime}(P_{2,4}^{-1}(z))}+\frac{Q_4(P_{3,4}^{-1}(z))}{P_4^{\prime}(P_{3,4}^{-1}(z))}
\end{equation}
explicitly.
From (\ref{map}) we can easily see that
\begin{equation}
\sqrt{2}P_{2,4}^{-1}(z) = -\sqrt{1-\sqrt{1-2(1-z)b^2(1-b^2)}}
\end{equation}
and
\begin{equation}
\sqrt{2}P_{3,4}^{-1}(z) = \sqrt{1-\sqrt{1-2(1-z)b^2(1-b^2)}}
\end{equation}
Substituting these directly into the expressions for $Q_4$ and $P_4^{\prime}$ we obtain
\begin{equation}
\frac{Q_4(P_{2,4}^{-1}(z))}{P_4^{\prime}(P_{2,4}^{-1}(z))}+\frac{Q_4(P_{3,4}^{-1}(z))}{P_4^{\prime}(P_{3,4}^{-1}(z))} = \frac{1}{2}\left(1 - \frac{1-2b\sqrt{1-b^2}}{\sqrt{1-2b^2+2b^4+[2b^2-2b^4]z}} \right).
\end{equation}
Therefore (\ref{int2}) becomes
\begin{equation}
(1-2b\sqrt{1-b^2})\int_{-1}^1\left(\frac{z+1}{2} \right)^{n+\frac{1}{2}}\left(\frac{1-z}{2} \right)^{\mu}(1-2b^2+2b^4+[2b^2-2b^4]z)^{-1/2} \frac{dz}{\sqrt{1-z^2}}
\end{equation}
By means of the subsitution $ u = \frac{z+1}{2} $ we obtain
\begin{equation}
\frac{1-2b\sqrt{1-b^2}}{2b^2-1} \int_0^1 u^{n}(1-u)^{\mu-\frac{1}{2}}\left(1-\frac{4b^2(b^2-1)}{(2b^2-1)^2}u \right)^{-1/2}du
\end{equation}
which can be written as
\begin{equation}
\frac{1-2b\sqrt{1-b^2}}{2b^2-1}\frac{(1)_n}{(\mu+\frac{1}{2})_{n+1}}\;_2F_1\left(\frac{1}{2}, n+1; \mu + n + \frac{3}{2} ; \frac{4b^2(b^2-1)}{(2b^2-1)^2} \right).
\end{equation}
Finally, by making use of Pfaff's transformation, \cite{AAR} page 68, the above expression can be written as
\begin{equation} \label{final-odd}
(1-2b\sqrt{1-b^2})\frac{(1)_n}{(\mu+\frac{1}{2})_{n+1}}\;_2F_1\left(\frac{1}{2}, \mu+1; \mu + n + \frac{3}{2} ; 4b^2(1-b^2) \right)
\end{equation}
\end{proof}
\noindent
From the representation for the odd moments derived above we can easily identify two limit cases. Firstly, as $b \to \frac{1}{\sqrt{2}}^+$, which corresponds to the gaps closing, it is clear that the expression (\ref{final-odd}) approaches zero. Similarly we can obtain the limiting case as the gaps approach their maximum size. By letting $b$ approach $1$ in (\ref{final-odd}) we find
\begin{equation}
\lim _{b \to 1^-} \int_{E_4} \left( \cos \phi_{E_4} \right)^{2n+1}\left(1-\cos^2 \phi_{E_4} \right)^{\mu} w_{E_4} \; dx = \frac{(1)_n}{(\mu+\frac{1}{2})_{n+1}}.
\end{equation}
The half range moments (\ref{halfeval2}) can be written in terms of the shifted factorial as
\begin{equation}
\begin{split}
\mu_{2n}^H &= \frac{1}{2} \frac{\Gamma(\frac{1}{2})\Gamma(\mu+\frac{1}{2})}{\Gamma(\mu+1)}\frac{\left(\frac{1}{2} \right)_n}{(\mu+1)_n} \\
\mu_{2n+1}^H &= \frac{1}{2} \frac{(1)_n}{(\mu+\frac{1}{2})_{n+1}}
\end{split}
\end{equation}
and therefore, as the gaps approach their maximum size the generalized moments tend towards the half range ultraspherical moments multiplied by a factor of $2$.
\bibliographystyle{plain}
| {
"timestamp": "2019-01-15T02:25:14",
"yymm": "1901",
"arxiv_id": "1901.04219",
"language": "en",
"url": "https://arxiv.org/abs/1901.04219",
"abstract": "Moment evaluations are important for the study of non-classical orthogonal polynomial systems for which explicit representations are not known. In this paper we compute, in terms of the hypergeometric function, the moments associated with a generalized ultraspherical weight on a collection of intervals with two symmetric gaps. These moments, parametrized by the endpoints of the gaps, are identified as a one parameter deformation between the full range ultraspherical moments and the half range ultraspherical moments.",
"subjects": "Classical Analysis and ODEs (math.CA)",
"title": "Ultraspherical moments on a set of disjoint intervals",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808690122163,
"lm_q2_score": 0.7217432062975979,
"lm_q1q2_score": 0.7075110574730726
} |
https://arxiv.org/abs/1605.02713 | The Avalanche Polynomial of a Graph | The (univariate) avalanche polynomial of a graph, introduced by Cori, Dartois and Rossin in 2006, captures the distribution of the length of (principal) avalanches in the abelian sandpile model. This polynomial has been used to show that the avalanche distribution in the sandpile model on a multiple wheel graph does not follow the expected power law function. In this article, we introduce the (multivariate) avalanche polynomial that enumerates the toppling sequences of all principal avalanches. This polynomial generalizes the univariate avalanche polynomial and encodes more information. In particular, the avalanche polynomial of a tree uniquely identifies the underlying tree. In this paper, the avalanche polynomial is characterized for trees, cycles, wheels, and complete graphs. | \section{Introduction}
The area of \emph{sandpile groups} is a flourishing area that started
in mathematical physics in 1987 with the seminal work of Bak, Tang and
Wiesenfeld \cite{BTW88}. Since then it has found many, often unexpected,
applications in diverse areas of mathematics, physics, computer
science and even some applications in the biological sciences and
economics. In thermodynamics, a \emph{critical point} is the end point of a
phase equilibrium curve. The most prominent example is the
liquid-vapor critical point, the end point of the pressure-temperature
curve at which the distinction between liquid and gas can no longer be
made. In order to drive this system to its critical point it is
necessary to tune certain parameters, namely pressure and temperature.
In nature, one can also observe different types of dynamical systems
that have a critical point as an attractor. The macroscopic behavior
of these systems displays the spatial and/or temporal scale-invariance
characteristic of the critical point of a phase transition, but
without the need to tune control parameters to precise values. Such a
system is said to display \emph{self-organized criticality}, a concept first
introduced in 1987 by Bak, Tang and
Wiesenfeld in their groundbreaking paper. This concept
is thought to be present in a large variety of physical systems like
earthquakes \cite{SS89, CBO91}, forest fires and in stock
market fluctuations \cite{B96}. Self-organized criticality is
considered to be one of the mechanisms by which \emph{complexity}
arises in nature \cite{BP95} and has been extensively studied
in the statistical physics literature during the last three decades
\cite{J98, P12, MG14}.
In \cite{BTW88}, Bak, Tang and Wiesenfeld conceived a \emph{cellular
automaton} model as a paradigm of self-organized criticality. This
model is defined on a rectangular grid of cells as shown in
Figure \ref{grid}. The system evolves in discrete time such that at
each time step a sand grain is
dropped onto a random grid cell. When a cell
amasses four grains of sand, it becomes \emph{unstable}. It relaxes by
\emph{toppling} whereby four sand grains leave the site, and each of the
four neighboring sites gets one grain. If the unstable cell is on the
boundary of the grid then, depending on whether the cell is a corner
or not, either
one or two sand grains fall off the edge and disappear. As the sand
percolates over the grid in this fashion, adjacent cells may
accumulate four grains of sand and become unstable causing an
\emph{avalanche}. This settling process continues until all cells are
\emph{stable}. Then another cell is picked randomly, the height of the sand
on that grid cell is increased by one, and the process is repeated.
\begin{figure}[!htb]
\begin{center}
\includegraphics[scale=.8]{grid}
\end{center}
\caption{The Bak, Tang and Wiesenfeld model of self-organized criticality.}
\label{grid}
\end{figure}
Imagine starting this process on an empty grid.
At first there is little
activity, but as time goes on, the size (the total number of topplings
performed) and extent of the avalanche
caused by a single grain of sand becomes hard to predict. Figure
\ref{soc} shows the distribution of avalanches in a computational
experiment performed on a $20\times 20$ grid. Starting with the
\emph{maximal stable sandpile}, i.e., the sandpile with three grains
of sand at each site, a total of $100,000$ sand grains were added at random,
allowing the sandpile to stabilize in between.
Many authors have studied the distribution of the sizes of the
avalanches for this model showing that it obeys a power law with
exponent $-1$ \cite{C91, DRSV95, KLGP00}.
\begin{figure}[!htb]
\begin{center}
\includegraphics[scale=.4]{soc}
\end{center}
\caption{Distribution of avalanches $D(s)$ as
a function of the size $s$.}
\label{soc}
\end{figure}
In 1990, Dhar generalized the Bak, Tang and Wiesenfeld model
replacing the rectangular grid with an arbitrary combinatorial graph
\cite{D90}. In this model, known as the \emph{(abelian) sandpile model},
the sand grains are placed
at the vertices of the graph and the toppling threshold depends on the
degree (outdegree) of each vertex. Certain conditions on the graph
(the existence of a \emph{global sink vertex}) ensure that any
avalanche terminates after a finite number of topplings.
The sandpile model was also considered by combinatorialists as a game on a
graph called the \emph{chip firing game} \cite{BLS91,
BL92, B99}.
The long-term behavior
of the abelian sandpile model on a graph is encoded by the \emph{critical
configurations}, also known as \emph{recurrent sandpiles}. These
critical configurations have
connections to \emph{parking functions} \cite{BCT10}, to the
\emph{Tutte polynomial} \cite{Merino05}, and to the lattices of \emph{flows}
and \emph{cuts} of a graph \cite{BDN97}. Among other properties, the
critical configurations have the structure of a
\emph{finite abelian group}. This group has been discovered in several
different contexts and received many
names: the {\it sandpile group} \cite{D90, HLMPPW08}, the {\it
critical group}~\cite{B99}, the {\it group of
bicycles} \cite{B86}, the {\it group of components}~\cite{L89}, and the
{\it jacobian} of a graph~\cite{BN07}.
The fact that the distribution of avalanches on the grid follows a
power law has been a focal point from the statistical physics
perspective. A natural question is what type of distributions do we
get in the more general setting introduced by Dhar. In this paper, we
focus on this question. In fact, we go one step beyond just finding such
distributions. We actually describe the combinatorial structure of each
avalanche for certain families of graphs.
Experiments on the distribution of sizes of the avalanches have been
mostly restricted to the cases of rectangular grids and some classes
of regular graphs. However, very
little is known for arbitrary graphs \cite{DaRoFPSAC03}.
In 2004, Cori, Dartois, and Rossin
introduced the (univariate)
\emph{avalanche polynomial} that encodes the sizes of \emph{principal
avalanches}, that is, avalanches resulting from adding a single grain
of sand to a recurrent sandpile \cite{CDR04}. These
authors completely describe the avalanche polynomials for
trees, cycles, complete and lollypop graphs. Moreover, using these
polynomials, they show that the resulting sandpile models on
these graphs no longer obey the power law observed in the rectangular
grid.
In 2003, Dartois and Rossin obtained exact results for the avalanche
distribution on wheel graphs
\cite{DaRoFPSAC03}.
In 2009, Cori, Micheli and Rossin studied further properties of
the avalanche polynomial on plane trees \cite{CMR09}. In particular,
they show that the avalanche polynomial of a tree does not uniquely
characterize the tree. They also give closed formulas for the average
and variance of the avalanche distribution on trees.
In this paper we introduce the \emph{(multivariate) avalanche
polynomial}, i.e., a multivariate polynomial encoding the toppling
sequences of all principal avalanches. This polynomial generalizes the
univariate avalanche polynomial and encodes more information.
In Section \ref{background}, we
describe the sandpile model on an undirected graph and
introduce both the univariate and multivariate avalanche
polynomials. We also present some particular evaluations of the latter
polynomial. In particular, one such evaluation gives rise to the
unnormalized distribution of burst sizes, that is, the number of
grains of sand that fall into the sink in a principal avalanche. In
Section \ref{trees}, we characterize the avalanche
polynomial of a tree. We also prove in Theorem \ref{uniqueness} that
this polynomial uniquely characterizes its underlying tree. In
Section \ref{cycles}, we compute the multivariate avalanche
polynomial for cycle graphs and in Section \ref{complete}, we compute
this polynomial for complete graphs using the bijection among
recurrent sandpiles and parking functions. Our
arguments fix some details in the proof of Proposition 5 in
\cite{CDR04} that enumerates the number of principal avalanches of
positive sizes in the complete graph. In Section \ref{wheel}, we
compute the avalanche polynomial for wheel graphs. Our methods
simplify the arguments in \cite{DaRoFPSAC03} where the authors use
techniques from \emph{regular languages}, \emph{automatas} and \emph{transducers} to characterize the recurrent
sandpiles and determine the exact distribution of avalanche lengths in the wheel graph.
\section{The Abelian Sandpile Model}\label{background}
The \emph{abelian sandpile model} is defined both on directed and
undirected graphs, but here we focus on families of undirected
multigraphs without loops.
\begin{defn} An \emph{(undirected) graph} $G$ is an ordered pair
$(V,E)$, where $V$ is a finite set and $E$ is a finite multiset of the
set of 2-element subsets of $V$. The elements of $V$ are called
\emph{vertices} and the elements of $E$ are called \emph{edges}.
Given an undirected graph $G = (V,E)$, the \emph{degree} $d_v$ of
a vertex $v \in V$ is the number of edges $e\in E$ with $v \in e$.
For a pair of vertices $u,v\in V$, the weight $\weight(u,v)$ is the
number of edges between $u$ and $v$. We say $u$ and $v$ are adjacent
if $\weight(u,v)>0$.
\end{defn}
Here, we will always assume that our graphs are
\emph{connected}. Moreover, given a graph $G = (V,E)$ we will
distinguish a vertex $s\in V$ and call it a \emph{sink}. The resulting
graph will be denoted $G = (V,E,s)$.
We will also denote the set of all non-sink vertices by
$\widetilde{V} = V\setminus \{s\}$.
\begin{defn} A \emph{sandpile} $c$ on $G = (V,E,s)$ is a
function $c: \widetilde{V} \to \mathbb{Z}_{\geq 0}$ from the non-sink vertices of
$G$ to the set of non-negative integers, where
$c(v)$ represents the number of grains of sand at vertex $v$.
We call $v$ \emph{unstable} if $c(v)
\geq d_{v}$. An unstable vertex $v$ can \emph{topple}, resulting in a
new sandpile $c'$ obtained by moving one grain of sand along each of
the $d_{v}$ edges emanating from $v$; that is, $c'(w) = c(w) +
\weight(v,w)$ for all $w\neq v$ and $c'(v) = c(v) - d_{v}$. A sandpile
is \emph{stable} if $c(v) < d_{v}$ for every non-sink vertex $v$ and
\emph{unstable} otherwise.
\end{defn}
The following proposition justifies the name ``abelian sandpile
model'' and it was first proved by Dhar in \cite{D90}.
\begin{prop}\label{prop:THEstabilization}
Given an unstable sandpile
$c$ on a graph $G = (V, E, s)$, any sequence of topplings of
unstable vertices will lead to the same stable sandpile.
\end{prop}
Given a sandpile $c$, if $c'$ is obtained from $c$ after a sequence of
sand additions and topplings, we say that $c'$ is \emph{accessible}
form $c$ and we call $c'$ a \emph{successor} of $c$. We denote this by
$c \leadsto c'$. Moreover, if $c'$ is obtained from $c$ by a sequence
of topplings, then the \emph{toppling vector} $f$ associated to the
stabilization $c \leadsto c'$ is the integer vector indexed by the
non-sink vertices of $G$ with $f(v)$ equal to the number of times
vertex $v$ appears in the vertex toppling sequence that sends $c$ to
$c'$. Finally, given a sandpile $c$, the \emph{unique stable} sandpile
obtained after a sequence of topplings is denoted by $c^{\circ}$ and
is called the \emph{stabilization} of $c$.
Let $G = (V, E, s)$ be a graph and $a, b$ be
two sandpiles on $G$. Then $a + b$ denotes the sandpile obtained
by adding the grains of sand vertex-wise, that is, $(a + b)(v) = a(v)
+ b(v)$ for each $v \in \widetilde{V}$. Note that even if $a$ and $b$
are stable, $a+b$ may not be. We denote the stabilization
of $a + b$ by $a \oplus b$, that is, $a \oplus b = (a +
b)^{\circ}$. The binary operator $\oplus$ is called \emph{stable
addition}.
\begin{defn} A sandpile $c$ is \emph{recurrent} if it is stable
and given any sandpile $a$, there exists a sandpile $b$ such that
$$a\oplus b = c.$$
\end{defn}
As an example, given a graph $G = (V, E, s)$, the sandpile $\mathrm{max}$
defined by $\mathrm{max}(v) = d_v-1$ for each $v$ in $\widetilde{V}$ is
recurrent. We call $\mathrm{max}$ the maximal stable sandpile on $G$.
The following well-known proposition gives a simpler way to compute
the recurrent
sandpiles.
\begin{prop}\label{prop:RecurrentCondition} A stable sandpile $c$
is recurrent if and only if there exists a sandpile $b$ with
$$\mathrm{max} \oplus b = c.$$
\end{prop}
As mentioned before, the set of recurrent sandpiles on a graph
$G$ under stable addition forms a finite abelian group denoted
$\mathscr{S}(G)$, see \cite{DRSV95}. Explicitly, the sandpile group of
$G$ is isomorphic to the cokernel of the \emph{reduced Laplacian}
matrix of $G$.
Moreover, this matrix
can be used to compute the stabilization of a sandpile algebraically.
\subsection{Graph Laplacians}
\begin{defn} Let $G$ be a graph with $n$ vertices
$v_{1},v_{2},\dots, v_{n}$. The \emph{Laplacian}
of $G$, denoted $L = L(G)$, is the $n\times n$ matrix defined by
\[L_{ij} =
\begin{cases} -\weight(v_{i},v_{j}) & \text{for } i\neq j,\\ d_{v_{i}} &
\text{for } i=j.
\end{cases}
\]
\end{defn}
The \emph{reduced Laplacian} of a graph $G = (V, E, s)$, denoted,
$\widetilde{L} = \widetilde{L}(G)$, is the matrix obtained by deleting the row and column
corresponding to the sink $s$ from the matrix $L$.
Kirchhoff's Matrix-Tree Theorem implies that the number of
recurrent sandpiles in $G$, that is, the determinant of reduced
Laplacian $\widetilde{L}$ equals the number of spanning trees of $G$.
From our definition
we also have that if $c
\leadsto c'$ by toppling vertex $v$, then $c' = c - \widetilde{L} 1_{v}$, where
$1_{v}$ denotes the (column) vector with
$1_{v}(v) = 1$ and $1_{v}(w) = 0$, for all $w\neq v$ in $\widetilde{V}$. The
previous observation leads to the following result.
\begin{prop}\label{prop:laplacian}
Given a graph $G = (V, E, s)$, and a
sandpile $c$. If $c \leadsto c'$ by a sequence of
topplings, then $c' = c - \widetilde{L} f$, where $f$ is the (column)
toppling vector associated to $c \leadsto c'$.
\end{prop}
The following result, known as Dhar's Burning Criterion, gives an
alternative and useful way to
characterize recurrent sandpiles in an undirected graph.
Given $G = (V,E,s)$, let $u$ denote the sandpile given by $u_j =
\weight(v_j,s)$ for each $v_j\in \widetilde{V}$. We will refer to the sandpile
$u$ as the sandpile obtained by `firing the sink'.
\begin{prop}[{\cite[Corollary 2.6]{CR00}}] \label{prop:Burn}
The sandpile $c$ is recurrent if and only if $u \oplus c =
c$. Moveover, the firing vector in the stabilization of $u \oplus c$
is $(1,\dots,1)$.
\end{prop}
\begin{cor}\label{lem:ToppleOnce} Let $G = (V,E,s)$ be a graph
and $c$ a recurrent sandpile on $G$.
When one grain of sand is added to a
vertex adjacent to the sink then every vertex can topple at most once.
\end{cor}
\subsection{Avalanche Polynomials} The (univariate) avalanche
polynomial was introduced
in \cite{CDR04}. This
polynomial enumerates the sizes of all principal avalanches.
\begin{defn} Let $G = (V,E,s)$ be a graph and let $v\in
\widetilde{V}$.
Let $c$ be a recurrent sandpile $c$ on
$G$ and $v\in \widetilde{V}$, the
\emph{principal avalanche} of $c$ at $v$ is the
sequence of vertex topplings
resulting from the stabilization of the sandpile $c+1_v$.
\end{defn}
\begin{defn} The avalanche polynomial for a graph $G$ is defined
as
\[ \mathscr{A}_{G}(x) = \sum \lambda_{m}x^{m}, \] where $\lambda_{m}$ is the
number of principal avalanches of size $m$.
\end{defn}
\begin{ex} \label{ex:CycleAvalanche}
In this example, we consider the 3-cycle $C_3$. The
recurrent sandpiles on $C_3$ are $(1,0), (0,1),$ and $(1,1)$.
The table in Figure \ref{table:C3} records
the size of the avalanche for the corresponding recurrent and vertex.
Therefore,
$\mathscr{A}_{C_{3}} (x) = 2x^{2} + 2x + 2$.
\begin{figure}[!htbp]
\begin{minipage}[c]{0.3\linewidth}
\centering
\begin{tikzpicture}
\tikzstyle{VertexStyle} = [shape = circle, draw = black,
inner sep = 2pt, outer sep = 0.5pt, minimum size = 2mm,
line width = 1pt]
\SetUpEdge[lw=1.5pt]
\SetGraphUnit{2}
\tikzset{EdgeStyle/.style={-}}
\Vertex[Math,L=v_1]{1}
\EA[Math,L=v_2](1){2}
\SO[Math,L=s](1){3}
\Edges(1,2,3,1)
\end{tikzpicture}
\end{minipage}
\begin{minipage}[c]{0.45\linewidth}
\centering
\begin{tabular}{c|cc}
{Recurrents} & $v_{1}$ \hspace{2pt} & $v_{2}$ \\ \hline
(1,0) & 1 & 0 \\
(0,1) & 0 & 1 \\
(1,1) & 2 & 2
\end{tabular
\end{minipage}
\caption{Principal avalanche sizes on $C_{3}$.}
\label{table:C3}
\end{figure}
\end{ex}
\subsection{Multivariate Avalanche Polynomial}
The (univariate) avalanche polynomial does not contain any information
regarding which vertices topple in a given principal avalanche. Here
we introduce the \emph{multivatiate avalanche polynomial} that encodes
this information.
\begin{defn} Let $G = (V,E,s)$ be a graph on $n+1$ vertices
and let $\widetilde{V} = \{v_1,\dots,v_n\}$. Given $k \in \{1,\dots, n\}$ and a
recurrent sandpile $c$, the \emph{avalanche monomial} of $c$ at $v_k$ is
$$\mu_{G} (c, v_k) = \bm{x}^{\nu(c,v_{k})} = \prod_{i=1}^n x_i^{f_i}$$
where $\nu(c,v_{k}) = (f_{1},\dots, f_{n})$ is the toppling vector of the
stabilization of $c+1_{v_{k}}$.
\end{defn}
\begin{defn} Let $G = (V,E,s)$ be a graph.
The \emph{multivariate avalanche polynomial} of $G$ is defined by
$$ \mathscr{A}_G(x_1,\dots,x_n)=\sum_{c\in \mathscr{S}(G)} \sum_{v\in\widetilde{V}}
\mu_{G} (c, v).$$
\end{defn}
Note that the multivariate avalanche polynomial is the sum of all
possible avalanche monomials. In what follows the term ``avalanche
polynomial'' will refer to the multivariate case.
\begin{ex}
\label{ex:CycleToppling}
As in Example \ref{ex:CycleAvalanche} we have recurrents $(1,0),
(0,1),$ and $(1,1)$ in $C_{3}$. The table in Figure
\ref{table:C3multi} records the toppling
vector $\nu(c,v_{i})$ for the
principal avalanche of $c$ at $v_i$.
\begin{figure}[!htbp]
\begin{minipage}[c]{0.3\linewidth}
\centering
\begin{tikzpicture}
\tikzstyle{VertexStyle} = [shape = circle, draw = black,
inner sep = 2pt, outer sep = 0.5pt, minimum size = 2mm,
line width = 1pt]
\SetUpEdge[lw=1.5pt]
\SetGraphUnit{2}
\tikzset{EdgeStyle/.style={-}}
\Vertex[Math,L=v_1]{1}
\EA[Math,L=v_2](1){2}
\SO[Math,L=s](1){3}
\Edges(1,2,3,1)
\end{tikzpicture}
\end{minipage}
\begin{minipage}[c]{0.45\linewidth}
\centering
\begin{tabular}{ c|c c}
Recurrents & $v_{1}$ \hspace{2pt} & $v_{2}$ \\ \hline
(1,0) & (1,0) & (0,0) \\
(0,1) & (0,0) & (0,1) \\
(1,1) & (1,1) & (1,1)
\end{tabular}
\end{minipage}
\caption{Toppling vectors on $C_{3}$.}
\label{table:C3multi}
\end{figure}
From this table we can easily read the avalanche monomials. For
example, $\mu_{C_3}((1,0),v_1) = x_1^1x_2^0 = x_1$. Adding all
avalanche monomials gives the avalanche polynomial
$$\mathscr{A}_{C_{3}} (x_1,x_2) = x_1^1 x_2^0 +2 x_1^0 x_2^0 + x_1^0 x_2^1 + 2
x_1^1 x_2^1 = 2x_1 x_2 + x_1 + x_2 + 2.$$
\end{ex}
Note that the univariate avalanche polynomial for a graph $G$ can
be recovered from the multivariate avalanche polynomial by
substituting each $x_{i}$ by $x$, i.e.,
\[\mathscr{A}_{G}(x) = \mathscr{A}_{G}(x,\dots, x).\]
It is a well-known fact that the sandpile group of an undirected graph
is independent of the choice of sink \cite[Proposition
1.1]{CR00}. Nevertheless, the structure of
the individual recurrent sandpiles may differ. This implies that the
avalanche polynomial of a graph is dependent on the choice of
sink. For this reason, we will always fix a sink before discussing the
avalanche polynomial of a graph. Nevertheless,
for certain graphs like cycles and complete graphs, the avalanche
polynomial does not depend on the choice of sink. For other families
such as wheel and fan graphs there is a natural choice of sink, namely
the dominating vertex. For trees, our method computes the avalanche
polynomial for any choice of sink.
\subsection{Burst Size}
There are other evaluations of the multivariate avalanche polynomial
$\mathscr{A}_{G}(x_{1},\dots, x_{n})$ that are
relevant in the larger field of sandpile groups.
In \cite{L15}, Levine introduces the concept of \emph{burst size} to
prove a conjecture of Poghosyan, Poghosyan, Priezzhev, Ruelle
\cite{PPPR11} on the relationship between the threshold state of the
fixed-energy sandpile and the stationary state of Dhar's abelian
sandpile.
\begin{defn}
Let $G = (V,E,s)$ be a graph and $c$ be a recurrent sandpile on $G$. Given $v\in
\widetilde{V}$, define the \emph{burst size} of $c$ at $v$ as
$$\text{av}(c,v):= |c'| - |c| + 1,$$
where the sandpile $c'$ is defined such that $c'\oplus 1_{v} = c$ and
$|c|$ denotes the number of grains of sand in $c$.
Equivalently, $\text{av}(c,v)$ is the number of grains that fall into
the sink $s$ during the stabilization of $c'+1_{v} \leadsto c$.
\end{defn}
Let $G = (V,E,s)$ be a \emph{simple graph} and $\mathscr{A}_{G}(x_{1},\dots,
x_{n})$ be its multivariate avalanche polynomial. Now, let $x_{i} = 1$
for each vertex $v_{i}$ that is not adjacent to the sink $s$. Also,
for each vertex $v_{j}$ adjacent to $s$, let $x_{j} = x$.
The resulting univariate polynomial $\mathscr{B}(x) = \sum_{k} b_{k}x^{k}$
satisifes the condition that $b_{k}$ is the number of principal
avalanches with burst size $k$.
\section{Toppling Polynomials of Trees}\label{trees}
Let $T$ be a tree on $n+1$ vertices
labelled $v_{1},\dots, v_{n},s$. Assume further that $T$ is
rooted at the sink $s$. It is a basic observation that $T$ has only
one recurrent, namely $\mathrm{max}_{T}$. So
$$\mathscr{A}_T(x_{1},\dots, x_{n}) = \sum_{v\in\widetilde{V}} \mu_T(\mathrm{max}_{T},v).$$
As noted in \cite{CDR04} any tree can be constructed from a
single vertex using two operations $\phi$ and $+$ defined below.
\begin{defn}
\label{defn:root} For two trees $T$ and $T'$ rooted at $s$ and $s'$
respectively, the operation $+$, called \emph{tree addition},
identifies $s$ and $s'$. For a tree $T$ rooted at $s$, the operation
$\phi$, called \emph{grafting} or \emph{root extension}, refers to
adding an edge from $s$ to a new root $s'$.
\end{defn}
The operations $\phi$ and $+$ can be seen in Figure \ref{fig:phi+}.
Theorem \ref{thm:ToppOfTrees} explains what happens to the toppling
polynomial of a tree under these operations.
\begin{figure}[!h]
\begin{center}
\begin{tikzpicture}[scale=1]
\tikzstyle{VertexStyle} = [shape = circle, draw = black,
inner sep = 2pt, outer sep = 0.5pt, minimum size = 2mm,
line width = 1pt]
\SetUpEdge[lw=1.5pt]
\SetGraphUnit{2}
\tikzset{EdgeStyle/.style={-}}
\Vertex[Math,L=s]{1}
\node at (0,.5) {$T_1$};
\node at (-2,.5) {$T_2$};
\SO[Math,L=v_2](1){2}
\node at (0,-2.75) {$\mathscr{A}_{T_1}=x_2$};
\Edges(1,2)
\WE[Math,L=s](1){3}
\SOWE[Math,L=v_1](1){4}
\node at (-2,-2.75) {$\mathscr{A}_{T_2}=x_1$};
\Edges(3,4)
\draw [->] (.5,-1) -- (2,-1);
\node at (1.25,-0.8) {+};
\coordinate (5) at (2,0);
\coordinate (8) at (1,0);
\EA[Math,L=s](8){7}
\SO[Math,L=v_1](5){6}
\EA[Math,L=v_2](6){9}
\Edges(6,7,9)
\node at (3,-2.75) {$\mathscr{A}_{T_1 + T_2}=x_1+x_2$};
\draw [->] (4,-1)--(5.5,-1);
\node at (4.75,-0.8) {$\phi$};
\coordinate (11) at (4.5,-1);
\coordinate (14) at (5.5,-1);
\EA[Math,L=v_3](11){12}
\NO[Math,L=s'](12){13}
\SO[Math,L=v_1](14){15}
\EA[Math,L=v_2](15){16}
\Edges(13,12,15,12,16)
\node at (6.5,-3.75) {$\mathscr{A}_{\phi(T_1 + T_2)}=x_1x_2x_3(x_1+x_2+1)$};
\end{tikzpicture}
\caption{Effect of $+$ and $\phi$ operations on the avalanche polynomial}
\label{fig:phi+}
\end{center}
\end{figure}
\begin{thm}
\label{thm:ToppOfTrees} Let $\mathscr{A}_{T}$, $\mathscr{A}_{T_1}$, and $\mathscr{A}_{T_2}$ be
the avalanche polynomials of trees $T$, $T_1$, and $T_2$,
respectively. Then
\begin{enumerate}
\item $\mathscr{A}_{T_1+T_2}= \mathscr{A}_{T_1} +\mathscr{A}_{T_2}$,
\item $\mathscr{A}_{\phi(T)}=x_1x_2 \cdots x_n(\mathscr{A}_{T}+1)$, where
$n = |\widetilde{V}(\phi(T))|$.
\end{enumerate}
\end{thm}
\begin{proof} Under tree addition, trees $T_1$ and $T_2$ are only
connected at the sink $s$. Since the sink never topples, a principal
avalanche at a vertex in $T_1$ will never affect the vertices in
$T_2$, and vice versa. Therefore, $\mathscr{A}_{T_1+T_2}= \mathscr{A}_{T_1}
+\mathscr{A}_{T_2}$.
For the second part, let $T$ be a tree on $n$ vertices with sink $s$.
Let $\mathrm{max}_{\phi(T)}$ and $\mathrm{max}_{T}$ be
the maximum stable sandpile on $\phi(T)$ and $T$, respectively. First
we consider $\mathrm{max}_{\phi(T)}
\oplus 1_s$. By Proposition \ref{prop:Burn},
$$\mu_{\phi(T)} (\mathrm{max}_{\phi(T)}, s) = x_1x_2\cdots x_n.$$
Now consider $\mathrm{max}_{\phi(T)}\oplus 1_{v_k}$ where $v_k\neq s$. Note
that when we apply the toppling sequence
$\nu_T(\mathrm{max}_T,v_k)$ to $\mathrm{max}_{\phi(T)} + 1_{v_k}$, we get the
sandpile $\mathrm{max}_{\phi(T)} + 1_s$. Thus,
each vertex will now topple once more. So for each $v_k$ with $v_k\neq
s$,
$$\mu_{\phi(T)}(\mathrm{max}_{\phi(T)},v_k) = (x_1x_{2}\cdots x_n ) \cdot
\mu_{T} (\mathrm{max}_T, v_k).$$
Therefore,
$$\mathscr{A}_{\phi(T)} = (x_1x_2 \cdots x_n) + (x_1x_2 \cdots x_n) \cdot
\mathscr{A}_{T} =x_1x_2 \cdots x_n(\mathscr{A}_{T}+1).$$
\end{proof}
Note that using Theorem \ref{thm:ToppOfTrees} we can
compute the multivariate avalanche polynomial of any tree.
Furthermore,
as noted in \cite{CDR04}, it is possible for two non-isomorphic trees
to have the same univariate avalanche polynomial. In contrast, the
multivariate avalanche polynomial distinguishes between labeled trees.
\begin{figure}[!hbpt]
\begin{center}
\begin{multicols}{2}
\begin{tikzpicture}[scale=.6]
\tikzstyle{VertexStyle} = [shape = circle, draw = black,
inner sep = 2pt, outer sep = 0.5pt, minimum size = 2mm,
line width = 1pt]
\SetUpEdge[lw=1.5pt]
\SetGraphUnit{2}
\tikzset{EdgeStyle/.style={-}}
\Vertex[Math,L=s]{0}
\node at (-2,0) {$T_1$};
\SOWE[Math,L=4](0){1}
\SOEA[Math,L=7](0){2}
\SO[Math,L=7](1){3}
\SO[Math,L=8](2){4}
\WE[Math,L=8](4){6}
\EA[Math,L=8](4){8}
\coordinate (5) at (3,-4);
\WE[Math,L=9](5){7}
\EA[Math,L=8](7){10}
\coordinate (9) at (-1,-6);
\WE[Math,L=8](9){11}
\EA[Math,L=8](11){12}
\SO[Math,L=10](7){13}
\Edges(0,1,3,11,3,12)
\Edges(0,2,4,2,10,2,6,2,8,2,7,13)
\end{tikzpicture}
\columnbreak
\begin{tikzpicture}[scale=.6]
\tikzstyle{VertexStyle} = [shape = circle, draw = black,
inner sep = 2pt, outer sep = 0.5pt, minimum size = 2mm,
line width = 1pt]
\SetUpEdge[lw=1.5pt]
\SetGraphUnit{2}
\tikzset{EdgeStyle/.style={-}}
\Vertex[Math,L=s]{0}
\node at (-2,0) {$T_2$};
\SOWE[Math,L=4](0){1}
\coordinate (2) at (-2,-1.5);
\SO[Math,L=7](2){3}
\coordinate (6) at (-2,-3);
\SO[Math,L=9](6){4}
\coordinate (7) at (-2,-4.5);
\SO[Math,L=10](7){5}
\SOEA[Math,L=7](0){8}
\coordinate (9) at (2.5,-4);
\WE[Math,L=8](9){10}
\EA[Math,L=8](10){11}
\EA[Math,L=8](11){12}
\coordinate (13) at (1.5,-4);
\WE[Math,L=8](13){14}
\EA[Math,L=8](14){15}
\EA[Math,L=8](15){16}
\Edges(0,1,3,4,5)
\Edges(0,8,16,8,15,8,14,8,12,8,11,8,10)
\end{tikzpicture}
\end{multicols}
\caption{$\mathscr{A}_{T_1}(x) = \mathscr{A}_{T_2}(x) = x^{10}+x^9+6x^8+2x^7+x^4$}
\label{fig:noniso}
\end{center}
\end{figure}
Figure \ref{fig:noniso} gives an example, first presented in
\cite{CDR04}, of two non isomorphic trees with the same univariate
avalanche polynomial. The vertices are labeled with the size of the
principal avalanche starting at that vertex. One can clearly see that
$T_1$ and $T_2$ have the same univariate avalanche
polynomial. However, they have different multivariate avalanche
polynomials. Let's examine the right subtrees of $T_1$ and $T_2$,
denoted by $R_{1}$ and $R_{2}$, respectively.
\begin{figure}[!h]
\centering
\begin{multicols}{2}
\begin{tikzpicture}[scale=.6]
\tikzstyle{VertexStyle} = [shape = circle, draw = black,
inner sep = 2pt, outer sep = 0.5pt, minimum size = 2mm,
line width = 1pt]
\SetUpEdge[lw=1.5pt]
\SetGraphUnit{2}
\tikzset{EdgeStyle/.style={-}}
\Vertex[Math,L=s]{s}
\node at (-2,0) {$R_1$};
\SOWE[Math,L=v_7](s){v7}
\SO[Math,L=v_8](s){v8}
\SOEA[Math,L=v_9](s){v9}
\EA[Math,L=v_{10}](v9){v10}
\WE[Math,L=v_{6}](v7){v6}
\SO[Math,L=v_5](v7){v5}
\Edges(s,v6,s,v7,v5)
\Edges(s,v8,s,v9,s,v10)
\end{tikzpicture}
\columnbreak
\begin{tikzpicture}[scale=.6]
\tikzstyle{VertexStyle} = [shape = circle, draw = black,
inner sep = 2pt, outer sep = 0.5pt, minimum size = 2mm,
line width = 1pt]
\SetUpEdge[lw=1.5pt]
\SetGraphUnit{2}
\tikzset{EdgeStyle/.style={-}}
\Vertex[Math,L=s]{s}
\node at (-2,0) {$R_2$};
\SO[Math,L=v_7](s){v7}
\SOEA[Math,L=v_8](s){v8}
\EA[Math,L=v_9](v8){v9}
\EA[Math,L=v_{10}](v9){v10}
\WE[Math,L=v_{6}](v7){v6}
\WE[Math,L=v_5](v6){v5}
\Edges(v5,s,v6,s,v7,s,v8,s,v9,s,v10)
\end{tikzpicture}
\end{multicols}
\label{F:subtrees}
\caption{Labelled right subtrees of $T_{1}$ and $T_{2}$.}
\end{figure}
The avalanche polynomial of $R_{1}$ is
$x_6 + x_5x_7(x_{5}+1) + x_8 + x_9 +x_{10}$
and the avalanche polynomial of $R_{2}$
is $x_5 + x_6 + x_7 + x_8 + x_9 + x_{10}$.
Note that the first polynomial has total degree 3 and the second polynomial
has total degree 1. The avalanche polynomials of the left
subtrees of $T_1$ and $T_2$ will be on a disjoint set of variables.
Thus, the avalanche polynomials for $T_1$ and $T_2$ must be distinct.
\begin{cor}\label{cor:PhiIsEffective}
Let $T$ and $T'$ be two trees on $n+1$
vertices. Then,
$\mathscr{A}_T(x_1,\dots,x_n) = \mathscr{A}_{T'}(x_1,\dots,x_n)$ if and only if
$\mathscr{A}_{\phi(T)}(x_1,\dots,x_{n+1}) = \mathscr{A}_{\phi(T')}(x_1,\dots,x_{n+1})$.
\end{cor}
\begin{proof}
Suppose that $\mathscr{A}_{T} = \mathscr{A}_{T'}$. Theorem \ref{thm:ToppOfTrees} implies
\[\mathscr{A}_{\phi(T)} = x_1x_2 \cdots x_n(\mathscr{A}_{T}+1) = x_1x_2 \cdots
x_n(\mathscr{A}_{T'}+1) = \mathscr{A}_{\phi(T')}.\]
Now assume $\mathscr{A}_{\phi(T)} = \mathscr{A}_{\phi(T')}$. Then $x_1x_2 \cdots x_n(\mathscr{A}_{T}+1) = x_1x_2 \cdots
x_n(\mathscr{A}_{T'}+1)$. This clearly implies
$\mathscr{A}_{T} = \mathscr{A}_{T'}$.
\end{proof}
\begin{thm}\label{uniqueness}
Let $T$ be a tree on $n+1$ vertices. If $\mathscr{A}_T(x_1,\dots, x_n) =
\mathscr{A}_{T'}(x_{1},\dots, x_{n})$ for some tree $T'$, then $T = T'$.
\end{thm}
\begin{proof}
We use induction on the height of $T$.
Recall that
the \emph{height} of a tree is the number of edges in the longest path
between the root and a leaf.
Suppose $T$ has height $0$, that is, $T$ consists of one vertex. Then
$\mathscr{A}_{T} = 0$. Clearly, if $T'$ has two or more vertices then
$\mathscr{A}_{T'} \neq 0$. Since $\mathscr{A}_{T} = \mathscr{A}_{T'}$, then $T'$ must also
consist of one vertex and $T=T'$.
Now suppose that $T$ has height $h>0$. In this case, the sink $s$ of $T$
must have at least one child. Assume $s$ has degree $d$.
Deleting $s$ creates
$d$ trees $T_1,T_2,\dots,T_{d}$. We have that $T = \phi(T_{1}) +
\phi(T_{2}) + \dotsm + \phi(T_{d})$. So $\mathscr{A}_{T}$ is the sum of $d$
multivariate polynomials with pairwise disjoint supports
\[\mathscr{A}_{T} = \mathscr{A}_{\phi(T_{1})} + \mathscr{A}_{\phi(T_{2})} + \dotsm +
\mathscr{A}_{\phi(T_{d})}. \]
Since
$\mathscr{A}_{T} = \mathscr{A}_{T'}$, then $\mathscr{A}_{T'}$ must also satisfy the same
condition. Hence the sink of $T'$ must also have degree $d$ and
$T' = \phi(T'_{1}) + \phi(T'_{2}) + \dotsm + \phi(T'_{d})$ for some
trees $T'_1,T'_2,\dots,T'_{d}$. Since the supports are pairwise
disjoint, we must also have that
$\mathscr{A}_{\phi(T_{1})} = \mathscr{A}_{\phi(T'_{j})}$, for some $j$. Corollary
\ref{cor:PhiIsEffective} implies
$\mathscr{A}_{T_{1}} = \mathscr{A}_{T'_{j}}$. But $T_{1}$ is a tree of height
$h-1$. By induction, $T_{1} = T'_{j}$. Therefore,
after relabeling the subtrees in $T'$, we must have $T_{i} = T'_{i}$
for all $1\leq i \leq d$ and $T = T'$.
\end{proof}
\section{Toppling Polynomials of Cycles}\label{cycles}
Now, we will compute the avalanche
polynomial of the cycle graph $C_{n+1}$ on $n+1$ vertices. Unless
otherwise stated, we will label the vertices $s, v_1, v_2, \dots, v_n$
in a clockwise manner. As shown in Example \ref{ex:CycleAvalanche},
$\mathscr{A}_{C_{3}}(x_{1},x_{2}) = 2x_1x_2 + x_1 + x_2 + 2$. We will
denote by $C_{2}$ the graph with two vertices and two edges
between these vertices.
It is clear that $\mathscr{A}_{C_{2}}(x_{1}) = x_{1} + 1$.
In this section, we will write sandpiles and toppling vectors
as strings instead of vectors. For example, the string
$1^{p-1}01^{n-p}$ denotes the sandpile with no grains of sand at
vertex $v_p$ and $1$ grain of sand at every other vertex. We will also
make the convention that a bit raised to the $0$ power does not appear
in the string, e.g., $0^{2}1^{0}0^{2} = 0^{4}$. In the
previous section, we saw that a tree has exactly one recurrent
sandpile. The cycle graph $C_{n+1}$ has exactly
$n+1$ recurrent sandpiles, namely, $\mathrm{max} = 1^n$ and $b_{p} =
1^{p-1}01^{n-p}$ for $p = 1, 2, \dots, n$, see \cite{CDR04}.
\subsection{Toppling Sequence for the Maximal Stable Sandpile} We
first focus our attention on understanding the toppling sequences for
$1^n + 1_{v_i}$ for $1\leq i \leq n$.
\begin{ex} \label{ex:cycle:1}
Figure \ref{cycletopple6} shows that
$\mu_{C_6}(1^5, v_2) = x_1x_2^2x_3^2x_4^2x_5$.
\end{ex}
\begin{figure}[!h]
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\caption{The principal avalanche created by adding a grain of sand to
$v_2$.}
\label{cycletopple6}
\end{figure}
\begin{lemma}
\label{lem:SinkVertices} Let $n\geq 1$. Then
$$\mu_{C_{n+1}}(1^n, v_1) = \mu_{C_{n+1}}(1^n,v_n) = x_1x_{2}\cdots x_n.$$
\end{lemma}
\begin{proof} Because $v_1$ and $v_n$ are adjacent to the sink, each
vertex topples at most once by Corollary \ref{lem:ToppleOnce}. Now
consider $\mathrm{max} + 1_{v_1}$.
Toppling $v_1$ results in $v_2$ being
unstable. Inductively, for $i\geq 2$, if $v_i$ becomes unstable it
will topple and $v_{i+1}$ will become unstable.
Thus each vertex topples exactly once. A similar argument works for
$\mu_{C_{n+1}}(1^n,v_n)$.
\end{proof}
Observe that the reduced Laplacian of the cycle $C_{n+1}$ is the
$n\times n$ matrix
$$\widetilde{L} = \begin{bmatrix*}[r]
2 &-1 & & & &\\ -1 &2 &-1 & & & \\ & \ddots &\ddots &\ddots & & \\ &
&-1 &2 &-1 & \\ & & &-1 &2
\end{bmatrix*},$$ with 2's on the diagonal, $-1$'s on the off
diagonals, and 0's elsewhere.
We will use the reduced Laplacian in the next two proofs. We will
abuse notation and write $\widetilde{L} b$ instead of $\widetilde{L} b^{t}$ to denote the
product of $\widetilde{L}$ times the vector $b$.
\begin{lemma}
\label{lem:FirstStage} Let $n\geq 3$, $2\leq i \leq n-1$ and consider the
sandpile $1^n + 1_{v_i}$ on $C_{n+1}$. We can legally topple each
vertex once and the resulting sandpile is $01^{i-2}21^{n-i-1}0$.
\end{lemma}
\begin{proof} The sequence $v_i,v_{i-1},v_{i-2},\dots,
v_1,v_{i+1},v_{i+2},\dots,v_{n}$ is a legal toppling sequence.
By Proposition
\ref{prop:laplacian}, $1^n +1_{v_i}$ accesses the sandpile
$$(1^n +1_{v_i}) - \widetilde{L}\cdot 1^n = (1^n + 1_{v_i}) -
10^{n-2}1 = 01^{i-2}21^{n-i-1}0.$$
\end{proof}
Lemma \ref{lem:FirstStage} states that
$1^n +1_{v_i}\leadsto 01^{i-2}21^{n-i-1}0$, another
unstable sandpile. Lemma \ref{lemma:OtherStages} will describe the
remaining toppling pattern.
\begin{lemma} \label{lemma:OtherStages}
Let $n\geq 3$, $2\leq i\leq n-1$, and $1\leq k
\leq \min\{i-1,n-i\}$. Let $c_{k}$ be the sandpile
$$c_k = 1^{k-1}01^{i-k-1}21^{n-i-k}01^{k-1}.$$
Then for each $1\leq k\leq \min\{i-2,n-i-1\}$ the sandpile
$c_{k}$ accesses the sandpile $c_{k+1}$
via toppling vertices $v_{k+1},v_{k+2},\dots,v_{n-k}$.
\end{lemma}
\begin{proof} The toppling sequence $v_i,v_{i-1},\dots, v_{k+1},
v_{i+1},\dots,v_{n-k}$ is a legal toppling sequence. Applying this
toppling sequence we obtain
$\widetilde{L}\cdot 0^{k}1^{n-2k}0^{k} =
0^{k-1}(-1)10^{n-2k-2}1(-1)0^{k-1}$. Thus, Proposition
\ref{prop:laplacian} implies $c_{k}$ accesses the sandpile
\[
1^{k-1}01^{i-k-1}21^{n-i-k}01^{k-1} - 0^{k-1}(-1)10^{n-2k-2}1(-1)0^{k-1}
= 1^k01^{i-k-2}21^{n-i-k-1}01^k = c_{k+1}.
\]
\end{proof}
\begin{thm}
\label{thm:CycleTopplingMonomialsMax} For $n\geq 1$, $1\leq i \leq n$,
let $m = \min\{i,n-i+1\}$. Then
$$\mu_{C_{n+1}} (1^n, v_i) = \prod_{j=1}^{m} x_j\cdots
x_{n-j+1}=(x_1\cdots x_n)(x_2\cdots x_{n-1})\cdots(x_m\cdots
x_{n-m+1}).$$
\end{thm}
\begin{proof}
The cases for $n=1$ and $n=2$ are discussed in the first paragraph of
this section. Now assume $n\geq 3$.
If $i = 1$ or $i=n$, the result follows from Lemma
\ref{lem:SinkVertices}. Suppose $2 \leq i \leq n-1$, by Lemma
\ref{lem:FirstStage} $\max+1_{v_i} \leadsto c_1$ via the toppling
of all vertices which gives the factor $x_1x_2\cdots
x_n$. By Lemma \ref{lemma:OtherStages},
$c_1 \leadsto c_2 \leadsto \cdots \leadsto c_{m-1}.$
Note that $c_{k}\leadsto c_{k+1}$ via
the toppling of vertices $v_{k+1},\dots, v_{n-k}$. This produces the factor
$x_{k+1}\cdots x_{n-k}$.
Suppose that $m=i=\min\{i,n-i+1\}$. Then
$c_{m-1} = c_{i-1} = 1^{i-2}021^{n-2i+1}01^{i-2}$. Only vertex $v_i$
is unstable, but when $v_i$ topples $v_{i-1}$ does not become
unstable. Vertex $v_{i+1}$ does become unstable and will topple. In
fact, vertices $v_i, v_{i+1}, \dots, v_{n-i+1}$ will topple resulting
in the stable sandpile $1^{n-i}01^{i-1}$. This gives the
last factor $x_m\cdots x_{n-m+1}$. So, if
$m= i = \min\{i,n-i+1\}$,
$$\mu_{C_{n+1}}(1^n,v_i) = \prod_{j=1}^m x_j\dotsm x_{n-j+1}.$$
The proof is similar for the case $m = n-i+1 = \min\{i,n-i+1\}$.
\end{proof}
\subsection{The toppling sequence for recurrents $1^{p-1}01^{n-p}$}
Theorem \ref{thm:CycleTopplingMonomialsMax} gives us the avalanche
monomials for the maximal stable sandpile at all vertices. Now we find
the avalanche monomials for recurrents of the form $1^{p-1}01^{n-p}$.
We will see that these monomials are closely related to the
avalanche monomials arising from $1^n$.
\begin{ex} \label{ex:cycle:2} In Example \ref{ex:cycle:1}, we saw
$\mu_{C_6} (1^5,v_2) = x_1x_2^2x_3^2x_4^2x_5$. Figure \ref{1p01q}
shows that $\mu_{C_{10}}(1^301^5,v_6) = x_5x_6^2x_7^2x_8^2x_9$. Notice
that the structure of these monomials.
The only difference is that there is a relabeling of the
variables $x_i \to x_{i+4}$.
\end{ex}
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\caption{The principal avalanche of $1^301^5 + 1_{v_{6}}$.}
\label{1p01q}
\end{figure}
Based on what we've seen in Example \ref{ex:cycle:2}, we may guess
that the toppling monomials associated to $1^{p-1}01^{n-p}$ are
related to the toppling monomials of $C_{p}$ and $C_{n-p+1}$. This
turns out to be true and will now formally state and prove this
idea. First, we must introduce some useful notation.
\begin{defn}
\label{def:ReindexNotation}
Let $q$ be an integer. Then $C_{n+1}^{q}$ will denote the cycle graph
on $n+1$ vertices labeled
$v_{q+1}, \dots, v_{q+n}, s$.
\end{defn}
\begin{thm}
\label{thm:CycleNonMaxRecurrents}
Let $b_p =1^{p-1}01^{n-p}$ be a
recurrent on $C_{n+1}$ such that $1 \leq p \leq n$.
\begin{enumerate}
\item[(1)]
If $1 \leq i \leq p-1$, then
$\mu_{C_{n+1}} (b_p,v_{i}) = \mu_{C_{p}}(1^{p-1}, v_i)$,
\item[(2)]
If $p+1 \leq i \leq n$, then
$\mu_{C_{n+1}} (b_p,v_{i}) = \mu_{C_{n-p+1}^{p}} (1^{n-p},v_{i})$,
\item[(3)]
$\mu_{C_{n+1}}(b_p,v_p) = 1$.
\end{enumerate}
\end{thm}
\begin{proof}
Let $1 \leq i \leq p-1$, this implies $p\geq 2$. If $p=2$, then
$i=1$, and $b_p + 1_{v_1} = 201^{n-2} \leadsto 0^21^{n-2}$. So
$\mu_{C_{n+1}}(b_p,v_{1}) = x_{1} = \mu_{C_{2}}(1, v_{1})$ and the
result holds.
Assume $p\geq 3$. If $i = 1$, then $v_{1},\dots, v_{p-1}$ is a legal
toppling sequence and $b_{p} + 1_{v_{1}} \leadsto
1^{p-2}01^{n-p+1}$. Lemma \ref{lem:SinkVertices} implies
$\mu_{C_{n+1}}(b_{p},v_{1}) = x_{1}x_{2}\dotsm x_{p-1} =
\mu_{C_{p}}(1^{p-1},v_{1})$. Similarly, if $i = p-1$, then
$v_{p-1},v_{p-2},\dots, v_{1}$ is a legal toppling sequence and
$b_{p} + 1_{v_{p-1}} \leadsto 01^{n-1}$. So again
$\mu_{C_{n+1}}(b_{p},v_{p-1}) =
x_{1}x_{2}\dotsm x_{p-1} = \mu_{C_{p}}(1^{p-1},v_{p-1})$.
If $p\geq 3$ and $2 \leq i \leq p-2$, we have
$b_p + 1_{v_i} = 1^{p-1}01^{n-p} + 1_{v_{i}} =
1^{i-1}21^{p-i-1}01^{n-p}$. Note that $v_i, v_{i-1},\dots, v_1, v_{i+1},
\dots, v_{p-1}$ is a legal toppling sequence. So $b_p + 1_{v_i} \leadsto
01^{i-2}21^{p-i-2}01^{n-p+1}$. From this computation we can deduce two
things. First, during the stabilization of $b_{p} + 1_{v_{i}}$, none
of the vertices $v_{p},\dots, v_{n}$ will topple. Second, the
sandpile $01^{i-2}21^{p-i-2}01^{n-p+1}$ is in fact the sandpile
$c_{1}$ defined in Lemma \ref{lemma:OtherStages} when $n+1 = p$ with
the string $1^{n-p+1}$ concatenated at the end. Therefore, from these
two observations we conclude that the
principal avalanche resulting from $b_{p} + 1_{v_{i}}$ in $C_{n+1}$
follows the pattern described in Lemma \ref{lemma:OtherStages}
for $C_{p}$. Hence $\mu_{C_{n+1}} (b_p,v_{i}) = \mu_{C_{p}}(1^{p-1}, v_i)$.
A similar argument works for the case $p+1 \leq i \leq n$ with the
exception that now the vertices that topple are $v_{p+1},\dots,
v_{n}$. So $\mu_{C_{n+1}} (b_p,v_{i}) = \mu_{C_{n-p+1}^{p}}
(1^{n-p},v_{i})$.
Finally, it is clear that $b_{p} + 1_{v_{p}}$ is stable. So
$\mu_{c_{n+1}}(1^{p-1}01^{n-p},v_{p}) = 1$.
\end{proof}
The following result follows immediately from Theorem
\ref{thm:CycleTopplingMonomialsMax} and Theorem
\ref{thm:CycleNonMaxRecurrents}.
\begin{cor}
\label{thm:Cycles}
The avalanche polynomial for $C_{n+1}$ for $n\geq 1$ is
$$\sum_{i=1}^n \mu_{C_{n+1}}(1^{n}, v_i)
+ \sum_{p=2}^{n} \sum_{i=1}^{p-1} \mu_{C_{p}}(1^{p-1}, v_i)
+ \sum_{p=1}^{n-1} \sum_{i=p+1}^{n} \mu_{C_{n-p+1}^{p}}(1^{n-p}, v_{i})
+ n,$$
where $\displaystyle \mu_{C_{q+1}}(1^{q}, v_i) = \prod_{j=1}^m
x_j\dotsm x_{q-j+1}$, $1\leq i\leq q$, and $m = \min\{i,q-i+1\}$.
\end{cor}
\section{Avalanche Polynomials of Complete Graphs}\label{complete}
In this section we will compute the avalanche polynomial of the
complete graph $K_{n+1}$ on $n+1$ vertices $v_1,v_{2},\dots,
v_{n},s$. In Example \label{ex:CycleToppling} we computed
$\mathscr{A}_{K_{3}}(x_{1},x_{2})$. Using SageMath \cite{sage}, we can compute
the avalanche polynomial of $K_{4}$:
$$\mathscr{A}_{K_4}(x_1,x_2,x_3)=9x_1x_2x_3+2x_1x_2+2x_1x_3+2x_2x_3+3x_1+3x_2+3x_3+24.$$
Note that $\mathscr{A}_{K_{4}}(x) = 9x^3+6x^2+9x+24$. So the set of principal
avalanches of size $m$ is evenly partitioned into $\binom{3}{m}$
subsets for $0\leq m \leq 3$. Moreover, $\mathscr{A}_{K_4}(x_1,x_2,x_3)$ is a
linear combination of elementary symmetric polynomials.
We will show that this characterizes the avalanche polynomial of $K_{n+1}$.
\begin{defn}
Let $m$ be an integer such that $0 \leq m \leq n$.
The \emph{elementary symmetric polynomial} of degree $m$ on variables
$x_1,x_2,\dots,x_n$ is
$$e_{m}(x_{1},\dots, x_{n}) = \sum_{\substack{A \subseteq [n] \\ |A| =
m}} \prod_{i \in A} x_{i}.$$
\end{defn}
Observe that $e_{0}(x_{1},\dots,x_{n}) = 1$ and the number of terms in
$e_{m}(x_{1},\dots, x_{n})$ is $\binom{n}{m}$.
As mentioned in Section \ref{background},
the number of recurrent sandpiles of a graph $G$ equals the number of
spanning trees of $G$. Cayley's formula implies that $K_{n}$ has
$n^{n-2}$ recurrent sandpiles. In order to study the principal
avalanches in this graph, we will use a beautiful result first proved
in \cite{CR00} that establishes a bijection between recurrent
sandpiles in $K_{n+1}$ and $n$-parking functions.
\begin{defn}
Given a function $p:\{0,1,\dots,n-1\} \to \{0,1,\dots,n-1\}$, let
$a_0\leq a_1\leq\cdots\leq a_{n-1}$
be the non-decreasing rearrangement of $p(0),\dots, p(n-1)$. We say
that $p$ is an \emph{$n$-parking function} provided that $a_i\leq i$
for $0 \leq i \leq n-1$.
\end{defn}
Note that the parking function $p$ can be represented by the vector
$(p(0),p(1),\dots,p(n-1))$.
\begin{prop}[{\cite[Proposition 2.8]{CR00}}]
\label{prop:park}
The sandpile $c$ is recurrent on $K_{n+1}$ if and only if
$\mathrm{max}_{K_{n+1}} - c $ is an $n$-parking function.
\end{prop}
It is clear from the definition that any permutation of a parking
function is also a parking function. We can concatenate parking
functions to obtain new parking functions.
\begin{lemma}
\label{lemma:PFConcatenate}
Let $p=(p_0,p_1,\dots,p_{m-1})$ and $q=(q_0,q_1,\dots,q_{n-1})$ be two
parking functions. Then
$(p_0,p_1,\dots,p_{m-1},q_0+m,q_1+m,\dots,q_{n-1}+m)$ is also a
parking function.
\end{lemma}
\begin{proof}
Let $a_0 \leq a_1 \leq \dotsm \leq a_{m-1}$ and $b_0 \leq b_1 \leq
\dotsm \leq b_{n-1}$ be non-decreasing rearrangements of $p$ and $q$,
respectively. Note
$b_0+m \leq b_1+m \leq \dotsm \leq b_{n-1}+m$ and $b_i + m \leq i +
m$ for each $i=0,\dots, n-1$, since $q$ is a parking function.
Moreover, $a_{m-1}\leq m-1 < m = b_0 + m$. So
$$a_0 \leq a_{1} \leq \cdots \leq a_{m-1} < b_0 + m\leq b_1 + m \leq
\cdots \leq b_{n-1} + m$$
and each term is less than or equal to its index.
\end{proof}
\subsection{The Avalanche Polynomial of $K_{n+1}$}
The following lemma gives a partial description of a sandpile $c$
given the size of a principal avalanche.
\begin{lemma}
\label{lem:WhatTopples}
Let $c$ be a sandpile on $K_{n+1}$.
Suppose that the principal avalanche resulting from stabilizing
$c+1_{v_{k}}$ has length $m\geq 1$. Let $w_0,w_1,\dots,w_{m-1}$ be the
associated toppling sequence and let $\{u_0,\dots,
u_{n-m-1}\}$ be the set of vertices that do not topple. Assume further,
perhaps after relabeling, that
$c(u_0)\geq c(u_1)\geq \cdots \geq c(u_{n-m-1})$. The following are
true:
\begin{enumerate}
\item $c(w_0) = c(v_k) = n-1$.
\item $n-i\leq c(w_i)\leq n-1$, for $i = 1,\dots, m-1$.
\item $n-m-i-1\leq c(u_i)\leq n-m-1$, for $i = 0,\dots, n-m-1$.
\end{enumerate}
\end{lemma}
\begin{proof}
First note that since $c$ is stable then $c(v) \leq n-1$.
Since $m \geq 1$, then $w_0 = v_{k}$ must topple. Thus, $c(w_0)=n-1$.
Corollary \ref{lem:ToppleOnce} implies that each $w_{j}$ appears exactly
once in the toppling sequence. Moreover, when a vertex topples, it adds
one grain of sand to every other non-sink vertex of $K_{n+1}$. Thus,
toppling vertices $w_0,\dots, w_{i-1}$ adds $i$ grains of sand to
$w_{i}$. Since this vertex must topple next, then $c(w_i)+i \geq n $. So,
$n-i \leq c(w_i) \leq n-1.$
On the other hand, since $u_{i}$ does not topple, then
$c(u_i) \leq n-m-1$ for $0\leq i \leq n-m-1$. By Proposition
\ref{prop:park}, $p=\mathrm{max}_{K_{n+1}} - c$ is an $n$-parking
function. Let $p'$ be its non-decreasing rearrangement.
Note that the first $m$ entries in $p'$ correspond to the $m$
vertices that topple and the last $n-m$ entries correspond to the
vertices that do not topple. So $p'(m+i)= n-1 - c(u_i)$. Since
$p'(m+i) \leq m+i$, then $c(u_i)
\geq n-m-i$. So, $n-m-i-1 \leq c(u_i) \leq n-m-1$.
\end{proof}
\begin{prop}
\label{lem:SymmPoly}
Let $\lambda_{m}$ denote the number of principal avalanches of size $m$
in $K_{n+1}$. Then
$$\mathscr{A}_{K_{n+1}}(x_{1},\dots,x_{n}) = \sum_{m=0}^n
\frac{\lambda_{m}}{\binom{n}{m}}e_{m}(x_{1},\dots,x_{n}).$$
\end{prop}
\begin{proof}
First note that each vertex can topple at most once in any principal
avalanche by Corollary \ref{lem:ToppleOnce}. So every monomial in
$\mathscr{A}_{K_{n+1}}$ is square-free and completely characterized by its support.
Fix an integer $m$ with $1\leq m\leq n$. Let $A \subseteq [n]$ with
$|A| = m$ and let $\mu_{A} = \prod_{i \in A} x_i$. Consider the sandpile
$c$ defined by
$$c(v_i) = \begin{cases}
n-1 & \text{if } i\in A,\\
n-1-m & \text{if } i\notin A.
\end{cases}$$
Note that $\mathrm{max} - c(v_{i}) = 0$ if $i\in A$ and $\mathrm{max}-c(v_{i}) = m$
if $i \notin A$.
The non-decreasing rearrangement of $\mathrm{max}-c$ is is
$(\underbrace{0,0,\dots,0}_m,\underbrace{m,m,\dots,m}_{n-m})$ which is a
parking function. Thus, by Proposition \ref{prop:park}, the sandpile $c$ is
recurrent. By Lemma \ref{lem:WhatTopples}, $\mu(c,v_{i}) = \mu_{A}$ for any
vertex $v_{i}$ with $i\in A$.
To complete the proof, we must show that if $A = (i_{1},\dots,i_{m})$
and $A' = (i_{1}',\dots, i_{m}')$ are two ordered
subsets of $[n]$ of cardinality $m$, then the number of principal
avalanches that produce the toppling sequence $(v_{i_{1}},\dots, v_{i_{m}})$
equals the number of principal avalanches that produce the toppling
sequence $(v_{i_{1}'},\dots, v_{i_{m}'})$. However, this follows directly from
the symmetry of $K_{n+1}$. Explicitly, if $\pi$ denotes the permutation that sends
$v_{i_{j}} \to v_{i_{j}'}$ for each $j=1,\dots, m$. Then $\mu_{K_{n+1}}(c,v_{i_{1}}) =
\mu_{A}$ if and only if $\mu_{K_{n+1}}(c',v_{i_{1}'}) = \mu_{A'}$, where
$c'$ is the sandpile obtained by permuting the entries of $c$
according to $\pi$. The explicit form of the coefficient of
$e_{m}(x_{1},\dots,x_{n})$ follows from the fact that this polynomial
has $\binom{n}{m}$ monomials.
\end{proof}
The coefficients $\lambda_{m}$ in Proposition \ref{lem:SymmPoly} were
computed in \cite[Propositions
4 and 5]{CDR04}. Explicitly, $\lambda_{0} = n(n-1)(n+1)^{n-2}$ and
\[\lambda_{m} = \binom{n}{m}m^{m-1}(n-m+1)^{n-m-1}, \text{ for } 1\leq
m \leq n.\]
We include a proof of the latter result in order to
correct a mistake in their original argument. However, we also want
to point out that the coefficient $\lambda_{m}$ is also the number of
principal avalanches with burst size $m$ since every non-sink vertex
in $K_{n+1}$ is adjacent to the sink.
\begin{defn}
\label{defn:phi}
Let $c\in \mathscr{S}(K_{n+1})$ and $v_i\in \widetilde{V}$ such that
when a grain of sand is added to $v_i$, an avalanche of size $m\geq 1$
occurs. Define the function
$$\phi: \mathscr{S}(K_{n+1}) \times \widetilde{V} \longrightarrow
\widetilde{V}\times
\binom{\widetilde{V}\setminus\{v_{i}\}}{m-1}\times
\mathscr{S}(K_{m})\times\mathscr{S}(K_{n-m+1}),$$
such that $\phi(c,v_i) = (v_i,J,c_1,c_2)$, where
$J = \{w_{1},\dots, w_{m-1}\}$ is the set of $m-1$ vertices that
topple other than $w_{0} = v_{i}$. The sandpile $c_{1}$ in
$K_{m}$ is defined by $$c_{1} = (c(w_{1})-(n-m+1),\dots,
c(w_{m-1})-(n-m+1)),$$ and the sandpile $c_{2}$ in $K_{n-m+1}$ is
defined by the values $c(v_{k})$ for $v_{k}\notin J\cup\{v_{i}\}$.
\end{defn}
\begin{ex}
\label{ex:phi}
Consider the recurrent sandpile $c = ( 8,7,8,1,0,3,7,2,4 )$ on
$K_{10}$. Note that adding a grain of sand at $v_1$ causes an
avalanche of size $m=4$. In this case $J = \{ v_2, v_3, v_7
\}$, $c_1 = ( 7-6,8-6,7-6 ) = (1,2,1)$ and $c_2 = (1,0,3,2,4)$. In
\cite{CDR04}, the authors define the sandpile $c_{1}$ by substracting
$m-2$ instead of $n-m+1$. In here, this would result in the sandpile
$(5, 6, 5)$ that is not even a stable sandpile on $K_{4}$.
\end{ex}
\begin{lemma}
The map $\phi$ described in Definition \ref{defn:phi} is a bijection.
\end{lemma}
\begin{proof}
First we need to show that the map $\phi$ above is well-defined, that
is, we need to show that $c_1$ and $c_2$ are, in fact, recurrent
sandpiles on $K_m$ and $K_{n-m+1}$, respectively.
To show $c_1$ is recurrent, let $J = \{w_1,\dots, w_{m-1}\}$ such that
$c(w_i)\leq c(w_{i+1})$ for $1\leq i \leq m-2$. By Lemma
\ref{lem:WhatTopples}, for $1\leq i \leq
m-1$, we have that $n-i\leq c(w_i)\leq n-1$. So, $$n-i - (n-m+1) \leq
c(w_{i}) - (n-m+1) \leq n-1 - (n-m+1)$$ and
$m-i-1\leq c_1(w_i) \leq m-2.$
This implies $c_1$ is a stable sandpile on $K_m$.
Consider $p_1 = \mathrm{max}_{K_m} - c_1$. For $1\leq i \leq m-1$,
$0 \leq p_1(w_i) \leq i-1$, so $p_1$ is a parking function and $c_1$
is recurrent.
To show $c_2$ is recurrent, let $\{u_0,\dots, u_{n-m-1}\} =
\widetilde{V}\setminus (J\cup \{v_i\})$ such that $c(u_i)\leq
c(u_{i+1})$ for $0\leq i \leq n-m-2$. By Lemma \ref{lem:WhatTopples},
$$n-m-i-1 \leq c(u_i)\leq n-m-1.$$
Since $c_2(u_i) = c(u_i)$ then $c_{2}$ is stable in $K_{n-m+1}$.
Consider $p_2 = \mathrm{max}_{K_{n-m+1}} - c_2$. For $0\leq i \leq n-m-2$,
we have $ 0\leq p_2(i) \leq i. $
Since $p_2$ is a parking function, $c_2$ is recurrent.
The fact that the map $\phi$ is injective follows immediately from the
definition of $c_{1}$ and $c_{2}$.
Finally, we will show that $\phi$ is onto. Given $(v,J,c_1,c_2)$
we define $c$ as follows. First, let $c(v) = n-1$. Now, for each
$w_{i} \in J$, define $c(w_{i})$ by adding $n-m+1$ to the $i$th entry
in $c_{1}$. The remaining $n-m$ entries in $c$ are filled with the
entries in $c_{2}$.
Since $c_1$ and $c_2$ are recurrent, Proposition \ref{prop:park} implies
$p_{1} = \mathrm{max}_{K_m}-c_1$ and $p_{2} = \mathrm{max}_{K_{n-m+1}}-c_2$ are
$(m-1)$ and $(n-m)$-parking functions, respectively.
By Lemma \ref{lemma:PFConcatenate}, concatenating $p_1$ and $p_2
+ \overline{m}$ defines an $(n-1)$-parking function
$p'$, where $\overline{m} = (m,\dots, m)$. Furthermore, concatenating $0$ and $p'$
gives an $n$-parking function $p$.
Moreover, $\mathrm{max}_{K_{n+1}}-p$ is a rearrangement
of $c$, so $c$ is a recurrent sandpile on $K_{n+1}$.
Clearly, $\phi(c,v) = (v,J,c_1,c_2)$ and this completes the proof.
\end{proof}
From the bijection $\phi$ we are able to compute the number
$\lambda_{m}$ of
principal avalanches of size $m>0$. Given $A \subseteq [n]$ with
$|A| = m$, Proposition \ref{lem:SymmPoly} states that $\lambda_{m}/\binom{n}{m}$ is the
number of principal avalanches with avalanche monomial $\mu_{A} =
\prod_{i \in A} x_i$. The bijection $\phi$ implies that this number
equals the number of four-tuples $(v_i,J,c_1,c_2)$ with $J\cup
\{v_{i}\} = A$. There are $m$ ways to pick $v_i$.
Cayley's formula implies that the number of recurrents on $K_m$ and
$K_{n-m+1}$ is $m^{m-2}$ and $(n-m+1)^{n-m-1}$,
respectively. Therefore,
\[\lambda_{m} = \binom{n}{m}\cdot m\cdot m^{m-2}(n-m+1)^{n-m-1}= \binom{n}{m} m^{m-1}(n-m+1)^{n-m-1}.\]
\section{The Avalanche Polynomial of the Wheel}\label{wheel}
The wheel graph, denoted $W_{n}$, is a cycle on $n \geq 3$ vertices
with an
additional \emph{dominating vertex}. Throughout, the vertices in the
cycle will be labeled clockwise as $v_0,\dots, v_{n-1}$, where the
indices are taken modulo $n$.
The
dominating vertex, denoted $s$, will always be assumed to be the sink.
The sandpile group of $W_{n}$ was first computed by Biggs in
\cite{B99}:
\[
\mathscr{S}(W_{n}) =
\begin{cases}
\mathbb{Z}_{l_{n}} \oplus \mathbb{Z}_{l_{n}} & \text{ if } n \text{ is
odd }\\
\mathbb{Z}_{f_{n}} \oplus \mathbb{Z}_{5f_{n}} & \text{ if } n \text{ is
even }
\end{cases}
\]
where $\{l_{n}\}$ is the Lucas sequence and $\{f_{n}\}$ is the
Fibonnaci sequence. These sequences are defined by initial
conditions $l_{0} = 2, l_{1} =1$ and $f_{0} = 1, f_{1} = 1$,
respectively, and the recursion $x_{n} = x_{n-1} + x_{n-2}$. There are
many relationships among these numbers. For example, $l_{n} = f_{n-1}
+ f_{n+1}$. Morever, the order of $\mathscr{S}(W_{n})$
equals the number of spanning trees $\tau(W_{n})$ in $W_{n}$. This
number equals $\tau(W_{n}) = l_{2n} - 2$, see \cite{H72, BY06}.
We have already computed the avalanche polynomial of $W_{3}$
since $W_{3} = K_{4}$. In this case,
\[\mathscr{A}_{W_{3}}(x_{0},x_{1},x_{2}) = 9x_{0}x_{1}x_{2} + 2(x_{0}x_{1} +
x_{1}x_{2} + x_{2}x_{0}) + 3(x_{0}+x_{1}+x_{2}) + 24.\]
Observe that the set of principal avalanches of size $0 < m < 3$ is
evenly partitioned into $n=3$ subsets. Moreover,
$\mathscr{A}_{W_3}(x_0,x_1,x_2)$ is a linear combination of \emph{cyclic
polynomials}. We will show that this characterizes
$\mathscr{A}_{W_n}(x_0,\dots, x_{n-1})$.
\begin{defn}
Let $m$ be an integer such that $1 \leq m \leq n-1$. We will denote by
$w_{m}(x_{0},\dots,x_{n-1})$ the
\emph{cyclic polynomial} of degree $m$ on variables
$x_0,\dots, x_{n-1}$ defined as
\[w_{m}(x_{0},\dots,x_{n-1}) = \sum_{i=0}^{n-1}x_{i}x_{i+1}\dotsm x_{i+m-1}.\]
\end{defn}
First note that for each $1\leq m \leq
n-1$, $w_{m}$ is the sum of $n$ terms of degree $m$.
For example, $w_{1} = x_{0} + \dotsm + x_{n-1}$.
For the case $m=n$, the above definition would give
\[w_{n} = \sum_{i=0}^{n-1}x_{i}x_{i+1}\dotsm x_{i+n-1} = nx_{0}\dotsm x_{n-1}. \]
However, we will remove the coefficient $n$ and define
\[w_{n}(x_{0},\dots,x_{n-1}) = x_{0}\dotsm x_{n-1}.\]
In \cite{DaRoFPSAC03}, Dartois and Rossin gave exact results on
the distribution of avalanches on $W_{n}$. Their approach
consisted in showing that the recurrents on $W_{n}$ can be
seen as words of a \emph{regular language}. They built an automaton
associated to this language and used the concept of \emph{transducers}
to determine the exact distribution of avalanche lengths in this
graph. Here we take a different approach focused solely on
the structure of the recurrent sandpiles.
Note that the degree of every non-sink vertex in $W_{n}$ is $3$. So
any stable sandpile on this graph can be written as a word of length $n$
in the alphabet $\{0,1,2\}$.
Applying Dhar's Burning Criterion (Proposition \ref{prop:Burn}),
Cori and Rossin \cite{CR00} showed that a sandpile on $W_{n}$ is
recurrent if and only if there is at least one vertex
with 2 grains of sand and between any two vertices with $0$ grains,
there is at least one vertex with 2 grains.
\begin{defn}
Let $m$ be an integer with $1\leq m \leq n-1$.
A sandpile $c$ in $W_{n}$ has a \emph{maximal 2-string of length $m$}
if there
are vertices $v_i,v_{i+1},\dots, v_{i+m-1}$, such that $c(v_i) =
\cdots = c(v_{i+m-1}) = 2$ and $c(v_{i-1}) \neq 2 \neq c(v_{i+m})$.
\end{defn}
Note that $\mathrm{max}_{W_{n}} = 2^{n}$ is the unique recurrent
with a maximal $2$-string of length $n$.
\begin{lemma}\label{lem:Toppling2strings}
Let $c\in \mathscr{S}(W_{n})$. The principal avalanche of $c$ at a non-sink
vertex $v$ has size $1\leq m \leq n-2$ if and only if $v$ is part of a
maximal $2$-string of length $m$.
\end{lemma}
\begin{proof}
Suppose a grain is added to a vertex $v$ that is part of a
maximal
$2$-string of length $m$. Since $m<n-1$, the two non-sink vertices
adjacent to the ends of the $2$-string are distinct. Thus,
exactly the $m$ vertices in the maximal $2$-string will topple. On the
other hand, if $v$ is part of a longer or
shorter maximal $2$-string, the avalanche will not have size $m$.
\end{proof}
Lemma \ref{lem:Toppling2strings} implies that for each $1\leq m \leq
n-2$,
the number $\lambda_{m}$ of principal avalanches of size $m$ equals
$m$ times the number of maximal $2$-strings of length $m$ over all
recurrents.
This is not the case for avalanches of size
$n-1$ or $n$ as the following simple lemma shows.
\begin{lemma}
\label{lem:SizeNAvalanche}
For any non-sink vertex $v$ in $W_{n}$,
$\mu(2^n,v) = x_0\cdots x_{n-1}$. Also, let $p$ be an integer with
$0\leq p \leq n-1$. For any non-sink vertex $v$ with $v\neq v_{p}$ we
have
\begin{align*}
\mu_{W_{n}}(2^p12^{n-p-1},v) &= x_0\cdots x_{n-1}, \\% \text{ and }\\
\mu_{W_{n}}(2^p02^{n-p-1}, v) &=
\frac{x_0\cdots x_{n-1}}{x_{p}}.
\end{align*}
This implies $\lambda_{n} = n^{2}$ and $\lambda_{n-1} = n(n-1)$.
\end{lemma}
\begin{proof}
Clearly the avalanche monomials for the given recurrents satisfy the
above claims.
Note also that the only avalanches of size $n$ occur on
recurrents of the form $2^p12^{n-p-1}$ and $2^n$. So there are
$n(n-1)+n = n^2$ avalanches of size $n$. The avalanches of size $n-1$
occur on recurrents of the form $2^p02^{n-p-1}$. Hence there are
$n(n-1)$ avalanches of size $n-1$.
\end{proof}
For each $1\leq m \leq n-2$,
we will count the maximal 2-strings of length $m$ by establishing
a map from the set of recurrents on $W_{n}$ with a given maximal
2-string of length $m$ into the set of recurrents on the \emph{fan
graph} $F_{n-m}$. Let $k\geq 2$,
the fan graph on $k+1$ vertices, denoted $F_{k}$, is a path on $k$
vertices, plus an additional dominating vertex $s$.
\begin{prop}
\label{prop:NonMaxRecurrents}
For each $1\leq m \leq n-2$,
there is a bijection between the set of recurrents on $W_{n}$ with
a maximal 2-string of length $m$ starting at $v_0$ and the set of
recurrents on $F_{n-m}$.
\end{prop}
\begin{proof}
Let $c$ be a recurrent sandpile on $F_{n-m}$.
Dhar's Burning Criterion (Proposition \ref{prop:Burn}) implies that
adding 1 grain of sand to each vertex must result in an avalanche
where every vertex topples exactly once. This
implies that at least one of the endpoint vertices has 1 grain of sand
or both endpoints have 0 grains of sand and there is an internal
vertex with 2 grains of sand. Moreover, if a
vertex has $0$ grains of sand, then its neighbors must topple before
it, hence there are no consecutive vertices with $0$ grains of
sand. For the same reason, between any two vertices with $0$ grains of
sand there cannot be a sequence of $1$'s. In summary, $c$
is a recurrent on $F_{n-m}$ if and only if between any two vertices
with $0$ grains of sand there is a vertex with $2$ grains of
sand.
Hence $c$ is a recurrent
sandpile on $F_{n-m}$ if and only if the sandpile obtained by
prepending a string of $m$ 2's to $c$ is recurrent on $W_{n}$.
\end{proof}
It is well-known that the number of spanning trees in the fan graph
$F_{k}$ is precisely the Fibonnaci number $f_{2k}$, see \cite{H72}.
So Proposition \ref{prop:NonMaxRecurrents} implies that for each
$1\leq m \leq n-2$, there are $f_{2(n-m)}$ recurrent sandpiles that
have a maximal $2$-string of length $m$ starting at $v_0$.
\begin{thm} \label{thm:wheel}
Given $n\geq 3$, the avalanche polynomial of the
wheel graph $W_{n}$ is
$$\mathscr{A}_{W_{n}} = n^2w_{n}(x_{0},\dots, x_{n-1}) + \sum_{m = 1}^{n-1} m\cdot f_{2(n-m)} w_{m}(x_{0},\dots, x_{n-1}) + 2n\left(f_{2n-1}-1\right).$$
\end{thm}
\begin{proof}
In Lemma \ref{lem:SizeNAvalanche} we showed that
$\lambda_{n} = n^{2}$.
This lemma also shows that
the avalanches of size $n-1$ are
caused by adding a grain of sand at any vertex with $2$ grains
in any
recurrent of the form $2^p02^{n-p-1}$ with $0\leq p \leq n-1$. Since
$\mu_{W_{n}}(2^p02^{n-p-1},v) = x_0\cdots x_{n-1}/x_{p}$,
for any $v\neq v_{p}$. Then
the degree $n-1$ part of
$\mathscr{A}_{W_{n}}$ equals $(n-1)w_{n-1}(x_{0},\dots, x_{n-1})$. Note that
when $m = n-1$ we have $f_{2(n-m)} = f_{2} = 1$.
Now let $1\leq m \leq n-2$.
Proposition \ref{prop:NonMaxRecurrents} implies
that there are $f_{2(n-m)}$ recurrents on $W_{n}$ with
a maximal 2-string of length $m$ starting at $v_0$.
So by Lemma \ref{lem:Toppling2strings},
there are $mf_{2(n-m)}$ principal avalanches
with avalanche monomial $x_{0}\cdots x_{m-1}$.
This lemma also shows that any avalanche of size $m$ must occur at a
maximal $2$-string of length $m$. So the only possible avalanche
monomials of degree $m$ are the monomials occuring in the cyclic
polynomial $w_{m}$. Moreover, the cyclic symmetry of $W_{n}$ implies
that the number of principal avalanches that produce the toppling
sequence $(v_{0},v_{1},\dots, v_{m-1})$ equals the number of
principal avalanches that produce the toppling sequence
$(v_{i},v_{i+1},\dots, v_{i+m-1})$ for any $0\leq i\leq
n-1$. Therefore, for any $1\leq m \leq n-2$, the degree $m$ part of
$\mathscr{A}_{W_{n}}$ equals $mf_{2(n-m)}w_{m}(x_{0},\dots, x_{n-1})$.
Lastly, note that an avalanche of size $0$ is produced by adding a
grain of sand to a vertex with $0$ or $1$ grains of sand. So
$\lambda_{0}$ equals the number of $0$'s and $1$'s in every recurrent.
Since there are $l_{2n} - 2$ recurrents on $W_{n}$, then
$\lambda_{0}$ equals $n(l_{2n} - 2)$ minus the total number of $2$'s
in every recurrent.
Recall that for $1\leq m \leq n-2$, the number $\lambda_{m}$ of
principal avalanches of size $m$ equals $m$ times the number of maximal
$2$-strings of length $m$ over all recurrents, that is, $\lambda_{m}$
equals the total number of $2$'s in every maximal
$2$-string of length $m$. Moreover, $\lambda_{n-1} + \lambda_{n} =
n^{2} + n(n-1) = 2n^{2}-n$ equals the number of principal avalanches
of size $\geq n-1$. But this number also equals the number
of $2$'s in every recurrent with a maximal $2$-string of size $\geq
n-1$.
Therefore, $\lambda_{1} + \dotsm + \lambda_{n}$
equals the number of $2$'s in every recurrent.
Hence
\begin{align*}
\lambda_{0} &= n(l_{2n} - 2) - (\lambda_{1} + \dotsm + \lambda_{n})
= n(l_{2n} - 2) - 2n^{2} + n - \sum_{m=1}^{n-2} nmf_{2(n-m)} \\
&= n\left[l_{2n} - 2n - 1 - \sum_{m=1}^{n-2}mf_{2(n-m)}\right]
= n\left[l_{2n} - 2n - 1 - \sum_{m=2}^{n-1}(n-m)f_{2m}\right]\\
&= n\left[l_{2n} - n - 2 - \sum_{m=1}^{n-1}(n-m)f_{2m}\right]
= n\left[l_{2n} - n - 2 - \sum_{m=1}^{n-1}\sum_{k=1}^{m}
f_{2k}\right]\\
&= n\left[l_{2n} - n - 2 - \sum_{m=1}^{n-1}\left(f_{2m+1}-1\right)\right]
= n\left[l_{2n} - 3 - \sum_{m=1}^{n-1}f_{2m+1}\right]\\
&= n(l_{2n} - 2 - f_{2n}) = n(f_{2n+1}+f_{2n-1}-f_{2n}-2)
= n(2f_{2n-1} - 2) = 2n(f_{2n-1}-1).
\end{align*}
\end{proof}
In this case, $\lambda_{m}$ is also the number of
principal avalanches with burst size $m$ since every non-sink vertex
in $W_{n}$ is adjacent to the sink. Note also that as $n\to \infty$,
the proportion of avalanches of size $0$ is
\[\lim_{n\to \infty} \frac{2n(f_{2n-1}-1)}{n(l_{2n}-2)} = 1-\frac{1}{\sqrt{5}}. \]
Thus, recovering the last result in \cite[Section 2]{DaRoFPSAC03}.
\section{Conclusions}\label{final}
In this paper, we introduce the \emph{multivariate avalanche
polynomial} of a graph $G$. This new combinatorial object enumerates the
toppling sequences of all principal avalanches generated by adding a
grain of sand to any recurrent sandpile on $G$. We also give explicit
descriptions of the multivariate avalanche polynomials for trees,
cycles, complete, and wheel graphs. Furthermore, we show that certain
evaluations of this polynomial recover some important information. In
particular from this polynomial we can compute the distribution of the
size of all principal avalanches, that is, we recover the
(univariate) avalanche polynomial first introduced in
\cite{CDR04}. Moreover, a different evaluation gives rise to the unnormalized distribution of burst sizes, that is, the number of
grains of sand that fall into the sink in a principal avalanche. The
burst size, introduced by Levine in \cite{L15}, is an important
statistic related to the relationship between the
threshold state of the
fixed-energy sandpile and the stationary state of Dhar's abelian
sandpile.
Of special interest is a description of the avalanche polynomial for
grids and the family of multiple wheel graphs introduced in
\cite{DaRoFPSAC03}.
\section{Acknowledgements}
The authors would like to thank David Perkinson for suggesting to study the multivariate version of the avalanche polynomial and for his invaluable guidance and support throughout this project. We would also like to thank Andrew Fry, Lionel Levine, Christopher O'Neill, and Gautam Webb for their kind input and suggestions related to this work.
| {
"timestamp": "2016-05-13T02:13:42",
"yymm": "1605",
"arxiv_id": "1605.02713",
"language": "en",
"url": "https://arxiv.org/abs/1605.02713",
"abstract": "The (univariate) avalanche polynomial of a graph, introduced by Cori, Dartois and Rossin in 2006, captures the distribution of the length of (principal) avalanches in the abelian sandpile model. This polynomial has been used to show that the avalanche distribution in the sandpile model on a multiple wheel graph does not follow the expected power law function. In this article, we introduce the (multivariate) avalanche polynomial that enumerates the toppling sequences of all principal avalanches. This polynomial generalizes the univariate avalanche polynomial and encodes more information. In particular, the avalanche polynomial of a tree uniquely identifies the underlying tree. In this paper, the avalanche polynomial is characterized for trees, cycles, wheels, and complete graphs.",
"subjects": "Combinatorics (math.CO)",
"title": "The Avalanche Polynomial of a Graph",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808684361289,
"lm_q2_score": 0.7217432062975979,
"lm_q1q2_score": 0.7075110570572855
} |
https://arxiv.org/abs/math/0603405 | Calculus proofs of some combinatorial inequalities | Using calculus we show how to prove some combinatorial inequalities of the type log-concavity or log-convexity. It is shown by this method that binomial coefficients and Stirling numbers of the first and second kinds are log-concave, and that Motzkin numbers and secondary structure numbers of rank 1 are log-convex. In fact, we prove via calculus a much stronger result that a natural continuous ``patchwork'' (i.e. corresponding dynamical systems) of Motzkin numbers and secondary structures recursions are increasing functions. We indicate how to prove asymptotically the log-convexity for general secondary structures. Our method also applies to show that sequences of values of some orthogonal polynomials, and in particular the sequence of central Delannoy numbers, are log-convex. | \section{Introduction}
In combinatorics the most prominent question is usually to find explicitly the size of certain finite set defined in an intricate way. It often happens that there
is no explicit expression for the size in question, but instead one can find
recursion, generating function or other gadgets which enable us to compute
concrete sizes or numbers. The next question then usually asks how the sequence
of numbers satisfying certain recursion behaves. By behavior of the sequence
$\left ( a_n \right )_{n \geq 0}$ of positive real numbers it is often meant
its log-concavity (or log-convexity). Recall that a sequence
$\left ( a_n \right )_{n \geq 0}$ of positive real numbers is {\bf log-concave}
if $a_n^2 \geq a_{n-1} a_{n+1}$ for all $n\geq 1$, and {\bf log-convex} if
$a_n^2 \leq a_{n-1} a_{n+1}$ for all $n\geq 1$. We say that a sequence
$\left ( a_n \right )_{n \geq 0}$ is {\bf log-straight} or {\bf geometric} if
$a_n^2 = a_{n-1} a_{n+1}$ for all $n\geq 1$. A (finite) sequence of positive
numbers $a_0, a_1, \ldots ,a_n$ is said to be {\bf unimodal} if, for some
$0 \leq j \leq n$ we have $a_0 \leq a_1 \leq \ldots \leq a_j \geq a_{j+1} \geq
\ldots \geq a_n$. This place $j$ is called a {\bf peak} of the sequence if it
is unique. If there are more such maximal values, we speak about a {\bf plateau} of
the sequence. It is easy to see that a log-concave positive sequence is
unimodal. The literature on log-concavity and unimodality is vast. We refer the
interested reader to the book \cite{karlin}. Combinatorial inequalities, and
in particular, the questions concerning log-concavity (or log-convexity) are
surveyed in \cite{brenti}, \cite{stanley00} and \cite{stanley89}. Some analytic
methods are described in \cite{bender}.
In combinatorics, a preferable way to prove a combinatorial inequality is to
give a {\bf combinatorial proof}. There are two basic ways to do it. Suppose
that we are given finite sets $A$ and $B$ with $|A|=a$ and $|B|=b$ and we want
to prove, say, $a \leq b$. One way to prove it is to construct an injection
$A \rightarrow B$ (or a surjection $B \rightarrow A$), and the other is to
show that the number $c=b-a$ is nonnegative, by showing that $c$ is cardinality
of certain set or that $c$ is the dimension of certain vector space (and hence
nonnegative) etc. As an example, let us show that binomial coefficients $
\binom {n}{k}$, $k=0,1,\ldots ,n$ are log-concave. It is trivial to check
algebraically that $\binom {n}{k} ^2 \geq \binom {n}{k-1} \binom {n}{k+1}$ by
using the standard formula $\binom {n}{k} = \frac {n!}{k!(n-k)!}$, but
combinatorially it goes as follows.
First define the {\bf Narayana numbers} $N(n,k)$ for integers $n, \quad k \geq
1$ as
$$N(n,k)=\frac {1}{n} \binom {n}{k} \binom {n}{k-1} = \frac {1}{k}\binom {n}{k-1}\binom{n-1}{k-1},$$
and $N(0,0):=1$. Next we note that
$$\binom {n}{k} ^2-\binom {n}{k-1} \binom {n}{k+1} = \left |
{{\binom {n}{k} \atop \binom {n}{k-1}} \quad {\binom {n}{k+1} \atop \binom {n}{k}}} \right | = N(n+1,k+1).$$
Finally, we need the fact that Narayana numbers have a combinatorial meaning, i.e.
they count certain finite sets (see below). Therefore we get $\binom {n}{k} ^2
- \binom {n}{k-1} \binom {n}{k+1} \geq 0.$ There are also other combinatorial
proofs of log-concavity of binomial coefficients, as well as log-concavity of
Stirling numbers (of both kinds) etc., but they are all rather involved and/or
tricky. In this paper we present a way to prove various combinatorial inequalities
by a straightforward method of calculus. Inductive and injective proofs of
log-convexity results are described in \cite{sagan}.
\section {Calculus proofs of log-concavity and log-convexity properties}
Let us first recall briefly calculus proofs of log-concavity of binomial
coefficients and Stirling numbers. Let $c(n,k)$ be the number of permutations
of the set $[n]:=\{1,2,\ldots ,n\}$ with exactly $k$ cycles and $S(n,k)$ the
number of partitions of $[n]$ into exactly $k$ parts (or blocks). The numbers
$c(n,k)$ and $S(n,k)$ are called {\bf Stirling numbers} of the {\bf first} and
{\bf second kind}, respectively. The following formulae are well known (see
\cite{stanley}).
\begin{equation}
(x+1)^n = \sum_{k=0}^n \binom {n}{k} x^k,
\end{equation}
\begin{equation}
x^{\bar n}=x(x+1)\ldots (x+n-1) = \sum_{k=0}^n c(n,k) x^k,
\end{equation}
\begin{equation}
x^n= \sum_{k=0}^n S(n,k) x^{\underline k},
\end{equation}
where $x^{\underline k}:=x(x-1)\ldots (x-k+1)$ is the $k$-th {\bf falling power}
and $x^{\bar k} = x(x+1)\ldots (x+k-1)$ the $k$-th {\bf rising power} of x.
The following Newton's lemma is a consequence of the Rolle's theorem from
calculus.
{\bf Lemma 1.}\\ Let $P(x)=\sum_{k=0}^n a_k x^k$ be a real polynomial whose
all roots are real numbers. Then its coefficients are log-concave, i.e.
$a_k^2 \geq a_{k-1} a_{k+1}, k=1, \ldots ,n-1$. (Moreover, $\frac {a_k}{\binom
{n}{k}} $ are log-concave). \qed
Now, from (1) and (2) we see that $(x+1)^n$ and $x^{\bar n}$ have only real roots
and by Lemma 1. we conclude that the sequences $\binom {n}{k}$ and $c(n,k)$ are
log-concave.
The case of the sequence $S(n,k)$ is a bit more involved. We claim that the
polynomial
\begin{equation}
P_n(x)=\sum_{k=0}^n S(n,k) x^k
\end{equation}
has all real roots (in fact non-positive and different). Namely, $P_0(x)=1$
and from the basic recursion $$S(n,k)=S(n-1,k-1)+kS(n-1,k)$$ it follows at once
that $$P_n(x)=x\left [P_{n-1}'(x)+P_{n-1}(x)\right ].$$
The function $Q_n(x)=P_n(x)e^x$ has the same roots as $P_n(x)$ and it is easy to
verify $Q_n(x)=xQ_n'(x)$. By induction on $n$ and by using the Rolle's theorem
it follows easily that $Q_n$ and hence $P_n$ have only real and non-positive
roots.
So, we have proved by calculus the following.
{\bf Theorem 1.}\\The sequences $\binom {n}{k}_{k \geq 0}, \left ( c(n,k) \right)_{k \geq 0}, \left (S(n,k) \right )_{k \geq 0}$ are log-concave. Hence they
are also unimodal. \qed
It is also well known that the peak of the sequence $\binom {n}{k}$ is at
$k=\lfloor n/2 \rfloor$, while the peak for the other two sequences is much
harder to determine. It is known that $S(n,k)$'s reach their peak for $k \approx
n/\log n$, if $n$ is large enough. (An inductive proof of Theorem 1. is given in
\cite{sagan}.)
Now we turn to a different kind of combinatorial entities. Recall that a {\bf
Dyck path} is a path in the coordinate $(x,y)$-plane from$(0,0)$ to $(2n,0)$
with steps $(1,1)$ and $(1,-1)$ never falling below the $x$-axis. Denote the set
of all such paths by ${\cal D}_n$. A {\bf peak} of a path $P \in {\cal D}_n$ is
a place at which the step $(1,1)$ is directly followed by the step $(1,-1)$.
Denote by ${\cal D}_{n,k} \subseteq {\cal D}_n$ the set of all Dyck paths of
length $2n$ with exactly $k$ peaks. Note that $1 \leq k \leq n$. The following
facts are also well known (see \cite{stanley}).
$$|{\cal D}_n| = \frac {1}{n+1} \binom {2n}{n} = C_n $$
$$|{\cal D}_{n,k}| = N(n,k),$$
where $C_n$ is $n$-th {\bf Catalan number}. The Catalan numbers are log-convex.
The Narayana numbers are log-concave in $k$ for fixed $n$. Both these facts can easily
be proved algebraically, but there are also combinatorial proofs, as well as
calculus proofs. We omit here these proofs, since we want to emphasize the
following more intricate combinatorial quantities, related to the above just
introduced.
A {\bf Motzkin path} is a path in the coordinate $(x,y)$-plane from $(0,0)$ to $(n,0)$
with steps $(1,1)$, $(1,0)$ and $(1,-1)$ never falling below the $x$-axis. Let
${\cal M}_n$ be the set of all such paths and let $M_n = |{\cal M}_n|$. The
number $M_n$ is called the $n$-th {\bf Motzkin number}.
Some basic properties of Motzkin numbers are as follows (\cite{stanley},
\cite{doslic1}).
{\bf Theorem 2.}\\
(a) $M_n=\sum_{k \geq 0} \binom {n}{2k}C_k, \quad C_{n+1}=\sum_{k \geq 0}\binom {n}{k} M_k;$\\
(b) $M_{n+1}=M_n+\sum_{k=0}^{n-1} M_kM_{n-k-1};$\\
(c) The generating function of $\left ( M_n \right )_{n\geq 0}$ is given by
$$M(x)=\sum_{n\geq 0}M_n x^n = \frac {1-x-\sqrt {1-2x-3x^2}}{2x^2};$$
(d) $(n+2)M_n=(2n+1)M_{n-1}+3(n-1)M_{n-2};$\\
(e) $M_n \sim \sqrt {\frac {3}{4 \pi }} 3^{n+1} n^{-3/2}.$ \qed
The log-convexity of the sequence of Motzkin numbers was first established
algebraically in \cite{aigner}, and shortly afterwards combinatorial proof
appeared in \cite{callan}.
We shall prove now by calculus that $\left ( M_n \right )_{n\geq 0}$ is a
log-convex sequence and some consequences of this property.
{\bf Theorem 3.}\\
(a) The sequence $\left ( M_n \right )_{n\geq 0}$ is log-convex;\\
(b) $M_n \leq 3 M_{n-1}$, for all $n\geq 1$;\\
(c) There exists $x=\lim_{n \rightarrow \infty} \frac {M_n}{M_{n-1}}$, and $x=3$.
{\bf Proof}\\
(a) Let us start from the short recursion in Theorem 2.(d):
$$M_n=\frac {2n+1}{n+2}M_{n-1} +\frac {3(n-1)}{n+2}M_{n-2}.$$
Divide this recursion by $M_{n-1}$ and denote $x_n:= \frac {M_n}{M_{n-1}}$. Then
we obtain the following recursion:
\begin{equation}
x_n=\frac {2n+1}{n+2} +\frac{3(n-1)}{n+2}\frac {1}{x_{n-1}}
\end{equation}
with initial condition $x_1=1$. The log-convexity $M_n^2 \leq M_{n-1}M_{n+1}$
is equivalent to $x_n \leq x_{n+1}$. To prove that $(x_n)_{n\geq 0}$ is an increasing
sequence, we shall prove a much stronger claim. To this end, define the following
function $f : [2,\infty) \rightarrow \R$. For $x\in [2,3]$, define $f(x)=2$.
For $x\geq 3$, let (by simulating (5))
\begin{equation}
f(x)=\frac {2x+1}{x+2} +\frac {3(x-1)}{x+2} \frac {1}{f(x-1)}.
\end{equation}
Note that $f(n)=x_n$. We shall prove that $f$ is an increasing function, and
consequently that $\left ( x_n \right )_{n\geq 0}$ is an increasing sequence.
Note first that the function $f$ is continuous ($f$ is, in fact, a dynamical
system), and on every open interval $(n,n+1)$, where $n\geq 2$ is an integer,
$f$ is a rational function, with no poles on it. Therefore, $f$ is smooth on
every open interval $(n,n+1)$, for $n\geq 2$. Note that, for example, $f(x)=
\frac {7x-1}{2(x+2)}$ for $x\in [3,4]$, $f(x)=\frac {20x^2-9x-14}{7x^2+6x-16}$
for $x\in [4,5]$, etc. It is trivial to check that $f(x) \geq 2$, for all
$x\geq 2$. Suppose inductively that $f$ is an increasing function on a segment
$[3,n]$. For $n=4$ it is (almost evidently) true. Let $n \geq 4$, and take a
point $x\in (n,n+1)$. By taking the derivative $f'(x)$ of (6), and plugging in
once more the term for $f'(x-1)$, we have:
$$\displaylines {
\qquad f'(x)=\frac {3}{[(x+2)f(x-1)]^2} \Bigg [f^2(x-1)+3f(x-1)- 3 \frac{(x-1)(x+2)} {(x+1)^2f(x-2)}[f(x-2)+3] \hfill \cr
\hfill +3 \frac{(x^2-1)(x^2-4)}{[(x+1)f(x-2)]^2} f'(x-2) \Bigg ] \qquad \cr }$$
By inductive hypothesis, $f$ is an increasing function on $[3,n]$ and hence
$f(x-1) \geq f(x-2) \geq 2$ and $f'(x-2) \geq 0$. So, it is enough to prove
that $f'(x)\geq 0$. However, this follows from the following.
The last term in square brackets is clearly positive, by the induction
hypothesis. We claim that the rest is positive, too. This claim is equivalent
with
$$\frac {[f^2(x-1)+3f(x-1)]f(x-2)}{f(x-2)+3} \geq 3 \frac {(x-1)(x+2)}{(x+1)^2}.$$
But this inequality is true, since by inductive hypothesis $f(x-1) \geq f(x-2)
\geq 2$, and hence the left hand side is at least equal to $f^2(x-2) \geq 4$,
while the right hand side has the maximum (for $x \geq 3$) equal to $3$.
Hence $f'(x) > 0$ for all $x \in (n,n+1)$. So, the function $f$ is
strictly increasing on $(n,n+1)$, and then, by continuity, also on $[3,n+1]$.
In particular, $f(n+1)=x_{n+1} \geq x_n = f(n)$.
This completes the step of induction.
(b) and (c) follow now simultaneously, because by (a), the sequence
$\left ( x_n \right )_{n\geq 0}$ is increasing and from (5) it follows easily
by induction on $n$ that $2 \leq x_n \leq 7/2$, i.e. $\left ( x_n \right )$ is
bounded. \qed
Closely related combinatorial structures to Motzkin paths are the so called
secondary structures. A {\bf secondary structure} is a simple planar graph
on vertex set $[n]$ with two kinds of edges: segments $[i,i+1]$, for $1\leq i
\leq n-1$ and arcs in the upper half-plane which connect some $i,j$, where
$i < j$ and $j-i > l$, for some fixed integer $l\geq -1$, such that the arcs
are totally disjoint. Such a structure
is called a secondary structure of {\bf size} $n$ and {\bf rank} $l$. The
importance for the study of these structures comes from biology. They are
crucial in understanding the role of RNA in the cell metabolism and in decoding
the hereditary information contained in DNA. Biologists call the vertices of a
secondary structure {\bf bases}, the segments they call {\bf p-bonds} (p stands
for phosphorus) and arcs they call {\bf h-bonds} (h stands for hydrogen). Let
${\cal S}^{(l)}(n)$ be the set of all secondary structures of rank $l$ on $n$
vertices and $S^{(l)}(n)=|{\cal S}^{(l)}(n)|$ the {\bf secondary structure numbers} of
rank $l$. In a sense, the Motzkin numbers are secondary structure numbers of rank $0$,
and the Catalan numbers are secondary structure numbers of the (degenerate) rank $-1$.
In these cases the corresponding graphs are not simple, but the other requirements
on secondary structures remain.
Now we shall apply our method of calculus to prove that in the case
$l=1$ the behavior of the numbers $S^{(1)}(n)$ is also log-convex. So, we have:
{\bf Theorem 4.}\\ The sequence $\left (S^{(1)}(n) \right )_{n\geq 0}$ is
log-convex.
{\bf Proof}\\ As for the Motzkin numbers, it turns out that for $S^{(1)}(n)$ the
following short recursion holds (see \cite{doslic1} and \cite{doslic}):
\begin{equation}
(n+2)S^{(1)}(n)=(2n+1)S^{(1)}(n-1)+(n-1)S^{(1)}(n-2)+(2n-5)S^{(1)}(n-3)-(n-4)
S^{(1)}(n-4)
\end{equation}
with initial conditions $S^{(1)}(0)=S^{(1)}(1)=S^{(1)}(2)=1, S^{(1)}(3)=2$.
By dividing this recursion with $S^{(1)}(n-1)$ and denoting
$$x_n=\frac {S^{(1)}(n)}{S^{(1)}(n-1)},$$
we get
\begin{equation}
x_n=\frac {1}{n+2} \left [2n+1 + \frac {n-1}{x_{n-1}} +\frac {2n-5}{x_{n-1}x_{n-2}}-\frac {n-4}{x_{n-1}x_{n-2}x_{n-3}}\right ],
\end{equation}
with initial conditions $x_3=x_4=x_5=2$ (note that $x_1=x_2=1$).
The log-convexity of $S^{(1)}(n)$'s is equivalent with the fact that $(x_n)$
is an increasing sequence.
Now define the function $f:[2,\infty ) \rightarrow \R$ by simulating (8) as:
\begin{equation}
f(x)= \cases {
2 &, if $x\in [2,5]$,\cr
\frac {1}{x+2}\left [2x+1 + \frac {x-1}{f(x-1)} +\frac {2x-5}{f(x-1)f(x-2)}-
\frac {x-4}{f(x-1)f(x-2)f(x-3)}\right ] &, if $x\geq 5.$\cr
}
\end{equation}
Clearly, for any integer $n\geq 3$, $f(n)=x_n$, and $f$ is continuous, and,
in fact, piecewise rational and smooth on any open interval $(n,n+1)$ for
$n\geq 2$. The basic idea is, as in the proof of Theorem 3.(a), to show that
$f$ is an increasing and bounded function, and hence $(x_n)$ is an increasing sequence.
In next few lemmas we proceed with details.
{\bf Lemma 2.}\\ For all $x\geq 2$, we have $2 \leq f(x) \leq 3$, while for
$x\geq 53$ we have even stronger bounds:\\ $2.5 \leq f(x) \leq 2.67$.
{\bf Proof} \\ We prove inductively that $2 \leq f(x) \leq 3$ for $x\in [2,n]$.
For $n\leq 11$ it can be checked directly. Let $n\geq 11$ and $x\in (n,n+1]$. Then
$$\displaylines{
\qquad f(x) \leq \frac {1}{x+2}\left [2x+1 + \frac {x-1}{f(x-1)} +\frac {2x-5}{f(x-1)f(x-2)}\right ] \hfill \cr
\hfill \leq \frac {1}{x+2}\left [2x+1 + \frac {x-1}{2}+\frac {2x-5}{4}\right ]
=\frac {12x-3}{4x+8} \leq 3. \qquad\cr
}$$
On the other hand, $$f(x) \geq \frac {1}{x+2}\left [2x+1 + \frac {x-1}{3}+\frac {2x-5}{9}-\frac {x-4}{8} \right ]
=\frac {175x+44}{72x+144} \geq 2,$$
for all $x\geq 8$. So, $2 \leq f(x) \leq 3$ on $(n,n+1]$ and the first claim is
proved.
The stronger bounds also follow by induction. By direct computation, (using
Mathematica) one can check that they hold on the interval $[53,56]$. Suppose
$2.5 \leq f(x) \leq 2.67$ on some interval $[53,n]$, where $n\geq 56$ and take
$x\in[n,n+1]$. From (8) we get
$$f(x) \leq \frac {1}{x+2}\left [2x+1 + \frac {x-1}{2.5}+\frac {2x-5}{2.5^2}
-\frac {x-4}{2.67^3}\right ] = \frac {2.6675x+0.010148}{x+2} \leq 2.67,$$
for all $x\geq 0$. On the other hand,
$$f(x) \geq \frac {1}{x+2}\left [2x+1 + \frac {x-1}{2.67}+\frac {2x-5}{2.67^2}
-\frac {x-4}{2.5^3}\right ] = \frac {2.59108x+0.0181}{x+2},$$
and this is greater than $2.5$ for $x \geq 53$ (since the right hand side is
equal to $2.5$ for $x=52.918$). So, Lemma 2. is proved. \qed
{\bf Lemma 3.}\\ The function $f$ is increasing.
{\bf Proof}\\ Suppose again inductively on $n \in \N$ that $f$ increases on $
(5,n]$. We shall prove that $f$ increases on $(n,n+1)$. One can check directly
(using, e.g. Mathematica) that $f$ increases on $(5,n_0]$, as far as $n_0=61$.
Namely, the function $f$ on interval $(n,n+1)$ is a rational function whose
both numerator and denominator are polynomials with integer coefficients of
degree $n-4$. The derivative of $f$ is also a rational function, and its
denominator is always positive. So, we need to show that the numerators of the
derivative of $f$ are positive on every interval $(n,n+1)$, for $n\leq n_0-1$.
An advanced computer algebra system, such as {\it Mathematica}, gives us
readily explicit expressions for $f(x)$ and $f'(x)$ on any given interval
$(n,n+1)$. Let us denote $f'(x)=\frac {N_n(x)}{D_n(x)}$ on interval $(n,n+1)$.
If we can find some $k \in \N$, $k\leq n$, such that all coefficients of
$N_n(x+k)$ are nonnegative, we are done, since then $f'(x)$ can not change its
sign on the considered interval. It turns out that $k=2$ works for all intervals
$(n,n+1)$ with $n \leq 60$. Hence, $f'(x)\geq 0$ for $x\in (n,n+1)$, $n \leq 60$
and $f(x)$ is increasing on $[5,61]$. It is important to note here that all
performed computations include only integer quantities, and no round-off errors
occur.
Take $x\in (n,n+1)$ for $n\geq n_0$. Then $f'(x)>0$ for $x\in (i,i+1), \quad
i=5, \ldots ,n-1$, and also $f(x)\geq f(x-1)$, for $4 \leq x \leq n$.
Denote for short $f_i=f(x-i)$, $i\geq 1$. Then (9) can be written as
$$(x+2)f_1f_2f_3f(x)=(2x+1)f_1f_2f_3+(x-1)f_2f_3+(2x-5)f_3-(x-4).$$
By taking derivative, we get
$$f'(x)=\frac {1}{D(x)}\left [F(x)+F_3(x)f_3'(x)-F_1(x)f_1'(x)-F_2(x)f_2'(x)\right ],$$
where
$$\eqalign{
D(x)& =(x+2)f_1f_2f_3,\cr
F(x) &=2f_1f_2f_3+f_2f_3+2f_3-1-f_1f_2f_3f(x),\cr
F_1(x) &=\left [(x+2)f(x)-(2x+1)\right ]f_2f_3,\cr
F_2(x) &=\left [(x+2)f_1f(x)-(2x+1)f_1 -(x-1)\right ]f_3,\cr
F_3(x) &=2x-5+(2x+1)f_1f_2+(x-1)f_2-(x+2)f_1f_2f(x).\cr
}$$
Using (9), let us express $D(x), F(x), F_i(x), i=1,2,3$ only in terms of $f_i$'s
and $x$: \\
$$\eqalign{
D(x) & =(x+2)f_1f_2f_3,\cr
F(x) & =\frac {3}{x+2}(f_1f_2f_3+f_2f_3+3f_2-2),\cr
F_1(x) & =\frac {1}{f_1}\left [(f_2f_3+2f_3-1)x-(f_2f_3+5f_3-4)\right ],\cr
F_2(x) & =\frac {1}{f_2}\left [(2f_3-1)x-(5f_3-4)\right ],\cr
F_3(x) & =\frac {1}{f_3} (x-4).\cr
}$$
Now plug in derivatives $f_1'$ and $f_2'$ by the same rule, to obtain
\begin{eqnarray}
f'(x) & = & \frac {1}{D(x)}\Bigg \{F(x) - \frac {F_1(x)F(x-1)}{D(x-1)}-
\frac {F_2(x)F(x-2)}{D(x-2)}+\frac {F_1(x)F_1(x-1)}{D(x-1)}f_2' \nonumber \\
& + & \left [ \frac {F_1(x)F_2(x-1)}{D(x-1)}+\frac {F_2(x)F_1(x-2)}{D(x-2)}+F_3(x)\right ]f_3' \cr
& + & \left [\frac {F_2(x)F_2(x-2)}{D(x-2)}-\frac {F_1(x)F_3(x-1)}{D(x-1)}\right ]f_4'
-\frac {F_2(x)F_3(x-2)}{D(x-2)}f_5' \Bigg \}.
\end{eqnarray}
The ``coefficients'' by $f_2'$ and $f_3'$ are positive. By further pumping in
$f_5'$, the terms $f_6'$ and $f_7'$ will appear with positive ``coefficients'',
while $f_8'$ will appear with negative ``coefficient'' and a ``free'' negative
term $$-\frac {F_2(x)F_3(x-2)F(x-5)}{D(x-2)D(x-5)}$$ also appears. Every further
pumping in for $f_{3k+2}'$ contributes positive terms by $f_{3k+3}'$ and
$f_{3k+4}'$, a negative term by $f_{3k+5}'$ and a negative ``free'' term. If
we continue to pump in long enough, the argument of the negative term will
be eventually ``trapped'' in the interval $(2,5)$, and there $f'=0$. So, to
prove that $f'(x)>0$ we only have to show that the ``coefficient'' of $f_4'$
is positive and that ``free'' term (i.e. the term without any $f_i'$) is also
positive. These two facts we prove in the next lemma.
{\bf Lemma 4.}\\The ``coefficient'' of $f_4'$ and the ``free'' term, obtained
by pumping in $f_5', f_8', \ldots$ in (10) are both positive. More precisely,
with previous notations we have:\\
$(a)$ \hskip 5cm $
L_4(x):=\frac {F_2(x)F_2(x-2)}{D(x-2)}-\frac {F_1(x)F_3(x-1)}{D(x-1)} \geq 0,
$\\
for $x\geq n_0$;
$$\displaylines{
(b) \quad\quad
\qquad L(x):=F(x)-\frac {F_1(x)F(x-1)}{D(x-1)}-\frac {F_2(x)F(x-2)}{D(x-2)} \hfill \cr
\hfill -\frac {F_2(x)F_3(x-2)}{D(x-2)} \frac{F(x-5)}{D(x-5)}\left [1+
\frac{F(x-8)}{D(x-8)}+\frac{F(x-8)}{D(x-8)}\frac{F(x-11)}{D(x-11)}+\ldots \right] \geq 0,\qquad\cr
}$$
for $x\geq n_0$, where $n_0$ can be taken in the worst case to be $n_0=61$.
{\bf Proof}\\ (a) The condition $L_4(x) \geq 0$ is easily seen to be equivalent
to
$$\displaylines{
\qquad (x+1)f_1\left [(2f_3-1)x-(5f_3-4)\right ]\left [(2f_5-1)(x-2)-(5f_5-4)\right ]\hfill\cr
\hfill -xf_5\left [(f_2f_3+2f_3-1)x-(f_2f_3+5f_3-4)\right ](x-5) \geq 0.\qquad\cr
}$$
If we leave out the factor $(x+1)f_1$ from the first term and the factor $xf_5$
from the second term, we obtain even stronger inequality (recall, we are still
under inductive hypothesis, and this implies that $f_1 \geq f_5$). By grouping
terms by powers of $x$, this stronger inequality can be written in the form
$$c_{24}(x)x^2+c_{14}(x)x+c_{04}(x)=[c_{24}(x)x+c_{14}(x)]x+c_{04}(x)\geq 0,$$
where \\
$$\eqalign{
c_{24}(x) & =4f_3f_5-f_2f_3-4f_3-2f_5+2,\cr
c_{14}(x) & =6f_2f_3+17f_5+32f_3-28f_3f_5-19,\cr
c_{04}(x) & =45f_3f_5-5f_2f_3-36f_5-55f_3+44.\cr
}$$
Now estimate $c_{24}(x), \quad c_{14}(x)$ and $c_{04}(x)$ using the bounds from
Lemma 2. We easily obtain $c_{24}(x) \geq 3.8516, \quad c_{14}(x) \geq -58.6042
\quad c_{04}(x) \geq 46.6355$ for $x\geq n_0$. For example, since $f_{min}=2.5$,
$f_{max}=2.67$ for $x\geq n_0$, we have then
$$c_{24}(x) \geq 4f_{min}^2-f_{max}^2-6f_{max}+2 = 3.8516.$$
These bounds then imply $c_{24}(x)x+c_{14}(x) \geq 0$, and hence
$[c_{24}(x)x+c_{14}(x)]x+c_{04}(x)\geq 0$, for $x\geq n_0$. So, $L_4(x) \geq 0$
for $x\geq n_0$ and the claim (a) is proved.
(b) First of all, the function $\frac {F(x)}{D(x)}$ is easily seen to be less
than $\frac {129}{8(x+2)^2}$ (by using $2 \leq f_1, f_2, f_3 \leq 3$). For
$x \geq 10$, it follows then that $$\frac {F(x)}{D(x)} \leq q,$$ where $q =
\frac {129}{1152}$. By using $\frac {F(x-i)}{D(x-i)}\leq q$ in the brackets
of (b), we see that this sum is less than the sum of the geometric series
$1+q+q^2+\ldots =\frac {1}{1-q} < 2$. Hence $L(x) \geq 0$ will be a consequence
of the stronger inequality:
$$F(x)-\frac {F_1(x)F(x-1)}{D(x-1)}-\frac {F_2(x)F(x-2)}{D(x-2)}-2\frac {F_2(x)F_3(x-2)}{D(x-2)} \frac{F(x-5)}{D(x-5)}\geq 0.$$
But, since we do not know which one of the quotients $\frac {F(x-1)}{D(x-1)}$,
$\frac {F(x-2)}{D(x-2)}$ and $\frac {F(x-5)}{D(x-5)}$ is the largest, the last
inequality will be a consequence of the three inequalities in the next Lemma.
\newpage
{\bf Lemma 5.}\\ Keeping the same notations as above, we have
$$\displaylines{
(a) \quad\quad
F(x)\geq \left [F_1(x)+F_2(x)+2\frac {F_2(x)F_3(x-2)}{D(x-2)}\right ]\frac{F(x-5)}{D(x-5)}, \quad x\geq n_0,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\cr
(b) \quad\quad
F(x)\geq \left [F_1(x)+F_2(x)+2\frac {F_2(x)F_3(x-2)}{D(x-2)}\right ]\frac{F(x-2)}{D(x-2)}, \quad x\geq n_0,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\cr
(c) \quad\quad
F(x)\geq \left [F_1(x)+F_2(x)+2\frac {F_2(x)F_3(x-2)}{D(x-2)}\right ]\frac{F(x-1)}{D(x-1)}, \quad x\geq n_0.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\cr
}$$
{\bf Proof}\\ We shall prove only (a) with substantial details. The other two
inequalities can be proved essentially in the same manner. The inequality (a)
is equivalent to
$$x(x-3)^2f_1f_2f_3f_4f_5^2f_6f_7f_8A\geq [xf_3f_4f_5^2(ax-b)+2f_1(cx-d)(x-6)](x+2)B,$$
where \\
$$\eqalign{
A & =f_1f_2f_3+f_2f_3+3f_3-2,\cr
B & =f_6f_7f_8+f_7f_8+3f_8-2,\cr
a & =f_2^2f_3+2f_1f_3+2f_2f_3-f_1-f_2,\cr
b & =f_2^2f_3+5f_1f_3+5f_2f_3-4f_1-4f_2,\cr
c & =2f_3-1, \cr
d & =5f_3-4.
}$$
By inductive hypothesis it follows that $A\geq B$, and so if we prove the stronger
inequality by leaving out $A$ and $B$ in the above inequality, we are done.
But this stronger inequality turns out to be (after grouping terms by powers
of $x$ and some manipulations):
$$c_{35}(x)x^3+c_{25}(x)x^2+c_{15}(x)x+c_{05}(x) \geq 0,$$
or, what is the same,
\begin{equation}
[c_{35}(x)x+c_{25}(x)]x^2+c_{15}(x)x+c_{05}(x) \geq 0,
\end{equation}
where
$$c_{35}(x)=f_1f_2f_3f_4f_5^2f_6f_7f_8-f_2^2f_3^2f_4f_5^2-2f_2f_3^2f_4f_5^2-
2f_1f_3^2f_4f_5^2+f_2f_3f_4f_5^2+f_1f_3f_4f_5^2-4f_1f_3+2f_1,$$
$$c_{25}(x)=-6f_1f_2f_3f_4f_5^2f_6f_7f_8-f_2^2f_3^2f_4f_5^2+f_2f_3^2f_4f_5^2
+f_1f_3^2f_4f_5^2-2f_2f_3f_4f_5^2-2f_1f_3f_4f_5^2+26f_1f_3-16f_1,$$
$$c_{15}(x)=9f_1f_2f_3f_4f_5^2f_6f_7f_8+2f_2^2f_3^2f_4f_5^2
+10f_2f_3^2f_4f_5^2+10f_1f_3^2f_4f_5^2-8f_2f_3f_4f_5^2-8f_1f_3f_4f_5^2+8f_1f_3+8f_1,$$
$$c_{05}(x)=96f_1-120f_1f_3.$$
Now we estimate the above functions $c_{i5}(x)$ by the bounds from Lemma 2.,
$f_{min}=2.5$ and $f_{max}=2.67$ for $x\geq n_0$. We have
$$c_{35}(x)\geq f_{min}^9-f_{max}^7-4f_{max}^6+2f_{min}^5-4f_{max}^2+2f_{min}=
1569.9574,$$
and similarly $c_{25}(x)\geq -42278.4392$, $c_{15}(x)\geq 38334.7087$ and
$c_{05}(x)\geq -615.468$. This altogether then yields $[c_{35}(x)x+c_{25}(x)]
\geq 0$ for $x\geq n_0$, and this in turn implies (11) for $x\geq n_0$. Thus we
have proved (a).
As we said earlier, the inequalities (b) and (c) can be proved in the same way,
and we omit their proofs. \qed
To conclude, by lemmas 4. and 5. and induction hypothesis $f_i'\geq 0$ we have
shown that $f'(x)\geq 0$ for $x\in (n,n+1)$. By continuity of $f$ it follows
that $f$ is increasing on $(5,n+1)$, hence on $(5,n+1]$ and by induction $f$
is increasing on the whole interval $(2,\infty )$. This finally proves Theorem
4. \qed
This proof of Theorem 4., although rather involved (mostly computationally), is
conceptually quite simple, and can be considered as a calculus proof. Once
again, our proofs of Theorems 1., 3. and 4. show the strong interference
between ``discrete'' and ``continuous'' mathematics.
We note finally that the proofs of Theorems 3 and 4 we have presented here
prove much stronger claims than actually stated in these theorems. Namely, they
show not only that sequences $(x_n)$ given by recursions (5) and (8) are
increasing, but also that their natural continuous ``patch-works'' are
increasing functions, too. Theorems 3 and 4 itself can be proved much simpler in
such a way that we interlace the sequences $(x_n)$ given by recursions (5) and (8)
with an increasing sequence $a_n$, i.e. $a_n \leq x_n \leq a_{n+1}$. In the
case (5), $a_n=\frac {6n}{2n+3}$ for $n\geq 3$, and in the case (8) $a_n=\frac
{2n\phi ^2}{2n+3}$, for $n\geq 6$, where $\phi = \frac {1 + \sqrt 5}{2}$
is the golden ratio.
This ``interlacing'' or ``sandwiching'' method can also be applied to prove
the log-convexity of sequences $S^{(l)}(n)$ for $l=2,3$ and $4$. The details
are rather involved and will appear elsewhere.
We are not aware of any combinatorial proofs of the log-convexity property of
the sequences $S^{(l)}(n)$.
It can be proved by geometric reasoning that the numbers $S^{(l)}(n)$
of rank $l$ secondary structures asymptotically behave as
$$S^{(l)}(n)\sim K_l \alpha _l^n n^{-3/2},$$
where $K_l$ and $\alpha _l$ are constants depending only on $l$, and $\alpha _l
\in [2,3]$ and $\alpha _l \searrow 2$ as $l \rightarrow \infty$. The constant
$\alpha _l$ is the largest real solution of $x^l(x-2)^2=1$. For instance,
$\alpha _0=3$, $\alpha _1=(3+\sqrt 5)/2$, $\alpha _2=1+\sqrt 2$, and $\alpha _3$,
$\alpha _4$, $\alpha _5$ and $\alpha _6$ can be also explicitly
computed (see \cite{doslic1}).
By taking the quotient $x^{(l)}_n=\frac {S^{(l)}(n)}{S^{(l)}(n-1)}$, we see
that
$$x^{(l)}_n=\frac {S^{(l)}(n)}{S^{(l)}(n-1)} \sim \alpha _l \left (1-\frac {1}{n}\right )^{3/2} := a^{(l)}_n.$$
Clearly, the sequence $\left ( a^{(l)}_n \right )_{n \geq 1}$ increasingly
tends to $\alpha _l$ as $n \rightarrow \infty$. This suggests that
$\left ( x^{(l)}_n \right )_{n \geq 1}$ should be interlaced with
$\left ( a^{(l)}_n \right )_{n \geq 1}$, at least asymptotically.
These and many other properties of general secondary structures will appear
elsewhere \cite{doslic1}. More on the biological background of secondary structures
the reader can find in \cite{waterm} and \cite{kruskal}.
Our ``calculus method'' can be applied to many other combinatorial quantities
as well. For example, it can be proved in this way (see \cite{doslic1}) that
{\bf big Schr\"oder numbers} $r_n$ are log-convex. Recall that $r_n$ is the number
of lattice paths from $(0,0)$ to $(n,n)$ with steps $(1,0)$, $(0,1)$ and $(1,1)$
that never rise above the line $y=x$.
As our final example, let us consider the sequence $P_n(t)$ of the values of
Legendre polynomials in some fixed real $t \geq 1$. We start from Bonnet's
recurrence (see \cite{szego}):
\begin{equation}
P_n(t)=\frac {2n-1}{n} tP_{n-1}(t) - \frac {n-1}{n} P_{n-2}(t), \quad n\geq 2,
\end{equation}
with $P_0(t)=1$, $P_1(t)=t$.
Dividing this by $P_{n-1}(t)$ and denoting the quotient $\frac {P_n(t)}{
P_{n-1}(t)}$ by $x_n(t)$, we get the following recursion for $x_n(t)$:
\begin{equation}
x_n(t)=t \frac {2n-1}{n} - \frac {n-1}{n} \frac {1}{x_{n-1}(t)}
\end{equation}
with initial condition $x_1(t)=t$. The log-convexity of the sequence $P_n(t)$
will follow if we show that the sequence $x_n(t)$ is increasing.
To this end we define the function $f_t(x) : [0,\infty) \rightarrow \R$ by
\begin{equation}
f_t(x)= \cases {
t &, if $x\in [0,1]$,\cr
t\frac{2x-1}{x} - \frac {x-1}{x}\frac {1}{f_t(x-1)} &, if $x\geq 1$\cr
}
\end{equation}
It is easy to show by induction on $n$ that $f_t$ is continuous and piecewise
rational function on any interval $[1,n]$. By the same method it easily follows
that $f_t$ is bounded, i.e. $1 \leq f_t(x) \leq 2t$ for all $x\geq 1$.
It is clear that $f_t(n)=x_n(t)$, for any integer $n\geq 1$.
{\bf Theorem 5.}\\ The sequence $P_n(t)$ of the values of Legendre polynomials
is log-convex for any fixed real $t \geq 1$.
{\bf Proof}\\ The claim will follow if we show that $f_t(x)$ is an increasing
function on $[1,\infty)$. From piecewise rationality and boundedness of $f_t$
it follows that $f_t$ is differentiable on every open interval $(n,n+1)$.
Suppose that $f_t$ is increasing on $[1,n]$ and take $x\in (n,n+1)$.
From (14) we have \\
$$\eqalign{
f_t'(x) & =t\frac {1}{x^2}-\frac {1}{x^2}\frac {1}{f_t(x-1)}+\left (1-\frac {1}{x}\right )
\frac {f_t'(x-1)}{f_t^2(x-1)} \cr
& = \frac {1}{x^2f_t(x-1)}\left [tf_t(x-1)-1 \right ] +\left (1-\frac {1}{x}\right )
\frac {f_t'(x-1)}{f_t^2(x-1)} \cr
}$$
The second term is positive by the induction hypothesis, and the first term
is positive because \\ $tf_t(x-1)-1 \geq f_t(x-1)-1 \geq 0$, for all $x\geq 1$.
So, the function $f_t(x)$ is increasing on the interval $(n,n+1)$, and then, by
continuity, also on $[1,n+1]$. This completes the step of induction. \qed
As a consequence, we get the log-convexity for the sequence of {\bf central Delannoy
numbers}. Recall that the $n$-th central Delannoy number counts the number of lattice
paths in $(x,y)$ coordinate plane from $(0,0)$ to $(n,n)$ with steps $(1,0)$,
$(0,1)$ and $(1,1)$. (Such paths are also known as {\bf king's paths}.)
{\bf Theorem 6}\\ (a) The sequence $D(n)$ of Delannoy numbers is log-convex.\\
(b) There exists $x=\lim_{n \rightarrow \infty} \frac {D(n)}{D(n-1)}$, and
$x=3+2 \sqrt 2$.
{\bf Proof}\\ (a) First note that the $n$-th central Delannoy number is the
value of the $n$-th Legendre polynomial at $t=3$, $D(n)=P_n(3)$. This
follows easily from the explicit expression for the
generating function of the sequence $D(n)$, $D(x)=\frac {1}{\sqrt {1-6x+x^2}}$.
Now apply Theorem 5.
(b) By (a) we know that $x_n(3)$ is increasing (and clearly bounded), and then
by passing to limit in (13) for $t=3$, the claim follows. \qed
| {
"timestamp": "2006-03-16T18:06:58",
"yymm": "0603",
"arxiv_id": "math/0603405",
"language": "en",
"url": "https://arxiv.org/abs/math/0603405",
"abstract": "Using calculus we show how to prove some combinatorial inequalities of the type log-concavity or log-convexity. It is shown by this method that binomial coefficients and Stirling numbers of the first and second kinds are log-concave, and that Motzkin numbers and secondary structure numbers of rank 1 are log-convex. In fact, we prove via calculus a much stronger result that a natural continuous ``patchwork'' (i.e. corresponding dynamical systems) of Motzkin numbers and secondary structures recursions are increasing functions. We indicate how to prove asymptotically the log-convexity for general secondary structures. Our method also applies to show that sequences of values of some orthogonal polynomials, and in particular the sequence of central Delannoy numbers, are log-convex.",
"subjects": "Combinatorics (math.CO)",
"title": "Calculus proofs of some combinatorial inequalities",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808759252645,
"lm_q2_score": 0.721743200312399,
"lm_q1q2_score": 0.7075110565953421
} |
https://arxiv.org/abs/2201.02530 | Semilinear Li & Yau inequalities | We derive an adaptation of Li & Yau estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative Ricci tensor. We then apply these estimates to obtain a Harnack inequality and to discuss monotonicity, convexity, decay estimates and triviality of ancient and eternal solutions. | \section{Introduction}
We show some Li~\&~Yau estimates (see~\cite{liyau}) for positive solutions of semilinear heat equations $u_t = \Delta u + u^p$ in ${{\mathbb R}}^n$, or in a complete Riemannian manifold $(M,g)$ with nonnegative Ricci tensor, where $p>0$ and we derive an associated Harnack inequality. Similar results can be found in the papers~\cite{jiayuli,wuyang}.
We will then discuss the application of these inequalities and methods to the analysis of positive classical {\em ancient} and {\em eternal} solutions.
\noindent We call a solution
\begin{itemize}
\item {\em ancient} if it is defined in $M\times(-\infty,T)$ for some $T\in{{\mathbb R}}$,
\item {\em immortal} if it is defined in $M\times(T,+\infty)$ for some $T\in{{\mathbb R}}$,
\item {\em eternal} if it is defined in $M\times{{\mathbb R}}$.
\end{itemize}
Furthermore, we call a solution $u$ {\em static} if is independent of time (\textit{i.e. }$u(x,t)=u(x)$, hence it satisfies $\Delta u + u^p=0$); {\em trivial} if it is constant in space (that is, $u(x,t)=u(t)$ and solves the ODE $u'=u^p$). We say that $u$ is simply {\em constant} if it is constant in space and time.\\
Notice that positive ancient (trivial) solutions always exist (by solving the corresponding ODE), while being eternal is quite restrictive: for instance, by an easy maximum principle argument, if $M$ is compact and $u$ is eternal then $u\equiv0$ (see~\cite[Corollary~2.2]{cama2}, for instance). In the noncompact situation, this is not true: the following ``Talenti's function''~\cite{tal}
$$
u(x)=\frac{24}{\bigl(1+|x|^2\bigr)^2}
$$
on ${{\mathbb R}}^6$ satisfies $\Delta u + u^2 = 0$, in particular $u$ is a nonzero eternal solution of $u_t=\Delta u + u^2$.
A reason for the interest in ancient or eternal solutions is that they typically arise as blow--up limits when the solutions of semilinear parabolic equations (in bounded intervals) develop a singularity at a certain time (i.e. the solution becomes unbounded). They also appear naturally and play a key role in the analysis of mean curvature flow and of Ricci flow (from which we get several suggestions), which are also described by (much more complicated systems of) parabolic PDEs. In such cases, the solutions are respectively, evolving hypersurfaces and abstract Riemannian manifolds. Analyzing their properties and eventually classifying them lead to understand the behavior of the solutions close to the singularity or even (in very lucky situations, notably the motion by curvature of embedded curves in the plane and the 2--dimensional Ricci flow) to actually {\em exclude} the formation (existence) of singularities. Moreover, this analysis can also be used to get uniform (universal) estimates on the ``rate'' a solution (or some related quantity) becomes unbounded at a singularity. Indeed (roughly speaking), typically a ``faster'' rate implies that performing a blow--up at a singularity, we obtain a bounded, nonzero, nonconstant, eternal solution, while with the slower ``standard'' natural rate, we get an ancient solution (immortal solutions are usually less significant). This has been done, for example, by Pol\'a\v cik, Quittner and Souplet through "universal estimates" for semilinear equations in ${{\mathbb R}}^n$ (see~\cite{pqs1,pqs2}) or by Hamilton through a technique of "smart point picking", which is more suitable for geometric flows (see~\cite{hamilton9}).\\
For instance, excluding the existence of bounded, positive, nonconstant, eternal solutions to the equation $u_t = \Delta u + u^p$ in ${{\mathbb R}}^n\times{{\mathbb R}}$, we have a (universal, up to a constant) $L^\infty$ bound from above on every solution, approaching the singular time $T$.\\
We will see how from the Li~\&~Yau estimates (holding for the exponent $p$ in a suitable range), one can exclude the existence of such eternal solutions. Instead, unfortunately, even if we obtain some strong properties (pointwise monotonicity and convexity in time) of ancient solutions on Riemannian manifolds with nonnegative Ricci tensor (for suitable exponents $p$), we are still not able to show the ``natural'' conjecture that they are trivial, constant in space, as it is known when the ambient is ${{\mathbb R}}^n$, by a result of Merle and Zaag~\cite{merlezaag}.
There is a quite large literature on the subject, for a rather complete account we refer the interested reader to the paper of Souplet and Zhang~\cite{souzha} as well as the book of Quittner and Souplet ~\cite{qsbook}. Very important developments for the Euclidean case (some of them very recent) are obtained in~\cite{pqs2,quittner,quittner2}, while some extensions to the case of Riemannian manifolds can be found in~\cite{jiayuli,wuyang,zhugrad}.
We mention that all the conclusions also hold for ancient solutions of $u_t = \Delta u + |u|^p$, without assuming their positivity, by the following theorem.
\begin{thmm}[Theorem~2.6 in~\cite{cama2}]
Let $(M,g)$ be a complete Riemannian manifold with Ricci tensor uniformly bounded below and let $u$ be an ancient solutions of $u_t = \Delta u + |u|^p$, then either $u\equiv 0$ or $u>0$ everywhere.
\end{thmm}
{\em In the whole paper the Riemannian manifolds $(M,g)$ will be smooth, complete, connected and without boundary. We will denote with $\Delta$ the associated Laplace--Beltrami operator. As it is standard, we will write ${\mathrm {Ric}}\geq0$ and we will say that the Ricci tensor is nonnegative with the meaning that all its eigenvalues are nonnegative. Finally, all the solutions we will consider are classical, $C^2$ in space and $C^1$ in time.}
\begin{ackn} We would like to thank Giacomo Ascione for his help with Mathematica\textsuperscript{TM}.
\end{ackn}
\section{Li~\&~Yau--type estimates for semilinear heat equations}\label{liyausem}
We now prove some Li~\&~Yau--type estimates for positive solutions of the semilinear heat equation $u_t = \Delta u + u^p$ with $p>1$, on a complete $n$--dimensional Riemannian manifold $(M,g)$ with nonnegative Ricci tensor ${\mathrm {Ric}}$.
For the sake of simplicity and more clarity we first assume that $M$ is compact.
Let $u:M\times[0,T)\to{{\mathbb R}}$ be a positive solution of $u_t = \Delta u + u^p$, then setting
$f=\log u$, we have
$$
\vert\nabla f\vert=\frac{\vert\nabla u\vert}{u}\qquad\qquad \Delta f=\frac{\Delta u}{u}-\frac{\vert\nabla u\vert^2}{u^2}=\frac{\Delta u}{u}-\vert\nabla f\vert^2
$$
$$
f_t=\frac{u_t}{u}=\frac{\Delta u}{u}+u^{p-1}=\Delta f+\vert\nabla f\vert^2+e^{(p-1)f}.
$$
We consider $\alpha,\beta\in[0,1]$ and after defining
\begin{align*}
F=&\,t(f_t-\alpha \vert\nabla f\vert^2-\beta e^{f(p-1)})\\
=&\,t\bigl(\frac{u_t}{u}-\alpha\frac{\vert\nabla u\vert^2}{u^2}-\beta u^{p-1}\bigr)\\
=&\,t\bigl(\frac{\Delta u}{u}-\alpha\frac{\vert\nabla u\vert^2}{u^2}+(1-\beta)u^{p-1}\bigr)\\
=&\,t(\Delta \log u+(1-\alpha)\frac{\vert\nabla u\vert^2}{u^2}+(1-\beta)u^{p-1}\bigr)\\
=&\,t(\Delta f+(1-\alpha)\vert\nabla f\vert^2+(1-\beta)e^{(p-1)f})
\end{align*}
we compute, for $t>0$,
\begin{align*}
(\partial_t-\Delta)F=&\,F/t+t(\partial_t-\Delta)\Delta f+t(p-1)(1-\beta)[(\partial_t-\Delta)f]e^{(p-1)f}\\
&\,-t(p-1)^2(1-\beta)\vert \nabla f\vert^2 e^{(p-1)f}+2t(1-\alpha)\nabla f\nabla f_t -t(1-\alpha)\Delta\vert\nabla f\vert^2\\
=&\,F/t+t\bigl\{\Delta (\vert\nabla f\vert^2+e^{(p-1)f})
+(p-1)(1-\beta)(\vert\nabla f\vert^2+e^{(p-1)f})e^{(p-1)f}\\
&\,-(p-1)^2(1-\beta)\vert \nabla f\vert^2 e^{(p-1)f}+2(1-\alpha)\nabla f\nabla f_t -(1-\alpha)\Delta\vert\nabla f\vert^2\bigr\}\\
=&\,F/t+t\bigl\{(p-1)\Delta fe^{(p-1)f}+(p-1)^2\vert\nabla f\vert^2 e^{(p-1)f}+(p-1)(1-\beta)e^{2f(p-1)}\\
&\,+(p-1)(1-\beta)\vert\nabla f\vert^2e^{(p-1)f}-(p-1)^2(1-\beta)\vert \nabla f\vert^2 e^{(p-1)f}\\
&\,+2(1-\alpha)\nabla f\nabla[\Delta f+\vert\nabla f\vert^2+e^{(p-1)f}]+\alpha\Delta\vert\nabla f\vert^2\bigr\}\\
=&\,F/t+t\bigl\{(p-1)Fe^{(p-1)f}/t+(p-1)[\alpha+\beta(p-2)]\vert \nabla f\vert^2 e^{(p-1)f}\\
&\,+2(1-\alpha)\nabla f\nabla[\Delta f+\vert\nabla f\vert^2+e^{(p-1)f}]+2\alpha\vert\mathrm{Hess}\, f\vert^2+2\alpha\nabla f\Delta\nabla f\bigr\}.
\end{align*}
By the nonnegativity of the Ricci tensor there holds $\nabla f\Delta\nabla f\geq \nabla f\nabla \Delta f $ and using the inequality $\vert\mathrm{Hess}\, f\vert^2\geq(\Delta f)^2/n$, we have, as
$$
\Delta f=F/t-(1-\alpha)\vert\nabla f\vert^2-(1-\beta)e^{(p-1)f},
$$
\begin{align}
(\partial_t-\Delta)F
\geq &\,F/t+t\bigl\{2\alpha(\Delta f)^2/n+(p-1)Fe^{(p-1)f}/t+(p-1)[\alpha+\beta(p-2)]\vert \nabla f\vert^2 e^{(p-1)f}\nonumber\\
&\,+2(1-\alpha)\nabla f\nabla[\Delta f+\vert\nabla f\vert^2+e^{(p-1)f}]+2\alpha\nabla f\nabla\Delta f\bigr\}\nonumber\\
=&\,F/t+t\bigl\{2\alpha(\Delta f)^2/n+(p-1)Fe^{(p-1)f}/t+(p-1)[\alpha+\beta(p-2)]\vert \nabla f\vert^2 e^{(p-1)f}\nonumber\\
&\,+2\nabla f\nabla[\Delta f+(1-\alpha)\vert\nabla f\vert^2+(1-\alpha)e^{(p-1)f}]\bigr\}\nonumber\\
=&\,F/t+t\bigl\{2\alpha(\Delta f)^2/n+(p-1)Fe^{(p-1)f}/t+2\nabla f\nabla F/t\nonumber\\
&\,+(p-1)(\beta p-\alpha)\vert \nabla f\vert^2 e^{(p-1)f}\bigr\}\nonumber\\
=&\,F/t+t\bigl\{2\alpha F^2/nt^2+2\alpha(1-\beta)^2e^{2f(p-1)}/n+[(p-1)-4\alpha(1-\beta)/n]Fe^{(p-1)f}/t\nonumber\\
&\,+2\alpha(1-\alpha)^2\vert\nabla f\vert^4/n-4\alpha(1-\alpha)\vert\nabla f\vert^2F/nt\\
&\,+2\nabla f\nabla F/t+[(p-1)(\beta p-\alpha)+4\alpha(1-\alpha)(1-\beta)/n]\vert \nabla f\vert^2 e^{(p-1)f}\bigr\}.\label{eq960}
\end{align}
Hence, at a point where $F\leq 0$ we conclude
\begin{align*}
(\partial_t-\Delta)F\geq &\,F/t+2\alpha F^2/nt+2t\alpha(1-\beta)^2e^{2f(p-1)}/n+[(p-1)-4\alpha(1-\beta)/n]Fe^{f(p-1)}\\
&\,+2\nabla f\nabla F+t[(p-1)(\beta p-\alpha)+4\alpha(1-\alpha)(1-\beta)/n]\vert \nabla f\vert^2 e^{(p-1)f}\\
\geq &\,F/t+2\alpha F^2/nt+2t\alpha(1-\beta)^2e^{2f(p-1)}/n+[(p-1)-4\alpha(1-\beta)/n]Fe^{(p-1)f}\\
&\,+2\nabla f\nabla F,
\end{align*}
provided that
\begin{equation}\label{cond2}
(p-1)(\beta p-\alpha)+4\alpha(1-\alpha)(1-\beta)/n\geq 0.
\end{equation}
Now we deal with the sum
$$
2t\alpha(1-\beta)^2w^2/n+[(p-1)-4\alpha(1-\beta)/n]Fw,
$$
where we set $w=e^{(p-1)f}\geq 0$, and we see that the minimum of this quadratic in $w$ is given by
$$
-F^2\frac{[(p-1)-4\alpha(1-\beta)/n]^2n}{8t\alpha(1-\beta)^2}=
-F^2\frac{[n(p-1)-4\alpha(1-\beta)]^2}{8nt\alpha(1-\beta)^2}
$$
Hence, after substituting in the above formula and using Hamilton's trick (see~\cite{hamilton2} or~\cite[Lemma~2.1.3]{Manlib}), being $M$ compact we get
\begin{align*}
\frac{d\,}{dt} F_{\min}(t)\geq
&\,\frac{F_{\min}(t)}{t}+\frac{F^2_{\min}(t)}{nt}\left[2\alpha-\frac{[n(p-1)-4\alpha(1-\beta)]^2}{8\alpha(1-\beta)^2}\right]\\
=&\,\,\frac{F_{\min}(t)}{t}+F^2_{\min}(t)\frac{n(p-1)}{8t\alpha(1-\beta)^2}\biggl[1+\frac{8\alpha(1-\beta)}{n}-p\biggr]\\
=&\,\,\frac{F_{\min}(t)}{t}+\frac{\varepsilon F^2_{\min}(t)}{t},
\end{align*}
at almost every $t>0$, where $F_{\min}\leq 0$ and we set
\begin{equation}\label{epseq}
\varepsilon=\varepsilon(n,p,\alpha,\beta)=\frac{n(p-1)}{8\alpha(1-\beta)^2}\biggl[1+\frac{8\alpha(1-\beta)}{n}-p\biggr].
\end{equation}
If now $\varepsilon$ is positive, that is,
\begin{equation}\label{cond1}
p< 1+\frac{8\alpha(1-\beta)}{n},
\end{equation}
when $F_{\min}\leq -1/\varepsilon$, we have $\frac{d\,}{dt}F_{\min}(t)>0$. Since $F_{\min}(0)=0$, this implies (``integrating'' this differential inequality) that
$F_{\min}\geq -1/\varepsilon$, hence
\begin{equation}
(f_t-\alpha \vert\nabla f\vert^2-\beta e^{(p-1)f})(x,t)=\frac{F(x,t)}{t}\geq F_{\min}(t)\geq -\frac{1}{\varepsilon t}\label{eq970}
\end{equation}
for every $x\in M$ and $t>0$. That is,
$$
\frac{u_t}{u}\geq \alpha \frac{\vert\nabla u\vert^2}{u^2}+\beta u^{p-1}-\frac{1}{\varepsilon t}
$$
or
\begin{equation}\label{eq980}
u_t\geq \alpha \frac{\vert\nabla u\vert^2}{u}+\beta u^p-\frac{u}{\varepsilon t}.
\end{equation}
Now the point is to find out for what exponents $p>1$ (depending on the dimension $n$) there exists constants $\alpha,\beta\in[0,1]$ satisfying conditions~\eqref{cond2} and~\eqref{cond1}, that is,
$$
(p-1)(\beta p-\alpha)+4\alpha(1-\alpha)(1-\beta)/n\geq 0
$$
$$
p< 1+\frac{8\alpha(1-\beta)}{n}.
$$
We find that these conditions are satisfied
\begin{itemize}
\item for $n\leq 3$, for every $p<8/n$ (choosing $\alpha=1$ and $\beta=1/p$\,) and for no $p\geq 8/n$,
\item for $n\geq 4$ (by using Mathematica\textsuperscript{TM}, see the Appendix), for every and only
$$
p<\frac{3n+4+3\sqrt{n(n+4)}}{2(3n-4)}
$$
and some $\alpha,\beta\in(0,1)$ (notice the right side is larger than $8/n$\,).
\end{itemize}
\begin{defn}\label{pienne}
Given $p>1$, we define $\overline{p}_n=8/n$ for $n\leq 3$ and
$$
\overline{p}_n=\frac{3n+4+3\sqrt{n(n+4)}}{2(3n-4)}
$$
for $n\geq 4$.\\
We say that $\alpha\in(0,1]$ and $\beta\in[0,1)$ are ``admissible'' for $p\in(1,\overline{p}_n)$ if they satisfy the two conditions~\eqref{cond2} and~\eqref{cond1}, in such case we define $\varepsilon(n,p,\alpha,\beta)$ as in formula~\eqref{epseq}.
\end{defn}
\begin{rem}\label{betanonzero}
It must be $\alpha>0$ and $\beta<1$, otherwise condition~\eqref{cond1} would imply $p<1$. Moreover, notice that for every $p\in(1,\overline{p}_n)$ there is always an admissible pair $(\alpha,\beta)$ with $\beta\not=0$.
\end{rem}
We deal now with the general case where $(M,g)$ is complete with ${\mathrm {Ric}}\geq 0$.\\
We consider $p\in(1,\overline{p}_n)$, with a relative admissible pair of constants $(\alpha,\beta)$ as above in the compact case. It is easy to see (by continuity) that if $a\in(0,\alpha)$ is sufficiently close to $\alpha$, the pair $(a,\beta)$ is still admissible for $p$, then we define a slightly perturbed function $F$,
\begin{align*}
F_a=&\,t(f_t-a \vert\nabla f\vert^2-\beta e^{(p-1)f})\\
=&\,t(\Delta f+(1-a )\vert\nabla f\vert^2+(1-\beta)e^{(p-1)f})
\end{align*}
with $0<a<\alpha$. Repeating the previous computations~\eqref{eq960} in the compact case, we have
\begin{align}
(\partial_t-\Delta)F_a
\geq &\,F_a/t+2a F_a^2/nt+2ta (1-\beta)^2e^{2f(p-1)}/n+[(p-1)-4a (1-\beta)/n]F_ae^{(p-1)f}\nonumber\\
&\,+2at(1-a )^2\vert\nabla f\vert^4/n-4a (1-a )\vert\nabla f\vert^2F_a/n\nonumber\\
&\,+2\nabla f\nabla F_a+t[(p-1)(\beta p-a )+4a (1-a )(1-\beta)/n]\vert \nabla f\vert^2 e^{(p-1)f}.\nonumber\\
\geq &\,F_a/t+2a F_a^2/nt+2ta (1-\beta)^2e^{2f(p-1)}/n+[(p-1)-4a (1-\beta)/n]F_ae^{(p-1)f}\nonumber\\
&\,-4a (1-a )\vert\nabla f\vert^2F_a/n+2\nabla f\nabla F_a,\label{eq950}
\end{align}
since condition~\eqref{cond2} holds, with $a$ in place of $\alpha$.\\
We now want to compute the evolution equation of a ``localization'' of the function $F_a$, for $a >1$. We consider the following cut--off functions (of Li and Yau~\cite{liyau}): let $\psi:[0,+\infty)\to{{\mathbb R}}$ be a smooth function satisfying:
\begin{enumerate}
\item $\psi(s)=1$ if $s\in[0,1]$ and $\psi(s)=0$ if $s\in[2,+\infty)$,
\item $-C_1\leq \psi'(s)\leq 0$ for every $s\in[0,+\infty)$, that is $\psi$ is nonincreasing,
\item $|\psi''(s)|\leq C_2$ for every $s\in[0,+\infty)$,
\item $|\psi'(s)|^2/\psi(s)\leq C_3$ for every $s\in[0,+\infty)$ such that $\psi(s)\not=0$,
\end{enumerate}
for some positive constants $C_1,C_2,C_3$. Then, fixed $x_0\in M$ and $R>0$, denoting with $r=r(x)$ the distance function from the point $x_0$ in $(M,g)$, we define $\varphi(x)=\psi\bigl(\frac{r(x)}{R}\bigr)$.\\
We clearly have
$\nabla\varphi(x)=\psi'\bigl(\frac{r(x)}{R}\bigr)\frac{\nabla r(x)}{R}$, hence
\begin{equation}\label{est2}
\vert \nabla\varphi(x)\vert^2=\Bigl\vert\psi'\Bigl(\frac{r(x)}{R}\Bigr)\Bigr\vert^2\frac{1}{R^2}
\leq C_3\psi\Bigl(\frac{r(x)}{R}\Bigr)\frac{1}{R^2}=\frac{C_3\varphi(x)}{R^2}\leq \frac{C_3}{R^2}.
\end{equation}
when $\nabla\varphi$ exists (almost everywhere), being $\vert\nabla r(x)\vert^2=1$.\\
Thanks to the nonnegativity assumption on the Ricci tensor, by the {\em Laplacian comparison theorem} (see~\cite[Chapter~9, Section~3.3]{petersen2} and also~\cite{sheng1}), if ${\mathrm {Ric}}\geq0$, we have
\begin{equation}\label{est1}
\Delta r(x)\leq\frac{n-1}{r(x)}
\end{equation}
for every $x\in M$ {\em in the sense of support functions} (or {\em in the sense of viscosity}, see~\cite{crisli1} -- check also~\cite[Appendix~A]{manmasura} for comparison of the two notions), in particular, this inequality can be used in maximum principle arguments, see again~\cite[Chapter~9, Section~3]{petersen2}, for instance. Hence, we have $\Delta\varphi(x)=0$ if $r(x)\leq R$ and
\begin{align}
\Delta \varphi(x)=&\,\psi''\Bigl(\frac{r(x)}{R}\Bigr)\frac{1}{R^2}+\psi'\Bigl(\frac{r(x)}{R}\Bigr)\frac{\Delta r(x)}{R}\\
\geq&\,-\frac{C_2}{R^2}-\frac{C_1(n-1)}{R\,r(x)}\\
\geq&\,-\frac{C_2}{R^2}-\frac{C_1(n-1)}{R^2}\\
=&\,-\frac{C_4}{R^2},
\end{align}
if $r(x)\geq R$. Thus, we have the general estimate on the whole $M$ (in the sense of support functions) $\Delta \varphi(x)\geq -{C_4}/{R^2}$.
Then, setting $Q (x,t) = \varphi(x) F_a (x,t)$, we compute (using formula~\eqref{eq950} and the inequalities above)
\begin{align*}
(\partial_t-\Delta)Q=&\,\varphi(\partial_t-\Delta)F_a -F_a \Delta\varphi-2
\nabla\varphi\nabla F_a \\
\geq&\,\frac{Q}{t}+\frac{2aQ^2}{nt\varphi}+\frac{2t\varphi a (1-\beta)^2}{n}e^{2f(p-1)}+Qe^{(p-1)f}\Bigl[(p-1)-\frac{4a (1-\beta)}{n}\Bigr]\\
&\,-\frac{4a(1-a)}{n}Q\vert\nabla f\vert^2+2\varphi\nabla f\nabla F_a-2\nabla\varphi\nabla F_a+F_a \frac{C_4}{R^2}.
\end{align*}
Arguing as before, setting $Q_{\min}(t)=\min_{x\in M}Q(x,t)$, we have $Q_{\min}(0)=0$ and
$$
\varphi>0,\qquad 0=\nabla Q=\varphi\nabla F_a +F_a \nabla\varphi,\qquad \Delta Q\geq 0,
$$
at every point $x$ where the minimum of $Q$ is achieved. Hence (recalling that $a<\alpha\leq 1$), by Hamilton's trick, at almost every $t>0$ such that $Q_{\min}(t)<0$ we have
\begin{align*}
\frac{d\,}{dt}Q_{\min}
\geq&\,\frac{Q_{\min}}{t}+\frac{2aQ_{\min}^2}{nt\varphi}+\frac{2t\varphi a (1-\beta)^2}{n}e^{2f(p-1)}+Q_{\min}e^{(p-1)f}\Bigl[(p-1)-\frac{4a (1-\beta)}{n}\Bigr]\\
&\,+Q_{\min }\frac{C_4}{\varphi R^2}-\frac{4a(1-a)}{n}Q_{\min}\vert\nabla f\vert^2+2\varphi F_a \frac{\vert\nabla\varphi\vert^2}{\varphi^2}-2\varphi F_a \frac{\nabla f\nabla\varphi}{\varphi}\\
\geq&\,\frac{Q_{\min}}{t}+\frac{2aQ_{\min}^2}{nt\varphi}+\frac{2t\varphi a (1-\beta)^2}{n}e^{2f(p-1)}+Q_{\min}e^{(p-1)f}\Bigl[(p-1)-\frac{4a (1-\beta)}{n}\Bigr]\\
&\,+Q_{\min}\biggl[\frac{C_4}{\varphi R^2}+\frac{2C_3}{\varphi R^2}\biggr]-\frac{4a(1-a)}{n}Q_{\min}\vert\nabla f\vert^2+2Q_{\min}\frac{C_5\vert\nabla f\vert}{R\sqrt{\varphi}}\\
=&\,\frac{Q_{\min}}{t}+\frac{2aQ_{\min}^2}{nt\varphi}+\frac{2t\varphi a (1-\beta)^2}{n}e^{2f(p-1)}+Q_{\min}e^{(p-1)f}\Bigl[(p-1)-\frac{4a (1-\beta)}{n}\Bigr]\\
&\,+Q_{\min}\biggl[\frac{C_4}{\varphi R^2}+\frac{2C_3}{\varphi R^2}\biggr]-Q_{\min}\frac{4a(1-a )}{na ^2\varphi}\biggl[\vert\nabla f\vert^2\varphi-\frac{2C_6\vert\nabla f\vert\sqrt{\varphi}}{R}\biggl],
\end{align*}
where we used the fact that $\vert \nabla\varphi\vert\leq \sqrt{C_3\varphi}/{R}$, by inequality~\eqref{est2}, and we set $C_5=\sqrt{C_3}$, $C_6=\frac{C_5na ^2}{4a(1-a)}>0$.\\
Setting $y=\vert\nabla f\vert\sqrt{\varphi}\geq 0$, we see that the difference inside the last square brackets is given by $y^2-2C_6y/R$, which is then larger or equal than $-C_6^2/R^2$, hence
\begin{align*}
\frac{d\,}{dt} Q_{\min}
\geq&\,\frac{Q_{\min}}{t}+\frac{2aQ_{\min}^2}{nt\varphi}+\frac{2t\varphi a (1-\beta)^2}{n}e^{2f(p-1)}+Q_{\min}e^{(p-1)f}\Bigl[(p-1)-\frac{4a (1-\beta)}{n}\Bigr]\\
&\,+Q_{\min}\biggl[\frac{C_4}{\varphi R^2}+\frac{2C_3}{\varphi R^2}\biggr]+Q_{\min}\frac{4a(1-a)C_6^2}{na ^2\varphi R^2}\\
=&\,\frac{Q_{\min}}{t}+\frac{2aQ_{\min}^2}{nt\varphi}+\frac{2t\varphi a (1-\beta)^2}{n}e^{2f(p-1)}+Q_{\min}e^{(p-1)f}\Bigl[(p-1)-\frac{4a (1-\beta)}{n}\Bigr]\\
&\,+\frac{C_7Q_{\min}}{\varphi R^2}
\end{align*}
with $C_7=C_4+2C_3+4a(1-a)C_6^2/na^2$. Now we deal with the sum
\begin{align*}
&\frac{2t\varphi a (1-\beta)^2}{n}e^{2f(p-1)}+Q_{\min}e^{(p-1)f}\Bigl[(p-1)-\frac{4a (1-\beta)}{n}\Bigr]\\
&=\frac{2t\varphi a (1-\beta)^2}{n}w^2+Q_{\min}\Bigl[(p-1)-\frac{4a (1-\beta)}{n}\Bigr]w,
\end{align*}
where we set $w=e^{(p-1)f}\geq 0$, as in the compact case. The minimum of this quadratic in $w$ is given by
$$
-Q_{\min}^2\frac{[n(p-1)-4a(1-\beta)]^2}{8nta(1-\beta)^2\varphi}=
\frac{\varepsilon Q_{\min}^2}{t\varphi}-\frac{2aQ_{\min}^2}{nt\varphi}
$$
where $\varepsilon=\varepsilon(n,p,a ,\beta)$ is defined by equation~\eqref{epseq} and it is positive, being the pair $(a,\beta)$ admissible for $p$ (see condition~\eqref{cond1}).\\
Then, substituting in the previous formula and collecting similar terms, we finally get
\begin{align*}
\frac{d\,}{dt} Q_{\min}
\geq&\,\frac{Q_{\min}}{t}+\frac{\varepsilon Q_{\min}^2}{t\varphi}+\frac{C_7Q_{\min}}{\varphi R^2}.
\end{align*}
It follows, arguing as in the compact case that
$$
Q_{\min}\geq -\frac{\varphi}{\varepsilon}-\frac{tC_7}{\varepsilon R^2}\geq -\frac{1}{\varepsilon}-\frac{tC_7}{\varepsilon R^2}
$$
where we used the fact that $\varphi\leq 1$, hence
$$
\varphi(x)\frac{F_a (x,t)}{t}\geq \frac{Q_{\min}(t)}{t}\geq-\frac{1}{\varepsilon t}-\frac{C_7}{\varepsilon R^2}
$$
for every $x\in M$ and $t>0$. Sending $R\to+\infty$ (so that $\varphi \equiv 1$ on $M$), we then have
$$
\frac{F_a (x,t)}{t}\geq -\frac{1}{\varepsilon t}
$$
that is,
$$
f_t-a\vert\nabla f\vert^2-\beta e^{(p-1)f}\geq -\frac{1}{\varepsilon(n,p,a,\beta)t}
$$
and finally sending $a \to\alpha^-$, being $a\mapsto\varepsilon(n,p,a,\beta)$ a continuous function, we get the same conclusion~\eqref{eq970}
$$
f_t-\alpha \vert\nabla f\vert^2-\beta e^{(p-1)f}\geq-\frac{1}{\varepsilon(n,p,\alpha,\beta)t}
$$
that is,
\begin{equation}\label{eq990}
u_t-\alpha\frac{\vert\nabla u\vert^2}{u}-\beta u^p\geq -\frac{u}{\varepsilon(n,p,\alpha,\beta)t}
\end{equation}
for every $x\in M$ and $t>0$, given any pair $(\alpha,\beta)$ admissible for $p\in(1,\overline{p}_n)$.
Hence, we obtained the following proposition.
\begin{prop}\label{main}
Let $u:M\times[0,T)\to{{\mathbb R}}$ a classical positive solution of the equation $u_t = \Delta u + u^p$ with $p\in(1,\overline{p}_n)$, on an $n$--dimensional complete Riemannian manifold
$(M,g)$ with nonnegative Ricci tensor. Then, for every pair $(\alpha,\beta)$ admissible for $p$, there holds
\begin{equation*}
u_t-\alpha\frac{\vert\nabla u\vert^2}{u}-\beta u^p\geq -\frac{u}{\varepsilon(n,p,\alpha,\beta)t}
\end{equation*}
for every $x\in M$ and $t>0$.
\end{prop}
\begin{rem}\label{remp1}
By carefully inspecting the proof, we can see that it works also if $p\in(0,1]$, moreover, we can choose $\alpha=\beta=1$ (check the last term of inequality~\eqref{eq960}), leading to the conclusion
\begin{equation*}
u_t-\frac{\vert\nabla u\vert^2}{u}-u^p\geq -\frac{2u}{nt}
\end{equation*}
for every $x\in M$ and $t>0$.
\end{rem}
\begin{rem}
Easily, considering a suitable multiple of the solution, a conclusion analogous to Proposition~\ref{main} also holds for the positive solutions of the equation $u_t = \Delta u + au^p$ with $a\in{{\mathbb R}}$ positive constant.
\end{rem}
\section{Harnack inequality}
We now derive the following Harnack inequality as a corollary of the semilinear Li~\&~Yau estimates.
\begin{prop}\label{harn}
Let $u:M\times[0,T)\to{{\mathbb R}}$ a classical positive solution of the equation $u_t = \Delta u + u^p$ with $p\in(1,\overline{p}_n)$, on an $n$--dimensional complete Riemannian manifold
$(M,g)$ with nonnegative Ricci tensor. Then, for every pair $(\alpha,\beta)$ admissible for $p$ and $\varepsilon =\varepsilon(n,p,\alpha,\beta)$ as in formula~\eqref{epseq}, given any $0<t_1<t_2\leq T$ and $x_1,x_2 \in M$, the following inequality holds
\begin{equation*}
u(x_1,t_1) \leq u(x_2,t_2) \left(\frac{t_2}{t_1}\right)^{\!\!1/\varepsilon}\!\!\!\exp(\rho_{\alpha,\beta}(x_1,x_2,t_1, t_2)),
\end{equation*}
where
\begin{equation*}
\rho_{\alpha,\beta} (x_1,x_2,t_1, t_2) = \inf_{\gamma \in \Gamma(x_1,x_2)} \int_{0}^{1} \left[\frac{|\dot{\gamma}(s)|^2}{4 \alpha (t_2-t_1)} - \beta (t_2-t_1) u^{p-1}(\gamma(s), (1-s)t_2+st_1) \right]\, ds,
\end{equation*}
with $\Gamma(x_1,x_2)$ given by all the paths in $M$ parametrized by $[0,1]$ joining $x_2$ to $x_1$. In particular, since $u \geq 0$, we have the following bound
\begin{equation*}
u(x_1,t_1) \leq u(x_2,t_2) \left(\frac{t_2}{t_1}\right)^{\!\!1/\varepsilon}\!\!\!\exp \left(\inf_{\gamma \in \Gamma(x_1,x_2)} \int_{0}^{1} \frac{|\dot{\gamma}(s)|^2}{4 \alpha (t_2-t_1)} \, ds \right).
\end{equation*}
\end{prop}
\begin{proof}
We start with the inequality for $\log u$ that we derived in Section~~\ref{liyausem},
\begin{equation}\label{harn1}
(\log u)_t-\alpha \vert\nabla \log u \vert^2-\beta u^{p-1} \geq -\frac{1}{\varepsilon t}.
\end{equation}
Then, let $\gamma:[0.1]\to M$ be any curve such that $\gamma(0) = x_2$ and $\gamma(1) = x_1$ and consider the path $\eta:[0,1]\to M\times[t_1,t_2]$ defined as $\eta(s)= (\gamma(s), (1-s)t_2+st_1)$. Notice that $\eta(0)= (x_2, t_2)$ and $\eta(1)= (x_1, t_1)$. Integrating along $\eta$, thanks to estimate~\eqref{harn1}, we have
\begin{align*}
\log& \left(\frac{u(x_1,t_1)}{u(x_2,t_2)}\right) = \log u(x_1,t_1) - \log u(x_2,t_2)\\
&\, = \int_{0}^{1} \left( \frac{d\,}{ds} \log u (\eta(s)) \right) \, ds = \int_{0}^{1} \left[\langle \nabla \log u, \dot{\gamma} \rangle - (t_2-t_1) (\log u)_s \right] \, ds\\
&\, \leq \int_{0}^{1} \left[ \vert \nabla \log u \vert \vert \dot{\gamma}\vert + (t_2-t_1) \left(\frac{1}{\varepsilon [(1-s)t_2+st_1]} -\alpha \vert\nabla \log u \vert^2-\beta u^{p-1}\right) \right] \, ds\\
&\, = \int_{0}^{1} \frac{t_2-t_1}{\varepsilon [(1-s)t_2+st_1]} \, ds + \int_{0}^{1} \left[ \vert \nabla \log u \vert \vert \dot{\gamma}\vert - \alpha (t_2-t_1) \vert\nabla \log u \vert^2-\beta (t_2-t_1) u^{p-1}\right] \, ds\\
&\, \leq -\frac{\log[(1-s)t_2+st_1]}{\varepsilon}\,\biggr\vert_{0}^{1} + \int_{0}^{1} \left[\frac{\vert \dot{\gamma}\vert^2}{4\alpha (t_2-t_1)} -\beta (t_2-t_1) u^{p-1} \right] \, ds\\
&\, = \frac{1}{\varepsilon} \log \left( \frac{t_2}{t_1}\right) + \int_{0}^{1} \left[\frac{\vert \dot{\gamma}\vert^2}{4\alpha (t_2-t_1)} -\beta (t_2-t_1) u^{p-1} \right]\,ds
\end{align*}
Taking the exponential of this estimate, we obtain the desired inequality.
\end{proof}
\section{Ancient and eternal solutions}\label{anci}
Even if in the analysis of singularities of positive solutions of equation $u_t = \Delta u + u^p$, which we briefly discuss in Section~\ref{blowupsec}, only the eternal/ancient solutions in ${{\mathbb R}}^n$ are relevant, it is clearly of interest discussing (and possibly determining all) such solutions also on a general Riemannian manifold.
Let $u:M\times(-\infty,T)\to{{\mathbb R}}$ be a classical, positive, {\em ancient} solution of $u_t = \Delta u + u^p$, with $p\in(0,\overline{p}_n)$, where $(M,g)$ is a complete $n$--dimensional Riemannian manifold with nonnegative Ricci tensor.\\
By repeating the argument of the previous sections (or ``translating'' the solution) with $t-T_0$ in place of $t$, for any $T_0<T$, in the time interval $[T_0,T)$, we conclude that at every $x\in M$ and $t>T_0$ we have
(estimate~\eqref{eq990})
$$
u_t\geq \alpha \frac{\vert\nabla u\vert^2}{u}+\beta u^p-\frac{u}{\varepsilon(t-T_0)}
$$
for every pair $(\alpha,\beta)$ admissible for $p$, where $\varepsilon>0$ is independent of $x$ and $t$.\\
Hence, sending $T_0\to-\infty$, we conclude
$$
u_t\geq \alpha \frac{\vert\nabla u\vert^2}{u}+\beta u^p>0
$$
for every $x\in M$ and $t\in(-\infty,T)$, if $\beta>0$ (which can always be chosen, by Remark~\ref{betanonzero}). Hence, we have the following consequence of Proposition~\ref{main}.
\begin{prop}\label{prop-mon}
Let $u:M\times(-\infty,T)\to{{\mathbb R}}$ a classical, positive, ancient solution of the equation $u_t = \Delta u + u^p$ with $p\in(0,\overline{p}_n)$, on an $n$--dimensional complete Riemannian manifold
$(M,g)$ with nonnegative Ricci tensor. Then, there exist admissible pairs $\alpha,\beta\in(0,1)$ for $p$ and for any of such pairs there holds
\begin{equation}\label{mono}
u_t\geq\alpha\frac{\vert\nabla u\vert^2}{u}+\beta u^p\geq\beta u^p>0,
\end{equation}
for every $x\in M$ and $t\in(-\infty,T)$. In particular, for every fixed $x\in M$ the function $t\mapsto u(x,t)$ is monotone increasing.
\end{prop}
The following conclusion for eternal solutions is then straightforward, since any (non--identically zero) solution must blow up in finite time.
\begin{cor} In the same hypotheses of this proposition, $u$ is uniformly bounded above (in space and time) in every $M\times(-\infty,T')$, for any $T'<T$. Moreover, the function $t\mapsto\sup_{x\in M}u(x,t)$ is monotone increasing and
$$
\lim_{t\to-\infty}\sup_{x\in M}u(x,t)=0.
$$
If the function $u$ is an eternal nonnegative solution of $u_t = \Delta u + u^p$, with $p\in(0,\overline{p}_n)$, where $(M,g)$ is a complete $n$--dimensional Riemannian manifold with nonnegative Ricci tensor, it must necessarily be identically zero.
\end{cor}
For $p>1$, by direct integration of inequality~\eqref{mono} we get the estimate
$$
u(x,t)\leq \frac{C}{(\overline{T}-t)^{\frac{1}{p-1}}}
$$
for some $\overline{T}\geq T$ and $C=(\beta(p-1))^{-\frac{1}{p-1}}$. This "universal" decay estimate for ancient solutions on manifolds with nonnegative Ricci tensor was previously known, thanks to Corollary 3.7 in~\cite{cama2}, in the case of Riemannian manifolds with bounded geometry, but with a different proof based on a blow--up argument (see Section~\ref{blowupsec}).
By the same blow--up argument (see~\cite[Theorem~3.6 and Corollaries~3.7, 3.8]{cama2}) it can be shown that if in ${{\mathbb R}}^n$, for a fixed $p>1$, there are no nonzero, bounded, positive, eternal solutions of $u_t = \Delta u + u^p$, then the same holds for every complete $n$--dimensional Riemannian manifold with bounded geometry (notice that nonnegativity of the Ricci tensor is not necessary).\\
In the Euclidean space, it is well known that nontrivial global radial (static) solutions on $\mathbb{R}^n \times \mathbb{R}$ exist for any supercritical exponent $p \geq p_S=\frac{n+2}{n-2}$, see for instance Proposition~B at page~1155 in~\cite{guiniwang}. Conversely, while the triviality of eternal {\em radial} solutions can be shown in the full range of subcritical exponents $1<p<\frac{n+2}{n-2}$ (see Theorem~B at page~882 in~\cite{pqs2}), the same expected result for general (not necessarily radial) solutions was known only in the range $1<p< \frac{n(n+2)}{(n-1)^2}$ (see Theorem A page 882 in~\cite{pqs2}). Indeed, such triviality when $\frac{n(n+2)}{(n-1)^2} \leq p< p_S$ was a quite long standing open problem (see~\cite{pqs1,pqs2}), which has been finally settled by Quittner for every $p>1$ when $n\leq 2$ in~\cite{quittner} and, very recently, for any $p< p_S$ when $n\geq 3$ in~\cite{quittner2}.\\
Then, by such optimal result of Quittner, in the general case of a Riemannian manifold, we can extend the triviality result for eternal solutions to Riemannian manifolds with bounded geometry (see Corollary~3.8 in~\cite{cama2}).
Unfortunately, our bound $\overline{p}_n$ on the exponent $p$ is actually smaller than $\frac{n(n+2)}{(n-1)^2}$, for every $n\geq 2$, but larger than $\frac{n}{n-2}$ (which appears in~\cite{quittner}) when $n\geq4$ (see the computation in the Appendix).\\
Turning our attention to the (only) ancient solutions of $u_t = \Delta u + u^p$ on a complete, $n$--dimensional Riemannian manifold with nonnegative Ricci tensor, we first notice that if $p\leq 1$, we have seen in Remark~\ref{remp1} that we can choose $\alpha=\beta=1$, hence
\begin{equation*}
u_t=\Delta u+u^p\geq\frac{\vert\nabla u\vert^2}{u}+u^p
\end{equation*}
that is,
\begin{equation*}
\Delta \log u\geq0.
\end{equation*}
If $M$ is compact, this clearly implies that every ancient, positive solution is trivial.\\
Not much is known, in a general Riemannian manifold, when $p>1$. By the work of Merle and Zaag~\cite{merlezaag} (and the above results of Quittner), it follows that in ${{\mathbb R}}^n$, if $1<p< p_S=\frac{n+2}{n-2}$ when $n\geq 3$, or for every $p>1$ if $n\leq 2$, any ancient, positive solution is trivial. We conjecture that this holds in general.
\begin{conge}
For any complete, $n$--dimensional Riemannian manifold $(M,g)$ with bounded geometry, if $p< p_S=\frac{n+2}{n-2}$ when $n\geq 3$, or for every $p>0$ if $n\leq 2$, any ancient, positive solution of $u_t = \Delta u + u^p$ is trivial.
\end{conge}
The line of~\cite{merlezaag} is (at least apparently) difficult to be generalized to manifolds, possibly requiring in the computations/estimates the (quite restrictive) assumptions of parallel Ricci tensor and nonnegative sectional curvature. Anyway, for a compact Riemannian manifold with ${\mathrm {Ric}}>0$, we were able to obtain the triviality of ancient solutions in the same range of exponents $0\leq p<p_S$ (by the discussion above and Theorem~1.2 in~\cite{cama2}, in combination with the results of Quittner). In the next section we are trying to find out some other properties of ancient solutions.
\section{Convexity in time and monotonicity of the gradient}\label{monot}
The aim of this section is to discuss further qualitative properties of ancient, positive solutions of $u_t = \Delta u + u^p$, that can be derived analogously as the previous Li~\&~Yau type inequalities.
We now show that any ancient solution, apart from being monotone in time, as proved in Proposition~\ref{prop-mon}, is also convex in time, i.e. $\partial^2_t u > 0$, provided $n\geq 5$. Without loss of generality, we assume that $M$ is compact, since the noncompact case can be treated, as in the second part of Section~\ref{liyausem}, through a cut--off argument.
Let $u:M\times(-\infty,T)\to{{\mathbb R}}$ a classical, positive, ancient solution of the equation $$u_t = \Delta u + u^p$$ with $p\in(0,\overline{p}_n)$, on an $n$--dimensional, $n\geq 5$, complete Riemannian manifold
$(M,g)$ with nonnegative Ricci tensor.
From Proposition~\ref{prop-mon}, for some admissible $0 <\alpha,\beta <1$, we proved
\begin{equation}\label{conv1}
u_t \geq \alpha \frac{|\nabla u|^2}{u}+ \beta u^p.
\end{equation}
In particular, the function
$$v=\partial_t u$$ is positive and satisfies
\begin{equation*}
\partial_t v - \Delta v = h v,
\end{equation*}
where we set $h=p u^{p-1}$.
Let now $f= \log v$ and, observing that
\begin{equation*}
\partial_t f = \frac{v_t}{v} \quad
\text{ and } \quad
\Delta f = \frac{\Delta v}{v} - |\nabla f|^2,
\end{equation*}
we easily see that $f$ satisfies
\begin{equation*}
(\partial_t - \Delta) f = |\nabla f|^2 + h.
\end{equation*}
We introduce, for every $T_0<t<T$, the function
\begin{equation}\label{conv2}
F = (t-T_0)\left( \Delta f + (1-\alpha) |\nabla f|^2 + (1-\beta) h\right).
\end{equation}
A simple computation shows the following
\begin{align*}
(\partial_t - \Delta) \Delta f &= 2|\nabla^2 f|^2 + 2 Ric(\nabla f,\nabla f)+2 \langle \nabla \Delta f, \nabla f \rangle + \Delta h\\
(\partial_t - \Delta) f_i &= 2f_{ik} f_k + h_i\\
(\partial_t - \Delta) |\nabla f|^2 &= 4 f_{ik} f_i f_k + 2 h_i f_i - 2|\nabla^2 f|^2.
\end{align*}
From the above equations, using ${\mathrm {Ric}}\geq 0$, we deduce
\begin{align*}
(\partial_t - \Delta) F \geq \frac{F}{(t-T_0)} + (t-T_0) &\bigl[ 2|\nabla^2 f|^2 + 2 \langle \nabla \Delta f, \nabla f \rangle + \Delta h\\
&\,\,+ 4(1-\alpha) f_{ik} f_i f_k + 2(1-\alpha) h_i f_i\\
&\,\,- 2(1-\alpha) |\nabla^2 f|^2 + (1-\beta)(\partial_t - \Delta) h \bigr].
\end{align*}
Notice that
\begin{equation*}
2 \frac{\langle \nabla F, \nabla f \rangle}{(t-T_0)} = 2 \langle \nabla \Delta f, \nabla f \rangle + 4(1-\alpha) f_{ik} f_i f_k + 2(1-\beta) h_i f_i,
\end{equation*}
which can be plugged above to get
\begin{equation}\label{monot3}
(\partial_t - \Delta) F \geq \frac{F}{(t-T_0)} + 2 \langle \nabla F, \nabla f \rangle + (t-T_0) \left[ 2 \alpha |\nabla^2 f|^2 + 2(\beta-\alpha) h_i f_i + \partial_t h -\beta(\partial_t - \Delta) h \right].
\end{equation}
Now we see that
\begin{align*}
n |\nabla^2 f|^2\geq&\, (\Delta f)^2\\
=&\, \Bigl(\frac{F}{(t-T_0)} - (1-\alpha) |\nabla f|^2 - (1-\beta) h\Bigr)^2\\
=&\, \frac{F^2}{(t-T_0)^2} + (1-\alpha)^2 |\nabla f|^4 + (1-\beta)^2 h^2\\
&\,-\frac{2(1-\alpha)}{(t-T_0)} F |\nabla f|^2 -\frac{2(1-\beta)}{(t-T_0)} F h + 2 (1-\alpha)(1-\beta) |\nabla f|^2 h.
\end{align*}
Moreover, by direct computation,
\begin{equation*}
\partial_t h = p(p-1) u^{p-2} \partial_t u,
\end{equation*}
and
\begin{equation*}
\Delta h = p(p-1) u^{p-2} \Delta u + p(p-1)(p-2) u^{p-3} |\nabla u|^2,
\end{equation*}
whence, using the equation for $u$ and inequality~\eqref{conv1}, we have
\begin{equation}
\begin{split}
\partial_t h - \beta (\partial_t -\Delta) h &= p(p-1) u^{p-2} \partial_t u - \beta p(p-1) u^{2(p-1)} + \beta p(p-1)(p-2) u^{p-3} |\nabla u|^2\\
&\geq \alpha p(p-1) u^{p-3} |\nabla u|^2 + \beta p(p-1)(p-2) u^{p-3} |\nabla u|^2\\
&= p(p-1) [\alpha + \beta(p-2)] u^{p-3} |\nabla u|^2\,.
\end{split}
\end{equation}
Hence, plugging everything in estimate~\eqref{monot3} we finally get
\begin{equation}\label{monot4}
\begin{split}
(\partial_t - \Delta) F \geq&\, \frac{F}{(t-T_0)} + 2 \langle \nabla F, \nabla f \rangle + \frac{2\alpha}{n} \frac{F^2}{(t-T_0)}\\
&\,+ \frac{2 \alpha}{n} (t-T_0) \Bigr[ (1-\alpha)^2 |\nabla f|^4 + (1-\beta)^2 h^2\\
&\,-\frac{2(1-\alpha)}{(t-T_0)} F |\nabla f|^2 -\frac{2(1-\beta)}{(t-T_0)} F h + 2 (1-\alpha)(1-\beta) |\nabla f|^2 h \Bigr]\\
&\,+ (t-T_0) \left[ 2(\beta-\alpha) h_i f_i + p(p-1) [\alpha + \beta(p-2)] u^{p-3} |\nabla u|^2 \right]\\
\geq&\, \frac{F}{(t-T_0)} + 2 \langle \nabla F, \nabla f \rangle + \frac{2\alpha}{n} \frac{F^2}{(t-T_0)}\\
&\,+ \frac{2 \alpha}{n} (t-T_0) \Bigl[ -\frac{2(1-\alpha)}{(t-T_0)} F |\nabla f|^2 -\frac{2(1-\beta)}{(t-T_0)} F h + 4 (1-\alpha)(1-\beta) |\nabla f|^2 h \Bigr]\\
&\,+ (t-T_0) \left[ 2(\beta-\alpha) h_i f_i + p(p-1) [\alpha + \beta(p-2)] u^{p-3} |\nabla u|^2 \right]\,.
\end{split}
\end{equation}
Let us first observe that $\alpha>\beta$ (see the Appendix). We claim that the quantity
\begin{equation*}
Q=-2(\alpha-\beta) h_i f_i + p(p-1) [\alpha + \beta(p-2)] u^{p-3} |\nabla u|^2 + \frac{8\alpha}{n} (1-\alpha)(1-\beta) |\nabla f|^2 h
\end{equation*}
is nonnegative. Notice that
\begin{align*}
2h_i f_i &= 2p(p-1) u^{p-2} u_i f_i = 2p(p-1) \frac{\sqrt{h}}{\sqrt{h}} u^{p-2} u_i f_i \\
&\leq p(p-1) \Bigl( \theta \frac{u^{2p-4}}{p u^{p-1}}|\nabla u|^{2}+\frac{1}{\theta} |\nabla f|^2 h \Bigr)\\
&= p(p-1) \Bigl( \frac{\theta}{p}u^{p-3}|\nabla u|^{2}+\frac{1}{\theta} |\nabla f|^2 h \Bigr)
\end{align*}
for all $\theta>0$. Thus we obtain
\begin{align*}
Q\geq &\,\left\{ p(p-1) [\alpha + \beta(p-2)]-\theta(p-1)(\alpha-\beta)\right\} u^{p-3} |\nabla u|^2 \\
&\,+\Bigr[\frac{8\alpha}{n} (1-\alpha)(1-\beta)-\frac{p(p-1)(\alpha-\beta)}{\theta} \Bigr] |\nabla f|^2 h.
\end{align*}
Choosing
$$
\theta = \frac{np(p-1)(\alpha-\beta)}{8\alpha(1-\alpha)(1-\beta)}
$$
we get
\begin{equation}\label{eqapp}
Q\geq p(p-1)\Bigl[ \alpha + \beta(p-2)-\frac{n(p-1)(\alpha-\beta)^{2}}{8\alpha(1-\alpha)(1-\beta)}\Bigr]u^{p-3} |\nabla u|^2.
\end{equation}
Using (again) Mathematica\textsuperscript{TM} (see the Appendix), we have that the coefficient in the right hand side is nonnegative, hence $Q\geq 0$, if $n\geq 5$, $\alpha$ and $\beta$ are admissible and $p<\overline{p}_{n}$. Thus, at a minimum point of $F$ with $F\leq 0$, we have
$$
(\partial_t - \Delta) F \geq \frac{F}{(t-T_0)} + \frac{2\alpha}{n} \frac{F^2}{(t-T_0)}
$$
and, by maximum principle,
$$
F\geq -\frac{n}{2\alpha}
$$
which implies
$$
f_t \geq \alpha \vert \nabla f\vert^2+\beta h -\frac{n}{2\alpha (t-T_0)}\,.
$$
Then, recalling that $f= \log u_t$, we finally obtain
\begin{equation}\label{monot6}
\frac{\partial^2_t u}{u_t}\geq \alpha \frac{\vert\nabla u_t \vert^2}{u_t^2}+\beta h -\frac{n}{2\alpha (t-T_0)}.
\end{equation}
In particular, since $\alpha,\beta > 0$, we have
\begin{equation}\label{monot7}
\frac{\partial^2_t u}{u_t}\geq -\frac{n}{2\alpha (t-T_0)}.
\end{equation}
Letting $T_0\to-\infty$, we conclude that at every $x\in M$ and $t<T$ we have $\partial^2_t u > 0$, for every pair $(\alpha,\beta)$ admissible for $p$. Therefore, we have showed the following proposition.
\begin{prop}\label{prop-conv}
Let $u:M\times(-\infty,T)\to{{\mathbb R}}$ a classical, positive, ancient solution of the equation $u_t = \Delta u + u^p$ with $p\in(0,\overline{p}_n)$, on an $n$--dimensional, $n\geq 5$, complete Riemannian manifold
$(M,g)$ with nonnegative Ricci tensor. Then, there exist admissible pairs $\alpha,\beta\in(0,1)$ for $p$ and for any of such pairs there holds
$$
\partial^2_t u\geq \alpha \frac{\vert\nabla u_t \vert^2}{u_t}+\beta p u^{p-1}u_t >0
$$
for every $x\in M$ and $t\in(-\infty,T)$. In particular, for every fixed $x\in M$ the function $t\mapsto u(x,t)$ is convex.
\end{prop}
\begin{rem}
If the ambient is ${{\mathbb R}}^n$ and $u:\mathbb{R}^n\times(-\infty,T)\to{{\mathbb R}}$ a classical, positive, ancient solution of the equation
$$
u_t = \Delta u + u^p
$$
on $\mathbb{R}^n$, with $p\in(0,\overline{p}_n)$, letting $u_m = \partial_{x_m} u$, $m=1,..,n$, be any spatial derivative of $u$, we observe that
\begin{equation*}
(\partial_t - \Delta) u_m = h u_m,
\end{equation*}
where $h= p u^{p-1}$, so that
\begin{equation*}
(\partial_t - \Delta) u_m^2 = 2 h u_m^2 - 2 |\nabla u_m|^2.
\end{equation*}
Thus, we consider, for $\varepsilon>0$, the function $f_\varepsilon = \frac12 \log(u_{m}^{2} + \varepsilon)$. By direct computation we see that $f_\varepsilon$ satisfies
\begin{equation*}
(\partial_t - \Delta) f_\varepsilon = (1+\psi_\varepsilon) h + |\nabla f_\varepsilon|^2 (1+\phi_\varepsilon),
\end{equation*}
whenever $u_m \ne 0$, with $\psi_\varepsilon = -\frac{\varepsilon}{u_m^2 + \varepsilon}$ and $\phi_\varepsilon = -\frac{\varepsilon}{u_m^2}$. We introduce
\begin{equation}\label{fave1}
F_\varepsilon = t\left( \Delta f_\varepsilon + (1-\alpha) |\nabla f_\varepsilon|^2 + (1-\beta) h\right).
\end{equation}
In order to prove that $F_\varepsilon \geq -C$ for some $C>0$, we notice that if $u_m = 0$, then $f_\varepsilon$ achieves a (global) minimum point, so that $\Delta f_\varepsilon \geq 0$, which in turn implies $F_\varepsilon \geq 0$ by this equation. Hence, without loss of generality, we can suppose that $u_m^2 >0$, which implies that the perturbations $\psi_\varepsilon$ and $\phi_\varepsilon$, as well as all of their derivatives, tend to zero uniformly as $\varepsilon \to 0$. In this way, if for $\varepsilon = 0$ we let $f=f_0$ and $F=F_0$, we have that
\begin{equation*}
(\partial_t - \Delta) f = h + |\nabla f|^2 + R(\varepsilon),
\end{equation*}
with $R(\varepsilon)\to 0$ as $\varepsilon\to 0$. Observe now that $R(\varepsilon)$ is smooth whenever $u_m \neq 0$ and that all of its derivatives tend to zero as $\varepsilon\to 0$. Thus, we can omit the remainder $R(\varepsilon)$ in the computations that we can perform as for the monotonicity of the time derivative $u_t$ proved above, only with the spatial derivative $\partial_{x_m} u$, $m=1,..,n$ in place of $u_t$. Following step by step the preceding computations, we then obtain the following fact.\\
\noindent{\em Let $u:\mathbb{R}^n\times(-\infty,T)\to{{\mathbb R}}$ a classical, positive, ancient solution of the equation $u_t = \Delta u + u^p$ with $p\in(0,\overline{p}_n)$, on $\mathbb{R}^n$, $n\geq 5$. Then, for every fixed $x\in \mathbb{R}^n$, the function $t\to|\nabla u|^2 (x,t)$ is monotone increasing.}\\
\noindent This could possibly be an initial step for an alternative line to show the Merle and Zaag result in~\cite{merlezaag} of triviality of ancient solutions in ${{\mathbb R}}^n$.
\end{rem}
\section{Singularities, blow--up and eternal/ancient solutions in ${{\mathbb R}}^n$}\label{blowupsec}
Let $u:M\times [0,T)\to{{\mathbb R}}$ a positive smooth solution of $u_t = \Delta u + u^p$ on an $n$--dimensional Riemannian manifold $(M,g)$, with $p>1$, such that $T<+\infty$ is the maximal time of existence. In this section we want to discuss, as in~\cite{pqs2}, the asymptotic behavior of a solution approaching the singular time $T$, by means of a slightly different blow--up technique, borrowed by the work of Hamilton about the Ricci flow~\cite{hamilton9} (and the mean curvature flow too).
Assuming that $M$ is compact and $u$ is uniformly bounded by some constant $A>0$, we have the evolution equations,
\begin{align*}
\frac{d\,\,}{dt} u^2=&\,2u u_t=2 u\Delta u+2pu^{p+1}=\Delta u^2-2\vert \nabla u\vert^2+2pu^{p+1}
\leq\Delta u^2+2pA^{p+1}\\
\frac{d\,\,}{dt}u_t^2=&\,2u_t\Delta u_t +2pu^{p-1}u_t^2
=\Delta u_t^2-2\vert \nabla u_t\vert^2+2pu^{p-1}u_t^2\leq\Delta u_t^2+2pA^{p-1}u_t^2\\
\frac{d\,\,}{dt}\vert\nabla u\vert^2
=&\,2\nabla u\nabla\Delta u+2p\vert\nabla u\vert^2u^{p-1}\\
=&\,2\nabla u\Delta \nabla u-2{\mathrm {Ric}}(\nabla u, \nabla u)+2p\vert\nabla u\vert^2u^{p-1}\\
=&\,\Delta\vert\nabla u\vert^2-2\vert D^2u\vert^2-2{\mathrm {Ric}}(\nabla u, \nabla u)+2p\vert\nabla u\vert^2u^{p-1}\\
\leq&\,\Delta\vert\nabla u\vert^2+C\vert\nabla u\vert^2\\
\end{align*}
which, by means of maximum principle, imply that $u_t$ is also uniformly bounded (and $\vert\nabla u\vert^2$ too), then, for every $x\in M$, the limit
$$
u_T(x)=\lim_{t\to T}u(x,t)
$$
exists and it is Lipschitz. Repeating the same (standard) argument for $\vert\nabla^k u\vert^2$, for every $k\in\mathbb N$, one can conclude that the limit map $u_T$ is smooth (and the convergence $u(\cdot,t)\to u_T$ also), hence we can ``restart'' the solution by standard methods (see~\cite{mantmart1}, for instance), getting a smooth solution in a larger time interval, in contradiction with the assumption that $T<+\infty$ was the maximal time of existence.
From this discussion, we conclude that
$$
\limsup_{t\to T}u_{\max}(t)=\limsup_{t\to T}\max_{x\in M}u(x,t)=+\infty,
$$
moreover, such function $u_{\max}:[0,T)\to{{\mathbb R}}$, which is locally Lipschitz, by maximum principle must satisfy distributionally (see~\cite[Section~2.1]{Manlib}, for instance)
$$
u'_{\max}(t)\leq u_{\max}^p(t)
$$
implying (after integration of this differential inequality) the estimate
\begin{equation}\label{maxestim0}
u_{\max}(t)\geq\frac{1}{\bigl[(p-1)(T-t)\bigr]^{\frac{1}{p-1}}}=\overline{u}(t)
\end{equation}
which is the unique nonzero solution of the ODE $u'=u^p$, hence the unique nonzero trivial ancient solution of $u_t=\Delta u+u^p$, defined in the maximal interval $(-\infty,T)$.\\
Notice, that by the same argument, applied to $u_{\min}(t)=\min_{x\in M}u(x,t)$, satisfying $u'_{\min}(t)\geq u_{\min}^p(t)$, excludes the possibility that $T=+\infty$, if $u$ is not identically zero, that is, there are no positive immortal solutions, if $M$ is compact.
Then, we know that the following superior limit, in the case $M$ is compact,
\begin{equation}
\limsup_{t\to T}\max_{x\in M} u(x,t)^{p-1}({T-t})
\end{equation}
must be at least $1/(p-1)$. Up to our knowledge, we do not know if the same estimate holds for {\em complete--only} manifold $M$ (possibly with bounded geometry).\\
Assume that
\begin{equation}\label{hyp}
\limsup_{t\to T}\max_{x\in M} u(x,t)^{p-1}({T-t})=+\infty\,.
\end{equation}
Then, let us choose a sequence of times $t_k\in[0,T-1/k]$ and points $x_k\in M$
such that
\begin{equation}\label{rateIIbis}
u(x_k,t_k)^{p-1}(T-1/k-t_k)=\max_{\genfrac{}{}{0pt}{}
{t\in[0,T-1/k]}{x\in M}}u(x,t)^{p-1}(T-1/k-t)\,.
\end{equation}
This maximum goes to $+\infty$ as $k\to\infty$, indeed, if it is
bounded by some constant $C$ on a subsequence $k_i\to\infty$, then for every $t\in[0,T)$ we have that definitely $t\in[0,T-1/k_i]$ and
\begin{equation*}
u(x,t)^{p-1}(T-t)=\lim_{i\to\infty}u(x,t)^{p-1}(T-1/k_i-t)\leq C
\end{equation*}
for every $x\in M$. This is in contradiction with the hypothesis~\eqref{hyp}.\\
This fact also forces the sequence $t_k$ to converge to $T$ as
$k\to\infty$. Indeed, if $t_{k_i}$ is a subsequence not converging to $T$, we
would have that the sequence $u(x_{k_i},t_{k_i})^{p-1}$ is
bounded, hence also
$$
\max_{\genfrac{}{}{0pt}{}
{t\in[0,T-1/{k_i}]}{x\in M}}u(x,t)^{p-1}(T-1/{k_i}-t)
$$
would be bounded.\\
Thus, we can choose an increasing (not relabeled) subsequence $t_k$
converging to $T$, such that $u(x_k,t_k)$ goes monotonically to
$+\infty$ and
$$
u(x_k,t_k)^{p-1}t_k\to+\infty\,,\qquad u(x_k,t_k)^{p-1}(T-1/k-t_k)\to+\infty\,,
$$
Moreover, we can also assume that $x_k\to \overline{x}$ for some $\overline{x}\in M$.\\
We rescale now {\em the Riemannian manifold $(M,g)$ and the solution $u$} as follows: for every $k\in\mathbb N$, we consider on $M$ the rescaled metric $g_k=u^p(x_k,t_k)g$ and the function
$u_k: M\times I_k\to{{\mathbb R}}^{n+1}$, where
$$
I_k=\bigl[-u(x_k,t_k)^{p-1}t_k,u(x_k,t_k)^{p-1}(T-1/k-t_k)\bigr]\,,
$$
and $u_k$ is the function given by
$$
u_k(x,s)=\frac{u\bigl(x,s/u(x_k,t_k)^{p-1}+t_k\bigr)}{u(x_k,t_k)}\,.
$$
Then, there holds
\begin{align*}
\frac{\partial\,}{\partial s} u_k(x,s)=&\,\frac{u_t\bigl(x,s/u(x_k,t_k)^{p-1}+t_k\bigr)}{u^p(x_k,t_k)}\\
=&\,\frac{\Delta u\bigl(x,s/u(x_k,t_k)^{p-1}+t_k\bigr)}{u^p(x_k,t_k)}+
\frac{u^p\bigl(x,s/u(x_k,t_k)^{p-1}+t_k\bigr)}{u^p(x_k,t_k)}\\
=&\,\Delta_k u_k(x,s)+u_k^p(x,s)
\end{align*}
where $\Delta_k$ is the Laplacian associated to the Riemannian manifold $(M,g_k)$.\\
Hence, every $u_k$ is still a solution of our equation on $(M,g_k)$ and on a different time interval $I_k$. Moreover the following
properties hold,
\begin{itemize}
\item $u_k(x_k,0)=1$,
\item for every $\varepsilon>0$ and $\omega>0$ there exists $\overline{k}\in\mathbb N$ such that
\begin{equation}\label{ccc}
\max_{x\in M} u_k(x,s)\leq 1+\varepsilon
\end{equation}
for every $k\geq \overline{k}$ and $s\in[-u(x_k,t_k)^2t_k,\omega]$,
\end{itemize}
indeed (the first point is immediate), by the choice of the minimizing pairs
$(x_k,t_k)$ we get
\begin{align*}
u_k(x,s)&\,=\frac{u(x,s/u(x_k,t_k)^{p-1}+t_k)}{u(x_k,t_k)}\\
&\,\leq \frac{u(x_k,t_k)}{u(x_k,t_k)}\,\frac{T-1/k-t_k}{T-1/k-t_k-s/u(x_k,t_k)^{p-1}}\\
&\,= \frac{T-1/k-t_k}{(T-1/k-t_k)-s}\,,
\end{align*}
if
$$
\frac{s}{u(x_k,t_t)^{p-1}}+t_k\in[0,T-1/k],
$$
that is, if $s\in I_k$. Then, assuming $s\leq\omega$ and $k$ large enough,
the claim follows as we know that $u(x_k,t_k)^{p-1}(T-1/k-t_k)\to+\infty$.
If we now take a (subsequential) limit of these {\em pairs} manifold--solution, clearly the manifolds $(M,g_k)$ under the {\em pointed convergence} in $x_k$ (see~\cite{petersen2}, for instance) converge to ${{\mathbb R}}^n$ with its flat metric and the solutions $u_k$ converge smoothly in every compact time interval of ${{\mathbb R}}$ (by standard local uniform parabolic estimates -- similar to the ones at the beginning of the section, for instance) to a smooth eternal solution $u_\infty:{{\mathbb R}}^n\times{{\mathbb R}}\to{{\mathbb R}}$, bounded with all its derivatives of the same semilinear equation $u_t=\Delta u+u^p$, notice indeed that the time interval of existence is
the whole ${{\mathbb R}}$, as $\lim_{k\to\infty} I_k=(-\infty,+\infty)$. Moreover, the function $u_\infty$ takes its absolute maximum, which is 1, at time $s=0$ at the origin of ${{\mathbb R}}^n$, hence the limit flow is nonzero.
Since we know that for $p<p_S=\frac{n+2}{n-2}$, by the works of Quittner~\cite{quittner,quittner2}, such eternal solutions in ${{\mathbb R}}^n$ do not exist, we can conclude that, for such range of exponents,
\begin{equation}\label{hyp2}
\limsup_{t\to T}\max_{x\in M}u(x,t)^{p-1}({T-t})=C<+\infty\,.
\end{equation}
Hence, there exists a constant $\overline{C}$ such that
$$
u(x,t)\leq\frac{\overline{C}}{(T-t)^{\frac{1}{p-1}}}
$$
for every $x\in M$ and $t\in[0,T)$.\\
In such situations, let us choose a sequence of times $t_k\nearrow$ and points $x_k\in M$ such that
\begin{equation}
u(x_k,t_k)=\max_{x\in M}u(x,t_k)\qquad\qquad\text{ and }\qquad\qquad u(x_k,t_k)^{p-1}({T-t_k})\to C
\end{equation}
(we can also assume that $x_k\to \overline{x}$, for some $\overline{x}\in M$). Repeating the above blow--up procedure, that is,
considering on $M$ the rescaled metric $g_k=u^p(x_k,t_k)g$ and the function
$$
u_k(x,s)=\frac{u\bigl(x,s/u(x_k,t_k)^{p-1}+t_k\bigr)}{u(x_k,t_k)},
$$
where
$$
I_k=\bigl[-u(x_k,t_k)^{p-1}t_k,u(x_k,t_k)^{p-1}(T-t_k)\bigr)\,,
$$
we get this time as a limit an ancient smooth solution $u_\infty:{{\mathbb R}}^n\times{{\mathbb R}}\to{{\mathbb R}}$ of $u_t=\Delta u+u^p$, defined in the time interval $(-\infty,C)=\lim_{k\to\infty} I_k$. As $u_k(x_k,0)=\max_{x\in M}u_k(x,0)=1$, we have $u_\infty(0,0)=\max_{x\in{{\mathbb R}}^n}u_\infty(x,0)=1$ and moreover, since by estimate~\eqref{maxestim0} we have
$$
\max_{x\in M}u_k(x,s)^{p-1}\geq\frac{1}{p-1}\,\frac{1}{u(x_k,t_k)^{p-1}(T-t_k) - s}
$$
passing to the limit, we conclude
$$
\max_{x\in{{\mathbb R}}^n}u_\infty(x,s)^{p-1}\geq\frac{1}{p-1}\frac{1}{C-s},
$$
since
\begin{equation}
u(x_k,t_k)^{p-1}({T-t_k})\to C\,.
\end{equation}
This clearly shows that the interval $(-\infty,C)$ is maximal for $u_\infty$ and $u_\infty$ is nonzero.\\
If now we consider any $\varepsilon>0$, there exists $\overline t\in[0,T)$ such that
\begin{equation}
u(x,t)^{p-1}({T-t})\leq C+\varepsilon
\end{equation}
for every $t\in(\overline{t},T)$ and $x\in M$, hence
\begin{align*}
u_k(x,s)^{p-1}&\,=\frac{u(x,s/u(x_k,t_k)^{p-1}+t_k)^{p-1}}{u(x_k,t_k)^{p-1}}\\
&\,\leq \frac{C+\varepsilon}{u(x_k,t_k)^{p-1}(T-s/u(x_k,t_k)^{p-1}-t_k)}\\
&\,= \frac{C+\varepsilon}{u(x_k,t_k)^{p-1}(T-t_k)-s}\,,
\end{align*}
if $s/u(x_k,t_t)^{p-1}+t_k\in(\overline{t},T)$, that is, if
$$
s\in\bigl( (\overline{t}-t_k)u(x_k,t_t)^{p-1},(T-t_k)u(x_k,t_t)^{p-1}\bigr)\,.
$$
Passing to the limit, we conclude that for every $s\in(-\infty,C)$ and $x\in{{\mathbb R}}^n$, there holds
$$
u_\infty(x,s)^{p-1}\leq \frac{C+\varepsilon}{C-s},
$$
hence, by the arbitrariness of $\varepsilon$
$$
u_\infty(x,s)^{p-1}\leq \frac{C}{C-s},
$$
for every $x\in{{\mathbb R}}^n$ and $s\in(-\infty,C)$. Notice that equality holds at $x=0$ and $s=0$.\\
Then, we have
$$
\frac{1}{p-1}\frac{1}{C-s}\leq\max_{x\in M}u_\infty(x,s)^{p-1}\leq\frac{C}{C-s}
$$
where the second inequality is an equality at $s=0$.\\
By means of the result of Merle and Zaag~\cite{merlezaag}, this solution $u_\infty$ is actually trivial (that is, constant in space), hence
$$
u_\infty(x,s)=\frac{1}{\bigl[(p-1)(C-s)\bigr]^{\frac{1}{p-1}}}\,.
$$
Being $u_\infty(0,0)=1$, it follows that $C=\frac{1}{p-1}$, hence
$$
u_\infty(x,s)=\frac{1}{\bigl[1-(p-1)s\bigr]^{\frac{1}{p-1}}},
$$
defined on $(-\infty,1/(p-1))$, moreover
\begin{equation*}
\limsup_{t\to T}\max_{x\in M}u(x,t)^{p-1}({T-t})=\frac{1}{p-1}\,.
\end{equation*}
This clearly gives a ``universal'' asymptotic profile at a singularity of any solution.
\newpage
| {
"timestamp": "2022-01-10T02:17:42",
"yymm": "2201",
"arxiv_id": "2201.02530",
"language": "en",
"url": "https://arxiv.org/abs/2201.02530",
"abstract": "We derive an adaptation of Li & Yau estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative Ricci tensor. We then apply these estimates to obtain a Harnack inequality and to discuss monotonicity, convexity, decay estimates and triviality of ancient and eternal solutions.",
"subjects": "Analysis of PDEs (math.AP); Differential Geometry (math.DG)",
"title": "Semilinear Li & Yau inequalities",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808759252645,
"lm_q2_score": 0.7217432003123989,
"lm_q1q2_score": 0.707511056595342
} |
https://arxiv.org/abs/2210.12423 | Large deviations for the volume of $k$-nearest neighbor balls | This paper develops the large deviations theory for the point process associated with the Euclidean volume of $k$-nearest neighbor balls centered around the points of a homogeneous Poisson or a binomial point processes in the unit cube. Two different types of large deviation behaviors of such point processes are investigated. Our first result is the Donsker-Varadhan large deviation principle, under the assumption that the centering terms for the volume of $k$-nearest neighbor balls grow to infinity more slowly than those needed for Poisson convergence. Additionally, we also study large deviations based on the notion of $\mathcal M_0$-topology, which takes place when the centering terms tend to infinity sufficiently fast, compared to those for Poisson convergence. As applications of our main theorems, we discuss large deviations for the number of Poisson or binomial points of degree at most $k$ in a geometric graph in the dense regime. | \section{Introduction}
The main theme of this paper is to develop the large deviations theory for the point process associated with the Euclidean volume of $k$-nearest neighbor balls.
We consider the unit cube $[0,1]^d$ equipped with the toroidal metric
$$
\mathsf{dist} (x,y) = \min_{z\in \protect{\mathbb Z}^d} \| x-y+z \|,
$$
where $\| \cdot \|$ denotes the Euclidean metric in $\reals^d$.
Then, the unit cube $[0,1]^d$ is considered as a flat torus with a periodic boundary.
Let $B_r(x) = \big\{ y\in [0,1]^d: \mathsf{dist} (x,y)\le r \big\}$, $r>0$, be the closed ball in $[0,1]^d$ of radius $r$ centered at $x\in[0,1]^d$.
Given a point measure $\omega$ in $[0,1]^d$ and a point $x\in [0,1]^d$, and a fixed integer $k\ge1$, define
\begin{equation} \label{e:def.k.NN.dist}
R_k(x,\omega) := \inf \big\{ r>0: \omega\big( B_r(x)\setminus \{x\} \big) \ge k \big\}
\end{equation}
to be the \emph{$k$-nearest neighbor distance} of $x$; it gives a radius $r$ for which $B_r(x)\setminus \{x\}$ contains exactly $k$ points of $\omega$ with one of those points lying on the boundary of $B_r(x)$.
Given a homogeneous Poisson point process $\mathcal P_n$ on $[0,1]^d$ with intensity $n$, we are interested in the stochastic behavior of the point process
\begin{equation} \label{e:def.L.kn}
L_{k,n} := \begin{cases}
\sum_{X\in \mathcal P_n}\delta_{(X, n\theta_d R_k(X,\mathcal P_n)^d -a_n)} & \text{ if } |\mathcal P_n|>k,\\
\emptyset & \text{ if } |\mathcal P_n| \le k,
\end{cases}
\end{equation}
where $\delta_{(x,y)}$ is the Dirac measure at $(x,y)\in [0,1]^d\times \reals$, and $\theta_d$ is volume of the unit ball in $\reals^d$, so that $\theta_d R_k(X,\mathcal P_n)^d$ represents the volume of a $k$-nearest neighbor ball centered at $X\in\mathcal P_n$.
Further, $\emptyset$ represents the null measure, i.e., the measure assigning zeros to all Borel measurable sets.
The process \eqref{e:def.L.kn} dictates the magnitude of the normalized volume of $k$-nearest neighbor balls, as well as the location of their centers.
There have been a number of studies on the asymptotics of the process \eqref{e:def.L.kn} or its similar variant, when the centering term $a_n$ is given as
\begin{equation} \label{e:Poisson.centering}
a_n = \log n + (k-1) \log \log n + \text{constant}.
\end{equation}
In this case, one may observe, asymptotically, at most \emph{finitely many} $k$-nearest neighbor balls whose volume are approximately $a_n/n$ up to the scale. As a consequence, the process \eqref{e:def.L.kn} will have a Poissonian structure in the limit \cite{penrose:1997, gyorfi:henze:walk:2019, chenavier:henze:otto:2022, otto:2020, bobrowski:schulte:yogeshwaran:2021}. In particular, the rate of Poisson convergence has recently been derived in terms of the Kantorovich-Rubinstein distance \cite{bobrowski:schulte:yogeshwaran:2021} and the total variation distance \cite{otto:2020}.
In addition to these results on Poisson convergence, there have also been many attempts at deriving other limit theorems for the functional of a $k$-nearest neighbor distance in \eqref{e:def.k.NN.dist}, among them, central limit theorems in \cite{penrose:2000, penrose:2007a} and laws of large numbers in \cite{penrose:yukich:2003, penrose:2007b, penrose:yukich:2011}. For example, Penrose \cite{penrose:2000} proved a (functional) central limit theorem for the number of inhomogeneous Poisson points $X_i$ with density $f$, such that $f(X_i)R_{k_n}(X_i,\mathcal P_n)^d$ does not exceed certain thresholds ($k_n$ is taken to be a function of $n$). Additionally, Penrose and Yukich \cite{penrose:yukich:2011} provided laws of large numbers for the sum of power weighted nearest neighbor distances. As for the large deviation results on the $k$-nearest neighbor distance, Schreiber and Yukich \cite{schreiber:yukich:2005} obtained the \emph{Donsker-Varadhan large deviation principle} (LDP) (see Section 1.2 in \cite{dembo:zeitouni:1998} for a precise definition) for the functional of the length of edges in the $k$-nearest neighbor graph in $\reals^d$. This was obtained as an application of a more general LDP for the spatial point process satisfying a weak dependence condition characterized by a radius of stabilization. Moreover, Hirsch et al.~\cite{hirsch:jahnel:tobias:2020} analyzed lower tail large deviations for general geometric functionals, including the power-weighted edge lengths in the $k$-nearest neighbor graph.
The primary objective of this paper is to provide comprehensive results on the asymptotics of the process \eqref{e:def.L.kn}, from the viewpoints of large deviations. We consider two distinct scenarios with respect to a divergence speed of the centering term $(a_n)$. The first scenario examined in this paper is that
\begin{equation} \label{e:cond1.an.intro}
a_n \to\infty, \ \ \ a_n - \log n - (k-1) \log \log n \to -\infty, \ \ \text{as } n\to\infty.
\end{equation}
In this case, $(a_n)$ grows to infinity more slowly than \eqref{e:Poisson.centering}. Intrinsically, there appear \emph{infinitely many} $k$-nearest neighbor balls as $n\to\infty$, whose volume are approximately $a_n/n$ up to the scale. Then, the process \eqref{e:def.L.kn} ``diverges" in the limit, in the sense that $L_{k,n}(A)\to\infty$ as $n\to\infty$, for all (nice) measurable sets $A$. Thus, in order to dictate its large deviation behavior, one has to scale the process \eqref{e:def.L.kn} by some growing sequence $u_n\to\infty$.
More concretely,
we aim to establish the Donsker-Varadhan LDP for the properly scaled process $(L_{k,n}/u_n)_{n\ge1}$.
In the second scenario of this paper, we consider the centering term $(a_n)$ satisfying
\begin{equation} \label{e:cond2.an.intro}
a_n - \log n - (k-1) \log \log n \to \infty, \ \ a_n=o(n), \ \ \text{as } n\to\infty.
\end{equation}
Then, $(a_n)$ tends to infinity more rapidly than \eqref{e:Poisson.centering}, so that the $k$-nearest neighbor balls centered around $\mathcal P_n$, whose volume are approximately $a_n/n$ up to the scale, are even less likely to occur. In other words, the occurrence of such $k$-nearest neighbor balls is a ``rare event", in the sense of $\mathbb{P}\big(L_{k,n}(A)\ge1\big)\to0$ as $n\to\infty$, for all measurable sets $A$. In this setup, we present the other type of large deviation results, by detecting a sequence $v_n\to\infty$, such that
\begin{equation} \label{e:M0.seq.intro}
\big( v_n \mathbb{P}(L_{k,n}\in \cdot), \, n\ge1 \big)
\end{equation}
converges to a (non-trivial) limit measure. The topology underlying the convergence of \eqref{e:M0.seq.intro} is \emph{$\mathcal M_0$-topology}. The notion of $\mathcal M_0$-topology was first developed by \cite{hult:lindskog:2006a}. Since then, it has been used mainly for the study of regular variation of stochastic processes \cite{hult:samorodnitsky:2010, lindskog:resnick:roy:2014, fasen:roy:2016, segers:zhao:meinguet:2017, owada:2022b}.
For the required LDP in Theorem \ref{t:LDP.L.kn} below, many of the techniques in our previous work \cite{hirsch:owada:2022} will be exploited. We first partition the unit cube $[0,1]^d$ into smaller cubes of equal volume, and define a collection of i.i.d.~point processes restricted to each of the small cubes. Next, using one of the main results in \cite{bobrowski:schulte:yogeshwaran:2021}, Proposition \ref{p:KR.conv1} proves that the law of these point processes restricted to small cubes converges to the law of certain Poisson point processes, in terms of the Kantorovich-Rubinstein distance. Subsequently, Proposition \ref{p:eta.zeta.negligible} justifies that this approximation is still feasible even for the version of the empirical measures. The main machinery here is the notion of maximal coupling, provided in \cite[Lemma 4.32]{kallenberg}. Other approximation arguments necessary for our proof will be completed in a series of results in Propositions \ref{p:exp.neg.eta.etap}--\ref{p:exp.neg.etap.Lp}. As a final note, we want to emphasize that the homogeneity assumption of $\mathcal P_n$ is crucial throughout our proof. We anticipate that the LDP still holds even when $\mathcal P_n$ in \eqref{e:def.L.kn} is replaced by an inhomogeneous Poisson point process. It seems, however, that unlike the previous studies \cite{penrose:1997, chenavier:henze:otto:2022, otto:2020, bobrowski:schulte:yogeshwaran:2021}, this extension should require much more involved machinery; this will be left as a topic of future research.
The rest of the paper is outlined as follows. In Section \ref{sec:LDP}, under the assumption \eqref{e:cond1.an.intro}, we give a precise setup for the point process \eqref{e:def.L.kn} and formalize the desired LDP. Section \ref{sec:M0} assumes condition \eqref{e:cond2.an.intro} and establishes the $\mathcal M_0$-convergence for the sequence \eqref{e:M0.seq.intro}. In both settings, we also consider the case that the point processes are generated by a binomial point process. Unfortunately, in the context of LDPs, there are no unified results on the De-Poissonization scheme, such as \cite[Section 2.5]{penrose:2003}, which may allow us to extend the LDPs with a Poisson input to those with a binomial input. Alternatively, we have proved directly the desired exponential equivalence (in terms of the total variation distance) between processes with a Poisson input and those with a binomial input (see Corollary \ref{c:LDP.LknB}). Corollary \ref{c:M0.LknB} applies a similar kind of De-Poissonization machinery to the $\mathcal M_0$-convergence in Theorem \ref{t:M0.L.kn}.
Finally, as an application of Theorem \ref{t:LDP.L.kn} and Corollary \ref{c:LDP.LknB}, we deduce the LDP for the number of Poisson (or binomial) points of degree at most $k$ in a geometric graph in the dense regime. Additional applications in a similar vein from the perspective of $\mathcal M_0$-convergence, can be found in Corollary \ref{c:M0.application}.
\medskip
\section{Large deviation principle} \label{sec:LDP}
Given a point measure $\omega$ in $[0,1]^d$ and $x\in [0,1]^d$, define the $k$-nearest neighbor distance $R_k(x,\omega)$ as in \eqref{e:def.k.NN.dist}.
Let $(a_n)_{n\ge1}$ be a sequence tending to infinity, such that
\begin{equation} \label{e:speed.an}
a_n - \log n -(k-1)\log \log n \to -\infty, \ \ \ n\to\infty.
\end{equation}
Given a homogeneous Poisson point process $\mathcal P_n$ on $[0,1]^d$ with intensity $n$, define the point process \eqref{e:def.L.kn}
on the space $M_p\big(E_0 \big)$ of point measures on $E_0 := [0, 1]^d\times [s_0,\infty)$, where $s_0\in \reals$ is a fixed real number.
Our aim is to explore the large deviation behavior of $(L_{k,n})_{n\ge1}$.
More specifically, we define the scaling constants
\begin{equation} \label{e:def.bn}
b_n = na_n^{k-1}e^{-a_n}, \ \ \ n\ge1,
\end{equation}
and establish the LDP for $(L_{k,n}/b_n)_{n\ge1}$ in the space $M_+(E_0)$ of Radon measures on $E_0$. The space $M_+(E_0)$ is equipped with the weak topology.
Under the assumption \eqref{e:speed.an}, it is elementary to show that $b_n\to\infty$ as $n\to\infty$.
To state our LDP more precisely, we introduce the measure
$$
\tau_k(\dif u) := \frac{ e^{-u} }{(k-1)!}\, {\mathbbm 1} \{ u\ge s_0 \}\dif u.
$$
Here, we provide two equivalent representations of the rate function. The first is based on the relative entropy. More precisely, writing $\mathsf{Leb} \otimes \tau_k$ for the product measure of the Lebesgue measure on $[0, 1]^d$ and $\tau_k$, define the relative entropy of $\rho \in M_+(E_0)$ with respect to $\mathsf{Leb} \otimes \tau_k$:
\begin{equation} \label{e:def.relative.entropy}
H_k(\rho \,| \, \mathsf{Leb} \otimes \tau_k) := \int_{E_0} \log\Big\{\frac{\dif \rho (x,u)}{\dif \, (\mathsf{Leb} \otimes \tau_k)}\Big\} \rho(\dif x, \dif u) - \rho\big(E_0\big) + (\mathsf{Leb} \otimes \tau_k)(E_0), \ \ \text{if } \rho \ll \mathsf{Leb}\otimes \tau_k,
\end{equation}
and $H_k(\rho \,| \, \mathsf{Leb} \otimes \tau_k)=\infty$ otherwise. The second representation of the rate function is given in terms of the Legendre transform. More concretely,
$$
\Lambda_k^*(\rho):=\sup_{f\in C_b(E_0)} \Big\{ \int_{E_0} f(x, u)\rho(\dif x, \dif u) - \int_{E_0} \big(e^{f(x, u)}-1 \big) \dif x\, \tau_k(\dif u) \Big\}, \ \ \rho\in M_+\big(E_0\big),
$$
where $C_b(E_0)$ is the space of continuous and bounded real-valued functions on $E_0$.
The proofs of the results in this section are all deferred to Section \ref{sec:proof.LDP}.
\begin{theorem} \label{t:LDP.L.kn}
The sequence $(L_{k,n}/b_n)_{n\ge1}$ satisfies an LDP on $M_+\big( E_0 \big)$ in the weak topology, with rate $b_n$ and rate function $H_k(\cdot\, | \, \mathsf{Leb} \otimes \tau_k)= \Lambda_k^*$.
\end{theorem}
The corollary below extends Theorem \ref{t:LDP.L.kn} to the case that the process is generated by a binomial point process $\mathcal B_n:=\{ X_1,\dots,X_n \}$. Precisely, we define, in the space $M_p(E_0)$,
$$
L_{k,n}^\mathsf{B} := \sum_{X\in \mathcal B_n} \delta_{(X, n\theta_d R_k(X, \mathcal B_n)^d-a_n)}, \ \ n > k,
$$
and $L_{k,n}^\mathsf{B} \equiv \emptyset$ for $n \le k$.
\begin{corollary} \label{c:LDP.LknB}
The sequence $(L_{k,n}^\mathsf{B}/b_n)_{n\ge1}$ satisfies an LDP on $M_+(E_0)$ in the weak topology, with rate $b_n$ and rate function $H_k(\cdot\, | \, \mathsf{Leb} \otimes \tau_k)= \Lambda_k^*$.
\end{corollary}
\medskip
As an application of Theorem \ref{t:LDP.L.kn}, we deduce the LDP for $(T_{k,n}/b_n)_{n\ge1}$, where
\begin{equation} \label{e:def.T.kn}
T_{k,n} := \sum_{X\in \mathcal P_n}{\mathbbm 1} \Big\{ \mathcal P_n \big( B_{r_n(s_0)}(X) \big)\le k \Big\}, \ \ n\ge1.
\end{equation}
The statistics \eqref{e:def.T.kn} represents the number of Poisson points of degree at most $k$ in a geometric graph of vertex set $\mathcal P_n$ and edges between $X_i$ and $X_j$ satisfying $\|X_i-X_j\| \le r_n(s_0)$, where
\begin{equation} \label{e:def.rn.s0}
r_n(s_0) := \Big( \frac{a_n+s_0}{n\theta_d} \Big)^{1/d}.
\end{equation}
The threshold radius in \eqref{e:def.rn.s0} ensures that the geometric graph under consideration is of the \emph{dense regime}, such that $nr_n(s_0)^d =(a_n+s_0)/\theta_d\to \infty$ as $n\to\infty$. Replacing $\mathcal P_n$ in \eqref{e:def.T.kn} with its binomial counterpart $\mathcal B_n$, we also derive the LDP for $(T_{k,n}^\mathsf{B}/b_n)_{n\ge1}$, where
$$
T_{k,n}^\mathsf{B} := \sum_{X\in \mathcal B_n}{\mathbbm 1} \Big\{ \mathcal B_n \big( B_{r_n(s_0)}(X) \big)\le k \Big\}, \ \ n\ge1.
$$
\begin{corollary} \label{c:LDP.application}
The sequence $(T_{k,n}/b_n)_{n\ge1}$ satisfies an LDP with rate $b_n$ and rate function
$$
I_k(x)=\begin{cases}
x\log \big( x/\alpha_k \big)-x+\alpha_k &\text{ if } x\ge0, \\
\infty &\text{if } x < 0,
\end{cases}
$$
where $\alpha_k=e^{-s_0}/(k-1)!$. Furthermore, $(T_{k,n}^\mathsf{B}/b_n)_{n\ge1}$ satisfies the same LDP as $(T_{k,n}/b_n)_{n\ge1}$.
\end{corollary}
In the above, $I_k(x)$ coincides with a rate function in the LDP for $\big( n^{-1}\sum_{i=1}^n Y_i \big)_{n\ge1}$ where the $Y_i$ are i.i.d.~Poisson with mean $\alpha_k$.
\medskip
\section{Large deviation under $\mathcal M_0$-topology} \label{sec:M0}
In this section, we explore the large deviation behavior of the process \eqref{e:def.L.kn}, in the case that $a_n$ tends to infinity more rapidly than in the last section. Namely, we assume that $(a_n)_{n\ge1}$ satisfies
$$
a_n -\log n -(k-1)\log \log n \to \infty, \ \ \ n\to\infty.
$$
We again introduce the sequence $b_n=na_n^{k-1}e^{-a_n}$ as in \eqref{e:def.bn}. However, unlike in the last section,
$b_n \to0$ as $n\to\infty$, because
\begin{align*}
b_n &= e^{-(a_n -\log n -(k-1)\log \log n )} \Big( \frac{a_n}{\log n} \Big)^{k-1} \\
&\le C e^{-(a_n -\log n -(k-1)\log \log n )} \big( a_n -\log n -(k-1)\log \log n \big)^{k-1} \to 0,
\end{align*}
for some $C>0$.
Our objective is to investigate large deviations for the sequence $(\mathbb{P}\circ L_{k,n}^{-1})_{n\ge1}$ of probability distributions of $(L_{k,n})_{n\ge1}$ on the space $M_p(E)$, where $E:= [0,1]^d\times (-\infty, \infty]$. A main challenge is that the space $M_p(E)$ is not locally compact, and therefore, the vague topology would no longer be applicable for the convergence of such probability distributions. To overcome this difficulty, we adopt the notion of \emph{$\mathcal M_0$-topology}. The main feature of $\mathcal M_0$-topology is that the corresponding test functions are continuous and bounded real-valued functions on $M_p(E)$ that vanish in the neighborhood of the origin. For the space $M_p(E)$, one can take the null measure $\emptyset$ as its origin. Let $B_{\emptyset, r}$ denote an open ball of radius $r>0$ centered at $\emptyset$ in the vague metric.
Denote by $\mathcal M_0 = \mathcal M_0\big( M_p(E) \big)$ the space of Borel measures on $M_p(E)$, the restriction of which to $M_p(E)\setminus B_{\emptyset, r}$ is finite for all $r>0$. Moreover, define $\mathcal C_0 = \mathcal C_0\big(M_p(E) \big)$ to be the space of continuous and bounded real-valued functions on $M_p(E)$ that vanish in the neighborhood of $\emptyset$. Given $\xi_n, \xi \in \mathcal M_0$, we say that $\xi_n$ converges to $\xi$ in the $\mathcal M_0$-topology, denoted as $\xi_n\to \xi$ in $\mathcal M_0$, if it holds that $\int_{M_p(E)}g(\eta)\xi_n(\dif \eta) \to \int_{M_p(E)}g(\eta)\xi(\dif \eta)$ for all $g\in \mathcal C_0$. For more information on $\mathcal M_0$-topology we refer to \cite{hult:lindskog:2006a}.
Before stating the main theorem, we will impose an additional condition that $a_n=o(n)$ as $n\to\infty$. To see the necessity of this assumption, suppose, to the contrary, that $a_n/n\to\infty$ as $n\to\infty$. Then,
it trivially holds that $L_{k,n}\big( [0,1]^d \times (-M,\infty] \big)=0$ a.s.~for large enough $n$ and any $M>0$. By putting the assumption $a_n=o(n)$ as above, one can exclude such triviality.
The proofs of the results below are all given in Section \ref{sec:proof.M0}.
\begin{theorem} \label{t:M0.L.kn}
In the above setting with $a_n = o(n)$, as $n\to\infty$,
\begin{equation} \label{e:M0.convergence}
b_n^{-1}\mathbb{P} (L_{k,n}\in \cdot ) \to \xi_k, \ \ \text{in } \mathcal M_0,
\end{equation}
where
$$
\xi_k(\cdot) := \frac{1}{(k-1)!} \int_E {\mathbbm 1} \{ \delta_{(x,u)} \in \cdot \} \, e^{-u} \dif x\dif u.
$$
\end{theorem}
Additionally, we present an analogous result for the process $(L_{k,n}^\mathsf{B})_{n\ge1}$ as well. For a precise statement, however, we need to put a more stringent condition on $(a_n)$ for the purpose of proving \eqref{e:Fnvep.M0} in Section \ref{sec:proof.M0}.
\begin{corollary} \label{c:M0.LknB}
In the above setting with $a_n=o(n^{1/3})$, as $n\to\infty$,
$$
b_n^{-1} \mathbb{P}(L_{k,n}^\mathsf{B}\in \cdot) \to \xi_k, \ \ \text{ in } M_0.
$$
\end{corollary}
Finally, certain asymptotic results on $(T_{k,n})_{n\ge1}$ in \eqref{e:def.T.kn}, as well as those on $(T_{k,n}^\mathsf{B})_{n\ge1}$, are presented as a corollary of the above results. This corollary gives the exact rate (up to the scale) of a probability that the number of Poisson (or binomial) points of degree at most $k$ becomes non-zero.
\begin{corollary} \label{c:M0.application}
$(i)$ If $a_n=o(n)$, then
$$
b_n^{-1}\mathbb{P}(T_{k,n}\ge1) \to \alpha_k \ \ \ n\to\infty,
$$
where $\alpha_k$ is given in Corollary \ref{c:LDP.application}. \\
$(ii)$ If $a_n=o(n^{1/3})$, then
$$
b_n^{-1}\mathbb{P}(T_{k,n}^\mathsf{B}\ge1) \to \alpha_k, \ \ \ n\to\infty.
$$
\end{corollary}
\medskip
\section{Proofs} \label{sec:proofs}
\subsection{Proofs of Theorem \ref{t:LDP.L.kn}, Corollary \ref{c:LDP.LknB}, and Corollary \ref{c:LDP.application}} \label{sec:proof.LDP}
First, let us generalize the radius \eqref{e:def.rn.s0} by
$$
r_n(u) := \Big(\frac{a_n+u}{n\theta_d}\Big)^{1/d}, \ \ u\in \reals.
$$
For $x\in [0,1]^d$ and $\omega \in M_p\big( [0,1]^d \big)$, define the following functions:
\begin{align*}
f(x,\omega) &:= n\theta_dR_k(x, \omega)^d-a_n, \quad\text{ and }\quad g(x,\omega) := {\mathbbm 1} \{ f(x, \omega)>s_0 \}.
\end{align*}
{Then, the process \eqref{e:def.L.kn} that is restricted here to the space $M_p(E_0)$, can be reformulated as}
\begin{equation} \label{e:def.Lkn2}
L_{k,n} = \begin{cases}
\sum_{X\in \mathcal P_n} g(X, \mathcal P_n)\, \delta_{(X, f(X,\mathcal P_n))} & \text{ if } |\mathcal P_n| > k, \\
\emptyset & \text{ if } |\mathcal P_n| \le k.
\end{cases}
\end{equation}
Next, the unit cube $[0,1]^d$ is partitioned into smaller cubes $Q_1,\dots,Q_{b_n}$, so that $\mathsf{Leb}(Q_\ell)=b_n^{-1}$ for all $\ell\in\{ 1,\dots,b_n \}$.
Here, it is assumed, without loss of generality, that $b_n$ takes only positive integers for all $n\ge1$. To avoid unnecessary technicalities, we will put the same assumption on many of the sequences and functions throughout the proof.
Fix a sequence $w_n\to \infty$ with $w_n=o(a_n)$ as $n\to\infty$. For each $\ell\in\{ 1,\dots,b_n \}$, define the ``boundary part" of $Q_\ell$ by
$$
M_\ell := \big\{ x\in Q_\ell: \inf_{y\in \partial Q_\ell} \| x-y\|\le r_n(w_n) \big\},
$$
while the ``internal region" of $Q_\ell$ is given as $K_\ell := Q_\ell \setminus M_\ell$.
We now consider the process
\begin{equation} \label{e:def.eta.kn}
\eta_{k,n} := \sum_{\ell=1}^{b_n} \sum_{X\in \mathcal P_n|_{K_\ell} }g\big(X, \mathcal P_n|_{Q_\ell}\big) \, \delta_{(X, f(X, \mathcal P_n|_{Q_\ell}))} \in M_p(E_0 ),
\end{equation}
where $\mathcal P_n|_{K_\ell}$ (resp.~$\mathcal P_n|_{Q_\ell}$) represents the Poisson point process restricted to $K_\ell$ (resp.~$Q_\ell$). Setting up a ``blocked" point process as in \eqref{e:def.eta.kn} is a standard approach in the literature (see, e.g., \cite{seppalainen:yukich:2001, schreiber:yukich:2005}). Clearly, the process \eqref{e:def.eta.kn} is different from \eqref{e:def.L.kn}. For example, \eqref{e:def.eta.kn} removes all the points $X\in \mathcal P_n$ lying within the distance of $r_n(w_n)$ from the boundary of $(Q_\ell)_{\ell=1}^{b_n}$, whereas those points are possibly counted by \eqref{e:def.L.kn}. Even when the center $X\in \mathcal P_n$ is chosen from the inside of $K_\ell$, i.e., $X\in \mathcal P_n|_{K_\ell}$, the processes \eqref{e:def.L.kn} and \eqref{e:def.eta.kn} may exhibit different $k$-nearest neighbor balls, whenever $R_k(X,\mathcal P_n)\neq R_k(X, \mathcal P_n|_{Q_\ell})$. Despite such differences, it is justified later in Propositions \ref{p:exp.neg.eta.etap} -- \ref{p:exp.neg.etap.Lp} that the process \eqref{e:def.eta.kn} can be used to approximate the large deviation behavior of \eqref{e:def.L.kn}.
For later analyses, it is convenient to express \eqref{e:def.eta.kn} as a superposition of i.i.d.~point processes on $E_0$, which themselves are transformed by some homeomorphisms.
For each $\ell\in\{ 1,\dots,b_n \}$, let $\kappa_{\ell,n}: [0,1]^d\to Q_\ell$ be the homeomorphism defined by $\kappa_{\ell,n} (x)=b_n^{-1/d}x+z_{\ell,n}$, where $z_{\ell, n}$ is a lower-left corner of $Q_\ell$. Further, define $\widetilde \kappa_{\ell,n}: E_0 \to Q_\ell \times (s_0,\infty]$ by $\widetilde \kappa_{\ell,n} (x,u)=\big( \kappa_{\ell,n}(x),u \big)$.
Using the homeomorphism $\widetilde \kappa_{\ell,n}$, one can express $\eta_{k,n}$ as
\begin{equation} \label{e:def.eta.kn2}
\eta_{k,n} = \sum_{\ell=1}^{b_n} \eta_{k,n}^{(\ell)} \circ \widetilde \kappa_{\ell,n}^{-1},
\end{equation}
where
$$
\eta_{k,n}^{(\ell)} := \sum_{X\in \mathcal P_n|_{K_\ell}} g\big(X, \mathcal P_n|_{Q_\ell}\big) \, \delta_{(\kappa_{\ell,n}^{-1}(X),\, f(X, \mathcal P_n|_{Q_\ell}))} \in M_p(E_0).
$$
Due to the spatial independence and homogeneity of $\mathcal P_n$, $(\eta_{k,n}^{(\ell)})_{\ell\ge 1}$ constitutes a sequence of i.i.d.~point processes on $E_0$.
Next, let $(\zeta_k^{(\ell)})_{\ell \ge 1}$ be a collection of i.i.d.~Poisson point processes on $E_0$ with intensity $\mathsf{Leb} \otimes \tau_k$.
Then, the proposition below claims that for each $\ell\ge1$, the law of $\eta_{k,n}^{(\ell)}$ converges to the law of $\zeta_k^{(\ell)}$ as $n\to\infty$, in terms of the \emph{Kantorovich-Rubinstein distance}. Recall that the Kantorovich-Rubinstein distance between the distributions of two point processes $\xi_i$, $i=1,2$, is defined as
\begin{equation} \label{e:def.KR.dist}
d_{\mathsf{KR}} \big( \mathcal L(\xi_1), \mathcal L(\xi_2)\big) := \sup_{h} \big| \mathbb{E}\big[ h(\xi_1) \big] - \mathbb{E}\big[ h(\xi_2) \big] \big|,
\end{equation}
where $\mathcal L(\xi_i)$ is a probability law of $\xi_i$, and
$h$ is taken over all measurable $1$-Lipschitz functions with respect to the total variation distance on the space of point measures; see \cite{bobrowski:schulte:yogeshwaran:2021} for more information on the Kantorovich-Rubinstein distance. As a related notion, the \emph{total variation distance} between two measures $\mu_1$ and $\mu_2$ on $E_0$ is defined as
$$
d_{\mathsf{TV}} (\mu_1, \mu_2) :=\sup_{A\subset E_0} \big| \mu_1(A)-\mu_2(A) \big|.
$$
As a final remark, we propose one key observation: for $x\in [0,1]^d$, $\omega \in M_p\big( [0,1]^d \big)$ with $x\in \omega$, and $u\in \reals$, the following conditions are equivalent.
\begin{equation} \label{e:K.and.Pn1}
f(x,\omega) >u \ \ \Leftrightarrow \ \ R_k(x,\omega) >r_n(u)\ \ \Leftrightarrow \ \ \omega \big( B_{r_n(u)}(x) \big) \le k.
\end{equation}
Throughout the proof, $C^*$ denotes a generic, positive constant, which is independent of $n$ but may vary from one line to another or even within the lines.
\begin{proposition} \label{p:KR.conv1}
For every $\ell\ge1$,
\begin{equation} \label{e:KR.dist.eta.zeta}
d_{\mathsf{KR}} \big( \mathcal L(\eta_{k,n}^{(\ell)}), \mathcal L(\zeta_k^{(\ell)})\big) \to 0, \ \ \text{as } n\to\infty.
\end{equation}
\end{proposition}
\begin{proof}
The proof is based on \cite[Theorem 6.4]{bobrowski:schulte:yogeshwaran:2021}. Before applying this theorem, we need some preliminary works. First, define for $x\in [0,1]^d$ and $\omega\in M_p\big([0,1]^d\big)$,
$$
\mathcal S(x,\omega) := B_{R_k(x,\omega)}(x);
$$
then, $f$ and $g$ are \emph{localized} to $\mathcal S$. Namely, for every $x\in \omega$ and all $S\supset \mathcal S(x,\omega)$, we have $g(x,\omega)=g(x,\omega\cap S)$, and also, $f(x,\omega) = f(x,\omega\cap S)$ if $g(x,\omega)=1$. Moreover, $\mathcal S(x,\omega)$ is a \emph{stopping set}; that is, for every compact $S\subset [0,1]^d$,
$$
\big\{ \omega: B_{R_k(x,\omega)}(x)\subset S \big\} = \big\{ \omega: B_{R_k(x,\omega\cap S)}(x)\subset S \big\}.
$$
Finally, we set $S_x :=B_{r_n(w_n)}(x)$ for $x\in K_\ell$.
According to \cite[Theorem 6.4]{bobrowski:schulte:yogeshwaran:2021}, \eqref{e:KR.dist.eta.zeta} can be obtained as a direct consequence of the following conditions. First, one needs to show that
\begin{equation} \label{e:TV.dist.mean.measures}
d_{\mathsf{TV}} \Big(\mathbb{E} \big[ \eta_{k,n}^{(\ell)}(\cdot) \big], \, \mathsf{Leb} \otimes \tau_k\Big) \to 0, \ \ \ n\to\infty,
\end{equation}
where $\mathbb{E} \big[ \eta_{k,n}^{(\ell)}(\cdot) \big]$ denotes the intensity measure of $\eta_{k,n}^{(\ell)}$.
In addition to \eqref{e:TV.dist.mean.measures}, we also have to show that as $n\to\infty$,
\begin{equation} \label{e:E1}
E_1 := 2n\int_{K_\ell} \mathbb{E} \big[ g(x,\mathcal P_n|_{Q_\ell}+\delta_x) \, {\mathbbm 1} \big\{ \mathcal S (x,\mathcal P_n|_{Q_\ell}+\delta_x) \not\subset S_x \big\} \big]\dif x\to0,
\end{equation}
\begin{equation} \label{e:E2}
E_2 := 2n^2 \int_{K_\ell}\int_{K_\ell} \hspace{-.2cm}{\mathbbm 1} \{ S_x \cap S_z \neq \emptyset \} \,\mathbb{E} \big[ g(x,\mathcal P_n|_{Q_\ell}+\delta_x) \big] \mathbb{E} \big[ g(z,\mathcal P_n|_{Q_\ell}+\delta_z) \big] \dif x \dif z \to 0,
\end{equation}
and
\begin{equation} \label{e:E3}
E_3 := 2n^2 \int_{K_\ell}\int_{K_\ell} \hspace{-.2cm}{\mathbbm 1} \{ S_x \cap S_z \neq \emptyset \} \,\mathbb{E} \big[ g(x,\mathcal P_n|_{Q_\ell}+\delta_x+\delta_z) \, g(z,\mathcal P_n|_{Q_\ell}+\delta_x+\delta_z) \big] \dif x \dif z \to 0.
\end{equation}
Our goal in the sequel is to prove \eqref{e:TV.dist.mean.measures} -- \eqref{e:E3}.
\medskip
\textit{Proof of \eqref{e:TV.dist.mean.measures}}: We begin with calculating the measure $\mathbb{E} \big[ \eta_{k,n}^{(\ell)}(\cdot) \big]$ more explicitly.
For $A \subset [0, 1]^d$ and $u>s_0$, by \eqref{e:K.and.Pn1} and the Mecke formula for Poisson point processes (see, e.g., Chapter 4 in \cite{last:penrose:2017}), together with the fact that $\mathcal P_n(Q_\ell)$ is Poisson distributed with mean $n\mathsf{Leb}(Q_\ell)=nb_n^{-1}$,
\begin{align*}
\mathbb{E}\big[ \eta_{k,n}^{(\ell)}\big(A \times (u,\infty)\big) \big] &= \mathbb{E}\Big[ \sum_{X\in \mathcal P_n|_{K_\ell}} {\mathbbm 1} \big\{\kappa_{\ell, n}^{-1}(X) \in A, \, f(X, \mathcal P_n|_{Q_\ell}) >u\big\} \Big] \\
&=\mathbb{E}\Big[ \sum_{X\in\mathcal P_n|_{K_\ell}} {\mathbbm 1} \big\{ \kappa_{\ell, n}^{-1}(X) \in A,\,\mathcal P_n|_{Q_\ell} \big( B_{r_n(u)}(X) \big) \le k\big\} \Big] \\
&=\mathbb{E}\Big[ \sum_{X\in\mathcal P_n|_{K_\ell}} {\mathbbm 1} \big\{ \kappa_{\ell, n}^{-1}(X) \in A,\,\mathcal P_n \big( B_{r_n(u)}(X) \big) \le k\big\} \Big] \\
&=nb_n^{-1}\mathbb{P} \Big( \kappa_{\ell,n}^{-1} (Y) \in A\setminus \kappa_{\ell,n}^{-1}(M_\ell), \, (\mathcal P_n +\delta_Y) \big( B_{r_n(u)}(Y) \big)\le k \Big),
\end{align*}
where $Y$ is a uniform random variable on $[0,1]^d$, independent of $\mathcal P_n$.
At the third equality above, we have dropped the restriction of $\mathcal P_n$, i.e., $\mathcal P_n|_{Q_\ell}=\mathcal P_n$, because $B_{r_n(w_n)}(X)\subset Q_\ell$ for all $X\in \mathcal P_n|_{K_\ell}$.
By the conditioning on $Y$, the last expression equals
\begin{align*}
&nb_n^{-1}\mathbb{E} \Big[ {\mathbbm 1} \big\{ \kappa_{\ell,n}^{-1}(Y)\in A \setminus \kappa_{\ell,n}^{-1}(M_\ell) \big\}\, \mathbb{P} \big( \mathcal P_n (B_{r_n(u)}(Y))\le k-1 \, \big|\, Y \big) \Big] \\
&=nb_n^{-1} \mathsf{Leb} \big( A \setminus \kappa_{\ell,n}^{-1}(M_\ell) \big)\sum_{i=0}^{k-1} e^{-(a_n+u)} \frac{(a_n+u)^i}{i!}.
\end{align*}
It thus turns out that $\mathbb{E} \big[ \eta_{k,n}^{(\ell)}(\cdot) \big]$ has the density
$$
q_k(x,u) := nb_n^{-1} {\mathbbm 1} \big\{ x\notin \kappa_{\ell,n}^{-1}(M_\ell) \big\}\, \frac{e^{-(a_n+u)} (a_n+u)^{k-1}}{(k-1)!}, \ \ x\in [0,1]^d, \, u >s_0.
$$
Therefore, we have
\begin{align*}
d_{\mathsf{TV}} \Big(\mathbb{E} \big[ \eta_{k,n}^{(\ell)}(\cdot) \big], \, \mathsf{Leb} \otimes \tau_k \Big) &\le \int_{E_0} \big|\, q_k(x,u)-\frac{e^{-u}}{(k-1)!} \, \big| \dif x \dif u \\
&\le \mathsf{Leb} \big( \kappa_{\ell,n}^{-1}(M_\ell) \big)\, \frac{e^{-s_0}}{(k-1)!} + \int_{s_0}^\infty \Big| \, \Big(1+\frac{u}{a_n} \Big)^{k-1}-1 \Big| \, e^{-u}\dif u.
\end{align*}
The first term above tends to $0$ as $n\to\infty$, because
$$
\mathsf{Leb} \big( \kappa_{\ell,n}^{-1}(M_\ell) \big) =b_n \mathsf{Leb} (M_\ell) = 1-\big( 1-b_n^{1/d} r_n(w_n) \big)^d \to 0, \ \ \ n\to\infty,
$$
while the second term vanishes by the dominated convergence theorem.
\medskip
\textit{Proof of \eqref{e:E1}}: It follows from \eqref{e:K.and.Pn1} that
\begin{align*}
{\mathbbm 1} \big\{ \mathcal S (x,\mathcal P_n|_{Q_\ell}+\delta_x) \not\subset S_x \big\} &= {\mathbbm 1} \big\{ B_{R_k(x, \mathcal P_n|_{Q_\ell}+\delta_x)}(x) \supset B_{r_n(w_n)}(x) \big\} \\
&= {\mathbbm 1} \big\{ (\mathcal P_n|_{Q_\ell}+\delta_x) \big( B_{r_n(w_n)}(x) \big)\le k \big\}\\
&= {\mathbbm 1} \big\{ \mathcal P_n|_{Q_\ell}\big( B_{r_n(w_n)}(x) \big) \le k-1 \big\}\\
&= {\mathbbm 1} \big\{ \mathcal P_n\big( B_{r_n(w_n)}(x) \big) \le k-1 \big\}.
\end{align*}
At the fourth equality above, we have dropped the restriction of $\mathcal P_n$, due to the fact that $B_{r_n(w_n)}(x)\subset Q_\ell$ for all $x\in K_\ell$.
Now, as $n\to\infty$,
\begin{align*}
E_1 &\le 2n \int_{K_\ell} \mathbb{P} \Big( \mathcal P_n \big( B_{r_n(w_n)} (x)\big) \le k-1 \Big)\dif x =2n \,\mathsf{Leb}(K_\ell) \sum_{i=0}^{k-1} e^{-(a_n+w_n)} \frac{(a_n+w_n)^i}{i!} \\
&\le 2e^{-w_n} \sum_{i=0}^{k-1} \frac{(a_n+w_n)^i}{i! a_n^{k-1}} \le C^*e^{-w_n}\to 0.
\end{align*}
\medskip
\textit{Proof of \eqref{e:E2}}: One can see that for $x,z\in K_\ell$,
\begin{equation} \label{e:Sx.Sz.indicator}
{\mathbbm 1} \{ S_x \cap S_z \neq \emptyset \} \le {\mathbbm 1} \big\{ \| x-z\| \le 2r_n(w_n) \big\},
\end{equation}
and from \eqref{e:K.and.Pn1},
$$
\mathbb{E} \big[ g(x,\mathcal P_n|_{Q_\ell}+\delta_x) \big] = \mathbb{P} \Big( \mathcal P_n \big( B_{r_n(s_0)} (x)\big) \le k-1 \Big) =\sum_{i=0}^{k-1} e^{-(a_n+s_0)}\frac{(a_n+s_0)^i}{i!}.
$$
Therefore, as $n\to\infty$,
\begin{align*}
E_2 &\le 2n^2 \Big\{ \sum_{i=0}^{k-1} e^{-(a_n+s_0)}\frac{(a_n+s_0)^i}{i!}\Big\}^2 \int_{Q_\ell}\int_{Q_\ell} {\mathbbm 1} \big\{ \| x-z\| \le 2r_n(w_n) \big\} \dif x \dif z \\
&\le C^*b_n^2 \int_{Q_\ell}\int_{Q_\ell} {\mathbbm 1} \big\{ \| x-z\| \le 2r_n(w_n) \big\} \dif x \dif z \\
&\le C^*b_n^2 \mathsf{Leb}(Q_\ell) r_n(w_n)^d = C^*b_nr_n(w_n)^d\to 0.
\end{align*}
\medskip
\textit{Proof of \eqref{e:E3}}: It follows from \eqref{e:K.and.Pn1} and \eqref{e:Sx.Sz.indicator} that $E_3$ can be split into two terms:
\begin{equation} \label{e:E3.split}
\begin{split}
E_3 &\le 2n^2 \int_{K_\ell}\int_{K_\ell}{\mathbbm 1} \big\{ \| x-z\| \le 2r_n(w_n) \big\} \\
&\qquad \qquad \qquad \times \mathbb{P} \Big( (\mathcal P_n+\delta_z) \big( B_{r_n(s_0)}(x) \big) \le k -1, \, (\mathcal P_n+\delta_x) \big( B_{r_n(s_0)}(z) \big) \le k -1\Big) \dif x \dif z \\
&= 2n^2 \int_{K_\ell}\int_{K_\ell}{\mathbbm 1} \big\{ \| x-z\| \le r_n(s_0) \big\} \\
&\qquad \qquad \qquad \times \mathbb{P} \Big( (\mathcal P_n+\delta_z) \big( B_{r_n(s_0)}(x) \big) \le k -1, \, (\mathcal P_n+\delta_x) \big( B_{r_n(s_0)}(z) \big) \le k -1\Big) \dif x \dif z \\
&\quad + 2n^2 \int_{K_\ell}\int_{K_\ell}{\mathbbm 1} \big\{r_n(s_0) < \| x-z\| \le 2r_n(w_n) \big\} \\
&\qquad \qquad \qquad \times \mathbb{P} \Big( (\mathcal P_n+\delta_z) \big( B_{r_n(s_0)}(x) \big) \le k -1, \, (\mathcal P_n+\delta_x) \big( B_{r_n(s_0)}(z) \big) \le k -1\Big) \dif x \dif z \\
&=: E_{3,1}+E_{3,2}.
\end{split}
\end{equation}
For $E_{3,1}$, if $\| x-z\|\le r_n(s_0)$ with $x,z \in K_\ell$, then $(\mathcal P_n+\delta_z) \big( B_{r_n(s_0)}(x) \big) = \mathcal P_n \big( B_{r_n(s_0)}(x) \big)+1$. Because of the spatial independence of $\mathcal P_n$,
\begin{align}
E_{3,1} &\le 2n^2 \int_{K_\ell}\int_{K_\ell} {\mathbbm 1} \big\{ \| x-z \|\le r_n(s_0) \big\}\, \mathbb{P}\Big(\mathcal P_n \big( B_{r_n(s_0)}(x) \big) \le k-2 \Big) \label{e:E31} \\
&\qquad \qquad \qquad \qquad \qquad \times \mathbb{P}\Big(\mathcal P_n \big( B_{r_n(s_0)}(z)\setminus B_{r_n(s_0)}(x) \big) \le k-2 \Big)\dif x \dif z. \notag
\end{align}
Then, it is easy to see that
$$
\mathbb{P}\Big(\mathcal P_n \big( B_{r_n(s_0)}(x) \big) \le k-2 \Big) \le C^*a_n^{k-2}e^{-a_n},
$$
and also,
$$
\mathbb{P}\Big(\mathcal P_n \big( B_{r_n(s_0)}(z)\setminus B_{r_n(s_0)}(x) \big) \le k-2 \Big) \le C^*e^{-\frac{n}{2}\mathsf{Leb}(B_{r_n(s_0)}(z)\setminus B_{r_n(s_0)}(x))}.
$$
Notice that
$$
\mathsf{Leb}\big(B_{r_n(s_0)}(z)\setminus B_{r_n(s_0)}(x)\big) \ge C^*r_n(s_0)^{d-1}\| x-z \|,
$$
whenever $\|x-z\| \le 2r_n(s_0)$ (see Equ.~(7.5) in \cite{penrose:goldstein:2010}); so, we have
$$
\mathbb{P}\Big(\mathcal P_n \big( B_{r_n(s_0)}(z)\setminus B_{r_n(s_0)}(x) \big) \le k-2 \Big) \le C^*e^{-\frac{n}{2}C^*r_n(s_0)^{d-1}\| x-z \|}.
$$
Referring these bounds back to \eqref{e:E31},
\begin{align*}
E_{3,1} &\le C^*n^2 \int_{Q_\ell}\int_{\reals^d}a_n^{k-2}e^{-a_n} e^{-\frac{n}{2}C^*r_n(s_0)^{d-1}\| x-z \|} \dif x \dif z \\
&=C^*n^2a_n^{k-2}e^{-a_n} b_n^{-1} \int_0^\infty e^{-\frac{n}{2}C^*r_n(s_0)^{d-1}\rho} \rho^{d-1}\dif \rho \\
&= \frac{C^*n^2a_n^{k-2}e^{-a_n} b_n^{-1} }{\big( nr_n(s_0)^{d-1} \big)^d} = C^*\Big( 1+\frac{s_0}{a_n} \Big)\, \frac{1}{(a_n+s_0)^d} \to 0, \ \ \ n\to\infty.
\end{align*}
By the spatial independence of $\mathcal P_n$,
\begin{align*}
E_{3,2} &\le 2n^2 \int_{K_\ell}\int_{K_\ell} \mathbb{P}\Big( \mathcal P_n\big( B_{r_n(s_0)}(x) \big)\le k-1 \Big) \mathbb{P}\Big( \mathcal P_n\big( B_{r_n(s_0)}(z)\setminus B_{r_n(s_0)}(x) \big)\le k-1 \Big) \\
&\qquad \qquad \qquad \qquad \qquad \qquad \times {\mathbbm 1} \big\{ r_n(s_0) < \| x-z\| \le 2r_n(w_n) \big\} \dif x \dif z
\end{align*}
Then,
$$
\mathbb{P}\Big( \mathcal P_n\big( B_{r_n(s_0)}(x) \big)\le k-1 \Big) \le C^*a_n^{k-1} e^{-a_n}.
$$
Moreover, if $\| x-z \|>r_n(s_0)$ with $x,z\in K_\ell$, then
$$
n\, \mathsf{Leb} \big( B_{r_n(s_0)}(z)\setminus B_{r_n(s_0)}(x) \big) \ge \frac{n}{2}\, \mathsf{Leb} \big( B_{r_n(s_0)}(z) \big) = \frac{a_n+s_0}{2},
$$
from which we have
$$
\mathbb{P}\Big( \mathcal P_n\big( B_{r_n(s_0)}(z)\setminus B_{r_n(s_0)}(x) \big)\le k-1 \Big) \le \sum_{i=0}^{k-1} e^{-\frac{a_n+s_0}{2}} \frac{(a_n+s_0)^i}{i!} \le C^*a_n^{k-1}e^{-\frac{a_n}{2}}.
$$
Appealing to these obtained bounds, as $n\to\infty$,
\begin{align*}
E_{3,2} &\le C^*n^2\int_{Q_\ell}\int_{\reals^d} a_n^{k-1}e^{-a_n} \cdot a_n^{k-1} e^{-\frac{a_n}{2}} {\mathbbm 1} \big\{ \|x-z\|\le 2r_n(w_n) \big\}\dif x \dif z \\
&=C^* n^2 a_n^{2(k-1)} e^{-\frac{3a_n}{2}} b_n^{-1} r_n(w_n)^d \le C^* a_n^k e^{-\frac{a_n}{2}} \to 0,
\end{align*}
as desired.
\end{proof}
Recall now that $(\zeta_k^{(\ell)})_{\ell\ge1}$ are i.i.d.~Poisson point processes on $E_0$ with intensity $\mathsf{Leb} \otimes \tau_k$.
The next proposition claims that the process
\begin{equation} \label{e:def.zeta.kn}
\zeta_{k,n} := \sum_{\ell=1}^{b_n} \zeta_k^{(\ell)} \circ \widetilde \kappa_{\ell,n}^{-1}
\end{equation}
satisfies the desired LDP in Theorem \ref{t:LDP.L.kn}. Let $(\Omega', \mathcal F', \mathbb{P}')$ be the probability space for which \eqref{e:def.zeta.kn} is defined.
\begin{proposition} \label{p:LDP.zeta.kn}
The sequence $(\zeta_{k,n}/b_n)_{n\ge1}$ satisfies an LDP in the weak topology, with rate $b_n$ and rate function $H_k(\cdot\, | \, \mathsf{Leb} \otimes \tau_k) = \Lambda_k^*$.
\end{proposition}
\begin{proof}
By the transformation theorem, $\zeta_k^{(\ell)} \circ \widetilde \kappa_{\ell,n}^{-1}$ becomes a Poisson point process on $E_0$ with intensity $(\mathsf{Leb} \otimes \tau_k) \circ \widetilde \kappa_{\ell,n}^{-1} =b_n (\mathsf{Leb}|_{Q_\ell}\otimes \tau_k)$. As $(\zeta_k^{(\ell)})_{\ell\ge1}$ are i.i.d., \eqref{e:def.zeta.kn} turns out to be a Poisson point process on $E_0$ with intensity $b_n (\mathsf{Leb} \otimes \tau_k)$; thus, there exists a sequence $(\xi_k^{(\ell)})_{\ell\ge1}$ of i.i.d.~Poisson point processes on $E_0$ with intensity $\mathsf{Leb} \otimes \tau_k$, so that
\begin{equation} \label{e:superposition.equality.dist}
\zeta_{k,n} \stackrel{d}{=} \sum_{\ell=1}^{b_n} \xi_k^{(\ell)}.
\end{equation}
For convenience, we assume that $(\xi_k^{(\ell)})_{\ell\ge1}$ are defined in the same probability space $(\Omega', \mathcal F', \mathbb{P}')$.
By applying the Poisson variant of Sanov's theorem to \eqref{e:superposition.equality.dist}, we conclude that $(\zeta_{k,n}/b_n)_{n\ge1}$ satisfies an LDP with rate $b_n$ and rate function $H_k(\cdot\, |\, \mathsf{Leb} \otimes \tau_k)$; see \cite[Proposition 3.6]{wireless3} and \cite[Theorem 6.2.10]{dembo:zeitouni:1998} for details.
To deduce the LDP with rate function $\Lambda_k^*$,
we compute the logarithmic Laplace functional of $\xi_k^{(1)}$: it follows from Theorem 5.1 in \cite{resnick:2007} that for every $f\in C_b(E_0)$,
$$
\Lambda_k(f):= \log \mathbb{E}' \big[ e^{\xi_k^{(1)}}(f) \big] =\int_{E_0} \big( e^{f(x,u)}-1 \big)\,\dif x \, \tau_k(\dif u).
$$
Now, Cram\'er's theorem in Polish spaces (see, e.g., Theorem 6.1.3 in \cite{dembo:zeitouni:1998}) can yield the required LDP, for which $\Lambda_k^*$ is obtained as the Legendre transform of $\Lambda_k$.
\end{proof}
By the maximal coupling argument (see \cite[Lemma 4.32]{kallenberg}), for every $\ell\ge1$ there exists a coupling $(\hat \eta_{k,n}^{(\ell)}, \hat \zeta_k^{(\ell)})$ defined on a probability space $(\hat \Omega_\ell, \mathcal{\hat F}_\ell, \hat \mathbb{P}_\ell)$, such that $\hat \eta_{k,n}^{(\ell)} \stackrel{d}{=} \eta_{k,n}^{(\ell)}$ and $\hat \zeta_k^{(\ell)} \stackrel{d}{=} \zeta_k^{(\ell)}$, and
\begin{equation} \label{e:max.coupling.conv}
\hat \mathbb{P}_\ell \big(\hat \eta_{k,n}^{(\ell)} \neq \hat \zeta_k^{(\ell)} \big) = d_{\mathsf{TV}} \big( \mathcal L (\eta_{k,n}^{(\ell)}), \mathcal L (\zeta_k^{(\ell)}) \big) \le d_{\mathsf{KR}} \big( \mathcal L (\eta_{k,n}^{(\ell)}), \mathcal L (\zeta_k^{(\ell)}) \big) \to 0, \ \ \ n\to\infty.
\end{equation}
In particular, $(\hat \eta_{k,n}^{(\ell)}, \hat \zeta_{k}^{(\ell)})_{\ell\ge1}$ constitutes a sequence of i.i.d.~random vectors on the probability space $(\hat \Omega, \hat{\mathcal F}, \hat \mathbb{P})$, where $\hat \Omega = \prod_{\ell=1}^\infty \hat \Omega_\ell$, $\hat{\mathcal F} = \bigotimes_{\ell=1}^\infty \hat{\mathcal F}_\ell$, and $\hat \mathbb{P}= \bigotimes_{\ell=1}^\infty \hat \mathbb{P}_\ell$.
Defining $\hat \eta_{k,n}$ and $\hat \zeta_{k,n}$ analogously to \eqref{e:def.eta.kn2} and \eqref{e:def.zeta.kn}, Proposition \ref{p:eta.zeta.negligible} below demonstrates that $(\hat \eta_{k,n}/b_n)_{n\ge1}$ and $(\hat \zeta_{k,n}/b_n)_{n\ge1}$ are exponentially equivalent (in terms of the total variation distance) under the coupled probability measure $\hat \mathbb{P}$. Since the LDP for $(\zeta_{k,n}/b_n)_{n\ge1}$ was already given by Proposition \ref{p:LDP.zeta.kn}, this exponential equivalence allows us to conclude that $(\eta_{k,n}/b_n)_{n\ge1}$ fulfills an LDP in Theorem \ref{t:LDP.L.kn}.
\begin{proposition} \label{p:eta.zeta.negligible}
For every $\delta>0$,
$$
\frac{1}{b_n}\log \hat \mathbb{P} \big( d_{\mathsf{TV}}( \hat\eta_{k,n}, \, \hat\zeta_{k,n}) \ge \delta b_n\big)\to -\infty, \ \ \ n\to\infty.
$$
\end{proposition}
\begin{proof}
The proof is highly related to \cite[Lemma 5.5]{hirsch:owada:2022}, but we still want to give a concise and self-contained proof.
By Markov's inequality and the fact that $(\hat \eta_{k,n}^{(\ell)}, \hat \zeta_{k}^{(\ell)})_{\ell\ge1}$ are i.i.d.~processes, we have, for every $a>0$,
\begin{align*}
\frac{1}{b_n} \log \hat \mathbb{P} \big( d_{\mathsf{TV}} ( \hat \eta_{k,n}, \hat \zeta_{k,n}) \ge \delta b_n\big) &\le \frac{1}{b_n} \log \hat \mathbb{P} \Big( \sum_{\ell=1}^{b_n} d_\mathsf{TV} (\hat \eta_{k,n}^{(\ell)}, \hat \zeta_k^{(\ell)}) \ge \delta b_n \Big)
\le -a\delta +\log \hat \mathbb{E} \big[e^{a d_\mathsf{TV} (\hat \eta_{k,n}^{(1)}, \, \hat \zeta_{k}^{(1)}) } \big],
\end{align*}
where $\hat \mathbb{E}$ denotes an expectation with respect to $(\hat \Omega, \hat{\mathcal F}, \hat \mathbb{P})$. Since $a>0$ is arbitrary, the desired result immediately follows if we can prove that for every $a>0$,
\begin{equation} \label{e:dTV.exp.conv}
\lim_{n\to\infty}\hat \mathbb{E} \Big[e^{a d_\mathsf{TV} (\hat \eta_{k,n}^{(1)}, \, \hat \zeta_{k}^{(1)}) } \Big] =1.
\end{equation}
By virtue of \eqref{e:max.coupling.conv}, $d_\mathsf{TV}(\hat \eta_{k,n}^{(1)}, \hat \zeta_k^{(1)})$ converges to $0$ in probability with respect to $\hat \mathbb{P}$. By the Cauchy-Schwarz inequality, \eqref{e:K.and.Pn1}, and the fact that $\zeta_k^{(1)}(E_0)$ is Poisson with mean $e^{-s_0}/(k-1)!$,
\begin{align*}
\hat \mathbb{E} \Big[e^{a d_\mathsf{TV} (\hat \eta_{k,n}^{(1)}, \, \hat \zeta_{k}^{(1)}) } \Big] &\le \bigg\{ \mathbb{E} \Big[ e^{2a \eta_{k,n}^{(1)}(E_0)} \Big] \bigg\}^{1/2} \bigg\{ \mathbb{E}' \Big[ e^{2a \zeta_k^{(1)}(E_0)} \Big] \bigg\}^{1/2} \\
&= \bigg\{ \mathbb{E} \Big[ e^{2a \sum_{X\in \mathcal P_n|_{K_1}} {\mathbbm 1} \big\{ \mathcal P_n(B_{r_n(s_0)}(X))\le k \big\} }\Big] \bigg\}^{1/2} \exp \Big\{ \frac{e^{-s_0}}{2(k-1)!} (e^{2a}-1) \Big\}.
\end{align*}
Now, the desired uniform integrability for \eqref{e:dTV.exp.conv} follows, provided that for every $a>0$,
\begin{equation} \label{e:UI1}
\limsup_{n\to\infty} \mathbb{E} \Big[ e^{a \sum_{X\in \mathcal P_n|_{K_1}} {\mathbbm 1} \big\{ \mathcal P_n(B_{r_n(s_0)}(X))\le k \big\} }\Big] <\infty.
\end{equation}
As in the proof of \cite[Lemma 5.5]{hirsch:owada:2022}, by utilizing the diluted family of cubes
$$
G=\big\{ 4r_n(s_0)z + \big[ 0,r_n(s_0)/\sqrt{d} \big]^d\subset Q_1: z\in \protect{\mathbb Z}^d \big\},
$$
it turns out that \eqref{e:UI1} is obtained as a consequence of
{
$$
\limsup_{n\to\infty} \bigg\{ \mathbb{E} \Big[ e^{a\sum_{X\in\mathcal P_n} {\mathbbm 1} \big\{ \mathcal P_n (B_{r_n(s_0)}(X)) \le k, \, X\in K_1 \cap J \big\} }\Big] \bigg\}^{1/(b_n(4r_n(s_0))^d)} < \infty,
$$
}
where $J=\big[ 0, r_n(s_0)/\sqrt{d} \big]^d$ {and $1/(b_n(4r_n(s_0))^d)$ represents the number of cubes in $G$}. Notice that
\begin{equation} \label{e:up.to.k}
\sum_{X\in\mathcal P_n} {\mathbbm 1} \big\{ \mathcal P_n (B_{r_n(s_0)}(X)) \le k, \, X\in K_1\cap J \big\}\in \{ 0,1,\dots,k \},
\end{equation}
because if there exist more than $k$ points inside $J$, these points never contribute to \eqref{e:up.to.k}. Therefore, by Markov's inequality,
{
\begin{align}
&\bigg\{ \mathbb{E} \Big[ e^{a\sum_{X\in\mathcal P_n} {\mathbbm 1} \big\{ \mathcal P_n (B_{r_n(s_0)}(X)) \le k, \, X\in K_1 \cap J \big\} }\Big] \bigg\}^{1/(b_n(4r_n(s_0))^d)} \label{e:suff.UI1}\\
&\le \bigg( 1 + \sum_{\ell=1}^k e^{a\ell} \mathbb{E} \Big[ \sum_{X\in\mathcal P_n} {\mathbbm 1} \big\{ \mathcal P_n (B_{r_n(s_0)}(X)) \le k, \, X\in K_1 \cap J \big\} \Big] \bigg)^{1/(b_n(4r_n(s_0))^d)}\notag \\
&= \bigg( 1 + \sum_{\ell=1}^k e^{a\ell} n \sum_{i=0}^{k-1} e^{-(a_n+s_0)} \frac{(a_n+s_0)^i}{i!} \mathbb{P} (X_1 \in K_1 \cap J )\bigg)^{1/(b_n(4r_n(s_0))^d)} \notag \\
&\le \big( 1+C^* e^{ak} b_nr_n(s_0)^d \big)^{1/(b_n(4r_n(s_0))^d)} \to e^{C^*e^{ak}/4^d} < \infty, \ \ \text{as } n\to\infty. \notag
\end{align}
}
\end{proof}
\medskip
As shown in the last two propositions, $(\eta_{k,n}/b_n)_{n\ge1}$ satisfies the LDP in Theorem \ref{t:LDP.L.kn}. Thus, our final task is to demonstrate that $(L_{k,n}/b_n)_{n\ge1}$ exhibits the same LDP as $(\eta_{k,n}/b_n)_{n\ge1}$. Although this can be done by
establishing exponential equivalence between $(L_{k,n}/b_n)_{n\ge1}$ and $(\eta_{k,n}/b_n)_{n\ge1}$ in terms of the total variation distance, proving directly this exponential equivalence seems to be difficult. Alternatively, we set up an additional sequence
\begin{equation} \label{e:def.eta.kn.p}
\eta_{k,n}' := \sum_{\ell=1}^{b_n} \sum_{X\in\mathcal P_n|_{K_\ell}}g\big(X, \mathcal P_n|_{Q_\ell}\big) \, {\mathbbm 1} \big\{ R_k(X, \mathcal P_n|_{Q_\ell}) \le r_n(w_n) \big\} \, \delta_{(X, \, f(X, \mathcal P_n|_{Q_\ell}))},
\end{equation}
and prove that $(\eta_{k,n}'/b_n)_{n\ge1}$ shows the same LDP as $(\eta_{k,n}/b_n)_{n\ge1}$ (see Proposition \ref{p:exp.neg.eta.etap}). Subsequently, we define
$$
L_{k,n}' := \begin{cases}
\sum_{X\in\mathcal P_n} g(X, \mathcal P_n)\, {\mathbbm 1} \big\{ R_k(X,\mathcal P_n)\le \sqrt{d}b_n^{-1/d} \big\}\, \delta_{(X, f(X, \mathcal P_n))} & \text{ if } |\mathcal P_n|>k,\\
\emptyset & \text{ if } |\mathcal P_n| \le k.
\end{cases}
$$
and prove that $(L_{k,n}'/b_n)_{n\ge1}$ satisfies the same LDP as $(L_{k,n}/b_n)_{n\ge1}$ (see Proposition \ref{p:exp.neg.L.Lp}). Finally, Proposition \ref{p:exp.neg.etap.Lp} gives exponential equivalence between $(L_{k,n}'/b_n)_{n\ge1}$ and $(\eta_{k,n}'/b_n)_{n\ge1}$. Combining Propositions \ref{p:exp.neg.eta.etap} -- \ref{p:exp.neg.etap.Lp} concludes the required exponential equivalence between $(L_{k,n}/b_n)_{n\ge1}$ and $(\eta_{k,n}/b_n)_{n\ge1}$.
\begin{proposition} \label{p:exp.neg.eta.etap}
The sequence $(\eta_{k,n}'/b_n)_{n\ge1}$ satisfies the same LDP as $(\eta_{k,n}/b_n)_{n\ge1}$.
\end{proposition}
\begin{proof}
For our purpose, we demonstrate that for every $\delta>0$,
\begin{equation} \label{e:diff.eta.kn.and.eta,kn.prime}
\frac{1}{b_n} \log \mathbb{P} \big( d_{\mathsf{TV}}(\eta_{k,n}, \, \eta_{k,n}') \ge \delta b_n\big) \to -\infty, \ \ \ n\to\infty.
\end{equation}
First, we see from \eqref{e:K.and.Pn1} that
\begin{align*}
d_{\mathsf{TV}}(\eta_{k,n}, \, \eta_{k,n}') &\le \sum_{\ell=1}^{b_n} \sum_{X\in \mathcal P_n|_{K_\ell}} {\mathbbm 1} \big\{ R_k(X, \mathcal P_n|_{Q_\ell}) >r_n(w_n) \big\} \\
&= \sum_{\ell=1}^{b_n} \sum_{X\in \mathcal P_n|_{K_\ell}} {\mathbbm 1} \Big\{ \mathcal P_n \big( B_{r_n(w_n)}(X) \big) \le k \Big\},
\end{align*}
from which one can bound \eqref{e:diff.eta.kn.and.eta,kn.prime} by
$$
-a\delta + \log \mathbb{E}\Big[ e^{a\sum_{X\in \mathcal P_n|_{K_1}} {\mathbbm 1} \{ \mathcal P_n( B_{r_n(w_n)}(X) ) \le k \} } \Big]
$$
for every $a>0$.
Now, one has to show that for every $a>0$,
\begin{equation} \label{e:UI2}
\limsup_{n\to\infty} \mathbb{E} \Big[ e^{a\sum_{X\in\mathcal P_n|_{K_1}} {\mathbbm 1} \{ \mathcal P_n( B_{r_n(w_n)}(X) ) \le k \} } \Big] \le 1.
\end{equation}
As in the proof of Proposition \ref{p:eta.zeta.negligible}, \eqref{e:UI2} can be implied by
$$
\limsup_{n\to\infty} \bigg\{ \mathbb{E} \Big[ e^{a\sum_{X\in\mathcal P_n} {\mathbbm 1} \big\{ \mathcal P_n (B_{r_n(w_n)}(X)) \le k, \, X\in K_1 \cap J \big\} }\Big] \bigg\}^{1/(b_nr_n(w_n)^d)} \le 1,
$$
for all $a>0$, where $J=\big[ 0, r_n(w_n)/\sqrt{d} \big]^d$. Now, instead of \eqref{e:suff.UI1}, we have that
\begin{align*}
&\limsup_{n\to\infty} \bigg\{ \mathbb{E} \Big[ e^{a\sum_{X\in\mathcal P_n} {\mathbbm 1} \big\{ \mathcal P_n (B_{r_n(w_n)}(X)) \le k, \, X\in K_1 \cap J \big\} }\Big] \bigg\}^{1/(b_nr_n(w_n)^d)} \\
&\le \limsup_{n\to\infty} \big( 1+C^* e^{ak} b_n r_n(w_n)^d e^{-w_n} \big)^{1/(b_nr_n(w_n)^d)} \\
&=\limsup_{n\to\infty} e^{C^*e^{ak} e^{-w_n}} = 1,
\end{align*}
as required.
\end{proof}
\begin{proposition} \label{p:exp.neg.L.Lp}
The sequence $(L_{k,n}'/b_n)_{n\ge1}$ satisfies the same LDP as $(L_{k,n}/b_n)_{n\ge1}$.
\end{proposition}
\begin{proof}
Throughout the proof, we assume $|\mathcal P_n|>k$. Using the bound
$$
d_{\mathsf{TV}} (L_{k,n},\, L_{k,n}') \le \sum_{X\in\mathcal P_n} {\mathbbm 1} \big\{ R_k(X,\mathcal P_n) > \sqrt{d}b_n^{-1/d} \big\},
$$
it is sufficient to show that, for every $\delta>0$,
$$
\frac{1}{b_n} \log \mathbb{P} \Big( \sum_{X\in\mathcal P_n} {\mathbbm 1} \big\{ R_k(X,\mathcal P_n) >\sqrt{d} b_n^{-1/d} \big\} \ge \delta b_n \Big) \to -\infty, \ \ \ n\to\infty.
$$
Suppose there exists a point $X \in \mathcal P_n \cap Q_\ell$ for some $\ell\in \{ 1,\dots,b_n\}$ so that $R_k(X, \mathcal P_n) > \sqrt{d}b_n^{-1/d}$. Then, $B_{R_k(X,\mathcal P_n)}(X)\cap Q_\ell$ contains at most $k+1$ points of $\mathcal P_n$ (including $X$ itself). Thus,
\begin{align*}
\sum_{X\in\mathcal P_n} {\mathbbm 1} \big\{ R_k(X,\mathcal P_n) >\sqrt db_n^{-1/d} \big\} &= \sum_{\ell=1}^{b_n} \sum_{X \in \mathcal P_n|_{ Q_\ell}} {\mathbbm 1} \big\{ R_k(X,\mathcal P_n) >\sqrt{d}b_n^{-1/d} \big\} \\
&\le (k+1) \sum_{\ell=1}^{b_n} {\mathbbm 1}\big\{ \mathcal P_n (Q_\ell) \le k+1\big\}.
\end{align*}
Now, we only have to demonstrate that
\begin{equation} \label{e:at.most.k+1.smaller.cube}
\frac{1}{b_n} \log \mathbb{P} \Big( \sum_{\ell=1}^{b_n} {\mathbbm 1}\big\{ \mathcal P_n (Q_\ell) \le k+1\big\} \ge \delta b_n\Big) \to -\infty, \ \ \ n\to\infty.
\end{equation}
By Markov's inequality, we have, for every $a>0$,
\begin{align*}
\frac{1}{b_n} \log \mathbb{P} \Big( \sum_{\ell=1}^{b_n} {\mathbbm 1}\big\{ \mathcal P_n (Q_\ell) \le k+1\big\} \ge \delta b_n\Big) &\le -a\delta + \log \mathbb{E} \Big[ e^{a{\mathbbm 1} \{ \mathcal P_n (Q_1) \le k+1 \}} \Big] \\
&\le -a\delta + \log \Big( 1 + e^a \sum_{i=0}^{k+1} e^{-n\mathsf{Leb}(Q_1)}\, \frac{(n\mathsf{Leb}(Q_1))^i}{i!} \Big) \\
&\le -a\delta + \log \Big( 1 + (k+2)e^ae^{-nb_n^{-1}} (nb_n^{-1})^{k+1} \Big) \\
&\to -a\delta, \ \ \text{as } n\to\infty.
\end{align*}
As $a>0$ is arbitrary, we have obtained \eqref{e:at.most.k+1.smaller.cube}.
\end{proof}
\begin{proposition} \label{p:exp.neg.etap.Lp}
The sequence $(L_{k,n}'/b_n)_{n\ge1}$ exhibits the same LDP as $(\eta_{k,n}'/b_n)_{n\ge1}$.
\end{proposition}
\begin{proof}
We prove exponential equivalence between the two sequences: for every $\delta>0$,
$$
\frac{1}{b_n} \log \mathbb{P} \big( d_{\mathsf{TV}}(L_{k,n}', \, \eta_{k,n}') \ge \delta b_n\big) \to -\infty, \ \ \ n\to\infty.
$$
For convenience, let us slightly change the formulation of $\eta_{k,n}'$ given at \eqref{e:def.eta.kn.p}. To begin, observe that if $X\in \mathcal P_n|_{K_\ell}$ with $R_k(X, \mathcal P_n|_{Q_\ell})\le r_n(w_n)$, then $R_k(X, \mathcal P_n|_{Q_\ell}) = R_k(X,\mathcal P_n)$. Hence, one can replace the restricted process $\mathcal P_n|_{Q_\ell}$ in \eqref{e:def.eta.kn.p} with $\mathcal P_n$; that is,
$$
\eta_{k,n}'= \sum_{\ell=1}^{b_n} \sum_{X\in\mathcal P_n|_{K_\ell} }g (X, \mathcal P_n ) \, {\mathbbm 1} \big\{ R_k(X, \mathcal P_n) \le r_n(w_n) \big\} \, \delta_{(X, \, f(X, \mathcal P_n))}.
$$
Using this representation and assuming $|\mathcal P_n|>k$ allows us to express the total variation distance in such a way that
\begin{align}
d_{\mathsf{TV}}(L_{k,n}', \,\eta_{k,n}') &= \frac{1}{b_n} \sup_{A\subset E_0} \Big\{ \sum_{X\in\mathcal P_n} {\mathbbm 1} \big\{ (X, f(X,\mathcal P_n))\in A, \, R_k(X,\mathcal P_n) \le \sqrt{d}b_n^{-1/d} \big\} \label{e:dTV.boundary.bound}\\
&\qquad - \sum_{\ell=1}^{b_n} \sum_{X\in\mathcal P_n|_{K_\ell}} {\mathbbm 1} \big\{ (X, f(X,\mathcal P_n))\in A, \, R_k(X,\mathcal P_n) \le r_n(w_n) \big\} \, \Big\} \notag \\
&=: \frac{1}{b_n} \sup_{A\subset E_0} \big( T_n^{(1)} - T_n^{(2)}\big). \notag
\end{align}
We now derive an upper bound of \eqref{e:dTV.boundary.bound}. We consider a $k$-nearest neighbor ball centered at $X\in \mathcal P_n$ with $(X,f(X,\mathcal P_n))\in A$. Suppose this ball is counted by $T_n^{(1)}$, but not counted by $T_n^{(2)}$. Then, this ball must be of either Type 1 or Type 2 as defined below.
\vspace{7pt}
\noindent \textbf{Type 1}: The center $X$ is located in $Q_\ell$ for some $\ell \in \{ 1,\dots,b_n \}$ such that
$$
r_n(w_n) <R_k(X,\mathcal P_n)\le \sqrt{d} b_n^{-1/d}.
$$
\noindent \textbf{Type 2}: The center $X$ is in $Q_\ell^\partial (r_n(w_n))$ for some $\ell\in \{ 1,\dots,b_n \}$, where
$$
Q_\ell^\partial(r) := \big\{ x\in Q_\ell: \inf_{y\in \partial Q_\ell} \| x-y\| \le r \big\}, \ \ r>0,
$$
so that $r_n(s_0) <R_k(X,\mathcal P_n) \le r_n(w_n)$.
\vspace{7pt}
\noindent Then, the number of $k$-nearest neighbor balls of Type 1 can be bounded by
$$
U_n^{(1)} := \sum_{\ell=1}^{b_n} \sum_{X\in \mathcal P_n|_{Q_\ell}} {\mathbbm 1} \big\{ R_k(X,\mathcal P_n)\in \big( r_n(w_n), \sqrt{d}b_n^{-1/d} \big] \big\},
$$
while the number of those of Type 2 is bounded by
$$
U_n^{(2)} := \sum_{\ell=1}^{b_n} \sum_{X\in \mathcal P_n|_{Q_\ell^\partial (r_n(w_n))}} {\mathbbm 1} \big\{ R_k(X,\mathcal P_n)\in \big( r_n(s_0), r_n(w_n) \big] \big\}.
$$
Hence, the desired result will follow if one can show that
for every $\delta>0$ and $j=1,2$,
\begin{equation} \label{e:Unj.negligibility}
\frac{1}{b_n} \log \mathbb{P} (U_n^{(j)} \ge \delta b_n) \to -\infty, \ \ \ n\to\infty.
\end{equation}
We first deal with the case $j=1$. Let $S_\ell$ be the collection of cubes $(Q_i)_{i=1}^{b_n}$ that intersect with
$$
\text{Tube} (Q_\ell, \sqrt{d}b_n^{-1/d}) := \big\{ x\in [0,1]^d: \inf_{y\in Q_\ell} \mathsf{dist} (x,y) \le \sqrt{d}b_n^{-1/d}\big\}.
$$
Then, the number $D_d$ of such cubes in $S_\ell$ is finite, depending only on $d$.
Now, we can offer the following bound:
$$
U_n^{(1)} \le \sum_{m=1}^{D_d} \sum_{\ell=1}^{\lfloor D_d^{-1}(b_n-m) \rfloor +1} \hspace{-20pt}\sum_{X \in \mathcal P_n} {\mathbbm 1} \{ X\in Q_{(\ell-1)D_d+m} \}\times {\mathbbm 1} \big\{ R_k(X, \mathcal P_n) \in \big( r_n(w_n), \sqrt{d}b_n^{-1/d} \big] \big\}.
$$
By the homogeneity of $\mathcal P_n$ on the torus, we consider only the case $m=D_d$ and obtain from Markov's inequality that, for every $a>0$,
\begin{align}
&\frac{1}{b_n} \log \mathbb{P} \bigg( \sum_{\ell=1}^{\lfloor D_d^{-1}b_n \rfloor } \hspace{-5pt}\sum_{X\in\mathcal P_n|_{ Q_{\ell D_d}}} \hspace{-10pt}{\mathbbm 1} \big\{ R_k(X, \mathcal P_n) \in \big( r_n(w_n), \sqrt{d}b_n^{-1/d} \big] \big\} \ge \delta b_n\bigg) \label{e:Un1.log}\\
&\le -a\delta + \frac{1}{b_n} \log \mathbb{E} \Big[ e^{a \sum_{\ell=1}^{\lfloor D_d^{-1}b_n \rfloor } \sum_{X\in\mathcal P_n|_{Q_{\ell D_d}}}{\mathbbm 1} \big\{ R_k(X, \mathcal P_n)) \in (r_n(w_n), \sqrt{d}b_n^{-1/d}] \big\} } \Big]. \notag
\end{align}
Here, a key observation is that if $X\in \mathcal P_n|_{Q_{\ell D_d}}$ with $R_k(X,\mathcal P_n)\le \sqrt{d}b_n^{-1/d}$, then $R_k(X,\mathcal P_n) = R_k (X, \mathcal P_n|_{S_{\ell D_d}})$, such that $\big( \mathcal P_n|_{S_{\ell D_d}}, \, \ell=1,\dots,\lfloor D_d^{-1}b_n \rfloor \big)$ are i.i.d.~Poisson point processes. Hence, \eqref{e:Un1.log} can be further bounded by
\begin{align*}
&-a\delta + D_d^{-1} \log \mathbb{E}\Big[ e^{a\sum_{X\in \mathcal P_n|_{Q_1}}{\mathbbm 1} \big\{ R_k(X,\mathcal P_n) \in (r_n(w_n), \sqrt{d}b_n^{-1/d}] \big\} } \Big] \\
&\le -a\delta + D_d^{-1} \log \mathbb{E}\Big[ e^{a\sum_{X\in \mathcal P_n|_{Q_1}}{\mathbbm 1} \big\{ \mathcal P_n (B_{r_n(w_n)}(X))\le k \big\} } \Big].
\end{align*}
For the inequality above, we have applied \eqref{e:K.and.Pn1} and dropped the condition $R_k(X, \mathcal P_n)\le \sqrt{d}b_n^{-1/d}$. Now, it remains to show that for every $a>0$,
$$
\limsup_{n\to\infty} \mathbb{E} \Big[ e^{a\sum_{X\in\mathcal P_n|_{Q_1}} {\mathbbm 1} \big\{ \mathcal P_n(B_{r_n(w_n)}(X)) \le k \big\} } \Big] \le 1.
$$
The proof is however a simple repetition of the argument for \eqref{e:UI2}, so we skip it here.
Returning to \eqref{e:Unj.negligibility}, we next work with the case $j=2$. To this aim, we exploit an argument similar to that in \cite[Proposition 5.6]{hirsch:owada:2022}. For $1\le j \le d$, define the collection of ordered $j$-tuples
$$
\mathcal I_j = \big\{ {\bm \ell}=(\ell_1,\dots,\ell_j): 1\le \ell_1 < \dots < \ell_j \le d \big\}.
$$
Given ${\bm \ell}=(\ell_1,\dots,\ell_j)\in \mathcal I_j$, define also the collection of hyper-rectangles by
\begin{align*}
J_n(r) &:= \bigg\{ \Big( b_n^{-1/d}z +\big[0,b_n^{-1/d}\big]^{\ell_1-1}\times [-r,r]\times \big[0,b_n^{-1/d}\big]^{\ell_2-\ell_1-1} \\
&\qquad \times [-r,r] \times \big[0,b_n^{-1/d}\big]^{\ell_3-\ell_2-1} \times \dots \times [-r,r] \\
&\qquad \times \big[0,b_n^{-1/d}\big]^{\ell_j-\ell_{j-1}-1} \times [-r,r] \times \big[0,b_n^{-1/d}\big]^{d-\ell_j} \Big) \cap [0,1]^d: z\in \protect{\mathbb Z}_+^d \bigg\}, \ \ \ r>0.
\end{align*}
By construction, all the rectangles in $J_n(r)$ are contained in $\bigcup_{\ell=1}^{b_n} Q_\ell^\partial (r)$, and the number of rectangles in $J_n(r)$ is $b_n$; hence, we can enumerate these rectangles as
$$
J_n(r)= \big( I_{p,n}^{\bm \ell}(r), \, p=1,\dots,b_n \big).
$$
In this setting, one can bound $U_n^{(2)}$ by
\begin{align*}
&\sum_{j=1}^d \sum_{{\bm \ell} \in \mathcal I_j} \sum_{p=1}^{b_n} \sum_{X\in\mathcal P_n} {\mathbbm 1} \Big\{ X \in I_{p,n}^{\bm \ell} \big( r_n(w_n) \big) \Big\}\, \\
&\qquad \qquad \qquad\qquad \times {\mathbbm 1} \Big\{ R_k(X,\mathcal P_n) > r_n(s_0), \, B_{R_k(X,\mathcal P_n)}(X)\subset I_{p,n}^{\bm \ell} \big( 2r_n(w_n) \big) \Big\}.
\end{align*}
Owing to this bound, we need to prove that for every $j\in \{ 1,\dots,d \}$, ${\bm \ell}\in \mathcal I_j$, and $\delta>0$,
\begin{align}
&\frac{1}{b_n}\log \mathbb{P} \bigg( \sum_{p=1}^{b_n} \sum_{X\in\mathcal P_n} {\mathbbm 1} \Big\{ X \in I_{p,n}^{\bm \ell} \big( r_n(w_n) \big) \Big\} \label{e:Un2.bound}\\
&\qquad \qquad \times {\mathbbm 1} \Big\{ R_k(X,\mathcal P_n) > r_n(s_0), \, B_{R_k(X,\mathcal P_n)}(X)\subset I_{p,n}^{\bm \ell} \big( 2r_n(w_n) \big) \Big\} \ge \delta b_n\bigg) \to -\infty, \ \ \ n\to\infty. \notag
\end{align}
In the above, $B_{R_k(X, \mathcal P_n)}(X)\subset I_{p,n}^{\bm \ell} \big( 2r_n(w_n) \big)$ with $X\in I_{p,n}^{\bm \ell} \big( r_n(w_n) \big)$, implies that $R_k(X,\mathcal P_n) = R_k\big( X,\mathcal P_n|_{I_{p,n}^{\bm \ell} (2r_n(w_n))} \big)$.
Additionally, $\big( I_{p,n}^{\bm \ell} (2r_n(w_n))\big)_{p=1}^{b_n}$ are disjoint sets, so $\big( \mathcal P_n|_{I_{p,n}^{\bm \ell} (2r_n(w_n))} \big)_{p=1}^{b_n}$ becomes a sequence of i.i.d.~Poisson point processes. Hence, by appealing to Markov's inequality as well as \eqref{e:K.and.Pn1}, one can bound \eqref{e:Un2.bound} by
\begin{align*}
&-a\delta +\log \mathbb{E}\Big[ e^{a \sum_{X\in\mathcal P_n} {\mathbbm 1} \big\{ X \in I_{1,n}^{\bm \ell}( r_n(w_n))\big\} \times {\mathbbm 1} \big\{ R_k(X,\mathcal P_n)>r_n(s_0), \, B_{R_k(X,\mathcal P_n)}(X)\subset I_{1,n}^{\bm \ell} (2r_n(w_n)) \big\} } \Big] \\
&\le -a\delta +\log \mathbb{E}\Big[ e^{a \sum_{X\in\mathcal P_n} {\mathbbm 1} \big\{ X \in I_{1,n}^{\bm \ell}( r_n(w_n)) \big\}\times {\mathbbm 1} \big\{ \mathcal P_n(B_{r_n(s_0)}(X)) \le k\big\} } \Big].
\end{align*}
It is now enough to demonstrate that, for every $a>0$,
$$
\limsup_{n\to\infty} \mathbb{E} \Big[ e^{a\sum_{X\in\mathcal P_n}{\mathbbm 1}\big\{ X \in I_{1,n}^{\bm \ell}( r_n(w_n))\big\} \times {\mathbbm 1} \big\{ \mathcal P_n( B_{r_n(s_0)}(X))\le k \big\} } \Big]=1.
$$
Since the required uniform integrability has already been proven by \eqref{e:UI1}, it suffices to show that as $n\to\infty$,
$$
\mathbb{E} \Big[ \sum_{X\in\mathcal P_n} {\mathbbm 1} \Big\{ X \in I_{1,n}^{\bm \ell} \big( r_n(w_n) \big) \Big\}\times {\mathbbm 1} \Big\{ \mathcal P_n \big( B_{r_n(s_0)}(X)\big)\le k \Big\}\Big]\to 0.
$$
Proceeding as before, we obtain that
\begin{align*}
&\mathbb{E} \Big[ \sum_{X\in\mathcal P_n} {\mathbbm 1} \Big\{ X \in I_{1,n}^{\bm \ell} \big( r_n(w_n) \big) \Big\}\times {\mathbbm 1} \Big\{ \mathcal P_n \big( B_{r_n(s_0)}(X)\big)\le k \Big\}\Big] \\
&= n e^{-(a_n+s_0)} \sum_{i=0}^{k-1} \frac{(a_n+s_0)^i}{i!}\, \mathbb{P} \big( X_1\in I_{1,n}^{\bm \ell} (r_n(w_n)) \big) \\
&= n e^{-(a_n+s_0)} \sum_{i=0}^{k-1} \frac{(a_n+s_0)^i}{i!} \times C^*r_n(w_n)^j b_n^{-(d-j)/d} \\
&\le C^*\big( b_nr_n(w_n)^d \big)^{j/d} \to 0, \ \ \ n\to\infty.
\end{align*}
\end{proof}
\begin{proof}[Proof of Corollary \ref{c:LDP.LknB}]
Since the desired LDP has already been shown for the case of a Poisson input, it is sufficient to demonstrate that for every $\varepsilon_0>0$,
\begin{equation} \label{e:diff.Poisson.binomial}
b_n^{-1} \log \mathbb{P} \big( d_{\mathsf{TV}} (L_{k,n}, L_{k,n}^\mathsf{B}) \ge \varepsilon_0 b_n \big) \to -\infty, \ \ \text{as } n\to\infty.
\end{equation}
Our proof is inspired by Corollary 2.3 in \cite{hirsch:owada:2022}. First, define
{
\begin{equation} \label{e:diluted.cubes}
G:= \big\{ 3r_n(w_n)z + [0, r_n(s_0)/\sqrt{d}]^d \subset [0,1]^d: z\in \protect{\mathbb Z}^d \big\},
\end{equation}
}
and consider \emph{finitely many} translates of $G$, denoted $G_1, G_2, \dots, G_M$ for some $M$, such that $[0,1]^d$ can be covered by the union of these translates. In particular, we set $G_1=G$ and denote it specifically as $G=\{ J_1,\dots,J_{b_n'} \}$, where $J_1=[0,r_n(s_0)/\sqrt{d}]^d$ {and $b_n' := \big( 3r_n(w_n) \big)^{-d}$} denotes the number of cubes in $G$. Since $M$ is a finite constant, \eqref{e:diff.Poisson.binomial} follows if one can show that
\begin{align*}
&b_n^{-1}\log \mathbb{P} \Big( \sup_{A\subset E_0} \Big| \, \sum_{X\in\mathcal P_n} g(X, \mathcal P_n)\, {\mathbbm 1} \Big\{ X \in \bigcup_{\ell=1}^{b_n'} J_\ell \Big\} \, \delta_{(X, f(X, \mathcal P_n))}(A) \\
&\qquad \qquad \qquad - \sum_{X\in \mathcal B_n} g(X, \mathcal B_n)\, {\mathbbm 1} \Big\{ X \in \bigcup_{\ell=1}^{b_n'} J_\ell \Big\} \, \delta_{(X, f(X, \mathcal B_n))}(A)\, \Big| \ge \varepsilon_0b_n \Big) \to -\infty.
\end{align*}
We say that $J_i$ is \emph{$n$-bad} if one of the following events happens.
\vspace{7pt}
\noindent $(i)$ There exists $X\in \mathcal P_n \cap J_i$ such that $g(X, \mathcal P_n)=1$ (equivalently, $\mathcal P_n \big( B_{r_n(s_0)}(X) \big) \le k$; see \eqref{e:K.and.Pn1}) and $X\notin \mathcal B_n$. \\
$(ii)$ There exists $X\in \mathcal B_n \cap J_i$ such that $g(X, \mathcal B_n)=1$ and $X\notin \mathcal P_n$. \\
$(iii)$ There exist $X\in \mathcal P_n \cap \mathcal B_n \cap J_i$ and $u\ge s_0$ such that $\min\big\{ \mathcal P_n \big( B_{r_n(u)}(X) \big), \mathcal B_n \big( B_{r_n(u)}(X) \big)\big\}\le k$ and
$\max\big\{\mathcal P_n \big( B_{r_n(u)}(X) \big), \mathcal B_n \big( B_{r_n(u)}(X) \big)\big\}> k$.
\vspace{7pt}
\noindent The key observation here is that
\begin{align*}
&\sup_{A\subset E_0} \Big| \, \sum_{X\in\mathcal P_n} g(X, \mathcal P_n)\, {\mathbbm 1} \Big\{ X \in \bigcup_{\ell=1}^{b_n'} J_\ell \Big\} \, \delta_{(X, f(X, \mathcal P_n))}(A) \\
&\qquad - \sum_{X\in \mathcal B_n} g(X, \mathcal B_n)\, {\mathbbm 1} \Big\{ X \in \bigcup_{\ell=1}^{b_n'} J_\ell \Big\} \, \delta_{(X, f(X, \mathcal B_n))}(A)\, \Big| \le (k+1)\sum_{i=1}^{b_n'} {\mathbbm 1} \{ J_i \text{ is } n\text{-bad} \}.
\end{align*}
Thus, it is enough to show that for every $\varepsilon_0>0$,
$$
\frac{1}{b_n} \log \mathbb{P} \Big( \sum_{i=1}^{b_n'} {\mathbbm 1} \{ J_i \text{ is } n\text{-bad} \} \ge \varepsilon_0b_n\Big) \to -\infty.
$$
For $\eta\in (0,1]$, let $\mathcal P_n^{(\eta)}$ be a homogeneous Poisson point process on $[0,1]^d$ with intensity $n\eta$. We take $\mathcal P_n^{(\eta)}$ to be independent of $\mathcal P_n$. Then, $\mathcal P_n^{(\eta, \mathsf{a})}:= \mathcal P_n \cup \mathcal P_n^{(\eta)}$ represents the \emph{augmented} Poisson point process with intensity $n(1+\eta)$. Moreover, let $\mathcal P_n^{(\eta, \mathsf{t})}$ denote a \emph{thinned} version of $\mathcal P_n$ obtained by removing each point of $\mathcal P_n$ with probability $\eta$.
If we denote by $\mathcal D_\eta(\mathcal P_n)$ a collection of deleted points of $\mathcal P_n$, one can write $\mathcal P_n^{(\eta, \mathsf{t})}=\mathcal P_n \setminus \mathcal D_\eta(\mathcal P_n)$.
Notice that $\mathcal P_n^{(\eta, \mathsf{a})}\stackrel{d}{=}\mathcal P_{n(1+\eta)}$ and $\mathcal P_n^{(\eta, \mathsf{t})}\stackrel{d}{=}\mathcal P_{n(1-\eta)}$. Subsequently, for $\varepsilon>0$ let
\begin{equation} \label{e:def.Fnvep}
F_{n,\varepsilon}= \{ \mathcal P_n^{(\varepsilon a_n^{-1}, \mathsf{t})} \subset \mathcal B_n \subset \mathcal P_n^{(\varepsilon a_n^{-1}, \mathsf{a})} \}
\end{equation}
and claim that
\begin{equation} \label{e:Fn.vep.negligible}
b_n^{-1}\log \mathbb{P}(F_{n,\varepsilon}^c)\to-\infty, \ \ \ n\to\infty.
\end{equation}
For the proof we use Lemma 1.2 in \cite{penrose:2003} to get that
\begin{equation} \label{e:lemma1.2.bound.Fnvep}
\mathbb{P}(F_{n,\varepsilon}^c) \le e^{-n(1+\varepsilon a_n^{-1})H( (1+\varepsilon a_n^{-1})^{-1} )} + e^{-n(1-\varepsilon a_n^{-1})H( (1-\varepsilon a_n^{-1})^{-1} )},
\end{equation}
where $H(x)=x\log x +1-x$, $x>0$. Applying the Taylor expansion to $H(\cdot)$, we have
$$
\limsup_{n\to\infty}\frac{1}{b_n} \log \mathbb{P} (F_{n,\varepsilon}^c) \le -\lim_{n\to\infty} \frac{e^{a_n}}{2a_n^{k-1}} \Big( \frac{\varepsilon a_n^{-1}}{1-\varepsilon a_n^{-1}} \Big)^2=-\infty.
$$
Suppose now that $J_i$ is an $n$-bad cube and $F_{n,\varepsilon}$ holds, {such that} one of the events in case $(i)$--$(iii)$ above occurs. Then, there exists $X\in \mathcal P_n^{(\varepsilon a_n^{-1}, \mathsf{a})}\cap J_i$, such that
\begin{equation} \label{e:necessary.cond.n-bad}
\mathcal P_n^{(\varepsilon a_n^{-1}, \mathsf{t})} \big( B_{r_n(s_0)}(X) \big) \le k, \ \ \text{ and } \ \ \big( \mathcal P_n^{(\varepsilon a_n^{-1}, \mathsf{a})} \setminus \mathcal P_n^{(\varepsilon a_n^{-1}, \mathsf{t})} \big) \big( B_{r_n(w_n)}(X) \big) \ge1.
\end{equation}
Since we work with the diluted cubes in \eqref{e:diluted.cubes}, it follows from \eqref{e:necessary.cond.n-bad} and the spatial independence of Poisson processes that
$\sum_{i=1}^{b_n'} {\mathbbm 1} \{ J_i \text{ is } n\text{-bad} \}$ is a binomial random variable. Below, we shall estimate its success probability $p_{n,\varepsilon}$ as follows:
\begin{align}
\begin{split} \label{e:estimate.pnvep}
p_{n,\varepsilon} &= \mathbb{P} \bigg( \bigcup_{X\in \mathcal P_n^{(\varepsilon a_n^{-1}, \mathsf{a})} \cap J_1} \hspace{-10pt}\Big\{ \mathcal P_n^{(\varepsilon a_n^{-1}, \mathsf{t})} \big( B_{r_n(s_0)}(X) \big) \le k, \\
&\qquad \qquad \qquad\qquad \qquad \qquad \big( \mathcal P_n^{(\varepsilon a_n^{-1}, \mathsf{a})} \setminus \mathcal P_n^{(\varepsilon a_n^{-1}, \mathsf{t})} \big) \big( B_{r_n(w_n)}(X) \big) \ge1\Big\} \bigg) \\
&\le \mathbb{E} \bigg[ \sum_{X\in \mathcal P_n \cap J_1} {\mathbbm 1} \Big\{ \big(\mathcal P_n \setminus \mathcal D_{\varepsilon a_n^{-1}}(\mathcal P_n) \big) \big( B_{r_n(s_0)}(X) \big) \le k, \\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \big( \mathcal P_n^{(\varepsilon a_n^{-1})} \cup \mathcal D_{\varepsilon a_n^{-1}}(\mathcal P_n)\big) \big( B_{r_n(w_n)}(X) \big) \ge1 \Big\} \bigg] \\
&\quad + \mathbb{E} \bigg[ \sum_{X\in \mathcal P_n^{(\varepsilon a_n^{-1})} \cap J_1} {\mathbbm 1} \Big\{ \big(\mathcal P_n \setminus \mathcal D_{\varepsilon a_n^{-1}}(\mathcal P_n) \big) \big( B_{r_n(s_0)}(X) \big) \le k \Big\} \bigg] \\
&=:A_n+B_n.
\end{split}
\end{align}
By the independence of $\mathcal P_n^{(\varepsilon a_n^{-1})}$ and $\mathcal P_n$,
\begin{align}
\begin{split} \label{e:calc.Bn}
B_n &= n\varepsilon a_n^{-1} \mathbb{P} \Big( Y\in J_1, \, \big( \mathcal P_n\setminus \mathcal D_{\varepsilon a_n^{-1}}(\mathcal P_n) \big) \big( B_{r_n(s_0)}(Y) \big) \le k\Big) \\
&= n\varepsilon a_n^{-1} \mathsf{Leb} (J_1) \mathbb{P} \Big( \mathcal P_{n(1-\varepsilon a_n^{-1})} \big( B_{r_n(s_0)}(Y) \big)\le k \Big) \\
&= n\varepsilon a_n^{-1} \mathsf{Leb} (J_1) \sum_{i=0}^k \frac{\big( a_n(1-\varepsilon a_n^{-1})(1+s_0a_n^{-1}) \big)^i}{i!}\, e^{-a_n(1-\varepsilon a_n^{-1})(1+s_0a_n^{-1}) }.
\end{split}
\end{align}
In the above $Y$ is a uniform random variable on $[0,1]^d$, independent of $\mathcal P_n$. Because of
\begin{equation} \label{e:s_0.cond}
a_n(1-\varepsilon a_n^{-1})(1+s_0a_n^{-1}) \ge a_n -C^*,
\end{equation}
we get that
$$
B_n \le C^* n\varepsilon a_n^{-1} \mathsf{Leb}(J_1) a_n^k e^{-a_n} = C^* \varepsilon \mathsf{Leb}(J_1) b_n.
$$
Applying the Mecke formula for Poisson point processes,
\begin{align*}
A_n &= n \mathbb{P} \Big( Y\in J_1, \, \big\{ (\mathcal P_n+\delta_Y) \setminus \mathcal D_{\varepsilon a_n^{-1}} (\mathcal P_n +\delta_Y) \big\} \big( B_{r_n(s_0)}(Y) \big) \le k, \\
&\qquad\qquad \qquad\qquad \big\{ \mathcal P_n^{(\varepsilon a_n^{-1} ) } \cup \mathcal D_{\varepsilon a_n^{-1}} (\mathcal P_n +\delta_Y) \big\} \big( B_{r_n(w_n)}(Y) \big) \ge 1 \Big).
\end{align*}
Denote $T(Y)=\{ Y \text{ is deleted by thinning}\}$. Note that if $T(Y)$ holds, then $\mathcal D_{\varepsilon a_n^{-1}}(\mathcal P_n +\delta_Y)=\mathcal D_{\varepsilon a_n^{-1}}(\mathcal P_n)+\delta_Y$ and if $T(Y)$ does not hold, we have $\mathcal D_{\varepsilon a_n^{-1}}(\mathcal P_n +\delta_Y) = \mathcal D_{\varepsilon a_n^{-1}}(\mathcal P_n)$. Hence,
\begin{align*}
A_n &\le n \mathbb{P} \Big( \Big\{ Y\in J_1, \, \big( \mathcal P_n \setminus \mathcal D_{\varepsilon a_n^{-1}}(\mathcal P_n) \big)\big( B_{r_n(s_0)} (Y)\big) \le k \Big\} \cap T(Y) \Big) \\
&\quad + n\mathbb{P} \Big( Y\in J_1, \, \big( \mathcal P_n \setminus \mathcal D_{\varepsilon a_n^{-1}}(\mathcal P_n) \big)\big( B_{r_n(s_0)} (Y)\big) \le k-1, \\
&\qquad \qquad \qquad \qquad \big( \mathcal P_n^{(\varepsilon a_n^{-1})} \cup \mathcal D_{\varepsilon a_n^{-1}} (\mathcal P_n)\big)\big( B_{r_n(w_n)} (Y)\big) \ge 1\Big) =: C_n + D_n.
\end{align*}
Repeating the same calculation as in \eqref{e:calc.Bn} and using \eqref{e:s_0.cond},
$$
C_n \le C^* na_n^k e^{-a_n} \mathbb{P} \big( \{ Y\in J_1\}\cap T(Y) \big) = C^* \varepsilon \mathsf{Leb}(J_1)b_n.
$$
Since $\mathcal P_n \setminus \mathcal D_{\varepsilon a_n^{-1}}(\mathcal P_n)$ and $\mathcal P_n^{(\varepsilon a_n^{-1})}\cup \mathcal D_{\varepsilon a_n^{-1}}(\mathcal P_n)$ are independent (see, e.g., Corollary 5.9 in \cite{last:penrose:2017}),
\begin{align*}
D_n &= n \mathsf{Leb}(J_1) \mathbb{P} \Big( \big( \mathcal P_n \setminus \mathcal D_{\varepsilon a_n^{-1}}(\mathcal P_n) \big) \big( B_{r_n(s_0)}(Y) \big) \le k-1 \Big) \\
&\qquad \qquad \times \mathbb{P}\Big(\big( \mathcal P_n^{(\varepsilon a_n^{-1})}\cup \mathcal D_{\varepsilon a_n^{-1}}(\mathcal P_n) \big) \big( B_{r_n(w_n)}(Y) \big) \ge 1 \Big) \\
&= n\mathsf{Leb}(J_1) \mathbb{P} \Big( \mathcal P_{n(1-\varepsilon a_n^{-1})} \big( B_{r_n(s_0)}(Y) \big) \le k-1 \Big) \mathbb{P}\Big(\mathcal P_{2n\varepsilon a_n^{-1}} \big( B_{r_n(w_n)}(Y) \big) \ge 1 \Big)
\end{align*}
By \eqref{e:s_0.cond} and Markov's inequality,
\begin{align*}
D_n &\le C^* n \mathsf{Leb}(J_1) \, a_n^{k-1} e^{-a_n} \mathbb{E} \Big[ \mathcal P_{2n\varepsilon a_n^{-1}} \big( B_{r_n(w_n)}(Y) \big) \Big] \\
&= C^* n \mathsf{Leb}(J_1) \, a_n^{k-1} e^{-a_n} 2 \varepsilon (1+w_na_n^{-1}) \le C^* \varepsilon \mathsf{Leb}(J_1) b_n.
\end{align*}
Combining all these calculations concludes that $p_{n, \varepsilon }\le C^* \varepsilon \mathsf{Leb}(J_1) b_n$. If one takes sufficiently small $\varepsilon\in (0,\varepsilon_0)$, we have that $b_n' p_{n,\varepsilon} \le C^* b_n' \varepsilon \mathsf{Leb}(J_1)b_n = C^* \varepsilon b_n \le \varepsilon_0 b_n$ for large $n$ enough. Therefore, one can exploit the binomial concentration inequality (see, e.g., Lemma 1.1 in \cite{penrose:2003}) to obtain that
\begin{align*}
\limsup_{n\to\infty} b_n^{-1} \log \mathbb{P} \Big( \text{Bin} (b_n', C^* \varepsilon \mathsf{Leb}(J_1)b_n) \ge \varepsilon_0 b_n \Big) &\le -\frac{\varepsilon_0}{2}\, \lim_{n\to\infty} \log \Big\{ \frac{\varepsilon_0}{C^* b_n' \varepsilon \mathsf{Leb}(J_1)} \Big\} \\
&= {-\frac{\varepsilon_0}{2}\, \log \Big\{ \frac{(3\sqrt{d})^d\varepsilon_0}{C^* \varepsilon} \Big\}.}
\end{align*}
The last term goes to $-\infty$ as $\varepsilon \to 0$. Combining this result with \eqref{e:Fn.vep.negligible} concludes the proof.
\end{proof}
\begin{proof}[Proof of Corollary \ref{c:LDP.application}]
Define the map $S: M_+(E_0) \to [0,\infty)$ by $S(\rho)=\rho(E_0)$. Since $S$ is continuous in the weak topology {and $H_k(\cdot\,|\, \mathsf{Leb}\otimes \tau_k)=\Lambda_k^*$ is a good rate function}, the contraction principle (see, e.g., \cite[Theorem 4.2.1]{dembo:zeitouni:1998}) is applied to the LDP in Theorem \ref{t:LDP.L.kn}. In conclusion,
$$
S\Big(\frac{L_{k,n}}{b_n}\Big) = \frac{1}{b_n}\sum_{X\in \mathcal P_n}g(X,\mathcal P_n) = \frac{T_{k,n}}{b_n}, \ \ n\ge1,
$$
shows an LDP with rate $b_n$ and rate function
\begin{equation} \label{e:rate.func.contraction.principle}
\inf_{\nu\in M_+(E_0), \, \nu(E_0)=x} H_k(\nu | \mathsf{Leb}\otimes \tau_k), \ \ \ x\in \reals,
\end{equation}
where $H_k$ is the relative entropy defined at \eqref{e:def.relative.entropy}. The rest of the argument must be devoted to verifying that \eqref{e:rate.func.contraction.principle} coincides with $I_k(x)$ for every $x\in \reals$; this is however an immediate result as an analogue of Equ.~(5.35) of \cite{hirsch:owada:2022}.
The LDP for $(T_{k,n}^\mathsf{B}/b_n)_{n\ge1}$ is obtained by applying the contraction principle to Corollary \ref{c:LDP.LknB}.
\end{proof}
\medskip
\subsection{Proofs of Theorem \ref{t:M0.L.kn}, Corollary \ref{c:M0.LknB}, and Corollary \ref{c:M0.application}} \label{sec:proof.M0}
\begin{proof}[Proof of Theorem \ref{t:M0.L.kn}]
Let $C_K^+(E)$ denote a collection of continuous and non-negative functions on $E$ with compact support.
Given two such functions $U_\ell \in C_K^+(E)$, $\ell=1,2$, together with $\varepsilon_1, \varepsilon_2 > 0$, we define $F_{U_1, U_2, \varepsilon_1, \varepsilon_2}: M_p(E) \to [0,1]$ by
\begin{equation} \label{e:def.F}
F_{U_1, U_2, \varepsilon_1, \varepsilon_2} (\eta) = \Big( 1-e^{-(\eta(U_1) - \varepsilon_1)_+} \Big)\Big( 1-e^{-(\eta(U_2) - \varepsilon_2)_+} \Big),
\end{equation}
where $\eta(U_\ell) = \int_E U_\ell(x,u) \eta (\dif x, \dif u)$, {and $(a)_+=a$ if $a\ge0$ and $0$ otherwise}.
Notice that $F_{U_1, U_2, \varepsilon_1, \varepsilon_2}\in \mathcal C_0$.
In what follows, we fix $U_1, U_2$ and $\varepsilon_1, \varepsilon_2$, and simply write $F=F_{U_1, U_2, \varepsilon_1, \varepsilon_2}$. Define
$\xi_{k,n} (\cdot):= b_n^{-1}\mathbb{P} ( L_{k,n} \in \cdot )$.
Then, according to Theorem A.2 in \cite{hult:samorodnitsky:2010}, \eqref{e:M0.convergence} follows if one can show that
$$
\xi_{k,n} (F) \to \xi_k (F), \ \ \text{as } n\to\infty.
$$
First, note that
$$
\xi_{k,n}(F) = \int_{M_p(E)} F(\eta) \xi_{k,n}(\dif \eta) = b_n^{-1} \mathbb{E} \big[ F(L_{k,n}) \big].
$$
Let $\zeta_{k,n}$ denote a Poisson point process on $E$ with mean measure
\begin{equation} \label{e:mean.meas.zeta.kn.original}
\frac{b_n}{(k-1)!}\, e^{-u} \dif x \dif u, \ \ x\in [0,1]^d, \ u\in \reals.
\end{equation}
In this setting, our proof breaks down into two parts:
\begin{align}
&b_n^{-1} \big|\, \mathbb{E}\big[ F(L_{k,n}) \big] - \mathbb{E} \big[ F(\zeta_{k,n}) \big]\, \big| \to 0, \ \ \ n\to\infty, \label{e:1st.M0} \\
&b_n^{-1} \mathbb{E} \big[ F(\zeta_{k,n}) \big] \to \xi_k(F), \ \ \ n\to\infty. \label{e:2nd.M0}
\end{align}
\textit{Proof of \eqref{e:1st.M0}}: Since $U_\ell$ has compact support on $E$, there exists $s_0\in \reals$, so that
$$
\text{supp} (U_1)\bigcup \text{supp} (U_2)\subset [0,1]^d \times (s_0,\infty],
$$
where $\text{supp}(U_\ell)$ represents the support of $U_\ell$.
Hence, we may assume, without loss of generality, that $L_{k,n}$ and $\zeta_{k,n}$ are both random elements of the \emph{restricted} state space $M_p\big( [0,1]^d \times (s_0, \infty] \big)$. Equivalently, one can reformulate $L_{k,n}$ by
\begin{equation} \label{e:Lkn.reformulate.M0}
L_{k,n} = \begin{cases}
\sum_{X\in\mathcal P_n} g(X,\mathcal P_n)\, \delta_{(X, f(X,\mathcal P_n))} & \text{ if } |\mathcal P_n| >k,\\
\emptyset & \text{ if } |\mathcal P_n| \le k,
\end{cases}
\end{equation}
in the same way as \eqref{e:def.Lkn2}. Similarly, $\zeta_{k,n}$ can be defined as the Poisson point process whose mean measure is given by the restricted version of \eqref{e:mean.meas.zeta.kn.original}; that is,
$$
(\mathsf{Leb}\otimes \tau_{k,n}) (\dif x, \dif u) :=\frac{b_n}{(k-1)!}\, e^{-u} {\mathbbm 1} \{ u\ge s_0 \}\dif x \dif u, \ \ x\in [0,1]^d, \ u \in \reals.
$$
Next, it is not hard to prove that $F$ in \eqref{e:def.F} is a $1$-Lipschitz function with respect to the total variation distance on the space of point measures. Namely, for $\eta_1, \eta_2 \in M_p(E)$,
$$
\big| F(\eta_1)-F(\eta_2) \big| \le 2 d_{\mathsf{TV}}(\eta_1,\eta_2).
$$
Thus, by \eqref{e:def.KR.dist},
$$
\big|\, \mathbb{E}\big[ F(L_{k,n}) \big] - \mathbb{E} \big[ F(\zeta_{k,n}) \big]\, \big| \le d_{\mathsf{KR}} \big( \mathcal L(L_{k,n}), \mathcal L(\zeta_{k,n}) \big).
$$
\begin{proposition} \label{p:KR.conv2}
We have, as $n\to\infty$,
$$
b_n^{-1} d_{\mathsf{KR}} \big( \mathcal L(L_{k,n}), \mathcal L(\zeta_{k,n}) \big)\to 0.
$$
\end{proposition}
\begin{proof}[Proof of Proposition \ref{p:KR.conv2}]
The proof is analogous to that of Proposition \ref{p:KR.conv1}. Precisely, we first need to show that
$$
b_n^{-1} d_{\mathsf{TV}} \big( \mathbb{E} [L_{k,n}(\cdot)], \mathsf{Leb} \otimes \tau_{k,n} \big) \to 0, \ \ \ n\to\infty,
$$
and verify also that $b_n^{-1}E_i \to 0$, $n\to\infty$, for $i=1,2,3$, where $E_i$'s are defined analogously to \eqref{e:E1}, \eqref{e:E2}, and \eqref{e:E3}. More concretely, they are respectively defined as
$$
E_1 := 2n\int_{[0,1]^d} \mathbb{E} \big[ g(x,\mathcal P_n+\delta_x) \, {\mathbbm 1} \big\{ \mathcal S (x,\mathcal P_n+\delta_x) \not\subset S_x \big\} \big]\dif x,
$$
$$
E_2 := 2n^2 \int_{[0,1]^d}\int_{[0,1]^d} \hspace{-.2cm}{\mathbbm 1} \{ S_x \cap S_z \neq \emptyset \} \,\mathbb{E} \big[ g(x,\mathcal P_n+\delta_x) \big] \mathbb{E} \big[ g(z,\mathcal P_n+\delta_z) \big] \dif x \dif z,
$$
and
$$
E_3 := 2n^2 \int_{[0,1]^d}\int_{[0,1]^d} \hspace{-.2cm}{\mathbbm 1} \{ S_x \cap S_z \neq \emptyset \} \,\mathbb{E} \big[ g(x,\mathcal P_n+\delta_x+\delta_z) \, g(z,\mathcal P_n+\delta_x +\delta_z) \big] \dif x \dif z,
$$
for which $\mathcal S(x,\omega)=B_{R_k(x,\omega)}(x)$ for $x\in [0,1]^d$ and $\omega\in M_p\big( [0,1]^d \big)$, and $S_x = B_{r_n(w_n)}(x)$ for some sequence $w_n\to\infty$ with $w_n=o(a_n)$, $n\to\infty$.
First, for $B\subset [0,1]^d$ and $u>s_0$, by the Mecke formula for Poisson point processes and \eqref{e:K.and.Pn1},
$$
\mathbb{E}\big[ L_{k,n}(B\times (u,\infty)) \big] = n\mathbb{P}\Big(Y\in B, \, (\mathcal P_n +\delta_Y ) \big( B_{r_n(u)}(Y) \big) \le k \Big),
$$
where $Y$ is a uniform random variable on $[0,1]^d$, independent of $\mathcal P_n$. By the conditioning on $Y$,
\begin{align*}
\mathbb{E}\big[ L_{k,n}(B\times (u,\infty)) \big] &= n\, \mathsf{Leb}(B)\sum_{i=0}^{k-1} e^{-(a_n+u)} \frac{(a_n+u)^i}{i!}.
\end{align*}
This means that $\mathbb{E}\big[ L_{k,n}(\cdot) \big]$ has the density
$$
n \, \frac{e^{-(a_n+u)}(a_n+u)^{k-1}}{(k-1)!}, \ \ x \in [0,1]^d, \ u >s_0,
$$
and hence, it follows from the dominated convergence theorem that
\begin{align*}
&b_n^{-1} d_{\mathsf{TV}} \big( \mathbb{E} [L_{k,n}(\cdot)], \, \mathsf{Leb}\otimes \tau_{k,n} \big) \\
&\le b_n^{-1} \int_{[0,1]^d \times (s_0,\infty)} \Big| \, ne^{-(a_n+u)}\frac{(a_n+u)^{k-1}}{(k-1)!} - b_n \frac{e^{-u}}{(k-1)!} \, \Big|\dif x \dif u \\
&= \frac{1}{(k-1)!}\, \int_{s_0}^\infty \Big| \, \Big( 1+\frac{u}{a_n} \Big)^{k-1}-1 \, \Big| e^{-u}\dif u \to0, \ \ \ n\to\infty.
\end{align*}
Subsequently,
\begin{align*}
b_n^{-1}E_1 &\le 2nb_n^{-1} \int_{[0,1]^d} \mathbb{P} \Big( \mathcal P_n \big( B_{r_n(w_n)}(x) \big) \le k-1 \Big)\dif x\\
&=2nb_n^{-1} \sum_{i=0}^{k-1}e^{-(a_n+w_n)} \frac{(a_n+w_n)^i}{i!} \le C^* e^{-w_n} \to 0, \ \ \ n\to\infty,
\end{align*}
while we also have
$$
b_n^{-1} E_2 \le 2n^2 b_n^{-1} \Big\{ \sum_{i=0}^{k-1} e^{-(a_n+s_0)} \frac{(a_n+s_0)^i}{i!} \Big\}^2 \le C^*b_n\to 0, \ \ \ n\to\infty.
$$
Similarly to \eqref{e:E3.split}, $E_3$ can be split into two additional terms:
\begin{align*}
E_3 &\le 2n^2 \int_{[0,1]^d}\int_{[0,1]^d}{\mathbbm 1} \big\{ \| x-z\| \le r_n(s_0) \big\} \\
&\qquad \qquad \qquad \times \mathbb{P} \Big( (\mathcal P_n+\delta_z) \big( B_{r_n(s_0)}(x) \big) \le k -1, \, (\mathcal P_n+\delta_x) \big( B_{r_n(s_0)}(z) \big) \le k -1\Big) \dif x \dif z \\
&\quad + 2n^2 \int_{[0,1]^d}\int_{[0,1]^d}{\mathbbm 1} \big\{r_n(s_0) < \| x-z\| \le 2r_n(w_n) \big\} \\
&\qquad \qquad \qquad \times \mathbb{P} \Big( (\mathcal P_n+\delta_z) \big( B_{r_n(s_0)}(x) \big) \le k -1, \, (\mathcal P_n +\delta_x) \big( B_{r_n(s_0)}(z) \big) \le k -1\Big) \dif x \dif z \\
&=: E_{3,1}+E_{3,2}.
\end{align*}
Although we shall skip detailed discussions, one can still demonstrate that $b_n^{-1}E_{3,1}\to0$ and $b_n^{-1}E_{3,2}\to0$, by the arguments nearly identical to those for Proposition \ref{p:KR.conv1}.
\end{proof}
Now, the proof of Proposition \ref{p:KR.conv2} has been completed, which in turn concludes \eqref{e:1st.M0}. Our next goal is to prove \eqref{e:2nd.M0}.
\textit{Proof of \eqref{e:2nd.M0}}: Note that $\zeta_{k,n}$ can be written as
$$
\zeta_{k,n} = \sum_{i=1}^{N_n} \delta_{(T_i, Z_i)},
$$
where $(T_i, Z_i)$ are i.i.d.~random variables on $E$ with density given by $e^{-(u-s_0)} {\mathbbm 1} \{ u\ge s_0 \} \dif x\dif u$, and $N_n$ is Poisson distributed with mean $b_ne^{-s_0}/(k-1)!$. Furthermore, $(T_i, Z_i)$ and $N_n$ are taken to be independent. Substituting this representation,
\begin{align*}
b_n^{-1}\mathbb{E} \big[ F(\zeta_{k,n}) \big] &= b_n^{-1} \mathbb{E} \Big[ \prod_{\ell=1}^2 \Big( 1-e^{-\big( \sum_{i=1}^{N_n} U_\ell (T_i, Z_i) -\varepsilon_\ell\big)_+} \Big) \Big] \\
&=b_n^{-1} \mathbb{E} \Big[ \prod_{\ell=1}^2 \Big( 1-e^{-\big( U_\ell (T_1, Z_1) -\varepsilon_\ell\big)_+} \Big) \, {\mathbbm 1} \{ N_n=1 \}\Big] \\
&\qquad \qquad + b_n^{-1} \mathbb{E} \Big[ \prod_{\ell=1}^2 \Big( 1-e^{-\big( \sum_{i=1}^{N_n} U_\ell (T_i, Z_i) -\varepsilon_\ell\big)_+} \Big) {\mathbbm 1} \{ N_n\ge2 \}\Big]\\
&=: A_n+B_n.
\end{align*}
Of the last two terms, one can immediately show that
$$
B_n \le b_n^{-1} \mathbb{P}(N_n\ge 2) \le \Big( \frac{e^{-s_0}}{(k-1)!} \Big)^2 b_n\to 0, \ \ \text{as } n\to\infty.
$$
By the independence of $(T_1, Z_1)$ and $N_n$, we have as $n\to\infty$,
\begin{align*}
A_n &= b_n^{-1} \mathbb{E} \Big[ \prod_{\ell=1}^2 \Big( 1-e^{-\big( U_\ell (T_1, Z_1) -\varepsilon_\ell\big)_+} \Big) \Big] \mathbb{P}(N_n=1) \\
&= \frac{e^{-s_0}}{(k-1)!}\, e^{-e^{-s_0}b_n/(k-1)!} \int_E \prod_{\ell=1}^2 \Big( 1-e^{-\big( U_\ell(x,u) -\varepsilon_\ell\big)_+} \Big) e^{-(u-s_0)} {\mathbbm 1} \{ u\ge s_0 \} \dif x \dif u \\
&\to \frac{1}{(k-1)!} \int_E \prod_{\ell=1}^2\Big( 1-e^{-\big( U_\ell(x,u) -\varepsilon_\ell\big)_+} \Big) e^{-u} \dif x \dif u = \xi_k(F).
\end{align*}
We thus conclude that $A_n+B_n \to \xi_k(F)$, $n\to\infty$, as required.
\end{proof}
\begin{proof}[Proof of Corollary \ref{c:M0.LknB}]
Because of \eqref{e:1st.M0} and \eqref{e:2nd.M0}, it is sufficient to show that
$$
b_n^{-1} \mathbb{E} \Big[ \, \big| F(L_{k,n})-F(L_{k,n}^\mathsf{B}) \big|\, \Big] \to 0, \ \ \ n\to\infty,
$$
where $F$ is defined at \eqref{e:def.F}.
Under the map $F$, one can represent $L_{k,n}$ as in \eqref{e:Lkn.reformulate.M0}. Clearly, $L_{k,n}^\mathsf{B}$ has the same representation as an element of $M_p\big( [0,1]^d\times (s_0,\infty] \big)$.
Since $F$ is bounded,
$$
b_n^{-1} \mathbb{E} \Big[ \, \big| F(L_{k,n})-F(L_{k,n}^\mathsf{B}) \big|\, \Big] \le 2b_n^{-1} \mathbb{P}( L_{k,n}\neq L_{k,n}^\mathsf{B}).
$$
We now claim that $b_n^{-1}\mathbb{P}( L_{k,n}\neq L_{k,n}^\mathsf{B} )\to 0$ as $n\to\infty$. The proof is analogous to that of \eqref{e:diff.Poisson.binomial} by borrowing the idea of $n$-bad cubes. Specifically, we say that $[0,1]^d$ is \emph{$n$-bad} if one of the following events occurs.
\vspace{7pt}
\noindent $(i)$ There exists $X\in \mathcal P_n$ such that $g(X, \mathcal P_n)=1$ and $X\notin \mathcal B_n$. \\
$(ii)$ There exists $X\in \mathcal B_n$ such that $g(X, \mathcal B_n)=1$ and $X\notin \mathcal P_n$. \\
$(iii)$ There exist $X\in \mathcal P_n \cap \mathcal B_n$ and $u\ge s_0$ such that $\min\big\{ \mathcal P_n \big( B_{r_n(u)}(X) \big), \mathcal B_n \big( B_{r_n(u)}(X) \big)\big\}\le k$ and
$\max\big\{\mathcal P_n \big( B_{r_n(u)}(X) \big), \mathcal B_n \big( B_{r_n(u)}(X) \big)\big\}> k$.
\vspace{7pt}
\noindent The key observation is that $[0,1]^d$ becomes $n$-bad whenever $L_{k,n}\neq L_{k,n}^\mathsf{B}$. Using this fact, we now need to show that
\begin{equation} \label{e:n.bad.cube.M0}
b_n^{-1} \mathbb{P}\big( [0,1]^d \text{ is } n\text{-bad} \big) \to 0, \ \ \ n\to\infty.
\end{equation}
The first step for the proof of \eqref{e:n.bad.cube.M0} is to demonstrate that
\begin{equation} \label{e:Fnvep.M0}
b_n^{-1}\mathbb{P}(F_{n,\varepsilon}^c) \to 0, \ \ \ n\to\infty,
\end{equation}
where $F_{n,\varepsilon}$ is given in \eqref{e:def.Fnvep}.
By virtue of the bound in \eqref{e:lemma1.2.bound.Fnvep}, together with an application of the Taylor expansion to $H(\cdot)$, and the assumption $a_n=o(n^{1/3})$, one can get \eqref{e:Fnvep.M0} as desired. Observe also that if $[0,1]^d$ is $n$-bad under $F_{n,\varepsilon}$, then there exists $X\in \mathcal P_n^{(\varepsilon a_n^{-1}, \mathsf{a})}$ such that \eqref{e:necessary.cond.n-bad} holds. Hence, by \eqref{e:Fnvep.M0} and Markov's inequality,
\begin{align*}
&b_n^{-1} \mathbb{P}\big( [0,1]^d \text{ is } n\text{-bad}\big) \le b_n^{-1} \mathbb{P}\big( \big\{[0,1]^d \text{ is } n\text{-bad}\big\}\cap F_{n,\varepsilon}\big) + o(1) \\
&\le b_n^{-1} \mathbb{P} \Big( \bigcup_{X\in \mathcal P_n^{(\varepsilon a_n^{-1}, \mathsf{a})} } \hspace{-10pt}\Big\{ \mathcal P_n^{(\varepsilon a_n^{-1}, \mathsf{t})} \big( B_{r_n(s_0)}(X) \big) \le k, \, \\
&\qquad \qquad \qquad \qquad\qquad \qquad\big( \mathcal P_n^{(\varepsilon a_n^{-1}, \mathsf{a})} \setminus \mathcal P_n^{(\varepsilon a_n^{-1}, \mathsf{t})} \big) \big( B_{r_n(w_n)}(X) \big) \ge1\Big\} \Big) +o(1)\\
&\le b_n^{-1}\bigg\{ \mathbb{E} \Big[ \sum_{X\in \mathcal P_n} {\mathbbm 1} \Big\{ \big(\mathcal P_n \setminus \mathcal D_{\varepsilon a_n^{-1}}(\mathcal P_n) \big) \big( B_{r_n(s_0)}(X) \big) \le k, \\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \big( \mathcal P_n^{(\varepsilon a_n^{-1})} \cup \mathcal D_{\varepsilon a_n^{-1}}(\mathcal P_n)\big) \big( B_{r_n(w_n)}(X) \big) \ge1 \Big\} \Big] \\
&\quad + \mathbb{E} \Big[ \sum_{X\in \mathcal P_n^{(\varepsilon a_n^{-1})}} {\mathbbm 1} \Big\{ \big(\mathcal P_n \setminus \mathcal D_{\varepsilon a_n^{-1}}(\mathcal P_n) \big) \big( B_{r_n(s_0)}(X) \big) \le k \Big\} \Big]\bigg\} +o(1)\\
&=:b_n^{-1}(A_n'+B_n') +o(1).
\end{align*}
where $\varepsilon\in (0,1)$ is an arbitrary constant.
Repeating the calculations very similar to those bounding $A_n, B_n$ in \eqref{e:estimate.pnvep}, one can see that $A_n'+B_n' \le C^* \varepsilon b_n$. Thus, $\limsup_{n\to\infty}b_n^{-1} \mathbb{P}\big( [0,1]^d \text{ is } n\text{-bad}\big) \le C^*\varepsilon$, and letting $\varepsilon \to0$ completes the proof of Corollary \ref{c:M0.LknB}.
\end{proof}
\begin{proof}[Proof of Corollary \ref{c:M0.application}]
We prove only the first statement. By a straightforward modification of Theorem \ref{t:M0.L.kn} by restricting the state space from $E$ to $E_0=[0,1]^d\times (s_0,\infty]$, we have, as $n\to\infty$,
$$
\xi_{k,n}(\cdot):=b_n^{-1}\mathbb{P}(L_{k,n}\in \cdot) \to \xi_k \ \ \text{in } \mathcal M_0.
$$
Due to the change of the state space, $L_{k,n}$ is now formulated as in \eqref{e:Lkn.reformulate.M0}, while the limit $\xi_k$ is taken to be
$$
\xi_k(\cdot)=\frac{1}{(k-1)!}\int_{E_0} {\mathbbm 1} \{ \delta_{(x,u)}\in \cdot \} e^{-u} \dif x \dif u.
$$
Now, we define a map $V:M_p(E_0)\to {\mathbb N}:=\{ 0,1,2,\dots \}$ by $V(\rho)=\rho(E_0)$. Here, ${\mathbb N}$ is equipped with the discrete topology.
Since $V$ is continuous in the weak topology,
it follows from \cite[Theorem 2.5]{hult:lindskog:2006a} that
\begin{equation} \label{e:another.M0.conv}
\xi_{k,n}\circ V^{-1} \to \xi_k \circ V^{-1}, \ \ \text{ in } \mathcal M_0, \ \ \ n\to\infty.
\end{equation}
Note that ${\mathbbm 1}_{[1,\infty)}(x)$ is continuous and bounded on ${\mathbb N}$ (in terms of the discrete topology), vanishing in the neighborhood of $0$ (i.e., the origin of ${\mathbb N}$). Thus, the $\mathcal M_0$-convergence in \eqref{e:another.M0.conv} implies that
$$
b_n^{-1}\mathbb{P}(T_{k,n}\ge1) = \int_{{\mathbb N}} {\mathbbm 1}_{[1,\infty)}(x) \xi_{k,n}\circ V^{-1}(\dif x) \to \int_{{\mathbb N}} {\mathbbm 1}_{[1,\infty)}(x) \xi_{k}\circ V^{-1}(\dif x) = \alpha_k,
$$
as desired.
\end{proof}
\noindent \textbf{Acknowledgment}: The third author is thankful for fruitful discussions with Yogeshwaran D.~and Z.~Wei, which has helped him to deduce the desired $\mathcal M_0$-convergence in Theorem \ref{t:M0.L.kn} by means of \cite[Theorem 6.4]{bobrowski:schulte:yogeshwaran:2021}.
| {
"timestamp": "2022-10-25T02:09:09",
"yymm": "2210",
"arxiv_id": "2210.12423",
"language": "en",
"url": "https://arxiv.org/abs/2210.12423",
"abstract": "This paper develops the large deviations theory for the point process associated with the Euclidean volume of $k$-nearest neighbor balls centered around the points of a homogeneous Poisson or a binomial point processes in the unit cube. Two different types of large deviation behaviors of such point processes are investigated. Our first result is the Donsker-Varadhan large deviation principle, under the assumption that the centering terms for the volume of $k$-nearest neighbor balls grow to infinity more slowly than those needed for Poisson convergence. Additionally, we also study large deviations based on the notion of $\\mathcal M_0$-topology, which takes place when the centering terms tend to infinity sufficiently fast, compared to those for Poisson convergence. As applications of our main theorems, we discuss large deviations for the number of Poisson or binomial points of degree at most $k$ in a geometric graph in the dense regime.",
"subjects": "Probability (math.PR)",
"title": "Large deviations for the volume of $k$-nearest neighbor balls",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808753491773,
"lm_q2_score": 0.7217432003123989,
"lm_q1q2_score": 0.707511056179555
} |
https://arxiv.org/abs/2102.09552 | Linear Functions to the Extended Reals | This note investigates functions from $\mathbb{R}^d$ to $\mathbb{R} \cup \{\pm \infty\}$ that satisfy axioms of linearity wherever allowed by extended-value arithmetic. They have a nontrivial structure defined inductively on $d$, and unlike finite linear functions, they require $\Omega(d^2)$ parameters to uniquely identify. In particular they can capture vertical tangent planes to epigraphs: a function (never $-\infty$) is convex if and only if it has an extended-valued subgradient at every point in its effective domain, if and only if it is the supremum of a family of "affine extended" functions. These results are applied to the well-known characterization of proper scoring rules, for the finite-dimensional case: it is carefully and rigorously extended here to a more constructive form. In particular it is investigated when proper scoring rules can be constructed from a given convex function. | \section{Introduction}
The extended real number line, denoted $\bar{\reals} = \mathbb{R} \cup \{\pm \infty\}$, is widely useful particularly in convex analysis.
But I am not aware of an answer to the question: What would it mean to have a ``linear'' $f: \mathbb{R}^d \to \bar{\reals}$?
The extended reals have enough structure to hope for a useful answer, but differ enough from a vector space to need investigation.
A natural approach is that $f$ must satisfy the usual linearity axioms of homogeneity and additivity whenever legal under extended-reals arithmetic.
Here is an example when $d=3$:
\[ f(x,y,z) = \begin{cases} \text{if $z > 0$} &\rightarrow~~ \infty \\
\text{if $z < 0$} &\rightarrow~~ -\infty \\
\text{if $z = 0$} &\rightarrow~~
\begin{cases} \text{if $y > 0$} &\rightarrow~~ \infty \\
\text{if $y < 0$} &\rightarrow~~ -\infty \\
\text{if $y = 0$} &\rightarrow~~ x . \end{cases} \end{cases} \]
We will see that all \emph{linear extended} functions (Definition \ref{def:linext}) on $\mathbb{R}^d$ have the above inductive format, descending dimension-by-dimension until reaching a finite linear function (Proposition \ref{prop:alg-correct}).
In fact, a natural representation of this procedure is parsimonious, requiring as many as $\Omega(d^2)$ real-valued parameters to uniquely identify an extended linear function on $\mathbb{R}^d$ (Proposition \ref{prop:parsim}).
This structure raises the possibility, left for future work, that linear extended functions are significantly nontrivial on infinite-dimensional spaces.
Linear extended functions arise naturally as \emph{extended subgradients} (Definition \ref{def:extsub}) of a convex function $g$.
These can be used to construct \emph{affine extended} functions (Definition \ref{def:affext}) capturing vertical supporting hyperplanes to convex epigraphs.
Along such a hyperplane, intersected with the boundary of the domain, $g$ can sometimes have the structure of an arbitrary convex function in $d-1$ dimensions.
For example, the above $f$ is an extended subgradient, at the point $(1,0,0)$, of the discontinuous convex function
\[ g(x,y,z) = \begin{cases} \text{if $z > 0$} &\rightarrow~~ \infty \\
\text{if $z < 0$} &\rightarrow~~ 0 \\
\text{if $z = 0$} &\rightarrow~~
\begin{cases} \text{if $y > 0$} &\rightarrow~~ \infty \\
\text{if $y < 0$} &\rightarrow~~ 0 \\
\text{if $y = 0$} &\rightarrow~~ \frac{1}{2}x^2 . \end{cases} \end{cases} \]
A proper function (one that is never $-\infty$) is convex if and only if it has an extended subgradient at every point in its effective domain (Proposition \ref{prop:subgrad-char}).
We also find it is convex if and only if it is the pointwise supremum of affine extended functions (Proposition \ref{prop:sup-char}).
\vskip1em
\paragraph{Proper scoring rules.}
The motivation for this investigation was the study of scoring rules: functions $S$ assigning a score $S(p,y)$ to any prediction $p$ (a probability distribution over outcomes) and observed outcome $y$.
This $S$ is called \emph{proper} if reporting the true distribution of the outcome maximizes expected score.
A well-known characterization states that proper scoring rules arise as and only as subgradients of convex functions~\citep{mccarthy1956measures,savage1971elicitation,schervish1989general}.
Modern works~\citep{gneiting2007strictly,frongillo2014general} generalize this characterization to allow scores of $-\infty$, corresponding to convex functions that are not subdifferentiable\footnote{They also consider infinite-dimensional outcome spaces, which are not treated here.}.
These works therefore replace subgradients with briefly-defined generalized ``subtangents'', but they slightly sidestep the questions tackled head-on here: existence of these objects and their roles in convex analysis.
So these characterizations are not fully constructive.
In particular, it may surprise even experts that rigorous answers are not available to the following questions, given a convex function $g$: \emph{(a)} under what conditions can one construct a proper scoring rule from it? \emph{(b)} when is the resulting scoring rule \emph{strictly} proper? \emph{(c)} do the answers depend on which subgradients of $g$ are used, when multiple choices are available?
Section \ref{sec:proper-scoring-rules} uses the machinery of extended linear functions to prove a ``construction of proper scoring rules'' that answers such questions for the finite-outcome case.
First, Theorem \ref{thm:proper-char} allows predictions to range over the entire simplex.
It implies a slightly-informal claim of \citet{gneiting2007strictly}, Section 3.1: \emph{Any convex function on the probability simplex gives rise to a proper scoring rule (using any choices of its extended subgradients).}
Furthermore, it shows that a strictly convex function can only give rise to \emph{strictly} proper scoring rules and vice versa.
These facts are likely known by experts in the community; however, I do not know of formal claims and proofs.
This may be because, when scoring rules can be $-\infty$, proofs seem to require significant formalization and investigation of ``subtangents''.
In this paper, this investigation is supplied by linear extended functions and characterizations of (strictly) convex functions as those that have (uniquely supporting) extended subgradients everywhere.
Next, Theorem \ref{thm:proper-subset-char} allows prediction spaces $\P$ to be any subset of the simplex.
It sharpens the characterizations of \citet{gneiting2007strictly,frongillo2014general} by showing $S$ to be proper if and only if it can be constructed from extended subgradients of a convex $g$ that is \emph{interior-locally-Lipschitz} (Definition \ref{def:inter-local}).
It also answers the construction questions \emph{(a)-(c)} above, e.g. showing that given such a $g$, some but not necessarily all selections of its extended subgradients give rise to proper scoring rules.
\paragraph{Preliminaries.}
Functions in this work are defined on Euclidean spaces of dimension $d\geq 0$.
I ask the reader's pardon for abusing notation slightly: I use $\mathbb{R}^d$ refer to any such space.
For example, I may argue that a certain function exists on domain $\mathbb{R}^d$, then refer to that function as being defined on a given $d$-dimensional subspace of $\mathbb{R}^{d+1}$.
Let $g: \mathbb{R}^d \to \bar{\reals}$.
The \emph{effective domain} $\effdom{g}$ of $g$ is $\{x \in \mathbb{R}^d \st g(x) \neq \infty\}$.
The function $g$ is \emph{convex} if its \emph{epigraph} $\{(x,y) \in \mathbb{R}^d \times \mathbb{R} : y \geq g(x)\}$ is a convex set.
Equivalently, it is convex if for all $x,x' \in \effdom{g}$ and all $0 < \rho < 1$, $g(\rho \cdot x + (1-\rho)x') \leq \rho g(x) + (1-\rho) g(x')$, observing that this sum may contain $-\infty$ but not $+\infty$.
It is \emph{strictly} convex on a convex set $\P \subseteq \effdom{g}$ if the previous inequality is always strict for $x,x' \in \P$ with $x \neq x'$.
We say a function $h$ \emph{minorizes} $g$ if $h(x) \leq g(x)$ for all $x$.
$\cl{A}$ denotes the closure of the set $A$ and $\inter{A}$ its interior.
For $a \in \bar{\reals}$, the sign function is $\sign(a) = 1$ if $a > 0$, $\sign(a) = 0$ if $a=0$, and $\sign(a) = -1$ if $a < 0$.
\paragraph{The extended reals.}
The extended reals, $\bar{\reals} = \mathbb{R} \cup \{\pm \infty\}$, have the following rules of arithmetic for any $\alpha,\beta \in \mathbb{R}$: $\beta + \infty = \infty$, $\beta - \infty = -\infty$, and
\[ \alpha \cdot \infty = \begin{cases} \infty & \alpha > 0 \\ 0 & \alpha = 0 \\ -\infty & \alpha < 0 . \end{cases} \]
Illegal and disallowed is addition of $\infty$ and $-\infty$.
Addition is associative and commutative as long as it is legal; multiplication of multiple scalars and possibly one non-scalar is associative and commutatitve.
Multiplication by a scalar distributes over legal sums.
$\bar{\reals}$ has the following rules of comparison: $-\infty < \beta < \infty$ for every $\beta \in \mathbb{R}$.
The supremum of a subset of $\bar{\reals}$ is $\infty$ if it contains $\infty$ or it contains an unbounded-above set of reals; the analogous facts hold for the infimum.
Also, $\inf \emptyset = \infty$ and $\sup \emptyset = -\infty$.
I will not put a topology on $\bar{\reals}$ in this work.
\section{Linear extended functions}
The following definition makes sense with a general real vector space $\mathcal{X}$ in place of $\mathbb{R}^d$, but it remains to be seen if all results can extend.
\begin{definition} \label{def:linext}
Call the function $f: \mathbb{R}^d \to \bar{\reals}$ a \emph{linear extended function} if:
\begin{enumerate}
\item (scaling) For all $x \in \mathbb{R}^d$ and all $\alpha \in \mathbb{R}$: $f(\alpha x) = \alpha f(x)$.
\item (additivity) For all $x,x' \in \mathbb{R}^d$: If $f(x) + f(x')$ is legal, i.e. $\{f(x),f(x')\} \neq \{\pm \infty\}$, then $f(x + x') = f(x) + f(x')$.
\end{enumerate}
\end{definition}
If the range of $f$ is included in $\mathbb{R}$, Definition \ref{def:linext} reduces to the usual definition of a linear function.
For clarity, this paper may emphasize the distinction by calling such $f$ \emph{finite linear}.
\vskip1em
To see that this definition can be satisfied nontrivially, let $f_1, f_2: \mathbb{R}^d \to \mathbb{R}$ be finite linear and consider $f(x) := \infty \cdot f_1(x) + f_2(x)$.
With a representation $f_1(x) = v_1 \cdot x$, we have
\[ f(x) = \begin{cases} \infty & v_1 \cdot x > 0 \\
-\infty & v_1 \cdot x < 0 \\
f_2(x) & v_1 \cdot x = 0 . \end{cases} \]
\begin{claim}
Any such $f$ is a linear extended function.
\end{claim}
\begin{myproof}
Multiplication by a scalar distributes over the legal sum $\infty \cdot f_1(x) + f_2(x)$, so $\alpha f(x) = \infty \cdot f_1(\alpha x) + f_2(\alpha x) = f(\alpha x)$.
To obtain additivity of $f$, consider cases on the pair $(v_1 \cdot x, ~ v_1 \cdot x')$.
If both are zero, $f(x+x') = f_2(x+x') = f_2(x) + f_2(x') = f(x) + f(x')$.
If they have opposite signs, then $\{f(x), f(x')\} = \{\pm \infty\}$ and the requirement is vacuous.
If one is positive and the other nonnegative, then $v_1 \cdot (x + x') > 0$, so we have $f(x+x') = \infty = f(x) + f(x')$.
The remaining case, one negative and the other nonpositive, is exactly analogous.
\end{myproof}
Such $f$ are not all the linear extended functions, because the subspace where $x \cdot v_1 = 0$ need not be entirely finite-valued.
As in the introduction's example, it can itself be divided into infinite and neg-infinite open halfspaces, with $f$ finite only on some further-reduced subspace.
We will show that all linear extended functions can be constructed in this recursive way, Algorithm \ref{alg:linext} (heavily relying on the setting of $\mathbb{R}^d$).
\begin{algorithm}
\caption{Computing a linear extended function $f: \mathbb{R}^d \to \bar{\reals}$}
\label{alg:linext}
\begin{algorithmic}
\STATE \textbf{Parameters:} $t \in \{0,\ldots,d\}$; finite linear $\hat{f}: \mathbb{R}^{d-t} \to \mathbb{R}$; if $t \geq 1$, unit vectors $v_1 \in \mathbb{R}^d, \ldots, v_t \in \mathbb{R}^{d-t+1}$
\STATE \textbf{Input:} $x \in \mathbb{R}^d$
\FOR{$j = 1, \ldots, t$}
\IF{$v_j \cdot x > 0$}
\STATE return $\infty$
\ELSIF{$v_j \cdot x < 0$}
\STATE return $-\infty$
\ENDIF
\STATE reparameterize $x$ as a vector in $\mathbb{R}^{d-j}$, a member of the subspace $\{x' \st v_j \cdot x' = 0\}$
\ENDFOR
\STATE return $\hat{f}(x)$
\end{algorithmic}
\end{algorithm}
For the following lemma, recall that a subset $S$ of $\mathbb{R}^d$ is a \emph{convex cone} if, when $x,x' \in S$ and $\alpha,\beta > 0$, we have $\alpha x + \beta x' \in S$.
A convex cone need not contain $\vec{0}$.
\begin{lemma}[Decomposition] \label{lemma:decomp}
$f: \mathbb{R}^d \to \bar{\reals}$ is linear extended if and only if: (1) the sets $S^+ = f^{-1}(\infty)$ and $S^- = f^{-1}(-\infty)$ are convex cones with $S^+ = - S^-$, and (2) the set $F = f^{-1}(\mathbb{R})$ is a subspace of $\mathbb{R}^d$, and (3) $f$ coincides with some finite linear function on $F$.
\end{lemma}
\begin{myproof}
$(\implies)$ Let $f$ be linear extended.
The scaling axiom implies $f(\vec{0}) = 0$, so $\vec{0} \in F$.
Observe that $F$ is closed under scaling (by the scaling axiom) and addition (the addition axiom is never vacuous on $x,x' \in F$).
So it is a subspace.
$f$ satisfies scaling and additivity (never vacuous) for all members of $F$, so it is finite linear there.
This proves (2) and (3).
Next: the scaling axiom implies that if $x \in S^+$ then $-x \in S^-$, giving $S^+ = - S^-$ as claimed.
Now let $x, x' \in S^+$ and $\alpha,\beta > 0$.
The scaling axiom implies $\alpha x$ and $\beta x'$ are in $S^+$.
The additivity axiom implies $\alpha x + \beta x' \in S^+$, proving it is a convex cone.
We immediately get $S^- = -S^+$ is a convex cone as well, proving (1).
$(\impliedby)$ Suppose $F$ is a subspace on which $f$ is finite linear and $S^+ = -S^-$ is a convex cone.
We prove $f$ satisfies the two axioms needed to be linear extended.
First, $F$ includes $\vec{0}$ by virtue of being a subspace (inclusion of $\vec{0}$ is also implied by $S^+ = -S^-$, because $S^+ \cap S^- = \emptyset$).
Because $f$ is finite linear on $F$, $f(\vec{0}) = 0$.
We now show $f$ satisfies the two axioms of linearity.
For scaling, if $x \in F$, the axiom follows from closure of $F$ under scaling and finite linearity of $f$ on $F$.
Else, let $\alpha \in \mathbb{R}$ and $x \in S^+$ (the case $x \in S^-$ is exactly analogous).
If $\alpha > 0$, then $\alpha x \in S^+$ because it is a convex cone, which gives $f(\alpha x) = \alpha f(x) = \infty$.
If $\alpha = 0$, then $f(\alpha x) = \alpha f(x) = 0$.
If $\alpha < 0$, then use that $-x \in S^-$ by assumption, and since $S^-$ is a cone and $-\alpha > 0$, we have $(-\alpha)(-x) \in S^-$.
In other words, $f(\alpha x) = -\infty = \alpha f(x)$, as required.
For additivity, let $x,x'$ be given.
\begin{itemize}
\item If $x,x' \in F$, additivity follows from closure of $F$ under addition and finite linearity of $f$ on $F$.
\item If $x,x' \in S^+$, then $x+x' \in S^+$ as required because it is a convex cone; analogously for $x,x' \in S^-$.
\item If $x \in S^+, x' \in S^-$ or vice versa, the axiom is vacuously satisfied.
\end{itemize}
The remaining case is, without loss of generality, $x \in S^+, x' \in F$ (the proof is identical for $x \in S^-, x' \in F$).
We must show $x + x' \in S^+$.
Because $F$ is a subspace and $x' \in F, x \not\in F$, we must have $x + x' \not\in F$.
Now suppose for contradiction that $x + x' \in S^-$.
We have $-x \in S^-$ because $S^- = -S^+$.
Because $S^-$ is a convex cone, it is closed under addition, so $x + x' + (-x) = x' \in S^-$, a contradiction.
So $x+x' \in S^+$.
\end{myproof}
\begin{lemma}[Recursive definition] \label{lemma:recursive}
$f: \mathbb{R}^d \to \bar{\reals}$ is linear extended if and only if one of the following hold:
\begin{enumerate}
\item $f$ is finite linear (this case must hold if $d=0$), or
\item There exists a unit vector $v_1$ and linear extended function $f_2$ on the $d-1$ dimensional subspace $\{x \st v_1 \cdot x = 0\}$ such that $f(x) = f_2(x)$ if $x \cdot v_1 = 0$, else $f(x) = \infty \cdot \sign(v_1 \cdot x)$.
\end{enumerate}
\end{lemma}
\begin{myproof}
$(\implies)$ Suppose $f$ is linear extended.
The case $d=0$ is immediate, as $f(\vec{0}) = 0$ by the scaling axiom.
So let $d \geq 1$ and suppose $f$ is not finite linear; we show case (2) holds.
Let $S^+ = f^{-1}(\infty)$, $S^- = f^{-1}(-\infty)$, and $F = f^{-1}(\mathbb{R})$.
Recall from Lemma \ref{lemma:decomp} that $F$ is a subspace, necessarily of dimension $< d$ by assumption that $f$ is not finite; meanwhile $S^+ = -S^-$ and both are convex cones.
We first claim that there is an open halfspace on which $f(x) = \infty$, i.e. included in $S^+$.
First, $\cl{S^+}$ includes a closed halfspace: if not, the set $\cl{S^+} \cup \cl{S^-} \neq \mathbb{R}^d$ and then its complement, an open set, would necessarily have affine dimension $d$ yet would be included in $F$, a contradiction.
Now, because $S^+$ is convex, it includes the relative interior of $\cl{S^+}$, so it includes an open halfspace.
Write this open halfspace $\{x \st v_1 \cdot x > 0\}$ for some unit vector $v_1$.
Because $S^- = -S^+$, we have $f(x) = -\infty$ on the complement $\{x \st v_1 \cdot x < 0\}$.
Let $f_2$ be the restriction of $f$ to the remaining subspace, $\{x \st v_1 \cdot x = 0\}$.
This set is closed under addition and scaling, and $f$ satisfies the axioms of a linear extended function, so $f_2$ is a linear extended function as well.
$(\impliedby)$ If case (1) holds and $f$ is finite linear, then it is immediately linear extended as well, QED.
In case (2), we apply Lemma \ref{lemma:decomp} to $f_2$.
We obtain that it is finite linear on a subspace of $\{ x \st v_1 \cdot x = 0\}$, which is a subspace of $\mathbb{R}^d$, giving that $f$ is finite linear on a subspace.
We also obtain $f_2^{-1}(\infty) = -f_2^{-1}(-\infty)$ and is a convex cone.
It follows directly that $f^{-1}(\infty) = - f^{-1}(-\infty)$.
In fact, $f^{-1}(\infty) = f_2^{-1}(\infty) \cup \{ x \st v_1 \cdot x > 0\}$.
The first set is a convex cone lying in the closure of the second set, also a convex cone.
So the union is a convex cone: scaling is immediate; any nontrivial convex combination of a point from each set lies in the second, giving convexity; scaling and convexity give additivity.
This shows that $f^{-1}(\infty)$ is a convex cone, the final piece needed to apply Lemma \ref{lemma:decomp} and declare $f$ linear extended.
\end{myproof}
\begin{proposition}[Correctness of Algorithm \ref{alg:linext}] \label{prop:alg-correct}
A function $f: \mathbb{R}^d \to \bar{\reals}$ is linear extended if and only if it is computed by Algorithm \ref{alg:linext} for some $t \in \{0,\ldots,d\}$, some $v_1 \in \mathbb{R}^d, \ldots, v_t \in \mathbb{R}^{d-t+1}$, and some finite linear $\hat{f}: \mathbb{R}^d \to \mathbb{R}$.
\end{proposition}
\begin{myproof}
$(\implies)$ Suppose $f$ is linear extended.
By Lemma \ref{lemma:recursive}, there are two cases.
If $f$ is finite linear, then take $t=0$ and $\hat{f} = f$ in Algorithm \ref{alg:linext}.
Otherwise, Lemma \ref{lemma:recursive} gives a unit vector $v_1$ so that $f(x) = \infty$ if $v_1 \cdot x > 0$ and $f(x) = -\infty$ if $v_1 \cdot x < 0$, as in Algorithm \ref{alg:linext}.
$f$ is linear extended on $\{ x \st v_1 \cdot x = 0\}$, so we iterate the procedure until reaching a subspace where $f$ is finite linear, setting $t$ to be the number of iterations.
$(\impliedby)$ Suppose $f$ is computed by Algorithm \ref{alg:linext}.
We will use the two cases of Lemma \ref{lemma:recursive} to show $f$ is linear extended.
If $t = 0$, then $f$ is finite linear, hence linear extended (case 1).
If $t \geq 1$, then $f$ is in case 2 with unit vector $v_1$ and function $f_2$ equal to the implementation of Algorithm \ref{alg:linext} on $t-1$, $\hat{f}$, and $v_2,\dots,v_t$.
This proves by induction on $t$ that, if $f$ is computed by Algorithm \ref{alg:linext}, then it satisfies one of the cases of Lemma \ref{lemma:recursive}, so it is linear extended.
\end{myproof}
\begin{proposition}[Parsimonious parameterization] \label{prop:parsim}
Each linear extended function has a unique representation by the parameters of Algorithm \ref{alg:linext}.
\end{proposition}
\begin{myproof}
For $i \in \{1,2\}$, let $f^{(i)}$ be the function computed by Algorithm \ref{alg:linext} with the parameters $t^{(i)}$, $\hat{f}^{(i)}$, $\{v_j^{(i)} \st j =1,\dots,t^{(i)}\}$.
We will prove that $f^{(1)}$ and $f^{(2)}$ are distinct if any of their parameters differ: i.e. if $t^{(1)} \neq t^{(2)}$, or else $\hat{f}^{(1)} \neq \hat{f}^{(2)}$, or else there exists $j$ with $v_j^{(1)} \neq v_j^{(2)}$.
By Lemma \ref{lemma:decomp}, each $f^{(i)}$ is finite linear on a subspace $F^{(i)}$ and positive (negative) infinite on a convex cone $S^{i,+}$ (respectively, $S^{i,-}$) with $S^{i,+} = -S^{i,-}$.
It follows that they are equal if and only if: $S^{1,+} = S^{2,+}$ (this implies $F^{(1)} = F^{(2)}$) and they coincide on $F^{(1)}$.
If $t^{(1)} \neq t^{(2)}$, then the dimensions of $F^{(1)}$ and $F^{(2)}$ differ, so they are nonequal, so $S^{1,+} \neq S^{2,+}$ and the functions are not the same.
So suppose $t^{(1)} = t^{(2)}$.
Now suppose the unit vectors are not the same, i.e. there is some smallest index $j$ such that $v^{(1)}_j \neq v^{(2)}_j$.
Observe that on iteration $j$ of the algorithm, the $d-j+1$ dimensional subspace under consideration is identical for $f^{(1)}$ and $f^{(2)}$.
But now there is some $x$ with $v^{(1)}_j \cdot x > 0$ while $v^{(2)}_j \cdot x < 0$, for example, $x = v^{(1)}_j - v^{(2)}_j$.
On this $x$ (parameterized as a vector in $\mathbb{R}^d$), $f^{(1)}(x) = \infty$ while $f^{(2)}(x) = -\infty$, so the functions differ.
Finally, suppose $t^{(1)} = t^{(2)}$ and all unit vectors are the same.
Observe that $F^{(1)} = F^{(2)}$.
But if $\hat{f}^{(1)} \neq \hat{f}^{(2)}$, then there is a point in $F^{(1)}$ where $\hat{f}^{(1)}$ and $\hat{f}^{(2)}$ differ.
On this point (parameterized as a vector in $\mathbb{R}^d$), $f^{(1)}$ and $f^{(2)}$ differ.
\end{myproof}
One corollary is the following definition:
\begin{definition}[Depth] \label{def:depth}
Say the \emph{depth} of a linear extended function $f$ is the value of $t$ in its parameterization of Algorithm \ref{alg:linext} (shown in Proposition \ref{prop:parsim} to be unique).
\end{definition}
Another is that, while finite linear functions on $\mathbb{R}^d$ are uniquely identified by $d$ real-valued parameters, linear extended functions require as many as ${d \choose 2} + 1 = \Omega(d^2)$: Unit vectors in $\mathbb{R}^k$ require $k-1$ parameters (for $k \geq 2$), so even assuming depth $t=d-1$, identifying the unit vectors takes $d-1 + \cdots + 1 = {d \choose 2}$ parameters, and one more defines $\hat{f}: \mathbb{R} \to \mathbb{R}$.
\vskip1em
By the way, we would be remiss in forgetting to prove:
\begin{proposition}[Convexity] \label{prop:lin-convex}
Linear extended functions are convex.
\end{proposition}
\begin{myproof}
We prove convexity by showing the epigraph of a linear extended function $f: \mathbb{R}^d \to \bar{\reals}$ is convex.
By induction on $d$:
If $d=0$, then $f$ is finite linear (Lemma \ref{lemma:recursive}), so it has a convex epigraph.
Otherwise, let $d \geq 1$.
If $f$ is finite linear, then again it has a convex epigraph, QED.
Otherwise, by Lemma \ref{lemma:recursive}, its epigraph is $A \cup B$ where: $A = \{x : v_1 \cdot x < 0\}$ for some unit vector $v_1 \in \mathbb{R}^d$; and $B$ is the epigraph (appropriately reparameterized) of a linear extended function $f_2: \mathbb{R}^{d-1} \to \bar{\reals}$ on the set $\{x : v_1 \cdot x = 0\}$.
We have $B \subseteq \cl{A}$, and both sets are convex (by inductive hypothesis), so their union is convex: any nontrivial convex combination of points lies in the interior of $A$.
\end{myproof}
The effective domain of a linear extended function at least includes an open halfspace.
If the depth is zero, it is $\mathbb{R}^d$; if the depth is $1$, it is a closed halfspace $\{ x \st v_1 \cdot x \leq 0\}$, and otherwise it is neither a closed nor open set.
\paragraph{Structure.}
It seems difficult to put useful algebraic or topological structure on the set of linear extended functions on $\mathbb{R}^d$.
For example, addition of two functions is typically undefined.
Convergence in the parameters of Algorithm \ref{alg:linext} does not seem to imply pointwise convergence of the functions, as for instance we can have a sequence $v_1^{(m)} \to v_1$ such that members of $\{x \st v_1 \cdot x = 0\}$ are always mapped to $\infty$.
But perhaps future work can put a creative structure on this set.
\section{Proper scoring rules} \label{sec:proper-scoring-rules}
This section will first define scoring rules; then recall their characterization and discuss why it is not fully constructive; then use linear extended functions to prove more complete and constructive versions.
\subsection{Definitions}
$\Y$ is a finite set of mutually exclusive and exhaustive outcomes, also called observations.
The probability simplex on $\Y$ is $\Delta_{\Y} = \{p \in \mathbb{R}^{\Y} \st (\forall y) ~ p(y) \geq 0 ~;~ \sum_{y \in \Y} p(y) = 1\}$.
The \emph{support} of $p \in \Delta_{\Y}$ is $\Supp{p} = \{y : p(y) > 0\}$.
The distribution with full mass on $y$ is $\delta_y \in \Delta_{\Y}$.
Proper scoring rules (e.g. \citet{gneiting2007strictly}) model an expert providing a forecast $p \in \Delta_{\Y}$, after which $y \in \Y$ is observed and the expert's score is $S(p,y)$.
The expert chooses $p$ to maximize expected score according to an internal belief $q \in \Delta_{\Y}$.
More generally, it is sometimes assumed that reports and beliefs are restricted to a subset $\P \subseteq \Delta_{\Y}$.
Two of the most well-known are the log scoring rule, where $S(p,y) = \log p(y)$; and the quadratic scoring rule, where $S(p,y) = -\|\delta_y - p\|_2^2$.
An example of a scoring rule that is not proper is $S(p,y) = p(y)$.
\begin{definition}[Scoring rule, regular] \label{def:scoring-rule}
Let $\Y$ be finite and $\P \subseteq \Delta_{\Y}$, nonempty.
A function $S: \P \times \Y \to \bar{\reals}$ is termed a \emph{scoring rule}.
It is \emph{regular} if $S(p,y) \neq \infty$ for all $p \in \P$, $y \in \Y$.
\end{definition}
Regular scoring rules are nice for two reasons.
First, calculating expected scores such as $\sum_y q(y) S(p,y)$ may lead to illegal sums if $S(p,y) \in \bar{\reals}$, but regular scores guarantee that the sum is legal (using that $q(y) \geq 0$).
Second, even if illegal sums were no problem (e.g. if we disallowed $-\infty$), allowing scores of $S(p,y) = \infty$ leads to strange and unexciting rules: $p$ is an optimal report for any belief $q$ unless $q(y) = 0$.
\begin{definition}[Expected score, strictly proper] \label{def:proper}
Given a regular scoring rule $S: \P \times \Y \to \mathbb{R} \cup \{-\infty\}$, the \emph{expected score} for report $p \in \P$ under belief $q \in \Delta_{\Y}$ is written $S(p;q) := \sum_{y \in \Y} q(y) S(p,y)$.
The regular scoring rule $S$ is \emph{proper} if for all $p,q \in \P$ with $p \neq q$, $S(p;q) \leq S(q;q)$, and it is \emph{strictly proper} if this inequality is always strict.
\end{definition}
Why use a general definition of scoring rules, taking values in $\bar{\reals}$, if I claim that the only interesting rules are regular?
It will be useful for technical reasons: we will consider constructions that obviously yield a well-defined scoring rule, then investigate conditions under which that scoring rule is regular.
Understanding these conditions is a main contribution of Theorem \ref{thm:proper-char} and particularly Theorem \ref{thm:proper-subset-char}.
\vskip1em
The well-known proper scoring rule characterization, discussed next, roughly states that all proper scoring rules are of the following form.
\begin{definition}[Subtangent rule] \label{def:subtangent-rule}
If a scoring rule $S: \P \times \Y \to \bar{\reals}$ satisfies
\begin{align}
S(p,y) &= g(p) + f_p(\delta_y - p) & \forall p \in \P, y \in \Y \label{eqn:subtan}
\end{align}
for some convex function $g: \mathbb{R}^{\Y} \to \mathbb{R} \cup \{\infty\}$ with $\P \subseteq \effdom{g}$ and some choices of its extended subgradients $f_p$ at each $p \in \P$, then call $S$ a \emph{subtangent rule (of $g$)}.
\end{definition}
The geometric intuition is that the affine extended function $q \mapsto g(p) + f_p(q - p)$ is a linear approximation of $g$ at the point $p$.
So a subtangent rule, on prediction $p$ and observation $y$, evaluates this linear approximation at $\delta_y$.
\subsection{Prior characterizations and missing pieces}
\paragraph{The classic characterization.}
The scoring rule characterization has appeared in many forms, notably in \citet{mccarthy1956measures, savage1971elicitation,schervish1989general,gneiting2007strictly}.
In particular, \citet{savage1971elicitation} proved:
\begin{theorem}[\citet{savage1971elicitation} characterization] \label{thm:savage-char}
A scoring rule $S$ on $\P = \Delta_{\Y}$ \emph{taking only finite values} is proper (respectively, strictly proper) if and only if it is of the form (\ref{eqn:subtan}) for some convex (respectively, strictly convex) $g$ with \emph{finite} subgradients $\{f_p\}$.
\end{theorem}
There are several nonconstructive wrinkles in this result.
By itself, it does not answer the following questions (and the fact that it does not may surprise even experts, especially if the answers feel obvious and well-known).
\begin{enumerate}
\item Let the finite-valued $S$ be proper, but not strictly proper.
Theorem \ref{thm:savage-char} asserts that it is of the form (\ref{eqn:subtan}) for some convex $g$.
Can $g$ be strictly convex?
\item Now let $S$ be strictly proper.
Theorem \ref{thm:savage-char} asserts that it is of the form (\ref{eqn:subtan}) for some strictly convex $g$.
Can it also be of the form (\ref{eqn:subtan}) for some other, \emph{non}-strictly convex $g$?
\item Let $g: \Delta_{\Y} \to \mathbb{R}$ be convex.
Theorem \ref{thm:savage-char} implies that \emph{if} there exists a finite-valued scoring rule $S$ of the form (\ref{eqn:subtan}), \emph{then} $S$ is proper.
But does such an $S$ exist?
\item Now let $g$ be strictly convex and suppose it has a finite-valued proper scoring rule $S$ of the form (\ref{eqn:subtan}).
Is $S$ necessarily strictly proper?
\end{enumerate}
For the finite-valued case, answers are not too difficult to derive, although I do not know of a citation.\footnote{%
The answer to the first two questions is \emph{no}, and the fourth (the contrapositive of the first) is \emph{yes}.
These follow because a convex subdifferentiable $g$ is strictly convex on $\Delta_{\Y}$ if and only if each subgradient appears at most at one point $p \in \Delta_{\Y}$; this is an easier version of Lemma \ref{lemma:strict-subgrad}.
The third question is the deepest, and the answer is \emph{not necessarily}.
Some convex $g$ are not subdifferentiable, i.e. they have no finite subgradient at some points, so it is not possible to construct a finite-valued scoring rule from them at all.}
But as discussed next, these questions are also unanswered by characterizations that allow scores of $-\infty$; although to an extent the answers may be known to experts.
Theorems \ref{thm:proper-char} and \ref{thm:proper-subset-char} will address them.
\paragraph{Extension to scores with $-\infty$.}
\citet{savage1971elicitation} disallows (intentionally, Section 9.4) the log scoring rule of \citet{good1952rational}, $S(p,y) = \log p(y)$, because it can assign score $-\infty$ if $p(y) = 0$.
The prominence of the log scoring rule has historically (e.g. \citet{hendrickson1971proper}) motivated including it, but doing so requires generalizing the characterization to include possibly-neg-infinite scores.
The modern treatment of \citet{gneiting2007strictly} (also \citet{frongillo2014general}) captures such scoring rules as follows: It briefly defines extended-valued \emph{subtangents}, analogues of this paper's extended subgradients (but possibly on infinite-dimensional spaces, so for instance requirements of ``legal sums'' are replaced with ``quasi-integrable'').
The characterization can then be stated: A regular scoring rule is (strictly) proper if and only if it is of the form (\ref{eqn:subtan}) for some (strictly) convex $g$ with \emph{subtangents} $\{f_p\}$.
However, these characterizations have drawbacks analogous to those enumerated above, and also utilize the somewhat-mysterious subtangent objects.
This motivates Theorem \ref{thm:proper-char} (for the case $\P = \Delta_{\Y}$) and Theorem \ref{thm:proper-subset-char} (for general $\P \subseteq \Delta_{\Y}$), which are more specific about when and how subtangent scoring rules can be constructed.
In particular, for $\P = \Delta_{\Y}$, Theorem \ref{thm:proper-char} formalizes a slightly-informal construction and claim of \citet{gneiting2007strictly}, Section 3.1: given \emph{any} convex function on $\Delta_{\Y}$, \emph{all} of its subtangent rules are proper.
I do not know if proving it would be rigorously possible without much of the development of extended subgradients in this paper.
This circles us back to first paper to state a characterization, \citet{mccarthy1956measures}, which asserts ``any convex function of a set of probabilities may serve'', but, omitting proofs, merely remarks, ``The derivative has to be taken in a suitable generalized sense.''
For general $\P$, the following question becomes interesting: Suppose we are given a convex function $g$ and form a subtangent scoring rule $S$.
Prior characterizations imply that if $S$ is regular, then it is proper.
But under what conditions is $S$ regular?
Theorem \ref{thm:proper-subset-char} gives an answer (see also Figure \ref{fig:scoring-gs}), namely, one must choose certain subgradients of a $g$ that is \emph{interior-locally-Lipschitz} (Definition \ref{def:inter-local}).
\paragraph{Aside: subgradient rules.}
For completeness, we will briefly connect to a different version of the characterization~\citep{mccarthy1956measures,hendrickson1971proper}: a regular scoring rule is proper if and only if it is a \emph{subgradient rule} of a \emph{positively homogeneous}\footnote{$g$ is positively homogeneous if $g(\alpha x) = \alpha g(x)$ for all $\alpha > 0, x \in \mathbb{R}^d$.} convex function $g$, where:
\begin{definition}[Subgradient scoring rule] \label{def:subgrad-score}
Let $\Y$ be finite and $\P \subseteq \Delta_{\Y}$.
A scoring rule $S: \P \times \Y \to \bar{\reals}$ is a \emph{subgradient scoring rule (of $g$)} if
\begin{align}
S(p,y) &= f_p(\delta_y) & \forall p \in \P, y \in \Y \label{eqn:subgrad-score}
\end{align}
for some set of extended subgradients, $f_p$ at each $p \in \P$, of some convex function $g: \mathbb{R}^{\Y} \to \mathbb{R} \cup \{\infty\}$ with $\effdom{g} \supseteq \P$.
\end{definition}
To understand the interplay of the two characterizations, consider the convex function $g(z) = z \cdot \vec{1} - 1 = \sum_{y \in \Y} z(y) - 1$.
In particular, it is zero for $z \in \Delta_{\Y}$.
Its subgradients are $f_p(z) = z \cdot \vec{1}$ for all $z$.
The proper scoring rule $S(p,y) = 0 ~ (\forall p,y)$ is a subtangent rule of $g$, since it is of the form $S(p,y) = g(p) + f_p(\delta_y - p) = 0 + 0 = 0$.
It is \emph{not} a subgradient rule of $g$, since each $f_p(\delta_y) = 1$, whereas $S(p,y) = 0$.
So $S$ instantiates the Savage characterization (subtangent rules) and not the McCarthy characterization (subgradient rules) with respect to this $g$.
However, this all-zero scoring rule is not a counterexample to the McCarthy characterization: it \emph{is} a subgradient rule of some \emph{other} convex function, which is in fact positively homogeneous, i.e. $g(z) = 0$.
Another example: The log scoring rule $S(p,y) = \log p(y)$ is typically presented as a \emph{subtangent} rule of the negative entropy $g(p) = \sum_{y \in \Y} p(y) \log p(y)$ (infinite if $p \not\in \Delta_{\Y}$), but it is not a \emph{subgradient} rule of that function, as pointed out by \citet{marschak1959remarks}.
But, as \citet{hendrickson1971proper} responds, it is a subgradient rule of the positively homogeneous function $g(p) = \sum_{y \in \Y} p(y) \log \left(p(y) / \sum_{y'} p(y')\right)$ defined on the nonnegative orthant.
So in general (Theorem \ref{thm:proper-char}), a proper scoring rule can be written as a subtangent rule of some convex function $g$; and it can also be written as a subgradient rule of a possibly-different one.
\subsection{Tool: extended expected scores}
Before proving the scoring rule characterization, we encounter a problem with the expected score function $S(p;q) := \sum_{y \in \Y} q(y) S(p,y)$ of a regular scoring rule.
Usual characterization proofs proceed by observing that $S(p;q)$ is an affine function of $q$, and that a pointwise supremum over these functions yields the convex $g(q)$ from Definition \ref{def:subtangent-rule} (subtangent rule).
However, $S(p;q)$ is not technically an affine nor affine extended function, as it is only defined on $q \in \Delta_{\Y}$.
Attempting to naively extend it to $q \in \mathbb{R}^{\Y}$ leads to illegal sums.
Therefore, this section formalizes a key fact about regular scoring rules (Lemma \ref{lemma:exp-score-exist}): for fixed $p$, $S(p;q)$ always coincides on $\Delta_{\Y}$ with at least one affine extended function; we call any such an extended expected score of $S$.
\begin{definition}[Extended expected score] \label{def:ext-exp-score}
Let $\Y$ be finite and $\P$ a subset of $\Delta_{\Y}$.
Given a regular scoring rule $S: \P \times \Y \to \mathbb{R} \cup \{-\infty\}$, for each $p \in \P$, an affine extended function $h: \mathbb{R}^{\Y} \to \bar{\reals}$ is an \emph{extended expected score function} of $S$ at $p$ if $h(\delta_y) = S(p,y)$ for all $y \in \Y$.
\end{definition}
The name is justified: $h(q)$ is the expected score for report $p$ under belief $q$.
In fact this holds for any $q \in \Delta_{\Y}$, not just $q \in \P$.
Recall that $S(p;q) := \sum_{y \in \Y} q(y) S(p,y)$
\begin{observation} \label{obs:ext-exp-score}
If $h$ is an extended expected score of a regular $S$ at $p$, then for all $q \in \Delta_{\Y}$, $h(q) = S(p;q)$.
\end{observation}
To prove it, write $h(q) = \beta + f(q-p)$ and recall by Definition \ref{def:ext-exp-score} that $S(p,y) = h(\delta_y)$.
So $S(p;q) = \beta + \sum_y q(y) f(\delta_y - p)$.
By regularity of $S$, the sum never contains $+\infty$, so it is a legal sum and, using the scaling axiom of $f$, it equals $\beta + f(q - p) = h(q)$.
\begin{lemma}[Existence of expected scores] \label{lemma:exp-score-exist}
Let $\Y$ be finite, $\P \subseteq \Delta_{\Y}$, and $S: \P \times \Y \to \mathbb{R} \cup \{-\infty\}$ a regular scoring rule.
Then $S$ has at least one extended expected score function $S_p$ at every $p \in \P$; in fact, it has an extended \emph{linear} one.
\end{lemma}
\begin{myproof}
Given $p$, let $Y_1 = \{y : S(p,y) = -\infty\}$ and $Y_2 = \Y \setminus Y_1$.
We define $S_p: \mathbb{R}^{\Y} \to \bar{\reals}$ via Algorithm \ref{alg:linext} using the following parameters.
Set $t = |Y_1|$.
Let $\hat{f}$ be defined on the subspace spanned by $\{\delta_y : y \in Y_2\}$ via $\hat{f}(x) = \sum_{y \in Y_2} x(y) S(p,y)$.
By definition, $\hat{f}$ is a finite linear function.
If $t \geq 1$, let $v_1,\dots,v_t = \{-\delta_y \st y \in Y_1\}$, in any order.
By Proposition \ref{prop:alg-correct}, $S_p$ is linear extended because it is implemented by Algorithm \ref{alg:linext}.
To show it is an extended expected score function, we calculate $S_p(\delta_y)$ for any given $y \in \Y$.
If $y \in Y_2$, then $v_j \cdot \delta_y = 0$ for all $j=1,\dots,t$ and $S_p(\delta_y) = \hat{f}(\delta_y) = S(p,y)$.
Otherwise, $v_j \cdot \delta_y = 0$ for all $j=1,\dots,t$ except for some $j^*$ where $v_{j^*} = -\delta_y$.
There, $v_{j^*} \cdot \delta_y = -1$, so $S_p(\delta_y) = -\infty = S(p,y)$.
\end{myproof}
\subsection{Scoring rules on the simplex} \label{sec:scoring-simplex}
This section considers $\P = \Delta_{\Y}$ and uses the machinery of linear extended functions to prove a complete characterization of proper scoring rules, including their construction from any convex function and set of extended subgradients.
This arguably improves little on the state of knowledge as in \citet{gneiting2007strictly} and folklore of the field, but formalizes some important previously-informal or unstated facts.
\begin{theorem}[Construction of proper scoring rules] \label{thm:proper-char}
Let $\Y$ be a finite set.
\begin{enumerate}
\item Let $g: \mathbb{R}^{\Y} \to \mathbb{R} \cup \{\infty\}$ be convex with $\effdom{g} \supseteq \Delta_{\Y}$.
Then: (a) it has at least one subtangent rule; and (b) all of its subtangent rules are regular and proper; and (c) if $g$ is strictly convex on $\Delta_{\Y}$, then all of its subtangent rules are strictly proper.
\item Let $S: \Delta_{\Y} \times \Y \to \mathbb{R} \cup \{-\infty\}$ be a regular, proper scoring rule.
Then: (a) it is a subtangent rule of some convex $g$ with $\effdom{g} \supseteq \Delta_{\Y}$; and (b) if $S$ is strictly proper, then any such $g$ is strictly convex on $\Delta_{\Y}$; and (c) in fact, it is a subgradient rule of some (possibly different) positively homogeneous convex $g$.
\end{enumerate}
\end{theorem}
\begin{myproof}
(1)
Given $g$, define any subtangent rule by $S(p,y) = g(p) + f_p(\delta_y - p)$ for any choices of extended subgradients $f_p: \mathbb{R}^{\Y} \to \bar{\reals}$ at each $p \in \Delta_{\Y}$.
Proposition \ref{prop:subgrad-exist} asserts that at least one choice of $f_p$ exists for all $p \in \Delta_{\Y}$, so at least one such function $S$ can be defined, proving \emph{(a)}.
And it is a regular scoring rule (i.e. never assigns score $+\infty$) because $\effdom{g} \supseteq \Delta_{\Y}$ and by definition of extended subgradient, $S(p,y) \leq g(\delta_y)$.
To show $S$ is proper: Immediately, the function $S_p(q) = g(p) + f_p(q-p)$ is an extended expected score of $S$ (Definition \ref{def:ext-exp-score}).
It is an affine extended function supporting $g$ at $p$.
So $S_q(q) \geq S_p(q)$ for all $p \neq q$, so $S$ is proper by definition, completing \emph{(b)}.
Furthermore, if $g$ is strictly convex on $\Delta_{\Y}$, then (for any choices of subgradients) each resulting $S_p$ must support $g$ at only one point in $\Delta_{\Y}$ by Lemma \ref{lemma:strict-subgrad}, so $S_q(q) > S_p(q)$ for all $p \neq q$ and $S$ is strictly proper.
This proves \emph{(c)}.
$(2)$
Given $S$, by Lemma \ref{lemma:exp-score-exist}, for each $p$ there exists a linear extended expected score $S_p$.
Define $g(q) = \sup_{p \in \Delta_{\Y}} S_p(q)$, a convex function by Proposition \ref{prop:sup-char}.
By definition of properness, $S_q(q) = \max_p S_p(q) = g(q)$.
So $S_q$ is an affine extended function supporting $g$ at $q \in \effdom{g}$.
So (e.g. Observation \ref{obs:affine}) it can be written $S_q(x) = g(q) + f_q(x-q)$ for some extended subgradient $f_q$ of $g$ at $q$.
In particular, by definition of extended expected score, $S(p,y) = S_q(\delta_y)$, so $S$ is a subtangent rule of $g$, whose effective domain contains $\Delta_{\Y}$.
This proves \emph{(a)}.
Furthermore, suppose $S$ is strictly proper and a subtangent rule of some $g$.
Then as above, it has affine extended expected score functions $S_q$ at each $q$ and, by strict properness, $S_q(p) < S_p(p) = g(p)$ for all $p \in \Delta_{\Y}$, $p \neq q$.
So at each $p \in \Delta_{\Y}$ there is a supporting affine extended function $S_p$ that supports $g$ nowhere else on $\Delta_{\Y}$.
By Lemma \ref{lemma:strict-subgrad}, this is a necessary and sufficient condition for strict convexity of $g$ on $\Delta_{\Y}$.
This proves \emph{(b)}.
Now looking closer, recall Lemma \ref{lemma:exp-score-exist} guarantees existence of a \emph{linear} extended $S_p$.
So $S_p(\vec{0}) = 0 = g(p) + f_p(-p)$, implying $f_p(p) = g(p) = S_p(p)$.
Then for all $x$, $S_p(x) = S_p(p) + f_p(x-p)$, which rearranges and uses the axioms of linearity (along with finiteness of $S_p(p)$) to get $S_p(x-p) = f_p(x-p)$, or $S_p = f_p$.
Since we have $S(p,y) = S_p(\delta_y) = f_p(\delta_y)$, we get that $S$ is a subgradient rule of $g$.
Furthermore, since each $S_p$ is linear extended, $\alpha g(q) = \alpha \sup_{p \in \Delta_{\Y}} S_p(q) = \sup_{p \in \Delta_{\Y}} \alpha S_p(q) = g(\alpha q)$, for $\alpha \geq 0$.
So $g$ is positively homogeneous, proving \emph{(c)}.
\end{myproof}
Observe that (as is well-known) if $S$ is a subtangent rule of $g$, then $g(q) = S(q;q)$ for all $q$, i.e. $g$ gives the expected score at belief $q$ for a truthful report.
This also implies that $g$ is uniquely defined on $\Delta_{\Y}$.
\vskip1em
One nontrivial example (besides the log scoring rule) is the scoring rule that assigns $k$ points to any prediction with support size $|\Y|-k$, assuming the outcome $y$ is in its support; $-\infty$ if it is not.
This is associated with the convex function $g$ taking the value $k$ on the relative interior of each $|\Y|-k-1$ dimensional face of the simplex; in particular it is zero in the interior of the simplex, $1$ in the relative interior of any facet, \dots, and $|\Y|-1$ at each corner $\delta_{y}$.
A generalization can be obtained from any set function $G: 2^{\Y} \to \mathbb{R}$ that is \emph{monotone}, meaning $X \subseteq Y \implies G(X) \leq G(Y)$.
We can set $g(p) := G(\Supp{p})$.
In other words, the score for prediction $p$ is $G(\Supp{p})$ if the observation $y$ is in $\Supp{p}$; the score is $-\infty$ otherwise.
We utilized this construction in \citet{chen2016informational}.
Of course neither of these is \emph{strictly} proper.
A similar, strictly proper approach is to construct $g$ from any strictly convex and bounded function $g_0$ on the interior of the simplex; then on the interior of each facet, let $g$ coincide with any bounded strictly convex function whose lower bound exceeds the upper bound of $g_0$; and so on.
\vskip1em
\begin{remark} \label{remark:set-notation-char}
In this remark, we restate these results using notation for the sets of regular, proper, strictly proper scoring rules, and so on.
Some readers may find this phrasing and comparison to prior work helpful.
Let $\Reg(\P)$, $\Proper(\P)$, and $\sProper(\P)$ be the sets of regular, proper, and strictly proper scoring rules on $\P$.
Let $\C(\P)$ be the set of convex functions, nowhere $-\infty$, with effective domain containing $\P$.
Let $\ST(G)$ be the subtangent scoring rules of all members of the set of functions $G$.
Then the characterization (prior work) is the statement $\Reg(\Delta_{\Y}) \cap \Proper(\Delta_{\Y}) = \Reg(\Delta_{\Y}) \cap \ST(\C(\Delta_{\Y}))$.
That is, regular rules are proper if and only if they are subtangent rules.
The construction (this work, Theorem \ref{thm:proper-char}) states that $\Proper(\Delta_{\Y}) = \ST(\C(\Delta_{\Y}))$.
In other words, it claims that all subtangent rules on $\Delta_{\Y}$ are regular.
Meanwhile, let $\sC(\P)$ be those $g \in \C(\P)$ that are strictly convex on $\P$, while $\nC(\P)$ is those that are not.
The strict characterization (prior work) is $\Reg(\Delta_{\Y}) \cap \sProper(\Delta_{\Y}) = \Reg(\Delta_{\Y}) \cap \ST(\sC(\Delta_{\Y}))$.
The construction, Theorem \ref{thm:proper-char}, states $\sProper(\Delta_{\Y}) = \ST(\sC(\Delta_{\Y}))$.
Theorem \ref{thm:proper-char} also states $\ST(\nC(\Delta_{\Y})) \cap \ST(\sC(\Delta_{\Y})) = \emptyset$.
Finally, for all $g \in \C(\P)$, it states $\ST(g) \neq \emptyset$.
\end{remark}
\subsection{General belief spaces}
We finally consider general belief and report spaces $\P \subseteq \Delta_{\Y}$.
Here proper scoring rules have been characterized by \citet{gneiting2007strictly} (when $\P$ is convex) and \citet{frongillo2014general} (in general), including for infinite-dimensional outcome spaces.
The statement is that a regular scoring rule $S: \P \times \Y \to \mathbb{R} \cup \{-\infty\}$ is (strictly) proper if and only if it is a subtangent rule of some (strictly) convex $g$ with $\effdom{g} \supseteq \P$.
Again, this characterization leaves open the question of exactly which convex functions produce proper scoring rules on $\P$.
Unlike in the case $\P = \Delta_{\Y}$, not all of them do, as Figure \ref{subfig:shrink-notok} illustrates: suppose $\P = \effdom{g} \subseteq \inter{\Delta_{\Y}}$ and it has only vertical supporting affine extended functions at some $p$ on the boundary of $\P$.
Then the construction $S(p,y) = g(p) + f_p(\delta_y - p)$ will lead to $S(p,y) = +\infty$ for some $y$.
So $S$ will not be regular, so it cannot be proper.
Therefore, the key question is: \textbf{for which convex $g$ are their subtangent rules possibly, or necessarily, regular?}
We next make some definitions to answer this question.
\begin{figure}[ht]
\caption{Let $\Y = \{0,1\}$; the horizontal axes are $\Delta_{\Y}$ parameterized by $p(1)$. Each subfigure gives the domain of $g$ and whether it is interior-locally-Lipschitz, i.e. produces a scoring rule. \textbf{(\ref{subfig:log})} plots the log scoring rule's associated $g(p) = \sum_y p(y) \log p(y)$; others shift/squeeze/truncate the domain. Vertical supports on the interior of the simplex violate interior-local-Lipschitzness and lead to illegal scoring rules, which would assign e.g. $S(p,0) = +\infty$ and $S(p,1) = -\infty$.}
\label{fig:scoring-gs}
\begin{subfigure}{0.24\linewidth}
\resizebox{\linewidth}{!}{\includegraphics{figures/logscore.pdf}}
\caption{\footnotesize $[0,1]$; yes.}
\label{subfig:log}
\end{subfigure}
\hfill
\begin{subfigure}{0.24\linewidth}
\resizebox{\linewidth}{!}{\includegraphics{figures/shrink-notok.pdf}}
\caption{\footnotesize $[0.25, 0.75]$; no.}
\label{subfig:shrink-notok}
\end{subfigure}
\hfill
\begin{subfigure}{0.24\linewidth}
\resizebox{\linewidth}{!}{\includegraphics{figures/shrink-ok.pdf}}
\caption{\footnotesize $(0.25, 0.75)$; yes.}
\end{subfigure}
\hfill
\begin{subfigure}{0.24\linewidth}
\resizebox{\linewidth}{!}{\includegraphics{figures/shift-ok.pdf}}
\caption{\footnotesize $[0.0, 0.5)$; yes.}
\end{subfigure}
\end{figure}
\vskip1em
The following definition will capture extended subgradients that lead to regular scoring rules.
Recall from Proposition \ref{prop:parsim} that a linear extended function has a unique parameterization in Algorithm \ref{alg:linext}.
Because we may have $t=0$ and $\Supp{p} = \Y$, the conditions can be vacuously true.
\begin{definition}[Interior-finite] \label{def:inter-finite}
Say a linear extended function $f: \mathbb{R}^{\Y} \to \bar{\reals}$ is \emph{$p$-interior-finite} for $p \in \Delta_{\Y}$ if its parameterization in Algorithm \ref{alg:linext} has: (1) for all pairs $y,y' \in \Supp{p}$, that $v_j(y) = v_j(y')$ for all $j=1,\dots,t$; and (2) for all $y \not\in \Supp{p}$ and $y' \in \Supp{p}$, that $v_j(y) = v_j(y')$ for some sequence $j=1,\dots,k$, with either $k=t$ or else $v_{k+1}(y) < v_{k+1}(y')$.
\end{definition}
If $f$ is $p$-interior-finite, then property (1) gives that $f(q-p)$ is finite for $q$ in the face of the simplex generated by the support of $p$.
(2) gives that $f(q-p)$ is either finite or $-\infty$ at all $q \in \Delta_{\Y}$ that are not in that face.
These are the key properties that we will need for regular scoring rules.
\begin{lemma}[Interior-finite implies quasi-integrable] \label{lemma:interior-quasi}
Let $p \in \Delta_{\Y}$.
A linear extended function $f: \mathbb{R}^{\Y} \to \bar{\reals}$ is $p$-interior-finite if and only if $f(\delta_y - p) \in \mathbb{R} \cup \{-\infty\}$ for all $y \in \Y$.
Furthermore, in this case $f(\delta_y - p) \in \mathbb{R}$ for any $y \in \Supp{p}$.
\end{lemma}
\begin{myproof}
Let $f$ be a linear extended function and $v_1,\dots,v_t$ its unit vectors in the parameterization of Algorithm \ref{alg:linext}.
$(\implies)$
Suppose $f$ is $p$-interior-finite.
If it is finite everywhere, then the conclusion is immediate.
Otherwise, let $y \in \Supp{p}$.
Using that $v_1(y) = v_1(y')$ for any $y' \in \Supp{p}$, we have
\begin{align*}
v_1 \cdot (\delta_y - p) &= v_1(y) - v_1 \cdot p \\
&= v_1(y) - \sum_{y' \in \Supp{p}} v_1(y') p(y') \\
&= v_1(y) - v_1(y) \sum_{y' \in \Supp{p}} p(y') \\
&= 0 .
\end{align*}
This applies to each successive $v_j$, so $f(\delta_y - p) \in \mathbb{R}$.
Now let $y \not\in \Supp{p}$.
Having just shown $v_1 \cdot (\delta_{y'} - p) = 0$ for any $y' \in \Supp{p}$, we have $v_1 \cdot (\delta_y - p) = v_1 \cdot (\delta_y - \delta_{y'}) = v_1(y) - v_1(y') \leq 0$ by assumption of $p$-interior-finite.
So either Algorithm \ref{alg:linext} returns $-\infty$ in the first iteration, or we continue to the second iteration.
The same argument applies at each iteration, so $f(\delta_y - p) \in \mathbb{R} \cup \{-\infty\}$.
$(\impliedby)$
Suppose $f(\delta_y - p) \in \mathbb{R} \cup \{-\infty\}$ for all $y \in \Y$.
Again if $f$ is finite everywhere, it is $p$-interior-finite, QED.
So suppose its depth is at least $1$.
We must have $v_1 \cdot (\delta_y - p) \leq 0$ for all $y \in \Y$.
This rearranges to $v_1(y) \leq v_1 \cdot p$ for all $y$, or $\max_{y \in \Y} v_1(y) \leq v_1 \cdot p$.
But $p$ is a probability distribution, so $v_1 \cdot p \leq \max_{y \in \Y} v_1(y)$.
So $v_1 \cdot p = \max_{y \in \Y} v_1(y)$.
This implies that $v_1(y) = v_1(y') = v_1 \cdot p$ for all $y,y' \in \Supp{p}$.
So $v_1 \cdot (\delta_y - p) = 0$ if $y \in \Supp{p}$.
Meanwhile, if $y \not\in \Supp{p}$, then $v_1(y) \leq v_1 \cdot p = v_1(y')$ for any $y' \in \Supp{p}$.
This argument repeats to prove that if $y \in \Supp{p}$, then for all $j=1,\dots,t$, $v_j(y) = \max_{y' \in \Y} v_j(y')$ and also $v_j \cdot (\delta_y - p) = 0$.
So it also gives $f(\delta_y - p) \in \mathbb{R}$.
Meanwhile, if $y \not\in \Supp{p}$, then we have a sequence $v_1(y) = \max_{y' \in \Y} v_1(y')$, \dots, $v_k(y) = \max_{y' \in \Y} v_k(y')$, where in each case $v_j \cdot (\delta_y - p) = 0$.
Then finally, either $k=t$ and $f(\delta_y - p) \in \mathbb{R}$, or we have $v_{k+1}(y) < \max_{y' \in \Y} v_{k+1}(y')$ and $f(\delta_y - p) = -\infty$.
\end{myproof}
We can now make the key definition for the scoring rule characterization.
\begin{definition}[Interior-locally-Lipschitz] \label{def:inter-local}
For $\P \subseteq \Delta_{\Y} \cap \effdom{g}$, say $g: \mathbb{R}^{d} \to \mathbb{R} \cup \{\infty\}$ is \emph{$\P$-interior-locally-Lipschitz} if $g$ has at every $p \in \P$ a $p$-interior-finite extended subgradient.
\end{definition}
An interior-locally-Lipschitz $g$ simply has two conditions on its vertical supporting hyperplanes.
First, they cannot cut faces of the simplex whose relative interior intersects $\effdom{g}$; they can only contain them.
This is enforced by $v_j(y) = v_j(y')$ for $y,y' \in \Supp{p}$.
In other words, the extended subgradients must be \emph{finite} (in particular bounded, an analogy of causing $g$ to be Lipschitz) relative to the affine hull of these faces.
Second, they must be oriented correctly so as not to cut that face from the rest of the simplex.
This is enforced by $v_j(y) \leq v_j(y')$ for $y \not\in \Supp{p}$, $y' \in \Supp{p}$.
Lemma \ref{lemma:interior-quasi} gives the following corollary, in which the word ``regular'' is the point:
\begin{corollary} \label{cor:lip-reg}
A convex $g: \mathbb{R}^{\Y} \to \mathbb{R} \cup \{\infty\}$ with $\effdom{g} \supseteq \P$ has a regular subtangent scoring rule if and only if it is $\P$-interior-locally-Lipschitz.
\end{corollary}
\begin{myproof}
$(\implies)$
Suppose $g$ has a regular subtangent scoring rule $S(p,y) = g(p) + f_p(\delta_y - p)$, where $f_p$ is an extended subgradient of $g$ at $p$.
By regularity, $f_p(\delta_y - p) \in \mathbb{R} \cup \{-\infty\}$, so by Lemma \ref{lemma:interior-quasi}, $f_p$ is $p$-interior-finite.
So $g$ has a $p$-interior-finite subgradient at every $p \in \P$, so it is $\P$-interior-locally-Lipschitz.
$(\impliedby)$
Suppose $g$ is $\P$-interior-locally-Lipschitz; then it has at each $p \in \P$ a $p$-interior-finite subgradient $f_p$.
Let $S(p,y) = g(p) + f_p(\delta_y - p)$; by definition of $p$-interior-finite, $f_p(\delta_y - p)$ is never $+\infty$, so $S$ is regular, and it is a subtangent score.
\end{myproof}
\begin{theorem}[Construction of scoring rules on $\P$] \label{thm:proper-subset-char}
Let $\Y$ be a finite set and $\P \subseteq \Delta_{\Y}$.
\begin{enumerate}
\item Let $g: \mathbb{R}^{\Y} \to \mathbb{R} \cup \{\infty\}$ be convex and $\P$-interior-locally-Lipschitz with $\effdom{g} \supseteq \P$.
Then: (a) $g$ has at least one subtangent scoring rule that is regular; and (b) all of its subtangent scoring rules that are regular are proper; and (c) if $g$ is strictly convex on $\P$, then all of its regular subtangent rules are strictly proper.
\item Let $S: \P \times \Y \to \mathbb{R} \cup \{-\infty\}$ be a regular proper scoring rule.
Then: (a) it is a subtangent rule of some convex $\P$-interior-locally-Lipschitz $g: \mathbb{R}^{\Y} \to \mathbb{R} \cup \{\infty\}$ with $\effdom{g} \supseteq \P$; and (b) if $S$ is strictly proper and $\P$ a convex set, then any such $g$ is strictly convex on $\P$.
\end{enumerate}
\end{theorem}
Observe that in (1), $g$ may have some subtangent rules that are regular and some that are not, depending on the choices of extended subgradients at each $p$.
(1) only claims that there exists at least one choice that is regular (and therefore proper).
An example is if $\P = \effdom{g}$ is a single point, say the uniform distribution; vertical tangent planes will not work, but finite subgradients will.
(This issue did not arise in the previous section where $\P = \Delta_{\Y}$; there, all of a convex $g$'s subtangent rules were regular.)
\begin{myproof}
(1)
Let such a $g$ be given.
We first prove that there exists a regular subtangent scoring rule of $g$.
At each $p \in \P$, let $f_p$ be a $p$-interior-finite extended subgradient of $g$, guaranteed by definition of interior-locally-Lipschitz.
Let $S: \P \times \Y \to \bar{\reals}$ be of the form $S(p,y) = g(p) + f_p(\delta_y - p)$.
Applying Lemma \ref{lemma:interior-quasi}, $f_p(\delta_y - p) \in \mathbb{R} \cup \{-\infty\}$, so $S(p,y) \neq \infty$, so it is a regular scoring rule, proving \emph{(a)}.
Now, let $S$ be \emph{any} regular subtangent scoring rule of $g$; we show $S$ is proper.
Immediately we have an affine expected score function at each $p$ satisfying $S_p(q) = g(p) + f_p(q-p)$ for all $p \in \P$ and all $q \in \Delta_{\Y}$.
We have $S_p(p) \geq S_q(p)$ for all $q$ because $S_p$ supports $g$ at $p$, so $S$ is proper, proving \emph{(b)}.
Furthermore, if $g$ is strictly convex, then each $S_q$ supports $g$ at just one point in its effective domain (Lemma \ref{lemma:strict-subgrad}).
So $S_p(p) > S_q(p)$ for $p,q \in \P$ with $q \neq p$, so $S$ is strictly proper, proving \emph{(c)}.
(2)
Let a regular proper scoring rule $S$ be given.
It has by Lemma \ref{lemma:exp-score-exist} an affine extended expected score function $S_p$ at each $p \in \P$.
Define $g: \mathbb{R}^{\Y} \to \mathbb{R} \cup \{\infty\}$ by $g(q) = \sup_{p \in \P} S_p(q)$.
This is convex by Proposition \ref{prop:sup-char}.
By definition of proper, for all $p \in \P$, we have $S_p(p) = \max_{q \in \P} S_q(p) = g(p)$.
So $S_p$ is an affine extended function supporting $g$ at $p$, so it can be written $S_p(q) = g(p) + f_p(q-p)$ for some subgradient $f_p$ of $g$ at $p$.
By definition of extended expected score, $S(p,y) = S_p(\delta_y) = g(p) + f_p(\delta_y - p)$, so it is a subtangent rule.
Next, since $S$ is a scoring rule, in particular $S_p(\delta_y) \in \mathbb{R} \cup \{-\infty\}$ for all $y$.
So by Lemma \ref{lemma:interior-quasi}, each $f_p$ is $p$-interior-finite, so $g$ is interior-locally-Lipschitz, proving \emph{(a)}.
Now suppose $S$ is strictly proper and $\P$ a convex set.
Then for any $q \in \P$, we have that $S_q$ supports $g$ at $q$ alone out of all points in $\P$, by definition of strict properness.
By Lemma \ref{lemma:strict-subgrad}, $g$ is strictly convex on $\P$, proving \emph{(b)}.
\end{myproof}
There remain open directions investigating interior-locally-Lipschitz functions.
Two important cases can be immediately resolved:
\begin{itemize}
\item If $\P \subseteq \inter{\Delta_{\Y}}$, then $g$ is $\P$-interior-locally-Lipschitz if $g$ is subdifferentiable on $\P$, i.e. has finite subgradients everywhere on $\P$.
This follows because every $p \in \P$ has full support.
More rigorously, this must be true of $g$ when restricted to the affine hull of the simplex; its subgradients can still be infinite perpendicular to the simplex, e.g. of the form $f(x) = \infty \cdot (x \cdot \vec{1}) + \hat{f}(x)$.
\item If $\P$ is an open set relative to the affine hull of $\Delta_{\Y}$, then every convex function $g$ with $\effdom{g} \supseteq \P$ is $\P$-interior-locally-Lipschitz.
This follows from the previous case along with the fact that convex functions are subdifferentiable on open sets in their domain.
\end{itemize}
One question for future work is whether one can optimize efficiently over the space of interior-locally-Lipschitz $g$, for a given set $\P \subseteq \Delta_{\Y}$.
This could be useful in constructing proper scoring rules that optimize some objective.
\vskip1em
Finally, we restate the general-$\P$ construction using the notation of Remark \ref{remark:set-notation-char}.
\begin{remark} \label{remark:set-notation-char-2}
Let $\L(\P)$ be the set of $\P$-interior-locally-Lipschitz convex $g$ with effective domain containing $\P$, with $\sL(\P)$ those that are strictly convex on $\P$ and $\nL(\P)$ those that are not.
The characterization (prior work) states $\Reg(\P) \cap \Proper(\P) = \Reg(\P) \cap \ST(\C(\P))$; the construction (Theorem \ref{thm:proper-subset-char}) states $\Proper(\P) = \Reg(\P) \cap \ST(\L(\P))$.
In other words, if a subtangent rule of $g$ is regular, then $g$ is $\P$-interior-locally-Lipschitz.
Similarly, the characterization states $\Reg(\P) \cap \sProper(\P) = \Reg(\P) \cap \ST(\sC(\P))$, while Theorem \ref{thm:proper-subset-char} states $\sProper(\P) = \Reg(\P) \cap \ST(\sL(\P))$.
Theorem \ref{thm:proper-subset-char} furthermore states that $\Reg(\P) \cap \ST(\nL(\P)) \cap \ST(\sL(\P)) = \emptyset$, i.e. strict properness exactly corresponds to strict convexity.
Finally, it also states that for any particular $g$, $\Reg(\P) \cap \ST(g) \neq \emptyset$ if and only if $g \in \L(\P)$, i.e. $g$ has a regular subtangent rule if and only if it is convex and $\P$-interior-locally-Lipschitz.
\end{remark}
\section{Extended subgradients}
Recall that a \emph{subgradient} of a function $g$ at a point $x_0$ is a (finite) linear function $f$ satisfying $g(x) \geq g(x_0) + f(x-x_0)$ for all $x$.
The following definition of extended subgradient replaces $f$ with a linear extended function; to force sums to be legal, we will only define it at points in the effective domain of a ``proper'' function, i.e. one that is never $-\infty$.
A main aim of this section is to show that a proper function is convex if and only if it has an extended subgradient everywhere in its effective domain (Proposition \ref{prop:subgrad-char}).
\begin{definition}[Extended subgradient] \label{def:extsub}
Given a function $g: \mathbb{R}^d \to \mathbb{R} \cup \{\infty\}$, a linear extended function $f$ is an \emph{extended subgradient} of $g$ at a point $x_0$ in its effective domain if, for all $x \in \mathbb{R}^d$, $g(x) \geq g(x_0) + f(x - x_0)$.
\end{definition}
Again for clarity, if this $f$ is a finite linear function, we may call it a \emph{finite} subgradient of $g$.
Note that the sum appearing in Definition \ref{def:extsub} is always legal because $g(x_0) \in \mathbb{R}$ under its assumptions.
\vskip1em
As is well-known, finite subgradients correspond to supporting hyperplanes of the epigraphs of convex functions $g$, but they do not exist at points where all such hyperplanes are vertical; consider, in one dimension,
\begin{equation}
g(x) = \begin{cases}
0 & x < 1 \\
1 & x = 1 \\
\infty & x > 1. \end{cases} \label{eqn:example-1d-g}
\end{equation}
We next show that every $g$ has an \emph{extended} subgradient everywhere in its effective domain.
For example, $f(z) = \infty \cdot z$ is an extended subgradient of the above $g$ at $x=1$.
\begin{proposition}[Existence of extended subgradients] \label{prop:subgrad-exist}
Each convex function $g: \mathbb{R}^d \to \mathbb{R} \cup \{\infty\}$ has an extended subgradient at every point in its effective domain.
\end{proposition}
\begin{myproof}
Let $x$ be in the effective domain $G$ of $g$; we construct an extended subgradient $f_x$.
The author finds the following approach most intuitive:
Define $g_x(z) = g(x+z) - g(x)$, the shift of $g$ that moves $(x,g(x))$ to $(\vec{0},0)$.
Let $G_x = \effdom{g_x}$.
Now we appeal to a technical lemma, Lemma \ref{lemma:conv-0-minorized} (next), which says there exists a linear extended $f_x$ that minorizes $g_x$.
This completes the proof: For any $x' \in \mathbb{R}^d$, we have $f_x(x'-x) \leq g_x(x'-x)$, which rearranges to give $g(x) + f_x(x'-x) \leq g(x')$ (using that $g(x) \in \mathbb{R}$ by assumption).
\end{myproof}
\begin{lemma} \label{lemma:conv-0-minorized}
Let $g_x : \mathbb{R}^d \to \mathbb{R} \cup \{\infty\}$ be a convex function with $g_x(\vec{0}) = 0$.
Then there exists a linear extended function $f_x$ with $f_x(z) \leq g_x(z)$ for all $z \in \mathbb{R}^d$.
\end{lemma}
\begin{myproof}
First we make Claim (*): if $\vec{0}$ is in the interior of the effective domain $G_x$ of $g_x$, then such an $f_x$ exists.
This follows because $g_x$ necessarily has at $\vec{0}$ a finite subgradient (\citet{rockafellar1970convex}, Theorem 23.4), which we can take to be $f_x$.
Now, we prove the result by induction on $d$.
If $d=0$, then it is only possible to have $f_x = g_x$.
If $d \geq 1$, there are two cases.
If $\vec{0}$ is in the interior of $G_x$, then Claim (*) applies and we are done.
Otherwise, $\vec{0}$ is in the boundary of the convex set $G_x$.
This implies\footnote{Definition and existence of a supporting hyperplane are referred to e.g. in \citet{hiriarturrut2001fundamentals}: Definition 1.1.2 of a hyperplane (there $v$ must merely be nonzero, but it is equivalent to requiring a unit vector); Definition 2.4.1 of a supporting hyperplane to $G_x$ at $x$, specialized here to the case $x = \vec{0}$; and Lemma 4.2.1, giving existence of a supporting hyperplane to any nonempty convex set $G_x$ at any boundary point.} that there exists a hyperplane supporting $G_x$ at $\vec{0}$.
That is, there is some unit vector $v \in \mathbb{R}^d$ such that $G_x \subseteq \{z \st v \cdot z \leq 0\}$.
Set $f_x(z) = \infty$ if $v \cdot z > 0$ and $f_x(z) = -\infty$ if $v \cdot z < 0$.
Note $f_x$ minorizes $g$ on these regions, in the first case because $g_x(z) = \infty = f_x(z)$ (by definition of effective domain), and in the second case because $f_x(z) = -\infty$.
So it remains to define $f_x$ on the subspace $S := \{ z \st v \cdot z= 0\}$ and show that it minorizes $g_x$ there.
But $g_x$, restricted to this subspace, is again a proper convex function equalling $0$ at $\vec{0}$, so by induction we have a minorizing linear extended function $\hat{f_x}$ on this $d-1$ dimensional subspace.
Define $f_x(z) = \hat{f_x}(z)$ for $z \in S$.
Then $f_x$ minorizes $g_x$ everywhere.
We also have that $f_x$ is linear extended by Lemma \ref{lemma:recursive}, as it satisfies the necessary recursive format.
\end{myproof}
This proof would have gone through if Claim (*) used ``relative interior'' rather than interior, because supporting hyperplanes also exist at the relative boundary.
That version would have constructed extended subgradients of possibly lesser depth (Definition \ref{def:depth}).
\vskip1em
It is now useful to define \emph{affine extended functions}.
Observe that a bad definition would be ``a linear extended function plus a constant.''
Both vertical and \emph{horizontal} shifts of linear extended functions must be allowed in order to capture, e.g.
\begin{equation}
h(x) = \begin{cases} -\infty & x < 1 \\
1 & x = 1 \\
\infty & x > 1 . \end{cases} \label{eqn:example-1d-h}
\end{equation}
\begin{definition}[Affine extended function] \label{def:affext}
A function $h: \mathbb{R}^d \to \bar{\reals}$ is \emph{affine extended} if for some $\beta \in \mathbb{R}$ and $x_0 \in \mathbb{R}^d$ and some linear extended $f: \mathbb{R}^d \to \bar{\reals}$ we have $h(x) = f(x - x_0) + \beta$.
\end{definition}
\begin{definition}[Supports] \label{def:supports}
An affine extended function $h$ \emph{supports} a function $g: \mathbb{R}^d \to \bar{\reals}$ at a point $x_0 \in \mathbb{R}^d$ if $h(x_0) = g(x_0)$ and for all $x \in \mathbb{R}^d$, $h(x) \leq g(x)$.
\end{definition}
For example, the above $h$ in Display \ref{eqn:example-1d-h} supports, at $x_0=1$, the previous example $g$ of Display \ref{eqn:example-1d-g}.
Actually, $h$ supports $g$ at all $x_0 \geq 1$.
\begin{observation} \label{obs:affine}
Given $\beta,x_0,f$, define the affine extended function $h(x) = f(x-x_0) + \beta$.
\begin{enumerate}
\item $h$ is convex: its epigraph is a shift of a linear extended function's.
\item Let $g: \mathbb{R}^d \to \mathbb{R} \cup \{\infty\}$ such that $x_0 \in \effdom{g}$.
Then $h$ supports $g$ at $x_0$ if and only if $f$ is an extended subgradient of $g$ at $x_0$.
\item If $h(x_1)$ is finite, then for all $x$ we have $h(x) = f(x-x_1) + \beta_1$ where $\beta_1 = f(x_1-x_0) + \beta$.
\end{enumerate}
\end{observation}
Point 2 follows because $h$ supports $g$ at $x_0$ if and only if $\beta = g(x_0)$ and $g(x) \geq \beta + f(x-x_0)$ for all $x$.
Point 3 follows because $h(x_1) = f(x_1 - x_0) + \beta$, so $f(x_1 - x_0)$ is finite: the sum $f(x-x_1) + f(x_1-x_0)$ is always legal and equals $f(x-x_0)$.
\vskip1em
An important fact in convex analysis is that a \emph{closed} (i.e. lower semicontinuous) convex function $g$ is the pointwise supremum of a family of affine functions.
We have the following analogue.
\begin{proposition}[Supremum characterization] \label{prop:sup-char}
A function $g: \mathbb{R}^d \to \mathbb{R} \cup \{\infty\}$ is convex if and only if it is the pointwise supremum of a family of affine extended functions.
\end{proposition}
\begin{myproof}
$(\impliedby)$
Let $H$ be a family of affine extended functions and let $g(x) := \sup_{h \in H} h(x)$.
The epigraph of each affine extended function is a convex set (Observation \ref{obs:affine}), and $g$'s epigraph is the intersection of these, hence a convex set, so $g$ is convex.
$(\implies)$ Suppose $g$ is convex; let $H$ be the set of affine extended functions minorizing $g$.
We show $g(x) = \sup_{h \in H} h(x)$.
First consider $x$ in the effective domain $G$ of $g$.
By Proposition \ref{prop:subgrad-exist}, $g$ has an extended subgradient at $x$, which implies (Observation \ref{obs:affine}) it has a supporting affine function $h_x$ there, and of course $h_x \in H$.
We have $g(x) = h_x(x)$ and by definition $g(x) \geq h(x)$ for all $h \in H$, so $g(x) = \max_{h \in H} h(x)$.
Finally, let $x \not\in G$.
We must show that $\sup_{h \in H} h(x) = \infty$.
We apply a technical lemma, Lemma \ref{lemma:sup-affine-indicates} (stated and proven next), to obtain a set $H'$ of affine extended functions that all equal $-\infty$ on $G$ (hence $H' \subseteq H$), but for which $\sup_{h \in H'} h(x) = \infty$, as required.
\end{myproof}
\begin{lemma} \label{lemma:sup-affine-indicates}
Let $G$ be a convex set in $\mathbb{R}^d$ and let $x \not\in G$.
There is a set $H'$ of affine extended functions such that $\sup_{h \in H'} h(x) = \infty$ while $h(x') = -\infty$ for all $h \in H$ and $x' \in G$.
\end{lemma}
For intuition, observe that the easiest case is if $x \not\in \cl{G}$, when we can use a strongly separating hyperplane to get a single $h$ that is $\infty$ at $x$ and $-\infty$ on $G$.
The most difficult case is when $x$ is an extreme, but not exposed, point of $\cl{G}$.
(Picture a convex $g$ with effective domain $\cl{G}$; now modify $g$ so that $g(x) = \infty$.)
Capturing this case requires the full recursive structure of linear extended functions.
\begin{myproof}
Let $G' = \{x' - x \st x' \in G\}$.
Note that $G'$ is a convex set not containing $\vec{0}$.
By Lemma \ref{lemma:linext-indicates}, next, there is a linear extended function $f$ equalling $-\infty$ on $G'$.
Construct $H = \{ h(x') = f(x'-x) + \beta \st \beta \in \mathbb{R}\}$.
Then $\sup_{h \in H} h(x) = \sup_{\beta \in \mathbb{R}} \beta = \infty$, while for $x' \in G$ we have $h(x') = f(x'-x) + \beta = -\infty$.
This proves Lemma \ref{lemma:sup-affine-indicates}.
\end{myproof}
\begin{lemma} \label{lemma:linext-indicates}
Given a convex set $G' \subseteq \mathbb{R}^d$ not containing $\vec{0}$, there exists a linear extended $f$ with $f(x') = -\infty$ for all $x' \in G'$.
\end{lemma}
\begin{myproof}
Note that if $G' = \emptyset$, then the claim is immediately true; and if $d=0$, then $G'$ must be empty.
Now by induction on $d$:
If $d=1$, then we can choose $f(z) = \infty \cdot z$ or else $f(z) = -\infty \cdot z$; one of these must be $-\infty$ on the convex set $G'$.
Now suppose $d \geq 2$; we construct $f$.
The convex hull of $G' \cup \{\vec{0}\}$ must be supported at $\vec{0}$, which is a boundary point\footnote{If $\vec{0}$ were in the interior of this convex hull, then it would be a convex combination of other points in the convex hull, so it would be a convex combination of points in $G'$, a contradiction. For the supporting hyperplane argument, see the above footnote, using reference \citet{hiriarturrut2001fundamentals}.}, by some hyperplane.
Let this hyperplane be parameterized by the unit vector $v_1$ where $G' \subseteq \{z \st v_1 \cdot z \leq 0\}$.
Set $f(z) = \infty$ if $v_1 \cdot z > 0$ and $-\infty$ if $v_1 \cdot z < 0$.
It remains to define $f$ on the subspace $S := \{z \st v_1 \cdot z = 0\}$.
Immediately, $G' \cap S$ is a convex set not containing $\vec{0}$.
So by inductive hypothesis, there is a linear extended $\hat{f}: \mathbb{R}^{d-1} \to \bar{\reals}$ that is $-\infty$ everywhere on $G' \cap S$.
Set $f(z) = \hat{f}(z)$ on $S$.
Now we have $f$ is linear extended because it satisfies the recursive hypotheses of Lemma \ref{lemma:recursive}.
We also have that $f$ is $-\infty$ on $G'$: each $z \in G'$ either has $v_1 \cdot z < 0$ or $z \in S$, and both cases have been covered.
\end{myproof}
Consider the convex function in one dimension,
\[ g(x) = \begin{cases}
0 & x < 1 \\
\infty & x \geq 1 . \end{cases} \]
For the previous proof to obtain $g$ as a pointwise supremum of affine extended functions, it needed a sequence of the form
\[ h(x) = \begin{cases}
-\infty & x < 1 \\
\beta & x = 1 \\
\infty & x > 1 . \end{cases} \]
Indeed, one can show that $g$ has no affine extended supporting function at $x=1$.
But it does everywhere else: in particular, $h$ supports $g$ at all $x > 1$.
So we might have hoped that each convex function is the pointwise \emph{maximum} of affine extended functions; but the example of $g$ and $x=1$ prevents this.
This does hold on $g$'s effective domain (implied by existence of extended subgradients there, Proposition \ref{prop:subgrad-exist}).
Speaking of which, we are now ready to prove the converse.
\begin{proposition}[Subgradient characterization] \label{prop:subgrad-char}
A function $g: \mathbb{R}^d \to \mathbb{R} \cup \{\infty\}$ is convex if and only if it has an extended subgradient at every point in its effective domain.
\end{proposition}
\begin{myproof}
$(\implies)$
This is Proposition \ref{prop:subgrad-exist}.
$(\impliedby)$
Suppose $g$ has an extended subgradient at each point in its effective domain $G$.
We will show it is the pointwise supremum of a family $H$ of affine extended functions, hence (Proposition \ref{prop:sup-char}) convex.
First, let $H^*$ be the set of affine extended functions that support $g$.
Then $H^*$ must include a supporting function at each $x$ in the effective domain (Observation \ref{obs:affine}).
Now we repeat a trick: for each $x \not\in G$, we obtain from Lemma \ref{lemma:sup-affine-indicates} a set $H_x$ of affine extended functions that are all $-\infty$ on $G$, but for which $\sup_{h \in H_x} h(x) = \infty$.
Letting $H = H^*$ unioned with $\bigcup_{x \not\in G} H_x$, we claim $g(x) = \sup_{h \in H} h(x)$.
First, every $h \in H$ minorizes $g$: this follows immediately for $h \in H^*$, and by definition of effective domain for each $H_x$.
For $x \in G$, the claim then follows because $H^*$ contains a supporting function with $h(x) = g(x)$.
For $x \not\in G$, the claim follows by construction of $H_x$.
So $g(x) = \sup_{h \in H} h(x)$.
\end{myproof}
Finally, characterizations of strict convexity will be useful.
Say an affine extended function is \emph{uniquely supporting} of $g$ at $b \in \P \subseteq \effdom{g}$, with respect to $\P$, if it supports $g$ at $b$ and at no other point in $\P$.
\begin{lemma}[Strict convexity] \label{lemma:strict-subgrad}
Let $g: \mathbb{R}^d \to \mathbb{R} \cup \{\infty\}$ be convex, and let $\P \subseteq \effdom{g}$ be a convex set.
The following are equivalent: (1) $g$ is strictly convex on $\P$; (2) no two distinct points in $\P$ share an extended subgradient; (3) every affine extended function supports $g$ in $\P$ at most once; (4) $g$ has at each $x \in \P$ a uniquely supporting affine extended function.
\end{lemma}
Although the differences between (2,3,4) are small, they may be useful in different scenarios.
For example, proving (4) may be the easiest way to show $g$ is strictly convex, whereas (3) may be the more powerful implication of strict convexity.
\begin{myproof}
We prove a ring of contrapositives.
$(\lnot 1 ~ \implies ~ \lnot 4)$
If $g$ is convex but not strictly convex on $\P \subseteq \effdom{g}$, then there are two points $a,c \in \P$ with $a \neq c$ and there is $0 < \rho < 1$ such that, for $b = \rho \cdot a + (1-\rho)c$, $g(b) = \rho g(a) + (1-\rho)) g(c)$.
We show $g$ has no uniquely supporting $h$ at $b$; intuitively, tangents at $b$ must be tangent at $a$ and $c$ as well.
Let $h$ be any supporting affine extended function at $b$ and write $h(x) = g(b) + f(x-b)$.
Because $h$ is supporting, $g(c) \geq h(c) = g(b) + f(c-b)$, implying with the axioms of linear extended functions that $f(b-c) \geq g(b) - g(c)$.
Now, $b-c = \rho(a-c) = \frac{\rho}{1-\rho}(a-b)$.
So $f(b-c) = \frac{\rho}{1-\rho}f(a-b)$.
Similarly, $g(b) - g(c) = \rho(g(a) - g(c)) = \frac{\rho}{1-\rho}(g(a) - g(b))$.
So $f(a-b) \geq g(a) - g(b)$, so $h(a) = g(b) + f(a-b) \geq g(a)$.
But because $h$ supports $g$, we also have $h(a) \leq g(a)$, so $h(a) = g(a)$ and $h$ supports $g$ at $a$.
$(\lnot 4 ~ \implies ~ \lnot 3)$
Almost immediate: $g$ has a supporting affine extended function at every point in $\P$ because it has an extended subgradient everywhere (Proposition \ref{prop:subgrad-exist}).
If $\lnot 4$, then one supports $g$ at two points in $\P$.
$(\lnot 3 ~ \implies ~ \lnot 2)$
Suppose an affine extended $h$ supports $g$ at two distinct points $a,b \in \P$.
Using Observation \ref{obs:affine}, we can write $h(x) = f(x - a) + g(a) = f(x-b) + g(b)$ for a unique linear extended $f$, so $f$ is an extended subgradient at two distinct points.
$(\lnot 2 ~ \implies ~ \lnot 1)$
Suppose $f$ is an extended subgradient at distinct points $a,c \in \P$.
By definition of extended subgradient, we have $g(a) \geq g(c) + f(a-c)$ and $g(c) \geq g(a) - f(a-c)$, implying $f(a-c) = g(a) - g(c)$.
Now let $b = \rho \cdot a + (1-\rho) c$ for any $0 < \rho < 1$.
By convexity and linear extended axioms, $g(b) \geq g(a) - f(a-b) = g(a) - f((1-\rho)(a-c)) = g(a) - (1-\rho)(g(a) - g(c)) = \rho g(a) + (1-\rho)g(c)$.
So $g$ is not strictly convex.
\end{myproof}
\paragraph{The extended subdifferential.}
Although its topological properties and general usefulness are unclear, we should probably not conclude without defining the \emph{extended subdifferential} of the function $g: \mathbb{R}^d \to \mathbb{R} \cup \{\infty\}$ at $x \in \effdom{g}$ to be the (nonempty) set of extended subgradients of $g$ at $x$.
On the interior of $\effdom{g}$, the extended subdifferential can only contain finite subgradients.
But note that if the effective domain has affine dimension smaller than $\mathbb{R}^d$, then the epigraph is contained in one of its vertical tangent supporting hyperplanes.
In this case, the relative interior, although it has finite sugradients, will have extended ones as well that take on non-finite values outside the effective domain.
| {
"timestamp": "2021-02-19T02:25:56",
"yymm": "2102",
"arxiv_id": "2102.09552",
"language": "en",
"url": "https://arxiv.org/abs/2102.09552",
"abstract": "This note investigates functions from $\\mathbb{R}^d$ to $\\mathbb{R} \\cup \\{\\pm \\infty\\}$ that satisfy axioms of linearity wherever allowed by extended-value arithmetic. They have a nontrivial structure defined inductively on $d$, and unlike finite linear functions, they require $\\Omega(d^2)$ parameters to uniquely identify. In particular they can capture vertical tangent planes to epigraphs: a function (never $-\\infty$) is convex if and only if it has an extended-valued subgradient at every point in its effective domain, if and only if it is the supremum of a family of \"affine extended\" functions. These results are applied to the well-known characterization of proper scoring rules, for the finite-dimensional case: it is carefully and rigorously extended here to a more constructive form. In particular it is investigated when proper scoring rules can be constructed from a given convex function.",
"subjects": "Statistics Theory (math.ST); Computer Science and Game Theory (cs.GT)",
"title": "Linear Functions to the Extended Reals",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.98028087477309,
"lm_q2_score": 0.7217432003123989,
"lm_q1q2_score": 0.7075110557637679
} |
https://arxiv.org/abs/1009.4227 | On piecewise linear cell decompositions | In this note, we introduce a class of cell decompositions of PL manifolds and polyhedra which are more general than triangulations yet not as general as CW complexes; we propose calling them PLCW complexes. The main result is an analog of Alexander's theorem: any two PLCW decompositions of the same polyhedron can be obtained from each other by a sequence of certain "elementary" moves.This definition is motivated by the needs of Topological Quantum Field Theory, especially extended theories as defined by Lurie. | \section{Introduction}\label{s:intro}
One of the main tools for studying piecewise-linear manifolds is the
notion of triangulation, or more generally, cell complexes formed by
convex cells. However, for many purposes this is too restrictive. For
example, for any explicit computation of state-sum invariants of
3-manifolds, triangulations turn out to be a very inefficient tool: the
number of simplices is necessarily quite large, a cylinder over a
triangulated manifold (or, more generally, a product of two triangulated
manifolds) does not have a canonical triangulation, etc. Allowing
arbitrary convex cells helps but does not solve all the problems: for example, a cell
decomposition shown below (which is quite useful for extended
topological field theories and 2-categories, as it illustrates a
2-morphism between two 1-morphisms) can not be realized using only convex
cells.
$$
\tzTwoGlobe
$$
In addition, for
many constructions it would be desirable to allow ``singular
triangulations'', where the different faces of the same cell are allowed
to be glued to each other (for example, this would allow a cell
decomposition of the torus $T^2$ obtained by gluing opposite sides of a
rectangle). On the other hand, CW complexes are too general and using them
creates other problems: for example, there is no analog of Alexander's
theorem describing simple moves necessary to obtain one CW cell decomposition
from another.
In this note, motivated by the author's earlier work with
Balsam~\cite{balsam-kirillov}, we
introduce a new notion of a cell decomposition of a compact polyhedron (in
particular, a PL manifold) which will address many of the problems mentioned
above. We propose calling such cell decompositions PLCW cell
decompositions. We also prove an analog of Alexander's theorem: any two
PLCW decompositions of the same polyhedron can be obtained from each other
by a sequence of certain ``elementary'' moves; these moves are special
cases of cell moves introduced by Oeckl~\cite{oeckl}.
\subsection*{Acknowledgments}
The authors would like to thank
Oleg Viro, Scott Morrison, Benjamin Balsam and Robert Oeckl
for helpful suggestions and discussions.
This work was partially suported by NSF grant DMS-0700589.
\section{Basic definitions}\label{s:basic}
In this section we recall some basic definitions and facts
of PL topology, following notation and terminology of Rourke and
Sanderson~\ocite{rourke}, where one can also find the proofs of all
results mentioned here.
Throughout this paper, the word ``map'' will mean ``piecewise linear map''.
We will write $X\simeq Y$ if there exists a PL homeomorphism $X\to Y$.
For a subset $X\subset \R^N$, we denote $\Int(X)$ the interior of $X$, by
$\cl(X)$ the closure of $X$ and by $\partial X$ the boundary of $X$. We will
also use the following standard notation:
$B^n=[-1,1]^n\subset \R^n$ --- the $n$-dimensional ball
$S^n=\partial B^{n+1}$ --- the $n$-sphere
$\Delta^n\subset \R^{n+1}$--- the $n$--dimensional simplex (note that
$\Delta^n\simeq B^n$)
For any polyhedra $X\subset \R^N$ and a point $a\in \R^N$, we denote $aX$
the cone over $X$. More generally, given two polyhedra $X,Y\subset \R^N$,
we denote by $XY$ the join of $X,Y$. When using this notation, we will
always assume that $X,Y$ are independent, i.e. that every $p\in XY$
can be uniquely written as $p=ax+by,$ $a,b\in \R$, $a+b=1$. For two
polyhedra $X\in \R^n$, $Y\in \R^m$, we denote by $X*Y\subset \R^{n+m+1}$
their external join.
We define a convex $n$-cell $C\subset \R^N$ as a convex compact polyhedron
generating an affine subspace of dimension $n$; in such a situation, we will
also write $\dim C=n$. In Rourke and Sanderson~\cite{rourke}, these are
called just cells; we prefer a more specific name to avoid confusion with
other types of cells to be introduced later.
For any such cell we can define the set $F(C)$ of faces of $C$ (of arbitrary codimension);
each face $F$ is itself a convex cell. We will write $F<C$ if $F, C$ are convex cells
and $F$ is a face of $C$.
Recall that each convex cell $C$ is homeomorphic to
a ball: $C=\varphi(B^n)$ for some homeomorphism $\varphi$. As usual, we denote
\begin{align*}
\odot{C}&=\Int(C)=\varphi(\Int(B^n))\\
\mdot{C}&=\partial C=\varphi(\partial B^n)
\end{align*}
if $\dim C>0$. If $\dim C=0$, i.e. $C$ is a point, then we let
$\odot{C}=C$, $\mdot{C}=\varnothing$.
Following \ocite{rourke}, we define a cell complex $K$ as a finite
collection of convex cells in $\R^N$ such that the following conditions are
satisfied:
\begin{enumerate}
\item If $A\in K$ and $B<A$, then $B\in K$
\item If $A,B\in K$, and $F=A\cap B\ne \varnothing$, then $F<A$, $F<B$.
\end{enumerate}
We define the support $|K|=\cup_{C\in K} C$; it is a compact polyhedron in $\R^N$.
Conversely, given a compact polyhedron $X$, a {\em cell decomposition} of $X$ is a
complex $K$ such that $|K|=X$; it is known that such a decomposition always exists.
We will denote by $\dim K$ the dimension of $K$ and by $K^n$ the $n$-skeleton of $K$.
Given a complex $K$ and a cell $C$, we will denote $K+C$ the complex obtained by
adding to $K$ the cell $C$ assuming that it does form a complex.
In particular, given a convex cell $C$, the set $F(C)$ of faces of $C$ is a
cell complex, with $|F(C)|=\mdot{C}$; by adding to it $C$ itself, we get a
cell decomposition of $C$.
\section{Generalized cells}\label{s:gen_cell}
Let $C$ be a convex cell in $\R^N$.
\begin{definition}
A map $f\colon C\to \R^m$ is called {\em regular} if the restriction
$f|_{\odot C}$ is injective.
\end{definition}
\begin{lemma}\label{l:regular1}
If $C$ is a convex cell and $f\colon C\to \R^m$ is regular, then $C$ admits
a cell decomposition $K$ such that for any cell $K_i\in K$, the
restriction $f|_{K_i}$ is injective.
\end{lemma}
\begin{proof}
By standard results of PL topology, $C$ admits a cell decomposition
such that $f|_{K_i}$ is linear, and a linear map which is injective on an
open set is injective.
\end{proof}
We can now define the generalization of the notion of a convex cell.
\begin{definition}\label{d:gen_cell}
A {\em generalized $n$-cell} is a subset $C\subset \R^N$ together with
decomposition $C=\odot{C}\sqcup \mdot{C}$ such that $\odot{C}=\varphi(\Int B^n)$,
$\mdot{C}=\varphi(\partial B^n)$ (and thus $C=\varphi(B^n)$) for some regular map $\varphi\colon B^n\to \R^N$.
In such a situation, the map $\varphi$ is called a {\em characteristic map}.
\end{definition}
Note that the definition implies that $C=\cl(\odot{C})$, so $C$
is completely determined by $\odot{C}$. It is also clear from
\leref{l:regular1} that any generalized cell is a compact polyhedron.
Clearly any convex cell is automatically a generalized cell.
Other examples of generalized cells are shown in \firef{f:gcc1} below.
Note that characteristic map $\varphi$ in the definition of generalized
cell is not unique. However, as the following theorem shows, it is unique
up to a PL homeomorphism of the ball.
\begin{theorem}\label{t:char_map}
Let $C\subset \R^N$ be a generalized cell and $\varphi_1,\varphi_2\colon B^n\to C$
be two characteristic maps. Then there exists a unique homeomorphism
$\psi\colon B^n\to B^n$ such that $\varphi_1=\varphi_2\circ \psi$.
\end{theorem}
\begin{proof}
Since restriction of $\varphi_i$ to $\Int(B^n)$ is injective, the composition
$\odot\psi= \varphi_2^{-1}\varphi_1$ is well defined as a map $\Int(B^n)\to
\Int(B^n)$. To show that it can be extended to the boundary, note that it
follows from \leref{l:regular1} that one can find a cell
decomposition $K$ of $B_n$ such that
$\odot\psi|_{K_i}$ is linear for every $n$-cell $K_i\in K$. This
immediately implies that $\odot\psi$ can be extended to a homeomorphism
$\psi\colon B^n\to B^n$.
\end{proof}
It is easy to show that cone and join of generalized cells is again a
generalized cell. Namely, if $C=\varphi(B^n)$ is a generalized cell, and $aC$
is the cone of $C$, then the map $\varphi$ can be in an obvious way lifted
to a map $\{pt\}*B^n\simeq B^{n+1}\to aC$, which is easily seen to be
regular. Thus, $aC$ is a generalized cell. In the similar way, using
homeomorphism $B^m*B^n\simeq B^{m+n+1}$, one shows that if $C_1, C_2$
are generalized cells that are independent, then the join $C_1C_2$ is
also a generalized cell.
\section{Generalized cell complexes}\label{s:gen_complex}
{\bf From now on, unless noted otherwise, the word ``cell'' stands for a
generalized cell.}
\begin{definition}\label{d:gen_complex}
A generalized cell complex (g.c.c.) is a finite collection $K$ of
generalized cells in $\R^N$ such that
\begin{enumerate}
\item for any distinct $A, B$ in $K$, we have
$$
\odot{A}\cap \odot{B}=\varnothing
$$
\item For any cell $C\in K$, $\mdot{C}$ is a union of cells.
\end{enumerate}
Support $|K|\subset \R^N$ of a generalized cell complex $K$ is defined by
$$
|K|=\bigcup_{C\in K}C
$$
A generalized cell decomposition of a compact polyhedron
$P\subset \R^N$ is a generalized complex $K$ such that $|K|=P$.
\end{definition}
We define the dimension $\dim K$ of a generalized cell complex
and the $n$-skeleton $K^n$ in the usual way. Also, if $A,B\in K$ are
cells such that $A\subset \mdot{B}$, we will say that $A$ is a face of $B$
and write $A<B$; clearly this is only possible if $\dim A<\dim B$.
If $K,L$ are g.c.c., we denote by $K+L$ the complex obtained by
taking all cells of $K$ and all cells of $L$, assuming that the
result is again a g.c.c.
\begin{example}\label{x:gcc}
\begin{enumerate}
\item Any cell complex is automatically a g.c.c.
\item A 0-dimensional g.c.c. is the same as finite collection of points. A
1-dimensional g.c.c. is the same as a finite collection of points
(vertices) and non-intersecting arcs (1-cells) with endpoints at these
vertices. Note that loops are allowed.
\item \firef{f:gcc1} shows some examples of 2-dimensional g.c.c.
\begin{figure}[ht]
\begin{tabular}{cccc}
\tzAnnulusWithRadius&
\tzCircleWithRadius&
\tzTwoGlobe&
\tzZigZagComplex\\
(a)&(b)&(c)&(d)
\end{tabular}
\caption{Examples of 2-dimensional generalized cell complexes.
The last one can be visualized as a sheet of paper with a fold,
with the lower edge glued back to itself. Note that it only has four 1-cells:
the lines showing where the paper was folded are not 1-cells.}
\label{f:gcc1}
\end{figure}
\item \firef{f:gcc2} shows a generalized cell decomposition of $S^1\times I\times I$
consisting of a single 3-cell, five 2-cells, eight 1-cells and 4 vertices.
\begin{figure}[ht]
\tzCylinderI
\caption{A generalized cell decomposition of $S^1\times I\times
I$}\label{f:gcc2}
\end{figure}
\end{enumerate}
\end{example}
\begin{definition}\label{d:regular_cellular}
Let $K,L$ be g.c.c. A {\em regular cellular map} $f\colon L\to K$ is a map
$f\colon |L|\to |K|$ such that for every cell $C\in L$, $C=\varphi(B^n)$,
there exists a cell $C'\in K$ such that $C'=f(C)$ and moreover, $f\circ
\varphi\colon B^n\to C' $ is a characteristic map for $C'$.
\end{definition}
In other words, such a map is allowed to identify different
cells of $L$ but is injective on the interior of each cell.
An example of a regular cellular map is shown in
\firef{f:regular_cellular}.
\begin{figure}[ht]
$\tzRectangle\to \tzAnnulusWithRadiusLabeled$
\caption{An example of a regular cellular map.
It identifies edges $a_1b_1$ and $a_2b_2$,
sending each of them to edge $ab$.}\label{f:regular_cellular}
\end{figure}
\section{PLCW complexes}\label{s:PLCW}
In this section, we give the central definition of the paper.
\begin{definition}\label{d:PLCW}
A generalized cell complex (respectively, a generalized cell
decomposition) $K$ will be called a {\em PLCW complex} (respectively, PLCW
decomposition) if $\dim K=0$, or $\dim K=n>0$ and the following
conditions holds:
\begin{enumerate}
\item $K^{n-1}$ is a PLCW complex
\item For any $n$-cell $A\in K$, $A=\varphi(B^n)$, there exists a PLCW
decomposition $L$ of $\partial B^n$ such that the restriction
$\varphi|_{\partial B^n}\colon L\to K^{n-1}$ is a regular cellular map.
(It follows from \thref{t:char_map} that this condition is independent
of the choice of characteristic map $\varphi$.)
\end{enumerate}
\end{definition}
In other words, a PLCW is obtained by successively attaching balls, and the attaching map
should be a regular cellular map for some PLCW decomposition of the boundary sphere.
Note that this definition is inductive: definition of an $n$-dimensional PLCW
complex uses definition of an $(n-1)$ dimensional PLCW complex.
\begin{example}\label{x:PLCW}
Among examples in \exref{x:gcc}, example 2(d) is not a PLCW complex. All
other are PLCW.
\end{example}
It is easy to show that for an $n$-cell $A\in K$ and fixed choice of characteristic
map $\varphi\colon B^n\to A$, the generalized cell decomposition $L$ of $\partial B^n$ used in
\deref{d:PLCW} is unique. Indeed, the cells of $L$ are closures of connected components
of $\varphi^{-1}(\odot{K_i})$, $K_i\in K^{n-1}$. We will call such an $L$ the
pullback of $K$ under the map $\varphi$ and denote it by
\begin{equation}\label{e:pullback}
L=\varphi^{-1}(K).
\end{equation}
The following properties of PLCW complexes are immediate from the definition.
\begin{enumerate}
\item $|K|=\sqcup_{C\in K}\odot{C}$
\item If $A,B\in K$ are two cells, then $A\cap B$ is a union of cells of $K$.
\item For any $n$-cell $C\in K$, $\mdot{C}$ is a union of $(n-1)$-cells of $K$.
\item Every PLCW complex is automatically a CW complex.
\end{enumerate}
Note that not every CW complex is a PLCW complex, even if its cells are polyhedra.
For example, property (3) could fail for more general CW complexes.
The following two lemmas, proof of which is straightforward and left to the reader,
show that product and join of PLCW complexes is a PLCW complex.
\begin{lemma}\label{l:PLCWproduct}
Let $K,L$ be PLCW complexes in $\R^M$, $\R^N$ respectively. Define the complex
$$
K\times L=\sum K_i\times L_j\subset \R^M\times \R^N.
$$
Then $K\times L$ is a PLCW complex with support $|K|\times |L|$.
\end{lemma}
\begin{lemma}\label{l:PLCWjoin}
Let $K,L$ be PLCW complexes in $\R^N$ such that $|K|$, $|L|$ are independent: every point
$p\in |K| |L|$ can be uniquely written in the form $p=ax+by$, $x\in |K|$, $y\in |L|$,
$a,b\ge 0$, $a+b=1$. Define the join of them by
$$
KL=K+L+\sum K_i L_j,
\qquad K_i\in K, \quad L_j
\in L
$$
Then $KL$ is a PLCW complex with support $|K| |L|$.
\end{lemma}
The proof is straightforward and left to the reader.
Note that in the case $K=\{a\}$ --- a point, we see that the cone
$$
aL=a+L+\sum aL_i,\quad L_i\in L
$$
of a PLCW complex is a PLCW complex.
\section{Subdivisions}\label{s:subdivision}
\begin{definition}\label{d:subdivision}
Let $K, L$ be PLCW complexes. We say that $L$ is a subdivision of $K$ (notation: $L\lhd K$)
if $|K|=|L|$ and for any cell $C\in K$, we have $\odot{C}=\cup \odot{L_i}$ for some
collection of cells $L_i\in L$.
\end{definition}
Note that this implies that any cell $L_i\in L$ is a subset of one of the cells of $K$
(which is the usual definition of subdivision of cell complexes). Moreover, it is easy to
see that if $K,L$ are cell complexes, then this definition is actually equivalent
to the usual definition of subdivision.
There is a special kind of subdivisions we will be interesed in.
\begin{definition}\label{d:radial_subdivision}
Let $K$ be a PLCW complex, $C=\varphi(B^n)$ an $n$-cell, $n>0$ and $L=\varphi^{-1}(K)$
the pullback cell decomposition on $\partial B^n$ (see \eqref{e:pullback}).
We define the {\em radial subdivision}
of $K$ to be the subdivision obtained by replacing the cell $C$ by the
cone PLCW complex $\varphi(O)+\varphi(OL_1)+\dots+\varphi(OL_k)$, where
$L=\{L_1,\dots, L_k\}$ and $O\in \Int(B^n)$ is the origin.
(Recall that a cone of a PLCW complex is PLCW complex, see \leref{l:PLCWjoin}.)
\end{definition}
\firef{f:radial_subdivision} shows examples of radial subdivisions.
\begin{figure}[ht]
\begin{tabular}{ccc}
\tzPentagon &\quad $\rhd$ \quad &\tzPentagonRadialDivision\\ \noalign{\smallskip\smallskip\smallskip}
\tzAnnulusWithRadius & \quad$\rhd$\quad & \tzAnnulusWithRadiusSubdivision\\ \noalign{\smallskip\smallskip\smallskip}
\tzTetrahedron &\quad$\rhd$ & \tzTetrahedronSubdivision
\end{tabular}
\caption{Examples of radial subdivisions.
Note that in the last example, we are subdividing a 1-cell.
}\label{f:radial_subdivision}
\end{figure}
Note that this is very closely related to the usual notion of stellar subdivision
for simplicial complexes but it is not identical to it. Namely, for radial
subdivision we are subdividing just one cell $C$ without changing the
higher dimensional cells adjacent to $C$
(see the last example in \firef{f:radial_subdivision}). Comparing it with the definition of the
stellar subdivision, we see that if $K$ is a simplicial complex, $C\in K$
--- an $n$-cell, and $L$--- the stellar subdivision of $K$ obtained by starring
at $a\in \odot{C}$, then $L$ can also be obtained by
\begin{enumerate}
\item Replacing $C$ by the radial subdivision $R$
\item Replacing every cell $A=CB$ in the star of $C$ by the complex $R_iB$, $R_i\in R$.
\end{enumerate}
\begin{theorem}\label{t:triangulation}
Any PLCW complex $K$ has a subdivision $T\lhd K$ which is a triangulation;
moreover, $T$ can be obtained from $K$ by a sequence of radial subdivisions.
\end{theorem}
\begin{proof}
Let $K'$ be obtained from $K$ by radially subdividing of each cell of $K$ of positive dimension
in order of increasing dimension. Then it is easy to see that $K'$ has the following property:
\begin{equation}
\text{For any $C\in K'$, the characteristic map $\varphi\colon B^n\to C$ is injective}
\end{equation}
Now, let $T$ be obtained by again doing the radial subdivision of each cell of $K'$ in order
of increasing dimension. It is easy to see that $T$ is a triangulation: this follows
by induction from the fact that given a triangulation $\mdot{T}$ of $S^{n-1}$, the radial subdivision
$a\mdot{T}$ of $B^n$ is a triangulation (which in turn follows from the fact that the cone
over a simplex is a simplex).
\end{proof}
\section{Elementary subdivisions}\label{e:elem_subdivisions}
The other type of subdivision will be called {\em elementary subdivision}.
Informally, these are obtained by dividing an $n$-cell into two $n$-cells
separated by an $(n-1)$-cell. To give a more formal definition, we need some notation.
Let $H_0\subset \R^n$ be hyperplane defined by equation $x_n=0$. It divides $\R^n$ into two subspaces:
\begin{equation}\label{e:halfspaces}
\begin{aligned}
H_+&=\{(x_1,\dots, x_n)\in \R^n\st x_n\ge 0\}\\
H_-&=\{(x_1,\dots, x_n)\in \R^n\st x_n\le 0\}
\end{aligned}
\end{equation}
For the $n$-ball $B^n\subset \R^n$, define upper and lower halfballs
\begin{equation}
B^n_+=B^n\cap H_+, \qquad B^n_-=B^n\cap H_-
\end{equation}
We also define the middle disk and the equator by
\begin{equation}
B^n_0=B^n\cap H_0\simeq B^{n-1}, \qquad E=S^{n-1}\cap H_0\simeq S^{n-2}
\end{equation}
\begin{lemma}\label{l:elem_subdivision}
Let $K$ be a PLCW and $C=\varphi(B^n)$ --- an $n$-cell. Assume that the pullback
decomposition $L=\varphi^{-1}(K)$ of $\partial B^n$ is such that the equator $E\subset \partial B^n$
is a union of cells of $L$. Let $K'$ be the g.c.c. obtained by replacing $C$ by the
collection of cells $C_+=\varphi(B^n_+)$, $C_-=\varphi(B^n_-)$, $C_0=\varphi(B^n_0)$.
Then $K'$ is a PLCW complex; moreover, $K'$ is a subdivision of $K$.
\end{lemma}
\begin{definition}\label{d:elem_subdivision}
Let $K,K'$ be as in \leref{l:elem_subdivision}. Then we say that $K'$ is obtained from $K$
by an elementary subdivision of cell $C$; we will also say that $K$ is obtained from
$K'$ by erasing cell $C_0$.
We will write $K\underset{\scriptstyle e}\sim L$ if $K$ can be obtained from $L$ by a finite sequence of elementary
subdivisions and their inverses.
\end{definition}
Note that elementary subdivisions are essentially the same as
$(n,n)$ moves in intruduced by Oeckl in \cite{oeckl1} and further studied Oeckl's book \cite{oeckl2}; in Oeckl's work, these moves are special case of a more general moves called $(n,k)$ moves.
An example of elementary subdivision is shown in \firef{f:elem_subdivision}.
\begin{figure}[ht]
\tzPentagonElemDivision\quad$\lhd$\quad \tzPentagon
\caption{An elementary subdivision}\label{f:elem_subdivision}
\end{figure}
\begin{remark}
Not every subdivision can be obtained by a sequence of elementary subdivisions. For example,
the subdivision shown in \firef{f:nonelem_subdivision} can not be obtained by a sequence of
elementary subdivisions.
\begin{figure}[ht]
\tzCircleWithRadius\quad $\lhd$\quad \tzCircle
\caption{A non-elementary subdivision}\label{f:nonelem_subdivision}
\end{figure}
However, it can be obtained by a sequence of elementary subdivisions
{\bf and their inverses} as shown in \firef{f:nonelem_subdivision2}.
\begin{figure}[ht]
\tzCircle $\rhd$ \tzCircleII$\rhd$\tzCircleIII$\rhd$\tzCircleIV
$\lhd$ \tzCircleV$\lhd$ \tzCircleWithRadius
\caption{Obtaining a non-elementary subdivision by a sequence of elementary
subdivisions and their inverses}\label{f:nonelem_subdivision2}
\end{figure}
\end{remark}
\begin{theorem}\label{t:elem_join}
If $M=KL$ is a join of two PLCW complexes and $K'\lhd K$ --- an elementary subdivision of $K$,
then $M'=K'L$ be obtained from $M$ by a sequence of elementary subdivisions.
\end{theorem}
\begin{proof}
If $C\in K$ is an $n$-cell and $C=C_++C_-+C_0$ its elementary subdivision as in
\leref{l:elem_subdivision}, and $D$ is a cell in $L$, then $CD=C_+D + C_-D + C_0D$
is an elementary subdivision of $CD$, which follows from existence
of a homeomorphism $\psi\colon B^n*B^m\isoto B^{m+n+1}$ such that
$\psi(B_0^n*B^m)=B_0^{m+n+1}$, $\psi(B_\pm^n*B^m)=B_\pm^{m+n+1}$. Repeating it for
every cell $D\in L$ in order of increasing dimension, we see that $K'L$ can be obtained
from $KL$ by a sequence of elementary subdivisions.
\end{proof}
\begin{corollary}\label{c:elem_join}
If $K\underset{\scriptstyle e}\sim K'$, then $KL\underset{\scriptstyle e}\sim K'L$.
\end{corollary}
\section{Main theorem}\label{s:main}
In this section, we formulate and prove the main theorem of this paper. Recall the notation
$K\underset{\scriptstyle e}\sim L$ from \deref{d:elem_subdivision}.
\begin{theorem}\label{t:main1}
Let $K, K'$ be two PLCW decompositions of a compact polyhedron $X$. Then $K\underset{\scriptstyle e}\sim K'$.
\end{theorem}
\begin{proof}
This proves (for PLCW decompositions) the conjecture of Oeckl \cite{oeckl2}:
that any any cell decompositions can be obtained form each other by a
sequence of $(,n,k)$ moves; in fact, it proves a stronger result, that
$(n,n)$ moves are already enough.
We proceed by induction in $n=\dim X$. If $n=0$, there is nothing to
prove. So from now on, we assume that $n>0$ and that the theorem is already
proved for all polyhedra of dimension less than $n$.
{\bf Step 1.}
Let $X=B^n$ be an $n$-ball, $\mdot{K}$ -- a PLCW decomposition of $S^{n-1}=\partial B^n$, and
$R=a\mdot{K}$ --- the corresponding radial cell decomposition of $X$, $a\in \Int(B^n)$. Then
$R\underset{\scriptstyle e}\sim B^n+\mdot{K}$.
Indeed, let $L$ be a PLCW decomposition of $S^{n-1}$ consisting of the upper and lower
hemispheres $S^{n-1}_\pm\simeq B^{n-1}$ and some PLCW decomposition $L_0$ of the equator $E$.
By induction assumption, $\mdot{K}\underset{\scriptstyle e}\sim L$; by \coref{c:elem_join}, this implies
$$
a\mdot{K}\underset{\scriptstyle e}\sim aL= B_+^n+B^n_- +S^{n-1}_+ +S^{n-1}_- + aL_0.
$$
By using the induction assumption again, $aL_0\underset{\scriptstyle e}\sim B^n_0 +L_0$, so
$$
a\mdot{K}\underset{\scriptstyle e}\sim B_+^n+B^n_- +B^n_0 +S^{n-1}_+ +S^{n-1}_- +L_0\underset{\scriptstyle e}\sim B^n+S^{n-1}_+ + S^{n-1}_- + L_0\underset{\scriptstyle e}\sim B^n+\mdot{K}
$$
{\bf Step 2.}
If $K'$ is obtained from $K$ by a sequence of radial subdivisions, then $K'\underset{\scriptstyle e}\sim K$.
This follows from the previous step and definition.
{\bf Step 3.}
For any PLCW decomposition $K$, there is a triangulation $T$ such that $K\underset{\scriptstyle e}\sim T$.
Indeed, it follows from the previous step and \thref{t:triangulation}.
{\bf Step 4.} If $T, T'$ are triangulations of $X$, then $T\underset{\scriptstyle e}\sim T'$.
By Alexander's theorem, $T$ can be obtained from $T'$ by a sequence of stellar moves, so it
suffices to prove the theorem in the case when $T'$ is obtained from $T$ by starring at
point $a\in \Int(C)$ for some simplex $C\in T$.
By the discussion in \seref{s:subdivision}, we can also describe $T'$ by replacing $C$ by the
radial subdivision $C'$ of $C$ and replacing every simplex $A=CB$ in the star of $C$ by $C'B$.
By step 2 and \coref{c:elem_join}, this implies that $T'\underset{\scriptstyle e}\sim K$.
Combining steps 3 and 4 above, we arrive at the statement of the theorem.
\end{proof}
\section{Orientations}\label{s:orientation}
Recall that the group of homeomorphisms of $B^n$ has a homomorphism
to $\Z_2$, called {\em orientation}. Using this, we can define the
notion of orientation of a cell.
\begin{definition}\label{d:orientation}
Let $C\subset \R^N$ be a generalized $n$-cell. An {\em orientation} of
$C$ is an equivalence class of characteristic maps $B^n\to C$, where two
characteristic maps $\varphi_1,\varphi_2\colon B^n\to C$ are equivalent if
$\psi=\varphi_2\varphi_1^{-1}\colon B^n\to B^n$ is orientation-preserving.
An oriented cell $\mathbf C=(C, [\varphi])$ is a pair consisting of a cell $C$
and an orientation $[\varphi]$.
\end{definition}
Note that any convex $n$-cell $C\subset \R^n$ has a canonical orientation. Moreover,
if $C\subset \R^n$ is is a convex $n$-cell, and $D\subset \partial C$ is a generalized
$(n-1)$-cell, then $D$ has a canonical orientation defined by the usual condition:
\begin{equation}\label{e:orientation_boundary}
\varepsilon(\mathbf{C}, \mathbf{D})=1
\end{equation}
where $\varepsilon(\mathbf{C}, \mathbf{D})$ is the incidence number, defined in
the same way as for CW cells (see, e.g., \cite{rourke}*{Appendix A.7}).
Thus, if $C$ is is a convex $n$-cell in $\R^n$, and $L$ -- a PLCW decomposition
of $\partial C$, then each of $(n-1)$-cells $L_i\in L$ has a canonical orientation.
The following definition generalizes this to an arbitrary oriented cell.
\begin{definition}\label{d:delC}
Let $K$ be a PLCW complex, and $\mathbf{C}=(C, [\varphi])$ --- an oriented cell.
Let $L=\varphi^{-1}(K)$ be the pullback decomposition of $\partial B^n$.
We define the boundary $\partial \mathbf{C}$ as a multiset (set with multiplicities)
of oriented $(n-1)$-cells
$$
\partial \mathbf{C}=\bigcup (\varphi(L_i), [\varphi\circ \varphi_i])
$$
where the union is over all $(n-1)$-cells $L_i\in L$, each taken
with the natural orientation $[\varphi_i]$ defined by \eqref{e:orientation_boundary}.
\end{definition}
It is easy to see, using \thref{t:char_map}, that this definition does not depend
on the choice of characteristic map $\varphi$ in the equivalence class.
Note that by definition of a PLCW,
for each $L_i\in L$, $\varphi(L_i)$ is an $(n-1)$-cell of $K$; however, the same $(n-1)$-cell $D\in K$
can appear in $\partial C$ more than once, and possibly with different
orientations. Note also that passing from the multisets to the abelian
group generated by oriented cells, we get the usual definition of the
boundary operator in the chain complex of a CW complex. However, for
applications to topological field theory, the definition of the boundary as a
multiset is much more useful.
\begin{example}
Let $C$ be the $2$-cell shown below. Then $\partial C=\{a,\bar a, b\}$, where $\bar a$
denotes $a$ with opposite orientation.
$$\tzCircleWithRadiusOrient
$$
\end{example}
The proof of the following lemma is left to the reader as an exercise.
\begin{lemma}
Let $X$ be an oriented PL manifold with boundary and $K$ --- a PLCW
decomposition of $X$. Then
$$
\cup_{\mathbf{C}} \partial \mathbf{C}=
\Bigl(\cup_{D} \mathbf{D}\Bigr) \cup
\Bigl(\cup_{F} \mathbf{F}\cup \mathbf{\bar F} \Bigr)
$$
where
\begin{itemize}
\item $C$ runs over all $n$-cells of $K$, each taken with orientation
induced by orientation of $X$
\item $D$ runs over all $(n-1)$ cells such that $D\subset \partial X$,
each taken with orientation induced by orientation of $\partial X$
\item $F$ runs over all \textup{(}unoriented\textup{)} $(n-1)$-cells such
that $\odot F\subset \Int(X)$; $\mathbf{F}$ and $\mathbf{\bar F}$ are the
two possible orientations of $F$.
\end{itemize}
\end{lemma}
\begin{bibdiv}
\begin{biblist}
\bib{balsam-kirillov}{article}{
author={Balsam, Benjamin },
author={Kirillov, Alexander, Jr},
title={Turaev-Viro invariants as an extended TQFT},
eprint={arXiv:1004.1533},
}
\bib{oeckl1}{article}{
author={Oeckl, Robert},
title={Renormalization of discrete models without background},
journal={Nuclear Phys. B},
volume={657},
date={2003},
number={1-3},
pages={107--138},
issn={0550-3213},
review={\MR{1969693 (2004j:83037)}},
doi={10.1016/S0550-3213(03)00145-7},
}
\bib{oeckl2}{book}{
author={Oeckl, Robert},
title={Discrete gauge theory},
note={From lattices to TQFT},
publisher={Imperial College Press},
place={London},
date={2005},
pages={xii+202},
isbn={1-86094-579-1},
review={\MR{2174961 (2006i:81142)}},
doi={10.1142/9781860947377},
}
\bib{rourke}{book}{
author={Rourke, Colin Patrick},
author={Sanderson, Brian Joseph},
title={Introduction to piecewise-linear topology},
series={Springer Study Edition},
note={Reprint},
publisher={Springer-Verlag},
place={Berlin},
date={1982},
pages={viii+123},
isbn={3-540-11102-6},
review={\MR{665919 (83g:57009)}},
}
\end{biblist}
\end{bibdiv}
\end{document}
| {
"timestamp": "2011-11-08T02:07:59",
"yymm": "1009",
"arxiv_id": "1009.4227",
"language": "en",
"url": "https://arxiv.org/abs/1009.4227",
"abstract": "In this note, we introduce a class of cell decompositions of PL manifolds and polyhedra which are more general than triangulations yet not as general as CW complexes; we propose calling them PLCW complexes. The main result is an analog of Alexander's theorem: any two PLCW decompositions of the same polyhedron can be obtained from each other by a sequence of certain \"elementary\" moves.This definition is motivated by the needs of Topological Quantum Field Theory, especially extended theories as defined by Lurie.",
"subjects": "Geometric Topology (math.GT)",
"title": "On piecewise linear cell decompositions",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808741970027,
"lm_q2_score": 0.7217432003123989,
"lm_q1q2_score": 0.7075110553479809
} |
https://arxiv.org/abs/math/0509603 | A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates | In this paper we obtain multifractal generalizations of classical results by Lévy and Khintchin in metrical Diophantine approximations and measure theory of continued fractions. We give a complete multifractal analysis for Stern--Brocot intervals, for continued fractions and for certain Diophantine growth rates. In particular, we give detailed discussions of two multifractal spectra closely related to the Farey map and the Gauss map. | \section{Introduction and statements of result}
In this paper we give a multifractal analysis for Stern--Brocot
intervals,
continued fractions and certain Diophantine growth rates. We apply
and extend the multifractal formalism for average growth rates of
\cite{KesseboehmerStratmann:04a} to obtain a complete multifractal
description
of two dynamical systems originating from the set of real numbers.
Recall that the process of writing an element $x$ of the unit interval
in its regular continued fraction expansion \[
x=[a_{1}(x),a_{2}(x),a_{3}(x),\ldots]=\cfrac{1}{a_{1}(x)+\cfrac{1}{a_{2}(x)+\cfrac{1}{a_{3}(x)+\cdots}}}\]
can be represented either by a uniformly hyperbolic dynamical system
which is based on an infinite alphabet and hence has infinite
topological
entropy, or by a non-uniformly hyperbolic dynamical system based on
a finite alphabet and having finite topological entropy. Obviously,
for these two systems the standard theory of multifractals (see e.g.
\cite{Pesin:97}) does not apply, and therefore it is an interesting task
to give a multifractal analysis for these two number theoretical
dynamical systems. There is a well known result which gives some
information
in the generic situation, that is for a set of full $1$-dimensional
Lebesgue measure $\lambda$. Namely with
$p_{n}(x)/q_{n}(x):=[a_{1}(x),a_{2}(x),\ldots,a_{n}(x)]$
referring to the $n$-th approximant of $x$, we have for $\lambda$-almost
every $x\in[0,1)$, \[
\ell_{1}(x):=\lim_{n\rightarrow\infty}\frac{2\log
q_{n}(x)}{\sum_{i=1}^{n}a_{i}(x)}=0.\]
Note that by employing the analogy between regular continued fraction
expansions of real numbers and geodesics on the modular surface, the
number $2\log q_{n}(x)$ can be interpreted as the 'hyperbolic length'
associated with the approximant $p_{n}(x)/q_{n}(x)$. Also,
the parameter $n$ represents the word length associated with
$p_{n}(x)/q_{n}(x)$
with respect to the dynamical system on the infinite alphabet, whereas
$\sum_{i=1}^{n}a_{i}(x)$ can be interpreted as the word length
associated
with $p_{n}(x)/q_{n}(x)$ with respect to the dynamical system on
the finite alphabet. There are two classical results by Khintchin and
L{\'e}vy \cite{Levy:29}, \cite{Levy:36}, \cite{Khinchin:35}, \cite{Khihtchine:36} which allow a
closer inspection of the limit $\ell_{1}$. That is, for
$\lambda$-almost every $x\in[0,1)$ we have, with
$\chi:=\pi^{2}/(6\log2)$,
\[
\ell_{2}(x):=\lim_{n\rightarrow\infty}\frac{\sum_{i=1}^{n}a_{i}(x)}{n}=\infty
\textrm{ and } \ell_{3}(x):=\lim_{n\rightarrow\infty}\frac{2\log
q_{n}(x)}{n}=\chi.\]
Clearly, dividing the sequence in $\ell_{3}$ by the sequence in
$\ell_{2}$ leads to the sequence in $\ell_{1}$. Therefore, if we
define the level sets \[
\mathcal{L}_{i}(s):=\left\{ x\in[0,1):\ell_{i}(x)=s\right\},
\textrm{ for } s\in\mathbb {R}\,,\]
then these classical results by L{\'e}vy and Khintchin imply for the
Hausdorff dimensions ($\dim_{H}$) of these level sets \[
\dim_{H}(\mathcal{L}_{1}(0))=\dim_{H}(\mathcal{L}_{2}(\infty)\cap\mathcal{L}_{3}(\chi))=1.\]
A natural question to ask is what happens to this relation between
these Hausdorff dimensions for prescribed non-generic limit behavior.
Our first main results in this paper will give an answer to this
question.
Namely, with $\gamma:=(1+\sqrt{5})/2$ referring to the Golden
Mean, we show that for each $\alpha\in[0,2\log\gamma]$ there exists
a number $\alpha^{\sharp}=\alpha^{\sharp}(\alpha)\in\mathbb {R}\cup\{\infty\}$
such that, with the convention $\alpha^{\sharp}(0):=\infty$ and
$0\cdot\alpha^{\sharp}(0):=\chi$,
\[
\dim_{H}(\mathcal{L}_{1}(\alpha))=\dim_{H}(\mathcal{L}_{2}(\alpha^{\sharp})\cap\mathcal{L}_{3}
(\alpha\cdot\alpha^{\sharp})).\]
Furthermore, for the dimension function $\tau$ given by \[
\tau(\alpha):=\dim_{H}(\mathcal{L}_{1}(\alpha)),\]
we show that $\tau$ can be expressed explicitly in terms of the
Legendre transform $\widehat{P}$ of a certain pressure function
$P$, referred to as the Stern--Brocot pressure. For the function $P$
we obtain the result that it is real-analytic on the interval
$(-\infty,1)$
and vanishes on the complement of this interval. We then show that
the dimension function $\tau$ is continuous and strictly decreasing
on $[0,2\log\gamma]$, that it vanishes outside the interval
$[0,2\log\gamma)$,
and that for $\alpha\in[0,2\log\gamma]$ we have \[
\alpha\cdot\tau(\alpha)=-\widehat{P}(-\alpha).\]
Before we state the main theorems, let us recall the following classical
construction of Stern--Brocot intervals (cf. \cite{Stern1858},
\cite{Brocot:1860}).
For each $n\in\mathbb {N}_{0}$, the elements
of the $n$-th member of the Stern--Brocot sequence \[
\mathfrak{T}_{n}:=\left\{
\frac{s_{n,k}}{t_{n,k}} : k=1,\ldots,2^{n}+1\right\} \]
are defined recursively as follows.
\begin{itemize}
\item $s_{0,1}:=0\,\,$ and $\,\, s_{0,2}:=t_{0,1}:=t_{0,2}:=1$;
\item $s_{n+1,2k-1}:=s_{n,k}\quad\textrm{and}\quad
t_{n+1,2k-1}:=t_{n,k},$
for $k=1,\ldots,2^{n}+1$;
\item $s_{n+1,2k}:=s_{n,k}+s_{n,k+1}\quad\textrm{and}\quad
t_{n+1,2k}:=t_{n,k}+t_{n,k+1}$,
for $k=1,\ldots2^{n}$.
\end{itemize}
With this ordering of the rationals in $[0,1]$ we define the set
$\mathcal{T}_{n}$ of Stern--Brocot intervals of order $n$ by \[
\mathcal{T}_{n}:=\left\{
T_{n,k}:=\left[\frac{s_{n,k}}{t_{n,k}},\frac{s_{n,k+1}}{t_{n,k+1}}\right):\,
k=1,\ldots,2^{n}\right\} .\]
Clearly, for each $n\in\mathbb {N}_{0}$ we have that $\mathcal{T}_{n}$ represents
a partition of the interval $[0,1)$. The first members in this sequence
of sets are the following, and it should be clear how to continue
this list using the well known method of mediants.
\begin{eqnarray*}
\mathcal{T}_{0}= & \left\{ \left[\frac{0}{1},\frac{1}{1}\right)\right\}
\\
\mathcal{T}_{1}= & \left\{
\left[\frac{0}{1},\frac{1}{2}\right),\left[\frac{1}{2},\frac{1}{1}\right)\right\}
\\
\mathcal{T}_{2}= & \left\{
\left[\frac{0}{1},\frac{1}{3}\right),\left[\frac{1}{3},\frac{1}{2}\right),
\left[\frac{1}{2},\frac{2}{3}\right),\left[\frac{2}{3},\frac{1}{1}\right)\right\}
\\
\mathcal{T}_{3}= & \left\{
\left[\frac{0}{1},\frac{1}{4}\right),\left[\frac{1}{4},\frac{1}{3}\right),
\left[\frac{1}{3},\frac{2}{5}\right),\left[\frac{2}{5},\frac{1}{2}\right),
\left[\frac{1}{2},\frac{3}{5}\right),\left[\frac{3}{5},\frac{2}{3}\right),
\left[\frac{2}{3},\frac{3}{4}\right),\left[\frac{3}{4},\frac{1}{1}\right)\right\}
\\
\vdots & \vdots\end{eqnarray*}
As already mentioned above, our multifractal analysis will make use
of the Stern--Brocot pressure function $P$. This function is defined for
$\theta\in\mathbb {R}$
by \[
P(\theta):=\lim_{n\rightarrow\infty}\frac{1}{n}\log\sum_{T\in\mathcal{T}_{n}}
\left|T\right|^{\theta}.\]
In here, $\left|T\right|$ refers to the Euclidean length of the interval $T$.
We will see that $P$ is a well--defined convex function (cf. Proposition
\ref{pro:AnalyticPropertiesP}). Also, note that we
immediately have that\[
P(\theta)=\lim_{n\rightarrow\infty}\frac{1}{n}\log\sum_{k=1}^{2^{n}}
\left(\frac{s_{n,k+1}}{t_{n,k+1}}-
\frac{s_{n,k}}{t_{n,k}}\right)^{\theta}=\lim_{n\rightarrow\infty}
\frac{1}{n}\log\sum_{k=1}^{2^{n}}\left(\frac{1}{t_{n,k}\cdot
t_{n,k+1}}\right)^{\theta}.\]
The following theorem gives the first main results of this paper.
In here, $\widehat{P}$ refers to the Legendre transform of $P$,
given for $t \in\mathbb {R}$ by
$\widehat{P}(t):=\sup_{\theta\in\mathbb {R}}\{\theta t-P(\theta)\}$.
\begin{thm}
\label{Thm:main}
{\em (see Fig. \ref{cap:The-Stern--Brocot-pressure})}
\begin{enumerate}
\item The Stern--Brocot pressure $P$ is convex, non-increasing and
differentiable
throughout $\mathbb {R}$. Furthermore, $P$ is real--analytic on the interval
$(-\infty,1)$ and is equal to $0$ on $[1,\infty)$.
\item For every $\alpha\in[0,2\log\gamma]$, there exist
$\alpha^{*}=\alpha^{*}(\alpha)\in\mathbb {R}$
and $\alpha^{\sharp}=\alpha^{\sharp}(\alpha)\in\mathbb {R}\cup\{\infty\}$
related by $\alpha\cdot\alpha^{\sharp}=\alpha^{*}$ such that, with
the conventions $\alpha^{*}(0):=\chi$ and $\alpha^{\sharp}(0):=\infty$,
\begin{eqnarray*}
\dim_{H}\left(\mathcal{L}_{1}(\alpha)\right) & = &
\dim_{H}\left(\mathcal{L}_{2}(\alpha^{\sharp})\cap\mathcal{L}_{3}(\alpha^{*})\right)
\,\,\left(=:\tau(\alpha)\right).\end{eqnarray*}
Furthermore, the dimension function $\tau$ is continuous and strictly
decreasing on $[0,2\log\gamma]$, it vanishes outside the interval
$[0,2\log\gamma)$, and for $\alpha\in[0,2\log\gamma]$ we have\[
\alpha\cdot\tau(\alpha)=-\widehat{P}(-\alpha),\]
where $\tau(0):=\lim_{\alpha\searrow0}-\widehat{P}(-\alpha)/\alpha=1$.
Also, for the left derivative of $\tau$ at $2\log\gamma$ we have
$\lim_{\alpha\nearrow2\log\gamma}\tau'\left(\alpha\right)=-\infty$.
\end{enumerate}
\end{thm}
Theorem \ref{Thm:main} has some interesting implications for other
canonical level sets. In order to
state these, recall that the elements
of $\mathcal{T}_{n}$ cover the interval $[0,1)$ without overlap.
Therefore, for each $x\in[0,1)$ and $n\in\mathbb {N}$ there exists a unique
Stern--Brocot interval $T_{n}(x)\in\mathcal{T}_{n}$ containing $x$.
The interval $T_{n}(x)$ is covered by two neighbouring intervals from
$\mathcal{T}_{n+1}$, a left and a right subinterval. If $T_{n+1}(x)$
is the left of these then we encode this event by the letter $A$,
otherwise we encode it by the letter $B$. In this way every
$x\in[0,1)$ can be described by a unique sequence of nested
Stern--Brocot intervals of any order that contain $x$, and therefore
by a unique infinite word in the alphabet $\{ A,B\}$. It is well
known that this type of coding is canonically associated with the
continued fraction expansion of $x$ (see Section 2
for the details). In particular,
this allows to relate the level sets $\mathcal{L}_{1}$ and
$\mathcal{L}_{3}$ to level sets given by means of the Stern--Brocot
growth rate $\ell_{4}$ of the nested sequences
$\left(T_{n}(x)\right)$, and to level sets of certain Diophantine
growth rates $\ell_{5}$ and $\ell_{6}$. These growth
rates are given by (assuming that these limits exist) \[
\ell_{4}(x):=\lim_{n\rightarrow\infty}\frac{\log\left|T_{n}(x)\right|}{-n},\,\,\]
\[
\ell_{5}(x):=\lim_{n\rightarrow\infty}\frac{2\log\left|x-\frac{p_{n}(x)}{q_{n}(x)}\right|}
{-\sum_{i=1}^{n}a_{i}(x)}\quad\textrm{and}\quad\ell_{6}(x):={\displaystyle
\lim_{n\rightarrow\infty}\frac{2\log\left|x-\frac{p_{n}(x)}{q_{n}(x)}\right|}{-n}.}\]
\begin{figure}
\psfrag{alpha-}{$-2\theta\log(\gamma)$}
\psfrag{DimFG}{$\dim_H({\mathcal{L}}_1(\alpha))$}
\psfrag{pss}{$P(\theta)$} \psfrag{2 log g}{$2\log(\gamma)$}
\psfrag{1}{\(1\)} \psfrag{a}{$\alpha$} \psfrag{s}{$\theta$}
\includegraphics[%
width=1.0\columnwidth,
keepaspectratio]{psl2.eps}
\caption{The Stern--Brocot pressure $P$ and the multifractal
spectrum $\tau$ for $\ell_1$.\label{cap:The-Stern--Brocot-pressure}}
\end{figure}
\begin{prop}
\label{Thm:2} Let $x \in [0,1)$ be given. If one of the limits in
$\{ \ell_{1}(x), \ell_{4}(x), \ell_{5}(x)\}$ exists then also the
other two do exist, and \[
\ell_{1}(x)=\ell_{4}(x)=\ell_{5}(x).\]
Furthermore, $\ell_{3}(x)$ exists if and only if $\ell_{6}(x)$ exists,
and if one of these exists then
\[ \ell_{3}(x)=\ell_{6}(x).\]
By Theorem \ref{Thm:main}, it therefore follows that for each
$\alpha\in[0,2\log\gamma]$,
\[
\dim_{H}\left(\mathcal{L}_{4}(\alpha)\right)=\dim_{H}\left(\mathcal{L}_{5}(\alpha)\right)=
\dim_{H}\left(\mathcal{L}_{2}(\alpha^{\sharp})\cap\mathcal{L}_{6}(\alpha^{*})\right)=\tau(\alpha).\]
\end{prop}
Note that the level sets $\mathcal{L}_{4}(\alpha)$ have already been
under consideration in
\cite{KesseboehmerStratmann:04}. There they were introduced in terms
of homological growth rates of hyperbolic geodesics (see Remark
\ref{rem:Vergleich} (2)). Clearly, Theorem \ref{Thm:main} and Proposition \ref{Thm:2} consider the
dynamical system associated with the finite alphabet, a system which is closely
related to the Farey map. Now, our second main result gives a multifractal
analysis for the system based on the infinite alphabet, and this system is
closely related
to the Gauss map. In here, the relevant pressure function is the
\emph{Diophantine
pressure} $P_{D}$, which is given by \[
P_{D}(\theta):=\lim_{k\rightarrow\infty}\frac{1}{k}\log\sum_{\left[a_{1},\ldots,a_{k}\right]}
q_{k}\left(\left[a_{1},\ldots,a_{k}\right]\right)^{-2\theta}, \textrm{
for } \theta>\frac{1}{2}.\]
We remark that a very detailed analysis of the function $P_{D}$ can be
found in \cite{Mayer}. Our second main result is the following.
\begin{thm}
\label{thm:main3} {\em (see Fig. \ref{cap:Diaophantine-pressure})} The function $P_{D}$ has a singularity
at $1/2$, and
$P_{D}$ is decreasing, convex and real-analytic on $\left(1/2,\infty\right)$.
Furthermore, for $\alpha\in[2\log\gamma,\infty)$ we have\[
\dim_{H}\left(\mathcal{L}_{3}(\alpha)\right)=\dim_{H}\left(\mathcal{L}_{6}(\alpha)\right)=
\frac{\widehat{P}_{D}(-\alpha)}{-\alpha}=:\tau_{D}(\alpha).\]
Also, the dimension function $\tau_{D}$ is real-analytic on
$(2\log\gamma,\infty)$,
it is increasing on $[2\log\gamma,\chi]$ and decreasing on
$[\chi,\infty)$.
In particular, $\tau_{D}$ has a point of inflexion at some point greater
than
$\chi$ and a unique maximum equal to $1$ at $\chi$. Additionally,
$\lim_{\alpha\to\infty}\tau_{D}\left(\alpha\right)=1/2$,
$\lim_{\alpha\searrow2\log\gamma}\tau_{D}\left(\alpha\right)=0$
and $\lim_{\alpha\searrow2\log\gamma}\tau_{D}'(\alpha)=\infty$.
\end{thm}
\begin{figure}
\psfrag{alpha-}{$-2\theta\log\gamma$}\psfrag{2loggam}{$2\log\gamma$}
\psfrag{DimFG}{$\dim_H({\mathcal{L}}_3(\alpha))$} \psfrag{pss}{$P_D
(\theta)$}
\psfrag{1}{\(1\)}\psfrag{1}{\(1\)}\psfrag{1/2}{\(\frac{1}{2}\)}\psfrag{a}{$\alpha$}
\psfrag{s}{$\theta$} \psfrag{chi}{$\chi:=\frac{\pi^2}{6\log2}$} \psfrag{d0}{$d_0$}
\includegraphics[%
width=1.0\columnwidth,
keepaspectratio]{psl2extra.eps}
\caption{The Diophantine pressure $P_{D}$ and the
multifractal
spectrum $\tau_{D}$ for $\ell_{3}$.\label{cap:Diaophantine-pressure}}
\end{figure}
The paper is organized as follows. In Section 2 we first recall two
ways of coding elements of the unit interval. One is based on a finite
alphabet and
the other on an infinite alphabet, and both are defined in terms of the modular group.
These codings are
canonically related to regular continued fraction expansions, and
we end the section by commenting on a 1-1 correspondence between
Stern--Brocot
sequences and finite continued fraction expansions. In Section 3 we
introduce certain cocycles which are relevant in our multifractal
analysis. In particular, we give various estimates relating these
cocycles with the geometry of the modular codings and with the sizes
of the Stern--Brocot intervals. This will then enable us to prove the
first part of Proposition \ref{Thm:2}. Section 4 is devoted to the
discussion
of several aspects of the Stern--Brocot pressure and its Legendre
transform.
In Section 5 we give the proof of Theorem \ref{Thm:main}, which we
have split into the parts \emph{The lower bound}, \emph{The upper
bound},
and \emph{Discussion of boundary points of the spectrum}. Finally, in Section 6 we
give the proof of Theorem \ref{thm:main3} by showing how to adapt
the multifractal formalism developed in Section 4 and 5 to the
situation here.
Throughout, we shall use the notation $f\ll g$ to denote that
for two non-negative functions $f$ and $g$ we have that $f/g$
is uniformly bounded away from infinity. If $f\ll g$ and $g\ll f$,
then we write $f\asymp g$.
\begin{rem}
One immediately verifies that the results of Theorem
\ref{Thm:main} and
Proposition \ref{Thm:2} can be expressed in terms of the Farey map $\mathfrak{f}$
acting on $[0,1]$, and then
$\tau$ represents the multifractal spectrum of
the measure of maximal entropy (see e.g. \cite{Nakaishi:00}).
Likewise, the results of Theorem \ref{thm:main3} can
be written in terms of the Gauss map $\mathfrak{g}$, and then in
this terminology $\tau_D$ describes the Lyapunov spectrum of $\mathfrak{g}$.
For the definitions of $\mathfrak{f}$ and $\mathfrak{g}$ and for a discussion
of their relationship we refer to Remark \ref{rem:FG}.
\end{rem}
\begin{rem}
Since the theory of multifractals started through essays of Mandelbrot
\cite{Mandelbrot:74} \cite{Mandelbrot:88}, Frisch and Parisi
\cite{FrischParisi:85},
and Halsey et al. \cite{Halsey:86}, there has been a steady increase of
the literature on multifractals and calculations of specific
multifractal
spectra. For a comprehensive account on the mathematical work we refer
to \cite{PesinWeiss:96} and \cite{Pesin:97}. Essays which are
closely related to the
work on multifractal number
theory in this paper are for instance \cite{Byrne:98},
\cite{FengOlivier:03}, \cite{KesseboehmerStratmann:04},
\cite{HanusMauldiUrban:02}, \cite{MesonVericat:04}, \cite{Nakaishi:00}
and \cite{PW}. We remark that brief sketches of some parts of Theorem \ref{thm:main3}
have already been given in \cite{KesseboehmerStratmann:04}. The
results there do for
instance not cover the boundary points of the spectra.
Furthermore, note that for the $\ell_{6}$--spectrum partial results have been
established in \cite{PW} (Corollary 2).
\end{rem}
\section{The Geometry of Modular Codings by Finite and Infinite
Alphabets\label{sec:coding}}
Let $\Gamma:=\textrm{PSL}_{2}\left(\mathbb{Z}\right)$ refer to the modular
group acting on the upper half-plane $\H$. It is well--known that
$\Gamma$ is generated by the two elements $P$ and $Q$, given by\[
P:z\mapsto z-1\,\,\,\textrm{and}\,\,\, Q:z\mapsto\frac{-1}{z}.\]
\begin{figure}[ht]
\psfrag{z0''}{$z_0'$} \psfrag{F}{$F$}\psfrag{R(F)}{$R(F)$}
\psfrag{R2(F)}{$R^2(F)$}\psfrag{i}{$i$} \psfrag{0}{$0$}\psfrag{1}{$1$}
\includegraphics[%
width=0.50\columnwidth,
keepaspectratio]{psl3.eps}
\caption{A fundamental domain $F$ for
$\textrm{PSL}{}_{2}\left(\mathbb{Z}\right)$
and the images under $R$ and $R^{2}$. \label{cap:FundDomain}}
\end{figure}
Defining relations for $\Gamma$ are $Q^{2}=(PQ)^{3}=\{\textrm{id}.\}$,
and a fundamental domain $F$ for $\Gamma$ is the hyperbolic
quadrilateral
with vertices at $i,1+i,\{\infty\}$ and $z_{0}':=(1+i\sqrt{3})/2$.
For $R:=QP$ such that $R:z\mapsto-1/(z-1)$, one easily verifies
that $\Gamma_{0}:=\Gamma/\left\langle R\right\rangle $ is a subgroup
of $\Gamma$ of index $3$ and that $F_{0}$ is a fundamental domain
for $\Gamma_{0}$, for $F_{0}:=F\cup R(F)\cup R^{2}(F)$ the ideal
triangle with vertices at $0,1$ and $\{\infty\}$ (see Fig.
\ref{cap:FundDomain}).
Consider the two elements $A,B\in\Gamma$ given by\[
A:=\left(Q^{-1}PQ\right):z\mapsto\frac{z}{z+1}\quad\textrm{and}\quad
B:=\left(P^{-1}A^{-1}P\right):z\mapsto\frac{-1}{z-2},\]
and let $G$ denote the free semi-group generated by $A$ and $B$.
It is easy to see that for $z_{0}:=A(z_{0}')=B(z_{0}')=(1+i/\sqrt{3})/2$
we have that the Cayley graph of $G$ with respect to $z_{0}$ coincides
with the restriction to \[\left\{
z\in\H{:}\,0\leq\Re(z)\leq1,\;0<\Im(z)\leq1/2\right\} \]
of the the Cayley graph of $\Gamma_{0}$ with respect to $z_{0}$
(see Fig. \ref{cap:fig1}).
\subsection{Finite Coding}
Let $\Sigma:=\left\{ A,B\right\} ^{\mathbb {N}}$ denote the full shift space
on the finite alphabet $\left\{ A,B\right\}$, for $A,B \in \Gamma$
given as above. Also, let
$\Sigma$ be equipped with the usual left-shift
$\sigma:\Sigma\rightarrow\Sigma$.
Then $\Sigma$ is clearly isomorphic to the completion of
$G$, where the completion is taken with respect to a suitable metric
on $G$ (see \cite{Floyd:80}). One immediately verifies that the
canonical map \begin{eqnarray*}
\pi:\qquad\Sigma\qquad & \rightarrow & \qquad[0,1],\\
(x_{1},x_{2},\ldots) & \mapsto & \lim_{n\rightarrow\infty}x_{1}\cdots
x_{n}(z_{0}),\end{eqnarray*}
is 1--1 almost everywhere, in the sense that it is 2--1 on the rationals in $[0,1]$ and 1--1 on $\mathbb {I}$,
for $\mathbb {I}$ referring to the irrational numbers in $\left[0,1\right]$. Note that
the Stern--Brocot sequence $\mathfrak{T}_{n+1}$ coincides
with the set of vertices at infinity of $\{ g(F_{0}):g\in
G$ of word length $n\} $, for each
$n\in\mathbb {N}$.
\begin{figure}[ht]
\psfrag{z0''}{ $z_0'$}\psfrag{z0}{ $z_0$} \psfrag{A(z0)} {$A(z_0)$}
\psfrag{AA(z0)}{ $AA(z_0)$} \psfrag{AAA(z0)}{ $AAA(z_0)$}
\psfrag{BB(z0)}{$BB(z_0)$}\psfrag{BBB(z0)}{$BBB(z_0)$}
\psfrag{AB(z0)}{\tiny $AB(z_0)$} \psfrag{BA(z0)}{\tiny $BA(z_0)$}
\psfrag{B(z0)}{ $B(z_0)$} \psfrag{1/2}{$1/2$}
\psfrag{1/3}{$1/3$} \psfrag{1/4}{$1/4$} \psfrag{2/3}{$2/3$}
\psfrag{1/4}{$1/4$} \psfrag{3/4}{$3/4$}
\psfrag{3/5}{$3/5$}\psfrag{2/5}{$2/5$} \psfrag{0}{$0$}
\psfrag{1}{$1$}
\psfrag{T}{$\times$}\psfrag{T22}{$T_{2,2}$}\psfrag{T21}{$T_{2,1}$}\psfrag{T23}{$T_{2,3}$}\psfrag{T24}{$T_{2,4}$}
\psfrag{T32}{$T_{3,2}$}\psfrag{T31}{$T_{3,1}$}\psfrag{T33}{$T_{3,3}$}\psfrag{T34}{$T_{3,4}$}
\psfrag{T35}{$T_{3,5}$}\psfrag{T36}{$T_{3,6}$}\psfrag{T37}{$T_{3,7}$}\psfrag{T38}{$T_{3,8}$}
\psfrag{F0}{$F_0$}\psfrag{ABF0}{$A(F_0)=B(F_0)$}
\includegraphics[%
width=0.98\columnwidth,
keepaspectratio]{PSL1.eps}
\caption{Part of the Cayley graph rooted at $z_{0}$, for
$\Gamma_0(z_0)$
restricted to $[0,1]\times\mathbb {R}^{+}$, and the
Stern--Brocot intervals of order $2$ and $3$. \label{cap:fig1}}
\end{figure}
\subsection{Infinite Coding}\label{2.2}
For the infinite alphabet $\left\{
X^{n}: n\in\mathbb {N}, X \in \{A,B\}\right\} $ we define the shift space of
finite type \[ \Sigma^{*}:=\left\{ \left(X^{n_{1}},
Y^{n_{2}},X^{n_{3}},\ldots\right):\{X,Y\}=\{A,B\}, n_i \in \mathbb {N} \hbox{
for all } i \in \mathbb {N}\right\}, \] which we assume to be equipped with the usual
left-shift $\sigma^{*}:\Sigma^{*}\rightarrow\Sigma^{*}$. Then there
exists a canonical bijection $\pi^{*}$, given by\begin{eqnarray*}
\pi^{*}:\qquad\Sigma^{*}\qquad & \rightarrow & \mathbb {I}\\
(y_{1},y_{2},\ldots) & \mapsto &
\lim_{k\rightarrow\infty}y_{1}y_{2}\cdots y_{k}(z_{0}).\end{eqnarray*}
This coding is closely related to the continuous fraction expansion.
Namely, if
$y=\left(X^{n_{1}},Y^{n_{2}},X^{n_{3}},\ldots\right)$ then \[
\pi^{*}\left(y\right)=\left\{ \begin{array}{lll}
[n_{1}+1,n_{2},n_{3},\ldots] & \,\,\textrm{for} &
X=A\\
{}[1,n_{1},n_{2},\ldots] & \,\,\textrm{for} &
X=B.\end{array}\right.\]
Also, if $S:[0,1]\rightarrow[0,1]$ and
$s:\Sigma^{*}\rightarrow\Sigma^{*}$
are given by, for $x \in [0,1]$ and $\left\{ X,Y\right\} =\left\{
A,B\right\}$,
\[ S(x) :=\left(1-x\right) \quad\textrm{and}\quad
s\left(X^{n_{1}},
Y^{n_{2}},X^{n_{3}},\ldots\right) :=\left(Y^{n_{1}},
X^{n_{2}},Y^{n_{3}},\ldots\right),\]
then by symmetry we have that $S\circ\pi^{*}=\pi^{*}\circ s$.
To overcome the fact that $\left(\Sigma^{*},\sigma^{*}\right)$ is
not topological transitive, we also require the full shift space
$\left(\overline{\Sigma},\overline{\sigma}\right)$ over $\mathbb {N}$. In here,
$\overline{\Sigma} := \mathbb {N}^{\mathbb {N}}$ and $\overline{\sigma}$ refers to
the left-shift map on $\overline{\Sigma} $. Clearly,
$\left(\overline{\Sigma},\overline{\sigma}\right)$
is \emph{finitely primitive} in the sense of \cite{MauldinUrbanski:03},
and we remark that this property is a necessary preliminary for
the thermodynamical formalism
used throughout this paper.
Note that the two shift spaces $(\Sigma^{*}, \sigma^{*})$ and
$\left(\overline{\Sigma},\overline{\sigma}\right)$ are related by the 2-1 factor
map $p$, which is given by
\begin{equation}
p:\Sigma^{*}\to\overline{\Sigma},\,
\,\,\,\left(X^{n_{1}},Y^{n_{2}},X^{n_{3}},\ldots\right)\mapsto\left(n_{1},n_{2},
n_{3},\ldots\right).\label{eq:FactorMap}\end{equation}
\begin{rem}\label{rem:FG}
Note that the finite coding is in 1-1 correspondence to the coding
of $[0,1]$ via the inverse branches $f_{1}$ and $f_{2}$ of the \emph{Farey
map} $\mathfrak{f}$. In here, $f_{1}$ and $f_{2}$ are given by $f_{1}(x)=x/(x+1)$ and
$f_{2}(x)= 1/(x+1)$, for $x \in [0,1]$.
One easily verifies that $f_{1}=A$ and $f_{2} \circ S=B$, and hence
$\Sigma$ can be interpreted as arising from a `twisted Farey
map'. Similarly, one notices that $\Sigma^{*}$ is closely related
to
the coding of $[0,1]$ via the infinitely many branches of the Gauss
map $\mathfrak{g}$, which is given by $\mathfrak{g}(x):=1/x \;\mathrm{mod }\;1$
for $x \in [0,1]$. More precisely, we have that the dynamical
system $(\mathbb {I},\mathfrak{g})$ is a topological 2--1 factor of the dynamical system
$(\Sigma^*,\sigma^*)$, where the factor map can be established either on the symbolic
level via $p$ or on the geometric level via ${\mathfrak{f}}$.
The situation is summarized in the following commuting diagram.
{\Large \begin{eqnarray*}\xymatrix{
\mathbb {I} \ar[d]_{\mathfrak{g}} && \ar[ll]_{\pi_{\mathrm{CF}}}\overline{\Sigma}\ar[d]_{\overline{\sigma}}&&
\ar[ll]_{p}\Sigma^* \ar[rr]^{\pi^*}\ar[d]_{\sigma^*} &&
\mathbb {I} \ar[d]^{{\mathfrak{g}_s}} \ar[rr]^{\mathfrak{f}} && \mathbb {I} \ar[d]^{{\mathfrak{g}}} \\
\mathbb {I} && \ar[ll]^{\pi_{\mathrm{CF}}} \overline{\Sigma} &&
\ar[ll]^{p}\Sigma^* \ar[rr]_{\pi^*} && \mathbb {I} \ar[rr]_{\mathfrak{f}} && \mathbb {I} \\}
\end{eqnarray*}}
In here, $\mathfrak{g}_s := \pi^*\circ\sigma^*\circ(\pi^*)^{-1}$ denotes the 'twisted Gauss map' and
$\pi_{\mathrm{CF}}$ is given by
$\pi_{\mathrm{CF}}(n_1,n_2,\ldots ):=[n_1,n_2,\ldots ]$ for $(n_1,n_2,\ldots )\in\overline\Sigma$.
Note that both $\pi_{\mathrm{CF}}$ and $\pi^*$ are bijections and that $\pi_{\mathrm{CF}}\circ p=\mathfrak{f}\circ\pi^*$.
\end{rem}
\subsection{Stern--Brocot sequences versus continued fractions}
We end this section by showing that there is a 1--1 correspondence
between
the elements of the Stern--Brocot sequence and finite continued fraction
expansions. This will turn out to be useful in the sequel.
For $n\geq2$, let $A_{k}^{n}$ refer to the set all $k$-tuples of
positive integers which add up to $n$ and whose $k$-th entry exceeds
$1$.
That is, \begin{equation}
A_{k}^{n}:=\left\{ \left(a_{1},a_{2},\ldots,
a_{k}\right)\in\mathbb{N}^{k}:\sum_{i=1}^{k}a_{i}=n,\,\,
a_{k}\not=1\right\} .\label{eq:DefAnk}\end{equation}
Since $a_{k}\not\neq1$, we can identify an element $\left(a_{1},
\ldots,a_{k}\right)\in A_{k}^{n}$
in a unique way with the finite continued fraction expansion
$\left[a_{1},a_{2},\ldots a_{k}\right]$.
Also, one easily verifies that for $1\leq k\leq n-1$,\begin{equation}
\card\left(A_{k}^{n}\right)={\binom{n-2}{
k-1}}.\label{eq:CardAkn}\end{equation}
\begin{lem}
\label{lem:TnbyCF} For all $n\geq2$ we have\[
\bigcup_{k=1}^{n-1}\bigcup_{(a_{1},a_{2},\ldots,a_{k}) \in A_{k}^{n}}\left[a_{1},a_{2},\ldots a_{k}
\right]=\mathfrak{T}_{n-1}\setminus\mathfrak{T}_{n-2}=
\left\{ \frac{s_{n-1,2\ell}}{t_{n-1,2\ell}}:1\leq\ell\leq2^{n-2}\right\}
.\]
Furthermore, if $\left(s_{n,k}/t_{n,k}\right)=\left[a_{1},a_{2},
\ldots,a_{m}\right]\in\mathfrak{T}_{n}\setminus\mathfrak{T}_{n-1}$ then its two siblings
in $\mathfrak{T}_{n+1}\setminus\mathfrak{T}_{n}$ are, for $\left\{ u,v\right\} =\left\{
2k,2k-2\right\} $,
\[
\frac{s_{n+1,u}}{t_{n+1,u}}=\left[a_{1},a_{2},\ldots,a_{m-1},a_{m}+1\right]
\quad \textrm{and} \quad \frac{s_{n+1,v}}{t_{n+1,v}}=\left[a_{1},a_{2},
\ldots,a_{m-1},a_{m}-1,2\right].\]
\end{lem}
\begin{proof}
For the first part of the lemma note that the second equality follows
by definition of $\mathfrak{T}_{n}$. The first equality is obtained
by induction as follows. We clearly have $\left\{
\left[2\right]\right\} =\mathfrak{T}_{1}\setminus\mathfrak{T}_{0}$.
Then assume that the assertion holds for $n-1$. Since the sets
$\mathcal{\mathfrak{T}}_{n}$
are $S$--invariant it follows for $n\geq 3$,\[
\mathfrak{T}_{n-1}\setminus\mathfrak{T}_{n-2}=\bigcup_{x\in\mathfrak{T}_{n-2}\setminus\mathfrak{T}_{n-3}}A(x)\cup
BS(x).\]
For $\left[a_{1},\ldots,a_{k}\right]
\in\mathfrak{T}_{n-2}\setminus\mathfrak{T}_{n-3}$ we have by the
inductive assumption that $
\sum_{i=1}^k a_i =n-1$, and hence
\begin{eqnarray*}
A\left(\left[a_{1},\ldots,a_{k}\right]\right) & = &
\frac{1}{1/\left[a_{1},\ldots,a_{k}\right]+1}=\left[a_{1}+1,a_{2},\ldots,a_{k}\right]\in
A_{k}^{n},\\
BS\left(\left[a_{1},\ldots,a_{k}\right]\right) & = &
\frac{1}{1+\left[a_{1},\ldots,a_{k}\right]}=\left[1,a_{1},a_{2},\ldots,a_{k}\right]\in
A_{k+1}^{n}.\end{eqnarray*}
By combining the two latter observation, we obtain
\[\mathfrak{T}_{n-1}\setminus\mathfrak{T}_{n-2}\subset\bigcup_{k=1}^{n-1}
\bigcup_{(a_{1},a_{2},\ldots,a_{k}) \in A_{k}^{n}}\left[a_{1},a_{2},\ldots,a_{k}\right].\]
Since \begin{eqnarray*}
\card\left(\mathcal{\mathfrak{T}}_{n-1}\setminus\mathfrak{T}_{n-2}\right)
& = &
\card\left(\mathfrak{T}_{n-1}\right)-\card\left(\mathfrak{T}_{n-2}\right)=2^{n-2}\\
& = & \sum_{k=1}^{n-1}{\binom{n-2}{
k-1}}=\card\left(\bigcup_{k=1}^{n-1}A_{k}^{n}\right),\end{eqnarray*}
the first part of the lemma follows.
For the second part note that by the above
\[\left[a_{1},a_{2},\ldots,a_{m}+1\right],\,
\left[a_{1},a_{2},\ldots,a_{m}-1,2\right]\in
\mathfrak{T}_{n+1}\setminus\mathfrak{T}_{n}.\]
Since $\left[a_{1},a_{2},\ldots,a_{m}+1\right]$,
$\left[a_{1},a_{2},\ldots,a_{m}\right]$, and
$\left[a_{1},a_{2},\ldots,a_{m}-1,2\right]$ are consecutive neighbours
in $\mathfrak{T}_{n+1}$, the lemma follows.
\end{proof}
\begin{rem}
Note that $P$ can be written alternatively also in terms of denominators
of approximants as follows.
\[
P(\theta)=\lim_{n\to\infty}\frac{1}{n}\log\sum_{k=1}^{n}\sum_{\left(a_{1},\ldots,a_{k}\right)\in
A_{k}^{n}}q_{k}\left(\left[a_{1},\ldots,a_{k}\right]\right)^{-2\theta}.\]
In order to see this, note that for $\theta\leq0$,
\[
\sum_{k=1}^{2^{n}}\left(t_{n,k}t_{n,k+1}\right)^{-\theta}\leq2\sum_{k=1}^{2^{n-1}}
\left(t_{n,2k}\right)^{-2\theta}\leq\sum_{k=1}^{2^{n+1}}\left(t_{n+1,k}t_{n+1,k+1}\right)^{-\theta}.\]
On the other hand, using the recursive definition
of $t_{n,k}$, we have for $\theta>0$,\[
\sum_{k=1}^{2^{n-1}}\left(t_{n-1,k}t_{n-1,k+1}\right)^{-\theta}\geq\sum_{k=1}^{2^{n-1}}
\left(t_{n,2k}\right)^{-2\theta}
\geq\frac{\left(n+1\right)^{-\theta}}{4}\sum_{k=1}^{2^{n+1}}\left(t_{n+1,k}t_{n+1,k+1}\right)^{-\theta}.\]
Therefore, by taking logarithms, dividing by $n$ and letting $n$
tend to infinity, we obtain \[
P\left(\theta\right)=\lim_{n\to\infty}\frac{1}{n}\log\sum_{k=1}^{2^{n-1}}\left(t_{n,2k}\right)^{-2\theta}.\]
Hence, using Lemma \ref{lem:TnbyCF}, the result follows.
\end{rem}
\section{Dynamical cocycles versus Stern--Brocot
sequences\label{sec:Dynamical-cocycles-versus}}
In this section we introduce the dynamical cocycles which will be
crucial in the multifractal analysis to come. We show that
these cocycles are closely related to Stern--Brocot intervals and
continued fractions. Finally, we give the proof
of the first part of Proposition \ref{Thm:2}. We remark that the results
in this section could be obtained alternatively by using elementary
estimates for countinued fractions only. Instead, we have put some
emphasis on obtaining these results by making use of the hyperbolic metric $d$
on ${\mathbb{H}}$. The intension here is that this should
make it easier to follow the later transfer of the results
of \cite{KesseboehmerStratmann:04a}, which were derived in terms of Kleinian
groups, into the language
of Stern--Brocot intervals and continued fractions.
Recall that the Poisson kernel $\mathfrak{P}$ for the upper half-plane is
given by\[
\mathfrak{P}:(z,\xi)\mapsto\frac{\Im\left(z\right)}{\left(\Re\left(z\right)-
\xi\right)^{2}+\Im\left(z\right)^{2}} \, , \textrm{ for all }
z\in\H, \xi\in\mathbb {R}.\]
With $z_{0}$ defined as in Section \ref{sec:coding}, the cocycle
$I:\Sigma\to \mathbb {R}$ associated with the finite alphabet is given
by\[
I(x):=\left|\log\left(\mathfrak{P}\left(x_{1}(z_{0}),\pi\left(x\right)\right)\right)
-\log\left(\mathfrak{P}\left(z_{0},\pi\left(x\right)\right)\right)\right| ,
\textrm{ for } x=\left(x_{1},x_{2},\ldots\right)\in\Sigma.\]
We remark that $I$ is continuous with respect to the standard metric.
Also, it is well--known that
$S_{n}I(x):=\sum_{i=0}^{n-1}I\left(\sigma^{i}\left(x\right)\right)$
is equal to the hyperbolic distance of $z_{0}$ to the horocycle
through $x_{1}x_{2}\cdots x_{n}\left(z_{0}\right)$ based at $\pi(x)$.
Furthermore, note that in terms of the theory of iterations of maps,
$I$ is equal to
the logarithm of the modulus of the derivative of the `twisted Farey
map', mentioned in Remark \ref{rem:FG}.
Similar,
we define the cocycle $I^{*}:\Sigma^{*}\to \mathbb {R}$ associated with the infinite
alphabet as follows. For
$y=\left(X^{n_{1}},Y^{n_{2}},\ldots\right)\in\Sigma^{*}$ such that
$\left\{ X,Y\right\} =\left\{ A,B\right\} $, let $I^{*}$ be given by\[
I^{*}(y):=\left|\log\left(\mathfrak{P}\left(X^{n_{1}}Y(z_{0}),\pi^{*}\left(y\right)\right)\right)
-\log\left(\mathfrak{P}\left(z_{0},\pi^{*}\left(y\right)\right)\right)\right|.\]
One immediately verifies that
$S_{k}I^{*}(y):=\sum_{i=0}^{k-1}I^{*}\left(\left(\sigma^{*}\right)^{i}(y)\right)$
is equal to the hyperbolic distance of $z_{0}$ to the horocycle
based at $\pi^{*}(y)$ containing either the point $X^{n_{1}}Y^{n_{2}}\cdots
X^{n_{k}}Y\left(z_{0}\right)$
(if $k$ is odd) or $X^{n_{1}}Y^{n_{2}}\cdots Y^{n_{k}}X\left(z_{0}\right)$
(if $k$ is even). Note that in terms of the theory of iterations of maps,
the function $I^{*}$ is clearly an analogue of
the logarithm of the modulus of the derivative of the Gauss map.
Throughout, we also require the potential function $N:\Sigma^{*}\to\mathbb {N}$,
which is given by
\[ N\left((X^{n_{1}},Y^{n_{2}},\ldots)\right):=n_{1}, \hbox{ for each }
(X^{n_{1}},Y^{n_{2}},\ldots)\in \Sigma^{*}.\]
Note
that $S_{k}N((X^{n_{1}},Y^{n_{2}},\ldots))=\sum_{i=1}^{k}n_{i}$.
Finally, the relevant potentials for the shift space $\left(\overline{\Sigma},\overline{\sigma}\right)$
are the functions
\begin{equation} \label{barIN} \overline{I}:= I^{*}\circ
p_{A}=I^{*}\circ p_{B}\;\;\;\mbox{and}\;\;\;
\overline{N}:=N\circ p_{A}=N\circ p_{B}.\end{equation}
In here,
$p_{X}$ refers to the inverse
branch with respect to $X\in\left\{ A,B\right\}$ of the $2$-$1$ factor map $p$ introduced in
Section \ref{2.2}. More precisely, we have for
$X,Y \in \left\{ A,B\right\} $ such that $X \neq Y$, \[
p_{X}\left(\left(n_{1},n_{2},n_{3},
\ldots\right)\right):=\left(X^{n_{1}},Y^{n_{2}},X^{n_{3}},\ldots\right).\]
The following lemma relates the Euclidean sizes of the Stern--Brocot
intervals to the hyperbolic distances of $z_{0}$ to the elements in the orbit
$G(z_{0})$.
\begin{lem}
\label{lem:UniformlyExcursion} For each $n\in\mathbb {N}$ and $x\in\mathbb {I}$
such that $\pi^{-1}(x)=(x_{1},x_{2},...)\in\Sigma$, we have
\[
\left|T_{n}(x)\right|\asymp m_{n}(x)\,
e^{-d\left(z_{0},x_{1}...x_{n}(z_{0})
\right)}.\]
In here, $m_{n}(x)$ is defined by $m_{n}(x):=\max\{ k:x_{n+1-i}=x_{n}\,\,\,\textrm{for}\,\,\,
i=1,...,k\}$.
\end{lem}
\begin{proof}
For $n=1$ the statement is trivial. For $n\geq2$, we first consider
the case $m_{n}(x)=1$. If $g:=x_{1}...x_{n-1}\in G$, then
$g^{-1}(T_{n}(x))$ is equal to either $T_{1,1}$ (for
$x_{n}=A$) or $T_{1,2}$ (for $x_{n}=B$). Also, note that
for the modulus of the conformal derivative
we have \[
\left|\left(g^{-1}\right)'(\xi)\right|\asymp e^{d(z_{0},g(z_{0}))} ,
\textrm{ for }
\xi \in T_{n}(x).\]
Combining these two observations, we obtain \[
\left|T_{n}(x)\right|\asymp\left|g'|_{[0,1]}\right|\asymp\left|\left(g^{-1}\right)'|_{T_{n}(x)}\right|^{-1}\asymp
e^{-d(z_{0},g(z_{0}))}\asymp e^{-d(z_{0},gx_{n}(z_{0}))}.\]
This proves the assertion for $m_{n}(x)=1$.
For the general situation we only consider the case
$x_{1}\cdots x_{n}=A^{y_{1}}B^{y_{2}}\cdots B^{y_{k}}$. The
remaining cases can be dealt with in a similar way. In this case
$m_{n}(x)=y_{k}$,
and the above implies, for $l:=\sum_{i=1}^{k-1}y_{i}$,
\[
\left|T_{l+1}(x)\right|\asymp e^{-d\left(z_{0},x_{1}\cdots
x_{l+1}(z_{0})\right)}.\]
Also, by using the well-known elementary fact that
$e^{d(z_{0},X^{k}(z_{0}))} \asymp \Im (X^{k}(z_{0})) \asymp
1/k^{2}$ for $X \in \{A,B\}$ and $k \in \mathbb {N}$,
one immediately obtains for $1<m \leq y_{k}$, \[
e^{d(z_{0},x_{1}\cdots x_{l+m}(z_{0}))}\asymp e^{d(z_{0},x_{1}\cdots
x_{l}(z_{0}))}e^{d(x_{1}\cdots x_{l}(z_{0}),x_{1}\cdots
x_{l+m}(z_{0}))}\asymp m^{2}e^{d(z_{0},x_{1}\cdots x_{l+1}(z_{0}))}.\]
Finally, one also immediately verifies that for $1\leq m \leq y_{k}$,
\begin{equation}
\left|T_{l+m}(x)\right|\asymp\sum_{k=m}^{\infty}k^{-2}\left|T_{l+1}(x)\right|\asymp
m^{-1}\left|T_{l+1}(x)\right|.\label{eq:AsympParabExcursion}\end{equation}
Combining the three latter observations, the statement of the lemma
follows.
\end{proof}
The previous lemma has the following immediate implication.
\begin{cor}
\label{cor:VergleichTSnI} For each $n\in\mathbb {N}$ and $x\in\mathbb {I}\,$
such that $\pi^{-1}(x) \in\Sigma$, we have
\[
\left|S_{n}I(\pi^{-1}(x))+\log\left|T_{n}(x)\right|\right|\ll\log n.\]
\end{cor}
The following lemma relates the cocycle $I^{*}$ to the
sizes of the Stern--Brocot intervals and to the denominators $q_{k}^{2}$ of
the approximants.
\begin{lem}
\label{lem:Vergleichbar} For each $k\in\mathbb {N}$ and $x\in\mathbb {I}$
we have, with $n_{k}(x):=S_{k}N\left(\left(\pi^{*}\right)^{-1}(x)
\right)$,
\[
\left|T_{n_{k}(x) +1}(x)\right|\asymp\exp\left(-S_{k}I^{*}\left(\left(\pi^{*}\right)^{-1}(x)\right)\right)\asymp
q_{k}(x)^{-2}.\]
\end{lem}
\begin{proof}
We only consider the case $k$ even and $X=A$. The remaining
cases can be obtained in a similar way. Let $g:=A^{y_{1}}B^{y_{2}}\cdots
A^{y_{k}}\in G$, and note that then
\[
q_{k}(x)^{-2}\asymp e^{-d(z_{0},g(z_{0}))}.\]
Combining this with the fact that for $\xi\in T_{n+1}(x)$ we have
\[
\exp(-d(z_{0},g(z_{0})))\asymp\exp\left(-S_{k}I^{*}\left(\left(\pi^{*}\right)^{-1}(\xi)\right)\right)\]
(which follows since on $T_{n+1}(x)$
we have that
$\exp\left(S_{k}I^{*}\circ\left(\pi^{*}\right)^{-1}\right)$
is comparable to $|\left(g^{-1}\right)'|$), we obtain \[
e^{-S_{k}I^{*}\left(\left(\pi^{*}\right)^{-1}(x)\right)}\asymp
q_{k}(x)^{-2}.\]
Finally, note that by Lemma \ref{lem:UniformlyExcursion} and since
$\exp\left(d\left(z_{0},gB(z_{0})\right)\right)\asymp\exp\left(d\left(z_{0},g(z_{0})\right)\right)$,
we have \[
\left|T_{n+1}(x)\right|\asymp e^{-d(z_{0},gB(z_{0}))}\asymp
e^{-d(z_{0},g(z_{0}))}.\]
Combining these estimates, the lemma follows.
\end{proof}
We are now in the position to prove the first part of Proposition
\ref{Thm:2}.
\begin{proof}
[Proof of first part of Proposition \ref{Thm:2}] The equalities
$\ell_{3}=\ell_{6}$ and $\ell_{1}=\ell_{5}$
are immediate consequences of the following well--known Diophantine
inequalities. For all
$x\in[0,1]$ and $k\in\mathbb {N}$, we have (see e.g. \cite{Khinchin:35})
\begin{equation}
\frac{1}{q_{k}(x)\left(q_{k+1}(x)+q_{k}(x)\right)}<\left|x-\frac{p_{k}(x)}{q_{k}\left(x\right)}
\right|<\frac{1}{q_{k}(x)q_{k+1}(x)}.\label{eq:CFInequality}\end{equation}
In order to show that $\ell_{1}=\ell_{4}$, let $n_{k}(x):=S_{k}N
\left(\left(\pi^{*}\right)^{-1}(x)\right)$, for $x\in\mathbb {I}$ and $k\in\mathbb {N}$.
Obviously, $\left(\log q_{k}(x)/n_{k}(x)\right)$ is a subsequence
of the sequence $\left(-\log |T_{n}(x)| / n\right)$. Therefore, if
$\ell_{4}(x)$ exists then so does $\ell_{1}(x)$, and both limits
must coincide. For the reverse, suppose that $\ell_{1}(x)$ exists
such that $\ell_{1}(x) = \alpha$. Let $m_{n}(x)$ be defined as in
the statement of Lemma \ref{lem:UniformlyExcursion}, and put
$k_n(x):=\sup\{k\in \mathbb {N}:n_k(x)\leq n\}$. By combining (\ref{eq:AsympParabExcursion}) and Lemma
\ref{lem:Vergleichbar}, we then have
\begin{eqnarray*}
\lim_{n\to\infty}\frac{-\log\left|T_{n}\left(x\right)\right|}{n} & \leq &
\lim_{n\to\infty}\frac{2\log q_{k_{n}\left(x\right)}\left(x\right)+
2\log (m_{n}\left(x\right)+1)}{n_{k_{n}\left(x\right)}\left(x\right)+m_{n}\left(x\right)}\\
& \leq & \lim_{n\to\infty}\frac{2\log q_{k_{n}\left(x\right)}\left(x\right)}{n_{k_{n}\left(x\right)}\left(x\right)}
+\lim_{n\to\infty}\frac{2\log (m_{n}\left(x\right)+1)}{n_{k_{n}\left(x\right)}\left(x\right)+m_{n}\left(x\right)}=
\alpha.\end{eqnarray*}
This gives the upper bound, and hence finishes the proof
in particular for $\alpha=0$.
For the opposite inequality we can therefore assume without loss of
generality that $\alpha>0$. First, observe that
\begin{eqnarray*}
\lim_{n\to\infty}\frac{-\log\left|T_{n}\left(x\right)\right|}{n} & = &
\lim_{n\to\infty}\frac{2\log q_{k_{n}\left(x\right)}\left(x\right)+
2\log m_{n}\left(x\right)}{n_{k_{n}\left(x\right)}\left(x\right)+m_{n}\left(x\right)}\\
& \geq & \lim_{n\to\infty}\frac{2\log q_{k_{n}\left(x\right)+1}
\left(x\right)}{n_{k_{n}\left(x\right)+1}\left(x\right)}-
\lim_{n\to\infty}\frac{2\log
a_{k_{n}\left(x\right)+1}\left(x\right)}{n_{k_{n}\left(x\right)}\left(x\right)} .
\end{eqnarray*}
Hence, it is now sufficient to show that $\lim_{k\to\infty}\log
a_{k+1}\left(x\right) / n_{k}\left(x\right)=0$,
or what is equivalent $\lim_{k\to\infty} \log
a_{k+1}\left(x\right) / \log q_{k}\left(x\right) =0$.
For this, observe that
\[ \lim_{k\to \infty} \frac{\log q_{k+1}(x)}{n_{k+1}(x)} =
\lim_{k\to \infty} \frac{\log q_{k}(x) +\log a_{k+1}(x)}{n_{k} (x)+
a_{k+1}(x)} =
\lim_{k\to \infty} \frac{\log q_{k}(x) \left(1+ \frac{\log
a_{k+1}(x)}{\log q_{k}(x)} \right)}{n_{k}(x) \left(1+
\frac{a_{k+1}(x)}{n_{k}(x) } \right) } .\]
If we would have that $\lim_{k\to\infty} \log
a_{k+1}\left(x\right) / \log q_{k}\left(x\right) \neq 0$, then
there exists a subsequence $(k_l)$ such that $\lim_{l\to\infty} \log
a_{k_l+1}\left(x\right) / \log q_{k_l}\left(x\right)=
c$, for some $c \in (0,\infty]$.
It follows that $\lim_{l \to\infty} a_{k_l+1} = \infty $, and hence by
combining this with the
calculation above, we obtain
\[
1=\lim_{l\to\infty}\frac{\log a_{k_{l}+1}(x)\cdot n_{k_l}(x)}{a_{k_{l}+1}(x) \cdot\log q_{k_l}(x)}=\frac{0}{\alpha}=0.
\]
This shows that $\ell_{1}(x)=\ell_{4}(x)$, and hence finishes the proof.
\end{proof}
\section{\label{sec:The-shapes-of-P} Analytic properties of $P$ and
$\widehat{P}$}
The main goal in this section is to derive various analytic properties
of the
Stern--Brocot pressure function $P$. These properties are derived by
considering the pressure functions associated with the systems $\Sigma, \Sigma^{*}$
and $\overline{\Sigma}$. In order to introduce these functions, let $\mathcal{C}_{n}:=\left\{ C_{n}(x):x\in\Sigma\right\} $ refer
to the set of all $n$--cylinders \[
C_{n}\left(x\right):=\left\{ y\in\Sigma:y_{i}=x_{i}, \hbox{for } i=1,\ldots,n\right\}
.\]
Likewise, let $\mathcal{C}^{*}_{n}$ (resp.
$\overline{\mathcal{C}}_{n}$) refer to the set of $n$-cylinders for
the system $(\Sigma^{*},\sigma^{*})$ (resp.
$(\overline{\Sigma},\overline{\sigma})$). The pressure function
$\mathcal{P}$ associated with $\Sigma$ is then given by \[
\mathcal{P}(\theta):=\lim_{n\to\infty}\frac{1}{n}\log\sum_{C\in\mathcal{C}_{n}}
\exp\left(\sup_{x\in
C}S_{n}\left(-\theta I\right)(x)\right), \textrm{ for }
\theta\in\mathbb {R}.\]
Also, for the system $\Sigma^{*}$ we define the pressure functions
$\P^{*}$ and $P^{*}$, for $\theta<1, q>0$ and $f: \Sigma^{*}
\to\mathbb {R}$ continuous, by
\[
\P^{*}(f):=\lim_{n\to\infty}\frac{1}{n}\log\sum_{C^{*}\in\mathcal{C}_{n}^{*}}\exp\left(\sup_{y\in
C^{*}}S_{n}f\left(y\right)\right) \hbox{ and } P^{*} (\theta, q):= \P^{*}(-\theta
I^{*}-qN).\]
Finally, the pressure functions $\overline{\P}$ and $\overline{P}$
associated with $(\overline{\Sigma}, \overline{\sigma})$
are given completely analogous, for $\theta<1, q>0$ and
$g: \overline{\Sigma} \to\mathbb {R}$ continuous, by
\[
\overline{\P}(g):=\lim_{n\to\infty}\frac{1}{n}\log\sum_{\overline{C}\in\overline{\mathcal{C}}_{n}}\exp\left(\sup_{y\in
\overline{C}}S_{n}g\left(y\right)\right)
\hbox{ and } \overline{P}(\theta, q):= \overline{\P}(-\theta
\overline{I}-q\overline{N}).\]
Clearly, by recalling the definitions of $\overline I, \overline N$
and $I^{*}, N$ in Section 3, we immediately have that $\overline{P} = P^*$.
\subsection{
Analytic Properties of
$P^{*}$ by Hanus, Mauldin and Urba\'nski}
In this subsection we employ important results of Hanus, Mauldin and
Urba\'nski obtained in \cite{MauldinUrban:01}
and \cite{HanusMauldiUrban:02}. The results here will be
crucial cornerstones in our
subsequent analysis of
the Stern--Brocot pressure.
Studies of analytic properties of
pressure functions are usually based on the existence
of certain Gibbs measures, here on $\Sigma^{*}$ and
$\overline{\Sigma}$. The existence of these measures in our situation here
is guaranteed by the following proposition, which essentially follows
from a result in
\cite{MauldinUrban:01}.
\begin{prop}\label{propHMU2}
For each $\theta <1$, $ q>0$, and for $(\theta,q)=(1,0)$,
there exists a unique completely ergodic $\overline{\sigma}$--invariant Gibbs measure
$\overline{\mu}_{\theta,q}$
associated with the potential $-\theta \overline{I}-q\overline{N}$.
That is, we have for all $n\in\mathbb {N}, \overline{C}\in
\overline{\mathcal{C}}_{n}$ and $y\in \overline{C}$,
\begin{equation}
\overline{\mu}_{\theta,q}(\overline{C})\asymp\exp\left(S_{n}\left(-\theta
\overline{I}(y)-q\overline{N}(y)\right)-n\,\overline{\P}(-\theta
\overline{I}-q\overline{N})\right).\label{eq:Gibbs*Property}
\end{equation}
In particular, the Borel measure
$\mu_{\theta,q}^{*}:=1/2\cdot \left(\overline{\mu}_{\theta,q}\circ
p_{A}^{-1}+\overline{\mu}_{\theta,q}\circ p_{B}^{-1}\right)$
is an ergodic $\sigma^{*}$--invariant Gibbs
measure on $\Sigma^{*}$ such that
for all $n \in \mathbb {N}, C^{*}\in \mathcal{C}_{n}^{*}$ and $y\in C^{*}$,
\begin{equation}
\mu^{*}_{\theta,q}(C^{*})\asymp\exp\left(S_{n}\left(-\theta
I^{*}(y)-q N (y)\right)-n\, \P^{*}(-\theta
I^{*}-q N)\right).\label{eq:Gibbs*Property2}\end{equation}
The measure $\mu_{\theta,q}^{*}$ is unique with respect
to this property, and $\overline{\mu}_{\theta,q}=\mu_{\theta,q}^{*}\circ
p^{-1}$.
\end{prop}
\begin{proof} By \cite{KesseboehmerStratmann:04a} (Lemma
3.4), the cocycle $I^{*}$ is H{\"o}lder continuous in the sense that there
exists $\kappa>0$ such that for each $n\in\mathbb {N}$, \[
\sup_{C\in\overline{\mathcal{C}}_{n}}\sup_{x,y\in
C}\left|\overline{I}(x)-\overline{I}(y)\right|\ll\exp(-\kappa\, n).\]
Clearly, we also immediately have that $\overline{N}$ is H\"older
continuous.
Furthermore, the following summability condition holds for $\theta <1$, $ q>0$, and for $(\theta,q)=(1,0)$,
\begin{equation}\label{summability}
\sum _{i\in \mathbb {N}}\exp(\sup\{-\theta \overline{I}(x)-q
\overline{N}(x):x_1=i\})\ll\sum _{i\in \mathbb {N}}
i^{-2\theta}\cdot \mbox{\rm e}^{-qi}<\infty .
\end{equation}
Hence, all preliminaries of \cite{MauldinUrban:01} (Corollary 2.10)
are fulfilled, which then gives the existence of a
unique invariant Gibbs measure $\overline{\mu}_{\theta,q}$ with properties as stated in the
proposition.
Immediate consequences of the definition of $N$ and the definition $\mu_{\theta,q}^{*}:=1/2\cdot \left(\overline{\mu}_{\theta,q}\circ
p_{A}^{-1}+\overline{\mu}_{\theta,q}\circ p_{B}^{-1}\right)$ are that
$\mu_{\theta,q}^{*}$
is $\sigma^*$--invariant, that $\mu_{\theta,q}^{*}$ fulfills the Gibbs property (\ref{eq:Gibbs*Property2}), and
that the equality
$\overline{\mu}_{\theta,q}=\mu_{\theta,q}^{*}\circ
p^{-1}$ is satisfied. To prove
ergodicity of $\mu^*_{\theta,q}$,
let $D\subset\Sigma^*$ such that ${\sigma^*}^{-1}(D)=D$.
We then have $\overline{\sigma}^{-1} p_X^{-1}(D)=p_Y ^{-1}(D)$,
for $X,Y\in \{A,B\}$ such that $X\neq Y$. This
gives $\overline{\sigma}^{-2}({p_X}^{-1}(D))={p_X}^{-1}(D)$.
Since $\overline{\mu}_{\theta,q}$ is completely ergodic, which by
definition means that $\overline{\mu}_{\theta,q}$ is ergodic with
respect to $\overline{\sigma}^n$ for all $n\in \mathbb {N}$, it follows
$\overline{\mu}_{\theta,q}({p_X}^{-1}(D))\in\{0,1\}$. The $\overline{\sigma}$-invariance of
$\overline{\mu}_{\theta,q}$ then implies that $\overline{\mu}_{\theta,q}({p_X}^{-1}(D))=
\overline{\mu}_{\theta,q}(\overline{\sigma}^{-1}({p_X}^{-1}(D)))=\overline{\mu}_{\theta,q}({p_Y}^{-1}(D))$.
Consequently, it follows that $\mu^*_{\theta,q}(D)\in \{0,1\}$.
\end{proof}
The following proposition employs yet another result of Hanus, Mauldin and
Urba\'nski, obtained in their spectral analysis of the Perron--Frobenius operator.
\begin{prop}\label{propHMU3}
The pressure function $P^{*}$ is a convex, decreasing and real-analytic
function with
respect to both coordinates In the second coordinate $P^{*}$ is strictly
decreasing to $(-\infty)$. In particular, there hence exists a positive
real--analytic function $\beta$ on $(-\infty,1)$ such that
$P^{*}(\theta,\beta(\theta))=0$. Furthermore, for the
derivative of $\beta$ at $\theta<1$ we have
\begin{equation}\label{derive} \beta'(\theta)=\frac{-\int I^{*}\,
d\mu_{\theta}^{*}}{\int N\, d\mu_{\theta}^{*}}=-\int I\, d\mu_{\theta}.
\end{equation}
In here,
$\mu_{\theta}^{*}:=\mu_{\theta,\beta(\theta)}^{*}$ refers to the
unique $\sigma^{*}$--invariant Gibbs measure
associated with the
potential $-\theta I^{*}-\beta(\theta)N$. Also, $\mu_{\theta}$
refers to the $\sigma$--invariant probability measure on $\Sigma$
absolutely continuous to $\mu_{\theta}^{*}$, whose existence is
guaranteed by {\em Kac's formula} (\cite{Kac:47}; see the proof).
\end{prop}
\begin{proof} First, note that it is sufficient to verify the
statements in the proposition for $\overline{\Sigma}$ only. Then note
that
$\overline{I}$ and $\overline{N}$ are H\"older continuous, that
the summability condition (\ref{summability}) is satisfied
for all
$(\theta,q)\in (-\infty,1)\times (0,\infty)$, and that
\[
\int \left( \theta \overline{I} + q \overline{N} \right)
d\overline{\mu}_{\theta,q}\ll\sum_{n\in\mathbb {N}}(2\theta\log n+q n) n^{-2\theta}\mbox{\rm e} ^{-q n}<\infty.
\]
Hence, we can apply \cite{HanusMauldiUrban:02} (Proposition 6.5)
(see also \cite{MauldinUrbanski:03} (Proposition 2.6.13)), from which
it follows that $\overline{P}(\theta,q)$ is real-analytic on
$(-\infty,1)\times (0,\infty)$. Next, note that for the partial derivatives of
$\overline{P}$
we have
\[
\frac{\partial \overline{P}(\theta,q)}{\partial \theta}=\int -\overline{I} \;d
\overline{\mu}_{\theta,q} \hbox{ and }
\frac{\partial \overline{P}(\theta,q)}{\partial q}=\int -\overline{N} \;d \overline{\mu}_{\theta,q}<0.
\]
This shows that $\overline{P}$ as a function in the second
coordinate is strictly decreasing, which then gives the existence of
a
real-analytic function $\beta:(-\infty,1)\to (0,\infty)$ for which
$\overline{P}(\theta,\beta(\theta))=0$, for all $\theta <1$.
Now, the first equality in (\ref{derive}) follows
from the Implicit Function Theorem. The second equality is a consequence
of {\em Kac's formula} (\cite{Kac:47}), which guarantees that there exists a
$\sigma$--invariant measure $\widetilde{\mu}_{\theta}$
on $\Sigma$,
given by \begin{equation}
\widetilde{\mu}_{\theta}(M):=\int\sum_{i=0}^{N(y)-1}\1_{M}\circ\sigma^{i}(\iota(y))\,\,
d\mu_{\theta}^{*}(y), \hbox{ for each Borel set } M \subset \Sigma .\label{eq:Kac}
\end{equation} In here,
$\iota:\Sigma^{*}\to\Sigma$ refers to the canonical injection which
maps an element of $\Sigma^{*}$ to its representation by means of the
finite alphabet of $\Sigma$. We remark that Kac's formula gives in
fact a 1--1 correspondence between $\sigma$--invariant measures on
$\Sigma$ and $\sigma^{*}$--invariant measures on $\Sigma^{*}$.
Now, one immediately verifies that \[
\widetilde{\mu}_{\theta}(\Sigma)=\sum_{\ell=1}^{\infty} \ell \mu^{*}_{\theta}\left(\left\{
N=\ell\right\}
\right)\asymp\sum_{\ell=1}^{\infty}\ell^{-2\theta+1}e^{-\beta(\theta)\ell}<\infty.\]
Hence, this allows to define
$\mu_{\theta}:=\widetilde{\mu}_{\theta}/\widetilde{\mu}_{\theta}
\left(\Sigma\right)$.
In particular, we then have
that $\int I\,
d\mu_{\theta}=\mu_{\theta}^{*}(N)^{-1}\int I^{*}\,
d\mu_{\theta}^{*}$, from which the second equality follows.
Finally, note that since $\overline{P}$
and $P^*$ coincide, in the integrals above we can replace
$\overline{I}$ and $\overline{N}$ by $I^{*}$ and $N$. This finishes proof
of the proposition.
\end{proof}
\begin{rem}
Note that the measure $\mu_{\theta}$ in Proposition \ref{propHMU3}
is in fact a weak Gibbs measure for
the potential $-\theta I$. Therefore, the results of
\cite{Kesseboehmer:01} are applicable, and hence in this way
one could immediately obtain some Large Deviation results for the
situation here.
\end{rem}
\subsection{Analytic properties of $P$ and
$\widehat{P}$}
In this subsection we employ the results of the previous subsection,
in order to derive analytic properties
of the Stern--Brocot pressure
$P$ and its Legendre transform $\widehat{P}$.
A key
preliminary observation is stated in the following proposition, which
shows that the Stern--Brocot pressure $P$ coincides with the function
$\beta$ obtained in
Proposition \ref{propHMU3}.
\begin{prop}\label{prop:P=beta}
For $\theta <1$, we have
\[ P(\theta)=\beta(\theta).\]
\end{prop}
\begin{proof}
Let $\mu_{\theta}^{*}$ be the measure obtained in Proposition
\ref{propHMU3}, for $\theta<1$ fixed.
First, recall from the proof of Proposition \ref{propHMU3}
that the measure class of $\mu^{*}_{\theta}$ contains a
$\sigma$--invariant probability measure $\mu_{\theta}$
on $\Sigma$ for which $\int I\,
d\mu_{\theta}=\mu_{\theta}^{*}(N)^{-1}\int I^{*}\,
d\mu_{\theta}^{*}$. Secondly, note that for the measure $\overline{\mu}_{\theta}:=
\mu_{\theta}^{*}\circ
p^{-1}$ we have by
{\em Abramov's formula} (\cite{Abramov:59}, \cite{Neveu:69}) that
the
measure theoretical entropies $h_{\mu_{\theta}}$ and
$h_{\overline{\mu}_{\theta}}$ are related by $h_{\mu_{\theta}} = h_{\overline{\mu}_{\theta}}
/\mu_{\theta}^{*}(N)$. Thirdly,
by applying {\em Pinsker's result on relative
entropies} (\cite{Rohlin:67}) to our situation here, we obtain that the relative entropy
$h_{\mu^{*}_{\theta}}\left(\sigma^{*}|\overline{\sigma}\right)$
of $\mu^{*}_{\theta}$ vanishes. This gives
$h_{\mu^{*}_{\theta}}-h_{\overline{\mu}_{\theta}}=
h_{\mu^{*}_{\theta}}\left(\sigma^{*}|\overline{\sigma}\right)=0$.
Combining these observations with the usual variational principle
(\cite{DenkerGrillenbergerSigmund:76}),
it now follows
\begin{eqnarray*}
P(\theta) & \geq & h_{\mu_{\theta}}-\int \theta I\,
d\mu_{\theta}
=
\left(\mu_{\theta}^{*}(N)\right)^{-1}\left(h_{\mu_{\theta}^{*}}-\int\theta
I^{*}\, d\mu_{\theta}^{*}\right)\\
& = &
\left(\overline{\mu}_{\theta}(\overline{N})\right)^{-1}
\left(h_{\overline{\mu}_{\theta}}-\int\theta
\overline{I} \,
d\overline{\mu}_{\theta}\right)
= \beta(\theta).\end{eqnarray*}
In here, the latter equality is obtained as follows. Note that
$\overline{\mu}_{\theta}$
is an equilibrium measure on $\left(\overline{\Sigma},\overline{\sigma}\right)$
for the potential $-\theta \overline{I}-\beta(\theta)\overline{N}$.
Also, $\overline{\mu}_{\theta}$ fulfills the Gibbs property
\ref{eq:Gibbs*Property}, and
$\overline{\mu}_{\theta}\left(\theta
\overline{I}+\beta(\theta)\overline{N}\right) < \infty$. Next, recall
\emph{Sarig's variational principle} (\cite{Sarig:99}) which
states that
for
$g: \overline{\Sigma} \to\mathbb {R}$ H\"older continuous,
\begin{equation}
\overline{\P}(g)=\sup\left\{ h_{\overline{\mu}} +\int g\,
d\overline{\mu} :\overline{\mu}\in\mathcal{M}\left(\overline{\Sigma}
,\overline{\sigma}\right)\,\,\textrm{such
that}\,\, -\int g\, d\mu < \infty\right\} .\label{eq:Sarig}\end{equation}
In here,
$\mathcal{M}(\overline{\Sigma},\overline{\sigma})$ refers to the set of
$\overline{\sigma}$--invariant Borel probability measures on
$\overline{\Sigma}$. Applying this to the situation here, we obtain
\[ h_{\overline{\mu}_{\theta}}-\int \left(\theta
\overline{I} + \beta(\theta) \overline{N} \right) \,
d\overline{\mu}_{\theta} = \overline{\P} (-\theta\overline{I} -
\beta(\theta) \overline{N})= \overline{P}(\theta, \beta(\theta)) =
0.\]
An elementary rearrangement then gives the result.
For the reverse inequality,
first note that
we can induce $\left(\Sigma,\sigma\right)$ on the set
$\mathcal{H}:=\left\{
\left(x_{1},x_{2},\ldots\right)\in\Sigma:x_{1}\not=x_{2}\right\} $.
For the resulting induced system the return
time to $\mathcal{H}$ of a point
$y=\iota(X,Y^{n_{1}},X^{n_{2}},\ldots)\in\mathcal{H}$ is given by
$n_{1}=N\left(\sigma^{*}(X,Y^{n_{1}},X^{n_{2}},\ldots)\right)$.
Define
$\mathcal{G}:=\iota\left(\Sigma^{*}\right)\cap\mathcal{H}$, and let
$m_{\theta}\in\mathcal{M}(\Sigma,\sigma)$
be an ergodic equilibrium measure for the potential $-\theta I$, that
is $P(\theta)=h_{m_{\theta}}-\theta\int I\, dm_{\theta}$.
In this situation we necessarily have that
$m_{\theta}(\mathcal{G})>0$, and this can be seen as follows.
First, we show that $m_{\theta}(\mathcal{G})=0$ implies that
$m_\theta$ is equal to either $\delta_{\overline{A}}$ or
$\delta_{\overline{B}}$, where $\delta_{\overline{A}}$ (resp. $\delta_{\overline{B}}$) refers to
the Dirac measure at the periodic point $\overline{A}=\pi^{-1}(0)$ (resp. $\overline{B}=\pi^{-1}(1)$).
Namely, if $m_\theta(\mathcal{H})=0$ then
we immediately have $m_\theta\in\{\delta_{\overline{A}},
\delta_{\overline{B}}\}$. On the other hand, if $m_\theta(\mathcal{H})>0$ then
ergodicity of $m_{\theta}$ gives $m_\theta(\Sigma\setminus\iota(\Sigma^*))=1$.
Now, since $I(x)\geq 0$ for all $x \in \Sigma$, where $I(x) = 0$ if
and only if
$x\in \{\overline{A},\overline{B}\}$, we have
$\lim n^{-1}S_n I(x)=0=\int I \;dm_\theta$ for $m_\theta$-almost
every $x$, and this again implies that $m_\theta\in\{\delta_{\overline{A}},
\delta_{\overline{B}}\}$. Thus, if
$m_{\theta}(\mathcal{G})=0$ then $m_\theta\in\{\delta_{\overline{A}},
\delta_{\overline{B}}\}$. This shows that
$h_{m_{\theta}}= $ $m_{\theta}\left(-\theta I\right)=0$, giving
$P(\theta)=0$, and hence contradicting the fact
$P(\theta)\geq\beta(\theta)>0$. Therefore, we can assume without loss
of generality that $m_{\theta}(\mathcal{G})>0$. We can now use Kac's formula
once more,
which
guarantees
that there exists a $\sigma^{*}$--invariant probability measure $m_\theta^{*}$
in the
measure class of $m_{\theta}$, such that $m_\theta^{*}:=
\frac{1}{m_\theta\left(\mathcal{G}\right)}
m_\theta|_{\mathcal{G}}\circ\sigma^{-1}\circ\iota$ and
\[
-\int\left(-\theta I^{*}-\beta(\theta)N\right)\,
dm_{\theta}^{*}=\left(m_{\theta}(\mathcal{G})\right)^{-1}\left(\int\theta
I\, dm_{\theta}+\beta(\theta)\right)<\infty.\]
For $\overline{m}_{\theta}:=m_{\theta}^{*}\circ p^{-1}$, we
argue similar as above and obtain
\begin{eqnarray*}
0 & \geq & h_{\overline{m}_{\theta}}-\int\left(\theta
\overline{I}+\beta(\theta)\overline{N}\right)\,
d\overline{m}_{\theta} \,\,\, \qquad\quad(\emph{Sarig's
variational principle})\\
& = & h_{m_{\theta}^{*}}-\int\left(\theta
I^{*}+\beta(\theta)N\right)\,
dm_{\theta}^{*}\,\quad\qquad(\emph{Pinsker's result on relative
entropies})\\
& = & m_{\theta}^{*}(N)\left(h_{m_{\theta}}-\int\theta I\,
dm_{\theta}-\beta(\theta)\right) \qquad(\emph{Abramov's
formula})\\
& = &
m_{\theta}^{*}(N)\left(P(\theta)-\beta(\theta)\right)
\,\,\,\,\,\,\,\qquad\quad\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\,(\emph{since} \,\, m_{\theta}\, \,\emph{is an equilibrium
state}).\end{eqnarray*}
\end{proof}
The following proposition collects the properties
of $P$ and $\widehat{P}$ which will be crucial in the analysis to come.
\newpage
\begin{prop}\label{pro:AnalyticPropertiesP}~\nopagebreak
\begin{enumerate}
\item The Stern--Brocot pressure function $P$ coincides with the
pressure function
$\mathcal{P}$ associated with $\Sigma$.
\item $P$ is convex and non-increasing on $\mathbb {R}$ and real-analytic on
$(-\infty,1)$.
\item $P(\theta)=0$, for all $\theta\geq1$.
\item $P$ is differentiable throughout $\mathbb {R}$.
\item The domain of $\widehat{P}$ is equal to $[-\alpha_{+},0]$, where
\[
-\alpha_{+}:=\lim_{\theta \to-\infty}\frac{P(\theta)}{\theta}=-2\log\gamma.\]
\item We have
$\lim_{\alpha\searrow0} \widehat{P}\left(-\alpha\right)/(-\alpha)=1$.
\item We have
$\lim_{\alpha\nearrow2\log\gamma}\left(-\widehat{P}\left(-\alpha\right)\right)=0.$
\item We have $\lim_{\theta\to-\infty}\left(P(\theta)
+2\theta\log\gamma\right)=0$.
\end{enumerate}
\end{prop}
For the proofs of (7) and (8) the following lemma will turn out
to be useful.
\begin{lem}
\label{lem:QGammaInequalitystrict} For each
$x:=[a_{1},a_{2},a_{3},\ldots]
\in(0,1)$
and $k \in\mathbb {N}_0$ we have, with $\tau_0:=0$, $\tau_{k}:=\sum_{i=1}^{k}a_{i}$ for
$k\in \mathbb {N}$,
and $\rho:=1-\gamma^{-6}$, \[
q_{k}(x)\leq\gamma^{\tau_{k}}\rho^{\tau_{k}-k-1}.\]
\end{lem}
\begin{proof}
We give a proof by complete induction of the slightly stronger inequality
\begin{equation}
q_{k}(x)\leq\gamma^{\tau_{k}}\rho^{\tau_{k}-k} \rho^{\delta_{1 ,
a_{k}}-1},\label{eq:slightlyStronger}\end{equation} in which
$\delta$ denotes the Kronecker symbol. \\
First note that $q_0\equiv 1$, $q_{1}([1,\ldots])=1\leq\gamma^{1}\rho^{1-1}$, and
if $a_{1}\geq2$ then one immediately verifies that
$q_{1}\left[a_{1},\ldots\right]=a_{1}\leq\gamma^{a_{1}}\rho^{a_{1}-1}\rho^{-1}$.
Also, for $k \in \mathbb {N}$ we have \begin{equation} q_k(\gamma-1)=q_k([1,1,1,\ldots])=f_k\leq \gamma^{k}=\gamma
^{\tau_k}\rho^{\tau_k-k},\label{eq:abschGoldenMean}\end{equation}
where $f_{k}$
refers to the $(k+1)$-th member of the Fibonacci sequence
$(f_{0},f_{1},f_{2},\ldots):= (0,1,1,2,\ldots)$, given by
$f_{k+1}:= f_{k-1}+f_{k}$ for all $k \in \mathbb {N}$.
Recall that $f_{k}=\left(\gamma^{k}-\left(-\gamma\right)^{-k}\right)/\sqrt{5}$.
Now suppose that (\ref{eq:slightlyStronger}) holds for some $k\in \mathbb {N}$ and
for all $0\leq m\leq k$. It is then sufficient to consider the
following
two cases.
\begin{enumerate}
\item \label{enu:case1} If $a_{k+1}=1$ such that $a_{n}\geq2$ and $a_{n+i}=1$, for all
$i=1,\ldots,l$
and some $n\leq k$ and $l\geq k-n+1$, then
$q_{n-1}(x)\leq\gamma^{\tau_{n-1}}\rho^{\tau_{n-1}-n+1} \rho^{-1}$
and
$q_{n}(x)\leq\gamma^{\tau_{n}}\rho^{\tau_{n}-n} \rho^{-1}$. Hence,
an elementary
calculation gives \begin{eqnarray*} q_{n+l}(x) & = &
f_{l+1}q_{n}(x)+f_{l}q_{n-1}(x)\\
& \leq &
f_{l+1}\gamma^{\tau_{n}}\rho^{\tau_{n}-n} \rho^{-1}+f_{l}\gamma^{\tau_{n-1}}
\rho^{\tau_{n-1}-n+1} \rho^{-1} \\
& \leq &
\gamma^{\tau_{n+l}}\rho^{\tau_{n+l}-n-l}\left(\rho^{-1}\left(\frac{f_{l+1}}
{\gamma^{l}}+\frac{f_{l}}
{\gamma^{a_{n}+l}\rho^{a_{n}-1}}\right)\right)\\
& \leq &
\gamma^{\tau_{n+l}}\rho^{\tau_{n+l}-n-l}\underbrace{\left(\rho^{-1}
\left(\frac{f_{l+1}}{\gamma^{l}}+
\frac{f_{l}}{\gamma^{l}\left(\gamma\rho\right)^{2}}\right)\right)}_{\leq1}.
\end{eqnarray*}
\item If $a_{k+1}=2$, then either $a_i=1$ for $i=1,\ldots,k$, or
there exists $n\leq k$ such that $a_{n}\geq2$ and $a_{i}=1$ for all $i$ with
$n <i\leq k$. In the first case we use (\ref{eq:abschGoldenMean}), whereas in
the second case we employ (\ref{enu:case1}), and obtain
\begin{eqnarray*}
q_{k+1}\left(\left[a_{1},\ldots,a_{k},2\right]\right) & = &
q_{k+2}\left(\left[a_{1},\ldots,a_{k},1,1\right]\right)\\
& \leq &
\gamma^{\tau_{k+1}}\rho^{\tau_{k+1}-k-1} \rho^{-1}.\end{eqnarray*}
For $a_{k+1}>2$, the inequality follows by induction over $a_{k+1}$,
using (\ref{enu:case1}) and the fact that
$q_{k+1}\left(\left[a_{1},\ldots,a_{k},a_{k+1}\right]\right)=q_{k+2}
\left(\left[a_{1},
\ldots,a_{k+1}-1,1\right]\right)$.
\end{enumerate}
\end{proof}
Before giving the proof of Proposition \ref{pro:AnalyticPropertiesP},
we remark that the statements (7) and (8) in Proposition \ref{pro:AnalyticPropertiesP}
are in fact equivalent.
Nevertheless, we shall prove these two statements separately, where
the proof of (7) primarily uses ergodic theory, whereas the
proof of (8) is of elementary number theoretical nature.
\begin{proof}
[Proof of Proposition \ref{pro:AnalyticPropertiesP}]~
\textbf{\emph{ad (1)}.} \, The assertion is an immediate consequence of
(\ref{eq:AsympParabExcursion}) and
Corollary \ref{cor:VergleichTSnI}.
\textbf{\emph{ad (2)}.} \, The assertion follows immediately by combining
Proposition \ref{propHMU3} and Proposition \ref{prop:P=beta}.
Alternatively, the statement can also be derived from
Proposition 2.1 in
\cite{KesseboehmerStratmann:04a}.
\textbf{\emph{ad (3)}.} \, By definition of $P$ we have $P(1)=0$.
Also,
by (2) we know that $P$ is non-increasing. Therefore, it is sufficient to show that $P$
is non-negative, and indeed this follows since\[
P(\theta)=\lim_{n\to\infty}\frac{1}{n}\log\sum_{k=1}^{2^{n}}\left|T_{n,k}\right|^{\theta}
\geq\lim_{n\to\infty}\frac{1}{n}\log\left|T_{n,1}\right|^{\theta}=\lim_{n\to\infty}
\frac{-\theta}{n}\log\left(n+1\right)=0.\]
\textbf{\emph{ad (4)}.} \, In order to determine the left derivative $P^{-}(1)$ of $P$
at $1$, recall from Proposition \ref{propHMU3} that $\mu_{\theta}^{*}$ refers to the unique Gibbs measure on
$\Sigma^{*}$
such that
$\mu_{\theta}^{*}\left(C_{n}^{*}\left(y\right)\right)\asymp\exp\left(
-\theta S_{n}I^{*}\left(y\right)-\beta(\theta) S_{n}N^{*}\left(y\right)\right)$,
for all $n \in {\mathbb{N}}, y \in \Sigma^{*}$. For each $n\in\mathbb {N}$,
let us fix an element $y_X^{(n)}\in\Sigma^{*}$
such that $y^{(n)}_X=\left(X^{n},\ldots\right)$, for $X\in \{A,B\}$. We then
have by Lemma
\ref{lem:Vergleichbar}, \begin{eqnarray*}
\int N\, d\mu_{\theta}^{*} & = &
\sum_{X\in \{A,B\}}\sum_{n=1}^{\infty} n \cdot \mu_{\theta}^{*}\left(C_{1}^{*}\left(y_X^{(n)}\right)\right)\\
&\asymp&\sum_{n=1}^{\infty}n
\cdot\exp\left(-\theta I^{*}\left(y_A^{(n)}\right)-\beta(\theta) N^{*}\left(y_A^{(n)}\right)\right)\\
& \gg & \sum_{n=2}^{\infty}n\cdot n^{-2\theta}
e^{-\beta(\theta)n}\to \infty, \;\; \mbox{ for }\theta\nearrow 1.\end{eqnarray*}
On the other hand, we have for all $\theta \in (1/2,1]$,
\begin{eqnarray*}
\int I^{*}\, d\mu_{\theta}^{*} & \asymp &
\sum_{X\in \{A,B\}}\sum_{n=1}^{\infty} \log n \, \mu_{\theta}^{*}\left(C_{1}^{*}\left(y_X^{(n)}\right)\right)\\
&\asymp&\sum_{n=1}^{\infty}\log n\exp\left(-\theta I^{*}\left(y_A^{(n)}\right)-\beta(\theta) N^{*}\left(y_A^{(n)}\right)\right)\\
& \ll & \sum_{n=1}^{\infty}n^{-2\theta}\log n < \infty.\end{eqnarray*}
This shows that $P^{-}(1)=0$, and hence $P$ is differentiable
everywhere.
\textbf{\emph{ad (5)}.} \, Since
$\lim_{\theta\to\infty} P\left(\theta\right)/\theta=0$,
the upper bound of the domain of $\widehat{P}$ is equal to $0$.
For
the lower bound $-\alpha_{+}$ of the domain we have
by \cite{KesseboehmerStratmann:04a} (Proposition 2.3), \begin{equation}
-\alpha_{+}=\lim_{\theta\to-\infty}\frac{P(\theta)}{\theta}=-\sup_{\nu\in\mathcal{M}
\left(\Sigma,\sigma\right)}\int I\, d\nu.\label{eq:alpha+}\end{equation}
We are left with to determine the actual value of
$\alpha_{+}$. For this, first note that for the linear combination
$m:=1/2
\left(
\delta_{\overline{AB}}+ \delta_{\overline{BA}}\right) \in \mathcal{M}
\left(\Sigma,\sigma\right)$
of the Dirac measures $\delta_{\overline{AB}}$
and $\delta_{\overline{BA}}$ at the periodic points
$\overline{AB}:=\pi^{-1}\left(2-\gamma\right)$ and
$\overline{BA}:=\pi^{-1}\left(\gamma-1\right)$, an elementary
calculation shows that $\int I\, dm=2\log\gamma$. This implies that
$\sup_{\nu\in\mathcal{M}\left(\Sigma,\sigma\right)}\int I\,
d\nu\geq2\log\gamma$. For the reverse inequality note that $\int I\,
d\nu\leq\sup_{x\in\Sigma}\limsup_{n\to\infty}(S_{n}I(x))/n$,
for all
$\nu\in\mathcal{M}\left(\Sigma,\sigma\right)$, where
$\mathcal{M}(\Sigma,\sigma)$ refers to the set of
$\sigma$--invariant Borel probability measures on $\Sigma$.
In order to calculate the right hand side of the latter inequality,
recall that the smallest interval in
$\mathcal{T}_{n}$ has the length $(f_{n+1} f_{n+2})^{-1}$.
Using this observation and Corollary
\ref{cor:VergleichTSnI},
we obtain \begin{eqnarray*}
\sup_{y\in\Sigma}\limsup_{n\to\infty}\frac{S_{n}I(y)}{n} &= &\!\!
\sup_{x\in[0,1)}\limsup_{n\to\infty}\frac{-\log\left|T_{n}\left(x\right)\right|}{n}=
\lim_{n\to\infty}
\frac{\log\left(f_{n+1}f_{n+2}\right)}{n}\\
& = &\!\!\!
\lim_{n\to\infty}\frac{\log\left(\gamma^{n+1}-\left(-\gamma\right)^{-(n+1)}\right)+\log
\left(\gamma^{n+2}-\left(-\gamma\right)^{-(n+2)}\right)}{n}\\
& = & \!\! 2\log\gamma.\end{eqnarray*}
Note that in here the supremum is achieved at for instance any noble
number in $(0,1)$, that is at numbers whose continued fraction expansion
eventually consists of $1$'s only.
\textbf{\emph{ad (6)}.} \, The result in (3) implies that
\[
\lim_{\alpha \searrow0}-\widehat{P}(-\alpha)/\alpha=\inf\left\{
t\in\mathbb {R}:P(t)=0\right\} .\]
Therefore, it is sufficient to show that $1$ is the least zero of $P$.
For this assume by way of contradiction that $P(s)=0$, for some
$s<1$. Since $P$ is non-increasing, it follows that $P$ vanishes
on the interval $(s,1)$. But this contradicts the fact that $P$ is
real-analytic
on $(-\infty,1)$ and positive at for instance $0$.
\textbf{\emph{ad (7)}.} \, For all $n\in\mathbb {N}$
and $\theta\leq 0$, we have
\begin{eqnarray*}
\left(\frac{\gamma^{n+1}-\left(-\gamma\right)^{-(n+1)}}{\sqrt{5}}\right)^{-2\theta}
& \leq & \left(f_{n+1}f_{n+2}\right)^{-\theta}\leq\sum_{k=1}^{2^{n}}\left|T_{n,k}
\right|^{\theta} \\
& \leq & 2^{n}\left(f_{n+1}f_{n+2}\right)^{-\theta}\leq2^{n}\gamma^{-2\theta
(n+2)}.\end{eqnarray*}
Therefore, \[
-2\theta\log\gamma\leq
P(\theta)\leq\log2-2\theta\log\gamma \textrm{ for all } \theta\leq0,\]
which implies that $\widehat{P}(-\alpha)\leq0$, for all
$\alpha\in[0,2\log\gamma]$.
Hence, in order to verify that
$\lim_{\alpha\nearrow2\log\gamma}\widehat{P}\left(-\alpha\right)=0$
it is sufficient to show that this limit is non-negative.
For this, let $t(\alpha):=\left(P'\right)^{-1}(-\alpha)$ and recall
that by the variational principle (cf.
\cite{DenkerGrillenbergerSigmund:76}) we have
for each $\alpha\in[0,2\log\gamma]$ that there exists
$m_{t(\alpha)}\in\mathcal{M}\left(\Sigma,\sigma\right)$
such that \[
P\left(t(\alpha)\right)=h_{m_{t(\alpha)}}-t(\alpha)\int I\,
dm_{t(\alpha)}.\]
Furthermore, by \cite{KesseboehmerStratmann:04a} (Proposition 2.3)
we have $\int I\, dm_{t(\alpha)}=\alpha$. Therefore, if
$\nu\in\mathcal{M}\left(\Sigma,\sigma\right)$
denotes a weak limit of some sequence $\left(\mu_{t(\alpha)}\right)$
for $\alpha$ tending to $2\log\gamma$ from below, then the lower semi-continuity of
the entropy (cf. \cite{DenkerGrillenbergerSigmund:76}) gives \[
h_{\nu}\geq\limsup_{\alpha\nearrow2\log\gamma}h_{m_{t(\alpha)}}=\limsup_{\alpha\nearrow2\log\gamma}
\left(P\left(t(\alpha)\right)+\alpha\cdot
t(\alpha)\right)=\limsup_{\alpha\nearrow2\log\gamma}\left(-\widehat{P}
\left(-\alpha\right)\right).\]
Note that we clearly have $\int I\, d\nu=2\log\gamma$. Now, the final step is to
show that for the discrete measure $m$ considered in the proof of
(5) we have \[
\left\{ \nu\in\mathcal{M}\left(\Sigma,\sigma\right):\int I\,
d\nu=2\log\gamma\right\} =\left\{ m\right\} .\]
This will be sufficient, since $h_{m}=0$.
Therefore, suppose by way of contradiction that there exists $\mu\not=m$
such that \[
\mu\in\left\{ \nu\in\mathcal{M}\left(\Sigma,\sigma\right):\int I\,
d\nu=2\log\gamma\right\} . \]
Let us first show that $\eta:= \mu\left(\left\{ x\in\Sigma:x_{1}=x_{2}=X\right\} \right)>0$,
for some $X \in \{A, B\}$. If this would not be the case,
then the $\sigma$--invariance of
$\mu$ would imply \[ \mu\left(\left\{ x\in\Sigma:x_{1}=A,x_{2}=B\right\}
\right)= \mu\left(\left\{ x\in\Sigma:x_{1}=B,x_{2}=A\right\} \right)=\frac{1}{2},\]
and hence we obtain by induction that $\mu =m$. This contradicts our assumption
$\mu \neq m$, showing that $\eta >0$. We can now continue the
above argument as follows.
Since $\{ \nu\in\mathcal{M}\left(\Sigma,\sigma\right):\int I\,
d\nu=2\log\gamma\} $
is convex, we can assume without loss of generality that $\mu$ is ergodic.
This then immediately implies that $\lim_{n\to\infty} (S_{n}I(y))/n=\int I\,
d\mu$ for $\mu$-almost every $y\in\Sigma$, and furthermore that for some $X \in \{A, B\}$ and
$n$ sufficiently large, \begin{equation}
S_{n}\1_{\left\{ x\in\Sigma:x_{1}=x_{2}=X \right\}
}(y)>\frac{n\eta}{2}.\label{eq:ErgodMean}\end{equation}
Let us fix $x\in \Sigma$ with this property, and define $\tau_k:=\sum _{i=1}^k a_i(x)$ as
in Lemma \ref{lem:QGammaInequalitystrict}.
Combining Lemma \ref{lem:TnbyCF} and inequality (\ref{eq:ErgodMean}),
it follows $\left(\tau_k-k-1)\right)\geq \tau_k\eta/2$. Hence, using Corollary
\ref{cor:VergleichTSnI}, Lemma \ref{lem:Vergleichbar} and Lemma
\ref{lem:QGammaInequalitystrict}, \begin{eqnarray*}
2\log\gamma & = & \int I\,
d\mu=\lim_{n\to\infty}\frac{S_{n}I(x)}{n}=\lim_{n\to\infty}\frac{-\log
\left|T_{n}(x)\right|}{n}\\
& = &
\lim_{k\to\infty}\frac{-\log
\left|T_{\tau_k +1}(x)\right|}{\tau_k}= \lim_{k\to\infty}\frac{2\log(q_k(x))}{\tau_k}\\
& \leq &
\limsup_{k\to\infty}\frac{2\log\left(\gamma^{\tau_k}\rho^{(\tau_k-k-1)}\right)}{\tau_k}\leq\limsup_{k\to\infty}
\frac{2\log\left(\gamma^{\tau_k}\rho^{\tau_k\eta/2}\right)}{\tau_k}\\
& = & 2\log\gamma+\eta\cdot\log\rho<2\log\gamma.\end{eqnarray*}
\textbf{\emph{ad (8)}.} \, First note that $t_{n,2\ell}>t_{n,2\ell\pm1}$,
for each $n\geq2$ and
$\ell=1,\ldots,2^{n-1}$. This implies that
$\left|T_{n,2\ell}\right|^{-1}=t_{n,2\ell}\cdot t_{n,2\ell+1}$
and $\left|T_{n,2\ell-1}\right|^{-1}=t_{n,2\ell-1}\cdot t_{n,2\ell}$
are both less than $\left(t_{n,2\ell}\right)^{2}$. Hence, using Lemma
\ref{lem:TnbyCF} and Lemma \ref{lem:QGammaInequalitystrict}, it
follows for $n>2$ and $\theta<0$, \begin{eqnarray*}
\sum_{k=1}^{2^{n}}|T_{n,k}|^{\theta} & \leq &
2\sum_{k=1}^{n}\sum_{A_{k}^{n+1}}q_{k}\left(\left[a_{1},\ldots,a_{k}\right]\right)^{-2\theta}\\
& \leq & 2\sum_{k=1}^{n}{\binom{n-1}{
k-1}}\left(\gamma^{n+1}\rho^{n+1-k-1}\right)^{-2\theta}\\
& = & 2\gamma^{-2\theta(n+1)}\sum_{k=0}^{n-1}{\binom{n-1}{
k}}\left(\rho^{n-1-k}\right)^{-2\theta}\\
& = & 2\gamma^{-2\theta(n+1)}\sum_{k=0}^{n-1}{\binom{n-1}{
k}}\left(\rho^{-2\theta}\right)^{n-1-k}\\
& \leq &
2\gamma^{-2\theta(n+1)}\left(1+\rho^{-2\theta}\right)^{n-1}.\end{eqnarray*}
Recalling the definition of $P$, we then obtain \[
P\left(\theta\right)\leq-2\theta\log\gamma+\log\left(1+\rho^{-2\theta}\right).\]
For the lower bound, note that \[
\sum_{k=1}^{2^{n}}|T_{n,k}|^{\theta}\geq(f_{n+1}f_{n+2})^{-\theta}.\]
Since $f_{n}=(\gamma^{n}-(-\gamma)^{-n})/\sqrt{5}$, it therefore
follows \[
P\left(\theta\right)\geq-2\theta\log\gamma.\]
By combining these two bounds for $P\left(\theta\right)$ and then letting $\theta$ tend to
$(-\infty)$,
the proposition follows.
\end{proof}
\section{Multifractal Formalism for continued fractions}
In this section we give the proof of Theorem \ref{Thm:main}, which
we have split up into the three separate parts
\emph{The lower bound}, \emph{The upper
bound} and \emph{Discussion of boundary points of the spectrum}. We begin with the
following important preliminary remarks.
\begin{rem}\label{rem:Vergleich}
$ \, $ \\
(1) \,
Note that by Corollary \ref{cor:VergleichTSnI} and Lemma
\ref{lem:Vergleichbar},
we have for $x\in\Sigma$ and $y\in\Sigma^{*}$ (assuming in each case that the limit
exists),
\begin{eqnarray*}
\ell_{1}\left(\pi^{*}(y)\right)=\lim_{n\to\infty}\frac{S_{n}I^{*}(y)}{S_{n}N(y)},
& &
\ell_{2}\left(\pi^{*}(y)\right)=\lim_{n\to\infty}\frac{S_{n}N(y)}{n},\\
\ell_{3}\left(\pi^{*}(y)\right)=\lim_{n\to\infty}\frac{S_{n}I^{*}(y)}{n},
& &
\ell_{4}\left(\pi(x)\right)=\lim_{n\to\infty}\frac{S_{n}I(x)}{n}.\end{eqnarray*}
(2) \,
Recall that in \cite{KesseboehmerStratmann:04a} and
\cite{KesseboehmerStratmann:04} we in particular
considered oriented geodesics
$\ell \subset {\mathbb {H}}^{2}$ from $\{\infty\}$ to [0, 1), and
coded these by means of their intersections
with the tesselation given by the $G$-orbit of the fundamental
domain $F$. More precisely, if $\ell$ ends at $\xi \in
[0,1) \cap \mathbb {I}$ such that $\ell$ intersects $ g_{\xi,1}(F), g_{\xi,2}(F),
g_{\xi,3}(F),\ldots$ in succession, with $g_{\xi,n} \in G$ for all $n
\in \mathbb {N}$, then $\xi$ is coded by the infinite word $(g_{\xi,1},
g_{\xi,2},
g_{\xi,3},\ldots)$. Clearly, this type of coding is
analogous to the finite coding represented by $\Sigma$. Hence,
the results of \cite{KesseboehmerStratmann:04a} and
\cite{KesseboehmerStratmann:04} for the
Hausdorff dimensions
of the level sets
\[ \mathcal{F}(\alpha):= \left\{ \xi\in [0,1): \frac{d(z_{0}, g_{\xi,n}
(z_{0}))}{n} = \alpha \right\}\]
can immediately be transfered to the situation in this paper, and in this
way we obtain that for each
$\alpha\in(0,2\log\gamma)$,
\begin{equation}
\dim_{H}\left(\mathcal{L}_{4}\left(\alpha\right)\right)= \dim_{H}\left(
\mathcal{F}(\alpha) \right) =
\frac{\widehat{P}(-\alpha)}{-\alpha}.\label{eq:OldSpec}\end{equation}
Therefore, the following proof of Theorem \ref{Thm:main} will
in particular also give an alternative proof of the
identity in (\ref{eq:OldSpec}).
Let us also emphasize that a straight forward inspection of the arguments in the general multifractal analysis
of \cite{KesseboehmerStratmann:04a}
shows that there we did not make full use of the group structure of the Kleinian group.
In fact, the arguments there exclusively
consider certain rooted sub-trees of the Cayley graph of the Kleinian group,
and therefore they continue to hold if the underlying algebraic
structure is only a semi-group acting on hyperbolic space, rather than a group.
Therefore, the main
results of this general multifractal analysis for growth rates can
be applied immediately to the setting in this paper. In this way
one also immediately obtains
that $P$ is differentiable everywhere, real-analytic on
$(-\infty,1)$
and equal to $0$ otherwise.
\end{rem}
\subsection{The lower bound}\label{section 5.1}
\begin{lem}
\label{lem:Inclusion} For each $\alpha\in(0,2\log\gamma)$ there
exists a unique Gibbs measure $\mu_{t(\alpha)}^{*}$ on $\Sigma^{*}$
such that for \begin{equation}
\alpha^{*}:=\int I^{*}\,
d\mu_{t(\alpha)}^{*}\,\;\textrm{ and }\;\alpha^{\sharp}:=\int N\,
d\mu_{t(\alpha)}^{*},\label{eq:Alpha*-Alpha}\end{equation}
we have \begin{equation}
\mathcal{L}_{2}\left(\alpha^{\sharp}\right)\cap\mathcal{L}_{3}\left(\alpha^{*}\right)\subset\mathcal{L}_{1}
\left(\alpha\right).\label{eq:SharpSternInclusion}\end{equation}
In here, the function $t$ is given by $t(\alpha):=\left(P'\right)^{-1}(-\alpha)$.
\end{lem}
\begin{proof}
The existence of the unique ergodic Gibbs measure $\mu_{t(\alpha)}^{*}$ has
already been obtained in Proposition \ref{propHMU2}.
As shown in Proposition \ref{propHMU3}, the significance of $\mu_{t(\alpha)}^{*}$
is that it allows to represent the Lyapunov exponent $\alpha$
in terms of $I^{*}$ and $N$ as follows.
\begin{equation}
\alpha=-P'(t(\alpha))=\frac{\int I^{*}\, d\mu_{t(\alpha)}^{*}}{\int N\,
d\mu_{t(\alpha)}^{*}}=\frac{\alpha^{*}}{\alpha^{\sharp}}.\label{eq:alpha-alphaalpha}\end{equation}
Using Remark \ref{rem:Vergleich} (1), it follows that
$\mathcal{L}_{2}\left(\alpha^{\sharp}\right)\cap\mathcal{L}_{3}\left(\alpha^{*}\right)\subset\mathcal{L}_{1}\left(\alpha\right)$.
\end{proof}
For the following lemma recall that the Hausdorff dimension
$\dim_{H}\left(\mu\right)$
of a probability measure $\mu$ on some metric space is given by \[
\dim_{H}\left(\mu\right):=\inf\left\{ \dim_{H}(K):\mu(K)=1\right\} .\]
\begin{lem}
For each $\alpha\in(0,2\log\gamma)$ we have, with
$\widetilde{\mu}_{t(\alpha)}:=\mu_{t(\alpha)}^{*}\circ\left(\pi^{*}\right)^{-1}$,
\[
\dim_{H}\left(\widetilde{\mu}_{t(\alpha)}\right)\leq\dim_{H}\left(\mathcal{L}_{2}\left(\alpha^{\sharp}\right)
\cap\mathcal{L}_{3}\left(\alpha^{*}\right)\right)\leq\dim_{H}\left(\mathcal{L}_{1}\left(\alpha\right)\right).\]
\end{lem}
\begin{proof}
The first inequality follows, since by ergodicity of $\mu_{t(\alpha)}^{*}$
we have\[
\widetilde{\mu}_{t(\alpha)}\left(\mathcal{L}_{2}\left(\alpha^{\sharp}\right)\cap
\mathcal{L}_{3}\left(\alpha^{*}\right)\right)=1.\]
The second inequality is an immediate consequence of Lemma
\ref{lem:Inclusion}.
\end{proof}
\begin{lem}
\label{lem:lowerBound} For each $\alpha\in(0,2\log\gamma)$ we have\[
\dim_{H}\left(\widetilde{\mu}_{t(\alpha)}\right)=\frac{\widehat{P}(-\alpha)}{-\alpha}.\]
\end{lem}
\begin{proof}
The aim is to show that the local dimension of
$\widetilde{\mu}_\alpha$ exists and
is equal to $\widehat{P}(-\alpha)/(-\alpha)$,
for each $\alpha \in(0,2\log\gamma)$.
For this, let $B(x,r):=[x-r,x+r]\cap\mathbb {I}$ for $0<r\leq1$ and $x\in\mathbb {I}$, and
define
\begin{eqnarray*}
m_{r}(x) & := & \max\left\{
n\in\mathbb {N}:\pi^{*}C_{n}^{*}\left(\left(\pi^{*}\right)^{-1}x\right)\supset
B(x,r)\right\} ,\\
n_{r}(x) & := & \min\left\{ n\in\mathbb {N}:\pi^{*}C_{n}^{*}
\left(\left(\pi^{*}\right)^{-1}x\right)\subset B(x,r)\right\}
.\end{eqnarray*}
We obviously have that $\left|m_{r}(x)-n_{r}(x)\right|$
is uniformly bounded from above, and hence
$\lim_{r\to 0}m_{r}\left(x\right)/n_{r}\left(x\right)=1$.
Combining the Gibbs property of $\mu_{t(\alpha)}^{*}$, Lemma
\ref{lem:Vergleichbar}, (\ref{eq:Alpha*-Alpha})
and (\ref{eq:alpha-alphaalpha}), it follows
for $\widetilde{\mu}_{t(\alpha)}$-almost every $x$,
\\
\\
${\displaystyle
\limsup_{r\to 0}\frac{\log\widetilde{\mu}_{t(\alpha)}\left(B(x,r)\right)}{\log
r}}$\begin{eqnarray*}
& \leq &
\limsup_{r\to 0}\frac{-t(\alpha)\left(S_{n_{r}\left(x\right)} I^{*}\left(\left(\pi^{*}\right)^{-1}x\right)
\right)-P\left(t\left(\alpha\right)\right)S_{n_{r}\left(x\right)} N\left(\left(\pi^{*}\right)^{-1}x\right)}
{-\left(S_{m_{r}\left(x\right)}I^{*}(x)\right)}\\
& = &
\limsup_{r\to 0}\frac{-t(\alpha)\frac{S_{n_{r}\left(x\right)}I^{*}\left(\left(\pi^{*}\right)^{-1}x\right)}
{S_{n_{r}\left(x\right)}N\left(\left(\pi^{*}\right)^{-1}x\right)}-P\left(t\left(\alpha\right)\right)}
{-\frac{S_{n_{r\left(x\right)}}I^{*}\left(\left(\pi^{*}\right)^{-1}x\right)}{S_{n_{r}\left(x\right)}N
\left(\left(\pi^{*}\right)^{-1}x\right)}\cdot\frac{S_{m_{r}\left(x\right)}I^{*}\left(\left(\pi^{*}
\right)^{-1}x\right)}{m_{r}(x)}\frac{n_{r}(x)}{S_{n_{r}\left(x\right)}
N\left(\left(\pi^{*}\right)^{-1}x\right)}\cdot\frac{m_{r}\left(x\right)}{n_{r}\left(x\right)}}\\
& = &
\frac{t(\alpha)\alpha+P(t(\alpha))}{\alpha}=\frac{\widehat{P}(-\alpha)}{-\alpha}
.\end{eqnarray*}
The reverse inequality for the `$\liminf$' is obtained by
similar means, and we omit its proof.
\end{proof}
\subsection{The upper bound}\label{section 5.2}
\begin{lem}
\label{lem:UpperBound} For each $\alpha\in(0,2\log\gamma)$ we
have\begin{eqnarray*}
\dim_{H}\left(\pi^{*}\left\{
x\in\Sigma^{*}:\liminf_{n\to\infty}\frac{S_{n}I^{*}(x)}{S_{n}N(x)}\geq\alpha\right\}
\right) & \leq & \frac{\widehat{P}(-\alpha)}{-\alpha}.\end{eqnarray*}
\end{lem}
\begin{proof}
Note that $\max\left\{
t\left(\alpha\right)+P(t\left(\alpha\right))/s:s\in[\alpha,2\log\gamma)\right\}
=t(\alpha)+P(t(\alpha))/\alpha$,
for each $\alpha\in(0,2\log\gamma)$. By combining this with
the Gibbs property of $\mu_{t(\alpha)}^{*}$,
it follows that for each $\varepsilon>0$ and $x\in\Sigma^{*}$ such
that
$\pi^{*}(x)\in\mathcal{L}_{4}\left(\alpha\right)$, we have for $n$
sufficiently large, \begin{eqnarray*}
\mu_{t(\alpha)}^{*}\left(C_{n}^{*}(x)\right) & \gg &
\exp\left(-t\left(\alpha\right)\, S_{n}I^{*}(x)-P(t(\alpha))\,
S_{n}N(x)\right)\\
& = &
\exp\left(-S_{n}I^{*}(x)\left(t\left(\alpha\right)+P(t\left(\alpha\right))\frac{S_{n}N(x)}{S_{n}I^{*}(x)}\right)\right)\\
& \gg &
\left(\exp\left(-S_{n}I^{*}(x)\right)\right)^{\frac{\widehat{P}(-\alpha)}{-\alpha}+\varepsilon}\\
& \gg &
\left|\pi^{*}\left(C_{n}^{*}(x)\right)\right|^{\frac{\widehat{P}(-\alpha)}{-\alpha}+\varepsilon}.\end{eqnarray*}
Therefore, for the sequence of balls
$\left(B\left(\pi(x),r_{n}\right)\right)$
with radii $r_{n}:=\left|\pi^{*}\left(C_{n}^{*}(x)\right)\right|$ and
centre $\pi(x)$, which tends to $\{\pi(x)\}$ for $n$ tending to infinity,
we have \[ \widetilde{\mu}_{t(\alpha)}\left(
B\left(\pi(x),r_{n}\right)\right)\gg\mu_{t(\alpha)}^{*}
\left(C_{n}^{*}(x)\right)\gg\left(r_{n}\right)^{\frac{\widehat{P}
(-\alpha)}{-\alpha}+\varepsilon}.\] Applying the mass distribution
principle, the proposition follows.
\end{proof}
\begin{cor}
\label{cor:DimUpperBound} For each $\alpha\in(0,2\log\gamma)$ we
have \[
\max\left\{
\dim_{H}\left(\mathcal{L}_{2}\left(\alpha^{\sharp}\right)\cap\mathcal{L}_{3}\left(\alpha^{*}\right)\right),\dim_{H}\left(\mathcal{L}_{1}\left(\alpha\right)\right)\right\}
\leq\frac{\widehat{P}(-\alpha)}{-\alpha}.\]
\end{cor}
\begin{proof}
The assertion is an immediate consequence of combining Lemma \ref{lem:UpperBound}
and the fact \[
\mathcal{L}_{2}\left(\alpha^{\sharp}\right)\cap\mathcal{L}_{3}\left(\alpha^{*}\right)\subset\mathcal{L}_{1}(\alpha)\subset\left\{
x\in\Sigma^{*}:\liminf_{n\to\infty}\frac{S_{n}I^{*}(x)}{S_{n}N(x)}\geq\alpha\right\}.\]
\end{proof}
\subsection{Discussion of boundary points of the
spectrum.\label{sub:Discussion-of-boundary-points}}
$ \, $\\
{\em The case $\alpha=0$}: \, Recall the two classical
results of L{\'e}vy and Khintchin mentioned in the introduction.
>From these we immediately deduce that $\tau(0)=1$. Also,
recall that by Proposition \ref{pro:AnalyticPropertiesP}
(6) we have that
$\lim_{\alpha\searrow0}\widehat{P}(-\alpha)/(-\alpha)=1$.
This shows that
$\tau(0)=\lim_{\alpha\searrow0} \widehat{P}(-\alpha)/(-\alpha) =1$,
and hence gives that the dimension function $\tau$ is continuous from
the right at $0$.\\
In order to show that $\alpha^{*}\left(0\right)=\chi$, we argue as
follows. For $\alpha=0$, we already know that $0=\int I^{*}\, d\mu_{1}^{*}/\int N\,
d\mu_{1}^{*}= \alpha^{*}\left(0\right)/\infty$ and that
$\lim_{k\to\infty} (2\log q_{k}(x)) / k=\alpha^{*}(0)$, for
$\mu_{1}^{*}\circ\left(\pi^{*}\right)^{-1}$-almost every
$x\in\left(0,1\right)$. Hence, L{\'e}vy's result gives that,
if $\mu_{1}^{*}\circ\left(\pi^{*}\right)^{-1}$ is absolutely
continuous with respect
to the Lebesgue measure $\lambda$ on $(0,1)$ then
$\alpha^{*}\left(0\right)=\chi$. Hence, it remains to show that $\mu_{1}^{*}\circ\left(\pi^{*}\right)^{-1}$
has this property. For this, consider some $T\in\mathcal{T}_{n}$ for
$n\in\mathbb {N}$, and fix $y\in\Sigma^{*}$ and $k\in\mathbb {N}$
such that $\pi^{*}\left(C_{k}^{*}(y)\right)=T\cap\mathbb {I}$. Using the Gibbs property of $\mu_{1}^{*}$
and
Lemma \ref{lem:Vergleichbar}, we obtain \begin{eqnarray*}
\mu_{1}^{*}\circ\left(\pi^{*}\right)^{-1}\left(T\right) & \asymp &
\mu_{1}^{*}\left(C_{k}^{*}\left(y\right)\right)\asymp\exp\left(-S_{k}\left(I^{*}(y)\right)\right)\\
& \asymp &
\left|\pi^{*}\left(C_{k}^{*}\left(y\right)\right)\right|\asymp\lambda\left(T\right).
\end{eqnarray*}
{\em The case $\alpha=2\log\gamma$}: \, In order to show that the dimension
function $\tau$ is continuous from the left at $2\log\gamma$, we proceed as follows. Proposition
\ref{pro:AnalyticPropertiesP}
(7) implies that
$\lim_{\alpha\nearrow2\log\gamma}\widehat{P}(-\alpha)/(-\alpha)=0$.
Using monotonicity of the Hausdorff
dimension together with Lemma \ref{lem:UpperBound},
it then follows \[
0\leq\tau\left(2\log\gamma\right)\leq\lim_{\alpha\nearrow2\log\gamma}\tau(\alpha)=0.\]
Hence, we have $\tau(2\log\gamma)=0$, which gives that $\tau$ is continuous
from the left at $2\log\gamma$.\\
Finally, for the left derivative of $\tau$ at $2\log\gamma$,
note that a straight forward computation of
the derivative of $\tau$ on the interval $(0,2\log\gamma)$ shows that $\tau'(\alpha)=-P\left(t\left(\alpha\right)\right)/\alpha^{2}$.
Since $t(\alpha)$ tends to $(-\infty)$ as $\alpha$ approaches $2\log\gamma$, it
follows
\[\lim_{\alpha\nearrow2\log\gamma}\tau'(\alpha)=-\infty.\]
\section{Multifractal formalism for approximants}
In this section we outline the necessary changes which have to be
implemented in the proof of Theorem \ref{Thm:main} in order to
derive Theorem \ref{thm:main3}.
The analytic properties of $P_{D}$ as stated in
Theorem \ref{thm:main3} can be obtained as follows. In
Section 4 replace the function
$N:\overline{\Sigma}\to\mathbb {N}$ (resp. $N:\Sigma^{*} \to \mathbb {N}$) by the
function $\overline{{\bf 1}}: \overline{\Sigma}\to \{1\}$ (resp.
${\bf 1}^{*}: \Sigma^{*} \to \{1\}$) constant equal to $1$.
In this way we
obtain for the pressure
function $\overline{\P}$
associated
with $\overline{\Sigma}$,
\[
\overline{\P}\left(-\theta \overline{I}-\overline{\P}
\left(-\theta
\overline{I}\right) \overline{{\bf 1}}\right)=0.\]
Also, note that by Lemma \ref{lem:Vergleichbar} we have \[
P_{D}(\theta)=\overline{\P} \left(-\theta \overline{I}\right).\]
(Below, we shall specify the domain of $P_D$). Hence, combining these observations with Proposition
\ref{propHMU3} adapted to the situation here,
the alleged analytic properties of $P_{D}$
follow. Also, using the same strategy in Section \ref{section 5.1} and
\ref{section 5.2}, that is replacing in there the function $N$ by
the function ${\bf
1}^{*}$, one immediately obtains
\[
\tau_{D}(\alpha)=
\frac{\widehat{P}_{D}(-\alpha)}{-\alpha}.\]
(Below, we shall specify the domain of $\tau_D$).
For clarifying the range of $P_D$ and of $\tau_D$, and for the discussion of the boundary points of $\tau_D$,
we first remark that $P_{D}$ has a singularity at $1/2$.
This follows, since for every approximant
$\left[a_{1},\ldots,a_{k}\right]$
we have (see e.g. \cite{Khihtchine:36}) \[ \prod_{i=1}^{k}a_{i}\leq
q_{k}\left(\left[a_{1},\ldots,a_{k}\right]\right)\leq2^{k}\prod_{i=1}^{k}a_{i},\]
which immediately gives
\[
0\leq\log\zeta\left(\theta\right)-P_{D}\left(\theta\right)\leq2\theta\log2,
\textrm{ for }\theta>1/2.\] Here, $\zeta$ refers to the Riemann
zeta-function $\zeta(\theta):= \sum_{n \in \mathbb {N}} n^{-2\theta}$. This shows that $P_{D}(\theta)$ and
$P_{D}'(\theta)$ both tend to infinity for $\theta$ tending to $1/2$ from
above. From this we deduce that
$\widehat{P}_{D}\left(-\alpha\right)$ is well defined for arbitrary
large values of $\alpha$, and also that
\[\lim_{\alpha \to \infty}\widehat{P}_{D}(-\alpha)/(-\alpha)=1/2.\]
In order to see that the domain of $\widehat{P}_{D}$ is the interval
$[2\log\gamma,\infty)$ and that $\lim_{\alpha\searrow
2\log\gamma}\widehat{P}_{D}(-\alpha)/(-\alpha)=0$, it is now sufficient to
show that \begin{equation}
\lim_{\theta\to\infty}\left|P_{D}\left(\theta\right)+2
\theta\log\gamma\right|=0.
\label{eq:toshow}\end{equation}
Indeed, on the one hand we have \begin{eqnarray*}
\lim_{k\to\infty}\frac{1}{k}\log\sum_{\left[a_{1},\ldots,a_{k}\right]}q_{k}
\left(\left[a_{1},\ldots,a_{k}\right]\right)^{-2\theta}
& \leq & \lim_{k\to\infty}-\frac{1}{k}2\theta\log
q_{k}\left(\gamma\right)=-2\theta\log\gamma.\end{eqnarray*} On the
other hand, using Lemma \ref{lem:QGammaInequalitystrict} and
\ref{lem:TnbyCF}, we observe for $N\in \mathbb {N}$ and $\theta>(1+\log
N)/(2\log\gamma)$, \\
$\quad{\displaystyle
\lim_{k\to\infty}\frac{1}{k}\log\sum_{\left[a_{1}
,\ldots,a_{k}\right]}q_{k}\left(\left[a_{1},\ldots
,a_{k}\right]\right)^{-2\theta}}$\begin{eqnarray*}
\qquad\qquad & \leq &
\limsup_{k\to\infty}\frac{1}{k}\log\sum_{n=k+1}^{\infty}\binom{n}{k}\gamma^{-2\theta n}\\
& = &
-2\theta\log\gamma+\limsup_{k\to\infty}\frac{1}{k}\log\sum_{n=1}^{\infty}\binom{n+k}{k}
\gamma^{-2\theta n}\\
& \leq &
-2\theta\log\gamma+\limsup_{k\to\infty}\frac{1}{k}\log\sum_{n=1}^{\infty}\frac{\left(n+k\right)^{\left(n+k\right)}}{k^{k}n^{n}}\gamma^{-2\theta
n}\\
& \leq &
-2\theta\log\gamma+\limsup_{k\to\infty}\frac{1}{k}\log\sum_{n=1}^{\infty}\left(1+\frac{k}{Nn}\right)^{n}N^{n}\left(1+\frac{n}{k}\right)^{k}\gamma^{-2\theta
n}\\
& \leq
&-2\theta\log\gamma+\limsup_{k\to\infty}\frac{1}{k}\log\sum_{n=0}^{\infty}e^{k/N}e^{n\left(1+\log
N-2\theta\log\gamma\right)}\\
& \leq & -2\theta\log\gamma+1/N.\end{eqnarray*}
A combination of these two observations gives the statement in (\ref{eq:toshow}).
In order to prove continuity of $\tau_{D}$ at $2\log\gamma$,
note that by arguing similar as in the proof of Lemma
\ref{lem:UpperBound}, we obtain
for $\alpha<\chi$, \begin{eqnarray*}
\dim_{H}\left(\pi_{\mathrm{CF}} \left\{
x\in\overline{\Sigma}:\limsup_{n\to\infty}\frac{S_{n}\overline{I}(x)}{n}\leq\alpha\right\}
\right) & \leq &
\frac{\widehat{P}_{D}(-\alpha)}{-\alpha}.\end{eqnarray*}
Combining this with the monotonicity of Hausdorff
dimension, it follows
\[ \dim_{H}\left(\mathcal{L}_{3}
\left(2\log\gamma\right)\right)\leq\lim_{\alpha\searrow2\log\gamma}
\widehat{P}_{D}(-\alpha)/(-\alpha)=0.\]
Finally, the same argument as used in
Section \ref{sub:Discussion-of-boundary-points} for
determining the limit behaviour of $\tau'$, gives that for the
left derivative of $\tau_{D}$ at $2\log\gamma$ we have
\[ \lim_{\alpha\searrow2\log\gamma}\tau_{D}'
\left(\alpha\right)=\infty.\]
This finishes the proof of Theorem \ref{thm:main3}.
\newcommand{\etalchar}[1]{$^{#1}$}
| {
"timestamp": "2005-12-09T20:17:08",
"yymm": "0509",
"arxiv_id": "math/0509603",
"language": "en",
"url": "https://arxiv.org/abs/math/0509603",
"abstract": "In this paper we obtain multifractal generalizations of classical results by Lévy and Khintchin in metrical Diophantine approximations and measure theory of continued fractions. We give a complete multifractal analysis for Stern--Brocot intervals, for continued fractions and for certain Diophantine growth rates. In particular, we give detailed discussions of two multifractal spectra closely related to the Farey map and the Gauss map.",
"subjects": "Number Theory (math.NT); Dynamical Systems (math.DS)",
"title": "A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808736209154,
"lm_q2_score": 0.7217432003123989,
"lm_q1q2_score": 0.7075110549321938
} |
https://arxiv.org/abs/2206.03007 | On Binomial coefficients of real arguments | As is well-known, a generalization of the classical concept of the factorial $n!$ for a real number $x\in {\mathbb R}$ is the value of Euler's gamma function $\Gamma(1+x)$. In this connection, the notion of a binomial coefficient naturally arose for admissible values of the real arguments.By elementary means, it is proved a number of properties of binomial coefficients $\binom{r}{\alpha}$ of real arguments $r,\,\alpha\in {\mathbb R}$ such as analogs of unimodality, symmetry, Pascal's triangle, etc. for classical binomial coefficients. The asymptotic behavior of such generalized binomial coefficients of a special form is established. | \section*{Introduction}
\hspace*{\parindent} We study binomial coefficients of real
arguments. The aim of the investigation is to obtain by elementary
methods analogs of the basic properties well known for the classic
binomial coefficients. Such properties are of independent interest
and, in addition, can simplify the work with binomial coefficients
of the form $\binom{n}{m}$ with integer non-negative arguments $n$
and $m$, given essentially by real values with considered rounding
to an integer (when, for example, floor and ceiling functions for a
real number are used, etc.). So, for example, the properties of
unimodality and symmetry allow passing from such binomial
coefficients $\binom{n}{m}$, $0\leq m \leq n$ to \/"close"\/ real
binomial coefficients of the form $\binom{r}{\alpha}$, $\alpha\in
(-1,r+1)$ and vice versa. This approach simplifies the evaluation of
expressions with discrete binomial coefficients with integer
arguments of the specified form.
Note that the binomial coefficients of the form $\binom{r}{n}$,
where $r\in {\mathbb R}$ and $n\in {\mathbb N}$, can be defined in
the standard way as
$$\binom{r}{n}=\frac{r(r-1)(r-2)\cdots (r-n+1)}{n!}.$$
\noindent This approach was discussed in [4], where a numerous
number of identities for such binomial coefficients is given. In
[3], D.~Fowler studied the graph of the function $\binom{r}{\alpha}$
of two real variables $r$ and $\alpha$, various slices of this graph
were constructed using a computer and their analysis was carried
out. It is also indicated there an explicit expression for the
binomial coefficient $\binom{n}{\alpha}$, where $n$ is a
non-negative integer, through elementary functions (see\/
Proposition $2$ in Section $2$). On the basis of this
representation, Stuart~T.~Smith investigated the binomial
coefficients of the form $\binom{n}{z}$ with complex variable $z\in
{\mathbb C}$ and fixed natural number $n\in {\mathbb N}$, a number
of properties of such a function of complex argument $z$ is
established in [6]. In particular, the derivatives of the first and
second orders are calculated, and for the real argument $z$,
increasing and decreasing intervals, zeros of the function, etc are
found. It is also noted there the nontriviality of the function
investigation $\binom{n}{\alpha}$ of real variable $\alpha$ exactly
on the interval $\alpha\in (-1,n+1)$, in contrast to the domain
outside this interval. In particular, the increasing and decreasing
of this function was established rather difficult.
In this paper, we prove by elementary means a number of properties
of the binomial coefficients $\binom{r}{\alpha}$ of real arguments
$r,\,\alpha\in {\mathbb R}$, $\alpha\in (-1,r+ 1)$ (analogs of the
properties of unimodality, symmetry, Pascal's triangle, etc. for
discrete binomial coefficients), which may be useful in further
research (see, for example, [1]).
\section{Preliminary information}
\hspace*{\parindent} The article uses the generally accepted
concepts and notation of real analysis [2], as well as the standard
concepts of combinatorial analysis [4]. Denote by $(a,b)$ the {\it
open real interval}\/ between the numbers $a,b\in {\mathbb R}$,
$o(1)$ is an {\it infinitesimal function} in a neighborhood of
$\infty$, $n!$ is the {\it factorial} of non-negative integer $n$,
i.e. $n!=n(n-1)\cdots 2\cdot 1$, and wherein we define $0!=1$,
$\binom{n}{m}$, where $0\leq m \leq n$, is the (standard) binomial
coefficient (with non-negative integer arguments $n$, $m$ ), i.e.
$$\binom{n}{m}=\frac{n!}{m!(n-m)!}.$$
To denote {\it asymptotic equality} of real-valued functions $f(x)$
and $g(x)$ as $x\rightarrow\infty$, we use the notation $f(x)\sim
g(x)$, which by definition means that $f(x)=g(x)(1+r(x))$ in some
neighborhood of $\infty$, where $r(x)=o(1)$, or, equivalently (for
functions positive in some neighborhood of $\infty$)
$$\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=1.$$
The standard approach is considered, according to which the concept
of factorial for non-negative integers extends to real (and even
complex) numbers by the {\it gamma function} $\Gamma(\alpha)$. We
will use its definition in the following {\it Euler-Gauss form}
[2, p. 393--394, 812]
\begin{equation}
\Gamma(\alpha)=\lim_{n\rightarrow\infty} \frac{(n-1)!\,n^
{\alpha}}{\alpha(\alpha +1)(\alpha +2)\cdots(\alpha
+n-1)},\,\alpha\in{\mathbb R}\setminus \{0,-1,-2,\ldots\},
\end{equation}
such a limit exists for any specified value of $\alpha$ (see, for
example, [5] or [2, p. 393]). In view of the problem statement, we
do not consider extensions of the gamma function outside its
standard domain of definition. Note that when defining the gamma
function in the form of the {\it Euler integral of the second kind}
$$\Gamma(\alpha)=\int_0^{\infty} x^{\alpha-1} e^{-x} dx,$$
converging for $\alpha >0$, we obtain an equivalent definition on
the interval $(0,\infty)$ [2, p. 811]. The gamma function
$\Gamma(\alpha)$ is continuous and has continuous derivatives of all
orders on $(0,\infty)$, has no real roots, and it is positive on
$(0,\infty)$. For any non-negative integer $n$ the following
equality holds
\begin{eqnarray}
\Gamma(1+n)=n!\,,
\end{eqnarray}
moreover, the next {\it reduction formula} is valid [2, p. 394]
\begin{eqnarray}
\Gamma(1+\alpha)=\alpha\Gamma(\alpha),\, \alpha\in {\mathbb
R}\setminus \{0,-1,-2,\ldots\}.
\end{eqnarray}
In addition, the series expansion of $\sin \pi \alpha$ and (1) imply
the following {\it reflection formula} (see, for example, [2, 5])
\begin{eqnarray}
\Gamma(\alpha)\,\Gamma(1-\alpha)=\frac{\pi}{\sin\pi\alpha}\,,\,
\alpha\in {\mathbb R}\setminus \{0,\pm1,\pm2,\ldots\}.
\end{eqnarray}
As a generalization of the discrete binomial coefficient
$\binom{n}{m}$, the binomial coefficient of real arguments is
defined as follows (see, for example, [3])
\begin{eqnarray}
\binom{r}{\alpha}=\frac{\Gamma(1+r)}{\Gamma(1+\alpha)\Gamma(1+r-\alpha)}\,.
\end{eqnarray}
Note that if $r\in(-1,+\infty)$ and $\alpha\in (-1,r+1)$, the
binomial coefficient $\binom{r}{\alpha}$ is defined correctly by the
equality (5).
\section{Binomial coefficients $\binom{r}{\alpha}$ for $r,\,\alpha\in {\mathbb R}$}
\begin{theorem}[properties of the binomial coefficient of real arguments]
Let $r\in(-1,+\infty)$ and $\alpha\in (-1,r+1)$. Then \\[-6pt]
{\rm (i)} $\binom{r}{\alpha}>0$, $\binom{r}{0}=1$ and $\binom{r}{r}=1;$\\[-3pt]
{\rm (ii)} $\binom{0}{\alpha}=\left\{
\begin{array}{ll}
\,1, & \mbox{if } \alpha=0,\\
\frac{\sin\pi\alpha}{\pi\alpha}\,, & \mbox{if } \alpha\neq 0 \mbox{
and } \alpha\in (-1,1);
\end{array} \right.$\\[1pt]
{\rm (iii)} $\binom{r}{r-\alpha}=\binom{r}{\alpha};$\\[-3pt]
{\rm (iv)} $\binom{r}{\alpha}=\binom{r-1}{\alpha -1} + \binom{r-1}{\alpha},$
if $r\in(0,+\infty)$ and $\alpha\in (0,r);$\\[-3pt]
{\rm (v)} binomial coefficient $\phi(\alpha)\!=\binom{r}{\alpha}$ is
strictly increasing on the interval \,$(-1,\frac{r}{2}]$ and
strictly decreasing on the interval \,$[\frac{r}{2},r+1);$\\[-3pt]
{\rm (vi)} binomial coefficient $\psi(r)\!=\binom{r}{\alpha}$ is
strictly increasing for $\alpha >0$, strictly decreasing for
$-1<\alpha <0$ and $\psi(r)\!\equiv1$ if $\alpha=0$.
\end{theorem}
\begin{proof}
Statement (i) follows from the relations (2), (5).
Prove (ii). If $\alpha =0$, the required equality follows from (i).
Further, we assume that $\alpha\neq 0$. Using the relations
(2)--(5), we obtain
$$\binom{0}{\alpha}=\frac{\Gamma(1)}{\Gamma(1+\alpha)\Gamma(1-\alpha)}=
\frac{1}{\alpha\Gamma(\alpha)\Gamma(1-\alpha)}=\frac{\sin\pi\alpha}{\pi\alpha}\,.$$
Note that if $\alpha\in (-1,r+1)$, then $r-\alpha\in (-1,r+1)$.
Therefore, the binomial coefficient $\binom{r}{r-\alpha}$ is defined
and the required equality from (iii) is satisfied due to (5). It is
also easy to prove (iv) from (3) and (5).
Prove statement (v). Let $\alpha$, $\beta\in (-1,r+1)$. From (1) we
obtain
\begin{equation*}
\Gamma(1+\alpha)\,\Gamma(1+r-\alpha)=\lim_{n\rightarrow\infty}
\frac{(n-1)!\,(n-1)!\,n^{2+r}}{\prod_{i=1}^n \, (\alpha
+i)(r-\alpha+i)}\,.
\end{equation*}
Hence,
\begin{equation}
\frac{\phi(\alpha)}{\phi(\beta)}=
\frac{\Gamma(1+\beta)\,\Gamma(1+r-\beta)}{\Gamma(1+\alpha)\,\Gamma(1+r-\alpha)}=
\lim_{n\rightarrow\infty} \prod_{i=1}^n \,\delta_i(\alpha,\beta),\,
\mbox{ where}
\end{equation}
$$\delta_i(\alpha,\beta)= \frac{(\alpha +i)(r-\alpha+i)}{(\beta
+i)(r-\beta+i)}\,.$$ Note that $\delta_i(\alpha,\beta)>0$ for every
$\alpha$, $\beta\in (-1,r+1)$ and $i=1,\ldots,n$. It is also easy to
prove that
\begin{equation}
\delta_i(\alpha,\beta)\geq 1 \Leftrightarrow f(\alpha)\geq f(\beta),
\end{equation}
where $f(x)=-x^2+xr$ and parabola $f(x)$ is strictly increasing on
$(-\infty,\frac{r}{2}]$ as well as strictly decreasing on $[\frac
{r}{2},+\infty)$. Moreover, it is directly established that
\begin{equation}
\delta_1(\alpha,\beta)=1+\varepsilon(\alpha,\beta), \mbox{ where }
\varepsilon(\alpha,\beta)=\frac{(r-\alpha-\beta)(\alpha-\beta)}{(\beta+1)(r-\beta+1)}.
\end{equation}
Let $-1<\beta <\alpha \leq \frac{r}{2}$. Then $f(\alpha)>f(\beta)$
and $\varepsilon(\alpha,\beta)>0$. By virtue of (7), we have
$\delta_i(\alpha,\beta)\geq 1$ for every $i=1,\ldots,n$. Hence, from
(6) and (8) we obtain
$$\frac{\phi(\alpha)}{\phi(\beta)}\geq
\delta_1(\alpha,\beta)=1+\varepsilon(\alpha,\beta)>1.$$
Similarly, if $\frac{r}{2}\leq \beta <\alpha < r+1$, then
$f(\alpha)<f(\beta)$ and $\varepsilon(\alpha,\beta)<0$. Therefore,
$0<\delta_i(\alpha,\beta)<1$, $i=1,\ldots,n$ and
$$\frac{\phi(\alpha)}{\phi(\beta)}\leq
\delta_1(\alpha,\beta)=1+\varepsilon(\alpha,\beta)<1.$$
Prove statement (vi). In view of statement (i), we can assume that
$\alpha\neq 0$. Let $r<r'$. Note that $1+r+i>0$, $1+r'+i>0$ and
$\alpha/(1+r+i)<1$, $\alpha/(1+r'+i)<1$ for every $i\geq 0$. From
(1) we obtain
$$\frac{\Gamma(1+r)}{\Gamma(1+r-\alpha)}=\Bigl(1-
\frac{\alpha}{1+r}\Bigr)\lim_{n\rightarrow\infty}
n^{\alpha}\prod_{i=1}^{n-1} \Bigl(1-
\frac{\alpha}{1+r+i}\Bigr)\,.$$ Hence,
$\Gamma(1+r)/\Gamma(1+r-\alpha) < \Gamma(1+r')/\Gamma(1+r'-\alpha)$
for $\alpha >0$ (and the reverse strict inequality holds for $\alpha
<0$). In view of (5), we conclude $\psi(r)<\psi(r')$ (respectively
$\psi(r)> \psi(r')$ for $\alpha <0$).
\end{proof}
\begin{proposition}
Let $r$ takes real values, $\alpha\in {\mathbb R}$ does not depend
on $r$ and $0<\alpha <1$. Then the following asymptotic equality is
valid as $r$ tends to infinity
\begin{equation}
\binom{r}{r\alpha}\sim \sqrt{\frac{1}{2\pi \alpha
(1-\alpha)r}}\,\,\Bigl(\frac{1}{\alpha}\Bigr)^{\alpha
r}\,\Bigl(\frac{1}{1-\alpha}\Bigr)^{(1-\alpha) r}.
\end{equation}
\end{proposition}
\begin{proof}
For $r>0$ the functions $\Gamma(1+r)$, $\Gamma(1+r\alpha)$,
$\Gamma(1+r-r\alpha)$ are defined correctly and positive. By virtue
of (5), we have
\begin{eqnarray}
\binom{r}{r\alpha}=\frac{\Gamma(1+r)}{\Gamma(1+r\alpha)\,\Gamma(1+r-r\alpha)}\,.
\end{eqnarray}
For the gamma function, the following generalized Stirling formula
is valid (see, for example, [2]):
$$\Gamma(1+x)\sim \sqrt{2\pi x}\,\Bigl(\frac{x}{e}\Bigr)^{x}\mbox{
as } x\rightarrow \infty.$$ In view of the condition $0<\alpha <1$,
we have $r\alpha\rightarrow \infty$ and $r-r\alpha\rightarrow
\infty$ as $r$ tends to infinity. Now, using the generalized
Stirling formula, we obtain by equivalent transformations the
asymptotic equality (9) from (10) .
\end{proof}
\begin{corollary}
Let $r$ takes non-negative integer values, $\alpha\in {\mathbb R}$
does not depend on $r$ and $0<\alpha <1$. Then the asymptotic
equality {\rm (9)} is valid as $r$ tends to infinity.
\end{corollary}
\begin{proof}
Let $g(r)$ be the function on the right side of the asymptotic
equality (9) and $f(r)=\binom{r}{r\alpha}/g(r)$. Then
$\lim_{r\rightarrow\infty} f(r)=1$ by Proposition $1$. Non-negative
integers $n$ form an infinitesimal subsequence of the values of real
variable $r$. Therefore, the existing limit value of function $f(r)$
of the real argument as $r\rightarrow \infty$ is preserved for
function $f(n)$ of the non-negative integer argument as
$n\rightarrow \infty$.
\end{proof}
As noted in [3], in the case of binomial coefficients of the form
$\binom{n}{\alpha}$ when $n$ is a non-negative integer, the binomial
coefficient is explicitly expressed in terms of elementary
functions. The following proposition formalizes this statement and
its justification is based on the properties of the gamma function
and binomial coefficients of real arguments.
\renewcommand{\theproposition}{\arabic{proposition} {\rm[3]}}
\begin{proposition}
Let $n$ be a non-negative integer and real number $\alpha\in
(-1,n+1)$. Then the following equality is valid
\[
\binom{n}{\alpha}=\left\{
\begin{array}{ll}
\frac{\sin \pi\alpha}{\pi\alpha}\,, & \mbox{if } n=0 \mbox{ and }
\alpha\neq 0,\\[5pt]
\frac{n!}{(n-\alpha)(n-1-\alpha)\cdots (1-\alpha)}\, \frac{\sin
\pi\alpha}{\pi\alpha}\,, & \mbox{if } n\geq 1 \mbox{ and } \alpha\notin\{0,1,\ldots,n\},\\[5pt]
\frac{n!}{\alpha!(n-\alpha)!}\,, & \mbox{if } \alpha\in
\{0,1,\ldots,n\}.
\end{array} \right.
\]
\end{proposition}
\renewcommand{\theproposition}{\arabic{proposition}}
\begin{proof}
The required equality is proved in Theorem $1$ for $n=0$, and for
$\alpha\in \{0,1,\ldots,n\}$ it follows from the property of the
gamma function (2). Let now $n\geq 1$ and
$\alpha\notin\{0,1,\ldots,n\}$. Suppose that $n-i-\alpha\in
\{0,-1,-2,\ldots\}$ for some $i\in {\mathbb N}$ and $0\leq i \leq
n-1$. Then $\alpha-(n-i)\in \{0,1,2,\ldots\}$. Hence, $\alpha\in
{\mathbb N}$ and therefore $\alpha\in \{0,1,\ldots,n\}$, got a
contradiction. Thus, $n-i-\alpha\notin \{0,-1,-2,\ldots\}$ for every
$i=0,1,\ldots,n-1$. By virtue of the reduction formula (3), we have
$$\Gamma(1+n-\alpha)=(n-\alpha)\Gamma(n-\alpha)=\ldots
=\prod_{i=0}^{n-1}(n-i-\alpha)\Gamma(1-\alpha).$$
Since $\alpha\in {\mathbb R}\setminus \{0,\pm1,\pm2,\ldots\}$, from
(3)--(5) we obtain the required expression for the binomial
coefficient $\binom{n}{\alpha}$.
\end{proof}
\bigskip
| {
"timestamp": "2022-06-08T02:08:24",
"yymm": "2206",
"arxiv_id": "2206.03007",
"language": "en",
"url": "https://arxiv.org/abs/2206.03007",
"abstract": "As is well-known, a generalization of the classical concept of the factorial $n!$ for a real number $x\\in {\\mathbb R}$ is the value of Euler's gamma function $\\Gamma(1+x)$. In this connection, the notion of a binomial coefficient naturally arose for admissible values of the real arguments.By elementary means, it is proved a number of properties of binomial coefficients $\\binom{r}{\\alpha}$ of real arguments $r,\\,\\alpha\\in {\\mathbb R}$ such as analogs of unimodality, symmetry, Pascal's triangle, etc. for classical binomial coefficients. The asymptotic behavior of such generalized binomial coefficients of a special form is established.",
"subjects": "Combinatorics (math.CO); Discrete Mathematics (cs.DM); Classical Analysis and ODEs (math.CA)",
"title": "On Binomial coefficients of real arguments",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808736209154,
"lm_q2_score": 0.7217432003123989,
"lm_q1q2_score": 0.7075110549321938
} |
https://arxiv.org/abs/0811.3599 | A second row Parking Paradox | We consider two variations of the discrete car parking problem where at every vertex of the integers a car arrives with rate one, now allowing for parking in two lines. a) The car parks in the first line whenever the vertex and all of its nearest neighbors are not occupied yet. It can reach the first line if it is not obstructed by cars already parked in the second line (screening). b) The car parks according to the same rules, but parking in the first line can not be obstructed by parked cars in the second line (no screening). In both models, a car that can not park in the first line will attempt to park in the second line. If it is obstructed in the second line as well, the attempt is discarded. We show that both models are solvable in terms of finite-dimensional ODEs. We compare numerically the limits of first and second line densities, with time going to infinity. While it is not surprising that model a) exhibits an increase of the density in the second line from the first line, more remarkably this is also true for model b), albeit in a less pronounced way. | \section{Introduction}
Car parking, first considered in a mathematical way by R\'{e}nyi \cite{Renyi} in 1958,
gives rise to interesting models that in several variations have been applied in
many fields of science. In the original car parking problem, unit
length cars are appearing with constant rate in time and with constant
density in space on the line where they try to park.
A new car is allowed to park only in case there is no intersection
with previously parked cars.
Otherwise the attempt is rejected. R\'{e}nyi proved that the density
of cars has the limit $0.7475...$, the so-called parking constant.
In the simplest discrete version of the car parking
problem, cars of length $2$ try to park at their midpoints randomly
on ${\Bbb Z}$. This model has been solved analytically as well
\cite{Co62}.
\par
This model belongs to a wider class of more complicated models of
deposition with exclusion interaction. Usually such models are not
analytically solvable. In physical chemistry ``cars'' become
particles which are deposited in layers on a substrate, a process
called random sequential adsorption (RSA). A variety of related
models are studied. For a review of recent developments see
\cite{Pri07}. Moreover, models with more complicated graphs e.g.
(random) trees have been investigated \cite{Pri04},\cite{Penrose},\cite{Gouet},\cite{Fl08}.
Multilayer variations of the model are used to describe
multilayer adsorption of particles on a substrate \cite{Privman}
and the sequential frequency assignment process \cite{De07} which
appears in telecommunication. In these papers it is also observed that the
density in higher layers increases up from the first layer, which at first seems
rather counterintuitive. Heuristic arguments for monotonicity of densities
were found in specific models \cite{Privman}, but no rigorous proofs could be given yet.
Moreover Privman finds numerically a
scaling behavior of the density in a similar RSA model \cite{Privman}
with slightly different adhesion rules which is notoriously difficult to explain
mathematically.
\par
In the present paper we aim for a rigorous investigation and
treat two versions of the discrete two-line
car-parking problem with cars of length 2. First we describe the
dynamics of the car parking process without screening and also with
screening. Then we provide the {\em solutions} of these models by
reducing them to closed finite dimensional systems of ODEs for
densities of local patterns, see Theorems \ref{Theo1} and
\ref{Theo2}. That it is possible to find a finite-dimensional dynamical
description is quite remarkable. It is not obvious, and in fact our
method ceases to work for a three-line extension of the model
without screening where an infinite system appears.
A second remarkable fact is that, even without screening, the second
line density is higher than the first. Cars do not communicate or
plan a common strategy and their arrival is random, but they seem to
use the resources in the second line more efficiently, once they
have been rejected in the first line.
\section{The Dynamics}
We will define a Markov jump process on the (suitably coded)
occupation numbers $m=(m_i)_{i\in {\Bbb Z}}\in \Omega=\{0,1,2,3\}^{\Bbb Z}$.
Here the {\it spin} $m_i$ denotes the joint occupation numbers at
vertex $i$ at height $1$ and $2$.
It is useful for short notation to interpret the occupation numbers
at various heights as binary digits and write ordinary
natural numbers. That is we write
\[ m_{i} = \left\{ \begin{array}{rl}
0 & \mbox{if vertex $i$ is vacant in the first and second line} \\
1 & \mbox{if vertex $i$ is occupied in the first but not in the second line} \\
2 & \mbox{if vertex $i$ is occupied in the second but not in the first line} \\
3 & \mbox{if vertex $i$ is occupied in the first and in the second
line}
\end{array} \right. \]
so that $m_i\in \{0,1,2,3\}$. The dynamics of the process is defined
in terms of the generator which is given by the right hand side of
the differential equation
\begin{equation}\begin{split}
\frac{d}{dt} \mathbb{E}^{m} f(m(t)) = \sum_{k,s} \left[ f(m^{s,k}) -
f(m) \right] c(s;m_{k-1},m_{k},m_{k+1}) \label{Generator}
\end{split} \end{equation}
with
\[ m^{s,k}_i = \left\{ \begin{array}{rl}
m_i & \mbox{if $k \neq i$ } \\
s & \mbox{if $k = i$ } \end{array} \right. \] denoting the
configuration which has been obtained by $m$ by changing the
configuration in $i$ to $s$. Here ${\Bbb E}^{m}$ denotes the expected
value with respect to the process, started at the initial
configuration $m$.
\subsubsection*{Two-line parking rates}
The rates are either equal to zero or one. They are $1$ precisely in
the following cases.
\begin{enumerate}
\item{$0\mapsto 1$}
Adding a car in the first line at site $i$. For the model without
screening we have
\begin{equation}\begin{split}
c(1;0,0,0)&=c(1;2,0,0)=c(1;0,0,2)=c(1;2,0,2)=1\cr
\end{split} \end{equation}
Indeed, this occurs when the site itself is empty on the first and
second line and the nearest neighbors are empty in the first line,
see figure \ref{fig:01} for an example.
\begin{figure}[h]
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\begin{picture}(150,30)(0,-7)
\multiput(20,5)(30,0){4}{\multiput(0,0)(5,0){5}{\line(0,1){10}}}
\multiput(20,5)(30,0){4}{\line(1,0){20}}
\multiput(20,10)(30,0){4}{\line(1,0){20}}
\put(55,10){\circle*{1.5}} \put(50,5){\circle*{1.5}}
\put(95,10){\circle*{1.5}} \put(100,5){\circle*{1.5}}
\put(115,10){\circle*{1.5}} \put(110,5){\circle*{1.5}}
\put(125,10){\circle*{1.5}} \put(130,5){\circle*{1.5}}
\multiput(51,11)(.1,0){80}{\line(0,1){3}}
\multiput(50,6)(.1,0){40}{\line(0,1){3}}
\multiput(91,11)(.1,0){80}{\line(0,1){3}}
\multiput(96,6)(.1,0){40}{\line(0,1){3}}
\multiput(111,11)(.1,0){80}{\line(0,1){3}}
\multiput(110,6)(.1,0){40}{\line(0,1){3}}
\multiput(121,11)(.1,0){80}{\line(0,1){3}}
\multiput(126,6)(.1,0){40}{\line(0,1){3}}
\multiput(22.5,0)(30,0){4}{\put(0,0){\small{$-1$}}}
\multiput(29.5,0)(30,0){4}{\put(0,0){\small{$0$}}}
\multiput(34.5,0)(30,0){4}{\put(0,0){\small{$1$}}}
\end{picture} \caption{Configurations of vertices $-1, 0,$ and $1$ that allow a transition from $m_0=0$ to $m_0=1$
in the model without screening. In the model with screening only the
most left configuration allows a transition to $m_0=1$. }
\label{fig:01}
\end{figure}
In the screening model however, cars in the second line will
obstruct cars from reaching the first line. Therefore, in the
screening model we have as the only nonvanishing rate
\begin{equation}\begin{split}
c^{\hbox{sc}}(1;0,0,0)&=1\cr
\end{split} \end{equation}
\item{$0\mapsto 2$} Adding a car in the second line at $i$ while the first line was empty at the site
\begin{equation}\begin{split}
c(2;1,0,0)&=c(2;0,0,1)=c(2;1,0,1)=1\cr
\end{split} \end{equation}
Indeed, this occurs when there was a supporting site $i-1$ or $i+1$
or both with one car in the first line. This is true for both
models.
\item{$1\mapsto 3$} Adding a car in the second line while the first line was full at the site
\begin{equation}\begin{split}
c(2;0,1,0)&=1\cr
\end{split} \end{equation}
Indeed, this occurs when there are no obstructing cars right and
left at height $2$. There can be no obstructing cars right and left
at height $1$ because there could not be a car in the first line at
$i$ otherwise. This is true for both models.
\end{enumerate}
All other transitions are impossible.
\par
This generator defines a Markov jump process on the infinite graph
by standard theory \cite{Liggett}, such that (\ref{Generator})
holds for any local function $f: \Omega\rightarrow{\Bbb R}$.
\section{Results}
We provide a closed system of differential equations for the
densities of occupied sites, involving densities of finitely many local patterns,
in both models. First we need some
definitions. Here and in the following we use for the densities at single sites,
and triples of sites the notation
\begin{equation}\begin{split}
D_t(s)&:=P_t(m_{0}=s)\cr
D_t(s_{-1},s_{0},s_{1})&:=P_t(m_{-1}=s_{-1},m_{0}=s_{0},m_{1}=s_{1})\cr
\label{def:Dt}
\end{split}\end{equation}
Further we need the following ``one-sided densities''
\begin{equation}\begin{split}
f_t (s)&:=P_t(m_{1}=s | N_0(t)=0), \quad \text{ for } s=0,1,2\cr
R_t &:=P_t(m_{1}=1, m_{2}=0 |
N_0(t)=0)
\label{osq}
\end{split}\end{equation}
where $N_j(t)$ denotes the Poisson counting process of events of car
arrivals at site $j$.
As our main result we show that the time-evolution of these
densities gives rise to a closed ODE.
\begin{theorem}
\label{Theo1}
Two-line Parking without Screening. \\ The time evolution of the probability vector
$(D_t(0),D_t(1),D_t(2),D_t(3))$ obeys the following system of
differential equations.
\begin{equation}\begin{split}
\frac{d}{dt}D_t(0)&= -(f_t(0) + f_t(2))^2e^{-t} - (2f_t(0)f_t(1)
+f_t(1)^2)e^{-t}\cr %
\frac{d}{dt}D_t(1)&= (f_t(0) + f_t(2))^2 e^{-t} - D_t(0,1,0) \cr %
\frac{d}{dt}D_t(2)&= (2f_t(0)f_t(1)
+f_t(1)^2)e^{-t}\cr %
\frac{d}{dt}D_t(3)&= D_t(0,1,0) \cr %
\end{split}\end{equation}
with initial conditions $D_{0}(s)=1_{s=0}$, where the vector
$(f_t(0),f_t(1),f_t(2),R_t)$ obeys the linear ODE
\begin{equation}\begin{split}\label{osqODE}
\frac{d}{dt}f_t(0)&= - f_t(0) e^{-t } - f_t(1) e^{-t }- f_t(2)e^{-t}\cr%
\frac{d}{dt}f_t(1)&= f_t(0)e^{-t} +f_t(2)e^{-t}- R_t\cr%
\frac{d}{dt}f_t(2)&= f_t(1)e^{-t}\cr %
\frac{d}{dt}R_t&= f_t(0)(e^{-t}-t e^{-2 t })-f_t(1) t e^{-2 t } -R_t
\end{split}\end{equation}
with initial conditions $(f_0(0),f_0(1),f_0(2),R_0)=(1,0,0,0)$, \\
and finally, $D_t(0,1,0)$ is obeying the equation
\begin{equation}\begin{split}
\frac{d}{dt}D_t(0,1,0)&= f_t(0)^2 e^{-t}- D_t(0,1,0) -2 R_t f_t(0)
e^{-t}-2 R_t f_t(1)e^{-t}\cr
\end{split}\end{equation}
with $D_0(0,1,0)=0$.
\end{theorem}
The system above can be solved numerically and the results are
depicted in figure \ref{fig:D}. As it can be seen in the right
figure, surprisingly the value of $D_t(2)$ has a slightly higher
limit than $D_t(1)$. This clearly means that the second line has a
higher limit density of cars than the first line. This result is
independently confirmed by simulations of the parking process
measuring the empirical densities.
\begin{figure}[htp]
\includegraphics[height=5.5cm]{densitiesline12all.pdf}
\includegraphics[height=5.5cm]{densitiesline12.pdf}
$\phantom{123}$
\caption{Numerical solution of the densities in the first and second line. The right figure
zooms in on the limits of $D_t(1)$ and $D_t(2)$ to show they tend to different values.
}
\label{fig:D}
\end{figure}
A similar system of equations holds for the model with screening.
Recall that in this model cars are not allowed to pass cars on the
second line to reach a void on the first line. This results in less
possibilities of filling voids of the first line than in the model
treated above. In fact we can derive the ODEs of this model by
simply deleting those terms in (\ref{osqODE}) that represent the
possibility of ``jumping'' over a car in the second line to reach a
void on the first line. So, we get
\begin{theorem}\label{Theo2} Two-line Parking with Screening\\
The time evolution of the probability vector $(D_t^{sc}(0),
D_t^{sc}(1), D_t^{sc}(2), D_t^{sc}(3))$ obeys the following system
of differential equations
\begin{equation}\begin{split}
\frac{d}{dt}D_t^{sc}(0)&= -(f_t^{sc}(0) + f_t^{sc}(1))^2 e^{-t}\cr %
\frac{d}{dt}D_t^{sc}(1)&= f_t^{sc}(0)^2 e^{-t} - D_t^{sc}(0,1,0) \cr %
\frac{d}{dt}D_t^{sc}(2)&= (2f_t^{sc}(0)f_t^{sc}(1)
+f_t^{sc}(1)^2)e^{-t}\cr %
\frac{d}{dt}D_t^{sc}(3)&= D_t^{sc}(0,1,0) \cr %
\end{split}\end{equation}
with initial conditions $D_0^{sc}(s)=1_{s=0}$, where the vector
$(f_t ^{sc}(0), f_t ^{sc}(1), f_t ^{sc}(2), R_t ^{sc})$ obeys the
linear ODE
\begin{equation}
\begin{split}
\frac{d}{dt}f_t^{sc}(0)&= - f_t^{sc}(0) e^{-t } - f_t^{sc}(1) e^{-t }\cr%
\frac{d}{dt}f_t^{sc}(1)&= f_t^{sc}(0)e^{-t} - R_t^{sc}\cr%
\frac{d}{dt}f_t^{sc}(2)&= f_t^{sc}(1)e^{-t}\cr %
\frac{d}{dt}R_t^{sc} &= f_t^{sc}(0)(e^{-t}-t e^{-2 t})-f_t^{sc}(1)
t e^{-2 t } -R_t^{sc} \cr
\end{split}
\end{equation}
with initial conditions $(f_0^{sc}(0),f_0^{sc}(1),f_0^{sc}(2),R_0^{sc})=(1,0,0,0)$, \\
and finally, $D_t^{sc}(0,1,0)$ is obeying the equation
\begin{equation}\begin{split}
\frac{d}{dt}D_t^{sc}(0,1,0)&= f_t^{sc}(0)^2 e^{-t}- D_t^{sc}(0,1,0)
-2 R_t^{sc} f_t^{sc}(0) e^{-t}-2 R_t^{sc} f_t^{sc}(1)e^{-t}\cr
\end{split}\end{equation}
with $D_0^{sc}(0,1,0)=0$.
\end{theorem}
\section{Proofs of Theorem 1 and Theorem 2}
The following lemmas are used to prove our theorems.
\begin{lem}
\label{D}
The probability vector $(D_t(0), D_t(1), D_t(2), D_t(3))$
obeys
\begin{eqnarray}
\frac{d}{dt}D_t(0)&=& -D_t(0,0,0) -2 D_t(2,0,0)-D_t(2,0,2)-2
D_t(1,0,0)-D_t(1,0,1) \label{D0} \\
\frac{d}{dt}D_t(1)&=& D_t(0,0,0) +2 D_t(2,0,0)+D_t(2,0,2)-D_t(0,1,0)
\label{D1}\\
\frac{d}{dt}D_t(2)&=& 2 D_t(1,0,0)+D_t(1,0,1) \\
\frac{d}{dt}D_t(3)&=& D_t(0,1,0)
\end{eqnarray}
\end{lem}
{\bf Remark: } Summing over the four right hand sides we get zero,
due to the fact that we have summed a probability vector. It is also
interesting to check that
\begin{equation}\begin{split}
\frac{d}{dt}D_t(1)+\frac{d}{dt}D_t(3)&= D_t(0,0,0) +2
D_t(2,0,0)+D_t(2,0,2)\cr
\end{split}\end{equation}
recovers the ODE for the density in the first line.\\
{\bf Proof: } Fix an arbitrary vertex. Let us call this vertex $0$.
Starting from the dynamics (\ref{Generator}) and using symmetries we
have
\begin{equation}\begin{split}
\frac{d}{dt}D_t(0)&=-D_t(0,0,0) -2 D_t(2,0,0)-D_t(2,0,2)-2
D_t(1,0,0)-D_t(1,0,1) \cr
\end{split}\end{equation}
Indeed, the first three terms correspond to adding a car in the
first line, the next two terms correspond to adding a car in the
second line, see figure \ref{fig:D0}.
\begin{figure}[h]
\setlength{\unitlength}{1mm}
\begin{picture}(150,30)(0,-7)
\multiput(10,5)(20,0){7}{\line(0,1){10}}
\multiput(10,10)(20,0){7}{\line(1,0){12}}
\multiput(16,5)(20,0){7}{\line(0,1){10}}
\multiput(22,5)(20,0){7}{\line(0,1){10}}
\multiput(10,5)(20,0){7}{\line(1,0){12}} %
\put(30,10){\circle*{1.5}} \put(62,10){\circle*{1.5}}
\put(70,10){\circle*{1.5}} \put(82,10){\circle*{1.5}}
\put(90,5){\circle*{1.5}} \put(122,5){\circle*{1.5}}
\put(130,5){\circle*{1.5}} \put(142,5){\circle*{1.5}}
\put(8.5,0){\small{$D_t(0,0,0)$}} \put(28.5,0){\small{$D_t(2,0,0)$}}
\put(48.5,0){\small{$D_t(0,0,2)$}}
\put(68.5,0){\small{$D_t(2,0,2)$}}
\put(88.5,0){\small{$D_t(1,0,0)$}}
\put(108.5,0){\small{$D_t(0,0,1)$}}
\put(128.5,0){\small{$D_t(1,0,1)$}}
\multiput(30,11)(.1,0){50}{\line(0,1){3}}
\multiput(57,11)(.1,0){50}{\line(0,1){3}}
\multiput(70,11)(.1,0){50}{\line(0,1){3}}
\multiput(77,11)(.1,0){50}{\line(0,1){3}}
\multiput(90,6)(.1,0){50}{\line(0,1){3}}
\multiput(117,6)(.1,0){50}{\line(0,1){3}}
\multiput(130,6)(.1,0){50}{\line(0,1){3}}
\multiput(137,6)(.1,0){50}{\line(0,1){3}}
\end{picture} \caption{List of all occupancy configurations of vertices -1, 0 and 1 in the first and second line
that (may) contribute to a reduction of the proportion of $D_t(0)$.}
\label{fig:D0}
\end{figure}
The other three differential equations are derived in a similar way.
\begin{lem} The triple site densities $D_t(s,0,s')$ and the one-sided densities as defined in
\ref{def:Dt} and \ref{osq} respectively are related in the following
way
\begin{equation}\begin{split}
D_t(s,0,s')&= f_t(s)f_t(s') e^{- t} \cr
\end{split}\end{equation}
for $(s,s')\in \{(0,0),(0,1),(1,0),(1,1),(2,2)\}$. \label{D&osq}
\end{lem}
{\bf Proof: } We note that for the mentioned choices of $(s,s')$
conditioning on non-arrival at zero does not change the probability,
that is
\begin{equation}\begin{split}
D_t(s,0,s')&= P_t(m_{-1}=s,m_{0}=0,m_{1}=s' , N_0(t)=0)\cr &=
P_t(m_{-1}=s,m_{0}=0,m_{1}=s' | N_0(t)=0) e^{- t} \cr
\end{split}\end{equation}
In the next step we note that, conditional on the event that no car
has arrived at the site $0$, the dynamics for the two sides that are
emerging from $0$ is independent. Consequently we have
\begin{equation}\begin{split}\label{2.7}
&P_t(m_{-1}=s,m_{0}=0,m_{1}=s' | N_0(t)=0)\cr &= P_t(m_{1}=s |
N_0(t)=0)P_t(m_{1}=s' | N_0(t)=0)\cr \cr
\end{split}\end{equation}
This concludes the proof of the Lemma.
$\hfill \Box$
Next we look at the time-evolution of the ``one-sided densities''.
\begin{lem}
\label{lem:osq} %
The vector $(f_t(0),f_t(1),f_t(2),R_t)$ obeys the ODE
\begin{equation}\begin{split}\label{2.7a}
\frac{d}{dt}f_t(0)&= - f_t(0) e^{-t } - f_t(1) e^{-t }- f_t(2)e^{-t}\cr%
\frac{d}{dt}f_t(1)&= f_t(0)e^{-t} +f_t(2)e^{-t}- R_t\cr%
\frac{d}{dt}f_t(2)&= f_t(1)e^{-t}\cr %
\frac{d}{dt}R_t&= f_t(0)(e^{-t}-t e^{-2 t })-f_t(1) t e^{-2 t } -R_t
\end{split}\end{equation}
with initial conditions $f_{t=0}(s)=1_{s=0}$ and $R_{t=0}=0$.
\end{lem}
{\bf Remark 1: } Note that combining the equations of
$\frac{d}{dt}f_t(0)$ and $\frac{d}{dt}f_t(2)$ readily gives
\begin{equation}
f_t(0) + f_t(2) = \exp(e^{-t}-1)
\end{equation}
which is a known result for the first line in a semi-infinite chain
\cite{Co62}.
{\bf Remark 2: } Note also that because $R_t := P_t(m_{1}=1, m_{2}=0
| N_0(t)=0)$ we have in fact $R_t = \frac{d}{dt} f_t(3)$.
{\bf Proof: } To derive ODEs for these densities we employ the
generator of the process, while putting the term at the site $0$ to
sleep, and correspondingly the spin at zero to be the constant
$m_0=0$. %
For the first quantity we get
\begin{equation}\begin{split}\label{2.7b}
\frac{d}{dt}f_t(0)&=\frac{d}{dt}P_t(m_{1}=0 | N_0(t)=0)\cr %
&= -P_t(m_{1}=0 , m_{2}=0| N_0(t)=0) - P_t(m_{1}=0 ,m_{2}=2|N_0(t)=0) \cr%
&~- P_t(m_{1}=0 , m_{2}=1| N_0(t)=0)\cr %
&= - P_t(m_{1}=0 , m_{2}=0|N_0(t)=0, N_1(t)=0)e^{-t} \cr%
&~-P_t(m_{1}=0 , m_{2}=2| N_0(t)=0, N_1(t)=0)e^{-t} \cr
&-P_t(m_{1}=1| N_0(t)=0)e^{-t }\cr %
&= - P_t(m_{1}=0| N_0(t)=0)e^{-t} - P_t(m_{1}=2 | N_0(t)=0)e^{-t}\cr%
&- P_t(m_{1}=1| N_0(t)=0)e^{-t }\cr &= - f_t(0) e^{-t } - f_t(1)
e^{-t }- f_t(2)e^{-t }\cr \cr
\end{split}\end{equation}
Next we have
\begin{equation}\begin{split}\label{2.7c}
\frac{d}{dt}f_t(1)&= \frac{d}{dt}P_t(m_{1}=1 | N_0(t)=0)\cr &=
P_t(m_{1}=0 , m_{2}=0| N_0(t)=0) +P_t(m_{1}=0 , m_{2}=2|
N_0(t)=0)\cr &- P_t(m_{1}=1 , m_{2}=0| N_0(t)=0)\cr &= f_t(0)e^{-t}
+f_t(2)e^{-t} - R_t\cr \cr
\end{split}\end{equation}
and
\begin{equation}\begin{split}\label{2.7d}
\frac{d}{dt}f_t(2)&= \frac{d}{dt}P_t(m_{1}=2 | N_0(t)=0)\cr &=
P_t(m_{1}=0 , m_{2}=1| N_0(t)=0)\cr &= P_t(m_{1}=1|
N_0(t)=0)e^{-t}\cr
\end{split}\end{equation}
Finally, we get
\begin{equation}\begin{split}
\frac{d}{dt}R_t&= \frac{d}{dt}P_t(m_{1}=1, m_{2}=0 | N_0(t)=0) \cr %
&= P_t(m_{1}=0, m_{2}=0 | N_0(t)=0) -P_t(m_{1}=1, m_{2}=0 |
N_0(t)=0)\cr &-P_t(m_{1}=1, m_{2}=0 , m_3=0| N_0(t)=0) \cr%
&-P_t(m_{1}=1, m_{2}=0 , m_3=1| N_0(t)=0)
\end{split}\end{equation}
Using conditioning on non-arrival again we get
\begin{equation}\begin{split}
&\frac{d}{dt}P_t(m_{1}=1, m_{2}=0 | N_0(t)=0)\cr &= P_t(m_{1}=0|
N_0(t)=0)e^{-t } -P_t(m_{1}=1, m_{2}=0 | N_0(t)=0)\cr &-P_t(m_{1}=1|
N_0(t)=0,N_2(t)=0 )P_t(m_{1}=0| N_0(t)=0)e^{-t }\cr &-P_t(m_{1}=1|
N_0(t)=0,N_2(t)=0 )P_t(m_{1}=1| N_0(t)=0)e^{-t } \cr
\end{split}\end{equation}
Clearly we have
\begin{equation}\begin{split}
&P_t(m_{1}=1| N_0(t)=0,N_2(t)=0 )= t e^{-t}
\end{split}\end{equation}
because there is precisely one car at $1$ if and only if precisely
one car arrived conditioning on no cars at $0$ and $2$. This shows
that the last ODE is correct and concludes the proof of the lemma.$\hfill \Box$
The only remaining term whose time-evolution we need to consider is $D_t(0,1,0)$.
\begin{lem} $D_t(0,1,0)$ is a solution of the
differential equation
\begin{equation}\begin{split}
\frac{d}{dt}D_t(0,1,0)&= f_t(0)^2 e^{-t}- D_t(0,1,0) -2 R_t f_t(0)
e^{-t}-2 R_t f_t(1)e^{-t}\cr
\end{split}\end{equation}
\label{D010}
\end{lem}
{\bf Proof: } We note that
\begin{equation}\begin{split}
\frac{d}{dt}D_t(0,1,0)&= D_t(0,0,0) - D_t(0,1,0)-2 D_t(0,1,0,0)- 2
D_t(0,1,0,1)\cr \cr
\end{split}\end{equation}
The first term is for adding a car at the central site from the
vacuum, the second for adding a car at the central site at height
one. The last two terms are for adding a car to the right
of the central site. As we already know we have
\begin{equation}\begin{split}
D_t(0,0,0) = P_t(m_{1}=0 | N_0(t)=0)^2 e^{-t} \cr
\end{split}\end{equation}
Using conditioning on non-arrival at $2$ we get, by reflection
invariance
\begin{equation}\begin{split}
D_t(0,1,0,0) &= P_t(m_{1}=1, m_{2}=0 | N_0(t)=0) P_t(m_{1}=0 |
N_0(t)=0) e^{-t} \cr &= R_t f_t(0) e^{-t}
\end{split}\end{equation}
For the last term we get in the same way
\begin{equation}\begin{split}
D_t(0,1,0,1) &= P_t(m_{1}=1, m_{2}=0 | N_0(t)=0) P_t(m_{1}=1 |
N_0(t)=0) e^{-t} \cr &= R_t f_t(1) e^{-t}\cr
\end{split}\end{equation}
$\hfill \Box$\\
{\bf Proof of Theorem \ref{Theo1}: } Combining the results of lemma
\ref{D}, \ref{D&osq}, \ref{lem:osq} and \ref{D010} proves Theorem 1. $\hfill \Box$ \\
{\bf Proof of Theorem \ref{Theo2}: } The proof follows from Theorem
\ref{Theo1} by deleting every term that represents the possibility
of skipping a second line car to reach a void in the first line.
This results in deleting $f_t(2)e^{-t}$ from the first two equations
of \ref{lem:osq}, and $D_t(2,0,0)$ and $D_t(2,0,2)$ from \ref{D0}
and \ref{D1}. $\hfill \Box$
\begin{figure}[htp]
\includegraphics[height=5.5cm]{densitylayer1.pdf}
\includegraphics[height=5.5cm]{densitylayer2.pdf}
$\phantom{123}$
\hfill
\caption{Densities of the first and second line.} \label{fig:DD}
\end{figure}
\section{Conclusion}
We introduced two extensions of the classic parking problem to a
two-line model i.e. a model with screening and a model without
screening. For both models we derived closed systems of finite-dimensional
ODEs from which the time-evolution of the densities in the first and second line
can be obtained.
Interestingly, the numerical solution of the ODE shows that the final densities in the second line
are higher than those in the first line, for both models. The increase factor in the
model without screening is approximately
\[ I = \lim_{t \rightarrow \infty} \frac{D_t(2) + D_t(3)}{D_t(1) + D_t(3)}
\approx \frac{0.434868 }{0.432332 } \approx 1.006 \]
It is known by analytical computations \cite{Co62} that $D_t(1) + D_t(3)$ approaches $(1-e^{-2})/2\approx 0.432332$ for large $t$,
which provides a checkup for the numerics.
In the screening model we find
\[ I_{sc} = \lim_{t \rightarrow \infty} \frac{D_t^{sc}(2) + D_t^{sc}(3)}{D_t^{sc}(1) + D_t^{sc}(3)}
\approx \frac{0.433896}{0.366475} \approx
1.184 \] %
In other words, in both models the cars seem to exploit the resources in the second line in a
(slightly) more efficient way than in the first line.
\section*{Acknowledgements}
The authors thank Aernout van Enter and Herold Dehling for interesting discussions.
| {
"timestamp": "2008-11-28T12:15:51",
"yymm": "0811",
"arxiv_id": "0811.3599",
"language": "en",
"url": "https://arxiv.org/abs/0811.3599",
"abstract": "We consider two variations of the discrete car parking problem where at every vertex of the integers a car arrives with rate one, now allowing for parking in two lines. a) The car parks in the first line whenever the vertex and all of its nearest neighbors are not occupied yet. It can reach the first line if it is not obstructed by cars already parked in the second line (screening). b) The car parks according to the same rules, but parking in the first line can not be obstructed by parked cars in the second line (no screening). In both models, a car that can not park in the first line will attempt to park in the second line. If it is obstructed in the second line as well, the attempt is discarded. We show that both models are solvable in terms of finite-dimensional ODEs. We compare numerically the limits of first and second line densities, with time going to infinity. While it is not surprising that model a) exhibits an increase of the density in the second line from the first line, more remarkably this is also true for model b), albeit in a less pronounced way.",
"subjects": "Mathematical Physics (math-ph)",
"title": "A second row Parking Paradox",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808724687407,
"lm_q2_score": 0.7217432003123989,
"lm_q1q2_score": 0.7075110541006195
} |
https://arxiv.org/abs/2012.08465 | Neural Collapse with Cross-Entropy Loss | We consider the variational problem of cross-entropy loss with $n$ feature vectors on a unit hypersphere in $\mathbb{R}^d$. We prove that when $d \geq n - 1$, the global minimum is given by the simplex equiangular tight frame, which justifies the neural collapse behavior. We also prove that as $n \rightarrow \infty$ with fixed $d$, the minimizing points will distribute uniformly on the hypersphere and show a connection with the frame potential of Benedetto & Fickus. | \section{Introduction and Results}
\subsection{Introduction}
We consider the following variational problem
\begin{equation}\label{eq:symvarp}
\min_{u} \mc{L}_\alpha(u) := \min_{u}\; \sum_{i=1}^n \log \Bigl( 1 + \sum_{j=1 \atop j\neq i }^n e^{\alpha (\average{u_j, u_i} - 1)} \Bigr) .
\end{equation}
where $\alpha >0 $ is a parameter and for $i = 1, \ldots, n$,
$u_i \in \mathbb{R}^d$ such that $\|u_i\| = 1$. Here and in the sequel, we
use $\|\cdot\|$ to denote the Euclidean norm of a vector, and
$\average{\cdot, \cdot}$ denotes the Euclidean inner product. The
question we would like to address in this note is the solution
structure of such variational problems. The problem has several
motivations from some recent works in the literature of machine
learning.\\
Our main motivation comes from the very nice recent paper
\cite{papyan2020prevalence}. In that paper, the authors proposed and
studied the neural collapse behavior of training of deep neural
networks for classification problems. Following the work
\cite{papyan2020prevalence} by choosing a cross-entropy loss, while
taking unconstrained features (i.e., not parametrized by some
nonlinear functions like neural networks) to be vectors on the unit sphere in $\mathbb{R}^d$, this
amounts to the study of the variational problem
\begin{equation}\label{eq:varp}
\min_{u, v} \mc{L}(u, v) := \min_{u, v}\; \sum_{i=1}^n \log \Biggl( \frac{\sum_{j=1}^n e^{\average{v_j, u_i}}}{e^{\average{v_i, u_i}}}\Biggr).
\end{equation}
where $u_i, v_i \in \mathbb{R}^d$ such that $\|u_i\| = 1$ for each $i$. Note
that the model in \cite{papyan2020prevalence} also contains a bias
vector, so that $\average{v_j, u_i}$ in \eqref{eq:varp} is replaced by
$\average{v_j, u_i} + b_j$, for $b \in \mathbb{R}^n$. We drop the bias to
remove some degeneracy of the problem for simplicity. Another, more
crucial, difference is that in actual deep learning, as considered in~\cite{papyan2020prevalence}, the feature vectors $u_i$ are given by
output of deep neural networks acting on the input data, this would
make the variational problem much harder to analyze and thus we will
only study the simplified scenario.
\smallskip
\subsection{Equiangular tight frame as minimizer.}
The connection between the two variational problems is evident, as
\eqref{eq:symvarp} can be viewed as a symmetric version of
\eqref{eq:varp}: In particular, if we choose $v_i = \alpha u_i$ for
some parameter $\alpha > 0$, then
$\mc{L}(\alpha u, u) = \mc{L}_{\alpha}(u)$. In fact, we will prove
that the minimum of \eqref{eq:varp} is indeed achieved by such
symmetric solutions. There is a small caveat though as one can take
the norm of $v$ to infinity (or $\alpha \to \infty$ for the symmetric
problem) to reduce the loss. Thus, in order to characterize better the
solution structure, we will consider the problem for a fixed scaling
of $v$, and in fact $\|v_i\| = 1$ (the case of $\|v_i\| = \alpha$
will be discussed below). We show that the solution of
the variational problem is given by a simplex equiangular tight frame
(ETF). This proves the neural collapse behavior for
\eqref{eq:varp}, which provides some justification to the observation
of such behavior in deep learning.
\begin{theorem}\label{thm:crossentropy}
Consider the variational problem
\begin{equation*}
\begin{aligned}
& \min_{u, v} \mc{L}(u, v) \\
& \text{such that } u_i, v_i \in \mathbb{R}^d, \, \|u_i\| = \|v_i\|
= 1, \quad i =1, \cdots, n.
\end{aligned}
\end{equation*}
If $d \geq n-1$, the global minimum of the problem corresponds to
the case where $\{u_i\}_{i=1}^{n}$ form a simplex equiangular tight frame and $u_i = v_i$ for all
$i = 1, \cdots, n$.
\end{theorem}
We remark that similar results have been proved for different loss
functions: for a large deviation type loss function in
\cite{papyan2020prevalence} and for a $L^2$-loss function in
\cite{mixon2020neural}, both for models with unconstrained feature
vectors (i.e., without neural network parametrization of $u_i$'s).
After making the first version of our paper available, we were informed that an
equivalent result has been obtained in \cite{ewojtowytsch}, based on a
quite different proof.
\smallskip
In Theorem~\ref{thm:crossentropy}, the restriction of the scale of
$\|v_i\| = 1$ does not in fact sacrifice generality. As we will
comment towards the end of the proof, if instead $\|v_i\| \leq \alpha$
is assumed, the solution would be given by $v_i = \alpha u_i$. This is
related to the following result for the symmetric problem
\eqref{eq:symvarp}.
\begin{theorem}\label{thm:symvarp}
Consider the variational problem
\begin{equation*}
\begin{aligned}
& \min_{u} \mc{L}_{\alpha}(u) \\
& \text{such that } u_i \in \mathbb{R}^d, \, \|u_i\|
= 1, \quad i =1, \cdots, n.
\end{aligned}
\end{equation*}
If $d \geq n-1$, then for any $\alpha > 0$, the global minimum of
the problem corresponds to the case where $\{u_i\}_{i=1}^{n}$ form a simplex equiangular tight frame.
\end{theorem}
We prove Theorem~\ref{thm:symvarp} first in
Section~\ref{sec:symmetric}; the idea of the proof extends to that of
Theorem~\ref{thm:crossentropy}, which will be presented in
Section~\ref{sec:crossentropy}.
\subsection{More vectors than dimensions.} In the above results, the assumption that $p \geq n-1$ (or
equivalently $n \leq p + 1$) is crucial as only then it is possible to
place $n$ vectors on the unit sphere in $\mathbb{\mathbb{R}}^p$ such that
these vectors form a simplex equiangular tight frame. It is natural to ask what happens
when $n \geq p + 2$. In this case, \eqref{eq:symvarp} is related to
loss functions used in unsupervised learning and self-supervised
learning, such as those used in Siamese networks
\cite{chopra2005learning} and word2vec \cite{ariel,mikolov2013efficient}. In
particular, the spherical contrastive loss considered in
\cite{wang2020understanding}, for which the goal is to embed many
points on a hypersphere such that the points are ``uniformly
distributed'', coincides with \eqref{eq:symvarp} when unconstrained
feature vectors are used.\\
For general $n \geq p + 2$, the study of the structure of the
minimizer seems difficult, but in the asymptotic regime
$n \to \infty$, we have the following theorem states that indeed the
points will uniformly distributed on the sphere.
\begin{theorem}\label{thm:uniform}
Consider the variational problem
\begin{equation*}
\begin{aligned}
& \min_{u} \mc{L}_{\alpha}(u) \\
& \text{such that } u_i \in \mathbb{R}^d, \, \|u_i\| = 1, \quad i =1,
\cdots, n.
\end{aligned}
\end{equation*}
Let $\mu_n$ be the probability measure on $\SS^d$ generated by a
minimizer
\begin{equation*}
\mu_n = \frac{1}{n} \sum_{i=1}^n \delta_{u_i},
\end{equation*}
then for any $\alpha > 0$, as $n \to \infty$, $\mu_n$
converges weakly to the uniform measure on $\SS^d$.
\end{theorem}
The proof of theorem will be presented in
Section~\ref{sec:uniform}. It uses the following proposition which
characterizes the minimizer for a relaxed version of the variational
problem defined for probability measures on $\SS^d$. With some abuse
of notation, for $\mu \in \mathcal{P}(\SS^d)$, we denote
\begin{equation}
\mc{L}_{\alpha}(\mu) := \int_{\SS^d} \log \left( \int_{\SS^d}
e^{\alpha ( \average{x, y} - 1)}
\mu(\mathrm{d} y)\right) \mu(\mathrm{d} x).
\end{equation}
It is easy to check that $\mc{L}_{\alpha}$ acting on
$\mu_n = \frac{1}{n} \sum_{i=1}^n \delta_{u_i}$ is equivalent (up to
some additive constants) to the objective function defined in
\eqref{eq:symvarp} evaluated at the point configuration $\{u_i\}$. The
following proposition states that the unique minimizer of
$\mc{L}_{\alpha}$ on $\mc{P}(\SS^d)$ is given by uniform probability
measure; this fact was established in \cite[Theorem
1]{wang2020understanding} using a rather different approach. Our
proof, deferred to Section~\ref{sec:uniform}, which is based on
variational arguments, seems simpler in comparison.
\begin{prop}\label{prop:uniformsphere}
The unique minimizer of the variational problem
\begin{equation*}
\inf_{\mu \in \mathcal{P}(\SS^d)} \mc{L}_{\alpha}(\mu)
\end{equation*}
is the uniform probability measure on $\SS^d$.
\end{prop}
One natural question is how quickly sets of $n$ points can have an energy that is comparable to the energy of the uniform measure. We will show that $n$ points can have an energy that is super-exponentially close to the energy of the flat distribution. Note first that
\begin{align*}
\mc{L}_{\alpha}(\mu) &= \int_{\SS^d} \log \left( \int_{\SS^d} e^{\alpha ( \average{x, y} - 1)} \mu(\mathrm{d} y)\right) \mu(\mathrm{d} x) \\
&= \int_{\SS^d} \log \left( e^{-\alpha} \int_{\SS^d} e^{\alpha \average{x, y} } \mu(\mathrm{d} y)\right) \mu(\mathrm{d} x) \\
&= - \alpha + \int_{\SS^d} \log \left( \int_{\SS^d} e^{\alpha \average{x, y} } \mu(\mathrm{d} y)\right) \mu(\mathrm{d} x)
\end{align*}
and that we can therefore study this equivalent but slightly more symmetric functional. In particular, when studying the case where $\mu$ is given by the sum of Dirac measure in a finite number of points, we can obtain an even more symmetric upper bound by applying Jensen's inequality
$$ \sum_{i =1}^{n} \log \left( \sum_{j=1}^{n} e^{\alpha \left\langle x_i, x_j \right\rangle} \right) \leq n \log\left( \frac{1}{n} \sum_{i, j=1}^{n} e^{\alpha \left\langle x_i, x_j \right\rangle} \right).$$
As it turns out, the particular structure of this upper bound allows us to obtain very precise bounds for the minimal energy of optimal configurations.
\begin{theorem} \label{upper} Let $\alpha > 0$ be fixed. For some $c_{\alpha, d} > 0$ and all $n$ sufficiently large, there exist sets of points $\bigl\{ x_1, \dots, x_n \bigr\} \subset \mathbb{S}^d$ for which
$$ 0 \leq \frac{1}{n^2} \sum_{i, j=1}^{n} e^{\alpha \left\langle x_i, x_j \right\rangle} - \frac{1}{|\mathbb{S}^d|^2} \int_{\mathbb{S}^d \times \mathbb{S}^d} e^{\alpha \left\langle x, y \right\rangle} dx dy \leq e^{- c_{\alpha,d} n \log{n}}.$$
\end{theorem}
This type of extremely rapid convergence might be an indicator that
the actual convergence for $\mathcal{L}_{\alpha}$ does indeed happen from below: our use of
Jensen's inequality applied to a concave function (the logarithm) in the
proof is a further indicator.
\subsection{The Frame Potential.}
We conclude with a simple observation: for $\alpha \rightarrow 0^+$, the functional $\mathcal{L}_{\alpha}$
has a Taylor expansion with quite excellent properties.
\begin{prop}\label{prop:frame} For a fixed set of point $\left\{u_1, \dots, u_n \right\} \subset \mathbb{S}^d$, we have, as $\alpha \rightarrow 0$,
\begin{align*} \sum_{i=1}^{n} \log \left( \sum_{j=1}^{n} e^{\alpha \left\langle u_i, u_j \right\rangle}\right) &= n\log{n} + \frac{\alpha}{n} \left\| \sum_{i=1}^{n}u_i \right\|^2 + \frac{\alpha^2}{2n} \sum_{i,j=1}^n \left\langle u_i, u_j\right\rangle^2 \\
&\qquad - \frac{\alpha^2}{2n} \sum_{i=1}^{n} \left\langle u_i, \sum_{j=1}^{n} u_j \right\rangle^2 + \mathcal{O}(\alpha^3).
\end{align*}
\end{prop}
This expansion has an interesting property: if $\alpha$ is quite small, then the linear term dominates and minimizers of the energy functional will be forced to have $\| \sum_{i=1}^{n}u_i \|$ quite small or possibly even 0.
This has implications for the third term which will then also be small.
As such we would expect that there is an emerging effective energy given by
$$ E(u_1, \dots, u_n) = \frac{\alpha}{n} \left\| \sum_{i=1}^{n}u_i \right\|^2 + \frac{\alpha^2}{2n} \sum_{i,j=1}^n \left\langle u_i, u_j\right\rangle^2.$$
This object function, however, is strongly connected to the
\textit{frame potential}
$$ F(u_1, \dots, u_n) = \sum_{i,j=1}^n \left\langle u_i, u_j\right\rangle^2.$$
The frame potential was introduced in the seminal work of Benedetto \& Fickus \cite{benedetto} and has since played an important role in frame theory. What is utterly remarkable is that the Frame Potential has a large number of highly structured minimizers (see, for example, Fig.~1). As shown by Benedetto \& Fickus, for any
$ \left\{u_1, \dots, u_n \right\} \subset \mathbb{S}^{d-1}$
$$ F(u_1, \dots, u_n) = \sum_{i,j=1}^n \left\langle u_i, u_j\right\rangle^2 \geq \frac{n^2}{d}$$
with equality if and only if the set of points form a tight frame, i.e. if
$$ \forall~u \in \mathbb{R}^d: \quad \sum_{i=1}^{n} \left\langle u,u_i \right\rangle^2 = \frac{n}{d}\left\| u \right\|^2.$$
\begin{center}
\begin{figure}[h!]
\includegraphics[width=0.3\textwidth]{pic1.pdf}
\caption{The 72 vertices of the Dodecahedron-Icosahedron compound form
a unit norm tight frame of $\mathbb{R}^3$. The point configuration is also a global
minimizer of the frame potential.}
\end{figure}
\end{center}
In fact, our effective energy $E(u_1, \dots, u_n)$ may be understood
as the frame potential with an additional strong incentive for the
point configuration to have mean value 0. It would be interesting to
have a better understanding whether $\mathcal{L}_{\alpha}$ inherits
some of the good properties of the Frame Potential for $\alpha$
small, for example, whether it is possible to say anything about minimal energy
configurations of $\mathcal{L}_{\alpha}$ when $n \geq d$ but not going
to $\infty$.
\medskip
\section{Proof for Theorem~\ref{thm:symvarp}}\label{sec:symmetric}
\begin{proof}
Recall the variational problem under consideration
\begin{equation*}
\min_u \mc{L}_{\alpha}(u) := \min_u \sum_{i=1}^n \log \Bigl(1 + \sum_{j=1 \atop j\neq i}^n e^{\alpha (\average{u_j, u_i}-1)} \Bigr).
\end{equation*}
Using Jensen's inequality, we have, for any fixed $1 \leq i \leq n$,
\begin{equation}\label{ineq:Jensen1}
\begin{aligned}
\frac{1}{n-1} \sum_{j=1 \atop j\neq i}^n e^{\alpha \average{u_j, u_i}} & \geq \exp \Bigl( \frac{1}{n-1} \sum_{j=1 \atop j\neq i}^n \alpha \average{u_j, u_i} \Bigr) \\
& = \exp \Bigl( \frac{\alpha}{n-1} \bigl(\average{U, u_i} - 1\bigr) \Bigr),
\end{aligned}
\end{equation}
where we introduce the sum
\begin{equation*}
U = \sum_{i=1}^n u_i.
\end{equation*}
Thus, since the logarithm is monotone,
\begin{equation}\label{ineq:1}
\begin{aligned}
\mc{L}_{\alpha}(u) & = \sum_{i=1}^n \log \Bigl( 1 + e^{-\alpha} \sum_{j=1 \atop j\neq i}^n e^{\alpha \average{u_j, u_i}} \Bigr) \\
& \geq \sum_{i=1}^n \log \Bigl(1 + (n-1) e^{-\frac{n\alpha}{n-1}}
e^{\frac{\alpha}{n-1} \average{U, u_i}} \Bigr).
\end{aligned}
\end{equation}
Note that for any $a, b > 0$, the function
$t \mapsto \log(1 + a e^{bt})$ is convex, applying Jensen's inequality again, we have
\begin{equation}\label{ineq:Jensen2}
\begin{aligned}
\frac{1}{n} \mc{L}_{\alpha}(u) &\geq
\frac{1}{n} \sum_{i=1}^n \log \left[1 + (n-1)
e^{-\frac{n\alpha}{n-1}} \exp \left(\frac{\alpha}{n-1} \average{U, u_i}\right)
\right] \\
& \geq \log \left[ 1 + (n-1)
e^{-\frac{n\alpha}{n-1}} \exp\left(\frac{\alpha}{n-1} \frac{1}{n} \sum_{i=1}^{n} \average{U, u_i}\right)
\right] \\
& = \log \left[1 + (n-1)
e^{-\frac{n\alpha}{n-1}} \exp\left(\frac{\alpha}{n-1} \frac{1}{n} \|U\|^2\right) \right] \\
& \geq \log \Bigl(1 + (n-1)
e^{-\frac{n\alpha}{n-1}} \Bigr),
\end{aligned}
\end{equation}
Therefore, we arrive at
\begin{equation}
\begin{aligned}
\mc{L}_{\alpha}(u) \geq n \log \Bigl(1 + (n-1)
e^{-\frac{n\alpha}{n-1}} \Bigr).
\end{aligned}
\end{equation}
To see when the minimum is achieved, note that in
\eqref{ineq:Jensen1}, due to strict convexity of the exponential function, equality only holds when, for $j \neq i$,
$$ \left\langle u_j, u_i \right\rangle = c_i \qquad \mbox{is independent of}~j.$$
Equality in \eqref{ineq:Jensen2} only holds if
$$ \left\langle U, u_i \right\rangle = c\qquad \mbox{independent of}~i.$$
Finally, inequality in the last part of \eqref{ineq:Jensen2} only holds when $U = 0$.
If $U=0$, then
$$ c = \left\langle U, u_i \right\rangle = \left\langle \sum_{j=1}^{n} u_j, u_i \right\rangle = 1 + \left\langle \sum_{j=1 \atop j \neq i}^{n} u_j, u_i \right\rangle = 1 + (n-1)c_i.$$
This shows that $c_i$ does not actually depend on $i$ and thus
$\average{u_i, u_j} \equiv c$ for some constant $c$ whenver $i$ is
different from $j$. Thus, we conclude
\begin{equation}
0 = \left\| \sum_{i=1}^{n} u_i \right\|^2 = n + \sum_{i\neq j} \average{u_i, u_j} = n + n (n-1) c.
\end{equation}
This implies that
$$\average{u_i, u_j} = -\frac{1}{n-1}.$$
Therefore,
the global minimum is achieved if and only if when $\{u_i\}_{i=1}^{n}$ form a simplex equiangular tight frame
(recall $d \geq n-1$ so it is achievable).
\end{proof}
\section{Proof of
Theorem~\ref{thm:crossentropy}}\label{sec:crossentropy}
\begin{proof}
The proof follows along similar lines as the proof of Theorem~\ref{thm:symvarp}. Recall
\begin{equation*}
\mc{L}(u, v) = \sum_{i=1}^n \log \Biggl(
\frac{\sum_{j=1}^n e^{\average{v_j, u_i} }}{e^{\average{v_i, u_i}}} \Biggr)= \sum_{i=1}^n \log \Biggl( 1 +
\sum_{j=1 \atop j \neq i }^n e^{\average{v_j - v_i, u_i} }
\Biggr)
\end{equation*}
Applying Jensen's inequality, we have, for fixed $1 \leq i \leq n$,
\begin{equation}\label{ineq:Jensen3}
\begin{aligned}
\sum_{j=1 \atop j\neq i}^n e^{\average{v_j - v_i, u_i}} &= e^{-\average{v_i, u_i }} \sum_{j=1 \atop j\neq i}^n e^{\average{v_j , u_i}}
\\
&\geq (n-1) e^{-\average{v_i, u_i }} \exp \Bigl(\frac{1}{n-1} \sum_{j=1 \atop j\neq i}^n \average{v_j, u_i} \Bigr) \\
& = (n-1) e^{-\average{v_i, u_i }} \exp \left(\frac{ \average{V, u_i} - \average{v_i, u_i} }{n-1} \right) \\
& = (n-1) \exp \left(\frac{ \average{V, u_i} - n \average{v_i, u_i} }{n-1} \right),
\end{aligned}
\end{equation}
where we denote the sum of $v_i$ as
\begin{equation*}
V = \sum_{i=1}^n v_i.
\end{equation*}
Thus, using the monotonicity of logarithm,
\begin{equation}
\begin{aligned}
\mc{L}(u, v) & = \sum_{i=1}^n \log \Bigl(1 + \sum_{j=1 \atop j\neq i}^n e^{\average{v_j- v_i, u_i} } \Bigr) \\
& \geq \sum_{i=1}^n \log \left[ 1 + (n-1) \exp\left( \frac{ \average{V, u_i} }{n-1}
- \frac{n}{n-1} \average{v_i, u_i} \right) \right]
\end{aligned}
\end{equation}
Applying Jensen's inequality to the convex function $t \mapsto \log(1 + a e^{bt})$ for $a, b > 0$, we have
\begin{equation}\label{ineq:Jensen4}
\begin{aligned}
\mc{L}(u, v) & \geq n \log \left[1 + (n-1) \exp\left( \frac{1}{n} \sum_{i=1}^{n} \left(\frac{\average{V, u_i}}{n-1} - \frac{n}{n-1} \average{v_i, u_i} \right) \right) \right] \\
& = n \log \left[1 + (n-1) \exp\left( \frac{1}{n} \left( \frac{\average{V, U} }{n-1} - \frac{n}{n-1} \sum_{i=1}^{n} \average{v_i, u_i} \right) \right) \right],
\end{aligned}
\end{equation}
where $U = \sum_i u_i$.
For the equalities to hold in the above inequalities
\eqref{ineq:Jensen3} and \eqref{ineq:Jensen4}, we require for some
constants $c_i$ and $c$ such that
\begin{align} \label{cond1}
& \average{v_j, u_i} = c_i, \qquad \forall\, j \neq i
\end{align}
and
\begin{align} \label{cond2}
& \frac{\average{V, u_i}}{n-1} - \frac{n}{n-1}\average{v_i, u_i} = c, \qquad \forall\, i.
\end{align}
Therefore, in order to find a lower bound on $\mathcal{L}$, we have to solve
$$ \frac{\average{V, U} }{n-1} - \frac{n}{n-1} \sum_{i=1}^{n} \average{v_i, u_i} \rightarrow \min.$$
If there is a minimizing configuration of this simpler problem that also satisfies \eqref{cond1} and \eqref{cond2}, then all
inequalities are actually equalities.
The above variational problem is equivalent to maximizing
\begin{equation}\label{eq:maxprob}
n \sum_{i=1}^{n} \average{v_i, u_i} - \left\langle \sum_{i=1}^{n} v_i, \sum_{i=1}^{n} u_i\right\rangle = \vec{v}^{\top} \bigl((n \mathbb{I}_n - \bd{1}_n\bd{1}_n^{\top}) \otimes \mathbb{I}_d \bigr) \vec{u},
\end{equation}
where $\otimes$ denotes the Kronecker product, $\mathbb{I}_d$ ($\mathbb{I}_n$) denotes a $d \times d$ ($n \times n$) identity matrix,
$\bd{1}_n$ denotes an all-$1$ $n$-vector, $\vec{u}$ denotes a long
$\mathbb{R}^{nd}$ column vector formed by concatinating $u_i \in \mathbb{R}^d$ for
$i = 1, \cdots, n$, and similarly for $\vec{v}$. We note that, being the concatenation of unit vectors, $\| \vec{u}\| = \sqrt{n} = \| \vec{v}\|.$\\
The eigenvalues of a Kronecker product $A \otimes B$ are given by $\lambda_i \mu_j$, where
$\lambda_i$ are the eigenvalues of $A$ and $\mu_j$ are the eigenvalues of $B$. The matrix
$n \mathbb{I}_n - \bd{1}_n\bd{1}_n^{\top}$ is acting like $n \mathbb{Id}_n$ on vectors having
mean value 0 while sending the constant vector to 0. Its spectrum is thus given by $n$ (with multiplicity
$n-1$) and 0.
It follows that $ \bigl((n \mathbb{I}_n - \bd{1}_n\bd{1}_n^{\top}) \otimes
\mathbb{I}_d \bigr)$ is symmetric, its largest eigenvalue is $n$ and its smallest eigenvalue is 0. Recalling that $\| \vec{u}\| = \sqrt{n} = \| \vec{v}\|,$ we
have that (without constraints \eqref{cond1} and \eqref{cond2})
\begin{equation} \label{universal}
n \sum_{i=1}^{n} \average{v_i, u_i} - \left\langle \sum_{i=1}^{n} v_i, \sum_{i=1}^{n} u_i\right\rangle \leq n^2.
\end{equation}
However, setting $\vec{u}$ to be the simplex and $\vec{v} = \vec{u}$, we see that $U = 0 = V$ and we have
equality in \eqref{universal} while simultaneously satisfying the constraints \eqref{cond1} and
\eqref{cond2} with
$$c_i = -\frac{1}{n-1} \qquad \mbox{and} \qquad c = - \frac{n}{n-1}.$$
We will now argue that this is the only extremal example. Using the Spectral Theorem, we see that equality in \eqref{universal}
can only occur if $\vec{u}$ is an eigenvector of the matrix corresponding to the eigenvalue $n$. In that case, we have
$$ \left\langle \vec{v}, \bigl((n \mathbb{I}_n - \bd{1}_n\bd{1}_n^{\top}) \otimes \mathbb{I}_d \bigr) \vec{u} \right\rangle = n \left\langle \vec{v}, \vec{u} \right\rangle \leq n \| \vec{v} \| \| \vec{u}\| \leq n^2.$$
We have equality in Cauchy-Schwarz if and only if $\vec{v} = \lambda \vec{u}$ for some $\lambda > 0$. For
$\alpha =1$, this implies that $\vec{v} = \vec{u}$ and we are back in the symmetric case and can argue as in the proof of Theorem 2. Conditions \eqref{cond1} and \eqref{cond2} simplify to
\begin{align*}
\average{u_j, u_i} = c_i, ~ \forall\, j \neq i \qquad \mbox{and} \qquad \frac{\average{U, u_i}}{n-1} - \frac{n}{n-1}\average{u_i, u_i} = c, ~ \forall\, i.
\end{align*}
Moreover, by summing over the second condition, we see that we want to minimize
$$ c \cdot n = \frac{\average{U, U} }{n-1} - \frac{n}{n-1} \sum_{i=1}^{n} \average{u_i, u_i} \rightarrow \min.$$
We are thus interested in minimizing $c$ which is given by
$$ c = \frac{\average{U, u_i}}{n-1} - \frac{n}{n-1}\average{u_i, u_i} = c_i - \left\langle u_i, u_i \right\rangle \geq c_i -1.$$
However, for any set of $n$ unit vectors, the largest inner product between any pair of distinct vectors satisfies
$$ 0 \leq \left\| \sum_{i=1}^{n} u_i \right\|^2 = n + n(n-1) \max_{i \neq j} \average{u_i, u_j}.$$
Thus
$$ c \geq \max_{1 \leq i \leq n} c_i -1 \geq - \frac{1}{n-1} - 1 = - \frac{n}{n-1}.$$
We see that equality is achieved for the simplex. Moreover, in the case of equality, we have to require that $\average{u_i, u_j} = -1/(n-1)$ for any pair of distinct vectors and this characterizes the simplex.
Moreover, if $\|v_i\| \leq \alpha$, then it is easy
to see that the maximum is achieved when $\vec{v} = \alpha \vec{u}$,
we can again conclude using Theorem~\ref{thm:symvarp}.
\end{proof}
\section{Proof of Theorem~\ref{thm:uniform}}\label{sec:uniform}
We first prove Proposition~\ref{prop:uniformsphere};
Theorem~\ref{thm:uniform} then follows from a limiting argument.
\begin{proof}[Proof of Proposition~\ref{prop:uniformsphere}]
Without loss of generality, we assume $\alpha = 1$ in the proof, and note that
the energy functional can be rewritten as
\begin{equation*}
\begin{aligned}
\mc{L}(\mu) & = \int_{\SS^d} \log \left( \int_{\SS^d} e^{\average
{x, y} - 1} \mu(\mathrm{d} y)\right) \mu(\mathrm{d} x) \\
& = \int_{\SS^d} \log \left( \int_{\SS^d} e^{\average{x, y} -
(\abs{x}^2 + \abs{y}^2)/2}
\mu(\mathrm{d} y)\right) \mu(\mathrm{d} x) \\
& = \int_{\SS^d} \log \left( \int_{\SS^d} e^{- \abs{x - y}^2/2}
\mu(\mathrm{d} y)\right) \mu(\mathrm{d} x).
\end{aligned}
\end{equation*}
Using the Gaussian convolution identity
\begin{equation*}
e^{-\abs{x - y}^2/2} = \frac{1}{(8\pi)^{d/2}} \int_{\mathbb{R}^d} e^{-\abs{x - z}^2 / 4} e^{-\abs{z - y}^2 / 4} \,\mathrm{d} z
\end{equation*}
and the Jensen's inequality, we have
\begin{equation*}
\begin{aligned}
\mc{L}(\mu) & = \int_{\SS^d} \log \left( \int_{\SS^d} \frac{1}{(8\pi)^{d/2}} \int_{\mathbb{R}^d} e^{-\abs{x - z}^2 / 4} e^{-\abs{z - y}^2 / 4} \,\mathrm{d} z \mu(\mathrm{d} y) \right) \mu(\mathrm{d} x) \\
& = \int_{\SS^d} \log \left( \int_{\SS^d} \frac{1}{(4\pi)^{d/2}} \int_{\mathbb{R}^d} e^{-\abs{x - z}^2 / 4} e^{-\abs{z - y}^2 / 4} \,\mathrm{d} z \mu(\mathrm{d} y) \right) \mu(\mathrm{d} x) - \frac{d}{2} \log 2 \\
& \geq \frac{1}{(4\pi)^{d/2}} \int_{\mathbb{R}^d} \int_{\SS^d} e^{-\abs{x - z}^2 / 4}\log \left( \int_{\SS^d} e^{-\abs{z - y}^2 / 4} \mu(\mathrm{d} y) \right) \mu(\mathrm{d} x) \,\mathrm{d} z - \frac{d}{2} \log 2 \\
& = \frac{1}{(4\pi)^{d/2}} \int_{\mathbb{R}^d} \left( \int_{\SS^d} e^{-\abs{x - z}^2 / 4}
\mu(\mathrm{d} x) \right) \log \left( \int_{\SS^d} e^{-\abs{z - y}^2 /
4} \mu(\mathrm{d} y) \right) \,\mathrm{d} z - \frac{d}{2} \log 2.
\end{aligned}
\end{equation*}
The above calculation shows that minimizing $\mc{L}(\mu)$ is equivalent to minimizing
\begin{equation*}
\mc{G}(\mu) := \int_{\mathbb{R}^d} f_{\mu}(z) \log f_{\mu}(z) \,\mathrm{d} z,
\end{equation*}
where we define the short-hand
\begin{equation*}
f_{\mu}(z) := \int_{\SS^d} e^{-\abs{x - z}^2 / 4} \mu(\mathrm{d} x).
\end{equation*}
Let us write the integral of $\mc{G}(\mu)$ in spherical coordinates
and get
\begin{equation*}
\mc{G}(\mu) = \int_0^{\infty} \left( \frac{1}{\abs{\SS^d}} \int_{\SS^d} f_{\mu}(r, \theta) \log f_{\mu}(r, \theta) \,\mathrm{d} \theta \right) r^{d-1} \,\mathrm{d} r.
\end{equation*}
\begin{lemma}
For any $r$, the integral
\begin{equation*}
\int_{\SS^d} f_{\mu}(r, \theta) \,\mathrm{d} \theta
\end{equation*}
is independent of $\mu \in \mathcal{P}(\SS^d)$.
\end{lemma}
\begin{proof}
By definition, we have (letting $z = (r, \theta)$ in the spherical coordinates)
\begin{equation*}
\begin{aligned}
\int_{\SS^d} f_{\mu}(r, \theta) \,\mathrm{d} \theta & = \int_{\SS^d} \int_{\SS^d} e^{-\abs{x - z}^2 / 4} \mu(\mathrm{d} x) \,\mathrm{d} \theta \\
& = \int_{\SS^d} \int_{\SS^d} e^{-\abs{x - z}^2 / 4} \,\mathrm{d} \theta \mu(\mathrm{d} x) \\
& = \int_{\SS^d} m(r) \mu(\mathrm{d} x) = m(r),
\end{aligned}
\end{equation*}
where
\begin{equation}\label{eq:defm}
m(r) := \int_{\SS^d} e^{-\abs{x - z}^2 / 4} \,\mathrm{d} \theta,
\end{equation}
which is independent of $x$ due to the spherical symmetry, and hence
is only a function of $r$.
\end{proof}
Now for each fixed $r$, we can consider the variational problem
\begin{equation*}
\begin{aligned}
& \arg \inf_{\mu} \; \mc{G}_r(\mu) := \int_{\SS^d} f_{\mu}(r, \theta) \log f_{\mu}(r, \theta) \,\mathrm{d} \theta\\
& \text{s.t.} \; \int f_{\mu}(r, \theta) = m(r),
\end{aligned}
\end{equation*}
where $m(r)$ is defined in \eqref{eq:defm}.
We note that $\mc{G}_r(\mu)$ in terms of $f_{\mu}(r, \cdot)$ is just
the entropy functional, which is strongly convex and is minimized if
and only if $f_{\mu}(r, \cdot)$ is uniform on $\SS^d$, which is
equivalent to the uniformity of $\mu$.
Since $\mc{G}(\mu) = \int_0^{\infty} \mc{G}_r(\mu) r^{d-1} \,\mathrm{d} r$ is a
positive linear combination of the energy functional $\mc{G}_r(\mu)$,
we conclude that the global minimum of $\mc{G}(\mu)$ and hence
$\mc{L}(\mu)$ corresponds to uniform probability distribution on $\SS^d$.
\end{proof}
We are now ready to prove Theorem~\ref{thm:uniform}.
\begin{proof}[Proof of Theorem~\ref{thm:uniform}]
Let $\mu_n$ be a sequence of probability measures corresponding to
minimizers for $n = 1, 2, \cdots$. Since $\SS^d$ is compact, it
suffices to prove that any weakly convergent subsequence $\mu_{n_k}$
would converge to the uniform measure. Denote $\mu$ the limit, and
define
\begin{equation*}
h_{n_k}(x) := \log \left( \int_{\SS^d} e^{-\abs{x-y}^2/2} \mu_{n_k}(\mathrm{d} y) \right),
\end{equation*}
and
\begin{equation*}
h(x) := \log \left( \int_{\SS^d} e^{-\abs{x-y}^2/2} \mu(\mathrm{d} y) \right).
\end{equation*}
As $e^{-\abs{x-y}^2/2}$ is a smooth function in $x$ and $y$ and is
bounded from below by $e^{-2}$ for $x, y \in \SS^d$,
$\mu_{n_k} \rightharpoonup \mu$ implies that $h_{n_k}$ converges to
$h(x)$ uniformly on $\SS^d$, and hence
\begin{equation*}
\mc{L}(\mu_{n_k}) = \int_{\SS^d} h_{n_k}(x) \mu_{n_k}(\mathrm{d} x) \to \int_{\SS^d} h(x) \mu(\mathrm{d} x) = \mc{L}(\mu).
\end{equation*}
Thus, the functional $\mc{L}$ is weakly continuous on
$\mc{P}(\SS^d)$. Since $\mu_{n_k}$ corresponds to a minimizer of the
variational problem for $n_k$ points, using the upper bound
Theorem~\ref{upper}, the limit $\mu$ minimizes $\mc{L}$ on
$\mc{P}(\SS^d)$, which implies by
Proposition~\ref{prop:uniformsphere} that $\mu$ is the uniform
probability measure on $\SS^d$. \end{proof}
\section{Proof of Theorem \ref{upper}}
\subsection{Outline.}
We start with Jensen's inequality: since the logarithm is concave, we obtain
$$ \sum_{i =1}^{n} \log \left( \sum_{j=1}^{n} e^{\alpha \left\langle x_i, x_j \right\rangle} \right) \leq n \log\left( \frac{1}{n} \sum_{i, j=1}^{n} e^{\alpha \left\langle x_i, x_j \right\rangle} \right).$$
For the rest of the proof, it suffices to understand this double sum. We will prove that there exists a sequence of positive $a_k > 0$ such that
$$ \sum_{i, j=1}^{n} e^{\alpha \left\langle x_i, x_j \right\rangle} = \sum_{k=0}^{\infty} a_k \left| \sum_{\ell = 1}^{n} \phi_k(x_{\ell})\right|^2,$$
where $\phi_k$ denotes the $k-$th spherical harmonic. Recalling that the $0-$th spherical harmonic is a constant normalized in $L^2$, we see that
$$ \phi_0(x) = \frac{1}{\sqrt{|\mathbb{S}^d|}}.$$
From this, we obtain
$$ \sum_{i, j=1}^{n} e^{\alpha \left\langle x_i, x_j \right\rangle} =\sum_{k=0}^{\infty} a_k \left| \sum_{\ell = 1}^{n} \phi_k(x_{\ell})\right|^2 \geq a_0 \frac{n^2}{|\mathbb{S}^d|}.$$
We can moreover determine the constant $a_0$: by plugging in randomly
chosen points (independently and identically distributed with respect to the uniform measure), we see that
$$ \lim_{n \rightarrow \infty} \frac{1}{n^2} \sum_{i, j=1}^{n} e^{\alpha \left\langle x_i, x_j \right\rangle} = \frac{1}{|\mathbb{S}^d|^2} \int_{\mathbb{S}^d \times \mathbb{S}^d} e^{\alpha \left\langle x, y \right\rangle} dx dy$$
while simultaneously
$$ \lim_{n \rightarrow \infty} \frac{1}{n^2} \sum_{k=0}^{\infty} a_k \left| \sum_{\ell = 1}^{n} \phi_k(x_{\ell})\right|^2 = \frac{a_0}{|\mathbb{S}^d|}$$
and therefore
$$ a_0 = \frac{1}{|\mathbb{S}^d|} \int_{\mathbb{S}^d \times \mathbb{S}^d} e^{\alpha \left\langle x, y \right\rangle} dx dy.$$
If we can prove that $a_k > 0$, then this would imply that
$$ \sum_{i, j=1}^{n} e^{\alpha \left\langle x_i, x_j \right\rangle} \geq \frac{n^2}{|\mathbb{S}^d|^2} \int_{\mathbb{S}^d \times \mathbb{S}^d} e^{\alpha \left\langle x, y \right\rangle} dx dy.$$
We will show that this is indeed the case and that one can find sets of $n$ points for which the expression is not much larger than that.
\subsection{The Expansion.}
We will now prove the desired expansion. We start by expanding the square
$$ \sum_{k=0}^{\infty} a_k \left| \sum_{\ell = 1}^{n} \phi_k(x_{\ell})\right|^2 = \sum_{k=0}^{\infty} a_k \sum_{i,j = 1}^{n} \phi_{k}(x_i) \phi_{k}( x_j).$$
At this point we start using a property of the sphere: by grouping spherical harmonics with respect to the Laplacian eigenvalue and prescribing that $a_k$ be constant for all spherical harmonics with the same eigenvalue, we can rewrite the sum as
$$ \sum_{k=0}^{\infty} a_k \sum_{i,j = 1}^{n} \phi_{\ell}(x_i) \phi_{\ell}( x_j) = \sum_{\ell =0}^{\infty} a_{\ell} \sum_{k \atop -\Delta \phi_k = \ell (\ell+1) \phi_k}\sum_{i,j = 1}^{n} \phi_{k}(x_i) \phi_{k}( x_j) .$$
However, on $\mathbb{S}^d$ the spherical harmonics are ordered in bands we have the addition formula valid for all $x, y \in \mathbb{S}^d$,
$$ \sum_{k \atop -\Delta \phi_{k} = \ell (\ell+1) \phi_{k}} \phi_{k}(x) \phi_{k}(y) = \frac{\ell+ \lambda}{\lambda} C_{k}^{\lambda} (\left\langle x, y\right\rangle),$$
where $C_{k}^{\lambda}$ are the Gegenbauer polynomials and
$$ \lambda = \frac{d-1}{2}.$$
This leads to
$$ \sum_{k=0}^{\infty} a_k \left| \sum_{\ell = 1}^{n} \phi_k(x_{\ell})\right|^2 = \sum_{\ell=0}^{\infty} a_{\ell} \frac{\ell + \lambda}{\lambda} \sum_{i,j=1}^{n} C_{\ell}^{(d-1)/2}\left( \left\langle x_i, x_j\right\rangle\right),$$
where, by an abuse of notation, we exploit that (by assumption) the $a_k$ coincide whenever the two spherical harmonics share the same Laplacian eigenvalue.
Using \cite[Prop. 2.2]{bilyk}, we see that if all the coefficients in the expansion
$$ e^{\alpha x} = \sum_{k=0}^{\infty} b_k C_k^{(d-1)/2}(x) \qquad \mbox{for}~x\in [-1,1]$$
are positive, then the function is positive on $\mathbb{S}^d$ and then \cite[Lemma 2.3]{bilyk} implies that the expansion converges uniformly. It suffices to show that $b_k > 0$. The Gegenbauer polynomials $C_k^{\alpha}$ are orthogonal on $[-1,1]$ with respect to the weight
$$ w(z) = (1-z^2)^{\alpha - \frac{1}{2}}.$$
Making an ansatz
$$ e^{\alpha x} = \sum_{k=0}^{\infty} b_k C_k^{(d-1)/2}(x),$$
we see that the coefficient $b_k$ is given by
$$ b_k = \frac{ \int_{-1}^{1} e^{\alpha x} (1-x^2)^{(d-2)/2} C_k^{(d-1)/2}(x) dx}{ \int_{-1}^{1} (1-x^2)^{(d-2)/2} \left(C_k^{(d-1)/2}(x)\right)^2 dx}.$$
The denominator has a closed form expression which we will abbreviate by $\alpha_1(k,d)$
$$ \alpha_1(k,d) = \int_{-1}^{1} (1-x^2)^{(d-2)/2} \left(C_k^{(d-1)/2}(x)\right)^2 dx = \frac{\pi 2^{2- d} \cdot \Gamma(k + d -1)}{k! \cdot (k+(d-1)/2) \cdot \Gamma\left(\frac{d-1}{2}\right)^2}.$$
It remains to understand the numerator.
Gegenbauer polynomials have a Rodrigues formula which is as follows:
$$ C_k^{(d-1)/2} = \frac{(-1)^k}{2^k \cdot k!} \frac{\Gamma(d/2) \cdot \Gamma(k + d-1)}{ \Gamma(d-1) \cdot \Gamma(d/2 + k)} (1-x^2)^{-(d-2)/2} \frac{d^k}{dx^k}\left[ (1-x^2)^{k + (d-2)/2}\right].$$
We will abbreviate the constant by $\alpha_2(k,d)$, i.e.
$$ C_k^{(d-1)/2}(x) = (-1)^k \alpha_2(k,d) (1-x^2)^{-(d-2)/2} \frac{d^k}{dx^k}\left[ (1-x^2)^{k + (d-2)/2}\right],$$
where
$$ \alpha_2(k,d) = \frac{1}{2^k \cdot k!} \frac{\Gamma(d/2) \cdot \Gamma(k + d-1)}{ \Gamma(d-1) \cdot \Gamma(d/2 + k)}.$$
Therefore
$$ \alpha_1(k,d) \cdot b_k = \alpha_2(k,d) \int_{-1}^{1} e^{\alpha x} (-1)^k \frac{d^k}{dx^k} \left[ (1-x^2)^{k+ (d-2)/2 } \right] dx.$$
It is easy to see that $b_0 > 0$. We now distinguish the cases $d=2$ and $d \geq 3$. Let us first assume that $d=2$.
We see that
$$ \frac{d^{\ell}}{dx^{\ell}} \left[ (1-x^2)^{k } \right] \big|_{x=-1,1} = 0 \qquad \mbox{for} \quad \ell=0,1,\dots,k-1.$$
We can thus use integration by parts and get
$$ \alpha_1(k,d) \cdot b_k = \alpha_2(k,d) \cdot \alpha^k \cdot \int_{-1}^{1} e^{\alpha x} \left[ (1-x^2)^{k+ (d-2)/2 } \right] dx > 0.$$
The same argument applies to $d \geq 3$. In that case we even have
$$ \frac{d^k}{dx^k} \left[ (1-x^2)^{k+ (d-2)/2 } \right] \big|_{x=-1,1} = 0$$
and can again integrate by parts $k$ times to obtain
$$ \alpha_1(k,d) \cdot b_k = \alpha_2(k,d) \cdot \alpha^k \int_{-1}^{1} e^{\alpha x} (1-x^2)^{k+ (d-2)/2 } dx > 0.$$
Altogether, we see that $b_k > 0$ for $\alpha > 0$ and thus there exists a sequence of positive $a_k$ such that
$$ \frac{1}{n^2} \sum_{i, j=1}^{n} e^{\alpha \left\langle x_i, x_j \right\rangle} = \sum_{k=0}^{\infty} a_k \left| \frac{1}{n} \sum_{\ell = 1}^{n} \phi_k(x_{\ell})\right|^2.$$
\subsection{Obtaining Quantitative Estimates.}
We will now go through the argument in the preceding section with the goal of getting quantitative estimates on $b_k$, where
$$ \alpha_1(k,d) \cdot b_k = \alpha_2(k,d) \cdot \alpha^k \cdot \int_{-1}^{1} e^{\alpha x} (1-x^2)^{k+ (d-2)/2 } dx > 0$$
and $\alpha_1(k,d)$ and $\alpha_2(k,d)$ are given above in closed form. We are interested in bounds from above, it therefore
suffices to estimate the integral. We see that the integral decays at a polynomial rate -- this is perhaps not all that relevant and
we bound very roughly
$$ \int_{-1}^{1} e^{\alpha x} (1-x^2)^{k+ (d-2)/2 } \leq 2 e^{\alpha}.$$
Therefore
$$ 0 \leq b_k \leq 2 e^{\alpha} \cdot \alpha^k \cdot \frac{\alpha_2(k,d)}{\alpha_1(k,d)}$$
We have, ignoring factors that depend solely on the dimension $d$,
\begin{align*}
\frac{\alpha_2(k,d)}{\alpha_1(k,d)} &= \frac{k! (k+(d-1)/2) \cdot \Gamma\left(\frac{d-1}{2}\right)^2}{\pi 2^{2- d} \cdot \Gamma(k + d -1)} \frac{1}{2^k \cdot k!} \frac{\Gamma(d/2) \cdot \Gamma(k + d-1)}{ \Gamma(d-1) \cdot \Gamma(d/2 + k)}\\
&\lesssim_d \frac{1}{2^k} \frac{ (k+(d-1)/2)}{ \Gamma(d/2 + k)}
\end{align*}
which decays faster than any exponential in $k$. We see that this is inherited by the coefficient $b_k$ which satisfies
$$ 0 \leq b_k \lesssim_d e^{\alpha} \cdot \frac{\alpha^k}{2^k} \frac{ (k+(d-1)/2)}{ \Gamma(d/2 + k)}.$$
We note that
$$ \lim_{k \rightarrow \infty} \log\left(\frac{1}{b_k} \right)\frac{1}{k \log{k}} =1.$$
\subsection{Proof of Theorem \ref{upper}}
The behavior of these coefficients, decaying faster than exponential
in $m$, has a number of interesting consequences. First and foremost,
it means that
$$ \sum_{k=0}^{\infty} a_k \left| \frac{1}{n} \sum_{\ell = 1}^{n} \phi_k(x_{\ell})\right|^2 \qquad \mbox{being small}$$
is really a statement about the distribution of the measure with respect to the first few spherical harmonics. Having a large error with regards to
some intermediate spherical harmonic is barely detectable -- in particular, deducing structural statements about the points via energy arguments
is presumably more difficult than it is for other kernels with slower decay in the coefficients. Another consequence is that we expect a fairly `flat' energy
landscape.
\begin{proof}[Proof of Theorem \ref{upper}] We use the representiation
$$ \frac{1}{n^2} \sum_{i, j=1}^{n} e^{\alpha \left\langle x_i, x_j \right\rangle} = \sum_{k=0}^{\infty} a_k \left| \frac{1}{n} \sum_{\ell = 1}^{n} \phi_k(x_{\ell})\right|^2$$
and the fact that the $k=0$ term corresponds to the integral. Thus
$$ \frac{1}{n^2} \sum_{i, j=1}^{n} e^{\alpha \left\langle x_i, x_j \right\rangle} - \frac{1}{|\mathbb{S}^d|^2} \int_{\mathbb{S}^d \times \mathbb{S}^d} e^{\alpha \left\langle x, y \right\rangle} dx dy = \sum_{k=1}^{\infty} a_k \left| \frac{1}{n} \sum_{\ell = 1}^{n} \phi_k(x_{\ell})\right|^2.$$
Now we pick the set $\left\{ x_1, \dots, x_n \right\} \subset \mathbb{S}^d$ to be an optimal spherical design: by a result of Bondarenko, Radchenko \& Viazovska \cite{bondarenko}, there exist $\left\{ x_1, \dots, x_n \right\} \subset \mathbb{S}^d$ such that the average of any polynomial of degree $\mbox{deg}(p) \leq c \cdot n^{1/d}$ evaluated in these points coincides with the global average of the polynomial on the sphere. In particular, the first $\sim_d n$ spherical harmonics are being evaluated exactly. Then, however,
$$ \sum_{k=1}^{\infty} a_k \left| \frac{1}{n} \sum_{\ell = 1}^{n} \phi_k(x_{\ell})\right|^2 = \sum_{k=c \cdot n}^{\infty} a_k \left| \frac{1}{n} \sum_{\ell = 1}^{n} \phi_k(x_{\ell})\right|^2.$$
We now use several rather crude bounds. We note that the $k-$th spherical harmonics has eigenvalue $\sim_d k^{2/d}$ (by Weyl's asymptotic) and use an old result of H\"ormander \cite{hor} to conclude that
$$ \| \phi_k\|_{L^{\infty}} \lesssim \lambda_k^{\frac{d-1}{4}} \sim k^{\frac{d-1}{2d}}.$$
This implies, for some $c_1 > 0$,
$$ \sum_{k=c \cdot n}^{\infty} a_k \left| \frac{1}{n} \sum_{\ell = 1}^{n} \phi_k(x_{\ell})\right|^2 \lesssim \sum_{k = c_1 \cdot n}^{\infty} a_k k^{\frac{d-1}{d}}.$$
By the previous result, we see that the superexponential decay of $a_k$ turns the sum essentially into its largest term and from this the desired bound follows.
\end{proof}
\section{Proof of Proposition~\ref{prop:frame}}
\begin{proof}
We are interested in asymptotics for
$$ \sum_{i=1}^{n} \log \biggl( \sum_{j=1}^{n} e^{\alpha \left\langle u_i, u_j \right\rangle}\biggr) \qquad \mbox{as}~\alpha \rightarrow 0.$$
We have the Taylor expansion
$$ \log{(1+x)} = x - \frac{x^2}{2} + \mathcal{O}(x^3)$$
and thus, as $\alpha \rightarrow 0$,
\begin{align*}
\sum_{i=1}^{n}\log \biggl( \sum_{j=1}^{n} e^{\alpha \left\langle u_i, u_j \right\rangle}\biggr) &= \sum_{i=1}^{n}\log \biggl( \sum_{j=1}^{n} 1 + \alpha \left\langle u_i, u_j \right\rangle + \frac{\alpha^2}{2}\left\langle u_i, u_j \right\rangle ^2 + \dots \biggr) \\
&= \sum_{i=1}^{n} \log \biggl( n \biggl( 1 + \frac{1}{n} \sum_{j=1}^{n} \alpha \left\langle u_i, u_j \right\rangle + \frac{\alpha^2}{2}\left\langle u_i, u_j \right\rangle ^2 + \dots \biggr) \biggr)\\
&= n \log{n} + \sum_{i=1}^{n} \log \biggl( 1 + \frac{1}{n} \sum_{j=1}^{n} \alpha \left\langle u_i, u_j \right\rangle + \frac{\alpha^2}{2}\left\langle u_i, u_j \right\rangle ^2 + \dots \biggr).
\end{align*}
Using the Taylor expansion of the logarithm and collecting all the terms that are constant, linear or quadratic in $\alpha$, we arrive at
\begin{align*}
\sum_{i=1}^{n}\log \biggl( \sum_{j=1}^{n} e^{\alpha \left\langle u_i, u_j \right\rangle}\biggr) &= n \log{n} +\sum_{i=1}^{n}\biggl( \frac{1}{n} \sum_{j=1}^{n} \alpha \left\langle u_i, u_j \right\rangle + \frac{\alpha^2}{2}\left\langle u_i, u_j \right\rangle ^2 \biggr) \\
& \qquad - \frac{1}{2}\sum_{i=1}^{n}\biggl( \frac{1}{n} \sum_{j=1}^{n} \alpha \left\langle u_i, u_j \right\rangle + \frac{\alpha^2}{2}\left\langle u_i, u_j \right\rangle ^2 \biggr)^2 + \mathcal{O}(\alpha^3).
\end{align*}
The first term simplifies to
$$ \sum_{i=1}^{n}\biggl( \frac{1}{n} \sum_{j=1}^{n} \alpha \left\langle u_i, u_j \right\rangle + \frac{\alpha^2}{2}\left\langle u_i, u_j \right\rangle ^2 \biggr)= \frac{\alpha}{n} \left\| \sum_{i=1}^{n}u_i \right\|^2+ \frac{\alpha^2}{2n} \sum_{i,j=1}^n \left\langle u_i, u_j\right\rangle^2.$$
The summand in the second term simplifies to, up to first and second order in $\alpha$
$$ -\frac{1}{2} \biggl( \frac{1}{n}\sum_{j=1}^{n} \alpha \left\langle u_i, u_j \right\rangle + \frac{\alpha^2}{2}\left\langle u_i, u_j \right\rangle ^2 \biggr)^2 = -\frac{\alpha^2}{2n^2} \biggl(\sum_j \left\langle u_i, u_j \right\rangle\biggr)^2 + \mathcal{O}(\alpha^3).$$
\end{proof}
| {
"timestamp": "2021-01-20T02:05:22",
"yymm": "2012",
"arxiv_id": "2012.08465",
"language": "en",
"url": "https://arxiv.org/abs/2012.08465",
"abstract": "We consider the variational problem of cross-entropy loss with $n$ feature vectors on a unit hypersphere in $\\mathbb{R}^d$. We prove that when $d \\geq n - 1$, the global minimum is given by the simplex equiangular tight frame, which justifies the neural collapse behavior. We also prove that as $n \\rightarrow \\infty$ with fixed $d$, the minimizing points will distribute uniformly on the hypersphere and show a connection with the frame potential of Benedetto & Fickus.",
"subjects": "Machine Learning (cs.LG); Classical Analysis and ODEs (math.CA)",
"title": "Neural Collapse with Cross-Entropy Loss",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.980280871316566,
"lm_q2_score": 0.7217432003123989,
"lm_q1q2_score": 0.7075110532690453
} |
https://arxiv.org/abs/1906.03512 | Group-theoretical origin of symmetries of hypergeometric class equations and functions | We show that properties of hypergeometric class equations and functions become transparent if we derive them from appropriate 2nd order differential equations with constant coefficients. More precisely, properties of the hypergeometric and Gegenbauer equation can be derived from generalized symmetries of the Laplace equation in 4, resp. 3 dimension. Properties of the confluent, resp. Hermite equation can be derived from generalized symmetries of the heat equation in 2, resp. 1 dimension. Finally, the theory of the ${}_1F_1$ equation (equivalent to the Bessel equation) follows from the symmetries of the Helmholtz equation in 2 dimensions. All these symmetries become very simple when viewed on the level of the 6- or 5-dimensional ambient space. Crucial role is played by the Lie algebra of generalized symmetries of these 2nd order PDE's, its Cartan algebra, the set of roots and the Weyl group. Standard hypergeometric class functions are special solutions of these PDE's diagonalizing the Cartan algebra. Recurrence relations of these functions correspond to the roots. Their discrete symmetries correspond to the elements of the Weyl group. | \section{Introduction}\label{s1}\init
These lecture notes
are
devoted to the properties of
the following equations:
\bigskip
\noindent the {\em Gauss hypergeometric equation}, called also the {\em ${}_2F_1$ equation},
\begin{equation} \Big(w(1-w) \partial_w^2+\big(c-(a+b+1)w\big) \partial_w-ab\Big)F(w)=0;
\label{hyp1}
\end{equation}
the {\em Gegenbauer equation}
\begin{equation} \Big((1-w^2) \partial_w^2-({a}+{b}+1)w \partial_w-{a}{b}\Big)F(w)=0;
\label{hyp2}\end{equation}
{\em Kummer's confluent equation}, called also the {\em${}_1F_1$ equation},
\begin{equation}\Big(w \partial_w^2+(c-w) \partial_w-a\Big)F(w)=0;
\label{hyp3}\end{equation}
the {\em Hermite equation}
\begin{equation} \Big( \partial_w^2-2w \partial_w-2{a}\Big)F(w)=0;
\label{hyp4}\end{equation}
and the {\em ${}_0F_1$ equation} (equivalent to the better known {\em Bessel
equation}, see eg. \cite{De})
\begin{equation} \Big(w \partial_w^2+c \partial_w-1\Big)F(w)=0.\label{hyp5}\end{equation}
Here, $w$ is a complex variable,
$\partial_w$ is the differentiation with respect to $w$,
and $a,b,c$ are arbitrary complex parameters.
These equations are typical representatives of the so-called
{\em hypergeometric class equations} \cite{NU}.
(Nikiforov and Uvarov call them {\em hypergeometric type equations}; following
\cite{SL}, we prefer in this context to use the word {\em class}, reserving {\em type} for narrower families of equations).
We refer the reader to Sect. \ref{s2}, where we discuss the terminology concerning
hypergeometric class equations and functions that we use.
The equations (\ref{hyp1})--(\ref{hyp5})
and their solutions belong to the most natural objects of mathematics and often appear in applications \cite{Fl,MF,WW}.
The aim of these notes is to elucidate the mathematical structure of a large class of identities satisfied by hypergeometric class equations and functions.
We believe that our approach brings order and transparency to this subject, usually considered to be complicated and messy.
We will restrict ourselves to {\em generic
parameters} $a,b,c$. We will not discuss special properties of two distingushed classes of parameters, when additional identities are true:
\begin{arabicenumerate}\item the {\em polynomial case} (which corresponds to negative integer values of $a$);
\item the {\em degenerate case} (which
corresponds to integer values of $c$).
\end{arabicenumerate}
The notes are to a large extent based on \cite{De} and \cite{DeMaj}, with some additions and improvements.
\subsection{From 2nd order PDE's with constant coefficients to hypergeometric class equations}\label{sub1.1}
In our approach, each of the equations (\ref{hyp1})--(\ref{hyp5}) is
derived from a certain
{\em complex 2nd order
PDE with constant coefficients}.
The identities satisfied by this PDE and their solutions
are very straightforward---they look obvious and symmetric.
After an appropriate change of variables,
we derive (\ref{hyp1})--(\ref{hyp5}) and identities satisfied by their solutions. They look much more complicated and messy.
We will argue that the main source of these identities are {\em generalized symmetries} of the parent PDE. Let us briefly recall this concept.
Suppose that we are given an equation
\begin{equation} {\cal K} f=0,\label{repo0}\end{equation}
where ${\cal K}$ is a linear differential operator.
Let $g$ be a Lie algebra and $G$ a group
equipped with pairs of representations
\begin{subequations}\begin{align}
g&\ni B\mapsto B^\flat,B^\#,\label{repo2}\\
G&\ni \alpha\mapsto \alpha^\flat,\alpha^\#,\label{repo1}
\end{align}\end{subequations}
where (\ref{repo2}) has its values in 1st order differential operators and
(\ref{repo1}) in point transformations with multipliers.
We say that (\ref{repo2}) and (\ref{repo1}) are generalized symmetries of (\ref{repo0}) if
\begin{subequations}\begin{align}
B^\flat {\cal K}&={\cal K} B^\#,\label{trans2}\\
\text{resp.}\qquad \alpha^\flat {\cal K}&={\cal K} \alpha^\#.\label{trans1}
\end{align}\end{subequations}
Note that (\ref{trans2}), resp. (\ref{trans1}) imply that $B^\#$ and $\alpha^\#$ preserve the space of solutions of (\ref{repo0}).
We will omit the word ``generalized'' if $B^\#=B^\flat$ and $\alpha^\flat=
\alpha^\#$.
We can distinguish 3 kinds of PDE's with constant coefficients in complex domain. Below we list these PDE's, together with the Lie algebra and group of their generalized symmetries:
\begin{arabicenumerate}
\item The {\em Laplace equation} on ${\mathbb C}^n$
\begin{equation} \Delta_n f=0,\quad n>2.\label{lapla1}\end{equation}
The {\em orthogonal Lie algebra} and {\em group} in $n{+}2$ dimensions, denoted $\so(n{+}2,{\mathbb C})$, resp. $\mathrm{O}(n{+}2,{\mathbb C})$, both acting conformally in $n$ dimensions. (For $n=1,2$ there are additional conformal symmetries).
\item The {\em heat equation} on ${\mathbb C}^{n-2}\oplus{\mathbb C}$:
\begin{equation}(\Delta_{n{-}2}+2\partial_s)f=0.\label{lapla2}\end{equation}
The
{\em Schr\"odinger Lie algebra} and {\em group}
in $n{-}2$ dimensions, denoted
${\rm sch}(n{-}2,{\mathbb C})$, resp. ${\rm Sch}(n{-}2,{\mathbb C})$.
\item The {\em Helmholtz equation} on ${\mathbb C}^{n-1}$,
\begin{equation} (\Delta_{n{-}1}-1)f=0.\label{lapla3}\end{equation}
The {\em affine orthogonal Lie algebra} and {\em group} in $n{-}1$ dimensions, denoted $\aso(n{-}1,{\mathbb C})$, resp. $\AO(n{-}1,{\mathbb C})$.
\end{arabicenumerate}
(The reason for the strange choice of dimensions in (\ref{lapla2}) and (\ref{lapla3}) will be explained later).
The basic idea of our approach is as follows. Let us start from the equation (\ref{repo0}), where ${\cal K}$ is appropriately chosen from among
(\ref{lapla1}), (\ref{lapla2}) and (\ref{lapla3}).
In the Lie algebra
of its generalized symmetries we fix
a certain maximal commutative algebra, which we will call the ``Cartan algebra''. Operators that are eigenvectors of the adjoint action of the ``Cartan algebra'' will be called ``root operators''.
In the group of generalized symmetries we fix a subgroup, which we call the ``Weyl group''. It is chosen in such a way, that its adjoint action fixes the
``Cartan algebra''.
Note that in some cases the Lie algebra of symmetries is simple, and then the names {\em Cartan algebra}, {\em root operators} amd {\em Weyl symmetries} correspond to the standard names. In other cases the Lie algebra is not semisimple, and then the names are less standard -- this is the reason for the quotation marks that we use above. In the sequel we drop the quotation marks.
Let us fix a basis of the Cartan algebra $N_1,\dots,N_k$.
Suppose that the dimension of the underlying space is by $1$ greater than the dimension of the Cartan algebra. Then we introduce new variables, say
$w,u_1,\dots,u_k$ such that $N_i=u_i\partial_{u_i}$.
Substituting a function of the form
\begin{equation} f=u_1^{\alpha_1}\cdots u_k^{\alpha_k}F(w),\label{subo1}\end{equation}
to the equation (\ref{repo0}), and using
\begin{equation}
N_i u^{\alpha_i}=\alpha_iu^{\alpha_i}\end{equation}
we obtain
the equation
\begin{equation}
{\cal F}_{\alpha_1,\dots,\alpha_k}F=0,\label{subo2}\end{equation}
which coincides with one of the equations (\ref{hyp1})--(\ref{hyp5}). The eigenvalues of the Cartan operators become the parameters of this equation.
Root operators shift the Cartan elements, typically by $1$ or $-1$ (like the well-known creation and annihilation operators). Therefore, root operators inserted into the relations (\ref{trans2}) lead to {\em transmutation relations} for
(\ref{hyp1})--(\ref{hyp5}).
Similarly, elements of the Weyl group
permute Cartan elements or change their signs.
Therefore, Weyl symmetries inserted into
(\ref{trans1}) leads to {\em discrete symmetries} of
(\ref{hyp1})--(\ref{hyp5}).
Of course, one can apply (\ref{trans1}) to elements of $G$ other than Weyl symmetries, obtaining interesting integral and addition identities for hypergeometric class functions. They are, however, outside of the scope of these notes.
There are five 2nd order PDE with constant coefficients where we can perform this procedure. They are all listed in the following table:
\[\begin{array}{ccccc}
\hbox{PDE}&\begin{array}{c}\hbox{Lie}\\ \hbox{algebra}\end{array}
&\begin{array}{c}\hbox{dimension of}\\ \hbox{Cartan algebra}\end{array}
&\begin{array}{c}\hbox{discrete}\\ \hbox{symmetries}\end{array}&
\hbox{equation}\\[1ex]
\hline\\[1ex]
\Delta_4&\so(6,{\mathbb C})&3&\hbox{cube}& {}_2F_1; \\[1.5ex]
\Delta_3&\so(5,{\mathbb C})&2&\hbox{square}&\hbox{Gegenbauer};\\[1.5ex]
\Delta_2+2 \partial_t&{\rm sch}(2,{\mathbb C})&2&{\mathbb Z}_2\times
{\mathbb Z}_2&
{}_1F_1\hbox{ or }{}_2F_0;\\[1.5ex]
\Delta_1+2 \partial_t&{\rm sch}(1,{\mathbb C})&1&{\mathbb Z}_4&
\hbox{Hermite};\\[1.5ex]
\Delta_2-1&\aso(2,{\mathbb C})&1&{\mathbb Z}_2&{}_0F_1.
\end{array}\]
Note that some other 2nd order PDE's have too few variables to be in the above list: this is the case of $\Delta_1$ and $\Delta_2$.
Others have too many variables: one can try to perform the above procedure,
however it leads to a differential equation in more than one variable.
\subsection{Conformal invariance of the Laplace equation}
The key tool of our approach is the conformal invariance of
the Laplace equation. Let us sketch a derivation of this invariance. For simplicity we restrict our attention to the complex case, for which
we do not need to distinguish between various signatures of the metric tensor.
In order to derive the conformal invariance of the Laplacian on ${\mathbb C}^n$, or on other complex manifolds with maximal conformal symmetry, it is convenient to start from the so-called {\em ambient space} ${\mathbb C}^{n+2}$, where the actions of $\so(n{+}2,{\mathbb C})$ and $\mathrm{O}(n{+}2,{\mathbb C})$ are obvious. In the next step these actions are restricted to the {\em null quadric}, and finally to the {\em projective null quadric}. Thus the dimension of the manifold goes down from $n{+}2$ to $n$. The null quadric can be viewed as a line bundle over the projective null quadric. By choosing an appropriate {\em section} we can identify the projective null quadric, or at least its open dense subset, with the flat space ${\mathbb C}^n$ or some other complex manifolds with a complex Riemannian structure, e.g. the product of two spheres.
The Lie algebra $\so(n{+}2,{\mathbb C})$ and the group $\mathrm{O}(n{+}2,{\mathbb C})$ act conformally on these manifolds.
What is more interesting, the above construction
leads to a definition of an invariantly defined operator, which we denote $\Delta^\diamond$, transforming functions on the null quadric homogeneous of degree $1-\frac{n}{2}$ onto functions homogeneous of degree $-1-\frac{n}{2}$. After fixing a section, this operator can be identified with the conformal Laplacian on the corresponding complex Riemannian manifold of dimension $n$. For instance, one obtains the Laplacian $\Delta_n$ on ${\mathbb C}^n$.
The representations of $\so(n{+}2,{\mathbb C})$ and $\mathrm{O}(n{+}2,{\mathbb C})$ on the level of the ambient space were true symmetries of $\Delta_{n+2}$. After the reduction to $n$ dimensions, they become generalized symmetries of the conformal Laplacian.
The fact that conformal transformations of the Euclidean
space are generalized symmetries of the Laplace equation was
apparently known already to Lord Kelvin.
Its explanation in terms of the null quadric first appeared
in \cite{Boc}, and is discussed e.g. in \cite{CGT}.
The reduction of $\Delta_{n+2}$ to $\Delta_n$ mentioned above, is based on a beautiful idea of
Dirac in \cite{Dir}, which was later rediscovered e.g. in \cite{HH,FG}---see a discussion by Eastwood \cite{East}.
The construction indicated above gives a rather special class of \hbox{(pseudo-)}Rie\-mannian manifolds---those having a conformal group of maximal dimension, see e.g. \cite{EMN}.
However, conformal invariance can be generalized to
arbitrary \hbox{(pseudo-)}Rie\-mannian manifolds. In fact, the
Laplace-Beltrami operator plus an appropriate multiple of the scalar
curvature, sometimes called the {\em Yamabe Laplacian}, is invariant in
a generalized sense with respect to
conformal maps, see e.g.
\cite{Tay,Or}.
\subsection{The Schr\"odinger Lie algebra and Lie algebra as generalized symmetries of the Heat equation}
\label{subsect-heat}
The heat equation
(\ref{lapla2}) possesses a large Lie algebra and group of generalized symmetries, which in the complex case, as we already indicated, we denote by
${\rm sch}(n{-}2,{\mathbb C})$ and ${\rm Sch}(n{-}2,{\mathbb C})$. Apparently, they were known already to
Lie \cite{L}. They were rediscovered (in the
essentially equivalent context of
the free
Schr\"odinger equation) by Schr\"odinger \cite{Sch}. They were then studied e.g. in \cite{Ha,Ni}.
By adding an additional variable, one can consider the heat equation as the Laplace equation acting on functions with an exponential dependence on one of the variables. This allows us to express generalized symmetries
of (\ref{lapla2}) by generalized symmetries of (\ref{lapla1}). They can be identified as a subalgebra of $\so(n{+}2,{\mathbb C})$, resp. a subgroup of
$\mathrm{O}(n{+}2,{\mathbb C})$ consisting of elements commuting with a certain distinguished element of $\so(n{+}2,{\mathbb C})$.
\subsection{Affine orthogonal group and algebra as symmetries of the Helmholtz equation}
\label{subsect-helm}
Recall that the affine orthogonal group $\AO(n{-}1,{\mathbb C})$ is generated by rotations and translations of ${\mathbb C}^{n-1}$.
It is obvious that elements of $\AO(n{-}1,{\mathbb C})$
commute with the Helmholtz operator $\Delta_{n-1}-1$. The same is true concerning the affine orthogonal Lie algebra $\aso(n{-}1,{\mathbb C})$. Therefore, they are symmetries of the Helmholtz equation (\ref{lapla3}).
The Helmholtz equation is conceptually simpler than that of the Laplace and heat equation,
because all generalized symmetries are true symmetries.
Note that one can embed the symmetries of the Helmholtz equation in conformal symmetries of the Laplace equation, similarly as was done with the heat equation. In fact,
$\aso(n{-}1,{\mathbb C})$ is a subalgebra of $\so(n{+}2,{\mathbb C})$, and $\AO(n{-}1,{\mathbb C})$ is a subgroup of $\mathrm{O}(n{+}2,{\mathbb C})$.
\subsection{Factorization relations}
\label{Factorization relations}
Another important class of identities satisfied by
hypergeometric class operators are
{\em factorizations} \cite{IH}. They come in pairs. They are identities of the form
\begin{subequations}\begin{align}
{\cal F}_1&={\cal A}_-{\cal A}_++c_1,\label{facto1-}\\
{\cal F}_2&={\cal A}_+{\cal A}_-+c_2,\label{facto-}
\end{align}\end{subequations}
where ${\cal A}_+$, ${\cal A}_-$ are 1st order differential operators, $c_1$, $c_2$ are numbers and ${\cal F}_1$, ${\cal F}_2$ are operators coming from
(\ref{hyp1})---(\ref{hyp5}) with slightly shifted parameters.
The number of such factorizations is the same as the number of roots of the Lie algebra of generalized symmetries. They can be derived from certain identities in the {\em enveloping algebra}. They are closely related to the {\em Casimir operators} of its subalgebras.
Factorizations imply {\em transmutation relations}.
In fact, it is easy to see that
(\ref{facto-}) and (\ref{facto1-}) imply
\begin{subequations}\begin{align}
{\cal A}_-{\cal F}_2&=({\cal F}_1+c_2-c_1){\cal A}_-,\label{darb1}\\
{\cal A}_+{\cal F}_1&=({\cal F}_2+c_1-c_2){\cal A}_+.\label{darb2}
\end{align}
\end{subequations}
Note that (\ref{darb1}) implies that
the operator ${\cal A}_-$ maps the kernel of ${\cal F}_2$ to the kernel of
${\cal F}_1+c_2-c_1$. Similarly, (\ref{darb2}) implies that the operator
${\cal A}_+$ maps the kernel of ${\cal F}_1$ to the kernel of
${\cal F}_2+c_1-c_2$. The above construction is usually called
the {\em Darboux transformation}.
\subsection{Standard solutions of hypergeometric class equations}
So far we discussed only identities satisfied by the operators corresponding to the equations
(\ref{hyp1})---(\ref{hyp5}). The approach discussed in these notes is also helpful in deriving and classifying the identities for their solutions.
The equations (\ref{hyp1})---(\ref{hyp5}) have at least 1 and at most 3 singular points on the Riemann sphere. One can typically find two solutions with a simple behavior at each of these points. We call them {\em standard solutions}. (If it is a regular--singular point, then the solutions are given by convergent power series, otherwise we have to use other methods to define them). The discrete symmetries map standard solutions on standard solutions. The best known example of this method of generating solutions is {\em Kummer's table} \cite{Ku}, which lists various possible expressions for solutions of the hypergeometric equation.
\subsection{Recurrence relations of hypergeometric class functions}
All transmutation relations have the form
\begin{equation}
{\cal A}{\cal F}_1={\cal F}_2{\cal A},\label{recu1}\end{equation}
where ${\cal A}$ is a first order differential operator and ${\cal F}_1$, ${\cal F}_2$ is a pair of hypergeometric class operators of the same type. Typically, some parameters of ${\cal F}_2$ differ from the corresponding parameters of ${\cal F}_1$ by $\pm1$. Clearly, if a function $F_1$ solves ${\cal F}_1F_1=0$, then ${\cal A} F_1$ solves
${\cal F}_2{\cal A} F_1=0$.
It turns out that if $F_1$ is a standard solution of ${\cal F}_1$, then ${\cal A} F_1$ is proportional to one of standard solutions of ${\cal F}_2$, say $F_2$. Thus we obtain
an identity
\begin{equation}
{\cal A} F_1=a F_2, \label{recu}\end{equation}
called a {\em recurrence relation}, or a {\em contiguity relation}.
The recurrence relation (\ref{recu}) is fixed by the transmutation relation (\ref{recu1}) except for the coefficient $a$. In practice it is not difficult to determine $a$.
\subsection{From wave packets to integral representations}
Hypergeometric class functions possess integral representations, where integrands are elementary functions. We show that integral representations come from certain natural solutions of the parent 2nd order PDE, which at the same time are eigenfunctions of Cartan operators. It will be convenient to have a name for this kind of solutions---we will call them {\em wave packets}.
Let us describe how to construct wave packets for the Laplace equation.
It is easy to see that each function depending only on variables from an isotropic subspace is harmonic, that is, satisfies the Laplace equation. By assuming that the function is homogeneous in appropriate variables we can make sure that it is an eigenfunction of Cartan operators.
Unfortunately, the above class of functions is too narrow for our purposes. There is still another construction that can be applied: we can rotate a function and integrate it (``smear it out'')
with respect to a weight. This procedure does not destroy the harmonicity. By choosing the weight appropriately, we can make sure that the resulting wave packet is an eigenfunction of Cartan perators.
(The ``smearing out'' is essentially a generalization of the Fourier (or Mellin) transformation to the complex domain.)
After substituting special coordinates to a wave packet, we obtain a
function of the form (\ref{subo1}) with $F$ solving (\ref{subo2}), and
having the form of an integral of an elementary function.
Wave packets for the heat and Helmholtz equation can be derived from wave packets for the Laplace equation.
\subsection{Plan of the lecture notes}
In Sect. \ref{s2} we give a concise introduction to hypergeometric class equations and functions. One can view this section as an extension of the introduction, concentrated on the terminology and classification of equations and functions we consider in these notes.
The remaining sections can be divided into two categories.
The first category consists of
Sects \ref{s3}, \ref{s5} and \ref{s8}. They have a general character and are devoted to basic geometric analysis in any dimension.
The most important one among them is Sect. \ref{s5}, devoted to the conformal invariance of the Laplace equation.
Of comparable importance is Sect. \ref{s8},
where
the Schr\"odinger Lie algebra and group are introduced.
In Subsect. \ref{Harmonic functions}---\ref{wave3} we explain how to construct ``wave packets''.
No special functions appear in Sects \ref{s3}, \ref{s5} and \ref{s8}. They can be read independently of the rest of the notes.
The second category consists of
Sects \ref{s6}, \ref{s7}, \ref{s9}, \ref{s10} and \ref{s11}. They are devoted to a detailed analysis of equations (\ref{hyp1}), (\ref{hyp2}),
(\ref{hyp3}), (\ref{hyp4}), resp. (\ref{hyp5}). Typically,
each section starts with
the ambient space corresponding to
the 2nd order PDE from the left column of the table in Subsect. \ref{sub1.1}.
In the ambient space these symmetries are very easy to describe.
Then we reduce the dimension and introduce special coordinates, which leads to the equation in the right
column of the table.
We made serious efforts to make Sects \ref{s6}, \ref{s7}, \ref{s9}, \ref{s10} and \ref{s11} as parallel as possible.
there is a one to one correspondence between subsections in all these 5 sections.
We try to use a uniform terminology and analogous conventions. This makes our text somewhat repetitive---we believe that this is helpful to the reader. Note also that these sections are to a large extent independent of one another.
We use various (minor but helpful) ideas to make our presentation as short and transparent
as possible. One of them is the use of two kinds of parameters. The parameters that appear in (\ref{hyp1}), (\ref{hyp2}),
(\ref{hyp3}), (\ref{hyp4}), and (\ref{hyp5}), denoted $a,b,c$, are called {\em classical parameters}. They are convenient when one defines ${}_kF_m$ functions by power series. However, in most of our text we prefer to use a different set of parameters, denoted by Greek letters $\alpha,\beta,\mu,\theta,\lambda$. They are much more convenient when we describe symmetries.
Another helpful idea is a consistent use of {\em split coordinates}
in ${\mathbb C}^n$ or ${\mathbb R}^n$.
In these coordinates root operators and Weyl symmetries have an especially simple form.
The notes are full of long lists of identities. We are convinced that most of them are easy to understand and appreciate without much effort. Typically, they are highly symmetric and parallel to one another.
We hesitated whether to use the complex or real setting for these notes.
The complex setting was e.g. in \cite{DeMaj}.
It offers undoubtedly some simplifications: there is no need to consider various signatures of the scalar product.
However, the complex setting can also be problematic: analytic functions are often multivalued, which causes issues with some global constructions.
Therefore, in these notes, except for the introduction, we use the real setting as the basic one. At the same time we keep in mind that all our formulas have obvious analytic continuations to appropriate complex domains.
In most of our notes, we do not make explicit the signature of the scalar product in our notation for Lie algebras and groups.
E.g. by writing $\so(n)$ we mean $\so(q,p)$ for some $n=q+p$ or $so(n,{\mathbb C})$.
Specifying each time the signature would be overly pedantic, especially since we usually want to complexify all objects, so that the signature loses its importance.
\subsection{Comparison with literature}
The literature about hypergeometric class functions is enormous---after all it is one of the oldest subjects of mathematics. Let us mention e.g. the books \cite{BE,SL,AAR,EMOT,Ho,MOS,NIST,R,WW}.
The relationship of special functions to Lie groups and algebras was noticed long time ago. For instance,
the papers by Weisner \cite{We1,We2} from the 50's describe
Lie algebras
associated with Bessel and Hermite functions.
The idea of studying hypergeometric class equations with help of Lie
algebras
was developed further by Miller. His early book \cite{M1}
considers mostly small Lie algebras/Lie groups, typically
$\mathrm{sl}(2,{\mathbb C})$/$\mathrm{SL}(2,{\mathbb C})$ and their contractions, and applies
them to obtain various identities about hypergeometric class functions. These Lie
algebras have
1-dimensional
Cartan algebras and a single pair of roots. This kind of analysis
is able to explain only a single pair of transmutation relations for each equation.
To explain bigger families of transmutation relations one
needs larger Lie algebras.
A Lie algebra strictly larger than $\mathrm{sl}(2,{\mathbb C})$ is $\so(4,{\mathbb C})$.
There exists a large literature on the relation of the
hypergeometric equation with $\so(4,{\mathbb C})$
and its
real forms,
see eg. \cite{KM,KMR}. This Lie algebra is however still too small to
account for all symmetries of the hypergeometric equation---its Cartan
algebra is only 2-dimensional, whereas the equation has three parameters.
An explanation of symmetries of the
Gegenbauer equation in terms of $\so(5,{\mathbb C})$ and of the hypergeometric
equation in terms of $\so(6,{\mathbb C})\simeq \mathrm{sl}(4,{\mathbb C})$ was first given by Miller,
see \cite{M4}, and especially \cite{M5}.
Miller and Kalnins wrote a series of papers where they studied the
symmetry approach to
separation of variables for various 2nd order partial differential
equations, such as the Laplace and wave equation, see eg. \cite{KM1}.
A large part of this research
is summed up in the book by Miller \cite{M3}. As an important
consequence of this study, one obtains detailed information about
symmetries of hypergeometric class equations.
The main tool that we use
to describe properties of hypergeometric class functions
are {\em generalized
symmetries} of 2nd order linear PDE's. Their theory is described in another
book by Miller \cite{M2}, and further developed in \cite{M3}.
A topic that is extensively treated in the literature on the relation
of special functions to group theory, such as \cite{V,Wa,M1, VK}, is
derivation of various
addition formulas. Addition formulas
say that a certain special function can be
written as a sum, often infinite, of some related functions.
As we mentioned above, they are outside of the scope of this text---we concentrate on the simplest identities.
The relationship of Kummer's table with the group of symmetries of a
cube (which is the Weyl group of $\so(6,{\mathbb C})$) was discussed in \cite{LSV}.
A recent paper, where symmetries of the hypergeometric equation play
an important role is \cite{Ko}.
The use of transmutation relations as a tool
to derive recurrence relations
for hypergeometric class functions is well known and can be found
eg. in the book by Nikiforov-Uvarov \cite{NU}, in the books by Miller
\cite{M1} or in older works such
as \cite{Tr,We1,We2}.
There exist various generalizations of hypergeometric class
functions. Let us mention the
class of $\mathcal{A}$-hypergeometric functions, which provides a
natural generalization of the usual hypergeometric
function to many-variable situations \cite{Be,Bod}.
Saito \cite{Sa} considers
generalized symmetries in the framework of $\mathcal{A}$-hypergeometric functions.
Another direction of generalizations of hypergeometric functions is the family of Gel’fand-Kapranov-Zelevinsky hypergeometric functions \cite{G,GKZ}. Similar constructions were explored by Aomoto and others \cite{A,AK,M-H}. The main idea is to generalize integral representations of hypergeometric functions, rather than hypergeometric equations. There exist also interesting confluent versions of these functions \cite{KHT}.
A systematic presentation and derivation of symmetries of hypergeometric class equations and functions from
2nd order PDE's with constant coefficients was given in \cite{De} and \cite{DeMaj}.
These papers consistently use Lie-algebraic parameters, describe transmutation relations, discrete symmetries and factorizations. \cite{De} describes integral representations and recurrence relations.
\cite{DeMaj} concentrates on the study of hypergeometric class operators, leaving out the properties of hypergeometric class functions.
These lecture notes are to a large extent based on \cite{De} and \cite{DeMaj}.
There are some corrections and minor changes of conventions. There are also some additions. A systematic derivation
of all integral representations from ``wave packets'' in higher dimensions
seems to be new.
There are a number of topics related to the hypergeometric class equation that we do not touch. Let us mention the question
whether hypergeometric functions can be expressed in terms of
algebraic functions. This topic, in the context of
$\mathcal{A}$-hypergeometric functions was considered eg. in the interesting
papers \cite{Be,Bod}.
We stick to a rather limited class of equations and functions
(\ref{hyp1})--(\ref{hyp5}). They have a surprisingly rich structure, which often seems to be lost
in more general classes. Nevertheless, it is natural to ask how far one can generalize the ideas of these notes to other equations and functions, such as higher hypergeometric functions, multivariable hypergeometric functions, Heun functions, $q$-hypergeometric functions, Painlev\'{e}{} equations.
\begin{acknowledgments}
The support of the National Science Center under the grant UMO-2014/15/B/ST1/00126 is gratefully acknowledged.
The author thanks P.~Majewski for collaboration at \cite{DeMaj}.
He is also grateful to A.~Latosi\'{n}ski, T.~Koornwinder, M.~Eastwood, S.-Y. Matsubara-Heo and Y. Haraoka for useful remarks.
\end{acknowledgments}
\section{Hypergeometric class equations}
\label{s2}
\init
In this short section we fix our terminology concerning
hypergeometric class equations and functions.
\subsection{Remarks on notation}
We use $\partial_w$ for the operator of differentiation in the variable $w$.
We will understand that the operator $ \partial_w$ acts on the whole expression on its right:
\begin{equation}
\partial_w f(w)g(w)= \partial_w \big(f(w)g(w)\big).\label{req0}\end{equation}
If we want to restrict the action of $\partial_w$ to the term immediately to the right, we will write $ f(w)_{,w}$, or simply $f'(w)$.
We use lhs and rhs as the abbreviations for the left hand side and right hand side.
\subsection{Generalized hypergeometric series}
For $a\in{\mathbb C}$ and $n\in{\mathbb N}$ we define the {\em Pochhammer symbol}
\[(a)_j:=a(a+1)\cdots(a+j-1).\]
For $a_1,\dots,a_k\in{\mathbb C}$, $c_1,\dots,c_m\in{\mathbb C}\backslash\{0,-1,-2,\dots\}$,
we define the {\em ${}_kF_m$
generalized hypergeometric series}, or for brevity the {\em ${}_kF_m$ series}:
\begin{equation}{}_kF_m(a_1,\dots, a_k;c_1,\dots,c_m;w):=
\sum_{j=0}^\infty\frac{(a_1)_j\cdots(a_k)_jw^j}{(c_1)_j\cdots(c_m)_jj!}.
\label{hiper}\end{equation}
By the d'Alembert criterion,
\begin{arabicenumerate}\item if $m+1>k$, the series (\ref{hiper}) is convergent for $w\in{\mathbb C}$;
\item if $m+1=k$, the series (\ref{hiper}) is convergent for $|w|<1$;
\item if $m+1<k$, the series (\ref{hiper}) is divergent, however sometimes
a certain function can be naturally associated with (\ref{hiper}).
\end{arabicenumerate}
The corresponding analytic function will be called the {\em ${}_kF_m$ function}.
The zeroth order term of the series (\ref{hiper}) is $1$. A different normalization of (\ref{hiper}) is often useful:
\begin{align}
{}_k{\bf F}_m(a_1,\dots, a_k;c_1,\dots,c_m;w)&:=
\frac{{}_kF_m(a_1,\dots, a_k;c_1,\dots,c_m;w)}{\Gamma(c_1)\cdots\Gamma(c_m)}
\nonumber\\
&=\sum\limits_{j=0}^\infty
\frac{(a_1)_j\cdots(a_k)_jw^j}{\Gamma(c_1+j)\cdots\Gamma(c_m+j)j!}.
\label{hiper1}\end{align}
In (\ref{hiper1}) we do not have to restrict the values of
$c_1,\dots,c_m\in{\mathbb C}$.
\subsection{Generalized hypergeometric equations}
\begin{theoreme} The ${}_kF_m$ function (\ref{hiper}) solves the dfferential equation
\begin{equation}
\begin{array}{rl}&
(c_1+w\partial_w)\cdots(c_m+w\partial_w)\partial_wF(a_1,\dots, a_k;c_1,\dots,c_m;w)\\[1ex]
=
&(a_1+w\partial_w)\cdots(a_k+w\partial_w)F(a_1,\dots, a_k;c_1,\dots,c_m;w).
\end{array}
\label{equa}\end{equation}
\end{theoreme}
{\bf Proof.}\ \
We check that both the left and right hand side of (\ref{equa}) are equal to
\[a_1\cdots a_kF(a_1+1,\dots, a_k+1;c_1,\dots,c_m;w).\]
$\Box$\medskip
We will call (\ref{equa}) the {\em ${}_k{\cal F}_m$ equation}. It has the order $\max(k,m+1)$.
Below we list all ${}_kF_m$ functions with equations of the order at most $2$.
\begin{itemize}
\item {\bf The ${}_2F_1$ function or the Gauss hypergeometric function}
\[\begin{array}{l}
F(a,b;c;w)=\sum\limits_{n=0}^\infty
\frac{(a)_n(b)_n}{n!(c)_n}w^n.
\end{array}\]
The series is convergent for $|w|<1$, and it extends to a multivalued
function on a covering of ${\mathbb C}\backslash\{0,1\}$.
It is a solution of
the {\em Gauss hypergeometric equation} or the {\em ${}_2{\cal F}_1$ equation}
\label{hipp}
\[\left(w(1-w) \partial_w^2+(c-(a+b+1)w) \partial_w-ab\right)f(w)=0.\]
\item {\bf The ${}_1F_1$ function or Kummer's confluent function}
\[\begin{array}{l}
F(a;c;w)=\sum\limits_{n=0}^\infty
\frac{(a)_n}{n!(c)_n}w^n.\end{array}\]
The series is convergent for all $w\in{\mathbb C}$.
It is a solution of {\em Kummer's
confluent equation} or the {\em ${}_1{\cal F}_1$ equation}
\[\big(w \partial_w^2+(c-w) \partial_w-a\big)f(w)=0.\]
\item {\bf The ${}_0F_1$ function}
\[\begin{array}{l}
F(-;c;w)= F(c;w)=\sum\limits_{n=0}^\infty
\frac{1}{n!(c)_n}w^n
.\end{array}\]
The series is convergent for all $w\in{\mathbb C}$.
It is a solution of the
{\em ${}_0{\cal F}_1$ equation} (related to the Bessel equation)
\[(w \partial_w^2+c \partial_w-1)f(w)=0.\]
\item {\bf The ${}_2F_0$ function}
For $\arg w\neq0$ we define
\[F(a,b;-;w):=\lim_{c\to\infty}F(a,b;c;cw).\]
It extends to an analytic function on the universal cover of
${\mathbb C}\backslash\{0\}$
with a branch point of an infinite order at 0.
It has the following divergent but asymptotic expansion:
\[
F(a,b;-;w)\sim\sum_{n=0}^\infty\frac{(a)_n(b)_n}{n!}w^n,
\ |\arg w-\pi|<\pi-\epsilon,\quad \epsilon>0.
\]
It is a solution of the ${}_2{\cal F}_0$ equation
\[\left(w^2 \partial_w^2+(-1+(a+b+1)w) \partial_w+ab\right)f(w)=0.\]
By a simple transformation described in Subsect. \ref{The ${}_2F_0$ function}
it is equivalent to the ${}_1{\cal F}_1$ equation.
\item {\bf The ${}_1F_0$ function or the power function}
\[\begin{array}{l}F(a;-;w)=(1-w)^{-a}=
\sum\limits_{n=0}^\infty
\frac{(a)_n}{n!}w^n
\end{array}\]
It solves
\[\big((w-1) \partial_w-a\big)f(w)=0.\]
\item {\bf The ${}_0F_0$ function or the exponential function}
\[\begin{array}{l}F(-;-;w)={\rm e}^w=
\sum\limits_{n=0}^\infty
\frac{1}{n!}w^n.
\end{array}\]
It solves
\[( \partial_w-1)f(w)=0.\]
\end{itemize}
\subsection{Hypergeometric class equations}
\label{sub-req}
Following \cite{NU}, equations of the form
\begin{equation}\left(\sigma(w) \partial_w^2+\tau(w) \partial_w+
\eta\right) f(w)=0,\ \ \
\label{req}\end{equation}
where
\begin{subequations}\begin{align} \sigma&\text{ is a polynomial of degree }\leq2,\\
\tau&\text{ is a polynomial of degree }\leq1,\\
\eta&\text{ is a number,}\end{align}\label{conb}\end{subequations}
will be called {\em hypergeometric class equations}.
Solutions of
(\ref{req})
will go under the name of {\em hypergeometric class functions}. Operators
$\sigma(w) \partial_w^2+\tau(w) \partial_w+
\eta$ with $\sigma,\tau,\eta$ satisfying (\ref{conb}) will be called {\em hypergeometric class operators}.
Let us review basic classes of hypergeometric class equations.
We will always assume that $\sigma(w)\neq0$.
Every class
will be simplified
by dividing by a constant and, except for (\ref{hy8}), by
an affine change of the complex variable $w$.
\medskip
\noindent
{\bf The ${}_2{\cal F}_1$
or Gauss hypergeometric equation}
\begin{equation}
\left(w(1-w) \partial_w^2+(c-(a+b+1)w) \partial_w-ab\right)f(w)=0.\label{hy1}\end{equation}
\noindent{\bf The ${}_2F_0$ equation}
\begin{equation}\left(w^2 \partial_w^2+(-1+(1+a+b)w) \partial_w+ab\right)f(w)=0.\label{hy2}\end{equation}
\noindent
{\bf The ${}_1{\cal F}_1$
or Kummer's confluent equation}
\begin{equation}
(w \partial_w^2+(c-w) \partial_w-a)f(w)=0.\label{hy3}\end{equation}
\noindent
{\bf The ${}_0{\cal F}_1$ equation
}
\begin{equation}
(w \partial_w^2+c \partial_w-1)f(w)=0.\label{hy4}\end{equation}
\noindent
{\bf The Hermite equation}
\begin{equation}
( \partial_w^2-2 w \partial_w-2a)f(w)=0.\label{hy5}\end{equation}
\noindent
{\bf 2nd order Euler equation}
\begin{equation}
\left(w^2 \partial_w^2+bw \partial_w+a\right)f(w)=0.\label{hy6}\end{equation}
\noindent
{\bf 1st order Euler equation for the derivative}
\begin{equation}
(w \partial_w^2+c \partial_w)f(w)=0.\label{hy7}\end{equation}
\noindent
{\bf 2nd order equation with constant coefficients}
\begin{equation}
( \partial_w^2+c \partial_w+a)f(w)=0.\label{hy8}\end{equation}
Note that the equations (\ref{hy6}), (\ref{hy7}) and (\ref{hy8}) are elementary.
The remaining ones
(\ref{hy1}), (\ref{hy2}), (\ref{hy3}), (\ref{hy4}) and (\ref{hy5})
are the subject of these lecture notes.
This is why they are contained in the list (\ref{hyp1})--(\ref{hyp5}) given at the beginning of these notes.
(Actually, (\ref{hy2}) is not explicitly mentioned in this list, however it is equivalent to (\ref{hy3}), so that these two equations are treated together).
This list contains also\\
\medskip
\noindent
{\bf The
Gegenbauer equation}
\begin{equation}\left((1-w^2) \partial_w^2-(a+b+1)w \partial_w-ab\right)f(w)=0,\end{equation}
which can be reduced to a subclass of ${}_2{\cal F}_1$ equations by a simple affine transformation. Its distinguishing property is the invariance with respect to the reflection. The Gegenbauer equation has special properties, which justify its separate treatment.
\section{(Pseudo-)Euclidean spaces}
\label{s3}
\init
In this section we introduce basic terminology and notation related to Lie algebras and groups acting on functions on ${\mathbb R}^n$ or, more generally, on manifolds. Lie algebras will be usually represented as 1st order differential operators. Lie groups will typically act as point transformations times multipliers.
We will discuss
various operators related to (pseudo-)orthogonal Lie algebras and groups.
In particular, we will introduce a convenient notation to describe their Cartan algebras, root operators and Weyl groups.
We will also discuss briefly the Laplacian and the Casimir operator.
We will show how to some special classes of harmonic functions---solutions of the Laplace equation. Of particular importance will be solutions that at the same time are eigenfunctions of the Cartan algebra. This construction will involve a contour integral, which can be viewed as a modification of the Fourier or Mellin transformation. These solutions will be informally called {\em wave packets}.
Finally, in the last subsection we will show how to construct a harmonic function in $n{-}1$ dimension from a harmonic function in $n$ dimensions.
\subsection{Basic notation}
We will write ${\mathbb R}^\times$ for ${\mathbb R}\backslash\{0\}$,
${\mathbb R}_+$ for $]0,\infty[$ and ${\mathbb R}_-$ for $]-\infty,0[$. We write ${\mathbb C}^\times$ for ${\mathbb C}\backslash\{0\}$.
We will treat ${\mathbb R}^n$ as a (real) subspace of ${\mathbb C}^n$.
If possible, we will often extend functions from real domains to holomorphic functions on complex domains.
In the following two subsections,
$\Omega,\Omega_1,\Omega_2$ are open subsets of ${\mathbb R}^n$, or more generally, manifolds.
Often it is advantageous to consider a similar formalism where $\Omega,\Omega_1,\Omega_2$ are open subsets of ${\mathbb C}^n$, or more generally, complex manifolds. We will usually stick to the terminology typical for the real case. The reader can easily translate it to the complex picture, if needed.
\subsection{Point transformations with multipliers}
Let $\alpha:\Omega_1\to\Omega_2$ be a diffeomorphism.
The {\em transport} of functions by the map
$\alpha$ will be also denoted by $\alpha$.\footnote{An alternative notation used often in mathematical literature for
the transport by $\alpha$ is $\alpha_*$ or $(\alpha^*)^{-1}$.} More precisely, for
$f\in C^\infty(\Omega_1)$ we define ${\alpha}f\in C^\infty(\Omega_2)$ by
\[({\alpha}f)(y):=f({\alpha}^{-1}(y)).\]
If $m\in C^\infty (\Omega_2)$, then we have a map
$m{\alpha}: C^\infty(\Omega_1)\to C^\infty(\Omega_2)$ given by
\begin{equation} (m{\alpha} f)(y):=m(y)f({\alpha}^{-1}(y)).\label{lie1}\end{equation}
Transformations of the form (\ref{lie1}) will be called {\em point transformations with a multiplier}.
Clearly, transformations of the form (\ref{lie1}) with $\Omega=\Omega_1=\Omega_2$ and $m$ everywhere nonzero
form a group.
\subsection{1st order differential operators}
A vector field $X$ on $\Omega$
will be identified with the differential operator
\[Xf(y)=\sum_i X^i(y)\partial_{y^i}f(y),\ \ f\in C^\infty(\Omega),\]
where $X^i\in C^\infty(\Omega)$, $i=1,\dots,n$.
More generally, we will often use 1st order differential operators
\begin{equation} (X+M)f(y):=\sum_i X^i(y)\partial_{y^i}f(y)+M(y)f(y),\label{lie2}\end{equation}
where $M\in C^\infty(\Omega)$.
Clearly, the set of operators of the form (\ref{lie2}) is a Lie algebra.
Let $\alpha:\Omega_1\to\Omega_2$ be a diffeomorphism.
If $X$ is a vector field on $\Omega_1$, then ${\alpha}(X)$ is the vector field on $\Omega_2$ defined as
\[{\alpha}(X):=\alpha X \alpha^{-1}.\]
\subsection{Affine linear transformations}
The {\em general linear group} is denoted $\GL({\mathbb R}^n)$.
It has a natural extension $\AGL({\mathbb R}^n):={\mathbb R}^n\rtimes \GL({\mathbb R}^n)$ called the {\em affine general linear group}. $(w,\alpha)\in\AGL({\mathbb R}^n)$ acts on ${\mathbb R}^n$ by
\[{\mathbb R}^n\ni y\mapsto w+\alpha y\in{\mathbb R}^n.\]
The permutation group $S_n$ can be naturally identified with a subgroup of $\GL({\mathbb R}^n)$.
If $\pi\in S_n$, then
\[(\pi y)^i:=y^{\pi_i^{-1}}.\]
On the level of functions, we have
\[\pi f(y^1,\dots,y^n)=f(y^{\pi_1},\dots,y^{\pi_n}).\]
The Lie algebra $\gl({\mathbb R}^n)$ represented by vector fields on ${\mathbb R}^n$ is spanned by $y^i\partial_{y^j}$.
The Lie algebra
$\agl({\mathbb R}^n):={\mathbb R}^n\rtimes \gl({\mathbb R}^n)$ is spanned by $\gl({\mathbb R}^n)$ and by $\partial_{y_i}$.
A special element of $\gl({\mathbb R}^n)$ is the {\em generator of dilations}, known also as the {\em Euler vector field},
\begin{equation}
A_n:=\sum_{i=1}^ny^i\partial_{y^i}.\label{euler}\end{equation}
We will often use the complex versions of the above groups, with ${\mathbb R}$ replaced with ${\mathbb C}$. We will write $\GL(n)$ and $\gl(n)$, where the choice of the field follows from the context.
\subsection{(Pseudo-)orthogonal group}
A {\em pseudo-Euclidean space} is ${\mathbb R}^n$ equipped with
a symmetric nondegenerate $n\times n$ matrix $g=[g_{ij}]$.
$g$ defines the {\em scalar product} of vectors $x,y\in{\mathbb R}^n$ and the {\em square} of a vector $x\in{\mathbb R}^n$:
\[\langle x|y\rangle:=\sum_{ij}x^i g_{ij}y^j,\quad
\langle x|x\rangle=\sum_{ij}x^i g_{ij}x^j.\]
The matrix $[g^{ij}]$ will denote the inverse of $[g_{ij}]$.
We will denote by ${\mathbb S}^{n-1}(R)$ the {\em sphere
in ${\mathbb R}^n$ of
squared radius $R\in{\mathbb R}$}:
\begin{equation}
{\mathbb S}^{n-1}(R):=\{y\in{\mathbb R}^n\ :\ \langle y|y\rangle=R\}.\label{sphere}\end{equation}
We will write ${\mathbb S}^{n-1}:={\mathbb S}^{n-1}(1)$.
Actually, ${\mathbb S}^{n-1}$ is the usual sphere only for the Euclidean signature. For non-Euclidean spaces it is a hyperboloid. Usually we will keep a uniform notation for all signatures. Occasionally, if we want to stress that ${\mathbb S}^{n-1}$ has a specific signature,
it will be denoted ${\mathbb S}^{q,p-1}$,
where the signature of the ambient space is $(q,p)$ (see (\ref{signa})).
We also introduce the {\em null quadric}
\begin{equation} {\cal V}^{n-1}:={\mathbb S}^{n-1}(0)\backslash\{0\}.\label{null}\end{equation}
The {\em (pseudo-)orthogonal} and the {\em special (pseudo-)orthogonal group of $g$}
is defined as
\begin{eqnarray*}
{\rm O}(g)&:=&\{{\alpha}\in \GL(n)\ :\ \langle {\alpha}y|{\alpha}x\rangle=\langle y|x\rangle,\ y,x\in {\mathbb R}^n\},\\
\SO(g)&:=&\{\alpha\in{\rm O}(g)\ :\ \det\alpha=1\}.\end{eqnarray*}
We also have
the {\em affine (special) orthogonal group} $\AO(g):={\mathbb R}^n\rtimes \O(g)$,
$\ASO(g):={\mathbb R}^n\rtimes \SO(g)$.
It is easy to see that the {\em pseudo-orthogonal Lie algebra}, represented by vector fields on ${\mathbb R}^n$, can be defined by
\[\so(g):=\{B\in \gl(n)\ :\ B\langle y|y\rangle =0\}.\]
For $i,j=1,\dots,n$, define
\[B_{ij}:=\sum_k(g_{ik}y^k \partial_{y^j}-g_{jk}y^k \partial_{y^i}).\]
$\{B_{ij}\ :\ i<j\}$ is a basis of $\so(g)$.
Clearly, $B_{ij}=-B_{ji}$ and $B_{ii}=0$.
The {\em affine pseudo-orthogonal Lie algebra} $\aso(g):={\mathbb R}^n\rtimes \so(g)$ is spanned by $\partial_{y^i}$ and $\so(g)$.
We will often use the complex versions of the above groups and Lie algebras.
In the real formalism we have to distinguish between various signatures of $g$---in the complex formalism there is only one signature and we can drop the prefix {\em pseudo}.
\subsection{Invariant operators}
Consider a pseudo-Euclidean space ${\mathbb R}^n$.
We define the Laplacian and the Casimir operator
\begin{eqnarray*}\Delta_n&:=&\sum_{i,j=1}^ng^{ij}\partial_{y^i}\partial_{y^j},\\
{\cal C}_n&:=&\frac{1}{2}\sum_{i,j,k,l=1}^ng^{ik}g^{jl}B_{ij}B_{kl}.
\end{eqnarray*}
The above definitions do not depend on the choice of a basis.
$\Delta_n$ commutes with
$\AO(g)$ and $\aso(g)$.
${\cal C}_n$ commutes with $\mathrm{O}(g)$ and $\so(g)$.
Note the identity
\begin{eqnarray}
\langle y|y\rangle\Delta_n
&=&A_n^2+(n-2)A_n+{\cal C}_n
,\label{polar}\end{eqnarray}
where $A_n$ is defined in (\ref{euler}).
\subsection{Orthonormal coordinates}
Suppose that $q+p=n$. Every scalar product of signature $(q,p)$ can be brought to the form
\begin{equation}
\langle y|y\rangle=-\sum_{i=1}^q y_i^2+\sum_{j=q+1}^{q+p}y_j^2.
\label{signa}\end{equation}
$\so(g)$ has a basis consisting of
\begin{subequations}\begin{align}
B_{ij}&=-y_i\partial_{y_j}+y_j\partial_{y_i},&&1\leq i<j\leq q\\
B_{ij}&=y_i\partial_{y_{j}}+y_j\partial_{y_i}&&1\leq i\leq q,\quad q< j\leq n;\\
B_{ij}&=y_i\partial_{y_j}-y_j\partial_{y_i},&&q< i<j\leq n.
\end{align}\end{subequations}
The Laplacian and the Casimir operator are
\begin{align}
\Delta_n&=-\sum_{1\leq i\leq q} \partial_{y_i}^2+\sum_{q<j\leq n}\partial_{y_j}^2,\\
{\cal C}_n&=\sum_{1\leq i<j\leq q}B_{ij}^2+\sum_{q< i<j\leq n}B_{ij}^2
-\sum_{\begin{array}{l}
\scriptstyle 1\leq i\leq q,\\ \scriptstyle q< j\leq n
\end{array}}B_{ij}^2.
\end{align}
We will rarely use orthonormal coordinates.
In the context of the signature $(q,p)$ the standard notation for the orthogoanl groups/Lie algebras is
$\mathrm{O}(q,p)$, $\AO(q,p)$, $\so(q,p)$, $\aso(q,p)$. We will however often use the notation $\mathrm{O}(n)$, $\AO(n)$, $\so(n)$, $\aso(n)$, without specifying the signature of the quadratic form, and even allowing for an arbitrary choice of the field (${\mathbb R}$ or ${\mathbb C}$).
\subsection{Split coordinates}
Suppose that $2m=n$. $(m,m)$ will be called the {\em split signature}.
If the scalar product has such a signature, we can find coordinates such that
\begin{equation}\langle y|y\rangle=\sum_{i=1}^m2y_{-i}y_i.
\label{spli1}\end{equation}
We will say that
(\ref{spli1}) is a scalar product in {\em split coordinates}.
$\so(2m)$ has a basis consisting of
\begin{subequations}\begin{eqnarray} N_i:=B_{-ii}
&=&-y_{-i} \partial_{y_{-i}}+y_{i} \partial_{y_{i}}, \ \ \ j=1,\dots,m,\label{posa1}\\
B_{ij}&=&y_{-i} \partial_{y_{j}}-y_{-j} \partial_{y_{i}},\ \ \ 1\leq|i|<|j|\leq m.\label{posa2}\end{eqnarray}\end{subequations}
The subalgebra of $\so(2m)$ spanned by (\ref{posa1})
is maximal commutative. It is called the {\em Cartan algebra} of $\so(2m)$.
(\ref{posa2}) are its {\em
root operators}. They satisfy
\begin{eqnarray*}
[N_k,B_{ij}]&=&-({\rm sgn}(i)\delta_{k,|i|}+{\rm sgn}(j)\delta_{k,|j|})B_{ij}.
\end{eqnarray*}
The Laplacian and the Casimir operator are
\begin{align}
\Delta_{2m}&=\sum_{i=1}^m 2\partial_{y_{-i}}\partial_{y_i},\\
{\cal C}_{2m}&=\sum_{1\leq |i|<|j|\leq m}B_{ij}B_{-i-j}-\sum_{i=1}^m N_i^2.
\label{casi1}\end{align}
Suppose now that $2m+1=n$. In this case, $(m,m+1)$ will be called the {\em split signature}.
Every scalar product of such signature can be brought to the form
\begin{equation} \langle y|y\rangle=y_0^2+\sum_{i=1}^m2y_{-i}y_i.
\label{spli2}\end{equation}
We will say that
(\ref{spli2}) is a scalar product in {\em split coordinates}.
$\so(2m+1)$ has then
a basis consisting of the above described basis of $\so(2m)$ and
\begin{eqnarray}
B_{0j}&=&
y_{0} \partial_{y_{j}}-y_{-j} \partial_{y_{0}},\ \ \ |j|=1,\dots,m.\label{posa3}
\label{posa22}\end{eqnarray}
The additional roots satisfy
\begin{align}
[N_k,B_{0j}]&=-{\rm sgn}(j)\delta_{k,|j|}B_{0j}.
\end{align}
The subalgebra spanned by (\ref{posa1})
is still maximal commutative in $\so(2m+1)$. It is called a {\em Cartan algebra} of $\so(2m+1)$.
We have
\begin{align}
\Delta_{2m+1}&=\partial_{y_0}^2+\sum_{i=1}^m 2\partial_{y_{-i}}\partial_{y_i},\\
{\cal C}_{2m+1}&=\sum_{|i|=1}^mB_{0i}B_{0-i}+
\sum_{1\leq |i|<|j|\leq m}B_{ij}B_{-i-j}-\sum_{i=1}^m N_i^2.\label{casi2}
\end{align}
In the real case we will most often consider the split signature, both in even and odd dimensions. In both real and complex cases we will usually prefer split coordinates.
We will often write (\ref{spli1}) and (\ref{spli2}) in the form
\begin{equation}\langle y|y\rangle=\sum_{|i|\leq m}y_{-i}y_i
\end{equation}
where it is understood that
$i\in\{-m,\dots,-1,1,\dots,m\}$ in the even case and
$i\in\{-m,\dots,-1,0,1,\dots,m\}$ in the odd case.
\subsection{Weyl group}
\label{Weyl symmetries}
In this subsection we introduce a certain finite subgroup of $\mathrm{O}(n)$, which will be called the {\em Weyl group}. We will also introduce a notation for elements of these groups.
The reader is referred to Subsects \ref{so(6) in 6 dimensions} and
\ref{sub-geg1}, for examples of application of this notation.
We will assume that the signature is split and split coordinates have been chosen.
Consider first dimension $2m$.
Permutations of $\{-1,\dots,-m\}\cup\{1,\dots,m\}$ that preserve the pairs
$\{-1,1\},\dots\{-m,m\}$ define elements of $\mathrm{O}(2m)$. They form a group, that we will call denote $D_m$. It is isomorphic to ${\mathbb Z}_2^m\rtimes S_m$. It is the Weyl group of $\mathrm{O}(2m)$.
The {\em flip} interchanging $-i,i$ will be denoted $\tau_i$.
The flips $\tau_i$, with $i=1,\dots,m$, generate a subgroup of $D_m$ isomorphic to ${\mathbb Z}_2^m$.
To every $\pi\in S_m$ there corresponds an element of $D_m$
denoted $\sigma_\pi$, that permutes pairs $(-i,i)$.
We have
\begin{eqnarray}\sigma_\pi f(y_{-1},y_{1},\dots,
y_{-m},y_m)&:=&
f(y_{-\pi_1},y_{\pi_1},\dots,
y_{-\pi_m},y_{\pi_m}).\label{per1}\end{eqnarray}
Let $\epsilon=(\epsilon_1,\dots,\epsilon_m)$ and
$\epsilon_1,\dots,\epsilon_m\in\{1,-1\}$. We will write
$\epsilon\pi$ as the shorthand for $\epsilon_1\pi_1,\dots,\epsilon_m\pi_m$.
We will
use the notation
\begin{equation}
\sigma_{\epsilon\pi}:=
\sigma_\pi\prod\limits_{\epsilon_j=-1}\tau_j
.\end{equation}
We have
\begin{eqnarray*}
\sigma_{\epsilon\pi}B_{ij}\sigma_{\epsilon\pi}^{-1}
&= B_{\epsilon_i\pi_i,\epsilon_j\pi_j};\quad
\sigma_{\epsilon\pi} N_j\sigma_{\epsilon\pi}^{-1}&=\epsilon_jN_{\pi_j}.
\end{eqnarray*}
Using ${\mathbb R}^{2m+1}={\mathbb R}\oplus{\mathbb R}^{2m}$, we embed
$D_m$ in ${\rm O}(2m+1)$.
We also introduce $\tau_0\in {\rm O}(2m+1)$
given by
\begin{equation} \tau_0f(y_0,y_{-1},y_1,\dots,y_{-m},y_m)
:=f(-y_0,y_{-1},y_1,\dots,y_{-m},y_m).\label{pasa}\end{equation}
Clearly, $\tau_0$ commutes with $D_m$.
The group $B_m$ is defined as the group generated by $D_m$ and
$\tau_0$. It is isomorphic to ${\mathbb Z}_2\times{\mathbb Z}_2^m\rtimes S_m$.
It is the Weyl group of $\mathrm{O}(2m+1)$.
We set
\[\tau_{\epsilon\pi}:=\tau_0\sigma_{\epsilon\pi}.\]
We have
\begin{eqnarray*}
\tau_{\epsilon\pi} B_{0j}\tau_{\epsilon\pi}^{-1}
=-B_{0,\epsilon_j\pi_j},\ \ &
\tau_{\epsilon\pi} B_{ij}\tau_{\epsilon\pi}^{-1}=B_{\epsilon_i\pi_i,\epsilon_j\pi_j},&\ \
\tau_{\epsilon\pi} N_j\tau_{\epsilon\pi}^{-1}=\epsilon_jN_{\pi_j}.\end{eqnarray*}
\subsection{Harmonic functions}
\label{Harmonic functions}
Suppose that ${\mathbb R}^{n}$ is equipped with a scalar product.
We say that a function $F$ on ${\mathbb R}^{n}$ is {\em harmonic} if
\begin{equation} \Delta_n F=0.\end{equation}
\begin{proposition} Let $e_1,\dots e_k\in{\mathbb R}^{n}$ satisfy
\[\langle e_i|e_j\rangle=0, \quad1\leq i,j\leq k.
\]
In other words, assume that $e_1,\dots,e_k$ span an {\em isotropic subspace} of ${\mathbb R}^{n}$.
Let $f$ be a function of $k$ variables. Then
\[F(z):=f\big(\langle e_1|z\rangle,\dots,\langle e_k|z\rangle\big)\]
is harmonic.\label{isotro}\end{proposition}
For instance, consider ${\mathbb R}^{n}$ with a split scalar product, where $n=2m$ or $n=2m+1$. Then any function $f(y_1,\dots,y_m)$ is harmonic, for instance
\begin{equation}
F_{\alpha_1,\dots\alpha_m}:=y_{1}^{\alpha_1}\cdots y_{m}^{\alpha_{m}},\label{fro}\end{equation}
which in addition satisfies
\begin{equation} N_jF_{\alpha_1,\dots\alpha_m}=\alpha_jF_{\alpha_1,\dots\alpha_m}.\label{eigen}\end{equation}
Harmonic functions satisfying in addition the eigenvalue equations (\ref{eigen}) will play an important role in our approach.
Unfortunately, functions of the form (\ref{fro}) constitute a rather narrow class. We need more general harmonic functions, which we will call {\em wave packets}. They are obtained by smearing a rotated
(\ref{fro}) with an appropriate weight, so that it is an eigenfunction of Cartan operators.
This construction will be explained in the Subsect.
\ref{wave1}--\ref{wave3}. It is essentially a version of the Fourier (or Mellin) transformation, possibly with a deformed complex contour of integration.
Note that the aim of Subsects \ref{wave1} and \ref{wave2} is to provide motivation, based on the concept of the Fourier transformation, for Subsect. \ref{wave3}, which contains the construction that will be used in what follows.
\subsection{Eigenfunctions of angular momentum I}
\label{wave1}
Suppose that ${\mathbb R}^{n}={\mathbb R}^2\oplus{\mathbb R}^{n-2}$, where we write
$z=(x,y,z')\in{\mathbb R}^{n}$ and
\[\langle x,y,z'|x,y,z'\rangle=x^2+y^2+\langle z'|z'\rangle.\]
Set
\[N_1:=-{\rm i}(x \partial_y-y \partial_x).\]
Let $m\in{\mathbb Z}$. Consider a function
$f(x,y,z')$. Then
\begin{equation}
F_m(x,y,z'):=\frac1{2\pi}\int_0^{2\pi} f(\cos\phi x-\sin\phi y,\sin\phi x+\cos\phi y,z'){\rm e}^{-{\rm i} m\phi}{\rm d}\phi,\label{porr1}
\end{equation}
\begin{equation}
\label{porr1.}
\text{satisfies }\quad N_1 F_m(x,y,z')= mF_m(x,y,z').\end{equation}
Note that if $f$ is harmonic, then so is $F_m$. This construction is essentially the Fourier transformation.
Introduce complex coordinates
\begin{equation}
z_{\pm1}:=\frac{1}{\sqrt2}(x\pm{\rm i} y).\label{porr2}\end{equation}
We will write $f(z_{-1},z_1,z')=f(x,y,z')$, $F_m(z_{-1},z_1,z')=F(x,y,z')$.
The operator $N_1$ takes the familiar form
\begin{equation} N_1=-z_{-1}\partial_{z_{-1}}+z_1\partial_{z_1},\label{kaj}\end{equation}
and the metric becomes
\begin{equation}\langle z_{-1},z_1,z'|z_{-1},z_1,z'\rangle=2z_{-1}z_1+\langle z'|z'\rangle.\label{porr3.}\end{equation}
Then (\ref{porr1}) and (\ref{porr1.}) can be rewritten as
\begin{align}
F_m(z_{-1},z_1,z')&:=\frac1{2\pi{\rm i}}\int_\gamma f(\tau^{-1} z_{-1},
\tau z_1,z')\tau^{-m-1}{\rm d}\tau,\label{porr1a}\\
N_1F_m(z_{-1},z_1,z')&= mF_m(z_{-1},z_1,z'),\end{align}
where $\gamma$ is the closed contour
$[0,2\pi[\ni\phi\mapsto \tau={\rm e}^{{\rm i}\phi}$.
\subsection{Eigenfunctions of angular momentum II}
\label{wave2}
We again consider ${\mathbb R}^{n}={\mathbb R}^2\oplus{\mathbb R}^{n-2}$, but we change the signature of the metric. We assume that the scalar product is given by
\begin{equation}\langle z_{-1},z_1,z'|z_{-1},z_1,z'\rangle=2z_{-1}z_1+\langle z'|z'\rangle.\label{porr3}\end{equation}
We start from a function $f(z_{-1},z_1,z')$. We would like to construct an eigenfunction of $N_1$ with a generic eigenvalue $\alpha$, and not only with an integer eigenvalues as (\ref{porr1a}). To do this we repeat a similar procedure as in the previous subsection. Now, however, we need to integrate over a half-line, so we need conditions at the ends: we assume that
\begin{equation} f(\tau^{-1}z_{-1},\tau z_1,z')\tau^{-\alpha}\Big|_{\tau=0}^{\tau=\infty}=0.\end{equation}
We set
\begin{align}
F_\alpha&:=\frac1{2\pi{\rm i}}
\int_0^\infty f(\tau^{-1}z_{-1},\tau z_1,z')\tau^{-\alpha-1}{\rm d}\tau.
\end{align}
Then, with $N_1$ given by (\ref{kaj}),
\begin{equation}
N_1 F_\alpha(z_{-1},z_1,z')=\alpha F_\alpha(z_{-1},z_1,z').\label{porr1a-}\end{equation}
Indeed,
\begin{align*}
& \partial_\tau
f(\tau^{-1}z_{-1},\tau z_1,z')\tau^{-\alpha}\\=&
-\alpha
f(\tau^{-1}z_{-1},\tau z_1,z')\tau^{-\alpha-1}\\
&-\tau^{-2}z_{-1}\partial_1f(\tau^{-1}z_{-1},\tau z_1,z')\tau^{-\alpha}
+z_1\partial_2f(\tau^{-1}z_{-1},\tau z_1,z')\tau^{-\alpha}\\
=&
\big( -\alpha-z_{-1} \partial_{z_{-1}}+z_{1} \partial_{z_{1}}\big)
f(\tau^{-1}z_{-1},\tau z_1,z')\tau^{-\alpha-1}.
\end{align*}
Hence
\begin{align}
0&=\frac{1}{2\pi{\rm i}}\int_0^\infty{\rm d}\tau
\partial_\tau
f(\tau^{-1}z_{-1},\tau z_1,z')\tau^{-\alpha}\,=\,(-\alpha+N_1)F_\alpha.
\end{align}
Note that $F_\alpha$ is the Mellin transform of $\tau\mapsto
f(\tau^{-1}z_{-1},\tau z_1,z')$. If $f$ is harmonic, then so is $F_\alpha$.
\subsection{Eigenfunctions of angular momentum III}
\label{wave3}
Assume now that $z_{-1},z_1,z'$ are complex variables and
$f$ is holomorphic. Then we can formulate a result that includes (\ref{porr1.}) and (\ref{porr1a-}), allowing for a greater flexibility of the choice of the contour of integration:
\begin{proposition}\label{cac0}
Suppose that $]0,1[\ni s\overset{\gamma}\mapsto \tau(s)$ is a contour on the Riemann surface of
\[\tau\mapsto
f(\tau^{-1}z_{-1},\tau z_1,z')\tau^{-\alpha}\] that satisfies
\begin{equation}
f(\tau^{-1}z_{-1},\tau z_1,z')\tau^{-\alpha}\Big|_{\tau(0)}^{\tau(1)}=0.\end{equation}
Then
\begin{equation}
F_\alpha:=\frac1{2\pi{\rm i}}
\int_\gamma f(\tau^{-1}z_{-1},\tau z_1,z')\tau^{-\alpha-1}{\rm d}\tau
\end{equation}
solves
\[N_1 F_\alpha=\alpha F_\alpha.\]
\end{proposition}
{\bf Proof.}\ \ We repeat the arguments of the previous subsection, where we replace $[0,\infty[$ with $\gamma$. $\Box$\medskip
\subsection{Dimensional reduction}
\label{Dimensional reduction}
In this subsection we describe how to construct harmonic functions in $n-1$ dimensions out of a harmonic function in $n$ dimensions.
Suppose that ${\mathbb R}^{n}$ is equipped with the scalar product
\[\langle z_{-1},z_{1},z'| z_{-1},z_{1},z'\rangle_{n}
=2z_{-1}z_{1}+\langle z'| z'\rangle_{n-2}.\]
As usual,
we write
\begin{align}
N_{1}&=-z_{-1} \partial_{z_{-1}}+z_{1} \partial_{z_{1}},\\
\Delta_{n}&=2 \partial_{z_{-1}} \partial_{z_{1}}+\Delta_{n-2}.
\end{align}
Introduce new variables and the Laplacian in $n{-}1$ dimensions.
\begin{align}
z_0:&=\sqrt{2z_{-1}z_{1}},\quad
u:=\sqrt{\frac{z_{1}}{z_{-1}}},\\
\Delta_{n-1}&:=\partial_{z_0}^2+\Delta_{n-2}.
\end{align}
In the new variables,
\begin{align}
N_1&=u \partial_u,\\
\Delta_n&=\partial_{z_0}^2+\frac1{z_0}\partial_{z_0}
-\frac1{z_0^2}(u\partial_u)^2+
\Delta_{n-2}. \end{align}
Consequently,
\begin{align}
z_0^{\frac12} \Delta_n z_0^{-\frac12}&=-\frac{1}{z_0^2}\Big(N_1-\frac12\Big)
\Big(N_1+\frac12\Big)+\Delta_{n-1}.
\end{align}
Therefore, if we set
\begin{align}
F_\pm(z_0,u,z')&=u^{\pm\frac12}z_0^{-\frac12}f_\pm(z_0,z'),\end{align}
then
\begin{align}
N_1F_\pm&=\pm\frac12F_\pm,\\
z_0^{\frac12}u^{\mp\frac12}\Delta_nF_\pm&=\Delta_{n-1} f_\pm.
\end{align}
Hence, the $n-1$-dimensional Laplace equation
$\Delta_{n-1} f=0$ is essentially equivalent to the $n$-dimensional Laplace equation $\Delta_{n} F=0$ restricted to the eigenspace of $N_1=\pm\frac12$.
\section{Conformal invariance of the Laplacian}
\label{s5}
\init
Conformal manifolds are manifolds equipped with a conformal stucture---a pseudo-Euclidean metric defined up to a positive multiplier. Conformal transformations are transformations that preserve the conformal structure.
The main objects of this section
are {\em projective null quadrics}. They possess a natural conformal structure with an exceptionally large group of conformal transformations.
In fact, on the
$n+2$ dimensional pseudo-Euclidean {\em ambient space} we have the obvious action of the pseudo-orthogonal Lie algebra and group. This action is inherited by the $n+1$ dimensional {\em null quadric} ${\cal V}$, and then by its $n$-dimensional projectivization ${\cal Y}$.
One can view ${\cal Y}$ as the base of the line bundle ${\cal V}\to{\cal Y}$. By choosing a section $\gamma$ of this bundle we can equip ${\cal Y}$ with a pseudo-Riemannian structure. Choosing various sections defines metrics that differ only by a positive multiple---thus ${\cal Y}$ has a natural conformal structure. If the signature of the ambient space is $(q+1,p+1)$, then the signature of ${\cal Y}$ is $(q,p)$.
We discuss a few examples of pseudo-Riemannian manifolds conformally equivalent to ${\cal Y}$ or to its open dense subset. The main example is the flat pseudo-Euclidean space. Another example is the product of two spheres ${\mathbb S}^q\times{\mathbb S}^p$, which is conformally equivalent to the entire
${\cal Y}$ of signature $(q,p)$.
Especially simple and important are the low dimensional cases:
in $1$ dimension ${\cal Y}\simeq{\mathbb S}^1$ and in $2$ dimensions ${\cal Y}\simeq{\mathbb S}^1\times{\mathbb S}^1$.
One should however remark that the dimensions 1 and 2 are somewhat special---
in these dimensions the full conformal Lie algebra is infinite dimensional, and the above construction gives only its subalgebra.
Conformal transformations are generalized symmetries of
the Laplacian. One can see this with help of a beautiful argument that goes back to Dirac.
Its first step is the construction of a certain geometrically defined operator denoted $\Delta_{n+2}^\diamond$, that transforms functions on ${\cal V}$ homogeneous of degree $1-\frac{n}{2}$ into functions homogeneous of degree $-1-\frac{n}{2}$.
After fixing a section $\gamma$ of the line bundle
${\cal V}\to{\cal Y}$, we can identify the somewhat abstract operator
$\Delta_{n+2}^\diamond$ with a concrete operator $\Delta_{n+2}^\gamma$ acting on fuctions on $\gamma({\cal Y})$. This operator turns out to be the Yamabe
Laplace-Beltrami operator for the corresponding pseudo-Riemannian structure.
On the $n+2$-dimensional ambient space the Laplacian $\Delta_{n+2}$
obviously
commutes with the pseudo-orthogonal Lie algebra and group. On the level of $\gamma({\cal Y})$
this commutation becomes a transmutation of $ \Delta_{n+2}^\gamma$ with two different representations---one corresponding to the
degree $1-\frac{n}{2}$, the other corresponding to the degree $-1-\frac{n}{2}$.
At the end of this section we consider in more detail the conformal action
of the pseudo-orthogonal Lie algebra and group
corresponding to the degree of homogeneity $\eta$
on the flat pseudo-Euclidean space.
In particular, we compute the representations for all elements of the pseudo-orthogonal Lie algebra. For the pseudo-orthogonal group, we compute the representations of Weyl symmetries.
\subsection{Pseudo-Riemannian manifolds}
We say that a manifold ${\cal Y}$ is {\em pseudo-Riemannian} if it is equipped with a nondegenerate symmetric covariant 2-tensor
\[{\cal Y}\ni y\mapsto g(y)=[g_{ij}(y)
,\]
called the {\em metric tensor}. For any vector field $Y$ it defines a function
$g(Y,Y)\in C^\infty({\cal Y})$:
\[{\cal Y}\ni y\mapsto g(Y,Y)(y):=g_{ij}(y)Y^i(y)Y^j(y).\]
Let $\alpha$ be a diffeomorphism of ${\cal Y}$.
As is well known, the tensor $g$ can be transported by $\alpha$. More precisely, $\alpha^*(g)$ is defined by
\[\alpha^*(g)( Y,Y):=g\big(\alpha(Y),\alpha(Y)\big),\]
where $Y$ is an arbitrary vector field.
We say that $\alpha$ is {\em isometric} if $\alpha^* g=g$.
Let $X$ be a vector field.
The Lie derivative in the direction of
$X$ can be applied to the tensor $g$. More precisely, ${\cal L}_Xg$ is defined by
\[({\cal L}_Xg)(Y,Y):=g\big([X,Y],Y\big)+
g\big(Y,[X,Y]\big).\]
We say that a vector field $X$ is {\em Killing} if
${\cal L}_Xg=0$.
\subsection{Conformal manifolds}
We say that the metric tensor $ g_1$ is {\em conformally equivalent to} $g$ if there exists a positive function
$m\in C^\infty({\cal Y})$ such that
\[m(y)g(y)= g_1(y).\]
Clearly, the conformal equivalence is an equivalence relation in the set of metric tensors.
We say that a manifold ${\cal Y}$ is equipped with a {\em conformal structure}, if it is equipped with an equivalence class of conformally equivalent metric tensors.
We say that a diffeomorphism $\alpha$ is {\em conformal} if for some metric tensor $g$ in the conformal class of ${\cal Y}$, $\alpha^* g$ is conformally equivalent to $g$. Clearly, this is equivalent to saying that for all $g$ in the conformal class of ${\cal Y}$, $\alpha^* g$ is conformally equivalent to $g$.
We say that a vector field $X$ is {\em conformal Killing} if for any metric tensors from the conformal class of ${\cal Y}$
there exists a smooth function
$M\in C^\infty({\cal Y})$
such that
\begin{equation} {\cal L}_Xg=Mg.\label{confi}
\end{equation}
Clearly, if (\ref{confi}) is true for one metric tensor $g$ from
the conformal class of ${\cal Y}$, it is true for all metric tensors conformally equivalent to $g$.
\subsection{Projective null quadric}
Consider a pseudo-Euclidean vector space $({\mathbb R}^{n+2},g)$
of signature $(q+1,p+1)$, which we will call the {\em ambient space}. Recall that
\[{\cal V}^{n+1}:=\{z\in{\mathbb R}^{n+2}\ :\ \langle z|z\rangle=0,\quad z\neq0\}.\]
is the {\em null quadric}. For simplicity, we will often write ${\cal V}$ for ${\cal V}^{n+1}$.
The scaling, that is the action of ${\mathbb R}^\times$, preserves ${\cal V}$. Let ${\cal Y}:={\cal V}/{\mathbb R}^\times$ be the
{\em projective null quadric}.
We obtain a {\em line bundle}
${\cal V}\to {\cal Y}$
with the base ${\cal Y}$ and the fiber ${\mathbb R}^\times$.
Let ${\cal Y}_i$ be an open subset of ${\cal Y}$ and ${\cal V}_i$ be the corresponding open subset of ${\cal V}$. Let
\[{\cal Y}_i\ni y\mapsto \gamma_i(y)\in{\cal V}_i\]
be a {\em section} of the bundle ${\cal V}_i\to{\cal Y}_i$, that is a smooth map satisfying $y={\mathbb R}^\times\gamma_i(y)$. Let $g_{\gamma_i}$ be the metric tensor $g$ restricted to $\gamma_i({\cal Y}_i)$ transported to ${\cal Y}_i$.
It is easy to prove the following fact:
\begin{proposition} Let $\gamma_i$, $i=1,2$, be sections of ${\cal V}_i\to{\cal Y}_i$. Then $g_{\gamma_i}$ are metrics on ${\cal Y}_i$ of signature $(q,p)$. The metrics $g_{\gamma_1}$ and $g_{\gamma_2}$ restricted to ${\cal Y}_1\cap{\cal Y}_2$ are conformally equivalent.
\label{wqeq} \end{proposition}
Prop. \ref{wqeq} equips ${\cal Y}$ with a conformal structure.
Choosing a section in the bundle ${\cal V}\to{\cal Y}$ endows ${\cal Y}$ with the structure
of a pseudo-Riemannian manifold.
For some special sections we obtain in particular various
{\em symmetric spaces}
together with an explicit description of their conformal structure.
In following subsections we present a few examples of this construction.
Instead of ${\cal Y}$ one can consider $\tilde{\cal Y}:={\cal V}/{\mathbb R}_+$. We obtain a bundle ${\cal V}\to\tilde{\cal Y}$ with fibre ${\mathbb R}_+$, which has similar properties as the bundle ${\cal V}\to{\cal Y}$. It is a double covering of ${\cal Y}$, which means that we have a canonical $2-1$ surjection
$\tilde{\cal Y}\to{\cal Y}$.
Let $\gamma$ be a section of ${\cal V}\to{\cal Y}$. Every $y\in{\cal Y}$ equals
${\mathbb R}^\times\gamma(y)$, and hence it is the disjoint union
of $\tilde y_+:={\mathbb R}_+\gamma(y)$ and $\tilde y_-:={\mathbb R}_-\gamma(y)$. Clearly $\{\tilde y_+,\tilde y_-\}\subset\tilde{\cal Y}$ is the preimage of $y$ under the canonical covering. Let us set
\begin{equation}\tilde\gamma(\tilde y_+):=\gamma(y),\quad
\tilde\gamma(\tilde y_-):=-\gamma(y).\end{equation}
Then $\tilde\gamma$ is a section of the bundle ${\cal V}\to\tilde{\cal Y}$. With help of $\tilde\gamma$
we can equip $\tilde{\cal Y}$ with a metric $\tilde g_{\tilde\gamma}$. Obviously, if ${\cal Y}$ is equipped with the metric
$g_{\gamma}$, the canonical surjection
$\tilde{\cal Y}\to{\cal Y}$ is isometric.
We would like to treat ${\cal Y}$ as the principal object, since it has a direct generalization to the complex case. However, for some purposes $\tilde{\cal Y}$ is preferable.
\subsection{Projective null quadric as a compactification of a pseudo-Euclidean space}
\label{Projective null quadric as a compactification of a (pseudo-)Euclidean space}
Consider a pseudo-Euclidean space $({\mathbb R}^n,g_n)$ of signature $(q,p)$ embedded in the pseudo-Euclidean space $({\mathbb R}^{n+2},g_{n+2})$ of signature $(q+1,p+1)$.
We assume that
the square of a vector $(z',z_-,z_+)\in{\mathbb R}^{n+2}={\mathbb R}^n\oplus{\mathbb R}^2$ is
\[\langle z',z_-,z_+|z',z_-,z_+\rangle_{n+2}:=
\langle z'|z'\rangle_n+2z_+z_-.\]
Set
\[{\cal V}_0:=\{(z',z_-,z_+)\in{\cal V}\ :\ z_-\neq0\},\quad {\cal Y}_0:={\cal V}_0/{\mathbb R}^\times.\]
${\cal Y}_0$ is dense and open in ${\cal Y}$.
We have a bijection and a section
\begin{equation}{\cal Y}_0\ni{\mathbb R}^\times \begin{bmatrix}y\\1\\-\frac{\langle y|y\rangle_n}{2}\end{bmatrix}\leftrightarrow \underset{{\mathbb R}^n}{\underset{\mathbin{\rotatebox[origin=c]{-90}{$\in$}}}{y}}\mapsto
\begin{bmatrix}y\\1\\-\frac{\langle y|y\rangle_n}{2}\end{bmatrix}\in{\cal V}_0.\label{bije}\end{equation}
Thus ${\mathbb R}^n$ is identified with ${\cal Y}_0$. The metric on ${\cal Y}_0$ given by the above section coincides with
the original metric on ${\mathbb R}^n$. We have thus embedded ${\mathbb R}^n$ with its conformal structure as a dense open subset of ${\cal Y}$.
\subsection{Projective null quadric as a sphere/compactification of a hyperboloid}
\label{Projective null quadric as a sphere/compactification of a hyperboloid}
Consider a Euclidean space $({\mathbb R}^{n+1},g_{n+1})$ embedded in a pseudo-Euclidean space $({\mathbb R}^{n+2},g_{n+2})$ of signature $(1,n+1)$. We assume that the square of a vector $(z',z_0)\in{\mathbb R}^{n+1}\oplus{\mathbb R}={\mathbb R}^{n+2}$ is
\[\langle z',z_0|z',z_0\rangle_{n+2}=\langle z'|z'\rangle_{n+1}-z_0^2.
\]
Recall that
\[{\mathbb S}^n:=\{\omega\in{\mathbb R}^{n+1}\ :\ \langle \omega|\omega\rangle=1\}
\]
is the unit sphere
of dimension $n$.
We have a bijection and a section
\begin{equation}{\cal Y}\ni{\mathbb R}^\times \begin{bmatrix}\omega\\1\end{bmatrix}
\leftrightarrow \underset{{\mathbb S}^n}{\underset{\mathbin{\rotatebox[origin=c]{-90}{$\in$}}}{y}}\mapsto \begin{bmatrix}\omega\\1\end{bmatrix}\in{\cal V}.\label{bije2+}\end{equation}
Thus ${\mathbb S}^n$ is identified with ${\cal Y}$.
The metric on ${\cal Y}$ given by the above section coincides with
the usual metric on ${\mathbb S}^n$.
$\tilde{\cal Y}$ is in this case simply the disjoint sum of two copies of ${\mathbb S}^n$.
The above construction can be repeated with minor changes for a general signature. Indeed, let the signature of $({\mathbb R}^{n+1},g_{n+1})$ be $(q,p+1)$,
so that the signature of $({\mathbb R}^{n+2},g_{n+2})$ is $(q+1,p+1)$.
Set
\[{\cal V}_0:=\{(z',z_0)\in{\cal V}\ :\ z_0\neq0\},\quad {\cal Y}_0:={\cal V}_0/{\mathbb R}^\times.\]
We have then the bijection and section
\begin{equation}{\cal Y}_0\ni{\mathbb R}^\times \begin{bmatrix}\omega\\1\end{bmatrix}
\leftrightarrow \underset{{{\mathbb S}}^{q,p}}{\underset{\mathbin{\rotatebox[origin=c]{-90}{$\in$}}}{\omega}}\mapsto \begin{bmatrix}\omega\\1\end{bmatrix}\in{\cal V}_0.\label{bije2++}\end{equation}
Note that now instead of the unit Euclidean sphere we have the unit hyperboloid of signature $(q,p)$, which has been identified with ${\cal Y}_0$, a dense open subset of ${\cal Y}$.
\subsection{Projective null quadric as the Cartesian product of spheres}
Consider now the space ${\mathbb R}^{n+2}$ of signature $(q+1,p+1)$.
The square of a vector $(\vec t,\vec x)=(t_0,\dots,t_q,x_0,\dots,x_p)$
is defined as
\begin{equation}
\langle \vec t,\vec x|\vec t,\vec x\rangle:=
-t_0^2-\cdots-t_q^2+x_0^2+\cdots+x_p^2.\label{signa1}\end{equation}
Note that ${\mathbb S}^q\times{\mathbb S}^p$ is contained in ${\cal V}$. It is easy to see that the map
\begin{equation}
{\cal Y}\ni {\mathbb R}^\times
(\vec\rho,\vec\omega)
\mathbin{\rotatebox[origin=c]{-180}{$\mapsto$}} (\vec\rho,\vec\omega)\in
{\mathbb S}^q\times{\mathbb S}^p\subset {\cal V}.\label{paio}\end{equation}
is a double covering. Indeed, we easily see that the map is onto and
\[{\mathbb R}^\times(\vec\rho,\vec\omega)={\mathbb R}^\times(-\vec\rho,-\vec\omega)
.\]
Thus
\[{\cal Y}\simeq{\mathbb S}^q\times{\mathbb S}^p/{\mathbb Z}_2,\qquad
\tilde{\cal Y}\simeq{\mathbb S}^q\times{\mathbb S}^p.\]
The map (\ref{paio}) can be interpreted as a section of ${\cal V}\to \tilde{\cal Y}$.
The corresponding metric tensor on ${\cal Y}$ is minus the standard metric tensor on ${\mathbb S}^q$ plus the standard metric tensor on ${\mathbb S}^p$. Its signature is $(q,p)$.
Again, similarly as in the previous subsection, the above construction can be generalized. Indeed,
replace (\ref{signa1}) with
\begin{align*}
\langle \vec t,\vec x|\vec t,\vec x\rangle:=& -t_0^2-\cdots-t_{q_1}^2+t_{q_1+1}^2+\cdots+t_{q_1+p_1}^2\\&+x_0^2+\cdots+x_{p_1}^2
-x_{p_1+1}^2-\cdots x_{p_2+q_2}^2.\end{align*}
We then obtain a map
\begin{equation}
{\cal Y}\ni {\mathbb R}^\times
(\vec\rho,\vec\omega)
\mathbin{\rotatebox[origin=c]{-180}{$\mapsto$}} (\vec\rho,\vec\omega)\in
{\mathbb S}^{p_1,q_1}\times{\mathbb S}^{q_2,p_2}\subset {\cal V}.\label{paios}\end{equation}
Unlike (\ref{paio}), the map (\ref{paios}) is in general not onto---it doubly covers only an open dense subset of ${\cal Y}$.
\subsection{Dimension $n=1$}
Consider now the dimension $n=1$ in more detail.
The ambient space is
${\mathbb R}^3$ with the split scalar product
\[\langle z|z\rangle=z_0^2+2z_{-1}z_{+1}.\]
The 1-dimensional projective quadric is isomorphic to ${\mathbb S}^1$ or, what is the same, the 1-dimensional projective space:
\[{\cal Y}^1\simeq{\mathbb S}^1\simeq{\mathbb R}\cup\{\infty\}=P^1{\mathbb R}.\]
Indeed, it is easy to see that
\[\phi:{\mathbb R}\cup\{\infty\}\to {\cal Y}^1\]
defined by
\begin{eqnarray*}
\phi(s)&:=&\Big(s,1,-\frac12s^2\Big){\mathbb R}^\times,\quad s\in{\mathbb R};\\
\phi(\infty)&:=&(1,0,0){\mathbb R}^\times
\end{eqnarray*}
is a homeomorphism.
The group $\mathrm{O}(1,2)$ acts on $P^1{\mathbb R}$ by homographies (M\"obius transformations).
The Lie algebra $\so(1,2)$ is spanned by
\[B_{0,1},\ B_{0,-1},\ N_1,\]
with the commutation relations
\begin{eqnarray}\nonumber
[B_{0,1},B_{0,-1}]&=&N_1,\\
\nonumber
[B_{0,1},N_1]&=&B_{0,1},\\{}
[B_{0,-1},N_1]&=&-B_{0,-1}.\nonumber\label{comrel}
\end{eqnarray}
Appying (\ref{casi2}) with $m=1$ we obtain its Casimir operator:
\begin{subequations}\begin{eqnarray}
{\cal C}_3&=&2B_{0,1}B_{0,-1}-N_1^2-N_1\\
&=&2B_{0,-1}B_{0,1}-N_1^2+N_1.
\end{eqnarray}\label{casimir1}\end{subequations}
\subsection{Dimension $ n=2$}
Consider finally the dimension $n=2$ in the signature $(1,1)$. The ambient space is
${\mathbb R}^4$ with the split scalar product
\[\langle z|z\rangle=2z_{-1}z_{+1}+2z_{-2}z_{+2}.\]
The 2-dimensional projective quadric is isomorphic to the product of two circles:
\[{\cal Y}^2\simeq P^1{\mathbb R}\times P^1{\mathbb R}
.\]
Indeed, define
\[\phi:\big({\mathbb R}\cup\{\infty\}\big)\times \big({\mathbb R}\cup\{\infty\}\big)\to{\cal Y}^2\]
by
\begin{subequations}\begin{eqnarray}
\phi(t,s)&:=&(-ts,1,t,s){\mathbb R}^\times,\label{kak1}\\
\phi(\infty,s)&:=&(-s,0,1,0){\mathbb R}^\times,\label{kak2}\\
\phi(t,\infty)&:=&(-t,0,0,1){\mathbb R}^\times,\label{kak3}\\
\phi(\infty,\infty)&:=&(-1,0,0,0){\mathbb R}^\times,\label{kak4}
\end{eqnarray}\end{subequations}
where $t,s\in{\mathbb R}$. We easily check that $\phi$ is a homeomorphism.
In fact, rewriting (\ref{kak1}) as
\begin{eqnarray*}
\phi(t,s)&=&\Big(-s,\frac1t,1,\frac{s}{t}\Big){\mathbb R}^\times\label{kak2a}\\
&=&\Big(-t,\frac1s,\frac{t}{s},1\Big){\mathbb R}^\times\label{kak3a}\\
&=&\Big(-1,\frac{1}{ts},\frac1s,\frac1t){\mathbb R}^\times,\label{kak4a}
\end{eqnarray*}
we see the continuity of $\phi$ at (\ref{kak2}), (\ref{kak3}), resp. (\ref{kak4}).
The Lie algebra $\so(2,2)$ is spanned by
\[N_1,\ N_2,\ B_{1,2},\ B_{1,-2},\ B_{-1,2},\ B_{-1,-2}.\]
Appying (\ref{casi2}) with $m=2$ we obtain its Casimir operator:
\[{\cal C}_4=2B_{1,2}B_{-1,-2}+2B_{1,-2}B_{-1,2}-N_1^2-N_2^2-2N_1.\]
As is well known, $\so(2,2)$ decomposes into a direct sum
of two copies of $\so(1,2)$. Concretely,
\[\so(2,2)=\so^+(1,2)\oplus\so^-(1,2),\]
where $\so^+(1,2)$, resp. $\so^-(1,2)$, both
isomorphic to $\so(1,2)$, are
spanned by
\begin{eqnarray*}
B_{1,2}, \ B_{-1,-2},\ N_1+N_2;&\text{resp.}&
B_{1,-2}, \ B_{-1,2},\ N_1-N_2.
\end{eqnarray*}
They have the commutation relations
\begin{eqnarray*}
\Big[\frac{B_{1,2}}{\sqrt2},\frac{B_{-1,-2}}{\sqrt2}\Big]=\frac{N_1+N_2}{2},&&\Big[\frac{B_{1,-2}}{\sqrt2},\frac{B_{-1,2}}{\sqrt{2}}\Big]=\frac{N_1-N_2}{2},
\\
{}\Big[\frac{N_1+N_2}{2},\frac{B_{-1,-2}}{\sqrt2}\Big]=\frac{B_{-1,-2}}{\sqrt2},&&\Big[\frac{N_1-N_2}{2},\frac{B_{-1,2}}{\sqrt2}\Big]=\frac{B_{-1,2}}{\sqrt2},
\\
{}\Big[\frac{N_1+N_2}{2},\frac{B_{1,2}}{\sqrt2}]=-\frac{B_{1,2}}{\sqrt2};&&\Big[\frac{N_1-N_2}{2},\frac{B_{1,-2}}{\sqrt2}\Big]=-2\frac{B_{1,-2}}{\sqrt2}.
\end{eqnarray*}
The corresponding Casimir operators are
\begin{eqnarray*}
{\cal C}_3^{+}&=&B_{1,2}B_{-1,-2}-\frac14(N_1+N_2)^2-\frac12N_1-\frac12N_2\\
&=&B_{-1,-2}B_{1,2}-\frac14(N_1+N_2)^2+\frac12N_1+\frac12N_2,\\
{\cal C}_3^{-}&=&B_{1,-2}B_{-1,2}-\frac14(N_1-N_2)^2-\frac12N_1+\frac12N_2\\
&=&B_{-1,2}B_{1,-2}-\frac14(N_1-N_2)^2+\frac12N_1-\frac12N_2.
\end{eqnarray*}
Thus
\[{\cal C}_4=2{\cal C}_3^{+}+2{\cal C}_3^{-}.\]
In the enveloping algebra of $\so(2,2)$ the operators ${\cal C}_3^{+}$ and
${\cal C}_3^{-}$ are distinct. They satisfy $\alpha({\cal C}_{-})={\cal C}_{+}$ for
$\alpha\in{\rm O}(2,2)\backslash\SO(2,2)$, for instance for
$\alpha=\tau_i$, $i=1,2$.
However, inside the associative algebra of differential operators on ${\mathbb R}^4$ we have the identity
\[B_{1,2}B_{-1,-2}-B_{-1,2}B_{1,-2}=N_1N_2+N_1,\]
which implies
\[{\cal C}_3^{+}={\cal C}_3^{-}\]
inside this algebra.
Therefore, represented in the algebra of differential operators we have
\begin{subequations}
\begin{eqnarray}
{\cal C}_4&=&4B_{1,2}B_{-1,-2}-(N_1+N_2)^2-2N_1-2N_2\\
&=&4B_{-1,-2}B_{1,2}-(N_1+N_2)^2+2N_1+2N_2\\
&=&4B_{1,-2}B_{-1,2}-(N_1-N_2)^2-2N_1+2N_2\\
&=&4B_{-1,2}B_{1,-2}-(N_1-N_2)^2+2N_1-2N_2.
\end{eqnarray}\label{casimir2}
\end{subequations}
\subsection{Conformal invariance of the projective null quadric}
\label{s-diamond1}
Obviously, $\mathrm{O}(n+2)$ and $\so(n+2)$ preserve ${\cal V}$.
They commute with the scaling (the action of ${\mathbb R}^\times$). Therefore,
we obtain the action on ${\cal Y}={\cal V}/{\mathbb R}^\times$, which we denote as follows:
\begin{subequations}\begin{eqnarray}
\so(n+2)\ni B&\mapsto &
B^{\diamond},\label{popo1a.}\\
{\rm O}(n+2) \ni {\alpha}&\mapsto&
{\alpha}^{\diamond}.\label{popo2a.} \end{eqnarray}\end{subequations}
Clearly, the vector fields $B^\diamond$ are conformal Killing and the diffeomorphisms
$\alpha^\diamond$ are conformal.
Let $\eta\in{\mathbb C}$.
We define $\Lambda_+^\eta({\cal V})$
to be the set of smooth
functions on ${\cal V}$ (positively) homogeneous of degree $\eta$, that is,
satisfying
\[f(ty)=t^\eta f(y),\quad t> 0, \quad y\in{\cal V}.\]
Clearly, $B\in \so(n+2)$ and ${\alpha}\in {\rm O}(n+2)$ preserve $\Lambda_+^\eta({\cal V})$. We will denote by $B^{\diamond,\eta}$, resp. $\alpha^{\diamond,\eta}$
the restriction of $B$, resp. $\alpha$ to $\Lambda_+^\eta({\cal V})$.
Thus we have representations
\begin{subequations}\begin{eqnarray}
\so(n+2)\ni B&\mapsto &
B^{\diamond,\eta},\label{popo1a}\\
{\rm O}(n+2) \ni {\alpha}&\mapsto&
{\alpha}^{\diamond,\eta},\label{popo2a} \end{eqnarray}\end{subequations}
acting on $\Lambda_+^\eta({\cal Y})$.
Clearly,
$\Lambda_+^0({\cal V})$ can be identified with $C^\infty(\tilde{\cal Y})$. Moreover,
(\ref{popo1a.}), resp. (\ref{popo2a.}) coincide with
(\ref{popo1a}), resp. (\ref{popo2a}) for $\eta=0$.
If $\eta\in{\mathbb Z}$ one can use another concept of homogeneity.
We define $\Lambda^\eta({\cal V})$ to be
the set of smooth
functions on ${\cal V}$ satisfying
\[f(ty)=t^\eta f(y),\quad t\neq 0, \quad y\in{\cal V}.\]
The properties of
$\Lambda^\eta({\cal V})$ are similar to $\Lambda_+^\eta({\cal V})$, except that
$\Lambda^0({\cal V})$ can be identified with $C^\infty({\cal Y})$.
\subsection{Laplacian on homogeneous functions}
\label{s-diamond2}
The following theorem
according to Eastwood \cite{East} goes back to Dirac \cite{Dir}. We find it
curious because it allows in some
situations to restrict a {\em second order} differential operator to a
submanifold.
\begin{theoreme} Let $\Omega\subset{\mathbb R}^{n+2}$ be an open conical set.
Let $ K\in C^\infty(\Omega)$ be homogeneous of degree
$1-\frac{n}{2}$
such that \[ K\Big|_{{\cal V}\cap\Omega}=0.\]
Then
\[\Delta_{n+2} K\Big|_{{\cal V}\cap\Omega}=0.\]
\label{wer}\end{theoreme}
Before we give two proofs of this theorem,
let us describe some of its consequences.
Let $k\in\Lambda_+^{1-\frac{n}{2}}({\cal V})$.
We can always find $\Omega$, a conical neighborhood of ${\cal V}$,
and $ K\in{\cal A}(\Omega)$ homogeneous of degree
$1-\frac{n}{2}$ such that
\[k= K\Big|_{{\cal V}}.\]
Note that $\Delta_{n+2} K$ is homogeneous of degree
$-1-\frac{n}{2}$.
We set
\begin{equation}
\Delta_{n+2}^\diamond k:=\Delta_{n+2} K\Big|_{{\cal V}}.\label{restic}\end{equation}
By Theorem \ref{wer},
the above definition (\ref{restic}) does not depend on the choice of $\Omega$ and $ K$.
We have thus defined a map
\begin{equation}
\Delta_{n+2}^\diamond:\Lambda_+^{1-\frac{n}{2}}({\cal V})\to
\Lambda_+^{-1-\frac{n}{2}}({\cal V}).\label{diam}\end{equation}
Obviously,
\begin{subequations}\label{dada}
\begin{eqnarray} B\Delta_{n+2}&=&
\Delta_{n+2} B,\ \ \ B\in \so(n+2),\label{so33.}\\
{\alpha}\Delta_{n+2}&=&
\Delta_{n+2} {\alpha},\ \ \ {\alpha}\in
{\rm O}(n+2).\label{SO33.}\end{eqnarray}
\end{subequations}
Restricting (\ref{dada})
to $\Lambda_+^{1-\frac{n}{2}}({\cal V})$ we obtain
\begin{subequations}\label{dada1}
\begin{eqnarray} B^{\diamond,-1-\frac{n}{2}}\Delta_{n+2}^\diamond&=&
\Delta_{n+2}^\diamond B^{\diamond,1-\frac{n}{2}},\ \ \ B\in \so(n+2),\label{so33}\\
{\alpha}^{\diamond,-1-\frac{n}{2}}\Delta_{n+2}^\diamond&=&
\Delta_{n+2}^\diamond {\alpha}^{\diamond,1-\frac{n}{2}},\ \ \ {\alpha}\in
{\rm O}(n+2).\label{SO33}\end{eqnarray}
\end{subequations}
\noindent {\bf 1st proof of Thm \ref{wer}.}
We use the decomposition
${\mathbb R}^{n+2}={\mathbb R}^n\oplus{\mathbb R}^2$ described in Subsect.
\ref{Projective null quadric as a compactification of a (pseudo-)Euclidean space}, with the distinguished coordinates denoted $z_-,z_+$.
We denote the square of a vector, the Laplacian, the Casimir, resp. the generator of dilations
on ${\mathbb R}^{n+2}$ by $R_{n+2}$, $\Delta_{n+2}$, ${\cal C}_{n+2}$, resp. $A_{n+2}$. Similarly,
we denote the square of a vector, the Laplacian, the Casimir, resp. the generator of dilations
on ${\mathbb R}^{n}$ by $R_n$, $\Delta_{n}$, ${\cal C}_n$ resp. $A_{n}$.
We will also write
\[N_{m+1}:=z_+\partial_{z_+}-z_-\partial_{z_-}.\]
We have
\begin{align*}
R_{n+2}&=
R_n+2z_+z_-,\\
\Delta_{n+2}&=\Delta_{n}+2\partial_{z_+}\partial_{z_-}
,\\
A_{n+2}&=A_{n}+z_+\partial_{z_+}+z_-\partial_{z_-}
.\end{align*}
The following identity is a consequence of (\ref{polar}):
\begin{eqnarray}\nonumber
R_n\Delta_{n+2}&=&
R_n\Delta_n+\big(R_{n+2}-2z_+z_-\big)2\partial_{z_+}\partial_{z_-}
\\\nonumber
&=&{\cal C}_n+\Bigl(A_n-1+\frac{n}{2}\Bigr)^2-
\Bigl(\frac{n}{2}-1\Bigr)^2\\\nonumber
&&+R_{n+2} 2\partial_{z_+}\partial_{z_-}-
(z_+\partial_{z_+}+z_-\partial_{z_-})^2+
N_{m+1}^2\\\notag
&=&R_{n+2} 2\partial_{z_+}\partial_{z_-}
\\\notag&&+
\Bigl(A_n-1+\frac{n}{2}-z_+\partial_{z_+}-z_-\partial_{z_-}
\Bigr)
\Bigl(A_{n+2}-1+\frac{n}{2}\Bigr)\\
&&-
\Bigl(\frac{n}{2}-1\Bigr)^2+{\cal C}_n+N_{m+1}^2.
\label{deq}\end{eqnarray}
$\bigl(\frac{n}{2}-1\bigr)^2$ is a scalar.
${\cal C}_n$ and $N_{m+1}^2$
are polynomials in elements of $\so(n+2)$, which are tangent to
${\cal V}$.
Therefore,
all operators in the last line of (\ref{deq}) can be
restricted to ${\cal V}$. The operator $ A_{n+2}-1+\frac{n}{2}$ vanishes
on functions in $\Lambda_+^{1-\frac{n}{2}}(\Omega)$.
The operator $R_{n+2} 2\partial_{z_+}\partial_{z_-}
$ is zero when restricted to ${\cal V}$ (because $R_{n+2}$ vanishes on ${\cal V}$).
Therefore, if $ K$ is homogeneous of degree
$1-\frac{n}{2}$ vanishing on ${\cal V}$, then
$R_n\Delta_{n+2} K$ vanishes on ${\cal V}$. We are free to choose different coordinates which give different $R_n$'s. Therefore we can conclude that
$\Delta_{n+2} K$ vanishes on ${\cal V}$.
$\Box$\medskip
\begin{corollary}
Using the operator $\Delta_{n+2}^\diamond$, we can write
\begin{eqnarray}
R_n\Delta_{n+2}^\diamond&=&
-\Bigl(\frac{n}{2}-1\Bigr)^2+{\cal C}_n^{\diamond,1-\frac{n}{2}}+\big(N_{m+1}^{\diamond,1-\frac{n}{2}}\big)^2.
\label{deq1a}\end{eqnarray}
\end{corollary}
\noindent {\bf 2nd proof of Thm \ref{wer}.}
We use the decomposition ${\mathbb R}^{n+2}={\mathbb R}^{n+1}\oplus{\mathbb R}$ with the distinguished
variable denoted by ${z_0}$, as in Subsect.
\ref{Projective null quadric as a sphere/compactification of a hyperboloid}.
We denote the square of a vector, the Laplacian, the Casimir, resp. the generator of dilations
on ${\mathbb R}^{n+1}$ by $R_{n+1}$, $\Delta_{n+1}$, ${\cal C}_{n+1}$, resp. $A_{n+1}$.
We have
\begin{align*}\label{2nd}
R_{n+2}&=R_{n+1}+z_0^2,\\
A_{n+2}&=A_{n+1}+z_0\partial_{z_0},\\
\Delta_{n+2}&=\Delta_{n+1}+\partial_{z_0}^2.\end{align*}
We have the following identity
\begin{eqnarray}\nonumber
R_{n+1}\Delta_{n+2}&=&
R_{n+1}\Delta_{n+1}+\big(R_{n+2}-z_0^2\big)\partial_{z_0}^2\\\nonumber
&=&{\cal C}_{n+1}+\Bigl(A_{n+1}+\frac{n-1}{2}\Bigr)^2-
\Bigl(\frac{n-1}{2}\Bigr)^2\\\nonumber
&&+R_{n+2}\partial_{z_0}^2- \Big(z_0\partial_{z_0}-\frac12\Bigr)^2+\Bigl(\frac12\Bigr)^2\\\notag
&=&R_{n+2} \partial_{z_0}^2+\Bigl(A_{ n+1}+\frac{n}{2}-z_0 \partial_{z_0}
\Bigr)
\Bigl(A_{n+2}+\frac{n}{2}-1\Bigr)\\
&&-\Bigl(\frac{n}{2}-1\Bigr)\frac{n}{2}+{\cal C}_{n+1}.
\label{deq3}\end{eqnarray}
Then we argue similarly as in the 1st proof. $\Box$\medskip
\begin{corollary}
Using the operator $\Delta_{n+2}^\diamond$, we can write
\begin{eqnarray}
R_{n+1}\Delta_{n+2}^\diamond&=&
-\Bigl(\frac{n}{2}-1\Bigr)\frac{n}{2}+{\cal C}_{n+1}^{\diamond,1-\frac{n}{2}}.
\label{deq3a}\end{eqnarray}
\end{corollary}
\subsection{Fixing a section}
For nonzero $\eta$, in order to identify functions from $\Lambda_+^\eta({\cal V})$ with functions on $\tilde{\cal Y}$ we need to fix a section of the line bundle ${\cal V}\to\tilde{\cal Y}$.
Let us describe this in detail.
Let ${\cal V}_0$ be an open homogeneous subset of ${\cal V}$ and $\tilde{\cal Y}_0:={\cal V}_0/{\mathbb R}_+$. Consider a section $\gamma:\tilde{\cal Y}_0\to{\cal V}_0$.
We then have the obvious identification
$\psi^{\gamma,\eta}:\Lambda_+^\eta({\cal V}_0)\to C^\infty(\tilde{\cal Y}_0)$: for
$k\in \Lambda_+^\eta({\cal V}_0)$ we set
\begin{equation}\big(\psi^{\gamma,\eta} k\big)(y):=k\big( \gamma(y)\big),\ \ y\in\tilde{\cal Y}_0
.\label{psi}\end{equation}
The map $\psi^{\gamma,\eta}$ is bijective and we can
introduce its
inverse, denoted $\phi^{\gamma,\eta}$, defined for any
$f\in C^\infty(\tilde{\cal Y}_0)$ by
\begin{equation} \big(\phi^{\gamma,\eta}f\big)\big(s\gamma(y)\big)=s^\eta f(y),\ \ s\in{\mathbb R}_+,
\ \ y\in\tilde{\cal Y}_0.\label{phi}\end{equation}
Let
$B\in \so(n+2)$ and $\alpha\in\mathrm{O}(n+2)$. As usual, $B$ and $\alpha$ are interpreted as transformations acting on functions on ${\mathbb R}^{n+2}$. Both $B$ and $\alpha$ preserve $\Lambda_+^\eta({\cal V}_0)$. Therefore, we can
define
\begin{subequations}\begin{eqnarray}
B^{\gamma,\eta}&:=&\psi^{\gamma,\eta} B\phi^{\gamma,\eta},\label{poiu}\\
{\alpha}^{\gamma,\eta}&:=&\psi^{\gamma,\eta} {\alpha}\phi^{\gamma,\eta}.\label{poiu2}\end{eqnarray}\end{subequations}
$B^{\gamma,\eta}$ is a 1st order differential operator on $\tilde{\cal Y}_0$.
$\alpha^{\gamma,\eta}$ maps
$C^\infty\big(\tilde{\cal Y}_0\cap(\alpha^\diamond)^{-1}(\tilde{\cal Y}_0)\big)$
onto $C^\infty\big(\tilde{\cal Y}_0\cap \alpha^\diamond(\tilde{\cal Y}_0)\big)$.
It is easy to see that for any $B\in \so(n+2)$ and $\alpha\in\mathrm{O}(n+2)$
there exist $M_B\in C^\infty(\tilde{\cal Y}_0)$
and $m_\alpha
\in C^\infty\big(\tilde{\cal Y}_0\cap \alpha^\diamond(\tilde{\cal Y}_0)\big)$
such that
\begin{subequations}\begin{align}
B^{\diamond,\eta}f(y)&=B^\diamond f(y)+\eta M_B(y) f(y),\label{popo1a..}\\
{\alpha}^{\diamond,\eta}f(y)&=m_\alpha^\eta(y)\alpha^\diamond f(y).\label{popo2a..} \end{align}\end{subequations}
We define also
\begin{equation}
\Delta_{n+2}^{\gamma}:=\psi^{\gamma,-1-\frac{n}{2}} \Delta_{n+2}^\diamond\phi^{\gamma,1-\frac{n}{2}}
.\end{equation}
This is a second order differential operator on $\tilde{\cal Y}_0$. It satisfies
\begin{subequations}\begin{eqnarray} B^{\gamma,-1-\frac{n}{2}}\Delta_{n+2}^\gamma&=&
\Delta_{n+2}^\gamma B^{\gamma,1-\frac{n}{2}},\ \ \ B\in \so(n+2),\label{so33-}\\
{\alpha}^{\gamma,-1-\frac{n}{2}}\Delta_{n+2}^\gamma&=&
\Delta_{n+2}^\gamma {\alpha}^{\gamma,1-\frac{n}{2}},\ \ \ {\alpha}\in
{\rm O}(n+2).\label{SO33-}\end{eqnarray}\end{subequations}
Note that for even $n$ the numbers $\pm1-\frac{n}{2}$ are
integers. Therefore, $\Lambda^{\pm1-\frac{n}{2}}({\cal V})$
are well defined. In the above construction, we can then use ${\cal Y}$ instead of its double cover $\tilde{\cal Y}$. We also do not have problems in the complex case.
For odd $n$ the numbers $\pm1-\frac{n}{2}$ are not integers, and so
$\Lambda^{\pm1-\frac{n}{2}}({\cal V})$ are ill defined.
Therefore,
we have to use $\Lambda_+^{\pm1-\frac{n}{2}}({\cal V})$
and $\tilde{\cal Y}$.
\subsection{Conformal invariance of the flat Laplacian}
\label{subsec-conf}
In this subsection we illustrate
the somewhat abstract theory of the previous subsections
with the example of the
{\em flat section} described
in (\ref{bije}). Recall that the flat section identifies an open subset of ${\cal Y}$ with ${\mathbb R}^n$. Therefore we obtain
an action of $\so(n+2)$ and $\mathrm{O}(n+2)$ on ${\mathbb R}^n$.
As a result we will obtain the invariance of the Laplacian on the flat pseudo-Euclidean space with respect to conformal
transformations.
The results of this subsection will be needed for our discussion of symmetries of the heat equation.
We will use the notation of
(\ref{poiu}) and (\ref{poiu2}),
where instead of $\gamma$ we
write
``${\rm fl}$'', for the
{\em flat section}.
We will describe conformal symmetries
on two levels:
\begin{arabicenumerate}\item[(a)] the ambient space
${\mathbb R}^{n+2}$
\item[(b)]
the space ${\mathbb R}^n$.\end{arabicenumerate}
We will use the split coordinates,
that is, $ z\in{\mathbb R}^{n+2}$ and $y\in {\mathbb R}^n$
have the square
\begin{subequations}\begin{eqnarray}
\langle z|z\rangle&=&\sum_{|j|\leq m+1}z_{-j}z_j,\\
\langle y|y\rangle &=&\sum_{|j|\leq m}y_{-j}y_j
.\end{eqnarray}
\end{subequations}
As a rule, if a given operator does not depend on $\eta$, we omit the subscript $\eta$.
Derivation of all the following identities will be sketched in Subsect. \ref{Computations}.
\medskip
\noindent
{\bf Cartan algebra of $\so(n+2)$}
\medskip
\noindent
Cartan operators of $\so(n)$, $i=1,\dots,m$:
\begin{subequations}\begin{eqnarray}
N_i&=&-z_{-i} \partial_{z_{-i}}+z_{i} \partial_{z_i},\\
N_i^{{\rm fl}}&=&-y_{-i} \partial_{y_{-i}}+y_{i} \partial_{y_i}.
\end{eqnarray}\end{subequations}
Generator of dilations:
\begin{subequations}\begin{eqnarray}
N_{m+1}&=&-z_{-m-1} \partial_{z_{-m-1}}+z_{m+1} \partial_{z_{m+1}},\\
N_{m+1}^{{\rm fl} ,\eta}&=&\sum\limits_{|i|\leq m}y_i \partial_{y_i}-\eta\ =\ A_n-\eta.
\end{eqnarray}\end{subequations}
\medskip
\noindent{\bf Root operators}
\medskip
\noindent
Roots of $\so(n)$,
$|i|<|j|\leq m$:
\begin{subequations}\begin{eqnarray}
B_{i,j}&=&z_{-i} \partial_{z_j}-z_{-j} \partial_{z_i},\\
B_{i,j}^{{\rm fl} }&
=&y_{-i} \partial_{y_j}-y_{-j} \partial_{y_i}.
\end{eqnarray}\end{subequations}
Generators of translations, $|j|\leq m$:
\begin{subequations}\begin{eqnarray}
B_{m+1,j}&=&z_{-m-1} \partial_{z_j}-z_{-j} \partial_{z_{m+1}},\\
B_{m+1,j}^{{\rm fl} }&=& \partial_{y_j}.
\end{eqnarray}\end{subequations}
Generators of special conformal transformations, $|j|\leq m$:
\begin{subequations}\begin{eqnarray}
B_{-m-1,j}&=&z_{m+1} \partial_{z_j}-z_{-j} \partial_{z_{-m-1}},\\
B_{-m-1,j}^{{\rm fl} ,\eta}&=&-\frac{1}{2} \langle y|y\rangle \partial_{y_j}
+y_{-j}\sum\limits_{|i|\leq m}y_i \partial_{y_i}-\eta y_{-j}.
\end{eqnarray}\end{subequations}
\medskip
\noindent{\bf Weyl symmetries}
\noindent We will write $K$ for a function on
${\mathbb R}^{n+2}$ and $f$ for a function on
${\mathbb R}^n$. We only give some typical elements that generate the whole Weyl group.
\medskip
\noindent
Reflection in the $0$th coordinate (for odd $n$):
\begin{subequations}\begin{eqnarray}
\tau_0
K(z_0,\dots)&=&
K(-z_0,
\dots),\\
\tau_0^{{\rm fl} }f(y_0,\dots)&&
=\ f(-y_0,\dots
).
\end{eqnarray}\end{subequations}
Flips, $j=1,\dots,m$:
\begin{subequations}\begin{align}
\tau_j
K(\dots,z_{-j},z_j,\dots,z_{-m-1},z_{m+1})&\notag\\
&\hspace{-20ex}=
K(
\dots,
z_{j},z_{-j},\dots,z_{-m-1},z_{m+1}),\\
\tau_j^{{\rm fl}}f(\dots,y_{-j},y_j,\dots)
=& f(\dots
y_{j},y_{-j},\dots).
\end{align}
\end{subequations}
Inversion:
\begin{subequations}\begin{align}
\tau_{m+1} K(\dots,z_{-m-1},z_{m+1})
&= K(\dots,z_{m+1},z_{-m-1}),\\
\tau_{m+1}^{{\rm fl},\eta} f(y)&=
\Bigl(-\frac{\langle y|y\rangle }{2}\Bigr)^\eta f\Bigl(-\frac{2y}{\langle y|y\rangle }\Bigr).
\end{align}\end{subequations}
Permutations, $\pi\in S_m$:
\begin{subequations}\begin{align}
\sigma_\pi
K(\dots,z_{-j},z_j,\dots,z_{-m-1},z_{m+1})&\notag\\
&\hspace{-14ex}
= K(
\dots,
z_{-\pi_j},z_{\pi_j},\dots,z_{-m-1},z_{m+1}),\\
\sigma_\pi^{{\rm fl} }f(\dots,y_{-j},y_j,\dots)&
=\ f(\dots
y_{-\pi_j},y_{\pi_j},\dots).
\end{align}\end{subequations}
Special conformal transformations, $j=1,\dots,m$:
\begin{subequations}\begin{align}
\sigma_{(j,m+1)} K(z_{-1},z_1,\dots,z_{-j},z_j,\dots,z_{-m-1},z_{m+1})&\notag\\
&\hspace{-30ex} =
K(z_{-1},z_1,\dots,z_{-m-1},z_{m+1},\dots,z_{-j},z_j),
\\
\sigma_{(j,m+1)}^{{\rm fl} ,\eta} f
(y_{-1},y_{1},\dots,y_{-j},y_j,\dots)&\notag\\
&\hspace{-30ex}=\quad y_{-j}^\eta
f\Bigl(
\frac{y_{-1}}{y_{-j}},\frac{y_{1}}{y_{-j}},\dots,\frac{1}{y_{-j}},-\frac{\langle y|y\rangle
}{2y_{-j}}
\dots\Bigr).\end{align}\end{subequations}
\medskip
\noindent{\bf Laplacian}
\medskip
\begin{subequations}\begin{eqnarray}
\Delta_{n{+}2}&=&\sum\limits_{|i|\leq m+1} \partial_{z_i} \partial_{z_{-i}},\\
\Delta_{n+2}^{\rm fl}&=&\sum\limits_{|i|\leq m} \partial_{y_i} \partial_{y_{-i}}\ =\ \Delta_n.
\end{eqnarray}\end{subequations}
We have the representations
on functions on ${\mathbb R}^n$:
\begin{subequations}\begin{eqnarray}
\so(n+2)\ni B&\mapsto&B^{{\rm fl}
,\eta
,\label{pos1}\\
{\rm O}(n+2)\ni
{\alpha}&\mapsto &{\alpha}^{{\rm fl}
,\eta}
\label{pos2}\end{eqnarray}
\end{subequations}
They yield generalized symmetries:
\begin{subequations}
\begin{eqnarray}
B^{{\rm fl} ,\frac{-2-n}{2}}\Delta_n&=&\Delta_n B^{{\rm fl} ,\frac{2-n}{2}},\ \ \ B\in \so(n+2),\\
{\alpha}^{{\rm fl} ,\frac{-2-n}{2}}\Delta_n&=&\Delta_n
{\alpha}^{{\rm fl} ,\frac{2-n}{2}},\ \ \ {\alpha}\in {\rm O}(n+2).
\end{eqnarray}
\end{subequations}
\medskip
\subsection{Computations}
\label{Computations}
Below we sketch explicit computations that lead to
the formulas on from the previous subsection.
Consider ${\mathbb R}^{n}\times{\mathbb R}^\times\times{\mathbb R}$ (defined by
$z_{-m-1}\neq0$),
which is an open dense subset of
${\mathbb R}^{n+2}$. Clearly, ${\cal V}_0$ is contained in
${\mathbb R}^{n}\times{\mathbb R}^\times\times{\mathbb R}$.
We will write $\Lambda^\eta({\mathbb R}^{n}\times{\mathbb R}^\times\times{\mathbb R})$ for the space
of functions {\em homogeneous} of degree $\eta$ on
${\mathbb R}^{n}\times{\mathbb R}^\times\times{\mathbb R}$.
Instead of using the maps $\phi^{{\rm fl} ,\eta}$ and $\psi^{{\rm fl} ,\eta}$,
as in (\ref{phi}) and (\ref{psi}),
we will prefer
$\Phi^{{\rm fl} ,\eta}:C^\infty({\mathbb R}^n)\to\Lambda^\eta({\mathbb R}^{n}\times{\mathbb R}^\times\times{\mathbb R})$ and
$\Psi^{{\rm fl} ,\eta}:\Lambda^\eta({\mathbb R}^{n}\times{\mathbb R}^\times\times{\mathbb R})\to C^\infty({\mathbb R}^n)$
defined below.
For $ K\in
\Lambda^\eta\bigl({\mathbb R}^{n}\times{\mathbb R}^\times\times{\mathbb R}\bigr)$, we define $\Psi^{{\rm fl},\eta} K\in C^\infty({\mathbb R}^n)$ by
\[\big(\Psi^{{\rm fl},\eta} K\big)(y)= K\Big(y,1,-\frac{\langle y|y\rangle }{2}\Big), \ \ y\in {\mathbb R}^n.\]
Let $f\in C^\infty({\mathbb R}^n)$.
Then there exists a unique function in
$\Lambda^\eta\bigl({\mathbb R}^{n}\times{\mathbb R}^\times\times{\mathbb R}\bigr)$
that extends $f$ and does not depend on $z_{m+1}$. It is given by
\[\big(\Phi^{{\rm fl} ,\eta} f\big)
(z,z_{-m-1},z_{m+1})
:=z_{-m-1}^\eta
f\Big(\frac{z}{z_{-m-1}}\Big),\quad z\in{\mathbb R}^n.\]
The map $\Psi^{{\rm fl},\eta}$ is a left inverse of $\Phi^{{\rm fl} ,\eta}$:
\begin{eqnarray*}\Psi^{{\rm fl},\eta}\Phi^{{\rm fl} ,\eta}& =&{\iota},\end{eqnarray*}
where ${\iota}$ denotes the identity. Clearly,
\begin{eqnarray*}\Phi^{{\rm fl} ,\eta}f\Big|_{{\cal V}_0}&=&\phi^{{\rm fl} ,\eta}f,\\
\Psi^{{\rm fl},\eta}K&=&\psi^{{\rm fl},\eta}\Big(K\Big|_{{\cal V}_0}\Big).\end{eqnarray*}
Moreover, functions in $\Lambda^\eta({\mathbb R}^{n}\times{\mathbb R}^\times\times{\mathbb R})$ restricted to ${\cal V}_0$ are in $\Lambda^\eta({\cal V}_0)$.
Therefore,
\begin{eqnarray*}
B^{{\rm fl} ,\eta}&=&\Psi^{{\rm fl},\eta} B\Phi^{{\rm fl} ,\eta},\ \ \ \ B\in \so({\mathbb R}^{n+2}),\\
{\alpha}^{{\rm fl} ,\eta}&=&\Psi^{{\rm fl},\eta} {\alpha}\Phi^{{\rm fl} ,\eta},\ \ \ {\alpha}\in
{\rm O}({\mathbb R}^{n+2}).\end{eqnarray*}
(Note that $\alpha,B$ preserve
$\Lambda^\eta({\mathbb R}^{n}\times{\mathbb R}^\times\times{\mathbb R})$).
Note also that
\[\Delta_{n+2}^{\rm fl}=\Psi^{{\rm fl},\eta} \Delta_{n+2}\Phi^{{\rm fl}
,\eta}=\Delta_n
.\]
In practice, the above idea can be implemented by
the following change
of coordinates on ${\mathbb R}^{n+2}$:
\[\begin{array}{l}
y_i:=\frac{z_i}{z_{-m-1}},\ |i|\leq m,\\[3mm]
R:=\sum\limits_{|i|\leq m+1}z_{i}z_{-i},\\[3mm]
p:=z_{-m-1}.
\end{array}\]
The inverse transformation is
\[\begin{array}{l}
z_i=py_i,\ |i|\leq m,\\[3mm]
z_{m+1}=\frac{1}{2}(\frac{R}{p}-p\sum\limits_{|i|\leq m}y_iy_{-i}),\\[3mm]
z_{-m-1}=p
\end{array}\]
The derivatives are equal to
\[\begin{array}{l}
\partial_{z_i}=z_{-m-1}^{-1} \partial_{y_i}+2z_{-i} \partial_R,\ |i|\leq m,\\[3mm]
\partial_{z_{m+1}}=2z_{-m-1} \partial_R,\\[3mm]
\partial_{z_{-m-1}}= \partial_p-z_{-m-1}^{-2}
\sum\limits_{|i|\leq m}z_i \partial_{y_i}+2 z_{m+1} \partial_R.
\end{array}\]
Note that these coordinates are defined on
${\mathbb R}^{n}\times{\mathbb R}^\times\times{\mathbb R}$.
The set ${\cal V}_0$ is given by
the condition $R=0$. The flat section is given by $p=1$.
For a function $y\mapsto f(y)$ we have
\[\big(\Phi^{{\rm fl} ,\eta} f\big)(y,R,p)=p^\eta f(y).\]
For a function
$(y,R,p)\mapsto K(y,R,p)$ we have
\[\big(\Psi^{{\rm fl},\eta} K\big)(y)= K(y,1,0).\]
Note also that on $\Lambda^\eta({\mathbb R}^{n}\times{\mathbb R}^\times\times{\mathbb R})$ we have
\[p \partial_p+2R \partial_R=\eta.\]
\section{Laplacian in 4 dimensions and the hypergeometric equation}
\label{s6}
\init
The goal of this section is to derive the ${}_2{\cal F}_1$ equation together with its symmetries from the Laplacian in $4$ dimensions, or actually from the Laplacian in $6$ dimensions, if one takes into account the ambient space. Let us describe the main steps of this derivation:
\be
\item\label{it1}
We start from the $4+2=6$ dimensional ambient space, with the obvious representations of $\so(6)$ and $\mathrm{O}(6)$, and the Laplacian $\Delta_6$.
\item\label{it2} As explained in Subsect. \ref{s-diamond1},
we introduce the representations
$\so(6)\ni B\mapsto B^{\diamond,\eta}$ and
$\mathrm{O}(6)\ni \alpha\mapsto \alpha^{\diamond,\eta}$. Besides,
as explained
in Subsect. \ref{s-diamond2},
we obtain the reduced Laplacian $\Delta_6^\diamond$.
The most relevant values of $\eta$ are $1-\frac{4}{2}=-1$ and $-1-\frac{4}{2}=-3$, which yield generalized symmetries of $\Delta_6^\diamond$.
\item\label{it3} We fix a section $\gamma$ of the null quadric.
It allows us to construct the representations $B^{\gamma,\eta}$, $\alpha^{\gamma,\eta}$ and the operator $\Delta_6^\gamma$, acting on a $4$ dimensional manifold whose pseudo-Riemannian structure depends on $\gamma$.
\item\label{it4} We choose coordinates $w,u_1,u_2,u_3$, so that the Cartan operators are expressed in terms of $u_1$, $u_2$, $u_3$. We compute
$\Delta_6^\gamma$, $B^{\gamma,\eta}$, and $\alpha^{\gamma,\eta}$ in the new coordinates.
\item\label{it5} We make an ansatz that diagonalizes the Cartan operators, whose eigenvalues, denoted by $\alpha$, $\beta$, $\mu$,
become parameters.
$\Delta_6^\gamma$, $B^{\gamma,\eta}$, and $\alpha^{\gamma,\eta}$ involve now only the single variable $w$.
$\Delta_6^\gamma$ turns out to be the ${}_2{\cal F}_1$ hypergeometric operator.
The generalized symmetries of
$\Delta_6^\gamma$ yield transmutation relations and discrete symmetries of the
${}_2{\cal F}_1$ operator.
\ee
Step \ref{it1} is described in Subsect. \ref{so(6) in 6 dimensions}.
We have a considerable freedom in the choice of the section $\gamma$
of Step \ref{it3}. For instance, it can be the flat section, which we described in Subsects
\ref{Projective null quadric as a compactification of a (pseudo-)Euclidean space} and \ref{subsec-conf}. However, to simplify computations we prefer to choose a different section, which we call the spherical section. (Both approaches are described in \cite{DeMaj}).
We perform Steps \ref{it2}, \ref{it3} and \ref{it4} at once.
They are described jointly in Subsect.
\ref{so(6) on the spherical section}. We choose coordinates $w,r,p,u_1,u_2,u_3$ in $6$ dimensions, so that the null quadric, the spherical section and the homogeneity of functions are expressed in a simple way.
In these coordinates, after the reductions of Steps \ref{it2} and \ref{it3}, the variables $r,p$ disappear. We are left with the variables $w,u_1,u_2,u_3$, and we are ready for Step \ref{it5}.
Step \ref{it5} is described in Subsects
\ref{Hypergeometric equation} and
\ref{Transmutation relations and discrete symmetries}.
Subsects \ref{Factorizations of the Laplacian} and
\ref{Factorizations of the hypergeometric operator}
are devoted to factorizations of the ${}_2{\cal F}_1$ operator. Again, we see that the additional dimensions make all the formulas more symmetric.
The role of factorizations is explained in Subsect.
\ref{Factorization relations}.
Subsects \ref{Transmutation relations and discrete symmetries} and
\ref{Factorizations of the hypergeometric operator} contain long lists of identities for the hypergeometric operator. We think that it is easy to appreciate and understand them at a glance, without studying them line by line.
Actually, the analogous lists of identities in the next sections, corresponding to other types of equations, are shorter but in a sense more complicated, because they correspond to ``less symmetric'' groups.
All the material so far has been devoted to the ${}_2{\cal F}_1$ operator
and its multidimensional ``parents''. Starting with Subsect. \ref{The ${}_2F_1$ hypergeometric function} we discuss the ${}_2F_1$ function and, more generally, distinguished solutions of the ${}_2{\cal F}_1$ equation. The symmetries of the
${}_2{\cal F}_1$ operator are helpful in deriving and organizing the identities concerning these solutions.
Subsects \ref{Wave packets in 6 dimensions},
\ref{Integral representations}, \ref{Integral representations of standard solutions} are devoted to integral representations of solutions of the
${}_2{\cal F}_1$ equation. In particular, Subsect. \ref{Wave packets in 6 dimensions}
shows that these representation are disguised ``wave packets'' solving the Laplace equation and diagonalizing Cartan operators.
In Subsect. \ref{Connection formulas} we derive connection formulas, where we use the pairs of solutions with a simple behavior at $0$ and at $\infty$ as two bases of solutions. The connection formulas follow easily from integral representations. These identities look symmetric when expressed in terms of the Lie-algebraic parameters.
\subsection{$\so(6)$ in 6 dimensions}
\label{so(6) in 6 dimensions}
We consider ${\mathbb R}^6$ with the split coordinates
\begin{equation}
z_{-1},z_1,z_{-2},z_2,z_{-3},z_3\label{sq0}\end{equation}
and the scalar product given by
\begin{equation}
\langle z|z\rangle=2z_{-1}z_1+2z_{-2}z_2+2z_{-3}z_3.\label{sq1}\end{equation}
The Lie algebra $\so(6)$ acts naturally on ${\mathbb R}^6$. Below we describe its natural basis. Then we consider its Weyl group, $D_3$, acting on functions on ${\mathbb R}^6$. For brevity, we list only elements
from its subgroup $D_3\cap \SO(6)$. Finally, we write down the Laplacian.
\medskip
\noindent{\bf Lie algebra $\so(6)$.}
Cartan algebra
\begin{subequations}\begin{align}
N_1&=-z_{-1}\partial_{z_{-1}}+z_{1}\partial_{z_{1}},\\
N_2&=-z_{-2}\partial_{z_{-2}}+z_{2}\partial_{z_{2}},\\
N_3&=-z_{-3}\partial_{z_{-3}}+z_{3}\partial_{z_{3}}.
\end{align}\end{subequations}
Root operators
\begin{subequations}
\begin{align}
B_{-2,-1}&=z_{2} \partial_{z_{-1}}-z_{1} \partial_{z_{-2}},\\
B_{2,1}&=z_{-2} \partial_{z_1}-z_{-1} \partial_{z_2},\\
B_{2,-1}&=z_{-2} \partial_{z_{-1}}-z_{1} \partial_{z_2},\\
B_{-2,1}&=z_{2} \partial_{z_1}-z_{-1} \partial_{z_{-2}};
\end{align}
\begin{align} B_{-3,-2}&=z_{3} \partial_{z_{-2}}-z_{2} \partial_{z_{-3}},\\
B_{3,2}&=z_{-3} \partial_{z_2}-z_{-2} \partial_{z_3},\\
B_{3,-2}&=z_{-3} \partial_{z_{-2}}-z_{2} \partial_{z_3},\\
B_{-3,2}&=z_{3} \partial_{z_2}-z_{-2} \partial_{z_{-3}};
\end{align}
\begin{align}
B_{-3,-1}&=z_{3} \partial_{z_{-1}}-z_{1} \partial_{z_{-3}},\\
B_{3,1}&=z_{-3} \partial_{z_1}-z_{-1} \partial_{z_3},\\
B_{3,-1}&=z_{-3} \partial_{z_{-1}}-z_{1} \partial_{z_3},\\
B_{-3,1}&=z_{3} \partial_{z_1}-z_{-1} \partial_{z_{-3}}.
\end{align}
\end{subequations}
\noindent{\bf Weyl symmetries}
\begin{subequations}
\begin{align}
\sigma_{123} K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=&K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3}),\\
\sigma_{-12-3}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=&K(z_{1},z_{-1},z_{-2},z_2,z_{3},z_{-3}),\\
\sigma_{1-2-3}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=
&K(z_{-1},z_{1},z_{2},z_{-2},z_{3},z_{-3}),\\
\sigma_{-1-23}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=&K(z_{1},z_{-1},z_{2},z_{-2},z_{-3},z_{3});\end{align}
\begin{align}
\sigma_{213}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=&K(z_{-2},z_2,z_{-1},z_1,z_{-3},z_3),\\
\sigma_{-21-3}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=&K(z_{2},z_{-2},z_{-1},z_1,z_{3},z_{-3}),\\
\sigma_{2-1-3}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=&K(z_{-2},z_2,z_{1},z_{-1},z_{3},z_{-3}),\\
\sigma_{-2-13}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=&K(z_{2},z_{-2},z_{1},z_{-1},z_{-3},z_3);\end{align}\begin{align}
\sigma_{321}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=&K(z_{-3},z_3,z_{-2},z_2,z_{-1},z_1),\\
\sigma_{-32-1}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=&K(z_{3},z_{-3},z_{-2},z_2,z_{1},z_{-1}),\\
\sigma_{3-2-1}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=&K(z_{-3},z_3,z_{2},z_{-2},z_{1},z_{-1}),\\
\sigma_{-3-21}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=&K(z_{3},z_{-3},z_{2},z_{-2},z_{-1},z_1);\end{align}\begin{align}
\sigma_{312}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=&K(z_{-3},z_3,z_{-1},z_1,z_{-2},z_2),\\
\sigma_{-31-2}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=&K(z_{3},z_{-3},z_{-1},z_1,z_{2},z_{-2}),\\
\sigma_{3-1-2}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=&K(z_{-3},z_3,z_{1},z_{-1},z_{2},z_{-2}),\\
\sigma_{-3-12}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=&K(z_{3},z_{-3},z_{1},z_{-1},z_{-2},z_2);\end{align}\begin{align}
\sigma_{231}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=
&K(z_{-2},z_2,z_{-3},z_3,z_{-1},z_1),\\
\sigma_{-23-1}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=
&K(z_{2},z_{-2},z_{-3},z_3,z_{1},z_{-1}),\\
\sigma_{2-3-1}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=
&K(z_{-2},z_2,z_{3},z_{-3},z_{1},z_{-1}),\\
\sigma_{-2-31}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=
&K(z_{2},z_{-2},z_{3},z_{-3},z_{-1},z_1);\end{align}\begin{align}
\sigma_{132}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=&K(z_{-1},z_1,z_{-3},z_3,z_{-2},z_2),\\
\sigma_{-13-2}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=&K(z_{1},z_{-1},z_{-3},z_3,z_{2},z_{-2}),\\
\sigma_{1-3-2}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=&K(z_{-1},z_1,z_{3},z_{-3},z_{2},z_{-2}),\\
\sigma_{-1-32}K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})=&K(z_{1},z_{-1},z_{3},z_{-3},z_{-2},z_2).
\end{align}\end{subequations}
\noindent{\bf Laplacian}
\begin{equation}
\Delta_6=2\partial_{z_{-1}}\partial_{z_{1}}+
2\partial_{z_{-2}}\partial_{z_{2}}+
2\partial_{z_{-3}}\partial_{z_{3}}
.\label{sq2}\end{equation}
\subsection{$\so(6)$ on the spherical section}
\label{so(6) on the spherical section}
In this subsection
we perform Steps \ref{it2}, \ref{it3} and \ref{it4} described in the introduction to this section. Recall that in Step \ref{it2}
we use the null quadric
\[{\cal V}^5:=\{z\in{\mathbb R}^6\backslash\{0\}\ :\ 2z_{-1}z_1+2z_{-2}z_2+2z_{-3}z_3=0\}.\]
Then, in Step \ref{it3}, we fix a section of the null quadric. We choose the section given by the equations
\[4 =2\left(z_{-1}z_1+z_{-2}z_2\right)=-2z_3z_{-3}.\]
We will call it the {\em spherical section}, because it coincides
with ${\mathbb S}^3(4)\times{\mathbb S}^1(-4)$. The superscript used for this section
will be ``${\rm sph}$'' for spherical.
In Step \ref{it4} we
introduce the coordinates
\begin{subequations}\label{coo}
\begin{align}
r& =\sqrt{2\left(z_{-1}z_1+z_{-2}z_2\right)}\;,&
w& =\frac{z_{-1}z_1}{z_{-1}z_1+z_{-2}z_{2}}\;,&
\\
u_1& =\frac{z_{1}}{\sqrt{z_{-1}z_{1}+z_{-2}z_2}} \;,&
u_2& =\frac{z_{2}}{\sqrt{z_{-1}z_1+z_{-2}z_{2}}}\;,\\
p& =\sqrt{-2z_3z_{-3}}\;,&
u_3& =\sqrt{-\frac{z_{3}}{z_{-3}}}.
\end{align}\label{subeq}
\end{subequations}
with the inverse transformation
\begin{subequations}\begin{align}
z_{-1}&=\frac{rw}{\sqrt2u_1},&z_1&=\frac{u_1r}{\sqrt2},\\
z_{-2}&=\frac{r(1-w)}{\sqrt2u_2},&z_2&=\frac{u_2r}{\sqrt2},\\
z_{-3}&=-\frac{p}{\sqrt2u_3},&z_3&=\frac{pu_3}{\sqrt2}. \end{align}\end{subequations}
The null quadric in these coordinates is given by
$r^2=p^2$. We will restrict ourselves to the sheet $r=p$.
The generator of dilations is
\[A_6=r\ddr+p\ddp.\]
The spherical section is given by the condition
$r^2=4$.
All the objects of the previous subsection will be now presented in the above coordinates after the reduction to the spherical section.
This reduction allows us to eliminate the variables $r,p$.
We omit the superscript $\eta$, whenever there is no dependence on this parameter.
\medskip
\noindent{\bf Lie algebra $\so(6)$.}
\noindent Cartan operators:
\begin{align*}
N_{1}^{\rm sph}& =u_1\ddu{1}\,,\\
N_{2}^{\rm sph}& =u_2\ddu{2}\,,\\
N_{3}^{\rm sph}& =u_3\ddu{3}\,.
\end{align*}
Roots:
\begin{align*}
B_{-2,-1}^{\rm sph}=u_1 u_2& \partial_w,\\
B_{2,1}^{\rm sph}=\frac{1}{u_1u_2}&\left((1-w)w \partial_w+(1-w)u_1 \partial_{u_1}-wu_2 \partial_{u_2}\right)\, ,\\
B_{2,-1}^{\rm sph}=\frac{u_1}{u_2}&\left((1-w) \partial_w-u_2 \partial_{u_2}\right)\, ,\\
B_{-2,1}^{\rm sph}=\frac{u_2}{u_1}&\left(w \partial_w+u_1 \partial_{u_1}\right)\, ;
\end{align*}
\begin{align*} B_{-3,-2}^{{\rm sph},{\eta}}{=}{-}u_2u_3&\left(w \partial_w+\frac12\big(u_1 \partial_{u_1}+u_2 \partial_{u_2}+u_3 \partial_{u_3}-\eta\big)\right)\, ,\\ B_{3,2}^{{\rm sph},{\eta}}{=}{-}\frac{1}{u_2u_3}&\left(w(w{-}1) \partial_w{+}\frac{(w{-}1)}{2}(u_1 \partial_{u_1}{+}u_2 \partial_{u_2}{-}u_3 \partial_{u_3}{-}\eta){+}u_2 \partial_{u_2}\right) ,\\
B_{3,-2}^{{\rm sph},{\eta}}=\frac{u_2}{u_3}&\left(w \partial_w+\frac12\big(u_1 \partial_{u_1}+u_2 \partial_{u_2}-u_3 \partial_{u_3}-\eta\big)\right)\, ,\\
B_{-3,2}^{{\rm sph},{\eta}}{=}\frac{ u_3}{u_2}&\left(w(w{-}1) \partial_w{+}\frac{(w{-}1)}{2}(u_1 \partial_{u_1}{+}u_2 \partial_{u_2}{+}u_3 \partial_{u_3}{-}\eta){+}u_2 \partial_{u_2}\right);
\end{align*}
\begin{align*}
B_{-3,-1}^{{\rm sph},{\eta}}{=} {-}u_1u_3&\left((w-1) \partial_w+\frac12\big(u_1 \partial_{u_1}+u_2 \partial_{u_2}+u_3 \partial_{u_3}-\eta\big)\right)\, ,\\ B_{3,1}^{{\rm sph},{\eta}}{=}\frac{1}{u_1u_3}&\left(w(w{-}1) \partial_w{+}\frac{w}{2}\big(u_1 \partial_{u_1}{+}u_2 \partial_{u_2}{-}u_3 \partial_{u_3}{-}\eta\big){-}u_1 \partial_{u_1}\right)\, ,\\
B_{3,-1}^{{\rm sph},{\eta}}=\frac{ u_1}{u_3}&\left((w-1) \partial_w+\frac12\big(u_1 \partial_{u_1}+u_2 \partial_{u_2}-u_3 \partial_{u_3}-\eta\big)\right)\, ;\\
B_{-3,1}^{{\rm sph},{\eta}}{=}{-}\frac{ u_3}{u_1}&\left(w(w{-}1) \partial_w{+}\frac{w}{2}\big(u_1 \partial_{u_1}{+}u_2 \partial_{u_2}{+}u_3 \partial_{u_3}{-}\eta\big)
{-}u_1 \partial_{u_1}\right).
\end{align*}
\noindent{\bf Weyl symmetries}
\begin{alignat*}{4}
\sigma_{123}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)& =&&f\left(w,u_1,u_2,u_3\right),\\
\sigma_{-12-3}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)&
=&&f\left(w,\frac{w}{u_1},u_2,\frac{1}{u_3}\right),\\
\sigma_{1-2-3}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)&
=&&f\left(w,u_1,\frac{1-w}{u_2},\frac{1}{u_3}\right),\\
\sigma_{-1-23}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)&
=&&f\left(w,\frac{w}{u_1},\frac{1-w}{u_2},u_3\right);\end{alignat*}\begin{alignat*}{4}
\sigma_{213}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)& =&&f\left(1-w,u_2,u_1,u_3\right),\\
\sigma_{-21-3}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)& =&&f\left(1-w,\frac{1-w}{u_2},u_1,\frac1{u_3}\right),\\
\sigma_{2-1-3}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)& =&&f\left(1-w,u_2,\frac{w}{u_1},\frac1{u_3}\right),\\
\sigma_{-2-13}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)& =&&f\left(1-w,\frac{1-w}{u_2},\frac{w}{u_1},u_3\right);\end{alignat*}\begin{alignat*}{4}
\sigma_{321}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)& =&
\left(\sqrt{-w}\right)^\eta&f\left(\frac{1}{w},\frac{u_3}{\sqrt{-w}}
,\frac{u_2}{\sqrt{-w}},\frac{u_1}{\sqrt{-w}}\right),\\
\sigma_{-32-1}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)& =&\left(\sqrt{-w}\right)^\eta&f\left(\frac{1}{w},\frac{1}{\sqrt{-w}u_3}
,\frac{u_2}{\sqrt{-w}},\frac{\sqrt{-w}}{u_1}\right),\\
\sigma_{3-2-1}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)& =&\left(\sqrt{-w}\right)^\eta&f\left(\frac{1}{w},\frac{u_3}{\sqrt{-w}}
,\frac{(w-1)}{\sqrt{-w} u_2},\frac{\sqrt{-w}}{u_1}\right),\\
\sigma_{-3-21}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)& =&\left(\sqrt{-w}\right)^\eta&f\left(\frac{1}{w},\frac{1}{\sqrt{-w}u_3}
,\frac{(w-1)}{\sqrt{-w}u_2},\frac{u_1}{\sqrt{-w}}\right);\end{alignat*}\begin{alignat*}{4}
\sigma_{312}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)& {=}&\left(\sqrt{w{-}1}\right)^\eta&f\left(\frac{1}{1{-}w},\frac{u_3}{\sqrt{w{-}1}},\frac{u_1}{\sqrt{w{-}1}},\frac{u_2}{\sqrt{w{-}1}}\right),\\
\sigma_{-31-2}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)& {=}&\left(\sqrt{w{-}1}\right)^\eta&f\left(\frac{1}{1{-}w},\frac{1}{\sqrt{w{-}1}u_3},\frac{u_1}{\sqrt{w{-}1}},\frac{\sqrt{w{-}1}}{u_2}\right),\\
\sigma_{3-1-2}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)& {=}&\left(\sqrt{w{-}1}\right)^\eta&f\left(\frac{1}{1{-}w},\frac{u_3}{\sqrt{w{-}1}},\frac{w}{\sqrt{w{-}1}u_1},\frac{\sqrt{w{-}1}}{u_2}\right),\\
\sigma_{-3-12}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)& {=}&\left(\sqrt{w{-}1}\right)^\eta&f\left(\frac{1}{1{-}w},\frac{1}{\sqrt{w{-}1}u_3},\frac{w}{\sqrt{w{-}1}u_1},\frac{u_2}{\sqrt{w{-}1}}\right);\end{alignat*}\begin{alignat*}{4}
\sigma_{231}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)&
=&\left(\sqrt{-w}\right)^\eta&f\left(\frac{w-1}{w},\frac{u_2}{\sqrt{-w}},\frac{u_3}{\sqrt{-w}},\frac{u_1}{\sqrt{-w}}\right),\\
\sigma_{-23-1}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)&
=&\left(\sqrt{-w}\right)^\eta&f\left(\frac{w-1}{w},\frac{(w-1)}{\sqrt{-w}u_2},\frac{u_3}{\sqrt{-w}},\frac{\sqrt{-w}}{u_1}\right),\\
\sigma_{2-3-1}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)&
=&\left(\sqrt{-w}\right)^\eta&f\left(\frac{w-1}{w},\frac{u_2}{\sqrt{-w}},\frac{1}{\sqrt{-w}u_3},\frac{\sqrt{-w}}{u_1}\right),\\
\sigma_{-2-31}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)&
=&\left(\sqrt{-w}\right)^\eta&f\left(\frac{w-1}{w},\frac{(w-1)}{\sqrt{-w}u_2},\frac{1}{\sqrt{-w}u_3},\frac{u_1}{\sqrt{-w}}\right);\end{alignat*}\begin{alignat*}{4}
\sigma_{132}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)& =&\left(\sqrt{w{-}1}\right)^\eta&f\left(\frac{w}{w{-}1},\frac{u_1}{\sqrt{w{-}1}},\frac{u_3}{\sqrt{w{-}1}},\frac{u_2}{\sqrt{w{-}1}}\right),\\
\sigma_{-13-2}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)& =&\left(\sqrt{w{-}1}\right)^\eta&f\left(\frac{w}{w{-}1},\frac{w}{\sqrt{w{-}1}u_1},\frac{u_3}{\sqrt{w{-}1}},\frac{\sqrt{w{-}1}}{u_2}\right),\\
\sigma_{1{-}3{-}2}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)& {=}&\left(\sqrt{w{-}1}\right)^\eta&f\left(\frac{w}{w{-}1},\frac{u_1}{\sqrt{w{-}1}},\frac{1}{\sqrt{w{-}1}u_3},\frac{\sqrt{w{-}1}}{u_2}\right),\\
\sigma_{{-}1{-}32}^{{\rm sph},{\eta}} f(w,u_1,u_2,u_3)& {=}&\left(\sqrt{w{-}1}\right)^\eta&f\left(\frac{w}{w{-}1},\frac{w}{\sqrt{w{-}1}u_1},\frac{1}{\sqrt{w{-}1}u_3},\frac{u_2}{\sqrt{w{-}1}}\right).
\end{alignat*}
\noindent{\bf Laplacian}
\begin{align}
\Delta_6^{\rm sph}=&\quad w(1-w) \partial_w^2-\big((1+u_1 \partial_{u_1})(w-1)+
(1+u_2 \partial_{u_2})w\big) \partial_w\notag\\
&-\frac14(u_1 \partial_{u_1}+u_2 \partial_{u_2}+1)^2
+\frac14(u_3 \partial_{u_3})^2.\label{lapiq2}
\end{align}
Let us give the computations that yield (\ref{lapiq2}). Using
\begin{align*} \partial_{z_{-1}}&=\frac{u_1}{\sqrt2r}\Big(-u_1 \partial_{u_1}-u_2 \partial_{u_2}+r \partial_r+2(1-w) \partial_w\Big),\\ \partial_{z_1}&=\frac{\sqrt2}{ru_1}\Big(\big(1-\frac{w}{2}\big)u_1 \partial_{u_1}-\frac{w}{2}u_2 \partial_{u_2}+\frac{w}{2}r \partial_r+w(1-w) \partial_w\Big),\\
\partial_{z_{-2}}&=\frac{u_2}{\sqrt2r}\Big(-u_1 \partial_{u_1}-u_2 \partial_{u_2}+r \partial_r-2w \partial_w\Big),\\ \partial_{z_2}&=\frac{\sqrt2}{ru_2}\Big(\frac{(w-1)}{2}u_1 \partial_{u_1}+\frac{(w+1)}{2}u_2 \partial_{u_2}+\frac{(1-w)}{2}r \partial_r+w(w-1) \partial_w\Big),\\
\partial_{z_{-3}}&=\frac{u_3}{\sqrt2 p}\Big(u_3 \partial_{u_3}-p \partial_p\Big),\\
\partial_{z_3}&=\frac{1}{\sqrt2 pu_3}\Big(u_3 \partial_{u_3}+p \partial_p\Big),
\end{align*}
we compute the Laplacian in coordinates (\ref{coo}):
\begin{align}
\Delta_6&
=\quad \frac1{r^2}\Big(4w(1-w) \partial_w^2-4\big((1+u_1 \partial_{u_1})(w-1)+
(1+u_2 \partial_{u_2})w\big) \partial_w\notag\\
&-(u_1 \partial_{u_1}+u_2 \partial_{u_2}+1)^2+(r \partial_r)^2+2r \partial_r+1\Big)\notag\\
&+\frac{1}{p^2}\Big((u_3 \partial_{u_3})^2-(p \partial_p)^2\Big).
\end{align}
Next we note that
\begin{equation}\frac1{r^2}\big((r \partial_r)^2+2r \partial_r-(p \partial_p)^2+1\big)=
\frac1{r^2}\big(r \partial_r-p \partial_p+1\big)\big(r \partial_r+p \partial_p+1\big).\label{lapiq3}\end{equation}
Using $p^2=r^2$ and $r \partial_r+p \partial_p=-1$, we see that (\ref{lapiq3}) is zero
on functions of degree $-1$.
Thus we obtain
\begin{align}
\Delta_6^\diamond ={}&\frac{4}{r^2}\Big(
w(1-w) \partial_w^2-\big((1+u_1 \partial_{u_1})(w-1)+
(1+u_2 \partial_{u_2})w\big) \partial_w\notag\\
&-\frac14(u_1 \partial_{u_1}+u_2 \partial_{u_2}+1)^2
+\frac14(u_3 \partial_{u_3})^2\Big).
\end{align}
To convert
$\Delta_6^\diamond$ into
the $\Delta_6^{\rm sph}$, we simply remove
the prefactor $\frac{4}{r^2}$.
\subsection{Hypergeometric equation}
\label{Hypergeometric equation}
Let us make the ansatz
\begin{equation}
f(u_1,u_2,u_3,w)=u_1^\alpha u_2^\beta u_3^\mu F(w).\label{form0}\end{equation}
Clearly,
\begin{subequations}\begin{eqnarray}
N_1^{\rm sph} f&=&\alpha f,\\
N_2^{\rm sph} f&=&\beta f,\\
N_3^{\rm sph} f&=&\mu f,\\
u_1^{-\alpha} u_2^{-\beta} u_3^{-\mu}\Delta_6^{\rm sph} f&
=& {\cal F}_{\alpha,\beta,\mu}(w, \partial_w)F(w),
\end{eqnarray}\end{subequations}
where
\begin{align}\notag
{\cal F}_{\alpha,\beta,\mu}(w, \partial_w):=
&w(1-w) \partial_w^2-\big((1+\alpha)(w-1)+
(1+\beta)w\big) \partial_w\\
& -\frac14(\alpha+\beta+1)^2
+\frac14\mu^2,\label{liea}\end{align}
which is the ${}_2{\cal F}_1$ {\em hypergeometric operator} in the Lie-algebraic parameters.
Traditionally, the {\em hypergeometric equation} is given by the operator
\begin{equation}
\mathcal{F}(a,b;c;w, \partial_w):=w(1-w) \partial_w^2+\big(c-(a+b+1)w\big) \partial_w-ab,\label{hy1-tra}\end{equation}
where $a,b,c\in{\mathbb C}$ will be called the
{
\em classical parameters}.
Here is the relationship between the Lie-algebraic and classical parameters:
\begin{subequations}\begin{alignat}{3}
\alpha:=c-1,&\ \ \beta: =a+b-c,&\ \ \mu:=a-b;\\[2ex]
\label{newnot}
a=\frac{1+\alpha+\beta +\mu}{2},&\ \ b=\frac{1+\alpha+\beta -\mu}{2},&\ \ c=1+\alpha.
\end{alignat}
\end{subequations}
Note that the Lie-algebraic parameters $\alpha,\beta,\mu$ are differences of the indices of the singular points $0,1,\infty$. For many purposes, they are more convenient than the traditional parameters $a,b,c$. They are used e.g. in Subsect. 2.7.2 of \cite{BE}, where they are called $\lambda,\nu,\mu$. In the standard notation for Jacobi Polynomials $P_n^{\alpha,\beta}$, the parameters $\alpha,\beta$ correspond to our $\alpha, \beta$ (where the singular points have been moved from $0,1$ to $-1,1$).
\subsection{Transmutation relations and discrete symmetries}
\label{Transmutation relations and discrete symmetries}
By \eqref{dada1}, we have the following generalized symmetries
\begin{subequations}
\begin{align}
B^{{\rm sph},-3}\Delta_6^{\rm sph}&=\Delta_6^{\rm sph} B^{{\rm sph},-1},\quad B\in \so(6),\label{siy1}\\
\alpha^{{\rm sph},-3}\Delta_6^{\rm sph}&=\Delta_6^{\rm sph} \alpha^{{\rm sph},-1},\quad \alpha\in \mathrm{O}(6).\label{siy2}
\end{align}
\end{subequations}
Applying (\ref{siy1}) to the roots of $\so(6)$ we obtain the {\em transmutation relations for the hypergeometric operator}:
\noindent
\begin{align*}
\partial_w&{\cal F}_{\alpha,\beta ,\mu }\notag\\
= {\cal F}_{\alpha+1,\beta +1,\mu } & \partial_w,\\
\big(w(1-w) \partial_w+(1-w)\alpha-w\beta \big)&{\cal F}_{\alpha,\beta ,\mu }\notag\\
= {\cal F}_{\alpha-1,\beta -1,\mu }&\big(w(1-w) \partial_w+(1-w)\alpha-w\beta \big),\\
\big((1-w) \partial_w -\beta \big)&{\cal F}_{\alpha,\beta ,\mu }\notag\\
= {\cal F}_{\alpha+1,\beta -1,\mu }& \big((1-w) \partial_w -\beta \big),\\
(w \partial_w+\alpha)&{\cal F}_{\alpha,\beta ,\mu }\notag
\\
= {\cal F}_{\alpha-1,\beta +1,\mu }&(w \partial_w+\alpha);
\end{align*}\begin{align*}
\Big(w \partial_w+\frac{1}{2} (\alpha+ \beta +\mu +1)\Big)&w{\cal F}_{\alpha,\beta ,\mu }\notag
\\
= w{\cal F}_{\alpha,\beta +1,\mu +1}&\Big(w \partial_w+\frac{1}{2} (\alpha+ \beta +\mu +1)\Big),\\
\Big(w(w{-}1) \partial_w{+}\frac12(w{-}1)(\alpha{+}\beta {-}\mu {+}1){-}\beta \Big)&w{\cal F}_{\alpha,\beta ,\mu }\notag\\
= w{\cal F}_{\alpha,\beta {-}1,\mu {-}1}\qquad&\hspace{-4ex}
\Big(w(w{-}1) \partial_w{+}\frac12(w{-}1)(\alpha{+}\beta {-}\mu {+}1){-}\beta \Big),\\
\Big( w \partial_w{+}\frac{1}{2}(\alpha{+}\beta {-}\mu {+}1)\Big)&w{\cal F}_{\alpha,\beta ,\mu }\notag\\
= w{\cal F}_{\alpha,\beta {+}1,\mu {-}1}&\Big( w \partial_w{+}\frac{1}{2}(\alpha{+}\beta {-}\mu {+}1\Big),
\\
\Big(w(w{-}1) \partial_w{-}\frac{1}{2}(1{-}w)(\alpha{+}\beta {+}\mu {+}1)
{+}\beta \Big)&w{\cal F}_{\alpha,\beta ,\mu }\notag\\
= w{\cal F}_{\alpha,\beta -1,\mu +1}\qquad&\hspace{-4ex}\Big(w(w{-}1) \partial_w{-}\frac{1}{2}(1{-}w)(\alpha{+}\beta {+}\mu {+}1)
{+}\beta \Big);\end{align*}\begin{align*}
\Big((w-1) \partial_w+\frac{1}{2}(\alpha+\beta +\mu +1)\Big)&(1-w){\cal F}_{\alpha,\beta ,\mu }\notag\\
= (1-w){\cal F}_{\alpha+1,\beta ,\mu +1}& \Big((w-1) \partial_w+\frac{1}{2}(\alpha+\beta +\mu +1\Big)
,
\\
\Big(w(w{-}1) \partial_w{+}\12w(\alpha{+}\beta {-}\mu {+}1)
{+}\alpha\Big)&(1-w){\cal F}_{\alpha,\beta ,\mu }\notag
\\
= (1-w){\cal F}_{\alpha-1,\beta ,\mu -1}&\Big(w(w{-}1) \partial_w{+}\12w(\alpha{+}\beta {-}\mu {+}1)
{+}\alpha\Big),
\\
\Big((w-1) \partial_w+\frac{1}{2}(\alpha+\beta -\mu +1)\Big)&
(1-w){\cal F}_{\alpha,\beta ,\mu }\notag
\\=
(1-w){\cal F}_{\alpha+1,\beta ,\mu -1}&\Big((w-1) \partial_w+\frac{1}{2}(\alpha+\beta -\mu +1)\Big),
\\
\Big(w(w{-}1) \partial_w{+}\12w(\alpha{+}\beta {+}\mu {+}1)
-\alpha\Big)&(1-w){\cal F}_{\alpha,\beta ,\mu }\notag
\\
= (1-w){\cal F}_{\alpha-1,\beta ,\mu +1}&\Big(w(w{-}1) \partial_w{+}\12w(\alpha{+}\beta {+}\mu {+}1)
{-}\alpha\Big).
\end{align*}
Applying (\ref{siy2}) to the Weyl group $D_3$ we obtain the {\em discrete symmetries of the hypergeometric operator}. We describe them below, restricting ourselves to $D_3\cap \SO(6)$.
All the operators below equal ${\cal F}_{\alpha,\beta ,\mu}(w, \partial_w)$ for the
corresponding $w$:
\noindent\[\begin{array}{rrcl}
w=v:&&
{\cal F}_{\alpha,\beta ,\mu}(v, \partial_v),&
\\[0ex]
&(-v)^{-\alpha}(v-1)^{-\beta }&{\cal F}_{-\alpha,-\beta ,\mu}(v, \partial_v)&(-v)^{\alpha}(v-1)^{\beta }
\\[0ex]
&(v-1)^{-\beta }&{\cal F}_{\alpha,-\beta ,-\mu}(v, \partial_v)&(v-1)^{\beta },\\[0ex]
&(-v)^{-\alpha}&{\cal F}_{-\alpha,\beta ,-\mu}(v, \partial_v)&
(-v)^{\alpha};\end{array}\]\[\begin{array}{rrcl}
w=1-v:&&{\cal F}_{\beta ,\alpha,\mu}(v, \partial_v),&
\\[0ex]
&(v-1)^{-\alpha}(-v)^{-\beta }&{\cal F}_{-\beta ,-\alpha,\mu}(v, \partial_v)&(v-1)^{\alpha}(-v)^{\beta },
\\[0ex]
&(v-1)^{-\alpha}&{\cal F}_{\beta ,-\alpha,-\mu}(v, \partial_v)&
(v-1)^{\alpha},\\[0ex]
&(-v)^{-\beta }&{\cal F}_{-\beta ,\alpha,-\mu}(v, \partial_v)
&(-v)^{\beta };\end{array}\]\[\begin{array}{rrcl}
&w=\frac{1}{v}:\quad(-v)^{\frac{\alpha+\beta +\mu +1}{2}}&(-v){\cal F}_{\mu ,\beta ,\alpha}(v, \partial_v)&
(-v)^{\frac{-\alpha-\beta -\mu -1}{2}},\\[0ex]
&(-v)^{\frac{\alpha+\beta -\mu +1}{2}}(v-1)^{-\beta }&
(-v){\cal F}_{-\mu ,-\beta ,\alpha}(v, \partial_v) &(-v)^{\frac{-\alpha-\beta +\mu -1}{2}}(v-1)^{\beta },
\\[0ex]
&(-v)^{\frac{\alpha+\beta +\mu +1}{2}}(v-1)^{-\beta }&
(-v)
{\cal F}_{\mu ,-\beta ,-\alpha}(v, \partial_v)& (-v)^{\frac{-\alpha-\beta -\mu -1}{2}}(v-1)^{\beta }
,\\[0ex]
&(-v)^{\frac{\alpha+\beta -\mu +1}{2}}&(-v){\cal F}_{-\mu ,\beta ,-\alpha}(v, \partial_v)
& (-v)^{\frac{-\alpha-\beta +\mu -1}{2}}
;\end{array}\]\[\begin{array}{rrcl}
&w=\frac{v-1}{v}:\quad(-v)^{\frac{\alpha+\beta +\mu +1}{2}}&
(-v){\cal F}_{\mu ,\alpha,\beta }(v, \partial_v)&(-v)^{\frac{-\alpha-\beta -\mu -1}{2}},\\[0ex]
&(-v)^{\frac{\alpha+\beta -\mu +1}{2}}(v-1)^{-\alpha}&
(-v){\cal F}_{-\mu ,-\alpha,\beta }(v, \partial_v) &(-v)^{\frac{-\alpha-\beta +\mu -1}{2}}(v-1)^{\alpha},\\[0ex]
&(-v)^{\frac{\alpha+\beta +\mu +1}{2}}(v-1)^{-\alpha}&
(-v) {\cal F}_{\mu ,-\alpha,-\beta }(v, \partial_v)& (-v)^{\frac{-\alpha-\beta -\mu -1}{2}}(v-1)^{\alpha},
\\[0ex]
&(-v)^{\frac{\alpha+\beta -\mu +1}{2}}&(-v){\cal F}_{-\mu ,\alpha,-\beta }(v, \partial_v)&
(-v)^{\frac{-\alpha-\beta +\mu -1}{2}};\end{array}\]\[\begin{array}{rrcl}
&w{=}\frac{1}{1{-}v}:\,
(v{-}1)^{\frac{\alpha+\beta +\mu +1}{2}}&
(v{-}1){\cal F}_{\beta ,\mu ,\alpha}(v, \partial_v)& (v{-}1)^{\frac{-\alpha-\beta -\mu -1}{2}},\\[0ex]
& (-v)^{-\beta }(v{-}1)^{\frac{\alpha+\beta -\mu +1}{2}}
&(v{-}1){\cal F}_{-\beta ,-\mu ,\alpha}(v, \partial_v) & (-v)^{\beta }(v{-}1)^{\frac{-\alpha-\beta +\mu -1}{2}},\\[0ex]
& (v{-}1)^{\frac{\alpha+\beta -\mu +1}{2}}
&(v{-}1){\cal F}_{\beta ,-\mu ,-\alpha}(v, \partial_v) & (v{-}1)^{\frac{-\alpha-\beta +\mu -1}{2}},\\[0ex]
&(-v)^{-\beta }(v{-}1)^{\frac{\alpha+\beta +\mu +1}{2}}&(v{-}1)
{\cal F}_{-\beta ,\mu ,-\alpha}(v, \partial_v)&
(-v)^{\beta }(v{-}1)^{\frac{-\alpha-\beta -\mu -1}{2}};
\end{array}\]\[\begin{array}{rrcl}
&w{=}\frac{v}{v{-}1}:\,
(v{-}1)^{\frac{\alpha+\beta +\mu+1}{2}}&(v{-}1){\cal F}_{\alpha,\mu,\beta }(v, \partial_v)&
(v{-}1)^{\frac{-\alpha-\beta -\mu-1}{2}}
,\\[0ex]
& (-v)^{-\alpha}(v{-}1)^{\frac{\alpha+\beta -\mu+1}{2}}&(v{-}1){\cal F}_{-\alpha,-\mu,\beta }(v, \partial_v) &
(-v)^{\alpha}(v{-}1)^{\frac{-\alpha-\beta +\mu-1}{2}}
,\\[0ex]
& (v{-}1)^{\frac{\alpha+\beta -\mu+1}{2}}&
(v{-}1){\cal F}_{\alpha,-\mu,-\beta }(v, \partial_v) & (v{-}1)^{\frac{-\alpha-\beta +\mu-1}{2}}
,\\[0ex]
& (-v)^{-\alpha}(v{-}1)^{\frac{\alpha+\beta +\mu +1}{2}}
&(v{-}1){\cal F}_{-\alpha,\mu ,-\beta }(v, \partial_v)&
(-v)^{\alpha}(v{-}1)^{\frac{-\alpha-\beta -\mu -1}{2}}.
\end{array}\]
\subsection{Factorizations of the Laplacian}
\label{Factorizations of the Laplacian}
In the Lie algebra $\so(6)$ represented on ${\mathbb R}^6$
we have 3 distinguished Lie subalgebras isomorphic to $\so(4)$:
\begin{equation}\so_{12}(4),\ \ \so_{23}(4),\ \ \so_{13}(4),\end{equation}
where we use a hopefully obvious notation.
By (\ref{casimir2}), the corresponding Casimir operators are
\begin{subequations}
\begin{eqnarray}
{\cal C}_{12}&=&
4B_{1,2}B_{-1,-2}-(N_1+N_2+1)^2+1\\
&=&4B_{-1,-2}B_{1,2}-(N_1+N_2-1)^2+1\\
&=&4B_{1,-2}B_{-1,2}-(N_1-N_2+1)^2+1\\
&=&4B_{-1,2}B_{1,-2}-(N_1-N_2-1)^2+1;\\
{\cal C}_{23}&=&
4B_{2,3}B_{-2,-3}-(N_2+N_3+1)^2+1\\
&=&4B_{-2,-3}B_{2,3}-(N_2+N_3-1)^2+1\\
&=&4B_{2,-3}B_{-2,3}-(N_2-N_3+1)^2+1\\
&=&4B_{-2,3}B_{2,-3}-(N_2-N_3-1)^2+1;\\
{\cal C}_{13}&=&
4B_{1,3}B_{-1,-3}-(N_1+N_3+1)^2+1\\
&=&4B_{-1,-3}B_{1,3}-(N_1+N_3-1)^2+1\\
&=&4B_{1,-3}B_{-1,3}-(N_1-N_3+1)^2+1\\
&=&4B_{-1,3}B_{1,-3}-(N_1-N_3-1)^2+1.
\end{eqnarray}\label{subu-}
\end{subequations}
Of course, for any $\eta$ we can append the superscript ${}^{\diamond,\eta}$ to all the operators in (\ref{subu-}).
After the reduction described in (\ref{deq1a}), we obtain the
identities
\begin{subequations}\label{facto}
\begin{eqnarray}\label{facto1--}
(2z_{-1}z_1+2z_{-2}z_2)\Delta_6^\diamond=-1+{\cal C}_{12}^{\diamond,-1}+(N_3^{\diamond,-1})^2,\\
\label{facto3--}
(2z_{-2}z_2+2z_{-3}z_3)\Delta_6^\diamond=-1+{\cal C}_{23}^{\diamond,-1}+(N_1^{\diamond,-1})^2,\\
\label{facto2--}(2z_{-1}z_1+2z_{-3}z_3)\Delta_6^\diamond=-1+{\cal C}_{13}^{\diamond,-1}+(N_2^{\diamond,-1})^2.
\end{eqnarray}
\end{subequations}
We insert (\ref{subu-}) with superscript ${}^{\diamond,-1}$ to
(\ref{facto}), obtaining
\begin{subequations}
\begin{align}\notag
&(2z_{-1}z_1+2z_{-2}z_2)\Delta_6^\diamond\\=&
4B_{1,2}B_{-1,-2}-(N_1+N_2+N_3+1)(N_1+N_2-N_3+1)\\
=&4B_{-1,-2}B_{1,2}-(N_1+N_2+N_3-1)(N_1+N_2-N_3-1)\\
=&4B_{1,-2}B_{-1,2}-(N_1-N_2+N_3+1)(N_1-N_2-N_3+1)\\
=&4B_{-1,2}B_{1,-2}-(N_1-N_2+N_3-1)(N_1-N_2-N_3-1);\\
&(2z_{-2}z_2+2z_{-3}z_3)\Delta_6^\diamond\notag\\
=&
4B_{2,3}B_{-2,-3}-(N_1+N_2+N_3+1)(-N_1+N_2+N_3+1)\\
=&4B_{-2,-3}B_{2,3}-(N_1+N_2+N_3-1)(-N_1+N_2+N_3-1)\\
=&4B_{2,-3}B_{-2,3}-(N_1+N_2-N_3+1)(-N_1+N_2-N_3+1)\\
=&4B_{-2,3}B_{2,-3}-(N_1+N_2-N_3-1)(-N_1+N_2-N_3-1);\\\notag
&(2z_{-1}z_1+2z_{-3}z_3)\Delta_6^\diamond\\=&
4B_{1,3}B_{-1,-3}-(N_1+N_2+N_3+1)(N_1-N_2+N_3+1)\\
=&4B_{-1,-3}B_{1,3}-(N_1+N_2+N_3-1)(N_1-N_2+N_3-1)\\
=&4B_{1,-3}B_{-1,3}-(N_1+N_2-N_3+1)(N_1-N_2-N_3+1)\\
=&4B_{-1,3}B_{1,-3}-(N_1+N_2-N_3-1)(N_1-N_2-N_3-1);
\end{align}\label{subu-.}
\end{subequations}
where for typographical reasons we omitted
the superscript ${}^{\diamond,{-}1}$ at all the operators $B$ and $N$.
If we use the coordinates \eqref{coo} and the spherical section, then we have to rewrite (\ref{subu-.})
by making the replacements
\begin{subequations}
\begin{align}\label{facto1b}
2z_{-1}z_1+2z_{-2}z_2&\to\quad1,\\
\label{facto3b}
2z_{-2}z_2+2z_{-3}z_3&\to\quad-w,\\
\label{facto2b}
2z_{-1}z_1+2z_{-3}z_3&\to\quad w-1,
\end{align}
\end{subequations}
as well as replacing the superscript ${}^{\diamond}$ with ${}^{{\rm sph}}$.
\subsection{Factorizations of the hypergeometric operator}
\label{Factorizations of the hypergeometric operator}
The factorizations of $\Delta_6^{\rm sph}$ described in Subsect.
\ref{Factorizations of the Laplacian} yield
the following factorizations of the hypergeometric operator:
\begin{align*}
&{\cal F}_{\alpha,\beta,\mu}\\=&
\Big(w(1-w)\partial_w+\big((1+\alpha)(1-w)-(1+\beta) w\big)\Big)\partial_w\\&-\frac14(\alpha+\beta+\mu+1)(\alpha+\beta-\mu+1)\\
=&
\partial_w\Big(w(1-w)\partial_w+\big(\alpha(1-w)-\beta w\big)\Big)\\&-\frac14(\alpha+\beta+\mu-1)(\alpha+\beta-\mu-1)\\
=&
\Big(w\partial_w+\alpha+1\Big)\Big((1-w)\partial_w-\beta\Big)\\
&-\frac14(\alpha-\beta+\mu+1)(\alpha-\beta-\mu+1)\\
=&
\Big((1-w)\partial_w-\beta-1\Big)\Big(w\partial_w+\alpha\Big)\\
&-\frac14(\alpha-\beta+\mu-1)(\alpha-\beta-\mu-1);
\end{align*}\begin{align*}
&w{\cal F}_{\alpha,\beta,\mu}\\=&
\Big(w\partial_w{+}\frac12(\alpha{+}\beta{+}\mu{-}1)\Big)
\Big(w(1{-}w)\partial_w{+}\frac12(1{-}w)(\alpha{+}\beta{-}\mu{+}1){-}\beta\Big)\\
&{-}\frac14(\alpha{+}\beta{+}\mu{-}1)(\alpha{-}\beta{-}\mu{+}1)\\
=&
\Big(w(1{-}w)\partial_w{+}\frac12(1{-}w)(\alpha{+}\beta{-}\mu{+}1){-}\beta{-}1\Big)
\Big(w\partial_w{+}\frac12(\alpha{+}\beta{+}\mu{+}1)\Big)
\\
&{-}\frac14(\alpha{+}\beta{+}\mu{+}1)(\alpha{-}\beta{-}\mu{-}1)\\
=&
\Big(w\partial_w{+}\frac12(\alpha{+}\beta{-}\mu{-}1)\Big)
\Big(w(1{-}w)\partial_w{+}\frac12(1{-}w)(\alpha{+}\beta{+}\mu{+}1){-}\beta\Big)\\
&{-}\frac14(\alpha{+}\beta{-}\mu{-}1)(\alpha{-}\beta{+}\mu{+}1)\\
=&
\Big(w(1{-}w)\partial_w{+}\frac12(1{-}w)(\alpha{+}\beta{+}\mu{+}1){-}\beta{-}1\Big)
\Big(w\partial_w{+}\frac12(\alpha{+}\beta{-}\mu{+}1)\Big)
\\
&{-}\frac14(\alpha{+}\beta{-}\mu{+}1)(\alpha{-}\beta{+}\mu{-}1);
\end{align*}\begin{align*}
&(w{-}1){\cal F}_{\alpha,\beta,\mu}\\
=&
\Big(w(w{-}1)\partial_w{+}\frac12w(\alpha{+}\beta{-}\mu{+}1){-}\alpha{-}1\Big)
\Big((w{-}1)\partial_w{+}\frac12(\alpha{+}\beta{+}\mu{+}1)\Big)
\\
&{-}\frac14(\alpha{+}\beta{+}\mu{+}1)(\alpha{-}\beta{+}\mu{+}1)\\
=&
\Big((w{-}1)\partial_w{+}\frac12(\alpha{+}\beta{+}\mu{-}1)\Big)
\Big(w(w{-}1)\partial_w{+}\frac12w(\alpha{+}\beta{-}\mu{+}1){-}\alpha\Big)\\
&{-}\frac14(\alpha{+}\beta{+}\mu{-}1)(\alpha{-}\beta{+}\mu{-}1)\\
=&
\Big(w(w{-}1)\partial_w{+}\frac12w(\alpha{+}\beta{+}\mu{+}1){-}\alpha{-}1\Big)
\Big((w{-}1)\partial_w{+}\frac12(\alpha{+}\beta{-}\mu{+}1)\Big)
\\
&{-}\frac14(\alpha{+}\beta{-}\mu{+}1)(\alpha{-}\beta{-}\mu{+}1)\\
=&
\Big((w{-}1)\partial_w{+}\frac12(\alpha{+}\beta{-}\mu{-}1)\Big)
\Big(w(w{-}1)\partial_w{+}\frac12w(\alpha{+}\beta{+}\mu{+}1){-}\alpha\Big)\\
&{-}\frac14(\alpha{+}\beta{-}\mu{-}1)(\alpha{-}\beta{-}\mu{-}1).
\end{align*}
\subsection{The ${}_2F_1$ hypergeometric function}
\label{The ${}_2F_1$ hypergeometric function}
$0$ is a regular singular point of
the ${}_2{\cal F}_1$ hypergeometric equation. Its indices are
$0$ and $1-c$.
For $c\neq 0,-1,-2,\dots$ the Frobenius method yields the unique solution of
the hypergeometric equation equal to $1$
at $0$, given by the series
\[F(a,b;c;w)=\sum_{j=0}^\infty
\frac{(a)_j(b)_j}{
(c)_j}\frac{w^j}{j!}\]
convergent for $|w|<1$. The function extends to the whole complex plane cut at $[1,\infty[$ and is
called the {\em hypergeometric function}.
Sometimes it is more convenient to consider the function
\[ {\bf F} (a,b;c;w):=\frac{F(a,b,c,w)}{\Gamma(c)}
=\sum_{j=0}^\infty
\frac{(a)_j(b)_j}{
\Gamma(c+j)}\frac{w^j}{j!}\]
defined for all $a,b,c\in{\mathbb C}$.
Another useful function proportional to $F$ is
\[ {\bf F}^{\rm\scriptscriptstyle I} (a,b;c;w):=\frac{\Gamma(b)\Gamma(c-b)}{\Gamma(c)}
F(a,b;c;w)
=\sum_{j=0}^\infty
\frac{\Gamma(b+j)\Gamma(c-b)(a)_j}{
\Gamma(c+j)}\frac{w^j}{j!}.
\]
We will usually prefer to parametrize all varieties of the
hypergeometric function with the Lie-algebraic parameters:
\begin{eqnarray*}
F_{\alpha,\beta ,\mu }(w)&=&F\Bigl(
\frac{1+\alpha+\beta +\mu}{2},\frac{1+\alpha+\beta -\mu}{2};1+\alpha;w\Bigr),\\
{\bf F}_{\alpha,\beta ,\mu }(w)&=&{\bf F} \Bigl(
\frac{1+\alpha+\beta +\mu}{2},\frac{1+\alpha+\beta -\mu}{2};1+\alpha;w\Bigr)\\
&=&
\frac{1}{\Gamma(\alpha+1)} F_{\alpha,\beta ,\mu }(w),\\
{\bf F}_{\alpha,\beta ,\mu }^{{\rm\scriptscriptstyle I}}(w)&=&{\bf F}^{{\rm\scriptscriptstyle I}}\Bigl(
\frac{1+\alpha+\beta +\mu}{2},\frac{1+\alpha+\beta -\mu}{2};1+\alpha;w\Bigr)\\
&=&
\frac{\Gamma\big(\frac{1+\alpha+\beta-\mu}{2}\big)\Gamma\big(\frac{1+\alpha-\beta+\mu}{2}\big)}{\Gamma(\alpha+1)}
F_{\alpha,\beta ,\mu }(w).
\end{eqnarray*}
\subsection{Standard solutions}
The hypergeometric equation has 3 singular points. With each of them we can associate two solutions with a simple behavior. Therefore, we obtain 6 standard solutions.
Applying the discrete symmetries from $D_3\cap \SO(6)$ to the hypergeometric function, we obtain 24 expressions for solutions of the hypergeometric equation, which go under the name of {\em Kummer's table}. Some of them coincide as functions, so that we obtain 6 standard solutions, each expressed in 4 ways:
\begin{align*}
\text{Solution $\sim1$ at $0$:}\quad& F_{\alpha,\beta ,\mu }(w)\\
=&(1-w)^{-\beta } F_{\alpha,-\beta ,-\mu }(w)\\
=&(1-w)^{\frac{-1-\alpha-\beta +\mu }{2}}
F_{\alpha,-\mu ,-\beta }(\frac{w}{w-1})\\
=&(1-w)^{\frac{-1-\alpha-\beta -\mu }{2}} F_{\alpha,\mu ,\beta }(\frac{w}{w-1});
\end{align*}\begin{align*}
\text{Solution $\sim w^{-\alpha}$ at $0$:}\quad&w^{-\alpha} F_{-\alpha,\beta ,-\mu }(w)\\
=&w^{-\alpha}(1-w)^{-\beta } F_{-\alpha,-\beta ,\mu }(w)\\
=&w^{-\alpha}(1-w)^{\frac{-1+\alpha-\beta +\mu }{2}}
F_{-\alpha,-\mu ,\beta }(\frac{w}{w-1})\\
=&w^{-\alpha}(1-w)^{\frac{-1+\alpha-\beta -\mu }{2}} F_{-\alpha,\mu ,-\beta }(\frac{w}{w-1});
\end{align*}\begin{align*}
\text{Solution $\sim1$ at $1$:}\quad
& F_{\beta ,\alpha,\mu }(1-w)\\
=&w^{-\alpha} F_{\beta ,-\alpha,-\mu }(1-w)\\
=&w^{\frac{-1-\alpha-\beta +\mu }{2}}
F_{\beta ,-\mu ,-\alpha}(1-w^{-1})\\
=&w^{\frac{-1-\alpha-\beta -\mu }{2}}
F_{\beta ,\mu ,\alpha}(1-w^{-1});
\end{align*}\begin{align*}
\text{Solution $\sim(1-w)^{-\beta }$ at $1$:}\quad&(1-w)^{-\beta } F_{-\beta ,\alpha,-\mu }(1-w)\\
=&w^{-\alpha}(1-w)^{-\beta } F_{-\beta ,-\alpha,\mu }(1-w)\\
=&w^{\frac{-1-\alpha+\beta -\mu }{2}}(1-w)^{-\beta }
F_{-\beta ,\mu ,-\alpha}(1-w^{-1})\\
=&w^{\frac{-1-\alpha+\beta +\mu }{2}}(1-w)^{-\beta }
F_{-\beta ,-\mu ,\alpha}(1-w^{-1});
\end{align*}\begin{align*}
\text{Solution $\sim w^{-a}$ at $\infty$:}\quad&(-w)^{\frac{-1-\alpha-\beta -\mu }{2}} F_{\mu ,\beta ,\alpha}(w^{-1})\\
=&(-w)^{\frac{-1-\alpha+\beta -\mu }{2}}(1-w)^{-\beta } F_{\mu ,-\beta ,-\alpha}(w^{-1})\\
=&(1-w)^{\frac{-1-\alpha-\beta -\mu }{2}} F_{\mu ,\alpha,\beta }((1-w)^{-1})\\
=&(-w)^{-\alpha}(1-w)^{\frac{-1+\alpha-\beta -\mu }{2}} F_{\mu ,-\alpha,-\beta }((1-w)^{-1});
\end{align*}\begin{align*}
\text{Solution $\sim w^{-b}$ at $\infty$:}\quad
&(-w)^{\frac{-1-\alpha-\beta +\mu }{2}} F_{-\mu ,\beta ,-\alpha}(w^{-1})\\
=&(-w)^{\frac{-1-\alpha+\beta +\mu }{2}}(1-w)^{-\beta } F_{-\mu ,-\beta ,\alpha}(w^{-1})\\
=&(1-w)^{\frac{-1-\alpha-\beta +\mu }{2}} F_{-\mu ,\alpha,-\beta }((1-w)^{-1})\\
=&(-w)^{-\alpha}(1-w)^{\frac{-1+\alpha-\beta +\mu }{2}} F_{-\mu ,-\alpha,\beta }((1-w)^{-1}).
\end{align*}
\subsection{Recurrence relations}
\label{s3.15}
To each root of $\so(6)$ there corresponds a recurrence relation:
\begin{align*}
\partial_w {\bf F}^{\rm\scriptscriptstyle I} _{\alpha,\beta ,\mu }(w)&=
\frac{1{+}\alpha{+}\beta {+}\mu }{2} {\bf F}^{\rm\scriptscriptstyle I} _{\alpha+1,\beta +1,\mu }(w),\\[0ex]
-(w(1{-}w) \partial_w{+}\alpha(1{-}w){-}\beta w) {\bf F}^{\rm\scriptscriptstyle I} _{\alpha,\beta ,\mu }(w)&=
\frac{1{-}\alpha{-}\beta{+}\mu}{2}{\bf F}^{\rm\scriptscriptstyle I} _{\alpha-1,\beta -1,\mu }(w),\\[0ex]
((1-w) \partial_w-\beta ) {\bf F}^{\rm\scriptscriptstyle I} _{\alpha,\beta ,\mu }(w)&=\frac{1{+}\alpha{-}\beta {-}\mu }{2}{\bf F}^{\rm\scriptscriptstyle I} _{\alpha{+}1,\beta {-}1,\mu }(w),\\[0ex]
- (w \partial_w+\alpha) {\bf F}^{\rm\scriptscriptstyle I} _{\alpha,\beta ,\mu }(w)&= \frac{1{-}\alpha{+}\beta{-}\mu}{2}{\bf F}^{\rm\scriptscriptstyle I} _{\alpha-1,\beta +1,\mu }(w);
\end{align*}
\begin{align*}
\left(w \partial_w+\frac{1+\alpha+\beta +\mu }{2}\right) {\bf F}^{\rm\scriptscriptstyle I} _{\alpha,\beta ,\mu }(w)&=
\frac{1{+}\alpha{+}\beta {+}\mu }{2} {\bf F}^{\rm\scriptscriptstyle I} _{\alpha,\beta +1,\mu +1}(w)
, \\[0ex]
{-}\! \left(\!w(w{-}1) \partial_w{+}\beta {+}\frac{1{+}\alpha{+}\beta {-}\mu }{2}(w{-}1)\!
\right)\! {\bf F}^{\rm\scriptscriptstyle I} _{\alpha,\beta ,\mu }(w)&=\frac{1{+}\alpha{-}\beta {-}\mu }{2} {\bf F}^{\rm\scriptscriptstyle I} _{\alpha,\beta -1,\mu -1}(w),\\[0ex]
- \left(w \partial_w+\frac{1+\alpha+\beta -\mu }{2}\right) {\bf F}^{\rm\scriptscriptstyle I} _{\alpha,\beta ,\mu }(w)&=
\frac{1{-}\alpha{+}\beta{-}\mu }{2} {\bf F}^{\rm\scriptscriptstyle I} _{\alpha,\beta +1,\mu -1}(w)
, \\[0ex]
\left(w(w{-}1) \partial_w{+}\beta {+}\frac{1{+}\alpha{+}\beta {+}\mu }{2}(w{-}1)\right) {\bf F}^{\rm\scriptscriptstyle I} _{\alpha,\beta ,\mu }(w)&=
\frac{1{-}\alpha{-}\beta{+}\mu }{2}
{\bf F}^{\rm\scriptscriptstyle I} _{\alpha,\beta -1,\mu +1}(w)
;\end{align*}
\begin{align*}
\left((w-1) \partial_w+\frac{1+\alpha+\beta +\mu }{2}\right) {\bf F}^{\rm\scriptscriptstyle I} _{\alpha,\beta ,\mu }(w)&=
\frac{1{+}\alpha{+}\beta {+}\mu }{2}
{\bf F}^{\rm\scriptscriptstyle I} _{\alpha{+}1,\beta ,\mu {+}1}(w)
, \\[0ex]
\left(w(w-1) \partial_w{-}\alpha{+}\frac{1{+}\alpha{+}\beta {-}\mu }{2}w\right)
{\bf F}^{\rm\scriptscriptstyle I} _{\alpha,\beta ,\mu }(w)&=\frac{1{-}\alpha{+}\beta{-}\mu}{2} {\bf F}^{\rm\scriptscriptstyle I} _{\alpha-1,\beta ,\mu -1}(w),
\\[0ex]
\left((w-1) \partial_w+\frac{1+\alpha+\beta -\mu }{2}\right)
{\bf F}^{\rm\scriptscriptstyle I} _{\alpha,\beta ,\mu }(w)&=\frac{1{+}\alpha{-}\beta {-}\mu }{2}
{\bf F}^{\rm\scriptscriptstyle I} _{\alpha{+}1,\beta ,\mu {-}1}(w)
, \\[0ex]
\left(w(w-1) \partial_w{-}\alpha{+}\frac{1{+}\alpha{+}\beta {+}\mu }{2}w\right) {\bf F}^{\rm\scriptscriptstyle I} _{\alpha,\beta ,\mu }(w)&=\frac{1{-}\alpha{-}\beta{+}\mu}{2} {\bf F}^{\rm\scriptscriptstyle I} _{\alpha-1,\beta ,\mu +1}(w)
.
\end{align*}
The recurrence relations are essentially fixed by the transmutation relations. The only missing piece of information is the coefficient on the right hand side, which can be derived by analyzing the behavior of both sides around zero. Another way to obtain these coefficients is to use the integral representations described in the following subsections.
\subsection{Wave packets in 6 dimensions}
\label{Wave packets in 6 dimensions}
We start with the following easy fact:
\begin{lemma} For any $\tau$, the following function is harmonic on ${\mathbb R}^6$:
\begin{equation}
(z_1-\tau^{-1}z_{-2})^{\alpha +\nu}(z_2+\tau^{-1}z_{-1})^{\beta +\nu}z_3^\mu
\label{cacy}\end{equation}
\label{cacko}\end{lemma}
{\bf Proof.}\ \
Set
$e_1:=(1,0,0,-\tau^{-1})$, $e_2:=(0,\tau^{-1},1,0)$.
Then
\[\langle e_1|e_1\rangle=\langle e_2|e_2\rangle=\langle e_2|e_1\rangle=0.\]
Hence, (\ref{cacy}) is harmonic by Prop. \ref{isotro}. $\Box$\medskip
Let us make a wave packet out of
(\ref{cacy}), which is an eigenfunction of the Cartan operators:
\begin{align}\notag
&K_{\alpha,\beta ,\mu,\nu}(z_{-1},z_1,z_{-2},z_2,z_{-3},z_{3})\\
:=&\int_\gamma(z_1-\tau^{-1}z_{-2})^{\alpha+\nu}(z_2+\tau^{-1}z_{-1})^{\beta +\nu}
z_3^{\mu}\tau^{\nu-1}\frac{{\rm d}\tau}{2\pi{\rm i}}.\label{wave}
\end{align}
\begin{proposition} \label{caca7}
Let the contour $]0,1[\ni s\overset{\gamma}\mapsto\tau(s)$
satisfy
\begin{align} (z_1-\tau^{-1}z_{-2})^{\alpha+\nu}(z_2+\tau^{-1}z_{-1})^{\beta +\nu}
\tau^{\nu-1}\Big|_{\tau(0)}^{\tau(1)}&=0.\label{condi2}
\end{align}
Then $ K_{\alpha,\beta ,\mu,\nu}$ is harmonic and
\begin{subequations}\begin{align}
\label{cac1.}
N_1K_{\alpha,\beta ,\mu,\nu}&=\alpha K_{\alpha,\beta ,\mu,\nu},\\
\label{cac2.} N_2K_{\alpha,\beta ,\mu,\nu}&=\beta K_{\alpha,\beta ,\mu,\nu},\\
\label{cac00.} N_3K_{\alpha,\beta ,\mu,\nu}&=\mu K_{\alpha,\beta ,\mu,\nu}.
\end{align}\end{subequations}
\end{proposition}
{\bf Proof.}\ \ $ K_{\alpha,\beta ,\mu,\nu}$ is harmonic by Lemma \ref{cacko}.
Writing
\begin{subequations} \begin{align}
K_{\alpha,\beta,\mu ,\nu}(z)&=
\int_\gamma(\tau z_1-z_{-2})^{\alpha+\nu}(z_2+\tau^{-1}z_{-1})^{\beta +\nu}z_3^\mu
\tau^{-\alpha-1}\frac{{\rm d}\tau}{2\pi{\rm i}}\\
&=
\int_\gamma( z_1-\tau^{-1}z_{-2})^{\alpha+\nu}(\tau z_2+z_{-1})^{\beta +\nu}z_3^\mu
\tau^{-\beta -1}\frac{{\rm d}\tau}{2\pi{\rm i}},
\end{align}\end{subequations}
we see that (\ref{cac1.}) and (\ref{cac2.}) follow from
assumption (\ref{condi2}) by
Prop. \ref{cac0}. (\ref{cac00.}) is obvious. $\Box$\medskip
\begin{proposition} If in addition to (\ref{condi2}) we assume that \begin{align}
(z_1-\tau^{-1}z_{-2})^{\alpha+\nu}(z_2+\tau^{-1}z_{-1})^{\beta +\nu}
\tau^{\nu}\Big|_{\tau(0)}^{\tau(1)}&=0,\label{condi1}\end{align}
and that we are allowed to differentiate under the integral sign, we obtain the recurrence relations
\begin{subequations}\begin{align}
\label{cac3.} B_{-12}K_{\alpha,\beta ,\mu,\nu}&=(\beta +\nu) K_{\alpha+1,\beta -1,\mu,\nu},\\
\label{cac4.} B_{1-2}K_{\alpha,\beta ,\mu,\nu}&=-(\alpha+\nu) K_{\alpha-1,\beta +1,\mu,\nu},\\
\label{cac5.} B_{12}K_{\alpha,\beta ,\mu,\nu}&=(\nu+1) K_{\alpha-1,\beta -1,\mu,\nu+1},\\
\label{cac6.} B_{-1-2}K_{\alpha,\beta ,\mu,\nu}&=-(\alpha+\beta +\nu+1) K_{\alpha+1,\beta +1,\mu,\nu-1},\\\label{cac7.}
B_{1-3}K_{\alpha,\beta ,\mu,\nu}&=-(\alpha+\nu)K_{\alpha-1,\beta ,\mu+1,\nu},\\
\label{cac8.}B_{-1-3}K_{\alpha,\beta ,\mu,\nu}&=-(\beta +\nu)K_{\alpha+1,\beta ,\mu+1,\nu-1},\\\label{cac9.}
B_{2-3}K_{\alpha,\beta ,\mu,\nu}&=-(\beta+\nu)K_{\alpha,\beta-1 ,\mu+1,\nu},\\
\label{cac10.}B_{-2-3}K_{\alpha,\beta ,\mu,\nu}&=(\alpha +\nu)K_{\alpha,\beta+1 ,\mu+1,\nu-1}.\end{align}
\label{caco} \end{subequations}
\end{proposition}
{\bf Proof.}\ \ Relations
(\ref{cac3.}), (\ref{cac4.}),
(\ref{cac7.}), (\ref{cac8.}), (\ref{cac9.}) and (\ref{cac10.})
are elementary. They follow by simple differentiation under the integral sign and do not need assumptions (\ref{condi1}) and (\ref{condi2}).
Relations (\ref{cac5.}) and (\ref{cac6.}) require assumption (\ref{condi2}) and follow by the following computations:
\begin{align}\label{caca1}
&B_{12}
(z_1-\tau^{-1}z_{-2})^{\alpha+\nu}(z_2+\tau^{-1}z_{-1})^{\beta +\nu}
\tau^{\nu+1}\\\notag
=\,&\partial_{\tau^{-1}}
(z_1-\tau^{-1}z_{-2})^{\alpha+\nu}(z_2+\tau^{-1}z_{-1})^{\beta +\nu}
\tau^{\nu+1}\\\notag
&+(\nu+1)(z_1-\tau^{-1}z_{-2})^{\alpha+\nu}(z_2+\tau^{-1}z_{-1})^{\beta +\nu}
\tau^{\nu},\\\label{caca2}
&B_{-1-2}
(\tau z_1-z_{-2})^{\alpha+\nu}(\tau z_2+z_{-1})^{\beta +\nu}
\tau^{-\alpha-\beta -\nu-1}\\\notag
=&-\partial_\tau
(\tau z_1-z_{-2})^{\alpha+\nu}(\tau z_2+z_{-1})^{\beta +\nu}
\tau^{-\alpha-\beta -\nu-1}\\\notag
&-(\alpha+\beta +\nu+1)
(\tau z_1-z_{-2})^{\alpha+\nu}(\tau z_2+z_{-1})^{\beta +\nu}
\tau^{-\alpha-\beta -\nu-2},
\end{align}
where in (\ref{caca2}) we used yet another representation:
\begin{equation}
K_{\alpha,\beta,\mu ,\nu}(z):=
\int_\gamma(\tau z_1-z_{-2})^{\alpha+\nu}(\tau z_2+z_{-1})^{\beta +\nu}z_3^\mu
\tau^{-\alpha-\beta -\nu-1}\frac{{\rm d}\tau}{2\pi{\rm i}}.
\end{equation}
$\Box$\medskip
If in addition
\[\nu=\frac{-\alpha-\beta-\mu-1}{2},\]
then (\ref{wave}) is homogeneous of degree $-1$, so that we can reduce
it to 4 dimensions.
Let us substitute the coordinates (\ref{subeq}), and then set $\tau=\frac{s}{u_1u_2}$, $s=t-w$:
\begin{equation}
K_{\alpha,\beta,\mu,\nu}(u_1,u_2,u_3,r,p,w)=
2^{\frac12} u_1^\alpha
u_2^\beta
u_3^\mu
r^{-\mu-1}p^\mu F(w),\end{equation}
\begin{align}\notag
F(w)&=\int_\gamma(s-1+w)^{\frac{\alpha-\beta-\mu-1}{2}}
(s+w)^{\frac{-\alpha+\beta-\mu-1}{2}}s^{\frac{-\alpha-\beta+\mu-1}{2}}{\rm d} s\\
&=\int_\gamma(t-1)^{\frac{\alpha-\beta-\mu-1}{2}}
t^{\frac{-\alpha+\beta-\mu-1}{2}}(t-w)^{\frac{-\alpha-\beta+\mu-1}{2}}{\rm d} t.
\label{cacy1}\end{align}
On the spherical section we can remove $r$ and $p$.
Therefore, the function $F$ given by (\ref{cacy1})
satisfies the hypergeometric equation: \begin{equation}
{\cal F}_{\alpha,\beta,\mu}(w, \partial_w)F(w)=0.\label{pwe}\end{equation}
From (\ref{caco}) we can also easily obtain the recurrence relations for $F$. Note that in this list the recurrence relations corresponding to
$B_{1,3}$, $B_{-1,3}$, $B_{2,3}$ and $B_{-2,3}$ are missing.
However, they can be obtained after the reduction to $4$ dimensions by an application of the factorization formulas.
\subsection{Integral representations}
\label{Integral representations}
Below we independently prove (\ref{pwe}),
without going through the additional variables. We will use the classical parameters.
\begin{theoreme}
Let $[0,1]\ni \tau \overset{\gamma}\mapsto t(\tau)$ satisfy
\[t^{a-c+1}(1-t)^{c-b}(t-w)^{-a-1}\Big|_{t(0)}^{t(1)}=0.\]
Then
\begin{equation}
{\cal F}(a,b;c;w, \partial_w)
\int_\gamma t^{a-c}(1-t)^{c-b-1}(t-w)^{-a}{\rm d} t=0.
\label{f4}\end{equation}
\label{intr}\end{theoreme}
{\bf Proof.}\ \
We check that for any contour $\gamma$
\[\text{lhs of (\ref{f4})}=-a \int_\gamma{\rm d} t\,
\partial_t t^{a-c+1}(1-t)^{c-b}(t-w)^{-a-1}
.\]
$\Box$\medskip
Analogous (and nonequivalent)
integral representations can be obtained by interchanging $a$ and $b$
in Theorem \ref{intr}.
The hypergeometric function with the type I normalization has
the integral representation
\begin{eqnarray}\label{eqa1}
&&\int_1^\infty t^{a-c}(t-1)^{c-b-1}(t-w)^{-a}{\rm d} t\\
&=&
{\bf F}^{\rm\scriptscriptstyle I} (a,b;c;w),\ \ \ \ \Re(c-b)>0,\ \Re b>0,\ \ \ w\not\in[1,\infty[.
\nonumber\end{eqnarray}
Indeed, by Theorem \ref{intr} the left hand side of (\ref{eqa1})
is annihilated by
the hypergeometric operator
(\ref{hy1-tra}). Besides, by Euler's identity it equals
$\frac{\Gamma(b)\Gamma(c-b)}{\Gamma(c)}$ at $0$. So does
the right hand side. Therefore, (\ref{eqa1}) follows by the uniqueness of the solution by the Frobenius method.
\subsection{Integral representations of standard solutions}
\label{Integral representations of standard solutions}
The integrand of \eqref{f4}
has four singularities: $\{0,1,\infty,w\}$. It is natural to chose
$\gamma$ as the interval joining a pair of singularities. This choice leads to $6$ standard solutions with the $\rm I$-type normalization:
\begin{align*}
\text{ $\sim1$ at $0$:}\quad&[1,\infty];\\
\text{ $\sim w^{-\alpha}$ at $0$:}\quad&[0,w];\\
\text{ $\sim1$ at $1$:}\quad
&[0,\infty];\\
\text{ $\sim(1-w)^{-\beta }$ at $1$:}\quad&[1,w];\\
\text{ $\sim w^{-a}$ at $\infty$:}\quad&[w,\infty];\\
\text{ $\sim w^{-b}$ at $\infty$:}\quad
&[0,1].
\end{align*}
Below we give explicit formulas.
To highlight their symmetry, we use Lie-algebraic parameters.
\begin{align}
\Re(1+\alpha)> |\Re(\beta -\mu )|:&\\
\int_1^\infty t^{\frac{-1-\alpha+\beta +\mu }{2}}
(t-1)^{\frac{-1+\alpha-\beta +\mu }{2}}(t-w)^{\frac{-1-\alpha-\beta -\mu }{2}}{\rm d} t&\notag\\
= {\bf F}^{\rm\scriptscriptstyle I} _{\alpha,\beta ,\mu }(w),&\quad
w\not\in[1,\infty[;\nonumber\end{align}\begin{align}
\Re(1-\alpha)> |\Re(\beta -\mu )|:&\\
\int_0^w t^{\frac{-1-\alpha+\beta +\mu }{2}}
(1-t)^{\frac{-1+\alpha-\beta +\mu }{2}}(w-t)^{\frac{-1-\alpha-\beta -\mu }{2}}{\rm d} t&\notag\\
=
w^{-\alpha} {\bf F}^{\rm\scriptscriptstyle I} _{-\alpha,\beta ,-\mu }(w),&\quad w\not\in]{-}\infty,0]{\cup}[1,\infty[
,\nonumber\notag\\
\int_w^0 (-t)^{\frac{-1-\alpha+\beta +\mu }{2}}
(1-t)^{\frac{-1+\alpha-\beta +\mu }{2}}(t-w)^{\frac{-1-\alpha-\beta -\mu }{2}}{\rm d} t&\notag\\
=
(-w)^{-\alpha} {\bf F}^{\rm\scriptscriptstyle I} _{-\alpha,\beta ,-\mu }(w),&\quad w\not\in[0,\infty[
;\nonumber\end{align}\begin{align}
\Re(1+\beta )> |\Re(\alpha-\mu )|:&\\
\int_{-\infty}^0 (-t)^{\frac{-1-\alpha+\beta +\mu }{2}}
(1-t)^{\frac{-1+\alpha-\beta +\mu }{2}}(w-t)^{\frac{-1-\alpha-\beta -\mu }{2}}{\rm d} t
&\notag\\=
{\bf F}^{\rm\scriptscriptstyle I} _{\beta ,\alpha ,\mu }(1-w),\quad
&w\not\in]-\infty,0];\notag\end{align}\begin{align}
\Re(1-\beta )> |\Re(\alpha+\mu )|:&\\
\int_w^1 t^{\frac{-1-\alpha+\beta +\mu }{2}}
(1-t)^{\frac{-1+\alpha-\beta +\mu }{2}}(t-w)^{\frac{-1-\alpha-\beta -\mu }{2}}{\rm d} t
&\notag\\=(1-w)^{-\beta } {\bf F}^{\rm\scriptscriptstyle I} _{-\beta ,\alpha,-\mu }(1-w),
&\quad w\not\in]-\infty,0]\cup[1,\infty[,\notag\end{align}\begin{align}
\int_1^w t^{\frac{-1-\alpha+\beta +\mu }{2}}
(t-1)^{\frac{-1+\alpha-\beta +\mu }{2}}(w-t)^{\frac{-1-\alpha-\beta -\mu }{2}}{\rm d} t\notag\\
=(w-1)^{-\beta } {\bf F}^{\rm\scriptscriptstyle I} _{-\beta ,\alpha,-\mu }(1-w),&\quad
w\not\in]-\infty,1];\notag\end{align}\begin{align}
\Re(1-\mu )> |\Re(\alpha+\beta )|:&\\
\int_w^\infty t^{\frac{-1-\alpha+\beta +\mu }{2}}
(t-1)^{\frac{-1+\alpha-\beta +\mu }{2}}(t-w)^{\frac{-1-\alpha-\beta -\mu }{2}}{\rm d} t&\notag\\
=
w^{\frac{-1-\alpha-\beta {+}\mu }{2}} {\bf F}^{\rm\scriptscriptstyle I} _{-\mu ,\beta ,-\alpha}(w^{-1})
,&\quad
&w\not\in]-\infty,1],\notag\end{align}\begin{align}
\int_{-\infty}^w (-t)^{\frac{-1-\alpha+\beta +\mu }{2}}
(1-t)^{\frac{-1+\alpha-\beta +\mu }{2}}(w-t)^{\frac{-1-\alpha-\beta -\mu }{2}}{\rm d} t&\notag\\
=
(-w)^{\frac{-1-\alpha-\beta {+}\mu }{2}} {\bf F}^{\rm\scriptscriptstyle I} _{-\mu ,\beta ,-\alpha}(w^{-1})
,&\quad
w\not\in]0,\infty];\notag\end{align}\begin{align}
\Re(1+\mu )> |\Re(\alpha-\beta )|:&\\
\int_0^1 t^{\frac{-1-\alpha+\beta -\mu }{2}}
(1-t)^{\frac{-1+\alpha-\beta +\mu }{2}}(t-w)^{\frac{-1-\alpha-\beta -\mu }{2}}{\rm d} t&\notag\\
=
(-w)^{\frac{-1-\alpha-\beta {-}\mu }{2}} {\bf F}^{\rm\scriptscriptstyle I} _{\mu ,\beta ,\alpha}(w^{-1}),
&\quad w\not\in[0,\infty[,\notag\\
\int_0^1 t^{\frac{-1-\alpha+\beta -\mu }{2}}
(1-t)^{\frac{-1+\alpha-\beta +\mu }{2}}(w-t)^{\frac{-1-\alpha-\beta -\mu }{2}}{\rm d} t&\notag\\=
w^{\frac{-1-\alpha-\beta {-}\mu }{2}} {\bf F}^{\rm\scriptscriptstyle I} _{\mu ,\beta ,\alpha}(w^{-1}),&
\quad w\not\in[-\infty,1[.\notag
\end{align}
\subsection{Connection formulas}
\label{Connection formulas}
Generically, each pair of standard solution is a basis of solutions to the hypergeometric equation.
For instance, we can use the pair of solutions $\sim 1$ and $\sim w^{-\alpha}$ at $0$
as one basis, and the pair $\sim w^{-a}$ and $\sim w^{-b}$ as another basis. We also assume that
$ w\not\in[0,\infty[$.
Introduce the matrix
\[
A_{\alpha,\beta,\mu}:= \frac{\pi}{\sin(\pi\mu)}\left[\begin{array}{cc}\frac{-1}
{\Gamma\left(\frac{1+\alpha+\beta -\mu }{2}\right)
\Gamma\left(\frac{1+\alpha-\beta -\mu }{2}\right)}&
\frac{1}
{\Gamma\left(\frac{1+\alpha+\beta +\mu }{2}\right)
\Gamma\left(\frac{1+\alpha-\beta +\mu }{2}\right)}\\[2.5ex]
\frac{-1}
{\Gamma\left(\frac{1-\alpha-\beta -\mu }{2}\right)
\Gamma\left(\frac{1-\alpha+\beta -\mu }{2}\right)}&
\frac{1}{\Gamma\left(\frac{1-\alpha-\beta +\mu }{2}\right)
\Gamma\left(\frac{1-\alpha+\beta +\mu }{2}\right)}
\end{array}\right].
\]
Then
\begin{align}\label{matri2}&\left[\begin{array}{c} {\bf F} _{\alpha,\beta ,\mu }(w)\\[2.5ex]
(-w)^{-\alpha} {\bf F} _{-\alpha,\beta ,-\mu }(w)
\end{array}\right]\\=\quad &A_{\alpha,\beta,\mu}
\left[\begin{array}{c}
(-w)^{\frac{-1-\alpha-\beta -\mu }{2}} {\bf F} _{\mu ,\beta ,\alpha}(w^{-1})\\[2.5ex]
(-w)^{\frac{-1-\alpha-\beta +\mu }{2}} {\bf F} _{-\mu ,\beta ,-\alpha}(w^{-1})
\end{array}\right]
.\notag
\end{align}
Note that in the Lie-algebraic parameters the matrix $A_{\alpha,\beta,\mu}$ has a very symmetric form. Here are some of its properties:
\begin{align}
A_{\alpha,\beta,\mu}=A_{\alpha,-\beta,\mu}&=-
\begin{bmatrix}0&1\\1&0
\end{bmatrix}A_{-\alpha,\beta,-\mu}
\begin{bmatrix}0&1\\1&0
\end{bmatrix}=A_{\mu,\beta,\alpha}^{-1},\\
\det A_{\alpha,\beta,\mu}&=-\frac{\sin(\pi\alpha)}{\sin(\pi\mu)}.
\end{align}
Relation (\ref{matri2}) can be derived from the integral representations. Indeed,
consider $\Im w<0$. Take the branches of the powers of $-t$ and $1-t$ and $w-t$ continued from the left clockwise onto the upper halfplane.
Then (under some conditions on $\alpha,\beta,\mu$) we can write
\begin{align*}
\Bigg(\int_{-\infty}^0+\int_0^1+\int_1^{+\infty}\Bigg)
(- t)^{\frac{-1-\alpha+\beta \pm\mu }{2}}
(1-t)^{\frac{-1+\alpha-\beta \pm\mu }{2}}(w-t)^{\frac{-1-\alpha-\beta \mp\mu }{2}}{\rm d} t&=0.\label{alberti1}
\end{align*}
We obtain
\begin{align*}
{\bf F}^{\rm\scriptscriptstyle I}_{\beta ,\alpha ,\pm\mu }(1{-}w)
-{\rm e}^{{\rm i}\pi\alpha}
( - w)^{\frac{-1-\alpha-\beta \mp\mu }{2}} {\bf F}^{\rm\scriptscriptstyle I} _{\pm\mu ,\beta ,\alpha}(w^{-1})
-{\rm i} {\rm e}^{{\rm i}\pi\frac{\alpha+\beta\mp\mu}{2}}{\bf F}^{\rm\scriptscriptstyle I}_{\alpha,\beta ,\pm\mu }(w)&=0.
\end{align*}
Using
\[{\bf F}_{\alpha,\beta,\mu}^{\rm\scriptscriptstyle I}(w)=
\Gamma\Big(\frac{1+\alpha+\beta-\mu}{2}\Big)\Gamma\Big(\frac{1+\alpha-\beta+\mu}{2}\Big)
{\bf F}_{\alpha,\beta ,\mu }(w),
\] we express everything in terms of ${\bf F}$. We eliminate $ {\bf F}_{\beta ,\alpha ,\mu }(1{-}w)={\bf F}_{\beta ,\alpha ,-\mu }(1{-}w)$.
We find
\begin{align*}
{\bf F} _{\alpha,\beta ,\mu }(w)\,=\,&
- \frac{\pi
(-w)^{\frac{-1-\alpha-\beta -\mu }{2}} {\bf F} _{\mu ,\beta ,\alpha}(w^{-1})
}{\sin(\pi\mu)
\Gamma\left(\frac{1+\alpha+\beta -\mu }{2}\right)
\Gamma\left(\frac{1+\alpha-\beta -\mu }{2}\right)}
\\
&
+\frac{\pi(-w)^{\frac{-1-\alpha-\beta +\mu }{2}} {\bf F} _{-\mu ,\beta ,-\alpha}(w^{-1})}
{\sin(\pi\mu)\Gamma\left(\frac{1+\alpha+\beta +\mu }{2}\right)
\Gamma\left(\frac{1+\alpha-\beta +\mu }{2}\right)}
,\end{align*}
which is the first line of
(\ref{matri2}).
A similar argument, starting with the integral $ \int_{-\infty}^0+\int_0^w+\int_w^{+\infty}$, yields the second line of (\ref{matri2}).
\section{Laplacian in 3 dimensions and the Gegenbauer equation}
\label{s7}
\init
The Gegenbauer equation
is equivalent to a subclass of the ${}_2{\cal F}_1$ equation.
Nevertheless, not all its symmetries are directly inherited from the symmetries of the ${}_2{\cal F}_1$ equation. Therefore it deserves a separate treatment, which is given in this section.
We start from the Laplacian in 5 dimensions, pass through 3 dimensions, and eventually we derive the
Gegenbauer equation.
This section is to a large extent parallel to the previous one, devoted to the ${}_2{\cal F}_1$ equation. The number of symmetries, parameters, etc. is now smaller than in the previous section, since we are in lower dimensions. Nevertheless, some things are here more complicated and less symmetric.
This is related to the fact that the number of dimensions is odd, which corresponds to a less symmetric orthogonal group and Lie algebra.
Let us describe the main steps of our derivation of the Gegenbauer equation, even though they are almost the same as for the ${}_2{\cal F}_1$ equation.
\be
\item
We start from the $3+2=5$ dimensional ambient space, with the obvious representation of $\so(5)$ and $\mathrm{O}(5)$, and the Laplacian $\Delta_5$.
\item\label{it2.} We go to the representations
$\so(5)\ni B\mapsto B^{\diamond,\eta}$ and
$\mathrm{O}(5)\ni \alpha\mapsto \alpha^{\diamond,\eta}$
and to the reduced Laplacian $\Delta_5^\diamond$. The most relevant values of $\eta$ are $1-\frac{3}{2}=-\frac12$ and $-1-\frac{3}{2}=-\frac52$.
\item\label{it3.} We fix a section $\gamma$ of the null quadric, obtaining the representations $B^{\gamma,\eta}$ and $\alpha^{\gamma,\eta}$, as well as the operator $\Delta_5^\gamma$, acting on an appropriate pseudo-Riemannian $3$ dimensional manifold.
\item\label{it4.} We choose coordinates $w,u_2,u_3$, so that the Cartan elements can be expressed in terms of $u_2$, $u_3$. We compute
$B^{\gamma,\eta}$, $\alpha^{\gamma,\eta}$ and $\Delta_5^\gamma$ in the new coordinates.
\item\label{it5.} We make an ansatz that diagonalizes the Cartan elements. The eigenvalues, denoted by $\alpha$, $\lambda$,
become parameters.
$B^{\gamma,\eta}$, $\alpha^{\gamma,\eta}$ and $\Delta_5^\gamma$ involve now only the single variable $w$.
$\Delta_5^\gamma$ turns out to be the Gegenbauer operator. We obtain its transmutation relations and discrete symmetries.
\ee
Again, we choose a special section which makes computations relatively easy.
We perform Steps \ref{it2.}, \ref{it3.} and \ref{it4.} at once, by choosing convenient coordinates $w,r,p,u_2,u_3$ in $5$ dimensions.
After the reductions of Steps \ref{it2.} and \ref{it3.}, we are left with the variables $w,u_2,u_3$, and we can perform Step \ref{it5.}.
The remaining material of this section is parallel to the analogous material of the previous section except for Subsect. \ref{Quadratic transformation},
which describes a quadratic relation reducing the Gegenbauer equation to the ${}_2{\cal F}_1$ equation. We describe a derivation of this relation starting from the level of the ambient space.
\subsection{$\so(5)$ in 5 dimensions}
\label{sub-geg1}
We consider ${\mathbb R}^5$ with the coordinates
\begin{equation} z_0,z_{-2},z_2,z_{-3},z_3\label{geg1}\end{equation}
and the scalar product given by
\begin{equation} \langle z|z\rangle=z_0^2+2z_{-2}z_2+2z_{-3}z_3.\label{geg5}\end{equation}
Note that we omit the indices $-1,1$; this makes it easier to compare
${\mathbb R}^5$ with ${\mathbb R}^6$.
The Lie algebra $\so(5)$ acts naturally on ${\mathbb R}^5$. Below we describe its natural basis. Then we consider the Weyl group $B_2$ acting on functions on ${\mathbb R}^5$. For brevity, we list only elements
from its subgroup $B_2\cap \SO(5)$. Finally, we write down the Laplacian.
\medskip
\noindent{\bf Lie algebra $\so(5)$.}
Cartan algebra
\begin{subequations}\begin{align}
N_2&=-z_{-2}\partial_{z_{-2}}+z_{2}\partial_{z_{2}},\\
N_3&=-z_{-3}\partial_{z_{-3}}+z_{3}\partial_{z_{3}}.
\end{align}\end{subequations}
Root operators
\begin{subequations}\label{coor5}
\begin{align}
B_{0,-2}&=z_{0} \partial_{z_{-2}}-z_{2} \partial_{z_0},\\
B_{0,2}&=z_{0} \partial_{z_2}-z_{-2} \partial_{z_0},\\
B_{0,-3}&=z_{0} \partial_{z_{-3}}-z_{3} \partial_{z_0},\\
B_{0,3}&=z_{0} \partial_{z_3}-z_{-3} \partial_{z_0};
\end{align}
\begin{align} B_{-3,-2}&=z_{3} \partial_{z_{-2}}-z_{2} \partial_{z_{-3}},\\
B_{3,2}&=z_{-3} \partial_{z_2}-z_{-2} \partial_{z_3},\\
B_{3,-2}&=z_{-3} \partial_{z_{-2}}-z_{2} \partial_{z_3},\\
B_{-3,2}&=z_{3} \partial_{z_2}-z_{-2} \partial_{z_{-3}}.
\end{align}
\end{subequations}
\noindent{\bf Weyl symmetries}
\begin{subequations}
\begin{align}
\sigma_{23} K(z_0,z_{-2},z_2,z_{-3},z_{3})=&K(z_0,z_{-2},z_2,z_{-3},z_{3}),\\
\tau_{2-3} K(z_0,z_{-2},z_2,z_{-3},z_{3})=&K(-z_0,z_{-2},z_2,z_{3},z_{-3}),\\
\sigma_{-2-3}K(z_0,z_{-2},z_2,z_{-3},z_{3})=&K(z_0,z_{2},z_{-2},z_{3},z_{-3}),\\
\tau_{-23}K(z_0,z_{-2},z_2,z_{-3},z_{3})=&K(-z_0,z_{2},z_{-2},z_{-3},z_{3});
\end{align}\begin{align}
\sigma_{32} K(z_0,z_{-2},z_2,z_{-3},z_{3})=&K(z_0,z_{-3},z_3,z_{-2},z_{2}),\\
\tau_{3-2} K(z_0,z_{-2},z_2,z_{-3},z_{3})=&K(-z_0,z_{-3},z_3,z_2,z_{-2}),\\
\sigma_{-3-2}K(z_0,z_{-2},z_2,z_{-3},z_{3})=&K(z_0,z_{3},z_{-3},z_{2},z_{-2}),\\
\tau_{-32}K(z_0,z_{-2},z_2,z_{-3},z_{3})=&K(-z_0,z_{3},z_{-3},z_{-2},z_{2}).
\end{align}\label{weylge1}
\end{subequations}
\noindent{\bf Laplacian}
\begin{equation}\Delta_{5}=\partial_{z_0}^2+2\partial_{z_{-2}}\partial_{z_{2}}+
2\partial_{z_{-3}}\partial_{z_{3}}
.\end{equation}
\subsection{$\so(5)$ on the spherical section}
In this subsection
we perform Steps \ref{it2.}, \ref{it3.} and \ref{it4.}, as described in the introduction to this section. Recall that Step \ref{it2.} involves
restricting to the null quadric
\[{\cal V}^4:=\{z\in{\mathbb R}^5\ :\ z_0^2+2z_{-2}z_2+2z_{-3}z_3=0\}.\]
To perform Step \ref{it3.} we need to fix a section of this quadric.
We choose the section given by the equations
\[1 =z_0^2+2z_{-2}z_2=-2z_3z_{-3}.\]
We will call it the {\em spherical section}, because it is
${\mathbb S}^2(1)\times{\mathbb S}^1(-1)$. The superscript used for this section
will be ``${\rm sph}$'' for spherical.
We introduce the coordinates $w,r,p,u_2,u_3$:
\begin{subequations}\begin{align}
r& =\sqrt{z_0^2+2z_{-2}z_2}\;,&&\\
w& =\frac{z_0}{\sqrt{2z_{-2}z_2+z_0^2}}\;,\ \ \
&u_2& =\frac{\sqrt2
z_2}{\sqrt{z_0^2+2z_{-2}z_{2}}}\;,\phantom{\sqrt{\frac{1}{2}}}\\
p& =\sqrt{-2z_3z_{-3}}\;,
&u_3& =\sqrt{-\frac{z_{3}}{z_{-3}}}\;.
\end{align}\label{cou}
\end{subequations}
Here is the inverse transformation:
\begin{subequations}\begin{align}
z_0&=wr,&\quad z_{-2}&=\frac{r(1-w^2)}{\sqrt2u_2},&\quad
z_2&=\frac{u_2r}{\sqrt2},\\
&&\quad z_{-3}&=-\frac{p}{\sqrt2 u_3},&\quad z_3&=\frac{pu_3}{\sqrt{2}}.
\end{align}\end{subequations}
Similarly as in the previous section,
the null quadric in these coordinates is given by
$r^2=p^2$. We choose the sheet $r=p$.
The generator of dilations is
\[A_{5}=r\ddr+p\ddp.\]
The spherical section is given by the condition
$r^2=1$.
\medskip
\noindent{\bf Lie algebra $\so(5)$.}
Cartan operators
\begin{align*}
N_{2}^{\rm sph}& =u_2\ddu{2}\,,\\
N_{3}^{\rm sph}& =u_3\ddu{3}\,.
\end{align*}
Roots
\begin{align*}
B_{0,-2}^{\rm sph}&=-\frac{u_2}{\sqrt2} \partial_w,\\
B_{0,2}^{\rm sph}&=\frac{1}{\sqrt2 u_2}\Big((w^2-1) \partial_w+2wu_2 \partial_{u_2}\Big),\\ B_{0,-3}^{{\rm sph},\eta}&=\frac{u_3}{\sqrt2}\Big((w^2-1) \partial_w+wu_2 \partial_{u_2}+wu_3 \partial_{u_3}-w\eta\Big),\\ B_{0,3}^{{\rm sph},\eta}&=\frac{1}{\sqrt2u_3}\Big((1-w^2) \partial_w-wu_2 \partial_{u_2}+wu_3 \partial_{u_3}+w\eta\Big);
\end{align*}
\begin{align*} B_{-3,-2}^{{\rm sph},{\eta}}&=\frac{u_2u_3}{2}
\left(-w \partial_w-u_2 \partial_{u_2}-u_3 \partial_{u_3}+\eta\right),\\
B_{3,2}^{{\rm sph},{\eta}}&=\frac{1}{2u_2u_3}\left(w(1-w^2) \partial_w-(1+w^2)u_2 \partial_{u_2}+(w^2-1)u_3 \partial_{u_3}+(w^2-1)\eta\right),\\
B_{3,-2}^{{\rm sph},{\eta}}&=\frac{u_2}{2u_3}\left(w \partial_w+
u_2 \partial_{u_2}-u_3 \partial_{u_3}-\eta\right),\\
B_{-3,2}^{{\rm sph},{\eta}}&=\frac{ u_3}{2u_2}\left(w(w^2-1) \partial_w+(1+w^2)u_2 \partial_{u_2}+(w^2-1)u_3 \partial_{u_3}+(1-w^2)\eta
\right).
\end{align*}
\noindent{\bf Weyl symmetries}
\begin{align*}
\sigma_{23}^{{\rm sph},\eta} f(w,u_2,u_3)=& f(w,u_2,u_3),\\
\tau_{2-3}^{{\rm sph},\eta} f(w,u_2,u_3)=& f\Big(-w,u_2,\frac{1}{u_3}\Big),\\
\sigma_{-2-3}^{{\rm sph},\eta} f(w,u_2,u_3)=& f\Big(w,\frac{1-w^2}{u_2},\frac{1}{u_3}\Big),\\
\tau_{-23}^{{\rm sph},\eta} f(w,u_2,u_3)=& f\Big(-w,\frac{1-w^2}{u_2},u_3\Big);
\end{align*}\begin{align*}
\sigma_{32}^{{\rm sph},\eta} f(w,u_2,u_3)=&(w^2-1)^{\frac{\eta}{2}}f\Big(\frac{w}{\sqrt{w^2-1}},\frac{u_3}{\sqrt{w^2-1}},\frac{u_2}{
\sqrt{w^2-1}}\Big),\\
\tau_{3-2}^{{\rm sph},\eta} f(w,u_2,u_3)=&(w^2-1)^{\frac{\eta}{2}}f\Big(\frac{-w}{\sqrt{w^2-1}},\frac{u_3}{\sqrt{w^2-1}},\frac{
\sqrt{w^2-1}}{u_2}\Big),\\
\sigma_{-3-2}^{{\rm sph},\eta} f(w,u_2,u_3)=&(w^2-1)^{\frac{\eta}{2}}f\Big(\frac{w}{\sqrt{w^2-1}},\frac{-1}{u_3\sqrt{w^2-1}},\frac{
\sqrt{w^2-1}}{u_2}\Big),\\
\tau_{-32}^{{\rm sph},\eta} f(w,u_2,u_3)=&(w^2-1)^{\frac{\eta}{2}}f\Big(\frac{-w}{\sqrt{w^2-1}},\frac{-1}{u_3\sqrt{w^2-1}},\frac{u_2}{
\sqrt{w^2-1}}\Big).
\end{align*}
\noindent
{\bf Laplacian}
\begin{align}\notag
\Delta_5^{\rm sph}
=&(1-w^2) \partial_w^2
-2(1+u_2 \partial_{u_2})w \partial_w
-\Big(u_2 \partial_{u_2}+\frac12\Big)^2+(u_3 \partial_{u_3})^2.
\label{lapiq} \end{align}
Let us sketch the computations that lead to (\ref{lapiq}). Using
\begin{align*}
\partial_{z_0}&=\frac1r\Big(wr \partial_r-wu_2 \partial_{u_2}+(1-w^2) \partial_w\Big),\\
\partial_{z_{-2}}&=\frac{u_2}{\sqrt{2}r}\Big(r \partial_r-u_2 \partial_{u_2}-w \partial_w\Big),\\
\partial_{z_2}&=\frac{1}{\sqrt{2}ru_2}\Big((1-w^2)r \partial_r+(1+w^2)u_2 \partial_{u_2}+(w^2-1)w \partial_w\Big)
,\\
\partial_{z_{-3}}&=\frac{u_3}{\sqrt2 p}\Big(u_3 \partial_{u_3}-p \partial_p\Big),\\
\partial_{z_3}&=\frac{1}{\sqrt2 pu_3}\Big(u_3 \partial_{u_3}+p \partial_p\Big).
\end{align*}
we change the variables in the Laplacian:
\begin{align}
\Delta_5=&\frac{1}{r^2}\Big((1-w^2) \partial_w^2-2(1+u_2 \partial_{u_2})w \partial_w
-(u_2 \partial_{u_2})^2-u_2 \partial_{u_2}\notag\\ &+(r \partial_r)^2+r \partial_r\Big)+\frac{1}{p^2}\Big(-(p \partial_p)^2+(u_3 \partial_{u_3})^2\Big).\end{align}
Now,
\begin{align*}
(r\ddr)^2+r\ddr-\frac{r^2}{p^2}(p\ddp)^2& =\biggl(r\ddr-p \partial_p+\frac{1}{2}\biggr)\biggl(r\ddr+p \partial_p+\frac{1}{2}\biggr)\\
&\quad+\biggl(1-\frac{r^2}{p^2}\biggr)(p \partial_p)^2-\frac{1}{4}.
\end{align*}
Therefore, using $r^2=p^2$, $r \partial_r+p \partial_p=-\frac12$, we obtain
\begin{align}\notag
\Delta_5^\diamond&
=\frac{1}{r^2}\Big((1-w^2)^2 \partial_w^2
-2(1+u_2 \partial_{u_2})w \partial_w\\
& -\big(u_2 \partial_{u_2}+\frac12\big)^2+(u_3 \partial_{u_3})^2\Big).
\end{align}
To obtain the Laplacian at the spherical section we drop $\frac1{r^2}$.
\subsection{The Gegenbauer operator}
Let us make the ansatz
\begin{equation}
f(u_2,u_3,w)= u_2^\alpha u_3^\lambda S(w).\label{form0-}\end{equation}
Clearly,
\begin{subequations}\begin{eqnarray}
N_2^{\rm sph} f&=&\alpha f,\\
N_3^{\rm sph} f&=&\lambda f,\\
u_2^{-\alpha} u_3^{-\lambda} \Delta_5^{\rm sph} f&
=& {\cal S}_{\alpha,\lambda}(w, \partial_w)S(w),\end{eqnarray}\end{subequations}
where
\begin{eqnarray}
{\cal S}_{\alpha ,\lambda }(w, \partial_w)
&:=&(1-w^2) \partial_w^2-2(1+\alpha )w \partial_w
+\lambda ^2-\Big(\alpha +\frac{1}{2}\Big)^2\end{eqnarray}
is the {\em Gegenbauer operator}.
Here is another parametrization of the Gegenbauer operator, which we call {\em classical}:
\begin{equation}
{\cal S}({a},{b};w, \partial_w):=(1-w^2) \partial_w^2-({a}+{b}+1)w \partial_w-{a}{b}.
\label{gego}\end{equation}
Here is the relationship between the classical and Lie-algebraic parameters:
\begin{subequations}\begin{align}
&\alpha :=\frac{{a}+{b}-1}{2},& \lambda: =\frac{{b}-{a}}{2},\\
&{a}=\frac12+\alpha -\lambda ,&{b}=\frac12+\alpha +\lambda .
\end{align}
\end{subequations}
The Gegenbauer operator is the ${}_2{\cal F}_1$ operator with its finite singular points moved to $-1$ and $1$, which in addition is reflection invariant.
Because of the reflection invariance, the third classical parameter can be obtained from the first two: $c=\frac{a+b+1}{2}$. Therefore, we use only ${a},{b}\in{\mathbb C}$
as the (classical) parameters of the Gegenbauer equation.
We can reduce the Gegenbauer equation to the ${}_2{\cal F}_1$ equation by
two affine transformations. They move the singular points from $-1$, $1$ to
$0$, $1$ or $1$, $0$:
\begin{equation}\begin{array}{l}
{\cal S}({a},{b};w, \partial_w)={\cal F}\big({a},{b};\frac{{a}+{b}+1}{2}
;v, \partial_v\big),\end{array}\label{ha2}\end{equation}
where
\begin{subequations}\begin{align}&v=\frac{1-w}{2},\ \ \ w=1-2v,\\
\hbox{or}\qquad&v=\frac{1+w}{2},\ \ \ w=-1+2v.
\end{align}\end{subequations}
In the Lie-algebraic parameters
\begin{equation}{\cal S}_{\alpha,\lambda}(w, \partial_w)={\cal F}_{\alpha,\alpha,2\lambda}(v, \partial_v).\end{equation}
\subsection{Quadratic transformation}
\label{Quadratic transformation}
Let us go back to 6 dimensions and the Laplacian
\begin{equation}\Delta_6=2\partial_{z_{-1}}\partial_{z_{1}}+
2\partial_{z_{-2}}\partial_{z_{2}}+
2\partial_{z_{-3}}\partial_{z_{3}}
.\label{sq20}\end{equation}
Let us use the reduction described in Subsect.
\ref{Dimensional reduction}.
Introduce new variables
\begin{equation} z_0:=\sqrt{2z_{-1}z_{1}},\quad
u:=\sqrt{\frac{z_{1}}{z_{-1}}}.\label{coo1}\end{equation}
In the new variables,
\begin{align}
N_1=&u \partial_u,\\\notag
\Delta_6
=&\Big(\partial_{z_0}+\frac1{2z_0}\Big)^2
-\frac1{z_0^2}\Big(u\partial_u-\frac12\Big)
\Big(u\partial_u+\frac12\Big)\\&+
2\partial_{z_{-2}}\partial_{z_{2}}+
2\partial_{z_{-3}}\partial_{z_{3}}.
\end{align}
Therefore,
\begin{subequations} \begin{align}
(uz_0)^{\frac12}\Delta_6 (uz_0)^{-\frac12}
& =
-\frac1{z_0^2}N_1
\Big(N_1- 1\Big)+
\Delta_5,\\
(u^{-1}z_0)^{\frac12}\Delta_6 (u^{-1}z_0)^{-\frac12}
& =
-\frac1{z_0^2}N_1
\Big(N_1+1\Big)+
\Delta_5.
\end{align}\end{subequations}
Compare the coordinates
the coordinates (\ref{coo}) for 6 dimensions and
(\ref{cou}) for 5 dimensions.
The coordinates $p$, $u_3$ are the same.
Taking into account $z_0:=\sqrt{2z_{-1}z_{1}}$, the coordinates $r,u_2$ also coincide. This is not the case of
$w$, so let us rename $w$ from (\ref{cou})
as $v$. We then have $w=v^2$.
We also have
\[uz_0=\sqrt2z_1=u_1r,\quad u^{-1}z_0=\sqrt 2z_{-1}=rw u_1^{-1}.\]
Hence on functions that do not depend on $u$ we obtain
\begin{subequations} \begin{align}
r^{\frac12}u_1^{\frac12}\Delta_6 r^{-\frac12}u_1^{-\frac12}&=
\Delta_5 ,\\
r^{\frac12}u_1^{-\frac12}v\Delta_6 r^{-\frac12}u_1^{\frac12}v^{-1}&=
\Delta_5 .
\end{align}
\end{subequations}
This implies that a quadratic substitution
transforms the ${}_2{\cal F}_1$ operator with $\alpha=\pm\frac12$ into the Gegenbauer operator.
Explicitly, if \[w=v^2,\ \ \ \
v=\sqrt w;\]
then in the classical parameters
\begin{subequations}\begin{align}{\cal S}({a},{b};v, \partial_v)&=
4{\cal F}\Big(\frac{{a}}{2},\frac{{b}}{2};\frac{1}{2};w, \partial_w\Big),\\[2ex]
v^{-1}{\cal S}({a},{b};v, \partial_v)v&=
4{\cal F}\Big(\frac{{a}+1}{2},\frac{{b}+1}{2};\frac32;w, \partial_w\Big),
\label{ha1}\end{align}\end{subequations}
and in the Lie-algebraic parameters
\begin{subequations}\begin{eqnarray}
\label{1/2}{\cal S}_{\alpha,\lambda}(v, \partial_v)&=&4{\cal F}_{-\frac{1}{2},\alpha,\lambda}(w, \partial_w),
\label{pol1}\\
\label{-1/2}v^{-1}{\cal S}_{\alpha,\lambda}(v, \partial_v)v&=&4{\cal F}_{\frac{1}{2},\alpha,\lambda}(w, \partial_w).
\label{pol2}\end{eqnarray}
\end{subequations}
\subsection{Transmutation relations and discrete symmetries}
\label{symcom}
We have the following generalized symmetries:
\begin{subequations}\begin{eqnarray}
B^{{\rm sph},-\frac52}\Delta_{5}^{\rm sph}&=&\Delta_{5}^{\rm sph} B^{{\rm sph},-\frac12},\ \ \ \ B\in \so(5);\label{vv1}\\
\alpha^{{\rm sph},-\frac52}\Delta_{5}^{\rm sph}&=&\Delta_{5}^{\rm sph}\alpha^{{\rm sph},-\frac12},\ \ \ \alpha\in \mathrm{O}(5).\label{vv2}\end{eqnarray}\end{subequations}
Equality (\ref{vv1}) applied to the roots of $\so(5)$ yield the following transmutation relations:
\[\hspace{-3ex}\begin{array}{rrl}
& \partial_w&
{\cal S}_{\alpha ,\lambda } \\[0.4ex]
&=\ \ \ {\cal S}_{\alpha +1,\lambda }& \partial_w,\\[0.5ex]
&((1-w^2) \partial_w-2\alpha w)&
{\cal S}_{\alpha ,\lambda }\\[0.4ex]
&=\ \ \ {\cal S}_{\alpha -1,\lambda }&((1-w^2) \partial_w-2\alpha w),\\[0.5ex]
&((1-w^2) \partial_w-
(\alpha +\lambda +\frac12 )w)&(1-w^2){\cal S}_{\alpha ,\lambda }\\[0.4ex]
&=\ \ \ (1-w^2){\cal S}_{\alpha ,\lambda +1}&
((1-w^2) \partial_w-
(\alpha +\lambda +\frac{1}{2})w),\\[0.5ex]
&((1-w^2) \partial_w-(\alpha -\lambda +\frac12)w)&
(1-w^2){\cal S}_{\alpha ,\lambda }\\[0.4ex]
&=\ \ \ (1-w^2){\cal S}_{\alpha ,\lambda -1}&((1-w^2) \partial_w-(\alpha -\lambda +\frac{1}{2})w);
\\[1ex]
&(w \partial_w+
\alpha -\lambda +\frac12)&w^2{\cal S}_{\alpha ,\lambda }\\[0.4ex]
&=\ \ \ w^2{\cal S}_{\alpha +1,\lambda -1}&(w \partial_w+
\alpha -\lambda +\frac{1}{2}),\\[0.5ex]
&(w(1{-}w^2) \partial_w{-}\alpha{-}\lambda{+}\frac12{-}(\alpha{-}\lambda{+}\frac12)w^2)
&w^2{\cal S}_{\alpha ,\lambda }\\[0.4ex]
&=\ \ \ w^2{\cal S}_{\alpha -1,\lambda +1}
&(w(1{-}w^2) \partial_w
{-}\alpha{-}\lambda{+}\frac12{-}(\alpha{-}\lambda{+}\frac12)w^2),
\\[0.5ex]
&(w \partial_w+
\alpha -\lambda +\frac12)&w^2{\cal S}_{\alpha ,\lambda }\\[0.4ex]
&=\ \ \ w^2{\cal S}_{\alpha +1,\lambda +1}&(w \partial_w+
\alpha -\lambda +\frac{1}{2}),\\[0.5ex]
&(w(1{-}w^2) \partial_w
{-}\alpha{+}\lambda{+}\frac12{-}(\alpha{+}\lambda{+}\frac12)w^2
)&
w^2{\cal S}_{\alpha ,\lambda }\\[0.4ex]
&=\ \ \ w^2{\cal S}_{\alpha -1,\lambda -1}&(w(1{-}w^2) \partial_w
{-}\alpha{+}\lambda{+}\frac12{-}(\alpha{+}\lambda{+}\frac12)w^2).
\end{array}\]
Next we describe discrete symmetries of the Gegenbauer operator, which follow from
Relation (\ref{vv2}) applied to Weyl symmetries.
All the operators below equal ${\cal S}_{\alpha ,\lambda }(w, \partial_w)$ for the
appropriate $w$:
\[\begin{array}{rrcl}
w=\pm v:
&&{\cal S}_{\alpha ,\pm\lambda }(v, \partial_v),&\\[1ex]
w=\pm v:
&(v^2-1)^{-\alpha }&{\cal S}_{-\alpha ,\mp\lambda }(v, \partial_v)&
(v^2-1)^{\alpha },\\[1ex]
w=\frac{\pm v}{(v^2-1)^{\frac{1}{2}}}:
& (v^2-1)^{\frac{1}{2}(\alpha +\lambda +\frac52)}
&{\cal S}_{\lambda ,\pm\alpha }(v, \partial_v)
& (v^2-1)^{\frac{1}{2}(-\alpha -\lambda -\frac{1}{2})},\\[1ex]
w=\frac{\pm v}{(v^2-1)^{\frac{1}{2}}}:
& (v^2-1)^{\frac{1}{2}(\alpha -\lambda +\frac52)}&
{\cal S}_{-\lambda ,\mp\alpha }(v, \partial_v)&
(v^2-1)^{\frac{1}{2}(-\alpha +\lambda -\frac{1}{2})}.
\end{array}\]
Note that we use $\pm$ to describe two symmetries at once. Therefore, the above list has all $2\times 4=8$ symmetries corresponding to the lists of Weyl symmetries (\ref{weylge1}).
\subsection{Factorizations of the Laplacian}
\label{Factorizations of the Laplacian2}
In the Lie algebra $\so(5)$ represented on ${\mathbb R}^5$
we have 3 distinguished Lie subalgebras:
two isomorphic to $\so(3)$ and
one isomorphic to $\so(4)$:
\begin{equation} \so_{02}(3),\; \; \so_{03}(3),\; \; \so_{23}(4),\end{equation}
where we use an obvious notation.
By (\ref{casimir1}) and
(\ref{casimir2}), the corresponding Casimir operators are
\begin{subequations}
\begin{align}
{\cal C}_{02}
&=2B_{0,-2}B_{0,2}-\Big(N_2-\frac12\Big)^2+\frac14\\
&=2B_{0,2}B_{0,-2}-\Big(N_2+\frac12\Big)^2+\frac14,\\
{\cal C}_{03}
&=2B_{0,-3}B_{0,3}-\Big(N_3-\frac12\Big)^2+\frac14\\
&=2B_{0,3}B_{0,-3}-\Big(N_3+\frac12\Big)^2+\frac14,\\
{\cal C}_{23}
&=4B_{2,3}B_{-2,-3}-(N_2+N_3+1)^2+1\\
&=4B_{-2,-3}B_{2,3}-(N_2+N_3-1)^2+1\\
&=4B_{2,-3}B_{-2,3}-(N_2-N_3+1)^2+1\\
&=4B_{-2,3}B_{2,-3}-(N_2-N_3-1)^2+1.
\end{align}\label{facto1}
\end{subequations}
After the reduction described in
(\ref{deq3a}) and (\ref{deq1a}), we obtain the
identities
\begin{subequations}
\begin{align}
\label{factor2}
(z_0^2+2z_{-2}z_2)\Delta_{5}^\diamond&=-\frac14+{\cal C}_{02}^{\diamond,-\frac12}
+(N_3^{\diamond,-\frac12})^2,
\\\label{factor3}
(z_0^2+2z_{-3}z_3)\Delta_{5}^\diamond&=-\frac14+{\cal C}_{03}^{\diamond,-\frac12}
+(N_2^{\diamond,-\frac12})^2,\\
\label{factor1}(2z_{-2}z_2+2z_{-3}z_3)\Delta_{5}^\diamond&=-\frac34+{\cal C}_{23}^{\diamond,-\frac12}.
\end{align}\label{facto2}
\end{subequations}
Inserting (\ref{facto1}) into (\ref{facto2}), we obtain
\begin{subequations}
\begin{align}
&(z_0^2+2z_{-2}z_2)\Delta_{5}^\diamond\notag\\
=&2B_{0,-2}B_{0,2}-\Big(N_2+N_3-\frac12\Big)\Big(N_2-N_3-\frac12\Big)\\
=&2B_{0,2}B_{0,-2}-\Big(N_2+N_3+\frac12\Big)\Big(N_2-N_3+\frac12\Big),
\\
&(z_0^2+2z_{-3}z_3)\Delta_{5}^\diamond\notag\\
=&2B_{0,-3}B_{0,3}-\Big(N_2+N_3-\frac12\Big)\Big(-N_2+N_3-\frac12\Big)
\\
=&2B_{0,3}B_{0,-3}-\Big(N_2+N_3+\frac12\Big)\Big(-N_2+N_3+\frac12\Big),\\
&(2z_{-2}z_2+2z_{-3}z_3)\Delta_{5}^\diamond\notag\\
=&4B_{2,3}B_{-2,-3}-\Big(N_2+N_3+\frac32\Big)\Big(N_2+N_3+\frac12\Big)
\\
=&4B_{-2,-3}B_{2,3}-\Big(N_2+N_3-\frac32\Big)\Big(N_2+N_3-\frac12\Big)\\
=&4B_{2,-3}B_{-2,3}-\Big(N_2-N_3+\frac32\Big)\Big(N_2-N_3+\frac12\Big)\\
=&4B_{-2,3}B_{2,-3}-\Big(N_2-N_3-\frac32\Big)\Big(N_2-N_3-\frac12\Big),
\end{align}\end{subequations}
where all the $B$ and $N$ operators need to have the superscript ${}^{\diamond,-\frac12}$.
If we use the spherical section,
we need to make the replacements
\begin{subequations}
\begin{align}
\label{factor2-}
z_0^2+2z_{-2}z_2&\quad\to\quad 1,
\\\label{factor3-}
z_0^2+2z_{-3}z_3&\quad\to\quad w^2-1,\\
\label{factor1-}
2z_{-2}z_2+2z_{-3}z_3&\quad\to\quad -w^2,
\end{align}\label{facto2-}
\end{subequations}
and replace the superscript ${}^\diamond$ with ${}^{\rm sph}$.
\subsection{Factorizations of the Gegenbauer equation}
\label{symcoma}
The factorizations of $\Delta_5^{\rm sph}$ of Subsect.
\ref{Factorizations of the Laplacian2} yield the following factorizations of the Gegenbauer operator:
\begin{align*}{\cal S}_{\alpha,\lambda}
=&\partial_w\Big((1-w^2)\partial_w
-2\alpha w\Big)\\
&+\Big(\alpha+\lambda-\frac12\Big)\Big(-\alpha+\lambda+\frac12\Big)\\
=&\Big((1-w^2)\partial_w
-2(1+\alpha)w\Big)\partial_w\\
&+\Big(\alpha+\lambda+\frac12\Big)\Big(-\alpha+\lambda-\frac12\Big),\\
(1{-}w^2){\cal S}_{\alpha,\lambda}=&\Big((1-w^2)\partial_w
-\big(\alpha+\lambda-\frac12\big)w\Big)\Big((1-w^2)\partial_w
-\big(\alpha-\lambda+\frac12\big)w\Big)\\
&+\Big(\alpha+\lambda-\frac12\Big)\Big(\alpha-\lambda+\frac12\Big)\\
=&\Big((1-w^2)\partial_w
-\big(\alpha-\lambda-\frac12\big)w\Big)\Big((1-w^2)\partial_w
-\big(\alpha+\lambda+\frac12\big)w\Big)\\
&+\Big(\alpha+\lambda+\frac12\Big)\Big(\alpha-\lambda-\frac12\Big),
\end{align*}
\begin{align*}w^2{\cal S}_{\alpha,\lambda}=&\Big(w(1-w^2)\partial_w
-\alpha-\lambda-\frac32+\big(-\alpha+\lambda-\frac12\big)w^2\Big)
\Big(w\partial_w+\alpha+\lambda+\frac12\Big)\\
&+\Big(\alpha+\lambda+\frac12\Big)\Big(\alpha+\lambda+\frac32\Big)\\
=&
\Big(w\partial_w+\alpha+\lambda-\frac32\Big)
\Big(w(1-w^2)\partial_w
-\alpha-\lambda+\frac12+\big(-\alpha+\lambda-\frac12\big)w^2\Big)\\
&+\Big(\alpha+\lambda-\frac12\Big)\Big(\alpha+\lambda-\frac32\Big)\\
=&\Big(w(1-w^2)\partial_w
-\alpha+\lambda-\frac32+\big(-\alpha-\lambda-\frac12\big)w^2\Big)
\Big(w\partial_w+\alpha-\lambda+\frac12\Big)\\
&+\Big(\alpha-\lambda+\frac12\Big)\Big(\alpha-\lambda+\frac32\Big)\\
=&
\Big(w\partial_w+\alpha-\lambda-\frac32\Big)
\Big(w(1-w^2)\partial_w
-\alpha+\lambda+\frac12+\big(-\alpha-\lambda-\frac12\big)w^2\Big)\\
&+\Big(\alpha-\lambda-\frac12\Big)\Big(\alpha-\lambda-\frac32\Big).
\end{align*}
\subsection{Standard solutions}
As usual, by standard solutions we mean solutions with a simple behavior around singular points. The singular points of the Gegenbauer equation are $\{1,-1,\infty\}$. The discussion of the point $-1$ can be easily reduced to that of $1$. Therefore, it is enough to discuss $2\times2=4$ solutions corresponding to two indices at $1$ and $\infty$.
The standard solutions can be expressed in terms of the function
\begin{align}\notag
S_{\alpha,\lambda}(w)=S(a,b;w)&:=F\Big({a},{b};\frac{{a}+{b}+1}{2};\frac{1-w}{2}\Big)\label{qqe}\\
&=
F\Big(\frac{{a}}{2},\frac{{b}}{2};\frac{{a}+{b}+1}{2};1-w^2\Big).
\end{align}
Here are the 4 standard solutions. We consistently use the Lie-algebraic parameters.
\begin{alignat*}{2}
\text{ $\sim1$ at $1$:}&\qquad S_{\alpha ,\lambda }(w)&\\
=&F_{\alpha ,\alpha ,2\lambda }
\Big(\frac{1-w}{2}\Big)
=&
F_{\alpha ,-\frac12,\lambda }(1-w^2),
\end{alignat*}
\begin{alignat*}{2}
&\text{ $\sim \frac{1}{2^{\alpha}(1-w)^{\alpha }}$ at $1$:}\qquad
(1-w^2)^{-\alpha }S_{-\alpha ,-\lambda }(w)\\=&2^{-\alpha}(1-w)^{-\alpha }F_{-\alpha ,\alpha ,-2\lambda }
\Big(\frac{1-w}{2}\Big)=
(1-w^2)^{-\alpha }F_{-\alpha ,-\frac12,-\lambda }(1-w^2),\end{alignat*}
\begin{alignat*}{2}
& \text{ $\sim w^{-{a}}$ at $\infty$:}\qquad
(w^2-1)^{\frac{-1-2\alpha +2\lambda }{4}}S_{-\lambda ,-\alpha }\Big(\frac{w}{\sqrt{w^2-1}}\Big)\\
=&(1+w)^{-\frac12-\alpha +\lambda }F_{-2\lambda ,\alpha ,-\alpha }
\Big(\frac{2}{1+w}\Big)=
w^{-\frac12-\alpha +\lambda }F_{-\lambda ,\alpha ,\frac12}(w^{-2}),
\end{alignat*}
\begin{alignat*}{2}
&\text{ $\sim w^{-{b}}$ at $\infty$:}\qquad(w^2-1)^{\frac{-1-2\alpha -2\lambda }{4}}S_{\lambda ,\alpha }\Big(\frac{w}{\sqrt{w^2-1}}\Big)\\
=&(1+w)^{-\frac12-\alpha -\lambda }F_{2\lambda ,\alpha ,\alpha }
\Big(\frac{2}{1+w}\Big)=
w^{-\frac12-\alpha -\lambda }F_{\lambda ,\alpha ,\frac12}(w^{-2}).
\end{alignat*}
\subsection{Recurrence relations}
We will use the following normalization to express recurrence relations:
\begin{align}
{\bf S}_{\alpha,\lambda}(w)&:=\frac{1}{\Gamma(\alpha+1)}S_{\alpha,\lambda}(w)\notag\\
&=\frac{1}{\Gamma(\frac{a+b+1}{2})}
F\Big({a},{b};\frac{{a}+{b}+1}{2};\frac{1-w}{2}\Big)\notag\\&=
{\bf F}_{\alpha,\alpha,2\lambda}\Big(\frac{1-w}{2}\Big).
\end{align}
To each root of $\so(5)$ there corresponds a recurrence relation:
\begin{align*}
& \partial_w {\bf S}_{\alpha ,\lambda }(w)=-\frac{1}{2}\Big(\frac12+\alpha -\lambda \Big)\Big(\frac12+\alpha +\lambda \Big)
{\bf S}_{\alpha +1,\lambda }(w),\\
&\left((1-w^2) \partial_w -2\alpha w\right)
{\bf S}_{\alpha ,\lambda }(w)
=-2{\bf S}_{\alpha -1,\lambda }(w) ,\\[2ex]
&\left((1-w^2) \partial_w -\Big(\frac12+\alpha +\lambda \Big) w\right){\bf S}_{\alpha ,\lambda }(w)
=-\Big(\frac12+\alpha +\lambda \Big) {\bf S}_{\alpha ,\lambda +1}(w),
\\
&\left((1-w^2) \partial_w -\Big(\frac12+\alpha -\lambda \Big) w\right){\bf S}_{\alpha ,\lambda }(w)
=-\Big(\frac12+\alpha -\lambda \Big){\bf S}_{\alpha ,\lambda -1}(w);
\\[2ex]
&\left(w \partial_w+\frac12+\alpha -\lambda \right){\bf S}_{\alpha ,\lambda }(w)
=\frac12\Big(\frac12+\alpha -\lambda \Big)\Big(\frac32+\alpha-\lambda\Big) {\bf S}_{\alpha +1,\lambda -1}(w),\\
&\left(w(1{-}w^2) \partial_w{+}\Big(\frac12{-}\alpha {+}\lambda \Big)(1{-}w^2){-}2\alpha w^2
\right){\bf S}_{\alpha ,\lambda }(w)
=-2{\bf S}_{\alpha -1,\lambda +1}(w),\\[2ex]
&\left(w \partial_w+\frac12+\alpha +\lambda \right){\bf S}_{\alpha ,\lambda }(w)
=\frac12\Big(\frac12+\alpha +\lambda \Big)\Big(\frac32+\alpha+\lambda\Big) {\bf S}_{\alpha +1,\lambda +1}(w),\\
&\left(w(1{-}w^2) \partial_w{+}\Big(\frac12{-}\alpha {-}\lambda \Big)(1{-}w^2){-}2\alpha w^2
\right){\bf S}_{\alpha ,\lambda }(w)
=-2{\bf S}_{\alpha -1,\lambda -1}(w).
\end{align*}
\subsection{Wave packets in 5 dimensions}
We easily check the following lemma:
\begin{lemma} For any $\tau$, the function
$z_2^\alpha\big(\sqrt2 z_0-\tau^{-1} z_{-3}+\tau z_3\big)^\nu$ is harmonic.
\end{lemma}
Let us make a wave packet from the above functions.
\begin{proposition} Let
the contour $]0,1[\ni s\overset{\gamma}\mapsto\tau(s)$
satisfy
\begin{equation}
\big(\sqrt2 z_0-\tau^{-1} z_{-3}+\tau z_3\big)^\nu\tau^{-\lambda}\Big|_{\tau(0)}^{\tau(1)}=0.\end{equation}
Then the function
\[
K_{\alpha,\nu,\lambda}(z_0,z_{-2},z_{2},z_{-3},z_3):=\int_\gamma z_2^\alpha\big(\sqrt2 z_0-\tau^{-1} z_{-3}+\tau z_3\big)^\nu\tau^{-\lambda-1}{\rm d}\tau
\]
is harmonic and
\begin{subequations} \begin{align}
N_2K_{\alpha,\nu,\lambda}&=\alpha K_{\alpha,\nu,\lambda},\label{caz1}\\
N_3K_{\alpha,\nu,\lambda}&=\lambda K_{\alpha,\nu,\lambda}.\label{caz2}
\end{align}\end{subequations}
\end{proposition}
{\bf Proof.}\ \ (\ref{caz1}) is obvious.
To obtain (\ref{caz2}) we use Prop. \ref{cac0}. $\Box$\medskip
If in addition
\[\nu=-\alpha-\frac12,\]
then $K_{\alpha,\nu,\lambda}$ is homogeneous of degree $-\frac12$. Therefore, we can reduce it to dimension $3$.
Let us express it in the coordinates $w,r,p,u_2,u_3$:
\begin{align*}
K
(w,r,p,u_2,u_3)&=\int u_2^\alpha r^\alpha
\Big(wr\sqrt2 +\frac{p}{\tau u_3\sqrt 2}+\frac{\tau pu_3}{\sqrt2}\Big)^{-\alpha-\frac12}\tau^{-\lambda-1}{\rm d}\tau\\
&=(\sqrt2)^{\alpha+\frac12}u_2^\alpha u_3^\lambda r^{-\frac12}
\int
\Big(2w\sigma+(1+\sigma^2)\frac{p}{r}\Big)^{-\alpha-\frac12}
\sigma^{\alpha-\lambda-\frac12}{\rm d}\sigma,
\end{align*}
where we set $\sigma:=u_3\tau$.
Noting that on the spherical section $p=r$, we see that
\begin{equation} S(w):=\int
\big(2w\sigma+1+\sigma^2\big)^{-\alpha-\frac12}
\sigma^{\alpha-\lambda-\frac12}{\rm d}\sigma
\end{equation}
satisfies the Gegenbauer equation.
\subsection{Integral representations}
In this subsection we describe two kinds of integral representations for solutions to the Gegenbauer equation. The first is essentially inherited
from the ${}_2{\cal F}_1$ equation. The second was derived using additional variables in the previous subsection. Here we give independent derivations.
We will use classical parameters.
\begin{theoreme}\begin{arabicenumerate}\item[a)]
Let $[0,1]\ni \tau\overset{\gamma}\mapsto t(\tau)$ satisfy
\[(t^2-1)^{\frac{{b}-{a}+1}{2}}(t-w)^{-{b}-1}\Big|_{t(0)}^{t(1)}=0.\]
Then
\begin{equation}
{\cal S}({a},{b};w, \partial_w)
\int_\gamma (t^2-1)^{\frac{{b}-{a}-1}{2}}(t-w)^{-{b}}{\rm d} t=0
.\label{dad9}\end{equation}
\item[b)]
Let $[0,1]\ni \tau\overset{\gamma}\mapsto t(\tau)$ satisfy
\begin{eqnarray*}&&
(t^2+2tw+1)^{\frac{-{b}-{a}}{2}+1}t^{b-2}
\Big|_{t(0)}^{t(1)}=0.\end{eqnarray*}
Then
\begin{equation}
{\cal S}({a},{b};w, \partial_w)
\int_\gamma(t^2+2tw+1)^{\frac{-{b}-{a}}{2}}t^{{b}-1}{\rm d} t=0
.\label{dad8}\end{equation}
\end{arabicenumerate}
\label{gqw}\end{theoreme}
{\bf Proof.}\ \
For any contour $\gamma$ we have
\begin{eqnarray*}
\text{lhs of (\ref{dad9})}&=&{a}
\int_\gamma {\rm d} t\, \partial_t(t^2-1)^{\frac{{b}-{a}+1}{2}}(t-w)^{-{b}-1},\\
\text{lhs of (\ref{dad8})}&=&\int\limits_\gamma{\rm d} t\,
\partial_t(t^2+2tw+1)^{\frac{-{b}-{a}}{2}+1}t^{b-2}.\end{eqnarray*}
$\Box$\medskip
Note that in the above theorem we can interchange $a$ and $b$. Thus we obtain four kinds of integral representations.
\subsection{Integral representations of the standard solutions}
As described in Thm \ref{gqw}, we have two types of integral representations of
solutions of Gegenbauer equations: a) and b).
It is natural to use singular points of the integrands as the endpoints of the contours of integration.
For the representations of type a) we have singular points at $\infty, -1,1, w$. For representations of type b) singular points are at $\infty,0$ and the two roots of $t^2+2tw+1=0$. Choosing an appropriate contour we obtain all standard solutions with both types of representations with some special normalizations. It is convenient to introduce special notation for these normalizations:
\begin{eqnarray}
{\bf S}_{\alpha,\lambda}^{\rm\scriptscriptstyle I}(w)&:=&2^{-\frac12-\alpha-\lambda}\frac{\Gamma(\frac{1+2\alpha+2\lambda}{2})\Gamma(\frac{1-2\lambda}{2})}{\Gamma(\alpha+1)}S_{\alpha,\lambda}
(w)
\\\notag
&=&
2^{-b}\frac{\Gamma(b)\Gamma(\frac{a-b+1}{2})}{\Gamma(\frac{a+b+1}{2})}
F\Big({a},{b};\frac{{a}+{b}+1}{2};\frac{1-w}{2}\Big)\\
\notag&=&2^{-\frac12-\alpha-\lambda}
{\bf F}_{\alpha,\alpha,2\lambda}^{\rm\scriptscriptstyle I}\Big(\frac{1-w}{2}\Big),\\[4ex]
{\bf S}_{\alpha,\lambda}^{{\rm\scriptscriptstyle {I}{I}}}(w)&:=&\frac{\Gamma(\frac{1+2\alpha-2\lambda}{2})\Gamma(\frac{1+2\alpha+2\lambda}{2})}{\Gamma(2\alpha+1)}S_{\alpha,\lambda}(w)\\
&=&\notag
\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}
F\Big({a},{b};\frac{{a}+{b}+1}{2};\frac{1-w}{2}\Big),\\[4ex]
{\bf S}_{\alpha,\lambda}^{{\rm\scriptscriptstyle 0}}(w)&:=
\sqrt{\pi}
\frac{\Gamma(\frac{1+2\alpha}{2})}{\Gamma(\alpha+1)}S_{\alpha,\lambda}(w)
\\\notag
&=&
\sqrt{\pi}
\frac{\Gamma(\frac{1+2\alpha}{2})}{\Gamma(\alpha+1)}
F\Big({a},{b};\frac{{a}+{b}+1}{2};\frac{1-w}{2}\Big).
\end{eqnarray}
In the following table we list all standard solutions together with the contours of integration and the corresponding normalizations.
\medskip
\noindent
\begin{tabular}{cllc}&a)&b)& \\[3ex]
$\sim 1$ at $1$:
&$\begin{array}{l}]-\infty,-1],\\ \mathrm{I};
\end{array}$&$\begin{array}{l}[0,\infty[,\\ \mathrm{II};\end{array}$& \\[2ex]
\hspace{-2ex}$\sim \frac{1}{2^{\alpha}(1{-}w)^{\alpha }}$ at $1$:
&$\begin{array}{l}]{-}1,w],\\ \mathrm{I};\end{array}$&$\begin{array}{l}[{-}{\rm i}\sqrt{1{-}w^2}-w,{\rm i}\sqrt{1{-}w^2}{-}w],\\ 0;\end{array}$&\\[2ex]
$\sim w^{-a}$ at $\infty$:
&$\begin{array}{l}]-1,1],\\ 0;\end{array}$&$\begin{array}{l}[\sqrt{w^2-1}-w,0[,\\ \mathrm{I};\end{array}$& \\[2ex]
$\sim w^{-b}$ at $\infty$:
&$\begin{array}{l}]w,\infty],\\
\mathrm{II};\end{array}$&$\begin{array}{l}]-\infty,-\sqrt{w^2-1}-w],\\
\mathrm{I}.\end{array}$&\\
\end{tabular}
\medskip
Here are representations of type a):
\begin{align}\frac12>\Re\lambda>-\frac12-\Re\alpha:\quad&\\
\int\limits_{-\infty}^{-1}(t^2-1)^{-\frac12-\lambda }(w-t)^{-\frac12-\alpha +\lambda }{\rm d}
t\nonumber \\
=
{\bf S}^{\rm\scriptscriptstyle I}_{\alpha ,\lambda }(w),&\quad w\not\in]-\infty,-1];\notag\end{align}
\begin{align}
\frac12>\Re\lambda>-\frac12+\Re\alpha:\quad&\\
\int\limits_{w}^1(1-t^2)^{-\frac12-\lambda }(w-t)^{-\frac12-\alpha +\lambda }{\rm d}
t\nonumber \\
=
(1-w^2)^{-\alpha } {\bf S}^{\rm\scriptscriptstyle I}_{-\alpha ,-\lambda }(w),&\quad w\not\in]-\infty,-1]\cup[1,\infty[;\notag\end{align}
\begin{align} \frac12>\Re\lambda:\quad\\
\int\limits_{-1}^1(1-t^2)^{-\frac12-\lambda }(w-t)^{-\frac12-\alpha +\lambda }{\rm d}
t\nonumber \\
=
(w^2-1)^{\frac{-1-2\alpha +2\lambda }{4}}{\bf S}_{-\lambda ,\alpha }^{{\rm\scriptscriptstyle 0}}\Big(\frac{w}{\sqrt{w^2-1}}\Big)
,&\quad w\not\in]-\infty,1];\notag\end{align}\begin{align}
\Re\lambda+\frac12>|\Re\alpha|:\quad&\\
\int\limits_w^\infty(t^2-1)^{-\frac12-\lambda }(t-w)^{-\frac12-\alpha +\lambda }{\rm d}
t\nonumber &\\
=(w^2-1)^{\frac{-1-2\alpha -2\lambda }{4}}{\bf S}_{\lambda ,\alpha }^{{\rm\scriptscriptstyle {I}{I}}}\Big(\frac{w}{\sqrt{w^2-1}}\Big),&\quad w\not\in]-\infty,1].\notag
\end{align}
Next we list representations of type b):
\begin{align}
\Re\alpha+\frac{1}{2}>|\Re\lambda|:\quad&\\
\int\limits_0^{\infty}(t^2+2tw+1)^{-\alpha-\frac12}t^{-\frac12+\alpha +\lambda }{\rm d} t&\notag\\
\nonumber=
{\bf S}_{\alpha ,\lambda }^{{\rm\scriptscriptstyle {I}{I}}}(w)&\quad w\not\in]-\infty,-1];\notag\end{align}\begin{align}
\frac{1}{2}>\Re\alpha:\quad&
\\
\int\limits_{-{\rm i}\sqrt{1-w^2}-w}^{{\rm i}\sqrt{1-w^2}-w}
(t^2+2tw+1)^{-\alpha -\frac12}(-t)^{-\frac12+\alpha +\lambda }{\rm d} t&
\nonumber\\
=
{\rm i}(1-w^2)^{-\alpha }
{\bf S}_{-\alpha ,-\lambda }^{{\rm\scriptscriptstyle 0}}(w),&\quad w\not\in]-\infty,-1]\cup[1,\infty[;\notag\end{align}\begin{align}
-\Re\lambda+\frac{1}{2}>-\Re\alpha>-\frac{1}{2}:\quad&\\
\int\limits_{\sqrt{w^2-1}-w}^{0}
(t^2+2tw+1)^{-\alpha-\frac12}(-t)^{-\frac12+\alpha -\lambda }{\rm d} t&\notag\\
=(w^2-1)^{\frac{-1-2\alpha +2\lambda }{4}}{\bf S}_{-\lambda ,\alpha }^{\rm\scriptscriptstyle I}\Big(\frac{w}{\sqrt{w^2-1}}\Big) ,&\quad w\not\in]-\infty,1];\notag
\end{align}\begin{align}
\Re\lambda+\frac{1}{2}>-\Re\alpha>-\frac{1}{2}:\quad&\\
\int\limits_{-\infty}^{-\sqrt{w^2-1}-w}
(t^2+2tw+1)^{-\alpha -\frac12}{(-t)}^{-\frac12+\alpha -\lambda }{\rm d} t&
\notag\\
=(w^2-1)^{-\frac14-\frac{\alpha}{2} -\frac{\lambda}{2}}{\bf S}_{\lambda ,\alpha }^{\rm\scriptscriptstyle I}\Big(\frac{w}{\sqrt{w^2-1}}\Big),&\quad w\not\in]-\infty,1].\notag
\end{align}
\section{The Schr\"odinger Lie algebra and the heat equation}
\label{s8}
\init
By the {\em heat equation} on ${\mathbb R}^n\oplus{\mathbb R}$ we mean the equation given by the {\em heat operator}
\begin{equation} {\cal L}_n:=\Delta_n+2\partial_t.\end{equation}
This operator has a large
family of generalized symmetries, the so-called {\em Schr\"odinger Lie algebra} and {\em group}. They can be derived from conformal symmetries of the Laplace equation. In this section we describe this derivation.
In order to be consistent with Sect. \ref{s5}, it is convenient to consider
${\cal L}_{n{-}2}$ instead of ${\cal L}_n$.
Then the starting point, just as in Sect. \ref{s5}, is the $n+2$-dimensional ambient space. The Schr\"odinger Lie algebra and group are naturally contained in the pseudo-orhogonal Lie algebra and group for $n+2$
dimensions. Then, as described in Sect. \ref{subsec-conf}, we descend to the (flat) $n$ dimensional space and the corresponding Laplacian $\Delta_n$. We assume that our functions depend on $y_m$ only through the factor ${\rm e}^{y_m}$. The variable $y_{-m}$ is renamed to $t$ (the ``time''). The Schr\"odinger Lie algebra and group respects functions of that form. The Laplacian $\Delta_n$ on such fuctions becomes
the heat operator ${\cal L}_{n{-}2}$. From the generalized symmetries of $\Delta_n$ we obtain generalized symmetries of ${\cal L}_{n{-}2}$.
\subsection{${\rm sch}(n{-}2)$ as a subalgebra of $\so(n{+}2)$}
We consider again the space ${\mathbb R}^{n+2}$ with the split scalar product.
A special role will be played by the operator
\[B_{m+1,m}=z_{-m-1} \partial_{z_{m}}-z_{-m} \partial_{z_{m+1}}\in\so(n+2).\]
We define the {\em Schr\"odinger Lie algebra} and the {\em
Schr\"odinger group} as the commutants (centralizers) of this element:
\begin{subequations}
\begin{align}{\rm sch}
(n-2)&:=\{B\in \so(n+2)\ :\ [B,B_{m+1,m}]=0\},\\
{\rm Sch}(n-2)&:=\{{\alpha}\in {\rm O}(n+2)\ :\
{\alpha}B_{m+1,m}=B_{m+1,m}\alpha\}.
\end{align}\end{subequations}
\subsection{Structure of ${\rm sch}(n{-}2)$}
Let us describe the structure of
${\rm sch}(n{-}2)$.
We will use our usual notation for elements of $\so(n{+}2)$ and ${\rm O}(n{+}2)$.
In particular,
\[N_{m}=-z_{-m} \partial_{z_{-m}}+z_{m} \partial_{z_{m}},\ \ \ \ \
N_{m+1}=-z_{-m-1} \partial_{z_{-m-1}}+z_{m+1} \partial_{z_{m+1}}.\]
Define
\begin{equation}
M:=-N_{m}+N_{m+1}.\end{equation}
Note that $M$ belongs to ${\rm sch}(n{-}2)$ and commutes with $\so(n{-}2)$, which is naturally embedded in ${\rm sch}(n{-}2)$.
The Lie algebra ${\rm sch}(n{-}2)$ is spanned by the following operators:
\begin{arabicenumerate}
\item
$B_{m+1,m}$, which spans the center of
${\rm sch}(n{-}2)$.
\item
$B_{m,j}$, $B_{m+1,j}$, $|j|=1,\dots,m-1$, which have the following nonzero commutator:
\begin{equation}[B_{m,j},B_{m+1,-j}]=B_{m+1,m}.\label{cent}\end{equation}
\item $B_{m+1,-m}$, $B_{-m-1,m}$, $M$,
which have the usual commutation relations of ${\rm sl}(2)\simeq \so(3)$:
\begin{subequations} \begin{align}
[B_{m+1,-m},B_{-m-1,m}]&=M,\\
[M,B_{m+1,-m}]&
=-2B_{m+1, -m},\\
[M,B_{-m-1,m}]&
=2B_{-m-1, m}
.\end{align}\end{subequations}
\item
$B_{i,j}$, $|i|<|j|\leq m-1$, $N_i$, $i=1,\dots,m-1,$ with the usual commutation
relations of $\so(n{-}2)$.
\end{arabicenumerate}
The span of (2) can be identified with ${\mathbb R}^{n-2}\oplus {\mathbb R}^{n-2}\simeq
{\mathbb R}^2\otimes {\mathbb R}^{n-2}$, which has a natural structure of a symplectic space.
The span of (1) and (2) is the central extension of
the abelian algebra ${\mathbb R}^2\otimes {\mathbb R}^{n-2}$ by (\ref{cent}). Such a Lie algebra is
usually called the {\em Heisenberg Lie algebra over }
${\mathbb R}^2\otimes {\mathbb R}^{n-2}$ and can be denoted by
\begin{equation} {\rm heis}(2(n{-}2))={\mathbb R}\rtimes({\mathbb R}^2\otimes {\mathbb R}^{n-2}).\end{equation}
Lie algebras ${\rm sl}(2)$ and $\so(n{-}2)$ act
in the obvious way on ${\mathbb R}^2$, resp.
${\mathbb R}^{n-2}$. Thus ${\rm sl}(2)\oplus \so(n-2)$ acts on ${\mathbb R}^2\otimes {\mathbb R}^{n-2}$. Thus
\begin{eqnarray}
{\rm sch}(n{-}2)&\simeq&
{\mathbb R}{\rtimes}( {\mathbb R}^2\otimes{\mathbb R}^{n-2})\rtimes \left({\rm sl}(2)\oplus \so(n{-}2)\right).
\end{eqnarray}
Note, in particular, that
${\rm sch}(n{-}2)$ is not semisimple.
The
subalgebra
spanned by the usual Cartan algebra of $\so(n{-}2)$, $M$ and $B_{-m-1,m}$ is a maximal commutative subalgebra of ${\rm sch}(n{-}2)$. It will be called
the {\em Cartan algebra} of ${\rm sch}(n{-}2)$.
Let us introduce $\kappa\in \SO(n{+}2)$:
\begin{equation} \kappa(\dots,z_{-m},z_{m},z_{-m-1},z_{m+1}):
=(\dots,z_{-m-1},z_{m+1},-z_{-m},-z_{m}).\end{equation}
Note that $\kappa^4={\iota}$ and $\kappa\in{\rm Sch}(n-2)$.
On the level of functions
\begin{equation}\kappa K(\dots,z_{-m},z_{m},z_{-m-1},z_{m+1}):
= K(\dots,-z_{-m-1},-z_{m+1},z_{-m},z_{m}).\end{equation}
The subgroup of ${\rm Sch}(n{-}2)$ generated by
the Weyl group of $ {\rm O}(n{-}2)$ and $\kappa$
will be called
the {\em Weyl group} of
${\rm sch}(n{-}2)$.
\subsection{${\rm sch}(n{+}2)$ in $n$ dimensions}
Recall from Subsect. \ref{subsec-conf}
that using the decomposition ${\mathbb R}^{n+2}={\mathbb R}^n\oplus{\mathbb R}^2$
we obtain the representations
\begin{subequations}\begin{eqnarray}
\so(n{+}2)\ni B&\mapsto &
B^{{\rm fl},\eta}
\label{repi1}\\
{\rm O}(n{+}2)\ni {\alpha}&\mapsto&
{\alpha}^{{\rm fl},\eta}
\label{repi2}\end{eqnarray}\end{subequations}
acting on functions on ${\mathbb R}^n$.
The Laplacian $\Delta_{n+2}$ becomes the Laplacian $\Delta_n$ and it satisfies
the generalized symmetry
\begin{subequations}
\begin{eqnarray}B^{{\rm fl},\frac{-2-n}{2}}
\Delta_n&=&
\Delta_nB^{{\rm fl},\frac{2-n}{2}},\ \ B\in
\so(n{+}2),\label{syme1}\\
{\alpha}^{{\rm fl},\frac{-2-n}{2}}
\Delta_n&=&
\Delta_n{\alpha}^{{\rm fl},\frac{2-n}{2}},\ \ {\alpha}\in {\rm O}
(n{+}2).\label{syme2}\end{eqnarray}
\label{subu} \end{subequations}
The operator $B_{m+1,m}$ becomes
\begin{equation} B_{m+1,m}^{{\rm fl} ,\eta}=\partial_{y_{m}}.\end{equation}
Therefore, all elements of ${\rm sch}(n{-}2)$
in the representation (\ref{repi1}) and all elements of ${\rm Sch}(n{-}2)$
in the representation (\ref{repi2})
have the form
\begin{subequations}\begin{align}
B^{{\rm fl} ,\eta}&=C+D\partial_{y_{m}},\label{invo1}\\
{\alpha}^{{\rm fl} ,\eta}f(\dots,y_{-m},y_m)&=\beta f\big(\dots,y_{-m},y_{m}+d(\dots,y_{-m})\big),\label{invo2}\end{align}
\label{invoo}
\end{subequations}
where $C$, $D$, $\beta$, $d$, do not involve the variable $y_{m}$.
\subsection{${\rm sch}(n{-}2)$ in $(n-2)+1$ dimensions}
\label{subsec-sch}
We consider now the space ${\mathbb R}^{n-2}\oplus{\mathbb R}$ with the generic
variables $(y,t)=(\dots,y_{m-1},t)$. Note that $t$ should be
understood as the new name for $y_{-m}$, and we keep the old names for
the first $n{-}2$ coordinates.
We define the map
$\theta: C^\infty({\mathbb R}^{n-2}\oplus{\mathbb R})\to C^\infty ({\mathbb R}^n)$
by setting
\begin{equation}
(\theta h)(\dots,y_{m-1},y_{-m},y_m):=h(\dots,y_{m-1},y_{-m}){\rm e}^{y_{m}}.
\label{the1}\end{equation}
We also define $\zeta: C^\infty ({\mathbb R}^n)\to C^\infty ({\mathbb R}^{n-2}\oplus{\mathbb R})$
\begin{equation}(\zeta f)(\dots,y_{m-1},t):=f(\dots,y_{m-1},t,0).\label{the2}\end{equation}
Clearly,
$\zeta$ is a left inverse of $\theta$:
\begin{equation}\zeta\circ\theta={\iota}.\end{equation}
Therefore, $\theta\circ\zeta={\iota}$ is true on the range of $\theta$.
The heat operator in $n-2$ spatial
dimensions
can be
obtained from the Laplacian in $n$ dimension:
\begin{equation}
{\cal L}_{n-2}:=\Delta_{n{-}2}+2 \partial_{t}=\zeta
\Delta_n\theta.\label{conta}\end{equation}
For $B\in {\rm sch}(n-2)\subset \so(n+2)$
and ${\alpha}\in {\rm Sch}(n-2)\subset {\rm O}(n+2)$ we define
\begin{subequations}\begin{align}
B^{{\rm sch},\eta}&:=\zeta B^{{{\rm fl}},\eta}\theta,\\
{\alpha}^{{\rm sch},\eta}&:=\zeta {\alpha}^{{\rm fl},\eta}\theta.
\end{align}\end{subequations}
It is easy to see, using
(\ref{invoo}), that
${\rm sch}(n{-}2)$, ${\rm Sch}(n{-}2)$ and $\Delta_n$
preserve the range of $\theta$.
Therefore, for any $\eta$ we obtain representations
\begin{subequations}\begin{eqnarray}{\rm sch}(n-2)\ni B&\mapsto& B^{{\rm sch},\eta},\\
{\rm Sch}(n-2)\ni {\alpha}&\mapsto &{\alpha}^{{\rm sch},\eta}\end{eqnarray}\end{subequations}
acting on functions on ${\mathbb R}^{n-2}\oplus{\mathbb R}$.
By (\ref{subu}), we also have generalized symmetries:
\begin{subequations}\begin{eqnarray}
B^{{\rm sch},\frac{-2-n}{2}}{\cal L}_{n-2}&=&{\cal L}_{n-2} B^{{\rm sch},\frac{2-n}{2}} ,\ \ \ \ B\in {\rm sch}(n{-}2),\\
{\alpha}^{{\rm sch},\frac{-2-n}{2}}{\cal L}_{n-2}&=&{\cal L}_{n-2} {\alpha}^{{\rm sch},\frac{2-n}{2}} ,\ \ \ \ {\alpha}\in {\rm Sch}(n{-}2).\end{eqnarray}\end{subequations}
\subsection{Schr\"odinger symmetries in coordinates}
In this subsection we sum up information about Schr\"odinger
symmetries on
3 levels described in the previous subsections.
We start with generic
names of the variables and the corresponding squares:
\begin{subequations}\begin{eqnarray}
z\in{\mathbb R}^{n+2},& \langle z|z\rangle_{n+2}=&\sum_{|j|\leq m+1}z_{-j}z_j,\\
y\in{\mathbb R}^n,&\langle y|y\rangle_n=&\sum_{|j|\leq m}y_{-j}y_j,\\
(y,t)\in{\mathbb R}^{n-2}\oplus{\mathbb R},&\langle y|y\rangle_{n-2}=&\sum_{|j|\leq m-1}y_{-j}y_j.
\end{eqnarray}\end{subequations}
\noindent{\bf Cartan algebra of ${\rm sch}(n{-}2)$.}
Central element
\begin{subequations}\begin{eqnarray}
B_{m+1,m}&=&z_{-m-1} \partial_{z_{m}}-z_{-m} \partial_{z_{m+1}},\\
B_{m+1,m}^{{{\rm fl}}}&=& \partial_{y_{m}},\\
B_{m+1,m}^{{\rm sch}}&=&1.
\end{eqnarray}\end{subequations}
\noindent Cartan algebra of $\so(n{-}2)$, $j=1,\dots,m-1$,
\begin{subequations}\begin{eqnarray}
N_j&=&-z_{-j} \partial_{z_{-j}}+z_j \partial_{z_j},\\
N_j^{\rm fl}&=&-y_{-j} \partial_{y_{-j}}+y_j \partial_{y_j},\\
N_j^{{\rm sch}}&=&-y_{-j} \partial_{y_{-j}}+y_j \partial_{y_j}.
\end{eqnarray}\end{subequations}
\noindent Generator of scaling
\begin{subequations}\begin{eqnarray}
M&=&z_{-m} \partial_{z_{-m}}{-}z_{m} \partial_{z_{m}}{-}
z_{-m-1} \partial_{z_{-m-1}}{+}z_{m+1} \partial_{z_{m+1}},\\
M^{{\rm fl},\eta}&=&\sum\limits_{|j|\leq m-1}y_j \partial_{y_j}+2y_{-m} \partial_{y_{-m}}-\eta,\\
M^{{\rm sch},\eta}&=&\sum\limits_{|j|\leq m-1}y_j \partial_{y_j}+2t \partial_t-\eta.
\end{eqnarray}\end{subequations}
\medskip
\noindent{\bf Root operators of ${\rm sch}(n{-}2)$.}
Roots of $\so(n{-}2)$, $|i|<|j|\leq m-1$,
\begin{subequations}\begin{eqnarray}
B_{i,j}&=&z_{-i} \partial_{z_j}-z_{-j} \partial_{z_i},\\
B_{i,j}^{{\rm fl}}&=&y_{-i} \partial_{y_j}-y_{-j} \partial_{y_i},\\
B_{i,j}^{{\rm sch}}&=&y_{-i} \partial_{y_j}-y_{-j} \partial_{y_i}.\end{eqnarray}\end{subequations}
\noindent Space translations, $ |j|\leq m-1$,
\begin{subequations}\begin{eqnarray}
B_{m+1,j}&=&z_{-m-1} \partial_{z_j}-z_{-j} \partial_{z_{m+1}},\\
B_{m+1,j}^{{\rm fl}}&=& \partial_{y_j},\\
B_{m+1,j}^{{\rm sch}}&=& \partial_{y_j}.\end{eqnarray}\end{subequations}
\noindent Time translation
\begin{subequations}\begin{eqnarray}
B_{m+1,-m}&=&z_{-m-1} \partial_{z_{-m}}-z_{m} \partial_{z_{m+1}},\\
B_{m+1,-m}^{{\rm fl}}&=& \partial_{y_{-m}},\\
B_{m+1,-m}^{{\rm sch}}&=& \partial_t.
\end{eqnarray}
\end{subequations}
\noindent Additional roots, $|j|\leq m-1$,
\begin{subequations}\begin{eqnarray}
B_{m,j}&=&z_{-m} \partial_{z_j}-z_{-j} \partial_{z_{m}},\\
B_{m,j}^{{\rm fl}}&=&y_{-m} \partial_{y_j}-y_{-j} \partial_{y_{m}},\\
B_{m,j}^{{\rm sch}}&=&t \partial_{y_j}-y_{-j};\end{eqnarray}\end{subequations}
\begin{subequations}\begin{eqnarray}
B_{-m-1,m}&=&z_{m+1} \partial_{z_{m}}-z_{-m} \partial_{z_{-m-1}},\\
B_{-m-1,m}^{{\rm fl},\eta}&=&y_{-m}\big(\sum\limits_{|j|\leq m-1}y_j \partial_{y_j}+y_{-m} \partial_{y_{-m}}
-\eta \big)\notag
\\&&-\frac{1}{2} \sum_{|j|\leq m-1}y_{-j}y_j \partial_{y_{m}},\\{}
B_{-m-1,m}^{{\rm sch},\eta}&=&t\big(\sum\limits_{|j|\leq m-1}y_j \partial_{y_j}+t \partial_t
-\eta \big)\notag\\&&-\frac{1}{2} \sum_{|j|\leq m-1}y_{-j}y_j.
\end{eqnarray}\end{subequations}
\medskip
\noindent{\bf Weyl symmetries.}
We present a representative selection of elements of the Weyl group of ${\rm Sch}(n{-}2)$.
We will write $K$ for a function on ${\mathbb R}^{n+2}$, $f$ for a function on
${\mathbb R}^n$, $h$ for a function on
${\mathbb R}^{n-2}\oplus {\mathbb R}$ in the coordinates $\big(\dots,y_{m-1},t\big)$.
\medskip
\noindent Reflection (for odd $n$)
\begin{subequations} \begin{align}
\tau_0 K(z_0,\dots,\dots,z_{-m},z_m,z_{-m-1},z_{m+1})\notag&\\
& \hspace{-20ex}=
K(-z_0,\dots,z_{-m},z_m,z_{-m-1},z_{m+1}),\\
\tau_0^{\rm fl} f(y_0,\dots,y_{-m},y_m)\notag\\
&\hspace{-10ex}=
f(-y_0,\dots,y_{-m},y_m),\\
\tau_0^{\rm sch} h(y_0,\dots,t)&=
h(-y_0,\dots,t).
\end{align}\end{subequations}
\noindent Flips, $j=1,\dots,m-1$,
\begin{subequations}\begin{align}
\tau_j K(\dots,z_{-j},z_j,\dots,z_{-m},z_m,z_{-m-1},z_{m+1})\notag&\\
&\hspace{-25ex}=
K(\dots,z_{j},z_{-j},\dots,z_{-m},z_m,z_{-m-1},z_{m+1}),\\
\tau_j^{\rm fl} f(\dots,y_{-j},y_j,
\dots,y_{-m},y_m)\notag\\&\hspace{-15ex}=
f(\dots,y_{j},y_{-j},\dots,y_{-m},y_m),\\
\tau_j^{\rm sch} h(\dots,y_{-j},y_j,\dots,t)&=
h(\dots,y_j,y_{-j},\dots,t).
\end{align}\end{subequations}
\noindent Permutations, $\pi\in S_{m-1}$,
\begin{subequations}\begin{align}
\sigma_\pi K(\dots,z_{-m+1},z_{m-1},z_{-m},z_m,z_{-m-1},z_{m+1})&\notag\\
&\hspace{-35ex}=
K(\dots,z_{-\pi_{m-1}},z_{\pi_{m-1}},z_{-m},z_m,z_{-m-1},z_{m+1}),\\
\sigma_\pi^{\rm fl} f(\dots,y_{-m+1},y_{m-1},y_{-m},y_m)&\notag\\
& \hspace{-15ex}=
f(\dots,y_{-\pi_{m-1}},y_{\pi_{m-1}},y_{-m},y_m),\\
\sigma_\pi^{\rm sch} h(\dots,y_{-m+1},y_{m-1},t)&\notag\\
& \hspace{-8ex}
=h(\dots,y_{-\pi_{m-1}},y_{\pi_{m-1}},t).
\end{align}\end{subequations}
\noindent Special transformation $\kappa$
\begin{subequations}\begin{align}
\kappa K(\dots,z_{m-1},z_{-m},z_{m},z_{-m-1},z_{m+1})&\notag\\
&\hspace{-30ex}= K(\dots,z_{m-1},-z_{-m-1},-z_{m+1},z_{-m},z_{m}),\\
\kappa^{{\rm fl},\eta}f(\dots,y_{m-1},y_{-m},y_m)&\notag\\
&\hspace{-28ex}= y_{-m}^\eta
f\Big(\dots,
\frac{y_{m-1}}{y_{-m}},-\frac{1}{y_{-m}},
\frac{1}{2y_{-m}}\sum_{|j|\leq m}y_{-j}y_j\Big),
\\
\kappa^{{\rm sch},\eta}h(\dots,y_{m-1},t)&\notag\\
&\hspace{-26ex}= t^\eta\exp\Big(\frac{1}{2t}\sum_{|j|\leq m-1}y_{-j}y_j\Big)
h\Big(\dots,\frac{y_{m-1}}{t},-\frac{1}{t}\Big).
\end{align}\end{subequations}
Square of $\kappa$
\begin{subequations}\begin{align}
\kappa^2 K(\dots,z_{m-1},z_{-m},z_{m},z_{-m-1},z_{m+1})
&\notag\\&\hspace{-30ex}= K(\dots,z_{m-1},-z_{-m},-z_{m},-z_{-m-1},-z_{m+1}),\\
(\kappa^{{\rm fl},\eta})^2 f(\dots,y_{m-1},y_{-m},y_{m})&\notag\\
&\hspace{-18ex}=f(\dots,-y_{m-1},y_{-m},y_{m}),\\
(\kappa^{{\rm sch},\eta})^2h(\dots,y_{m-1},t)&=
h(\dots,-y_{m-1},t).
\end{align}\end{subequations}
\medskip
\noindent{\bf Laplacian/Laplacian / Heat operator}
\begin{subequations}\begin{eqnarray}
\Delta_{n+2}
&=&\sum\limits_{|j|\leq m+1} \partial_{z_{-j}} \partial_{z_j},\\
\Delta_{n}
&=&\sum\limits_{|j|\leq m} \partial_{y_{-j}} \partial_{y_j},\\
{\cal L}_{n-2}&=&\sum\limits_{|j|\leq m-1} \partial_{y_{-j}} \partial_{y_j}+2 \partial_t.
\end{eqnarray}\end{subequations}
\subsection{Special solutions of the heat equation}
\label{Special solutions of the heat equation}
Let us describe how to obtain solutions of the heat equation from solutions of the Laplace equation.
Consider first a function on the level of ${\mathbb R}^{n+2}$
\begin{align} K(z)&=z_{-m}^{1-\frac{n}{2}}g\Big(\frac{z_{1}}{z_{-m}},\dots,\frac{z_{m-1}}{z_{-m}}\Big)\exp\Big(-\frac{z_{m+1}}{z_{-m}}\Big),
\end{align}
where $g$ is a harmonic function on ${\mathbb R}^{n-2}$.
It is easy to see that $K$ is harmonic and satisfies
\begin{align}
B_{m+1,m}K&=K.
\end{align}
Besides, $K$ is homogeneous of degree $1-\frac{n}{2}$. Therefore, we can descend on the level of dimension $n$, obtaining the function
\begin{align} k(y)&=y_{-m}^{1-\frac{n}{2}}g\Big(\frac{y_{1}}{y_{-m}},\dots,\frac{y_{m-1}}{y_{-m}}\Big) \exp\Big(\sum_{|i|\leq m-1}\frac{y_{-i}y_i}{y_{-m}}+y_{m} \Big).
\end{align}
It is harmonic and satisfies
\begin{align}
B_{m+1,m}^{\rm fl} k&=k.
\end{align}
Descending on the level of ${\mathbb R}^{n-2}\oplus{\mathbb R}$ we obtain
\begin{equation}
h(y,t)=
t^{1-\frac{n}{2}}g\Big(\frac{y_{1}}{t},\dots,\frac{y_{m-1}}{t}\Big) \exp\Big(\sum_{|i|\leq m-1}\frac{y_{-i}y_i}{t} \Big).
\end{equation}
which solves the heat equation:
\begin{equation}
{\cal L}_{n-2}h=0.
\end{equation}
\subsection{Wave packets for the heat equation}
Let us use the coordinates $(y,t)\in{\mathbb R}^{n-2}\oplus{\mathbb R}$.
Recall that
\begin{align}
M^{{\rm sch},\eta}&=\sum\limits_{|j|\leq m-1}y_j \partial_{y_j}+2t \partial_t-\eta.
\end{align}
The following proposition is proven by analogous arguments as Prop.
\ref{cac0}. It allows us to form wave packets that are eigenfunctions of $M$:
\begin{proposition}
Suppose that
$]0,1[\ni s\overset{\gamma}\mapsto \tau(s)$ is a contour satisfying
\begin{equation}
f(\tau y,\tau^2 t)\tau^{-\nu}\Big|_{\tau(0)}^{\tau(1)}=0.\end{equation}
Set
\begin{equation}
h_\nu(y,t):=\int_\gamma f(\tau y,\tau^2 t)\tau^{-1-\nu}{\rm d}\tau.\end{equation}
Then
\begin{equation}
M^{{\rm sch},\eta}
h_\nu=(\nu-\eta) h_\nu.\end{equation}
\label{analo} \end{proposition}
\section{Heat equation in 2 dimensions and the confluent equation}
\label{s9}
\init
The goal of this section is to derive the ${}_1{\cal F}_1$ equation together with its symmetries from the heat equation in $2$ dimensions, which in turn comes from the Laplace equation in $6$ and $4$ dimensions. Let us describe the main steps of this derivation:
\be
\item\label{it1h}
We start from the Schr\"odinger Lie algebra ${\rm sch}(2)$ and group ${\rm Sch}(2)$
considered as a subalgebra of $\so(6)$, resp. a subgroup of $\mathrm{O}(6)$, acting in $6$ dimensions. The main initial operator is the Laplacian $\Delta_6$.
\item\label{it3h} We descend onto $4$ dimensions. The 6-dimensional Laplacian $\Delta_6$ becomes the 4-dimensional Laplacian $\Delta_4$.
\item We assume that the variable $y_2$ appears only in the exponential ${\rm e}^{y_2}$ and the variable $y_{-2}$ is renamed $t$. The Laplacian $\Delta_4$ becomes the heat operator ${\cal L}_2$.
The representations $B^{{\rm sch},\eta}$ and $\alpha^{{\rm sch},\eta}$ preserve our class of functions. With $\eta=-1$ and $\eta=-3$ they are generalized symmetries of the heat operator.
\item\label{it4h} We choose coordinates $w,s,u_1$, so that the Cartan operators are expressed in terms of $s$, $u_1$. We compute
${\cal L}_2$, $B^{{\rm sch},\eta}$, and $\alpha^{{\rm sch},\eta}$ in the new coordinates.
\item\label{it5h} We make an ansatz that diagonalizes the Cartan operators, whose eigenvalues, denoted by $-\theta$ and $\alpha$,
become parameters. The operators
${\cal L}_2$, $B^{{\rm sch},\eta}$, and $\alpha^{{\rm sch},\eta}$ involve now only the single variable $w$.
The operator $\frac{s^2}{2}{\cal L}_2$ becomes the ${}_1{\cal F}_1$ operator.
Generalized symmetries of
${\cal L}_2$ yield transmutation relations and discrete symmetries of the
${}_1{\cal F}_1$ operator.
\ee
The first part of this section is devoted to a description of the above steps, except for Step \ref{it3h}, discussed in detail in Sect. \ref{s8}.
The remaining part of this section is devoted to the theory of the ${}_1{\cal F}_1$ equation
and its solutions. Its organization is parallel
to that of Sect. \ref{s6} on the ${}_2{\cal F}_1$ equation.
The main additional complication is the fact that besides
the ${}_1{\cal F}_1$ equation
and the ${}_1F_1$ function,
it is useful to discuss the closely related
${}_2{\cal F}_0$ equation
and the ${}_2F_0$ function.
In fact, some of the standard solutions of the ${}_1{\cal F}_1$ equation are expressed in terms of
the ${}_1F_1$ function, others in terms of the
${}_2F_0$ function.
\subsection{${\rm sch}(2)$ in $6$ dimensions}
We again consider ${\mathbb R}^6$ with the coordinates (\ref{sq0})
and the product given by (\ref{sq1}):
\[
\langle z|z\rangle=2z_{-1}z_1+2z_{-2}z_2+2z_{-3}z_3.\]
We describe various object related to the Lie algebra
${\rm sch}(2)$ treated as a subalgebra of $\so(6)$. We also list
a few typical Weyl symmetries of ${\rm Sch}(2)$.
\noindent{\bf Lie algebra ${\rm sch}(2)$.}
Cartan algebra
\begin{subequations}\begin{eqnarray}
M&=&z_{-2} \partial_{z_{-2}}-z_{2} \partial_{z_{2}}-z_{-3} \partial_{z_{-3}}+z_{3} \partial_{z_{3}},\\
N_1&=&-z_{-1} \partial_{z_{-1}}+z_1 \partial_{z_1},\\
B_{3,2}&=&z_{-3} \partial_{z_2}-z_{-2} \partial_{z_{3}}.
\end{eqnarray}\end{subequations}
Root operators
\begin{subequations}\begin{eqnarray}
B_{3,-1}&=&z_{-3} \partial_{z_{-1}}-z_1 \partial_{z_{3}},\\
B_{2,1}&=&z_{-2} \partial_{z_1}-z_{-1} \partial_{z_{2}},\\
B_{3,1}&=&z_{-3} \partial_{z_{1}}-z_{-1} \partial_{z_{3}},\\
B_{2,-1}&=&z_{-2} \partial_{z_{-1}}-z_1 \partial_{z_2},\\
B_{3,-2}&=&z_{-3} \partial_{z_{-2}}-z_2 \partial_{z_{3}},\\
B_{-3,2}&=&z_{3} \partial_{z_{2}}-z_{-2} \partial_{z_{-3}}.
\end{eqnarray}\end{subequations}
\noindent{\bf Weyl symmetries}
\begin{subequations}\begin{align}{\iota}
K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_3)&=K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_3),\\
\tau_1
K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_3)&=K(z_{1},z_{-1},z_{-2},z_2,z_{-3},z_3),\\
\kappa
K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_3)&=
K(z_{-1},z_1,-z_{-3},-z_{3},z_{-2},z_{2}),\\
\tau_1\kappa
K(z_{-1},z_1,z_{-2},z_2,z_{-3},z_3)&=
K(z_1,z_{-1},-z_{-3},-z_{3},z_{-2},z_{2}).
\end{align}\end{subequations}
\noindent{\bf Laplacian}
\begin{equation}
\Delta_6=2\partial_{z_{-1}}\partial_{z_{1}}+
2\partial_{z_{-2}}\partial_{z_{2}}+
2\partial_{z_{-3}}\partial_{z_{3}}
.\label{sq2-}\end{equation}
\subsection{${\rm sch}(2)$ in $4$ dimensions}
We descend on the level of ${\mathbb R}^4$, with the coordinates $(y_{-1},y_1,y_{-2},y_2)$
and the scalar product given by
\[
\langle y|y\rangle=2y_{-1}y_1+2y_{-2}y_2.\]
\noindent{\bf Lie algebra ${\rm sch}(2)$.}
Cartan algebra
\begin{subequations}\begin{eqnarray*}
M^{{\rm fl} ,\eta}&=&y_{-1} \partial_{y_{-1}}+y_{1} \partial_{y_{1}}+2y_{-2} \partial_{y_{-2}}-
\eta,\\
N_1^{\rm fl} &=&-y_{-1} \partial_{y_-1}+y_{1} \partial_{y_{1}},\\
B_{3,2}^{\rm fl}&=& \partial_{y_2}.
\end{eqnarray*}\end{subequations}
\noindent Root operators
\begin{subequations}\begin{eqnarray*}
B_{3,-1}^{{\rm fl}}&=& \partial_{y_{-1}},\\
B_{2,1}^{{\rm fl}}&=&y_{-2} \partial_{y_1}-y_1 \partial_{y_{2}},\\
B_{3,1}^{{\rm fl}}&=& \partial_{y_{1}},\\
B_{2,-1}^{{\rm fl}}&=&y_{-2} \partial_{y_{-1}}-y_1 \partial_{y_2},\\
B_{3,-2}^{{\rm fl}}&=& \partial_{y_{-2}},\\
B_{-3,2}^{{\rm fl},\eta}&=&
y_{-2}(y_{-1} \partial_{y_{-1}}+y_{1} \partial_{y_{1}}+y_{-2} \partial_{y_{-2}}-\eta)-y_{-1}y_1 \partial_{y_2}.
\end{eqnarray*}\end{subequations}
\noindent{\bf Weyl symmetries}
\begin{subequations}\begin{align*}{\iota}
f(y_{-1},y_1,y_{-2},y_2)&=f(y_{-1},y_1,y_{-2},y_2),\\
\tau_1^{\rm fl} f(y_{-1},y_{1},y_{-2},y_2)&=f(y_1,y_{-1},y_{-2},y_2),\\
\kappa^{{\rm fl},\eta}f(y_{-1},y_{1},y_{-2},y_2)
&=y_{-2}^\eta
f\Big(\frac{y_{-1}}{y_{-2}},\frac{y_1}{y_{-2}},-\frac1{y_{-2}},
\frac{y_{-1}y_1+y_{-2}y_2}{y_{-2}}\Big),\\
\tau_1\kappa^{{\rm fl},\eta}f(y_{-1},y_{1},y_{-2},y_2)
&=y_{-2}^\eta
f\Big(\frac{y_{1}}{y_{-2}},\frac{y_{-1}}{y_{-2}},-\frac1{y_{-2}},
\frac{y_{-1}y_1+y_{-2}y_2}{y_{-2}}\Big).
\end{align*}
\end{subequations}
\subsection{${\rm sch}(2)$ in $2+1$ dimensions}
We apply the ansatz involving the exponential ${\rm e}^{y_2}$.
We rename $y_{-2}$ to $t$.
\noindent{\bf Lie algebra ${\rm sch}(2)$.}
Cartan algebra
\begin{subequations}
\begin{eqnarray}
M^{{\rm sch},\eta}&=&y_{-1} \partial_{y_{-1}}+y_{1} \partial_{y_{1}}+2t \partial_{t}-
\eta,\label{heat1}\\
N_1^{{\rm sch}}&=&-y_{-1} \partial_{y_{-1}}+y_{1} \partial_{y_{1}}
,\\B_{32}^{\rm sch}&=&1.\label{heat2}
\end{eqnarray}
\end{subequations}
\noindent
Root operators
\begin{subequations}\begin{eqnarray}
B_{3,-1}^{{\rm sch}}&=& \partial_{y_{-1}},\\
B_{2,1}^{{\rm sch}}&=&t \partial_{y_1}-y_{-1},\\
B_{3,1}^{{\rm sch}}&=& \partial_{y_{1}},\\
B_{2,-1}^{{\rm sch}}&=&t \partial_{y_{-1}}-y_1,\\
B_{3,-2}^{{\rm sch}}&=& \partial_{t},\\
B_{-3,2}^{{\rm sch},\eta}&=&
t(y_{-1} \partial_{y_{-1}}+y_{1} \partial_{y_{1}}+t \partial_{t}-\eta)-y_{-1}y_1.
\end{eqnarray}\end{subequations}
\noindent
{\bf Weyl symmetries}
\begin{subequations}
\begin{align}
{\iota} g(y_{-1},y_1,t)&=g(y_{-1},y_1,t),\\
\tau_1^{\rm sch} h(y_{-1},y_{1},t)&=h(y_1,y_{-1},t),\\
\kappa^{{\rm sch},\eta}h(y_{-1},y_1,t)
&=t^\eta\exp\Big(\frac{y_{-1}y_1}{t}\Big)
h\Big(\frac{y_{-1}}{t},\frac{y_1}{t},-\frac1{t}\Big),\\
\tau_1\kappa^{{\rm sch},\eta}h(y_{-1},y_1,t)
&=t^\eta\exp\Big(\frac{y_{-1}y_1}{t}\Big)
h\Big(\frac{y_{1}}{t},\frac{y_{-1}}{t},-\frac1{t}\Big).
\end{align}
\end{subequations}
\noindent{\bf Heat operator}
\begin{equation}
{\cal L}_2=2 \partial_{y_{-1}} \partial_{y_1}+2 \partial_{t}.\label{heat0}\end{equation}
\subsection{${\rm sch}(2)$ in the coordinates $w,s,u_1$}
We introduce new coordinates $w,s,u_1$
\begin{align}
w =\frac{y_{-1}y_1}{t}\;,\ \ \ \
&u_1 =\frac{y_{1}}{\sqrt{t}} \;,\ \ \ \
s =\sqrt{t}\,,\phantom{\frac{1}{2}}\label{cor}\\
\intertext{with the reverse transformations}
y_{-1} =\frac{ sw}{u_1} \;,\ \ \ \
&y_{1} =u_1s\;,\ \ \ \
t =s^2\,.\phantom{\frac{1}{2}}
\end{align}
\noindent{\bf Lie algebra ${\rm sch}(2)$.} Cartan algebra
\begin{subequations}
\begin{align*}
M^{{\rm sch},{\eta}}& =s\dds-\eta,\\
N_1^{\rm sch}& =u_1 \partial_{u_1}
,\\B_{32}^{\rm sch}&=1.
\end{align*}
\end{subequations}
\noindent Root operators
\begin{subequations}
\begin{align*}
B_{3,-1}^{\rm sch}& =
\frac{u_1}{s} \partial_w,\\
B_{2,1}^{\rm sch}& =\frac{s}{u_1}(w \partial_w+u_1 \partial_{u_1}-w),\\
B_{3,1}^{\rm sch}& =\frac{1}{{u_1}s}(w \partial_w+{u_1} \partial_{u_1}),\\
B_{2,-1}^{\rm sch}& =
s{u_1}( \partial_w-1),\\
B_{3,-2}^{{\rm sch}}& =\frac{1}{s^2}\big(-w \partial_w-\frac12{u_1} \partial_{u_1}+\frac12s \partial_s\big),\\
B_{-3,2}^{{\rm sch},\eta}&=s^2\big(w \partial_w+\frac12{u_1} \partial_{u_1}+\frac12s \partial_s-w-\eta\big).
\end{align*}
\end{subequations}
\noindent{\bf Weyl symmetries}
\begin{subequations}\begin{align*}
{\iota} h(w,{u_1},s)&=h(w,{u_1},s),\\
\tau_1^{\rm sch} h\Big(w,{u_1},s\Big)&=h\Big(w,\frac{w}{{u_1}},s\Big),\\
\kappa^{{\rm sch},\eta}h(w,{u_1},s)&
=s^{2\eta}{\rm e}^w
h\Big(-w,-{\rm i} {u_1},\frac{{\rm i}}s\Big),\\
\tau_1\kappa^{{\rm sch},\eta}h(w,{u_1},s)&
=s^{2\eta}{\rm e}^w
h\Big(-w,-\frac{{\rm i} w}{{u_1}},\frac{{\rm i}}s\Big).
\end{align*}\end{subequations}
\noindent{\bf Heat operator}
\begin{equation}
\label{con:1}
{\cal L}_{2}
=\frac{2}{s^2}\Big(w \partial_w^2+({u_1} \partial_{u_1}+1-w) \partial_w+\frac12(-{u_1} \partial_{u_1}+s \partial_s)\Big).
\end{equation}
\subsection{Confluent operator}
Let us make the ansatz
\begin{equation} h(w, {u_1},s)={u_1}^\alpha s^{-\theta-1} F(w).\label{ansatz}\end{equation}
Clearly,
\begin{subequations}\begin{eqnarray}
M^{{\rm sch},-1} h&=&-\theta h,\\
N_1^{{\rm sch}} h&=&\alpha h,\\
u_1^{-\alpha}s^{\theta+1}\frac{s^2}{2}{\cal L}_2 h&=&
{\cal F}_{\theta ,\alpha}(w, \partial_w)F(w),
\end{eqnarray}\end{subequations}
where we have introduced the {\em ${}_1{\cal F}_1$ operator}
\begin{eqnarray}
{\cal F}_{\theta ,\alpha}(w, \partial_w)
&=&w \partial_w^2+(1+\alpha-w) \partial_w-\frac{1}{2}(1+\theta +\alpha)
.\end{eqnarray}
Let us also define the closely related {\em ${}_2{\cal F}_0$ operator}
\begin{eqnarray}
\tilde{\cal F}_{\theta,\alpha}(w, \partial_w)&=&
w^2 \partial_w^2+(-1+(2+\theta)w) \partial_w+\frac14(1+\theta)^2-\frac14\alpha^2.\label{closely}
\end{eqnarray}
It is equivalent to the ${}_1{\cal F}_1$ operator. In fact, if $z=-w^{-1}$, then
\begin{equation}
(-z)^{\frac{3+\alpha+\theta}{2}}\tilde{\cal F}_{\theta,\alpha}(z, \partial_z)(-z)^{-\frac{1+\alpha+\theta}{2}}
={\cal F}_{\theta,\alpha}(w, \partial_w).
\label{g8a}\end{equation}
We will treat ${\cal F}_{\theta ,\alpha}(w, \partial_w)$ as the principal operator.
Traditionally, one uses the {\em classical parameters} $a,b,c$:
\begin{subequations}\begin{align}
&\alpha:=c-1=a-b,&& \theta: =-c+2a=-1+a+b;&\\
&a=\frac{1+\alpha+\theta }{2},&& b=\frac{1 -\alpha+\theta}{2},\ \ \ &
c=1+\alpha.
\end{align}\end{subequations}
Here are the traditional forms of the ${}_1{\cal F}_1$ and ${}_2{\cal F}_0$ operators:
\begin{align}
{\cal F}(a;c;w, \partial_w)&:=w \partial_w^2+(c-w) \partial_w-a,
\label{f1c}\\
{{\cal F}}(a,b;-;w,\partial_w)&:=w^2\partial_w^2 +( -1+(1+a+b)w)\partial_w +ab. \label{g8}\end{align}
\subsection{Transmutation relations and discrete symmetries}
The heat operator satisfies the following generalized symmetries:
\begin{subequations}\begin{align}
B^{{\rm sch},-3}{\cal L}_2&={\cal L}_2 B^{{\rm sch},-1},\quad B\in {\rm sch}(2),\label{simmi1}\\
\alpha^{{\rm sch},-3}{\cal L}_2&={\cal L}_2 \alpha^{{\rm sch},-1},\quad \alpha\in {\rm Sch}(2).
\label{simmi2} \end{align}\end{subequations}
Applying (\ref{simmi1}) to the roots of ${\rm sch}(2)$ we obtain the following transmutation relations
of the confluent operator:
\[\begin{array}{rrl}
& \partial_w&{\cal F}_{\theta ,\alpha}\\[0ex]
&=\ \ \ {\cal F}_{\theta +1,\alpha+1}& \partial_w,\\[0.6ex]
&(w \partial_w+\alpha-w)&{\cal F}_{\theta ,\alpha}\\[0ex]
&=\ \ \ {\cal F}_{\theta -1,\alpha-1}&(w \partial_w+\alpha-w),\\[0.6ex]
&(w \partial_w+\alpha)&
{\cal F}_{\theta ,\alpha}\\[0ex]
&=\ \ \ {\cal F}_{\theta +1,\alpha-1}&(w \partial_w+\alpha),\\[0.6ex]
& ( \partial_w-1)&{\cal F}_{\theta ,\alpha},\\[0ex]
&=\ \ \ {\cal F}_{\theta -1,\alpha+1}&( \partial_w-1);\\[0.6ex]
&\big( w \partial_w+\frac{1}{2}(\theta + \alpha+1)\big)&w{\cal F}_{\theta ,\alpha}\\[0ex]
&=\ \ \ w{\cal F}_{\theta +2,\alpha}&\big( w \partial_w+\frac{1}{2}(\theta + \alpha+1)\big),\\[0.6ex]
&\big(w \partial_w+\frac{1}{2}(-\theta +\alpha+1)-w)
&w{\cal F}_{\theta ,\alpha}\\[0ex]
&=\ \ \ w{\cal F}_{\theta -2,\alpha}&\big(w \partial_w+\frac{1}{2}(-\theta +\alpha+1)-w\big).
\end{array}\]
Applying (\ref{simmi2})
to the Weyl symmetries of ${\rm sch}(2)$ yields discrete symmetries of the confluent operator, described below.
The following operators equal ${\cal F}_{\theta ,\alpha}(w, \partial_w)$ for the appropriate $w$:
\[\begin{array}{rrcl}
w=v:
&&{\cal F}_{\theta ,\alpha}(v, \partial_v),&\\[0.2ex]
&v^{-\alpha}&{\cal F}_{\theta ,-\alpha}(v, \partial_v)&v^{\alpha},\\
w=-v:
&-{\rm e}^{-v}&{\cal F}_{-\theta ,\alpha}(v, \partial_v)&{\rm e}^v,
\\[0.2ex]
&-{\rm e}^{-v}v^{-\alpha}&{\cal F}_{-\theta ,-\alpha}(v, \partial_v)&{\rm e}^vv^{\alpha}
.\end{array}\label{newnot1}\]
The third symmetry is sometimes called the {\em 1st Kummer transformation}.
\subsection{Factorizations of of the heat operator}
\label{Factorizations of of the heat operator}
Special role is played by three distinguished subalgebras in ${\rm sch}(2)$: two isomorphic to ${\rm heis}(2)$ and one isomorphic to $\so(3)$.
First note
the commutation relations
\begin{equation}
[B_{2,-1},B_{3,1}] = [B_{2,1}\,,B_{3,-1}] = B_{3,2}.\label{heis:1}\end{equation}
Therefore, the following subalgebras in ${\rm sch}(5)$ are isomorphic to ${\rm heis}(2)$:
\begin{subequations}\begin{eqnarray}
{\rm heis}_-(2)& \hbox{spanned by}&B_{2,-1},\ B_{3,1},\ B_{3,2},\\
{\rm heis}_+(2)& \hbox{spanned by}&B_{2,1},\ B_{3,-1},\ B_{3,2}.
\end{eqnarray}\end{subequations}
Note that the flip of $(1,-1)$, denoted $\tau_1$,
belongs to ${\rm Sch}(5)$ and satisfies
\begin{equation}\tau_1B_{2,-1}\tau_1=B_{2,1},\quad
\tau_1B_{3,1}\tau_1=B_{3,-1},\quad \tau_1B_{3,2}\tau_1=B_{3,2}.\label{heis:1-}\end{equation}
Hence,
\begin{equation} \tau_1{\rm heis}_-(2)\tau_1={\rm heis}_+(2).\end{equation}
Let us define
\begin{subequations}
\begin{align}
{\cal C}_{-}& =2\,B_{2,-1} B_{3,1}+M+N_1-B_{3,2}\,\\
& =2\,B_{3,1} B_{2,-1}+M+N_1+B_{3,2},\\
{\cal C}_{+}& =2\,B_{2,1} B_{3,-1}+M-N_1-B_{3,2}\,\\
& =2\,B_{3,-1} B_{2,1}+M-N_1+B_{3,2}.
\end{align}\label{dire1}
\end{subequations}
${\cal C}_+$ and ${\cal C}_-$ can be viewed as the Casimir operators
for ${\rm heis}_+(2)$, resp. for ${\rm heis}_-(2)$. Indeed, ${\cal C}_+$, resp. ${\cal C}_-$
commute with all operators in ${\rm heis}_+(2)$, resp. ${\rm heis}_-(2)$.
We also have
\begin{equation}\tau_1{\cal C}_-\tau_1={\cal C}_+.\end{equation}
On the level of ${\mathbb R}^2\oplus{\mathbb R}$, the two operators ${\cal C}_+$ and ${\cal C}_-$ coincide. Indeed, a
direct calculation yields
\begin{eqnarray}
{\cal C}_{+}^{{\rm sch},\eta}={\cal C}_{-}^{{\rm sch},\eta}=2t( \partial_{y_{-1}} \partial_{y_1}+ \partial_{t})
-\eta-1.
\label{dire}\end{eqnarray}
Second,
note
the commutation relations
\begin{equation}
[B_{-3,2},B_{3,-2}] = N_2-N_3=-M.\label{commu}
\end{equation}
Therefore, the following of ${\rm sch}(2)$ is isomorphic to $\so(3)$:
\begin{equation} \so_{23}(3)\quad\text{ spanned by }\quad
B_{-3,2}, B_{3,-2}, M.\end{equation}
The Casimir operator
for $\so_{23}(3)$ is
\begin{subequations}
\begin{align}
{\cal C}_{23}& =4\,B_{3,-2} B_{-3,2}-(M-1)^2+1\,\\
& =4\,B_{-3,2} B_{3,-2}-(M+1)^2+1.
\end{align}\label{dire2}
\end{subequations}
By (\ref{deq3a}) we have
\begin{equation} (2z_{-2}z_2+2z_{-3}z_3)\Delta_6^\diamond=-1+{\cal C}_{23}^{\diamond,-1}+(N_1^{\diamond,-1})^2.
\label{dire.}\end{equation}
Inserting (\ref{dire2})
into (\ref{dire.}) we obtain
\begin{subequations}\begin{align}\notag
& (2z_{-2}z_2+2z_{-3}z_3)\Delta_6^\diamond\\=&4B_{2,-3}B_{-2,3}
-(N_1+M+1)(-N_1+M+1)\\
=&4B_{-2,3}B_{2,-3}
-(N_1+M-1)(-N_1+M-1),
\end{align}\label{suu1}\end{subequations}
where the $B$, $N_1$ and $M$ operators should be equipped with the superscript ${}^{\diamond,-1}$.
Let us sum up the factorizations
in the variables $y_{-1}y_1,t$
obtained with the help of the three subalgebras:
\begin{subequations}
\begin{align}
t{\cal L}_2
& =2\,B_{2,-1} B_{3,1}+M+N_1-1\,\label{su1}\\
& =2\,B_{3,1} B_{2,-1}+M+N_1+1\label{su2}\\
& =2\,B_{2,1} B_{3,-1}+M-N_1-1\,\label{su3}\\
& =2\,B_{3,-1} B_{2,1}+M-N_1+1,\label{su4}\\
2y_{-1}y_1{\cal L}_2&=-4B_{2,-3}B_{-2,3}
-(N_1+M+1)(N_1-M-1)\label{su5}\\
&=-4B_{-2,3}B_{2,-3}
-(N_1+M-1)(N_1-M+1),\label{su6}
\end{align}\end{subequations}
where the $B$, $N_1$ and $M$ operators should be equipped with the superscript ${}^{{\rm sch},-1}$.
Indeed, to obtain (\ref{su1})--(\ref{su4}) we insert (\ref{dire1}) into (\ref{dire}). To obtain (\ref{su5})--(\ref{su6}) we rewrite (\ref{suu1}), multiplying it by $-1$.
In the variables $w,u,s$, we need to make the replacements
\begin{subequations}
\begin{eqnarray}
y_{-1}y_1&\quad\to\quad ws^2,
\label{faca-1.}\\
t&\quad\to\quad s^2.\label{faca-2.}
\end{eqnarray}
\end{subequations}
\subsection{Factorizations of the confluent operator}
Factorizations of ${\cal L}_2$ described in Subsect.
\ref{Factorizations of of the heat operator}
yield the following factorizations of the confluent operator:
\begin{eqnarray*}
{\cal F}_{\theta,\alpha}
&=&\Big(\partial_w-1\Big)\Big(w\partial_w+\alpha\Big)-\frac12(\theta-\alpha+1)\\
&=&\Big(w\partial_w+1+\alpha\Big)\Big(\partial_w-1\Big)-\frac12(\theta-\alpha-1)\\
&=&\partial_w\Big(w\partial_w+\alpha-w\Big)-\frac12(\theta+\alpha-1)\\
&=&\Big(w\partial_w+1+\alpha-w\Big)\partial_w-\frac12(\theta+\alpha+1),
\end{eqnarray*}
\begin{eqnarray*}
w{\cal F}_{\theta,\alpha}
&=&\Big(
w\partial_w+\frac12(-\theta+\alpha-1)-w\Big)
\Big(w\partial_w+\frac12(\theta+\alpha+1)\Big)
\\&&-\frac14(-\theta+\alpha-1)(\theta+\alpha+1)\\
&=&\Big(w\partial_w+\frac12(\theta+\alpha-1)\Big)\Big(
w\partial_w+\frac12(-\theta+\alpha+1)-w\Big)
\\&&-\frac14(-\theta+\alpha+1)(\theta+\alpha-1).
\end{eqnarray*}
\subsection{The ${}_1F_1$ function}
The ${}_1{\cal F}_1$ equation (\ref{f1c})
has a regular singular point at $0$.
Its indices at $0$ are equal to $0$, $1-c$.
For $c\neq 0,-1,-2,\dots$, the unique solution of the confluent equation analytic
at $0$ and equal to 1 at 0 is called
the {\em ${}_1F_1$ function} or
{\em Kummer's confluent function}. It is equal to
\[F(a;c;w):=\sum_{n=0}^\infty
\frac{(a)_n}{
(c)_n}\frac{w^n}{n!}.\]
It is defined for
$c\neq0,-1,-2,\dots$.
Sometimes it is more convenient to consider
the functions
\begin{align*} {\bf F} (a;c;w)&:=\frac{F(a;c;w)}{\Gamma(c)}=
\sum_{n=0}^\infty
\frac{(a)_n}{
\Gamma(c+n)}\frac{w^n}{n!},\\
{\bf F}^{\rm\scriptscriptstyle I} (a;c;w)&:=\frac{\Gamma(a)\Gamma(c-a)}{\Gamma(c)}F(a;c;w).
\end{align*}
In the Lie-algebraic parameters:
\begin{eqnarray*}
F_{\theta ,\alpha}(w)&:=&F\Bigl(\frac{1+\alpha+\theta }{2};1+\alpha;w\Bigr)
,\\
{\bf F} _{\theta ,\alpha}(w)&:=&
{\bf F} \Bigl(\frac{1+\alpha+\theta }{2};1+\alpha;w\Bigr)
=
\frac{F_{\theta ,\alpha}(w)}{\Gamma(\alpha+1)},\\
{\bf F}^{\rm\scriptscriptstyle I} _{\theta ,\alpha}(w)&:=&
{\bf F}^{\rm\scriptscriptstyle I} \Bigl(\frac{1+\alpha+\theta }{2};1+\alpha;w\Bigr)=
\frac{\Gamma(\frac{1+\alpha+\theta}{2})
\Gamma(\frac{1+\alpha-\theta}{2})F_{\theta ,\alpha}(w)}{\Gamma(\alpha+1)}
.\end{eqnarray*}
\subsection{The ${}_2F_0$ function}
\label{The ${}_2F_0$ function}
Recall from (\ref{g8a}) that in parallel with the ${}_1{\cal F}_{1}$ operator
it is useful to consider the ${}_2{\cal F}_0$ operator.
The ${}_2{\cal F}_0$ operator
does not have a regular singular point at zero, hence to construct its solutions having a simple behavior at zero we cannot use the Frobenius method. One of such solutions is the {\em ${}_2F_0$ function}. For $w\in{\mathbb C}\backslash[0,+\infty[$ it can be defined by
\[F(a,b;-;w):=\lim_{c\to\infty}F(a,b;c;cw),\]
where $|\arg c-\pi|<\pi-\epsilon$, $\epsilon>0$.
It extends to an analytic function on the universal cover of
${\mathbb C}\backslash\{0\}$
with a branch point of an infinite order at 0.
It has the following asymptotic expansion:
\[
F(a,b;-;w)\sim\sum_{n=0}^\infty\frac{(a)_n(b)_n}{n!}w^n,
\ |\arg w-\pi|<\pi-\epsilon.
\]
Sometimes instead of ${}_2F_0$ it is useful to consider the function
\begin{eqnarray*}
F^{\rm\scriptscriptstyle I} (a,b;-;w)&:=&\Gamma(a)F(a,b;-;w).
\end{eqnarray*}
When we use the Lie-algebraic parameters, we denote the ${}_2F_0$ function by
$\tilde F$ and $\tilde F^{\rm\scriptscriptstyle I}$. The tilde is needed to avoid the confusion with the ${}_1F_1$ functions:
\begin{eqnarray*}
\tilde F_{\theta ,\alpha}(w)&:=&F\Bigl(\frac{1+\alpha+\theta }{2},\frac{1-\alpha+\theta }{2};-;w\Bigr),\\
\tilde F^{\rm\scriptscriptstyle I} _{\theta ,\alpha}(w)&:=& F^{\rm\scriptscriptstyle I} \Bigl(\frac{1+\alpha+\theta }{2},
\frac{1 -\alpha+\theta}{2};-;w\Bigr)
\,=\, \Gamma\Big(\frac{1-\alpha+\theta}{2}\Big)\tilde F_{\theta ,\alpha}(w) .
\end{eqnarray*}
\subsection{Standard solutions}
\label{Standard solutions11}
The ${}_1F_1$ equation has two singular points. $0$ is a regular
singular point and with each of its two indices we can associate the
corresponding solution. $\infty$ is not a regular singular point. However we can define two solutions with a simple behavior around $\infty$. Therefore, we obtain 4 {\em standard solutions}.
The solutions that have a simple behavior at zero are expressed in terms of the function $F_{\theta,\alpha}$. Using $4$ discrete symmetries yields 4 distinct expressions. Taking into account Kummer's identity we obtain $2$ pairs of standard solutions.
The solutions with a simple behavior at $\pm\infty$ are expressed in terms of $\tilde F_{\theta,\alpha}$. Again, 4 discrete symmetries yield 4 distinct expressions. Taking into account the trivial identity
$\tilde F_{\theta,\alpha}= \tilde F_{\theta,-\alpha}$ we obtain $2$ pairs of standard solutions.
\begin{align*}
\text{ $\sim1$ at $0$}:\quad&\quad
F_{\theta ,\alpha}(w)\\
=&{\rm e}^wF_{-\theta ,\alpha}(-w);\\
\text{ $\sim w^{-\alpha}$ at $0$}:\quad&\quad
w^{-\alpha} F _{\theta ,-\alpha}(w)\\=&w^{-\alpha}
{\rm e}^w F _{-\theta ,-\alpha}(-w);\end{align*}\begin{align*}
\text{ $\sim w^{-a}$ at $+\infty$}:\quad&\quad
w^{\frac{-1-\theta -\alpha}{2}}\tilde F_{\theta , \alpha}(-w^{-1})\\=& w^{\frac{-1-\theta -\alpha}{2}}\tilde F_{\theta , -\alpha}(-w^{-1});\\
\text{ $\sim (-w)^{b-1}{\rm e}^w$ at $-\infty$}:\quad&\quad
{\rm e}^w(-w)^{\frac{-1+\theta -\alpha}{2}}\tilde F_{-\theta ,\alpha}(w^{-1})\\=&{\rm e}^w(-w)^{\frac{-1+\theta -\alpha}{2}}\tilde F_{-\theta ,-\alpha}(w^{-1})
.
\end{align*}
The solution $\sim w^{-a}$ at $+\infty$ is often called {\em Tricomi's confluent function}.
\subsection{Recurrence relations}
Recurrence relations for the confluent function
correspond to roots of the Lie algebra ${\rm sch}(2)$:
\begin{eqnarray*}
\partial_w {\bf F} _{\theta ,\alpha}(w)&=&\frac{1+\theta +\alpha}{2} {\bf F} _{\theta +1,\alpha+1}(w),
\\
\left(w \partial_w+\alpha-w\right) {\bf F} _{\theta ,\alpha}(w)&=& {\bf F} _{\theta -1,\alpha-1}(w),\\[0.6ex]
\left(w \partial_w+\alpha\right) {\bf F} _{\theta ,\alpha}(w)&=& {\bf
F} _{\theta +1,\alpha-1}(w),\\
\left( \partial_w-1\right) {\bf F} _{\theta ,\alpha}(w)&=&\frac{-1+\theta -\alpha}{2} {\bf F} _{\theta -1,\alpha+1}(w),
\\[0.6ex]
\left(w \partial_w+\frac{1+\theta +\alpha}{2}\right) {\bf F} _{\theta ,\alpha}(w)&=&\frac{1+\theta +\alpha}{2} {\bf F} _{\theta +2,\alpha}(w),
\\
\left(w \partial_w+\frac{1-\theta +\alpha}{2}-w\right) {\bf F} _{\theta ,\alpha}(w)&
=&\frac{1-\theta +\alpha}{2} {\bf F} _{\theta
-2,\alpha}(w).
\end{eqnarray*}
\subsection{Wave packets for the heat equation in 2 dimensions}
Consider the space ${\mathbb R}^2\oplus{\mathbb R}$ and the heat equation given by the operator ${\cal L}_2=2\partial_{y_{-1}}\partial_{y_1}+2\partial_t$.
Recall that \begin{align*}
M^{{\rm sch},-1}&= y_{-1} \partial_{y_{-1}}+y_1 \partial_{y_1}+2t \partial_t+1,\\
N_1^{\rm sch}&=-y_{-1} \partial_{y_{-1}}+y_1 \partial_{y_1}.\end{align*}
Set
\begin{subequations} \begin{align}\notag&
G^a_{\theta,\alpha}(y_{-1},y_1,t)\\:=& \int_{\gamma^a}\tau^{-\alpha-1}t^{\frac{-1-\theta+\alpha}{2}}(\tau^{-1}y_{-1}-1)^{\frac{-1+\theta-\alpha}{2}}\exp\Big(\frac{(y_{-1}-\tau)y_1}{t}\Big){\rm d}\tau,\\
\notag & G_{\theta,\alpha}^b(y_{-1},y_1,t)\\
:= &\int_{\gamma^b}\tau^{-\alpha-1}t^{\frac{-1-\theta-\alpha}{2}}(\tau y_1-1)^{\frac{-1+\theta+\alpha}{2}}\exp\Big(\frac{y_{-1}(y_1-\tau^{-1})}{t}\Big){\rm d}\tau. \end{align}\end{subequations}
(The superscripts $a$ and $b$ denote two kinds of wave packets, and not parameters $a$, $b$).
\begin{proposition} If the contours $\gamma^a$ and $\gamma^b$ are appropriately chosen, then
\begin{align}
{\cal L}_2 G_{\theta,\alpha}^a&=0,\quad&{\cal L}_2G^b_{\theta,\alpha}&=0,
\label{ko1}\\
M^{{\rm sch},-1} G_{\theta,\alpha}^a&=-\theta
G_{\theta,\alpha}^a,\quad& M^{{\rm sch},-1} G_{\theta,\alpha}^b&=-\theta G_{\theta,\alpha}^b,
\label{ko2}\\
N_1 G_{\theta,\alpha}^a&=\alpha
G_{\theta,\alpha}^a,\quad & N_1 G_{\theta,\alpha}^b&=\alpha
G_{\theta,\alpha}^b.\label{ko3}
\end{align}
\end{proposition}
{\bf Proof.}\ \
By the analysis of Subsect.
\ref{Special solutions of the heat equation},
the following functions
\begin{subequations}\begin{align}
g_\nu^a(y_{-1},y_1,t)&:=
t^{-1-\nu}y_{-1}^\nu\exp\Big(\frac{y_{-1}y_1}{t}\Big),\label{sd1}\\
g_\nu^b(y_{-1},y_1,t)&:=
t^{-1-\nu}y_1^\nu\exp\Big(\frac{y_{-1}y_1}{t}\Big)\label{sd2}
\end{align}\end{subequations}
solve the heat equation. They still solve the heat equation after translating and rotating. Therefore,
\begin{subequations}\begin{align}
G_{\theta,\alpha}^a(y_{-1},y_1,t)
&=\int_{\gamma^a} g_{\frac{-1+\theta-\alpha}{2}}^a(\tau^{-1}(y_{-1}-1),\tau y_1,t)\tau^{-\alpha-1}{\rm d}\tau,
\label{ka3}\\
G_{\theta,\alpha}^b(y_{-1},y_1,t)
&=\int_{\gamma^b} g_{\frac{-1+\theta+\alpha}{2}}^b(\tau^{-1}y_{-1},\tau (y_1-1),t)\tau^{-\alpha-1}{\rm d}\tau
\label{ka1}
\end{align}\label{kaa}\end{subequations}
also solve the heat equation. This proves (\ref{ko1}).
If the contours satisfy the requirements of Prop. \ref{cac0}, then (\ref{kaa})
imply (\ref{ko3}).
We can rewrite (\ref{kaa}) in a somewhat different way:
\begin{subequations}\begin{align}
(\ref{ka3})
&=\int_{\gamma^a} g_{\frac{-1+\theta-\alpha}{2}}^a(\tau^{-1}( y_{-1}-1),\tau^{-1} y_1,\tau^{-2}t)(\tau^{-1})^\theta{\rm d}(\tau^{-1}),
\label{ka4}\\
(\ref{ka1}) &=\int_{\gamma^b} g_{\frac{-1+\theta+\alpha}{2}}^b(\tau y_{-1},\tau (y_1-1),\tau^2t)\tau^\theta{\rm d}\tau.
\label{ka2}
\end{align}\label{kab}\end{subequations}
If the contours satisfy the requirements of Prop. \ref{analo}, then (\ref{kab}) imply (\ref{ko2}).
$\Box$\medskip
Now we express the above wave packets in the coordinates $w,s,{u_1}$:
\begin{subequations}\begin{align}
(\ref{ka4})&=
\int s^{-1-\theta+\alpha}\Big(\frac{ws}{\tau {u_1}}-1\Big)^{\frac{-1+\theta-\alpha}{2}}\exp\Big(w-\frac{\tau {u_1}}{s}\Big)\tau^{-\alpha-1}{\rm d}\tau,\label{ki4}\\
(\ref{ka2})&=\int s^{-1-\theta-\alpha}(\tau {u_1}s-1)^{\frac{-1+\theta+\alpha}{2}}\exp\Big(w\big(1-\frac{1}{\tau {u_1}s}\big)\Big)\tau^{-\alpha-1}{\rm d}\tau.
\label{ki1}
\end{align}\end{subequations}
In (\ref{ki4}) we make the substitution $\sigma:=w-\frac{\tau {u_1}}{s}$, or $\tau=\frac{s}{{u_1}}(w-\sigma)$.
In (\ref{ki1}) we make the substitution $\sigma:=\frac{1}{1-\frac{1}{\tau {u_1}s}}$, or $\tau=\frac{\sigma}{{u_1}s(\sigma-1)}$.
We obtain
\begin{subequations}\begin{align}
G_{\theta,\alpha}^a(w,s,{u_1})
&=
s^{-1-\theta}{u_1}^\alpha
F_{\theta,\alpha}^a(w), \label{ki6}\\
G_{\theta,\alpha}^b(w,s,{u_1})
&= s^{-1-\theta}{u_1}^\alpha
F_{\theta,\alpha}^b(w),
\label{ki3}
\end{align}\end{subequations}
where
\begin{subequations}\begin{align}
F_{\theta,\alpha}^a(w)&:=\int_{\gamma^a}\sigma^{\frac{-\alpha+\theta-1}{2}}
(w-\sigma)^{\frac{-\alpha-\theta-1}{2}}{\rm e}^\sigma{\rm d}\sigma, \label{ki6.}\\ F_{\theta,\alpha}^b(w)&:=\int_{\gamma^b}\exp\Big(\frac{w}{\sigma}\Big)\sigma^{-\alpha-1}(\sigma-1)^{\frac{\alpha+\rho-1}{2}}{\rm d}\sigma. \label{ki3.}
\end{align}\end{subequations}
The above analysis shows that (for appropriate contours) the functions
(\ref{ki6.}) and (\ref{ki3.}) satisfy the confluent equation.
\subsection{Integral representations}
Let us prove directly that integral (\ref{ki6.}) and (\ref{ki3.}) solve the confluent equation.
\begin{theoreme}\begin{arabicenumerate} \item[a)]
Let $[0,1]\ni \tau\overset{\gamma}\mapsto t(\tau)$ satisfy
$t^{a-c+1}{\rm e}^t(t-w)^{-a-1}\Big|_{t(0)}^{t(1)}=0.$
Then
\begin{equation}
{\cal F}(a;c;w, \partial_w)
\int_\gamma t^{a-c}{\rm e}^t(t-w)^{-a}{\rm d} t=0.\label{dad1}\end{equation}
\item[b)]
Let $[0,1]\ni \tau\overset{\gamma}\mapsto t(\tau)$ satisfy
${\rm e}^{\frac{w}{t}}t^{-c}(1-t)^{c-a}\Big|_{t(0)}^{t(1)}=0.$
Then
\begin{equation}
{\cal F}(a;c;w, \partial_w)\int_\gamma{\rm e}^{\frac{w}{t}}t^{-c}(1-t)^{c-a-1}{\rm d} t
=0.
\label{dad}
\end{equation}
\end{arabicenumerate}\label{dad4}\end{theoreme}
{\bf Proof.}\ \
We check that for any contour $\gamma$
\begin{eqnarray*}
\text{lhs of (\ref{dad1})}&=&
-a\int_\gamma{\rm d} t\, \partial_t
t^{a-c+1}{\rm e}^t(t-w)^{-a-1},\\
\text{lhs of (\ref{dad}) }&=&-\int_\gamma{\rm d} t\, \partial_t
{\rm e}^{\frac{w}{t}}t^{-c}(1-t)^{c-a}.
\end{eqnarray*}
$\Box$\medskip
\subsection{Integral representations of standard solutions}
Using the integral representations of type a)
and attaching contours to
$-\infty$, $0$ and $w$
we can obtain all standard solutions.
Similarly, using the integral representations of type b)
and attaching contours to
$0-0$, $1$ and $\infty$ we can obtain all standard solutions.
Here is the list of contours:
\medskip
\noindent
\begin{tabular}{lll}&a)&b)\\
$\sim1$ at $0$:\qquad&
$]-\infty,(0,w)^+,-\infty[$,&$[1,+\infty[$;\\
\text{ $\sim w^{-\alpha}$ at $0$}:\qquad&
$[0,w]$,&$(0-0)^+$;\\
\text{ $\sim w^{-a}$ at $+\infty$}:\qquad&
$ ]-\infty,0]$,&$]-\infty,0]$;\\
\text{ $\sim (-w)^{b-1}{\rm e}^w$ at $-\infty$}:\qquad&
$[w,-\infty[$,&$[0,1]$.\\
\end{tabular}
\medskip
($0,w)^+$ means that we bypass $0$ and $w$ counterclockwise. $(0-0)^+$ means that the contour departs from $0$ on the negative side, encircles it and then comes back again from the negative side.
Here are the explicit formulas for a)-type integral representations:
\begin{subequations}
\begin{align}\text{all $\theta,\alpha$:}&\label{sub1}\\
\underset{]-\infty,(0,w)^+-\infty[}{\frac{1}{2\pi {\rm i}}\int}
t^{\frac{-1+\theta -\alpha}{2}}{\rm e}^t(t-w)^{\frac{-1-\theta -\alpha}{2}}{\rm d} t&\notag\\
= {\bf F} _{\theta ,\alpha}(w),\notag\end{align}
\begin{align}
\Re(1-\alpha)>|\Re \theta |:&\label{sub2}\\
\int_0^w
t^{\frac{-1+\theta -\alpha}{2}}{\rm e}^t(w-t)^{\frac{-1-\theta -\alpha}{2}}{\rm d} t\notag
&\\
=
w^{-\alpha} {\bf F}^{\rm\scriptscriptstyle I} _{\theta ,-\alpha}(w),&\quad w\not\in]-\infty,0];\notag\\
\int_w^0
(-t)^{\frac{-1+\theta -\alpha}{2}}{\rm e}^t(t-w)^{\frac{-1-\theta -\alpha}{2}}{\rm d} t\notag
&\\
=
(-w)^{-\alpha} {\bf F}^{\rm\scriptscriptstyle I} _{\theta ,-\alpha}(w),&\quad w\not\in[0,\infty[;\notag\end{align}
\begin{align}
\Re(1+\theta -\alpha)>0:&\label{sub3}\\
\int_{-\infty}^0
(-t)^{\frac{-1+\theta -\alpha}{2}}{\rm e}^t(w-t)^{\frac{-1-\theta -\alpha}{2}}{\rm d} t
&\notag\\=
w^{\frac{-1-\theta -\alpha}{2}} \tilde F^{\rm\scriptscriptstyle I} _{\theta ,\alpha}(-w^{-1}),&\quad
w\not\in]-\infty,0];\notag\end{align}\begin{align}xs
\Re(1-\theta -\alpha)>0:&\label{sub4}\\
\int_{-\infty}^w
(-t)^{\frac{-1+\theta -\alpha}{2}}{\rm e}^t(w-t)^{\frac{{-}1-\theta -\alpha}{2}}{\rm d} t
&\notag\\=
{\rm e}^w(-w)^{\frac{-1+\theta -\alpha}{2}} \tilde F^{\rm\scriptscriptstyle I} _{-\theta ,\alpha}(w^{-1}),&\quad w\not\in[0,\infty[\notag.\end{align}
\end{subequations}
We also present explicit formulas for b)-type integral representations:
\begin{subequations}
\begin{align}
\Re(1+\alpha)>|\Re \theta |:&\label{subi1}\\
\int\limits_{[1,+\infty[}{\rm e}^{\frac{w}{t}}t^{-1-\alpha}(t-1)^{\frac{-1-\theta +\alpha}{2}}
{\rm d} t&\notag\\
= {\bf F}^{\rm\scriptscriptstyle I} _{\theta ,\alpha}(w);&
\notag\end{align}\begin{align}
\text{ all $\theta,\alpha$:}&\label{subi2}\\
\frac{1}{2\pi{\rm i}}
\int\limits_{(0-0)^+}
{\rm e}^{\frac{w}{t}}t^{-1-\alpha}(1-t)^{\frac{-1-\theta +\alpha}{2}}
{\rm d} t&\notag\\
=w^{-\alpha} {\bf F} _{\theta ,-\alpha}(w),&\quad \Re w>0;\notag\end{align}\begin{align}
\Re(1+\theta +\alpha)>0:&\label{subi3}\\
\int_{-\infty}^0
{\rm e}^{\frac{w}{t}}(-t)^{-1-\alpha}(1-t)^{\frac{-1{ -}\theta +\alpha}{2}}
{\rm d} t&\notag\\=
w^{\frac{-1-\theta -\alpha}{2}}
\tilde F^{\rm\scriptscriptstyle I} _{\theta ,-\alpha}(-w^{-1}),&\quad \Re w>0;\notag\end{align}\begin{align}
\Re(1-\theta +\alpha)>0:&\label{subi4}\\
\int_0^1
{\rm e}^{\frac{w}{t}}t^{-1-\alpha}(1-t)^{\frac{-1{ -}\theta +\alpha}{2}}
{\rm d} t&\notag\\
={\rm e}^w(-w)^{\frac{ -1+\theta -\alpha}{2}}
\tilde F^{\rm\scriptscriptstyle I} _{-\theta ,-\alpha}(w^{-1}),&\quad \Re w<0.\notag
\end{align}
\end{subequations}
\subsection{Connection formulas}
The two solutions with a simple behavior at infinity
can be expressed as linear combination of the solutions with a simple behavior at zero:
\begin{subequations}\begin{eqnarray}\label{pqd1}
w^{\frac{-1-\theta -\alpha}{2}}\tilde F_{\theta ,\pm \alpha}(-w^{-1})
&=&\frac{\pi{\bf F} _{\theta ,\alpha}(w)}{\sin{\pi(-\alpha)}\Gamma\left(\frac{1+\theta -\alpha}{2}\right)}
\\
&&+\frac{\pi w^{-\alpha} {\bf F} _{\theta ,-\alpha}(w)}{\sin\pi \alpha\Gamma\left(\frac{1+\theta +\alpha}{2}\right)}
,
\quad w\not\in]-\infty,0];\notag
\\\label{pqd2}
{\rm e}^w(-w)^{\frac{-1 +\theta -\alpha}{2}}\tilde F_{-\theta ,\pm\alpha}(w^{-1})
&=&\frac{\pi {\bf F} _{\theta ,\alpha}(w)}{\sin{\pi(-\alpha)}
\Gamma\left(\frac{1-\theta -\alpha}{2}\right)}
\\
&&+\frac{\pi(-w)^{-\alpha} {\bf F}_{\theta ,-\alpha}(w)}
{\sin\pi \alpha\Gamma\left(\frac{1-\theta +\alpha}{2}\right)}
,\quad w\not\in[0,+\infty[.\notag
\end{eqnarray}\label{pqd0}
\end{subequations}
Note that (\ref{pqd1}) uses a different domain from (\ref{pqd2}). This is natural, however it is inconvenient when we want to rewrite
(\ref{pqd0}) in the matrix form, because on the right hand side of (\ref{pqd1}) and (\ref{pqd2}) the second standard solutions differ by a phase factor.
Let us introduce the matrix
\begin{align*}
A_{\theta,\alpha}&:=
\frac{\pi}{\sin(\pi\alpha)}\left[\begin{array}{cc}
\frac{-1}{\Gamma\left(\frac{1+\theta -\alpha}{2}\right)}
& \frac{{\rm e}^{-\frac{{\rm i}\pi}{2}\alpha}}{\Gamma\left(\frac{1+\theta +\alpha}{2}\right)}
\\[2.5ex]
\frac{-1}{\Gamma\left(\frac{1-\theta -\alpha}{2}\right)}
&\frac{{\rm e}^{\frac{{\rm i}\pi}{2}\alpha}}{\Gamma\left(\frac{1-\theta +\alpha}{2}\right)}
\end{array}\right],\end{align*}
satisfying
\begin{align}
A_{\theta,\alpha}^{-1}&=
\frac{{\rm i}{\rm e}^{\frac{{\rm i}\pi}{2}\theta}}{2}
\left[\begin{array}{cc}
\frac{{\rm e}^{\frac{{\rm i}\pi\alpha}{2}}}{\Gamma\left(\frac{1-\theta +\alpha}{2}\right)}
& \frac{-{\rm e}^{-\frac{{\rm i}\pi\alpha}{2}}}{\Gamma\left(\frac{1+\theta +\alpha}{2}\right)}
\\[2.5ex]
\frac{1}{\Gamma\left(\frac{1-\theta -\alpha}{2}\right)}
&\frac{-1}{\Gamma\left(\frac{1+\theta -\alpha}{2}\right)}
\end{array}\right],\\
\det A_{\theta,\alpha}&= -\frac{{\rm i}\pi{\rm e}^{-\frac{{\rm i}\pi}{2}\theta}}{2\sin(\pi\alpha)}.
\end{align}
Then we have for $\Im w>0$
\begin{align}&
\left[\begin{array}{c}
w^{\frac{-1-\theta -\alpha}{2}}\tilde F_{\theta ,\pm \alpha}(-w^{-1})\\[2ex]
{\rm e}^w(-w)^{\frac{-1+\theta -\alpha}{2}}\tilde F_{-\theta ,\pm\alpha}(w^{-1})
\end{array}\right]
=
A_{\theta,\alpha}
\left[\begin{array}{c}
{\bf F} _{\theta ,\alpha}(w)\\[2.5ex]
(-{\rm i} w)^{-\alpha} {\bf F}_{\theta ,-\alpha}(w)
\end{array}\right].
\end{align}
Let us show how to derive connection
formulas from integral representations of type a).
We have
\begin{subequations} \begin{align}\notag
& \Bigg(
\int\limits_{]-\infty,0-{\rm i} 0]}+\int\limits_{[0-\i0,w]}
- \int\limits_{]-\infty,0+{\rm i} 0]}-\int\limits_{[0+\i0,w]}\Bigg) t^{\frac{-1+\theta -\alpha}{2}}{\rm e}^t(t-w)^{\frac{-1-\theta -\alpha}{2}}{\rm d} t\\
=&
\underset{]-\infty,(0,w)^+,-\infty[}{\int}t^{\frac{-1+\theta -\alpha}{2}}{\rm e}^t(t-w)^{\frac{-1-\theta -\alpha}{2}}{\rm d} t ,\qquad w\not\in[-\infty,0[;
\\[1ex]\notag
& \Bigg(
\int\limits_{]-\infty,w-{\rm i} 0]}+\int\limits_{[w-\i0,0]}
- \int\limits_{]-\infty,w+{\rm i} 0]}-\int\limits_{[w+\i0,0]}\Bigg) t^{\frac{-1+\theta -\alpha}{2}}{\rm e}^t(t-w)^{\frac{-1-\theta -\alpha}{2}}{\rm d} t\\
=&
\underset{]-\infty,(0,w)^+,-\infty[}{\int}t^{\frac{-1+\theta -\alpha}{2}}{\rm e}^t(t-w)^{\frac{-1-\theta -\alpha}{2}}{\rm d} t ,
\qquad w\not\in]0,+\infty]
.\end{align}\end{subequations}
We obtain
\begin{subequations} \begin{align}\notag
&-\sin(\pi\alpha) w^{\frac{-1-\theta -\alpha}{2}} \tilde F^{\rm\scriptscriptstyle I} _{\theta ,\alpha}(-w^{-1})
+
\cos\frac{\pi(\theta+\alpha)}{2}w^{-\alpha} {\bf F}^{\rm\scriptscriptstyle I} _{\theta ,-\alpha}(w)\\=&\pi {\bf F} _{\theta ,\alpha}(w),\qquad w\not\in[-\infty,0[
;\label{exw2}\\[1ex]\notag
& -\sin(\pi\alpha) {\rm e}^w(-w)^{\frac{-1+\theta -\alpha}{2}} \tilde F^{\rm\scriptscriptstyle I} _{-\theta ,\alpha}(w^{-1})+
\cos\frac{\pi(\theta-\alpha)}{2}(-w)^{-\alpha} {\bf F}^{\rm\scriptscriptstyle I} _{\theta ,-\alpha}(w)\\=&\pi {\bf F} _{\theta ,\alpha}(w),\qquad w\not\in]0,+\infty].\label{exw1}
\end{align}\end{subequations}
This implies (\ref{pqd0}).
\section{Heat equation in 1 dimension and the Hermite equation}
\label{s10}
\init
The goal of this section is to derive the Hermite equation together with its symmetries from the heat equation in $1$ dimension, which in turn comes from the Laplace equation in $5$ and $3$ dimensions.
The first part of this section describes main steps of the derivation of the Hermite equation. They are parallel to those of the derivation of the ${}_1{\cal F}_1$ equation:
\be
\item\label{it1hh}
We start from the Schr\"odinger Lie algebra ${\rm sch}(1)$ and group ${\rm Sch}(1)$
considered as a subalgebra of $\so(5)$, resp. a subgroup of $\mathrm{O}(5)$, acting in $5$ dimensions. The main initial operator is the Laplacian $\Delta_5$.
\item\label{it3hh} We descend onto $3$ dimensions. The 5-dimensional Laplacian $\Delta_5$ becomes the 3-dimensional Laplacian $\Delta_3$.
\item We descend on $1+1$ dimensions. The Laplacian $\Delta_3$ becomes the heat operator ${\cal L}_1$.
The representations $B^{{\rm sch},\eta}$ and $\alpha^{{\rm sch},\eta}$ with $\eta=-\frac12$ and $\eta=-\frac52$ are generalized symmetries of ${\cal L}_1$.
\item\label{it4hh} We choose coordinates $w,s$, so that the Cartan operator is expressed in terms of $s$. We compute
${\cal L}_1$, $B^{{\rm sch},\eta}$ and $\alpha^{{\rm sch},\eta}$ in the new coordinates.
\item\label{it5hh} We make an ansatz that diagonalizes the Cartan operator, whose eigenvalue
becomes a parameter, denoted by $\lambda$.
${\cal L}_1$, $B^{{\rm sch},\eta}$ and $\alpha^{{\rm sch},\eta}$ involve now only the single variable $w$.
$2s^2{\cal L}_1$ turns out to be the Hermite operator.
The generalized symmetries of
${\cal L}_1$ yield transmutation relations and discrete symmetries of the
Hermite operator.
\ee
(As in the previous section, in our presentation we omit the step \ref{it3hh}).
In the remaining part of this section
we develop the theory of the Hermite equation
and its solutions. Its organization is parallel to that of all other sections on individual equations, and especially of Sect. \ref{s7} on the Gegenbauer equation.
In particular, the Gegenbauer equation can be derived by a quadratic relation from the ${}_2{\cal F}_1$ equation in essentially the same way as
the Hermite equation can be derived from the ${}_1{\cal F}_1$ equation.
\subsection{${\rm sch}(1)$ in 5 dimensions}
We again consider ${\mathbb R}^5$ with the coordinates
\begin{equation} z_0,z_{-2},z_2,z_{-3},z_3\label{geg1-}\end{equation}
and the scalar product given by
\begin{equation} \langle z|z\rangle=z_0^2+2z_{-2}z_2+2z_{-3}z_3.\label{geg5-}\end{equation}
We
keep the notation from $\so(5)$---remember that ${\rm sch}(1)$ is a subalgebra of $\so(5)$.
\noindent
{\bf Lie algebra ${\rm sch}(1)$.}
The Cartan algebra
\begin{subequations}\begin{eqnarray}
M&=&
z_{-2} \partial_{z_{-2}}-z_{2} \partial_{z_{2}}-z_{-3} \partial_{z_{-3}}+z_{3} \partial_{z_{3}},\\
B_{3,2}& =&z_{-3} \partial_{z_2}-z_{-2} \partial_{z_{3}}.
\end{eqnarray}\end{subequations}
Root operators
\begin{subequations}\begin{eqnarray}
B_{3,0}&=&z_{-3} \partial_{z_0}-z_0 \partial_{z_{3}},\\
B_{2,0}&=&z_{-2} \partial_{z_0}-z_0 \partial_{z_2},\\
B_{3,-2}&=&z_{-3} \partial_{z_{-2}}-z_2 \partial_{z_{3}},\\
B_{-3,2}&=&z_{3} \partial_{z_2}-z_{-2} \partial_{z_{-3}}.
\end{eqnarray}\end{subequations}
\noindent{\bf Weyl symmetries}
\begin{subequations}\begin{align} {\iota} K(z_0,z_{-2},z_2,z_{-3},z_3)
&=K(z_0,z_{-2},z_2,z_{-3},z_3),\\
\kappa K(z_0,z_{-2},z_2,z_{-3},z_3)
&=K(z_0,-z_{-3},-z_{3},z_{-2},z_{2}),\\
\kappa^2 K(z_0,z_{-2},z_2,z_{-3},z_3)
&=K(z_0,-z_{-2},-z_2,-z_{-3},-z_3),\\
\kappa^3 K(z_0,z_{-2},z_2,z_{-3},z_3)
&=K(z_0,z_{-3},z_{3},-z_{-2},-z_{2}).
\end{align}\end{subequations}
\noindent{\bf Laplacian}
\begin{equation} \Delta_5=\partial_{z_0}^2+2\partial_{z_{-2}}\partial_{z_{2}}+
2\partial_{z_{-3}}\partial_{z_{3}}
.\end{equation}
\subsection{${\rm sch}(1)$ in 3 dimensions}
We descend on the level of ${\mathbb R}^3$,
with the variables $ y_0,y_{-2},y_2$
and the scalar product given by
\[ \langle y|y\rangle=y_0^2+2y_{-2}y_2.\label{geg5--}\]
\noindent{\bf Lie algebra ${\rm sch}(1)$.}
Cartan algebra
\begin{eqnarray*}
M^{{\rm fl},\eta}&=&y_{0} \partial_{y_0}+2y_{-2} \partial_{y_{-2}}-\eta,\\
B_{3,2}^{\rm fl}& =& \partial_{y_2}.
\end{eqnarray*}
\noindent Root operators
\begin{eqnarray*}
B_{3,0}^{{\rm fl}}&=& \partial_{y_0},\\
B_{2,0}^{{\rm fl}}&=&y_{-2} \partial_{y_0}-y_0 \partial_{y_2},\\
B_{3,-2}^{{\rm fl}}&=& \partial_{y_{-2}},\\
B_{-3,2}^{{\rm fl},\eta}
&=&y_{-2}\big(y_0 \partial_{y_0}+y_{-2} \partial_{y_{-2}}-\eta\big)-\frac12y_0^2 \partial_{y_2}.
\end{eqnarray*}
\noindent{\bf Weyl symmetries}
\begin{align*}
{\iota}^{{\rm fl},\eta}f(y_0,y_{-2},y_2)&=f(y_0,y_{-2},y_2),\\
\kappa^{{\rm fl},\eta}f(y_0,y_{-2},y_2)&
=y_{-2}^\eta
f\Big(\frac{y_0}{y_{-2}},-\frac1{y_{-2}},\frac{y_0^2+2y_{-2}y_2}{2y_{-2}}\Big),\\
\big(\kappa^{{\rm fl},\eta}\big)^2f(y_0,y_{-2},y_2)&=(-1)^\eta f(-y_0,y_{-2},y_2),\\
\big(\kappa^{{\rm fl},\eta}\big)^3f(y_0,y_{-2},y_2)&
=(-y_{-2})^\eta
f\Big(-\frac{y_0}{y_{-2}},-\frac1{y_{-2}},\frac{y_0^2+2y_{-2}y_2}{2y_{-2}}\Big).
\end{align*}
\noindent{\bf Laplacian}
\[\Delta_5^{\rm fl}=\partial_{y_0}^2+2\partial_{y_{-2}}\partial_{y_{2}}
.\]
\subsection{${\rm sch}(1)$ in $1+1$ dimensions}
We descend onto the level of ${\mathbb R}\oplus{\mathbb R}$, as described in
Subsect. \ref{subsec-sch}.
We rename $y_{-2}$ to $t$.
\noindent{\bf Lie algebra ${\rm sch}(1)$.} Cartan algebra:
\begin{subequations}\begin{align}
M^{{\rm sch},\eta}&={y_0} \partial_{y_0}+2t \partial_t-\eta,\\
B_{3,2}&=1.
\end{align}\end{subequations}
Root operators
\begin{subequations}\begin{eqnarray}
B_{3,0}^{{\rm sch}}&=& \partial_{y_0},\\
B_{2,0}^{{\rm sch}}&=&t \partial_{y_0}-{y_0},\\
B_{3,-2}^{{\rm sch}}&=& \partial_t,\\
B_{-3,2}^{{\rm sch},\eta}&=&t({y_0} \partial_{y_0}+t \partial_t-\eta)-\frac12y_0^2.\end{eqnarray}\end{subequations}
\noindent{\bf Weyl symmetry}
\begin{subequations}\begin{align}
{\iota}^{{\rm sch},\eta}h({y_0},t)&=
h({y_0},t),\\
\kappa^{{\rm sch},\eta}h({y_0},t)&= t^{\eta}\exp(\tfrac{y_0^2}{2t})
h(\tfrac{{y_0}}{t},-\tfrac{1}{t}),\\
\big(\kappa^{{\rm sch},\eta}\big)^2h({y_0},t)&=
(-1)^\eta h(-{y_0},t),\\
\big(\kappa^{{\rm sch},\eta}\big)^3h({y_0},t)&= (-t)^{\eta}\exp(\tfrac{y_0^2}{2t})
h(-\tfrac{{y_0}}{t},-\tfrac{1}{t}).
\end{align}\end{subequations}
\noindent{\bf Heat operator}
\begin{equation}\Delta_5^{\rm sch}={\cal L}_1=\partial_{y_0}^2+2\partial_t.\end{equation}
\subsection{${\rm sch}(1)$ in the coordinates $w,s$}
Let us define new coordinates
\begin{align}\label{coo2}
w =\frac{{y_0}}{\sqrt{2\,t}}\;,\ \ \ \ \ \ \ \ \
&s =\sqrt{t}\,,\phantom{\frac{1}{2}}
\intertext{with the reverse transformation}
{y_0} =\sqrt{2}\,s\,w\;,\ \ \ \ \ \ \ \
&t =s^2\,.\phantom{\frac{1}{2}}
\end{align}
\noindent{\bf Lie algebra ${\rm sch}(1)$.} Cartan algebra
\begin{eqnarray*}
M^{{\rm sch},{\eta}}&=&s\dds-\eta,\\
B_{3\,2}& =&1.
\end{eqnarray*}
Root operators
\begin{align*}
B_{3,0}^{\rm sch}& =\frac{1}{\sqrt{2}\,s}\,\ddw,\\
B_{2,0}^{\rm sch}& =\frac{s}{\sqrt{2}}\,\left(\ddw-2w\right),\\
B_{3,-2}^{\rm sch}& =\frac{1}{2\,s^2}\,\left(-w\ddw+s \partial_s\right),\\
B_{-3,2}^{{\rm sch},{\eta}}& =\frac{s^2}{2}\,\left(w\ddw+s \partial_s-2\eta-2w^2\right).
\end{align*}
\noindent{\bf Weyl symmetries}
\begin{align*}
{\iota}^{{\rm sch},{\eta}} h(w,s)&= h(w,s),\\
\kappa^{{\rm sch},{\eta}} h(w,s)&=s^{2\eta}{\rm e}^{w^2}h({\rm i} w,-\tfrac{{\rm i}}{s}),\\
(\kappa^{{\rm sch},{\eta}})^2 h(w,s)&=(-1)^\eta h(-w,s),\\
(\kappa^{{\rm sch},{\eta}})^3 h(w,s)&=(-s^2)^\eta{\rm e}^{w^2}h(-{\rm i} w,-\tfrac{{\rm i}}{s}) .\end{align*}
\noindent{\bf Heat operator}
\begin{equation}
{\cal L}_1=\frac{1}{2s^2}\left(\ddw^2-2w\ddw+2s\dds\right)\,.\end{equation}
\subsection{Hermite operator}
Let us set $\eta=-\frac12$ and use
the ansatz
\begin{equation} h(w,s)=s^{-\lambda-\frac12} S(w).\label{ansatz1}\end{equation}
Clearly,
\begin{eqnarray}
M^{{\rm sch},-\frac12} h&=&-\lambda h,\\
s^{\lambda+\frac12}2s^2{\cal L}_1 h&=&{\cal S}_\lambda(w, \partial_w)S(w),
\end{eqnarray}
where we have introduced the {\em Hermite operator}
\begin{equation}
S_\lambda(w, \partial_w):= \partial_w^2-2w \partial_w-2\lambda-1.
\end{equation}
We will also use an alternative notation
\begin{equation}
S(a;w, \partial_w):= \partial_w^2-2w \partial_w-2a,
\end{equation}
so that
\begin{equation}
\lambda={a}-\frac12,\ \ \ a=\lambda+\frac12.\end{equation}
\subsection{Quadratic transformation}
\label{Quadratic transformation-}
Let us go back to 2+1 dimensions and the heat operator
\begin{equation}{\cal L}_2=2\partial_{y_{-1}}\partial_{y_{1}}+
2 \partial_t
.\label{sq20=}\end{equation}
Let us use the reduction described in Subsect.
\ref{Dimensional reduction}, and then applied in Subsect. \ref{Quadratic transformation}:
\begin{equation} y_0:=\sqrt{2y_{-1}y_{1}},\quad
u:=\sqrt{\frac{y_{1}}{y_{-1}}}.\label{coo1=}\end{equation}
In the new variables,
\begin{align}
N_1&=u \partial_u,\\
{\cal L}_2
&=\Big(\partial_{y_0}+\frac1{2y_0}\Big)^2
-\frac1{y_0^2}\Big(u\partial_u-\frac12\Big)
\Big(u\partial_u+\frac12\Big)+
2 \partial_t.
\end{align}
Therefore,
\begin{subequations} \begin{align}
(uy_0)^{\frac12}{\cal L}_2 (uy_0)^{-\frac12}
& =
-\frac1{y_0^2}N_1
\Big(N_1- 1\Big)+
{\cal L}_1,\\
(u^{-1}y_0)^{\frac12}{\cal L}_2 (u^{-1}y_0)^{-\frac12}
& =
-\frac1{y_0^2}N_1
\Big(N_1+1\Big)+
{\cal L}_1.
\end{align}\end{subequations}
Compare the coordinates (\ref{cor}) for 2+1 dimensions and
the coordinates (\ref{coo2}) for 1+1 dimensions.
The coordinate $s$ are the same.
This is not the case of
$w$, so let us rename $w$ from (\ref{coo2})
as $v$. We then have $w=v^2$.
We also have
\[uy_0=\sqrt2su_1,\quad u^{-1}y_0=\sqrt 2w u_1^{-1}s.\]
Hence on functions that do not depend on $u$ we obtain
\begin{subequations} \begin{align}
s^{\frac12}u_1^{\frac12}{\cal L}_2 s^{-\frac12}u_1^{-\frac12}&=
{\cal L}_1 ,\\
s^{\frac12}u_1^{-\frac12}v{\cal L}_2 s^{-\frac12}u_1^{\frac12}v^{-1}&=
{\cal L}_1 .
\end{align}
\end{subequations}
Thus by a quadratic
transformation we can transform the Hermite equation into a special case of the confluent equation:
\begin{subequations}\begin{eqnarray}
{\cal S}_\lambda(v, \partial_v)&=&4{\cal F}_{\lambda,-\frac12}(w, \partial_w),\\
v^{-1}{\cal S}_\lambda(v, \partial_v)v&=&4{\cal F}_{\lambda,\frac12}(w, \partial_w),
\end{eqnarray}\label{ada1}\end{subequations}
where
\[w=v^2,\ \ \ \ v=\sqrt w.\]
\subsection{Transmutation relations and discrete symmetries}
\label{symcom5}
The heat operator satisfies the
generalized symmetries
\begin{subequations}\begin{eqnarray}
B^{{\rm sch},-\frac52}{\cal L}_1&=&{\cal L}_1B^{{\rm sch},-\frac12},\quad B\in {\rm sch}(1);\label{torr1}\\
\alpha^{{\rm sch},-\frac52}{\cal L}_1&=&{\cal L}_1\alpha^{{\rm sch},-\frac12},\quad
\alpha\in {\rm Sch}(1).\label{torr3}
\end{eqnarray}\end{subequations}
Equation (\ref{torr1}) applied to the roots of ${\rm sch}(1)$ implies the transmutation relations of the Hermite operator:
\[\begin{array}{rrll}
\partial_w&{\cal S}_\lambda &
=\ \ {\cal S}_{\lambda +1}& \partial_w,\\[0.3ex]
( \partial_w-2w)
&{\cal S}_\lambda &
=\ \ \ {\cal S}_{\lambda -1}&( \partial_w-2w),\\[0.6ex]
(w \partial_w+\lambda +\frac12)&w^2{\cal S}_\lambda&
=\ \ w^2{\cal S}_{\lambda +2}& (w \partial_w+\lambda +\frac{1}{2}),\\[0.3ex]
(w \partial_w-\lambda +\frac12-2w^2)&w^2{\cal S}_\lambda &
=\ \ w^2{\cal S}_{\lambda -2}&(w \partial_w-\lambda +\frac{1}{2}-2w^2).
\end{array}\]
Relation (\ref{torr1}) applied to the Weyl symmetries of ${\rm sch}(1)$ implies the discrete symmetries of the Hermite operator, described below.
The following operators equal ${\cal S}_\lambda (w, \partial_w)$ for an
appropriate $w$:
\begin{subequations} \begin{align}
w=\pm v:\qquad
&&&{\cal S}_\lambda (v, \partial_v),\\
w=\pm{\rm i} v:\qquad
&&-\exp(-v^2)&{\cal S}_{-\lambda }(v, \partial_v)\exp(v^2).\label{siu2}
\end{align}
\end{subequations}
\subsection{Factorizations of the heat operator}
\label{Factorizations of the heat operator2}
Special role is played by two distinguished subalgebras of ${\rm sch}(2)$.
First note the commutation relations
\begin{equation} [B_{2,0},B_{3,0}]=B_{3,2}.\end{equation}
Therefore, we have the following distinguished subalgebra in ${\rm sch}(1)$ isomorphic to ${\rm heis}(2)$:
\begin{eqnarray}
{\rm heis}_0(2)& \hbox{spanned by}&B_{2,0},\ B_{3,0},\ B_{3,2}.
\end{eqnarray}
Let us define
\begin{subequations}\begin{align}
{\cal C}_0& =2\,B_{2,0} B_{3,0}+2M-B_{3,2}\,\\
& =2\,B_{3,0} B_{2,0}+2M+B_{3,2}.
\end{align}\end{subequations}
We have the commutation relations
\begin{align*}
[{\cal C}_0,B_{2,0}]&=-2B_{2,0}(B_{3,2}-1),\\
[{\cal C}_0,B_{3,0}]&=2B_{3,0}(B_{3,2}-1),\\
[{\cal C}_0,B_{3,2}]&=0.\end{align*}
But $B_{3,2}^{{\rm sch},\eta}=1$. Therefore,
on the level of ${\mathbb R}\oplus{\mathbb R}$ the operator ${\cal C}_0^{{\rm sch},\eta}$
can be treated as a kind of a Casimir operator of ${\rm heis}_0(2)$:
it commutes with all elements of
${\rm heis}_0(2)$.
Note the identity
\begin{equation} 2t{\cal L}_1={\cal C}_{0}^{{\rm sch},{-\tfrac{1}{2}}}.\end{equation}
Second,
consider
$B_{-3,2}$, $B_{3,-2}$, $M$. They are contained both in ${\rm sch}(6)$ and in ${\rm sch}(5)$. Therefore, the subalgebra $\so_{23}(3)$, described
in Sect. \ref{Factorizations of of the heat operator} in the context of
${\rm sch}(6)$, is also contained in ${\rm sch}(5)$.
Recall that its
Casimir operator is
\begin{subequations}
\begin{align}
{\cal C}_{23}& =4\,B_{3,-2} B_{-3,2}-(M+1)^2+1\,\\
& =4\,B_{-3,2} B_{3,-2}-(M-1)^2+1.
\end{align}\label{dire2.}
\end{subequations}
By (\ref{deq3a}) we have
\begin{equation} (2z_{-2}z_2+2z_{-3}z_3)\Delta_5^\diamond
={\cal C}_{23}^{\diamond,{-\tfrac{1}{2}}}-\frac{3}{4}.
\label{dire..}\end{equation}
Inserting (\ref{dire2.})
into (\ref{dire..}) we obtain
\begin{subequations}\begin{align}
& (2z_{-2}z_2+2z_{-3}z_3)\Delta_5^\diamond\notag\\=&4B_{2,-3}B_{-2,3}
-\Big(M+\frac32\Big)\Big(M+\frac12\Big),\\
=&4B_{-2,3}B_{2,-3}
-\Big(M-\frac32\Big)\Big(M-\frac12\Big),
\end{align}\label{suu1-}\end{subequations}
where the $B$, $N_1$ and $M$ operators should be decorated with the superscript ${}^{\diamond,-\frac12}$.
Let us sum up the factorizations
in the variables $y_0,t$
obtained with the help of the two subalgebras:
\begin{subequations}
\begin{align}
2t{\cal L}_1& =2\,B_{2,0} B_{3,0}-(-2M+1)\,\\
& =2\,B_{3,0} B_{2,0}-(-2M-1),\\
-y_0^2{\cal L}_1&=4B_{2,-3}B_{-2,3}
-\Big(M+\frac32\Big)\Big(M+\frac12\Big)\\
&=4B_{-2,3}B_{2,-3}
-\Big(M-\frac32\Big)\Big(M-\frac12\Big),
\end{align}
\end{subequations}
where the $B$, $N_1$ and $M$ operators should be equipped with the superscript ${}^{{\rm sch},-\frac12}$.
In the coordinates $w,s$ we need to make the replacements
\begin{subequations}
\begin{align}\label{facb-2-}
t&\quad\to\quad s^2,\\
y_0^2&\quad\to\quad 2w^2s^2
\label{facb-1-}.\end{align}
\end{subequations}
\subsection{Factorizations of the Hermite operator}
\label{symcom5a}
The factorizations
of ${\cal L}_1$ described in Subsect.
\ref{Factorizations of the heat operator2}
yield the following factorizations of the Hermite operator:
\begin{eqnarray*}
{\cal S}_\lambda&=&\big( \partial_w-2w\big) \partial_w-2\lambda-1\\
&=& \partial_w\big( \partial_w-2w\big)-2\lambda+1,\\[0.5ex]
w^2{\cal S}_\lambda&=&
\Big(w \partial_w+\lambda-\frac32\Big)\Big(w \partial_w-\lambda+\frac12-2w^2\Big)+\Big(\lambda-\frac32\Big)\Big(\lambda-\frac12\Big)\\
&=&
\Big(w \partial_w-\lambda-\frac32-2w^2\Big)
\Big(w \partial_w+\lambda+\frac12\Big)
+\Big(\lambda+\frac32\Big)\Big(\lambda+\frac12\Big).
\end{eqnarray*}
\subsection{Standard solutions}
The Hermite equation has only one singular point, $\infty$. One can define two kinds of solutions with a simple asymptotics at $\infty$. They can be derived from the expressions of Subsect. \ref{Standard solutions11},
using (\ref{ada1}) and (\ref{siu2})
\begin{align*}
\text{ $\sim w^{-{a}}$ for $w\to+\infty$:}&&\quad
S_\lambda (w)&:=w^{-\lambda -\frac12}\tilde F_{\lambda,\frac12 }(-w^{-2})\\
&&=&
w^{-{a}}F\Big(\frac{{a}}{2},\frac{{a}+1}{2};-;-w^{-2}\Big),\\
\text{${\sim}({-}{\rm i} w)^{{a}-1}{\rm e}^{w^2}$ for $w\to{+}{\rm i}\infty$:}&&\quad
{\rm e}^{w^2}S_{-\lambda }(-{\rm i} w)&=
(-{\rm i} w)^{\lambda -\frac12}{\rm e}^{w^2}\tilde F_{-\lambda,\frac12}(w^{-2})\\
&&&
\hspace{-14ex}=(-{\rm i} w)^{{a}-1}{\rm e}^{w^2}F\Big(\frac{1-a}{2},\frac{2-a}{2};-;-w^{-2}\Big).
\end{align*}
\subsection{Recurrence relations}
Each of the following recurrence relations corresponds to a root of ${\rm sch}(1)$:
\begin{eqnarray*}
\partial_w S_\lambda(w)&=&-\Big(\frac12+\lambda \Big) S_{\lambda +1}(w),\\
( \partial_w -2w)S_\lambda (w)&=&-2S_{\lambda -1}(w),\\[0.6ex]
\Big(w \partial_w+\frac12+\lambda \Big)S_\lambda (w)&=&\frac12\Big(\frac12+\lambda \Big)
\Big(\frac32+\lambda\Big) S_{\lambda +2}(w),\\
\Big(w \partial_w+\frac12-\lambda -2w^2\Big) S_\lambda (w)&=&-2
S_{\lambda -2}(w).
\end{eqnarray*}
The first pair corresponds correspond to the celebrated annihilation and creation operators in the theory of quantum harmonic oscillator. The second pair are the double annihilation and creation operators.
\subsection{Wave packets for the heat equation in 1 dimensions}
Consider the space ${\mathbb R}\oplus{\mathbb R}$ and the heat equation given by the operator ${\cal L}_1=\partial_{y}^2+2\partial_t$.
Recall that
\begin{equation} M^{{\rm sch},-\frac12}=y \partial_y+2t \partial_t+\frac12.\end{equation}
Set
\begin{subequations}\begin{align}
G_\lambda^a(y,t)&:=\int_{\gamma^a}
t^{-\frac12}\exp\Big(\frac{(y-\tau^{-1})^2}{2t}\Big)
\tau^{-\frac32+\lambda}{\rm d}\tau,\\
G_\lambda^b(y,t)&:=
\int_{\gamma^b}{\rm e}^{-\sqrt{2}y\tau-t\tau^2}\tau^{-\frac12+\lambda}{\rm d}\tau.
\end{align}\end{subequations}
\begin{proposition} We have
\begin{subequations}\begin{align}
{\cal L}_1 G_\lambda^a&=0,&\quad {\cal L}_1 G_\lambda^b&=0;\\
M^{{\rm sch},-\frac12} G_\lambda^a&=-\lambda G_\lambda^a,&\quad M^{{\rm sch},-\frac12} G_\lambda^b &=-\lambda G_\lambda^b.\label{ko2.}
\end{align}\end{subequations}
\end{proposition}
{\bf Proof.}\ \
Set
\begin{subequations}\begin{align} g^a(y,t)&:= t^{-\frac12}\exp\frac{(y-1)^2}{2t},\\
g^b(y,t)&:={\rm e}^{-\sqrt{2} y-t}.\end{align}\end{subequations}
We have
\begin{subequations}\begin{align}
G_\lambda^a&=\int_{\gamma^a}\tau^{-1+\frac12+\lambda}g^a(\tau y,\tau^2 t) {\rm d}\tau,\\
G_\lambda^b&=\int_{\gamma^b}\tau^{-1+\frac12+\lambda}g^b(\tau y,\tau^2 t) {\rm d}\tau.\label{kaka}\end{align}\end{subequations}
Clearly, $g^a$ and $g^b$ solve the heat equation.
By (\ref{kaka}), $G_{\lambda}^a$, resp.
$G_{\lambda}^b$ are wave packets made out of rotated $g^a$, resp. $g^b$. Therefore, they also solve the heat equation.
If the contours satisfy the requirements of Prop. \ref{analo}, then (\ref{kaka}) implies (\ref{ko2.}).
$\Box$\medskip
Let us express these wave packets in the coordinates $w,s$:
\begin{subequations}\begin{align}
G_\lambda^a(w,s)&
=\int s^{-1}\exp\Big(\Big(w-\frac{1}{\sqrt2\tau s}\Big)^2\Big)\tau^{-2+\frac12+\lambda}{\rm d}\tau,\label{poe1}\\
G_\lambda^b(w,s)
&=\int{\rm e}^{-2sw\tau-s^2\tau^2}\tau^{-1+\frac12+\lambda}{\rm d}\tau.\label{poe2}
\end{align}\end{subequations}
In (\ref{poe1})
we set $\sigma:=w-\frac{1}{\sqrt2\tau s}$,
so that $\tau=\frac{1}{(w-\sigma)\sqrt2 s}$.
In (\ref{poe2})
we set $\sigma:= s\tau$, so that $\tau=\frac{\sigma}{s}$. We obtain
\begin{subequations}\begin{align}
G_\lambda^a(w,s)&
=(\sqrt2)^{\frac12-\lambda} s^{-\frac12-\lambda}
F_\lambda^a(w),\\
G_\lambda^b(w,s) &=s^{-\frac12-\lambda}F_\lambda^b(w),
\end{align}\end{subequations}
where
\begin{subequations}\begin{align}
F_\lambda^a(w)&:=
\int_{\gamma^a}{\rm e}^{\sigma^2}(w-\sigma)^{-\frac12-\lambda}{\rm d}\sigma,
\label{poee1}\\
F_\lambda^b(w)&:=\int_{\gamma^b}{\rm e}^{-2\sigma w-\sigma^2}\sigma^{-1+\frac12+\lambda}{\rm d}\sigma.\label{poee2}
\end{align}\end{subequations}
The above analysis shows that for appropriate contours
(\ref{poee1}) and (\ref{poee2}) are solutions of the Hermite equation.
\subsection{Integral representations}
Below we directly describe the two kinds of integral representations of solutions, without passing through additional variables.
\begin{theoreme}\begin{arabicenumerate}\item[a)]
Let $[0,1]\ni \tau\overset{\gamma}\mapsto t(\tau)$ satisfy
${\rm e}^{t^2}(t-w)^{-{a}-1}\Big|_{t(0)}^{t(1)}=0.$
Then
\begin{equation}{\cal S}({a};w, \partial_w)\int_\gamma{\rm e}^{t^2}(t-w)^{-{a}}{\rm d} t=0.\label{dad10}\end{equation}
\item[b)]
Let $[0,1]\ni \tau\mapsto t(\tau)$ satisfy
${\rm e}^{-t^2-2wt}t^{{a}}\Big|_{t(0)}^{t(1)}=0.$
Then\begin{equation}
{\cal S}({a};w, \partial_w)
\int_\gamma
{\rm e}^{-t^2-2wt}t^{a-1}{\rm d} t=0.\label{dad11}\end{equation}
\end{arabicenumerate}\label{dad12}\end{theoreme}
{\bf Proof.}\ \
We check that for any contour $\gamma$
\begin{eqnarray*}
\text{lhs of (\ref{dad10})} &=&-{a} \int_\gamma{\rm d} t\, \partial_t
{\rm e}^{t^2}(t-w)^{-{a}-1},\\
\text{lhs of (\ref{dad11})}&=&-2\int_\gamma{\rm d} t\,
\partial_t{\rm e}^{-t^2-2wt}t^{{a}}.
\end{eqnarray*}
We can also deduce the second representation from the first
by the discrete symmetry (\ref{siu2}).
$\Box$\medskip
\subsection{Integral representations of standard solutions}
In type a) representations the integrand has a singular point at $0$ and goes to zero as $t\to \pm \infty$. We can thus use contours with such endpoints. We will see that they give all standard solutions.
In type b) representations the integrand has a singular point at $w$ and goes
to zero as $t\to \pm {\rm i}\infty$. Using contours with such
endpoints, we will also obtain all standard solutions.
\medskip
\noindent
\begin{tabular}{lll}&a)&b)\\
$\sim w^{-{a}}$ for $w\to+\infty$:\qquad&$[0,\infty[$,&$]-{\rm i}\infty,w^-,-{\rm i}\infty[$;\\
$\sim(-{\rm i} w)^{{a}-1}{\rm e}^{w^2}$ for $w\to+{\rm i}\infty$:\qquad&
$]-\infty,0^+,-\infty[,$&$[w,{\rm i}\infty[$.\\
\end{tabular}
\medskip
It is convenient to introduce
alternatively normalized solutions:
\begin{eqnarray*}
S_\lambda^{\rm\scriptscriptstyle I} (w)&:=&2^{-\lambda-\frac12}\Gamma\Big(\lambda+\frac12\Big)S_\lambda(w).
\\
\end{eqnarray*}
Here are integral representations of type a):
\begin{align}\text{all $\lambda$:}&&\\
\underset{]-{\rm i}\infty,w^-,{\rm i}\infty[}{-{\rm i}\int}{\rm e}^{t^2}
(w-t)^{-\lambda -\frac12}
{\rm d}
t
&=\sqrt\pi S_\lambda (w),&\quad w\not\in]-\infty,0];\notag\\
\Re\lambda<\frac12:&&\\
\underset{[w,{\rm i}\infty[}{-{\rm i}\int}{\rm e}^{t^2}(-{\rm i} (t-w))^{-\lambda -\frac12}{\rm d} t
&= {\rm e}^{w^2} S_{-\lambda }^{\rm\scriptscriptstyle I}(-{\rm i} w),&\quad w\not\in[0,\infty[.\notag
\end{align}
And here are integral representations of type b):
\begin{align} -\frac12<\Re\lambda:&&\\
\int\limits_0^\infty{\rm e}^{-t^2-2tw}t^{\lambda -\frac12}{\rm d} t
&= S_\lambda ^{\rm\scriptscriptstyle I}(w),&\quad w\not\in]-\infty,0];\notag\\
\text{all $\lambda$:}&&\\
\int\limits_{]-\infty,0^+,\infty[}{\rm e}^{-t^2-2tw}({\rm i} t)^{\lambda -\frac12}{\rm d} t&=
\sqrt\pi{\rm e}^{w^2} S_{-\lambda }(-{\rm i} w),&\quad w\not\in[0,\infty[.\notag
\end{align}
\section{The Helmholtz equation in 2 dimensions and the ${}_0{\cal F}_1$ equation}
\label{s11}
\init
The goal of this section is to derive the ${}_0{\cal F}_1$ equation together with its symmetries from the Helmoltz equation in $2$ dimensions.
The symmetries of these equations, together with its derivation, are the simplest and the best known. In particular, we do not need to consider {\em generalized symmetries}.
Here are the main steps from the derivation:
\be
\item\label{it1he}
We start from the Helmholtz operator $\Delta_2-1$. The Lie algebra $\aso(2)$ and group $\ASO(2)$
acting in $2$ dimensions, are the obvious symmetries of this operator.
\item\label{it4he} We choose coordinates $w,u$, so that the Cartan element is expressed in terms of $u$. We compute
$\Delta_2-1$ and the representations
of $\aso(2)$ and $\ASO(2)$ in the new coordinates.
\item\label{it5he} We make an ansatz diagonalizing the Cartan element,
whose eigenvalue $\alpha$
becomes a parameter.
The only variable left is $w$.
The Helmholtz operator $\Delta_2-1$ becomes the ${}_0{\cal F}_1$ operator.
The symmetries of
$\Delta_2-1$ yield transmutation relations and discrete symmetries of the
${}_0{\cal F}_1$ operator.
\ee
The remaining part of this section is to a large extent parallel to their analogs in Sects
\ref{s6}, \ref{s7}, \ref{s9} and \ref{s10}. Essentially all subsections have their counterparts there.
The only exception is Subsect.
\ref{Equivalence with a subclass of the confluent equation}
on the equivalence of the ${}_0{\cal F}_1$ equation with a subclass of the ${}_1{\cal F}_1$ equation, and its many-dimensional unravelling.
This equivalence is obtained by a quadratic transformation, which
is quite different from the quadratic transformations for the Gegenbauer and Hermite equation considered in
Subsects \ref{Quadratic transformation}, resp.
\ref{Quadratic transformation-}.
\subsection{$\aso(2)$}
We consider ${\mathbb R}^2$ with split coordinates $x_-,x_+$ and the scalar product
\begin{equation}
\langle x|x\rangle=2x_-x_+.
\end{equation}
\noindent{\bf Lie algebra $\aso({\mathbb C}^2)$.}
Cartan operator
\begin{equation} N=-x_{-}\partial_{x_{-}}+x_{+}\partial_{x_+}. \label{helm1}\end{equation}
Root operators
\begin{subequations}\begin{align}
B_- &= \partial_{x_-},\\
B_+ &= \partial_{x_+}.
\end{align}\end{subequations}
\noindent{\bf Weyl symmetry}
\begin{equation}
\tau f(x_-,x_+)=f(x_+,x_-).
\end{equation}
\noindent{\bf Helmholtz operator}
\begin{equation}\Delta_2-1=
2 \partial_{x_-} \partial_{x_+}-1.\label{helm2}\end{equation}
\subsection{Variables $w,u$}
\label{Variables $w,u$}
We introduce the coordinates
\begin{align}
w =\frac{x_-x_+}{2}\,,\qquad
u =x_+.
\label{coor3}\end{align}
\noindent{\bf Lie algebra $\aso(2)$.}
Cartan operator
\[ N=u \partial_u.\]
Root operators
\begin{align*}
B_+& =\frac{u}{2}\partial_w,\\
B_-& =\frac{1}{u}(w\partial_w+u\partial_u).
\end{align*}
\noindent{\bf Weyl symmetry}
\[
\tau
f(w,u)=f\Big(w,\frac{w}{u}\Big).
\]
\noindent{\bf Helmholtz operator}
\begin{equation}
\Delta_2-1=w\partial_w^2+(1+u\partial_u)\partial_w-1.
\end{equation}
\subsection{The ${}_0{\cal F}_1$ operator}
Let us make the ansatz
\begin{equation} f(w,u)=u^\alpha F(w).\end{equation}
Clearly,
\begin{align} Nf&=\alpha f,\\
u^{-\alpha} (\Delta_2-1)f&={\cal F}_\alpha(w,\partial_w)F,\end{align}
where we have introduced the {\em ${}_0{\cal F}_1$ operator}
\begin{equation}
{\cal F}_\alpha(w,\partial_w):=w\partial_w^2+(1+\alpha)\partial_w-1.
\end{equation}
Instead of the {\em Lie-algebraic parameter} $\alpha$ one could also use the {\em classical parameter}~$c$
\begin{equation} \alpha:=c-1,\ \ \ c=\alpha+1,\label{newnot3}\end{equation}
so that the ${}_0{\cal F}_1$ operator becomes
\begin{equation} {\cal F}(c;w, \partial_w):=w \partial_w^2+c \partial_w-1.\end{equation}
\subsection{Equivalence with a subclass of the confluent equation}
\label{Equivalence with a subclass of the confluent equation}
The ${}_0{\cal F}_1$ equation is equivalent to a subclass of the ${}_1{\cal F}_1$ equation by a quadratic transformation. This quadratic transformation is however quite different from transformations described in Subsect.
\ref{Dimensional reduction}, and then applied to derive the Gegenbauer quation
and the Hermite equation. In this subsection we derive this equivalence starting from the heat equation in 2 dimensions.
First let us recall some elements of our derivation of the ${}_1{\cal F}_1$ operator.
As described in Sect. \ref{s9}, it was obtained from the heat operator
(\ref{heat0})
together with Cartan operators (\ref{heat1}), (\ref{heat2}):
\begin{subequations} \begin{align}
\frac{t}{2}{\cal L}_2&=\frac{t}{2}\big(2\partial_t+2\partial_{y_{-1}}\partial_{y_1}\big),
\label{hhet} \\
M&=y_{-1} \partial_{y_{-1}}+y_{1} \partial_{y_{1}}+2t \partial_t+1,\\
N_1&=-y_{-1} \partial_{y_{-1}}+y_{1} \partial_{y_{1}}.
\end{align}\end{subequations}
(We set $\eta=-1$ and dropped the superscript ${}^{{\rm sch},-1}$).
Recall that substituting the coordinates (\ref{cor})
\begin{equation} w =\frac{y_{-1}y_1}{t}\;,\ \ \ \
u_1 =\frac{y_{1}}{\sqrt{t}} \;,\ \ \ \
s =\sqrt{t}\end{equation}
we obtain
\begin{subequations} \begin{align}
\frac{t}{2}{\cal L}_2&= w \partial_w^2+(u \partial_u+1-w) \partial_w+\frac12(-u \partial_u+s \partial_s),
\label{pocor}\\
M&=s \partial_s+1,\\
N_1&=u_1 \partial_{u_1}.
\end{align}\end{subequations}
After we set $M=-\theta$, $N_1=\alpha$, (\ref{pocor}) becomes ${\cal F}_{\theta,\alpha}(w, \partial_w)$.
Consider now
\begin{align}\notag
& \frac{2t^2}{y_{-1}y_{1}}{\rm e}^{-\frac{y_{-1}y_1}{2t}}
{\cal L}_2
{\rm e}^{\frac{y_{-1}y_1}{2t}}\\
=&\frac{2t}{y_{-1}y_1}\big( y_{-1} \partial_{y_{-1}}+y_{1} \partial_{y_{1}}+2t \partial_t+1\big)+\frac{4t^2}{y_{-1}y_1} \partial_{y_{-1}} \partial_{y_1}-1\notag\\
=& \frac{2t}{y_{-1}y_1}M+2 \partial_{x_-} \partial_{x_+}-1,\label{helmo}\\
& {\rm e}^{-\frac{y_{-1}y_1}{2t}}
N_1
{\rm e}^{\frac{y_{-1}y_1}{2t}}=N_1\notag
=-2x_- \partial_{x_-}+2x_+ \partial_{x_+},
\end{align}
where we
introduced new variables
\begin{equation}
x_-=\frac{y_{-1}^2}{2\sqrt2 t},\quad x_+=\frac{y_{1}^2}{2\sqrt2 t}.
\end{equation}
Therefore, on the subspace $M=0$ we have
\begin{align}\notag
\frac{2t^2}{y_{-1}y_{1}}{\rm e}^{-\frac{y_{-1}y_1}{2t}}
{\cal L}_2
{\rm e}^{\frac{y_{-1}y_1}{2t}}
=&\Delta_2-1,\\
{\rm e}^{-\frac{y_{-1}y_1}{2t}}
N_1
{\rm e}^{\frac{y_{-1}y_1}{2t}}&=2N,
\end{align}
where $\Delta_2-1$ is the Helmholtz operator
(\ref{helm2})
and $N$ the Cartan operator (\ref{helm1}).
Remember, that in Subsect. \ref{Variables $w,u$}
we express these operators
in the coordinates (\ref{coor3}). To avoid a clash of symbols, we rename $w$ from (\ref{coor3}) into $v$:
\begin{equation}
v=\frac{y_-y_+}{2},\quad
u=y_+.\end{equation}
Recall that in the $v,u$ coordinates we have
\begin{subequations} \begin{align}
\Delta_2-1&=v\partial_v^2+(1+u\partial_u)\partial_v-1,\label{wqe}\\
N&=u \partial_u,
\end{align}\end{subequations}
so that (\ref{wqe}) on $N=\alpha$ becomes ${\cal F}_\alpha(v, \partial_v)$.
Now we can compare the coordinates $w,u_1$ and $v,u$
\begin{equation}
v=\frac{y_{-1}^2y_1^2}{16 t^2}=\Big(\frac{w}{4}\Big)^2,\quad
u=\frac{y_1^2}{2\sqrt2t}=\frac{u_1^2}{2\sqrt2}.\end{equation}
This leads to the so-called {\em Kummer's 2nd transformation}, which reduces
the ${}_0{\cal F}_1$ equation to a special class of
the confluent equation by a quadratic transformation:
\begin{equation}
{\cal F}_\alpha (v, \partial_v)
=\frac{4}{w}{\rm e}^{-w/2}{\cal F}_{0,2\alpha }(w, \partial_w){\rm e}^{w/2},\label{gas1}\end{equation}
or, in classical parameters
\begin{eqnarray}
{\cal F}(c;v, \partial_v)
=\frac{4}{w}{\rm e}^{-w/2}{\cal F}\Big(c-\frac{1}{2};2c-1;w, \partial_w\Big){\rm e}^{w/2},
\label{gas}\end{eqnarray}
where $w=\pm 4\sqrt{v}$, $v=\big(\frac{w}{4}\big)^2$.
\subsection{Transmutation relations and symmetries}
\label{symcom2}
The following symmetries of the Helmholtz operator are obvious:
\begin{subequations}
\begin{align}
B(\Delta_2-1)&=(\Delta_2-1)B;\quad B\in \aso(2);\label{lorr3+}\\
\alpha (\Delta_2-1)&=(\Delta_2-1)\alpha;\quad \alpha\in\ASO(2).\label{lorr3}
\end{align}
\end{subequations}
Applying (\ref{lorr3+}) to the roots of $\aso(2)$ we obtain the trasmutation relations
\[\begin{array}{rrll}
\partial_w&{\cal F}_\alpha &
=\ {\cal F}_{\alpha +1}& \partial_w,\\[3ex]
(w \partial_w+\alpha )&{\cal F}_\alpha
&=\ \ {\cal F}_{\alpha -1}&(w \partial_w+\alpha ).\end{array}\]
Applying (\ref{lorr3}) to the Weyl symmetry of $\aso(2)$ we obtain the symmetry
\[\begin{array}{lcr}w^{-\alpha }\ {\cal F}_{-\alpha }\ w^\alpha &=&{\cal F}_\alpha .\end{array}\]
\subsection{Factorizations}
The factorizations
\begin{subequations}\begin{align}
\Delta_2-1&=2B_-B_+-1\\
&=2B_+B_--1,
\end{align}\end{subequations}
are completely obvious. They yield the factorizations of the ${}_0{\cal F}_1$
operator:
\begin{eqnarray*}
{\cal F}_\alpha&=&
\big(w\partial_w+\alpha+1\big)\partial_w-1\\
&=& \partial_w
\big(w\partial_w+\alpha\big)-1.
\end{eqnarray*}
\subsection{The ${}_0F_1$ function}
The ${}_0{\cal F}_1$ equation
has a regular singular point at $0$.
Its indices at $0$ are equal to $0$, $\alpha=1-c$.
If $c\neq0,-1,-2,\dots$, then the only solution of the ${}_0F_1$ equation
$\sim1$ at 0
is called
the {\em ${}_0F_1$ function}.
It is
\begin{equation} F(c;w):=\sum_{j=0}^\infty
\frac{1}{
(c)_j}\frac{w^j}{j!}.\end{equation}
It is defined for $c\neq0,-1,-2,\dots$.
Sometimes it is more convenient to consider
the function
\begin{equation} {\bf F} (c;w):=\frac{F(c;w)}{\Gamma(c)}=
\sum_{j=0}^\infty
\frac{1}{
\Gamma(c+j)}\frac{w^j}{j!}\end{equation}
defined for all $c$.
Using (\ref{gas}), we can express the ${}_0F_1$ function in terms of the confluent function
\begin{subequations}\begin{eqnarray}F(c;w)&=&
{\rm e}^{-2\sqrt{w}}F\Big(\frac{2c-1}{2};
2c-1;4\sqrt{w}\Big)\\&=&
{\rm e}^{2\sqrt{w}}F\Big(\frac{2c-1}{2};
2c-1;-4\sqrt{w}\Big).
\end{eqnarray}\end{subequations}
We will usually prefer to use the Lie-algebraic parameters:
\begin{subequations}\begin{eqnarray}
F_\alpha (w)&:=&F(\alpha+1;w),\\
{\bf F} _\alpha (w)&:=& {\bf F} (\alpha+1;w)
.
\end{eqnarray}\end{subequations}
\subsection{Standard solutions}
We have two standard solutions corresponding to two indices
of the regular singular point $w=0$.
Besides, using Tricomi's function described in Subsect. \ref{Standard solutions11}, we have an additional solution with a special behavior at $\infty$:
\begin{align*}
\text{$\sim1$ at 0:}&&\quad
F_\alpha (w)&=
{\rm e}^{-2\sqrt{w}}F_{0,2\alpha }\big(4\sqrt{w}\big)\notag\\&&&=
{\rm e}^{2\sqrt{w}}F_{0,2\alpha }\big(-4\sqrt{w}\big);\\[1ex]
\text{$\sim w^{-\alpha }$ at 0:}&&\quad
w^{-\alpha }F_{-\alpha} (w)&=
w^{-\alpha }{\rm e}^{-2\sqrt{w}}F_{0,-2\alpha }\big(4\sqrt{w}\big)\notag\\&&&=
w^{-\alpha }{\rm e}^{2\sqrt{w}}F_{0,-2\alpha }\big(-4\sqrt{w}\big);
\end{align*}
\begin{align*}
\text{
$\sim{\rm e}^{- 2\sqrt{w}} w^{-\frac{ \alpha }2-\frac14}$,
$w\to+\infty$:}&&\quad
\tilde F_\alpha (w)&:={\rm e}^{-2\sqrt w} w^{-\frac{\alpha}{2} -\frac14}
\tilde F_{0,2\alpha }\Big(-\frac{1}{4\sqrt w}\Big)\notag\\&&
&={\rm e}^{-2\sqrt w} w^{-\frac{\alpha}{2} -\frac14}
\tilde F_{0,-2\alpha }\Big(-\frac{1}{4\sqrt w}\Big).
\end{align*}
Note that the third standard solution is a new function closely related to the MacDonald function. It satisfies
the identity
\begin{equation}\tilde F_\alpha (w)=w^{-\alpha }\tilde F_{-\alpha }(w).\end{equation}
Its asymptotics
\begin{equation}
\tilde F_\alpha (w)\sim{\rm exp}(- 2w^\frac{1}{2}) w^{-\frac{\alpha }2-\frac14}
\label{saddle1}\end{equation}
is valid
in the sector
$|\arg w|<\pi/2-\epsilon$ for
$|w|\to\infty$.
\subsection{Recurrence relations}
The following recurrence relations follow from the transmutation relations
\begin{eqnarray*}
\partial_w {\bf F} _\alpha (w)&=& {\bf F} _{\alpha +1}(w),
\\[3mm]
\left(w \partial_w+\alpha\right) {\bf F} _\alpha (w)&=& {\bf F} _{\alpha -1}(w).
\end{eqnarray*}
\subsection{Wave packets}
Obviously, for any $\tau$
the function $\exp\Big(\frac{x_-}{\sqrt 2\tau}+\frac{\tau x_+}{\sqrt{2}}\Big)$ solves the Helmholtz equation. Therefore, for appropriate contours $\gamma$,
\begin{equation}
f(x_-,x_+):=\frac{1}{2\pi{\rm i}}\int_\gamma
\exp\Big(\frac{x_-}{\sqrt 2\tau}+\frac{\tau x_+}{\sqrt{2}}\Big)\tau^{-\alpha-1}{\rm d}\tau\end{equation}
solves
\begin{align}
(\Delta_2-1)f&=0,\\
Nf&=\alpha f.
\end{align}
Substituting the coordinates $w,u$ we obtain
\begin{align}\notag
f(w,u)&=
\int_\gamma \exp\Big(\frac{w}{\tau u\sqrt2}+\frac{\tau u}{\sqrt2}\Big)\tau^{-\alpha-1}{\rm d}\tau
\\
&=u^\alpha 2^{-\frac{\alpha}{2}}\int_\gamma \exp\Big(\frac{w}{s}+s\Big)s^{-\alpha-1}{\rm d} s,\end{align}
where we made the substitution $s=\frac{\tau u}{\sqrt2}$.
Therefore,
\begin{equation}
F(w)=\int_\gamma \exp\Big(\frac{w}{s}+s\Big)s^{-\alpha-1}{\rm d} s.
\end{equation}
solves the ${}_0F_1$ equation.
\subsection{Integral representations}
There are three kinds of integral representations of solutions to the
${}_0F_1$ equation. The first is suggested by the previous subsection. Representations of the first kind
will be called {\em
Bessel-Schl\"afli type representations}.
The next two are inherited from the confluent equation by 2nd Kummer's identity. We will call them
{\em
Poisson-type representations}.
\begin{theoreme}\label{schl}
\begin{arabicenumerate}\item[i)] {\bf Bessel-Schl\"afli type representations.}
Suppose that $[0,1]\ni t\mapsto\gamma(t)$ satisfies
\[
{\rm e}^t{\rm e}^{\frac{w}{t}}t^{-c}\Big|_{\gamma(0)}^{\gamma(1)}=0.\]
Then
\begin{equation}{\cal F}(c;w, \partial_w)
\int_\gamma{\rm e}^t{\rm e}^{\frac{w}{t}}t^{-c}{\rm d} t=0.\label{dad5}\end{equation}
\item[ii)] {\bf Poisson type a) representations.} Let the contour $\gamma$ satisfy
\[(t^2-w)^{-c+3/2}{\rm e}^{2t}\Big|_{\gamma(0)}^{\gamma(1)}=0.\]
Then
\begin{equation}
{\cal F}(c;w, \partial_w)\int_\gamma(t^2-w)^{-c+1/2}{\rm e}^{2t}{\rm d} t=0.\label{in2}\end{equation}
\item[iii)] {\bf Poisson type b) representations.}
Let the contour $\gamma$ satisfy
\[(t^2-1)^{c-1/2}{\rm e}^{2t\sqrt w}\Big|_{\gamma(0)}^{\gamma(1)}=0.\]
Then
\begin{equation}
{\cal F}(c;w, \partial_w)\int_\gamma(t^2-1)^{c-3/2}{\rm e}^{2t\sqrt w}{\rm d} t=0.\label{in3}\end{equation}
\end{arabicenumerate}
\end{theoreme}
{\bf Proof.}\ \
We check that for any contour $\gamma$
\[\text{lhs of (\ref{dad5})}=-\int_\gamma{\rm d} t
\partial_t{\rm e}^t{\rm e}^{\frac{w}{t}}t^{-c}.\]
This proves i).
To prove both Poisson type representations we use the quadratic relation
(\ref{gas}). Using the type a) representation for solutions of ${}_1{\cal F}_1$ (\ref{dad1}), for appropriate contours $\gamma$ and $\gamma'$, we see that
\begin{eqnarray*}
&&{\rm e}^{-2\sqrt w}\int_\gamma{\rm e}^s s^{-c+\frac12}(s-4\sqrt w)^{-c+\frac12}{\rm d} s\\
&=&2^{-2c+2}\int_{\gamma'}{\rm e}^{2t}(t^2-w)^{-c+\frac12}{\rm d} t
\end{eqnarray*}
is annihilated by ${\cal F}(c)$, where we set
$t=\frac{s}{2}-\sqrt w$. This proves ii).
Similarly, by the type b) representation for solutions of ${}_1{\cal F}_1$ (\ref{dad}),
\begin{eqnarray*}
&&{\rm e}^{-2\sqrt w}\int_\gamma{\rm e}^{\frac{4\sqrt w}{s}} s^{-2c+1}(1-s)^{c-\frac32}{\rm d} s\\
&=&-2^{-2c+2}\int_{\gamma'}{\rm e}^{2t\sqrt w}(1-t^2)^{c-\frac32}{\rm d} t
\end{eqnarray*}
is annihilated by ${\cal F}(c)$, where we set
$t=\frac2s-1$. This proves iii).
$\Box$\medskip
\subsection{Integral representations of standard solutions}
In Bessel-Schl\"afli type representations the integrand goes to zero as $t\to-\infty$ and $t\to0-0$ (the latter for $\Re w>0$). Therefore, contours ending at these points yield solutions. We will see that in this way we can obtain all 3 standard solutions.
We can also obtain all solutions using Poisson type representations
(which are actually special cases of representations for solutions of the confluent equation).
\medskip
\noindent
\begin{tabular}{llll}
&Bessel-Schl\"afli&Poisson type a) &Poisson type b)\\[2ex]
$ \sim1\text{ at }0$:&$]-\infty,0^+,\infty[$&
$[-1,1]$&
\\
$\sim w^{-\alpha }\text{ at }0$:&$(0-0)^+$&&$[-\sqrt{w},\sqrt{w}]$\\
$\sim{\rm e}^{- 2\sqrt{w}} w^{-\frac{ \alpha }2-\frac14}
\text{ for }w\to+\infty$:&$]-\infty,0]$&
$]-\infty,-1]$&$]-\infty,-\sqrt{w}]$\\
\end{tabular}
\bigskip
Here are Bessel-Schl\"afli type representations. They are valid for all values of $\alpha$ and $\Re w>0$:
\begin{align}
\frac{1}{2\pi {\rm i}}\int\limits_{]-\infty,0^+,-\infty[}
{\rm e}^t{\rm e}^{\frac{w}{t}}t^{-\alpha -1}{\rm d} t
&= {\bf F} _\alpha (w),\\
\frac{1}{2\pi {\rm i}}\int\limits_{[(0-0)^+]}{\rm e}^t{\rm e}^{\frac{w}{t}}t^{-\alpha -1}{\rm d} t
&=w^{-\alpha } {\bf F} _{-\alpha }(w),\\
\int_{-\infty}^{0}{\rm e}^t{\rm e}^{\frac{w}{t}}(-t)^{-\alpha -1}{\rm d} t
&=\pi^{\frac{1}{2}}\tilde F_\alpha (w).\end{align}
Next we give Poisson type representations, valid for
$w\not\in]-\infty,0]$:
\begin{align}
\Re \alpha >-\frac12:&\\
\int_{-1}^1(1-t^2)^{\alpha -\frac12}{\rm e}^{
2t\sqrt w}{\rm d} t&=\Gamma(\alpha +\frac12)\sqrt\pi {\bf F} _\alpha (w),\notag\\
\frac12>\Re\alpha:&\\
\int_{-\sqrt w}^{\sqrt w}(w-t^2)^{-\alpha -\frac12}{\rm e}^{
2t}{\rm d} t&=\Gamma\Big({-}\alpha +\frac12\Big)\sqrt\pi w^{-\alpha }
{\bf F} _{-\alpha }(w);\notag\end{align}
\begin{align}
\Re \alpha >-\frac12:&\\
\int_{-\infty}^{-1}(t^2-1)^{\alpha -\frac12}{\rm e}^{
2t\sqrt w}{\rm d} t&=
\frac12\Gamma\Big(\alpha +\frac12\Big) \tilde F_\alpha (w),\notag\\
\Re \alpha <\frac12:&\\
\int_{-\infty}^{-\sqrt w}(t^2-w)^{-\alpha -\frac12}{\rm e}^{
2t}{\rm d} t&=\frac12\Gamma\Big(-\alpha +\frac12\Big) \tilde F_\alpha (w).\notag
\end{align}
\subsection{Connection formulas}
From integral representations we easily obtain connection formulas.
As the basis we can use the solutions with a simple behavior at zero:
\begin{eqnarray*}
\tilde F_\alpha (w)
&=&\frac{\sqrt\pi}{\sin\pi (-\alpha )} {\bf F} _\alpha (w)
+\frac{\sqrt \pi}{\sin\pi \alpha }
w^{-\alpha } {\bf F} _{-\alpha }(w).
\end{eqnarray*}
Alternatively, we can use the basis conisting of the $\tilde F$ function and its clockwise or anti-clockwise analytic
continuation around $0$:
\begin{eqnarray*}
{\bf F}_\alpha(w)&=&\frac{1}{ 2\sqrt\pi}\left({\rm e}^{\pm{\rm i}\pi
( \alpha+\frac{1}{2})}
\tF_\alpha(w)+
{\rm e}^{\mp{\rm i}\pi (\alpha+\frac{1}{2})}\tF_\alpha({\rm e}^{\mp{\rm i} 2\pi}w)\right),\\
w^{-\alpha}{\bf F}_{-\alpha}(w)
&=&\frac{1}{ 2\sqrt\pi}\left({\rm e}^{\mp{\rm i}\pi (\alpha-\frac{1}{2})}\tF_\alpha(w)-
{\rm e}^{\mp{\rm i}\pi( \alpha-\frac{1}{2})}\tF_\alpha({\rm e}^{\mp{\rm i} 2\pi}w)\right).
\end{eqnarray*}
| {
"timestamp": "2019-06-11T02:12:32",
"yymm": "1906",
"arxiv_id": "1906.03512",
"language": "en",
"url": "https://arxiv.org/abs/1906.03512",
"abstract": "We show that properties of hypergeometric class equations and functions become transparent if we derive them from appropriate 2nd order differential equations with constant coefficients. More precisely, properties of the hypergeometric and Gegenbauer equation can be derived from generalized symmetries of the Laplace equation in 4, resp. 3 dimension. Properties of the confluent, resp. Hermite equation can be derived from generalized symmetries of the heat equation in 2, resp. 1 dimension. Finally, the theory of the ${}_1F_1$ equation (equivalent to the Bessel equation) follows from the symmetries of the Helmholtz equation in 2 dimensions. All these symmetries become very simple when viewed on the level of the 6- or 5-dimensional ambient space. Crucial role is played by the Lie algebra of generalized symmetries of these 2nd order PDE's, its Cartan algebra, the set of roots and the Weyl group. Standard hypergeometric class functions are special solutions of these PDE's diagonalizing the Cartan algebra. Recurrence relations of these functions correspond to the roots. Their discrete symmetries correspond to the elements of the Weyl group.",
"subjects": "Classical Analysis and ODEs (math.CA); Group Theory (math.GR)",
"title": "Group-theoretical origin of symmetries of hypergeometric class equations and functions",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808678600415,
"lm_q2_score": 0.7217432003123989,
"lm_q1q2_score": 0.7075110507743222
} |
https://arxiv.org/abs/1202.6182 | A canonical form for Gaussian periodic processes | This article provides a representation theorem for a set of Gaussian processes; this theorem allows to build Gaussian processes with arbitrary regularity and to write them as limit of random trigonometric series. We show via Karhunen-Love theorem that this set is isometrically equivalent to l2. We then prove that regularity of trajectory path of anyone of such processes can be detected just by looking at decrease rate of l2 sequence associated to him via isometry. | \section{Introduction}
The aim of this article is to provide a simple and general method for constructing continuous and periodic Gaussian
processes of arbitrary regularity.
Periodic real processes arise as natural tool in analyzing continuous processes on the circle
(e.g., in image analysis, when processing noises of closed lines). More recently,
they found a great development in the theory of Random Fields on the sphere (see
\cite{Mar-Pec} and the reference therein).
The Brownian Bridge $B_t$ is a well known example of a continuous and periodic Gaussian
process on $[0,1]$. Although it is not stationary, it becomes stationary if we remove the path-integral
$\int_0^1 B_s ds$, see \cite{Darling}.
If we model a noise with such a process,
its supremum may
be used to test the null hypothesis. Moreover,
this supremum is the limiting distribution of an optimal test statistic for the uniformity of the
distribution on a circle, \cite{Darling}; in Section~\ref{BB}, we show that this process is strongly related to one
generated by $B_t$ by randomly choosing the starting point on $[0,1]$.
To generalize this example, we propose an approach which is linked to
Karhunen-Lo\`eve's expansion that gives uncorrelated coefficient.
This expansion is optimal in regression functional studies, as shown in \cite{Cohen}.
In fact, periodic processes are easily decomposed
with Fourier basis, and stationarity will cause the coefficients of the $\sin$ and $\cos$
of the same frequency to be equal. Thus, we underline a natural isometry between this
representation of Gaussian processes and $\ell^2$.
The asymptotic decay of the Karhunen-Lo\`eve's coefficients will be related to
the regularity of the paths, as a consequence of Fourier analysis.
Thus, it will be possible to define a
``periodic fractional Brownian motion'' by choosing
an appropriate asymptotic decay (see also \cite{Adler,Flandrin}) of the coefficients.
For what concerns notations, $s,t,\ldots$ relates to time variables, and will often belong
to $[0,1]$. We denote by $\{x_t\}_{t\in[0,1]},
\{y_t\}_{t\in[0,1]}, \ldots$ stochastic adapted process
defined on a given filtered space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\in[0,1]}, \mathbb{P})$,
while $ (X_n)_n,(Y_n)_n,(Z_n)_n,\ldots$ are sequences of random variables. $C(s,t)$ is a
positive semidefinite function (it will be the correlation function of a stochastic process).
When a process is stationary, its covariance function will often be replaced by
the associated covariogram function $\tilde{C}(t)$. The sequence $(e_k(t))_{k\in\mathbb{N}}$
denotes a sequence of orthogonal function on $L^2([0,1])$.
Finally, we denote by $\fracPart{t}$ the fractional part sawtooth function of the real number $t$,
which is defined by the formula
$\fracPart{t} = t - \operatorname{floor}(t)$.
\section{A canonical form for Gaussian periodical processes}
The first example of signal theory usage in the description of stochastic processes can be found in
\cite{Karhunen}, where is exposed a theorem that allows to represent Gaussian processes as limits of stochastic Fourier series. The classical
statement of Karhunen-Loève's theorem is the following, as described in \cite{ash}.
\begin{teo}[Karhunen-Loève]\label{teo:KL}
Let $\{x_t\}_{t\in [a,b]} $, $a,b<\infty$ , such that
$
E[x_t] = 0,$ $\forall t\in [a,b],
$
and
$
Cov(x_t,x_s) = C(t,s),
$
continuous in both variables. Then
$$
x_t = \sum_{k=1}^{\infty}Z_ke_k(t),\qquad a\leq t\leq b,
$$
where $e_k$ are the eigenfunction of following integral operator from $L^2[a,b]$ in itself
$$
f\in L^{2}[a,b]\to g(t) = \int_{a}^{b}{C(t,\tau)f(\tau)d\tau},\,\,\,\,\,\,a\leq t\leq b,
$$
and $e_k$ form an orthonormal bases for the space spanned by eigenfunctions corresponding to nonzero eigenvalues. The $Z_k$ are given by
$$
Z_k = \int_{a}^{b}x_te_k(t)dt
$$
and are orthogonal random variables ($ E(Z_kZ_j) = 0$ for $k\neq j$), with zero mean and variance $\lambda_k^2$, where $\lambda_k^2$ is
the eigenvalue corresponding to $e_k$.
The series $ \sum_{k=1}^{\infty}Z_ke_k(t) $ converges in mean square to $x_t$, uniformly in $t$,
that is
$$
E\Big((x_t - \sum_{k=1}^{\infty}Z_ke_k(t))^2\Big) \mathop{\longrightarrow}_{n\to\infty}0
$$
uniformly for $t\in[a,b]$.
Moreover if $x_t$ is Gaussian, the $Z_k$ in expansion are real independent Gaussian random variables.
\end{teo}
In this paper we provide a result that allows to build Gaussian processes of arbitrary regularity, proceeding on
the way tracked by Adler (see \cite{Adler}).
The result will be based on Karhunen-Loève's decomposition theorem, and will deal with following set of processes.
\begin{defi}
$ \H$ is the set of real Gaussian stochastic processes $ \{x_t\}_{t\in[0,1]}$ such that they are continuously stationary (so if $ C(s,t) $ is the covariance
function then there exist a real continuous function $ \tilde{C}(s-t) = C(s,t) \,\, \forall s,t,\,\R$), periodical (i.e. $ x_{0} = x_{1} , a.s.$) with
$E(x_t) = 0,\,\forall t\in\R $.
\end{defi}
The set $ \H$ is a Banach space, when it is equipped with the inner product given by
$$
(\{x_t\}_{t\in[0,1]},\{y_t\}_{t\in[0,1]}) = \int_{0}^{1}{E(x_ty_t)dt}\in\R_+.
$$
\begin{oss}\label{rem:1.1}
We remark that if $ \{x_t\}_{t\in[0,1]}\in\H $ and if $ \tilde{C}(s-t) = C(s,t) $ is its covariogram function, then $ \tilde{C}(t_0) = \tilde{C}(t_0+1)$.
\end{oss}
We are going to specialize Karhunen-Loève's decomposition theorem to $\H$, showing that a process is in $\H$ if and only if it can be written as
limit of a canonical trigonometric random series. From this result we will show that $\H$
may be seen as a Hilbert space, isometrically equivalent to the space of the coefficients $ \ell^2 $;
via this isometry it will be easy to create Gaussian stationary processes with arbitrary regularity,
by looking at decreasing speed rate of canonical series
coefficients. Moreover, this result allows also to detect information about regularity of a process, since it relates it with the regularity of its covariogram function.
\begin{teo}\label{teo:2.1}
Let $ \{x_t\}_{t\in[0,1]} \in \H $ with covariance $C(s,t) = \tilde{C}(t-s)$; then in mean square, uniformly in $t$,
$$
x_t = c_0 Y'_0 + \sum_{k=1}^{\infty}c_k(Y_k\sqrt{2}\sin(2k\pi t) + Y'_k\sqrt{2}\cos(2k\pi t))
$$
where
$
(Y_n)_n,(Y'_n)_n
$ are two independent sequence of independent standard Gaussian variables,
and
$
(c_k)_k\in \ell^2
$
is such that
$$
c_n^2 = \int_{0}^{1}{\tilde{C}(s)\cos(2n\pi s)ds},\qquad n=0,1,2,\ldots
$$
\end{teo}
\proof
By Mercer Theorem (see, e.g., \cite{ash}) we know that if $ (e_n)_n $ is an orthonormal bases for the space spanned by the
eigenfunctions corresponding to nonzero eigenvalues of integral operator
$$
x\in L^{2}[0,1]\to y(t) = \int_{0}^{1}{C(t,\tau)x(\tau)d\tau},\qquad a\leq t\leq b,
$$
then, uniformly, absolutely and in $ L^2[0,1]\times [0,1] $,
$
C(s,t) = \sum_{k=0}^{\infty}{e_k(t)e_k(s)\lambda_k}
$ ,
where $\lambda_k$ is the eigenvalue corresponding to $e_k$.
By Remark~\ref{rem:1.1} we are going to
see that $ (\cos(2n\pi s),\sin(2n\pi s))_n $ are
eigenfunctions relative to operator whose kernel is $ C(s,t) $.
In fact, let
$
a_n= \int_{0}^{1}{\tilde{C}(s)\cos(2n\pi s)ds}
$,
then
$$
\int_{0}^{1}{\cos(2n\pi t)\tilde{C}(t-\tau)dt} = a_n\cos(2n\pi \tau),
$$
the same relation holding when $\cos$ is replaced by $\sin$.
It follows from Mercer Theorem that
$$
C(s,t)=a_0+ \sum_{k=1}^{\infty}{2a_k\cos(2k\pi(s- t))}
$$
uniformly, absolutely and in $ L^2[0,1]\times [0,1] $ and that $ (a_n)_n\in\ell^1 $,
and hence the sequence $ (c_n)_n $ formed by $ c_n = \sqrt{|a_n|} $ lays in $ \ell^2 $.
From Theorem~\ref{teo:KL} and Theorem~\ref{teo:2.1} we deduce the existence
of two independent sequence of independent standard Gaussian variables
$
(Y_n)_n,(Y'_n)_n
$
such that
in mean square, uniformly in $ t $
$$
x_t = c_0Y'_0+\sum_{k=1}^{\infty}c_k(Y_k\d\sin(2k\pi t) + Y'_k\d\cos(2k\pi t)).
$$
\endproof
\begin{teo}\label{teo:2.2}
Let $
(Y_n)_n,(Y'_n)_n
$ be two independent sequence of independent standard Gaussian variables,
and $(c_k)_k\in \ell^2 $.
Then the sequence
$$
y^{(n)}_t = c_0Y'_0+\sum_{k=1}^{n}c_k(Y_k\d\sin(2k\pi t) + Y'_k\d\cos(2k\pi t))
$$
converges in mean square, uniformly in $t$ to $\{y_t\}_{t\in[0,1]}\in\H$.
Moreover if $C(s,t)$ is the $y_t$ covariance function,
then uniformly, absolutely and in $ L^2[0,1]\times [0,1] $,
$$
C(s,t) = c_0^2+\sum_{k=1}^{\infty}{2c_k^2\cos(2k\pi(s- t))}.
$$
\end{teo}
\proof
First of all we notice that Gaussian process $ y^{(n)}_t $ converges to a periodical
$ \{y_t\}_{t\in[0,1]} $ in mean square uniformly in $t$, because
$$
\sup_{t\in[0,1]}E[| y^{(n)}_t - y^{(m)}_t |^{2}] =2 \sum_{k=n}^{m}c_k^2\underbrace{\to}_{m,n} 0.
$$
Let's look at $ \{y_t\}_{t\in[0,1]} $ properties: it is a calculation to show that $ E[y_t] = 0 $ for all $ t $, and
that
$$
Cov(y_t,y_s) = c_0^2+2\sum_{k=1}^{\infty}{c_k^2\cos(2k\pi (s-t))},
$$
which is a continuous function, and that
$$
E[y_t^2] = c_0^2+2\sum_{k=1}^{\infty}c_k^2 = 2\|(c_n)_n\|^2-c_0^2.
$$
Moreover $\{y_t\}_{t\in[0,1]} $ is a Gaussian process, because the
two sequences $(Y_n)_n$ and$,(Y'_n)_n$ are Gaussians.
\endproof
\begin{cor}\label{cor:isometry}
Let us consider a couple $ Z=((\bar{Y}_n)_n,(\bar{Y'}_n)_n) $ of independent sequence of
independent standard Gaussian variables. For each $ \{z_t\}_{t\in[0,1]}\in \H $,
there exists an $ \{x_t\}_{t\in[0,1]}\in \H_{Z} $ having the same law, where
\begin{multline*}
\H_{Z} = \Big\{ \{x_t\}_{t\in[0,1]} \in \H : \\
x_t =a_0\bar{Y'}_0+\d\sum_{k=1}^{\infty}a_k(\bar{Y}_k\sin(2k\pi t) + \bar{Y'}_k\cos(2k\pi t))
, (a_n)_n\in \ell^2 \Big\}
\end{multline*}
and the limit is in mean square and uniformly in $t$.
\end{cor}
\subsubsection*{Geometry of $ \H_{Z} $: isometry with $\ell^2$}
For $(a_n)_n\in\ell^2$, define
$$
x_t =a_0\bar{Y'}_0 + \d\sum_{k=1}^{\infty}a_k(\bar{Y}_k\sin(2k\pi t) + \bar{Y'}_k\cos(2k\pi t)).
$$
From Theorem~\ref{teo:2.1} and Theorem~\ref{teo:2.2} it follows that
$\|x_t\|_{\H} = \sqrt{a_0^2+2\sum_n a_n^2}$, and hence it is naturally defined an isometry
between the representative space $\H_Z$ and $\ell^2$.
\subsubsection*{Regularity of the paths}
We have seen that to each stochastic process in $\H$
can be associated a sequence in $ \ell^2 $. We are now showing how are related the decrease rate of such sequence
with the regularity of the process trajectory path. We first recall a classic regularity theorem.
\begin{teo}[see \cite{Revuz}]\label{teo:2.3}
Let $ \{x_t\}_{t\in[0,1]} $ a real stochastic process such that there exist three positive constants $ \gamma $, $c$ and $ \epsilon $ so that
$$
E\Big(|x_t-x_s|^{\gamma}\Big) \leq c|t-s|^{1+\epsilon};
$$
so there exists a modification $ \{\tilde{x}_{t}\}_{t\in[0,1]} $ of $\{x_t\}_{t\in[0,1]}$, such that
$$
E((\sup_{s\neq t}{\frac{|\tilde{x}_t-\tilde{x}_s|}{|t-s|^{\alpha}}})^{\gamma})<\infty
$$
for all $ \alpha\in [0,\frac{\epsilon}{\gamma}) $; in particular the trajectories of $ \{\tilde{x}_{t}\}_{t\in[0,1]} $
are Holder continuous of order $ \alpha $.
\end{teo}
It is simple to apply this last theorem to processes staying in $ \H $.
It is well known that if $ Y\approx N(0,\sigma^2) $, then
$
E(|Y|^p) = \sigma^p \frac{2^{\frac{p}{2}}\Gamma\big(\frac{p+1}{2}\big)}{\sqrt{\pi}} ,
$ (see, e.g., \cite{normale}).
From this fact we deduce the following result.
\begin{teo}\label{teo:2.4}
Assume that $ \{x_t\}_{t\in [0,1]}\in\H $ and
let $C(s,t) = \tilde{C}(s-t)$ be its covariance function. If
$\tilde{C}$ is Holder continuous of order $ \alpha $, then
almost all trajectories of $ \{x_t\}_{t\in[0,1]} $ are Holder continuous of order $ \beta < \frac{\alpha}{2} $.
\end{teo}
\proof
Since
$$
E(|x_{t+h} - x_t|^2) = E(x_t^2 + x_{t+h}^2 - 2x_{t+h}x_{t}) = 2(\tilde{C}(0) - \tilde{C}(h)) \leq M|h|^{\alpha},
$$
we deduce that
$$
E(|x_{t+h} - x_t|^{2p}) = 2^pC_p(\tilde{C}(0) - \tilde{C}(h))^p \leq \tilde{M}|h|^{p\alpha}
$$
so, by Theorem~\ref{teo:2.3}, if $ p\alpha = 1+\epsilon $, for all $p>\frac{1}{\alpha}$, almost all trajectories of
$ \{x_t\}_{t\in[0,1]} $ are Holder continuous of order $\beta $, with $\beta < \frac{p\alpha-1}{2p}$;
but this is true for each $ p > \frac{1}{\alpha} $, we conclude that almost all trajectory path of $ \{x_t\}_{t\in[0,1]} $ are Holder continuous
of order $\beta < \frac{\alpha}{2}$.
\endproof
A very useful result for our analysis will be the following one, whose proof may be found in
\cite{Boas}.
\begin{teo}[Boas' Theorem]
Let $f\in L^1[0,1]$ be a function whose Fourier expansion has only nonnegative cosine terms,
and let $ (a_n)_n $ be the sequence of its cosine coefficient. Then
$$
f\text{is Holder continuous of order }\alpha \Longleftrightarrow a_k =
O\Big(\frac{1}{k^{\alpha+1}}\Big).
$$
\end{teo}
Boas' Theorem may be used in connection with
Theorem~\ref{teo:2.1} and Theorem~\ref{teo:2.2} to deduce
more regularity properties of the processes in $ \H $.
In fact, take $(c_n)_n$ as in Theorem~\ref{teo:2.1} and Theorem~\ref{teo:2.2}.
From Boas' Theorem we have that
if
$
k^2c_k^2 = O(\frac{1}{k^{1+\alpha}})
$
for $ 0<\alpha\leq 1 $, then $ \partial^2 \tilde{C} $ is Holder continuous of order $ \alpha $.
This link between the regularity of $\tilde{C}$ and the paths of $ \{x_t\}_{t\in[0,1]} $
is underlined in the following theorem.
\begin{teo}
With the notations of Theorem~\ref{teo:2.2},
if $c_k^2 = O(\frac{1}{k^{3+\alpha}})$, then
almost all trajectories of $ \{x_t\}_{t\in[0,1]} $ are Lipshitz continuous, and, as function of $t$, $ \{x_t\}_{t\in[0,1]} $ have a continuous derivative
$ \{x'_t\}_{t\in[0,1]}$ Holder continuous of order $ \beta < \frac{\alpha}{2} $.
\end{teo}
\proof
It is clear that
$$
\partial^2 \tilde{C}(\delta) = 2\partial^2 \sum_{k=1}^{\infty}{c^2_k\cos(2k\pi(\delta))} = -2\sum_{k=1}^{\infty}{(2\pi)^2k^2c^2_k\cos(2k\pi(\delta))}
$$
and that $ \partial^2 \tilde{C} $ is Holder continuous of
order $ \alpha $, for some $ 0<\alpha\leq1 $.
Moreover we have that uniformly in $t$ and in mean square
$$
x_t = c_0Y'_0 + \d\sum_{k=1}^{\infty}c_k(Y_k\sin(2k\pi t) + Y'_k\cos(2k\pi t)).
$$
and, from Theorem~\ref{teo:2.1}, there also exist a stochastic process in $ \H $ such that uniformly in $t$ and in mean square
$$
\tilde{x}_t = 2\d\pi\sum_{k=1}^{\infty}kc_k(Y_k\cos(2k\pi t) - Y'_k\sin(2k\pi t)),
$$
which has covariogram function Holder continuous of order $ \alpha $ given by
$$
\tilde{\bar C}(\delta) = 2\sum_{k=1}^{\infty}{(2\pi)^2k^2c^2_k\cos(2k\pi(\delta))}.
$$
If we define
$$
\begin{aligned}
y^{(n)}_t &:= c_0Y'_0 + \d\sum_{k=1}^{n}c_k(Y_k\sin(2k\pi t) + Y'_k\cos(2k\pi t))
\\
\tilde{y}^{(n)}(t) &:= 2\d\pi\sum_{k=1}^{n}kc_k(Y_k\cos(2k\pi t) - Y'_k\sin(2k\pi t)),
\end{aligned}
$$
than $
y^{(n)}_t = {y}^{(n)}_0 + \int_{0}^{t}\tilde{y}^{(n)}_{\tau}d\tau
$,
a.s. for any $n$, while
for each fixed $t$, in mean square we have
$
\int_{0}^{t}\tilde{y}^{(n)}_{\tau}d\tau \to \int_{0}^{t}\tilde{x}_{\tau}d\tau.
$
Since
\begin{multline*}
\sqrt{E\Big((x_t - x_0 - \int_{0}^{t}\tilde{x}_{\tau}d\tau)^2\Big)}
\leq \sqrt{E\Big((x_t - y^{(n)}_t)^2\Big)}
\\
+ \sqrt{E\Big((y^{(n)}_0
+ \int_{0}^{t}\tilde{y}^{(n)}_{\tau}d\tau- x_0 - \int_{0}^{t}\tilde{x}_{\tau}d\tau)^2\Big)}
\mathop{\longrightarrow}_{n\to\infty} 0,
\end{multline*}
it follows that a.s.
$
x_t = x_0 + \int_{0}^{t}\tilde{x}_{\tau}d\tau .
$
By Theorem~\ref{teo:2.4} we know that almost all trajectory path of $ \tilde{x}_t$ are Holder continuous of order $ \beta < \frac{\alpha}{2} $,
and thesis follows.
\endproof
A natural generalization of this result is the following:
\begin{cor}
If, in previous notation, $c_k^2 = O(\frac{1}{k^{1+2m+\alpha}})$
then almost all trajectory path of $ \{\partial^kx_t\}_{t\in[0,1]} $, with $ k<m $, are Lipschitz continuous, and admit, as function of $t$,
a continuous derivative $ \{\partial^mx_t\}_{t\in[0,1]}$ Holder continuous of order $\beta $, for all $ \beta < \frac{\alpha}{2} $.
\end{cor}
\subsection{The centered Brownian bridge}\label{BB}
Take a Brownian bridge $ \{x_t\}_{t\in[0,1]} $.
This process is Gaussian, periodic but not stationary, since $x_0=x_1 \equiv 0$.
If we randomize the starting point of the process, by shifting the $t$-axis of
a $[0,1]$-uniform random variable $U$, we obtain the process
\begin{equation}\label{eq:def_sBB}
\hat{x}_t = x_{\fracPart{t-U}}
\end{equation}
which is expected to belong to $\H$.
A process with the law of $\{\hat{x}_t\}_{t\in[0,1]}$ is called \emph{centered Brownian
bridge}.
Let us recall that
a Brownian bridge may also be represented as
\[
x_t = \d\sum_{k=1}^\infty W_k \frac{\sin(k \pi t)}{k \pi}.
\]
One may expect that the periodic process
\[
y_t = aY_0 +\d\sum_{k=1}^\infty \frac{b}{k \pi}(Y_k\sin(2k\pi t) + Y'_k\cos(2k\pi t)),
\]
which shares the same asymptotic behavior of the coefficients of $\{x_t\}_{t\in[0,1]}$ is
closely related to $\{\hat{x}_t\}_{t\in[0,1]}$. The next theorem shows this facts.
\begin{teo}
A centered Brownian bridge $\{\hat{x}_t\}_{t\in[0,1]}$ given in \eqref{eq:def_sBB}
belongs to $\H$,
with covariogram function $\tilde{C}(\delta) = \frac{(|\delta|-1/2)^2}{2} + \frac{1}{24}$.
It may be represented as
\[
\hat{x}_t = \frac{1}{\sqrt{12}}
Y_0' +\d\sum_{k=1}^\infty \frac{1}{2k \pi}(Y_k\sin(2k\pi t) + Y'_k\cos(2k\pi t)).
\]
\end{teo}
\begin{proof}
By conditioning on $U$, it is simple to prove that $E(\hat{x}_t)=0$ and
$E(\hat{x}_s\hat{x}_t) = \frac{(|t-s|-1/2)^2}{2} + \frac{1}{24} = \tilde{C}(|t-s|) $.
A straightforward calculation gives
\[
c^2_0 = \int_0^1 \tilde{C}(s) ds = \frac{1}{12},
\qquad c^2_n = \int_0^1 \tilde{C}(s)\cos(2n\pi s) ds = \frac {1}{(2\pi n)^2}, n\geq 1,
\]
and hence the Karhunen-Loève theorem gives
\[
\hat{x}_t = \frac{1}{\sqrt{12}}
Y_0' +\sum_{k=1}^\infty \frac{1}{2k \pi}(Y_k\d\sin(2k\pi t) + Y'_k\d\cos(2k\pi t)).
\]
What remains to prove is that all the $(Y_k,Y'_k)$s are Gaussian. To sketch the proof for
$Y_k$ (the same arguments apply to $Y'_k$),
we first recall that, conditioned on $U=s$,
\begin{align*}
Z_k := \frac{Y_k}{2\pi k} &= \int_0^1 \hat{x}_t \d\sin(2 k\pi t) dt =
\int_0^1 \d\sum_{n=1}^\infty W_n \frac{\sin(n \pi \fracPart{t-s})}{n \pi}
\d\sin(2 k\pi t) dt \\
& = \cos(2k\pi s) \frac{W_{2k}}{2k\pi} + \frac{2 \sin(2k\pi s) }{\pi^2}
\sum_{n=1,n\neq 2k}^\infty W_n
\frac{(-1)^n-1}{4k^2-n^2}.
\end{align*}
Therefore, the characteristic function $\Phi_{s,Z_k}(t)$ of $Z_k$ conditioned on $U=s$ is
\[
\Phi_{s,Z_k}(t) = e^{-\frac{t^2}{2} \frac{1}{4k^2\pi^2}\cos^2(2k\pi s) }
e^{-\frac{t^2}{2} \frac{4 \sin^2(2k\pi s) }{\pi^4}
\sum_{n=1,n\neq 2k}^\infty
\big(\frac{(-1)^n-1}{4k^2-n^2}\big)^2}
\]
and hence the characteristic function $\Phi_{Z_k}(t)$ of $Z_k$ is
\[
\Phi_{Z_k}(t) = \int_0^1
e^{-\frac{t^2}{2} \frac{1}{4k^2\pi^2}\cos^2(2k\pi s) }
e^{-\frac{t^2}{2} \frac{4 \sin^2(2k\pi s) }{\pi^4}
\sum_{n=1,n\neq 2k}^\infty
\big(\frac{(-1)^n-1}{4k^2-n^2}\big)^2}
ds.
\]
Now, since
\[
\frac{4 }{\pi^4}
\sum_{n=1,n\neq 2k}^\infty
\Big(\frac{(-1)^n-1}{4k^2-n^2}\Big)^2
=
\frac{1}{k^2\pi^4}\sum_{m=0}^\infty \Big(
\frac{1}{2k+2m+1}+\frac{1}{2k-2m-1}\Big)^2
= \frac{1}{4k^2\pi^2}
\]
we get
\(
\Phi_{Z_k}(t) =
e^{-\frac{t^2}{2} \frac{1}{4k^2\pi^2}}
\), which concludes the proof.
\end{proof}
The word ``centered'' in the definition of $\{\hat{x}_t\}_{t\in[0,1]}$ is clearly related
to the randomization of the starting point of the underlying Brownian bridge.
In fact, we can say more: $\{\hat{x}_t\}_{t\in[0,1]}$ is strongly related to the
$y$-centralization of the Brownian bridge, i.e. to the following process
\begin{equation}\label{eq:def_CCBB}
\check{x}_t = x_t - \int_0^1 x_t \,dt,
\end{equation}
where $\{x_t\}_{t\in[0,1]}$ is a Brownian bridge.
This last process stays in $\H$,
and his sup is the limiting distribution of an optimal
test statistic for the uniformity of the distribution on a circle, see \cite{watson,Darling}.
\begin{cor}
If $\{\hat{x}_t\}_{t\in[0,1]}$ is a centered Brownian Bridge, then it holds
$$
\hat{x}_t = \check{x}_t + Z
$$
where $Z$ is an independent random variable with null expectation and variance $ \frac{1}{12} $, and $\{ \check{x}_t\}_{t\in[0,1]} $ is defined as in \eqref{eq:def_CCBB}. Furthermore
\begin{equation}\label{proc_start}
\check{x}_t = \d\sum_{k=1}^\infty \frac{1}{2k \pi}(Y_k\sin(2k\pi t) + Y'_k\cos(2k\pi t)).
\end{equation}
\end{cor}
\begin{proof}
The covariogram function of $\{\check{x}_t \}_{t\in[0,1]}$ is
$\tilde{C}(\delta) = \frac{(|\delta|-1/2)^2}{2} - \frac{1}{24}$ (see \cite{Darling}).
It is sufficient to calculate the covariance function of $ \check{x}_t + Z $
and to use the Corollary~\ref{cor:isometry} to complete the proof.
\end{proof}
\subsection{A computational parametric model for smoothing}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=13cm, height=10cm]{ConfrontoLisce}
\caption{A comparison between trajectories of process $x_t$ referred to same event and varying the value of $p$ in $\{1,1.6,2.1,3.1\}$}\label{fig:1}
\end{center}
\end{figure}
Results provided in this paper allows to create a Gaussian parametric family of stationary
and periodic processes of arbitrary
regularity. In fact, let us consider the following family of processes in $\H$ that extends \eqref{proc_start}
\begin{equation}\label{model}
x_t = \sum_{k=1}^{\infty}\frac{a}{k^{p}}(Y_k\sin(2k\pi t) + Y'_k\cos(2k\pi t)).
\end{equation}
Theorem~\ref{teo:2.4} states that
the paths become more regular as $p$ increases.
This property is shown in Figure~\ref{fig:1}, which
suggests how to smooth a process by changing $p$.
Summing up, model \eqref{model} gives a family of Gaussian processes whose
trajectories are arbitrarily regular.
In application, maximum likelihood estimates of $a$ and $p$ is a straightforward consequence of a FFT of the
observed discretized process $\{x_t\}_{t\in[0,1]}$.
| {
"timestamp": "2012-02-29T02:02:43",
"yymm": "1202",
"arxiv_id": "1202.6182",
"language": "en",
"url": "https://arxiv.org/abs/1202.6182",
"abstract": "This article provides a representation theorem for a set of Gaussian processes; this theorem allows to build Gaussian processes with arbitrary regularity and to write them as limit of random trigonometric series. We show via Karhunen-Love theorem that this set is isometrically equivalent to l2. We then prove that regularity of trajectory path of anyone of such processes can be detected just by looking at decrease rate of l2 sequence associated to him via isometry.",
"subjects": "Probability (math.PR)",
"title": "A canonical form for Gaussian periodic processes",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808759252645,
"lm_q2_score": 0.7217431943271999,
"lm_q1q2_score": 0.7075110507281659
} |
https://arxiv.org/abs/1711.05169 | On the Complex Cayley Grassmannian | We define a torus action on the (complex) Cayley Grassmannian $X$. Using this action, we prove that $X$ is a singular variety. We also show that the singular locus is smooth and has the same cohomology ring as that of $\mathbb{CP}^5$. Furthermore, we identify the singular locus with a quotient of $G_2^\mathbb{C}$ by a parabolic subgroup. | \section{Introduction}
On octonions $\mathbb{O}\cong\mathbb{C}^8$, one can define a 4-fold cross product which we denote by \linebreak $(\cdot\times\cdot\times\cdot\times\cdot)\in\Lambda^4\mathbb{O}^*$ \cite{HL82}.
A 4-plane $\xi= \langle u,v,w,z\rangle$ is called a Cayley plane if it satisfies $\text{Im}(u\times v\times w\times z)=0$.
We call the space of all Cayley planes the Cayley Grassmannian denoted by $X$.
It is well known that using octonions over the real numbers with a positive definite metric, the Cayley Grassmannian is isomorphic to the ordinary Grassmannian of 3-planes in 7-space, Gr$(3,7)$.
However, the complex analogue of this space is not well studied.
Using the Pl\"ucker relations and the above description of Cayley planes one can show that $X$ is a 12-dimensional subvariety of the complex 4-planes in 8-space, Gr$(4,\mathbb{O})$, see Section~\ref{sec:charts}.
The group Spin$(7,\mathbb{C})$ admits an action on $X$ and by restriction to a maximal torus, we get a $(\mathbb{C}^*)^3$ action on $X$.
It turns out this action has finitely many fixed points, see Theorem~\ref{thm:fixpnts}.
Furthermore, some of the fixed points are singular showing that $X$ is a singular variety, see Theorem~\ref{thm:fxdpnts}.
The singular locus $\Sigma=Sing(X)$ is, however, a smooth projective variety.
Indeed, it can be identified as $G_2^\mathbb{C}/P_2$, see Theorem~\ref{thm:parabolicquotient}.
Here, $P_2$ is a parabolic subgroup of $G_2^\mathbb{C}$ corresponding to a long root of $G_2^\mathbb{C}$.
The structure of this paper mainly follows that of \cite{AC15}. Whereas we investigate the Cayley Grassmannian, \cite{AC15} investigates associative Grassmannian. The main difficulty in our case is that the Cayley Grassmannian is not a smooth variety.
In Section \ref{sec:oct}, we give an overview of composition algebras. Then, we write down a concrete description of octonions $\mathbb{O}$ and define various cross product operations using octonionic multiplication.
In Section \ref{sec:cayley}, we define associative and Cayley calibration forms. Then, we express the Cayley calibration and the imaginary part of a four-fold cross product in coordinates.
In Section \ref{sec:charts}, we describe charts of Gr(4,8) when viewed as a subvariety of $\mathbb{P}(\Lambda^4\mathbb{C}^8)$. We also give seven linear equations on $\mathbb{P}(\Lambda^4\mathbb{C}^8)$ that the Cayley Grassmannian satisfy. Then, we localize these equations to a chart, using Pl\"ucker relations.
In Section \ref{sec:spinsevenandthreesls}, we give a definition of Spin(7,$\mathbb{C}$), describe three of its subgroups each of which is isomorphic to SL(2,$\mathbb{C}$) and also describe a maximal torus.
In Section \ref{sec:torusfxd}, we diagonalize the maximal torus and list the eigenvalues and corresponding eigenvectors of the induced action on $\Lambda^4\mathbb{C}$. Then, we find which eigenvectors lie in $X$. We also prove that the fixed point set of the torus action is finite.
In section \ref{sec:mincompt}, we show that all but six of the torus fixed points are regular, proving $X$ is singular.
In section \ref{sec:singloc}, we prove that the singular locus is smooth and has the same cohomology ring as that of $\mathbb{C}\mathbb{P}^5$. Furthermore, we express the singular locus as a quotient of $G_2^{\mathbb{C}}$ by a parabolic subgroup.
In section \ref{sec:toruslocalsmooth}, we describe the torus action near regular fixed points using a one parameter subgroup of the maximal torus.
\subsection*{\bf Acknowledgement.} I would like to thank Mahir Bilen Can for many inspirational ideas and discussions that not only improved this work but also me as a mathematician.
\section{Octonions and multiple cross products}\label{sec:oct}
In this section, we recall the definition of octonions and multiple cross products defined using octonionic multiplication. Although octonions are not necessary to define these cross products, they give an alternative way to verify their properties using octonions. Further details can be found in \cite{SV13,HL82,AC15,Bae02,SW10}.
\begin{definition}[\cite{SV13}]
A composition algebra $C$ over a field $k$ is an algebra over $k$ with identity element and a nondegenerate quadratic form $N$ such that
\begin{equation*}
N(uv)=N(u)N(v)
\end{equation*}
for $u,v\in C$.
The quadratic form $N$ is often referred to as the norm on $C$, and the associated bilinear form $B(\cdot,\cdot)$ is called the inner product.
\end{definition}
In this paper, we take the base field $k$ to be $\mathbb{C}$.
However, it is not necessary for some parts of the discussion.
A four-dimensional composition algebra is called a quaternion algebra.
Let $\mathbb{H}$ be a copy of $\mathbb{C}^4$ generated by $1,\textbf{i}, \textbf{j}$ and $\textbf{k}$ with the relations $\textbf{i}^2=\textbf{j}^2=\textbf{k}^2=\textbf{i}\textbf{j}\textbf{k}= -1$, and $N$ be the quadratic form given by
\begin{equation*}
N(a1+b\textbf{i}+c\textbf{j}+d\textbf{k})=a^2+b^2+c^2+d^2.
\end{equation*}
Then, $\mathbb{H}$ is a quaternion algebra over $\mathbb{C}$.
Following ~\cite{SV13}, we use Cayley-Dickinson doubling to construct an eight-dimensional composition algebra $\mathbb{O}$ (called octonion algebra) with the new generator $\textbf{l}$ whose norm is $1$, i.e., $N(\textbf{l})=1$.
We use the following figure to describe the multiplication table.
\begin{center}
\includegraphics{octMultTable}
\end{center}
For each (oriented) line from $x$ to $y$ to $z$, we have the relations
\begin{equation*}
xy=z, \qquad
yz=x, \qquad
zx=y, \qquad \text{and} \qquad
x^2=y^2=z^2=-1.
\end{equation*}
To ease notation later on, we set $e_0 = 1, e_1 = \textbf{i}, e_2 = \textbf{j},$ $e_3 = \textbf{k}, e_4=\textbf{l}, e_5=\textbf{li}, e_6=\textbf{lj}$ and $e_7=\textbf{lk}$.
We also set $e^{pqrs}=e^p\wedge e^q\wedge e^r\wedge e^s$ where $\left\{ e^p \right\}$ is the dual basis of $\{e_p\}$.
The octonions are non-associative but they are alternative, i.e., the subalgebra generated by any two elements is associative.
We denote the projection map from $\mathbb{O}$ to the span of $1$ by $\text{Re}$, and
projection to the orthogonal complement $1^{\perp}$ by $\text{Im}$.
This allows us to define an involution $$u\mapsto\conj u = \text{Re}(u)-\text{Im}(u).$$
The bilinear form $B$ associated to $N$ can be expressed as $B(u,v)=\text{Re}(\conj u v)$.
So $N(u)=B(u,u)=\text{Re}(\conj u u)$.
A key fact one can verify on the basis elements is
\begin{equation}
\conj {uv} = \conj v\, \conj u.
\label{eq:conjofproduct}
\end{equation}
Note that (\ref{eq:conjofproduct}) implies
\begin{equation*}
\conj{\conj u u} = \conj u u
\end{equation*}
that is $\conj u u\in\text{Re}(\mathbb{O})$.
Thus,
\begin{equation*}
N(uv)=\text{Re}(\conj{uv} uv ) =\text{Re}(\conj v(\conj u u) v) = \text{Re}(\conj u u) \text{Re}(\conj vv) = N(u)N(v)
\end{equation*}
proving directly that the above multiplication table defines an eight-dimensional composition algebra.
Over $\mathbb{C}$, there is only one such algebra, namely the octonions.
\begin{lemma}
For $u,v,v'\in\mathbb{O}$,
\begin{equation}
N(u)B(v,v') = B(uv,uv') = B(vu,v'u).
\end{equation}
In particular, for unit $u$, (left or right) multiplication by $u$ is an orthogonal transformation of $\mathbb{O}$.
\label{lem:orth}
\end{lemma}
\begin{proof}
Since $B(v,v')=\frac{1}{2}\left( N(v+v')-N(v)-N(v') \right)$, we have
\begin{eqnarray*}
N(u)B(v,v') &=& \frac{1}{2}\left( N(u)N(v+v')-N(u)N(v)-N(u)N(v') \right) \\
&=& \frac{1}{2}\left( N(uv+uv')-N(uv)-N(uv') \right) \\
&=& B(uv,uv').
\end{eqnarray*}
The second equality can be proved similarly.
\end{proof}
Next, we recall the definition of an $r$-fold cross product.
\begin{definition}
\label{defn:crsprd}
Let $(V,B)$ be a vector space with a (non-degenerate) symmetric bilinear form.
A multilinear map $L:V^r\to V$ is called an $r$-fold cross product if
\begin{equation}
N(L(v_1,\dots,v_n)) = N(v_1\wedge\dots\wedge v_n)
\label{eq:nfldpro1}
\end{equation}
with the induced norm on $\Lambda^nV$ and
\begin{equation}
B(L(v_1,\dots,v_n),v_i) = 0 \qquad \text{for all } i.
\label{eq:nfldpro2}
\end{equation}
\end{definition}
\begin{remark}
If $L$ is an alternating multilinear map, then it is enough to check (\ref{eq:nfldpro1}) on orthogonal vectors in which case (\ref{eq:nfldpro1}) becomes
\begin{equation}
N(L(v_1,\dots,v_n))=N(v_1)\dots N(v_n).
\label{eq:nfldpro3}
\end{equation}
\label{rm:altpro}
\end{remark}
\begin{remark}
The usual cross product operation on $\mathbb{R}^3$ is naturally a (two-fold) cross product according to this definition.
\end{remark}
In \cite{BG67}, Brown and Gray proved that an $r$-fold cross product exists on an $n$-dimensional vector space only in the following cases:
\begin{enumerate}
\item $n$ is even, $r=1$
\item $n$ is arbitrary, $r=n-1$
\item $n=3$ or 7, $r=2$
\item $n=4$ or 8, $r=3$.
\end{enumerate}
Below, we give concrete description of the ``exceptional'' two-fold and three-fold cross products using octonions.
Then, we introduce a ``four-fold cross product'' operation on $\mathbb{O}$.
Although it is not a cross product according to Definition \ref{defn:crsprd}, it is conventionally called so \cite{HL82,SW10}.
A two-fold cross product (or simply a cross product) can be defined as the restriction of octonionic multiplication to the imaginary part, $\text{Im}(\mathbb{O})$:
\begin{definition}
For $u,v\in\text{Im}(\mathbb{O})$, let
\begin{equation}
u\times v = \text{Im}(uv).
\end{equation}
\end{definition}
\begin{remark}
By restricting to $u,v\in\text{Im}(\mathbb{H})$, one gets the two-fold cross product on the three-dimensional space $\text{Im}(\mathbb{H})$.
\end{remark}
\begin{proposition}
The map $(u,v)\mapsto u\times v = \text{Im}(uv)$ is a two-fold cross product on $\text{Im}(\mathbb{O})$.
\end{proposition}
\begin{proof}
Since $\conj u u\in\text{Re}(\mathbb{O})$, $u\times u = \text{Im}(uu)=-\text{Im}(\conj uu)=0$.
So, $u\times v$ is an alternating map.
By Remark \ref{rm:altpro}, we may assume $u,v\in\text{Im}(\mathbb{O})$ are orthogonal, that is, $B(u,v)=0$.
Then, by Lemma \ref{lem:orth} we get
\begin{eqnarray*}
0 &=& N(u)B(u,v) \\
&=& -B(\conj uu, uv) \\
&=& -B(N(u), uv).
\end{eqnarray*}
Thus, $uv\in\text{Im}(\mathbb{O})$.
This gives us
\begin{eqnarray*}
N(u\times v) &=& N(\text{Im}(uv)) \\
&=& N(uv) \\
&=& N(u)N(v).
\end{eqnarray*}
To prove (\ref{eq:nfldpro2}), we once again use Lemma \ref{lem:orth}.
\begin{eqnarray*}
B(u\times v,u) &=& B(\text{Im}(uv),u) \\
&=& B(uv-\text{Re}(uv),u) \\
&=& B(uv,u)-B(\text{Re}(uv),u) \\
&=& N(u)B(v,1) \\
&=& 0
\end{eqnarray*}
since $u$ and $v$ are orthogonal to $1$.
\end{proof}
\begin{remark}
There are a number of ways to express the (two-fold) cross product.
It is easy to show that $\text{Im}(uv)=\text{Im}(\conj vu)=\text{Alt}(\conj vu)$ where $\text{Alt}(L)$ is alternation of the multilinear map $L:V^r\to V$. More precisely,
\begin{equation*}
\text{Alt}(L)(v_1,\dots,v_r) = \frac{1}{r!}\sum_{\sigma\in S_r} \text{sign}(\sigma)L(v_{\sigma(1)},\dots,v_{\sigma(r)})
\end{equation*}
where $S_r$ is the symmetric group on the set $\left\{ 1,\dots,r \right\}$.
\end{remark}
This cross product operation is of vital importance in $G_2$ geometry.
Next, following \cite{SW10} we define three-fold and four-fold cross product operations as follows:
\begin{definition}
For $u,v,w\in\mathbb{O}$, let
\begin{equation}
u\times v \times w = \frac{1}{2}\left( (u\conj v)w - (w\conj v)u \right).
\label{eq:tcrpr}
\end{equation}
\end{definition}
\begin{definition}
For $u,v,w,x\in\mathbb{O}$, let
\begin{equation}
x \times u\times v \times w = -\frac{1}{4}\left[(x \times u\times v)\conj w
- (w \times x\times u)\conj v
+ (v \times w\times x)\conj u
- (u \times v\times w)\conj x \right].
\label{eq:qcrpr}
\end{equation}
\end{definition}
\begin{remark}
The four-fold cross product operation (\ref{eq:qcrpr}) does not satisfy (\ref{eq:nfldpro2}).
However, it is alternating and satisfies (\ref{eq:nfldpro1}).
Hence, for orthogonal vectors $x,u,v,$ and $w$ we have
\begin{equation}
N(x \times u \times v \times w) = N(x)N(u)N(v)N(w).
\label{eq:qdcrprnorm}
\end{equation}
\end{remark}
\section{Cayley four-planes}\label{sec:cayley}
A $k$-form $\omega\in\Lambda^kV^*$ is called a calibration form if for every orthonormal set of vectors $\left\{ v_1,\dots,v_k \right\}$ we have $|\omega(v_1,\dots,v_n)|\le 1$.
Given a $k$-plane $\xi$ generated by an orthonormal basis $\left\{ v_1,\dots,v_k \right\}$, $\xi$ is called calibrated if $\omega(v_1,\dots,v_k)=\pm 1$.
Using the two-fold (resp. three-fold) cross product, we define a calibration three-form (resp. four-form) $\phi$ (resp. $\Phi$) called associative (resp. Cayley) calibration on $\text{Im}(\mathbb{O})$ (resp. $\mathbb{O}$) as follows:
\begin{definition}
For $u,v,w\in\text{Im}(\mathbb{O})$, let
\begin{equation}
\phi(u,v,w)=B(u, v\times w)
\label{eq:phi}
\end{equation}
and for $x,u,v,w\in\mathbb{O}$, let
\begin{equation}
\Phi(x,u,v,w)=B(x,u\times v\times w).
\label{eq:Phi}
\end{equation}
\end{definition}
For a proof of the following proposition see \cite{SW10}.
\begin{proposition} The equations (\ref{eq:phi}) and (\ref{eq:Phi}) define calibration forms and they satisfy
\begin{equation}
\phi(u,v,w)=\text{Re}(u\times v\times w)
\label{eq:phicrpr}
\end{equation}
and
\begin{equation}
\Phi(x,u,v,w)=\text{Re}(x\times u\times v\times w).
\label{eq:Phicrpr}
\end{equation}
\end{proposition}
By (\ref{eq:qdcrprnorm}) and (\ref{eq:Phicrpr}), it is clear that $\Phi(x,u,v,w)=\pm1$ if and only if $\Xi(x,u,v,w):=\text{Im}(x \times u \times v \times w)=0$ for orthonormal $x,u,v,$ and $w$.
\begin{definition}
A four-plane $\xi$ generated by $\left\{x,u,v,w\right\}$ is called a Cayley plane if \linebreak $\Xi(x,u,v,w)=0$.
The set of all Cayley planes is called the Cayley Grassmannian and denoted $X$.
We denote the set of all planes $\xi$ generated by orthonormal $\left\{ x,u,v,w \right\}$ satisfying $\Phi(x,u,v,w)=\pm 1$ by $X^0$.
\end{definition}
\begin{remark}
If a four-plane $\xi$ is generated by orthonormal vectors $\left\{ x,u,v,w \right\}$, then \linebreak $\Xi(x,u,v,w)=0$ is equivalent to $\Phi(x,u,v,w)=\pm 1$.
Therefore, over $\mathbb{R}$ with a positive definite metric, there is no distinction between the two conditions and $X^0=X$.
In fact, they show $X^0=Gr(3,7)$ which is a 12-dimensional variety in \cite{HL82}.
However, over $\mathbb{C}$, not every four-plane is generated by an orthonormal basis (with respect to $B$).
Therefore, we have $X^0\subsetX$ in general.
Furthermore, $X^0$ can be thought of as the set of generic points in $X$.
\end{remark}
It is helpful to express $\phi$, $\Phi$ and $\Xi$ in coordinates.
The associative calibration form is given by
\begin{equation*}
\phi = e^{123}-e^{145}-e^{167}-e^{246}+e^{257}-e^{347}-e^{356}
\end{equation*}
the Cayley calibration form is given by
\begin{eqnarray*}
\Phi&=& e^{0123}-e^{0145}-e^{0167}-e^{0246}+e^{0257}-e^{0347}-e^{0356}
\\ & & -e^{1247}-e^{1256}+e^{1346}-e^{1357}-e^{2345}-e^{2367}+e^{4567}
\end{eqnarray*}
and the imaginary part of the four-fold cross product is given by
\begin{eqnarray}
\Xi =
\nonumber
& \left( -e^{0247} -e^{0256} +e^{0346} -e^{0357} +e^{1246} -e^{1257} +e^{1347} +e^{1356}\right)e_{1}\\
\nonumber
& + \left( +e^{0147} +e^{0156} -e^{0345} -e^{0367} -e^{1245} -e^{1267} +e^{2347} +e^{2356}\right)e_{2}\\
\nonumber
& + \left( -e^{0146} +e^{0157} +e^{0245} +e^{0267} -e^{1345} -e^{1367} -e^{2346} +e^{2357}\right)e_{3}\\
\label{eq:xi}
& + \left( -e^{0127} +e^{0136} -e^{0235} +e^{0567} +e^{1234} -e^{1467} +e^{2457} -e^{3456}\right)e_{4}\\
\nonumber
& + \left( -e^{0126} -e^{0137} +e^{0234} -e^{0467} +e^{1235} -e^{1567} +e^{2456} +e^{3457}\right)e_{5}\\
\nonumber
& + \left( +e^{0125} -e^{0134} -e^{0237} +e^{0457} +e^{1236} -e^{1456} -e^{2567} +e^{3467}\right)e_{6}\\
\nonumber
& + \left( +e^{0124} +e^{0135} +e^{0236} -e^{0456} +e^{1237} -e^{1457} -e^{2467} -e^{3567}\right)e_{7}.
\nonumber
\end{eqnarray}
Using these expressions, one can immediately see that
\begin{equation}
\Phi = e^0\wedge \phi + *\phi
\label{eq:AssocAndCayleyCal}
\end{equation}
where $*$ is the Hodge star operator.
\section{Charts of Gr(4,\texorpdfstring{$\mathbb{O}$}{\bf O})}\label{sec:charts}
Set $p_{ijkl}=e^{ijkl}$ so they are coordinate functions on $\Lambda^4\mathbb{O}$.
We also view $p_{ijkl}$ as homogeneous coordinates on $\mathbb{P} (\Lambda^4\mathbb{O})$.
The Grassmannian of four-planes in eight dimensions Gr(4,$\mathbb{O}$) is the variety given by the Pl\"ucker relations (see \cite{KL72}):
\begin{equation}
p_{i_1i_2i_3j_1}p_{j_2j_3j_4j_5}= p_{i_1i_2i_3j_2}p_{j_1j_3j_4j_5}-p_{i_1i_2i_3j_3}p_{j_1j_2j_4j_5}+p_{i_1i_2i_3j_4}p_{j_1j_2j_3j_5}-p_{i_1i_2i_3j_5}p_{j_1j_2j_3j_4}
\label{eq:plucker}
\end{equation}
We consider the intersection of $\mt{Gr}(4,\mathbb{O})$ and the zero locus of $\Xi$.
By (\ref{eq:xi}), the zero locus of $\Xi$ is given by these seven linear equations:
\begin{eqnarray}
\label{eq:plucker1}
f_1:= -p_{0247} -p_{0256} +p_{0346} -p_{0357} +p_{1246} -p_{1257} +p_{1347} +p_{1356} &=& 0 \\
f_2:= +p_{0147} +p_{0156} -p_{0345} -p_{0367} -p_{1245} -p_{1267} +p_{2347} +p_{2356} &=& 0 \\
f_3:= -p_{0146} +p_{0157} +p_{0245} +p_{0267} -p_{1345} -p_{1367} -p_{2346} +p_{2357} &=& 0 \\
f_4:= -p_{0127} +p_{0136} -p_{0235} +p_{0567} +p_{1234} -p_{1467} +p_{2457} -p_{3456} &=& 0 \\
f_5:= -p_{0126} -p_{0137} +p_{0234} -p_{0467} +p_{1235} -p_{1567} +p_{2456} +p_{3457} &=& 0 \\
f_6:= +p_{0125} -p_{0134} -p_{0237} +p_{0457} +p_{1236} -p_{1456} -p_{2567} +p_{3467} &=& 0 \\
f_7:= +p_{0124} +p_{0135} +p_{0236} -p_{0456} +p_{1237} -p_{1457} -p_{2467} -p_{3567} &=& 0 .
\label{eq:plucker7}
\end{eqnarray}
This intersection is the Cayley Grassmannian.
Once we choose a chart $U_{stun}=\left\{ x\in\mathbb{P}(\Lambda^4\mathbb{O}) \;|\; p_{stun}(x)\neq 0 \right\}$, we use the following notation for local coordinates (suppressing the indices $s,t,u,n$).
\begin{equation*}
q_{ijkl} = \frac{p_{ijkl}}{p_{stun}}.
\end{equation*}
For example, over $U_{0123}$, using Pl\"ucker relations (\ref{eq:plucker}), we have
\begin{equation}
\frac{p_{4567}}{p_{0123}} = \frac{p_{0456}}{p_{0123}}\frac{p_{1237}}{p_{0123}} - \frac{p_{1456}}{p_{0123}}\frac{p_{0237}}{p_{0123}} + \frac{p_{2456}}{p_{0123}}\frac{p_{0137}}{p_{0123}}-\frac{p_{3456}}{p_{0123}}\frac{p_{0127}}{p_{0123}}
\label{eq:pluckerexample}
\end{equation}
or, more concisely,
\begin{equation}
q_{4567} = q_{0456}q_{1237}-q_{1456}q_{0237}+q_{2456}q_{0137}-q_{3456}q_{0127}.
\label{eq:pluckerexampleq}
\end{equation}
Fix a chart $U_{stun}$.
Then, one can show that any coordinate function can be expressed only in terms of the variables $q_{ijkl}$ with $|\{i,j,k,l\}\cap\{s,t,u,n\}| = 3$ by using (\ref{eq:plucker}) (repeatedly if necessary).
There are exactly 16 such variables corresponding to the fact that $\dim(\text{Gr}(4,\mathbb{O}))=16$ and they give us the charts of Gr(4,$\mathbb{O}$), see \cite{KL72,AC15}.
For example, on $U_{0123}$, the local (Grassmannian) variables are
$$\left\{ q_{0124},q_{0125},q_{0126},q_{0127},q_{0134},q_{0135},q_{0136},q_{0137},q_{0234},q_{0235},q_{0236},q_{0237},q_{1234},q_{1235},q_{1236},q_{1237} \right\}.$$
We localize the defining equations (\ref{eq:plucker1})-(\ref{eq:plucker7}) of Cayley planes to these coordinates.
\begin{eqnarray*}
f_1 &=& - q_{0124} q_{0237} + q_{0124} q_{1236} - q_{0125} q_{0236} - q_{0125} q_{1237} + q_{0126} q_{0235} - q_{0126} q_{1234} + q_{0127} q_{0234} \\
& & + q_{0127} q_{1235} + q_{0134} q_{0236} + q_{0134} q_{1237} - q_{0135} q_{0237} + q_{0135} q_{1236} - q_{0136} q_{0234} - q_{0136} q_{1235} \\
& & + q_{0137} q_{0235} - q_{0137} q_{1234} \\
f_2 &=& q_{0124} q_{0137} - q_{0124} q_{1235} + q_{0125} q_{0136} + q_{0125} q_{1234} - q_{0126} q_{0135} - q_{0126} q_{1237} - q_{0127} q_{0134} \\
& & + q_{0127} q_{1236} - q_{0134} q_{0235} + q_{0135} q_{0234} - q_{0136} q_{0237} + q_{0137} q_{0236} + q_{0234} q_{1237} + q_{0235} q_{1236} \\
& & - q_{0236} q_{1235} - q_{0237} q_{1234}\\
f_3 &=& - q_{0124} q_{0136} + q_{0124} q_{0235} + q_{0125} q_{0137} - q_{0125} q_{0234} + q_{0126} q_{0134} + q_{0126} q_{0237} - q_{0127} q_{0135} \\
& & - q_{0127} q_{0236} - q_{0134} q_{1235} + q_{0135} q_{1234} - q_{0136} q_{1237} + q_{0137} q_{1236} - q_{0234} q_{1236} + q_{0235} q_{1237} \\
& & + q_{0236} q_{1234} - q_{0237} q_{1235}
\end{eqnarray*}
\begin{eqnarray*}
f_4 &=& - q_{0124} q_{0136} q_{1237} + q_{0124} q_{0137} q_{1236} + q_{0124} q_{0235} q_{1237} - q_{0124} q_{0237} q_{1235} \\
& & + q_{0125} q_{0136} q_{0237} - q_{0125} q_{0137} q_{0236} - q_{0125} q_{0234} q_{1237} + q_{0125} q_{0237} q_{1234} \\
& & + q_{0126} q_{0134} q_{1237} - q_{0126} q_{0135} q_{0237} + q_{0126} q_{0137} q_{0235} - q_{0126} q_{0137} q_{1234} \\
& & - q_{0127} - q_{0127} q_{0134} q_{1236} + q_{0127} q_{0135} q_{0236} - q_{0127} q_{0136} q_{0235} + q_{0127} q_{0136} q_{1234}\\
& & + q_{0127} q_{0234} q_{1235} - q_{0127} q_{0235} q_{1234} - q_{0134} q_{0235} q_{1236} + q_{0134} q_{0236} q_{1235} \\
& & + q_{0135} q_{0234} q_{1236} - q_{0135} q_{0236} q_{1234} + q_{0136} - q_{0136} q_{0234} q_{1235} + q_{0136} q_{0235} q_{1234} - q_{0235} + q_{1234} \\
f_5 &=& - q_{0124} q_{0136} q_{0237} + q_{0124} q_{0137} q_{0236} + q_{0124} q_{0235} q_{1236} - q_{0124} q_{0236} q_{1235} \\
& & - q_{0125} q_{0136} q_{1237} + q_{0125} q_{0137} q_{1236} - q_{0125} q_{0234} q_{1236} + q_{0125} q_{0236} q_{1234} \\
& & - q_{0126} + q_{0126} q_{0134} q_{0237} + q_{0126} q_{0135} q_{1237} - q_{0126} q_{0137} q_{0234} - q_{0126} q_{0137} q_{1235} \\
& & + q_{0126} q_{0234} q_{1235} - q_{0126} q_{0235} q_{1234} - q_{0127} q_{0134} q_{0236} - q_{0127} q_{0135} q_{1236} \\
& & + q_{0127} q_{0136} q_{0234} + q_{0127} q_{0136} q_{1235} + q_{0134} q_{0235} q_{1237} - q_{0134} q_{0237} q_{1235} \\
& & - q_{0135} q_{0234} q_{1237} + q_{0135} q_{0237} q_{1234} - q_{0137} + q_{0137} q_{0234} q_{1235} - q_{0137} q_{0235} q_{1234} + q_{0234} + q_{1235} \\
f_6 &=& q_{0124} q_{0135} q_{0237} - q_{0124} q_{0135} q_{1236} + q_{0124} q_{0136} q_{1235} - q_{0124} q_{0137} q_{0235} \\
& & + q_{0125} - q_{0125} q_{0134} q_{0237} + q_{0125} q_{0134} q_{1236} - q_{0125} q_{0136} q_{1234} + q_{0125} q_{0137} q_{0234} \\
& & - q_{0125} q_{0236} q_{1237} + q_{0125} q_{0237} q_{1236} - q_{0126} q_{0134} q_{1235} + q_{0126} q_{0135} q_{1234} \\
& & + q_{0126} q_{0235} q_{1237} - q_{0126} q_{0237} q_{1235} + q_{0127} q_{0134} q_{0235} - q_{0127} q_{0135} q_{0234} \\
& & - q_{0127} q_{0235} q_{1236} + q_{0127} q_{0236} q_{1235} - q_{0134} + q_{0134} q_{0236} q_{1237} - q_{0134} q_{0237} q_{1236} \\
& & - q_{0136} q_{0234} q_{1237} + q_{0136} q_{0237} q_{1234} + q_{0137} q_{0234} q_{1236} - q_{0137} q_{0236} q_{1234} - q_{0237} + q_{1236} \\
f_7 &=& q_{0124} - q_{0124} q_{0135} q_{0236} - q_{0124} q_{0135} q_{1237} + q_{0124} q_{0136} q_{0235} + q_{0124} q_{0137} q_{1235} \\
& & - q_{0124} q_{0236} q_{1237} + q_{0124} q_{0237} q_{1236} + q_{0125} q_{0134} q_{0236} + q_{0125} q_{0134} q_{1237} \\
& & - q_{0125} q_{0136} q_{0234} - q_{0125} q_{0137} q_{1234} - q_{0126} q_{0134} q_{0235} + q_{0126} q_{0135} q_{0234} \\
& & + q_{0126} q_{0234} q_{1237} - q_{0126} q_{0237} q_{1234} - q_{0127} q_{0134} q_{1235} + q_{0127} q_{0135} q_{1234} \\
& & - q_{0127} q_{0234} q_{1236} + q_{0127} q_{0236} q_{1234} + q_{0135} - q_{0135} q_{0236} q_{1237} + q_{0135} q_{0237} q_{1236} \\
& & + q_{0136} q_{0235} q_{1237} - q_{0136} q_{0237} q_{1235} - q_{0137} q_{0235} q_{1236} + q_{0137} q_{0236} q_{1235} + q_{0236} + q_{1237}
\end{eqnarray*}
\section{Spin(7,\texorpdfstring{$\mathbb{C}$}{\bf C}) and Three SL(2,\texorpdfstring{$\mathbb{C}$}{\bf C}) actions}\label{sec:spinsevenandthreesls}
In this section, we first give an unusual definition of the group Spin(7,$\mathbb{C}$) and $G_2^{\mathbb{C}}$ following Bryant \cite{Bry87}.
It is well known that Spin(7,$\mathbb{C}$) is of rank three, that is, its maximal tori are three-dimensional.
Then, we describe a maximal torus of Spin(7,$\mathbb{C}$) and three subgroups of Spin(7,$\mathbb{C}$) each of which is isomorphic to SL(2,$\mathbb{C}$).
\begin{definition}
Spin$(7,\mathbb{C})$ is the identity component of $\{M\in \mt{SO}(8,\mathbb{C})\ |\ M^* \Phi = \Phi\}$ and $G_2^\mathbb{C} = \left\{ M\in \mt{SO}(7,\mathbb{C}) \ |\ M^*\phi = \phi \right\}$.
\end{definition}
Given $A\in G_2^\mathbb{C}$, we can extend it linearly so that it fixes $1\in \mathbb{O}$.
Then, it is easy to see that $G_2^\mathbb{C}$ is a subgroup of $\mt{Spin}(7,\mathbb{C})$ using (\ref{eq:AssocAndCayleyCal}).
By definition, an element of Spin(7,$\mathbb{C}$) acts on $\mathbb{O}$ preserving orthonormality and the values of $\Phi$.
Thus, it takes a Cayley plane to a Cayley plane.
In other words, it defines an action on the Cayley Grassmannian.
Consider the following matrix
\begin{eqnarray*}
L_\lambda = \left(
\begin{matrix}
P_\lambda & -iM_\lambda \\
iM_\lambda & P_\lambda
\end{matrix}
\right)
\end{eqnarray*}
where $P_\lambda=\frac{\lambda+\lambda^{-1}}{2}, M_\lambda=\frac{\lambda-\lambda^{-1}}{2},$ and $\lambda\in\mathbb{C}^*$.
Note that its determinant is $1$.
In fact, it has eigenvalues $\lambda$ and $\lambda^{-1}$ with eigenvectors
\[
\begin{pmatrix}
1 \\
i
\end{pmatrix}
\quad \mt{ and }
\quad
\begin{pmatrix}
1 \\
-i
\end{pmatrix}
\]
respectively.
Considering $L_\lambda$ as a block matrix, we define the following $8\times8$ matrices $A_\lambda = L_\lambda \oplus L_{\lambda}\oplus L_\lambda \oplus L_{\lambda},$ $B_\mu = L_\mu\oplus L_{\mu^{-1}} \oplus I_4$ and $C_\gamma = I_4\oplus L_\gamma\oplus L_{\gamma^{-1}} $ and view them as transformations of $\mathbb{O}$ with respect to the standard basis $\left\{ e_i \right\}$ where $I_n$ is the $n\times n$ identity matrix.
\begin{lemma}
The image of $h:(\mathbb{C}^*)^3\to $SL(8,$\mathbb{C}$) defined by
\begin{equation*}
h(\lambda,\mu,\gamma) = A_\lambda B_\mu C_\gamma
\end{equation*}
is a maximal torus $T$ of Spin(7,$\mathbb{C}$).
\label{lm:maxtor}
\end{lemma}
\begin{proof}
It is easy to prove that $L_\lambda L_\mu = L_{\lambda\mu}$ and if $L_\lambda = I_2$ then $\lambda=1$.
It follows that $A_\lambda,B_\mu$ and $C_\gamma$ commute with each other.
Hence, $h$ is a well defined homomorphism.
Furthermore, the kernel of $h$ is given by $\left\{ \pm(1,1,1) \right\}$ and thus, the image $T \cong (\mathbb{C}^*)^3/\mathbb{Z}_2 $ is isomorphic to $(\mathbb{C}^*)^3$.
Since the rank of Spin(7,$\mathbb{C}$) is $3$, we only need to show that $T\subset$ Spin(7,$\mathbb{C}$).
A simple computation shows that $(L_\lambda)^{-1}=L_{\lambda^{-1}}=L_\lambda^T$.
In other words, $L_\lambda\in$ SO$(2,\mathbb{C})$ which implies $T\subset$ SO$(8,\mathbb{C})$.
Finally, we need to show that for $M\in T$, $M^*\Phi=\Phi$.
We verify this by a direct computation with the help of a software.
\end{proof}
From our discussion above for $L_\lambda$, it is easy to find eigenvalues and eigenvectors for $h(\lambda,\mu,\gamma)$.
They are given in the following table.
\begin{center}
\begin{equation}
\label{tbl:eiva}
\begin{tabular}[]{c|c}
eigenvalue & eigenvector \\ \hline
$\lambda\mu$ & $1+i\textbf{i}$ \\ \hline
$\lambda^{-1}\mu^{-1}$ & $1-i\textbf{i}$ \\ \hline
$\lambda\mu^{-1}$ & $\textbf{j}+i\textbf{k}$ \\ \hline
$\lambda^{-1}\mu$ & $\textbf{j}-i\textbf{k}$ \\ \hline
$\lambda\gamma$ & $\textbf{l}+i\textbf{li}$ \\ \hline
$\lambda^{-1}\gamma^{-1}$ & $\textbf{l}-i\textbf{li}$ \\ \hline
$\lambda\gamma^{-1}$ & $\textbf{lj}+i\textbf{lk}$ \\ \hline
$\lambda^{-1}\gamma$ & $\textbf{lj}-i\textbf{lk}$
\end{tabular}
\end{equation}
\end{center}
We identify $\mt{SL}(2,\mathbb{C})$ as the subgroup of the multiplicative group of $\mathbb{H}$ with $N=1$.
More explicitly, $u\in\mathbb{H}$ is identified with the matrix $A_u$ given by
\[
u=a1+b\textbf{i}+c\textbf{j}+d\textbf{k} \mapsto
A_u =
\begin{pmatrix}
a - id & -b + ic\\
b + ic & a + id
\end{pmatrix}.
\]
Note that $A_u:\mathbb{H}\to \mathbb{C}^{2\times2}$ is a linear isomorphism and it satisfies
\[
A_uA_v = A_{uv}.
\]
Moreover, $N(u) = \det(A_u)$.
Hence, $(\mathbb{C}^{2\times 2}, \det)$ is a quaternion algebra isomorphic to $(\mathbb{H},N)$ via $(u\mapsto A_u)$.
Thus, $\mt{SL}(2,\mathbb{C})$ can be identified with the unit sphere of $\mathbb{H}$, i.e., $\left\{ v\in \mathbb{H}\;|\; N(v)=1 \right\}$.
\begin{proposition}
There are three SL(2,$\mathbb{C}$) actions on $\mathbb{O}$ which preserve $B$ and $\Phi$.
To describe these actions we express $\mathbb{O}$ as a direct sum $\mathbb{O} = \mathbb{H} \oplus \textbf{l}\mathbb{H} $.
Let $v=(x,y)\in\mathbb{O}$.
\begin{enumerate}
\item $g\cdot v = (xg^{-1},y)$
\label{actone}
\item $g\cdot v = (x,yg^{-1})$
\item $g\cdot v = (gx,gy)$
\end{enumerate}
\end{proposition}
\begin{proof}
Since $g\in$ SL(2,$\mathbb{C}$) is identified with an element of $\mathbb{H}$ with norm 1, multiplication by $g$ is an orthogonal transformation by Lemma \ref{lem:orth}.
Thus, $B$ is preserved in all three actions.
To show that $\Phi$ is also preserved, we instead look at the corresponding action of the Lie algebra $\mathfrak{sl}(2,\mathbb{C})\cong \text{Im}(\mathbb{H})=\langle\textbf{i},\textbf{j},\textbf{k}\rangle$.
It is enough to show that $\textbf{i}\cdot\Phi=\textbf{j}\cdot\Phi=\textbf{k}\cdot\Phi=0$.
We verify this by a direct computation for all three actions.
\end{proof}
\begin{remark}
Note that all three actions are faithful and thus, provide three different embeddings of SL$(2,\mathbb{C})$ into Spin(7,$\mathbb{C}$).
Furthermore, we can define an action of the group $(\mt{SL}(2,\mathbb{C}))^3$ on $\mathbb{O}$ by
\begin{equation}
(a,b,c)\cdot(x,y) = (cxa^{-1}, cyb^{-1})
\label{eq:allactionscombined}
\end{equation}
for $a,b,c\in \mt{SL}(2,\mathbb{C})$ and $(x,y)\in\mathbb{O}$.
The kernel of this action is $\left\{ \pm(1,1,1) \right\}$.
\label{rm:slembeddings}
\end{remark}
The matrices corresponding to the unipotent element $\begin{pmatrix} 1 & u \\ 0 & 1 \end{pmatrix}\in SL(2,\mathbb{C})$ (under three actions) are $\begin{bmatrix} A & 0 \\ 0 & I_4 \end{bmatrix}$, $\begin{bmatrix} I_4 & 0 \\ 0 & A \end{bmatrix}$ and $\begin{bmatrix} B & 0 \\ 0 & B \end{bmatrix}$ where
\begin{equation*}
A =
\begin{pmatrix}
1 & 0 & \frac{-u}{2} & \frac{iu}{2} \\
0 & 1 & \frac{iu}{2} & \frac{u}{2} \\
\frac{u}{2} & \frac{-iu}{2} & 1 & 0 \\
\frac{-iu}{2} & \frac{-u}{2} & 0 & 1
\end{pmatrix}
\end{equation*}
and
\begin{equation*}
B =
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & iu & u \\
0 & -iu & 1+\frac{u^2}{2} & \frac{-iu^2}{2} \\
0 & -u & \frac{-iu^2}{2} & 1-\frac{u^2}{2}
\end{pmatrix}.
\end{equation*}
Clearly, the SL(2,$\mathbb{C}$) actions preserve the splitting $\mathbb{O} = \mathbb{H} \oplus \textbf{l}\mathbb{H}$.
Hence, the points $[e_{0123}],[e_{4567}]\in X$ are fixed by all three of these actions.
Restricting to the unipotent group inside SL(2,$\mathbb{C}$), we conclude that fixed point sets (of the unipotent subgroup) are positive dimensional by a result of Horrocks \cite{Hor69}.
\section{Torus fixed points}\label{sec:torusfxd}
We set $\wil e_0= 1+i\textbf{i}, \wil e_1= 1-i\textbf{i}, \wil e_2= \textbf{j}+i\textbf{k}, \wil e_3= \textbf{j}-i\textbf{k}, \wil e_4= \textbf{l}+i\textbf{li}, \wil e_5= \textbf{l}-i\textbf{li}, \wil e_6= \textbf{lj}+i\textbf{lk}, $ and $\wil e_7 = \textbf{lj} -i \textbf{lk}$.
Recall from (\ref{tbl:eiva}) that these vectors are the eigenvectors of the matrix $h(\lambda,\mu,\gamma)$.
We also set $\wil e_{pqrs}=\wil e_p\wedge \wil e_q\wedge\wil e_r\wedge\wil e_s$.
If we denote by $\wil p_{pqrs}$ the transformed Pl\"ucker coordinates, the equations (\ref{eq:plucker1})-(\ref{eq:plucker7}) can be rewritten as
\begin{eqnarray}
\label{eq:transformedplucker1}
\wil f_1 &:=& \wil p_{0257} - \wil p_{1346} =0\\
\wil f_2 &:=& \wil p_{0146} - \wil p_{0157} - \wil p_{0245} - \wil p_{0267} + \wil p_{1345} + \wil p_{1367} + \wil p_{2346} - \wil p_{2357} =0\\
\wil f_3 &:=& \wil p_{0146} + \wil p_{0157} - \wil p_{0245} - \wil p_{0267} - \wil p_{1345} - \wil p_{1367} + \wil p_{2346} + \wil p_{2357} =0\\
\wil f_4 &:=& \wil p_{0127} - \wil p_{0136} + \wil p_{0235} - \wil p_{0567} - \wil p_{1234} + \wil p_{1467} - \wil p_{2457} + \wil p_{3456} =0\\
\wil f_5 &:=& \wil p_{0127} + \wil p_{0136} - \wil p_{0235} + \wil p_{0567} - \wil p_{1234} + \wil p_{1467} - \wil p_{2457} - \wil p_{3456} =0\\
\wil f_6 &:=& \wil p_{0125} - \wil p_{0134} - \wil p_{0237} + \wil p_{0457} + \wil p_{1236} - \wil p_{1456} - \wil p_{2567} + \wil p_{3467} =0\\
\wil f_7 &:=& \wil p_{0125} + \wil p_{0134} + \wil p_{0237} - \wil p_{0457} + \wil p_{1236} - \wil p_{1456} - \wil p_{2567} - \wil p_{3467} =0
\label{eq:transformedplucker7}
\end{eqnarray}
Now that we diagonalized the action, it is easy to describe the eigenvalues and the corresponding eigenvectors of the action of $h(\lambda,\mu,\gamma)$ on $\Lambda^4 \mathbb{O}$.
\begin{center}
\begin{tabular}[]{c|c}
eigenvalue & eigenvector \\ \hline
$1$ & $\wil e_{0123}, \wil e_{0145}, \wil e_{0167}, \wil e_{0257}, \wil e_{1346}, \wil e_{2345}, \wil e_{2367}, \wil e_{4567}$ \\ \hline
$\gamma^{-2}$ & $\wil e_{0156}, \wil e_{2356}$ \\ \hline
$\gamma^{-2} \lambda^{-2}$ & $\wil e_{1356}$ \\ \hline
$\gamma^{-2} \lambda^{2}$ & $\wil e_{0256}$ \\ \hline
$\gamma^{-2} \mu^{-2}$ & $\wil e_{1256}$ \\ \hline
$\gamma^{-2} \mu^{2}$ & $\wil e_{0356}$ \\ \hline
$\gamma^{-1} \lambda^{-2} \mu^{-1}$ & $\wil e_{1235}, \wil e_{1567}$ \\ \hline
$\gamma^{-1} \lambda^{-2} \mu$ & $\wil e_{0135}, \wil e_{3567}$ \\ \hline
$\gamma^{-1} \lambda^{2} \mu^{-1}$ & $\wil e_{0126}, \wil e_{2456}$ \\ \hline
$\gamma^{-1} \lambda^{2} \mu$ & $\wil e_{0236}, \wil e_{0456}$ \\ \hline
$\gamma^{-1} \mu^{-1}$ & $\wil e_{0125}, \wil e_{1236}, \wil e_{1456}, \wil e_{2567}$ \\ \hline
$\gamma^{-1} \mu$ & $\wil e_{0136}, \wil e_{0235}, \wil e_{0567}, \wil e_{3456}$ \\ \hline
$\gamma \lambda^{-2} \mu^{-1}$ & $\wil e_{1237}, \wil e_{1457}$ \\ \hline
$\gamma \lambda^{-2} \mu$ & $\wil e_{0137}, \wil e_{3457}$ \\ \hline
$\gamma \lambda^{2} \mu^{-1}$ & $\wil e_{0124}, \wil e_{2467}$ \\ \hline
$\gamma \lambda^{2} \mu$ & $\wil e_{0234}, \wil e_{0467}$ \\ \hline
$\gamma \mu^{-1}$ & $\wil e_{0127}, \wil e_{1234}, \wil e_{1467}, \wil e_{2457}$ \\ \hline
$\gamma \mu$ & $\wil e_{0134}, \wil e_{0237}, \wil e_{0457}, \wil e_{3467}$ \\ \hline
$\gamma^{2}$ & $\wil e_{0147}, \wil e_{2347}$ \\ \hline
$\gamma^{2} \lambda^{-2}$ & $\wil e_{1347}$ \\ \hline
$\gamma^{2} \lambda^{2}$ & $\wil e_{0247}$ \\ \hline
$\gamma^{2} \mu^{-2}$ & $\wil e_{1247}$ \\ \hline
$\gamma^{2} \mu^{2}$ & $\wil e_{0347}$ \\ \hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}[]{c|c}
eigenvalue & eigenvector \\ \hline
$\lambda^{-4}$ & $\wil e_{1357}$ \\ \hline
$\lambda^{-2}$ & $\wil e_{0157}, \wil e_{1345}, \wil e_{1367}, \wil e_{2357}$ \\ \hline
$\lambda^{-2} \mu^{-2}$ & $\wil e_{1257}$ \\ \hline
$\lambda^{-2} \mu^{2}$ & $\wil e_{0357}$ \\ \hline
$\lambda^{2}$ & $\wil e_{0146}, \wil e_{0245}, \wil e_{0267}, \wil e_{2346}$ \\ \hline
$\lambda^{2} \mu^{-2}$ & $\wil e_{1246}$ \\ \hline
$\lambda^{2} \mu^{2}$ & $\wil e_{0346}$ \\ \hline
$\lambda^{4}$ & $\wil e_{0246}$ \\ \hline
$\mu^{-2}$ & $\wil e_{1245}, \wil e_{1267}$ \\ \hline
$\mu^{2}$ & $\wil e_{0345}, \wil e_{0367}$
\end{tabular}
\end{center}
\begin{theorem}
The following eigenvectors of $h(\lambda,\mu,\gamma)$ lie in $X$.
\label{thm:eive}
\begin{center}
\begin{tabular}[]{c|c}
eigenvalue & eigenvector \\ \hline
$1$ & $\wil e_{0123}, \wil e_{0145}, \wil e_{0167}, \wil e_{2345}, \wil e_{2367}, \wil e_{4567}$ \\ \hline
$\gamma^{-2}$ & $\wil e_{0156}, \wil e_{2356}$ \\ \hline
$\gamma^{-2} \lambda^{-2}$ & $\wil e_{1356}$ \\ \hline
$\gamma^{-2} \lambda^{2}$ & $\wil e_{0256}$ \\ \hline
$\gamma^{-2} \mu^{-2}$ & $\wil e_{1256}$ \\ \hline
$\gamma^{-2} \mu^{2}$ & $\wil e_{0356}$ \\ \hline
$\gamma^{-1} \lambda^{-2} \mu^{-1}$ & $\wil e_{1235}, \wil e_{1567}$ \\ \hline
$\gamma^{-1} \lambda^{-2} \mu$ & $\wil e_{0135}, \wil e_{3567}$ \\ \hline
$\gamma^{-1} \lambda^{2} \mu^{-1}$ & $\wil e_{0126}, \wil e_{2456}$ \\ \hline
$\gamma^{-1} \lambda^{2} \mu$ & $\wil e_{0236}, \wil e_{0456}$ \\ \hline
$\gamma \lambda^{-2} \mu^{-1}$ & $\wil e_{1237}, \wil e_{1457}$ \\ \hline
$\gamma \lambda^{-2} \mu$ & $\wil e_{0137}, \wil e_{3457}$ \\ \hline
$\gamma \lambda^{2} \mu^{-1}$ & $\wil e_{0124}, \wil e_{2467}$ \\ \hline
$\gamma \lambda^{2} \mu$ & $\wil e_{0234}, \wil e_{0467}$ \\ \hline
$\gamma^{2}$ & $\wil e_{0147}, \wil e_{2347}$ \\ \hline
$\gamma^{2} \lambda^{-2}$ & $\wil e_{1347}$ \\ \hline
$\gamma^{2} \lambda^{2}$ & $\wil e_{0247}$ \\ \hline
$\gamma^{2} \mu^{-2}$ & $\wil e_{1247}$ \\ \hline
$\gamma^{2} \mu^{2}$ & $\wil e_{0347}$ \\ \hline
$\lambda^{-4}$ & $\wil e_{1357}$ \\ \hline
$\lambda^{-2} \mu^{-2}$ & $\wil e_{1257}$ \\ \hline
$\lambda^{-2} \mu^{2}$ & $\wil e_{0357}$ \\ \hline
$\lambda^{2} \mu^{-2}$ & $\wil e_{1246}$ \\ \hline
$\lambda^{2} \mu^{2}$ & $\wil e_{0346}$ \\ \hline
$\lambda^{4}$ & $\wil e_{0246}$ \\ \hline
$\mu^{-2}$ & $\wil e_{1245}, \wil e_{1267}$ \\ \hline
$\mu^{2}$ & $\wil e_{0345}, \wil e_{0367}$
\end{tabular}
\end{center}
\end{theorem}
\begin{proof}
Once we evaluate all the eigenvectors on the transformed defining equations (\ref{eq:transformedplucker1})-(\ref{eq:transformedplucker7}), we see that exactly the above list of vectors satisfy them.
\end{proof}
\begin{theorem}
The fixed point set $X^{T}$ of the maximal torus action is only the above set of points.
\label{thm:fixpnts}
\end{theorem}
\begin{proof}
We only need to verify that in eigenspaces of dimension greater than one, there are no other fixed points.
We prove this for the eigenspace associated to the eigenvalue 1.
Let
$$\wil x = \wil c_{0123}\wil e_{0123}+\wil c_{0145}\wil e_{0145}+\wil c_{0167}\wil e_{0167}+\wil c_{2345}\wil e_{2345}+\wil c_{2367}\wil e_{2367}+\wil c_{4567}\wil e_{4567}\in X$$
be a torus fixed point that is different from
$\wil e_{0123}, \wil e_{0145}, \wil e_{0167}, \wil e_{2345}, \wil e_{2367},$ and $\wil e_{4567}$.
Then, at least two of the coordinates $\wil c_I,$ and $\wil c_J$ are nonzero for index sets $I\neq J$.
Let $I=\{i_1i_2i_3j_1\}$, and $J=\{j_2j_3j_4j_5\}$.
If $\wil x\inX \subset$ Gr(4,$\mathbb{O}$) than it has to satisfy the Pl\"ucker relation (\ref{eq:plucker})
\[
\wil c_{i_1i_2i_3j_1}\wil c_{j_2j_3j_4j_5}= \wil c_{i_1i_2i_3j_2}\wil c_{j_1j_3j_4j_5}-\wil c_{i_1i_2i_3j_3}\wil c_{j_1j_2j_4j_5}+\wil c_{i_1i_2i_3j_4}\wil c_{j_1j_2j_3j_5}-\wil c_{i_1i_2i_3j_5}\wil c_{j_1j_2j_3j_4}.
\]
Now, the left hand side of this relation is nonzero by choice.
However, the right hand side has to be zero as the index sets of eigenvectors associated to eigenvalue 1 differ by at least two elements.
This means that $\wil x$ does not belong to $\mt{Gr}(4,\mathbb{O})$ and hence, $\wil x\notin X$.
This argument also works for other eigenspaces of dimension greater than one.
\end{proof}
\section{The Cayley Grassmannian}\label{sec:mincompt}
We cite the following well known theorem, here.
\begin{theorem}[Jacobian Criterion for Smoothness]\label{thm:jacobian}
Let $I=(f_1,\dots,f_m)$ be an ideal from $\mathbb{C}[x_1,\dots,x_n]$ and let $x\in V(I)$ be a point from the vanishing locus of $I$ in $\mathbb{C}^n$.
Suppose $d=\dim V(I)$.
If the rank of the Jacobian matrix $(\partial f_i/\partial x_j)_{i=1,\dots,m,\ j=1,\dots,n}$ at $x$ is equal to $n-d$, then $x$ is a smooth point of $V(I)$.
\end{theorem}
Torus action takes singular locus to singular locus.
Since the singular locus is also a projective variety, it must contain a torus fixed point by Borel fixed point theorem.
Therefore, we can check whether $X$ is smooth or not, by using the Jacobian criterion on the torus fixed points.
\begin{theorem}\label{thm:fxdpnts}
Among the fixed points listed in Theorem \ref{thm:eive}, all but the following six of them are smooth points.
\[
\wil e_{0246}, \wil e_{0347}, \wil e_{0356}, \wil e_{1247}, \wil e_{1256}, \wil e_{1357}
\]
\end{theorem}
\begin{proof}
We need to analyze neighborhoods of fixed points by using affine charts.
We start with the fixed point $m=\wil e_{0123}$, which lies on the open chart $\wil U_{0123}$ as its origin.
Recall that $X$ is cut-out on $\wil U_{0123}$ by the vanishing of the seven linear equations (\ref{eq:transformedplucker1})-(\ref{eq:transformedplucker7}).
At first, it may seem that it is necessary to express these equations in local variables $\wil q_{0124},\wil q_{0125},\wil q_{0126},\wil q_{0127},\wil q_{0134},\wil q_{0135},\wil q_{0136},\wil q_{0137},$ $\wil q_{0234}, \wil q_{0235},\wil q_{0236},\wil q_{0237},\wil q_{1234},\wil q_{1235},\wil q_{1236},$ and $\wil q_{1237}$.
However, the Pl\"ucker relations (in local coordinates) replace linear terms with higher order terms (see, for example, (\ref{eq:pluckerexampleq})) and we compute Jacobian at the origin.
So, there will be no contribution to Jacobian matrix from other variables.
The Jacobian matrix at $m$ is given by
\begin{eqnarray*}
\wil J_{0123} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0
\end{pmatrix}
\end{eqnarray*}
\begin{comment}
\begin{eqnarray*}
\wil J_{0124} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\wil J_{0126} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0
\end{pmatrix} \\
\wil J_{0135} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\wil J_{0137} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\
0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\wil J_{0145} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & 0 \\
0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 0
\end{pmatrix} \\
\end{eqnarray*}
\begin{eqnarray*}
\wil J_{0147} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\wil J_{0156} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0
\end{pmatrix} \\
\wil J_{0167} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\
0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 \\
0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\wil J_{0234} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\end{eqnarray*}
\begin{eqnarray*}
\wil J_{0236} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0
\end{pmatrix} \\
\wil J_{0246} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\wil J_{0247} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\wil J_{0256} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1
\end{pmatrix}
\end{eqnarray*}
\begin{eqnarray*}
\wil J_{0345} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\
-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\wil J_{0346} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\
-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1
\end{pmatrix} \\
\wil J_{0347} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1
\end{pmatrix} \\
\wil J_{0356} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & -1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\wil J_{0357} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\wil J_{0367} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0
\end{pmatrix}
\end{eqnarray*}
\begin{eqnarray*}
\wil J_{0456} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 0
\end{pmatrix} \\
\wil J_{0467} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & 0
\end{pmatrix} \\
\wil J_{1235} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\wil J_{1237} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\wil J_{1245} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\
0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\wil J_{1246} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\end{eqnarray*}
\begin{eqnarray*}
\wil J_{1247} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & 0 \\
0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\wil J_{1256} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 \\
1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1
\end{pmatrix} \\
\wil J_{1257} &=&
\begin{pmatrix}
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1
\end{pmatrix} \\
\wil J_{1267} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1
\end{pmatrix} \\
\wil J_{1347} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1
\end{pmatrix} \\
\wil J_{1356} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\end{eqnarray*}
\begin{eqnarray*}
\wil J_{1357} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\wil J_{1457} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\wil J_{1567} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0
\end{pmatrix} \\
\wil J_{2345} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 \\
0 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \\
\wil J_{2347} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1
\end{pmatrix} \\
\wil J_{2356} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0
\end{pmatrix} \\
\end{eqnarray*}
\begin{eqnarray*}
\wil J_{2367} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & 0
\end{pmatrix} \\
\wil J_{2456} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0
\end{pmatrix} \\
\wil J_{2467} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & 0
\end{pmatrix} \\
\wil J_{3457} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0
\end{pmatrix} \\
\wil J_{3567} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 0
\end{pmatrix} \\
\wil J_{4567} &=&
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 \\
0 & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 0
\end{pmatrix}
\end{eqnarray*}
\end{comment}
which is of rank four. So, it is at most 12-dimensional. However, it contains the 12-dimensional $X^0$. So, it must be 12-dimensional, that is, codimension four in Gr(4,$\mathbb{O}$). Hence, by Theorem~\ref{thm:jacobian}, $\wil e_{0123}$ is a smooth point of $X$.
We repeat this computation for the other points and see that their Jacobian matrices are all rank four, except for the six points we have listed above.
Thus, they are all smooth points of $X$.
\end{proof}
\section{Singular Locus}\label{sec:singloc}
Next, we turn our attention to the singular locus $\Sigma:=\text{Sing($\xmin$)}$ of $X$.
The torus action on $X$ restricts to $\Sigma$.
By Theorem~\ref{thm:fxdpnts}, we know that there are six points in $\Sigma^T$.
We quote the following lemma from \cite{BCM02}.
\begin{lemma}
\label{lem:cntFxdPnts}
If $Y\subset \mathbb P(V)$ is a projective $T$-variety, then $Y^T$ contains at least dim $Y+1$ points.
\end{lemma}
Therefore, we have the following corollary.
\begin{corollary}
The singular locus $\Sigma$ is at most five-dimensional.
\end{corollary}
As we did with $X$, we can check whether $\Sigma$ is singular or not, by using Jacobian criterion on these six torus fixed points $\Sigma^T$.
\begin{theorem}
\label{thm:singLocSmooth}
The singular locus $\Sigma$ is smooth and five-dimensional.
\end{theorem}
\begin{proof}
We start with the point $\wil e_{0246}$.
This point lies at the origin of the chart $\wil U_{0246}=\left\{x\in\mathbb{P}(\Lambda^4\mathbb{O})\;|\; \wil p_{0246}(x)\neq 0\right\}$.
On this chart the local variables are $\wil q_{0124},$ $\wil q_{0126},$ $\wil q_{0146},$ $\wil q_{0234},$ $\wil q_{0236},$ $\wil q_{0245},$ $\wil q_{0247},$ $\wil q_{0256},$ $\wil q_{0267},$ $\wil q_{0346},$ $\wil q_{0456},$ $\wil q_{0467},$ $\wil q_{1246},$ $\wil q_{2346},$ $\wil q_{2456},$ and $\wil q_{2467}$ where $\wil q_{ijkl}=\wil p_{ijkl}/\wil p_{0246}$.
Since the codimension of $X$ in $\text{Gr}(4,\mathbb{O})$ is four, $\Sigma$ is locally the vanishing locus of the equations (\ref{eq:transformedplucker1})-(\ref{eq:transformedplucker7}) localized to $\wil U_{0246}$ and all $4\times 4$ minors of the Jacobian of these localized equations.
These equations by themselves do not generate a radical ideal, so we take the radical ideal generated by those equations with the help of a software called Singular.
It turns out the ideal is generated by
\begin{eqnarray*}
& & \wil q_{1246},\quad
\wil q_{0346},\quad
\wil q_{0267}- \wil q_{2346},\quad
\wil q_{0256},\quad
\wil q_{0247},\quad
\wil q_{0245}- \wil q_{2346},\quad
\wil q_{0236}- \wil q_{0456},\\
& & \wil q_{0234}- \wil q_{0467},\quad
\wil q_{0146}- \wil q_{2346},\quad
\wil q_{0126}- \wil q_{2456},\quad
\text{and}\quad \wil q_{0124}- \wil q_{2467}.
\end{eqnarray*}
So, $\Sigma$ is just cut out by some hyperplanes in $\wil U_{0246}\cap$Gr(4,$\mathbb{O}$).
Therefore, it is clearly smooth at the origin and of dimension five.
We repeat this computation for the other fixed points and see they are all smooth points of $\Sigma$. Hence, $\Sigma$ is smooth and five-dimensional.
\end{proof}
\begin{theorem}
The singular locus $\Sigma$ has the same cohomology ring (over $\mathbb{Q}$) as $\mathbb{C}\mathbb{P}^5$.
\label{thm:cohomp5}
\end{theorem}
\begin{proof}
By Theorem \ref{thm:singLocSmooth}, we see that $\Sigma$ is a smooth projective variety and hence, it is K\"ahler.
Thus, $2i^{th}$ Betti number is at least one for $i=0,\dots,5$.
So, the sum of its Betti numbers is at least six.
On the other hand, there is a torus action on $\Sigma$ (induced from $T$-action on $X$) with exactly six fixed points.
Thus, by Bia\l ynicki-Birula decomposition \cite{BB73}, the sum of Betti numbers is exactly six.
\end{proof}
Next, we restrict the action of Spin(7,$\mathbb{C}$) on $\Sigma$ to the subgroup $G_2^\mathbb{C}$.
A maximal torus for $G_2^\mathbb{C}$ is given by $T\cap G_2^\mathbb{C}$.
Let $\mf g$ denote the Lie algebra of $G_2^\mathbb{C}$ and $\mf h$ denote the Cartan subalgebra corresponding to our choice of maximal torus.
Choose a set of positive roots $S^{+}$ so that $\wil e_{0246}$ has the highest weight and let $\Delta=\left\{ \alpha_1,\alpha_2 \right\}$ be the set of simple roots corresponding to this choice where $\alpha_2$ is the longer root.
Let $P_i$ be the parabolic subgroup of $G_2^{\mathbb{C}}$ whose Lie algebra is $\mf g_{-\alpha_i}\bigoplus \mf h \bigoplus\left( \oplus_{\alpha\in S^+} \mf g_{\alpha} \right)$ where $\mf g_{\alpha}=\left\{ X\in \mf g \ |\ [H,X]=\alpha(H)X \ \mt{for all } H\in \mf h\right\}$.
A straight-forward, albeit lengthy, calculation shows that the stabilizer subgroup of $\wil e_{0246}$ is $P_2$.
Hence, we get the following theorem.
\begin{theorem}
$G_2^\mathbb{C}$ acts on the singular locus $\Sigma$ and the stabilizer group of $\wil e_{0246}$ is $P_2$.
So, by dimensional reasons, $\Sigma = G_2^\mathbb{C}/P_2$.
\label{thm:parabolicquotient}
\end{theorem}
\section{Torus action near smooth fixed points}\label{sec:toruslocalsmooth}
Note that torus action induces an action on the tangent spaces at the fixed points.
In this section, we choose a regular one parameter subgroup $\tau(\lambda)$ of $T$ and then describe the induced action on these tangent spaces.
The subgroup $\tau(\lambda)$ is regular in the sense that the fixed point set of $T$ and of $\tau(\lambda)$ are the same.
We choose the one parameter subgroup $\tau(\lambda)=h(\lambda,\lambda^{10},\lambda^{100})$ for our computations.
This is regular since it pairs to a non-trivial homomorphism ($\mathbb{C}^*\to \mathbb{C}^*$) with any character.
Below, we compute a weight space decomposition for each tangent space.
We list the weight vectors with their corresponding weights and give the number of positive weights.
On a chart $\wil U_{stun}=\left\{ x\in\mathbb{P}(\Lambda^4\mathbb{O})\;|\; \wil p_{stun}(x)\neq 0 \right\}$, we denote the vector field $\frac{\partial}{\partial \wil q_{ijkl}}$ by $\partial_{ijkl}$.
At $\wil e_{0247}$ in $\wil U_{0247}$, the kernel of $\wil J_{0247}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0124}$ & $\partial_{0147}$ & $\partial_{0234}$ & $\partial_{0246}$ & $\partial_{0347}$ & $\partial_{0467}$\\\hline
weight & $\lambda^{-110}$ & $\lambda^{-2}$ & $\lambda^{-90}$ & $\lambda^{-198}$ & $\lambda^{18}$ & $\lambda^{-90}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{1247}$ & $\partial_{2347}$ & $\partial_{2467}$ & $\partial_{0127} + \partial_{2457}$ & $\partial_{0237} + \partial_{0457}$ & $\partial_{0245} - \partial_{0267}$\\\hline
weight & $\lambda^{-22}$ & $\lambda^{-2}$ & $\lambda^{-110}$ & $\lambda^{-112}$ & $\lambda^{-92}$ & $\lambda^{-200}$
\end{tabular}$$
and the number of positive weights is 1.
At $\wil e_{0147}$ in $\wil U_{0147}$, the kernel of $\wil J_{0147}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0124}$ & $\partial_{0137}$ & $\partial_{0145}$ & $\partial_{0167}$ & $\partial_{0247}$ & $\partial_{0347}$\\\hline
weight & $\lambda^{-108}$ & $\lambda^{-92}$ & $\lambda^{-200}$ & $\lambda^{-200}$ & $\lambda^{2}$ & $\lambda^{20}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0467}$ & $\partial_{1247}$ & $\partial_{1347}$ & $\partial_{1457}$ & $\partial_{0127} - \partial_{1467}$ & $\partial_{0134} + \partial_{0457}$\\\hline
weight & $\lambda^{-88}$ & $\lambda^{-20}$ & $\lambda^{-2}$ & $\lambda^{-112}$ & $\lambda^{-110}$ & $\lambda^{-90}$
\end{tabular}$$
and the number of positive weights is 2.
At $\wil e_{2347}$ in $\wil U_{2347}$, the kernel of $\wil J_{2347}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0234}$ & $\partial_{0247}$ & $\partial_{0347}$ & $\partial_{1237}$ & $\partial_{1247}$ & $\partial_{1347}$\\\hline
weight & $\lambda^{-88}$ & $\lambda^{2}$ & $\lambda^{20}$ & $\lambda^{-112}$ & $\lambda^{-20}$ & $\lambda^{-2}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{2345}$ & $\partial_{2367}$ & $\partial_{2467}$ & $\partial_{3457}$ & $\partial_{0237} + \partial_{3467}$ & $\partial_{1234} - \partial_{2457}$\\\hline
weight & $\lambda^{-200}$ & $\lambda^{-200}$ & $\lambda^{-108}$ & $\lambda^{-92}$ & $\lambda^{-90}$ & $\lambda^{-110}$
\end{tabular}$$
and the number of positive weights is 2.
At $\wil e_{0234}$ in $\wil U_{0234}$, the kernel of $\wil J_{0234}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0123}$ & $\partial_{0124}$ & $\partial_{0236}$ & $\partial_{0246}$ & $\partial_{0247}$ & $\partial_{0345}$\\\hline
weight & $\lambda^{-112}$ & $\lambda^{-20}$ & $\lambda^{-200}$ & $\lambda^{-108}$ & $\lambda^{90}$ & $\lambda^{-92}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0346}$ & $\partial_{0347}$ & $\partial_{2345}$ & $\partial_{2347}$ & $\partial_{0134} - \partial_{0237}$ & $\partial_{0245} + \partial_{2346}$\\\hline
weight & $\lambda^{-90}$ & $\lambda^{108}$ & $\lambda^{-112}$ & $\lambda^{88}$ & $\lambda^{-2}$ & $\lambda^{-110}$
\end{tabular}$$
and the number of positive weights is 3.
At $\wil e_{0467}$ in $\wil U_{0467}$, the kernel of $\wil J_{0467}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0147}$ & $\partial_{0167}$ & $\partial_{0246}$ & $\partial_{0247}$ & $\partial_{0346}$ & $\partial_{0347}$\\\hline
weight & $\lambda^{88}$ & $\lambda^{-112}$ & $\lambda^{-108}$ & $\lambda^{90}$ & $\lambda^{-90}$ & $\lambda^{108}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0367}$ & $\partial_{0456}$ & $\partial_{2467}$ & $\partial_{4567}$ & $\partial_{0146} + \partial_{0267}$ & $\partial_{0457} - \partial_{3467}$\\\hline
weight & $\lambda^{-92}$ & $\lambda^{-200}$ & $\lambda^{-20}$ & $\lambda^{-112}$ & $\lambda^{-110}$ & $\lambda^{-2}$
\end{tabular}$$
and the number of positive weights is 3.
At $\wil e_{1347}$ in $\wil U_{1347}$, the kernel of $\wil J_{1347}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0137}$ & $\partial_{0147}$ & $\partial_{0347}$ & $\partial_{1237}$ & $\partial_{1247}$ & $\partial_{1357}$\\\hline
weight & $\lambda^{-90}$ & $\lambda^{2}$ & $\lambda^{22}$ & $\lambda^{-110}$ & $\lambda^{-18}$ & $\lambda^{-202}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{1457}$ & $\partial_{2347}$ & $\partial_{3457}$ & $\partial_{0134} + \partial_{3467}$ & $\partial_{1234} + \partial_{1467}$ & $\partial_{1345} - \partial_{1367}$\\\hline
weight & $\lambda^{-110}$ & $\lambda^{2}$ & $\lambda^{-90}$ & $\lambda^{-88}$ & $\lambda^{-108}$ & $\lambda^{-200}$
\end{tabular}$$
and the number of positive weights is 3.
At $\wil e_{0124}$ in $\wil U_{0124}$, the kernel of $\wil J_{0124}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0123}$ & $\partial_{0126}$ & $\partial_{0145}$ & $\partial_{0147}$ & $\partial_{0234}$ & $\partial_{0246}$\\\hline
weight & $\lambda^{-92}$ & $\lambda^{-200}$ & $\lambda^{-92}$ & $\lambda^{108}$ & $\lambda^{20}$ & $\lambda^{-88}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0247}$ & $\partial_{1245}$ & $\partial_{1246}$ & $\partial_{1247}$ & $\partial_{0127} + \partial_{1234}$ & $\partial_{0146} + \partial_{0245}$\\\hline
weight & $\lambda^{110}$ & $\lambda^{-112}$ & $\lambda^{-110}$ & $\lambda^{88}$ & $\lambda^{-2}$ & $\lambda^{-90}$
\end{tabular}$$
and the number of positive weights is 4.
At $\wil e_{0137}$ in $\wil U_{0137}$, the kernel of $\wil J_{0137}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0123}$ & $\partial_{0135}$ & $\partial_{0147}$ & $\partial_{0167}$ & $\partial_{0347}$ & $\partial_{0357}$\\\hline
weight & $\lambda^{-108}$ & $\lambda^{-200}$ & $\lambda^{92}$ & $\lambda^{-108}$ & $\lambda^{112}$ & $\lambda^{-90}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0367}$ & $\partial_{1237}$ & $\partial_{1347}$ & $\partial_{1357}$ & $\partial_{0134} - \partial_{0237}$ & $\partial_{0157} + \partial_{1367}$\\\hline
weight & $\lambda^{-88}$ & $\lambda^{-20}$ & $\lambda^{90}$ & $\lambda^{-112}$ & $\lambda^{2}$ & $\lambda^{-110}$
\end{tabular}$$
and the number of positive weights is 4.
At $\wil e_{0346}$ in $\wil U_{0346}$, the kernel of $\wil J_{0346}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0234}$ & $\partial_{0236}$ & $\partial_{0246}$ & $\partial_{0345}$ & $\partial_{0347}$ & $\partial_{0356}$\\\hline
weight & $\lambda^{90}$ & $\lambda^{-110}$ & $\lambda^{-18}$ & $\lambda^{-2}$ & $\lambda^{198}$ & $\lambda^{-202}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0367}$ & $\partial_{0456}$ & $\partial_{0467}$ & $\partial_{0134} + \partial_{3467}$ & $\partial_{0136} + \partial_{3456}$ & $\partial_{0146} - \partial_{2346}$\\\hline
weight & $\lambda^{-2}$ & $\lambda^{-110}$ & $\lambda^{90}$ & $\lambda^{88}$ & $\lambda^{-112}$ & $\lambda^{-20}$
\end{tabular}$$
and the number of positive weights is 4.
At $\wil e_{2467}$ in $\wil U_{2467}$, the kernel of $\wil J_{2467}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0246}$ & $\partial_{0247}$ & $\partial_{0467}$ & $\partial_{1246}$ & $\partial_{1247}$ & $\partial_{1267}$\\\hline
weight & $\lambda^{-88}$ & $\lambda^{110}$ & $\lambda^{20}$ & $\lambda^{-110}$ & $\lambda^{88}$ & $\lambda^{-112}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{2347}$ & $\partial_{2367}$ & $\partial_{2456}$ & $\partial_{4567}$ & $\partial_{0267} + \partial_{2346}$ & $\partial_{1467} + \partial_{2457}$\\\hline
weight & $\lambda^{108}$ & $\lambda^{-92}$ & $\lambda^{-200}$ & $\lambda^{-92}$ & $\lambda^{-90}$ & $\lambda^{-2}$
\end{tabular}$$
and the number of positive weights is 4.
At $\wil e_{3457}$ in $\wil U_{3457}$, the kernel of $\wil J_{3457}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0345}$ & $\partial_{0347}$ & $\partial_{0357}$ & $\partial_{1347}$ & $\partial_{1357}$ & $\partial_{1457}$\\\hline
weight & $\lambda^{-88}$ & $\lambda^{112}$ & $\lambda^{-90}$ & $\lambda^{90}$ & $\lambda^{-112}$ & $\lambda^{-20}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{2345}$ & $\partial_{2347}$ & $\partial_{3567}$ & $\partial_{4567}$ & $\partial_{0457} - \partial_{3467}$ & $\partial_{1345} + \partial_{2357}$\\\hline
weight & $\lambda^{-108}$ & $\lambda^{92}$ & $\lambda^{-200}$ & $\lambda^{-108}$ & $\lambda^{2}$ & $\lambda^{-110}$
\end{tabular}$$
and the number of positive weights is 4.
At $\wil e_{0345}$ in $\wil U_{0345}$, the kernel of $\wil J_{0345}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0135}$ & $\partial_{0145}$ & $\partial_{0234}$ & $\partial_{0346}$ & $\partial_{0347}$ & $\partial_{0356}$\\\hline
weight & $\lambda^{-112}$ & $\lambda^{-20}$ & $\lambda^{92}$ & $\lambda^{2}$ & $\lambda^{200}$ & $\lambda^{-200}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0357}$ & $\partial_{0456}$ & $\partial_{2345}$ & $\partial_{3457}$ & $\partial_{0134} + \partial_{0457}$ & $\partial_{0235} - \partial_{3456}$\\\hline
weight & $\lambda^{-2}$ & $\lambda^{-108}$ & $\lambda^{-20}$ & $\lambda^{88}$ & $\lambda^{90}$ & $\lambda^{-110}$
\end{tabular}$$
and the number of positive weights is 5.
At $\wil e_{0367}$ in $\wil U_{0367}$, the kernel of $\wil J_{0367}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0137}$ & $\partial_{0167}$ & $\partial_{0236}$ & $\partial_{0346}$ & $\partial_{0347}$ & $\partial_{0356}$\\\hline
weight & $\lambda^{88}$ & $\lambda^{-20}$ & $\lambda^{-108}$ & $\lambda^{2}$ & $\lambda^{200}$ & $\lambda^{-200}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0357}$ & $\partial_{0467}$ & $\partial_{2367}$ & $\partial_{3567}$ & $\partial_{0136} - \partial_{0567}$ & $\partial_{0237} + \partial_{3467}$\\\hline
weight & $\lambda^{-2}$ & $\lambda^{92}$ & $\lambda^{-20}$ & $\lambda^{-112}$ & $\lambda^{-110}$ & $\lambda^{90}$
\end{tabular}$$
and the number of positive weights is 5.
At $\wil e_{1237}$ in $\wil U_{1237}$, the kernel of $\wil J_{1237}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0123}$ & $\partial_{0137}$ & $\partial_{1235}$ & $\partial_{1247}$ & $\partial_{1257}$ & $\partial_{1267}$\\\hline
weight & $\lambda^{-88}$ & $\lambda^{20}$ & $\lambda^{-200}$ & $\lambda^{92}$ & $\lambda^{-110}$ & $\lambda^{-108}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{1347}$ & $\partial_{1357}$ & $\partial_{2347}$ & $\partial_{2367}$ & $\partial_{0127} + \partial_{1234}$ & $\partial_{1367} + \partial_{2357}$\\\hline
weight & $\lambda^{110}$ & $\lambda^{-92}$ & $\lambda^{112}$ & $\lambda^{-88}$ & $\lambda^{2}$ & $\lambda^{-90}$
\end{tabular}$$
and the number of positive weights is 5.
At $\wil e_{1457}$ in $\wil U_{1457}$, the kernel of $\wil J_{1457}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0145}$ & $\partial_{0147}$ & $\partial_{1245}$ & $\partial_{1247}$ & $\partial_{1257}$ & $\partial_{1347}$\\\hline
weight & $\lambda^{-88}$ & $\lambda^{112}$ & $\lambda^{-108}$ & $\lambda^{92}$ & $\lambda^{-110}$ & $\lambda^{110}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{1357}$ & $\partial_{1567}$ & $\partial_{3457}$ & $\partial_{4567}$ & $\partial_{0157} + \partial_{1345}$ & $\partial_{1467} + \partial_{2457}$\\\hline
weight & $\lambda^{-92}$ & $\lambda^{-200}$ & $\lambda^{20}$ & $\lambda^{-88}$ & $\lambda^{-90}$ & $\lambda^{2}$
\end{tabular}$$
and the number of positive weights is 5.
At $\wil e_{0123}$ in $\wil U_{0123}$, the kernel of $\wil J_{0123}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0124}$ & $\partial_{0126}$ & $\partial_{0135}$ & $\partial_{0137}$ & $\partial_{0234}$ & $\partial_{0236}$\\\hline
weight & $\lambda^{92}$ & $\lambda^{-108}$ & $\lambda^{-92}$ & $\lambda^{108}$ & $\lambda^{112}$ & $\lambda^{-88}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{1235}$ & $\partial_{1237}$ & $\partial_{0125} - \partial_{1236}$ & $\partial_{0127} + \partial_{1234}$ & $\partial_{0134} - \partial_{0237}$ & $\partial_{0136} + \partial_{0235}$\\\hline
weight & $\lambda^{-112}$ & $\lambda^{88}$ & $\lambda^{-110}$ & $\lambda^{90}$ & $\lambda^{110}$ & $\lambda^{-90}$
\end{tabular}$$
and the number of positive weights is 6.
At $\wil e_{0145}$ in $\wil U_{0145}$, the kernel of $\wil J_{0145}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0124}$ & $\partial_{0135}$ & $\partial_{0147}$ & $\partial_{0156}$ & $\partial_{0345}$ & $\partial_{0456}$\\\hline
weight & $\lambda^{92}$ & $\lambda^{-92}$ & $\lambda^{200}$ & $\lambda^{-200}$ & $\lambda^{20}$ & $\lambda^{-88}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{1245}$ & $\partial_{1457}$ & $\partial_{0125} + \partial_{1456}$ & $\partial_{0134} + \partial_{0457}$ & $\partial_{0146} + \partial_{0245}$ & $\partial_{0157} + \partial_{1345}$\\\hline
weight & $\lambda^{-20}$ & $\lambda^{88}$ & $\lambda^{-110}$ & $\lambda^{110}$ & $\lambda^{2}$ & $\lambda^{-2}$
\end{tabular}$$
and the number of positive weights is 6.
At $\wil e_{0167}$ in $\wil U_{0167}$, the kernel of $\wil J_{0167}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0126}$ & $\partial_{0137}$ & $\partial_{0147}$ & $\partial_{0156}$ & $\partial_{0367}$ & $\partial_{0467}$\\\hline
weight & $\lambda^{-108}$ & $\lambda^{108}$ & $\lambda^{200}$ & $\lambda^{-200}$ & $\lambda^{20}$ & $\lambda^{112}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{1267}$ & $\partial_{1567}$ & $\partial_{0127} - \partial_{1467}$ & $\partial_{0136} - \partial_{0567}$ & $\partial_{0146} + \partial_{0267}$ & $\partial_{0157} + \partial_{1367}$\\\hline
weight & $\lambda^{-20}$ & $\lambda^{-112}$ & $\lambda^{90}$ & $\lambda^{-90}$ & $\lambda^{2}$ & $\lambda^{-2}$
\end{tabular}$$
and the number of positive weights is 6.
At $\wil e_{0357}$ in $\wil U_{0357}$, the kernel of $\wil J_{0357}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0135}$ & $\partial_{0137}$ & $\partial_{0345}$ & $\partial_{0347}$ & $\partial_{0356}$ & $\partial_{0367}$\\\hline
weight & $\lambda^{-110}$ & $\lambda^{90}$ & $\lambda^{2}$ & $\lambda^{202}$ & $\lambda^{-198}$ & $\lambda^{2}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{1357}$ & $\partial_{3457}$ & $\partial_{3567}$ & $\partial_{0157} - \partial_{2357}$ & $\partial_{0235} + \partial_{0567}$ & $\partial_{0237} + \partial_{0457}$\\\hline
weight & $\lambda^{-22}$ & $\lambda^{90}$ & $\lambda^{-110}$ & $\lambda^{-20}$ & $\lambda^{-108}$ & $\lambda^{92}$
\end{tabular}$$
and the number of positive weights is 6.
At $\wil e_{1246}$ in $\wil U_{1246}$, the kernel of $\wil J_{1246}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0124}$ & $\partial_{0126}$ & $\partial_{0246}$ & $\partial_{1245}$ & $\partial_{1247}$ & $\partial_{1256}$\\\hline
weight & $\lambda^{110}$ & $\lambda^{-90}$ & $\lambda^{22}$ & $\lambda^{-2}$ & $\lambda^{198}$ & $\lambda^{-202}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{1267}$ & $\partial_{2456}$ & $\partial_{2467}$ & $\partial_{0146} - \partial_{2346}$ & $\partial_{1234} + \partial_{1467}$ & $\partial_{1236} + \partial_{1456}$\\\hline
weight & $\lambda^{-2}$ & $\lambda^{-90}$ & $\lambda^{110}$ & $\lambda^{20}$ & $\lambda^{108}$ & $\lambda^{-92}$
\end{tabular}$$
and the number of positive weights is 6.
At $\wil e_{2345}$ in $\wil U_{2345}$, the kernel of $\wil J_{2345}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0234}$ & $\partial_{0345}$ & $\partial_{1235}$ & $\partial_{1245}$ & $\partial_{2347}$ & $\partial_{2356}$\\\hline
weight & $\lambda^{112}$ & $\lambda^{20}$ & $\lambda^{-112}$ & $\lambda^{-20}$ & $\lambda^{200}$ & $\lambda^{-200}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{2456}$ & $\partial_{3457}$ & $\partial_{0235} - \partial_{3456}$ & $\partial_{0245} + \partial_{2346}$ & $\partial_{1234} - \partial_{2457}$ & $\partial_{1345} + \partial_{2357}$\\\hline
weight & $\lambda^{-108}$ & $\lambda^{108}$ & $\lambda^{-90}$ & $\lambda^{2}$ & $\lambda^{90}$ & $\lambda^{-2}$
\end{tabular}$$
and the number of positive weights is 6.
At $\wil e_{2367}$ in $\wil U_{2367}$, the kernel of $\wil J_{2367}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0236}$ & $\partial_{0367}$ & $\partial_{1237}$ & $\partial_{1267}$ & $\partial_{2347}$ & $\partial_{2356}$\\\hline
weight & $\lambda^{-88}$ & $\lambda^{20}$ & $\lambda^{88}$ & $\lambda^{-20}$ & $\lambda^{200}$ & $\lambda^{-200}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{2467}$ & $\partial_{3567}$ & $\partial_{0237} + \partial_{3467}$ & $\partial_{0267} + \partial_{2346}$ & $\partial_{1236} + \partial_{2567}$ & $\partial_{1367} + \partial_{2357}$\\\hline
weight & $\lambda^{92}$ & $\lambda^{-92}$ & $\lambda^{110}$ & $\lambda^{2}$ & $\lambda^{-110}$ & $\lambda^{-2}$
\end{tabular}$$
and the number of positive weights is 6.
At $\wil e_{4567}$ in $\wil U_{4567}$, the kernel of $\wil J_{4567}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0456}$ & $\partial_{0467}$ & $\partial_{1457}$ & $\partial_{1567}$ & $\partial_{2456}$ & $\partial_{2467}$\\\hline
weight & $\lambda^{-88}$ & $\lambda^{112}$ & $\lambda^{88}$ & $\lambda^{-112}$ & $\lambda^{-108}$ & $\lambda^{92}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{3457}$ & $\partial_{3567}$ & $\partial_{0457} - \partial_{3467}$ & $\partial_{0567} + \partial_{3456}$ & $\partial_{1456} - \partial_{2567}$ & $\partial_{1467} + \partial_{2457}$\\\hline
weight & $\lambda^{108}$ & $\lambda^{-92}$ & $\lambda^{110}$ & $\lambda^{-90}$ & $\lambda^{-110}$ & $\lambda^{90}$
\end{tabular}$$
and the number of positive weights is 6.
At $\wil e_{0236}$ in $\wil U_{0236}$, the kernel of $\wil J_{0236}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0123}$ & $\partial_{0126}$ & $\partial_{0234}$ & $\partial_{0246}$ & $\partial_{0256}$ & $\partial_{0346}$\\\hline
weight & $\lambda^{88}$ & $\lambda^{-20}$ & $\lambda^{200}$ & $\lambda^{92}$ & $\lambda^{-110}$ & $\lambda^{110}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0356}$ & $\partial_{0367}$ & $\partial_{2356}$ & $\partial_{2367}$ & $\partial_{0136} + \partial_{0235}$ & $\partial_{0267} + \partial_{2346}$\\\hline
weight & $\lambda^{-92}$ & $\lambda^{108}$ & $\lambda^{-112}$ & $\lambda^{88}$ & $\lambda^{-2}$ & $\lambda^{90}$
\end{tabular}$$
and the number of positive weights is 7.
At $\wil e_{0456}$ in $\wil U_{0456}$, the kernel of $\wil J_{0456}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0145}$ & $\partial_{0156}$ & $\partial_{0246}$ & $\partial_{0256}$ & $\partial_{0345}$ & $\partial_{0346}$\\\hline
weight & $\lambda^{88}$ & $\lambda^{-112}$ & $\lambda^{92}$ & $\lambda^{-110}$ & $\lambda^{108}$ & $\lambda^{110}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0356}$ & $\partial_{0467}$ & $\partial_{2456}$ & $\partial_{4567}$ & $\partial_{0146} + \partial_{0245}$ & $\partial_{0567} + \partial_{3456}$\\\hline
weight & $\lambda^{-92}$ & $\lambda^{200}$ & $\lambda^{-20}$ & $\lambda^{88}$ & $\lambda^{90}$ & $\lambda^{-2}$
\end{tabular}$$
and the number of positive weights is 7.
At $\wil e_{1245}$ in $\wil U_{1245}$, the kernel of $\wil J_{1245}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0124}$ & $\partial_{0145}$ & $\partial_{1235}$ & $\partial_{1246}$ & $\partial_{1247}$ & $\partial_{1256}$\\\hline
weight & $\lambda^{112}$ & $\lambda^{20}$ & $\lambda^{-92}$ & $\lambda^{2}$ & $\lambda^{200}$ & $\lambda^{-200}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{1257}$ & $\partial_{1457}$ & $\partial_{2345}$ & $\partial_{2456}$ & $\partial_{0125} + \partial_{1456}$ & $\partial_{1234} - \partial_{2457}$\\\hline
weight & $\lambda^{-2}$ & $\lambda^{108}$ & $\lambda^{20}$ & $\lambda^{-88}$ & $\lambda^{-90}$ & $\lambda^{110}$
\end{tabular}$$
and the number of positive weights is 7.
At $\wil e_{1267}$ in $\wil U_{1267}$, the kernel of $\wil J_{1267}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0126}$ & $\partial_{0167}$ & $\partial_{1237}$ & $\partial_{1246}$ & $\partial_{1247}$ & $\partial_{1256}$\\\hline
weight & $\lambda^{-88}$ & $\lambda^{20}$ & $\lambda^{108}$ & $\lambda^{2}$ & $\lambda^{200}$ & $\lambda^{-200}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{1257}$ & $\partial_{1567}$ & $\partial_{2367}$ & $\partial_{2467}$ & $\partial_{0127} - \partial_{1467}$ & $\partial_{1236} + \partial_{2567}$\\\hline
weight & $\lambda^{-2}$ & $\lambda^{-92}$ & $\lambda^{20}$ & $\lambda^{112}$ & $\lambda^{110}$ & $\lambda^{-90}$
\end{tabular}$$
and the number of positive weights is 7.
At $\wil e_{0126}$ in $\wil U_{0126}$, the kernel of $\wil J_{0126}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0123}$ & $\partial_{0124}$ & $\partial_{0156}$ & $\partial_{0167}$ & $\partial_{0236}$ & $\partial_{0246}$\\\hline
weight & $\lambda^{108}$ & $\lambda^{200}$ & $\lambda^{-92}$ & $\lambda^{108}$ & $\lambda^{20}$ & $\lambda^{112}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0256}$ & $\partial_{1246}$ & $\partial_{1256}$ & $\partial_{1267}$ & $\partial_{0125} - \partial_{1236}$ & $\partial_{0146} + \partial_{0267}$\\\hline
weight & $\lambda^{-90}$ & $\lambda^{90}$ & $\lambda^{-112}$ & $\lambda^{88}$ & $\lambda^{-2}$ & $\lambda^{110}$
\end{tabular}$$
and the number of positive weights is 8.
At $\wil e_{0135}$ in $\wil U_{0135}$, the kernel of $\wil J_{0135}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0123}$ & $\partial_{0137}$ & $\partial_{0145}$ & $\partial_{0156}$ & $\partial_{0345}$ & $\partial_{0356}$\\\hline
weight & $\lambda^{92}$ & $\lambda^{200}$ & $\lambda^{92}$ & $\lambda^{-108}$ & $\lambda^{112}$ & $\lambda^{-88}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0357}$ & $\partial_{1235}$ & $\partial_{1356}$ & $\partial_{1357}$ & $\partial_{0136} + \partial_{0235}$ & $\partial_{0157} + \partial_{1345}$\\\hline
weight & $\lambda^{110}$ & $\lambda^{-20}$ & $\lambda^{-110}$ & $\lambda^{88}$ & $\lambda^{2}$ & $\lambda^{90}$
\end{tabular}$$
and the number of positive weights is 8.
At $\wil e_{1257}$ in $\wil U_{1257}$, the kernel of $\wil J_{1257}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{1235}$ & $\partial_{1237}$ & $\partial_{1245}$ & $\partial_{1247}$ & $\partial_{1256}$ & $\partial_{1267}$\\\hline
weight & $\lambda^{-90}$ & $\lambda^{110}$ & $\lambda^{2}$ & $\lambda^{202}$ & $\lambda^{-198}$ & $\lambda^{2}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{1357}$ & $\partial_{1457}$ & $\partial_{1567}$ & $\partial_{0125} + \partial_{2567}$ & $\partial_{0127} + \partial_{2457}$ & $\partial_{0157} - \partial_{2357}$\\\hline
weight & $\lambda^{18}$ & $\lambda^{110}$ & $\lambda^{-90}$ & $\lambda^{-88}$ & $\lambda^{112}$ & $\lambda^{20}$
\end{tabular}$$
and the number of positive weights is 8.
At $\wil e_{2456}$ in $\wil U_{2456}$, the kernel of $\wil J_{2456}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0246}$ & $\partial_{0256}$ & $\partial_{0456}$ & $\partial_{1245}$ & $\partial_{1246}$ & $\partial_{1256}$\\\hline
weight & $\lambda^{112}$ & $\lambda^{-90}$ & $\lambda^{20}$ & $\lambda^{88}$ & $\lambda^{90}$ & $\lambda^{-112}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{2345}$ & $\partial_{2356}$ & $\partial_{2467}$ & $\partial_{4567}$ & $\partial_{0245} + \partial_{2346}$ & $\partial_{1456} - \partial_{2567}$\\\hline
weight & $\lambda^{108}$ & $\lambda^{-92}$ & $\lambda^{200}$ & $\lambda^{108}$ & $\lambda^{110}$ & $\lambda^{-2}$
\end{tabular}$$
and the number of positive weights is 8.
At $\wil e_{3567}$ in $\wil U_{3567}$, the kernel of $\wil J_{3567}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0356}$ & $\partial_{0357}$ & $\partial_{0367}$ & $\partial_{1356}$ & $\partial_{1357}$ & $\partial_{1567}$\\\hline
weight & $\lambda^{-88}$ & $\lambda^{110}$ & $\lambda^{112}$ & $\lambda^{-110}$ & $\lambda^{88}$ & $\lambda^{-20}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{2356}$ & $\partial_{2367}$ & $\partial_{3457}$ & $\partial_{4567}$ & $\partial_{0567} + \partial_{3456}$ & $\partial_{1367} + \partial_{2357}$\\\hline
weight & $\lambda^{-108}$ & $\lambda^{92}$ & $\lambda^{200}$ & $\lambda^{92}$ & $\lambda^{2}$ & $\lambda^{90}$
\end{tabular}$$
and the number of positive weights is 8.
At $\wil e_{0256}$ in $\wil U_{0256}$, the kernel of $\wil J_{0256}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0126}$ & $\partial_{0156}$ & $\partial_{0236}$ & $\partial_{0246}$ & $\partial_{0356}$ & $\partial_{0456}$\\\hline
weight & $\lambda^{90}$ & $\lambda^{-2}$ & $\lambda^{110}$ & $\lambda^{202}$ & $\lambda^{18}$ & $\lambda^{110}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{1256}$ & $\partial_{2356}$ & $\partial_{2456}$ & $\partial_{0125} + \partial_{2567}$ & $\partial_{0235} + \partial_{0567}$ & $\partial_{0245} - \partial_{0267}$\\\hline
weight & $\lambda^{-22}$ & $\lambda^{-2}$ & $\lambda^{90}$ & $\lambda^{88}$ & $\lambda^{108}$ & $\lambda^{200}$
\end{tabular}$$
and the number of positive weights is 9.
At $\wil e_{1235}$ in $\wil U_{1235}$, the kernel of $\wil J_{1235}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0123}$ & $\partial_{0135}$ & $\partial_{1237}$ & $\partial_{1245}$ & $\partial_{1256}$ & $\partial_{1257}$\\\hline
weight & $\lambda^{112}$ & $\lambda^{20}$ & $\lambda^{200}$ & $\lambda^{92}$ & $\lambda^{-108}$ & $\lambda^{90}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{1356}$ & $\partial_{1357}$ & $\partial_{2345}$ & $\partial_{2356}$ & $\partial_{0125} - \partial_{1236}$ & $\partial_{1345} + \partial_{2357}$\\\hline
weight & $\lambda^{-90}$ & $\lambda^{108}$ & $\lambda^{112}$ & $\lambda^{-88}$ & $\lambda^{2}$ & $\lambda^{110}$
\end{tabular}$$
and the number of positive weights is 9.
At $\wil e_{1567}$ in $\wil U_{1567}$, the kernel of $\wil J_{1567}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0156}$ & $\partial_{0167}$ & $\partial_{1256}$ & $\partial_{1257}$ & $\partial_{1267}$ & $\partial_{1356}$\\\hline
weight & $\lambda^{-88}$ & $\lambda^{112}$ & $\lambda^{-108}$ & $\lambda^{90}$ & $\lambda^{92}$ & $\lambda^{-90}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{1357}$ & $\partial_{1457}$ & $\partial_{3567}$ & $\partial_{4567}$ & $\partial_{0157} + \partial_{1367}$ & $\partial_{1456} - \partial_{2567}$\\\hline
weight & $\lambda^{108}$ & $\lambda^{200}$ & $\lambda^{20}$ & $\lambda^{112}$ & $\lambda^{110}$ & $\lambda^{2}$
\end{tabular}$$
and the number of positive weights is 9.
At $\wil e_{0156}$ in $\wil U_{0156}$, the kernel of $\wil J_{0156}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0126}$ & $\partial_{0135}$ & $\partial_{0145}$ & $\partial_{0167}$ & $\partial_{0256}$ & $\partial_{0356}$\\\hline
weight & $\lambda^{92}$ & $\lambda^{108}$ & $\lambda^{200}$ & $\lambda^{200}$ & $\lambda^{2}$ & $\lambda^{20}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0456}$ & $\partial_{1256}$ & $\partial_{1356}$ & $\partial_{1567}$ & $\partial_{0125} + \partial_{1456}$ & $\partial_{0136} - \partial_{0567}$\\\hline
weight & $\lambda^{112}$ & $\lambda^{-20}$ & $\lambda^{-2}$ & $\lambda^{88}$ & $\lambda^{90}$ & $\lambda^{110}$
\end{tabular}$$
and the number of positive weights is 10.
At $\wil e_{2356}$ in $\wil U_{2356}$, the kernel of $\wil J_{2356}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0236}$ & $\partial_{0256}$ & $\partial_{0356}$ & $\partial_{1235}$ & $\partial_{1256}$ & $\partial_{1356}$\\\hline
weight & $\lambda^{112}$ & $\lambda^{2}$ & $\lambda^{20}$ & $\lambda^{88}$ & $\lambda^{-20}$ & $\lambda^{-2}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{2345}$ & $\partial_{2367}$ & $\partial_{2456}$ & $\partial_{3567}$ & $\partial_{0235} - \partial_{3456}$ & $\partial_{1236} + \partial_{2567}$\\\hline
weight & $\lambda^{200}$ & $\lambda^{200}$ & $\lambda^{92}$ & $\lambda^{108}$ & $\lambda^{110}$ & $\lambda^{90}$
\end{tabular}$$
and the number of positive weights is 10.
At $\wil e_{1356}$ in $\wil U_{1356}$, the kernel of $\wil J_{1356}$ is generated by the following vectors
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{0135}$ & $\partial_{0156}$ & $\partial_{0356}$ & $\partial_{1235}$ & $\partial_{1256}$ & $\partial_{1357}$\\\hline
weight & $\lambda^{110}$ & $\lambda^{2}$ & $\lambda^{22}$ & $\lambda^{90}$ & $\lambda^{-18}$ & $\lambda^{198}$
\end{tabular}$$
$$\begin{tabular}{c|c|c|c|c|c|c}
vector & $\partial_{1567}$ & $\partial_{2356}$ & $\partial_{3567}$ & $\partial_{0136} + \partial_{3456}$ & $\partial_{1236} + \partial_{1456}$ & $\partial_{1345} - \partial_{1367}$\\\hline
weight & $\lambda^{90}$ & $\lambda^{2}$ & $\lambda^{110}$ & $\lambda^{112}$ & $\lambda^{92}$ & $\lambda^{200}$
\end{tabular}$$
and the number of positive weights is 11.
\nocite{AK16,AS08,AS10,AS08Cal,Kar06,CG83,Zho05,FH13,Kna13,Hum12,Bor12,Joy00}
\bibliographystyle{alpha}
| {
"timestamp": "2019-03-01T02:03:42",
"yymm": "1711",
"arxiv_id": "1711.05169",
"language": "en",
"url": "https://arxiv.org/abs/1711.05169",
"abstract": "We define a torus action on the (complex) Cayley Grassmannian $X$. Using this action, we prove that $X$ is a singular variety. We also show that the singular locus is smooth and has the same cohomology ring as that of $\\mathbb{CP}^5$. Furthermore, we identify the singular locus with a quotient of $G_2^\\mathbb{C}$ by a parabolic subgroup.",
"subjects": "Algebraic Geometry (math.AG); Differential Geometry (math.DG); Representation Theory (math.RT)",
"title": "On the Complex Cayley Grassmannian",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808759252646,
"lm_q2_score": 0.7217431943271998,
"lm_q1q2_score": 0.7075110507281659
} |
https://arxiv.org/abs/1712.00875 | Ollivier Ricci curvature for general graph Laplacians: Heat equation, Laplacian comparison, non-explosion and diameter bounds | Discrete time random walks on a finite set naturally translate via a one-to-one correspondence to discrete Laplace operators. Typically, Ollivier curvature has been investigated via random walks. We first extend the definition of Ollivier curvature to general weighted graphs and then give a strikingly simple representation of Ollivier curvature using the graph Laplacian. Using the Laplacian as a generator of a continuous time Markov chain, we connect Ollivier curvature with the heat equation which is strongly related to continuous time random walks. In particular, we prove that a lower bound on the Ollivier curvature is equivalent to a certain Lipschitz decay of solutions to the heat equation. This is a discrete analogue to a celebrated Ricci curvature lower bound characterization by Renesse and Sturm. Our representation of Ollivier curvature via the Laplacian allows us to deduce a Laplacian comparison principle by which we prove non-explosion and improved diameter bounds. | \section*{Preface}
\pagestyle{plain}
\begin{abstract}
Discrete time random walks on a finite set naturally translate via a one-to-one correspondence to discrete Laplace operators.
Typically, Ollivier curvature has been investigated via random walks.
We first extend the definition of Ollivier curvature to general weighted graphs and then
give a strikingly simple representation of Ollivier curvature using the graph Laplacian.
Using the Laplacian as a generator of a continuous time Markov chain, we connect Ollivier curvature with the heat equation which is strongly related to continuous time random walks. In particular, we prove that a lower bound on the Ollivier curvature is equivalent to a certain Lipschitz decay of solutions to the heat equation.
This is a discrete analogue to a celebrated Ricci curvature lower bound characterization by Renesse and Sturm.
Our representation of Ollivier curvature via the Laplacian allows us to deduce a Laplacian comparison principle by which we prove non-explosion and improved diameter bounds.
\end{abstract}
\tableofcontents
\section{Introduction}
Ricci curvature is strongly related to the heat equation.
In particular, lower Ricci curvature bounds can be characterized via gradient estimates as in the following theorem by Renesse and Sturm
(see \cite[Theorem~1.3 and Corollary~1.4]{renesse2005transport}).
\begin{theorem}[Ricci curvature and gradient estimates]\label{thm:Sturm}
For any smooth connected Riemannian manifold $M$ and any $K \in {\mathbb{R}}$ the following properties are equivalent:
\begin{enumerate}
\item[(i)]
$Ric(M) \geq K$.
\item[(ii)]
For all $f \in C_c^{\infty}(M)$ and all $t > 0$
$$
\|\nabla P_t f \|_\infty \leq e^{-Kt} \|\nabla f \|_\infty.
$$
\item[(iii)]
For all bounded $f \in C^{Lip}(M)$ and all $t>0$
$$
Lip(P_t f) \leq e^{-Kt} Lip(f).
$$
\item[(iv)]
For all $x,y \in M$ and all $t>0$
\begin{align*}
W(p^x_t,p^y_t) \leq e^{-Kt} d(x,y)
\end{align*}
\end{enumerate}
where $P_t$ denotes the heat semigroup generated by the Laplace-Beltrami operator, $p_t^x$ denotes the heat kernel and $W$ denotes the $L_1$-Wasserstein distance.
\end{theorem}
We prove that the same holds true on graphs (see Theorem~\ref{thm:gradientGraphs}).
To do so, we employ a new method intertwining the heat semigroup with a cutoff function, which we call the perpetual cutoff method.
Our curvature notion will be a generalized Ollivier curvature based on its modification by Lin, Lu and Yau (see \cite{lin2011ricci}) which we extend to the case of general graph Laplacians. In particular, we now apply this curvature notion to graphs with unbounded vertex degree.
For an introduction to Ollivier curvature, see \cite{ollivier2007ricci, ollivier2009ricci}.
A relation between curvature and the number of triangles in a graph is given in \cite{jost2014ollivier}.
Methods to compute the curvature can be found in \cite{loisel2014ricci}.
Ollivier curvature has been applied to describe the internet topology \cite{ni2015ricci,wang2016interference}, wireless network theory \cite{wang2014wireless}, economic market analysis \cite{sandhu2016ricci} and cancer networks \cite{sandhu2015graph,tannenbaum2015ricci,sandhu2015analytical}.
The breakthrough paper of Renesse and Sturm mentioned above motivated a generalization of Ollivier curvature to semiproups compatible with Lipschitz functions and Wasserstein metrics. Approaches in this direction have been made in \cite{bass1981markov,joulin2007poisson,joulin2009new,veysseire2012coarse}). However, the problem suggested by Ollivier (see Problem~D in \cite{ollivier2010survey}), namely, if a lower curvature bound implies non-explosion (also known as stochastic completeness), is still open.
Non-explosion in this context means that the process remains in the state space for all time.
We prove non-explosion for all locally finite graph Laplacians with Ollivier curvature decaying not faster than $-\log R$ (see Theorem~\ref{thm:StochComplete}). Therefore, this theorem can be seen as an initial step to solve Ollivier's problem in a general setting. Moreover, we prove that the curvature decay rate $-\log R$ is optimal.
One of the main observations of our paper is that, on graphs, the limit expression for Lin, Lu and Yau's Ollivier curvature simplifies to the limit-free expression
\begin{align*}
\kappa(x,y) = \inf_{\substack {f\in Lip(1) \\ \nabla_{yx}f=1}} \nabla_{xy} \Delta f
\end{align*}
where $\nabla_{xy} f = \frac{f(x)-f(y)}{d(x,y)}$, $d$ is the usual combinatorial graph distance, $Lip(1)$ denotes the functions with Lipschitz constant 1 with respect to this metric and $\Delta$ is the graph Laplacian (see Theorem~\ref{thm:nablaDelta}).
Furthermore, the curvature admits another limit-free expression in terms of transport costs (see Proposition~\ref{pro:CharTransport}) which simplifies to give an explicit formula in the case of combinatorial graph (see Theorem~\ref{thm:TransprtCombinatorial}).
These simplifications give the starting point for proving the semigroup characterization of a lower Ricci curvature bound in analogy to the work of Renesse and Sturm.
Our gradient estimates for the continuous time heat equation seem to be the first result of this kind for Ollivier curvature.
Indeed, these gradient estimates have been previously used as a globally defined Wasserstein curvature bound (see \cite[Definition~2.1]{joulin2007poisson}).
In contrast to Ollivier curvature, there are various gradient estimates on graphs under Bakry-Emery curvature bounds \cite{horn2014volume,gong2015properties,munch2014li,lin2015equivalent,bauer2015li} and under entropic Ricci curvature \cite[Theorem~3.1]{erbar2016poincar}.
Using a modification of Ollivier curvature, gradient estimates have been established for continuous time Markov processes in \cite{veysseire2012coarse}. In Section~\ref{sec:MarkovProcesses}, we prove that this modification coincides with our curvature notion on locally finite graphs with a lower curvature bound (see Corollary~\ref{cor:MarkovChains}).
The control of the Lipschitz constant of the semigroup yields stochastic completeness
for all graphs with a constant lower Ollivier curvature bound (see Lemma~\ref{lem:StochComplete}). However, as already mentioned above, one can get even better results by employing different techniques.
To do so, we first establish a Laplacian comparison principle which seems to be the first of its kind for any discrete Ricci curvature notion.
The simplest version (Theorem~\ref{thm:LaplaceCompare}) states that under the assumption of a lower curvature bound $K \in {\mathbb{R}}$, we have
\begin{align*}
\Delta d(x_0,\cdot) \leq \operatorname{Deg}(x_0) -Kd(x_0,\cdot)
\end{align*}
where $\operatorname{Deg}(x_0)$ is the weighted vertex degree of a fixed vertex $x_0$.
This Laplacian comparison can be extended to the case of decaying curvature (see Theorem~\ref{thm:LaplaceComparisonNonConst}).
Via the Laplacian comparison, we compare the curvature of a given graph to the curvature of the birth-death chain associated to the graph (see Corollary~\ref{cor:CurvCompare}).
Birth-death chains are Markov processes on weighted path graphs. For our purposes, we identify the birth-death chain with its associated weighted path graph, see Section~\ref{sec:OllivierBirthDeath}.
The comparison to birth-death chains allows us to reduce many problems to the case of weighted path graphs where the curvature can be easily calculated (see Theorem~\ref{thm:line}).
Using the Laplacian comparison principle and Khas'minskii's criterion (see \cite{huang2011stochastic}), we will prove stochastic completeness under the assumption that the Ollivier curvature does not decay to $-\infty$ faster than $-\log R$ (see Theorem~\ref{thm:StochComplete}).
This result is remarkable when comparing to known stochastic completeness results for graphs which use the Bakry-Emery curvature instead and require a constant lower bound
as well as additional assumptions, such as a non-local completeness condition and a lower bounded vertex measure (see \cite[Theorem~1.2]{hua2017stochastic}).
As such, this article may also give inspiration to transfer the new methods presented here to discrete Bakry-Emery theory.
As another application of the Laplacian comparison principle, we can prove finiteness and improved diameter bounds.
For diameter bounds under uniformly positive Ollivier curvature, see \cite{lin2011ricci, ollivier2009ricci}. Diameter bounds under a positive average Ollivier curvature can be found in \cite{paeng2012volume}. Diameter bounds under uniformly positive Bakry-Emery curvature are proven in \cite{liu2016bakry,fathi2015curvature,horn2014volume}.
In this article we show that if the vertex degree is bounded and the curvature decays not faster than $1/R$, then the graph is finite (see Theorem~\ref{thm:ImprovedDiamBound} and Corollary~\ref{cor:finite}).
Surprisingly, uniformly positive Ricci curvature alone does not imply finiteness (see Example~\ref{Ex:positiveCurvInfiniteDiam}). However, if we additionally assume a lower bound on the vertex measure, then uniformly positive Ricci curvature indeed implies finiteness (see Corollary~\ref{cor:FiniteDiamBoundedMeasure}).
\subsection{Discussion and comparison to manifolds}
The reader familiar with the manifold case might be surprised at the optimal curvature decay rates: $- \log R$ for stochastic completeness and $1/R$ for compactness in the case of graphs.
In \cite[Theorem~15.4]{grigor1999analytic}, it is shown that, for manifolds, the optimal curvature decay rate for stochastic completeness is $-R^2$ which was proven in \cite{varopoulos1983potential, hsu1989heat}.
One tempting explanation for the discrepancy of the decay rate between manifolds and graphs is the choice of the metric. Frequently, intrinsic metrics introduced in \cite{frank2014intrinsic} are used to describe the geometry of graphs with unbounded vertex degrees and to give analogues to results on manifolds (see, for example, \cite{bauer2012cheeger,huang2013note,folz2015volume,keller2015intrinsic}).
However, we give an example of a stochastically incomplete graph with curvature decaying like $-(\log \sigma)^{1+\varepsilon}$ with respect to an intrinsic metric $\sigma$, even if the curvature is defined by incorporating the intrinsic metric, see Example~\ref{ex:incompleteIntrinsic}.
The optimal decay rate on manifolds to guarantee compactness is $C/R^2$ for some constant $C$. Interestingly, for $C>\frac{n-1}4$, compactness holds, but for $C \leq \frac{n-1}4$, non-compact manifolds are known with the corresponding Ricci curvature decay (see \cite{cheeger1982finite,holcman2005boundary}).
The discrepancy of the decay rate between manifolds and graphs here also cannot be explained via intrinsic metrics since we assume a bounded vertex degree for our result and, therefore, the combinatorial distance is intrinsic up to a factor. Hence, it might be interesting to ask for the deeper reasons for these two discrepancies.
Before introducing the setup and notations, we give a brief summary of the subsequent sections.
In Section~\ref{sec:Ollivier}, we prove the limit-free simplifications of the curvature formula and compute the curvature of combinatorial graphs and birth-death chains.
In Section~\ref{sec:GradEstimates}, we introduce the perpetual cutoff method and non-linear cutoff semigroups which turn out to perfectly intermesh with Ollivier curvature, yielding the desired gradient estimate for the semigroup.
In Section~\ref{sec:LaplaceCompare}, we present the Laplacian comparison theorem and, as applications, we prove a birth-death chain reduction, stochastic completeness and improved diameter bounds which lead to our finiteness results.
Finally in Section~\ref{sec:MarkovProcesses}, we prove that on graphs with a lower curvature bound, our curvature notion coincides with the curvature introduced in \cite{veysseire2012coarse}.
\subsection{Setup and Notation}
A triple $G=(V,w,m)$ is called a \emph{graph} if $V$ is a countable set, $w:V^2 \to [0,\infty)$ is symmetric and zero on the diagonal and $m:V \to (0,\infty)$. In the following, we only consider \emph{locally finite} graphs, i.e., for every $x \in V$ there are only finitely many $y \in V$ with $w(x,y) >0$.
We call $V$ the \emph{vertex set} with elements of $V$ called \emph{vertices}, $w$ the \emph{edge weight} and $m$ the \emph{vertex measure}.
We will write $x \sim y$ if $w(x,y)>0$ and say that $(x,y)$ is an \emph{edge} in the graph.
We say that $G$ is a \emph{combinatorial graph} if $w(x,y) \in \{0,1\}$ for all $x,y \in V$ and if $m \equiv 1$.
We define the \emph{graph Laplacian} $\Delta: {\mathbb{R}}^V \to {\mathbb{R}}^V$ via
$$ \Delta f(x) := \frac 1 {m(x)} \sum_{y\in V} w(x,y)(f(y) - f(x)). $$
We define the function spaces
\begin{align*}
C(V)&:=\{f:V \to {\mathbb{R}}\}= {\mathbb{R}}^V, \\
\ell_\infty(V)&:=\{f \in C(V): f \mbox{ is bounded}\}, \\
C_c(V) &:= \{f \in C(V): f \mbox{ is finitely supported}\},
\end{align*}
all endowed with the supremum norm $\|\cdot \|_\infty$.
We let $\ell^p(V,m)$ denote the $\ell^p$ spaces with respect to $m$, that is,
$\ell^p(V,m) = \{ f \in C(V) : \sum_{x \in V} |f(x)|^p m(x) <\infty\}.$
We let
$$\operatorname{Deg}(x) := \frac1 {m(x)} \sum_{y \in V} {w(x,y)}$$
denote the \emph{vertex degree} and let $\operatorname{Deg}_{\max} := \sup_x \operatorname{Deg}(x) \in (0,\infty]$. We remark that the Laplace operator is bounded on $\ell_\infty(V)$ and $\ell^p(V,m)$ for any $p\geq 1$ if and only if $\operatorname{Deg}_{\max} < \infty$ (see \cite[Theorem 11]{keller2010unbounded}, \cite[Theorem 9.3]{haeseler2011laplacians}).
In this case, we will say that $G$ has \emph{bounded vertex degree}.
For a non-negative $f \in \ell_\infty(V)$, we denote by $P_t f$ the smallest non-negative bounded continuous solution $u(x,t)$ to the heat equation
\begin{align*}
\begin{cases}
\begin{tabular}{rll}
$\Delta u(x,t)$ & $=\partial_t u(x,t)$ & $x \in V$, $t\geq0$ \\
$u(x,0)$ & $= f(x)$ &$x \in V$.
\end{tabular}
\end{cases}
\end{align*}
A proof of the existence and uniqueness of $P_t f$ and further details can be found in \cite{wojciechowski2008heat, keller2012dirichlet}. Note, in particular, that $P_{s+t} f = P_s P_t f$ which is referred to as the \emph{semigroup property} and that $P_t$ is \emph{positivity preserving}, i.e., $P_t f\geq0$ for $f\geq0$.
A graph is called \emph{stochastically complete} or \emph{non-explosive} if $P_t \mathbf 1 = \mathbf 1$ for all $t>0$ where $\mathbf 1$ is the constant function which is 1 on $V$.
We define the \emph{combinatorial graph distance} $d$ on $V\times V$ via
$d(x,y):= \inf\{n:x=x_0\sim \ldots \sim x_n=y\}$. A graph is said to be \emph{connected} if $d(x,y) < \infty$ for all $x,y$ in $V$. We will always assume that graphs are connected. We write $B_r(x) = \{ y \in V \ | \ d(x,y) \leq r \}$ and $S_r(x) = \{ y \in V \ | \ d(x,y)=r \}$.
We note that $G$ is connected if and only if $P_t$ is a positivity improving semigroup, that is, $P_t f >0$ if $f \geq 0$ for $t>0$, see \cite{keller2012dirichlet}.
We write $f \in Lip(1)$ if $|f(x)-f(y)| \leq d(x,y)$ for all $x,y \in V$.
The \emph{Wasserstein distance} $W(\mu,\nu)$ for probability measures $\mu$ and $\nu$ on $V$ is given by
\begin{align*}
W(\mu,\nu) &:= \sup_{f \in Lip(1) \cap \ell_\infty(V)} \int f d\mu - \int f d\nu \\
&= \sup_{f \in Lip(1) \cap \ell_\infty(V)} \sum_{x \in V} f(x) (\mu(x) - \nu(x)).
\end{align*}
We note that the supremum is well defined due to the boundedness of the functions and that it suffices to take the supremum over functions in $Lip(1)$ when the measures are finitely supported.
Equivalently (see e.g. \cite[Theorem 1.14]{villani2003topics}), the Wasserstein metric can be defined as
$$
W(\mu,\nu) := \inf_{\rho} \sum_{x,y \in V} \rho(x,y) d(x,y)
$$
where the infimum is taken over all $\rho: V^2 \to [0,1]$ which satisfy $\sum_{y \in V} \rho(x,y) = \mu(x)$ and $\sum_{x \in V} \rho(x,y) = \nu(y)$ for all $x,y \in V$. We call such a $\rho$ a \emph{coupling} between $\mu$ and $\nu$.
\section{Ollivier curvature and graph Laplacians}\label{sec:Ollivier}
Ollivier curvature is a powerful and easy to calculate tool used to study analytic and geometric properties of a space. Until now, Ollivier curvature for graphs has only been used in the case of bounded Laplacians. In this section, we extend the definition of Ollivier curvature to the case of unbounded graph Laplacians. We then present a strikingly simple expression for calculating the curvature using only the Laplacian as well as a formula involving transport costs. Along the way, we illustrate how to calculate the curvature in a variety of situations including graphs without cycles, combinatorial graphs and birth-death chains.
\bigskip
For $\varepsilon>0$, we let
\begin{align*}
m_x^\varepsilon(y) := 1_y(x) + \varepsilon \Delta 1_y (x)
\end{align*}
which is a finitely supported probability measure and, in particular, non-negative if $\varepsilon$ is sufficiently small.
This can be seen as
\begin{align*}
m_x^\varepsilon (y) = \begin{cases}
1 - \varepsilon \operatorname{Deg}(x) &: y=x \\
\varepsilon w(x,y)/m(x) &: \mbox{otherwise}.
\end{cases}
\end{align*}
In particular,
\begin{align*}
\int f dm_x^\varepsilon = \sum_{y \in V} f(y) m_x^\varepsilon(y) = (f + \varepsilon \Delta f) (x).
\end{align*}
We remark that $m_x^\varepsilon$ can be seen as a first order approximation to the heat kernel $P_\varepsilon 1_x$. This connection will be further explored in Section~\ref{sec:MarkovProcesses}.
In the case of the normalized Laplacian, that is, when $w: V^2 \to \{ 0,1\}$ and $m(x) = d_x := \# \{y \sim x \}$, for $\alpha := 1-\varepsilon$ one has
\begin{align*}
m_x^\varepsilon (y) = \begin{cases}
\alpha &: y=x \\
(1-\alpha)/d_x&: y \sim x \\
0 &: \mbox{otherwise}
\end{cases}
\end{align*}
which corresponds to the definition of Lin, Lu and Yau (see \cite{bauer2011ollivier,lin2011ricci}). Note that, in this case, $\operatorname{Deg}=1$ so that the normalized Laplacian is always a bounded operator.
Following the standard definition, we let, for $x \not = y$
\begin{align*}
\kappa_\varepsilon(x,y) := 1 - \frac{W(m_x^\varepsilon,m_y^\varepsilon)}{d(x,y)}
\end{align*}
where $W$ denotes the Wasserstein distance.
In \cite{bourne2017ollivier} it is shown that for the normalized Laplacian and $x \sim y$, the function $\kappa_\varepsilon(x,y)$ is concave and piecewise linear in $\varepsilon \in [0,1]$ with at most three linear parts.
Analogous to Lin, Lu and Yau, one can prove the existence of
\begin{align*}
\kappa(x,y) := \lim_{\varepsilon \to 0^+} \frac 1 \varepsilon \kappa_\varepsilon(x,y)
\end{align*}
by which we extend Lin, Lu and Yau's curvature definition to arbitrary graph Laplacians.
Using standard theory, the curvature $\kappa(x,y)$ is uniquely determined by the induced subgraph $B_1(x) \cup B_1(y)$ for $x \sim y$.
We write $Ric(G) \geq K$ if $\kappa(x,y) \geq K$ for all $x,y$.
We remark that to show $Ric(G) \geq K$ it suffices to show that $\kappa(x,y) \geq K$ for adjacent vertices $x \sim y$ as in \cite{lin2011ricci}.
As a first example, we mention that it is well-known that, in the case of the normalized Laplacian, Abelian Cayley graphs have non-negative Ollivier curvature (see e.g. \cite[Theorem~2]{lin2014ricci}). We will give further examples later in this section.
\subsection{Bypassing the limit}
Our first aim is to express the curvature without the limit which turns out to be surprisingly simple. To do so, we introduce the notation of the gradient
$$
\nabla_{xy} f :=
\frac{f(x) - f(y)}{d(x,y)}
$$
for $x\neq y \in V$ and $f \in C(V)$ and the associated Lipschitz constant
$$
\|\nabla f \|_\infty := \sup_{x \neq y} |\nabla_{xy} f| = \sup_{x \sim y} |\nabla_{xy} f| \in [0,\infty].
$$
For $K \geq 0$, we let $Lip(K) = \{f \in C(V): \|\nabla f \|_\infty \leq K \},$ that is, the set of functions with Lipschitz constant $K$ or $K$-Lipschitz functions.
We are now prepared to present our limit-free curvature formula.
\begin{theorem}[Curvature via the Laplacian]\label{thm:nablaDelta}
Let $G=(V,w,m)$ be a graph and let $x \neq y$ be vertices. Then,
\begin{align*}
\kappa(x,y) = \inf_{\substack {f\in Lip(1) \cap C_c(V) \\ \nabla_{yx}f=1}} \nabla_{xy} \Delta f.
\end{align*}
\end{theorem}
\begin{proof}
By definition, one has
\begin{align*}
W(m_x^\varepsilon,m_y^\varepsilon) &= \sup_{f \in Lip(1)} \sum_z f(z)(m_y^\varepsilon(z) - m_x^\varepsilon(z))\\
&= \sup_{f \in Lip(1)} [(f(y) + \varepsilon \Delta f(y)) - (f(x) + \varepsilon \Delta f(x))]\\
&= d(x,y)\sup_{f \in Lip(1)} \nabla_{yx}(f + \varepsilon \Delta f).
\end{align*}
Hence,
\begin{align*}
\frac 1 \varepsilon \kappa_\varepsilon(x,y)
&= \frac 1 \varepsilon \left( 1- \frac{W(m_x^\varepsilon,m_y^\varepsilon)}{d(x,y)} \right) \\
&= \frac 1 \varepsilon \left( \inf_{f \in Lip(1)} ( 1- \nabla_{yx}(f + \varepsilon \Delta f) ) \right)\\
& = \inf_{f \in Lip(1)} \left(\frac 1 \varepsilon(1- \nabla_{yx}f) +\nabla_{xy} \Delta f \right)\\
&\leq \inf_{\substack {f\in Lip(1) \cap C_c(V) \\ \nabla_{yx}f=1}} \nabla_{xy} \Delta f
\end{align*}
To prove the other inequality, we first show the existence of a minimizer $f_\varepsilon \in Lip(1) \cap C_c(V)$ of the expression $\frac 1 \varepsilon(1- \nabla_{yx}f) +\nabla_{xy} \Delta f$ found above for every $\varepsilon >0$ satisfying $f_\varepsilon(x)=0$.
This follows as, for every $f \in Lip(1)$, we construct $\widetilde f \in Lip(1)$ supported on $B_{2r}(x)$ with $r:=d(x,y)+1$ which satisfies
\begin{align}
\frac 1 \varepsilon(1- \nabla_{yx}f) +\nabla_{xy} \Delta f = \frac 1 \varepsilon(1- \nabla_{yx}\widetilde f) +\nabla_{xy} \Delta \widetilde f. \label{eq:fwidetildefMaxExist}
\end{align}
By adding a constant to $f$, we can assume that $f(x)=0$.
This yields that $|f(z)|\leq r$ for all $z \in B_1(x) \cup B_1(y)$ since $f \in Lip(1)$.
Let $\phi:V \to {\mathbb{R}}$ be given by
\[
\phi(z) = \left[r \wedge (2r - d(x,z)) \right]_+.
\]
Observe that $\phi(z)=r$ for all $z \in B_1(x)\cup B_1(y)$.
Therefore,
$\widetilde f := -\phi \vee f \wedge \phi$
satisfies (\ref{eq:fwidetildefMaxExist}) as it agrees with $f$ on $B_1(x)\cup B_1(y)$. Moreover, $\phi$ and thus $\widetilde f$ are supported on $B_{2r}(x)$.
This construction of $\widetilde f$ shows that we can restrict the infimum to functions supported on the compact set $B_{2r}(x)$ which yields the existence of a minimizer $f_\varepsilon$ with $f_\varepsilon(x)=0$ for all $\varepsilon>0$ due to continuity.
Due to the compactness of $B_{2r}(x)$ and since $f_\varepsilon(x)=0$ and $f_\varepsilon \in Lip(1)$ for all $\varepsilon>0$, there exists a sequence $\varepsilon_n \to 0$ such that $f_0 := \lim_{n\to \infty} f_{\varepsilon_n}$ exists.
Since $\frac 1 \varepsilon \kappa_\varepsilon(x,y) = \frac{1}{\varepsilon} (1-\nabla_{yx}f_\varepsilon)+\nabla_{xy}\Delta f_\varepsilon$ and $\lim_{\varepsilon \to 0^+} \frac 1 \varepsilon \kappa_\varepsilon(x,y)$ exists,
we get that $\nabla_{yx}f_\varepsilon \to 1$ as $\varepsilon \to 0^+$.
Therefore, $f_0 \in Lip(1) \cap C_c(V)$, $\nabla_{yx}f_0 = 1$ and since $\nabla_{yx} f_\varepsilon \leq 1$, we get
\begin{align*}
\kappa(x,y) &= \lim_{\varepsilon \to 0^+} \frac 1 \varepsilon(1- \nabla_{yx}f_\varepsilon) +\nabla_{xy} \Delta f_\varepsilon \\
&\geq \lim_{n \to \infty} \nabla_{xy} \Delta f_{\varepsilon_n} \\
& = \nabla_{xy} \Delta f_0\\
&\geq \inf_{\substack {f\in Lip(1) \cap C_c(V) \\ \nabla_{yx}f=1}} \nabla_{xy} \Delta f.
\end{align*}
Putting together the upper and lower estimates yields
\begin{align*}
\kappa(x,y) = \inf_{\substack {f\in Lip(1) \cap C_c(V) \\ \nabla_{yx}f=1}} \nabla_{xy} \Delta f
\end{align*}
as desired.
\end{proof}
Following \cite[Lemma~2.2]{bhattacharya2015exact}, it suffices to optimize over all integer valued Lipschitz functions $f$ which yields the following corollary.
\begin{corollary}
Let $G=(V,w,m)$ be a graph and let $x \neq y$ be vertices. Then,
\begin{align*}
\kappa(x,y) = \inf_{\substack {f: B_1(x) \cup B_1(y) \to {\mathbb{Z}} \\ f\in Lip(1) \\ \nabla_{yx}f=1}} \nabla_{xy} \Delta f
\end{align*}
Moreover, on combinatorial graphs, the curvature $\kappa(x,y)$ is integer valued for all $x\sim y$.
\end{corollary}
\begin{proof}
The proof of the first part follows \cite[Lemma~2.2]{bhattacharya2015exact}. For the integrality of the curvature in the case of combinatorial graphs, observe that $\nabla_{xy}\Delta f \in {\mathbb{Z}}$ whenever $f$ is integer valued, $x\sim y$ and $\Delta$ is the combinatorial graph Laplacian.
\end{proof}
We now explicitly calculate the curvature of large girth graphs in our setting by using Theorem~\ref{thm:nablaDelta}.
\begin{example}\label{ex:NoCycles}
Let $G=(V,w,m)$ be a graph and let $x\sim y$ be vertices.
Suppose that the edge $(x,y)$ is not contained in any 3-,4- or 5-cycles. Then, an optimal 1-Lipschitz function $f$ is given by an extension of
\[
f(z) = \begin{cases}
0 &: z \sim x \mbox{ and } z \neq y\\
1 &: z=x\\
2 &: z=y\\
3 &: z \sim y \mbox{ and } z \neq x
\end{cases}
\]
yielding the curvature
\[
\kappa(x,y) = 2w(x,y) \left(\frac 1{m(x)} + \frac 1 {m(y)} \right) - \operatorname{Deg}(x) - \operatorname{Deg}(y).
\]
\end{example}
\bigskip
We now give another limit-free expression of our extension of Lin-Lu-Yau's Ollivier curvature via transport costs.
\begin{proposition}[Curvature via transport cost]\label{pro:CharTransport}
Let $G=(V,w,m)$ be a graph and let $x_0 \neq y_0$ be vertices. Then,
\begin{align}
\kappa(x_0,y_0) &= \sup_{\rho} \sum_{\substack{x \in B_1(x_0) \\ y \in B_1(y_0)}}\rho(x,y) \left[1 - \frac{d(x,y)}{d(x_0,y_0)}\right] \label{eq:PropTransport}
\end{align}
where the supremum is taken over all $\rho: B_1(x_0) \times B_1(y_0) \to [0,\infty)$ such that
\begin{align}
\sum_{y \in B_1(y_0)} \rho(x,y) &= \frac {w(x_0,x)}{m(x_0)} \qquad \mbox{ for all } x \in S_1(x_0) \mbox{ and} \label{eq:rhoXProp}\\
\sum_{x \in B_1(x_0)} \rho(x,y) &= \frac {w(y_0,y)}{m(y_0)} \qquad \mbox{ for all } y \in S_1(y_0) \label{eq:rhoYProp}.
\end{align}
\end{proposition}
\begin{rem}
We remark that $\rho$ is defined on balls, but we only require the coupling property on spheres. We additionally do not assume anything concerning
$\sum_{x,y} \rho(x,y)$.
\end{rem}
\begin{proof}
We will write
\[ F(\rho) = \sum_{x \in B_1(x_0)} \sum_{y \in B_1(y_0)} \rho(x,y) \left[1 - \frac{d(x,y)}{d(x_0,y_0)}\right] \]
for any coupling $\rho$. We wish to show that $\kappa(x_0,y_0) = \sup_\rho F(\rho)$ where the supremum is taken over all couplings $\rho$ satisfying (\ref{eq:rhoXProp}) and (\ref{eq:rhoYProp}).
Using that $\sum_{x,y}\rho(x,y)=1$ for all couplings considered in the transport definition of $W$, we have
\begin{align*}
\kappa_\varepsilon(x_0,y_0) = 1 - \frac {W(m_{x_0}^\varepsilon,m_{y_0}^\varepsilon)}{d(x_0,y_0)} = 1 - \frac{\inf_\rho \sum_{x,y} \rho(x,y)d(x,y)}{d(x_0,y_0)} = \sup_\rho F(\rho)
\end{align*}
where the supremum is taken over all $\rho: B_1(x_0) \times B_1(y_0) \to [0,\infty)$ such that
\begin{align*}
\sum_{y \in B_1(y_0)} \rho(x,y) &= m_{x_0}^\varepsilon(x) = 1_x(x_0) + \varepsilon \Delta 1_x(x_0) \qquad \mbox{ for all } x \in B_1(x_0) \mbox{ and} \\
\sum_{x \in B_1(x_0)} \rho(x,y) &= m_{y_0}^\varepsilon(y) = 1_y(y_0) + \varepsilon \Delta 1_y(y_0) \qquad \mbox{ for all } y \in B_1(y_0).
\end{align*}
Dividing $\rho$ by $\varepsilon$ yields
\begin{align*}
\frac{1}{\varepsilon} \kappa_\varepsilon(x_0,y_0) = \frac 1 {\varepsilon} \left(1 - \frac {W(m_{x_0}^\varepsilon,m_{y_0}^\varepsilon)}{d(x_0,y_0)} \right) =
\sup_\rho F(\rho)
\end{align*}
with the supremum taken over all $\rho: B_1(x_0) \times B_1(y_0) \to [0,\infty)$ such that
\begin{align}
\sum_{y \in B_1(y_0)} \rho(x,y) &= \frac 1 \varepsilon 1_x(x_0) + \Delta 1_x(x_0) \qquad \mbox{ for all } x \in B_1(x_0) \mbox{ and} \label{eq:rhoXProof} \\
\sum_{x \in B_1(x_0)} \rho(x,y) &= \frac 1 \varepsilon 1_y(y_0) + \Delta 1_y(y_0) \qquad \mbox{ for all } y \in B_1(y_0) \label{eq:rhoYProof}.
\end{align}
We remark that (\ref{eq:rhoXProp}) and (\ref{eq:rhoYProp}) hold for all $\rho$ satisfying (\ref{eq:rhoXProof}) and (\ref{eq:rhoYProof}) as
$\Delta 1_x (x_0)= \frac {w(x_0,x)}{m(x_0)}$ for $x \not = x_0$.
Therefore, $\frac 1 \varepsilon \kappa_\varepsilon(x_0,y_0)$ is less than or equal to the right hand side of (\ref{eq:PropTransport}).
We now show that if we modify $\rho$ satisfying (\ref{eq:rhoXProp}) and (\ref{eq:rhoYProp}) appropriately, then the value of $F(\rho)$ in the right hand side of (\ref{eq:PropTransport}) does not change and the modified $\rho$ satisfies (\ref{eq:rhoXProof}) and (\ref{eq:rhoYProof}) which will show that $\frac 1 \varepsilon \kappa_\varepsilon(x_0,y_0)$ is larger than or equal to the right hand side of (\ref{eq:PropTransport}) for small $\varepsilon$.
Suppose that $\rho$ satisfies (\ref{eq:rhoXProp}) and (\ref{eq:rhoYProp}).
We define
\begin{align*}
\rho_\varepsilon(x,y) := \rho(x,y) + 1_{x}(x_0)1_{y}(y_0) \left( \frac 1 \varepsilon - \sum_{u,v} \rho(u,v) \right)
\end{align*}
which is non-negative if $\varepsilon$ is small.
We observe that
\begin{align*}
F(\rho)= \sum_{x,y} \rho(x,y) \left[1 - \frac{d(x,y)}{d(x_0,y_0)} \right] = \sum_{x,y} \rho_\varepsilon(x,y) \left[1 - \frac{d(x,y)}{d(x_0,y_0)} \right] = F(\rho_\varepsilon)
\end{align*}
since $\rho_\varepsilon(x,y)$ and $\rho(x,y)$ only differ at $(x_0,y_0)$ where the latter factor in the sums vanishes.
Moreover,
\begin{align*}
\sum_{x \in B_1(x_0)} \sum_{y \in B_1(y_0)} \rho_\varepsilon(x,y) = \frac 1 \varepsilon.
\end{align*}
We now show that $\rho_\varepsilon$ satisfies (\ref{eq:rhoXProof}).
Since $\frac 1 \varepsilon 1_x(x_0) = 0$ on $S_1(x_0)$, we see that (\ref{eq:rhoXProp}) implies (\ref{eq:rhoXProof}) for $x\in S_1(x_0)$.
For the remaining case $x=x_0$, equation (\ref{eq:rhoXProof}) follows since by (\ref{eq:rhoXProp}),
\begin{align*}
\sum_{y\in B_1(y_0)} \rho_\varepsilon(x_0,y) = \frac 1 \varepsilon - \sum_{x \in S_1(x_0)} \sum_{y \in B_1(y_0)} \rho(x,y) = \frac 1 \varepsilon - \sum_{x \in S_1(x_0)} \frac{w(x_0,x)}{m(x_0)} = \frac 1 \varepsilon + \Delta 1_{x_0} (x_0).
\end{align*}
Due to an analogous argument, $\rho_\varepsilon$ also satisfies (\ref{eq:rhoYProof}).
Putting everything together proves that $\frac 1 \varepsilon \kappa_\varepsilon(x_0,y_0)$ equals the right hand side of (\ref{eq:PropTransport}) for small $\varepsilon$.
Taking $\varepsilon \to 0^+$ finishes the proof.
\end{proof}
\subsection{Ollivier curvature on combinatorial graphs}
We now show how the transport cost expression for the curvature simplifies on combinatorial graphs.
We first describe the curvature on combinatorial graphs intuitively. We note how this case complements Example~\ref{ex:NoCycles}
which considered the case of graphs with no cycles.
\begin{itemize}
\item
Given an edge $x \sim y$, we have initial curvature $\kappa(x,y)=2$.
\item
Every triangle containing $x,y$ increases $\kappa(x,y)$ by one.
\item
Adding 4-cycles containing $x,y$ does not change $\kappa(x,y)$.
\item
Adding 5-cycles containing $x,y$ decreases $\kappa(x,y)$ by one.
\item
Every additional neighbor of both $x$ and $y$ decreases $\kappa(x,y)$ by one.
\end{itemize}
The following theorem gives a precise expression for the curvature of combinatorial graphs making the above intuition explicit.
\begin{theorem}[Transport and combinatorial graphs]\label{thm:TransprtCombinatorial}
Let $G=(V,w,m)$ be a combinatorial graph and let $x_0 \sim y_0$ be adjacent vertices.
Let $B_{x_0y_0}:= B_1(x_0) \cap B_1(y_0)$, $B_{x_0}^{y_0}:=B_1(x_0) \setminus B_1(y_0)$ and $B_{y_0}^{x_0}:=B_1(y_0) \setminus B_1(x_0).$
Let $\Phi_{x_0y_0} := \{\phi: D(\phi) \subseteq B_{x_0}^{y_0} \to R(\phi) \subseteq B_{y_0}^{x_0} : \phi \mbox{ bijective}\}. $
For $\phi \in \Phi_{x_0y_0}$ write $D(\phi)^c := B_{x_0}^{y_0} \setminus D(\phi)$ and
$R(\phi)^c := B_{y_0}^{x_0} \setminus R(\phi).$
Then,
\begin{align*}
\kappa(x_0,y_0) = \# B_{x_0y_0} - \inf_{\phi \in \Phi_{x_0y_0}} \left( \#D(\phi)^c + \# R(\phi)^c + \sum_{x\in D(\phi)} [d(x,\phi(x)) - 1] \right).
\end{align*}
\end{theorem}
We remark that $\Phi_{x_0y_0} \neq \emptyset$ since $\Phi_{x_0y_0}$ always contains the empty function.
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.8, transform shape]
\node[vertex][label={[name=lx] below:{$x_0$}}](x)at (-1,-1) {};
\node[vertex][label={[name=ly] below:{$y_0$}}](y) at (1,-1) {};
\node(common)[draw,rectangle] at (0,0.5) {$S_1(x_0) \cap S_1(y_0)$};
\Edge(x)(y)
\Edge(x)(common)
\Edge(y)(common)
\node[draw, circle, fit=(x)(y)(common)(lx)(ly),minimum size=4cm, label=below:{$B_{x_0y_0}$}](BI){};
\node[draw, rectangle](D) at (-3,2) {$D(\phi)$};
\node(Dc) at (-3,3) {$D(\phi)^c$};
\node[label=left:{$B_{x_0}^{y_0}$}, fit=(D)(Dc)](Bxy)[draw, circle, minimum size=2.8cm] {};
\node[draw, rectangle](R) at (3,2) {$R(\phi)$};
\node(Rc) at (3,3) {$R(\phi)^c$};
\node[label=right:{$B_{y_0}^{x_0}$}, fit=(R)(Rc)](Byx)[draw, circle, minimum size=2.8cm] {};
\Edge(x)(y)
\Edge(x)(common)
\Edge(y)(common)
\Edge(x)(Bxy)
\Edge(y)(Byx)
\draw[->,>=stealth,line width = 1.5pt] (D) -- node[label=$\phi$] {}(R);
\end{tikzpicture}
\caption{The figure is a scheme of the terms used in Theorem~\ref{thm:TransprtCombinatorial} providing a simple method to compute the curvature for combinatorial graphs.}
\label{fig:transportCombinatorial}
\end{figure}
To prove the theorem, we first show that the coupling function $\rho$ which gives $\kappa(x_0,y_0)$ via the expression found in Proposition~\ref{pro:CharTransport} can be assumed to be integer valued for combinatorial graphs. As in the proof of Proposition~\ref{pro:CharTransport} and since we assume that $x_0 \sim y_0$, we let
\[ F(\rho)= \sum_{x,y} \rho(x,y)(1 - d(x,y)) \]
for $\rho: B_1(x_0) \times B_1(y_0) \to [0,\infty)$.
We note that in the case of combinatorial graphs, (\ref{eq:rhoXProp}) and (\ref{eq:rhoYProp}) become
\begin{equation}\label{eq:TransportCombinatorial}
\sum_{y \in B_1(y_0)} \rho(x,y) = 1 \quad \forall x \sim x_0 \qquad \mbox{ and} \qquad \sum_{x \in B_1(x_0)} \rho(x,y) = 1 \quad \forall y \sim y_0.
\end{equation}
In particular, as we assume that $x_0 \sim y_0$, we have $\sum_x \rho(x,x_0)= \sum_y \rho(y_0,y) =1$.
\begin{lemma}\label{lem:01lemma}
Let $G=(V,w,m)$ be a combinatorial graph and let $x_0 \sim y_0$. Then, there exists $\rho:B_1(x_0) \times B_1(y_0) \to \{0,1\}$ satisfying \eqref{eq:TransportCombinatorial} such that
$\kappa(x_0,y_0) = F(\rho)$.
Furthermore, $\rho$ can be chosen to satisfy $\rho(z,z)=1$ for all $z \in B_{x_0y_0}$.
\end{lemma}
\begin{proof}
We first show that $\rho$ can be chosen to take values in $\{0,1\}$.
Suppose not. Let $\rho$ be a coupling which satisfies \eqref{eq:TransportCombinatorial} such that $\kappa(x_0,y_0) = F(\rho)$ and so that $\rho$ has the minimal number of non-$\{0,1\}$ entries.
Denote by
\[ M=\{(x,y) \in B_1(x_0) \times B_1(y_0):\rho(x,y) \notin \{0,1\}\}. \]
By assumption $M \not = \emptyset$.
We first note that $(x_0,y_0) \not \in M$ as, if $(x_0,y_0) \in M$, then we could replace $\rho$ by a coupling whose value at $(x_0,y_0)$ is 0 without changing the value of $F(\rho)$.
By using \eqref{eq:TransportCombinatorial} repeatedly,
we can then construct a maximal sequence $S=((x_A,y_A),\ldots,(x_B,y_B))$ in $M$ with $B\geq A \geq 0$ which has the following properties:
\begin{enumerate}
\item
$x_{2n+1} = x_{2n} \neq x_k$ for all $k \notin \{2n,2n+1\}$.
\item
$y_{2n}=y_{2n-1} \neq y_k$ for all $k \notin \{2n,2n-1\}$.
\end{enumerate}
Without loss of generality, we may assume that either $x_B=x_{B-1}$ or $A=B$.
Now, suppose that $y_B \neq y_0$.
Then, by \eqref{eq:TransportCombinatorial} there exists $(x_{B+1},y_B) \in M$ with $x_{B+1} \neq x_B$ since $\sum_x \rho(x,y_B) = 1$.
Due to the maximality of $S$, we cannot add $(x_{B+1},y_B)$ to $S$ and the only possible reason for this is that there exists $A'<B$ with $x_{A'} = x_{B+1}$ where we choose $A'$ to be maximal.
In this case, we replace $S$ by the loop $L=((x_{A'},y_{A'}), \ldots, (x_{B},y_B),(x_{B+1},y_B))$.
We proceed analogously if $y_B=y_0$ and $x_A \neq x_0$ and replace $S$ by the loop $L=((x_{A-1},y_A),\ldots,(x_{B'},y_{B'}))$.
In case we do not replace $S$ by a loop, the sequence starts with $(x_0,y_A)$ or $(x_A,y_0)$ and ends with $(x_0,y_B)$ or $(x_B,y_0)$.
Given a sequence $S$ or a loop $L$ as constructed above, we can change $\rho$ on $S$ or $L$ while preserving \eqref{eq:TransportCombinatorial}. We do this by letting $\rho_C(x_n,y_n) := \rho(x_n,y_n) + C(-1)^n$ and $\rho_C(x,y) := \rho(x,y)$ otherwise.
It is easy to check that \eqref{eq:TransportCombinatorial} also holds for $\rho_C$.
The objective function $F$ is linear. Therefore, $F(\rho_C) \geq F(\rho)$ for all negative or all positive $C$. Without loss of generality, we assume that $F(\rho_C) \geq F(\rho)$ for all positive $C$.
We choose $C$ maximal such that $\rho_C \geq 0$. Then, there exists $(x,y)$ in the sequence with $\rho_C(x,y) = \rho(x,y)-C = 0$ so that $\rho_C(x,y) \in \{0,1\}$ but $\rho(x,y) \notin \{0,1\}$.
This contradicts the minimality of the number of $\{0,1\}$ entries of $\rho$.
The contradiction finishes the proof of the first part of the statement.
We now show the furthermore statement, that is, that $\rho$ can additionally be chosen so that $\rho(z,z) =1$ for all $z \in B_{x_0y_0}=B_1(x_0) \cap B_1(y_0)$. Suppose that $\rho(z,z) \neq 1$ for some $z \in B_{x_0y_0}$. Then, $\rho(z,z)=0$.
Case 1.
We first assume that $z \in S_1(x_0) \cap S_1(y_0)$.
Then, there exists $x_z \in B_1(x_0)$ and $y_z \in B_1(y_0)$ with $\rho(x_z,z)=1=\rho(z,y_z)$ and thus $\rho(x_z,y_z)=0$ by \eqref{eq:TransportCombinatorial}.
Define $\widetilde\rho(z,z)=\widetilde\rho (x_z,y_z)=1$ and $\widetilde \rho(x_z,z) = \widetilde \rho (z,y_z)=0$ and $\widetilde\rho(x,y) = \rho(x,y)$ otherwise.
Then, $\widetilde \rho$ also satisfies \eqref{eq:TransportCombinatorial}.
Moreover, $F(\tilde \rho) = F(\rho) + d(x_z,z) + d(z,y_z) - d(x_z,y_z) \geq F(\rho)$.
Case 2.
If $z=x_0$, there exists $x_z\sim x_0$ with $\rho(x_z,x_0)=1$.
Now, set $\widetilde \rho(x_0,x_0)=\widetilde \rho(x_z,y_0) = 1$ and $\widetilde \rho(x_z,x_0)=0$ and $\widetilde{\rho}(x,y)=\rho(x,y)$ otherwise.
Then, $\widetilde \rho$ also satisfies \eqref{eq:TransportCombinatorial}.
Moreover, $F(\tilde \rho) = F(\rho) + 2 - d(x_z,y_0) \geq F(\rho)$. An analogous argument works in the case $z=y_0$.
Therefore, in both cases, $\widetilde \rho$ is also a $\{0,1\}$-valued function satisfying \eqref{eq:TransportCombinatorial} such that $\kappa(x_0,y_0) = F(\widetilde \rho)$ and $\widetilde{\rho}(z,z)=0$. Repeating the argument yields the existence of a $\widetilde \rho$ such that
$\widetilde\rho(z,z) = 1$ for all $z \in B_{x_0y_0}$.
\end{proof}
\begin{rem}
One can also prove the integrality of the transport function $\rho$ in Lemma~\ref{lem:01lemma} by using the theory of linear programming. In particular, the constraint matrix is a submatrix of the constraint matrix of a classical assignment problem and, therefore, totally unimodular. By standard theory and due to the integrality of all parameters, this implies the existence of an integral optimal solution $\rho$.
\end{rem}
We are now prepared to prove Theorem~\ref{thm:TransprtCombinatorial} expressing the curvature for combinatorial graphs via transport costs.
\begin{proof}[Proof of Theorem~\ref{thm:TransprtCombinatorial}]
Due to Lemma~\ref{lem:01lemma}, we can assume that the optimizing function $\rho$ satisfying \eqref{eq:TransportCombinatorial} and $\kappa(x_0,y_0)=F(\rho)=\sum_{x,y} \rho(x,y)(1-d(x,y))$ takes values in $\{0,1\}$ and satisfies
$\rho(z,z) = 1$ for all $z \in B_{x_0y_0}$. Therefore, $\rho(x,x_0)=0$ for all $x \sim x_0$, $\rho(y_0,y)=0$ for all $y \sim y_0$ and $\rho(z,y)=\rho(x,z)=0$ for all $z \in S_1(x_0) \cap S_1(y_0)$ where $x,y \not = z$.
Thus,
\begin{align*}
F(\rho) &= \sum_{x,y} \rho(x,y)(1 - d(x,y)) \\
&= \sum_{x\neq x_0, y\neq y_0} \rho(x,y) (1-d(x,y)) + \rho(x_0,x_0) - \sum_{y \in B_{y_0}^{x_0}} \rho(x_0,y) + \rho(y_0,y_0) - \sum_{x \in B_{x_0}^{y_0}} \rho(x,y_0) \\
&=\# B_{x_0y_0} - \sum_{x \in B_{x_0}^{y_0}} \rho(x,y_0) - \sum_{y \in B_{y_0}^{x_0}} \rho(x_0,y)
+ \sum_{x\in B_{x_0}^{y_0}, y \in B_{y_0}^{x_0}} \rho(x,y)(1-d(x,y)). \end{align*}
If $x \in B_{x_0}^{y_0}$, then $x \sim x_0$ so that $\sum_y \rho(x, y)=1$ by \eqref{eq:TransportCombinatorial}. Therefore, as $\rho(x,z)=0$ for all $z \in S_1(x_0) \cap S_1(y_0)$, either $\rho(x,y_0)=1$ or there exists a unique $y \in B_{y_0}^{x_0}$ such that $\rho(x,y)=1$. In the second case, $y \in B_{y_0}^{x_0}$ is unique as $\sum_x \rho(x,y)=1$ by \eqref{eq:TransportCombinatorial}. Hence, $\rho$ can be uniquely associated with a bijection $\phi_\rho \in \Phi_{x_0y_0}$ by letting
\[D(\phi_\rho) = \{ x \in B_{x_0}^{y_0} : \mbox{ there exists a unique } y \in B_{y_0}^{x_0} \mbox{ such that } \rho(x,y)=1 \} \]
and $\phi_\rho(x) = y$ for $x \in D(\phi)$.
Note, by the dichotomy above, that $D(\phi_\rho)^c = \{ x \in B_{x_0}^{y_0} : \rho(x,y_0) =1 \}$ and $R(\phi_\rho)^c=\{ y \in B_{y_0}^{x_0} : \rho(x_0,y)=1\}$. Therefore,
\begin{align*}
\kappa(x_0,y_0) = F(\rho) &= \# B_{x_0y_0} - \left( \#D(\phi_\rho)^c + \#R(\phi_\rho)^c + \sum_{x \in D(\phi_\rho)} [d(x, \phi_\rho(x))-1] \right) \\
&\leq \# B_{x_0y_0} - \inf_{\phi \in \Phi_{x_0y_0}} \left( \#D(\phi)^c + \# R(\phi)^c + \sum_{x\in D(\phi)} [d(x,\phi(x)) - 1] \right).
\end{align*}
On the other hand, if $\phi \in \Phi_{x_0y_0}$, we can reverse the process above to define $\rho_\phi:B_1(x_0) \times B_1(y_0) \to \{0,1\}$ by letting $\rho_\phi(z,z)=1$ for all $z \in B_{x_0y_0}$, $\rho_\phi(x, \phi(x))=1$ for all $x \in D(\phi)$, $\rho_\phi(x,y_0)=1$ for all $x \in D(\phi)^c$, $\rho(x_0,y)=1$ for all $y \in R(\phi)^c$ and $\rho_\phi(x,y)=0$ otherwise. As above, it follows that $\rho_\phi$ satisfies \eqref{eq:TransportCombinatorial} and that $F(\rho_\phi) = \# B_{x_0y_0} - \left( \#D(\phi)^c + \#R(\phi)^c + \sum_{x \in D(\phi)} [d(x, \phi(x))-1] \right).$
Therefore,
\begin{align*}
\kappa(x_0,y_0) = \sup_\rho F(\rho) \geq F(\rho_\phi) &= \# B_{x_0y_0} - \left( \#D(\phi)^c + \#R(\phi)^c + \sum_{x \in D(\phi)} [d(x, \phi(x))-1] \right)
\end{align*}
for all $\phi \in \Phi_{x_0y_0}$.
Combining the two inequalities completes the proof.
\end{proof}
\subsection{Ollivier curvature on birth-death chains}\label{sec:OllivierBirthDeath}
The curvature of birth-death chains is easy to compute. Moreover, as we will see later, many problems of interest concerning Ollivier curvature can be reduced to the case of birth-death chains.
\begin{defn}
A graph $G=({\mathbb{N}}_0,w,m)$ is called a \emph{birth-death chain} if
$$
w(m,n) =0 \quad \mbox{ whenever } \quad |m-n|\neq 1.
$$
\end{defn}
\begin{theorem}[Curvature of a birth-death chain]\label{thm:line}
Let $G=({\mathbb{N}}_0,w,m)$ be a birth-death chain and let $f(r):=d(0,r)=r$. Then for $0\leq r<R$,
\begin{align*}
\kappa(r,R) &= \nabla_{rR}\Delta f = \frac{\Delta f(r) - \Delta f(R)}{R-r} \\&= \frac{w(r,r+1) - w(r,r-1)}{(R-r)m(r)} - \frac{w(R,R+1) - w(R,R-1)}{(R-r)m(R)}
\end{align*}
where we set $w(r,r-1) :=0$ if $r=0$.
\end{theorem}
\begin{proof}
The last equality is a straightforward computation. We now prove the first equality.
Due to Theorem~\ref{thm:nablaDelta}, as $f \in Lip(1)$ and $\nabla_{Rr}f=1$, it is clear that
\begin{align*}
\kappa(r,R) \leq \nabla_{rR} \Delta f = \frac{\Delta f(r) - \Delta f(R)}{R-r}.
\end{align*}
We will now show the other inequality to complete the proof.
Let $g \in Lip(1)$ be such that $\nabla_{Rr}g=1$, i.e., $g(R)-g(r)=R-r$. Therefore, $g(n+1)-g(n)=1$ for all $r \leq n \leq R-1$ so that, in particular,
$g(r+1)-g(r) = 1 = g(R)-g(R-1)$.
Moreover, $a:=g(r)-g(r-1) \leq 1$ and $b:=g(R+1)-g(R) \leq 1$ since $g \in Lip(1)$.
As
$$
m(r)\Delta g(r) = w(r,r+1) - a w(r,r-1) \geq m(r)\Delta f(r)
$$
and
$$
m(R)\Delta g(R) = bw(R,R+1) - w(R,R-1) \leq m(R)\Delta f(R)
$$
it follows that
\begin{align*}
\frac{\Delta g(r) - \Delta g(R)}{R-r} \geq \frac{\Delta f(r) - \Delta f(R)}{R-r}.
\end{align*}
Therefore, Theorem~\ref{thm:nablaDelta} yields that
$$
\kappa(r,R) \geq \frac{\Delta f(r) - \Delta f(R)}{R-r}
$$
which implies the claim of the theorem.
\end{proof}
\begin{rem}\label{rem:line}
We note that it is easy to see from the above that
\[ \kappa(0,r) = \frac{1}{r} \sum_{n=0}^{r-1} \kappa(n,n+1).\]
In particular, $\kappa(r-1,r)=K$ if and only if $\kappa(0,r)=K$ for all $r \geq 1$.
\end{rem}
\section{Gradient estimates}\label{sec:GradEstimates}
Our proof of the gradient estimate of the semigroup under a Ricci curvature bound deeply relies on the maximum principle which requires taking maxima over compact sets. For applying this technique to infinite, and hence, non-compact graphs, we employ a cutoff method.
However, standard cutoff techniques like taking Dirichlet boundary conditions on a finite subgraph do not work since the gradient of a function may leave the subgraph.
Also cutting off with a finitely supported function after taking the semigroup appears to be not successful since we do not have control over the semigroup before taking the cutoff.
The idea to overcome these difficulties is to deeply intertwine the semigroup with a finitely supported cutoff function.
We call this the perpetual cutoff method which will result in a non-linear cutoff semigroup whose general properties we first develop below.
For general theory on non-linear semigroups, see e.g. \cite{barbu1976nonlinear,kato1967nonlinear, miyadera1992nonlinear}.
We will then apply this general theory to prove our main characterization which connects a lower Ricci curvature bound with a gradient decay of the semigroup.
\subsection{The perpetual cutoff method}
The intuition of the non-linear cutoff semigroup presented below is that it behaves exactly as the heat semigroup whenever the heat does not surpass the cutoff threshold.
The name perpetual cutoff method comes from the fact that the cutoff threshold is not only applied once, but perpetually for all times $t>0$.
\begin{defn}[Cutoff semigroup]
Let $\phi \in C_c(V)$ be a non-negative function and let $f \in [0,\phi] := \{g \in C_c(V) : 0 \leq g \leq \phi\}$.
For $t\geq0$, we define
\[
Q_t^\phi f := P_t f \wedge \phi
\]
and the \emph{cutoff semigroup}
\[
P_t^\phi f := \inf_{t_1 + \ldots + t_n = t} Q_{t_1}^\phi \ldots Q_{t_n}^\phi f.
\]
\end{defn}
We note, by checking cases, that $Q_t^\phi Q_s^\phi \leq Q_{t+s}^\phi$ and, as $P_t$ is positivity preserving, the infimum exists.
Let $W \subset V$ be finite.
We will show that $P_t^\phi f$ is a generalization of the semigroup $e^{t \Delta_W}$ with $\Delta_W f := 1_W \Delta (1_W f)$ corresponding to the Dirichlet problem $\partial_t u = \Delta u$ on $W$ and $u=0$ on $V \setminus W$. In particular, $P_t^\phi = e^{t \Delta_W}$ when we take $\phi = 1_W$ as the cutoff function. Furthermore, $P_t^\phi$ solves the heat equation at all vertices $x$ where $P_t^\phi(x) < \phi(x)$.
We collect these and some other useful properties of $P_t^\phi$ in the following theorem.
We write
\[\overline {\partial_t^\pm} G(t) := \limsup_{h \to 0^\pm} \frac{G(t+h) -G(t)} h\] and \[\underline {\partial_t^\pm} G(t) := \liminf_{h \to 0^\pm} \frac{G(t+h) -G(t)} h\] for a function $G$ depending on $t$.
\begin{theorem}[Properties of the cutoff semigroup]\label{thm:Cutoff}
Let $G=(V,w,m)$ be a graph and let $\phi \in C_c(V)$ be non-negative.
The family $P_t^\phi : [0,\phi] \to [0,\phi]$ is a nonlinear contraction semigroup with respect to $\|\cdot\|_p$ for all $p \in [1,\infty]$ and $t\geq0$. In particular, for $f,g \in [0,\phi]$ and $s,t \geq0$, we have:
\begin{enumerate}[(i)]
\item $P_t^\phi P_s^\phi = P_{t+s}^\phi$,
\item $\| P_t^\phi f - P_t^\phi g \|_p \leq \|f-g\|_p$,
\item $P_0^\phi f = f$,
\item
$
P_t^\phi f \geq P_t^\psi g \quad \mbox{ whenever } \quad \phi \geq \psi \geq f \geq g,
$
\item $e^{-t \operatorname{Deg}} f \leq P_t^\phi f \leq P_t f$,
\item $P_t^\phi f$ is Lipschitz in $t$,
\item
$
\overline{\partial_t^\pm} P_t^\phi f \leq \Delta P_t^\phi f,
$
\item
$
\partial_t P_t^\phi f (x) = \Delta P_t^\phi f (x) \quad \mbox{ whenever } \quad P_t^\phi f (x) < \phi(x),
$
\item
$
P_t^\phi f = e^{t\Delta_W} f \quad \mbox{ whenever } \quad \phi = 1_W \mbox{ for } W \subset V \mbox{ finite}.
$
\end{enumerate}
\end{theorem}
\begin{proof}
By definition, $Q_t^\phi$ maps $[0,\phi]$ to $[0,\phi]$, and so does $P_t^\phi$.
We prove the semigroup property $(i)$ by observing that
\begin{align*}
P_t^\phi P_s^\phi f = \inf_{t_1 + \ldots + t_n = t} Q_{t_1}^\phi \ldots Q_{t_n}^\phi \inf_{s_1 + \ldots + s_m = s} Q_{s_1}^\phi \ldots Q_{s_m}^\phi f &=
\inf_{\substack{t_1 + \ldots + t_n = t \\ s_1 + \ldots + s_m = s}} Q_{t_1}^\phi \ldots Q_{t_n}^\phi Q_{s_1}^\phi \ldots Q_{s_m}^\phi
\\
&= \inf_{t_1 + \ldots + t_{n+m} = t+s} Q_{t_1}^\phi \ldots Q_{t_{n+m}}^\phi f\\
&=P_{t+s}^\phi f.
\end{align*}
where the second equality follows from the montone convergence of $Q_t^\phi$ and the
third equality follows from $Q_t^\phi Q_s^\phi f \leq Q_{t+s}^\phi f$.
To prove the contraction property $(ii)$, observe that for $p \in [1,\infty]$, $P_t$ is contracting on $\ell^p(V,m)$ so that
\[
\| Q_t^\phi f - Q_t^\phi g \|_p \leq \| P_t f - P_t g \|_p \leq \|f-g\|_p
\]
implying that
\[
\| P_t^\phi f - P_t^\phi g \|_p \leq \|f-g\|_p.
\]
It is clear that $P_0^\phi f = f$ since $f \leq \phi$. This proves $(iii)$.
To prove $(iv)$, observe that $Q_t^\phi f \geq Q_t^\psi g$ whenever $\phi \geq \psi \geq f \geq g$ as $P_t$ is positivity preserving. This property is immediately transmitted to $P_t^\phi$.
To prove $(v)$, i.e., the lower and upper estimate of $P_t^\phi f$, we use
\[
e^{-t \operatorname{Deg}} f \leq Q_t^\phi f \leq P_t f
\]
which implies that
\[
e^{-t \operatorname{Deg}} f \leq P_t^\phi f \leq P_t f.
\]
as desired.
By using the estimates directly above and applying Taylor's theorem to $P_t f$ at $t=0$, we can deduce the existence of a constant $C_\phi>0$ such that for all $f \in [0,\phi]$ and $t\geq0$,
\begin{align*}
-C_\phi t \leq 1_{\operatorname{supp} \phi} (e^{-t\operatorname{Deg}} - 1) f \leq P^\phi_{t} f - f \leq 1_{\operatorname{supp} \phi} (P_t -1) f \leq C_\phi t
\end{align*}
implying that $P_t^\phi f$ is Lipschitz in $t$ by using the semigroup property $(i)$, thus proving $(vi)$.
Furthermore, due to Taylor's theorem again, there exists a constant $C_\phi'>0$, such that for all $t>0$ and all $f \in [0,\phi]$,
\[
\frac 1 t \left( P^\phi_{t} f - f \right) \leq 1_{\operatorname{supp} \phi} \frac 1 t (P_t f -f) \leq \Delta f + C_\phi't
\]
since $1_{\operatorname{supp} \phi} \Delta f \leq \Delta f$ for $f \in [0,\phi]$. This directly implies that
\[ \overline{\partial_t^+} P_t^\phi f = \limsup_{\varepsilon \to 0^+} \frac 1 \varepsilon \left({P_{\varepsilon}^\phi P_t^\phi f - P_t^\phi f} \right) \leq \Delta P_t^\phi f \]
by using the semigroup property and the fact that $P_t^\phi f \in [0,\phi]$.
Similarly,
\[ \overline{\partial_t^-} P_t^\phi f = \limsup_{\varepsilon \to 0^-} \frac{1}{-\varepsilon} \left( {P_{-\varepsilon}^\phi P_{t+\varepsilon}^\phi f - P_{t+\varepsilon}^\phi f} \right) \leq \limsup_{\varepsilon \to0^-} \left(\Delta P_{t+\varepsilon}^\phi f - C_\phi' \varepsilon \right) = \Delta P_{t}^\phi f. \]
\eat{
Therefore,
\begin{align}
\frac1 \varepsilon \left(P_t^\phi f - P_{t - \varepsilon}^\phi f \right) \leq 1_{\operatorname{supp} \phi} \Delta P_{t-\varepsilon}^\phi f + C_\phi'\varepsilon \stackrel{\varepsilon \to 0}{\longrightarrow} \Delta P_t^\phi f
\end{align}
This implies for $t>0$ and $f \in [0,\phi]$,
\[
\overline{\partial_t^-} P_t^\phi f \leq \Delta P_t^\phi f.
\]
}
Putting these two inequalities together yields $(vii)$.
In order to do prove $(viii)$, we first define $\Delta^x: C(V)\to C(V)$ via
\[
\Delta^x f (y) := \begin{cases} \Delta f(x) & \mbox{if } y=x \\
-\operatorname{Deg}(y) f(y) & \mbox{otherwise}
\end{cases}
\]
and let $P_t^x := e^{t\Delta^x}$.
We remark that $\Delta^x$ is an asymmetrization of $\Delta$ and that $P_t^x u$ does \emph{not} give a solution to the Dirichlet problem $\partial_t u(x)=\Delta u(x)$ and $u=0$ on $V\setminus \{x\}$. We also note that $P_t^x$ is positivity preserving.
\begin{lemma} \label{lem:Ptx}
Let $t>0$ and let $f \in [0,\phi]$.
If $P_s f(x) \leq \phi(x)$ for $0\leq s \leq t$, then
\[P_t^x f \leq P_t^\phi f. \]
\end{lemma}
\begin{proof}
Obviously, $P_s^x f \leq P_s f$.
Observe that $P_s^x f \leq \phi$ since
$P_s^x f(y) = e^{-s\operatorname{Deg}(y)}f(y) \leq f(y) \leq \phi(y)$ for $y \not = x$
and since $P_s^x f (x) = P_s f (x) \leq \phi(x)$ by assumption. Hence, $P_t^x f \leq Q_t^\phi f$. Induction over $n$ for $s_1 + \ldots + s_n = t$ yields
\[
P_t^x f= P_{s_1 +\ldots + s_n}^xf = P_{s_1}^x \ldots P_{s_n}^x f\leq Q_{s_1}^\phi \ldots Q_{s_n}^\phi f \leq Q_t^\phi f. \]
Taking the infimum over all such $s_1, \ldots, s_n$ finishes the proof of the lemma.
\end{proof}
We now prove $(viii)$. Since we already proved $(vii)$,
it suffices to show that
\[
\underline{\partial_t^\pm} P_t^\phi f(x) \geq \Delta P_t^\phi f(x).
\]
whenever $P_t^\phi f(x) < \phi(x)$.
Due to Taylor's theorem with Lagrange remainder term, there exists a constant $C_\phi''>0$ such that for all $g \in [0,\phi]$ and all $\varepsilon \in(0,t]$, there exists $\delta \in [0,\varepsilon]$ such that
\begin{align}\label{eq:Taylor}
\frac{1}{\varepsilon} (P_\varepsilon^x g - g)(x) &= \Delta^x g(x) + \frac \varepsilon 2 \partial_s^2 P_s^x g(x)|_{s=\delta} \nonumber
\\
&\geq \Delta g(x) - C_\phi'' \varepsilon.
\end{align}
since $\partial_s^2 P_s^x g(x) = \Delta^x \Delta^x P^x_s g(x)$ is uniformly bounded on $[0,t]\times [0,\phi]$ and $x$ is fixed.
Choose $g=P_t^\phi f$. Since we have assumed that $g(x)=P_t^\phi f(x) < \phi(x)$, by continuity of $P_s$, there exists $\varepsilon \in (0,t]$ such that $P_sg(x) \leq \phi(x)$ for all $s \in [0,\varepsilon]$.
By Lemma~\ref{lem:Ptx}, we then have that $P_\varepsilon^x g \leq P_\varepsilon^\phi g$
proving that $\underline{\partial_t^+} P_t^\phi f(x) \geq \Delta P_t^\phi f(x)$ by using \eqref{eq:Taylor}.
We next prove the same inequality for the left derivative.
We note that for $\varepsilon<0$ small enough, we have that $P_{t+\varepsilon}^\phi f(x)<\phi(x)$ so by using continuity of $P_s$ as above, we may apply Lemma~\ref{lem:Ptx} and \eqref{eq:Taylor} again to get
\begin{align*}
\underline{\partial_t^-} P_t^\phi f(x) &\geq \liminf_{\varepsilon \to 0^-} \frac{1}{-\varepsilon} \left({P_{-\varepsilon}^x P_{t+\varepsilon}^\phi f(x) - P_{t+\varepsilon}^\phi f(x)}\right) \\
& \geq \liminf_{\varepsilon \to 0^-} \left( \Delta P_{t +\varepsilon}^\phi f(x) - C_\phi'' \varepsilon \right) = \Delta P_{t}^\phi f(x).
\end{align*}
\eat{
Choose $\varepsilon$ small enough s.t. $P_s P_{t-\delta}^\phi f(x) < \phi(x)$ for $0< \delta,s < \varepsilon$.
At vertex $x$, we calculate
\begin{align}
\frac{1}{\varepsilon} \left( P_t^\phi f - P_{t-\varepsilon}^\phi f \right) \geq \frac{1}{\varepsilon} (P_\varepsilon^x - 1) P_{t-\varepsilon}^\phi f = \Delta P_{t-\varepsilon}^\phi f - C_\phi''(\varepsilon) \stackrel{\varepsilon \to 0}{\longrightarrow} \Delta P_t^\phi f
\end{align}
where the estimate follows from Lemma~\ref{lem:Ptx}.
This proves $\underline{\partial_t^-} P_t^\phi f(x) \geq \Delta P_t^\phi f(x)$. }
Putting this together with $\underline{\partial_t^+} P_t^\phi f(x) \geq \Delta P_t^\phi f(x)$ and $(vii)$ yields $(viii)$.
We finally prove $(ix)$.
Let $\phi=1_W$.
First, we suppose that $f\leq \mathbf 1-\varepsilon$. Then, $P_t^\phi f$ solves the Dirichlet problem $\partial_t u = \Delta u$ on $W$ and $u=0$ on $V \setminus W $ due to $(viii)$ as $P_sf(x) < 1 = 1_W(x)$ for $x \in W$ and $s \in [0,t]$ implies that $P_s^\phi f(x) < \phi(x)$ for all $x \in W$.
This shows that $P_t^\phi f = e^{t\Delta_W} f$ since $e^{t\Delta_W} f$ is the unique solution to the Dirichlet problem.
For a general function $f \in [0,\phi]$, the claim follows by approximation since both $P_t^\phi$ and $e^{t\Delta_W}$
are contraction semigroups with respect to $\|\cdot\|_\infty$.
This proves $(ix)$ and finishes the proof of the theorem.
\end{proof}
\subsection{Cutoff semigroups and Ricci curvature}
Using the above observations, we can deduce a Lipschitz decay of the cutoff semigroup under lower curvature bounds.
We observe that the cutoff semigroup $P_t^\phi$ defined on $[0, \phi]=\{g \in C_c(V) : 0 \leq g \leq \phi \}$ canonically extends to functions $f: V \to [0,\infty)$ via $P_t^\phi f := P_t^\phi (f \wedge \phi)$. In particular, $P_0^\phi f = f \wedge \phi$ whenever we do not assume that $f \leq \phi$.
\begin{lemma}\label{lem:CutoffRicci}
Let $G=(V,w,m)$ be a graph with $Ric(G) \geq K$.
Let $f:V \to [0,1]$ be non-constant, $T>0$ and $\phi:V \to [0,1]$ be compactly supported such that $\|\nabla \phi\|_\infty < \|\nabla f\|_\infty (1 \wedge e^{-KT})$. Then, for $t \in [0,T]$,
\[
\|\nabla P_t^\phi f\|_\infty \leq e^{-Kt} \|\nabla f\|_\infty.
\]
\end{lemma}
\begin{proof}
Without loss of generality, we can assume that $\kappa(x,y) > K$ for all $x,y \in V$ instead of $\kappa(x,y) \geq K$. Furthermore, as we assume that $f$ is non-constant, it follows that $\|\nabla f\|_\infty>0$.
For $t \in [0,T], x,y \in V$ with $x \sim y$ we define
\begin{align*}
F(t,x,y) := e^{Kt} \nabla_{yx} P_t^\phi f.
\end{align*}
We aim to show that $F \leq \|\nabla f\|_\infty$.
Suppose not.
Since the support of $P_t^\phi f$ is contained in the finite support of $\phi$, the continuous function $F$ attains its maximum $F_{\max}$ at some $(t_0,x_0,y_0)$ where $y_0$ is in the support of $\phi$. Therefore, $F(t_0,x_0,y_0) = F_{\max} > \| \nabla f \|_\infty$.
Since \[F(0,x_0,y_0) = \nabla_{y_0x_0}(f \wedge \phi) \leq \|\nabla f\|_\infty \vee \|\nabla \phi\|_\infty = \|\nabla f\|_\infty < F(t_0,x_0,y_0),\]
we obtain that $t_0>0$.
Furthermore, observe that
\[
P_{t_0}^\phi f(x_0) < \phi(x_0)
\]
since otherwise
\[
\nabla_{y_0x_0} P_{t_0}^\phi f \leq \nabla_{y_0x_0} \phi < \|\nabla f\|_\infty (1 \wedge e^{-KT}) \leq \|\nabla f\|_\infty e^{-Kt_0}
\]
which would imply that $F(t_0,x_0,y_0) < \|\nabla f\|_\infty$.
This yields $\partial_t P_{t}^\phi f(x_0)|_{t=t_0} = \Delta P_{t_0}^\phi f(x_0)$ due to Theorem~\ref{thm:Cutoff} $(viii)$.
Moreover at $y_0$, Theorem~\ref{thm:Cutoff} $(vii)$ gives that $\overline{\partial_t^-} P_{t}^\phi f(y_0)|_{t=t_0} \leq \Delta P_{t_0}^\phi f(y_0)$.
Subtracting yields
\begin{align*}
\underline{\partial_t^-} \nabla_{x_0y_0} P_t^\phi f |_{t=t_0} \geq \nabla_{x_0y_0} \Delta P_{t_0}^\phi f.
\end{align*}
Observe that $\|\nabla P_{t_0}^\phi f\|_\infty \leq F_{\max}e^{-Kt_0}$ and $\nabla_{y_0x_0}P_{t_0}^\phi f = F_{\max}e^{-Kt_0}$ due to maximality. Hence, due to Theorem~\ref{thm:nablaDelta}, we get that $F_{\max}e^{-Kt_0} \cdot \kappa(x_0,y_0) \leq \nabla_{x_0y_0} P_{t_0}^\phi f.$ Therefore, by our curvature assumption,
\begin{align*}
\underline{\partial_t^-} \nabla_{x_0y_0} P_t^\phi f |_{t=t_0} \geq \nabla_{x_0y_0}\Delta P_{t_0}^\phi f \geq F_{\max}e^{-Kt_0} \cdot \kappa(x_0,y_0) > F_{\max}e^{-Kt_0} K.
\end{align*}
Thus,
\begin{align*}
\overline{\partial_t^-} F(t_0,x_0,y_0) &= \overline{\partial_t^-} \left(e^{Kt}\nabla_{y_0x_0} P_{t}^\phi f\right)|_{t=t_0} \\
&= KF_{\max} - e^{Kt_0} \underline{\partial_t^-} \nabla_{x_0y_0} P_t^\phi f |_{t=t_0} \\
&< KF_{\max} - KF_{\max} =0.
\end{align*}
Due to maximality in time of $F$ at $(t_0,x_0,y_0)$, since $t_0>0$, we have
$\overline{\partial_t^-} F(t_0,x_0,y_0) \geq 0$ which contradicts the above inequality.
Hence, $F\leq \|\nabla f\|_\infty$ which finishes the proof.
\end{proof}
\begin{lemma}\label{lem:RicImpliesGradient}
Let $G=(V,w,m)$ be a graph with $Ric(G) \geq K$.
Let $f : V \to [0,1]$ be non-constant. Then, for all $t>0$,
\[
\|\nabla P_t f\|_\infty \leq e^{-Kt} \|\nabla f\|_\infty.
\]
\end{lemma}
\begin{proof}
Let $T>0$. We prove the statement for all $t \in [0,T]$ which will prove the lemma.
Let $W_1 \subset W_2 \subset \ldots$ be finite subsets of $V$ such that $\bigcup W_n = V$.
Let $\phi_n :V \to [0,1]$ be functions such that $\phi_n = 1$ on $W_n$ and such that $\|\nabla \phi_n\|_\infty < \|\nabla f\|_\infty(1 \wedge e^{-KT})$.
Let $x\neq y \in V$ and $t \in [0,T]$. For all $n \in {\mathbb{N}}$, Lemma~\ref{lem:CutoffRicci} yields
\[
\nabla_{xy} P_t^{\phi_n} f \leq e^{-Kt} \|\nabla f\|_\infty.
\]
Due to Theorem~\ref{thm:Cutoff} $(ix)$, we have $e^{t\Delta_{W_n}}= P_t^{1_{W_n}}$ on $[0,1_{W_n}]$, and since $1_{W_n} \leq \phi_n$, Theorem~\ref{thm:Cutoff} $(iv)$ yields
\[
e^{t\Delta_{W_n}}f = P_t^{1_{W_n}} f\leq P_t^{\phi_n} f \leq P_t f.
\]
Since $e^{t\Delta_{W_n}}f$ converges to $P_t f$ pointwise as $n \to \infty$, we infer that
\[
\nabla_{xy} P_t f = \lim_{n \to \infty} \nabla_{xy} P_t^{\phi_n} f \leq e^{-Kt} \|\nabla f\|_\infty.
\]
Now the claim follows immediately since $x,y$ and $t$ are arbitrary.
\end{proof}
Using semigroup methods, we can now show that a lower curvature bound implies stochastic completeness. We want to point out that we will later independently prove stochastic completeness under even weaker assumptions using the Laplacian comparison principle (see Theorem~\ref{thm:StochComplete}).
\begin{lemma}\label{lem:StochComplete}
If $G=(V,w,m)$ is a graph with $Ric(G) \geq K$, then $G$ is stochastically complete.
\end{lemma}
\begin{rem}
We note that the proof closely follows the proof of stochastic completeness under a Bakry-Emery curvature bound in (\cite[Theorem~1.2]{hua2017stochastic}).
\end{rem}
\begin{proof}
Let $\eta_i:V \to [0,1]$ be non-constant such that $\eta_i \to 1$ pointwise and $\|\nabla \eta_i\|_\infty \to 0$ as $i \to \infty$. Then for all $x\neq y$ and $t>0$, Lemma~\ref{lem:RicImpliesGradient} implies that
\begin{align*}
\nabla_{xy} P_t \mathbf{1} = \lim_{i \to \infty} \nabla_{xy} P_t \eta_i \leq \lim_{i \to \infty} e^{-Kt}\|\nabla \eta_i\|_\infty =0.
\end{align*}
Hence, $\|\nabla P_t \mathbf{1}\|_\infty = 0$ which implies stochastic completeness as $P_0 \mathbf{1}=\mathbf{1}$.
\end{proof}
\subsection{Semigroup characterization}
Using Theorem~\ref{thm:nablaDelta} and Lemma~\ref{lem:RicImpliesGradient}, we now give a heat semigroup characterization of lower curvature bounds.
\begin{theorem}[Gradient of the semigroup]\label{thm:gradientGraphs}
Let $G=(V,w,m)$ be a graph and let $K\in{\mathbb{R}}$. The following statements are equivalent:
\begin{enumerate}[(1)]
\item $Ric(G) \geq K$
\item For all $f \in C_c(V)$ and all $t>0$ $$\|\nabla P_t f\|_\infty \leq e^{-Kt}\|\nabla f\|_\infty.$$
\item For all $f \in \ell_\infty(V)$ and all $t>0$ $$\| \nabla P_t f \|_\infty \leq e^{-Kt} \| \nabla f \|_\infty.$$
\item $G$ is stochastically complete and for all $x,y \in V$ and all $t>0$
\begin{align*}
W(p^x_t,p^y_t) \leq e^{-Kt} d(x,y)
\end{align*}
where $p_t^x := \frac m {m(x)} P_t 1_x$ denotes the heat kernel.
\end{enumerate}
\end{theorem}
\begin{rem}
We note that stochastic completeness is needed to state (4) since the Wasserstein distance $W$ is only defined on probability measures and $p_t^x$ is a probability measure only in the case of stochastic completeness.
\end{rem}
\begin{proof}
We first prove $(3) \Leftrightarrow (4)$.
For all bounded $1-$Lipschitz functions $f$, we have
\begin{align}
\int f dp^x_t - \int f dp^y_t = \sum_{z \in V}f(z)\left(p_t^x(z)-p_t^y(z)\right) = P_t f(x) - P_t f(y) \label{eq:Wassertstein-Lipschitz}.
\end{align}
By definition, assertion $(4)$ is equivalent to
\begin{align*}
\int f dp^x_t - \int f dp^y_t \leq e^{-Kt} d(x,y)
\end{align*}
for all bounded Lipschitz functions which is equivalent to assertion $(3)$ due to
(\ref{eq:Wassertstein-Lipschitz}).
It is also clear that $(3)$ implies stochastic completeness by noting that $(3)$ implies that $\| \nabla P_t \mathbf{1} \|_\infty =0$.
The implication $(1)\Rightarrow (3)$ follows from Lemma~\ref{lem:RicImpliesGradient} if $f$ is non-constant and Lemma~\ref{lem:StochComplete} if $f$ is constant.
The implication $(3)\Rightarrow (2)$ is trivial.
We finally prove $(2) \Rightarrow (1)$.
Fix $x\sim y \in V$.
By Theorem~\ref{thm:nablaDelta},
it suffices to show that
$$\inf_{\substack {f\in Lip(1) \cap C_c(V)\\ \nabla_{yx}f=1}} \nabla_{xy} \Delta f \geq K.$$
Let $f \in Lip(1)\cap C_c(V)$ be such that $\nabla_{yx}f =1$.
By assertion $(2)$, we have
$$\nabla_{yx} P_t f \leq e^{-Kt}.$$
Hence, by taking the time derivative at $t=0$,
\begin{align*}
\nabla_{xy} \Delta f = - \partial^+_t \nabla_{yx} P_t f |_{t=0}
= \lim_{t\to 0^+} \frac 1 t \left( \nabla_{yx} f - \nabla_{yx} P_t f \right)
\geq \limsup_{t\to 0^+} \frac 1 t \left(1 - e^{-Kt} \right)
= K
\end{align*}
which proves assertion $(1)$ of the theorem since $f$ is arbitrary.
\end{proof}
\section{Laplacian comparison principle}\label{sec:LaplaceCompare}
The classical Laplacian comparison theorem on manifolds compares the Laplacian of the distance function on the manifold to that of a model space with constant curvature. This means, for a given Riemannian manifold $M$ with Ricci curvature bounded from below by $K$ and for the model space $H$ with constant Ricci curvature $K$, one has
$$
\Delta^M d(x_0^M,\cdot) \leq \Delta^H d(x_0^H,\cdot).
$$
For a survey of comparison geometry of Ricci curvature on manifolds see \cite{zhu1997comparison}.
We give a discrete analogue of the above theorem in the sense that we upper bound the Laplacian of the distance function.
As a replacement of a model space, we will associate a birth-death chain to a given graph having the same sphere measure (see Section~\ref{sec:LaplaceCompareAndLineGrpahs}).
We will also introduce a new quantity called the sphere curvature which depends only on the distance to a fixed vertex instead of considering all curvatures between neighbors.
We first give a discrete Laplacian comparison principle without a model space by explicitly estimating $\Delta d(x_0,\cdot)$.
Even though the proof is a one-liner in light of Theorem~\ref{thm:nablaDelta}, the following discrete Laplacian comparison theorem, and its extension to the case of decaying curvature, turns out to be a foundation of a variety of applications, such as results concerning stochastic completeness and improved diameter bounds.
\begin{theorem}[Laplacian comparison]\label{thm:LaplaceCompare}
Let $G=(V,w,m)$ be a graph. Let $x_0 \in V$ and suppose that $\kappa(x_0,\cdot) \geq K$ for some $K\in {\mathbb{R}}$. Then,
\begin{align*}
\Delta d(x_0,\cdot) \leq \operatorname{Deg}(x_0) - Kd(x_0,\cdot).
\end{align*}
\begin{proof}
Let $y \in V$, $y \not=x_0$, and set $f:=d(x_0,\cdot)$. Note that $f \in Lip(1)$ and $\nabla_{yx_0}f=1$ so that due to Theorem~\ref{thm:nablaDelta}, we have
\begin{align*}
K \leq \kappa(x_0,y) \leq \nabla_{x_0y} \Delta f = \frac{\Delta f(x_0)- \Delta f(y)}{d(x_0,y)} = \frac{\operatorname{Deg}(x_0) - \Delta f(y)}{d(x_0,y)}.
\end{align*}
Rearranging yields the claim.
\end{proof}
\end{theorem}
We next give a Laplacian comparison principle for decaying curvature.
To do so, we need to measure the minimal curvature in terms of the distance to some fixed vertex $x_0$.
\begin{defn}[Sphere curvatures]\label{def:sphereCurvature}
Let $x_0 \in V$ be a fixed vertex. By abuse of notation, we denote $S_{r} := S_{r}(x_0)$ and $B_r := B_r(x_0)$.
For $r\geq 1$, we let the \emph{sphere curvatures} be given by
\begin{align*}
\kappa(r) := \min_{y\in S_{r}} \max_{\substack{x \in S_{r-1} \\ x\sim y}} \kappa(x,y).
\end{align*}
\end{defn}
\begin{rem}
We remark that
$$
\kappa(r) \geq \min_{x,y \in B_r} \kappa(x,y)
$$
which describes the the curvature decay in a simpler way. However, for all of our results it will suffice to have a lower bound on $\kappa(r)$.
\end{rem}
\begin{theorem}[Laplacian comparison and decaying curvature]\label{thm:LaplaceComparisonNonConst}
Let $G=(V,w,m)$ be a graph, $x_0 \in V$ and $f:= d(x_0,\cdot)$. Then,
\begin{align*}
\Delta f \leq \Phi(f)
\end{align*}
with
\begin{align*}
\Phi(R) := \operatorname{Deg}(x_0) - \sum_{r=1}^{R} \kappa(r)
\end{align*}
for $R\geq1$ and $\Phi(0) = \operatorname{Deg}(x_0)$.
The inequality is sharp for birth-death chains where we take $x_0=0$ so that $f(r)=d(0,r)=r$.
\end{theorem}
\begin{rem}
Note, in particular, that if $G$ is a graph with $Ric(G)\geq K$ and $H$ is a birth-death chain with $Ric(H)=K$ satisfying $\operatorname{Deg}_G(x_0)=\operatorname{Deg}_H(0)$, then
\[ \Delta^G d(x_0, x) \leq \Delta^H d(0,R) \]
for all $x \in S_R$. This makes the analogy to the statement concerning manifolds mentioned above precise.
\end{rem}
\begin{proof}
We prove the result via induction over the radius $R$. The claim is clear for $R=0$ since $\Delta f(x_0) = \operatorname{Deg}(x_0)$.
Let $R>0$ and let $y \in S_R$.
Let $x \in S_{R-1}$ with $x\sim y$ be such that $\kappa(x,y)$ is maximal on $\{ (z,y) \ | \ z \in S_{R-1}, z \sim y \}$.
Due to the definition of $\kappa(R)$ and Theorem~\ref{thm:nablaDelta}, we have
\begin{align*}
\kappa(R) \leq \kappa(x,y) \leq \nabla_{xy}\Delta f= \Delta f(x) - \Delta f(y).
\end{align*}
By the induction assumption, we have
\begin{align*}
\Delta f(x) \leq \operatorname{Deg}(x_0) - \sum_{r=1}^{R-1} \kappa(r).
\end{align*}
Rearranging and combining these yields
\begin{align*}
\Delta f(y) \leq \Delta f(x) -\kappa(R) \leq \operatorname{Deg}(x_0) - \sum_{r=1}^{R} \kappa(r)
\end{align*}
which proves the first statement.
For birth-death chains, due to Theorem~\ref{thm:line}, we have for $r\geq 1$,
\begin{align*}
\kappa(r) = \kappa(r-1,r) = \Delta f(r-1) - \Delta f(r).
\end{align*}
Summing this up yields
\begin{align*}
\Delta f(R) =\operatorname{Deg}(0) - \sum_{r=1}^R \Delta f(r) = \Phi(R) = \Phi(f(R)).
\end{align*}
for all $R\geq 1$ which finishes the proof.
\end{proof}
\subsection{Curvature comparison and associated birth-death chains}\label{sec:LaplaceCompareAndLineGrpahs}
We now prove that the Laplacian comparison principle is compatible with the transition to birth-death chains.
\begin{defn}[Associated birth-death chain]\label{def:AssociatedlineGraph}
Let $G=(V,w,m)$ be a graph with $x_0 \in V$ called the root vertex and let $S_r:=S_r(x_0)$. We define the \emph{associated birth-death chain} $ \widetilde G = ({\mathbb{N}}_0,\widetilde w, \widetilde m)$ via
\begin{align*}
\widetilde m(r) &:= m(S_r) \qquad \mbox{ and }\\
\widetilde w(r,r+1)&:= w(S_r,S_{r+1}) := \sum_{\substack{x \in S_r \\y \in S_{r+1}}} w(x,y).
\end{align*}
\end{defn}
\begin{theorem}[Associated birth-death chain and Laplacian comparison]\label{thm:AssociatedLaplaceComparisonLine}
Let $G=(V,w,m)$ be a graph, $x_0 \in V$ and $f:=d(x_0,\cdot)$. Let $\widetilde G$ be the associated birth-death chain with Laplacian $\widetilde \Delta$ and $\widetilde f:=d(0,\cdot)$. Let $\Phi :{\mathbb{N}}_0 \to {\mathbb{R}}$ be a function. Then,
\begin{align*}
\Delta f \leq \Phi(f) \qquad \mbox{ implies } \qquad \widetilde \Delta \widetilde f \leq \Phi(\widetilde f).
\end{align*}
\end{theorem}
\begin{proof}
We first note that $\Delta f(x_0) = \operatorname{Deg}(x_0)=\widetilde{\operatorname{Deg}}(0) = \widetilde{\Delta}\widetilde{f}(0)$.
Next, we let $r \in {\mathbb{N}}$ and integrate $\Delta f \leq \Phi(f)$ over the sphere $S_r:=S_r(x_0)$. For $x \in S_r$, we note that $m(x) \Delta f(x)= \sum_{y \in S_{r+1}} w(x,y) - \sum_{y \in S_{r-1}} w(x,y) \leq m(x) \Phi(r)$ so that
\begin{align*}
\Phi(r)\widetilde m(r) = \Phi(r) m(S_r) &= \sum_{x \in S_r} \Phi(r) m(x) \\&\geq \sum_{x \in S_r} \left( \sum_{y \in S_{r+1}} w(x,y) - \sum_{y \in S_{r-1}} w(x,y) \right) \\
&=\widetilde{w}(r,r+1) - \widetilde w(r,r-1).
\end{align*}
Hence,
\begin{align*}
\widetilde \Delta \widetilde f(r) = \frac{\widetilde{w}(r,r+1) - \widetilde w(r,r-1)}{\widetilde{m}(r)} \leq \Phi(r) = \Phi(\widetilde f(r))
\end{align*}
which finishes the proof.
\end{proof}
\eat{
We now prove that the Laplacian comparison is sharp on birth-death chains.
\begin{theorem}[Sharp Laplacian comparison and birth-death chains]\label{thm:LaplaceComparisonLine}
Let $G=({\mathbb{N}}_0,w,m)$ be a birth-death chain and let $x_0=0 \in {\mathbb{N}}_0$ be the root vertex. Let $f := d(x_0,\cdot)$. Then,
\begin{align*}
\Delta f = \Phi(f)
\end{align*}
with
\begin{align*}
\Phi(R) := \operatorname{Deg}(x_0) - \sum_{r=1}^{R} \kappa(r).
\end{align*}
\end{theorem}
\begin{proof}
Due to Theorem~\ref{thm:line}, we have for $r\geq 1$,
\begin{align*}
\kappa(r) = \kappa(r,r-1) = \Delta f(r-1) - \Delta f(r).
\end{align*}
Summing this up yields
\begin{align*}
\Delta f(R) =\Delta f(0) - \sum_{r=1}^R \Delta f(r) = \Phi(R) = \Phi(f(R)).
\end{align*}
for all $R\geq 0$ which finishes the proof.
\end{proof}
}
Combining this with the sharp Laplacian comparison for birth-death chains allows us to compare the curvature between a graph and its associated birth-death chain.
\begin{corollary}[Associated birth-death chain and curvature comparison]\label{cor:CurvCompare}
Let $G=(V,w,m)$ be a graph, $x_0 \in V$ be a root vertex and $\kappa(r)$ be the sphere curvatures with respect to $x_0$. Let $\widetilde G= ({\mathbb{N}}_0, \widetilde w, \widetilde m)$ be the associated birth-death chain with root vertex $\widetilde x_0 = 0$ and sphere curvatures $\widetilde \kappa(r) = \widetilde \kappa(r,r-1)$.
Then,
\begin{align*}
\sum_{r=1}^R \widetilde \kappa(r) \geq \sum_{r=1}^R \kappa(r).
\end{align*}
\end{corollary}
\begin{proof}
Let $f:=d(x_0,\cdot)$ on $G$ and $\widetilde f := d(0,\cdot)$ on $\widetilde G$.
Let
\begin{align*}
\Phi(R) := \operatorname{Deg}(x_0) - \sum_{r=1}^{R} \kappa(r) \qquad \mbox{ and} \qquad
\widetilde\Phi(R) := \widetilde \operatorname{Deg}(0) - \sum_{r=1}^{R} \widetilde \kappa(r) .
\end{align*}
Due to Theorem~\ref{thm:LaplaceComparisonNonConst}, we have
\begin{align*}
\Delta f \leq \Phi(f) \qquad \mbox{ and } \qquad \widetilde \Delta \widetilde f = \widetilde \Phi(\widetilde f)
\end{align*}
Now, Theorems~\ref{thm:AssociatedLaplaceComparisonLine} yields
\begin{align*}
\widetilde \Phi(\widetilde f) = \widetilde \Delta \widetilde f \leq \Phi(\widetilde f)
\end{align*}
so that $\widetilde \Phi(R) \leq \Phi(R)$. The fact that $\widetilde \operatorname{Deg}(0) = \operatorname{Deg}(x_0)$ completes the proof.
\end{proof}
One might be tempted to think that the sphere curvatures can also be compared without summation, i.e., $\widetilde \kappa(r) \geq \kappa(r)$ for all $r$. But this turns out to be wrong as demonstrated by the following example.
\begin{example}[Graph with $\widetilde\kappa(r) < \kappa(r)=0$]
Let $G=({\mathbb{Z}},w,m)$ with root $x_0 = 0$ be given by
\begin{align*}
w(z,z+1) := m(z) := 2^z
\end{align*}
and $w(m,n)=0$ if $|m-n| \neq 1$.
It is easy to see using the same techniques as in the proof of Theorem~\ref{thm:line} that $G$ has curvature $\kappa(r)= \kappa(r-1,r) = \Delta f(r-1) - \Delta f(r) = 0$ everywhere.
The associated birth-death chain $\widetilde{G} =({\mathbb{N}}_0, \widetilde w, \widetilde m)$ is then given by
\begin{align*}
\widetilde w(n,n+1) &= 2^n + 2^{-n-1} \mbox{ for } n\geq 0 \\
\widetilde m(n)&=2^n + 2^{-n} \quad \mbox{ for } n\geq 1 \qquad \mbox{ and } \qquad \widetilde m(0)=1.
\end{align*}
Let $\widetilde f := d(0,\cdot)$ on $\widetilde G$. Thus, for $n\geq 1$,
$$
\widetilde \Delta \widetilde f(n) = \frac{\widetilde{w}(n,n+1) - \widetilde{w}(n,n-1)}{\widetilde{m}(n)} = \frac{2^{n-1} - 2^{-n-1}}{2^n + 2^{-n}}
$$
which is strictly increasing in $n$.
Hence for $r\geq 2$,
$$
\widetilde \kappa (r) = \widetilde \Delta \widetilde f (r-1) - \widetilde \Delta \widetilde f (r) < 0.
$$
\end{example}
\subsection{Stochastic completeness}
To prove stochastic completeness,
we will use the Khas'minskii criterion on graphs established by Huang in \cite[Theorem~3.3]{huang2011stochastic} which we restate now using our notation.
\begin{theorem}[Khas'minskii's criterion]\label{thm:Huang3.3}
Let $G=(V,w,m)$ be a graph. If there exists a non-negative function $f \in C(V)$
with
\begin{align*}
f(x) \to \infty \mbox{ as } \operatorname{Deg}(x) \to \infty
\end{align*}
satisfying
\begin{align*}
\Delta f \leq \Psi(f)
\end{align*}
outside of a set of bounded vertex degree for some positive, increasing function $\Psi \in C^1([0,\infty))$ with
$$
\int_0^\infty \frac{dr}{\Psi(r)} = \infty,
$$
then $G$ is stochastically complete.
\end{theorem}
Combining the Laplacian comparison with the Khas'minskii's criterion using $f = d(x_0, \cdot)$ yields an optimal stochastic completeness result.
\begin{theorem}[Stochastic completeness]\label{thm:StochComplete} \ \\
\begin{enumerate}[(i)]
\item If $G=(V,w,m)$ is a graph with
$$\kappa(r) \geq -C \log r$$ for some constant $C>0$ and large $r$, then
$G$ is stochastically complete.
\item For $\varepsilon>0$, let $G_\varepsilon=({\mathbb{N}}_0, w,m)$ be a birth-death chain with $m\equiv1$ and
$$w(R,R+1) = 1 + \sum_{r=1}^R\sum_{k=1}^r \left(\log k\right)^{1+\varepsilon}.$$
Then $G_\varepsilon$ is stochastically incomplete and satisfies
$$\kappa(r) \geq - (\log r)^{1+\varepsilon}$$
for all $r\geq 2$.
\end{enumerate}
\end{theorem}
\begin{rem}
We note that the second statement shows that the first statement is optimal in the sense that the decay rate $-\log r$ cannot be replaced by the faster decay rate $-( \log r )^{1+\varepsilon}$.
\end{rem}
\begin{proof}
For the proof of $(i)$, let $f := d(x_0,\cdot)$.
Using the Laplacian comparison, Theorem~\ref{thm:LaplaceComparisonNonConst}, we have
$$
\Delta f \leq \Phi(f)
$$
with
$$\Phi(R) = \operatorname{Deg}(x_0) - \sum_{r=1}^{R} \kappa(r) \leq \Psi(R)\in O(R \log (R))$$
since $-\kappa(r) \in O(\log(R))$, where $\Psi \in C^1([0,\infty))$ is some positive increasing function to which we can apply the Khas'minskii's criterion.
In particular,
$$
\int_0^\infty \frac {dr}{\Psi(r)} = \infty,
$$
so that Theorem~\ref{thm:Huang3.3} yields stochastic completeness as desired.
To prove $(ii)$, we let $f :=d(0,\cdot)$.
We first observe that for $R\geq1$,
\begin{align*}
\Delta f(R) = w(R,R+1) - w(R,R-1) = \sum_{k=1}^{R} (\log k)^{1+\varepsilon}.
\end{align*}
Since $G_\varepsilon$ is a birth-death chain, Theorem~\ref{thm:line} yields
$$\kappa(R) = \kappa(R-1,R) = \Delta f(R-1) - \Delta f(R) = -(\log R)^{1+\varepsilon}$$
for $R \geq 2$ as desired.
Since $\iint (\log x)^{1+\varepsilon} \in \Theta(x^2 (\log x)^{1+\varepsilon})$, by definition of $w$, we have
\begin{align}
w(R,R+1) \in \Theta(R^2 (\log R)^{1+\varepsilon}). \label{eq:wThetaR}
\end{align}
Observe that as $G$ is a birth-death chain, it is weakly spherically symmetric with respect to $x_0 = 0 \in V= {\mathbb{N}}_0$ in the sense of \cite[Definition~2.3]{keller2013volume}.
Hence, due to \cite[Theorem~5]{keller2013volume}, we know that $G$ is stochastically complete if and only if
$$
\sum_r \frac{r+1}{w(r,r+1)} = \infty.
$$
Due to (\ref{eq:wThetaR}), we have
$$
\frac {r+1}{w(r,r+1)} \in \Theta\left( \frac 1 {r (\log r)^{1+\varepsilon}}\right)
$$
and since
$$\sum_r \frac 1 {r (\log r)^{1+\varepsilon}} < \infty$$
we have
$$
\sum_r \frac{r+1}{w(r,r+1)} < \infty
$$
which implies stochastic incompleteness.
\end{proof}
As mentioned in the introduction, the optimal curvature decay rate on Riemannian manifolds is of the order $-r^2$.
As the use of intrinsic metrics has resolved various discrepancies between the manifold and graph settings in the past,
one might think that using an intrinsic metric $\sigma$ instead of the combinatorial graph metric might give stochastic completeness when assuming $\kappa(r) \geq -C \sigma(0,r)^2$ in line with the manifolds case. This turns out to be wrong as we give an example of a stochastically incomplete graph with $\kappa(r) \sim -(\log \sigma(0,r))^{1+\varepsilon} $ for an intrinsic metric $\sigma$ where $f(n) \sim g(n)$ means $cf(n)<g(n)<Cf(n)$ for all $n \in {\mathbb{N}}$ and some $C>c>0$.
We recall that a metric $\sigma$ on $V$ is called \emph{intrinsic} if
\[ \Delta \sigma(x,\cdot)^2(x)=\frac 1 {m(x)}\sum_{y \in V} w(x,y)\sigma(x,y)^2 \leq 2 \]
for all $x \in V$. For various uses the intrinsic metrics in the graph setting, see \cite{keller2015intrinsic}.
\begin{example}\label{ex:incompleteIntrinsic}
Let $G=({\mathbb{N}}_0,w,m)$ be a birth-death chain with
$m(r)=2^r$ and $w(r-1,r)=(\log r)^{1+\varepsilon} \cdot r \cdot 2^r$ for $\varepsilon>0$.
By Theorem~\ref{thm:line}, we obtain that $\kappa(r) \sim -(\log r)^{1+\varepsilon}$.
Moreover, one can check that
\[ \sigma(r,R) := \sum_{k=r}^{R-1} \operatorname{Deg}_+(k)^{-1/2} \]
gives an intrinsic metric where $\operatorname{Deg}_+(r) := w(r,r+1)/m(r) \sim r(\log r)^{1+\varepsilon}$. In particular,
$\sigma(0,r) \sim \sqrt{r/(\log r)^{1+\varepsilon}}$ and, thus,
$\kappa(r) \sim -(\log \sigma(0,r))^{1+\varepsilon}$.
An objection to the example above is that the definition of the spherical curvature $\kappa$ depends on the combinatorial graph distance function $d$.
However, in analogy to Theorem~\ref{thm:nablaDelta}, we can also define a curvature $\kappa_\sigma$ with respect to the intrinsic metric $\sigma$ via
\[
\kappa_\sigma(x,y)= \inf\left\{\nabla_{xy}^\sigma \Delta f : \nabla_{yx}^\sigma f=1, \; \|\nabla^\sigma f\|_\infty = 1 \right\}
\]
where $\nabla_{xy}^\sigma f := \frac{f(x)-f(y)}{\sigma(x,y)}$.
On birth-death chains and intrinsic path metrics $\sigma$, for $x<y$ this simplifies to
\[
\kappa(x,y)= \nabla_{xy}^\sigma\Delta \sigma(0,\cdot).
\]
In our example, we have
\[
\Delta \sigma(0,\cdot)(r) \sim \sqrt{\operatorname{Deg}_+(r)} \sim \sqrt{r \cdot (\log r)^{1+\varepsilon}}
\]
and, by using the mean value theorem to estimate the difference,
\begin{align*}
\kappa_\sigma(r,r+1) &=\sqrt{\operatorname{Deg}_+(r)} \cdot \left(\Delta \sigma(0,\cdot)(r)- \Delta \sigma(0,\cdot)(r+1) \right) \\
&\sim - \sqrt{\operatorname{Deg}_+(r)} \cdot \sqrt{\frac {(\log r)^{1+\varepsilon}}r} \\
&\sim - (\log r)^{1+\varepsilon}.
\end{align*}
In particular, we also have $\kappa_\sigma(r) \sim -(\log \sigma(0,r))^{1+\varepsilon}$.
We are left to show stochastic incompleteness.
Due to \cite[Theorem~5]{keller2013volume}, $G$ is stochastically complete if and only if
\[
\sum_r \frac{m(\{1,...,r\})}{w(r,r+1)} = \infty.
\]
However,
\[
\frac{m(\{1,...,r\})}{w(r,r+1)} \sim \frac 1{r (\log r)^{1+\varepsilon}}
\]
which is summable. Therefore, $G$ is stochastically incomplete.
\end{example}
\subsection{Improved diameter bounds}
We prove that a graph with bounded degree and sphere curvatures decaying not faster than $1/R$ must be finite (Corollary~\ref{cor:finite}). We also show that this decay rate is optimal (Theorem~\ref{thm:FiniteOptimal}). For various diameter bounds on finite graphs see \cite{paeng2012volume}.
On the other hand, we show that in the case of unbounded degree, even a uniform positive lower curvature bound does not imply finiteness (see Example~\ref{Ex:positiveCurvInfiniteDiam}).
In contrast, if we assume that the measure is bounded from below, then a uniform positive lower curvature bound implies finiteness even in the case of unbounded degree (see Corollary~\ref{cor:FiniteDiamBoundedMeasure}).
As a warm-up, we start with the following diameter bound from \cite[Theorem~4.1]{lin2011ricci} transferred to our setting.
\begin{proposition}\label{prop:FiniteDiamBoundedDegree}
Let $G=(V,w,m)$ be a graph and let $x, y \in V$ with $x \not = y$. If $\kappa(x,y)>0$, then
$$
d(x,y) \leq \frac{\operatorname{Deg}(x) + \operatorname{Deg}(y)} {\kappa(x,y)}.
$$
\end{proposition}
\begin{proof}
It is easy to see that $W(1_x,1_y) = d(x,y)$. Furthermore,
observe that for sufficiently small $\varepsilon$,
$$
W(1_x,m^\varepsilon_x) = \varepsilon \operatorname{Deg}(x).
$$
This follows as $W(1_x,m_x^\varepsilon) = \sup_{f \in Lip(1)} - \varepsilon \Delta f(x) \leq \varepsilon \operatorname{Deg}(x)$ with equality for $f=1_x$.
Hence, by the triangle inequality,
\begin{align*}
W(m_x^\varepsilon, m_y^\varepsilon) &\geq W(1_x,1_y) - W(1_x,m^\varepsilon_x) -W(1_y,m^\varepsilon_y) \\
&= d(x,y) - \varepsilon(\operatorname{Deg}(x) + \operatorname{Deg}(y)).
\end{align*}
Thus,
\begin{align*}
\kappa_\varepsilon(x,y)
&= 1 - \frac{W(m_x^\varepsilon, m_y^\varepsilon)}{d(x,y)} \\
&\leq \varepsilon \cdot \frac{ \operatorname{Deg}(x) + \operatorname{Deg}(y)} {d(x,y)}.
\end{align*}
This yields the claim since $\kappa(x,y) = \lim_{\varepsilon\to 0^+} \frac{1} {\varepsilon}\kappa_\varepsilon(x,y)$.
\end{proof}
In particular, if the degree is bounded and the curvature is uniformly positive, then the graph is finite.
More specifically, if we let $\operatorname{diam}(G)= \sup_{x,y \in V} d(x,y)$ denote the diameter of $G$, then if $\operatorname{Deg}(x)\leq M$ and $Ric(G)\geq K>0$, then
$$\operatorname{diam}(G) \leq \frac{2M}{K}.$$
We now improve this result in the sense that we only lower bound the sphere curvatures, which allows for some negative curvature, and consider part of the vertex degrees.
For a fixed vertex $x_0 \in V$, we let for $x \in S_r:= S_r(x_0)$,
\[ \operatorname{Deg}_{\pm}(x) = \frac{1}{m(x)} \sum_{y \in S_{r\pm1}} w(x,y) \]
denote the \emph{outer} and \emph{inner} degree of $x$.
Using the Laplacian comparison principle for non-constant curvature, we immediately obtain the following improved diameter bound.
\begin{theorem}[Improved diameter bound]\label{thm:ImprovedDiamBound}
Let $G=(V,w,m)$ be a graph with $x_0 \in V$.
If $S_R \not = \emptyset$ for $R>0$, then
\begin{align*}
\sum^R_{r=1} \kappa(r) \leq \operatorname{Deg}(x_0) + \min_{x \in S_R} \left(\operatorname{Deg}_-(x) - \operatorname{Deg}_+(x) \right).
\end{align*}
In particular, if $\min_{x \in S_r} \left(\operatorname{Deg}_-(x) - \operatorname{Deg}_+(x) \right) \leq M$ and $\kappa(r) \geq K >0$ for all $r\geq1$, then
\[ \operatorname{diam}(G) \leq \frac{2(\operatorname{Deg}(x_0) +M)}{K}. \]
\end{theorem}
\begin{proof}
We recall that the Laplacian comparison, Theorem~\ref{thm:LaplaceComparisonNonConst}, gives that
\[ \Delta f(x) \leq \operatorname{Deg}(x_0) - \sum_{r=1}^R \kappa(r) \]
for $x \in S_R$ where $f(x) = d(x,x_0)$. Now, the first statement follows as $\Delta f(x) = \operatorname{Deg}_+(x) - \operatorname{Deg}_-(x)$ by an easy calculation.
The second statement is an immediate consequence of the first statement and the triangle inequality.
\eat{
We prove that there is no $x \in B_{R} = B_R(x_0)$ which will prove the result by the triangle inequality. Let $f:=d(x_0,\cdot)$.
Suppose there exists an $x \in B_{R}$. Then, due to the Laplacian comparison principle for non-constant curvature (Theorem~\ref{thm:LaplaceComparisonNonConst}), we have
\begin{align*}
-\operatorname{Deg}(x) \leq \Delta f(x) \leq \operatorname{Deg}(x_0) - \sum_{r=1}^{R} \kappa(r) &< \operatorname{Deg}(x_0) -\left(\operatorname{Deg}(x_0) + \max_{y \in S_R} \operatorname{Deg}(y) \right) \\&\leq -\operatorname{Deg}(x).
\end{align*}
This is a contradiction which finishes the proof.}
\end{proof}
The theorem immediately gives us the following corollary.
\begin{corollary}\label{cor:finite}
If $G=(V,w,m)$ is a graph with bounded degree, then
$$\limsup_{R \to \infty} \sum_r^R \kappa(r) < \infty.$$
Consequently, there is no infinite graph with bounded vertex degree satisfying
$$\limsup_{R \to \infty} \sum_r^R \kappa(r)=\infty.$$
\end{corollary}
We show that the results above are optimal in the sense that whenever we have a given summable positive sequence $k_r$, we can find an infinite graph with bounded degree and
summable sphere curvatures $\kappa(r)$ larger than $k_r$.
\begin{theorem}\label{thm:FiniteOptimal}
For every positive sequence $(k_r)_{r \in {\mathbb{N}}}$ such that
$ \sum_r k_r < \infty$
there exists an infinite graph $G=(V,w,m)$ with bounded degree such that
$$ \kappa(r) \geq k_r \qquad \mbox{ and } \qquad \sum_r \kappa(r) <\infty.$$
\end{theorem}
\begin{proof}
We define a birth-death chain $G=({\mathbb{N}}_0,w,m)$ inductively with $w$ symmetric and $m$ satisfying $m(0)=1$, $w(0,1) = 2 \sum_{i>0} k_i$
and for $r\geq 1$,
\begin{align*}
m(r) = \frac{w(r,r-1)}{k_{r+1}}\qquad \mbox{ and} \qquad w(r,r+1) = 2m(r) \sum_{i>r} k_i.
\end{align*}
Note, in particular, that $\frac{w(r,r-1)}{m(r)} = k_{r+1}$ while $\frac{w(r-1, r)}{m(r-1)}= 2 \sum_{i >r-1}k_i$.
Due to Theorem~\ref{thm:line}, for $r>1$,
\begin{align*}
\kappa(r) &= \kappa(r-1,r) \\&= \frac{w(r-1,r)-w(r-1,r-2)}{m(r-1)}- \frac{w(r,r+1)-w(r,r-1)}{m(r)} \\
&= 2 \sum_{i>r-1} k_i - k_r - 2 \sum_{i>r} k_i + k_{r+1} \\
&=k_r + k_{r+1} \geq k_r
\end{align*}
which also shows that $\sum_r \kappa(r)<\infty$.
Similarly, $\kappa(1)= 2k_1 + k_2 \geq k_1$.
It is left to show that the graph has bounded degree.
We have
\begin{align*}
\operatorname{Deg}(r) = \frac{w(r,r-1)}{m(r)} + \frac{w(r,r+1)}{m(r)} = k_{r+1} + 2 \sum_{i>r}k_i \leq 3 C
\end{align*}
with $C:= \sum_r k_r < \infty$ by assumption. This finishes the proof.
\end{proof}
\begin{example} \label{Ex:positiveCurvInfiniteDiam}
In contrast to Theorem~\ref{thm:ImprovedDiamBound}, we now show that there exist graphs with uniformly positive curvature which are infinite. We note that all such graphs must have unbounded vertex degree.
We construct an infinite birth-death chain $({\mathbb{N}}_0,w,m)$ such that $\kappa(x,y) = K>0.$
We first let $w(r,r+1)$ be strictly positive and decreasing in $r \in {\mathbb{N}}_0$. By Theorem~\ref{thm:line} and Remark~\ref{rem:line}, it suffices to find a choice of measure $m$ such that $\kappa(0,r)=K$, that is, for $f=d(0,\cdot)$
$$
\Delta f(r) = \Delta f(0) - Kr = \operatorname{Deg}(0)-Kr.
$$
Choose $m(0)$ such that $\operatorname{Deg}(0) < Kr$ for all $r\geq 1$. For this it suffices that $m(0) > \frac{w(0,1)}{K}$.
Then, for $r\geq 1$, choose
$$
m(r) := \frac{w(r,r-1) - w(r,r+1)}{Kr - \operatorname{Deg}(0)}
$$
guaranteeing
$$
\Delta f(r) = \frac 1 {m(r)} (w(r,r+1) - w(r,r-1)) = \operatorname{Deg}(0) - Kr.
$$
We remark that $m(r) >0$ since $w(r,r+1)$ is strictly decreasing.
\end{example}
\subsection{Finiteness of the measure}
In this section, we show that a suited positive lower bound on the curvature implies finite measure, that is, $m(V):= \sum_{x\in V} m(x) < \infty$.
\begin{theorem}[Curvature and finite measure]
Let $G=(V,w,m)$ be a graph. If
\[
\liminf_{R\to \infty}\sum_{r=1}^R \kappa(r)>\operatorname{Deg}(x_0),
\]
then $m(V) < \infty$.
\end{theorem}
\begin{proof}
We first show that it suffices to prove the theorem for birth-death chains.
Let $\widetilde G = ({\mathbb{N}}_0,\widetilde w, \widetilde m)$ be the birth-death chain associated to $G$.
Due to Corollary~\ref{cor:CurvCompare}, we also have
\[
\liminf_{R\to \infty}\sum_{r=1}^R \widetilde\kappa(r)>\operatorname{Deg}(x_0)
\]
where $\widetilde \kappa(r)$ are the sphere curvatures of $\widetilde G$. Assuming that the theorem is proven for birth-death chains, we obtain that $m(V)=\widetilde m({\mathbb{N}}_0) < \infty$ which would finish the proof.
Now we prove the theorem for birth-death chains.
Let $f=d(0,\cdot)$. Due to Theorem~\ref{thm:LaplaceComparisonNonConst} and since $\liminf_{R\to \infty} \sum_r^R \kappa(r)> \operatorname{Deg}(0)$, we get
\[ \limsup_{R \to \infty} \Delta f(R) = \limsup_{R \to \infty} \left(\operatorname{Deg}(0) - \sum_{r=1}^R \kappa(r) \right) = \operatorname{Deg}(0) - \liminf_{R \to \infty} \sum_{r=1}^R \kappa(r) <0\]
so that
there exists $\varepsilon>0$ and $R>0$ such that $\Delta f(r)\leq -\varepsilon$ for all $r\geq R$.
This implies that
\[
\varepsilon m(r) \leq w(r,r-1) - w(r,r+1)
\]
for $r\geq R$. Summing up, we obtain
\[
\varepsilon \sum_{r=R}^\infty m(r) \leq w(R,R-1)
\]
which yields the finiteness of the measure of the birth-death chain.
This finishes the proof.
\end{proof}
The theorem immediately gives the following corollary.
\begin{corollary}\label{cor:FiniteDiamBoundedMeasure}
Let $G=(V,w,m)$ be a graph. If $\liminf_{R\to \infty}\sum_{r=1}^R \kappa(r) = \infty$, then $m(V)$ is finite. If, additionally, $\inf_{x \in V} m(x) >0$, then $G$ is finite.
\end{corollary}
Combining this with Corollary~\ref{cor:finite} we get the following dichotomy.
\begin{corollary}
Let $G=(V,w,m)$ be a graph and suppose that
\[ \kappa(r)\geq \frac{C}{r} \]
for some $C>0$ and all large $r$. Then either
$G$ is finite or $G$ is infinite with unbounded vertex degree and finite measure.
\end{corollary}
\section{Ricci curvature for continuous-time Markov processes}\label{sec:MarkovProcesses}
In this section, we compare our curvature notion to the curvature defined in \cite{veysseire2012coarse} for continuous time Markov processes, which generalize both locally finite graphs and Riemannian manifolds.
To make the comparison clear, we recall our curvature definition
\[
\kappa(x,y) = \lim_{t \to 0^+} \frac 1 t \left(1 - \frac{W(m_x^t, m_y^t)}{d(x,y)} \right)
\]
where the discrete time
Markov kernel $m_x^t$ with laziness parameter $t \in(0,\infty)$ is given by
\[
\int{ f dm_x^t} = (f + t \Delta f)(x).
\]
Note that $m^t$ is only non-negative if the vertex degree is bounded and if $t$ is sufficiently small. By abuse of notation, we call $m^t$ a Markov kernel in any case.
The idea to define curvature in \cite{veysseire2012coarse} is to replace the measure $m_x^t$ by the continuous time heat kernel $p_x^t$ which has already appeared in Theorem~\ref{thm:gradientGraphs} and is
given by
\[
\int{ f dp_x^t} := P_t f (x).
\]
We note that this is equivalent to
\[
p_x^t (y)= P_t 1_y(x) = \frac{m(y)}{m(x)}P_t 1_x (y).
\]
Due to Taylor's theorem, it is reasonable to hope that $m_x^t$ is a good approximation for $p_x^t$ as $t \to 0^+$. Criteria for this approximation will be investigated in the next subsection.
Corresponding to \cite[Definition~6]{veysseire2012coarse}, the coarse Ricci curvature on stochastically complete, continuous time Markov processes
is defined by
\[
\overline \kappa(x,y) := \limsup_{t \to 0^+} \frac 1 t \left( 1 - \frac{W(p_x^t, p_y^t)}{d(x,y)} \right)
\]
and
\[
\underline \kappa(x,y) := \liminf_{t \to 0^+} \frac 1 t \left( 1 - \frac{W(p_x^t, p_y^t)}{d(x,y)} \right).
\]
We recall that $\overline \kappa$ and $\underline \kappa$ do not coincide in general (see e.g. \cite[Example~8]{veysseire2012coarse}).
Furthermore, the above definition only makes sense in the stochastically complete case since, otherwise, $p_x^t$ is not a probability measure and, therefore, the Wasserstein distance is not well defined.
The main result of \cite{veysseire2012coarse} is the equivalence of the lower curvature bound $\overline \kappa(x,y) \geq K$ and the Wasserstein contraction property
\[
W(p_x^t,p_y^t) \leq d(x,y)e^{-Kt}.
\]
We note that the same statement for a lower bound on $\kappa$ was shown in Theorem~\ref{thm:gradientGraphs}.
We will show in Corollary~\ref{cor:MarkovChains} that $\kappa=\underline\kappa=\overline \kappa$ when assuming that $\kappa$ is uniformly bounded from below. Therefore, the result in \cite{veysseire2012coarse} combined with this equality gives an alternative proof of Theorem~\ref{thm:gradientGraphs}.
\subsection{Discrete and continuous time Markov kernels}
We will next give conditions guaranteeing that discrete and continuous time Markov kernels approximate each other.
As a convenient notation, we extend the definition of the semigroup $P_t$ to possibly unbounded non-negative functions.
\begin{defn}
For $f\geq 0$, we define \[P_t f := \sup_{\substack{0\leq g\leq f \\ g \in \ell_\infty(V)}} P_t g. \]
\end{defn}
The aim of this subsection is to prove that $W(p_x^t,m_x^t)=O(t^2)$ if and only if $P_t d(x,\cdot)<\infty$ for small $t>0$.
As a first step, we show a uniform boundedness property of the semigroup when applied to unbounded functions.
\begin{lemma}\label{lem:uniformFinitePt}
Let $x \in V$ and let $f \geq 0$. If $P_T f(x) < \infty$ for some $T>0$, then
\begin{enumerate}[(i)]
\item $
\sup_{t \in [0,T]} P_t f(x) < \infty.
$
\item $P_t f < \infty$ for all $t < T$.
\end{enumerate}
\end{lemma}
\begin{proof}
We first prove $(i)$.
Let $t \in [0,T]$ and let $g \in \ell_\infty(V)$ be such that $0\leq g \leq f$.
Then, \[P_t g(x) \leq e^{(T-t)\operatorname{Deg}(x)}P_T g(x) \leq e^{T\operatorname{Deg}(x)} P_T f(x) < \infty\] independently of $t$ and $g$.
Taking the supremum over $t \in [0,T]$ and $g$ yields $(i)$.
We now prove $(ii)$. Let $t<T$ and $y \in V$.
Let $g \in \ell_\infty(V)$ be such that $0\leq g \leq f$.
Observe that $P_T g(x) \geq P_{T-t}1_y(x) \cdot P_t g(y)$ where $P_{T-t}1_y(x)>0$ due to connectedness.
Hence, taking the supremum over $g$ yields
\[
P_t f(y) \leq \frac{P_T f(x)}{P_{T-t}1_y(x)} < \infty
\]
due to assumption. This proves $(ii)$ and finishes the proof of the lemma.
\end{proof}
We now characterize when the ball measure $m_x^t$ approximates the heat kernel measure $p_x^t$ as $t \to 0^+$.
\begin{proposition}\label{pro:PtdANDWpm} Suppose that $G=(V,w,m)$ is stochastically complete and
let $x \in V$.
The following statements are equivalent:
\begin{enumerate}[(1)]
\item $P_t d(x,\cdot) < \infty$ for some $t>0$.
\item $P_t f < \infty$ for all $f\geq 0$ with $\|\nabla f \|_\infty < \infty$ and some $t>0$.
\item $W(p_x^t,m_x^t) = O(t^2)$ as $t \to 0^+$.
\item $W(p_x^t,m_x^t) < \infty$ for some $t>0$.
\end{enumerate}
\end{proposition}
\begin{rem}
We remark that the above properties also play an important role as a standing assumption in \cite{joulin2007poisson} denoted by
$P_t (x,\cdot) \in \mathscr{P}_1(E)$.
\end{rem}
\begin{proof}
The implication (1) $\Rightarrow$ (2) follows since $f \leq d(x,\cdot) + f(x)$ implies that $P_t f \leq f(x) + P_td(x,\cdot) < \infty$.
The implication (2) $\Rightarrow$ (1) is obvious.
We now show that (1) $\Rightarrow$ (3).
Due to Kantorovich duality, we have
\begin{align*}
W(p_x^t,m_x^t) &= \sup_{f \in Lip(1) \cap \ell_\infty(V)} \sum_{y \in V} f(y) \left( p_x^t(y) - m_x^t(y) \right) \\
&= \sup_{f \in Lip(1) \cap \ell_\infty(V)} \sum_{y\in V} f(y) \left( P_t 1_y(x) - 1_y(x) - t \Delta 1_y(x) \right)\\
&= \sup_{f \in Lip(1) \cap \ell_\infty(V)} \left( P_t f - f - t \Delta f \right)(x).
\end{align*}
When optimizing, we can assume that $f(x)=1$ without loss of generality. Since $f \in Lip(1)$, replacing $f$ by its positive part does not change the values of $f$ on $B_1(x)$ and does not decrease the values on $V\setminus B_1(x)$.
Since $p_x^t - m_x^t$ is non-negative on $V \setminus B_1(x)$, the objective function $\sum_y f(y) \left( p_x^t(y) - m_x^t(y) \right)$ is not decreased when replacing $f$ by its positive part. Therefore, we can assume that $f(x)=1$ and $f\geq 0$ when optimizing. This gives
\begin{align}\label{eq:WPepsmEps}
W(p_x^t,m_x^t) & = \sup_{\substack{f \in Lip(1) \cap \ell_\infty(V) \\ f\geq 0, f(x)=1}} (P_t f - f - t \Delta f)(x).
\end{align}
Therefore, let $f \in Lip(1) \cap \ell_\infty(V)$ with $f\geq0$ and $f(x)=1$.
Then, $0 \leq f \leq 1 + d(x,\cdot).$
Due to Lemma~\ref{lem:uniformFinitePt} $(i)$ and by assumption, there exists $C>0$ such that $P_t(1 + d(x,\cdot)) \leq C$ on $B_2(x)$ for all $t \in [0,T]$.
Thus, we also have $0 \leq P_t f \leq C$ on $B_2(x)$ for all $t \in [0,T]$. This yields the existence of $C'$ independent of $f$ and $t \in [0,T]$ such that
\[
|\Delta \Delta P_t f| \leq C'.
\]
Due to Taylor's theorem, there exists $\delta \in [0,t]$ such that
\begin{align*}
(P_t f - f - t \Delta f)(x) = \frac {t^2} 2 \Delta \Delta P_\delta f(x) \leq \frac {t^2} 2 C' = O(t^2).
\end{align*}
Putting this together with (\ref{eq:WPepsmEps}) proves that (1) $\Rightarrow$ (3).
The implication (3) $\Rightarrow$ (4) is obvious.
We now show that (4) $\Rightarrow$ (1).
Let $f = d(x,\cdot)+1$ and $f_n := f \wedge n \in \ell_\infty(V)$.
Due to (\ref{eq:WPepsmEps}), we have
\[
\infty > W(p_x^t,m_x^t) \geq (P_t f_n - f_n -t \Delta f_n)(x)
\]
yielding
\[
P_t f(x) = \sup_n P_t f_n(x) \leq W(p_x^t,m_x^t) + f(x) + t \Delta f(x) <\infty.
\]
Thus by Lemma~\ref{lem:uniformFinitePt} $(ii)$,
$P_s f < \infty$ for all $s < t$ as desired.
This finishes the proof.
\end{proof}
\subsection{Another Ricci curvature characterization}
We now prove that on locally finite graphs with Ricci curvature bounded from below, our definition of $\kappa$ coincides with $\overline{\kappa}$ and $\underline{\kappa}$ as defined in \cite[Definition~6]{veysseire2012coarse}.
This will yield another characterization of lower Ricci curvature bounds by combining with \cite[Theorem 9]{veysseire2012coarse}.
As a preparation, we show the subexponential behavior of non-negative $\lambda$-subharmonic functions under the heat equation.
\begin{lemma}\label{lem:subharmonicPt}
Let $f \geq 0$ satisfy $\Delta f \leq \lambda f$ for some $\lambda >0$. Then, $P_t f \leq e^{\lambda t} f$.
\end{lemma}
\begin{proof}
Let $W \subset V$ be finite.
Let $f_W:= f 1_W$ and let $e^{t\Delta_W}$ be the semigroup corresponding to $\Delta_W$ with $\Delta_W g := 1_W \Delta (g 1_W)$ representing Dirichlet boundary conditions.
First, we observe that $\Delta_W f_W \leq \lambda f_W$ since $\Delta f \leq \lambda f$ and $f \geq 0$.
Let $\phi := e^{t\left(\Delta_W-\lambda \right)} f_W$. Then,
\[
\partial_t \phi = e^{-\lambda t} \left( \Delta_W -\lambda \right) e^{t\Delta_W} f_W = e^{-\lambda t} e^{t\Delta_W} \left( \Delta_W -\lambda \right) f_W \leq 0
\]
showing that $\phi(t)=e^{-\lambda t}e^{t \Delta_W} f_W \leq f_W = \phi(0)$.
Since $e^{t_W} f_W \to P_t f$ pointwise as $W\to V$, we obtain the desired claim that $P_t f \leq e^{\lambda t} f$.
\end{proof}
\begin{rem}
We remark that the step in the proof above where we take Dirichlet boundary conditions is necessary to ensure that $\Delta P_t f = P_t \Delta f$ which generally only holds true on the domain $D(\Delta) \subseteq \ell^2(V)$ on which $\Delta$ is self-adjoint.
\end{rem}
We next prove that a lower Ricci curvature bound implies that $P_t d(x,\cdot)<\infty$.
\begin{lemma}\label{lem:Ptd}
Let $x \in V$ and $f := d(x,\cdot)$. If $Ric(G) \geq -K$ for some $K >0$,
then
\[P_t f \leq e^{Kt} (f + \operatorname{Deg}(x)/K)<\infty.\]
\end{lemma}
\begin{proof}
Due to the Laplacian comparison principle, Theorem~\ref{thm:LaplaceCompare}, we have that
\[
\Delta(f + \operatorname{Deg}(x)/K) = \Delta f \leq K(f + \operatorname{Deg}(x)/K).
\]
Thus, Lemma~\ref{lem:subharmonicPt} yields
\[
P_t f \leq P_t (f + \operatorname{Deg}(x)/K) \leq e^{Kt} (f + \operatorname{Deg}(x)/K)\]
as desired.
\end{proof}
We now present the main theorem of this section.
\begin{theorem}[Continuous and discrete time curvature]\label{thm:MarkovChains}
Let $G=(V,w,m)$ be a stochastically complete graph. Suppose that $P_t d(x_0,\cdot) < \infty$ for some $x_0 \in V$ and some $t>0$. Then, for all $x \neq y$,
\[
\kappa(x,y) = \lim_{t \to 0^+} \frac 1 t \left(1 - \frac{W(p_x^t,p_y^t)}{d(x,y)} \right) = \overline\kappa(x,y) = \underline \kappa(x,y).
\]
\end{theorem}
\begin{proof}
Let $x \neq y \in V$.
Due to the triangle inequality and Proposition~\ref{pro:PtdANDWpm}, we have
\[
W(m_x^t,m_y^t) = W(p_x^t,p_y^t) + O(t^2)
\]
as $t\to 0^+$.
By definition,
\begin{align*}
\kappa(x,y) = \lim_{t\to 0^+} \frac 1 t \left(1 - \frac{W(m_x^t,m_y^t)}{d(x,y)} \right) &=\lim_{t\to 0^+} \frac 1 t \left(1 - \frac{W(p_x^t,p_y^t) + O(t^2)}{d(x,y)} \right)\\
&=\lim_{t \to 0^+} \frac 1 t \left(1 - \frac{W(p_x^t,p_y^t)}{d(x,y)} \right).
\end{align*}
This finishes the proof.
\end{proof}
Since both stochastic completeness and $P_t d(x_0,\cdot)<\infty$ are implied by a lower Ricci curvature bound (see Theorem~\ref{thm:StochComplete} and Lemma~\ref{lem:Ptd}), we immediately obtain the following corollary.
\begin{corollary}\label{cor:MarkovChains}
Let $G=(V,w,m)$ be a graph with $Ric(G) \geq K$ for some $K \in {\mathbb{R}}$. Then, for all $x \neq y$,
\[
\kappa(x,y) = \lim_{t \to 0^+} \frac 1 t \left(1 - \frac{W(p_x^t,p_y^t)}{d(x,y)} \right).
\]
\end{corollary}
Combining with \cite[Theorem~9]{veysseire2012coarse}, we immediately obtain that $W(p_t^x,p_t^y) \leq e^{-Kt} d(x,y)$ whenever $Ric(G)\geq K$.
We remark that this gives an alternative method for proving Theorem~\ref{thm:gradientGraphs}.
\addtocontents{toc}{\ignorethis}
\subsection*{Acknowledgments}
F.M. wants to thank the German National Merit Foundation for financial support.
R.K.W. gratefully acknowledges financial support from PSC-CUNY Awards, jointly funded by the Professional Staff Congress and the City University of New York, and the Collaboration Grant for Mathematicians, funded by the Simons Foundation.
R.K.W. would also like to thank Fudan and Hokkaido Universities for their generous hospitality while parts of this work were completed.
Furthermore, both authors want to thank the Harvard University Center of Mathematical Sciences and Applications for their hospitality.
\printbibliography
Florentin M\"unch, \\
Department of Mathematics,
University of Potsdam, Potsdam, Germany\\
Currently: Center of Mathematical Sciences and Applications, Harvard University, Cambridge MA, USA \\
\texttt{chmuench@uni-potsdam.de}\\
\\
Rados{\l}aw K. Wojciechowski,\\
York College and the Graduate Center of the City University of New
York, New York, USA\\
\texttt{rwojciechowski@gc.cuny.edu}
\addtocontents{toc}{\endignorethis}
\end{document}
| {
"timestamp": "2017-12-05T02:13:48",
"yymm": "1712",
"arxiv_id": "1712.00875",
"language": "en",
"url": "https://arxiv.org/abs/1712.00875",
"abstract": "Discrete time random walks on a finite set naturally translate via a one-to-one correspondence to discrete Laplace operators. Typically, Ollivier curvature has been investigated via random walks. We first extend the definition of Ollivier curvature to general weighted graphs and then give a strikingly simple representation of Ollivier curvature using the graph Laplacian. Using the Laplacian as a generator of a continuous time Markov chain, we connect Ollivier curvature with the heat equation which is strongly related to continuous time random walks. In particular, we prove that a lower bound on the Ollivier curvature is equivalent to a certain Lipschitz decay of solutions to the heat equation. This is a discrete analogue to a celebrated Ricci curvature lower bound characterization by Renesse and Sturm. Our representation of Ollivier curvature via the Laplacian allows us to deduce a Laplacian comparison principle by which we prove non-explosion and improved diameter bounds.",
"subjects": "Differential Geometry (math.DG)",
"title": "Ollivier Ricci curvature for general graph Laplacians: Heat equation, Laplacian comparison, non-explosion and diameter bounds",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.980280867283954,
"lm_q2_score": 0.7217432003123989,
"lm_q1q2_score": 0.707511050358535
} |
https://arxiv.org/abs/1307.5094 | Double-interval societies | Consider a society of voters, each of whom specify an approval set over a linear political spectrum. We examine double-interval societies, in which each person's approval set is represented by two disjoint closed intervals, and study this situation where the approval sets are pairwise-intersecting: every pair of voters has a point in the intersection of their approval sets. The approval ratio for a society is, loosely speaking, the popularity of the most popular position on the spectrum. We study the question: what is the minimal guaranteed approval ratio for such a society? We provide a lower bound for the approval ratio, and examine a family of societies that have rather low approval ratios. These societies arise from double-n strings: arrangements of n symbols in which each symbol appears exactly twice. | \section{Introduction}
Consider the voting model of Berg et.\,al.\cite{NSTW}\ in which a political spectrum $X$ is viewed as a continuum, with liberal positions on the left and conservative positions on the right, and in which each voter $v$ ``approves'' an interval of positions along this line. For example, a tolerant moderate might approve a wide interval near the middle of the line, while an intolerant partisan may approve a narrower interval near one of the ends.
More formally, a \emph{society} is a spectrum $X$ together with a set of voters $V$ and a
collection of approval sets $\{ A_{v} \}$, one for each voter. A point on the spectrum $X$ is called a \emph{platform}.
In our situation, we imagine $X$ to be $\mathbb{R}$, and each approval set $A_{v}$ is a closed interval that represents the set of all platforms that $v$ approves.
Now suppose that every pair of people can agree on some platform; that is, their intervals overlap. In this situation, Helly's Theorem \cite{helly} implies that there exists a point on the line that lies in everyone's approval set, i.e., there is a platform that everyone approves. Thus a strong hypothesis (pairwise intersecting sets) produces a strong conclusion (a point in all the sets). However, in voting theory, we are usually not looking for unanimity, but may be satisfied with a platform that has high \emph{approval ratio}: the fraction of voters that approve this platform.
Various authors have relaxed the hypotheses. Berg et.\,al.\cite{NSTW}\ define a linear $(k,m)$-agreeable society in which voter preferences again are modeled by closed intervals in $\mathbb{R}^1$. In this society, given any set of $m$ voters, there exists a subset of $k$ voters whose approval intervals mutually intersect. They prove that there must exist some platform with approval ratio $\frac{k-1}{m-1}$.
Another generalization by Hardin \cite{hardin} looks at approval intervals on a circle rather than a line, and finds that with $(k,m)$-agreeability, the approval ratio of the society is at least $\frac{k-1}{m}$.
\begin{figure}[h]
\centering
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\begin{picture}(15,2)
\linethickness{2pt}
\color{darkred}
\put(1,1.8){\line(1,0){3}}
\put(7,1.8){\line(1,0){8}}
\color{darkblue}
\put(3,1.3){\line(1,0){5}}
\put(11,1.3){\line(1,0){3}}
\color{darkyellow}
\put(2,.8){\line(1,0){4}}
\put(10,.8){\line(1,0){2}}
\color{darkgreen}
\put(5,0.3){\line(1,0){4}}
\put(13,0.3){\line(1,0){1.5}}
\color{black}
\put(-0.5,1.65){A, A':}
\put(-0.5,1.15){B, B':}
\put(-0.5,.65){C, C':}
\put(-0.5,0.15){D, D':}
\put(2,-1.5){}
\end{picture}
\caption{A pairwise-intersecting society of size 4 with approval number 3.}
\label{examplesociety}
\end{figure}
We generalize the one-interval model to a society in which every member is identified with two disjoint approval intervals and call such a society a {\it double-interval society}. This situation may arise naturally in the context of voting to account for voters who do not place candidates along a linear spectrum in exactly the same order, or to account for voters who find disjoint sets of platforms appealing for entirely different reasons (e.g., for being a party purist, or having the ability to work across party lines). In a scheduling context, such intervals might model a society of workers, each of whom has two different work shifts.
Figure \ref{examplesociety} illustrates a double-interval society with four voters. The approval sets of each voter have been separated vertically so they are easier to see. Note that the approval sets are pairwise-intersecting: each voter overlaps every other voter in one or both of their approval intervals. In this example, there are several platforms approved by three voters, but no platform is approved by all four. The {\it approval number} of a platform $a(p)$ is the number of voters (in a society $S$) who approve of platform $p$. The {\it approval number} of a society $a(S)$ is the maximum approval number over all platforms in the spectrum $X$. That is
$$ a(S) = \max_{p \in X} a(p).$$
Finally, define the {\it approval ratio} of a society
to be the approval number of $S$ divided by the number of voters in $S$.
The main question we address in this paper is: what is the minimal approval ratio of a pairwise-intersecting, double-interval society with $n$ voters?
Examples suggest that the minimal approval ratio of such societies is $1/3$; that is, there is always a platform that will get at least a third of the votes. Our results in this paper attempt to clarify this intuition.
We will first examine a family of double-interval societies with low approval ratios that have regular patterns of interval overlap.
These arise from the construction of what we call \emph{double-$n$} strings, defined in the next Section.
The combinatorics of such strings are quite nifty and provide a lower bound for the approval ratio of societies in this family
(Theorem \ref{klowerbound}) as well as an upper bound (Theorem \ref{klaweupperbound}) for societies in this family.
Roughly speaking, the double-$n$ strings produce societies with asymptotic approval ratios between $0.348$ and $0.385$.
We will also prove a general lower bound for the approval ratio of any pairwise-intersecting, double-interval society in Theorem \ref{lowerbound}, which shows the approval ratio is always greater than $0.268$.
Then we ask if we can find specific societies with lower approval ratios than the ones arising from double-$n$ strings, and discover that there are such examples. We find them by modifying the construction that comes from double-$n$ strings. See Table \ref{boundresults}. However, all of these examples have approval ratio greater than or equal to 1/3.
\section{Double-$n$ String Societies}
Double-interval societies with regular patterns of interval overlap can be represented by \emph{double-$n$ strings}, that is, strings of length $2n$ containing exactly two occurrences of each of $n$ symbols. At times we will also represent double-$n$ strings as strings of the symbols $1, \ldots, n$.
We define the \emph{distance} between two distinct symbols in a double-$n$ string to be the minimum distance between a pair of occurrences of the symbols, where the distance between two adjacent symbols is taken as 1. The \emph{diameter} of a double-$n$ string is the maximum over all $1\leq i<j\leq n$ of the distance between $i$ and $j$. We will call two entries in the list \emph{adjacent} if their positions in the list differ by no more than the diameter of the string.
Let $\delta(n)$ be the minimum diameter over all double-$n$ strings.
\begin{figure}[h]
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\mysetb{2.8}{8.4}{3}{9.8}{2.4}{3.1}{darkyellow}
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\mysetb{5.6}{6.0}{3 } {8.4} {3.6} {3.1 } {orange}
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\mylabel{2.8}{9.8}{8.4}{5}{B}
\mylabel{4.2}{8.6}{11.2}{2.6}{C}
\mylabel{5.6}{7.4}{14}{0.2}{D}
\mylabel{7} {6.2 } {9.8} {3.8} {E}
\end{picture}
\caption{A society represented by the double-$5$ string $ABCDEBECAD$.}
\label{double-n}
\end{figure}
We can construct a pairwise-intersecting double-interval society from a double-$n$ string with diameter $d$ by assigning intervals of equal width to the symbols, long enough so that each interval overlaps the intervals of the $d$ symbols to its right and left.
For example, consider the double-5 string $ABCDEBECAD$. This string has diameter 2, since any pair of symbols $A$ through $E$ appear somewhere in this list separated by at most one other symbol (e.g., the second occurrences of $A$ and $E$ in this string are distance 2 apart).
We build a society from this string by assigning intervals of equal width as in Figure \ref{double-n}.
This society has approval number 3 as can be seen
since the right endpoint of $A$'s first interval intersects the left endpoint of $C$'s first interval, and both intersect $B$'s first interval. Hence we see that $\delta(5) \leq 2$ (and in fact $\delta(5)=2$). Note that in general the approval number of a
society with an underlying double-$n$ string is one more than the diameter, that is, $a(S) = d+1$.
\section{Asymptotic approval ratios for double-$n$ string societies}
If $S$ is arises from a double-$n$ string with diameter $d$, then since $a(S)=d+1$, we see that the minimal approval ratio of such a society is
$(\delta(n)+1)/n$. By taking limits, we see that
$$
\Delta = \lim_{n \rightarrow \infty} \frac{\delta(n)+1}{n} = \lim_{n \rightarrow \infty} \frac{\delta(n)}{n}
$$
is the asymptotic approval ratio for societies arising from double-$n$ strings. In this section, we will show that
$$8/23 \leq \Delta \leq 5/13.$$
It is clear that for $n > 1$ we have $\delta(n-1) \leq \delta(n)$ since for any double-$n$ string we can form a double-$(n-1)$ string of no larger diameter by deleting both occurrences of the $n$-th symbol. Given a double-$n$ string $S$ we label the symbols as $1,2, \ldots, n$ according to the left to right order of their first occurrence within $S$.
It is easy to see that $\Delta \leq 1/2$ since the double-$n$ string $1,2,\ldots , n, 1, 2, \ldots, n$ shows that $\delta(n) \leq n/2$ for $n$ at least 2. In fact, we can show $\delta(n) < n/2$ for $n$ at least 3. Although this does not change the upper bound on $\Delta$, we will need this result for the lower bound $8/23 \leq \Delta$.
\begin{lemma} If $n>2$, then $\delta(n) < n/2$.
\label{lem:n/2}
\end{lemma}
\begin{proof}
The case $\delta(3) = 1$ follows from the double-3 string 1,2,3,1,2,3 and the case $\delta(4) = 1$ follows from the double-4 string 1,2,3,4,1,3,2,4. The proof for the general case $n>4$ is based on this double-4 string. Let $r = \lfloor n/4 \rfloor$.
We will partition 1,2,\ldots,$n$ into four strings $S_1$, $S_2$, $S_3$, $S_4$ where each is of length $ r$ or $r+1$ depending on the value of $n \mathbin{\mathrm{mod}} 4$. We prove the result by looking at the diameter of the double-$n$ string $T(n) = S_1, S_2, S_3, S_4, S_1, S_3, S_2, S_4$.
Suppose $n = 4r$ for some positive integer $r$. We let $S_i$ be the string $(i-1)r+1, (i-1)r+2,\ldots, ir $ of length $r$ and it is easy to check that the diameter of $T(n)$ is $2r - 1$ and we have $2r -1 < n/2$. For $n = 4r+1$, let $S_1$, $S_2$, $S_3$ be as before and let $S_4$ be the string $3r+1$,\ldots,$4r+1$ of length $r+1$. Now the diameter of $T(n)$ is $2r$, and again we have $2r < n/2$ as desired. For $n = 4r+2$ we set $S_1$ and $S_4$ to have length $r$ and $S_2$ and $S_3$ to have length $r+1$. Consider $b$ in $S_i$ and $c$ in $S_j$ with $i < j$. It is easy to see that unless $i = 2$ and $j = 3$, the distance between $b$ and $c$ in $T(n)$ is at most $2r$ since at least one of $S_i$ and $S_j$ has length $r$ and the other has length at most $r+1$. Moreover, for $b$ in $S_2$ and $c$ in $S_3$ the distance between $b$ and $c$ in $T(n)$ is at most $r+1$ since both of the substrings $S_2$, $S_3$ and $S_3$, $S_2$ occur in $T(n)$. Thus the diameter of $T(n) = 2r$ and we have $2r < 2r+1 = n/2$. Finally for $n = 4r+3$ we set $S_1$ to have length $r$ and $S_2$, $S_3$, $S_4$ to have length $r+1$. In this case it is easy to see that the diameter of $T(n)$ is $2r+1$ and we have $2r+1 < 2r+3/2 = n/2$.
\end{proof}
For simplicity, without loss of generality assume that if the first occurrence of symbol $m$ occurs at position $i$ in a double-$n$ string, then all symbols at positions $1 \leq j < i$ are less than $m$ (otherwise this condition can be satisfied by a permutation of the symbols in the double-$n$ string). From Lemma \ref{lem:n/2}, it is sufficient to consider double-$n$ strings that have diameter less than $n/2$. It is also easy to obtain the lower bound $\Delta \geq 1/3$ as shown in the following lemma.
\begin{lemma}
Let $r$ be a positive integer. We have
$\delta(3r+1) \geq r$.
\label{lem:3r+1}
\end{lemma}
\begin{proof}
Let $n=3r+1$. In any double-$n$ string of diameter $d$, the first occurrence of the symbol 1 can be adjacent to at most $d$ other symbols while the second occurrence can be adjacent to at most $2d$. Because 1 must be adjacent to all $n-1$ other symbols, $d+2d \geq n-1=3r$, and so $d \geq r$.\end{proof}
\begin{lemma}
\label{distinctness}
In a double-$n$ string with diameter $d$, the first $n-d$ symbols are distinct (and hence in the order $1,2,\dots,n-d$).
\end{lemma}
\begin{proof}
Assume that there exists some symbol $x$ both of whose occurrences are within the first $n-d$ entries. Thus the first occurrence of $n$ must be at position at least $n+1$, so the distance between $x$ and $n$ is at least $d+1$, a contradiction.
\end{proof}
\begin{lemma}
\label{closeones}
Let $d<n/2$ be the diameter of a double-$n$ string, and let $r_i$ be the number of symbols both of whose occurrences are within $d$ of either occurrence of $i$ for $1 \leq i \leq d+1$. Then $r_i \leq 3d+i-n$.
\end{lemma}
\begin{proof} As $d <n/2$, Lemma \ref{distinctness} gives that the first $d+1$ symbols of such a double-$n$ string are $1, 2, \dots,d+1$. For $1 \leq i \leq d+1$, there are only $i-1$ symbols before the first occurrence of $i$, so there are at most $3d+i-1$ symbols adjacent to $i$, of which $r_i$ of them are repeats. Hence $n-1 \leq 3d+i-1-r_i$.
\end{proof}
\begin{cor}
\label{cordistinct}
For $1 \leq i \leq d+1$, and $d < n/2$, at most $3d+i-n$ of the symbols $1, 2,\dots, \hat{i}, \ldots ,d+1$, are within $d$ of the second occurrence of $i$. (Here $\hat{i}$ means omit $i$).
\end{cor}
\begin{proof}
This follows directly from Lemma~\ref{closeones} since each of the symbols $1,\ldots, \hat{i},$ $\ldots, d+1$ occurs within $d$ of the first occurrence of $i$.
\end{proof}
We are now ready to prove the lower bound.
\begin{theorem}
\label{klowerbound}
Let $r$ be a positive integer. Then $\delta(23r) \geq 8r$. Thus the asymptotic approval ratio for double-$n$ strings is bounded below by $8/23$.
\end{theorem}
\begin{proof}
Let $n = 23r$ and let $S$ be a double-$n$ string with diameter $d$. Suppose $d < 8r$. Since $d$ is an integer we have $d \leq 8r - 1$.
Note that $d \geq \delta(23r) \geq \delta(21r + 1) \geq 7r$ by Lemma \ref{lem:3r+1}.
By Lemma \ref{distinctness} the first $n-d \geq 23r -8r + 1 =15r + 1$ symbols in $S$ are distinct (and in order). Now since $d < 8r$ the first occurrence of the symbol labeled $15r+1$ is not within $d$ of the first occurrence of $i$ for $1 \leq i \leq 7r+1$. Thus for any such $i$ we must have the second occurrence of $i$ occurring in one of three sets of positions, namely the block $B_1$ of length $d$ following the first occurrence of $15r+1$, the block $B_2$ of length $d$ ahead of the second occurrence of $15r+1$, or the block $B_3$ of length $d$ following the second occurrence of $15r+1$. These blocks are illustrated in Figure~\ref{klawelowerbounddiagram}. Let $k_j$ be the number of symbols in $1 \leq i \leq 7r+1$ with their second occurrence in block $B_j$. From the preceding observation we have $k_1 + k_2 + k_3 \geq 7r + 1$ (conceivably such a second occurrence of $i$ could be in both $B_1$ and $B_2$ if they overlap).
Note that any pair of symbols in $B_j$ lie within $d$ of each other. Suppose without loss of generality that the second occurrence of 1 lies in $B_1$. For any $i$ with $1 < i \leq 7r+1$ with the second occurrence of $i$ in $B_1$, both occurrences of $i$ lie within $d$ of an occurrence of 1, since $d \geq 7r$. By Corollary \ref{cordistinct}, the number of such $i$ is at most
$$3d + 1 - n \leq 3(8r-1) + 1 - 23r = r - 2,$$ giving $k_1 \leq 1 + r - 2 = r - 1$.
Let $x$ be the minimal number such that the second occurrence of $x$ is not in $B_1$. Then $x \leq r$ since $k_1 \leq r - 1$. Without loss of generality suppose the second occurrence of $x$ is in $B_2$. Again, by Corollary \ref{cordistinct} there are at most
\[3d+r-n \leq 3(8r-1)+r-23r = 2r-3\]
symbols $i$ with $1 \leq i \leq 7r+1$ other than $x$ in $B_2$, so $k_2 \leq 2r-2$.
Similarly, let $y$ be the smallest symbol (in value) whose second occurrence is in $B_3$ (i.e., is not in $B_1$ or $B_2$). There are at most $k_1+k_2$ symbols in $B_1 \cup B_2$, so $y \leq 3r-2$. Using Corollary \ref{cordistinct} one last time, we see that there are at most $3d+(3r-2)-n\leq 3(8r-1)+(3r-2)-23r = 4r-5$ symbols $i \neq y$ with $1 \leq i \leq 7r+1$ in $B_3$, so $k_3 \leq 4r-4$. However, this is a contradiction: we needed $k_1+k_2+k_3 \geq 7r+1$, but
\[k_1+k_2+k_3 \leq (r-1)+(2r-2)+(4r-4) = 7r-7.\]
Therefore we could not have $d<8r$, proving the theorem.
\end{proof}
A general argument showing $\delta(br)\geq ar$, for large $r$, leads to the inequalities $b<3a$ and $23a\leq 8b$. Thus the lower bound of Theorem \ref{klowerbound} is the best possible asymptotic bound using this argument.
Now we turn to the upper bound.
\begin{figure}
\centering
\includegraphics[width=4in]{klawelowerbounddiagram2.pdf}
\caption{$B_1$, $B_2$, and $B_3$ (in the case that they are disjoint).}
\label{klawelowerbounddiagram}
\end{figure}
\begin{theorem}
\label{klaweupperbound}
For any $n>0$, there exists a double-$n$ string with diameter $d \leq 5\ceil{\frac{n}{13}}-1$. Hence the asymptotic approval ratio for double-$n$ strings is bounded above by $5/13$.
\end{theorem}
\begin{proof}
Note that the double-13 string
\[1,2,3,4,5,6,7,8,9,10,1,11,6,12,13, 5, 4, 7, 11,10, 9, 2, 3,13,12, 8\]
has diameter 4, meaning that any two symbols in it appear somewhere in the double-13 string separated by no more than three other elements. This yields a general construction for double-$n$ strings of any length. Let $k=\ceil{\frac{n}{13}}$. Then replacing each symbol $i$ in the above string with the substring
\[k(i-1)+1, k(i-1)+2,\dots, ki,\]
and removing any symbols in the resulting string that are greater than $n$, yields a double-$n$ string. An example of this string for $n=34$ ($k=3$) is shown in Figure~\ref{klaweupperboundexample}. Because the diameter of the above double-13 string is 4, any two symbols $1 \leq i<j \leq n$ are within substrings that are separated by at most three substrings of length $k$. Also, $i$ and $j$ are at worst on the far ends of their substrings, giving a maximum total distance between $i$ and $j$ in the new string of\\
\[3\ceil{\frac{n}{13}}+\left(2\ceil{\frac{n}{13}}-1\right) =5\ceil{\frac{n}{13}}-1. \qedhere \]
\end{proof}
\begin{figure}
\centering
{\fontsize{10}{8}
\[(1,2,3)(4,5,6)(7,8,9)(10,11, 12)(13,14, 15)(16,17, 18)(19,20, 21)(22,23, 24)\]
\[(25,26, 27)(28,29, 30)(1,2, 3)(31,32,33)(16,17,18)(34)(13,14,15)(10,11,12)\]
\[(19,20, 21)(31,32, 33)(28,29, 30)(25,26, 27)(4,5, 6)(7,8, 9)(34)(22,23, 24)\]}
\caption{A double-34 string with diameter $\leq 14$ constructed as in Theorem~\ref{klaweupperbound}. Symbols are grouped together by parentheses to elucidate its construction. Some groupings have fewer than three elements since symbols larger than 34 in value are removed. Empty groupings are also omitted.}
\label{klaweupperboundexample}
\end{figure}
\section{A double-interval society lower bound}
In the previous section we considered double-$n$ strings as examples of societies with low approval ratios. These examples give upper bounds for the minimal guaranteed approval ratio for any society. In this section, we give lower bounds for the minimal guaranteed approval ratio, by considering how general pairwise-intersecting double-interval societies force conditions on the number of intervals that can intersect a given interval at its endpoints. This approach largely ignores the geometry of the approval sets and considers only combinatorial constraints.
\begin{theorem}
\label{lowerbound}
The approval number $a(S)$ of any $n$-voter society $S$ satisfies
\begin{equation}
\label{nicebound}
a(S) \geq \left\lceil 2n+\frac{1}{2}-\sqrt{3n^2-n+\frac{1}{4}} \right\rceil .
\end{equation}
Then the approval ratio satisfies
\begin{equation}
\label{niceapproval}
\frac{a(S)}{n} \geq 2-\sqrt{3} + \frac{3+\sqrt{3}}{6n} - \frac{\sqrt{3}}{24n^{2}}
\approx 0.268 + \frac{0.789}{n} - \frac{1.732}{24n^{2}}.
\end{equation}
Alternatively, the size $n$ of a society achieving a given approval number $a(S)$ is bounded above by
\begin{equation}
\label{nicen}
n \leq \left\lfloor 2a(S) - \frac32 + \sqrt{3(a(S))^2 - 5 a(S) + \frac94} \right\rfloor.
\end{equation}
\end{theorem}
\begin{proof}
Let $A_i$ and $A_i'$ represent the left and right intervals, respectively, of voter $i$'s approval set in the $n$-voter society $S$. Without loss of generality we may assume no two interval endpoints coincide.
For any interval $I$,
define numbers $L(I)$, $R(I)$, $B(I)$, and $C(I)$ to keep track of the number of other intervals that intersect $I$ in various ways.
Let $L(I)$ count the number of other intervals that, of two endpoints of $I$, contain only the \emph{left} endpoint.
Let $R(I)$ count the number of other intervals that, of two endpoints of $I$, contain only the \emph{right} endpoint.
Let $B(I)$ count the number of other intervals that contain \emph{both} endpoints of $I$.
Let $C(I)$ count the number of other intervals that intersect $I$ but contain \emph{neither} endpoint of $I$, and are hence in the ``center'' of $I$.
For example, in Figure~\ref{examplesociety}, we see that $L(A')=2$, $R(A')=0$, $C(A')=3$, and $B(A')=0$. Also $L(C')=0$, $R(C')=1$, $C(C')=0$, and $B(C')=1$. Since each set must intersect all $n-1$ other sets,
$$
L(A_i)+L(A_i')+R(A_i)+R(A_i')+C(A_i)+C(A_i')+B(A_i)+B(A_i') \geq n-1.
$$
Then clearly
\begin{eqnarray}
\sum\limits_{i=1}^{n} \left[ L(A_i)+L(A_i')+R(A_i)+R(A_i')+C(A_i')+C(A_i)+B(A_i)+B(A_i') \right] \nonumber \\
\geq n(n-1).
\label{sumbound}
\end{eqnarray}
Note that an interval $J$ covers both endpoints of another interval $I$ and contributes $1$ to the count $B(I)$ exactly when $I$ is the in the center of $J$ and contributes $1$ to the count $C(J)$.
This implies:
\begin{equation}
\label{BC}
\sum\limits_{i=1}^{n} \left[ B(A_i)+B(A_i') \right] = \sum\limits_{i=1}^{n} \left [ C(A_i)+C(A_i') \right].
\end{equation}
Notice that given an approval number $a(S)$, each interval may have at most $a(S)-1$ other sets intersecting its left endpoint. This gives an initial bound
\[\sum\limits_{i=1}^{n} \left[ L(A_i)+L(A_i')+B(A_i)+B(A_i') \right] \leq 2n(a(S)-1).\]
and similarly, considering right endpoints:
\[\sum\limits_{i=1}^{n} \left[ R(A_i)+R(A_i')+B(A_i)+B(A_i') \right] \leq 2n(a(S)-1).\]
However, if the $2n$ intervals are ordered by the left endpoint, then the $k$th interval under this ordering from left to right can have at most $k-1$ intervals intersecting its left endpoint, not $a(S)-1$. Thus we need to adjust the formulas above, to obtain:
\[\sum\limits_{i=1}^{n} \left[ L(A_i)+L(A_i')+B(A_i)+B(A_i') \right] \leq 2n(a(S)-1)-\frac{a(S)(a(S)-1)}{2}, \]
\[\sum\limits_{i=1}^{n} \left[ R(A_i)+R(A_i')+B(A_i)+B(A_i') \right] \leq 2n(a(S)-1)-\frac{a(S)(a(S)-1)}{2}.\]
Adding these equations and applying equation~(\ref{BC}) yields\\
\begin{eqnarray}
\sum\limits_{i=1}^{n} \left[ L(A_i)+L(A_i')+R(A_i)+R(A_i')+C(A_i')+C(A_i)+B(A_i)+B(A_i') \right] \nonumber \\
\leq 4n(a(S)-1)-a(S)(a(S)-1). \nonumber
\end{eqnarray}
So by equation~(\ref{sumbound}), we see\\
\[ \label{niceeqnbound}(4n-a(S))(a(S)-1) \geq n(n-1).\]
Solving this quadratic inequality for $a(S)$, and rounding up to the nearest integer gives the conclusion (\ref{nicebound}). Using $(1-x)^{1/2} \leq 1-(1/2)x$ gives conclusion (\ref{niceapproval}). Solving the quadratic inequality for $n$ and rounding down gives the conclusion (\ref{nicen}).
\end{proof}
Values of $a(S)$ and the corresponding bounds on $n$ and the approval ratio derived from equation~(\ref{nicen}) are given in Table~\ref{boundresults}.
\begin{table}
\centering
\begin{tabular}{c|cc|cc}
\hline
$a(S)$ & $n$ & Approval & Observed & Observed
\\
& & Ratio& $n$ & Approval Ratio\\
\hline
$2$ & $\leq 4$ & $\geq 0.500$ & 4 & 0.500\\
$3$ & $\leq 8$ & $\geq 0.375$ & 8 & 0.375\\
$4$ & $\leq 12$ & $\geq 0.333$ & 12 & 0.333\\
$5$ & $\leq 15$ & $\geq 0.333$ & 15 & 0.333\\
$6$ & $\leq 19$ & $\geq 0.316$ & 18 & 0.333\\
$7$ & $\leq 23$ & $\geq 0.304$ & 21 & 0.333\\
$8$ & $\leq 26$ & $\geq 0.308$ & 24 & 0.333\\
$9$ & $\leq 30$ & $\geq 0.300$ & 27 & 0.333\\
$10$ & $\leq 34 $ & $\geq 0.294$ & 30 & 0.333\\
$11$ & $\leq 38$ & $\geq 0.289$ & 32 & 0.344\\
$12$ & $\leq 41 $ & $\geq 0.293$ & 35 & 0.343\\
\hline
\end{tabular}
\bigskip
\caption{On the left, this table shows for a given approval number the largest $n$ that is given by inequality (\ref{nicen}) as well as the resulting bound on the approval ratio derived from inequality (\ref{nicebound}). On the right, this table shows, for a given approval number, known examples of the largest $n$ that has this approval number and the observed approval ratio in that case, obtained by a modification of a double-$n$ string construction.}
\label{boundresults}
\end{table}
\section{Modifying double-$n$ string societies}
In this section we give an example of a double-interval society with an approval ratio lower than the bound given by Theorem \ref{klowerbound}, thus showing that
double-$n$ strings do not always provide examples of societies with minimal approval ratios.
We will require a new notation, called the \emph{endpoint representation} of a society. We will encode a society as a sequence of symbols (corresponding to the approval sets) representing the order of the endpoints of all the approval sets, each prefixed by a ``+'' or a ``$-$' to denote a left or right endpoint respectively. For example, the society in Figure~\ref{examplesociety} is represented as
\[+A+C+B-A+D-C+A-B-D+C+B-C+D-B-D-A.\]
\begin{prop}
\label{counterexample}
There exists a society of size $n=8$ with approval number 3. Hence there exist $n$ for which double-$n$ strings do not produce the lowest possible approval numbers.
\end{prop}
\begin{figure}
\centering
\setlength{\unitlength}{11pt}
\begin{picture}(32,20)
\linethickness{2pt}
\mysetb{1}{19.8}{3}{15}{10.2}{7}{darkred}
\mysetb{2}{18.6}{4}{23}{5.4}{8}{darkblue}
\mysetb{3}{17.4}{5}{25}{4.2}{5}{darkyellow}
\mysetb{7}{15}{3}{19}{7.8}{5}{darkgreen}
\mysetb{5}{16.2}{7}{17}{9}{1}{orange}
\mysetb{11}{12.6}{3}{21}{6.6}{5}{violet}
\mysetb{9}{13.8}{7}{27}{3}{1}{teal}
\mysetb{13}{11.4}{7}{29}{1.8}{3}{brown}
\mylabel{2.25}{20}{18.25}{10.4}{A}
\mylabel{3.75}{18.8}{26.75}{5.6}{B}
\mylabel{5.25}{17.6}{27.25}{4.4}{C}
\mylabel{8.25}{15.2}{21.25}{8}{D}
\mylabel{8.25}{16.4}{17.25}{9.2}{E}
\mylabel{12.25}{12.8}{23.25}{6.8}{F}
\mylabel{12.25}{14}{27.25}{3.2}{G}
\mylabel{16.25}{11.6}{30.25}{2}{H}
\end{picture}
\caption{A society of size 8 with approval number 3.}
\label{3-8counterexample}
\end{figure}
The society shown in Figure~\ref{3-8counterexample} provides such a society. This example was
derived from the double-8 string
\[ABCDEFGHEADFCGBH.\]
If each interval in the string overlaps two intervals on each side, this arrangement
is missing the adjacencies $AG$, $BE$, $BF$, $CA$, $DG$, $DG$ and $CH$ and has duplicate adjacencies $BC$, $CD$, $DC$, $DE$, $DF$, $EG$, $FG$, and $GH$.
By doing a series of moves that interchanges endpoints in such a way as to introduce missing adjacencies (at the expense of duplicate adjacencies) without increasing the approval number, we arrive at the society
\begin{eqnarray*}
&&+A+B+C-A+E-B+D-C+G-D+F-E+H-F+A-G\\
&&+E-E+D-H+F-A+B-D+C-F+G-G+H-C-B-H.
\end{eqnarray*}
We note that an example like this with $n=8$ and $a(S)=3$ cannot be achieved by a double-$n$ string since the first symbol in a double-$n$ string with diameter $d$ is adjacent to at most $3d$ other symbols. Thus, as in Lemma \ref{lem:3r+1}, we have
$n \leq 3d+1 = 3(a(S)-1) +1$, and so the approval number of a double-8 string must be at least 4.
It is not clear how to systematically interchange endpoints to achieve all possible adjacencies. However, an algorithm which aims at making ``smart'' swaps produced societies with approval ratios given in Table \ref{boundresults}. A description of the algorithm can be found in \cite{Scott}.
The results of the hill-climbing algorithm in Figure~\ref{hillclimbingoutput} suggest that the asymptotic approval ratio should be $1/3$.
\begin{figure}
\begin{verbatim}
a(S)=3, n=8, AR=0.375: +A+B+F-F+G-A+F-B+C-C+D-G+E-D+H-F+G-G+A-E+D-H
+C-A+B-D+E-E+H-H-B-C
a(S)=4, n=12, AR=0.333: +A+B+C+F-C+H-B+L-L+G-G+I-F+D-A+C-H+L-D+E-E+J
-I+G-J+K-C+B-L+I-I+D-K+E-G+J-B+F-D+K-F+A-A+H
-K-E-H-J
a(S)=5, n=15, AR=0.333: +A+B+C+D+E-C+G-A+O-G+F-D+K-F+J-J+N-E+C-B+A-O
+D-K+H-H+M-N+J-M+L-L+I-D+F-A+G-C+N-I+L-N+H-J
+M-F+K-G+I-K+B-B+E-E+O-H-M-I-L-O
a(S)=6, n=18, AR=0.333: +A+B+C+D+E+G-A+M-C+K-G+Q-Q+P-P+O-O+J-D+F-E+A
-B+C-F+P-K+I-I+R-M+O-R+H-H+L-J+Q-L+N-A+G-C+J
-J+F-P+I-O+R-N+H-Q+L-G+D-F+N-D+K-K+B-B+E-R+M
-N-M-E-H-L-I
a(S)=7, n=21, AR=0.333: +A+B+C+D+E+F+I-A+K-K+N-N+R-R+G-E+T-B+J-T+U-G
+Q-D+B-I+P-Q+M-C+H-F+L-J+S-U+O-B+D-D+C-C+I-I
+F-F+G-P+Q-M+R-L+J-S+N-O+K-H+A-G+E-J+T-Q+P-P
+M-M+U-R+S-S+O-U+L-N+H-T-H-K-L-E-O-A
a(S)=8, n=24, AR=0.333: +A+B+C+D+E+F+G+L-G+N-C+O-F+Q-O+M-N+T-B+H-H+K
-Q+I-I+U-U+X-D+J-L+F-A+O-E+C-J+G-M+P-T+U-P+H
-K+W-X+I-W+R-R+V-V+S-O+Q-C+J-F+N-G+B-U+X-S+P
-H+R-I+V-X+W-Q+K-B+D-J+T-N+S-D+L-T+M-K+A-L+E
-P-A-W-S-E-M-V-R
\end{verbatim}
\caption{Output pairwise-intersection double-interval societies with given sizes and approval ratios found by a heuristic
algorithm. Here $AR$ denotes the approval ratio.}
\label{hillclimbingoutput}
\end{figure}
\bigskip
\section{Conclusion and Open Questions}
We have studied pairwise-intersecting double-interval societies, and determined bounds for the minimum guaranteed approval ratio for such societies. Such questions naturally motivated the study of double-$n$ strings, which represent certain special double-interval societies with low approval ratios. Although these do not necessarily provide the smallest such ratios, all of the known examples that provide smaller ratios come from modifying the double-$n$ string construction.
There are numerous open questions.
\begin{itemize}
\item For double-$n$ strings, is there a systematic way to construct strings of the smallest diameter?
\item
Beyond double-$n$ strings, is there a better general construction that yields societies with the lowest approval ratios?
\item
With double-$n$ strings, we currently have $\Delta$ bounded by $0.348 \leq \Delta \leq 0.385$. Can we tighten the bounds on $\Delta$?
\item
What results can be obtained for triple-interval societies?
\item
What about higher-dimensional approval sets? What can be said if each voter's approval set consists of two convex sets in the plane?
\end{itemize}
Finally, we end with our initial conjecture, which now has more evidence as support.
\begin{conjecture}
For all pairwise-intersecting double-interval societies $S$, the approval ratio $$\frac{a(S)}{n} \geq \frac13.$$
\end{conjecture}
\bibliographystyle{amsplain}
| {
"timestamp": "2013-07-22T02:00:56",
"yymm": "1307",
"arxiv_id": "1307.5094",
"language": "en",
"url": "https://arxiv.org/abs/1307.5094",
"abstract": "Consider a society of voters, each of whom specify an approval set over a linear political spectrum. We examine double-interval societies, in which each person's approval set is represented by two disjoint closed intervals, and study this situation where the approval sets are pairwise-intersecting: every pair of voters has a point in the intersection of their approval sets. The approval ratio for a society is, loosely speaking, the popularity of the most popular position on the spectrum. We study the question: what is the minimal guaranteed approval ratio for such a society? We provide a lower bound for the approval ratio, and examine a family of societies that have rather low approval ratios. These societies arise from double-n strings: arrangements of n symbols in which each symbol appears exactly twice.",
"subjects": "Combinatorics (math.CO)",
"title": "Double-interval societies",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808736209155,
"lm_q2_score": 0.7217431943271999,
"lm_q1q2_score": 0.7075110490650177
} |
https://arxiv.org/abs/1705.10490 | Goussarov-Polyak-Viro's $n$-equivalence and the pure virtual braid group | In the context of finite type invariants, Stanford introduced a family of equivalence relations on knots defined by the lower central series of the pure braid groups and characterized the finite type invariants in terms of the structure of the braid groups. It is known that this equivalence and Ohyama's equivalence defined by a local move are equivalent. On the other hand, in the virtual knot theory, the concept of Ohyama's equivalence was extended by Goussarov-Polyak-Viro, which called an $n$-equivalence. In this paper we extend Stanford's equivalence to virtual knots and virtual string links by using the lower central series of the pure virtual braid group, and call it an $L_n$-equivalence. We then prove that the $L_n$-equivalence is equal to the $n$-equivalence on virtual string links. Moreover we directly prove that two virtual string links are not distinguished by any finite type invariants of degree $n-1$ if they are $L_n$-equivalent, for any positive integer $n$. | \section{Introduction}
The theory of finite type invariants of knots and links was introduced by Vassiliev \cite{V} and Goussarov \cite{G1, G2} and developed by Birman-Lin \cite{BL}.
Goussarov \cite{G3, G4} and Habiro \cite{H0, H} independently introduced theories of surgery along embedded graphs in 3-manifolds, called $Y$-graphs or variation axes by Goussarov, and claspers by Habiro.
An $n$-variation equivalence (called $n$-equivalence in \cite{G4}) or $C_n$-equivalence for links is generated by n-variations \cite{G4} or $C_n$-moves \cite{H}, respectively.
Goussarov proved in \cite{G4} that for string links and knots in $S^3$, the n-variation (or $C_n$-) equivalence coincides with the Goussarov-Ohyama $n$-equivalence \cite{G1, O}.
Stanford proved in \cite{S2} that two links are not distinguished by any finite type invariant of degree $n$ if one is obtained from the other by inserting an element of the $(n+1)$-th lower central series subgroup of the pure braid group.
Goussarov \cite{G4} and Habiro \cite{H0, H} independently proved that two knots are not distinguished by any finite type invariant of degree $n$ if and only if they are related by a finite sequence of $C_n$-moves and ambient isotopies.
Moreover Stanford \cite{S3} translated Habiro's result for $C_n$-moves into the pure braid setting.
On the other hand, a {\it $($long$)$ virtual knot} is defined by a (long) knot diagram with virtual crossings module Reidemeiser moves, introduced by Kauffman \cite{K}.
Goussarov-Polyak-Viro \cite{GPV} showed that the (long) virtual knot can be redefined as {\it Gauss diagram} and also gave the theory of finite type invariants on Gauss diagrams.
They also defined an $n$-equivalence on (long) virtual knots and notioned that the value of a finite type invariant of degree less than or equal to $n$ depended only the $n$-equivalence class.
In this paper, we extend a $C_n$-move to (long) virtual knots, called an $L_n$-move, by using Stanford's method.
The $L_n$-moves generate the $L_n$-equivalence on (long) virtual knots.
We prove that $L_n$-equivalence coincides with $n$-equivalence on long virtual knots.
Moreover, we directly prove that, for any non-negative integer $n$, if two long virtual knots are $L_n$-equivalent, then they are not distinguished by any finite type invariants of degree $n-1$.
Their extensions and results are also establish on virtual string links.
\section*{Acknowledgements}
The author thanks Professor Kazuo Habiro for a lot of comments, discussions and suggestions.
The author also thanks Professor Vassily Manturov for comments and suggestions.
\section{Gauss diagram}
A {\it Gauss diagram on the interval} is an oriented interval with several oriented chords having disjoint end points and equipped with sign as in Figure \ref{gaussdiagram}.
Here, we call the chord an {\it arrow}.
\begin{figure}[h]
\begin{center}
\includegraphics[width=.7\linewidth]{gaussdiagram.eps}
\put(-225,30){$+$} \put(-185,46){$-$} \put(-120,30){ $-$} \put(-85,30){ $+$} \put(-35,30){ $+$}
\caption{A Gauss diagram}
\label{gaussdiagram}
\end{center}
\end{figure}
{Reidemeister moves} among Gauss diagrams are the following three moves in Figure \ref{Reidemeistermoves}:
First Reidemeister move (RI) is in the top row. Second Reidemeister move (RII) is in the second row.
Third Reidemeister move (RIII) is in the remain two rows.
\begin{figure}[h]
\begin{center}
\includegraphics[width=.8\linewidth]{Reid.eps}
\put(-207,160){$RI$} \put(-95,160){$RI$} \put(-207,112){$RII$} \put(-95,112){$RII$} \put(-150,58){$RIII$} \put(-150,8){$RIII$}
\put(-263,175){$\epsilon$} \put(-35,175){$\epsilon$} \put(-263,130){$-\epsilon$} \put(-263,115){$\epsilon$}\put(-50,125){$\epsilon$} \put(-20,125){$-\epsilon$}
\put(-263,65){$-$} \put(-233,80){$-$} \put(-190,70){$-$} \put(-100,65){$-$} \put(-80,85){$-$} \put(-60,70){$-$}
\put(-263,20){$+$} \put(-235,22){$+$} \put(-210,22){$+$} \put(-110,20){$+$} \put(-80,30){$+$} \put(-40,20){$+$}
\caption{The Reidemeister moves}
\label{Reidemeistermoves}
\end{center}
\end{figure}
\begin{definition}
Two Gauss diagrams $D$ and $D'$ are said to be {\it equivalent} if $D$ and $D'$ are related by Reidemeister moves.
By $D {\sim} D'$ we mean that $D$ and $D'$ are equivalent.
We define a {\it long virtual knot} to be the equivalence class of a Gauss diagram $D$, which is denoted by $[D]$.
Similarly, the equivalence class of Gauss diagram on circle ($k$ intervals) is virtual knot ($k$-component virtual string, respectively).
\end{definition}
\section{Finite type invariant of virtual knot}
Goussarov Polyak and Viro defined a finite type invariant for (long) virtual knots in \cite{GPV}.
Similar way to classical knots, we can define Vassiliev-Goussarov filtration on $\mathbb{Z}$-module generated by the set of (long) virtual knots.
\begin{definition}
Let $\mathcal{LVK}$ be the set of long virtual knots.
For each $n \geq 0$, let $\mathcal{SLVK}^{n}$ denote the set of equivalence classes of Gauss diagrams with $n$ dashed arrows equipped with sign with fixing dashed arrows.
We construct a map $\varphi : \mathcal{SLVK}^{n} \rightarrow \mathbb{Z}\mathcal{LVK}$ as follows.
Let $D$ be a Gauss diagram with $n$ dashed arrows.
Let $a_1, \cdots , a_n$ be the dashed arrows of $D$.
For $\epsilon_1, \cdots , \epsilon_n$ in $\{\pm 1\}$, let $D_{\epsilon_1, \cdots , \epsilon_n}$ denote the Gauss diagram obtained from $D$ by replacing each dashed arrow $a_i$ with an arrow if $\epsilon_i =1$ and removing each dashed arrow $a_i$ if $\epsilon_i =-1$.
We then define
$$\varphi([D]) = \sum_{\epsilon_1, \cdots , \epsilon_n \in \{\pm1\}} \epsilon_1 \cdot \dots \cdot \epsilon_n [D_{\epsilon_1, \cdots , \epsilon_n}].$$
Let $f$ be an invariant of $\mathcal{LVK}$ with values in an abelian group $A$.
We extend it to $\mathbb{Z}\mathcal{LVK}$ by linearly. Then $f$ is said to be a {\it finite type invariant of degree $n$} if $f \circ \varphi$ vanishes for any long virtual knot with more than $n$ dashed arrows.
\end{definition}
\begin{definition}
Denote by $J_n$ the subgroup of $\mathbb{Z}\mathcal{LVK}$ generated by the set consisting of the element $\varphi([D])$, where $[D]$ is in $\mathcal{SLVK}^{n}$.
It is easy to see that the $J_n$'s form a descending filtration of two-sided ideals of the monoid ring $\mathbb{Z}\mathcal{LVK}$ under the composition:
$$\mathbb{Z}\mathcal{LVK}=J_0 \supset J_1 \supset J_2 \supset \cdots , $$
which we call the Vassiliev-Goussarov filtration on $\mathbb{Z}\mathcal{LVK}$.
Here for Gauss diagrams (or virtual knots) $D$ and $D'$ ($K$ and $K'$), we denote by $D \cdot D'$ ($K \cdot K'$) their composition.
\end{definition}
Later, we will redefine $J_n$ by using claspers.
\begin{remark
Let $A$ be an abelian group and $n$ a positive integer.
The following two conditions are equivalent.
A map is an $A$-valued finite type invariant of degree $n$ on $\mathcal{LVK}$ and the map is a homomorphism of $\mathbb{Z}\mathcal{LVK}$ into $A$ which vanishes on $J_{n+1}$
\end{remark}
\begin{definition}
For $n \geq 0$, two long virtual knots $K$ and $K'$ are said to be {\it $V_n$-equivalent} if $K$ and $K'$ are not distinguished by any finite type invariants of degree $n$ with values in any abelian group, equivalently, $K- K' \in J_{n+1}$.
\end{definition}
\section{Definition of $L_n$-equivalence}
By using the pure virtual braid group, we introduce a new equivalence relation on Gauss diagrams, called $L_n$-equivalence.
Because the pure braid group is a subgroup of the pure virtual braid group (see \cite{F, M}), this is an extension of $C_n$-equivalence.
We then give properties of the set of $L_n$-equivalence classes.
\begin{definition}[\cite{B, KL}]
A {\it pure virtual braid group} $PV_k$ on $k$ strands is a group represented by the following group representation.
\begin{align*}
PV_k =
\left<
\begin{array}{l}
\mu_{ij} \ (1 \leq i, j \leq k, i \neq j)
\end{array}
\right.
\left|
\begin{array}{l}
\mu_{ij}\mu_{il}\mu_{jl} =\mu_{jl}\mu_{il}\mu_{ij} \\
\mu_{ij}\mu_{lm} =\mu_{lm}\mu_{ij}
\end{array}
\right.
\left.
\begin{array}{l}
\text{(for all distinct } i,j,l) \\
(\{i,j\} \cap \{l,m\}= \emptyset)
\end{array}
\right>.
\end{align*}
\end{definition}
Here, the element of the pure braid group is represented by a diagram as in Figure \ref{purevirtualbraid},
where $\mu_{ij}^\epsilon$ is correspondence with a horizontal arrow equipped with sign $\epsilon$ from the $i$-th strand to the $j$-th strand,
and we determine that the orientation of the strand is from top to bottom.
For example, the diagram in Figure \ref{purevirtualbraid} is correspondence with $\mu_{12}\mu_{31}^{-1}\mu_{23}\mu_{12}^{-1} \in PV_3$.
\begin{figure}[h]
\begin{center}
\includegraphics[width=.15\linewidth]{purebraid.eps}
\put(-55,53){$1$} \put(-30,53){$2$} \put(-6,53){$3$}
\put(-45,48){$+$} \put(-20,38){$-$} \put(-20,27){$+$} \put(-45,16){$-$}
\caption{An element of pure virtual braid group}
\label{purevirtualbraid}
\end{center}
\end{figure}
Let $h \in PV_k$ and $h' \in PV_{k'}$.
We denote the {\it composition} and {\it tensor product} of two elements of the pure virtual braid group as $h \cdot h' = {\includegraphics[width=.04\linewidth]{composition.eps}\put(-10,10){$h$} \put(-10,1){$h'$}} \in PV_{k}$ if $k=k'$ and $h \otimes h' = {\includegraphics[width=.08\linewidth]{product.eps} \put(-25,1){$h$} \put(-10,1){$h'$}} \in PV_{k+k'}$ for any $k$, $k'$, respectively.
By $\Gamma_n(G)$ we mean the $n$-th lower central subgroup of the group $G$, that is, $\Gamma_1(G)= G$ and $\Gamma_n(G)= [\Gamma_{n-1}(G), G]$, which is the commutator of $\Gamma_{n-1}(G)$ and $G$, that is $<[a,b] \mid a \in \Gamma_{n-1}(G), b \in G >$ where $[a,b]=aba^{-1}b^{-1}$.
\begin{definition}
Two Gauss diagrams $D$ and $D'$ are related by an {\it $L_n$-move} if there are a positive integer $k$, an element $h$ in the $n$-th lower central subgroup $\Gamma_n(PV_k)$ of the pure virtual braid group $PV_k$ on $k$ strands and not in $\Gamma_{n+1}(PV_k)$, and an embedding $e$ of $k$ strands such that $D^{(h,e)}=D'$,
where $D^{(h,e)}$ is obtained from $D$ by attaching $h$ by an embedding $e$ of $k$ strands of $h$ in the interval of $D$ except for the end points of all arrows of $D$ as in Figure \ref{clasperdef}.
By $D \overset{L_n}{\rightarrow} D'$ we mean that $D'$ is obtained from $D$ by $L_n$-move.
In particular, we write $D \overset{(h,e)}{\rightarrow} D'$ if $D'=D^{(h,e)}$.
\begin{figure}[h]
\begin{center}
\includegraphics[width=.8\linewidth]{clasperdef.eps}
\put(-235,98){$e$} \put(-275,70){$h$} \put(-0,73){$D$} \put(-145,-5){$1$} \put(-90,-5){$2$} \put(-30,-5){$3$}
\put(-245,5){$D^{(h,e)}=$}
\put(-290,118){$1$} \put(-275,118){$2$} \put(-260,118){$3$}
\put(-283,110){{\small $+$}} \put(-269,98){$-$} \put(-283,80){$+$} \put(-269,88){$-$}
\put(-190,100){$+$} \put(-130,130){$+$}
\put(-190,25){$+$} \put(-130,55){$+$}
\put(-40,20){$-$} \put(-40,35){$+$} \put(-105,8){$-$} \put(-80,8){$+$}
\caption{A clasper for a Gauss diagram}
\label{clasperdef}
\end{center}
\end{figure}
We call a pair $(h,e)$ for $D$ a {\it clasper} for $D$.
We define that a clasper $(h,e)$ is of {\it degree} $n$ if $h \in \Gamma_n(PV_k)$ and $h \notin \Gamma_{n+1}(PV_k)$,
where $k$ is a positive integer, and denote the degree of the clasper $(h,e)$ by deg$((h,e))$.
Hereinafter, we omit the number $k$ of strands if it is not important.
In particular, we call a pair $(h,e)$ a {\it tree clasper} for $D$ if $h \in \Gamma_n(PV_k)$ is an $n$-th commutator $[a_1, [a_2, \cdots [a_n, a_{n+1}] \cdots ] ]$ where $a_i \in PV_k$
and a forest clasper otherwise.
Two claspers for $D$ are {\it disjoint} if the embeddings of all strands of claspers are disjoint in the interval of $D$.
For disjoint claspers $(h_1,e_1)$ and $(h_2,e_2)$ for $D$, $D^{\{(h_1,e_1), (h_2,e_2) \}}$ means $(D^{(h_1,e_1)})^{(h_2,e_2)}$ or equivalently $(D^{(h_2,e_2)})^{(h_1,e_1)}$.
\end{definition}
\begin{definition}
An {\it $L_n$-equivalence} is an equivalence relation on Gauss diagrams generated by the $L_n$-moves and Reidemeister moves.
By $D \overset{L_n}{\sim} D'$ we mean that $D$ and $D'$ are $L_n$-equivalent.
\end{definition}
\begin{proposition}\label{
The $L_n$-equivalence is an equivalence relation on Gauss diagrams.
\end{proposition}
\begin{proof}
First of all, we show the reflexive relation. For any $k \geq 1$ and $n \geq 1$, the identity element $1 \in \Gamma_n(PV_k)$ and $D^{(1,e)} = D$ for any embedding $e$. Therefore $D \overset{L_n}{\sim} D$.
Secondly, we show the symmetric relation. Let $D'=D^{(h,e)}$ where $h \in \Gamma_n(PV_k)$. Then there is an embedding $e'$ of $h^{-1} \in \Gamma_n(PV_k)$
such that $D'^{(h^{-1},e')} = D^{(h \cdot {h^{-1}},e)}$. Since the Gauss diagram $D^{(h \cdot h^{-1},e)}$ is $D$ up to a sequence of second Reidemeister moves, we have that $D' \overset{L_n}{\sim} D$.
Finally, the case of transitive relation is obvious.
\end{proof}
\begin{proposition}\label{
If $1 \leq n \leq n'$, then an $L_{n'}$-move is achieved by an $L_n$-move. Therefore $L_{n'}$-equivalence implies $L_{n}$-equivalence.
\end{proposition}
\begin{proof}
By the property of the lower central series, $\Gamma_n(PV_k) < \Gamma_{n-1}(PV_k)$.
\end{proof}
\begin{proposition}\label{anotherdef}
Two Gauss diagrams $D$ and $D'$ are $L_{n}$-equivalent if and only if there exists a clasper $(h,e)$ of degree $n$ such that $D^{(h,e)}$ equals to $D'$ up to a sequence of Reidemeister moves.
\end{proposition}
\begin{proof}
A necessary condition is obvious.
To prove a sufficient condition, we will show the following three statements (1), (2) and (3).
(1) If $D_2$ is obtained from $D_1$ by a first (second or third, respectively) Reidemeister move and then an $L_{n}$-move $(h_1,e_1)$ ($(h_2,e_2)$ or $(h_3,e_3)$, respectively) ($n' \geq n$), then there is an $L_{n'}$-move $(h'_1,e'_1)$ ($(h'_2,e'_2)$ or $(h'_3,e'_3)$, respectively) and a sequence of Reidemeister moves such that $D_2$ is obtained from $D_1$ by the $L_{n'}$-move $(h'_1,e'_1)$ ($(h'_2,e'_2)$ or $(h'_3,e'_3)$, respectively) and then the sequence of Reidemeister moves.
(2) If $D_2$ is obtained from $D_1$ by an $L_{n}$-move $(h,e)$ and then another $L_n$-move $(h',e')$, then there is an $L_{n'}$-move $(h'',e'')$ ($n' \geq n$) and two sequences of second Reidemeister moves such that $D_2$ is obtained from $D_1$ by one sequence of the second Reidemeister moves and then the $L_{n'}$-move $(h'',e'')$ and then the other sequence of the second Reidemeister moves.
(3) For any clasper $(h,e)$ of degree more than or equal to $n$ for $D$, there exists a clasper $(h',e')$ of degree $n$ for $D$ such that $D^{(h,e)}=D^{(h',e')}$, because any $h \in \Gamma_{n+1}$ is represented by the product of elements in $\Gamma_{n}$.
By (1), (2) and (3), if $D$ and $D'$ are $L_{n}$-equivalent, there is an $L_n$-move and a sequence of Reidemeister moves such that $D'$ is obtained from $D$ by the $L_n$-move and then the sequence of Reidemeister moves.
We show (1). We consider the case of the first Reidemeister move RI.
In Figure \ref{$L_n$ and RI}, these Gauss diagrams are identical except in a local place of RI represented by this figure.
By gray line we mean a clasper.
Given a clasper $(h_1,e_1)$, we can move the ends of chords of clasper out the arrow derived from RI by a sequence of second Reidemeister moves.
We denote the obtained clasper by $(h'_1,e'_1)$ (See Figure \ref{$L_n$ and RI}).
Moreover similar considerations apply to the other first Reidemeister move.
\begin{figure}[h]
\begin{center}
\includegraphics[height=3.3cm, scale=3]{changeReid1andclasper.eps}
\put(-248,85){$RI$} \put(-128,88){$(h_1,e_1)$} \put(-295,55){$(h'_1,e'_1)$} \put(-20,68){$(h_1,e_1)$} \put(-40,55){$RII's$} \put(-190,5){$RI$} \put(-30,-5){$(h'_1,e'_1)$}
\caption{Change of an $L_n$-move and a first Reidemeister move}
\label{$L_n$ and RI}
\end{center}
\end{figure}
Similar way to RI, in the case of RII and RIII, we give claspers $(h'_2,e'_2)$ and $(h'_3,e'_3)$ as in Figure \ref{$L_n$ and RII} and \ref{$L_n$ and RIII}, which are one of RII and RIII.
Here, in Figure \ref{$L_n$ and RI} for simplicity we draw only one strand is embedding in each interval between endpoints of arrows derived from RIII.
\begin{figure}[h]
\begin{center}
\includegraphics[height=4.3cm, scale=4]{changeReid2andclasper.eps}
\put(-260,105){$RII$} \put(-133,105){$(h_2,e_2)$} \put(-300,70){$(h'_2,e'_2)$} \put(-20,88){$(h_2,e_2)$} \put(-50,68){$RII's$} \put(-190,5){$RII$} \put(-30,-5){$(h'_2,e'_2)$}
\caption{Change of an $L_n$-move and a second Reidemeister move}
\label{$L_n$ and RII}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[height=4.8cm, scale=4]{changeReid3andclasper.eps}
\put(-260,105){$RIII$} \put(-131,105){$(h_3,e_3)$} \put(-300,70){$(h'_3,e'_3)$} \put(-20,88){$(h_3,e_3)$} \put(-48,65){$RII's$} \put(-258,5){$RIII$} \put(-128,5){$RII's$}
\caption{Change of an $L_n$-move and a third Reidemeister move}
\label{$L_n$ and RIII}
\end{center}
\end{figure}
We show (2). Given a Gauss diagram $D_1$ and a clasper $(h,e)$ for $D_1$, we can transform $D_1$ to $D_1^{(h \cdot h^{-1},e)}$ by a sequence of the second Reidemeister moves.
Let $h \in \Gamma_n(PV_k)$ and $h' \in \Gamma_n(PV_{k'})$.
There is an embedding $\bar{e}$ of $h \otimes h'$ in $\Gamma_n(PV_{k+k'})$ such that $(D^{(h \cdot h^{-1},e)})^{(h \otimes h',\bar{e})}=(D_1^{(h \cdot h^{-1} \cdot h,e)})^{(h',\bar{e})}$.
Moreover, $(D_1^{(h \cdot h^{-1} \cdot h,e)})^{(h',\bar{e}|_{h'})}$ is $(D_1^{(h,e)})^{(h',\bar{e}|_{h'})}=D_2$ up to a sequence of the second Reidemeister moves.
We set $(h'',e'')=(h \otimes h',\bar{e})$.
\end{proof}
\begin{remark}
It is obvious that Proposition \ref{anotherdef} is equivalent to the following statement.
There exists the union $H$ of disjoint claspers of degree $n$ such that $D^{H}$ equals to $D'$ up to a sequence of the Reidemeister moves.
\end{remark}
\begin{remark}
In \cite{MY}, Meilhan and Yasuhara also extended the concept of the clasper to welded knots, which is a quotient of virtual knot.
\end{remark}
\begin{lemma}\label{clasperproperty}
Let $n \geq 1$. Let $D$ be a Gauss diagram and $(h,e)$ a clasper of degree $n$ for $D$.
Then for any Gauss diagram $D'$ which is equivalent to $D$ there is a clasper $(h',e')$ of degree $n$ for $D'$ such that $D^{(h,e)}$ is equivalent to $D'^{(h',e')}$.
\end{lemma}
\begin{proof}
Since $D' \sim D$ and $D \overset{L_n}{\sim} D^{(h,e)}$, we have that $D' \overset{L_n}{\sim} D^{(h,e)}$.
It is from Proposition \ref{anotherdef} that there is a clasper $(h',e')$ of degree $n$ for $D'$ such that $D^{(h,e)} {\sim} D'^{(h',e')}$.
\end{proof}
\begin{remark}
We can show Lemma \ref{clasperproperty} directly.
If $D'$ be obtained from $D$ by Reidemeister move RI, RII or RIII, then given a clasper $(h,e)$ for $D$ we can construct a clasper $(h',e')$ such that $D^{(h,e)} \sim D'^{(h',e')}$ by similar method of Figure \ref{$L_n$ and RI}, \ref{$L_n$ and RII} and \ref{$L_n$ and RIII} in the proof of Proposition \ref{anotherdef}.
\end{remark}
\begin{definition}
A Gauss diagram $D$ is {\it $L_n$-trivial} if $D$ is $L_{n}$-equivalent to the trivial Gauss diagram $D_0$.
\end{definition}
The next proposition is well-known fact of group theory.
\begin{proposition}\label{2.25
Let $G$ be a group. Let $x$ and $y$ be elements in the $n$-th and $n'$-th lower central subgroup of $G$, respectively.
Then the commutator $[x, y]$ of $x$ and $y$ is in $(n+n')$-th lower central subgroup of $G$.
\end{proposition}
\begin{lemma}\label{slidedef}
Let $D$ be a Gauss diagram.
Let $n_1$, $n_2 \geq 1$. Let $(h_1,e_1)$ be a clasper of degree $n_1$ and $(h_2,e_2)$ a clasper of degree $n_2$ for $D$,
where they are disjoint.
Let $s_i$ be the $i$-th strand of $h_1$ and $t_j$ the $j$-th one of $h_2$.
Suppose that there is no end point of arrows and no embedding of another strands of claspers on the interval between embeddings $e_1(s_i)$ and $e_2(t_j)$.
Then, these embeddings may replace each other up to $L_{n_1+n_2}$-equivalence as in Figure \ref{slide}.
Let $(h_1,e'_1)$ and $(h_2,e'_2)$ be claspers of degree $n_1$ and $n_2$ obtained from $(h_1,e_1)$ and $(h_2,e_2)$ by replacing $e_1(s_i)$ and $e_2(t_j)$ as in Figure \ref{slide}.
Then, there exists a clasper $(h,e)$ of degree $n_1+n_2$ such that $(D^{\{(h_1,e'_1), (h_2,e'_2)\}})^{(h,e)}$ is equivalent to $D^{\{(h_1,e_1),(h_2,e_2)\}}$.
\begin{figure}[h]
\begin{center}
\includegraphics[height=2.cm, scale=1.5]{slide.eps}
\put(-285,-10){$D$} \put(-140,35){$(h_1,e'_1)$} \put(-285,35){$(h_1,e_1)$} \put(-0,35){$(h_2,e'_2)$} \put(-190,35){$(h_2,e_2)$} \put(-120,-10){$D$}
\put(-250,-20){$t_j$} \put(-205,-20){$s_i$} \put(-75,-20){$s_i$} \put(-45,-20){$t_j$}
\caption{A sliding}
\label{slide}
\end{center}
\end{figure}
\end{lemma}
We call the transformation between two claspers a {\it sliding}.
\begin{proof}
For $h_1 \in \Gamma_{n_1}(PV_{k_1})$ and $h_2 \in \Gamma_{n_2}(PV_{k_2})$, we construct $h \in \Gamma_{n_1+n_2}(PV_{k_1+k_2-1})$ and its embedding $e$.
Let $\tilde{h}_1$ be an element of pure virtual braid $\Gamma_{n_1}(PV_{k_1+k_2-1})$ from $h_1$ by adding $j-1$ strands before 1st strand of $h_1$ and $k_2-j$ strands after $k_1$-th strand of $h_1$.
Let $\tilde{h}_2$ be an element of $\Gamma_{n_2}(PV_{k_1+k_2-1})$ from $h_2$ by adding $i-1$ strands between $(j-1)$-th and $j$-th strand of $h_2$ and,
$k_1-i$ strands between $j$-th and $(j+1)$-th strand of $h_2$.
Then the $i$-th strand of $h_1$ and $j$-th strand of $h_2$ are the same order in $\tilde{h}_1$ and $\tilde{h}_2$.
If both of the orientations of $s_i$ and $t_j$ are compatible or not with the orientation of the interval of $D$, the product $\tilde{h}_1 \cdot \tilde{h}_2$ has a natural embedding $e$ induced by $e_1$ and $e_2$.
Let $h=[\tilde{h}_2, \tilde{h}_1]$.
Then $h \in \Gamma_{n_1+n_2}(PV_{k_1+k_2-1})$ by Proposition \ref{2.25}.
Since $h \cdot \tilde{h}_1 \cdot \tilde{h_2} = \tilde{h_2} \cdot \tilde{h_1}$, we may replace $s_i$ and $t_j$ each other and leave other embeddings up to $L_{n_1+n_2}$-equivalence.
If only one of the orientations of $s_i$ and $t_j$ is compatible with that of the interval of D, to adjust $s_i$ and $t_j$ the orientation we set $h=[\bar{h}_2, \tilde{h}_1]$, where $\bar{h}_2$ is the mirror image of $\tilde{h}_2$ for a horizontal line and its embedding $e$ is induced by $e_1$ and $e_2$.
\end{proof}
\begin{proposition}\label{5.8
Let $n, n' \geq 1$.
Let $D$ be an $L_n$-trivial Gauss diagram and $D'$ be an $L_{n'}$-trivial one.
Then the Gauss diagram $D \cdot D'$ is $L_{n+n'}$-equivalent to $D' \cdot D$.
\end{proposition}
\begin{proof}
By assumption, there are two claspers $(h,e)$ and $(h',e')$ with $h \in \Gamma_n(PV)$ and $h' \in \Gamma_{n'}(PV)$ such that $D_0^{(h,e)} \sim D$ and $D_0^{(h',e')} \sim D'$.
Then by Lemma \ref{slidedef} we have $D \cdot D' \sim D_0^{(h,e)} \cdot D_0^{(h',e')} \overset{L_{n+n'}}{\sim} D_0^{(h',e')} \cdot D_0^{(h,e)} \sim D' \cdot D$.
\end{proof}
\begin{proposition}\label{1.39
For any $L_n$-trivial Gauss diagram $D$, there is an $L_n$-trivial Gauss diagram $D'$ such that both $D \cdot D'$ and $D' \cdot D$ are $L_{2n}$-trivial.
\end{proposition}
\begin{proof}
By assumption, there is a clasper $(h,e)$ with $h \in \Gamma_n(PV_k)$ such that $D_0^{(h,e)} \sim D$.
We define $D'=D_0^{(h^{-1},e)}$.
Then by Lemma \ref{slidedef} we have $D \cdot D' \sim D_0^{(h,e)} \cdot D_0^{(h^{-1},e)} \overset{L_{2n}}{\sim} D_0^{(h \cdot h^{-1},e)} \sim D_0$.
\end{proof}
\begin{notation}
The set $\mathcal{LVK}$ of equivalence classes of Gauss diagrams has a monoid structure under the composition.
For $n \geq 1$, let $\mathcal{LVK}_n$ denote the submonoid of $\mathcal{LVK}$ consisting of the equivalence classes of Gauss diagrams which are $L_n$-trivial.
There is a descending filtration of monoids
\[ \mathcal{LVK} = \mathcal{LVK}_1 \supset \mathcal{LVK}_2 \supset \mathcal{LVK}_3 \supset \cdots. \]
For $l \geq n$, $\mathcal{LVK}_n / {L_l}$ denotes the quotient of $\mathcal{LVK}_n$ by $L_l$-equivalence.
It is easy to see that the monoid structure on $\mathcal{LVK}_n$ induces that of $\mathcal{LVK}_n / {L_l}$.
There is a filtration on $\mathcal{LVK}_n / {L_l}$ of finite length
\[ \mathcal{LVK}/ {L_l} = \mathcal{LVK}_1/ {L_l} \supset \mathcal{LVK}_2/ {L_l} \supset \mathcal{LVK}_3/ {L_l} \supset \cdots \supset \mathcal{LVK}_l/ {L_l}=\{1\}. \]
\end{notation}
\begin{lemma}\label{abelian}
For $n \geq 1$, the monoid $\mathcal{LVK}_n/ {L_{l}}$ $(1 \leq l \leq 2n)$ is an abelian group.
\end{lemma}
\begin{proof}
By Proposition \ref{1.39}, for any $K \in \mathcal{LVK}_n$ there exists $K' \in \mathcal{LVK}_n$ such that both $K \cdot K'$ and $K \cdot K'$ are trivial up to $L_{2n}$-equivalence,
and the monoid $\mathcal{LVK}_n/ {L_{2n}}$ is a group.
By Proposition \ref{5.8}, for any $K_1, K_2 \in \mathcal{LVK}_n$, $K_1 \cdot K_2 $ is $K_2 \cdot K_1$ up to $L_{2n}$-equivalence, and the group $\mathcal{LVK}_n/ {L_{2n}}$ is abelian.
\end{proof}
\begin{proposition}\label{5.4
Let $1 \leq n \leq l$. We then have as follows. \\
(1) The monoid $\mathcal{LVK}_n/ {L_{l}}$ is a group. \\
(2) $[\mathcal{LVK}_n/ {L_{l}}, \mathcal{LVK}_{n'}/ {L_{l}}] \subset \mathcal{LVK}_{n+n'}/ {L_{l}}$ for $n, n' \geq 1$ with $n+n' \leq l$. \\
(3) The group $\mathcal{LVK}_n/ {L_{l}}$ is nilpotent. \\
\end{proposition}
\begin{proof}
(1) We fix $l$ and prove it by induction on $n$.
If $n=l$, it is obvious.
Assume that $\mathcal{LVK}_{n+1}/ {L_{l}}$ is a group for some $n$ with $1 \leq n \leq l$. We then have a shot exact sequence of monoids:
\[1 \rightarrow \mathcal{LVK}_{n+1}/ {L_{l}} \rightarrow \mathcal{LVK}_{n}/ {L_{l}} \rightarrow \mathcal{LVK}_{n}/ {L_{n+1}} \rightarrow 1 \]
Here, $\mathcal{LVK}_{n+1}/ {L_{l}}$ and $\mathcal{LVK}_{n}/ {L_{n+1}} $ are groups by the assumption of induction and Lemma \ref{abelian}.
Therefore $\mathcal{LVK}_{n}/ {L_{l}}$ is also a group. \\
(2)
It is from Proposition \ref{5.8},
$a \in \mathcal{LVK}_{n}/ {L_{l}}$ and $b \in \mathcal{LVK}_{n'}/ {L_{l}}$ are commute up to $L_{n+n'}$-equivalent. Here $[a,b] \overset{L_{n+n'}}{\sim} 1$.
Therefore $[a,b] \in \mathcal{LVK}_{n+n'} / {L_{l}}$. \\
(3) From (2), it is easy to check.
\end{proof}
\section{$L_n$-equivalence and $n$-equivalence}
In this section, we prove that $L_n$-equivalence coincides with $n$-equivalence defined by Goussarov-Polyak-Viro \cite{GPV}.
\begin{definition}\cite{GPV}\label{trivial}
Let $n \geq 0$. A Gauss diagram $D$ on $k$ strands is said to be {\it $n$-trivial} if the Gauss diagram satisfies the following condition.
There exist $n+1$ non-empty disjoint subsets $A_1, A_2, \cdots , A_{n+1}$ of the set of arrows of $D$ such that
for any non-empty subfamily of the set $\{A_1, A_2, \cdots , A_{n+1}\}$ the Gauss diagram obtained from $D$ by removing all arrows which belongs to the subfamily is trivial up to a sequence of second Reidemeister moves.
Two Gauss diagrams $D$ and $D'$ are related by {\it $n$-variation} if $D'$ is obtained from $D$ by attaching an $(n-1)$-trivial Gauss diagram on several strands to segments of $D$ without endpoints of any arrow.
Two Gauss diagrams are said to be {\it $n$-equivalent} if they are related by $(n+1)$-variations and Reidemeister moves.
\end{definition}
\begin{theorem}\label{L_nandn-equ
For any $n \geq 1$, $L_{n}$-equivalence and ${(n-1)}$-equivalence on long virtual knots are equal.
\end{theorem}
\begin{proof}
It is obvious that if two Gauss diagrams are $L_{n}$-equivalent, then they are ${(n-1)}$-equivalent.
Therefore it suffices to prove that if Gauss diagrams $D$ and $D'$ are related by an $n$-variation then they are $L_{n}$-equivalent.
Let $D_t$ be an $(n-1)$-trivial Gauss diagram such that $D'$ is obtained from $D$ by attaching $D_t$.
Let $A_1, A_2, \cdots , A_{n}$ be disjoint sets of arrows of $D_t$ satisfying the condition in Definition \ref{trivial}.
By the property of $(n-1)$-triviality, $D$ coinsids with the Gauss diagram obtained from $D'$ by removing all arrows in $A_1\cup\cdots\cup A_n$ up to second Reidemeister move.
Therefore, by the method of (1) in proof of Proposition \ref{anotherdef} it is sufficient to consider the case that all arrows of $D_t$ belong to $A_1\cup\cdots\cup A_n$.
Let $H$ be the set of tree claspers of degree 1 corresponding to the arrows in $A_1\cup\cdots\cup A_n$.
We define a weight for a clasper $h$, which is a subset of $\mathbb{N}$, and denote it by $w(h)$.
We consider $H$ as a set of tree claspers each clasper of which assigns $i$ as weight if the clasper corresponds with an arrow of $A_i$.
Let $I$ be a finite subset of $\mathbb{N}$. Then $H(I)$ denote the subset of $H$ each clasper of which has a subset of $I$ as weight,
and $H_I$ denote the subset of $H(I)$ each clasper of which has $I$ as weight.
Let $N=\{1,2, \cdots ,n\}$. We then can regard $D'$ as $D^{H(N)}$.
Moreover, by the property of $(n-1)$-triviality if $I$ is a proper subset of $N$, then $D^{H(I)}=D$ up to a sequence of second Reidemeister moves.
We show the following claim, which proves the theorem.
\begin{claim}
Suppose that $D$ and $H$ are as above.
Then there exists a set $H'$ of tree claspers of degree $\geq n$ with weight $N$ such that $D^{H'}$ is equivalent to $D^{H}$.
\end{claim}
Let us first prove the case that $D$ is equivalent to the trivial Gauss diagram $D_0$.
Then by Lemma \ref{clasperproperty}, it is sufficient to show that the case that $D$ is trivial.
To prove this claim, we prove the following statement depending on a positive integer $s$.
(A)
There exists a set $G = \{ (h_1,e_1),\dots, (h_k,e_k) \}$ of tree claspers $(h_1,e_1),\dots, (h_k,e_k)$ ($k\geq0$) for $D_0$ such that for each $i=1,\dots,k$ $s \leq |w((h_i,e_i))| \leq$ deg($(h_i,e_i)$) where $|\cdot|$ means the number of a set, and $D_0^ {G(I)} \sim D_0^{H(I)}$ for every subset $I$ of $N$.
We prove it by induction on $s$ for $s=1, 2, \cdots, n$.
For $s=1$, we can set $G=H$.
Under the assumption of the claim, assuming the statement (A) to hold for $s(<n)$, we will prove it for $s+1$.
Let $G$ be a set of tree claspers for $D_0$ satisfying (A) for $s$.
We take a subset $I$ of $N$ such that $|I|=s$ and $G_I$ is not empty.
Then we shift all tree clasper in $G_I$ to the ahead with fixing claspers in $G \setminus G_I$ by sliding of claspers (Lemma \ref{slidedef})
until all end points of all clasper in $G_I$ are completely to the ahead of those in $G \setminus G_I$.
We denote the obtained set of tree claspers for $D_0$ by $G'$.
Here we define the weight of new tree claspers obtained by sliding in Lemma \ref{slidedef} as follows.
If two claspers have the weight $w_1$ and $w_2$, then the new tree clasper $(h,e)$ has the weight $w_1 \cup w_2$.
We remark that $D_0^{G(I)} \sim D_0^{G'(I)}$ for every $I \subset N$ and $s < |w((h,e))| \leq$ deg($(h,e)$) for any new tree $(h,e) \in G'$.
Let $G''$ be a set $G' \setminus G'_{I}$ of tree claspers for the Gauss diagram $D_0^{G'_I}$.
We consider a subset $J$ of $N$.
If $J \supset I$, then it is clear that $D_0^{G'(J)} = (D_0^{G_{I}'})^{G''(J)}$.
If not, then the new tree claspers do not contain in $G_{J}''$.
Hence $(D_0^{G'_I})^{G''(J)}=D_0^{G'(J)} \cdot D_0^{G'_I}$.
By the assumption of claim, $D_0^{G_{I}'}$ is equivalent to $D_0$.
Therefore the Gauss diagrams $D_0^{G'(J)}$ and $(D_0^{G'_I})^{G''(J)}$ are equivalent for every $J \subset N$.
Repeating this procedure for $I$ such that $|I|=s$, we obtain a set of tree claspers $(h,e)$'s with $d((h,e)) \geq |w((h,e))| \geq s+1$ for a Gauss diagram which is equivalent to $D_0$.
By Lemma \ref{clasperproperty}, we obtain a set of tree claspers preserving above condition for $D_0$, which is the required set satisfying (A) for $s+1$.
This proves the claim for the case that $D \sim D_0$.
Next we prove the case that $D$ is not equivalent to the trivial one.
Since the set of $L_n$-equivalence classes has a group structure, there is an inverse ${D}^{-1}$ of $D$ up to $L_n$-equivalence.
Then $D$ is $L_n$-equivalent to $D \cdot {D}^{-1} \cdot D$.
It is $L_n$-equivalent to $D' \cdot {D}^{-1} \cdot D$, since $D \cdot {D}^{-1}$ and $D' \cdot {D}^{-1}$ are $L_n$-equivalent.
Because $D'\cdot {D}^{-1}$ is $(D\cdot {D}^{-1})^{H}$. By Proposition \ref{anotherdef}, there exists a clasper $h$ of degree $\geq n$ such that $(D\cdot {D}^{-1})^{h} \sim D_0$. Therefore it follows from the case $D \sim D_0$ that there exists a set $H'$ of claspers of degree $\geq n$ such that $((D \cdot {D}^{-1})^{h})^{H} \sim ((D \cdot {D}^{-1})^{h})^{H'}$.
Hence $D$ is $L_n$-equivalent to $D'$.
\end{proof}
\begin{remark}
Even though we change``second Reidemeister moves" into``Reidemeister moves" in the definition of the $n$-trivial in Definition \ref{trivial},
we can show Theorem \ref{L_nandn-equ} similarly.
Therefore it is concluded that these two $n$-equivalences coincide.
\end{remark}
\section{$L_n$-equivalence and $V_n$-equivalence}
Goussarov-Polyak-Viro \cite{GPV} mentioned that the value of a finite type invariant of degree less than or equal to $n$ depends only on the $n$-equivalence classes.
Therefore it follows from Theorem \ref{L_nandn-equ} that $L_{n+1}$-equivalence implies $V_n$-equivalence, indirectly.
In this section, we give this relation directly, by redefining the two-sided ideal $J_n$ of the monoid ring $\mathbb{Z}\mathcal{LVK}$ by using claspers.
\begin{definition
Let $l \geq 1$.
A {\it scheme} of size $l$, $H=\{ (h_1,e_1), (h_2,e_2), \dots ,(h_l,e_l) \}$, for a Gauss diagram $D$ is the set of disjoint claspers for $D$.
Denote an element $[D,H]$ of $\mathbb{Z}\mathcal{LVK}$ by
$$ [D, H] = \sum_{G \subset H} (-1)^{l - \#G} [D^{G}],$$
where $G$ runs over all $2^l$ subsets of $H$.
The {\it degree} of a scheme $H=\{ (h_1,e_1), (h_2,e_2), \dots ,(h_l,e_l) \}$ is the sum of the degree of its elements, denoted by deg($H$).
\end{definition}
\begin{lemma}\label{clasperproperty2}
Let $D$ be a Gauss diagram and $H$ a scheme of size $l$ for $D$ of degree $n$.
Then for any Gauss diagram $D'$ which is equivalent to $D$ there is a scheme $H'$ of size $l$ for $D'$ of degree $n$ such that $[D, H]$ is equal to $[D', H']$ in $\mathbb{Z}\mathcal{LVK}$.
\end{lemma}
\begin{proof}
If $D'$ be obtained from $D$ by Reidemeister move RI, RII or RIII, we can construct a scheme $H'$ such that $D^H \sim D'^{H'}$ by similar method of Figure \ref{$L_n$ and RI}, \ref{$L_n$ and RII} and \ref{$L_n$ and RIII} in the proof of Proposition \ref{anotherdef}.
From the construction of $H'$, for each $G \subset H$, there is the corresponding $G' \subset H'$ such that $D^G \sim D'^{G'}$.
Therefore $[D, H] = [D', H']$ in $\mathbb{Z}\mathcal{LVK}$.
\end{proof}
\begin{lemma}\label{schemeproperty}
(1) $[D,\emptyset] = [D].$ \\
(2) $ [D, \{(h_1,e_1), (h_2,e_2), \dots ,(h_l,e_l) \}] \\
\hspace{.5cm}=[D^{(h_1,e_1)}, \{ (h_2,e_2), \dots ,(h_l,e_l)\}] - [D, \{(h_2,e_2), \dots ,(h_l,e_l)\}]$. \\
(3) $[D, \{ (h_{1,1} \otimes \dots \otimes h_{1,m},e_1), (h_2,e_2), \dots ,(h_l,e_l)\}] \\
\hspace{.5cm}=\sum_{i=1}^{m}[D^{(h_{1,1} \otimes \dots \otimes h_{1,i-1},e|_{h_{1,1} \otimes \dots \otimes h_{1,i-1}})}, \{(h_{1,i},e|_{h_{1,i}}), (h_2,e_2), \dots ,(h_l,e_l) \}]$.
\end{lemma}
\begin{proof}
It is easy to check.
\end{proof}
\begin{definition}
Let $n$, $l$ be integers with $1 \leq l \leq n$. Let $J_{n,l}$ denote the two-sided ideal of $\mathbb{Z}\mathcal{LVK}$ generated by the elements $[D, H]$ under the composition,
where $D$ is any Gauss diagram and $H$ is any scheme of size $l$ for $D$ of degree $n$.
\end{definition}
\begin{remark}
By Lemma \ref{schemeproperty}(3), $J_{n,l}$ can be generated by schemes $[D, H]$ where $H$ is the set of tree claspers for $D$.
\end{remark}
\begin{remark}\label{ringiso
The natural homomorphism $i : \mathbb{Z}\mathcal{LVK} \rightarrow \mathbb{Z}(\mathcal{LVK}/ {L_n})$ induces the ring isomorphism $\mathbb{Z}\mathcal{LVK}/ J_{n,1} \cong \mathbb{Z}(\mathcal{LVK}/{L_n})$.
\end{remark}
\begin{lemma}\label{Thm6.7
Let $D$ be a Gauss diagram. We then have the following properties. \\
(1) For any integer $n \geq 1$ $J_{n, n} =J_n$ \\
(2) For any integers $n$, $l$, $l'$ with $1 \leq l \leq l' \leq n$, $J_{n, l} \subset J_{n,l'}$ \\
(3) For any integers $n$, $n'$ $l$ with $1 \leq l \leq n \leq n'$, $J_{n', l} \subset J_{n,l}$
\end{lemma}
\begin{proof}
(1) We show that
$$\varphi([\includegraphics[height=.4cm, scale=.8]{somesemicrossing.eps}])=[\includegraphics[height=.4cm, scale=.8]{some1claspers.eps}]$$
where the left-hand side of the equation means the image of $n$ dashed arrows by $\varphi$ and the right-hand side of the equation means a Gauss diagram with a scheme of size $n$ of degree $n$, which consists of $n$ clasper of degree 1.
If $n=1$, $\varphi([ \includegraphics[height=.4cm, scale=.8]{1semicrossing.eps} ])= [\includegraphics[height=.4cm, scale=.8]{1crossing.eps}] - [\includegraphics[height=.07cm, scale=.2]{0crossing.eps}] = [\includegraphics[height=.4cm, scale=.8]{1clasper.eps}]$.
Assume the formula holds less than or equal to $n$, it is easy to check that the formula holds $n+1$.
\\ (2) It suffices to show that $J_{n, l} \subset J_{n,l+1}$ for $n$, $l$ with $1 \leq l \leq n-1$.
Let $[D,H] \in J_{n, l}$. By assumption, there is a clasper of degree $d$ in $H$, say to $(h_1,e_1)$, where $d \geq 2$.
Then $h_1$ can be represented by a pure virtual braid $[h_{1,1}, h_{1,2}] \cdot [h_{2,1}, h_{2,2}] \cdot \cdots \cdot [h_{m,1}, h_{m,2}]$ where deg$(h_{i,j})=d_j$ for any $i, j$ and $d_1 + d_2 = d$.
Let $h_1^j=h_{1,j} \otimes h_{1,j}^{-1} \otimes h_{2,j} \otimes h_{2,j}^{-1} \otimes \cdots \otimes h_{m,j} \otimes h_{m,j}^{-1}$ for $j=1,2$.
Then deg$(h_1^j)=d_j$ and $D_0^{(h^j_1,e_1^j)} \sim D_0$, where $e^j_1$ is induced by $e_1$. Therefore we have
\begin{align*}
[D_0, \{(h_1,e_1)\}]
&= [D_0^{(h_1,e_1)}] - [D_0] \\
&= [D_0^{(h_1,e_1)}] - [D_0^{(h_1^1,e_1^1)}] - [D_0^{(h_1^2,e_1^2)}] + [D_0] \\
&= [D_0, \{(h_1^1,e_1^1), (h_1^2,e_1^2)\}].
\end{align*}
Hence
\begin{align*}
[D,H] &=[D, \{(h_1,e_1), \cdots , (h_l, e_l)\}] \\
&= [D, \{(h_1^1,e_1^1), (h_1^2,e_1^2), (h_2,e_2), \cdots , (h_l,e_l) \}] \\
& \in J_{n,l+1}.
\end{align*}
\\ (3) It suffices to show that $J_{n+1, l} \subset J_{n, l}$ for $n$, $l$ with $1 \leq l \leq n$.
Let $[D,H] \in J_{n+1, l}$. By assumption, there is a clasper of degree $d$ in $H$, say to $(h_1,e_1)$, where $d \geq 2$.
Since $h_1 \in \Gamma_d(PV_k) \lhd \Gamma_{d-1}(PV_k)$ for some $k$, we have $[D,H] \in J_{n,l}$.
\end{proof}
By Lemma \ref{Thm6.7}, we can redefine $J_n$ as the ideal of $\mathbb{Z}\mathcal{LVK}$ generated by elements $[D, H]$ where $D$ is any Gauss diagram and $H$ is any scheme for $D$ of degree $n$.
\begin{proposition}\label{L_{n+1}V_n
For any $n \geq 0$, if $K$ and $K'$ are $L_{n+1}$-equivalent, then $K$ and $K'$ are $V_{n}$-equivalent.
\end{proposition}
\begin{proof}
By Remark \ref{ringiso} and Lemma \ref{Thm6.7}, if $K$ and $K'$ are $L_{n+1}$-equivalent, then $K-K' \in J_{n+1,1} \subset J_{n+1,n+1} = J_{n+1}$.
It is equivalent to that $K$ and $K'$ are $V_{n}$-equivalent.
\end{proof}
| {
"timestamp": "2017-12-19T02:08:43",
"yymm": "1705",
"arxiv_id": "1705.10490",
"language": "en",
"url": "https://arxiv.org/abs/1705.10490",
"abstract": "In the context of finite type invariants, Stanford introduced a family of equivalence relations on knots defined by the lower central series of the pure braid groups and characterized the finite type invariants in terms of the structure of the braid groups. It is known that this equivalence and Ohyama's equivalence defined by a local move are equivalent. On the other hand, in the virtual knot theory, the concept of Ohyama's equivalence was extended by Goussarov-Polyak-Viro, which called an $n$-equivalence. In this paper we extend Stanford's equivalence to virtual knots and virtual string links by using the lower central series of the pure virtual braid group, and call it an $L_n$-equivalence. We then prove that the $L_n$-equivalence is equal to the $n$-equivalence on virtual string links. Moreover we directly prove that two virtual string links are not distinguished by any finite type invariants of degree $n-1$ if they are $L_n$-equivalent, for any positive integer $n$.",
"subjects": "Geometric Topology (math.GT)",
"title": "Goussarov-Polyak-Viro's $n$-equivalence and the pure virtual braid group",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808730448281,
"lm_q2_score": 0.7217431943271999,
"lm_q1q2_score": 0.7075110486492305
} |
https://arxiv.org/abs/1906.03385 | Applications of Gaussian Binomials to Coding Theory for Deletion Error Correction | We present new applications on $q$-binomials, also known as Gaussian binomial coefficients. Our main theorems determine cardinalities of certain error-correcting codes based on Varshamov-Tenengolts codes and prove a curious phenomenon relating to deletion sphere for specific cases. | \section{Introduction}
$q$-binomials \cite{stanley2013algebraic},
also known as Gaussian
binomial coefficients
\cite{cohn2004projective}
are $q$-analogs of the binomial coefficients.
They are well-known and well-studied, with important
combinatorial implications and have properties
analogous to binomial coefficients
\cite{dhand2014combinatorial,kim2008q-bernoulli,pak2013strict,stanley2016some}.
However, to the best of the authors' knowledge,
they have not been considered from
the perspective of coding theory for deletion errors.
Terminology is defined precisely in subsequent sections,
but here we give an informal description of the descent
moment distribution. First, a descent vector
(also studied in \cite{levenshtein1992perfect})
is a binary $01$-vector
that indicates the indices of
descent in an associated vector.
The moment of a $01$-vector
(also studied in
\cite{levenshtein1966binary,levenshtein1967asymptotically,levenshtein1992perfect,tenengolts1965correction}) is a
summation of the product of the index
by the value of the $01$-vector.
A descent moment is simply the amalgam of these
two concepts, and the ``descent moment distribution"
of a set of vectors is the polynomial whose
coefficients indicate the number of vectors
having a particular descent moment in the given set.
The main contributions of this paper
relate to a class of deletion codes.
This provides implications for calculating the
cardinality of sets that are of interest in the theory of error-correcting codes.
The descent moment distribution in the formula of the main theorem
above is taken over certain sets of interest.
These sets are related to
a well-known class of sets studied by R. P. Stanley
known as VT (Varshamov-Tenengolts) codes \cite{stanley1972study}
(also known as special cases of Levenshtein codes
\cite{tenengolts1965correction}).
In his study, Stanley obtained an exact formula for the cardinality of
VT codes by considering a certain moment distribution in
conjunction with the Hamming weight.
The formula was non-trivial, and involved the sum of
M\"obius functions and Euler functions.
His formula was for the original VT codes,
but other related sets, in particular permutation and
multi-permutation codes based on VT codes,
do not have such formulas. Moreover, only the moment
distribution was considered, not the descent moment
distribution nor any relation to $q$-binomials.
Partial results were presented at the IEEE International
Symposium on Information Theory (ISIT) 2018 \cite{DBLP:conf/isit/HagiwaraK18}.
\section{Preliminaries and Remarks}\label{Contribution}
\subsection{Descent Moment Distribution}
Let $\mathbf{x}:= x_1 x_2 \dots x_l$ be an element of $\{A, B\}^l $,
where $\{A, B\}$ is a binary ordered set with $A < B$.
Instead of $(x_1, x_2, \dots, x_l)$, the notation $x_1 x_2 \dots x_l$ is used in this paper.
A 01-vector $\mathbf{y} := y_1 y_2 \dots y_{l-1} \in \{ 0,1 \}^{l-1}$ is called
a \textit{descent vector} of $\mathbf{x}$ if
\[
y_i =
\begin{cases}
1 & (x_i > x_{i+1}), \\
0 & (\text{otherwise}).
\end{cases}
\]
We denote
the descent vector of $\mathbf{x}$ by $\delta(\mathbf{x})$.
Sets considered in this paper are often
defined via conditions with descent vectors.
For a 01-vector $\mathbf{y}$,
the \textit{moment} of $\mathbf{y} := y_1 y_2 \dots y_{l-1}$ is defined as
$y_1 + 2 y_2 + \dots + (l-1) y_{l-1}$.
The moment is denoted by $\rho ( \mathbf{y})$.
Note that the moment does not belong to a binary field
but rather it is defined as an integer, whereas $\mathbf{y}$ is a 01 vector.
For a set $C$ of binary sequences,
we introduce the following polynomial $\mathrm{DM}(C)$ of $q$ as
our primary interest:
\[
\mathrm{DM}(C) :=
\sum_{ \mathbf{x} \in C} q^{ \rho \circ \delta ( \mathbf{x} ) },
\]
where $\rho \circ \delta( \mathbf{x}) := \rho( \delta (\mathbf{x}) )$.
\begin{remark}\label{rem:major}
It is easy to see that
$$\rho \circ \delta( \mathbf{x}) = \sum_{1 \le i < n, x_{i} > x_{i+1} } i.$$
The right hand side is well-known
as the \textbf{major index} of $\mathbf{x}$.
In this paper, we call $\mathrm{DM}()$ the \textbf{descent moment distribution}
for connecting coding theory,
while it is the statistic of major index.
\end{remark}
The Hamming weight distribution is an object similar to $\mathrm{DM}$
\cite{macwilliams1963theorem}.
If $A = 0$ and $B = 1$
the distribution is
defined as
$
\sum_{ \mathbf{x} \in C} Y^{\mathrm{wt}( \mathbf{x} )},
$
where $\mathrm{wt}( \mathbf{x} )$ is the Hamming weight of $\mathbf{x}$,
i.e., the number of non-zero entries of $\mathbf{x}$.
Both distributions may be applied to obtain the cardinality $\# C$ by substituting $1$ for their variable:
\[
\# C =
\sum_{ \mathbf{x} \in C} 1
=
\sum_{ \mathbf{x} \in C} 1^{ \rho \circ \delta ( \mathbf{x} ) }
=
\sum_{ \mathbf{x} \in C} 1^{\mathrm{wt}( \mathbf{x} )}.
\]
Another related distribution with both the moment and Hamming weight is:
$$
\sum_{ \mathbf{x} \in C} q^{\rho( \mathbf{x} )} Y^{ \mathrm{wt}( \mathbf{x} ) },
$$
which is used to obtain the Hamming weight distribution for VT (Varshamov-Tenengolts)
codes (see \ref{subsec:Levenshtein} in \cite{stanley1972study}).
Notice that the descent moment distribution for the union of
disjoint sets is equal to the sum of their descent moment distributions.
\begin{lemma}\label{DM_SUM}
For $C_1, C_2 \subset \{ A, B \}^l$,
if $C_1 \cap C_2 = \emptyset$,
$$
\mathrm{DM}( C_1 \cup C_2 ) = \mathrm{DM}(C_1) + \mathrm{DM}( C_2 ).
$$
\end{lemma}
\begin{proof}
\begin{align*}
\mathrm{DM}( C_1 \cup C_2) &=
\sum_{ \mathbf{x} \in C_1 \cup C_2 } q^{ \rho \circ \delta ( \mathbf{x} ) }\\
&=
\sum_{ \mathbf{x} \in C_1 } q^{ \rho \circ \delta ( \mathbf{x} ) }
+
\sum_{ \mathbf{x} \in C_2 } q^{ \rho \circ \delta ( \mathbf{x} ) }\\
&=
\mathrm{DM}(C_1) + \mathrm{DM}(C_2)
\end{align*}
\end{proof}
\subsection{$q$-integer, $q$-factorial and $q$-binomial}
The notion of a $q$-analogue is a general notion in pure mathematics
for generalizing or extending a mathematical object.
For a mathematical object $f$,
another mathematical object $F(q)$ is called a $q$-analogue
of $f$ if $f = F(1)$ or $f = \lim_{q \rightarrow 1} F(q)$.
For a positive integer $i$,
the \textit{$q$-integer} $[i]$ is defined as
\[
[i] := \frac{q^i - 1}{q - 1}
= 1 + q + \dots + q^{i-1},
\]
and the \textit{$q$-factorial} $[i]!$ is defined as
\begin{align*}
[i]! &:= [i] [i-1] \dots [2] [1],\\
[0]! &:= 1.
\end{align*}
Using $q$-factorials, for non-negative integers $i$ and $j$,
we define the \textit{$q$-binomial} as
\[
\begin{bmatrix} i \\ j \end{bmatrix}
:=
\begin{cases}
\dfrac{[i]!}{[j]![i-j]!} & (i \ge j \ge 0),\\
0 & (\textrm{otherwise}).\\
\end{cases}
\]
$q$-binomial is also called a \textbf{Gaussian binomial coefficient}.
It is easy to see that the $q$-integer $[i]$, the $q$-factorial $[i]!$, and the $q$-binomial$ \begin{bmatrix} i \\ j \end{bmatrix}$
are $q$-analogues of the integer $i$, the factorial $i!$, and the binomial $\binom{i}{j}$ respectively.
$q$-binomials are known to correlate to certain weight distributions of lattice paths.
Let us consider the set $L$ of lattice paths from $(0,0)$ to $(j, i-j)$.
As is well-known, its cardinality $\# L$ is given by $\binom{i}{j}$.
By defining the weight $S(p)$ of a path $p$ as the number of squares
which are on the north-western side of the path,
the following is also well-known \cite{stanley2013algebraic} (see Example \ref{example:path22} below):
\begin{align}
\sum_{p \in L} q^{S(p)} =
\begin{bmatrix} i \\ j \end{bmatrix}.\label{known fact}
\end{align}
\begin{example}[Paths from $(0,0)$ to $(2,2)$]\label{example:path22}
There are 6 paths from $(0,0)$ to $(2, 2)$ (see Figure \ref{figure:smallPath}).
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=11cm,bb=0 0 348 203]{smallPath.jpg}
\caption{lattice paths from $(0,0)$ to $(2,2)$ }
\label{figure:smallPath}
\end{center}
\end{figure}
Their weights are $0$, $1$, $2$, $2$, $3$, and $4$.
Hence the weight distribution is
\[
1 + q + 2 q^2 + q^3 + q^4.
\]
On the other hand, the $q$-binomial with $i=4$ and $j=2$ is
\begin{align*}
\begin{bmatrix} 4 \\ 2 \end{bmatrix}
&=\frac{[4][3][2][1]}{[2][1][2][1]}\\
&=\frac{[4] [3]}{[2] [1]}\\
&=\frac{(1+q+q^2+q^3)(1+q+q^2)}{(1+q)(1)}\\
&=1+q+2 q^2 + q^3 + q^4.
\end{align*}
\end{example}
The following is used in the proof of Corollary \ref{theorem:Card}.
\begin{lemma}\label{app:2}
Let $\zeta$ be the $d$th primitive root of $1$
and $\zeta^{\alpha + \beta} = 1$.
\[
\lim_{q \rightarrow \zeta} \begin{bmatrix} \alpha + \beta \\ \beta \end{bmatrix}
=
\begin{cases}
\displaystyle \binom{ \frac{\alpha + \beta}{d} }{\frac{\beta}{d} } & d | \langle \alpha, \beta \rangle,\\
0 &\text{otherwise,}
\end{cases}
\]
where $\langle \alpha, \beta \rangle$ is the greatest common divisor of $\alpha$ and $\beta$.
\end{lemma}
\begin{proof}
The assumption $\zeta^{\alpha + \beta} = 1$ implies that $d | \alpha + \beta$,
and in particular $1 \le d \le \alpha + \beta$.
The number of zero factors of $[\alpha + \beta]!$ for substituting $\zeta$ to $q$
is $\lfloor (\alpha + \beta) / d \rfloor$ and the number of zero factors of $[\alpha]! [\beta]!$ is
$\lfloor \alpha /d \rfloor + \lfloor \beta / d \rfloor$.
If $d$ does not divide $\langle \alpha, \beta \rangle$,
it implies $\lfloor (\alpha + \beta) / d \rfloor > \lfloor \alpha /d \rfloor + \lfloor \beta / d \rfloor$.
Hence $\lim_{q \rightarrow \zeta} \begin{bmatrix} \alpha + \beta \\ \beta \end{bmatrix} = 0$.
If $d$ divides $\langle \alpha, \beta \rangle$,
it implies $\lfloor (\alpha + \beta) / d \rfloor = \lfloor \alpha /d \rfloor + \lfloor \beta / d \rfloor$.
Note that
\begin{align*}
\lim_{q \rightarrow \zeta} \begin{bmatrix} cd \\ bd\end{bmatrix} &= \frac{c}{b},
&
\text{and }
\lim_{q \rightarrow \zeta} \begin{bmatrix} a + cd \\ a + bd\end{bmatrix}
&= 1,
\end{align*}
for $0 < a < d$.
Hence
\begin{align*}
\lim_{q \rightarrow \zeta} \begin{bmatrix} \alpha + \beta \\ \beta \end{bmatrix}
&=
\lim_{q \rightarrow \zeta}
\frac{[\alpha + \beta] [\alpha + \beta - d] \dots [d] }{([\alpha] [\alpha - d] \dots [d] )( [\beta] [\beta - d] \dots [d] ) }\\
&=
\frac{(\alpha + \beta) (\alpha + \beta - d) \dots (d) }{( (\alpha) (\alpha - d) \dots (d) ) ( (\beta) (\beta - d) \dots (d) )}\\
&=
\frac{(\frac{\alpha + \beta}{d}) (\frac{\alpha + \beta}{d} - 1) \dots (1) }{( (\frac{\alpha}{d}) (\frac{\alpha}{d}-1) \dots (1) ) ( (\frac{\beta}{d}) (\frac{\beta}{d}-1) \dots (1) ) }\\
&=
\binom{ \frac{\alpha + \beta}{d} }{ \frac{\beta}{d} }.
\end{align*}
\end{proof}
\subsection{Major Index and $q$-binomial}
For positive integers $\alpha$ and $\beta$, let $C_{\alpha, \beta}$ be the set of vectors
with $\alpha$ entries of $A$ and $\beta$ entries of $B$.
Hence $C_{\alpha, \beta}$ consists of $\binom{\alpha + \beta}{\alpha}$ elements
that are obtained by all permutations to $AA \dots A BB \dots B$.
The following is well-known for major index.
\begin{fact}[See \cite{stanley1997enumerative}]\label{theorem:DM}
For any positive integers $\alpha$ and $\beta$,
\begin{align*}
\mathrm{DM}( C_{\alpha, \beta} )
(
=
\sum_{0 \le m < \alpha + \beta}
\mathrm{DM}( C_{\alpha, \beta, m} )
)
=
\begin{bmatrix} \alpha + \beta \\ \beta \end{bmatrix}
\end{align*}
\end{fact}
We partition $C_{\alpha, \beta}$ into $\alpha + \beta$ subsets as follows:
\[
C_{\alpha, \beta, m}:= \{ \mathbf{x} \in C_{\alpha, \beta} \mid \rho \circ \delta ( \mathbf{x} ) \equiv m \pmod{\alpha + \beta}\}.
\]
Since $\{ C_{\alpha, \beta, m} \}_{0 \le m < \alpha + \beta}$ partitions $C_{\alpha, \beta}$,
by Lemma \ref{DM_SUM},
we remark the following:
\begin{remark}
\[
\mathrm{DM}( C_{\alpha, \beta} ) = \sum_{0 \le m < \alpha + \beta} \mathrm{DM}( C_{\alpha, \beta, m} ).
\]
\end{remark}\label{DM_remark}
From Fact \ref{theorem:DM} and the definition of $C_{\alpha, \beta, m}$,
we have the following:
\begin{cor}
\begin{align}
\sum_{0 \le m < \alpha + \beta}
\# C_{\alpha, \beta, m} q^m
\equiv
\begin{bmatrix} \alpha + \beta \\ \beta \end{bmatrix}
\pmod{ q^{\alpha + \beta} - 1 }
.
\label{eq:main2}
\end{align}
\end{cor}
\begin{example}[$\alpha = \beta = 2$]
\label{example:alpha2M2qbinom}
As seen in Example \ref{example:path22},
\[
\begin{bmatrix} 2 + 2 \\ 2 \end{bmatrix}
=
1 + q + 2 q^2 + q^3 + q^4.\]
On the other hand,
$C_{2,2,0} = \{AABB, BABA \}$,
$C_{2,2,1} = \{BAAB \}$,
$C_{2,2,2} = \{ABAB, BBAA \}$,
and
$C_{2,2,3} = \{ABBA \}$.
Hence we verify
\[
\sum_{0 \le m < 4} \# C_{2,2, m} q^m
=
2 + q + 2 q^2 + q^3
\equiv
\begin{bmatrix} 2 + 2 \\ 2 \end{bmatrix}
\pmod{q^4 - 1}.
\]
\end{example}
\subsection{Coding Theoretic Remarks: Deletions and Partitions via VT Codes}
\label{subsec:Levenshtein}
Deletion is a combinatorial operation for a sequence.
Single deletions shorten a given sequence.
For example, a sequence $AAAA$ of length $4$ changes to the sequence $AAA$ of length $3$
after a single deletion.
Note that a single deletion that occurs in a string of
consecutive repeated entries results in the same sequence
regardless of where the deletion occurs.
Indeed, the deletions in either the 1st entry or the 2nd entry from the sequence $AABAAA$
result in the same sequence $ABAAA$.
Hence a sequence $AABAAA$ of length $6$ may be changed by a single deletion
to one of three possible sequences of length $5$:
$ABAAA$, $AAAAA$, or $AABAA$.
For a set $C$ of vectors,
we define the set $\mathrm{dS}(C)$ as
the set of sequences
obtained by a single deletion in $C$,
and call it the \textit{deletion sphere} of $C$.
For example, for $C := \{ AABAABB \}$,
$$
\mathrm{dS}(C) = \{ABAABB, AAAABB, AABABB, AABAAB \}.
$$
A maximal consecutive subsequence of repetitions of the same entry
is called a \textit{run}.
For a vector $\mathbf{x}$, the number of runs is denoted by $|| \mathbf{x} ||$
and is called the run number in this paper.
For example, $|| AAAA || = 1$ and $||AABAAA|| = 3$.
The run number $|| \mathbf{x} ||$ is equal to the number of sequences
that are obtained by single deletions to $\mathbf{x}$:
\begin{fact}[\cite{levenshtein1967asymptotically}]
\label{fact:sphere_run}
For any vector $\mathbf{x}$,
$$
\# \mathrm{dS}( \{ \mathbf{x} \}) = || \mathbf{x} ||.
$$
\end{fact}
Hence, the cardinality of $\mathrm{dS}( C )$ for a singleton
depends on its element.
A set $C$ is called a \textit{single deletion correcting code} if
$$
\# \mathrm{dS} (C) = \sum_{\mathbf{x} \in C} \# \mathrm{dS} ( \{ \mathbf{x} \}).
$$
This definition is equivalent to
$$
\mathrm{dS}(C) = \sqcup_{ \mathbf{x} \in C} \mathrm{dS} (\{ \mathbf{x} \}).
$$
Levenshtein showed that the following sets $\mathrm{VT}_{l-1, m}$ are single deletion correcting codes for any positive integer $l$ and
any integer $m$
\cite{levenshtein1966binary}:
$$
\mathrm{VT}_{l-1, m} := \{ \mathbf{x} \in \{0,1 \}^{l-1} \mid \rho( \mathbf{x} ) \equiv m \pmod{l} \}.
$$
This code $\mathrm{VT}_{l-1, m}$ is called a \textit{VT code}.
The set $C_{\alpha, \beta, m}$ is written by using VT codes:
$$
C_{\alpha, \beta, m} =
\{ \mathbf{x} \in \{A,B \}^{\alpha + \beta} \mid \delta( \mathbf{x} ) \in \mathrm{VT}_{\alpha + \beta-1, m} \}.
$$
The following statement strengthens our motivation to investigate
$C_{\alpha, \beta, m}$.
The proof is a direct corollary of Lemma 3.2 in \cite{levenshtein1992perfect}.
\begin{theorem}\label{theorem:single_deletion_error_correction}
\label{section:single_deletion_error_correction}
The set $C_{\alpha, \beta, m}$ is a single deletion correcting code.
\end{theorem}
\section{Main Contributions}
\label{Corollaries}
Our main contributions of this paper are the properties of $C_{\alpha, \beta, m}$.
Theorem \ref{theorem:Card}
and Corollary \ref{cor:maxC0} are enumerative combinatorial results
and
Theorem \ref{theorem:binarySphere}
is a coding theoretic result.
\subsection{Cardinality of $C_{\alpha, \beta, m}$}
\begin{theorem}\label{theorem:Card}
For any $\alpha, \beta$,
\[
\# C_{\alpha, \beta, m}
=
\frac{1}{\alpha + \beta}
\sum_{ d | \langle \alpha, \beta \rangle } \binom{ \frac{\alpha + \beta}{d} }{ \frac{\beta}{d} }
\mu( \frac{ d }{ \langle d, m \rangle } ) \frac{ \phi(d) }{ \phi( d / \langle d, m \rangle )},
\]
where $\mu$ is the m\"obius function, $\phi$ is the Euler function,
and $\langle d, m \rangle$ is the greatest common divisor of $d$ and $m$.
In particular,
\[
\# C_{\alpha, \beta, 0} =
\frac{1}{\alpha + \beta} \sum_{ d | \langle \alpha, \beta \rangle } \binom{ \frac{\alpha + \beta}{d} }{ \frac{\beta}{d} }
\phi(d).
\]
\end{theorem}
\begin{proof}
Applying Eq. (\ref{eq:main2}), the second half of
Fact \ref{theorem:DM},
we analyze the $q$-binomial
$
Q(q):=
\begin{bmatrix}
\alpha + \beta\\
\beta
\end{bmatrix}\pmod{q^{\alpha + \beta} -1}.
$
Since the polynomial $Q(q)$
is of degree at most $\alpha + \beta$,
$Q(q)$ is determined by $\alpha + \beta$ different points of a complex field $\mathbb{C}$,
for example
the elements of the set $Z := \{1, \zeta, \zeta^2, \cdots, \zeta^{\alpha + \beta -1} \}$ of $\alpha + \beta$th roots of 1.
By Lemma \ref{app:2},
$Q(q)$ may be written as
\[
Q(q)=\displaystyle \sum_{d| \langle \alpha, \beta \rangle }
\binom{\frac{\alpha + \beta }{d}}{\frac{\beta}{d}}
h_d(q),
\]
where $h_t(q)$ is a polynomial such that:
1) the degree is at most $\alpha + \beta -1$,
2) $h_t(z)=1$ for a primitive $t$th root of $1$,
3) $h_t(z)=0$
for $z \in Z$ but not a primitive $t$th root. Indeed,
\[
h_t(q)=\dfrac{1}{\alpha + \beta} \cdot \displaystyle \sum_{z_0 \in Z_t}\dfrac{q^{\alpha + \beta}-1}{q-r_0}\cdot z_0^{-1},
\]
where $Z_t$ is the set of primitive $t$th roots of $1$.
Since
\[
\displaystyle\sum_{z_0\in Z_t}\dfrac{q^{\alpha + \beta}-1}{q-r_0}\cdot z_0^{-1}
=\displaystyle\sum_{z_0\in Z_t}\displaystyle\sum_{ 0 \leq m < \alpha + \beta}z^m_{0}q^m
\]
and
\[
\displaystyle\sum_{z_0\in Z_t} z_0^m
= \mu \left(\dfrac{t}{ \langle t,m \rangle }\right)\dfrac{\phi(t)}{\phi ( t / \langle t,m \rangle )},
\]
we have
\begin{align*}
&Q(q)=\\
&\displaystyle\sum_{0 \leq m < \alpha + \beta}
\left\{
\dfrac{1}{\alpha + \beta}\displaystyle\sum_{d| \langle \alpha, \beta \rangle }
\binom{\frac{\alpha + \beta}{d}}
{\frac{\beta}{d}}
\mu\left(\dfrac{d}{ \langle d,m \rangle }\right)\dfrac{\phi(d)}{\phi ( d / \langle d,m \rangle )}
\right\}q^m.
\end{align*}
Hence by observing the coefficient of $q^m$,
\[
\# C_{\alpha, \beta, m}=\dfrac{1}{\alpha + \beta}\displaystyle \sum_{d | \langle \alpha, \beta \rangle}
\binom{\frac{\alpha + \beta}{d}}
{\frac{\beta}{d}}
\mu\left(\dfrac{d}{ \langle d,m \rangle }\right)\dfrac{\phi(d)}{\phi\left(\frac{d}{\langle d,m \rangle }\right)}.
\]
Setting $m := 0$ implies
$
\mu\left(\dfrac{d}{ \langle d,m \rangle }\right) = 1
$
and
$
\phi\left(\frac{d}{\langle d,m \rangle }\right) = 1.
$
Thus we obtain the formula for $\# C_{\alpha, \beta, 0}$.
\end{proof}
\begin{cor}
\label{cor:maxC0}
For any $\alpha, \beta$ and $m$,
$$
\# C_{\alpha, \beta, m} \le \# C_{\alpha, \beta, 0}.
$$
\end{cor}
\begin{proof}
Note that
\[
\mu( d / \langle d,a \rangle ) = -1, 0, 1
\text{ and }
\mu( d / \langle d,0 \rangle ) = 1.
\]
This implies $\mu ( d / \langle d,m \rangle ) \leq 1 = \mu ( d / \langle d,0 \rangle )$.
Similarly
$\dfrac{\phi(d)}{\phi ( d / \langle d,m \rangle )} \leq
\phi(d) =
\dfrac{\phi(d)}{\phi ( d / \langle d, 0 \rangle) )}$ holds.
Therefore
\[
\# C_{ \alpha, \beta, m} \le \dfrac{1}{\alpha + \beta}\displaystyle \sum_{d | \langle \alpha, \beta \rangle}
\binom{\frac{\alpha + \beta}{d}}
{\frac{\beta}{d}}
\phi(d)
=\# C_{\alpha, \beta, 0}.
\]
\end{proof}
\subsection{Deletion Sphere in the Case $\alpha = \beta$}\label{subsec:deletionSphere}
In this subsection we briefly discuss a curious phenomenon
relating the cardinality of $C_{\alpha,\beta, m}$ and deletion spheres
when $\alpha = \beta$.
As is mentioned in Fact \ref{fact:sphere_run},
the cardinality of $\mathrm{dS}( C )$ for a singleton
depends on its element.
However, we have the following:
\begin{theorem}\label{theorem:dS_a2_beta}\label{theorem:binarySphere}\label{fact:permutationSphere}
Assume $\alpha = \beta$, and set $\gamma := \alpha (= \beta)$. Then
\[
\# \mathrm{dS} ( C_{\gamma, \gamma, m} )
= (\gamma + 1) \# C_{\gamma, \gamma, m}.\]
\end{theorem}
\begin{example}[Case $\alpha = \beta = 2$]
As we have seen in Example \ref{example:alpha2M2qbinom},
$C_{2,2,0} = \{ AABB, BABA \}$.
Hence $\mathrm{dS}( C_{2,2,0} ) = || AABB || + || BABA || = 2 + 4 = 3 \# C_{2,2,0}$.
Similarly, $\mathrm{dS}( C_{2,2,1} ) = || BAAB || = 3 = 3 \# C_{2,2,1}$.
\end{example}
\begin{definition}[$R_r(q)$]
For an integer $r$,
let us define
\begin{align*}
R_r (q) &:= \displaystyle \sum_{\substack{\mathbf{x}\in C_{\alpha, \beta} \\ || \mathbf{x} || =r}}q^{\rho \circ \delta(\mathbf{x}) }\\
&= \mathrm{DM}( \{ \mathbf{x} \in C_{\alpha, \beta} \mid || \mathbf{x} || = r \} ),
\end{align*}
where $|| \mathbf{x} ||$ is the run number of $\mathbf{x}$.
\end{definition}
\begin{lemma}\label{lemma:DM,R_r}
\begin{align}
\mathrm{DM}( C_{\alpha, \beta}) &= \sum_{2 \le r \le \alpha + \beta} R_r (q),
\label{eq:DM1}
\end{align}
and
\begin{align}
\sum_{0 \le m < \alpha + \beta} \# \mathrm{dS}( C_{\alpha, \beta, m} ) q^m &\equiv
\sum_{2 \le r \le \alpha + \beta} r R_r (q) \pmod{ q^{\alpha + \beta} - 1}.
\label{eq:DM2}
\end{align}
\end{lemma}
\begin{proof}
Since the run number of an element of $C_{\alpha, \beta}$ is greater than or equal to $2$ and
is at most $\alpha + \beta$,
Eq. (\ref{eq:DM1}) holds.
\begin{align*}
\sum_{0 \le m < \alpha + \beta} \# \mathrm{dS}( C_{\alpha, \beta, m} ) q^m
&= \sum_{0 \le m < \alpha + \beta} \big( \sum_{ \mathbf{x} \in C_{\alpha, \beta, m} } || \mathbf{x} || q^m \big) \\
&\equiv \sum_{0 \le m < \alpha + \beta} ( \sum_{ \mathbf{x} \in C_{\alpha, \beta, m} } || \mathbf{x} || q^{\rho \circ \delta( \mathbf{x} )} )\\
&= \sum_{ \mathbf{x} \in C_{\alpha, \beta}} || \mathbf{x} || q^{\rho \circ \delta( \mathbf{x} )} \\
&=\sum_{2 \le r < \alpha + \beta}( \sum_{ \mathbf{x} \in C_{\alpha, \beta }, || x || = r } r q^{\rho \circ \delta( \mathbf{x} )} )\\
&=\sum_{2 \le r < \alpha + \beta}r ( \sum_{ \mathbf{x} \in C_{\alpha, \beta }, || x || = r } q^{\rho \circ \delta( \mathbf{x} )} )\\
&=
\sum_{2 \le r \le \alpha + \beta} r R_r (q) \pmod{ q^{\alpha + \beta} - 1}.
\end{align*}
Hence Eq. (\ref{eq:DM2}) holds.
\end{proof}
These two relations above will be used for the proof of
Theorem \ref{theorem:binarySphere}.
As preparation, we show the following:
\begin{lemma}
\label{lemma:R_r}
\[
R_r(q) =
q^{\lfloor \frac{(r-1)^2}{4}\rfloor}
\begin{bmatrix}
\alpha-1\\
\lfloor \frac{r-2}{2}\rfloor
\end{bmatrix}
\begin{bmatrix}
\beta-1\\
\lfloor \frac{r-1}{2}\rfloor
\end{bmatrix}
+
q^{\beta}q^{\lfloor \frac{(r-1)^2}{4}\rfloor}
\begin{bmatrix}
\alpha -1 \\ \lfloor \frac{r-1}{2}\rfloor
\end{bmatrix}
\begin{bmatrix}
\beta -1 \\ \lfloor \frac{r-2}{2} \rfloor
\end{bmatrix}
,
\]
where $\lfloor \epsilon \rfloor$ denotes the maximal integer that does not exceed $\epsilon$.
\end{lemma}
\begin{proof}
For the sake of brevity, we only show the case when $r$ is even. The odd case is similarly proven.
Any element of $C_{\alpha,\beta}$ with $r$ runs, where $r$ is even, has one of the following two forms:
\[
\overbrace{AA \dots A}^{a_1}
\overbrace{BB \dots B}^{b_1}
\overbrace{AA \dots A}^{a_2}
\overbrace{BB \dots B}^{b_2}
\dots \dots
\overbrace{AA \dots A}^{a_{r/2}}
\overbrace{BB \dots B}^{b_{r/2}},
\eqno(*)
\]
\[
\overbrace{BB \dots B}^{b_1}
\overbrace{AA \dots A}^{a_1}
\overbrace{BB \dots B}^{b_2}
\overbrace{AA \dots A}^{a_2}
\dots \dots
\overbrace{BB \dots B}^{b_{r/2}},
\overbrace{AA \dots A}^{a_{r/2}},
\eqno(**)
\]
where $a_j$ ($b_j$) denotes the length of the $j$th run with entry A (B), and
$1 \le j \le r/2$.
For case (*), the descent moment is
\[
(a_1+b_1)+(a_1+a_2+b_1+b_2)+\cdots +(a_1+\cdots +a_{\frac{r}{2}-1}+b_1+\cdots+b_{\frac{r}{2}-1}),
\]
and for case ($\ast\ast$), the descent moment is
\[
(b_1)+(a_1+b_1+b_2)+\cdots +(a_1+\cdots +a_{\frac{r}{2}-1}+b_1+\cdots+b_{\frac{r}{2}}).
\]
Hence
\begin{align*}
R_r(q)
&=\displaystyle \sum_{\substack{1\leq a_1,a_2,\cdots,a_{\frac{r}{2}}\\a_1+\cdots a_{\frac{r}{2}}=\alpha}}
\displaystyle\sum_{\substack{1\leq b_1,b_2,\cdots ,b_{\frac{r}{2}}\\b_1+\cdots +b_{\frac{r}{2}}=\beta}}
(
q^{\mathcal{A}^{*}}
q^{\mathcal{B}^{*}}
+
q^{\mathcal{A}^{**}}
q^{\mathcal{B}^{**}}
) \\
&=
\displaystyle\sum
q^{\mathcal{A}^{*}}
\displaystyle\sum
q^{\mathcal{B}^{*}}
+
\displaystyle\sum
q^{\mathcal{A}^{**}}
\displaystyle\sum
q^{\mathcal{B}^{**}},
\end{align*}
where
$\mathcal{A}^{*} = a_1 + (a_1 + a_2) + \dots + (a_1 + a_2 + \dots + a_{r/2 - 1})$,
$\mathcal{A}^{**} = \mathcal{A}^{*}$, \\
$\mathcal{B}^{*} = b_1 + (b_1 + b_2) + \dots + (b_1 + b_2 + \dots + b_{r/2 - 1})$,
and
$\mathcal{B}^{**} = b_1 + (b_1 + b_2) + \dots + (b_1 + b_2 + \dots + b_{r/2})$.
For calculating $\sum q^{\mathcal{A}^{*}}$,
let us
define a bijection,
depicted by Figure \ref{figure:shiftedPath},
from the set of sequences $a_1, a_2, \dots, a_{r/2} \ge 1$
to the set of lattice paths $p$ from $(0,0)$ to $(\alpha - r/2, r/2 - 1)$.
Note that $\mathcal{A}^* = S(p) + (1 + 2 + \dots + (r/2 -1) ) = (r/2) (r/2 - 1)/ 2 + S(p)$.
\begin{figure}[htbp]
\includegraphics[width=11cm,bb=0 0 383 267]{shiftedPath.jpg}
\caption{bijection between the set of lattice paths from $(0,0)$ to $(\alpha-r/2,r/2 -1)$
and the set of sequences $a_1, a_2, \dots, a_{r/2} \ge 1$}
\label{figure:shiftedPath}
\end{figure}
Therefore by Eq. (\ref{known fact}),
\begin{align*}
\displaystyle \sum
q^{\mathcal{A}^* }
=
q^{\frac{(r/2)(r/2 - 1)}{2}}
\begin{bmatrix}
\alpha-1\\
r/2-1
\end{bmatrix}
=
q^{\frac{(r/2)(r/2 - 1)}{2}}
\begin{bmatrix}
\alpha-1\\
\lfloor \frac{r-2}{2}\rfloor
\end{bmatrix}.
\end{align*}
Similarly
\begin{align*}
\displaystyle \sum
q^{\mathcal{B}^* }
\; = \;
q^{(r/2)(r/2 - 1)/2}
\begin{bmatrix}
\beta-1\\
r/2-1
\end{bmatrix}
\;=\;
q^{\frac{(r/2)(r/2 - 1)}{2}}
\begin{bmatrix}
\beta-1\\
\lfloor \frac{r-1}{2}\rfloor
\end{bmatrix}.
\end{align*}
Note that in the previous two equations we have used
both $\lfloor (r-2)/2\rfloor$ and $\lfloor (r-1)/2\rfloor$ to
represent $r/2-1$, which is permissible since $r$ is even.
The choices were made so that the end result is consistent
with the case when $r$ is odd.
The previous two equations imply that
\begin{align*}
\displaystyle \sum
q^{\mathcal{A}^* }
\displaystyle \sum
q^{\mathcal{B}^* }
&\; = \;
q^{(r/2)(r/2 - 1)}
\begin{bmatrix}
\alpha-1\\
\lfloor \frac{r-2}{2}\rfloor
\end{bmatrix}
\begin{bmatrix}
\beta-1\\
\lfloor \frac{r-1}{2}\rfloor
\end{bmatrix} \\
&\; = \;
q^{ \lfloor \frac{(r-1)^2}{4} \rfloor }
\begin{bmatrix}
\alpha-1\\
\lfloor \frac{r-2}{2}\rfloor
\end{bmatrix}
\begin{bmatrix}
\beta-1\\
\lfloor \frac{r-1}{2}\rfloor
\end{bmatrix}.
\end{align*}
By a similar argument, we can show
\[
\displaystyle \sum
q^{\mathcal{A}^{**} }
\displaystyle \sum
q^{\mathcal{B}^{**} }\\
=
q^{\beta} q^{ \lfloor \frac{(r-1)^2}{4} \rfloor }
\begin{bmatrix}
\alpha-1\\
\lfloor \frac{r-1}{2}\rfloor
\end{bmatrix}
\begin{bmatrix}
\beta-1\\
\lfloor \frac{r-2}{2}\rfloor
\end{bmatrix}.
\]
Hence for even $r$,
\begin{align*}
R_r (q)
&=
q^{ \lfloor \frac{(r-1)^2}{4} \rfloor }
\begin{bmatrix}
\alpha-1\\
\lfloor \frac{r-2}{2}\rfloor
\end{bmatrix}
\begin{bmatrix}
\beta-1\\
\lfloor \frac{r-1}{2}\rfloor
\end{bmatrix}
+
q^{\beta} q^{ \lfloor \frac{(r-1)^2}{4} \rfloor }
\begin{bmatrix}
\alpha-1\\
\lfloor \frac{r-1}{2}\rfloor
\end{bmatrix}
\begin{bmatrix}
\beta-1\\
\lfloor \frac{r-2}{2}\rfloor
\end{bmatrix}.
\end{align*}
As mentioned at the beginning of the proof, the case
when $r$ is odd is similarly proven.
\end{proof}
The following is the key lemma to prove
Theorem \ref{theorem:dS_a2_beta}.
It states that
a sort of symmetry of $R_r (q)$ on $r$ holds
by
the assumption $\alpha = \beta$ and
considering $\pmod{q^{2\gamma} - 1}$.
\begin{lemma}\label{app:1}
For $2 \le r \le 2\gamma$,
$$
R_{r}(q)\equiv R_{2\gamma + 2 - r}(q) \pmod{q^{2 \gamma }-1}.
$$
\end{lemma}
\begin{proof}
By Lemma \ref{lemma:R_r},
\begin{align*}
R_{2 \gamma + 2 - r}(q)
&=(1+q^\gamma)
q^{\lfloor \frac{(2 \gamma + 2 - r-1)^2}{4}\rfloor}
\begin{bmatrix}
\gamma-1\\
\lfloor \frac{2 \gamma + 2 - r-1}{2}\rfloor
\end{bmatrix}
\begin{bmatrix}
\gamma-1\\
\lfloor \frac{2 \gamma + 2 - r-2}{2}\rfloor
\end{bmatrix}\\
&=
(1+q^\gamma)
q^{\gamma^2 - \gamma(r-1) + \lfloor \frac{(r-1)^2}{4}\rfloor}
\begin{bmatrix}
\gamma-1\\
\gamma-1 - \lfloor \frac{r-1}{2}\rfloor
\end{bmatrix}
\begin{bmatrix}
\gamma-1\\
\gamma-1
-
\lfloor \frac{r-2}{2}\rfloor
\end{bmatrix}\\
&=
(q^{\gamma (\gamma-r+1)} + q^{\gamma (\gamma-r+2)})
q^{\lfloor \frac{(r-1)^2}{4}\rfloor}
\begin{bmatrix}
\gamma-1\\
\lfloor \frac{r-1}{2}\rfloor
\end{bmatrix}
\begin{bmatrix}
\gamma-1\\
\lfloor \frac{r-2}{2}\rfloor
\end{bmatrix}.
\end{align*}
$q^{\gamma (\gamma-r+1)} + q^{\gamma (\gamma-r+2)} \equiv 1 + q^\gamma \pmod{q^{2\gamma} - 1}$ holds
from
$\{ \gamma-r+1, \gamma-r+2 \} \equiv \{ 0, 1 \} \pmod{2}$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{theorem:dS_a2_beta}]
Define $G_m (q)$ as
$$
G_m (q) :=\displaystyle \sum_{\mathbf{x}\in C_{\gamma, \gamma, m}} || \mathbf{x}|| q^{\rho \circ \delta(\mathbf{x})},
\text{ for }0 \le m < 2 \gamma.$$
Then
\begin{align}
\#{\rm dS}(C_{\gamma, \gamma, m})\cdot q^m
&=
\displaystyle \sum_{\mathbf{x}\in C_{\gamma, \gamma, m}} || \mathbf{x}|| q^{m} \nonumber \\
&\equiv
G_m(q) \pmod{q^{2 \gamma} - 1}. \label{eq:lastEq}
\end{align}
Hence the proof is done by showing
$$
G_m (q) \equiv (\gamma+1) \mathrm{DM}( C_{\gamma, \gamma, m}) \pmod{q^{2 \gamma}-1}.
$$
To this end, it is enough to show
\[
\sum_{0 \leq m < 2 \gamma} G_m (q)
\equiv (\gamma + 1){\rm DM}( C_{\gamma, \gamma}) \pmod{q^{2 \gamma}-1}.
\]
Finally we have
\begin{align*}
&\sum_{0 \leq m < 2 \gamma} G_m (q)\\
&\equiv \sum_{0 \leq m < 2 \gamma}
\#{\rm dS}(C_{\gamma, \gamma, m})\cdot q^m & (\text{by Eq. (\ref{eq:lastEq})})\\
&=
\displaystyle \sum_{2 \leq r \leq 2 \gamma} r R_r(q)
& (\text{by Eq. (\ref{eq:DM2})})\\
&=
\frac{1}{2}
\displaystyle \sum_{2 \leq r \leq 2 \gamma} r R_r(q)
+
\frac{1}{2}
\displaystyle \sum_{2 \leq r \leq 2 \gamma} r R_r(q)\\
&\equiv
\frac{1}{2} \displaystyle \sum_{2 \leq r \leq 2 \gamma} r R_r (q)
+ \frac{1}{2} \displaystyle \sum_{2 \leq r \leq 2 \gamma} r R_{2 \gamma + 2 - r} (q)
& (\text{by Lemma \ref{app:1}})\\
&=
\frac{1}{2} \displaystyle \sum_{2 \leq r \leq 2 \gamma} r R_r(q)
+ \frac{1}{2} \displaystyle \sum_{2 \leq r \leq 2 \gamma} (2 \gamma + 2-r) R_r(q) \\
&=
\frac{1}{2} \sum_{2 \leq r \leq 2 \gamma} (2\gamma +2) R_r(q)\\
&=
(\gamma+1) \sum_{2 \leq r \leq 2 \gamma} R_r(q)\\
&=
(\gamma+1) \mathrm{DM}( C_{\gamma, \gamma} ) \pmod{ q^{2 \gamma} - 1 }.
& (\text{by Eq. (\ref{eq:DM1})})
\end{align*}
\end{proof}
\section{Conclusion}
\label{Conclusion}
In this paper we proved a relationship between
descent moment distributions and $q$-binomials.
To accomplish this, we employed a lattice-path approach
to prove pertinent lemmas.
The relationship between descent moment distributions and
$q$-binomials was then applied to determine the
cardinality of $C_{\alpha, \beta, m}$.
We have seen how the descent moment distribution
has some interesting properties and may provide
insights into other problems.
Thus further investigation into descent moment distributions,
especially as it relates to combinatorics,
is a logical future research direction.
Below we state two open questions
regarding the subject.
The natural open question
is to extend the main results of this paper
to ternary (or more) and then arbitrary $q$-multinomials.
That is, the initial part of this open question is to prove a similar relationship
for the descent moment distributions of ternary subsets of $\{A, B, C\}^l$
with fixed multiplicities of $A$, $B$, and $C$.
\bibliographystyle{plain}
| {
"timestamp": "2019-06-13T02:06:37",
"yymm": "1906",
"arxiv_id": "1906.03385",
"language": "en",
"url": "https://arxiv.org/abs/1906.03385",
"abstract": "We present new applications on $q$-binomials, also known as Gaussian binomial coefficients. Our main theorems determine cardinalities of certain error-correcting codes based on Varshamov-Tenengolts codes and prove a curious phenomenon relating to deletion sphere for specific cases.",
"subjects": "Information Theory (cs.IT)",
"title": "Applications of Gaussian Binomials to Coding Theory for Deletion Error Correction",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808701643912,
"lm_q2_score": 0.7217431943271999,
"lm_q1q2_score": 0.7075110465702948
} |
https://arxiv.org/abs/1111.6832 | Gaussian Probabilities and Expectation Propagation | While Gaussian probability densities are omnipresent in applied mathematics, Gaussian cumulative probabilities are hard to calculate in any but the univariate case. We study the utility of Expectation Propagation (EP) as an approximate integration method for this problem. For rectangular integration regions, the approximation is highly accurate. We also extend the derivations to the more general case of polyhedral integration regions. However, we find that in this polyhedral case, EP's answer, though often accurate, can be almost arbitrarily wrong. We consider these unexpected results empirically and theoretically, both for the problem of Gaussian probabilities and for EP more generally. These results elucidate an interesting and non-obvious feature of EP not yet studied in detail. | \section{Introduction}
This paper studies approximations to definite integrals of Gaussian distributions $p_0({\mathbf x}) = \mathcal{N}({\mathbf x};{\mathbf m},K)$. Despite the battery of convenient analytic characteristics of Gaussian \emph{densities}, Gaussian \emph{probabilities} are difficult to calculate: the cumulative distribution function (cdf) has no closed-form expression and is numerically challenging in high-dimensional spaces.
Here we consider a generalisation of the cdf: the probability $F(\mathcal{A}) = \mathrm{Prob}\left\{{\mathbf x}\in \mathcal{A}\right\}$ that a draw from $p_0({\mathbf x})$ falls in a (possibly unbounded) \emph{polyhedral} region $\mathcal{A}$.
Applications of these probabilities are widespread, {\it e.g.} \cite{Gibbons1996, lesaffre1991, thiebautCMPB2004, liaoICML2007, gassmann2002}. Three machine learning problems that can be cast as Gaussian probabilities include the Bayes Point Machine \cite[]{herbrichBook}, Gaussian Process classification \cite[]{rasmussenBook, KussRasmussen2005}, and probit regression \cite[]{ochi1984}.
Univariate Gaussian probabilities can be calculated with machine-level precision, but no similarly powerful algorithm exists for the multivariate case. There are special cases where analytic decompositions \cite[]{Placket1954,Huguenin2009} or sampling methods \cite[]{LermanManski1981,Pakes1989}
are possible though very costly, but the most efficient general, known method is numerical integration (the recent book \cite{genzBook} gives a good overview). The Genz methods represent the state of the art, but achieving high accuracy invokes substantial computational cost. Also, applications such as Bayesian model selection require \emph{analytic} approximations of $F(\mathcal{A})$, usually because the goal is to optimise $F(\mathcal{A})$ with respect to the parameters $\{{\mathbf m},K\}$ of the Gaussian, which requires the corresponding derivatives. These derivatives and other features are not currently offered by numerical integration methods. Thus, there remains significant work to be done to address this important problem.
Here we develop an analytic approximation to $F(\mathcal{A})$ using Expectation Propagation (EP) \cite[]{minkaUAI01, minka01phd, minkaMSFTTR2005, opperTAP2000, opperGPMF2000} as an approximate integration method. In its most general form, this algorithm is applicable to polyhedral integration regions, which include the most commonly required hyperrectangular case (which, in turn, generalises the cdf). For hyperrectangular $\mathcal{A}$, the approximations are of high quality. For polyhedral $\mathcal{A}$, they are sometimes considerably worse. This shortcoming of EP will come as a surprise to many readers, because the differences to the rectangular case are inconspiciously straightforward. We study this interesting result, which has implications for some important applications where EP is often used.
In particular, the geometric interpretation of the Gaussian probability problem provides an insightful testbed for analysing properties of EP, which has importance for approximate inference well beyond Gaussian probabilities.
\section{EP for Gaussian Probabilities}
\label{sec:ep}
A prototypical aspect of Bayesian inference is that an intractable $p({\mathbf x})$ can be written as a product of a prior $p_0({\mathbf x})$ and likelihood factors $t_i({\mathbf x})$ such that $p({\mathbf x}) = p_0({\mathbf x}) \prod_i t_i({\mathbf x})$. Our Gaussian probability problem is the normalisation constant of such a distribution:
\begin{equation}
\label{eqn:cumdensityEP}
F(\mathcal{A}) = \int_\mathcal{A} p_0({\mathbf x})\d{\mathbf x} = \int p({\mathbf x})\d{\mathbf x} = \int p_0({\mathbf x}) \overset{m}{\prod_{i=1}} t_i({\mathbf x}) \d{\mathbf x}
\end{equation}
where $m$ can be any number of terms, greater than or less than or equal to the dimensionality $n$ of $p_0({\mathbf x})$. The factor $t_i({\mathbf x})$ is an indicator function
defined in a particular direction ${\mathbf c}_i$, namely a ``box function":
\begin{equation}
\label{eqn:truefactor}
t_i({\mathbf x}) = \mathbb{I} \bigl\{ l_i < {\mathbf c}_i^T{\mathbf x} < u_i \bigr\} =
\begin{cases}
1 & ~~ l_i < {\mathbf c}_i^T{\mathbf x} < u_i \\
0 & ~~ \mathrm{otherwise}.
\end{cases}
\end{equation}
Note that either $l_i$ or $u_i$ can be infinite or inactive (not supporting the polyhedron), and thus this polyhedral definition is general. Using box functions instead of step-functions for this definition may be non-standard, but this choice will be useful in our analyses, and it reduces to hyperrectangular $\mathcal{A}$ by setting $m=n$ and ${\mathbf c}_i$ to the cardinal axes.
EP approximates the intractable $p({\mathbf x})$ by iteratively replacing true factors $t_i({\mathbf x})$ with approximate factors $\tilde{t}_i({\mathbf x})$ from an exponential family, yielding the tractable approximate unnormalised distribution $q({\mathbf x}) = p_0({\mathbf x})\prod_i \tilde{t}_i({\mathbf x})$.
The EP objective function is involved, but EP is \emph{motivated} by the idea of minimising the Kullback-Leibler divergence $D_{KL}(p \| q)$. For $q({\mathbf x})$ in the exponential family, this is equivalent to global moment matching (including the zeroth moment) between $q({\mathbf x})$ and $p({\mathbf x})$.
The details of EP are by now standard in our community, so we give only a short description in the interest of brevity.
At any point in the
iteration, EP tracks a current approximate $q({\mathbf x})$ in an
exponential family, and a set of \emph{approximate factors} (or \emph{messages}) $\tilde{t}_i$, also
in the family. The factors are updated by constructing a
\emph{cavity} distribution
\begin{equation}
\label{eq:5}
q^{\setminus i}({\mathbf x}) = \frac{q({\mathbf x})}{\tilde{t}_i({\mathbf x})},
\end{equation}
and then
\emph{projecting} into the exponential family
\begin{equation}
\label{eq:6}
\tilde{t}_i({\mathbf x})q^{\setminus i}({\mathbf x}) = \operatorname{proj} [t_i({\mathbf x})q^{\setminus i}({\mathbf x})],
\end{equation}
where this projection operation (an M-projection from information geometry \cite[]{kollerBook}) is defined as setting $\tilde{t}_i({\mathbf x})$ to the unnormalised member of the exponential family minimizing $D_\text{KL}(t_i q^{\setminus i}\| \tilde{t}_i q^{\setminus i})$ \cite[]{PowerEP}. This projection entails matching the sufficient statistics of $\tilde{t}_i({\mathbf x})q^{\setminus i}({\mathbf x})$ to those of $t_i({\mathbf x}) q^{\setminus i}({\mathbf x})$, which for Gaussian $\tilde{t}_i({\mathbf x})$ is equivalent to matching \emph{zeroth}, first and second moments.
Our use of EP is nonstandard in two ways. First, our factors $t_i({\mathbf x})$ are unnormalised and thus so is $p({\mathbf x})$ and $q({\mathbf x})$ - here we use EP not to construct parametric marginal distributions, but to calculate log partition functions. Second, we note that the moment matching projection in Equation \ref{eq:6} is tractable for \emph{any} rank-one factor $t_i({\mathbf x})$, not simply the usual axis-aligned likelihood terms where ${\mathbf c}_i$ are the cardinal axes. Simple rank-one updates make this algorithm computationally efficient. Critically, this generality allows us to consider probabilities $F(\mathcal{A})$ both in axis-aligned hyperrectangular cases and arbitrary polyhedra of any orientation and any number $m$ of constraints/faces, more or less than the dimensionality $n$ of $p_0({\mathbf x})$.
\begin{figure}
\centering
\hspace{0.0cm}
\epsfig{figure=white_aistats.pdf,width=3.2in}
\caption{\small{The effect of whitening the Gaussian probability based on the mean and covariance of the Gaussian.}}
\label{fig:white}
\end{figure}
It will be useful to consider whitening the space of the problem via a substitution ${\mathbf y} = L^{-1}({\mathbf x} - {\mathbf m})$ where $L$ is a Cholesky factor of the covariance $K$. In this transformed space, we have a standard $\mathcal{N}(0,I)$ Gaussian, and we have a transformed set of polyhedral faces $L^T{\mathbf c}_i$ as in Figure \ref{fig:white}. To understand sources of error, we can then attend to properties of this transformed region. Intuition suggests that the more (less) hyperrectangular this transformed region is, the better (worse) the performance is, as hyperrectangular cases (in a white space) decompose into a product of solvable univariate problems. We then see that hyperrectangular cases are in fact not a fundamental distinction other than having the same number of constraints $m$ as dimensions $n$. Considering the whitened space and transformed integration region is conceptually useful and theoretically sound, as EP with Gaussians is invariant to this linear transformation \cite[]{seeger08epexpfam}.
The resulting algorithm, EP for multivariate Gaussian probabilities (EPMGP), yields a nice geometric interpretation. We want to integrate a Gaussian over a polyhedron defined by several box functions, but this operation is intractable. Instead, EP allows us to replace each of those intractable truncations functions with soft Gaussian truncations $\tilde{t}_i({\mathbf x})$ for which multiplication is simple.
There are a number of attractive features of the EP approach which may motivate its use in many applied settings, including fast runtime, low computational overhead, and
a natural ability to calculate derivatives with respect to the distribution's parameters, tail probabilities, and first and second moments of $p({\mathbf x})$.
\section{Experiments and Results}
\label{sec:exp}
To evaluate EP for this application, we need to establish a baseline for calculating these probabilities, and we need to choose test cases.
Our baseline will be the Genz method, currently the gold standard for these calculations, which makes a series of transformations of the region $\mathcal{A}$ and the Gaussian $p_0({\mathbf x})$ to enable accurate numerical integration. The integrand of Equation \ref{eqn:cumdensityEP} is transformed to the unit cube, and heuristic choices on integration order are made before quasi-random integration or lattice-point rules
are used \cite[]{genzBook}. For our comparisons to EPMGP, we used the Genz method with $5 \times 10^5$ lattice points.
We note that Monte Carlo methods are not particularly well suited to this problem: rejection samplers from $p_0({\mathbf x})$, importance samplers over the polyhedron $\mathcal{A}$ (``pinball" or ``hit-and-run" MCMC algorithms \cite[]{herbrichBook}), and elliptical slice sampling \cite[]{murrayESS} do not perform favorably (the latter is best among samplers, but still well beneath EP and Genz). In light of the accuracy of the Genz algorithms, we will not consider samplers further here.
We next describe choosing our problem cases. To define a Gaussian $p_0({\mathbf x})$, we draw eigenvalues from an exponential distribution and eigenvectors uniformly from the unit hypersphere to form the covariance $K$. This procedure produces a more interesting range of $K$ - in particular a better spread of condition numbers - than using a Wishart or similar. We set the mean ${\mathbf m} = 0$ without loss of generality.
In the hyperrectangular integration cases (where dimension $n$ equals number of constraints $m$), $\mathcal{A}$ can be defined by the upper and lower bounds ${\mathbf u}$ and ${\mathbf l}$, which we defined by taking a draw from $p_0({\mathbf x})$ and adding and subtracting uniform random lengths that scale with dimensionality $n$. Having the region size scale with the dimensionality $n$ helps to prevent the probabilities $F(\mathcal{A})$ from becoming vanishingly small with increasing dimension (which is handled fine by EP but problematic for the Genz method). In the arbitrary polyhedral case, the same procedure is repeated for drawing ${\mathbf u}$ and ${\mathbf l}$ for all $m$ constraints. We also choose axes of orientation, which are unit-norm vectors ${\mathbf c}_i \in \mathsf{I\!R}^n$ for each of the $m$ factors, again uniformly from the unit hypersphere.
It is also useful to consider special cases where we can analytically find the true probability $F(\mathcal{A})$. Orthants generalise quadrants to high dimension, for example $\{ {\mathbf x} \in \mathsf{I\!R}^n : x_i > 0~ \forall i \}$. For zero-mean Gaussians in $\mathsf{I\!R}^2$, calculating the probability $F(\mathcal{A})$ when $\mathcal{A}$ is an orthant is a simple geometry problem, and there are also geometric solutions in $\mathsf{I\!R}^3, \mathsf{I\!R}^4,$ and $\mathsf{I\!R}^5$ \cite[]{genzBook, sinnKeller2010}.
\subsection{EP results for hyperrectangular and polyhedral integration regions}
\label{sec:1results}
\begin{figure*}[t]
\centering
\hspace{0.0cm}
\epsfig{figure=results_ALL_aistats.pdf,width=\textwidth}
\caption{\small{Empirical results for EPMGP with hyperrectangular and polyhedral integration regions. See text.}}
\label{fig:results}
\end{figure*}
Figure \ref{fig:results}A-C shows the empirical results for hyperrectangular integration regions. For each dimension $n = \{2,3,4,5,10,20,50,100\}$, we chose 250 random Gaussians $p_0({\mathbf x})$ and random regions $\mathcal{A}$, and we calculated the relative error between the EPMGP and Genz estimates. Each of these errors is plotted against dimensionality $n$ as a point in Figure \ref{fig:results}A, with added horizontal jitter to show the distribution of points. The blue line and error bars represent the median and $\{25\%,75\%\}$ quartiles. The black line below the EP results is the median error estimate given by the Genz method. Since this line is almost always a few orders of magnitude below the distribution of EP errors, it is safe to use the high accuracy Genz method as a proxy to ground truth. Figure \ref{fig:results}A demonstrates that the EP method gives a reliable estimate of Gaussian probabilities with hyperrectangular integration regions, with relative errors typically on the order of $10^{-4}$ and with individual errors rarely in excess of 1\%.
Figure \ref{fig:results}B plots the errors by the value of $\log Z$ itself, where we use the $\log Z$ calculated from Genz. On this panel we can also plot a $1\%$ error for $Z\approx F(\mathcal{A})$, as the log scale can suggest misleadingly large or small errors. Though it is a common intuition that this method should have high error in tail probabilities, there is instead a weakly positive or nonexistent correlation of error with $\log Z$. Panel C repeats the same errors but plotted by the condition number of the covariance $K$ and shows pronounced trend, which is expected by considering Figure \ref{fig:white}.
Figure \ref{fig:results}D-F has the same setup as \ref{fig:results}A-C, except we test over polyhedral integration regions, which amounts to additionally drawing $m=n$ random polyhedral faces ${\mathbf c}_i$. The data tell a similar story but with surprisingly higher errors.
The black line in Panel D shows the median EP errors in the axis aligned case ({\it i.e.}, the blue line in Figure \ref{fig:results}A). The purpose of this line is to show that indeed for the same Gaussian cases, the error rate of polyhedral EPMGP is typically an order of magnitude or two higher than in the hyperrectangular case. Genz errors are not shown. Again Figure \ref{fig:results}E shows a modest positive effect in log$Z$. The only other change is Figure \ref{fig:results}F, where we use the concept of whitening the space to consider the transformed region integrated over a standard Gaussian as in Figure \ref{fig:white}. We consider this region as a matrix $C'$ with columns equal to $L^T{\mathbf c}_i$. To test the transformed region's proximity to $I$ (a decomposable problem with no EP error), we use the condition number of $C'$. Conveniently, in the hyperrectangular case this metric is the square root of the condition number of the covariance $K$. This allows us to compare Figures \ref{fig:results}C and \ref{fig:results}F, since this square root is just a constant scaling on the log scale shown. Figure \ref{fig:results}F confirms our intuition that less hyperrectangular transformed integration regions will have higher error\footnote{There are a number of other sensible metrics on the transformed region, and we have produced similar plots with Frobenius and $l_1$ norms of the scaled Gram matrix $\frac{1}{n}C'^TC'$. The error trend is the same.}.
The sensitivity of error to the number of polyhedral constraints $m$ is also an important question. Figure \ref{fig:results}G-I gives these empirical results: Panel G plots errors by number of polyhedral face constraints $m$ instead of by Gaussian dimension $n$. In this row of the figure, all cases use $n=10$ dimensions, and we show polyhedral sizes of $m = \{2,4,8,10,12,16,32,64\}$. Larger numbers of constraints/polyhedral faces do imply larger error, trending up by roughly an order of magnitude or two. The errors shown in Figure \ref{fig:results}H seem largely invariant to the value $\log Z$ itself. However, Panel I still indicates some upward trend, as we would expect.
Figure \ref{fig:results}D-I, namely the arbitrary polyhedral cases, show error rates that may in many cases be unacceptable. Though EP still performs well often, the reliability of the algorithm is not what it is in the hyperrectangular case of Figure \ref{fig:results}A-C.
\begin{figure*}
\centering
\hspace{0.0cm}
\epsfig{figure=results_scenario3_aistats.pdf,width=5in}
\caption{\small{Empirical results for orthant probabilities. See text for description.}}
\label{fig:specialresults}
\end{figure*}
Next, we calculate true orthant probabilities analytically: Figure \ref{fig:specialresults} shows errors for both the EP method (colour) and the Genz method (grayscale). The two panels are the usual errors plotted as a function of condition number of covariance (Panel A) and the true probability $\log Z$ (Panel B). The four cases shown are orthant probabilities at $n = \{2,3,4,5\}$. First, we note that there is a clear separation between the Genz errors and the EP errors, which helps to solidify the earlier claim that the Genz numerical answer can be used as a reasonable proxy to ground truth.
Second, it is interesting to note that there is significant structure in the EP errors with orthant probabilities when plotted by $\log Z$ in Panel B. Each dimension has a ``V" shape of errors. This can be readily explained by reviewing what an orthant probability is. For a zero-mean Gaussian, an orthant probability in $\mathsf{I\!R}^2$ is simply the mass of one of the quadrants. If that quadrant has a mass of $\log Z = \log 0.25$, then the correlation must be white, and hence EP will produce a highly accurate answer (the problem decomposes). Moving away from white correlation, EP will produce error. This describes the shape of the red curve for $n=2$, which indeed is minimised at $\log 0.25 \approx -1.39$. The same argument can be extended for why there should be a minimum in $\mathsf{I\!R}^3$ at $\log Z = \log 0.125$, and so on.
In summary, the empirical results of Figures \ref{fig:results} and \ref{fig:specialresults} indicate that EPMGP is a successful candidate algorithm for Gaussian probabilities. The error rate is non-zero but generally quite low, with median errors less $10^{-4}$ and individual errors rarely in excess of 1\% across a range of dimensions, which may be acceptable in many applied settings.
\subsection{Pathological cases}
\label{sec:pathologicalresults}
\begin{figure*}[t]
\centering
\hspace{0.0cm}
\epsfig{figure=results_pathology.pdf,width=6.4in}
\caption{\small{Empirical results from intentionally pathological cases. See text for description.}}
\label{fig:resultspathology}
\end{figure*}
Figure \ref{fig:resultspathology} shows pathological cases designed to illustrate the shortcomings of EP. In all cases, we want to integrate the $\mathcal{N}(0,I)$ Gaussian in $n=2$ dimensions over the $[-1,1] \times [-1,1]$ box, which is simply the product of two axis-aligned univariate box functions ($t_1(x_1)= \mathbb{I}\{-1 < x_1 < 1\}$ and $t_2(x_2)= \mathbb{I}\{-1 < x_2 < 1\}$). The $y$-axis shows the errors as usual, and the $x$-axis is a feature chosen to create large errors.
First, when looking at a transformed integration region, EP error may derive from approximate factors $\tilde{t}_i({\mathbf x})$ being not orthogonal, and thus overcounting the mass when forming the EP approximation $q({\mathbf x})$. An extreme, contrived example of this is adding two repeats of identical factors $t_i({\mathbf x})$. Though the integration region is unchanged, EP is not invariant to this change, as it still must make an approximate factor for each true factor. Figure \ref{fig:resultspathology}A shows this {\bf redundancy} pathology. We use EP to solve the same probability with increasing numbers of redundant factors, adding up to 1000 copies of the same integration region. Mathematically, we go from using EP to solve $\int p_0({\mathbf x}) t(x_1)t(x_2) d{\mathbf x}$, to using EP to solve $\int p_0({\mathbf x}) t(x_1)^{1000}t(x_2)^{1000} d{\mathbf x}$. Though the true $F(\mathcal{A})$ is unchanged, EP becomes arbitrarily bad. EP in this redundancy case is \emph{underestimating} the true probability, for reasons explained in the next section.
Second, EP error may derive from the approximate factors accounting for mass inappropriately. When two $t_i({\mathbf x})$ are not orthogonal, moment matching will include mass that is outside the integration region. Figure \ref{fig:resultspathology}B shows an integration region that is still the $[-1,1] \times [-1,1]$ box, but that can be described by the intersection of two boxes of any size, as the cartoon shows. However, EP must consider the mass that exists in each true box factor \emph{individually}. Hence we expect and indeed see that EP will not be invariant to this change, and further that EP should overestimate the true probability when there is {\bf extra mass}, which is the term we use to describe this pathology of EP. Note that the case where there is no extra mass - Panel B at left - is still a case where there are two redundant boxes, and hence we expect the underestimation error from Panel A (corresponding points circled in blue). Lastly, Figure \ref{fig:resultspathology}C shows that preprocessing the constraints (by removing redundant factors or tightening factor limits) can not fix these pathologies: increasing numbers of slightly rotated boxes minimally describe $\mathcal{A}$, but the same magnitude of errors is seen.
These redundancy and extra mass issues are of course two effects of a single underlying problem: EP enforces a Gaussian approximation to a hard box-function factor, which is a highly non-Gaussian form (but better than a conventional halfspace step function). This effect is particularly pronounced for sharp box factors, but not limited to them: factors with weaker or heavier tails, and non-symmetric factors, all have similar issues. Thus redundancy and extra mass generally go hand in hand: if there are likelihood terms that are not orthogonal with respect to the prior covariance $K$, they will each have to consider both mass already considered by another factor (redundancy) and mass that lies outside the actual polyhedron (extra mass).
\section{Correcting EP Errors}
\label{sec:discussion}
These empirical results suggest two ways to improve EP for multivariate Gaussian probabilities, and they shed further light on the underpinnings of EP's failure modes.
\subsection{$\boldsymbol\alpha$-divergence perspective}
The redundancy and extra mass problems can be viewed as changing the effective $\alpha$-divergence that EP minimises. The $\alpha$-divergence $D_{\alpha}(p \parallel q)$ is a generalisation of the KL-divergence, and its relationship to EP and other variational schemes has been studied \cite[]{minkaMSFTTR2005}.
The Power-EP algorithm is an extension of EP where the factors are raised to a specific power $\alpha_i$ before moment matching \cite[]{PowerEP}. One of the motivations for this algorithm is that raising the factor $t_i$ to a power can make the projection more tractable. That paper also notes that running EP on a model where each factor $t_i$ is repeated $n_i$ times is equivalent to running Power-EP (with the factors unrepeated) with $\alpha_i = 1/n_i$. Importantly, the Power-EP algorithm can also be derived as the minimisation of local $\alpha_i$-divergences. This repetition of factors is precisely the ``redundancy" issue that we created by repeating constraints $n_i$ times in Figure~\ref{fig:resultspathology}A. Our EP algorithm in the pure redundant case can thus be interpreted as running Power-EP with $\alpha_i = 1/n_i$.
First, we note that the zeroth moment $Z$ is always \emph{underestimated} by $\alpha$-divergence with $\alpha<1$. Also, $Z$ is only correctly estimated for $\alpha=1$ (KL-divergence), and it is \emph{overestimated} for \emph{inclusive} $\alpha$-divergences which have $\alpha>1$. This provides a theoretical explanation for the systematic underestimation of $Z$ in the redundancy case of Figure~\ref{fig:resultspathology}A: EP in this case is the same as Power-EP with $\alpha_i = 1/n_i < 1$. As in regular EP, a subtle aspect here is that Power-EP minimises the $\alpha$-divergence \emph{locally}, whereas the overestimation/underestimation of the zeroth moment $Z$ holds for \emph{global} $\alpha$-divergence minimisation, and the relationship between the two is not yet fully understood. However, our construction in Figure \ref{fig:resultspathology}A was a fully factorised case, so there is direct correspondence between global and local divergences, and as such we did systematically observe underestimation in these experiments. Our experiments indicate similar underestimation even when there is not direct global-local correspondence.
Second, this relationship between EP and $\alpha$-divergence gives us a notion of \emph{effective} $\alpha$-divergence $\alpha_\text{eff}$. In particular, running EP with the factor $t_i$ repeated $m_i$ times is the same as local $\alpha$-divergence minimisation by Power-EP with $\alpha_\text{eff} = 1/m_i$. When the constraints are almost-repeated (they have a large dot product, but are not fully colinear, such as Figure~\ref{fig:resultspathology}C), we lose the exact correspondence between EP and Power-EP, but we could still imagine that EP corresponds to Power-EP with a suitable $\frac{1}{m_i} < \alpha_\text{eff}<1$ in this case also (using a continuity argument).
\begin{figure}
\centering
\hspace{0.0cm}
\epsfig{figure=alphaCcorr.pdf,width=3.2in}
\caption{\small{$\alpha'$ corrections to the rotated box example. The line $\alpha' = 1$ (no correction) is the same curve as Figure \ref{fig:resultspathology}C.}}
\label{fig:alpharotate}
\end{figure}
With these facts, we can try to correct for this $\alpha_\text{eff}$ by using Power-EP with $\alpha' = 1/\alpha_\text{eff}$ instead of standard EP. Thus we can use the above arguments directly in Figure \ref{fig:resultspathology}A to remove the errors entirely. For box factors that are repeated $m_i$ times, we correct with $\alpha_i' = 1/\alpha_\text{eff} = m_i$. Empirical findings confirm this theoretical treatment, and the errors in Figure \ref{fig:resultspathology}A are entirely removed by this Power-EP correction. In Figure \ref{fig:alpharotate} we repeat the experiment of Figure \ref{fig:resultspathology}C, but for a range of $\alpha'$ corrections. We use a single correction for all factors $t_i({\mathbf x})$, though in general each factor may be corrected individually. The $\alpha'=1$ curve in Figure \ref{fig:alpharotate} is the same as Figure \ref{fig:resultspathology}C (up to a sign change). We then plot the same curve for many different corrections, and those black curves indicate that indeed there is an optimal $\alpha'$ correction that allows the EP estimate to be error free. In the inset panel we plot those optimal $\alpha'$ values, which shows, for a given number of rotated boxes ($x$-axis, log scale), the corrective $\alpha'$ required ($y$ axis, log scale) to remove EP error entirely from the result. Here the curve is not linear, nor are the optimal $\alpha' = m$ (though reasonably close, since slightly rotated boxes are nearly repeats). These results are not surprising, since there should be some error-free value in many cases. Nonetheless we see that even choosing a sensible $\alpha'$ correction can considerably improve the EP result in this redundancy example.
Next, the extra mass issue can also be viewed in terms of $\alpha$-divergences, though the connection is not as rigorous as the redundancy issue. The extra mass problem is one of inclusivity: the Gaussian approximate factor is including mass that it should not (as that mass lies outside the true polyhedron), which is a property of \emph{inclusive} $\alpha$-divergences as mentioned previously, {\it i.e.}, divergences with $\alpha > 1$. These we would expect to overestimate the true probability, and indeed this is what we see in Figure \ref{fig:resultspathology}B. We can also consider correcting for these $\alpha_{\text{eff}}$ with Power EP. However, the situation is slightly more complicated here as the extra mass problem involves understanding how $\alpha_{\text{eff}}$ changes with the decay of the Gaussian distribution. Results showing this are shown in Figure \ref{fig:alphaextramass}. This figure tells the same story as Figure \ref{fig:alpharotate}: a sensible choice of correction term will remove much of the EP error.
\begin{figure}
\centering
\hspace{0.0cm}
\epsfig{figure=alphaBcorr.pdf,width=3.2in}
\caption{\small{$\alpha'$ corrections to the extra mass example. The line $\alpha' = 1$ (no correction) is the same curve as Figure \ref{fig:resultspathology}B.}}
\label{fig:alphaextramass}
\end{figure}
Of course, the challenge is to come up with a theory or principled heuristics for choosing $\alpha_i'$ in general polyhedral problems, as intuition tells us there should be no straightforward answer here. For almost-repeated constraints and polyhedra in general, we have attempted to similarly improve EP by calculating a suitable $\alpha_\text{eff}$ term from problem parameters (the region and the Gaussian), and using Power-EP. Our early investigation of this correction has given some promising results, often reducing errors significantly. However, because often $\alpha_\text{eff} < 1$ and hence the correction term is greater than unity (unlike standard Power-EP), this EP algorithm encounters frequent convergence and stability issues, even with significant damping on the outbound messages (which does \emph{not} alter the solution or $\alpha_\text{eff}$). This obstacle suggests two areas of future work: first, better characterisation of $\alpha_\text{eff}$ from geometric quantities such as dot products of constraints. Second, addressing convergence issues via a provably convergent double-loop approach to EP \cite[]{opper2005, seeger2011}.
To summarise, both the redundancy and extra mass problems can be cast in the language of effective $\alpha$-divergence. We know that minimising $D_{\alpha}$ only returns moment matching when $\alpha = 1$ (KL-divergence), and thus any efforts to drive $\alpha_{\text{eff}}$ closer to 1 should improve all the results seen here.
\subsection{Perturbation perspective}
Another important direction of work in correcting EP is approximations to the remainder term $R = \frac{\log Z}{\log Z_\text{EP}}$ \cite[]{paquet2009, opper2009, cseke2011}. By making various expansions to this remainder term, we might hope to generally reduce EP errors. A particularly relevant expansion for the purposes of Gaussian probability calculations is from \cite{paquet2009} and uses the error terms $\epsilon_i({\mathbf x}) = \frac{t_i({\mathbf x})}{Z_i \tilde{t}_i({\mathbf x})} - 1$, such that the correction term $R$ becomes a finite series of moments over products of increasing numbers of true factors and approximating factors. Of course this is intractable in general. However, in our problem setting these moments are again multivariate probabilities of increasing order, which allows us to consider a variety of interesting schemes.
Our current direction of work is to use the Genz method in low-dimensions as a refinement to EP. This effectively allows us to capture all the benefits of the EP method, which scales well with dimension and $\log Z$, while using numerical quadrature where it is fast and extremely accurate, namely in the low dimensional case. This creates a hybrid EP-Genz method, where we can trade between the two approaches on a spectrum determined by the depth of the expansion. An important caveat of these expansion methods is that they rely on an already-good approximation from EP. However, our results from Figure \ref{fig:results} indicate that this is likely acceptable in most all cases, as EP is already highly accurate. At this time we do not have preliminary results to report on this approach.
\section{Discussion}
We have presented an EP algorithm constructing analytic approximations to polyhedral integrals on multivariate Gaussian distributions. Such approximations highlight interesting connections between approximate inference and approximate integration, and these approximations are of value for a number of applications in machine learning and throughout science and applied statistics. We show that under a variety of circumstances, relative EP errors rarely exceed $1\%$, and median relative errors are typically two to four orders of magnitude smaller. While we spent much of the results exploring the regions of poor performance and possible corrections, the general strength of the method should not be neglected. EPMGP also has some nice features such as analytical derivatives, fast runtime, and a natural ability to compute tail probabilities. On the other hand, existing numerical methods, in particular the Genz method, typically demonstrate high numerical accuracy.
Our results put a qualifier on the results of previous work using EP for evidence estimation, such as the Bayes Point Machine \cite[]{herbrichBook, minkaUAI01, minka01phd}, which was thoroughly investigated only for a relatively low dimensional case ($m=3$, $n=2$ in our notation) and for prediction accuracy, not ground truth. Our results caution that in such applications, EP should be used with care in problems where constraints outnumber dimensions, particularly $m \gg n$.
Another important existing connection point is Gaussian Process classification \cite[]{rasmussenBook, KussRasmussen2005}. When a probit likelihood (namely a univariate Gaussian cdf) is used, the data likelihood becomes a Gaussian probability problem. The same is true of probit regression \cite[]{ochi1984}, where changing the representation of the more standard prior and likelihood reveals a high-dimensional Gaussian probability. Though these problems are usually not considered Gaussian probabilities, it is another point of broad importance for EP and for Gaussian probabilities. Furthermore, as EP is often used in a different way in these problems, exploring the similarities between EP for Gaussian probabilities and EP for problems that can be cast as Gaussian probabilities is an interesting subject for future investigation.
Here we have investigated properties for EP as applied to Gaussian integration, but understanding how much of these issues stems from EP itself and how much from the application is an important clarification. Gaussian probabilities are particularly instructive, as they encourage the geometric interpretation that we have maintained throughout this paper. However, the issues of redundancy and extra mass are not specific to strict box functions, and the interpretation in terms of $\alpha_{\text{eff}}$ is no less applicable in the case of smooth functions.
Perhaps the most interesting aspect of this work is that the Gaussian probability problem, by its simple geometric interpretation, allows clear investigation into the strengths and weaknesses of EP,
which bears relevance to applications well beyond just Gaussian probabilities.
As EP becomes a more popular approximate inference method, this cautionary result suggests that EP should not be trusted blindly in new applications.
\subsubsection*{Acknowledgements}
We thank Tom Minka, Thore Graepel, Ralf Herbrich, Carl Rasmussen, Zoubin Ghahramani, Ed Snelson, Ulrich Paquet, and David Knowles for helpful discussions. JPC was supported by the UK Engineering and Physical Sciences Research Council (EPSRC EP/H019472/1); PH and SLJ were supported by a grant from Microsoft Research Ltd.
\bibliographystyle{apalike}
| {
"timestamp": "2011-11-30T02:04:51",
"yymm": "1111",
"arxiv_id": "1111.6832",
"language": "en",
"url": "https://arxiv.org/abs/1111.6832",
"abstract": "While Gaussian probability densities are omnipresent in applied mathematics, Gaussian cumulative probabilities are hard to calculate in any but the univariate case. We study the utility of Expectation Propagation (EP) as an approximate integration method for this problem. For rectangular integration regions, the approximation is highly accurate. We also extend the derivations to the more general case of polyhedral integration regions. However, we find that in this polyhedral case, EP's answer, though often accurate, can be almost arbitrarily wrong. We consider these unexpected results empirically and theoretically, both for the problem of Gaussian probabilities and for EP more generally. These results elucidate an interesting and non-obvious feature of EP not yet studied in detail.",
"subjects": "Machine Learning (stat.ML)",
"title": "Gaussian Probabilities and Expectation Propagation",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9802808690122163,
"lm_q2_score": 0.7217431943271998,
"lm_q1q2_score": 0.7075110457387204
} |
https://arxiv.org/abs/1901.06678 | A study on random permutation graphs | For a given permutation $\pi_n$ in $S_n$, a random permutation graph is formed by including an edge between two vertices $i$ and $j$ if and only if $(i - j) (\pi_n(i) - \pi_n (j)) < 0$. In this paper, we study various statistics of random permutation graphs. In particular, the degree of a given node, the number of nodes with a given degree, the number of isolated vertices, and the number of cliques are analyzed. Further, explicit formulas for the probabilities of having a given number of connected components and isolated vertices are obtained. | \section{Introduction}\label{sec:intro}
Let $S_n$ be the set of all permutations on $[n] := \{1,2,\ldots,n\}$. For a given permutation $\pi_n$ in $S_n$, we may form a corresponding graph by including an edge between the given two vertices $i$ and $j$ if and only if $(i - j) (\pi_n(i) - \pi_n (j)) < 0$. In other words, there turns out to be an edge between two vertices if and only if the corresponding values in the permutation form an inversion. The graph resulting from this process is known to be a \emph{permutation graph}. If we consider the uniform distribution over $S_n$, then the resulting model is known to be the \emph{random permutation graph model}. The following figure shows a sample from this model when $n = 5$.
\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.4]{RPGeg}
\caption{The permutation graph corresponding to the permutation $\binom{1 \hspace{0.1in} 2 \hspace{0.1in} 3 \hspace{0.1in} 4 \hspace{0.1in} 5}{5 \hspace{0.1in} 2 \hspace{0.1in} 3 \hspace{0.1in} 1 \hspace{0.1in} 4}$.}
\end{center}
\end{figure}
Permutation graphs were first introduced by \cite{ELP:1972} and \cite{PLE:71}. These two works provide a characterization for being a permutation graph in terms of transitive orientability. In particular, it is shown that a graph is a permutation graph if and only if both the graph and its complement are transitively orientable, which means that whenever the edges are assigned directions the presence of the edges $(x,y)$ and $(y,z)$ implies the existence of the edge $(x,z)$. In the cited works, they also provided a polynomial time algorithm to find a transitive orientation when it exists. It is also known that testing whether a graph is a permutation graph can be done in linear time.
One particular interest in permutation graphs stems from the fact that they are perfect graphs. This enhances their applications in several fields such as channel routing, scheduling, memory allocation, genomics and bioinformatics because various optimization problems become polynomial. See \cite{BP:96} as an exemplary work. In contrast to computational advantages in certain cases - and simplicity of sampling such permutations, developing the theory for certain parts of permutation graphs turn out to be quite challenging. One special topic we will emphasize below in terms of toughness is the connectivity related problems. See \cite{AP:2013} for some recent results in this direction for permutation graphs with a given number of edges. Also, note that a more recent work \cite{AH:2016} discusses the random trees obtained from permutations graphs.
In \cite{AH:2016}, they emphasize that various features of random permutation graphs can be understood by turning the problem into a random permutation problem. That will be the path we follow below. For example, we will be interested in the number of inversions, the level, the cycle structure, number of increasing sequences of (random) permutations among several others. Most of the statistics we will work on will be related to the descent structure of the underlying permutation - for a certain reason related to independence. We refer to \cite{Bona:2016} as a general reference of combinatorics on permutations. \cite{BDJ:1999}, \cite{DFG:2008} and \cite{O:2000} are some important works on statistics of random permutations, but the field is so huge that it is hard to include enough references here.
We will now briefly sketch what we discuss below. First, let us note that from here on we will be using one line notation for permutations; for example, $\binom{1 \hspace{0.1in} 2 \hspace{0.1in} 3 \hspace{0.1in} 4}{3 \hspace{0.1in} 1 \hspace{0.1in} 2 \hspace{0.1in} 4} = (3,1,2,4)$. Let $\mathcal{G}_n$ be a random permutation graph with $n$ vertices. Our first focus will be on the number of $m$-cliques $K_m$, $m \in\mathbb{N}$ fixed, and we will prove that the following central limit theorem holds by making use of the $U$-statistics theory:
$$\frac{K_m - \mathbb{E}[K_m] }{\sqrt{Var(K_m)}}
\longrightarrow_d \mathcal{Z}, \qquad \text{as} \; \; n\rightarrow \infty,$$ with certain expressions for
$\mathbb{E}[K_m]$ and $Var(L_m)$, where $\mathcal{Z}$ is a standard normal random variable.
Later, we also see that the same central limit theorem also holds for the number of cycles in $\mathcal{G}_n$ of length at least $m$. Both the number of cliques and the number of cycles arguments will be related to the number of increasing subsequences in random permutations of a given length. Exploiting this relation a little bit more, we also determine the limiting distribution of the largest cycle in $\mathcal{G}_n$ as the Tracy-Widom distribution. We do not go into computations related to the exact distribution of the number of cycles here which require a good understanding of partitions and a lot more pages. But we hope to return back to such problems in an upcoming work.
Afterwards, we focus on the asymptotic distribution of the degree of a given node. In particular, we provide a very simple proof for a central limit theorem for the mid-node which was previously proven in \cite{BM:2017} by using different techniques. We do not restrict ourselves here just to the mid-node, but we also prove a central limit theorem for any given fixed node $k$ as the number of nodes $n$ grows - actually $k$ is allowed to grow as well, see below for the exact statement.
Another interest for us will be on the number of isolated vertices, and also the number of vertices with a given degree $d$. The latter problem turns out to be challenging as we will see below, and its details will be given in a separate work. Another tough issue we briefly discuss is the connectivity. Recently, in \cite{AP:2013} when we choose a random permutation graph on $n$ vertices with $m$ edges, it was shown that the resulting graph is connected with high probability when we have a certain restriction on the growth rate of the number of edges. However, the general problem with no constraints or related problems such as the distribution of the number of connected components for a given $n$ turn out to be quite tough. We hope that we will be able to demonstrate this in the relevant section with the combinatorics which we use in a much simpler problem.
The last statistics of interest for us will be the extremal statistics where we will focus on the nodes with the fewest and largest degrees. Here, for symmetry reasons studying the minima and the maxima turn out to be equivalent. In this part, we will sketch a central limit theorem for the maximal degree using a probabilistic argument - or equivalently, as we shall see, for the minimal degree. Such a central limit theorem was previously proven in \cite{BM:2017} via more analytic techniques.
The paper also discusses two variations of random permutation graphs. The first one will be on introducing directed random permutation graphs. For this case, it will turn out that analyzing the extremal statistics become trivial due to the existing literature on random permutations. In particular, the maximal degree in a random permutation coincides in distribution with the level of a uniformly random permutation. Secondly, we will introduce two generalizations/variations of standard random permutation graphs, which preserve the role of independence that is crucial in many of our arguments. These two generalizations will be built on biased riffle shuffles and generalized unfair permutations.
We fix some notation now. First, $=_d$, $\rightarrow_d$ and
$\rightarrow_{\mathbb{P}}$ are used for equality in distribution,
convergence in distribution and convergence in probability,
respectively. $\mathcal{Z}$ denotes a standard normal random
variable, and $C$ is used for constants (which may differ in each
line) that do not depend on any of the parameters. The notations $d_K$, $d_W$ and $d_{TV} $ are reserved for the Kolmogorov, Wasserstein and total variation distances between probability measures, respectively. Finally, for two
sequences $a_n$ and $b_n$, we write $a_n \sim b_n$ if $\lim_{n
\rightarrow \infty} a_n / b_n =1$.
\section{Number of $m$-cliques}
Let $K_m$ be the number of $m$-cliques, $m \in \mathbb{N}$, in a random permutation graph $\mathcal{G}_n$ with $n$ vertices. Let $\pi_n$ be the corresponding permutation representation in $S_n$. Clearly, $K_1 = n$ and $K_2$ is the number of edges in $\mathcal{G}_n$ which of course is just the number of inversions in $\pi_n$, denoted by $Inv(\pi_n)$. The permutation statistic $Inv(\pi_n)$ is well-studied in the literature, and there are various different proofs that it satisfies a central limit theorem when $\pi_n$ is uniformly random. Let us just refer to \cite{fulman} where not only $$\frac{Inv(\pi_n) - \binom{n}{2}}{\sqrt{\frac{n(n-1) (2n +5 )}{72}}} \rightarrow_d \mathcal{Z}$$ as $n \rightarrow \infty$ is proven, but it is also shown that a convergence rate of order $1 \sqrt{n}$ can be obtained with respect to $d_K$. In this section, we focus on $K_m$ for a given $m \geq 2$ and show that it satisfies a central limit theorem. Our main tools will be the random permutation interpretation and the theory of $U$-statistics.
\begin{theorem}\label{thm:Km}
Let $\mathcal{G}_n$ be a random permutation graph and $K_m$ be the number of $m$-cliques in $\mathcal{G}_n$. We then have $$\frac{K_m - \mathbb{E}[K_m] }{\sqrt{Var(K_m)}}
\longrightarrow_d \mathcal{Z}, \qquad \text{as} \; \; n\rightarrow \infty,$$ where $$\mathbb{E}[K_m] = \binom{n}{m} \frac{1}{m!},$$ and where $Var(K_m) = \mathbb{E}[K_m^2] - \left( \mathbb{E}[K_m] \right)^2$, with \begin{equation}\label{eqn:2ndmomentKm}
\mathbb{E}[K_m^2]= \sum_{t+s \leq m} [(2 m - t)!]^{-1} 4^{m-t} \binom{n}{2m-t}\binom{m-t-s-1/2}{m-t-s} \binom{s+(t+1)/2-1}{s} \binom{2m-t}{2m-2t-2s}.
\end{equation}
\end{theorem}
\begin{remark}
(i) It is further shown in \cite{IO:2018} that $$Var(K_m) \sim_{n \rightarrow \infty} \frac{1}{2 ((2 m - 1)!)^2} \left( \binom{4m - 2}{2m -1} - 2 \binom{2m - 1}{m}^2 \right) n^{2m - 1},$$ therefore an application of Slutsky' theorem yields a slightly more compact form
$$\frac{K_m - \binom{n}{m} \frac{1}{m!}}{ \frac{1}{2 ((2 m - 1)!)^2} \left( \binom{4m - 2}{2m -1} - 2 \binom{2m - 1}{m}^2 \right) n^{2m - 1}} \rightarrow_{d} \mathcal{Z}, \quad n \rightarrow \infty.$$
(ii) The theory of $U-$statistics is well developed, and indeed we may also obtain a convergence rate of order $1 / \sqrt{n}$ with respect to the Kolmogorov metric. See, for example, \cite{CS:2007} for the necessary background on $U$-statistics. But the constant term, which will depend on $m$, will probably be far from the optimal one even after cumbersome computations. So we do not go into details obtaining converging rates here and in the rest of this paper.
\end{remark}
\begin{proof}
Let $\pi_n$ be the corresponding random permutation to $\mathcal{G}_n$. We first start by observing that a subset $\mathcal{S} = \{i_1, i_2,\ldots,i_m\}$, $i_1< i_2 < \cdots < i_m$ forms an $m$-clique in $\mathcal{G}_n$ if and only if $$\pi_{i_1} > \pi_{i_2} > \cdots > \pi_{i_m}.$$ Using this, we then write $$K_m = \sum_{1 \leq i_1 < i_2 < \cdots < i_m \leq n} \mathbf{1}(\pi_{i_1} > \pi_{i_2} > \cdots > \pi_{i_m}).$$ In other words, $K_m$ is merely the number of decreasing subsequences of length $m$ in $\pi_n$. Noting that this equals in distribution to the number of increasing subsequences of length $m$ in $\pi_n$, and denoting the latter by $I_{n,m}$, \cite{IO:2018} shows that $\mathbb{E}[I_{n,m}] = \binom{n}{m} \frac{1}{m!}$ and $Var(I_{n,m}) = \mathbb{E}[I_{n,m}^2] - \left( \mathbb{E}[I_{n,m}] \right)^2$ where $\mathbb{E}[I_{n,m}^2]$ is as given in \eqref{eqn:2ndmomentKm}.
From these observations, the moments given in the statement of Theorem \ref{thm:Km} are clear.
Now we move on to proving the central limit theorem. Although a brief sketch for the number of increasing subsequences of a given length in a random permutation was given in \cite{IO:2018}, we include all the details here for the sake of completeness\footnote{The reason why it was only a sketch in \cite{IO:2018} is that a similar argument on random words was detailed there.}. First, the proof is based on $U$-statistics, and a general reference for such statistics is \cite{KT:2008}. Now, remember that $$K_m = \sum_{1 \leq i_1 < i_2 < \cdots < i_m \leq n} \mathbf{1}(\pi_{i_1} > \pi_{i_2} > \cdots > \pi_{i_m})=_d \sum_{1 \leq i_1 < i_2 < \cdots < i_m \leq n} \mathbf{1}(\pi_{i_1} < \pi_{i_2} < \cdots < \pi_{i_m}).$$ Letting then $X_1,\ldots,X_n$ be i.i.d.
random variables from some continuous distribution, a result often attributed to R\'enyi tells us that
$$ \sum_{1 \leq i_1 < i_2 < \cdots < i_m \leq n} \mathbf{1}(\pi_{i_1} < \pi_{i_2} < \cdots < \pi_{i_m}) =_d \sum_{1 \leq i_1 < i_2 < \cdots < i_m \leq n} \mathbf{1}(X_{i_1} < X_{i_2} < \cdots < X_{i_m}).$$
Let now $Y_1,\ldots,Y_n$ be another i.i.d. sequence of random variables from some continuous distribution. Assume further that the families $\{X_1,\ldots,X_n\}$ and $\{Y_1,\ldots,Y_n\}$ are independent as well. Define the (random) permutation $\gamma \in S_n$ with the property $$Y_{\gamma(1)}< Y_{\gamma(2)} < \cdots < Y_{\gamma(n)}.$$ Clearly, the i.i.d. assumption yields $(X_1,\ldots,X_n) =_d (X_{\gamma(1)},\ldots,X_{\gamma(n)})$. But then, letting $\mathcal{R} = \{(i_1,\ldots,i_m): \text{each } i_j \in [n], i_j\text{'s are distinct from each other}\}$
\begin{eqnarray*}
K_m &=_d& \sum_{1 \leq i_1 < i_2 < \cdots < i_m \leq n} \mathbf{1}(X_{i_1} < X_{i_2} < \cdots < X_{i_m}) \\ &=_d& \sum_{1 \leq i_1 < i_2 < \cdots < i_m \leq n} \mathbf{1}(X_{\gamma(i_1)} < X_{\gamma(i_2)} < \cdots < X_{\gamma(i_m)}) \\
&=& \sum_{\mathcal{R}} \mathbf{1}(X_{\gamma(i_1)} < X_{\gamma(i_2)} < \cdots < X_{\gamma(i_m)}, i_1 < i_2 < \cdots < i_m) \\
&=& \sum_{\mathcal{R}} \mathbf{1}(X_{\gamma(i_1)} < X_{\gamma(i_2)} < \cdots < X_{\gamma(i_m)}, Y_{\gamma(i_1)} <Y_{\gamma(i_2)} < \cdots < Y_{\gamma(i_m)}) \\
&=_d& \sum_{\mathcal{R}} \mathbf{1}(X_{i_1} < X_{i_2} < \cdots < X_{i_m}, Y_{i_1} <Y_{i_2} < \cdots < Y_{i_m}).
\end{eqnarray*}
Here, in the last step we used the fact that $\gamma$ is a bijection, being a member of the symmetric group.
Now we define the function $$h((x_{i_1},y_{i_1}),\ldots,(x_{i_m},y_{i_m})) = \sum f((x_{j_1},y_{j_1}),\ldots,(x_{j_m},y_{j_m})),$$ where the summation is over all permutations of $i_1,\ldots,i_m$ and where $$ f((x_{i_1},y_{i_1}),\ldots,(x_{i_m},y_{i_m})) = \frac{1}{\binom{n}{m}} \mathbf{1}(x_{i_1} < x_{i_2} < \cdots < x_{i_m}, y_{i_1} <y_{i_2} < \cdots < y_{i_m}).$$ Note that $h$ is clearly symmetric. Therefore, we are able to express $$K_m =_d \frac{1}{\binom{n}{m}} \sum_{1 \leq i_1 < i_2 < \cdots < i_m \leq n} h((X_{i_1}, Y_{i_1}),\ldots,(X_{i_m}, Y_{i_m}))$$ But then $K_m$ is a U-statistic because
\begin{itemize}
\item[i.] $h$ is symmetric;
\item[ii.] $h \in L^2$ for each $n$, since it is just a finite sum of indicators;
\item[iii.] $h$ is a function of independent random variables (or, vectors, to be more precise).
\end{itemize}
Result follows.
\hfill $\square$
\end{proof}
\section{Asypmtotic distribution of the mid-node}
The purpose of this section is to prove that the mid-node\footnote{We will write $n/2$ instead of $\lfloor n /2 \rfloor$ as our results are mostly asymptotic. In non-asymptotic cases, let us assume that we are taking $n$ even, and that otherwise we would have the floor function somewhere around.} $n/2$ satisfies a central limit theorem. The result we prove was Theorem 3.4 in a recent work proven in \cite{BM:2017}. But we believe that our treatment is much elementary and has a little bit more probabilistic flavour. One other advantage is that it extends to other nodes as we shall see in next section
\begin{theorem}
Let $\mathcal{G}_n$ be a random permutation graph on $n$ vertices. Then the degree $d(n/2)$ of the mid-vertex satisfies $$\frac{d(n/2) - n / 2}{2 \sqrt{n}} \rightarrow_d \mathcal{N}(0, U,(1-U)),$$ where $U$ is a uniform random variable over $(0,1)$ independent of $\mathcal{G}_n$ and $\mathcal{N}( U,(1-U))$ is a normal random variable with random parameters $U$ and $1 - U$.\footnote{The expression $\frac{d(n/2) - n / 2}{2 \sqrt{n}}$ in \cite{BM:2017} is $\frac{d(n/2) - n}{2 \sqrt{n}}$, we believe that there is a typo there.}
\end{theorem}
\noindent {\bf Proof:}
First note that expectation and variance will be special cases of results from next section, and so we skip these computations here. Let now $X_1,X_2,\ldots$ be an i.i.d. sequence of uniform random variables over $(0,1)$. Let $X_1^*,X_2^*,\ldots$ be i.i.d. copies of $X_1,X_2,\ldots$ and assume that the sequences are independent within themselves. Let $\alpha \in \mathbb{R}$. Expressing $d(n/2)$ in terms of the underlying random permutation $\pi_n$, expressing the order relations via i.i.d. random variables, and
integrating over $X_{n/2}$, we then have
\begin{eqnarray*}
\mathbb{P} \left( \frac{d(n/2) - n}{2 \sqrt{n}} \leq \alpha \right) &=& \mathbb{P} \left( \frac{\sum_{j=1}^{n/2 -1 } \mathbf{1}(X_j > X_{n/2}) + \sum_{j = n/2 +1}^n \mathbf{1}(X_{n/2} > X_j) - \frac{n}{2}}{2 \sqrt{n}} \leq \alpha \right) \\
&=& \int_0^1 \mathbb{P} \left( \frac{\sum_{j=1}^{n/2 -1 } \mathbf{1}(X_j > u) + \sum_{j = n/2 +1}^n \mathbf{1}(u > X_j) - \frac{n}{2}}{2 \sqrt{n}} \leq \alpha \right) du \\
&=& \int_0^1 \mathbb{P} \left( \frac{\sum_{j=1}^{n/2 -1 } \mathbf{1}(X_j > u) + \sum_{j = n/2 +1}^n(1 - \mathbf{1}(u< X_j)) - \frac{n}{2}}{2 \sqrt{n}} \leq \alpha \right) du \\
&=& \int_0^1 \mathbb{P} \left( \frac{\sum_{j=1}^{n/2 -1 } \mathbf{1}(X_j > u) - \sum_{j = n/2 +1}^n \mathbf{1}(u < X_j)}{2 \sqrt{n}} \leq \alpha \right) du \\
&=& \int_0^1 \mathbb{P} \left( \frac{\sum_{j=1}^{n/2 -1} \mathbf{1}(X_j > u) - \sum_{j = 1}^{n/2} \mathbf{1}(u < X_j^*) }{2 \sqrt{n} } \leq \alpha \right) du \\
&=& \int_0^1 \mathbb{P} \left( \frac{\left( \sum_{j=1}^{n/2 -1 } ( \mathbf{1}(X_j > u) - \mathbf{1}(X_j^* > u)) \right) - \mathbf{1}(X_{n/2}^* > u) }{2 \sqrt{n}} \leq \alpha \right) du \\
\end{eqnarray*}
Now the probability in the integrand for given $u$ converges to $\int_{- \infty}^{\alpha} \frac{1}{\sqrt{2 \pi} \sqrt{u ( 1 - u)}} e^{-\frac{x^2}{2 \sqrt{u (1 - u)}}} dx $ by using the standard central limit theorem for i.i.d. random variables. Integration over $u$ from 0 to 1, and an application of dominated convergence theorem then yields $$\mathbb{P} \left( \frac{d(n/2) - n}{2 \sqrt{n}} \leq \alpha \right) \rightarrow \int_0^1 \int_{- \infty}^{\alpha} \frac{1}{\sqrt{2 \pi} \sqrt{u ( 1 - u)}} e^{-\frac{x^2}{2 \sqrt{u (1 - u)}}} dx du.$$ But the last expression equals $\mathbb{P}(N(0,U(1-U)) \leq \alpha)$ where $U$ is uniform over $(0,1)$ independent of the random graph - we have the product measure. Therefore, we have convergence in Kolmogorov metric which in particular implies convergence in distribution.
\hfill $\Box$
\section{Degrees of other nodes}
In the previous section, we focused on the mid-node, and proved a central limit theorem for it. Can we say anything about the other nodes? Firstly, since the variance of $d(n / 2)$ is $\Theta(n)$, $d(m)$ will still satisfy a CLT whenever $m = n/2 + o(n)$. Let us next look at one other particular case where $m$ is "small" compared $n$. We begin by finding the first two moments of $d(k)$ where $k$ is some positive integer at most $n$.
\begin{theorem}\label{thm:expvardeggen}
Let $\mathcal{G}_n$ be a random permutation graph on $n$ vertices, and let $d(k)$ be the degree of node $k$. Then we have
\begin{itemize}
\item[i.] $\mathbb{E}[d(k)] = \frac{n-1}{2}$;
\item[ii.] $Var(d(k)) = \frac{n - 1 + (n - 2k +1)^2}{6}$.
\end{itemize}
\end{theorem}
\noindent {\bf Proof:} Recall that $d(k) =_d \sum_{j =1 }^{k-1} \mathbf{1}(X_j > X_k) + \sum_{j = k + 1 }^{n} \mathbf{1}(X_j < X_k)$, where $X_i$'s are i.i.d. uniform random variables over $(0,1)$.
(i) We have $\mathbb{E}[d(k)] = \sum_{j =1 }^{k-1} \mathbb{P}(X_j > X_k) + \sum_{j = k + 1 }^{n} \mathbb{P}(X_j < X_k) = \frac{n-1}{2}$, since $X_i$'s are continuous random variables.
(ii) For the variance recall the law of total variance which in our case reads as
\begin{equation}\label{vardkcomp1}
Var(d(k)) = \mathbb{E}[Var(d(k) \mid X_k)] + Var(\mathbb{E} [d(k) \mid X_k]).
\end{equation}
Beginning with the first term on the right-hand side of \eqref{vardkcomp1}, observing $$Var(d(k) \mid X_k) = \sum_{j=1}^{k-1} (1 - X_k) X_k + \sum_{j = k +1}^n X_k (1 - X_k),$$ we have
$$ \mathbb{E}[Var(d(k) \mid X_k)] = \mathbb{E} \left[ \sum_{j=1}^{k-1} (1 - X_k) X_k + \sum_{j = k +1}^n X_k (1 - X_k)\right] = \frac{n-1}{6},$$ where we used $\mathbb{E}[X_k] = 1 /2$ and $\mathbb{E}[X_k^2] = 1 /3$.
For the second term on right-hand side of \eqref{vardkcomp1}, we first note that $$\mathbb{E}[d(k)\mid X_k] = \sum_{j=1}^{k-1} (1 - X_k) + \sum_{j=k+1}^n X_k = k - 1 + (n - 2k +1) X_k.$$ Therefore, $$ Var(\mathbb{E} [d(k) \mid X_k]) = Var ( k - 1 + (n - 2k +1) X_k) = (n-2k +1)^2 \frac{1}{6}.$$ Using \eqref{vardkcomp1}, (ii) now follows.
\hfill $\Box$
\begin{rmk}
(i) $\mathbb{E}[d(k)]$ is constant independent of $n$ due to symmetry reasons.
(ii) Variance of $d(k)$ is minimized at $k = n /2$, and is maximized at $k = 1$ and $k =n$.
\end{rmk}
When we use $k = n /2$ in Theorem \ref{thm:expvardeggen}, we obtain $\mathbb{E}[d(n /2)] = \frac{n-1}{2}$ and $Var(d(n/2)) = \frac{n}{6}$. Given these, we may prove an alternative central limit theorem for the mid-node with a deterministic limit. The proof technique is very similar to the one in the previous section, so we just state it skipping the proof.
\begin{theorem}
Let $\mathcal{G}_n$ be a random permutation graph on $n$ vertices. Then $$\frac{d(n) - n/2}{\sqrt{n / 6}} \rightarrow_d \mathcal{Z}, \quad \text{as } n \rightarrow \infty. $$
\end{theorem}
How about other $k$'s? Let us this time consider the case $k = h(n)$ where $h(n) = o(n)$ as $n \rightarrow \infty$.
\begin{theorem}
Let $\mathcal{G}_n$ be a random permutation graph on $n$ vertices and $h(n) = o (n)$ be an integer valued function. Then, $$\frac{d(h(n)) - n/2}{\sqrt{Var(d(h(n)))}} \rightarrow_d \mathcal{Z}, \quad \text{as } n \rightarrow \infty,$$ where $Var(h(n)) = \frac{n^2 - n (1 + 4 h(n)) + 4 h(n) (h(n) +1)}{6} \sim \frac{n^2}{6}$.
\end{theorem}
\noindent {\bf Proof:}
As always $X_i$'s are i.i.d. uniform random variables over $(0,1)$. Let us use the convention $h = h(n)$. Write
\begin{eqnarray}\label{eqn:hn1}
\nonumber d(h(n)) &=& \sum_{j=1}^{h(n) - 1} \mathbf{1}(X_j > X_h) + \sum_{j=h(n) +1}^{n} \mathbf{1}(X_j < X_h) \\
\nonumber &=& \sum_{j=1}^{h(n) - 1} \mathbf{1}(X_j > X_h) + \sum_{j=h(n) +1}^{n} (1 - \mathbf{1}(X_j > X_h) ) \\
\nonumber &=& n + h + \sum_{j=1}^{h(n) - 1} \mathbf{1}(X_j > X_h) - \sum_{j=h(n) +1}^{n} \mathbf{1}(X_j > X_h) \\
&=& n - \sum_{j = h(n)}^{n-h(n)} \mathbf{1}(X_j > X_h) + h + \sum_{j=1}^{h(n) - 1} \mathbf{1}(X_j > X_h) - \sum_{j=n-h(n)+1}^n \mathbf{1}(X_j > X_h).
\end{eqnarray}
Now, recalling $\sqrt{Var(h(n))} \sim n / \sqrt{6} = \Theta(n)$ and since $h(n) = o(n)$, we have $h/n \rightarrow 0$. It is also easy to see that the right-most two terms in \eqref{eqn:hn1}
$$\frac{\sum_{j=1}^{h(n) - 1} \mathbf{1}(X_j > X_h) - \sum_{j=n-h(n)}^n \mathbf{1}(X_j > X_h) }{n} \rightarrow 0,$$ with probability one as $n \rightarrow \infty$. This, in particular, implies the distributional convergence of $\frac{\sum_{j=1}^{h(n) - 1} \mathbf{1}(X_j > X_h) - \sum_{j=n-h(n)}^n \mathbf{1}(X_j > X_h) }{n}$ to zero.
Also, using symmetry techniques similar to ones in the previous section, the standard central limit theorem and the use of Slutsky's theorem imply that $$\frac{\frac{n}{2} - \sum_{j = h(n) +1}^{n-h(n) - 1} \mathbf{1}(X_j > X_k) }{n \sqrt{6}}=_d \frac{\sum_{j = h(n) +1}^{n-h(n) - 1} \mathbf{1}(X_j > X_k) - \frac{n}{2}}{n/ \sqrt{6}} \rightarrow_d \mathcal{Z}$$ as $n \rightarrow \infty$. With our observations, we conclude that
\begin{eqnarray*}
\nonumber \frac{d(h(n)) - \frac{n}{2}}{n / \sqrt{6}} &=&
\frac{ n - \sum_{j = h(n)}^{n-h(n)} \mathbf{1}(X_j > X_h) + h + \sum_{j=1}^{h(n) - 1} \mathbf{1}(X_j > X_h) - \sum_{j=n-h(n)}^n \mathbf{1}(X_j > X_h) - \frac{n}{2}}{n / \sqrt{6}} \\
&& \longrightarrow_d \mathcal{Z}
\end{eqnarray*}
as claimed.
\hfill $\Box$
\bigskip
This completes the picture as we have shown that the CLT holds in both cases when $k$ is close to $n/2$ and is far from $n$. The following corollary is now immediate.
\begin{cor}
Let $\mathcal{G}_n$ be a random permutation graph on $n$ vertices and $k \in \mathbb{N}$ be fixed. Then, $$\frac{d(k) - n/2}{\sqrt{\frac{n^2 - 4nk + 2n + 4k^2 -2}{12}}} \rightarrow_d \mathcal{Z}, \quad \text{as } n \rightarrow \infty. $$
\end{cor}
\section{Number of isolated vertices}
Let $I_n$ be the number of isolated vertices in a random permutation graph $\mathcal{G}_n$ with $n$ vertices. Let $\pi_n$ be the corresponding random permutation representation of $\mathcal{G}_n$.
Using the random permutations point of view, it is not hard to see that
$$I_n = \sum_{k = 1}^n \mathbf{1}(\pi_n(k) = \max\{\pi_n(1), \ldots, \pi_n(k-1)\}, \pi_n(k) = \min\{\pi_n(k+1), \ldots, \pi_n(n)\}).$$
Let us compute $\mathbb{E}[I_n]$. First, note that the simultaneous occurrence of the events $\pi_n(k) = \max\{\pi_n(1), \ldots, \pi_n(k-1)\}$ and $\pi_n(k) = \min\{\pi_n(k+1), \ldots, \pi_n(n)\}$ necessarily implies that there is fixed point at $k$, i.e. $\pi_n(k) = k$. Further, for this event to occur $\pi_n(1), \ldots, \pi_n(k-1)$ should all be in $\{1,\ldots,k-1\}$ with any ordering, and $\pi_n(k+1), \ldots, \pi_n(n)$ should all be in $\{k+1, \ldots,n\}$. Using these observations, we then obtain the probability of node $k$ being isolated to be $$p_k :=\mathbb{P}(\text{node } k \text{ is isolated}) = \frac{1}{n} (k-1)! (n - k)! \frac{1}{n!} $$ which after some manipulation gives $$p_k = \frac{1}{k n \binom{n}{k}}.$$ Now we may compute $\mathbb{E}[I_n]$:
\begin{eqnarray*}
\mathbb{E}[I_n] = \sum_{k=1}^n \mathbb{P}(\text{node } k \text{ is isolated}) &=& \frac{1}{n} \sum_{k=1}^n \frac{1}{k \frac{n (n-1)!}{k (k-1)! (n-k)!}} \\
&=& \frac{1}{n^2} \sum_{k=1}^n \frac{1}{\binom{n-1}{k-1}} \\
&=& \frac{1}{n^2} \sum_{k=0}^{n-1} \frac{1}{\binom{n-1}{k}}.
\end{eqnarray*}
In order to understand this last expression, we need a computational proposition.
\begin{propn}
Let $n \in \mathbb{N}$. Then $$2 \leq \sum_{k=0}^{n} \frac{1}{\binom{n}{k}} \leq 2 + \frac{2}{n } + \frac{2 (n - 3)}{n (n - 1)} \leq 2 + \frac{4}{n}.$$ In particular, $$\lim\limits_{n \rightarrow \infty} \sum_{k=0}^{n} \frac{1}{\binom{n}{k}} = 2.$$
\end{propn}
\noindent {\bf Proof:}
Note that when $2 \leq k \leq n-2$, we have $\frac{1}{\binom{n}{k}} \leq \frac{1}{\binom{n}{2}} = \frac{2}{n (n-1)}$. But then $$1 + \frac{1}{n} + \frac{1}{n} + 1 \leq \sum_{k=0}^{n} \frac{1}{\binom{n}{k}} \leq 1 + \frac{1}{n} + \frac{1}{n} + 1 + \sum_{k=2}^{n-2} \frac{2}{n(n-1)}.$$ So $$2 + \frac{2}{n} \leq \sum_{k=0}^{n} \frac{1}{\binom{n}{k}} \leq 2 + \frac{2}{n} + \frac{2 (n-3)}{n (n-1)} \leq 2 + \frac{4}{n}.$$ The limit result now follows from sandwich principle.
\hfill $\Box$
Now we are ready to give the expected number of isolated vertices in $\mathcal{G}_n$.
\begin{theorem}
Let $I_n$ be the number of isolated vertices in a random permutation graph $\mathcal{G}_n$. Then we have $$\frac{2n+2}{n^3} \leq \mathbb{E}[I_n] \leq \frac{2n+4}{n^3}.$$ In particular, $$\frac{n^2 \mathbb{E}[I_n]}{2} \rightarrow 1, \quad \text{as } n \rightarrow \infty.$$
\end{theorem}
\begin{remark}
If $\mathcal{G}_n'$ is an Erdös-R\'enyi graph with parameters $n$ and $p \in (0,1)$, and if $I_n'$ is the number of isolated vertices in $\mathcal{G}_n'$, then $\mathbb{E}[I_n'] = n (1 -p)^{n-1}$. An elementary computation shows that $\mathcal{G}_n'$ will behave like a random permutation graph in terms of the expected number of isolated vertices exactly when $p \sim 1- \exp(2 n^{-4})$ as $n \rightarrow \infty$.
\end{remark}
Finding the exact distribution of the number of isolated nodes in $\mathcal{G}_n$ turns out to be a nice problem, but we will not go into it here as it is much involved than one may think at first instance. Instead, letting $i \in \{1,2,\ldots,n-k+1\}$, let us have a look at the easier problem where we are interested in the probability that $i,i+1,i+2,\ldots,i + k$ are all isolated points. In this case all of $i,i+1,\ldots,i+k$ are necessarily fixed points. Besides, for each $j < i$, we should have $\pi_n(j) <\pi_n(i) = i$, and for each $j > i+k$, we should have $\pi_n(j) > \pi_n(i+k) = i+k$. Similar arguments to our previous result then yield:
\begin{propn}
Consider a random permutation graph $\mathcal{G}_n$. The probability that the nodes $i,i+1,\ldots,i+k$ will all be isolated at the same time is given by $$p = \frac{1}{n(n-1)\cdots (n-k +1) } (i-1)! (n - k)! \frac{1}{n!} = \left( \frac{(n-k)!}{n!} \right)^2 (i-1)!.$$
\end{propn}
\section{Connectivity issues}
As it is discovered easily, connectivity issues in random permutation graphs turn out to be very challenging. The literature here is rather scarce, see \cite{AH:2016}, \cite{AP:2013} for related work. When dealing with Erdös-R\'enyi graphs, say with parameters $n$ and $p= p(n) \in (0,1)$, the standard way to observe that the random graph is with high probability connected in certain regions where $p(n)$ is "large" is based on establishing $$\mathbb{P}(\text{connected}) = \mathbb{P}(\text{no isolated vertices} ) + o(1),$$ \cite{Hofstadt:2009}. Such an approach in our case turns out to be hard as it can be felt from the study on the isolated vertices, where one will realize that an inclusion-exclusion formula with tough combinatorial computations will be necessary in evaluating certain probabilities. We hope to dive into this sort of a general result in a future work, but let us leave some relevant questions here:
\bigskip
\textbf{Question:} (Q1) Under what conditions (such as the ones in \cite{AP:2013}) is the random permutation graph model connected with high probability?
(Q2) Can you find the exact distribution of the number of connected components in random permutation graph with a given number of vertices?
(Q3) Can you find the exact distribution of the number of isolated vertices in a random permutation graph?
\bigskip
Here, we focus on a simpler question, just to give a feeling why these problems tend to be challenging. We already know that two given nodes $i$ and $j$ are adjacent with probability $1/2$. How about the following question? What is the probability that $i$ and $j$, say $i < j$, will be in the same component and that there is some other node $k$ connecting these two vertices to each other? Even this question turns out to involve various computations as we shall see below. In order to answer this question we will focus on the complementary event. Let us start by defining the following event which is equivalent to the complementary event we are trying to understand $$E = \{\text{There is no } k \in [n] - \{i,j\} \text{ so that } i \leftrightarrow k \text{ and } j \leftrightarrow k \}.$$
\begin{comment}
We observe that $E$ can occur if and only if the following three events occur simultaneously:
$$E_1= \{\nexists k \in \{1,\ldots,i-1\}: \pi_n(k) > \pi_n(i) \text{ and } \pi_n(k) > \pi_n(j)\},$$
$$E_2= \{\nexists k \in \{i+1,\ldots,j-1\}: \pi_n(i) > \pi_n(k) \text{ and } \pi_n(j) < \pi_n(k)\},$$
$$E_3 = \{\nexists k \in \{j+1,\ldots,n\}: \pi_n(i) > \pi_n(k) \text{ and } \pi_n(k) > \pi_n(j)\}.$$ That is, $$E = E_1 \cap E_2 \cap E_3.$$
Now, in terms of these events the probability we are looking for is $\mathbb{P}(E^c) = 1 - \mathbb{P}(E)$.
\end{comment}
Now we write $E$ in the form $$E = \bigcap_{k \notin \{i,j\}} A_k,$$ where $$A_k = \{k\text{th node is not connected both } i < j \}.$$ Assume that the graph is generated via the permutation $\pi_n$, and the corresponding i.i.d. sequence uniform over $(0,1)$ is given by $X_1,X_2,\ldots,X_n$. Observe that given $X_i$ and $X_j$, $A_k$'s are independent.
We then have
\begin{eqnarray*}
\mathbb{P}(E) &=& \mathbb{P} \left( \bigcap_{k \notin \{i,j\}} A_k \right) \\
&=& 2 \int_0^1 \int_0^{x_i} \mathbb{P}_{x_i, x_j} \left( \bigcap_{k \notin \{i,j\}} A_k \right) dx_j dx_i,
\end{eqnarray*}
where here we use the notation $ \mathbb{P}_{x_i, x_j} $ in order to emphasize that we are conditioning on $X_j = x_j$ and $X_i = x_i$. The 2 term in front of the integrals comes from an obvious symmetry.
Using conditional independence,
\begin{eqnarray*}
\mathbb{P}(E) &=& 2 \int_0^1 \int_0^{x_i} \prod_{k=1}^{i-1} \mathbb{P}_{x_i, x_j} (X_k < x_j) \prod_{k=i+1}^{j-1} \mathbb{P}_{x_i, x_j} (X_k < x_j \text{ or } X_k > x_i ) \prod_{k=j+1}^n \mathbb{P}_{x_i, x_j} (X_k > x_j) dx_j dx_i \\
&=& 2 \int_0^1 \int_0^{x_i} \left(x_i^{i-1} (1 - x_i + x_j)^{j - i - 1} (1 - x_j )^{n -j} \right)dx_j dx_i \\
&=& 2 \int_0^1 x_i^{i-1} \int_0^{x_i} \left((1 - x_i + x_j)^{j - i - 1} (1 - x_j )^{n -j} \right)dx_j dx_i.
\end{eqnarray*}
Now doing the substitution $1 - x_j = u$, the right-most term equals
\begin{eqnarray*}
2 \int_0^1 x_i^{i-1} \left( \int_0^{1- x_i} (2 - x_i + u)^{j - i -1} u^{n- j} du \right) d x_i.
\end{eqnarray*}
Binomial expansion then gives
\begin{eqnarray*}
\mathbb{P}(E) &=& 2 \int_0^1 x_i^{i-1} \left( \int_0^{1- x_i} \sum_{\ell= 0 }^n \binom{n}{\ell} u^{(j-i-1)(n - \ell)} u^{n-j} du \right)dx_i \\
&=& 2 \sum_{\ell= 0 }^n \int_0^1 x_i^{i-1} (2 - x_i)^{\ell} \binom{n}{\ell} \left( \int_0^{1- x_i} u^{(j-i-1)(n - \ell)+ (n - j)} du \right)dx_i \\
&=& 2 \sum_{\ell= 0 }^n \int_0^1 x_i^{i-1} (2 - x_i)^{\ell} \binom{n}{\ell} \frac{(1 - x_i)^{(j-i-1)(n - \ell)+ (n - j) +1}}{(j-i-1)(n - \ell)+ (n - j) +1} dx_i\\
&=& 2 \sum_{\ell= 0 }^n \frac{\binom{n}{\ell}}{(j-i-1)(n - \ell)+ (n - j) +1} \int_0^1 x_i^{i-1} (2 - x_i)^{\ell} (1 - x_i)^{(j-i-1)(n - \ell)+ (n - j) +1} dx_i.
\end{eqnarray*}
Writing $2 - x_i = 1 + 1 - x_i$, another binomial expansion gives
\begin{eqnarray*}
\mathbb{P}(E) &=& 2 \sum_{\ell= 0 }^n \frac{\binom{n}{\ell}}{(j-i-1)(n - \ell)+ (n - j) +1} \int_0^1 x_i^{i-1} (2 - x_i)^{\ell} (1 - x_i)^{(j-i-1)(n - \ell)+ (n - j) +1} dx_i \\
&=& 2 \sum_{\ell= 0 }^n \frac{\binom{n}{\ell}}{(j-i-1)(n - \ell)+ (n - j) +1} \int_0^1 x_i^{i-1} \sum_{r=0}^{\ell} \binom{\ell}{r} (1 - x_i)^{(j-i-1)(n - \ell)+ (n - j) +1 + \ell - r} dx_i \\
&=&2 \sum_{\ell= 0 }^n \sum_{r=0}^{\ell} \frac{\binom{n}{\ell} \binom{\ell}{r} }{(j-i-1)(n - \ell)+ (n - j) +1} \int_0^1 x_i^{i-1} (1 - x_i)^{(j-i-1)(n - \ell)+ (n - j) +1 + \ell - r} dx_i \\
&=& \sum_{\ell= 0 }^n \sum_{r=0}^{\ell} \frac{\binom{n}{\ell} \binom{\ell}{r} B(i, (j-i-1)(n - \ell)+ (n - j) +2 + \ell - r)}{((j-i-1)(n - \ell)+ (n - j) +1)},
\end{eqnarray*}
where $B(\cdot, \cdot)$ is the standard beta function.
What we proved is summarized in the following proposition.
\begin{proposition}
Given two nodes $1 \leq i < j \leq n$ in a random permutation graph with $n$ vertices, a distinct node $k \notin \{i,j\}$ will be connected to both of them with probability
$$1 - \sum_{\ell= 0 }^n \sum_{r=0}^{\ell} \frac{\binom{n}{\ell} \binom{\ell}{r} B(i, (j-i-1)(n - \ell)+ (n - j) +2 + \ell - r)}{((j-i-1)(n - \ell)+ (n - j) +1) }.$$
\end{proposition}
It seems that generating functions will help in such computations, and we plan to approach harder problems by trying so.
\section{Extremal degrees}
For a given random permutation graph $\mathcal{G}_n$, we write $\delta(\mathcal{G}_n)$ and $\Delta (\mathcal{G}_n)$ for the minimal and maximal degrees in $\mathcal{G}_n$. We begin by quoting a result from \cite{BM:2017} related to $\delta (\mathcal{G}_n)$.
\begin{theorem}\label{thm:minasymptote}
\cite{BM:2017} For a sequence of random permutation graphs $\mathcal{G}_n$, we have $$\frac{\delta (\mathcal{G}_n)}{\sqrt{n}} \rightarrow_d \Gamma,$$ as $n \rightarrow \infty$, where $\Gamma$ has a Rayleigh distribution with parameter $1 / \sqrt{2}$, i.e. $\mathbb{P}(\Gamma > \gamma) = e^{- \gamma^2}$ for all $\gamma > 0$.
\end{theorem}
\bigskip
How about $\Delta(\mathcal{G}_n)$? The following symmetry result relating $\Delta(\mathcal{G}_n)$ to $\delta(\mathcal{G}_n)$ will answer this question.
\begin{propn}
For a random permutation graph $\mathcal{G}_n$, we have $$\delta(\mathcal{G}_n) =_d n - 1 - \Delta(\mathcal{G}_n).$$
\end{propn}
\noindent {\bf Proof:}
Let $X_1,X_2,\ldots$ be a sequence of i.i.d. random variables uniformly distributed over $(0,1)$. We have
\begin{eqnarray*}
\Delta(\mathcal{G}_n) &=_d& \max_{k \in [n]} \left( \sum_{j=1}^{k-1} \mathbf{1} (X_j > X_k) + \sum_{j=k+ 1}^{n} \mathbf{1} (X_j< X_k)\right) \\ &=& \max_{k \in [n]} \left( \sum_{j=1}^{k-1}( 1 - \mathbf{1} (X_j < X_k) ) + \sum_{j=k+ 1}^{n} ( 1 - \mathbf{1} (X_j > X_k)) \right) \\
&=& \max_{k \in [n]} \left( n - 1 - \sum_{j=1}^{k-1} \mathbf{1} (X_j < X_k) - \sum_{j=k+ 1}^{n} \mathbf{1} (X_j> X_k)\right) \\
&=_d& n - 1 - \min_{k \in [n]} \left( \sum_{j=1}^{k-1} \mathbf{1} (X_j > X_k) + \sum_{j=k+ 1}^{n} \mathbf{1} (X_j< X_k)\right) \\
&=& n- 1 - \delta(\mathcal{G}_n),
\end{eqnarray*}
as claimed.
\hfill $\Box$
\begin{corollary}
For a sequence of random permutation graphs $\mathcal{G}_n$, we have $$\frac{n - 1 - \Delta (\mathcal{G}_n)}{\sqrt{n}} \rightarrow_d \Gamma,$$ as $n \rightarrow \infty$, where $\Gamma$ has a Rayleigh distribution with parameter $1 / \sqrt{2}$.
\end{corollary}
Focusing on the maximal, or equivalently on the minimal as shown above, degree we now give a sketch of an alternative proof for the distributional convergence to the Rayleigh distribution. The flavour of the below sketch is much more probabilistic than the analytic proof given in \cite{BM:2017} in our perspective, though we do not attempt any rigor here .
Let $\mathcal{G}_n$ be a random permutation graph with the corresponding uniform permutation $\pi_n$. Let $X_1,X_2,\ldots$ be a sequence of i.i.d. random variables each of which is uniformly distributed over $(0,1)$. Then, a little bit of thought gives that
\begin{eqnarray*}
\Delta(\mathcal{G}_n) &=_d& \max_{k \in [n]} \left( \sum_{j=1}^{k-1} \mathbf{1} (X_j > X_k) + \sum_{j=k+ 1}^{n} \mathbf{1} (X_j< X_k)\right) \\
&=_d& \max_{k \in [n]} \sum_{j \neq k}\mathbf{1} (X_j > X_k).
\end{eqnarray*}
But then
$$\frac{\Delta(\mathcal{G}_n) - n}{\sqrt{n}} =_d \max_{k \in [n]} \frac{ \sum_{j \neq k}\mathbf{1} (X_j > X_k) - \frac{n}{2}}{\sqrt{n}} - \frac{\sqrt{n}}{2}.$$ When we look at each of $ \frac{ \sum_{j \neq k}\mathbf{1} (X_j > X_k) - \frac{n}{2}}{\sqrt{n}}$, it can be easily verified that $$ \frac{ \sum_{j \neq k}\mathbf{1} (X_j > X_k) - \frac{n}{2}}{\sqrt{n}} \rightarrow_d \mathcal{Z}$$ as $n \rightarrow \infty$, and it is easy intuitively clear that the dependence among these is very weak. Therefore, for large $n$, $$\max_{k \in [n]} \frac{ \sum_{j \neq k}\mathbf{1} (X_j > X_k) - \frac{n}{2}}{\sqrt{n}}$$ can be "considered" as the maximum of "independent" standard Gaussian random variables which can be computed via standard methods, and a translate of it will eventually will give the Rayleigh distribution with parameter $1 / \sqrt{2}$. This of course is not a proof, but gives a probabilistic flavour of what is going on underneath.
\section{Discussion of a directed version}
The purpose of this section is to give a directed model for random permutation graphs, and to discuss the advantage of having directions for the edges.
\bigskip
\noindent \textbf{Directed Model.} For a given permutation $\pi_n$ in $S_n$, form a corresponding graph by including an outward edge from vertex $i$ to vertex $j$ if and only if $i < j $ and $(i - j) (\pi_n(i) - \pi_n (j)) < 0$ (or, equivalently, include an inward edge between $i$ and $j$ if and only if $i > j $ and $(i - j) (\pi_n(i) - \pi_n (j)) < 0$.
\bigskip
\begin{example}
An example with 5 vertices:
\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.4]{RPGdirectedeg}
\caption{The directed permutation graph corresponding to the permutation $\binom{1 \hspace{0.1in} 2 \hspace{0.1in} 3 \hspace{0.1in} 4 \hspace{0.1in} 5}{5 \hspace{0.1in} 2 \hspace{0.1in} 3 \hspace{0.1in} 1 \hspace{0.1in} 4}$.}
\end{center}
For this particular example, vertex 1 has in degree 4. Vertex 4 has in degree 1 and out degree 2. \end{figure}
\end{example}
Focusing on this directed model, it is easy for us to understand the extremal statistics thanks to the work of \cite{R:2012} on random permutations. Now let us recall a definition from the permutations literature.
\begin{defn}
The \textbf{level of a permutation} $\pi_n \in S_n$ is defined to be $$\max_{1 \leq k \leq n} \left\{ \sum_{1 \leq j \leq k} \mathbf{1}(\pi_n(j) >\pi_n(k) )\right\}.$$ We denote it by $L (\pi_n)$.
\end{defn}
\begin{example}
Let $\pi_5 = (3,4,1,5,2)$. Then it can be easily checked that $L(\pi_5) = 3$.
\end{example}
Now, in our case it can be easily seen that the maximal outer (or, inner) degree of a random permutation graph generated has the same distribution as the level of a permutation chosen uniformly at random. Therefore we may interpret the results of \cite{R:2012} to obtain various properties of the extremal statistics in the directed case.
\begin{theorem} \cite{R:2012}
Let $\mathcal{G}_n^d$ be a directed random permutation graph, and let $\Delta_n$ be the maximal out degree in $\mathcal{G}_n^d$. Then, we have
\begin{itemize}
\item[a.] $\mathbb{E}[\Delta_n] = n - 1 - \sum_{\ell = 1}^{n - 1} \frac{ \ell ! \ell^{n-1} }{n!}$
\item[b.] $Var (\Delta_n) = n (n - 1) - 2\sum_{\ell = 1}^{n - 1} \frac{\ell! \ell^{n - \ell +1} }{n!}- \left( n - 1 - \sum_{\ell = 1}^{n - 1} \left( \frac{\ell ! \ell^{n-1}}{n!} \right) \right)^2$.
\item[c] $\mathbb{P}(|L_n - \mathbb{E}[L_n]| \geq t) \leq 2\exp \left(-\frac{t^2}{4 \log n} \right)$, $t > 0$.
\end{itemize}
\end{theorem}
Moreover, the following asymptotic relations hold:
\begin{theorem} \cite{R:2012}
Let $\mathcal{G}_n^d$ be a directed random permutation graph, and let $\Delta_n$ be the maximal out degree in $\mathcal{G}_n^d$. Then we have
\begin{itemize}
\item[a.] $\mathbb{E}[\Delta_n] \sim n -\sqrt{\frac{\pi n}{2}}$ as $n \rightarrow \infty$.
\item[b.] $Var (\Delta_n) \sim \left( 2 - \frac{\pi}{2}\right) n$ as $n \rightarrow \infty$.
\item[c] $\frac{\Delta_n}{n} \rightarrow_{\mathbb{P}} 1$ as $n \rightarrow \infty$.
\item[d.] $\frac{\Delta_n - n}{\sqrt{n}} \rightarrow_d - \mathcal{R}$, where $\mathcal{R}$ is the Rayleigh distribution whose density function is given by $$f(x) = x e^{-x^2/2}, x > 0.$$
\end{itemize}
\end{theorem}
Since all these follow from the corresponding results of \cite{R:2012}, so we do not go into any more details here.
\begin{rmk}
(i) As before, we may provide a symmetry argument here, and we can give similar results for the minimal degree, say $\delta_n$, in $\mathcal{G}_n^d$.
(ii) It would be interesting to use the advantage of directed versions to understand standard random permutation graphs, again by using certain symmetries. We have not worked on this yet.
(iii) The Mallows distribution $\mu$ on $S_n$ is parametrized by a parameter $\beta$, and is defined by $$\mu(\pi_n) = \frac{\beta^{Inv(\pi_n)}}{Z},$$ where $Inv(\pi_n)$ is as before the number of inversions in $\pi_n$, and $Z$ is a normalizing constant depending on $\beta$ and $n$. This distribution specializes to the uniform distribution when we take $\beta = 1$. The level in Mallows permutations was studied in \cite{R:2012} along with uniform permutations. This allows one to, for example, understand the extremal degree statistics in random permutation graphs formed with Mallows permutations via the discussions above. \cite{BM:2017} also has degree related results in random permutation graphs with the Mallows distribution. It would be interesting to see the intersection of these, and we will do it later on.
\end{rmk}
\section{Number of $m$-cycles}
Let $C_m$ be the number cycles of size at least $m$ in a random permutation graph corresponding to the random permutation $\pi_n$. Three observations:
\begin{enumerate}
\item $C_m$ is equal in distribution to the number of decreasing subsequences in a random permutation of length at least $m$.
\item The distribution of the number of decreasing subsequences in $\pi_n$ is the same as the distribution of the number of increasing subsequences in $\pi_n$;
\item Letting $X_1,X_2,\ldots$ be i.i.d. random variables uniform over the interval $(0,1)$ $$C_m =_d \sum_{1 \leq i_1 < i_2 \leq \cdots < i_m \leq n} \mathbf{1}(X_{i_1}< X_{i_2} < \cdots < X_{i_m}).$$
\end{enumerate}
But we are already familiar with the the last statistic on the right-hand side from our discussion on $m$-cliques, and the exact arguments there show that
\begin{theorem}
For a sequence of random permutation graphs $\mathcal{G}_n$, the number of cycles of length at least $m $ satisfies the central limit theorem: $$\frac{C_m - \binom{n}{m} \frac{1}{m!}}{ \frac{1}{2 ((2 m - 1)!)^2} \left( \binom{4m - 2}{2m -1} - 2 \binom{2m - 1}{m}^2 \right) n^{2m - 1}} \rightarrow_{d} \mathcal{Z}, \quad n \rightarrow \infty.$$
\end{theorem}
\bigskip
Since $\#(\text{of cycles exactly of length } m) = C_{m} - C_{m+1}$, it also possible to infer information about cycles of a given exact length. We do not go into this here.
\bigskip
Let us note that the literature on the number of increasing subsequences of a random permutation is vast. The central limit just given was proven for random permutations as well as random words was proven in \cite{IO:2018}. We will mention a few more pointers here for an analysis of the number of cycles of a random permutation graph. As a first result, if we are not only interested in cycles of given sizes, but in all cycles of a random permutation graph, then the results of Lifschitz and Pittel \cite{LP:1981} are really useful. Letting $C_0 = 1$ by definition, denoting the number of all cycles in $\mathcal{G}_n$ by $C_n^*$, and interpreting results of Lifschitz and Pittel in our setting, we obtain
\begin{equation*}\label{eqn:ECn}\mathbb{E}[C_n^*] =
\sum_{m=0}^n \frac{1}{m!} \binom{n}{m}
\end{equation*} and
\begin{equation*}\label{eqn:VarCn}\mathbb{E}[(C_n^*)^2] = \sum_{m+\ell \leq n} 4^{\ell} ((m+\ell)!)^{-1} \binom{n}{m+\ell} \binom{(m+1)/2+\ell-1}{\ell}.
\end{equation*}
Moreover, in the same paper, they prove certain asymptotic relations which in our case read as
$$\mathbb{E}[C_n^*] \sim (2 \sqrt{\pi e} )^{-1} n^{-1/4}
\exp(2n^{1/2}),$$ and
$$\mathbb{E}[(C_n^*)^2] \sim cn^{-1/4} \exp\left(2 \sqrt{2 + \sqrt{5}}
n^{1/2}\right)$$ as $n \rightarrow \infty$, where $c \approx 0.0106$. It is not hard to see that with these moment asymptotics, a classical central limit theorem does not hold for $C_n^*$ which is slightly disappointing, but of course maybe there is some other distributional convergence we do not know yet.
Another question that could be understood here is the length of the longest cycle in a given random permutation graph $\mathcal{G}_n$. Let us denote this statistic by $L_n = L_n(\mathcal{G}_n)$ By our discussions above, if we let $X_1,X_2,\ldots,$ be an i.i.d. sequence of uniform random variables over $(0,1)$, it should be clear that $L_n $ has the same distribution as the largest $k$ so that there exists some $1 \leq i_1 < i_2 \leq \cdots < i_k \leq n$ that satisfies $X_{i_1}< X_{i_2} < \cdots < X_{i_k}$. But then our problem just reduces to the standard longest increasing subsequence problem which was finally completely solved by Baik, Deift and Johansson \cite{BDJ:1999}. Their results imply that
\begin{theorem} Let $\mathcal{G}_n$ be a sequence of random permutation graphs. Then the length of the longest cycle $L_n$ of $\mathcal{G}_n$ satisfies
$$\frac{L_n - 2 \sqrt{n}}{n^{1/6}}
\longrightarrow_d TW \qquad \text{as} \quad n \rightarrow \infty,
$$ where $TW$ is the Tracy-Widom distribution whose cumulative distribution function is given by $$F(t) =
\exp\left(-\int_t^{\infty} (x-t) u^2(x) dx \right)$$ where $u(x)$ is the solution of the Painlev\'e II equation $$u_{xx}=2u^3+xu \qquad \text{with} \qquad
u(x) \sim -Ai(x)\quad \text{as} \quad x \rightarrow \infty, $$ and $Ai(x)$ is the Airy's function.
\end{theorem}
One can go further from here, obtain concentration inequalities for $C_m$, prove Poisson approximations for the number of large cycles among various possible other results. We leave the cycle related statistics for now, and leave the harder combinatorial problems such as obtaining exact distributions related to cycles to some other future work.
\begin{comment}
\section{Comparisons to the Erdös R\'enyi graphs}
Compare triangles
Number of isolated vertices .
cycle structure
maximal degree
minimal degree
\end{comment}
\section{Extensions/Variations?}
There are two directions that can generalize the proof techniques above, the underlying idea being preserving the independence. After discussing these two, we will conclude the paper with a question which we find important as the answer may extend the results of this paper to much larger graph families.
\subsection{Unfair permutations}
Consider $n$ players where player $i$ picks $i$ independent random numbers $\{X_j^{(i)}\}_{j=1}^i$ each of which is uniform over $(0,1)$. Set $Z_i = \max\{X_j^{(i)}: j =1,\ldots, i\}$, \footnote{From here on, we use the notation $Z_i$ for the maximum of $i$ i.i.d. random numbers without further mention whenever it is clear from the context.} and $R_1,\ldots,R_n$ be the ranks of $Z_1,\ldots,Z_n$, respectively. As an example, let us consider the samples, $\{X_j^{(1)}\}_{j=1}^1 = \{0.75\}$ $\{X_j^{(2)}\}_{j=1}^2 = \{0.15, 0.95\}$, $\{X_j^{(3)}\}_{j=1}^3 = \{0.12, 0.31, 0.72\}$ and $\{X_j^{(4)}\}_{j=1}^4 = \{0.03, 0.27, 0.34, 0.52\}$. Then the corresponding rank sequence is
$R_1 = 3, \quad R_2 = 4, \quad R_3 = 2, \quad \text{and} \quad R_4 = 1$,
Define now a random permutation $\rho_n$ by setting $\rho_n = (R_1,R_2,\ldots,R_n)$, and call it an unfair permutation. Here, $\rho_n$ is unfair in the sense that when $i$ is large, $\rho_n(i)$ favors having larger values, and vice versa. The motivation of \cite{PSW:2011} for introducing unfair permutations is related to the theory of partitions, see the cited work for a relevant discussion. If we continue our example in the previous paragraph, then the resulting unfair permutations corresponding to the given samples turn out to be $\rho_4 = (3,4,2,1)$.
Moving on to why unfair permutations could be useful for a permutation graph model, let us begin with an elementary observation. This will give the reader a feeling how the results in previous sections can be extended to the case of unfair permutations. First, letting $\rho_n$ be an unfair permutation in $S_n$, what is the probability that $\rho_n(i) < \rho_n(j)$ for some given $i\neq j$ ? Clearly, this probability would be merely $1/2$ if $\rho_n$ were a uniformly random permutation. In our case, letting $X_1,\ldots,X_i,Y_1,\ldots,Y_j$ be i.i.d. uniform random variables over $(0,1)$,
\begin{eqnarray*}\label{usefulproperty}
\nonumber \mathbb{P}(\rho_n(i) < \rho_n(j))
&=& \mathbb{P}(\max \{X_1.\ldots,X_i \} < \max \{Y_1,\ldots,Y_j\}) \\
\nonumber &=& \mathbb{P}\left(\bigcup_{\ell=1}^{j} \{\max\{X_1,\ldots,X_i,Y_1,\ldots,Y_j\} = Y_{\ell}\}\right) \\
&=& \sum_{\ell=1}^{j} \mathbb{P}\left(\max\{X_1,\ldots,X_i,Y_1,\ldots,Y_j\} = Y_{\ell} \right) = \sum_{\ell=1}^{j} \frac{1}{i+j} = \frac{j}{i+j}.
\end{eqnarray*}
Here, $ \mathbb{P}\left(\max\{X_1,\ldots,X_i,Y_1,\ldots,Y_j\} = Y_{\ell} \right) = 1/ (i + j)$ since
$X_1,\ldots,X_i,Y_1,\ldots,Y_j$ are i.i.d. random variables. So due to the underlying independence, we are able to understand the distributions of descents and descent related statistics such as the number of inversions, the number of local extrema, the number of increasing sequences, etc. in such permutations.
More importantly, unfair permutations admit a very natural generalization. In the original setting $i$th player chooses $i$ i.i.d. uniform numbers over $(0,1)$ instead of just one and picks the maximum. What if the $i^{th}$ player chooses $\phi_i$ random numbers for some function $\phi$? When $\phi$ is identically equal to $1$ and $\phi$ is the identity function, we recover the standard uniformly random and the unfair permutation cases, respectively. However, there are several other interesting regimes some of which are $\phi(x)$ could be logarithmic, exponential, polynomial, and so on. Note still that whatever the $\phi$ function is, the independence arguments we discussed previously, will remain true. Therefore, this naturally defined random $\phi$-permutations graph model provides a rich family of random graphs, and depending on the $\phi$ sequence, one may obtain various extremal graphs worth studying. Also see the relevant question in next subsection.
\bigskip
\subsection{Riffle shuffles}
A second variation of random permutation graphs can be given by replacing the underlying uniform distribution on $S_n$ with (biased) riffle shuffles. Riffle shuffles is the method most often used to shuffle a deck of cards: first, cut the deck into two piles, and then riffle the
piles together, that is, drop the cards from the bottom of each pile
to form a new pile. The first mathematical models for riffle
shuffles were introduced in \cite{gilbert:1955} and \cite{reeds}, and they were later
further investigated in \cite{AD:1992, fulman:1998}. The mathematics behind riffle shuffles was popularized by Persi Diaconis. See \cite{D:2003} for a survey with relations to various other fields. Now, following
\cite{fulman:1998}, we give two equivalent rigorous descriptions of biased riffle shuffles.
\vspace{0.1in}
\noindent \textbf{Description 1: } Cut the $n$ card deck into $a$
piles by picking pile sizes
according to the $mult(a;\textbf{p})$ distribution, where $\textbf{p}=(p_1,\ldots,p_a)$.
That is, choose $b_1,\ldots,b_a$ with probability
$$\binom{n}{b_1,\ldots,b_a} \prod_{i=1}^a p_i^{b_i} .$$ Then choose
uniformly one of the $\binom{n}{b_1,\ldots,b_a}$ ways of interleaving
the packets, leaving the cards in each pile in their original order.
\vspace{0.1in}
\begin{definition}
The probability distribution on $S_n$ resulting from Description 1
will be called the \emph{riffle shuffle distribution} and will be
denoted by $P_{n,a,\textbf{p}}.$ When
$\textbf{p}=(1/a,1/a,\ldots,1/a)$, the shuffle is said to be
\emph{unbiased} and the resulting probability measure is denoted by
$P_{n,a}.$ Otherwise, the shuffle is said to be \emph{biased.}
\end{definition}
Note that the usual way of shuffling $n$ cards with two hands corresponds to $P_{n,2, \mathbf{p}}$. Before moving on to Description
2, let us give an example via unbiased 2-shuffles. The permutation
$$\rho_{n,2}=\left(
\begin{array}{ccccccc}
1 & 2 & 3 & 4 & 5 & 6 & 7 \\
1 & 2 & 5 & 3 & 6 & 7 & 4 \\
\end{array}
\right)$$
is a possible outcome of the $P_{n,2}$ distribution. Here, the first
four cards form the first pile, the last three form the second one
and these two piles are riffled together. The following alternative description will be important in the
sequel because it introduces the necessary independence required for our arguments.
\vspace{0.1in}
\noindent \textbf{Description 2:} (Inverse $a$-shuffles) The
inverse of a biased $a$-shuffle has the following description. Assign
independent random digits from
$\{1,\ldots,a\}$ to each card with distribution $\textbf{p}=(p_1,\ldots,p_a)$. Then sort according to the digits,
preserving relative order for cards with the same digit.
\vspace{0.1in}
In other words, if $\sigma$ is generated according to Description 2,
then $\sigma^{-1} \sim P_{n,a,\textbf{p}}.$ A proof of the
equivalence of these two descriptions for unbiased shuffles can be found in \cite{AD:1992}. Extension to
the biased case is then a standard work. Here is an example of these two shuffles when $n=7$, $a = 2$, and when the shuffle is uniform - so, we consider $P_{7,2}$. We will to shuffle the deck using inverse shuffles. Let
$\mathbf{X}=(X_1,\ldots,X_n)=(1,1,2,1,2,2,1)$ be a sample from the uniform distribution over
$\{1,2\}^7$. Then, sorting according to digits preserving
relative order for cards with the same digit gives the shuffled arrangement of cards as $(1,2,4,7,3,5,6)$. The resulting permutation after the inverse
shuffle is then
$$\sigma= \left(
\begin{array}{ccccccc}
1 & 2 & 3 & 4 & 5 & 6 & 7 \\
1 & 2 & 4 & 7 & 3 & 5 & 6 \\
\end{array}
\right),
$$
and the resulting sample from $P_{7,2}$ turns out to be
$$\rho_{7,2}:=\sigma^{-1}= \left(
\begin{array}{ccccccc}
1 & 2 & 3 & 4 & 5 & 6 & 7 \\
1 & 2 & 5 & 3 & 6 & 7 & 4 \\
\end{array}
\right).
$$
Letting $\rho_{n,a,\textbf{p}}$ be a random permutation with
distribution $P_{n,a,\textbf{p}}$ that is generated using inverse
shuffles with the random word $\mathbf{X}=(X_1,\ldots,X_n)$,
observe that
\begin{equation}\label{eqn:rif}
\rho_{n,a,\textbf{p}}(i)= \#\{j:X_j < X_i\}+\#\{j\leq i : X_j = X_i\}.
\end{equation}
The proofs of the following proposition and its corollary whose straightforward proofs based on the observation in \eqref{eqn:rif} can be found in \cite{I:2018}. These two show why the riffle shuffles can be useful in providing variations of random permutation graphs.
\begin{lemma}\label{import} Let $\mathbf{X}=(X_1,\ldots,X_n)$ where $X_i$'s are independent with distribution $\textbf{p}=(p_1,\ldots,p_a)$.
Also let $\rho_{n,a,\textbf{p}}$ be the corresponding random permutation having distribution
$P_{n,a,\textbf{p}}.$ Then for $i<k$, $\rho_{n,a,\textbf{p}}(i)>
\rho_{n,a,\textbf{p}}(k)$ if and only if $X_i > X_{k}.$
\end{lemma}
\begin{corollary}\label{inverse}
Consider the setting in Lemma \ref{import} and let $S \subset
\{(i,j) \in [n]\times [n]: i < j \}.$ Then
\begin{equation}\label{eq:wordconnection}
\sum_{(i,j) \in S}
\mathbf{1}(\rho_{n,a,\textbf{p}}(i)> \rho_{n,a,\textbf{p}}(j)) =
\sum_{(i,j) \in S} \mathbf{1} (X_i > X_j).
\end{equation}
\end{corollary}
Therefore, again permutation statistics that are related to descents can be understood via independent random variables, though with random words this time. This observation was previously used in \cite{AI:2017} and was further developed in \cite{HI:2018}. Also, see \cite{H:2017}. However, at this point an extension of random permutation graphs with generalized unfair permutation seems more plausible for two reasons: (1) Unfair permutations strictly generalize uniform permutations, whereas in riffle shuffles the generalization only occurs when the size of the alphabet size tends to infinity, i.e. $a \rightarrow \infty$, (2) Working with continuous random variables is easier when dealing with descent related statistics as in this case no repetitions are present with probability one.
In whichever setting one would like to move on with, one fundamental question for us is the following
\bigskip
\textbf{Question:} Can we prove that either the distributions arising from generalized unfair permutations or biased riffle shuffles dense in the space of all distributions on random permutations. If so, with respect to which metric/distance? If a positive result in this direction can be given, then via limiting processes, we hope to be able to understand random permutation graphs with any arbitrary distribution on the permutations (via certain limiting procedures).
\bigskip
Our next step will be on investigating this question \cite{GIY:2019}, and focusing on various difficult combinatorial problems mentioned throughout the paper.
\bigskip
\bigskip
\bigskip
\noindent \textbf{Acknowledgements:} The second author is supported by the Scientific and Research
Council of Turkey [TUBITAK-117C047]. We would like to thank T{\i}naz Ekim A\c{s}{\i}c{\i} for introducing us random permutation graphs.
| {
"timestamp": "2019-01-23T02:14:51",
"yymm": "1901",
"arxiv_id": "1901.06678",
"language": "en",
"url": "https://arxiv.org/abs/1901.06678",
"abstract": "For a given permutation $\\pi_n$ in $S_n$, a random permutation graph is formed by including an edge between two vertices $i$ and $j$ if and only if $(i - j) (\\pi_n(i) - \\pi_n (j)) < 0$. In this paper, we study various statistics of random permutation graphs. In particular, the degree of a given node, the number of nodes with a given degree, the number of isolated vertices, and the number of cliques are analyzed. Further, explicit formulas for the probabilities of having a given number of connected components and isolated vertices are obtained.",
"subjects": "Combinatorics (math.CO); Probability (math.PR)",
"title": "A study on random permutation graphs",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754447499795,
"lm_q2_score": 0.7185944046238982,
"lm_q1q2_score": 0.7075104055274213
} |
https://arxiv.org/abs/1311.5179 | Sparse PCA via Covariance Thresholding | In sparse principal component analysis we are given noisy observations of a low-rank matrix of dimension $n\times p$ and seek to reconstruct it under additional sparsity assumptions. In particular, we assume here each of the principal components $\mathbf{v}_1,\dots,\mathbf{v}_r$ has at most $s_0$ non-zero entries. We are particularly interested in the high dimensional regime wherein $p$ is comparable to, or even much larger than $n$. In an influential paper, \cite{johnstone2004sparse} introduced a simple algorithm that estimates the support of the principal vectors $\mathbf{v}_1,\dots,\mathbf{v}_r$ by the largest entries in the diagonal of the empirical covariance. This method can be shown to identify the correct support with high probability if $s_0\le K_1\sqrt{n/\log p}$, and to fail with high probability if $s_0\ge K_2 \sqrt{n/\log p}$ for two constants $0<K_1,K_2<\infty$. Despite a considerable amount of work over the last ten years, no practical algorithm exists with provably better support recovery guarantees.Here we analyze a covariance thresholding algorithm that was recently proposed by \cite{KrauthgamerSPCA}. On the basis of numerical simulations (for the rank-one case), these authors conjectured that covariance thresholding correctly recover the support with high probability for $s_0\le K\sqrt{n}$ (assuming $n$ of the same order as $p$). We prove this conjecture, and in fact establish a more general guarantee including higher-rank as well as $n$ much smaller than $p$. Recent lower bounds \cite{berthet2013computational, ma2015sum} suggest that no polynomial time algorithm can do significantly better. The key technical component of our analysis develops new bounds on the norm of kernel random matrices, in regimes that were not considered before. |
\section{Introduction}
In the spiked covariance model proposed by \cite{johnstone2004sparse}, we are given data
${\mathbf x}_1,{\mathbf x}_2,\dots,{\mathbf x}_n$ with ${\mathbf x}_i\in \mathbb{R}^p$ of the form\footnote{Throughout the paper, we follow the convention of denoting
scalars by lowercase, vectors by lowercase boldface, and
matrices by uppercase boldface letters.}:
\begin{align}
{\mathbf x}_i &= \sum_{q=1}^{r}\sqrt{\beta_{q}}\, u_{q,i} \,{\mathbf v} _q+ {\mathbf z}_i\,, \label{eq:model}
\end{align}
Here ${\mathbf v}_1,\dots, {\mathbf v}_r \in\mathbb{R}^p$ is a set of orthonormal vectors, that we want to
estimate, while $u_{q,i} \sim{\sf N}(0, 1)$ and ${\mathbf z}_i \sim{\sf N}(0, {\rm I}_p)$
are independent and identically distributed. The quantity $\beta_q\in
\mathbb{R}_{>0}$ quantifies the signal-to-noise ratio.
We are interested in the high-dimensional limit $n, p\to\infty$ with
$\lim_{n\to\infty}p/n= \alpha\in(0,\infty)$. In the rest of this
introduction we will refer to the rank one case, in order to simplify
the exposition, and drop the subscript $q=\{1,2,\dots,r\}$. Our
results and proofs hold for general bounded rank.
The standard method of principal component analysis
involves computing the
sample covariance matrix ${\mathbf G} = n^{-1}\sum_{i=1}^n{\mathbf x}_i{\mathbf x}_i^{{\sf T}}$ and estimates
${\mathbf v}={\mathbf v}_1$ by its principal eigenvector ${\mathbf v}_{\mbox{\tiny{\sc PC}}}({\mathbf G})$. It is a well-known
fact that, in the high dimensional asymptotic $p/n\to \alpha > 0$, this yields an inconsistent estimate
\cite{johnstone2009consistency}.
Namely $\|{\mathbf v}_{\mbox{\tiny{\sc PC}}}-{\mathbf v}\|_2\not\to 0$ in the high-dimensional asymptotic limit,
unless $\alpha\to 0$ (i.e. $p/n\to 0$).
Even worse, Baik, Ben-Arous and P\'ech\'e \cite{baik2005phase} and Paul \cite{paul2007asymptotics} demonstrate a phase transition
phenomenon: if $\beta< \sqrt{\alpha}$ the estimate is asymptotically
orthogonal to the signal $\<{\mathbf v}_{\mbox{\tiny{\sc PC}}},{\mathbf v}\>\to 0$. On the other hand, for $\beta>\sqrt{\alpha}$,
$\<{\mathbf v}_{\mbox{\tiny{\sc PC}}},{\mathbf v}\>$ remains strictly positive as $n,p\to\infty$.
This phase transition phenomenon has attracted considerable attention
recently within random matrix theory
\cite{feral2007largest,capitaine2009largest,benaych2011eigenvalues,knowles2013isotropic}.
These inconsistency results motivated several efforts to exploit
additional structural information on the signal ${\mathbf v}$.
In two influential papers, Johnstone and Lu
\cite{johnstone2004sparse,johnstone2009consistency} considered the
case of a signal ${\mathbf v}$ that is sparse in a suitable basis, e.g. in the
wavelet domain. Without loss of generality, we will assume here that ${\mathbf v}$
is sparse in the canonical basis ${\mathbf e}_1$, \dots ${\mathbf e}_p$.
In a nutshell, \cite{johnstone2009consistency} proposes the following:
\begin{enumerate}
\item Order the diagonal entries of the Gram matrix
${\mathbf G}_{i(1),i(1)}\ge {\mathbf G}_{i(2),i(2)}\ge\dots\ge{\mathbf G}_{i(p),i(p)}$, and let
$J\equiv \{i(1),i(2),\dots,i(k)\}$ be the set of indices
corresponding to the $k$ largest entries.
\item Set to zero all the entries ${\mathbf G}_{i,j}$ of ${\mathbf G}$ unless $i,j\in
J$, and estimate ${\mathbf v}$ with the principal eigenvector of the
resulting matrix.
\end{enumerate}
Johnstone and Lu formalized the sparsity assumption by requiring that ${\mathbf v}$ belongs to a weak $\ell_q$-ball
with $q\in (0,1)$. Instead, here we consider a
strict sparsity constraint where ${\mathbf v}$
has exactly $k$ non-zero entries, with magnitudes bounded
below by $\theta/\sqrt{k}$ for some constant $\theta >0$.
Amini and Wainwright \cite{amini2009high} studied
the more restricted case when every entry of ${\mathbf v}$ has equal
magnitude of $1/\sqrt{k}$.
Within this model, it was proved that
diagonal thresholding successfully recovers the support of ${\mathbf v}$
provided ${\mathbf v}$ is sparse enough, namely $k\le C\sqrt{n/\log p}$ with
$C= C(\alpha, \beta)$ a constant \cite{amini2009high}. (Throughout the paper
we denote by $C$ constants that can change from point to point.)
This result is a striking improvement over vanilla
PCA. While the latter requires a number of samples scaling as the
number of parameters\footnote{Throughout the introduction, we write
$f(n)\gtrsim g(n)$ as a shorthand of \emph{`$f(n)\ge C\, g(n)$ for a
some constant $C = C(\beta,\alpha)$'.} Further $C$
denotes a constant that may change from point to point.} $n\gtrsim p$, sparse PCA via diagonal thresholding achieves the
same objective with a number of samples scaling as the
number of \emph{non-zero} parameters, $n\gtrsim k^2\log p$.
At the same time, this result is not as strong as might have
been expected. By searching exhaustively over all possible supports
of size $k$ (a method that has complexity of order $p^k$) the correct
support can be identified with high probability as soon as $n\gtrsim
k\log p$. On the other hand, no method can succeed for much smaller
$n$, because of information theoretic obstructions \cite{amini2009high}.
Over the last ten years, a significant effort has been devoted to developing practical
algorithms that outperform diagonal thresholding, see e.g.
\cite{moghaddam2005spectral,zou2006sparse,d2007direct,d2008optimal,witten2009penalized}.
In particular, d'Aspremont et al. \cite{d2007direct} developed a
promising M-estimator based on a semidefinite programming (SDP)
relaxation. Amini and Wainwright \cite{amini2009high} carried out an
analysis of this method and proved that, if \emph{(i)} $k\le C(\beta)\, n/\log
p$, and \emph{(ii)} if the SDP solution has rank one, then the SDP
relaxation provides a
consistent estimator of the support of ${\mathbf v}$.
At first sight, this appears as a satisfactory solution of the
original problem.
No procedure can estimate the support of ${\mathbf v}$ from less than $k\log
p$ samples, and the SDP relaxation succeeds in doing it from --at most-- a constant
factor more samples.
This picture was upset by a recent, remarkable result by Krauthgamer,
Nadler and Vilenchik \cite{KrauthgamerSPCA} who showed that
the rank-one condition assumed by Amini and Wainwright
does not hold for $ \sqrt{n}\lesssim k\lesssim (n/\log p)$. This
result is consistent with recent work of Berthet and Rigollet
\cite{berthet2013computational} demonstrating that sparse PCA cannot be
performed in polynomial time in the regime $k\gtrsim \sqrt{n}$, under
a certain computational complexity conjecture for the so-called
planted clique problem.
In summary, the sparse PCA problem demonstrates a fascinating
interplay between computational and statistical barriers.
\begin{description}
\item[From a statistical perspective,] and disregarding computational
considerations, the support of ${\mathbf v}$ can be estimated consistently
if and only if $k\lesssim n/\log p$. This can be done, for instance,
by exhaustive search over all the $\binom{p}{k}$ possible supports of
${\mathbf v}$. (See \cite{vu2012minimax,cai2013sparse} for a minimax analysis.)
\item[From a computational perspective,] the problem appears to be
much more difficult. There is rigorous evidence
\cite{berthet2013computational, ma2013computational}
that no polynomial
algorithm can reconstruct the support unless $k\lesssim \sqrt{n}$.
On the positive side, a very simple algorithm (Johnstone and Lu's
diagonal thresholding) succeeds for $k\lesssim \sqrt{n/\log p}$.
\end{description}
Of course, several elements are still missing in this emerging
picture. In the present paper we address one of them, providing
an answer to the following question:
\begin{quote}
\emph{Is there a polynomial time algorithm that is guaranteed to solve
the sparse PCA problem with high probability for $\sqrt{n/\log
p}\lesssim k\lesssim \sqrt{n}$?}
\end{quote}
We answer this question positively by analyzing a covariance
thresholding algorithm that proceeds, briefly, as follows.
(A precise, general definition, with some technical changes is given
in the next section.)
\begin{enumerate}
\item Form the empirical covariance matrix ${\mathbf G}$ and set to zero all its entries that
are in modulus smaller than $\tau/\sqrt{n}$, for $\tau$ a suitably
chosen constant.
\item Compute the principal eigenvector $\mathbf{\widehat{v}}_1$ of this thresholded
matrix.
\item Denote by ${\sf B}\subseteq \{1,\dots,p\}$ be the set of indices
corresponding to the $k$ largest entries of $\mathbf{\widehat{v}}_1$.
\item Estimate the support of ${\mathbf v}$ by `cleaning' the set ${\sf B}$.
(Briefly, ${\mathbf v}$ is estimated by thresholding ${\mathbf G}\mathbf{\widehat{v}}_{{\sf B}}$
with $\mathbf{\widehat{v}}_{{\sf B}}$ obtained by zeroing the entries outside ${\sf B}$.)
\end{enumerate}
Such a covariance thresholding approach was proposed in
\cite{KrauthgamerSPCA}, and is in turn related to earlier work by
Bickel and Levina \cite{bickel2008regularized}. The formulation
discussed in the next section presents some technical differences that
have been introduced to simplify the analysis. Notice that, to
simplify proofs, we assume $k$ to be known: This issue is discussed in
the next two sections.
The rest of the paper is organized as follows. In the next section we
provide a detailed description of the algorithm and state our main
results. Our theoretical results assume full knowledge of problem
parameters for ease of proof. In light of this, in Section \ref{sec:practical} we discuss a practical implementation
of the same idea that does not require prior knowledge of problem parameters, and is
entirely data-driven. We also illustrate the method through simulations.
The complete proofs are available in the accompanying supplement, in
Sections \ref{sec:prelim}, \ref{sec:proofmain} and \ref{sec:proofcorr} respectively.
\section{Algorithm and main result}
\begin{algorithm}
\caption{Covariance Thresholding}
\label{alg:ct}
\begin{algorithmic}[1]
\State {\bf Input:} Data $({\mathbf x}_i)_{1\le i\le 2n}$, parameters
$k_q\in {\mathbb N}$, $\tau,\rho\in \mathbb{R}_{\ge 0}$;
\State Compute the empirical covariance matrices ${\mathbf G}\equiv \sum_{i=1}^n{\mathbf x}_i{\mathbf x}_i^{{\sf T}}/n$ , ${\mathbf G}' \equiv \sum_{i=n+1}^n%
{\mathbf x}_i{\mathbf x}_i^{\sf T}/n$;
\State Compute ${\mathbf{\widehat{\Sigma}}} = {\mathbf G} - {\rm I}_p$ (resp. ${\mathbf{\widehat{\Sigma}}}' = {\mathbf G}'-{\rm I}_p$);
\State Compute the matrix $\eta({\mathbf{\widehat{\Sigma}}})$ by soft-thresholding
the entries of ${\mathbf{\widehat{\Sigma}}}$:
\begin{align*}
\eta({\mathbf{\widehat{\Sigma}}})_{ij}
&= \begin{cases}
{\mathbf{\widehat{\Sigma}}}_{ij}-\frac{\tau}{\sqrt{n}} & \mbox{if
${\mathbf{\widehat{\Sigma}}}_{ij}\ge \tau/\sqrt{n}$,}\\
0& \mbox{if $-\tau/\sqrt{n}<{\mathbf{\widehat{\Sigma}}}_{ij}< \tau/\sqrt{n}$,}\\
{\mathbf{\widehat{\Sigma}}}_{ij}+\frac{\tau}{\sqrt{n}} & \mbox{if
${\mathbf{\widehat{\Sigma}}}_{ij}\le -\tau/\sqrt{n}$,}
\end{cases}
\end{align*}
\State Let $(\mathbf{\widehat{v}}_{q})_{q\le r}$ be the first $r$ eigenvectors of $\eta({\mathbf{\widehat{\Sigma}}})$;
\State Define ${\mathbf s}_q\in\mathbb{R}^p$ by $s_{q,i} = \widehat{v}_{q,i}\mathbb{I}(\abs{\widehat{v}_{q, i} \ge \theta/2\sqrt{k_q}})$;
\State {\bf Output:} ${\widehat{\sf Q}} = \{i\in [p]: \;\exists\, q \text{ s.t. } |({\mathbf{\widehat{\Sigma}}}'{\mathbf s}_q)_i|\ge \rho \}$.
\end{algorithmic}
\end{algorithm}
For notational convenience, we shall assume hereafter that $2n$ sample
vectors are given (instead of $n$): $\{{\mathbf x}_i\}_{1\le i\le2n}$. These
are distributed according to the model (\ref{eq:model}).
The number of spikes $r$ will be treated as a
known parameter in the problem.
We will make the following
assumptions:
\begin{enumerate}
\item[{\sf A1}] The number of spikes $r$ and the signal strengths
$\beta_1,\dots,\beta_r$ are fixed as $n,p\to\infty$.
Further $\beta_1>\beta_2>\dots\beta_r$ are all \emph{distinct}.
\item[{\sf A2}] Let ${\sf Q}_q$ and $k_q$ denote the support of ${\mathbf v}_q$ and its size respectively.
We let ${\sf Q} = \cup_q{\sf Q}_q$ and $k = \sum_q k_q$ throughout.
Then the non-zero
entries of the spikes satisfy $|v_{q,i}|\ge
\theta/\sqrt{k_q}$ for all $i\in {\sf Q}_q$ for some $\theta >0$. Further, for any $q, q'$
we assume $\abs{v_{q, i}/v_{q', i}} \le \gamma$ for every $i\in{\sf Q}_q\cap{\sf Q}_{q'}$, for
some constant $\gamma$.
\end{enumerate}
As before, we are interested in
the high-dimensional limit of $n, p\to\infty$ with $p/n \to
\alpha$. A more detailed description of the covariance
thresholding algorithm for the general model (\ref{eq:model}) is given in Table \ref{alg:ct}.
We describe the basic intuition for the simpler rank-one case
(omitting the subscript $q\in\{1,2,\dots,r\}$), while stating results
in generality.
We start by splitting the data into
two halves: $({\mathbf x}_i)_{1\le i\le n}$ and
$({\mathbf x}_{i})_{n< i\le 2n}$ and compute the respective sample covariance matrices ${\mathbf G}$ and
${\mathbf G}'$ respectively. As we will see, the matrix ${\mathbf G}$ is used to obtain a
good estimate for the spike ${\mathbf v}$. This estimate, along with the (independent) second part
${\mathbf G}'$, is then used to construct a consistent estimator for the supports of ${\mathbf v}$.
Let us focus on the first phase of the algorithm, which aims to obtain a good
estimate of ${\mathbf v}$. We first compute ${\mathbf{\widehat{\Sigma}}} = {\mathbf G} - {\rm I}$.
For $\beta>\sqrt{\alpha}$, the
principal eigenvector of ${\mathbf G}$, and hence of ${\mathbf{\widehat{\Sigma}}}$ is positively
correlated with ${\mathbf v}$, i.e. $\lim_{n\to\infty}\<\mathbf{\widehat{v}}_1({\mathbf{\widehat{\Sigma}}}),{\mathbf v}\> >0$.
However, for $\beta<\sqrt{\alpha}$, the noise component in ${\mathbf{\widehat{\Sigma}}}$ dominates
and the two vectors become asymptotically
orthogonal, i.e. for instance $\lim_{n\to\infty}\<\mathbf{\widehat{v}}_1({\mathbf{\widehat{\Sigma}}}),{\mathbf v}\> =0$.
In order to reduce the noise level, we must exploit the sparsity of
the spike ${\mathbf v}$.
Denoting by ${\mathbf X}\in\mathbb{R}^{n\times p}$ the matrix with rows ${\mathbf x}_1$,
\dots ${\mathbf x}_n$, by ${\mathbf Z}\in\mathbb{R}^{n\times p}$ the matrix with rows ${\mathbf z}_1$,
\dots ${\mathbf z}_n$, and letting ${\mathbf u} = (u_1,u_2,\dots,u_n)$, the model
(\ref{eq:model}) can be rewritten as
\begin{align}\label{eq:model2}
{\mathbf X} &= \sqrt{\beta}\, {\mathbf u} \,{\mathbf v}^{{\sf T}} + {\mathbf Z}\, .
\end{align}
Hence, letting $\beta' \equiv \beta\|u\|^2/n\approx\beta$, and ${\mathbf w}
\equiv \sqrt{\beta}{\mathbf Z}^{{\sf T}}{\mathbf u}/n$
\begin{align}
{\mathbf{\widehat{\Sigma}}} &= \beta'\,{\mathbf v}\bv^{{\sf T}} + {\mathbf v}\,{\mathbf w}^{{\sf T}}+{\mathbf w} \, {\mathbf v}^{{\sf T}} + \frac{1}{n}{\mathbf Z}^{{\sf T}}{\mathbf Z}\;\;
- {\rm I}_p, . \label{eq:SigmaDef}
\end{align}
For a moment, let us neglect the cross terms $({\mathbf v}{\mathbf w}^{{\sf T}}+{\mathbf w}
{\mathbf v}^{{\sf T}})$. The `signal' component $\beta'\,{\mathbf v}\bv^{{\sf T}}$ is sparse
with $k^2$ entries of magnitude $\beta/k$, which (in the regime of
interest $k =\sqrt{n}/C$) is equivalent to $C\beta/\sqrt{n}$. The `noise' component
${\mathbf Z}^{{\sf T}}{\mathbf Z}/n -{\rm I}_p$ is dense with entries of order $1/\sqrt{n}$.
Assuming
$k/\sqrt{n}$ a small enough constant, it should be possible to remove
most of the noise by thresholding the entries at level of order
$1/\sqrt{n}$. For technical reasons, we use the soft thresholding function
$\eta:\mathbb{R}\times\mathbb{R}_{\ge 0}\to \mathbb{R}, \, \eta(z; \tau) = {\operatorname{\rm{sgn}}}(z)(\abs{z}-\tau)_+$.
We will omit the second argument from $\eta(\cdot; \cdot)$ wherever it is clear from context. Classical denoising theory \cite{DJ94a,johnstone2013function} provides upper bounds the
estimation error of such a procedure. Note however that these results fall short of our goal.
Classical theory measures estimation error by (element-wise)
$\ell_p$ norm, while here we are interested in the resulting
principal eigenvector. This would require bounding, for instance,
the error in operator norm.
Since the soft thresholding function $\eta(z; \tau/\sqrt{n})$ is affine
when $z \gg \tau/\sqrt{n}$, we would expect that the following decomposition
holds approximately (for instance, in operator norm):
\begin{align} \label{eq:heurDecom}
\eta({\mathbf{\widehat{\Sigma}}}) &\approx \eta\left( \beta'{\mathbf v}\bv^{\sf T} \right) + \eta\left( \frac{1}{n}{\mathbf Z}^{\sf T}{\mathbf Z} -{\rm I}_p\right).
\end{align}
The main technical challenge now is to control the operator norm of
the perturbation $\eta({\mathbf Z}^{\sf T}{\mathbf Z}/n - {\rm I}_p)$.
It is easy to see that $\eta({\mathbf Z}^{\sf T}{\mathbf Z}/n -{\rm I}_p)$ has entries of
variance $\delta(\tau)/n$, for $\delta(\tau)\to 0$ as
$\tau\to\infty$. If entries were independent with mild decay, this would imply --with high probability--
\begin{align}
\norm{\eta\left( \frac{1}{n}{\mathbf Z}^{\sf T}{\mathbf Z}-{\rm I}_p \right)}_2 \lesssim C\delta(\tau),\label{eq:kernRMnorm}
\end{align}
for some constant $C$. Further, the first component
in the decomposition (\ref{eq:heurDecom})
is still approximately rank
one with norm of the order of $\beta'\approx \beta$. Consequently, with standard linear algebra
results on the perturbation of eigenspaces \cite{davis1970sin}, we obtain an error bound $\norm{\mathbf{\widehat{v}} -{\mathbf v}}\lesssim \delta(\tau)/C'\beta$.
Our first theorem formalizes this intuition and provides a bound
on the estimation error in the principal components of $\eta({\mathbf{\widehat{\Sigma}}})$.
\begin{theorem}\label{thm:corr}
Under the spiked covariance model \myeqref{eq:model} satisfying Assumption
{\sf A1}, let $\mathbf{\widehat{v}}_q$ denote the $q^\text{th}$
eigenvector of $\eta({\mathbf{\widehat{\Sigma}}})$
using threshold $\tau$.
For every $\alpha, (\beta_q)_{q=1}^r \in (0, \infty)$, integer $r$ and every ${\varepsilon} >0$
there exist constants, $\tau = \tau({\varepsilon},\alpha, (\beta_q)_{q=1}^r, r, \theta)$ and $ 0 < c_*=c_*({\varepsilon},\alpha, (\beta_q)_{q=1}^r, r, \theta)< \infty$
such that, if $\sum_q k_q = \sum_q|{\rm supp}({\mathbf v}_q)| \le c_*\sqrt{n})$, then
\begin{align}
{\mathbb P}\Big\{\min( \norm{\mathbf{\widehat{v}}_q - {\mathbf v}_q}, \norm{\mathbf{\widehat{v}}_q + {\mathbf v}_q}) \le
{\varepsilon}\;\; \forall q\in \{1,\dots,r\}\Big\}\ge 1-\frac{\alpha}{n^4}\, .
\end{align}
\end{theorem}
It is clear from the discussion above that the proof of Theorem \ref{thm:corr}
requires a formalization of \myeqref{eq:kernRMnorm}. Indeed, the spectral properties
of
random matrices of the type $f({\mathbf Z}^{\sf T}{\mathbf Z}/n - {\rm I}_p)$ , called
inner-product
kernel random matrices, have attracted
recent interest within probability theory \cite{el2010information,el2010spectrum,cheng2012spectrum}.
In this literature, the asymptotic
eigenvalue distribution of a matrix $f({\mathbf Z}^{\sf T}{\mathbf Z}/n-{\rm I}_p)$ is the object of
study. Here $f:\mathbb{R}\to\mathbb{R}$ is a kernel function and is applied entry-wise
to the matrix ${\mathbf Z}^{\sf T}{\mathbf Z}/n-{\rm I}_p$, with ${\mathbf Z}$ a matrix
as above. Unfortunately, these results do not suffice to prove Theorem \ref{thm:corr} for the
following reasons:
\begin{itemize}
\item The results \cite{el2010information,el2010spectrum} are
perturbative and provide conditions under which the spectrum of $f({\mathbf Z}^{\sf T} {\mathbf Z}/n-{\rm I}_p)$
is close to a rescaling of the spectrum of $({\mathbf Z}^{\sf T} {\mathbf Z}/n-{\rm I}_p)$ (with
rescaling factors depending on the Taylor expansion of $f$ close to $0$).
We are interested instead in a non-perturbative regime in which the
spectrum of $f({\mathbf Z}^{\sf T} {\mathbf Z}/n-{\rm I}_p)$ is very different from the one of
$({\mathbf Z}^{\sf T} {\mathbf Z}/n-{\rm I}_p)$ (and the Taylor expansion is trivial).
\item The authors of \cite{cheng2012spectrum} consider $n$-dependent kernels, but their results are
asymptotic and concern the weak limit of the empirical spectral distribution of $f({\mathbf Z}^{\sf T}{\mathbf Z}/n-{\rm I}_p)$.
This does not yield an upper bound on the spectral norm\footnote{Note that \cite{cheng2012spectrum}
also provide a non-asymptotic bound for the spectral norm of $f({\mathbf Z}^{\sf T}{\mathbf Z}/n-{\rm I}_p)$ via the moment method, but
this bound diverges with $n$ and does not give a result of the type
of \myeqref{eq:kernRMnorm}.}
of $f({\mathbf Z}^{\sf T}{\mathbf Z}/n-{\rm I}_p)$.
\end{itemize}
Our approach to prove Theorem \ref{thm:corr} follows instead the
so-called ${\varepsilon}$-net method: we develop
high probability bounds on the maximum Rayleigh quotient:
\begin{align*}
\max_{{\mathbf y}\in S^{p-1}} \<{\mathbf y}, \eta({\mathbf Z}^{\sf T}{\mathbf Z}/n-{\rm I}_p){\mathbf y}\> &= \max_{{\mathbf y}\in S^{p-1}} \sum_{i, j }\eta\left( \frac{\<{\mathbf{\tilde{z}}}_i, {\mathbf{\tilde{z}}}_j\>}{n}; \frac{\tau}{\sqrt{n}} \right)y_i y_j,
\end{align*}
where $S^{p-1} = \{{\mathbf y}\in\mathbb{R}^p:\|{\mathbf y}\|=1\}$ is the unit sphere.
Since $\eta({\mathbf Z}^{\sf T}{\mathbf Z}/n -{\rm I}_p)$ is not Lipschitz continuous in the
underlying Gaussian variables ${\mathbf Z}$, concentration does not follow
immediately from Gaussian isoperimetry. We have to develop more
careful (non-uniform) bounds on the gradient of $\eta({\mathbf Z}^{\sf T}{\mathbf Z}/n-{\rm I}_p)$
and show that they imply concentration as required.
While Theorem \ref{thm:corr} guarantees that $\mathbf{\widehat{v}}$ is a reasonable
estimate of the spike ${\mathbf v}$ in $\ell_2$ distance (up to a sign flip),
it does not provide a consistent estimator of its support. This brings us
to the second phase of our algorithm. Although $\mathbf{\widehat{v}}$ is not even expected
to be sparse, it is easy to see that
the largest entries of $\mathbf{\widehat{v}}$ should have significant overlap
with ${\rm supp}({\mathbf v})$. Steps 6, 7 and 8 of the
algorithm exploit this property to construct a consistent estimator
${\widehat{\sf Q}}$ of the support of the spike ${\mathbf v}$. Our second theorem guarantees
that this estimator is indeed consistent.
\begin{theorem}\label{thm:main}
Consider the spiked covariance model of \myeqref{eq:model} satisfying
Assumptions {\sf A1}, {\sf A2}. For any
$\alpha, (\beta_q)_{q\le r} \in (0, \infty)$, $\theta, \gamma>0$ and integer $r$, there exist constants $c_*, \tau, \rho$ dependent on
$\alpha, (\beta_q)_{q\le r}, \gamma, \theta, r$, such that, if $\sum_q k_q = |{\rm supp}({\mathbf v}_q)|\le c_*\sqrt{n}$, the Covariance
Thresholding algorithm of Table \ref{alg:ct} recovers the union of supports of ${\mathbf v}_q$
with high probability.
Explicitly, there exists a constant $C>0$ such that
\begin{align}
{\mathbb P}\Big\{{\widehat{\sf Q}} =\cup_q{\rm supp}({\mathbf v}_q) \Big\} \ge 1-\frac{C}{n^4}\, .
\end{align}
\end{theorem}
Before passing to the proofs of Theorem \ref{thm:corr} and Theorem
\ref{thm:main} (respectively in Sections \ref{sec:proofcorr} and
\ref{sec:proofmain} of the Supplementary Material), it is useful to pause for a few remarks.
\begin{remark}
We focus on a consistent estimation of the union of the supports
$\cup_q{\rm supp}({\mathbf v}_q)$ of the spikes. In the rank-one case, this
obviously corresponds to the standard support recovery. In the general case,
once the union is correctly estimated,
estimating the individual supports poses no additional difficulty: indeed, since
$|\cup_q{\rm supp}({\mathbf v}_q))|=O(\sqrt{n})$ an extra step with $n$ fresh samples ${\mathbf x}_i$
restricted to ${\widehat{\sf Q}}$ yields consistent estimates for ${\mathbf v}_q$, hence ${\rm supp}({{\mathbf v}_q})$.
\end{remark}
\begin{remark}
Recovering the signed supports ${\sf Q}_{q,+} = \{i\in[p] : v_{q, i} > 0\}$ and
${\sf Q}_{q,-} = \{i\in[p]: v_{q,i} <0\}$ is possible using the same technique
as recovering the supports ${\rm supp}({\mathbf v}_q)$ above, and poses no additional difficulty.
\end{remark}
\begin{remark}
Assumption {\sf A2} requires $|v_{q,i}|\ge
\theta/\sqrt{k_q}$ for all $i\in {\sf Q}_q$.
This is a standard requirement in the support recovery literature \cite{wainwright2009sharp, meinshausen2006high}.
The second part of assumption {\sf A2} guarantees that when the supports of two spikes overlap, their
entries are roughly of the same order. This is necessary for our proof technique
to go through. Avoiding such an assumption altogether remains an open
question.
\end{remark}
Our covariance thresholding algorithm assumes knowledge of the correct
support sizes $k_q$.
Notice that the same assumption is made in earlier theoretical, e.g. in
the analysis of SDP-based reconstruction by Amini and Wainwright \cite{amini2009high}.
While this assumption is not realistic in applications, it helps to
focus our exposition on the most challenging aspects of the problem.
Further, a ballpark estimate of $k_q$ (indeed of $\sum_{q}k_q$) is actually sufficient.
Indeed consider the algorithm obtained by replacing steps 7 and 8 as
following.
\begin{itemize}
\item[{\sc 7:}] Define ${\mathbf s}'_q\in\mathbb{R}^p$ by
\begin{align} \label{eq:Sprime}
s'_{q, i} = \begin{cases}
\widehat{v}_{q, i} & \mbox{ if } |\widehat{v}_{q, i}| > \theta/(2\sqrt{k_{0}})\\
0 & \mbox{ otherwise.}
\end{cases}
\end{align}
\item[{\sc 8:}] Return
\begin{align} \label{eq:Cprime}
{\widehat{\sf Q}} = \cup_q\{i\in [p]: \; |({\mathbf{\widehat{\Sigma}}}'{\mathbf s}'_q)_i|\ge \rho\}\, .\end{align}
\end{itemize}
The next theorem shows that this procedure is effective even if $k_0$
overestimates $\sum_q k_q$ by an order of magnitude. Its proof is deferred to
Section \ref{sec:proofmain}.
\begin{theorem}\label{thm:main3}
Consider the spiked covariance model of \myeqref{eq:model}. For any
$\alpha, \beta \in (0, \infty)$, let constants $c_*, \tau, \rho$
be given as in Theorem \ref{thm:main}. Further assume $k =
\sum_{q}|{\rm supp}({\mathbf v}_q)|\le c_*\sqrt{n}$, and $\sum_{q}k\le k_0\le 20\, \sum_q k_q$.
Then, the Covariance
Thresholding algorithm of Table \ref{alg:ct}, with the definitions
in Eqs.~(\ref{eq:Sprime}) and (\ref{eq:Cprime}),
recovers the union of supports of ${\mathbf v}_q$ successfully, i.e.
\begin{align}
{\mathbb P}\Big({\widehat{\sf Q}} = \cup_q{\rm supp}({\mathbf v}_q)\Big) \ge 1-\frac{C}{n^4}\, .
\end{align}
\end{theorem}
\section{Practical aspects and empirical results}\label{sec:practical}
Specializing to the rank one case, Theorems \ref{thm:corr} and \ref{thm:main} show that
Covariance Thresholding succeeds with high probability for a number of
samples $n\gtrsim k^2$, while Diagonal Thresholding requires $n\gtrsim
k^2\log p$. The reader might wonder whether eliminating the $\log p$ factor
has any practical relevance or is a purely conceptual improvement.
Figure \ref{fig:supportRecovery} presents simulations on synthetic data under the
strictly sparse model, and the Covariance Thresholding algorithm of
Table \ref{alg:ct}, used in the proof of Theorem \ref{thm:main}. The
objective is to check whether the $\log p$ factor has an impact at
moderate $p$. We compare this with Diagonal Thresholding.
\begin{figure}[t]
\includegraphics[width=0.33\linewidth]{supportRecoveryDT.pdf}
\includegraphics[width=0.33\linewidth]{supportRecoveryCT.pdf}
\includegraphics[width=0.33\linewidth]{supportRecoveryCTdata.pdf}
\caption{The support recovery phase transitions for Diagonal Thresholding (left) and
Covariance Thresholding (center) and the data-driven version
of Section \ref{sec:practical} (right). For Covariance Thresholding, the
fraction of support recovered correctly \emph{increases} monotonically with
$p$, as long as $k \le c\sqrt{n}$ with $c\approx 1.1$. Further, it
appears to converge to one throughout this region. For Diagonal
Thresholding, the fraction of support recovered correctly
\emph{decreases} monotonically with $p$ for all $k$ of order
$\sqrt{n}$.
This confirms that Covariance Thresholding (with or without knowledge
of the support size $k$) succeeds with high
probability for $k \le c\sqrt{n}$, while Diagonal Thresholding
requires a significantly sparser principal component.
\label{fig:supportRecovery}}
\end{figure}
We plot the empirical
success probability as a function of $k/\sqrt{n}$ for several values
of $p$, with $p=n$. The empirical success probability was computed by
using $100$ independent instances of the problem.
A few observations are of interest: $(i)$ Covariance
Thresholding appears to have a significantly larger success
probability in the `difficult' regime where Diagonal Thresholding
starts to fail; $(ii)$ The curves for Diagonal
Thresholding appear to decrease monotonically with $p$ indicating that
$k$ proportional to $\sqrt{n}$ is not the right scaling for this
algorithm (as is
known from theory); $(iii)$ In contrast, the curves for Covariance
Thresholding become steeper for larger $p$, and, in particular, the
success probability increases with $p$ for $k\le 1.1\sqrt{n}$. This
indicates a sharp
threshold for $k ={\rm const}\cdot\sqrt{n}$, as suggested by our theory.
In terms of practical applicability, our algorithm in Table \ref{alg:ct}
has the shortcomings of requiring knowledge of problem parameters
$\beta_q, r, k_q$. Furthermore, the thresholds $\rho, \tau$ suggested by theory need
not be optimal.
We next describe a
principled approach to estimating (where possible) the parameters
of interest and running the algorithm in a purely data-dependent manner.
Assume the following model, for $i\in [n]$
\begin{align*}
{\mathbf x}_i &= {\boldsymbol{\mu}} + \sum_q\sqrt{\beta_q}u_{q,i}{\mathbf v}_q + \sigma{\mathbf z}_i,
\end{align*}
where ${\boldsymbol{\mu}}\in\mathbb{R}^p$ is a fixed mean vector, $u_{q, i}$ have mean $0$
and variance $1$, and ${\mathbf z}_i$ have mean $0$ and covariance ${\rm I}_p$.
Note that our focus in this section is not on rigorous analysis, but instead to demonstrate
a principled approach to applying covariance thresholding in practice.
We proceed as follows:
\begin{description}
\item [Estimating ${\boldsymbol{\mu}}$, $\sigma$:]
We let $\widehat{\boldsymbol{\mu}} = \sum_{i=1}^n {\mathbf x}_i/n$ be the empirical mean estimate
for $\mu$. Further letting $\overline{\mathbf{X}}={\mathbf X}-\mathbf{1}\widehat{{\boldsymbol{\mu}}}^{\sf T}$ we
see that $pn-(\sum_q k_q)n \approx pn$ entries of $\overline{\mathbf{X}}$ are mean $0$ and variance $\sigma^2$.
We let $\widehat{\sigma} = {{\rm MAD}(\overline{\mathbf{X}})}/{\nu}$
where ${\rm MAD}(\,\cdot\,)$ denotes the median absolute
deviation of the entries of the matrix in the argument, and $\nu$
is a constant scale factor. Guided by the
Gaussian case, we take $\nu = \Phi^{-1}(3/4) \approx 0.6745$.
\item[Choosing $\tau$:]
Although in the statement of the theorem, our choice of $\tau$ depends on the SNR $\beta/\sigma^2$,
we believe this is an artifact of our analysis. Indeed it is reasonable to
threshold `at the noise level', as follows. The noise component of
entry $i,j$ of the sample covariance (ignoring lower order
terms) is given by $\sigma^2\<{\mathbf z}_i, {\mathbf z}_j\>/n$. By the central limit theorem,
$\<{\mathbf z}_i, {\mathbf z}_j\>/\sqrt{n} {\,\stackrel{\mathrm{d}}{\Rightarrow} \,} {\sf N}(0, 1)$. Consequently, $\sigma^2\<{\mathbf z}_i, {\mathbf z}_j\>/n \approx
{\sf N}(0, \sigma^4/n)$, and we need to choose the (rescaled) threshold proportional
to $\sqrt{\sigma^4} = \sigma^2$. Using previous estimates, we let
$\tau = \nu'\cdot \widehat{\sigma}^2$
for a constant $\nu'$. In simulations, a choice $3\lesssim \nu' \lesssim 4$ appears to work well.
\item[Estimating $r$:]
We define ${\mathbf{\widehat{\Sigma}}} = \overline{\mathbf{X}}^{\sf T}\overline{\mathbf{X}}/n-\sigma^2{\rm I}_p$ and soft threshold it
to get $\eta({\mathbf{\widehat{\Sigma}}})$ using $\tau$ as above.
Our proof of Theorem \ref{thm:corr} relies on the fact
that $\eta({\mathbf{\widehat{\Sigma}}})$ has $r$ eigenvalues that are separated from the
bulk of the spectrum. Hence, we estimate $r$ using $\widehat{r}$: the number of
eigenvalues separated from the bulk in $\eta({\mathbf{\widehat{\Sigma}}})$. The edge of the
spectrum can be computed numerically using the Stieltjes transform method as
in \cite{cheng2012spectrum}.
\item[Estimating ${\mathbf v}_q$:]
Let $\mathbf{\widehat{v}}_q$ denote the $q^{\text{th}}$
eigenvector of $\eta({\mathbf{\widehat{\Sigma}}})$. Our theoretical analysis indicates that $\mathbf{\widehat{v}}_q$ is
expected to be close to ${\mathbf v}_q$. In order to denoise $\mathbf{\widehat{v}}_q$, we assume
$\mathbf{\widehat{v}}_q\approx (1-\delta){\mathbf v}_q + {\boldsymbol{\eps}}_q$, where ${\boldsymbol{\eps}}_q$ is additive random noise.
We then threshold ${\mathbf v}_q$ `at the
noise level' to recover a better estimate of ${\mathbf v}_q$. To do this, we
estimate the standard deviation of the ``noise'' ${\boldsymbol{\eps}}$ by
$\widehat{\sigma_{{\boldsymbol{\eps}}}}
= {{\rm MAD}({\mathbf v}_q)}/{\nu}$. Here we set --again guided by the Gaussian heuristic-- $\nu \approx 0.6745$. Since ${\mathbf v}_q$ is sparse,
this procedure returns a good estimate for the size of the noise
deviation. We let $\eta_{H}(\mathbf{\widehat{v}}_q)$ denote the vector obtained by hard
thresholding $\mathbf{\widehat{v}}_q$: set $(\eta_H(\mathbf{\widehat{v}}_q))_i = \mathbf{\widehat{v}}_{q,i} \text{ if }
\abs{\widehat{v}_{q,i}} \ge \nu' \widehat{\sigma}_{{\boldsymbol{\eps}}_q}$
and $
0 \text{ otherwise.}$
We then let $\mathbf{\widehat{v}}^*_q = \eta(\mathbf{\widehat{v}}_q)/\norm{\eta(\mathbf{\widehat{v}}_q)}$ and return $\mathbf{\widehat{v}}^*_q$
as our estimate for ${\mathbf v}_q$.
\end{description}
Note that --while different in several respects-- this empirical approach
shares the same philosophy of the algorithm in Table
\ref{alg:ct}.
On the other hand, the data-driven algorithm presented in this section is less
straightforward to analyze, a task that we defer to future work.
Figure \ref{fig:supportRecovery} also shows results of a support recovery
experiment using the `data-driven' version of this section. Covariance thresholding
in this form also appears to work for supports of size $k \le \text{const}\sqrt{n}$.
Figure \ref{fig:threePeak} shows the performance of vanilla PCA, Diagonal Thresholding
and Covariance Thresholding on the ``Three Peak'' example of Johnstone and Lu \cite{johnstone2004sparse}.
This signal is sparse in the wavelet domain and the simulations employ the data-driven
version of covariance thresholding. A similar experiment with the ``box'' example of Johnstone
and Lu is provided in the supplement. These experiments demonstrate that, while for large values of $n$ both Diagonal
Thresholding and Covariance Thresholding perform well, the latter
appears superior for smaller values of $n$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.45]{threePeak.pdf}
\caption{The results of Simple PCA, Diagonal Thresholding and Covariance Thresholding (respectively) for
the ``Three Peak'' example of Johnstone and Lu \cite{johnstone2009consistency} (see Figure 1
of the paper). The signal is sparse in the `Symmlet 8' basis. We use $\beta = 1.4, p=4096$, and the rows correspond to sample sizes $n=1024, 1625, 2580, 4096$
respectively. Parameters for Covariance Thresholding are chosen as in Section \ref{sec:practical},
with $\nu' = 4.5$. Parameters for Diagonal Thresholding are from \cite{johnstone2009consistency}. On each curve,
we superpose the clean signal (dotted). \label{fig:threePeak}}
\end{figure}
\section{Proof preliminaries}\label{sec:prelim}
In this section we review some notation and preliminary facts
that we will use throughout the paper.
\subsection{Notation}
We let $[m] = \{1,2,\dots,m\}$ denote the set of first $m$ integers.
We will represent vectors using boldface lower case letters,
e.g. ${\mathbf u}, {\mathbf v}, {\mathbf x}$. The entries of a vector ${\mathbf u}\in\mathbb{R}^n$ will be
represented by $u_i, i\in[n]$.
Matrices are represented using boldface upper
case letters e.g. ${\mathbf A}, {\mathbf X}$. The entries of a matrix ${\mathbf A} \in\mathbb{R}^{m\times n}$ are
represented by ${\mathbf A}_{ij}$ for $i\in[m], j\in[n]$.
Given a matrix ${\mathbf A}\in\mathbb{R}^{m\times n}$, we generically let ${\mathbf a}_1$,
${\mathbf a}_2, \dots, {\mathbf a}_m$ denote its rows, and ${\mathbf{\tilde{a}}}_1$,
${\mathbf{\tilde{a}}}_2, \dots, {\mathbf{\tilde{a}}}_n$ its columns.
For $E\subseteq [m]\times [n]$, we define the projector operator
${\cal P}_E:\mathbb{R}^{m\times n}\to \mathbb{R}^{m\times n}$ by letting
${\cal P}_E({\mathbf A})$ be the matrix with entries
\begin{align}
{\cal P}_{E}({\mathbf A})_{ij} = \begin{cases}
{\mathbf A}_{ij} & \mbox{if $(i,j)\in E$,}\\
0 & \mbox{otherwise.}
\end{cases}
\end{align}
If $E = E_1\times E_2$, we write ${\cal P}_{E_1, E_2}$ for
${\cal P}_{E_1\times E_2}$. In the case $E=E_1\times E_2$ we
also define a projection operator ${\widetilde{\cal P}}_{E_1, E_2}:\mathbb{R}^{m\times n}\to
\mathbb{R}^{|E_1|\times|E_2|}$ that returns the $E_1\times E_2$ submatrix.
If $m=n$, and $E$ is the diagonal,
we write ${\mathcal{P}_{\sf d}}$ for ${\cal P}_E$. If instead $E$ is the complement of
the diagonal, we write ${\mathcal{P}_{\sf nd}}$.
For a matrix ${\mathbf A}\in\mathbb{R}^{m\times n}$, and a set $E\subseteq[n]$, we
define its column restriction ${\mathbf A}_{E}\in\mathbb{R}^{m\times n}$ to be the
matrix obtained by setting to $0$ columns outside $E$:
\begin{align*}
({\mathbf A}_{E})_{ij} &= \begin{cases}
{\mathbf A}_{ij} &\text{ if }j\in E,\\
0 &\text{otherwise. }
\end{cases}
\end{align*}
Similarly ${\mathbf y}_E$ is obtained from ${\mathbf y}$ by setting to zero all
indices outside $E$.
The operator norm of a matrix ${\mathbf A}$ is denoted by $\norm{{\mathbf A}}$ (or
$\norm{{\mathbf A}}_2$)
and its Frobenius norm by $\norm{{\mathbf A}}_F$. We write $\norm{{\mathbf x}}$ for the standard $\ell_2$
norm of a vector ${\mathbf x}$.
We let ${\sf Q}_q$ denotes the support of the $q^{\text{th}}$
spike ${\mathbf v}_q$. Also, we denote the union of the supports of ${\mathbf v}_q$ by ${\sf Q}=\cup_q{\sf Q}_q$.
The complement of a set $E\in[n]$ is denoted by $E^c$.
We write $\eta(\cdot; \cdot)$ for the soft-thresholding function.
By $\partial\eta(\cdot ; \tau)$ we denote the derivative of $\eta(\cdot; \tau)$ with respect
to the \emph{first} argument, which exists Lebesgue almost everywhere.
In the statements of our results, consider the limit of large $p$ and large
$n$ with $p/n\to\alpha$. This limit will be referred to either as ``$n$ large enough''
or ``$p$ large enough'' where the phrase ``large enough'' indicates dependence of $p$ (and
thereby $n$) on specific problem parameters.
\subsection{Preliminary facts}
Let $S^{n-1}$ denote the unit sphere in $n$ dimensions, i.e. $S^{n-1} = \{{\mathbf x} : \norm{{\mathbf x}} = 1\}$.
We use the following definition (see \cite{Vershynin-CS}) of
the ${\varepsilon}$-net of a set $X\subseteq\mathbb{R}^n$:
\begin{definition}[Nets, Covering numbers]\label{def:nets}
A subset $T^{\varepsilon}(X)\subseteq X$ is called an ${\varepsilon}$-net of $X$ if every point in
$X$ may be approximated by one in $T^{\varepsilon}(X)$ with error at most ${\varepsilon}$. More precisely:
\begin{align*}
\forall x\in X,\quad \inf_{y\in T^{\varepsilon}(X)} \norm{x - y} &\le {\varepsilon}.
\end{align*}
The minimum cardinality of an ${\varepsilon}$-net of $X$, if finite, is called its covering number.
\end{definition}
The following two facts are useful while using ${\varepsilon}$-nets to bound the
spectral norm of a matrix. For proofs, we refer the reader to \cite{Vershynin-CS}.
\begin{lemma}
Let $S^{n-1}$ be the unit sphere in $n$ dimensions. Then there exists an ${\varepsilon}$-net
of $S^{n-1}$, $T^{\varepsilon}(S^{n-1})$ satisfying:
\begin{align*}
|T^{\varepsilon}(S^{n-1})| \le \left( 1+ \frac{2}{{\varepsilon}} \right)^n.
\end{align*}
\label{lem:epsnetcard}
\end{lemma}
\begin{lemma}
Let ${\mathbf A}\in\mathbb{R}^{n\times n}$ be a symmetric matrix. Then:
\begin{align*}
\norm{{\mathbf A}}_2 = \sup_{{\mathbf x}\in S^{n-1}}|\<{\mathbf x}, {\mathbf A}{\mathbf x}\>| \le (1-2{\varepsilon})^{-1}\sup_{{\mathbf x}\in T^{\varepsilon}(S^{n-1})} |\<{\mathbf x}, {\mathbf A}{\mathbf x}\>|.
\end{align*}
In particular, if ${\mathbf A}$ is a random matrix, then for $\Delta>0$ we have:
\begin{align*}
{\mathbb P}\left\{ \norm{{\mathbf A}}_2 \ge \Delta \right\} &\le \left( 1+\frac{2}{{\varepsilon}} \right)^n \sup_{{\mathbf x}\in T^{\varepsilon}(S^{n-1}) }{\mathbb P}\left\{ \abs{\<{\mathbf x}, {\mathbf A}{\mathbf x}\>} \ge \Delta(1-2{\varepsilon}) \right\}.
\end{align*}
\label{lem:specnormbnd}
\end{lemma}
Throughout the paper we will denote by $T^{\varepsilon}_n$ the \emph{minimum cardinality} ${\varepsilon}$-net on the unit sphere
$S^{n-1}$, which naturally satisfies Lemma \ref{lem:epsnetcard}. Further, for a non-zero vector ${\mathbf y}\in\mathbb{R}$, we define
the set $S^{n-1}_{\mathbf y} = \{{\mathbf x} : \<{\mathbf x}, {\mathbf y}\>=0, \norm{{\mathbf x}}=1\}$ and let its minimum cardinality ${\varepsilon}$-net
be denoted by $T^{\varepsilon}_n({\mathbf y})$. Since $S^{n-1}_{\mathbf y}$ is isometric to $S^{n-2}$, Lemma \ref{lem:epsnetcard}
holds for $T^{\varepsilon}_{n}({\mathbf y})$ as well.
We now state some measure concentration results that we will use
at various points in the proofs of Theorems \ref{thm:corr} and \ref{thm:main}.
\begin{lemma}
Consider ${\mathbf z}\sim{\sf N}(0, {\rm I}_N)$ be a vector of $N$ i.i.d. standard normal
random variables on a probability space $(\Omega,{\cal F}, {\mathbb P})$. Suppose
$F:\mathbb{R}^N\to\mathbb{R}$ is a $\mathbb{R}$-valued, continuous, a.e. differentiable function
and $G\in{\mathcal{B}}_{\mathbb{R}^N}$ is a closed convex set satisfying:
\begin{align*}
\norm{{\nabla} F({\mathbf z})}\mathbb{I}({\mathbf z}\in G)&\le L \quad {\mathbb P}\emph{-a.e.} \\
{\mathbb P}\left\{ G \right\} &\ge 1-q.
\end{align*}
Then, there exists a function $F_L:\mathbb{R}^N\to \mathbb{R}$ such that $F_L$ is
$L$-Lipschitz throughout and $F_L$ coincides with $F$ on the set $G$. Further
for each $\Delta>0$ we have that:
\begin{align*}
{\mathbb P}\left\{ |F({\mathbf z}) - \E F({\mathbf z})| \ge \Delta \right\} &\le q + 2\exp\left( -\frac{\widetilde{\Delta}^2}{2L^2} \right),
\end{align*}
where $\widetilde{\Delta} = \Delta - |\E F({\mathbf z}) - \E F_L({\mathbf z})|$.
\label{lem:basicConc}
\end{lemma}
\begin{proof}
For any ${\mathbf y}, {\mathbf y}' \in G$ we have that:
\begin{align*}
F({\mathbf y}') &= F({\mathbf y}) + \int_{0}^{1} \<{\nabla} F(t {\mathbf y}' + (1-t){\mathbf y}), {\mathbf y}' - {\mathbf y}\>
\mathrm{d} t .
\end{align*}
From this we obtain that $|F({\mathbf y}') - F({\mathbf y})| \le L\norm{{\mathbf y}' - {\mathbf y}}$ using the bound
on ${\nabla}{F}$ in $G$ and the convexity of $G$. By Kirszbraun's theorem, there
exists an $L$-Lipschitz extension $F_L$ of $F$ to $\mathbb{R}^N$. Indeed we may
take $F_L({\mathbf y}) = \inf_{{\mathbf y}'\in G} F({\mathbf y}) + L\norm{{\mathbf y} - {\mathbf y}'}$. Then:
\begin{align*}
{\mathbb P}\left\{ |F({\mathbf z}) - \E F({\mathbf z})| \ge \Delta \right\} &= %
{\mathbb P}\left\{ |F({\mathbf z}) - \E F({\mathbf z})| \ge \Delta ; {\mathbf z} \in G \right\} + %
{\mathbb P}\left\{ |F({\mathbf z}) - \E F({\mathbf z})| \ge \Delta ; {\mathbf z}\in G^c \right\} \\
&\le {\mathbb P}\{ |F_L({\mathbf z}) -\E F_L({\mathbf z})| \ge \widetilde{\Delta}\} + {\mathbb P}\{G^c\}
\end{align*}
Applying Gaussian concentration of measure \cite{Ledoux} to $F_L$ finishes the proof.
\end{proof}
For further reference, we define the following:
\begin{definition}
For a function $F:\mathbb{R}^N\to\mathbb{R}$, a constant $L>0$ and a measurable set $G$, we call
$F_L(\cdot)$ the \emph{$G, L$-Lipschitz extension} of $F(\cdot)$. It is given by:
\begin{align*}
F_{L}\left({\mathbf y} \right) &= \inf_{{\mathbf y}' \in G} \left( F({\mathbf y}') + L\norm {{\mathbf y} - {\mathbf y}'}\right).
\end{align*}
\end{definition}
\begin{lemma}
Let ${\mathbf A}\in\mathbb{R}^{M\times N}$ be a matrix with i.i.d. standard normal
entries, i.e. ${\mathbf A}_{ij}\sim{\sf N}(0, 1)$. Then, for every $t\ge 0$:
\begin{align*}
{\mathbb P}\left\{ \norm{{\mathbf A}}_2 \ge \sqrt{M} + \sqrt{N} + t \right\} &\le \exp\left( -\frac{t^2}{2} \right).
\end{align*}
\label{lem:gaussianmatnorm}
\end{lemma}
The proof of this result can be found in \cite{Vershynin-CS}.
\section{Proof of Theorems \ref{thm:main} and \ref{thm:main3}}\label{sec:proofmain}
In this section we prove Theorem \ref{thm:main}
and Theorem \ref{thm:main3}, assuming that Theorem
\ref{thm:corr} holds. The proof of the latter can be found in Section
\ref{sec:proofcorr}.
\subsection{Proof of Theorem \ref{thm:main} }
For any fixed ${\varepsilon}>0$, and assume $\sum_q k_q \le \sqrt{n\log\tau/\tau^3}$,
where $\tau = \tau({\varepsilon}, {\underline{\beta}}, \alpha)$ as per Theorem \ref{thm:corr}.
Then we have for every $q$, $\norm{\mathbf{\widehat{v}}_q - {\mathbf v}_q} \le {\varepsilon}$
with probability at least $1-C/n^4$ for some constant $C>0$.
Throughout the proof, we will work on this favorable event of Theorem
\ref{thm:corr}, namely use
\begin{align}
{\mathbb P}\Big({\widehat{\sf Q}} \neq \cup_q{\rm supp}({\mathbf v}_q)\Big) \le
{\mathbb P}\Big({\widehat{\sf Q}} \neq \cup_q{\rm supp}({\mathbf v}_q) ;\;\; \norm{\mathbf{\widehat{v}}_q - {\mathbf v}_q}^2 \le {\varepsilon}^2\Big) +\frac{C}{n^4}
\, ,
\end{align}
hence focusing on bounding the first term on the right hand side.
It is convenient to isolate the following lemma.
\begin{lemma}\label{lem:AboutS}
Assume $\|\mathbf{\widehat{v}}_q-{\mathbf v}_q\|^2\le {\varepsilon}^2$ and that $\abs{v_{q, i}} \ge \theta/\sqrt{k_q}$.
Let ${\sf B}_q \equiv {\rm supp}({\mathbf s}_q)$ with ${\mathbf s}_q$
defined as per Algorithm \ref{alg:ct}, step 7. Then $|{\sf B}_q\triangle{\sf Q}_q|
\le 4{\varepsilon}^2 k_q/\theta^2$ and hence $|{\sf B}_q\cap{\sf Q}_q|\ge (1-4{\varepsilon}^2/\theta^2)k_q$.
(Here $\triangle$ denotes
the symmetric set-difference.) Further $\min(\norm{{\mathbf s}_q - {\mathbf v}_q}^2, \norm{{\mathbf s}_q+{\mathbf v}_q}^2) \le 5{\varepsilon}^2$.
\end{lemma}
\begin{proof}
Recall that $s_{q, i} = \widehat{v}_{q, i}\mathbb{I}(\abs{\widehat{v}_{q,i}}\ge \theta/2\sqrt{k_q})$.
Since $\abs{v_{q, i}} \ge \theta/\sqrt{k_q}$:
\begin{align*}
{\sf B}_q \triangle {\sf Q}_q &\subseteq \left\{i: \abs{v_{q, i} - \widehat{v}_{q, i}} \ge \frac{\theta}{2\sqrt{k_q}}\right\}.
\end{align*}
Thus $\abs{{\sf B}_q\triangle{\sf Q}_q} \le 4k_q\norm{\mathbf{\widehat{v}}_q -{\mathbf v}_q}^2/\theta^2 \le 4{\varepsilon}^2k_q/\theta^2$.
Now we bound the error $\norm{{\mathbf s}_q - {\mathbf v}_q}$, assuming that
$\norm{\mathbf{\widehat{v}}_q - {\mathbf v}_q} \le {\varepsilon}$. The other case is handled in
an analogous fashion:
\begin{align*}
\norm{{\mathbf s}_q - {\mathbf v}_q}^2 &= \sum_{i\in{\sf Q}_q} (\widehat{v}_{q, i}\mathbb{I}(|\widehat{v}_{q, i}|\ge\theta/2\sqrt{k_q}) - v_{q, i})^2 %
+ \sum_{i\in{\sf Q}_q^c} (\widehat{v}_{q, i} )^2\mathbb{I}(\abs{\widehat{v}_{q, i}}\ge \theta/2\sqrt{k_q})\\
&=\sum_{i\in{\sf Q}_q}v_{q, i}^2 \mathbb{I}(\abs{\widehat{v}_{q, i}} \le \theta/2\sqrt{k_q})
+ \sum_{i\in{\sf Q}_q} (\widehat{v}_{q, i} - v_{q, i})^2 \mathbb{I}(\abs{\widehat{v}_{q, i}} \ge \theta/2\sqrt{k_q}) %
+ \sum_{i\in{\sf Q}_q^c} (\widehat{v}_{q, i} )^2\mathbb{I}(\abs{\widehat{v}_{q, i}}\ge \theta/2\sqrt{k_q}) \\
&\le \sum_{i\in{\sf Q}_q} v_{q, i}^2 \mathbb{I}(\abs{\widehat{v}_{q, i} -v_{q, i}} \ge |{v_{q, i}| - \theta/(2\sqrt{k_q})}) + \norm{\mathbf{\widehat{v}}_q - {\mathbf v}_q}^2 \\
&\le \sum_{i\in{\sf Q}_q} \frac{v_{q, i}^2}{(\abs{v_{q, i}} - \theta/2\sqrt{k_q})^2} (\widehat{v}_{q, i} - v_{q, i})^2 + \norm{\mathbf{\widehat{v}}_q -{\mathbf v}_q}^2 \\
&\le 5\norm{\mathbf{\widehat{v}}_q -{\mathbf v}_q}^2 \le 5{\varepsilon}^2.
\end{align*}
The first inequality above follows from triangle inequality as
$\abs{\widehat{v}_{q, i}} \ge \abs{v_{q, i}} - \abs{\widehat{v}_{q, i} - v_{q, i}}$.
The second inequality employs $\mathbb{I}(z\ge z') \le (z/z')^2$. The final
inequality uses the fact that $\abs{v_{q, i}} \ge \theta/2\sqrt{k_q}$
implies $\abs{v_{q, i}}/(\abs{v_{q, i}} - \theta/2\sqrt{k_q}) \le 2$.
\end{proof}
Now we are in position to prove the main theorem. Without loss of
generality, we will
assume that $\<\mathbf{\widehat{v}}_q, {\mathbf v}_q\> >0$ for every $q$. The other case is treated in the same way.
Recall that ${\mathbf{\widehat{\Sigma}}}'$ was formed from the samples $({\mathbf x}_i)_{n< i\le 2n}$, which
are independent of $\mathbf{\widehat{v}}_q$ and hence ${\sf B}_q$. We let ${\mathbf X}'\in\mathbb{R}^{n\times p}$
denote the matrix with rows $({\mathbf x}_i)_{n<i\le 2n}$ we have, in the same fashion
as \myeqref{eq:model2}, ${\mathbf X}' = \sum_q\sqrt{\beta_q}{\mathbf u}^{\prime}_q({\mathbf v}_q)^{\sf T} + {\mathbf Z}'$. We let ${\mathbf{\tilde{z}}}'_i , 1\le i\le p$
denote the columns of ${\mathbf Z}'$.
For any $i$:
\begin{align*}
({\mathbf{\widehat{\Sigma}}}'{\mathbf s}^1)_i &= \frac{\beta_1\norm{{\mathbf u}^{\prime}_1}^2\<{\mathbf v}_1, {\mathbf s}_1\>v_{1,i}}{n} %
+ \sum_{q\ne 1} \frac{\beta_q\norm{{\mathbf u}^{\prime}_q}_2^2\<{\mathbf v}_q, {\mathbf s}_q\>v_{q,i}}{n} %
+ \sum_{q\ge 1} \frac{\sqrt{\beta_q}}{n} (\<{\mathbf Z}^{\prime {\sf T}}{\mathbf u}^{\prime}_q, {\mathbf s}_1\> v_{q,i} + \<{\mathbf v}_q, {\mathbf s}_1\>({\mathbf Z}^{\prime {\sf T}}{\mathbf u}_q)_i) \\%
& + \sum_{q'> q} \frac{\sqrt{\beta_q\beta_{q'}}}{n}\<{\mathbf u}^{\prime}_q, {\mathbf u}^{\prime}_{q'}\> (v_{q, i} \<{\mathbf v}_{q'}, {\mathbf s}_1\> + v_{q',i} \<{\mathbf v}_q, {\mathbf s}_1\>) %
+\frac{1}{n} \sum_{j\in{\sf B}^1, j\ne i} \<{\mathbf{\tilde{z}}}^{\prime}_j, {\mathbf{\tilde{z}}}^{\prime}_i\>s_{1,j} %
+ \bigg( \frac{\lVert{\mathbf{\tilde{z}}}^{\prime}_i\rVert^2}{n} - 1 \bigg)s_{1, i}
\end{align*}
Let $T_1, T_2 \dots T_5$ denote the terms above.
Firstly, by a standard calculation $n/2\le \norm{{\mathbf u}^{\prime}_q}_2^2 \le 2n$
and $\max_{q\ne q'} |\<{\mathbf u}^{\prime}_q, {\mathbf u}^{\prime}_q\>| \le \sqrt{Cn\log n}$ with probability
at least $1-rn^{-10}$ for some constant $C$. Further, using Lemma \ref{lem:AboutS}
and Cauchy-Schwarz we have that $\<{\mathbf v}_1, {\mathbf s}_1\> \ge (1 - 5{\varepsilon}^2)$ and $|\<{\mathbf v}_q, {\mathbf s}_1\>| \le \norm{{\mathbf v}_1 - {\mathbf s}_1} \le 3{\varepsilon}$.
This
implies that:
\begin{align*}
\abs{T_1} &\ge \frac{\beta_1(1-5{\varepsilon}^2)\abs{v_{1,i}}}{2},\\
\abs{T_2} &\le 6{\varepsilon}\sum_{q>1}\beta_q\abs{v_{q, i}}, \\
\abs{T_4} &\le C((\beta_q)_{q\le r})\sqrt{\frac{\log n}{n}}.\\
\end{align*}
Now consider the term
$T_5 = \sum_{j\in{\sf B}_1\backslash i }\<{\mathbf{\tilde{z}}}'_i, {\mathbf{\tilde{z}}}'_j\>s_{1, j}/n = \<{\mathbf{\tilde{z}}}'_i,\sum_{j\in{\sf B}_1\backslash i}s_{1, j}{\mathbf{\tilde{z}}}'_j\>/n$.
Thus, $T_5 {\,\stackrel{\mathrm{d}}{=} \,} Y_{ij}\equiv \<{\mathbf{\tilde{z}}}_i', {\mathbf{\tilde{z}}}_j'\norm{{\mathbf s}_1}\>/n$ for $j\ne i$. Conditional on ${\mathbf{\tilde{z}}}'_j$,
$Y_{ij}\sim{\sf N}(0, \lVert{{\mathbf{\tilde{z}}}'_j}\rVert^2\norm{{\mathbf s}_1}^2/n^2)$. Using the Chernoff bound ,
$\norm{{\mathbf{\tilde{z}}}'_i}^2\le 2n$ with probability at leat $1-\exp(-n/8)$ and, conditional on this event,
$\abs{Y_{ij}} \le \sqrt{C'\log n/n}$ with probability at least $1 -n^{-10}$ for some absolute constant $C'$. It
follows from the union bound that $\abs{T_5} \le \sqrt{C'\log n/n}$ with probability at least $1-2n^{-10}$
for $n$ large enough. Using a similar calculation $\abs{T_3} \le \sqrt{C'( (\beta_q)q )\log n/n}$ with
probability exceeding $1-n^{-10}$. Finally using Proposition \ref{prop:diag} below, we have that
\begin{align*}
\abs{T_5} &\le \norm{{\mathbf s}_1} \max_i \bigg(\frac{\norm{{\mathbf{\tilde{z}}}_i}^2}{n} - 1\bigg) \\
&\le \sqrt\frac{C''\log n}{n},
\end{align*}
with probability at least $1-n^{-10}$. Here we used the fact that $\norm{{\mathbf s}_1} \le \norm{\mathbf{\widehat{v}}_1} = 1$.
By Assumption {\sf A2}, and the above estimates, we have with probability
at least $1-n^{-9}$:
\begin{align*}
\text{For }i\in{\sf Q}_1, \quad\abs{({\mathbf{\widehat{\Sigma}}}{\mathbf s}_1)_i} &\ge \frac{\beta_1}{2} (1-5{\varepsilon}^2 - 12{\varepsilon}\gamma \sum_q \beta_q)\abs{v_{1, i}} - \sqrt{C\log n/n} \\
&\ge \frac{\beta_1(1-5{\varepsilon}^2 - 12{\varepsilon}\gamma \sum_q \beta_q)\theta}{4\sqrt{k_1}} - \sqrt{\frac{C\log n}{n}}, \\
\text{For }i\in [p]\backslash(\cup_q{\sf Q}_q), \quad \abs{({\mathbf{\widehat{\Sigma}}}{\mathbf s}_1)_i} &\le \sqrt\frac{C\log n}{n}.
\end{align*}
Choosing ${\varepsilon} = {\varepsilon}((\beta_q)_{q\le r}, r, \theta, \gamma)$
small enough and using threshold $\rho = \min_q(\beta_q\theta/4\sqrt{k_q})$ we
have that ${\sf Q}_1\subseteq {\widehat{\sf Q}}$ and ${\widehat{\sf Q}}\subseteq\cup_q{{\sf Q}_q}$.
The analogous guarantees for all $1\le q\le r$ imply Theorem \ref{thm:main}.
\subsection{Proof of Theorem \ref{thm:main3}}
Analogously to the previous proof, we fix ${\varepsilon}>0$, and observe that
$\sum_q{k_q} \le \sqrt{n\log\tau/\tau^3}$,
where $\tau = \tau({\varepsilon}, {\underline{\beta}}, \alpha, \theta)$, and per
Theorem \ref{thm:corr}. Then we have that $\norm{\mathbf{\widehat{v}}_q - \mathbf{\widehat{v}}}^2 \le {\varepsilon}/20$
with probability at least $1-C/n^4$ for some constant $C>0$.
We then use
\begin{align}
{\mathbb P}\Big({\widehat{\sf Q}} \neq \cup_q{\rm supp}({\mathbf v}_q)\Big) \le
{\mathbb P}\Big({\widehat{\sf Q}} \neq \cup_q{\rm supp}({\mathbf v}_q) ;\;\; \norm{\mathbf{\widehat{v}}_q - {\mathbf v}}^2 \le \frac{{\varepsilon}}{20r}\Big) +\frac{C}{n^4}
\, ,
\end{align}
and bound the first term.
The key change with respect to the proof of theorems \ref{thm:main}
is that we need to replace Lemma \ref{lem:AboutS} with the following lemma,
whose proof follows exactly the same argument as that of Lemma \ref{lem:AboutS}.
\begin{lemma}
Assume $\|{\mathbf v}_q-\mathbf{\widehat{v}}_q\|^2\le {\varepsilon}/20$, and let ${\sf B}'\equiv {\rm supp}({\mathbf s}')$ with ${\mathbf s}'$
defined as per Eq. (\ref{eq:Sprime}). Further assume $k\le k_0\le 20\,k$. Then $\norm{{\mathbf s}_q -{\mathbf v}_q}^2 \le 5{\varepsilon}^2$.
\end{lemma}
The rest of the proof of Theorem \ref{thm:main3} is identical to the
one of Theorem \ref{thm:main} in the previous section.
\section{Proof of Theorem \ref{thm:corr}}\label{sec:proofcorr}
Since ${\mathbf{\widehat{\Sigma}}} = {\mathbf X}^{{\sf T}}{\mathbf X}/n - {\rm I}_p$, we have: %
\begin{align}\label{eq:empcov}
{\mathbf{\widehat{\Sigma}}} &= \sum_{q=1}^r \left\{\frac{\beta_q\norm{{\mathbf u}_q}^2}{n}{\mathbf v}_q({\mathbf v}_q)^{\sf T} + \frac{\sqrt{\beta_q}}{n}({\mathbf v}_q({\mathbf u}_q)^{\sf T}{\mathbf Z} + {\mathbf Z}^{\sf T}{\mathbf u}_q{\mathbf v}^{\sf T})\right\} \nonumber\\
&\quad+ \sum_{q\ne q'} \left\{\frac{\sqrt{\beta_q\beta_{q'}}\<{\mathbf u}_q,
{\mathbf u}_{q'}\>}{n}{\mathbf v}_q({\mathbf v}_{q'})^{\sf T} \right\} + \frac{{\mathbf Z}^{\sf T}{\mathbf Z}}{n} - {\rm I}_p.
\end{align}
We let ${\sf D} =\{(i, i) : i\in[p]\backslash\cup_q{\sf Q}_q\}$ be the diagonal
entries not included in any support and ${\sf Q}=\cup_q{\sf Q}_q$ denote the
union of the supports. Further let ${\sf E} = \cup_{q}({\sf Q}_q\times{\sf Q}_q)$, ${\sf F} = ({\sf Q}^c\times{\sf Q}^c)\backslash{\sf D}$,
and ${\sf G} = [p]\times[p]\backslash({\sf D}\cup{\sf E}\cup{\sf F})$. Since these are disjoint we have:
\begin{align}%
\eta({\mathbf{\widehat{\Sigma}}}) &= \underbrace{{\cal P}_{{\sf E}}\left\{ \eta({\mathbf{\widehat{\Sigma}}}) \right\}}_{{\mathbf S}} %
+\underbrace{{\cal P}_{{\sf F}}\left\{ \eta\left( \frac{1}{n}{\mathbf Z}^{\sf T} {\mathbf Z} \right) \right\}}_{{\mathbf N}} %
+\underbrace{{\cal P}_{{\sf G}}\left\{ \eta({\mathbf{\widehat{\Sigma}}}) \right\}}_{{\mathbf R}_1}%
+\underbrace{{\cal P}_{{\sf D}}\left\{ \eta( {\mathbf{\widehat{\Sigma}}}) \right\}}_{{\mathbf R}_2}. \label{eq:decomp}
\end{align}
The first term corresponds to the `signal' component while
the last three terms correspond to the `noise' component.
Theorem
\ref{thm:corr} is a direct consequence of the next four propositions.
The first of these proves that the signal component is preserved, while
the others demonstrate that the noise components are small.
\begin{proposition}\label{prop:signal}
Let ${\mathbf S}$ denote the first term in
\myeqref{eq:decomp}:
\begin{align}
{\mathbf S} &= {\cal P}_{\sf E}\left\{\eta({\mathbf{\widehat{\Sigma}}})\right\}.
\end{align}
Then with probability at least $1-3\exp(-n^{2/3}/4)$:
\begin{align*}
\norm{ {\mathbf S} - \sum_{q=1}^r \beta_q {\mathbf v}_q({\mathbf v}_q)^{\sf T} }_2 &\le \frac{\tau\sum_q{k_q}}{\sqrt{n}}+ \kappa_n.
\end{align*}
Here $\kappa_n = 16(\sqrt{r\alpha}+ r\sqrt{\beta_1})n^{-1/6}$.
\end{proposition}
\begin{proposition} \label{prop:noise}
Let ${\mathbf N}$ denote the second term
of \myeqref{eq:decomp}:
\begin{align*}
{\mathbf N} &= {\cal P}_{{\sf F}}\left\{\eta\left( \frac{1}{n}{\mathbf Z}^{\sf T} {\mathbf Z} \right)\right\}.
\end{align*}
Then there exists $\tau_1=\tau_1(\alpha)$ such that
for any $\tau \ge\tau_1$ and all $p$ large enough, we have
\begin{align}
\norm{{\mathbf N}}_2 \le C_1(\alpha)\sqrt{\frac{\log \tau}{\tau}}\, ,
\end{align}
with probability at least $1-2\exp(-c_1(\tau) p)$. The constants
can be taken as $\tau_1 = 100\max(1, \alpha^2\log\alpha)$, $c_1(\tau) =
1/4\tau$ and $C_1(\alpha) = 5000\max(1, \alpha^{3/2})$.
\end{proposition}
\begin{proposition}\label{prop:cross}
Let ${\mathbf R}_1$ denote the matrix corresponding to the third term
of \myeqref{eq:decomp}:
\begin{align*}
{\mathbf R}_1 &= {\cal P}_{{\sf G}}\left\{\eta({\mathbf{\widehat{\Sigma}}})\right\}.
\end{align*}
Then there exists $\tau_2 = \tau_2(\alpha, \beta_1, r)$ such that for $\tau\ge \tau_2$
and every $p$ large enough we have:
\begin{align}
\norm{{\mathbf R}_1}_2 &\le C_2(\alpha, r, \beta_1)\sqrt\frac{\log \tau}{{\tau}} .
\end{align}
with probability at least $1-\exp(-c_2(\tau) p)$. Here we may take
$c_2(\tau) = c_1(\tau)=1/4\tau$.
\end{proposition}
\begin{proposition}\label{prop:diag}
Let ${\mathbf R}_2$ denote the matrix corresponding to the third term
of \myeqref{eq:decomp}:
\begin{align*}
{\mathbf R}_2 &= {\cal P}_{{\sf D}}\left\{\eta({\mathbf{\widehat{\Sigma}}})\right\}.
\end{align*}
Then with probability at least $1-\alpha n^{-C/6 +1}$ for every $n$ large enough:
\begin{align}
\norm{{\mathbf R}_2}_2 &\le \sqrt{\frac{C\log n}{n}}.
\end{align}
\end{proposition}
We defer the proofs of Propositions \ref{prop:signal},
\ref{prop:noise}, \ref{prop:cross} and \ref {prop:diag} to
Sections \ref{subsec:proofsignal}, \ref{subsec:proofnoise},
\ref{subsec:proofcross} and \ref{subsec:proofdiag}
respectively.
\begin{proof}[Proof of Theorem \ref{thm:corr}]
Using these results we now proceed to prove Theorem \ref{thm:corr}.
We will assume that the events in these proposition hold, and control the probability of their complement
via the union bound.
Denote by $k$ the sum of the support sizes, i.e. $\sum_q k_q$.
From Propositions \ref{prop:signal}, \ref{prop:noise}, \ref{prop:cross}, \ref{prop:diag}
and the triangle inequality we have:
\begin{align*}
\norm{\eta({\mathbf{\widehat{\Sigma}}}) - \sum_q \beta_q{\mathbf v}_q({\mathbf v}_q)^{\sf T}} &\le \frac{k\tau}{\sqrt{n}} + \max(C_1, C_2)\sqrt\frac{\log\tau}{\tau},
\end{align*}
for every $\tau \ge \max(\tau_1, \tau_2)$ with probability at least $1 - \alpha n^{-4}$.
Setting $k \le \sqrt{n \log\tau /\tau^3}$, the right hand side above is bounded by $\delta(\tau) = 2\max(C_1, C_2)\sqrt{\log\tau/\tau}$.
Further define ${\underline{\beta}} \equiv \min_{q \ne q' \le r} (\beta_q, \abs{\beta_q - \beta_{q'}})$.
Employing the Davis-Kahan $\sin\theta$ theorem
\cite{davis1970sin} we have:
\begin{align*}
\min(\norm{\mathbf{\widehat{v}}_q - {\mathbf v}_q}, \norm{\mathbf{\widehat{v}}_q +{\mathbf v}_q}) &\le \sqrt{2}\sin\theta(\mathbf{\widehat{v}}_q,{\mathbf v}_q) \\
&\le \frac{\sqrt{2}\delta(\tau)}{{\underline{\beta}} - \delta(\tau)}.
\end{align*}
Choosing $\tau \ge (8\max(C_1, C_2)/{\underline{\beta}}{\varepsilon})^4$ yields that $\delta(\tau)/({\underline{\beta}}-\delta(\tau)) \le {\varepsilon}$.
Letting $\tau$ be the largest of $\tau_1$, $\tau_2$ and $(8\max(C_1, C_2)/{\underline{\beta}}{\varepsilon})^4)$ gives the desired result.
\end{proof}
\subsection{Proof of Proposition \ref{prop:signal}}\label{subsec:proofsignal}
The proof proceeds in two steps. In the first lemma we bound $\norm{\E\{{\mathbf S}\} - \sum_{q}\beta_q {\mathbf v}_q({\mathbf v}_q)^{\sf T}}$
and in the second we control $\norm{{\mathbf S} - \E\{{\mathbf S}\}}$.
\begin{lemma}
\label{lem:exptrq2}
Consider ${\mathbf S}$ as defined in Proposition \ref{prop:signal}. Then
\begin{align*}
\norm{\E\{{\mathbf S}\} - \sum_{q}\beta_q{\mathbf v}_q({\mathbf v}_q)^{\sf T}} &\le \frac{\tau\sum_{q}k_q}{\sqrt{n}}.
\end{align*}
\end{lemma}
\begin{proof}
Notice that $\E\{{\mathbf S}\}$ is supported on a set of indices
$\cup_q{\sf Q}_q\times\cup_q{\sf Q}_q$ which has
size at most $(\sum_{q}k_q)^2$. Hence
\begin{align*}
\norm{\E\{{\mathbf S}\} - \sum_q {\beta_q} {\mathbf v}_q({\mathbf v}_q)^{\sf T}} &\le (\sum_q k_q)\norm{\E\{{\mathbf S}\} - \sum_{q}\beta_q{\mathbf v}_q({\mathbf v}_q)^{\sf T}}_\infty,
\end{align*}
where the last term denotes the entrywise $\ell_\infty$ norm of the matrix. Since ${\mathbf S}$ and
$\sum_{q}\beta_q{\mathbf v}_q({\mathbf v}_q)^{\sf T}$ have common support and since $|\eta(z;
\tau/\sqrt{n}) -z|\le \tau/\sqrt{n}$
we obtain that:
\begin{align*}
\norm{\E\{{\mathbf S}\} - \sum_{q}\beta_q{\mathbf v}_q({\mathbf v}_q)^{\sf T} }_\infty &\le\norm{\E\{{\cal P}_{{\sf E}}(\eta({\mathbf{\widehat{\Sigma}}}))\} - \sum_q \beta_q {\mathbf v}_q {\mathbf v}_q^{\sf T}}_{\infty}\\
&\le \frac{\tau}{\sqrt{n}}.
\end{align*}
The thesis then follows directly.
\end{proof}
\begin{lemma}
\label{lem:sigconc}
Let ${\mathbf S}$ be as defined in Proposition \ref{prop:signal}. Then:
\begin{align*}
\norm{{\mathbf S} - \E\{{\mathbf S}\}} &\le \kappa_n,
\end{align*}
with probability at least $1 - \exp(-n^{2/3}/4)$ where we define $\kappa_n \equiv 16(\sqrt{r\alpha} + r\sqrt{\beta_1})n^{-1/6}$.
\end{lemma}
Proposition \ref{prop:signal} follows directly from these two lemmas
since we have by triangle inequality:
\begin{align*}
\norm{{\mathbf S} -\sum_{q}\beta_q{\mathbf v}_q({\mathbf v}_q)^{\sf T}} &\le \norm{{\mathbf S} - \E\{{\mathbf S}\}} + \norm{\E\{{\mathbf S}\} - \sum_q\beta_q{\mathbf v}_q({\mathbf v}_q)^{\sf T}}.
\end{align*}
This completes the proof of Proposition \ref{prop:signal} conditional
on Lemma \ref{lem:sigconc}. In the next subsection we prove Lemma
\ref{lem:sigconc}.
\subsubsection{Proof of Lemma \ref{lem:sigconc}}
Let ${\mathbf y} \in\mathbb{R}^p$ denote a vector supported on
$\cup_q{\sf Q}_q$. Recall that
${\sf Q} = \cup_{q}{\sf Q}_q$. Fix an $\ell \in{\sf Q}$. The gradient of the Rayleigh quotient
${\nabla}_{{\mathbf{\tilde{z}}}_\ell}\<{\mathbf y}, {\mathbf S}{\mathbf y}\>$ reads:
\begin{align*}
{\nabla}_{{\mathbf{\tilde{z}}}_\ell}\<{\mathbf y}, {\mathbf S}{\mathbf y}\> &= \frac{1}{n}\sum_{i:(i, \ell)\in
\cup_q{\sf Q}_q\times{\sf Q}_q}2\partial\eta\left({\mathbf{\widehat{\Sigma}}}_{i\ell} ; \frac{\tau}{\sqrt{n}}\right)%
({\mathbf{\tilde{z}}}_i + \sum_q\sqrt{\beta_q}v^q_i {\mathbf u}_q)y_iy_\ell.
\end{align*}
Define the vector ${\boldsymbol{\sigma}}^{\ell}({\mathbf y})\in\mathbb{R}^p$ as follows:
\begin{align*}
\sigma^{\ell}_i({\mathbf y}) &= \begin{cases}
\partial\eta\left( {\mathbf{\widehat{\Sigma}}}_{i\ell}; \frac{\tau}{\sqrt{n}} \right)y_i, &\text{
if } (i, \ell)\in\cup_q({\sf Q}_q\times{\sf Q}_q) \\
0 \text{ otherwise.}
\end{cases}
\end{align*}
where the left hand side denotes the $i^{\text{th}}$ entry of ${\boldsymbol{\sigma}}^{\ell}({\mathbf y})$.
Recall that ${\mathbf Z}_E$ is the matrix obtained from ${\mathbf Z}$ by setting to zero all columns with indices
outside $E\subseteq[p]$. Using this, we can now rewrite the
gradient in the following form:
\begin{align*}
{\nabla}_{{\mathbf{\tilde{z}}}_\ell}\<{\mathbf y}, {\mathbf S}^q{\mathbf y}\> &= \frac{2y_\ell}{n} ({\mathbf Z}_{{\sf Q}} + \sum_q\sqrt{\beta_q}{\mathbf u}_q({\mathbf v}_q)^{\sf T}){\boldsymbol{\sigma}}^\ell({\mathbf y}). %
\end{align*}
Since $\partial\eta(\cdot;\cdot)\in\{0, 1\}$,
we see that $\norm{{\boldsymbol{\sigma}}^\ell({\mathbf y})}\le \norm{{\mathbf y}} =1$.
Consequently, we have that:
\begin{align*}
\norm{{\nabla}_{{\mathbf{\tilde{z}}}_\ell}\<{\mathbf y}, {\mathbf S}^q{\mathbf y}\>} &\le \frac{\abs{2y_\ell}}{n} \norm{{\mathbf Z}_{{\sf Q}} + %
\sum_q\sqrt{\beta_q}{\mathbf u}_q({\mathbf v}_q)^{\sf T}} \norm{{\boldsymbol{\sigma}}^\ell({\mathbf y})} \\
&\le \frac{2\abs{y_\ell}}{n} \left(\norm{{\mathbf Z}_{{\sf Q}}} + \sum_q\sqrt{\beta_q}\norm{{\mathbf u}_q({\mathbf v}_q)^{\sf T}}\right) \\
&= \frac{2\abs{y_\ell}}{n} \left(\norm{{\mathbf Z}_{{\sf Q}}} + \sum_q\sqrt{\beta_q}\norm{{\mathbf u}_q}\right),
\end{align*}
Squaring and summing over $\ell$:
\begin{align*}
\norm{{\nabla}_{{\mathbf Z}_{\sf Q}}\<{\mathbf y}, {\mathbf S}{\mathbf y}\>}^2 &\le \frac{4}{n^2} (\norm{{\mathbf Z}_{{\sf Q}}} + \sum_{q}\beta_q\norm{{\mathbf u}_q})^2.
\end{align*}
The gradient above is with respect to all the variables ${\mathbf{\tilde{z}}}_\ell, \ell\in{\sf Q}_q$ and
the norm is the standard vector $\ell_2$ norm.
Let $G: \{{\mathbf Z}, ({\mathbf u}_q)_{q\le r} : \norm{{\mathbf Z}_{{\sf Q}}} \le
(2+\sqrt{r\alpha})\sqrt{n}, \norm{{\mathbf u}_q} \le 4\sqrt{n}\}$. Clearly $G$ is a
closed, convex set. Further, using Lemma \ref{lem:gaussianmatnorm} we can bound
the probability of $G^c$: $\norm{{\mathbf Z}_{{\sf Q}}} \le (\sqrt{n} + \sqrt{\sum_qk_q} + \sqrt{n}) \le (2+\sqrt{r\alpha})\sqrt{n}$
with probability at least $1-\exp(-n/2)$. Also, with probability at least $1-r\exp(-n/2)$, for every $q$ $\norm{{\mathbf u}_q} \le 4\sqrt{n}$.
Thus, on the set $G$ we have:
\begin{align*}
\norm{{\nabla}\<{\mathbf y}, {\mathbf S}{\mathbf y}\>}^2\mathbb{I}\{({\mathbf Z}, {\mathbf u}_1 \cdots {\mathbf u}_r)\in G\} &\le \frac{64}{n}(2 +\sqrt{r\alpha} + \sqrt{\beta})^2\\
{\mathbb P}\{G^c\} &\le 2\exp\left( -\frac{n}{4} \right).
\end{align*}
Define $L$ and $\kappa_n$ as follows:
\begin{align*}
L &\equiv \frac{8(2+\sqrt{r\alpha}+r\sqrt{\beta_1})}{\sqrt{n}}\\
\kappa_n &\equiv 16(2+\sqrt{r\alpha}+r\sqrt{\beta_1})n^{-1/6} = 2Ln^{1/3}.
\end{align*}
Also let $F_L({\mathbf Z}_{{\sf Q}})$ denote the $G, L$-Lipschitz extension of $\<{\mathbf y}, {\mathbf S}{\mathbf y}\>$.
We prove the following remark in Appendix \ref{sec:App}:
\begin{remark} \label{rem:exptsignal}
For every $n$ large enough, $|\E\left\{ \<{\mathbf y}, {\mathbf S}{\mathbf y}\> - F_L({\mathbf Z}_{{\sf Q}}) \right\}| \le n^{-1}$.
\end{remark}
Now employing Lemma \ref{lem:basicConc}:
\begin{align*}
{\mathbb P}\left\{ |\<{\mathbf y}, {\mathbf S}{\mathbf y}\> - \E\<{\mathbf y}, {\mathbf S}{\mathbf y}\>| \ge \kappa_n/2 \right\} &\le 2\exp\left( -\frac{n^{2/3}}{2} \right) + 2r\exp\left( -\frac{n}{4} \right) \\
&\le 3\exp\left( -\frac{n^{2/3}}{2} \right),
\end{align*}
for every $n$ large enough. Then using ${\mathbf y}$ as a vector in the $1/4$-net $T^{1/4}_{|{\sf Q}|}$ embedded
in $\mathbb{R}^p$ via the union of supports ${\sf Q}$, we use Lemma \ref{lem:specnormbnd} to obtain that:
\begin{align*}
\norm{{\mathbf S} - \E\{{\mathbf S}\}} \le \kappa_n,
\end{align*}
with probability at least $1 - 3\cdot 9^{\abs{{\sf Q}}} \exp (-n^{2/3}/2 ) \ge 1- \exp(-n^{2/3}/4)$
since $\abs{{\sf Q}} \le \sum_q{k_q} = O(\sqrt{n}) \le n^{2/3}/2$ for large enough
$n$.
\subsection{Proof of Proposition \ref{prop:noise}}\label{subsec:proofnoise}
It suffices to bound the norm of ${\mathbf{\widetilde{N}}}$ defined as
\begin{align*}
{\mathbf{\widetilde{N}}} &= {\mathcal{P}_{\sf nd}}\left\{\eta\left( \frac{1}{n}{\mathbf Z}^{\sf T}{\mathbf Z} \right)\right\}.
\end{align*}
We use a variant of the ${\varepsilon}$-net argument.
For a set of indices $E\subseteq [p]$, recall that
${\mathbf y}_E\in\mathbb{R}^p$ denotes the vector coinciding
with ${\mathbf y}$ on $E$, and zero outside $E$. By decomposing
the
Rayleigh quotient:
\begin{align*}
{\mathbb P}\left\{ \norm{{\mathbf{\widetilde{N}}}}_2 \ge \Delta \right\}
&\le {\mathbb P}\left\{\sup_{{\mathbf y}\in T^{\varepsilon}_p} \<{\mathbf y}, {\mathbf{\widetilde{N}}}{\mathbf y}\> \ge \Delta(1-2{\varepsilon})\right\} \\
&\le {\mathbb P}\left\{ \sup_{{\mathbf y}\in T^{\varepsilon}_p} \<{\mathbf y}_E, {\mathbf{\widetilde{N}}}{\mathbf y}_E\> + %
\<{\mathbf y}_{E^c}, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\> + 2\<{\mathbf y}_{E}, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\> \ge \Delta(1-2{\varepsilon}) \right\}.
\end{align*}
We let $E = \{i\in [p] : |y_i| > \sqrt{A/p}\}$ for the
constant $A = A(\tau) = \tau\log\tau$. Since $\norm{{\mathbf y}} =1$, it follows that
$|E| \le p/A$. The following lemma allows to bound the term
$\<{\mathbf y}_E, {\mathbf{\widetilde{N}}}, {\mathbf y}_E\>$ uniformly over all subsets $E$ smaller
than $p/A$.
\begin{lemma}
Fix $A\ge180\max(\sqrt{\alpha}, 1)$. Then, for every $p$ large enough,
the following holds with probability at least $1 - \exp(-p\log A/4A)$:
\begin{align*}
\sup_{E\subseteq[p], |E|\le p/A}\norm{ {\widetilde{\cal P}}_{E, E}({\mathbf{\widetilde{N}}})}_2 &\le 32\sqrt{\alpha\frac{\log A}{A}}.
\end{align*}
\label{lem:smallset}
\end{lemma}
The proof of this lemma is provided in subsection \ref{sec:proofsmallset}.
Denoting by ${\cal E}$ the favorable event
of Lemma \ref{lem:smallset}, we obtain:
\begin{align*}
{\mathbb P}\left\{\norm{{\mathbf{\widetilde{N}}}}_2 \ge \Delta \right\} &\le {\mathbb P}\left\{ {\cal E}^c \right\} + %
{\mathbb P}\left\{ \sup_{{\mathbf y}\in T^{\varepsilon}_p}\left( \<{\mathbf y}_E, {\mathbf{\widetilde{N}}}{\mathbf y}_E\> + \<{\mathbf y}_{E^c}, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\> + 2\<{\mathbf y}_{E}, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\>\right) \ge \Delta(1-2{\varepsilon}) , {\cal E} \right\}\nonumber\\
&\le {\mathbb P}\left\{ {\cal E}^c \right\} + {\mathbb P}\left\{\sup_{{\mathbf y}\in T^{\varepsilon}_p} \left(\<{\mathbf y}_{E^c}, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\> + 2\<{\mathbf y}_{E}, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\> \right)\ge \widetilde{\Delta} \right\},
\end{align*}
where $\widetilde{\Delta} = \Delta(1-2{\varepsilon}) - 16\sqrt{2\alpha\log A/A}$. Further, using
the union bound and Lemma \ref{lem:epsnetcard}:
\begin{align}
{\mathbb P}\left\{ \norm{{\mathbf{\widetilde{N}}}}_2 \ge \Delta \right\} &\le {\mathbb P}\left\{ {\cal E}^c \right\} + %
\abs{T^{\varepsilon}_p}\sup_{{\mathbf y}\in T^{\varepsilon}_p}{\mathbb P}\left\{ \<{\mathbf y}_{E^c}, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\> \ge \frac{\widetilde{\Delta}}{3} \right\} \nonumber \\ %
&\quad+\abs{T^{\varepsilon}_p}\sup_{{\mathbf y}\in T^{\varepsilon}_p}{\mathbb P}\left\{ \<{\mathbf y}_{E}, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\> \ge \frac{\widetilde{\Delta}}{3} \right\}. \label{eq:baseZbnd}%
\end{align}
${\mathbb P}\left( {\cal E}^c \right)$ is bounded in Lemma \ref{lem:smallset}. We
now proceed to bound the latter two terms. For the second term,
the gradient ${\nabla}_{{\mathbf{\tilde{z}}}_\ell}\<{\mathbf y}_{E^c}, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\>$
reads, for any fixed $\ell\in E^c$:
\begin{align*}
{\nabla}_{{\mathbf{\tilde{z}}}_\ell}\<{\mathbf y}_{E^c}, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\> &= %
\frac{2y_\ell}{n}\sum_{i\in E^c\backslash\ell}\partial \eta\left( \frac{\<{\mathbf{\tilde{z}}}_i, {\mathbf{\tilde{z}}}_\ell\>}{n}; \frac{\tau}{\sqrt{n}} \right)y_i {\mathbf{\tilde{z}}}_i.
\end{align*}
Let ${\boldsymbol{\sigma}}^\ell({\mathbf y})\in\mathbb{R}^p$ be a vector defined by:
\begin{align*}
\sigma_i^\ell({\mathbf y}) &= \begin{cases}
\partial \eta\left( \frac{\<{\mathbf{\tilde{z}}}_i, {\mathbf{\tilde{z}}}_\ell\>}{n}; \frac{\tau}{\sqrt{n}} \right)y_i &\text{ if } i\in E^c\backslash\ell,\\
0 &\text{ otherwise.}
\end{cases}
\end{align*}
With this definition we can represent the norm of the gradient as:
\begin{align*}
\norm{{\nabla}_{{\mathbf{\tilde{z}}}_\ell}\<{\mathbf y}_{E^c}, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\>} &= \frac{2\abs{y_\ell}}{n}\norm{{\mathbf Z}_{E^c}{\boldsymbol{\sigma}}^\ell({\mathbf y})} \\
&\le \frac{2\abs{y_\ell}}{n}\norm{{\mathbf Z}_{E^c}}\norm{{\boldsymbol{\sigma}}^\ell({\mathbf y})} \\
&\le \frac{2\abs{y_\ell}}{n}\norm{{\mathbf Z}}\norm{{\boldsymbol{\sigma}}^\ell({\mathbf y})}.
\end{align*}
For ${\boldsymbol{\sigma}}^\ell({\mathbf y})$:
\begin{align*}
\norm{{\boldsymbol{\sigma}}^\ell({\mathbf y})}^2 &= \sum_{i\in E^c\backslash\ell}\partial \eta\left( \frac{\<{\mathbf{\tilde{z}}}_i, {\mathbf{\tilde{z}}}_\ell\>}{n}; \frac{\tau}{\sqrt{n}} \right)^2 y_i^2 \\
&\le \sum_{i\in E^c\backslash\ell} \frac{\<{\mathbf{\tilde{z}}}_i, {\mathbf{\tilde{z}}}_\ell\>^2}{n\tau^2} y_i^2
\\
&\le\frac{A}{\tau^2 np} \<{\mathbf{\tilde{z}}}_\ell, {\mathbf Z}_{E^c\backslash\ell}^{\sf T}
{\mathbf Z}_{E^c\backslash\ell}{\mathbf{\tilde{z}}}_\ell\> \\
&\le \frac{A \norm{{\mathbf{\tilde{z}}}_\ell}^2 \norm{{\mathbf Z}}^2}{np\tau^2}.
\end{align*}
Here the first line follows from $\partial \eta(x ; y) = \mathbb{I}(\abs{x}\ge y ) \le
\abs{x}/y$. The second line follows from the choice of $E$ whereby
$\abs{y_i}\le \sqrt{A/p}$ and the last line from Cauchy-Schwarz.
For any $\ell \in E$, ${\nabla}_{{\mathbf{\tilde{z}}}_\ell}\<{\mathbf y}_{E^c}, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\> = 0$.
Now,
fix $\Gamma=5$, $\gamma = \Gamma\max(\alpha^{-1}, 1)\ge \Gamma$
and let $G = \{{\mathbf Z} : \norm{{\mathbf Z}} \le 2\sqrt{\gamma p}, \forall \ell,
\norm{{\mathbf{\tilde{z}}}_\ell} \le \sqrt{2\gamma p}, \}$. Clearly, $G$ is a closed,
convex set. Furthermore, on the set $G$, we obtain
from the gradient and ${\boldsymbol{\sigma}}^\ell({\mathbf y})$ estimates above that:
\begin{align}
\norm{{\nabla}_{{\mathbf Z}}\<{\mathbf y}_{E^c}, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\>}^2 &= \sum_{\ell\in E^c} \norm{{\nabla}_{{\mathbf{\tilde{z}}}_\ell}\<{\mathbf y}_{E^c}, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\>}^2 \nonumber\\
&\le \sum_{\ell\in E^c}\frac{4y_\ell^2}{n^2}\norm{{\mathbf Z}}^2\norm{{\boldsymbol{\sigma}}^\ell({\mathbf y})}^2 \nonumber\\
&\le \frac{4\norm{{\mathbf Z}}^2}{n^2}\max_{\ell\in E^c} \norm{{\boldsymbol{\sigma}}^\ell({\mathbf y})}^2 \nonumber\\
&\le \frac{4A\norm{{\mathbf Z}}^4 \max_{\ell \in E^c}\norm{{\mathbf{\tilde{z}}}_\ell}^2}{n^3 p
\tau^2} \\
&\le\frac{128 A\gamma^3\alpha^3}{p\tau^2}. \label{eq:lipbnd}
\end{align}
Here we treat ${\nabla}_{{\mathbf Z}}\<{\mathbf y}_{E^c}, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\>$ as a vector in $\mathbb{R}^{np}$,
hence the norm above is the standard $\ell_2$ norm on vectors. We also write the
gradient as ${\nabla}_{({\mathbf{\tilde{z}}}_{\ell})_{\ell\in[p]}}\<{\mathbf y}_{E^c}, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\>$
to avoid ambiguity in specifying the norm.
We now bound ${\mathbb P}\{G^c\}$ as follows.
Lemma \ref{lem:gaussianmatnorm} implies that with probability at least $1-\exp(-\Gamma p/2)$:
\begin{align}
\norm{{\mathbf Z}}_2 &\le (1+ \sqrt{\Gamma} + \alpha^{-1/2})\sqrt{p}\nonumber\\
&\le 2\sqrt{\gamma p}, \label{eq:Znormbnd}
\end{align}
since $\gamma\ge (1+\alpha^{-1/2})^2$.
Further, the standard Chernoff bound
implies that, for a fixed $\ell$, $\norm{{\mathbf{\tilde{z}}}_\ell}^2 \le 2\gamma\alpha n = 2\gamma p$ with
probability at least $1 - \exp(-\gamma p/2)$. By the union bound,
we then obtain that ${\mathbb P}\{G^c\} \le p\exp(-\gamma p/2)+\exp(-\Gamma p/2) \le (p+1)\exp(-\Gamma p/2)$.
Define $K = \sqrt{128 A\gamma^3 \alpha^3 / p \tau^2}$.
Let $F_K({\mathbf Z})$ denote the $G, K$-Lipschitz extension of $F({\mathbf Z}) = \<{\mathbf y}_{E^c},{\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\>$. We
have the following remark for $F_{K}({\mathbf Z})$ which is proved in Appendix \ref{sec:App}.
\begin{remark}\label{rem:exptnoise1}
We have $\E\{\<{\mathbf y}_{E^c}, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\>\}=0$. Further, for every $p$ large enough, $|\E\{F_K({\mathbf Z})\}| \le p^{-1}$.
\end{remark}
We can now use Lemma \ref{lem:basicConc} for $F({\mathbf Z})$,
Thus for any $\Delta_2\ge 2/p$:
\begin{align}
{\mathbb P}\left\{ \<{\mathbf y}_{E^c}, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\> \ge \Delta_2 \right\} &\le %
\exp\left(-\frac{\Delta_2^2}{4K^2} \right) + %
2p\exp\left( -\frac{\Gamma p}{2} \right).
\end{align}
Using $\Delta_2 = \sqrt{2\Gamma p}K = 16\sqrt{A\Gamma\gamma^3\alpha^3}/\tau$ we obtain:
\begin{align}
{\mathbb P}\left\{ \<{\mathbf y}_{E^c}, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\> \ge 16\frac{\sqrt{A\Gamma\gamma^3\alpha^3}}{\tau}\right\} &\le %
(2p+2)\exp(-\Gamma p/2).
\label{eq:quadformbnd2}
\end{align}
Now we can use the essentially same strategy on the term $\<{\mathbf y}_E, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\>$.
For $\ell \in E$ we have as before:
\begin{align*}
{\nabla}_{{\mathbf{\tilde{z}}}_{\ell}}\<{\mathbf y}_E, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\> &= \frac{y_{\ell}}{n}\sum_{i\in E^c} \partial\eta\left( \frac{\<\widetilde{z}_i, \widetilde{z}_\ell\>}{n}; \frac{\tau}{\sqrt{n}} \right)y_i{\mathbf{\tilde{z}}}_i, \\
\norm{{\nabla}_{{\mathbf{\tilde{z}}}_\ell}\<{\mathbf y}_{E}, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\>}^2 &\le \frac{y_\ell^2 A\norm{{\mathbf Z}}^{4}\max_{i\in E^c}\norm{{\mathbf{\tilde{z}}}_i}^2}{\tau^2pn^{3}}.
\end{align*}
Hence:
\begin{align}
\sum_{\ell\in E}\norm{{\nabla}_{{\mathbf{\tilde{z}}}_{\ell}}\<{\mathbf y}_E, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\>}^2 &\le \frac{A\norm{{\mathbf Z}}^4 \max_{i}\norm{{\mathbf{\tilde{z}}}_i}^2}{\tau^2pn^3}. \label{eqn:EEcomppartgrad1}
\end{align}
Analogously, for $\ell\in E^c$:
\begin{align*}
{\nabla}_{{\mathbf{\tilde{z}}}_{\ell}}\<{\mathbf y}_E, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\> &= \frac{y_\ell}{n}\sum_{i\in E} \partial\eta\left( \frac{\<{\mathbf{\tilde{z}}}_i, {\mathbf{\tilde{z}}}_\ell\>}{n}; \frac{\tau}{\sqrt{n}} \right)y_i {\mathbf{\tilde{z}}}_i \\
&= \frac{y_\ell}{n}{\mathbf Z}_{E}{\boldsymbol{\sigma}}^\ell_E({\mathbf y}),
\end{align*}
where we define the vector ${\boldsymbol{\sigma}}^\ell_E({\mathbf y}) \in \mathbb{R}^{E}$ as:
\begin{align*}
\forall i\in E, \quad \sigma^\ell_E({\mathbf y})_i &= y_i\partial\eta\left( \frac{\<{\mathbf{\tilde{z}}}_i, {\mathbf{\tilde{z}}}_\ell\>}{n}; \frac{\tau}{\sqrt{n}} \right).
\end{align*}
By Cauchy-Schwarz:
\begin{align*}
\norm{{\nabla}_{{\mathbf{\tilde{z}}}_\ell}\<{\mathbf y}_E, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\>}^2 &\le \frac{y_\ell^2}{n^2}\norm{{\mathbf Z}_E}^2 \norm{{\boldsymbol{\sigma}}^\ell_E({\mathbf y})}^2.
\end{align*}
Summing over $\ell \in E^c$:
\begin{align*}
\sum_{\ell\in E^c } \norm{{\nabla}_{{\mathbf{\tilde{z}}}_\ell}\<{\mathbf y}_E, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\>}^2 &\le \frac{\norm{{\mathbf Z}_E}^2}{n^2}\sum_{\ell\in E^c}y_\ell^2 \norm{{\boldsymbol{\sigma}}^\ell_E({\mathbf y})^2}^2 \\
&\le \frac{A\norm{{\mathbf Z}}^2}{pn^2}\sum_{\ell\in E^c} \norm{{\boldsymbol{\sigma}}^\ell_E({\mathbf y})}^2\\
&= \frac{A\norm{{\mathbf Z}}^2}{pn^2} \sum_{\ell \in E^c}\sum_{i\in E} y_i^2 \partial\eta\left( \frac{\<{\mathbf{\tilde{z}}}_i, {\mathbf{\tilde{z}}}_\ell\>}{n}; \frac{\tau}{\sqrt{n}} \right)^2 \\
&\le \frac{A\norm{{\mathbf Z}}^2}{pn^2} \sum_{i\in E} y_i^2 \sum_{\ell \in E^c} \frac{\<{\mathbf{\tilde{z}}}_i, {\mathbf{\tilde{z}}}_\ell\>^2}{\tau^2 n} \\
&= \frac{A\norm{{\mathbf Z}}^2}{\tau^2 pn^3} \sum_{i\in E}y_i^2 \<{\mathbf{\tilde{z}}}_i, {\mathbf Z}_{E^c}^{\sf T} {\mathbf Z}_{E^c}{\mathbf{\tilde{z}}}_i\> \\
&\le \frac{A\norm{{\mathbf Z}}^2 \norm{{\mathbf Z}_{E^c}}^2 \max_{i\in [E]}\norm{{\mathbf{\tilde{z}}}_i}^2}{\tau^2 np^3} \\
&\le \frac{A\norm{{\mathbf Z}}^{4}\max_{i\in [p]}\norm{{\mathbf{\tilde{z}}}_i}^2}{\tau^2 pn^3}.
\end{align*}
This bound along with \myeqref{eqn:EEcomppartgrad1} gives:
\begin{align*}
\norm{{\nabla}_{({\mathbf{\tilde{z}}}_{\ell})_{\ell\in[p]}}\<{\mathbf y}_E, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\>}^2 &\le \frac{2A\norm{{\mathbf Z}}^4\max_{i\in [p]}\norm{{\mathbf{\tilde{z}}}_i}^2}{\tau^2 np^3}.
\end{align*}
On the set $G$ defined before, we have that:
\begin{align*}
\norm{{\nabla}_{({\mathbf{\tilde{z}}}_{\ell})_{\ell\in[p]}}\<{\mathbf y}_E, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\>}^2
&\le\frac{64 A\gamma^3\alpha^3}{p\tau^2 }.
\end{align*}
Proceeding as before, applying Lemma \ref{lem:basicConc} we have:
\begin{align}
{\mathbb P}\left\{ \<{\mathbf y}_{E}, {\mathbf{\widetilde{N}}}{\mathbf y}_{E^c}\> \ge 16\frac{\sqrt{A\Gamma\gamma^2\alpha^3}}{\tau} \right\} &\le %
2p\exp\left( -\frac{\Gamma p}{2} \right).
\label{eq:quadformbnd3}
\end{align}
We can now use Eqs.\eqref{eq:quadformbnd2}, \eqref{eq:quadformbnd3} in
\myeqref{eq:baseZbnd}:
\begin{align*}
{\mathbb P}\left\{ \norm{{\mathbf{\widetilde{N}}}}_2 \ge (1-2{\varepsilon})^{-1}\left( 32\sqrt{\frac{\alpha\log A}{A}} + 48\sqrt{\frac{{A\Gamma\gamma^2\alpha^3}}{\tau^2}}\right) \right\} %
&\le \exp\left( -\frac{p\log A}{4A} \right)\\
&\quad+|T^{\varepsilon}_p| (4p+4)\exp\left( \frac{-\Gamma p}{2} \right) %
\end{align*}
We first simplify the probability bound. Since $A = \tau\log \tau$, $\log A/A \ge 1/\tau$ when $\tau\ge \exp(1)$. Further,
choosing ${\varepsilon}=1/4$, with Lemma \ref{lem:epsnetcard} we get that $|T^{{\varepsilon}}_p| \le (1+2/{\varepsilon})^p= 9^p$.
Since $\log 9 = 2.19\dots < \Gamma/2 = 5/2$, we
have $(4p+4)|T^{\varepsilon}_p|\exp(-\Gamma p/2) \le \exp(-p/20)$ for large enough $p$. Thus the right hand side is bounded
above by $ 2\exp\left( -p/4\max(\tau, 5) \right)$ for every $p$ large enough.
Now we simplify the operator norm bound. As $A = \tau\log\tau$, $\log A/A \le \log \tau/\tau$
since $\log z/z$ is decreasing. Further $\alpha \le \max(1, \alpha^3)$ and $\Gamma=5$ imply:
\begin{align*}
(1-2{\varepsilon})^{-1}\left( 32\sqrt{\frac{\alpha\log A}{A}} +
64\sqrt\frac{{A\Gamma\gamma^3\alpha^3}}{\tau^2}\right)
&\le 2(32+64\Gamma^{2})\sqrt{\frac{\max(1, \alpha^3)\log\tau}{\tau}}\\
&\le 5000\sqrt{\frac{\max(1, \alpha^3)\log\tau}{\tau}}.
\end{align*}
Our conditions on $\tau$ and $A$ were: $(i)$ $\tau \ge \max(4\sqrt{\Gamma\gamma\alpha}, \exp(1)) = 20\max(1, \sqrt{\alpha})$
and $(ii)$ $A\ge 180\max(\sqrt{\alpha}, 1)$. Using $\tau \ge 100\max(1, \alpha^2\log \alpha)$ satisfies both conditions.
\subsubsection{Proof of Lemma \ref{lem:smallset}}\label{sec:proofsmallset}
This proof also follows an ${\varepsilon}$-net argument. Let $a$ denote the size of
the set $E$. For notational simplicity, we will permute the rows and columns of
${\mathbf{\widetilde{N}}}$ to ensure $E = [a]$ (i.e. $E$ is the first $a$ entries of $[p]$).
For a fixed ${\mathbf y}\in T^{{\varepsilon}}_a$, we bound the Rayleigh quotient $\<{\mathbf y},{\widetilde{\cal P}}_{E,E}({\mathbf{\widetilde{N}}}){\mathbf y}\>$ with
high probability. Note that $\<{\mathbf y},{\widetilde{\cal P}}_{E, E}({\mathbf{\widetilde{N}}}){\mathbf y}\>$ is a function of ${\mathbf{\tilde{z}}}_\ell, \ell\in E$.
The gradient of this function with respect to ${\mathbf{\tilde{z}}}_\ell$ is:
\begin{align*}
{\nabla}_{{\mathbf{\tilde{z}}}_\ell}\<{\mathbf y}, {\widetilde{\cal P}}_{E, E}({\mathbf{\widetilde{N}}}){\mathbf y}\> &= \frac{2y_\ell}{n}\sum_{i\in E\backslash\ell} \partial \eta\left( \frac{\<{\mathbf{\tilde{z}}}_i, {\mathbf{\tilde{z}}}_\ell\>}{n};\frac{\tau}{\sqrt{n}} \right)y_i {\mathbf{\tilde{z}}}_i ,
\end{align*}
where ${\boldsymbol{\sigma}}^\ell({\mathbf y})\in\mathbb{R}^p$ is the vector defined as:
\begin{align*}
\sigma_i^\ell({\mathbf y}) &= \begin{cases}
y_i\partial \eta\left(\frac{\<{\mathbf{\tilde{z}}}_i, {\mathbf{\tilde{z}}}_\ell\>}{n}; \frac{\tau}{\sqrt{n}}\right) &\text{ when } i\in E\backslash\ell\\
0 & \text{ otherwise.}
\end{cases}
\end{align*}
The (square of the) total gradient is thus given by:
\begin{align*}
\norm{{\nabla} \<{\mathbf y}, {\widetilde{\cal P}}_{E, E}({\mathbf{\widetilde{N}}}){\mathbf y}\>}^2 &= \frac{4}{n^2}\sum_{\ell\in E} \norm{ {\mathbf Z}_E {\boldsymbol{\sigma}}^\ell({\mathbf y})}_2^2 y_\ell^2 \\
&\le \frac{4}{n^2}\sum_{\ell\in E } \norm{{\mathbf Z}_{E}}_2^2 \norm{{\boldsymbol{\sigma}}^\ell({\mathbf y})}^2 y_\ell^2\\
&\le \left( \frac{2\norm{{\mathbf Z}_E}_2}{n} \right)^2\sum_{\ell\in E\backslash\ell}\norm{{\boldsymbol{\sigma}}^\ell({\mathbf y})}^2 y_\ell^2,
\end{align*}
Since $|\partial\eta(\cdot; \tau/\sqrt{n})|\le1$ we have that $\norm{{\boldsymbol{\sigma}}^\ell({\mathbf y})}^2 \le \norm{{\mathbf y}}^2 \le 1$.
Consequently we obtain the bound:
\begin{align*}
\norm{{\nabla}\<{\mathbf y}, {\widetilde{\cal P}}_{E, E}({\mathbf{\widetilde{N}}}){\mathbf y}\>}^2 &\le \left( \frac{2\norm{{\mathbf Z}_E}_2}{n} \right)^2.
\end{align*}
From Lemma \ref{lem:gaussianmatnorm} we have that:
\begin{align*}
\norm{{\mathbf Z}_E}_2 &\le \sqrt{n} + \sqrt{a} + t\sqrt{p},
\end{align*}
with probability at least $1 - \exp(-pt^2/2)$. Let $G = \{{\mathbf Z}_E: \norm{{\mathbf Z}_E}_2 \le \sqrt{n} + \sqrt{a}+ t\sqrt{p}\}$.
Then:
\begin{align}
\norm{{\nabla}\<{\mathbf y}, {\widetilde{\cal P}}_{E, E}({\mathbf{\widetilde{N}}}){\mathbf y}\>}^2 &\le \frac{4\alpha}{p}\Bigg( 1 + \sqrt{\frac{a\alpha}{p}} + t\sqrt{\alpha} \Bigg)^2 \equiv L^2 \label{eq:Glipschitz}\\
\text{and } {\mathbb P}(G^c) &\le e^{-pt^2/2}. \label{eq:Gprob}
\end{align}
We let $F_L({\mathbf Z}_E)$ denote the $G,L$-Lipschitz extension of ${\nabla}\<{\mathbf y}, {\widetilde{\cal P}}_{E, E}({\mathbf{\widetilde{N}}}){\mathbf y}\>$.
The following remark is proved in Appendix \ref{sec:App}:
\begin{remark}\label{rem:exptsmallset}
Firstly, $\E\{\<{\mathbf y},{\widetilde{\cal P}}_{E, E}({\mathbf{\widetilde{N}}}){\mathbf y}\>\}=0$. Secondly, for every $p$ large enough: $|\E(F_L({\mathbf Z}))| \le p^{-1}$.
\end{remark}
Let $\widetilde{\Delta} = \Delta(1-2{\varepsilon})$ and $\nu = 1+ \sqrt{\alpha a/p}$.
We choose $t= \left(\sqrt{\nu^2 + \widetilde{\Delta}/2\sqrt{\alpha}} - \nu\right)/2$
and apply Lemma \ref{lem:basicConc} and Remark \ref{rem:exptsmallset}. This choice
of $t$ ensures that the two unfavorable events of Lemma \ref{lem:basicConc} are
both bounded above by $\exp(-pt^2/2)$. Thus,
\begin{align*}
{\mathbb P}\{\<{\mathbf y}, {\widetilde{\cal P}}_{E, E}({\mathbf{\widetilde{N}}}){\mathbf y}\> \ge \widetilde{\Delta}\} &\le 2e^{-pt^2/2},
\end{align*}
for $p$ large enough. Further, our choice of $t$ implies:
\begin{align*}
t^2 &= \frac{1}{4}\left( \sqrt{\nu^2 + \frac{\widetilde{\Delta}}{2\sqrt{\alpha}}} - \nu \right)^2 \\
&= \frac{\nu^2}{2}\left( 1 + \frac{\widetilde{\Delta}}{4\nu^2\sqrt{\alpha}} - \sqrt{1+ \frac{\widetilde{\Delta}}{2\nu^2\sqrt{\alpha}}} \right)\\
&\ge \frac{\widetilde{\Delta}^2}{128\nu^2\alpha},
\end{align*}
where the last inequality follows from the fact that $g(x) = 1+x/2 - \sqrt{1+x} \ge x^2/16$ when
$x \le 2$. This requires $\widetilde{\Delta} \le 4\nu^2\sqrt{\alpha}$). Now, Lemma \ref{lem:epsnetcard}
and \ref{lem:specnormbnd} imply:
\begin{align*}
{\mathbb P}\left\{\norm{{\widetilde{\cal P}}_{E, E}({\mathbf{\widetilde{N}}})}_2 \ge \Delta\right\} %
&\le 2\left( 1+\frac{2}{{\varepsilon}} \right)^a\exp\left( -\frac{p\widetilde{\Delta}^2}{256\nu^2\alpha} \right) \\
&\le \exp\left( -\frac{p\widetilde{\Delta}^2}{256\alpha\nu^2} + a \log\left(2+ \frac{4}{{\varepsilon}} \right) \right).
\end{align*}
There are $\binom{p}{a} \le (pe/a)^a$ possible choices for the set $E$. Using the union bound we
have that:
\begin{align*}
{\mathbb P}\left\{\sup_{E\subseteq [p], |E| = a} \norm{{\widetilde{\cal P}}_{E, E}({\mathbf{\widetilde{N}}})}_2 \ge \Delta\right\} &\le %
\exp\left\{ -\frac{p\widetilde{\Delta}^2}{256\alpha\nu^2} + a \log\left(2+ \frac{4}{{\varepsilon}} \right) + a\log\left( \frac{pe}{a} \right) \right\}.
\end{align*}
Since $a \le p/A$, $\nu = 1 + \sqrt{a\alpha/p} \le 2$ when $A\ge\max(\sqrt{\alpha}, 1)$. Using ${\varepsilon}=1/4$ we obtain that
\begin{align*}
{\mathbb P}\left\{\sup_{E\subseteq [p], |E| = a} \norm{{\widetilde{\cal P}}_{E, E}({\mathbf{\widetilde{N}}})}_2 \ge \Delta\right\} &\le %
\exp\left( - p\left(\frac{\Delta^2}{1024\alpha} - \frac{\log\left(18eA \right)}{A}\right) \right).
\end{align*}
We required $\widetilde{\Delta} \le 4\nu^2\sqrt{\alpha}$, and $\widetilde{\Delta} = \Delta/2$.
Hence we require $\Delta\le 8\sqrt{\alpha} \le 8\nu^2\sqrt{\alpha}$.
Choosing $\Delta=32\sqrt{\alpha\log A/A}$, where $A\ge 180\max(\sqrt{\alpha}, 1)$ satisfies
this condition. Further, with this choice of $A$, $\log(18eA)\le 1.75\log A$.
Consequently:
\begin{align*}
P\left\{\sup_{E\subseteq [p], |E| = a} \norm{{\widetilde{\cal P}}_{E, E}(N)}_2 \ge 32\sqrt{\frac{\alpha\log A}{A}}\right\} &\le \exp\left(- \frac{p\log A}{4 A}\right).%
\end{align*}
\subsection{Proof of Proposition \ref{prop:cross}} \label{subsec:proofcross}
We explicitly write the $(i, j)^\mathrm{th}$ entry of ${\mathbf R}_1$ (when $(i, j)\in
{\sf G}$) as:
\begin{align*}
(R_1)_{ij} &= \eta\left( \frac{\<\sum_{a} \sqrt{\beta_q}{\mathbf u}_q(v_q)_i + {\mathbf{\tilde{z}}}_i,
\sum_{q} {\mathbf u}_q (v_q)_j + {\mathbf{\tilde{z}}}_j \>}{n} ; \frac{\tau}{\sqrt{n}}\right)
\end{align*}
Since ${\sf G}$ is a symmetric set of entries excluding the diagonal, it suffices to consider the case
$i < j$ above. Denote by ${\mathbf R}$ the upper triangle of ${\mathbf R}_1$.
Let $g$ denote the number of nonzero rows in ${\mathbf R}$. By the definition of $g$, $ g \le \sum_q
\abs{{\sf Q}_q} = k$. We wish to bound (with slight abuse of notation) the quantity:
$ \sup_{{\mathbf x}\in {S}^{g-1}}\sup_{{\mathbf y}\in {S}^{p-1}} \<{\mathbf x}, {\mathbf R}{\mathbf y}\>$.
The proof follows an epsilon net argument entirely analogous to
the proof of Proposition \ref{prop:noise}. The only difference is the
further dependence on the Gaussian random vectors ${\mathbf u}_q$. Hence we only
give a proof sketch, highlighting the difference with the proof of Proposition
\ref{prop:noise}.
Fix a vector ${\mathbf y}\in T^{1/4}_{p}$ and ${\mathbf x} \in T^{1/4}_{g}$, and let $E$ be the subset of indices
$E = \{i \in [p] : \abs{y_i} \ge \sqrt{A/p}\}$ for some constant $A$ to
be fixed later in the proof. As before, we split the Rayleigh quotient
$\<{\mathbf x}, {\mathbf R}_3{\mathbf y}\> = \<{\mathbf x}, {\mathbf R}{\mathbf y}_E\> + \<{\mathbf x}, {\mathbf R}{\mathbf y}_{E^c}\> \le
\norm{{\cal P}_{[p]\times E}({\mathbf R})} + \<{\mathbf x}, {\mathbf R}{\mathbf y}_{E^c}\>$. By the condition on
$E$, we have that $\abs{E} \le p/A$. Consequently:
\begin{align*}
{\mathbb P}\left\{ \norm{{\mathbf R}} \ge \Delta \right\} &\le
\sup_{{\mathbf x}\in T^{1/4}_g, {\mathbf y} \in T^{1/4}_{p}} {\mathbb P}\left\{ \<{\mathbf x}, {\mathbf R}{\mathbf y}\> \ge \Delta/2 \right\} \\
&\le
{\mathbb P}\left\{ \max_{\abs{E}\le p/A } \norm{{\cal P}_{[p]\times E}\left( {\mathbf R} \right)} \ge \frac{\Delta}{4} \right\} +
\sup_{{\mathbf x}\in T^{1/4}_g, {\mathbf y}\in T^{1/4}_p}{\mathbb P}\left\{ \<{\mathbf x}, {\mathbf R}{\mathbf y}_{E^c}\> \ge \frac{\Delta}{4} \right\}.
\end{align*}
We first concentrate on the second term, whose gradient with respect to a fixed ${\mathbf{\tilde{z}}}_i$ is given by:
\begin{align*}
{\nabla}_{{\mathbf{\tilde{z}}}_i}\<{\mathbf x}, {\mathbf R}{\mathbf y}_{E^c}\> &= \frac{x_i}{n}\sum_{j> i, (i, j)\in
{\sf G}} (y_{E^c})_i \partial\eta\left( \<\sum_q\sqrt{\beta_q}{\mathbf u}_q (v_q)_i+ {\mathbf{\tilde{z}}}_i,
\sum_{q}\sqrt{\beta_q}{\mathbf u}_q(v_q)_j + {\mathbf{\tilde{z}}}_j \>; \tau\sqrt{n}\right) \left(
\sum_{q}\sqrt{\beta_q}{\mathbf u}_q(v_q)_j + {\mathbf{\tilde{z}}}_j \right) \\
&\quad+ \frac{(y_{E^c})_i}{n}\sum_{j< i, (i, j)\in
{\sf G}} x_j \partial\eta\left( \<\sum_q\sqrt{\beta_q}{\mathbf u}_q (v_q)_i+ {\mathbf{\tilde{z}}}_i,
\sum_{q}\sqrt{\beta_q}{\mathbf u}_q(v_q)_j + {\mathbf{\tilde{z}}}_j\> ; \tau\sqrt{n}\right) \left(
\sum_{q}\sqrt{\beta_q}{\mathbf u}_q(v_q)_j + {\mathbf{\tilde{z}}}_j \right).
\end{align*}
Defining ${\boldsymbol{\sigma}}^i({\mathbf y})$ and ${\boldsymbol{\sigma}}^i({\mathbf x})$ similar to Proposition
\ref{prop:noise}, we have by
Cauchy Schwarz:
\begin{align*}
\norm{{\nabla}_{{\mathbf{\tilde{z}}}_i}\<{\mathbf x}, {\mathbf R}{\mathbf y}_{E^c}\>}^2 &\le
\frac{2\norm{\sum_{q}\sqrt{\beta_q}{\mathbf u}_q{\mathbf v}_q^{\sf T} + {\mathbf Z}}^2}{n^2} \left( x_i^2
\norm{{\boldsymbol{\sigma}}^i({\mathbf y})}^2 + (y_{E^c})^2_i \norm{{\boldsymbol{\sigma}}^i({\mathbf x})}^2
\right).
\end{align*}
Summing over $i$:
\begin{align*}
\sum_{i} \norm{{\nabla}_{{\mathbf{\tilde{z}}}_i}\<{\mathbf x}, {\mathbf R}{\mathbf y}_{E^c}\>}^2 &\le
\frac{2\norm{{\mathbf X}}^2}{n^2}\sum_{i}\left(x_i^2 \norm{{\boldsymbol{\sigma}}^i({\mathbf y})^2} +
(y_{E^c})^2_i \norm{{\boldsymbol{\sigma}}^i({\mathbf x})}^2\right)\\
&\le \frac{2\norm{{\mathbf X}}^2}{n^2} \sup_{i}\norm{{\boldsymbol{\sigma}}^i({\mathbf y})}^2 +
\frac{2\norm{{\mathbf X}}^2}{n}\sum_i(y_{E^c})_i^2 \norm{{\boldsymbol{\sigma}}^i({\mathbf x})}^2
\end{align*}
Let $G = \{({\mathbf u})_{q\le r}, {\mathbf Z}: \forall q \norm{{\mathbf u}_q}\le
C'\sqrt{n}, \norm{{\mathbf Z}} \le C'(\sqrt{p}+\sqrt{n}), \forall i \norm{{\mathbf{\tilde{z}}}_i}\le C'\sqrt{n}\}$. It is clear that
$G$ is convex, and that ${\mathbb P}\{G^c\} \le p\exp(-C''p)$ for some $C''$ dependent on $C'$.
It is not hard to show that:
\begin{align}
\sum_{i} \norm{{\nabla}_{{\mathbf{\tilde{z}}}_i}\<{\mathbf x}, {\mathbf R}{\mathbf y}_{E^c}\>}^2 &\le \frac{AC(\alpha,
(\beta)_{q\le r}, r)}{p\tau^2}, \label{eq:gradboundz}
\end{align}
for some constant $C$, when $C'$ is large enough.
Similarly, taking derivatives with respect to ${\mathbf u}_q$ for a fixed $q$, we have:
\begin{align*}
{\nabla}_{{\mathbf u}_q}\<{\mathbf x}, {\mathbf R}{\mathbf y}_{E^c}\> &= \frac{1}{n}\sum_{(i, j)\in
{\sf G}} \partial\eta\left( \<\sum_q\sqrt{\beta_q}{\mathbf u}_q (v_q)_i+ {\mathbf{\tilde{z}}}_i,
\sum_{q}\sqrt{\beta_q}{\mathbf u}_q(v_q)_j + {\mathbf{\tilde{z}}}_j \>; \tau\sqrt{n}\right)
\quad \cdot \quad \\
&\quad \left(x_i (y_{E^c})_j \sqrt{\beta_q} (v_q)_i(\sum_{q'
}\sqrt{\beta_{q'}}{\mathbf u}_{q'}(v_{q'})_j
+ {\mathbf{\tilde{z}}}_j) +
x_j (y_{E^c})_i \sqrt{\beta_q} (v_q)_j(\sum_{q'
}\sqrt{\beta_{q'}}{\mathbf u}_{q'}(v_{q'})_i
+ {\mathbf{\tilde{z}}}_i)
\right)\\
&= \frac{{\mathbf X}({\boldsymbol{\sigma}}^1_{{\sf G}}({\mathbf x}, {\mathbf y}) + {\boldsymbol{\sigma}}^2_{{\sf G}}({\mathbf x},
{\mathbf y}))}{n},
\end{align*}
where we define the vectors ${\boldsymbol{\sigma}}_{\sf G}^1({\mathbf x}, {\mathbf y}), {\boldsymbol{\sigma}}_{\sf G}^2({\mathbf x},
{\mathbf y})$ appropriately.
By Cauchy Schwarz:
\begin{align*}
\norm{{\nabla}_{{\mathbf u}_q}\<{\mathbf x}, {\mathbf R}{\mathbf y}_{E^c}\>}^2 &\le \frac{2\norm{{\mathbf X}}^2}{n^2}
(\norm{{\boldsymbol{\sigma}}_{\sf G}^1({\mathbf x}, {\mathbf y})}^2 + \norm{{\boldsymbol{\sigma}}_{\sf G}^2({\mathbf x}, {\mathbf y})}^2).
\end{align*}
We now bound the first term above, and the second term follows from a similar
argument.
\begin{align*}
\norm{{\boldsymbol{\sigma}}^1({\mathbf x}, {\mathbf y})}^2 &= \sum_{j}(y_{E^c})_j^2 \left( \sum_{i: (i, j)\in
{\sf G}} \sqrt{\beta_q}x_i(v_q)_i\partial\eta\left(
\<\sum_q\sqrt{\beta_q}{\mathbf u}_q (v_q)_i+ {\mathbf{\tilde{z}}}_i,
\sum_{q}\sqrt{\beta_q}{\mathbf u}_q(v_q)_j + {\mathbf{\tilde{z}}}_j \>; \tau\sqrt{n}
\right) \right)^2
\end{align*}
For simplicity of notation, define $D_{ij} = \partial\eta\left( \<\sum_q\sqrt{\beta_q}{\mathbf u}_q (v_q)_i+ {\mathbf{\tilde{z}}}_i,
\sum_{q}\sqrt{\beta_q}{\mathbf u}_q(v_q)_j + {\mathbf{\tilde{z}}}_j \>; \tau\sqrt{n}
\right)$.
The above sum then can be reduced to:
\begin{align*}
\norm{{\boldsymbol{\sigma}}^1({\mathbf x}, {\mathbf y})}^2 &= \sum_{i_1, i_2}\beta_q x_{i_1}x_{i_2}
(v_q)_{i_1}(v_q)_{i_2}\sum_{j: (i_1, j)\in {\sf G}\text{ or } (i_2, j)\in {\sf G}} (y_{E^c})_j^2
D_{i_1 j}D_{i_2 j}.
\end{align*}
We first bound the inner summation uniformly in $i_1, i_2$ as follows:
\begin{align*}
\sum_{j: (i_1, j)\in {\sf G} \text{ or } (i_2, j)\in {\sf G}} (y_{E^c})_j^2 D_{i_1, j}D_{i_2, j}
&\le \frac{A}{p} \sum_{j} \frac{\abs{\<{\mathbf{\tilde{x}}}_{i_1}, {\mathbf{\tilde{x}}}_j\>\<{\mathbf{\tilde{x}}}_{i_2}, {\mathbf{\tilde{x}}}_j\>}}{n\tau^2} \\
&\le \frac{A}{p}\sum_{j} \frac{\<{\mathbf{\tilde{x}}}_{i_1}, {\mathbf{\tilde{x}}}_j\>^2 + \<{\mathbf{\tilde{x}}}_{i_2},{\mathbf{\tilde{x}}}_j\>^2 }{2n\tau^2} \\
&\le \frac{A}{n\tau^2 p} \norm{{\mathbf X}}^2 (\norm{{\mathbf{\tilde{x}}}_{i_1}}^2 + \norm{{\mathbf{\tilde{x}}}_{i_2}}^2)
\end{align*}
Employing a similar strategy for the other term, it is not hard to show that:
\begin{align*}
\norm{{\boldsymbol{\sigma}}^1({\mathbf x}, {\mathbf y})} ^2&\le \frac{A\beta_q \norm{{\mathbf X}}^2 \sup_{i} \norm{{\mathbf{\tilde{x}}}_{i}}^2}{pn\tau^2}.
\end{align*}
Thus, on the set $G$, we obtain that:
\begin{align}
\sum_{q}\norm{{\nabla}_{{\mathbf u}_q} \<{\mathbf x}, {\mathbf R}{\mathbf y}_{E^c}\>}^2 &\le \frac{AC(\alpha, \beta_1, r)}{p\tau^2},\label{eq:gradboundu}
\end{align}
for every $\tau$ sufficiently large. Indeed the same bound, with a modified value for $C$ holds
for the gradient with respect to all the variables $( ({\mathbf u}_q)_{q\le r}, ({\mathbf{\tilde{z}}}_i)_{i\le p})$ using
Eqs.\eqref{eq:gradboundz}, \eqref{eq:gradboundu}. Lemma \ref{lem:basicConc} then
implies that
\begin{align*}
\sup_{{\mathbf x}\in T^{1/4}_g, {\mathbf y}\in T^{1/4}_{p}}{\mathbb P}\left\{ \<{\mathbf x}, {\mathbf R}{\mathbf y}\> \ge \sqrt{A C(\alpha, \beta_1, r)}{\tau^2} \right\} \le \exp(-c p),
\end{align*}
for an appropriate $c$.
We omit the proof of the following remark that uses similar techniques as above,
followed by a union bound.
\begin{remark}
For every $A\ge A_0(\alpha, \beta_1, r)$ we have that:
\begin{align*}
{\mathbb P}\left\{ \sup_{\abs{E} \le p/A} {\cal P}_{[p]\times E}({\mathbf R}) \ge C(\alpha, \beta_1, r)\sqrt{\frac{\log A}{A}} \right\} &\le \exp(-c_2(\tau) p).
\end{align*}
Here $c_2(\tau) = 1/4\tau$ suffices.
\end{remark}
Using $A = \tau\log \tau$ for $\tau$ large enough completes the proof.
\subsection{Proof of Proposition \ref{prop:diag}}\label{subsec:proofdiag}
Since ${\mathbf R}_2$ is a diagonal matrix, its spectral norm is bounded by the maximum of its entries.
This is easily done as, for every $i\in {\sf Q}^c$:
\begin{align*}
\abs{({\mathbf R}_2)_{ii}} &= \abs{\eta\left( \frac{\norm{{\mathbf{\tilde{z}}}_i}^2}{n} - 1;\frac{\tau}{\sqrt{n}} \right)} \\
&\le \abs{\frac{\norm{{\mathbf{\tilde{z}}}_i}^2-n}{n}}.
\end{align*}
By the Chernoff bound for $\chi$-squared random variables followed by the union bound
we have that:
\begin{align*}
\max_{i}\abs{\frac{\norm{{\mathbf{\tilde{z}}}_i}^2}{n}-1} \ge t,
\end{align*}
with probability at most $p(\exp(n(-t + \log(1+t))/2) + \exp(n(t - \log(1-t))/2))$. Setting
$t = \sqrt{C\log n/n}$ and using $\log(1+t) \le t - t^2/3, \log(1-t) \ge -t -t^2/3$ for
every $t$ small enough we obtain the probability bound of $pn^{-C/6} = \alpha n^{-C/6 +1}$.
\section*{Acknowledgements}
We are grateful to David Donoho for his feedback on this manuscript.
This work was
partially supported by the NSF CAREER award CCF-0743978, the NSF grant CCF-1319979, and
the grants AFOSR/DARPA FA9550-12-1-0411 and FA9550-13-1-0036.
| {
"timestamp": "2014-11-05T02:07:25",
"yymm": "1311",
"arxiv_id": "1311.5179",
"language": "en",
"url": "https://arxiv.org/abs/1311.5179",
"abstract": "In sparse principal component analysis we are given noisy observations of a low-rank matrix of dimension $n\\times p$ and seek to reconstruct it under additional sparsity assumptions. In particular, we assume here each of the principal components $\\mathbf{v}_1,\\dots,\\mathbf{v}_r$ has at most $s_0$ non-zero entries. We are particularly interested in the high dimensional regime wherein $p$ is comparable to, or even much larger than $n$. In an influential paper, \\cite{johnstone2004sparse} introduced a simple algorithm that estimates the support of the principal vectors $\\mathbf{v}_1,\\dots,\\mathbf{v}_r$ by the largest entries in the diagonal of the empirical covariance. This method can be shown to identify the correct support with high probability if $s_0\\le K_1\\sqrt{n/\\log p}$, and to fail with high probability if $s_0\\ge K_2 \\sqrt{n/\\log p}$ for two constants $0<K_1,K_2<\\infty$. Despite a considerable amount of work over the last ten years, no practical algorithm exists with provably better support recovery guarantees.Here we analyze a covariance thresholding algorithm that was recently proposed by \\cite{KrauthgamerSPCA}. On the basis of numerical simulations (for the rank-one case), these authors conjectured that covariance thresholding correctly recover the support with high probability for $s_0\\le K\\sqrt{n}$ (assuming $n$ of the same order as $p$). We prove this conjecture, and in fact establish a more general guarantee including higher-rank as well as $n$ much smaller than $p$. Recent lower bounds \\cite{berthet2013computational, ma2015sum} suggest that no polynomial time algorithm can do significantly better. The key technical component of our analysis develops new bounds on the norm of kernel random matrices, in regimes that were not considered before.",
"subjects": "Statistics Theory (math.ST); Machine Learning (stat.ML)",
"title": "Sparse PCA via Covariance Thresholding",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754515389344,
"lm_q2_score": 0.7185943985973772,
"lm_q1q2_score": 0.7075104044723617
} |
https://arxiv.org/abs/1811.00384 | Encoding and Visualization in the Collatz Conjecture | The Collatz conjecture is one of the easiest mathematical problems to state and yet it remains unsolved. For each $n\ge 2$ the Collatz iteration is mapped to a binary sequence and a corresponding unique integer which can recreate the iteration. The binary sequence is used to produce the Collatz curve, a 2-D visualization of the iteration on a grid, which, besides the aesthetics, provides a qualitative way for comparing iterations. Two variants of the curves are explored, the r-curves and on-change-turn-right curves. There is a scarcity of acyclic r-curves and only three r-curves were found having a cycle of minimum length greater than 4. | \section{Introduction}
\IEEEPARstart{T}{he} Collatz Conjecture states that starting with a positive integer $n$, the iteration of the function
\begin{equation}
\label{eq:1}
C(n) = \twocases{\frac{n}{2},}{\text{if $n\equiv 0 \pmod 2 $}}{3n+1,}{\text{if $n\equiv 1 \pmod 2$,}}
\end{equation}
eventually reaches 1; that is to say that there exists a finite integer $k$ such that $C^{k}(n) =1$. For example, starting with $n=3$, $C^0(3)=3$, we obtain the sequence $3 \rightarrow 10 \rightarrow 5 \rightarrow 16 \rightarrow 8 \rightarrow 4 \rightarrow 2 \rightarrow 1$. Continuing the iteration, it is noted that the sequence is caught in the cycle $1 \rightarrow 4 \rightarrow 2 \rightarrow 1$. The Collatz conjecture can be stated in another way: starting the iteration with any positive integer $n$ there is no other cycle than this one and the iteration does not diverge to infinity.
The origins of the Collatz Conjecture can be traced back to the 1930's to the German mathematician Luther Collatz \cite{Lagarias1985}. Although known in the 1950's among mathematicians, it first appeared in print in 1971 \cite{Coxeter1971,Lagarias1985}. Despite the simplicity with which it can be stated and despite the multitude of mathematicians who studied it and attempted to solve it, its proof has remained elusive. Empirically, it has been verified to be true for $n < 5 \times 2^{60} \approx 5.7646 \times 10^{18}$ \cite{OliveiraeSilva2010}.
The consensus seems to be that a proof is not within reach at this time. In general, randomness and unpredictability govern the sequences produced by the iterations from one number to the next. Paul Erdős famously commented ``Mathematics may not be ready for such problems'' \cite{Lagarias1985}. The conjecture, as it circulated in many campuses through many mathematicians, it came to be known by various names such as the \emph{$3x+1$ problem,} the \emph{Syracuse Problem}, \emph{Hasse's Algorithm}, \emph{Kakutani's Problem} and \emph{Ulam's Problem} \cite{Lagarias1985,Lagarias2006}. The sequence of numbers produced by the Collatz iteration is known as \emph{hailstone numbers} \cite{Hayes1984}.
It is noted that in Equation~\ref{eq:1}, when the second rule is applied, the resulting number $3n+1$ is necessarily an even number, and hence the iteration can be written more compactly thusly \cite{Terras1976,Terras1979,Lagarias1985}:
\begin{equation}
\label{eq:2}
T(n) =
\begin{cases}
\frac{n}{2}, & {\text{if $n\equiv 0 \pmod 2 $}}\\
\frac{3n+1}{2}, & {\text{if $n\equiv 1 \pmod 2$.}}
\end{cases}
\end{equation}
This version of the iteration is widely used and it will be used for the rest of this paper. Using the same example $n=3$, with iteration $T^k(n)$ the sequence produced is $3 \rightarrow 5 \rightarrow 8 \rightarrow 4 \rightarrow 2 \rightarrow 1$, and the cycle becomes $1 \rightarrow 2 \rightarrow 1$.
The \emph{total stopping time} $\sigma_\infty(n)$ of $n$ is the least $k$ applications of $T$ that the sequence of the iteration will reach 1 for the first time:
\begin{equation}
\label{eq:3}
\sigma_\infty(n) = \inf\:\{k:T^k(n) =1\}.
\end{equation}
For example, $\sigma_\infty(3)= 5.$
In this paper, the iteration $T^k(n)$ of each number $n$ is encoded as a binary number (or string), and in turn as a unique positive integer $m$. The binary string is mapped to a curve in 2-D in an analogous way that the binary string of the Dragon Curve is mapped to the corresponding curve.
\section{The Binary Encoding}
\label{sec:binary-string-}
The conversion of a decimal number $n$ to its corresponding binary representation involves successive divisions by 2 and looking at the parity of the result. We note that the process is similar to the Collatz iteration: each time $T(n)$ is applied a decision is made based on whether $n$ is even or odd. We choose to have a ``1'' if $T^k(n)$ is even and ``0'' if odd. For example, $n=3$ is encoded as 11100. The bits are produced, just like in the usual decimal-to-binary conversion, from right to left. Table~\ref{tab:1} shows the binary encoding and decimal equivalent.
For all $n \geq 2$, assuming that the Collatz Conjecture is true, the last application of $T$ is on $2$, an even number, before reaching $1$. Making the same assumption, the binary encoding of all $n \geq 5$ begins with $111$, noting that eventually the iteration will end with $8 \rightarrow 4 \rightarrow 2 \rightarrow 1$. The length of the binary encoding of $n$ is $\sigma_\infty(n)$ since the number of digits is exactly the number of applications of $T$ needed to reach $1$ in the sequence of iterations. The choice of when to use ``1'' or ``0'' was made so that the binary encoding of $n$, the original number, starts with a ``1'', to avoid having any leading zeros. Had we had any leading zeros, the binary string would not have been recoverable from its corresponding decimal number, since leading zeros are not significant and are ignored when converting to decimal. This sequence was independently published as sequence A304715 in OEIS \cite{A304715}, which we noticed after a draft of this paper had been released.
Given the binary encoding of $n$, it is a simple matter to find the original $n$. It can be done by the following process: Scan its binary encoding $B=b_lb_{l-1}\ldots{}b_1b_0$, where $b_l$ is the most significant bit, from left to right. Starting with initial $n=1$ and $i=0$, iteratively apply $R^i(n,b)$, for $i = 0, 1, \ldots, l$:
\begin{equation}
\label{eq:5}
R^i(n,b) =
\begin{cases}
2n, & {\text{if $b_{l-i} = 1$}}\\
\frac{2n-1}{3}, & {\text{if $b_{l-i}=0$}}.
\end{cases}
\end{equation}
As an example, given the encoding $B=11100$, starting with $n=1$ and iteratively applying $R^i(n,b)$ we obtain the sequence $1 \rightarrow 2 \rightarrow 4 \rightarrow 8 \rightarrow 5 \rightarrow 3$, and thus we obtain $n=3$. This is the reverse sequence produced by Equation~\ref{eq:2}.
\begin{table}[ht]
\centering\small
\begin{tabular}{|r|r|r|}\hline
$n$&\makecell[cc]{Binary encoding}&\makecell[cc]{Decimal}\\\hline
2 & 1 & 1 \\
3 & 11100 & 28 \\
4 & 11 & 3 \\
5 & 1110 & 14 \\
6 & 111001 & 57 \\
7 & 11101101000 & 1896 \\
8 & 111 & 7 \\
9 & 1110110100010 & 7586 \\
10 & 11101 & 29 \\
11 & 1110110100 & 948 \\
12 & 1110011 & 115 \\
13 & 1110110 & 118 \\
14 & 111011010001 & 3793 \\
15 & 111011110000 & 3824 \\
16 & 1111 & 15 \\
17 & 111011010 & 474 \\
18 & 11101101000101 & 15173 \\
19 & 11101101001100 & 15180 \\
20 & 111011 & 59 \\
21 & 111110 & 62 \\
22 & 11101101001 & 1897 \\
23 & 11101111000 & 1912 \\
24 & 11100111 & 231 \\
25 & 1110110100110010 & 60722 \\
26 & 11101101 & 237 \\
27 & \makecell[tl]{%
111011110001101110\\
101011100001100000\\
010010001101000010\\
\hphantom{01}0010010100000100} & 1102691417057682138372 \\
28 & 1110110100011 & 7587 \\
29 & 1110110100110 & 7590 \\
30 & 1110111100001 & 7649\\\hline
\end{tabular}
\caption{The binary and corresponding decimal encoding.}\label{tab:1}
\end{table}
\section{Collatz Curves: Visualizing the Binary Encoding}
\label{sec:visualizing-n}
For each positive integer $n\ge 2$, we will visualize the Collatz sequence generated by the iteration $T^k(n)$ by making use of its binary encoding (Table~\ref{tab:1}). The binary encoding is viewed as a random walk where each of its digits generates a unit length line segment in a certain direction.
The manner of generating the Collatz curve corresponding to $n$ is similar to that of generating the Dragon Curve (Figure~\ref{fig:1}) \cite{Davis1970,Gardner1978a}.
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.5\linewidth,keepaspectratio]{dragon-10.pdf}
\caption{The Dragon Curve of order $10$.}
\label{fig:1}
\end{figure}
The string $S_k$ of the Dragon Curve of order $k$ is generated as follows. There are two symbols, R (right) and L (left). In the first iteration (order $k=1$) the string $S_1$ is just the symbol R. In each successive iteration the new string $S_{n+1}$ is recursively produced:
\begin{equation}
S_{n+1}=S_n + R + \overline{S}_n,
\end{equation}
where $+$ is string concatenation and $\overline{S}_n$ is $S_n$ reversed with each of its symbols flipped. Table~\ref{tab:2} shows the first four iterations.
\begin{table}[ht]
\centering\small
\begin{tabular}{|c|c|}\hline
Iteration ($k$)&\makecell[cc]{String $S_k$}\\\hline
1&R\\
2&R R L\\
3&R R L R R L L\\
4&R R L R R L L R R R L L R L L\\\hline
\end{tabular}
\caption{The first four iterations of the Dragon Curve.}\label{tab:2}
\end{table}
Let $S_k=s_ls_{l-1}\ldots s_1s_0.$ From $S_k$ a curve can be produced such as the one in Figure~\ref{fig:1} ($k=10$) by making use of turtle graphics \cite{Abelson1986,dragon}: string $S_k$ is scanned from left to right:\\
\begin{center}
\begin{tabular}[h]{ll}
if $s_i = R$, &turn right $90^\circ$, move forward one unit\\
if $s_i=L$,& turn left $90^\circ$, move forward one unit.
\end{tabular}
\end{center}
The initial orientation of the turtle in all figures in this paper is facing east.
The curve of each Collatz sequence is produced in an analogous way. The digits of the binary encoding $B$ of $n$ are scanned from right to left.
\begin{center}
\begin{tabular}[h]{ll}
if $b_i = 1$, &turn right $90^\circ$, move forward one unit\\
if $b_i=0$,& turn left $90^\circ$, move forward one unit.
\end{tabular}
\end{center}
The curve for $n=3$ appears in Figure~\ref{fig:2}. Its binary encoding is $11100$. The initial point, that is the initial position of the turtle, is at the bottom of the curve. A more involved curve is generated for $n=27$ (Figure~\ref{fig:3}), a number known for its relatively long binary encoding, which is of length $70$. A still more involved curve, for $n=75,128,138,247$, is shown in Figure~\ref{fig:4}, which bears a resemblance of the figure of a lion. Its binary sequence is of length $767$.
\begin{figure}[!htbp]
\centering
\includegraphics[height=1.0cm,keepaspectratio]{3c}
\caption{The curve for $n=3$.}
\label{fig:2}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.3\linewidth,keepaspectratio]{27c.pdf}
\caption{The curve for $n=27$.}
\label{fig:3}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.5\linewidth,keepaspectratio]{75128138247.pdf}
\caption{The ``lion curve'' for $n=75,128,138,247$.}
\label{fig:4}
\end{figure}
The curve for $n=6$ (Figure~\ref{fig:5}, also in Figure~\ref{fig:6}), whose binary encoding is $111001$, coincidentally forms the shape of $6$! The orientation is arbitrary since it only depends on the starting orientation of the turtle.
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.10\linewidth,keepaspectratio]{6c.pdf}
\caption{The curve for $n=6$.}
\label{fig:5}
\end{figure}
Figure~\ref{fig:6} shows the Collatz curves for $2\le n \le 89$ and Figure~\ref{fig:7} those for $90\le n \le 177$.
One important difference between the Dragon Curve and the curves we generate using the Collatz iteration is the former does not intersect itself \cite{Davis1970}, whereas the latter ones do. It is possible, and it often happens, that the curve as it is generated writes over previously generated segments. Hence, although the binary encoding is unique for each $n$, as discussed in Section~\ref{sec:binary-string-}, its corresponding curve is not. As an example, the curves for $n=2^k, k \ge 4$ appear identical: a unit square, which can be seen in Figures~\ref{fig:6} and \ref{fig:7} for $n=16, 32, 64, 128.$ But powers of two are not the only ones that produce a square, e.g. consider the figures of $n= 5, 21, 85.$
\begin{theorem}\label{theo:1}
Assuming that the Collatz Conjecture is true, all curves for $n \ge 2$, except those for $n = 2,4, \text{or } 8$, end with a cycle, i.e. a unit square.
\end{theorem}
\begin{IEEEproof}
For $n=2,4,8$ the binary encoding is of length three or less (Table~\ref{tab:1}), which precludes forming a cycle, since a cycle requires at least 4 segments. For all other $n$ the binary encoding is of length 4 or more and the iteration ends in the cycle $4\rightarrow 2\rightarrow 1$. Necessarily, the sequence before reaching $1$ for the first time is $8\rightarrow 4\rightarrow 2\rightarrow 1$, which corresponds to $111$. Right before $111$ is drawn, the turtle is at the end of an existing segment. Turning right and drawing 3 times creates a final cycle.
\end{IEEEproof}
\begin{definition}
The \emph{girth} $g(G)$ of a graph $G$ is the minimum length of a cycle contained in $G$ \cite{Diestel2017}. If $G$ has no cycles, $g(G)=\infty$.
\end{definition}
\begin{corollary}
When the Collatz curve is viewed as a graph $G(n)=(V,E)$, $g(G(n))=4$ for all $n > 8$.
\end{corollary}
The curves allow for easy identification of similar patterns in the Collatz iteration. As an example, visually inspecting the curves in Figure~\ref{fig:6}, we note that those for $n=27$, $31$, $41$, $47$, $54$, $55$, $62$, $63$, $71$, $73$, $82$, and $83$ bear a strong resemblance to each other. A closer look at the binary encodings reveals that indeed when compared pairwise they have a long common substring (subsequence). Table~\ref{tab:4} shows the lengths of the longest common substrings, all of which happen to start from the leftmost bit position (see Table~\ref{tab:5}). Each common substring is of length at least 60. In Table~\ref{tab:4}, on the diagonal is the length of the binary encoding of the corresponding $n$.
\begin{table}[ht]
\centerin
\resizebox{\columnwidth}{!}{\begin{tabular}{c|cccccccccccc}
&27&31&41&47&54&55&62&63&71&73&82&83\\\hline
27&\textbf{70}&67&69&66&70&66&68&60&65&66&69&66\\
31&67&\textbf{67}&67&66&67&66&67&60&65&66&67&66\\
41&69&67&\textbf{69}&66&69&66&68&60&65&66&69&66\\
47&66&66&66&\textbf{66}&66&66&66&60&65&66&66&66\\
54&70&67&69&66&\textbf{71}&66&68&60&65&66&69&66\\
55&66&66&66&66&66&\textbf{71}&66&60&65&71&66&70\\
62&68&67&68&66&68&66&\textbf{68}&60&65&66&68&66\\
63&60&60&60&60&60&60&60&\textbf{68}&60&60&60&60\\
71&65&65&65&65&65&65&65&60&\textbf{65}&65&65&65\\
73&66&66&66&66&66&71&66&60&65&\textbf{73}&66&70\\
82&69&67&69&66&69&66&68&60&65&66&\textbf{70}&66\\
83&66&66&66&66&66&70&66&60&65&70&66&\textbf{70}
\end{tabular}}
\caption{The pairwise length of the longest common substring of the binary encodings. On the diagonal is the length of the binary encoding of the corresponding $n$.}\label{tab:4}
\end{table}
\begin{table*}[ht]
\centering\normalsiz
\begin{tabular}{c|l}
n&\makecell[cc]{Binary encoding}\\\hline
27&11101111000110111010101110000110000001001000110100001000100\textbf{1}0100000100\\
31&11101111000110111010101110000110000001001000110100001000100\textbf{1}0100000\\
41&11101111000110111010101110000110000001001000110100001000100\textbf{1}010000010\\
47&11101111000110111010101110000110000001001000110100001000100\textbf{1}010000\\
54&11101111000110111010101110000110000001001000110100001000100\textbf{1}01000001001\\
55&11101111000110111010101110000110000001001000110100001000100\textbf{1}01000011000\\
62&11101111000110111010101110000110000001001000110100001000100\textbf{1}01000001\\
63&11101111000110111010101110000110000001001000110100001000100\textbf{1}11000000\\
71&11101111000110111010101110000110000001001000110100001000100\textbf{1}01000\\
73&11101111000110111010101110000110000001001000110100001000100\textbf{1}0100001100010\\
82&11101111000110111010101110000110000001001000110100001000100\textbf{1}0100000101\\
83&11101111000110111010101110000110000001001000110100001000100\textbf{1}0100001100\\
\end{tabular
\caption{Numbers with similar binary encodings, i.e. similar trajectories in the Collatz iteration. They have been identified visually via their Collatz curves. Pairwise they have a common substring, leftmost bits, of length at least 60. The 60th bit from the left is in bold.}\label{tab:5}
\end{table*}
\section{Reverse Curves}
\label{sec:reverse-curves}
Another version of the Collatz curves can be obtained by applying the same rules except that the bits in the encoding of $n$ are scanned in reverse, from left-to-right. The resulting reverse curves, or \emph{r-curves}, are very similar to the Collatz curves. In general they are not affine transformations of each other. Theorem~\ref{theo:1} does not apply, and thus the curves no longer necessarily have a cycle for $n>8$. As an example, compare the two curves for $n=33$ in Figure~\ref{fig:8}. The r-curve has no cycle. Figure~\ref{fig:13} shows the r-curves for $2\le n \le 89$ and Figure~\ref{fig:14} those for $90\le n \le 177$.
\begin{figure}[!htbp]
\centering
\fbox{\includegraphics[height=2.0cm,keepaspectratio]{33c}}\hspace*{2.0cm}
\fbox{\includegraphics[height=2.0cm,keepaspectratio]{33r}}
\caption{For $n=33$, the Collatz curve (left) and the r-curve (right). Note the absence of a cycle in the r-curve.}
\label{fig:8}
\end{figure}
Two questions arise: (1) how many r-curves are devoid of cycles, and (2) how many r-curves have finite girth $g(G(n))>4$, i.e. they have a cycle of minimum length other than the unit square. Using Python 3 and the software library \emph{NetworkX} \cite{Hagberg2008}, we have checked all r-curves up to $n \le 10^8$ and we found that there are only 40 r-curves with no cycles, with the largest $n=308$. All such curves can be found in Figure~\ref{fig:9}.
In the same range, i.e. up to $n \le 10^8$, it was found that there are only three r-curves having finite girth $g(G(n))>4$, and those are for $n=273, 410, \text{and } 820$. In all three cases $g(G(n))=12$. The only cycle in these curves is the cross pattern which can be seen in Figure~\ref{fig:10}. We call these curves \emph{miracle curves}. The ones for $410$ and $820$ are identical due to overwriting.
\begin{figure}[!tbhp]
\captionsetup[subfigure]{labelformat=empty}
\centering
\subfloat[273]{\fbox{\includegraphics[height=2.5cm]{273l}}}
\hfil
\subfloat[410]{\fbox{\includegraphics[height=2.5cm]{410l}}}
\hfil
\subfloat[820]{\fbox{\includegraphics[height=2.5cm]{820l}}}
\caption{Miracle curves: The only three r-curves found having a cycle of minimum length greater than $4$, verified up to $n \le 10^8$. Their only cycle, the cross pattern, has length $12$. The curves for $410$ and $820$ are identical due to overwriting.}
\label{fig:10}
\end{figure}
\section{Alternative Curves}
\label{sec:alternative-curves}
It is possible to use alternative schemes to draw the curves that correspond to the binary encoding of $n$ by modifying the rules. For example, we can use this rule for the turtle:\\
\begin{tabular}[h]{ll}
if $b_i = b_{i-1}$,& move forward one unit\\
if $b_i \ne b_{i-1}$,& turn right $90^\circ$, move forward one unit.\\
\end{tabular}\\
\noindent In other words, as long as there is no change from one bit to the next, scanning the bits from right to left, either from 0 to 1 or vice versa, the turtle keeps moving forward. If there is a change, the turtle turns right $90^\circ$ and moves forward. The \emph{on-change-turn-right} curves produced with this rule appear in Figures~\ref{fig:15} and \ref{fig:16}. Visual identification of similar curves, e.g. those similar to the curve of $n=27$, is easily done. These curves are characterized by longer straight lines compared to the Collatz curves, Figures~\ref{fig:6} and \ref{fig:7}. For comparison with Figure~\ref{fig:3}, the on-change-turn-right curve for $n=27$ is shown in Figure~\ref{fig:11}. For purely aesthetic reasons, we show the picture of the same curve 3-D printed in Figure~\ref{fig:12}.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.18\linewidth,keepaspectratio]{27cc.pdf}
\captionof{figure}{The on-change-turn-right curve for $n=27$.}
\label{fig:11}
\end{center}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.3\linewidth,keepaspectratio]{27-3-d.jpg}
\caption{The on-change-turn-right curve for ${n=27}$ 3-D printed.}
\label{fig:12}
\end{figure}
\section{Conclusion}
\begin{figure*}[!hbt]
\centering
\includegraphics[width=0.8\textwidth,keepaspectratio]{fig-no-cycles-cropped}
\caption{The r-curves without cycles. Verified up to $n \le 10^8$.}\label{fig:9}
\end{figure*}
The Collatz iteration for each $n\ge 2$ is mapped to a binary sequence, which in turn is mapped to an integer. The binary sequence is translated to a 2-D curve on a grid, the Collatz curve, in an analogous way that the Dragon Curve is drawn from a binary sequence. Two variants of the curves were explored: the reverse curves (\emph{r-curves}) and the the \emph{on-change-turn-right} curves. There is a scarcity of acyclic r-curves, only 40 were found, and only three r-curves were found having a shortest length cycle of length greater than four, the \emph{miracle curves}. This was verified up to $n\le 10^8$. The Collatz curves, the r-curves, and the on-change-turn-right curves provide a way to visualize and qualitatively compare Collatz iterations.
\section*{Acknowledgment}
\label{sec:acknowledgement}
I would like to thank Rémy Sigrist for suggesting the exploration of the reverse curves and Arusyak Hovhannesyan for 3-D printing the on-change-turn-right curve for $n=27$.
\begin{figure*}[ht]
\centering
\includegraphics[width=0.95\textwidth,keepaspectratio]{2-89}
\caption{The Collatz curves for $n=2, 3, \ldots, 89.$}
\label{fig:6}
\end{figure*}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.95\textwidth,keepaspectratio]{90-177}
\caption{The Collatz curves for $n=90, 91, \ldots, 177.$}
\label{fig:7}
\end{figure*}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.9\textwidth,keepaspectratio]{rev2-89}
\caption{The r-curves for $n=2, 3, \ldots, 89.$}
\label{fig:13}
\end{figure*}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.9\textwidth,keepaspectratio]{rev90-177}
\caption{The r-curves for $n=90, 91, \ldots, 177.$}
\label{fig:14}
\end{figure*}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.95\textwidth,keepaspectratio]{memory-2-67}
\caption{The on-change-turn-right curves for $n=2, 3, \ldots, 67.$}
\label{fig:15}
\end{figure*}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.95\textwidth,keepaspectratio]{memory-68-144}
\caption{The on-change-turn-right curves for $n=68, 69, \ldots, 144.$}
\label{fig:16}
\end{figure*}
\FloatBarrier
\renewcommand*{\UrlFont}{\rmfamily}
| {
"timestamp": "2019-01-04T02:04:43",
"yymm": "1811",
"arxiv_id": "1811.00384",
"language": "en",
"url": "https://arxiv.org/abs/1811.00384",
"abstract": "The Collatz conjecture is one of the easiest mathematical problems to state and yet it remains unsolved. For each $n\\ge 2$ the Collatz iteration is mapped to a binary sequence and a corresponding unique integer which can recreate the iteration. The binary sequence is used to produce the Collatz curve, a 2-D visualization of the iteration on a grid, which, besides the aesthetics, provides a qualitative way for comparing iterations. Two variants of the curves are explored, the r-curves and on-change-turn-right curves. There is a scarcity of acyclic r-curves and only three r-curves were found having a cycle of minimum length greater than 4.",
"subjects": "History and Overview (math.HO)",
"title": "Encoding and Visualization in the Collatz Conjecture",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754501811438,
"lm_q2_score": 0.7185943985973772,
"lm_q1q2_score": 0.707510403496661
} |
https://arxiv.org/abs/1705.01199 | Four Edge-Independent Spanning Trees | We prove an ear-decomposition theorem for $4$-edge-connected graphs and use it to prove that for every $4$-edge-connected graph $G$ and every $r\in V(G)$, there is a set of four spanning trees of $G$ with the following property. For every vertex in $G$, the unique paths back to $r$ in each tree are edge-disjoint. Our proof implies a polynomial-time algorithm for constructing the trees. | \section{Introduction}
\noindent
If $r$ is a vertex of a graph $G$, two subtrees $T_1,T_2$ of $G$ are \textit{edge-independent with root $r$} if each tree contains $r$, and for each $v\in V(T_1)\cap V(T_2)$, the unique path in $T_1$ between $r$ and $v$ is edge-disjoint from the unique path in $T_2$ between $r$ and $v$. Larger sets of trees are called \textit{edge-independent with root $r$} if they are pairwise edge-independent with root $r$.
Itai and Rodeh~\cite{ItaiRodeh} posed the Edge-Independent Tree Conjecture, that for every $k$-edge-connected graph $G$ and every $r\in V(G)$, there is a set of $k$ edge-independent spanning trees of $G$ rooted at $r$. Here, we prove the case $k=4$ of the Edge-Independent Tree Conjecture. That is, we prove the following:
\begin{theorem}\label{thmTrees}
If $G$ is a $4$-edge-connected graph and $r\in V(G)$, then there exists a set of four edge-independent spanning trees of $G$ rooted at $r$.
\end{theorem}
There is a similar conjecture which has been studied in parallel, concerning vertices rather than edges. If $r$ is a vertex of $G$, two subtrees $T_1,T_2$ of $G$ are \textit{independent with root $r$} if each tree contains $r$, and for each $v\in V(T_1)\cap V(T_2)$, the unique path in $T_1$ between $r$ and $v$ is internally vertex-disjoint from the unique path in $T_2$ between $r$ and $v$. Larger sets of trees are called \textit{independent with root $r$} if they are pairwise independent with root $r$.
Itai and Rodeh~\cite{ItaiRodeh} also posed the Independent Tree Conjecture, that for every $k$-connected graph $G$ and for every $r\in V(G)$, there is a set of $k$ independent spanning trees of $G$ rooted at $r$.
The case $k=2$ of each conjecture was proven by Itai and Rodeh~\cite{ItaiRodeh}. The case $k=3$ of the Independent Tree Conjecture was proven by Cheriyan and Maheshwari~\cite{CheriMahesh}, and then independently by Zehavi and Itai~\cite{ZehaviItai}. Huck~\cite{Huck} proved the Independent Tree Conjecture for planar graphs (with any $k$). Building on this work and that of Kawarabayashi, Lee, and Yu~\cite{klylovasz}, the case $k=4$ of the Independent Tree Conjecture was proven by Curran, Lee, and Yu across two papers~\cite{clychain,cly}. The Independent Tree Conjecture is open for nonplanar graphs with $k>4$.
In 1992, Khuller and Schieber~\cite{Khuller} published a later-disproven argument that the Independent Tree Conjecture implies the Edge-Independent Tree Conjecture. Gopalan and Ramasubramanian~\cite{Gopalan} demonstrated that Khuller and Schieber's proof fails, but salvaged the technique, and proved the case $k=3$ of the Edge-Independent Tree Conjecture by reducing it to the case $k=3$ of the Independent Tree Conjecture. Schlipf and Schmidt~\cite{Schlipf} provided an alternate proof of the case $k=3$ of the Edge-Independent Tree Conjecture, which does not rely on the Independent Tree Conjecture. The case $k=4$ of the Edge-Independent Tree Conjecture is proven here, while the case $k>4$ remains open.
By adapting the technique of Schlipf and Schmidt~\cite{Schlipf}, we prove an edge analog of the planar chain decomposition of Curran, Lee, and Yu~\cite{clychain}. We then use this decomposition to create two edge numberings which define the required trees.
The conjectures are related to network communication with redundancy. If $G$ represents a communication network, one can wonder if information can be broadcast through the entire network with resistance to edge failures (i.e. it would require $k$ simultaneous edge failures to disconnect a client from every broadcast). The Edge-Independent Tree Conjecture implies that the absence of edge bottlenecks of size less than $k$ is necessary and sufficient for a redundant broadcast to be possible from any source $r$. The Independent Tree Conjecture answers the analogous problem where vertex failures are the concern, rather than edge failures.
\section{The Chain Decomposition}
\noindent
In this paper, a \textit{graph} will refer to what is commonly called a multigraph. That is, there may be multiple edges between the same pair of vertices (``parallel edges") and an edge may connect a vertex to itself (a ``loop"). All paths and cycles are simple, meaning they have no repeated vertices or edges. We consider a loop to induce a cycle of length one and a pair of parallel edges to induce a cycle of length two. Also, the presence of a loop increases the degree of a vertex by two. We will use the overline notation $\overline{H}$ to name specific subgraphs, rather than for the graph complement.
Throughout this section, fix a graph $G$ with $\abs{V(G)}\geq1$ and a vertex $r\in V(G)$. We begin by defining a decomposition analogous to the planar chain decomposition in~\cite{clychain}.
\begin{definition*}
An \textit{up chain} of $G$ with respect to a pair of edge-disjoint subgraphs ($H$, $\overline{H}$) is a subgraph of $G$, edge-disjoint from $H$ and $\overline{H}$, which is either:
\begin{enumerate}[label=\roman*]
\item A path with at least one edge such that every vertex is either $r$ or has degree at least two in $\overline{H}$, and the ends are either $r$ or are in $H$, OR
\item A cycle such that every vertex is either $r$ or has degree at least two in $\overline{H}$, and some vertex $v$ is either $r$ or has degree at least two in $H$. We will consider $v$ to be both ends of the chain, and all other vertices in the chain to be internal vertices.
\end{enumerate}
Chains which are paths will be called \textit{open} and chains which are cycles will be called \textit{closed}, analogous to the standard ear decomposition.
\end{definition*}
\begin{definition*}
A \textit{down chain} of $G$ with respect to a pair of edge-disjoint subgraphs ($H$, $\overline{H}$) is an up chain with respect to ($\overline{H}$, $H$).
\end{definition*}
\begin{definition*}
A \textit{one-way chain} of $G$ with respect to the pair of edge-disjoint subgraphs ($H$, $\overline{H}$) is a subgraph of $G$, induced by an edge $e\notin H\cup\overline{H}$ with ends $u$ and $v$, such that $u$ is either $r$ or has degree at least two in $H$, and $v$ is either $r$ or has degree at least two in $\overline{H}$. We call $u$ the \textit{tail} of the chain and $v$ the \textit{head}.
\end{definition*}
\begin{definition*}
Let $G_0,G_1,\ldots,G_m$ be a sequence of subgraphs of $G$. Denote $H_i=G_0\cup G_1\cup\cdots\cup G_{i-1}$ and $\overline{H_i}=G_{i+1}\cup G_{i+2}\cup\cdots\cup G_m$, so that $H_0$ and $\overline{H_m}$ are the null graph. We say that the sequence $G_0,G_1,\ldots,G_m$ is a \textit{chain decomposition} of $G$ rooted at $r$ if:
\begin{enumerate}
\item The sets $E(G_0),E(G_1),\ldots,E(G_m)$ partition $E(G)$, AND
\item For $i=0,\ldots,m$, the subgraph $G_i$ is either an up chain, a down chain, or a one-way chain with respect to the subgraphs $(H_i,\overline{H_i})$.
\end{enumerate}
\end{definition*}
\begin{figure}[h]
\centering
\includegraphics[scale=0.6]{arXivPNG1.png}
\caption{An illustration of an up chain of length 4, a down chain of length 3, and a one-way chain. The red/dashed edges are in earlier chains, while the blue/dotted edges are in later chains.}
\label{figChains}
\end{figure}
\begin{definition*}
The \textit{chain index} of $e\in E(G)$, denoted $CI(e)$, is the index of the chain containing $e$.
\end{definition*}
\begin{definition*}
An up chain $G_i$ is \textbf{minimal} if no internal vertex of $G_i$ is in $\{r\}\cup V\left(H_i\right)$.
\end{definition*}
\begin{definition*}
A down chain $G_i$ is \textbf{minimal} if no internal vertex of $G_i$ is in $\{r\}\cup V\left(\overline{H_i}\right)$.
\end{definition*}
\begin{definition*}
A chain decomposition is \textbf{minimal} if all of its up chains and down chains are minimal.
\end{definition*}
\begin{remarks*}
\ \\\vspace{-15pt}
\begin{enumerate}
\item A minimal up chain is a special case of an ear in the standard ear decomposition.
\item The chain decomposition is symmetric in the following sense. If $G_0,G_1,\ldots,G_m$ is a chain decomposition rooted at $r$, then $G_m,G_{m-1},\ldots,G_0$ is a chain decomposition rooted at $r$, with the up and down chains switched and the heads and tails of one-way chains switched. Throughout this paper, we will refer to this fact as ``symmetry".
\item $G_0$ is either a closed up chain ending at $r$ or a one-way chain with $r$ as the tail, and $G_m$ is either a closed down chain ending at $r$ or a one-way chain with $r$ as the head.
\item In the planar chain decomposition in~\cite{clychain}, up chains and down chains are analogous to the corresponding open chains. The elementary chain is analogous to a one-way chain.
\end{enumerate}
\end{remarks*}
\begin{remark}\label{rmkMin}
An up chain or down chain may be subdivided into several minimal chains by breaking at the offending internal vertices. These minimal chains may then be inserted consecutively to the decomposition at the index of the old chain. In this way, one can easily obtain a minimal chain decomposition from any chain decomposition.
\end{remark}
We will prove Theorem~\ref{thmTrees} by combining the following results:
\begin{theorem}\label{thmChains}
If $G$ is a $4$-edge-connected graph and $r\in V(G)$, then $G$ has a chain decomposition rooted at $r$.
\end{theorem}
\begin{theorem}\label{thmTreesFromChains}
Suppose $G$ is a graph with no isolated vertices. If $G$ has a chain decomposition rooted at some $r\in V(G)$, then there exists a set of four edge-independent spanning trees of $G$ rooted at $r$.
\end{theorem}
\section{Preliminary Results}
While not needed for our main results, the following proposition demonstrates how the chain decomposition fits in with the various decompositions used in other cases of the Independent Tree Conjecture and Edge-Independent Tree Conjecture. A partial chain decomposition and its complement are ``almost $2$-edge-connected" in the following sense.
\begin{proposition}\label{prp2con}
Suppose $G_0,G_1,\ldots,G_m$ is a chain decomposition of a graph $G$ rooted at $r$. Then for $i=1,\ldots,m$, $H_i$ and $\overline{H_{i-1}}$ are connected. Further, if $e$ is a cut edge of $H_i$ (resp. $\overline{H_{i-1}}$), then $e$ induces a one-way chain and one component of $H_i-e$ (resp. $\overline{H_{i-1}}-e$) contains one vertex and no edges.
\end{proposition}
\begin{proof}
By symmetry, we need only prove the result for the $H_i$'s. The connectivity follows from the fact that every type of chain is connected and contains at least one vertex in an earlier chain.
Suppose $e$ is a cut edge of some $H_i$. Since $e$ is an edge in $H_i$, we have $CI(e)<i$ and $H_{CI(e)}\subset H_i$. We also know that $H_{CI(e)}$ is connected by the previous paragraph. Then $e$ cannot be part of an up chain, or else $e$ would be part of a cycle formed by the chain $G_{CI(e)}$ and a path in $H_{CI(e)}$ between the ends of $G_{CI(e)}$ (if $G_{CI(e)}$ is open; else the chain itself is a cycle). Also, $e$ cannot be part of a down chain, or else $e$ would be part of a cycle formed by $e$ and a path in $H_{CI(e)}$ between the ends of $e$. Therefore, $e$ induces a one-way chain.
Let $C$ be the component of $H_i-e$ not containing $r$, and suppose for the sake of contradiction that $C$ contains an edge. Let $e'$ be an edge of $C$ with minimal chain index. Consider $G_{CI(e')}$, the chain containing $e'$. Regardless of the chain type, some vertices in $V(G_{CI(e')})$ are incident to at least two edges in $H_{CI(e')}\subset H_i$ since $r\notin C$, so one of these edges is not $e$. This contradicts the minimality of $CI(e')$.
\end{proof}
The next lemma and its corollary will allow us to ignore the possibility of loops in the graph when convenient.
\begin{lemma}\label{lmaNoLoops}
Suppose $G_0,G_1,\ldots,G_m$ is a chain decomposition of $G$ rooted at $r$. If $v\neq r$ is in $H_i$ (resp $\overline{H_i}$), then $v$ is incident to a non-loop edge in $H_i$ (resp $\overline{H_i}$). If $v$ has degree at least two in $H_i$ (resp. $\overline{H_i}$), then $v$ is incident to two distinct non-loop edges in $H_i$ (resp. $\overline{H_i}$).
\end{lemma}
\begin{proof}
Note that the second claim in the lemma implies the first, since a loop increases the degree of a vertex by 2, so it suffices to prove the second claim in the lemma.
Suppose $v$ is incident to a loop, which by symmetry we may assume is in $H_i$. Of all loops incident to $v$, choose the one with minimal chain index $j<i$. Consider the chain classification of $G_j$. The chain definitions all coincide for a loop, and require that $v$($\neq r$) has degree at least two in $H_j$. By the minimality of $j$, $v$ is not incident to any loops in $H_j$. It follows that $v$ is incident to two distinct non-loop edges in $H_j\subset H_i$.
\end{proof}
\begin{corollary}\label{corNoLoops}
Suppose $G_0,G_1,\ldots,G_m$ is a chain decomposition of $G$ rooted at $r$, and $e\in E(G_i)$ is a loop. Then $G_0,G_1,\ldots,G_{i-1},G_{i+1},\ldots,G_m$ is a chain decomposition of $G-e$ rooted at $r$. Further, if $G$ has no isolated vertices, then $G-e$ has no isolated vertices.
\end{corollary}
\begin{proof}
The first claim follows from the preceding lemma. For the second, observe that if $e$ is the only edge incident to its end, then it fails the conditions for every chain definition.
\end{proof}
Next, we prove the following useful fact about minimal chain decompositions.
\begin{lemma}\label{lmaDeg2}
Suppose $G$ is a graph with no isolated vertices, $G_0,G_1,\ldots,G_m$ is a minimal chain decomposition of $G$ rooted at $r$, and $v\in V(G)$ with $v\neq r$. Then there are indices $i,j$ so that $v$ has degree exactly two in $H_i$ and $\overline{H_j}$.
\end{lemma}
\begin{proof}
By symmetry, we need only find $i$. Since $G$ has no isolated vertices, $v$ is in some chain. Consider the chain $G_{i_0}$ containing $v$ so that $i_0$ is minimal. Note that $v\notin V(H_{i_0})$.
If $G_{i_0}$ is an up chain, then $v$ is an internal vertex of $G_{i_0}$ since $v\notin V(H_{i_0})$, so $v$ has degree two in $G_{i_0}$ and degree at least two in $\overline{H_{i_0}}$. Therefore $\overline{H_{i_0}}$ is not null, so $i_0<m$. Then $i=i_0+1$ completes the proof.
The chain $G_{i_0}$ is not a down chain since $v\notin V(H_{i_0})$.
So we may assume that $G_{i_0}$ is a one-way chain, and $v$ must be the head since $v\notin V(H_{i_0})$. Therefore $v$ has degree at least two in $\overline{H_{i_0}}$, so we may consider the next chain to contain $v$, say $G_{i_1}$. Note that $v$ has degree one in $H_{i_1}$ by the definition of $i_1$.
If $G_{i_1}$ is an up chain, then it is open and $v$ is an end of the chain, since the chain decomposition is minimal and $v$ has degree one in $H_{i_1}$. The chain $G_{i_1}$ is not a down chain since $v$ has degree one in $H_{i_1}$. If $G_{i_1}$ is a one-way chain, then $v$ is the head since $v$($\neq r$) does not have degree at least two in $H_{i_1}$. In all cases, $v$ has degree one in $G_{i_1}$ and degree at least two in $\overline{H_{i_1}}$. Therefore $\overline{H_{i_1}}$ is not null, so $i_1<m$. Then $i=i_1+1$ completes the proof.
\end{proof}
Finally, we show that the chain decomposition implies a minimum degree result.
\begin{lemma}\label{lmaMinDeg4}
Suppose $G$ is a graph with no isolated vertices, $G_0,G_1,\ldots,G_m$ is a chain decomposition of $G$ rooted at $r$, and $v\in V(G)$ with $v\neq r$. Then $v$ has degree at least $4$.
\end{lemma}
\begin{proof}
By Corollary \ref{corNoLoops}, we may assume that there are no loops in $G$. If $v$ is in an up chain $G_i$, then $v$ has degree at least $2$ in $\overline{H_i}$, and either degree $2$ in $G_i$ (if $v$ is internal) or degree at least $1$ in $G_i$ and degree at least $1$ in $H_i$ (if $v$ is an end). Either way, $v$ has degree at least $4$ in $G$. By symmetry, the same is true if $v$ is in a down chain.
So we may assume that the only chains containing $v$ are one-way chains. Since $G$ has no isolated vertices, there is at least one such chain $G_j$. Then $v$ has degree $1$ in $G_j$ and degree at least $2$ in $H_j$ (if $v$ is the tail) or $\overline{H_j}$ (if $v$ is the head). We conclude that $v$ has degree at least $3$ in $G$.
Assume for the sake of contradiction that $v$ does not have degree at least $4$. Then $v$ has degree $3$ and is in exactly three one-way chains, say $G_{\ell_1}$, $G_{\ell_2}$, $G_{\ell_3}$ with $\ell_1<\ell_2<\ell_3$. Consider $G_{\ell_2}$. Since we know all of the chains containing $v$, we can say that $v$ has degree $1$ in $H_{\ell_2}$ and degree $1$ in $\overline{H_{\ell_2}}$. This contradicts the definition of a one-way chain, as $v$ can be neither the head nor the tail of the chain $G_{\ell_2}$. We conclude that $v$ has degree at least $4$ as desired.
\end{proof}
\begin{remark*}
If $\abs{V(G)}\geq2$ in addition to $G$ having a chain decomposition and no isolated vertices, then $G$ is $4$-edge-connected so $r$ has degree at least $4$ as well. However, we will not need this result, and it will follow from Corollary \ref{corSummary}.
\end{remark*}
\section{The Mader Construction}
We will adapt the strategy of Schlipf and Schmidt~\cite{Schlipf} in order to construct a chain decomposition. In particular, we will use a construction method for $k$-edge-connected graphs due to Mader~\cite{Mader}. We limit our description of the construction to the needed case $k=4$, since the method is more complicated for odd $k$.
\begin{definition*} A \textit{Mader operation} is one of the following operations:
\begin{enumerate}
\item Add an edge between two (not necessarily distinct) vertices.
\item Consider two distinct edges, say $e_1$ with ends $x$, $y$ and $e_2$ with ends $z$, $w$, and ``pinch" them as follows. Delete the edges $e_1$ and $e_2$, add a new vertex $v$, then add the new edges $e_x,e_y,e_z,e_w$ with one end $v$ and the other end $x,y,z,w$ respectively. While $e_1$ and $e_2$ must be distinct, the ends $x,y,z,w$ need not be. In this case, $v$ will have parallel edges to any repeated vertex.
\end{enumerate}
\end{definition*}
\begin{theorem}[{\cite[Corollary~14]{Mader}}]\label{thmMader}
A graph $G$ is $4$-edge-connected if and only if, for any $r\in V(G)$, one can construct $G$ in the following way. Begin with a graph $G^0$ consisting of $r$ and one other vertex of $G$, connected by four parallel edges. Then, repeatedly perform Mader operations to obtain $G$.
\end{theorem}
\begin{remark*}
Mader does not explicitly state that one can include a fixed vertex $r$ in $G^0$, but it follows from his work. His proof starts with $G$, and then reverses one of the Mader operations while maintaining $4$-edge-connectivity. An edge can be deleted unless $G$ is minimally $4$-edge-connected, in which case he finds two vertices of degree $4$ in his Lemma 13. He then shows that any degree $4$ vertex can be ``split off" (the reverse of a pinch) in his Lemma 9, so we can always split off a vertex not equal to $r$.
\end{remark*}
\section{Proof of Theorem~\ref{thmChains}}
Due to Theorem~\ref{thmMader}, it suffices to prove that a chain decomposition can be maintained through a Mader operation. The decomposition in the starting graph $G^0$ is as follows. Two of the edges form a closed up chain. The remaining two edges form a closed down chain.
Suppose the graph $G'$ is obtained from the graph $G$ by a Mader operation, with both graphs $4$-edge-connected. Assume that we have a chain decomposition $G_0,G_1,\ldots,G_m$ of $G$. By Remark~\ref{rmkMin}, we may assume that we have a minimal chain decomposition. We wish to create a new chain decomposition of $G'$.
\subsection{Adding an Edge}
Suppose $G'$ is obtained from $G$ by adding an edge with ends $u$, $v$. If one of the ends is the root $r$, we can classify the new edge as a one-way chain with tail $r$ at, say, the very beginning of the chain decomposition. The head must have at least two incident edges in later chains, since all chains are later.
If neither end is $r$, choose the minimal index $i$ such that $u$ or $v$ has degree exactly two in $H_i$, guaranteed to exist by Lemma~\ref{lmaDeg2}. Note that $i\geq1$ since $H_0$ is null. Without loss of generality, $u$ has degree exactly two in $H_i$. By the definition of $i$, $v$ has degree at most two in $H_i$, and therefore degree at least two in $\overline{H_{i-1}}$. We classify the new edge as a one-way chain with tail $u$ and head $v$, between the chains $G_{i-1}$ and $G_i$.
We consider the impact of these changes on other chains in the graph. A new chain was added, but none of the other chains changed index relative to each other. Vertices may have increased degree in the $H_i$'s or the $\overline{H_i}$'s due to the new edge, but increasing degree does not invalidate any chain types. Note that some chains may no longer be minimal, so the new chain decomposition in $G'$ is not necessarily minimal.
\subsection{Pinching Edges}
Suppose $G'$ is obtained from $G$ by pinching the edges $e_1$ with ends $x$, $y$ and $e_2$ with ends $z$, $w$, replacing them with edges $e_x,e_y,e_z,e_w$. We will use the notation $J_1=G_{CI(e_1)}=P_xe_1P_y$ for the chain containing $e_1$, where $P_x$ is the subpath between $x$ and an end of $J_1$ so that $e_1\notin E(P_x)$, and $P_y$ is defined similarly. Note that $P_x$ (resp. $P_y$) may have no edges if $x$ (resp. $y$) is an end of $J_1$. In the same way, we will use the notation $J_2=G_{CI(e_2)}=P_ze_2P_w$ for the chain containing $e_2$.
We now prove several claims to deal with all possible chain classification and chain index combinations for $J_1$ and $J_2$.
\begin{claim}\label{clm1}
If $CI(e_1)=CI(e_2)$, then $G'$ has a chain decomposition rooted at $r$.
\end{claim}
\begin{proof}
If $CI(e_1)=CI(e_2)$, then $J_1=J_2$. Without loss of generality, $e_1\in E(P_z)$ and $e_2\in E(P_y)$, so that the chain can be written as $J_1=J_2=P_xe_1(P_y\cap P_z)e_2P_w$ (where $P_y\cap P_z$ may have no edges if $y=z$). Recall that $e_1$ and $e_2$ are distinct, so $J_1=J_2$ is not a one-way chain.
By symmetry, we may assume $J_1=J_2$ is an up chain. In $G'$, we replace the chain $J_1=J_2$ with the following chains (in the listed order); see Figure~\ref{figClm1} for an illustration:
\begin{enumerate}
\item $P_xe_xe_wP_w$. This is an up chain. Since the edges $e_y$ and $e_z$ have not yet been used, the new vertex $v$ is incident to two edges in later chains.
\item $e_y$. This is a one-way chain with tail $v$ and head $y$. The tail $v$ is incident to two edges in earlier chains, namely $e_x$ and $e_w$. The head $y$ is incident to two edges in later chains since it was an internal vertex in the old up chain $J_1=J_2$.
\item $e_z$. This is a one-way chain with tail $v$ and head $z$. The tail $v$ is incident to two edges in earlier chains, namely $e_x$ and $e_w$. The head $z$ is incident to two edges in later chains since it was an internal vertex in the old up chain $J_1=J_2$.
\item $(P_y\cap P_z)$. Only add this chain if $P_y\cap P_z$ contains an edge. This is an up chain. The new ends $y,z$ are each incident to an edge in an earlier chain ($e_y$ and $e_z$, respectively) and are each incident to two edges in later chains since they were interior vertices of the old up chain $J_1=J_2$.
\end{enumerate}
\begin{figure}[h]
\centering
\includegraphics[scale=0.6]{arXivPNG2.png}
\caption{An illustration of the procedure in Claim 1. The original up chain $J_1=J_2$ is on the left, while its replacements in $G'$ are on the right. The red/dashed edges are in earlier chains than $J_1=J_2$, while the blue/dotted edges are in later chains than $J_1=J_2$. The black/dashed-and-dotted segments represent paths which may have any length (including 0).}
\label{figClm1}
\end{figure}
We consider the impact of these replacements on other chains in the graph. We inserted most of the edges of the old chain $J_1=J_2$ at the same chain index $CI(e_1)=CI(e_2)$, preventing any changes. The exception is the pinched edges $e_1$ and $e_2$ which were deleted, but the ends each received new incident edges $e_x,e_y,e_z,e_w$ inserted at the same chain index $CI(e_1)=CI(e_2)$. Thus, we have maintained the chain decomposition. This proves Claim \ref{clm1}.
\end{proof}
Without loss of generality, we assume the following for the remainder of the proof:
\begin{itemize}
\item $CI(e_1)<CI(e_2)$.
\item If $J_1$ is a one-way chain, then $x$ is the tail and $y$ the head.
\item If $J_2$ is a one-way chain, then $z$ is the tail and $w$ the head.
\end{itemize}
\begin{claim}\label{clm2}
Suppose that either $J_1$ is a one-way chain whose head $y$ has degree one in $H_{CI(e_2)}$, or $J_2$ is a one-way chain whose tail $z$ has degree one in $\overline{H_{CI(e_1)}}$. Then $G'$ has a chain decomposition rooted at $r$.
\end{claim}
\begin{proof}
By symmetry, we may assume $J_1$ is a one-way chain whose head $y$ has degree one in $H_{CI(e_2)}$.
First, we replace $J_1$ with $e_x$. This is a one-way chain with tail $x$ and head $v$. The tail $x$ was the tail of the old one-way chain $J_1$. The head $v$ has two (in fact three) incident edges in later chains, namely $e_y$, $e_z$, $e_w$.
\begin{itemize}
\item Case 1: $J_2$ is an up chain. Since $y$ has degree one in $H_{CI(e_2)}$, if $J_2$ is closed then $y$ is not the end of $J_2$. By swapping $z$ and $w$ if necessary, we may assume that $y$ is not the end of $J_2$ in $P_z$. Thus, the end of $J_2$ in $P_z$ is still either $r$ or incident to an edge in an earlier chain, despite having not placed $e_y$ yet. We use the edges of $J_2$ and $e_y$, $e_z$, $e_w$ to construct chains at the index $CI(e_2)$ as follows:
\begin{enumerate}
\item $P_ze_z$. This is an up chain. The new end, $v$, has one incident edge in an earlier chain ($e_x$) and two incident edges in later chains ($e_y$, $e_w$). By assumption, the old end in $P_z$ is still either $r$ or incident to an edge in an earlier chain.
\item $e_y$. This is a one-way chain with tail $v$ and head $y$. The tail $v$ is incident two edges in earlier chains ($e_x$, $e_z$). The head $y$ is either $r$ or incident to two edges in later chains, since $y$ has degree one in $H_{CI(e_2)}$ by assumption.
\item $e_w$. This is a one-way chain with tail $v$ and head $w$. The tail $v$ has two (in fact three) incident edges in earlier chains ($e_x$, $e_y$, $e_z$). The head $w$ is either $r$ or incident to two edges in later chains, since it was part of the old up chain $J_2$.
\item $P_w$. Only add this if $P_w$ contains an edge. This is an up chain. The new end, $w$, has one incident edge in an earlier chain ($e_w$) and two incident edges in later chains since it was an internal vertex of the old up chain $J_2$. Since we placed $e_y$ above, the end of $J_2$ in $P_w$ has is either $r$ or incident to an end in an earlier chain, even if the end is $y$.
\end{enumerate}
\item Case 2: $J_2$ is a down chain. Since $y$ has degree one in $H_{CI(e_2)}$, $y\notin V(J_2)$, so each vertex of $J_2$ is still either $r$ or incident to two edges in earlier chains, despite having not placed $e_y$ yet. We use the edges of $J_2$ and $e_y$, $e_z$, $e_w$ to construct chains at the index $CI(e_2)$ as follows:
\begin{enumerate}
\item $P_w$. Only add this if $P_w$ contains an edge. This is a down chain. The new end, $w$, has one incident edge in a later chain ($e_w$) and two incident edges in earlier chains since it was an internal vertex of the old down chain $J_1$.
\item $e_w$. This is a one-way chain with tail $w$ and head $v$. The tail $w$ is either $r$ or incident to two edges in earlier chains since it was part of the old down chain $J_2$. The head $v$ is incident to two edges in later chains ($e_y$, $e_z$).
\item $P_ze_z$. This is a down chain. The new end, $v$, has one incident edge in a later chain ($e_y$) and two incident edges in earlier chains ($e_x$, $e_w$).
\item $e_y$. This is a one-way chain with tail $v$ and head $y$. The tail $v$ has two (in fact three) incident edges in earlier chains ($e_x$, $e_z$, $e_w$). The head $y$ is either $r$ or incident to two edges in later chains since $y$ has degree one in $H_{CI(e_2)}$ and $y\notin V(J_2)$ by assumption, so $y$ has degree at least three in $\overline{H_{CI(e_2)}}$ unless it is $r$.
\end{enumerate}
\item Case 3: $J_2$ is a one-way chain. Since $y$ has degree one in $H_{CI(e_2)}$, $y\neq z$ so the tail $z$ is still either $r$ or incident to two edges in earlier chains, despite having not placed $e_y$ yet. We use the edges $e_y$, $e_z$, $e_w$ to construct chains at the index $CI(e_2)$ as follows:
\begin{enumerate}
\item $e_z$. This is a one-way chain with tail $z$ and head $v$. The tail $z$ is either $r$ or incident to two edges in earlier chains as discussed above. The head $v$ is incident to two edges in later chains ($e_y$, $e_w$).
\item $e_w$. This is a one-way chain with tail $v$ and head $w$. The tail $v$ is incident to two edges in earlier chains ($e_x$, $e_z$). The head $w$ is either $r$ or incident to two edges in later chains since it was the head of $J_2$.
\item $e_y$. This is a one-way chain with tail $v$ and head $y$. The tail $v$ has two (in fact three) incident edges in earlier chains ($e_x$, $e_z$, $e_w$). The head $y$ is either $r$ or incident to two edges in later chains since $y$ has degree one in $H_{CI(e_2)}$ and $y\notin V(J_2)$ by assumption, so $y$ has degree at least three in $\overline{H_{CI(e_2)}}$ unless it is $r$.
\end{enumerate}
\end{itemize}
We consider the impact of these replacements on other chains in the graph. As before, most of the edges of the old chains $J_1$ and $J_2$ were inserted at the same chain indices $CI(e_1)$ and $CI(e_2)$ respectively, preventing any changes. The pinched edges $e_1$ and $e_2$ were deleted, but the ends $x$, $z$, $w$ each received new incident edges $e_x$, $e_z$, $e_w$ inserted at the same chain indices ($CI(e_1)$, $CI(e_2)$, and $CI(e_2)$ respectively). However, $e_y$ was inserted at a different chain index than the deleted edge $e_1$ since $e_1$ was at $CI(e_1)$ while $e_y$ is at $CI(e_2)$. By the claim assumptions, $y$ has degree one in $H_{CI(e_2)}$, so there are no chains containing $y$ between $CI(e_1)$ and $CI(e_2)$, and so no chains were affected by the change. Thus, we have maintained the chain decomposition. This proves Claim \ref{clm2}.
\end{proof}
We may now assume the following for the remaining cases:
\begin{itemize}
\item If $J_1$ is a one-way chain, then $y$ has degree at least two in $H_{CI(e_2)}$.
\item If $J_2$ is a one-way chain, then $z$ has degree at least two in $\overline{H_{CI(e_1)}}$.
\end{itemize}
We also make the following conditional definitions, which will aid in distinguishing the remaining cases:
\begin{itemize}
\item If $J_1$ is a one-way chain and $y$ is not in $H_{CI(e_1)}$, then define the minimal index $i$ such that $y\in V(G_i)$ and $CI(e_1)<i<CI(e_2)$. Since $i$ is minimal, $y$ has degree one in $H_i$ (incident only to the pinched edge $e_1$). From this and the fact that $G_i$ is a minimal chain, it follows that either $y$ is one of two distinct ends of the up chain $G_i$, or $y$ is the head of the one-way chain $G_i$ which is not a loop.
\item If $J_2$ is a one-way chain and $z$ is not in $\overline{H_{CI(e_2)}}$, then define the maximal index $j$ such that $z\in V(G_j)$ and $CI(e_1)<j<CI(e_2)$. Since $j$ is maximal, $z$ has degree one in $\overline{H_j}$ (incident only to the pinched edge $e_2$). From this and the fact that $G_j$ is a minimal chain, it follows that either $z$ is one of two distinct ends of the down chain $G_j$, or $z$ is the tail of the one-way chain $G_j$ which is not a loop.
\end{itemize}
\begin{claim}\label{clm3}
Suppose that either one of $i,j$ is not defined, or $i<j$. Then $G'$ has a chain decomposition rooted at $r$.
\end{claim}
\begin{proof}
The chains replacing $J_1$ will have indices adjacent to $CI(e_1)$ and $i$ (if it is defined). Likewise, the chains replacing $J_2$ will have indices adjacent to $CI(e_2)$ and $j$ (if it is defined). Thus, by the assumptions of this claim, the chains replacing $J_1$ will have lower chain index than the chains replacing $J_2$. This fact will be needed when confirming that the new chains are valid. We begin by replacing $J_1$ as follows:
\begin{itemize}
\item Case 1: $J_1$ is an up chain. We replace it with $P_xe_xe_yP_y$. This is an up chain. The new vertex $v$ has two incident edges in later chains, namely $e_z$ and $e_w$.
\item Case 2: $J_1$ is a down chain. We replace it with the following chains (in the listed order):
\begin{enumerate}
\item $P_x$. Only add this chain if $P_x$ contains an edge. This is a down chain. The new end $x$ has an incident edge in a later chain, namely $e_x$.
\item $P_y$. Only add this chain if $P_y$ contains an edge. This is a down chain. The new end $y$ has an incident edge in a later chain, namely $e_y$.
\item $e_x$. This is a one-way chain with tail $x$ and head $v$. The tail $x$ is either $r$ or incident to two edges in earlier chains since it was in the old down chain $J_1$. The head $v$ has two incident edges in later chains, namely $e_z$ and $e_w$.
\item $e_y$. This is a one-way chain with tail $y$ and head $v$. The tail $y$ is either $r$ or incident to two edges in earlier chains since it was in the old down chain $J_1$. The head $v$ has two incident edges in later chains, namely $e_z$ and $e_w$.
\end{enumerate}
\item Case 3: $J_1$ is a one-way chain whose head $y$ is in $H_{CI(e_1)}$. We replace it with the following chains (in the listed order):
\begin{enumerate}
\item $e_x$. This is a one-way chain with tail $x$ and head $v$. The tail $x$ was the tail of the old one-way chain $J_1$. The head $v$ has two (in fact three) incident edges in later chains, namely $e_y, e_z, e_w$.
\item $e_y$. This is an up chain. The vertex $y$ is either $r$ or incident to two edges in later chains since it was the head of the old one-way chain $J_1$, and it has an incident edge in an earlier chain by assumption. The vertex $v$ has two incident edges in later chains, namely $e_z$ and $e_w$, and is incident to $e_x$ from the previous chain.
\end{enumerate}
\item Case 4: $J_1$ is a one-way chain whose head $y$ is not in $H_{CI(e_1)}$. Then $i$ is defined as above.
First, we replace $J_1$ with $e_x$. This is a one-way chain with tail $x$ and head $v$. The tail $x$ was the tail of the old one-way chain $J_1$. The head $v$ has two (in fact three) incident edges in later chains, namely $e_y, e_z, e_w$.
\begin{itemize}
\item Subcase 1: $y$ is one of two distinct ends of the up chain $G_i$. Replace $G_i$ with $G_ie_y$. This is an up chain. Since $G_i$ was a path and $v$ is a new vertex, this new chain is a path. The new end $v$ is adjacent to one edge in an earlier chain ($e_x$) and two edges in later chains ($e_z$ and $e_w$).
\item Subcase 2: $y$ is the head of the one-way chain $G_i$ which is not a loop. Then $y$ is not required to be in $H_i$ for $G_i$ to be a valid chain. In fact, $y$ is not required to be in any of $H_0,H_1,\ldots,H_i$ by the definition of $i$ and the assumptions of this case. Thus, we can leave $G_i$ as is and insert the chain $e_y$ immediately after $G_i$. This is an up chain. The vertex $y$ is incident to an edge in the previous chain $G_i$, and is either $r$ or incident to two edges in later chains since it is the head of $G_i$. The vertex $v$ is adjacent to one edge in an earlier chain ($e_x$) and two edges in later chains ($e_z$ and $e_w$).
\end{itemize}
\end{itemize}
The procedure for replacing $J_2$ is symmetric, by following the above steps in the reversed chain decomposition.
We consider the impact of these replacements on other chains in the graph. In most cases, we replaced the old chain $J_1$ with new chains inserted at the same chain index $CI(e_1)$, preventing any changes. The pinched edge $e_1$ was deleted, but the end $x$ received a new incident edge $e_x$ at the same chain index $CI(e_1)$. In Cases 1-3, the same is true for $y$. In Case 4, $y$ received a new incident edge $e_y$ either at or immediately after the chain index $i$. However, by the definition of $i$ and the claim assumptions, no chains were affected by the new chain index except $G_i$, which was specifically considered and shown to be valid in Case 4. By similar arguments, the changes caused by replacing $J_2$ also did not invalidate any chains. Thus, we have maintained the chain decomposition. This proves Claim \ref{clm3}.
\end{proof}
\begin{claim}\label{clm4}
Suppose that both of $i,j$ are defined and $i=j$. Then $G'$ has a chain decomposition rooted at $r$.
\end{claim}
\begin{proof}
Recall that $G_i$ is either an up chain or a one-way chain with head $y$, and $G_j$ is either a down chain or a one-way chain with tail $z$. Since $i=j$, we conclude that $G_i=G_j$ must be a one-way chain with tail $z$ and head $y$, and $y\neq z$ since $i$ and $j$ are defined. We can replace $J_1$ and $J_2$ with the following chains, in the listed order. The first two will be placed immediately before index $i=j$, and the last two immediately after index $i=j$; see Figure~\ref{figClm4} for an illustration:
\begin{enumerate}
\item $e_x$. This is a one-way chain with tail $x$ and head $v$. The tail $x$ was the tail of the old one-way chain $J_1$ and we are placing this chain after index $CI(e_1)$. The head $v$ has two (in fact three) incident edges in later chains, namely $e_y, e_z, e_w$.
\item $e_z$. This is a one-way chain with tail $z$ and head $v$. By the definition of $j$, the tail $z$ is either $r$ or incident to two edges in earlier chains than $G_j$, and we are placing this chain immediately before index $j$. The head $v$ has two incident edges in later chains, namely $e_y$ and $e_w$.
\item $e_y$. This is a one-way chain with tail $v$ and head $y$. The tail $v$ has two incident edges in earlier chains, namely $e_x$ and $e_z$. By the definition of $i$, the head $y$ is either $r$ or incident to two edges in later chains than $G_i$, and we are placing this chain immediately after index $i$.
\item $e_w$. This is a one-way chain with tail $v$ and head $w$. The tail $v$ has two (in fact three) incident edges in earlier chains, namely $e_x, e_y, e_z$. The head $w$ was the head of the old one-way chain $J_2$, and we are placing this chain before $CI(e_2)$.
\end{enumerate}
\begin{figure}[h]
\centering
\includegraphics[scale=0.6]{arXivPNG3.png}
\caption{An illustration of the procedure in Claim 4. The original chains $J_1$ and $J_2$ are on the left, while their replacements in $G'$ are on the right. The red/dashed edges are in earlier chains, while the blue/dotted edges are in later chains, with the particular meanings of ``earlier" and ``later" in the corresponding labels.}
\label{figClm4}
\end{figure}
We consider the impact of these replacements on other chains in the graph. The deleted edge $e_1$ was replaced by two edges with chain index greater than $CI(e_1)$, so we must be careful. The edge $e_x$ was inserted before index $i$, but $x$ had degree at least two in $H_{CI(e_1)}$, so losing a degree in later $H$ subgraphs will not invalidate any chains. The edge $e_y$ was inserted immediately after index $i$, so by the definition of $i$, the only chain affected is $G_i$. Since $G_i$ has $y$ as a head, losing a degree in $H_i$ will not invalidate the chain. By a symmetric argument, the changes caused by $e_z$ and $e_w$ do not invalidate any chains.
This proves Claim \ref{clm4}.
\end{proof}
\begin{claim}\label{clm5}
Suppose that both of $i,j$ are defined, and $i>j$. Then $G'$ has a chain decomposition rooted at $r$.
\end{claim}
\begin{proof}
We can replace $J_1$ and $J_2$ with the following chains, at the indicated chain indices; see Figure~\ref{figClm5} for an illustration:
\begin{enumerate}
\item $e_x$. Add this chain at index $CI(e_1)$. This is a one-way chain with tail $x$ and head $v$. The tail $x$ was the tail of the old one-way chain $J_1$ and we are placing this chain at index $CI(e_1)$. The head $v$ has two (in fact three) incident edges in later chains, namely $e_y, e_z, e_w$.
\item $e_z$. Add this chain immediately after $G_j$. This is a one-way chain with tail $z$ and head $v$. By the definition of $j$, the tail $z$ is either $r$ or incident to two edges in earlier chains than $G_j$, and we are placing this chain after index $j$. The head $v$ has two incident edges in later chains, namely $e_y$ and $e_w$.
\item $e_y$. Add this chain immediately before $G_i$. This is a one-way chain with tail $v$ and head $y$. The tail $v$ has two incident edges in earlier chains, namely $e_x$ and $e_z$. By the definition of $i$, the head $y$ is either $r$ or incident to two edges in later chains than $G_i$, and we are placing this chain before index $i$.
\item $e_w$. Add this chain at index $CI(e_2)$. This is a one-way chain with tail $v$ and head $w$. The tail $v$ has two (in fact three) incident edges in earlier chains, namely $e_x, e_y, e_z$. The head $w$ was the head of the old one-way chain $J_2$, and we are placing this chain at index $CI(e_2)$.
\end{enumerate}
\begin{figure}[h]
\centering
\includegraphics[scale=0.6]{arXivPNG4.png}
\caption{An illustration of the procedure in Claim 5. The original chains $J_1$ and $J_2$ are on the left, while their replacements in $G'$ are on the right. The red/dashed edges are in earlier chains, while the blue/dotted edges are in later chains, with the particular meanings of ``earlier" and ``later" in the corresponding labels. The black/dashed-and-dotted segments represent paths which may have any length (including 0).}
\label{figClm5}
\end{figure}
We consider the impact of these replacements on other chains in the graph. The edge $e_1$ was deleted, but $x$ received a new incident edge $e_x$ at the same chain index $CI(e_1)$. The edge $e_y$ was inserted before index $i$, but the index is still smaller than $i$, so by the definition of $i$, no chains are affected. By a symmetric argument, the changes caused by $e_z$ and $e_w$ also do not invalidate any chains. This proves Claim \ref{clm5}.
\end{proof}
The claims cover all possibilities of pinching edges. The proof of Theorem~\ref{thmChains} is complete. The proof also implies a polynomial-time algorithm to construct a chain decomposition.\qed
\section{Proof of Theorem~\ref{thmTreesFromChains}}
Assume that we have a chain decomposition $G_0,G_1,\ldots,G_m$ of $G$. By Remark \ref{rmkMin}, we may assume that the chain decomposition is minimal. We will adapt the strategy of Curran, Lee, and Yu~\cite{cly} to prove Theorem~\ref{thmTreesFromChains}. In particular, we will construct two partial numberings of the edges of $G$ using the chain decomposition. We will then construct four spanning trees in two pairs, with one pair associated with each numbering. Within each pair, paths back to the root $r$ will be monotonic in the associated numbering to ensure independence. Between pairs, paths back to the root $r$ will be monotonic in chain index to ensure independence.
Using Corollary \ref{corNoLoops}, we may assume that there are no loops in $G$. By Lemma \ref{lmaDeg2}, for each vertex $v\neq r$, there are two distinct non-loop edges incident to $v$ whose chain indices are strictly smaller than the chain index of any other edge incident to $v$. Likewise there are two distinct edges whose chain indices are strictly larger than the chain index of any other edge adjacent to $v$. We will name these edges as follows:
\begin{definition*}
For each vertex $v\neq r$, the two \textit{$f$-edges} of $v$ are the two incident edges with the lowest chain index. Similarly, the two \textit{$g$-edges} of $v$ are the two incident edges with the highest chain index.
\end{definition*}
\begin{remark}\label{rmkfDown}
By the definition of a down chain, the edges of down chains are never $f$-edges. Likewise, by the definition of an up chain, the edges of up chains are never $g$-edges.
\end{remark}
Next, we will iteratively define a numbering $f$, which will assign distinct values in $\mathbb{R}$ to all edges in up chains and one-way chains. Here, two ``consecutive'' edges in a chain will refer to two edges in the chain which are incident to an internal vertex of the chain, so the two edges incident to the end of a closed chain are not consecutive, despite being adjacent.
We begin by numbering the edges in $E(G_0)$, and then number the edges of each up chain and one-way chain in order of chain index. When we reach a chain $G_i$, we may assume that all edges in $E(H_i)$ belonging to up chains and one-way chains have been numbered, which includes all $f$-edges in $E(H_i)$ by Remark \ref{rmkfDown}. We use the following procedure to number the edges in $E(G_i)$:
\begin{itemize}
\item If $G_i$ is a closed up chain containing $r$, then number the edges in $E(G_i)$ so that the values change monotonically between consecutive edges in the chain. The particular numbers used are arbitrary.
\item If $G_i$ is a closed up chain not containing $r$, then both $f$-edges of the common end have already been numbered. Call these two $f$-edges \textit{numbering edges} of $G_i$. Say the numbering edges of $G_i$ have $f$-values $a$ and $b$. Number the edges in $E(G_i)$ so that the values change monotonically between consecutive edges in the chain, and all values are between $a$ and $b$.
\item If $G_i$ is an open up chain containing $r$, then $r$ is an end and the other end is some $u\neq r$. At least one $f$-edge of $u$ has already been numbered. Choose an $f$-edge which has already been numbered and call it a \textit{numbering edge} of $G_i$. Say that $a$ is the $f$-value of the numbering edge. Number the edges in $E(G_i)$ so that the values increase between consecutive edges in the chain when moving from $u$ to $r$, and all values are larger than $a$.
\item If $G_i$ is an open up chain not containing $r$, then at least one $f$-edge of each end has been numbered. If the ends are $u$ and $v$, we can choose two distinct edges $e_u, e_v\in E(H_i)$ so that $e_u$ is an $f$-edge of $u$ and $e_v$ is an $f$-edge of $v$. We can choose these two distinct edges because otherwise, the only $f$-edge of $u$ or $v$ in $E(H_i)$ would be a single edge between $u$ and $v$, and then $H_i$ would not be connected. Call the edges $e_u,e_v$ \textit{numbering edges} of $G_i$. Without loss of generality, $f(e_u)=a<b=f(e_v)$. Number the edges in $E(G_i)$ so that the values increase between consecutive edges in the chain when moving from $u$ to $v$, and all values are between $a$ and $b$.
\item If $G_i$ is a one-way chain whose tail is $r$, then number the edge of $G_i$ arbitrarily.
\item If $G_i$ is a one-way chain whose tail is not $r$, then both $f$-edges of the tail are already numbered, say with $f$-values $a$ and $b$. Number the edge of $G_i$ between $a$ and $b$.
\end{itemize}
We symmetrically define a numbering $g$, which assigns distinct values in $\mathbb{R}$ to the edges of down chains and one-way chains, by using the above procedure in the reversed chain decomposition.
We are finally ready to construct the trees. Define the subgraphs $T_1,T_2,T_3,T_4$ as follows. For each $v\neq r$, consider the two $f$-edges of $v$. Assign the edge with the lower $f$-value to $T_1$ and the edge with the higher $f$-value to $T_2$. Similarly, consider the two $g$-edges of $v$. Assign the edge with the lower $g$-value to $T_3$ and the edge with the higher $g$-value to $T_4$.
Several properties of $T_1,T_2,T_3,T_4$ will follow from the following claim.
\begin{claim*}
For any $v\neq r$, consider the edge $e_1$ assigned to $T_1$ at $v$. Let $v'$ be the other end of $e_1$. If $v'\neq r$, let $e'_1$ be the edge assigned to $T_1$ at $v'$. Then $CI(e'_1)\leq CI(e_1)$ and $f(e'_1)<f(e_1)$.
\end{claim*}
\begin{proof}
Let $e_2$ be the edge assigned to $T_2$ at $v$. The edge $e_1$ is not in a down chain by Remark \ref{rmkfDown}. We break into two cases.
\begin{itemize}
\item Suppose $e_1$ is in an up chain $G_i$. Since the chain decomposition is minimal and $v'\in V(G_i)$, its $f$-edges are either in $E(G_i)$, or else have chain index less than $i$. In either case, $CI(e'_1)\leq i=CI(e_1)$ as desired.
Note that $e_2$ is either in $E(G_i)$, or else is the numbering edge of $G_i$ at the end $v$. By the numbering procedure, we know that $f(e_1)$ is between $f(e_2)$ and the $f$-value of one of the $f$-edges of $v'$, say $e^*$. By the definition of $T_1$, $f(e_1)<f(e_2)$, so it follows that $f(e^*)<f(e_1)$. Again by the definition of $T_1$, $f(e'_1)\leq f(e^*)$, so $f(e'_1)<f(e_1)$ as desired.
\item Suppose $e_1$ induces a one-way chain $G_i$. Since $e_1$ is an $f$-edge, $v$ has degree at most one in $H_i$, so $v$ must be the head of $G_i$. Then $v'$ is the tail of $G_i$, so the $f$-edges of $v'$ have chain indices smaller than $i$, which means $e'_1\neq e_1$ and $CI(e'_1)<CI(e_1)$ as desired.
From the numbering procedure, we know that $f(e_1)$ is between the $f$-values of the two $f$-edges of $v'$, with $f(e'_1)$ being the smaller by the definition of $T_1$. So, $f(e'_1)<f(e_1)$ as desired.
\end{itemize}
In both cases we have $CI(e'_1)\leq CI(e_1)$ and $f(e'_1)<f(e_1)$. This proves the claim.
\end{proof}
With the claim proven, it follows that the edges assigned to $T_1$ are all distinct, there are no cycles in $T_1$, and following consecutive edges assigned to $T_1$ produces a path which is decreasing in chain index, strictly decreasing in $f$-value, and can only end at $r$. Thus, $T_1$ is connected and is a spanning tree of $G$. A similar argument shows that $T_2$ is a spanning tree of $G$ where paths to $r$ are decreasing in chain index and strictly increasing in $f$-value. Due to the opposite trends in $f$-values, $T_1$ and $T_2$ are edge-independent with root $r$.
By symmetry, we obtain analogous results for $T_3$ and $T_4$. It remains to show that a tree from $\{T_1,T_2\}$ and a tree from $\{T_3,T_4\}$ are edge-independent. The paths back to $r$ from a vertex $v\neq r$ are decreasing in chain index in one tree and increasing in chain index in the other tree, but not strictly. The first edges in these paths are an $f$-edge and a $g$-edge of $v$, respectively. By Lemmas \ref{lmaDeg2} and \ref{lmaMinDeg4}, there is a positive difference in chain index between these initial edges, so the paths are in fact edge-disjoint. The proof of Theorem \ref{thmTreesFromChains} is complete. The proof also implies a polynomial-time algorithm to construct the edge-independent spanning trees.\qed
\section{Summary of Results}
With Theorems \ref{thmChains} and \ref{thmTreesFromChains} proven, we obtain Theorem \ref{thmTrees}. In fact, we can examine the argument more carefully to extract a stronger, summarizing result.
\begin{corollary}\label{corSummary}
Suppose $G$ is a graph with no isolated vertices and $V(G)\geq2$. Then the following statements are equivalent.
\begin{enumerate}
\item $G$ is $4$-edge-connected.
\item There exists $r\in V(G)$ so that $G$ has a chain decomposition rooted at $r$.
\item For all $r\in V(G)$, $G$ has a chain decomposition rooted at $r$.
\item There exists $r\in V(G)$ so that $G$ has four edge-independent spanning trees rooted at $r$.
\item For all $r\in V(G)$, $G$ has four edge-independent spanning trees rooted at $r$.
\end{enumerate}
\end{corollary}
\begin{proof}
Theorem \ref{thmChains} gives us $(1)\Rightarrow(3)$. Theorem \ref{thmTreesFromChains} gives us $(2)\Rightarrow(4)$ and $(3)\Rightarrow(5)$. Trivially, we have $(3)\Rightarrow(2)$ and $(5)\Rightarrow(4)$. Therefore, we need only show $(4)\Rightarrow(1)$.
Assume for the sake of contradiction that $G$ has four edge-independent spanning trees rooted at some $r\in V(G)$, but is not $4$-edge-connected. Suppose $S\subseteq E(G)$ is an edge cut with $\abs{S}<4$. Consider a vertex $v$ in a component of $G-S$ not containing $r$. Using the paths in each of the edge-independent spanning trees, we find that there exist four edge-disjoint paths between $v$ and $r$. This contradicts the existence of $S$.
\end{proof}
| {
"timestamp": "2017-11-23T02:00:28",
"yymm": "1705",
"arxiv_id": "1705.01199",
"language": "en",
"url": "https://arxiv.org/abs/1705.01199",
"abstract": "We prove an ear-decomposition theorem for $4$-edge-connected graphs and use it to prove that for every $4$-edge-connected graph $G$ and every $r\\in V(G)$, there is a set of four spanning trees of $G$ with the following property. For every vertex in $G$, the unique paths back to $r$ in each tree are edge-disjoint. Our proof implies a polynomial-time algorithm for constructing the trees.",
"subjects": "Combinatorics (math.CO)",
"title": "Four Edge-Independent Spanning Trees",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754497285468,
"lm_q2_score": 0.7185943985973772,
"lm_q1q2_score": 0.7075104031714272
} |
https://arxiv.org/abs/1503.08004 | Residues and the Combinatorial Nullstellensatz | We interpret the Combinatorial Nullstellensatz of Noga Alon as a multidimensional residue formula, describe some consequences of this interpretation and related open problems. | \section{Introduction}
The Combinatorial Nullstellensatz of Noga~Alon~\cite{alon1999} turned out to be an efficient tool to obtain results in combinatorics and discrete geometry. This is an almost elementary algebraic statement, but it has not so elementary consequences and generalizations.
In the recent papers~\cite{lason2010,karpet2011} a version of the Combinatorial Nullstellensatz was expressed as a certain formula, which turned out to be useful in several problems (see~\cite{kn2012,klw2012}, for example):
\begin{theorem}[The Combinatorial Nullstellensatz]
\label{cn-prod}
Suppose a multivariate polynomial $f(x_1,x_2,\dots,x_n)$ over a field $\mathbb{F}$ has degree at most $c_1+c_2+\dots+c_n$, where $c_i$ are non-negative integers. Denote by $C$ the coefficient at $x_1^{c_1}\dots x_n^{c_n}$ in $f$. Let $A_1$, $A_2$, \ldots, $A_n$ be arbitrary subsets of $\mathbb{F}$ such that $|A_i|=c_i+1$ for any $i$. Put $\varphi_i(x)=\prod_{\alpha\in A_i}(x-\alpha)$. Then
\begin{equation}
\label{cn-eq}
C=\sum_{\alpha_i\in A_i} \frac{f(\alpha_1,\dots,\alpha_n)}
{\varphi_1'(\alpha_1)\dots \varphi_n'(\alpha_n)}.
\end{equation}
In particular, if $C\ne 0$, then there exists a system of representatives $\alpha_i\in A_i$ such that $f(\alpha_1,\alpha_2,\dots,\alpha_n)\ne 0$.
\end{theorem}
The general way to apply this theorem, developed by Fedor Petrov in~\cite{karpet2011}, is as follows: Express a combinatorial statement in the from that a certain polynomial $f$ of appropriate degree attains a nonzero value on the product $A_1\times \dots \times A_n$. In order to prove this, by the theorem, we need to show that $C$ is nonzero. Then we try to modify the polynomial $f$ without changing $C$, usually it corresponds to a special choice of the parameters of the initial combinatorial problem, and obtain another polynomial $g$ such that the right hand side of (\ref{cn-eq}) contains one (or slightly more) summands that are easy to calculate.
In~\cite{karpet2011} a simple proof of this theorem was given, using the Lagrange interpolation formula, see the review~\cite{gassa2000} for more information about interpolation.
The emphasis of this note is that this formula can be viewed, less elementary, as a multidimensional residue formula. In what follows we explain the meaning of this and try to show other situations when this point of view may be useful. In principle, this allows, with some care, to consider the case when the sets $A_i$ are multisets (sets with some multiplicities). We also show the relation between the Combinatorial Nullstellensatz and the old Cayley--Bacharach theorem about incidence of intersection of hypersufraces.
\section{Residues on the product of projective lines}
Let us interpret the Combinatorial Nullstellensatz (Theorem~\ref{cn-prod}) as a corollary of the residue theorem~\cite[Ch.~5, \S~1]{gh1978}:
\begin{theorem}[The residue theorem]
\label{residue-theorem}
Let $D_1,\ldots, D_n$ be a set of divisors on a compact analytic $n$-dimensional manifold $M$, having a zero-dimensional intersection. Then for any holomorphic $\omega\in \Omega^n(M\setminus \bigcup_{i=1}^n D_i)$ we have:
$$
\sum_{x\in D_1\cap\dots\cap D_n} \Res_x \omega = 0.
$$
\end{theorem}
\begin{remark}
Note that the value $\Res_x \omega$ actually depends on the set of divisors $(D_1,\ldots, D_n)$. In particular it changes sign if the divisors are permuted by an odd permutation. To keep the things clear, we restrict ourselves to ``geometric'' divisors, that is combinations of prime divisors with unit coefficients.
\end{remark}
\begin{remark}
The algebraic version of Theorem~\ref{residue-theorem} is valid for any algebraically closed field of coefficients, but let us restrict ourselves to $\mathbb C$ here.
\end{remark}
Now we deduce Theorem~\ref{cn-prod} from the residue theorem. Take the product of projective lines $M=\underbrace{\mathbb CP^1\times\dots\times \mathbb CP^1}_n$. Consider a grid subset:
$$
X = X_1\times\dots \times X_n \subseteq \underbrace{\mathbb C\times\dots\times \mathbb C}_n,
$$
where $|X_i|=k_i$, and a polynomial $f\in \mathbb C[z_1,\ldots,z_n]$. Denote
$$
g_i(z) = \prod_{x\in X_i} (z-x),
$$
and apply the residue theorem to the differential form
$$
\omega = \frac{f(z_1,\ldots, z_n)dz_1\wedge\dots\wedge dz_n}{g_1(z_1)\dots g_n(z_n)}.
$$
The singularities of this differential form are at sets
$$
D_i = \{(z_1,\ldots, z_n)\in (\mathbb CP^1)^{\times n} : z_i\in X_i\ \text{or}\ z_i = \infty\},
$$
that we consider as divisors. The intersection of these divisors is
$$
D_1\cap\dots\cap D_n = (X_1\cup\{\infty\})\times\dots\times (X_n\cup\{\infty\}),
$$
and applying the residue formula yields:
\begin{equation}
\label{res-form}
\sum_{(z_1,\ldots, z_n)\in D_1\cap\dots\cap D_n} \Res_{(z_1,\ldots, z_n)} \binom{\omega}{D_1 D_2 \dots D_n} = 0.
\end{equation}
The residue at $(\infty, \ldots, \infty)$ with respect to $t_1=\frac{1}{z_1}, \ldots, t_n=\frac{1}{z_n}$ is
\begin{multline*}
\Res_{(\infty,\ldots,\infty)} \omega = (-1)^n\Res_{(0,\ldots, 0)} \frac{f(\frac{1}{t_1},\ldots, \frac{1}{t_n})dt_1\wedge\dots\wedge dt_n}{t_1^2g_1(\frac{1}{t_1})\dots t_n^2g_n(\frac{1}{t_n})} = \\
= (-1)^n\Res_{(0,\ldots, 0)} f\left(\frac{1}{t_1},\ldots, \frac{1}{t_n}\right)dt_1\wedge\dots\wedge dt_n \prod_{i=1}^n\left(t_i^{n_i-2}\prod_{x\in X_i} \frac{1}{1 - t_ix})\right),
\end{multline*}
if the total degree $\deg f \le \sum_{i=1}^n (k_i-1)$, then we have a simple formula
$$
\Res_{(\infty,\ldots,\infty)} \binom{\omega}{D_1 D_2 \dots D_n} = (-1)^n c_{k_1-1,\ldots, k_n-1},
$$
where $c_{k_1-1,\ldots, k_n-1}$ is a coefficient at $z_1^{k_1-1}\dots z_n^{k_n-1}$ in $f(z_1,\ldots, z_n)$.
The equation~\ref{res-form} would give the desired result (up to sign), but the intersection $D_1\cap\dots\cap D_n$ has points with some coordinates $\infty$, and some finite. Fortunately, this issue is resolved by considering ``rearranged'' divisors
$$
D'_i = \{(z_1,\ldots, z_n)\in (\mathbb CP^1)^{\times n} : z_i\in X_i\ \text{or}\ z_{i+1} = \infty\},
$$
where the indexes of $z_i$ are taken modulo $n$. Now the intersection of divisors becomes what we want:
$$
D'_1\cap\dots\cap D'_n = (X_1\times\dots\times X_n)\cup\{(\infty,\infty,\ldots,\infty)\},
$$
and the result follows. Because of the cyclic shift the residue at infinity becomes $-c_{k_1-1,\ldots, k_n-1}$ and the formula is correct.
\section{Observation on the residue formula for two sets of divisors}
The trick with rearranging the divisors may be replaced by the following version of the residue formula:
\begin{theorem}[Gelfond--Khovanskii, 2002]
\label{residue-theorem2}
Let $D_1,\ldots, D_n$ and $D'_1,\ldots, D'_n$ be two sets of divisors on a compact analytic $n$-dimensional manifold $M$, each having a zero-dimensional intersection. Assume $D_i\cap D'_i=\emptyset$ for every $i$ and put $Z = \bigcup_{i=1}^n D_i \cup \bigcup_{i=1}^n D'_i$. Then for any holomorphic $\omega\in \Omega^n(M\setminus Z)$ we have:
$$
\sum_{p\in D_1\cap\dots\cap D_n} \Res_p \omega = (-1)^n \sum_{q\in D'_1\cap\dots\cap D'_n} \Res_q \omega.
$$
Here the residues on the left are considered with respect to the set of divisors $(D_1,\ldots, D_n)$ and the residues on the right use the set of divisors $(D'_1,\ldots, D'_n)$.
\end{theorem}
This theorem in the analytic case was established by Gelfond and Khovanskii in~\cite[Theorem~2]{gekh2002}. The algebraic analogue of this theorem for algebraically closed fields follows from the ordinary residue theorem (like Theorem~\ref{residue-theorem}) by the same rearranging trick: put $D''_i = D_i\cup D'_{i+1}$ (the indices understood mod $n$) and note that the points of the intersection $D''_1\cap\dots\cap D''_n$ split into two subsets $D_1\cap\dots\cap D_n$ and $D'_1\cap\dots\cap D'_n$.
In view of Theorem~\ref{residue-theorem2} the Combinatorial Nullstellensatz is easily obtained by taking $M=\underbrace{\mathbb CP^1\times\dots\times \mathbb CP^1}_n$,
$$
\omega = \frac{f(z_1,\ldots, z_n)dz_1\wedge\dots\wedge dz_n}{g_1(z_1)\dots g_n(z_n)},
$$
$$
D_i = \{(z_1,\ldots, z_n) : z_i \in X_i\}\quad\text{and}\quad D'_i=\{(z_1,\ldots, z_n) : z_i =\infty\}.
$$
Again, the sum of residues at finite points turns out to be equal up to sign to the residue at $(\infty,\ldots, \infty)$.
\begin{remark}
Another observation is that the assumption that the degree of $f$ is at most $c_1+c_2+\dots+c_n$ in Theorem~\ref{cn-prod} is not actually needed. What is really needed is that besides the monomial $C z_1^{c_1}\dots z_n^{c_n}$ all other monomials $C' z_1^{d_1}\dots z_n^{d_n}$ of $f$ have $d_i<c_i$ for \emph{at least one} index $i$.
\end{remark}
\section{Toric version of the Combinatorial Nullstellensatz}
Continuing to follow the results of~\cite{gekh2002}, we observe that Theorem~\ref{cn-prod} can also be viewed as a very particular case of the toric residue formula in~\cite{gekh2002}. Let us show this in more detail. We are going to deal with Laurent polynomials $f\in \mathbb C[z_1,\ldots, z_n,z_1^{-1},\ldots, z_n^{-1}]$ and their Newton polytopes $N(f)\in\mathbb Z^n$, that is convex hulls of the degrees of nonzero monomials in $f$.
Take some $n$ Laurent polynomials $g_1,\ldots, g_n$, the Newton polynomial of their product $N(g_1\dots g_n)$ equals the Minkowski sum $N(g_1)+\dots + N(g_n)$. Following Gelfond and Khovanskii~\cite{gekh2002} the system $N(g_1), \ldots, N(g_n)$ is called \emph{unfolded} if in their Minkowski sum every face $F$ of positive codimension in its unique decomposition $F=F_1+\ldots+F_n$ into the sum of faces of the polytopes $N(g_1), \ldots, N(g_n)$ has at least one zero-dimensional $F_i$. This is a certain requirement of general position and it is easy to check that for polynomials $g_1(z_1),\ldots,g_n(z_n)$ each depending on its respective one variable (as in Theorem~\ref{cn-prod}) this assumption is satisfied.
\begin{theorem}[Gelfond--Khovanskii, 2002]
Consider a differential form
$$
\omega = \frac{f}{g_1\dots g_n} dz_1\wedge\dots\wedge dz_n
$$
where the system of Newton polytopes $N(g_1), \ldots, N(g_n)$ is unfolded. Let $Z$ be the set of common zeros of $g_1,\ldots, g_n$ in $T={\mathbb C^*}^n$ and $V$ be the set of vertices of the polytope $N=N(g_1)+\dots + N(g_n)$, then
\begin{equation}
\label{toric-res-eqn}
\sum_{z\in Z} \Res_z \omega = (-1)^n \sum_{v\in V} k_v \Res_v \omega.
\end{equation}
\end{theorem}
Definitely, this formula needs some explanations. The left hand side of (\ref{toric-res-eqn}) is the ordinary sum of residues over the ``finite'' points of the toric variety, that is points lying in $T$. The right hand side is the sum of residues in the ``infinite'' points of the compactification of $T$ that gives the toric variety. The coefficients $k_v$ are integers depending on the combinatorial structure of $N$ near its vertex $v$ and the value $\Res_v \omega$ is calculated explicitly as the constant term in the Laurent series expansion of $\frac{z_1^{v_1}\dots z_n^{v_n}}{g_1\dots g_n}$ multiplied by the Laurent polynomial $\frac{f}{z_1^{v_1-1}\dots z_n^{v_n-1}}$. Here it is convenient to denote $z_1^{v_1}\dots z_n^{v_n}$ by $z^v$ for $z=(z_1,\ldots, z_n)$ and $v=(v_1,\ldots, v_n)$; also denote by $\mathbf{e}=(1,\ldots, 1)\in \mathbb Z^n$ the all-unit vector.
Let us describe a particular case when everything has a very explicit form. We make the important assumption: For every vertex $v\in V$ there exists an outer support halfspace $H$ to $N$ at $v$ such that $H\cap N = \{v\}$ and the polytope $N(f)+\mathbf{e}$ does not intersect $\inte H$ (in Theorem~\ref{cn-prod} this corresponds to the degree upper bound).
The set of vertices of $N$ thus splits into two parts $V=V_+\cup V_0$ depending on whether they are outside $N(f) + \mathbf{e}$ or on its boundary. It is easy to see that for $v\in V_+$ the value $\Res_v \omega$ is zero and for $v\in V_0$ it equals the coefficient at $z^{v-\mathbf{e}}$ in $f$, divided by the product of coefficients in $g_i$ at the monomials corresponding to the unique representation of $v$ as a sum of vertices of $N(g_1), \ldots, N(g_n)$. Finally we obtain:
\begin{corollary}
\label{toric-cn}
Under above assumptions $\sum_{z\in Z} \Res_z \omega$ equals a linear combination of the coefficients of $f$ at monomials $z^{v-\mathbf{e}}$ for $v\in V_0$ with integer coefficients $k_v$.
\end{corollary}
\begin{remark}
In some cases one may guarantee that the coefficient $k_v$ for $v\in V_0$ is nonzero. For example, this is the case when exactly $n$ facets of $N$ meet at $v$. It is easy to check that this is the case in Theorem~\ref{cn-prod}.
\end{remark}
\begin{remark}
As it was already discussed, when all zeros in $Z$ are simple then on the left hand side of (\ref{toric-res-eqn}) we have a sum of values of $f$ in the points of $Z$ with certain nonzero coefficients.
\end{remark}
\begin{remark}
Corollary~\ref{toric-cn} formally requires the points of $Z$ to have only nonzero coordinates (they have to lie in $T$), but it is easy to see that Theorem~\ref{cn-prod} follows in its full generality by a translation of the sets $A_i$ so that they avoid zero.
\end{remark}
\section{Residues on $\mathbb CP^n$ and the Cayley--Bacharach theorem}
Another version of the proof for Combinatorial Nullstellensatz arises if we consider the form
$$
\omega = \frac{f(z_1,\ldots, z_n)dz_1\wedge\dots\wedge dz_n}{g_1(z_1)\dots g_n(z_n)}.
$$
over the projective space $\mathbb CP^n$. Compared to the previous section, this approach allows to make the results more flexible and independent on the Newton polynomials of $f$ and $g_i$.
Suppose first that $\deg f \le \sum_{i=1}^n k_i - n - 1$. In this case a simple calculation shows that $\omega$ has no singularity over the hyperplane at infinity, and we obtain the equality (the residues are with respect to the divisors corresponding to $g_1, \ldots, g_n$)
$$
\sum_{(z_1,\ldots, z_n)\in X_1\times\dots X_n} \Res_{(z_1,\ldots, z_n)} \omega = \sum_{(z_1,\ldots, z_n)\in X_1\times\dots X_n} \frac{f(z_1,\ldots, z_n)}{g'_1(z_1)\dots g'_n(z_n)} = 0,
$$
which leads to the Cayley--Bacharach theorem (see~\cite{bach1886,cayley1889} and the textbook~\cite[Ch.~5, \S~2]{gh1978}): If $f$ is zero at all but one points of $X$, then it should be zero at the remaining point. We give here the general statement of the Cayley--Bacharach theorem:
\begin{theorem}[Cayley--Bacharach, XIXth century]
\label{cb}
If the system of equations
\begin{eqnarray*}
g_1(x) &=& 0\\
&\ldots&\\
g_n(x) &=& 0
\end{eqnarray*}
of degrees $k_1,\ldots, k_n$ has $k=k_1k_2\dots k_n$ isolated solutions $X=\{x_1,\ldots, x_k\}$, then there exists a linear dependence with nonzero coefficients:
\begin{equation}
\label{cb-rel}
\sum_{i=1}^k \alpha_i f(x_i) = 0
\end{equation}
between values of every polynomial of degree $\deg f \le \sum_{i=1}^n k_i - n - 1$. In particular, the polynomial should be zero on $X$ if and only if it is zero on all but one points of $X$.
\end{theorem}
This theorem holds over arbitrary field if all the points of $X$ are defined over this field. Let us list some recent nontrivial uses of this theorem:
\begin{itemize}
\item
An interesting application of the Cayley--Bacharach relations (\ref{cb-rel}) is distinguishing between nonnegative polynomials and sums of squares, see~\cite{ble2010} for further details.
\item
The least nontrivial case of the Caylet--Bacharach theorem, for intersection of two triples of lines, was used in the recent paper~\cite{greentao2012} about Sylvester type problems.
\end{itemize}
It is curious that different particular cases of the Cayley--Bacharach theorem have their own names. For example, Miquel's six circle theorem~\cite{wikimiq} asserts that if $7$ out of $8$ vertices of a combinatorial cube $C$ in $\mathbb R^3$ lie on a quadratic surface $S$ then the remaining vertex of $C$ also must lie on $S$. Another particular case of the Cayley--Bacharach theorem is the result about cutting the integer points in a cube by hyperplanes (see~\cite[Theorem~6.3]{alon1999} and Problem~6 at \href{http://www.imo-official.org/}{IMO}~2007), which we state in a bit more general, than usual, form here:
\begin{corollary}
\label{cover-hyperpl}
Suppose we have $n$ families of hyperplanes $\mathcal H_1,\ldots,\mathcal H_n$ in $\mathbb CP^n$ with respective cardinalities $k_1,\ldots, k_n$. Define the intersection set
$$
X=\{H_1\cap\dots\cap H_n : H_1\in\mathcal H_1,\ldots, H_n\in\mathcal H_n\}
$$
and assume that it is discrete and has the maximum possible cardinality $k=k_1k_2\dots k_n$. If $x\in X$ is any point, then the set $X\setminus x$ cannot be covered by less than $\sum_{i=1}^n k_i - n$ hyperplanes that do not pass through $x$.
\end{corollary}
Using the projective duality we obtain another consequence:
\begin{corollary}
Let $n$ finite point sets $X_1,\ldots, X_n\subset \mathbb CP^n$ have cardinalities $k_1,\ldots, k_n$. Assume that any system of representatives $x_i\in X_i$ defines a unique hyperplane $H(x_1,\ldots, x_n)$ containing $\{x_i\}_{i=1}^n$ and all these hyperplanes are distinct. Then one needs at least $\sum_{i=1}^n k_i - n$ points to pierce all such hyperplanes $H(x_1,\ldots, x_n)$ but one $H(x^0_1, \ldots, x^0_n)$ without touching this one.
\end{corollary}
Now return to the original statement of the Combinatorial Nullstellensatz, where $\deg f = \sum_{i=1}^n k_i - n$. In this case $\omega$ has the singularity at the hyperplane at infinity, and we should include this hyperplane to a divisor in the definition of the residues. Finite singularity hyperplanes are
$$
\mathcal H_i = \{H : H=\{z_i = x\},\ x\in X_i\}.
$$
The hyperplane at infinity can be added to the first family of hyperplanes $\mathcal H_1$ for example, to give $\mathcal H_1^*$. The corresponding set
$$
X^*=\{H_1\cap\dots\cap H_n : H_1\in\mathcal H_1^*,H_2\in\mathcal H_2,\ldots, H_n\in\mathcal H_n\}
$$
will contain all the points of $X$, and the point $x^*$ at the infinite direction of $(1,0,\ldots,0)$ axis. Note that the form $\omega$ has a bad singularity in $x^*$, and the residue formula is hard to apply at this point. But this can be corrected, if we perturb the families $\mathcal H_i$ ($i=2,\ldots, n$) so that the point $x^*$ becomes a set of $k_2\dots k_n$ points with simple singularities, lying on the hyperplane at infinity. For these points the formula can be proved by induction, by putting the sum of residues to the hyperplane at infinity and applying the inductive assumption.
This proof is good, but it is much longer than the original proof without residues. In order to justify this we may generalize the Combinatorial Nullstellensatz in some way, for example:
\begin{theorem}
Suppose we have $n$ hypersurfaces $S_1, \ldots, S_n\subset \mathbb C^n$ with respective degrees $k_1,\ldots, k_n$, and their equations have the form
$$
g_i(z_1,\ldots, z_n) = z_i^{k_i} + \text{terms of less degree}.
$$
Assume that they intersect in a discrete set $X$ of cardinality $k=k_1k_2\dots k_n$. If a polynomial $f(z_1,\ldots, z_n)$ has degree $\le \sum_{i=1}^n k_i - n$ and a nonzero coefficient at $z_1^{k_1-1}\dots z_n^{k_n-1}$, then its zero set cannot contain $X$.
\end{theorem}
It seems that for arbitrary $g_i(z_1,\ldots, z_n)$ the condition ``coefficient at $z_1^{k_1-1}\dots z_n^{k_n-1}$ is nonzero'' should be replaced by some other condition, depending on the maximal degree parts of $f, g_1,\ldots, g_n$.
\section{Further similar problems}
The first question is: Does the two-dimensional case of Theorem~\ref{cover-hyperpl} admit a simpler proof? Its elementary statement reads as follows:
\begin{problem}
Suppose $n$ red and $m$ blues lines in the plane have $nm$ points of transversal red-blue intersection, denote this intersection set by $X$. Prove that if a family of green lines covers all points of $X$ but one then there are at least $n+m-2$ green lines.
\end{problem}
Another question is related to some algebraic constructions of hypergraphs in~\cite{fr2011}. We believe that the residues may help to answer it, but cannot tell anything particular at the moment.
\begin{problem}
\label{grid-lines}
Suppose $n$ red and $n$ blue lines in the plane have $n^2$ points of transversal red-blue intersection, again denote this intersection set by $X$. Describe all cases when $X$ can be covered by $n$ green lines, distinct from the original blue and red lines.
\end{problem}
There are nontrivial examples for Problem~\ref{grid-lines}: In $\mathbb F_p\times \mathbb F_p$ we may consider all vertical lines red, all horizontal lines blue, and all lines with a fixed slope green. Here $n=p$ is the characteristic of the field.
Another example is: Let $U\subset\mathbb F^*$ be a finite multiplicative subgroup of order $n$, which necessarily coincides with the $n$-th roots of unity. Consider the blue lines $\{x-uy\}_{u\in U}$, the red lines $\{y = u\}_{u\in U}$, and the green lines $\{x=u\}_{u\in U}$. This is a valid configuration in Problem~\ref{grid-lines} and an important observation is that all three color families of lines are concurrent.
Actually, the case of interest in~\cite{fr2011} is when $n<p$ (in $\mathbb F_p$) and the green lines form the (concurrent) family of vertical lines $x=0, x=1,\ldots, x = n-1$. In~\cite[Lemma~2.9]{fr2011} it is shown that no such configurations (with vertical green lines) exists for $n>3$ over the field $\mathbb R$, the proof using combinatorics of pseudolines. The case of finite characteristic with this selection of vertical lines is reduced to the real case (see~\cite{fr2011}) for $p>n^{4n}$ using the Dirichlet theorem on approximation by rational numbers.
We have a couple of observations on Problem~\ref{grid-lines}, with no use of residues, considering concurrent families of lines:
\begin{claim}
\label{grid-conc}
In terms of Problem~\ref{grid-lines}, let $r_i(x)=0$ be the equations of the red lines, let $b_i(x)=0$ and $g_i(x)=0$ be the equations of blues and green lines respectively. If all the green lines are concurrent then there is a linear dependence between the products $R(x) = \prod_i r_i(x)$, $B(x) = \prod_i b_i(x)$, and $G(x) = \prod_i g_i(x)$.
\end{claim}
\begin{proof}
We denote by the same letter the straight line and its corresponding linear function. Let $x_0$ be the common point of the green lines. Note that on every line $g_i$ there must be at most $n$ points of $X$, because it meets at most $n$ red lines. Hence every $g_i$ contains exactly $n$ points of $X$ and these $n$-tuples are pairwise disjoint. Hence the common point $x_0$ cannot be in $X$.
Now choose coefficients $\alpha$ and $\beta$ so that $Z(x) = \alpha R(x) + \beta B(x)$ vanishes on $x_0$; it also vanishes on $X$. For every line $g_i$ the function $Z(x)$ vanishes on $g_i$ at $x_0$ and at $n$ intersection points $X\cap g_i$. Since $Z(x)$ has degree $\le n$ it must vanish on every $g_i$ and therefore it must be proportional to the product $G(x)$.
\end{proof}
\begin{claim}
\label{grid-conc2}
If we assume in Problem~\ref{grid-lines} that the red lines are concurrent and the green lines are concurrent, and also assume that $n$ is coprime with the characteristic of $\mathbb F$, then the example with roots of unity becomes unique up to projective transformation.
\end{claim}
\begin{proof}
After a projective transformation we assume that the red lines are $\{x = u\}_{u\in U}$ and the blue lines are $\{y = v\}_{v\in V}$. Then every green line $g_i$ is a graph of a linear bijection $U\to V$. Hence we have a set of linear transforms $g_j^{-1}g_i$ for the set $U$. These linear transforms must preserve the mass center $\frac{1}{n}(u_1+\dots + u_n)$ of $U$, and after another shift of the coordinates we assume that this mass center is zero and all the transforms $g_j^{-1}g_i$ are multiplications by a constant $c_{j i}$. Let us also rescale so that $U$ contains $1$. Then every $c_{j i}$ is contained in $U$, and since there must be at least $n$ distinct constants corresponding to $g_1^{-1}g_1, g_2^{-1}g_1, \ldots, g_n^{-1}g_1$ then we conclude that $U$ is a multiplicative subgroup and the transforms are multiplications by elements of this groups. After an appropriate shift and rescaling of the $y$ axis the set $V$ becomes equal to $U$.
\end{proof}
Finally we mention a problem from~\cite{gassa2000} related to the polynomial interpolation, which is in the spirit of the present discussion:
\begin{problem}
Suppose $X$ is a set of $\binom{n+2}{2}$ points in the plane such that for any $x\in X$ there exist $n$ lines covering $X\setminus \{x\}$ and not touching $x$. Describe such sets $X$ or, at least, prove that some $n+1$ points of $X$ lie on a single line.
\end{problem}
| {
"timestamp": "2015-03-30T02:06:10",
"yymm": "1503",
"arxiv_id": "1503.08004",
"language": "en",
"url": "https://arxiv.org/abs/1503.08004",
"abstract": "We interpret the Combinatorial Nullstellensatz of Noga Alon as a multidimensional residue formula, describe some consequences of this interpretation and related open problems.",
"subjects": "Combinatorics (math.CO); Algebraic Geometry (math.AG)",
"title": "Residues and the Combinatorial Nullstellensatz",
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"lm_label": "1. YES\n2. YES",
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https://arxiv.org/abs/2101.00096 | Generalized Hardy Type and Caffarelli-Kohn-Nirenberg Type Inequalities on Finsler Manifolds | In this paper we derive both local and global geometric inequalities on general Riemannnian and Finsler manifolds and prove generalized Caffarelli-Kohn-Nirenberg type and Hardy type inequalities on Finsler manifolds, illuminating curvatures of both Riemannian and Finsler manifolds influence geometric inequalities. | \section[#1]{#2}}
\numberwithin{equation}{section}
\begin{document}
\title
[Inequalities on Finsler Manifolds] {Generalized Hardy Type and Caffarelli-Kohn-Nirenberg Type Inequalities on Finsler Manifolds}
\author
[Shihshu Walter Wei and Bingye Wu]{Shihshu Walter Wei$^*$ and Bing Ye Wu$^{**}$}
\address[Shihshu Walter Wei]
{Department of Mathematics, University of Oklahoma, Norman, Oklahoma, 73019-0315, U.S.A. }
\email{wwei@ou.edu}
\address[Bing-Ye Wu]
{Department of Mathematics, Minjiang University,
Fuzhou, Fujian, 350108, China} \email{wubingye@mju.edu.cn}
\thanks{Key words: Finsler manifold, radial flag curvature, radial Ricci curvature, Hardy inequality, Caffarelli-Kohn-Nirenberg Inequality. }
\subjclass{Primary 53C60; Secondary 53B40 }\date{}
\maketitle
\begin{abstract}
In this paper we derive both local and global geometric inequalities on general Riemannnian and Finsler manifolds and prove generalized Caffarelli-Kohn-Nirenberg type and Hardy type inequalities on Finsler manifolds, illuminating curvatures of both Riemannian and Finsler manifolds influence geometric inequalities.
\end{abstract}
\section{Introduction}
It is well-known that Hardy type inequalities have been widely used in analysis and differential equations. In \cite{CKN} Caffarelli, Kohn and Nirenberg proved rather general interpolation inequalities with weights. Recently in \cite{WL}, Wei and Li used comparison theorems in Riemannian geometry to prove some sharp generalized Hardy type and Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds.
Some applications of generalized Hardy type inequalities in $p$-harmonic geometry have been studied in \cite{CLW1}.
Finsler geometry, as the natural generalization of Riemannian geometry, has been a very active field in differential geometry and appears in a broad spectrum of contexts. (e.g., two different Finsler metrics, the Kobayashi metric and the Carath\'eodory metric appear very naturally in the theory of several complex variables.) The main purpose of the present paper is to, on the one hand, give a local and two $L^p$ versions of the results in \cite{WL} so that the inequalities work on every Riemannian manifold and in a wider class, and on the other hand, generalize their results from Riemannian manifolds to Finsler manifolds. We use Hessian and Laplacian comparison theorems in Finsler Geometry by constructing appropriate vector fields. It should be pointed out here that
the volume form on a Riemannian manifold is uniquely determined by the given Riemannian metric,
while there are different choices of volume forms for Finsler
metrics. The frequently used volume forms in Finsler geometry are the so-called Busemann-Hausdorff volume form and Holmes-Thompson volume form, and in \cite{Wu1,Wu2} we introduce the extreme volume forms $dV_{\rm ext}$ (include the maximal and minimal volume forms, cf. \eqref{2.8} and \eqref{2.9}) for Finsler manifolds
which also play the important role in Finsler geometry. In this paper we shall mainly use the extreme volume forms.
To state our results we need some notions from Finsler geometry, for details see \S 2. Throughout this paper, unless otherwise stated, we let $(M,F)$ be a complete Finsler manifold with finite uniformity constant $\mu_F$ (cf. \eqref{2.11}), Cut($x_0$) be the cut locus of a fixed point $x_0$, and $\Omega\subset M\backslash{\rm Cut}(x_0)$ be a domain in $M.$ It should be pointed out here that in general there are three completeness for Finsler manifolds: forward complete, backward complete and complete (i.e., both forward and backward complete), and they are equivalent when $\mu_F<\infty$. In this situation, the distance function $r=d_F(x_0,\cdot):\Omega\rightarrow\mathbb{R}$ from $x_0$ is smooth on $\Omega\backslash\{x_0\}$, and thus the gradient vector field $\nabla r$ of $r$ (with respect to Finsler metric $F$) is also smooth on $\Omega\backslash\{x_0\}$. We usually call $\nabla r$ the {\it radial vector field with respect to} $x_0$. We call $x_0\in M$ a {\it pole}, if the exponential map exp$_{x_0}:T_xM\rightarrow M$ is a diffeomorphism. We say that $M$ {\it has nonpositive} (resp. {\it nonnegative}) {\it radial flag curvature at} $x_0$ if flag curvature {\bf K}$(\nabla r;P)$ of flag $(\nabla r;P)$ whose flag pole is a radial vector is nonpositive (resp. nonnegative) for every plane $P$ (cf. \eqref{2.5}). Similarly, we say that $M$ {\it has nonpositive} (resp. {\it nonnegative}){\it radial Ricci curvature} {\bf Ric} $(\nabla r) $ at $x_0$ if {\bf Ric} $(\nabla r) \leqslant0$ (resp. $\geqslant0$) (cf. \eqref{2.6}).
In this paper we first derive both local and global
{\bf Geometric Inequalities 4.1 and 4.2} on every Riemannnian manifold and Finsler manifold $\big ($cf. $(4.1.a), (4.1.b), (4.2.a), (4.2.b)\big ).$
We then prove generalized Caffarelli-Kohn-Nirenberg type and Hardy type inequalities on Finsler manifolds.
The main results of this paper are the following:
\noindent
{\bf Theorem 1.1} {\it Let $(M,F)$ be an $n$-dimensional complete Finsler manifold with finite uniformity constant $\mu_F$. Let ${\rm Cut}(x_0)$ be the cut locus of a fixed point $x_0$, and $\Omega\subset M\backslash{\rm Cut}(x_0)$ be a domain in $M\, .$ Suppose that the radial flag curvature ${\bf K}(\nabla r;\,\cdot)$
or radial Ricci curvature ${\bf Ric}(\nabla r)$ of $M$ satisfies one of the following three conditions:
$($i$)$ $0\leqslant {\bf Ric}(\nabla r)$ and $n \leqslant a+b+1$;
$($ii$)$ ${\bf K}(\nabla r;\,\cdot) \leqslant 0$ and $a+b+1\leqslant n$;
$($iii$)$ ${\bf K}(\nabla r;\,\cdot) = 0$ and $a,b\in
\mathbb{R}$ are any constants.
Then for any $u\in C_0^{\infty}(\Omega\backslash \{x_0\})$, the following Caffarelli-Kohn-Nirenberg type inequality holds:
\begin{equation}\label{11}
\int_{\Omega}\frac{|u|^2}{r^{a+b+1}}dV_{\rm ext}\leqslant\hat{\mu}_F^{\frac{n+1}{2}}\cdot
\left(\int_{\Omega}\frac{|u|^{p}}{r^{ap}}dV_{\rm ext}\right)^{\frac {1}{p}}\left(\int_{\Omega}\frac{(F(\nabla u))^{q}}{r^{bq}}dV_{\rm ext}\right)^{\frac {1}{q}}.
\end{equation} In particular, if $M$ has a pole $x_0$ or $\operatorname{Cut}(x_0)$ is empty in $($i$)$, or $M$ is simply connected in $($ii$)$ or $($iii$)$, then for any $u\in C_0^{\infty}(M\backslash \{x_0\})$,
\begin{equation}\label{12}
\int_{M}\frac{|u|^2}{r^{a+b+1}}dV_{\rm ext}\leqslant\hat{\mu}_F^{\frac{n+1}{2}}\cdot
\left(\int_{M}\frac{|u|^{p}}{r^{ap}}dV_{\rm ext}\right)^{\frac {1}{p}}\left(\int_{M}\frac{(F(\nabla u))^{q}}{r^{bq}}dV_{\rm ext}\right)^{\frac {1}{q}},
\end{equation}
where $\hat{\mu}_F^{\frac{n+1}{2}}= \mu_F^{\frac{n+1}{2}} \cdot |\frac{2}{n-a-b-1}\big |$, $p\in [1, \infty]$, $\frac 1p + \frac 1q = 1,$ and if $p=\infty$, $\left(\int_{M}\frac{|u|^{p}}{r^{ap}}dV_{\rm ext}\right)^{\frac {1}{p}}$ stands for the supreme of $\frac{|u|}{r^{a}}.$}
This result is new, even when $M$ is a Riemannian manifold:
\noindent
{\bf Corollary 1.1} {\it Let $M$ be a complete $n$-dimensional Riemanian manifold with the volume element $dv$. Fix $x_0\in M$ let $\Omega\subset M\backslash{\rm Cut}(x_0)$ be a domain in $M\, .$ Suppose that radial curvature $K_r$, or radial Ricci curvature $\text {Ric}_{rad}$ of $\Omega$ satisfies one of the following three conditions:
$($i$)$ $0\leqslant \text {Ric}_{rad}$ and $n \leqslant a+b+1$;
$($ii$)$ $K_r \leqslant 0$ and $a+b+1\leqslant n$;
$($iii$)$ $K_r = 0$ and $a,b\in
\mathbb{R}$ are any constants.
Then for any $u\in C_0^{\infty}(\Omega\backslash\{x_0\})$, the following Caffarelli-Kohn-Nirenberg type inequality holds:
\begin{equation}\label{13}
C\cdot\int_{\Omega}\frac{|u|^2}{r^{a+b+1}}dv \leqslant
\left(\int_{\Omega}\frac{|u|^{p}}{r^{ap}}dv\right)^{\frac {1}{p}}\left(\int_{\Omega}\frac {|\nabla u|^{q}}{r^{bq}}dv\right)^{\frac {1}{q}}.
\end{equation} In particular, if in $($i$)$ $\operatorname{Cut}(x_0) = \emptyset$, or in $($ii$)$ or in $($iii$)$ $\pi_1(M)=0$, then for any $u\in C_0^{\infty}(M\backslash \{x_0\})$,
\begin{equation}\label{14}
C\cdot\int_{M}\frac{|u|^2}{r^{a+b+1}}dv \leqslant
\left(\int_{M}\frac{|u|^{p}}{r^{ap}}dv\right)^{\frac {1}{p}}\left(\int_{M}\frac {|\nabla u|^{q}}{r^{bq}}dv\right)^{\frac {1}{q}},
\end{equation} where $C=C(a,b)=\big |\frac{n-a-b-1}{2}\big |$, $p\in [1, \infty]$, and $\frac 1p + \frac 1q = 1.$}
\noindent
{\bf Theorem 1.2} {\it Let $(M,F), \Omega$ and curvature conditions as in Theorem 1.1 $(i), (ii), (iii)$ Then for any $u\in C_0^\infty(\Omega\backslash\{x_0\})$, the following Caffarelli-Kohn-Nirenberg type inequality holds:
\begin{equation}\label{15}
\int_{\Omega}\frac{|u|^p}{r^{a+b+1}}dV_{\rm ext}\leqslant\check{\mu}_F^{\frac{n+1}{2}}\cdot
\left(\int_{\Omega}\frac{|u|^{p}}{r^{aq}}dV_{\rm ext}\right)^{\frac {1}{q}}\left(\int_{\Omega}\frac{(F(\nabla u))^{p}}{r^{bp}}dV_{\rm ext}\right)^{\frac {1}{p}}.
\end{equation} In particular, if $M$ has a pole $x_0$ or $\operatorname{Cut}(x_0)$ is empty in $($i$)$, or $M$ is simply connected in $($ii$)$ or $($iii$)$, then for any $u\in C_0^\infty(M\backslash\{x_0\})$,
\begin{equation}\label{16}
\int_{M}\frac{|u|^p}{r^{a+b+1}}dV_{\rm ext}\leqslant\check{\mu}_F^{\frac{n+1}{2}}\cdot
\left(\int_{M}\frac{|u|^{p}}{r^{aq}}dV_{\rm ext}\right)^{\frac {1}{q}}\left(\int_{M}\frac{(F(\nabla u))^{p}}{r^{bp}}dV_{\rm ext}\right)^{\frac {1}{p}},
\end{equation}
where $\check{\mu}_F^{\frac{n+1}{2}}=\mu_F^{\frac{n+1}{2}}\cdot \big |\frac{p}{n-a-b-1}\big |$, $p\in (1, \infty)$ and $\frac 1p + \frac 1q = 1.$}
Similarly, when $M$ is a Riemannian manifold, we have the following new result:
\noindent
{\bf Corollary 1.2} {\it Let $M, \Omega$ and curvature conditions as in Corollary 1.1 $(i), (ii), (iii)$ Then for any $u\in C_0^\infty(\Omega\backslash\{x_0\})$, the following Caffarelli-Kohn-Nirenberg type inequality holds:
\begin{equation}\label{17}
\tilde{C}\cdot\int_{\Omega}\frac{|u|^p}{r^{a+b+1}}dv \leqslant
\left(\int_{\Omega}\frac{|u|^{p}}{r^{aq}}dv\right)^{\frac {1}{q}}\left(\int_{\Omega}\frac {|\nabla u|^{p}}{r^{bp}}dv\right)^{\frac {1}{p}}.
\end{equation} In particular, if in $($i$)$ $\operatorname{Cut}(x_0) = \emptyset$, or in $($ii$)$ or in $($iii$)$ $\pi_1(M)=0$, then for any $u\in C_0^\infty(M\backslash\{x_0\})$,
\begin{equation}\label{18}
\tilde{C}\cdot\int_{M}\frac{|u|^p}{r^{a+b+1}}dv \leqslant
\left(\int_{M}\frac{|u|^{p}}{r^{aq}}dv\right)^{\frac {1}{q}}\left(\int_{M}\frac {|\nabla u|^{p}}{r^{bp}}dv\right)^{\frac {1}{p}},
\end{equation} where $\tilde{C}=\tilde{C}(a,b)=\big |\frac{n-a-b-1}{p}\big |$, $p\in (1, \infty)$ and $\frac 1p + \frac 1q = 1.$}
When $p=q=2,$ the above two Theorems $1.1$ and $1.2$ meet and give their $L^2$ version:
\noindent
{\bf Theorem 1.3} {\it Let $(M,F)$ be an $n$-dimensional complete Finsler manifold with finite uniformity constant $\mu_F$. Fix $x_0\in M$ let $\Omega\subset M\backslash{\rm Cut}(x_0)$ be a domain. Suppose that the radial flag curvature ${\bf K}(\nabla r;\,\cdot)$
or radial Ricci curvature ${\bf Ric}(\nabla r)$ satisfies one of the following three conditions:
$($i$)$ $0\leqslant {\bf Ric}(\nabla r)$ and $n \leqslant a+b+1$;
$($ii$)$ ${\bf K}(\nabla r;\,\cdot) \leqslant 0$ and $a+b+1\leqslant n$;
$($iii$)$ ${\bf K}(\nabla r;\,\cdot) = 0$ and $a,b\in
\mathbb{R}$ are any constants.
Then for any $u\in C_0^{\infty}(\Omega\backslash\{x_0\})$, the following Caffarelli-Kohn-Nirenberg type inequality holds:
\begin{equation}\label{19}
\int_{\Omega}\frac{|u|^2}{r^{a+b+1}}dV_{\rm ext}\leqslant\hat{\mu}_F^{\frac{n+1}{2}}\cdot
\left(\int_{\Omega}\frac{|u|^{2}}{r^{2a}}dV_{\rm ext}\right)^{\frac {1}{2}}\left(\int_{\Omega}\frac{(F(\nabla u))^{2}}{r^{2b}}dV_{\rm ext}\right)^{\frac {1}{2}}.
\end{equation} In particular, if $M$ has a pole $x_0$ or $\operatorname{Cut}(x_0)$ is empty in $($i$)$, or $M$ is simply connected in $($ii$)$ or $($iii$)$, then for any $u\in C_0^{\infty}(M\backslash\{x_0\})$,
\begin{equation}\label{110}
\int_{M}\frac{|u|^2}{r^{a+b+1}}dV_{\rm ext}\leqslant\hat{\mu}_F^{\frac{n+1}{2}}\cdot
\left(\int_{M}\frac{|u|^{2}}{r^{2a}}dV_{\rm ext}\right)^{\frac {1}{2}}\left(\int_{M}\frac{(F(\nabla u))^{2}}{r^{2b}}dV_{\rm ext}\right)^{\frac {1}{2}}.
\end{equation}
where $\hat{\mu}_F^{\frac{n+1}{2}}=\mu_F^{\frac{n+1}{2}} \cdot \big |\frac{2}{n-a-b-1}\big |$.}
\noindent
{\bf Remark} When $F$ is Riemannian, one has $\mu_F=1$, and we recapture the corresponding results for Riemannian manifolds \cite{WL, CLW1, CLW2, W}.
As applications, we obtain embedding theorems for weighted Sobolev spaces of functions on Finsler manifolds (cf. Theorem 7.1) and geometric differential-integral inequalities on Finsler manifolds (cf. Theorem 7.2), generalizing the work in \cite{WL} in Riemannian manifolds. We then focus our study on generalized Hardy type inequalities on Finsler manifolds by using the {\it double} limiting technique in \cite {WL}, and extend the density argument in \cite{CLW2}. We introduced the notion of the space $W_{F,0}^{1,p}(M)$ on a Finsler manifold to be the completion of smooth compactly supported functions $u\in C_0^{\infty}(M)$
with respect to the ``norm"
\begin{equation}\label{1.11}
\|u\|_{W_{F,0}^{1,p}(M)}:= \left(\int_M \big ( |u|^p +
(F(\nabla u))^p \big )\, dV_{\rm ext} \right)^{\frac{1}{p}}.
\end{equation}
It is easy to verify that $\|\cdot\|_{W_{F,0}^{1,p}(M)}$ satisfies the following properties:\\\indent
(i) (Positive definiteness) $\|u\|_{W_{F,0}^{1,p}(M)}\geqslant0, \forall u\in W_{F,0}^{1,p}(M)$, and $\|u\|_{W_{F,0}^{1,p}(M)}=0$ if and only if $u=0$ almost everywhere.\\\indent
(ii) (Positive homogeneity) $\|\lambda u\|_{W_{F,0}^{1,p}(M)}=\lambda\|u\|_{W_{F,0}^{1,p}(M)},\forall \lambda>0$ and $u\in W_{F,0}^{1,p}(M)$.\\\indent
(iii) (Triangle inequality) $\|u+v\|_{W_{F,0}^{1,p}(M)}\leqslant \|u\|_{W_{F,0}^{1,p}(M)}+\|v\|_{W_{F,0}^{1,p}(M)}, \forall u,v\in W_{F,0}^{1,p}(M)$.\\\indent
We note here that $\|\lambda u\|_{W_{F,0}^{1,p}(M)}=|\lambda|\cdot\|u\|_{W_{F,0}^{1,p}(M)}$ does not hold for general Finsler metric, that is to say, $\|\cdot\|_{W_{F,0}^{1,p}(M)}$ is not a genuine norm for general Finsler manifold. Nevertheless we have $\|\lambda u\|_{W_{F,0}^{1,p}(M)}\leqslant\mu_F^\frac12|\lambda|\cdot\|u\|_{W_{F,0}^{1,p}(M)}$, and since we assume $\mu_F<\infty$, we may define Cauchy sequence $\{u_i\}\subset C_0^\infty(M)$ with respect to $\|\cdot\|_{W_{F,0}^{1,p}(M)}$ in the usual way, and thus $W_{F,0}^{1,p}(M)$ is well-defined.
We say {\it $\frac {u}{r} \in L^p(M)$ in a Finsler sense}, denoted by $\frac {u}{r} \in L_{F}^p(M)$ if $\int_M\frac{|u|^p}{r^p}dV_{\rm ext} < \infty\, .$
In particular, we have
\noindent
{\bf Theorem 1.4} {\it Let $(M,F)$ be an $n$-dimensional complete Finsler manifold with nonpositive radial flag curvature at the pole $x_0\in M$ and with finite uniformity constant $\mu_F$. Then for any $u\in W_{F,0}^{1,p}(M)$ and $1<p<n$, the following Hardy type inequality holds:
\begin{equation}\label{111}
\int_M\frac{|u|^p}{r^p}dV_{\rm ext}\leqslant\acute{\mu}_F^{\frac{n+p}{2}}\cdot \int_M(F(\nabla u))^p \, dV_{\rm ext},
\end{equation}
where $\acute{\mu}_F^{\frac{n+p}{2}} = \mu_F^{\frac{n+p}{2}} \cdot \left(\frac{p}{n-p}\right)^p\, .$ Furthermore, $\frac {u}{r} \in L^p(M)\, $ in a Finsler sense.}
This recaptures a result of Wei-Li \cite[Theorem 1, Corollary 1.2]{WL} (cf. Corollary 8.2), when $M$ is a Riemannian manifold.
\noindent
{\bf Theorem 1.5} {\it Let $(M,F)$ be an $n$-dimensional complete Finsler manifold with nonnegative radial Ricci curvature at the pole $x_0\in M$, and with finite uniformity constant $\mu_F$. Then for any $u\in W_{F,0}^{1,p}(M),$ $\frac {u}{r} \in L_{F}^p(M)$ and $p>n$, the following Hardy type inequality holds:}
\begin{equation}\label{112}
\int_M\frac{|u|^p}{r^p}dV_{\rm ext}\leqslant\grave{\mu}_F^{\frac{n+p}{2}}\cdot \int_M(F(\nabla u))^p\, dV_{\rm ext},
\end{equation}
where $\grave{\mu}_F^{\frac{n+p}{2}} = \mu_F^{\frac{n+p}{2}} \cdot \left(-\frac{p}{n-p}\right)^p $
This recaptures a theorem of Chen-Li-Wei \cite[Theorem 5]{CLW2} (cf. Corollary 8.4), when $M$ is a Riemannian manifold. Furthermore, the assumption $\frac {u}{r} \in L_{F}^p(M)$ cannot be dropped, or a counter-example is constructed in Section 5 in Chen-Li-Wei \cite{CLW2}.
\noindent
{\bf Corollary 1.3} {\it Let $(M,F)$ be an $n$-dimensional complete Finsler manifold with vanishing flag curvature at the pole $x_0\in M$, and with finite uniformity constant $\mu_F$. Then $(i)$ for any $u\in W_{F,0}^{1,p}(M),$ $\frac {u}{r} \in L_{F}^p(M)$ and $1 < p < \infty$, or (ii) for any $u\in W_{F,0}^{1,p}(M),$ and $p < n$ the following Hardy type inequality holds:
\begin{equation}\label{113}
\int_M\frac{|u|^p}{r^p}dV_{\rm ext}\leqslant\dot{\mu}_F^{\frac{n+p}{2}}\cdot \int_M(F(\nabla u))^pdV_{\rm ext},
\end{equation}
where $\dot{\mu}_F^{\frac{n+p}{2}} = \mu_F^{\frac{n+p}{2}} \cdot \left|\frac{p}{n-p}\right|^p\, .$}
Theorem 1.5 is in contrast to Theorem 1.4, in which $\frac {u}{r} \in L^p(M)$ in a Finsler sense is a conclusion, rather than an assumption.
The above theorems illuminate that curvatures of both Riemannian and Finsler manifolds influence geometric inequalities such as generalized Hardy Type and Caffarelli-Kohn-Nirenberg Type inequalities.
\section{Finsler Geometry}
In this section we shall recall some basic notations and formulas in Finsler geometry, for details we refer to \cite{Wu1,Wu3,WX}.
Let $(M,F)$ be a
Finsler $n$-manifold with Finsler metric $F:TM\rightarrow
[0,\infty)$, where $TM$ is the tangent bundle of $M\, .$ Let $(x,y)=(x^i,y^j)$ be local coordinates on $TM$,
and $\pi:TM\backslash\{0\}\rightarrow M$ be the natural projection. Unlike
in the Riemannian case, in general Finsler quantities are functions defined on
$TM$ rather than $M$. The {\it fundamental tensor} $g_{ij}$ and the
{\it Cartan tensor} $C_{ijk}\, , 1 \leqslant i, j, k \leqslant n$ are defined by
$$g_{ij}(x,y):=\frac{1}{2}\frac{\partial^2F^2(x,y)}{\partial y^i\partial
y^j},\quad
C_{ijk}(x,y):=\frac{1}{4}\frac{\partial^3F^2(x,y)}{\partial
y^i\partial y^j\partial y^k}.$$
According to \cite{Chern}, the pulled-back bundle $\pi^*TM$ admits a
unique affine connection, called the {\it Chern connection}. Its
connection forms $\omega^i_j$ are characterized by the following two structure equations:
\noindent
$\bullet$ Torsion freeness:
$$dx^j\wedge\omega^i_j=0;$$
$\bullet$ Almost $g$-compatibility:
$$dg_{ij}-g_{kj}\omega^k_i-g_{ik}\omega^k_j=2C_{ijk}(dy^k+N^k_ldx^l),$$
where $N^k_l$ are real-valued functions determined by $N^k_ldx^l=y^l\omega^k_l$.
It is easy to know that torsion freeness is equivalent to the
absence of $dy^k$ terms in $\omega^i_j$; namely,
$$\omega^i_j=\Gamma^i_{jk}dx^k,$$
together with the symmetry
$$\Gamma^i_{jk}=\Gamma^i_{kj}.$$
Let $V=V^i\partial/\partial x^i$ be a non-vanishing vector field on
an open subset $\mathcal{U}\subset$ $M$. One can introduce a
Riemannian metric $\widetilde{g}^{V} (\cdot,\cdot) =\langle\cdot,\cdot\rangle_V$ in the direction of $V\, ,$ and a linear connection
$\nabla^V$ on the tangent bundle over $\mathcal{U}$ as follows:
$$\widetilde{g}^{V} (X,Y)=\langle X,Y\rangle_V:=X^iY^jg_{ij}(x,V),\quad \forall X=X^i\frac{\partial}{\partial
x^i},Y=Y^j\frac{\partial}{\partial x^j};$$
$$\nabla^V_{\frac{\partial}{\partial x^i}}\frac{\partial}{\partial
x^j}:=\Gamma^k_{ij}(x,V)\frac{\partial}{\partial x^k}.$$ From the
torsion freeness and almost $g$-compatibility of Chern connection we
have
\begin{equation}\label{21}
\nabla^V_XY-\nabla^V_YX=[X,Y],
\end{equation}
\begin{equation}\label{22}X\cdot \langle Y,Z\rangle_V=\langle \nabla^V_XY,Z\rangle_V+\langle Y,\nabla^V_XZ\rangle_V+2{\bf C}_V(\nabla^V_XV,Y,Z),
\end{equation}
here ${\bf C}_V$ is defined by
$${\bf C}_V(X,Y,Z)=X^iY^jZ^kC_{ijk}(x,V).$$By definition {\bf C}$_V(X,X,Z)$ is totally symmetric with respect to $X,Y,Z$, and by Euler's lemma it also satisfies
\begin{equation}\label{23}
{\bf C}_V(V,X,Y)=0.
\end{equation} In view of \eqref{21}-\eqref{23} we see that
the Chern connection $\nabla^V$ and the Levi-Civita connection
$\widetilde{\nabla}^V$ of $\widetilde{g}^{V}(\cdot,\cdot)=\langle\cdot,\cdot\rangle_V$ are related by
\begin{equation}\label{24}
\begin{aligned}
\langle \nabla^V_XY,Z\rangle_V& =\langle\widetilde{\nabla}^V_XY,Z\rangle_V-{\bf
C}_V(\nabla^V_XV,Y,Z)\\
& \quad -{\bf C}_V(\nabla^V_YV,X,Z)+{\bf
C}_V(\nabla^V_ZV,X,Y).
\end{aligned}
\end{equation}
The {\it Chern curvature}
${\bf R}^V(X,Y)Z$ for vector fields $X,Y,Z$ on $\mathcal{U}$ is
defined by
$${\bf R}^V(X,Y)Z:=\nabla^V_X\nabla^V_YZ-\nabla^V_Y\nabla^V_XZ-\nabla^V_{[X,Y]}Z.$$
In the Riemannian case this curvature does not depend on $V$ and
coincides with the Riemannian curvature tensor. Let $P\subset T_xM$ be a 2-plane and $V\in P$ be a nonzero vector. We call the pair $(V;P)$ a {\it flag with pole} $V$.
The {\it flag
curvature} ${\bf K}(V;P)$ of given flag is defined as follows:
\begin{equation}\label{2.5}{\bf K}(V;P)={\bf K}(V;W):=\frac{\langle{\bf R}^V(V,W)W,V\rangle_V}{\langle V,V\rangle_V\langle W,W\rangle_V-\langle V,W\rangle_V^2} .\end{equation}
Here $W$ is a tangent vector such that $V,W$ span the 2-plane $P$
and $V\in T_xM$ is extended to a geodesic field, i.e.,
$\nabla^V_VV=0$ near $x$. The {\it Ricci curvature}\index{Ricci curvature} ${\bf Ric}(V)$ of $V\in T_xM$ is defined by
\begin{equation}\label{2.6}{\bf Ric}(V)=\sum_{i=1}^{n}{\bf K}(V;E_i),\end{equation}
here $E_1,\cdots,E_n$ is the $\widetilde{g}^V$-orthonormal basis for $T_xM$.
Let $f:M\rightarrow \mathbb{R}$ be a smooth function on $M$. The
{\it gradient} $\nabla f$ of $f$ is defined by
$$ df(X)=\langle\nabla f,X\rangle_{\nabla f},\quad \forall X\in \Gamma(TM)$$ whenever $df\ne0$, and $\nabla f=0$ where $df=0$.
Let $\mathcal{U}=$ $\{ x\in M:\nabla f\, |_x\, \ne 0\}$. We define the
{\it Hessian} $Hess(f)$ of $f$ on $\mathcal{U}$ as follows:
$$Hess(f)(X,Y):=X(Y(f))-\nabla^{\nabla f}_XY(f),\quad \forall X,Y\in
\Gamma(T\mathcal{U}).$$ It is known that $Hess(f)$ is symmetric, and it
can be rewritten as (see \cite{WX})
\begin{equation}\label{25}
Hess(f)(X,Y)=\langle \nabla^{\nabla f}_X\nabla
f,Y\rangle_{\nabla f}.
\end{equation} It should be noted that the notion of Hessian
defined here is different from that in \cite{Sh1}. In that case
$Hess(f)$ is in fact defined by
$$Hess(f)(X,X)=X(X(f))-\nabla^X_XX(f),$$ and there is no definition for
$Hess(f)(X,Y)$ if $X\ne Y$. The advantage of our definition is that
$Hess(f)$ is a symmetric bilinear form and we can treat it by using the
theory of symmetric matrix.
The following Hessian comparison theorem for distance function $r=d_F(x_0,\cdot)$ from $x_0$ first was proved in \cite{WX} with pointwise curvature bounds, and it is easy to see from the proof that the pointwise curvature bounds can be weakened to radial curvature bounds. More precisely, we have the following:
\noindent
{\bf Theorem 2.1 $($Hessian Comparison Theorem under Radial Curvature Assumptions$)$}
{\it Let $(M,F)$ be an $n$-dimensional complete Finsler
manifold, and $x_0\in M$.\\\indent
$(1)$ Suppose that $M$ has nonpositive radial flag curvature ${\bf K}(\nabla r;\,\cdot)\leqslant 0$, then for any tangent vector field $X$ on $M$ the following inequality holds whenever $r$ is smooth:
$$Hess(r)(X,X)\geqslant\frac1r\left(\langle X,X\rangle_{\nabla r}-\left\langle\nabla r,X\right\rangle_{\nabla r}^2\right);$$\indent
$(2)$ Suppose that $M$ has nonnegative radial Ricci curvature ${\bf Ric}(\nabla r)\geqslant 0$, then the following inequality holds whenever $r$ is smooth:
$$\sum_{i=1}^nHess(r)(E_i,E_i)\leqslant\frac{n-1}r,$$ here $E_1,\cdots,E_n$ is the local $\langle\cdot,\cdot\rangle_{\nabla r}$-orthonormal frame on $M$.}
Define
\begin{equation}\label{2.8}
\begin{aligned}dV_{\rm max}&=\sigma_{\rm max}(x)dx^1\wedge\cdots\wedge dx^n\\ {\rm and} \qquad
dV_{\rm min}&=\sigma_{\rm min}(x)dx^1\wedge\cdots\wedge dx^n,\\ \operatorname{where}\qquad
\sigma_{\max} (x):&=\max_{y\in T_xM\setminus
\{0\}}\sqrt{\det\left(\left\langle\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right\rangle_y\right)},\\
\sigma_{\rm min}(x):&=\min_{y\in
T_xM\setminus \{0\}}\sqrt{\det\left(\left\langle\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right\rangle_y\right)}.\end{aligned}
\end{equation} $\big ($Since $F(x,\lambda y) = \lambda F(x,y)$ for $\forall \lambda > 0\, $ $\Rightarrow$ $\left\langle\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right\rangle_y := g_{ij}(x,y):=\frac{1}{2}\frac{\partial^2F^2(x,y)}{\partial y^i\partial
y^j}$ is homogeneous of degree $0$ in $y$, both $\max_{y\in T_xM\setminus
\{0\}}$ and $\min_{y\in T_xM\setminus
\{0\}}$ in \eqref{2.8} are taken over a compact set and hence can be realized.$\big)$ Then it is easy to check
that the $n$-forms $dV_{\rm max}$ and $dV_{\rm min}$ as well as the function
$\frac{\sigma_{\rm max}}{\sigma_{\rm min}}$ are well-defined on $M$.
We call $dV_{\rm max}$ and $dV_{\rm min}$ the {\it maximal volume form}
and the {\it minimal volume form} of $(M,F)$, respectively\cite{Wu1,Wu3}. Both
maximal volume form and minimal volume form are called {\it extreme
volume form}, and we shall denote by
\begin{equation}\label{2.9}dV_{\rm ext}\, \, \, = \quad \rm{either} \quad {\it d}V_{\rm max}\quad \rm{or}\quad {\it d}V_{\rm min}\end{equation}
The {\it uniformity function} $\mu:M\rightarrow\mathbb{R}$ is
defined by
\begin{equation}\label{2.10}
\begin{aligned}\mu(x)&=\max_{y,z,w\in T_xM\backslash\{0\}}\frac{\langle w,w\rangle_y}{\langle w,w\rangle_z}.\end{aligned}\end{equation}
Then
\begin{equation}\label{2.11}
\begin{aligned}\mu_F&=\sup_{x\in M}\mu(x)\end{aligned}\end{equation}
is called the {\it
uniformity constant}. We always assume $\mu_F<\infty$ throughout this paper. It is known that
\begin{equation}\label{26}
\mu^{-1}_FF^2(w)\leqslant\langle w,w\rangle_y\leqslant\mu_F F^2(w).
\end{equation} Furthermore, we have \cite{Wu1,Wu2}
$$\frac{\sigma_{\rm max}}{\sigma_{\rm min}}\leqslant\mu_F^{\frac{n}{2}}, $$and thus
\begin{equation}\label{27}dV_{\rm max}\leqslant\mu_F^{\frac{n}{2}}dV_{\rm min}.\end{equation}
\section{The Induced Riemannian Metric $\widetilde{g}$ in the Radial Direction $T$}
In this section we introduce the induced Riemannian metric $\widetilde{g}$ by the Finsler metric $F$ in the radial direction $T$. This will play an important role in the proof of main theorems. Let $(M,F)$ be a complete Finsler manifold, and $x_0\in M$. Then the distance function $r=r(x)=d_F(x_0,x)$ is smooth on $\breve{M}$, where \begin{equation}\breve{M}:=M\backslash\big \lbrace \{x_0\}\cup{\rm Cut}(x_0) \big \rbrace\label{3.0}\end{equation}
and the radial vector field $T=\nabla r$ is also smooth on $\breve{M}$. It is also well-known that $T$ is the unit geodesic field, i.e., $F(T)=1$ and $\nabla^T_TT=0$. Now we define \begin{equation}\label{3.1}\widetilde{g}(\cdot,\cdot) = \langle\cdot,\cdot\rangle_T\, .\end{equation} Then $\widetilde{g}$ is the Riemannian metric on $\breve{M}$ induced by $F$ in the radial direction $T$. It is clear from the definition of gradient that the gradient $\widetilde{\nabla}r$ of $r$ with respect to $\widetilde{g}$ is just the radial vector field, i.e., $$\widetilde{\nabla}r=\nabla r=T.$$ Thus by \eqref{23},\eqref{24} and \eqref{25} the Laplacian $\widetilde{\Delta}r$ of $r$ with respect to $\widetilde{g}$ is
\begin{equation}
\begin{aligned}{\widetilde{\Delta}}r= & \widetilde{\rm div}(\widetilde{\nabla}r)=\sum_{i=1}^n\widetilde{Hess}(r)(E_i,E_i)=\sum_{i=1}^n\langle\widetilde{\nabla}_{E_i}T,E_i\rangle_T\\
\label{31}= &\sum_{i=1}^n\langle\nabla^T_{E_i}T,E_i\rangle_T=\sum_{i=1}^n Hess(r)(E_i,E_i),
\end{aligned}
\end{equation}
here $\widetilde{\rm div}$ and $\widetilde{Hess}$ are the divergence and Hessian with respect to $\widetilde{g}$, and $E_1,\cdots,E_n$ is the local $\widetilde{g}$-orthonormal field on $\breve{M}$. Therefore, by Theorem 2.1 and \eqref{31} we clearly have the following
\noindent
{\bf Theorem 3.1 (Laplacian Comparison Theorem) } {\it With the same notations as above.\\ \indent
$(1)$ If $M$ has nonpositive radial flag curvature at $x_0$, then $\widetilde{\Delta}r\geqslant\frac{n-1}{r}$.\\ \indent
$(2)$ If $M$ has nonnegative radial Ricci curvature at $x_0$, then $\widetilde{\Delta}r\leqslant\frac{n-1}{r}$.}
Let $u:M\rightarrow\mathbb{R}$ be a smooth function on $M$. By definition, the gradient $\nabla u$ of $u$ (with respect to $F$) and the gradient $\widetilde{\nabla}u$ of $u$ with respect to $\widetilde{g}$ are related by
$$du(X)=\langle\nabla u,X\rangle_{\nabla u}=\langle\widetilde{\nabla} u,X\rangle_{T},\quad\forall X\in\Gamma(T\breve{M}),$$ which together with \eqref{26} and Schwartz inequality yields
$$\langle\widetilde{\nabla}u,\widetilde{\nabla}u\rangle_T=\langle\widetilde{\nabla}u,\nabla u\rangle_{\nabla u}\leqslant F(\nabla u)F(\widetilde{\nabla}u)\leqslant\mu_F^{\frac{1}{2}}F(\nabla u)\langle\widetilde{\nabla}u,\widetilde{\nabla}u\rangle_T^{\frac{1}{2}},$$namely,
\begin{equation}\label{32}|\widetilde{\nabla}u|_{\widetilde{g}}\leqslant\mu_F^{\frac{1}{2}}F(\nabla u).
\end{equation}Here $|\cdot|_{\widetilde{g}}=\langle\cdot,\cdot\rangle_T^{\frac{1}{2}}$ denotes the norm of vector field with respect to $\widetilde{g}$. Let $dV_{\widetilde{g}}$ be the Riemannian volume form of $\widetilde{g}$. It is clear from \eqref{27} that on $\breve{M}$ we have
\begin{equation}\label{33}dV_{\rm min}\leqslant dV_{ \widetilde{g}}\leqslant dV_{\rm max}\leqslant\mu_F^{\frac{n}{2}}dV_{\rm min}.
\end{equation}
\smallskip
A positive {\it Radon measure} $\nu$ is a linear functional on the space $C_0(\mathcal H)$ of real-valued continuous functions on a locally compact Hausdorff space $\mathcal H$ with compact support, $\nu: f \mapsto \nu (f) \in \mathbb R\, ,$ such that $\nu(f) \geqslant 0$, for any $f \geqslant 0$. We define an {\it upper integral} for the nonnegative functions as folloows.
If $\xi \geqslant 0$ is lower semicontinuous: $$\nu^{\ast}(\xi) = \sup \nu (f)\, \operatorname{for}\, \operatorname{all}\, \operatorname{nonnegative-valued}\, f \in C_0(\mathcal H)\, \operatorname{satisfying}\, f \leqslant \xi\, ,$$
and for any function $\eta \geqslant 0\, :$
$\nu^{\ast}(\eta) = \inf \nu (\xi)\,$ for all lower semicontinuous functions $\xi \, \operatorname{satisfying}\, \eta \leqslant \xi\, .$
A function $f$ is said to be $\nu$-{\it integrable} (or integrable if without ambiguity) if there exists a sequence $\{f_n\} \in C_0(\mathcal H)$ such that $\nu^{\ast}(|f-f_n|) \to 0 $ as $n \to \infty\, .$
A subset $A \subset \mathcal H$ is {\it measurable} with finite measure $\nu (A)\, ,$ if its characteristic function $\chi _A$ is integrable.
We set $\nu(A) = \int \chi_A\, d\nu\, .$
A function $f$ is said to be $\nu$-{\it measurable} (or measurable if without ambiguity) if for all compact sets $K$ and for all $\epsilon > 0\, ,$ there exists a compact set $K_{\epsilon} \subset K\, ,$ such that $\nu (K - K_{\epsilon}) < \epsilon$ and such that the restriction $f \big |_{K_{\epsilon}}$ is continuous on $K_{\epsilon}\, $ (cf. \cite {A}).
\noindent
{\bf Theorem 3.2.} {\it Let $\breve{M}$ be as in \eqref{3.0}. Then
\noindent
$(1)$ The volume element $dV_{\widetilde{g}}$ on Riemannian manifold $(\breve{M}, \widetilde{g})$ induces a positive Radon measure $\nu$ on a locally compact Hausdorff space $M$.
\noindent
$(2)$ The set $\{x_0\}\cup{\rm Cut}(x_0) $ has measure
$\nu \big ( \{x_0\}\cup{\rm Cut}(x_0) \big )
= 0$.}
\begin{proof}
$(1)$ We first note that a Finsler manifold is a locally compact Hausdorff space. For $f \in C_0 (M), $ we define
\begin{equation}\label{3.5}
\nu(f) = \int _{M\backslash\lbrace \{x_0\}\cup{\rm Cut}(x_0) \big \rbrace} f\, dV_{\widetilde{g}} = \int _{\breve{M}} f\, dV_{\widetilde{g}}.\end{equation}
Then $\nu$ is a linear functional on $C_0 (M)$ with $\nu(f) \geqslant 0$, for any $f \geqslant 0\, .$ Thus $\nu$ is a positive Radon measure on $M$.
\noindent
$(2)$ In view of the definition of the measure of a subset of $M$ and \eqref{3.5}, we have
the measure $$\begin{aligned}\nu \big ( \{x_0\}\cup{\rm Cut}(x_0) \big ) & = \nu (\chi _{\{x_0\}\cup{\rm Cut}(x_0)}) \\
& = \int _{\lbrace \{x_0\}\cup{\rm Cut}(x_0) \rbrace \backslash \lbrace \{x_0\}\cup{\rm Cut}(x_0) \rbrace} \chi _{\{x_0\}\cup{\rm Cut}(x_0)} \, dV_{\widetilde{g}} \\
& = \int _{\emptyset} 0\, dV_{\widetilde{g}}, \qquad \operatorname{where}\, \emptyset\, \operatorname{is}\, \operatorname{the}\, \operatorname{empty}\, \operatorname{set}\, \\
& = 0 . \end{aligned}$$
\end{proof}
Assume $1 \leqslant p \leqslant \infty\, $ for the remaining of this section.
\noindent
{\bf Definition 3.1.} A measurable function $u : M \to \mathbb R$ is said to {\it belong to $L^p(M)$ with respect to Riemannian metric $\widetilde{g}$ }, denoted by $u \in L^p(M, \widetilde{g})$ if $ \nu(|u|^p) = \int_{\breve{M}} |u|^p dV_{\widetilde{g}} < \infty\, ,$ and {\it $u$ is said to belong to $W_0^{1,p}(M)$ with respect to Riemannian metric $\widetilde{g}$}, denoted by $u \in W_0^{1,p}(M, \widetilde{g})$ if there exists a sequence $\{u_i\}$ in $C_0^{\infty}(M)$ such that $\left(\int_{\breve{M}} |u-u_i|^p + |\widetilde{\nabla} (u- u_i)|^p dV_{\widetilde{g}}\right)^{\frac 1p} \to 0$, as $i \to \infty\, .$
\smallskip
\noindent
{\bf Theorem 3.3.} {\it $\, L^p(M, \widetilde{g})$ is complete, i.e. every Cauchy sequence $\{u_i\}$ in $L^p(M, \widetilde{g})$ converges $($This means that if for every $\epsilon > 0$, there exists $N$ such that $\big (\nu ^{\ast} (|u_{i} - u_{j}| ^p)\big )^{\frac {1}{p}} = \left(\int_{\breve{M}} |u_{i} - u_{j}| ^p dV_{\widetilde{g}}\right)^{\frac 1p} < \epsilon$, when $i > N$ and $j > N$, then there exists a unique function $u\in L^p(M, \widetilde{g})$, such that $\big (\nu ^{\ast} (|u_{i} - u| ^p)\big )^{\frac {1}{p}}=\left(\int_{\breve{M}} |u_{i} - u| ^p dV_{\widetilde{g}}\right)^{\frac 1p} \to 0$, as $i \to \infty$ $)$}.
\begin{proof}
In view of Theorem 3.2.(1), $M$ is a measure space with a Radon measure $\nu$ as defined in \eqref{3.5}. To prove that every Cauchy sequence in $L^p(M, \widetilde{g})$ converges, it suffices to prove that for every Cauchy sequence $\{u_i\}$ in $L^p(M, \widetilde{g})$, there exists a subsequence $\{u_{i_k}\}$ which converges strongly to a function $u$ in $L^p(M, \widetilde{g})$ as $k \to \infty$, by the triangle inequality. Indeed,
$$\begin{aligned}\left(\int_{\breve{M}} |u_{i} - u| ^p dV_{\widetilde{g}}\right)^{\frac 1p} & \leqslant \left(\int_{\breve{M}} |u_{i} - u_{i_k}| ^p dV_{\widetilde{g}}\right)^{\frac 1p} + \left(\int_{\breve{M}} |u_{i_k} - u| ^p dV_{\widetilde{g}}\right)^{\frac 1p} \\
& < \frac {\epsilon}{2} + \frac {\epsilon}{2}\quad \operatorname{for}\, \operatorname{sufficiently}\, \operatorname{large}\, i, i_k \quad \operatorname{if}\, \{u_{i_k}\}\, \operatorname{converges}\, \operatorname{to}\, u.
\end{aligned}$$
This also proves that if the limit $u$ exists, then it is unique. The subsequence can be obtained by choosing $u_{i_k}$
so that $\left(\int_{\breve{M}} |u_{i_k} - u_n| ^p dV_{\widetilde{g}}\right)^{\frac 1p} < \frac {1}{2^k}$ for all $n > i_k$. (This is the definition of Cauchy sequence). In particular, $\left(\int_{\breve{M}} |u_{i_k} - u_{i_{k+1}}| ^p dV_{\widetilde{g}}\right)^{\frac 1p} < \frac {1}{2^k}$ for $k=1,2, \dots$. Furthermore, this subsequence $\{u_{i_k}\}$ gives rise to a bounded monotone sequence of positive functions$$U_{\ell} = |u_{i_1}| +\sum_{k=1}^{\ell} |u_{i_{k+1}} - u_{i_{k}}| .$$
Indeed, by the triangle inequality $U_{\ell}$ is bounded in $L^p(M, \widetilde{g})$ as
$$\left(\int_{\breve{M}} U_{\ell}^p dV_{\widetilde{g}}\right)^{\frac 1p} \leqslant \left(\int_{\breve{M}} |u_{i_1}|^p dV_{\widetilde{g}}\right)^{\frac 1p} +\sum_{k=1}^{\ell} \frac {1}{2^k}\, = \left(\int_{\breve{M}} |u_{i_1}|^p dV_{\widetilde{g}}\right)^{\frac 1p} + \left(1 - \frac {1}{2^{\ell}}\right).$$ By the monotone convergence theorem, $U_{\ell} $ converges pointwise a.e. to a positive function $U$ which is in $L^p(M, \widetilde{g})$ and hence is finite almost everywhere. The sequence $$u_{i_{\ell +1}} = u_{i_1} +\sum_{k=1}^{\ell} \left( u_{i_{k+1}} - u_{i_{k}}\right )$$
thus converges absolutely for almost every $x$ and hence it converges for the same $x$ to some function $u(x)$. Since $|u_{i_k} (x)| \leqslant U_{k-1} (x)\leqslant U(x)$ a.e. and $U \in L^p(M, \widetilde{g})$, by dominated convergence theorem (applying to the sequence $\{|u_{i_k}|^p\}$), $u \in L^p(M, \widetilde{g})$. Since $ |u_{i_k}(x) - u(x)| \leqslant U(x) + |u(x)| \in
L^p(M, \widetilde{g})\, ,$ again by dominated convergence theorem (applying to the sequence $\{|u_{i_k} - u|^p\}$), we conclude
$$\lim _{k \to \infty}\left(\int _{\breve{M}} |u_{i_k} - u|^p dV_{\widetilde{g}}\right)^{\frac 1p} = \left(\int _{\breve{M}} (\lim _{k \to \infty}|u_{i_k} - u|)^p dV_{\widetilde{g}}\right)^{\frac 1p} = 0. $$
That is the desired $(\int _{\breve{M}} |u_{i_k} - u|^p dV_{\widetilde{g}})^{\frac 1p} \to 0 $ as $k \to \infty$, (cf. \cite {LL} for real analysis.)\end{proof}
The proof of Theorem 3.3 yields the following domination and pointwise convergence properties:
\noindent
{\bf Theorem 3.4.} {\it If $\{u_i\}$ is a Cauchy sequence in $L^p(M, \widetilde{g})\, ,$ then there exists a subsequence $\{u_{i_k}\}$ and a nonnegative function $U$ in $L^p(M, \widetilde{g})\, ,$ such that
\noindent
$(1)$ $|u_{i_k}| \leqslant U$ almost everywhere in $M .$
\noindent
$(2)$ $\lim _{k \to \infty} u_{i_k} = u$ almost everywhere in $M .$}
\section{Geometric Inequalities on General Manifolds}
In this section, we begin with the following geometric inequalities on general Riemannian manifolds and on Finsler Manifolds:
\noindent
{\bf Local and Global Geometric Inequalities 4.1}\label{P:3.1} {\it \, Let $\Omega\subset M\backslash{\rm Cut}(x_0)$ be a domain in a Finsler manifold $(M, F).$ For every $u\in C_{0}^{\infty}(\Omega\backslash\{x_0\}),$ and every $a,b\in \mathbb{R},$ the following inequality holds:
\[
\frac 12 \bigg |\int_{\Omega}\frac{|u|^2}{r^{a+b+1}}\left(r\widetilde{\Delta}r-(a+b)\right)dV_{\widetilde{g}} \bigg |\leqslant \left(\int_{\Omega}\frac{|u|^{p}}{r^{ap}}dV_{\widetilde{g}}\right)^{\frac {1}{p}}
\left(\int_{\Omega}\frac{|\widetilde{\nabla}u|^{q}_{\widetilde{g}}}{r^{bq}}dV_{\widetilde{g}}\right)^{\frac {1}{q}},\tag{4.1.a}
\]
where $p \in [1, \infty]$ with
$\frac 1p+\frac 1q=1$. In particular, if $\operatorname{Cut}(x_0)$
is empty, or $x_0$ is a pole, then}
\[
\frac 12 \bigg |\int_{M}\frac{|u|^2}{r^{a+b+1}}\left(r\widetilde{\Delta}r-(a+b)\right)dV_{\widetilde{g}}\bigg |\\
\leqslant \left(\int_{M}\frac{|u|^{p}}{r^{ap}}dV_{\widetilde{g}}\right)^{\frac {1}{p}}
\left(\int_{M}\frac{|\widetilde{\nabla}u|^{q}_{\widetilde{g}}}{r^{bq}}dV_{\widetilde{g}}\right)^{\frac {1}{q}}.\tag{4.1.b}
\]
\begin{proof} We observe that by (\ref{31}) on any domain $\Omega\subset M\backslash{\rm Cut}(x_0)$,
$$
\begin{aligned}
\widetilde{\rm div}\left(\frac{|u|^2}{r^{a+b}}T\right) & =\frac{|u|^2}{r^{a+b}}\widetilde{\Delta}r+2\widetilde{g}\left(\frac{u}{r^{a+b}}T,\widetilde{\nabla} u\right)+\widetilde{g}\left(|u|^2T\, , \widetilde{\nabla} r^{-a-b}\right)\\
& = \frac{|u|^2}{r^{a+b}}\widetilde{\Delta}r+2\left\langle\frac{uT}{r^{a+b}},\widetilde{\nabla}u\right\rangle_T-(a+b)\frac{|u|^2}{r^{a+b+1}}.
\end{aligned}
$$ Hence, by the divergence theorem,
\[
\begin{aligned}
\frac 12\bigg |\int_{\Omega}\frac{|u|^2}{r^{a+b+1}}\left(r\widetilde{\Delta}r-(a+b)\right)dV_{\widetilde{g}}\bigg | & =
\bigg |\int_{\Omega}\left\langle\frac{uT}{r^{a+b}},\widetilde{\nabla}u\right\rangle_TdV_{\widetilde{g}}\bigg |\\
& =
\bigg |\int_{\Omega}\left\langle\frac{uT}{r^{a}},\frac{\widetilde{\nabla}u}{r^{b}}\right\rangle_TdV_{\widetilde{g}}\bigg |,
\end{aligned}
\]
(cf. \cite[p.409]{WL}). Now applying the H\"older inequality to the right side of the above formula we obtain the desired inequalities.
\end{proof}
\noindent
{\bf Local and Global Geometric Inequalities 4.2} {\it \, Let $\Omega\subset M\backslash{\rm Cut}(x_0)$ be a domain in a Finsler manifold $(M, F).$ For every $u\in C_{0}^{\infty}(\Omega\backslash\{x_0\}),$
and every $a,b\in
\mathbb{R},$ the following inequality holds:
\
\frac 1p\bigg |\int_{\Omega}\frac{|u|^p}{r^{a+b+1}}\big (r\widetilde{\Delta}r-(a+b)\big )dV_{\widetilde{g}} \bigg |\leqslant \left(\int_{\Omega}\frac{|u|^{p}}{r^{aq}}dV_{\widetilde{g}}\right)^{\frac {1}{q}}
\left(\int_{\Omega}\frac{|\widetilde{\nabla}u|^{p}_{\widetilde{g}}}{r^{bp}}dV_{\widetilde{g}}\right)^{\frac {1}{p}},
\tag{4.2.a}
\]where $p \in (1, \infty)$ and
$\frac 1p+\frac 1q=1$. In particular, if $\operatorname{Cut}(x_0)$
is empty, then}
\[
\frac 1p\bigg |\int_{M}\frac{|u|^p}{r^{a+b+1}}\big (r\widetilde{\Delta}r-(a+b)\big )dV_{\widetilde{g}} \bigg |\leqslant \left(\int_{M}\frac{|u|^{p}}{r^{aq}}dV_{\widetilde{g}}\right)^{\frac {1}{q}}
\left(\int_{M}\frac{|\widetilde{\nabla}u|^{p}_{\widetilde{g}}}{r^{bp}}dV_{\widetilde{g}}\right)^{\frac {1}{p}}.
\tag{4.2.b}
\]
\begin{proof}
First consider the case that $1<p<2$. For every $u\in
C_0^{\infty}(\Omega\backslash \left\{ x_{0}\right\})$, given $\epsilon>0$, consider $I:=p\underset{\Omega}{\displaystyle\int }\left\langle
(|u|^2+\epsilon)^\frac{p-2}{2}u\dfrac{T}{r^{a+b}},\widetilde{\nabla} u \right\rangle_{T} dV_{\widetilde{g}}\, $. Then it follows from the Guass lemma that
\[
I=\underset{\Omega}{\int }\widetilde{\rm div}\left( (|u|^2+\epsilon)^\frac{p}{2} \frac{T }{r^{a+b}}
\right) dV_{\widetilde{g}}
-\underset{\Omega}{\int
}\frac{\widetilde{\rm div}\left(T \right)}{r^{a+b}}(\left\vert
u\right\vert^2+\epsilon) ^\frac{p}{2}
dV_{\widetilde{g}}+\int_{\Omega}\frac{a+b}{r^{a+b+1}}(\left\vert
u\right\vert^2+\epsilon) ^\frac{p}{2}dV_{\widetilde{g}}.\tag{4.3}
\]
Hence by the Divergence Theorem and \eqref{31},
\begin{equation}
\begin{aligned}\label{4.4}
I &= \int_{\partial V}\left\langle
\frac{T}{r^{a+b}} (|u|^2+\epsilon)^\frac{p}{2},\xi\right\rangle_{T} dS-\int _{\Omega}\, \frac{
r\widetilde{\Delta}r-(a+b) }{r^{a+b+1}}(|u|^2+\epsilon)^\frac{p}{2}dV_{\widetilde{g}} \\
& =\epsilon^\frac{p}{2} \int_{\partial V}\left\langle
\frac{T}{r^{a+b}},\xi\right\rangle_{T} dS-\int _{\Omega}\, \frac{
r\widetilde{\Delta}r-(a+b) }{r^{a+b+1}}(|u|^2+\epsilon)^\frac{p}{2}dV_{\widetilde{g}},
\end{aligned}\tag{4.4}
\end{equation}
where $V$ is an open set with $supp\left\{ u\right\} \subset
V\subset \subset \Omega\backslash \left\{ x_{0}\right\}$, $\xi$ is the
outward unit normal vector of $\partial V$, $dS$ is the area
element induced from $dV_{\widetilde{g}}$.
\noindent
Now the triangle inequality, \eqref{4.4}, and the H\"{o}lder inequality imply that
\begin{equation}
\begin{aligned}\label{4.5}
&\left|\int_\Omega\frac{(|u|^2+\epsilon)^\frac{p}{2}}{r^{a+b+1}}(r \widetilde{\Delta} r - a - b) dV_{\widetilde{g}}\right|-\left|\epsilon^\frac{p}{2} \int_{\partial V}\left\langle
\frac{T}{r^{a+b}},\xi\right\rangle_{T} dS\right|\\
\leqslant & \left| \epsilon^\frac{p}{2} \int_{\partial V}\left\langle
\frac{T}{r^{a+b}},\xi\right\rangle_{T} dS - \int_\Omega\frac{(|u|^2+\epsilon)^\frac{p}{2}}{r^{a+b+1}}(r \widetilde{\Delta} r - a - b) dV_{\widetilde{g}}\right|\\
= & |I| \\
\leqslant& p\left( \int _{\Omega}\, \frac{\left\vert(\left\vert
u\right\vert^2+\epsilon)^\frac{p-2}{2}u
\right\vert^\frac{p}{p-1}}{r^{a\frac{p}{p-1}}}dV_{\widetilde{g}}\right)^{\frac
{p-1}{p}}\left( \int _{\Omega}\, \frac{| \widetilde{\nabla} u|_{\widetilde{g}}^{p}}{r^{bp}}dV_{\widetilde{g}}\right)^{\frac 1p}.
\end{aligned}\tag{4.5}
\end{equation}
Since $1<p<2$, we have $(|u|^2+\epsilon)^\frac{p-2}{2}<
(|u|^2)^\frac{p-2}{2}$. Thus, for every $1<p<2\, $ with $\frac
1p+\frac 1q=1$, one has via \eqref{4.5}
\begin{equation}
\begin{aligned}\label{4.6} & \left|\int_\Omega\frac{(|u|^2+\epsilon)^\frac{p}{2}}{r^{a+b+1}}(r \widetilde{\Delta} r - a - b) dV_{\widetilde{g}}\right|\\
\leqslant &
p \left( \int _{\Omega}\, \frac{\left\vert u\right\vert
^{p}}{r^{aq}}dV_{\widetilde{g}}\right)^{\frac 1q}\left( \int _{\Omega}\, \frac{|
\widetilde{\nabla} u|_{\widetilde{g}}^{p}}{r^{bp}}dV_{\widetilde{g}}\right)^{\frac 1p} + \left|\epsilon^\frac{p}{2} \int_{\partial V}\left\langle
\frac{T}{r^{a+b}},\xi\right\rangle_{T} dS\right|.
\end{aligned}\tag{4.6}
\end{equation}
Since \eqref{4.6} holds for every sufficiently small $\epsilon > 0\, ,$ we have
\begin{equation}
\begin{aligned}\label{4.7} \left|\int_\Omega\frac{(|u|^2+\epsilon)^\frac{p}{2}}{r^{a+b+1}}(r \widetilde{\Delta} r - a - b) dV_{\widetilde{g}}\right|
\leqslant
p \left( \int _{\Omega}\, \frac{\left\vert u\right\vert
^{p}}{r^{aq}}dV_{\widetilde{g}}\right)^{\frac 1q}\left( \int _{\Omega}\, \frac{|
\widetilde{\nabla} u|_{\widetilde{g}}^{p}}{r^{bp}}dV_{\widetilde{g}}\right)^{\frac 1p}.
\end{aligned}\tag{4.7}
\end{equation}
Let $\epsilon \to 0,$ the Monotone Convergence Theorem gives the
desired result.
For the case $p\geqslant 2$, consider $I:=p\displaystyle\int_\Omega\left\langle |u|^{p-2}u\frac{T }{r^{a+b}}, \widetilde{\nabla}
u\right\rangle_T dV_{\widetilde{g}}\, $. Then it follows from the Guass lemma and \eqref{31} that
\begin{equation}
\begin{aligned}\label{4.8}I=&\int_\Omega\widetilde{\rm div}\left( \frac{|u|^p}{r^{a+b}}T \right) dV_{\widetilde{g}}-\int_\Omega \frac{\widetilde{\rm div}\left(T\right)}{r^{a+b}}|u|^{p} dV_{\widetilde{g}} +\int_{\Omega}\frac{a+b}{r^{a+b+1}}|u|^{p}dV_{\widetilde{g}}\\
=&\int_\Omega\widetilde{\rm div}\left( \frac{|u|^p}{r^{a+b}} T
\right) dV_{\widetilde{g}}-\int_{\Omega}\frac{|u|^p}{r^{a+b+1}}\big (r\widetilde{\Delta}r-(a+b)\big )dV_{\widetilde{g}}
\end{aligned}\tag{4.8}
\end{equation}
for every $u\in C_0^\infty(\Omega\backslash \left\{ x_{0}\right\})$. Hence
by the Divergence Theorem and the H\"{o}lder inequality we easily get the desired result.
\end{proof}
\section{Proof of Theorems 1.1 and 1.2 - Generalized Caffarelli-Kohn-Nirenberg type inequalities on Finsler manifolds}
\begin{proof}[Proof of Theorem 1.1]
Case $($i$)$ $0\leqslant {\bf Ric}(\nabla r)$ and $n \leqslant a+b+1$ :
\noindent
In view of Laplacian Comparison Theorem 3.1.(2),
\begin{equation}\label{91}
\begin{aligned}
\frac{a+b+1-n}{2}\int_{\Omega}\frac{|u|^2}{r^{a+b+1}} dV_{\widetilde{g}} & \leqslant \frac 12 \bigg |\int_{\Omega}\frac{|u|^2}{r^{a+b+1}}\left(r\widetilde{\Delta}r-(a+b)\right)dV_{\widetilde{g}} \bigg |\\
& \leqslant \left(\int_{\Omega}\frac{|u|^{p}}{r^{ap}}dV_{\widetilde{g}}\right)^{\frac {1}{p}}
\left(\int_{\Omega}\frac{|\widetilde{\nabla}u|^{q}_{\widetilde{g}}}{r^{bq}}dV_{\widetilde{g}}\right)^{\frac {1}{q}}\\
& \leqslant \left(\int_{\Omega}\frac{|u|^{p}}{r^{ap}}dV_{\rm max}\right)^{\frac {1}{p}}
\left(\int_{\Omega}\frac{|\widetilde{\nabla}u|^{q}_{\widetilde{g}}}{r^{bq}}dV_{\rm max}\right)^{\frac {1}{q}}\\
& \leqslant \mu_F^{\frac 12}\left(\int_{\Omega}\frac{|u|^{p}}{r^{ap}}dV_{\rm max}\right)^{\frac {1}{p}}
\left(\int_{\Omega}\frac{(F({\nabla}u))^{q}}{r^{bq}}dV_{\rm max}\right)^{\frac {1}{q}}.\\
\end{aligned}
\end{equation}
The second, third and last steps follow from Geometric Inequality 4.1.(a), \eqref{33}, and \eqref{32} respectively. On the other hand, \eqref{33} implies that
\begin{equation}\label{92}
\begin{aligned}
\frac{a+b+1-n}{2}\mu_F^{-\frac {n}{2}}\int_{\Omega}\frac{|u|^2}{r^{a+b+1}} dV_{\rm {max}} & \leqslant \frac{a+b+1-n}{2}\int_{\Omega}\frac{|u|^2}{r^{a+b+1}} dV_{\rm {min}}\\
& \leqslant \frac{a+b+1-n}{2}\int_{\Omega}\frac{|u|^2}{r^{a+b+1}} dV_{\widetilde{g}}.
\end{aligned}
\end{equation}
Combining \eqref{91} and \eqref{92}, we have proved \eqref{11} when the extreme volume form $dV_{\rm {ext}}$ is $dV_{\rm {max}}$.
Similarly, we can prove \eqref{11} when $dV_{\rm {ext}}$ is the minimum volume form $dV_{\rm {min}}$ and hence \eqref{12} holds if in addition, $\operatorname{Cut}(x_0)$
is empty or $x_0$ is a pole. This completes the proof of Case $($i$)$. Similarly, by considering the fact that a simply connected flat Finsler manifold does not have a cut point in Case $($ii$)$, and so does a simply connected Finsler manfold with nonpositive flag curvature by a comparison theorem in Case $($iii$)$, the assertions follow.
\end{proof}
\begin{proof}[Proof of Theorem 1.2] Proceeding as in the proof of Theorem 1.1 by applying Theorem 3.1.(2), \eqref{32}, and \eqref{33} to Geometric Inequality 4.2.(a), the assertions follow.\end{proof}
\section{Proof of Theorem 1.3 }
{\rm We will give two methods:
\noindent
{\bf First Method:} This follows at once from substituting $p=q=2$ into Theorem $1.1$ or Theorem $1.2.$
\noindent
{\bf Second Method:} We first follow \cite{WL}. For every $u\in W^{1,2}_0(\Omega\backslash\{x_0\}, \widetilde{g})$ (cf. Denfinition 3.1, where $p=2$) and any $a,b,t\in\mathbb{R}$, we have, analogous to \cite[(4.1)]{WL}
$$\int_{\Omega}\left\langle\frac{\widetilde{\nabla}u}{r^b}+t\frac{u}{r^a}T,\frac{\widetilde{\nabla}u}{r^b}+t\frac{u}{r^a}T\right\rangle_TdV_{\widetilde{g}}
\geqslant0,$$namely,
\begin{equation}\label{51}\int_{\Omega}\frac{|\widetilde{\nabla}u|^2_{\widetilde{g}}}{r^{2b}}dV_{\widetilde{g}}+t^2\int_{\Omega}\frac{|u|^2}{r^{2a}}
dV_{\widetilde{g}}+2t\int_{\Omega}\left\langle\frac{uT}{r^{a+b}},\widetilde{\nabla}u\right\rangle_TdV_{\widetilde{g}}\geqslant0.
\end{equation} Observing that
$$
\begin{aligned}\widetilde{\rm div}\left(\frac{|u|^2}{r^{a+b}}T\right) & =\frac{|u|^2}{r^{a+b}}\widetilde{\Delta}r+2\widetilde{g}\left(\frac{u}{r^{a+b}}T, \widetilde{\nabla}u\right)+\widetilde{g}\left(|u|^2T, \widetilde{\nabla} r^{-a-b}\right)\\
&=\frac{|u|^2}{r^{a+b}}\widetilde{\Delta}r+2\left\langle\frac{uT}{r^{a+b}},\widetilde{\nabla}u\right\rangle_T-(a+b)\frac{|u|^2}{r^{a+b+1}},
\end{aligned}
$$ we have by divergence theorem,
\begin{equation}\label{52}2\int_{\Omega}\left\langle\frac{uT}{r^{a+b}},\widetilde{\nabla}u\right\rangle_TdV_{\widetilde{g}}
=-\int_{\Omega}\frac{|u|^2}{r^{a+b+1}}\left(r\widetilde{\Delta}r-(a+b)\right)dV_{\widetilde{g}}.
\end{equation}Let
$$A=\int_{\Omega}\frac{|u|^2}{r^{2a}}
dV_{\widetilde{g}},\quad B=2\int_{\Omega}\left\langle\frac{uT}{r^{a+b}},\widetilde{\nabla}u\right\rangle_TdV_{\widetilde{g}},\quad
C=\int_{\Omega}\frac{|\widetilde{\nabla}u|^2_{\widetilde{g}}}{r^{2b}}dV_{\widetilde{g}},$$ then \eqref{51} takes the form
$$At^2+Bt+C\geqslant0\, ,\, A > 0$$ for every $t\in\mathbb{R}$ which implies that $B^2-4AC\leqslant0$. Thus by \eqref{52} one has
\begin{equation}\label{53}\frac12\left|\int_{\Omega}\frac{|u|^2}{r^{a+b+1}}\left(r\widetilde{\Delta}r-(a+b)\right)dV_{\widetilde{g}}\right|\leqslant
\left(\int_{\Omega}\frac{|u|^2}{r^{2a}}dV_{\widetilde{g}}\right)^{\frac12}
\left(\int_{\Omega}\frac{|\widetilde{\nabla}u|^2_{\widetilde{g}}}{r^{2b}}dV_{\widetilde{g}}\right)^{\frac12}.\end{equation}
Suppose now that $M$ has nonnegative radial Ricci curvature at $x_0$. Then it follows from Theorem 3.1 that $r\widetilde{\Delta}r\leqslant n-1$. Thus if in addition, $n\leqslant a+b+1$, then \eqref{53} becomes
\begin{equation}\label{54} \frac12\int_\Omega\frac{|u|^2}{r^{a+b+1}}\bigg( a+b+1-n \bigg )dV_{\widetilde{g}}\leqslant
\left(\int_\Omega\frac{|u|^2}{r^{2a}}dV_{\widetilde{g}}\right)^{\frac12}
\left(\int_\Omega\frac{|\widetilde{\nabla}u|^2_{\widetilde{g}}}{r^{2b}}dV_{\widetilde{g}}\right)^{\frac12}.\end{equation}Hence, \eqref{19} follows directly from \eqref{32}, \eqref{33} and \eqref{54}. This proves the case $($i$)$. Analogously, we can prove the cases $($ii$)$ and $($iii$)$. In particular, if $M$ has a pole $x_0$ or $\operatorname{Cut}(x_0)$ is empty in $($i$)$, or $M$ is simply connected in $($ii$)$ or $($iii$)$, then
we can choose $\Omega=M$ and \eqref{19} becomes \eqref{110}.
This completes the proof.
\noindent
{\bf Remark:} The case that $M = \mathbb R^n$ is due to Costa $($c.f. \cite {C}$)$, The case that $M$ is a Riemannian manfiold is due to Wei-Li. $($\cite{WL}$).$
\section{Applications - Embedding Theorems for Weighted Sobolev Spaces and Differential-Integral Inequalities on Finsler manifolds}
As in the Riemannian case \cite{WL}, for giving Finsler manifold $(M,F)$ we let $L^2_{F,a}(M)$ be the completion of $C^\infty_{0}(M\backslash\{x_0\})$ with respect to the norm
$$ \|u\|_{L^2_{F,a}(M)}:=\left(\int_M\dfrac{|u|^2}{r^{2a}}dV_{\rm ext}\right)^{\frac12}, $$ $D^{1,2}_{F}(M)$ be the completion of $C^\infty_{0}(M\backslash\{x_0\})$ with respect to the ``norm"
$$ \|u\|_{D^{1,2}_{F}(M)}:=\left(\int_M(F(\nabla u))^2dV_{\rm ext}\right)^{\frac12},$$
and $H^1_{F,a,b}(M)$ be the completion of $C^\infty_{0}(M\backslash\{x_0\})$ with respect to the ``norm"
$$ \|u\|_{H^1_{F,a,b}(M)}:=\left(\int_M\left[\dfrac{|u|^2}{r^{2a}}+\dfrac{(F(\nabla u))^2}{r^{2b}}\right]dV_{\rm ext}\right)^{\frac12}.$$
It should be pointed out here that in general $\|k\cdot u\|_{H^1_{F,a,b}(M)}=|k|\cdot\|u\|_{H^1_{F,a,b}(M)}$ holds only when $k\geqslant0$, thus $\|\cdot\|_{H^1_{F,a,b}(M)}$ although satisfies the triangle inequality is not a genuine norm. Neverthless, by Theorems 1.3 we clearly have
\noindent
{\bf Theorem 7.1} {\it Let $(M,F)$ be an $n$-dimensional complete Finsler manifold with nonpositive radial flag curvature or nonnegative radial Ricci curvature at the pole $x_0\in M$. Suppose also that $M$ has finite uniformity constant $\mu_F$. Then the following embeddings hold
\begin{equation}
H_{F,a,b}^{1}(M) \subset L_{F, \frac {a+b+1}{2}}^{2}(M)\qquad and \qquad H_{F, b,a}^{1}(M) \subset L_{F, \frac {a+b+1}{2}}^{2}(M).\label{5.7}
\end{equation}}
As a consequence, we have differential-integral inequalities on Finsler manifolds:
\noindent
{\bf Theorem 7.2} {\it Let $M$ be as in Theorem 7.1. Then
\noindent
\textbf{i$)$ }For any $u\in D^{1,2}_{F}(M),$
\begin{equation}\label{5.99}
\underset{M}{\int }\frac{|u|^2}{r^{2}}dV_{\rm ext}\leqslant\left(\frac {2}{n-2}\right)^2 \cdot \mu_F^{n+1} \underset{M}{\int }|F(\nabla u)|^2 dV_{\rm ext};
\end{equation}
\noindent
\textbf{ii$)$ }For any $u\in H_{F, b+1,b}^{1}(M),$
\begin{equation}\label{5.8}
\underset{M}{\int }\frac{|u|^2}{r^{2(b+1)}}dV_{\rm ext}\leqslant\left(\frac n2-(b+1)\right)^{-2} \cdot \mu_F^{n+1} \underset{M}{\int }\frac{|F(\nabla u)|^2}{r^{2b}}dV_{\rm ext};
\end{equation}
\noindent
\textbf{iii$)$ }For any $u\in H_{F, a+1,a}^{1}(M),$
\begin{equation}\label{5.10}
\left( \underset{M}{\int }\frac{\left\vert
u\right\vert ^{2}}{r^{2(a+1)}}dV_{\rm ext}\right)^2\leqslant \left(\frac n2-(a+1)\right)^{-2} \cdot \mu_F^{n+1}\left( \underset{M}{\int }\frac{%
\left\vert u\right\vert ^{2}}{r^{2a}}dV_{\rm ext}\right) \left( \underset%
{M}{\int }\frac{\left\vert F(\nabla u)\right\vert ^{2}}{r^{2(a+1)}}dV_{\rm ext}\right);
\end{equation}
\noindent
\textbf{iv$)$ }If $u\in H_{F,-(b+1),b}^{1}(M)$ then $u$ $\in L_F^{2}(M)$
and
\begin{equation}\label{5.12}
\left( \underset{M}{\int }\left\vert u\right\vert
^{2}dV_{\rm ext}\right) ^2 \leqslant\frac{4}{n^2} \cdot \mu_F^{n+1} \left( \underset{M}{\int }r^{2(b+1)}\left\vert u\right\vert
^{2}dV_{\rm ext}\right) \left( \underset{M}{\int }\frac{\left\vert
F(\nabla u)\right\vert ^{2}}{r^{2b}}dV_{\rm ext}\right) ;
\end{equation}
\noindent
\textbf{v$)$ } If $u\in H_{F,0,1}^{1}(M),$ then $u$ $\in L_{F,1}^{2}(M)$
and
\begin{equation}\label{5.14}
\left( \underset{M}{\int }\frac{\left\vert u\right\vert ^{2}}{r^{2}}%
dV_{\rm ext}\right)^2\leqslant \frac{4}{(n-2)^2} \cdot \mu_F^{n+1} \left( \underset{M}{\int }\left\vert u\right\vert ^{2}dV_{\rm ext}\right)\left( \underset{M}{\int }\frac{\left\vert F(\nabla u)\right\vert
^{2}}{r^{2}}dV_{\rm ext}\right);
\end{equation}
\noindent
\textbf{vi$)$ }If $u\in H_{F,-1,1}^{1}(M),$ then $u$ $\in
L_{F,\frac{1}{2}}^{2}(M)$ and
\begin{equation}\label{5.16}
\left( \underset{M}{\int }\frac{\left\vert u\right\vert
^{2}}{r}dV_{\rm ext}\right) ^2 \leqslant \frac{4}{(n-1)^2} \cdot \mu_F^{n+1} \left( \underset{M}{\int }r^{2}\left\vert u\right\vert
^{2}dV_{\rm ext}\right) \left( \underset{M}{\int }\frac{\left\vert
F(\nabla u)\right\vert ^{2}}{r^{2}}dV_{\rm ext}\right) ;
\end{equation}
\noindent
\textbf{vii$)$ }If $u\in H_F^{1}(M)=$ $H_{F,0,0}^{1}(M),$ then $u$ $\in L_{F,\frac{1%
}{2}}^{2}(M)$ and
\begin{equation}\label{5.18}
\left( \underset{M}{\int }\frac{\left\vert u\right\vert
^{2}}{r}dV_{\rm ext}\right)^2 \leqslant \frac{4}{(n-1)^2} \cdot \mu_F^{n+1} \left( \underset{M}{\int }\left\vert u\right\vert
^{2}dV_{\rm ext}\right) \left( \underset{M}{\int }\left\vert F(\nabla
u)\right\vert ^{2}dV_{\rm ext}\right).
\end{equation}}
\noindent
{\bf Remark}
The case that $M= \mathbb{R}^n$ is due to \cite{C}. The case that $M$ is a Riemannian manfiold is due to \cite{WL}. Item $(i)$ is a generalized Hardy's inequality. In the next Section we will discuss its generalizations.
\begin{proof} We make special choices in Theorems 1.3 as follows:
\noindent
\textbf{i)} \ \ \ Let $a=1, b=0;$
\noindent
\textbf{ii)} \ \ \ Let $a=b+1;$
\noindent
\textbf{iii)} \ \ Let $b=a+1;$
\noindent
\textbf{iv) }\ Let $a=-b-1;$
\noindent
\textbf{v) }\ Let $a=0,b=1;$
\noindent
\textbf{vi)} \ \ Let $a=-1,b=1;$
\noindent
\textbf{vii) } Let $a=0,b=0. $
\end{proof}
\section{Proof of Theorems 1.4 and 1.5 - Generalized Hardy inequalities on Finsler manifolds}
Employing the {\it double} limiting technique in [7] , we prove the following:
\noindent
{\bf Geometric Inequality 8.1} (cf. \cite[(1.3)]{WL}, \cite[(3)]{CLW2}) {\it Let $M$ be a Finsler manifold, $u\in C_{0}^{\infty}(M)$ and $\partial B_{\delta}(x_0)$ be the $C^1$ boundary of the geodesic ball $B_{\delta}(x_0)$ centered
at $x_0$ with radius $\delta >0$. Let $V$ be an open set with smooth boundary $\partial V$ such that $V\subset\subset M$, and $u=0$ off $V$. We choose a sufficiently small $\delta>0$ so that $\partial V\cap\partial B_{\delta}(x_0)=\emptyset$. Then for every $\epsilon>0$ and $p>1$, we have
$$\left|-\int_{V \cap \partial B_{\delta}(x_0)}\frac{r}{r^p+\epsilon}|u|^p\langle T,\xi\rangle_T dS
+\int_{\breve{M}\backslash B_{\delta}(x_0)}\frac{(r^p+\epsilon)(1+r\widetilde{\Delta}r)-pr^p}{(r^p+\epsilon)^2}|u|^pdV_{\widetilde{g}}\right|$$
\begin{equation}\label{44} \leqslant p\left(\int_{\breve{M}\backslash B_{\delta}(x_0)}\left(\frac{|u|^{p-1}r}{r^p+\epsilon}\right)^{\frac{p}{p-1}}dV_{\widetilde{g}}\right)^{\frac{p-1}{p}}
\left(\int_{\breve{M}\backslash B_{\delta}(x_0)}|\widetilde{\nabla}u|^p_{\widetilde{g}}dV_{\widetilde{g}}\right)^{\frac{1}{p}},\end{equation}
where $\breve{M}$ is as in \eqref{3.0}, $\xi$ is the outward unit normal vector field on $\partial B_{\delta}(x_0)$ with respect to $\widetilde{g}$, and $dS$ is the volume form on $\partial B_{\delta}(x_0)$ induced from $dV_{\widetilde{g}}$.}
\begin{proof}
Observing via Gauss Lemma
$$\begin{aligned}\widetilde{\rm div}\left(\frac{rT}{r^p+\epsilon}|u|^p\right)&=\frac{\widetilde{\rm div}(rT)}{r^p+\epsilon}|u|^p+\widetilde{g}\left(rT, \widetilde{\nabla}\left(\frac{|u|^p}{r^p+\epsilon}\right)\right) \\
&=\frac{\widetilde{\rm div}(rT)}{r^p+\epsilon}|u|^p-\left\langle rT, \frac{\widetilde{\nabla} r^p}{(r^p+\epsilon)^2}|u|^p\right\rangle_T +\frac{r}{r^p+\epsilon}\langle T, \widetilde{\nabla}|u|^p\rangle_T \\
&=\frac{\widetilde{\rm div}(rT)}{r^p+\epsilon}|u|^p-\frac{pr^p}{(r^p+\epsilon)^2}|u|^p+\frac{pru|u|^{p-2}}{r^p+\epsilon}\langle T, \widetilde{\nabla}u\rangle_T,
\end{aligned} $$ we have
\begin{equation}\label{41}\begin{aligned}&p\int_{\breve{M}\backslash B_{\delta}(x_0)}\left\langle|u|^{p-2}u\frac{rT}{r^p+\epsilon},\widetilde{\nabla}u\right\rangle_T dV_{\widetilde{g}}\\
=&\int_{\breve{M}\backslash B_{\delta}(x_0)}\widetilde{\rm div}\left(\frac{rT}{r^p+\epsilon}|u|^p\right)dV_{\widetilde{g}}
-\int_{\breve{M}\backslash B_{\delta}(x_0)}\frac{\widetilde{\rm div}(rT)}{r^p+\epsilon}|u|^pdV_{\widetilde{g}}
+\int_{\breve{M}\backslash B_{\delta}(x_0)} \frac{pr^p}{(r^p+\epsilon)^2}|u|^pdV_{\widetilde{g}}.\end{aligned}\end{equation} By the divergence theorem it follows that for sufficiently small $\delta > 0$,
\begin{equation}\begin{aligned}\label{42}\int_{\breve{M}\backslash B_{\delta}(x_0)}\widetilde{\rm div}\left(\frac{rT}{r^p+\epsilon}|u|^p\right)dV_{\widetilde{g}}
&=\int_{V\backslash B_{\delta}(x_0)}\widetilde{\rm div}\left(\frac{rT}{r^p+\epsilon}|u|^p\right)dV_{\widetilde{g}}\\
&=\int_{V \cap \partial B_{\delta}(x_0)}\frac{r}{r^p+\epsilon}|u|^p\langle T,\xi\rangle_T dS.\end{aligned}\end{equation} On the other hand,
\begin{equation}\label{43} \widetilde{\rm div}(rT)=r\widetilde{\rm div}(T)+\widetilde{g}(T, \widetilde{\nabla} r)=1+r\widetilde{\Delta}r.\end{equation}
Substituting \eqref{42} and \eqref{43} into \eqref{41} and using the H\"{o}lder inequality one has
the desired \eqref{44}.
\end{proof}
\begin{proof}[Proof of Theorem 1.4]
We first assume $u \in C_0^{\infty}(M)\, .$ Since $1<p<n$ and $M$ has nonpositive radial flag curvature at $x_0$, by Theorem 3.1 we have $r\widetilde{\Delta}r+1\geqslant n>p$, and thus
\begin{equation}\label{8.5} \begin{aligned} \int_{\breve{M}\backslash B_{\delta}(x_0)}\frac{(r^p+\epsilon)(1+r\widetilde{\Delta}r)-pr^p}{(r^p+\epsilon)^2}|u|^pdV_{\widetilde{g}}
& \geqslant \int_{\breve{M}\backslash B_{\delta}(x_0)}\frac{(n-p)r^p+(n-p)\epsilon}{(r^p+\epsilon)^2}|u|^pdV_{\widetilde{g}}\\
& \geqslant(n-p)\int_{\breve{M}\backslash B_{\delta}(x_0)}\frac{(r^p+\epsilon)^{\frac{1}{p-1}}}{(r^p+\epsilon)^{\frac{p}{p-1}}}|u|^pdV_{\widetilde{g}}\\
& \geqslant(n-p)\int_{\breve{M}\backslash B_{\delta}(x_0)}\frac{(r^p)^{\frac{1}{p-1}}}{(r^p+\epsilon)^{\frac{p}{p-1}}}|u|^pdV_{\widetilde{g}}.
\end{aligned}\end{equation}
Substituting \eqref{8.5} into \eqref{44}, we have for sufficiently small $\delta>0$,
$$-\int_{V\cap\partial B_{\delta}(x_0)}\frac{r}{r^p+\epsilon}|u|^p\langle T,\xi\rangle_T dS
+(n-p)\int_{\breve{M}\backslash B_{\delta}(x_0)}\frac{(r^p)^{\frac{1}{p-1}}}{(r^p+\epsilon)^{\frac{p}{p-1}}}|u|^pdV_{\widetilde{g}}$$
\begin{equation}\label{45} \leqslant p\left(\int_{\breve{M}\backslash B_{\delta}(x_0)}\frac{(r^p)^{\frac{1}{p-1}}}{(r^p+\epsilon)^{\frac{p}{p-1}}}|u|^pdV_{\widetilde{g}}\right)^{\frac{p-1}{p}}
\left(\int_{\breve{M}\backslash B_{\delta}(x_0)}|\widetilde{\nabla}u|^p_{\widetilde{g}}dV_{\widetilde{g}}\right)^{\frac{1}{p}}.\end{equation}
For sufficiently small $\delta>0$, one has
\begin{equation}\label{8.7}\int_{\partial B_{\delta}(x_0)}\frac{r}{r^p+\epsilon}|u|^p\langle T,\xi\rangle_T dS=0\quad \text{if}\quad x_0\notin V\end{equation} and
\begin{equation}\label{8.8}\left|\int_{\partial B_{\delta}(x_0)}\frac{r}{r^p+\epsilon}|u|^p\langle T,\xi\rangle_T dS\right|
\to 0\quad \text{as}\quad \delta\to 0\quad \text{if}\quad x_0\in V,\end{equation}
Indeed, $\frac{r}{r^p+\epsilon}$ is a continuous, nondecreasing function for $r \in [0, \delta _0]$, where $\delta _0 = (\frac{\epsilon}{p-1})^{\frac 1p}$ and $u$ is bounded in $M$. Hence, for $\delta < \delta _0\, ,$
\begin{equation}\label{8.9}\left|\int_{\partial B_{\delta}(x_0)}\frac{r}{r^p+\epsilon}|u|^p\langle T,\xi\rangle_T dS\right|
\leqslant\frac{\delta}{\delta ^p+\epsilon} \int _{\partial B_{\delta}(x_0)} \max_M |u|^p dS.
\end{equation} This implies \eqref{8.8}.
It follows from \eqref{45}, via \eqref{8.7} or \eqref{8.8}
that for every $\epsilon>0$,
\begin{equation}\label{46}\begin{aligned} (n-p)\left(\int_{\breve{M}\backslash B_{\delta}(x_0)}\frac{(r^p)^{\frac{1}{p-1}}}{(r^p+\epsilon)^{\frac{p}{p-1}}}|u|^pdV_{\widetilde{g}}\right)^{\frac{1}{p}}&\leqslant p
\left(\int_{\breve{M}\backslash B_{\delta}(x_0)}|\widetilde{\nabla}u|^p_{\widetilde{g}}dV_{\widetilde{g}}\right)^{\frac{1}{p}}\\
& \leqslant p \left(\int_{\breve{M}}|\widetilde{\nabla}u|^p_{\widetilde{g}}dV_{\widetilde{g}}\right)^{\frac{1}{p}}.
\end{aligned}\end{equation}
\noindent
Monotone convergence theorem and \eqref{33} imply that as $\epsilon \to 0$, for every $u \in C_0^{\infty}(M)$,
\begin{equation}\label{8.11} \bigg (\frac {n-p}{p}\bigg ) \left(\int_{\breve{M}}\bigg | {\frac{u}{r}}\bigg |^p dV_{\widetilde{g}}\right)^{\frac 1p}\leqslant
\left(\int_{\breve{M}}|\widetilde{\nabla}u|^p_{\widetilde{g}}dV_{\widetilde{g}}\right)^{\frac {1}{p}},\end{equation}
\begin{equation}\label{8.12}\quad \text{and}\quad \frac {u}{r} \in L^p(M, \widetilde{g}).
\end{equation}
Now we {\it extend \eqref{8.11} from $u \in C^{\infty}_{0} (M)$ to $u \in W_{F,0}^{1,p}(M)\, .$} Let $\{u_i\}$ be a sequence of functions in $ C_0^{\infty}(M)$ tending to $u \in W_{F,0}^{1,p}(M)\, $ in $
\|\, \cdot \,\|_{W_{F,0}^{1,p}(M)}$ as in \eqref{1.11}.
Applying the inequality \eqref{8.11} to difference
$u_{i_m} - u_{i_n}\, ,$ we have via \eqref{32} and \eqref{33}
\begin{equation}\begin{aligned}
\left(\int_{\breve{M}}\bigg | {\frac{|u_{i_m} - u_{i_n}}{r}}\bigg |^p dV_{\widetilde{g}}\right)^{\frac 1p}&\leqslant \bigg (\frac {p}{n-p}\bigg )
\left(\int_{\breve{M}}|\widetilde{\nabla}(u_{i_m} - u_{i_n})|^p_{\widetilde{g}}dV_{\widetilde{g}}\right)^{\frac {1}{p}}\\
& \leqslant \bigg (\frac {p}{n-p}\bigg ) \mu _F^{\frac{1}{2}}
\left(\int_{M}\bigg (F\big (\nabla(u_{i_m} - u_{i_n})\big )\bigg )^p dV_{\text{max}}\right)^{\frac {1}{p}}.\end{aligned}
\label{8.13}
\end{equation}
\noindent
Hence $\{\frac {u_i}{r}\}$ is a Cauchy sequence in $L^p(M, \widetilde{g})\, .$ By Theorem 3.3, there exists a limiting function $f(x) \in L^p(M,
\widetilde{g})\, $ satisfying,
via \eqref{32} and \eqref{33},
\begin{equation}
\begin{aligned}
\int _{\breve{M}} |f(x)|^p\, dV_{\widetilde{g}} = \lim_{i\to \infty} \int _{\breve{M}} \frac {|u_i(x)|^p}{r^p}\, dV_{\widetilde{g}}
& \leqslant \bigg (\frac {p}{n-p}\bigg )^p \lim_{i\to \infty} \int _{\breve{M}} |\widetilde{\nabla} u_{i} |^p_{\widetilde{g}} \, dV_{\widetilde{g}}\\
& \leqslant \bigg (\frac {p}{n-p}\bigg )^p \mu _F^{\frac{p}{2}}\lim_{i\to \infty} \int _{M} (F(\nabla u_{i}))^p \, dV_{\text{max}}\\
& \leqslant \bigg (\frac {p}{n-p}\bigg )^p \mu _F^{\frac{p}{2}} \int _{M} (F(\nabla u))^p \, dV_{\text{max}}\\
& \leqslant \bigg (\frac {p}{n-p}\bigg )^p \mu _F^{\frac{n+p}{2}} \int _{M} (F(\nabla u))^p \, dV_{\text{min}}.
\label{8.14}
\end{aligned}
\end{equation}
On the other hand, since $\frac {1}{r^p}$ is bounded in $M\backslash B_{\epsilon}(x_0)\, ,$ where $ B_{\epsilon}(x_0)$ is the open geodesic ball of radius $\epsilon > 0\, ,$ centered at $x_0$, and the pointwise convergence in Theorem 3.4.(2), we have for every $\epsilon > 0\, ,$
\begin{equation}
\begin{aligned}
\int _{\breve{M}\backslash B_{\epsilon}(x_0)} |f(x)|^p\, dV_{\widetilde{g}} = \lim_{i\to \infty} \int _{\breve{M}\backslash B_{\epsilon}(x_0)} \frac {|u_i(x)|^p }{r^p}\, dV_{\widetilde{g}} &= \int _{\breve{M}\backslash B_{\epsilon}(x_0)} \frac {|u|^p }{r^p}\, dV_{\widetilde{g}}\\
& = \int _{\breve{M}} \chi _{\breve{M}\backslash B_{\epsilon}(x_0)} \frac {|u|^p }{r^p}\, dV_{\widetilde{g}},
\label{8.15}
\end{aligned}
\end{equation}
where $\chi _{\breve{M}\backslash B_{\epsilon}(x_0)}$ is the characteritic function on $ \breve{M}\backslash B_{\epsilon}(x_0)$.
As $\epsilon \to 0\, ,$ monotone convergence theorem and \eqref{33} imply that
\begin{equation}
\begin{aligned}
\int _{\breve{M}} |f(x)|^p\, dV_{\widetilde{g}} = \lim_{i\to \infty} \int _{\breve{M}} \frac {|u_i|^p }{r^p}\, dV_{\widetilde{g}} & = \int _{\breve{M}} \frac {|u|^p }{r^p}\, dV_{\widetilde{g}}\\
& \geqslant \int _{\breve{M}} \frac {|u|^p }{r^p}\, dV_{\text{min}}\\
& \geqslant \mu _F^{-\frac {n}{2}}\int _{\breve{M}} \frac {|u|^p }{r^p}\, dV_{\text{max}}.
\end{aligned}
\label{8.16}\end{equation}
Substituting \eqref{8.16} into \eqref{8.14} we have for every $u \in W_{F,0}^{1,p}(M)$,
$$\int _{\breve{M}} \frac {|u|^p }{r^p}\, dV_{\rm ext} \leqslant \bigg (\frac {p}{n-p}\bigg )^p
\mu _F^{\frac{n+p}{2}} \int _{M} (F(\nabla u))^p \, dV_{\rm ext}, $$ which implies that
$$ \int _{M} \chi _{M\backslash B_{\epsilon}(x_0)}\frac {|u|^p }{r^p}\, dV_{\rm ext}=
\int_{M\backslash B_{\epsilon}(x_0)} \frac {|u|^p }{r^p}\, dV_{\rm ext}
=\int_{\breve{M}\backslash B_{\epsilon}(x_0)} \frac {|u|^p }{r^p}\, dV_{\rm ext} $$
$$ \leqslant \bigg (\frac {p}{n-p}\bigg )^p
\mu _F^{\frac{n+p}{2}} \int _{M} (F(\nabla u))^p \, dV_{\rm ext}, \;\forall \epsilon>0. $$
As $\epsilon \to 0\, ,$ again by monotone convergence theorem we obtain the desired $\frac {u}{r} \in L_{F}^p(M)$ and inequality \eqref{111}.
\end{proof}
The proof of Theorem 1.4 yields
\noindent
{\bf Corollary 8.1} {\it Let $(M,F)$ be an $n$-dimensional complete Finsler manifold with nonpositive radial flag curvature at the pole $x_0\in M$ and with finite uniformity constant $\mu_F$. Then for any $u\in W_{0}^{1,p}(M,\tilde{g})$ and $1<p<n$, the following Hardy type inequality holds:}
\[ \bigg (\frac {n-p}{p}\bigg ) \left(\int_{\breve{M}}\bigg | {\frac{u}{r}}\bigg |^p dV_{\widetilde{g}}\right)^{\frac 1p}\leqslant
\left(\int_{\breve{M}}|\widetilde{\nabla}u|^p_{\widetilde{g}}dV_{\widetilde{g}}\right)^{\frac {1}{p}}.\]
{\it Furthermore, $\frac {u}{r} \in L^p(M, \tilde{g})\, .$}\smallskip
Either Theorem 1.4 or Corollary 8.1 recaptures the following, when $M$ is a Riemannian manifold.\smallskip
\noindent
{\bf Corollary 8.2} (\cite[Theorem 1, Corollary 1.2]{WL}) {\it Let $M$ be an $n$-dimensional complete Riemannian manifold of nonpositive radial curvature with the volume element $dv$. Then for any $u\in W_{0}^{1,p}(M)$ and $1 < p < n$, the following Hardy type inequality holds:
\begin{equation}\label{8.21}
\left(\frac{n-p}{p}\right)^p \int_M\frac{|u|^p}{r^p} dv\leqslant \int_M |\nabla u|^p \, dv.
\end{equation}
Furthermore, $\frac {u}{r} \in L^p(M)\, .$}
\begin{proof}[Proof of Theorem 1.5.] We first assume $u\in C_0^{\infty} (M)\, .$ When $p>n$ and $M$ has nonnegative radial Ricci curvature at $x_0$, by Theorem 3.1 we have $r\widetilde{\Delta}r+1\leqslant n<p$. In view of this inequality
and the triangle inequality, \eqref{44} implies
\begin{equation}\label{8.17} \begin{aligned} \int_{\breve{M}\backslash B_{\delta}(x_0)}\frac{(p-n)r^p-n\epsilon}{(r^p+\epsilon)^2}|u|^pdV_{\widetilde{g}}
& \leqslant p\left(\int_{\breve{M}\backslash B_{\delta}(x_0)}\left(\frac{|u|^{p-1}r}{r^p+\epsilon}\right)^{\frac{p}{p-1}}dV_{\widetilde{g}}\right)^{\frac{p-1}{p}}
\left(\int_{\breve{M}\backslash B_{\delta}(x_0)}|\widetilde{\nabla}u|^p_{\widetilde{g}}dV_{\widetilde{g}}\right)^{\frac{1}{p}} \\ & \quad+ \left|\int_{\partial B_{\delta}(x_0)}\frac{r}{r^p+\epsilon}|u|^p\langle T,\xi\rangle_T dS\right|.
\end{aligned}\end{equation}
Applying \eqref{8.7} or \eqref{8.8}, and letting $\delta \to 0$ in \eqref{8.17}, one has
\begin{equation}\label{8.18} \int_{\breve{M}}\frac{(p-n)r^p-n\epsilon}{(r^p+\epsilon)^2}|u|^pdV_{\widetilde{g}}
\leqslant p\left(\int_{\breve{M}}\left(\frac{|u|^{p-1}r}{r^p+\epsilon}\right)^{\frac{p}{p-1}}dV_{\widetilde{g}}\right)^{\frac{p-1}{p}}
\left(\int_{\breve{M}}|\widetilde{\nabla}u|^p_{\widetilde{g}}dV_{\widetilde{g}}\right)^{\frac{1}{p}}.\end{equation}
We observe that the integrands in the left, and in the first factor in the right of \eqref{8.18} are monotone and uniformly bounded above by a positive constant multiple of $\big |\frac {u}{r}\big |^p$ on $M\, .$ Since $\frac {u}{r} \in L_F^p(M)\, , \frac {u}{r} \in L^p(M, \widetilde{g})$. By the dominated convergent theorem, as $\epsilon\to 0\, ,$
\begin{equation}\label{47} (p-n) \left(\int_{\breve{M}}\left|\frac {u}{r}\right|^p dV_{\widetilde{g}}\right)
\leqslant p\left(\int_{\breve{M}}\left|\frac {u}{r}\right|^p dV_{\widetilde{g}}\right)^{\frac{p-1}{p}}
\left(\int_{\breve{M}}|\widetilde{\nabla}u|^p_{\widetilde{g}}dV_{\widetilde{g}}\right)^{\frac{1}{p}}.\end{equation}
Simplifying and raising to the $p$-th power,
\begin{equation}\label{48} \left(\frac{p-n}{p}\right)^p \left(\int_{\breve{M}}\left|\frac {u}{r}\right|^p dV_{\widetilde{g}}\right)
\leqslant
\left(\int_{\breve{M}}|\widetilde{\nabla}u|^p_{\widetilde{g}}dV_{\widetilde{g}}\right).\end{equation}
Now analogously we {\it extend \eqref{48} from $u \in C^{\infty}_{0}(M)$ to $u \in W_{F,0}^{1,p}(M)\, .$} Let $\{u_i\}$ be a sequence of functions in $ C_0^{\infty}(M)$ tending to $u \in W_{F,0}^{1,p}(M)\, .$ Applying the inequality \eqref{48} to difference
$u_{i_m} - u_{i_n}\, ,$ employing \eqref{32} and \eqref{33} and proceeding as in the proof of Theorem 1.4, we obtain the desired inequality \eqref{112} for every $u \in W_{F,0}^{1,p}(M)\, .$
\end{proof}
Similarly, the proof of Theorem 1.4 yields
\noindent
{\bf Corollary 8.3} {\it Let $(M,F)$ be an $n$-dimensional complete Finsler manifold with nonnegative radial Ricci curvature at the pole $x_0\in M$, and with finite uniformity constant $\mu_F$. Then for any $u\in W_{0}^{1,p}(M, \tilde{g}),$ $\frac {u}{r} \in L^p(M,\tilde{g})$ and $p>n$, the following Hardy type inequality holds:}
\[ \left(\frac{p-n}{p}\right)^p \left(\int_{\breve{M}}\left|\frac {u}{r}\right|^p dV_{\widetilde{g}}\right)
\leqslant
\left(\int_{\breve{M}}|\widetilde{\nabla}u|^p_{\widetilde{g}}dV_{\widetilde{g}}\right).\]
\smallskip
Either Theorem 1.5 or Corollary 8.3 recaptures the following, when $M$ is a Riemannian manifold.\smallskip
\noindent
\noindent
{\bf Corollary 8.4} (\cite[Theorem 5]{CLW2}) {\it Let $M$ be an $n$-dimensional Riemannian manifold with a pole, nonnegative radial Ricci curvature and the volume element $dv$. Then for any $u\in W_{0}^{1,p}(M)\, ,$ $\frac {u}{r} \in L^p(M)$ and $p>n$, the following Hardy type inequality holds:}
\begin{equation}\label{8.22}
\left(\frac{p-n}{p}\right)^p \int_M\frac{|u|^p}{r^p} dv\leqslant \int_M |\nabla u|^p \, dv.
\end{equation}
\section*{Acknowledgments}
\noindent
$^*$ The first author is supported in part by NSF (DMS-1447008), and the OU Arts and Sciences Travel Assistance Program;\\
$^{**}$ The corresponding author is supported in part by the National Science Foundation of China (No. 12001259) and the National Science Foundation of Fujian province of China (No. 2020J01131142).\\
The authors wish to thank the referees for their helpful comments and suggestions.
| {
"timestamp": "2021-01-05T02:01:53",
"yymm": "2101",
"arxiv_id": "2101.00096",
"language": "en",
"url": "https://arxiv.org/abs/2101.00096",
"abstract": "In this paper we derive both local and global geometric inequalities on general Riemannnian and Finsler manifolds and prove generalized Caffarelli-Kohn-Nirenberg type and Hardy type inequalities on Finsler manifolds, illuminating curvatures of both Riemannian and Finsler manifolds influence geometric inequalities.",
"subjects": "Differential Geometry (math.DG); Analysis of PDEs (math.AP)",
"title": "Generalized Hardy Type and Caffarelli-Kohn-Nirenberg Type Inequalities on Finsler Manifolds",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754488233529,
"lm_q2_score": 0.7185943985973772,
"lm_q1q2_score": 0.70751040252096
} |
https://arxiv.org/abs/1411.4159 | The Annihilating-Ideal Graph of a Ring | Let $S$ be a semigroup with $0$ and $R$ be a ring with $1$. We extend the definition of the zero-divisor graphs of commutative semigroups to not necessarily commutative semigroups. We define an annihilating-ideal graph of a ring as a special type of zero-divisor graph of a semigroup. We introduce two ways to define the zero-divisor graphs of semigroups. The first definition gives a directed graph ${\Gamma}(S)$, and the other definition yields an undirected graph $\overline{\Gamma}(S)$. It is shown that $\Gamma(S)$ is not necessarily connected, but $\overline{\Gamma}(S)$ is always connected and ${\rm diam}(\overline{\Gamma}(S))\leq 3$. For a ring $R$ define a directed graph $\Bbb{APOG}(R)$ to be equal to $\Gamma(\Bbb{IPO}(R))$, where $\Bbb{IPO}(R)$ is a semigroup consisting of all products of two one-sided ideals of $R$, and define an undirected graph $\overline{\Bbb{APOG}}(R)$ to be equal to $\overline{\Gamma}(\Bbb{IPO}(R))$. We show that $R$ is an Artinian (resp., Noetherian) ring if and only if $\Bbb{APOG}(R)$ has DCC (resp., ACC) on some special subset of its vertices. Also, It is shown that $\overline{\Bbb{APOG}}(R)$ is a complete graph if and only if either $(D(R))^{2}=0$, $R$ is a direct product of two division rings, or $R$ is a local ring with maximal ideal $\mathfrak{m}$ such that $\Bbb{IPO}(R)=\{0,\mathfrak{m},\mathfrak{m}^{2}, R\}$. Finally, we investigate the diameter and the girth of square matrix rings over commutative rings $M_{n\times n}(R)$ where $n\geq 2$. | \section{introduction}
In \cite{Be88}, I. Beck associated to a commutative ring $R$ its zero-divisor graph
$G(R)$ whose vertices are the zero-divisors of $R$ (including $0$), and two distinct vertices
$a$ and $b$ are adjacent if $ ab = 0$. In \cite{AL99}, Anderson and Livingston introduced and studied the subgraph
$\Gamma(R)$ (of $G(R)$) whose vertices are the nonzero zero-divisors of $R$. This graph turns out
to best exhibit the properties of the set of zero-divisors of $R$, and the ideas and
problems introduced in \cite{AL99} were further studied in \cite{AM04, AB13, ALS03}. In \cite{Re01}, Redmond extended the definition of zero-divisor graph to non-commutative rings. Some fundamental results concerning zero-divisor graph for a
non-commutative ring were given in \cite{AM041,AM042,Wu05}.
For a commutative ring $R$ with $1$, denoted by $\Bbb{A}(R)$, the set of ideals with nonzero annihilator. The annihilating-ideal graph of
$R$ is an undirected graph $\Bbb{AG}(R)$ with vertices $\Bbb{A}(R)^{*} = \Bbb{A}(R)
\setminus \{0\}$, where distinct vertices $I$ and $J$ are
adjacent if $IJ = (0)$. The concept of the
annihilating-ideal graph of a commutative ring was introduced in \cite{BRI11,BRII11}. Several fundamental results concerning $\Bbb{AG}(R)$ for a
commutative ring were given in \cite{AAB,AANS,AAN13,AB12}. For a ring $R$, let $D(R)$ be the set of one-sided zero-divisors of $R$ and $\Bbb{IPO}(R) = \{{\rm {\it A \subseteq R : A = IJ} ~where~ {\it I} ~and ~{\it J} ~are~ left~ or~ right~ ideals~ of}~R\}$. Let $S$ be a semigroup with $0$, and $D(S)$ be the set of one-sided zero-divisors of $S$.
The zero-divisor graph of a commutative semigroup is an undirected graph with vertices $Z(S)^{*}$ (the set of non-zero zero-divisors) and two distinct vertices $a$ and $b$ are adjacent if $ab=0$. The zero-divisor graph of a commutative semigroup was introduced in \cite{DMS02} and further studied in \cite{DD05,WL07,WL06,WQL09}.
\\
Let $\Gamma$ be a graph. For vertices $x$ and $y$ of $\Gamma$, let $d(x, y)$ be the length of a shortest path
from $x$ to $y$ ($d(x, x) = 0$ and $d(x, y) = \infty$ if there is
no such a path). The diameter of $\Gamma$ is defined as $ {\rm diam}(\Gamma) = sup\{d(x, y) |$ $x$
and $y$ are vertices of $\Gamma\}$.
The girth of $\Gamma$, denoted by
${\rm gr}(\Gamma)$, is the length of a shortest cycle in $\Gamma$ (${\rm gr}(\Gamma) =
\infty$ if $\Gamma$ contains no cycles).
\\
In Section 2, we introduce a directed graph $\Gamma(S)$ for a semigroup $S$ with 0. We show that $\Gamma(S)$ is not necessarily connected. Then we find a necessarily and sufficient condition for $\Gamma(S)$ to be connected. After that we extend the annihilating-ideal graph to a (not necessarily commutative) ring. It is shown that $\Bbb{IPO}(R)$ is a semigroup. We associate to a ring $R$ a directed graph (denote by $\Bbb{APOG}(R)$) the zero-divisor graph of $\Bbb{IPO}(R)$, i.e., $\Bbb{APOG}(R)=\Gamma(\Bbb{IPO}(R))$. Then we show that $R$ is an Artinian (resp., Noetherian) ring if and only if $\Bbb{APOG}(R)$ has DCC (resp., ACC) on some subset of its vertices.
In Section 3, we introduce an undirected graph $\overline{\Gamma}(S)$ for a semigroup $S$ with 0. We show that $\overline{\Gamma}(S)$ is always connected and ${\rm diam}(\overline{\Gamma}(S))\leq 3$. Moreover, if $\overline{\Gamma}(S)$ contains a cycle, then ${\rm gr}(\overline{\Gamma}(S))\leq 4$. After that we define an undirected graph which extends the annihilating-ideal graph to a not necessarily commutative ring. We associate to a ring $R$ an undirected graph (denoted by $\overline{\Bbb{APOG}}(R)$) the undirected zero-divisor graph of $\Bbb{IPO}(R)$, i.e., $\overline{\Bbb{APOG}}(R)=\overline{\Gamma}(\Bbb{IPO}(R))$. Finally, we characterize rings whose undirected annihilating-ideal graphs are complete graphs.
In Section 4, we investigate the undirected annihilating-ideal graphs of matrix rings over commutative rings. It is shown that ${\rm diam}((\overline{\Bbb{APOG}}(M_{n}(R)))\geq 2$ where $n\geq 2$. Also, we show that ${\rm diam}(\overline{\Bbb{APOG}}(M_{n}(R))\geq {\rm diam}(\overline{\Bbb{APOG}}(R))$.
\section{Directed Annihilating-Ideal Graph of a Ring}
Let $S$ be a semigroup with $0$ and $D(S)$ denote the set of one-sided zero-divisors of $S$. We associate to $S$ a directed graph $\Gamma(S)$ with vertices set $D(S)^{*}=D(S)\setminus \{0\}$ and $a\rightarrow b$ if $ab=0$. In this section, we investigate the properties of $\Gamma(S)$ and we first show the following result.
\begin{ppro}
Let $R$ be a ring. Then $\Bbb{IPO}(R)$ is a semigroup.
\end{ppro}
\begin{pproof}
Let $A,B \in \Bbb{IPO}(R)$. Then
there exist left or right ideals $I_{1}, J_{1}, I_{2}, J_{2}$ of $R$ such that
$A=I_{1}J_{1}$ and $B=I_{2}J_{2}$. We show that
$AB=(I_{1}J_{1})(I_{2}J_{2})\in \Bbb{IPO}(R)$.
$Case$ 1: $J_{1}$ is a left
ideal. Then $AB=I_{1}(J_{1}I_{2}J_{2})
\in \Bbb{IPO}(R)$ (as $J_{1}I_{2}J_{2}$ is a left ideal of $R$).
$Case$ 2: $J_{1}$ is a right ideal and either $I_{2}$ is a left ideal or
$J_{2}$ is a right ideal. Then $AB=(I_{1}J_{1})(I_{2}J_{2})\in
\Bbb{IPO}(R)$.
$Case$ 3: $J_{1}$ is a right ideal, $I_{2}$ is a right ideal, and
$J_{2}$ is a left ideal. Then $AB=(I_{1}J_{1}I_{2})J_{2}\in
\Bbb{IPO}(R)$.
Thus $\Bbb{IPO}(R)$ is closed multiplicatively. Since the multiplication is associative, $\Bbb{IPO}(R)$ is a semigroup.\hfill $\square$
\end{pproof}
\\
It was shown in \cite[Theorem 1.2]{DMS02} that the zero-divisor graph of a commutative semigroup $S$ is connected and ${\rm diam} (\Gamma(S))\leq 3$ . In the following example we show that $\Gamma(S)$ is not necessarily connected when $S$ is a non-commutative semigroup.
\begin{example} Let $K$ be a field and $V =
\oplus_{i=1}^{\infty}K$. Then $R = HOM_{K}(V,V)$, under
the point-wise addition and the multiplication taken to be the composition of
functions, is an infinite non-commutative ring with identity.
Let $\pi_{1}: V \rightarrow V$ be defined by $(a_{1},a_{2}, ...)
\mapsto (a_{1},0, ...)$ and $f : V \rightarrow V$ be defined by
$(a_{1},a_{2}, ...) \mapsto (0, a_{1},a_{2}, ...)$. Then $\pi_{1}, f \in R$. Note that $(R\pi_{1})(fR) = 0 $, so $\Gamma(\Bbb{IPO}(R)) \neq\emptyset$. However, $\Gamma(\Bbb{IPO}(R))$ is not connected as there is no
path leading from the vertex $(fR)$ to any other vertex of
$\Gamma(\Bbb{IPO}(R))$. This is because there exists $g : V \rightarrow V$ given by
$(a_{1},a_{2}, ...) \mapsto (a_{2},a_{3}, ...)$ and $g \in R$
such that $gf = 1_{R}$.\hfill $\square$
\end{example}
For a semigroup $S$, let $$A^{l}(S) =\{a \in D(S)^{*}:~{\rm there ~exists}~ b \in D(R)^{*}
{\rm ~such~ that~} ba = 0\}$$ and
$$A^{r}(S) = \{a \in D(S)^{*}:~ {\rm there ~exists}~ b \in D(R)^{*}~ {\rm such~ that} ~ab = 0 \}.$$
Next we show that $\Gamma(S)$ is connected if and only if $A^{l}(S)=A^{r}(S)$. Moreover, if $\Gamma(S)$ is connected, then ${\rm diam}(\Gamma(S))\leq 3$.
\begin{ttheo}\label{diam33}
Let $S$ be a semigroup. Then $\Gamma(S)$ is connected if
and only if $A^{l}(S) = A^{r}(S)$. Moreover, if $\Gamma(S)$ is
connected, then $diam(\Gamma(S)) \leq 3$.
\end{ttheo}
\begin{pproof}
Suppose that $A^{l}(S) = A^{r}(S)$.\\
Let $a$ and $b$ be distinct vertices of $\Gamma(S)$. Then $a \neq
0$ and $b \neq 0$. We show that there is always a path with length at most 3 from $a$ to $b$.
$Case ~1$: $ab = 0$. Then $a \rightarrow b$ is a desired path.
$Case ~2$: $ab \neq 0$. Then since $A^{l}(S) = A^{r}(S)$,
there exists $c \in D(S) \setminus \{0\}$ such that
$ac = 0$ and $d \in D(S) \setminus \{0\}$ such
that $db= 0$.
$Subcase~ 2.1$: $c= d$. Then $a \rightarrow c \rightarrow b$ is a desired
path.
$Subcase ~2.2$: $c \neq d$. If $cd = 0$, then $a \rightarrow c \rightarrow d \rightarrow b$ is a desired
path. If $cd \neq 0$, then $a \rightarrow cd \rightarrow b$ is
a desired path.
Thus $\Gamma(S)$ is connected and ${\rm diam}(\Gamma(S)) \leq 3$.
Conversely, if $\Gamma(S)$ is connected, then it is easy to show that $A^{l}(S) = A^{r}(S)$.\hfill $\square$
\end{pproof}
\\
Now, we define a directed graph which extends the annihilating-ideal graph to an arbitrary ring. We associate to a ring $R$ a directed graph (denoted by $\Bbb{APOG}(R)$) the zero-divisor graph of $\Bbb{IPO}(R)$, i.e., $\Bbb{APOG}(R)=\Gamma(\Bbb{IPO}(R))$.
\begin{ccoro}
Let $R$ be a ring. Then $\Bbb{APOG}(R)$ is connected if and only if $A^{l}(\Bbb{IPO}(R))=A^{r}(\Bbb{IPO}(R))$. Moreover, if $\Bbb{APOG}(R)$ is connected, then ${\rm diam}(\Bbb{APOG}(R))\leq 3$.
\end{ccoro}
\begin{pproof}
Since $\Bbb{APOG}(R)$ is equal to $\Gamma(\Bbb{IPO}(R))$, it follows from Theorem \ref{diam33} that $\Bbb{APOG}(R)$ is a connected if and only if $A^{l}(\Bbb{IPO}(R))=A^{r}(\Bbb{IPO}(R))$. Also, if $\Bbb{APOG}(R)$ is connected, then ${\rm diam}(\Bbb{APOG}(R))\leq 3$.
\end{pproof}
\\
Recall that a Duo ring is a ring in which every one-sided ideal is a two-sided ideal.
\begin{ppro}\label{Duo}
Let $R$ be an Artinian Duo ring. Then $A^{l}(\Bbb{IPO}(R))=A^{r}(\Bbb{IPO}(R)) = \Bbb{IPO}(R)\setminus \{0, R\}$. Moreover, $\Bbb{APOG}(R)$ is connected and ${\rm diam}(\Bbb{APOG}(R))\leq 3$.
\end{ppro}
\begin{pproof}
Let $R$ be a Duo ring. Then by \cite[Lemma 4.2]{KK10}, $R= (R_{1},\mathfrak{m}_{1}) \times (R_{2},\mathfrak{m}_{2}) \times \cdots (R_{n},\mathfrak{m}_{n})$, where each $R_{i} (1\leq i\leq n)$ is an Artinian local ring with unique maximal ideal $\mathfrak{m}_{i}$. Let $A \in \Bbb{IPO}(R)\setminus \{0, R\}$. Then $A = (I_{1} \times I_{2} \times \cdots \times I_{n})$
$(J_{1} \times J_{2} \times ... \times J_{n})$, where every $I_{i} (1\leq i\leq n$) is an one-sided ideal, so is every $J_{j}(1\leq j \leq n)$. Since $A \neq R$, there exists
$I_{i}$ (or $J_{j}$) such that $I_{i}\neq R$ (or $J_{j} \neq R$). Without loss of generality we may assume that $I_{i} \neq R$. So $A = (I_{1} \times I_{2} \times \cdots \times I_{n})$ $(J_{1} \times J_{2} \times \cdots \times J_{n})$
$ \subseteq $ $(R_{1} \times \cdots\times I_{i} \times \cdots \times R_{n})$ $(R_{1} \times \cdots \times R_{i} \times \cdots \times
R_{n})$. Suppose $k$ is the smallest positive integer such that ${I_{i}}^{k} =
0 $. Thus $(0 \times \cdots \times I_{i}^{k - 1} \times ... \times
0) ((R_{1} \times \cdots \times I_{i} \times \cdots \times R_{n})(R_{1} \times \cdots \times R_{i} \times \cdots \times
R_{n})) = 0 $ and $((R_{1} \times \cdots \times I_{i} \times \cdots \times R_{n})(R_{1} \times \cdots \times R_{i} \times \cdots \times
R_{n}))(0 \times \cdots \times I_{i}^{k - 1} \times \cdots \times
0) = 0$. Therefore $A \in A^{l}(\Bbb{IPO}(R))$ and $A\in A^{r}(\Bbb{IPO}(R))$. Thus $\Bbb{IPO}(R)\setminus \{0, R\}\subseteq A^{r}(\Bbb{IPO}(R))$ and $\Bbb{IPO}(R)\setminus \{0, R\}\subseteq A^{l}(\Bbb{IPO}(R))$. We conclude that $A^{r}(\Bbb{IPO}(R))=\Bbb{IPO}(R)\setminus \{0, R\}=A^{l}(\Bbb{IPO}(R))$.
The second part follows from Theorem \ref{diam33}. \hfill $\square$
\end{pproof}
\\
It is well known that if $|D(R)|\geq 2$ is finite, then $|R|$
is finite. Let $A,B$ be vertices of $\Bbb{APOG}(R)$. We use $A\rightleftharpoons B$ if $A\rightarrow B$ or $A\leftarrow B$. For any vertices $C$ and $D$ of $\Bbb{APOG}(R)$, let ${\rm ad}(C)$ = $\{A$ is a vertex of $\Bbb{APOG}(R)$ : $C = A$ or $C \rightleftharpoons A$ or there exists a vertex $B$ of $\Bbb{APOG}(R)$ such that $C \rightleftharpoons B \rightleftharpoons A$ $\}$ and ${\rm adu}(D)=\bigcup_{C\subseteq D}{\rm ad}(C)$. We know that $ {\rm ad}(C)\subseteq D(R)$. The following proposition shows that if a principal left or right ideal $I$ of $R$ is a vertex of $\Bbb{APOG}(R)$ and all left and right ideals of ${\rm ad}(I)$ have finite cardinality, then $R$ has finite cardinality.
\begin{ppro}
Let $R$ be a ring and $I$ be a
principal left or right ideal of $R$ such that $I$ is a vertex of $\Bbb{APOG}(R)$. If all left and right ideals of ${\rm ad}(I)$ have finite cardinality, then $R$ has finite cardinality.
\end{ppro}
\begin{pproof}
Without loss of generality, we may assume that $I$ is a left principal ideal. Thus
$I = Rx$ for some non-zero $x \in R$. If $Ann_{l}(x) = 0$, then $| R | = | I| <\infty$. So we may always assume that $Ann_{l}(x) \not= 0$.
$ Case ~1$: $I = Ann_{r}(x)$ and $Ann_{r}(x)Ann_{l}(x) = 0$. Then
$$I \rightarrow Ann_{l}(x)$$ and so $Ann_{l}(x) \in {\rm ad}(I)$.
Therefore, $Ann_{l}(x)$ is finite. Since $I \cong R/
Ann_{l}(x)$, $| R | = | I| | Ann_{l}(x)|<\infty$.
$ Case ~2$: $I \neq Ann_{r}(x)$ and $Ann_{r}(x)Ann_{l}(x) = 0$. If $Ann_{r}(x) \not= 0$,
then $$I \rightarrow Ann_{r}(x) \rightarrow Ann_{l}(x)$$ and so $Ann_{l}(x) \in {\rm ad}(I)$.
Therefore, $Ann_{l}(x)$ is finite. Since $I \cong R/
Ann_{l}(x)$, $| R | = | I| | Ann_{l}(x)|<\infty$. If $Ann_{r}(x) = 0$, then since $Rx$ is a vertex of $\Bbb{APOG}(R)$, there exists a (nonzero right ideal) $J$ such that $JRx=0$ (replace $J$ by $JR$ if necessary). Since $Ann_{r}(x) = 0$, we have $xJ$ is a nonzero right ideal and so $$ Ann_{l}(x) \rightarrow xJ \rightarrow I.$$ Thus $Ann_{l}(x) \in {\rm ad}(I)$, so $Ann_{l}(x)$ is finite. Again, we have $| R | = | I| | Ann_{l}(x)|<\infty$.
$ Case ~3$: $I \neq Ann_{r}(x)$ and $ Ann_{r}(x)Ann_{l}(x) \neq 0 $.
Then $$Ann_{r}(x)\leftarrow I \rightarrow Ann_{r}(x)Ann_{l}(x) \rightarrow (xR)$$
and so $(xR), Ann_{r}(x) \in {\rm ad}(I)$. Therefore, $(xR)$ and $Ann_{r}(x)$
are finite. Since $(xR) \cong R/
Ann_{r}(x)$, $| R | = | (xR)| | Ann_{r}(x)|<\infty$. This completes the proof. \hfill $\square$
\end{pproof}\\
Here is our main result in this section.
\begin{ttheo}\label{Artin}
Let $R$ be a ring such that $\Bbb{APOG}(R)\neq \emptyset$. Then $R$ is Artinian (resp., Noetherian) if and only if for a left or right ideal $I$ in the vertex set of $\Bbb{APOG}(R)$, ${\rm adu}(I)$ has DCC (resp., ACC) on both its left and right ideals.
\end{ttheo}
\begin{proof}
If $R$ is Artinian, then $\Bbb{IPO}(R)$ has DCC on both its left ideals and right ideals. Thus for every left or right ideal of the vertex set of $\Bbb{APOG}(R)$, ${\rm adu}(I)$ has DCC on both its left and right ideals as ${\rm adu}(I)\subseteq \Bbb{IPO}(R)$.
Conversely, without loss of generality let $I$ be a left ideal of vertex set of $\Bbb{APOG}(R)$ such that ${\rm adu}(I)$ has DCC on its left and right ideals. Assume that $x\in I$. We have the following cases:
$Case~1$: $xRx\neq \{0\}$, $Ann_l(x)\neq 0$, and $Ann_r(x)\neq 0$. Then $$(xR)\leftarrow Ann_l(x)\leftarrow xRx\rightarrow Ann_{r}(x)\leftarrow (Rx).$$
Therefore $(xR),Ann_r(x), Ann_l(x), (Rx)\in {\rm ad}(xRx)$. Since ${\rm ad}(xRx)\subseteq {\rm adu}(I)$ and ${\rm adu}(I)$ has DCC on its left and right ideals, we conclude that $(Rx)$ and $Ann_l(x)$ are left Artinian $R$-modules, and $(xR)$ and $Ann_r(x)$ are right Artinian $R$-modules. Since $(Rx)\cong R/Ann_l(x)$ and $(xR)\cong R/Ann_r(x)$, by \cite[(1.20)]{La91} we conclude that $R$ is Artinian.
$Case~2$: $xRx=\{0\}$, $Ann_l(x)\neq 0$, and $Ann_r(x)\neq 0$. Then $$Ann_l(x)\rightarrow(xR)\rightarrow(Rx)\rightarrow Ann_r(x).$$ Since ${\rm ad}(Rx)\subseteq {\rm adu}(I)$ and ${\rm adu}(I)$ has DCC on its left and right ideals, we conclude that $(Rx)$ and $Ann_l(x)$ are left Artinian $R$-modules, and $(xR)$ and $Ann_r(x)$ are right Artinian $R$-modules. Since $(Rx)\cong R/Ann_l(x)$ and $(xR)\cong R/Ann_r(x)$, by \cite[(1.20)]{La91} we conclude that $R$ is Artinian.
$Case~3$: $Ann_l(x)=\{0\}$. Then $Rx\cong R$. Therefore, $R$ is a left Artinian module. Since $Rx$ is a vertex of $\Bbb{APOG}(R)$, we have $Ann_r(x)\neq \{0\}$. So there exists $y\in D(R)\setminus \{0\}$ such that $xy=0$.
$Subcase~3.1$: $yRy\neq \{0\}$. If $Ann_r(y)=\{0\}$, then since $$Rx\rightarrow yR,$$ we have $yR\in {\rm adu}(I)$, so $yR$ is a Artinian right $R$-module. Note that $yR\cong R$. Therefore, $R$ is a right Artinian module. If $Ann_r(y)\neq\{0\}$, then $$Ann_r(y)\leftarrow yRy \leftarrow yRx \rightarrow yR.$$ Therefore $(yR),Ann_{r}(y)\in {\rm ad }(yRx)\subseteq {\rm adu}(I)$. Since ${\rm adu}(I)$ has DCC on its right ideals, we conclude that $(yR)$ and $Ann_r(y)$ are right Artinian $R$-modules. Note that $(yR)\cong R/Ann_r(y)$, by \cite[(1.20)]{La91} we conclude that $R$ is a right Artinian module.
$Subcase~3.2$: $yRy= \{0\}$. Then $$yR\leftarrow yRx\leftarrow Ry\rightarrow Ann_r(y).$$ Since $(yR), Ann_r(y)\in {\rm ad}(yRx)\subseteq {\rm adu}(I)$, we conclude that $(yR)$ and $Ann_r(y)$ are right Artinian $R$-modules. Note that $(yR)\cong R/Ann_r(y)$, by \cite[(1.20)]{La91} we conclude that $R$ is a right Artinian module.
$Case~4$: $Ann_r(x)=\{0\}$. Then $xRx\neq \{0\}$ and since $Rx$ is a vertex of $\Bbb{APOG}(R)$, we have $Ann_l(x)\neq \{0\}$. Therefore, $$(xR)\leftarrow Ann_l(x)\rightarrow xRx.$$ We conclude that $xR,Ann_l(x)\in {\rm ad}(xRx)\subseteq {\rm adu}(I)$. Since $xR,Rx,Ann_l(x)\in {\rm adu}(I)$, we have $Rx$ and $Ann_l(x)$ are left Artinian modules and $xR$ is a right Artinian module. Note that $(Rx)\cong R/Ann_l(x)$ and $(xR)\cong R/Ann_r(x)$. Again by \cite[(1.20)]{La91} we conclude that $R$ is Artinian.\hfill $\square$
\end{proof}
\\
\begin{ccoro}\label{Artin2}
Let $R$ be a ring such that $\Bbb{APOG}(R)\neq \emptyset$. Then $R$ is Artinian (resp., Noetherian) if and only if $\Bbb{APOG}(R)$ has DCC (resp., ACC) on left and right ideals of its vertex set.
\end{ccoro}
\begin{pproof}
Since vertex set of $\Bbb{APOG}(R)$ is a subset of $\Bbb{IPO}(R)$, As in the proof of Theorem \ref{Artin}, if $R$ is Artinian (resp., Noetherian), then $\Bbb{APOG}(R)$ has DCC (resp., ACC) on left and right ideals of its vertex set. \\
Conversely, since for a left or right ideal $I$ of the vertex set of $\Bbb{APOG}(R)$, ${\rm adu}(I)$ is a subset of the vertex set of $\Bbb{APOG}(R)$, it follows from Theorem \ref{Artin} that $R$ is Artinian.\hfill $\square$
\end{pproof}
\\
A directed graph $\Gamma$ is called a
tournament if for every two distinct vertices $x$ and $y$ of $\Gamma$
exactly one of $xy$ and $yx$ is an edge of $\Gamma$. In other words, a
tournament is a complete graph with exactly one direction assigned
to each edge.
\begin{ppro} Let $R$ be a ring such that
$A^{2}\neq \{0\}$ for every non-zero $A \in \Bbb{IPO}(R)$ and
$A^{l}(\Bbb{IPO}(R))\cap A^{r}(\Bbb{IPO}(R)) \neq \emptyset$. Then $\Bbb{APOG}(R)$ is not a tournament.
\end{ppro}
\begin{pproof}
Assume $\Bbb{APOG}(R)$ is a tournament. Since
$A^{l}(\Bbb{IPO}(R)) \cap A^{r}(\Bbb{IPO}(R)) \neq \emptyset$, there exists $B \in
A^{l}(\Bbb{IPO}(R))\cap A^{r}(\Bbb{IPO}(R))$, that is, there exist distinct non-zero
$A, C \in \Bbb{IPO}(R)$ such that $A\rightarrow B\rightarrow C$
is a path in $\Bbb{APOG}(R)$. If $CA\neq \{0\}$, then $B(CA)=(BC)A=\{0\}$ and
$(CA)B=C(AB)=\{0\}$, which is a contradiction. So $CA=\{0\}$ and
therefore $AC\neq \{0\}$ since $\Bbb{APOG}(R)$ is a tournament. Also, $AC
\neq A$ (otherwise $A^{2}= (ACAC)=A(CA)C = \{0\}$) and similarly,
$AC\neq C$. Let $a,a_1\in A$ and $c,c_1\in C$. Then we have $B\rightarrow
C\rightarrow ((a-a_1c)R)$ and $(R(c-ac_1))\rightarrow A\rightarrow B$. As the above $((a-a_1c)R)B=\{0\}$ and $B(R(c-ac_1))=\{0\}$. Let $b\in B$ be an arbitrary element. Then $-acb=a_1b-acb\in ((a-a_1c)R)B=\{0\}$ and $bac=bc_1-bac\in B(R(c-ac_1))=\{0\}$. Therefore, $ACB=\{0\}$ and $BAC=\{0\}$. Thus both $AC\rightarrow B$ and $B\rightarrow AC$ are edges of
$\Bbb{APOG}(R)$. This is a contradiction, hence, $\Bbb{APOG}(R)$ cannot be a
tournament.\hfill $\square$
\end{pproof}
\section{Undirected Annihilating-Ideal Graph of a Ring}
Let $S$ be a semigroup with $0$ and recall that $D(S)$ denotes the set of one-sided zero-divisors of $S$. We associate to $S$ an undirected graph $\overline{\Gamma}(S)$ with vertices set $D(S)^{*}=D(S)\setminus \{0\}$ and two distinct vertices $a$ and $b$ are adjacent if $ab=0$ or $ba=0$.
Similarly, we associate to a ring $R$ an undirected graph (denoted by $\overline{\Bbb{APOG}}(R)$) the undirected zero-divisor graph of $\Bbb{IPO}(R)$, i.e., $\overline{\Bbb{APOG}}(R)=\overline{\Gamma}(\Bbb{IPO}(R))$.
The only difference between $\Bbb{APOG}(R)$ and $\overline{\Bbb{APOG}}(R)$
is that the former is a directed graph and the latter is
undirected (that is, these graphs share the same vertices and the
same edges if directions on the edges are ignored). If $R$ is a
commutative ring, this definition agrees with the previous
definition of the annihilating-ideal graph. In this section we study the properties of $\overline{\Gamma}(R)$. We first show that $\overline{\Gamma}(R)$ is always connected with diameter at most 3.
\begin{ttheo}\label{diam}
Let $S$ be a semigroup. Then
$\overline{\Gamma}(S)$ is a connected graph and
${\rm diam}(\overline{\Gamma}(S)) \leq 3.$
\end{ttheo}
\begin{pproof}
Let $a$ and $b$ be distinct vertices of
$\overline{\Gamma}(S)$. If $ab = 0$ or $ba = 0$, then $a - b$ is a path. Next assume that $ab \neq 0$ and $ba \neq 0$.
$Case ~1$: $a^{2} = 0$ and $b^{2} = 0$. Then $a - ab - b$ is
a path.
$Case ~2$: $a^{2} = 0 $ and $b^{2} \neq 0$. Then there is a
some $c \in D(S) \setminus \{a, b , 0\}$ such that either
$cb = 0$ or $bc = 0$. If either $ac = 0$ or $ca = 0$, then
$a - c - b$ is a path. If $ac \neq 0$ and $ca\neq 0$, then $a
- ca - b$ is a path if $bc =
0$ and $ a - ac - b$ is a path if $cb = 0$.
$Case ~3$: $a^{2} \neq 0$ and $b^{2} = 0$. We can use an
argument
similar to that of the above case to obtain a path.
$Case ~4$: $a^{2} \neq 0$ and $b^{2} \neq 0$. Then there
exist $c, d \in D(S) \setminus \{a, b , 0\}$ such that
either $ca = 0$ or $ac = 0$ and either $db = 0$ or $bd =
0$. If $bc = 0$ or $cb=0$, then $a - c - b$ is a path. Similarly, if $ad=0$ or $ da=0$, $a - d - b$ is a path. So we may assume that $c\neq d$. If $cd=0$ or $dc=0$, then $a - c - d - b$ is a path. Thus we may further assume that $cd \neq 0, dc \neq 0$, $bc \neq 0, cb \neq 0 $, $ad \neq 0$ and $da \neq 0.$ We divide the proof into 4 subcases.
$Subcase ~4.1$: $ac = 0$ and $db = 0$. Then $a - cd - b$ is a path.
$Subcase ~4.2$: $ac = 0$ and $bd = 0$. Then $a - cb - d - b$ is a path.
$Subcase ~4.3$: $ca = 0$ and $bd = 0$. Then $a - dc - b$ is a path.
$Subcase ~4.4$: $ca = 0$ and $db = 0$. $a - bc - d - b$ is a path.
Thus $\overline{\Gamma}(S)) $ is connected and
${\rm diam}(\overline{\Gamma}(S)) \leq 3.$ \hfill $\square$
\end{pproof}\\
In \cite {AL99}, Anderson and Livingston proved that if $\Gamma(R)$ (the zero-divisor graph of a commutative ring $R$) contains a cycle, then ${\rm gr} (\Gamma(R))\leq 7$. They also proved that ${\rm gr}(\Gamma(R)) \leq 4$ when $R$ is Artinian and
conjectured that this is the case for all commutative rings $R$. Their conjecture was proved
independently by Mulay \cite{Mu02} and DeMeyer and Schneider \cite{DS02}. Also, in \cite{Re01}, Redmond proved that if $\overline{\Gamma}(R)$ (the undirected zero-divisor graph of a non-commutative ring) contains a cycle, then ${\rm gr}(\overline{\Gamma}(R))\leq 4$. The following is our first main result in this section which shows that for a (not necessarily commutative) semigroup $S$, if $\overline{\Gamma}(S)$ contains a cycle, then $gr(\overline{\Gamma}(S))\leq 4$.
\begin{ttheo}\label{girth}
Let $S$ be a semigroup. If $\overline{\Gamma}(S)$ contains a cycle, then $gr(\overline{\Gamma}(S))\leq 4$.
\end{ttheo}
\begin{pproof}
Let $ a_{1}-a_{2}-\cdots -a_{n-1}-a_{n}-a_{1}$ be a cycle of shortest length
in $\overline{\Gamma}(S)$. Assume that
$gr(\overline{\Gamma}(S)) > 4$, i.e., assume $n\geq 5$. Note that $a_2a_{n-1} \neq 0$ and $a_{n-1}a_2 \neq 0$ ( as $n\geq 5$). If $a_2 a_{n-1}\not \in \{a_1, a_n\}$, then $a_1-a_2a_{n-1}-a_n-a_1$ is a cycle of length 3, yielding a contradiction. Also, if $a_{n-1}a_2\not \in \{a_1, a_n\}$, then $a_1-a_{n-1}a_2-a_n-a_1$ is a cycle of length 3, yielding a contradiction. We have the following cases:
$Case ~1:$ $a_2a_{n-1}=a_1$ and $a_{n-1}a_2=a_n$. If $a_2a_3=0$, then $a_n a_3=(a_{n-1}a_2)a_3=0$. Therefore, $a_1-a_2-a_3-a_n-a_1$ is a cycle of length 4, yielding a contradiction. So, $a_3a_2=0$. Thus, $a_3a_1=a_3(a_2 a_{n-1})=0$. Therefore, $a_1-a_3-a_4-\cdots - a_{n-1}-a_n-a_1$ is a cycle of length $n-1$, yielding a contradiction.
$Case ~2:$ $a_2a_{n-1}=a_1$ and $a_{n-1}a_2=a_1$. If $a_2a_3=0$, then $a_1 a_3=(a_{n-1}a_2)a_3=0$. Therefore, $a_1-a_3-a_4-\cdots - a_{n-1}-a_n-a_1$ is a cycle of length $n-1$, yielding a contradiction. So, $a_3a_2=0$. Thus, $a_3a_1=a_3(a_2 a_{n-1})=0$. Therefore, $a_1-a_3-a_4-\cdots - a_{n-1}-a_n-a_1$ is a cycle of length $n-1$, yielding a contradiction.
$Case ~3:$ $a_2a_{n-1}=a_n$ and $a_{n-1}a_2=a_1$. If $a_2a_3=0$, then $a_1 a_3=(a_{n-1}a_2)a_3=0$. Therefore, $a_1-a_3-a_4-\cdots - a_{n-1}-a_n-a_1$ is a cycle of length $n-1$, yielding a contradiction. So, $a_3a_2=0$. Thus, $a_3a_n=a_3(a_2 a_{n-1})=0$. Therefore, $a_1-a_2-a_3-a_n-a_1$ is a cycle of length 4, yielding a contradiction.
$Case ~4:$ $a_2a_{n-1}=a_n$ and $a_{n-1}a_2=a_n$. If $a_2a_3=0$, then $a_n a_3=(a_{n-1}a_2)a_3=0$. If $a_3a_2=0$, then $a_3a_n=a_3(a_2 a_{n-1})=0$. Therefore, $a_1-a_2-a_3-a_n-a_1$ is a cycle of length $4$, yielding a contradiction.
Since in all cases we have found contradictions, we conclude that if $\overline{\Gamma}(S)$ contains a cycle, then $gr(\overline{\Gamma}(S)) \leq 4$.\hfill $\square$
\end{pproof}
\begin{ccoro}\label{diam3}
Let $R$ be a ring. Then
$\overline{\Bbb{APOG}}(R)$ is a connected graph and
${\rm diam}(\overline{\Bbb{APOG}}(R)) \leq 3.$ Moreover, If
$\overline{\Bbb{APOG}}(R)$ contains a cycle, then ${\rm gr}(\overline{\Bbb{APOG}}(R)) \leq 4$.
\end{ccoro}
\begin{pproof}
Note that $\overline{\Bbb{APOG}}(R)$ is equal to $\overline{\Gamma}(\Bbb{IPO}(R))$. So by Theorem \ref{diam}, $\overline{\Bbb{APOG}}(R)$ is a connected graph and ${\rm diam}(\overline{\Bbb{APOG}}(R)) \leq 3$. Also, by Theorem \ref{girth}, if
$\overline{\Bbb{APOG}}(R)$ contains a cycle, then ${\rm gr}(\overline{\Bbb{APOG}}(R)) \leq 4$.\hfill $\square$
\end{pproof}
\\
For a not necessarily commutative ring $R$, we define a simple undirected graph $\overline{\Gamma}(R)$ with vertex set $D(R)^*$ (the set of all non-zero zero-divisors of $R$) in which two distinct
vertices $x$ and $y$ are adjacent if and only if either $xy = 0$ or $yx = 0$ (see \cite{Re01}). The Jacobson radical of $R$, denoted by $J(R)$, is equal to the intersection of all maximal right ideals of $R$. It is well-known that $J(R)$ is also equal to the intersection of all maximal left ideals of $R$. In our second main theorem in this section we characterize rings whose undirected annihilating-ideal graphs are complete graphs.
\begin{ttheo}
Let $R$ be a ring. Then $\overline{\Bbb{APOG}}(R)$ is a complete graph if and only if either $(D(R))^{2}=0$, or $R$ is a direct product of two division rings, or $R$ is a local ring with maximal ideal $\mathfrak{m}$ such that $\Bbb{IPO}(R)=\{0,\mathfrak{m},\mathfrak{m}^{2}, R\}$.
\end{ttheo}
\begin{pproof}
Assume that $\overline{\Bbb{APOG}}(R)$ is a complete graph. If $\overline{\Gamma}(R)$ is a complete graph, then by \cite[Theorem 5]{AM042}, either $R\cong \Bbb{Z}_2 \times \Bbb{Z}_2$ or $D(R)^2=\{0\}$. So the forward direction holds. Next assume that $\overline{\Gamma}(R)$ is not a complete graph. So there exist different vertices $x$ and $y$ of $\overline{\Gamma}(R)$ such that $x$ and $y$ are not adjacent. We have the following cases:
$Case~1$: $x\in A^{r}(R)$. Without loss of generality assume that $y\in A^r(R)$. If $Rx\neq Ry$, then since $\Bbb{APOG}(R)$ is a complete graph, we have $Rx$ is adjacent to $Ry$ in $\overline{\Bbb{APOG}}(R)$, so $x$ and $y$ are adjacent in $\overline{\Gamma}(R)$, yielding a contradiction. Thus $Rx=Ry$. Since $x\in A^{r}(R)$, there exists non-zero element $z\in D(R)$ such that $xz=0$. If $Rx\subseteq zR$, then $(Rx)^2=\{0\}$. So $(Rx)(Ry)=\{0\}$, and $x$ and $y$ are adjacent in $\overline{\Gamma}(R)$, yielding a cintradiction. Therefore, $Rx \nsubseteq zR$. If there exists a left or right ideal $I$ of $R$ expect $zR$ such that $I\nsubseteq Rx$, then there exists nonzero element $s\in I\setminus Rx$. Then $(Rs+Rx) (zR)=\{0\}$. Since $\overline{\Bbb{APOG}}(R)$ is a complete graph $Rx$ is adjacent to $(Rs+Rx)=\{0\}$. Thus $(Rx)^{2}=\{0\}$, and so $x$ and $y$ are adjacent in $\overline{\Gamma}(R)$, yielding a contradiction. Therefore, $\{zR,Rx\}$ is the set of nonzero proper left or right ideals of $R$. Thus by Corollary \ref{Artin2}, $R$ is an Artinian ring. We have the following subcases:
$Subcase~1$: $zR\nsubseteq Rx$. Then $zR$ and $Rx$ are maximal ideals. If $zR$ or $Rx$ is not a two-sided ideal, then $zR=J(R)=Rx$, yielding a contradiction. Therefore, $Rx$ and $zR$ are two-sided ideals. Also, $Rx$ and $zR$ are minimal ideals and so $Rx\cap zR=\{0\}$. Thus by Brauer's Lemma (see \cite[10.22]{La91}), $(Rx)^2=0$ or $Rx=Re$, where $e$ is a idempotent in $R$. If $(Rx)^2=\{0\}$, then $x$ is adjacent to $y$ in $\overline{\Gamma}(R)$, yielding a contradiction. So $Rx=Re$, where $e$ is an idempotent in $R$. Therefore, $R=eRe\oplus eR(1-e)\oplus (1-e)Re\oplus (1-e)R(1-e)$. Since $\{zR,Rx\}$ is the set of nonzero proper left or right ideals of $R$ and $Rx\cap zR=\{0\}$, we conclude that $Re=Rx=eR$ and $(1-e)R=zR=R(1-e)$. Therefore, $(1-e)Re=(1-e)eR=\{0\}$ and $eR(1-e)=e(1-e)R=\{0\}$.
So $R=eRe\oplus(1-e)R(1-e)$. Since $R$ is an Artinian ring with two nonzero left or right ideals, we conclude that $eRe$ and $(1-e)R(1-e)$ are division rings.
$Subcase~2$: $zR\subseteq Rx$. Then $Rx=D(R)$. If $(Rx)^2=\{0\}$, then $x$ is adjacent to $y$ in $\overline{\Gamma}(R)$, yielding a contradiction. If $D(R)^2\neq 0$, then $D(R)^2=zR$. Therefore, $R$ is a local ring with maximal ideal $\mathfrak{m}$ such that $\Bbb{IPO}(R)=\{0,\mathfrak{m},\mathfrak{m}^{2}, R\}$.
In summary, we obtain that either $R$ is a direct product of two division rings, or $R$ is a local ring with maximal ideal $\mathfrak{m}$ such that $\Bbb{IPO}(R)=\{0,\mathfrak{m},\mathfrak{m}^{2}, R\}$. Thus the forward direction holds.
$Case~2$: $x\in A^{l}(R)$. Similar to Case 1, we conclude that either $R$ is a direct product of two division rings, or $R$ is a local ring with maximal ideal $\mathfrak{m}$ such that $\Bbb{IPO}(R)=\{0,\mathfrak{m},\mathfrak{m}^{2}, R\}$. So the forward direction holds.\\
The converse is obvious. \hfill $\square$
\end{pproof}
\section{Undirected Annihilating-Ideal Graphs for Matrix Rings Over Commutative Rings }
In this section we investigate the undirected annihilating-ideal graphs of matrix rings over commutative rings. By Theorem \ref{diam3}, ${\rm diam}(\overline{\Bbb{APOG}}(R)) \leq 3$ for any ring $R$. In Proposition~\ref{mdiam} we show that ${\rm diam}((\overline{\Bbb{APOG}}(M_{n}(R)))\geq 2$ where $n\geq 2$. A natural question is whether or not ${\rm diam}(\overline{\Bbb{APOG}}(M_{n}(R))\geq {\rm diam}(\overline{\Bbb{APOG}}(R))$. We show that the answer to this question is affirmative.
\begin{ppro}\label{mdiam}
Let $R$ be a commutative ring. Then ${\rm diam}(\overline{\Bbb{APOG}}(M_{n}(R))\geq 2$ where $n\geq 2$.
\end{ppro}
\begin{pproof}
Let $$A=(M_{n}(R)\begin{bmatrix}
1 & 0 & 0 & \cdots & 0\\
0 & 0 & 0 & \cdots & 0\\
\vdots & \vdots & \vdots & \cdots & \vdots \\
0 & 0 & 0 & \cdots & 0
\end{bmatrix}) ~{\rm and}~ B=(\begin{bmatrix}
1 & 0 & 0 & \cdots & 0\\
0 & 0 & 0 & \cdots & 0\\
\vdots & \vdots & \vdots & \cdots & \vdots \\
0 & 0 & 0 & \cdots & 0
\end{bmatrix}M_{n}(R)).$$
Since
$$A(\begin{bmatrix}
0 & 0 & 0 & \cdots & 0\\
1 & 0 & 0 & \cdots & 0\\
\vdots & \vdots & \vdots & \cdots & \vdots \\
0 & 0 & 0 & \cdots & 0
\end{bmatrix}M_{n}(R))=0 ~{\rm and} ~(M_{n}(R)\begin{bmatrix}
0 & 0 & 0 & \cdots & 0\\
1 & 0 & 0 & \cdots & 0\\
\vdots & \vdots & \vdots & \cdots & \vdots \\
0 & 0 & 0 & \cdots & 0
\end{bmatrix})B=0, $$
we conclude that $A$ and $B$ are vertices in $(\overline{\Bbb{APOG}}(M_{n}(R))$.
Note that
$$\begin{bmatrix}
1 & 0 & 0 & \cdots & 0\\
0 & 0 & 0 & \cdots & 0\\
\vdots & \vdots & \vdots & \cdots & \vdots \\
0 & 0 & 0 & \cdots & 0
\end{bmatrix}^{2}\neq 0 ~{\rm and}~
\begin{bmatrix} 1 & 0 & 0 & \cdots & 0\\
0 & 0 & 0 & \cdots & 0\\
\vdots & \vdots & \vdots & \cdots & \vdots \\
0 & 0 & 0 & \cdots & 0
\end{bmatrix}\in A\cap B,$$ so $AB\neq 0$. Therefore, ${\rm diam}(\overline{\Bbb{APOG}}(M_{n}(R))\geq 2$.\hfill $\square$
\end{pproof}
\begin{ttheo}
Let $R$ be a commutative ring. Then ${\rm diam}(\overline{\Bbb{APOG}}(M_{n}(R))\geq {\rm diam}(\Bbb{AG}(R))={\rm diam}(\overline{\Bbb{APOG}}(R))$.
\end{ttheo}
\begin{pproof}
By \cite[Theorem 2.1]{BRI11}, ${\rm diam} (\Bbb{AG}(R))\leq 3$.
$Case$ 1: $diam(\Bbb{AG}(R))\leq 2$. By Proposition \ref{mdiam}, ${\rm diam}(\overline{\Bbb{APOG}}(M_{n}(R))\geq 2$. Thus ${\rm diam} (\overline{\Bbb{APOG}}(M_{n}(R))\geq {\rm diam}(\Bbb{AG}(R))$.
$Case$ 2: ${\rm diam}(\Bbb{AG}(R))=3$. Then there exist vertices $I,J,K$, and $L$ of $\Bbb{AG}(R)$ such that $I-K-L-J$ is a shortest path between $I$ and $J$. So $d(I, J)=3$. Since $I$ and $J$ are vertices of $\Bbb{AG}(R)$, $M_n(I)$ and $M_n(J)$ are vertices of $\overline{\Bbb{APOG}}(M_n(R))$. Suppose that ${\rm diam}(\overline{\Bbb{APOG}}(M_{n}(R))=2$. So we can assume that there exists $\alpha=[a_{ij}]\in M_{n}(R)$ such that $M_{n}(I)\alpha=\alpha M_{n}(J)=0$. Without loss of generality, we may assume that $a_{11}\neq 0$. For every $a\in I$,
$$
\begin{bmatrix}
a & 0 & 0 & \cdots & 0\\
0 & 0 & 0 & \cdots & 0\\
\vdots & \vdots & \vdots & \cdots & \vdots \\
0 & 0 & 0 & \cdots & 0
\end{bmatrix}A=0,$$ so $aa_{11}=0$. Therefore $I(a_{11}R)=0$. For every $b\in J$,
$$A\begin{bmatrix}
b & 0 & 0 & \cdots & 0\\
0 & 0 & 0 & \cdots & 0\\
\vdots & \vdots & \vdots & \cdots & \vdots \\
0 & 0 & 0 & \cdots & 0
\end{bmatrix}=0.$$ Therefore $(a_{11}R)J=0$. Thus $I-(a_{11}R)-J$ is a path of length 2 in $\Bbb{AG}(R)$, and so $d(I,J) \leq 2$, yielding a contradiction. Therefore, ${\rm diam}(\overline{\Bbb{APOG}}(M_{n}(R))=3$ and we are done. \hfill $\square$
\end{pproof}
\\
It was shown in Corollary \ref{diam3} that ${\rm gr}(\overline{\Bbb{APOG}}(R)) \leq 4$. We now show that ${\rm gr}(\overline{\Bbb{APOG}}(M_{n}(R)))=3$ where $n\geq 2$.
\begin{ppro}
Let $R$ be a commutative ring. Then ${\rm gr}(\overline{\Bbb{APOG}}(M_{n}(R))=3$ where $n\geq 2$.
\end{ppro}
\begin{pproof}
Let
$$
A=\begin{bmatrix}
1 & 1 & 0 & \cdots & 0\\
0 & 0 & 0 & \cdots & 0\\
\vdots & \vdots & \vdots & \cdots & \vdots \\
0 & 0 & 0 & \cdots & 0
\end{bmatrix}
,
B=\begin{bmatrix}
1 & -1 & 0 & \cdots & 0\\
-1 & 1 & 0 & \cdots & 0\\
\vdots & \vdots & \vdots & \cdots & \vdots \\
0 & 0 & 0 & \cdots & 0
\end{bmatrix}
,$$ and
$$
C=\begin{bmatrix}
0 & 1 & 0 & \cdots & 0\\
0 & 1 & 0 & \cdots & 0\\
\vdots & \vdots & \vdots & \cdots & \vdots \\
0 & 0 & 0 & \cdots & 0
\end{bmatrix}.$$
Then $(AM_{n}(R)A) - (BM_{n}(R)B) - (CM_{n}(R)C)$ is a cycle in $(\overline{\Bbb{APOG}}(M_{n}(R))$, so ${\rm gr}(\overline{\Bbb{APOG}}(M_{n}(R))=3$.\hfill $\square$
\end{pproof}
| {
"timestamp": "2014-11-18T02:09:57",
"yymm": "1411",
"arxiv_id": "1411.4159",
"language": "en",
"url": "https://arxiv.org/abs/1411.4159",
"abstract": "Let $S$ be a semigroup with $0$ and $R$ be a ring with $1$. We extend the definition of the zero-divisor graphs of commutative semigroups to not necessarily commutative semigroups. We define an annihilating-ideal graph of a ring as a special type of zero-divisor graph of a semigroup. We introduce two ways to define the zero-divisor graphs of semigroups. The first definition gives a directed graph ${\\Gamma}(S)$, and the other definition yields an undirected graph $\\overline{\\Gamma}(S)$. It is shown that $\\Gamma(S)$ is not necessarily connected, but $\\overline{\\Gamma}(S)$ is always connected and ${\\rm diam}(\\overline{\\Gamma}(S))\\leq 3$. For a ring $R$ define a directed graph $\\Bbb{APOG}(R)$ to be equal to $\\Gamma(\\Bbb{IPO}(R))$, where $\\Bbb{IPO}(R)$ is a semigroup consisting of all products of two one-sided ideals of $R$, and define an undirected graph $\\overline{\\Bbb{APOG}}(R)$ to be equal to $\\overline{\\Gamma}(\\Bbb{IPO}(R))$. We show that $R$ is an Artinian (resp., Noetherian) ring if and only if $\\Bbb{APOG}(R)$ has DCC (resp., ACC) on some special subset of its vertices. Also, It is shown that $\\overline{\\Bbb{APOG}}(R)$ is a complete graph if and only if either $(D(R))^{2}=0$, $R$ is a direct product of two division rings, or $R$ is a local ring with maximal ideal $\\mathfrak{m}$ such that $\\Bbb{IPO}(R)=\\{0,\\mathfrak{m},\\mathfrak{m}^{2}, R\\}$. Finally, we investigate the diameter and the girth of square matrix rings over commutative rings $M_{n\\times n}(R)$ where $n\\geq 2$.",
"subjects": "Rings and Algebras (math.RA)",
"title": "The Annihilating-Ideal Graph of a Ring",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754488233528,
"lm_q2_score": 0.7185943985973772,
"lm_q1q2_score": 0.7075104025209599
} |
https://arxiv.org/abs/1102.0712 | Matchings on infinite graphs | Elek and Lippner (2010) showed that the convergence of a sequence of bounded-degree graphs implies the existence of a limit for the proportion of vertices covered by a maximum matching. We provide a characterization of the limiting parameter via a local recursion defined directly on the limit of the graph sequence. Interestingly, the recursion may admit multiple solutions, implying non-trivial long-range dependencies between the covered vertices. We overcome this lack of correlation decay by introducing a perturbative parameter (temperature), which we let progressively go to zero. This allows us to uniquely identify the correct solution. In the important case where the graph limit is a unimodular Galton-Watson tree, the recursion simplifies into a distributional equation that can be solved explicitly, leading to a new asymptotic formula that considerably extends the well-known one by Karp and Sipser for Erdös-Rényi random graphs. | \section{Introduction}
A \textit{matching} on a finite graph $G=(V,E)$ is a subset of
pairwise non-adjacent edges $M\subseteq E$. The $|V|-2|M|$ isolated
vertices of $(V,M)$ are said to be \textit{exposed} by $M$. We let
${\mathbb M}(G)$ denote the set of all possible matchings on $G$. The
\textit{matching number} of $G$ is defined as
\begin{equation}
\label{eq:mn}
\nu(G)=\max_{M\in{\mathbb M}(G)}|M|,
\end{equation}
and those $M$ which achieve this maximum -- or equivalently, have the
fewest exposed vertices -- are called \textit{maximum matchings}. The \textit{normalized matching number} of $G$ is simply $\nu(G)/|V|$.
\iffalse
The \textit{matching number} $\nu(G)$ of a finite graph $G=(V,E)$ is the largest possible cardinality of a \textit{matching} (a subset of mutually non-adjacent edges) on $G$.
This simple invariant plays an important role in graph theory \cite{godsil, lovaszplummer}.
\fi
Our results belong to the theory of convergent graph sequences. Convergence of bounded degree graph sequences was defined
by Benjamini and Schramm \cite{bensch}, Aldous and Steele
\cite{aldste}, see also Aldous and Lyons \cite{aldlyo}. The notion of local weak convergence has
then inspired a lot of work \cite{BenjaminiSS08}, \cite{res10}, \cite{borgs2010left},
\cite{elek}, \cite{lyotree}. In \cite{eleklippner}, it is shown that the convergence of a sequence of bounded-degree graphs guarantees the existence of a limit for their normalized matching numbers. We provide a characterization of the limiting parameter via a local recursion defined directly on the limit of the graph sequence. In the important case where the graph limit is a unimodular Galton-Watson tree, the recursion simplifies into a distributional equation that can be solved explicitly, leading to a new asymptotic formula.
A classical example in this context is the Erd\H{o}s-R\'enyi random graph with average degree $c$ on $n$ vertices, denoted by $G(n,c/n)$ : as $n\to\infty$, $G(n,c/n)$ converges in the local weak sense to a Galton-Watson tree with degree distribution Poisson with parameter $c$. In this case, Karp and Sipser \cite{karpsipser} showed that
\begin{equation}
\label{eq:KS}
\frac{\nu(G(n,c/n))}{n}\xrightarrow[n\to\infty]{}1-\frac{t_c+e^{-ct_c}+ct_ce^{-ct_c}}{2},
\end{equation}
where $t_c\in(0,1)$ is the smallest root of $t=e^{-ce^{-ct}}$ (we will see in the sequel that the convergence is almost sure).
The explicit formula (\ref{eq:KS}) rests on the analysis of a heuristic
algorithm now called Karp-Sipser algorithm.
The latter is based on the
following observation : if $e\in E$ is a pendant edge (i.e. an edge incident to a vertex of degree one) in $G=(V,E)$, then
there is always a maximum matching that contains $e$, so all edges that are
adjacent to $e$ may be deleted without affecting $\nu(G)$. The first stage
of the algorithm consists in iterating this until no more pendant edge is
present. This is the \textit{leaf-removal process}.
$G$ is thus simplified into a sub-graph with only isolated
vertices, matched pairs, and a so-called \textit{core} with minimum degree
at least $2$. As long as that core is non-empty, one of its edges is
selected uniformly at random, the adjacent edges are deleted, and the whole
process starts again. When the algorithm stops, the remaining edges clearly
form a matching on $G$, but its size may be far below $\nu(G)$ due to the
sub-optimal removals on the core.
On $G(n,c/n)$, the dynamics of the deletion process can be approximated in the $n\to \infty$ limit by differential equations which can be explicitly solved.
In particular, the asymptotic size of both the optimal part
constructed in the first stage, and the sub-optimal part constructed on the
core can be evaluated up to an $o(n)$ correcting term (which has been later
refined, see \cite{ksrevisited}). Moreover, the second part happens to be almost
perfect, in the sense that only $o(n)$ vertices are exposed in the
core. This guarantees that the overall construction is asymptotically
optimal, and the asymptotic formula for $\nu(G_n)$ follows.
More recently, the same technique has
been applied to another class of random graphs with a fixed log-concave degree profile \cite{friezebohman}, resulting in the asymptotical existence of an \textit{almost perfect matching} on these graphs :
\begin{equation}
\label{eq:BF}
\frac{\nu(G_{n})}{|V_n|}\xrightarrow[n\to\infty]{}\frac{1}{2}.
\end{equation}
In both cases, the proof of optimality -- and hence
the asymptotic formula for $\nu(G_n)$ -- relies on the fact that the second stage exposes only $o(n)$ vertices, which is bound to fail as soon as one considers more general graph ensembles where the core does
not necessarily admits an almost-perfect matching. We give
simple examples in the Appendix. By using a completely different approach -- namely establishing and solving an appropriate \textit{recursive distributional equation} (a usual ingredient of the objective method, see \cite{aldousbandyopadhya}) --, we manage to obtain a general formula that considerably extends the above results.
The rest of our paper is organized as follows: we state our main results in Section \ref{sec:res}.
In Section \ref{sec:z}, we extend the Boltzmann-Gibbs distribution over matchings on a finite graph to infinite graphs. This will allow us to derive our Theorem \ref{th:main} in Section \ref{sec:zero}.
We deal with the specific cases of trees (and random graphs) in Section \ref{sec:rdet}.
We end the paper with an Appendix presenting simple examples of graphs for which the limiting local recursion admit multiple solutions.
\section{Results}\label{sec:res}
Let us start with a brief recall on local weak convergence (see \cite{bensch, aldste} for details). A \textit{rooted graph} $(G,\circ)$ is a graph $G=(V,E)$ together with the
specification of a particular vertex $\circ\in V$, called the
\textit{root}. We let ${\mathcal G}$ denote the set of all locally finite connected rooted graphs considered up to \textit{rooted isomorphism}, i.e. $(G_1,\circ_1)\equiv(G_2,\circ_2)$ if there exists a bijection $\gamma\colon V_1\to V_2$ that preserves roots ($\gamma(\circ_1)=\circ_2$) and
adjacency ($uv\in E_1\Longleftrightarrow \gamma(u)\gamma(v)\in E_2$). In the space ${\mathcal G}$, a sequence $\left\{(G_n,\circ_n);n\in {\mathbb N}\right\}$ \textit{converges locally} to $(G,\circ)$ if for every radius $k\in{\mathbb N}$, there is $n_k\in{\mathbb N}$ such that
$$n\geq n_k\Longrightarrow [G_n,\circ_n]_k\equiv [G,\circ]_k.$$
Here, $[G,\circ]_k$ denotes the finite rooted subgraph induced by the vertices lying at graph-distance at
most $k$ from $\circ$. It is not hard to construct a distance which metrizes this notion of convergence and turns ${\mathcal G}$ into a complete separable metric space. We can thus import the usual machinery of weak convergence of probability measures on Polish spaces (see e.g. \cite{billingsley}). We define ${\mathcal P}(G)$ as the set of probability measure on ${\mathcal G}$. There is a natural procedure for turning a finite deterministic
graph $G=(V,E)$ into a random element of ${\mathcal G}$ : one simply chooses uniformly at random a vertex
$\circ\in V$ to be the root, and then restrains $G$ to the
connected component of $\circ$. The resulting law is denoted by ${\mathcal U}(G) \in {\mathcal P}({\mathcal G})$. If $(G_n)_{n\in{\mathbb N}}$ is a sequence of finite graphs
such that $({\mathcal U}(G_n))_{n\in{\mathbb N}}$ admits a weak limit $\rho\in{\mathcal P}({\mathcal G})$, we call $\rho$ the \textit{random weak limit} of the sequence $(G_n)_{n\in{\mathbb N}}$. Finally, for any $d\geq 0$, we define ${{\mathcal G}_d}$ as the space of all rooted connected graphs with maximal degree no more than $d$.
Rather than just graphs $G=(V,E)$, it will be sometimes convenient to
work with \textit{discrete networks} $G=(V,E,{\mathcal M})$, in which the additional
specification of a \textit{mark map} ${\mathcal M}\colon E\to{\mathbb N}$ allows to attach useful local
information to edges, such as their absence/presence in a certain matching. We then simply require the isomorphisms in the
above definition to preserve these marks.
The first main implication of our work is that the local weak convergence of a sequence of graphs is enough to guarantee the convergence of their normalized matching numbers to a quantity that can be described directly on the random weak limit of the graph sequence.
\begin{theorem}
\label{th:main}
Let $G_n=(V_n,E_n), n\in{\mathbb N},$ be a sequence of finite graphs admitting a random weak limit $\rho$. Then,
\begin{equation}
\label{lim:g}\frac{\nu(G_n)}{|V_n|} \xrightarrow[n\to\infty]{} \gamma,
\end{equation}
where $\gamma\in [0,\frac{1}{2}]$ is defined by a recursion defined directly on the random weak limit $\rho$.
\end{theorem}
Since the work of Heilmann and Lieb \cite{heilmannlieb}, it is known
that the thermodynamic limit for monomer-dimer systems exists and
basic properties of this limit are derived for lattices. In
particular, \cite[Lemma 8.7]{heilmannlieb} shows the convergence of
the normalized matching number when the underlying graph is a lattice. More recently, Elek and Lippner \cite{eleklippner} extended this result by using the framework of local weak convergence for bounded degree graphs. Here we remove the bounded degree assumption. More importantly, the approach in \cite{eleklippner} is non-constructive. In contrast, we provide a characterization of $\gamma$ in terms of a local recursion defined directly on the random weak limit $\rho$. We postpone the discussion on how to actually compute $\gamma$ from $\rho$ to Subsection \ref{subsec:algo}. Our approach starts as in \cite{heilmannlieb} with the introduction of a natural family of probability distributions on the set of matchings parametrized by a single parameter $z>0$ called the Boltzmann-Gibbs distribution. The analysis in \cite{heilmannlieb} concentrates on the properties of the partition function, also known as the matching polynomial, from which a result like (\ref{lim:g}) can be deduced. Our analysis differs from this approach and concentrates on the analysis of the local marginals of the Boltzmann-Gibbs distribution, in a similar spirit as in the (non-rigorous) work of Zdeborov\'a and M\'ezard \cite{zdeborovamezard}. As in \cite{zdeborovamezard} and \cite{god81}, we start from an elementary formal recursion satisfied by the matching polynomials, and deduce an exact recursion for the local marginals of the Boltzmann-Gibbs distribution on any finite graph. A careful analysis of the contractivity properties of this recursion allows us to define the monomer-dimer model on infinite graphs (see Theorem \ref{co:cvweak}), and to define their "normalized matching number" (see Theorem \ref{co:monotony}). We should stress that the analysis of the marginal probabilities is essential for our second main result, namely the explicit computation of the matching number when the local weak limit is a Galton-Watson tree. Although simple adaptations of the argument in \cite{heilmannlieb} would yield a result like Theorem \ref{th:main}, the limit would be given in an implicit way which would not be sufficient to get our second main result.
As many other classical graph sequences, Erd\H{o}s-R\'enyi graphs and random graphs with a prescribed degree profile admit almost surely a particularly simple random weak limit, namely a \textit{unimodular Galton-Watson (UGW) tree} (see Example 1.1 in \cite{aldlyo}). This random rooted tree is parametrized by a probability distribution $\pi\in{\mathcal P}({\mathbb N})$ with finite mean, called its \textit{degree distribution}. It is obtained by a Galton-Watson branching process where the root has offspring distribution $\pi$ and all other genitors have offspring distribution $\widehat{\pi}\in{\mathcal P}({\mathbb N})$ defined by
\begin{equation}\label{eq:F}
\forall n\in {\mathbb N}, \widehat{\pi}_{n} = \frac{(n+1) \pi_{n+1}}{\sum_{k} k \pi_k}.
\end{equation}
Thanks to the markovian nature of the branching process, the recursion defining $\gamma$ simplifies into a \textit{recursive distributional equation}, which has been explicitly solved by the authors in a different context \cite{rankarxiv}.
\begin{theorem}
\label{th:KS}
With the notation of Theorem \ref{th:main}, if the random weak limit $\rho$ is a UGW tree with degree distribution $\pi$, we have the explicit formula
\begin{eqnarray*}
\gamma = \frac{1-\max_{t\in [0,1]} F(t)}{2},
\end{eqnarray*}
where
\begin{eqnarray*}
F(t)=t\phi'(1-t)+\phi(1-t)+\phi\left(1-\frac{\phi'(1-t)}{\phi'(1)}\right)-1,
\end{eqnarray*}
and $\phi(t)=\sum_{n}\pi_nt^n$ is the moment generating function of the degree distribution $\pi$.
\end{theorem}
Differentiating the above expression, we see that any $t$ achieving the maximum must satisfy
\begin{equation}
\label{eq:diff}
\phi'(1)t=\phi'\left(1-\frac{\phi'(1-t)}{\phi'(1)}\right).
\end{equation}
For Erd\H{o}s-R\'enyi random graphs with connectivity $c$, the degree
of the limiting UGW tree is Poisson with parameter $c$
(i.e. $\phi(t)=\exp(ct-c)$), so that (\ref{eq:diff}) becomes
$t=e^{-ce^{-ct}}$. We thus recover precisely Karp and Sipser's formula
(\ref{eq:KS}). Similarly, for random graphs with a prescribed degree
sequence, the log-concave assumption made by Bohmann and Frieze
guarantees that the above maximum is achieved at $t=0$ with $F(0)=0$, hence (\ref{eq:BF}) follows automatically.
A classical area of combinatorial optimization is formed by bipartite matching \cite{lovaszplummer}.
We end this section, with a specialization of our results to bipartite graphs $G=(V=V^a\cup V^b, E)$.
The natural limit for a sequence of bipartite graphs is the following
hierarchal Galton-Watson tree parameterized by two distributions
on ${\mathbb N}$ with finite first moment,
$\pi^a$ and $\pi^b$ and a parameter $\lambda\in [0,1]$.
We denote $\widehat{\pi}^a$ and
$\widehat{\pi}^b$ the corresponding distributions given by the transformation
(\ref{eq:F}).
We also denote $\phi^a$ and $\phi^b$ the generating functions of
$\pi^a$ and $\pi^b$.
The hierarchal Galton-Watson tree is then defined as follows: with probability $\lambda$, the
root has offspring distribution $\pi^a$, all odd generation genitors
have offspring distribution $\widehat{\pi}^b$ and all even generation genitors
have offspring distribution $\widehat{\pi}^a$; similarly with probability
$1-\lambda$, the root has offspring distribution $\pi^b$, all odd
generation genitors have offspring distribution $\widehat{\pi}^a$ and all even
generation genitors have offspring distribution $\widehat{\pi}^b$. In the first
(resp. second) case, we say that the root and all even generations are
of type $a$ (resp. $b$) and all the odd generations are of type $b$
(resp. $a$). To get a {\em unimodular hierarchal Galton-Watson (UHGW) tree} with degree distributions $\pi^a$ and $\pi^b$, we need to have:
$\lambda \phi^a{}'(1) =(1-\lambda)\phi^b{}'(1)$, so that
\begin{eqnarray}
\label{eq:contrlamb}\lambda = \frac{\phi^b{}'(1)}{\phi^a{}'(1)+\phi^b{}'(1)}.
\end{eqnarray}
\begin{theorem}
\label{th:KSbi}
With the notation of Theorem \ref{th:main}, assume that the random weak limit $\rho$ is a UHGW tree with degree distributions
$\pi^a,\pi^b$.
If $\pi^a$ and $\pi^b$ have finite first moment, then
\begin{equation}
\label{eq:KSbi}
\gamma=\frac{\phi^b{}'(1)}{\phi^a{}'(1)+\phi^b{}'(1)}(1-\max_{t\in[0,1]}F^a(t)),
\end{equation}
where $F^a$ is defined by:
\begin{eqnarray*}
F^a(t) &=& \phi^a\left(1-\frac{\phi^b{}'(1-t)}{\phi^b{}'(1)}\right)-\frac{\phi^a{}'(1)}{\phi^b{}'(1)}\left(1-\phi^b(1-t)-t\phi^b{}'(1-t)\right).
\end{eqnarray*}
\end{theorem}
Note that if $\phi^a(x)=\phi^b(x)$, we find the same limit as in
Theorem \ref{th:KS}. Note that it is not obvious from formula
\eqref{eq:KSbi} that our expression for $\gamma$ is symmetric in $a$
and $b$ as it should. In the forthcoming Section \ref{sec:tlgwt},
Equation \eqref{eq:KSbi2} gives an alternative symmetric formula for
$\gamma$ which simplifies to (\ref{eq:KSbi}) thanks to (\ref{eq:contrlamb}).
Note also that our Theorem \ref{th:KSbi} computes the independence
number of random bipartite graphs.
Recall that a set of vertices in a graph $G$ is said to be
independent if no two of them are adjacent. The cardinality of any
largest independent set of points in $G$ is known as the independence
number of $G$ or the stability number of $G$ and is denoted by
$\alpha(G)$.
By K\H{o}nig's theorem, we know that for any bipartite graph
$G$ with vertex set $V$, $\alpha(G)+\nu(G)=|V|$.
The fact that a limit for $\frac{\alpha(G_n)}{{|V_n|}}$ exists,
has been proved recently in \cite{bayati-2010} for Erd\H{o}s-R\'enyi and
random regular graphs. The actual value for this limit is unknown except
for Erd\H{o}s-R\'enyi graphs with mean degree $c<e$. In this case, the
leaf-removal algorithm allows to compute explicitly the limit which agrees with
(\ref{eq:KSbi}) with $\phi^a(x)=\phi^b(x)=\exp(cx-x)$.
Motivated by some applications for Cuckoo Hashing \cite{fopa09},
\cite{kxor}, recent results have been obtained in the particular case
where $\pi^a(k)=1$ for some $k\geq 3$ and $\pi^b$ is a Poisson
distribution with parameter $\alpha k$. These degree distributions arise if one consider a sequence of bipartite graphs with $\lfloor \alpha m\rfloor$ nodes of type $a$ (called the items), $m$ nodes of type $b$ (called the locations) and each node of type $a$ is connected with $k$ nodes of type $b$ chosen uniformly at random (corresponding to the assigned locations the item can be stored in).
The result in this domain, obtained in \cite{frpa09} follows (see Section
\ref{sec:corfr}) from our Theorem \ref{th:KSbi}, namely:
\begin{corollary}\label{cor:fr}
Under the assumption of Theorem \ref{th:KSbi} and with $\pi^a(k)=1$ for some $k\geq 3$ and $\pi^b$ is a Poisson
distribution with parameter $\alpha k$.
Let $\xi$ be the unique solution of the equation:
\begin{eqnarray*}
k=\frac{\xi(1-e^{-\xi})}{1-e^{-\xi}-\xi e^{-\xi}},
\end{eqnarray*}
and $\alpha_c = \frac{\xi}{k(1-e^{-\xi})^{k-1}}$.
\begin{itemize}
\item for $\alpha\leq \alpha_c$, all (except $o_p(n)$) vertices of type
$a$ are covered, i.e. $\frac{\nu(G_n)}{|V^a_n|} \xrightarrow[n\to\infty]{}1$.
\item for $\alpha>\alpha_c$,
we have:
\begin{eqnarray}
\label{eq:min}\frac{\nu(G_n)}{|V^a_n|}
\xrightarrow[n\to\infty]{}1-\frac{1}{\alpha}\left(e^{-\xi^*}+\xi^*e^{-\xi^*}+\frac{\xi^*}{k}(1-e^{-\xi^*})-1\right),
\end{eqnarray}
where $\xi^*= k\alpha x^*$ and $x^*$ is the largest solution of
$x=\left(1-e^{-k\alpha x}\right)^{k-1}$.
\end{itemize}
\end{corollary}
In words, $\alpha_c$ is the load threshold: if $\alpha\leq \alpha_c$,
there is an assignment of the $\lfloor \alpha m\rfloor$ items to a
table with $m$ locations that respects the choices of all items,
whereas for $\alpha>\alpha_c$, such an assignement does not exist and
(\ref{eq:min}) gives the maximal number of items assigned without
collision. Note that results in \cite{fopa09}, \cite{kxor} are
slightly different in the sense that for the specific sequence of
random graphs described above (i.e. uniform hypergraphs), they show that for $\alpha<
\alpha_c$ all vertices of type $a$ are covered with high probability.
It is shown in \cite{lel12} how to get such results from Corollary
\ref{cor:fr} under the additional assumption that the sequence of
graphs are uniform hypergraphs.
\section{The Monomer-Dimer model}
\label{sec:z}
We start with the case of a finite graph $G=(V,E)$.
Consider a natural family of probability
distributions on the set of matchings ${\mathbb M}(G)$,
parameterized by a single parameter $z>0$ called the
\textit{temperature} (note that the standard temperature $T$ in physics would correspond to $z=e^{-1/T}$ but this will not be important here): for any $M\in{\mathbb M}(G)$,
\begin{equation}
\label{eq:gibbs}
\mu^z_{G}(M)=\frac{z^{|V|-2|M|}}{P_G(z)},
\end{equation}
where $P_G$ is the \textit{matching polynomial},
$P_G(z)=\sum_{M\in{{\mathbb M}(G)}}z^{|V|-2|M|}$.
In statistical physics, this is called the \textit{monomer-dimer model} at temperature $z$ on $G$ (see \cite{heilmannlieb} for a complete treatment).
We let ${\mathcal M}^z_G$ denote a random element of ${\mathbb M}(G)$ with
law $\mu^z_G$, and we call it a \textit{Boltzmann random matching at
temperature $z$} on $G$.
Note that the lowest degree coefficient of $P_G$ is precisely the number of largest matchings on $G$. Therefore, $\mathcal M^z_G$ converges in law to a uniform largest matching as the temperature $z$ tends to zero.
We define the \textit{root-exposure probability (REP)} of the
rooted graph $(G,\circ)$ as
\begin{eqnarray}
\label{eq:mpe}{\mathcal R}_z{[G,\circ]} = \mu^z_{G}\left(\circ\textrm{ is
exposed}\right).
\end{eqnarray}
\subsection{Local recursions to compute $\gamma$}\label{subsec:algo}
Before starting with the proof, we explain (whithout proofs) how to
compute $\gamma$ in (\ref{lim:g}).
For a finite graph, our computation of $\gamma$ follows exactly the approach
of Godsil \cite{god81}.
We recall Godsil's notion of
the \textit{path-tree} associated with a rooted graph $G$:
if $G$ is any rooted graph with root $\circ$, we define
its path-tree $T_G$ as the rooted tree whose vertex-set consists of all finite simple paths starting at the
root $\circ$; whose edges are the pairs $\{P,P'\}$ of the
form $P=v_1\ldots v_n$, $P'=v_1\ldots v_nv_{n+1} (n\geq 1)$; whose root is the single-vertex path $\circ$.
By a \textit{finite simple path}, we mean here a finite sequence of
distinct vertices $v_1\ldots v_n$ ($n\geq 1$) such that $v_iv_{i+1}\in E$
for all $1\leq i < n$.
It is well-known since Godsil's result \cite{god81} that path-trees
capture considerable information about matchings in general graph and
are easier to work with than the graph itself.
For a rooted graph $[G,\circ]$, let $T_{[G,\circ]}$ be the associated
path-tree and consider the corresponding system of equations (where $u\succ
v$ if $u$ is a child of $v$):
\begin{equation}
\label{eq:fpzeroi}
\forall v\in T_{[G,\circ]}, \quad x_v=\frac {1}
{1+\sum_{u\succ v}{\left({\sum_{w\succ u}x_w}\right)^{-1}}}.
\end{equation}
For any finite rooted graph $[G,\circ]$, (\ref{eq:fpzeroi}) has a
unique solution in $[0,1]^{T_{[G,\circ]}}$ and we denote the value taken at the
root by $x_\circ(G)$. Then $x_\circ(G)$ is exactly the probability for
the root $\circ$ of being exposed in a uniform maximal matching. In
particular, we have
\begin{eqnarray*}
\nu(G) = \sum_{v\in V}\frac{1-x_v(G)}{2}.
\end{eqnarray*}
This argument follows from \cite{god81} and will be a special case of our
analysis
For an infinite graph with bounded degree, it turns out that it is not
always possible to make sense of the local recursions
(\ref{eq:fpzeroi}). However, our analysis will show that for any $z>0$, the
infinite set of equations:
\begin{equation*}
\forall v\in T_{[G,\circ]}, \quad x_v(z)=\frac {1}
{1+\sum_{u\succ v}{\left(z^2+{\sum_{w\succ u}x_w(z)}\right)^{-1}}},
\end{equation*}
has a unique solution in $[0,1]^{T_{[G,\circ]}}$ and the value taken
by the root is exactly ${\mathcal R}_z[G,\circ]$ (which is the probability for the root
$\circ$ of being exposed in a Boltzmann random matching at temperature
$z$ when the graph $G$ is finite). Then our Theorem \ref{co:cvMN} will
imply that for any sequence of
finite graphs $(G_n=(V_n,E_n))_{n\in{\mathbb N}}$ satisfying $|E_n|=O(|V_n|)$ and having $\rho$ as a random weak limit,
\begin{eqnarray*}
\frac{\nu(G_n)}{{|V_n|}}\xrightarrow[n\to\infty]{}\frac{1-{\mathbb E}_\rho\left[\lim_{z\to
0}{\mathcal R}_z\right]}{2},
\end{eqnarray*}
and $\lim_{z\to 0}{\mathcal R}_z[G,\circ]$ is actually the largest solution to (\ref{eq:fpzeroi}).
From a practical point of view, it is possible to compute an
approximation of ${\mathcal R}_z[G,\circ]$ by looking at a sufficient large
ball centered at the root $\circ$.
Moreover our analysis will show that the quantity
${\mathbb E}_\rho\left[{\mathcal R}_z\right]$ is a good approximation of ${\mathbb E}_\rho\left[\lim_{z\to
0}{\mathcal R}_z\right]$ as soon as $|E_n|=O(|V_n|)$ (see Lemma \ref{lm:unifctrl}).
\subsection{Extension of the model on infinite graphs with bounded degree}
Let $G-\circ$ be the graph obtained from $G$ by removing its root $\circ$.
Since the matchings of $G$ that expose $\circ$ are exactly the matchings of
$G-\circ$, we have the identity
\begin{equation}
{\mathcal R}_z{[G,\circ]}= \frac{z P_{G-\circ}(z)}{P_G(z)},
\end{equation}
which already shows that the REP is an analytic function of the
temperature. The remarkable fact that its domain of analyticity contains the
right complex half-plane $${\mathbb H}_+=\{z\in{\mathbb C};\Re(z)>0\}$$
is a consequence of the powerful Heilmann-Lieb theorem \cite[Theorem 4.2]{heilmannlieb}
(see \cite{sokalwagner} for generalizations).
The key to the study of the REP is the following elementary but fundamental local recursion :
\begin{equation}
\label{eq:recg}
{\mathcal R}_z{[G,\circ]}=z^2\left(z^2+\sum_{v\sim \circ}{\mathcal R}_z{[G-\circ,v]}\right)^{-1}.
\end{equation}
Clearly, this recursion determines uniquely the functional ${\mathcal R}_z$ on the
class of finite rooted graphs, and may thus be viewed as an inductive
definition of the REP. Remarkably enough, this alternative characterization
allows for a continuous extension to infinite graphs with bounded degree, even though the above recursion never ends. We let ${\mathcal H}$ denote the space of analytic functions on ${\mathbb H}_+$, equipped with its usual topology of uniform
convergence on compact sets. Our fundamental lemma is as follows :
\begin{theorem}[The fundamental local lemma]\mbox{}
\label{th:mainb}
\begin{enumerate}
\item For every fixed $z\in{\mathbb H}_+$, the local recursion (\ref{eq:recg})
determines a unique ${\mathcal R}_z\colon {{\mathcal G}_d} \to z{\mathbb H}_+$.
\item For every fixed $[G,\circ]\in{{\mathcal G}_d}$, $z\mapsto {\mathcal R}_{z}[G,\circ]$ is
analytic.
\item The resulting mapping $[G,\circ] \in{{\mathcal G}_d}\longmapsto
{\mathcal R}_{(\cdot)}[G,\circ]\in{\mathcal H}$ is continuous.
\end{enumerate}
\end{theorem}
This local lemma has strong implications for the monomer-dimer model, which
we now list. The first one is the existence of an infinite volume limit for
the Gibbs-Boltzmann distribution.
\begin{theorem}[Monomer-dimer model on infinite graphs]
\label{co:cvweak}
Consider a graph $G\in{{\mathcal G}_d}$ and a temperature $z>0$. For any finite matching $M$ of $G$, the cylinder-event marginals defined by
\begin{equation*}
\mu_G^z(M\subseteq {\mathcal M}) = z^{-2|M|}\prod_{k=1}^{2|M|}{\mathcal R}_z{[G - \{v_1,\ldots,v_{k-1}\},v_k]},
\end{equation*}
are consistent and independent of the ordering $v_1,\ldots v_{2|M|}$ of the
vertices spanned by $M$. They thus determine a unique probability
distribution $\mu_G^z$ over the matchings of $G$. It coincides with the former definition in the case where $G$ is
finite, and extends it continuously in the following sense : for any $\circ\in V$ and any sequence $([G_n,\circ_n])_{n\in{\mathbb N}}\in{{\mathcal G}_d}^{\mathbb N}$ converging to $[G,\circ]$,
$$[G_n,\circ_n,{\mathcal M}_n]\xrightarrow[n\to\infty]{d}[G,\circ,{\mathcal M}],$$
in the local weak sense for random networks, where ${\mathcal M}_n$ has law $\mu^z_{G_n}$ and ${\mathcal M}$ has law $\mu_G^z$.
\end{theorem}
Although it is not our concern here, we obtain as a by-product the strong convergence of the logarithm of the matching polynomial, also called \textit{free energy} in the monomer-dimer model :
\begin{corollary}
\label{co:fe}
Let $(G_n)_{n\in{\mathbb N}}$ be a sequence of finite graphs with bounded degree admitting a random weak limit $[G,\circ]$. The following convergence holds in the analytic sense on ${\mathbb H}_+$,
\begin{equation*}
\frac{1}{|V_n|}\log \frac{P_{G_n}(z)}{P_{G_n}(1)}\xrightarrow[n\to\infty]{}\int_1^{z}\frac{{\mathbb E}_{\rho}[\mathcal{R}_s[G,\circ]]}{s}ds,
\end{equation*}
where ${\mathbb E}_\rho[\mathcal{R}_s[G,\circ]]$ denotes the expectation under the measure $\rho$ of the variable $\mathcal{R}_s[G,\circ]$.
\end{corollary}
A similar result was established in \cite{heilmannlieb} for the lattice case, and in \cite{bayatinair} under a restrictive large girth assumption.
\subsection{Proof of Theorem \ref{th:mainb} : the fundamental lemma}
\label{sec:finite}
The local recursion (\ref{eq:recg}) involves mappings of the form :
$$\phi_{z,d}\colon \left(x_1,\ldots,x_d\right) \mapsto z^2\left(z^2+\sum_{i=1}^d
x_i\right)^{-1},$$
where $d\in{\mathbb N}$. In the following lemma, we gather a few elementary
properties of this transformation, which are immediate to check but will be of constant use throughout
the paper.
\begin{lemma}[Elementary properties]
\label{lm:propphi}
For any $d\in{\mathbb N}$ and $z\in{\mathbb H}_+$,
\begin{enumerate}
\item $\phi_{z,d}$ maps analytically $z{\mathbb H}_+\times\ldots\times z{\mathbb H}_+$ into $z{\mathbb H}_+$
\item $|\phi_{z,d}|$ is uniformly bounded by $|z|/\Re(z)$ on
$z{\mathbb H}_+\times\ldots\times z{\mathbb H}_+$.
\end{enumerate}
\end{lemma}
From part $1$, it follows that the
REP of a finite rooted graph belongs to ${\mathcal H}$, when viewed as a function of the temperature $z$. Part $2$ and Montel's theorem guarantee that the family of all those REPs
is relatively compact in ${\mathcal H}$. Note that relative compactness also plays a central role in \cite{heilmannlieb}. Combined with the following
uniqueness property at high temperature, it will quickly lead to the proof of
Theorem \ref{th:mainb}.
The local recursion (\ref{eq:recg}) also involves graph
transformations of the form $[G,\circ]\mapsto[G-\circ,v]$, where
$v\sim\circ$. Starting from a given $[G,\circ]\in{{\mathcal G}_d}$, we let
$\rm{Succ}^*[G,\circ]\subseteq{{\mathcal G}_d}$ denote the (denumerable) set of all
rooted graphs that can be obtained by successively applying finitely many
such transformations.
\begin{lemma}[Uniqueness at high temperature]
\label{lm:uniqueness}
Let $[G,\circ]\in{{\mathcal G}_d}$ and $z\in {\mathbb H}_+$ such that $\Re(z)> \sqrt{ d}$. If
$${\mathcal R}^1_z,{\mathcal R}_z^2\colon \rm{Succ}^*[G,\circ]\to z{\mathbb H}_+$$ both satisfy the local recursion
(\ref{eq:recg}) then ${\mathcal R}^1_z={\mathcal R}^2_z$.
\end{lemma}
\bp{}Set $\alpha=2{|z|}/{\Re(z)}$ and $\beta=\Re(z)^{-2}$. From
(\ref{eq:recg}) and part 2 of Lemma \ref{lm:propphi} it is clear that the absolute difference
$\Delta=|{\mathcal R}^1_z-{\mathcal R}^2_z|$ must satisfy
$$\Delta[G,\circ]\leq \alpha\qquad\textrm{ and }\qquad\Delta[G,\circ]\leq \beta\sum_{v\sim\circ}\Delta[G-\circ,v].$$
In turn, each $\Delta[G-\circ,v]$ appearing in the second upper-bound may
be further expanded into
$\beta\sum_{w\sim v,w\neq\circ}\Delta[G-\circ-v,w]$. Iterating this procedure $k$ times, one obtains
$\Delta[G,\circ]\leq \beta^{k}d^k\alpha$.
Taking the infimum over all $k$
yields $\Delta[G,\circ]=0$, since the assumption $\Re (z ) > \sqrt d$ means precisely $\beta d<1$.
\hfill $\Box$
\bp{ of Theorem \ref{th:mainb}} For clarity, we divide the proof in three parts : we first define a specific solution which satisfies (\ref{eq:recg}). We will then prove its unicity and check its continuity. This will prove parts 1-3 of Theorem \ref{th:mainb}.
\textbf{Analytic existence. }
Fix $[G,\circ]\in{{\mathcal G}_d}$, and consider an
arbitrary collection of ${\mathbb H}_+\to z{\mathbb H}_+$ analytic functions
$z\mapsto {\mathcal R}^0_z[H,i]$, indexed by the elements $[H,i]\in\rm{Succ}^*[G,\circ]$. For every $n\geq 1$, define recursively
\begin{equation}
\label{eq:def}
{\mathcal R}^{n}_z[H,i]=z^2\left(z^2+\sum_{j\sim i}{\mathcal R}^{n-1}_{z}[H-i,j]\right)^{-1},
\end{equation}
for all $z\in{\mathbb H}_+$ and $[H,i]\in\rm{Succ}^*[G,\circ]$. By Lemma
\ref{lm:propphi}, each sequence $\left(z\mapsto {\mathcal R}^n_{z}[H,i]\right)_{n\in{\mathbb N}}$
is relatively compact in ${\mathcal H}$. Consequently, their joint collection
as $[H,i]$ varies in the denumerable set $\rm{Succ^*}[G,\circ]$ is
relatively compact in the product space
${\mathcal H}^{\rm{Succ^*}[G,\circ]}$. Passing to the limit in (\ref{eq:def}), we
see that any pre-limit ${\mathcal R}_z\colon\rm{Succ^*}[G,\circ]\to z{\mathbb H}_+$ must automatically satisfy (\ref{eq:recg})
for each $z\in{\mathbb H}_+$. By Lemma \ref{lm:uniqueness}, this determines uniquely the
value of ${\mathcal R}_z[G,\circ]$ for $z$ with sufficiently large real part, and hence
everywhere in ${\mathbb H}_+$ by analyticity. To sum up, we have just proved the following : for every $[G,\circ]\in{{\mathcal G}_d}$,
the limit
\begin{equation}
\label{eq:recsol}
{\mathcal R}_z{[G,\circ]}:=\lim_{n\to\infty}{\mathcal R}^n_z{[G,\circ]}
\end{equation}
exists in ${\mathcal H}$, satisfies the recursion (\ref{eq:recg}), and does not
depend upon the choice of the initial condition ${\mathcal R}^0_z\colon
\rm{Succ^*}[G,\circ]\to z{\mathbb H}_+$ (provided that the latter is analytic
in $z\in{\mathbb H}_+$).
\textbf{Pointwise uniqueness. }Let us now show that any
${\mathcal S}\colon\rm{Succ^*}[G,\circ]\to z{\mathbb H}_+$ satisfying the recursion (\ref{eq:recg}) at a fixed value $z=z_0\in{\mathbb H}_+$ must coincide with the
$z=z_0$ specialization of the analytic solution constructed above. For
each $[H,i]\in\rm{Succ^*}[G,\circ]$, the constant initial function ${\mathcal R}^{0}_{z}[H,i]:={\mathcal S}[H,i]$ is trivially analytic from ${\mathbb H}_+$ to $z{\mathbb H}_+$,
so the iteration (\ref{eq:def}) must converge to the analytic solution
${\mathcal R}_z$. Since ${\mathcal R}^n_{z_0}={\mathcal S}$ for all $n\in{\mathbb N}$, we obtain ${\mathcal R}_{z_0}=
{\mathcal S}$, as desired.
\textbf{Continuity. }Finally, assume that $\left([G_n,\circ]\right)_{n\geq
1}\in{{\mathcal G}_d}^{\mathbb N}$ converges locally to $[G,\circ]$, and let us show that
\begin{equation}
\label{eq:continuity}
{\mathcal R}_z{[G_n,\circ]}\xrightarrow[n\to\infty]{{\mathcal H}}{\mathcal R}_z[G,\circ].
\end{equation}
It is routine that,
up to rooted isomorphisms, $G,G_1,G_2,\ldots$ may be represented on a common vertex set, in such a
way that for each fixed $k\in{\mathbb N}$, $[G_n,\circ]_k=[G,\circ]_k$ for all $n\geq n_k$. By construction, any
simple path $v_1\ldots v_{k}$
starting from the root in $G$ is now also a simple path starting from the root
in each $G_n,n\geq n_k$, so the ${\mathcal H}-$valued sequence
$\left(z\mapsto {\mathcal R}_{z}[G_n-\{v_1,\ldots,v_{k-1}\},v_k]\right)_{n\geq n_k}$ is
well defined, and relatively compact (Lemma \ref{lm:propphi}). Again, the denumerable
collection of all sequences obtained by letting the simple path
$v_1 \ldots v_{k}$ vary in $[G,\circ]$ is relatively compact for the product topology,
and any pre-limit must by construction satisfy (\ref{eq:recg}). By
pointwise uniqueness, the convergence (\ref{eq:continuity}) must hold. \hfill $\Box$
\subsection{Proof of Theorem \ref{co:cvweak} : convergence of the Boltzmann distribution}
\label{sec:cvweak}
Consider an infinite $[G,\circ]\in{{\mathcal G}_d}$, and let
$\left([G_n,\circ]\right)_{n\geq 1}$ be a sequence of finite rooted
connected graphs converging locally to $[G,\circ]$. As above, represent $G,G_1,G_2,\ldots$ on a common vertex set, in such a
way that for each $k\in{\mathbb N}$, $[G_n,\circ]_k=[G,\circ]_k$ for all $n\geq
n_k$.
Now fix an arbitrary finite matching $M$ in $G$, and denote by
$v_1,\ldots,v_{2|M|}$ the vertices spanned by $M$, in any order. By
construction, $M$ is also a matching of $G_n$ for large enough $n$. But the matchings of $G_n$ that contain $M$ are in $1-1$ correspondence with the matchings of $G_n -
\{v_1,\ldots,v_{2|M|}\}$, and hence
$$\mu^z_{G_n}\left(M\subseteq{\mathcal M}\right)=\frac{P_{G_n-\{v_1,\ldots,v_{2|M|}\}}(z)}{P_{G_n}(z)}=z^{-2M}\prod_{k=1}^{2M}{\mathcal R}_z[G_n-\{v_1,\ldots,v_{k-1}\},v_k].$$
But $[G_n-\{v_1,\ldots,v_{k-1}\},v_k]$ converges locally to
$[G-\{v_1,\ldots,v_{k-1}\},v_k]$, so by continuity of ${\mathcal R}_z$,
$$\mu^z_{G_n}\left(M\subseteq{\mathcal M}\right)\xrightarrow[n\to\infty]{}z^{-2M}\prod_{k=1}^{2M}{\mathcal R}_z[G-\{v_1,\ldots,v_{k-1}\},v_k].$$
\bp{ of Corollary \ref{co:fe}}
Analytic convergence of the free energy follows from Theorem \ref{co:cvweak} and Lebesgue dominated convergence Theorem, since for any finite graph $G=(V,E)$ we have
$$(\log P_G)'(z) = \frac{P_G'(z)}{P_G(z)} = \frac{1}{|V|} \sum_{\circ \in V}\frac{{\mathcal R}_z[G,\circ]}{z} = \frac{\rho[{\mathcal R}_z[G,\circ]]}{z}.$$
The uniform domination $\left|\frac{\rho[{\mathcal R}_z[G,\circ]]}{z}\right|\leq \frac{1}{\Re(z)}$ is provided by Lemma \ref{lm:propphi}.
\hfill $\Box$
\section{The zero-temperature limit}\label{sec:zero}
Motivated by the asymptotic study of maximum matchings, we now let the
temperature $z\to 0$.
\subsection{The case of bounded degree}
We first use the results from the previous section to prove a version of Theorem \ref{th:main} for graphs with bounded degree.
\begin{theorem}[The zero temperature limit in graphs with bounded degree]
\label{co:monotony}
For any $[G,\circ]\in{{\mathcal G}_d}$, the zero temperature limit
$${\mathcal R}_*[G,\circ]=\lim_{z\to 0}\downarrow {\mathcal R}_z[G,\circ]$$
exists. Moreover, ${\mathcal R}_*\colon {{\mathcal G}_d} \to [0,1]$
is the largest solution to the recursion
\begin{equation}
\label{eq:recgi}
{\mathcal R}_*[G,\circ] = \left(1+\sum_{v\sim\circ}\left(\sum_{w\sim
v}{\mathcal R}_*[G-\circ-v,w]\right)^{-1}\right)^{-1},
\end{equation}
with the conventions $0^{-1}=\infty$, $\infty^{-1}=0$. When $G$ is
finite, ${\mathcal R}_*[G,\circ]$ is the probability that $\circ$ is exposed
in a uniform maximum matching.
\end{theorem}
\bp{}
Fix $[G,\circ]\in{{\mathcal G}_d}$. First, we claim that $z\mapsto {\mathcal R}_z[G,\circ]$ is
non-decreasing on ${\mathbb R}_+$. Indeed, this is obvious if $G$ is reduced to
$\circ$, since in that case the REP
is simply $1$. It then inductively extends to any finite graph $[G,\circ]$,
because iterating twice (\ref{eq:recg}) gives
\begin{equation}
\label{eq:recgtwice}
{\mathcal R}_z[G,\circ] = \left(1+\sum_{v\sim\circ}\left(z^2+\sum_{w\sim
v}{\mathcal R}_z[G-\circ-v,w]\right)^{-1}\right)^{-1}.
\end{equation}
For the infinite case, $[G,\circ]$ is the local limit of the sequence
of finite truncations $\left([G,\circ]_n\right)_{n\in{\mathbb N}}$, where, for $n \geq 1$, $[G,\circ]_n$ denotes the finite rooted subgraph induced by the vertices lying at graph-distance at
most $n$ from $\circ$. So by
continuity of the REP, ${\mathcal R}_z[G,\circ]=\lim_{n\to\infty}{\mathcal R}_z[G,\circ]_n$
must be non-decreasing in $z$ as well.
This guarantees the existence of the $[0,1]-$valued limit
$${\mathcal R}_*[G,\circ]=\lim_{z\to 0}\downarrow{\mathcal R}_z[G,\circ].$$
Moreover, taking the $z\to 0$ limit in (\ref{eq:recgtwice}) guarantees
the recursive formula (\ref{eq:recgi}).
Finally, consider
${\mathcal S}_*\colon\rm{Succ}^*[G,\circ]\to[0,1]$ satisfying the recursion
(\ref{eq:recgi}). Let us show by induction over $n\in{\mathbb N}$ that for every
$[H,i]\in\rm{Succ}^*[G,\circ]$ and $z>0$,
\begin{equation}
\label{eq:compare}
{\mathcal S}_*[H,i]\leq {\mathcal R}_z[H,i]_{2n}.
\end{equation}
The statement is trivial when $n=0$ (${\mathcal R}_z[H,i]_0=1$), and is preserved from $n$ to $n+1$
because
\begin{eqnarray*}
{\mathcal R}_z[H,i]_{2n+2} & = & \left(1+\sum_{j\sim i}\left(z^{2}+\sum_{k\sim
j}{\mathcal R}_z[H-i-j,k]_{2n}\right)^{-1}\right)^{-1} \\
& \geq & \left(1+\sum_{j\sim
i}\left(\sum_{k\sim j}{\mathcal S}_*[H-i-j,k]\right)^{-1}\right)^{-1} =\, {\mathcal S}_*[H-i,j].
\end{eqnarray*}
Letting $n\to\infty$ and then $z\to 0$ in (\ref{eq:compare}) yields ${\mathcal S}_*\leq {\mathcal R}_*$, which completes the proof
\hfill $\Box$
This naturally raises the following question : may the
\textit{zero temperature limit} be interchanged with the \textit{infinite
volume limit}, as suggested by the diagram below ?
\begin{figure}[h!]
$$\xymatrix @!0 @R=2.5cm @C=4cm{
{\mathcal R}_z[G_n,\circ_n]\ar[d]_{z\to 0 } \ar[r]^{n\to\infty}
& {\mathcal R}_z[G,\circ]\ar[d]^{z\to 0} \\
{\mathcal R}_*[G_n,\circ_n] \ar@{.>}[r]_{n\to\infty} & {\mathcal R}_*[G,\circ]
}$$
\label{fig:proof}
\end{figure}
Unfortunately, the recursion (\ref{eq:recgi}) may admit several distinct
solutions, and this translates as follows : in the limit of zero temperature, \textit{correlation decay breaks for the monomer-dimer model}, in the precise sense that
the functional ${\mathcal R}_*\colon{{\mathcal G}_d}\to[0,1]$ is no longer continuous with respect to local convergence. For example, one can easily construct an infinite rooted tree $[T,\circ]$ with
bounded degree such that
\begin{eqnarray*}
\lim_{n\to\infty}\downarrow {\mathcal R}_*[T,\circ]_{2n} & \neq &
\lim_{n\to\infty}\uparrow {\mathcal R}_*[T,\circ]_{2n+1}.
\end{eqnarray*}
Indeed, consider the case of $T$ being the graph on ${\mathbb N}$ rooted at $0 = \circ$, where two integers share an edge if they differ by $1$. Then, a straightforward computation gives ${\mathcal R}_*[T,\circ]_{2n} =1/2$ while ${\mathcal R}_*[T,\circ]_{2n+1} = 0$.
Despite this lack of correlation decay,
the interchange of limits turns out to be valid ``on
average'', i.e. when looking at a uniformly chosen vertex
$\circ$.
\begin{theorem}[The limiting matching number of bounded-degree graph sequences]
\label{co:cvMN}
Let $\rho$ be a probability distribution over ${{\mathcal G}_d}$. For any sequence of
finite graphs $(G_n=(V_n,E_n))_{n\in{\mathbb N}}$ satisfying $|E_n|=O(|V_n|)$ and having $\rho$ as a random weak limit,
\begin{eqnarray*}
\frac{\nu(G_n)}{{|V_n|}}\xrightarrow[n\to\infty]{}\frac{1-{\mathbb E}_\rho\left[{\mathcal R}_*\right]}{2}.
\end{eqnarray*}
\end{theorem}
In order to get our Theorem \ref{th:main}, we need to remove the bounded degree assumption. This is done below.
In the case where the limit $\rho$ is a (two-level) Galton-Watson tree, the recursion
(\ref{eq:recgi}) simplifies into a \textit{recursive distributional
equation} (RDE). The computations for these cases are done in Section \ref{sec:rdet}.
\bp{ of Theorem \ref{co:cvMN}}
Let $G=(V,E)$ be a finite graph and $M$ be any maximal matching of $G$. Then
$$
\sum_{ v \in V} {\mathbf{1}} (v \hbox{ is exposed in $M$}) = |V| - 2 \sum_{ e \in E} {\mathbf{1}} (e \in M).
$$
In particular, if $\rho={\mathcal U}(G)$, we have the
elementary identity
\begin{equation}\label{eq:R*nu}
{\mathbb E}_\rho\left[{\mathcal R}_*\right] = 1-\frac{2\nu(G)}{|V|}.
\end{equation} The proof of Theorem \ref{co:cvMN} will easily follow from the following
uniform control:
\begin{lemma}[Uniform continuity around the zero-temperature point]
\label{lm:unifctrl}
Let $G=(V,E)$ be a finite graph. For any $0<z<1$,
\begin{eqnarray}
\label{eq:lemma}
{\mathbb E}_\rho\left[{\mathcal R}_z\right] + \frac{|E|}{|V|}\frac{\log 2}{\log z} \leq
{\mathbb E}_\rho\left[{\mathcal R}_*\right] \leq {\mathbb E}_\rho\left[{\mathcal R}_z\right].
\end{eqnarray}
\end{lemma}
Indeed, let $\rho$ be a probability distribution on ${{\mathcal G}_d}$, and let
$(G_n=(V_n,E_n))_{n\in{\mathbb N}}$ be a sequence of finite graphs with
$|V_n|=O(|E_n|)$, whose random weak limit is $\rho$. For each $n\in{\mathbb N}$, set
$\rho_n={\mathcal U}(G_n)$. With these notations, proving Theorem \ref{co:cvMN}
amounts to establish :
\begin{equation}
\label{eq:cvrepi}
{\mathbb E}_{\rho_n}\left[{\mathcal R}_*\right]\xrightarrow[n\to\infty]{}{\mathbb E}_\rho\left[{\mathcal R}_*\right].
\end{equation}
However, since $\rho_n\Longrightarrow \rho$, and since each ${\mathcal R}_z,z>0$ is continuous and
bounded, we have for every $z>0$,
$${\mathbb E}_{\rho_n}[{\mathcal R}_z] \xrightarrow[n\to\infty]{} {\mathbb E}_\rho[{\mathcal R}_z].$$
Thus, setting $C=\sup_{n\in{\mathbb N}}\frac{|E_n|}{|V_n|}$ and letting $n\to\infty$ in (\ref{eq:lemma}), we see that for any $z<1$,
\begin{eqnarray*}
{\mathbb E}_\rho\left[{\mathcal R}_z\right] + C\frac{\log 2}{\log z} \leq \liminf_{n\to\infty}{\mathbb E}_{\rho_n}\left[{\mathcal R}_*\right] \leq \limsup_{n\to\infty}{\mathbb E}_{\rho_n}\left[{\mathcal R}_*\right] \leq {\mathbb E}_{\rho}\left[{\mathcal R}_z\right].
\end{eqnarray*}
Letting finally $z\to 0$, we obtain exactly (\ref{eq:cvrepi}), and it
only remains to show Lemma \ref{lm:unifctrl}.
\bp{ of Lemma \ref{lm:unifctrl}} Fix $0<z<1$. Since $z\mapsto
{\mathbb E}_\rho\left[{\mathcal R}_z\right]$ is non-decreasing, we have
$$ {\mathbb E}_\rho\left[{\mathcal R}_*\right] \leq {\mathbb E}_\rho\left[{\mathcal R}_z\right] \leq \frac{-1}{\log z}\int_{z}^1s^{-1} {\mathbb E}_\rho\left[{\mathcal R}_s\right]ds.$$
Use ${\mathbb E}_\rho\left[{\mathcal R}_s\right]= \frac{s P'_G(s)}{|V| P_G(s)}$ to rewrite this as
$${\mathbb E}_\rho\left[{\mathcal R}_*\right] \leq {\mathbb E}_\rho\left[{\mathcal R}_z\right] \leq \frac{1}{|V|\log z}\log
\frac{P_G(z)}{P_G(1)}.$$
Now, $P_G(1)$ is the total number of matchings and is thus clearly at
most $2^{|E|}$, while $P_G(z)$ is at least $z^{|V|-2\nu(G)}$. Using \eqref{eq:R*nu}, these two bounds yield to
$$
{\mathbb E}_\rho\left[{\mathcal R}_*\right] \leq {\mathbb E}_\rho\left[{\mathcal R}_z\right] \leq \frac{1}{|V|\log z} \left( |V|{\mathbb E}_\rho\left[{\mathcal R}_*\right] \log z - |E| \log 2 \right).
$$
This gives (\ref{eq:lemma}). \hfill $\Box$
\subsection{The case of unbounded degree}
\label{sec:deg}
In this section, we establish Theorem \ref{th:main} in full generality, removing the restriction of bounded degree from Theorem \ref{co:cvMN}. To this end, we introduce the $d-$\textit{truncation} $G^d$ ($d\in {\mathbb N}$) of a graph $G=(V,E)$, obtained from $G$ by \textit{isolating} all vertices with degree more than $d$, i.e. removing any edge incident to them. This transformation is clearly continuous with respect to local convergence. Moreover, its effect on the matching number can be easily controlled :
\begin{equation}
\label{eq:trunc}
\nu(G^d)\leq \nu(G)\leq \nu(G^d)+\#\{v\in V;{\mathrm{deg}}_G(v)>d\}.
\end{equation}
Now, consider a sequence of finite graphs $(G_n)_{n\in{\mathbb N}}$ admitting a random weak limit $(G,\circ)$.
First, fixing $d\in{\mathbb N}$, we may apply Theorem \ref{co:cvMN} to the sequence $(G^d_n)_{n\in{\mathbb N}}$ to obtain :
\begin{equation*}
\frac{\nu(G^d_n)}{|V_n|}\xrightarrow[n\to\infty]{}\frac{1-{\mathbb E}_{\rho_d}\left[{\mathcal R}_*\right]}2,
\end{equation*}
where $\rho_d$ is the $d$-truncation of $\rho$.
Second, we may rewrite (\ref{eq:trunc}) as
\begin{equation*}
\left|\frac{\nu(G^d_n)}{|V_n|}-\frac{\nu(G_n)}{|V_n|}\right|\leq \frac{\#\{v\in V_n;{\mathrm{deg}}_{G_n}(v)>d\}}{|V_n|}.
\end{equation*}
Letting $n\to\infty$, we obtain
\begin{equation*}
\limsup_{n\to\infty}\left|\frac{1-\rho_d\left[{\mathcal R}_*\right]}2-\frac{\nu(G_n)}{|V_n|}\right|\leq \rho\left({\mathrm{deg}}(\circ)>d\right),
\end{equation*}
This last line is, by an elementary application of Cauchy criterion, enough to guarantee the convergence promised by Theorem \ref{th:main}, i.e.
\begin{equation}
\label{eq:toshow}
\frac{\nu(G_n)}{|V_n|}\xrightarrow[n\to\infty]{}\gamma,\qquad\textrm{ where }\qquad\gamma:=\lim_{d\to\infty}\frac{1-{\mathbb E}_{\rho_d}\left[{\mathcal R}_*\right]}2.
\end{equation}
Note that because of the possible absence of correlation decay, the largest solution ${\mathcal R}_*[G,\circ]$ is not a continuous function of $(G,\circ)\in{\mathcal G}$. In particular, we do not know whether it is always the case that
\begin{equation}
\label{eq:question}
\gamma=\frac{1-{\mathbb E}_\rho[{\mathcal R}_*]}2,
\end{equation} as established in Theorem \ref{co:cvMN} for graphs with bounded degree. However, (\ref{eq:question}) holds in the particular cases where we have an explicit formula for ${\mathbb E}_\rho[{\mathcal R}_*]$ which depends continuously upon the degree distribution as will be the case in Section \ref{sec:rdet}.
\section{Computations on (hierarchal) Galton-Watson trees}\label{sec:rdet}
\subsection{The case of Galton-Watson trees}
\label{sec:gwt}
We now investigate the special case where the limiting random graph is
a UGW tree $T$. Specifically, we fix a distribution
$\pi\in{\mathcal P}({\mathbb N})$ with finite support (we will relax this
assumption in the sequel) and we consider
a UGW tree $T$ with degree
distribution $\pi$ as defined in Section \ref{sec:res}.
The random matchings ${\mathcal M}^z_T, z\geq 0$ are perfectly
well-defined, and all the previously established results for graphs with bounded degree hold almost
surely. However, the self-similar
recursive structure of $T$ gives to the fixed-point characterizations
(\ref{eq:recg}) and (\ref{eq:recgi}) a very special form that is worth
making explicit.
Before we
start, let us insist on the fact that ${\mathcal R}_z[T] $ ($z >0$) is random : it
is the quenched probability that the root is exposed at temperature $z$,
given the random tree $T$. In light of Theorem \ref{th:main}, it becomes important to ask for its distribution.
Let ${\mathcal P}\left([0,1]\right)$ denote the space of Borel probability measures on $[0,1]$. Given $z>0$, $\nu\in{\mathcal P}\left({\mathbb N}\right)$ and $\mu\in{\mathcal P}\left([0,1]\right)$, we denote by $\Theta_{\nu,z}(\mu)$ the law of the $[0,1]-$valued r.v.
\begin{equation*}
Y{=}\frac {z^2}
{z^2+\sum_{i=1}^{{\mathcal{N}}}{X}_i},
\end{equation*}
where $\mathcal{N}\sim \nu$ and ${X}_1,{X}_2,\ldots \sim \mu$, all of them being
independent. This defines an operator $\Theta_{\nu,z}$ on
${\mathcal P}\left([0,1]\right)$. The corresponding fixed point equation
$\mu=\Theta_{\nu,z}(\mu)$ belongs to the general class of \textit{recursive distributional equations}, or RDE.
Equivalently, it can be rewritten as
\begin{equation*}
\label{rde}
X\stackrel{d}{=}\frac {z^2}
{z^2+\sum_{i=1}^{{\mathcal{N}}}{X}_i},
\end{equation*}
where ${X}_1,{X}_2,\ldots$ are i.i.d. copies of the unknown random variable ${X}$. Note that the same RDE appears in the analysis of the spectrum and rank of adjacency matrices of random graphs \cite{res10}, \cite{rankarxiv}.
With this notations in hands, the infinite system of equations
(\ref{eq:recg}) defining ${\mathcal R}_z[T]$ clearly leads to the following
distributional characterization:
\begin{lemma}
For any $z>0$, ${\mathcal R}_z[T]$ has distribution $\Theta_{\pi,z}(\mu_z)$,
where $\mu_z$ is solution to the RDE $\mu_z=\Theta_{\widehat{\pi},z}(\mu_z)$.
\end{lemma}
The same program can be carried out in the zero temperature limit. Specifically, given $\nu,{\nu'}\in{\mathcal P}({\mathbb N})$ and $\mu\in{\mathcal P}\left([0,1]\right)$, we
define $\Theta_{\nu,{\nu'}}(\mu)$ as the law of the $[0,1]-$valued r.v.
\begin{equation}
\label{eq:operator}
Y=\frac{1}{1+\sum_{i=1}^{\mathcal{N}}\left(\sum_{j=1}^{{\mathcal{N}_i}'}X_{ij}\right)^{-1}},
\end{equation}
where ${\mathcal{N}}\sim\nu$, ${\mathcal{N}_i}'\sim{\nu'}$, and ${X}_{ij}\sim\mu$, all of
them being independent. This defines an operator $\Theta_{\nu,{\nu'}}$ on
${\mathcal P}\left([0,1]\right)$ whose fixed points will play a crucial role in our
study.
Then, Theorem \ref{co:monotony} implies:
\begin{lemma}
\label{th:mtzero}
The random variable ${\mathcal R}_*[T]$ has law
$\Theta_{\pi,\widehat{\pi}}(\mu_*)$, where $\mu_*$ is the largest
solution to the RDE $\mu_*=\Theta_{\widehat{\pi},\widehat{\pi}}(\mu_*)$.
\end{lemma}
Recall that the mean of $\Theta_{\pi,\widehat{\pi}}(\mu_*)$
gives precisely the asymptotic size of a maximum matching for
any sequence of finite random graphs whose random weak limit is
$T$ (Theorem \ref{co:cvMN}). We will solve this RDE in the next section in the more general set-up of UHGW trees. Combined with Theorem \ref{th:main} and a simple continuity argument to remove the bounded degree assumption, this will prove Theorem \ref{th:KS}.
\subsection{The case of hierarchal Galton-Watson trees}
\label{sec:tlgwt}
As in previous section, we first assume that both $\pi^a$ and $\pi^b$
have a finite support.
We can define a RDE but with some care
about the types $a$ and $b$. The corresponding results read as
follows:
\begin{lemma}
\label{lem:RDEbi}
For any $z>0$, conditionally on the root being of type $b$
(resp. $a$), ${\mathcal R}_z[T]$ has distribution $\Theta_{\pi^b,z}(\mu^a_z)$
(resp. $\Theta_{\pi^b,z}(\mu^b_z)$),
where $\mu^a_z$ is solution to the RDE:
$$\mu^a_z=\Theta_{\widehat{\pi}^a,z}\circ
\Theta_{\widehat{\pi}^b,z}(\mu^a_z),$$
and $\mu^b_z=\Theta_{\widehat{\pi}^b,z}(\mu^a_z)$.
For $z=0$: conditionally on the root being of type $a$
(resp. $b$), the random variable ${\mathcal R}_*[T]$ has law
$\Theta_{\pi^a,\widehat{\pi}^b}(\mu_*^a)$ (resp. $\Theta_{\pi^b,\widehat{\pi}^a}(\mu_*^b)$), where $\mu^a_*$ is the largest solution to the RDE
\begin{equation}
\label{eq:rdezeroa}
\mu^a_*=\Theta_{\widehat{\pi}^a,\widehat{\pi}^b}(\mu^a_*),
\end{equation}
and $\mu^b_*$ is the largest solution to the RDE
$\mu^b_*=\Theta_{\widehat{\pi}^b,\widehat{\pi}^a}(\mu^b_*)$.
\end{lemma}
We now analyze the RDE (\ref{eq:rdezeroa}).
We define:
\begin{eqnarray}
\label{eq:defa}F^a(x) &=& \phi^a\left(
1-\widehat{\phi}^b(1-x)\right)-\frac{\phi^a{}'(1)}{\phi^b{}'(1)}\left(1-\phi^b(1-x)-x\phi^b{}'(1-x).
\right)
\end{eqnarray}
Observe that
\begin{eqnarray*}
F^a{}'(x) = \frac{\phi^a{}'(1)}{\phi^b{}'(1)}\phi^b{}''(1-x)\left(\widehat{\phi}^a(1-\widehat{\phi}^b(1-x)) -x\right).
\end{eqnarray*}
Hence any $x$ where $F^a$ admits a local maximum must satisfy
$x=\widehat{\phi}^a(1-\widehat{\phi}^b(1-x))$. We define the historical records of $F^a$
as the set of $x \in [0,1]$ such that $x=\widehat{\phi}^a(1-\widehat{\phi}^b(1-x))$ and for any $0 \leq y < x$, $F^a (x) > F^a (y)$ (the latter condition being empty if $x = 0$).
\begin{theorem}
\label{th:rdea}
If $p_1<\ldots<p_r$ are the locations of the historical records of $F^a$, then the RDE (\ref{eq:rdezeroa}) admits exactly $r$ solutions ; moreover, these solutions can be stochastically ordered, say $\mu_{1}{<}\ldots{<}\mu_{r}$, and for any $i\in\{1,\ldots,r\}$,
\begin{itemize}
\item $\mu_{i}((0,1])=p_i$ ;
\item $\Theta_{\pi^a,\widehat{\pi}^b}(\mu_{i})$ has mean $F^a(p_i)$.
\end{itemize}
\end{theorem}
The proof of Theorem \ref{th:rdea} relies on two lemmas.
\begin{lemma}
\label{lm:contincra}
The operators $\Theta_{\pi^a,\widehat{\pi}^b}$ and
$\Theta_{\widehat{\pi}^a,\widehat{\pi}^b}$ are continuous (with
respect to weak convergence) and strictly increasing (with respect to
stochastic ordering) on ${\mathcal P}\left([0,1]\right)$.
\end{lemma}
\bp{ of Lemma \ref{lm:contincra}}
It follows directly from the fact that, for any $n\geq 0$ and any
$n_1,\ldots,n_n\geq 0$, the
mapping $$x\mapsto\frac{1}{1+\sum_{i=1}^n\left(\sum_{j=1}^{n_i}x_{ij}\right)^{-1}}$$
is continuous and increasing from $[0,1]^{n_1+\ldots+n_n}$ to $[0,1]$.
\hfill $\Box$
\begin{lemma}
\label{lm:pFa(p)}
For any $\mu\in{\mathcal P}\left([0,1]\right)$, letting $p=\mu\left((0,1]\right)$, we have
\begin{enumerate}
\item $\Theta_{\widehat{\pi}^a,\widehat{\pi}^b}(\mu)\left((0,1]\right)=\widehat{\phi}^a(1-\widehat{\phi}^b(1-p))$
\item if $\Theta_{\widehat{\pi}^a,\widehat{\pi}^b}(\mu)\leq\mu$, then the mean of $\Theta_{{\pi}^a,\widehat{\pi}^b}(\mu)$ is at least $F^a(p).$
\item if $\Theta_{\widehat{\pi}^a,\widehat{\pi}^b}(\mu)\geq\mu$, then the mean of $\Theta_{{\pi}^a,\widehat{\pi}^b}(\mu)$ is at most $F^a(p)$;
\end{enumerate}
In particular, if $\mu$ is a fixed point of $\Theta_{\widehat{\pi}^a,\widehat{\pi}^b}$, then $p=\widehat{\phi}^a(1-\widehat{\phi}^b(1-p))$ and $\Theta_{{\pi}^a,\widehat{\pi}^b}(\mu)$ has mean $F^a(p)$.
\end{lemma}
\bp{ of Lemma \ref{lm:pFa(p)}}
In equation (\ref{eq:operator}) it is clear that $Y>0$ if and only if
for any $i\in\{1,\ldots,\mathcal{N}\}$, there exists
$j\in\{1,\ldots,{\mathcal{N}_i}'\}$ such that $X_{ij}>0$. With the notation
introduced above, this rewrites:
$$\Theta_{\widehat{\pi}^a,\widehat{\pi}^b}(\mu)\left( (0,1]\right)=\widehat{\phi}^a\left(1-\widehat{\phi}^b\left(1-\mu\left((0,1]\right)\right)\right),$$
hence the first result follows.
Now let $X\sim\mu$, $Y\sim
\Theta_{\widehat{\pi}^a,\widehat{\pi}^b}(\mu)$, $\mathcal{N}^a\sim\pi^a$,
$\widehat{\mathcal{N}}^a\sim\widehat{\pi}^a$, and let $S,S_1,\ldots$ have the
distribution of the sum of a $\widehat{\pi}^b-$distributed number of
i.i.d. copies of $X$, all these variables being independent. Observe that
\begin{eqnarray*}
\frac{1}{1+\sum_{i=1}^{\mathcal{N}^a} {S_i}^{-1}}
& = & \left(1-\frac{\sum_{i=1}^{\mathcal{N}^a} {S_i}^{-1}}{1+\sum_{i=1}^{\mathcal{N}^a} {S_i}^{-1}}\right) {\mathbf{1}}_{\left\{\forall i=1\ldots\mathcal{N}^a, S_i>0\right\}} \\
& = & {\mathbf{1}}_{\left\{\forall i=1\ldots\mathcal{N}^a, S_i>0\right\}} -\sum_{j=1}^{\mathcal{N}^a} \frac{ {S_j}^{-1}}{1+S_{j} ^{-1} + \sum_{ 1 \leq i \leq \mathcal{N}^a , i \ne j} {S_i}^{-1}} {\mathbf{1}}_{ \left\{\forall i=1\ldots\mathcal{N}^a, S_i>0\right\}}
\end{eqnarray*}
Then, $\Theta_{\pi^a,\widehat{\pi}^b}(\mu)$ has mean
\begin{eqnarray*}
{\mathbb E}\left[\frac{1}{1+\sum_{i=1}^{\mathcal{N}^a} {S_i}^{-1}}\right] & = & {\mathbb P} \left( \forall i=1\ldots\mathcal{N}^a, S_i>0\right) - \sum_{k = 1 } ^\infty k \pi^a_k {\mathbb E}\left[\frac {S^{-1}}{S^{-1}+1+\sum_{i=1}^{k-1}{S_i}^{-1}}\mathbf 1_{\left\{S>0,\forall i=1\ldots k -1, S_i>0\right\}}\right] \\
& = & \phi^a(1-\widehat{\phi}^b(1-p))-\phi^a{}'(1){\mathbb E}\left[\frac {S^{-1}}{S^{-1}+1+\sum_{i=1}^{\widehat{\mathcal{N}}^a}{S_i}^{-1}}\mathbf 1_{\left\{S>0,\forall i=1\ldots\widehat{\mathcal{N}}^a, S_i>0\right\}}\right]\\
& = &
\phi^a(1-\widehat{\phi}^b(1-p))-\phi^a{}'(1){\mathbb E}\left[\frac{Y}{Y+S}\mathbf 1_{\left\{S>0\right\}}\right],
\end{eqnarray*}
where the second and last lines follow from (\ref{eq:F}) and
$Y\sim\Theta_{\widehat{\pi}^a,\widehat{\pi}^b}(\mu)$, respectively.
Now, for any $s>0$, $x\mapsto {x}/{x+s}$ is increasing and hence, depending on whether $\Theta_{\widehat{\pi},\widehat{\pi}}(\mu)\geq\mu$ or $\Theta_{\widehat{\pi}^a,\widehat{\pi}^b}(\mu)\leq\mu$, $\Theta_{\pi^a,\widehat{\pi}^b}(\mu)$ has mean at most/least:
\begin{eqnarray*}
\label{eq:calc}
\phi^a(1-\widehat{\phi}^b(1-p))-\phi^a{}'(1){\mathbb E}\left[\frac{X}{X+S}\mathbf
1_{\left\{S>0\right\}}\right] =
\phi^a(1-\widehat{\phi}^b(1-p))-\phi^a{}'(1){\mathbb E}\left[\frac{X }{X +\sum_{ i = 1} ^{\widehat{\mathcal{N}}^b} X_i }{\mathbf{1}}_{\left\{\mathcal{N}^*\geq 1\right\}} \right],
\end{eqnarray*}
with $X_i$ are i.i.d. copies of $X$ independent of $\widehat{\mathcal{N}}^b\sim\widehat{\pi}^b$ and $\mathcal{N}^*=\sum_{i=1}^{\widehat{\mathcal{N}}^b}\mathbf 1_{\{X_i>0\}}$. Now if $X'$ is the law of $X$ conditioned on $\{X > 0\}$, and $X'_i$ are i.i.d. copies of $X'$, by exchangeability, we find
$$
{\mathbb E}\left[\frac{X }{X +\sum_{ i = 1} ^{\widehat{\mathcal{N}}^b} X_i }{\mathbf{1}}_{\left\{\mathcal{N}^*\geq 1\right\}} \right] = p {\mathbb E}\left[\frac{X' }{X'+\sum_{ i = 1} ^{ \mathcal{N}^*} X'_i }{\mathbf{1}}_{\left\{\mathcal{N}^*\geq 1\right\}} \right] = p {\mathbb E}\left[\frac{1}{1+\mathcal{N}^*}\mathbf 1_{\left\{\mathcal{N}^*\geq 1\right\}}\right].
$$
Hence finally, depending on whether $\Theta_{\widehat{\pi},\widehat{\pi}}(\mu)\geq\mu$ or $\Theta_{\widehat{\pi}^a,\widehat{\pi}^b}(\mu)\leq\mu$, $\Theta_{\pi^a,\widehat{\pi}^b}(\mu)$, $\Theta_{\pi^a,\widehat{\pi}^b}(\mu)$ has mean at most/least:
\begin{eqnarray*}
\phi^a(1-\widehat{\phi}^b(1-p))-p\phi^a{}'(1){\mathbb E}\left[\frac{1}{1+\mathcal{N}^*}\mathbf 1_{\left\{\mathcal{N}^*\geq 1\right\}}\right]
\end{eqnarray*}
But using the definition (\ref{eq:F}) and the combinatorial identity $(n+1){n\choose d}=(d+1){n+1\choose d+1}$, one easily derive :
\begin{eqnarray*}
&&\phi^a(1-\widehat{\phi}^b(1-p))-p\phi^a{}'(1){\mathbb E}\left[\frac{1}{1+\mathcal{N}^*}\mathbf 1_{\left\{\mathcal{N}^*\geq 1\right\}}\right]\\
&=& \phi^a(1-\widehat{\phi}^b(1-p))- p\phi^a{}'(1)\sum_{n\geq 1}\widehat{\pi_n}^b\sum_{d=1}^n{n\choose d}\frac {p^d(1-p)^{n-d}}{d+1}
= F^a(p).
\end{eqnarray*}
\hfill $\Box$
\bp{ of Theorem \ref{th:rdea}}
Let $p\in[0,1]$ such that $\widehat{\phi}^a(1-\widehat{\phi}^b(1-p))=p$, and define
$\mu_0=\textrm{Bernoulli}(p)$.
From Lemma \ref{lm:pFa(p)} we know that
$\Theta_{\widehat{\pi}^a,\widehat{\pi}^b}\left(\mu_0\right)\left((0,1]\right)=p$,
and since \textrm{Bernoulli(p)} is the largest element of ${\mathcal P}([0,1])$
putting mass $p$ on $(0,1]$, we have
$\Theta_{\widehat\pi^a,\widehat\pi^b}\left(\mu_0\right)\leq\mu_0$.
Immediately, Lemma \ref{lm:contincra} guarantees that the limit
$$\mu_\infty=\lim_{k\to\infty}\searrow\Theta^k_{\widehat\pi^a,\widehat\pi^b}\left(\mu_0\right)$$
exists in ${\mathcal P}\left([0,1]\right)$ and is a fixed point of
$\Theta_{\widehat\pi^a,\widehat\pi^b}$. Moreover, by Fatou's lemma,
the number $p_\infty=\mu_\infty\left((0,1]\right)$ must satisfy
$p_\infty\leq p$. But then the mean of
$\Theta_{\pi^a,\widehat\pi^b}(\mu_\infty)$ must be both
\begin{itemize}
\item equal to $F^a(p_\infty)$ by Lemma \ref{lm:pFa(p)} with $\mu=\mu_\infty$ ;
\item at least $F^a(p)$ since this holds for all
$\Theta_{\pi^a,\widehat\pi^b}\circ
\Theta^k_{\widehat\pi^a,\widehat\pi^b}(\mu_0)$, $k\geq 1$ (Lemma \ref{lm:pFa(p)} with $\mu=\Theta^k_{\widehat\pi^a,\widehat\pi^b}(\mu_0)$).
\end{itemize}
We have just shown both $F^a(p)\leq F^a(p_\infty)$ and $p_\infty\leq
p$. From this, we now deduce the one-to-one correspondence between
historical records of $F^a$ and fixed points of
$\Theta_{\widehat\pi^a,\widehat\pi^b}$.
We treat each inclusion separately:
\begin{enumerate}
\item If $F^a$ admits an historical record at $p$, then clearly $p_\infty=p$, so $\mu_\infty$ is a fixed point satisfying $\mu_\infty\left((0,1]\right)=p$
\item Conversely, considering a fixed point $\mu$ with
$\mu\left((0,1]\right)=p$, we want to deduce that $F^a$ admits an
historical record at $p$. We first claim that $\mu$ is the above
defined limit $\mu_\infty$. Indeed, $\mu\leq\textrm{Bernoulli(p)}$
implies $\mu\leq\mu_\infty$ ($\Theta_{\widehat\pi^a,\widehat\pi^b}$
is increasing), and in particular $p\leq p_\infty$. Therefore,
$p=p_\infty$ and $F^a(p)=F^a(p_\infty)$. In other words, the two
ordered distributions
$\Theta_{\pi^a,\widehat\pi^b}(\mu)\leq\Theta_{\pi^a,\widehat\pi^b}(\mu_\infty)$
share the same mean, hence are equal. This ensures
$\mu=\mu_\infty$.
Now, if $q<p$ is any historical record location, we know from part 1 that
$$\nu_\infty=\lim_{k\to\infty}\searrow
\Theta_{\widehat\pi^a,\widehat\pi^b}^{k}\left(\textrm{Bernoulli(q)}\right)$$
is a fixed point of $\Theta_{\widehat\pi^a,\widehat\pi^b}$ satisfying
$\nu_\infty\left((0,1]\right)=q$. But $q<p$, so
$\textrm{Bernoulli(q)}<\textrm{Bernoulli(p)}$, hence
$\nu_\infty\leq\mu_\infty$. Moreover, this limit inequality is strict
because $\nu_\infty\left((0,1]\right) = q < p =
\mu_\infty\left((0,1]\right)$. Consequently,
$\Theta_{\pi^a,\widehat\pi^b}(\nu_\infty)<
\Theta_{\pi^a,\widehat\pi^b}(\mu_\infty)$ and taking expectations, $F^a(q)<
F^a(p)$. Thus, $F^a$ admits an historical record at $p$.
\end{enumerate}
\hfill $\Box$
We may now finish the proof of Theorem \ref{th:KSbi}.
\noindent{\em Proof of Theorem \ref{th:KSbi} : case of bounded degrees. }
We assume that $\pi_a$ and $\pi_b$ have bounded support. Recall
(\ref{eq:contrlamb}), so that we have
$\lambda=\frac{\phi^b{}'(1)}{\phi^a{}'(1)+\phi^b{}'(1)}$, where $\lambda$ is the probability that the root is of type $a$.
Theorems \ref{co:cvMN} and \ref{th:rdea} and Lemma \ref{lem:RDEbi} give:
\begin{equation}
\label{eq:KSbi2}
\frac{\nu(G_n)}{{|V_n|}}\xrightarrow[n\to\infty]{}
\frac{\lambda(1-\max_{x\in[0,1]}F^a(x))+(1-\lambda)(1-\max_{x\in[0,1]}F^b(x))}{2},
\end{equation}
where $F^a$ is defined in (\ref{eq:defa}) and $F^b$ is defined similarly by
\begin{eqnarray}
\label{eq:defb}F^b(x) &=& \phi^b\left(1-\widehat{\phi}^a(1-x)\right)-\frac{\phi^b{}'(1)}{\phi^a{}'(1)}\left(1-\phi^a(1-x)-x\phi^a{}'(1-x)
\right).
\end{eqnarray}
For any $x$ which is an historical record of $F^a$, we define
$y=\widehat{\phi}^b(1-x)$ so that $\widehat{\phi}^a(1-y)=x$. Then we have:
\begin{eqnarray*}
\lambda(1-F^a(x)) &=& \lambda\left(1-\phi^a(1-y)+\phi^a{}'(1)\left(
\frac{1}{\phi^b{}'(1)}-\frac{\phi^b(1-\widehat{\phi}^a(1-y))}{\phi^b{}'(1)}-y\widehat{\phi}^a(1-y)\right)\right)\\
&=& (1-\lambda)(1-F^b(y)).
\end{eqnarray*}
By symmetry, this directly
implies that
$\lambda(1-\max_{x\in[0,1]}F^a(x))=(1-\lambda)(1-\max_{x\in[0,1]}F^b(x))$
so that (\ref{eq:KSbi2}) is equivalent to (\ref{eq:KSbi}). This proves Theorems \ref{th:KS} and \ref{th:KSbi} for distributions with bounded support. \hfill $\Box$
\noindent{\em Proof of Theorem \ref{th:KSbi} : general case. } To keep notation simple, we only prove Theorem \ref{th:KS}. The following proof clearly extends to the case of UHGW trees. Let $G_1,G_2,\ldots$ be finite random graphs whose local weak limit is a
Galton-Watson tree $T$, and assume that the degree distribution $\pi$ of
$T$ (with generating function $\phi$) has a finite mean : $\phi'(1)=\sum_{n}n\pi_n<\infty$.
For any rooted graph $G$ and any fixed integer $d\geq 1$, recall that $G^d$ is the graph obtained from $G$ by deleting all edges adjacent to a vertex $v$ whenever ${\mathrm{deg}}(v)>d$. Hence $T^d$ is a Galton-Watson tree whose degree distribution $\pi^{d}$ is defined by
$$\forall i\geq 0, \pi^{d}_i=\pi_i{\bf
1}_{i\leq d} + {\mathbf{1}}_{i = 0} \sum_{k \geq d+1} \pi_k.$$
By Theorem \ref{th:main}, Equation \eqref{eq:toshow} and our weaker version of
Theorem \ref{th:KS} for distributions with bounded support,
\begin{equation}
\label{eq:arg1}
\frac{\nu(G_n)}{|V_n|}\xrightarrow[n\to\infty]{} \lim_{d \to \infty} \min_{x\in[0,1]}g^d(x),
\end{equation}
with $\phi_d (x)=\sum_{k=0}^{d}\pi_kx^k$
and $$g^d(x)=1-\frac{1}{2}(1-x)\phi_d'(x)-\frac{1}{2}\phi_d(x)-\frac{1}{2}\phi_d\left(1-\frac{\phi_d'(x)}{\phi_d'(1)}\right).$$
Also, as $d\to\infty$, we have $\phi_d\to\phi$ and
$\phi'_d\to\phi'$ uniformly on $[0,1]$, so
\begin{equation}
\label{eq:arg3}
\min_{x\in[0,1]}g^d(x)\xrightarrow[n\to\infty]{}\min_{x\in[0,1]}g(x),
\end{equation}
with $g(x)=1-\frac{1}{2}(1-x)\phi'(x)-\frac{1}{2}\phi(x)-\frac{1}{2}\phi\left(1-\frac{\phi'(x)}{\phi'(1)}\right).$
Finally, combining (\ref{eq:arg1}) and (\ref{eq:arg3}), we
easily obtain the desired
\begin{equation*}
\label{eq:arg4}
\frac{\nu(G_n)}{|V_n|}\xrightarrow[n\to\infty]{}\min_{x\in[0,1]}g(x).
\end{equation*}
\hfill $\Box$
\subsection{Proof of Corollary \ref{cor:fr}}\label{sec:corfr}
Note that in Corollary \ref{cor:fr}, we divide $\nu(G_n)$ by
$|V^a_n|=\lfloor \alpha m\rfloor$ instead of $|V^a_n|+|V^b_n|=\lfloor \alpha m\rfloor+m$, so
that by Theorem \ref{th:KSbi}, we have
$\frac{\nu(G_n)}{|V^a_n|} \xrightarrow[n\to\infty]{}\min_{t\in [0,1]} 1-F^a(t)$.
We have $\phi^a(x)=x^k$, $\phi^b(x)=e^{\alpha k
(x-1)}$ so that we have:
\begin{eqnarray*}
F^a(x) &=& \left( 1-e^{-k\alpha
x}\right)^k-\frac{1}{\alpha}\left(1-e^{-k\alpha x}-k\alpha
xe^{-k\alpha x} \right)\\
F^a{}'(x)&=& k^2\alpha e^{-k\alpha x}\left( \left(1-e^{-k\alpha
x}\right)^{k-1}-x\right).
\end{eqnarray*}
Let $x^*$ be defined as in Corollary \ref{cor:fr} as the largest
solution to $x=\left(1-e^{-k\alpha x}\right)^{k-1}$.
It is easy to check (see Section 6 in \cite{lel12} for a more general
analysis) that
\begin{eqnarray*}
\min_{t\in [0,1]} 1-F^a(t)=\min\{1,1-F^a(x^*)\}.
\end{eqnarray*}
Setting $\xi^*=k\alpha x^*$, we have
$\frac{\xi^*}{k\alpha}=(1-e^{-\xi^*})^{k-1}$, so that
\begin{eqnarray*}
\min_{t\in [0,1]}
1-F^a(t)=\min\left\{1,1-\frac{1}{\alpha}\left(e^{-\xi^*}+\xi^*
e^{-\xi^*}+\frac{\xi^*}{k}(1-e^{-\xi^*})-1\right)\right\}.
\end{eqnarray*}
Since $z\mapsto \frac{z(1-e^{-z})}{1-e^{-z}-z e^{-z}}$ is
increasing in $z$, we see that $\xi^*\geq \xi$ if and only if
$\alpha\geq \alpha_c$ and we get
\begin{eqnarray*}
\min_{t\in [0,1]}
1-F^a(t)=1-{\mathbf{1}}(\alpha\geq \alpha_c)\frac{1}{\alpha}\left(e^{-\xi^*}+\xi^*
e^{-\xi^*}+\frac{\xi^*}{k}(1-e^{-\xi^*})-1\right).
\end{eqnarray*}
\section*{Acknowledgement}
We would like to thank Andrea Montanari and Guilhem Semerjian for explaining us a
key idea for the proof of Lemma \ref{lm:unifctrl}, as well as Nikolaos
Fountoulakis, David Gamarnik, James Martin and Johan W\"astlund for
interesting discussions
The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-11-JS02-005-01 (GAP project)
\bibliographystyle{abbrv}
| {
"timestamp": "2012-04-12T02:03:30",
"yymm": "1102",
"arxiv_id": "1102.0712",
"language": "en",
"url": "https://arxiv.org/abs/1102.0712",
"abstract": "Elek and Lippner (2010) showed that the convergence of a sequence of bounded-degree graphs implies the existence of a limit for the proportion of vertices covered by a maximum matching. We provide a characterization of the limiting parameter via a local recursion defined directly on the limit of the graph sequence. Interestingly, the recursion may admit multiple solutions, implying non-trivial long-range dependencies between the covered vertices. We overcome this lack of correlation decay by introducing a perturbative parameter (temperature), which we let progressively go to zero. This allows us to uniquely identify the correct solution. In the important case where the graph limit is a unimodular Galton-Watson tree, the recursion simplifies into a distributional equation that can be solved explicitly, leading to a new asymptotic formula that considerably extends the well-known one by Karp and Sipser for Erdös-Rényi random graphs.",
"subjects": "Probability (math.PR)",
"title": "Matchings on infinite graphs",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754470129648,
"lm_q2_score": 0.7185943985973772,
"lm_q1q2_score": 0.7075104012200253
} |
https://arxiv.org/abs/1207.4884 | A short proof for the polyhedrality of the Chvátal-Gomory closure of a compact convex set | Recently Schrijver's open problem, whether the Chvátal--Gomory closure of an irrational polytope is polyhedral was answered independently in the affirmative by Dadush, Dey, and Vielma (even for arbitrarily compact convex set) as well as by Dunkel and Schulz. We present a very short, easily accesible proof that the Chvátal--Gomory closure of a compact convex set is a polytope. | \section{Introduction}
\label{sec:introduction}
The Chvátal-Gomory procedure was the first cutting-plane procedure
introduced (in \cite{GY1,GY2,GY3,Chvatal}) and has been studied thoroughly from a
theoretical as well as a practical point of view (see e.g., \cite{CCH,CF1,BEHS,letchford2002strengthening,ES,fischetti2007optimizing,C,bonami2008projected}).
Recall that the Chvátal-Gomory closure \(K'\) of a polyhedron or
a compact convex set \(K \subseteq \mathbb{R}^n\) is defined as
\[K' \coloneqq \bigcap_{\substack{(c,\delta) \in \mathbb{Z}^{n} \times \mathbb{R}, \\
K \subseteq \face{cx \leq \delta}}} \face{cx \leq
\ffloor{\delta}},\]
where we use \(\face{ax \leq b}\) as a shorthand for \(\face{x
\mid ax \leq b}\); for brevity we refer to it as CG\xspace closure and
to the defining inequalities as CG\xspace cuts.
One of the fundamental
questions in cutting-plane theory is whether the closure arising from a
cutting-plane procedure (i.e., adding all potential cuts that can be
derived from valid inequalities) is a polyhedron. Clearly, we
add an infinite number of cuts here and thus it is not clear \emph{a
priori} whether \(K'\) is a polyhedron. However, for the case where
\(K\) is a rational polyhedron it is well-known that the CG\xspace closure is a rational
polyhedron again (see \cite{schr1980},
\cite{Chvatal}). As a natural consequence, in \cite{schr1980} the question was raised
whether the CG\xspace closure of an irrational
polytope \(P\) is a polytope. This important question was answered
in the affirmative independently in the works by
\cite{ds2010} and \cite{DDV2011} (the latter established the even more
general case of arbitrarily compact convex
set). The relevance of this result is many-fold, from the convergence
of adding cutting planes of the CG\xspace type to Mixed-Integer Nonlinear Programming over compact convex
sets to the theory of proof systems where we consider proofs of
assertions with infinitely many defining sentences.
\subsection*{Contribution}
\label{sec:contribution}
We provide a short proof of the more general result \cite{DDV2011}. In
contrast to \cite{DDV2011}, we do not rely so much on convex
analysis but take a rather direct topological approach while
maintaining the overall high-level strategy. Key is here a
strengthened, quantitative homogeneity lemma (see
Lemma~\ref{lem:homogeneity}) from which many required properties
follow immediately. Before, these had been established separately using
different strategies. We believe that the proposed proof
lends itself to potential applications to
many more classes of cutting planes.
The proof consists of three steps:
\begin{enumerate}
\item \emph{Continuity:}
Faces and
implied cuts deform smoothly when
perturbing the coefficients. This is crucial for the
actual finiteness argument via compactness, and is valid for
inequalities in general. See Lemma~\ref{lem:nearby-dir}.
\item \emph{Homogeneity} for a procedure \(M\):
The cutting-plane closure \(M\) commutes with intersections with faces, i.e.,
\[M(K \cap F) = M(K) \cap F,\]
where \(F\) is a face-defining hyperplane of the convex body
\(K\). See Lemma~\ref{lem:homogeneity}.
Not only does homogeneity hold for many
cutting-plane procedures \(M\) (such as the split closure, the
Lovász-Schrijver closures, Sherali-Adams hierarchies, and the
Lift-and-Project closure, see \cite{PS20091}) in the case of rational
polyhedra, but
homogeneity also allows a very clean, inductive approach
to polyhedrality. See Theorem~\ref{lem:boundary}.
Moreover, it is also homogeneity that ensures
that finitely many CG\xspace cuts suffice to restrict the CG\xspace closure
to a rational subspace of the affine space spanned by the convex set
(which is necessary for
polyhedrality here). See Corollary~\ref{cor:bringIntoRationalSpace}.
\item \emph{Locality:} Informally,
every point \(x\) in the relative interior of a
\emph{polytope} \(P\) can be cut out by a finite number of
CG\xspace cuts.
This step is contained in the proof
of Theorem~\ref{thm:cg-closure-polytope}.
\end{enumerate}
From all those properties the hardest one to establish and the
cornerstone of our proof is the quantitative version of the
\hyperref[lem:homogeneity]{homogeneity lemma}
in Section~\ref{sec:homogeneity}.
In fact, for the proof we need a generalization
of a famous theorem due to Kronecker and Weyl provided in
Lemma~\ref{lem:vectorSpace}. Once homogeneity is established, the
conclusion of polyhedrality follows naturally in our
framework and it is actually very similar to the proof for rational
polyhedra given in \cite{schr1980}.
\section{Preliminaries}
\label{sec:preliminaries}
In the following, we only consider exposed faces of convex sets
and for the sake of
brevity we refer to them as \emph{faces}.
In other words, a face \(F\) of a compact convex set \(K\)
is a subset of the form \(F = K \cap \face{\pi x = \pi_0}\)
for some supporting hyperplane \(\pi x = \pi_0\),
i.e., \(K \subseteq \face{\pi x \leq
\pi_0}\), and there exists \(x_0 \in K\) with \(\pi x_0 = \pi_0\);
we will call the face \(F\) the \emph{\(\pi\)-face of \(K\)}.
In particular, \(F = K\) is allowed if \(K\) is lower
dimensional. Recall that a compact convex set is uniquely
determined by its exposed proper faces
(i.e., exposed faces different from \(K\)).
All facts that we mention without pointers to the literature can be
found in \cite{schrijver1998theory} and \cite{barvinok2002course}.
To formalize continuity of directions,
we identify a direction with the unit vector pointing to that
direction,
therefore
for any non-zero vector \(\pi\),
let \(\unitvector{\pi} \coloneqq \pi \mathbin{/} \tnorm{\pi}\).
The next lemma shows that compact \(\pi\)-faces
change \textquote{upper semi-continuously} in the direction of
\(\pi\). In the following we denote the dimension of the ambient space
\(\mathbb{R}^n\) by \(n\) and we use the shorthand \([k] \coloneqq \face{1, \dots,k}\).
Let \(\partial K\) denote the relative boundary of
a closed convex set \(K\).
\begin{lem}[Continuity]
\label{lem:nearby-dir}
Let \(K\) be a closed convex set in \(\mathbb{R}^{n}\).
Let \(\pi\) be a non-zero vector,
and let the \(\pi\)-face \(F\) of \(K\) be compact.
Then for every neighbourhood \(U\) of \(F\)
there exists an \(\varepsilon > 0\) such that
whenever
\(\tnorm{\unitvector{\pi'} - \unitvector{\pi}}
< \varepsilon\),
the \(\pi'\)-face of \(K\) is a subset of \(U\),
i.e.,
all the maximizers of
the function \(x \mapsto \pi' x\) on \(K\) lie in \(U\).
\begin{proof}
Without loss of generality,
we may assume that \(U\) is compact and convex.
By assumption, for some constant \(c\),
we have \(\pi x = c\) for all \(x \in F\),
and \(\pi x < c\) for all \(x \in K \setminus F\),
in particular for all \(x \in \partial U \cap K\).
As \(\partial U \cap K\) is compact, actually
\(\pi x < c_{1}\) for some \(c_{1} < c\)
and all \(x \in \partial U \cap K\).
As \(\pi' x\) is continuous in \(\pi'\) and \(x\),
and \(F\) and \(\partial U \cap K\)
are compact,
for all \(\pi'\) in a neighbourhood of \(\pi\),
we have \(\pi' x > c_{1}\) for all \(x \in F\),
but \(\pi' x < c_{1}\) for all \(x \in \partial U \cap K\).
In particular,
all maximizers of \(x \mapsto \pi' x\) on \(K\) lie in \(U\).
This is obvious if \(K \subseteq U\).
If \(K \nsubseteq U\),
then \(\pi' x\) is everywhere smaller on \(K \setminus U\)
than on \(F\):
take arbitrary points \(x_{0} \in K \setminus U\)
and \(x_{1} \in F\).
There is an \(x_{2} \in \partial U \cap K\)
in the line segment \([x_{0}, x_{1}]\).
As \(\pi' x_{2} < \pi' x_{1}\),
we obtain \(\pi' x_{0} < \pi' x_{1}\).
This finishes the proof of the lemma.
\end{proof}
\end{lem}
\begin{rem}
If \(K\) is a polyhedron one can show more:
the \(\pi'\)-face is contained in \(F\).
In
particular, there is no need for \(U\). To see this we choose \(U\)
to be a polytope in the proof.
Then \(U \cap K\) is also a polytope,
and \(x \mapsto \pi' x\) is everywhere larger
on the vertices
of the \(\pi\)-face than on the other vertices,
when the direction of \(\pi'\) is close to that of \(\pi\).
Hence the \(\pi'\)-face is contained in the \(\pi\)-face,
as claimed.
\end{rem}
We will use a well-known approximation theorem due to Kronecker.
We state a version suitable for our needs,
which we derive from Weyl’s criterion.
\begin{lem}[{\citep{kronecker1884},\citep[Satz~3]{weyl1916}}]
\label{lem:vectorSpace}
Let \(n, N_{0} \in \mathbb{N}\) and \(\pi \in \mathbb{R}^{n}\) with \(\pi \neq 0\).
Then
\(\mathbb{Z}^{n} + \pi \mathbb{Z}_{> N_{0}}\) contains a dense subset of
a linear subspace \(V\) of \(\mathbb{R}^{n}\).
In particular, \(\mathbb{Z}^{n} + \pi \mathbb{Z}_{> N_{0}}\) contains points
arbitrarily close to \(0\),
i.e., for every \(\epsilon > 0\)
there exists \(N > N_{0}\) and \(a \in \mathbb{Z}^{n}\)
with \(\tnorm{a - N \pi} < \epsilon\).
\begin{proof}
When the components of \(\pi\) together with \(1\)
are linearly independent over \(\mathbb{Q}\),
this is a special case of Weyl’s criterion with \(V = \mathbb{R}^{n}\).
We reduce the general case to this one.
First we define \(V\).
Let \(\pi_{1}, \dots, \pi_{n}\) denote the components of \(\pi\).
We can assume without loss of generality that
a linear basis of \(1, \pi_{1}, \dots, \pi_{n}\) over \(\mathbb{Q}\)
is \(1, \pi_{1}, \pi_{2}, \dots, \pi_{k}\).
Thus for \(j > k\) there are integers \(n_{j,i}\) and \(n_{j}\)
together with a positive integer \(m\)
such that
\begin{align*}
m
\pi_{j} &= n_{j} + \sum_{i=1}^{k} n_{j,i} \pi_{i}, & j &> k.
\intertext{We use these as
the defining equations of \(V\),
i.e., \(V\) is defined by}
m
x_{j} &= \sum_{i=1}^{k} n_{j,i} x_{i}, & j &> k.
\intertext{Let \(e_{1}, \dots, e_{n}\) denote
the canonical basis of \(\mathbb{Z}^{n}\).
The following elements lie in \(V\):}
\widetilde{e_{i}} &\coloneqq m e_{i} + \sum_{j=k+1}^{n} n_{j,i} e_{j},
& i &< k, \\
\widetilde{\pi} &\coloneqq m \pi - \sum_{j=k+1}^{n} n_{j} e_{j}.
\end{align*}
By Weyl’s criterion,
\(\mathbb{Z}^{k} + (\pi_{1}, \dots, \pi_{k}) \mathbb{Z}_{> N_{0}}\)
is dense in \({\mathbb{R}}^{k}\),
and hence also
\(m \mathbb{Z}^{k} + m (\pi_{1}, \dots, \pi_{k}) \mathbb{Z}_{> N_{0}}\)
is dense in \({\mathbb{R}}^{k}\).
We reformulate this for \(V\)
via the projection to the first \(k\) coordinates,
which is obviously an isomorphism
between \(V\) and \({\mathbb{R}}^{k}\):
a dense subset of \(V\) is
\(\sum_{i=1}^{k} \mathbb{Z} \widetilde{e_{i}}
+ \widetilde{\pi} \mathbb{Z}_{> N_{0}}\),
which is a subset of \(\mathbb{Z}^{n} + \pi \mathbb{Z}_{> N_{0}}\).
This finishes the proof.
(For \(k = 0\) the argument above is overkill,
as \(\pi\) is rational and hence
\(\mathbb{Z}^{n} + \pi \mathbb{Z}_{> N_{0}}\) contains \(0\).)
\end{proof}
\end{lem}
\section{Homogeneity}
\label{sec:homogeneity}
In this section we compare \(K'\) with the CG\xspace closure \(F'\)
of a face \(F\).
\begin{lem}[Homogeneity for compact faces]
\label{lem:homogeneity}
Let \(K \subseteq \mathbb{R}^n\) be a closed convex set and let
\[
F \coloneqq K \cap \face{ \pi x = \pi_0}
\]
be a compact \(\pi\)-face of \(K\)
for some \(\pi \in \mathbb{R}^{n}\) and \(\pi_0 \in \mathbb{R}\)
with \(K \subseteq \face{\pi x \leq \pi_0 }\).
Further, assume that \(F\) satisfies \(c x \leq \delta\) with \(c \in
\mathbb{Z}^{n}\) (and hence \(F'\) satisfies \(cx \leq \ffloor{\delta}\)).
Then there are finitely many CG\xspace cuts of \(K\)
defining a polyhedron \(P\) satisfying
\((c + \alpha \pi) x \leq \ffloor{\delta} + \alpha \pi_{0}\)
for some \(\alpha > 0\).
\begin{proof}
Rescaling \((\pi_0,\pi)\) we may assume, without loss of generality, that \(\pi_{0} \in \mathbb{Z}\).
By increasing \(\delta\) a little if necessary,
we may assume that \(\delta\) is not an integer.
Note that a small increase of \(\delta\)
does not change \(\ffloor{\delta}\).
Let \(0 < \varepsilon < \delta - \ffloor{\delta}\) be
a small positive number.
Choose a small compact neighbourhood \(U\) of \(F\)
such that \(c x \leq \delta + \varepsilon\) for \(x \in U\).
There is clearly an \(\varepsilon_{1} > 0\)
such that for all \(y \in \mathbb{R}^{n}\)
with \(\tnorm{y} < \varepsilon_{1}\),
we have
\(\abs{y x} < \varepsilon\) for all \(x \in U\).
By Lemma~\ref{lem:nearby-dir},
there exists an \(\varepsilon_{2} > 0\) such that
whenever
\(\tnorm{\unitvector{\pi'} - \unitvector{\pi}}
< \varepsilon_{2}\),
all maximizers of the function \(x \mapsto \pi' x\) on \(K\)
lie in \(U\).
There is a large positive integer \(N\)
such that for all positive integer \(m \geq N\)
and \(a \in \mathbb{Z}^{n}\)
with \(\tnorm{a - m \pi} < \varepsilon_{1}\)
we have
\(\tnorm{\unitvector{c+a} - \unitvector{\pi}} = \tnorm{\unitvector{\frac{c+a}{m}} - \unitvector{\pi}}
< \varepsilon_{2}\).
In particular,
all maximizers of \((c + a) x\) on \(K\) lie in \(U\)
for all such \(m\) and \(a\).
All in all,
for all positive integer \(m \geq N\)
and \(a \in \mathbb{Z}^{n}\)
with \(\tnorm{a - m \pi} < \varepsilon_{1}\),
all maximizers of the function \(x \mapsto (c + a) x\)
on \(K\) lie in \(U\),
and we have \(\tnorm{(a - m \pi) x} < \varepsilon\)
for all \(x \in U\).
Now we choose a finite collection of such pairs \((m, a)\).
By Lemma~\ref{lem:vectorSpace}
the collection \(\mathbb{Z}^{n} - \mathbb{Z}_{{} \geq N} \pi\)
contains a dense subset of
a linear subspace \(V\) of \(\mathbb{R}^{n}\).
Let \(v_{1}, \dotsc, v_{k}\) be the vertices of a small simplex in \(V\)
containing \(0\) in its relative interior,
with \(\tnorm{v_{i}} < \varepsilon_{1}\) for all \(i\).
We choose the simplex to be full dimensional in \(V\),
i.e., here \(k-1\) is the dimension of \(V\).
By slightly perturbing the \(v_{i}\) in \(V\),
the vertices remain in the \(\varepsilon_{1}\)-ball,
and \(0\) in the interior of the simplex.
As \(\mathbb{Z}^{n} - \mathbb{Z}_{{} \geq N} \pi\) contain a dense subset of \(V\),
by a slight perturbation we can even move the vertices
inside \(\mathbb{Z}^{n} - \mathbb{Z}_{{} \geq N} \pi\),
obtaining a new simplex with vertices \(a_{i} - m_{i} \pi\),
with \(0\) still contained
in the relative interior of the new simplex:
i.e., there exist coefficients \(\lambda_{i}\) satisfying
\begin{equation}
\label{eq:5}
\sum_{i \in [k]} \lambda_{i} (a_{i} - m_{i} \pi) = 0,
\qquad \text{where }\lambda_{1},\dots,\lambda_k > 0 \text{ and } \sum_{i \in [k]}\lambda_i = 1
\end{equation}
with some \(a_{i} \in \mathbb{Z}^{n}, \ m_i \in \mathbb{N},\ m_{i} \geq N\)
satisfying \(
\tnorm{a_{i} - m_{i} \pi} < \varepsilon_{1}\).
As a consequence, for all \(x \in U \cap K\), one has
\begin{equation}
\label{eq:2}
(c + a_{i}) x = c x + m_{i} \pi x + (a_{i} - m_{i} \pi) x
\leq (\delta + \varepsilon) + m_{i} \pi_{0} + \varepsilon,
\end{equation}
which is also valid for all \(x \in K\) as \((c + a_{i})x\)
attains its maximum in \(U\).
Hence
\begin{equation*}
(c + a_{i}) x \leq \ffloor{\delta + m_{i} \pi_{0} + 2 \varepsilon}
= \ffloor{\delta} + m_{i} \pi_{0}
\end{equation*}
is a CG\xspace cut for \(K\) for \(i \in [k]\).
Let \(P\) be the polyhedron defined by these CG\xspace cuts.
The convex combination of the CG\xspace cuts
with coefficients \(\lambda_{i}\)
is valid for \(P\),
which is exactly the claimed inequality
\begin{equation*}
(c + \alpha \pi) x \leq \ffloor{\delta} + \alpha \pi_{0},
\end{equation*}
for \(P\) with \(\alpha \coloneqq \sum_{i \in [k]} \lambda_{i} m_{i} > 0\) as
\eqref{eq:5} can be rewritten to
\begin{equation*}
\sum_{i \in [k]} \lambda_{i} (c + a_{i}) =
c + \underbrace{\sum_{i \in [k]} \lambda_{i} m_{i}}_{\alpha} \pi.
\qedhere
\end{equation*}
\end{proof}
\end{lem}
Apart from establishing the basis for the later induction, the main
advantage of Lemma~\ref{lem:homogeneity} is that many
important properties of the CG\xspace closure follow as simple corollaries.
\begin{cor}
\label{cor:face}
Let \(K \subseteq \mathbb{R}^n\) be a compact convex set.
Then \(K' \subseteq K\) and we have \(K' \cap F = F'\) for every face \(F\) of \(K\).
\begin{proof}
Applying Lemma~\ref{lem:homogeneity} to \(c = 0\) and \(\delta =0\),
we obtain that \(K'\) satisfies
every inequality \(\pi x \leq \pi_{0}\) satisfied by \(K\).
For a face \(F\), Lemma~\ref{lem:homogeneity} implies that \(K' \cap F\)
satisfies the CG\xspace cuts defining \(F'\),
hence \(K' \cap F \subseteq F'\).
The inclusion in the other direction \(F' \subseteq K' \cap F\)
is obvious.
\end{proof}
\end{cor}
\begin{rem}
If one merely
wants to establish \(K' \cap F \subseteq (K \cap F)'\) and one is not interested in
the finiteness statement of Lemma~\ref{lem:homogeneity}
then it suffices to consider a single vector \(c + a_1\)
in the proof of Lemma~\ref{lem:homogeneity}
instead of a finite family.
From \eqref{eq:2} the proof can then be concluded as
follows: For every \(x \in K' \cap F\)
\begin{equation*}
c x = (c + a_1) x + (m_1 \pi - a_1) x - m_1 \pi_{0}
\leq \ffloor{\delta} + \varepsilon
\end{equation*}
for every \(\varepsilon > 0\) small enough.
Thus \(c x \leq \ffloor{\delta}\) is valid for \(K' \cap F\).
Also note that \(K' \subseteq K\) alternatively follows with \cite[Lemma
2]{DeyP2010} (see \cite{Dadush:de:vi:10} for a similar result).
\end{rem}
We further obtain
\begin{cor}
\label{cor:bringIntoRationalSpace}
Let \(K\) be a compact convex set.
Then finitely many CG\xspace cuts of \(K\) define
a polyhedron in a rational affine subspace \(V\)
with \(V \subseteq \aff{K}\).
\begin{proof}
The affine subspace \(\aff{K}\) is defined by
finitely many inequalities \(a_i x \leq b_i\)
with \(i \in [\ell]\) for some
\(\ell \in \mathbb{N}\).
These are consequences of finitely many CG\xspace cuts
via Lemma~\ref{lem:homogeneity} with \(\pi = a_i\), \(\pi_0 = b_i\)
and \(c = 0\), \(\delta = 0\).
Therefore the polyhedron defined by these CG\xspace cuts spans
a rational affine subspace \(V\) of \(\aff{K}\).
\end{proof}
\end{cor}
\section{The CG\xspace closure of a compact convex set}
\label{sec:gc-closure-compact}
We will now prove the main theorem:
\begin{thm}
\label{thm:cg-closure-polytope}
Let \(K\) be a compact convex set.
Then \(K'\) is a rational polytope
defined by finitely many CG\xspace cuts of \(K\).
\end{thm}
The proof will proceed via induction on the dimension of \(K\) using
the following step lemma.
\begin{lem}
\label{lem:boundary}
Let \(K\) be a compact convex set.
Let us assume that for every proper face \(F\) of \(K\),
the CG\xspace closure \(F'\) is defined by finitely many CG\xspace cuts of \(F\)
(i.e., Theorem~\ref{thm:cg-closure-polytope} holds for \(F\)).
Then
there is a polytope \(P\) defined by finitely many CG\xspace cuts of \(K\),
which is contained in \(K\)
and \(P \cap \partial K = K' \cap \partial K\).
\begin{proof}
For all unit vectors \(\pi\) in the lineality space of \(\aff{K}\),
let \(\pi x \leq \pi_{0}\) define
the associated supporting hyperplane of \(K\).
Now \(F_{\pi} = K \cap \face{ \pi x = \pi_0}\)
is a proper face, the \(\pi\)-face of \(K\),
and hence \(F_{\pi}'\) is defined
by finitely many CG\xspace cuts of \(F_{\pi}\) by our assumption.
By Lemma~\ref{lem:homogeneity} there are finitely many CG\xspace cuts
of \(K\) defining a polyhedron \(P_{\pi}\)
satisfying \(\pi x \leq \pi_{0}\)
and \(P_{\pi} \cap \face{ \pi x = \pi_0} = F_{\pi}'\).
For \(F_{\pi}' \neq \emptyset\) this means exactly
that the \(\pi\)-face of \(P_{\pi}\) is \(F_{\pi}'\).
By adding finitely many CG\xspace cuts,
we may assume that \(P_{\pi}\) is a polytope.
We claim that
for vectors \(\pi'\) in a neighbourhood \(U_{\pi}\) of \(\pi\),
the polytope \(P_{\pi}\) still satisfies
\(\pi' x \leq \pi'_{0}\)
and \(P_{\pi} \cap \face{ \pi' x = \pi'_0} = F_{\pi'}'\).
This is immediate
if \(F_{\pi}' = \emptyset\),
as \(F_{\pi'}\) is disjoint from \(P_{\pi}\)
by Lemma~\ref{lem:nearby-dir}.
If \(F_{\pi}' \neq \emptyset\),
let \(F_{P_{\pi}, \pi'}\) denote the \(\pi'\)-face of \(P_{\pi}\).
The inequality \(\pi' x \leq \pi'_{0}\)
is satisfied by \(P_{\pi}\) as \(F_{P_{\pi}, \pi'}\)
is also the \(\pi'\)-face of \(F'_{\pi}\).
If \(F_{P_{\pi}, \pi'}\) is not contained
in the \(\pi'\)-face \(F_{\pi'}\) of \(K\),
then \(P_{\pi}\) satisfies \(\pi' x < \pi'_{0}\) and
\(F_{\pi'}' = \emptyset\).
However,
if \(F_{P_{\pi}, \pi'}\) is contained in the \(\pi'\)-face \(F_{\pi'}\),
then clearly
\(F_{\pi'}' = F_{P_{\pi}, \pi'} = P_{\pi} \cap \face{ \pi' x = \pi'_{0}}\),
proving the claim.
We obtain an open cover of the unit
sphere of \(\aff{K}\) with neighborhoods \(U_\pi\)
such that for each \(\pi' \in U_\pi\)
we have
\(P_{\pi} \cap \face{ \pi' x = \pi'_0} = F_{\pi'}'\)
and \(\pi' x \leq \pi'_{0}\) for all \(x \in P_{\pi}\).
Since the unit sphere is compact,
it follows by choosing a finite subcover that
finitely many CG\xspace cuts define a polytope \(Q\)
with
\(Q \cap \face{ \pi x = \pi_0} = F_{\pi}'\)
and \(\pi x \leq \pi_{0}\) for all \(x \in Q\).
By Corollary~\ref{cor:bringIntoRationalSpace},
by adding finitely many cuts
we obtain a polytope \(P\) in a rational affine subspace of \(\aff{K}\),
which is contained in \(Q\).
In particular, it lies in \(K\) and
\(P \cap \partial K = K' \cap \partial K\).
\end{proof}
\end{lem}
Finally, we are ready to prove the main theorem.
\begin{proof}[Proof of Theorem~\ref{thm:cg-closure-polytope}]
The proof proceeds via induction on the dimension of \(K\).
By the induction hypothesis, the Theorem holds
for proper faces of \(K\).
From Lemma~\ref{lem:boundary} we know that
finitely many CG\xspace cuts define a polytope \(P\) in \(K\)
with \(P \cap \partial K = K' \cap \partial K\).
The polytope \(P\) spans
a rational affine subspace \(V\) of \(\aff{K}\).
Let \(D\) denote the orthogonal projection
of \(\mathbb{Z}^{n}\) onto the lineality space \(W\) of \(V\).
As \(W\) is rational, the orthogonal projection \(D\)
is a lattice.
We claim that there are only finitely many \(d \in D\)
with a preimage \(c \in \mathbb{Z}^{n}\),
for which a CG\xspace cut \(c x \leq \ffloor{\delta}\)
cuts out something from \(P\)
i.e., at least one vertex \(v\) of \(P\).
As vertices on the boundary of \(K\) belong to \(K'\),
these cannot be cut out. Therefore \(v\) has to be contained in the relative interior of
\(K\) and so does a small ball \(U\) in \(V\) around \(v\).
Let \(r\) denote the radius of \(U\).
Now whenever \(d \in D\) is too long, i.e., \(\tnorm{d} \geq 1/r\),
we have \(\max_{x \in K} c x \geq \max_{x \in U} cx \geq cv + 1\) as
\(x-v \in W\) and so \(cx-cv = dx - dv\). Hence \(cx \leq \ffloor{\max_{x
\in K} cx}\) cannot cut off \(v\). As there are only a finite number
of vertices \(v\) of \(P\),
there is a global upper bound on the length of the \(d\)
which could cut out a vertex in the relative interior of \(K\).
As \(D\) is discrete, there are only finitely many such vectors \(d\),
and adding these CG\xspace cuts to \(P\) we obtain \(K'\).
Actually, for every \(d\) we need to add only one cut,
thus defining \(K'\) by finitely many CG\xspace cuts, as claimed.
To prove this,
we consider all the CG\xspace cuts \(c x \leq \ffloor{\delta}\)
where \(c\) is a preimage of a fixed \(d \in D\).
We claim that unless \(P = \emptyset\)
(when \(P = K'\) and the theorem holds),
there is a deepest cut among these
This will be the only cut
we need to add to \(P\) for the vector \(d\).
To prove the last claim, let \(x_{0}\) be a rational point of \(V\).
Restricted to \(V\), every CG\xspace cut \(c x \leq \ffloor{\delta}\)
can be rewritten to \(d x \leq \ffloor{\delta} - (c - d) x_{0}\).
As \(\ffloor{\delta}\) is an integer,
\(c - d\) is rational with bounded denominator
(as \(D\) is discrete),
and \(x_{0}\) is rational,
therefore
the right-hand side can take only a discrete set of values,
and \(d x\) is a lower bound on the set of values for every
\(x \in P\).
Therefore there is a cut with \(\ffloor{\delta} - (c - d) x_{0}\)
minimal, which is obviously a deepest cut.
\end{proof}
\section*{Acknowledgements}
\label{sec:acknowledgements}
The authors are grateful to Daniel Dadush and Santanu Dey for
helpful comments and discussions and to the anonymous reviewers whose
comments improved the presentation significantly.
\bibliographystyle{abbrvnat}
| {
"timestamp": "2014-05-20T02:01:51",
"yymm": "1207",
"arxiv_id": "1207.4884",
"language": "en",
"url": "https://arxiv.org/abs/1207.4884",
"abstract": "Recently Schrijver's open problem, whether the Chvátal--Gomory closure of an irrational polytope is polyhedral was answered independently in the affirmative by Dadush, Dey, and Vielma (even for arbitrarily compact convex set) as well as by Dunkel and Schulz. We present a very short, easily accesible proof that the Chvátal--Gomory closure of a compact convex set is a polytope.",
"subjects": "Combinatorics (math.CO); Computational Geometry (cs.CG); Discrete Mathematics (cs.DM); Metric Geometry (math.MG)",
"title": "A short proof for the polyhedrality of the Chvátal-Gomory closure of a compact convex set",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754465603679,
"lm_q2_score": 0.7185943985973772,
"lm_q1q2_score": 0.7075104008947917
} |
https://arxiv.org/abs/2103.01338 | Acceleration via Fractal Learning Rate Schedules | In practical applications of iterative first-order optimization, the learning rate schedule remains notoriously difficult to understand and expensive to tune. We demonstrate the presence of these subtleties even in the innocuous case when the objective is a convex quadratic. We reinterpret an iterative algorithm from the numerical analysis literature as what we call the Chebyshev learning rate schedule for accelerating vanilla gradient descent, and show that the problem of mitigating instability leads to a fractal ordering of step sizes. We provide some experiments to challenge conventional beliefs about stable learning rates in deep learning: the fractal schedule enables training to converge with locally unstable updates which make negative progress on the objective. |
\section{Introduction}
In the current era of large-scale machine learning models, a single deep neural network can cost millions of dollars to train. Despite the sensitivity of gradient-based training to the choice of learning rate schedule, no clear consensus has emerged on how to select this high-dimensional hyperparameter, other than expensive end-to-end model training and evaluation. Prior literature indirectly sheds some light on this mystery, showing that the learning rate schedule governs tradeoffs between accelerated convergence and various forms of algorithmic stability.
In this work, we highlight the surprising consequences of these tradeoffs in a very simple setting: first-order optimization of a convex quadratic function. We start by pointing out the existence of a non-adaptive step size schedule, derived from the roots of Chebyshev polynomials, which allows plain gradient descent to obtain accelerated convergence rates without momentum. These learning rates overshoot the region of guaranteed local progress, resulting in unstable optimization trajectories. Extending a relatively obscure line of work motivated by numerical imprecision in PDE solvers \cite{lebedev1971order}, we show that stable acceleration is achieved by selecting a \emph{fractal} permutation of the Chebyshev step sizes.
Acceleration via large step sizes may provide an useful alternative to momentum: it is less stable according to our worst-case bounds, but inherits the memory-efficiency and statelessness of vanilla gradient descent. More broadly, we discuss how this form of acceleration might implicitly present itself in settings like deep learning, introducing hidden entanglements and experimental confounds. We hope that these ideas will lead to new adaptive algorithms which overstep the ``edge of stability'' (the largest constant learning rate at which model training converges) \citep{giladi2019stability,cohen2020gd}, and accelerate training via carefully scheduled negative progress. We provide some supporting experiments towards bridging the theory-practice gap, as well as open questions for future investigation.
\begin{figure}
\centering
\ifdefined\arxiv
\includegraphics[height=0.2\linewidth]{figures/semicircle.pdf}
\includegraphics[height=0.2\linewidth]{figures/sched_monotone.pdf}
\includegraphics[height=0.2\linewidth]{figures/sched_sigma.pdf}
\else
\includegraphics[height=0.26\linewidth]{figures/semicircle.pdf}
\includegraphics[height=0.26\linewidth]{figures/sched_monotone.pdf}
\includegraphics[height=0.26\linewidth]{figures/sched_sigma.pdf}
\fi
\caption{Visualization of the Chebyshev nodes $\gamma_t$, their corresponding step sizes $\gamma_t^{-1}$, and the fractal permutation \cite{lebedev1971order} studied in this paper.}
\label{fig:cheb-lr-overview}
\end{figure}
\subsection{Our contributions}
\paragraph{Provably stable acceleration without momentum.} We revisit an oft-neglected variant of the Chebyshev iteration method for accelerating gradient descent on convex quadratics. In lieu of momentum, it uses a recursively-defined sequence of large step sizes derived from Chebyshev polynomials, which we call the fractal Chebyshev schedule. We prove a new stability guarantee for this algorithm: under bounded perturbations to all the gradients, no iterate changes by more than $O(\mathrm{poly}(\kappa))$, where $\kappa$ is the condition number of the problem. We also some provide theoretically-grounded practical variants of the schedule, and negative results for function classes beyond convex quadratics.
\paragraph{Empirical insights on stable oscillating schedules.} We demonstrate empirically that the fractal Chebyshev schedule stabilizes gradient descent on objectives beyond convex quadratics. We observe accelerated convergence on an instance of multiclass logistic regression, and convergent training of deep neural networks at unstable learning rates. These experiments highlight the power of optimizing the ``microstructure'' of the learning rate schedule (as opposed to global features like warmup and decay). We discuss how these findings connect to other implicit behaviors of SGD and learning rate schedules.
\subsection{Related work}
The predominant algorithms for accelerated first-order optimization are the momentum methods of \citet{polyak1964some} and \citet{nesterov1983method}. The former, known as the \emph{heavy-ball} method, only achieves provable acceleration on quadratic objectives. The latter achieves minimax optimal convergence rates for general smooth convex objectives. Both are widely used in practice, far beyond their theoretical scope; for instance, they are the standard options available in deep learning frameworks.
\paragraph{Empirical challenges and tradeoffs.} \citep{bottou2007tradeoffs} discuss the competing objectives of stability, acceleration, and computation in large-scale settings, where one cannot afford to consider a single asymptotically dominant term. \citet{devolder2014first,chen2018stability,agarwal2020stochastic} study this specifically for acceleration. Optimizing the learning rate schedule remains a ubiquitous challenge; see Section~\ref{subsec:experiments-dl} and Appendix~\ref{subsec:appendix-lr-schedule-practice} for references.
\paragraph{Numerical methods and extremal polynomials.} There are many connections between algorithm design and approximation theory \citep{vishnoi2012laplacian,sachdeva2013faster}. We emphasize that the beautiful idea of the fractal permutation of Chebyshev nodes is an innovation by \citet{lebedev1971order,lebedev1973solution,lebedev1976utilization}; our technical results are generalizations and refinements of the ideas therein.
We give an overview of this line of work in Appendix~\ref{subsec:appendix-lebedev-literature}.
\paragraph{Learning rate schedules in stochastic optimization.} Bias-variance tradeoffs in optimization are studied in various theoretical settings, including quadratics with additive and multiplicative noise \citep{lan2012optimal,ge2019step,gorbunov2020unified}. Many of them also arrive at theoretically principled learning rate schedules; see Appendix~\ref{subsec:appendix-lr-schedule-theory}. On the more empirical side, \citet{zhang2019algorithmic} use a noisy quadratic model to make coarse predictions about the dynamics of large-scale neural net training. Cyclic learning rate schedules have been employed in deep learning, with various heuristic justifications \cite{loshchilov2016sgdr,smith2017cyclical,fu2019cyclical}. In parallel work, \cite{oymak2021super} considers a cyclic ``1 high, $n$ low'' schedule, which gives $\log(\kappa)$ convergence rates in the special case of convex quadratics whose Hessians have bimodal spectra. We discuss in Appendix~\ref{subsec:appendix-vanilla-spiky} why this approach does not provide acceleration in the general case; the MNIST experiments in Appendix~\ref{subsec:appendix-deeplearning} include a comparison with this schedule.
\section{Preliminaries}
\subsection{Gradient descent}
We consider the problem of iterative optimization of a differentiable function $f : \mathbb{R}^d \rightarrow \mathbb{R}$, with a first-order oracle $\nabla f : \mathbb{R}^d \rightarrow \mathbb{R}^d$ which computes the gradient of $f$ at a query point. The simplest algorithm in this setting is gradient descent, which takes an arbitrary initial iterate $x_1 \in \mathbb{R}^d$ and executes $T$ update steps
\begin{equation}
\label{eq:gd}
\{ x_{t+1} \leftarrow x_t - \eta_t \nabla f(x_t) \}_{t=1}^{T}
\end{equation}
according to a learning rate schedule $(\eta_1, \ldots, \eta_T)$, producing a final iterate $x_{\mathrm{out}} := x_{T+1}$. When the $\{\eta_t\}$ do not depend on $T$, an analogous infinite sequence of iterates $\{x_t\}_{t \in \mathbb{N}}$ can be defined.
There are many ways to choose the learning rate schedule, depending on the structure of $f$ and uncertainty in the gradient oracle. Some schedules are static (non-adaptive): $(\eta_1, \ldots, \eta_T)$ are chosen before the execution of the algorithm. For instance, when $f$ is an $M$-smooth convex function, $\eta_t = 1/M$ achieves the classical convergence rates.
Adaptive choices of $\eta_t$ are allowed to depend on the observed feedback from the current execution (including $x_t$ and $\nabla f(x_t)$), and are considerably more expressive. For example, $\eta_t$ can be chosen adaptively via line search, adaptive regularization, or curvature estimation.
\subsection{The special case of quadratics}
\label{subsec:prelims-quadratic-opt}
Consider the case where the objective is of the form
\[f(x) = \frac{1}{2} x^\top A x - b^\top x,\]
where $A \in \mathbb{R}^{d\times d}$ is symmetric and positive definite, and $b \in \mathbb{R}^d$,
so that $\nabla f(x) = Ax - b$ is an affine function of the query point $x$. Then, the mapping $\mathcal{G} : x_t \mapsto x_{t+1}$ induced by gradient descent is also affine. Let $x^* := \min f$ (a fixed point of $\mathcal{G}$). Then,
\begin{align*}
x_{t+1} - x^* &= \mathcal{G}(x_t) - x^* = \mathcal{G}(x_t) - \mathcal{G}(x^*) \\
&= (I - \eta_t A)(x_t - x^*).
\end{align*}
By induction, we conclude that
\begin{align*}
x_{\mathrm{out}} - x^* = \bra{ \prod_{t=1}^T (I - \eta_t A) }(x_1 - x^*).
\end{align*}
Thus, the residual after $T$ steps of gradient descent is given by a degree-$T$ matrix polynomial times the initial residual:
\begin{definition}[Residual polynomial]
Fix a choice of non-adaptive $(\eta_1, \ldots, \eta_T)$. Then, define the \emph{residual polynomial} $p : \mathbb{R}^{d \times d} \rightarrow \mathbb{R}^{d \times d}$ as
\[ p(A) := \prod_{t=1}^T (I - \eta_t A). \]
\end{definition}
When clear, we will interchange to denote scalar and matrix polynomials with the same coefficients. Thus, overloading $p : \mathbb{R} \rightarrow \mathbb{R}$, we have $p(0) = 1$, and $p(1/\eta_t) = 0$ for each $t$.
\begin{remark}
The matrices in the above product all commute. Thus, when $f$ is quadratic, $p(A)$ (and thus $x_{\mathrm{out}}$ given $x_1$) does not depend on the permutation of $(\eta_1, \ldots, \eta_T)$.
\end{remark}
\subsection{Chebyshev polynomials and Chebyshev methods}
\label{subsec:prelims-cheb}
The problem of choosing $p(A)$ to optimize convergence for least-squares has roots in numerical methods for differential equations \cite{richardson1911ix}. The Chebyshev polynomials, which appear ubiquitously in numerical methods and approximation theory \cite{chebyshev1853theorie,mason2002chebyshev}, provide a minimax-optimal solution \cite{flanders1950numerical,gavurin1950use,young1953richardson}\footnote{For a modern exposition, see the blogpost \url{http://fa.bianp.net/blog/2021/no-momentum/}.}: choose positive real numbers $m \leq M$, and set
\[ p(\lambda) = \frac{ \mathcal{T}_T\pa{ z } }{ \mathcal{T}_T(\theta) }, \]
where $z := \frac{M+m-2\lambda}{M-m}$, $\theta := \frac{M+m}{M-m} = 1 + \frac{2m}{M-m}$, and $\mathcal{T}_n(\cdot)$ is the degree-$n$ Chebyshev polynomial of the first kind. One of many equivalent definitions is $\mathcal{T}_n(z) = \cos (n \arccos z)$ for $|z| \leq 1$. From this definition it follows that the roots of $p$ occur at the \emph{Chebyshev nodes}
\[ \gamma_t := \frac{M+m}{2} - \frac{M-m}{2} \cos \frac{(t-\frac{1}{2})\pi}{T}, \; t = 1, \ldots, T. \]
Setting $\{\eta_t\}$ to be any permutation of $\{1/\gamma_t\}$ suffices to realize this choice of $p$. Note that $1/\gamma_t$ is decreasing in $t$.
The limiting case $m = M$ is gradient descent with a constant learning rate, and $p(\lambda) = (1 - \lambda/m)^T$.
Let $\lambda_{\min}, \lambda_{\max}$ denote the smallest and largest eigenvalues of $A$, so that the \emph{condition number} of $A$ is $\kappa := \lambda_{\max}/\lambda_{\min}$. Viewing $m,M$ as estimates for the spectrum, we define
\[\widehat{\kappa} := \frac{M}{m} \geq \frac{\lambda_{\max}}{\lambda_{\min}} = \kappa.\]
We state a classic end-to-end convergence rate for Chebyshev iteration (proven in Appendix~\ref{sec:appendix-cheb-background} for completeness):
\begin{theorem}[Convergence rate of Chebyshev iteration]
\label{thm:cheb-convergence-rate}
Choose spectral estimates $m \leq M$ such that $0 < m \leq \lambda_{\min} \leq \lambda_{\max} \leq M$. Then, setting $\{\eta_t\}$ to be any permutation of $\{1/\gamma_t\}$, the final iterate of gradient descent $x_{\mathrm{out}}$ satisfies the following:
\begin{align*}
\norm{x_{\mathrm{out}} - x^*} &\leq \frac{2\rho^T}{1+\rho^{2T}} \norm{x_1 - x^*} \\
&\leq e^{-\Omega(T)/\sqrt{\widehat{\kappa}}} \norm{x_1 - x^*},
\end{align*}
where $\rho := \frac{ \sqrt{M} - \sqrt{m} }{ \sqrt{M} + \sqrt{m} } \leq 1 - \Omega\pa{\frac{1}{\sqrt{\widehat{\kappa}}}}$.
\end{theorem}
Thus, accelerated methods like Chebyshev iteration get $\epsilon$-close to the minimizer in $O(\sqrt{\widehat{\kappa}} \log (1/\epsilon))$ iterations, a quadratic improvement over the $O(\widehat{\kappa} \log (1/\epsilon))$ rate of gradient descent with a constant learning rate. Theorem~\ref{thm:cheb-convergence-rate} is proven using approximation theory: show that $|p(\lambda)|$ is small on an interval containing the spectrum of $A$.
\begin{definition}[Uniform norm on an interval]
Let $p : \mathbb{R} \rightarrow \mathbb{R}$, and $m \leq M \in \mathbb{R}$. Define the norm
\[\norm{p}_{[m,M]} := \norm{p}_{L_\infty([m,M])} = \max_{\lambda \in [m,M]} |p(\lambda)|. \]
\end{definition}
Then, any upper bound on this norm gives rise to a convergence rate like Theorem~\ref{thm:cheb-convergence-rate}:
\[ \norm{x_{\mathrm{out}} - x^*} \leq \norm{p}_{[m,M]} \cdot \norm{x_1 - x^*}. \]
These can be converted into optimality gaps on $f$ by considering the polynomial $\lambda \, p^2(\lambda)$.
Moving beyond infinite-precision arithmetic, the optimization literature typically takes the route of \citet{stiefel1958kernel}, establishing a higher-order recurrence which ``semi-iteratively'' (iteratively, but keeping some auxiliary state) constructs the same final polynomial $p$. This is the usual meaning of the Chebyshev iteration method, and coincides with Polyak's momentum on quadratics.
This is where we depart from the conventional approach.\footnote{For instance, this is not found in references on acceleration \citep{bubeck2017convex,d2021acceleration}, or in textbooks on Chebyshev methods \citep{gottlieb1977numerical,higham2002accuracy}.} We revisit the idea of \emph{working directly with the Chebyshev step sizes}, giving a different class of algorithms with different trajectories and stability properties.
\section{The fractal Chebyshev schedule}
\label{sec:quadratic}
In this section, we work in the strongly\footnote{Accelerated rates in this paper have $O(1/T^2)$ analogues when $\lambda_{\min} = 0$ \citep{allen2016optimal}.} convex quadratic setting from Section~\ref{subsec:prelims-quadratic-opt}.
Our new contributions on top of the existing theory address the following questions:
\begin{enumerate}
\item[(1)] How noise-tolerant is gradient descent with Chebyshev learning rates, beyond numerical imprecision?
\item[(2)] How do we choose the ordering of steps?
\end{enumerate}
We first introduce the construction originally motivated by numerical error, which provides an initial answer to (2). Then, our extended robustness analysis provides an answer to (1), and subsequently a more refined answer to (2).
\subsection{Construction}
We begin with the construction from \cite{lebedev1971order}, defined below and visualized in Figure~\ref{fig:scheds}.
\begin{definition}[Fractal Chebyshev schedule]
Let $\sigma_1 := [1]$, and for each $T \geq 1$ a power of 2, define
\[\sigma_{2T} := \mathrm{interlace}(\sigma_T, 2T+1-\sigma_T), \]
where
\[\mathrm{interlace}([a_1 \ldots a_n], [b_1 \ldots b_n]) := [a_1 \; b_1 \; a_2 \; b_2 \ldots a_n \; b_n].\]
Then, for given $m \leq M$, and $T$ a power of 2, the \emph{fractal Chebyshev schedule} is the sequence of learning rates
\[ \eta_t := 1/\gamma_{\sigma_{T}(t)}, \quad t = 1, \ldots, T. \]
\end{definition}
Below are the first few nontrivial permutations $\sigma_T$:
\[\sigma_2 = [1\;2],\]
\[\sigma_4 = [1\;4\;2\;3],\]
\[\sigma_8 = [1\;8\;4\;5\;2\;7\;3\;6],\]
\[\sigma_{16} = [1\;16\;8\;9\;4\;13\;5\;12\;2\;15\;7\;10\;3\;14\;6\;11].\]
\begin{figure}
\centering
\ifdefined\arxiv
\includegraphics[width=0.8\linewidth]{figures/scheds_wide.pdf}
\else
\includegraphics[width=\linewidth]{figures/scheds.pdf}
\vspace{-5mm}
\fi
\caption{Shapes of the Chebyshev step sizes and fractal permutations. \emph{Left:} Step sizes in sorted order for $M=1$, and $m=1,\frac{1}{2},\ldots,\frac{1}{20}$ (black to blue). \emph{Right:} Permuted schedule with $M=1, m=\frac{1}{20}, T=128$ (red). Subsequences with strides $\{1, 4, 16, 64\}$ are overlaid, demonstrating self-similarity arising from the interlacing construction.}
\label{fig:scheds}
\end{figure}
\subsection{Basic properties}
We first list some basic facts about the unordered step sizes:
\begin{proposition}
\label{prop:cheb-basics}
For all $m < M$ and $T$, the fractal Chebyshev step sizes $\{\gamma_t^{-1}\}$ satisfy the following:
\begin{enumerate}
\item[(i)] $\frac{1}{M} < \gamma_t^{-1} < \frac{1}{m} = \frac{\widehat{\kappa}}{M}$.
\item[(ii)] The number of step sizes greater than $\frac{2}{M}$ is $\pa{ \frac{1}{2} - \epsilon }T$, where $0 \leq \epsilon \leq O(1/\widehat{\kappa})$ as $\widehat{\kappa} \rightarrow \infty$.
\item[(iii)] For $t \leq \frac{T}{2}$, we have $\gamma_t^{-1} < \frac{1}{m + \frac{2(M-m)t^2}{T^2}}$, and
\hspace{-3mm}$\frac{1}{T}\sum_{t=1}^T \gamma_t^{-1} = \frac{\tanh\pa{ T \, \mathrm{acosh}\left(\frac{2m}{M-m}\right)}}{\sqrt{Mm}} < \frac{1}{\sqrt{Mm}} = \frac{\sqrt{\widehat{\kappa}}}{M}.$
\end{enumerate}
\end{proposition}
Interpreting $m, M$ as estimates for $\lambda_{\min}, \lambda_{\max}$:
\begin{enumerate}
\item[(i)] \emph{Every} step size in the schedule exceeds the classic fixed learning rate of $1/\lambda_{\max}$. As $T$ gets large, the largest step approaches $1/\lambda_{\min}$, a factor of $\kappa$ larger.
\item[(ii)] For large $\kappa$, close to half of the step sizes \emph{overshoot} the stable regime $\eta \in [0, 2/\lambda_{\max}]$, where local progress on $f$ is guaranteed.
\item[(iii)] The large steps are neither highly clustered nor dispersed. The largest $\gamma_t^{-1}$ overshoots the stable regime by a factor of $\Theta(\kappa)$, but the average factor is only $O(\sqrt{\kappa})$.
\end{enumerate}
Next, some basic observations about the fractal schedule:
\begin{proposition}[Hierarchy and self-similarity]
\label{prop:cheb-fractal-basis}
For all $m,M,T$ and $0 \leq i \leq \log_2 T$:
\begin{enumerate}
\item[(i)] The largest $\frac{T}{2^i}$ steps $\eta_t$ in the fractal Chebyshev schedule occur when $t = 1 + 2^i(\tau-1)$, with $\tau = 1, \ldots, \frac{T}{2^i}$.
\item[(ii)] The subsampled sequence $\{\eta_{1+2^i(\tau-1)}\}$ has the same ordering as the fractal permutation of the same length:
\[ \eta_{1+2^i\tau} = \gamma^{-1}_{1+2^i(\tau' - 1)}, \quad \text{ where } \tau' = \sigma_{T/2^i}(\tau). \]
\end{enumerate}
\end{proposition}
Figure~\ref{fig:scheds} visualizes these observations, while Appendix~\ref{subsec:appendix-basics} contains formal statements and proofs.
\subsection{Self-stabilization via infix polynomial bounds}
Now, let us examine why the fractal ordering is needed. As discussed, in the noiseless infinite-precision setting, the final iterate $x_{\mathrm{out}}$ is invariant to the permutation of $\{\eta_t\}$. However, the intermediate iterates $x_t$ depend on a sequence of \emph{partial} products, which depend very sensitively on the permutation; Figure~\ref{fig:perm-stability} illustrates these tradeoffs; details are found in Appendix~\ref{subsec:appendix-perm-stability}.
\begin{figure}
\centering
\ifdefined\arxiv
\includegraphics[width=0.8\linewidth]{figures/perm_stability_wide.pdf}
\else
\includegraphics[width=\linewidth]{figures/perm_stability.pdf}
\vspace{-5mm}
\fi
\caption{The optimization trajectories of various permutations of the Chebyshev step sizes. \emph{Left:} In the noiseless case, the final iterates coincide, but $x_t$ can wander exponentially far away. \emph{Right:} With (i.i.d. Gaussian) noise, there is a tradeoff between $\norm{x_t}$ and the stability of $x_{\mathrm{out}}$.}
\label{fig:perm-stability}
\end{figure}
We motivate our first new results using an additive noise model; this is a refinement of \cite{lebedev1971order,lebedev1973solution,lebedev1976utilization}, which are only concerned with preventing exponential blowup of negligible perturbations at the numerical noise floor. We consider adding a sequence of perturbations $(\xi_1, \ldots, \xi_T)$ to gradient descent (Equation~\ref{eq:gd}):
\begin{equation}
\label{eq:perturbed-gd}
\{ x_{t+1} \leftarrow x_t - \eta_t \nabla f(x_t) + \xi_t \}_{t=1}^{T}.
\end{equation}
Note that this captures an inexact (e.g. stochastic) gradient oracle $\widetilde{\nabla f}(\cdot)$, in which case
\begin{equation}
\label{eq:gradient-noise}
\xi_t = \eta_t( \nabla f(x_t) - \widetilde{\nabla f}(x_t) ).
\end{equation}
Unrolling the recursion, we get:
\begin{gather*}
x_2 - x^* = (I - \eta_1 A) (x_1 - x^*) + \xi_1, \\
x_3 - x^* = (I - \eta_2 A)\bra{ (I - \eta_1 A)(x_1 - x^*) + \xi_1 } + \xi_2, \\
\cdots \\
x_t - x^* = p_{1:t-1}(A)(x_1 - x^*) + \sum_{t'=2}^{t} p_{t':t-1}(A) \xi_{t' - 1},
\end{gather*}
where we have defined the \emph{infix polynomial} as the (possibly empty) product
\[p_{s:t}(A) := \prod_{\tau=s}^t (I - \eta_\tau A).\]
\citet{lebedev1971order} give bounds on the norms of the \emph{prefix polynomials} $p_{1:t}$ and \emph{suffix polynomials} $p_{s:T}$:
\begin{theorem}[Prefix and suffix bounds]
\label{thm:lebedev-main}
For a fractal Chebyshev schedule with $m,M,T$, and all $1 \leq s \leq t \leq T$:
\begin{enumerate}
\item[(i)] $\norm{p_{1:t}}_{[m,M]} \leq \frac{ \widehat{\kappa} - 1 }{ 4^{\min(\mathrm{bits}(t))} } \prod_{j \in \mathrm{bits}'(t)} \frac{2}{1+\mathcal{T}_{2^j}(\theta)}$;
\item[(ii)] $\norm{p_{s:T}}_{[m,M]} \leq \prod_{j \in \mathrm{bits}(T+1-s)} \frac{2}{1+\mathcal{T}_{2^j}(\theta)}$,
\end{enumerate}
where $\mathrm{bits}(n)$ denotes the sequence $j_1 > j_2 > \ldots > j_k$ of indices in the binary expansion of $n$, and $\mathrm{bits}'(n) := \mathrm{bits}(n) \setminus j_k $. For example, when $n = 6 = 2^2 + 2^1$, $\mathrm{bits}(n) = \{2,1\}$, and $\mathrm{bits}'(n) = \{2\}$.
\end{theorem}
Let $\mathcal{V}(\cdot), \mathcal{V}'(\cdot)$ denote the bounds from Theorem~\ref{thm:lebedev-main},
so that $\norm{p_{1:t}}_{[m,M]} \leq \mathcal{V}'(t)$, and $\norm{p_{s:T}}_{[m,M]} \leq \mathcal{V}(T+1-s)$. Notice that $\mathcal{V}(t) \leq \frac{2}{1+\mathcal{T}_{\lfloor t/2 \rfloor}(\theta)} \leq e^{-\Omega(t)/\sqrt{\widehat{\kappa}}}$ for all $t \geq 1$,
and $\mathcal{V}'(t) \leq \widehat{\kappa} \mathcal{V}(t)$.
To fully understand the propagation of $\xi_t$ through Equation~\ref{eq:perturbed-gd}, we provide bounds on the infix polynomial norms:
\begin{theorem}[Infix polynomial bounds]
\label{thm:infix-bound}
For the fractal Chebyshev schedule with $m,M,T$, and all $1 \leq s \leq t \leq T$:
\[ \norm{p_{s:t}}_{[m,M]} \leq \mathcal{V}(\zeta+1-s) \cdot \mathcal{V}'(t-\zeta), \]
where $\zeta$ is the index such that $s-1 \leq \zeta \leq t$ and $\zeta, \zeta+1$ differ at the most significant bit.
\end{theorem}
Then, analyzing the decay of $\mathcal{V}, \mathcal{V}'$, we derive cumulative error bounds:
\begin{theorem}[Infix series bounds]
\label{thm:infix-series-bound}
For a fractal Chebyshev schedule with $m,M,T$, and all $1 \leq s \leq t \leq T$:
\[ \sum_{t'=s}^t \norm{p_{t':t}}_{[m,M]} \leq O(\widehat{\kappa}^{1 + \frac{1}{\ln 4}} \log \widehat{\kappa}) = o\pa{ \widehat{\kappa}^{1.73} }. \]
This bound, a sum of up to $T$ terms, is independent of $T$.
\end{theorem}
These require generalizations of the combinatorial proofs for Theorem~\ref{thm:lebedev-main},
presented (along with more precise statements) in Appendices~\ref{subsec:appendix-infix-proof} and \ref{subsec:appendix-infix-series-proof}.
\subsection{Implications for gradient descent}
Theorem~\ref{thm:infix-series-bound} translates to the following end-to-end statement about gradient descent with the fractal schedule:
\begin{corollary}
\label{thm:cheb-noise-stability}
Suppose $0 < m \leq \lambda_{\min} \leq \lambda_{\max} \leq M$. Then, gradient descent
with the fractal Chebyshev schedule of length $T$, and perturbations (as in Equation~\ref{eq:perturbed-gd}) such that $\norm{\xi_t} \leq \epsilon$, outputs iterates $x_t$ satisfying
\[ \norm{x_{t+1} - x^*} \leq \norm{p_{1:t}}_{[m,M]} \cdot \norm{x_1 - x^*} + o(\widehat\kappa^{1.73}) \cdot \epsilon. \]
Recall that Theorems~\ref{thm:lebedev-main} and \ref{thm:cheb-convergence-rate} guarantee
\ifdefined\arxiv
\[ \norm{p_{1:t}}_{[m,M]} \leq e^{-\Omega(T) \cdot \log(\widehat{\kappa})/\sqrt{\widehat{\kappa}}} \; ; \qquad \norm{p_{1:T}}_{[m,M]} \leq e^{-\Omega(T)/\sqrt{\widehat{\kappa}}}. \]
\else
\begin{align*} \norm{p_{1:t}}_{[m,M]} &\leq e^{-\Omega(T) \cdot \log(\widehat{\kappa})/\sqrt{\widehat{\kappa}}} \; ; \\
\norm{p_{1:T}}_{[m,M]} &\leq e^{-\Omega(T)/\sqrt{\widehat{\kappa}}}.
\end{align*}
\fi
\end{corollary}
The fractal schedule allows the stability factor to be independent of $T$. When the perturbations arise from noisy gradients (as in Equation~\ref{eq:gradient-noise}), so that each $\xi_t$ is $\eta_t \epsilon$-bounded, this factor becomes $o(\widehat{\kappa}^{2.73})$.
\paragraph{Provable benefit of negative progress.}
A striking fact about the fractal Chebyshev schedule is that this \emph{non-adaptive} method provably beats the minimax convergence rate of line search, the most fundamental \emph{adaptive} algorithm in this setting \citep{boyd2004convex}:
\begin{equation}
\label{eq:line-search}
\eta_{t}^{\mathrm{(ls)}} := \argmin_{\eta \geq 0} f(x_t - \eta \nabla f(x_t)).
\end{equation}
\begin{proposition}[No acceleration from line search]
\label{prop:line-search-bad}
On a strongly convex quadratic objective $f(x) = \frac{1}{2} x^\top A x + b^\top x$,
let $\{x_t\}$ be the sequence of iterates of gradient descent with the adaptive learning rate schedule $\eta_{t}^{\mathrm{(ls)}}$ from Equation~\ref{eq:line-search}. Then, for each $A,b$, there exists a setting of $x_1$ such that
\[ \norm{x_{t+1} - x^*} \geq \pa{1-\frac{1}{\Omega(\kappa)}}^T \!\!\!\! \cdot \norm{x_1 - x^*}, \quad \forall t \geq 1. \]
\end{proposition}
This is a classic fact; for a complete treatment, see Section~3.2.2 of \cite{kelley1999iterative}. In the context of our results, it shows that greedily selecting the locally optimal learning rates is provably suboptimal, even compared to a feedback-independent policy.
Adaptive estimation of the local loss curvature is an oft-attempted approach, amounting to finding the best conservative step size $\frac{1}{M}$. Proposition~\ref{prop:line-search-bad} suggests that although such methods have numerous advantages, greedy local methods can miss out on acceleration. The fact that acceleration can be obtained from carefully scheduled overshooting is reminiscent of simulated annealing \cite{aarts1989simulated}, though we could not find any rigorous connections.
\paragraph{Comparison with momentum.} We stress that this form of acceleration does not replace or dominate momentum. The dependence of the stability term on $\widehat{\kappa}$ is suboptimal \citep{devolder2014first}. In exchange, we get a \emph{memoryless} acceleration algorithm: gradient descent has no auxiliary variables or multi-term recurrences, so that $x_t$ fully specifies the state. This bypasses the subtleties inherent in restarting stateful optimizers \citep{o2015adaptive,loshchilov2016sgdr}.
Finally, our theory (especially Theorem~\ref{thm:underoverstepping}) implies that experiments attempting to probe the acceleration benefits of momentum might be confounded by the learning rate schedule, even in the simplest of settings (thus, certainly also in more complicated settings, like deep learning).
\subsection{Brief overview of proof ideas}
Figure~\ref{fig:perm-stability} suggests that there is a tradeoff between taking large $\Omega(1/m)$ steps for acceleration vs. small $O(1/M)$ steps for stability. To get acceleration, we must take all of the large steps in the schedule. However, we must space them out: taking $k = o(T)$ of the largest steps consecutively incurs an exponential blowup in the infix polynomial:
\begin{align*}
\prod_{i=1}^k \norm{ \pa{1 - \frac{\lambda}{\gamma_i}} }_{[m,M]} \!\!\!\! \approx \norm{ \pa{1 - \frac{\lambda}{m}}^k }_{[m,M]} \!\!\!\! = \pa{\widehat{\kappa} - 1}^k\!\!.
\end{align*}
The difficulty arises from the fact that there are not enough small steps in the schedule, so that a large step will need to be stabilized by \emph{internal copies of Chebyshev iteration}. This is why the fractal schedule is necessary. Theorem~\ref{thm:infix-bound} shows that this is surprisingly possible: the fractal schedule is only as unstable as the largest single step.
This intuition does not get us very far towards an actual proof: the internal copies of Chebyshev iteration, which form a complete binary tree, are ``skewed'' in a way that is sometimes better, sometimes worse. Isolating a combinatorial \emph{tree exchange lemma} used to prove Theorem~\ref{thm:lebedev-main}, we can iteratively swap two special infix polynomials with two others, and localize ``bad skewness'' to only one large step. Theorem~\ref{thm:infix-bound} follows from decomposing each infix into two infixes amenable to the tree exchange procedure. Theorem~\ref{thm:infix-series-bound} follows by combining Theorem~\ref{thm:infix-bound} with sharpened generalizations of the original paper's series bounds.
The proofs involve delicate trigonometric inequalities and various interesting facts about the geometry of polynomials. Appendices~\ref{sec:appendix-cheb-background}, \ref{sec:appendix-lebedev}, and \ref{sec:appendix-proofs} build up to self-contained proofs.
\section{Extensions and variants}
\label{sec:extensions}
Next, we explore some theoretically justified variants.
\subsection{Useful transformations of the fractal schedule}
\paragraph{Reversing the schedule.} Notice that the first step $\eta_1$ is the largest step in the schedule. This might not be desirable when $\xi_t$ is proportional to $\norm{x-x^*}$ (like in linear regression with minibatch SGD noise). It is a simple consequence of the symmetries in the main theorems that reversing the fractal Chebyshev schedule produces a contractive variant:
\begin{proposition}
\label{prop:cheb-sched-reverse}
Suppose we run gradient descent with the reversed fractal Chebyshev schedule
$\sigma_{T}(T+1-t)$. Then:
\begin{enumerate}
\item[(i)] For any $1 \leq t < t' \leq T$, we have
\[ \overline{\norm{p_{1:t}}}_{[m,M]} \leq \overline{\norm{p_{1:t'}}}_{[m,M]} \leq 1, \]
where $\overline{\norm{\cdot}}$ denotes the corresponding suffix norm bound from Theorem~\ref{thm:lebedev-main} (ii).
\item[(ii)] The bounds from Theorem~\ref{thm:lebedev-main} are swapped: replace $(p_{1:t},p_{s:T}) \rightarrow (p_{T+1-t:T}, p_{1:T+1-s})$.
\item[(iii)] Theorem~\ref{thm:infix-bound} holds, swapping $\mathcal{V} \leftrightarrow \mathcal{V}'$. Theorem~\ref{thm:infix-series-bound} holds.
\end{enumerate}
\end{proposition}
\paragraph{Concatenating schedules.} One can also repeat the fractal Chebyshev schedule indefinitely.\footnote{This is known as a cyclic iterative method, and was in fact the original motivation for \citep{lebedev1971order}.} Note that each infix polynomial of a repeated schedule can be written as a product of one prefix $p_{1:t}$, one suffix $p_{s:T}$, and a power of $p_{1:T}$, so stability bounds analogous to Theorems~\ref{thm:infix-bound} and \ref{thm:infix-series-bound} follow straightforwardly. It is also possible to concatenate schedules with different lengths $T$. Choosing $T$ to be successive powers of 2, one obtains an infinitely long schedule suitable for unknown time horizons.
\subsection{Conservative overstepping and partial acceleration}
\label{subsec:underoverstepping}
In this section, we decouple the eigenvalue range $[\lambda_{\min}, \lambda_{\max}]$ from the Chebyshev node range $[m,M]$ used in constructing the schedule. This can simply arise from an incorrect estimation of the eigenvalue range. However, more interestingly, if we think of $[m, M]$ as purposefully omitting the lower spectrum of $A$ (and thus taking smaller large steps), this allows us to interpolate between the fractal Chebyshev schedule and the vanilla constant learning rate.
\paragraph{Easy cases.} If $m < \lambda_{\min}$ or $M > \lambda_{\max}$, then $[m,M]$ is still an interval containing the spectrum of $A$; it is simply the case that convergence rates and stability bounds will depend on a worse $\widehat{\kappa} > \kappa$. On the other hand, if $M < \lambda_{\max}$, the residual blows up exponentially.
The subtle case is when $m > \lambda_{\min}$, when we are overstepping with restraint, trading off acceleration for stability via more conservative step sizes. This requires us to reason about $\norm{p}_{[\lambda_{\min}, M]}$ when $p$ was constructed to shrink $\norm{p}_{[m, M]}$. Analyzing this case, we get \emph{partial} acceleration:
\begin{theorem}
\label{thm:underoverstepping}
Given a quadratic objective with matrix $A$ and $0 < \lambda_{\min} \leq m \leq \lambda_{\max} \leq M$, gradient descent with the Chebyshev step sizes results in the following convergence guarantee:
\[
\|x_{\mathrm{out}} - x^*\| \leq 2\left(1 - \phi^{-1}(\lambda_{\min},m,M) \right)^T \cdot \|x_1 - x^*\|,
\]
with
\ifdefined\arxiv
\[
\phi^{-1}(\lambda_{\min},m,M) :=
2 \cdot \frac{ \lambda_{\min} + \sqrt{Mm} - \sqrt{(M-\lambda_{\min})(m-\lambda_{\min})}}{ (\sqrt{M} + \sqrt{m})^2 }.
\]
\else
\begin{multline*}
\phi^{-1}(\lambda_{\min},m,M) \\ :=
2 \cdot \frac{ \lambda_{\min} + \sqrt{Mm} - \sqrt{(M-\lambda_{\min})(m-\lambda_{\min})}}{ (\sqrt{M} + \sqrt{m})^2 }.
\end{multline*}
\fi
\end{theorem}
This is an interpolation between the standard and accelerated convergence rates of $O(\kappa \log(1/\epsilon))$ and $O(\sqrt{\kappa} \log(1/\epsilon))$. Figure~\ref{fig:understepping} shows the shape of $\phi$ for $m \in [\lambda_{\min},M]$, as it ranges from $\sim\sqrt{\kappa} \rightarrow \kappa$.
\begin{figure}
\centering
\ifdefined\arxiv
\includegraphics[width=0.8\linewidth]{figures/underoverstep_wide.pdf}
\else
\includegraphics[width=\linewidth]{figures/underoverstep.pdf}
\vspace{-5mm}
\fi
\caption{Summary of the discussion in Section~\ref{subsec:underoverstepping}. Suboptimal decay times $\phi(\lambda_{\min}=0.01,m,M=1)$ interpolate between the standard and accelerated rates. Green curves correspond to settings of $m < \lambda_{\min}$ where Theorem~\ref{thm:cheb-convergence-rate} applies; notice the distorted horizontal scale.}
\label{fig:understepping}
\end{figure}
\subsection{Existence of clairvoyant non-adaptive schedules}
\label{subsec:clairvoyant}
Finally, we present one more view on the provable power of tuning (i.e. searching globally for) a learning rate schedule on a fixed problem instance. An ambitious benchmark is the conjugate gradient method \cite{hestenes1952methods}, which is optimal for \emph{every} (rather than the worst-case) choice of $A, b$. That is, at iteration $t$, it outputs
\[x_{t+1} := \argmin_{\substack{\deg p \leq t \\ p(0) = 1}} \norm{ p(A) (x_1 - x^*) }_A ,\]
where $\norm{x}_A := \sqrt{x^\top A x}$. This can be much stronger than the guarantee from Theorem~\ref{thm:cheb-convergence-rate} (e.g. when the eigenvalues of $A$ are clustered). In Appendix~\ref{subsec:appendix-cg}, we prove that there are non-adaptive (but instance-dependent) learning rate schedules that compete with conjugate gradient:
\begin{theorem}[Conjugate gradient schedule; informal]
\label{thm:cg-sched}
For every problem instance $(A, b)$, there is a learning rate schedule $\{\eta_t\}$ for gradient descent, with each $\eta_t \in [\frac{1}{\lambda_{\max}}, \frac{1}{\lambda_{\min}}]$, such that $x_{\mathrm{out}}$ is the output of conjugate gradient.
\end{theorem}
\section{Beyond convex quadratics}
\subsection{General convex objectives: a counterexample}
\label{subsec:logcosh}
A mysterious fact about acceleration is that some algorithms and analyses transfer from the quadratic case to general convex functions, while others do not. \cite{lessard2016analysis} exhibit a smooth and strongly convex non-quadratic $f$ for which Polyak's momentum gets stuck in a limit cycle.
For us, $f(x) = \log \cosh (x) + 0.01 x^2$ serves as a one-dimensional ``proof by simulation'' that gradient descent with the fractal Chebyshev schedule can fail to converge. This is shown in Appendix~\ref{subsec:appendix-logcosh}; note that this is a tiny instance of ridge logistic regression.
\subsection{Non-convex objectives: a no-go}
None of this theory carries over to worst-case non-convex $f$: the analogue of Theorem~\ref{thm:cg-sched} is vacuously strong. We point out that global optimization of the learning rate schedule is information-theoretically intractable.
\begin{proposition}[Non-convex combination lock; informal]
\label{prop:nonconvex-combination-lock}
For every ``passcode'' $\{\eta_1^*, \ldots, \eta_T^*\}$ and $\delta > 0$, there is a smooth non-convex optimization problem instance $(f(\cdot), x_1)$ for which the final iterate $x_{\mathrm{out}}$ of gradient descent is an $1$-approximate global minimum only if
\[ |\eta_t - \eta^*_t| \leq \delta, \quad \forall t = 1, \ldots, T. \]
\end{proposition}
A formal statement and proof are given in Appendix~\ref{subsec:appendix-nonconvex}.
\subsection{More heuristic building blocks}
\label{subsec:heuristics}
With Polyak momentum as the most illustrious example, an optimizer can be very useful beyond its original theoretical scope. We present some more ideas for heuristic variants (unlike the theoretically justified ones from Section~\ref{sec:extensions}):
\paragraph{Cheap surrogates for the fractal schedule.} The worst-case guarantees for Chebyshev methods depend sensitively on the choice of nodes. However, beyond worst-case objectives, it might suffice to replace $\{\gamma_t^{-1}\}$ with any similarly-shaped distribution (like the triangular one considered by \cite{smith2017cyclical}), and $\sigma$ with any sequence that sufficiently disperses the large steps. We show in Appendix~\ref{subsec:appendix-vanilla-spiky} that acceleration cannot arise from the simple cyclic schedule from \cite{oymak2021super}. An intriguing question is whether adaptive gradient methods or the randomness of SGD implicitly causes partial acceleration, alongside other proposed ``side effect'' mechanisms \cite{keskar2016large,jin2017escape,staib2019escaping}.
\paragraph{Inserting slow steps.}
We can insert any number of steps $\eta \in [0, \frac{2}{M}]$ at any point in a schedule without worsening stability or convergence, because $\norm{(1-\eta \lambda)}_{[m,M]} \leq 1$. That is, $\norm{p_{s':t'}}$ in the supersequence is bounded by the corresponding $\norm{p_{s:t}}$ in the original schedule, and Theorems~\ref{thm:infix-bound} and \ref{thm:infix-series-bound} apply.
A special case of this is \emph{warmup} or \emph{burn-in}: take any number of small steps at the beginning.
Another option is to insert the small steps cyclically: notice from Propositions~\ref{prop:cheb-basics} (ii) and \ref{prop:cheb-fractal-basis} (i) that the steps $\{\eta_t\}$ come in ``fast-slow'' pairs: an odd step overshoots, and an even step corrects it. This suggests further heuristics, like the following ``Chebyshevian waltz'': in minibatch SGD, run triplets of iterations with step sizes $(\eta_{2t-1}, \eta_{2t}, \frac{1}{M})$.\footnote{In non-GPU-bound regimes \citep{choi2019faster,agarwal2020disentangling} and deep RL, one can sometimes take these steps for free, without causing a time bottleneck.} In theory, this degrades the worst-case convergence rate by a constant factor, but improves stability by a constant factor.
\section{Experiments}
\subsection{Convex problems and non-local progress}
In spite of the simple negative result in Section~\ref{subsec:logcosh}, we find that the fractal Chebyshev schedule can exhibit accelerated convergence beyond quadratic objectives. Figure~\ref{fig:convex-preview} shows training curves for logistic regression for MNIST classification; details are in Appendix~\ref{subsec:appendix-convex-experiments}. We leave a theoretical characterization of the schedule's acceleration properties on general convex functions to future work; this may require further assumptions on ``natural'' problem instances beyond minimax bounds.
\begin{figure}
\centering
\ifdefined\arxiv
\includegraphics[width=0.8\linewidth]{figures/convex_wide.pdf}
\else
\includegraphics[width=\linewidth]{figures/convex.pdf}
\vspace{-5mm}
\fi
\caption{Logistic regression/MNIST training loss curves. \emph{Left:} Standard algorithms, with constant (more opaque = larger) learning rates. \emph{Right:} A fractal Chebyshev schedule.}
\label{fig:convex-preview}
\end{figure}
\subsection{Beyond the edge of stability in deep learning}
\label{subsec:experiments-dl}
We provide a small set of deep learning experiments, finding that the fractal Chebyshev schedule can overstep the empirical ``edge of stability'' (i.e. the largest constant multiplier on the learning rate for which training does not diverge). Figure~\ref{fig:cifar-preview} gives an overview of these findings; details are in Appendix~\ref{subsec:appendix-deeplearning}.
\begin{figure}
\centering
\ifdefined\arxiv
\includegraphics[width=0.8\linewidth]{figures/cifar10-small_wide.pdf}
\else
\includegraphics[width=\linewidth]{figures/cifar10-small.pdf}
\vspace{-5mm}
\fi
\caption{ResNet-18/CIFAR-10 training with batch size $8192$ and a repeated $T=8$ fractal Chebyshev schedule. \emph{Left:} Training loss curves. \emph{Right:} Learning rates; the schedule pokes through the edge of stability (magenta and red) without destabilizing training.}
\label{fig:cifar-preview}
\end{figure}
Estimating the scale of $\lambda_{\max}(\nabla^2 f)$ is an old paradigm for selecting learning rates \cite{lecun1993automatic,schaul2013no}; there are many proposed mechanisms for the success of larger learning rates. Our theory (especially Theorem~\ref{thm:underoverstepping}) and experiments point to the possibility of \emph{time-varying} schedules to enable larger learning rates, on a much finer scale than cyclic restarts \cite{loshchilov2016sgdr,smith2017cyclical,fu2019cyclical}.
A nascent line of work also challenges the classical $\eta_t \sim 1/\lambda_{\max}$ wisdom from an empirical angle \citep{cohen2020gd}, finding a phenomenon dubbed \emph{progressive sharpening} during normal (smooth $\eta_t$) training.
End-to-end improvements on training benchmarks are outside the scope of this work: the learning rate schedule interacts with generalization \citep{jiang2020characterizing}, batch normalization + weight decay \citep{li2019exponential}, batch size \citep{smith2018don}, adaptive preconditioners \citep{agarwal2020disentangling} and now (from this work) acceleration. This adds yet one more perspective on why it is so difficult to standardize experimental controls and ablations in this space. Analogously, it has been proposed that momentum acts as a variance reduction mechanism \citep{li2017stochastic,cutkosky2019momentum}, alongside its classical role in acceleration.
As an invitation to try these ideas in various experimental settings, we provide in Appendix~\ref{sec:appendix-code} some Python code to generate Chebyshev learning rates and fractal schedules.
\section{Conclusion}
We have revisited a lesser-known acceleration algorithm which uses a fractal learning rate schedule of reciprocal Chebyshev nodes, proved a stronger stability guarantee for its iterates, and developed some practical variants. Our experiments demonstrate promising empirical behaviors of the schedule beyond low-noise quadratics. We hope that this work provides new foundations towards investigating local optimization algorithms which take carefully scheduled ``leaps of faith''.
\paragraph{Open questions.} We conclude with some natural follow-up questions for future work:
\begin{itemize}
\item Find ``reasonable''\footnote{One example which is unreasonable in every way: run conjugate gradient ahead of time, maintaining monomial-basis expansions of the $A$-orthogonal basis. Compute the roots of the final polynomial, and use their inverses as a learning rate schedule.} (computationally efficient, oracle-efficient, and perturbation-stable) adaptive learning rate schedulers with accelerated convergence rates. What are the acceleration properties of commonly-used adaptive step size heuristics \citep{duchi2011adaptive,kingma2014adam,ward2019adagrad}?
\item Do there exist learning rate schedules (adaptive or non-adaptive) which obtain the accelerated rate for general strongly convex $f$, as opposed to only quadratics?
\end{itemize}
\section*{Acknowledgments}
We are grateful to Sham Kakade for helpful discussions and pointers to prior literature.
Special thanks go to Maria Ratskevich for helping with the translation of \cite{lebedev1971order}.
\section{Code snippets}
\label{sec:appendix-code}
Below, we provide some Python code to compute the Chebyshev step sizes $\{1/\gamma_t\}$, and the permutation $\sigma_T$ that generates the fractal Chebyshev schedule $\{\eta_t\}$.
\begin{minted}{python}
import numpy as np
def cheb_steps(m, M, T):
C, R = (M+m)/2., (M-m)/2.
thetas = (np.arange(T)+0.5)/T * np.pi
return 1./(C - R*np.cos(thetas))
def cheb_perm(T):
perm = np.array([0])
while len(perm) < T:
perm = np.vstack([perm, 2*len(perm)-1-perm]).T.flatten()
return perm
steps = cheb_steps(0.1, 1, 8) # [9.20, 5.69 ... 1.01]
perm = cheb_perm(8) # [0, 7, 3, 4, 1, 6, 2, 5]
steps[perm] # [9.20, 1.01 ... 1.25]
\end{minted}
\section{Notation and background on Chebyshev polynomials}
\label{sec:appendix-cheb-background}
First, we gather the notation and classic results on Chebyshev polynomials that will be useful for the proofs in Appendices~\ref{sec:appendix-lebedev}, \ref{sec:appendix-proofs}, and \ref{sec:appendix-proofs-misc}.
\subsection{Definitions}
To review the notation in Section~\ref{subsec:prelims-cheb}, for given values of $m, M, T$, we have defined the following:
\begin{itemize}
\item The shifted and scaled Chebyshev polynomial construction: set \[ p(\lambda) := \frac{ \mathcal{T}_T\pa{ z } }{ \mathcal{T}_T(\theta) } \qquad\text{ where } z := \frac{M+m-2\lambda}{M-m}, \quad \theta := \frac{M+m}{M-m} = 1 + \frac{2m}{M-m}.\]
We will keep using the auxiliary notation from above, which is useful in switching between different coordinate systems using the bijection $\lambda \leftrightarrow z$ which allows us to switch between the horizontal scales of $p(\lambda)$ and its corresponding $\mathcal{T}(z)$. This bijection maps $\lambda \in [m, M]$ to $z \in [-1, 1]$.
Note that $\theta$ is the value of $z$ corresponding to applying the above bijection to $\lambda=0$.
\item A characterization of $p$ by its roots: \[ \gamma_t := \frac{M+m}{2} - \frac{M-m}{2} \cos \frac{(t-\frac{1}{2})\pi}{T}, \quad t = 1, \ldots, T. \]
\end{itemize}
If $m = \lambda_{\min}(A)$ and $M = \lambda_{\max}(A)$, then $M/m$ coincides with $\kappa := \lambda_{\max}(A)/\lambda_{\min}(A)$, the condition number of the matrix $A$. Then, we can think of $\theta = 1 + \frac{2}{\kappa-1}$.
Next, we review some well-known facts about Chebyshev polynomials beginning with the definition.
\begin{definition}[Chebyshev polynomials \cite{chebyshev1853theorie}]
For each $n \geq 0$, the Chebyshev polynomials of the first kind $\mathcal{T}_n(z)$ are defined by the following recurrence:
\begin{itemize}
\item $\mathcal{T}_0(z) = 1$.
\item $\mathcal{T}_1(z) = z$.
\item $\mathcal{T}_n(z) = 2z T_{n-1}(z) - T_{n-2}(z)$, for all $n \geq 2$.
\end{itemize}
\end{definition}
The equivalence of the above with the alternate definition ($\mathcal{T}_n(z) = \cos (n \arccos z)$ for $|z| \leq 1$) follows by verifying the base cases and applying the cosine sum-of-angles formula.
\subsection{Basic lemmas}
We gather some basic lemmas used to prove the results in \cite{lebedev1971order}, which are classic.
\begin{lemma}[Alternative characterizations of the Chebyshev polynomials]
\label{lem:cheb-alt}
The following are true for all non-negative integers $n$ and $|z| \geq 1$:
\begin{enumerate}
\item[(i)] $T_n(z) = \pm \cosh( n \; \mathrm{acosh}(z) ).$ The sign is $+1$ when $z$ is positive, and $(-1)^n$ when $z$ is negative.
\item[(ii)] We have
\[\mathcal{T}(z) = \frac{\pa{z - \sqrt{z^2 - 1}}^n + \pa{z + \sqrt{z^2 - 1}}^n}{2}.\]
\end{enumerate}
\end{lemma}
\begin{proof}
(i) follows from recursively applying the identities $\cosh(a+b) = \sinh(a)\sinh(b) + \cosh(a)\cosh(b)$ and $\sinh^2(a) = \cosh^2(a) - 1$. (ii) follows from (i), performing the hyperbolic substitution $z = \cosh(a)$, noticing that all odd-powered $\sinh(a)$ terms cancel, and verifying the base cases and recurrence relation.
\end{proof}
Combining the $\cos$ and $\cosh$ characterizations of $\mathcal{T}_n(z)$, we obtain the composition property: for all integers $k, n \geq 0$ and all $z \in \mathbb{R}$,
\[ \mathcal{T}_{kn}(z) = \mathcal{T}_k( \mathcal{T}_n(z) ). \]
The half-angle cosine formulas will be useful: for any $\alpha, z \in \mathbb{R}$, and positive even $n$,
\[ \cos(\theta) \; = \; 2\cos^2(\theta/2) - 1 \; = \; 1 - 2\sin^2(\theta/2); \qquad \mathcal{T}_n(z) = 2\mathcal{T}^2_{n/2}(z) - 1. \]
The third statement is true for all $z$, since it is the composition property with $k=2$.
The key reason why we are interested in the Chebyshev polynomials is their \emph{extremal} property: outside the range of their roots, they expand faster than any other polynomial of the same degree. There are various ways to formalize this. We will only need the following:
\begin{lemma}[Expansion lower bounds]
\label{lem:cheb-expand}
For all $\delta \geq 0$, each $\mathcal{T}_n$ satisfies the following:
\begin{enumerate}
\item[(i)] $\mathcal{T}_n(1+\delta) = \frac{(1 + \delta + \sqrt{2\delta+ \delta^2})^{2n} + 1}{ 2(1 + \delta + \sqrt{2\delta+ \delta^2})^n } \geq \frac{(1 + \sqrt{2\delta})^n}{2} .$
\item[(ii)] $\mathcal{T}_n(1+\delta) \geq 1 + n^2 \delta.$
\end{enumerate}
\end{lemma}
\begin{proof}
To prove (i) note that, using Lemma \ref{lem:cheb-alt}(ii), we have that
\[\mathcal{T}_n(1+\delta) = \frac{(1 + \delta + \sqrt{2\delta+ \delta^2})^n + (1 + \delta - \sqrt{2\delta+ \delta^2})^n}{2} = \frac{(1 + \delta + \sqrt{2\delta+ \delta^2})^{2n} + 1}{ 2(1 + \delta + \sqrt{2\delta+ \delta^2})^n }, \]
where the last equality follows by noticing that $(1 + \delta - \sqrt{2\delta+ \delta^2})^{-1} = (1 + \delta + \sqrt{2\delta+ \delta^2})$.
The inequality in (i) is concluded by noticing that
\[\frac{(1 + \delta + \sqrt{2\delta+ \delta^2})^n + (1 + \delta - \sqrt{2\delta+ \delta^2})^n}{2} \geq \frac{(1 + \sqrt{2\delta})^n}{2},\]
by dropping the positive terms. To conclude (ii), we perform a series expansion upto degree 2 to get,
\begin{align*}
\frac{(1 + \delta + \sqrt{2\delta+ \delta^2})^n + (1 + \delta - \sqrt{2\delta+ \delta^2})^n}{2} &\geq \frac{2 + 2n\delta + \frac{n(n-1)}{2}(4\delta + 4\delta^2) }{2}\\
&= 1 + n^2 \delta.
\end{align*}
\end{proof}
\paragraph{Classic convergence rate of Chebyshev iteration.} Though Theorem~\ref{thm:cheb-convergence-rate} is classic, the exact statement of the convergence rate has several variants. For sake of completeness, we give a quick proof of the convergence rate of Chebyshev iteration implied by the exact formula in Lemma~\ref{lem:cheb-expand} (i):
\begin{manualtheorem}{\ref{thm:cheb-convergence-rate}}
Choose spectral estimates $m \leq M$ such that $0 < m \leq \lambda_{\min} \leq \lambda_{\max} \leq M$. Then, setting $\{\eta_t\}$ to be any permutation of $\{1/\gamma_t\}$, the final iterate of gradient descent $x_{\mathrm{out}}$ satisfies the following:
\begin{align*}
\norm{x_{\mathrm{out}} - x^*} &\leq \frac{2\rho^T}{1+\rho^{2T}} \norm{x_1 - x^*} \leq e^{-\Omega(T)/\sqrt{\widehat{\kappa}}} \norm{x_1 - x^*},
\end{align*}
where $\rho := \frac{ \sqrt{M} - \sqrt{m} }{ \sqrt{M} + \sqrt{m} } \leq 1 - \Omega\pa{\frac{1}{\sqrt{\widehat{\kappa}}}}$.
\end{manualtheorem}
\begin{proof}
First, assume $m<M$. Then, we have
\[ \|x_{\mathrm{out}} - x^*\| \leq \|p\|_{[\lambda_{\min}, \lambda_{\max}]} \cdot \|x_{\mathrm{out}} - x^*\| \leq \|p\|_{[m,M]} \cdot \|x_{\mathrm{out}} - x^*\|. \]
So, we need to bound $\|p\|_{[m,M]}$.
Setting $\delta = \theta-1 = \frac{2m}{M-m}$, notice that $1+\delta+\sqrt{2\delta + \delta^2} = 1/\rho$. Then, using Lemma~\ref{lem:cheb-expand} (i), we have
\[\|p\|_{[m,M]} = \max_{|z|\leq 1} \frac{\mathcal{T}_T(z)}{\mathcal{T}_T(\theta)}
= \frac{1}{\mathcal{T}_T(\theta)}
= \frac{2\rho^{-T}}{1+\rho^{-2T}}
= \frac{2\rho^{T}}{1+\rho^{2T}},
\]
as desired.
When $m = M$, the inequality is trivially true because $\rho$ and $\norm{p}_{[m,M]}$ are both 0.
\end{proof}
\section{Theorems and proofs from \cite{lebedev1971order}}
\label{sec:appendix-lebedev}
In the hope of bridging old algorithmic ideas from numerical methods with modern optimization for machine learning, we present a self-contained exposition of the results and proofs from \cite{lebedev1971order} used in this paper. This is far from an exact translation from the original Russian-language manuscript, whose exposition is somewhat terse. We provide some more intuitive proofs, fix some small (inconsequential) typos, change some notation to match this paper, isolate lemmas which are useful for proving our other results, and omit some weaker and irrelevant results.
\subsection{Skewed Chebyshev polynomials}
First, we show an equivalent divide-and-conquer root-partitioning construction of the fractal permutation $\sigma_T$. To review, this construction defines $\sigma_1 := [1]$, and for each $T \geq 1$ a power of 2, uses the recurrence
\[\sigma_{2T} := \mathrm{interlace}(\sigma_T, 2T+1-\sigma_T), \]
where
\[\mathrm{interlace}([a_1 \ldots a_n], [b_1 \ldots b_n]) := [a_1 \; b_1 \; a_2 \; b_2 \ldots a_n \; b_n].\]
Some examples are below:
\[\sigma_2 = [1\;2],\]
\[\sigma_4 = [1\;4\;2\;3],\]
\[\sigma_8 = [1\;8\;4\;5\;2\;7\;3\;6],\]
\[\sigma_{16} = [1\;16\;8\;9\;4\;13\;5\;12\;2\;15\;7\;10\;3\;14\;6\;11].\]
Let us formalize the sense in which $\sigma$ contains internal copies of Chebyshev iteration. For positive integers $n$ and $\alpha \in (0, \pi)$, let us define the \emph{skewed Chebyshev polynomials}
\[ \mathcal{P}_{n,\alpha}(z) := \frac{\mathcal{T}_n(z) - \cos(\alpha)}{\mathcal{T}_n(\theta) - \cos(\alpha)}, \]
noting that $\mathcal{P}_{T,\frac{\pi}{2}}(z) = p(\lambda)$. If $\alpha \in (0, \frac{\pi}{2}]$, then call $\mathcal{P}_{n,\alpha}$ \goodcolor{good} and if $\alpha \in [\frac{\pi}{2}, \pi)$, then call $\mathcal{P}_{n,\alpha}$ \badcolor{bad}. We use the colours blue and red to highlight good and bad polynomials for clarity in our proofs. Note that $p(\lambda)$ is both good and bad. Next, we note some additional facts:
\begin{lemma}[Properties of skewed Chebyshev polynomials]
\label{lem:cheb-skew}
The following are true for all $n \in \mathbb{N}$ and $\alpha \in (0, \pi)$:
\begin{enumerate}
\item[(i)] $\mathcal{P}_{n,\alpha}(z)$ has $n$ real roots.
\item[(ii)] If $\mathcal{P}_{n,\alpha}$ is good, then
\[ \norm{\mathcal{P}_{n,\alpha}}_{L_\infty([-1,1])} \leq \frac{2}{1 + \mathcal{T}_n(\theta)}. \]
\item[(iii)] If $\mathcal{P}_{n, \alpha}$ is bad, then
\[ \norm{\mathcal{P}_{n,\alpha}}_{L_\infty([-1,1])} \leq \frac{2}{n^2 (\theta - 1)}. \]
\item[(iv)] If $n$ is even, then for all $-1 \leq z \leq 1$,
\[ \mathcal{P}_{n,\alpha}(z) = \mathcal{P}_{\frac{n}{2},\frac{\alpha}{2}}(z) \cdot \mathcal{P}_{\frac{n}{2},\pi-\frac{\alpha}{2}}(z). \]
\end{enumerate}
\end{lemma}
\begin{proof}
(i) follows from the fact that $\mathcal{T}_n(z_i) = (-1)^i$ at $z_i = \arccos(i\pi/n), i = 0, \ldots, n$.
(ii) follows from the fact that $|\mathcal{T}(z)| \leq 1$, $\mathcal{T}(\theta) > 1$, and $\cos(\alpha) \leq 0$, so that for $u = 1 + \cos(\alpha) \geq 0$,
\[ \norm{\mathcal{P}_{n,\alpha}}_{L_\infty([-1,1])} \leq \frac{1 - \cos(\alpha)}{\mathcal{T}_n(\theta) - \cos(\alpha)} \leq \frac{1 - \cos(\alpha) + u}{\mathcal{T}_n(\theta) - \cos(\alpha) + u} = \frac{2}{1+\mathcal{T}_n(\theta)}, \]
where the last inequality uses the mediant inequality.
(iii) is a weaker bound than the above, because we can't use the mediant inequality, as $-\cos(\alpha)$ is negative. Using part (ii) of Lemma~\ref{lem:cheb-expand} and $\cos(\alpha) \leq 1$, we get
\[ \norm{\mathcal{P}_{n,\alpha}}_{L_\infty([-1,1])} \leq \frac{2}{\mathcal{T}_n(\theta) - \cos(\alpha)} \leq \frac{2}{(1+n^2(\theta - 1))-1}. \]
(iv) follows from half-angle formulas and factorizing differences of squares:
\begin{align*}
\mathcal{P}_{n,\alpha}(z) &= \frac{\mathcal{T}_n(z) - \cos(\alpha)}{\mathcal{T}_n(\theta) - \cos(\alpha)} = \frac{(2\mathcal{T}^2_{n/2}(z)-1) - (2\cos^2(\frac{\alpha}{2})-1)}{(2\mathcal{T}^2_{n/2}(\theta)-1) - (2\cos^2(\frac{\alpha}{2})-1)} = \frac{\mathcal{T}^2_{n/2}(z) - \cos^2(\frac{\alpha}{2})}{\mathcal{T}^2_{n/2}(\theta) - \cos^2(\frac{\alpha}{2})} \\
&= \frac{\mathcal{T}_{n/2}(z) - \cos(\frac{\alpha}{2})}{\mathcal{T}_{n/2}(\theta) - \cos(\frac{\alpha}{2})} \cdot \frac{\mathcal{T}_{n/2}(z) + \cos(\frac{\alpha}{2})}{\mathcal{T}_{n/2}(\theta) + \cos(\frac{\alpha}{2})}
= \mathcal{P}_{\frac{n}{2},\frac{\alpha}{2}}(z) \cdot \mathcal{P}_{\frac{n}{2},\pi-\frac{\alpha}{2}}(z),
\end{align*}
as claimed.
\end{proof}
\subsection{The fractal schedule splits skewed Chebyshev polynomials} In this section, we connect the skewed polynomials $\mathcal{P}_{n,\alpha}$ to the construction of the fractal permutation $\sigma_T$, obtained via recursive binary splitting. This construction will provide the basis for all the proofs regarding the fractal schedule. The starting point for the construction is Lemma \ref{lem:cheb-skew}, which shows that when $T \geq 2$ is a power of 2,
\[ p(\lambda) = \mathcal{P}_{T,\frac{\pi}{2}}(z) = \mathcal{P}_{\frac{T}{2},\frac{\pi}{4}}(z) \cdot \mathcal{P}_{\frac{T}{2},\frac{3\pi}{4}}(z). \]
The above splitting procedure can be recursively repeated (since $T$ is a power of 2) on the pieces produced till we reach degree 1 polynomials of the form $\mathcal{P}_{1,\alpha}$ for some $\alpha$. Such splitting can easily be visualized via the construction of a complete binary tree of depth $\log_2(T)$ (see Figure \ref{fig:tree_construct}), by associating to every node a polynomial of the form $\mathcal{P}_{n,\alpha}$ and setting its left child to be $\mathcal{P}_{n/2,\alpha/2}$ and right child to be $\mathcal{P}_{n/2,\pi - \alpha/2}$.
Note that every non-leaf node is a product of its children by Lemma \ref{lem:cheb-skew}. The root node corresponds to the polynomial $p(\lambda)$ and the leaf nodes correspond to one degree polynomials which can be equivalently identified by its root, which are by construction the roots of polynomial $p(\lambda)$.
The key fact regarding this construction is the following.
\begin{fact}
The fractal schedule corresponds to the ordering of the roots as generated by a pre-order traversal of the tree.
\end{fact}
To see this note that every time a split is made in the tree, a constraint on the pre-order traversal is placed, i.e. the roots of the left child polynomial $\mathcal{P}_{n/2,\alpha/2}$ precede that of the right child polynomial $\mathcal{P}_{n/2,\pi - \alpha/2}$. It can be easily verified that the procedure for generating $\sigma_T$ produces an ordering of the roots $\gamma_t$ satisfying all of these constraints; the corresponding learning rate schedule $\eta_t$ is by definition the fractal Chebyshev schedule for each $T$, a power of 2.
Using the above, it can be seen that every node in the tree also corresponds to a particular infix polynomial $p_{s:t}(x)$ which includes all the roots corresponding to all the leaves underneath the node.
\begin{figure}[h!]
\centering
\begin{tikzpicture}[font=\small, level/.style={sibling distance=40mm/#1}, align=center ]
\node [circle,draw] (z){$\mathcal{P}_{T,\frac{\pi}{2}}$}
child {node [circle,draw=red] (a){\badcolor{$\mathcal{P}_{\frac{T}{2},\frac{\pi}{4}}$}}
child {node {$\vdots$}
child{node[circle,draw=red] (b){\badcolor{$\mathcal{P}_{1,\alpha_1}$}}
child { node [above=4mm](n ){$\sigma_T(1)$} edge from parent[draw=none]
}
}
child{ node[circle,draw=blue] (c){\goodcolor{$\mathcal{P}_{1,\alpha_2}$}}
child { node[above=4mm] (o){$\sigma_T(2)$} edge from parent[draw=none]
}
}
}
child {node {$\vdots$}}
}
child {node [circle,draw=blue] (g) {\goodcolor{$\mathcal{P}_{\frac{T}{2},\frac{3\pi}{4}}$}}
child {node {$\vdots$}}
child {node {$\vdots$}
child{ node[circle,draw=red, scale=0.8] (d){\badcolor{$\mathcal{P}_{1,\alpha_{T-1}}$}}
child { node [above=4mm](n ){$\sigma_T(T-1)$}
edge from parent[draw=none]}
}
child{ node[circle,draw=blue] (e){\goodcolor{$\mathcal{P}_{1,\alpha_{T}}$}}
child { node [above=4mm](n ){$\sigma_T(T)$}
edge from parent[draw=none]
}
}
}
};
\path (c) -- (d) node [midway] {$\ldots$};
\end{tikzpicture}
\caption{The binary tree construction for the decomposition of skewed Chebyshev polynomials. Each non-leaf node corresponds to the product of the polynomials corresponding to its children. The root corresponds to the entire polynomial $p_{1:T}(\lambda)$. Leaf nodes correspond to 1 degree polynomials and the pre-order traversal on the leaves induces the ordering given by $\sigma_T$.}
\label{fig:tree_construct}
\end{figure}
\subsection{Tree partitions and tree exchanges}
In this section we collect some observations regarding the tree construction, which are essential to the proofs for the bounds on the substring polynomials.
Firstly note that in the binary tree, a node/polynomial is \goodcolor{good} (\badcolor{bad}) if and only if it is the left (right) child of its parent. We begin by analyzing the special case of suffix polynomials $p_{s:T}$ and prefix polynomials $p_{1:s}$ respectively.
\paragraph{Suffix polynomials:}
The following key observation follows from the tree construction.
\begin{fact}
Every suffix polynomial $p_{s:T}$ (for any $s$) can be written as a product of good polynomials
\end{fact}
To see the above, consider the binary expansion of the number $T+1-s$, $\mathrm{bits}(T+1-s) = \{s_1, s_2 \ldots s_k\}$ defined by the unique decomposition, $T+1-s = 2^{s_1} + 2^{s_2} + \ldots + 2^{s_k}$ such that $s_1 > s_2 > \ldots s_k \ge 0$. We now perform the following iterative decomposition of the polynomial $p_{s:T}$,
\begin{align*}
p_{s:T} &= p_{s:T_1} \cdot \goodcolor{p_{T_1+1:T}} &\text{where } T_1 := T - 2^{s_1}, \\
&= p_{s:T_2} \cdot \goodcolor{p_{T_2:T_1}} \cdot \goodcolor{p_{T_1+1:T}} &\text{where } T_2 := T_1 - 2^{s_2}, \\
&\ldots,
\end{align*}
until we reach $s:T_k$, which is the empty interval. It can be seen that every intermediate polynomial $p_{T_i + 1:T_{i-1}}$ produced is a good polynomial because each one is the rightmost node at level $s_i$ (i.e. with distance $\log_2 T - s_i$ from the root node), restricted to the subtree rooted at the lowest common ancestor of roots $s$ through $T_{i-1}$ (setting $T_0 := T$). An example of the above decomposition is highlighted in Figure \ref{fig:suffix_decomp}. Combining with statement (ii) in Lemma~\ref{lem:cheb-skew}, we get
\begin{equation}
\norm{p_{s:T}}_{L_\infty([-1,1])} \le \prod_{i=1}^k \frac{2}{1 + \mathcal{T}_{2^{s_i}}(\theta)}
\end{equation}
\begin{figure}
\captionsetup[subfigure]{justification=centering}
\centering
\begin{subfigure}[b]{0.5\linewidth}
\begin{tikzpicture}
[font=\small, level/.style={sibling distance=40mm/#1}, align=center ]
\node [circle,draw=gray,fill=gray] (z){}
child {node [circle,draw=red, fill=red] (a){}
child {node [circle,draw=red, fill=red]{}
child{node[circle,draw=red, fill=red] (b){}
}
child{ node[circle,draw=blue, fill=blue] (c){} node[circle,draw=black, scale=1.7, line width=0.6mm] (c){}
child { node[above=4mm] (o){$\sigma(2)$} edge from parent[draw=none]
}
}
}
child {node [circle,draw=blue, fill=blue]{} node[circle,draw=black, scale=1.7, line width=0.6mm] {}
child{ node[circle,draw=red, fill=red] (d){}
}
child{ node[circle,draw=blue, fill=blue] (e){}
}
}
}
child {node [circle,draw=blue, fill=blue] (g){} node[circle,draw=black, scale=1.7, line width=0.6mm] {}
child {node[circle,draw=red, fill=red] {}
child{ node[circle,draw=red, fill=red] (d){}
}
child{ node[circle,draw=blue, fill=blue] (e){}
}
}
child {node[circle,draw=blue, fill=blue] (r){}
child{ node[circle,draw=red, fill=red] (d){}
}
child{ node[circle,draw=blue, fill=blue] (e){}
child { node [above=4mm] (n){$\sigma(8)$}
edge from parent[draw=none]
}
}
}
};
\draw[<->](o) to node [above, midway] {$p_{2:8}$}(n);
\end{tikzpicture}
\caption{Suffix decomposition (eg. $p_{2:8}$) into good polynomials}
\label{fig:suffix_decomp}
\end{subfigure}%
~
\begin{subfigure}[b]{0.5\linewidth}
\begin{tikzpicture}
[font=\small, level/.style={sibling distance=40mm/#1}, align=center ]
\node [circle,draw=gray,fill=gray] (z){}
child {node [circle,draw=red, fill=red] (a){} node[circle,draw=black, scale=1.7, line width=0.6mm] {}
child {node [circle,draw=red, fill=red]{}
child{node[circle,draw=red, fill=red] (b){}
child { node[above=4mm] (o){$\sigma(1)$} edge from parent[draw=none]
}
}
child{ node[circle,draw=blue, fill=blue] (c){}
}
}
child {node [circle,draw=blue, fill=blue]{}
child{ node[circle,draw=red, fill=red] (d){}
}
child{ node[circle,draw=blue, fill=blue] (e){}
}
}
}
child {node [circle,draw=blue, fill=blue] (g) {}
child {node[circle,draw=red, fill=red] {} node[circle,draw=black, scale=1.7, line width=0.6mm] {}
child{ node[circle,draw=red, fill=red] (d){}
}
child{ node[circle,draw=blue, fill=blue] (e){}
child { node [above=4mm] (n){$\sigma(6)$}
edge from parent[draw=none]
}
}
}
child {node[circle,draw=blue, fill=blue] (r){}
child{ node[circle,draw=red, fill=red] (d){}
}
child{ node[circle,draw=blue, fill=blue] (e){}
}
}
};
\draw[<->](o) to node [above, midway] {$p_{1:6}$}(n);
\end{tikzpicture}
\caption{Prefix decomposition (e.g. $p_{1:6}$) into bad polynomials before exchange}
\label{fig:prefix_decomp}
\end{subfigure}
\end{figure}
\paragraph{Prefix polynomials:}
The prefix polynomials $p_{1:s}$ are more challenging to analyze, as the immediate approach of reversing the above construction gives a decomposition into bad polynomials only.
To this end, consider the binary expansion of $s = 2^{s_1} + 2^{s_2} + \ldots + 2^{s_k}$ such that $s_1 > s_2 > \ldots s_1 \ge 0$. We decompose $p_{1:s}$ into products in the following manner: starting with $\{1, \ldots, s-1\}$, we iteratively partition
\begin{align*}
p_{1:s} &= \badcolor{p_{1:T_1}} \cdot p_{T_1+1:s} &\text{where } T_1 := 2^{s_1}, \\
&= \badcolor{p_{1:T_1}} \cdot \badcolor{p_{T_1 + 1:T_2}} \cdot p_{T_2+1:s} &\text{where } T_2 := T_1 + 2^{s_2}, \\
&\ldots,
\end{align*}
until we reach $T_k+1:s$, which is the empty interval. Note that this partition results in all bad polynomials. An example of the above decomposition is highlighted in Figure \ref{fig:prefix_decomp}. We can in fact exactly characterize these polynomials. Define the angle recurrence $\alpha_1 = \frac{2^{s_1}}{T} \cdot \frac{\pi}{2}$ and $\alpha_{i+1} = \frac{\pi - \alpha_i}{2^{s_i - s_{i+1}}}$. It can be seen that
\begin{equation}
\label{eqn:prefixdecomp}
p_{1:s}= \prod_{i=1}^k \mathcal{P}_{2^{s_i}, \alpha_i}.
\end{equation}
To get a tight bound for the norms of these polynomials, we require another innovation from \cite{lebedev1971order}. This innovation can be captured as a \emph{tree exchange property} that allows us to switch two bad polynomials for a good and bad polynomial. Starting with a partition of the roots of the polynomial we want to analyze, applying this tree switching repeatedly allows us to convert a partition with multiple bad polynomials into a set containing only one bad polynomial (and the rest good).
To further elucidate this tree exchange trick, let us establish some notation. We will be manipulating upper bounds for norms of $\mathcal{P}_{n,a}$, the product of which will serve as an upper bound for a prefix or suffix polynomial. Let
\[ \mathcal{B}_{n, \alpha} := \norm{\mathcal{P}_{n,\alpha}}_{L_\infty([-1,1])} = \max_{z \in [-1, 1]} \abs{ \frac{\mathcal{T}_n(z) - \cos(\alpha)}{\mathcal{T}_n(\theta) - \cos(\alpha)} }. \]
Note that the denominator of this fraction is positive independent of $z$. In the numerator, the maximum is achieved when $\mathcal{T}_n(z)$ is either $+1$ or $-1$, depending on the sign of $\cos(\alpha)$. When $\mathcal{P}_{n,\alpha}$ is good, $\cos(\alpha) \le 0$, thus we have
\[
\mathcal{B}_{n, \alpha} = \frac{1 - \cos(\alpha)}{\mathcal{T}_n(\theta) - \cos(\alpha)} = \frac{2 \sin^2\left(\frac{\alpha}{2}\right)}{\mathcal{T}_n(\theta) - \cos(\alpha)}.
\]
When $\mathcal{P}_{n,\alpha}$ is bad, $\cos(\alpha) \ge 0$, thus we have
\[
\mathcal{B}_{n, \alpha} = \frac{1 + \cos(\alpha)}{\mathcal{T}_n(\theta) - \cos(\alpha)} = \frac{2 \cos^2\left(\frac{\alpha}{2}\right)}{\mathcal{T}_n(\theta) - \cos(\alpha)}.
\]
With this notation, using \eqref{eqn:prefixdecomp}, we get that,
\[
\norm{p_{1:s}}_{L_\infty([-1,1])} \le \prod_{i=1}^k \mathcal{B}_{2^{s_i}, \alpha_i}.
\]
Now, we introduce the key tool which will allow us to handle products of bad polynomials.
\begin{lemma}[Tree Exchange Property] \label{lem:tree-exchange} For any $0 < \alpha < \frac{\pi}{2}$, and integers $n \ge 2$, $r \geq 1$, we have
\[
\badcolor{\mathcal{B}_{nr,\alpha}} \cdot \badcolor{\mathcal{B}_{r,\frac{\pi - \alpha}{n}}} \leq \goodcolor{\mathcal{B}_{nr, \pi-\alpha}} \cdot \badcolor{\mathcal{B}_{r, \frac{\alpha}{n}}}.
\]
\end{lemma}
If we view the arguments as indexing the corresponding subtrees in our construction, then the right hand side can be viewed as exchanging a (bad) $\badcolor{\mathcal{P}_{nr,\alpha}}$ with its (good) sibling $\goodcolor{\mathcal{P}_{nr,\pi - \alpha}}$ at the cost of exchanging $\badcolor{\mathcal{P}_{r,\frac{\pi - \alpha}{n}}}$ with the leftmost degree-$r$ polynomial $\badcolor{\mathcal{P}_{r,\frac{\alpha}{n}}}$ under $\badcolor{\mathcal{P}_{nr,\alpha}}$ (see Figure \ref{fig:exchanges}).
\begin{proof}
Using the derived bounds on $\mathcal{B}$, our claim reduces to proving the inequality
\begin{align*}
\frac{\cos^2\left(\frac{\alpha}{2}\right)}{\mathcal{T}_{nr}(\theta) - \cos(\alpha)} \cdot \frac{ \cos^2\left(\frac{\pi - \alpha}{2n}\right)}{\mathcal{T}_{r}(\theta) - \cos\left(\frac{\pi - \alpha}{n}\right)} &\le \frac{ \cos^2\left(\frac{\alpha}{2}\right)}{\mathcal{T}_{nr}(\theta) + \cos(\alpha)} \cdot \frac{\cos^2\left(\frac{\alpha}{2n}\right)}{\mathcal{T}_{r}(\theta) - \cos\left(\frac{\alpha}{n}\right)},
\end{align*}
which is equivalent to the following (since all terms are positive).
\begin{align*}
\frac{\mathcal{T}_{nr}(\theta) - 1 + 2 \sin^2\left(\frac{\alpha}{2}\right)}{\cos^2\left(\frac{\alpha}{2}\right)} \cdot \frac{\mathcal{T}_{r}(\theta) - 1 + 2\sin^2\left(\frac{\pi - \alpha}{2n}\right)}{\cos^2\left(\frac{\pi - \alpha}{2n}\right)} &\ge \frac{\mathcal{T}_{nr}(\theta) - 1 + 2\cos^2\left(\frac{\alpha}{2}\right)}{ \cos^2\left(\frac{\alpha}{2}\right)} \cdot \frac{\mathcal{T}_{r}(\theta) - 1 + 2 \sin^2\left(\frac{\alpha}{2n}\right)}{\cos^2\left(\frac{\alpha}{2n}\right)}.
\end{align*}
For ease of exposition, let us set $\Delta_{i}(\theta) = \mathcal{T}_{i}(\theta) - 1$. Note that by definition of $\theta$, we have $\Delta_i(\theta) \ge 0$. Observe that the second inequality holds if the following three inequalities are true,
\begin{align}
&\frac{\Delta_{nr}(\theta)\Delta_{r}(\theta)}{\cos^2\left(\frac{\alpha}{2}\right)}\left(\frac{1}{\cos^2\left(\frac{\pi - \alpha}{2n}\right)} - \frac{1}{\cos^2\left(\frac{\alpha}{2n}\right)}\right) \ge 0 \label{eq:prod1}\\
&\tan^2\left(\frac{\alpha}{2}\right)\tan^2\left(\frac{\pi - \alpha}{2n}\right) - \tan^2\left(\frac{\alpha}{2n}\right) \ge 0\label{eq:prod2}\\
&\Delta_{nr}(\theta)\left(\frac{\sin^2\left(\frac{\pi - \alpha}{2n}\right) }{\cos^2\left(\frac{\pi - \alpha}{2n}\right)\cos^2\left(\frac{\alpha}{2}\right)} - \frac{ \sin^2\left(\frac{\alpha}{2n}\right)}{ \cos^2\left(\frac{\alpha}{2n}\right)\cos^2\left(\frac{\alpha}{2}\right)}\right)+ \Delta_{r}(\theta)\left(\frac{ \sin^2\left(\frac{\alpha}{2}\right)}{\cos^2\left(\frac{\alpha}{2}\right)\cos^2\left(\frac{\pi - \alpha}{2n}\right)} - \frac{1}{\cos^2\left(\frac{\alpha}{2n}\right)}\right) \ge 0 \label{eq:prod3}
\end{align}
Note that Equation \eqref{eq:prod1} follows from the fact that $\cos^2\left(\frac{\alpha}{2n}\right) \ge \cos^2\left(\frac{\pi - \alpha}{2n}\right) > 1/2$ since $\frac{\pi}{4} \ge \frac{\pi - \alpha}{2n} \ge \frac{\alpha}{2n} \ge 0$.
To prove Equation \eqref{eq:prod2}, observe that it is equivalent to proving
\begin{align*}
\sin\left(\frac{\alpha}{2}\right) \sin\left(\frac{\pi - \alpha}{2n}\right) \cos\left(\frac{\alpha}{2n}\right) - \cos\left(\frac{\alpha}{2}\right) \cos\left(\frac{\pi - \alpha}{2n}\right) \sin\left(\frac{\alpha}{2n}\right) \ge 0
\end{align*}
Further simplifying the left hand side, we get
\begin{align*}
&2\left(\sin\left(\frac{\alpha}{2}\right) \sin\left(\frac{\pi - \alpha}{2n}\right) \cos\left(\frac{\alpha}{2n}\right) - \cos\left(\frac{\alpha}{2}\right) \cos\left(\frac{\pi - \alpha}{2n}\right) \sin\left(\frac{\alpha}{2n}\right) \right)\\
&= \sin\left(\frac{\alpha}{2}\right)\left(\sin\left(\frac{\pi}{2n}\right) + \sin\left(\frac{\pi - 2\alpha}{2n}\right)\right) - \cos\left(\frac{\alpha}{2}\right)\left(\sin\left(\frac{\pi}{2n}\right) - \sin\left(\frac{\pi - 2\alpha}{2n}\right)\right) \\
&= \sin\left(\frac{\pi}{2n}\right) \left(\sin\left(\frac{\alpha}{2}\right) - \cos\left(\frac{\alpha}{2}\right)\right) + \sin\left(\frac{\pi - 2\alpha}{2n}\right) \left(\sin\left(\frac{\alpha}{2}\right) + \cos\left(\frac{\alpha}{2}\right)\right)\\
&= \sqrt{2}\left(\sin\left(\frac{\pi - 2\alpha}{2n}\right)\cos\left(\frac{\pi - 2\alpha}{4}\right)-\sin\left(\frac{\pi}{2n}\right)\sin\left(\frac{\pi - 2\alpha}{4}\right)\right).
\end{align*}
Observe that $\frac{\sin(x)}{x}$ is a decreasing function for $0 < x \le \frac{\pi}{4}$, therefore we have $\frac{\sin\left(\frac{\pi - 2\alpha}{2n}\right)}{\frac{\pi - 2\alpha}{2n}} \ge \frac{\sin\left(\frac{\pi}{2n}\right)}{\frac{\pi}{2n}}$. Substituting this back, we get
\begin{align*}
&\sqrt{2}\left(\sin\left(\frac{\pi - 2\alpha}{2n}\right)\cos\left(\frac{\pi - 2\alpha}{4}\right)-\sin\left(\frac{\pi}{2n}\right)\sin\left(\frac{\pi - 2\alpha}{4}\right)\right)\\
&\ge \sqrt{2}\sin\left(\frac{\pi}{2n}\right)\left(\frac{\pi - 2 \alpha}{\pi} \cos\left(\frac{\pi - 2\alpha}{4}\right) - \sin\left(\frac{\pi - 2\alpha}{4}\right)\right) \ge 0.
\end{align*}
Here the last inequality follows from observing that $\tan(x) \le \frac{4x}{\pi}$ for $0 \le x \le \frac{\pi}{4}$. This proves Equation \eqref{eq:prod2}. Now it remains to prove Equation \eqref{eq:prod3}. Further simplifying the equation, it is equivalent to,
\begin{align*}
\frac{\Delta_{nr}(\theta)}{\Delta_{r}(\theta)}\left(\sin^2\left(\frac{\pi - \alpha}{2n}\right)\cos^2\left(\frac{\alpha}{2n}\right) - \sin^2\left(\frac{\alpha}{2n}\right)\cos^2\left(\frac{\pi - \alpha}{2n}\right) \right) + \sin^2\left(\frac{\alpha}{2}\right)\cos^2\left(\frac{\alpha}{2n}\right)- \cos^2\left(\frac{\alpha}{2}\right)\cos^2\left(\frac{\pi - \alpha}{2n}\right)\ge 0
\end{align*}
Let us prove this inequality. We have
\begin{align*}
& \frac{\Delta_{nr}(\theta)}{\Delta_{r}(\theta)}\left(\sin^2\left(\frac{\pi - \alpha}{2n}\right)\cos^2\left(\frac{\alpha}{2n}\right) - \sin^2\left(\frac{\alpha}{2n}\right)\cos^2\left(\frac{\pi - \alpha}{2n}\right) \right) + \sin^2\left(\frac{\alpha}{2}\right)\cos^2\left(\frac{\alpha}{2n}\right)- \cos^2\left(\frac{\alpha}{2}\right)\cos^2\left(\frac{\pi - \alpha}{2n}\right)\\
& = \frac{\Delta_{nr}(\theta)}{\Delta_{r}(\theta)}\left(\sin^2\left(\frac{\pi - \alpha}{2n}\right) - \sin^2\left(\frac{\alpha}{2n}\right) \right) + \cos^2\left(\frac{\alpha}{2}\right)\left(\cos^2\left(\frac{\alpha}{2n}\right)- \cos^2\left(\frac{\pi - \alpha}{2n}\right)\right) + \left(\sin^2\left(\frac{\alpha}{2}\right) - \cos^2\left(\frac{\alpha}{2}\right)\right)\cos^2\left(\frac{\alpha}{2n}\right)\\
& = \frac{\Delta_{nr}(\theta)}{\Delta_{r}(\theta)}\sin\left(\frac{\pi - 2\alpha}{2n}\right)\sin\left(\frac{\pi}{2n}\right) + \cos^2\left(\frac{\alpha}{2}\right)\left(\cos^2\left(\frac{\alpha}{2n}\right)- \cos^2\left(\frac{\pi - \alpha}{2n}\right)\right) - \cos(\alpha)\cos^2\left(\frac{\alpha}{2n}\right)\\
&\ge \frac{\Delta_{nr}(\theta)}{\Delta_{r}(\theta)}\sin\left(\frac{\pi - 2\alpha}{2n}\right)\sin\left(\frac{\pi}{2n}\right) - \sin\left(\frac{\pi - 2\alpha}{2}\right).
\end{align*}
Here the last inequality follows from the fact that $\cos\left(\frac{\alpha}{2n}\right) \ge \cos\left(\frac{\pi - \alpha}{2n}\right)$ since $\alpha \le \frac{\pi}{2}$ and $\cos(\alpha)\cos^2\left(\frac{\alpha}{2n}\right) \le \cos(\alpha) = \sin\left(\frac{\pi - 2\alpha}{2}\right)$.
By the composition property and part (ii) of Lemma~\ref{lem:cheb-expand}, we have
\begin{align*}
\frac{\Delta_{nr}(\theta)}{\Delta_{r}(\theta)} =
\frac{\mathcal{T}_n( \mathcal{T}_{r}(\theta) ) - 1}{\mathcal{T}_{r}(\theta) - 1} \leq
\frac{1 + n^2 ( \mathcal{T}_{r}(\theta) - 1 ) - 1}{\mathcal{T}_{r}(\theta) - 1} \leq n^2.
\end{align*}
We also know from before that $\frac{\sin(x)}{x}$ is decreasing in $0 < x \le \frac{\pi}{4}$, therefore, we have $\frac{\sin\left(\frac{\pi}{2n}\right)}{\frac{\pi}{2n}} \ge \frac{\sin\left(\frac{\pi}{4}\right)}{\frac{\pi}{4}}$ and $\frac{\sin\left(\frac{\pi - 2\alpha}{2n}\right)}{\frac{\pi - 2\alpha}{2n}} \ge \frac{\sin\left(\frac{\pi - 2\alpha}{2}\right)}{\frac{\pi - 2\alpha}{2}}$. Combining these and substituting back, we get
\begin{align*}
\frac{\Delta_{nr}(\theta)}{\Delta_{r}(\theta)}\sin\left(\frac{\pi - 2\alpha}{2n}\right)\sin\left(\frac{\pi}{2n}\right) - \sin\left(\frac{\pi - 2\alpha}{2}\right) &\ge n^2 \cdot \frac{\sin\left(\frac{\pi - 2\alpha}{2}\right)}{n} \cdot \frac{2\sin\left(\frac{\pi}{4}\right)}{n} - \sin\left(\frac{\pi - 2\alpha}{2}\right)\\
&= (\sqrt{2} - 1)\sin\left(\frac{\pi - 2\alpha}{2}\right) \ge 0.
\end{align*}
This completes the proof of the tree exchange lemma.
\end{proof}
\subsection{Completing the main theorems}
\begin{manualtheorem}{\ref{thm:lebedev-main}}[Prefix and suffix bounds]
For a fractal Chebyshev schedule with $m,M,T$, and all $1 \leq s, t \leq T$:
\begin{enumerate}
\item[(i)] $\norm{p_{1:t}}_{L_\infty([m,M])} \leq \pa{ \frac{M}{m} - 1 } \prod_{j \in \mathrm{bits}'(t)} \frac{2}{1+\mathcal{T}_{2^j}(\theta)}$;
\item[(ii)] $\norm{p_{s:T}}_{L_\infty([m,M])} \leq \prod_{j \in \mathrm{bits}(T+1-s)} \frac{2}{1+\mathcal{T}_{2^j}(\theta)}$,
\end{enumerate}
where $\mathrm{bits}(n)$ denotes the indices in the binary expansion of $n$, formally defined as the unique sequence $\{j_1, j_2 \ldots \}$ with $j_1 > j_2 > j_3 \ldots$ such that $n = \sum_{j_k \in \mathrm{bits}(n)} 2^{j_k}$. Further we define $\mathrm{bits}'(n) := \mathrm{bits}(n) \setminus j_1 $. For example, when $n = 6 = (110)_2$, $\mathrm{bits}(n) = \{2,1\}$, and $\mathrm{bits}'(n) = \{1\}$.
\end{manualtheorem}
Starting with a prefix decomposition and repeatedly applying Lemma~\ref{lem:tree-exchange}:
\begin{align*}
\norm{p_{1:s}}_{L_\infty([-1,1])} &= \max_{-1 \le z \le 1}\left| \prod_{i=1}^k \badcolor{\mathcal{P}_{2^{s_i}, \alpha_i}}(z)\right| \tag{Using \eqref{eqn:prefixdecomp}}\\
& \le \prod_{i=1}^k \max_{-1 \le z \le 1}\left| \badcolor{\mathcal{P}_{2^{s_i}, \alpha_i}}(z)\right|\\
& = \prod_{i=1}^k \badcolor{\mathcal{B}_{2^{s_i}, \alpha_i}}\\
&= \left(\prod_{i=1}^{k-2}\badcolor{\mathcal{B}_{2^{s_i}, \alpha_i}}\right) \cdot \badcolor{\mathcal{B}_{2^{s_{k-1}}, \alpha_{k-1}}} \cdot \badcolor{\mathcal{B}_{2^{s_{k}}, \frac{\pi - \alpha_{k-1}}{2^{s_{k-1} - s_{k}}}}}\\
&\le \goodcolor{\mathcal{B}_{2^{s_{k-1}}, \pi - \alpha_{k-1}}} \cdot \left( \prod_{i=1}^{k-2}\badcolor{\mathcal{B}_{2^{s_i}, \alpha_{k-1}}} \right)\cdot \badcolor{\mathcal{B}_{2^{s_{k}},\frac{ \alpha_{k-1}}{2^{s_{k-1} - s_{k}}}}} \tag{using Lemma \ref{lem:tree-exchange}}\\
&= \goodcolor{\mathcal{B}_{2^{s_{k-1}}, \pi - \alpha_{k-1}}} \cdot \left( \prod_{i=1}^{k-2}\badcolor{\mathcal{B}_{2^{s_i}, \alpha_{k-1}}} \right) \cdot \badcolor{\mathcal{B}_{2^{s_{k}},\frac{ \pi - \alpha_{k-2}}{2^{s_{k-2} - s_{k}}}}} \tag{using recurrence angle relation}\\
&= \goodcolor{\mathcal{B}_{2^{s_{k-1}}, \pi - \alpha_{k-1}}} \cdot \left( \prod_{i=1}^{k-3}\badcolor{\mathcal{B}_{2^{s_i}, \alpha_{k-1}}} \right) \cdot \badcolor{\mathcal{B}_{2^{s_{k-2}}, \alpha_{k-2}}} \cdot \badcolor{\mathcal{B}_{2^{s_{k}},\frac{ \pi - \alpha_{k-2}}{2^{s_{k-2} - s_{k}}}}} \\
&= \goodcolor{\mathcal{B}_{2^{s_{k-1}}, \pi - \alpha_{k-1}}}\cdot \goodcolor{\mathcal{B}_{2^{s_{k-2}}, \pi - \alpha_{k-2}}} \left( \prod_{i=1}^{k-3}\badcolor{\mathcal{B}_{2^{s_i}, \alpha_{k-1}}} \right) \cdot \badcolor{\mathcal{B}_{2^{s_{k}},\frac{ \alpha_{k-2}}{2^{s_{k-2} - s_{k}}}}} \tag{using Lemma \ref{lem:tree-exchange}}\\
&\qquad\vdots \tag{repeating this switching argument iteratively}\\
&\le \left(\prod_{i=1}^{k-1}\goodcolor{\mathcal{B}_{2^{s_{i}}, \pi - \alpha_{i}}} \right)\cdot \badcolor{\mathcal{B}_{2^{s_{k}}, \frac{\alpha_{1}}{2^{s_{1} - s_{k}}}}}\\
&= \left(\prod_{i=1}^{k-1}\goodcolor{\mathcal{B}_{2^{s_{i}}, \pi - \alpha_{i}}}\right) \cdot \badcolor{\mathcal{B}_{2^{s_{k}}, \frac{2^{s_k}\pi}{2T}}}.
\end{align*}
\begin{figure}
\captionsetup[subfigure]{justification=centering}
\centering
\begin{subfigure}{0.3\linewidth}
\begin{tikzpicture}
[scale=0.65, font=\small, level/.style={sibling distance=37mm/#1}, align=center ]
\node [circle,draw=gray,fill=gray] (a){}
child {node [circle,draw=red, fill=red] (b){} node[circle,draw=black, scale=1.4, line width=0.3mm] {}
child {node [circle,draw=red, fill=red] (c){}
child{node[circle,draw=red, fill=red] (d){}
}
child{ node[circle,draw=blue, fill=blue] (e){}
}
}
child {node [circle,draw=blue, fill=blue] (f){}
child{ node[circle,draw=red, fill=red] (g){}
}
child{ node[circle,draw=blue, fill=blue] (h){}
}
}
}
child {node [circle,draw=blue, fill=blue] (i) {}
child {node[circle,draw=red, fill=red] (j){} node[circle,draw=black, scale=1.4, line width=0.3mm] {}
child{ node[circle,draw=red, fill=red] (k){}
}
child{ node[circle,draw=blue, fill=blue] (l){}
}
}
child {node[circle,draw=blue, fill=blue] (m){}
child{ node[circle,draw=red, fill=red] (n){} node[circle,draw=black, scale=1.4, line width=0.3mm] {}
}
child{ node[circle,draw=blue, fill=blue] (o){}
}
}
};
\draw[->, line width=0.3mm](j) to (m);
\draw[->, line width=0.3mm](n.south) to [out=210,in=330] (k.south);
\end{tikzpicture}
\end{subfigure}%
~$\quad$
\begin{subfigure}{0.3\linewidth}
\begin{tikzpicture}
[scale=0.65, font=\small, level/.style={sibling distance=37mm/#1}, align=center ]
\node [circle,draw=gray,fill=gray] (a){}
child {node [circle,draw=red, fill=red] (b){} node[circle,draw=black, scale=1.4, line width=0.3mm] {}
child {node [circle,draw=red, fill=red] (c){}
child{node[circle,draw=red, fill=red] (d){}
}
child{ node[circle,draw=blue, fill=blue] (e){}
}
}
child {node [circle,draw=blue, fill=blue] (f){}
child{ node[circle,draw=red, fill=red] (g){}
}
child{ node[circle,draw=blue, fill=blue] (h){}
}
}
}
child {node [circle,draw=blue, fill=blue] (i) {}
child {node[circle,draw=red, fill=red] (j){}
child{ node[circle,draw=red, fill=red] (k){} node[circle,draw=black, scale=1.4, line width=0.3mm] {}
}
child{ node[circle,draw=blue, fill=blue] (l){}
}
}
child {node[circle,draw=blue, fill=blue] (m){} node[circle,draw=black, scale=1.4, line width=0.3mm] {}
child{ node[circle,draw=red, fill=red] (n){}
}
child{ node[circle,draw=blue, fill=blue] (o){}
}
}
};
\draw[->, line width=0.3mm](b) to (i);
\draw[->, line width=0.3mm](k.south) to [out=210,in=330] (d.south);
\end{tikzpicture}
\end{subfigure}~$\quad$
\begin{subfigure}{0.3\linewidth}
\begin{tikzpicture}
[scale=0.65, font=\small, level/.style={sibling distance=35mm/#1}, align=center ]
\node [circle,draw=gray,fill=gray] (a){}
child {node [circle,draw=red, fill=red] (b){}
child {node [circle,draw=red, fill=red] (c){}
child{node[circle,draw=red, fill=red] (d){} node[circle,draw=black, scale=1.4, line width=0.3mm] {}
}
child{ node[circle,draw=blue, fill=blue] (e){}
}
}
child {node [circle,draw=blue, fill=blue] (f){}
child{ node[circle,draw=red, fill=red] (g){}
}
child{ node[circle,draw=blue, fill=blue] (h){}
}
}
}
child {node [circle,draw=blue, fill=blue] (i) {} node[circle,draw=black, scale=1.4, line width=0.3mm] {}
child {node[circle,draw=red, fill=red] (j){}
child{ node[circle,draw=red, fill=red] (k){}
}
child{ node[circle,draw=blue, fill=blue] (l){}
}
}
child {node[circle,draw=blue, fill=blue] (m){} node[circle,draw=black, scale=1.4, line width=0.3mm] {}
child{ node[circle,draw=red, fill=red] (n){}
}
child{ node[circle,draw=blue, fill=blue] (o){}
}
}
};
\end{tikzpicture}
\end{subfigure}
\caption{An example of successive exchanges to fix a prefix polynomial ($p_{1:7}$). In every exchange, the product of two bad nodes are exchanged with a product of a good node and a bad node. Eventually, one is left with exactly one bad node and the remaining good nodes.}
\label{fig:exchanges}
\end{figure}
Note that this repeated exchange leads to only one bad polynomial and rest all good polynomials. Using (ii) and (iii) of Lemma \ref{lem:cheb-skew}, we get
\begin{align*}
\norm{p_{1:s}}_{L_\infty([-1,1])} &\le \prod_{i=1}^{k-1}\frac{2}{1 + \mathcal{T}_{2^{s_{i}}}(\theta)} \cdot \frac{2}{4^{s_{k}}(\theta - 1)}\\
& \le \frac{\frac{M}{m} - 1}{4^{s_k}} \cdot \prod_{i=1}^{k-1}\frac{2}{1 + \mathcal{T}_{2^{s_{i}}}(\theta)} \tag{using the definition of $\theta$}
\end{align*}
\section{Proofs for Section~\ref{sec:quadratic}}
\label{sec:appendix-proofs}
\subsection{Basic facts about the schedule}
\label{subsec:appendix-basics}
We provide full statements and proofs of Proposition~\ref{prop:cheb-basics}:
\begin{manualproposition}{\ref{prop:cheb-basics}}
For all $m, M, T$, the fractal Chebyshev step sizes $\gamma_t^{-1}$ satisfy the following:
\begin{enumerate}
\item[(i)] $\frac{1}{M} < \gamma_t^{-1} < \frac{1}{m}$.
\item[(ii)] The number of step sizes greater than $\frac{2}{M}$ is $\pa{ \frac{1}{2} - \epsilon }T$, where $0 \leq \epsilon \leq O(1/\widehat{\kappa})$ as $\widehat{\kappa} \rightarrow \infty$.
\item[(iii)] For $t \leq \frac{T}{2}$, we have $\gamma_t^{-1} < \frac{1}{m + \frac{2(M-m)t^2}{T^2}}$. Further,
$\frac{1}{T}\sum_{t=1}^T \gamma_t^{-1} = \frac{\tanh\pa{ T \mathrm{acosh}\left(\frac{2m}{M-m}\right)}}{\sqrt{Mm}} < \frac{1}{\sqrt{Mm}}.$
\end{enumerate}
\end{manualproposition}
(i) is obvious from the construction of $\gamma_t$, keeping in the mind the fact that $-1 < \cos(x) < 1$ for $x \in (0,\pi)$.
\begin{proof}[Proof of (ii).]
It is obvious that $\gamma_t^{-1} \geq 2/M$ for at most half of the indices $t = 1, \ldots, T$: the nodes $\gamma_t$ are symmetric with respect to reflection around the axis $\frac{M+m}{2} < \frac{M}{2}$, so at least half of them are greater than or equal to $\frac{M}{2}$. Now, let us establish the lower bound on the number of these steps. We have $\gamma_t^{-1} \leq 2/M$ if and only if
\begin{align*}
&\gamma_t = \frac{M+m}{2} - \frac{M-m}{2} \cos \frac{(t - 1/2) \pi}{T} \geq \frac{M}{2} \\
\Leftrightarrow \qquad &\cos \frac{(t - 1/2) \pi}{T} \leq \frac{m}{M-m} \\
\Leftrightarrow \qquad &t \geq \frac{1}{2} + \frac{\arccos(\frac{m}{M-m})}{\pi} \cdot T.
\end{align*}
Call the right hand side the \emph{threshold} $t^*$. Then, using the fact that $\arccos(x) \leq \frac{\pi}{2} - x$ for $x \in [0, 1]$, we have
\[ t^* \leq \frac{T}{2} + \frac{T}{\pi} \cdot \frac{m}{M-m} + \frac{1}{2}, \]
as required.
\end{proof}
\begin{proof}[Proof of (iii).]
The first statement follows from the upper bound $\cos \alpha \leq 1 - (\frac{2\alpha}{\pi})^2$, which is valid for $\alpha \leq \frac{\pi}{2}$:
\begin{align*}
\gamma_t^{-1} &= \frac{1}{\frac{M+m}{2} - \frac{M-m}{2} \cos\pa{ \frac{t-\frac{1}{2}}{T}\pi }}
< \frac{1}{\frac{M+m}{2} - \frac{M-m}{2} \cos\pa{ \frac{t}{T}\pi }}
\leq \frac{1}{\frac{M+m}{2} - \frac{M-m}{2} \pa{1 - \pa{\frac{2t}{T}}^2} },
\end{align*}
from which the claim follows.
A cheap version of the second statement, which is tight up to a constant, can be proven by viewing the summation above as a Riemann sum for a continuous integral. Interestingly, this also provides a way to prove bounds on all moments: for example, using the identity
\[ \int_{0}^1 \frac{1}{(c + x^2)^2} dx = \frac{1}{2} \pa{ \frac{\mathrm{arccot}(\sqrt{c})}{c^{3/2}} + \frac{1}{c^2+c} } \leq O(1/c^{3/2}), \]
one can verify that the root-mean-square $\sqrt{\frac{1}{T}\sum_{t=1}^T \gamma_t^{-2}}$ is bounded by $O(\widehat{\kappa}^{3/4}/M)$.
The exact statement in (iii) is subtler. We suspect that this is known in the literature on Chebyshev spectral methods, but could not find a reference. Consider $q(\lambda) = x^T p(1/\lambda)$, which is $p(\lambda)$ with its coefficients reversed. Then, the sum of reciprocal roots of $p(\lambda)$ we want is the sum of roots of $q(\lambda)$. By Vi\`ete's formula, this is $-a_{T-1}/a_T$, where $a_i$ is the $\lambda^i$ coefficient of $q(\lambda)$. By the coefficient reversal, $a_T$ is the constant term of $p(\lambda)$, and $a_{T-1}$ is its linear term. Thus, we have:
\[\sum_{t=1}^T \gamma_t^{-1} = \frac{\frac{d}{d\lambda} p(\lambda) |_{\lambda = 0}}{p(0)} = \frac{2}{M-m}\frac{\frac{d}{dz}\mathcal{T}_{T}(z)|_{z=\theta}}{\mathcal{T}_T(\theta)}. \]
To reason about the derivatives, we need to introduce the Chebyshev polynomials of the \emph{second} kind, $\mathcal{U}_n(z)$. They can be defined as the unique polynomial satisfying
\[ \mathcal{U}_n( \cos \alpha ) \sin \alpha = \sin( (n+1)\alpha ).\]
From this, the cosine characterization of $\mathcal{T}_n(z)$, and the trigonometric substitution $z = \cos \alpha$, we have
\[ \frac{d}{dz} \mathcal{T}_n(z) = n \, \mathcal{U}_{n-1}(z). \]
By definition we have that
\[\mathcal{U}_{n-1}(z) = \frac{(z + \sqrt{z^2 - 1})^n - (z - \sqrt{z^2 - 1})^n}{2 \sqrt{z^2 - 1}}, \quad \mathcal{T}_{n-1}(z) = \frac{(z + \sqrt{z^2 - 1})^n + (z - \sqrt{z^2 - 1})^n}{2}.\]
Therefore,
\[\frac{\mathcal{U}_{n-1}(z)}{\mathcal{T}_{n}(z)} = \frac{(z + \sqrt{z^2 - 1})^n - (z - \sqrt{z^2 - 1})^n}{(z + \sqrt{z^2 - 1})^n + (z - \sqrt{z^2 - 1})^n } \cdot \frac{1}{\sqrt{z^2 - 1}} = \frac{\tanh(n\,\mathrm{acosh}(z))}{\sqrt{z^2 - 1}}. \]
Substituting these back, we get that
\[\sum_{t=1}^T \gamma_t^{-1} = \frac{2}{M-m}\frac{\frac{d}{dz}\mathcal{T}_{T}(z)|_{z=\theta}}{\mathcal{T}_T(\theta)} = \frac{2T\tanh(T\,\mathrm{acosh}(\theta))}{(M-m)\sqrt{\theta^2 - 1}}
= \frac{T\tanh(T\,\mathrm{acosh}(\theta))}{\sqrt{Mm}}.\]
\end{proof}
Finally, we prove the simple observations about the fractal permutation:
\begin{manualproposition}{\ref{prop:cheb-fractal-basis}}
For all $m,M,T$ and $0 \leq i \leq \log_2 T$:
\begin{enumerate}
\item[(i)] The largest $\frac{T}{2^i}$ steps $\eta_t$ in the fractal Chebyshev schedule occur when $t = 1 + 2^i(\tau-1)$, with $\tau = 1, \ldots, \frac{T}{2^i}$.
\item[(ii)] The subsampled sequence $\{\eta_{1+2^i(\tau-1)}\}$ has the same ordering as the fractal permutation of the same length:
\[ \eta_{1+2^i\tau} = \gamma^{-1}_{1+2^i(\tau' - 1)}, \quad \text{ where } \tau' = \sigma_{T/2^i}(\tau). \]
\end{enumerate}
\end{manualproposition}
\begin{proof}
(i) and (ii) are true by the recursive interlacing construction. Let $T_i := \frac{T}{2^i}$. The interlacing step makes it true that the indices in $\sigma_{T_i}$ are every other element of $\sigma_{2T_i}$, the indices in $\sigma_{2T_i}$ are every other element in $\sigma_{4T_i}$, and so forth. Note that $\gamma_t^{-1}$ are decreasing in $t$.
\end{proof}
\subsection{Infix polynomial bounds}
\label{subsec:appendix-infix-proof}
\begin{manualtheorem}{\ref{thm:infix-bound}}
For the fractal Chebyshev schedule with $m,M,T$, and all $1 \leq s \leq t \leq T$:
\[ \norm{p_{s:t}}_{[m,M]} \le \left(\frac{M}{m} - 1\right) \cdot \prod_{i \in \mathrm{bits}(\zeta+1-s)} \frac{2}{1 + \mathcal{T}_{2^{i}}(\theta)} \cdot \prod_{i \in \mathrm{bits}^\prime(t-\zeta)}\frac{2}{1 + \mathcal{T}_{2^{i}}(\theta)}, \]
where $\zeta$ is the index such that $s-1 \leq \zeta \leq t$ and $\mathrm{lca}(\zeta, \zeta+1)$ is maximized, where
\[ \mathrm{lca}(a,b) := \max \{ j : j \in \mathrm{bits}(a) \text{ xor } j \in \mathrm{bits}(b) \} \]
is the index of the most significant bit at which the binary decompositions of $a,b$ differ.
\end{manualtheorem}
\begin{figure}
\centering
\begin{tikzpicture}
[font=\small,
level 1/.style ={sibling distance=80mm},
level 2/.style ={sibling distance=40mm},
level 3/.style ={sibling distance=20mm},
level 4/.style ={sibling distance=10mm},align=center ]
\node [circle,draw=gray,fill=gray] {}
child {node [circle,draw=red, fill=red] (a){}
node[circle,draw=black, scale=1.7, line width=0.3mm] {}
child {node [circle,draw=red, fill=red]{}
child{node[circle,draw=red, fill=red] {}
child {node[circle,draw=red, fill=red] {}
}
child {node[circle,draw=blue, fill=blue] {}
child { node[above=4mm] (o){$\sigma(2)$} edge from parent[draw=none]
}
}
}
child{ node[circle,draw=blue, fill=blue] {}
child {node[circle,draw=red, fill=red] {}
}
child {node[circle,draw=blue, fill=blue] {}
}
}
}
child {node [circle,draw=blue, fill=blue]{}
child{ node[circle,draw=red, fill=red] {}
child {node[circle,draw=red, fill=red] {}
}
child {node[circle,draw=blue, fill=blue] {}
}
}
child{ node[circle,draw=blue, fill=blue] {}
child {node[circle,draw=red, fill=red] {}
child { node[above=4mm] (n){$\sigma(7)$} edge from parent[draw=none]
}
}
child {node[circle,draw=blue, fill=blue] {}
}
}
}
}
child {node [circle,draw=blue, fill=blue] {}
child {node[circle,draw=red, fill=red] {}
child{ node[circle,draw=red, fill=red] {}
child {node[circle,draw=red, fill=red] {}
}
child {node[circle,draw=blue, fill=blue] {}
}
}
child{ node[circle,draw=blue, fill=blue] {}
child {node[circle,draw=red, fill=red] {}
}
child {node[circle,draw=blue, fill=blue] {}
}
}
}
child {node[circle,draw=blue, fill=blue] {}
child{ node[circle,draw=red, fill=red] {}
child {node[circle,draw=red, fill=red] {}
}
child {node[circle,draw=blue, fill=blue] {}
}
}
child{ node[circle,draw=blue, fill=blue] {}
child {node[circle,draw=red, fill=red] {}
}
child {node[circle,draw=blue, fill=blue] {}
}
}
}
};
\draw[<->](o) to node [above, midway] (b){$p_{2:7}$}(n);
\draw[dashed](a)+(0, -4mm) to (b);
\node[] at (-40mm,-74mm) (q) {};
\node[] at (-55mm,-15mm) {LCA of 1 and 8};
\draw[dashed](b)+(0, -4mm) to (q);
\draw[<-](q) to node [above] {suffix} +(-25mm, 0);
\draw[->](q) to node [above] {prefix} +(25mm, 0);
\end{tikzpicture}
\caption{A schematic for the decomposition of the infix $p_{2:7}$ into a suffix and a prefix polynomial corresponding to the child subtrees of the Lowest Common Ancestor}
\label{fig:infix_decomp}
\end{figure}%
\begin{proof}
We have previously shown how to bound the norms of the prefix and suffix polynomial. In this section, we will extend these arguments to bound any infix polynomial $p_{s:t}$ for $1< s < t < T$. To bound the norm of this polynomial, we will decompose the polynomial into two polynomials lying in disjoint subtrees. On one part we will use the suffix argument whereas on the other part we will use the prefix argument.
To formalize this, we split based on the lowest common ancestor of node $s-1$ and $t+1$.\footnote{Note that we include $s-1$ and $t+1$ when doing this split to ensure that the polynomials do not cover the corresponding subtrees completely and are in fact prefixes and suffixes.} Consider the binary expansions of $s - 1 = 2^{s_1} + 2^{s_2} + \ldots + 2^{s_k}$ such that $s_1 > s_2 > \ldots s_k \ge 0$ and $t + 1 = 2^{t_1} + 2^{t_2} + \ldots + 2^{t_l}$ such that $t_1 > t_2 > \ldots t_l \ge 0$. Let $h$ be the minimum index such that $s_{h} \ne t_{h}$ and let $\zeta = \sum_{i = 1}^{h} 2^{t_h}$. Note that $h$ is the level of the lowest common ancestor in the tree, and $\zeta,\zeta+1$ are the indices splitting the infix between the lowest common ancestor's two subtrees. We will perform the following decomposition based on this,
\[
p_{s:t} = p_{s:\zeta} \cdot p_{\zeta+1:t}.
\]
It is not hard to see that this decomposition puts the two polynomials in disjoint subtrees: a subtree corresponding to $p_{\zeta-2^{t_h} + 1:\zeta}$ and $p_{\zeta + 1: \zeta+2^{t_h}}$. Observe that $p_{s:\zeta}$ is a suffix in the left subtree and $p_{\zeta+1:t}$ is a prefix in the right subtree. See Figure \ref{fig:infix_decomp} for a schematic depiction of the above decomposition.
Let us first analyze $p_{s:\zeta}$. Note that $\zeta$ can be $< s$, in that case the polynomial is the empty product with norm $1$. Thus, let us assume $\zeta \ge s$. More formally, consider the binary expansions of $M \stackrel{\text{def}}{=} \zeta+1-s = 2^{r_1} + 2^{r_2} + \ldots + 2^{r_j}$ such that $r_1 > r_2 > \ldots r_j \ge 0$. We now perform the following iterative decomposition of the polynomial $p_{s:M}$,
\begin{align*}
p_{s:M} &= p_{s:T_1} \cdot \goodcolor{p_{T_1+1:\zeta}} &\text{where } T_1 := M - 2^{r_1}, \\
&= p_{s:T_2} \cdot \goodcolor{p_{T_2:T_1}} \cdot \goodcolor{p_{T_1+1:M}} &\text{where } T_2 := T_1 - 2^{r_2}, \\
&\ldots,
\end{align*}
As in the suffix argument, we will decompose the polynomial into good polynomials starting from the right. Recall that the right child of every node is a good polynomial. It can be seen that every intermediate polynomial $p_{T_i + 1:T_{i-1}}$ produced is a good polynomial because each one is the rightmost node at level $r_i$ (i.e. with distance $\log_2 \zeta - r_i$ from the root node of the subtree), restricted to the subtree rooted at the lowest common ancestor of roots $s$ through $T_{i-1}$ (setting $T_0 := \zeta$). Combining with statement (ii) in Lemma~\ref{lem:cheb-skew}, we get
\begin{equation}
\norm{p_{s:\zeta}}_{[m,M]} \le \prod_{i=1}^j \frac{2}{1 + \mathcal{T}_{2^{r_i}}(\theta)} = \prod_{i \in \mathrm{bits}(\zeta+1-s)} \frac{2}{1 + \mathcal{T}_{2^{i}}(\theta)}
\end{equation}
Let us now look at polynomial $p_{\zeta+1:t}$. We will use the prefix argument as before on this polynomial. Consider the binary expansion of $t - \zeta = 2^{q_1} + 2^{q_2} + \ldots + 2^{q_j}$ such that $q_1 > q_2 > \ldots q_j \ge 0$. We decompose $p_{\zeta+1:t}$ into products in the following manner: starting with $\{\zeta + 1, \ldots, t\}$, we iteratively partition
\begin{align*}
p_{\zeta+1:t} &= \badcolor{p_{\zeta + 1:T_1}} \cdot p_{T_1+1:t} &\text{where } T_1 := 2^{q_1}, \\
&= \badcolor{p_{\zeta + 1:T_1}} \cdot \badcolor{p_{T_1 + 1:T_2}} \cdot p_{T_2+1:t} &\text{where } T_2 := T_1 + 2^{q_2}, \\
&\ldots,
\end{align*}
until we reach $T_j+1:t$, which is the empty interval. Note that this partition results in all bad polynomials. We can in fact exactly characterize these polynomials. Define the angle recurrence with $\alpha_1$ being the angle corresponding to the subtree $p_{\zeta + 1: \zeta-2^{t_j}}$ and $\alpha_{i+1} = \frac{\pi - \alpha_i}{2^{q_i - q_{i+1}}}$. It can be seen that
\begin{equation*}
p_{\zeta+1:t}= \prod_{i=1}^j \mathcal{P}_{2^{q_i}, \alpha_i} \implies \norm{p_{\zeta+1:t}}_{[m,M]} \le \prod_{i=1}^j \mathcal{B}_{2^{q_i}, \alpha_i}.
\end{equation*}
Using Lemma \ref{lem:tree-exchange} iteratively on this, we can see that
\begin{align*}
\norm{p_{\zeta+1:t}}_{[m,M]} &\le \badcolor{\mathcal{B}_{2^{q_{j}}, \frac{\alpha_1}{2^{q_1 - q_j}}}} \cdot \prod_{i=1}^{k-1}\goodcolor{\mathcal{B}_{2^{q_{i}}, \pi - \alpha_{i}}}\\
&\le \frac{2}{4^{q_{j}}(\theta - 1)} \cdot \prod_{i=1}^{j-1}\frac{2}{1 + \mathcal{T}_{2^{q_{i}}}(\theta)} \tag{using Lemma \ref{lem:cheb-skew} (ii) and (iii)}\\
& \le \left(\frac{M}{m} - 1\right) \cdot \prod_{i \in \mathrm{bits}^\prime(t-\zeta)}\frac{2}{1 + \mathcal{T}_{2^{i}}(\theta)} \tag{using the definition of $\theta$}.
\end{align*}
This gives us the final bound on the infix as,
\begin{align}
\norm{p_{s:t}}_{[m,M]} \le \left(\frac{M}{m} - 1\right) \cdot \prod_{i \in \mathrm{bits}(\zeta+1-s)} \frac{2}{1 + \mathcal{T}_{2^{i}}(\theta)} \cdot \prod_{i \in \mathrm{bits}^\prime(t-\zeta)}\frac{2}{1 + \mathcal{T}_{2^{i}}(\theta)}. \label{eq:infix}
\end{align}
\end{proof}
\subsection{Infix series bounds}
\label{subsec:appendix-infix-series-proof}
First, we provide a useful bound for an infix series contained entirely within a subtree:
\begin{lemma}
\label{lem:series-sum-main}
For any $N = 2^k$ for some $k > 0$ and any $\delta > 0$,
\[
\sum_{i=1}^N \prod_{i \in \mathrm{bits}(N+1-i)} \frac{2}{1 + \mathcal{T}_{2^{i}}(1 + \delta)} \le \exp\left(\frac{1}{1 + \delta}\right)\cdot \left(\frac{1 + \delta}{\delta}\right)^{1/\log(4)}.
\]
\end{lemma}
\begin{proof}
Define $P_{N} := \sum_{i=1}^N \prod_{i \in \mathrm{bits}(N+1-i)} \frac{2}{1 + \mathcal{T}_{2^{i}}(\theta)}$. Since $N$ is a power of 2, it is not hard to see that
\[
P_{N} = \left(\frac{2}{1 + T_{N/2}(\theta)}\right) \cdot P_{N/2} + P_{N/2} = \left(1 + \frac{2}{1 + T_{N/2}(\theta)}\right) \cdot P_{N/2}.
\]
Recursively applying the above, we have
\[
P_{N} = \prod_{i=0}^{k-1} \left(1 + \frac{2}{1 + T_{2^i}(\theta)}\right).
\]
To bound the above, let us take $\log$ on both sides. This gives us
\begin{align*}
\log(P_{N}) &= \sum_{i=0}^{k-1} \log \left(1 + \frac{2}{1 + T_{2^i}(\theta)}\right)\\
&\le \sum_{i=0}^{k-1}\frac{2}{1 + T_{2^i}(\theta)} \\
&\le \sum_{i=0}^{k-1}\frac{1}{1 + 4^i\delta}\\
&\le \frac{1}{1 + \delta} + \int_0^{k-1} \frac{1}{1 + 4^x\delta} dx \\
&\le \frac{1}{1 + \delta} + \frac{\log(1 + \delta) - \log(\delta)}{\log 4}.
\end{align*}
Substituting back gives us the desired result.
\end{proof}
Now, we are ready to prove the main theorem about infix series sums.
\begin{manualtheorem}{\ref{thm:infix-series-bound}}
For a fractal Chebyshev schedule with $m,M,T$, and all $1 \leq s \leq t \leq T$:
\[ \sum_{t'=s}^t \norm{p_{t':t}}_{[m,M]} \leq 18\left(\frac{M}{m} - 1 \right) \left(\left(\frac{M+m}{2m}\right)^{1/\log(4)} \right) \left(1 + \log\left(\frac{M+m}{2m}\right) \right). \]
\end{manualtheorem}
\begin{proof}
We prove the theorem only for $s=1$ which subsumes all the cases. To prove the statement we will first consider the binary expansion of $t$, $\mathrm{bits}(t)$ which is the unique sequence of numbers $\{t_1 \ldots t_k\}$ such that $t = \sum_{j=1}^k 2^{t_i}$ and $t_1 > t_2 > t_3 \ldots$. Further, define the following the sequence
\[ \bar{t}_0 = 0; \qquad \qquad \bar{t}_j = \sum_{j' = 1}^{j} 2^{t_{j'}} \;\;\forall j \in [k].\]
We will break the sum and analyze it in the following manner:
\[
\sum_{j=1}^{k} \underbrace{ \sum_{t'=\bar{t}_{j-1}+1}^{\bar{t}_{j}} \norm{p_{t':t}}_{[m,M]}}_{:= T_j}.
\]
Before analyzing $T_j$, we establish some calculations, which will be useful. Firstly, note that for all $j$, the roots in the range $[\bar{t}_{j}+1, t]$ are all contained inside a subtree of height $2^{t_j}$ in the tree representation. Therefore the polynomial $p_{\bar{t}_{j}+1, t}$ forms a prefix polynomial within the subtree for which we can apply the bounds in Theorem \ref{thm:lebedev-main} (see usage in the proof of Theorem \ref{thm:infix-bound} on how Theorem \ref{thm:lebedev-main} applies to prefix of any subtree). Using the above, we get the following bounds:
\begin{equation}
\label{eqn:subeqn1}
\|p_{\bar{t}_{j}+1 :t}\|_{[m,M]} \leq \left(\frac{M}{m} - 1 \right) \prod_{r=j+2}^{k} \frac{2}{1 + \mathcal{T}_{2^{t_r}}(\theta)}.
\end{equation}
Note that for the above and the rest of this section if the sum $\Sigma$ and product $\prod$ is over an empty set then they are assumed to be $0$ and $1$ respectively. Furthermore, note that for any $j$ and $t' \in [\bar{t}_{j-1}+2, \bar{t}_j]$, the range of roots $[t', \bar{t}_j]$, belongs to a subtree of height $2^{t_j}$ in the tree representation. In particular the infix polynomial $p_{t', \bar{t}_j}$ is a suffix polynomial within the tree and the bound from Theorem \ref{thm:lebedev-main} can be invoked to give
\begin{equation}
\label{eqn:subeqn2}
\|p_{t', \bar{t}_j}\|_{[m,M]} \leq \prod_{r \in \mathrm{bits}(\bar{t}_j+1-t')} \frac{2}{1+\mathcal{T}_{2^r}(\theta)}.
\end{equation}
We are now ready to analyze the terms $T_j$ for any $j$ as follows.
\begin{align*}
T_j &\leq \|p_{\bar{t}_{j-1}+1 :t}\|_{[m,M]} + \sum_{t'=\bar{t}_{j-1}+2}^{\bar{t}_{j}} \norm{p_{t':\bar{t}_j}}_{[m,M]}\norm{p_{\bar{t}_j+1:t}}_{[m,M]} \\
&\leq \left(\frac{M}{m} - 1 \right) \left( \prod_{r=j+1}^{k} \frac{2}{1 + \mathcal{T}_{2^{t_r}}(\theta)} + \prod_{r=j+2}^{k} \frac{2}{1 + \mathcal{T}_{2^{t_r}}(\theta)} \left(\sum_{t'=\bar{t}_{j-1}+2}^{\bar{t}_{j}} \left( \prod_{r \in \mathrm{bits}(\bar{t}_j+1-t')} \frac{2}{1+\mathcal{T}_{2^r}(\theta)} \right) \right) \right) \\
&\leq \left(\frac{M}{m} - 1 \right) \left( \prod_{r=j+1}^{k} \frac{2}{1 + \mathcal{T}_{2^{t_r}}(\theta)} + \prod_{r=j+2}^{k} \frac{2}{1 + \mathcal{T}_{2^{t_r}}(\theta)} \left(3\left(\frac{\theta}{\theta - 1}\right)^{1/\log(4)} \right) \right). \\
&\leq 6\left(\frac{M}{m} - 1 \right) \left( \prod_{r=j+2}^{k} \frac{2}{1 + \mathcal{T}_{2^{t_r}}(\theta)} \left(\left(\frac{\theta}{\theta - 1}\right)^{1/\log(4)} \right)\right).
\end{align*}
In the above, the first inequality follows from triangle inequality, the second from \eqref{eqn:subeqn1},\eqref{eqn:subeqn2} and the third from Lemma \ref{lem:cheb-expand}. Summing over $j$ now gives us the bound as follows:
\begin{align*}
\sum_{j=1}^k T_j \leq & 6\left(\frac{M}{m} - 1 \right) \left( \sum_{j=1}^{k-2} \left( \prod_{r=j+2}^{k} \frac{2}{1 + \mathcal{T}_{2^{t_r}}(\theta)} \right) + 2 \right) \left(\left(\frac{\theta}{\theta - 1}\right)^{1/\log(4)} \right). \\
& \leq 6\left(\frac{M}{m} - 1 \right) \left( \sum_{j=1}^{k-2} \left( \frac{2}{1 + \mathcal{T}_{2^{t_{j+2}}}(\theta)} \right) + 2 \right) \left(\left(\frac{\theta}{\theta - 1}\right)^{1/\log(4)} \right). \\
& \leq 6\left(\frac{M}{m} - 1 \right) \left( \log\left(\frac{\theta}{\theta-1}\right) + 3 \right) \left(\left(\frac{\theta}{\theta - 1}\right)^{1/\log(4)} \right). \\
&\leq 18\left(\frac{M}{m} - 1 \right) \left(\left(\frac{M+m}{2m}\right)^{1/\log(4)} \right) \left(1 + \log\left(\frac{M+m}{2m}\right) \right).
\end{align*}
This concludes the theorem.
\end{proof}
\section{Proofs for Section~\ref{sec:quadratic}}
\label{sec:appendix-proofs-misc}
\subsection{Modifications to the fractal schedule}
\begin{proof}[Proof of Proposition~\ref{prop:cheb-sched-reverse} (i)]
For any $1 \leq t \leq T-1$, consider $p_{1:t}$ and $p_{1:t+1}$ (under the reversed permutation). Both decompose into a product of good polynomials, whose levels in the tree are given by $\mathrm{bits}(t)$ and $\mathrm{bits}(t+1)$.
Notice that $\mathrm{bits}(t+1) \setminus \mathrm{bits}(t)$ contains exactly one element (the index where the carrying operation stops in binary addition); call it $c$. Then, $\mathrm{bits}(t) \setminus \mathrm{bits}(t+1) = \{1, \ldots, c-1\}$. Thus, it suffices to prove that
\[ \frac{2}{1+\mathcal{T}_{2^c}(\theta)} \leq \prod_{i=0}^{c-1} \frac{2}{1+\mathcal{T}_{2^i}(\theta)}. \]
Notice that when $i \geq 1$, we have
\[ \frac{2}{1+\mathcal{T}_{2^i}(\theta)} = \frac{1}{\mathcal{T}_{2^{i-1}}^2(\theta)}, \]
so when $c \geq 1$, the statement we wish to prove reduces to
\[ \mathcal{T}_{2^{c-1}}^2(\theta) \geq \frac{1+\theta}{2} \cdot \prod_{i=0}^{c-2} \mathcal{T}_{2^i}^2(\theta). \]
This is true because $\frac{1+\theta}{2} \leq \theta^2 = \mathcal{T}_1^2(\theta)$, and we can recursively apply the following inequality:
\[\mathcal{T}_{2n}(\theta) = 2\mathcal{T}_n^2(\theta) - 1 \geq \mathcal{T}_n^2(\theta), \]
which holds because $\theta \geq 1$.
\end{proof}
The other proofs in Section~\ref{sec:quadratic} follow immediately from the definitions.
\subsection{Overstepping with conservative parameters}
\begin{proof}[Proof of Theorem \ref{thm:underoverstepping}]
With $p = p_{1:T}$ as the shifted Chebyshev polynomial with parameters $m,M$, we wish to bound $\norm{p}_{[\lambda_{\min}, M]}.$ In the range $[m,M]$, the bounds from Theorem~\ref{thm:cheb-convergence-rate} hold; moreover, the cosine formula for the Chebyshev polynomials implies that the inequality is tight at the boundary: the maximum is achieved at $p(m)$. In the remaining part $[\lambda_{\min}, M]$, which is outside the range of the roots of $p$, $p(\lambda)$ grows monotonically as $\lambda$ decreases in this interval. Thus, it will suffice to derive a bound for $p(\lambda_{\min})$.
When $m<M$, we use the notation $\theta = \frac{M+m}{M-m}$, and define $z_{\max} := \frac{M+m-2\lambda_{\min}}{M-m}$ (the image of $\lambda_{\min}$ under the bijection).
\begin{align*}
\frac{\mathcal{T}_T(z_{\max})}{\mathcal{T}_n(\theta)} &= \frac{\cosh(T \, \mathrm{acosh}(z_{\max}))}{\cosh(T \,\mathrm{acosh}(\theta))} = \frac{e^{T \, \mathrm{acosh}(z_{\max})} + e^{-T \, \mathrm{acosh}(z_{\max})}}{e^{T \, \mathrm{acosh}(\theta)} + e^{-T \, \mathrm{acosh}(\theta)}} \leq \frac{2e^{T \, \mathrm{acosh}(z_{\max})}}{e^{T \, \mathrm{acosh}(\theta)}} \\
&= 2\left(\frac{z_{\max} + \sqrt{z_{\max}^2 - 1}}{\theta + \sqrt{\theta^2 - 1}}\right)^T
= 2\left(1 - \frac{\theta + \sqrt{\theta^2 - 1} - z_{\max} - \sqrt{z_{\max}^2 - 1}}{\theta + \sqrt{\theta^2 - 1}} \right)^T.
\end{align*}
The quantity in the fraction is equal to
\begin{align*}
\frac{\frac{M+m+2\sqrt{Mm}}{M-m} - \frac{M+m+2\lambda_{\min} + 2\sqrt{(M-\lambda_{\min})(m-\lambda_{\min})} }{M-m}}{ \frac{M+m+2\sqrt{Mm}}{M-m} }
=
2 \cdot \frac{\lambda_{\min} + \sqrt{Mm} - \sqrt{(M-\lambda_{\min})(m-\lambda_{\min})}}{ (\sqrt{M} + \sqrt{m})^2 },
\end{align*}
as required. The same is concluded for $m=M$ by taking the limit $m \rightarrow M$.
\end{proof}
\subsection{Conjugate gradient schedule}
\label{subsec:appendix-cg}
The simplest definition of the conjugate gradient algorithm, without having to worry about how to implement the iterations in linear time, is the non-iterative formula
\begin{equation}
\label{eq:cg}
x_{t+1} := \min_{\substack{\deg p \leq t \\ p(0) = 1}} \norm{ p(A) (x_1 - x^*) }_A,
\end{equation}
where $\norm{x}_A := \sqrt{x^\top A x}$, and the minimization is over polynomials with real coefficients.
\begin{manualtheorem}{\ref{thm:cg-sched}}[Conjugate gradient schedule]
For all positive definite matrices $A \in \mathbb{R}^{d\times d}$ and $b \in \mathbb{R}^d$, there exists a multiset of real numbers $\{\eta_t\}$, all in the interval $[\frac{1}{\lambda_{\max}(A)}, \frac{1}{\lambda_{\min}(A)}]$, such that $x_{T+1}$ as defined by the conjugate gradient algorithm (Equation~(\ref{eq:cg})) is equal to $x_{T+1}$ as defined by gradient descent (Equation~(\ref{eq:gd})).
\end{manualtheorem}
\begin{proof}
Let
\[ p^* \in \argmin_{\substack{\deg p \leq t \\ p(0) = 1}} \norm{ p(A) (x_1 - x^*) }_A. \]
By the fundamental theorem of algebra applied to $p^* : \mathbb{R} \rightarrow \mathbb{R}$, and noting that 0 cannot be a root of $p^*$, this is obviously true if the step sizes $\eta_t$ are allowed to be arbitrary complex numbers. We will show that there exists a minimal real-rooted polynomial that achieves the minimum in Equation~(\ref{eq:cg}), with all roots lying in the specified interval. To do this, we will start with a $p^*$ with possibly complex roots, and transform it to fit our conditions, without increasing the residual norm.
Let $\mathcal{F}(p)$ denote the functional that returns the squared residual of a residual polynomial:
\[\mathcal{F}(p) := \norm{ p(A) (x_1 - x^*) }_A^2 = \sum_{ (\lambda_i, u_i) \in \mathrm{eigs}(A) } \lambda_i \bra{ p(\lambda_i) }^2 (u_i ^\top (x_1 - x^*))^2,\]
where $\mathrm{eigs}(A)$ denotes the eigendecomposition of $A$.
Define a partial ordering on functions $p : [\lambda_{\min}(A), \lambda_{\max}(A)] \rightarrow \mathbb{R}$:
\[p \succcurlyeq q \quad \Leftrightarrow \quad |p(\lambda)| \geq |q(\lambda)| \quad \forall \lambda \in [\lambda_{\min}(A), \lambda_{\max}(A)]. \]
Notice that $\mathcal{F}(p)$ is monotone with respect to $(\succcurlyeq)$. That is,
\[p \succcurlyeq q \quad \rightarrow \quad \mathcal{F}(p) \succcurlyeq \mathcal{F}(q).\]
Now we can complete the proof.
\paragraph{Roots are real w.l.o.g.} By the complex conjugate root theorem, if $p^*$ has any complex roots, they come in conjugate pairs $(a \pm bi)$ with matching multiplicities. Multiplying these in pairs gives us quadratic factors $q(a,b) := (x-a)^2 + b^2$. But $|q(a,0)| \leq |q(a,b)|$, so we can construct a real-rooted polynomial $p'$ with the same degree as $p^*$ such that $p' \preccurlyeq p^*$, by deleting the complex parts of each root.
\paragraph{Roots lie within the eigenvalue range w.l.o.g.} By the above, $p^*$ is real-rooted; write $p^*(\lambda) = \prod_{i=1}^{\deg p} (1 - \lambda/\alpha_i)$. Split the real line into intervals $I_1 = (-\infty, 0)$, $I_2 = (0, \lambda_{\min}(A))$, $I_3 = [\lambda_{\min}(A), \lambda_{\max}(A)]$, and $I_4 = (\lambda_{\max}(A), \infty)$. We will show that we can move all the roots of $p^*$ into $I_3$ without increasing $\mathcal{F}$. For roots $\alpha \in I_1$ and $\alpha \in I_4$, notice that $(1 - \lambda/\lambda_{\max}(A)) \preccurlyeq (1 - \lambda/\alpha)$, so we can change those roots to $\lambda_{\max}(A)$. For roots $\alpha \in I_2$, notice that $(1 - \lambda/\lambda_{\min}(A)) \preccurlyeq (1 - \lambda/\alpha)$, so that we can change those roots to $\lambda_{\min}(A)$. By making these changes, we have obtained a polynomial $p''$ with the desired properties, such that $\mathcal{F}(p'') \leq \mathcal{F}(p^*)$.
Finally, the roots of $p^*$ give the reciprocal step sizes needed for the final iterate of gradient descent to match that of conjugate gradient. If $\deg p < T$, then we have more step sizes to assign than roots, and we can simply assign the remaining $T - \deg p$ step sizes to 0. This completes the proof.
\end{proof}
\subsection{Non-convex combination lock}
\label{subsec:appendix-nonconvex}
This construction is a simple variant of the ``needle-in-haystack'' construction for global non-convex optimization with a first-order oracle. This statement can be strengthened, but we optimize for brevity.
\begin{manualproposition}{\ref{prop:nonconvex-combination-lock}}
Let $(\eta_1^*, \ldots, \eta_T^*)$ be any sequence of positive real numbers, and $0 < \delta \leq \frac{1}{2} \min_t \eta_t^*$. Then, there exists a function $f : \mathbb{R}^T \rightarrow \mathbb{R}$ for which:
\begin{itemize}
\item $f$ is infinitely differentiable. All of its derivatives are $O(1/\delta)$.
\item $-1 \leq f(x) \leq 2$ for all $x \in \mathbb{R}^T$, and $\min_{x \in \mathbb{R}^T} f(x) = -1$, where $\eta_{\min} := \min_t \eta^*_t$. The minimizer is unique.
\item Let $x_{\mathrm{out}}$ be the final iterate of gradient descent, starting from $x_1 = 0$ and with learning rate schedule $(\eta_1, \ldots, \eta_t)$. Then, if $\eta_t = \eta^*_t$ for each $t$, then $x_{\mathrm{out}} = -1$. Furthermore, for any $t$ we have
$|\eta_t - \eta^*_t| \geq \delta,$
then
$f(x_{\mathrm{out}}) \geq 0.$
\end{itemize}
\end{manualproposition}
\begin{proof}
We will start by constructing a non-smooth such function.
For all $z \in \mathbb{R}$, $\eta > 0$, define
\[g^{(T)}_{\eta^*_T}(z) = \begin{cases}
2 & z \in (-\infty, -\delta/2] \\
1-z & z \in [-\delta/2, \eta^*_T-\delta/2) \\
-1 & z \in [\eta^*_T-\delta/2, \eta^*_T+\delta/2] \\
0 & z \in (\eta^*_T + \delta/2, \infty)
\end{cases}.\]
Starting at $z=0$, one step of gradient descent on $g^{(T)}$ with learning rate $\eta$ reaches a global minimizer only if $\eta = \eta^*_T$.
Now, for each $t = T-1, \ldots, 1$, define
\[g_{\eta^*_t}^{(t)}(z,z_{t+1},\ldots,z_T) = \begin{cases}
2 & z \in (-\infty, -\delta/2] \\
1-z & z \in [-\delta/2, \eta^*_t-\delta/2) \\
g^{(t+1)}_{\eta^*_{t+1}}(z_{t+1},\ldots,z_T) & z \in [\eta^*_t-\delta/2, \eta^*_t+\delta/2] \\
0 & z \in (\eta^*_t + \delta/2, \infty)
\end{cases},\]
and so forth. Then let $g := g_{\eta^*_1}^{(1)}$ be our unsmoothed function of choice: define $f = g * \psi(2x/\delta)$, where
\[\psi(x) = \begin{cases}
\frac{1}{Z} e^{-\frac{1}{1-\norm{x}^2}} & \norm{x} < 1 \\
0 & \text{ otherwise}
\end{cases},\]
where $Z = \int_{x \in \mathbb{R}^T} \psi(2x/\delta) \; dx \leq O(1/\delta)$. Then:
\begin{itemize}
\item $f$ is infinitely differentiable because $g$ is bounded and $\psi(2x/\delta)$ is infinitely differentiable.
\item $-1 \leq f(x) \leq 2$ by Young's inequality. Since the support of $\psi(2x/\delta)$ is $\delta/2$ times the unit sphere, and $g = -1$ exactly on the $\ell_\infty$ ball of radius $\delta/2$ centered at $(\eta_1^*, \ldots, \eta_T^*)$, the $f$ has a unique minimum at $(\eta_1^*, \ldots, \eta_T^*)$.
\item By the construction, gradient descent with learning rates $\{\eta_t^*\}$, starting at $0$, encounters the gradient sequence $\{e_t\}$, the elementary unit vectors, so it outputs the minimizer. At each iteration $t$, $x_{t+1}$ must lie in the span of $\{e_1, \ldots, e_t\}$ in order for $x_{\mathrm{out}}$ to reach the minimizer. If $|\eta_{t'} - \eta^*_{t'}| \geq \delta$ at any iteration $t$, this invariant cannot hold, since the next gradient is in the span of $e_t$.
\end{itemize}
\end{proof}
It may not be overly pessimistic to think of tuning the learning rate schedule in deep learning as a ``needle-in-haystack'' search problem. Learning rate schedules have been observed to affect generalization behavior in practice \citep{jiang2020characterizing,agarwal2020disentangling}, so that restarting training with a new schedule is the only way to escape poor local optima.
\subsection{No acceleration from the simple spiky schedule}
\label{subsec:appendix-vanilla-spiky}
A natural choice of self-stabilizing learning rate schedule is that which takes one large step of size $\eta^+$ to make progress in directions with shallow curvature, then several small steps of size $\eta^-$ to correct for the overshooting of the large step. This is the cyclic schedule considered by \cite{oymak2021super}, which is shown to obtain a $\log(\kappa)$ ``super-convergent'' rate under the assumption that the eigenvalues of $A$ lie in two clusters.
In this section, we provide a brief note on why this cannot obtain the $\sqrt{\kappa}$ rate on general strongly convex quadratics.
\begin{proposition}
Let $\eta^+, \eta^- \in [1/\lambda_{\max}, 1/\lambda_{\min}]$, and suppose $\eta^+ \geq 10\eta^-$. Let $n$ be a positive integer. Consider the polynomial
\[p(\lambda) := (1 - \eta^+ \lambda) (1 - \eta^- \lambda)^n.\]
Then, if $n \leq 0.1\eta^+ / \eta^-$, it must be true that
\[\norm{p^m}_{[\lambda_{\min}, \lambda_{\max}]} > 1.34^m \]
for all positive integers $m$.
\end{proposition}
\begin{proof}
We have
\[\frac{dp}{d\lambda} = -\eta^+(1-\eta^-\lambda)^n - n\eta^-(1-\eta^+\lambda)(1-\eta^-\lambda)^{n-1}
= -(1-\eta^-\lambda)^{n-1}\pa{n\eta^-(1-\eta^+\lambda) + \eta^+(1-\eta^-\lambda)},
\]
which has a root at $\lambda^* := \frac{\eta^+ + n\eta^-}{(n+1)\eta^+\eta^-}$.
Then,
\begin{align*}
\norm{p}_{[\lambda_{\min}, \lambda_{\max}]} &\geq |p(\lambda^*)|
= \frac{\frac{\eta^+}{\eta^-} - 1}{n+1} \pa{ \pa{1-\frac{\eta^-}{\eta^+}}\pa{1-\frac{1}{n+1}} }^n \\
&\geq \frac{1}{e} \frac{\frac{\eta^+}{\eta^-} - 1}{n+1} \pa{1 - \frac{\eta^-}{\eta^+}} e^{-n\eta^-/\eta^+}
\geq \frac{1}{e} \frac{\frac{\eta^+}{\eta^-} - 1}{0.1 \frac{\eta^+}{\eta^-} + 1} \pa{1 - \frac{\eta^-}{\eta^+}} e^{-n\eta^-/\eta^+} \\
&\geq \frac{1}{e} \cdot \frac{9}{2} \cdot 0.9 \cdot e^{-0.1} > 1.34.
\end{align*}
\end{proof}
Note that $p^m$ is the residual polynomial associated with repeating this cyclic schedule $m$ times. Thus, if the unstable step size in this schedule is $\kappa^\alpha$ times larger than the stable step size, the number of small steps required to prevent exponential blowup of the residual polynomial norm is $\Omega(\kappa^\alpha)$. For any $\alpha \in [0, 1]$, we have $\norm{p^m} \geq |p(\lambda_{\min})|^m \geq \pa{\exp\pa{-O(\sqrt{\kappa})} }^m$, so that $m \geq \Omega(\kappa^{1-\alpha} \log(1/\epsilon))$ cycles are required to make the residual norm at most $\epsilon$. But we have shown that each cycle requires $\Omega(\kappa^\alpha)$ steps; thus, no choice of $\eta^+, \eta^-, n$ can get a better unconditional convergence rate than $O(\kappa \log(1/\epsilon))$.
\section{Experimental details and supplements}
\label{sec:appendix-experiments}
\subsection{Visualization of the quadratic (theoretical) setting}
\label{subsec:appendix-perm-stability}
In Figure~\ref{fig:perm-stability}, we provide a simple illustrative numerical experiment visualizing the tradeoffs; details are below.
This is an instance in dimension $d=100$ with $A = L/\lambda_{\max}(L) + 0.1I$, where $L$ is the Laplacian matrix of the path graph on $100$ vertices; this objective is $2.2$-smooth and $0.2$-strongly convex, $b$ was sampled from $\mathcal{N}(0,I_{100})$, and $x_1=0$. Gradient descent (the non-accelerated baseline) was run with a learning rate of $0.9$, determined via grid search on $0.1$ increments (convergence was not significantly improved with a finer grid).
The Chebyshev nodes were chosen with $m = 0.2, M = 2.2, T = 32$, resulting in the four learning rate schedules shown.
This experiment was run with 80-bit (long double) precision, for illustrative purposes. At 32 or even 64 bits, or with larger $T$, the increasing schedule exhibits exponential blowup of numerical noise, even in this small setting. In the plot to the right, i.i.d. spherical Gaussian noise $\sim \mathcal{N}(0, 0.0005I)$ was added.
\subsection{One-dimensional counterexample}
\label{subsec:appendix-logcosh}
In Section~\ref{subsec:logcosh}, we noted $\log \cosh (x) + 0.01 x^2$ as a ``counterexample by numerical simulation'' to the hypothesis that gradient descent with the fractal Chebyshev schedule converges on general convex functions. This function is $1.02$-smooth and $0.02$-strongly convex.
To refute the possibility that Theorem~\ref{thm:cheb-convergence-rate} holds, it simply suffices to show that there is some setting $m \leq 0.02, M \geq 1.02$ and $T$ such that the theoretical bound does not hold. We chose $m = 0.01$, $M = 5$, $T=32$ (noting that it was quite easy to generate counterexamples). The initial iterate was set to $x_1=2$. The trajectory compared to the theoretical bound is shown in Figure~\ref{fig:logcosh}.
Results are shown in Figure~\ref{fig:logcosh}. Gradient descent (constant step size $1/M$) and Nesterov's accelerated gradient (constant step size $1/M$; momentum parameter $\gamma = 1 - \sqrt{1/0.02}$) are shown for comparison. Of course, none of these parameters are optimized; in the one-dimensional case, it is possible to reach the exact minimizer in one iteration.
\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{figures/logcosh.pdf}
\caption{Non-convergent behavior of the fractal Chebyshev schedule on $f(x) = \log \cosh (x) + 0.01x^2$. The final iterate fails to follow the convergence bound from Theorem~\ref{thm:cheb-convergence-rate}.}
\label{fig:logcosh}
\end{figure}
We conjecture that stronger negative results (say, an infinite family of counterexamples for all $T$) can be constructed.
\subsection{Convex experiments}
\label{subsec:appendix-convex-experiments}
To examine the empirical behavior of gradient descent with the fractal Chebyshev learning rate schedule on a deterministic higher-dimensional convex (but not quadratic) loss, we used the benchmark of logistic regression (with trainable biases, thus totaling $d=7850$ trainable parameters) on the MNIST dataset \cite{lecun1989backpropagation} with normalized raw pixel features, with an $\ell_2$ regularization coefficient of $10^{-3}$. The initial iterate was set to zero during all runs, for a completely deterministic setting. To measure the global minimum, we ran L-BFGS \cite{liu1989limited} until convergence to the 64-bit numerical precision floor.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{figures/convex_large.pdf}
\caption{Convex deterministic MNIST experiments: comparison of classic non-accelerated and accelerated algorithms, and the fractal Chebyshev schedule. Most opaque curves correspond to the optimal tuned constant multiplier; lower opacity corresponds to shrinking the steps in equally spaced increments, down from $0.5$ to $0.1$. Gradient descent with the fractal schedule makes non-local progress and converges at the accelerated rate in practice.}
\label{fig:convex-large}
\end{figure}
We compared three iterative algorithms: gradient descent with a constant learning rate (known in theory to get the slow rate), Nesterov's accelerated gradient descent with a constant learning rate and momentum parameter $0.9$ (known in theory to get the accelerated rate), and gradient descent with the reversed fractal schedule $\eta_{T:1}$ (no known theoretical guarantees in this setting). We tuned the constant learning rates in increments of $0.1$ until divergence (arriving at $0.5$ as the largest stable learning rate). For the fractal schedule, we chose $m=0.0006,M=5$, and tuned the global learning rate multiplier in increments of $0.1$ (arriving also at $0.5$). The step sizes in the schedule were in the range $[0.05, 408.65]$, with a mean of $4.56$ (\emph{much} larger than the maximum stable constant learning rate).
Figure~\ref{fig:convex-large} shows our results: in this setting, gradient descent can achieve accelerated convergence by overstepping the threshold of guaranteed local progress. We used the reverse schedule here (largest step last), as suggested for parameter stability in the noiseless setting. The forward schedule converged with accelerated rates for some choices of hyperparameters, but convergence on this non-quadratic objective was sensitive to initial large steps.
It is not the purpose of Figure~\ref{fig:convex-large} to demonstrate a comparison between Nesterov's acceleration and the fractal schedule; this is a somewhat brittle comparison and is sensitive to hyperparameter choice and floating-point precision. The quantitative comparison from this experiment is between gradient descent with the optimal constant learning rate and any fractal Chebyshev schedule which outperforms it. The Nesterov training loss curves are provided as an illustration only.
\subsection{Deep learning experiments}
\label{subsec:appendix-deeplearning}
We present some simple experiments for the fractal Chebyshev schedule on deep neural networks.
The purpose of this preliminary study is to demonstrate that the constant learning rate ``edge of stability'' can be overstepped without causing training to diverge, using a carefully designed schedule. We do not make claims about end-to-end performance improvements that are robust under ablation and tuning other hyperparameters.
A more systematic examination of the behavior of ``spiky'' learning rate schedules in deep learning is left for future work.
In the deep learning experiments, it is most convenient to think of a learning rate schedule as a time-varying multiplier on top of an existing baseline. Thus, it is most helpful to set the scaling hyperparameters to let the fractal schedule act as a ``learning rate bonus'': set $M = 1$, so that the smallest multiplier is approximately $1$, the largest $\approx 1/m$, and the mean $\approx 1/\sqrt{m}$.
\paragraph{CIFAR-10/ResNet experiments.} The experiments were conducted on the CIFAR-10 dataset on a pre-activation ResNet-18 model \cite{he2016identity} with $d\approx 11\mathrm{M}$ parameters. As a baseline, we trained the network with vanilla minibatch SGD with batch size 8192; the choice of a large batch was made to reduce confounding factors arising from stochasticity, and we omitted the usual practice of momentum in order to remove temporal correlations between step sizes. To find the edge of stability for training with a constant learning rate, we searched over the fixed learning rate parameter on an exponential grid of powers of 2, as depicted in Figure~\ref{fig:cifar-large} (right): the learning rate of $0.05$ leads to stable and the best results; at $0.1$, training is subject to destabilizing outliers, and at $0.2$, the model does not train at all; this is summarized in the fainter training loss curves in Figure~\ref{fig:cifar-large} (left).
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{figures/cifar10-large.pdf}
\caption{ResNet-18/CIFAR-10 training with batch size $8192$ and a repeated $T=8$ fractal Chebyshev schedule. \emph{Left:} Training loss curves. \emph{Right:} Learning rates; the schedule pokes through the edge of stability (magenta and red) without destabilizing training.}
\label{fig:cifar-large}
\end{figure}
We applied a cyclic fractal schedule with $m=0.05$, $M=1$, and $T=8$, as a periodic multiplier on top of the constant learning rate $0.125$; this is pictured in Figure~\ref{fig:cifar-large} (right) as the blue curve. Although this schedule uses large learning rates that would cause unstable training, the fractal Chebyshev schedule periodically surpasses these learning rates while maintaining stable training.
We did not evaluate the model based on generalization performance (indeed, we have removed the usual practices of momentum, random cropping image augmentation, and a decaying learning rate schedule), but in this set of experiments we found the test accuracy to be slightly higher (83\%) than the best constant learning rate baseline (81\%). The stability results were consistent over 5 trials. These experiments were run in PyTorch with an $8\times$ NVIDIA Tesla V100 GPU machine, and each run took less than 30 minutes for 200 epochs.
\paragraph{MNIST experiments with a small neural network.} We chose a simpler and cheaper-to-train model to present a few more empirical insights on the behavior of fractal Chebyshev schedules beyond known theory. Namely, we use the model for MNIST classification from the PyTorch tutorial\footnote{\texttt{https://pytorch.org/tutorials/recipes/recipes/defining\_a\_neural\_network.html}}: two convolutional layers, followed by two fully-connected layers, with a total of $\sim1.2\textrm{M}$ parameters. The model was trained with SGD with batch size 1024.
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{figures/mnist.pdf}
\caption{MNIST experiments, to show that baselines fail to stabilize training as successfully as the fractal Chebyshev schedule. The constant learning rate edge of stability is surpassed at a constant learning rate of 0.8, while the fractal schedule can take steps of up to 1.99. Random permutations of the same schedule cause divergent training, as does a simple ``spiky'' schedule which only oversteps once per cycle.}
\label{fig:mnist}
\end{figure}
In this setup, the same methodology as the larger-scale experiments was used on a finer grid (linearly spaced between 0.1 and 0.8) to determine a stable constant learning rate (0.1) and an unstable one (0.8). A fractal Chebyshev schedule with $m=1/20, M=1, T=64$ accelerated convergence when applied to the stable constant learning rate baseline. However, randomly permuting this schedule caused divergent training. Furthermore, applying a periodic multiplier of $(20,1,1,1,1,1,1,1)$ resulted in \emph{worse} convergence. This exploratory study suggests that some of the self-stabilizing behavior of the fractal schedule in the theoretical setting (where large steps are stabilized by internal copies of Chebyshev iteration, which also consist of large steps) may hold, even for deep networks. Results were consistent over 10 trials.
These experiments were run in PyTorch on a $1\times$ NVIDIA Tesla P100 GPU machine, and each run took around less than 1 minute for 10 epochs.
\section{Additional discussion on related work}
\label{sec:appendix-literature}
\subsection{Fractal cyclic Chebyshev iterative methods}
\label{subsec:appendix-lebedev-literature}
We provide a review of the line of work that serves as the origin of these fractal permutations. These were motivated by the setting of \emph{cyclic iteration methods} for solving linear equations by least-squares in Banach spaces, a primitive in finite element methods for solving partial differential equations. All citations we could find for this line of work have been in the context of numerical methods for least-squares; much of it is in the Russian language, untranslated. We have not encountered prior work linking these methods to machine learning.
\citet{lebedev1971order} construct the fractal permutation seen in this paper, and proved the prefix and suffix bounds, as well a series bound for all prefixes (as opposed to infixes). This remarkable paper is the starting point for us (as well as the authors, evidently). Appendix~\ref{sec:appendix-lebedev} is an attempt to make that paper more accessible (it is far longer than the original paper).
\citet{lebedev1973solution} consider generalized versions of the construction, where $T$ is any positive integer, and the polynomial splitting is performed with the prime factors. They describe general conditions under which stability of a cyclic method (thus only prefix, suffix, and series bounds) can be achieved, and prove stability theorems like the previous work about constructions where the only prime factors of $T$ and 2 and 3.
\citet{lebedev1976utilization}, working in this generalized setting, also analyze the stability of a single cycle of a fractal Chebyshev schedule. They consider series sums of infixes where the series terminate at indices $d_n$ which form a divisor chain of $T$, rather than general indices.
\citet{lebedev2002construction,lebedev2004construction} provide English-language overviews of selected methods and theorems from this line of work.
In all of these works, the authors were motivated by high-precision settings in computational physics, rather than statistical or model error. This is perhaps why they were motivated to generalize the results of \citep{lebedev1971order} in their follow-up papers in a different way than ours. Thus, even though the ideas and motivations overlap with the ones considered in our work, especially in \citep{lebedev1976utilization}, the authors might not have found it important to bound the noise stability of every intermediate iteration of the algorithm. However, thinking of the perturbations as arising from statistical error or model misspecification, this is a natural notion for our setting. We could not find a way to immediately derive our estimates from any theorem or intermediate lemma in \citep{lebedev1976utilization}.
It remains an interesting direction for future work to find efficient algorithms to compute stable schedules for general $T$, and analyze their every-iterate stability like in our work. We could not see immediate ways to extend Theorems~\ref{thm:infix-bound} and \ref{thm:infix-series-bound} to their more general classes of schedules.
\subsection{Learning rate schedules and tradeoffs in practice}
\label{subsec:appendix-lr-schedule-practice}
State-of-the-art models do not show any signs of consensus towards principled or fully-understood learning rate schedules, adaptive or otherwise. A common practice has been to use a cosine learning rate schedule, originally proposed for cyclic warm restarts \cite{loshchilov2016sgdr} but widely adopted in its one-cycle form. For example, GPT-3 \citep{brown2020language} was trained with a cosine schedule. Large-scale empirical studies \cite{shallue2019measuring} indicate that the optimal choice of learning rate schedule is sensitive to the batch size. See the discussion on learning rate schedules in \cite{you2019large} for a discussion of recent empirical observations in pretraining large-scale language models.
Several papers study the theoretical tradeoffs between stability and acceleration in large-scale stochastic optimization: \cite{bottou2007tradeoffs,devolder2014first,chen2018stability,agarwal2020stochastic}. A common message throughout these papers is that the best choice of iterative optimization algorithm depends in general on the data, model, and computational resources.
\cite{cohen2020gd} provide an empirical account of the insufficiency of second-order Taylor approximations of the loss function in deciding the correct learning rate. \cite{agarwal2020disentangling} point out that learning rate schedules are entangled with adaptive gradient methods.
\subsection{Learning rate schedules in theory}
\label{subsec:appendix-lr-schedule-theory}
While learning rate schedules while ubiquitously used in practice, the diversity of existing practical learning rate schedules has received little theoretical treatment. In convex optimization, learning rate schedules have primarily been employed in stochastic settings, in particular to correctly average the zero-mean noise. It is well known in the stochastic and online optimization literature that a step schedule akin to $t^{-1/2}$ is necessary for the convergence of stochastic gradient descent. In the case of zero-mean bounded variance stochastic noise, the AC-SA algorithm proposed by \cite{lan2012optimal} which achieves optimal convergence rates employs an effective step decay schedule of $t^{-3/2}$. In a complementary line of work, \cite{ge2019step} show that for the streaming least squares regression problems, no polynomial decay of learning rates achieves the minimax rate; on the other hand the rate is achieved by the geometric decay learning rate schedule which is very popular in practice.
An alternative point of view towards the power of learning rate schedules arises from the Polyak step size \cite{polyak1987introduction, hazan2019revisiting}, which is a single learning rate per step which generalizes the classical gradient descent oblivious to the smoothness/strong convexity properties of the function. The Polyak step size requires the knowledge of the optimality gap at any point. The vanilla version of the Polyak step size is unable to provide accelerated rates; an extension of these ideas to momentum has been carried out by \citet{barre2020complexity}.
Practical deep learning models due to the presence of normalization layers lead to homogeneous models. For such models, \citet{li2019exponential} perhaps surprisingly show that the standard training algorithm which includes weight decay and momentum is equivalent to performing an exponentially increasing learning rate schedule. \citet{li2020reconciling} further explore the intricate interaction of weight decay and learning rates in such models proposing the notion of an intrinsic learning rate. The practice of using a large initial learning rate in optimization from the point of view of better generalization has been theoretically investigated in \cite{li2019towards} (see references herein for a detailed treatment of the topic).
A line of work \citep{orabona2017training,cutkosky2018black} derives parameter-free algorithms for selecting learning rates which are optimal in the noise-dominated (as opposed to curvature-dominated) regime. These algorithms are shown to be practical for training deep neural networks with small batch size (e.g. a convolutional network for CIFAR-10 with batch size 128). The theory presented in this paper is only applicable to large batch/curvature-dominated settings which is the regime, where one might hope to isolate the benefits of acceleration. In small-batch/noise-dominated settings, the precise role of acceleration/learning rate schedules is muddled with confounding factors (e.g. variance reduction); see the next section for a discussion of this point. Designing adaptive algorithms which interpolate between these results and ours, like the analysis of Nesterov's acceleration under additive noise \cite{lan2012optimal}, is an interesting direction for future work; we hope that this will lead to new practical algorithms for large-scale settings.
\subsection{Acceleration methods and momentum}
The phenomenon of acceleration in numerical analysis and optimization is a classical concept which has manifested through a large variety of viewpoints , algorithms, and analyses over the years. We provide a very short and limited summary of these manifestations, focusing on more modern machine learning focused developments. For an in-depth treatment, we strongly recommend the reader to refer the recent monograph \cite{d2021acceleration}. Possibly the earliest works on non-linear acceleration in numerical analyses date back to Aitken's $\Delta^2$ \cite{aitken1927xxv}, Wynn's epsilon algorithm \cite{wynn1956device}, and Anderson acceleration \cite{anderson1965iterative} (see \cite{sidi1986acceleration} for an in-depth survey, or the blogpost \cite{BachBlog} for a condensed description). The recent work of \cite{li2020fast} establishes an optimum rate for an Anderson acceleration method based on Chebyshev polynomials. The more standard suite of acceleration algorithms applied in machine learning arise from the direct acceleration algorithms like Polyak's momentum (also known as the heavy ball algorithm) \cite{POLYAK19641} and Nesterov's breakthrough result \cite{nesterov1983method} which established the optimal rates for general smooth convex optimization.
More recently, various acceleration algorithms \cite{allen2014linear, bubeck2015geometric} have been proposed, with more intuitive analyses than Nesterov's. Another line of work stemming from the work of \cite{su2014differential, wibisono2015accelerated} derives Nesterov-like methods via discretizations of appropriate continuous-time differential equations. A lesser known (but relevant to our work) version of direct acceleration is Nemirovski's acceleration based on a line search (\emph{not} a search over $\eta_t$ like the greedy steepest descent method); see \cite{BubeckBlogNemirovski} for a concise exposition. An alternative methodology for acceleration \cite{monteiro2013accelerated, lin2018catalyst} comes about via iteratively solving appropriate (strongly convex) proximal point problems using classical iterative methods. This latter line of work has been influential in deriving optimal accelerated versions of higher-order methods \cite{nesterov2008accelerating, bubeck2019near}.
In stochastic and/or non-convex settings (including deep learning), the role of acceleration is not fully clear. In the general convex case, worst case theory \cite{lan2012optimal} suggests that acceleration leads to benefits only in curvature-dominated regime (as opposed to the noise dominated regime). Nevertheless, heavy-ball momentum and Nesterov acceleration are part of the core toolkit in state-of-the-art optimization for optimization in various batch size regimes \cite{sutskever2013importance, kingma2014adam, dozat2016incorporating}. Recent theoretical work \cite{cutkosky2019momentum} suggests that momentum can implicitly perform variance reduction (akin to a low-pass filter), leading to improved convergence rates for stochastic optimization in non-convex problems. Understanding the variance-reducing mechanisms of the fractal schedules (or any learning rate schedule in general) is an interesting direction for future work.
In a recent orthogonal line of inquiry into momentum methods, \citet{pmlr-v119-pedregosa20a, pmlr-v119-scieur20a} analyze an average case setting of the quadratic model, and establish the universality of Polyak momentum as an optimal algorithm. The analysis of globally-optimized learning rate schedules in average-case settings is an interesting direction for future work.
\subsection{Optimization as a dynamical system}
Our approach to analyzing stability is most similar to the view of optimization algorithms as dynamical systems \cite{lessard2016analysis,li2017stochastic}. Of course, beyond the simplest objectives and noise models, optimization algorithms are nonlinear dynamical systems; thus, theory under this very general abstraction is very limited. \citet{bousquet2002stability} define related but stronger notions of stability, which can lead to generalization properties \cite{hardt2016train,chen2018stability,agarwal2020stochastic}.
In the dynamical systems view, our work shows that stable acceleration is obtained by treating the learning rate schedule as a long-horizon planning problem, accounting for the interactions between the choices of $\eta_t$ at different times and the global curvature of the loss. Even an open-loop control sequence (i.e. non-adaptive schedule) designed with global objectives has a provable benefit over a naive closed-loop controller (i.e. adaptive line search) which only uses instantaneous feedback (i.e. $x_t, g_t$). Thus, it may be beneficial for any closed-loop controller for the learning rate schedule to depend on global context or curvature, and possibly make negative local progress. In light of this, neural and reinforcement learning-based optimizer search \citep{bello2017neural} may be an enticing solution to the empirical problem of scheduling the learning rate with awareness of global curvature.
| {
"timestamp": "2021-06-15T02:04:23",
"yymm": "2103",
"arxiv_id": "2103.01338",
"language": "en",
"url": "https://arxiv.org/abs/2103.01338",
"abstract": "In practical applications of iterative first-order optimization, the learning rate schedule remains notoriously difficult to understand and expensive to tune. We demonstrate the presence of these subtleties even in the innocuous case when the objective is a convex quadratic. We reinterpret an iterative algorithm from the numerical analysis literature as what we call the Chebyshev learning rate schedule for accelerating vanilla gradient descent, and show that the problem of mitigating instability leads to a fractal ordering of step sizes. We provide some experiments to challenge conventional beliefs about stable learning rates in deep learning: the fractal schedule enables training to converge with locally unstable updates which make negative progress on the objective.",
"subjects": "Machine Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)",
"title": "Acceleration via Fractal Learning Rate Schedules",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754456551736,
"lm_q2_score": 0.7185943985973773,
"lm_q1q2_score": 0.7075104002443242
} |
https://arxiv.org/abs/1810.05759 | Topological Inference of Manifolds with Boundary | Given a set of data points sampled from some underlying space, there are two important challenges in geometric and topological data analysis when dealing with sampled data: reconstruction -- how to assemble discrete samples into global structures, and inference -- how to extract geometric and topological information from data that are high-dimensional, incomplete and noisy. Niyogi et al. (2008) have shown that by constructing an offset of the samples using a suitable offset parameter could provide reconstructions that preserve homotopy types therefore homology for densely sampled smooth submanifolds of Euclidean space without boundary. Chazal et al. (2009) and Attali et al. (2013) have introduced a parameterized set of sampling conditions that extend the results of Niyogi et al. to a large class of compact subsets of Euclidean space. Our work tackles data problems that fill a gap between the work of Niyogi et al. and Chazal et al. In particular, we give a probabilistic notion of sampling conditions for manifolds with boundary that could not be handled by existing theories. We also give stronger results that relate topological equivalence between the offset and the manifold as a deformation retract. | \section{Introduction}
\label{sec:introduction}
In \emph{manifold learning}, a topic of high interest is to understand the structure of low-dimensional objects embedded in high-dimensional space.
Such objects are typically assumed to be (sub)manifolds of Euclidean spaces.
In recent years, it is becoming clear that the offsets of sampled data points on a manifold can reflect the geometric and topological structure of the manifold itself (e.g.~\cite{AmentaBern1999, ChengDeyRamos2005, NiyogiSmaleWeinberger2008}).
In particular, given sampled points drawn from a probability distribution that has support on or near a submanifold without boundary, Niyogi et al.~\cite{NiyogiSmaleWeinberger2008} have shown that one can learn the homology of the submanifold with high confidence.
More specifically, for a compact manifold $\mathcal{M}$ embedded in Euclidean space $\mm{{\mathbb R}}^N$ and a set of randomly sampled data points $\overline{x}=\{x_1,...,x_n\}$ on $\mathcal{M}$, let $U=\bigcup_{x\in\overline{x}}B_{\epsilon}(x)$ be the offset of the data set $\overline{x}$, where $\epsilon$ is chosen to be small relative to the minimum local feature size of $\mathcal{M}$. Then for any $p\in (0,1)$, there is a number $m$ such that for all $n>m$, with probability $p$, $\mathcal{M}$ is a deformation retract of $U$. Therefore the homology of $U$
equals the homology of $\mathcal{M}$ (see \cite[Theorem 3.1]{NiyogiSmaleWeinberger2008} for details).
Data that arise from smooth compact manifolds have been well-studied.
However, the study of more complex spaces that are not necessarily manifolds via data samples seems much more difficult.
When samples arise not from manifolds but from mixtures of manifolds with possible singularities, we are dealing with the notion of \emph{stratification learning}.
Roughly speaking, a \emph{stratified space} is a space that can be decomposed in to manifold pieces (referred to as \emph{strata}) that are glued together in a nice way.
The study of stratified spaces is a classic topic in pure mathematics~\cite{GoreskyMacPherson1988, Weinberger1994}.
Statistical approaches rely on inferences of mixture models, local dimension estimation and subspace clustering~\cite{HaroRandallSapiro2005, LermanZhang2010, VidalMaSastry2005}.
In geometric and topological data analysis, progress has been made recently in the study of stratified spaces in the discrete and noisy settings~\cite{BendichCohen-SteinerEdelsbrunner2007, BendichWangMukherjee2012, SkrabaWang2014, BelkinQueWang2012, Nanda2017, BrownWang2018},
which draw inspirations from computational topology~\cite{EdelsbrunnerHarer2010}, intersection homology~\cite{Bendich2008, BendichHarer2011, GoreskyMacPherson1982}, graph theory and sheaf theory.
Among stratified spaces, manifolds with boundary is one of the simplest forms.
A manifold with boundary is a stratified space: one stratum is its boundary, and the other stratum is the complement.
In this paper, we study the topology of offsets of data points on compact differentiable manifolds with boundary.
We give a probabilistic notion of sampling conditions for manifolds with boundary that could not be handled by existing theories.
In particular, we show that, with some care, a similar statement as \cite[Theorem 3.1]{NiyogiSmaleWeinberger2008} holds for manifolds with boundary.
We also demonstrate via simple examples how our sampling lower bounds could be derived in practice.
The main result of this paper, Theorem \ref{main}, is proved by following the framework of~\cite{NiyogiSmaleWeinberger2008}.
First, we prove that the offset of data points deformation retracts to the manifold if the sample is sufficiently dense with respect to the local feature size of the manifold and the radius of the offset (Theorem \ref{deformretrth}).
Second, we show that such density is achieved with high confidence when the data points are sufficiently abundant (Theorem \ref{probabestim}). However, our proof in detail is very different from that of \cite{NiyogiSmaleWeinberger2008}:
particular efforts have been made to overcome the complexity caused by the boundary.
When a data point is near or on the boundary, the local geometry around it is more complicated, and the original ways~\cite{NiyogiSmaleWeinberger2008} of performing deformation retract and estimating distances and volumes no longer work. We found that, through our arguments, such issues can be resolved by imposing concise and reasonable requirements regarding the minimum local feature size of \emph{both} the manifold and its boundary and the (local) projection of the manifold to its tangent spaces (see Condition \eqref{condition1} and \eqref{condition2}).
It is worth noting that Chazal et al.~\cite{ChazalCohen-SteinerLieutier2009} and Attali et al.~\cite{AttaliLieutierSalinas2013} have extended the results of Niyogi et al. to a large class of compact subsets of Euclidean space.
Specifically, \cite{ChazalCohen-SteinerLieutier2009} shows that the offset $K^{\beta}$ of a compact set $K$ is homotopy equivalent to the offset $K'^{\alpha}$ of another compact set $K'$, for sufficiently small $\beta>0$, if $\alpha$ satisfies some inequality involving the $\mu$-reach of $K'$ and the Hausdorff distance between $K$ and $K'$; \cite{AttaliLieutierSalinas2013} shows that the offset $X^{\eta}$ of a compact set $X$ is homotopy equivalent to the \v{C}ech (respectively the Rips) complex of certain radius of a point cloud $P$ in $X$ if some inequality involving $\eta$, the radius and the $\mu$-reach of $X$ holds. Compared to the result of~\cite{ChazalCohen-SteinerLieutier2009} and \cite{AttaliLieutierSalinas2013}, the one in this paper only deals with compact manifolds with boundary. However, our result does have its own advantages. First of all, the topological equivalence between the offset of data points and the manifold we get here is a deformation retract, which is stronger than homotopy equivalence derived in \cite{ChazalCohen-SteinerLieutier2009} and \cite{AttaliLieutierSalinas2013}. Next, we get a probability estimate for the topological equivalence which is not provided in \cite{ChazalCohen-SteinerLieutier2009} and \cite{AttaliLieutierSalinas2013}. Last but not least, it appears that there are elementary cases of data on manifolds with boundary where the result in \cite{ChazalCohen-SteinerLieutier2009} does not apply, as the parameters associated with the data are completely outside the scope of \cite[Theorem 4.6]{ChazalCohen-SteinerLieutier2009} and \cite[Theorem 13 and 14]{AttaliLieutierSalinas2013}. A scenario is discussed in Section \ref{sec:results} (Example \ref{exbeyondccsl}), where our Theorems \ref{main} and \ref{deformretrth}
become applicable and work well.
\para{Result at a glance.}
In short, this paper improves our understanding of topological inference for manifolds with boundary and
therefore enriches the toolbox for topological data analysis.
Given a sample of $n$ points from a differentiable manifold $\mathcal{M}$ with boundary
in a high-dimensional Euclidean space, for a sufficiently large $n$,
the $\epsilon$-offset of the sample points is shown to have the same homotopy type as and deformation retracts to $\mathcal{M}$.
The homotopy equivalence result has been proved by Niyogi et al.~\cite{NiyogiSmaleWeinberger2008} for manifolds without boundary.
Chazal et al.~\cite{ChazalCohen-SteinerLieutier2009} have extended the result of Niyogi et al.~to manifolds with boundary,
however with weaker conclusions than the original paper.
The current paper, instead, reaches the same conclusion as Niyogi et al.~\cite{NiyogiSmaleWeinberger2008} for manifold with boundary,
but under two mild conditions for the boundary (i.e.,~regarding the minimum feature size of the boundary and the uniform smoothness of the tangent bundle).
A specific example shows that the new method is more powerful than that of Chazal et al.~\cite{ChazalCohen-SteinerLieutier2009}.
Therefore, our results fill a gap between the original work of Niyogi et al.~\cite{NiyogiSmaleWeinberger2008} and
the broader but weaker result of Chazal et al.~\cite{ChazalCohen-SteinerLieutier2009}.
Our proof techniques are nontrivial, although they frequently use the results of Niyogi et al.~and make necessary adjustments.
The theoretical results are complemented by experiments that confirm the theoretical findings.
\section{Notations and preliminaries}
\label{sec:prelim}
\subsection{Basics on manifolds}
In this paper, for two points $p,q\in \mm{{\mathbb R}}^N$, we use $|pq|$ to denote the line segment connecting $p$ and $q$, $\overrightarrow{pq}$ to denote the vector from $p$ to $q$, and $\|p-q\|$ to denote the Euclidean distance between $p$ and $q$. For a set $K\subset \mm{{\mathbb R}}^N$, $d(p,K):=\inf\{\|p-q\|: q\in K\}$ denotes the distance from $p \in \mm{{\mathbb R}}^N$ to $K$. Moreover, for a non-negative real number $\alpha \in \mm{{\mathbb R}}_{\geq 0}$, we use $K^{\alpha}$ to denote the offset of $K$ with radius $\alpha$,
$K^{\alpha} = \{ p \in \mm{{\mathbb R}}^N \mid d(p,K) \leq \alpha\}$.
$B_{r}(p)$ denotes the open ball with center $p$ and radius $r$.
Let $\mathcal{M}\subseteq\mm{{\mathbb R}}^N$ be a compact, \emph{differentiable}, $k$-dimensional manifold possibly with boundary. Let $\partial\mathcal{M}$ denote the boundary of $\mathcal{M}$. Then $\partial\mathcal{M}$ is a compact manifold. Let $\mathcal{M}^{\circ}$ denote the interior of $\mathcal{M}$.
The \emph{local feature size} of $\mathcal{M}$ is the function $\lfs: \mathcal{M} \to \mm{{\mathbb R}}_{\geq 0}$ defined by the distance from a point $x \in \mathcal{M}$ to its medial axis.
The infimum of $\lfs$ is the reach of $\mathcal{M}$, $\reach(\mathcal{M})$.
For every number $0<r<\reach(\mathcal{M})$, the normal bundle about $\mathcal{M}$ of radius $r$ is embedded in $\mm{{\mathbb R}}^N$. In the same way we define $\reach(\partial\mathcal{M})$, and since $\partial\mathcal{M}$ is also a compact manifold, $\reach(\partial\mathcal{M})$ is well-defined.
We use $\varphi_{p, \mathcal{M}}$ to denote the natural projection from $\mathcal{M}$ to $T_p(\mathcal{M})$, the tangent space to $\mathcal{M}$ at the point $p$. That is,
$\varphi_{p, \mathcal{M}}: \mathcal{M} \to T_p(\mathcal{M})$.
Conversely for any point $q\in T_p(\mathcal{M})$ we use $\varphi_{p, \mathcal{M}}^{-1}(q)$ to denote the set of points in $\mathcal{M}$ which maps to $q$ via $\varphi_{p, \mathcal{M}}$.
We take $\delta=\delta(\mathcal{M}) \in \mm{{\mathbb R}}_{\geq 0}$ to be any non-negative real number such that for any $p\in\mathcal{M}$, the following Condition~\ref{condition1} and Condition~\ref{condition2} are satisfied:
\begin{ceqn}
\begin{align}
& \delta <\min\{\reach(\mathcal{M}), \reach(\partial\mathcal{M})\}, \label{condition1}\\
& \varphi_{p, \mathcal{M}}|_{B_{\delta}(p)\cap\mathcal{M}} \textnormal{ is a diffeomorphism onto its image. } \label{condition2}
\end{align}
\end{ceqn}
Finally, suppose $\overline{x}=\{x_1,...,x_n\}$ is a set of sampled data points from a compact, differentiable manifold $\mathcal{M}$ with boundary.
$\overline{x}$ is \emph{$\epsilon$-dense} if for any $p \in \mathcal{M}$, there is a point $x \in \overline{x}$ such that $x \in B_{\epsilon}(p)$.
Let $U=\bigcup_{x\in\overline{x}}B_{\epsilon}(x)$ denote the offset of $\overline{x}$.
We also define the canonical map $\pi:U\to \mathcal{M}$ by
$$\pi(x):={\rm arg}\min_{p\in\mathcal{M}}\|x-p\|.$$
By Condition (1) and the definition of $\reach$, $\pi$ is well-defined.
\subsection{Volume of a hyperspherical cap}
\label{sechypersphecap}
Let $S$ be a $k$-dimensional hypersphere of radius $r$. Let $H$ be a hyperplane that divides $S$ into two parts. We take the smaller part as a hyperspherical cap. Let $a$ be the radius of the base of the cap, and $\phi:=\arcsin(\dfrac{a}{r})$. Then by \cite{Li2011}, the volume of the hypershperical cap is
$$V=\frac{2\pi^{\frac{k-1}{2}}r^k}{\Gamma(\frac{k+1}{2})}=\frac{1}{2}V_k(r)I_{\sin^2\phi}(\frac{k+1}{2},\frac{1}{2}),$$
where $\Gamma(x)$ is the Gamma function, $V_k(r)$ is the volume of the $k$-dimensional sphere with radius $r$ and $I_x(a,b)$ is the regularized incomplete beta function.
\section{The main results}
\label{sec:results}
The main theorem of this paper (Theorem~\ref{main}) is centered around a probabilistic notion of sampling conditions for manifolds with boundary, which relates the topological equivalence between the offset (of samples) and the manifold as a deformation retract.
To the best of our knowledge, such a result has not been addressed by existing theories.
\begin{theorem} \label{main}
Let $\mathcal{M}\subseteq\mathbb{R}^N$ be a compact differentiable $k$-dimensional manifold possibly with boundary. Let $\overline{x}=x_1, x_2, ..., x_n$ be drawn by sampling $\mathcal{M}$ in i.i.d fasion according to the uniform probability measure on $\mathcal{M}$. Let $\epsilon\in (0,\dfrac{1}{2}\delta(\mathcal{M}))$ and $U=\bigcup_{x\in\overline{x}}B_{\epsilon}(x)$. Then for all
$$n> \beta(\epsilon)(\ln \beta(\frac{\epsilon}{2})+\ln(\frac{1}{\gamma})),$$
$U$ deformation retracts to $\mathcal{M}$ with probability $>1-\gamma$.
In particular, with such confidence, the homology of $U$ is the same as that of $\mathcal{M}$.
Here, $\displaystyle\beta(x):=\frac{{\rm vol}(\mathcal{M})}{\frac{\cos^k\theta}{2^{k+1}}I_{1-\frac{x^2\cos^2\theta}{16\delta^2}}(\frac{k+1}{2}, \frac{1}{2}){\rm vol}(B_{x}^k)}$ and $\theta=\arcsin(\frac{x}{4\delta})$.
\end{theorem}
Theorem~\ref{main} is implied by combining Theorems \ref{deformretrth} and \ref{probabestim} below.
\begin{theorem}
\label{deformretrth}
Let $\bar{x}$ be any finite collection of points $x_1, ..., x_n\in \mathbb{R}^{N}$ such that it is $\dfrac{\epsilon}{2}$-dense in $\mathcal{M}$. Then for any $\epsilon<\dfrac{\delta}{2}$, we have that $U$ deformation retracts to $\mathcal{M}$.
\end{theorem}
\begin{theorem}
\label{probabestim}
Let $\overline{x}$ be drawn by sampling $\mathcal{M}$ in i.i.d fasion according to the uniform probability measure on $\mathcal{M}$. Then with probability $1-\gamma$, we have that $\overline{x}$ is $\dfrac{\epsilon}{2}$-dense ($\epsilon<\frac{\delta}{2})$ in $\mathcal{M}$ provided
$$|\overline{x}|\ge \beta(\epsilon)(\ln \beta(\frac{\epsilon}{2})+\ln(\frac{1}{\gamma})).$$
\end{theorem}
As is mentioned in Section~\ref{sec:introduction}, compared to \cite[Theorem 4.6]{ChazalCohen-SteinerLieutier2009} and \cite[Theorem 13]{AttaliLieutierSalinas2013}, Theorem \ref{deformretrth} only deals with manifolds with boundary, but it establishes a criterion for deformation retract, which is stronger than homotopy equivalence as in \cite{ChazalCohen-SteinerLieutier2009}. The following is an example where neither \cite[Theorem 4.6]{ChazalCohen-SteinerLieutier2009} or \cite[Theorem 13]{AttaliLieutierSalinas2013} applies but Theorem \ref{deformretrth} does.
\begin{example}
\label{exbeyondccsl}
Let $C$ be a semi-unit-circle. Then $C$ is a manifold with boundary and its boundary consists of the two end points which are also the end points of a diameter. Let the data be the points $A_1,A_2,...,A_8$ which divide the semi-circle evenly into 7 arcs, with $A_1$ and $A_8$ being the end points. Denote $A:=\{A_1,...,A_8\}$.
We first treat this case with the result in \cite{ChazalCohen-SteinerLieutier2009}. We adopt the notations there. \cite[Theorem 4.6]{ChazalCohen-SteinerLieutier2009} is the main reconstruction theorem, and it requires the inequality
\begin{ceqn}
\begin{align}
\label{resultccsl}
\frac{4d_H(K,K')}{\mu^2}\le \alpha<r_{\mu}(K')-3d_H(K,K')
\end{align}
\end{ceqn}
in order for the offset $K^{\alpha}$ to be homotopy equivalent to $K'$. Here we let $K'=C$ and $K=A$. It is easy to see that $d_H(K,K')=2\sin\dfrac{\pi}{28}\approx 0.223928$, and
\begin{ceqn}
\begin{align*}
r_{\mu}(K')=r_{\mu}(C)=\begin{cases}
0 & \mu> 1 \\
1 & 0<\mu\le 1.
\end{cases}
\end{align*}
\end{ceqn}
If $\mu>1$, then as $r_{\mu}(K')=0$, the right half of \eqref{resultccsl} does not make sense. If $0<\mu\le 1$, then
$$\dfrac{4d_H(K,K')}{\mu^2}\ge 4d_H(K,K')=8\sin\dfrac{\pi}{28}>1-6\sin\dfrac{\pi}{28}=r_{\mu}(K')-3d_H(K,K').$$
This is a contradiction to \eqref{resultccsl}. Therefore in this case, \cite[Theorem 4.6]{ChazalCohen-SteinerLieutier2009} does not apply.
Next we try \cite[Theorem 13 and 14]{AttaliLieutierSalinas2013}. \cite[Theorem 13]{AttaliLieutierSalinas2013} requires that
$$d_H(A,C)\le \epsilon <\lambda^{\rm cech}(\mu)r_{\mu}(C),$$
where
$$\lambda^{\rm cech}(\mu)=\dfrac{-3\mu+3\mu^2-3+\sqrt{-8\mu^2+4\mu^3+18\mu+2\mu^4+9+\mu^6-4\mu^5}}{-7\mu^2+22\mu+\mu^4-4\mu^3+1}$$
We have $d_H(K,K')=0.223928$ as well as the value of $r_{\mu}(C)$ as is deduced above. On the other hand, from \cite[Fig. 9]{AttaliLieutierSalinas2013} we see that $\lambda^{\rm cech}(\mu)$ is increasing on (0,1], hence $\lambda^{\rm cech}(\mu)\le \lambda^{\rm cech}(1)=\dfrac{-3+\sqrt{22}}{13}\approx 0.130032$. So it is easy to see that $d_H(A,C)>\lambda^{\rm cech}(\mu)r_{\mu}(C)$ for all $\mu\in \mathbb{R}_+$. Therefore \cite[Theorem 13]{AttaliLieutierSalinas2013} does not apply. \cite[Theorem 14]{AttaliLieutierSalinas2013} requires that
$$d_H(A,C)\le \epsilon <\lambda_{n}^{\rm rips}(\mu)r_{\mu}(C).$$
But by \cite[Fig. 9]{AttaliLieutierSalinas2013}, $\lambda_{n}^{\rm rips}(\mu)$ is always smaller than $\lambda^{\rm cech}(\mu)$, so \cite[Theorem 14]{AttaliLieutierSalinas2013} does not apply either.
Finally we try to apply Theorem \ref{deformretrth} to this case. It is easy to see that $\reach(C)=\reach(\partial C)=1$, and it also satisfies Condition \eqref{condition2} to let $\delta(C)=1$. Now let $\epsilon=0.48<\dfrac{1}{2}=\dfrac{\delta(C)}{2}$. Since $d_H(A,C)=2\sin\dfrac{\pi}{28}\approx 0.223928<0.24=\dfrac{\epsilon}{2}$ as is calculated above, $A$ is $\dfrac{\epsilon}{2}$-dense. Therefore Theorem \ref{deformretrth} applies.
\end{example}
\section{Proofs of the main results}
\label{sec:proofs}
To prove our main results, we begin with a series of lemmas. Recall the canonical map $\pi:U\to \mathcal{M}$ is defined by
$\pi(x):={\rm arg}\min_{p\in\mathcal{M}}\|x-p\|$. Lemmas \ref{allintubneighbor}-\ref{star} contribute to the proof of Theorem \ref{deformretrth}.
Roughly speaking, they show that $\pi^{-1}(p)$ is star-shaped for every $p\in\mathcal{M}$, hence the deformation retract is well-defined.
Lemmas \ref{localvolest}-\ref{volest} contribute to the proof of Theorem \ref{probabestim} by giving a lower bound to the volume of $\mathcal{M}\cap B_{\epsilon}(p)$ for every $p\in\mathcal{M}$ in terms of $\epsilon$. Then by directly applying results in \cite{NiyogiSmaleWeinberger2008}, we get the estimation of the number of data points as in Theorem \ref{probabestim}.
\begin{lemma} \label{allintubneighbor}
Choose $\delta=\delta(\mathcal{M})>0$ as in Section~\ref{sec:prelim}. Then for any $\lambda\in(0,\delta)$, any point $q$ such that $\|q-\pi(q)\|<\lambda$ and any $q'\in |q\pi(q)|$, we have $\pi(q')=\pi(q)$.
\end{lemma}
\begin{proof}
Suppose that this is not the case. Then
$$\|q-\pi(q)\|=\|q-q'\|+\|q'-\pi(q)\|>\|q-q'\|+\|q'-\pi(q')\|\ge \|q-\pi(q')\|.$$
This is a contradiction to Condition \ref{condition1}.
\end{proof}
Suppose a certain point $q \in U$ deformation retracts to a point $\pi(q) \in \mathcal{M}$.
Let $|q \pi(q)|$ be the path of the deformation retract for $q$.
Lemma \ref{allintubneighbor} tells us that along the path $|q \pi(q)|$, all the points deformation retracts to $\pi(q)$.
From now on we set $\epsilon<\dfrac{\delta}{2}$. For a point $p\in \mathcal{M}$ and a point $p'\in \partial\mathcal{M}$ we define $\myst_{\mathcal{M}}(p)$ and $\myst_{\partial\mathcal{M}}(p')$ as
\begin{ceqn}
\begin{align*}
& \myst_{\mathcal{M}}(p):=\bigcup_{x\in\bar{x};x\in B_{\epsilon}(p)}(B_{\epsilon}(x)\cap T_p(\mathcal{M})^{\perp}). \\
& \myst_{\partial\mathcal{M}}(p'):=\bigcup_{x\in\bar{x};x\in B_{\epsilon}(p')}(B_{\epsilon}(x)\cap T_{p'}(\partial\mathcal{M})^{\perp}).
\end{align*}
\end{ceqn}
For convenience, we also define $\myst_{\partial\mathcal{M}}(p'):=\emptyset$ if $p' \in \mathcal{M}^{\circ}$.
We present the next lemma, whose proof is exactly the same as those corresponding ones in \cite{NiyogiSmaleWeinberger2008}, although the statements are somewhat different.
\begin{lemma} [\cite{NiyogiSmaleWeinberger2008}, Proposition 4.1] \label{ststarshaped}
$\myst_{\mathcal{M}}(p)$ (resp.~$\myst_{\partial\mathcal{M}}(p)$) is star-shaped for any $p\in\mathcal{M}$ (resp.~$p\in\partial\mathcal{M}$).
\end{lemma}
\begin{lemma}
\label{pi-1contained}
$\pi^{-1}(p)\subseteq \myst_{\mathcal{M}}(p)$ (resp. $\pi^{-1}(p)\subseteq \myst_{\partial\mathcal{M}}(p)$) for any $p\in\mathcal{M}$ (resp. $p\in\partial\mathcal{M}$).
\end{lemma}
\begin{proof}
If $p \in \mathcal{M}^{\circ}$,
this is already proven in \cite[Proposition 4.2]{NiyogiSmaleWeinberger2008}. So for the rest of the proof we assume that $p\in\partial\mathcal{M}$.
Let $v$ be an arbitrary point in $\pi^{-1}(p)$. By the definition of $\myst_{\partial\mathcal{M}}$, we only need to consider the case where there is a point $q\in \bar{x}$ such that $q\notin B_{\epsilon}(p)$ and $v\in B_{\epsilon}(q)$. In this case, the distance between $v$ and $p$ is at most $\dfrac{\epsilon^2}{\delta}$, and the proof is exactly the same as that of \cite[Lemma 4.1]{NiyogiSmaleWeinberger2008}. Now by the $\dfrac{\epsilon}{2}$-dense condition, there is a point $x\in\overline{x}$ such that $\|x-p\|\le \dfrac{\epsilon}{2}$. Therefore
$$\|v-x\|\le \|v-p\|+\|p-x\|\le \dfrac{\epsilon^2}{\delta}+\dfrac{\epsilon}{2}< \dfrac{\epsilon}{2}+\dfrac{\epsilon}{2}=\epsilon.$$
\end{proof}
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=0.45\linewidth]{lemma7.pdf}
\end{tabular}
\vspace{-12mm}
\caption{The worse case for Lemma~\ref{pi-1contained}, where $||v-p|| = ||p-x|| = \frac{\epsilon}{2}$.}
\label{fig:lemma7}
\end{center}
\end{figure}
\begin{remark}
We note that the worst case in Lemma \ref{pi-1contained} can happen when $p$ lies on the boundary of $\mathcal{M}$. This is illustrated in Fig.~\ref{fig:lemma7}, where $M$ is a plane region with boundary, $v$ is also inside the plane, and $|vp|$ and $|xv|$ are both perpendicular to $\partial\mathcal{M}$.
This is why we require that $\epsilon<\dfrac{\delta}{2}$, which is more restrictive than the requirement $\epsilon<\sqrt{\dfrac{5}{3}}\delta$ in \cite{NiyogiSmaleWeinberger2008}.
\end{remark}
\begin{lemma} \label{star}
For any $p\in\mathcal{M}$, $\pi^{-1}(p)$ is star-shaped with respect to $p$.
\end{lemma}
\begin{proof}
If $p \in \mathcal{M}^{\circ}$,
we have by Lemma \ref{ststarshaped} and \ref{pi-1contained} that $\pi^{-1}(p)=st_{\mathcal{M}}(p)$ which is star-shaped. So for the rest of the proof we assume that $p\in\partial\mathcal{M}$. By Lemma \ref{ststarshaped}, $st_{\partial\mathcal{M}}(p)$ is star-shaped; by Lemma \ref{allintubneighbor}, for any ray $l\subset T_p(\partial\mathcal{M})^{\perp}$ starting from $p$, either $\pi^{-1}(p)\cap l=\{p\}$ or $st_{\partial\mathcal{M}}(p)\cap l\subseteq \pi^{-1}(p)\cap l$; by Lemma \ref{pi-1contained} we know that $\pi^{-1}(p)\subseteq st_{\partial\mathcal{M}}(p)$. So if $q\in\pi^{-1}(p)$, then $|pq|\subseteq\pi^{-1}(p)$.
\end{proof}
Then Theorem~\ref{deformretrth} generalizes \cite[Proposition 3.1]{NiyogiSmaleWeinberger2008} to compact manifolds with boundaries.
Its proof is as follows.
\begin{proof}
We define the deformation retract $F(x,t): U\times [0,1]\to U$ as $F(x,t)=tx+(1-t)\pi(x)$. By Lemma \ref{allintubneighbor} and Lemma \ref{star}, this deformation retract is well-defined. Moreover since $U$ is contained in $\mathcal{M}^{\delta}$, there is no critical point for distance functions, we get that $U$ deformation retracts to $\mathcal{M}$.
\end{proof}
\begin{lemma} \label{localvolest}
Let $p\in\partial\mathcal{M}$ and $A=\mathcal{M}\cap B_{\epsilon}(p)$, where $\epsilon\in(0,\delta)$. Then
$${\rm vol}(A)> \frac{\cos^k\theta}{2}I_{1-\frac{\epsilon^2\cos^2\theta}{4\delta^2}}(\frac{k+1}{2}, \frac{1}{2}){\rm vol}(B_{\epsilon}^k(p)),$$
where $I$ is the regularized incomplete beta function, $B_{\epsilon}^k(p)$ is the $k$-dimensional ball in $T_p$ centered at $p$, and $\theta=\arcsin(\dfrac{\epsilon}{2\delta})$.
\end{lemma}
\begin{proof}
Let $p'\in T_p(\mathcal{M})$ be the point such that $\overrightarrow{pp'}$ is perpendicular to $T_p(\partial\mathcal{M})$ and points to the inside of $\mathcal{M}$, and $\|p-p'\|=\delta$. We first want to show that
\begin{ceqn}
\begin{align} \label{inclintersball}
B_{\epsilon\cos\theta}^k(p)\cap B_{\delta}^k(p')\subseteq \varphi_{p,\mathcal{M}}(A).
\end{align}
\end{ceqn}
By Condition \eqref{condition2}, $\varphi_{p, \mathcal{M}}|_{A}$ is a homeomorphism onto its image. In particular,
$$\varphi_{p,\mathcal{M}}(\partial A)=\partial (\varphi_{p,\mathcal{M}}(A)).$$
It is easy to see that $B_{s}^k(p)\cap B_{\delta}^k(p')\cap \varphi_{p,\mathcal{M}}(A)\ne \emptyset$ for any $s>0$. Let $q\in\mathcal{M}$ be a point on the boundary of $A$. Then $q\in \partial\mathcal{M}\cup\partial B_{\epsilon}(p)$. We prove \eqref{inclintersball} by proving the claim that no matter whether $q\in \partial\mathcal{M}$ or $q\in \partial B_{\epsilon}(p)$, $\varphi_{p,\mathcal{M}}(q)$ is outside $B_{\epsilon\cos\theta}^k(p)\cap B_{\delta}^k(p')$. Indeed, if there exists a point $q'\in B_{\epsilon\cos\theta}^k(p)\cap B_{\delta}^k(p')$ such that $q'\not\in \varphi_{p,\mathcal{M}}(A)$, we choose a point $o\in B_{\epsilon\cos\theta}^k(p)\cap B_{\delta}^k(p')\cap \varphi_{p,\mathcal{M}}(A)$. Then the line segment connecting $o$ and $q'$ must intersect with the $\partial(\varphi_{p,\mathcal{M}}(A))$. But on the other hand, by the convexity of $B_{\epsilon\cos\theta}^k(p)\cap B_{\delta}^k(p')$, any intersection point is inside $B_{\epsilon\cos\theta}^k(p)\cap B_{\delta}^k(p')$. This is a contradiction.
To prove the claim, we first suppose that $q\in \partial\mathcal{M}$. Let $p''$ be the point where $\overrightarrow{pp''}$ is in the same direction with $\overrightarrow{\varphi_{p,\partial\mathcal{M}}(q)q}$ and $\|p''-p\|=\delta$ (if $\varphi_{p,\partial\mathcal{M}}(q)=q$, then set $p''=p'$). By Condition \eqref{condition1}, $q$ is outside $B_{\delta}(p'')$. Now $q$, $\varphi_{p,\mathcal{M}}(q)$ and $\varphi_{p,\partial\mathcal{M}}(q)$ form a right-angled triangle where $|q\varphi_{p,\partial\mathcal{M}}(q)|$ is the hypotenuse, so $\|\varphi_{p,\mathcal{M}}(q)-\varphi_{p,\partial\mathcal{M}}(q)\|\le \|q-\varphi_{p,\partial\mathcal{M}}(q)\|$. Therefore $\varphi_{p,\mathcal{M}}(q)$ is certainly outside $B_{\delta}^k(p')$. This case is illustrated in Fig.~\ref{fig:lemma9a}.
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=0.5\linewidth]{lemma9a.pdf}
\end{tabular}
\vspace{-2mm}
\caption{An illustration for Lemma~\ref{localvolest} for the case when $q \in \partial \mathcal{M}$, where $||p' - p|| = \delta$, $||p'' - p|| = \delta$, $||p^{''} - q|| > \delta$, $||q - \varphi_{p, \partial \mathcal{M}}(q)|| \leq \delta$, therefore $||p' - \varphi_{p, \mathcal{M}}(q)|| \geq ||p^{''} - q|| > \delta$.}
\label{fig:lemma9a}
\end{center}
\end{figure}
Next suppose that $q\in \partial B_{\epsilon}(p)$, which implies that $\|q-p\|=\epsilon$. We have that $\|\varphi_{p,\mathcal{M}}(q)-p\|=\epsilon\cos(\phi)$ where $\phi$ is the (smallest nonnegative) angle between $\overrightarrow{pq}$ and $\overrightarrow{p\varphi_{p,\mathcal{M}}(q)}$. Let $p''$ be the point where $\overrightarrow{pp''}$ is in the same direction with $\overrightarrow{\varphi_{p,\mathcal{M}}(q)q}$ (if $\varphi_{p,\mathcal{M}}(q)=q$, then choose an arbitrary $p''$ such that $pp''\bot T_{p}(\mathcal{M})$) and $\|p''-p\|=\delta$. Then by Condition \eqref{condition1}, $\|p''-q\|>\delta$. So by the definition of $\theta$ we see that $\phi<\theta$, and $\|\varphi_{p,\mathcal{M}}(q)-p\| = \epsilon\cos (\phi) > \epsilon\cos(\theta)$. Hence $\varphi_{p,\mathcal{M}}(q)$ is outside $B_{\epsilon\cos\theta}^k(p)$. This is illustrated in Fig.~\ref{fig:lemma9b}.
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=0.5\linewidth]{lemma9b.pdf}
\end{tabular}
\vspace{-2mm}
\caption{An illustration for Lemma~\ref{localvolest} for the case when $q \in \partial B_{\epsilon}(p)$, where $||p-q|| = \epsilon$,
$\phi < \theta$, $|p^{''} - p|| = \delta$ and $||p^{''} - q|| > \delta$.}
\label{fig:lemma9b}
\end{center}
\end{figure}
So we have ${\rm vol}(A)\ge {\rm vol}(B_{\epsilon\cos\theta}^k(p)\cap B_{\delta}^k(p'))$. The right-hand side consists of two hyperspherical caps. For convenience we choose the lower bound of the right side to be the hyperspherical cap that belongs to $B_{\epsilon\cos\theta}^k(p)$. By \ref{sechypersphecap}, the volume of this hyperspherical cap is $\displaystyle\frac{1}{2}I_{1-\frac{\epsilon^2\cos^2\theta}{4\delta^2}}(\frac{k+1}{2}, \frac{1}{2}){\rm vol}(B_{\epsilon\cos\theta}^k(p))$. Moreover we know that ${\rm vol}(B_{\epsilon\cos\theta}^k(p))=\cos^k\theta{\rm vol}(B_{\epsilon}^k(p))$, so we are done.
\end{proof}
Using the same argument as in the third paragraph of the proof of the last lemma, we actually have
\begin{lemma} \label{estnointersectionwithbound}
Let $p\in\mathcal{M}$ and $\epsilon>0$ such that $\partial\mathcal{M}\cap B_{\epsilon}(p)=\emptyset$. Let $A:=\mathcal{M}\cap B_{\epsilon}(p)$. Then
$${\rm vol}(A)\ge {\rm vol}(B_{\epsilon\cos\theta}^k(p))=\cos^k\theta{\rm vol}(B_{\epsilon}^k(p)).$$
\end{lemma}
Combining Lemma \ref{localvolest} and \ref{estnointersectionwithbound}, we have
\begin{lemma} \label{volest}
Let $p\in\mathcal{M}$ and $A=\mathcal{M}\cap B_{\epsilon}(p)$, where $\epsilon\in(0,\delta)$. Then
$${\rm vol}(A)\ge\frac{\cos^k\theta'}{2^{k+1}}I_{1-\frac{\epsilon^2\cos^2\theta'}{16\delta^2}}(\frac{k+1}{2}, \frac{1}{2}){\rm vol}(B_{\epsilon}^k(p)),$$
where $\theta':=\arcsin(\dfrac{\epsilon}{4\delta})$.
\end{lemma}
\begin{proof}
If $d(p,\partial\mathcal{M})>\dfrac{\epsilon}{2}$, then $\partial\mathcal{M}\cap B_{\frac{\epsilon}{2}}(p)=\emptyset$. So by Lemma \ref{estnointersectionwithbound},
$${\rm vol}(A)\ge \cos^k\theta'{\rm vol}(B_{\frac{\epsilon}{2}}^k(p))$$
and we are done. If $d(p,\partial\mathcal{M})\le \dfrac{\epsilon}{2}$, let $p'$ be a point on $\partial\mathcal{M}$ that has minimum distance from $p$. Then $B_{\frac{\epsilon}{2}}(p')\subset B_{\epsilon}(p)$. So
\begin{align*}
{\rm vol}(A)\ge{\rm vol}(B_{\frac{\epsilon}{2}}(p')\cap \mathcal{M})& \ge \frac{\cos^k\theta'}{2}I_{1-\frac{\epsilon^2\cos^2\theta'}{16\delta^2}}(\frac{k+1}{2}, \frac{1}{2}){\rm vol}(B_{\frac{\epsilon}{2}}^k(p)) \\
& =\frac{\cos^k\theta'}{2^{k+1}}I_{1-\frac{\epsilon^2\cos^2\theta'}{16\delta^2}}(\frac{k+1}{2}, \frac{1}{2}){\rm vol}(B_{\epsilon}^k(p)),
\end{align*}
where the last inequality is by Lemma \ref{localvolest}.
\end{proof}
We observe that the right side of the inequality in Lemma \ref{volest} is $\dfrac{{\rm vol}(\mathcal{M})}{\beta(\epsilon)}$, where the function $\beta$ is defined in Theorem~\ref{main} (note that the $\theta'$ here corresponds to the $\theta$ in Theroem~\ref{main}). By \cite[Lemma 5.1 and 5.2]{NiyogiSmaleWeinberger2008}, a satisfactory number of points $|\overline{x}|$ as in Theorem~\ref{probabestim} is of the form $\dfrac{1}{\alpha}(\ln l+\ln \dfrac{1}{\gamma})$, where $\alpha$ is any lower bound of $\dfrac{{\rm vol}(A)}{{\rm vol}(\mathcal{M})}$ and $l$ is any upper bound of $\dfrac{\epsilon}{2}$-packing-number.
So by Lemma \ref{volest}, it suffices to take $\alpha$ and $l$ to be $\dfrac{1}{\beta(\epsilon)}$ and $\beta(\dfrac{\epsilon}{2})$, respectively. Therefore we obtain Theorem~\ref{probabestim}.
Finally, combining Theorems \ref{deformretrth} and \ref{probabestim}, we arrive at Theorem~\ref{main}.
\section{Experiments}
\label{sec:experiments}
In this section, we work on two typical examples of manifolds with boundary. The first example is a cylindrical surface, referred to as the \emph{cylinder} dataset, which has radius $1$ and height $1$. More precisely, it can be expressed as
$$\{(x,y,z)\in \mathbb{R}^3: x^2+y^2=1, z\in [0,1]\}.$$
The second example is a torus with a cap chopped off, referred to as the \emph{torus} dataset. In $\mathbb{R}^3$, it can be expressed as the torus with inner circle $x^2+y^2=1$ and the outer circle $x^2+y^2=9$, and the part with $x\ge 2$ is chopped off.
\para{Sampling parameters.}
As stated in the main Theorem~\ref{main}, the lower bound of sampling that guarantees deformation retraction with probability $1-\gamma$ can be expressed as $$n^* = \beta(\epsilon)(\ln \beta(\frac{\epsilon}{2})+\ln(\frac{1}{\gamma})),$$
where $\displaystyle\beta(x):=\frac{{\rm vol}(\mathcal{M})}{\frac{\cos^k\theta}{2^{k+1}}I_{1-\frac{x^2\cos^2\theta}{16\delta^2}}(\frac{k+1}{2}, \frac{1}{2}){\rm vol}(B_{x}^k)}$ and $\theta=\arcsin(\frac{x}{4\delta})$.
For \emph{cylinder}, ${\rm vol}(\mathcal{M}) = 2\pi$, $k=2$, ${\rm vol}(B_{x}^2) = \pi x^2$, $\delta = 1$.
For instance, setting $\epsilon=0.49$, $\gamma = 0.1$ gives rise to $n^* = 638$, as illustrated in Fig.~\ref{fig:data} (a).
For \emph{torus}, ${\rm vol}(\mathcal{M}) = (8-0.522) \cdot 2 \pi$, $k=2$, ${\rm vol}(B_{x}^2) = \pi x^2$, $\delta = 1$.
For instance, setting $\epsilon=0.49$, $\gamma = 0.1$ gives rise to $n^* = 9809$, as illustrated in Fig.~\ref{fig:data} (b).
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.45\linewidth]{cylinder.pdf} &
\includegraphics[width=0.45\linewidth]{torus.pdf}\vspace{-2mm}\\
{\bf (a)} & {\bf (b)}\\
\end{tabular}
\vspace{-2mm}
\caption{Point cloud samples for \emph{cylinder} {\bf (a)} and \emph{torus} {\bf (b)}.}
\label{fig:data}
\end{center}
\end{figure}
\para{Distribution of lower bounds.}
For a fixed sample quality $\epsilon$, we demonstrate the distribution of lower bounds $n^*$ as $\gamma$ increases from $0.05$ to $0.95$ (that is, confidence ranges from $95\%$ to $5\%$). This is shown in Fig.~\ref{fig:nstar}. Intuitively, for a fixed sample quality, we need more point samples in order to obtain higher confidence in topological inference.
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.4\linewidth]{cylinder-nstar.pdf} &
\includegraphics[width=0.4\linewidth]{torus-nstar.pdf}\vspace{-2mm}\\
{\bf (a)} & {\bf (b)}\\
\end{tabular}
\vspace{-2mm}
\caption{Lower bounds for \emph{cylinder} {\bf (a)} and \emph{torus} {\bf (b)} for a fixed $\epsilon =0.49$, x-axis corresponds to $\gamma$ while y-axis corresponds to $n^*$.}
\label{fig:nstar}
\end{center}
\end{figure}
Meanwhile, for a fixed $\gamma = 0.1$, which corresponds to a confidence of $90\%$, we illustrate the distribution of lower bounds $n^*$ as $\epsilon$ increases from $0.15$ to $0.5$.
By Theorem~\ref{main}, it is rather obvious that we need more points to have higher quality samples for a fixed confidence level. This is shown in Fig.~\ref{fig:samplequality}.
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.4\linewidth]{cylinder-epsilon.pdf} &
\includegraphics[width=0.4\linewidth]{torus-epsilon.pdf}\vspace{-2mm}\\
{\bf (a)} & {\bf (b)}\\
\end{tabular}
\vspace{-2mm}
\caption{Lower bounds for \emph{cylinder} {\bf (a)} and \emph{torus} {\bf (b)} for a fixed $\gamma =0.1$, x-axis corresponds to the $\epsilon$ while y-axis corresponds to $n^*$.}
\vspace{-2mm}
\label{fig:samplequality}
\end{center}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.47\linewidth]{cylinder-p-0.pdf} &
\includegraphics[width=0.47\linewidth]{cylinder-epsilon-0.pdf} \vspace{-2mm}\\
{\bf (a)} & {\bf (d)}\\
\includegraphics[width=0.47\linewidth]{cylinder-p-1.pdf} &
\includegraphics[width=0.47\linewidth]{cylinder-epsilon-1.pdf} \vspace{-2mm}\\
{\bf (b)} & {\bf (e)}\\
\includegraphics[width=0.47\linewidth]{cylinder-p-2.pdf} &
\includegraphics[width=0.47\linewidth]{cylinder-epsilon-2.pdf} \vspace{-2mm}\\
{\bf (c)} & {\bf (f)}\\
\end{tabular}
\vspace{-2mm}
\caption{Persistent barcodes for \emph{cylinder}. Each plot only shows the top 20 most persistent (longest) cycles.
For a fixed $\epsilon = 0.49$, $r \in [0, 2.2]$: {\bf (a)} $\gamma = 0.1$, $n^* = 638$; {\bf (b)} $\gamma = 0.1$, $n^* = 583$; and {\bf (c)} $\gamma = 0.3$, $n^* = 551$.
For a fixed $\gamma=0.1$ ($90\%$ confidence), $r \in [0, 0.5]$: {\bf (d)} $\epsilon = 0.2$, $n^* = 4160 $; {\bf (e)} $\epsilon = 0.3$, $n^* = 1763$; and {\bf (f)} $\epsilon = 0.4$, $n^* = 967$. Notice different scales between the plots on the left and on the right.
}
\vspace{-2mm}
\label{fig:cylinder-ph}
\end{center}
\end{figure}
\para{Homology computation.}
Finally we can perform homology computation on the above point clouds; in particular, for a given sample $\overline{x}$ and its corresponding $U$, we show that the homology of $U$ equals the homology of $\mathcal{M}$. Admittedly, homology is a very weak verification of our main sampling theorem. In fact, if one's goal is only to recover the same homology of a manifold with point samples, our estimation from Theorem~\ref{main} is an obvious overestimation.
In other words, our estimation of the lower bound $n^*$ has to account for the boundary condition and to guarantee deformation retract (not just homology or homotopy equivalence).
Nevertheless, we show the results of homology inference across multiple $\gamma$ with a fixed $\epsilon$, as well as the results across multiple $\epsilon$ with a fixed $\gamma$. We rely on the computation of persistent homology to recover the homological information of a point cloud sample.
Persistent homology, roughly speaking, operates on a point cloud sample $\overline{x}$ and tracks how the homology of $U(r)=\bigcup_{x\in\overline{x}}B_{r}(x)$ changes as $r$ increases (where typically $r \in [r_0=0, r_k]$, for some positive real value $r_k$).
Specifically, it applies the homology functor $\Hgroup$ to a sequence of topological spaces connected by inclusions,
$$U(r_0) \to \cdots \to U(r_i) \to U(r_{i+1}) \to \cdots \to U(r_k),$$
and studies a multi-scale notion of homology,
$$\Hgroup(U(r_0)) \to \cdots \to \Hgroup(U(r_i)) \to \Hgroup(U(r_{i+1})) \to \cdots \to \Hgroup(U(r_k)),$$
see~\cite{EdelsbrunnerHarer2008, EdelsbrunnerHarer2010, Ghrist2008} for introduction to persistent homology.
We use the software package Ripser~\cite{Bauer2016} for the computation of persistent homology.
Given a point cloud sample $\overline{x}$, Ripser computes its persistent homology using Vietoris--Rips complexes formed
on $\overline{x}$ and encodes the homological information using persistence barcodes.
In a nutshell, each bar in the persistence barcodes captures the time when a homology class appears and disappears as $r$ increases.
As the homology of a union of balls is guaranteed (by the Nerve Lemma) to be the one of a \v{C}ech complex,
the results of~\cite{AttaliLieutierSalinas2013} could be utilized to deduce results on a Vietoris--Rips complex from a \v{C}ech complex.
For \emph{cylinder} dataset, the $1$-dimensional homology of its underlying manifold should be of rank one, as the dataset contains one significant cycle (tunnel).
For a fixed $\epsilon = 0.49$, we compute the $1$-dimensional persistent homology of the point clouds at parameter $\gamma = 0.1, 0.2, 0.3$ respectively. Their persistent barcodes are shown in Fig.~\ref{fig:cylinder-ph}(a)-(c) respectively. For each plot, the longest bar corresponds to the most significant $1$-dimensional cycle, which clearly corresponds to the true homological feature of the underlying manifold.
Meanwhile, the $1$-dimensional homology of the manifold underlying the \emph{torus} dataset (with a cap chopped off) should be of rank two, as the dataset contains two significant cycles (same as the classic torus dataset).
We have similar results as in the case of \emph{cylinder} dataset. For simplicity, we give the persistent barcodes for $\epsilon = 0.49$, $\gamma = 0.2$, $n^* = 9157$ in Fig.~\ref{fig:torus-ph}.
Here, the first two longest bars correspond to the two most significant $1$-dimensional cycles, which again clearly correspond to the true homological features of the underlying manifold.
\begin{figure}[!ht]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=0.8\linewidth]{torus-p-0.pdf}
\end{tabular}
\vspace{-2mm}
\caption{Persistent barcodes for \emph{torus}, for $\epsilon = 0.49$,
$\gamma = 0.2$, $n^* = 9157$, $r \in [0,0.8]$. Only the top 20 most persistent (longest) cyclesare shown.}
\vspace{-2mm}
\label{fig:torus-ph}
\end{center}
\end{figure}
\section{Discussions}
\label{sec:discussions}
Given a point cloud sample of a compact, differentiable manifold with boundary, we give a probabilistic notion of sampling condition that is not handled by existing theories.
Our main results relate topological equivalence between the offset and the manifold as a deformation retract, which is stronger than homological or homotopy equivalence. Many interesting questions remain.
First, while our sampling condition considers differentiable manifolds with boundary, it cannot be trivially extended to handle manifolds with corners.
The fundamental difficulty arises because the $\reach$ becomes zero in the case of manifolds with corners. We suspect that deriving practical sampling conditions for manifolds with corners, and in general, for stratified spaces, is challenging and requires new way of thinking.
Second, we have conducted experiments that verify homological equivalence between the offset of samples and the underlying manifold. However, such an experiment is a very weak verification of our main inference theorem.
Experimentally computing or verifying deformation retract in the point cloud setting (as stated in Theorem~\ref{main}), possibly via the study of discrete gradient fields, remains an open question.
\bibliographystyle{abbrv}
| {
"timestamp": "2018-10-16T02:04:31",
"yymm": "1810",
"arxiv_id": "1810.05759",
"language": "en",
"url": "https://arxiv.org/abs/1810.05759",
"abstract": "Given a set of data points sampled from some underlying space, there are two important challenges in geometric and topological data analysis when dealing with sampled data: reconstruction -- how to assemble discrete samples into global structures, and inference -- how to extract geometric and topological information from data that are high-dimensional, incomplete and noisy. Niyogi et al. (2008) have shown that by constructing an offset of the samples using a suitable offset parameter could provide reconstructions that preserve homotopy types therefore homology for densely sampled smooth submanifolds of Euclidean space without boundary. Chazal et al. (2009) and Attali et al. (2013) have introduced a parameterized set of sampling conditions that extend the results of Niyogi et al. to a large class of compact subsets of Euclidean space. Our work tackles data problems that fill a gap between the work of Niyogi et al. and Chazal et al. In particular, we give a probabilistic notion of sampling conditions for manifolds with boundary that could not be handled by existing theories. We also give stronger results that relate topological equivalence between the offset and the manifold as a deformation retract.",
"subjects": "Computation (stat.CO); Data Structures and Algorithms (cs.DS)",
"title": "Topological Inference of Manifolds with Boundary",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754506337406,
"lm_q2_score": 0.7185943925708562,
"lm_q1q2_score": 0.7075103978883299
} |
https://arxiv.org/abs/2001.01154 | One-Sided Derivative of Distance to a Compact Set | We give a complete and self-contained proof of a folklore theorem which says that in an Alexandrov space the distance between a point $\gamma(t)$ on a geodesic $\gamma$ and a compact set $K$ is a right-differentiable function of $t$. Moreover, the value of this right-derivative is given by the negative cosine of the minimal angle between the geodesic and any shortest path to the compact set (Theorem 4.3). Our treatment serves as a general introduction to metric geometry and relies only on the basic elements, such as comparison triangles and upper angles. | \section{Introduction}
Let \((X,d)\) be a metric space. Given a compact set \(K \subseteq X\) and a geodesic \(\gamma:[0,T]\to X\), the distance from \(\gamma\) to \(K\) at any given time is defined by the function
\[\ell(t) = d(\gamma(t),K) . \]
In an Alexandrov space (see Section \ref{alexandrov spaces} for definition), if we replace the compact set with a point, \(K = \{p\}\), it is well known that
\begin{equation}\label{limit}
\lim_{t\to 0^+} \frac{\ell(t) -\ell(0)}{t} = -\cos(\angle_{\min})
\end{equation}
where \(\angle_{\min}\) is the infimum of angles between \(\gamma\) and any distance minimizing path connecting \(\gamma(0)\) to \(p\). This result is commonly known as the First Variation Formula (after the similar result for Riemannian manifolds) and can be found in \cite[Proposition 3.3]{abn86}, \cite[Corollary II.3.6]{bridson}, \cite[Corollary 4.5.7]{bbi2001}, and \cite[Corollary 62]{plaut}.
It is asserted, in publications such as \cite[Exercise 4.5.11]{bbi2001} and \cite[Example 11.4]{bgp92}, that the first variation formula (\ref{limit}) still holds for the distance to an arbitrary compact set \(K\), with \(\angle_{\min}\) representing the infimum of angles between \(\gamma\) and any distance minimizing path connecting \(\gamma(0)\) to \(K\).
However, neither of these sources (\cite{bbi2001} and \cite{bgp92}) provide a proof.
On the other hand, there is a proof of the first variation formula in \cite[Proposition 9.4]{lytchak}, which is further generalized to hold in a class of geometric metric spaces broader than Alexandrov spaces. This generalization, however, is achieved at the cost of some very technical machinery.
Namely, it involves constructing a tangent space at every point \(p\) via the ultralimit of blow-ups of the pointed space \((X,p)\).
Further, the angle between geodesics may not be well defined in this context, so Lytchak is required to make use of Busemann functions to form a metric on this tangent space, much like the law of cosines in the Euclidean case.
In this more general formulation, the right-hand side of (\ref{limit}) is replaced by a Busemann function.
Our goal is to present a complete and self-contained proof of (\ref{limit}), which relies solely on the fundamentals of metric geometry. As such, this article may also serve as a gentle yet rigorous introduction to the theory of Alexandrov spaces.
Our approach is largely based on techniques presented in \cite{bbi2001}, with insights taken from \cite{bridson}, \cite{plaut}, \cite{shiohama}, and others.
\subsection{Background and Motivation}
Loosely speaking, an Alexandrov space is metric space which satisfies enough structural requirements for some classic geometric notions such as geodesics, angles, and curvature make sense. In this context, curvature is based on a local bound, which is obtained through comparison to one of the two-dimensional space forms - hyperbolic, spherical, or Euclidean space.
The importance of Alexandrov spaces can be seen through a few examples. First and foremost, all Riemannian manifolds are in fact Alexandrov spaces.
On the other hand, limits of Riemannian manifolds (in the Gromov-Hausdorff metric) with lower-bounded curvature and upper-bounded diameter are not necessarily Riemannian manifolds, but are always Alexandrov spaces (this is Gromov's compactness theorem). A simple example is the surface of an \(n\)-dimensional cube in \(\mathbb{R}^{n+1}\). The surface of a cube is clearly not a smooth manifold, as it has sharp corners and edges; however, it is an Alexandrov space and can be attained as the limit as \(q\) tends to infinity of the smooth spherical \(n\)-manifolds \(\{x\in \mathbb{R}^{n+1} : |x_1|^q + \cdots + |x_{n+1}|^q = 1\}\), each of which has non-negative curvature.
Another reason that Alexandrov spaces are significant, is that certain general characteristics or properties that we generally associate with smoothness - or smooth structures - in fact hold in a more general context. This is somewhat akin to observing that at a local extremum of a smooth function from \(\mathbb{R}\) to itself, this function admits a horizontal tangent line. With a careful definition of tangent line, this is true in a more general setting. Similarly, in this case we have that (\ref{limit}) is not only true for Riemannian manifolds (with their smooth metric), but also in a broader set of spaces.
Studying geometry in a `weaker' setting gives insight into both the properties in question, and the smooth structures themselves.
The first variation formula, in particular, is a fundamental property of distances in Alexandrov space, and has found numerous applications in geometry. We name a few applications here. In \cite{shiohama1996cut}, the authors use the first variation formula to study cut-loci on spheres in Alexandrov space.
In the paper \cite{perelman1994quasigeodesics}, the authors use this theorem to prove that the length of convex curves is preserved when taking limits of Alexandrov spaces under suitable conditions.
Another consequence of note is that the first variation formula allows one to introduce a metric on the space of directions emanating from a point to a compact set, which is fundamental in studying tangent cones (see \cite{busemann1970recent}).
Finally, in \cite{petrunin} a version of the first variation formula
is used to prove a deep glueing theorem for Alexandrov spaces with boundary (informally: if \(X\) and \(Y\) are Alexandrov spaces with boundary and of curvature \(\geq k\), then gluing \(X\) and \(Y\) along their boundaries produces an Alexandrov space of curvature \(\geq k\)).
Our own motivation is twofold. It is well known (\cite{Veer} and references therein) that if $K$ is a
convex set in $\mathbb{R}^n$, then the derivative of $\ell(t)$ equals the negative cosine of the angle between \(\gamma\) and the distance minimizing path connecting \(\gamma(0)\) to \(K\). The result considered here is a strong generalization of that fact.
Additionally, the result is crucial for the study of mediatrices; that is, for fixed points \(p,q\in X\), the set \(\{x\in X : d(x,p) = d(x,q)\}\). In \cite{HPV}, the first variation formula (for distance to a point) is used to show that mediatrices on compact Riemannian surfaces have a Lipschitz structure. There is a natural generalization of mediatrices as the equidistant set between disjoint compact subsets, and the result we prove here is a necessary step in extending \cite{HPV} to this case.
\section{Fundamentals of Metric Geometry}
This section begins with two of the most basic notions of metric geometry: length and comparison configurations; followed by a survey of (upper) angles between geodesics. Although the use of angles has somewhat faded in the modern theory (for generalizations beyond Alexandrov spaces), it can aid geometric intuition, especially to the beginner. The definitions and results presented in this section are well-established, and can also be found in the introductory chapters of \cite{aleksandrovzalgaller}, \cite{bbi2001}, \cite{reshetnyak}, and \cite{bridson}.
\subsection{Length Spaces}
In order to establish a synthetic geometry in a metric space, we rely on paths in the space to get from one point to another. To that end, the metric needs to align with our intuitive idea of how distance is measured; the distance between any two points is the length of a `straight line' connecting them. Here we build the vocabulary and structure for these kinds of metric spaces, known as length spaces.
Let \((X,d)\) be a metric space. A path (or curve) in \(X\) is a continuous injective function \(\gamma: [a,b] \to X\) where \([a,b]\) is an interval of \(\mathbb{R}\) (possibly degenerate). We define the length of any path \(\gamma\) as the supremum of the distance along finite partitions of the path:
\[ L(\gamma) = \sup \bigg\{ \sum_{k=1}^{n-1} d\big( \gamma(t_{k}), \gamma(t_{k+1}) \big) : a=t_1 < t_2<\cdots < t_n = b \bigg\} . \]
If for all \(x,y\in X\),
\[ d(x,y) = \inf \{ L(\gamma) : \gamma \text{ is a path connecting } x \text{ and } y \} \]
then the metric \(d\) is said to be intrinsic. A path-connected metric space with an intrinsic metric is known as a \emph{length space}.
Two paths, \(\gamma:[0,T] \to X\) and \(\eta: [0,S] \to X\), which have the same image but are not the same function are said to have different \emph{parameterizations}. A path \(\gamma: [0,T] \to X\) is parameterized by arc-length (or \emph{unit-speed}, for short) if for any \(t,t' \in [0,T]\),
\[ L(\gamma|_{[t,t']}) = |t' - t| . \]
A sequence of paths \(\{\gamma_n\}_{n=1}^\infty\) is said to converge uniformly to a path \(\gamma\) if each \(\gamma_n\) admits a parameterization such that \(\{\gamma_n\}_{n=1}^\infty\) converges uniformly to some parameterization of \(\gamma\).
Finally, a \emph{shortest path} is a unit-speed curve \(\gamma: [0,T] \to X\) such that the length of \(\gamma\) is precisely the distance between its endpoints; \(L(\gamma) = d(\gamma(0),\gamma(T)) = T\). Any curve which is locally a shortest path is known as a \emph{geodesic}.
In a length space which is both complete and locally compact, we make use of the Hopf-Rinow Theorem, although we will not reference it directly.
\begin{theorem}[Hopf-Rinow]
If \(X\) is a complete and locally compact length space, then every closed and bounded subset of \(X\) is compact; and any two points in \(X\) can be connected by a shortest path.
\end{theorem}
\subsection{Comparison Triangles}
We denote by \(M^2_k\) the \(2\)-dimensional simply-connected space form\footnote{A space form is a complete Riemannian manifold of constant sectional curvature.} of curvature \(k\), equipped with intrinsic metric \(d_k\) induced by the Riemannian metric. The diameter of the space \(M^2_k\) is denoted \(D_k\) and defined by
\[
D_k = \sup \{ d_k(x,y) : x,y \in M^2_k\} = \begin{cases} \pi/\sqrt{k} & \text{ for } k > 0 \\ \infty & \text{ for } k\leq 0 .\end{cases}
\]
Given any three points \(x,y,z\in X\) with \(d(x,y) + d(y,z) + d(x,z) < D_k\), we can fix three points \(\bar{x}\), \(\bar{y},\) and \(\bar{z}\) in \(M^2_k\) such that
\begin{equation}\label{triangle sides}
d(x,y) = d_k( \bar{x} , \bar{y} ) , \ d(x,z) = d_k( \bar{x} , \bar{z} ) , \ \text{and } d(y,z) = d_k( \bar{y},\bar{z} ) .
\end{equation}
The points \(\bar{x}\), \(\bar{y}\), and \(\bar{z}\), together with the shortest paths joining them,
form a geodesic triangle in \(M^2_k\), which we call the \emph{comparison triangle} and denote it
\(\overline{\Delta}(x,y,z)\). Such a comparison triangle is unique up to isometry. The interior angle
in the geodesic triangle \(\overline{\Delta}(x,y,z)\) (in \(M^2_k\)) with vertex \(\bar{x}\) is denoted
\(\angle^k_x(y,z)\)
and referred to as the \emph{\(k\)-comparison angle}.
\subsection{Upper Angles}
If \(\gamma:[0,T] \to X\) and \(\eta:[0,S]\to X\) are shortest paths in \(X\) with \(\gamma(0) = \eta(0)\), then for any sufficiently small\footnote{Sufficiently small meaning the inequality \(d(\gamma(0),\gamma(t)) + d(\gamma(0),\eta(s)) + d(\gamma(t),\eta(s)) < 2D_k\) is satisfied. } \(t\in (0,T]\) and \(s\in (0,S]\), we can consider the comparison triangle \(\overline{\Delta}\big(\gamma(0),\gamma(t),\eta(s)\big)\). The \emph{upper angle} between \(\gamma\) and \(\eta\) is defined as
\begin{align*}
\angle^+_{\gamma(0)}(\gamma,\eta) & = \limsup_{t,s\to0^+} \angle^k_{\gamma(0)} (\gamma(t),\eta(s))
\\ & = \lim_{\varepsilon\to 0} \sup \big\{ \angle^k_{\gamma(0)} (\gamma(t),\eta(s)) : 0<s,t\leq \varepsilon \big\} .
\end{align*}
When it is understood that \(\gamma(0)\) is the point at which we are measuring the angle, the subscript for the vertex is often omitted (i.e. \(\angle^+_{\gamma(0)}(\gamma,\eta) = \angle^+(\gamma,\eta)\) and \(\angle^k_{\gamma(0)}(\gamma(t),\eta(s)) = \angle^k(\gamma(t),\eta(s))\)).
Besides the upper angle, one may also consider the \emph{lower angle} between two shortest paths, which is defined as \(\angle^-(\gamma,\eta) = \liminf_{s,t\to 0^+} \angle^k (\gamma(t),\eta(s))\). If the upper angle and lower angle are equal, then we say the angle exists and denote it by \(\angle(\gamma, \eta)\).
We note that the upper and lower angles are indeed independent of the curvature of the space form chosen (as per \cite{plaut}, all space forms are infinitesimally Euclidean; see also Appendix \ref{sec: inf-eucl}).
The following proposition is commonly referred to as the triangle inequality for angles. The proof given here the same as that found in \cite{bridson}.
\begin{proposition}\label{triangle inequality angles}
Let \(X\) be a length space and let \(\gamma\), \(\eta\), and \(\sigma\) be shortest paths in \(X\) with \(\gamma(0)=\eta(0)=\sigma(0)\). Then \(\angle^+ (\gamma,\eta) \leq \angle^+(\gamma,\sigma)+\angle^+(\sigma,\eta)\).
\end{proposition}
\begin{proof}
If \(\angle^+(\gamma,\sigma) + \angle^+(\sigma,\eta)\geq\pi\), then the result is trivial, so we assume that \(\angle^+(\gamma,\sigma) + \angle^+(\sigma,\eta)<\pi\). By way of contradiction, suppose that there is an \(\varepsilon>0\) such that
\begin{equation}\label{eq: triangle contradiction}
\angle^+(\gamma,\eta) > \angle^+(\gamma,\sigma) + \angle^+(\sigma,\eta) + \varepsilon .
\end{equation}
By the definition of \(\limsup\) there is a \(\delta>0\) such that
\begin{align}
\angle^k(\gamma(t),\eta(r)) & > \angle^+(\gamma,\eta) -\varepsilon/3 \quad \text{for some } t,r<\delta \label{eq: triangle 1}
\\ \angle^k(\gamma(t),\sigma(s)) & < \angle^+(\gamma,\sigma) +\varepsilon/3 \quad \text{for all } t,s < \delta \label{eq: triangle 2}
\\ \angle^k(\sigma(s),\eta(r)) & < \angle^+(\sigma,\eta)+\varepsilon/3 \quad \text{for all } s,r<\delta . \label{eq: triangle 3}
\end{align}
Fix \(t\) and \(r\) satisfying (\ref{eq: triangle 1}) and let \(\bar{p},\bar{t},\bar{r}\in M^2_k\) be such that
\(t = d_k(\bar{t},\bar{p})\), \(r = d_k(\bar{r},\bar{p})\), and
\[\angle^k(\gamma(t),\eta(r)) > \theta_{\bar{t},\bar{r}} > \angle^+(\gamma,\eta) -\varepsilon/3 \]
where \(\theta_{\bar{t},\bar{r}}\) is the angle between \(\bar{p}\bar{t}\) and \(\bar{p}\bar{r}\) in \(M^2_k\). The left side of he above ineqality tells us that \(d(\gamma(t),\eta(r)) > d_k(\bar{t},\bar{r})\). Combining the right side of the above inequality with (\ref{eq: triangle contradiction}), we have
\[ \theta_{\bar{t},\bar{r}} > \angle^+(\gamma,\sigma) + \angle^+(\sigma,\eta) + 2\varepsilon/3 . \]
Therefore, we can fix \(\bar{s}\in M^2_k\) along the path \(\bar{t}\bar{r}\) such that
\[ \theta_{\bar{t},\bar{s}} > \angle(\gamma,\sigma)+\varepsilon/3 \quad \text{and} \quad \theta_{\bar{s},\bar{r}} > \angle(\sigma,\eta)+\varepsilon/3 . \]
Set \(s = d_k(\bar{s},\bar{p})\). Since \(d_k(\bar{s},\bar{p})\leq \max\{d_k(\bar{t},\bar{p}),d_k(\bar{r},\bar{p})\} < \delta\), by (\ref{eq: triangle 2}) and (\ref{eq: triangle 3}) we have
\[ \theta_{\bar{t},\bar{s}} > \angle^k(\gamma(t),\sigma(s)) \quad \text{and} \quad \theta_{\bar{s},\bar{r}} > \angle^k(\sigma(s),\eta(r)) . \]
It follows that \(d_k(\bar{t},\bar{s}) > d(\gamma(t),\sigma(s))\) and \(d_k(\bar{s},\bar{r})>d(\sigma(s),\eta(r))\).
Thus, we have
\[d(\gamma(t),\eta(r)) > d_k(\bar{t},\bar{r}) = d_k(\bar{t},\bar{s})+d_k(\bar{s},\bar{r}) > d(\gamma(t),\sigma(s)) + d(\sigma(s),\eta(r)) \]
which contradicts the triangle inequality.
\end{proof}
\subsection{Two Results for Thin Triangles}
The next lemma is arguably the crux of this work. As observed above, it is clear that small
triangles in space forms are essentially Euclidean. However, what we need here are the properties of long, thin triangles, that is: triangles with only one small side (and two long sides). The
surprising --- and perhaps counter-intuitive --- fact is that these also behave like Euclidean triangles!
\begin{lemma}\label{lem: cosine inequality}
Let \(X\) be a length space and let \(\gamma:[0,T]\to X\) and \(\eta: [0,S] \to X\) be shortest paths such that \(\gamma(0) = \eta(0)\). Then for fixed \(s\) such that \(0<s<D_k\),
\[
\lim_{t\to 0^+} \left| \cos\Big(\angle^k (\gamma(t),\eta(s))\Big) - \dfrac{s - d(\gamma(t),\eta(s))}{t}\right| = 0.
\]
\end{lemma}
\begin{proof}
We first look at \(k=0\) and summarize the proof found in \cite[Lemma 4.5.5]{bbi2001}.
For simplicity of notation, let \(\theta = \angle^k (\gamma(t),\eta(s))\) and \(d = d(\gamma(t),\eta(s))\). Recall that \(t=d(\gamma(0),\gamma(t))\) and \(s=d(\gamma(0),\eta(s))\). Employing the Euclidean law of cosines, we find
\[
d^2=s^2+t^2-2st\cos ( \theta ) .
\]
A trivial computation confirms that
\[
\left| \cos ( \theta ) - \dfrac{s-d}{t}\right|=\left| \dfrac{s-d}{t}\cdot \dfrac{d-s}{2s} +
\dfrac{t}{2s}\right|.
\]
By the triangle inequality, $|s-d|=|d-s| \leq t$, which gives the desired result.
We next consider the case \(k>0\). If we radially project
the triangle with sides of lengths $s$, $t$, and $d$ to the unit-sphere, we can use
the spherical law of cosines to derive
\begin{align*}
\cos(\theta) & = \frac{\cos(d\sqrt{k}) - \cos(t\sqrt{k}) \cos(s\sqrt{k})}{\sin(t\sqrt{k})\sin(s\sqrt{k})}
\\ & = \frac{\cos(d\sqrt{k}) - \cos(s\sqrt{k})}{\sin(t\sqrt{k})\sin(s\sqrt{k})} + \frac{\cos(s\sqrt{k}) (1 - \cos(t\sqrt{k}))}{\sin(t\sqrt{k})\sin(s\sqrt{k})} .
\end{align*}
Recall from the trigonometric relations that
\begin{align*}
\cos(d\sqrt{k})-\cos(s\sqrt{k}) & = 2\sin\bigg(\frac{(s+d)\sqrt{k}}{2} \bigg) \sin\bigg(\frac{(s-d)\sqrt{k}}{2}\bigg) , \\
1 - \cos(t\sqrt{k}) & = 2\sin^2\bigg( \frac{t\sqrt{k}}{2} \bigg) , \\
\text{ and } \sin(t\sqrt{k}) & = 2\sin\bigg( \frac{t\sqrt{k}}{2} \bigg) \cos\bigg( \frac{t\sqrt{k}}{2} \bigg) .
\end{align*}
Combining all of the above, we get
\[ \cos(\theta) = \bigg( \frac{\sin\big(\frac{(s+d)\sqrt{k}}{2} \big)}{\sin(s\sqrt{k})} \bigg) \bigg( \frac{2\sin\big(\frac{(s-d)\sqrt{k}}{2}\big)}{\sin(t\sqrt{k})} \bigg) + \frac{\cos(s\sqrt{k})\sin\big( \frac{t\sqrt{k}}{2} \big)}{\sin(s\sqrt{k})\cos\big( \frac{t\sqrt{k}}{2} \big)}. \]
Note that \(d \to s\) as \(t\to 0\). Using the limit of $\frac{\sin x}{x}$, we find
\begin{align*}
& \lim_{t\to 0^+} \frac{\sin\big(\frac{(s+d)\sqrt{k}}{2} \big)}{\sin(s\sqrt{k})} = 1 , \\
& \lim_{t\to 0^+} \frac{2\sin\big(\frac{(s-d)\sqrt{k}}{2}\big)}{\sin(t\sqrt{k})} = \lim_{t\to 0^+} \frac{s-d}{t} , \\
\text{and } & \lim_{t\to 0^+} \frac{\cos(s\sqrt{k})\sin\big( \frac{t\sqrt{k}}{2} \big)}{\sin(s\sqrt{k})\cos\big( \frac{t\sqrt{k}}{2} \big)} = 0
\end{align*}
which gives us \( \lim_{t\to 0^+} \left| \cos (\theta) - \frac{s-d}{t} \right| = 0\).
The proof for \(k<0\) follows from the relationships \(\cos(ix) = \cosh(x)\) and \(\sin(ix) = i\sinh(x)\) and is very similar. It can be found in \cite[p. 11]{alexandrov51} and \cite[Lemma 4.1]{shiohama}.
\end{proof}
\begin{lemma}
If \(X\) is a length space, then for all shortest paths \(\gamma:[0,T]\to X\) and \(\eta:[0,S]\to X\) with \(\gamma(0)=\eta(0)\), for every fixed \(s>0\), we have
\[\limsup_{t\to 0^+} \angle^k (\gamma(t),\eta(s)) \leq \angle^+ (\gamma,\eta) .
\]
\label{lem:strong angle}
\end{lemma}
\begin{proof} If \(s' < s\), then by the triangle inequality
\[ s-s' \geq d \big( \gamma(t),\eta(s) \big) - d \big( \gamma(t),\eta(s') \big) \]
which gives us
\[ s - d \big( \gamma(t),\eta(s) \big) \geq s' - d \big( \gamma(t),\eta(s') \big) . \]
Substituting this into Lemma \ref{lem: cosine inequality}, we see that
\[ \liminf_{t\to 0^+} \cos \big( \angle^k (\gamma(t),\eta(s)) \big) \geq \liminf_{t\to 0^+} \cos \big( \angle^k (\gamma(t),\eta(s')) \big) . \]
As cosine is nonincreasing on \([0,\pi]\), we have
\[ \cos \bigg( \limsup_{t\to0^+} \angle^k (\gamma(t),\eta(s)) \bigg) \geq \cos\bigg( \limsup_{s,t \to 0^+} \angle^k (\gamma(t),\eta(s)) \bigg) . \]
The right hand equals $\cos ( \angle^+(\gamma,\eta) )$.
\end{proof}
\section{Alexandrov Spaces}\label{alexandrov spaces}
Here we give an account of metric spaces which exhibit bounded curvature in the sense of Alexandrov, which most nearly resemble Riemannian maifolds with bounded sectional curvature. The idea of bounded curvature in a metric space is certainly not unique to Alexandrov spaces; another common example is that developed by Busemann for non-positive curvature (see for example \cite{papadopoulos}). Even recently, new characterizations for spaces of bounded curvature, such as in \cite{jost1} and \cite{jost2}, have given rise to further possibilities in the application of metric geometry.
\subsection{Bounded Curvature}
A length space \(X\) is said to be of \emph{curvature bounded above} (or \emph{curvature \(\leq k\)}) if there is a \(k\in \mathbb{R}\) for which the following holds: At every point in \(X\) there is a neighborhood \(U\) such that for every geodesic triangle \(\Delta\subseteq U\) with comparison triangle \(\overline{\Delta}\subseteq M_k^2\),
\begin{equation}\label{curvature bound}
d(u,v) \leq d_k(\bar{u},\bar{v})
\end{equation}
for all \(u,v\in \Delta\) and their comparison points \(\bar{u},\bar{v} \in \overline{\Delta}\) (see Figure \ref{fig:comparison-triangles}).
Similarly, \(X\) is said to be of \emph{curvature bounded below} (or \emph{curvature
\(\geq k\)}) if \(d(u,v) \geq d_k(\bar{u},\bar{v})\).
In either case, the neighborhood \(U\) is referred to as a \emph{region of bounded curvature}.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.80\linewidth]{comparisontriangles.eps}
\caption{A geodesic triangle with vertices \(x\), \(y\), and \(z\) in a length space (left) and the respective comparison triangle \(\overline{\Delta}(x,y,z)\) in \(M^2_0\) (right). The points \(\bar{u}\) and \(\bar{v}\) are chosen to satisfy \(d(x,u) = d_k(\bar{x},\bar{u})\) and \(d(x,v) = d_k(\bar{x},\bar{v})\). }
\label{fig:comparison-triangles}
\end{center}
\end{figure}
An \emph{Alexandrov space} is a complete and locally compact length space with curvature bounded either above or below.
It should be noted that this definition of Alexandrov space (which comes from \cite{shiohama}) is not necessarily the uniformly accepted definition, but is necessary for our main theorem. In \cite{bbi2001}, an Alexandrov space is simply a length space with curvature bounded above or below. Due to the Hopf-Rinow Theorem, our additional requirement that the space be complete and locally compact allows us to avoid continually addressing the existence of shortest paths in the hypothesis of every proposition; however, it does limit the scope of some of the following preliminary results. In any case, a more general treatment of spaces of bounded curvature (in the sense of Alexandrov) can be found in \cite{bbi2001} or \cite{bridson}.
\subsection{Properties of Alexandrov Spaces}
It is well known that for Alexandrov spaces, the angle between two geodesics emanating from a common point always exists.
\begin{lemma}\label{angle exists}
Let \(X\) be an Alexandrov space. If \(\gamma:[0,T]\to X\) and \(\eta:[0,S]\to X\) are shortest paths with \(\gamma(0) = \eta(0)\) then the angle \(\angle(\gamma,\eta)\) exists and
\[ \angle(\gamma,\eta) = \lim_{t\to 0^+} \angle^k (\gamma(t),\eta(t)) . \]
\end{lemma}
\begin{proof}
Suppose that \(X\) is of curvature \(\leq k\). Fix \(s\in (0,S]\) and \(a,b\in (0,T]\) such that \(a<b\). We will consider two distinct comparison triangles in \(M^2_k\). For simplicity of notation, we will denote them
\[ \overline{\Delta}(a) := \overline{\Delta}\big(\gamma(a),\gamma(0),\eta(s)\big) \quad \text{ and } \quad \overline{\Delta}(b) := \overline{\Delta}\big( \gamma(b),\gamma(0),\eta(s)\big) . \]
From the definition of \(\overline{\Delta}(a)\) we have (see Figure \ref{fig:monotoneangles})
\[
d_k(\overline{\gamma(a)}, \overline{\eta(s)})=d({\gamma(a)}, {\eta(s)}) .
\]
Let \(\tilde a\) be the comparison point of $\gamma(a)$ in \(\overline{\Delta}(b)\) (as opposed to \(\overline{\gamma(a)}\), which is the comparison point in \(\overline{\Delta}(a)\)). The upper bound $k$ for
the curvature gives
\[
d({\gamma(a)}, {\eta(s)}) \leq d_k(\tilde a, \overline{\eta(s)}) .
\]
Thus\footnote{Note that the left distance is in $\overline{\Delta}(a)$ while the
right distance is in $\overline{\Delta}(b)$.}
\(d_k \big(\overline{\gamma(a)} , \overline{\eta(s)} \big) \leq d_k\big(\tilde{a} ,
\overline{\eta(s)}\big) \), which in turn implies
\[
\angle^k \big(\gamma(a),\eta(s)\big) \leq \angle^k \big( \gamma(b) , \eta(s) \big) .
\]
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.90\linewidth]{monotoneangles.eps}
\caption{An illustration of the comparison triangles \(\overline{\Delta}(\gamma(0),\gamma(b),\eta(s))\) and \(\overline{\Delta}(\gamma(0),\gamma(b),\eta(s))\) from Lemma \ref{angle exists}. }
\label{fig:monotoneangles}
\end{center}
\end{figure}
Thus, for any fixed \(s_0\in (0,S]\), the map \(t \mapsto \angle^k(\gamma(t),\eta(s_0))\) is
monotonically nondecreasing. By the same reasoning the map \(s\mapsto \angle^k(\gamma(t_0),\eta(s))\)
is nondecreasing for any fixed \(t_0\in (0,T]\). It follows from the monotonicity in both
coordinates\footnote{For clarification on monotonicity in functions of two variables, see Proposition \ref{prop: monotone two} of the Appendix.} that
\[ \angle^+(\gamma,\eta) = \limsup_{s,t\to0^+} \angle^k (\gamma(t),\eta(s)) = \liminf_{s,t\to0^+} \angle^k (\gamma(t),\eta(s)) = \angle^-(\gamma,\eta) . \]
We conclude that the angle \(\angle(\gamma,\eta)\) exists and is equal to \(\lim_{t\to 0^+}\angle^k(\gamma(t),\eta(t))\).
If \(X\) is of curvature \(\geq k\), the same method of proof applies, but the inequalities are reversed and the maps \(t\mapsto \angle^k(\gamma(t),\eta(s_0))\) and \(s\mapsto \angle^k(\gamma(t_0),\eta(s))\) are monotonically nonincreasing.
\end{proof}
\begin{corollary}\label{cor: angle condition}
Let \(X\) be an Alexandrov space of curvature \(\leq k\) (resp. \(\geq k\)). If the shortest paths \(\gamma:[0,T]\to X\) and \(\eta:[0,S]\to X\) (with \(\gamma(0) = \eta(0)\)) are contained in a region of bounded curvature, then
\[\angle(\gamma,\eta) \leq \angle^k(\gamma(t),\eta(s)) \quad \big(\text{ resp. } \angle(\gamma,\eta) \geq \angle^k(\gamma(t),\eta(s)) \ \big)\]
for any \(s,t>0\).
\end{corollary}
\begin{proof}
By Lemma \ref{angle exists}, if \(X\) is of curvature \(\leq k\) (resp. \(\geq k\)) the map \(t\mapsto \angle^k(\gamma(t),\eta(t))\) is nondecreasing (resp. nonincreasing). It follows immediately that \(\angle(\gamma,\eta)\leq \angle^k(\gamma(t),\eta(s))\) (resp. \(\angle(\gamma,\eta)\geq \angle^k(\gamma(t),\eta(s))\)) for any \(t\in (0,T]\) and \(s\in (0,S]\).
\end{proof}
While spaces of curvature bounded above and below share many properties, the following lemma gives an example of a property of spaces of curvature \(\geq k\) which is not valid in spaces of curvature \(\leq k\). This lemma also makes use of notation we shall need again, so we introduce it here. Let \(\gamma:[0,T]\to X\) be a path and fix \(t\in (0,T)\). The path \(\gamma|_{[t,0]}\) is defined by \(\gamma|_{[t,0]}(s) = \gamma(t-s)\) for \(s\in [0,t]\). In other words, \(\gamma|_{[t,0]}\) is the path that runs backwards along \(\gamma\) from \(\gamma(t)\) to \(\gamma(0)\).
\begin{lemma}\label{supplementary angles}
If \(X\) is an Alexandrov space of curvature bounded below, \(\gamma: [0,T]\to X\) is a shortest path, \(0 < t < T\), and \(\sigma_t:[0,S] \to X\) is a shortest path with \(\sigma_t(0) = \gamma(t)\) then
\[ \angle_{\gamma(t)} \big( \gamma |_{[t,T]} , \sigma_t \big) + \angle_{\gamma(t)} \big( \gamma |_{[t,0]} , \sigma_t \big) = \pi . \]
In other words, adjacent angles along a shortest path sum to \(\pi\).
\end{lemma}
\begin{proof}
By Proposition \ref{triangle inequality angles}, we know that
\[\angle_{\gamma(t)} \big( \gamma |_{[t,T]} , \sigma_t \big) + \angle_{\gamma(t)} \big( \gamma |_{[t,0]} , \sigma_t \big) \geq \angle_{\gamma(t)} \big( \gamma|_{[t,T]}, \gamma|_{[t,0]} \big) = \pi \]
so it suffices to prove the reverse inequality.
Fix a small \(\delta>0\). We will consider a configuration of comparison points in \(M^2_k\) for the points \(\gamma(t-\delta)\), \(\gamma(t)\), \(\gamma(t+\delta)\), and \(\sigma_t(\delta)\).
First, consider the comparison triangle \(\overline{\Delta}(\gamma(t-\delta),\sigma_t(\delta),\gamma(t+\delta)\) with the comparison point \(\overline{\gamma(t)}\). Second, consider the comparison triangle \(\overline{\Delta}(\gamma(t),\sigma_t(\delta),\gamma(t+\delta))\) with the points \(\overline{\gamma(t)}\) and \(\overline{\gamma(t+\delta)}\) aligned as in Figure \ref{fig:supplementary}. Given that each triangle has a vertex representing \(\sigma_t(\delta)\), we have labeled them \(\overline{\sigma_t(\delta)}\) and \(\widehat{\sigma_t(\delta)}\) respectively, to distinguish them.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.6\linewidth]{supplementary.eps}
\caption[width=0.5\linewidth]{The comparison point construction of Lemma \ref{supplementary angles}. }
\label{fig:supplementary}
\end{center}
\end{figure}
By the definition of curvature \(\geq k\), we know that
\[d_k\big( \overline{\sigma_t(\delta)},\overline{\gamma(t)} \big) \leq d_k \big( \widehat{\sigma_t(\delta)},\overline{\gamma(t)} \big) = d\big( \sigma_t(\delta),\gamma(t) \big) . \]
Considering that
\begin{align*}
& d_k\big( \overline{\sigma_t(\delta)},\overline{\gamma(t+\delta)} \big) = d_k \big( \widehat{\sigma_t(\delta)},\overline{\gamma(t+\delta)} \big) \\
\text{and } & d_k\big( \overline{\gamma(t)},\overline{\gamma(t+\delta)} \big) = d_k \big( \overline{\gamma(t)},\overline{\gamma(t+\delta)} \big) ,
\end{align*}
we have an inequality between the interior angles at \(\overline{\gamma(t)}\);
\[ \angle^k_{\gamma(t)} \big(\sigma_t(\delta),\gamma(t+\delta) \big) = \angle^k_{\overline{\gamma(t)}} \big(\widehat{\sigma_t(\delta)},\overline{\gamma(t+\delta)} \big) \leq \angle^k_{\overline{\gamma(t)}} \big( \overline{\sigma_t(\delta)},\overline{\gamma(t+\delta)} \big) . \]
Applying the analogous argument to \(\overline{\Delta}(\gamma(t-\delta),\sigma_t(\delta),\gamma(t))\), we see that
\[\angle^k_{\gamma(t)} \big(\sigma_t(\delta),\gamma(t-\delta) \big) + \angle^k_{\gamma(t)} \big(\sigma_t(\delta),\gamma(t+\delta) \big) \leq \angle^k_{\gamma(t)} \big(\gamma(t-\delta),\gamma(t+\delta) \big) . \]
Taking the limit as \(\delta\to 0^+\) yields the result.
\end{proof}
\begin{proposition}[Semi-continuity of angles]\label{prop: semi-continuity}
Let \(X\) be an Alexandrov space of curvature bounded above (resp. below). Suppose that the sequences of shortest paths \(\{\gamma_n\}_{n=1}^\infty\) and \(\{\sigma_n\}_{n=1}^\infty\), with \(\gamma_n(0) = \sigma_n(0)\) for all \(n\), converge uniformly to shortest paths \(\gamma\) and \(\sigma\) respectively. Then \(\angle (\gamma,\sigma) \geq \limsup_{n\to\infty} \angle (\gamma_n,\sigma_n)\) (resp. \(\angle(\gamma,\sigma) \leq \liminf_{n\to\infty} \angle (\gamma_n,\sigma_n)\)).
\end{proposition}
\begin{proof}
First, suppose that \(X\) is of curvature \(\leq k\). For any \(t\in [0,T]\), since \(\gamma_n \to \gamma\) uniformly, \(\gamma_n(t) \to \gamma(t)\); and the same can be said for the path \(\sigma\). By Lemma \ref{angle exists} and Corollary \ref{cor: angle condition},
\begin{align}
\angle (\gamma,\sigma) & = \lim_{t\to 0^+} \angle^k_{\gamma(0)} \big( \gamma(t) , \sigma(t) \big) \label{line1}
\\ & = \lim_{t\to 0^+} \bigg( \lim_{n\to\infty} \angle^k_{\gamma_n(0)} \big( \gamma_n(t) , \sigma_n(t) \big) \bigg) \label{line2}
\\ & \geq \lim_{t\to 0^+} \bigg( \limsup_{n\to\infty} \angle ( \gamma_n, \sigma_n) \bigg) . \label{line3}
\end{align}
As the final quantity above is independent of \(t\), we have \(\angle (\gamma,\sigma) \geq \limsup_{n\to\infty} \angle (\gamma_n,\sigma_n)\).
Alternatively, if we suppose that \(X\) is of curvature \(\geq k\). Then (\ref{line1}) and (\ref{line2}) above still hold, but in (\ref{line3}) we make use of the other inequality of Corollary \ref{cor: angle condition} to obtain \(\angle (\gamma,\sigma) \leq \liminf_{n\to\infty} \angle (\gamma_n,\sigma_n)\).
\end{proof}
\section{Right-Derivative of Distance to a Compact Set}\label{sec: derivative}
In any metric space, it is an easy application of the triangle inequality to show that distance to a set is 1-Lipschitz.
It follows that distance along a geodesic is differentiable almost everywhere in the domain of the geodesic. In this section, we prove an explicit value for this (one-sided) derivative, when the distance is taken to a compact set.
Even more than finding the derivative of distance along a geodesic, if we think of the geodesic \(\gamma\) as representing a direction in an Alexandrov space, this formula is akin to the directional derivative of distance in the direction \(\gamma\). This idea serves as a precursor to developing gradients of functions on Alexandrov spaces, which is explored in more detail in \cite[Section 7.4]{plaut}, and in a more general setting of `geometric' metric spaces beyond Alexandrov spaces in \cite{lytchak}.
\begin{lemma}\label{limsup}
If \(X\) is an Alexandrov space, \(\gamma:[0,T]\to X\) is a shortest path, and \(p\) is an element of \(X\) such that \(\gamma(0)\neq p\), then
\[\limsup_{t\to 0^+} \frac{ d(\gamma(t),p) - d(\gamma(0),p)}{t} \leq - \cos \big( \angle_{\min} \big) \]
where \(\angle_{\min}\) is the infimum of angles between \(\gamma\) and shortest paths from \(\gamma(0)\) to \(p\).
\end{lemma}
\begin{proof}
Let \(\eta :[0,S]\to X\) be a shortest path connecting \(\gamma(0)\) to \(p\). Using
Lemma \ref{lem:strong angle} and the fact that \(-\cos\) is nondecreasing on \([0,\pi]\),
\begin{align*}
-\cos \big( \angle (\gamma,\eta) \big) & \geq \limsup_{t \to 0^+} \bigg( - \cos\big( \angle^k(\gamma(t),p)\big) \bigg)
\\ & = \limsup_{t\to 0^+} \frac{- d(\gamma(0),p) + d(\gamma(t),p)}{t} ,
\end{align*}
where the last equality comes from Lemma \ref{lem: cosine inequality}. Therefore,
\[ \limsup_{t\to 0^+} \frac{d(\gamma(t),p) - d(\gamma(0),p)}{t} \leq -\cos \big( \angle (\gamma,\eta) \big) . \]
Given that this holds for any shortest path \(\eta\) connecting \(\gamma(0)\) to \(p\), we can replace \(\angle (\gamma,\eta)\) above with \(\angle_{\min}\).
\end{proof}
\begin{lemma}\label{lem:pi angle}
Let \(X\) be an Alexandrov space, \(K\) a compact set in \(X\), and \(\gamma:[0,T] \to X\) a shortest path such that \(\gamma(0)\notin K\). For each \(t\in [0,T]\) let \(\sigma_t\) be a shortest path connecting \(\gamma(t)\) to \(K\). If there is a sequence \(\{t_n\}_{n=1}^\infty\) such that \(t_n\to 0\) and the sequence of shortest paths \(\{\sigma_{t_n}\}_{n=1}^\infty\) converges to \(\sigma_0\), then
\[\limsup_{n\to\infty} \angle^k_{\gamma(t_n)} \big( \gamma(0),\sigma_{t_n}(s) \big) \leq \pi - \angle(\gamma,\sigma_0) \]
for all sufficiently small \(s>0\). (See Figure \ref{fig:firstvar}.)
\end{lemma}
\begin{proof}
Begin by fixing \(N\in \mathbb{N}\) and \(s'\in (0,S]\) such that \(\gamma(0)\), \(\gamma(t_n)\), and \(\sigma_{t_n}(s)\) all lie in a region of bounded curvature whenever \(n\geq N\) and \(s\leq s'\). For simplicity of notation let \(s_n = \sigma_{t_n}(s)\) for some fixed \(s>0\).
Suppose \(X\) is of curvature \(\geq k\). Then
\begin{align*}
\limsup_{n\to\infty} \angle^k_{\gamma(t_n)} (\gamma(0),s_n) & \leq \limsup_{n\to\infty} \angle_{\gamma(t_n)} (\gamma|_{[t_n,0]}, \sigma_{t_n})
\\ & = \pi - \liminf_{n\to\infty} \angle_{\gamma(t_n)}(\gamma|_{[t_n,T]},\sigma_{t_n})
\\ & \leq \pi - \angle (\gamma,\sigma_0)
\end{align*}
by Corollary \ref{cor: angle condition}, Lemma \ref{supplementary angles}, and Proposition \ref{prop: semi-continuity}, respectively.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.8\linewidth]{distcompactset.eps}
\caption[width=0.5\linewidth]{An illustration of the paths, points, and angles in the proof of Lemma \ref{lem:pi angle} and Theorem \ref{thm:compact set derivative}. }
\label{fig:firstvar}
\end{center}
\end{figure}
Next, suppose \(X\) is of curvature \(\leq k\).
For each \(n\), let \(\eta_n\) be a shortest path connecting \(\gamma(0)\) to \(s_n\) (see Figure
\ref{fig:firstvar2}). By Corollary \ref{cor: angle condition}, \(\angle(\eta_n,\sigma_0) \leq \angle^k_{\gamma(0)} (s_n,\sigma_0(s))\) for all \(n\). Since \(\sigma_{t_n} \to \sigma_0\), we have \(\angle^k_{\gamma(0)}(s_n,\sigma_0(s)) \to 0\) and so \(\angle(\eta_n,\sigma_0) \to 0\).
By Proposition \ref{triangle inequality angles} twice,
\[ \angle (\gamma,\sigma_0) \leq \angle (\gamma,\eta_n) + \angle (\eta_n,\sigma_0) \leq \angle (\gamma,\sigma_0) + 2 \angle (\eta_n,\sigma_0) . \]
So, as \(\angle (\eta_n,\sigma_0) \to 0\), we have \(\angle (\gamma,\eta_n) \to \angle (\gamma,\sigma_0)\).
Thus, using Corollary \ref{cor: angle condition} again
\begin{equation}
\angle (\gamma,\sigma_0) = \liminf_{n\to\infty} \angle (\gamma,\eta_n) \leq
\liminf_{n\to\infty} \angle^k_{\gamma(0)} (\gamma(t_n),s_n) .
\label{eq:inequality}
\end{equation}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.60\linewidth]{compactseteta.eps}
\caption[width=0.5\linewidth]{The relationship between the paths \(\gamma\), \(\eta_n\), and \(\sigma_0\).}
\label{fig:firstvar2}
\end{center}
\end{figure}
Now let \(\delta >0\) be given. By the fact that infinitesimal triangles are Euclidean, we know
that in the comparison triangle \(\overline{\Delta}(\gamma(t_n),s_n,\gamma(0))\), we have for $n$ large enough
\begin{equation}\label{sum angles}
\angle^k_{\gamma(0)}(\gamma(t_n) , s_n) + \angle^k_{\gamma(t_n)}(\gamma(0) , s_n) + \angle^k_{s_n}(\gamma(t_n) , \gamma(0)) < \pi + \delta .
\end{equation}
Since \(\angle^k_{s_n} (\gamma(t_n),\gamma(0)) \to 0\), with (\ref{eq:inequality}) this gives
\[ \limsup_{n\to\infty} \angle^k_{\gamma(t_n)}(\gamma(0) , s_n) < \pi + \delta - \liminf_{n\to\infty} \angle^k_{\gamma(0)}(\gamma(t_n),s_n) \leq \pi + \delta - \angle(\gamma,\sigma_0)\]
Letting \(\delta>0\) go to zero gives the desired result.
\end{proof}
\begin{theorem}\label{thm:compact set derivative}
Let \(X\) be an Alexandrov space, \(\gamma:[0,T] \to X\) a shortest path, and \(K\) a compact set not containing \(\gamma(0)\). If \(\ell(t) = d(\gamma(t), K)\), then
\[\lim_{t\to 0^+} \frac{\ell(t) - \ell(0)}{t} = -\cos(\angle_{\min})\]
where \(\angle_{\min}\) is the infimum of angles between \(\gamma\) and any shortest path of length \(\ell(0)\) which connects \(\gamma(0)\) to \(K\).
\end{theorem}
\begin{proof}
First, let \(\eta_0\) be a shortest path connecting \(\gamma(0)\) to \(K\) and let \(a\in K\) be the endpoint of \(\eta_0\). Note that for each \(t>0\), \(\ell(t) \leq d(\gamma(t),a)\). Therefore, by Lemma \ref{limsup},
\[\limsup_{t\to 0^+} \frac{\ell(t) - \ell(0)}{t} \leq \limsup_{t\to 0^+} \frac{ d(\gamma(t),a) - d(\gamma(0),a)}{t} \leq - \cos \big( \angle_{\min} \big) . \]
To get the reverse estimate, let \(\{t_n\}_{n=1}^\infty\) be a sequence in \((0,T]\) such that \(t_n \to 0\) and
\[ \lim_{n\to\infty} \frac{\ell(t_n) - \ell(0)}{t_n} = \liminf_{t\to 0^+} \frac{\ell(t) - \ell(0)}{t} . \]
Similar to Lemma \ref{lem:pi angle}, for each \(n\) let \(\sigma_{t_n}\) be a shortest path connecting \(\gamma(t_n)\) to \(K\). Since \(K\) is compact, the length of each path in the sequence \(\{\sigma_{t_n}\}_{n=1}^\infty\) is uniformly bounded. Therefore, by the Arzela-Ascoli Theorem,\footnote{see Appendix \ref{sec: arzela-ascoli} for clarification on how Arzela-Ascoli is used here.} \(\{\sigma_{t_n}\}_{n=1}^\infty\) contains a subsequence which converges uniformly to a shortest path \(\sigma_0\) connecting \(\gamma(0)\) to \(K\). Without loss of generality, we assume that the sequence \(\{\sigma_{t_n}\}_{n=1}^\infty\) is this uniformly convergent subsequence.
Fix \(s\) sufficiently small to satisfy the hypothesis of Lemma \ref{lem:pi angle}. For simplicity of notation let \(p_n\in K\) be the endpoint of \(\sigma_{t_n}\), let \(p\in K\) be the endpoint of \(\sigma_0\), and let \(s_n = \sigma_{t_n}(s)\) (see Figures \ref{fig:firstvar} and \ref{fig:firstvar2}).
By Lemma \ref{lem: cosine inequality},
\begin{equation}\label{eq:cosine}
\liminf_{n\to\infty} \frac{s-d(s_n,\gamma(0))}{t_n} = \liminf_{n\to\infty} \cos\big( \angle^k_{\gamma(t_n)} (\gamma(0),s_n) \big) .
\end{equation}
Note that \(\ell(t_n) = s + d(s_n,p_n)\) and
\[\ell(0) \leq d(\gamma(0),p_n) \leq d(\gamma(0),s_n) + d(s_n,p_n) .\]
Combining these observations with (\ref{eq:cosine}), we get
\begin{align*}
\liminf_{n\to\infty} \frac{\ell(t_n) - \ell(0)}{t_n} & \geq \liminf_{n\to\infty} \cos\big( \angle^k_{\gamma(t_n)} (\gamma(0),s_n) \big)
\\ & = \cos \big( \limsup_{n\to\infty} \angle^k_{\gamma(t_n)} (\gamma(0),s_n) \big) .
\end{align*}
Then by Lemma \ref{lem:pi angle},
\[ \cos \big( \limsup_{n\to\infty} \angle^k_{\gamma(t_n)} (\gamma(0),s_n) \big) \geq \cos \big( \pi - \angle(\gamma,\sigma_0) \big) = - \cos \big( \angle(\gamma,\sigma_0)\big) .\]
Thus,
\begin{align*}
\liminf_{t\to 0^+} \frac{\ell(t) - \ell(0)}{t} & = \liminf_{n\to\infty} \frac{\ell(t_n) - \ell(0)}{t_n}
\\ & \geq - \cos \big( \angle(\gamma,\sigma_0)\big)
\\ & \geq -\cos(\angle_{\min})
\end{align*}
which is the desired reverse estimate.
\end{proof}
| {
"timestamp": "2020-09-21T02:02:02",
"yymm": "2001",
"arxiv_id": "2001.01154",
"language": "en",
"url": "https://arxiv.org/abs/2001.01154",
"abstract": "We give a complete and self-contained proof of a folklore theorem which says that in an Alexandrov space the distance between a point $\\gamma(t)$ on a geodesic $\\gamma$ and a compact set $K$ is a right-differentiable function of $t$. Moreover, the value of this right-derivative is given by the negative cosine of the minimal angle between the geodesic and any shortest path to the compact set (Theorem 4.3). Our treatment serves as a general introduction to metric geometry and relies only on the basic elements, such as comparison triangles and upper angles.",
"subjects": "Metric Geometry (math.MG)",
"title": "One-Sided Derivative of Distance to a Compact Set",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754506337406,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.7075103978883298
} |
https://arxiv.org/abs/1403.5060 | A Discrete Method to Solve Fractional Optimal Control Problems | We present a method to solve fractional optimal control problems, where the dynamic depends on integer and Caputo fractional derivatives. Our approach consists to approximate the initial fractional order problem with a new one that involves integer derivatives only. The latter problem is then discretized, by application of finite differences, and solved numerically. We illustrate the effectiveness of the procedure with an example. | \section{Introduction}
The fractional calculus deals with differentiation
and integration of arbitrary (noninteger) order \cite{book:samko}.
It has received an increasing interest due to the fact that fractional
operators are defined by natural phenomena \cite{Benson}.
A particularly interesting, and very active, research area is that of
the fractional optimal control, where the dynamic control system involves
not only integer order derivatives but fractional operators as well \cite{book:frac,pooseh:press}.
Such fractional problems are difficult to solve and numerical methods are often applied,
which have proven to be efficient and reliable \cite{sug:r3,ford1,ford2,Lotfi,Almeida1}.
Typically, using an approximation for the fractional operators, the fractional optimal control
problem is rewritten into a new one, that depends on integer derivatives only \cite{Atanackovic,Almeida}.
Then, using necessary optimality conditions, the problem is reduced
to a system of ordinary differential equations and, by finding its solution,
one approximates the solution to the original fractional problem \cite{Almeida}.
Our approach is different here: after replacing the fractional operator,
we consider the augmented functional and apply an Euler-like method.
The text is organized as follows. In Section~\ref{sec:2}
we briefly recall the necessary concepts and results.
Our method is presented in Section~\ref{sec:3}
and then illustrated, with an example, in Section~\ref{sec:4}.
\section{Preliminaries}
\label{sec:2}
We start with some definitions needed in the sequel.
For more on the subject we refer the reader
to \cite{Kilbas,book:frac,Podlubny,book:samko}.
\begin{definition}
\label{def:DF:IF}
Let $x:[a,b]\to\mathbb R$ and $\alpha>0$
be the order of the integral or the derivative.
For $t \in[a,b]$, we define
\begin{enumerate}
\item the left Riemann--Liouville fractional integral by
$$
{_aI_t^{\alpha}}x(t)=\frac{1}{\Gamma(\alpha)}
\int_a^t (t-\tau)^{\alpha-1}x(\tau) d\tau,
$$
\item the right Riemann--Liouville fractional integral by
$$
{_tI_b^{\alpha}}x(t)=\frac{1}{\Gamma(\alpha)}
\int_t^b (\tau-t)^{\alpha-1}x(\tau) d\tau,
$$
\item the left Riemann--Liouville fractional derivative by
$$
{_aD_t^{\alpha}}x(t)=\frac{1}{\Gamma(n-\alpha)}
\frac{d^n}{dt^n}\int_a^t (t-\tau)^{n-\alpha-1}x(\tau) d\tau,
$$
\item the right Riemann--Liouville fractional derivative by
$$
{_tD_b^{\alpha}}x(t)=\frac{(-1)^n}{\Gamma(n-\alpha)}
\frac{d^n}{dt^n}\int_t^b (\tau-t)^{n-\alpha-1}x(\tau) d\tau,
$$
\item the left Caputo fractional derivative by
$$
{_a^CD_t^{\alpha}}x(t)=\frac{1}{\Gamma(n-\alpha)}
\int_a^t (t-\tau)^{n-\alpha-1}x^{(n)}(\tau) d\tau,
$$
\item the right Caputo fractional derivative by
$$
{_t^CD_b^{\alpha}}x(t)=\frac{(-1)^n}{\Gamma(n-\alpha)}
\int_t^b (\tau-t)^{n-\alpha-1}x^{(n)}(\tau) d\tau,
$$
\end{enumerate}
where $n=[\alpha]+1$ for the definitions of
Riemann--Liouville fractional derivatives, and
\begin{equation}
\label{def:n}
n=\left\{\begin{array}{lll}
[\alpha]+1 & \mbox{ if } \, &\alpha \notin \mathbb N_0,\\
\alpha & \mbox{ if } \, &\alpha \in \mathbb N_0,\\
\end{array}\right.
\end{equation}
for the Caputo fractional derivatives.
\end{definition}
When $\alpha=n$ is an integer, the operators
of Definition~\ref{def:DF:IF}
reduce to standard ones:
\begin{equation*}
\begin{split}
{_aI_t^{n}}x(t)
&= \displaystyle \int_a^t d\tau_1 \int_a^{\tau_1} d\tau_2
\ldots \int_a^{\tau_{n-1}}x(\tau_n) d\tau_n,\\
{_tI_b^{n}}x(t) &= \displaystyle\int_t^b d\tau_1
\int_{\tau_1}^b d\tau_2 \ldots \int_{\tau_{n-1}}^b x(\tau_n) d\tau_n,\\
{_aD_t^{n}}x(t) &= \displaystyle {_a^CD_t^{n}}x(t) \,
= \, x^{(n)}(t),\\
{_tD_b^{n}}x(t) &= \displaystyle {_t^CD_b^{n}}x(t)
\, = \, (-1)^n x^{(n)}(t).
\end{split}
\end{equation*}
There is an useful relation between Riemann--Liouville and Caputo
fractional derivatives:
\begin{equation}
\label{relation}
{_a^CD_t^\alpha}x(t)={_aD_t^\alpha}x(t)-\sum_{k=0}^{n-1}
\frac{x^{(k)}(a)}{\Gamma(k-\alpha+1)}(t-a)^{k-\alpha}.
\end{equation}
For numerical purposes, approximations are used to deal with these fractional operators.
The Riemann--Liouville derivatives are expandable in a power series involving integer order derivatives only.
If $x$ is an analytic function, then (cf. \cite{Kilbas})
\begin{equation}
\label{app1}
{_aD_t^{\alpha}} x(t)=\sum_{k=0}^\infty \binom{\alpha}{k}
\frac{(t-a)^{k-\alpha}}{\Gamma(k+1-\alpha)}x^{(k)}(t),
\end{equation}
where $$\binom{\alpha}{k}=\frac{(-1)^{k-1}\alpha\Gamma(k-\alpha)}{\Gamma(1-\alpha)\Gamma(k+1)}.$$
The obvious disadvantage of using \eqref{app1} in numerical computations
is that in order to have a small error, one has to sum a large number
of terms and thus the function has to possess higher-order derivatives,
which is not suitable for optimal control. To address this problem,
a second approach was carried out in \cite{Atanackovic,Almeida},
where a good approximation is obtained without the requirement
of such higher-order smoothness on the admissible functions.
The method can be explained, for left-derivatives, in the following way.
Let $\alpha\in(0,1)$ and $x\in C^2[a,b]$. Then,
\begin{equation*}
{_aD_t^\a} x(t)=A(\alpha)(t-a)^{-\alpha}x(t)+B(\alpha)(t-a)^{1-\alpha}\dot{x}(t)
-\sum_{p=2}^{\infty}C(\alpha,p)(t-a)^{1-p-\alpha}V_p(t),
\end{equation*}
where
$V_p(t)$ is the solution of the system
$$
\begin{cases}
\dot{V}_p(t)=(1-p)(t-a)^{p-2}x(t),\\
V_p(a)=0,
\end{cases}
$$
for $p=2,3,\ldots,$ and $A$, $B$ and $C$ are given by
\begin{equation*}
\begin{split}
A(\alpha) &= \frac{1}{\Gamma(1-\alpha)}
\left[1+\sum_{p=2}^{\infty}\frac{\Gamma(p-1+\alpha)}{\Gamma(\alpha)(p-1)!}\right],\\
B(\alpha) &= \frac{1}{\Gamma(2-\alpha)}
\left[1+\sum_{p=1}^{\infty}\frac{\Gamma(p-1+\alpha)}{\Gamma(\alpha-1)p!}\right],\\
C(\alpha,p) &= \frac{1}{\Gamma(2-\alpha)\Gamma(\alpha-1)}\frac{\Gamma(p-1+\alpha)}{(p-1)!}.
\end{split}
\end{equation*}
Using \eqref{relation}, a similar formula can be deduced
for the Caputo fractional derivative. When we consider
finite sums only, that is, when we use the approximation
\begin{equation}
\label{approximation}
{_aD_t^\a} x(t)\approx A(\alpha,K)(t-a)^{-\alpha}x(t)+B(\alpha,K)(t-a)^{1-\alpha}\dot{x}(t)
-\sum_{p=2}^{K}C(\alpha,p)(t-a)^{1-p-\alpha}V_p(t),
\end{equation}
where $K\geq 2$ and
\begin{equation*}
\begin{split}
A(\alpha,K) &= \frac{1}{\Gamma(1-\alpha)}\left[1+\sum_{p=2}^{K}
\frac{\Gamma(p-1+\alpha)}{\Gamma(\alpha)(p-1)!}\right],\\
B(\alpha,K) &= \frac{1}{\Gamma(2-\alpha)}\left[1+\sum_{p=1}^{K}
\frac{\Gamma(p-1+\alpha)}{\Gamma(\alpha-1)p!}\right],
\end{split}
\end{equation*}
the error is bounded by
\begin{equation*}
|E_{tr}(t)|\leq \max_{\tau \in [a,t]}\left|\ddot{x}(\tau)\right|
\frac{\exp((1-\alpha)^2+1-\alpha)}{\Gamma(2-\alpha)(1-\alpha)K^{1-\alpha}}(t-a)^{2-\alpha}.
\end{equation*}
See \cite{Almeida} for proofs and other details.
\section{Problem Statement and the Approximation Method}
\label{sec:3}
The fractional optimal control problem that we consider here is the following one.
Let $\alpha\in(0,1)$ be the fixed fractional order, and consider
two differentiable functions $L$ and $f$ with domain
$[a,b]\times \mathbb{R}^2$. Minimize the functional
$$
J(x,u)=\int_a^b L(t,x(t),u(t))\,dt
$$
subject to the fractional dynamic constraint
$$
M \dot{x}(t) + N{_a^CD_t^\a} x(t)
= f\left(t,x(t),u(t)\right), \quad t\in[a,b],
$$
and the initial condition
$$
x(a)=x_a,
$$
where $(M,N)\not=(0,0)$ and $x_a$ is a fixed real number.
Two situations are considered: $x(b)$ fixed or free.
Sufficient and necessary conditions to obtain solutions
for this problem were studied in \cite{pooseh:press}.
Here we proceed with a different approach. First, we replace
the operator ${_a^CD_t^\a} x(t)$ with the approximation given in \eqref{approximation}.
With relation \eqref{relation} we get
\begin{multline*}
M \dot{x}(t) + N\Bigg[A(t-a)^{-\alpha}x(t)+B(t-a)^{1-\alpha}\dot{x}(t)\\
-\sum_{p=2}^{K}C_p(t-a)^{1-p-\alpha}V_p(t)-\frac{x_a(t-a)^{-\alpha}}{\Gamma(1-a)}\Bigg]
= f\left(t,x(t),u(t)\right),
\end{multline*}
where, for simplicity,
$$
A=A(\alpha,K)\, , \quad B= A(\alpha,K) \, \quad \mbox{and} \quad C_p=C(\alpha,p).
$$
Thus, one has
$$
\dot{x}(t)
=\frac{f(t,x(t),u(t))-NA(t-a)^{-\alpha}x(t)
+\displaystyle\sum_{p=2}^K NC_p(t-a)^{1-p-\alpha}V_p(t)
+\displaystyle\frac{Nx_a(t-a)^{-\alpha}}{\Gamma(1-a)}}{M+NB(t-a)^{1-\alpha}}.
$$
Define the vector
$$
\overline V(t)=\left(V_2(t),V_3(t),\ldots,V_K(t)\right)
$$
and the new function
$$
F(t,x,\overline V,u)
=\frac{f(t,x,u)-NA(t-a)^{-\alpha}x
+\displaystyle\sum_{p=2}^K NC_p(t-a)^{1-p-\alpha}V_p
+\displaystyle\frac{Nx_a(t-a)^{-\alpha}}{\Gamma(1-a)}}{M+NB(t-a)^{1-\alpha}}.
$$
We construct a new optimal control problem: minimize the functional
\begin{equation}
\label{new1}
\overline{J}(x,\overline V,u)=\int_a^b L(t,x(t),u(t))\,dt
\end{equation}
\label{new2}
subject to the dynamic constraints
\begin{equation}
\begin{cases}
\dot{x}(t)=F\left(t,x(t),\overline V(t),u(t)\right),\\[0.25cm]
\dot{V}_p(t)=(1-p)(t-a)^{p-2}x(t), \quad p=2,\ldots,K,
\end{cases}
\end{equation}
and the initial conditions
\begin{equation}
\label{new3}
\begin{cases}
x(a)=x_a,\\
V_p(a)=0, \quad p=2,\ldots,K.
\end{cases}
\end{equation}
To solve the problem \eqref{new1}--\eqref{new3},
one can consider the Hamiltonian function
$$
H(t,x,\overline V,\overline\lambda,u)=L(t,x,u)
+\lambda_1 F(t,x,\overline V,u)+\sum_{p=2}^K \lambda_p (1-p)(t-a)^{p-2}x,
$$
where $\overline\lambda$ denotes the Lagrange multipliers,
$$
\overline\lambda(t)=(\lambda_1(t),\lambda_2(t),\ldots,\lambda_K(t)).
$$
By the Pontryagin maximum principle \cite{Pontryagin},
to solve the problem one should solve the following system of ODEs:
$$
\begin{cases}
\displaystyle\frac{\partial H}{\partial u}=0,\\[0.25cm]
\displaystyle\frac{\partial H}{\partial \lambda_1}=\dot{x},\\[0.25cm]
\displaystyle\frac{\partial H}{\partial \lambda_p}=\dot{V}_p, \quad p=2,\ldots,K,\\[0.25cm]
\displaystyle\frac{\partial H}{\partial x}=-\dot{\lambda}_1,\\[0.25cm]
\displaystyle\frac{\partial H}{\partial V_p}=-\dot{\lambda}_p,\quad p=2,\ldots,K,
\end{cases}
$$
subject to the boundary conditions
$$
\left\{ \begin{array}{l}
x(a)=x_a,\\
V_p(a)=0, \quad p=2,\ldots,K,\\
\lambda_p(b)=0, \quad p=1,\ldots,K,
\end{array}\right.
$$
if $x(b)$ is free, or
$$
\left\{ \begin{array}{l}
x(a)=x_a,\\
V_p(a)=0, \quad p=2,\ldots,K,\\
\end{array}\right.
$$
otherwise. Instead of this indirect approach, we apply here a direct method,
based on an Euler discretization, to obtain a finite-dimensional approximation
of the continuous problem \eqref{new1}--\eqref{new3}.
We summarize briefly the method. Consider a finite grid
$$
a=t_0<t_1<\cdots<t_n=b
$$
where, for simplicity, $t_{i+1}-t_{i}:=\Delta t$ is assumed constant
for all $i\in\{0,\ldots,n-1\}$. Each dynamic constraint is approximated by
$$
x_{i+1}=x_i+\Delta t F(t_i,x_i,\overline V_i,u_i)
$$
and
$$
V_{p,i+1}=V_{p,i}+\Delta t (1-p)(t_i-a)^{p-2}x_i,
\quad p=2,\ldots,K,
$$
where
$$
x_i=x(t_i), \quad u_i=u(t_i), \quad \overline V_i
=(V_2(t_i),V_3(t_i),\ldots,V_K(t_i))
\quad \mbox{and} \quad V_{p,i}=V_p(t_i).
$$
The integral
$$
\int_a^b L\left(t,x(t),u(t)\right)\,dt
$$
in replaced by the Riemann sum
$$
\Delta t \sum_{i=0}^{n-1} L\left(t_i,x_i,u_i\right).
$$
The finite version of problem \eqref{new1}--\eqref{new3}
is the following one: minimize
\begin{equation*}
\Delta t \sum_{i=0}^{n-1} L\left(t_i,x_i,u_i\right)
\end{equation*}
subject to the dynamic constraints
\begin{equation*}
\begin{cases}
x_{i+1}=x_i+\Delta t F\left(t_i,x_i,\overline V_i,u_i\right),\\[0.25cm]
V_{p,i+1}=V_{p,i}+\Delta t (1-p)(t_i-a)^{p-2}x_i, \quad p=2,\ldots,K,
\end{cases}
\end{equation*}
and the initial conditions
\begin{equation*}
\begin{cases}
x_0=x_a,\\
V_{p,0}=0, \quad p=2,\ldots,K
\end{cases}
\end{equation*}
($x(b)$ fixed or free). In the next section we illustrate the method with an example.
\section{An Illustrative Example}
\label{sec:4}
In this section we exemplify the procedure of Section~\ref{sec:3} with a concrete example.
To start, we recall the Caputo fractional derivative of a power function
(cf. Property~2.16 of \cite{Kilbas}). Let $\alpha>0$ be arbitrary, $\beta>n$
with $n$ given by \eqref{def:n}, and define $x(t)=(t-a)^{\beta-1}$. Then,
$$
{_a^CD_t^\a} x(t) = \frac{\Gamma(\beta)}{\Gamma(\beta-\alpha)}(t-a)^{\beta-\alpha-1}.
$$
Assume that we wish to minimize the functional
\begin{equation}
\label{ex1}
J(x,u)=\int_0^1 (u^2(t)-4x(t))^2\,dt
\end{equation}
subject to the dynamic constraint
\begin{equation}
\label{eq:dc:ex}
\dot{x}(t) +{_0^CD_t^{0.5}} x(t)
=u(t)+\frac{2}{\Gamma(2.5)}t^{1.5}, \quad t\in[0,1],
\end{equation}
and the boundary conditions
\begin{equation}
\label{eq:bc:ex}
x(0)=0 \quad \mbox{and} \quad x(1)=1.
\end{equation}
The solution is the pair
\begin{equation}
\label{eq:sol:ex}
(\overline{x}(t),\overline{u}(t))=(t^2,2t).
\end{equation}
Indeed, \eqref{eq:sol:ex} satisfies both constraints \eqref{eq:dc:ex} and \eqref{eq:bc:ex}
with $J(\overline{x},\overline{u}) = 0$ while functional \eqref{ex1} is non-negative:
$J(x,u) \ge 0$ for all admissible pairs $(x,u)$. After replacing the fractional
derivative by the appropriate approximation, and fixing $K\geq2$,
we get the following problem: minimize the functional
\begin{equation}
\label{eq:J:ap}
\overline{J}(x,\overline V,u)=\int_0^1 (u^2(t)-4x(t))^2\,dt
\end{equation}
subject to the dynamic constraints
\begin{equation}
\label{eq:dc:ap}
\begin{cases}
\dot{x}(t)=\displaystyle \frac{u(t)+\frac{2}{\Gamma(2.5)}t^{1.5}
-At^{-0.5}x(t)+\sum_{p=2}^K C_pt^{0.5-p}V_p(t)}{1+Bt^{0.5}},\\[0.25cm]
\dot{V}_p(t)=(1-p)t^{p-2}x(t), \quad p=2,\ldots,K,
\end{cases}
\end{equation}
and the boundary conditions
\begin{equation}
\label{eq:bc:ap}
\begin{cases}
x(0)=0,\\
V_p(0)=0, \quad p=2,\ldots,K,\\
x(1)=1.\\
\end{cases}
\end{equation}
We apply a direct method to the previous problem, also called a ``first discretize then optimize method''.
More precisely, the objective functional \eqref{eq:J:ap} and the system of ODEs \eqref{eq:bc:ap}
were discretized by a simple Euler method with fixed step size, as explained in Section~\ref{sec:3}.
Then we solved the resulting nonlinear programming problem by using the AMPL
modeling language for mathematical programming \cite{AMPL}
in connection with IPOPT of W\"achter and Biegler \cite{IPOPT}.
In Figure~\ref{plotX} we show the results for the state function $x$
and in Figure~\ref{plotU} we show the results for the control function $u$,
where we have used $K=3$ and $n=100$.
\begin{figure}[!htb]
\centering
\subfloat[\footnotesize{State function $x(t)$.}]{\label{plotX}
\includegraphics[scale=0.30,angle=270]{plotX.eps}}
\subfloat[\footnotesize{Control function $u(t)$.}]{\label{plotU}
\includegraphics[scale=0.30,angle=270]{plotU.eps}}
\caption{Exact solution to \eqref{ex1}--\eqref{eq:bc:ex} (continuous line)
versus numerical approximation to solution of \eqref{eq:J:ap}--\eqref{eq:bc:ap}
with $K=3$ and $n=100$ (dot line).}
\label{fig:X:U}
\end{figure}
\section*{Acknowledgments}
This article was supported by Portuguese funds through the
\emph{Center for Research and Development in Mathematics and Applications} (CIDMA),
and \emph{The Portuguese Foundation for Science and Technology} (FCT),
within project PEst-OE/MAT/UI4106/2014. Torres was also supported
by the FCT project PTDC/EEI-AUT/1450/2012,
co-financed by FEDER under POFC-QREN
with COMPETE reference FCOMP-01-0124-FEDER-028894.
| {
"timestamp": "2014-03-21T01:07:04",
"yymm": "1403",
"arxiv_id": "1403.5060",
"language": "en",
"url": "https://arxiv.org/abs/1403.5060",
"abstract": "We present a method to solve fractional optimal control problems, where the dynamic depends on integer and Caputo fractional derivatives. Our approach consists to approximate the initial fractional order problem with a new one that involves integer derivatives only. The latter problem is then discretized, by application of finite differences, and solved numerically. We illustrate the effectiveness of the procedure with an example.",
"subjects": "Optimization and Control (math.OC)",
"title": "A Discrete Method to Solve Fractional Optimal Control Problems",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754492759499,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.707510396912629
} |
https://arxiv.org/abs/1007.4018 | Expressiveness and Closure Properties for Quantitative Languages | Weighted automata are nondeterministic automata with numerical weights on transitions. They can define quantitative languages~$L$ that assign to each word~$w$ a real number~$L(w)$. In the case of infinite words, the value of a run is naturally computed as the maximum, limsup, liminf, limit-average, or discounted-sum of the transition weights. The value of a word $w$ is the supremum of the values of the runs over $w$. We study expressiveness and closure questions about these quantitative languages. We first show that the set of words with value greater than a threshold can be non-$\omega$-regular for deterministic limit-average and discounted-sum automata, while this set is always $\omega$-regular when the threshold is isolated (i.e., some neighborhood around the threshold contains no word). In the latter case, we prove that the $\omega$-regular language is robust against small perturbations of the transition weights. We next consider automata with transition weights Weighted automata are nondeterministic automata with numerical weights ontransitions. They can define quantitative languages~$L$ that assign to eachword~$w$ a real number~$L(w)$. In the case of infinite words, the value of arun is naturally computed as the maximum, limsup, liminf, limit-average, ordiscounted-sum of the transition weights. The value of a word $w$ is thesupremum of the values of the runs over $w$. We study expressiveness andclosure questions about these quantitative languages. We first show that the set of words with value greater than a threshold canbe non-$\omega$-regular for deterministic limit-average and discounted-sumautomata, while this set is always $\omega$-regular when the threshold isisolated (i.e., some neighborhood around the threshold contains no word). Inthe latter case, we prove that the $\omega$-regular language is robust againstsmall perturbations of the transition weights. We next consider automata with transition weights $0$ or $1$ and show thatthey are as expressive as general weighted automata in the limit-average case,but not in the discounted-sum case. Third, for quantitative languages $L_1$ and~$L_2$, we consider the operations$\max(L_1,L_2)$, $\min(L_1,L_2)$, and $1-L_1$, which generalize the booleanoperations on languages, as well as the sum $L_1 + L_2$. We establish theclosure properties of all classes of quantitative languages with respect tothese four operations.$ or $ and show that they are as expressive as general weighted automata in the limit-average case, but not in the discounted-sum case. Third, for quantitative languages $L_1$ and~$L_2$, we consider the operations $\max(L_1,L_2)$, $\min(L_1,L_2)$, and -L_1$, which generalize the boolean operations on languages, as well as the sum $L_1 + L_2$. We establish the closure properties of all classes of quantitative languages with respect to these four operations. | \section{Introduction}
A boolean language $L$ can be viewed as a function that assigns to each word $w$ a boolean value,
namely, $L(w) = 1$ if the word $w$ belongs to the language, and $L(w) = 0$ otherwise.
Boolean languages model the computations of reactive programs. The verification
problem ``does the program~$A$ satisfy the specification~$B$?''
then reduces to the language-inclusion problem ``is $L_A \subseteq L_B$?'',
or equivalently, ``is $L_{A}(w)\leq L_{B}(w)$ for all words~$w$?'',
where $L_A$ represents the behaviors of the program, and $L_B$ contains
all behaviors allowed by the specification. When boolean languages are defined
by finite automata, this framework
is called the \emph{automata-theoretic approach} to model-checking~\cite{VardiW86}.
In a natural generalization of this framework, a cost function
assigns to each word a real number instead of a boolean value.
For instance, the value of a word (or behavior) can be interpreted as the amount of some resource
(e.g., memory consumption, or power consumption) that the program needs to produce it,
and a specification may assign a maximal amount of available resource to each
behavior, or bound the long-run average available use of the resource.
Weighted automata over semirings (i.e., finite automata with transition weights
in a semiring structure) have been used to define cost functions,
called formal power series for finite words~\cite{Wautomata,KuichS86}
and $\omega$-series for infinite words~\cite{CulikK94,DrosteK03,EsikK04}.
In~\cite{CDH08}, we study new classes of cost functions using
operations over rational numbers that do not form a semiring.
We call them \emph{quantitative languages}.
We set the value of a (finite or infinite) word $w$ as the supremum value of all runs over~$w$
(if the automaton is nondeterministic, then there may be many runs over~$w$),
and the value of a run $r$ is a function of the (finite or infinite) sequence
of weights that appear along~$r$. We consider several
functions, such as $\mathsf{Max}$
and $\mathsf{Sum}$ of weights for finite runs,
and $\mathsf{Sup}$, $\mathsf{LimSup}$, $\mathsf{LimInf}$,
limit-average, and discounted sum of weights for infinite runs.
For example, peak power consumption can be modeled as the maximum of a sequence
of weights representing power usage; energy use can be modeled as the sum;
average response time as the limit-average \cite{CCHK+05,CAHS03}.
Quantitative languages can also be used to specify and verify reliability
requirements: if a special symbol $\bot$ is used to denote failure and has
weight~$1$, while the other symbols have weight $0$, one can use a limit-average
automaton to specify a bound on the rate of failure in the long run~\cite{CGHIKPS08}.
The discounted sum can be used to specify that failures happening
later are less important than those happening soon~\cite{AHM03}.
The \emph{quantitative language-inclusion problem} ``given two automata~$A$ and~$B$,
is $L_{A}(w)\leq L_{B}(w)$ for all words~$w$?'' can then be used to check, say, if for each behavior,
the peak power used by the system lies below the bound given by the specification;
or if for each behavior, the long-run average response time of the system
lies below the specified average response requirements.
In~\cite{CDH08}, we showed that the quantitative language-inclusion problem
is PSPACE-complete for $\mathsf{Sup}$-, $\mathsf{LimSup}$-, and $\mathsf{LimInf}$-automata, while
the decidability is unknown for (nondeterministic) limit-average and discounted-sum automata.
We also compared the expressive power of the different classes of quantitative languages
and showed that
nondeterministic automata are strictly more expressive than deterministic automata in the
limit-average and discounted-sum cases.
In this paper, we investigate alternative ways of comparing the \emph{expressive power}
of weighted automata. First, we consider the cut-point languages of weighted
automata, a notion borrowed from the theory of probabilistic automata~\cite{Rabin63}.
Given a threshold $\eta \in {\mathbb R}$, the cut-point language of a quantitative
language $L$ is the set of all words $w$ with value $L(w) \geq \eta$, thus
a boolean language.
We show that deterministic limit-average and
discounted-sum automata can define cut-point languages that are not
$\omega$-regular.
Note that there also exist $\omega$-regular languages that cannot be expressed
as a cut-point language of a limit-average or discounted-sum automaton~\cite{CDH08}.
Then, we consider the special case where the threshold~$\eta$ is isolated, meaning
that there is no word with a value in the neighborhood of~$\eta$. We argue that
isolated cut-point languages are robust, by showing that they
remain unchanged under small perturbations of the transition weights.
Furthermore, we show that every discounted-sum automaton with isolated cut-point defines
an $\omega$-regular language, and the same holds for deterministic limit-average
automata. This question is open for nondeterministic limit-average automata.
Finally, we consider a boolean counterpart of limit-average and discounted-sum
automata in which all transitions have weight~$0$ or~$1$.
Of special interest is a proof that for every limit-average automaton with rational weights in
the interval $[0,1]$ there is an equivalent limit-average automaton with boolean weights.
Therefore, the restriction to boolean weights does not change the class of quantitative
languages definable by limit-average automata; on the other hand, we show that it reduces
the expressive power of discounted-sum automata.
In the second part of this paper, we study the \emph{closure properties} of quantitative languages.
It is natural and convenient to decompose a specification or a design
into several components, and to apply composition operators
to obtain a complete specification.
We consider a natural generalization of the classical operations
of union, intersection, and complement of boolean languages. We define the \emph{maximum},
\emph{minimum}, and \emph{sum} of two quantitative languages $L_1$ and $L_2$ as the quantitative language
that assigns $\max(L_1(w),L_2(w))$, $\min(L_1(w),L_2(w))$, and $L_1(w) + L_2(w)$
to each word $w$.
The \emph{complement}~$L^c$ of a quantitative language~$L$ is defined by $L^c(w) = 1-L(w)$
for all words $w$.\footnote{One can define $L^c(w) = k-L(w)$ for any rational constant $k$ without changing
the results of this paper.}
The sum is a natural way of composing two automata
if the weights represent costs (e.g., energy consumption).
We give other examples in Section~\ref{sec:definitions} to illustrate
the composition operators and the use of quantitative languages
as a specification framework.
\begin{table}
\begin{center}
\begin{tabular}{|l|*{4}{c|}}
\hline
& max. & min. & comp. & sum \\
\hline
$\mathsf{Max}$ & \raisebox{0.2em}{$\sqrt{}$} & \raisebox{0.2em}{$\sqrt{}$} & $\times$ & \raisebox{0.2em}{$\sqrt{}$} \\
\hline
${\sf Last}$ & \raisebox{0.2em}{$\sqrt{}$} & \raisebox{0.2em}{$\sqrt{}$} & \raisebox{0.2em}{$\sqrt{}$} & \raisebox{0.2em}{$\sqrt{}$} \\
\hline
Det. $\mathsf{Sum}$ & $\times$ & \raisebox{0.2em}{$\sqrt{}$} & \raisebox{0.2em}{$\sqrt{}$} & \raisebox{0.2em}{$\sqrt{}$} \\
\hline
Nondet. $\mathsf{Sum}$ & \raisebox{0.2em}{$\sqrt{}$} & $\times$ & $\times$ & \raisebox{0.2em}{$\sqrt{}$} \\
\hline
\multicolumn{5}{c}{(a) Finite words}
\end{tabular}\hfill
\begin{tabular}{|l|*{4}{c|}}
\hline
& max. & min. & comp. & sum \\
\hline
{\sc \slopefrac{N}{D}Sup} & \raisebox{0.2em}{$\sqrt{}$} & \raisebox{0.2em}{$\sqrt{}$} & $\times$ & \raisebox{0.2em}{$\sqrt{}$} \\
\hline
{\sc \slopefrac{N}{D}Linf} & \raisebox{0.2em}{$\sqrt{}$} & \raisebox{0.2em}{$\sqrt{}$} & $\times$ & \raisebox{0.2em}{$\sqrt{}$} \\
\hline
{\sc DLsup} & \raisebox{0.2em}{$\sqrt{}$} & \raisebox{0.2em}{$\sqrt{}$} & $\times$ & \raisebox{0.2em}{$\sqrt{}$} \\
\hline
{\sc NLsup} & \raisebox{0.2em}{$\sqrt{}$} & \raisebox{0.2em}{$\sqrt{}$} & \raisebox{0.2em}{$\sqrt{}$} & \raisebox{0.2em}{$\sqrt{}$} \\
\hline
{\sc DLavg} & $\times$ & $\times$ & $\times$ & $\times$ \\
\hline
{\sc NLavg} & \raisebox{0.2em}{$\sqrt{}$} & $\times$ & $\times$ & $\times$ \\
\hline
{\sc DDisc} & $\times$ & $\times$ & \raisebox{0.2em}{$\sqrt{}$} & \raisebox{0.2em}{$\sqrt{}$} \\
\hline
{\sc NDisc} & \raisebox{0.2em}{$\sqrt{}$} & $\times$ & $\times$ & \raisebox{0.2em}{$\sqrt{}$} \\
\hline
\multicolumn{5}{c}{(b) Infinite words}
\end{tabular}\hfill
\end{center}
\caption{Closure properties. The meaning of the acronyms is described on p.\pageref{page:notation}.\label{tab:closure-properties}}
\end{table}
We give a complete picture of the closure properties of
the various classes of quantitative languages (over finite and infinite words) under maximum,
minimum, complement and sum (see Table~\ref{tab:closure-properties}).
For instance, (non)deterministic limit-average automata are not closed under sum and complement,
while nondeterministic discounted-sum automata are closed under sum but not under complement.
All other classes of weighted automata are closed under sum.
For infinite words, the closure properties of $\mathsf{Sup}$-, $\mathsf{LimSup}$-, and $\mathsf{LimInf}$-automata
are obtained as a direct extension of the results for boolean finite automata,
while for limit-average and discounted-sum automata, the proofs require the analysis of the structure
of the automata cycles and properties of the solutions of polynomials with rational coefficients.
Note that the quantitative language-inclusion problem
``is $L_{A}(w)\leq L_{B}(w)$ for all words~$w$?'' reduces to
closure under sum and complement, because it is equivalent
to the question of the non-existence of a word $w$ such that $L_{A}(w) + L^c_{B}(w) > 1$,
an \emph{emptiness} question which is decidable for all classes of quantitative languages~\cite{CDH08}.
Also note that deterministic limit-average and discounted-sum automata are not
closed under maximum, which implies
that nondeterministic automata are strictly more expressive in these cases
(because the maximum can be obtained by an initial nondeterministic choice).
\smallskip\noindent{\it Related work.} Functions such as limit-average (or mean-payoff) and discounted
sum have been studied extensively in the branching-time context of game theory~\cite{Sha53,EM79,Condon92,ZP96,CAHS03}.
It is therefore natural to use the same functions in the linear-time context of languages
and automata.
Weighted automata with discounted sum have been considered in~\cite{DrosteR07},
with multiple discount factors and a boolean acceptance condition (Muller or B\"uchi);
they are shown to be equivalent to a weighted monadic second-order logic with
discounting. Several other works have considered quantitative generalizations of
languages, over finite words~\cite{DrosteGastin07}, over trees~\cite{DrosteKR08},
or using finite lattices~\cite{GurfinkelC03}, but none of these works
has addressed the expressiveness questions and closure properties for quantitative
languages that are studied here.
The lattice automata of~\cite{KL07} map finite words to values from a finite
lattice. The lattice automata with B\"uchi condition are analogous to our
$\mathsf{LimSup}$ automata, and their closure properties are established there.
However, the other classes of quantitative automata ($\mathsf{Sum}$, limit-average,
discounted-sum) are not studied there as they cannot be defined using lattice
operations and finite lattices.
\section{Quantitative Languages}\label{sec:definitions}
A \emph{quantitative language} $L$ over a finite alphabet $\Sigma$
is either a mapping $L: \Sigma^{+} \to {\mathbb R}$ or a mapping $L: \Sigma^{\omega} \to {\mathbb R}$,
where ${\mathbb R}$ is the set of real numbers.
\paragraph{\bf Weighted automata.}
A \emph{weighted automaton} is a tuple $A=\tuple{Q,q_I,\Sigma,\delta,\gamma}$,
where
\begin{enumerate}[$\bullet$]
\item $Q$ is a finite set of states, $q_I \in Q$ is the initial state, and $\Sigma$ is a finite alphabet;
\item $\delta \subseteq Q \times \Sigma \times Q$ is a finite set of labelled transitions.
We assume that $\delta$ is \emph{total}, i.e., for all
$q \in Q$ and $\sigma \in \Sigma$, there exists $q'$ such that
$(q,\sigma,q') \in \delta$;
\item $\gamma: \delta \to {\mathbb Q}$ is a \emph{weight} function, where ${\mathbb Q}$ is the
set of rational numbers. We assume that rational numbers are encoded as pairs of
integers in binary.
\end{enumerate}
\noindent We say that $A$ is \emph{deterministic} if for all
$q \in Q$ and $\sigma \in \Sigma$, there exists $(q,\sigma,q') \in \delta$
for exactly one $q' \in Q$. We sometimes call automata \emph{nondeterministic}
to emphasize that they are not necessarily deterministic.
A \emph{run} of $A$ over a finite (resp. infinite) word $w=\sigma_1 \sigma_2 \dots$
is a finite (resp. infinite) sequence $r = q_0 \sigma_1 q_1 \sigma_2 \dots $
of states and letters such that
\begin{compressEnum}
\stepcounter{compressEnum}{(\thecompressEnum) } $q_0 = q_I$, and
\stepcounter{compressEnum}{(\thecompressEnum) } $(q_i,\sigma_{i+1},q_{i+1}) \in \delta$ for all $0 \leq i < \abs{w}$.
\end{compressEnum}
We denote by $\gamma(r) = v_0 v_1 \dots$ the sequence of weights that occur in~$r$
where $v_i = \gamma(q_i,\sigma_{i+1},q_{i+1})$ for all $0 \leq i < \abs{w}$.
Given a \emph{value function} $\mathsf{Val}: {\mathbb Q}^+ \to {\mathbb R}$ (resp. $\mathsf{Val}: {\mathbb Q}^{\omega} \to {\mathbb R}$),
we say that the $\mathsf{Val}$-automaton~$A$ defines the quantitative language $L_A$ such
that for all $w \in \Sigma^{+}$ (resp. $w \in \Sigma^{\omega}$):
$$L_A(w) = \sup \{\mathsf{Val}(\gamma(r)) \mid r \text{ is a run of } A \text{ over } w\}.$$
We assume that $\mathsf{Val}(v)$ is bounded when the numbers in $v$ are taken from a finite set (namely,
the set of weights in $A$), and since weighted automata are total, every word has
at least one run and thus $L_A(w)$ is not infinite.
We consider the following value functions to define quantitative languages (they all satisfy
the boundedness assumption above).
Given a finite sequence $v= v_1 \dots v_n$ of rational numbers, define
\begin{enumerate}[$\bullet$]
\item $\mathsf{Max}(v) = \max \{v_i \mid 1 \leq i \leq n\}$;\medskip
\item ${\sf Last}(v) = v_n$;\medskip
\item $\mathsf{Sum}(v) = \displaystyle\sum_{i=1}^{n} v_i$;
\end{enumerate}
Given an infinite sequence $v=v_0 v_1 \dots$ of rational numbers, define
\begin{enumerate}[$\bullet$]
\item $\mathsf{Sup}(v) = \sup \{v_n \mid n \geq 0\}$;\medskip
\item $\mathsf{LimSup}(v) = \displaystyle\limsup_{n\to\infty} \ v_n = \lim_{n\to\infty} \sup \{v_i \mid i \geq n\}$;\medskip
\item $\mathsf{LimInf}(v) = \displaystyle\liminf_{n\to\infty} \ v_n = \lim_{n\to\infty} \inf \{v_i \mid i \geq n\}$;\medskip
\item $\mathsf{LimAvg}(v) = \displaystyle\liminf_{n\to\infty} \ \frac{1}{n} \cdot \sum_{i=0}^{n-1} v_i$;\medskip
\item for $0 < \lambda < 1$, $\mathsf{Disc}_{\lambda}(v) = \displaystyle \sum_{i=0}^{\infty} \lambda^i \cdot v_i$;
\end{enumerate}
Intuitively for a sequence $v=v_0 v_1 \dots$ of rational numbers from the finite set $V$,
the $\mathsf{Sup}$ function chooses the maximal number that appear in $v$;
the $\mathsf{LimSup}$ function chooses the maximal number that appear infinitely often in $v$;
the $\mathsf{LimInf}$ function chooses the minimal number that appear infinitely often in $v$;
the $\mathsf{LimAvg}$ functions gives the long-run average of the numbers in $v$;
and the $\mathsf{Disc}_{\lambda}$ gives the discounted sum of the numbers in $v$.
Note that $\mathsf{LimAvg}(v)$ is defined using $\liminf$ and is therefore well-defined;
all results of this paper hold also if the limit-average of $v$ is
defined instead as
$\limsup_{n\to\infty} \ \frac{1}{n}\cdot \sum_{i=0}^{n-1} v_i$.
One could also consider the value function $\inf \{v_n \mid n \geq 0\}$
and obtain results analogous to the $\mathsf{Sup}$ value function.
Note that the classical finite-word acceptance condition of finite automata (defining regular languages)
can be encoded by ${\sf Last}$-automata with weights in $\{0,1\}$, while
B\"uchi and coB\"uchi automata are special cases of respectively
$\mathsf{LimSup}$- and $\mathsf{LimInf}$-automata, with weights in $\{0,1\}$.
The class of languages defined by nondeterministic B\"uchi automata is called
$\omega$-regular.
\smallskip\noindent{\em Significance of value functions.}
The value functions provide natural generalizations of the classical boolean
languages, they are complete for different levels of the Borel hierarchy, and
they have been well studied in the context of game theory.
\begin{enumerate}[(1)]
\item The $\mathsf{Sup}$ value function is the natural quantitative generalization of
the reachability condition and is complete for the first level of the Borel
hierarchy ($\Sigma_1$ complete).
\item The $\mathsf{LimSup}$ and $\mathsf{LimInf}$ objectives are the natural quantitative
generalizations of the classical B\"uchi and coB\"uchi conditions.
Moreover, the $\mathsf{LimSup}$ and $\mathsf{LimInf}$ objectives are complete for the second
level of the Borel hierarchy, and hence important and canonical quantitative
functions ($\mathsf{LimSup}$ and $\mathsf{LimInf}$ objectives are
$\Pi_2$ and $\Sigma_2$ complete, respectively) (see~\cite{Wadge,MannaPnueliBook}
for details related to completeness and reducibility of objectives
in the Borel hierarchy).
\item The $\mathsf{LimAvg}$ and $\mathsf{Disc}_{\lambda}$ value functions have been studied in
many different contexts in game theory.
Discounted functions on graph games were introduced in the seminal work of
Shapley~\cite{Sha53}, and have been extensively studied in economics.
Discounted conditions have also been studied for discounting the future in
systems theory~\cite{AHM03}.
The $\mathsf{LimAvg}$ function has also been studied extensively
in the context of games on graphs: the works of Everett~\cite{Eve57},
Liggett-Lippman~\cite{LigLip69},
Hopfman-Karp~\cite{HofKar66}, Ehrenfeucht-Mycielski~\cite{EM79},
Mertens-Neyman~\cite{MN81}, Zwick-Paterson~\cite{ZP96} have studied different
classes of games with $\mathsf{LimAvg}$ objective.
Also see the books~\cite{FV97,Puterman} for applications of discounted
and limit-average value functions in the context of games on graphs.
Moreover, the $\mathsf{LimAvg}$ value function is complete for the third level of the
Borel hierarchy ($\Pi_3$-complete)~\cite{ChaTCS07}.
\end{enumerate}
Hence the value functions considered are classical, canonical, and well-studied
in the bran\-ching-time framework of games on graphs, and we study them in the
linear-time framework of weighted automata.
\paragraph{\bf Notation.} \label{page:notation}
Classes of weighted automata over infinite words are denoted with acronyms of the form
$xy$ where $x$ is either {\sc N}(ondeterministic), {\sc D}(eterministic),
or {\sc \slopefrac{N}{D}}\/ (when deterministic and nondeterministic automata have the same expressiveness),
and $y$ is one of the following: {\sc Sup}, {\sc Lsup}(LimSup), {\sc Linf}(LimInf), {\sc Lavg}(LimAvg), or {\sc Disc}.
For B\"uchi and coB\"uchi condition, we use {\sc BW} and {\sc CW} respectively.
\paragraph{\bf Reducibility.}
A class $\mathcal{C}$ of weighted automata is \emph{reducible}
to a class $\mathcal{C}'$ of weighted automata if for every $A \in \mathcal{C}$ there exists
$A' \in \mathcal{C}'$ such that $L_A=L_{A'}$, i.e., $L_{A}(w)=L_{A'}(w)$ for all (finite or infinite) words $w$.
In particular, a class of weighted automata \emph{can be determinized}
if it is reducible to its deterministic counterpart.
Reducibility relationships for (non)deterministic weighted automata
are given in~\cite{CDH08}.
\begin{figure*}[!tb]
\begin{center}
\subfigure[Limit-average automaton $A$. \label{fig:left}]{%
\begin{tikzpicture}[node distance=1.8cm,auto,shorten >=1 pt,>=latex']
\node[rdbox,initial,initial text={}] (0) at (150:2) {OFF} ;
\node[rdbox] (1) at ( 30:2) {ON} ;
\node[rdbox] (2) at (270:2) {SLOW} ;
\draw[->] (0) edge[loop above,out=120, in=60,looseness=8]
node[above] {off, 0}(0);
\draw[->] (1) edge[loop above,out=120, in=60,looseness=8]
node[above] {on, 2}(1);
\draw[->] (2) edge[loop below,out=300, in=240,looseness=8]
node[below] {slow, 1}(2);
\draw[->] (0) edge[bend left=20] node{on, 10} (1);
\draw[->] (1) edge[bend left= 0] node{off, 10} (0);
\draw[->] (0) edge[bend left= 0] node[sloped]{\mlap{slow, 5\quad}} (2);
\draw[->] (2) edge[bend left=20] node[sloped]{\mlap{\quad off, 5}} (0);
\draw[->] (1) edge[bend left=20] node[sloped]{\mlap{slow, 5\quad}} (2);
\draw[->] (2) edge[bend left= 0] node[sloped]{\mlap{\quad on, 5}} (1);
\end{tikzpicture}\quad
}
\subfigure[Limit-average automaton $B$. \label{fig:right}]{%
\quad\begin{tikzpicture}[node distance=1.8cm,auto,shorten >=1 pt,>=latex']
\node[rdbox,initial,initial text={}] (0) at (0,0) {OFF} ;
\node[rdbox] (1) at (4,0) {ON} ;
\node at(2,-2.5){};
\draw[->] (0) edge[loop above,out=120, in=60,looseness=8]
node[above] {off, 0}(0);
\draw[->] (1) edge[loop above,out=120, in=60,looseness=8]
node[above] {on, 2}(1);
\draw[->] (0) edge[bend left=10] node{on, slow, 10} (1);
\draw[->] (1) edge[bend left=10] node{off, slow, 10} (0);
\end{tikzpicture}
}
\end{center}
\caption{Specifications for the energy consumption of a motor:
$A$ refines $B$, i.e., $L_A \leq L_{B}$.\label{fig:motor-spec}}
\end{figure*}
\paragraph{\bf Composition.}
Given two quantitative languages $L$ and $L'$ over $\Sigma$,
and a rational number $c$,
we denote by $\max(L,L')$ (resp. $\min(L,L')$, $L+L'$, $c+L$, and $cL$)
the quantitative language that assigns $\max\{L(w),L'(w)\}$
(resp. $\min\{L(w),L'(w)\}$, $L(w) + L'(w)$, $c+L(w)$, and $c\cdot L(w)$) to each word
$w \in \Sigma^{+}$ (or $w \in \Sigma^{\omega}$). We say that $c+L$ is the \emph{shift by $c$}
of $L$ and that $cL$ is the \emph{scale by $c$} of $L$.
The language $1-L$ is called the \emph{complement} of $L$.
The $\max$, $\min$ and complement operators for quantitative languages
generalize respectively the union, intersection and complement
operator for boolean languages. For instance, De Morgan's laws hold
(the complement of the max of two languages is the min of their complement, etc.)
and complementing twice leave languages unchanged.
\smallskip\noindent{\it Example 1.} We consider a simple illustration of the use of limit-average automata
to model the energy consumption of a motor. The automaton $B$ in Figure~\ref{fig:right}
specifies the maximal energy consumption to maintain the motor on or off, and the
maximal consumption for a mode change. The specification abstracts away that a mode
change can occur smoothly with the $\mathit{slow}$ command. A refined specification $A$ is
given in Figure~\ref{fig:left} where the effect of slowing down is captured
by a third state. One can check that $L_{A}(w)\leq L_{B}(w)$
for all words~$w \in \{\mathit{on},\mathit{off},\mathit{slow}\}^\omega$.
Given two limit-average automata that model the energy consumption of
two different motors, one needs to define composition operations for
weighted automata to obtain the maximal, minimal, and sum of the average
energy consumption of the motors.
\smallskip\noindent{\it Example 2.} Consider an investment of 100~dollars
that can be made in two banks~$A_1$ and~$A_2$ as follows:
(a)~100~dollars to bank~$A_1$, (b)~100~dollars to bank $A_2$, or
(c)~50~dollars to bank~$A_1$ and 50~dollars to bank~$A_2$.
The banks can be either in a good state (denoted $G_1$, $G_2$)
or in a bad state (denoted $B_1$, $B_2$).
If it is in a good state, then~$A_1$ offers 8\% reward while~$A_2$ offers 6\% reward.
If it is in a bad state, then~$A_1$ offers 2\% reward while~$A_2$ offers 4\% reward.
The change of state is triggered by the input symbols~$b_1, b_2$ (from a good
to a bad state) and~$g_1, g_2$ (from a bad to a good state).
The rewards received earlier weight more than rewards received later
due to inflation represented by the discount factor.
The automata~$A_1$ and~$A_2$ in Figure~\ref{fig:bank-spec} specify the behavior
of the two banks for an investment of 100~dollars, where the input alphabet is
$\set{g_1,b_1} \times \set{g_2,b_2}$ (where the notation $(g_1,\cdot)$
represents the two letters $(g_1,g_2)$ and $(g_1,b_2)$, and similarly for the other symbols).
If 50~dollars are invested in each bank, then we obtain automata~$C_1$ and~$C_2$
from~$A_1$ and~$A_2$ where each reward is halved.
The combined automaton is obtained as the composition of~$C_1$ and~$C_2$
under the sum operator.
\begin{figure*}[!t]
\begin{center}
\subfigure[100 dollars invested in bank $A_1$. \label{fig:bank1}]{%
\begin{tikzpicture}[node distance=1.8cm,auto,shorten >=1 pt,>=latex']
\node[rdbox,initial,initial text={}] (0) at (0,0) {$G_1$} ;
\node[rdbox] (1) at (3,0) {$B_1$} ;
\draw[->] (0) edge[loop above,out=120, in=60,looseness=8]
node[above] {$(g_1,\cdot),8$}(0);
\draw[->] (1) edge[loop above,out=120, in=60,looseness=8]
node[above] {$(b_1,\cdot),2$}(1);
\draw[->] (0) edge[bend left=15] node{$(b_1,\cdot),2$} (1);
\draw[->] (1) edge[bend left=15] node{$(g_1,\cdot),8$} (0);
\end{tikzpicture}\quad
}
\subfigure[100 dollars invested in bank $A_2$. \label{fig:bank2}]{%
\quad\begin{tikzpicture}[node distance=1.8cm,auto,shorten >=1 pt,>=latex']
\node[rdbox,initial,initial text={}] (0) at (0,0) {$G_2$} ;
\node[rdbox] (1) at (3,0) {$B_2$} ;
\draw[->] (0) edge[loop above,out=120, in=60,looseness=8]
node[above] {$(\cdot,g_2),6$}(0);
\draw[->] (1) edge[loop above,out=120, in=60,looseness=8]
node[above] {$(\cdot,b_2),4$}(1);
\draw[->] (0) edge[bend left=15] node{$(\cdot,b_2),4$} (1);
\draw[->] (1) edge[bend left=15] node{$(\cdot,g_2),6$} (0);
\end{tikzpicture}
}
\end{center}
\caption{The discounted-sum automaton models of two banks.\label{fig:bank-spec}}
\end{figure*}
\section{Expressiveness Results}
The expressive power of weighted automata can be compared by mean of the
reducibility relation, saying that a class $\mathcal{C}$ of weighted automata is at least
as expressive as a class $\mathcal{C}'$ if every quantitative language definable
by some automaton in $\mathcal{C}'$ is also definable by some automaton in $\mathcal{C}$.
The comparison includes boolean languages, considering them as a special case of quantitative languages
of the form $L:\Sigma^{\omega} \to \{0,1\}$.
It was shown in~\cite{CDH08} that a wide variety of classes of quantitative languages
can be defined by the different types of weighted automata, depending on the
value function and whether they are deterministic or not. This contrasts with
the situation for boolean languages where most of the classes of automata define
$\omega$-regular languages.
In this section, we investigate alternative ways of comparing the expressive power
of weighted automata and of classical finite automata.
First, we use the cut-point languages of weighted automata to compare with
the class of $\omega$-regular languages, and then we use weighted automata
with boolean weights, i.e. all transitions have weight $0$ or $1$, to compare
with general weighted automata.
\subsection{Cut-point languages}
Let $L$ be a quantitative language over infinite words and let $\eta \in {\mathbb R}$ be a threshold.
The \emph{cut-point language} defined by $(L,\eta)$ is the (boolean) language
$$L^{\geq \eta} = \{w \in \Sigma^{\omega} \mid L(w) \geq \eta\}.$$
Cut-point languages for finite words are defined analogously. They
have been first defined for probabilistic automata~\cite{Rabin63},
then generalized to inverse image recognition for semiring automata over finite words (see e.g.~\cite{KuichS86,CortesM00}).
It is easy to see that the cut-point languages of $\mathsf{Max}$- and ${\sf Last}$-automata are regular
(they have the same acceptance condition as finite automata),
those of $\mathsf{Sum}$-automata are context-free (using a stack to simulate accumulated weights),
and those of $\mathsf{Sup}$-, $\mathsf{LimSup}$-, and $\mathsf{LimInf}$-automata are $\omega$-regular
(they have the same acceptance condition as B\"uchi and coB\"uchi automata).
We show that the classes of cut-point languages definable by (non)deterministic limit-average and
discounted-sum automata are incomparable with the $\omega$-regular languages.
One direction of this result follows from Theorem~\ref{theo:cut-point-language},
and the other direction follows from~\cite[Theorems 13 and 14]{CDH08} where
$\omega$-regular languages are given that are not definable as cut-point language
of nondeterministic limit-average and discounted-sum automata.
\begin{thm}\label{theo:cut-point-language}
There exist deterministic limit-average and discounted-sum automata
whose cut-point language is not $\omega$-regular.
\end{thm}
\proof
Consider the alphabet $\Sigma=\set{a,b}$, and consider the languages $L_1$
that assigns to each word its long-run average number of $a$'s,
and $L_2$ that assigns the discounted sum of $a$'s. Note that $L_1$
is definable by a deterministic limit-average automaton,
and $L_2$ by a deterministic discounted-sum automaton.
It was shown in~\cite{Cha-TCS} that the cut-point language
$L_1^{\geq 1}$ is complete for the third level of the Borel hierarchy,
and therefore is not $\omega$-regular.
We show that $L_2^{\geq 1}$ is not $\omega$-regular.
Given a finite word $w \in \Sigma^*$, let $v_a(w) = \sum_{i\mid w_i = a} \lambda^{i-1}$
be the discounted sum of $a$'s in $w$. We say that $w$ is \emph{ambiguous}
if $1- \frac{\lambda^{\abs{w}}}{1-\lambda} \leq v_a(w) < 1$. The ambiguity lies
in that some continuations of $w$ (namely $w.a^{\omega}$) are in $L_2^{\geq 1}$ and some are not
(namely $w.b^{\omega}$).
We show that for all $\lambda > \frac{1}{2}$, if $w$ is ambiguous,
then either $w.a$ or $w.b$ is ambiguous, which entails
that there exists an infinite word $\hat{w}$ all of whose
finite prefixes are ambiguous (and $L_2(\hat{w}) = 1$).
To do this, assume that $1- \frac{\lambda^{\abs{w}}}{1-\lambda} \leq v_a(w) < 1$,
and show that either
$1- \frac{\lambda^{1+\abs{w}}}{1-\lambda} \leq v_a(w.a) < 1$
or $1- \frac{\lambda^{1+\abs{w}}}{1-\lambda} \leq v_a(w.b) < 1$.
Since $v_a(w.a) = v_a(w) + \lambda^{\abs{w}}$ and $v_a(w.b) = v_a(w)$,
we have to show that
$1- \frac{\lambda^{\abs{w}}}{1-\lambda} \leq v_a(w) < 1 - \lambda^{\abs{w}}$
or $1- \frac{\lambda^{1+\abs{w}}}{1-\lambda} \leq v_a(w) < 1$.
This holds if $1- \frac{\lambda^{1+\abs{w}}}{1-\lambda} < 1 - \lambda^{\abs{w}}$,
which is equivalent to $\lambda > \frac{1}{2}$.
Now, we show that if there exists a nondeterministic B\"uchi automaton~$A$ for~$L_2^{\geq 1}$,
then the set of states $S_n$ reached in~$A$ by reading the first~$n$ letters
of $\hat{w}$ (which we denote by $\hat{w}_{[1\dots n]}$) should be different for each~$n$,
i.e., $n \neq m$ implies $S_n \neq S_m$.
Towards a contradiction, assume that $S_n = S_m$ for $n < m$.
Then for all continuations $w' \in \Sigma^\omega$, we have $\hat{w}_{[1\dots n]}.w' \in L_2^{\geq 1}$
if and only if $\hat{w}_{[1\dots m]}.w' \in L_2^{\geq 1}$ $(\star)$.
In particular, consider the continuations $\hat{w}_{[n+1\dots]}$ and $\hat{w}_{[m+1\dots]}$,
and for each $i \geq 1$, let $\gamma^i = v_a(\hat{w}_{[1\dots i]})$ and
$K^i = L_2(\hat{w}_{[i+1\dots]})$. Then, we have $\gamma^i + \lambda^i \cdot K^i = 1$,
and thus $\gamma^m + \lambda^m \cdot K^n \leq 1$ iff $K^n \leq K^m$. Since either $K^n \leq K^m$
or $K^m \leq K^n$, we have either $L_2(\hat{w}_{[1\dots m]}.\hat{w}_{[n+1\dots]}) \leq 1$ or
$L_2(\hat{w}_{[1\dots n]}.\hat{w}_{[m+1\dots]}) \leq 1$. By $(\star)$,
this implies that either $L_2(\hat{w}_{[1\dots m]}.\hat{w}_{[n+1\dots]}) = 1$, or
$L_2(\hat{w}_{[1\dots n]}.\hat{w}_{[m+1\dots]}) = 1$, and in both cases since
$L_2(\hat{w}) = 1$, we get
$$\frac{1-\gamma^m}{\lambda^m} = \frac{1-\gamma^n}{\lambda^n}.$$
This implies $\lambda^{m-n} (1-P(\lambda)) = 1 - Q(\lambda)$
where $P(\lambda) = v_a(\hat{w}_{[1\dots n]})$ and $Q(\lambda) = v_a(\hat{w}_{[1\dots m]})$
are polynomials of respective degree $n-1$ and $m-1$, and with coefficients in
the set $\{0,1\}$. First, observe that the equation is not identically $0$
because the coefficient of the term of degree $0$ is not $0$ (as the first letter
of $\hat{w}$ must be $b$ since $a$ is not ambiguous). Second, every
coefficient in the equation is in the set $\{-2,-1,0,1,2\}$, and a classical
result shows that if $\frac{p}{q}$ is a solution of a polynomial equation
with $p$ and $q$ mutually prime, then $p$ divides the coefficient of degree $0$,
and $q$ divides the coefficient of highest degree. Therefore, no rational number
in the interval $]\frac{1}{2},1[\,$ can be a solution.
This shows that $n \neq m$ implies $S_n \neq S_m$, and thus the automaton $A$ cannot have
finitely many states.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
We note that cut-point languages are not stable under arbitrarily small
perturbations of the transition weights, nor of the value of the cut-point.
Consider the quantitative languages $L_1$, $L_2$ from the proof of Theorem~\ref{theo:cut-point-language}.
If for instance a limit-average automaton $A$ assigns weight $1+\epsilon$
to the $a$'s and $0$ to the $b$'s, its cut-point language $L_A^{\geq 1}$
is clearly different from $L_1^{\geq 1}$, no matter the value of $\epsilon > 0$.
The same holds with respect to $L_2$ if $A$ is interpreted as a discounted-sum automaton.
In the theory of probabilistic automata, where finite words are assigned a probability
of acceptance, the cut-point languages may also be non-regular. Therefore,
one considers the special case where the cut-point is isolated, and
shows that the cut-point languages are then regular~\cite{Rabin63}.
A number $\eta$ is an \emph{isolated cut-point} of a quantitative language $L$
if there exists $\epsilon > 0$ such that
$$\abs{L(w) - \eta} > \epsilon \text{ for all } w \in \Sigma^{\omega}.$$
We argue that isolated cut-point languages are robust, in that
they remain unchanged under small perturbations of
the transition weights. This follows from a more general result about
the robustness of weighted automata.
A class of weighted automata is robust if a small (syntactical) perturbation
in the weights of an automaton induces only a small (semantical) perturbation
in the values of the words in the quantitative language of the automaton,
and the semantical perturbation tends to $0$ when the syntactical perturbation tends to $0$.
To formally define robustness, we need $\epsilon$-approximations of automata,
and distance between quantitative languages.
Let $A=\tuple{Q,q_I,\Sigma,\delta,\gamma}$ be a (nondeterministic) weighted automaton,
and let $\epsilon \in {\mathbb R}^{\geq 0}$. We say that a weighted automaton $B=\tuple{Q',q'_I,\Sigma,\delta',\gamma'}$
is an \emph{$\epsilon$-approximation} of $A$ if
\begin{enumerate}[$\bullet$]
\item $Q' = Q$, $q'_I = q_I$, $\delta' = \delta$, and
\item $\abs{\gamma'(q,\sigma,q') - \gamma(q,\sigma,q')} \leq \epsilon$ for all $(q,\sigma,q') \in \delta$.
\end{enumerate}
The \emph{$\sup$-distance} between two quantitative languages $L_1,L_2: \Sigma^{\omega} \to {\mathbb R}$
is defined by
$$D_{\sup}(L_1,L_2) = \sup_{w \in \Sigma^{\omega}} \abs{L_1(w) - L_2(w)}.$$
We say that a class $\mathcal{C}$ of weighted automata is \emph{uniformly robust} if for all $\eta \in {\mathbb R}^{> 0}$,
there exists $\epsilon \in {\mathbb R}^{> 0}$ such that for all automata $A,B \in \mathcal{C}$ such that
$B$ is an $\epsilon$-approximation of $A$, we have $D_{\sup}(L_A,L_B) \leq \eta$.
Note that uniform robustness implies a weaker notion of robustness where
a class $\mathcal{C}$ of weighted automata is called \emph{robust} if for all automata $A \in \mathcal{C}$
and for all $\eta \in {\mathbb R}^{> 0}$, there exists $\epsilon \in {\mathbb R}^{> 0}$ such that
for all $\epsilon$-approximations $B$ of $A$ (with $B \in \mathcal{C}$), we have $D_{\sup}(L_A,L_B) \leq \eta$
(here the value of~$\epsilon$ can depend for instance on the weights of the automaton~$A$).
\begin{thm}\label{theo:uniformly-robust}
The classes of (non)deterministic $\mathsf{Sup}$-, $\mathsf{LimSup}$-, $\mathsf{LimInf}$-, $\mathsf{LimAvg}$- and $\mathsf{Disc}$-automata
are uniformly robust.
\end{thm}
\proof
Let $A,B$ be two weighted automata with $B$ an $\epsilon$-approximation of $A$.
It is easy to see that for $\mathsf{Sup}$-, $\mathsf{LimSup}$-, $\mathsf{LimInf}$- and $\mathsf{LimAvg}$-automata,
the value of a run $r$ of $B$ differs by at most $\epsilon$ from the value of
the same run in $A$. Therefore, $D_{\sup}(L_A,L_B) \leq \epsilon$ and we can take
$\epsilon = \eta$.
For $\mathsf{Disc}$-automata, the value of a run of $B$ differs by at most $\frac{\epsilon}{1-\lambda}$
from the value of the same run in $A$, where $\lambda$ is the discount factor.
Therefore, we can take $\epsilon = \eta (1 - \lambda)$.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
As a corollary of Theorem~\ref{theo:uniformly-robust}, for an isolated cut-point $\eta$,
the cut-point language $L^{\geq \eta}$ remains unchanged under small
perturbations of the weights.
\begin{cor}
Let $L_A$ be the quantitative language defined by a weighted automaton $A$,
and let $\eta$ be an isolated cut-point of $L_A$. There exists a rational $\epsilon > 0$
such that for all $\epsilon$-approximations $B$ of $A$, we have $L_A^{\geq \eta} = L_B^{\geq \eta}$
(where $L_B$ is the quantitative language defined by $B$).
\end{cor}
Now, we show that the isolated cut-point languages of
deterministic discounted-sum and limit-average automata are $\omega$-regular.
For nondeterministic automata, the same property holds in the discounted-sum case,
but the question is open for limit-average.
\begin{thm}
Let $L$ be the quantitative language defined by a $\mathsf{Disc}$-automaton.
If $\eta$ is an isolated cut-point of $L$, then the cut-point language $L^{\geq \eta}$
is $\omega$-regular.
\end{thm}
\proof
Let~$\lambda$ be the discount factor of the $\mathsf{Disc}$-automaton~$A$ that defines~$L$.
Since~$\eta$ is an isolated cut-point of $L$, let $\epsilon > 0$ such that
$\abs{L(w) - \eta} > \epsilon$ for all $w \in \Sigma^{\omega}$.
Let $n \in \nat$ such that $u_n = \frac{V\cdot \lambda^n}{1-\lambda} < \epsilon $
where $V = \max_{(q,\sigma,q') \in \delta_A} \abs{\gamma(q,\sigma,q')}$ is the largest weight in $A$.
Note that $u_n$ is a bound on the difference between the $\lambda$-discounted sum of the weights
in any infinite run $\hat{r}$ of $A$ and the $\lambda$-discounted sum of the weights in the prefix
of length $n$ of $\hat{r}$, and $u_n \to 0$ when $n \to \infty$.
\noindent Consider an arbitrary run $r$ in $A$ of length $n$, and let $\gamma(r)$ be the
$\lambda$-discounted sum of the weights along $r$. Then, it should be clear that
$\gamma(r) \not\in [\eta - \epsilon + u_n, \eta + \epsilon - u_n]$, because
otherwise, the value of any (infinite) continuation of $r$ would lie in the
interval $[\eta - \epsilon, \eta + \epsilon]$, which would be a contradiction
to the fact that $\eta$ is an isolated cut-point of $L$.
Moreover, if $\gamma(r) \leq \eta - \epsilon + u_n$, then any (infinite) continuation of $r$
has value less than $\eta - \epsilon + 2 u_n < \eta + \epsilon$, and therefore
less than $\eta$, while if $\gamma(r) \geq \eta + \epsilon - u_n$,
then any (infinite) continuation of $r$ has value greater than $\eta$.
Therefore, the cut-point language $L^{\geq \eta}$ can be defined by the
unfolding up to length $n$ of the $\mathsf{Disc}$-automaton that defines $L$,
in which the states that are reached via a path with value at least
$\eta + \epsilon - u_n$ are declared to be accepting (for B\"uchi condition),
and have a self-loop on $\Sigma$.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\begin{thm}
Let $L$ be the quantitative language defined by a deterministic $\mathsf{LimAvg}$-automaton.
If $\eta$ is an isolated cut-point of $L$, then the cut-point language $L^{\geq \eta}$
is $\omega$-regular.
\end{thm}
\proof
Let $A$ be a deterministic $\mathsf{LimAvg}$-automaton, defining the language $L$.
Consider the SCC-decomposition $C_1,C_2,\dots,C_k$ of the underlying graph of $A$.
For each $1 \leq i \leq k$, let $m_i$ and $M_i$ be the minimal and maximal
average weight of a cycle in $C_i$ (those values can be computed with
Karp's algorithm~\cite{Karp78}).
It is easy to see that for every $1 \leq i \leq k$, for every $v \in [m_i, M_i]$,
there exists a word $w \in \Sigma^{\omega}$ such that $L(w) = v$.
Therefore, since $\eta$ is an isolated cut-point of $L$, we have $\eta \not\in [m_i, M_i]$
for all $1 \leq i \leq k$.
A deterministic B\"uchi automaton ({\sc DBW}) for $L^{\geq \eta}$ is obtained from $A$ by declaring to be accepting all
states $q$ of $A$ such that $q \in C_i$ and $m_i > \eta$.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\subsection{Boolean weights}
We consider weighted automata with boolean set of weights, i.e. all transitions have weight $0$ or $1$.
The aim is to have a boolean counterpart to limit-average and discounted-sum
automata, and compare the expressive power.
We show that the restriction does not change the class of quantitative
languages definable by limit-average automata, but does reduce the expressive
power of discounted-sum automata.
\begin{thm}
The class of nondeterministic (resp., deterministic) $\mathsf{LimAvg}$-automata with rational weights in $[0,1]$
is reducible to the class of nondeterministic (resp., deterministic) $\mathsf{LimAvg}$-automata with weights $0$ and $1$ only.
\end{thm}
\proof
Given a {\sc NLavg}\/ $A=\tuple{Q,q_I,\Sigma,\delta,\gamma}$ with weights in $[0,1]$,
we construct a {\sc NLavg}\/ $B$ with weights in $\{0,1\}$ such that $L_A = L_B$.
First, let $W = \{\gamma(q,\sigma,q') \mid (q,\sigma,q') \in \delta\}$
be the set of weights that occur in $A$, and let $n_A$ be the smallest integer $n$
such that for all $v \in W$, there exists $p \in \nat$ such that $v = \frac{p}{n}$
(i.e., $\frac{1}{n_A}$ is the greatest common divisor of the weights of~$A$).
We define $B=\tuple{Q',q'_I,\Sigma,\delta',\gamma'}$ as follows:
\begin{enumerate}[$\bullet$]
\item $Q' = Q \times [n_A]$ (where $[n_A]$ denotes the set $\{0,1,\dots,n_A-1\}$).
Intuitively, when we reach a state $(q,i)$ in $B$, it means that the state $q$
was reachable in $A$ and that the sum of the weights to reach $q$ is of the form
$k + \frac{i}{n_A}$ for some integer $k$. In $B$ however, the sum of the weights
to reach $(q,i)$ will then be $k$, and we store in the discrete state the information
that the remainder weight is $\frac{i}{n_A}$. Whenever this remainder exceeds $1$,
we introduce a weight $1$ and decrement the remainder.
\item $q'_I = (q_I,0)$;
\item for each transition $(q,\sigma,q') \in \delta$ and each value $i \in [n_A]$,
the following transitions are in $\delta'$ (where $v = \gamma(q,\sigma,q')$):
\begin{enumerate}[$-$]
\item $((q,i),\sigma,(q',j))$ for $j=i+v\cdot n_A$ if $\frac{i}{n_A} + v < 1$;
the weight of such a transition is $0$ in $\gamma'$,
\item $((q,i),\sigma,(q',j))$ for $j=i+(v-1)\cdot n_A$ if $\frac{i}{n_A} + v \geq 1$;
the weight of such a transition is $1$ in $\gamma'$.
\end{enumerate}
Note that in the above, $v\cdot n_A$ is an integer and $j \in [n_A]$.
\end{enumerate}
There is a straightforward correspondence between the runs in $A$ and the runs in $B$.
Moreover, if the average weight of a prefix of length $n$ of a run in $A$ is $\frac{S}{n}$,
then the average weight of the prefix of length $n$ of the corresponding run in $B$
is between $\frac{S}{n}$ and $\frac{S+1}{n}$. Hence the difference tends to $0$
when $n \to \infty$. Therefore, the value of a run in $A$ is the same as the
value of the corresponding run in $B$, and therefore $L_A = L_B$.
Finally, note that if $A$ is deterministic, then $B$ is deterministic.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\begin{thm}
The class of deterministic $\mathsf{Disc}$-automata with rational weights in $[0,1]$
is not reducible to the class of (even nondeterministic) $\mathsf{Disc}$-automata with weights $0$ and $1$ only.
\end{thm}
\proof
Given a discount factor $0 < \lambda <1$, consider the {\sc DDisc}\/ over $\Sigma = \{a,b\}$
that consists of a single state with a self-loop over $a$ with weight $\frac{1+\lambda}{2}$
and a self-loop over $b$ with weight $0$.
Let $L_{\lambda}$ be the quantitative language defined by this automaton.
Towards a contradiction, assume that this language is defined by a {\sc NDisc}\/ $A$ with weights in $\{0,1\}$.
First, consider the word $a b^{\omega}$ whose value in $L_{\lambda}$ is $\frac{1+\lambda}{2} < 1$.
This entails that $A$ cannot have a transition from the initial state over $a$ with weight $1$
(as this would imply that $L_A(a b^{\omega}) \geq 1$).
Now, the maximal value that $L_A$ can assign to the word $a^{\omega}$ is
$\lambda + \lambda^2 + \lambda^3 + \cdots = \frac{\lambda}{1-\lambda}$ which is
strictly smaller than $L_{\lambda}(a^{\omega}) = \frac{1+\lambda}{2(1-\lambda)}$.
This shows that $A$ cannot exist.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\begin{comment}
{\bf Old proof}
\proof
Consider the {\sc NDisc}${}_{[0,1]}$ over $\Sigma = \{a,b\}$ that consists of a single
state with a self-loop over $a$ with weight $\frac{1}{2}$ and a self-loop over $b$
with weight $0$. Let $L_{\lambda}$ be the quantitative language defined by this automaton
where $\lambda$ is the (rational) discount factor.
Towards a contradiction, assume that this language is defined by a {\sc NDisc}${}_{\{0,1\}}$ $A$
with (finite) set of states $Q_A$, i.e., $L_A = L_{\lambda}$.
First, assume that $\lambda < \frac{1}{2}$. Consider the word $a b^{\omega}$
whose value in $L$ is $\frac{1}{2}$. This entails that $A$ cannot have a transition
from the initial state over $a$ with weight $1$ (as this would mean that $L_A(a b^{\omega}) \geq 1$).
Now, the maximal value that $L_A$ can assign to the word $a^{\omega}$ is
$\lambda + \lambda^2 + \lambda^3 + \cdots = \frac{\lambda}{1-\lambda}$ which is
strictly smaller than $L(a^{\omega}) = \frac{1}{2(1-\lambda)}$ since $\lambda < \frac{1}{2}$.
This shows that $A$ cannot exist in this case.
Second, assume that $\lambda > \frac{1}{2}$ and $\lambda \neq \frac{2}{3}$.
Consider two words $w_1.b^{\omega}$ and $w_2.b^{\omega}$ where $w_1,w_2 \in \Sigma^n$
are finite words of length $n$. Let $r_1$ and $r_2$ be the runs of $A$ over
$w_1.b^{\omega}$ and $w_2.b^{\omega}$ respectively, with maximal discounted sum
of weights. Let $q_1,q_2 \in Q_A$ be the $n$-th states in $r_1$ and $r_2$
(i.e., the states reached after reading $w_1$ and $w_2$ respectively).
If $q_1 = q_2$, then the sequences of weights in $r_1$ and $r_2$ after the $n$-th state
have to be the same (because the suffix of $r_1$ from $q_1$ has to be a run
with maximal discounted sum of weights over $b^{\omega}$, independently of $w_1$,
and similarly for the suffix of $r_2$ from $q_2$). Therefore, if we call $K_1,K_2$
the discounted sum of the first $n$ weights in $r_1,r_2$ respectively, we have
\begin{equation}\label{eq:weights}
L(w_1.b^{\omega}) - L(w_2.b^{\omega}) = K_1 - K_2
\end{equation}
which can be written as
$\sum_{i=1}^{n} c_i\cdot \lambda^i = \sum_{i=1}^{n} d_i\cdot \lambda^i$
where $c_i \in \{0, \frac{1}{2}, -\frac{1}{2}\}$ and $d_i \in \{0, 1, -1\}$
for all $1 \leq i \leq n$, or equivalently as $P(\lambda) = 0$ where $P(\lambda)$
is a polynomial with coefficients in the set
$\{0, 1,-1,2,-2,3,-3\}$.
The only possible rational solutions of such a polynomial equation
in the interval $[\frac{1}{2},1]$ are $\frac{1}{2}$ and $\frac{2}{3}$ (since the denominator
and numerator of a rational solution have to divide the coefficient of the term
of respectively the highest and the lowest degree in~$P$).
Therefore, Equation~\eqref{eq:weights} cannot hold, and thus $q_1 \neq q_2$.
This means that for each word of length $n$, the automaton $A$ has to be able
to reach a different state.
Since this holds for all $n$, the automaton $A$ cannot have finitely many states.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\end{comment}
\section{Closure Properties}
\label{sec:closure-properties}
We study the closure properties of weighted automata
with respect to $\max$, $\min$, complement and sum. We say that a class $\mathcal{C}$ of weighted
automata is \emph{closed} under a binary operator $\op(\cdot,\cdot)$
(resp. a unary operator $\op'(\cdot)$) if for all $A_1,A_2 \in \mathcal{C}$,
there exists $A_{12} \in \mathcal{C}$ such that $L_{A_{12}} = \op(L_{A_1},L_{A_2})$
(resp. $L_{A_{12}} = \op'(L_{A_1})$).
All closure properties that we present in this paper are constructive:
when $\mathcal{C}$ is closed under an operator,
we can always construct the automaton $A_{12} \in \mathcal{C}$ given $A_1,A_2 \in \mathcal{C}$.
We say that the \emph{cost} of the closure property of $\mathcal{C}$ under a binary operator $\op$
is at most $O(f(n_1,m_1,n_2,m_2))$ if
for all automata $A_1,A_2 \in \mathcal{C}$ with $n_i$ states and $m_i$ transitions (for $i=1,2$ respectively),
the constructed automaton $A_{12} \in \mathcal{C}$ such that $L_{A_{12}} = \op(L_{A_1},L_{A_2})$
has at most $O(f(n_1,m_1,n_2,m_2))$ many states.
Analogously, the \emph{cost} of the closure property of $\mathcal{C}$ under a unary operator $\op'$
is at most $O(f(n,m))$ if
for all automata $A_1 \in \mathcal{C}$ with $n$ states and $m$ transitions,
the constructed automaton $A_{12} \in \mathcal{C}$ such that $L_{A_{12}} = \op'(L_{A_1})$
has at most $O(f(n,m))$ many states.
For all reductions presented, the size of the largest weight in $A_{12}$
is linear in the size~$p$ of the largest weight in~$A_1,A_2$ (however,
the time needed to compute the weights is quadratic in $p$,
as we need addition, multiplication, or comparison, which are quadratic
in $p$).
Notice that every class of weighted automata is closed under shift by~$c$ and
under scale by~$\abs{c}$ for all~$c \in {\mathbb Q}$. For $\mathsf{Sum}$-automata
and discounted-sum automata, we can define the shift by~$c$ by making a copy of the initial
states and adding $c$ to the weights of all its outgoing transitions.
For the other automata, it suffices to add~$c$
to (resp. multiply by~$\abs{c}$) all weights of an automaton to obtain the
automaton for the shift by~$c$ (resp. scale by~$\abs{c}$) of its language.
Therefore, all closure properties also hold if the complement of a
quantitative language~$L$ was defined as~$k-L$ for any constant~$k$.
Our purpose is the study of quantitative languages over infinite words.
For the sake of completeness we first give an overview of the closure
properties for finite words. Table~\ref{tab:closure-properties}(a)
summarizes the closure properties for finite words,
and Table~\ref{tab:closure-properties}(b) for infinite words.
\subsection{Closure properties for finite words}
For finite words, we consider closure under $\max$, $\min$, complement,
and sum for $\mathsf{Max}$-, ${\sf Last}$- and $\mathsf{Sum}$-automata.
\begin{thm}\label{theo:max-closure-finite}
Deterministic $\mathsf{Max}$- and ${\sf Last}$-automata are closed under $\max$,
with cost $O(n_1 \cdot n_2)$.
Nondeterministic $\mathsf{Max}$-, ${\sf Last}$- and $\mathsf{Sum}$-automata are closed under $\max$,
with cost $O(n_1 + n_2)$.
Deterministic $\mathsf{Sum}$-automata are not closed under $\max$.
\end{thm}
\proof
For the nondeterministic automata, the result follows from the fact that
the $\max$ operator can be obtained by an initial nondeterministic choice
between two quantitative automata. For deterministic $\mathsf{Max}$- and ${\sf Last}$-automata,
the result is obtained using a standard synchronized product construction,
where the weight of a transition in the product is the maximum of the corresponding
transition weights in the two automata.
Finally, deterministic $\mathsf{Sum}$-automata are not closed under the $\max$ operator
because the language over $\Sigma = \{a,b\}$ that assigns to each finite word $w \in \Sigma^{+}$
the number $\max\{L_a(w),L_b(w)\}$ where $L_{\sigma}(w)$ is the number of occurrences of $\sigma$ in $w$ (for $\sigma = a,b$)
is definable by the max of two deterministic-$\mathsf{Sum}$ languages, but not
by a deterministic $\mathsf{Sum}$-automaton (Theorem~2 in~\cite{CDH08}).
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\begin{thm}\label{theo:min-closure-finite}
Deterministic and nondeterministic $\mathsf{Max}$-automata
are closed under $\min$, with cost $O(n_1\cdot m_1 \cdot n_2 \cdot m_2)$.
Deterministic and nondeterministic ${\sf Last}$-automata
are closed under $\min$, with cost $O(n_1 \cdot n_2)$.
Deterministic and nondeterministic $\mathsf{Sum}$-automata are not closed
under $\min$.
\end{thm}
\proof
Given two ${\sf Last}$-automata $A_1$ and $A_2$ (over the same alphabet),
we use the classical synchronized product $A_{12} = A_1 \times A_2$,
where the weight of a transition in $A_{12}$ is the minimum of the corresponding
transition weights in $A_1$ and $A_2$. It is easy to see that $L_{A_{12}} = \min(L_{A_1}, L_{A_2})$.
If $A_1$ and $A_2$ are deterministic, then so is $A_{12}$.
The construction for $\mathsf{Max}$-automata is the same as for $\mathsf{Sup}$-automata
over infinite words given in the proof of Theorem~\ref{theo:max-closed-under-min}.
Finally, for $\mathsf{Sum}$-automata, consider the language $L_m$ over $\Sigma = \{a,b\}$
that assigns to each finite word $w \in \Sigma^{+}$ the value $\min\{L_a(w),L_b(w)\}$
where $L_{\sigma}(w)$ is the number of occurrences of $\sigma$ in $w$ (for $\sigma = a,b$).
We claim that $L_m$ is not definable by a nondeterministic $\mathsf{Sum}$-automaton.
Indeed, assume that the $\mathsf{Sum}$-automaton $A$ with state space $Q$ defines $L_m$.
First, the sum of weights in every reachable cycle of $A$ over $a$'s
must be at most $0$. Otherwise, we can reach the cycle with a finite word $w_1$
and obtain an arbitrarily large value for the word $w_1 a^i$ for $i$ sufficiently
large, while for any such $i$ the value of $w_1 a^i$ is the number of $b$'s
in $w_1$ which is independent of $i$. Analogously, the sum of weights in
every reachable cycle of $A$ over $b$'s must be at most $0$.
Now, let $\beta= \max_{e \in \delta} \abs{\gamma(e)}$ be the maximal
weight in $A$, and consider the word $w = a^n b^n$ for $n > 2\beta \cdot \abs{Q}$.
Every run of $A$ over $a^n$ (or over $b^n$) can be decomposed in possibly nested cycles
(since $A$ is nondeterministic) and a remaining non-cyclic path of length
at most $\abs{Q}$. Hence, the value of any run over $w$ is at most $2\beta \cdot \abs{Q}$.
However, the value of $w$ should be $n$, thus $A$ cannot exist.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\begin{thm}\label{theo:closure-under-complement-finite}
Deterministic ${\sf Last}$- and $\mathsf{Sum}$-automata are closed under complement,
with cost $O(n)$.
Nondeterministic ${\sf Last}$-automata are closed under complement,
with cost $O(2^n)$.
Nondeterministic $\mathsf{Sum}$ automata, and both deterministic and nondeterministic
$\mathsf{Max}$-auto\-mata are not closed under complement.
\end{thm}
\proof
To define the complement of the language of a deterministic $\mathsf{Sum}$- (or ${\sf Last}$-) automaton,
it suffices to multiply all the weights by $-1$, and then shift the language by $1$.
For the class of nondeterministic ${\sf Last}$-automata, the result follows from the fact that
it is reducible to its deterministic counterpart.
The negative result for $\mathsf{Max}$-automata follows from an analogous in the boolean
case (consider the language $L$ over $\{a,b\}$ such that $L(a^i) = 0$
for all $i \geq 1$, and $L(w) = 1$ for all words containing the letter $b$).
Finally, according to the proof of Theorem~\ref{theo:min-closure-finite},
the language $\min(L_a,L_b)$ where $L_{\sigma}(w)$ is the number of occurrences
of $\sigma$ in $w$ (for $\sigma = a,b$) is not definable by a nondeterministic
$\mathsf{Sum}$-automaton. Since $\min(L_a,L_b) = 1-\max(1-L_a,1-L_b)$ and
\begin{compressEnum}
\stepcounter{compressEnum}{(\thecompressEnum) } $1-L_a$ and $1-L_b$ are definable by $\mathsf{Sum}$-automata, and
\stepcounter{compressEnum}{(\thecompressEnum) } nondeterministic $\mathsf{Sum}$-automata are closed under $\max$ (Theorem~\ref{theo:max-closure-finite}),
\end{compressEnum}
the language $\max(1-L_a,1-L_b)$ is definable by a nondeterministic $\mathsf{Sum}$-automaton,
but not its complement and the result follows.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\begin{thm}\label{theo:closure-under-sum-finite}
Every class of weighted automata over finite words is closed under sum.
The cost is $O(n_1\cdot n_2)$ for ${\sf Last}$- and $\mathsf{Sum}$-automata,
and $O(n_1\cdot m_1 \cdot n_2 \cdot m_2)$ for $\mathsf{Max}$-automata.
\end{thm}
\proof
It is easy to see that the synchronized product of two ${\sf Last}$-automata (resp. $\mathsf{Sum}$-automata)
defines the sum of their languages if the weight of a joint transition
is defined as the sum of the weights of the corresponding transitions in the two
${\sf Last}$-automata (resp. $\mathsf{Sum}$-automata).
We give the construction for two $\mathsf{Max}$-automata
$A_1=\tuple{Q_1,q_I^1,\Sigma,\delta_1,\gamma_1}$ and $A_2=\tuple{Q_2,q_I^2,\Sigma,\delta_2,\gamma_2}$.
We construct a $\mathsf{Max}$-automaton $A_{12}=\tuple{Q,q_I,\Sigma,\delta,\gamma}$ such that
$L_{A_{12}} = L_{A_1} + L_{A_2}$.
Let $V_i = \{\gamma_i(e) \mid e \in \delta_i\}$ be the set of weights that appear in $A_i$ (for $i=1,2$),
and define:
\begin{enumerate}[$\bullet$]
\item $Q = Q_1 \times V_1 \times Q_2 \times V_2$. Intuitively, we remember in
a state $(q_1, v_1, q_2, v_2)$ the largest weights $v_1,v_2$ seen so far in the corresponding
runs of $A_1$ and $A_2$;
\item $q_I = (q_I^1, v_{\min}^1, q_I^2, v_{\min}^2)$ where $v_{\min}^i$ is the minimal weight in $V_i$ (for $i=1,2$);
\item For each $\sigma \in \Sigma$, the set $\delta$ contains all the triples
$\tuple{(q_1, v_1, q_2, v_2),\sigma,(q'_1,v'_1,q'_2,v'_2)}$ such that
$v_i \in V_i$, $(q_i,\sigma,q'_i) \in \delta_i$, and $v'_i = \max\{v_i,\gamma(q_i,\sigma,q'_i)\}$,
for $i=1,2$;
\item $\gamma$ is defined by \[\gamma(\tuple{(q_1, v_1, q_2, v_2),\sigma,(q'_1,v'_1,q'_2,v'_2)}) = v'_1 + v'_2\]
for each $\tuple{(q_1, v_1, q_2, v_2),\sigma,(q'_1,v'_1,q'_2,v'_2)} \in \delta$.
\end{enumerate}
If $A_1$ and $A_2$ are deterministic, then $A_{12}$ is deterministic.
The result for deterministic $\mathsf{Max}$-automata follows.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\subsection{Closure under $\max$ for infinite words}
The maximum of two quantitative languages defined by nondeterministic automata
can be obtained by an initial nondeterministic choice between the two automata.
This observation was also made in~\cite{DrosteR07} for discounted-sum automata.
For deterministic automata, a synchronized product can be used
for $\mathsf{Sup}$ and $\mathsf{LimSup}$, while for $\mathsf{LimInf}$ we use the fact that {\sc NLinf}\/
is determinizable with an exponential blow-up~\cite{CDH08}.
\begin{thm}\label{theo:max-closure}
The nondeterministic $\mathsf{Sup}$-, $\mathsf{LimSup}$-, $\mathsf{LimInf}$-, $\mathsf{LimAvg}$- and $\mathsf{Disc}$-automata
are closed under $\max$, with cost $O(n_1+n_2)$,
the deterministic $\mathsf{Sup}$- and $\mathsf{LimSup}$-automata with cost $O(n_1 \cdot n_2)$,
the deterministic $\mathsf{LimInf}$-automata with cost $O((m_1+m_2)^{n_1 + n_2})$.
\end{thm}
\proof[Sketch]
For all the nondeterministic quantitative automata, the result
follows from the fact that the $\max$ operator can be achieved
with an initial nondeterministic choice between two weighted
automata.
For {\sc DLinf}, the result follows from the reducibility of {\sc NLinf}\/ to {\sc DLinf}\/
with an exponential blow-up~\cite{CDH08}.
We now prove that {\sc DLsup}\/ and {\sc DSup}\/ are closed under $\max$ with cost $O(n_1 \cdot n_2)$.
Given two {\sc DLsup}\/ (or {\sc DSup}) $A_1$ and $A_2$ over the same alphabet, we construct the usual
synchronized product $A_{12} = A_1 \times A_2$, where the weight of a
transition in $A_{12}$ is the maximum of the corresponding
transition weights in $A_1$ and $A_2$.
It is easy to see that $L_{A_{12}} = \max(L_{A_1}, L_{A_2})$ in both cases.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\begin{thm}
The deterministic $\mathsf{LimAvg}$- and $\mathsf{Disc}$-automata are not closed under $\max$.
\end{thm}
\proof
The fact that {\sc DDisc}\/ is not closed under $\max$ follows
from the proof of Theorem~16 in~\cite{CDH08}, where it is shown that the quantitative language
$\max(L_1,L_2)$ cannot be defined by a {\sc DDisc}, where $L_1$ (resp. $L_2$) is the language
defined by the {\sc DDisc}\/ that assigns weight $1$ (resp. $0$) to $a$'s
and weight $0$ (resp. $1$) to $b$'s.
We now show that {\sc DLavg}\/ is not closed under $\max$.
Consider the alphabet $\Sigma=\set{a,b}$ and the quantitative languages
$L_a$ and $L_b$ that assign the value of long-run average
number of $a$'s and $b$'s, respectively.
There exists {\sc DLavg}\/ for $L_a$ and $L_b$.
We show that $L_m=\max(L_a,L_b)$ cannot be expressed by
a {\sc DLavg}. By contradiction, assume that $A$ is a {\sc DLavg}\/
with set of states $Q$ that defines $L_m$.
Consider any reachable cycle $C$ over $a$'s in $A$. The sum of the weights
of the cycle must be its length $\abs{C}$, as if we consider the
word $w^*=w_C \cdot (a^{\abs{C}})^\omega$ where $w_C$
is a finite word whose run reaches $C$, the value
of $w^*$ in $L_m$ is $1$. It follows that the sum of the weights
of the cycle $C$ must be $\abs{C}$. Hence, the sum of the weights
of all the reachable cycles $C$ over $a$'s in $A$ is $\abs{C}$.
\newcommand{\widehat}{\widehat}
Consider the infinite word $w_\infty=(a^{\abs{Q}} \cdot b^{2\abs{Q}})^\omega$,
and let $w_j=(a^{\abs{Q}} \cdot b^{2\abs{Q}})^j$.
Since $L_m(w_\infty)=\frac{2}{3}$, the run of $A$ over $w_\infty$
has value $\frac{2}{3}$.
It follows that for all $\varepsilon>0$, there is an integer $j_\varepsilon$, such that
for all $j\geq j_\varepsilon$, we have
\[
\frac{\gamma(w_j)}{\abs{w_j}} \geq \frac{2}{3} -\varepsilon
\]
where $\gamma(w_j)$ is the sum of the weights of the run of $A$ over $w_j$.
Consider a word $\widehat{w}_\infty$ constructed as follows. We start
with the empty word $\widehat{w}_0$ and the initial state $q_0$ of $A$, and for all $j\geq 0$, we construct
$(\widehat{w}_{j+1}, q_{j+1})$ from $(\widehat{w}_j,q_j)$ as follows: the state $q_{j+1}$
is the last state of the run of $A$ from $q_j$ over $a^{\abs{Q}} \cdot b^{2\abs{Q}}$.
This run has to contain a cycle $C_{j+1}$ over $a$'s. We set
$\widehat{w}_{j+1} = \widehat{w}_j \cdot a^{\abs{Q} + \abs{C_{j+1}}} \cdot b^{2\abs{Q}}$.
Observe that for all $j \geq 1$, the run of $A$ over $w_\infty$
in the segment between $w_j$ and $w_{j+1}$ is identical to
the run from $q_j$ to $q_{j+1}$ up to the repetition of the cycle $C_{j+1}$
once more.
The word $\widehat{w}_\infty$ is the limit of this construction ($\widehat{w}_j$ is
a prefix of $\widehat{w}_\infty$ for all $j \geq 0$).
Let $\alpha_j=\sum_{i=1}^j \abs{C_i}$.
Since $1\leq \abs{C_i} \leq \abs{Q}$ we have $j \leq \alpha_j \leq j \cdot \abs{Q}$.
Hence we have the following equality:
$\frac{\gamma(\widehat{w}_j)}{\abs{\widehat{w}_j}}
=\frac{\gamma(w_j) + \alpha_j}{\abs{w_j}+ \alpha_j}$.
Hence for all $\varepsilon>0$, there exists $j_\varepsilon$ such that
for all $j \geq j_\varepsilon$ we have
\[
\begin{array}{rcl}
\displaystyle
\frac{\gamma(\widehat{w}_j)}{\abs{\widehat{w}_j}}
& \geq &
\displaystyle
\frac{ \frac{2}{3}\cdot \abs{w_j} - \varepsilon \cdot \abs{w_j} + \alpha_j }{ \abs
{w_j} + \alpha_j} \\[2ex]
& \geq &
\displaystyle
\frac{2}{3} -\varepsilon + \frac{1}{3} \cdot \frac{\alpha_j}{\abs{w_j} + \alpha_j}
\\[1ex]
& \geq &
\displaystyle
\frac{2}{3} -\varepsilon + \frac{1}{3} \cdot \frac{j}{j \cdot(3 \abs{Q} + \abs
{Q})} \\[1ex]
& \geq &
\displaystyle
\frac{2}{3} -\varepsilon + \frac{1}{12 \abs{Q}}. \\[1ex]
\end{array}
\]
This shows that $\liminf_{j \to \infty} \frac{\gamma(\widehat{w}_j)}{\abs{\widehat{w}_j}} \geq \frac{2}{3} + \frac{1}{12 \abs{Q}}$
and thus we have $L_A(\widehat{w}_\infty) \geq \frac{2}{3} + \frac{1}{12 \abs{Q}}$.
Since $1 \leq \abs{C_i} \leq \abs{Q}$ for all $i\geq 1$, we have $L_m(\widehat{w}_\infty)\leq\frac{2}{3}$
which is a contradiction.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\subsection{Closure under $\min$ for infinite words}
The positive results about closure properties under $\min$ for quantitative languages
generalize the closure properties of boolean languages under intersection.
The constructions are straightforward extensions of the standard constructions
for finite, B\"uchi, and coB\"uchi automata (see e.g.~\cite{Vardi96}).
\begin{thm}\label{theo:max-closed-under-min}
The (non)deterministic $\mathsf{Sup}$-automata are closed under $\min$, with cost $O(n_1\cdot m_1 \cdot n_2 \cdot m_2)$,
\end{thm}
\proof
The construction in the proof of Theorem~\ref{theo:closure-under-sum-finite}
can be adapted by defining the weight $\gamma(\tuple{(q_1, v_1, q_2, v_2),\sigma,(q'_1,v'_1,q'_2,v'_2)})$
as $\min\{v'_1, v'_2\}$ for each $\tuple{(q_1, v_1, q_2, v_2),\sigma,(q'_1,v'_1,q'_2,v'_2)} \in \delta$.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\begin{thm}
The deterministic $\mathsf{LimSup}$-automata are closed under $\min$ with cost $O(n_1 \cdot n_2 \cdot 2^{m_1 + m_2})$.
\end{thm}
\proof
Let $A_1=\tuple{Q_1,q_I^1,\Sigma,\delta_1,\gamma_1}$ and $A_2=\tuple{Q_2,q_I^2,\Sigma,\delta_2,\gamma_2}$
be two {\sc DLsup}. We construct a {\sc DLsup}\/ $A=\tuple{Q,q_I,\Sigma,\delta,\gamma}$ such that
$L_A = \min\{L_{A_1},L_{A_2}\}$. Let $V_i = \{\gamma_i(e) \mid e \in \delta_i\}$
be the set of weights that occur in $A_i$ (for $i=1,2$). For each weight
$v \in V_1 \cup V_2 = \{v_1,\dots,v_n\}$, we construct a {\sc DBW}\/ $A^v_{12}$ that
consists of a copy of $A_1$ and a copy of $A_2$.
We switch from one copy to the other whenever an edge with weight at least $v$
is crossed. All such switching edges are accepting in $A^v_{12}$ (i.e., they have
weight~$1$ while all other edges have weight~$0$). The automaton $A$ then
consists of the synchronized product of these {\sc DBW}, where the weight of a joint
edge is the largest weight $v$ for which the underlying edge in $A^v_{12}$ is accepting.
Formally, let
\begin{enumerate}[$\bullet$]
\item $Q = Q_1 \times Q_2 \times \{1,2\}^m$ where $m = \abs{V_1 \cup V_2}$ (and assume $V_1 \cup V_2= \{v_1,\dots,v_m\}$);
\item $q_I = (q_I^1, q_I^2, b_1, \dots, b_m)$ where $b_i=1$ for all $1 \leq i \leq m$;
\item $\delta$ contains all the triples
$(\tuple{q_1,q_2,b_1,\dots,b_m},\sigma, \tuple{q'_1,q'_2,b'_1,\dots,b'_m})$ such that
$\sigma \in \Sigma$ and
\begin{enumerate}[$-$]
\item $(q_i,\sigma,q'_i) \in \delta_i$ for $i=1,2$;
\item for all $1\leq j \leq m$, we have $b'_{j} = 3-b_j$ if
$\gamma_{b_j}(q_{b_j},\sigma,q'_{b_j}) \geq v_j$, and $b'_j = b_j$ otherwise.
\end{enumerate}
\item $\gamma$ assigns to each transition
$(\tuple{q_1,q_2,b_1,\dots,b_m},\sigma, \tuple{q'_1,q'_2,b'_1,\dots,b'_m}) \in \delta$
the weight $v=\max(\{v_{\min}\}\cup \{v_j \mid b_j\neq b'_j\})$
where $v_{\min}$ is the minimal weight in $V_1 \cup V_2$.{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\end{enumerate}
\begin{thm}
The (non)deterministic $\mathsf{LimInf}$-automata are closed under $\min$ with cost $O(n_1 \cdot n_2)$, and
the nondeterministic $\mathsf{LimSup}$-automata with cost $O(n_1\cdot n_2 \cdot (m_1 + m_2))$.
\end{thm}
\proof
Let $A_1=\tuple{Q_1,q_I^1,\Sigma,\delta_1,\gamma_1}$ and $A_2=\tuple{Q_2,q_I^2,\Sigma,\delta_2,\gamma_2}$
be two {\sc NLsup}. We construct a {\sc NLsup}\/ $A=\tuple{Q,q_I,\Sigma,\delta,\gamma}$ such that
$L_A = \min\{L_{A_1},L_{A_2}\}$. Let $V_i = \{\gamma_i(e) \mid e \in \delta_i\}$
be the set of weights that appear in $A_i$ (for $i=1,2$). Let $V_1 \cup V_2 = \{v_1,\dots,v_n\}$
and define
\begin{enumerate}[$\bullet$]
\item $Q = \{q_I\} \cup Q_1 \times Q_2 \times \{1,2\} \times (V_1 \cup V_2)$ (where $q_I \not\in Q_1 \cup Q_2$
is a new state). Initially, a guess is made of the value $v$ of the input word.
Then, we check that both $A_1$ and $A_2$ visit a weight at least $v$ infinitely often.
In a state $\tuple{q_1,q_2,j,v}$ of $A$, the guess is stored in $v$ (and will never change
along a run) and the value of the index $j$ is toggled to $3-j$ as soon as $A_j$
does visit a weight at least $v$;
\item For each $\sigma \in \Sigma$, the set $\delta$ contains all the triples
\begin{enumerate}[$-$]
\item $(q_I,\sigma,\tuple{q_1,q_2,1,v})$ such that
$v\in V_1 \cup V_2$ and for all $i \in \{1,2\}$,
we have $(q_I^{i}, \sigma, q_i) \in \delta_i$.
\item $(\tuple{q_1,q_2,j,v}, \sigma, \tuple{q'_1,q'_2,j',v'})$ such that
$v'=v$, $(q_i,\sigma,q'_i) \in \delta_i$ ($i=1,2$), and $j' = 3-j$ if $\gamma_j(q_j,\sigma,q'_j) \geq v$,
and $j' = j$ otherwise.
\end{enumerate}
\item $\gamma$ is defined by $\gamma(q_I,\sigma,\tuple{q_1,q_2,1,v}) = 0$ and $\gamma(\tuple{q_1,q_2,j,v}, \sigma, \tuple{q'_1,q'_2,j',v'})$
is $v$ if $j \neq j'$ and $v_{\min}$ otherwise, where $v_{\min}$ is the minimal weight in $V_1 \cup V_2$.
\end{enumerate}
For the case of $\mathsf{LimInf}$-automata $A_1,A_2$, we can use the synchronized product
$A_{12} = A_1 \times A_2$, where the weight of a joint transition in $A_{12}$ is the minimum
of the corresponding transition weights in $A_1$ and $A_2$. It is easy to see that
$L_{A_{12}} = \min(L_{A_1}, L_{A_2})$ in both cases, and $A_{12}$ is deterministic
when $A_1$ and $A_2$ are deterministic. This case is simpler also because for
$\mathsf{LimInf}$-automata, deterministic and nondeterministic automata have the same
expressive power.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
On the negative side, the (deterministic or not) limit-average and discounted-sum
automata are not closed under $\min$.
\begin{comment}
The following lemma establishes the result for limit-average.
\begin{lemma}\label{lem:limavg-min-comp}
Consider the alphabet $\Sigma=\set{a,b}$, and consider the languages
$L_a$ and $L_b$ that assign the long-run average number
of $a$'s and $b$'s, respectively.
Then the following assertions hold.
\begin{enumerate}
\item There is no {\sc NLavg}\/ for the language $L_m=\min\set{L_a,L_b}$.
\item There is no {\sc NLavg}\/ for the language $L^*= 1 -\max\set{L_a,L_b}$.
\end{enumerate}
\end{lemma}
\proof
To obtain a contradiction, assume that there exists a {\sc NLavg}\/ $A$ (for either $L_m$ or $L^*$).
We first claim that there must be either an $a$-cycle or a
$b$-cycle $C$ that is reachable in $A$ such that the sum of the weights in $C$ is positive.
Otherwise, if for all $a$-cycles and $b$-cycles we have that
the sum of the weights is zero or negative, then we fool the automaton
as follows. Let $\beta$ be the maximum of the absolute values of the
weights in $A$, and let $\alpha=\lceil \beta \rceil$.
Then consider the word
$w=(a^{5\cdot \alpha \cdot \abs{Q}} \cdot b^{5\cdot\alpha \cdot \abs{Q}})^\omega$.
For a run $r$ of $A$ over $w$, the long-run average of the
weights is bounded as follows:
\[
\frac{4 \cdot \beta \cdot \abs{Q}}{10\cdot \alpha\cdot \abs{Q}} \leq \frac{2}{5}.
\]
The above bound is as follows: in the run over $a^{5 \cdot\alpha \cdot \abs{Q}}$, there
can be a prefix of size at most $\abs{Q}$ with sum of weights at most
$\abs{Q}\cdot \beta$, and then there would be $a$-cycles, and then a trailing
prefix of size at most $\abs{Q}$ with sum of weights at most $\abs{Q}\cdot \beta$.
Similar argument holds for the segment of $b^{5 \cdot \alpha \cdot \abs{Q}}$.
Hence $L_{A}(w)\leq \frac{2}{5}$, however, $L_m(w)=L^*(w)=\frac{1}{2}$,
i.e., we have a contradiction.
W.l.o.g., we assume that there is an $a$-cycle $C$ such that the sum of weights of $C$
is positive.
Then we present the following word $w$: a finite word $w_C$ to reach
the cycle $C$, followed by $a^\omega$; the answer of the automaton is positive,
{\it i.e.}, $L_{A}(w)>0$, while $L_m(w)=L^*(w)=0$.
Hence the result follows.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\end{comment}
\begin{thm}\label{theo:dla-nla-not-closed-under-min}
The (non)deterministic $\mathsf{LimAvg}$-automata are not closed under $\min$.
\end{thm}
\proof
Consider the alphabet $\Sigma=\set{a,b}$, and consider the languages
$L_a$ and $L_b$ that assign the long-run average number
of $a$'s and $b$'s, respectively. Note that there exist {\sc DLavg}\/ for the languages $L_a$ and $L_b$.
We show that there is no {\sc NLavg}\/ for the language $L_m=\min\set{L_a,L_b}$.
To obtain a contradiction, assume that there exists a {\sc NLavg}\/ $A$ for $L_m$.
We first claim that there must be either an $a$-cycle or a
$b$-cycle $C$ that is reachable in $A$ such that the sum of the weights in $C$ is positive.
Otherwise, if for all $a$-cycles and $b$-cycles we have that
the sum of the weights is zero or negative, then we fool the automaton
as follows. Let $\beta$ be the maximum of the absolute values of the
weights in $A$, and let $\alpha=\lceil \beta \rceil$.
Then consider the word
$w=(a^{5\cdot \alpha \cdot \abs{Q}} \cdot b^{5\cdot\alpha \cdot \abs{Q}})^\omega$.
For a run $r$ of $A$ over $w$, the long-run average of the
weights is bounded as follows:
\[
\frac{4 \cdot \beta \cdot \abs{Q}}{10\cdot \alpha\cdot \abs{Q}} \leq \frac{2}{5}.
\]
The above bound is as follows: in the run over $a^{5 \cdot\alpha \cdot \abs{Q}}$, there
can be a prefix of size at most $\abs{Q}$ with sum of weights at most
$\abs{Q}\cdot \beta$, and then there would be $a$-cycles, and then a trailing
prefix of size at most $\abs{Q}$ with sum of weights at most $\abs{Q}\cdot \beta$.
Similar argument holds for the segment of $b^{5 \cdot \alpha \cdot \abs{Q}}$.
Hence $L_{A}(w)\leq \frac{2}{5}$, however, $L_m(w)=\frac{1}{2}$,
i.e., we have a contradiction.
W.l.o.g., we assume that there is an $a$-cycle $C$ such that the sum of weights of $C$
is positive.
Then we present the following word $w$: a finite word $w_C$ to reach
the cycle $C$, followed by $a^\omega$; the answer of the automaton is positive,
{\it i.e.}, $L_{A}(w)>0$, while $L_m(w)=0$.
Hence the result follows.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\noindent Finally, we show that discounted-sum automata are not closed under $\min$.
\newcommand{\mathit{discount}}{\mathit{discount}}
\begin{thm}\label{theo:disc-min}
The (non)deterministic $\mathsf{Disc}$-automata are not closed under $\min$.
\end{thm}
\proof
Let $\lambda$ be a non-algebraic number in $]\frac{1}{2},1[$.
We consider the quantitative languages
$L_a^\lambda$ and $L_b^\lambda$ that assign the $\lambda$-discounted sum
of $a$'s and $b$'s, respectively.
Formally, given a (finite or infinite) word $w = w_0 w_1 \dots \in \Sigma^* \cup \Sigma^{\omega}$, let
$$v_a(w) = \sum_{i\mid w_i = a}^{\abs{w}} \lambda^i \quad \text{ and } \quad
v_b(w) = \sum_{i\mid w_i = b}^{\abs{w}} \lambda^i $$
be the $\lambda$-discounted sum of the $a$'s (resp. $b$'s) of $w$.
Then, $L_a^\lambda(w) = v_a(w)$ and $L_b^\lambda(w) = v_b(w)$.
These languages are definable by {\sc DDisc}.
We show that the language $L_m =\min(L_a^\lambda, L_b^\lambda)$ is not definable by a {\sc NDisc}.
Assume towards contradiction that there is a {\sc NDisc}\/ $A$ for $L_m$.
By Lemmas~2 and~3 in~\cite{CDH08}, there exists an infinite word $\hat{w}$
such that $v_a(\hat{w}) = v_b(\hat{w})$.
Since $v_a(\hat{w}) + v_b(\hat{w}) = \frac{1}{1-\lambda}$,
we have $L_m(\hat{w}) = \frac{1}{2(1-\lambda)}$ and this is the maximal
value of a word in $L_m(\cdot)$.
The maximal value in the automaton $A$ can be obtained for a lasso-word
of the form $w_1.(w_2)^{\omega}$ (where $w_1,w_2$ are finite words
and $w_2$ is nonempty), as pure memoryless strategies exist in games
over finite graphs with the objective to maximize the discounted sum
of payoffs. Since the language of $A$ is $L_m$, the value of $w_1.(w_2)^{\omega}$
is $\frac{1}{2(1-\lambda)}$, and thus $v_a(w_1.(w_2)^{\omega}) = v_b(w_1.(w_2)^{\omega})$ by
a similar argument as above. This last condition can be written as
$$p_a(\lambda) + \frac{\lambda^{n_1}\cdot q_a(\lambda)}{1-\lambda^{n_2}} = p_b(\lambda) + \frac{\lambda^{n_1}\cdot q_b(\lambda)}{1-\lambda^{n_2}}$$
for some polynomials $p_a, p_b, q_a, q_b$ and integers $n_1 \geq 0$ and $n_2 > 0$,
or more simply as
\begin{equation}
(1-\lambda^{n_2})\cdot p(\lambda) + \lambda^{n_1}\cdot q(\lambda) = 0 \label{eq:polynomial-lambda}
\end{equation}
for some polynomials $p$ of degree $n_1-1$ and $q$ of degree $n_2-1$,
all of whose coefficients are either $1$ or $-1$. Equation~\eqref{eq:polynomial-lambda}
is not identically zero as either $(i)$ $n_1 = 0$ and it reduces to $q(\lambda) = 0$
or $(ii)$ $n_1 > 0$ and then $p$ has degree at least $0$ so that the term of degree
zero is not null in~\eqref{eq:polynomial-lambda}.
Therefore, $\lambda$ must be algebraic, a contradiction.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\begin{comment}
\proof
Given $0<\lambda<1$, we consider the quantitative languages
$L_a^\lambda$ and $L_b^\lambda$ that assigns the $\lambda$-discounted sum
of $a$'s and $b$'s, respectively.
Let $L_m^\lambda =\min\set{L_a^\lambda, L_b^\lambda}$, and
we show that there is no {\sc NDisc} for $L_m^\lambda$.
Assume towards contradiction that there is a non-deterministic automata $A$
for $L_m^\lambda$.
Let $A$ be a non-deterministic automata with $n$ states.
Consider a non-algebraic $\lambda> \frac{1}{2}$.
Then there exists a real $\alpha>0$ such that for all finite words $w'$ of
length at most $n$ we have
\[
\abs{v_a(w') -v_b(w')} \geq \alpha.
\]
Consider an infinite word $w$ such that the following conditions hold:
(i)~$v_a(w)>v_b(w)$ and
(ii)~$v_a(w)-v_b(w) < \alpha\cdot \lambda^{3n}$.
Such a word $w$ can be easily constructed, since given $\lambda>\frac{1}{2}$
we can construct words $w$ such that the difference of $\abs{v_a(w) -v_b(w)}$
is arbitrarily close to $0$.
Consider a run $r$ of $A$ over $w$ such that the value of the run $r$ (i.e., the
$\lambda$-discounted sum of the weights in $r$) is $v_b(w)$ (since we assumed that
$A$ returns the value of the min of $L_a$ and $L_b$, the output must be
$v_b(w)$ for some run $r$).
Consider a cycle $C$ in $r$ and let the corresponding segment in $w$ be $w_c$.
We first consider the following cases.
We split the word $w$ as $w=w_1 \cdot w_c \cdot w_2$ and the run $r=r_1 \cdot
C \cdot r_2$.
Observe that the $\lambda$-discounted sum of $a$'s and the $\lambda$-discounted
sum of $b$'s in $w_c$ cannot be equal (otherwise we would have a polynomial
equation with a non-algebraic root $\lambda$).
In the following we denote by $\mathit{discount}_{w_c}(a)$ and $\mathit{discount}_{w_c}(b)$
the $\lambda$-discounted sum of $a$'s and $b$'s in $w_c$, respectively, and we
denote by $\mathit{discount}(C)$ the $\lambda$-discounted sum of the weights in $C$.
\begin{enumerate}
\item We first consider the case when $\mathit{discount}_{w_c}(b) > \mathit{discount}_{w_c}(a)$, i.e., the
discounted sum of $b$'s in $w_c$ is greater than that of $a$'s.
We have the following two sub-cases.
\begin{enumerate}
\item If $\mathit{discount}_{w_c}(b) > \mathit{discount}(C)$, (i.e., the discounted sum of $b$'s in
$w_c$ exceeds the discounted sum of weights of $C$), then
consider the word $w'=w_1 \cdot w_2$ (i.e., we omit the segment $w_c$).
Since $\mathit{discount}_{w_c}(b) > \mathit{discount}_{w_c}(a)$, it follows that we have
$v_a(w') > v_b(w')$, and for the run $r'=r_1 \cdot r_2$ we have the value of
the run $r'$ exceeds $v_b(w')$ (since $\mathit{discount}_{w_c}(b) > \mathit{discount}(C)$).
Hence we would have $L_{A}(w') > L_m^\lambda(w')$, this would be
a contradiction.
\item If $\mathit{discount}_{w_c}(b) < \mathit{discount}(C)$, then consider the word
$w'=w_1 \cdot w_c \cdot w_c \cdot w_2$ (i.e., we pump $w_c$ once more).
Consider the run $r'=r_1 \cdot C \cdot C \cdot r_2$, i.e., we pump the
cycle $C$ once more.
Since $\mathit{discount}_{w_c}(b) < \mathit{discount}(C)$, the value of the run $r'$ exceeds $v_b(w')$.
Hence we have $L_{A}(w') > v_b(w') \geq \min \set{v_a(w'),v_b(w')}$.
Hence we would have a contradiction.
\end{enumerate}
Hence we conclude that for any cycle $C$ and the corresponding word $w_c$
we must have
$\mathit{discount}_{w_c}(b)=\mathit{discount}(C)$.
\item We now consider the case when $\mathit{discount}_{w_c}(b) < \mathit{discount}_{w_c}(a)$.
We consider two sub-cases.
\begin{enumerate}
\item If $\mathit{discount}_{w_c}(b) < \mathit{discount}(C)$, then consider the word
$w'=w_1 \cdot w_c \cdot w_c \cdot w_2$ (i.e., we pump $w_2$ once more).
Since $\mathit{discount}_{w_c}(b) < \mathit{discount}_{w_c}(a)$, we have $v_a(w')> v_b(w')$, and
since $\mathit{discount}_{w_c}(b) < \mathit{discount}(C)$, for the run $r'=r_1 \cdot C \cdot C \cdot r_2$
we have the value of $r'$ exceeds $v_b(w')$.
Hence this would be contradiction.
\item If $\mathit{discount}_{w_c}(b) > \mathit{discount}(C)$, then consider the word $w'=w_1 \cdot
w_2$, (i.e., we omit $w_c$).
The value of the run $r'=r_1 \cdot r_2$ exceeds $v_b(w')$.
Hence we have $L_{A}(w') > v_b(w') \geq \min \set{v_a(w'),v_b(w')}$,
i.e., we would have a contradiction.
\end{enumerate}
Hence we conclude that for all cycles $C$ and the corresponding word $w_c$
of $w$ we have $\mathit{discount}_{w_c}(b)=\mathit{discount}(C)$.
\end{enumerate}
Consider the first cycle $C$ in the run $r$, and the cycle must start with
in distance $n$ from the starting point of the word.
From the above analysis we have that $\mathit{discount}_{w_c}(b) =\mathit{discount}(C)$.
We now consider two cases to obtain the contradiction.
\begin{enumerate}
\item If $\mathit{discount}_{w_c}(a) >\mathit{discount}_{w_c}(b)$, then if we omit $w_c$,
i.e., we consider the word $w'=w_1 \cdot w_2$.
Then for the run $r'=r_1 \cdot r_2$ we have the value of the run
is $v_b(w')$.
But since $v_a(w) - v_b(w) < \alpha \cdot \lambda^{3n}$, we have
$v_a(w')<v_b(w')$.
Thus we have $L_{A}(w') > L_m(w')$.
Hence we obtain a contradiction.
\item If $\mathit{discount}_{w_c}(b) > \mathit{discount}_{w_c}(a)$, then we pump $w_c$
again, i.e., consider the word $w'=w_1 \cdot w_C \cdot w_C \cdot w_2$.
The run $r'=r_1 \cdot C \cdot C \cdot r_2$ yields a value of
$v_b(w')$.
However, since $v_a(w) -v_b(w) \leq \alpha \cdot \lambda^{3n}$
we have $v_a(w') < v_b(w')$.
Thus we have $L_{A}(w') > L_m(w')$.
We again have a contradiction.
\end{enumerate}
The result follows.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\end{comment}
\subsection{Closure under complement for infinite words}
Most of the weighted automata are not closed under complement.
The next result is a direct extension of the boolean case.
\begin{thm}\label{theo:max-liminf-limsup-not-closed-under-complement}
The (non)deterministic $\mathsf{Sup}$- and $\mathsf{LimInf}$-automata, and the deterministic
$\mathsf{LimSup}$-automata are not closed under complement.
\end{thm}
\proof
The result follows from a similar result for the boolean version of these classes.
For {\sc DSup}\/ and {\sc NSup}, consider the language $L_1$ over $\Sigma=\{a,b\}$
such that $L_1(a^{\omega}) = 0$ and $L_1(w) = 1$ for all $w \neq a^{\omega}$.
For {\sc DLinf}\/ and {\sc NLinf}, consider the language $L_2$ over $\Sigma=\{a,b\}$
such that $L_2(\Sigma^*.a^{\omega}) = 1$ and $L(w) = 0$ for all words $w$ containing
infinitely many $b$'s, and for {\sc DLsup}, consider $L_3$ the complement of $L_2$.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
The next theorem is a positive result of closure under complementation
for {\sc NLsup}. It reduces to the complementation of nondeterministic B\"uchi automata.
\begin{thm}\label{theo:nls-closed-under-complement}
The nondeterministic $\mathsf{LimSup}$-automata are closed under complement, with cost $O(m \cdot 2^{n \log n})$.
\end{thm}
\proof
Let $A=\tuple{Q,q_{0},\Sigma,\delta,\gamma}$ be a {\sc NLsup},
and let $V = \{\gamma(e) \mid e \in \delta\}$
be the set of weights that appear in $A$.
For each $v \in V$, it is easy to construct a {\sc NBW}\/ $A_v$
whose (boolean) language is the set of words $w$
such that $L_A(w) \geq v$, by declaring to be accepting
the edges with weight at least $v$.
We then construct for each $v \in V$ a {\sc NBW}\/ $\bar{A}_v$ (with accepting edges)
that accepts the (boolean) complement of the language accepted by $A_v$.
Finally, assuming that $V= \{v_1,\dots,v_n\}$ with $v_1 < v_2 < \dots < v_n$,
we construct the {\sc NLsup}\/ $B_i$ for $i=2,\dots,n$ where $B_i$ is obtained
from $\bar{A}_{v_i}$ by assigning weight $1-v_{i-1}$ to each accepting edges,
and $1-v_n$ to all other edges. The complement of $L_A$ is then
$\max\{L_{B_2},\dots,L_{B_n}\}$ which is accepted by a {\sc NLsup}\/ by Theorem~\ref{theo:max-closure}.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\begin{thm}
The deterministic $\mathsf{Disc}$-automata are closed under complement, with cost $O(n)$.
\end{thm}
\proof[sketch]
It suffices to replace each weight $v$ of a {\sc DDisc}\/ by $1-\lambda-v$
(where $\lambda$ is the discount factor) to obtain the {\sc DDisc}\/ for
the complement.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\begin{thm}
The deterministic $\mathsf{LimAvg}$-automata are not closed under complement.
\end{thm}
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}[node distance=1.8cm,auto,shorten >=1 pt,>=latex']
\node[rdbox,initial,initial text={}] (0) at (0,0) {} ;
\draw[->] (0) edge[loop,out=120,in=60,looseness=8]
node[above] {$a,1$}(0);
\draw[->] (0) edge[loop,out=240, in=300,looseness=8]
node[below] {$b,0$}(0);
\end{tikzpicture}
\end{center}
\caption{Deterministic Limit-average Automaton.}
\label{figure:aut1}
\end{figure}
\proof
Consider the {\sc DLavg}\/ $A$ over alphabet $\Sigma= \{a,b\}$
(shown in Figure~\ref{figure:aut1}) that consists of
a single self-loop state with weight $1$ for $a$ and $0$ for $b$.
Notice that $L_A(w.a^\omega) = 1$ and $L_A(w.b^\omega) = 0$ for all $w \in \Sigma^*$.
To obtain a contradiction, assume that there exists a {\sc DLavg}\/ $B$
whose language is $L_B = 1-L_A$. For all finite words $w \in \Sigma^*$,
let $L^{\mathsf{Avg}}_B(w)$ be the average weight of the unique (finite) run of $B$ over $w$.
Fix $0 < \epsilon < \frac{1}{2}$. For all finite words $w$, there exists
a number $n_w$ such that the average number of $a$'s in $w.b^{n_w}$ is at most $\epsilon$,
and there exists a number $m_w$ such that $L^{\mathsf{Avg}}_B(w.a^{m_w}) \leq \epsilon$
(since $L_B(w.a^\omega) = 0$). Hence, we can construct a word
$w = b^{n_1} a^{m_1} b^{n_2} a^{m_2} \dots$ such that
$L_A(w) \leq \epsilon$ and $L_B(w) \leq \epsilon$.
Since $L_B = 1-L_A$, this implies that $1 \leq 2\epsilon$, a contradiction.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\begin{thm}
The nondeterministic $\mathsf{LimAvg}$- and $\mathsf{Disc}$-automata are not closed under complement.
\end{thm}
\proof
The fact that {\sc NLavg}\/ are not closed under complementation is
as follows. Consider the quantitative language $L^*=1-\max\set{L_a,L_b}$
where $L_a$ and $L_b$ assign the long-run average number of $a$'s and $b$'s, respectively.
Exactly the same argument as in the proof of Theorem~\ref{theo:dla-nla-not-closed-under-min}
shows that $L^*$ cannot be expressed as a
{\sc NLavg}, while the language $\max\set{L_a,L_b}$ can be expressed
as {\sc NLavg}\/ by Theorem~\ref{theo:max-closure}.
That {\sc NDisc}\/ are not closed under complement can be
obtained as follows: given $0<\lambda<1$, consider the
language $L_a^\lambda$ and $L_b^\lambda$ that assigns to words the
$\lambda$-discounted sum of $a$'s and $b$'s, respectively.
The language $L_a^\lambda$ and $L_b^\lambda$ can be expressed
as {\sc DDisc}, and the max of them can be defined by {\sc NDisc}.
Observe that $L_a^\lambda(w)+L_b^\lambda(w) = \frac{1}{1-\lambda}$
for all $w \in \Sigma^{\omega}$. Therefore,
$\min\set{L_a^\lambda,L_b^\lambda}=\frac{1}{1-\lambda}-\max\set{L_a^\lambda,L_b^\lambda}$.
Since {\sc NDisc}\/ is not closed under min (Theorem~\ref{theo:disc-min}), we
immediately obtain that {\sc NDisc}\/ are not closed under complementation.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\subsection{Closure under sum for infinite words}
All weighted automata are closed under sum,
except {\sc DLavg}\/ and {\sc NLavg}.
\begin{thm}\label{theo:max-closed-under-sum}
The (non)deterministic $\mathsf{Sup}$-automata are closed under sum, with cost $O(n_1\cdot m_1 \cdot n_2 \cdot m_2)$.
\end{thm}
\proof
The construction is the same as for $\mathsf{Max}$-automata over finite words
given in the proof of Theorem~\ref{theo:closure-under-sum-finite}.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\begin{thm}\label{theo:nls-closed-under-sum}
The nondeterministic $\mathsf{LimSup}$-automata are closed under sum, with cost $O(n_1\cdot m_1 \cdot n_2 \cdot m_2)$.
\end{thm}
\proof[Sketch]
Given two {\sc NLsup}\/ $A_1$ and $A_2$, we construct a {\sc NLsup}\/ $A$
for the sum of their languages as follows. Initially, we make a guess of
a pair $(v_1,v_2)$ of weights ($v_i$ in $A_i$, for $i=1,2$) and we branch
to a copy of the synchronized product of $A_1$ and $A_2$. We attach a bit $b$
whose range is $\{1,2\}$ to each state to remember that we expect $A_b$ to
visit the guessed weight $v_b$. Whenever this occurs, the bit $b$ is set to $3-b$,
and the weight of the transition is $v_1 + v_2$. All other transitions ({\it i.e.}
when $b$ is unchanged) have weight $\min\{v_1 + v_2 \mid v_1 \in V_1 \land v_2 \in V_2\}$.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\begin{thm}
The deterministic $\mathsf{LimSup}$-automata are closed under sum, with cost $O(n_1\cdot n_2 \cdot 2^{m_1 \cdot m_2})$.
\end{thm}
\proof
Let $A_1=\tuple{Q_1,q_I^1,\Sigma,\delta_1,\gamma_1}$ and $A_2=\tuple{Q_2,q_I^2,\Sigma,\delta_2,\gamma_2}$
be two {\sc DLsup}. We construct a {\sc DLsup}\/ $A=\tuple{Q,q_I,\Sigma,\delta,\gamma}$ such that
$L_A = L_{A_1} + L_{A_2}$. Let $V_i = \{\gamma_i(e) \mid e \in \delta_i\}$
be the set of weights that appear in $A_i$ (for $i=1,2$).
The automaton $A$ implements the synchronized product of $A_1$ and $A_2$,
and keeps one bit $b(v_1,v_2)$ for each pair $(v_1,v_2)$ of weights $v_1 \in V_1$ and $v_2 \in V_2$.
For $i=1,2$, if $b(v_1,v_2)=i$, then $A_i$ is expected to cross a transition with weight
$v_i$. Whenever this occurs, the bit is set to $3-i$.
The weight of a transition in $A$ is the largest value of $v_1 + v_2$
such that the corresponding bit $b(v_1,v_2)$ has changed in the transition.
Formally, we define:
\begin{enumerate}[$\bullet$]
\item $Q = Q_1 \times Q_2 \times [V_1 \times V_2 \to \{1,2\}]$;
\item $q_I = \tuple{q_I^1,q_I^2,b_I}$ where $b_I(v_1,v_2) = 1$ for all $(v_1,v_2) \in V_1 \times V_2$;
\item For each $\sigma \in \Sigma$, the set $\delta$ contains all the triples
$(\tuple{q_1,q_2,b}, \sigma, \tuple{q'_1,q'_2,b'})$ such that
$(q_i,\sigma,q'_i) \in \delta_i$ ($i=1,2$), and for all $(v_1,v_2) \in V_1 \times V_2$,
we have $b'(v_1,v_2) = 3-b(v_1,v_2)$
if $\gamma_i(\tuple{q_i,\sigma,q'_i}) = v_i$ for $i=b(v_1,v_2)$,
and otherwise $b'(v_1,v_2) = b(v_1,v_2)$.
\item $\gamma$ is defined by $\gamma(\tuple{q_1,q_2,b}, \sigma, \tuple{q'_1,q'_2,b'}) =
\max(\{v_{\min} \cup \{v_1 + v_2 \mid b'(v_1,v_2) \neq b(v_1,v_2) \})$
where $v_{\min}$ is the minimal weight in $V_1 + V_2 = \{v_1 + v_2 \mid v_1 \in V_1 \land v_2 \in V_2\}$.{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\end{enumerate}
\begin{thm}
The (non)deterministic $\mathsf{LimInf}$-automata are closed under sum with cost $O(n_1\cdot n_2 \cdot 2^{m_1 \cdot m_2})$.
\end{thm}
\proof
Let $A_1=\tuple{Q_1,q_I^1,\Sigma,\delta_1,\gamma_1}$ and $A_2=\tuple{Q_2,q_I^2,\Sigma,\delta_2,\gamma_2}$
be two {\sc NLinf}. We construct a {\sc NLinf}\/ $A=\tuple{Q,q_I,\Sigma,\delta,\gamma}$ such that
$L_A = L_{A_1} + L_{A_2}$. Let $V_i = \{\gamma_i(e) \mid e \in \delta_i\}$
be the set of weights that appear in $A_i$ (for $i=1,2$).
The automaton $A$ implements the synchronized product of $A_1$ and $A_2$,
and keeps one bit $b(v_1,v_2)$ for each pair $(v_1,v_2)$ of weights $v_1 \in V_1$ and $v_2 \in V_2$.
If a transition in $A_i$ for some $i \in \{1,2\}$ has weight less than $v_i$,
then the bit $b(v_1,v_2)$ is set to $\bot$, otherwise is set to $\top$.
The weight of a transition in $A$ is the largest value of $v_1 + v_2$
such that the corresponding bit $b(v_1,v_2)$ is $\top$.
Formally, we define:
\begin{enumerate}[$\bullet$]
\item $Q = Q_1 \times Q_2 \times [V_1 \times V_2 \to \{\top,\bot\}]$;
\item $q_I = \tuple{q_I^1,q_I^2,b_I}$ where $b_I(v_1,v_2) = \bot$ for all $(v_1,v_2) \in V_1 \times V_2$;
\item For each $\sigma \in \Sigma$, the set $\delta$ contains all the triples
$(\tuple{q_1,q_2,b}, \sigma, \tuple{q'_1,q'_2,b'})$ such that
$(q_i,\sigma,q'_i) \in \delta_i$ ($i=1,2$), and for all $(v_1,v_2) \in V_1 \times V_2$,
we have $b'(v_1,v_2) = \top$
if $\gamma_i(\tuple{q_i,\sigma,q'_i}) \geq v_i$ for $i=1,2$,
and otherwise $b'(v_1,v_2) = \bot$.
\item $\gamma$ is defined by $\gamma(\tuple{q_1,q_2,b}, \sigma, \tuple{q'_1,q'_2,b'}) =
\max(\{v_{\min} \cup \{v_1 + v_2 \mid b'(v_1,v_2) = \top \})$
where $v_{\min}$ is the minimal weight in $V_1 + V_2 = \{v_1 + v_2 \mid v_1 \in V_1 \land v_2 \in V_2\}$.
\end{enumerate}
The result for {\sc DLinf}\/ follows from the fact $A$ is deterministic if
$A_1$ and $A_2$ are deterministic.
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\begin{thm}\label{theo:ddi-ndi-closed-under-sum}
The (non)deterministic $\mathsf{Disc}$-automata are closed under sum, with cost $O(n_1 \cdot n_2)$.
\end{thm}
\proof[Sketch]
It is easy to see that the synchronized product of two {\sc NDisc}\/ (resp. {\sc DDisc})
defines the sum of their languages, if the weight of a joint transition
is defined as the sum of the weights of the corresponding transitions in the two {\sc NDisc}\/ (resp. {\sc DDisc}).
{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\begin{thm}
The (non)deterministic $\mathsf{LimAvg}$-automata are not closed under sum.
\end{thm}
\proof
Consider the alphabet $\Sigma=\set{a,b}$, and consider the {\sc DLavg}-definable languages
$L_a$ and $L_b$ that assigns to each word $w$ the long-run average number of $a$'s and $b$'s in $w$
respectively.
Let $L_{+}=L_a + L_b$. Assume that $L_{+}$ is defined by a {\sc NLavg}\/ $A$ with set
of states $Q$ (we assume w.l.o.g that every state in $Q$ is reachable).
First, we claim that from every state $q \in Q$, there is a run of $A$ over $a^{\abs{Q}}$
that visit a cycle $C^*$ with average weight $1$. To see this, notice that from every state $q \in Q$,
there is an infinite run $\rho$ of $A$ over $a^\omega$ whose value is $1$
(since $L_{+}(w_q\cdot a^\omega) =1$ for all finite words $w_q$).
Consider the following decomposition of $\rho$.
Starting with an empty stack, we push the states of $\rho$ onto the stack
as soon as all the states on the stack are different. If the next state is already
on the stack, we pop all the states down to the repeated state thus removing a simple
cycle of $\rho$. Let $C_1$, $C_2, \dots$ be the cycles that are successively
removed. Observe that the height of the stack is always at most $\abs{Q}$.
Let $\beta$ be the largest average weight of the cycles $C_i$, $i\geq 1$,
and let $\alpha_{\max}$ be the largest weight in $A$.
Assume towards contradiction that $\beta < 1$. Then, for all $n > 0$,
the value of the prefix of length $n$ of $\rho$ is at most:
$$ \frac{\alpha_{\max} \cdot \abs{Q} + \beta \cdot \sum_{i=1}^{k_n} \abs{C_i}}{n} $$
where $k_n$ is the number of cycles that have been removed from the stack
when reading the first $n$ symbols of $\rho$. Hence, the value of $\rho$ is at most $\beta < 1$,
which is a contradiction. Therefore, the average weight of some cycle $C^* = C_i$
is exactly\footnote{It cannot be greater than $1$ since $L_{+}(w\cdot a^\omega)=1$ for all finite
words $w$.} $1$ (there are finitely many different cycles as they are simple cycles).
Since the height of the stack is at most $\abs{Q}$, the cycle $C^*$ is reachable
in at most $\abs{Q}$ steps.
Second, it can be shown analogously that from every state $q \in Q$, there is a
run over $b^{\abs{Q}}$ that visit a cycle $C^*$ with average weight $1$.
Third, for arbitrarily small $\epsilon >0$, consider the word $w$ and the run $\rho$ of $A$ over $w$ generated inductively
by the following procedure: $w_0$ is the empty word and $\rho_0$ is the initial state of $A$
We generate $w_{i+1}$ and $\rho_{i+1}$ from $w_i$ and $\rho_i$ as follows:
\begin{compressEnum}
\stepcounter{compressEnum}{(\thecompressEnum) } generate a long enough sequence $w_{i+1}'$ of $a$'s after $w_i$ such that the
average number of $b$'s in $w_i \cdot w_{i+1}'$ falls below $\epsilon$
and we can continue $\rho_i$ and reach within at most $\abs{Q}$ steps (and then repeat $k$ times)
a cycle $C$ of average weight $1$ and such that the average weight of this run
prolonged by $\abs{Q}$ arbitrary transitions is at least $1-\epsilon$, {\it i.e.}
$$ \frac{\gamma(\rho_i)+k\cdot\abs{C} + 2 \alpha_{\min} \cdot \abs{Q} }{\abs{\rho_i} + k\cdot \abs{C} + 2 \cdot \abs{Q}}
\geq 1-\epsilon$$
where $\alpha_{\min}$ is the least weight in $A$. This is possible since $k$ can be chosen
arbitrarily large. Let $\rho'_i$ be the prolongation of $\rho_i$ over $w_{i+1}'$;
\stepcounter{compressEnum}{(\thecompressEnum) } then generate a long enough sequence $w_{i+1}''$ of $b$'s such that the average number of
$a$'s in $w_{i} \cdot w_{i+1}' \cdot w_{i+1}''$ falls below $\epsilon$
and as above, we can construct a continuation $\rho''_i$ of $\rho'_i$ whose
average weight is at least $1-\epsilon$ (even if prolonged by $\abs{Q}$ arbitrary transitions);
\stepcounter{compressEnum}{(\thecompressEnum) } the word $w_{i+1}=w_i \cdot w_{i+1}' \cdot w_{i+1}''$ and the run
$\rho_{i+1}$ is $\rho''_i$.
\end{compressEnum}
The word $w$ and the run $\rho$ are the limit of these sequences.
We have $L_{a}(w) = L_{b}(w) = 0$ and thus $L_{+}(w) = 0$, while
the value of $\rho$ is at least $1-\epsilon$, a contradiction.{\nobreak\leavevmode\endof\eopsymbol}\medskip}\global\def\qed{}
\paragraph{{\bf Acknowledgment.}} We thank Wolfgang Thomas for pointing out the isolated cut-point problem.
| {
"timestamp": "2010-09-03T02:00:53",
"yymm": "1007",
"arxiv_id": "1007.4018",
"language": "en",
"url": "https://arxiv.org/abs/1007.4018",
"abstract": "Weighted automata are nondeterministic automata with numerical weights on transitions. They can define quantitative languages~$L$ that assign to each word~$w$ a real number~$L(w)$. In the case of infinite words, the value of a run is naturally computed as the maximum, limsup, liminf, limit-average, or discounted-sum of the transition weights. The value of a word $w$ is the supremum of the values of the runs over $w$. We study expressiveness and closure questions about these quantitative languages. We first show that the set of words with value greater than a threshold can be non-$\\omega$-regular for deterministic limit-average and discounted-sum automata, while this set is always $\\omega$-regular when the threshold is isolated (i.e., some neighborhood around the threshold contains no word). In the latter case, we prove that the $\\omega$-regular language is robust against small perturbations of the transition weights. We next consider automata with transition weights Weighted automata are nondeterministic automata with numerical weights ontransitions. They can define quantitative languages~$L$ that assign to eachword~$w$ a real number~$L(w)$. In the case of infinite words, the value of arun is naturally computed as the maximum, limsup, liminf, limit-average, ordiscounted-sum of the transition weights. The value of a word $w$ is thesupremum of the values of the runs over $w$. We study expressiveness andclosure questions about these quantitative languages. We first show that the set of words with value greater than a threshold canbe non-$\\omega$-regular for deterministic limit-average and discounted-sumautomata, while this set is always $\\omega$-regular when the threshold isisolated (i.e., some neighborhood around the threshold contains no word). Inthe latter case, we prove that the $\\omega$-regular language is robust againstsmall perturbations of the transition weights. We next consider automata with transition weights $0$ or $1$ and show thatthey are as expressive as general weighted automata in the limit-average case,but not in the discounted-sum case. Third, for quantitative languages $L_1$ and~$L_2$, we consider the operations$\\max(L_1,L_2)$, $\\min(L_1,L_2)$, and $1-L_1$, which generalize the booleanoperations on languages, as well as the sum $L_1 + L_2$. We establish theclosure properties of all classes of quantitative languages with respect tothese four operations.$ or $ and show that they are as expressive as general weighted automata in the limit-average case, but not in the discounted-sum case. Third, for quantitative languages $L_1$ and~$L_2$, we consider the operations $\\max(L_1,L_2)$, $\\min(L_1,L_2)$, and -L_1$, which generalize the boolean operations on languages, as well as the sum $L_1 + L_2$. We establish the closure properties of all classes of quantitative languages with respect to these four operations.",
"subjects": "Logic in Computer Science (cs.LO)",
"title": "Expressiveness and Closure Properties for Quantitative Languages",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754492759498,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.7075103969126288
} |
https://arxiv.org/abs/1311.3867 | The Robber Locating game | We consider a game in which a cop searches for a moving robber on a graph using distance probes, studied by Carragher, Choi, Delcourt, Erickson and West, which is a slight variation on one introduced by Seager. Carragher, Choi, Delcourt, Erickson and West show that for any fixed graph $G$ there is a winning strategy for the cop on the graph $G^{1/m}$, obtained by replacing each edge of $G$ by a path of length $m$, if $m$ is sufficiently large. They conjecture that the cop does not have a winning strategy on $K_n^{1/m}$ if $m<n$; we show that in fact the cop wins if and only if $m\geqslant n/2$, for all but a few small values of $n$. They also show that the robber can avoid capture on any graph of girth 3, 4 or 5, and ask whether there is any graph of girth 6 on which the cop wins. We show that there is, but that no such graph can be bipartite; in the process we give a counterexample for their conjecture that the set of graphs on which the cop wins is closed under the operation of subdividing edges. We also give a complete answer to the question of when the cop has a winning strategy on $K_{a,b}^{1/m}$. | \section{Introduction}
Pursuit and evasion games on graphs have been widely studied. Perhaps the most significant variant is the Cops and Robbers game, an instance of which is a graph $G$ together with a fixed number of cops. The cops take up positions on vertices of $G$ and a robber then starts on any unoccupied vertex. The cops and the robber take turns: the robber chooses either to remain at his current vertex or to move to any adjacent vertex, and then the cops simultaneously make moves of the same form. The game is played with perfect information, so that at any time each of the players knows the location of all others. The cops win if at any point one of them is at the same location as the robber. The cop number of a graph is the minimum number of cops required for the cops to have a winning strategy. Early results on this game include those obtained by Nowakowski and Winkler \cite{NW83}, who categorise the graphs of cop number 1, and Aigner and Fromme \cite{AF84}, who show that every planar graph has cop number at most 3. An important open problem is Meyniel's conjecture, published by Frankl \cite{Fra87}, that the cop number of any $n$-vertex connected graph is at most $O(\sqrt{n})$ -- this has been shown to be true up to a $\log(n)$ factor for random graphs by Bollob{\'a}s, Kun and Leader \cite{BLK13}, following which \L uczak and Pra\l at improved the error term \cite{LP10}. More recently, several variations on the game have been analysed by Clarke and Nowakowski (e.g. \cite{CN00}).
In this paper we consider the Robber Locating game, introduced in a slightly different form by Seager \cite{Sea12}, and further studied by Carragher, Choi, Delcourt, Erickson and West \cite{CCDEW}, in which a cop probes a vertex at each turn and is told the current distance to the robber. For ease of reading we shall refer to the cop as female and the robber as male. In this setting the cop is not on the graph herself, and can probe vertices without restriction; she wins if at any point she is able to determine the robber's current location. Clearly the cop can win eventually with probability 1 on a finite graph against a robber who has no knowledge of her future moves, simply by probing random vertices until she hits the current location of the robber. This naturally leads to a different emphasis: we consider the question of whether the cop has a strategy which is guaranteed to win in bounded time, or equivalently whether she can catch an omniscient robber. We say that a graph is \textit{locatable} if such a strategy exists. A similar game phrased in terms of a cat and mouse, in which the cat wins only if it probes the current location of the mouse, and receives no information otherwise, but the mouse must move at each turn, was recently analysed by one of the authors \cite{Has13}.
In the Robber Locating game each round consists of a move for the robber, in which he either moves to an adjacent vertex or stays where he is, followed by a probe of a particular vertex by the cop. The cop then receives a response giving the current distance of the robber from the vertex probed. She wins if she is then able to identify the robber's location. In the game as introduced by Seager there was an additional rule that the robber cannot move to the vertex probed in the previous round (the \textit{no-backtrack condition}). Carragher, Choi, Delcourt, Erickson and West consider the game without this restriction, as do we.
The authors of \cite{CCDEW} write $G^{1/m}$ for the graph obtained by replacing each edge of $G$ by a path of length $m$ through new vertices. Each such path is called a \textit{thread}, and an \textit{original vertex} in $G^{1/m}$ is a vertex which corresponds to a vertex of $G$. The main result of \cite{CCDEW} is that $G^{1/m}$ is locatable provided $m\geqslant \min\{n(G),1+\max\{\mu(G)+2^{\mu(G)},\Delta(G)\}\}$, where $\mu(G)$ is the metric dimension of $G$. The notion of metric dimension was introduced independently by Slater \cite{Sla75}, and by Harary and Melter \cite{HM76}. The metric dimension of $G$ is the size of the smallest set $S$ of vertices such that for every $x,y\in V(G)$ with $x\neq y$ there is some $z\in S$ with $d(x,z)\neq d(y,z)$.
The authors of \cite{CCDEW} give better bounds on $m$ for complete bipartite graphs, and in this case we will find the critical value of $m$ exactly. They also conjecture that their bound is tight for complete graphs, i.e. that $K^{1/m}_n$ is locatable if and only if $m\geqslant n$. We show that in fact, except for a few small values of $n$, the actual threshold is $n/2$. They also prove that no graph of girth 3, 4 or 5 is locatable. The cycle $C_6$ is not locatable, and so they ask whether there is a locatable graph of girth 6. We give an example of such a graph, but show that no bipartite graph of girth 6 is locatable. In the process we give a counterexample to their conjecture that if $G$ is locatable then so is any graph obtained by subdividing a single edge of $G$.
\section{Graphs of girth 6}
\label{Girth6Section}
In this section we first give an example of a locatable graph of girth 6, together with an explicit strategy for the cop. Define $H$ to be the graph obtained from the cycle $v_1v_2\cdots v_{11}$ by adding the edge $v_3v_9$. $H$ consists of a 6-cycle and a 7-cycle with an edge in common. We include an illustration of $H$ in Figure \ref{robbergirth6graph}.
\begin{figure}[ht!] \centering
\begin{tikzpicture}[scale=0.6]
\tikzstyle{vertex}=[draw,shape=ellipse,minimum size=15pt,inner sep=0pt, fill=white]
\tikzstyle{smallvertex}=[draw,shape=ellipse,minimum size=5pt,inner sep=0pt, fill=white]
\foreach \x/\y/\name/\alabel in {-3.464/1/$v_1$/V1, -1.732/2/$v_2$/V2, 0/1/$v_3$/V3, 1.732/2/$v_4$/V4, 3.732/2/$v_5$/V5, 5.464/0/$v_6$/V6, 3.732/-2/$v_7$/V7, 1.732/-2/$v_8$/V8, 0/-1/$v_9$/V9, -1.732/-2/$v_{10}$/V10, -3.464/-1/$v_{11}$/V11}
{\node[vertex] (\alabel) at (\x, \y) {\name};}
\foreach \start/\end in {V1/V2, V2/V3, V3/V4, V4/V5, V5/V6, V6/V7, V7/V8, V8/V9, V9/V10, V10/V11, V11/V1, V3/V9}
{\draw (\start) -- (\end);}
\end{tikzpicture}
\caption{A cycle of length 6 and one of length 7 sharing an edge.}
\label{robbergirth6graph}
\end{figure}
\begin{theorem}\label{c78}The graph $H$ as defined above is locatable.
\end{theorem}
\begin{proof}
We first give several situations from which the cop can either win or reduce to an earlier situation, and then show how she can reach a winning situation.
\begin{enumerate}
\item If the robber is known to be at $v_2$ or $v_4$ then the cop wins by probing $v_1$.
\item If the robber is known to be at $v_3$ or $v_4$ then the cop probes $v_9$, winning or reducing to (i).
\item If the robber is known to be at $v_3$ or $v_8$ then the cop probes $v_7$, winning or reducing to (ii).
\item If the robber is known to be at $v_3$ or $v_9$ then the cop probes $v_{10}$, winning or reducing to (ii) or (iii).
\item If the robber is known to be at $v_4$ or $v_5$ then the cop wins by probing $v_6$.
\item If the robber is known to be at $v_5$ or $v_7$ then the cop probes $v_8$, winning or reducing to (v).
\item If the robber is known to be at $v_6$ or $v_8$ then the cop probes $v_6$, winning or reducing to (vi).
\item If the robber is known to be at $v_4$ or $v_7$ then the cop probes $v_7$, winning or reducing to (vii) or (ii).
\item If the robber is known to be at $v_4$ or $v_8$ then the cop probes $v_9$, winning or reducing to (iii) or (viii).
\item If the robber is known to be at $v_1$ or $v_{11}$ then the cop wins by probing $v_2$.
\end{enumerate}
The cop starts by probing $v_6$. If the answer is 0 she has won, and if it is 1, 2, 3, or 5 she has reduced to (vi), (ix), (iv), or (x) respectively. Otherwise the answer must be 4, in which case she probes $v_2$. This locates him unless the answer is 1 (when the robber must be at $v_1$ or $v_3$) or 2 (when he must be at $v_{11}$ or $v_9$). These two cases are equivalent by the symmetry of $H$, so assume the former. Now the cop probes $v_6$. If the answer is 4 the robber must be at $v_2$. If not the cop has reduced to (iv) or (x).
\end{proof}
We have shown that there is a locatable graph of girth 6, answering a question of \cite{CCDEW}. Next we show that a significant class of graphs of girth 6 are non-locatable.
\begin{theorem}Any bipartite graph of girth 6 is non-locatable.
\end{theorem}
\begin{proof}Let $G$ be a bipartite graph of girth 6 and let $C$ be a 6-cycle of $G$. We show that the robber can win even if he is restricted to $V(C)$, by proving that if there are two non-adjacent possible robber locations in $V(C)$ after the $t^\text{th}$ probe, then no matter what vertex the cop probes next, some answer will leave two non-adjacent possible robber locations.
Suppose the robber may be at either of two non-adjacent vertices in $V(C)$ after the cop's $t^\text{th}$ probe. There are at least 5 vertices in $V(C)$ which the robber may have reached before the $(t+1)^\text{st}$ probe. Suppose the cop's $(t+1)^\text{st}$ probe is at some vertex $v$, and consider the distances from $v$ to these 5 vertices. Writing $d$ for the minimum of these distances, each one must be either $d$, $d+1$, $d+2$ or $d+3$. Since there are 5 vertices, some two must be at the same distance from $v$, so if that distance is returned there are two vertices in $V(C)$ which are possible robber locations after the $(t+1)^\text{st}$ probe. Since these two vertices are at the same distance from $v$, and $G$ is bipartite, they cannot be adjacent.
\end{proof}
Write $H'$ for the graph obtained by subdividing the edge $v_5v_6$ of $H$. Since $H'$ is bipartite (it consists of a 6-cycle and an 8-cycle with an edge in common), the robber wins on $H'$, but the cop wins on $H$ by Theorem \ref{c78}. Consequently these two graphs give a counterexample to the conjecture of \cite{CCDEW} that subdividing an edge of any cop-win graph gives another cop-win graph.
\section{Subdivisions of complete graphs}
\label{CompleteSection}
In this section we consider graphs of the form $K^{1/m}_n$. We show that if $m < n/2$ the robber wins and if $m \geqslant n/2$ for $n \geqslant 14$ the graph is locatable. For the remaining cases when $n$ is small we note the few exceptional cases that do not follow this behaviour. Consequently for each $n$ we shall have determined the winning player in all cases.
If $x$ and $y$ are original vertices of $G^{1/m}$ which correspond to adjacent vertices of $G$ we will write $x\cdots y$ for the thread of length $m$ between them. We use ``a vertex on $x\cdots y$'' to mean any of the $m+1$ vertices of the thread, but ``a vertex inside $x\cdots y$'' will exclude $x$ and $y$. When $m$ is even we will use the term ``midpoint'' for the central vertex of a thread, and when $m$ is odd we will use the term ``near-midpoint'' for either vertex of the central edge of a thread.
We will present the proofs separately for the robber and cop winning conditions. We begin with the proofs that the robber wins for $m \leqslant (n-1)/2$, which will rely on him being able to move between original vertices without being located by the cop.
\begin{theorem}\label{CARmsmallrobberwin}Let $m < n/2$. Then the robber wins on the graph $K_n^{1/m}$.
\end{theorem}
\begin{proof}We prove this by giving an explicit strategy for the robber that achieves the following. Assuming at some time he could be in a set of two original vertices, then we claim he can either remain in this pair of original vertices, or reach another pair without being located, and hence he can evade capture indefinitely. We will denote the set of original vertices $\{v_1, \ldots, v_n\}$.
Let us first assume that following a probe by the cop (which we will refer to as the $0^\text{th}$ probe) the robber reveals that he could be in the pair of original vertices $\{v_1, v_2\}$, but that the cop does not know which of them he is in. After this he can move to anywhere in $(v_1 \cup v_2 \cup N(v_1) \cup N(v_2))$. Firstly we will separately consider the result of the cop's first probe, which can be in one of two places.
\begin{enumerate}
\item If her probe was equidistant to $v_1$ and $v_2$ then the robber can claim to have remained in $\{v_1, v_2\}$, and thus still be in $(v_1 \cup v_2 \cup N(v_1) \cup N(v_2))$ after the probe. If the cop always probes vertices that are equidistant from $v_1$ and $v_2$ then the robber can repeat this, evading capture indefinitely.
\item If her probe was not equidistant to $v_1$ and $v_2$ then it was on a thread incident to at least one of them. Let us call the vertex she probes here $p$. Without loss of generality we may assume both that this probe is her first probe (ignoring any that were equidistant to $\{v_1, v_2\}$ and came before it), and that it is in the span of $v_1$. Following this probe the robber will now adopt his motivating strategy of moving towards a new original vertex. He can thus return the distance $(d(a, p) + 1)$, claiming that he was in $v_1$, and so moved to the neighbourhood of $v_1$ at the previous step. He will then continue moving down some thread towards another original vertex. Given that the robber now commits to follow this strategy the cop only needs to determine his destination before he reaches it. We will show that this is not possible by keeping a count of how many threads the cop has not yet eliminated. This first probe only eliminates the thread that $p$ is on, so following it the cop knows that the robber was at distance 1(and is now at distance 2) from $a$ and is moving along one of $(n-2)$ possible threads.
\end{enumerate}
Each subsequent probe can eliminate at most 2 threads for the robber, since probing anywhere on a thread from $v_1$ eliminates only that thread and probing inside $v_i \cdots v_j$ eliminates only $v_1 \cdots v_i$ and $v_1 \cdots v_j$. The robber can then remove those from his possible destinations and continue moving away from $a$. Hence after $t$ steps the robber is at distance $(t+1)$ from $a$ and at most $2t-1$ threads have been eliminated. After the $(m-1)^\text{st}$ step the robber reaches the remaining possible original vertices that he could have been heading towards. There were initially $n-1$ threads that he could have been heading down, and so after $(m-1)$ steps he could be on any of at least $n - 1 - (2(m-1) - 1) = n - 2m + 2 \geqslant 3$ possible threads.
There are two possible scenarios to consider. Firstly, if as described above, the cop eliminates 2 threads on every probe except the first, then he would be unable to determine if the robber had gone halfway down a thread (pausing at the first near midpoint for a step if $m$ is odd) and then returned to $a$. Hence in this case after the $(m-1)^\text{st}$ probe the robber could move into any of at least 4 original vertices (those at either end of the uneliminated threads). If the cop did check to see if the robber turned around he would have to do so by probing on a vertex on a thread of $a$, and this would only eliminate one thread on that turn. This would mean she would eliminate one fewer thread, leaving him at least 4 threads he could be on after $(m-1)$ steps and thus at least 4 original vertices he could reach. In either case he can move into a set of at least 4 original vertices. The next probe by the cop must lie on some thread between at most 2 of them, so at least 2 will be equidistant to the next probe. The robber can now claim to have moved into that pair, and so can reach another pair of original vertices as required. Repeating this process lets him avoid capture indefinitely.
\end{proof}
We now turn our attention to the bound for the cop winning. We shall show that if $m > (n-1)/2$ then the cop can follow a simple strategy to locate the robber, which proceeds in three stages. This argument also requires $m \geqslant 7$, but that only leaves a few small cases to check manually. The second stage of this strategy works slightly differently depending on if $m$ is odd or even, but the motivating idea is the same so we present it in a single proof.
\begin{theorem}\label{CARlargemcopwin}
Let $m \geqslant n/2$ and $m \geqslant 7$. Then the cop wins on the graph $K_n^{1/m}$.
\end{theorem}
\begin{proof}
Our strategy for the cop runs in three stages. In the first stage she forces the robber to enter some original vertex, although she does not attempt to control which. In the second stage she narrows down the set of original vertices that he could be in to a set of size 2. In the final stage she locates him.
In Stage 1 the cop probes all the original vertices in any order until she either gets an answer equal to $m$ or finds two original vertices at distance less than $m$ from the robber. If she gets an answer equal to $m$ then she knows he has entered an original vertex, and moves to Stage 2. If this does not happen then he must have remained on a single thread. When probing either end of it she would get an answer less than $m$, and by noting which two original vertices this occurs on she can identify which thread he is on, and locate him. Thus either the robber is located or the cop moves to Stage 2.
In Stage 2 the cop wishes to narrow down the set of possible original vertices the robber could be in to a set of size 2. She will do this by eliminating candidates, so let us now re-order the original vertices as $v_1, \ldots, v_n$ such that $v_1$ is the last original vertex that she probed in Stage 1 -- hence the robber is known not to be in $v_1$ at the start of Stage 2. Throughout she will track the candidates she has eliminated by maintaining a counter $r$ which is the index of the last vertex that she eliminated. Hence we set $r=1$ initially, and throughout this stage having eliminated the vertices up to $v_r$ she will be trying to eliminiate $v_{r+1}$ and thus increment $r$. We can assume throughout that $r < (n-2)$, as once she has eliminated $v_{n-2}$ there are only two vertices left, and she can proceed to Stage 3.
To eliminate $v_{r+1}$ the cop begins by probing this vertex, which can give one of five possible responses. Three of these are simple to deal with:
\begin{enumerate}
\item \emph{The distance is $0$}. The cop has found the robber and wins the game.
\item \label{CARStage2Casem-1}\emph{The distance is $m-1$}. The cop then knows that the robber was in an original vertex of higher index, and that he has left it, moving towards $v_{r+1}$. The cop can now force the robber to return to the original vertex that he came from by alternatingly probing $v_{r+1}$ and the remaining original vertices with indices higher than $r+1$ in order. If the robber moves into $v_{r+1}$ the cop will detect this and thus locate him easily, and if he does not return then she will eventually find the vertex he came from, and thus locate him. Hence he must return, which she will detect when she gets distance $m$. Along this process she will potentially eliminate not just $v_{r+1}$ but possibly many more candidates -- she proceeds by setting $r$ to the highest index that she has eliminated, and probing the next original vertex.
\item \emph{The distance is $m$}. The cop concludes that the robber is still in an original vertex of higher index than $(r+1)$. She increases $r$ by 1, and repeats the process by probing the next original vertex.
\item The most complicated case to deal with is when \emph{the distance is $m+1$}. The cop now concludes that the robber was in an original vertex of higher index, say $v_i$, and has left it moving towards another original vertex, say $v_j$. She now has two situations to consider. If $j \leqslant r$ then identifying $v_j$ before he reaches it will let her force him back into $v_i$ as in case (ii) above. If $j > r$ (and thus $j > (r+1)$ as if $j = (r+1)$ then the distance would have been $m-1$ which was case (ii) above), then she is less concerned with finding $v_j$, it suffices for her to force him into either $v_i$ or $v_j$, as then she can continue with the above process having eliminated all the original vertices up to $v_{r+1}$ as required. She will therefore address these situations sequentially.
Firstly the cop establishes whether $j > r$ by checking all the vertices in $v_1, \ldots, v_r$ to see if they are the destination for the robber. She can do this by a similar strategy to the worst case in Theorem \ref{CARmsmallrobberwin}. Ideally at each step she would check two possible destinations by probing the midpoints of the threads linking the first $r$ original vertices. In general she will not be able to do this for the first step, but she can begin by probing $v_1$ which eliminates that as a destination. If the robber announces distance $m$ then he has returned to $v_i$, and the cop can continue Stage 2 with $v_{r+1}$ eliminated. If he gives distance $m+1$ then he is still at distance 1 from $v_i$, and she can continue to probe through the set $\{v_2, \ldots, v_r\}$ until he moves in either direction or she eliminates all of them -- in the latter case we move to the next paragraph which outlines what to do once they have all been eliminated. If at some point the robber answers $m+2$ then she knows he was not heading to the vertex just probed but has moved to the second layer of vertices from $v_i$. From this point she can eliminate two vertices from $\{v_2, \ldots, v_r\}$ at each step by either probing midpoints if $m$ is even or near-midpoints if $m$ is odd. Either way she can tell whether he moves back towards $v_i$, in which case she moves back to probing single vertices once he gets back to the first layer to identify the exact moment he returns to $v_i$, or keeps eliminating pairs if he does not. If he continues to head away from $v_i$ by eliminating two vertices at each step she can eliminate $2(m-3) + 1$ before he reaches another original vertex. But as $m > (n-1)/2$ and there were only at most $(n-3)$ original vertices in $\{v_1, \ldots, v_r\}$ this leaves only two vertices that he can reach. By probing a vertex on the thread between these last two vertices she can distinguish whether he is in this pair, allowing her to move to Stage 3 if he is. Hence if he tries to move towards $\{v_1, \ldots, v_r\}$ she will either locate him, move to Stage 3 or force him back into $v_i$ successfully.
If he left the $v_i \cdots v_j$ thread either by reaching $v_i$ during the above probes or $v_j$ on the last the cop will detect this during them, allowing her to repeat Stage 2 having eliminated $v_{r+1}$.
If he has not left this thread she could then probe original vertices with indices higher than $(r+1)$ to eliminate those directly until she found either of $v_i$ or $v_j$ -- in which case she would proceed as in Case (ii) to force him into the other end of the thread, and repeat Stage 2 with more vertices eliminated.
\item The final case to consider is when \emph{the distance is 1}, which means the cop has found the original vertex that the robber was on, and he has moved 1 away from it. The strategy here is very similar to that above, she first makes sure that he is not moving towards $\{v_1, \ldots, v_r\}$, making sure to note if he returns to $v_i$, and then afterwards continues checking the remaining vertices in pairs. Carrying out the same analysis shows that in most cases he will be located when he returns to $v_i$, the only case when he is not is when he either moves halfway down a thread towards $\{v_{n-2}, v_{n-1}, v_n\}$ and then moves back to $v_i$ or goes all the way to $v_n$. But as this is the only case where the cop does not locate him directly if it occurs she will know, and thus be able to move to Stage 3 knowing he is in either of $v_i$ or $v_n$.
\end{enumerate}
We now move to Stage 3, which starts after the cop makes some probe and knows the robber is in one of two original vertices. We will label them as $\{a, b\}$, and note that he can move to the neighbourhood of them before the cop's first probe in Stage 3. For her first probe the cop probes the vertex at distance 1 from $a$ on the $a \cdots b$ thread. This allows her to distinguish whether the robber was on $a$ or $b$ before, and whether he is on the $a \cdots b$ thread now or another one. The cop wins immediately unless the robber answers distance 2 or distance m, in which case he has left the initial vertex he was on and moved towards an initial vertex other than $a$ or $b$. Without loss of generality we shall assume he was on $a$ and is thus now moving to one of the other $(n-2)$ possible locations, noting that this first probe reveals him to be at distance 1 from $a$.
Her strategy now reduces to finding which thread he is on before he can reach the other end of it, being sure to note if he returns to $a$. The second probe varies according to whether $m$ is even or odd. If $m$ is even then the cop probes a midpoint of a thread between two original vertices that have not been eliminated yet, whereas if it is odd then the cop probes a neighbour of a near-midpoint, say the vertex on the $c \cdots d$ thread that is distance 3 further from $d$ than $c$. In either case we can distinguish whether the robber is heading to that pair or not -- the one slightly complicated case is if $m$ is odd and he responds with $(m-1)/2+m$ in which case he could have remained at distance 1 from $a$ or be distance 2 from $a$ heading specifically towards $d$. If following this probe the robber uses the fact that he could have been distance 2 from $a$ to move to the vertex distance 3 from $a$ the cop will notice on her next probe and locate him. In this case the cop can therefore assume that the robber is at distance 1 and effectively eliminate $c$ from the possible destinations, doing so without him having moved closer to another original vertex so effectively for free. Thus the cop's second probe can always eliminate two possible destinations for the robber -- and by probing at midpoints if $m$ is even or near-midpoints if $m$ is odd this also holds for the subsequent probes.
After $t$ probes he will be within distance $t$ of $a$, and she will have eliminated $1 + 2(t-1)$ possible destinations. Thus after $(m-1)$ probes there are only at most three original vertices left that he could be moving between, two possible destinations (which we shall refer to as $v$ and $w$) and $a$. Including the possibility that he turned around at the midpoint or near-midpoint (according to the parity of $m$), and assuming at each step he continued to move (as otherwise it is easier to locate him), this means that following the $(m-1)^\text{st}$ probe he is either distance $0, 1, 2, (m-2), (m-1)$ or $m$ from $a$ along either the $a \cdots v$ thread or the $a \cdots w$ thread. However, in this case he can be located by probing the vertex at distance 2 from $v$ along the $a \cdots v$ thread, provided $m \geqslant 7$, so hence he can be located even in this worst case scenario, completing the proof.
\end{proof}
This answers the question of who wins on $K_n^{1/m}$ for all but a small number of cases, which we summarise in the conclusion. We now turn our attention to bipartite graph in the following section.
\section{Subdivisions of complete bipartite graphs}
\label{BipartiteSection}
We now turn our attention to complete bipartite graphs, where we are able to determine the winning player on $K_{a,b}^{1/m}$ for any $a$, $b$, and $m$. In \cite{CCDEW} it is shown that the cop wins for $m\geqslant \max\{a,b\}$, but in fact the cop wins if and only if $m \geqslant \min \{ a,b \} - 1$, provided $a,b \geqslant 4$. Throughout this section we shall write $A$ and $B$ for the sets of original vertices in $K_{a,b}^{1/m}$ corresponding to the two vertex classes of $K_{a,b}$, with $|A|=a$ and $|B|=b$.
\begin{theorem}\label{biplower}If $a,b\geqslant 3$ and $m \leqslant \min\{a,b\}-2$ then $K_{a,b}^{1/m}$ is not locatable.
\end{theorem}
\begin{proof}We will prove the stronger statement that the robber wins even if he is required to be at an original vertex for every $m^\text{th}$ probe, alternating between $A$ and $B$, so that he is in $A$ at the time of the $km^\text{th}$ probe for every even $k$. We show that, provided the cop has not won after the $km^\text{th}$ probe, the robber can ensure that she has not won by the $(k+1)m^\text{th}$ probe. For ease of writing, we assume that $k$ is even.
Suppose that the robber is at $u\in A$ for the $km^\text{th}$ probe, but that the $km^\text{th}$ probe does not locate him uniquely. We show that, no matter which vertices the cop probes, there are two possible threads for the robber to travel along between the $km^\text{th}$ and $(k+1)m^\text{th}$ probes, which the cop is unable to distinguish between, so that she will not be able to win by time $(k+1)m$. Suppose her $(km+l)^\text{th}$ probe (for some $1\leqslant l\leqslant m$) is at vertex $z$, which is on the thread $x \cdots y$ for some $x\in A$ and $y\in B$. For each $v\in B$, write $w_{v,l}$ for the vertex on the thread $u\cdots v$ at distance $l$ from $u$. If $x\neq u$ then for any $v\neq y$ we have $d(z,w_{v,l})=\min\{d(z,x)+2m-l,d(z,y)+m+l\}$, whereas if $x=u$ then, again for any $v\neq y$, we have $d(z,w_{v,l})=d(z,x)+l$. Suppose that for each $l$ with $1\leqslant l \leqslant m$ the answer consistent with the robber being at any one of the vertices $w_{v,l}$ for $v\neq y$ is received from the $(km+l)^\text{th}$ probe. Then each probe eliminates at most one of the threads leaving $u$, and since $m$ probes have been made, and $m \leqslant \min\{a,b\}-2$, at least 2 remain, so the cop has not yet won.
\end{proof}
Note that if $a, b\geqslant 2$ then the robber can win on $K_{a,b}$ by ensuring he is in the opposite part to the vertex the cop probes at every time. In the case where $\min\{a, b\} = 3$, Theorem \ref{biplower} can be strengthened to say that the robber will win for $m=2$.
\begin{lemma}If $\min\{a,b\}=3$ then the graph $K_{a,b}^{1/2}$ is not locatable.
\end{lemma}
\begin{proof}Suppose that after the cop's $t^\text{th}$ probe there are two possible locations for the robber which are both in $A$ or both in $B$, say $u$ and $v$ with $u, v\in A$. We show that the robber can ensure either that there are still two possible locations, both in $A$ or both in $B$, either after the $(t+1)^\text{st}$ probe or after the $(t+2)^\text{nd}$. If the $(t+1)^\text{st}$ probe is equidistant from $u$ and $v$ this is trivial, as the robber can return the distance to $u$ or $v$. If the $(t+1)^\text{st}$ probe is $u$ or $v$ at time $t+1$ then all neighbours of $u$ will be equidistant, so the robber can claim to be at one of them. Any vertex in $A$ is equidistant from all vertices in $B$, and any other vertex is equidistant from all but one of the vertices in $B$, so no matter what vertex the cop chooses for her $(t+2)^\text{nd}$ probe, there will be at least $b-1\geqslant 2$ vertices in $B$ at the same distance from it. By this point the robber can have reached any of these without being caught. The only remaining case is for the $(t+1)^\text{st}$ probe to be at a vertex which is adjacent to either $u$ or $v$, say the vertex $w$ between $u$ and $x$ with $x\in B$. Let $y$ and $z$ be two other vertices in $B$. The midpoints of the threads $u\cdots y$, $u\cdots z$ and $v\cdots x$ are all at distance 2 from $w$, so if the robber moves to one of these the cop cannot determine which. Then no matter which vertex the cop probes at time $t+2$, some two of $x$, $y$ and $z$ are at the same distance, and so the robber can ensure there are two possible locations in $B$ after this probe.
\end{proof}
We have shown that $K_{a,b}^{1/m}$ is not locatable for $m \leqslant \min\{a,b\}-2$ when $\min\{a,b\}>3$, or for $m \leqslant \min\{a,b\}-1$ when $2 \leqslant \min\{a,b\}\leqslant 3$. Next we show that in all other cases $K_{a,b}^{1/m}$ is locatable. Note that the cop can win on the star $K_{1,b}$ by probing leaves in turn. This covers the case $\min\{a,b\}=1$. Next we deal with the case $\min\{a,b\}=2$.
\begin{lemma}The graph $K_{2,b}^{1/2}$ is locatable for any $b\ge 2$.
\end{lemma}
\begin{proof}Write $x$ and $y$ for the two vertices in $A$. Let the cop start by probing $x$. If she receives the answer 2 the robber is in $B$. If the answer is 0 or 4 she has won. If it is 1 or 3 she knows that the robber is adjacent to $x$ or $y$ respectively.
Now we show that the cop can win from a position in which she knows that the robber is in a particular subset of the neighbourhood of $x$ (or, equivalently, if she knows the robber is in a particular subset of the neighbourhood of $y$), and she can win from a position in which she knows that the robber is in a fixed subset of $B$. We prove both simultaneously by induction on the size of the subset, $k$. In each case if $k=1$ she has already won.
If the robber was at one of $k$ neighbours of $x$, the cop probes one of the $k$ adjacent vertices of $B$. If the answer is at most 2 then the robber is caught. If the answer is 3 then he is known to be at one of $k-1$ neighbours of $x$ and if it is 4 he is known to be at one of $k-1$ vertices of $B$; in either case we are done by induction.
If the robber was at one of $k$ vertices in $B$, the cop probes one of these. An answer of 2 is impossible, and if the answer is 1 then she can win by next probing $x$. If the answer is 4 then she knows the robber is at one of $k-1$ vertices of $B$, and we are done by induction. If the answer is 3 then she probes $x$ next; now if the answer is 0 or 4 she has won, and if it is 1, 2, or 3 she has reduced to one of $k-1$ vertices adjacent to $x$, in $B$, or adjacent to $y$ respectively, so we are done by induction.
\end{proof}
Finally we show that, provided $m \geqslant 3$, the only non-locatable graphs of the form $K_{a,b}^{1/m}$ are those given in Theorem \ref{biplower}.
\begin{theorem}Let $a,b\geqslant 3$. If $m \geqslant \min \{a,b\}-1$ and $m\geqslant 3$ then $K_{a,b}^{1/m}$ is locatable.
\end{theorem}
\begin{proof}Suppose $a\leqslant b$. Again we give a two-stage winning cop strategy. In the first stage we show that the cop can win or establish that the robber is in $B$, and in the second stage we show that she may win once he knows that the robber is in $B$.
In the first stage, the cop probes vertices in $A$ in turn until she receives an answer of $m$ (indicating that the robber is in $B$) or less than $m$. This must eventually happen, since if the robber does not reach $B$ he must remain nearer one particular vertex in $A$ than any other, and when the cop probes this vertex she will get an answer of less than $m$. In this case write $x$ for the vertex in question. Once the cop has found $x$, the robber cannot leave his current thread without moving either to $x$ or to some vertex in $B$, so the cop then probes vertices in $B$ until she receives an answer of $2m$ (indicating that the robber is in $B$) or at most $m$ (in which case she can determine his location).
In the second stage we show that the cop may win from a position where the robber is known to be in a fixed subset of $B$, by induction on the size of the subset, $k$. This is true for $k=1$ as she has already won. If $k>1$ then write $B'$ for the set of $k$ vertices in question. The cop starts by probing the vertex adjacent to $B'$ on the thread $x\cdots y$ for some $x\in A$ and $y\in B'$. The possible answers are 0 (if the robber is at that vertex), 1 (if he is at $y$), 2 (if he is at some other neighbour of $y$), $2m-2$ (if he is on another thread leading to $x$), $2m-1$ (if he is at a vertex of $B'$ other than $y$), and $2m$ (if he is on a thread which does not include $x$ or $y$). Since $m\geqslant 3$, these are all different. An answer of 0 or 1 is an immediate win for the cop, and after an answer of $2m-1$ she wins by the induction hypothesis. After an answer of $2m-2$ the cop probes vertices of $B'$ until either she receives an answer of at most $m$, winning, or she receives an answer of $2m$, in which case she knows the robber is at one of at most $k-2$ vertices of $B'$ and she wins by the induction hypothesis. After an answer of $2$ or $2m$, the robber must be in a thread which does not reach $x$. The cop now probes vertices of $A$, other than $x$, in turn, until she receives an answer of $m$, $2m$, or less than $m$. One of these must eventually happen since either the robber reaches one end of the thread he is currently on, or he remains in the same thread until such time as the cop probes its end in $A$. If the answer $2m$ occurs first, the cop knows that the robber has reached some vertex $u\in A$ which is neither $x$ nor one she has probed since the robber left $B$. Since the robber has taken at least $m$ steps to reach $A$, she has probed at least $m-1$ vertices in $A$, and together with $x$ she has eliminated at least $m\geqslant a-1$ vertices of $A$, so there is only one possibility and the robber is caught. If the answer $m$ occurs first then the robber is at a vertex of $B'$, and, since the cop knows whether or not this is $y$, she has either caught the robber or reduced to a set of $k-1$ vertices, so wins by the induction hypothesis. If an answer less than $m$ occurs first, say when probing $u$, then the cop has won if that answer is 0, or if the robber was initially known to be on a thread meeting $y$. Otherwise, she knows that the robber is on some thread $u\cdots v$ for $v\in B'\setminus\{y\}$; now she proceeds by probing vertices of $B'\setminus\{y\}$ in turn until she receives an answer of at most $m$ (in which case she has won) or of $2m$ (in which case she knows that the robber is at some vertex in $B'\setminus\{y\}$, and so wins by induction).
\end{proof}
We can now completely determine which graphs of the form $K^{1/m}_{a,b}$ are locatable. If $\min\{a,b\}\geqslant 4$ then $K^{1/m}_{a,b}$ is locatable if and only if $m\geqslant \min\{a,b\}-1$, whereas if $\min\{a,b\}\leqslant 3$ then $K^{1/m}_{a,b}$ is locatable if and only if $m\geqslant \min\{a,b\}$.
\section{Conclusion and Open Problems}
We note that in the proof of Theorem \ref{CARlargemcopwin} the condition that $m \geqslant 7$ is only required for the final part of Stage 3, which only arises when $m = n/2$. If $m \geqslant n/2 + 1$ then this condition is not necessary, and the result still holds that the cop wins for all values of $n$. This answers the question of who wins for which $m$ on $K_n^{1/m}$ in almost all cases, except for a few small values. These are small enough to be checked by hand, we note that almost all of them obey the same relationship of the cop winning if $m \geqslant n/2$ and the robber winning if $m < n/2$, with the exceptions that the robber can also win in the following cases: $m=2, n=3$ or $4$; $m=3, n=6$; and $m=5, n=10$.
In Section \ref{Girth6Section} we have shown that it is not necessarily true that subdividing a single edge of a locatable graph yields another locatable graph. It remains an open conjecture that it is however true that subdividing every edge of a locatable graph yields another locatable graph. Another natural question is whether for every graph $G$ there is some $m_G$ for which $G^{1/m}$ is locatable if and only if $m\geqslant m_G$. We have shown that this is the case for complete graphs and complete bipartite graphs by finding exact values of $m_G$ in those cases in Sections \ref{CompleteSection} and \ref{BipartiteSection} respectively. The question remains open in generality, although the authors believe it to be true.
Moreover, the authors note that the ideas developed in Section \ref{BipartiteSection} can also be used to prove similar results for complete $ r $-partite graphs. More precisely, if $G$ is such a graph with parts of sizes $ a_{1}, \ldots, a_{r} $, where $ a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{r} $, then the lengths of subdivision required and sufficient to make $G^{1/m}$ locatable are each about $ \max \{ (a_{1} + \cdots + a_{r-1})/2 , a_{r-1} \} $. This generalises the results on complete bipartite graphs. In the case of balanced $r$-partite graphs this gives a threshold of $(n/2)(1-1/r) + O(1)$.
Finally, we note that for all graphs considered in this paper, subdividing the edges of an $ n $-vertex-graph about $ n/2 $ times is sufficient to make it locatable, and we conjecture that this is indeed the case for all finite graphs.
\begin{conjecture}
For all sufficiently large $n$, if $ G $ is a graph on $ n $ vertices then $ G^{1/m} $ is locatable for every $ m \geqslant n/2 $.
\end{conjecture}
\section{Acknowledgements}
The first author acknowledges support from the European Union through funding under FP7--ICT--2011--8 project HIERATIC (316705), and is grateful to Douglas B. West for drawing his attention to this problem. The second author acnowledges support through funding from NSF grant DMS~1301614 and MULTIPLEX grant no. 317532, and is grateful to the organisers of the $8^\text{th}$ Graduate Student Combinatorics Conference at the University of Illinois at Urbana-Champaign for drawing his attention to the problem. The third author acknowledges support through funding from the European Union under grant EP/J500380/1 as well as from the Studienstiftung des Deutschen Volkes. The second and third authors would also like to thank Yuval Peres and the Theory Group at Microsoft Research Redmond for hosting them while some of this research was conducted.
| {
"timestamp": "2014-01-17T02:08:54",
"yymm": "1311",
"arxiv_id": "1311.3867",
"language": "en",
"url": "https://arxiv.org/abs/1311.3867",
"abstract": "We consider a game in which a cop searches for a moving robber on a graph using distance probes, studied by Carragher, Choi, Delcourt, Erickson and West, which is a slight variation on one introduced by Seager. Carragher, Choi, Delcourt, Erickson and West show that for any fixed graph $G$ there is a winning strategy for the cop on the graph $G^{1/m}$, obtained by replacing each edge of $G$ by a path of length $m$, if $m$ is sufficiently large. They conjecture that the cop does not have a winning strategy on $K_n^{1/m}$ if $m<n$; we show that in fact the cop wins if and only if $m\\geqslant n/2$, for all but a few small values of $n$. They also show that the robber can avoid capture on any graph of girth 3, 4 or 5, and ask whether there is any graph of girth 6 on which the cop wins. We show that there is, but that no such graph can be bipartite; in the process we give a counterexample for their conjecture that the set of graphs on which the cop wins is closed under the operation of subdividing edges. We also give a complete answer to the question of when the cop has a winning strategy on $K_{a,b}^{1/m}$.",
"subjects": "Combinatorics (math.CO); Discrete Mathematics (cs.DM)",
"title": "The Robber Locating game",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.984575448370756,
"lm_q2_score": 0.7185943925708562,
"lm_q1q2_score": 0.7075103962621618
} |
https://arxiv.org/abs/1407.1869 | Topological properties of the unitary group | We show that the strong operator topology, the weak operator topology and the compact-open topology agree on the space of unitary operators of a infinite dimensional separable Hilbert space. Moreover, we show that the unitary group endowed with any of these topologies is a Polish group. | \section*{Introduction}\label{Introduction}
The purpose of this short note is to settle some topological properties of the unitary group ${U(\mathcal{H})}$ of an infinite dimensional separable Hilbert space $\mathcal{H}$,
whenever the group is endowed with the compact open topology.
When dealing with equivariant Hilbert bundles and its
relation with its associated unitary principal equivariant bundles (see \cite{AtiyahSegal}), one is obliged to
consider the compact-open topology on the structural group ${U(\mathcal{H})}$.
In one of the foundational papers for twisted equivariant K-theory, Atiyah and Segal claimed that the unitary
group endowed with the compact-open topology was not a topological group \cite[Page 40]{AtiyahSegal}, based on
the fact that the inverse map on $GL(\mathcal{H})$ is not continuous when $GL(\mathcal{H})$ is endowed with the strong operator topology.
This unfortunate claim obliged Atiyah and Segal to device a set of ingenious constructions in order to make ${U(\mathcal{H})}$ into a topological
group with the desired topological properties suited for the classification of Fredholm bundles. Nevertheless, these ingenious constructions
of Atiyah and Segal added difficulties on the quest of finding local cross sections for equivariant projective bundles, and therefore a
clarification on the veracity of the claim was due.
The purpose of this note is to show that the unitary group ${U(\mathcal{H})}$ endowed with the compact-open topology is indeed a topological group,
moreover a Polish group, and that this topology agrees with the strong operator topology, as the weak operator topology; this is the content of Theorem \ref{thm:topologies-on-U(H)-agree} which is the main result of this note.
\section{Operator topologies on the unitary group}
Let $\mathcal{H}$ be a separable Hilbert space with inner product $\langle,\rangle$ inducing the norm $|x|:=\sqrt{\langle x,x \rangle}$
for $ x \in \mathcal{H}$. Let $L(\mathcal{H},\mathcal{H})$
denote the vector space of linear operators from $\mathcal{H}$ to $\mathcal{H}$
and denote by ${B(\mathcal{H})}$ the vector space of bounded linear operators
$${B(\mathcal{H})}:= \{T \in L(\mathcal{H},\mathcal{H}) \colon \exists M >0 \ \ \forall x \in \mathcal{H} \ \ |Tx| \leq M|x| \}.$$
The space of bounded linear operators ${B(\mathcal{H})}$ endowed with the norm
$$|T| = \sup_{|x| \leq 1} |Tx|$$
becomes a Banach space. Denote the adjoint operator $T^*$ as the operator defined by the equality
$\langle Tx,y \rangle = \langle x,T^*y \rangle$
for all $x,y \in \mathcal{H}$.
The space ${B(\mathcal{H})}$ can be endowed with several operator topologies and we will list the ones that interest us in this work.
They all can be defined by specifying which are the convergent sequences. Then let $\{T_k\}$ be a sequence of linear operators on ${B(\mathcal{H})}$ and denote by $T_k \to_? T$ the statement that $T_k$ converges to $T$ in the topology ?, and denote ${B(\mathcal{H})}_?$ the space
of bounded operators endowed with the topology ?. In what follows the notation $x_k \to x$ denotes
that the sequence $\{ x_k \}_k$ in $\mathcal{H}$ converges to $x$.
\begin{itemize}
\item{\bf{Norm Topology (Uniform convergence):} }\\ $T_k \to_n T$ if $\sup_{ |x| \leq 1 }\{ |T_kx -Tx| \} \to 0$.
\item{\bf{Strong Operator Topology (Pointwise convergence):}}\\ $T_k \to_s T$ if for all $x \in \mathcal{H}$, $T_kx \to Tx$.
\item{\bf{Weak Operator Topology:}}\\ $T_k \to_w T$ if for all $x,y \in \mathcal{H}$, $\langle T_kx,y \rangle \to \langle Tx,y \rangle$
\item{\bf{Compact Open Topology (Uniform convergence on compact sets):}}\\ $T_k \to_{co} T$ if for all compact subsets $C \subset \mathcal{H}$, the restricted sequence
$\{T_k|_C\}_k$ converges uniformly to $T|_C$.
\item{\bf{Strong* Operator Topology:}}\\ $T_k \to_{s^*} T$ if both $T_k \to_{s} T$ and $T_k^* \to_{s} T^*$.
\item{\bf{Weak* Operator Topology:}}\\ $T_k \to_{w^*} T$ if both $T_k \to_{w} T$ and $T_k^* \to_{w} T^*$.
\item{\bf{Compact Open* Topology:}}\\ $T_k \to_{co^*} T$ if both $T_k \to_{co} T$ and $T_k^* \to_{co} T^*$.
\end{itemize}
The identity map on ${B(\mathcal{H})}$ induces the following commutative diagram of continuous maps
\begin{equation}
\xymatrix{
{B(\mathcal{H})}_n \ar[r] & {B(\mathcal{H})}_{co^*} \ar[d] \ar[r] & {B(\mathcal{H})}_{s^*} \ar[d] \ar[r] & {B(\mathcal{H})}_{w^*} \ar[d] \\
& {B(\mathcal{H})}_{co} \ar[r] & {B(\mathcal{H})}_{s} \ar[r] & {B(\mathcal{H})}_{w}
} \label{diagram_B(H)}
\end{equation}
with the property that none of the maps is a homeomorphism.
Let ${U(\mathcal{H})}$ denote the group of unitary operators on $\mathcal{H}$, i.e.
$${U(\mathcal{H})}=\{ U \in {B(\mathcal{H})} \colon U U^*=U^*U= \rm{Id}_\mathcal{H} \},$$
and note that $U^{-1}=U^*$ and that
$\langle x,y \rangle = \langle Ux,Uy \rangle$ for all $x,y \in \mathcal{H}$. Since ${U(\mathcal{H})} \subset {B(\mathcal{H})}$, the group ${U(\mathcal{H})}$
can be endowed with any of the operator topologies previously defined. The group ${U(\mathcal{H})}_n$ endowed with the
norm topology is a topological group and it is the prototypical example of what is known as a Banach Lie group, see
\cite{Neeb}. The fact that ${U(\mathcal{H})}$ endowed with any of the other topologies defined above is also a topological group
is the main result of this section
\begin{theorem} \label{thm:topologies-on-U(H)-agree}
The operator topologies: compact open, strong, weak and their * counterparts, all agree on the group ${U(\mathcal{H})}$, i.e.
$${U(\mathcal{H})}_{co^*} = {U(\mathcal{H})}_{s^*} = {U(\mathcal{H})}_{w^*} = {U(\mathcal{H})}_{co} = {U(\mathcal{H})}_{s} = {U(\mathcal{H})}_{w}.$$
Moreover, the group ${U(\mathcal{H})}$ endowed with any of these topologies is a Polish group, i.e. a completely metrizable
topological group.
\end{theorem}
The proof of the theorem is built out from various Lemmas:
\begin{lemma} \label{lem:adjoint-continuous}
The map ${U(\mathcal{H})}_w \to {U(\mathcal{H})}_w$, $T \mapsto T^*$ is continuous. In particular ${U(\mathcal{H})}_{w^*}={U(\mathcal{H})}_w$.
\end{lemma}
\begin{proof}
Consider $T_k \to_wT$ and $x,y \in \mathcal{H}$. Then
$$\langle x, T_k^*y \rangle = \langle T_kx,y \rangle \to \langle Tx,y \rangle = \langle x, T^*y \rangle,$$
and therefore $T^*_k \to_w T^*$. This proves that $T \mapsto T^*$ is continuous. Hence we conclude
that the weak and the weak* topologies agree on ${U(\mathcal{H})}$.
\end{proof}
\begin{lemma} \label{lem:composition-continuous}
The composition of operators ${U(\mathcal{H})}_s \times {U(\mathcal{H})}_s \to {U(\mathcal{H})}_s$, $(T,S)\mapsto TS$, is continuous in the strong operator topology.
\end{lemma}
\begin{proof}
Consider a convergent sequence $(T_k,S_k) \to_s (T,S)$ in ${U(\mathcal{H})}_s\times {U(\mathcal{H})}_s$. For $x \in \mathcal{H}$ we have
\begin{align*}
|T_kS_kx - TSx| & = |T_kS_kx - T_kSx + T_kSx -TSx|\\
& \leq |T_kS_kx - T_kSx| + |T_kSx -TSx|\\
& = |T_k(S_kx-Sx) | + |(T_k-T)Sx|\\
& = |S_kx-Sx| + |(T_k-T)Sx|,
\end{align*}
and since $S_kx \to Sx$ and $T_k(Sx) \to T(Sx)$, then we have that $(T_kS_k)x \to (TS)x$.
Therefore the composition of operators is continuous in the strong operator topology.
\end{proof}
\begin{lemma}\label{lem:weak=strong}
The weak and the strong operator topologies agree on ${U(\mathcal{H})}$. Therefore we have that
${U(\mathcal{H})}_s$ is a topological group and moreover that ${U(\mathcal{H})}_{s^*}={U(\mathcal{H})}_s={U(\mathcal{H})}_w$.
\end{lemma}
\begin{proof}
Consider a convergent sequence $T_k \to_w T$ in ${U(\mathcal{H})}_w$. To prove that $T_k \to_s T$ it is enough to show the convergence
$T_kx \to Tx$ for $x$ a unit vector in $\mathcal{H}$. We have then that $|x|=1$ and therefore $|T_kx|=1=|Tx|$. Now we compute
\begin{align*}
|T_kx-Tx|^2 = |T_kx|^2 -2{\rm{Re}}\langle T_kx,Tx \rangle + |Tx|^2
= 2 -2{\rm{Re}}\langle T_kx,Tx \rangle
\end{align*}
and since $T_k\to_wT$, we have that $\langle T_kx,Tx \rangle \to \langle Tx,Tx \rangle=1$. Therefore
$|T_kx-Tx|^2 \to 0$ and hence we conclude that $T_kx \to Tx$. This shows that the identity map ${U(\mathcal{H})}_w \to {U(\mathcal{H})}_s$ is continuous, and
therefore ${U(\mathcal{H})}_w={U(\mathcal{H})}_s$.
Now, by Lemma \ref{lem:adjoint-continuous} we know that the inverse map $T \mapsto T^*=T^{-1}$ is continuous, and by Lemma
\ref{lem:composition-continuous} we know that the composition is continuous. Then we have that ${U(\mathcal{H})}$ with the strong (or weak) operator
topology is a topological group; the fact that the strong* topology agrees with the strong topology follows from the continuity of the inverse map.
\end{proof}
The proofs of the previous lemmas follow the proofs that appear in \cite[Cor. 9.4]{Hilgert-Neeb}.
\begin{lemma}
The topological group ${U(\mathcal{H})}_s$ is metrizable.
\end{lemma}
\begin{proof}
Consider an orthonormal basis $\{e_j\}_{j \in \mathbb{N}}$ of $\mathcal{H}$. Consider the map
$$\Psi : {U(\mathcal{H})}_s \to \mathcal{H}^\mathbb{N}, \ \ T \mapsto (Te_j)_{j \in \mathbb{N}}$$
where $\mathcal{H}^\mathbb{N}$ is endowed with the product topology. For a convergent sequence
$T_k \to_s T$ we have that $T_ke_j \to Te_j$ for all $j \in \mathbb{N}$, this implies that
the maps ${U(\mathcal{H})}_s \to \mathcal{H}, \ T\mapsto Te_j$ are all continuous and therefore by the universal properties of the
product topology we obtain that the map $\Psi$ is continuous. Since the basis $\{e_j\}_{j \in \mathbb{N}}$
generates a dense subset of $\mathcal{H}$, we have that any two operators which agree on the basis $\{e_j\}_{j \in \mathbb{N}}$
must be equal; hence we have that the map $\Psi$ is injective.
Now let us show that the map $\Psi$ is an embedding. For this purpose let us take a convergent
sequence $\Psi(T_k) \to \Psi(T)$ in the image of $\Psi$. Since $T_ke_j \to Te_j$ for all basis vectors,
we have that the sequence $\{T_k\}_k$ converges pointwise to $T$ in the dense subset of $\mathcal{H}$ generated
by the basis $\{e_j\}_{j \in \mathbb{N}}$, and since the operators are unitary we can conclude that $T_k \to_s T$. Let
us see in more detail this argument: take $x \in \mathcal{H}$ and let $\{x_j\}_{j \in \mathbb{N}}$ with each $x_j$ belonging
to the dense subset generated by $\{e_j\}_{j \in \mathbb{N}}$ and such that $x_j \to x$. We compute
\begin{align*}
|T_kx-Tx| & \leq |T_kx - T_kx_j| + |T_kx_j - Tx_j| + |Tx_j -Tx| \\
&= |T_k(x-x_j)| + |T_kx_j - Tx_j| + |T(x_j -x)| \\
&= 2|x-x_j| + |T_kx_j - Tx_j|,
\end{align*}
and since $T_kx_j \to Tx_j$, we have that $T_kx \to Tx$ and therefore $T_k \to_s T$. This implies that the map $\Psi$
induces a homeomorphism with its image, and hence it is an embedding.
The Hilbert space $\mathcal{H}$ is a metric space, and the product $\mathcal{H}^\mathbb{N}$ of countable copies of $\mathcal{H}$ can be endowed
with a metric. Since $\Psi: {U(\mathcal{H})}_s \to \mathcal{H}^\mathbb{N}$ is an embedding, then ${U(\mathcal{H})}_s$ inherits the induced metric and hence it is
metrizable.
\end{proof}
The previous argument follows the proof that appears in \cite[Prop II.1]{Neeb}.
In \cite[Prop II.1]{Neeb} it is also shown that the the map $\Psi$ also provides an embedding
${\rm{Iso}}(\mathcal{H}) \to \mathcal{H}^\mathbb{N}$ where ${\rm{Iso}}(\mathcal{H}):= \{T \in {B(\mathcal{H})} \colon T^*T=\rm{Id}_\mathcal{H} \}$ is the monoid of
all isometries of $\mathcal{H}$, which is moreover complete with respect to the induced metric. It is furthermore shown
that ${U(\mathcal{H})} = \{T \in {\rm{Iso}}(\mathcal{H}) \colon T^* \in {\rm{Iso}}(\mathcal{H}) \}$ is a $G_\delta$ set in ${\rm{Iso}}(\mathcal{H})$, and since
$G_\delta$ sets on a complete metrizable space are complete in the induced metric \cite[Thm 1, p. 93]{Schwartz} this implies that
${U(\mathcal{H})}_s$ is a complete metrizable space.
\begin{lemma}[{\cite[Prop II.1]{Neeb}}] \label{lem:U(H)-polish} The unitary group ${U(\mathcal{H})}_s$ is a Polish group.
\end{lemma}
So far we have shown that ${U(\mathcal{H})}_s$ is a Polish group and we have that ${U(\mathcal{H})}_s={U(\mathcal{H})}_{s^*}={U(\mathcal{H})}_w={U(\mathcal{H})}_{w^*}$. Note in particular that
${U(\mathcal{H})}_s$ is compactly generated since it is a metrizable space.
Let us now see the relation with the compact open topology on ${U(\mathcal{H})}$.
\begin{lemma} \label{lem:compact-open=strong}
The compact open topology and the strong operator topology agree on ${U(\mathcal{H})}$, i.e. ${U(\mathcal{H})}_{co}={U(\mathcal{H})}_s$.
\end{lemma}
\begin{proof}
Let us start by recalling the retraction functor $k$ defined in \cite[Def. 3.1]{Steenrod}. For $X$ a Hausdorff topological space the associated
compactly generated space $k(X)$ is the set $X$ whose topology is defined as follows: a set in $k(X)$ is closed, if its intersection
with every compact set in $X$ is itself closed. By \cite[Thm 3.2]{Steenrod} we know that the identity map $k(X) \to X$ is a continuous map,
that $k(X)$ is compactly generated, that $k(X)$ and $X$ have the same compact sets and that $X=k(X)$ whenever $X$ is compactly generated.
Let $C$ be a compact subset in ${U(\mathcal{H})}_s$. Take a sequence of operators $\{T_n\}_n$ in $C$, since $C$ is compact in ${U(\mathcal{H})}_s$ there exists a convergent subsequence $\{T_{n_k}\}_k$, and since ${U(\mathcal{H})}_s$ is completely metrizable and hence $C$ is furthermore closed, this convergent subsequence
$T_{n_k} \to_s T$ converges to an operator $T \in C$. By the Bannach-Steinhaus Theorem (see \cite[Cor. of Thm 33.1, p. 348]{Treves}), we have that the
sequence $\{T_{n_k}\}_k$ converges uniformly on every compact set of $\mathcal{H}$, and therefore we have that $T_{n_k} \to_{co} T$. Thus
the space $C$ is also compact in ${U(\mathcal{H})}_{co}$, and by the same argument as before, the induced topology of $C$ in ${U(\mathcal{H})}_{co}$ agrees with the induced topology of $C$ in ${U(\mathcal{H})}_s$.
We conclude that the spaces ${U(\mathcal{H})}_{co}$ and ${U(\mathcal{H})}_s$ have the same compact sets with the same induced topologies. This implies that
the retraction functor $k$ applied on the map ${U(\mathcal{H})}_{co} \to {U(\mathcal{H})}_s$ induces a homeomorphism
$$k({U(\mathcal{H})}_{co}) \stackrel{\cong}{\to} k({U(\mathcal{H})}_s).$$
Then we have the commutative diagram
$$\xymatrix{
k({U(\mathcal{H})}_{co}) \ar[r] \ar[d]^\cong &{U(\mathcal{H})}_{co} \ar[d]\\
k({U(\mathcal{H})}_{s}) \ar[r]^\cong & {U(\mathcal{H})}_s
}$$
which implies that ${U(\mathcal{H})}_{co}={U(\mathcal{H})}_s$.
\end{proof}
\begin{proof}[proof of Theorem \ref{thm:topologies-on-U(H)-agree}]
By Lemma \ref{lem:weak=strong} we know that ${U(\mathcal{H})}_{s^*}={U(\mathcal{H})}_s={U(\mathcal{H})}_w$, so the vertical arrow in the middle and the bottom right horizontal arrow of diagram \eqref{diagram} are homeomorphisms; by Lemma \ref{lem:compact-open=strong} we know that ${U(\mathcal{H})}_{co}={U(\mathcal{H})}_s$, and then the bottom left horizontal arrow is also a homeomorphism. The proof of Lemma \ref{lem:compact-open=strong}
can also be used to show that ${U(\mathcal{H})}_{co^*}={U(\mathcal{H})}_{s^*}$. Finally, the right vertical arrow is also a homeomorphism because of Lemma \ref{lem:adjoint-continuous}. Hence diagram \eqref{diagram_B(H)} restricted to ${U(\mathcal{H})}$ becomes
\begin{equation} \xymatrix{
{U(\mathcal{H})}_n \ar[r] & {U(\mathcal{H})}_{co^*} \ar[d] \ar[r]^{\cong} & {U(\mathcal{H})}_{s^*} \ar[d]^{\cong} \ar[r] & {U(\mathcal{H})}_{w^*} \ar[d]^{\cong} \\
& {U(\mathcal{H})}_{co} \ar[r]^{\cong} & {U(\mathcal{H})}_{s} \ar[r]^{\cong} & {U(\mathcal{H})}_{w} } \label{diagram}
\end{equation}
Therefore, besides ${U(\mathcal{H})}_n$, every topological group in diagram \eqref{diagram} can be connected using a homeomorphism with the Polish group ${U(\mathcal{H})}_s$. This finishes the proof of Theorem \ref{thm:topologies-on-U(H)-agree}.
\end{proof}
The norm topology on ${U(\mathcal{H})}$ has strictly more open sets than the strong operator topology as can be easily checked with
the following sequence of operators.
Consider $\mathcal{H}:= l^2({\mathbb{N}})$ and take the sequence of operators
$T_k:\mathcal{H} \to \mathcal{H}$ such that for all $x=(\xi_n)\in\mathcal{H}$ the $n$-th coordinate of $T_kx$ is defined by
$$ (T_k x)_n = \begin{cases}
\xi_n & n \neq k \\
\sqrt{-1}\xi_n & n = k.
\end{cases}
$$ It follows that $T_k^{*}=T_k^{-1}$ and therefore $\{ T_k \}_k \subset {U(\mathcal{H})}$.
In the strong operator topology the sequence $\{T_k\}_k$ converges to the identity operator $\rm{Id}: \mathcal{H} \to \mathcal{H}$, since we have that
$$|T_k x - {\rm Id}x | =|i\xi_k - \xi_k| = |\xi_k||i - 1| \to 0 $$ for all $x=(\xi_n)\in\mathcal{H}$.
On the other hand, if $ x_k = (\xi_n^{(k)}) \in \mathcal{H}$ is defined by
$$ \xi_n^{(k)} = \begin{cases}
0 & n \neq k \\
i & n = k,
\end{cases}
$$ then $|T_kx_k - {\rm Id}x_k| = |-1-i| = \sqrt{2}$ and it follows that $\sup_{|x|\leq 1} \{ |T_kx - {\rm Id}x| \} \not\to 0$ whenever $k\to \infty$. Hence we have that $T_k \to_{s} {\rm Id}$ but $T_k \not \to_n {\rm Id}$.
Therefore we conclude that the operator
topologies defined at the beginning of the chapter reduced to only two once restricted to the unitary group. The unitary
group with the norm topology, making ${U(\mathcal{H})}_n$ into a Banach Lie group, and the strong operator topology, making ${U(\mathcal{H})}_s$
into a Polish group.
\typeout{-------------------------------------- References ---------------------------------------}
\bibliographystyle{plain}
| {
"timestamp": "2014-07-09T02:01:35",
"yymm": "1407",
"arxiv_id": "1407.1869",
"language": "en",
"url": "https://arxiv.org/abs/1407.1869",
"abstract": "We show that the strong operator topology, the weak operator topology and the compact-open topology agree on the space of unitary operators of a infinite dimensional separable Hilbert space. Moreover, we show that the unitary group endowed with any of these topologies is a Polish group.",
"subjects": "Algebraic Topology (math.AT)",
"title": "Topological properties of the unitary group",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754483707558,
"lm_q2_score": 0.7185943925708562,
"lm_q1q2_score": 0.7075103962621617
} |
https://arxiv.org/abs/1209.6444 | Large free sets in universal algebras | We prove that for each universal algebra $(A,\mathcal A)$ of cardinality $|A|\ge 2$ and an infinite set $X$ of cardinality $|X|\ge|\mathcal A|$, the $X$-th power $(A^X,\mathcal A^X)$ of the algebra $(A,\mathcal A)$ contains a free subset $\mathcal F\subset A^X$ of cardinality $|\mathcal F|=2^{|X|}$. This generalizes the classical Fichtenholtz-Kantorovitch-Hausdorff result on the existence of an independent family $\mathcal I\subset\mathcal P(X)$ of cardinality $|\mathcal I|=|\mathcal P(X)|$ in the Boolean algebra $\mathcal P(X)$ of subsets of an infinite set $X$. | \section*{Introduction}
It is well-known that each maximal linearly independent subset of a linear space $X$ generates $X$. A natural notion of independence can be also defined for other algebraic structures, in particular, for universal algebras. Unfortunately, even for relative simple universal algebras (like groups) the notion of independence does not work as good as for linear spaces. For example, any infinite linear space $V$ over the two-element field $\IF_2=\{0,1\}$ can be embedded into a non-commutative group $G$ of cardinality $|G|=2^{|V|}$ so that each maximal linearly independent subset $B\subset V$ remains maximal independent in $G$ and hence does not generate the whole group $G$. This fact (proved in Proposition~\ref{p1.3}) shows that the problem of constructing large independent sets in
general algebraic setting is not trivial.
A general theory of algebraic independence was created by Marczewski \cite{M1}--\cite{M3} and further developed by many mathematicians (see \cite{Glazek} and references therein). A subset $B$ of an abstract algebra $(A,\A)$ is called M-independent provided each function from $B$ in $A$ can be extended to a homomorphism defined on the algebra generated by $B$. This notion is a generalization of the linear and algebraic independence of numbers, linear
independence of vectors, independence of sets and elements of Boolean algebras and
many others. In our paper, Marczewski's independent sets will be called ``free''. The notion of a ``hull'' (or ``closure'') was introduced by Birkhoff \cite{B}
and also considered by Marczewski (e.g. in \cite{M2}). Other concepts of
independence considered in our paper can be also
found in \cite{M2} and \cite{Glazek}. For the reader's convenience we recall some proofs and
considerations in the modern notation.
Our main result is Theorem~\ref{main} saying that for each universal algebra $\IA=(A,\A)$ of cardinality $|A|\ge 2$ and each infinite set $X$ of cardinality $|X|\ge|\A|$, the $X$-power $\IA^X=(A^X,\A^X)$ of the algebra $\IA$ contains a free set $\F\subset A^X$ of cardinality $|\F|=2^{|X|}$. This generalizes a classical Fichtenholtz-Katorovitch-Hausdorff result on the existence of an independent family $\F\subset\mathcal P(X)$ of cardinality $|\F|=2^{|X|}$ in the power-set $\mathcal P(X)$ of $X$. {The other famous generalization of Fichtenholtz-Katorovitch-Hausdorff theorem is the Balcar-Fran\v ek theorem on the number of independent elements in complete Boolean algebras \cite{BF}.}
\section{Hull operators and (strongly) independent sets}
By a {\em hull operator} on a set $X$ we understand a function $\C:\mathcal P(X)\to\mathcal P(X)$ defined on a family of all subsets of $X$, which is {\em monotone} in the sense that for any subsets $A\subseteq B\subseteq X$ we get $A\subseteq \C(A)\subseteq \C(B)\subseteq X$.
Let $\C:\mathcal P(X)\to\mathcal P(X)$ be a hull operator on a set $X$. A subset $B\subset X$ is called
\begin{itemize}
\item {\em $\C$-independent} if $b\notin \C(B\setminus\{b\})$ for each $b\in B$;
\item {\em strongly $\C$-independent} if $B\cap \C(\emptyset)=\emptyset$ and $\C(B_1)\cap \C(B_2)=\C(B_1\cap B_2)$ for any subsets $B_1,B_2\subset B$.
\end{itemize}
In \cite{Schmidt} strongly $\C$-independent sets were called $\C_3$-independent.
\begin{proposition}\label{p1.1} Let $X$ be a set and $\C$ be a hull operator on $X$. Each strongly $\C$-independent set $B\subset X$ is $\C$-independent.
\end{proposition}
\begin{proof} Assuming that a strongly $\C$-independent set $B$ is not $\C$-independent, we can find a point $b\in B$ such that $b\in \C(B\setminus\{b\})$. Then $b\in \C(\{b\})\cap \C(B\setminus\{b\})=\C(\{b\}\cap (B\setminus\{b\}))=\C(\emptyset)$, which is not possible as $B\subset X\setminus \C(\emptyset)$.
\end{proof}
Proposition~\ref{p1.1} can be reversed for hull operators of matroid type.
We say that a hull operator $\C:\mathcal P(X)\to\mathcal P(X)$
\begin{itemize}
\item is {\em a closure operator} if $\C(\C(A))=\C(A)$ for each subset $A\subset X$;
\item has {\em finite supports} if for each subset $A\subset X$ and a point $x\in \C(A)$ there is a finite subset $F\subset A$ with $x\in \C(F)$;
\item has {\em the MacLane-Steinitz exchange property} if for any subset $A\subset X$
and points $x, y \in X \setminus \C(A)$ the inclusion $x\in \C(A\cup\{y\})$ is equivalent
to $y\in C(A\cup\{x\})$;
\item is of {\em matroid type} if $\C$ is a closure operator with finite supports and the Maclane-Steinitz exchange property.
\end{itemize}
It is well-known (and easy to see) that the operator of taking the linear hull in a linear space is of matroid type. The following proposition is well-known in the theory of independence (see \cite{Matroids}, \cite{Schmidt}, and \cite{Glazek}). We present a short proof for the convenience of the reader.
\begin{proposition}\label{p1.2} Let $\C:\mathcal P(X)\to\mathcal P(X)$ be a hull operator of matroid type on a set $X$.
\begin{enumerate}
\item A subset $B\subset X$ is $\C$-independent if and only if it is strongly $\C$-independent.
\item Each $\C$-independent subset lies in a maximal $\C$-independent subset of $X$.
\item Each maximal $\C$-independent subset $B\subset X$ has hull $\C(B)=X$.
\end{enumerate}
\end{proposition}
\begin{proof} 1. Proposition~\ref{p1.1} implies that each strongly $\C$-independent subset of $X$ is $\C$-independent.
Now assume that a set $B$ is $\C$-independent. Then for each $b\in B$ we get $b\notin \C(B\setminus\{b\})\supset \C(\emptyset)$ and hence $B\cap \C(\emptyset)=\emptyset$. To show that $B$ is strongly $\C$-independent, it remains to check that $\C(B_1)\cap \C(B_2)\subset \C(B_1\cap B_2)$ for any subsets $B_1,B_2\subset B$. This inclusion is trivial if $B_1\subset B_2$ or $B_2\subset B_1$. So, we assume that $B_1\not\subset B_2$ and $B_2\not\subset B_1$. Given any point $c\in \C(B_1)\cap \C(B_2)$, we should prove that $c\in \C(B_1\cap B_2)$. Assume conversely that $c\notin \C(B_1\cap B_2)$.
Since the closure operator $\C$ has finite supports, there are minimal finite subsets $F_1\subset B_1$ and $F_2\subset B_2$ such that $c\in \C(F_1)\cap \C(F_2)$. It follows from $F_1\cap F_2\subset \C(F_1\cap F_2)\subset \C(B_1\cap B_2)\not\ni h$ that $c\notin F_1\cap F_2$ and hence $c\notin F_1$ or $c\notin F_2$. Without loss of generality, $c\notin F_1$. {It follows from $\C(F_1)\cap \C(F_2)\not\subset \C(F_1\cap F_2)$, that $F_1\not\subset F_2$ and $F_2\not\subset F_1$.}
Choose any point $b\in F_1\setminus F_2$ and let $A=F_1\setminus\{b\}$. The $\C$-independence of the set $B$ implies $b\notin \C(B\setminus \{b\}){\supset} \C(A)$ and the minimality of the set $F_1$ implies $c\notin \C(A)$. Since $c\in \C(F_1)=\C(A\cup\{b\})$, the MacLane-Steinitz exchange property of $\C$ guarantees that $$b\in \C(A\cup\{c\})\subset \C(A\cup \C(F_2))\subset \C(\C(A\cup F_2))=\C(A\cup F_2)\subset \C(B\setminus\{b\}),$$
which contradicts the $\C$-independence of $B$.
\smallskip
2. The second statement can be easily proved using Zorn's Lemma and the fact that the closure operator $\C$ has finite supports.
\smallskip
3. Let $B$ be a maximal $\C$-independent subset in $X$. If $\C(B)\ne X$, then we can choose a point $x\in X\setminus \C(B)$. By the maximality of $B$ the set $B_x=B\cup\{x\}$ is not $\C$-independent. Consequently, there is a point $b\in B_x$ such that $b\in \C(B_x\setminus\{b\})$. If $b\ne x$, then consider the set $A=B\setminus\{b\}$ and observe that $b\in \C(B_x\setminus\{b\})=\C(A\cup\{x\})$. Then the MacLane-Steinitz exchange property implies $x\in \C(A\cup\{b\})=\C(B)$, which contradicts the choice of $x$. So, $b=x$ and $x=b\in \C(B_x\setminus\{b\})=\C(B)$, which contradicts the choice of $x$.
\end{proof}
Unfortunately many hull operators that naturally appear in algebra are not of matroid type.
The simplest example is the group hull operator. It assigns to each subset $A$ of a group $G$ the subgroup $\C(A)\subset G$ generated by $A$. Observe that the set $\{2,3\}$ is $\C$-independent in the group of integers $\IZ$ but fails to be strongly $\C$-independent in $\IZ$.
Each maximal strongly $\C$-independent subset $B$ in the group of rational $\IQ$ is a singleton and hence $\C(B)\ne \IQ$. The following proposition yields an example of a (non-commutative) group $G$ of cardinality continuum containing a maximal $\C$-independent subset which is countable and hence does not generate the whole group.
\begin{proposition}\label{p1.3} Each infinite linear space $V$ over the two-element field $\IF_2=\{0,1\}$ embeds into a group $G$ of cardinality $|G|=2^{|V|}$ such that for the operator of group hull $\C:\mathcal P(G)\to\mathcal P(G)$ each maximal $\C$-independent subset of $V$ remains maximal $\C$-independent in $G$.
\end{proposition}
\begin{proof} Since the field $\IF_2$ is finite, the linear space $V$ has a Hamel basis of cardinality $|V|$. Since each permutation of points of the Hamel basis induces a linear automorphism of the linear space $V$, the linear automorphism group $GL(V)$ of $V$ has cardinality $|GL(V)|=2^{|V|}$.
Let $G=V\rtimes GL(V)$ be the semidirect product of the groups $(V,+)$ and $GL(V)$.
Elements of the group $G$ are ordered pairs $(v,f)\in V\times GL(V)$ and for two pairs $(v,f),(u,g)\in V\rtimes GL(V)$ their product is defined by the formula
$$(v,f)\cdot(u,g)=(v+f(u),f\circ g).$$
The inverse element to a pair $(v,f)\in G$ is the pair $({-}f^{-1}(v),f^{-1})$.
Identify the group $V$ with the normal subgroup $V\times\{\id_V\}$ of $G$. Here by $\id_V:V\to V$ we denote the identity automorphism of $V$.
Let $M\subset V$ be a maximal $\C$-independent subset of the additive group $V$. Since for each subset $B$ of $V$ the group hull $\C(B)$ of $B$ coincides with its linear hull, the operator of group (=linear) hull in $V$ is of matroid type.
We need to check that $M$ remains maximal $\C$-independent in the group $G$. Given any element $g\in G\setminus M$ we need to show that the union $M\cup \{g\}$ is not $\C$-independent in $G$.
If $g\in V$, then the set $M\cup\{g\}\subset V$ is not $\C$-independent in $G$ by the maximality of the $\C$-independent set $M$ in $V$.
So, we assume that $g\notin V$. In this case $g=(u,f)\in V\times GL(V)$ for some non-identity automorphism $f$ of $V$.
Observe that $g^{-1}=(-f^{-1}({v}),f^{-1})$ and for each $v\in V$ we get
$$\begin{aligned}
gvg^{-1}&=(u,f)\cdot (v,\id_V)\cdot(-f^{-1}(u),f^{-1})=(f(v),{\id_V})=f(v)\mbox{ and }\\
g^{-1}vg&=(-f^{-1}(u),f^{-1})\cdot (v,{\id_V})\cdot (u,f)=(f^{-1}(v),{\id_V})=f^{-1}(v).
\end{aligned}
$$
Being a maximal linearly independent subset of $V$, the set $M$ is a Hamel basis in $V$. Since $f\ne\id_V$, there is a point $a\in M$ with $f(a)\ne a$. Since $M$ is a Hamel basis of the linear space $V$, each point $v\in V$ can be written as the sum $\Sigma B=\sum_{b\in B}b$ of a unique finite subset $B\subset M$. In particular, $a=\Sigma A$ and $f(a)=\Sigma F$ for some finite subsets $A,F\subset M$. Since $F\ne A$, there is a point $v\in (A\setminus F)\cup(F\setminus A)$.
If $v\in A\setminus F$, then consider the set $A_v=A\setminus\{v\}\subset M\setminus\{v\}$ and observe that
$$v=a-\Sigma A_v=f^{-1}(f(a))-\Sigma A_v= g^{-1}\cdot(\Sigma F,{\id_A})\cdot g\cdot(\Sigma {A_v,\id_A})^{-1}{\in} \C(\{g\}\cup M\setminus\{v\}),$$
which implies that $M\cup\{g\}$ is not $\C$-independent in $G$.
If $v\in F\setminus A$, then consider the set $F_v=F\setminus\{v\}\subset M\setminus\{v\}$ and observe that$$v=\Sigma F-\Sigma F_v=f(a)-\Sigma F_v=g\cdot (\Sigma A,{\id_V})\cdot g^{-1}\cdot (\Sigma F_v,{id_V})^{-1}{\in} \C(\{g\}\cup M\setminus\{v\}),$$which implies that $M\cup\{g\}$ is not $\C$-independent in $G$.
\end{proof}
So, in general, hull operators generated by algebraic structures need not be of matroid type, which makes the problem of constructing large (strongly) independent sets non-trivial. We shall be interested in hull operators induced by the structure of universal algebra (which includes as partial cases the structures of group, linear space, linear algebra, etc.) Such hull operators will be defined and studied in the next section.
\section{Universal algebras}\label{s:algebras}
A {\em universal algebra} is a pair $\IA=(A,\A)$ consisting of a set $A$ and a family $\A$ of algebraic operations on $A$. An {\em algebraic operation} on a set $A$ is any function $\alpha:A^{S_\alpha}\to A$ defined on a finite power $A^{S_\alpha}$ of $A$, where $S_\alpha$ is a finite subset of $\w$ called {\em the support} of the operation $\alpha$.
An algebraic operation $\alpha:A^{S_\alpha}\to A$ is {\em constant} if $\alpha(A^{S_\alpha})$ is a singleton. In particular, each algebraic operation $\alpha:A^\emptyset\to A$ with empty support is constant.
Observe that any function $\sigma:F\to E$ between finite subsets of $\w$ induces a dual function $\sigma^*:A^E\to A^F$, $\sigma^*:f\mapsto f\circ\sigma$, called a {\em substitution operator}. Then for any algebraic operation $\alpha:A^{S_\alpha}\to A$ with support $S_\alpha=F$ the composition $\beta=\alpha\circ\sigma^*$ is a well-defined algebraic operation on $A$ with support $S_\beta=E$.
A family $\A$ of operations on a set $A$ is called
\begin{itemize}
\item {\em unital} if $\A$ contains the identity operation $\id_A:A^1\to A$, $\id_A:(a_0)\mapsto a_0$;
\item {\em $\emptyset$-regular} if for each constant operation $\alpha\in\A$ there is an operation $\beta\in\A$ with empty support $S_\beta=\emptyset$ such that $\beta(A^\emptyset)=\alpha(A^{S_\alpha})$;
\item {\em stable under substitutions} (briefly, {\em substitution-stable}) if for any function $\sigma:F\to E$ between finite subsets of $\w$ and any algebraic operation $\alpha\in\A$ with $\supp(\alpha)=F$ the algebraic operation $\alpha\circ\sigma^*:A^E\to A$ belongs to $\A$;
\item {\em stable under compositions} if for any finite subset $S\subset\w$ and algebraic operations $\alpha\in\A$ and $\alpha_i\in\A$, $i\in S_\alpha$, with supports $\supp(\alpha_i)=S$ for all $i\in E$ the composition $\alpha\circ(\alpha_i)_{i\in S_\alpha}:A^S\to A$ of the diagonal product $(\alpha_i)_{i\in S_\alpha}:A^S\to A^{S_\alpha}$ and $\alpha$ belongs to $\A$;
\item a {\em clone} if $\A$ is unital, $\emptyset$-regular, and stable under substitutions and compositions.
\end{itemize}
The {\em clone} of a universal algebra $\IA=(A,\A)$ is the universal algebra $\bar\IA=(A,\bar\A)$
endowed with the smallest clone $\bar\A$ that contains the operation family $\A$.
The clone $\bar\A$ is equal to the union $\bar\A=\bigcup_{n\in\w}\A_n$ of operation families $\A_n$, $n\in\w$, defined by induction. Let $\A_0=\{\id_A\}$ and
$$
\begin{aligned}
\A_{n+1}=\A_n&\cup\{\alpha\in A^{A^\emptyset}:\exists \beta\in\A_n\;\;\alpha(A^\emptyset)=\beta(A^{S_\beta})\}\;\cup \\
&\cup \{\alpha\circ\sigma^*:\alpha\in{\A_n},\;\sigma:S_\alpha\to F\mbox{ is a function into a finite subset $F\subset\w$}\}\;\cup \\
&\cup\{\alpha\circ(\alpha_i)_{i\in {S_\alpha}}:\alpha\in\A,\;(\alpha_i)_{i\in S_\alpha}\in(\A_n)^{S_\alpha},\;\;\forall i,j\in S_\alpha \;\;S_{\alpha_i}=S_{\alpha_j}\}
\end{aligned}
$$for $n\in\w$.
This implies that the clone $\bar\A$ of $\A$ has cardinality $|\bar\A|\le\max\{|\A|,\aleph_0\}$.
Each universal algebra $\IA=(A,\A)$ possesses the {\em canonical hull operator} $\A(\cdot):\mathcal P(A)\to\mathcal P(A)$ assigning to each subset $B\subset A$ its {\em $\A$-hull} $$\A(B)=B\cup\bigcup_{\alpha\in\A}\alpha(B^{S_\alpha}).$$
The definition implies that this hull operation has finite supports.
If the operation family $\A$ is $\emptyset$-regular, then for each constant algebraic operation $\alpha\in\A$, the singleton $\alpha(A^{n_\alpha})$ lies in the $\A$-hull $\A(\emptyset)$ of the empty subset of $A$.
\begin{proposition}\label{p2.1} Let ${\IA=}(A,\A)$ be a universal algebra whose operation family $\A$ is unital and stable under substitutions. Then for each subset $B\subset A$ and a point $a\in\A(B)$ there is an algebraic operation $\alpha\in \A$ and an injective function $x:S_\alpha\to B$ such that $a=\alpha(x)$.
\end{proposition}
\begin{proof} Since $\A$ is unital, $\A(B)=\bigcup_{\beta\in\A}\alpha(B^{S_\beta})$ and we can find an operation $\beta\in\A$ and a function $z:S_\beta\to B$ such that $a=\beta(z)$. Let $x:S\to z(S_\beta)\subset B$ be any bijective map defined on a finite subset $S\subset\w$. Consider the function $\sigma=x^{-1}\circ z:S_\beta\to S$, which induces the substitution operator $\sigma^*:A^S\to A^{S_\beta}$. Since $\A$ is stable under substitutions, the operation $\alpha=\beta\circ\sigma^*:A^S\to A$ belongs to $\A$ and has support $S_\alpha=S$.
Moreover, $\alpha(x)=\beta\circ \sigma^*(x)=\beta(x\circ \sigma)=\beta(z)=a$.
\end{proof}
A subset $B\subset A$ is called a {\em subalgebra} of a universal algebra $\IA=(A,\A)$ if $\A(B)\subset B$, i.e., $B$ coincides with its $\A$-hull $\A(B)$. Since the intersection of subalgebras is a subalgebra, for each subset $B\subset A$ there is the smallest subalgebra of $\IA$ that contains $B$. This subalgebra is called the {\em subalgebra generated by $B$} and admits the following simple description:
\begin{proposition}\label{p2.2} Let $\IA=(A,\A)$ be a universal algebra and $\bar\IA=(\IA,\bar\A)$ be its clone. For each subset $B\subset A$ the subalgebra generated by $B$ coincides with the $\bar\A$-hull $\bar\A(B)$ of $B$.
\end{proposition}
\begin{proof} Let $\langle B\rangle$ denote the subalgebra of $\IA$ generated by $B$. The inclusion $\langle B\rangle\subset\bar{\A}(B)$ will follow as soon as we check that the $\bar{\A}$-hull $\bar\A(B)$ of $B$ is a subalgebra of the universal algebra $\IA$, that is $\bar{\A}(B)$ contains all constants and it is closed under all operations from $\A$. We need to check that $\A(\bar{\A}(B))\subset\bar{\A}(B)$. Take any element $y\in \A(\bar\A(B))$ and find an operation $\alpha\in\A$ and a function $x:S_\alpha\to \bar \A(B)$ such that $y=\alpha(x)$. For every $i\in S_\alpha$ the point $x(i)$ belongs to the $\bar\A$-hull $\bar\A(B)$ of $B$ and hence can be written as $x(i)=\alpha_i(z_i)$ for some algebraic operation $\alpha_i\in\bar\A$ and some function $z_i:S_{\alpha_i}\to B$. Choose a finite subset $S\subset\w$ of cardinality $|S|=\sum_{i\in S_\alpha}|S_{\alpha_i}|$ and for every $i\in S_\alpha$ choose an injective function $\sigma_i:S_{\alpha_i}\to S$ so that $\sigma_i(S_{\alpha_i})\cap \sigma_j(S_{\alpha_j})=\emptyset$ for $i\ne j$. Each function $\sigma_i$ induces the surjective substitution operators $\sigma_i^*:A^S\to A^{S_{\alpha_i}}$, $\sigma_i^*:f\mapsto f\circ\sigma_i$. Consider a unique function $z:S\to B$ such that $z\circ\sigma_i=z_i$ for all $i\in S_\alpha$.
Since $\bar \A$ is a clone, it is closed under substitutions. Consequently, for every $i\in S_\alpha$ the operation $\beta_i=\alpha_i\circ\sigma_i^*:A^S\to A$ belongs to $\bar \A$. It follows that $\beta_i(z)=\alpha_i\circ\sigma^*_i(z)=\alpha(z\circ\sigma_i)=\alpha_i(z_i)=x(i)$. Since the function family $\bar\A$ is closed under compositions, the operation $\beta=\alpha\circ(\beta_i)_{i\in S_\alpha}:A^S\to A$ belongs to $\bar\A$. Since $$\beta(z)=\alpha\big((\beta_i(z))_{i\in S_\alpha}\big)=\alpha\big((x(i))_{i\in S_\alpha}\big)=\alpha(x)=y,$$
the point $y=\beta(z)$ belongs to $\bar\A(B)$. So, $\A(\bar\A(B))\subset\bar\A(B)$ and $\bar\A(B)$ is a subalgebra of $\IA$, which implies $\langle B\rangle\subset\bar\A(B)$.
\smallskip
To prove that $\bar\A(B)\subset\langle B\rangle$, we use the decomposition $\bar\A=\bigcup_{n\in\w}\A_n$ of $\bar\A$ into the countable union of the operation families $\A_n$, $n\in\w$, defined at the beginning of Section~\ref{s:algebras} right after the definition of clone. Since $\bar\A(B)=\bigcup_{n\in\w}\A_n(B)$, it suffices to check that $\A_n(B)\subset\langle B\rangle$ for every $n\in\w$. This will be done by induction on $n\in\w$.
Since $\A_0=\{\id_A\}$, $\A_0(B)=B\subset\langle B\rangle$. Assume that for some $n\in\w$ we have proved that $\A_n(B)\subset\langle B\rangle$. The inclusion $\A_{n+1}(B)\subset\langle B\rangle$ will follow as soon as we check that $\beta(x)\in\langle B\rangle$ for each operation $\beta\in\A_{n+1}$ and a function $x:{S_\beta}\to B$. If $\beta\in\A_n$, then $\beta(f)\in\A_n(B)\subset\langle B\rangle$.
If $\beta\in\A_{n+1}\setminus\A_n$, then by the definition of the operation family $\A_{n+1}$, the following three cases are possible:
\smallskip
1) $S_\beta=\emptyset$ and there is a constant operation $\alpha\in\A_n$ such that $\beta(A^\emptyset)=\alpha(A^{S_\alpha})$. If $B=\emptyset$, then $x\in B^{S_\alpha}$ implies $S_\alpha=\emptyset=S_\beta$ and hence $\beta(x)=\alpha(x)\in \A_n(B)\subset\langle B\rangle$.
\smallskip
If $B\not=\emptyset$, then we can take any function $y:S_\alpha\to B$ and conclude that $\beta(x)=\alpha(y)\in\A_n(B)\subset\langle B\rangle$.
\smallskip
2) $\beta=\alpha\circ\sigma^*$ for some operation $\alpha\in\A_n$ and some function $\sigma:S_\beta\to S_\alpha$. Consider the function $y=x\circ\sigma:S_\beta\to B$ and observe that $\beta(x)=\alpha\circ\sigma^*(x)=\alpha(x\circ\sigma)\in\A_n(B)\subset\langle B\rangle$.
\smallskip
3) $\beta=\alpha\circ(\alpha_i)_{i\in S_\alpha}$ for some operations $\alpha\in\A$ and $(\alpha_i)_{i\in S_\alpha}\in\A_n^{S_\alpha}$ with $S_{\alpha_i}=S_\beta$ for all $i\in S_\alpha$.
Since $x\in B^{S_\beta}$, the inductive assumption $\A_n(B)\subset\langle B\rangle$ guarantees that for every $i\in S_\alpha$
the point $y(i)=\alpha_i(x)\in \A_n(B)$ belongs to $\langle B\rangle$. Consider the function $y:{S_\alpha}\to\langle B\rangle$, $y:i\mapsto y(i)=\alpha_i(x)$. Then $\beta(x)=\alpha\big((\alpha_i(x))_{i\in {S_\alpha}}\big)=\alpha((y(i))_{i\in {S_\alpha}})=\alpha(y)\in\A(\langle B\rangle)\subset\langle B\rangle$. The last inclusion $\A(\langle B\rangle)\subset\langle B\rangle$ follows from the fact that $\langle B \rangle$ is a subalgebra of $\IA$.
\end{proof}
Let $\IA=(A,\A)$ be a universal algebra, $X$ be a non-empty set, and $A^X$ be the set of all functions from $X$ to $A$. For every $x\in X$ denote by $\delta_x:A^X\to A$, $\delta_x:f\mapsto f(x)$, the projection onto $x$-th coordinate.
Observe that each algebraic operation $\alpha:A^{S_\alpha}\to A$ induces an algebraic operation $\alpha^X:(A^X)^{S_\alpha}\to A^X$ on the set $A^X$ of all functions from $X$ to $A$. The operation $\alpha^X$ assigns to
each function $f:S_\alpha\to A^X$ the function $\alpha^X(f):X\to A$ defined by $\alpha^X(f):x\mapsto \alpha(\delta_x\circ f)$ for $x\in X$. Writing the function $f$ is coordinates as $f=(f_i)_{i\in S_\alpha}$, we get that $\alpha^X(f)=\alpha^X\big((f_i)_{i\in S_\alpha}\big)$ is the function assigning to each $x\in X$ the point $\alpha\big((f_i(x))_{i\in S_{\alpha}}\big).$
For a universal algebra $\IA=(A,\A)$ its {\em $X$-th power} is the pair $\IA^X=(A^X,\A^X)$ consisting of the $X$-th power of $A$ and the operation family $\A^X=\{\alpha^X\}_{\alpha\in\A}$.
Now let us consider an important example of a universal algebra $\mathbf 2=(2,\mathcal B)$, called the {\em Boolean clone}. It consists of the doubleton $2=\{0,1\}$ and the family $\mathcal B$ of all possible algebraic operations on $2$. In the next section, we shall see that the powers $\mathbf 2^X$ of the Boolean clone play an important role in studying independent and free sets in powers of arbitrary universal algebras.
\section{Independent and free sets in universal algebras}
In this section we shall be interested in three independence notions in universal algebras.
A subset $B\subset A$ of a universal algebra $\IA=(A,\A)$ is called
\begin{itemize}
\item {\em $\A$-independent} if $b\notin\A(B\setminus\{b\})$ for all $b\in B$;
\item {\em strongly $\A$-independent} if $B\cap\A(\emptyset)=\emptyset$ and $\A(B_1)\cap\A(B_2)=\A(B_1\cap B_2)$ for any subsets $B_1,B_2\subset B$;
\item {\em $\A$-free} if for any function $f:B\to A$ there is a function $\bar f:\A(B)\to A$ such that $\bar f\circ \alpha(x)=\alpha(f\circ x)$ for any algebraic operation $\alpha\in \A$ and any function $x\in B^{S_\alpha}\subset A^{S_\alpha}$.
\end{itemize}
The notion of a (strongly) $\A$-independent set is induced by the operator of $\A$-hull. On the other hand, the notion of an $\A$-free set is specific for universal algebras and has no hull counterpart.
In fact, $\A$-free sets in universal algebras were introduced by E.Marczewski \cite{M1}, \cite{M2}, \cite{M3}. Because of that such set are sometimes called {\em $M$-independent}, see \cite{Glazek} and references therein.
The definitions imply that the notions of $\A$-independent and $\A$-free sets are ``monotone'' which respect to $\A$:
\begin{proposition} If a subset $B\subset A$ of a universal algebra $(A,\A)$ is $\A$-independent ($\A$-free), then it is $\A'$-independent ($\A'$-free) for each operation family $\A'\subset\A$.
\end{proposition}
On the other hand, the strong $\A$-independence is not monotone with respect to $\A$.
\begin{example} Take a set $A$ of cardinality $|A|\ge 2$ and consider an operation family $\A=\{\alpha,\beta\}$ consisting of two constant operations $\alpha:A^1\to A$, $\beta:A^0\to A$ with $\alpha(A^1)=\beta(A^0)$. Observe that each subset of $A$ is strongly $\A$-independent while each subset $B\subset A$ of cardinality $|B|\ge 2$ fails to be strongly $\A'$-independent for the subfamily $\A'=\{\alpha\}$. Indeed, take two non-empty disjoint subsets $B_1,B_2\subset B$ and observe that $\A'(B_1)\cap\A'(B_2)=\{\alpha(A^1)\}\ne\emptyset=\A'(\emptyset)$.
\end{example}
Now we shall characterize $\A$-free sets in universal algebras. Our first characterization follows immediately from the definition and was noticed by Marczewski in \cite{M2}.
\begin{proposition}\label{p3.3} A subset $B\subset A$ of a universal algebra $(A,\A)$ is $\A$-free if and only if for any algebraic operations $\alpha,\beta\in\A$ and functions $x\in B^{S_\alpha}$, $y\in B^{S_\beta}$ the equality $\alpha(x)=\beta(y)$ implies that $\alpha(f\circ x)=\beta(f\circ y)$ for any function $f:B\to A$.
\end{proposition}
For unital substitution-stable universal algebras this characterization can be improved as follows.
\begin{proposition}\label{p3.4} A subset $B\subset A$ of a unital substitution-stable universal algebra $(A,\A)$ is $\A$-free if and only if for any distinct algebraic operations $\alpha,\beta\in\A$ with $S_\alpha=S_\beta$ the inequality $\alpha(x)\ne\beta(x)$ holds for each injective function $x\in B^{S_\alpha}=B^{S_\beta}$.
\end{proposition}
\begin{proof} To prove the ``only if'' part, assume that the set $B$ is $\A$-free. Fix two distinct algebraic operations $\alpha,\beta\in\A$ such that $S_\alpha=S_\beta=S$ for some finite set $S\subset\w$. We need to show that $\alpha(x)\ne\beta(x)$ for each injective function
$x\in B^S$.
Since $\alpha\ne \beta$, there is a function $y\in B^S$ such that $\alpha(y)\ne\beta(y)$.
Using the injectivity of $x$, choose a function $f:B\to A$ such that $f|x(S)=y\circ x^{-1}|x(S)$. Since $B$ is $\A$-free, there is a function $\bar f:\A(B)\to A$ such that $\bar f(\gamma(z))=\gamma(f\circ z)$ for any $\gamma\in\A$ and $z\in A^{S_\gamma}$. In particular,
${\bar f(\alpha(x))}=\alpha(f\circ x)=\alpha(y)\ne\beta(y)=\beta(f\circ x)=\bar f(\beta(x))$, which implies $\alpha(x)\ne\beta(x)$.
\smallskip
To prove the ``if'' part, assume that the set $B$ is not $\A$-free. Applying Proposition~\ref{p3.3}, find algebraic operations $\alpha,\beta\in\A$ and functions $x\in B^{S_\alpha}$, $y\in B^{S_\beta}$ such that $\alpha(x)=\beta(y)$ and $\alpha(f\circ x)\ne \beta(f\circ y)$ for some function $f:B\to A$. Fix any bijective function $z:S\to x(S_\alpha)\cup y(S_\beta)\subset B$ defined on a finite subset $S\subset\w$.
Consider the functions $\sigma_\alpha=z^{-1}\circ x:S_\alpha\to S$ and $\sigma_\beta:z^{-1}\circ y:S_\beta\to S$, which induce the substitution operators $\sigma_\alpha^*:A^S\to A^{S_\alpha}$ and $\sigma_\beta^*:A^S\to A^{S_\beta}$. Since the operation family $\A$ is stable under substitutions, the algebraic operations $\tilde \alpha=\alpha\circ \sigma_\alpha^*:A^S\to A$ and $\tilde\beta=\beta\circ \sigma_\beta^*:A^S\to A$ belong to the family $\A$. Observe that $\tilde\alpha(z)=\alpha\circ\sigma_\alpha^*(z)=\alpha(z\circ\sigma_\alpha)= \alpha(x)=\beta(y)=\beta(z\circ\sigma_\beta)=\tilde\beta(z)$.
On the other hand, $$\tilde \alpha(f\circ z)=\alpha\circ\sigma_\alpha^*(f\circ z)=\alpha(f\circ z\circ \sigma_\alpha)=\alpha(f\circ x)\ne \beta(f\circ y)=\beta(f\circ z\circ\sigma_\beta)=\tilde \beta(f\circ z)$$implies that $\tilde\alpha\ne\tilde \beta$.
\end{proof}
By Proposition~\ref{p1.1}, each strongly $\A$-independent subset $B\subset A$ of a universal algebra $(A,\A)$ is $\A$-independent.
\begin{proposition}\label{p3.5} Let $(A,\A)$ be a unital $\emptyset$-regular substitution-stable universal algebra of cardinality $|A|\ge 2$. Each $\A$-free subset $B\subset A$ is strongly $\A$-independent.
\end{proposition}
\begin{proof} First we prove that $B\cap\A(\emptyset)=\emptyset$. Assume conversely that $B\cap\A(\emptyset)$ contains some point $b$.
Since $|A|\ge 2$, we can find two functions $f_1,f_2:B\to A$ such that $f_1(b)\ne f_2(b)$. Since $B$ is $\A$-free, for every $i\in\{1,2\}$, there is a function $\bar f_i:\A(B)\to A$ such that $\bar f_i\circ\alpha(x)=\alpha(f_i\circ x)$ for any $\alpha\in \A$ and $x\in B^{S_\alpha}$.
The universal algebra $(A,\A)$ is unital and hence contains the identity algebraic operation $\alpha:A^1\to A$, $\alpha:(a)\mapsto a$. For this operation we get $\bar f_i(b)=\bar f_i\circ\alpha(b)=\alpha(f_i(b))=f_i(b)$ and hence $\bar f_1(b)\ne\bar f_2(b)$.
On the other hand, the inclusion $b\in\A(\emptyset)$ yields a 0-ary operation $\beta\in\A$ such that $b=\beta(\emptyset)$ where $\emptyset:\emptyset\to B$ is the unique element of the power $B^\emptyset=B^{S_\beta}$.
Then for every $i\in\{1,2\}$, the choice of $\bar f_i$ guarantees that $\bar f_i(b)=\bar f_i\circ\beta(\emptyset)=\beta(f_i\circ\emptyset)=\beta(\emptyset)=b$ and hence $\bar f_1(b)=b=\bar f_2(b)$, which contradicts the inequality $\bar f_1(b)\ne\bar f_2(b)$ proved earlier.
Hence $B\cap\A(\emptyset)=\emptyset$.
Next, we prove that $\A(B_1)\cap\A(B_2)=\A(B_1\cap B_2)$ for any subsets $B_1,B_2\subset B$.
This equality is trivial if $B_1\subset B_2$ or $B_2\subset B_1$. So we assume that both complements $B_1\setminus B_2$ and $B_2\setminus B_1$ are not empty.
Assume that $\A(B_1)\cap\A(B_2)\ne\A(B_1\cap B_2)$ and find a point $a\in\A(B_1)\cap\A(B_2)\setminus\A(B_1\cap B_2)$. {For the point $a\in\A(B_1)\cap\A(B_2)$, there are algebraic operations $\alpha,\beta\in\A$ and functions $x\in B_1^{S_\alpha}$, $y\in B_2^{S_\beta}$ such that $\alpha(x)=a=\beta(y)$. Using a similar reasoning as in the proof of Proposition~\ref{p3.4} we may assume that $x$ and $y$ are injective}
Let us show that the operation $\alpha$ is not constant. In the opposite case, the $\emptyset$-regularity of $\A$, yields a 0-ary operation $\gamma\in\A$ such that $\alpha(A^{S_\alpha})=\gamma(A^\emptyset)$. Then $a=\alpha(x)=\gamma(\emptyset)\in\A(\emptyset)\subset \A(B_1\cap B_2)$, which contradicts the choice of $a$.
Next, we prove that the intersection $x(S_\alpha)\cap y(S_\beta)\subset B_1\cap B_2$ is not empty. Assuming the converse and using the fact that the operation $\alpha$ is not constant,
find a function $x'\in A^{S_\alpha}$ such that $\alpha(x')\ne\beta(y)$. Since the subsets $x(S_\alpha)$ and $y(S_\beta)$ of $B$ are disjoint, we can find a function $f:B\to A$ such that $f|x(S_\alpha)=x'\circ x^{-1}$ and $f|y(S_\beta)=y\circ y^{-1}$. Then
$$\bar f(a)=\bar f(\alpha(x))=\alpha(f\circ x)=\alpha(x')\ne\beta(y)=\beta(f\circ y)=\bar f(\beta(y))=\bar f(a),$$
which is a contradiction.
Thus the intersection $x(S_\alpha)\cap y(S_\beta)\subset B_1\cap B_2$ is not empty and we can
choose a function $f:B\to A$ such that $f|y(S_\beta)={\id_{y(S_\beta)}}$ and $f(x(S_\alpha)\setminus y(S_\beta))\subset B_1\cap B_2$. Such a choice of $f$ guarantees that $f\circ x\in B_1\cap B_2$ and then $\alpha(f\circ x)\in \A(B_1\cap B_2)\not\ni a$ implies that $a\ne \alpha(f\circ x)$. Since the set $B$ is $\A$-free, the function $f$ can be extended to a function $\bar f:\A(B)\to A$ such that $\bar f(\gamma(z))=f\circ \gamma(z)$ for each $\gamma\in\A$ and $z\in B^{S_\gamma}$. In particular,
$$\bar f(a)=\bar f(\beta(y))=\beta(f\circ y)=\beta(y)=a\ne \alpha(f\circ x)=\bar f(\alpha(x))=\bar f(a).$$
This contradiction completes the proof of the strong $\A$-independence of the set $B$.
\end{proof}
Proposition~\ref{p3.5} shows that for a subset of a unital $\emptyset$-regular substitution-stable universal algebra $(A,\A)$ we have the following implications:{
$$\mbox{$\A$-free $\Ra$ strongly $\A$-independent $\Ra$ $\A$-independent.}
$$
The first} implication cannot be reversed as shown by the following simple example.
\begin{example} There is a clone $(A,\bar\A)$ containing an infinite strongly $\bar\A$-independent subset $B\subset X$ such that each $\bar\A$-free subset of $A$ is empty.
\end{example}
\begin{proof} Consider the linear algebra $c_{00}$ consisting of all functions $x:\w\to\IR$ with finite support $\supp(x)=\{n\in\w:x(n)\ne 0\}\subset\w$. This algebra is endowed with the operation family $\A=\{+,\cdot\}\cup\{\alpha_t:t\in\IR\}$ consisting of two binary operations (of addition and multiplication) and continuum many unary operations $\alpha_t:\vec x\mapsto t\cdot\vec x$ of multiplication by a real number $t$. The clone $\bar\A$ of $\A$ contains the subfamily $\A'$ consisting of all polynomials $p(x)=\sum_{i=0}^n\lambda_ix^{i+1}$ of one variable, equal to zero at the zero function.
\begin{claim} The set
$B=\{x\in c_{00}:x(\w)\subset\{0,1\},\;|\supp(x)|=1\}$ of characteristic functions of singletons is strongly $\bar \A$-independent.
\end{claim}
\begin{proof} It is easy to check that for each subset $B'\subset B$ its $\bar\A$-hull $\bar\A(B')$ is equal to the linear subspace $\IR^{\supp(B')}=\{x\in c_{00}:\supp(x)\subset \supp(B')\}$ where $\supp(B')=\bigcup_{b\in B'}\supp(b)$. It follows that $B\cap\bar\A(\emptyset)=B\cap\{0\}=\emptyset$ and for any two subsets $B_1,B_2\subset B$ we get
$$\bar\A(B_1)\cap\bar\A(B_2)=\IR^{\supp(B_1)}\cap\IR^{\supp(B_2)}=\IR^{\supp(B_1)\cap\supp(B_2)}=\IR^{\supp(B_1\cap B_2)}=\bar\A(B_1\cap B_2),$$which means that the subset $B$ is strongly independent.
\end{proof}
\begin{claim} Any non-empty subset $B\subset A$ is not $\A'$-free and hence is not $\bar\A$-free.
\end{claim}
\begin{proof} Fix any function $b\in B$. This function has finite support $F=\supp(b)$. Then $b$ and all its finite powers $b^n$, $n>0$, belong to the finite-dimensional linear subspace $\IR^F=\{x\in c_{00}:\supp(x)\subset F\}$ of $c_{00}$. Consequently, the set $\{b^{n+1}:0\le n\le |F|\}$ is linearly dependent, which allows us to find a non-zero vector $(\lambda_0,\dots,\lambda_{|F|})\in\IR^{|F|+1}$ such that $\sum_{i=0}^{|F|}\lambda_{i}b^{i+1}=0$. This means that $p(b)=0$ for the non-zero polynomial $p(x)=\sum_{i=0}^{|F|}\lambda_i x^{i+1}$. Since the polynomial $p\in\A'$ is non-zero, there is a vector $y\in \IR^F$ such that $p(y)\ne 0$. Let $f:B\to c_{00}$ be any function such that $f(b)=y$.
Assuming that the set $B$ is $\A'$-free, we could find a function $\bar f:\A'(B)\to c_{00}$ such that $\bar f(p(b))=p(f\circ b)$. But $\bar f(p(b))=0\ne p(y)=p(f(b))$. This contradiction completes the proof.
\end{proof}
\end{proof}
As we saw in Proposition~\ref{p1.2}, the $\C$-independence is equivalent to the strong $\C$-independence for closure operators of matroid type. In contrast, the notions of an $\A$-independent set and a $\A$-free set are not be equivalent (even in presence of the MacLane-Steinitz exchange property). The following simple example is taken from \cite[p.297]{Gould}.
\begin{example} Consider the 2-element set $A=\{0,1\}$ and the set $\A=\{\cdot\}$ containing the binary operation $\cdot:A\times A\to A$ of multiplication. It follows that each subset of $A$ is an $\A$-subalgebra, which implies that the set $A$ is $\A$-independent. By a trivial reason the $\A$-hull operator on $X$ has the MacLane-Steinitz exchange property and hence is a closure operator of matroid type. However, the set $\A$ is not $\A$-free, as the unique non-identity bijection $f:A\to A$ is not a homomorphism of $A$.
\end{example}
According to \cite{Gould}, a universal algebra $\IA=(A,\A)$ is called an {\em independence algebra} if the operator of $\bar\A$-hull has the MacLane-Steinitz exchange property and each $\bar\A$-independent subset of $A$ is $\bar\A$-free. Independent algebras were thoroughly studied in \cite{Gould}, \cite{AF} and much earlier in \cite{Nar1}, \cite{Nar2} (as $v^*$-algebras).
\section{Free sets in powers of universal algebras}
In this section we shall construct free sets of large cardinality in powers of universal algebras.
We start with studying free sets in a power $\mathbf 2^X=(2^X,\mathcal B^X)$ of the Boolean clone $\mathbf 2=(2,\mathcal B)$ consisting of the doubleton $2=\{0,1\}$ and the family $\mathcal B$ of all possible algebraic operations on $2$.
It turns out that $\mathcal B^X$-free subsets of $2^X$ coincide with independent sets, well studied in Set Theory \cite[\S17]{JW}.
Let us recall \cite[p.83]{JW} that a family $\F$ of subsets of a set $X$ is called {\em independent} if for any finite disjoint sets $\F_1,\F_2\subset \F$ the intersection
$$\bigcap_{F\in\F_1}F\cap\bigcap_{F\in\F_2}(X\setminus F)$$is not empty.
Identifying each subset $F\subset X$ with its characteristic function $\chi_F:X\to 2=\{0,1\}$, we can reformulate the notion of an independent family of sets in the language of an independent function family. Namely, a function family $\F\subset 2^X$ is {\em independent} if for any pairwise distinct functions $f_0,\dots,f_{n-1}\in\F$ their diagonal product $(f_i)_{i<n}:X\to 2^n$ is surjective.
\begin{proposition}\label{p4.1} For any set $X$ a function family $\F\subset 2^X$ is independent if and only it is $\mathcal B^X$-free in the clone $\mathbf 2^X$.
\end{proposition}
\begin{proof} To prove the ``if'' part, assume that $\F$ is $\mathcal B^X$-free but not independent. Then there are pairwise distinct functions $\xi_0,\dots,\xi_{n-1}\in\F\subset 2^X$ whose diagonal product $\delta=\Delta_{i<n}\xi_i:X\to 2^n$, $\delta:x\mapsto (\xi_i(x))_{i<n}$ is not surjective and hence the set $F=2^n\setminus \delta(X)$ is not empty. Let $\alpha,\beta:2^n\to 2$ be two algebraic operations on $2$ defined by $\alpha^{-1}(1)=F$ and $\beta^{-1}(1)=\emptyset$. They induce the algebraic operations $\alpha^X,\beta^X:(2^X)^n\to 2^X$ on $2^X$. It follows that for the function $\xi:n\to \F$, $\xi:i\mapsto \xi_i$, we get $\alpha^X(\xi)=\beta^X(\xi)$.
On the other hand, Since $\alpha^X\ne\beta^X$, there is a function $\tilde\xi:n\to 2^X$ such that $\alpha^X(\tilde\xi)\ne\beta^X(\tilde \xi)$. Now choose any function $f:\F\to 2^X$ such that $f|\xi(n)=\tilde\xi\circ\xi^{-1}$. Since $\F$ is $\mathcal B^X$-free, there is a function $\bar f:\mathcal B^X(\F)\to 2^X$ such that $\bar f(\gamma^X(\xi))=\gamma^X(f\circ\xi)$ for any algebraic operation $\gamma\in\mathcal B$ with $S_\gamma=n$. In particular,
$\bar f(\alpha^X(\xi))=\alpha^X(f\circ\xi)=\alpha^X(\tilde\xi)\ne\beta^X(\tilde \xi)=\beta^X(f\circ\xi)=\bar f(\beta^X(\xi))$, which contradicts $\alpha^X(\xi)=\beta^X(\xi)$. This contradiction completes the proof of the ``if'' part.
\smallskip
To prove the ``only if'' part, assume that $\F$ is independent but not $\mathcal B^X$-free.
By Proposition~\ref{p3.4}, there are two distinct operations $\alpha,\beta\in\mathcal B$ such that $S_\alpha=S=S_\beta$ for some finite set $S\subset\w$ and $\alpha^X(\xi)=\beta^X(\xi)$ for some injective function $\xi:S\to \F$. The function $\xi$ can be written in the form $(\xi_i)_{i\in S}$ where $\xi_i=\xi(i)\in\F$. The independence of $\F$ guarantees that the diagonal product $(\xi_i)_{i\in S}:X\to 2^S$, $\delta=(\xi_i)_{i\in S}:x\mapsto (\xi_i(x))_{i\in S}$ of these functions is surjective.
Then $\alpha\ne\beta$ implies $\alpha\circ\delta\ne\beta\circ\delta$, which yields a point $x\in X$ such that $\alpha\circ\delta(x)\ne\beta\circ\delta(x)$. Now observe that
$$\alpha^X(\xi)(x)=\alpha((\xi_i(x))_{i\in S})=\alpha\circ\delta(x)\ne\beta\circ\delta(x)=\beta((\xi_i(x))_{i\in S})=\beta^X(\xi)(x)$$which contradicts $\alpha^X(\xi)=\beta^X(\xi)$.
\end{proof}
By a classical Fichtenholtz-Katorovitch-Hausdorff Theorem 17.20 \cite{JW}, the power-set $\mathcal P(X)$ of each infinite set $X$ contains an independent family $\F\subset\mathcal P(X)$ of cardinality $|\F|=|2^{X}|=|\mathcal P(X)|$. Reformulating this result with help of Proposition~\ref{p4.1}, we get the following result:
\begin{corollary}\label{c4.2} For each infinite set $X$ the power-clone $\mathbf 2^X=(2^X,\mathcal B^X)$ contains a $\mathcal B^X$-free subset $B\subset 2^X$ of cardinality $|B|=|2^X|$.
\end{corollary}
In fact, Corollary~\ref{c4.2} is a partial case of the following theorem, which is the main result of this paper.
\begin{theorem}\label{main} Assume that a universal algebra $\IA=(A,\A)$ has cardinality $|A|\ge 2$. For each infinite set $X$ of cardinality $|X|\ge |\A|$ the function algebra $\IA^X=(A^X,\A^X)$ contains an $\A^X$-free subset $\F\subset A^X$ of cardinality $|\F|=2^{|X|}$.
\end{theorem}
\begin{proof} Let $\bar\A$ be the clone of the operation family $\A$. It has cardinality $|\bar\A|\le\max\{|\A|,\aleph_0\}\le|X|$. Since each $\bar\A^X$-free subset of $A^X$ is $\A^X$-free, we lose no generality assuming that $\A$ is a clone. In particular, $\A$ is unital, $\emptyset$-regular and substitution-stable.
For each finite subset $S\subset \w$, consider the family
$$\Tau_S=\{(\alpha,\beta,s)\in\A\times\A\times X^S:S_\alpha=S=S_\beta,\;\alpha\ne\beta\}$$and observe that it has cardinality $|\Tau_S|\le |\A\times \A\times X^S|\le |X|$.
Then the union $\Tau=\bigcup_{S\in[\w]^{<\w}}\Tau_S$ where $S$ runs over all finite subsets of $\w$ also has cardinality $|\Tau|\le|X|$ and
hence admits an enumeration $\Tau=\{(\alpha_x,\beta_x,s_x):x\in X\}$ by points of the set $X$. By the definition of the family $\Tau$, for every $x\in X$ the algebraic operations $\alpha_x$ and $\beta_x$ are distinct. Consequently, we can choose a function $p_x:S_x\to A$ defined on the set $S_x=S_{\alpha_x}=S_{\beta_x}$ such that $\alpha_x(p_x)\ne \beta_x(p_x)$.
Using Fichtenholtz-Katorovitch-Hausdorff Theorem~17.20 \cite{JW}, fix an independent subfamily $\U\subset \mathcal P(X)$ of cardinality $|\U|=2^{|X|}$.
For each set $U\in\U$ define a function $f_U:X\to A$ assigning to a point $x\in X$ the point
$p_x(i)$ where $i$ is a unique point of the set $s_x^{-1}(U)\subset {U_x}$ if this set is a singleton, and an arbitrary point of $A$ otherwise.
We claim that the set $\F=\{f_U\}_{U\in\U}\subset A^X$ has cardinality $|\F|=2^{|X|}$ and is $\A^X$-free.
\begin{claim} The function $f:\U\to \F$, $f:U\mapsto f_U$, is bijective and hence $|\F|=|\U|=2^{|X|}$.
\end{claim}
\begin{proof} Given two distinct sets $U,V\in\U$, we should prove that ${f_U\ne f_V}$. By the unitality, the operation family $\A$ contains the identity operation $\id:A^1\to A$, $\id:(a)\mapsto a$.
Consider the embeddings $$\sigma_0:1\to 2\;\;\sigma_0:0\mapsto 0\mbox{ \ \ and \ \ }\sigma_1:1\to 2,\;\; \sigma_1:0\mapsto 1.$$ The substitution-stability of $\A$ implies that the algebraic operations $\alpha=\id\circ\sigma^*_0:A^2\to A$, $\alpha:(a,b)\mapsto a$, and $\beta=\id\circ\sigma^*_1:A^2\to A$, $\beta:(a,b)\mapsto b$, belong to the operation family $\A$. It follows from $|A|\ge 2$ that $\alpha\ne \beta$.
By the independence of $\U\ni U,V$, there is a function $s:2\to X$ such that $s(0)\in U\setminus V$ and $s(1)\in V\setminus U$. The triple $(\alpha,\beta,s)$ belongs to the family $\Tau_2\subset\Tau$ and hence is equal to $(\alpha_x,\beta_x,s_x)$ for some $x\in X$. Then $p_x\in A^2$ is a function such that $p_x(0)=\alpha_x(p_x)\ne \beta_x(p_x)=p_x(1)$. The definition of the functions $f_U$ and $f_V$ guarantees that $f_U(x)=p_x(0)\ne p_x(1)=f_V(x)$ and hence $f_U\ne f_V$.
\end{proof}
\begin{claim} The set $\F$ is $\A$-free.
\end{claim}
\begin{proof} Assuming that $\F$ is not $\A$-free and applying Proposition~\ref{p3.4}, find a finite subset $S\subset\w$, an injective function $\xi:S\to \F$ and two distinct algebraic operations $\alpha,\beta\in\A$ such that $S_\alpha=S=S_\beta$ and $\alpha^X(\xi)=\beta^X(\xi)$.
For every $i\in {S}$ find a set $U_i\in\U$ such that $\xi(i)=f_{U_i}$.
The independence of the family $\U$ guarantees the existence of a function $s:S\to X$ such that $s(i)\in U_i\cap\bigcap_{j\in {S}\setminus\{i\}}(X\setminus U_j)$. It follows that $s^{-1}(U_i)=\{i\}$ for each $i\in {S}$.
The triple $(\alpha,\beta,s)$ belongs to the family $\Tau_{S}\subset\Tau$ and hence is equal to $(\alpha_x,\beta_x,s_x)$ for some point $x\in X$. For every $i\in {S}$ the definition of the function $f_{U_i}$ guarantees that $f_{U_i}(x)=p_x(i)$. Let $\delta_x:A^X\to A$, $\delta_x:g\mapsto g(x)$, denote the $x$-th coordinate projection. We claim that $\delta_x\circ\xi=p_x$. Indeed, for each $i\in S$ we get $\delta_x\circ \xi(i)=\delta_x(f_{U_i})=f_{U_i}(x)=p_x(i)$.
Applying the function $\delta_x$ to the equality $\alpha^X(\xi)=\beta^X(\xi)$, we get
$\delta_x\circ\alpha^X(\xi)=\delta_x\circ\beta^X(\xi)$. On the other hand,
$\delta_x\circ\alpha^X(\xi)=\alpha(\delta_x\circ\xi)=\alpha(p_x)\ne\beta(p_x)=\beta(\delta_x\circ\xi)=\delta_x\circ \beta^X(\xi)$, which is a desired contradiction.
\end{proof}
\end{proof}
\begin{remark} For some concrete universal algebras, in particular, for the linear algebra $\mathbb F^X$ of all functions from an infinite set $X$ to a field $\mathbb F\in\{\IR,\IC\}$, a free subalgebra $A\subset\mathbb F^X$ with $2^{|X|}$ generators (and some additional properties) has been constructed in \cite{BGP}.
\end{remark}
\begin{problem} Let $\IA=(A,\A)$ be a universal algebra with $|A|\ge 2$, $X$ be an infinite set of cardinality $|X|\ge|\A|$, and $\F\subset A^X$ be a maximal $\A^X$-free subset. Is $|\F|\ge |2^X|$?
\end{problem}
\section{Acknowledgment}
The authors express their sincere thanks to Jo\~ao Ara\'ujo for pointing us relevant references in Independence Theory.
| {
"timestamp": "2013-01-10T02:02:18",
"yymm": "1209",
"arxiv_id": "1209.6444",
"language": "en",
"url": "https://arxiv.org/abs/1209.6444",
"abstract": "We prove that for each universal algebra $(A,\\mathcal A)$ of cardinality $|A|\\ge 2$ and an infinite set $X$ of cardinality $|X|\\ge|\\mathcal A|$, the $X$-th power $(A^X,\\mathcal A^X)$ of the algebra $(A,\\mathcal A)$ contains a free subset $\\mathcal F\\subset A^X$ of cardinality $|\\mathcal F|=2^{|X|}$. This generalizes the classical Fichtenholtz-Kantorovitch-Hausdorff result on the existence of an independent family $\\mathcal I\\subset\\mathcal P(X)$ of cardinality $|\\mathcal I|=|\\mathcal P(X)|$ in the Boolean algebra $\\mathcal P(X)$ of subsets of an infinite set $X$.",
"subjects": "Logic (math.LO); Rings and Algebras (math.RA)",
"title": "Large free sets in universal algebras",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754479181588,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.7075103959369279
} |
https://arxiv.org/abs/2209.05677 | Connectivity of a Family of Bilateral Agreement Random Graphs | Bilateral agreement based random undirected graphs were introduced and analyzed by La and Kabkab in 2015. The construction of the graph with $n$ vertices in this model uses a (random) preference order on other $n-1$ vertices and each vertex only prefers the top $k$ other vertices using its own preference order; in general, $k$ can be a function of $n$. An edge is constructed in the ensuing graph if and only if both vertices of a potential edge prefer each other. This random graph is a generalization of the random $k^{th}$-nearest neighbor graphs of Cooper and Frieze that only consider unilateral preferences of the vertices. Moharrami \emph{et al.} studied the emergence of a giant component and its size in this new random graph family in the limit of $n$ going to infinity when $k$ is finite. Connectivity properties of this random graph family have not yet been formally analyzed. In their original paper, La and Kabkab conjectured that for $k(t)=t \log n$, with high probability connectivity happens at $t>1$ and the graph is disconnected for $t<1$. We provide a proof for this conjecture. We will also introduce an asymptotic for the average degree of this graph. | \section{Introduction}\label{sec:intro}
Graphs are powerful mathematical tools used for modeling objects in a wide variety of topics~\cite{easley2010networks,jackson2010social,newman2018networks}: from the Internet and chemistry to social science and economics. These mathematical objects have been studied using algebraic, combinatorial, and stochastic methods. Studying graphs using a stochastic framework~\cite{van2016random} has a long history starting from when Erdos and Renyi introduced the well-known Erdos-Renyi model in their celebrated paper \cite{erdHos1960evolution} in 1960. This fundamental random graph model has been extensively analyzed over the last 50 years.
An Erdos-Renyi random graph with $n$ vertices is characterized by a parameter $0\leq p(n) \leq 1$ indicating the probability of existence of each potential undirected edge between two vertices. Edges then appear independently based on Bernoulli coin tosses with no intrinsic preference between the edges on the part of the vertices. One question that immediately arises by specifying the parameter $p(n)$ is whether the Erdos-Renyi graphs are connected when $n\rightarrow\infty$. In particular, Erdos and Renyi~\cite{erdHos1960evolution} prove that if $p(n)=t \log(n)/n$, with high probability, the graph is connected for $t>1$ and disconnected for $t<1$; note that the mean degree in this setting is $t\log(n)$.
Cooper and Frieze in \cite{cooper1995connectivity} described a family of new random graphs based on preferences constructed using a distance measure. They considered the complete graph $K_n$ and assigned independent uniform $[0,1]$ random variables as distances to all edges. They then keep the $k(n)$ shortest edges incident to each vertex. In this model, the concept of distance induces a preference order on all edges. Indeed, the shorter an edge is the higher its preference is. Since the distances are \emph{i.i.d.}, the preference order will be chosen uniformly over the set of all permutations. We say a vertex agrees on an edge if it belongs to the set of the $k(n)$ most-preferred edges incident to it. Note that in this model, an edge is kept if at least one of its endpoints agrees on it which is a unilateral perspective.
Preference relations of the sort used by Cooper and Frieze create a complicated structure on any resulting graph. For example, keeping edges based on their ranking even in the unilateral mode as done by Cooper and Frieze, induces an edge dependency which is not the case for Erdos-Renyi model. Cooper and Frieze studied the connectivity of their model and proved that the graph is connected with high probability for $k\geq 3$. They also provided upper and lower bounds for the probability of connectivity when $k=2$ as $n\rightarrow\infty$.
Just like the Cooper-Frieze unilateral agreement random graphs, bilateral agreement random graphs introduced by La and Kabkab \cite{la2015new} are constructed based on the parameter $k(n)$ which is the maximum number of other vertices each vertex wants as its neighbors. Considering vertices as agents, once again, all agents have their own preferences on the potential edges with others via a priority or preference order over other agents. Then, in contrast to both the Erdos-Renyi and the Cooper-Frieze models, an edge is drawn if and only if each end vertex has the other vertex in its $k(n)$ preferred vertices.
One can interpret the bilateral agreement model as a network formation process conducted via a game among agents. Assume there are $n$ agents are aware of their own benefits from the potential pairwise contracts with others. Then each agent agrees on its own $k(n)$ most profitable contracts. Finally, each contract will be concluded if and only if both parties have mutual agreement. The interpretation above highlights the importance of bilateral agreement random graphs from the game-theoretic (individual agent oriented) point of view. These kinds of network formation processes~\cite{easley2010networks,jackson2010social,newman2018networks} have been studied in social science and economics.
A core question in the La and Kabkab paper~\cite{la2015new} is to determine when the graphs produced are connected. The authors showed the following results hold with high probability: 1) if $k(n) > C \log(n)$ for $C=2.4625$, then the graph is connected; and 2) $k(n)< c \log(n)$ for $c=0.5$, then the graph has isolated vertices and so is not connected. Furthermore, using extensive simulation-based experimentation, they also conjectured that the connectivity threshold was exactly $k(n)=\log(n)$, surprisingly, which is the same threshold for connectivity of the Erdos-Renyi random graphs family in terms of the mean degree. In this paper, we prove this conjecture by La and Kabkab.
Moharrami \emph{et al.} \cite{moharrami2020erlang} introduce a new branching process called the Erlang Weighted Tree (EWT) as the local weak limit of the bilateral agreement random graphs of La and Kabkab. In \cite{moharrami2020erlang} the parameter $k(n)$ is a (finite) random parameter $k$ for each vertex that is independently chosen and identically distributed with a distribution on $\mathbb{N}$ with finite mean. The authors then studied the degree distribution of the root vertex and its average. We will return to this specific result of \cite{moharrami2020erlang} in Section~\ref{sec:avgdegree}. They also discuss the probability of extinction of an EWT, and conjecture its relevance to the probability of a giant component~\cite{erdHos1960evolution,van2016random} arising.
\section{Mathematical Model}\label{sec:model}
Consider the complete undirected graph $K_n$ with vertices $[n]=\{1,2,\cdots,n\}$ and edges $\{\{i,j\}: 1\leq i,j\leq n, i\neq j\}$. Let $k(n)$ be an integer such that $1\leq k(n)\leq n$; henceforth, to avoid cumbersome notation, we will use $k$ instead of $k(n)$. We assign independently and identically distributed random variables called priority scores to all edges of $K_n$. For any explicit calculations, one can assume they are uniformly distributed in $[0,1]$ or exponentially distributed with parameter $1$; this holds because we will only be interested in the order statistics. We denote the score of edge $\{i,j\}$ by $V(i,j)=V(j,i)$. The set of all scores of the edges of vertex $i\in[n]$ is denoted by $\mathcal{V}_i=\{V(i,1),\cdots,V(i,i-1), V(i,i+1),\cdots, V(i,n)\}$, and all the associated edges by $\mathcal{E}_i=\{\{i,1\},\cdots, \{i,i-1\},\{i,i+1\},\cdots,\{i,n\}\}$. Without loss of generality, we can also assume that the scores are distinct, then $(R_i^j)_{1\leq j\leq n-1}$ represents an order on $[n]\setminus\{i\}=\{1,2,\cdots,i-1,i+1,\cdots,n\}$ based on $V(i,j)$ values. In other words, for each $1\leq i\leq n$, the random vector $\mathcal{R}_i=(R_i^1,R_i^2,\cdots,R_i^{n-1})$ is a permutation of $[n]\setminus\{i\}$ in which
\[V(i,R_i^1)> V(i,R_i^2)>\cdots> V(i,R_i^{n-1}).\]
As the scores are chosen \emph{i.i.d.}, the distribution of the random vector $\mathcal{R}_i$ is uniform among all permutations of $[n]\setminus\{i\}$ as it only depends on the order-statistics. In general, the scores also impose a permutation over all the edges. This plays an important role in defining the bilateral agreement random graphs.
Let $V(i,j)$ be realized for all edges $\{i,j\}$ and parameter $k$ be fixed. Then we can declare two different classes of random graphs on vertices $[n]$. We first define the notion of agreement. If $V(i,j)$ is among the $k$ largest scores in $\mathcal{V}_i$, i.e., $j\in \mathcal{R}_i^{\leq k}:=\{R_i^1,R_i^2,\cdots,R_i^k\}$ or $\{i,j\}\in\mathcal{E}_i^{\leq k} := \{\{i,R_i^1\},\cdots,\{i,R_i^k\}\}$, we say that vertex $i$ agrees on edge $\{i,j\}$. The first model introduced by Cooper and Frieze \cite{cooper1995connectivity} let the edge $\{i,j\}$ be present if at least one of them agrees on $\{i,j\}$. We denote this class of random graphs by $\mathbb{F}(n,k)$. The second model described by La and Kabkab \cite{la2015new} requires the agreement of both vertices for an edge to appear. We denote this class by $\mathbb{G}(n,k)$ and we call it the class of bilateral agreement random graphs.
Both $\mathbb{F}(n,k),\mathbb{G}(n,k)$ only depend on the order of $V(i,j)$ not the values. This is the consequence of the scores being \emph{i.i.d.}, which results in a uniformly drawn permutation among all permutations of edges of $K_n$. Assigning a random variable to each edge only helps us to understand the extent of independence in this problem. However, in the analysis of any underlying probability, we will always refer to permutation viewpoint of those random graphs.
In the following sections, we will analyze the connectivity of $\mathbb{G}(n,k)$ when $k$ is around $\log n$ as $n\rightarrow\infty$. More precisely, we will prove that when $k=t\log n$ with high probability this graph will be connected for $t>1$ and disconnected for $t<1$.
\section{Asymptotically Equivalent Distribution}\label{sec:asympdist}
In our analysis we will use the abstraction of an infinite urn model. The next two lemmas allow us to link the probability of an event in finite permutations space to an event in the infinite urn model. In essence, we will show that the appropriate probabilities converge to a negative multinomial distribution. Before delving into details, we will consider the set of all permutations with order restrictions over some elements. The notation $x\succ y$ states that $x$ appears earlier than $y$ in the permutation. We need to work on the set of all permutations with some order restrictions. For instance, whenever we say the set of permutations of $\{1,2,3\}$ with order restrictions $\{1\succ 2, 1\succ 3\}$, this narrows down the the set of all $3!=6$ permutations to $\{(1,2,3),(1,3,2)\}$.
\begin{lem}
Let $n\in\mathbb{Z}$ grow to infinity. Assume $s=o(n^{1/4})$ is a positive integer depending on $n$. Moreover, $m,m_0,m_1,\cdots,m_s$ are positive integer variables depending on $n$ and $M=m_0+m_1+\cdots+m_s$. Suppose there are $m_i$ distinct objects of type $i$ for $0\leq i\leq s$. Furthermore, $m=o(n^{1/4})$ and for each $i$, $m_i=h_i n+o(n^{1/4})$ where $1\leq h_i$.
We consider a uniformly random permutation of all $M$ objects with some order restrictions which are only within type 0. Let denote the number of objects of type 0 that lie at the first $m$ places of this permutation by $X$. The law of $X$ for $j<m$ is given by the following formula:
\begin{equation}
\begin{split}
p_j := \mathbb{P}\{X=j\} = & \binom{m}{j} m_0(m_0-1)\cdots(m_0-j+1)\times \\
& \frac{(M-m_0)(M-m_0-1)\cdots(M-m_0-m+j+1)\times (M-m)!}{M!}
\end{split}
\end{equation}
which is asymptotically:
\[ \binom{m}{j}\frac{h_0^j(h_1+\cdots+h_s)^{m-j}}{(h_0+h_1+\cdots+h_s)^m}(1+o(n^{-1/2}))\]
as $n\rightarrow\infty$.
\label{common-lem1}
\end{lem}
\begin{proof}
To compute $p_j$ we will count the number of permutations with $j$ elements of type 0 at the first $m$ place then divide it by the number all permutations. Any given order restrictions on objects of type 0 make both numerator and denominator of this fraction divided by the number of symmetries. As a result, we can assume no restriction on objects of type 0.
First part is straightforward by choosing those $j$ places at the first $m$ observations for objects of type 0 and $m_0 - j$ places in the remaining part. Next, counting the number of desired arrangements for objects of type 0 and other types we arrive at the following formula for the probability
\[p_j=\frac{\binom{m}{j}\binom{M-m}{m_0 - j}m_0!(M-m_0)!}{M!}.\]
That obtains the first part by a few lines of algebra.
For the second part, we can write
\begin{equation*}
\begin{split}
p_j & = \frac{\prod_{l_1=0}^{j-1} (m_0-l_1) \prod_{l_2=0}^{m-j-1}(M-m_0-l_2)\times (M-m)!}{M!} \\
& = \left(\prod_{i=1}^j \frac{m_0-j+i}{M-m+i}\right)\left(\prod_{i=1}^{m-j} \frac{M-m_0-m+j+i}{M-m+j+i} \right) \\
& = \frac{\left(h_0 n + o(n^{1/4})\right) ^ j \left((h_1+\cdots +h_s)n+ o(n^{1/4})\right)^{m-j}}{\left((h_0+h_1+\cdots +h_s)n+ o(n^{1/4})\right)^{m}}\\
& = \left( \frac{h_0^j(h_1+\cdots+h_s)^{m-j}}{(h_0+h_1+\cdots+h_s)^m}\right) \\
& \qquad \times \frac{\left(1 + (1/h_0)o(n^{-3/4})\right) ^ j \left(1+ (1/(h_1+\cdots+h_s))o(n^{-3/4})\right)^{m-j}}{\left(1+ (1/(h_0+h_1+\cdots+h_s))o(n^{-3/4})\right)^{m}}
\end{split}
\end{equation*}
Using the facts that $h_0,h_0+\cdots+h_s\geq 1$ implies
\begin{equation*}
\begin{split}
p_j& = \left( \frac{h_0^j(h_1+\cdots+h_s)^{m-j}}{(h_0+h_1+\cdots+h_s)^m}\right) (1+o(n^{-3/4}))^{m}\\
& = \left( \frac{h_0^j(h_1+\cdots+h_s)^{m-j}}{(h_0+h_1+\cdots+h_s)^m}\right) (1+o(n^{-3/4}))^{o(n^{1/4})}\\
& = \left( \frac{h_0^j(h_1+\cdots+h_s)^{m-j}}{(h_0+h_1+\cdots+h_s)^m}\right) (1+o(n^{-1/2})).
\end{split}
\end{equation*}
This completes the proof.
\end{proof}
\begin{lem}
Under the same assumption of Lemma \ref{common-lem1} let $i_j=o(n^{1/4})<m_j$ be positive integers for each $j$. Suppose vector $X$ represents the number of appearance of each type before the first observation of type 0 in a uniformly chosen permutation. Then
\begin{align*}
& \mathbb{P}\{X=(i_1,\cdots,i_s)\} \\
& = \frac{m_0\binom{m_1}{i_1}\cdots\binom{m_s}{i_s}(i_1+\cdots+i_s)!(M-i_1-\cdots-i_s-1)!}{M!}
\end{align*}
which asymptotically is
\[\frac{h_0 h_1 ^{i_1}\cdots h_s^{i_s} }{( h_0+h_1+\cdots+h_s)^{i_1+\cdots+i_s+1}}\frac{(i_1+i_2+\cdots+i_s)!}{i_1!i_2!\cdots i_s!} (1+o(n^{-1/4})).\]
\label{common-lem2}
\end{lem}
\begin{proof}
It is easy to check that the number of permutations of $M$ objects where at the first $i_1+i_2+\cdots+i_s$ places there are exactly $i_j$ elements of type $j$ for each $1\leq j\leq k$, and the first observation of type 0 happens at the $(i_1+\cdots+i_s+1)$-th position is as follows
\[m_0\binom{m_1}{i_1}\cdots\binom{m_s}{i_s}(i_1+\cdots+i_s)!\binom{M-i_1-\cdots-i_s-1}{m_0-1}(M-m_0-i_1-\cdots-i_k)!\]
Therefore, using $h_i\geq 1, h_0+h_1+\cdots+h_s\geq s$ the probability of this event is as follows:
\begin{equation*}
\begin{split}
P_{i_1,\cdots,i_s} & = \frac{m_0\binom{m_1}{i_1}\cdots\binom{m_s}{i_s}(i_1+\cdots+i_s)!\binom{M-i_1-\cdots-i_s-1}{m_0-1}(M-m_0-i_1-\cdots-i_s)!}{M!}\\
& = \frac{m_0\binom{m_1}{i_1}\cdots\binom{m_s}{i_s}(i_1+\cdots+i_s)!(M-i_1-\cdots-i_s-1)!}{M!}\\
& = \frac{(h_0 n + o(n^{1/4}))(h_1 n + o(n^{1/4}))^{i_1}\cdots(h_s n + o(n^{1/4}))^{i_s}(i_1+\cdots+i_s)!}{ (( h_0+h_1+\cdots+h_s)n+so(n^{1/4}))^{i_1+\cdots+i_s+1}i_1!\cdots i_s!}\\
& = \left(\frac{h_0 h_1^{i_1}\cdots h_s^{i_s}(i_1+\cdots+i_s)!}{( h_0+h_1+\cdots+h_s)^{i_1+\cdots+i_s+1}i_1!\cdots i_s!}\right)\left(\frac{1+o(n^{-3/4})}{1+(\frac{s}{h_1+\cdots +h_s})o(n^{-3/4})}\right)^{i_1+\cdots+i_s+1}\\
& = \left(\frac{h_0 h_1^{i_1}\cdots h_s^{i_s}(i_1+\cdots+i_s)!}{( h_0+h_1+\cdots+h_s)^{i_1+\cdots+i_s+1}i_1!\cdots i_s!}\right)\left(1 + o(n^{-3/4})\right)^{i_1+\cdots+i_s+1}\\
& = \left(\frac{h_0 h_1^{i_1}\cdots h_s^{i_s}(i_1+\cdots+i_s)!}{( h_0+h_1+\cdots+h_s)^{i_1+\cdots+i_s+1}i_1!\cdots i_s!}\right)\left(1 + o(n^{-3/4})\right)^{o(n^{1/2})}\\
& = \left(\frac{h_0 h_1^{i_1}\cdots h_s^{i_s}(i_1+\cdots+i_s)!}{( h_0+h_1+\cdots+h_s)^{i_1+\cdots+i_s+1}i_1!\cdots i_s!}\right)\left(1 + o(n^{-1/4})\right),\\
\end{split}
\end{equation*}
This finishes the proof.
\end{proof}
Lemmas \ref{common-lem1} and \ref{common-lem2} demonstrate that a negative multinomial distribution is the limit for the distribution of number of other types before the first appearance of a given type. As mentioned earlier, this distributional convergence will help us greatly in our analysis.
\section{Disconnectivity of graph for $t<1$}\label{sec:disconnectivity}
This section begins by recalling the negative multinomial distribution. In the special case that we need here, we present an urn model to interpret the distribution. Suppose we have a sequence of \emph{i.i.d.} random variables with $k+1$ possible outcomes. We describe these outcomes as the type of objects. Therefore, there are $k+1$ types. This sequence stops as soon as the first object of type $k+1$ occur in the sequence. The probability that there are $i_1$ objects of type 1, $i_2$ objects of type 2, ... , $i_k$ objects of type $k$, and finally the $n-$th observation which is of type $k+1$, follows the negative multinomial distribution below
\begin{equation}
f(i_1,i_2,\cdots, i_k)=\frac{1}{(k+1)^{i_1+i_2+\cdots+i_k+1}}\frac{(i_1+i_2+\cdots+i_k)!}{i_1!i_2!\cdots i_k!},
\label{negative-multi}
\end{equation}
in which $i_t\geq 0$ for $1\leq t\leq k$ and $i_1+i_2+\cdots+i_k=n-1$. We now introduce some lemmas that help with any required computation.
\begin{lem}
With the definition (\ref{negative-multi})
\[\sum_{i_1+i_2+\cdots+i_k=n-1} f(i_1,i_2,\cdots, i_k)=\frac{1}{k+1}(\frac{k}{k+1})^{n-1}\]
\label{binom}
\end{lem}
\begin{proof}
There is an algebraic proof to this lemma using the multinomial expansion formula. However, we give a simpler probabilistic way. This sum is over the probability of all possible arrangements of $n$ items in which all the first $n-1$ items are of type $1,2,\cdots,k$ and the last one is of type $k+1$. Hence, the probability of this event is $(\frac{k}{k+1})^{n-1}(\frac{1}{k+1})$ due to the independent choice made in each position.
\end{proof}
\begin{lem}
Let $0<\delta<1$. Then:
\[\sum_{k^2(1+\delta)<i_1+i_2+\cdots+i_k<4(k+1)^2} f(i_1,i_2,\cdots, i_k)\geq \sum_{k^2(1+\delta)<i_1+i_2+\cdots+i_k} f(i_1,i_2,\cdots, i_k) - e^{-4(k+1)}\]
\label{remind}
\end{lem}
\begin{proof}
\begin{equation*}
\begin{split}
r & = \sum_{k^2(1+\delta)<i_1+i_2+\cdots+i_k} f(i_1,i_2,\cdots, i_k) - \sum_{k^2(1+\delta)<i_1+i_2+\cdots+i_k<4(k+1)^2} f(i_1,i_2,\cdots, i_k)\\
& = \sum_{4(k+1)^2\leq i_1+i_2+\cdots+i_k} f(i_1,i_2,\cdots, i_k)\\
& = \sum_{n = 4(k+1)^2}^{\infty} \frac{1}{k+1}(\frac{k}{k+1})^n\\
& = (\frac{k}{k+1})^{4(k+1)^2}\\
& = \left((1 - \frac{1}{k+1})^{(k+1)}\right)^{4(k+1)}\\
& \leq e^{-4(k+1)}
\end{split}
\end{equation*}
\end{proof}
\begin{lem}
\[\sum_{k^2(1+\delta)<i_1+i_2+\cdots+i_k<4(k+1)^2,\; \forall j,\, i_j\geq k} f(i_1,i_2,\cdots, i_k)\geq\]\[ (1 - k e^{\frac{-k\delta^2}{7}}) \sum_{k^2(1+\delta)<i_1+i_2+\cdots+i_k<4(k+1)^2} f(i_1,i_2,\cdots, i_k)\]
\label{bern}
\end{lem}
\begin{proof}
Assume $n=i_1+\cdots+i_k+1>k^2(1+\delta)$ is a fixed number. Then, at each step prior to the $n$-th observation, an outcome of type $m$ has a Bernoulli distribution with parameter $\frac{1}{k}$. Denote this random variable by $X_m^{(j)}$ where $j$ refers to the step number. For an arbitrary $1\leq m \leq k$, we have $\mathbb{E}(X_m^{(j)})=\frac{1}{k}$ and $\mathrm{Var}(X_m^{(j)})=\frac{k-1}{k^2}$. If $i_m<k$, then
\[\frac{X_m^{(1)}+\cdots+X_m^{(n-1)}}{n-1}<\frac{k}{k^2(1+\delta)}=\frac{1}{k(1+\delta)}.\]
However, the expected value of the average above is $\frac{1}{k}$, and $\frac{1}{k(1+\delta)}-\frac{1}{k}=-\frac{\delta}{k(1+\delta)}$, so this is a (lower) tail event. Now, we apply the Bernstein inequality:
\begin{equation*}
\begin{split}
\mathbb{P}\{i_m<k|i_1+\cdots+i_k=n-1\} & \leq \mathbb{P}\left\{\frac{X_m^{(1)}+\cdots+X_m^{(n-1)}}{n-1} < \frac{1}{k(1+\delta)}\right\}\\
& \leq\exp\left(\frac{-(n-1)\times \frac{\delta^2}{k^2(1+\delta)^2}}{\frac{2(k-1)}{k^2}+\frac{2\delta}{3k(\delta+1)}}\right) \\
& \leq\exp\left(\frac{-k^2(1+\delta)\times \frac{\delta^2}{k^2(1+\delta)^2}}{\frac{2(k-1)}{k^2}+\frac{2\delta}{3k(\delta+1)}}\right) \\
& = \exp\left(\frac{-3k^2\delta^2}{6(k-1)(\delta+1) + 2\delta k}\right)\\
& \leq \exp\left(\frac{-k\delta^2}{7}\right)
\end{split}
\end{equation*}
Using the union bound yields:
\[\mathbb{P}\left\{\forall j:i_j\geq k|\sum_{m=1}^k i_m=n-1\right\} \geq 1 - k\exp\left(\frac{-k\delta^2}{7}\right).\]
Therefore,
\begin{align*}
& \sum_{k^2(1+\delta)<\sum i_m<4(k+1)^2,\; \forall j,\, i_j\geq k} f(i_1,i_2,\cdots, i_k) \\ & =
\sum_{ k^2(1+\delta)< n < 4(k+1)^2}\sum_{\sum i_m=n-1,\; \forall j,\, i_j\geq k}f(i_1,i_2,\cdots, i_k) \\
& =
\sum_{ k^2(1+\delta)< n < 4(k+1)^2}\mathbb{P}\Big\{\forall j:i_j\geq k,\sum i_m=n-1\Big\} \\
& =
\sum_{ k^2(1+\delta)< n < 4(k+1)^2}\mathbb{P}\Big\{\sum i_m=n-1\Big\}\mathbb{P}\Big\{\forall j:i_j\geq k\big|\sum i_m=n-1\Big\} \\
& \geq
\sum_{ k^2(1+\delta)< n < 4(k+1)^2}\mathbb{P}\Big\{\sum i_m=n-1\Big\}\left(1-k\exp\left(\frac{-k\delta^2}{7}\right)\right) \\
& =
\left(1-k\exp\left(\frac{-k\delta^2}{7}\right)\right) \sum_{k^2(1+\delta)<\sum i_m<4(k+1)^2} f(i_1,i_2,\cdots, i_k).
\end{align*}
This completes the proof.
\end{proof}
\begin{lem}
Let $n$ be a positive integer and $k=\lfloor t\log n\rfloor $. Then for any $0<\delta<1$ we have
\[\left(\sum_{k^2(1+\delta)<i_1+i_2+\cdots+i_k<4(k+1)^2,\; \forall j,\, i_j\geq k} f(i_1,i_2,\cdots, i_k)\right) \geq O(n^{-t(\delta + 1)})\]
as $n\rightarrow\infty$.
\label{infinite-exp}
\end{lem}
\begin{proof}
Using Lemmas \ref{bern}, \ref{remind}, \ref{binom}, and the fact that $n\geq e^{k/t}$ :
\begin{equation*}
\begin{split}
& \sum_{k^2(1+\delta)<i_1+i_2+\cdots+i_k<4(k+1)^2,\; \forall j,\, i_j\geq k} f(i_1,i_2,\cdots, i_k) \\
& \geq
\Big(1 - k e^{\frac{-k\delta^2}{7}}\Big) \sum_{k^2(1+\delta)<i_1+i_2+\cdots+i_k<4(k+1)^2} f(i_1,i_2,\cdots, i_k)\\
& \geq
\Big(1 - ke^{\frac{-k\delta^2}{7}}\Big)\left(\sum_{k^2(1+\delta)<i_1+i_2+\cdots+i_k} f(i_1,i_2,\cdots, i_k) - e^{-4(k+1)}\right) \\
& =
\Big(1 - ke^{\frac{-k\delta^2}{7}}\Big)\left(\sum_{N=k^2(1+\delta)+1}^{\infty}\frac{1}{k+1}(\frac{k}{k+1})^{N-1}-e^{-4(k+1)}\right)\\
& =
\Big(1 - ke^{\frac{-k\delta^2}{7}}\Big)\left((1-\frac{1}{k+1})^{k^2(\delta + 1)} - e^{-4(k+1)}\right).
\end{split}
\end{equation*}
We know that $(1-\frac{1}{k+1})^k\approx e^{-1}$ (more precisely $(1-\frac{1}{k+1})^k> e^{-1}$) and $1 - ke^{\frac{-k\delta^2}{7}}\rightarrow 1$ as $k\rightarrow\infty$, which holds when $n\rightarrow\infty$. Hence, the mentioned sum is asymptotically greater than:
\[e^{-k(\delta+1)} - e^{-4(k+1)}\]
Furthermore, $0<\delta<1$ causes $e^{-4(k+1)}$ to vanish faster than $e^{-k(\delta+1)}$. Assumption $k=[t\log n]$ then completes the proof.
\end{proof}
We are now in a position to relate the lemmas above to the disconnectivity result. Assume $I_j$ represents the event of vertex $j$ being isolated. We want to show that with high probability there exists a $j$ such that $I_j=1$.
\begin{lem}
For the random graph model $\mathbb{G}(n,k)$ where $k=\lfloor t\log n\rfloor$ we have
\[\mathbb{P}\{I_1 = 1\} \geq O(n^{-t(1+\delta)})\]
for each $0<\delta<1$.
\label{prob-iso-lower-bound}
\end{lem}
\begin{proof}
As mentioned earlier, we can look at each configuration of independent random variables $\{V(i,j)\}_{i,j}$ as a permutation over all edges; $\mathcal{E}$. Without loss of generality, one can assume vertex labels are sorted according to the scores of edges connecting to vertex 1. In other words, $V(1,2)>V(1,3)>\cdots>V(1,n)$, or equivalently $R_1^j=j+1$ for $1\leq j\leq n-1$. Then, for vertex 1 to be isolated, it is sufficient that for each $j\in\{2,3,\cdots,k+1\}$ there exist at least $k$ elements of $\mathcal{E}_j$ (except $\{1,j\}$) appearing before all elements of $\mathcal{E}_1$ in the permutation. Strictly speaking, if $k$ elements of $\mathcal{E}_j$ appear before $\{1,j+1\}$, then that would be necessary and sufficient, but we consider a stricter condition which insists that $k$ elements of $\mathcal{E}_j$ appear before all elements of $\mathcal{E}_1$. This way, vertex $j$ does not agree on edges $\{1,j\}$ for $2\leq j \leq k+1$ and vertex $1$ also does not agree upon edges $\{1,j\}$ for $k+2 \leq j \leq n$.
To make this condition stronger, we exclude the intersections. Define for each $2\leq i\leq k+1$: $\mathcal{E'}_i=\mathcal{E}_i \setminus \{ \{i,1\} , \cdots, \{i,k+1\}\}$. As a result, $\{\mathcal{E}_1,\mathcal{E}'_2,\mathcal{E}'_3,\cdots,\mathcal{E}'_{k+1}\}$ is a pairwise disjoint collection with $|\mathcal{E}_1|=n-1, |\mathcal{E}'_i|=n-k-1$. Note that we only determined the order over $\mathcal{E}_1$. Thus, any configuration of edge scores induces a uniformly random permutation on $\mathcal{E}_1\cup \mathcal{E'}_2\cup\cdots\cup\mathcal{E'}_{k+2}$ with the order restriction $\{1,2\} \succ \{1,3\}\succ\cdots\succ \{1,n\}$ on $\mathcal{E}_1$.
Assume $\mathcal{E}_1$ are objects of type 0 and $\mathcal{E'}_i$ are objects of type $i-1$ for $2\leq i\leq k+2$. This satisfies all the conditions of Lemma \ref{common-lem2} since $k=[t\log n]$. Therefore, the probability of having $i_j$ elements of type $j$ before the first observation of type 0 when $k^2(\delta + 1) < i_1+\cdots+i_k < 4(k+1)^2=O(\log n ^2)$ can be approximated by the distribution $f(\cdot)$ from \eqref{negative-multi}. Hence, for large enough $n$ we have:
\[\mathbb{P}\{X=(i_1,\cdots,i_k)\}\geq \frac{1}{2} f(i_1,\cdots,i_k)\]
From what we explained earlier, the extra condition $i_j\geq k$ guarantees that vertex 1 is isolated. Applying Lemma \ref{infinite-exp} then allows us to conclude that
\[\mathbb{P}\{I_1 = 1\}\geq O(n^{-t(1+\delta)}).\]
\end{proof}
\begin{lem}
For the random graph model $\mathbb{G}(n,k)$ where $k=\lfloor t\log n \rfloor $ and $t<1$, we have
\[\mathbb{E}\{\text{number of isolated vertices}\}\rightarrow \infty\]
\label{lem-second-method-1}
\end{lem}
\begin{proof}
Note that
\[\mathbb{E}\{\textrm{number of isolated vertices}\} = \mathbb{E}\{\sum_{i=1}^n I_i\}= n\mathbb{E}\{I_1\}=n\mathbb{P}\{I_1=1\}\]
Choosing $\delta$ so that $0<\delta<\min(\frac{1}{t}-1,1)$ guarantees $n^{-t (1+\delta) + 1}\rightarrow\infty$. As a result, Lemma \ref{prob-iso-lower-bound} completes the proof.
\end{proof}
There is one more step needed to prove the desired result of there being at least one isolated vertex (with high probability) as the mean of a non-negative random variable being unbounded doesn't necessarily imply that the probability of it being $0$ is $0$. For this we will employ the second moment method \cite{alon2016probabilistic}. To apply the second moment method we need to study the correlation between pairs of indicator random variables corresponding to the isolation of a vertex. Consider two arbitrary vertices, which could be without loss of generality chosen to be vertices 1 and 2. We want to study $\mathbb{P}\{I_1=1, I_2=1\}=\mathbb{P}\{I_1 I_2=1\}=\mathbb{E}\{I_1 I_2\}$. From now on, we consider $k=[t \log n]$ for a given $0<t<1$.
Define the following events:
\begin{itemize}
\item $B_1:=\{2\in\mathcal{R}^{\leq k}_1\}$.
\item $\hat{B}_1:=\{1\in\mathcal{R}^{\leq k}_2\}$.
\item $B_2:=\{\mathcal{R}^{\leq k}_1\cap\mathcal{R}^{\leq k}_2\neq \phi\}$.
\item $B_3:=\{\exists i\in\mathcal{R}^{\leq k}_1 \textrm{ such that } (\mathcal{R}^{\leq k}_1 \cup \mathcal{R}^{\leq k}_2) \cap \mathcal{R}^{\leq k}_i \neq \phi\}$.
\item $\hat{B}_3:=\{\exists i\in\mathcal{R}^{\leq k}_2 \textrm{ such that } (\mathcal{R}^{\leq k}_1 \cup \mathcal{R}^{\leq k}_2) \cap \mathcal{R}^{\leq k}_i \neq \phi\}$.
\end{itemize}
For ease of presentation, we omit the value 1 from $I_1=1, I_2=1$ or $I_1I_2=1$. For example $\mathbb{P}\{I_1I_2\}=\mathbb{P}\{I_1I_2=1\}$.
\begin{lem}
For each $0<\alpha<1$ there is an integer $s$ such that $\mathbb{P}\{I_1I_2\cap B_1\} \leq O(k^{s} / n^{1+t\alpha})$. Using symmetry, this bound also holds for $\hat{B_1}$.
\label{boundB1}
\end{lem}
\begin{proof}
Suppose $B_1$ holds. Let $s$ be a positive integer. Assume $n$ is sufficiently large so that $k\geq s+2$. Define $\mathcal{E'}_1 = \mathcal{E}_1 \setminus \{\{1,2\}\}$ and for $i\in \mathcal{R}^{\leq s}_1 \setminus \{2\}$, $\mathcal{E'}_{i} = \mathcal{E}_{i} \setminus \{\{i,j\}:j\in\mathcal{R}^{\leq s}_1\}\setminus\{\{i,2\}\}$. Therefore, if we consider the following union
\[\mathcal{E}' = \mathcal{E'}_1 \bigcup_{i\in \mathcal{R}^{\leq s}_1 \setminus \{2\}} \mathcal{E'}_i \]
It will be a partition in which $|\mathcal{E'}_1|=n-2$, $|\mathcal{E'}_i|=n-s-1$ or $n-s-2$ (depending on whether vertex $2$ belongs to $\mathcal{R}^{\leq s}_1$ or not). Consider $\mathcal{E'}_1$ the objects of type 0 and $\mathcal{E'}_i$ the objects of type $i$. On can easily see that each realization of $\mathbb{G}(n,k)$ induces a uniformly random permutation over $\mathcal{E}'$ which has only a fixed order on the first $s$ or $s+1$ elements of $\mathcal{E}'_1$.
A necessary condition for vertex 1 to be isolated is that for each $i\in \mathcal{R}^{\leq s}_1 \setminus \{2\}$ there should be at least $k$ edges of $\mathcal{E}_{i}$ appearing before the $s^{th}$ element of $\mathcal{E}_{1}$. This implies there must be at least $k-s-1$ edges of $\mathcal{E'}_{i}$ before the $s^{th}$ element of $\mathcal{E'}_{1}$. As a result, in the first $(s-1)(k-s-1)$ elements of this permutation there are at most $s-1$ elements of type 0.
It follows from Lemma \ref{common-lem1} with $h_i=1$ for each $1\leq i\leq s$ that the probability of this event is eventually bounded above by
\begin{equation*}
\begin{split}
& \sum_{j=0}^{s-1} \binom{(s-1)(k-s-1)}{j} \frac{s^{(s-1)(k-s-1)-j}}{(s+1)^{(s-1)(k-s-1)}} \\
& \leq s \binom{(s-1)(k-s-1)}{s-1} \frac{s^{(s-1)(k-s-1)-s+1}}{(s+1)^{(s-1)(k-s-1)}} \\
& = \binom{(s-1)(k-s-1)}{s-1} \frac{1}{s^{s-2}}\left(1-\frac{1}{s+1}\right)^{(s-1)(k-s-1)}.
\end{split}
\end{equation*}
One can see $\left(1-\frac{1}{s+1}\right)^{(s-1)}$ is a decreasing function converging to $e^{-1}$. Hence, for sufficiently large $s$, one has $\left(1-\frac{1}{s+1}\right)^{(s-1)}<e^{-\alpha}$ and the bound above becomes
\begin{equation*}
\begin{split}
& \binom{(s-1)(k-s-1)}{s-1} \frac{1}{s^{s-2}}\left(1-\frac{1}{s+1}\right)^{(s-1)(k-s-1)} \\
& \quad \leq \binom{(s-1)(k-s-1)}{s-1} \frac{1}{s^{s-2}}e^{-\alpha (k-s-1)} \\
& \quad = \binom{(s-1)(k-s-1)}{s-1} \frac{e^{\alpha(s+1)}}{s^{s-2}}e^{-k\alpha}.
\end{split}
\end{equation*}
Fixing $s$ and increasing $n$ (which increases $k$ as well), the term $\binom{(s-1)(k-s-1)}{s-1} \frac{e^{\alpha(s+1)}}{s^{s-2}}$ is a polynomial of degree $s-1$ of $k$. Therefore, $e^{-k\alpha}$ is dominant, and more precisely, the bound becomes
\[O\left(\frac{k^{s-1}}{n^{\alpha t}}\right).\]
Finally, we have
\begin{equation*}
\begin{split}
\mathbb{P}\{I_1 I_2 \cap B_1\} & = \mathbb{P}\{B_1\} \mathbb{P}\{I_1 I_2 | B_1\} \\
& \leq \mathbb{P}\{B_1\} \mathbb{P}\{I_1 | B_1\}\\
& = \left(\frac{k}{n-1}\right)O\left(\frac{k^{s-1}}{n^{\alpha t}}\right) \\
& = O(k^s / n^{1+\alpha t})
\end{split}
\end{equation*}
\end{proof}
\begin{lem}
We have $\mathbb{P}\{ B_2 \} \leq O(k^2 / n)$.
\label{boundB2-0}
\end{lem}
\begin{proof}
First assume both $B^c_1, \hat{B}_1^c$ occur which holds with probability $1-O(k/n)$. We estimate the probability of $B_2^c$. Indeed, for each $i$, $\mathcal{R}_i^{\leq k}$ is uniform among all subsets of size $k$. It alters the problem to finding the probability of having an intersection for two completely random subsets of $\{3,\cdots,n\}$ of size $k$.
Counting the number of ways to choose a pair of two disjoint subsets of size $k$ yields:
\begin{equation}
\frac{1}{2}\binom{n-2}{k}\binom{n-k-2}{k},
\end{equation}
and for choosing two arbitrary subsets the number is:
\begin{equation}
\frac{1}{2}\binom{n-2}{k}^2.
\end{equation}
Therefore, the probability of not having intersection is
\begin{equation}
\begin{split}
\mathbb{P}\{B_2^c\} = &\frac{\binom{n-k-2}{k}}{\binom{n-2}{k}}\\
= & \frac{(n-k-2)!^2}{(n-2)!(n-2k-2)!}\\
= & \frac{(n-k-2)\cdots(n-2k-1)}{(n-2)\cdots(n-k-1)}\\
\geq & \left(1-\frac{2k-1}{n-2}\right)^k\\
= & 1 - O(k^2/n)
\end{split}
\end{equation}
which completes the proof.
\end{proof}
\begin{lem}
For each $0<\alpha<1$ there is an integer $s$ such that $\mathbb{P}\{I_1I_2\cap B_2\} \leq O(k^{s+1} / n^{1+t\alpha})$.
\label{boundB2}
\end{lem}
\begin{proof}
We can assume both $B^c_1, \hat{B}_1^c$ hold due to Lemma \ref{boundB1}. In order to bound the value of $\mathbb{P}\{I_1 I_2 | B_2\}$, we bound $\mathbb{P}\{I_1 | B_2\}\geq \mathbb{P}\{I_1 I_2 | B_2\}$. We apply the same technique as Lemma \ref{boundB1}. Consider a constant $s$. Assume $n$ is sufficiently large so that $k\geq s+2$. Define the types 0 as $\mathcal{E'}_1 = \mathcal{E}_1 - \{\{1,2\}\}$ and for $i\in \mathcal{R}^{\leq s}_1$, types $i$ as $\mathcal{E'}_{i} = \mathcal{E}_{i} - \{\{i,j\}:j\in\mathcal{R}^{\leq s}_1\}-\{\{i,2\}\}$.
A necessary condition for vertex $1$ to be isolated is that for each $i\in \mathcal{R}^{\leq s}_1$ there should be at least $k$ edges of $\mathcal{E}_{i}$ appearing before the $s^{th}$ edge of $\mathcal{E}_{1}$. This implies there must be at least $k-s-1$ elements of $\mathcal{E'}_{i}$ before the $s^{th}$ element of $\mathcal{E'}_{1}$. As a result, in the first $(s-1)(k-s-1)$ elements of the induced permutation on $\mathcal{E}' = \mathcal{E'}_1 \bigcup_{i\in \mathcal{R}^{\leq s}_1} \mathcal{E'}_i $ there are at most $s-1$ elements of type 0. One can use Lemma \ref{common-lem1} for the types above with $h_i=1$. Repeating the same arguments as in the proof of Lemma \ref{boundB1} results in a $O(k^{s-1}/n^{\alpha t})$ bound for $\mathbb{P}\{I_1 | B_2\}$.
Therefore, Lemma \ref{boundB2-0} implies
\begin{equation}
\begin{split}
\mathbb{P}\{ I_1 I_2 \cap B_2\} & = \mathbb{P}\{B_2\} \mathbb{P}\{I_1 I_2|B_2\} \\
& \leq O(k^2/ n)\mathbb{P}\{I_1 |B_2\}\\
& \leq O(k^2/ n)O(k^{s-1} / n^{\alpha t}) \\
& = O(k^{s+1}/n^{\alpha t+1}),
\end{split}
\end{equation}
and the proof holds.
\end{proof}
\begin{lem}
We have $\mathbb{P}\{B_3\}= \mathbb{P}\{\hat{B}_3\}\leq O(k^3/n)$
\label{boundB3-0}
\end{lem}
\begin{proof}
With probability at least $1-O(k^2/n)$, $B_1^c, \hat{B}_1^c, B_2^c$ holds. Let $i\in\mathcal{R}_1^{\leq k}$. The probability that for all $z\in\mathcal{R}_i^{\leq k} $ we have $z\notin\mathcal{R}_1^{\leq k} \cup \mathcal{R}_2^{\leq k} $ is bounded below by $(1-\tfrac{2k}{n-k-1})^k$ (look at the proof of Lemma \ref{boundB2-0}). As a result, with probability at most $k(1-(1-\tfrac{2k}{n-k-1})^k)$ there exists $i\in\mathcal{R}_1^{\leq k}$ without this property. Therefore
\begin{equation}
\mathbb{P}\{B_3\} \leq k\left(1-\left(1-\frac{2k}{n-k-1}\right)^k\right) = O(k^3/n)
\end{equation}
\end{proof}
\begin{lem}
For each $0<\alpha<1$ there is an integer $s$ such that $\mathbb{P}\{I_1I_2\cap B_3\} \leq O(k^{s+2} / n^{1+t\alpha})$. The same result also holds if $B_3$ is replaced by $\hat{B_3}$.
\label{boundB3}
\end{lem}
\begin{proof}
We can assume $B_2^c$, $B_1^c$, and $\hat{B}_1^c$ hold due to Lemmas \ref{boundB1} and \ref{boundB2}. Then then proof is quite similar to that of Lemma \ref{boundB2}. Note that $\mathbb{P}\{I_1|B_3\}\geq \mathbb{P}\{I_1 I_2|B_3\}$. The next step is fixing $s$ and repeating the same idea used in Lemmas \ref{boundB1} and \ref{boundB2}. This will provide $O(k^{s-1}/n^{\alpha t})$ bound for $\mathbb{P}\{I_1|B_3\}$. Thereafter we have
\begin{equation}
\begin{split}
\mathbb{P}\{ I_1 I_2 \cap B_3\} & = \mathbb{P}\{B_3\} \mathbb{P}\{I_1 I_2|B_3\} \\
& \leq O(k^3/ n)\mathbb{P}\{I_1 |B_3\}\\
& \leq O(k^3/ n)O(k^{s-1} / n^{\alpha t}) \\
& = O(k^{s+2}/n^{\alpha t+1}).
\end{split}
\end{equation}
\end{proof}
\begin{lem}
We have
\[ \limsup_{n\rightarrow\infty} \frac{\mathbb{E}\{I_1 I_2\}}{\mathbb{E}\{I_1\}^2} \leq 1\]
\label{lem-second-method-2}
\end{lem}
\begin{proof}
Define $B=B_1\cup \hat{B}_1 \cup B_2 \cup B_3 \cup \hat{B}_3$. Lemma \ref{boundB2-0} and \ref{boundB3-0} imply
\begin{equation}
\begin{split}
\mathbb{P}\{I_1 \cap B \} & \leq \mathbb{P}\{B_1\cup \hat{B}_1 \cup B_2 \cup B_3 \cup \hat{B}_3\} \\
& \leq \mathbb{P}\{ B_1\} + \mathbb{P}\{ \hat{B}_1\} + \mathbb{P}\{ B_2\} + \mathbb{P}\{ B_3 \}+ \mathbb{P}\{ \hat{B}_3 \}\\
& \leq O(k^3 / n). \\
\end{split}
\label{bbb}
\end{equation}
We now have
\begin{equation}
\begin{split}
\frac{\mathbb{P}\{I_1\}}{\mathbb{P}\{B^c\}} & = \frac{\mathbb{P}\{I_1\cap B^c\}}{\mathbb{P}\{B^c\}} + \frac{\mathbb{P}\{I_1 \cap B\}}{\mathbb{P}\{B^c\}}
\end{split}
\end{equation}
Hence,
\begin{equation}
\begin{split}
\frac{1}{\mathbb{P}\{B^c\}} & = \frac{\mathbb{P}\{I_1\cap B^c\}}{\mathbb{P}\{I_1\}\mathbb{P}\{B^c\}} + \frac{\mathbb{P}\{I_1 \cap B\}}{\mathbb{P}\{I_1\}\mathbb{P}\{B^c\}}\\
& = \frac{\mathbb{P}\{I_1|B^c\}}{\mathbb{P}\{I_1\}} + \frac{\mathbb{P}\{I_1 \cap B\}}{\mathbb{P}\{I_1\}\mathbb{P}\{B^c\}}.
\label{bbb2}
\end{split}
\end{equation}
We choose $0<\delta<\min((\frac{1}{t}-1)/2,1)$ in Lemma \ref{prob-iso-lower-bound} so that $-t (1+\delta) > -1$. Therefore, $\mathbb{P}\{I_1\}\geq O(n^{-t(1+\delta)})$ and with \eqref{bbb}, one has
\begin{equation}
\begin{split}
\frac{\mathbb{P}\{I_1 \cap B\}}{\mathbb{P}\{I_1\}\mathbb{P}\{B^c\}} \rightarrow 0.
\end{split}
\end{equation}
Considering the fact $\frac{1}{\mathbb{P}\{B^c\}}\rightarrow 1$, equation \eqref{bbb2} implies that
\begin{equation}
\frac{\mathbb{P}\{I_1|B^c\}}{\mathbb{P}\{I_1\}}\rightarrow 1.
\label{single-asym}
\end{equation}
On the other hand, if $B_1\cup\hat{B}_1\cup B_2\cup B_3 \cup \hat{B}_3$ does not happen, then $I_1$ and $I_2$ become independent. Indeed, $I_1$ holds whenever for each $1\leq i\leq k$ there are at least $k$ elements $j\in[n]\setminus\{i, R_1^i\}$ such that the score of $\{R_1^i,j\}$ is higher than $V(1,R_1^i)$. According to our definition of $R_i^j$ this implies $V(R_1^i, R_{R_1^i}^j)>V(1,R_1^i)$ for $1\leq j\leq k$. Additionally, $B_3$ implies that these $k$ elements are from $[n]\setminus(\mathcal{R}_1^k\cup \mathcal{R}_2^k)$. As a result, we can ignore common edges and there should only exist $k$ elements $j\in[n]\setminus(\mathcal{R}_1^k\cup \mathcal{R}_2^k\cup\{i,R_1^i\})$ where $V(R_1^i,j)$ is higher than $V(1,R_1^i)$. The scores of edges are chosen independently and there is no extra information on $V(R_1^i,j)$ for $j\in[n]\setminus(\mathcal{R}_1^k\cup \mathcal{R}_2^k\cup\{i,R_1^i\})$. Therefore,
\begin{equation}
\begin{split}
\mathbb{P}\{I_1 I_2 | B^c \} & = \mathbb{P}\{I_1| B^c \} \mathbb{P}\{ I_2 | B^c \}.
\end{split}
\label{independ}
\end{equation}
It follows from Lemma \ref{boundB1}, \ref{boundB2}, \ref{boundB3}, and (\ref{independ}) that for any $0<\alpha<1$ there is a constant $s$ so that:
\begin{equation}
\begin{split}
\mathbb{P}\{I_1 I_2\} = & \mathbb{P}\{I_1 I_2 \cap B^c \} + \mathbb{P}\{I_1 I_2 \cap B \}\\
\leq & \mathbb{P}\{I_1 I_2 \cap B^c \} + \mathbb{P}\{I_1 I_2 \cap B_1\} + \mathbb{P}\{I_1 I_2 \cap \hat{B}_1\} + \mathbb{P}\{ I_1 I_2 \cap B_2\} + \\
& \mathbb{P}\{ I_1 I_2 \cap B_3 \} + \mathbb{P}\{ I_1 I_2 \cap \hat{B}_3 \}\\
\leq & \mathbb{P}\{I_1 I_2 \cap B^c \} + O(k^{s+2}/n^{1+t\alpha})\\
\leq & \mathbb{P}\{I_1 I_2 | B^c \} + O(k^{s+2}/n^{1+t\alpha})\\
= & \mathbb{P}\{I_1| B^c \} \mathbb{P}\{ I_2 | B^c \} + O(k^{s+2}/n^{1+t\alpha})
\end{split}
\label{bbb3}
\end{equation}
Choosing $\delta<(\frac{1}{t}-1)/2$, enables us to find an $0<\alpha<1$ so that $1+t\alpha > 2t(1+\delta)$. Dividing both sides of (\ref{bbb3}) by $\mathbb{P}\{I_1\}\mathbb{P}\{I_2\}=\mathbb{P}\{I_1\}^2$ we have
\begin{equation}
\frac{\mathbb{P}\{I_1 I_2\}}{\mathbb{P}\{I_1\}\mathbb{P}\{I_2\}}\leq \frac{\mathbb{P}\{I_1| B^c \} \mathbb{P}\{ I_2 | B^c \}}{\mathbb{P}\{I_1\}\mathbb{P}\{I_2\}} + \frac{O(k^{s+2}/n^{1+t\alpha})}{{\mathbb{P}\{I_1\}\mathbb{P}\{I_2\}}}
\label{final-step}
\end{equation}
Equation \eqref{single-asym} guarantees the following for the right hand side: the first term converges to 1, and the second term is less than $O(k^{s+2}/n^{1+t\alpha-2t(\delta + 1)})$ due to Lemma \ref{prob-iso-lower-bound} which vanishes based on the choice of $\alpha$ and $\delta$, and the fact that $k\sim t\log n$. This completes the proof.
\end{proof}
It is worth mentioning that in the proof above, we first find $\delta$ according to $t$ which is a fixed number. Next, we determine $\alpha$ based on $\delta$, and finally $\alpha$ provides a constant $s$ due to Lemma \ref{boundB1}, \ref{boundB2}, and \ref{boundB3}. Now having $t, \delta, \alpha, s$ fixed, if $n$ goes to infinity, equation \eqref{final-step} yields what we need.
\begin{thm}
Let $0<t<1$ be a real number. The random graph model $\mathbb{G}(n,k)$ with $k=\lfloor t\log n \rfloor$ is not connected with high probability.
\end{thm}
\begin{proof}
Having both Lemmas \ref{lem-second-method-1} and \ref{lem-second-method-2} one can apply the second moment method. We begin with
\begin{equation}
\begin{split}
& \mathbb{E}\left\{\left(\sum_{i=1}^n I_i \right)^2\right\} - \mathbb{E}\left\{\sum_{i=1}^n I_i \right\}^2 \\
& \quad = \mathrm{Var}\left\{\sum_{i=1}^n I_i \right\} \geq \left(0 - \mathbb{E}\left\{\sum_{i=1}^n I_i \right\} \right)^2 \mathbb{P}\left\{\sum_{i=1}^n I_i = 0 \right\} \\
\end{split}
\end{equation}
Therefore,
\begin{equation}
\begin{split}
\mathbb{P}\left\{\sum_{i=1}^n I_i = 0\right\} & \leq \frac{\mathbb{E}\{(\sum_{i=1}^n I_i )^2\} - \mathbb{E}\{\sum_{i=1}^n I_i \}^2}{\mathbb{E}\{\sum_{i=1}^n I_i \} ^2 } \\
& \leq \frac{\mathbb{E}\{(\sum_{i=1}^n I_i )^2\}}{\mathbb{E}\{\sum_{i=1}^n I_i \} ^2 } - 1 \\
& = \frac{\sum_{i=1}^n \mathbb{E}\{I_i^2\}+\sum_{i,j,i\neq j}\mathbb{E}\{I_i I_j\}}{n^2 \mathbb{E}\{I_1\}^2} - 1 \\
& = \frac{n \mathbb{E}\{I_1\} + n(n-1) \mathbb{E}\{I_1 I_2\}}{n^2 \mathbb{E}\{I_1\}^2} - 1 \\
& = \frac{1}{n \mathbb{E}\{I_1\}}+\frac{(n-1) \mathbb{E}\{I_1 I_2\}}{n \mathbb{E}\{I_1\}^2} - 1 \\
\label{ineq1}
\end{split}
\end{equation}
Now it follows from Lemma \ref{lem-second-method-1} that the first fraction vanishes and from Lemma \ref{lem-second-method-2} that the second fraction will be sufficiently close to 1 (or less than 1) for large $n$. Hence, the probability of having no isolated vertex converges to 0.
\end{proof}
\section{Connectivity for $t>1$}\label{sec:connectivity}
A common method to tackle the connectivity of random graphs in limit is by locating (with high-probability) a Erdos-Renyi sub-graph consisting of all the vertices. Then, the celebrated result of Erdos and Renyi \cite{erdHos1960evolution} guarantees the connectivity if the connectivity probability is $p = \frac{t\log n}{n}$ where $t>1$. This method used in \cite{la2015new} proves the connectivity of $\mathbb{G}(n,k)$ for $k=t\log n$ where $t>C=2.4625$. The connection between Erdos-Renyi and $\mathbb{G}(n,k)$ is made through a concentration property of order statistics. The authors of \cite{la2015new} assume that the edges are independently scored by $\exp(1)$ random variables which results in having a connection for all edges of scores greater than equal $\log((n-1)/(t\log n)) + \sqrt{2/t}$ with high probability. Therefore, if $L_{i,j}$ is a random variable of law $\exp(1)$, then
\[\mathbb{P}\Big\{L_{i,j}>\log(\frac{n-1}{t\log n}) + \sqrt{\frac{2}{t}}\Big\}= \frac{t\log n}{n-1} e^{-\sqrt{\frac{2}{t}}}.\]
It can be easily shown that $te^{-\sqrt{\frac{2}{t}}}>1$ for $t>2.4625$. This provides an independent possibility of connection for all edges with the probability of $t'\log n/ (n-1)$ for a $t'=te^{-\sqrt{\frac{2}{t}}}>1$. Thereafter, the connectivity result of Erdos-Renyi implies the connectivity of $\mathbb{G}(n,k)$.
We prove the connectivity of $\mathbb{G}(n,k)$ for all $t>1$ in two steps. First, we rule out the possibility of having components of size $O(1)$. Second, we apply the idea from \cite{la2015new} described above to find an Erdos-Renyi graph with all vertices contained in $\mathbb{G}(n,k)$. As opposed to \cite{la2015new}, we do not restrict $t$ to find a $t'>1$. Hence, the resulting Erdos-Renyi graph is not necessarily connected. However, by modifying the original proof of the connectivity of Erdos-Renyi graphs for $t'>1$, we can prove a weaker result for an arbitrary $t'<1$ which then helps us deduce the connectivity result.
\begin{thm}
Consider the random graphs of class $\mathbb{G}(n,k)$ where $k=\lfloor t\log n\rfloor $ and $t>1$. Let $\kappa$ be a constant non-negative integer. Then
\[\mathbb{P}\{\exists \textrm{ a vertex of degree less than } \kappa\}\rightarrow 0\]
as $n\rightarrow\infty$.
\label{exclude-o(1)}
\end{thm}
\begin{proof}
We can assume vertex $1$ has degree less than $\kappa$. We will use the same technique as Lemmas \ref{boundB1}, \ref{boundB2}, and \ref{boundB3}.
It is straightforward that for non-negative integer $\kappa$, $f(s)=(1-\frac{1}{s+1})^{s-\kappa}$ is a decreasing function converging to $e^{-1}$ (when $s\geq \kappa$). As a result, for a given $\kappa,t$, we can find a $s>\kappa$ such that $f(s)=e^{-\alpha}$ where $\frac{1}{t}<\alpha<1$.
For $i\in \mathcal{R}^{\leq s}_1$ define $\mathcal{E'}_{i} = \mathcal{E}_{i} \setminus \{\{i,j\}:j\in\mathcal{R}^{\leq s}_1\}$. Therefore, if we consider the following union
\[\mathcal{E}' = \mathcal{E}_1 \bigcup_{i\in \mathcal{R}^{\leq s}_1} \mathcal{E'}_i, \]
it will be a partition in which $|\mathcal{E}_1|=n-1$, $|\mathcal{E'}_i|=n-s-1$. Define $\mathcal{E}_1$ objects of type 0 and $\mathcal{E'}_i$ objects of type $i$ for $i\in\mathcal{R}_1^{\leq s}$. One can easily see that each realization of $\mathbb{G}(n,k)$ induces a uniformly random permutation over $\mathcal{E}'$ which has only a fixed order on the first $s$ or $s+1$ elements of $\mathcal{E}_1$, i.e., type 0.
A necessary condition for vertex 1 to be of degree at most $\kappa$ is that for at least $s-\kappa$ elements $i\in \mathcal{R}^{\leq s}_1$, there should be at least $k$ edges of $\mathcal{E}_{i}$ appearing before the $s^{th}$ element of $\mathcal{E}_{1}$. This implies there must be at least $k-s$ edges of $\mathcal{E'}_{i}$ before the $s^{th}$ element of $\mathcal{E'}_{1}$. As a result, in the first $(s-\kappa)(k-s)$ elements of this permutation there are at most $s-1$ elements of type 0.
With all $h_i=1$ in the setting of Lemma \ref{common-lem1} it follows that the probability of the event above is eventually bounded above by
\begin{equation*}
\begin{split}
\sum_{j=0}^{s-1} \binom{(s-\kappa)(k-s)}{j} \frac{s^{(s-\kappa)(k-s)-j}}{(s+1)^{(s-\kappa)(k-s)}} & \leq s \binom{(s-\kappa)(k-s)}{s-1} \frac{s^{(s-\kappa)(k-s)-s+1}}{(s+1)^{(s-\kappa)(k-s)}} \\
& = \binom{(s-\kappa)(k-s)}{s-1} \frac{1}{s^{s-1}}\left(1-\frac{1}{s+1}\right)^{(s-\kappa)(k-s)}\\
& = \binom{(s-\kappa)(k-s)}{s-1} \frac{1}{s^{s-1}}e^{-\alpha(k-s)}\\
& = O(n^{-t\alpha}) \quad (\, s\textrm{ is a constant})
\end{split}
\end{equation*}
Then, using the union bound we have:
\begin{equation*}
\begin{split}
\mathbb{P}\{\exists \textrm{ a vertex of degree less than } c\} & \leq n\times \mathbb{P}\{\textrm{vertex $1$ is of degree less than } c\} \\
& \leq n O(n^{-t\alpha}) \\
&= O(n^{1-t\alpha})
\end{split}
\end{equation*}
which vanishes due to the assumption on $\alpha$.
\end{proof}
Theorem \ref{exclude-o(1)} excludes the possibility of having components of size $O(1)$.
\begin{cor}
Consider the random graphs model of class $\mathbb{G}(n,k)$ where $k=\lfloor t\log n \rfloor$ and $t>1$. Let $c$ be a constant non-negative integer. Then
\[\mathbb{P}\{\exists \textrm{ a component of size less than } \kappa\}\rightarrow 0\]
as $n\rightarrow\infty$.
\label{finite-comp}
\end{cor}
\begin{proof}
In case of having a component of size $\kappa$ there most be a vertex of degree at most $\kappa$ which is impossible with high probability due to Theorem \ref{exclude-o(1)}.
\end{proof}
Next, we establish a proof based on finding an Erdos-Renyi sub-graph and super-graph of $\mathbb{G}(n,k)$ with both containing all the vertices. Recall that the order statistics of a set of \emph{i.i.d.} random variables $\{X_1,X_2,\cdots,X_n\}$ is defined as their non-increasing rearrangement $X^{(1)}\geq X^{(2)}\geq \cdots \geq X^{(n)}$. Recall that the $k^{th}$ order statistic of $\mathcal{V}_i$ is denoted by $V(i,R_i^k)$.
\begin{thm}
For the random graph model $\mathbb{G}(n,k)$ with $k=\lfloor t \log n \rfloor $ where $t>1$ there is no component of size $10\leq r \leq \lfloor n/2\rfloor$ with high probability.
\label{ER-estim}
\end{thm}
\begin{proof}
First, we note Lemma A.1 in \cite{la2015new} which states that if we choose scores $V(i,j)$ from $\exp(1)$ (independently distributed as exponential random variables with parameter $1$), and define:
\[A_n = \left\{\forall \; 1\leq i\leq n: \;V(i,R_i^k)\in \left(\log(\frac{n-1}{t\log n})-\sqrt{2} , \log(\frac{n-1}{t\log n})+\sqrt{2} \right) \right\},\]
then $\mathbb{P}\{A_n\}\rightarrow 1$ as $n\rightarrow\infty$. We let $\underline{l}=\log(\frac{n-1}{t\log n})-\sqrt{2}$ and $\bar{l} = \log(\frac{n-1}{t\log n})+\sqrt{2}$. Next, note that
\[\bar{p} := \mathbb{P}\{V(i,j)> \bar{l}\} = \frac{t\log n}{n-1}e^{-\sqrt{2}}\approx\frac{0.24t\log n}{n-1}\]
\[\underline{p} := \mathbb{P}\{V(i,j)> \underline{l}\} = \frac{t\log n}{n-1}e^{\sqrt{2}}\approx \frac{4.11t\log n}{n-1}\]
Moreover, if the graph satisfies $A_n$ condition (which holds whp), then $V(i,j)>\bar{l}$ implies that the edge $\{i,j\}$ exists and $V(i,j)<\underline{l}$ implies that the edge $\{i,j\}$ does not exist.
Let the graph satisfy $A_n$. In order to have a component of size $r$, one must choose $r$ vertices. The connectivity within this component implies that it should contains at least a tree. Moreover, according to Cayley's formula there are $r^{r-2}$ trees with $r$ vertices. Hence, the $r-1$ edges of the tree must have scores not less than $\underline{l}$ which happens with probability $\underline{p}^{r-1}$ due to independence. On the other hand, we require those $r$ vertices not to be connected to the rest of graph. This implies the scores for the intermediate edges between the component and the rest of graph must be less than $\bar{l}$. Note that both of these are necessary conditions. Counting the number of possible components yields the following:
\[\Pi=\mathbb{P}(\exists \textrm{ a component of size between } 10 \textrm{ and } \lfloor n/2 \rfloor)\leq\sum_{r=10}^{\lfloor n/2\rfloor}\binom{n}{r}r^{r-2}\underline{p}^{r-1}(1-\bar{p})^{r(n-r)}\]
Substituting $\binom{n}{r}<\frac{n^r}{r!}$, $r!\approx\sqrt{2\pi r}(r/e)^r$ implies
\begin{equation*}
\begin{split}
\Pi & < \frac{1}{\sqrt{2\pi}} \sum_{r=10}^{\lfloor n/2\rfloor}n^r r^{-r-1/2} e^r r^{r-2}\underline{p}^{r-1}(1-\bar{p})^{r(n-r)}\\
& < \frac{1}{\sqrt{2\pi}} \sum_{r=10}^{\lfloor n/2\rfloor}e^r \frac{n^r}{\underline{p}} r^{-5/2}\underline{p}^{r}e^{-r(n-r)\bar{p}}\quad (\textrm{using}\;1-x<e^{-x})\\
& < \frac{n}{\sqrt{2\pi}} \sum_{r=10}^{\lfloor n/2\rfloor}n^r e^r \underline{p}^{r}e^{-r\bar{p}n/2}\quad (\textrm{using}\;1/\underline{p}<n, \; n-r>n/2, \; r^{-5/2}<1)\\
& < \frac{n}{\sqrt{2\pi}} \sum_{r=10}^{\lfloor n/2\rfloor} e^{r(1 + \log(n\underline{p}) - n\bar{p}/2)}.
\end{split}
\end{equation*}
As $t$ is a fixed number, the dominant term in $1 + \log(n\underline{p}) - n\bar{p}/2$ is $- n\bar{p}/2\approx -0.12 t\log n$. As a result, for sufficiently large $n$, $1 + \log(n\underline{p}) - n\bar{p}/2\leq -\frac{t\log n}{9}<\frac{-\log n}{9}$, Thus
\begin{equation*}
\begin{split}
\Pi & < \frac{n}{\sqrt{2\pi}} \sum_{r=10}^{\infty} e^{\frac{-r\log n}{9}}\\
& = \frac{n}{\sqrt{2\pi}} \sum_{r=10}^{\infty} n^{\frac{-r}{9}}\\
& = \frac{n}{\sqrt{2\pi}}\left(\frac{n^{\frac{-10}{9}}}{1- n^{\frac{1}{9}}}\right)\\
& = \frac{n^{\frac{-1}{9}}}{\sqrt{2\pi}(1-n^{\frac{-1}{9}})}\rightarrow 0
\end{split}
\end{equation*}
Therefore, with high probability there is no component of size between 10 and $n/2$.
\end{proof}
Having Theorem \ref{ER-estim} and Corollary \ref{finite-comp}, we conclude the main theorem.
\begin{thm}
Let $t>1$ be a real number. The random graph model $\mathbb{G}(n,k)$ with $k=\lfloor t\log n \rfloor$ is connected with high probability.
\end{thm}
\begin{proof}
If a graph from class $\mathbb{G}(n,k)$ is disconnected, then it should have at least a component of size $s$ where $1\leq s \leq \lfloor n/2 \rfloor $. Theorem \ref{ER-estim} rules out the possibility of having components of size between 10 and $\lfloor n/2 \rfloor$ with high probability. Theorem \ref{finite-comp} guarantees that the probability of having components of size $O(1)$ which includes components of size less than 10 vanishes when $n$ grows to infinity.
\end{proof}
\section{Average Degree}\label{sec:avgdegree}
Here, we discuss another set of results for $\mathbb{G}(n,k)$ random graphs. The degree sequence is bounded above by $k$. Therefore, average degree is also bounded by $k$. However, we will show that this number is very close to $k$. We also specify the error term to some extent. Finally we compare our result to the sparse case in \cite{moharrami2020erlang}. Before presenting the main theorem we recall negative binomial distribution. Consider an urn having infinite number of red and blue balls and each time we draw red with probability $p$. The probability of having $j$ blue balls appeared before the $k^{th}$ red ball is a negative binomial distribution given by the following formula $\mathbb{P}\{X_k=j\}=\binom{k+j-1}{j}p^k(1-p)^{j}$. The theorems below provide an approximation for average degree.
\begin{thm}\label{thm:genasymptotic}
In the model $\mathbb{G}(n,k)$ suppose $D$ represents the degree of a randomly chosen vertex. If $k=o(\sqrt{n})$ then we have the following asymptotic
\begin{equation}
\mathbb{E}\{D\}= k - \left[\sum_{j=0}^{k-1}\binom{k+j-1}{j}\frac{k-j}{2^{k+j}}\right](1+O(k^2/n))
\label{original-ave-deg-eq}
\end{equation}
\label{original-ave-deg}
\end{thm}
\begin{proof}
Without loss of generality assume the randomly chosen vertex is vertex 1 and $R_1^i = i+1$ for all $1\leq i\leq n-1$. Denote the indicator function of vertex 1 being connected to vertex $i+1$ by $E_i$. In order for vertex 1 not to be connected to vertex $i+1$ there must be at least $k$ elements of $\mathcal{E}_{i+1}$ before the $i^{th}$ element of $\mathcal{E}_1$ which is $\{1,i+1\}$. Hence, there could be $0, 1,\cdots, i-1$ elements of $\mathcal{E}_1$ before the $k^{th}$ element of $\mathcal{E}_{i+1}$. With the same idea used in Lemma \ref{common-lem1} one can prove that the probability of $j$ elements of $\mathcal{E}_1$ appear before the $k^{th}$ element of $\mathcal{E}_{i+1}$ for $j\leq k$ is:
\[\binom{k+j-1}{j} \left(\frac{n+O(k)}{2n+O(k)}\right)^{k+j}= \binom{k+j-1}{j} \left(\frac{1}{2}\right)^{k+j}(1 + O(k^2/n))\]
Varying $0\leq j \leq i-1$ we conclude that for each $1\leq i \leq k$ one has
\[\mathbb{P}\{ E_i = 0 \} = \left[\sum_{j=0}^{i-1} \binom{k+j-1}{j} \frac{1}{2^{k+j}}\right](1+O(k^2/n))\]
We now compute $\mathbb{E}\{D\}$ as follows
\begin{align}
\begin{split}
\mathbb{E}\{D\} & = \sum_{i=1}^{n-1} \mathbb{E}\{E_i\} \\
& = \sum_{i=1}^{n-1} \mathbb{P}\{E_i = 1 \} \\
& = \sum_{i=1}^{k} \mathbb{P}\{E_i = 1 \} \\
& = k - \sum_{i=1}^{k} \mathbb{P}\{E_i = 0 \}\\
& = k - \sum_{i=1}^{k} \left[\sum_{j=0}^{i-1} \binom{k+j-1}{j} \frac{1}{2^{k+j}}\right](O(k^2/n)+1) \\
& = k - \left[\sum_{j=0}^{k-1} \binom{k+j-1}{j} \frac{k-j}{2^{k+j}}\right](O(k^2/n)+1).
\end{split}
\end{align}
This completes the proof.
\end{proof}
\begin{thm}
In the model $\mathbb{G}(n,k)$, assume that $k=o(\sqrt{n})$ grows to infinity as $n$ goes to infinity. Suppose $D$ represents the degree of a randomly chosen vertex. We have the following asymptotic
\[\mathbb{E}\{D\}= k - \sqrt{k/\pi}+o(\sqrt{k}).\]
\end{thm}
\begin{proof}
The term $p_j = \binom{k+j-1}{j} \frac{1}{2^{k+j}}$ in (\ref{original-ave-deg-eq}) is the distribution of negative binomial random variable $X_k$ with parameter $p=1/2$. Reference \cite{bagui2016convergence} provides a normal approximation (in the weak sense) of a negative binomial distribution with mean $\mu=k$, and standard deviation $\sigma=\sqrt{2k}$: $Z_k:=(X_k-k)/\sqrt{2k}\stackrel{k\rightarrow\infty}{\Longrightarrow} N(0,1)$. Then, from Theorem~\ref{thm:genasymptotic} we have
\begin{equation}
\begin{split}
\mathbb{E}\{D\} & = k - \left[\sum_{j=0}^{k-1}\binom{k+j-1}{j}\frac{k-j}{2^{k+j}}\right](1+O(k^2/n))\\
& = k + \sqrt{2k}\left[\sum_{j=0}^{k-1} \frac{1}{2^{k+j}}\binom{k+j-1}{j} \left(\frac{j-k}{\sqrt{2k}}\right)\right](1+O(k^2/n))\\
\end{split}
\label{rewrite-the-ave-deg}
\end{equation}
The term within brackets above is $\mathbb{E}\{(\frac{X_k - k}{\sqrt{2k}})1_{X_k<k}\}=\mathbb{E}\{Z_k 1_{Z_k<0}\}$. Set $Y_k=-Z_k 1_{Z_k < 0}$ which is a non-negative random variable. Therefore, it follows from Fubini's theorem that
\begin{equation}
\begin{split}
\mathbb{E}\{Y_k\} & = \int_0^\infty \mathbb{P}\{Y_k > x\} dx \\
& = \int_0^\infty \mathbb{P}\{Z_k < -x\} dx \\
& = \int_{-\infty}^0 \mathbb{P}\{Z_k < x\} dx \\
& = \int_{-\infty}^0 \mathbb{P}\{Z_k \leq x\} dx - \int_{-\infty}^0 \mathbb{P}\{Z_k = x\} dx \\
& = \int_{-\infty}^0 \mathbb{P}\{Z_k \leq x\}dx \quad (\forall k: \textrm{at a finite number of points}\; Z_k=x)\\
& = \int_{-\infty}^0 F_{Z_k}(x) dx.
\label{fubini}
\end{split}
\end{equation}
Random variable $Z_k$ is normalized to mean 0 and standard deviation 1. Using Chebyshev's inequality we have the following upper bound for its (lower) tails
\begin{equation}
\begin{split}
\mathbb{P}\{Z_k \leq t\} & \leq 1/t^2
\end{split}
\label{bound-for-dominated}
\end{equation}
We define $g(t):=1/t^2$ for $t\leq -1$, and $g(t):=1$ for $-1<t\leq 0$. This function is integrable over $(-\infty, 0]$. It follows from (\ref{bound-for-dominated}) that all $F_{Z_k}(t)$ are bounded above by $g(t)$ for $t\in(-\infty,0]$. In addition, through weak convergence, for each $t\in(-\infty, 0]$, $F_{Z_k}(t)\stackrel{k\rightarrow\infty}{\longrightarrow}F_Z(t)$ where $Z\sim N(0,1)$ (since $F_Z(t)$ is continuous for all $t\in(-\infty,\infty)$). Therefore, dominated convergence theorem (see \cite{royden1988real}) implies that $\int_{-\infty}^0 F_{Z_k}(t)dt$ converges to $\int_{-\infty}^0 F_{Z}(t)dt$ as $k\rightarrow\infty$. The identity (\ref{fubini}) results in $\mathbb{E}\{Z_k 1_{Z_k<0}\}\stackrel{k\rightarrow\infty}{\longrightarrow}\mathbb{E}\{Z 1_{Z<0}\}=\frac{-1}{2}\sqrt{\frac{2}{\pi}}$. Hence, substituting this in (\ref{rewrite-the-ave-deg}), $\mathbb{E}\{D\}$ is asymptotically given by
\[k - \frac{\sqrt{2k}}{2} \sqrt{\frac{2}{\pi}}=k-\sqrt{\frac{k}{\pi}}\]
\end{proof}
\paragraph{Remark:} As mentioned earlier, the work in \cite{moharrami2020erlang} considers the preference threshold to be a random variable independently chosen per vertex using a distribution $P$ over $\mathbb{N}^0$ with finite mean instead of just a fixed $k$ for the potential number of neighbors of a randomly chosen vertex. Further, \cite[Theorem 5.1]{moharrami2020erlang} specifies the following formula for the average degree in the limit of $n$ going to infinity
\begin{equation}
\mathbb{E}\{D\} = \sum_{i=1}^{\infty}\sum_{j=1}^{\infty} P(i)P(j)\int_0^{\infty}\bar{F}_i(x)\bar{F}_j(x)dx
\label{ave-deg-mehrdad}
\end{equation}
where $\bar{F}_i$ denotes the complementary
cumulative distribution function of $\textrm{Erlang}(. ;i,1)$ (the Erlang distribution of shape $i$ and rate $1$).
In the case of bilateral agreement with preference threshold parameter to be a fixed $k$ the probability distribution $P$ becomes a delta mass function on $k$, i.e., $P(i)=1$ if $i=k$, and $P(i)=0$ otherwise. In addition, the mean being finite as $n$ goes to infinity, implies that $k$ must be finite and cannot grow to infinity. The sum \ref{ave-deg-mehrdad} now simplifies to
\begin{equation}
\mathbb{E}\{D\} = \int_0^{\infty}\bar{F}_k(x)^2dx
\end{equation}
Then using $\bar{F}_k(x)=\sum_{i=0}^{k-1}\frac{1}{i!}e^{-x}x^i$, distributing the square, and exchanging the finite sum with integral, we obtain
\begin{equation}
\begin{split}
\mathbb{E}\{D\} & = \sum_{0\leq i,j\leq k-1} \int_0^{\infty} \frac{x^{i+j}}{i!j!}e^{-2x}dx\\
& = \sum_{0\leq i,j\leq k-1} \int_0^{\infty} \frac{(2x)^{i+j}}{2^{i+j}i!j!}e^{-2x} dx\\
& = \sum_{0\leq i,j\leq k-1} \frac{\Gamma(i+j+1)}{2^{i+j+1} i! j!}\\
& = \frac{1}{2}\sum_{0\leq i,j\leq k-1} \frac{1}{2^{i+j}} \binom{i+j}{i}\\
\end{split}
\end{equation}
We substitute $s=i+j$ and use the notation of $X_k$ for a negative binomial random variable with parameter $p=1/2$ representing the probability of appearing $k^{th}$ red ball after observing $X_k$ blue balls. Then we arrive at
\begin{equation}
\begin{split}
\mathbb{E}\{D\} & = \frac{1}{2} \left[ \sum_{s=0}^{k-1} \sum_{i=0}^s \frac{1}{2^{s}} \binom{s}{i} + \sum_{s=k}^{2k-2}\sum_{i=s-k+1}^{k-1} \frac{1}{2^s}\binom{s}{i}\right]\\
& = \frac{1}{2} \left[ k + \sum_{s=0}^{k-2}\left(1-2\sum_{i=0}^{s}\frac{1}{2^{s+k}}\binom{s+k}{i}\right)\right]\\
& = \frac{1}{2} \left[ k + \sum_{s=0}^{k-2}\left(1-2\mathbb{P}\{X_k\leq s \}\right)\right]\\
& = \frac{1}{2} \left[ k + \sum_{s=0}^{k-2}\left(2\mathbb{P}\{X_k> s \}-1\right)\right]\\
& = \frac{1}{2} \left[ 1 + 2\mathbb{E}\{X_k 1_{X_k\leq k-1}\}+2(k-1)\mathbb{P}\{X_k\geq k\}\right]\\
& = \frac{1}{2} \left[ 1 + \sum_{j=0}^{k-1}\binom{k+j-1}{j}\frac{2j}{2^{k+j}} +2(k-1)\mathbb{P}\{X_k\geq k\}\right]\\
\end{split}
\label{ave-deg-mehrdad-to-mine}
\end{equation}
In the second of line of (\ref{ave-deg-mehrdad-to-mine}), we use the fact that $\sum_{i=0}^{s} \frac{1}{2^{s+k}}\binom{s+k}{i}$ represents the probability of appearing at most $s$ blue balls in the first $s+k$ observation which is equivalent to observing at least $k$ red balls in this segment. Now, $\mathbb{P}\{X_k<k\}=\sum_{j=0}^{k-1} \binom{k+j-1}{j} \frac{1}{2^{k+j}}$ represents the probability of observing the $k^{th}$ red ball earlier than the $k^{th}$ blue ball in the infinite urn model. According to symmetry and since $p=1/2$, this value must be $1/2$. Using this fact, we have
\begin{equation}
\begin{split}
\mathbb{E}\{D\}& = \frac{1}{2} \left[ 1 + \sum_{j=0}^{k-1}\binom{k+j-1}{j}\frac{2j}{2^{k+j}} +2(k-1)\mathbb{P}\{X_k\geq k\}\right]\\
& = \frac{k}{2} + \sum_{j=0}^{k-1}\binom{k+j-1}{j}\frac{j}{2^{k+j}}\\
& = k - \sum_{j=0}^{k-1}\binom{k+j-1}{j}\frac{k-j}{2^{k+j}}\\
\end{split}
\end{equation}
Therefore, we reproduce the same formula for the average degree as in Theorem \ref{original-ave-deg} using the result in \cite{moharrami2020erlang}. We discussed this remark with the assumption of sparseness. However, Theorem \ref{original-ave-deg} only needs $k=o(\sqrt{n})$ which covers both the sparse and non-sparse regimes.
\section{Conclusions}
We have studied the bilateral random graphs and presented a complete proof of their connectivity threshold. This threshold is $k=\log n$ as conjectured by La and Kabkab~\cite{la2015new}. The proof for disconnectivity is established using second moment method in which we proved the expected number of isolated vertices grows to infinity and showed that asymptotically any pair of vertices isolated in an independent fashion, i.e. $\mathbb{E}\{I_1I_2\}\approx \mathbb{E}\{I_1\}\mathbb{E}\{I_2\}$. Therefore, with high probability there exists an isolated vertex which yield the disconnectivity result. For the connectivity part, we took the same approach as in La and Kabkab~\cite{la2015new} to find Erdos-Reyni sub and super random graphs with all vertices associated with the original graph. This excludes the possibility of having components with $r$ vertices for $10\leq r\leq n/2$. Independently, we also ruled out the existence of components of size $O(1)$. At the end, we discussed the average degree and presented an asymptotic for it.
\printbibliography
\end{document}
| {
"timestamp": "2022-09-14T02:09:22",
"yymm": "2209",
"arxiv_id": "2209.05677",
"language": "en",
"url": "https://arxiv.org/abs/2209.05677",
"abstract": "Bilateral agreement based random undirected graphs were introduced and analyzed by La and Kabkab in 2015. The construction of the graph with $n$ vertices in this model uses a (random) preference order on other $n-1$ vertices and each vertex only prefers the top $k$ other vertices using its own preference order; in general, $k$ can be a function of $n$. An edge is constructed in the ensuing graph if and only if both vertices of a potential edge prefer each other. This random graph is a generalization of the random $k^{th}$-nearest neighbor graphs of Cooper and Frieze that only consider unilateral preferences of the vertices. Moharrami \\emph{et al.} studied the emergence of a giant component and its size in this new random graph family in the limit of $n$ going to infinity when $k$ is finite. Connectivity properties of this random graph family have not yet been formally analyzed. In their original paper, La and Kabkab conjectured that for $k(t)=t \\log n$, with high probability connectivity happens at $t>1$ and the graph is disconnected for $t<1$. We provide a proof for this conjecture. We will also introduce an asymptotic for the average degree of this graph.",
"subjects": "Probability (math.PR); Combinatorics (math.CO)",
"title": "Connectivity of a Family of Bilateral Agreement Random Graphs",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754474655618,
"lm_q2_score": 0.7185943925708562,
"lm_q1q2_score": 0.7075103956116944
} |
https://arxiv.org/abs/math/0510584 | Hilbert series of subspace arrangements | The vanishing ideal I of a subspace arrangement is an intersection of linear ideals. We give a formula for the Hilbert polynomial of I if the subspaces meet transversally. We also give a formula for the Hilbert series of a product J of the linear ideals without any assumptions on the subspace arrangement. It turns out that the Hilbert series of J is a combinatorial invariant of the subspace arrangement: it only depends on the intersection lattice and the dimension function. The graded Betti numbers of J are determined by the Hilbert series, so they are combinatorial invariants as well. The results can be applied to Generalized Principal Component Analysis (GPCA), a tool that is useful for computer vision and image processing. | \section{Introduction}
Suppose that $V$ is an $n$-dimensional $K$-vector space. A subspace
arrangement is a union
$$
{\mathcal A}=V_1\cup \cdots \cup V_m
$$
where $V_i$ is a subspace of $V$ for all $i$.
Interestingly, various algebraic and topological properties of
the arrangement ${\mathcal A}$ only depend on the dimensions
$n_S:=\dim_K\bigcap_{i\in S} V_i, S\subseteq \{1,2,\dots,m\}$.
Such properties are called {\it
combinatorial invariants\/} of the subspace arrangement.
For example, if $K={\mathbb R}$,
then the topological Betti numbers of the complement $V\setminus {\mathcal A}$
are combinatorial invariants (see~\cite{GM}).
If $K={\mathbb C}$, then the cohomology ring of $V\setminus {\mathcal A}$
is a combinatorial invariant (see~\cite{LS,DGM}).
For more on subspace arrangements and hyperplane arrangements, see~\cite{OT}.
Let $I_j\subseteq K[V]$ be the vanishing ideal $V_j\subseteq V$
for $j=1,2,\dots,m$. The vanishing ideal $I$ of ${\mathcal A}$
is equal to the intersection $I_1\cap I_2\cap \cdots \cap I_m$.
We also define $J=I_1I_2\cdots I_m$ as the product ideal.
We give a formula for the
Hilbert series of $J$ (Theorem~\ref{thmmain1}).
We also will give a formula
for the Hilbert polynomial of $I$ if all subspaces meet transversally
(Theorem~\ref{theo1}).
The Hilbert series of $J$ is a combinatorial
invariant (Corollary~\ref{cor5}), but the Hilbert polynomial of $I$ is not (Example~\ref{exlast}).
The Betti numbers and graded Betti numbers of $J$ are also combinatorial invariants
(Corollary~\ref{corComb}).
The original motivation for this paper comes from computer vision.
A generalization of Principal Component Analysis naturally
leads to the question of recovering the dimensions
$n_i:=\dim V_i$, $i=1,2,\dots,m$, given the Hilbert polynomial
of the subspace arrangement. For more on Generalized
Principal Component Analysis, see~\cite{VMS}.
{\bf Acknowledgement.}
I would like to thank Robert Fossum for interesting discussions,
for explaining me the results in computer vision, and for suggesting
to study the Hilbert function of a general subspace arrangement.
I also thank Allen Yang and Yi Ma for useful references.
\section{Hilbert functions, series and polynomials}
Suppose that $V$ is an $n$-dimensional vector space over a field $K$.
We identify the coordinate ring $R:=K[V]$
with the polynomial ring $K^{[n]}:=K[x_1,x_2,\dots,x_n]$ in $n$ variables
by choosing a basis in $V$.
There is
a natural grading $R=\bigoplus_{d\in {\mathbb N}} R_d$ where
${\mathbb N}=\{0,1,2,\dots\}$ is the set of natural numbers and
$R_d$ denotes the space of homogeneous polynomials of degree $d$.
Let ${\mathbb Z}$ be the integers and suppose that $M=\bigoplus_{d\in {\mathbb Z}} M_d$ is a finitely
generated graded $R$-module. We have
$M_d=0$ for $d\ll 0$ because $M$ is finitely generated.
The {\it Hilbert function\/} $h_M$ of $M$ is
$$
h_M(d)=h(M,d)=\dim_K M_d,\quad d\in {\mathbb Z}.
$$
The {\it Hilbert series\/} of $M$ is defined by
$$
{\mathcal H}(M,t):=\sum_{d\in{\mathbb Z}}h(M,d)t^d.
$$
It is a Laurent series because $h(M,d)=0$ for $d\ll 0$.
\begin{example}
For $M=R$ we get
$h(R,d)={n-1+d\choose n-1}$ for $d\geq 0$ and $h(R,d)=0$ for $d<0$.
So we have
$$
{\mathcal H}(R,t)=\sum_{d=0}^\infty {\textstyle {n-1+d\choose n-1}}t^d=\frac{1}{(1-t)^n}.
$$
Define the polynomial $\widetilde{h}_R$ by
$$
\widetilde{h}_R(d)=\widetilde{h}(R,d)=\frac{(d+n-1)(d+n-2)\cdots (d+1)}{(n-1)!}.
$$
One easily checks that $h_R(d)=\widetilde{h}_R(d)$ for $d\geq 1-n$.
\end{example}
Let $M$ be again a finitely generated graded $R$-module.
For $r\in {\mathbb Z}$ we
define the shifted module $M[r]=\bigoplus_{d\in {\mathbb Z}} M[r]_d$ by
$M[r]_d:=M_{r+d}$, $d\in {\mathbb Z}$.
Shifting the degrees affects the Hilbert function and the Hilbert series
as follows:
$$
h(M[r],d)=h(M,d+r),\quad d,r\in{\mathbb Z},
$$
$$
{\mathcal H}(M[r],t)=t^{-r}{\mathcal H}(M,t).
$$
The module $M$ has a minimal finite free graded resolution
\begin{equation}\label{eq1}
0\to \bigoplus_{j\in {\mathbb Z}} R[-j]^{\beta_{r,j}}\to\bigoplus_{j\in {\mathbb Z}}
R[-j]^{\beta_{1,j}}\to\cdots\to
\bigoplus_{j\in {\mathbb Z}}R[-j]^{\beta_{0,j}}\to M\to 0.
\end{equation}
by Hilbert's Syzygy Theorem (see for example \cite{Hilbert},\cite[\S 13]{ZSII},\cite[\S 19.2]{Eisenbud}).
The nonnegative integers $\beta_{i,j}$ are
called the {\it graded Betti numbers}. For all but finitely many pairs $(i,j)$
we have $\beta_{i,j}=0$.
The {\it Betti
numbers\/} are defined by $\beta_i=\sum_{j\in {\mathbb Z}}
\beta_{i,j}$ (not to be confused with the topological Betti numbers
of the complement of the subspace arrangement mentioned earlier).
Without loss of generality we may assume that $\beta_r\neq 0$.
The nonnegative integer $\operatorname{cd}(M):=r$
is the {\it cohomological dimension\/} of the module $M$
and is at most $n$.
The {\it Castelnuovo-Mumford regularity\/} of $M$ is
$$
\operatorname{reg}(M):=\max\{j-i\mid 0\leq i\leq r,\beta_{i,j}\neq 0\}.
$$
From the exactness of (\ref{eq1})
follows that
\begin{equation}\label{eq11}
h(M,d)=
\sum_{i=0}^r(-1)^i\sum_{j\in {\mathbb Z}}\beta_{i,j}h(R[-j],d)=
\sum_{i=0}^r(-1)^i\sum_{j\in {\mathbb Z}}\beta_{i,j}h(R,d-j)
\end{equation}
and
$$
{\mathcal H}(M,t)=\frac{\sum_{i=0}^r (-1)^i\sum_{j\in {\mathbb Z}}\beta_{i,j}t^{j}}{(1-t)^n}.
$$
We define the {\it Hilbert polynomial\/} $\widetilde{h}_M$ of M by
\begin{equation}\label{eq12}
\widetilde{h}_M(d)=\widetilde{h}(M,d)=\sum_{i=0}^r(-1)^i\sum_{j\in {\mathbb Z}}
\beta_{i,j}\widetilde{h}(R,d-j).
\end{equation}
\begin{corollary}\label{cor1}
If
$d\geq
1-n+\operatorname{reg}(M)+\operatorname{cd}(M)$,
then we have
$$
h_M(d)=\widetilde{h}_M(d).
$$
\end{corollary}
\begin{proof}
If $\beta_{i,j}\neq 0$, then
$$
d-j=d-i-(j-i)\geq d-\operatorname{cd}(M)-\operatorname{reg}(M)\geq 1-n,
$$
hence
$h_R(d-j)=\widetilde{h}_R(d-j)$
(see Example~\ref{ex1}).
The corollary follows from (\ref{eq11}) and (\ref{eq12}).
\end{proof}
\section{Subspace arrangements}
For the remainder of this paper, let $V$ be an $n$-dimensional
vector space and suppose that $V_1,V_2,\dots,V_m$
are subspaces of $V$.
For a subset $X\subseteq V$, let ${\mathcal I}(X)\subseteq R=K[V]$
be its vanishing ideal.
Define $I_j={\mathcal I}(V_j)$ for $j=1,2,\dots,m$.
The union
$$
{\mathcal A}=V_1\cup V_2\cup \cdots\cup V_m
$$
is a subspace arrangement. Its vanishing ideal is
$$
I:={\mathcal I}({\mathcal A})=I_1\cap I_2\cap \cdots \cap I_m.
$$
Define
$$
J:=I_1I_2\cdots I_m.
$$
\begin{theorem}\label{theoregularity}
The regularity of $I$ and $J$ are bounded by
$\operatorname{reg}(I)\leq m$ and
$\operatorname{reg}(J)\leq m$.
\end{theorem}
\noindent For $\operatorname{reg}(J)\leq m$, see~\cite{CH}.
The bound $\operatorname{reg}(I)\leq m$ was proven in \cite{DS1,DS2}.
For a ideal ${\mathfrak a}\subseteq R$ we have
$\operatorname{cd}({\mathfrak a})=\operatorname{cd}(R/{\mathfrak a})-1\leq n-1$.
In particular, we get $\operatorname{cd}(I)\leq n-1$ and $\operatorname{cd}(J)\leq n-1$.
\begin{corollary}\label{cor5}
We have $h_I(d)=\widetilde{h}_I(d)$
and
$h_J(d)= \widetilde{h}_J(d)$
for $d\geq m$.
\end{corollary}
\begin{proof}
By Corollary~\ref{cor1}, $h_{I}(d)=\widetilde{h}_{I}(d)$ and
$h_J(d)=\widetilde{h}_J(d)$
for
$$
d\geq m= (1-n)+m+(n-1)\geq 1-n+\operatorname{reg}(I)+\operatorname{cd}(I).
$$
\end{proof}
\section{Main results}
For $S\subseteq\{1,2,\dots,m\}$, define
$I_S=\bigcap_{s\in S}I_s$ and
$J_S=\prod_{s\in S}I_s$.
Note that
$I=I_{\{1,2,\dots,m\}}$ and $J=J_{\{1,2,\dots,m\}}$.
We use the convention $I_\emptyset=J_{\emptyset}=R$.
For $S\subseteq\{1,2,\dots,m\}$ define
$V_S=\bigcap_{i\in S}V_i$, $n_S=\dim V_S$ and $c_S=n-n_S$
is the codimension of $V_S$ in $V$. We also set $n_i=n_{\{i\}}=\dim V_i$
and $c_i=c_{\{i\}}=n-n_i$ for $i=1,2,\dots,m$.
We define polynomials $p_S(t)$ recursively as follows. First we define
$$
p_\emptyset(t)=1.
$$
If $S\neq\emptyset$ and $p_X(t)$ is already defined for all proper
subsets $X\subset S$, then
$p_S(t)$ is uniquely determined by
$$
\sum_{X\subseteq S}(-t)^{|X|}p_X(t) \equiv 0 \bmod (1-t)^{c_S},\quad
\deg(p_S(t))<c_S.
$$
Here $\deg(p_X(t))$ is the degree of the polynomial $p_X(t)$
and $|X|$ is the cardinality of the set $X$.
\begin{theorem}\label{thmmain1}
We have
$$
{\mathcal H}(J,t)=\frac{p(t)t^m}{(1-t)^n},
$$
where $p(t)=p_{\{1,2,\dots,m\}}(t)$.
\end{theorem}
\begin{corollary}
The Hilbert series ${\mathcal H}(J,t)$ depends only on the numbers
$n_S, S\subseteq \{1,2,\dots,m\}$.
\end{corollary}
\begin{proof} This follows immediately from the recursive formulas
for $p_S(t)$, $S\subseteq \{1,2,\dots,n\}$ and the observations
that $n=n_{\emptyset}$ and $c_S=n-n_S$ for all $S$.
\end{proof}
\begin{proposition}\label{prop43}
Let $\{\beta_{i,j}\}$ be the graded Betti numbers of the ideal $J$. If $\beta_{i,j}\neq 0$
then $j=m+i$ and $\beta_i:=\sum_{j}\beta_{i,j}=\beta_{i,m+i}$ is the $i$-th Betti
number.
We have
$$
{\mathcal H}(J,t)=\frac{\sum_{i=0}^r(-1)^i\beta_it^{i+m}}{(1-t)^n}=\frac{t^mp(t)}{(1-t)^n},
$$
where
$$
p(t)=\beta_0-\beta_1t+\beta_2t^2-\cdots+(-1)^{r}\beta_{r}t^{r},
$$
and $r=\operatorname{cd}(J)\leq n-1$.
The Betti numbers and graded Betti numbers of $J$ are uniquely
determined by the Hilbert series of $J$, so they are combinatorial invariants.
\end{proposition}
\begin{proof}
By Theorem~\ref{theoregularity} we have
$$
\operatorname{reg}(J)\leq m.
$$
The ideal $J=I_1I_2\cdots I_m$ is generated in degree $m$.
This means that $J$ has a {\it linear\/} minimal free resolution
(see~\cite[Proposition]{EisenbudGoto}):
$$
0\to R[-m-r]^{\beta_r}\to \cdots \to R[-m-1]^{\beta_1}\to R[-m]^{\beta_0}\to J\to 0.
$$
The proposition follows.
\end{proof}
\begin{corollary}\label{corComb}
The Betti numbers and graded Betti numbers of $J$ are uniquely determined
by the Hilbert series of $J$, so they are combinatorial invariants.
\end{corollary}
\begin{definition}
The subspaces $V_1,\dots,V_m$ are called {\it transversal\/}
if
$$
c_S=\min\big(n,\textstyle\sum_{i\in S}c_i\big)
$$
for all $S\subseteq\{1,2,\dots,m\}$, where $\min$ denotes the minimum.
\end{definition}
Note that we always have $c_S\leq \min\big(n,\sum_{i\in S}c_i\big)$.
So the subspaces are transversal if any intersection of some of
the subspaces has the smallest possible dimension.
\begin{theorem}\label{theo1}
Suppose that $V_1,\dots,V_m$ are transversal.
Then ${\mathcal H}(I,t)-f(t)$ and ${\mathcal H}(J,t)-f(t)$ are polynomials in $t$,
where
$$
f(t)=\frac{\prod_{i=1}^d \big(1-(1-t)^{c_i}\big)}{(1-t)^n}.
$$
\end{theorem}
\begin{corollary}
If $V_1,\dots,V_m$ are transversal, then
$$
h_I(d)=\widetilde{h}_I(d)=\widetilde{h}_J(d)=h_J(d)
$$
for all $d\geq m$.
\end{corollary}
\begin{proof}
From Theorem~\ref{theo1} and the exact sequence
$$
0\to J\to I\to I/J\to 0
$$
follows that
${\mathcal H}(I/J,t)={\mathcal H}(I,t)-{\mathcal H}(J,t)$ is a polynomial. So
$I/J$ has a finite dimension.
It follows that $\widetilde{h}_I(t)=\widetilde{h}_J(t)$.
We have
$$
h_I(d)=\widetilde{h}_I(d)=\widetilde{h}_J(d)=h_J(d)
$$
for $d\geq (1-n)+m+(n-1)=m$ by Corollary~\ref{cor5}.
\end{proof}
\begin{corollary}
If $V_1,\dots,V_m$ are transversal, then
$$
h_I(d)=h_J(d)=
\sum_{S}(-1)^{|S|}
{d+n-1-c_S\choose n-1-c_S}.
$$
where $c_S=\sum_{i\in S}c_i$ and the sum is over all $S\subseteq\{1,2,\dots,m\}$
for which $c_S<n$.
\end{corollary}
\section{Examples}
\begin{example}\label{ex1}
Suppose that $m=3$ and $n=3$ and that $V_1,V_2,V_3$ are one-dimensional
and distinct. Now $V_1,V_2,V_3$ correspond to 3 points
$P_1,P_2,P_3\in {\mathbb P}^2$.
Suppose that $P_1,P_2,P_3$ are not colinear. We can change coordinates
such that
$$P_1=(1:0:0),\ P_2=(0:1:0),\ P_3=(0:0:1).
$$
We have
$$
I=I_1\cap I_2\cap I_3=(x_2,x_3)\cap (x_1,x_3)\cap (x_1,x_2)=
(x_2x_3,x_1x_3,x_1x_2)\subseteq K[x_1,x_2,x_3]
$$
and
$$
J=I_1I_2I_3=(x_1x_2x_3,x_1^2x_2,x_1^2x_3,x_2^2x_3,x_2^2x_1,x_3^2x_1,x_3^2x_2)\subseteq
K[x_1,x_2,x_3].
$$
The minimal free resolutions are as follows
$$
0\to R[-3]^2\to R[-2]^3\to I\to 0,
$$
$$
0\to R[-5]^3\to R[-4]^9\to R[-3]^7 \to J\to 0.
$$
So we have $\operatorname{cd}(I)=1$, $\operatorname{cd}(J)=2$, $\operatorname{reg}(I)=2$, $\operatorname{reg}(J)=3$,
$$
{\mathcal H}(I,t)=\frac{3t^2-2t^3}{(1-t)^3}\mbox{ and }
{\mathcal H}(J,t)=\frac{7t^3-9t^4+3t^5}{(1-t)^3}.
$$
The Hilbert functions, which can be found as the
coefficients of the Hilbert series, are given in the following table.
$$
\begin{array}{c||c|c|c|c|c|c}
d & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline\hline
h_{I}(d) & 0 & 0 & 3 & 7 & 12 & 18\\ \hline
h_J(d) & 0 & 0 & 0 & 7 & 12 & 18
\end{array}
$$
The Hilbert polynomials $\widetilde{h}_I(d)$ and $\widetilde{h}_J(d)$
are both equal to
$$
\frac{d^2+3d-4}{2}.
$$
Let us verify Theorems~\ref{thmmain1} and \ref{theo1}. The subspaces $V_1,V_2,V_3$
are transversal. Let
\begin{equation}\label{eq55}
f(t)=\prod_{i=1}^3\frac{(1-(1-t)^{c_i})}{(1-t)^n}=\frac{(1-(1-t)^2)^3}{(1-t)^3}=
\frac{-2+6t-3t^2}{(1-t)^3}+2-3t^2+t^3.
\end{equation}
We can write
$$
{\mathcal H}(I,t)=\frac{3t^2-2t^3}{(1-t)^3}=\frac{-2+6t-3t^2}{(1-t)^2}+2
$$
and
$$
{\mathcal H}(J,t)=\frac{7t^3-9t^4+3t^5}{(1-t)^3}=\frac{-2+6t-3t^2}{(1-t)^2}+2-3t^2.
$$
As Theorem~\ref{theo1} predicts, ${\mathcal H}(I,t)-f(t)$ and ${\mathcal H}(J,t)-f(t)$ are polynomials.
Using the recursive definitions of $p_S$ we get $p_{\emptyset}(t)=1$,
$$
p_{\emptyset}(t)-tp_{\{1\}}(t)\equiv 1-t p_{\{2\}}(t)\equiv 1-tp_{\{3\}}(t)\equiv 0 \bmod (1-t)^2,
\quad \deg(p_{\{1\}}),\deg(p_{\{2\}}),\deg(p_{\{3\}})<2,
$$
so $p_{\{1\}}(t)=p_{\{2\}}(t)=p_{\{3\}}(t)=2-t$. Furthermore,
$$
1-t(p_{\{1\}}(t)+p_{\{2\}}(t))+t^2p_{\{1,2\}}(t)\equiv 0\bmod (1-t)^3,
$$
so we have
$$
t^2p_{\{1,2\}}(t)\equiv -1+2t(2-t)\equiv -1+4t-2t^2 \bmod (1-t)^3
$$
and $\deg(p_{\{1,2\}})<3$.
It follows that
$$
p_{\{1,2\}}(t)=p_{\{1,3\}}(t)=p_{\{2,3\}}(t)=4-4t+t^2
$$
Finally we have
$$
1-t(p_{\{1\}}(t)+p_{\{2\}}(t)+p_{\{3\}}(t))+
t^2(p_{\{1,2\}}(t)+p_{\{1,3\}}(t)+p_{\{2,3\}}(t))-t^3p_{\{1,2,3\}}(t)\equiv
0 \bmod (1-t)^3,
$$
so
$$
t^3p_{\{1,2,3\}}(t)\equiv 1-3t(2-t)+3t^2(4-4t+t^2)\equiv -2+6t-3t^2\bmod (1-t)^3.
$$
From this follows that
$$
p_{\{1,2,3\}}(t)=7-9t+3t^2.
$$
Theorem~\ref{thmmain1} correctly gives
$$
{\mathcal H}(J,t)=\frac{t^3p_{\{1,2,3\}}(t)}{(1-t)^3}=
\frac{7t^3-9t^4+3t^5}{(1-t)^3}.
$$
\end{example}
\begin{example}\label{ex2}
If $P_1,P_2,P_3$ are colinear, then, after a change of coordinates,
we may assume
$$
P_1=(1:0,0),\ P_2=(0:1:0),\ P_3=(1:1:0).
$$
We have
$$
I=I_1\cap I_2\cap I_2=(x_2,x_3)\cap (x_1,x_3)\cap (x_1-x_2,x_3)=
(x_1x_2(x_1-x_2),x_3)\subseteq K[x_1,x_2,x_3]
$$
and
$$
J=I_1I_2I_3=(x_1x_2(x_1-x_2),x_1^2x_3,x_1x_2x_3,x_2^2x_3,x_1x_3^2,x_2x_3^2,x_3^3)\subseteq
K[x_1,x_2,x_3].
$$
We have minimal free resolutions
$$
0\to R[-4]\to R[-1]\oplus R[-3]\to I\to 0,$$
$$
0\to R[-5]^3\to R[-4]^6\to R[-3]^7\to J\to 0.
$$
We get $\operatorname{cd}(I)=1$, $\operatorname{cd}(J)=2$, $\operatorname{reg}(I)=\operatorname{reg}(J)=3$,
$$
{\mathcal H}(I,t)=\frac{t+t^3-t^4}{(1-t)^3}\mbox{ and }
{\mathcal H}(J,t)=\frac{7t^3-9t^4+3t^5}{(1-t)^3}.
$$
The Hilbert functions are
$$
\begin{array}{c||c|c|c|c|c|c}
d & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline\hline
h_{I}(d) & 0 & 1 & 3 & 7 & 12 & 18\\ \hline
h_J(d) & 0 & 0 & 0 & 7 & 12 & 18
\end{array}
$$
We can compute $f(t)$ and it is the same as in Example~\ref{ex1}, (\ref{eq55}).
Now
$$
{\mathcal H}(I,t)=\frac{t+t^3-t^4}{(1-t)^3}=\frac{-2+6t-3t^2}{(1-t)^3}+2+t,
$$
so ${\mathcal H}(I,t)-f(t)$ is indeed a polynomial as in Theorem~\ref{theo1}.
Theorem~\ref{thmmain1} gives the same result for ${\mathcal H}(J,t)$ as in
Example~\ref{eq1}.
\end{example}
\begin{example}\label{ex3}
Let $n=4$ and $m=3$.
Consider the ideals $I_1=(x_2,x_3)$, $I_2=(x_1,x_3)$ and $I_3=(x_1,x_2)$
in the polynomial ring $K[x_1,x_2,x_3,x_4]$ in 4 variables
and let again $I=I_1\cap I_2\cap I_3$.
The ideals $I_1,I_2,I_3$ correspond to 2-dimensional
subspaces $V_1,V_2,V_3$ of the 4-dimensional space $V$ such that
$V_1\cap V_2\cap V_3$ is $1$-dimensional, and $V_1+V_2+V_3=V$.
We get similar free resolutions of $I$ as in Example~\ref{ex1}:
$$
0\to R[-3]^2\to R[-2]^3\to I\to 0,
$$
We have
$$
{\mathcal H}(I,t)=\frac{3t^2-2t^3}{(1-t)^4}.
$$
\end{example}
\begin{example}\label{exlast}
Let $n=4$ and $m=3$.
Consider the ideals $I_1=(x_1,x_3)$, $I_2=(x_2,x_3)$ and $I_3=(x_1-x_2,x_3)$
in the polynomial ring $K[x_1,x_2,x_3,x_4]$ in 4 variables
and let $I=I_1\cap I_2\cap I_3$, $J=I_1I_2I_3$.
The ideals $I_1,I_2,I_3$ correspond to 2-dimensional
subspaces $V_1,V_2,V_3$ of the 4-dimensional space $V$ such that
$V_1\cap V_2\cap V_3$ is $1$-dimensional, and $V_1+V_2+V_3$ is 3-dimensional.
In projective space ${\mathbb P}^3$, we have 3 lines lying an a plane
and going through 1 point.
We get similar free resolutions of $I$ and $J$ as in Example~\ref{ex2}.
$$
0\to R[-4]\to R[-1]\oplus R[-3]\to I\to 0,
$$
$$
0\to R[-5]^3\to R[-4]^9\to R[-3]^7\to J\to 0.
$$
We obtain
$$
{\mathcal H}(I,t)=\frac{t+t^3-t^4}{(1-t)^4}\mbox{ and
}{\mathcal H}(J,t)=\frac{7t^3-9t^4+3t^5}{(1-t)^4}.
$$
Taking the difference gives
$$
{\mathcal H}(I,t)-{\mathcal H}(J,t)=\frac{3t^2+t}{1-t},
$$
which is not a polynomial. Note that Theorem~\ref{theo1} does not apply
because $V_1,V_2,V_3$ are not transversal.
We have
$$
{\mathcal H}(I,t)=\frac{t+t^3-t^4}{(1-t)^4},
$$
but in Example~\ref{ex3} we got
$$
{\mathcal H}(I,t)=\frac{3t^2-2t^3}{(1-t)^4}.
$$
This shows that ${\mathcal H}(I,t)$ is not a combinatorial invariant.
The difference of both Hilbert series is not even a polynomial.
This implies that the Hilbert {\it polynomial\/} $h_I(d)$ is not
a combinatorial invariant either.
\end{example}
\section{Complexes of product ideals and intersection ideals}
\begin{theorem}[See Chapter IV of \cite{Sidman}]\label{theoSidman}
There exists complexes
$$
0\to I\to \bigoplus_{|S|=m-1}I_{S}\to
\bigoplus_{|S|=m-2}I_S\to \cdots \to\bigoplus_{|S|=1}I_S\to R\to 0
$$
and
$$
0\to J\to \bigoplus_{|S|=m-1}J_{S}\to
\bigoplus_{|S|=m-2}J_S\to \cdots \to \bigoplus_{|S|=1}J_S\to R\to 0.
$$
whose homologies are killed by ${\mathfrak a}=\sum_{j=1}^m I_j$.
\end{theorem}
To describe the the maps in the complexes in Theorem~\ref{theoSidman}
it suffices to define maps $I_T\to I_S$ and $J_T\to J_S$
for all subsets $S,T\subseteq \{1,2,\dots,m\}$ with $|T|=|S|+1$.
If $T=\{i_1,i_2,\dots,i_r\}$ with $i_1<i_2<\cdots<i_r$
and $S=\{i_1,i_2,\dots,i_{s-1},i_{s+1},\dots,i_r\}$
then the maps $I_T\to I_S$ and $J_T\to J_S$
in the complexes in Theorem~\ref{theoSidman} are given
by $f\mapsto (-1)^sf$. All other maps are equal to $0$.
\begin{corollary}\label{corHilbertseries}
If $V_{\{1,2,\dots,m\}}=\bigcap_{i=1}^m V_i=(0)$, then
$$
\sum_{S\subseteq \{1,2,\dots,m\}}(-1)^{|S|}{\mathcal H}(I_S,t)
$$
and
$$
\sum_{S\subseteq \{1,2,\dots,m\}}(-1)^{|S|}{\mathcal H}(J_S,t)
$$
are polynomials in $t$.
\end{corollary}
\begin{proof}
The ideal
$$
\sum_{j=1}^m I_j={\mathcal I}\big({\textstyle \bigcap_{i=1}^m V_i}\big)=
{\mathcal I}(\{0\})={\mathfrak m}
$$
is the maximal homogeneous ideal of $R$.
Suppose that
$$
\xymatrix{
0\ar[r]^{\partial_{r+1}} & C_{r}\ar[r]^{\partial_r} &
C_{r-1}\ar[r]^{\partial_{r-1}} &\cdots\ar[r]^{\partial_1} &
C_{0}\ar[r]^{\partial_0} & 0}
$$
is a complex of finitely generated graded $R$-modules.
The $i$-th homology group is
$$H_i=\ker(\partial_{i})/\operatorname{im}(\partial_{i+1}).
$$
We have
$$
\sum_{i=0}^r (-1)^i {\mathcal H}(C_i,t)=\sum_{i=0}^r(-1)^i {\mathcal H}(H_i,t).
$$
If ${\mathfrak m}H_i=0$, then $H_i$ is
finite dimensional, and ${\mathcal H}(H_i,t)$ is a polynomial for all~$i$.
We now apply this to the complexes in Theorem~\ref{theoSidman}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thmmain1}]
By Proposition~\ref{prop43} we can write
$$
{\mathcal H}(J,t)=\frac{\sum_{i=0}^r(-1)^i\beta_it^{i+m}}{(1-t)^n}=\frac{t^mp(t)}{(1-t)^n},
$$
where
\begin{equation}\label{eqBetti}
p(t)=\beta_0-\beta_1t+\cdots+(-1)^r\beta_rt^r
\end{equation}
is a polynomial of degree $r\leq \operatorname{cd}(J)\leq n-1$. Similarly we
can write
$$
{\mathcal H}(J_S,t)=\frac{t^{|S|}p_S(t)}{(1-t)^n}
$$
with
\begin{equation}\label{eqdegbound}
\deg(p_S(t))\leq n-1
\end{equation}
for all $S\subseteq \{1,2,\dots,m\}$.
Let $W=V/V_{S}$ and define $W_X=V_X/V_{S}$ for $X\subseteq S$.
Let $\overline{J}_i\subseteq K[W]\cong K^{[n-n_S]}$ be the vanishing ideal
of $W_i$ for all $i\in S$. Define $\overline{J}_X=\prod_{i\in
X}\overline{J}_i$ for all $X\subseteq S$.
We have
$$
\overline{J}_X\otimes K^{[n_S]}=J_X
$$
inside $K^{[n]}=K^{[n-n_S]}\otimes K^{[n_S]}$.
From this follows that
$$
\frac{t^{|X|}p_X(t)}{(1-t)^n}=
{\mathcal H}(J_X,t)=\frac{{\mathcal H}(\overline{J}_X,t)}{(1-t)^{n_S}}.
$$
In particular, we have
$$
{\mathcal H}(\overline{J}_S,t)=\frac{t^{|S|}p_S(t)}{(1-t)^{n-n_S}}.
$$
From this it follows that $\deg(p_S(t))\leq \dim W-1=n-n_S-1$
(see~(\ref{eqdegbound})).
Since $\bigcap_{i\in S}W_i=0$ in $W$, Corollary~\ref{corHilbertseries} implies
that
$$
\sum_{X\subseteq S}(-1)^{|X|}{\mathcal H}(\overline{J}_X,t)=\sum_{X\subseteq S}
\frac{(-t)^{|X|}p_X(t)}{(1-t)^{n-n_S}}
$$
is a polynomial in $t$.
Multiplying with $(1-t)^{n-n_S}$ gives
$$
\sum_{X\subseteq S}(-t)^{|X|}p_X(t)\equiv 0 \bmod (1-t)^{n-n_S}.
$$
\end{proof}
\begin{proof}[Proof of Theorem~\ref{theo1}]\
{\bf Special case:} Suppose that $c_1+\cdots+c_m<n$.
After a change of coordinates (and using that the
arrangement is transversal) we may identify
$K^{[n]}$
with
$$
K^{[c_1]}\otimes K^{[c_2]}\otimes \cdots \otimes K^{[c_m]}\otimes
K^{[n-c_1-\cdots-c_m]}
$$
and
$I_k$ with
$$
K^{[c_1]}\otimes\cdots\otimes K^{[c_{k-1}]}\otimes {\mathfrak m}_{c_k}\otimes
K^{[c_{k+1}]}\otimes \cdots\otimes K^{[c_{m}]}\otimes
K^{[n-c_1-\cdots-c_m]}
$$
for
$k=1,2,\dots,m$.
Here ${\mathfrak m}_r$ is the homogeneous maximal ideal of $K^{[r]}$.
We get
$$
I=
{\mathfrak m}_{c_1}\otimes {\mathfrak m}_{c_2}\otimes \cdots\otimes
{\mathfrak m}_{c_m}\otimes K^{[n-c_1-\cdots-c_m]}=J.
$$
Note that ${\mathcal H}(K^{[r]},t)=(1-t)^{-r}$ and ${\mathcal H}({\mathfrak m}_r,t)=(1-t)^{-r}-1$
for
all $r$.
Therefore, we get
$$
{\mathcal H}(I,t)=
{\mathcal H}(J,t)=H({\mathfrak m}_{c_1},t){\mathcal H}({\mathfrak m}_{c_2},t)\cdots
{\mathcal H}({\mathfrak m}_{c_m},t){\mathcal H}(K^{[n-c_1-\cdots-c_m]},t)=
$$
$$
=\left(\prod_{i=1}^m \big((1-t)^{-c_i}-1\big)\right) \cdot (1-t)^{c_1+\cdots+c_m-n}=
\frac{\prod_{i=1}^m \big(1-(1-t)^{c_i})}{(1-t)^n}.
$$
\noindent {\bf The general case.} We prove the Theorem by induction on $m$.
The base case $m=1$ follows from the special case above. If $c_1+\cdots+c_m<n$
then we are also done. Let us assume that $c_1+\cdots+c_m \geq n$.
In particular we have $c_{\{1,2,\dots,m\}}=\min(n,\sum_{i=1}^mc_i)=n$, so
$\bigcap_{i=1}^mV_i=(0)$. By Corollary~\ref{corHilbertseries} we have that
$$
\sum_{S\subseteq \{1,2,\dots,m\}}(-1)^{|S|}{\mathcal H}(I_S,t)
$$
is a polynomial.
By induction we have that
$${\mathcal H}_{I_S}(t)-w\prod_{i\in S}v_i
$$
is a polynomial for all strict subsets $S\subset \{1,2,\dots,m\}$,
where $v_i=1-(1-t)^{c_i}$ and $w=(1-t)^{-n}$.
To show that
$$
{\mathcal H}_I(t)-w\prod_{i=1}^mv_i
$$
is a polynomial, it suffices to show that
\begin{equation}\label{eqbigsum}
\sum_{S\subseteq \{1,2,\dots,m\}}(-1)^{|S|}\big({\mathcal H}(I_S,t)-w\prod_{i\in
S}v_i\big)
\end{equation}
is a polynomial.
Now
$$
\sum_{S\subseteq\{1,2,\dots,m\}}(-1)^{|S|}{\mathcal H}(I_S,t)
$$
is a polynomial by Corollary~\ref{corHilbertseries}, and
$$
\sum_{S\subseteq\{1,2,\dots,m\}}(-1)^{|S|}w\prod_{i\in S}v_i=
w\prod_{i=1}^m(1-v_i)=(1-t)^{-n}\prod_{i=1}^m(1-t)^{c_i}=(1-t)^{c_1+\cdots+c_m-n}
$$
is a polynomial as well.
Therefore (\ref{eqbigsum}) is a polynomial.
\end{proof}
\section{Application to Generalized Pricipal Component Analysis}
The object of Principal Component Analysis (PCA) is to
approximate a data set inside a vector space $V$ by a subspace
of smaller dimension. In Generalized Principal Component
Analysis (GPCA) one tries to approximate a data set inside a vector space
$V$ by a {\it union\/} of subspaces spaces (in other words, a {\it subspace arrangement}).
Some applications of GPCA are
motion segmentation (see~\cite{VidalR2002,VidalR2004}),
image segmentation (see~\cite{VidalR2004E}),
image compression (see~\cite{HongW}) and hybrid control
systems (\cite{MaY}). For an overview of GPCA, see~\cite{VMS}.
A first start in GPCA is to decide on the number of subspaces
and the dimensions of the subspaces of the subspaces arrangement
that will approximate the data.
Suppose that $v_1,v_2,\dots,v_r\in V$ are data points.
Here $r$ is fairly large.
Suppose that $v_1,\dots,v_r$ are contained in some
subspace arrangement ${\mathcal A}=V_1\cup \cdots \cup V_m$,
unknown to us.
We would like to recover $n_1,\dots,n_m$ where $n_i=\dim V_i$.
Let ${\mathfrak a}_j$ be the vanishing ideal of the ray through $v_j$.
Then we have that
$$
h({\mathfrak a}_1\cap \cdots\cap {\mathfrak a}_r,d)=
h(I,d)
$$
for small values of $d$, where $I={\mathcal I}({\mathcal A})$ as before.
Now
$$
h({\mathfrak a}_1\cap \cdots\cap {\mathfrak a}_r,d)
$$
can be computed using linear algebra for small values of $d$.
Therefore, we can determine $h(I,d)$ for small values
of $d$. So an important question is,
given $h(I,d)$ for small values of $d$,
can we determine the dimensions $n_1,n_2,\dots,n_m$?
Proposition~\ref{prop18} gives an affirmative answer if the
subspaces are transversal. Of course, in real applications
the data is approximated by the subspaces arrangement, but
not contained in it. In that case, using the PCA method
in $K[V]_d$ one can still can estimate the value
$h({\mathcal I}({\mathcal A}),d)$.
\begin{proposition}\label{prop18}
Assume that the arrangement is transversal.
Suppose that $c_1,\dots,c_m$ are unknown, but we know the values of the
Hilbert polynomial
$$
h_I(d)
$$
for $d=m,m+1,\dots,m+n-1$, then we can recover $c_1,\dots,c_m$.
\end{proposition}
\begin{proof}
Note that $h_I(d)=\widetilde{h}_I(d)$ for $d\geq m$.
Since we know $\widetilde{h}_I(d)$ for $d=m,m+1,\dots,m+n-1$
and $\widetilde{h}_I$ has degree $\leq n-1$, $\widetilde{h}_I$ is
uniquely determined. From this, we can determine ${\mathcal H}(I,t)$, up to a polynomial.
Suppose that
${\mathcal H}(I,t)$
is equal to $a(t)/(1-t)^n$ up to a polynomial.
Let $b(t)$ be the remainder of division of $a(t)$ by $(1-t)^n$.
Then $b(t)$ has degree $<n$ and ${\mathcal H}(I,t)$ is equal to $b(t)/(1-t)^n$
modulo a polynomial. So we have that
$$
b(t)\equiv \prod_{i=1}^d(1-(1-t)^{c_i})\bmod (1-t)^n
$$
and
$$
b(1-t)\equiv \prod_{i=1}^d(1-t^{c_i})\bmod t^n.
$$
Let $r_i$ be the number of the $c_j$'s equal to $i$.
Then we have
$$
b(1-t)\equiv \prod_{i=1}^d(1-t^i)^{r_i}\bmod t^n.
$$
From this we can easily determine $r_1,r_2,\dots,r_{n-1}$ (in that order).
Indeed, if we already know $r_1,\dots,r_s$, then the Taylor series
of
$$
\frac{b(1-t)}{\prod_{i=1}^{s}(1-t^i)^{r_i}}
$$
is
$$
1-r_{s+1}t^{s+1}+\mbox{higher order terms.}
$$
So we find the value of $r_{s+1}$.
\end{proof}
| {
"timestamp": "2005-10-27T05:03:56",
"yymm": "0510",
"arxiv_id": "math/0510584",
"language": "en",
"url": "https://arxiv.org/abs/math/0510584",
"abstract": "The vanishing ideal I of a subspace arrangement is an intersection of linear ideals. We give a formula for the Hilbert polynomial of I if the subspaces meet transversally. We also give a formula for the Hilbert series of a product J of the linear ideals without any assumptions on the subspace arrangement. It turns out that the Hilbert series of J is a combinatorial invariant of the subspace arrangement: it only depends on the intersection lattice and the dimension function. The graded Betti numbers of J are determined by the Hilbert series, so they are combinatorial invariants as well. The results can be applied to Generalized Principal Component Analysis (GPCA), a tool that is useful for computer vision and image processing.",
"subjects": "Commutative Algebra (math.AC); Combinatorics (math.CO)",
"title": "Hilbert series of subspace arrangements",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754470129648,
"lm_q2_score": 0.7185943925708562,
"lm_q1q2_score": 0.7075103952864608
} |
https://arxiv.org/abs/2009.08960 | Polychromatic colorings of 1-regular and 2-regular subgraphs of complete graphs | If $G$ is a graph and $\mathcal{H}$ is a set of subgraphs of $G$, we say that an edge-coloring of $G$ is $\mathcal{H}$-polychromatic if every graph from $\mathcal{H}$ gets all colors present in $G$ on its edges. The $\mathcal{H}$-polychromatic number of $G$, denoted $\operatorname{poly}_\mathcal{H} (G)$, is the largest number of colors in an $\mathcal{H}$-polychromatic coloring. In this paper we determine $\operatorname{poly}_\mathcal{H} (G)$ exactly when $G$ is a complete graph on $n$ vertices, $q$ is a fixed nonnegative integer, and $\mathcal{H}$ is one of three families: the family of all matchings spanning $n-q$ vertices, the family of all $2$-regular graphs spanning at least $n-q$ vertices, and the family of all cycles of length precisely $n-q$. There are connections with an extension of results on Ramsey numbers for cycles in a graph. | \section{Introduction}
\label{sec:introduction}
If $G$ is a graph and $\ensuremath{\mathcal{H}}$ is a set of subgraphs of $G$, we say that an edge-coloring of $G$ is $\ensuremath{\mathcal{H}}$-{\it polychromatic } if every graph from $\ensuremath{\mathcal{H}}$ has all colors present in $G$ on its edges. The $\ensuremath{\mathcal{H}}$-polychromatic number of $G$, denoted $\poly_\ensuremath{\mathcal{H}} (G)$ is the largest number of colors in an $\ensuremath{\mathcal{H}}$-polychromatic coloring.
If an $\ensuremath{\mathcal{H}}$-polychromatic coloring of $G$ uses $\ensuremath{\poly_{\sH}(G)}$ colors, it is called an {\it optimal } $\ensuremath{\mathcal{H}}$-polychromatic coloring of $G$.
Alon \emph{et. al.} \cite{Alon:2007cd} found a lower bound for $\ensuremath{\poly_{\sH}(G)}$ when $G=Q_n$, the $n$-dimensional hypercube, and $\ensuremath{\mathcal{H}}$ is the family of all subgraphs isomorphic to $Q_d$, where $d$ is fixed. Offner \cite{Offner:2008vb} showed this lower bound is, in fact, the exact value for all $d$ and sufficiently large $n$. Bialostocki \cite{Bialostocki:1983wo} showed that if $d=2$, then the polychromatic number is $2$ and that any optimal coloring uses each color about half the time. Goldwasser \emph{et. al.} \cite{group_paper} considered the case when $\ensuremath{\mathcal{H}}$ is the family of all subgraphs isomorphic to $Q_d$ minus an edge or $Q_d$ minus a vertex.
Bollobas \emph{et. al.} \cite{BPRS} treated the case where $G$ is a tree and $\ensuremath{\mathcal{H}}$ is the set of all paths of length at least $r$, where $r$ is fixed. Goddard and Henning \cite{Goddard:2018} considered vertex colorings of graphs such that each open neighborhood gets all colors.
For large $n$, it makes sense to consider $\ensuremath{\poly_{\sH}(K_n)}$ only if $\ensuremath{\mathcal{H}}$ consists of sufficiently large graphs. Indeed, if the graphs from $\ensuremath{\mathcal{H}}$ have at most a fixed number $s$ of vertices, then $\ensuremath{\poly_{\sH}(K_n)} =1$ for sufficiently large $n$ by Ramsey's theorem, since even with only two colors there exists a monochromatic clique with $s$ vertices.
Axenovich \emph{et. al.} \cite{previous_paper} considered the case where $G=K_n$ and $\ensuremath{\mathcal{H}}$ is one of three families of spanning subgraphs: perfect matchings (so $n$ must be even), $2$-regular graphs, and Hamiltonian cycles. They determined $\ensuremath{\poly_{\sH}(K_n)}$ precisely for the first of these and to within a small additive constant for the other two. In this paper, we determine the exact $\ensuremath{\mathcal{H}}$-polychromatic number of $K_n$, where $q$ is a fixed nonnegative integer and $\ensuremath{\mathcal{H}}$ is one of three families of graphs: matchings spanning precisely $n-q$ vertices, $(n-q)$-cycles, and $2$-regular graphs spanning at least $n-q$ vertices (so $q=0$) gives the results of Axenovich \emph{et. al.} in \cite{previous_paper} without the constant.)
This paper is organized as follows. We give a few definitions and state the main results in Section \ref{sec:main_results}. We give some more definitions in Section \ref{Definitions}. The optimal polychromatic colorings in this paper are all based on a type of ordering, and in Section \ref{Lemmas} we state and prove the technical ordering lemmas we will need. In Section \ref{sec:optimal_polychromatic_colorings} we describe precisely the various ordered optimal polychromatic colorings of $K_n$. In Section \ref{sec:proof_of_theorem:one} we prove Theorem \ref{theorem:one}, a result about matchings. In Section \ref{sec:_c_q_polychromatic_numbers_1_and_2} we use some classical results on Ramsey numbers for cycles to take care of polychromatic numbers 1 and 2 for cycles. In Section \ref{sec:proofs_of_theorem_and_lemmas_on_long_cycles} we prove Theorem \ref{theorem:six}, a result about coloring cycles, and use some results on long cycles in the literature to prove a lemma we need. In Section \ref{Theorems} we give the rather long proofs of the three main lemmas we need. In Section \ref{sec:polychromatic_cyclic_ramsey_numbers} we show how our results can be reconstituted in a context which generalizes the classical results on Ramsey numbers of cycles presented in Section \ref{sec:_c_q_polychromatic_numbers_1_and_2}. In Section \ref{sec:conjectures_and_closing_remarks} we state a general conjecture of which most of our results are special cases.
\section{Main Results}
\label{sec:main_results}
We call an edge coloring $\varphi$ of $K_n$ {\it ordered} if there exists an ordering $v_1,v_2,\ldots,v_n$ of $V(K_n)$ such that $\varphi(v_i v_j)=\varphi(v_i v_m)$ for all $1\leq i<j<m\leq n$. Moreover this coloring is {\it simply-ordered} if for all $i<j<m$, $\varphi(v_i v_m)=\varphi(v_j v_m) =a$ implies that $\varphi(v_tv_m)=a$ for all $i\leq t\leq j$. Simply-ordered colorings play a fundamental role in this paper. An ordered edge coloring $\varphi$ induces a vertex coloring $\varphi'$ on $V(K_n)$ called the {\it $\varphi$-inherited coloring}, defined by $\varphi'(v_i)=\varphi(v_i v_m)$ for $i<m\leq n$ and $\varphi'(v_n)=\varphi'(v_{n-1})$. We can represent the induced vertex coloring $\varphi'$ by the sequence $c_1,c_2,\ldots,c_n$ of colors, where $c_i=\varphi'(v_i)$ for each $i$.
A \emph{block} in this sequence is a maximal set of consecutive vertices of the same color. If $\varphi$ is simply-ordered then the vertices in each color class appear in a single block, so in that case, the number of blocks equals the number of colors.
Let $q$ be a fixed nonnegative integer. We define four families of subgraphs of $K_n$ as follows.
\begin{enumerate}
\item $F_q(n)$ is the family of all matchings in $K_n$ spanning precisely $n-q$ vertices (so $n-q$ must be even)
\item $C_q(n)$ is the family of all cycles of length precisely $n-q$
\item $R_q(n)$ is the family of all $2$-regular subgraphs spanning at least $n-q$ vertices.
\item $C_q^*(n)$ is the family of all cycles of length precisely $n-q$ where $n$ and $q$ are such that $\poly_{C_q(n)}(K_n)\geq 3$.
\end{enumerate}
Further, let $\pfq(n)=\poly_{F_q(n)}(K_n)$, $\pcq(n)=\poly_{C_q(n)}(K_n)$, and $\prq(n)=\poly_{R_q(n)}(K_n)$. Our main result is that for $F_q(n), R_q(n)$, and $C^*_q(n)$ there exist optimal polychromatic colorings which are simply ordered, or almost simply ordered (except for $C_q(n)$ if $\pcq(n)=2$). Once we know there exists an optimal simply ordered (or nearly simply ordered) coloring, it is easy to find it and to determine a formula for the polychromatic number. Our main results are the following.
\begin{theorem}\label{theorem:one}
For all integers $q$ and $n$ such that $q$ is nonnegative and $n-q$ is positive and even, there exists an optimal simply-ordered $F_q$-polychromatic coloring of $K_n$.
\end{theorem}
\begin{theorem}\label{theorem:two}
\cite{previous_paper} If $n\geq 3$, then there exist optimal $R_0$-polychromatic and $C_0$-polychromatic colorings of $K_n$ which can be obtained from simply-ordered colorings by recoloring one edge.
\end{theorem}
\begin{theorem}\label{theorem:three}
If $n\geq 4$, then there exist optimal $R_1$-polychromatic and $C_1$-polychromatic colorings of $K_n$ which can be obtained from simply-ordered colorings by recoloring two edges.
\end{theorem}
\begin{theorem}\label{theorem:four}
Let $q\geq 2$ be an integer. If $n\geq q+3$, then there exists an optimal simply-ordered $R_q$-polychromatic coloring of $K_n$. If $n\geq q+4$, then there exists an optimal simply-ordered $C_q$-polychromatic coloring except if $n\in[2q+2,3q+2]$ and $n-q$ is odd.
\end{theorem}
\begin{theorem}\label{theorem:five}
Suppose $q\geq 2$ and $n\geq 6$
\begin{enumerate}[label=\alph*)]
\item If $n-q$ is even then there exists a $C_q$-polychromatic 2-coloring of $K_n$ if and only if $n\geq 3q+3$.
\item If $n-q$ is odd then there exists a $C_q$-polychromatic 2-coloring of $K_n$ if and only if $n\geq 2q+2$.
\end{enumerate}
\end{theorem}
Theorem \ref{theorem:five} follows from results of Bondy and Erd{\H{o}}s \cite{Bondy:1972} and Faudree and Schelp \cite{Faudree:1974}.
The following result, which is needed for the proof of Theorem \ref{theorem:four}, may be of independent interest, so we state it as a theorem:
\begin{theorem}\label{theorem:six}
Let $n$ and $j$ be integers with $4\leq j\leq n$, and let $\varphi$ be an edge-coloring of $K_n$ with at least three colors so that every $j$-cycle gets all colors. Then every cycle of length at least $j$ gets all colors under $\varphi$.
\end{theorem}
The statements about cycles in Theorems \ref{theorem:two}--\ref{theorem:five} can be used to get an extension of the result of Faudree and Schelp \cite{Faudree:1974} in the following manner. Let $s$ and $t$ be integers with $t\geq2, s\geq 3$, and $s\geq t$. The $t$-polychromatic cyclic Ramsey number $\pr_t(s)$ is the smallest integer $N\geq s$ such that in any $t$-coloring of the edges of $K_N$ there exists an $s$-cycle whose edges do not contain all $t$ colors. Note that in the special case $t=2$, this is the classical Ramsey number for cycles, the smallest integer $N$ such that in any $2$-coloring of the edges of $K_N$ there exists a monochromatic $s$-cycle. These numbers were determined for all $s$ by Faudree and Schelp \cite{Faudree:1974}, confirming a conjecture of Bondy and Erd{\H{o}}s \cite{Bondy:1972}.
\begin{theorem}\label{extension}
Let $\pr_t(s)$ be the smallest integer $n\geq s\geq 3$ such that in any $t$-coloring of the edges of $K_n$ there exists an $s$-cycle whose edges do not contain all $t$ colors. If $t\geq 3$,
\[
\pr_t(s)=\begin{cases}
s, & \mathrm{if\ } 3<s\leq 3\cdot 2^{t-3}\\
s+1, & \mathrm{if\ } s\in [3\cdot2^{t-3}+1,5\cdot2^{t-2}-2]\\
s+2, & \mathrm{if\ } s\in [5\cdot 2^{t-2}-1, 5\cdot 2^{t-1}-4]\\
s + \round\left(\frac{s-2}{2^t-2}\right), & \mathrm{if\ } s\geq 5\cdot2^{t-1}-3
\end{cases}
\]
\end{theorem}
\noindent where $\round\left(\frac{s-2}{2^t-2}\right)$ is the closest integer to $\frac{s-2}{2^t - 2}$, rounding up if it is $\frac{1}{2}$ more than an integer.
\section{Definitions}
\label{Definitions}
Recall that if $\varphi$ is an ordered edge coloring of $K_n$ with respect to the ordering $v_1, \ldots, v_n$ of its vertices, we say that $\varphi'$ is the {\it $\varphi$-inherited coloring} (or just {\it inherited coloring}) if it is the vertex coloring of $K_n$ defined by $\varphi'(v_i)=\varphi(v_i v_j)$ for $1\leq i<j\leq n$ and $\varphi'(v_n)=\varphi'(v_{n-1})$. Given an ordering of $V(K_n)$, any vertex coloring $\varphi'$ such that $\varphi'(v_{n-1})=\varphi'(v_n)$ uniquely determines a corresponding ordered coloring. We define a \emph{color class $M_i$ of color $i$} to be the set of all vertices $v$ where $\varphi'(v)=i$. In this paper, we shall always think of the ordered vertices as arranged on a horizontal line with $v_i$ to the left of $v_j$ if $i<j$. We say that an edge $v_iv_m$, $i<m$ goes from $v_i$ to the right and from $v_m$ to the left. If $X$ is a (possibly empty) subset of $V(K_n)$, we say that the edge-coloring $\varphi$ of $K_n$ is
\begin{itemize}
\item \emph{$X$-constant} if for any $v\in X$, $\varphi(v u)=\varphi(v w)$ for all $u, w\in V\setminus X$,
\item \emph{$X$-ordered} if it is $X$-constant and the vertices of $X$ can be ordered $x_1, \ldots, x_m$ such that for each $i = 1,\ldots, m$,
$\varphi(x_i x_p ) = \varphi(x_i x_m) = \varphi(x_i w)$ for all $i<p\leq m$ and all $w\in V\setminus X$,
\end{itemize}
If $Z$ is a nonempty subset of $V(K_n)$ we say $\varphi$ is
\begin{itemize}
\item \emph{$Z$-quasi-ordered} if
\begin{enumerate}
\item $\varphi$ is $Z$-constant
\item Each vertex $v_i$ in $Z$ is incident to precisely $n-2$ edges of one color, which we call the \emph{main color} of $v_i$, and one edge $v_i v_j$ of another color, where $v_j\in Z$. If that other color is $t$, then $v_j$ is incident to precisely $n-2$ edges of color $t$.
\end{enumerate}
\end{itemize}
It is not hard to show that there are only two possibilities for the set $Z$ in a $Z$-quasi-ordered coloring:
\begin{enumerate}
\item $\abs{Z}=3$, the three vertices in $Z$ have different main colors, and there is one edge in $Z$ of each of these colors
\item $\abs{Z}=4$, with two vertices $u,v$ in $Z$ with one main color, say $i$ and two vertices $y,z$ in $Z$ with another main color, say $j$, and $\varphi(uv)=\varphi(uy)=\varphi(vz)=i,\varphi(yz)=\varphi(yv)=\varphi(zu)=j$.
\end{enumerate}
\begin{itemize}
\item \emph{quasi-ordered} if it is $Z$-quasi-ordered and $\varphi$ restricted to $V\setminus Z$ is ordered
\item \emph{quasi-simply ordered} if it is $Z$-quasi-ordered and $\varphi$ restricted to $V\setminus Z$ is simply ordered.
\item \emph{nearly $X$-ordered} if it is $Z$-quasi-ordered and the restriction of $\varphi$ to $V(K_n)\setminus Z$ is $T$-ordered for some (possibly empty) subset $T$ of $V(K_n)\setminus Z$ and $X=Z\cup T$. (If $\varphi$ is nearly $X$-ordered then one or two edges could be recolored to get an $X$-ordered coloring.)
\end{itemize}
It is easy to check that if $\varphi$ is quasi-ordered (quasi-simply-ordered) for some set $Z$ then if $\abs{Z}=3$ one edge can be recolored, and if $\abs{Z}=4$, then two edges can be recolored to get an ordered (simply-ordered) coloring.
The \emph{maximum monochromatic degree} of an edge coloring of $K_n$ is the maximum number of edges of the same color incident with a single vertex. If the maximum monochromatic degree of a coloring is $d$, and the vertex $v$ is incident with $d$ edges of color $t$, and the other $n-1-d$ edges incident with $v$ have color $s$, we say $v$ is a $t$-max vertex and also a \emph{$(t,s)$-max vertex} with \emph{majority color $t$} and \emph{minority color $s$}.
We extend the notion of inherited to quasi-ordered colorings as follows. If $\varphi$ is a quasi-ordered coloring with $\psi$ the ordered coloring which is a restriction of $\varphi$ to $V\setminus Z$, we define $\varphi'$, the $\varphi$-inherited coloring, by letting $\varphi'(x)$ equal the main color of $x$ if $x\in Z$ and $\varphi(y)=\psi'(y)$ if $y\not\in Z$. We think of the vertices in $Z$ preceding those not in $Z$, in the order left to right, and if $\abs{Z}=4$ we list two vertices in $Z$ with the same main color first, then the other two vertices with the same main color.
\section{Ordering Lemmas}
\label{Lemmas}
Let $\varphi$ be an ordered edge coloring of $K_n$ with vertex order $v_1,v_2,\ldots,v_n$, colors $1, \ldots, k$, and $\varphi'$ be the inherited coloring of $V(K_n)$. For each $t\in[k]$ and $j\in[n]$, let $M_t$ be a color class $t$ of $\varphi'$ and $M_t(j)=M_t\cap\{v_1,v_2,\ldots,v_j\}$. The next Lemma is a key structural lemma that characterizes ordered polychromatic colorings.
\begin{lem}\label{orderedPClemma}
Let $\varphi:E(K_n) \to[k]$ be an ordered or quasi-ordered coloring with vertex order $v_1,v_2 \ldots, v_n$.
Then the following statements hold:
\begin{enumerate}[label=(\Roman*)]
\item\label{OPC-1F} $\varphi$ is $F_q$-polychromatic $\Longleftrightarrow$ $\forall t\in [k]$ $\exists j\in[n]$ such that $\abs{M_t(j)} >\frac{j+q}{2}$,
\item\label{OPC-HC} $\varphi$ is $C_q$-polychromatic $\Longleftrightarrow$ $\forall t\in [k]$ either
\begin{enumerate}[label=(\alph*)]
\item\label{HCone} $\exists j\in[q+1,n-1]$ such that $\abs{M_t(j)} \geq \frac{j+q}{2}$ or
\item\label{HCtwo} $q=0$, $\varphi$ is $Z$-quasi-ordered with $\abs{Z}=3$ and $t$ is the color of some edge in $Z$ or
\item\label{HCthree} $q=1$, $\varphi$ is $Z$-quasi-ordered with $\abs{Z}=4$ and $t$ is the color of some edge in $Z$.
\end{enumerate}
\item\label{OPC-2F} $\varphi$ is $R_q$-polychromatic $\Longleftrightarrow$ $\forall t\in [k]$ either
\begin{enumerate}
\item\label{2Fone} $\exists j\in[n]$ such that
\begin{enumerate}[label=(\roman*)]
\item\label{one} $\abs{M_t(j)}>\frac{j+q}{2}$ or
\item\label{two} $\abs{M_t(j)}=\frac{j+q}{2}$ and $j\in\{2+q,n-2\}$ or
\item\label{three} $\abs{M_t(j)}=\frac{j+q}{2}$ and $\abs{M_t(j+2)}=\frac{j+q+2}{2}$ where $j\in[4+q,n-3].$
\end{enumerate}
\item\label{2Ftwo} $q=0$, $\varphi$ is $Z$-quasi-ordered and $t$ is the color of some edge in $Z$
\item\label{2Fthree} $q=1$, $\varphi$ is $Z$-quasi-ordered with $\abs{Z}=3$ and $t$ is the color of some edge in $Z$
\end{enumerate}
\end{enumerate}
\end{lem}
\begin{proof}
Note that to prove the lemma, it is sufficient to consider an arbitrary color $t$ and show for $\ensuremath{\mathcal{H}} \in \{ F_q, C_q, R_q \}$ and for each $H\in \ensuremath{\mathcal{H}}$, that the given respective conditions are equivalent to $H$ containing an edge of color $t$.
\emph{\ref{OPC-1F}}
Let $j$ be an index such that $\abs{M_t(j)}=m_j>(j+q)/2$ and let $H$ be a $1$-factor. Let $x_1, \ldots, x_{m_j}$ be the vertices of $M_t$ in order and let $y_1,\ldots y_{j-{m_j}}$ be the other vertices of $\{v_1,v_2,\ldots,v_{j}\}$ in order. Since $j-m_j<\frac{j-q}{2}$ and $m_j-q>\frac{j-q}{2}$, then at least one edge of $H$ with an endpoint in $M_t(j)$ must go to the right, and thus, have color $t$.
On the other hand, by way of contradiction, assume that for each $j\in[n]$, $\abs{M_t(j)}\leq (j+q)/2$. Letting $m=\abs{M_t}$, we have $m\leq (n+q)/2$. Consider a $1$-factor that spans all vertices except for $q$ vertices in $M_t$. Let $x_1, \ldots, x_{m-q}$ be the $m-q$ vertices remaining from $M_t$ in order and let $y_1,\ldots y_{n-m}$, be the vertices outside of $M_t$ in order. Note that since $m\leq (j+q)/2$, it follows that $n-m\geq m-q$ since if $n-m<m-q$ then $n<2m-q$ and so $j>n$ which is impossible. Now, let $H$ consist of the edges $x_1y_1, x_2y_2, \ldots, x_{m-q}y_{m-q}$ and a perfect matching on $\{y_{m-q+1}, \ldots, y_{n-m}\}$ (if this set is non-empty). We will show that $y_i$ precedes $x_i$ in the order $v_1,v_2,\ldots,v_n$ for each $i\in[m-q]$, so $H$ has no edge of color $t$.
By way of contradiction, assume $x_i$ precedes $y_i$ for some $i\in[m-q]$. Letting $j=2i-1+q$, $y_i$ cannot be among the first $j$ vertices in the order $v_1,v_2,\ldots,v_n$, because if it were there would be at least $i+q$ vertices of color $t$ among these $j$ vertices, so a total of at least $2i+q>j$ vertices. Hence
\[
\frac{j+q}{2}=\frac{2i+2q-1}{2}<i+q\leq \abs{M_t(j)}\leq \frac{j+q}{2}
\]
which is impossible. Hence $y_i$ precedes $x_i$ for each $i$ and $\varphi$ is not $F_q$ polychromatic.
\emph{\ref{OPC-HC}}
If $t$ is a color such that \ref{HCone} holds with strict inequality, the argument in \ref{OPC-1F} shows there is an edge of $H$ with color $t$. If $\abs{M_t(j)}=\frac{j+q}{2}$ for some $j\in[q+1,n-1]$ and every edge in $H$ incident to a vertex in $M_t(j)$ goes to the left then, since each of these edges has its other vertex not in $M_t(j)$, $H$ contains $\frac{j-q}{2}$ vertices in $M_t(j)$ and the same number not in $M_t(j)$. If $\frac{j-q}{2}=1$, then the vertex in $M_t(j)$ is incident with at least one edge which goes to the right, and if $\frac{j-q}{2}\geq 2$ then $H$ contains a $2$-regular subgraph, which is impossible because an $n-q$ cycle can't have a $2$-regular subgraph on less than $n-q$ vertices.
If $t$ is such that \ref{HCtwo} holds, then note that $t$ must be the main color of a vertex in $Z$ and that the cycle must contain 2 edges incident with each vertex in $Z$. Any choice of these edges will contain an edge of color $t$ since only one edge incident with each vertex in $Z$ is not the main color of that vertex.
If $t$ is such that \ref{HCthree} holds, then note that $t$ must be the main color of a vertex in $Z$ and any cycle on $n-1$ vertices must contain 2 edges incident with at least three of the four vertices in $Z$. Any choice of these edges will contain an edge of color $t$ since only one edge incident with each vertex in $Z$ is not the main color of that vertex.
On the other hand, suppose that for each $j\in [q+1,n-1]$, $\abs{M_t(j)}=m<\frac{j+q}{2}$ and $\varphi$ is not $Z$-quasi-ordered with $t$ a main color. In particular, when $j=n-2$, we have that $\abs{M_t(j)}=m<\frac{n+q}{2}-1$. Consider a cycle that spans all vertices except for $q$ vertices in $M_t$. Let $x_1,\ldots,x_{m-q}$ be the other $m-q$ vertices in $M_t$ in order and $y_1,\ldots,y_{n-m}$ be the vertices outside of $M_t$ in order. Note that if $m<\frac{j+q}{2}$, then $n-m>m-q$ since $n-m\leq m-q \implies j>n$ which is impossible. Consider the cycle $y_1 x_1 y_2 x_2 \cdots y_{m-q} x_{m-q} y_{m-q+1}\cdots y_{n-m} y_1$. Suppose $y_i$ is to the right of $x_i$ for some $i$. Then at most $i$ of the first $j=2i+q$ vertices are not in $M_t(j)$, so $\abs{M_t(j)}\geq i+q=\frac{j+q}{2}$, which is impossible. Hence $y_i$ and $y_{i+1}$ are to the left of $x_i$ for each $1\leq i\leq m$, all edges of $H$ incident to $M_t$ go to the left, and thus are not of color $t$.
\begin{obs}
If $H$ is a $2$-regular subgraph that has no edge of color $t$, and $M$ is any subset of $M_t$, then all edges of $H$ incident to $M$ go to the left, so at most half the vertices in $H$ are in $M_t$ and if $\abs{M_t(j)}=\frac{j+q}{2}$, then of the first $j$ vertices, precisely $j-q$ are in $H$, precisely half of these in $M_t$, and if $j-q\geq 4$ then these $j-q$ vertices induce a 2-regular subgraph of $H$.
\end{obs}
\emph{\ref{OPC-2F}}
Let $j$ be an index such that \ref{2Fone}~\ref{one}, \ref{two}, or \ref{three} holds. Assume first that \ref{one} holds, i.e., that $\abs{M_t(j)}>\frac{j+q}{2}$ and let $H$ be a $2$-factor. Then the argument given in \ref{OPC-1F} shows that at least one edge of $H$ with an endpoint in $M_t(j)$ must go to the right, and thus, have color $t$. Assume that \ref{two} holds. If $j=2+q$, then $M_t$ contains $q+1$ of the first $q+2$ vertices, so $H$ contains a vertex in $M_t$ which has an edge that goes to the right, so there is an edge of color $t$ in $H$. If $j=n-2$ and $H$ has no edges of color $t$, then (by the previous observation) the subgraph of $H$ induced by $[n-2]$ is a $2$-factor. Since the remaining two vertices do not form a cycle, $H$ is not a $2$-factor, a contradiction. Finally, assume that \ref{three} holds. If $H$ does not have an edge of color $t$, then by the previous observation, $H$ has a 2-regular subgraph spanning $j-q+2$ vertices, which has a 2-regular subgraph spanning $j-q$ vertices, which is impossible.
If \ref{2Ftwo} or \ref{2Fthree} holds, by the argument for \ref{OPC-HC}, $H$ has an edge of color $t$.
On the other hand, suppose that none of \ref{2Fone}, \ref{2Ftwo}, or \ref{2Fthree} hold. We shall construct a $2$-factor that does not have an edge of color $t$. If $\abs{M_t(j)}<\frac{j+q}{2}$ for each $j\in[q+1,n-1]$, then there is a cycle with no color $t$ edge as described in \ref{OPC-HC}. If not, let $i_1,i_2,\ldots,i_k$ be the values of $j$ in $[4+q,n-3]$ for which $\abs{M_t(j)}=\frac{j+q}{2}$. Since \ref{2Fone}\ref{three} is not satisfied, $i_{q+1}-i_q$ is at least 4 and even for $q=1,2,\ldots,k-1$. As before, suppose there are $m$ vertices of color $t$. Let $x_1,x_2,\ldots,x_{m-q}$ be the last $m-q$ of these, in order, and let $y_1,y_2,\ldots,y_{n-m}$ be the other vertices, in order. Note that since $m\leq \frac{n+q}{2}$ we have $m-q\leq\frac{n-q}{2}$ and $n-m\geq\frac{n-q}{2}$. For each $q$ in $[1,k-1]$, moving left to right within the interval $[i_q+1,i_{q+1}]$, there are always more $y$'s than $x$'s (except an equal number of each at the end of the interval), since otherwise there would have been another value of $j$ between $i_q$ and $i_{q+1}$ where $\abs{M_t(j)}=\frac{j+q}{2}$. Form an $(i_{q+1}-i_q)$-cycle by alternately taking $y$'s and $x$'s, starting with the $y$ with the smallest subscript. Also form an $i_1-q$ cycle using the first $\frac{i_1-q}{2}$ $y$'s and the same number of $x$'s, and an $n-i_k$ cycle at the end, first alternating the $y$'s and $x$'s, putting any excess $y$'s at the end.
\end{proof}
\begin{lem}\label{O2SO}
Let $\ensuremath{\mathcal{H}} \in \{F_q, R_q, C_q\}$.
If there exists an ordered (quasi-ordered) $\ensuremath{\mathcal{H}}$-polychromatic coloring of $K_n$ with $k$ colors, then there exists one which is simply-ordered (quasi-simply-ordered) with $k$ colors.
\begin{proof}
Let $V(K_n) =[n]$ with the natural order. If $c'$ is a coloring of $[n]$, a {\it block} of $c'$ is a maximal interval of integers from $[n]$ which all have the same color. So a simply-ordered $k$-polychromatic coloring has precisely $k$ blocks. We define a {\it block shift operation} as follows. Assume that $t\in[k]$ is a color for which there are at least $2$ blocks. Let $j(t)=j$ be the smallest integer so that $M_t(j)>(j+q)/2$ if such exists. If there is a block $[m,s]$ in $M_t$ where $m>j$, delete this block, then take the color of the last vertex in the remaining sequence, and add $s-m+1$ more vertices with this color at the end of the sequence. If each block of color $t$ has its smallest element less than or equal to $j$, consider the block $B$ of color $t$ that contains $j$ and consider another block $B_1$ of color $t$ that is strictly to the left of $B$. Form a new coloring by ``moving'' $B_1$ next to $B$. We see that the resulting coloring has at least one less block.
Let $c$ be a ordered (quasi-ordered) $F_q$-polychromatic coloring of $K_n$ on vertex set $[n]$ with $k$ colors such that the inherited vertex coloring $c'$ has the smallest possible number of blocks. Assume that color $t$ has at least $2$ blocks. Let $j(t)=j$ be the smallest integer so that $M_t(j)>(j+q)/2$.
Such $j$ exists by Lemma \ref{orderedPClemma}$\ref{OPC-1F}$, and the color of $j$ is $t$. Apply the block shifting operation. The condition from part $\ref{OPC-1F}$ of Lemma \ref{orderedPClemma} is still valid for all color classes, so the new coloring is $F_q$-polychromatic using $k$ colors. This contradicts the choice of $c$ having the smallest number of blocks.
If $c$ is an ordered (quasi-ordered) $C_q$-polychromatic coloring of $K_n$, an argument very similar to the one above shows if \ref{OPC-HC}\ref{HCone}, \ref{HCtwo}, or \ref{HCthree} hold, there exists a simply-ordered (quasi-simply-ordered) coloring that uses the same number of colors and that is $C_q$-polychromatic.
Finally, let $c$ be an ordered ($X$-quasi-ordered) $R_q$-polychromatic coloring of $K_n$ on vertex set $[n]$ with $k$ colors such that the inherited vertex coloring $c'$ has the minimum possible number of blocks. Assume that $t\in[k]$ is a color for which there are at least $2$ blocks. If \ref{2Ftwo} or \ref{2Fthree} hold, then the block shifting operation gives a coloring that is still $R_q$-polychromatic with the same number of colors and fewer blocks.
Thus, by Lemma \ref{orderedPClemma}\ref{OPC-2F} there exists $j$ such that
\begin{enumerate}[label=($\arabic*$)]
\item\label{first} $\abs{M_t(j)}>(j+q)/2$ or
\item\label{seconda} $\abs{M_t(2+q)}=1+q$ or
\item\label{secondb} $\abs{M_t(n-2)}= (n+q-2)/2$ or
\item\label{secondc} $\abs{M_t(n-1)}= (n+q-1)/2$ or
\item\label{third} $\abs{M_t(j)} = (j+q)/2$ and $\abs{M_t(j+2)} = (j+q+2)/2$ and $4+q\leq j\leq n-3.$
\end{enumerate}
If \ref{first} holds, then we apply the block shifting operation and observe, as in the case of $F_q$, that the resulting coloring is still $R_q$-polychromatic with the same number of colors and fewer blocks. The case when \ref{seconda} applies is similar.
Assume neither \ref{first} nor \ref{seconda} holds. If \ref{secondb} holds then, since $c'(v_{n-1})=c'(v_n)$, neither $v_{n-1}$ nor $v_n$ can have color $t$. Hence there is another block of color $t$ vertices to the left of the one containing $v_{n-2}$, so we can do a block shift operation ot reduce the number of blocks, a contradiction.
The same argument works if \ref{secondc} holds.
Finally, assume that none of \ref{first}--\ref{secondc} holds, but \ref{third} holds. This implies that $c'(j)=c'(j+2)=t$ and $c'(j+1)=u\neq t$. Now define $c''$ by $c''(i)=c'(i)$ if $i\not\in\{j+1, j+2\}, c''(j+1)=t$, and $c''(j+2)=u$. Clearly $c''$ has at least one fewer block than $c'$. Since $j+q+1$ is odd, the only situation where $c''$ would not be $R_q$-polychromatic is if $M_u(j+1)>\frac{j+q+1}{2}$. However, then $\abs{M_u(j-1)}=\abs{M_u(j+1)}-1>\frac{j+q-1}{2}$, so $c''$ is $R_q$-polychromatic after all.
\end{proof}
\end{lem}
\section{Optimal Polychromatic Colorings}
\label{sec:optimal_polychromatic_colorings}
The seven following colorings are all optimal $F_q,R_q$, or $C_q$ polychromatic colorings for various values of $q$ and $n$. Each of them is simply-ordered or quasi-simply-ordered. We describe the color classes for each, and give a formula for the polychromatic number $k$ in terms of $q$ and $n$.
\subsection{$F_q$-polychromatic coloring $\pfq$ of $E(K_n)$ (even $n-q\geq 2$).}
\label{subsec:_k_1f_polychromatic}
Let $q$ be nonnegative and $n-q$ positive and even with $k$ a positive integer such that
\begin{equation}\label{n_eq_F}
(q+1)(2^k-1)\leq n<(q+1)(2^{k+1}-1).
\end{equation}
Let $\varphi_{F_q}$ be the simply-ordered edge $k$-coloring with colors $1,2,\ldots,k$ and inherited vertex $k$ coloring of $\varphi'_{F_q}$ with successive color classes $M_1, M_2,\ldots, M_k$, moving left to right such that $\abs{M_i}=2^{i-1}(q+1)$ if $i<k$ and $\abs{M_k}=n-\sum_{i=1}^{k-1}\abs{M_i}=n-(2^{k-1}-1)(q+1)$. We have $k\leq \log_2\frac{n+q+1}{q+1}<k+1$ so $\pfq=k=\floor{\log_2\frac{n+q+1}{q+1}}$.
\subsection{$R_q$-polychromatic coloring $\varphi_{R_q}$ ($q\geq 2$)}
\label{sec:_prq_polychromatic_qgeq_2_}
If $q\geq 2$, $n\geq q+3$ and $n$ and $k$ are such that \eqref{n_eq_F} is satisfied, we let $\varphi_{R_q}=\varphi_{F_q}$ (same color classes), giving us the same formula for $k$ in terms of $n$.
\subsection{$C_q$-polychromatic coloring $\varphi_{C_q}$, ($q\geq 2$).}
\label{subsec:_k_hc_polychromatic_coloring}
If $q\geq 2$, $n\geq q+3$ and
\begin{equation}\label{n_eq_C}
(2^{k}-1)q+2^{k-1}<n\leq (2^{k+1}-1)q+2^k
\end{equation}
let $\varphi_{C_q}$ be the simply-ordered edge $k$-coloring with colors $1,2,\ldots,k$ and inherited vertex $k$ coloring $\varphi'_{C_q}$ with successive color classes $M_1,M_2,\ldots,M_k$ of sizes given by:
\begin{align*}
\abs{M_1}&=q+1\\
\abs{M_i}&=2^{i-1}q+2^{i-2} \rm{\ if\ }i\in[2,k-1]\\
\abs{M_k}&=n-\sum_{i=1}^{k-1}\abs{M_i}=n-2^{k-1}q-2^{k-2}
\end{align*}
From equation \eqref{n_eq_C} we get $\pcq = k=\floor{\log_2\frac{2(n+q-1)}{2q+1}}$.
\subsection{$R_0$-polychromatic coloring $\varphi_{R_0}$ ($q=0$).}
\label{subsec:_k_2f_polychromatic_coloring}
If $n\geq 3$ and $2^{k-1}-1\leq n<2^{k-1}$ let $\varphi_{R_0}$ be the quasi-simply-ordered coloring with $\abs{X}=3$ and color class sizes $\abs{M_1}=\abs{M_2}=1$ and $\abs{M_3}=n-2$ if $3\leq n\leq 6$, and if $n\geq 7$:
\begin{align*}
\abs{M_1}&=\abs{M_2}=\abs{M_3}=1\\
\abs{M_i}&=2^{i-2} \rm{\ if\ }i\in[4,k-1]\\
\abs{M_k}&=n-\sum_{i=1}^k-1\abs{M_i}=n-2^{k-2}+1
\end{align*}
From this, we get $\operatorname{P_{R_0}}=k=1+\floor{\log_2(n+1)}$ where $n\geq 3$.
\subsection{$C_0$-polychromatic coloring $\varphi_{C_0}$ ($q=0$)}
\label{sec:_c_0_polychromatic_q_0_}
If $n\geq 3$ and $3\cdot 2^{k-3}<n\leq 3\cdot 2^{k-2}$ let $\varphi_{C_0}$ be the quasi-simply-ordered coloring with $\abs{X}=3$ and color class sizes $\abs{M_1}=\abs{M_2}=1$ and $\abs{M_3}=n-2$ if $3\leq n\leq 6$, and if $n\geq 7$:
\begin{align*}
\abs{M_1}&=\abs{M_2}=\abs{M_3}=1\\
\abs{M_i}&=3\cdot 2^{i-4} \rm{\ if\ }i\in[4,k-1]\\
\abs{M_k}&=n-\sum_{i=1}^{k-1}\abs{M_i}=n-3\cdot 2^{k-4}
\end{align*}
From this, we get $\operatorname{P_{C_0}}=k=\floor{\log_2\frac{8(n-1)}{3}}$ where $n\geq 4$.
\subsection{$R_1$-polychromatic colring $\varphi_{R_1}$ ($q=1$)}
\label{sec:_r_1_polychromatic_q_1_}
If $n\geq 4$ and $3\cdot 2^{k-1}-2\leq n<3\cdot 2^k-2$ let $\varphi_{R_1}$ be the quasi-simply-ordered coloring with $\abs{X}=4$ and color class sizes $\abs{M_1}=2$ and $\abs{M_2}=n-2$ if $4\leq n\leq 9$, and if $n\geq 10$:
\begin{align*}
\abs{M_1}&=\abs{M-2}=2\\
\abs{M_i}&=3\cdot 2^{i-2} \rm{\ if\ }i\in[3,k-1]\\
\abs{M_k}&=n-\sum_{i=1}^{k-1}\abs{M_i}=n-3\cdot2^{k-2}+2
\end{align*}
From this, we get $\operatorname{P_{R_1}}=k=\floor{\log_2\frac{2(n+2)}{3}}$ where $n\geq 4$.
\subsection{$C_1$-polychromatic coloring $\varphi_{C_1}$ ($q=1$)}
\label{sec:_c_1_polychromatic_q_1_}
If $n\geq 4$ and $5\cdot 2^{k-2}\leq n< 5\cdot 2^{k-1}$ let $\varphi_{C_1}$ be the quasi-simply-ordered coloring with $\abs{X}=4$ and color class sizes $\abs{M_1}=\abs{M_2}=2$ and $\abs{M_3}=n-4$ if $4\leq n\leq 9$ and change every edge of color $3$ to color $2$, and if $n\geq 10$:
\begin{align*}
\abs{M_1}&=\abs{M_2}=2\\
\abs{M_i}&=5\cdot 2^{i-3} \rm{\ if\ }i\in[3,k-1]\\
\abs{M_k}&=n-\sum_{i=1}^{k-1}\abs{M_i}=n-5\cdot 2^{k-3}+1
\end{align*}
From this, we get $\operatorname{P_{C_1}}=k=\floor{\log_2\frac{4n}{5}}$ where $n\geq 4$.
\section{Proof of Theorem \ref{theorem:one} on Matchings}
\label{sec:proof_of_theorem:one}
We prove Theorem \ref{theorem:one}. This proof is similar to the proof of Theorem 1 in \cite{previous_paper}. Let $k=\pfq(n)$ be the polychromatic number for $1$-factors spanning $n-q$ vertices in $G=K_n=(V,E)$. Among all $F_q$-polychromatic colorings of $K_n$ with $k$ colors we choose ones that are $X$-ordered for a subset $X$ (possibly empty) of the largest possible size, and, of these, choose a coloring $c$ whose restriction to $V\setminus X$ has the largest possible maximum monochromatic degree. Let $v$ be a vertex of maximum monochromatic degree, $r$, in $c$ restricted to $G[V\setminus X]$, let the majority color on the edges incident to $v$ in $V\setminus X$ be color $1$. By the maximality of $\abs{X}$, there is a vertex $u$ in $V\setminus X$ such that $c(uv)\neq 1$. Assume $c(uv)=2$. If every $1$-factor spanning $n-q$ vertices containing $uv$ had another edge of color $2$, then the color of $uv$ could be changed to $1$, resulting in a $F_q$-polychromatic coloring where $v$ has a larger maximum monochromatic degree in $V\setminus X$, a contradiction. Hence, there is a $1$-factor $F$ spanning $n-q$ vertices in which $uv$ is the only edge with color $2$ in $c$.
Let $c(vy_i)=1$, $y_i\in V\setminus X$, $i=1, \ldots, r$. Note that for each $k\in[r]$, $y_k$ must be in $F$. If not, then $F-{uv}+{vy_k}$ is a $1$-factor spanning $n-q$ vertices with no edge of color $2$ (since $uv$ was the unique edge of color $2$ in $F$ and $vy_k$ is color $1$). For each $i\in[r]$, let $y_i w_i$ be the edge of $F$ containing $y_i$ (perhaps $w_i=y_j$ for some $j\neq i$). See Figure \ref{fig:1Fswitch}. We can get a different $1$-factor $F_i$ by replacing the edges $uv$ and $y_i w_i$ in $F$ with edges $v y_i$ and $u w_i$. Since $F_i$ must have an edge of color $2$ and $c(v y_i)=1$, we must have $c(u w_i)=2$ for each $i\in[r]$.
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[every text node part/.style={align=center},scale=1,inner sep=1.75mm]
\node[circle,ultra thick,draw=black,fill=white] (v) at (5,4) {$v$};
\node[circle,ultra thick,draw=black,fill=white] (u) at (7,4) {$u$};
\node[circle,ultra thick,draw=black,fill=white] (y1) at (1,2) {$y_1$};
\node[circle,ultra thick,draw=black,fill=white] (y2) at (3,2) {$y_2$};
\node[circle,ultra thick,draw=black,fill=white,inner sep=.3mm] (y3) at (5,2) {$y_3$\\$w_4$};
\node[circle,ultra thick,draw=black,fill=white,inner sep=.3mm] (y4) at (7,2) {$y_4$\\$w_3$};
\node (dots1) at (8,2) {\large $\ldots$};
\node[circle,ultra thick,draw=black,fill=white] (yr) at (9,2) {$y_r$};
\node[circle,ultra thick,draw=black,fill=white] (w1) at (1,0) {$w_1$};
\node[circle,ultra thick,draw=black,fill=white] (w2) at (3,0) {$w_2$};
\node (dots2) at (8,0) {\large $\ldots$};
\node[circle,ultra thick,draw=black,fill=white] (wr) at (9,0) {$w_r$};
\draw[ultra thick, red] (v) -- node[above] {2} (u);
\draw[ultra thick, blue,dotted] (v) -- node[above left] {1} (y1);
\draw[ultra thick, blue,dotted] (v) -- node[right] {1} (y2);
\draw[ultra thick, blue,dotted] (v) -- node[right] {1} (y3);
\draw[ultra thick, blue,dotted] (v) -- node[right] {1} (y4);
\draw[ultra thick, blue,dotted] (v) -- node[above right] {1} (yr);
\draw[ultra thick, black] (y3) -- (y4);
\foreach \i in {1,2,r} {
\draw[ultra thick, black] (y\i) -- (w\i);
}
\end{tikzpicture}
\caption{Maximum polychromatic degree in an $F_q$-polychromatic coloring}
\label{fig:1Fswitch}
\end{figure}
If $w_i\in X$ for some $i$ then, since $c$ is $X$-constant, $c(w_iy_i) = c(w_iu) =2$, so $y_i w_i$ and $uv$ are two edges of color $2$ in $F$, a contradiction. So, $w_i\in V\setminus X$.
Thus $c(u v)=c(uw_1) = \cdots =c(uw_r) = 2$, and the monochromatic degree of $u$ in $V\setminus X$ is at least $r+1$, larger than that of $v$, a contradiction.
Hence $X=V$, $c$ is ordered, and, by Lemma \ref{O2SO}, there exists a simply-ordered $F_1$-polychromatic coloring $c_s$ with $k$ colors. By Lemma \ref{orderedPClemma}$\ref{OPC-1F}$, if $M_1,M_2,\ldots,M_k$ are the successive color classes, moving left to right, of the inherited vertex coloring $c'_s$, then $\abs{M_t}\geq 2^{t-1}(q+1)$ for $t=1,2,\ldots,k$. Since this inequality holds with equality for $t=1,2,\ldots,k-1$ for the inherited vertex-coloring $\pfq$, the number of color classes of $c_s$ cannot be greater than that of $\pfq$, so $k\leq \floor{\log_2 \frac{n+q+1}{q+1}}$.
\qed
\section{$C_q$-polychromatic Numbers 1 and 2}
\label{sec:_c_q_polychromatic_numbers_1_and_2}
The following theorem is a special case of a theorem of Faudree and Schelp.
\begin{theorem}\cite{Faudree:1974}\label{FS}
Let $s\geq 5$ be an integer and let $c(s)$ denote the smallest integer $n$ such that in any 2-coloring of the edges of $K_n$ there is a monochromatic $s$-cycle. Then $c(s)=2s-1$ if $s$ is odd and $c(s)=\frac{3}{2}s-1$ if $s$ is even.
\end{theorem}
Faudree and Schelp actually determined all values of $c(r,s)$, the smallest integer $n$ such that in any coloring of the edges of $K_n$ with red and blue, there is either a red $r$-cycle or a blue $s$-cycle. Their theorem extended partial results and confirmed conjectures of Bondy and Erd\H{o}s \cite{Bondy:1972} and Chartrand and Schuster \cite{ChartrandSchuster} (who showed $c(3)=c(4)=6$). The coloring of $K_{2s-2}$ to prove the lower bound for $s$ odd is a copy of $K_{s-1,s-1}$ of red edges with all other edges blue, while for $s$ even it's a red $K_{\frac{s}{2}-1,s-1}$ with all other edges blue.
\begin{proof}[Proof of Theorem \ref{theorem:five}]
By Theorem \ref{FS}, if $s\geq 5$ is odd then there is a polychromatic 2-coloring of $K_n$ if and only if $n\leq 2s-2=2(n-q)-2$, so if and only if $n\geq 2q+2$. If $s\geq 5$ is even then there is a polychromatic 2-coloring if and only if $n\leq \frac{3}{2}s-2=\frac{3}{2}(n-q)-2$, so if and only if $n\geq 3q+4$. Hence if $n\in[2q+2,3q+2]$ then $\pcq(n)=1$ if $n-q$ is even and $\pcq(n)=2$ if $n-q$ is odd. The smallest value of $n$ for which there is a simply ordered $C_q$-polychromatic 2-coloring is $n=3q+3$, so there does not exist one if $n-q$ is odd and $n\leq 3q+2$.
\end{proof}
We remark that the only values for $q\geq2$ and $n$ such that there is no optimal simply-ordered $C_q(n)$-polychromatic coloring of $K_n$ are the ones given in Theorem \ref{theorem:five} ($n\in[2q+2,3q+2]$ and $n-q$ is odd), and $q=2$, $n=5$ (two monochromatic $C_5$'s is a coloring of $K_5$ with no monochromatic $C_3$'s).
\section{Proofs of Theorem \ref{theorem:six} and Lemmas on Long Cycles}
\label{sec:proofs_of_theorem_and_lemmas_on_long_cycles}
We will need some results on the existence of long cycles in bipartite graphs.
\begin{theorem}[Jackson \cite{Jackson:1985}]\label{Jackson}
Let $G$ be a connected bipartite graph with bipartition $V(G)=S\cup T$ where $\abs{S}=s$, $\abs{T}=t$, and $s\leq t$. Let $m$ be the minimum degree of a vertex in $S$ and $p$ be the minimum degree of a vertex in $T$. Then $G$ has a cycle with length at least $\min\{2s,2(m+p-1)\}$.
\end{theorem}
\begin{theorem}[Rahman, Kaykobad, Kaykobad \cite{Rahman:2013}]\label{Rahman}
Let $G$ be a connected $m$-regular bipartite graph with $4m$ vertices. Then $G$ has a Hamiltonian cycle.
\end{theorem}
\begin{lem}\label{disjoint_union}
Let $B$ be a bipartite graph with vertex bipartition $S,T$ where $\abs{S}=s$, $\abs{T}=t$, and $s\leq t$. Suppose each vertex in $T$ has degree $m$ and each vertex in $S$ has degree $t-m$. Then $B$ has a $2s$-cycle unless $s=t=2m$ and $B$ is the disjoint union of two copies of $K_{m,m}$.
\begin{proof}
Suppose $s<t$. Summing degrees in $S$ and $T$ gives us $s(t-m)=tm$, so
\[
m=\frac{st}{s+t}>\frac{st}{2t}=\frac{s}{2}
\]
so $B$ is connected. By Theorem \ref{Jackson}, $B$ has a $2s$-cycle, since $2[m+(t-m)-1]=2(t-1)\geq 2s$. If $s=t$, then $B$ is an $m$-regular graph with $4m$ vertices. If $B$ is connected then, by Theorem \ref{Rahman}, it has a $2s$-cycle. If $B$ is not connected then clearly it is the disjoint union of two copies of $K_{m,m}$.
\end{proof}
\end{lem}
We say that a cycle $H'$ of length $n-q$ is obtained from a cycle $H$ of length $n-q$ by a {\it twist} of disjoint edges $e_1$ and $e_2$ of $H$ if $E(H)\setminus \{e_1, e_2\} \subseteq E(H')$, i.e. we remove $e_1, e_2$ from $H$ and introduce two new edges to make the resulting graph a cycle. Note that the choice of the two edges to add is unique (due to connectedness), however, both choices would result in a $2$-regular subgraph.
One main difference between the definitions of $C_q(n)$ and $R_q(n)$ is that for the former, we consider only cycles of length precisely $n-q$, whereas, in the latter, we consider all $2$-regular subgraphs spanning \emph{at least} $n-q$ vertices. This is because we can prove Theorem \ref{extension} for cycles, however, a similar result for $2$-regular subgraphs remains elusive (see Conjecture \ref{2-regular_conjecture}).
\subsection{Proof of Theorem \ref{theorem:six}}
\label{subsec:proof_of_theorem_theorem_six}
Suppose not. Let $m$ be an integer in $[j,n-1]$ such that every $m$-cycle gets all colors but there is an $(m+1)$-cycle $H$, $v_1v_2,\ldots,v_{m+1}v_1$ which does not have an edge of color $t$. Then $c(v_i v_{i+2})=t$ for all $i$, where the subscripts are read mod $(m+1)$, because otherwise, there is an $m$-cycle with no edge of color $t$.
\begin{case}
If $m+1$ is odd, then $v_1 v_3 v_5\cdots v_{m+1} v_2 v_4\cdots v_{m-2} v_1$ is an $m$-cycle with at most two colors, since all edges except possibly $v_{m-2} v_1$ have color $t$. This is impossible.
\end{case}
\begin{case}
Suppose $m+1$ is even. Then $c_E = v_2 v_4\cdots v_{m+1} v_2$ and $c_O=v_1 v_3\cdots v_m v_1$ are $\frac{m+1}{2}$-cycles with all edges of color $t$. Suppose $H$ has a chord $v_j v_{j+r}$ with color $t$ for some $j$ and odd integer $r$ in $[3,m-2]$. Then $v_{j+2} v_{j+4} \cdots v_{j-2} v_j v_{j+r} v_{j+r+2}\cdots v_{j+r-4}$ is a path with $m$ vertices (missing $v_{j+r-2}$) and all edges of color $t$, so there is an $m$-cycle with at most two colors, which is impossible. Hence if $v_i$ is a vertex in $c_E$ and $v_j$ is a vertex in $c_O$, then $v(v_i v_j)\neq t$.
We claim that for each $j$ and even integer $s$, $c(v_j v_{j+s})=t$. If not, then $v_j v_{j+s} v_{j+s+1} \cdots \allowbreak v_{j-3} v_{j-2} v_{j+s-1} v_{j+s-2}\cdots\allowbreak v_{j+1}v_j$ is an $m$-cycle (missing $v_{j-1}$) with no edge of color $t$ (note $c(v_{j-2}v_{j+s-1})\allowbreak\neq t$ because $j-2$ and $j+s-1$ have different parities). Hence, the vertices of $c_E$ and $c_O$ each induce a complete graph with $\frac{m+1}{2}$ vertices and all edges of color $t$, and there are no other edges of color $t$ in $K_n$.
If there is a color $w$, different than $t$, such that there exist two disjoint edges of color $w$, then it is easy to find an $m$-cycle with two edges of color $w$ and the rest of color $t$. If there do not exist two such edges of color $w$, then all edges of color $w$ are incident to a single vertex $x$, so any $m$-cycle with $x$ incident to two edges of color $t$ does not contain an edge of color $w$ (these exist since $\frac{m+1}{2}\geq 3$).\hfill\qedsymbol
\end{case}
We remark that the statement in Theorem \ref{theorem:six} would be false without the requirement that there be at least three colors. If $m\geq 3$ is odd, then two vertex disjoint complete graphs each with $\frac{m+1}{2}$ vertices and all edges of color $t$ with all edges between them of color $w$ has an $(m+1)$-cycle with all edges of color $w$, while every $m$-cycle has edges of both colors. This is the reason for the difference between odd and even values of $n-q$ in Theorem \ref{theorem:five}. The statement would also be false with three colors if $j=3$ and $n=4$.
\section{Main Lemmas and Proofs of Theorems}
\label{Theorems}
We now state and prove the three main lemmas needed for the proofs of Theorems \ref{theorem:two}, \ref{theorem:three}, and \ref{theorem:four}.
\begin{lem}\label{max-vertex}
\hfill
\begin{enumerate}[label=(\alph*)]
\item\label{mvert1} Let $\ensuremath{\mathcal{H}}\in\{R_q(n),C^*_q(n)\}$. Of all optimal $\ensuremath{\mathcal{H}}$-polychromatic colorings, let $\varphi$ be one which is $X$-ordered on a (possibly empty) subset $X$ of $V(K_n)$ of maximum size and, of these, such that $G_m=K_n[Y]$ has a vertex $v\in Y$ of maximum possible monochromatic degree $d$ in $G_m$ where $Y=V(K_n)\setminus X$, $\abs{Y}=m$, and $d<(m-1)$. If $v$ is incident in $G_m$ to $d$ edges of color 1 and $u\in Y$ is such that $\varphi(vu)=2$, then $v$ is a $(1,2)$-max vertex in $G_m$ and $u$ is a $(2,t)$-max vertex in $G_m$ for some color $t$ (possibly $t=1$).
\item\label{mvert2} The same is true if $X\neq \emptyset$ and $\varphi$ is nearly $X$-ordered.
\end{enumerate}
\begin{proof}[Proof of \ref{mvert1}]
Let $y_1,y_2,\ldots,y_d\in Y$ be such that $\varphi(vy_i)=1$. Let $H\in C_q^*$ or $H\in R_q$ be such that $uv$ is the only edge of color 2. There must be such an $H$ otherwise we could change the color of $uv$ from $2$ to $1$, giving an $\ensuremath{\mathcal{H}}$-polychromatic coloring with monochromatic degree greater than $d$ in $G_m$. \Lightning. Orient the edges of $H$ to get a directed cycle or $2$-regular graph $H'$ where $\vv{uv}$ is an arc.
If $y_i\in H'$ then the predecessor $w_i$ of $y_i$ in $H'$ must be such that $\varphi(w_i u)=2$, because otherwise we can twist $uv$ and $w_i y_i$ to get an $(n-q)$-cycle (if $H\in C_q^*$) or a $2$-regular graph (if $H\in R_q$) with no edge of color 2. Note that $w_i$ must be in $Y$ because otherwise, since $\varphi$ is $X$-constant, $\varphi(w_i u)=\varphi(w_i y_i)=2$, contradicting the assumption that $uv$ is the only edge in $H$ of color 2.
Suppose $y_i\not\in H$ for some $i\in[d]$. If $\varphi(y_i u)\neq 2$, then $J=(H\setminus\{uv\})\cup\{vy_i,y_iu\}$ has no edge of color $2$. This is impossible if $H\in R_q$, because $J$ is a $2$-regular graph spanning $n-q+1$ vertices. If $H\in C_q^*$, then $J$ is an $(n-q+1)$-cycle with no edge of color $2$, so by Theorem \ref{theorem:six}, since the polychromatic number of $H$ is at least $3$, there exists an $(n-q)$-cycle which is not polychromatic, a contradiction. Hence $\varphi(y_i u)=2$ in either case.
Thus, for each $i\in[d]$, either $y_i\not\in H$ and $\varphi(y_i u)=2$, or $y_i\in H$ and $\varphi(w_i u)=2$ where $w_i$ is the predecessor of $y_i$ in $H'$. That gives us $d$ edges in $G_m$ of color $2$ which are incident to $u$. Since $v$ has maximum monochromatic degree in $G_m$, it follows that $v=w_i$ for some $i$ (otherwise $uv$ is a different edge of color $2$ incident to $u$) and it also follows that no edge in $G_m$ incident to $v$ can have color $t$ where $t\not\in\{1,2\}$. This is because if $vz$ were such an edge, as shown above, then either $z\in H$ and $\varphi(w'u=2)$ where $w'$ is the predecessor of $z$ in $H'$, or $z\not\in H$ and $\varphi(zu)=2$. In either case we get $d+1$ edges of color $2$ in $G_m$ incident to $u$, a contradiction. So $v$ is a $(1,2)$-max-vertex and $u$ is a $(2,t)$-max-vertex for some color $t$.
The proof of \ref{mvert2} is exactly the same.
\end{proof}
\end{lem}
\begin{lem}\label{structurelemma}Let $n\geq 7$ and $\ensuremath{\mathcal{H}}\in\{R_q(n),C_q(n)\}$. If there does not exist an optimal {\ensuremath{\mathcal{H}}}-polychromatic coloring of $K_n$ with maximum monochromatic degree $n-1$, then one of the following holds.
\begin{enumerate}[label=\alph*)]
\item\label{structure-one} $\ensuremath{\mathcal{H}}=C_q(n)$, $n-q$ is odd and $n\in[2q+2,3q+2]$ (and $\pcq(n)=2$).
\item $q=0$ and there exists an optimal $\ensuremath{\mathcal{H}}$-polychromatic coloring which is $Z$-quasi-ordered with $\abs{Z}=3$.
\item $q=1$ and there exists an optimal $\ensuremath{\mathcal{H}}$-polychromatic coloring which is $Z$-quasi-ordered with $\abs{Z}=4$.
\end{enumerate}
\begin{proof}\let\qed\relax
First assume that $\ensuremath{\mathcal{H}}=C_q(n)$ and that $q\geq 2$ and $n$ are such that $\pcq(n)\leq 2$. If $n-q$ is even then, by Theorem \ref{theorem:five}, there is a $C_q$-polychromatic 2-coloring if and only if $n\geq 3q+3$. Since $3q+3$ is the smallest value of $n$ such that the simply-ordered $C_q$-polychromatic coloring $\varphi_{C_q}$ uses two colors, if $\pcq(n)\leq 2$ and $n-q$ is even, then there is an optimal simply-ordered $C_q$-polychromatic coloring, and this coloring has a vertex (in fact $q+1$ of them) with monochromatic degree $n-1$.
If $n-q$ is odd then, by Theorem \ref{theorem:five}, there is a $C_q$-polychromatic 2-coloring if and only if $n\geq 2q+2$. Since there is a simply-ordered $C_q$-polychromatic 2-coloring if $n\geq 3q+3$, that means that if $n-q$ is odd, $\pcq(n)\leq 2$ and $n\not\in[2q+2,3q+2]$ then there is a simply-ordered $C_q$-polychromatic coloring. Thus if $\pcq(n)\leq 2$, there is an optimal simply-ordered $C_q$-polychromatic coloring, and hence one with maximum monochromatic degree $n-1$, unless $n-q$ is odd and $n\in[2q+2,3q+2]$, which are the conditions for \ref{structure-one}.
Now let $\ensuremath{\mathcal{H}}\in\{R_q(n),C^*_q(n)\}$ and suppose there does not exist an optimal $\ensuremath{\mathcal{H}}$-polychromatic coloring of $K_n$ with maximum monochromatic degree $n-1$. Of all optimal $\ensuremath{\mathcal{H}}$-polychromatic colorings of $K_n$, let $\varphi$ be the one with maximum possible monochromatic degree $d$ (so $d<n-1$).
\end{proof}
\end{lem}
\begin{claim}\label{dsize}
$d>\frac{n-1}{2}$.
\begin{proof}\let\qed\relax
Since there are only two colors at a max-vertex, certainly $d\geq \frac{n-1}{2}$. Assume $d=\frac{n-1}{2}$ (so n is odd) and that $x$ is a max-vertex where colors $i$ and $j$ appear. Then $x$ is both an $i$-max and $j$-max vertex so, by Lemma \ref{max-vertex}, each vertex in $V$ is a max-vertex.
Suppose there are more than 3 colors, say colors $i,j,s,t$ are all used. If $i$ and $j$ appear at $x$ then no vertex $y$ can have colors $s$ and $t$, because there is no color for $xy$. So the sets of colors on the vertices is an intersecting family of $2$-sets. Since there are at least 4 colors, the only way this can happen is if some color, say $i$, appears at every vertex. Let $n_{ij}, n_{is}$, and $n_{it}$ be the number of $(i,j)$-max, $(i,s)$-max, and $(i,t)$-max vertices with $n_{ij}\leq n_{is} \leq n_{it}$. Then $n_{ij}<\frac{n}{2}$ (in fact, $n_{ij}\leq \frac{n}{3}$). If $x$ is an $(i,j)$-max vertex and $y$ is an $(i,s)$-max vertex, then $c(xy)=i$. Hence the number of edges of color $j$ incident to $x$ is at most $n_{ij}-1<\frac{n-2}{2}<d$, a contradiction.
Now suppose there are precisely 3 colors. Let $A, B, C$ be the set of all $(1,2)$-max, $(2,3)$-max, and $(1,3)$-max vertices, respectively, with $\abs{A}=a, \abs{B}=b$, and $\abs{C}=c$. All edges from a vertex in $A$ to a vertex in $B$ have color 2, from $B$ to $C$ have color 3, from $A$ to $C$ have color 1; internal edges in $A$ have color 1 or 2, in $B$ have color 2 or 3, in $C$ have color 1 or 3. We clearly cannot have $a,b,$ or $c$ greater than $\frac{n-1}{2}$ so, without loss of generality, we can assume $a\leq b\leq c\leq \frac{n-1}{2}$ and $a+b+c=n$.
Consider the graph $F$ formed by the edges of color 1 or 2. Vertices of $F$ in $B$ or $C$ have degree $\frac{n-1}{2}$, while vertices in $A$ have degree $n-1$. Since $a\leq c$ we have $a\leq \frac{n-b}{2}$. The internal degree in $F$ of each vertex in $B$ is $\frac{n-1}{2}-a\geq \frac{n-1}{2}-\frac{n-b}{2}=\frac{b-1}{2}$. As is well known (Dirac's theorem), that means there is a Hamiltonian path within $B$. Similarly there is one within $C$. If $a\geq 2$, that makes it easy to construct a Hamiltonian cycle in $F$. If $a=1$ we must have $b=c=\frac{n-1}{2}$, so $F$ is two complete graphs of size $\frac{n+1}{2}$ which share one vertex. This graph has a spanning 2-regular subgraph if $n\geq 7$ (a 3-cycle and a 4-cycle if $n=7$), so no $R_q$-polychromatic coloring with 3 colors for any $q\geq 0$ if $n\geq 7$.
If $a=1$ and $b=c=\frac{n-1}{2}$ consider the subgraph of all edges of colors 1 or 3. It consists of a complete bipartite graph with vertex parts $A\cup B$ and $C$, with sizes $\frac{n+1}{2}$ and $\frac{n-1}{2}$, plus internal edges in $C$. Clearly this graph has an $(n-1)$-cycle, but no Hamiltonian cycle. Hence there can be a $C_q$-polychromatic 3-coloring only if $q=0$. However, the $C_0$-polychromatic coloring $\varphi_{C_0}$ uses at least 4 colors if $n\geq 7$, so there is no optimal one with maximum monochromatic degree $\frac{n-1}{2}$.
\end{proof}
\end{claim}
\begin{claim}
If $q=0$, then, up to relabeling the colors, there is a $(1,2)$-max-vertex, a $(2,3)$-max-vertex and a $(3,1)$-max-vertex.
\begin{proof}\let\qed\relax
Assume that every max-vertex has majority color either $1$ or $2$. Then $u$ must be a $(2,1)$-max-vertex. This is because by Lemma \ref{max-vertex}, if it were a $(2,t)$-max-vertex for some third color $t$, and $c(u z)=t$, then $z$ would have to be a $t$-max-vertex, a contradiction. Hence, every max-vertex is either a $(1,2)$-max-vertex or a $(2,1)$-max-vertex. Let $S$ be the set of all $(1,2)$-max-vertices, $T$ be the set of all $(2,1)$-max-vertices, and $W=V\setminus(S\cup T)$. Edges within $S$ and from $S$ to $W$ must have color $1$ (because any minority color edge at a max-vertex is incident to a max-vertex of that color), edges within $T$ and from $T$ to $W$ must have color $2$, and all edges between $S$ and $T$ must have color $1$ or $2$. If $\abs{S}=s$ and $\abs{T}=t$ and $m=n-1-d$, then each vertex in $S$ is adjacent to $m$ vertices in $T$ by edges of color $2$ (and adjacent to $t-m$ vertices in $T$ by edges of color $1$), and each vertex in $T$ is adjacent to $m$ vertices in $S$ by edges of color $1$.
Suppose $s<t$ and consider any edge $ab$ from $S$ to $T$ of color 2. As before, there is an $H\in\ensuremath{\mathcal{H}}$ which contains $ab$, but no other edges of color 2. Hence $H$ has no edges from $T$ to $W$. Since $s<t$ there must be an edge of $H$ with both vertices in $T$, so it does have another edge of color 2 after all, a contradiction. The same argument works if $t<s$ with an edge with color 1. To avoid this, we must have $s=t=2m$. If there is an edge from $S$ to $W$ then, again, $H$ has an internal edge in $T$, which is impossible. Hence if $\ensuremath{\mathcal{H}}=C^*_0$ then $W=\emptyset$ and every edge has color 1 or 2, which is impossible since $H$ has at least 3 colors. If $\ensuremath{\mathcal{H}}=R_0$ then the subgraph of $H$ induced by $S\cup T$ is the union of cycles. If $m=1$ then $S\cup T$ induces a 4-cycle in $H$, two edges of each color, so $ab$ is not the only edge with color 2. If $m\geq 2$ then two applications of Hall's Theorem gives two disjoint perfect matchings of edges of color 1 between $S$ and $T$, whose union is a 2-factor of edges of color 1 spanning $S\cup T$, which together with the subgraph of $H$ induced by $W$, produces a 2-factor $H'\in R_0$ with no edge of color 2.
We have shown that $u$ is not a $(2,1)$-max vertex, so it must be a $(2,3)$-max vertex for some other color 3. Say $\varphi(uz)=3$. Then, by Lemma \ref{max-vertex}, $z$ is a $3$-max vertex. If $\varphi(vz)=2$, then $z$ would be a $2$-max vertex. So $z$ would be both a $2$-max and a $3$-max vertex, and so $d=\frac{n-1}{2}$, a contradiction to Claim \ref{dsize}. Hence $\varphi(vz)=1$, which means $z$ must be a $(3,1)$-max vertex.
\end{proof}
\end{claim}
\begin{claim}\label{structure}
If $q=0$ then $V$ can be partitioned into sets $A,B,D,E$ where the following properties hold (see Figure \ref{fig:graphfigure}).
\begin{enumerate}
\item\label{C5one} All vertices in $A$ are $(1,2)$-max-vertices.
\item\label{C5two} All vertices in $B$ are $(2,3)$-max-vertices.
\item\label{C5three} All vertices in $D$ are $(3,1)$-max-vertices.
\item\label{C5four} No vertex in $E$ is a max-vertex.
\item\label{C5five} All edges within $A$, from $A$ to $D$, and from $A$ to $E$ are color 1.
\item\label{C5six} All edges within $B$, from $B$ to $A$, and from $B$ to $E$ are color 2.
\item\label{C5seven} All edges within $D$, from $D$ to $B$, and from $D$ to $E$ are color 3.
\item\label{C5eight} $\abs{A}=\abs{B}=\abs{D}=m=n-1-d$.
\end{enumerate}
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[every text node part/.style={align=center},scale=3,inner sep=1mm]
\node[circle,ultra thick,draw=black,fill=white,inner sep=6.5mm] (e) at (0,0) {$E$};
\node[circle,ultra thick,draw=green!50!black,fill=white] (d) at (0,1.1547) {$D$\\$(3,1)$-max};
\node[circle,ultra thick,draw=blue,fill=white] (a) at (-1,1.73205) {$A$\\$(1,2)$-max};
\node[circle,ultra thick,draw=red,fill=white] (b) at (1,1.73205) {$B$\\$(2,3)$-max};
\draw [ultra thick,blue] (a) .. controls (-1.85,1.85) and (-1.4,2.5) .. (a) {node [above left,pos=.5] {\large 1}};
\draw [ultra thick,loosely dashed,red] (b) .. controls (1.4,2.5) and (1.85,1.85) .. (b) {node [above right,pos=.5] {\large 2}};
\draw [ultra thick,dotted,green!50!black] (d) .. controls (-.5,2) and (.5,2) .. (d) {node [below,pos=.5] {\large 3}};
\draw [ultra thick,blue] (a) -- node[below] {\large 1} (d);
\path (a) edge [ultra thick,blue,bend right] node[below left] {\large 1} (e);
\path (b) edge [ultra thick,loosely dashed,red,bend right] node[above] {\large 2} (a);
\path (b) edge [ultra thick,loosely dashed,red,bend left] node[below right] {\large 2} (e);
\draw [ultra thick,dotted,green!50!black] (d) -- node[below] {\large 3} (b);
\draw [ultra thick,dotted,green!50!black] (d) -- node[left] {\large 3} (e);
\end{tikzpicture}
\caption{}
\label{fig:graphfigure}
\end{figure}
\begin{proof}\let\qed\relax
Let $A=\{x : x\textrm{ is a }(1,2)\textrm{-max vertex}\}$, $B=\{x : x\textrm{ is a }(2,3)\textrm{-max vertex}\}$, $D=\{x : x\textrm{ is a }(3,1)\textrm{-max vertex}\}$ and $E=V\setminus (A\cup B \cup D)$. Let $x\in A$. If $y\in A$, then $\varphi(xy)=1$ because if $\varphi(xy)=2$, then $y$ would be a $2$-max vertex. If $y\in B$, then $\varphi(xy)=2$ because that is the only possible color for an edge incident to $x$ and $y$ and, similarly, if $y\in D$, then $\varphi(xy)=1$.
Suppose $w$ is a max-vertex in $E$. Then the two colors on edges incident to $w$ must be a subset of $\{1,2,3\}$, because, otherwise, it would be disjoint from $\{1,2\}$, $\{2,3\}$, or $\{1,3\}$, so there would be an edge incident to $w$ for which there is no color. Say $1$ and $2$ are the colors at $w$. Since $w\not\in A$, $w$ is a $(2,1)$-max vertex. Let $z$ be a $(3,1)$-max vertex. Then the edge $wz$ must have color 1 so, by Lemma \ref{max-vertex}, $z$ is a $1$-max vertex, a contradiction. We have now verified \eqref{C5one}--\eqref{C5four}. If $x\in A$ and $w\in E$ then $\varphi(xw)=1$ because if $\varphi(xw)=2$ then $w$ would be a $2$-max vertex. Similar arguments show that if $y\in B$ then $\varphi(yw)=2$ and if $y\in D$ then $\varphi(yw)=3$. We have now verified \eqref{C5one}--\eqref{C5seven}.
We have shown that if $x$ is in $A$ then $\varphi(xy)=2$ if and only if $y\in B$. That means $\abs{B}=m$, and by the same argument $\abs{A}=\abs{C}=m$ as well, completing the proof of Claim \ref{structure}.
\end{proof}
\end{claim}
\begin{claim}\label{q=0_optimal-quasi-ordered}
If $\ensuremath{\mathcal{H}}\in\{C_0^*,R_0\}$, and there exists an optimal $\ensuremath{\mathcal{H}}$-polychromatic coloring satisfying \eqref{C5one}--\eqref{C5eight} with $m>1$, then there exists one with $m=1$, i.e. one that is $Z$-quasi-ordered with $\abs{Z}=3$.
\begin{proof}
Let $A=\{a_i:i\in[m]\}, B=\{b_i:i\in[m]\}, D=\{d_i:i\in[m]\}$. Define an edge coloring $\gamma$ by
\begin{align*}
\gamma(a_1 b_i)&=1\mathrm{\ if\ }i>1\\
\gamma(b_1 d_i)&=2\mathrm{\ if\ }i>1\\
\gamma(d_1 a_i)&=3\mathrm{\ if\ }i>1\\
\gamma(u v)&=\varphi(u v)\mathrm{\ for\ all\ other\ }u,v\in V.\\
\end{align*}
It is easy to check that $\gamma$ has the structure described above with $m=1$. We have essentially moved $m-1$ vertices from each of $A$, $B$, and $D$, to $E$. Since $a_1, b_1,$ and $c_1$ each have monochromatic degree $n-2$, any 2-factor must have edges of colors 1,2, and 3 under the coloring $\gamma$, so if it had all colors under $\varphi$, it still does under $\gamma$.
\end{proof}
\end{claim}
We remark that the coloring $\gamma$ with $m=1$ in Claim \ref{q=0_optimal-quasi-ordered} is $Z$-quasi-ordered with $\abs{Z}=3$. As we have shown, if there exists such an $R_0$-polychromatic coloring $\varphi$ with $m>1$, then there exists one with $m=1$. However, if $m>1$ and $n>6$, a coloring $\varphi$ satisfying properties \eqref{C5one}--\eqref{C5eight} might not be $R_0$-polychromatic. This is because if $E$ has no internal edges with color $1$, then any $2$-factor with a $2m$-cycle consisting of alternating vertices from $A$ and $B$ has no edge with color $1$. However, the modified coloring $\gamma$ (with $m=1$) is an $R_0$-polychromatic coloring because then colors $1$, $2$, and $3$ must appear in any $2$-factor.
\begin{claim}\label{max-vertex-q=1}
If $q\geq 1$ then, up to relabelling colors, every max vertex is a $(1,2)$-max vertex or a $(2,1)$-max vertex.
\begin{proof}\let\qed\relax
As before, we assume $v$ is a $(1,2)$-max vertex, that $\varphi(uv)=2$ and that $H\in R_q$ (or $H\in C_q^*$) is such that $uv$ is the only edge of color 2. We know that $u$ is a $(2,t)$-max vertex for some color $t$. By way of contradiction, suppose $u$ is a $(2,3)$-max vertex. Then we have the configuration of Figure \ref{fig:graphfigure}, with $\abs{A}=\abs{B}=\abs{D}=m$. If $uw$ is also an edge of $H$ then $w\in D$, since otherwise $\varphi(uw)=2$. Let $Q$ be the set of vertices not in $H$ (so $\abs{Q}=q>0$) and suppose $p\in Q$ but $p\not\in B$. Then we can replace $u$ in $H$ with $p$ to get a 2-regular graph (cycle) with no edge of color 2. Hence $Q\subseteq B$. Orient the edges of $H$ to get a directed graph $H'$ where $\vv{uv}$ is an arc. Since $\abs{B\setminus Q}<\abs{D}$, and every vertex in $D$ appears in $H'$, for some $d\in D$ and $e\not\in B$, $\vv{de}$ is an arc in $H'$. Since $\varphi(du)=3$ and $\varphi(ev)=1$, when you twist $uv$ and $de$ you get a 2-regular graph (cycle) with no edge of color 2, a contradiction. Hence every max-vertex is a $(1,2)$-max vertex or $(2,1)$-max vertex.
\end{proof}
\end{claim}
\begin{claim}\label{q>1-XnotEmpty}
If $q=1$ then, up to relabelling colors, the vertex set can be positioned into $S,T,W$ such that
\begin{enumerate}
\item\label{q>1-first} $S$ is the set of all $(1,2)$-max vertices
\item $T$ is the set of all $(2,1)$-max vertices
\item $W$ has no max vertices
\item All internal edges in $S$ and all edges from $S$ to $W$ have color 1; all internal edges in $T$ and all edges from $T$ to $W$ have color 2
\item\label{q>1-last} The edges of color 1 between $S$ and $T$ form two disjoint copies of $K_{m,m}$, as do the edges of color 2 (so $\abs{S}=\abs{T}=2m$, where $n-m-1$ is the maximum monochromatic degree)
\end{enumerate}
\begin{proof}\let\qed\relax
By Claim \ref{max-vertex-q=1}, if $q\geq 1$, then every max vertex is a $(1,2)$ or $(2,1)$-max vertex.
Let $S$ be the set of all $(1,2)$-max vertices and $T$ be the set of all $(2,1)$-max vertices, with $\abs{S}=s$ and $\abs{T}=t$, $s\leq t$, and let m be the maximum monochromatic degree. Let $W=V(G)\setminus (S\cup T)$ and let $B$ be the complete bipartite graph with vertex bipartition $S,T$ and edges colored as they are in $G$. So each vertex of $B$ in $S$ is incident with $m$ edges of color 2 and $t-m$ edges of color 1, and each vertex of $B$ in $T$ is incident with $m$ edges of color 1 and $s-m$ edges of color 2. All edges of $G$ within $S$ and between $S$ and $W$ have color 1 (otherwise there would be a $(2,1)$-max vertex not in $T$) and all edges within $T$ and between $T$ and $W$ have color 2.
We note that the edges of color 1 in $B$ satisfy the conditions of Lemma \ref{disjoint_union}, so $B$ has a $2s$-cycle of edges of color 1 unless $s=t=2m$ and the edges of color 1 (and those of color 2) form two disjoint copies of $K_{m,m}$.
Again, let $v\in S$ and $u\in T$ be such that $c(uv)=2$, and let $H\in C_q^*(n)$ (or $H\in R_q(n)$), $q\geq 1$, be such that $uv$ is the only edge of color 2. If $uw$ is also an edge of $H$ then $w\in S$, because otherwise $c(uw)=2$. Hence if $z$ is a vertex of $G$ not in $H$ then $z\in T$, because otherwise we can replace $u$ with $z$ in $H$ to get $H''\in C_q^*(n)$ (or $H''\in R_q(n)$) with no edge of color 2. That means that if $Q$ is the set of vertices of $G$ not in $H$, then $Q\subseteq T$. Since $uv$ is the only edge in $H$ with color 2, each vertex in $T\setminus Q$ is adjacent in $H$ to two vertices in $S$, so there are $2(t-q)$ edges in $H$ between $S$ and $T$, where $q=\abs{Q}\geq t-s$.
Let $M$ be the subgraph of $H$ remaining when the $2(t-q)$ edges in $H$ between $S$ and $T$ have been removed (along with any remaining isolated vertices). If $q=t-s$ then, since every edge in $H$ incident to a vertex in $T$ goes to $S$, either $H$ is a $2s$-cycle and $W=\emptyset$ (if $H\in C_q^*(n)$) or the union of the components of $H$ which have a vertex in $T$ is a 2-regular graph spanning $S$ and $s=t-q$ vertices in $T$. In either case, since $s<t$, we can replace the components of $H$ which intersect $T$ with the $2s$-cycle of edges of color 1 promised by Theorem \ref{Jackson}, to get an $H''\in C_q^*(n)$ (or $H''\in R_q(n)$) with no edge of color 2. Hence $q>t-s$.
Each component of $M$ is a path with at least one edge, both endpoints in $S$ with interior points in $S$ or $W$. If a component has $j>2$ vertices in $S$, we split it into $j-1$ paths which each have their endpoints in $S$ with all interior points in $W$. If a vertex of $S$ is an interior point in a component then it is an endpoint of two of these paths. The number of such paths is $\frac{2(s-(t-q))}{2}=s-(t-q)>0$.
We denote the paths by $P_1,P_2,\ldots,P_r$ where $r=s-(t-q)$. For each $i$ in $[r]$ where $P_i$ has more than 2 vertices, we remove the edges containing the two endpoints (which are both in $S$), leaving a path $W_i$ whose vertices are all in $W$ (the union of the vertices in all the $W_i$'s is equal to $W$).
We will now show that there cannot be a $2s$-cycle of edges of color 1 in $B$. Suppose $J$ is such a $2s$-cycle. Let $R=\{x_1,x_2,\ldots,x_r\}$ be the set of any $r$ vertices in $T\cap V(J)$ and let $K$ be the subgraph of $J$ obtained by removing the $r$ vertices in $R$. For each $i\in [r]$ let $y_{ia}$ and $y_{ib}$ be the vertices adjacent to $x_i$ in $J$. Both are in $S$ and possibly $y_{ib}=y_{ja}$ if $i\neq j$. Now, for each $i\in[r]$, attach $W_i$ to $y_{ia}$ and $y_{ib}$ ($R_i$ can be oriented either way). More precisely, if $W_i$ is the path $w_{i1},w_{i2},\ldots,w_{id}$ in $W$, we attach it to $K$ by adding the edges $y_{ia}w_{i1}$ and $y_{ib}w_{id}$, while if $W_i$ is empty (meaning the $i^{\rm{th}}$ component of $M$ has only two vertices, so none in $W$) we add the edge $y_{ia}y_{ib}$. The resulting graph $H''$ has no edge of color 2, since we constructed it using only edges from $J$ and edges from $H$ within $S\cup W$. Since $V(H'')=V(G)\setminus R$, $H''$ has $n-q$ vertices. Clearly $H''$ is 2-regular and, if $H$ is a cycle, so is $H''$ (if $H$ is not a cycle, $H''$ will still be a cycle if $H$ does not have any components completely contained in $W$). Thus $H''\in R_q(n)$ ($H''\in C_q^*(n)$) and has no edge of color 2, a contradiction. Hence there is no $2s$-cycle of edges of color 1 in $B$.
By Lemma \ref{disjoint_union} it follows that $s=t=2m$ with the edges of color 1 forming two vertex-disjoint copies of $K_{m,m}$. (If these two disjoint copies have vertex sets $S_1\cup T_1$ and $S_2\cup T_2$, where $S_1\cup S_2=S$ and $T_1\cup T_2=T$, then $S_1\cup T_2$ and $S_2\cup T_1$ are the vertex sets which induce two disjoint copies of $K_{m,m}$ with edges of color 2.) We have now verified that properties \eqref{q>1-first}--\eqref{q>1-last} hold if $q\geq 1$. We will now show we get a contradiction if $q\geq 2$.
Assume $q\geq 2$. Let $T_1$ and $T_2$ be the sets of vertices in $T$ in the two $s$-cycles of edges of color 1 ($\abs{T_1}=\abs{T_2}=\frac{s}{2}$, $T_1\cup T_2=T$). Recall that $v\in S$, $u\in T$, and $uv$ is the only edge of $H$ of color 2. The subgraph $M$ of $H$ defined earlier still consists of paths which can be split into paths $P_1,P_2,\ldots,P_q$ (since $r=s-t+q=q$) with endpoints in $S$ and interior points in $W$. Let $J$ be the union of the two $s$-cycles of edges of color 1. Choose the subset $Q$ of size $q$ so that it has at least one vertex in each of $T_1$ and $T_2$, say $Q=\{x_1,x_2,\ldots,x_q\}$ where $x_1\in T_1$ and $x_q\in T_2$. Again, let $K$ be the subgraph obtained from $J$ by removing the vertices in $Q$. Then, as before, the paths $W_1,W_2,\ldots,W_q$ (perhaps some of them empty) can be stitched into $K$. We attach $W_i$ to $y_{ia}$ and $y_{ib}$ if $i\in[2,q-1]$ (just adding the edge $y_{ia}y_{ib}$ if $W_i$ is empty). We attach $W_1$ to $y_{1a}$ and $y_{qb}$ and $W_q$ to $y_{1b}$ and $y_{qa}$, creating an $(n-q)$-cycle if no component of $H$ is contained in $W$, and a 2-regular graph spanning $n-q$ vertices if $H$ has a component contained in $W$. There is no edge of color 2 in this graph contradicting the assumption that if $q\geq 2$ and $\mathcal{H}\in\{R_q(n),C_q^*(n)\}$ then the maximum monochromatic degree in all optimal $\mathcal{H}$-polychromatic colorings is less than $n-1$.
\end{proof}
\end{claim}
\begin{claim}
If $\ensuremath{\mathcal{H}}\in\{C_1^*,R_1\}$ and there exists an $\ensuremath{\mathcal{H}}$-polychromatic coloring satisfying \eqref{q>1-first}--\eqref{q>1-last} in Claim \ref{q>1-XnotEmpty} with $m>1$, then there exists one with $m=1$, i.e. one that is $Z$-quasi-ordered with $\abs{Z}=4$.
\begin{proof}
Assume there is an $R_1$-polychromatic coloring ($C_1^*$-polychromatic coloring) $c$ with $q=1$ satisfying \eqref{q>1-first} -- \eqref{q>1-last} of Claim \ref{q>1-XnotEmpty} where $s=t>2$. Let $v$ and $x$ be vertices in $S$ and $u$ and $y$ be vertices in $T$ such that $c(vu)=c(xy)=2$ and $c(xu)=c(vy)=1$. Let $c'$ be the coloring obtained from $c$ by recoloring the following edges (perhaps they are recolored the same color they had under $c$):
\begin{center}
\begin{tabular}{lll}
$c'(vp)=1$ & for all & $p\in T\setminus\{u,y\}$\\
$c'(xp)=1$ & for all & $p\in T\setminus\{u,y\}$\\
$c'(zu)=2$ & for all & $z\in S\setminus\{v,x\}$\\
$c'(zy)=2$ & for all & $z\in S\setminus\{v,x\}$\\
$c'(zp)=3$ & for all & $p\in T\setminus\{u,y\}$ and $z\in S\setminus\{v,x\}$
\end{tabular}
\end{center}
Since all but one edge incident to $v$ and $x$ have color 1 under $c'$, certainly every $(n-1)$-cycle contains an edge of color 1. Similarly for $u$ and $y$ and edges of color 2. Every edge which was recolored had color 1 or 2 under $c$, so $c'$ must be a polychromatic coloring with the same number of colors. It has the desired form with $\abs{S}=\abs{T}=2$, so, in fact, is $Z$-quasi-ordered with $Z=\{v,x,u,y\}$.
\end{proof}
\end{claim}
%
We remark that a coloring $c$ satisfying properties \eqref{q>1-first}--\eqref{q>1-last} of Claim \ref{q>1-XnotEmpty} with $s=t>2$ is actually not $R_1$-polychromatic. To see this, let $S_1 \cup T_1$ and $S_2\cup T_2$ be the vertex sets of the two copies of $K_{m,m}$ of edges of color 1 ($S_1\cup S_2 = S$, $T_1\cup T_2=T$) where $v\in S_1, u\in T_2$ and $uv$ is the only edge of color 2 in $H\in R_1$. The subgraph $M$ of $H$ in the proof of Claim \ref{q>1-XnotEmpty} has only one component (since $s-(t-q)=1$), a path $d w_1 w_2\ldots w_e z$ where $d\in S_1$, $z\in T_1$, and $\{w_1,w_2,\ldots,w_e\}\subseteq W$. To construct a 2-regular subgraph with no edges of color 2 spanning $n-1$ vertices, remove a vertex $x$ in $T_2$ from one of the two $s$-cycles of edges of color 1. If $y_a$ and $y_b$ are the two vertices in $S_2$ adjacent to $x$ in the $s$-cycle, attach the path $w_1 w_2 \ldots w_e$ to $y_a$ and $y_b$ to get a 2-regular subgraph with no edge of color 2 spanning $n-1$ vertices. However, this construction cannot be done when $m=1$, so in this case you do get an $R_1$-polychromatic coloring.
\begin{lem}\label{XoToO}
Let $\ensuremath{\mathcal{H}}\in\{R_q(n),C^*_q(n)\}$.
\begin{enumerate}[label=\alph*)]
\item Suppose for some $X\neq \emptyset$ there exists an optimal $X$-ordered $\ensuremath{\mathcal{H}}$-polychromatic coloring of $K_n$. Then there is one which is ordered.
\item Suppose there exists an optimal $Z$-quasi-ordered $\ensuremath{\mathcal{H}}$-polychromatic coloring of $K_n$. Then there is one which is quasi-ordered
\end{enumerate}
\begin{proof}
Among all such $\ensuremath{\mathcal{H}}$-polychromatic colorings we assume $\varphi$ is one such that
\begin{enumerate}[label=\alph*)]
\item\label{XoToO-a} if $\varphi$ is $X$-ordered then $X$ has maximum possible size
\item\label{XoToO-b} if $\varphi$ is $Z$-quasi-ordered then the restriction of $\varphi$ to $V(K_n)\setminus Z$ is $T$-ordered for the largest possible subset $T$ of $V(K_n)\setminus Z$. In this case, we let $X=Z\cup T$ so $\varphi$ is nearly $X$-ordered (one or two edges could be recolored to make it $X$-ordered).
\end{enumerate}
For both \ref{XoToO-a} and \ref{XoToO-b} we assume that $\varphi$ is such that its restriction to $G_m=K_n[Y]$ has a vertex $v$ of maximum possible monochromatic degree in $G_m$, where $Y=V(K_n)\setminus X$, $\abs{Y}=m$, and the degree of $v$ in $G_m$ is $d<m-1$ (if $d=m-1$ then $\abs{X}$ is not maximal).
Since $v$ has maximum monochromatic degree $d$ in $G_m$, by Lemma \ref{max-vertex} it is a $(1,2)$-max vertex in $G_m$, for some colors 1 and 2, and if $u\in Y$ is such that $\varphi(uv)=2$, then $u$ is a $(2,t)$-max vertex for some color $t$ (perhaps $t=1$).
As before, let $y_1,y_2,\ldots,y_d$ be vertices in $Y$ such that $c(vy_i)=1$ for $i=1,2,\ldots,d$. As before, let $H\in\ensuremath{\mathcal{H}}$ be such that $uv$ is its only edge with color 2. Let $H'$ be a cyclic orientation of the edges of $H$ such that $\vv{uv}$ is an arc, and let $w_i$ be the predecessor of $y_i$ in $H'$ for $i=1,2,\ldots,d$. As shown before, $c(w_i v)=2$ for $i=1,2,\ldots,d$.
Suppose there is an edge of $H$ which has one vertex in $X$ and one in $Y$. Then there exist $w\in Y$ and $x\in X$ such that $\vv{wx}\in H'$. Certainly $w$ is not the predecessor in $H'$ of any $y_i$ in $Y$. Since $\varphi$ is $X$-constant and $uv$ is the only edge of color 2 in $H$, $\varphi(xv)=\varphi(xw)\neq 2$. Now twist $xw,uv$ in $H$. Since $\varphi(xv)\neq 2$, we must have $\varphi(wu)=2$, so $u$ is incident in $G_m$ to at least $d+1$ vertices of color 2, a contradiction \Lightning. Hence $H$ cannot have an edge with one vertex in $X$ and one in $Y$.
Now suppose $x\in X$ and $x\not\in H$. If $\varphi(xv)=\varphi(xu)\neq 2$ then $H\setminus\{uv\}\cup\{ux,xv\}$ is an $(n-q+1)$-cycle with no edge of color 2, which is clearly impossible if $\ensuremath{\mathcal{H}}=R_q(n)$, and is impossible if $\ensuremath{\mathcal{H}}=C^*_q(n)$ by Theorem \ref{theorem:six}. Hence $\varphi(xv)=\varphi(xu)=2$ for each $x\in X$.
Since $u$ is a $(2,t)$-max vertex for some color $t\neq 2$, we can repeat the above argument with $u$ in place of $v$. That shows that $\varphi(xv)=\varphi(xu)=t$ for each $x\in X$, which is clearly impossible.
It remains to consider the possibility that $\ensuremath{\mathcal{H}}=R_q(n)$ and $X$ is spanned by a union of cycles in $H$. Suppose $xz$ is an edge of $H$ contained in $X$. Then we can twist $xz$ and $uv$ to get another subgraph in $R_q$ and, unless either $x$ or $z$ has main color $2$, this subgraph has no edge of color $2$. Hence at least half the vertices in $X$ have main color $2$ (and more than half would if $H$ had an odd component in $X$).
The above argument can be repeated with $u$ in place of $v$. If $u$ is a $(2,t)$-max-vertex then that would show that at least half the vertices in $X$ have main color $t\neq 2$. So each vertex in $X$ has main color $2$ or $t$. Since $\varphi$ is $X$-ordered or nearly $X$-ordered, some vertex $x\in X$ has monochromatic degree $n-2$ or $n-1$ and the main color of $x$ must be 2 or $t$. Assume it is $2$. Then every cycle containing $x$ has an edge with color 2, contradicting the assumption that $H$ has only one edge with color 2. Similarly, we get a contradiction if the main color of $x$ is $t$. We have shown there is no vertex $v$ with monochromatic degree $d<m-1$, so $\varphi$ is ordered or quasi-ordered.
\end{proof}
\end{lem}
Now there is not much left to do to prove Theorems \ref{theorem:two}, \ref{theorem:three}, and \ref{theorem:four}.
\subsection{Proof of Theorem \ref{theorem:four}}
\label{sec:proof_of_theorem_ref_theorem_four}
Theorem \ref{theorem:five} takes care of the case of $C_q$-polychromatic colorings when $q\geq 2$ and $n\in[2q+2,3q+2]$. The smallest value of $n$ for which there is a simply-ordered $C_q$-polychromatic 2-coloring is $n=3q+3$ (the coloring $\varphi_{C_q}$ in Section \ref{subsec:_k_hc_polychromatic_coloring}). Hence if $q\geq 2$ and $\pcq\leq2$ then there exists an optimal simply-ordered $C_q$-polychromatic coloring except if $n-q$ is odd and $n\in[2q+2,3q+2]$, or if $q=2$ and $n=5$ (the coloring of $K_5$ with two monochromatic 5-cycles has no monochromatic 3-cycle). So we need only consider $\ensuremath{\mathcal{H}}\in\{R_q(n),C^*_q(n)\}$ (when $q\geq 2$). Since \ref{structure-one} is not satisfied in Lemma \ref{structurelemma}, there exists an optimal $\ensuremath{\mathcal{H}}$-polychromatic coloring with maximum monochromatic degree $n-1$. That means it is $X$-ordered, for some nonempty set $X$, so by Lemma \ref{XoToO} there exists an optimal $\ensuremath{\mathcal{H}}$-polychromatic coloring which is ordered, and then, by Lemma \ref{O2SO}, one which is simply-ordered.\hfill\qedsymbol
\subsection{Proof of Theorem \ref{theorem:two}}
\label{sec:proof_of_theorem_ref_theorem_two}
If $\ensuremath{\mathcal{H}}\in\{R_0(n),C_0(n)\}$ then, by Lemma \ref{structurelemma}, there exists an optimal $\ensuremath{\mathcal{H}}$-polychromatic coloring which is $Z$-quasi-ordered with $\abs{Z}=3$. Then, by Lemma \ref{XoToO}, there exists one which is quasi-ordered and then, by Lemma \ref{O2SO}, one which is quasi-simply-ordered with $\abs{Z}=3$, so recoloring one edge would give a simply-ordered coloring.\qedsymbol
\subsection{Proof of Theorem \ref{theorem:three}}
\label{sec:proof_of_theorem_ref_theorem_three}
Exactly the same as the proof of Theorem \ref{theorem:two}, except now $\abs{Z}=4$, so two edges need to be recolored to get a simply-ordered coloring.
\section{Polychromatic cyclic Ramsey numbers}
\label{sec:polychromatic_cyclic_ramsey_numbers}
Let $s, t$, and $j$ be integers with $t\geq 2, s\geq 3, s\geq t$, and $1\leq j\leq t-1$. We define $\cyram(s,t,j)$ to be the smallest integer $n$ such that in any $t$-coloring of the edges of $K_n$ there exists an $s$-cycle that uses at most $j$ colors. Erd{\H{o}}s and Gy\'{a}rf\'{a}s \cite{Erdos:1997} defined a related function for cliques instead of cycles. So $\cyram(s,t,1)$ is the classical $t$-color Ramsey number for $s$-cycles and $\cyram(s,2,1)=c(s)$, the function in Theorem \ref{FS}. While it may be difficult to say much about the function $\cyram(s,t,j)$ in general, if $j=t-1$ we get $\cyram(s,t,t-1)=\pr_t(s)$ the smallest integer $n\geq s$ such that in any $t$-coloring of $K_n$ there exists an $s$-cycle that does not contain all $t$ colors. This is the function of Theorem \ref{extension} if $t\geq 3$, while $\pr_2(s)=c(s)$.
\subsection{Proof of Theorem \ref{extension}}
\label{sec:proof_of_theorem_extension}
Let $q\geq 0, s\geq 3$, and $n$ be integers with $n=q+s$. Assume $q\geq 2$. By Theorem \ref{theorem:four} and the properties of the coloring $\varphi_{C_q}$ (see Section \ref{subsec:_k_hc_polychromatic_coloring}), there exists a $C_q$-polychromatic $t$-coloring of $K_n$ if and only if
\begin{align*}
q+s&=n\geq (2^t -1)q+2^{t-1}+1,\\
s&\geq(2^t-2)q+2^{t-1}+1,\\
q&\leq\frac{s-2^{t-1}-1}{2^t -2} = \frac{s-2}{2^t -2}-\frac{1}{2}
\end{align*}
Since $q\geq 2$, we want to choose $s$ so that the right-hand side of the last inequality is at least 2, so
\begin{align*}
s-2&\geq \frac{5}{2}(2^t -2) = 5\cdot 2^{t-1}-5\\
s&\geq 5\cdot 2^{t-1}-3
\end{align*}
So if $s\geq 5\cdot 2^{t-1}-3$, then the smallest $n$ for which there does not exist a $C_q$-polychromatic $k$-coloring is $n=q+s$ where $q>\frac{s-2}{2^t-2}-\frac{1}{2}$, so $n=s+\left\lfloor \frac{s-2}{2^t-2} +\frac{1}{2} \right\rfloor=s+\round\left(\frac{s-2}{2^t-2}\right)$.
We note that if $s\geq 5\cdot 2^{t-1}-3$ then $\round\left( \frac{s-2}{2^t-2} \right)\geq \round\left( \frac{5}{2} \right)=3$, so $\pr_t(s)\geq s+3$ if $s\geq 5\cdot 2^{t-1}-3$.
Now we assume that $\pr_t(s)=s+2$. So $s+2$ is the smallest value of $n$ for which in any $t$-coloring of the edges of $K_n$ there is an $s$-cycle which does not have all colors, which means there is a polychromatic $t$-coloring when $n=s+1$. Since $q=1$ in such a coloring, by Theorem \ref{theorem:three} and the properties of the coloring $\varphi_{C_1}$, $n\geq 5\cdot 2^{t-2}$. Hence if $s\in[5\cdot 2^{t-2}-1,5\cdot 2^{t-1}-4]$, then $\pr_t(s)=s+2$.
Now we assume that $\pr_t(s)=s+1$. So $n-s$ is the largest value of $n$ such that in any $t$-coloring of $K_n$, every $s$-cycle gets all colors. So $q=n-s=0$ and, by Theorem \ref{theorem:two} and properties of the coloring $\varphi_{C_0}$, $n\geq 3\cdot 2^{t-3}+1$.
Finally, since the $t$-coloring $\varphi_{C_0}$ requires $n\geq 3\cdot 2^{t-3}+1$ where $t\geq 4$ if $n\leq 3\cdot 2^{t-3}$ and $t\geq 4$, then in any $t$-coloring of $K_n$, some Hamiltonian cycle will not get all colors, so $\pr_t(s)=s$ if $3<s\leq3\cdot2^{t-3}$.
\section{Conjectures}
\label{sec:conjectures_and_closing_remarks}
We mentioned that we have been unable to prove a result for 2-regular graphs analogous to Theorem \ref{theorem:six} for cycles. In fact we think it even holds for two colors, except for a few cases with $j$ and $n$ small.
\begin{conj}\label{2-regular_conjecture}
Let $n\geq 6$ and $j$ be integers such that $3\leq j <n$, and if $j=5$ then $n\geq 9$, and let $\varphi$ be an edge-coloring of $K_n$ so that every 2-regular subgraph spanning $j$ vertices gets all colors. Then every 2-regular subgraph spanning at least $j$ vertices gets all colors under $\varphi$.
\end{conj}
This does not hold for $j=3, n=4$, and 3 colors; $n=5, j=3$, and 2 colors.
We can extend the notions of $Z$-quasi-ordered, quasi-ordered, and quasi-simply-ordered to sets $Z$ of larger size, allowing a main color to have degree less than $n-2$. Let $q\geq 0$ and $r\geq 1$ be integers such that $q\leq 2r-3$. Hence $\frac{2r-2}{q+1}\geq 1$, and we let $k=\left\lfloor \frac{2r-2}{q+1} \right\rfloor+1\geq 2$ and $z=k(q+1)$. Let $Z$ be a set of $z$ vertices. We define a \emph{seed-coloring} $\varphi$ with $k$ colors on the edges of the complete graph $K_z$ with vertex set $Z$ as follows. Partition the $z$ vertices into $k$ sets $S_1,S_2,\ldots,S_k$ of size $q+1$. For $j=1,2,\ldots,k$, all edges within $S_j$ have color $j$, all edges between $S_i$ and $S_j$ ($i\neq j$) have color $i$ or $j$, and for each $j$ and each vertex $v$ in $S_j$, $v$ is incident to $\left\lceil \frac{(q+1)(k-1)}{2} \right\rceil$ or $\left\lfloor \frac{(q+1)(k-1)}{2} \right\rfloor$ edges with colors other than $j$ (so, within round off, half of the edges from each vertex in $S_j$ to vertices in other parts have color $j$). We say each vertex in $S_j$ has main color $j$.
If $n\geq z$, we get a $Z$-quasi-ordered coloring $c$ of $K_n$ which is an extension of the coloring $\varphi$ on $Z$ if for each $j$ and each $v\in S_j$, $c(vy)=j$ for each $y\in V(K_n)\setminus Z$. If $c$ is $Z$-quasi-ordered then it is quasi-ordered if $c$ restricted to $V(K_n)\setminus Z$ is ordered, and quasi-simply-ordered if $c$ restricted to $V(K_n)\setminus Z$ is simply-ordered.
If $r>0$ and $q\geq 0$ are integers we let $\mathscr{R}(n,r,q)$ be the set of all $r$-regular subgraphs of $K_n$ spanning precisely $n-q$ vertices (assume $n-q$ is even if $r$ is odd, so the set is nonempty), and if $r\geq 2$ let $\mathscr{C}(n,r,q)$ be the set of all such subgraphs which are connected.
Since $k-1=\left\lfloor \frac{2r-2}{q+1} \right\rfloor\leq \frac{2r-2}{q+1}$, we have $r\geq\frac{(q+1)(k-1)}{2}+1>\left\lceil \frac{(q+1)(k-1)}{2} \right\rceil$. So if $H$ is in $\mathscr{R}(n,r,q)$ or $\mathscr{C}(n,r,q)$, then $H$ contains an edge with each of the $k$ colors on edges within $Z$, because it contains at least one vertex in $S_j$ for each $j$, and fewer than $r$ of the edges incident to this vertex have colors other than $j$. We can get an $\mathscr{R}(n,r,q)$-polychromatic or $\mathscr{C}(n,r,q)$-polychromatic quasi-simply-ordered coloring of $K_n$ with $m>k$ colors by making the color classes $M_t$ on the vertices in $V(K_n)\setminus Z$ for $t=k+1,k+2,\ldots,m$ sufficiently large. If $H\in\mathscr{R}(n,r,q)$, for each $t\in[k+1,m]$ we will need the size of $M_t$ to be at least $q+1$ more than the sum of the sizes of all previous color classes, while if $H\in\mathscr{C}(n,r,q)$ we will need the size of $M_t$ to be at least $q$ more than the sum of the sizes of all previous classes, with an extra vertex in $M_m$. To try to get optimal polychromatic colorings we make the sizes of these color classes as small as possible, yet satisfying these conditions.
For example, if $r=2$ and $q=0$ then $k=\left\lfloor \frac{2r-2}{q+1} \right\rfloor +1=3$ and $z=k(q+1)=3$, and we get the quasi-simply-ordered colorings $\varphi_{R_0}$ and $\varphi_{C_0}$ with $\abs{Z}=3$ of Theorem \ref{theorem:two}. If $r=2$ and $q=1$ then $k=2$ and $z=4$, and we get the colorings $\varphi_{R_1}$ and $\varphi_{C_1}$ with $\abs{Z}=4$ of Theorem \ref{theorem:three}.
\begin{example}[$r=3,q=0$, so $k=5,z=5$]\label{previous-example}
Let $\varphi$ be the edge coloring obtained where $\{v_1,v_2,v_3,v_4,v_5\}=Z$ such that $v_i v_{i+1}$ and $v_i v_{i+2}$ ($\operatorname{mod} 5$) have color $i$. The edges connecting $v_i$ to the remaining vertices in $V(K_n)\setminus Z$ are color $i$. See Figure \ref{fig:kFrpoly}.
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[mystyle/.style={draw,shape=circle,fill=white}]
\def5{5}
\node[regular polygon,regular polygon sides=5,minimum size=3cm] (p) {};
\foreach\x in {1,...,5}{\node[mystyle] (p\x) at (p.corner \x){$v_{\x}$};}
\foreach\i in {1,...,5} {
\foreach\t in {1,2}{
\pgfmathsetmacro{\j}{int(mod(\i-1+\t,5)+1)}
\draw (p\i) --node {\i} (p\j);
}
}
\foreach \i in {2,...,5}{
\pgfmathsetmacro{\angle}{int(18+72*\i)}
\draw (p\i) --node {\i} (\angle:2.5);
}
\draw (p1) --node[left] {1} (90:2.5);
\path [draw=black,fill=gray,fill opacity=.25,even odd rule] (0,0) circle (3.5) (0,0) circle (2.5);
\node (graphlabel) at (0,3) {$V(K_n)\setminus Z$};
\end{tikzpicture}
\caption{The coloring for Example \ref{previous-example}.}
\label{fig:kFrpoly}
\end{figure}
\end{example}
\begin{example}[$r=3,q=3,k=2,z=8$]
$Z$ has two color classes, 4 vertices in each. The complete bipartite graph between these two sets of vertices could have two vertex disjoint copies of $K_{2,2}$ of one color and also of the other color, or could have an 8-cycle of each color.
\end{example}
\begin{example}[$r=4,q=2,k=3,z=9$]
So $S_1,S_2,S_3$ each have size $q+1=3$. One way to color the edges between parts is for $j=1,2,3$, each vertex in $S_j$ is incident with 2 edges of color $j$ to vertices in $S_{j+1}$ and 1 edge of color $j$ to a vertex in $S_{j-1}$ (so is incident with one edge of color $j+1$ and two edges of color $j-1$, cyclically). The smallest value of $n$ for which this seed can generate a quasi-simply-ordered $\mathscr{R}(n,4,2)$-polychromatic coloring with 5-colors is $n=45$ (the 4$^{\rm{th}}$ and 5$^{\rm{th}}$ color classes would have sizes $9+2+1=12$ and $21+2+1=24$ respectively), while to get a simply-ordered $\mathscr{R}(n,4,2)$-polychromatic coloring with 5 colors you would need $n\geq 69$ (color class sizes $3,3,9,18,36$ works).
\end{example}
\begin{conj}
Let $r\geq 1$ and $q\geq 0$ be integers such that $q\leq 2r-3$. Let $k=\left\lfloor \frac{2r-2}{q+1} \right\rfloor+1\geq 2$ and $z=k(q+1)$. If $n\geq z$ and $n-q$ is even if $r$ is odd, then there exist optimal quasi-simply-ordered $\mathscr{R}(n,r,q)$ and $\mathscr{C}(n,r,q)$-polychromatic colorings with seed $Z$ with parameters $r,q,k,z$.
\end{conj}
It is not hard to check that each of these quasi-simply-ordered colorings does at least as well as a simply-ordered coloring for those values of $r$ and $q$. The only quesiton is whether some other coloring does better and the conjecture says no.
What if $\frac{2r-2}{q+1}<1$? Then $k=\left\lfloor \frac{2r-2}{q+1} \right\rfloor+1=1$, which seems to be saying no seed $Z$ exists with at least 2 colors.
\begin{conj}
Let $r\geq 1$ and $q\geq 0$ be integers with $q\geq 2r-2$, $n\geq q+r+1$, and not both $r$ and $n-q$ are odd. Then there exists an optimal simply-ordered $\mathscr{R}(n,r,q)$-polychromatic coloring of $K_n$. If $r\geq 2$ then there exists a $\mathscr{C}(n,r,q)$-polychromatic coloring of $K_n$ (unless $r=2$, $q\geq 2$, $n-q$ is odd, and $n\in[2q+2,3q+1]$).
\end{conj}
Theorem \ref{theorem:one} says this conjecture is true for $r=1$. Theorem \ref{theorem:four} says it is true for $\mathscr{C}(n,r,q)$ for $r=2$ and that it would be true for $\mathscr{R}(n,r,q)$ for $r=2$ if Theorem \ref{theorem:six} held for 2-regular graphs.
\bibliographystyle{plain}
| {
"timestamp": "2020-09-21T02:18:16",
"yymm": "2009",
"arxiv_id": "2009.08960",
"language": "en",
"url": "https://arxiv.org/abs/2009.08960",
"abstract": "If $G$ is a graph and $\\mathcal{H}$ is a set of subgraphs of $G$, we say that an edge-coloring of $G$ is $\\mathcal{H}$-polychromatic if every graph from $\\mathcal{H}$ gets all colors present in $G$ on its edges. The $\\mathcal{H}$-polychromatic number of $G$, denoted $\\operatorname{poly}_\\mathcal{H} (G)$, is the largest number of colors in an $\\mathcal{H}$-polychromatic coloring. In this paper we determine $\\operatorname{poly}_\\mathcal{H} (G)$ exactly when $G$ is a complete graph on $n$ vertices, $q$ is a fixed nonnegative integer, and $\\mathcal{H}$ is one of three families: the family of all matchings spanning $n-q$ vertices, the family of all $2$-regular graphs spanning at least $n-q$ vertices, and the family of all cycles of length precisely $n-q$. There are connections with an extension of results on Ramsey numbers for cycles in a graph.",
"subjects": "Combinatorics (math.CO)",
"title": "Polychromatic colorings of 1-regular and 2-regular subgraphs of complete graphs",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754461077707,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.7075103946359932
} |
https://arxiv.org/abs/1612.00421 | Bulk Universality for Generalized Wigner Matrices With Few Moments | In this paper we consider $N \times N$ real generalized Wigner matrices whose entries are only assumed to have finite $(2 + \varepsilon)$-th moment for some fixed, but arbitrarily small, $\varepsilon > 0$. We show that the Stieltjes transforms $m_N (z)$ of these matrices satisfy a weak local semicircle law on the nearly smallest possible scale, when $\eta = \Im (z)$ is almost of order $N^{-1}$. As a consequence, we establish bulk universality for local spectral statistics of these matrices at fixed energy levels, both in terms of eigenvalue gap distributions and correlation functions, meaning that these statistics converge to those of the Gaussian Orthogonal Ensemble (GOE) in the large $N$ limit. | \section{Introduction}
\label{Introduction}
Since the seminal work of Wigner \cite{CVBMWID} over sixty years ago, the spectral analysis of random matrices has been a topic of intense study. A central phenomenon that has guided significant effort in this field has been that of \emph{universality}, also called the \emph{Wigner-Dyson-Mehta conjecture}. This approximately states (see Conjecture 1.2.1 and Conjecture 1.2.2 of \cite{RM}) that the bulk local spectral statistics of an $N \times N$ real symmetric (or complex Hermitian) Wigner matrix should become independent of the explicit laws of its entries as $N$ tends to $\infty$.
Over the past decade, this conjecture has seen remarkable progress. In particular, the Wigner-Dyson-Mehta conjecture has been proved \cite{FEUGM, SSGESEE, UM, BUMSD, URMLRF, LRFULSRM, BUCM, BUGM, BUSM, FEUM, RCMULS, ULES, UCM} for Wigner matrices whose entries are ``restrained from being too large.'' In the past \cite{UM, BUMSD, URMLRF, LRFULSRM, BUGM, ULES}, this condition had been typically quantified by imposing that the laws of its entries exhibit some type of subexponential decay. This was later \cite{FEUGM, SSGESEE, BUCM, FEUM, RCMULS, UCM} replaced by the less stringent constraint that the entries have finite $C$-th moment for some sufficiently large constant $C > 0$; until now, the smallest value of $C$ one could take had been $4 + \varepsilon$, for any $\varepsilon > 0$ \cite{SSGESEE, FEUM}.
Although the known proofs of the Wigner-Dyson-Mehta conjecture appear to rely quite strongly on such growth assumptions, it is widely believed that these constraints are irrelevant. Namely, the bulk local spectral statistics of a Wigner matrix should be universal only under the assumption that its entries have finite $(2 + \varepsilon)$-moment for some fixed, but arbitrary small, $\varepsilon > 0$ (for our model, having finite second moment will not suffice; see Remark \ref{second}).
The purpose of this paper is to prove this statement, which we do by following and extending upon parts of what is known as the ``three-step strategy'' for establishing bulk universality in random matrices, as set forth in the papers \cite{UM, URMLRF, ERLRM, LRFULSRM, DRM}. Before explaining this in more detail, we will explain the model and our results in Section \ref{Model}. In Section \ref{Outline1} we provide some context for our results. In Section \ref{LocalCircleLaw} we state a version of the heavy-tailed local semicircle law, which is the main new estimate needed to prove universality in the low-moment setting.
\subsection{The Model and Results}
\label{Model}
We begin this section by defining the random matrix model that we will study.
\begin{definition}
\label{momentassumption}
For each positive integer $N$, let $\textbf{H} = \textbf{H}_N = \{ h_{ij} \} = \{ h_{i, j}^{(N)} \}$ be a real symmetric $N \times N$ random matrix whose entries are centered, mutually independent random variables, subject to the symmetry constraint $h_{ij} = h_{ji}$. We call the set of matrices $\{ \textbf{H}_N \}$ a \emph{family of generalized Wigner matrices} if there exist constants $0 < \varepsilon < 1$; $0 < c_1 < 1 < C_1$; and $C_2 > 1$ (all independent of $N$) satisfying the following three assumptions.
\begin{enumerate}
\item[\namedlabel{generalized}{\textbf{A1}}\textbf{.}]{ Denote $s_{ij} = \mathbb{E} \big[ |h_{ij}|^2 \big]$. Then, $c_1 < N s_{ij} < C_1$ for all $i, j$.}
\item[\namedlabel{stochastic}{\textbf{A2}}\textbf{.}]{ For each $i$, set $t_i = \sum_{j = 1}^N s_{ij} - 1$; then, $|t_i| < C_1 N^{- \varepsilon}$ for each $i$. }
\item[\namedlabel{moments}{\textbf{A3}}\textbf{.}]{ For each $i, j$, we have that $\mathbb{E} \big[ |h_{ij} \sqrt{N}|^{2 + \varepsilon} \big] < C_2$. }
\end{enumerate}
\noindent We refer to each individual $\textbf{H} = \textbf{H}_N$ as a \emph{generalized Wigner matrix}.
\end{definition}
When $\Var h_{ij} = N^{-1}$ for each $i, j$, generalized Wigner matrices become \emph{Wigner matrices}, which were first analyzed in \cite{CVBMWID}. In that work, Wigner studied the large $N$ limiting profile of the \emph{empirical spectral distribution} of $\textbf{H}$, defined by $\mu_{\textbf{H}} = N^{-1} \sum_{j = 1}^N \delta_{\lambda_j}$, where $\lambda_1, \lambda_2, \ldots , \lambda_N$ denote the eigenvalues of $\textbf{H}$. He showed, if one assumes that all moments of $|h_{ij} \sqrt{N}|$ are finite, then $\mu_{\textbf{H}}$ converges weakly to the \emph{semicircle law}
\begin{flalign}
\label{rhodefinition}
\rho_{\semicircle} (x) = (2 \pi)^{-1} \textbf{1}_{|x| < 2} \sqrt{4 - x^2},
\end{flalign}
\noindent as $N$ tends to $\infty$. This is an example of the convergence of \emph{global spectral statistics} of the random matrix $\textbf{H}$ to a deterministic limit shape.
Also of interest, and in fact the original impetus for Wigner to initiate his study on random matrices, are the \emph{local spectral statistics} of $\textbf{H}$; these concern the behavior of (nearly) neighboring eigenvalues of $\textbf{H}$ close to a fixed \emph{energy level} $E \in \mathbb{R}$. There are two ways in which this behavior is typically quantified.
The first is in terms of the joint distribution of the (normalized) gaps $\{ N (\lambda_i - \lambda_{i + j_r}) \}_{1 \le r \le k}$ of the eigenvalues; here, $j_1, j_2, \ldots , j_k$ are bounded independently of $N$ and $i$ can grow linearly with $N$. In fact, we will be interested in these statistics in the \emph{bulk}, meaning that we take $i \in [\kappa N, (1 - \kappa) N]$ for some fixed $\kappa > 0$ independent of $N$; this corresponds to imposing that the energy level $E$ be inside the interval $(-2, 2)$ and uniformly bounded away from its endpoints.
The second is in terms of the correlation functions of $\textbf{H}$, which are defined as follows.
\begin{definition}
\label{correlation}
Let $N$ be a positive integer and $\textbf{H}$ be an $N \times N$ real symmetric random matrix. Denote by $p_{\textbf{H}}^{(N)} (\lambda_1, \lambda_2, \ldots , \lambda_N)$ the joint eigenvalue distribution of $\textbf{H}$.
For each integer $k \in [1, N]$, define the \emph{$k$-th correlation function} of $\textbf{H}$ by
\begin{flalign*}
p_{\textbf{H}}^{(k)} (x_1, x_2, \ldots , x_k) = \displaystyle\int_{\mathbb{R}^{N - k}} p_{\textbf{H}}^{(N)} (x_1, x_2, \ldots , x_k, y_{k + 1}, y_{k + 2}, \ldots , y_N) \displaystyle\prod_{j = k + 1}^N d y_j.
\end{flalign*}
\end{definition}
One form of the Wigner-Dyson-Mehta conjecture essentially states that, in the bulk of the spectrum, the local gap statistics and correlation functions of a Wigner matrix should be independent of the explicit laws of the matrix entries satisfying assumption \ref{moments}, in the large $N$ limit.
\begin{rem}
\label{second}
Observe that this universality can become false if we remove assumption \ref{moments}. Indeed, suppose that the $h_{ij}$ are independent, identically distributed random variables (up to the symmetry constraint $h_{ij} = h_{ji}$) with $h_{ij} \in \{ -1, 0, 1 \}$, equal to $-1$ and $1$ each with probability $(2N)^{-1}$ and equal to $0$ otherwise. Then $\Var h_{ij} = N^{-1}$, so $\textbf{H}$ is a Wigner matrix.
However, a given row contains no nonzero entries with probability $(1 - N^{-1})^N \ge 1 / 4$. Therefore we expect with high probability to see, for example, at least $N / 5$ rows $\textbf{H}$ whose entries are all equal to $0$. Thus, with high probability, $\textbf{H}$ has the eigenvalue $0$ with very large multiplicity. This violates universality of both the gap statistics and correlation functions near $0$; it also violates the macroscopic Wigner semicircle law around $0$.
\end{rem}
\begin{rem}
\label{smallmoments}
On a different note, one can choose a more restrictive family of matrix entries $h_{ij}$, which do not satisfy assumption \ref{moments}, and still expect universality to hold. For instance, let $X$ be a random variable with variance $1$ (but infinite $(2 + \varepsilon)$-th moment for any $\varepsilon > 0$), take $\binom{N + 1}{2}$ mutually independent copies $\{ X_{ij} \}_{1 \le i \le j \le N}$ of $X$, and set $h_{ij} = h_{ji} = N^{-1 / 2} X_{ij}$ for each $1 \le i, j \le N$. Then it is plausible that the local spectral statistics of the resulting matrix $\textbf{H}$ are universal.
In fact, one could also consider matrices whose entries have infinite variance; this leads to the study of \emph{L\'{e}vy matrices} \cite{SHTRM, TM, LSLTM}. For these matrices, the semicircle law no longer governs the limit shape of the empirical spectral density \cite{SHTRM}. However, it is still predicted \cite{LSLTM} that the local statistics of these matrices should be universal at sufficiently small energy levels $E \in \mathbb{R}$. We will not pursue this here but refer to the papers \cite{DSERM, LDEHTRM} for partial progress in that direction.
\end{rem}
In particular, one can consider the generalized Wigner matrices given by the \emph{Gaussian Orthogonal Ensemble} (GOE). This is defined to be the $N \times N$ real symmetric random matrix $\textbf{GOE}_N = \{ g_{ij} \}$, where $g_{ij}$ is a Gaussian random variable with variance $2 N^{-1}$ if $i = j$ and $N^{-1}$ otherwise. This particular ensemble of matrices is exactly solvable through the framework of orthogonal polynomials and Pfaffian point processes, and the local gap statistics and correlation functions can be evaluated explicitly in the large $N$ limit; we will not state these results here, but they can be found in Chapter 6 of \cite{RM} or Chapter 3.9 of \cite{TRM}.
That said, the Wigner-Dyson-Mehta conjecture can be rephrased by stating that the gap statistics and correlation functions of an $N \times N$ generalized Wigner matrix converge to those of $\textbf{GOE}_N$, as $N$ tends to $\infty$. This can be written more precisely as follows.
\begin{definition}
\label{gapscorrelations}
For each integer $N \ge 1$, let $\textbf{H} = \textbf{H}_N$ be an $N \times N$ real symmetric random matrix and let $i_N$ be a positive integer. We say that \emph{the gap statistics of $\textbf{\emph{H}}$ are universal near the $i$-th eigenvalue} if the following holds. Fix positive integers $k$ and $j_1, j_2, \ldots , j_k$. For any compactly supported smooth function $F \in \mathcal{C}_0^{\infty} (\mathbb{R}^k)$, we have that
\begin{flalign}
\label{universality1}
\begin{aligned}
\displaystyle\lim_{N \rightarrow \infty} \bigg| & \mathbb{E}_{\textbf{H}} \Big[ F \big( N (\lambda_i - \lambda_{i + j_1}), N (\lambda_i - \lambda_{i + j_2}), \ldots , N (\lambda_i - \lambda_{i + j_k} )\big) \Big] \\
& - \mathbb{E}_{\textbf{GOE}_N} \Big[ F \big( N (\lambda_i - \lambda_{i + j_1}), N (\lambda_{i} - \lambda_{i + j_2}), \ldots , N (\lambda_i - \lambda_{i + j_k} ) \big) \Big] \bigg| = 0.
\end{aligned}
\end{flalign}
Furthermore, for a fixed real number $E \in \mathbb{R}$, we say that \emph{the correlation functions of $\textbf{\emph{H}}$ are universal at energy $E$} if the following holds. For any positive integer $k$ and any compactly supported smooth function $F \in \mathcal{C}_0^{\infty} (\mathbb{R}^k)$, we have that
\begin{flalign}
\label{universality2}
\begin{aligned}
\displaystyle\lim_{N \rightarrow \infty} \Bigg| \displaystyle\int_{\mathbb{R}^k} & F (a_1, a_2, \ldots , a_k) \bigg( p_{\textbf{H}}^{(k)} \Big( E + \displaystyle\frac{a_1}{N \rho_{\semicircle} (E)}, E + \displaystyle\frac{a_2}{N \rho_{\semicircle} (E)}, \ldots , E + \displaystyle\frac{a_k}{N \rho_{\semicircle} (E)} \Big) \\
& \qquad - p_{\textbf{GOE}_N}^{(k)} \Big( E + \displaystyle\frac{a_1}{N \rho_{\semicircle} (E)}, E + \displaystyle\frac{a_2}{N \rho_{\semicircle} (E)}, \ldots , E + \displaystyle\frac{a_k}{N \rho_{\semicircle} (E)} \Big) \bigg) \displaystyle\prod_{j = 1}^k d a_j \Bigg| = 0.
\end{aligned}
\end{flalign}
\end{definition}
The purpose of this paper is to establish the following two results, which establish that both the gap statistics and correlation functions of generalized Wigner matrices are universal in the bulk.
\begin{thm}
\label{gapsfunctions}
Fix real numbers $\varepsilon, \kappa, c_1, C_1, C_2 > 0$. Let $\{ \textbf{\emph{H}} = \textbf{\emph{H}}_N \}_{N \in \mathbb{Z}_{\ge 1}} $ denote a family of generalized Wigner matrices, as in Definition \ref{momentassumption} (with parameters $\varepsilon, c_1, C_1, C_2$), and let $i = i_N \in [\kappa N, (1 - \kappa) N]$ be a positive integer. Then, the gap statistics of $\textbf{\emph{H}}$ are universal near the $i$-th eigenvalue as in \eqref{universality1} of Definition \ref{gapscorrelations}.
\end{thm}
\begin{thm}
\label{bulkfunctions}
Fix real numbers $\varepsilon, \kappa, c_1, C_1, C_2 > 0$ and a real number $E \in [\kappa - 2, 2 - \kappa]$. Let $\{ \textbf{\emph{H}} = \textbf{\emph{H}}_N \}_{N \in \mathbb{Z}_{\ge 1}} $ denote a family of generalized Wigner matrices, as in Definition \ref{momentassumption} (with parameters $\varepsilon, c_1, C_1, C_2$). Then, the correlation functions of $\textbf{\emph{H}}$ are universal at energy level $E$ as in \eqref{universality2} of Definition \ref{gapscorrelations}.
\end{thm}
Observe that Theorem \ref{gapsfunctions} and Theorem \ref{bulkfunctions} above are only stated for real symmetric matrices. However, after minor modification, our methods and results should also apply to complex Hermitian random matrices (whose local statistics will instead converge to those of the GUE as $N$ tends to $\infty$); in order to avoid complicated notation later in the paper, we will not pursue this further.
We conclude this section by mentioning that it is necessary to take the index $i$ and the energy level $E$ (from Theorem \ref{gapsfunctions} and Theorem \ref{bulkfunctions}, respectively) to be in the bulk of the spectrum. Indeed, it is possible for the \emph{edge} local spectral statistics of Wigner matrices satisfying assumption \ref{moments} to be non-universal \cite{CLEHTRM}. In fact, in \cite{CEUM} Lee and Yin showed that edge universality of Wigner matrices holds if and only if the $h_{ij} \sqrt{N}$ have finite weak fourth moment.
\subsection{Context}
\label{Outline1}
In this section we provide some context for Theorem \ref{gapsfunctions} and Theorem \ref{bulkfunctions} by explaining their relationship with some previous results in the field.
Although the local spectral statistics of the GUE (Gaussian Unitary Ensemble) and GOE were found explicitly by Mehta-Gaudin \cite{ODERM} and Mehta \cite{RM} in the early 1960s, the question of universality for Wigner matrices had seen few results until the work of Johansson \cite{ULSDCM} in 2001. In that paper Johansson considered Hermitian \emph{Gaussian divisible matrices}, that is, matrices of the form $\textbf{H} + t \textbf{GUE}_N$, where $\textbf{H}$ is an $N \times N$ Hermitian Wigner matrix, $\textbf{GUE}_N$ is an $N \times N$ independent GUE matrix, and $t$ is a constant of order $1$. Through asymptotic analysis of the Br\'{e}zin-Hikami identity, Johansson showed \cite{ULSDCM} that the correlation functions of these matrices are universal.
There were two limitations to this method. The first is that the Br\'{e}zin-Hikami identity is only valid for complex Hermitian matrices and thus gave no results for real symmetric matrices. The second is that the Gaussian perturbation $t \textbf{GUE}_N$ happens to not be so immediately removed.
Now these issues have been overcome through what is known as the \emph{three-step strategy} for establishing bulk universality in random matrices, developed in the papers \cite{UM, URMLRF, ERLRM, LRFULSRM, DRM, ULES}, almost 10 years after Johansson's work \cite{ULSDCM}. Since this is the route we will eventually follow, we briefly outline it below; for a detailed review of the method, we refer to the survey \cite{URM} or the more comprehensive book \cite{DRM}.
\begin{enumerate}
\item{ \label{localcircle} The first step is to establish a \emph{local semicircle law} for the generalized Wigner matrix $\textbf{H}$, meaning that the spectral density of $\textbf{H}$ asymptotically follows that of the semicircle law \eqref{rhodefinition} on scales nearly of order $N^{-1}$. }
\item{ \label{perturb} The second step is to consider a perturbation $\textbf{H} + t \textbf{GOE}_N$ of the original random matrix $\textbf{H}$, where $t$ is small (optimally nearly of order $N^{-1}$). Using the local semicircle law from step \ref{localcircle}, one shows that the local statistics of the perturbed matrix are universal. }
\item{ \label{originalmatrix} The third step is to compare the local statistics of the original matrix $\textbf{H}$ and the perturbed matrix $\textbf{H} + t \textbf{GOE}_N$, and show that they are asymptotically the same if $t$ is small. }
\end{enumerate}
The first proofs of the local semicircle law appeared in the papers \cite{ERLRM, LSLCDRM}, although it has seen several improvements \cite{SSG, LSLGCRM, BUGM, LSLUMC, LSLM, CEUM} since then.
The first proof of the second step appeared in \cite{UM} for complex Hermitian Wigner matrices, this time by combining the Br\'{e}zin-Hikami identity with the local semiciricle law. Later, however, through a very different and more analytic method, the second step was extended to real symmetric Wigner matrices \cite{URM, LRFULSRM}, under a slightly weaker topology than stated in \eqref{universality2}. The originally stringent rigidity conditions under which this universality could be proven were later weakened \cite{URMFTD, CLSM}. More recently \cite{FEUGM, FEUM}, the topology under which this universality held was strengthened to what was stated in \eqref{universality2} above.
When the laws of the entries of the matrix $\textbf{H}$ are not smooth, the third step was originally performed by Tao-Vu \cite{RME, ULES}, in which works they developed the \emph{Four Moment Theorem}, which essentially states that if $\textbf{H}$ and $\widetilde{\textbf{H}}$ are complex Hermitian (or real symmetric) matrices with mutually independent entries whose first four moments are finite and coincide, then the local spectral statistics of $\textbf{H}$ and $\widetilde{\textbf{H}}$ converge in the large $N$ limit. This result had in the past been used to establish universality for a wide class of models \cite{SSGESEE, CEUM, RCMULS, ULES, UCM}. However, it was later \cite{FEUGM, SSSG, BUSM} realized that this method could be significantly simplified through an application of It\^{o}'s Lemma, if the value of $t$ from step \ref{perturb} is sufficiently small.
This three-step strategy is remarkably general; it has been applied to establish universality for local statistics of Wigner matrices in many different contexts \cite{FEUGM, SSGESEE, UM, BUMSD, URMLRF, LRFULSRM, BUCM, BUGM, BUSM, FEUM, RCMULS, ULES, UCM}. In fact, it has also been recently used to establish bulk universality of random matrices whose entries exhibit various forms of correlation \cite{LESRMWGSRC, LSLRRG, BESRRG, URMCE}.
However, until now, all known proofs of these universality results required a growth hypothesis on the entries of the matrix \textbf{H} that is stronger than assumption \ref{moments}. Originally \cite{UM, BUMSD, URMLRF, LRFULSRM, BUGM, ULES}, this took the form of a subexponential decay condition on the entries $h_{ij}$ of $\textbf{H}$, which stipulated that $\mathbb{P} \big[ |h_{ij} \sqrt{N}| > r \big] < C \exp(-c r^c)$ for all $i, j$ and some constants $c, C > 0$.
The reason for this comes from the proof the local semicircle law, which is needed to proceed with the second and third steps of the three-step strategy. Specifically, the proof of the local semicircle law requires large deviation estimates that are obtained by taking very large moments of (functionals of) the entries of $\textbf{H}$. If higher moments of $h_{ij} \sqrt{N}$ are infinite, it is no longer immediately apparent that these large deviation estimates remain valid.
There have been several attempts to weaken this decay condition, typically through different truncation arguments. For instance, in \cite{RCMULS}, Tao-Vu establish universality for Wigner matrices (and also covariance matrices) under the assumption that $h_{ij} \sqrt{N}$ has finite $C$-th moment for sufficiently large $C$; they took $C = 10^4$ and made no attempt to optimize, but it seems unlikely that their method could take $C$ close to $4$.
Later, in \cite{UCMMC}, Johansson used a more refined analysis of the Br\'{e}zin-Hikami identity to establish bulk universality of complex Hermitian Gaussian divisible matrices $\textbf{H}$ whose entries $|h_{ij} \sqrt{N}|$ have only two moments. Again, this only applied to complex Hermitian random matrices and therefore did not yield results on real symmetric Wigner matrices. Furthermore, removing the Gaussian component remained troublesome, particularly since the Four Moment Theorem of Tao and Vu could no longer be applied (the $|h_{ij} \sqrt{N}|$ do not have four moments).
Moreover, in \cite{SSGESEE}, Erd\H{o}s-Knowles-Yau-Yin implemented a new truncation procedure, based on the local semicircle law for sparse graphs \cite{SSG}, to prove bulk universality for $\textbf{H}$ when the $h_{ij} \sqrt{N}$ have $C = 4 + \varepsilon$ moments. Before this work, $4 + \varepsilon$ had been the lowest value of $C$ one could take.
Again, the main part in the three-step strategy that requires the moment condition is the proof of the local semicircle law. Partly for that reason, there have been several works \cite{LSLUMC, LSLM, LENM} analyzing the extent to which the local semicircle law remains valid under perturbed moment conditions. For example, the recent work of G\"{o}tze-Naumov-Tikhomirov \cite{LSLUMC} and G\"{o}tze-Naumov-Timushev-Tikhomirov \cite{LSLM} establishes a strong local semicircle law for Wigner matrices whose entries $h_{ij} \sqrt{N}$ again have at least $C = 4 + \varepsilon$ moments; as before, this had until now been the smallest value of $C$ one could take in order to prove a local semicircle law.
Hence, each of the methods mentioned above appears to exhibit a block preventing verification of bulk universality for (generalized) Wigner matrices whose entries have less than four moments. One possible reason for this is that the qualitative behavior of a matrix $\textbf{H}$ with infinite $(4 - \varepsilon)$-th moment is different from that of a matrix $\widetilde{\textbf{H}}$ with finite $(4 + \varepsilon)$-moment. In particular, the entries of $\widetilde{\textbf{H}}$ are expected to decay with $N$; one can show that, with high probability, the largest entries of $\widetilde{\textbf{H}}$ are at most of order $N^{-\varepsilon / 4}$. Until now, this decay of the entries seemed to be what was needed in the proof of a local semicircle law (see, for example, Theorem 3.1 of \cite{CEUM}). However, it is possible (and in some cases expected) that several entries of the more heavy-tailed matrix $\textbf{H}$ will grow with $N$. This poses issues in all known proofs \cite{SSG, LSLGCRM, ERLRM, LSLCDRM, LSLUMC, LSLM, CEUM} of local semicircle laws.
Our purpose here is to overcome these issues and establish bulk universality for generalized Wigner matrices only subject to the (essentially weakest possible) assumption \ref{moments}. The main novelty of this paper that allows us to do this is the proof of a local semicircle law for Wigner matrices whose entries have less than four (and in fact only $2 + \varepsilon$) moments.
\subsection{The Local Semicircle Law}
\label{LocalCircleLaw}
The local semicircle law for random matrices can be stated in several different ways. The formulation of interest to us is in terms of the \emph{Stieltjes transform} of empirical spectral distribution $\mu_{\textbf{H}}$; this is defined by the function
\begin{flalign}
\label{mn}
m_N = m_N (z) = m_{N; \textbf{H}} (z) = \displaystyle\frac{1}{N} \displaystyle\sum_{j = 1}^N \displaystyle\frac{1}{\lambda_j - z} = \displaystyle\frac{1}{N} \Tr \big( \textbf{H} - z \big)^{-1},
\end{flalign}
\noindent for any $z \in \mathbb{H}$, where $\mathbb{H} = \{ z \in \mathbb{C} : \Im z > 0 \}$ denotes the upper half plane.
In view of the fact that $\mu_{\textbf{H}}$ converges weakly to the semicircle law $\rho_{\semicircle}$ \eqref{rhodefinition} as $N$ tends to $\infty$, one expects $m_N (z)$ to converge to $m_{\semicircle} = m_{\semicircle} (z) = \int_{\mathbb{R}} \rho_{\semicircle} (x) dx / (x - z)$. It can be quickly verified that $m_{\semicircle} (z)$ is the unique solution $m$ with positive imaginary part to the quadratic equation
\begin{flalign}
\label{mquadratic}
m^2 + zm + 1 = 0.
\end{flalign}
Denoting $E = \Re z$ and $\eta = \Im z$ (so that $z = E + \textbf{i} \eta$), observe that $\lim_{\eta \rightarrow 0} \Im m_N (z)$ converges weakly to the distribution $(\pi N)^{-1} \sum_{j = 1}^N \delta_{\lambda_j = E}$, which provides information about the spectral behavior of $\textbf{H}$ near the energy level $E$. This suggests that, in order to understand the local spectral properties of $\textbf{H}$ near $\eta$, one might try to understand the behavior of $m_N (z)$ when $\eta$ is very small.
A \emph{local semicircle law} for $\textbf{H}$ is an estimate on $\big| m_N (z) - m_{\semicircle} (z) \big|$ when $\eta$ is nearly of order $N^{-1}$. Establishing local semicircle laws is also often a task of independent interest \cite{LSSRRGFD, LSLRRG, SSG, LSLGCRM, LSLCDRM, LSLUMC, LSLM, BCESDSL, LENM}, since it quantifies the convergence of the spectral distribution of the random matrix $\textbf{H}$ to the semicircle law on very small sub-intervals (approximately of size $N^{-1}$) of $[-2, 2]$. It also has other consequences, such as complete eigenvector delocalization (to be discussed below).
The following theorem provides such an estimate for generalized Wigner matrices. In what follows, we define the domain
\begin{flalign}
\label{dkappanr}
\mathscr{D}_{\kappa; N; r} = \big\{ z = E + \textbf{i} \eta \in \mathbb{H} : E \in [\kappa - 2, 2 - \kappa], \eta \in [r, 5] \big\},
\end{flalign}
\noindent for any integer $N > 0$ and real numbers $\kappa, r = r_N > 0$. Further define $\mathscr{D}_{\kappa; N} = \mathscr{D}_{\kappa; N; \varphi}$, where $\varphi = \varphi_N = (\log N)^{8 \log \log N} N^{-1}$.
\begin{thm}
\label{localmoments}
Fix $\kappa > 0$, let $N > 0$ be a positive integer, and let $\textbf{\emph{H}} = \textbf{\emph{H}}_N$ be an $N \times N$ generalized Wigner matrix as in Definition \ref{momentassumption}. Then, there exist constants $C, c, \xi > 0$ (that only depend on $\varepsilon, c_1, C_1, C_2, \kappa$) such that
\begin{flalign}
\label{localmomentsequation}
\mathbb{P} \Bigg[ \bigcup_{z \in \mathscr{D}_{\kappa; N}} \bigg\{ \big| m_N (z) - m \big| > C (\log N)^{\xi} \Big( \displaystyle\frac{1}{\sqrt{N \eta}} + N^{-c \varepsilon } \Big) \bigg\} \Bigg] < C N^{-c \log \log N}.
\end{flalign}
\end{thm}
\begin{rem}
\label{stronglaw}
Theorem \ref{localmoments} is known as a \emph{weak local semicircle law}, since the error term in \eqref{localmomentsequation} is not optimal in terms of $N \eta$. A \emph{strong local semicircle law} \cite{LSLGCRM} would correspond to the estimate \eqref{localmomentsequation}, with the term $(N \eta)^{-1 / 2}$ replaced by the smaller error $(N \eta)^{-1}$. It is possible to establish this strong version of Theorem \ref{localmoments} by suitably combining our method with Theorem 5.6 of \cite{SSG}. However, this will not be necessary for us, so we do not pursue it.
\end{rem}
\begin{rem}
Observe the term $N^{-c \varepsilon}$ appearing as an error in \eqref{localmomentsequation}; it is also present in the local semicircle law for sparse Erd\H{o}s-Renyi graphs and sparse regular random graphs (see Theorem 2.8 of \cite{SSG} and Theorem 1.1 of \cite{LSLRRG}, respectively). Although this additional term has little impact when $\eta = \Im z$ is small (nearly of order $N^{-1}$), it indicates a possible obstruction of convergence from $m_N (z)$ to $m_{\semicircle}$ when $\eta = \Im z$ is large (nearly of order $1$).
In particular, it suggests that $m_N (z)$ might converge to $m_{\semicircle}$ at rate $N^{-c \varepsilon}$, instead of at the fastest possible rate $N^{-1}$, which was established by G\"{o}tze-Tikhomirov \cite{BCESDSL} in the case when the fourth moments of $h_{ij} \sqrt{N}$ are bounded. In view of the results of \cite{LSHTRM}, we do not believe that \eqref{localmomentsequation} is optimal for small $\varepsilon$, when the $h_{ij} \sqrt{N}$ only have $(2 + \varepsilon)$ moments. Instead, we find it plausible that the error term $N^{-c \varepsilon}$ in \eqref{localmomentsequation} should replaced by $N^{-1 / 2 - c \varepsilon}$. However, the weaker estimate \eqref{localmomentsequation} will suffice to establish bulk universality, so we do not pursue these improvements any further.
\end{rem}
To the best of our knowledge, all known proofs of local semicircle laws rely on a detailed understanding of the \emph{resolvent} of $\textbf{H}$, defined to be the $N \times N$ matrix $\textbf{G} = \textbf{G} (z) = \textbf{G} (z, \textbf{H}) = \big( \textbf{H} - z \big)^{-1} = \big\{ G_{ij} (z) \big\} = \big\{ G_{ij} \big\}$. Indeed, since $m_N = N^{-1} \Tr \textbf{G}$, it suffices to estimate the diagonal entries of $\textbf{G}$. In most known cases (with the exception of the very recent work \cite{LSSRRGFD} on regular random graphs of finite degree), it happens that all entries $G_{ij}$ of the resolvent will be close to $\textbf{1}_{i = j} m_{\semicircle}$ with very high probability.
In the low-moment setting, this will not quite be the case. Similar to in \cite{LSSRRGFD}, the following will instead hold. For ``most'' pairs $i, j \in [1, N]$, we will have that $G_{ij}$ is close to $\textbf{1}_{i = j} m_{\semicircle}$; however, there will be a small fraction of index pairs $(i, j)$ for which this will not be true. Still, it will hold that these few entries remain uniformly bounded with very high probability as $N$ tends to $\infty$.
The result is more specifically stated as follows.
\begin{thm}
\label{estimate1gij}
Under the same assumptions as in Theorem \ref{localmoments}, there exist constants $C, c, \xi > 0$ (that only depend on $\varepsilon, c_1, C_1, C_2, \kappa$) such that the following two estimates hold.
First, we have that
\begin{flalign}
\label{gijestimate}
\mathbb{P} \Bigg[ \displaystyle\max_{1 \le i, j \le N} \displaystyle\sup_{z \in \mathscr{D}_{\kappa; N}} \big| G_{ij} (z) \big| > C \Bigg] < C N^{- c \log \log N}.
\end{flalign}
Second, if we set $s = s_N = \lfloor C N^{1 - c \varepsilon} \rfloor$, then
\begin{flalign}
\label{gijestimate2}
\begin{aligned}
\mathbb{P} \Bigg[ \bigcup_{\substack{I \subseteq [1, N] \\ |I| \ge s}} & \bigg\{ \displaystyle\min_{i \in I} \displaystyle\max_{1 \le j \le N} \displaystyle\sup_{z \in \mathscr{D}_{\kappa; N}} \big| G_{i j} (z) - \textbf{\emph{1}}_{i = j} m_{\semicircle} (z) \big| > C (\log N)^{\xi} \Big( \displaystyle\frac{1}{\sqrt{N \eta}} + N^{-c \varepsilon} \Big) \bigg\} \Bigg] \\
& < C N^{- c \log \log N}.
\end{aligned}
\end{flalign}
\end{thm}
Observe that Theorem \ref{localmoments} is a direct consequence of Theorem \ref{estimate1gij} and the estimate
\begin{flalign}
\label{gdt}
\big| m_N - m_{\semicircle} \big| = \big| N^{-1} \Tr \textbf{G} - m_{\semicircle} \big| \le N^{-1} \displaystyle\sum_{i \in I} \big| G_{ii} - m_{\semicircle} \big| + N^{-1} \displaystyle\sum_{i \in [1, N] \setminus I} \big| G_{ii} - m_{\semicircle} \big|,
\end{flalign}
\noindent which holds for any subset $I \subseteq [1, N]$. In particular, we apply \eqref{gdt} with $I$ equal to the set of indices $i$ such that $G_{ij}$ is not close to $\textbf{1}_{i = j} m_{\semicircle}$ for some $j$ (in the sense of \eqref{gijestimate2}), which has cardinality at most $s_N$ with high probability due to \eqref{gijestimate2}. We then use \eqref{gijestimate} to estimate the first sum on the right side of \eqref{gdt} and \eqref{gijestimate2} to estimate the second sum; this yields Theorem \ref{localmoments} assuming Theorem \ref{estimate1gij}. Thus, it suffices to establish the latter theorem.
Another consequence of Theorem \ref{estimate1gij} is the \emph{complete delocalization of eigenvectors} of $\textbf{H}$ corresponding to eigenvalues in the bulk; this is stated precisely by the following corollary. We mention that, in the low-moment case given by assumption \ref{moments}, this delocalization does not always hold for eigenvectors whose eigenvalues are at the edge of the spectrum of $\textbf{H}$ \cite{CLEHTRM, CEUM}.
\begin{cor}
\label{delocalization}
Under the same assumptions as in Theorem \ref{localmoments}, there exist constants $C, c, \xi > 0$ (that only depend on $\varepsilon, c_1, C_1, C_2, \kappa$) such that the following holds. The probability that there exists an eigenvalue $\lambda \in [\kappa - 2, 2 - \kappa]$ of $\textbf{\emph{H}}$ with eigenvector $\textbf{\emph{v}}$ such that $\| \textbf{\emph{v}} \|_2 = 1$ and $\| \textbf{\emph{v}} \|_{\infty} > C (\log N)^{\xi} N^{- 1 / 2}$ is less than $C N^{- c \log \log N}$.
\end{cor}
Given Theorem \ref{estimate1gij}, the proof of Corollary \ref{delocalization} is very similar to that of Theorem 2.21 of \cite{URM} and is therefore omitted.
Thus, in order to establish the local semicircle law Theorem \ref{localmoments} and complete eigenvector delocalization Corollary \ref{delocalization}, it remains to establish Theorem \ref{estimate1gij}. We will outline the proof of this result in Section \ref{ProofOutline}, explaining why previous proofs of semicircles no longer seem to directly apply and also indicating some of the new elements needed to show Theorem \ref{estimate1gij}. In Sections \ref{DefineIndices}, \ref{LawLarge}, \ref{LawSmallNotDeviant}, \ref{LawSmallDeviant}, and \ref{ProofLocalLaw}, we implement this outline in detail. In Section \ref{LocalH}, we proceed with the remaining two parts of the three-step strategy and establish the bulk universality results Theorem \ref{gapsfunctions} and Theorem \ref{bulkfunctions}; given Theorem \ref{localmoments}, this will mainly involve recalling what was done in the recent works \cite{FEUM, BUSM} that individually address these second and third steps, respectively.
\subsection*{Acknowledgements}
The author heartily thanks Horng-Tzer Yau for proposing this question, for several valuable discussions, and for looking through an early version of this paper. The author is also very grateful to Jiaoyang Huang and Benjamin Landon for many fruitful conversations and helpful explanations. This work was funded by the NSF Graduate Research Fellowship under grant number DGE1144152 and partially by the Eric Cooper and Naomi Siegel Graduate Student Fellowship I.
\section{Outline of the Proof of the Local Semicircle Law}
\label{ProofOutline}
In this section we outline a proof of Theorem \ref{estimate1gij} and explain in what ways it differs from known proofs of the local semicircle law for Wigner matrices with less singular entries. We begin in Section \ref{DiagonalResolvent} with some matrix identities used for the analysis in both the original and heavy-tailed setting. In Section \ref{LargeMoment} we recall the idea of the proof when the $h_{ij} \sqrt{N}$ have all moments and explain the issues that arise when this restriction is no longer assumed. Then, in Section \ref{SmallMoment} we outline how to resolve these issues. Throughout this section, we adopt the notation of Theorem \ref{estimate1gij}.
\subsection{Estimating the Diagonal Resolvent Entries}
\label{DiagonalResolvent}
We begin by collecting several matrix identities that will be useful for us. In what follows, for any $N \times N$ matrix $\textbf{M} = \{ M_{ij} \}$ and subset $\mathcal{S} \subset [1, N]$, let $\textbf{M}^{(\mathcal{S})} = \{ M_{ij}^{(\mathcal{S})} \}$ denote the $(N - |\mathcal{S}|) \times (N - |\mathcal{S}|)$ matrix formed from removing the $j$-th row and column from $\textbf{M}$, for each $j \in \mathcal{S}$. If $S = \{ i \}$ consists of one element, we abbreviate $\textbf{M}^{(\{ i \})} = \textbf{M}^{(i)}$.
\begin{lem}
Let $\textbf{\emph{H}}$ be an $N \times N$ (deterministic or random) matrix, $z \in \mathbb{H}$, and $\eta = \Im z$. Denote $\textbf{\emph{G}} = (\textbf{\emph{H}} - z)^{-1}$.
\label{matrixidentities}
\begin{enumerate}[align=left]
\item[\emph{1. Schur complement identity:} ]{ Suppose that $\textbf{\emph{A}}$, $\textbf{\emph{B}}$, $\textbf{\emph{C}}$, and $\textbf{\emph{D}}$ are generic $k \times k$, $k \times m$, $m \times k$, and $m \times m$ matrices, respectively. Then,
\begin{flalign}
\label{blockinverse}
\left[ \begin{array}{cc} \textbf{\emph{A}} & \textbf{\emph{B}} \\ \textbf{\emph{C}} & \textbf{\emph{D}} \end{array} \right]^{-1} = \left[ \begin{array}{cc} (\textbf{\emph{A}} - \textbf{\emph{B}} \textbf{\emph{D}}^{-1} \textbf{\emph{C}})^{-1} & (\textbf{\emph{B}} \textbf{\emph{D}}^{-1} \textbf{\emph{C}} - \textbf{\emph{A}})^{-1} \textbf{\emph{B}} \textbf{\emph{D}}^{-1} \\ \textbf{\emph{D}}^{-1} \textbf{\emph{C}} (\textbf{\emph{B}} \textbf{\emph{D}}^{-1} \textbf{\emph{C}} - \textbf{\emph{A}})^{-1} & \textbf{\emph{D}}^{-1} + \textbf{\emph{D}}^{-1} \textbf{\emph{C}} (\textbf{\emph{A}} - \textbf{\emph{B}} \textbf{\emph{D}}^{-1} \textbf{\emph{C}}) \textbf{\emph{B}} \textbf{\emph{D}}^{-1} \end{array} \right].
\end{flalign}
\noindent In particular, for any $i \in [1, N]$ we have that
\begin{flalign}
\label{gii}
\displaystyle\frac{1}{G_{ii}} = h_{ii} - z - \sum_{\substack{1 \le j, k \le N \\ j, k \ne i }} h_{ij} G_{jk}^{(i)} h_{ki},
\end{flalign}}
\item[\emph{2. Resolvent identity:}]{If $\textbf{\emph{A}}$ and $\textbf{\emph{B}}$ are square matrices of the same dimension, then
\begin{flalign}
\label{resolvent}
\textbf{\emph{A}}^{-1} - \textbf{\emph{B}}^{-1} = \textbf{\emph{A}}^{-1} \big( \textbf{\emph{B}} - \textbf{\emph{A}} \big) \textbf{\emph{B}}^{-1}.
\end{flalign}
\noindent In particular, for any distinct $i, j \in [1, N]$, we have that
\begin{flalign}
\label{gij}
G_{ij} & = - G_{ii} \displaystyle\sum_{\substack{k \in [1, N] \\ k \ne i}} h_{ik} G_{kj}^{(i)} = -G_{jj} \displaystyle\sum_{\substack{k \in [1, N] \\ k \ne j}} h_{kj} G_{ik}^{(j)}.
\end{flalign}
\noindent Thus, for all $i, j, k \in [1, N]$ such that $i \notin \{ j, k \}$, we have that
\begin{flalign}
\label{gkj}
G_{kj} = G_{kj}^{(i)} + \displaystyle\frac{G_{ki} G_{ij}}{G_{ii}}.
\end{flalign}}
\item[\emph{3. Deterministic estimates:}]{For any $i, j \in [1, N]$, we have that
\begin{flalign}
\label{gijeta}
\big| G_{ij} \big| < \eta^{-1}.
\end{flalign}}
\item[\emph{4. Ward identity:}]{Let $\mathcal{S} \subset [1, N]$. For any $j \in [1, N] \setminus \mathcal{S}$, we have that
\begin{flalign}
\label{sumgij}
\displaystyle\sum_{j \in [1, N] \setminus \mathcal{S}} \big| G_{jk}^{(\mathcal{S})} \big|^2 = \displaystyle\frac{\Im G_{jj}^{(\mathcal{S})}}{\eta}.
\end{flalign}}
\end{enumerate}
\end{lem}
Each of the statements above can be found in the book \cite{DRM}. Specifically, \eqref{blockinverse} can be found as (7.2) and (7.3) there; \eqref{gii} as (7.7); \eqref{resolvent} as (8.4); \eqref{gij} as (8.2); \eqref{gkj} as (8.1); \eqref{gijeta} as (8.34); and \eqref{sumgij} as (8.3).
The proof of the local semicircle law is based on the identity \eqref{gii}; let us explain why it is useful. To that end, observe that sum on the left side of \eqref{gii} can be rewritten as
\begin{flalign}
\label{sumedmf}
\sum_{\substack{1 \le j, k \le N \\ i \notin \{ j , k \} }} h_{ij} G_{jk}^{(i)} h_{ki} = F_i + E_i + D_i + M_i + m_{\semicircle} + m_{\semicircle} t_i
\end{flalign}
\noindent where
\begin{flalign}
\label{edmf}
\begin{aligned}
F_i & = \sum_{\substack{1 \le j \ne k \le N \\ j, k \ne i }} h_{ij} G_{jk}^{(i)} h_{ki}; \qquad \quad E_i = \sum_{\substack{1 \le j \le N \\ j \ne i }} \big( | h_{ij} |^2 - s_{ij} \big) G_{jj}^{(i)}; \\
D_i & = \sum_{\substack{1 \le j \le N \\ j \ne i }} s_{ij} \big( G_{jj}^{(i)} - G_{jj} \big); \qquad M_i = \sum_{\substack{1 \le j \le N \\ j \ne i }} s_{ij} \big( G_{jj} - m_{\semicircle} \big),
\end{aligned}
\end{flalign}
\noindent where we recalled from assumption \ref{generalized} that $s_{ij} = \Var h_{ij}$ and from assumption \ref{stochastic} that $t_i = \sum_{j = 1}^N s_{ij} - 1$. Thus, if we denote
\begin{flalign}
\label{vj}
v_j = G_{jj} - m_{\semicircle},
\end{flalign}
\noindent for each $j \in [1, N]$ and insert \eqref{sumedmf} into \eqref{gii}, we deduce that
\begin{flalign*}
\displaystyle\frac{1}{v_i + m_{\semicircle}} + z + m_{\semicircle} = h_{ii} - F_i - E_i - D_i - M_i - m_{\semicircle} t_i.
\end{flalign*}
\noindent Applying \eqref{mquadratic}, it follows that
\begin{flalign}
\label{vmfed}
\displaystyle\frac{v_i}{1 + m_{\semicircle}^{-1} v_i} - m_{\semicircle}^2 \displaystyle\sum_{j = 1}^N s_{ij} v_i = m_{\semicircle}^2 \big( F_i + E_i + D_i - h_{ii} + m_{\semicircle} t_i \big).
\end{flalign}
\noindent Now, recall from assumption \ref{stochastic} that $|t_i| < C_1 N^{-varespilon}$. Suppose that we additionally knew that
\begin{flalign}
\label{vmfedsmall}
|F_i|, |E_i|, |D_i|, |h_{ii}| = \mathcal{O} \Bigg( (\log N)^{C_3} \left( \displaystyle\frac{1}{\sqrt{N \eta}} + N^{- c_2 \varepsilon} \right) \Bigg); \qquad |v_i| = o (1).
\end{flalign}
\noindent Then, \eqref{vmfed} would imply that
\begin{flalign}
\label{vsmall}
\big( \Id - m_{\semicircle}^2 \textbf{S} \big) \textbf{v} = \mathcal{O} \Bigg( (\log N)^{C_3} \left( \displaystyle\frac{1}{\sqrt{N \eta}} + N^{- c_2 \varepsilon} \right) + \displaystyle\max_i |v_i|^2 \Bigg) ,
\end{flalign}
\noindent where the $N \times N$ matrix $\textbf{S} = \{ s_{ij} \}$ and the $N$-dimensional vector $\textbf{v} = (v_1, v_2, \ldots , v_N)$. It can be shown that $\| \Id - m_{\semicircle}^2 \textbf{S} \|^{-1} = \mathcal{O} (\log N)$, from which we would obtain that
\begin{flalign}
\label{largevsmall}
\displaystyle\max_i |v_i| = \mathcal{O} \Bigg( (\log N)^{C_4} \left( \displaystyle\frac{1}{\sqrt{N \eta}} + N^{- c_2 \varepsilon} \right) \Bigg),
\end{flalign}
\noindent from which it would follow that each $|G_{ii} - m_{\semicircle}|$ is small as in \eqref{gijestimate2}. Thus, we would like to establish the two estimates \eqref{vmfedsmall}.
\subsection{When the \texorpdfstring{$h_{ij}$}{} Have All Moments}
\label{LargeMoment}
Let us first restrict to a case that is well understood, namely when the laws of the $h_{ij}$ exhibit subexponential decay (in this case, we can take $\varepsilon = \infty$). For simplicity, we also assume in this section that $s_{ij} = N^{-1}$ for all $i, j$. Under these assumptions, the estimates \eqref{vmfedsmall} will hold with very high probability (greater than $1 - N^{-10}$, for instance) for all indices $i \in [1, N]$; let us provide a heuristic as to why.
The second estimate in \eqref{vmfedsmall} (on $|v_i|$) will be a consequence of what is known as a \emph{multiscale argument} on $\eta$. First, it can be shown directly (see, for instance, Section \ref{LawLarge}) that the estimate \eqref{largevsmall} holds deterministically when $\Im z = \eta = \eta_0$ is relatively large, of order $1$; this is sometimes referred to as an \emph{initial estimate}. Then, we will slowly decrement $\eta$ through a sequence $\eta_0 > \eta_1 > \cdots > \eta_r \approx N^{-1}$; here, $r$ is some large integer (which will be of order $(\log N)^2$).
We will have that $\eta_k \approx \eta_{k + 1}$ for all $k$, so that $v_i (\eta_k) \approx v_i (\eta_{k + 1})$ for all $i$. Combining this with the estimate \eqref{largevsmall} on $\big| v_i (\eta_k) \big|$ will yield $\big| v_i (\eta_{k + 1}) \big| = o(1)$, which is the second estimate of \eqref{vmfedsmall}. Thus, if we could establish the first estimate (on $F_i$, $E_i$, and $D_i$) of \eqref{vmfedsmall}, we would deduce \eqref{vsmall} and thus \eqref{largevsmall} for $v_i (\eta_{k + 1})$. This allows us to increment $k$ from $0$ to $r$.
Now, let us explain the first estimate in \eqref{vmfedsmall}. As an example, we provide a heuristic as to why one would expect (with very high probability) the estimate on $|F_i|$ to hold. To that end, we take the second moment of $F_i$ to obtain that
\begin{flalign}
\label{fi21}
\begin{aligned}
\mathbb{E} \big[ |F_i|^2 \big] & = \sum_{\substack{1 \le j \ne k \le N \\ j, k \ne i }} \sum_{\substack{1 \le j' \ne k' \le N \\ j', k' \ne i }} \mathbb{E} \big[ h_{ij} G_{jk}^{(i)} h_{ki} h_{ij'} G_{j'k'}^{(i)} h_{k'i} \big] \\
& = 2 N^{-2} \sum_{\substack{1 \le j \ne k \le N \\ j, k \ne i }} \mathbb{E} \Big[ \big| G_{jk}^{(i)} \big|^2 \Big] = \displaystyle\frac{2}{N^2 \eta} \sum_{\substack{1 \le j \le N \\ j \ne i }} \mathbb{E} \Big[ \Im G_{jj}^{(i)} \Big] = \displaystyle\frac{1}{N \eta} \Big( 2 N^{-1} \Im \mathbb{E} \big[ \Tr \textbf{G}^{(i)} \big] \Big),
\end{aligned}
\end{flalign}
\noindent where we used the mutual independence of the $h_{ij}$, the independence of $\textbf{G}^{(i)}$ from the $h_{ij}$, and the Ward identity \eqref{sumgij}. The estimate $|v_i| = o(1)$ yields that $N^{-1} \Im \Tr \textbf{G}^{(i)} = m_{\semicircle} + o (1) = \mathcal{O} (1)$.
Inserting this into \eqref{fi21} yields $\mathbb{E} \big[ |F_i|^2 \big] = \mathcal{O} \big( (N \eta)^{-1} \big)$, which suggests that $|F_i|$ should be of order $(N \eta)^{-1 / 2}$. By taking very large moments (instead of only the second moment), one can show that this in fact holds with very high probability. Observe here that bounding higher moments of $|F_i|$ requires that the $h_{ij}$ have all moments. The estimates on $E_i$ and $D_i$ can either be done similarly or through other matrix identities.
The above procedure essentially describes the framework for establishing local semicircle laws when the matrix entries $h_{ij}$ are quite regular. However, this method does not seem to immediately work in the same way when the $h_{ij}$ have few (for instance, less than four) moments. In the next section, we explain why and outline how to resolve this.
\subsection{When the \texorpdfstring{$h_{ij}$}{} Have Few Moments}
\label{SmallMoment}
In this section we again assume that $s_{ij} = N^{-1}$ for all $i, j$, but we no longer require that the $h_{ij}$ have all moments. Instead, we only require that they have $2 + \varepsilon$ moments, as in assumption \ref{moments}. Let us explain which parts of Section \ref{LargeMoment} still apply and which do not.
The multiscale argument for the bound on $|v_i|$ would still essentially be valid, assuming that one were able to establish the first estimate of \eqref{vmfedsmall} with high probability. However, this does not appear to be possible. Indeed, the high-moment method used to establish these estimates (outlined in the previous section) no longer applies since the $h_{ij}$ have very few moments.
Instead, for fixed $i$, these bounds only hold with probability $1 - \mathcal{O} \big( N^{1 - c \varepsilon} \big)$, for some constant $c > 0$. This tends to $1$ as $N$ tends to $\infty$, but not as quickly as one may like. In particular, one cannot apply a union estimate to deduce that \eqref{vmfedsmall} likely holds for all $i \in [1, N]$ simultaneously.
In fact, we generally expect to see $\mathcal{O} \big( N^{1 - c \varepsilon} \big)$ indices $i \in [1, N]$ for which \eqref{vmfedsmall} is false. This is the phenomenon that makes Wigner matrices whose entries only have $2 + \varepsilon$ moments different from Wigner matrices whose entries have $4 + \varepsilon$ moments; in the latter model, a truncation procedure can be applied \cite{CEUM} to deduce that \eqref{vmfedsmall} holds for all indices $i$ with high probability.
In our setting, the set of indices $i$ for which this estimate does not hold will fall into an often non-empty class of what we call \emph{deviant} indices. Indices that are not deviant will be called \emph{typical}; both estimates \eqref{vmfedsmall} will hold for each typical index $j$, which will comprise the majority of $[1, N]$.
Thus, our task is essentially four-fold.
\begin{enumerate}
\item{ \label{definetd} We must give a precise definition of deviant and typical indices. Denoting the set of typical indices by $\mathcal{T}$ and the set of deviant indices by $\mathcal{D}$, we must also show that $\big| \mathcal{D} \big| = \mathcal{O} \big( N^{1 - c \varepsilon} \big)$, for some constant $c > 0$.}
\item{\label{initial} We require an initial estimate on $v_i$, in the case $\eta \approx 1$ is relatively large.}
\item{ \label{vmfedsmallt} Given the definitions from step \ref{definetd}, we must establish \eqref{vmfedsmall} for all $i \in \mathcal{T}$. After this, it will be possible to essentially ``restrict" \eqref{vsmall} to the set of all typical indices, which will suffice to establish \eqref{largevsmall}, where the maximum in that estimate instead ranges over all $i \in \mathcal{T}$. This will suffice to establish the estimate \eqref{gijestimate2} of Theorem \ref{estimate1gij}. }
\item{ \label{vmfedd} In order to establish the estimate \eqref{gijestimate} of Theorem \ref{estimate1gij}, we must also show that $|v_i| = \mathcal{O} (1)$ for each (deviant and typical) index $i$.}
\end{enumerate}
Combining results of step \ref{definetd}, step \ref{initial}, step \ref{vmfedsmallt}, and step \ref{vmfedd}, and the procedure outlined in Section \ref{DiagonalResolvent} will lead to the proof of Theorem \ref{estimate1gij}. The next several sections will go through these four steps in more detail. In particular, we implement step \ref{definetd}, step \ref{initial}, step \ref{vmfedsmallt}, and step \ref{vmfedd} in Section \ref{DefineIndices}, Section \ref{LawLarge}, Section \ref{LawSmallNotDeviant}, and Section \ref{LawSmallDeviant}, respectively. We will then conclude the proof of Theorem \ref{estimate1gij} in Section \ref{ProofLocalLaw}.
\section{Typical and Deviant Indices}
\label{DefineIndices}
In this section, we define and give properties of the sets of \emph{typical} and \emph{deviant} indices. The stimulus for these definitions comes from a comparison between heavy-tailed random matrices and adjacency matrices of sparse random graphs, for which a local semicircle law has already been established \cite{SSG}.
In particular, the authors of \cite{SSG} established a local semicircle law for random matrices $\textbf{H}$ whose entries satisfy a more stringent constraint than assumption \ref{moments}. We will not state their constraint in full generality, but for our purposes it was essentially that $\big| h_{ij} \big| < N^{-c}$ holds deterministically (this can be mildly weakened), for some constant $c > 0$ independent of $N$.
This assumption does not always hold in our setting. In particular, a Markov estimate shows (for example) that
\begin{flalign}
\label{pijestimate1}
\mathbb{P} \big[ |h_{ij}| \ge N^{-\varepsilon / 10} \big] \le \displaystyle\frac{\mathbb{E} \big[ |h_{ij} \sqrt{N}|^{2 + \varepsilon} \big]}{\big| N^{1 / 2 - \varepsilon / 10} \big|^{2 + \varepsilon}} < \displaystyle\frac{C_2}{N^{1 + 3 \varepsilon / 10 - \varepsilon^2 / 10}} \le C_2 N^{-1 - \varepsilon / 10},
\end{flalign}
\noindent if $\varepsilon \le 2$. Thus, we expect there to exist $\mathcal{O} \big( N^{1 - \varepsilon / 10} \big)$ pairs $(i, j) \in [1, N]^2$ for which $|h_{ij}| > N^{-\varepsilon / 10}$. Such pairs $(i, j)$ will be called \emph{big}; pairs $(i, j)$ satisfying $|h_{ij}| \le N^{-\varepsilon / 10}$ are called \emph{amenable}.
Informally, an index $i \in [1, N]$ will be \emph{deviant} if there exists a $j \in [1, N]$ for which the pair $(i, j)$ is big; otherwise, $i$ will be \emph{typical}. Unfortunately, the sets of deviant and typical indices are random subsets of $[1, N]$; this will complicate the analysis in future sections.
Thus, in Section \ref{EntryTruncation}, we first resample the entries of $\textbf{H}$, essentially by conditioning on which pairs $(i, j) \subset [1, N]^2$ are amenable or big. This produces a symmetric $N \times N$ array, each of entry of which is either $A$ (amenable) or $B$ (big); we refer to this array as the \emph{$AB$ label} $\textbf{L} (\textbf{H})$ of $\textbf{H}$. Conditioning on $\textbf{L} (\textbf{H})$, the deviant and typical indices of $\textbf{H}$ become deterministic. We give a definition (as well as some properties) of these indices in Section \ref{IndicesGraph}. In Section \ref{deviantnondeviantlocallaw}, we explain how the notions introduced in Section \ref{EntryTruncation} and Section \ref{IndicesGraph} can be used to provide a reformulation of Theorem \ref{estimate1gij}.
\subsection{Resampling }
\label{EntryTruncation}
As outlined above, we first define the $AB$ label associated with a matrix $\textbf{M}$.
\begin{definition}
\label{ml}
Let $\textbf{M} = \{ m_{ij} \}$ be an $N \times N$ matrix. The \emph{$AB$ label} of $\textbf{M}$, denoted $\textbf{L} (\textbf{M}) = \{ L_{ij} \}$, is the $N \times N$ array, whose entries are either equal to $A$ or $B$, such that $L_{ij} = A$ if $|m_{ij}| \le N^{-\varepsilon / 10}$ and $L_{ij} = B$ otherwise.
\end{definition}
Now, we can resample $\textbf{H}$ by first choosing its $AB$ label $\textbf{L}$ and then by sampling the entries conditioned on $\textbf{L}$. Let us explain this in more detail. In what follows, we assume that the densities of the matrix entries $h_{ij}$ are smooth and nonzero everywhere. This is primarily for notational convenience and can be arranged by adding a small Gaussian component to $\textbf{H}$ (of order $e^{-N}$, for instance); using \eqref{resolvent} and \eqref{gijeta}, one can quickly verify that (with very large probability) this perturbation does not affect the asymptotics of the entries of $\textbf{G}$, as $N$ tends to $\infty$.
To explain this resampling further, we require some additional terminology. In what follows, we denote $p_{ij} = \mathbb{P} \big[ |h_{ij}| < N^{-\varepsilon / 10} \big]$ for each $1 \le i, j \le N$; our assumption implies that $p_{ij} \notin \{ 0, 1 \}$.
\begin{definition}
\label{hl}
We say that a random $N \times N$ symmetric $AB$ label $\textbf{L} = \{ L_{ij} \}$ is \emph{$\textbf{\emph{H}}$-distributed} if its upper triangular entries $\{ L_{ij} \}_{1 \le i \le j \le N}$ are mutually independent, $\mathbb{P} \big[ L_{ij} = A \big] = p_{ij}$, and $\mathbb{P} \big[ L_{ij} = B \big] = 1 - p_{ij}$, for each $i, j$.
\end{definition}
We next provide notation for the random variables $h_{ij}$, conditioned on the event that $L_{ij} = A$ or that $L_{ij} = B$.
\begin{definition}
\label{abdefinition}
For each $1 \le i \le j \le N$, let $a_{ij}$ denote the random variable such that $\mathbb{P} [a_{ij} \in I] = p_{ij}^{-1} \mathbb{P} [h_{ij} \in I \cap (-N^{- \varepsilon / 10}, N^{-\varepsilon / 10})]$, for each interval $I \subset \mathbb{R}$; equivalently, $a_{ij}$ is the random variable $h_{ij}$ conditioned on the event that $|h_{ij}| < N^{-\varepsilon / 10}$.
Furthermore, let $b_{ij}$ denote the random variable such that $\mathbb{P} [b_{ij} \in I] = (1 - p_{ij})^{-1} \mathbb{P} \big[ h_{ij} \in I \cap \big( ( - \infty, -N^{- \varepsilon / 10}] \cup [ N^{- \varepsilon / 10}, \infty) \big) \big]$ for each $I \subset \mathbb{R}$; equivalently, $b_{ij}$ is the random variable $h_{ij}$, conditioned on the event that $|h_{ij}| \ge N^{-\varepsilon / 10}$. Here, $\big\{ a_{ij} \} \cup \{ b_{ij} \}$ are mutually independent.
\end{definition}
Using the previous definition, we can sample $\textbf{H}$ conditioned on its $AB$ label $\textbf{L} (\textbf{H})$.
\begin{definition}
\label{hl1}
Fix an $N \times N$ symmetric $AB$ label $\textbf{L}$. Let $\textbf{H} (\textbf{L})$ denote the random symmetric matrix, which is sampled as follows. For each $1 \le i \le j \le N$, place the random variable $a_{ij}$ at entry $(i, j)$ if and only if $L_{ij} = A$; otherwise, place the random variable $b_{ij}$ at this entry. The lower triangular entries of $\textbf{H} (\textbf{L})$ (corresponding to entries $(i, j)$ with $i > j$) are then determined by symmetry. We call the matrix $\textbf{H} (\textbf{L})$ an \emph{$\textbf{\emph{L}}$-distributed symmetric random matrix}.
\end{definition}
\begin{sampling}
\label{lh}
To sample the random matrix $\textbf{H}$, we perform the following steps. First, sample an $\textbf{H}$-distributed $AB$ label, denoted $\textbf{L}$. Then, given $\textbf{L}$, sample an $\textbf{L}$-distributed symmetric random matrix $\textbf{H} = \textbf{H} (\textbf{L})$.
\end{sampling}
It is quickly verified that the distribution of $\textbf{H}$ resulting from Sampling \ref{lh} coincides with the original distribution of $\textbf{H}$, so the above procedure indeed yields a resampling.
We conclude this section with the following lemma, which provides some statistics on the $a_{ij}$.
\begin{lem}
\label{entrysmaller}
If $\varepsilon \le 2$ and $N > 2 C_2$, we have that
\begin{flalign*}
1 - p_{ij} \le C_2 N^{-1 - \varepsilon / 10}; \qquad \big| \mathbb{E} [a_{ij}] \big| \le 2 C_2 N^{-1 - \varepsilon / 10}; \qquad \big| \mathbb{E} [|a_{ij}|^2] - s_{ij} \big| \le 3 C_2 N^{-1 - \varepsilon / 10}.
\end{flalign*}
\end{lem}
\begin{proof}
Each of these statements follows from a Markov estimate; the first one was verified in \eqref{pijestimate1}. To establish the second one, observe that
\begin{flalign*}
\big| \mathbb{E} [a_{ij}] \big| \le p_{ij}^{-1} \mathbb{E} \big[ | h_{ij} | \textbf{1}_{h_{ij} < N^{- \varepsilon / 10}} \big] & \le p_{ij}^{-1} N^{\varepsilon (1 + \varepsilon) / 10} \mathbb{E} \big[ | h_{ij}|^{2 + \varepsilon} \big] < 2 C N^{-1 - \varepsilon / 10},
\end{flalign*}
\noindent where we have used \eqref{pijestimate1} and the fact that $N > 2 C_2$ to deduce that $p_{ij} > 1 / 2$. The proof of the third estimate is very similar and is thus omitted.
\end{proof}
\subsection{Deviant and Typical Indices}
\label{IndicesGraph}
In this section we give a precise definition of typical and deviant indices, which were informally introduced at the start of Section \ref{DefineIndices}. To that end, we begin with the following preliminary notion.
\begin{definition}
\label{linkedunlinkedconnected}
Fix an $N \times N$ $AB$ label $\textbf{L} = \{ L_{ij} \} $. We call $i, j \in [1, N]$ \emph{linked} (with respect to $\textbf{L}$) if $L_{ij} = B$; otherwise we call them \emph{unlinked}. If there exists a sequence of indices $i = i_1, i_2, \ldots , i_r = j$ such that $i_j$ is linked to $i_{j + 1}$ for each $j \in [1, r - 1]$, then we call $i$ and $j$ \emph{connected}; otherwise, they are \emph{disconnected}.
\end{definition}
Using Definition \ref{linkedunlinkedconnected}, we can define typical and deviant indices.
\begin{definition}
\label{deviantnondeviant}
Fix an $N \times N$ $AB$ label $\textbf{L}$. We call an index $i \in [1, N]$ \emph{deviant} (with respect to $\textbf{L}$) if there exists some index $j \in [1, N]$ such that $i$ and $j$ are linked. Otherwise, $i$ is called \emph{typical} (with respect to $\textbf{L}$). Let $\mathcal{D} = \mathcal{D}_{\textbf{L}} \subseteq [1, N]$ denote the set of deviant indices, and let $\mathcal{T} = \mathcal{T}_{\textbf{L}} \subseteq [1, N]$ denote the set of typical indices.
\end{definition}
Now fix an $N \times N$ symmetric $AB$ label $\textbf{L}$. Let us investigate how the resolvent $\textbf{G}$ of an $\textbf{L}$-distributed symmetric random matrix looks. Equivalently, by Sampling \ref{lh}, we consider how the resolvent $\textbf{G}$ of the generalized Wigner matrix $\textbf{H}$ looks, after conditioning on the event that $\textbf{L} (\textbf{H}) = \textbf{L}$. In view of Theorem \ref{estimate1gij}, we would hope that $\textbf{G} \approx m_{\semicircle} \Id$ for ``most'' $AB$ labels $\textbf{L}$. This is indeed true; the following definition clarifies the meaning of ``most'' $AB$ labels in our context.
\begin{definition}
\label{admissiblelabel}
Fix an $N \times N$ symmetric $AB$ label $\textbf{L}$.
\begin{itemize}
\item{\label{delta1} We call $\textbf{L}$ \emph{deviant-inadmissible} if there exist at least $N^{1 - \varepsilon / 20}$ deviant indices.}
\item{\label{delta3} We call $\textbf{L}$ \emph{connected-inadmissible} if there exist distinct indices $j_1, j_2, \ldots , j_r$ that are are pairwise connected, where $r = \lceil \log \log N \rceil$. }
\end{itemize}
We call $\textbf{L}$ \emph{inadmissible} if it is either deviant-inadmissible or connected-inadmissible. Otherwise, $\textbf{L}$ is called \emph{admissible}. Let $\mathcal{A} = \mathcal{A}_N$ denote the set all admissible $N \times N$ $AB$ labels.
\end{definition}
\begin{definition}
\label{delta}
Let $\textbf{H}$ be a generalized Wigner matrix in the sense of Definition \ref{momentassumption}, and let $\textbf{L}$ be an $\textbf{H}$-distributed $AB$ label. Define $\Delta_1$ and $\Delta_2$ to be the events on which $\textbf{L}$ deviant-inadmissible and connected-inadmissible, respectively.
Let $\Delta = \Delta_1 \cup \Delta_2$ denote the event on which $\textbf{L}$ is inadmissible, and let $\overline{\Delta}$ denote the complementary event on which $\textbf{L}$ is admissible. Furthermore, let $D(i)$ denote the event on which $i \in \mathcal{D}$, and let $T(i)$ denote the complementary event on which $i \in \mathcal{T}$.
\end{definition}
The following lemma shows that the event $\Delta$ occurs with small probability.
\begin{lem}
\label{probabilitydelta}
There exist constants $c, C > 0$ (only dependent on $C_2$ and $\varepsilon$ from Definition \ref{momentassumption}) such that $\mathbb{P} \big[ \Delta \big] < C N^{- c \log \log N}$.
\end{lem}
\begin{proof}
To establish this lemma we individually estimate $\mathbb{P} [\Delta_1]$ and $\mathbb{P} [\Delta_2]$.
We begin with the former. To that end, first observe that if there exist $N^{1 - \varepsilon / 20}$ deviant indices, then there must exist at least $R = N^{1 - \varepsilon / 20} / 2$ indices $(i, j) \in [1, N]^2$ such that $1 \le i \le j \le N$ and each $L_{ij} = B$ or such that $1 \le j \le i \le N$ and each $L_{ij} = B$.
The two cases are equivalent, so assume that the first holds; then, the $L_{ij}$ are independent. Furthermore, in view of \eqref{pijestimate1}, we have that $\mathbb{P} \big[ L_{ij} = B \big] \le C_2 N^{-1 - \varepsilon / 10}$. Therefore, the independence of the $L_{ij}$ implies that
\begin{flalign}
\label{delta1estimate}
\mathbb{P} \big[ \Delta_1 \big] \le \displaystyle\sum_{j = R}^N \binom{N^2}{j} \big( C_2 N^{-1 - \varepsilon / 10} \big)^j \le C_3 N^{- c_3 \log \log N},
\end{flalign}
\noindent for some constants $c_3, C_3 > 0$.
To bound $\mathbb{P} [\Delta_2]$, observe that the event $\Delta_2$ is contained in the event that there exists a sequence $S = \{ i_1, i_2, \ldots , i_r \} \subset [1, N]$ of indices such that there are at least $r - 1$ pairs of distinct indices $(i_j, i_k)$ that are linked.
There are $\binom{N}{r}$ ways to select such a sequence, and there are less than $\binom{r^2}{r - 1}$ to select $r - 1$ pairs of indices to link in this sequence. Furthermore, the event that $i_j$ and $i_k$ are linked is independent of the event that $i_{j'}$ and $i_{k'}$ is linked, unless $j = j'$ and $k = k'$ or $j = k'$ and $k = j'$. Thus,
\begin{flalign}
\label{delta3estimate}
\mathbb{P} \big[ \Delta_3 \big] \le \binom{N}{r} \binom{r^2}{r - 1} \big( C_2 N^{- \varepsilon / 10 - 1} \big)^{r - 1} \le C_4 N^{- c_4 \log \log N},
\end{flalign}
\noindent for some constants $c_4, C_4 > 0$; here, we have again used \eqref{pijestimate1}.
Now, the lemma follows from summing \eqref{delta1estimate} and \eqref{delta3estimate}.
\end{proof}
\subsection{Results and Reductions}
\label{deviantnondeviantlocallaw}
Our next goal is to analyze individual entries of the resolvent $\textbf{G} = \{ G_{ij} \}$, after conditioning on its $AB$ label $\textbf{L} (\textbf{H})$ of $\textbf{H}$. To that end, for any $c, C, \xi > 0$ and integers $i, j \in [1, N]$, we define the events
\begin{flalign}
\label{omegaij}
\begin{aligned}
\Omega_{C; \xi}^{(c)} (i, j) & = \Omega_{C; \xi}^{(c)} (i, j; \textbf{H}, z) = \bigg\{ \big| G_{ij} (z) - \textbf{1}_{i = j} m_{\semicircle} \big| \ge C (\log N)^{3 \xi} \left( \displaystyle\frac{1}{\sqrt{N \eta}} + N^{-c \varepsilon} \right) \bigg\}; \\
\Omega_C (i, j) & = \Omega_C (i, j; \textbf{H}, z) = \Big\{ \big| G_{ij} (z) \big| \ge C \Big\} .
\end{aligned}
\end{flalign}
As outlined in Section \ref{SmallMoment}, to prove Theorem \ref{estimate1gij} we will show that $\Omega_{C; \xi}^{(c)} (i, j)$ holds with very high probability when $i$ or $j$ is typical (with respect to $\textbf{L}$) and that $\Omega_C (i, j)$ holds with very high probability if both $i$ and $j$ are deviant (with respect to $\textbf{L}$). This is stated more carefully in the following theorem. In what follows $\mathbb{P}_{\textbf{H} (\textbf{L})}$ denotes the probability distribution with respect to the symmetric random matrix $\textbf{H} (\textbf{L})$ (from Definition \ref{hl1}).
\begin{thm}
\label{localmomentsdeltalabel}
Fix $\kappa > 0$ and $E \in [\kappa - 2, 2 - \kappa]$. Let $N$ be a positive integer, and take $\eta \in \mathbb{R}_{> 0}$ such that $N \eta > (\log N)^{8 \log \log N}$; denote $z = E + \textbf{\emph{i}} \eta$. Fix an admissible $N \times N$ $AB$ label $\textbf{\emph{L}} \in \mathcal{A}$. Let $\textbf{\emph{H}}$ be an $N \times N$ real generalized Wigner matrix (as in Definition \ref{momentassumption}).
Then, there exist constants $c, C, \xi > 0$ (only dependent on $\kappa$, $c_1$, $C_1$, $C_2$, and $\varepsilon$ from Definition \ref{momentassumption}) such that the following estimates hold.
\begin{enumerate}
\item{\label{nondeviantlocalmoments1} If $i \in [1, N]$ is typical with respect to $\textbf{\emph{L}}$, then for each $j \in [1, N]$ we have that
\begin{flalign}
\label{typicalomega}
\mathbb{P}_{\textbf{\emph{H}} (\textbf{\emph{L}})} \big[ \Omega_{C; \xi}^{(c)} (i, j; \textbf{\emph{H}}, z) \big] \le C \big( -c (\log N)^{\xi} \big).
\end{flalign}
}
\item{\label{deviantlocalmoments1} If $i \in [1, N]$ is deviant with respect to $\textbf{\emph{L}}$, then for each $j \in [1, N]$ we have that
\begin{flalign}
\label{deviantomega}
\mathbb{P}_{\textbf{\emph{H}} (\textbf{\emph{L}})} \big[ \Omega_C (i, j; \textbf{\emph{H}}, z) \big] \le C \exp \big( - c (\log N)^{\xi} \big) .
\end{flalign}
}
\end{enumerate}
\end{thm}
Let us see how Theorem \ref{estimate1gij} can be established, assuming Theorem \ref{localmomentsdeltalabel}.
\begin{proof}[Proof of Theorem \ref{estimate1gij} Assuming Theorem \ref{localmomentsdeltalabel}]
To establish Theorem \ref{estimate1gij}, we first show that the estimates \eqref{gijestimate} and \eqref{gijestimate2} hold when $z \in \mathscr{D}_{\kappa; N}$ is fixed; then, we use a union estimate to establish these estimates after taking the supremum over $z \in \mathscr{D}_{\kappa; N}$, as originally stated above.
To implement the first part, fix $z \in \mathscr{D}_{\kappa; N}$ and sample the generalized Wigner matrix $\textbf{H}$ according to Sampling \ref{lh}, that is, first sample an $\textbf{H}$-distributed $AB$ label $\textbf{L}$ and then sample an $\textbf{L}$-distributed symmetric random matrix $\textbf{H}$. Denote by $\widetilde{c}, \widetilde{C}, \widetilde{\xi}$ the constants $c, C, \xi$ from Theorem \ref{localmomentsdeltalabel}, respectively. Let us restrict to the complement $\overline{\Omega} (z)$ of the event
\begin{flalign*}
\Omega (z) = \bigcup_{i, j \in \mathcal{T}} \Omega_{\widetilde{C}; \widetilde{\xi}}^{(\widetilde{c})} (i, j) \cup \bigcup_{i, j \in [1, N]} \Omega_{\widetilde{C}} (i, j).
\end{flalign*}
\noindent By Lemma \ref{probabilitydelta}, \eqref{typicalomega}, and \eqref{deviantomega} we have that
\begin{flalign}
\label{omegaprobability}
\mathbb{P} [\Delta] \le \widetilde{C} N^{-\widetilde{c} \log \log N}; \qquad \mathbb{P}_{\textbf{H} (\textbf{L})} \big[ \Omega (z) \big] \le \widetilde{C} N^2 \exp \big( - \widetilde{c} (\log N)^{\widetilde{\xi}} \big),
\end{flalign}
\noindent after altering $\widetilde{c}, \widetilde{C}, \widetilde{\xi}$ if necessary. Restricting to $\overline{\Omega} (z)$, we have that
\begin{flalign}
\label{gijestimatel}
\begin{aligned}
& \big| G_{ij} (z) - \textbf{1}_{i = j} m_{\semicircle} (z) \big| < \widetilde{C} (\log N)^{3 \widetilde{\xi}} \left( \displaystyle\frac{1}{\sqrt{N \eta}} + N^{- \widetilde{c} \varepsilon} \right) \quad \text{if $i$ is typical}; \\
& \big| G_{ij} (z) \big| < \widetilde{C} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \text{if $i$ is deviant}.
\end{aligned}
\end{flalign}
Further restricting to the complement $\overline{\Delta}$ of the event $\Delta$, we have that $\textbf{L}$ is admissible. Therefore, there exist less than $N^{1 - \varepsilon / 20}$ deviant indices with respect to $\textbf{L}$. Combining this with the estimates \eqref{omegaprobability} and \eqref{gijestimatel} yields
\begin{flalign}
\label{zgijestimate}
\begin{aligned}
& \mathbb{P}_{\textbf{H}} \Bigg[ \displaystyle\max_{1 \le i, j \le N} \big| G_{ij} (z) \big| > \widetilde{C} \Bigg] < \widetilde{C} N^{- \widetilde{c} \log \log N}; \\
& \mathbb{P}_{\textbf{H}} \Bigg[ \bigcup_{\substack{I \subseteq [1, N] \\ |I| \ge s}} \bigg\{ \displaystyle\min_{i \in I} \displaystyle\max_{1 \le j \le N} \big| G_{i j} (z) - \textbf{1}_{i = j} m_{\semicircle} (z) \big| > \widetilde{C} (\log N)^{3 \widetilde{\xi}} \Big( \displaystyle\frac{1}{\sqrt{N \eta}} + N^{-\widetilde{c} \varepsilon} \Big) \bigg\} \Bigg] < \widetilde{C} N^{- \widetilde{c} \log \log N},
\end{aligned}
\end{flalign}
\noindent where we set $s = s_N = \lceil N^{1 - \varepsilon / 20} \rceil $.
The estimates \eqref{zgijestimate} hold for each fixed $z \in \mathscr{D}_{\kappa; N}$. To establish the stronger results claimed in Theorem \ref{estimate1gij}, we must take the supremum over all $z \in \mathscr{D}_{\kappa; N}$. To that end, define the sublattice $\mathbb{L}_{\kappa; N} = \big\{ z \in \mathscr{D}_{\kappa; N} : N^{10} \Re z \in \mathbb{Z}, N^{10} \Im z \in \mathbb{Z} \big\}$. Then, from a union estimate, it follows that
\begin{flalign}
\label{zgijestimate1}
\begin{aligned}
& \mathbb{P}_{\textbf{H}} \Bigg[ \displaystyle\max_{1 \le i, j \le N} \displaystyle\sup_{z \in \mathbb{L}_{\kappa; z}} \big| G_{ij} (z) \big| > \widetilde{C} \Bigg] < 100 \widetilde{C} N^{20 - \widetilde{c} \log \log N}; \\
& \mathbb{P}_{\textbf{H}} \Bigg[ \bigcup_{\substack{I \subseteq [1, N] \\ |I| \ge s}} \bigg\{ \displaystyle\min_{i \in I} \displaystyle\max_{1 \le j \le N} \displaystyle\sup_{z \in \mathbb{L}_{\kappa; z}} \big| G_{i j} (z) - \textbf{1}_{i = j} m_{\semicircle} (z) \big| > \widetilde{C} (\log N)^{3 \widetilde{\xi}} \Big( \displaystyle\frac{1}{\sqrt{N \eta}} + N^{-\widetilde{c} \varepsilon} \Big) \bigg\} \Bigg] \\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad < 100 \widetilde{C} N^{20 - \widetilde{c} \log \log N}.
\end{aligned}
\end{flalign}
\noindent Now \eqref{gijestimate} and \eqref{gijestimate2} follow from \eqref{zgijestimate1} and the fact that
\begin{flalign}
\label{ztildez}
\big| G_{ij} (z) - G_{ij} (\widetilde{z}) \big| \le N^{-1},
\end{flalign}
\noindent if $z, \widetilde{z} \in \mathbb{H}$ satisfy $|z - \widetilde{z}| < N^{-4}$ and $\Im z, \Im \widetilde{z} > N^{-1}$; \eqref{ztildez} is a consequence of the resolvent identity \eqref{resolvent} and the deterministic estimate \eqref{gijeta}.
\end{proof}
Thus, it suffices to establish Theorem \ref{localmomentsdeltalabel}. This will be the topic of the next several sections.
\section{The Initial Estimate}
\label{LawLarge}
The purpose of this section is to establish Theorem \ref{localmomentsdeltalabel} when $\eta$ is relatively large, of order $1$; this is the content of Proposition \ref{locallargeeta}.
To establish this proposition, we follow the outline from Section \ref{DiagonalResolvent}. To that end, in accordance with \eqref{vmfedsmall}, we first begin with a high probability estimate on $|F_i|$, $|E_i|$, $|D_i|$, and $|h_{ii}|$ (recall \eqref{edmf}) for deviant indices $i$. This estimate is provided by the following lemma, under the assumption that each resolvent entry $G_{jk}^{(i)}$ is already bounded by some $U > 0$.
\begin{lem}
\label{edfhsmalletalarge}
Fix $\kappa > 0$, $U > 1$, and $E \in [\kappa - 2, 2 - \kappa]$; also, let $\eta \in \mathbb{R}_{> 0}$, and denote $z = E + \textbf{\emph{i}} \eta$. Fix a positive integer $N$ and an admissible $N \times N$ $AB$ label $\textbf{\emph{L}} \in \mathcal{A}$; fix a typical index $i \in \mathcal{T}_{\textbf{\emph{L}}}$. Let $\textbf{\emph{H}}$ be an $N \times N$ generalized Wigner matrix. Recall the definition of $t_i$ from assumption \ref{stochastic} and of $F_i$, $E_i$, and $D_i$ from \eqref{edmf}.
Then, there exist constants $C, \nu > 0$ (only dependent on $C_1$ and $C_2$) such that
\begin{flalign}
\label{uedfhsmall}
\begin{aligned}
\mathbb{P}_{\textbf{\emph{H}} (\textbf{\emph{L}})} \Bigg[ |\Gamma_i| \displaystyle\prod_{\substack{1 \le j, k \le N \\ j, k \ne i}} \textbf{\emph{1}}_{|G_{jk}^{(i)}| \le U} \ge C U & (\log N)^{2 \xi} \Big( N^{-\varepsilon / 10} + \displaystyle\frac{1}{\sqrt{N \eta}} \Big) \Bigg] \le \exp \big( - \nu (\log N)^{\xi} \big),
\end{aligned}
\end{flalign}
\noindent for any $2 \le \xi \le \log \log N$, where for each $i$ we have set
\begin{flalign}
\label{gammai}
\Gamma_i = F_i + E_i + D_i - h_{ii} + m_{\semicircle} t_i.
\end{flalign}
\end{lem}
\begin{proof}
We will establish this lemma by individually bounding each term $|F_i|$, $|E_i|$, $|D_i|$, $|h_{ii}|$, and $|m_{\semicircle} t_i|$ with large probability. To that end first observe that $\big| m_{\semicircle} t_i \big| < C_1 |m_{\semicircle}| N^{-\varepsilon}$ by assumption \ref{stochastic}. Furthermore, observe that, since $i$ is typical with respect to $\textbf{L}$, we have that $|h_{ii}| < N^{- \varepsilon / 10}$ also holds deterministically.
Next, consider $D_i$. From \eqref{gkj}, it follows that
\begin{flalign}
\label{etalargedi}
|D_i| \le \displaystyle\sum_{\substack{1 \le j \le N \\ j \ne i}} s_{ij} \Big| G_{jj}^{(i)} - G_{jj} \Big| = \displaystyle\sum_{j = 1}^N \displaystyle\frac{s_{ij} \big| G_{ij} \big|^2}{\big| G_{ii} \big|} \le \displaystyle\frac{C_1}{N |G_{ii}| } \displaystyle\sum_{j = 1}^N \big| G_{ij} \big|^2 \le \displaystyle\frac{C_1}{N \eta},
\end{flalign}
\noindent where we applied assumption \ref{generalized} to deduce the third estimate in \eqref{etalargedi}, and we applied Ward's identity \eqref{sumgij} to deduce the fourth estimate. Thus, \eqref{etalargedi} provides a deterministic bound on $|D_i|$.
We will bound the remaining terms $|E_i|$ and $|F_i|$ with very high probability, using \eqref{bi2} and \eqref{bij2} from Corollary \ref{largeprobability2}.
We first address $|F_i|$. To that end, recall the definitions from Section \ref{EntryTruncation} (in particular, the definitions of $a_{ij}$ and $b_{ij}$ from Definition \ref{abdefinition}, the definition of $\textbf{H} (\textbf{L})$ from Definition \ref{hl1}, and Sampling \ref{lh}). Since $i \in \mathcal{T}$ is typical, the $(i, j)$ entry of $\textbf{H} (\textbf{L})$ is $a_{ij}$ for each $j \in [1, N]$.
Now, from Lemma \ref{entrysmaller} we have that $\big| \mathbb{E} [a_{ij}] \big| \le 2 C_2 N^{-1 - \varepsilon / 10}$. Moreover,
\begin{flalign}
\label{qhijsmallmoments}
\mathbb{E} \big[ |a_{ij}|^p \big] \le q^{2 - p} \mathbb{E} \big[ |a_{ij}|^2 \big] \le \displaystyle\frac{C_1}{N q^{p - 2}} \le \displaystyle\frac{q^2}{N} \left( \displaystyle\frac{C_1}{q} \right)^p,
\end{flalign}
\noindent for any $p \ge 2$, where we have set $q = N^{\varepsilon / 10}$. To establish the first estimate in \eqref{qhijsmallmoments}, we used the fact that $|a_{ij}| < q^{-1}$ deterministically; to establish the second estimate, we used the fact that $\Var a_{ij} \le \Var h_{ij} \le C_1 N^{-1}$; and to establish the third estimate, we used the fact that $C_1 > 1$.
Thus, we can apply \eqref{bij2} with $R_{jk} = G_{jk}^{(i)}$; $X_j = a_{ij}$; and $N^{- \delta} = N^{- \varepsilon / 10} = q^{-1}$. This yields
\begin{flalign*}
\mathbb{P} \Bigg[ \bigg| \displaystyle\sum_{\substack{ 1 \le j \ne k \le N \\ j, k \ne i}} a_{ij} G_{jk}^{(i)} a_{ki} \bigg| \ge (\log N)^{2 \xi} \bigg( (20 C_2^2 + 1) q^{-1} \displaystyle\max_{\substack{1 \le j \ne k \le N \\ j, k \ne i}} \big| G_{jk}^{(i)} \big| & + \Big( \displaystyle\frac{1}{N^2} \displaystyle\sum_{\substack{1 \le j \ne k \le N\\ j, k \ne i}} \big| G_{jk}^{(i)} \big|^2 \Big)^{1 / 2} \bigg) \Bigg] \\
& \le \exp \big( -\widetilde{\nu} (\log N)^{\xi} \big),
\end{flalign*}
\noindent where $\widetilde{\nu} = \nu (C_1, 2C_2)$ from Proposition \ref{largeprobability2}. Applying Ward's identity \eqref{sumgij} and the definition \eqref{edmf} of $F_i$ yields
\begin{flalign}
\label{flargeeta1}
\begin{aligned}
\mathbb{P} \Bigg[ |F_i| \ge (\log N)^{2 \xi} \bigg( (20 C_2^2 + 1) N^{-\varepsilon / 10} \displaystyle\max_{\substack{1 \le j \ne k \le N \\ j, k \ne i}} \big| G_{jk}^{(i)} \big| + \displaystyle\frac{1}{\sqrt{N \eta}} & \Big( \displaystyle\frac{1}{N} \displaystyle\sum_{\substack{1 \le j \le N\\ j, k \ne i}} \big| G_{jj}^{(i)} \big| \Big)^{1 / 2} \bigg) \Bigg] \\
& \le \exp \big( -\widetilde{\nu} (\log N)^{\xi} \big),
\end{aligned}
\end{flalign}
\noindent for all $2 \le \xi \le \log \log N$. In particular, combining \eqref{flargeeta1} with the estimate $\big| G_{jk}^{(i)} \big| \le U$ (due to the factors of $\textbf{1}_{|G_{jk}^{(i)}| \le U}$ on the left side of \eqref{uedfhsmall}), and the fact that $C_2, U > 1$, we deduce that
\begin{flalign}
\label{flargeeta}
\mathbb{P} \left[ |F_i| > 21 C_2^2 U (\log N)^{2 \xi} \bigg( \displaystyle\frac{1 }{N^{\varepsilon / 10}} + \displaystyle\frac{1}{\sqrt{N \eta}} \bigg) \right] \le \exp \big( - \widetilde{\nu} (\log N)^{\xi} \big),
\end{flalign}
\noindent which provides an estimate on $|F_i|$.
To estimate $|E_i|$, we use \eqref{bi2} with $R_j = G_{jj}^{(i)}$; $s_j = s_{ij}$; $X_j = h_{ij}$; and $N^{- \delta} = N^{- \varepsilon / 10} = q^{-1}$ to deduce that
\begin{flalign}
\label{elargeeta1}
\begin{aligned}
\mathbb{P} \Bigg[ \bigg| \displaystyle\sum_{\substack{1 \le j \le N \\ j \ne i}} G_{jj}^{(i)} \big( |a_{ij}|^2 - s_{ij} \big) \bigg| \ge (20 C_2^2+ 1) (\log N)^{\xi} N^{-\varepsilon / 10} \displaystyle\max_{\substack{1 \le j \le N \\ j \ne i}} \big| G_{jj}^{(i)} \big| \Bigg] \le \exp \big( - \widetilde{\nu} (\log N)^{\xi} \big).
\end{aligned}
\end{flalign}
\noindent Combining \eqref{elargeeta1} with the definition \eqref{edmf} of $E_i$, the estimate $\big| G_{jk}^{(i)} \big| \le U$, and the fact that $C_2 > 1$, we deduce that
\begin{flalign}
\label{elargeeta}
\begin{aligned}
\mathbb{P} \left[ \big| E_i \big| \ge 21 C_2^2 U (\log N)^{\xi} N^{-\varepsilon / 10} \right] \le \exp \big( - \widetilde{\nu} (\log N)^{\xi} \big).
\end{aligned}
\end{flalign}
Now, the existence of $C$ and $\nu$ (only dependent on $C_1$ and $C_2$) satisfying \eqref{uedfhsmall} follows from summing \eqref{etalargedi}, \eqref{flargeeta}, \eqref{elargeeta}, and the deterministic estimates $|t_i m_{\semicircle} | < C_1 |m_{\semicircle} N^{-\varepsilon}$ and $|h_{ii}| \le N^{- \varepsilon / 10}$.
\end{proof}
Using Proposition \ref{edfhsmalletalarge}, we can establish Theorem \ref{localmomentsdeltalabel} in the case when $\eta$ is sufficiently large.
\begin{prop}
\label{locallargeeta}
Fix $\kappa > 0$, and let $E \in [\kappa - 2, 2 - \kappa]$; also, let $\eta \in \mathbb{R}_{> 0}$, and denote $z = E + \textbf{\emph{i}} \eta$. Fix a positive integer $N$ and an admissible $N \times N$ $AB$ label $\textbf{\emph{L}} \in \mathcal{A}$. Let $\textbf{\emph{H}}$ be an $N \times N$ generalized Wigner matrix. Recall the definitions of $F_i$, $E_i$, and $D_i$ from \eqref{edmf}, and the definitions of $\Omega_{C; \xi}^{(c)} (i, j)$ and $\Omega_C (i, j)$ from \eqref{omegaij}.
Then, there exist constants $C, \nu > 0$ (only dependent on $C_1$ and $C_2$) such that the following estimates hold for any $2 \le \xi \le \log \log N$ and sufficiently large $N$ (in comparison to $C_1$ and $\varepsilon^{-1}$).
\begin{enumerate}
\item{\label{nondeviantlocalmoments} If $\eta > C$ and $i \in [1, N]$ is typical with respect to $\textbf{\emph{L}}$, then for each $j \in [1, N]$ we have that
\begin{flalign}
\label{typicalomegaetalarge}
\mathbb{P} \big[ \Omega_{C; \xi}^{(1 / 20)} (i, j) \big| \textbf{\emph{L}} (\textbf{\emph{H}}) = \textbf{\emph{L}} \big] \le \big( - \nu (\log N)^{\xi} \big).
\end{flalign}
}
\item{\label{deviantlocalmoments} If $\eta > C$ and $i \in [1, N]$ is deviant with respect to $\textbf{\emph{L}}$, then for each $j \in [1, N]$ we have that
\begin{flalign}
\label{deviantomegaetalarge}
\mathbb{P} \big[ \Omega_1 (i, j) \big| \textbf{\emph{L}} (\textbf{\emph{H}}) = \textbf{\emph{L}} \big] \le \exp \big( - \nu (\log N)^{\xi} \big) .
\end{flalign}
}
\end{enumerate}
\end{prop}
\begin{proof}
First, observe that if $C > 1$, then we have from the deterministic estimate \eqref{gijeta} that $\big| G_{ij} \big| < \eta^{-1} < 1$; this implies \eqref{deviantomegaetalarge}.
Hence, it suffices to establish \eqref{typicalomegaetalarge}, so let $i \in [1, N]$ be typical with respect to $\textbf{L}$. Denoting $\Gamma_i$ as in \eqref{gammai}, we deduce from \eqref{vmfed} that
\begin{flalign}
\label{vi1}
v_i = \displaystyle\frac{m_{\semicircle} \sum_{j = 1}^N s_{ij} v_j + m_{\semicircle} \Gamma_i}{m_{\semicircle}^{-1} - \Gamma_i - \sum_{j = 1}^N s_{ij} v_j}.
\end{flalign}
We will next show that the denominator on the right side of \eqref{vi1} is large and that its numerator is small. To that end, we first require an estimate on $\Gamma_i$. Let $\widetilde{C}$ denote the constant $C$ from \eqref{uedfhsmall}. Setting $C > \widetilde{C}$ yields $\big| G_{jk}^{(i)} \big| < \eta^{-1} < \widetilde{C}^{-1}$. Thus, applying \eqref{uedfhsmall} with $U = \widetilde{C}^{-1}$ yields
\begin{flalign}
\label{gammaietalarge}
\mathbb{P} \left[ \big| \Gamma_i \big| \le (\log N)^{2 \xi} \bigg( N^{-\varepsilon / 10} + \displaystyle\frac{1}{\sqrt{N \eta}} \bigg) \right] \le \exp \big( - \widetilde{\nu} \big( \log N \big)^{\xi} \big),
\end{flalign}
\noindent where $\widetilde{\nu}$ is the constant $\nu$ from Proposition \ref{uedfhsmall}.
Assume further that $C > 30$. Then, it is quickly derived from \eqref{mquadratic} that $| m_{\semicircle} | \le 2 \eta^{-1} < 1 / 10$, from which it follows that $|v_i| \le \big| G_{ii} \big| + |m_{\semicircle}| \le 3 \eta^{-1} < 1 / 10$ from \eqref{gijeta}.
Hence, if we restrict to the event that $\Gamma_i$ is small in the sense of \eqref{gammaietalarge} and assume that $N$ is sufficiently large in comparison to $\varepsilon^{-1}$ such that
\begin{flalign*}
\left| (\log N)^{2 \log \log N} \bigg( N^{-\varepsilon / 10} + \displaystyle\frac{1}{\sqrt{N \eta}} \bigg) \right| \le 1,
\end{flalign*}
\noindent then it follows that
\begin{flalign*}
\left| m_{\semicircle}^{-1} - \Gamma_i - \sum_{j = 1}^N s_{ij} v_j \right| \ge \big| m_{\semicircle}^{-1} \big| - \big| \Gamma_i \big| - \left| \sum_{j = 1}^N s_{ij} v_j \right| \ge 10 - \big| \Gamma_i \big| - (1 + C_1 N^{-\varepsilon} ) \displaystyle\max_{1 \le j \le N} |v_j| > 1,
\end{flalign*}
\noindent where we have used the fact that $\big| \sum_{j = 1}^N s_{ij} - 1 \big| = |t_i| \le C_1 N^{-\varepsilon}$ by assumption \ref{stochastic}. Inserting this (and the estimate $|m_{\semicircle}| < 1 / 10$) into \eqref{vi1} yields
\begin{flalign}
\label{vilargeeta1}
\big| v_i \big| \le \left| \displaystyle\frac{1}{10} \displaystyle\sum_{j = 1}^N s_{ij} v_j + m_{\semicircle} \Gamma_i \right| \le \displaystyle\frac{1}{10} \displaystyle\sum_{j = 1}^N s_{ij} | v_j | + \big| \Gamma_i \big|,
\end{flalign}
\noindent for each $i \in \mathcal{T}_{\textbf{L}}$. Hence,
\begin{flalign}
\label{vilargeeta2}
\big| v_i \big| & \le \displaystyle\frac{1}{10} \displaystyle\sum_{j \in \mathcal{T}} s_{ij} | v_j | + \displaystyle\frac{1}{10} \displaystyle\sum_{j \in \mathcal{D}} s_{ij} | v_j | + \big| \Gamma_i \big| \le \displaystyle\frac{1}{10} \displaystyle\sum_{j \in \mathcal{T}} s_{ij} | v_j | + C_1 N^{-\varepsilon / 20} + \big| \Gamma_i \big|,
\end{flalign}
\noindent where we recalled the facts that $\big| \mathcal{D} \big| < N^{1 - \varepsilon / 20}$ (since $\textbf{L}$ is admissible), that $s_{ij} < C_1 N^{-1}$ (from assumption \ref{generalized}), and that $|v_j| < 1 / 10 < 1$ for all $j \in [1, N]$.
Using \eqref{vilargeeta2} and the fact that $\big| \sum_{j = 1}^N s_{ij} - 1 \big| = |t_i| < C_1 N^{-\varepsilon}$, we deduce that
\begin{flalign}
\label{vilargeeta3}
\displaystyle\max_{j \in \mathcal{T}} \big| v_j \big| & \le \displaystyle\frac{1}{5} \displaystyle\max_{j \in \mathcal{T}} |v_j| + C_1 N^{- \varepsilon / 20} + \displaystyle\max_{j \in \mathcal{T}} \big| \Gamma_j \big|.
\end{flalign}
Using \eqref{vilargeeta3} and applying \eqref{gammaietalarge} for all $i \in \mathcal{T}$ and a union estimate, we obtain that
\begin{flalign*}
\mathbb{P} \left[ \displaystyle\max_{j \in \mathcal{T}} |v_j| \le 2 C_1 (\log N)^{2 \xi} \bigg( N^{-\varepsilon / 20} + \displaystyle\frac{1}{\sqrt{N \eta}} \bigg) \right] \le N \exp \big( - \widetilde{\nu} (\log N)^{\xi} \big),
\end{flalign*}
\noindent from which \eqref{typicalomegaetalarge} quickly follows.
\end{proof}
\section{The Multiscale Argument for Typical Indices}
\label{LawSmallNotDeviant}
Our next goal is to provide a framework for establishing the local semicircle law for typical indices, given by \eqref{typicalomega}. This is done through the following two propositions, whose proofs are similar to that of the local semicircle for sparse graphs from \cite{SSG}.
What these lemmas approximately yield are estimates on the probability of the event that the local semicircle law \eqref{typicalomega} does not hold for some typical entry of the resolvent $\textbf{G} (z)$, in terms of probabilities of the events the local semicircle law does not hold for some entry of a different resolvent $\textbf{G} (z')$, where $\Im z' > \Im z$. In terms of the notation from \eqref{omegaij}, this can be restated as an estimate on $\mathbb{P} \big[ \Omega_{C; \xi}^{(c)} (i, j; \textbf{H}, z) \big]$ in terms of $\mathbb{P} \big[ \Omega_{C; \xi}^{(c)} (i, j; \textbf{H}, z') \big]$ and $\mathbb{P} \big[ \Omega_C (i, j; \textbf{H}, z') \big]$, conditional on some admissible $AB$ label $\textbf{L}$ of $\textbf{H}$, if $i \in [1, N]$ is typical with respect to $\textbf{L}$.
Since $\Im z' > \Im z$, this suggests that repeated application of these two propositions might estimate $\mathbb{P} \big[ \Omega_{C; \xi}^{(c)} (i, j; \textbf{H}, z) \big]$ in terms of $\mathbb{P} \big[ \Omega_{C; \xi}^{(c)} (i, j; \textbf{H}, z') \big]$ and $\mathbb{P} \big[ \Omega_C (i, j; \textbf{H}, z') \big]$, where $\Im z'$ is very large; then, we could apply the initial estimate Proposition \ref{locallargeeta}. Indeed, this is what we will do in Section \ref{ProofLocalLaw}, after obtaining an analog of the two propositions below for deviant indices in Section \ref{LawSmallDeviant}.
In what follows, we recall that $\mathbb{P}_{\textbf{H} (\textbf{L})}$ is the probability measure the random matrix $\textbf{H} (\textbf{L})$ from Definition \ref{hl1}. Furthermore, for any subset $\mathcal{S} \subseteq [1, N]$, we let $\mathbb{P}_{\textbf{H}^{(\mathcal{S})} (\textbf{L}^{(\mathcal{S})})}$ denote the probability measure with respect to the symmetric random matrix $\textbf{H}^{(\mathcal{S})} \big( \textbf{L}^{(\mathcal{S})} \big)$, obtained from $\textbf{H} (\textbf{L})$ by removing all rows and columns indexed by some $j \in \mathcal{S}$.
\begin{prop}
\label{incrementnotdeviant}
Fix $\kappa > 0$ and $U_1, U_2 > 1$, and let $E \in [\kappa - 2, 2 - \kappa]$; also, let $\eta \in \mathbb{R}_{> 0}$ such that $N \eta > (\log N)^{8 \log \log N}$, and denote $z = E + \textbf{\emph{i}} \eta$. Fix a positive integer $N$, let $\textbf{\emph{H}}$ be an $N \times N$ generalized Wigner matrix. Furthermore, fix an admissible $N \times N$ $AB$ label $\textbf{\emph{L}} \in \mathcal{A}$. Recall the definitions of $v_i$ from \eqref{vj}, and of $\Omega_{C; \xi}^{(c)} (i, j)$ and $\Omega_C (i, j)$ from \eqref{omegaij}. Denote
\begin{flalign}
\label{probabilitynotdeviantevent}
\begin{aligned}
P_i & = \displaystyle\sum_{\substack{1 \le j, k \le N \\ j, k \ne i}} \mathbb{P}_{\textbf{\emph{H}}^{(i)} (\textbf{\emph{L}}^{(i)})} \Bigg[ \Omega_{U_2} \bigg( j, k; \textbf{\emph{H}}^{(i)}; E + \textbf{\emph{i}} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big) \bigg) \Bigg] \\
& \qquad + \displaystyle\sum_{j = 1}^N \mathbb{P}_{\textbf{\emph{H}} (\textbf{\emph{L}})} \Bigg[ \Omega_{U_1; \xi}^{(1 / 20)} \bigg( i, j; \textbf{\emph{H}}, E + \textbf{\emph{i}} \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big) \bigg)\Bigg] \\
& \qquad + \displaystyle\sum_{1 \le j, k \le N} \mathbb{P}_{\textbf{\emph{H}} (\textbf{\emph{L}})} \Bigg[ \Omega_{U_2} \bigg( j, k; \textbf{\emph{H}}, E + \textbf{\emph{i}} \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big) \bigg) \Bigg].
\end{aligned}
\end{flalign}
\noindent Then, there exist constants $C, \nu > 0$ (only dependent on $C_1$, $C_2$, and $U_2$) such that
\begin{flalign}
\label{linearvnotdeviantsmall4}
\mathbb{P}_{\textbf{\emph{H}} (\textbf{\emph{L}})} \Bigg[ \displaystyle\max_{j \in \mathcal{T}} |v_j| & \ge C (\log N)^{3 \xi} \left( N^{- \varepsilon / 20 } + \displaystyle\frac{1}{\sqrt{N \eta}} \right) \Bigg] \le \exp \big( - \nu (\log N)^{\xi} \big) + \displaystyle\sum_{i = 1}^N P_i,
\end{flalign}
\noindent for any $2 \le \xi \le \log \log N$ and sufficiently large $N$ (in comparison to $C_1$, $C_2$, $U_1$, and $U_2$).
\begin{rem}
\label{u1u2c}
Observe in the above proposition that the constants $C$ and $\mu$ are dependent on $U_2$ but independent of $U_1$; however, the minimal value of $N$ for which the result holds depends on both $U_1$ and $U_2$.
\end{rem}
\end{prop}
\begin{proof}
In what follows, we fix a typical index $i \in \mathcal{T}_{\textbf{L}}$. Recall the definitions of $F_i$, $D_i$, and $E_i$ from \eqref{edmf}. Let us restrict to the event
\begin{flalign}
\label{notdeviantevent}
\begin{aligned}
\overline{\Omega_i} & = \bigcap_{\substack{j, k \in [1, N] \\ j, k \ne i}} \overline{\Omega}_{U_2} \bigg( j, k; \textbf{H}^{(i)}; E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big) \bigg) \cap \bigcap_{j = 1}^N \overline{\Omega}_{U_1; \xi}^{(1 / 20)} \bigg( i, j; \textbf{H}, E + \textbf{i} \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big) \bigg) \\
& \qquad \cap \bigcap_{j, k \in [1, N]} \overline{\Omega}_{U_2} \bigg( j, k; \textbf{H}, E + \textbf{i} \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big) \bigg),
\end{aligned}
\end{flalign}
\noindent where $\overline{E}$ denotes the complement of any event $E$; observe that $\mathbb{P}_{\textbf{H} (\textbf{L})} \big[ \Omega_i \big] \le P_i$.
We will first obtain a uniform estimate on $G_{jk}^{(i)} (E + \textbf{i} \eta)$. Since we are restricting to the event $\overline{\Omega_i}$, we have that
\begin{flalign}
\label{notdeviantgjk1}
\Bigg| G_{jk}^{(i)} \bigg( E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big) \bigg) \Bigg| \le U_2,
\end{flalign}
\noindent for all $j, k \in [1, N]$ with $j, k \ne i$. Thus, assuming that $N > 10$, \eqref{gjjzgjj} implies that
\begin{flalign}
\label{notdeviantgjj1}
\big| G_{jj}^{(i)} ( E + \textbf{i} \eta ) \big| \le 2 U_2,
\end{flalign}
\noindent for all $j \in [1, N]$ not equal to $i$. Inserting \eqref{notdeviantgjj1} into \eqref{gjkzgjk} yields
\begin{flalign}
\label{notdeviantgjk2}
\Bigg| G_{jk}^{(i)} \bigg( E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big) \bigg) - G_{jk}^{(i)} (E + \textbf{i} \eta) \Bigg| \le \displaystyle\frac{2 U_2}{(\log N)^2},
\end{flalign}
\noindent for all $j, k \in [1, N]$ with $j, k \ne i$. In view of the fact that $\log N > 2$, \eqref{notdeviantgjk1} and \eqref{notdeviantgjk2} together imply that
\begin{flalign}
\label{notdeviantgjk3}
\big| G_{jk}^{(i)} ( E + \textbf{i} \eta ) \big| \le \Bigg| G_{jk}^{(i)} \bigg( E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big) \bigg) \Bigg| + U_2 \le 2 U_2,
\end{flalign}
\noindent for all $j, k \in [1, N]$ with $j, k \ne i$.
Having obtained this estimate on $G_{jk}^{(i)}$, we now we apply Proposition \ref{edfhsmalletalarge} with $U$ in that statement replaced by $2 U_2$. In particular, let us insert \eqref{notdeviantgjk3} into \eqref{uedfhsmall}; denote by $\widetilde{C}$ the constant $C$ from the left side of \eqref{uedfhsmall}; and denote by $\widetilde{\nu}$ the constant $\nu$ from the right side of \eqref{uedfhsmall}. Using the fact that $\mathbb{P}_{\textbf{H} (\textbf{L})} \big[ \Omega_i \big] \le P_i$, we deduce that
\begin{flalign}
\label{notdeviantedfhsmall}
\mathbb{P}_{\textbf{H} (\textbf{L})} \Bigg[ \big| \Gamma_i \big| \ge 2 \widetilde{C} U_2 & (\log N)^{2 \xi} \Big( N^{-\varepsilon / 10} + \displaystyle\frac{1}{\sqrt{N \eta}} \Big) \Bigg] \le \exp \big( - \widetilde{\nu} (\log N)^{\xi} \big) + P_i,
\end{flalign}
\noindent where we recall the definition of $\Gamma_i$ from \eqref{gammai}. This bounds the right side of \eqref{vmfed} with very high probability.
We will now attempt to establish some version of the estimate \eqref{vsmall}. To that end, observe that
\begin{flalign}
\label{vismall1notdeviant}
\left| \displaystyle\frac{v_i}{1 + m_{\semicircle}^{-1} v_i} - v_i \right| = \left| \displaystyle\frac{v_i^2}{m_{\semicircle} + v_i} \right| \le \displaystyle\frac{2 |v_i|^2}{|m_{\semicircle}|}, \qquad \text{if} \quad |v_i| \le \displaystyle\frac{|m_{\semicircle}|}{2}.
\end{flalign}
\noindent Let us show that $|v_i| < |m_{\semicircle}| / 2$ indeed holds. Since we are restricting to the event $\overline{\Omega_i}$, we have that
\begin{flalign}
\label{giigjjsmall1notdeviant1}
\begin{aligned}
\Bigg| G_{ii} \bigg( E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{ (\log N)^2} \Big) \bigg) - m_{\semicircle} \Bigg| & \le U_1 (\log N)^{3 \xi} \left( N^{- \varepsilon / 20} + \displaystyle\frac{1}{\sqrt{N \eta}} \right); \\
\Bigg| G_{jj} \bigg( E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{ (\log N)^2} \Big) \bigg) \Bigg| & \le U_2,
\end{aligned}
\end{flalign}
\noindent for all $j \in [1, N]$. As in \eqref{notdeviantgjk3} (or from the first estimate in \eqref{giigjjsmall1notdeviant1}), we can show that $\big| G_{ii} (E + \textbf{i} \eta) \big| \le 2 U_2$ using the second estimate in \eqref{giigjjsmall1notdeviant1} and \eqref{gjjzgjj}. Inserting this and \eqref{giigjjsmall1notdeviant1} into \eqref{gjkzgjk} yields
\begin{flalign}
\label{giismall1notdeviant1}
\big| G_{ii} (E + \textbf{i} \eta ) - m_{\semicircle} \big| \le U_1 (\log N)^{3 \xi} \left( N^{- \varepsilon / 20} + \displaystyle\frac{1}{\sqrt{N \eta}} \right) + \displaystyle\frac{2 U_2}{(\log N)^2}, \qquad \text{if $i$ is typical.}
\end{flalign}
Assuming that $N$ is sufficiently large (in a way that only depends on $U_1$ and $U_2$, since $\xi \le \log \log N$ and $N \eta > (\log N)^{8 \log \log N}$), \eqref{giismall1notdeviant1} implies that $2 |v_i| \le |m_{\semicircle}|$, so that the estimate \eqref{vismall1notdeviant} applies. Inserting \eqref{vismall1notdeviant} into \eqref{vmfed} and applying \eqref{notdeviantedfhsmall}, we deduce that
\begin{flalign}
\label{linearvnotdeviantsmall}
\begin{aligned}
\mathbb{P} \Bigg[ \bigg| v_i - m_{\semicircle}^2 \displaystyle\sum_{j = 1}^N s_{ij} v_j \bigg| \ge 2 |m_{\semicircle}|^2 \widetilde{C} U_2 (\log N)^{2 \xi} \Big( N^{- \varepsilon / 10 } + & \displaystyle\frac{1}{\sqrt{N \eta}} \Big) + \displaystyle\frac{2 |v_i|^2}{|m_{\semicircle}|} \Bigg] \\
& \le \exp \big( - \widetilde{\nu} (\log N)^{\xi} \big) + P_i,
\end{aligned}
\end{flalign}
\noindent for any typical index $i \in \mathcal{T}$.
Now, we would like to restrict the sum $\sum_{j = 1}^N s_{ij} v_j$ in \eqref{linearvnotdeviantsmall} to range over the typical indices $j \in \mathcal{T}$ instead of over all indices $j \in [1, N]$. To that end, we may use similar reasoning as applied to deduce \eqref{notdeviantgjk3} to find that
\begin{flalign}
\label{notdeviantgij1}
\big| G_{ij} (E + \textbf{i} \eta) \big| \le 2 U_2; \qquad \big| G_{ij} (E + \textbf{i} \eta) - m_{\semicircle} \big| \le 2 U_2 + |m_{\semicircle}|.
\end{flalign}
\noindent Using \eqref{notdeviantgij1}, the fact that $|s_{ij}| \le C_1 N^{-1}$, and the fact that $\big| \mathcal{D} \big| \le N^{1 - \varepsilon / 20}$, we find that
\begin{flalign}
\label{notdeviantgij2}
\left| \displaystyle\sum_{j = 1}^N s_{ij} v_j - \displaystyle\sum_{j \in \mathcal{T}} s_{ij} v_j \right| \le \big( 2 U_2 + |m_{\semicircle}| \big) C_1 N^{- \varepsilon / 20}.
\end{flalign}
\noindent Inserting \eqref{notdeviantgij2} into \eqref{linearvnotdeviantsmall}, we deduce that
\begin{flalign}
\label{linearvnotdeviantsmall1}
\begin{aligned}
\mathbb{P} \Bigg[ \bigg| v_i - m_{\semicircle}^2 \displaystyle\sum_{j \in \mathcal{T}} s_{ij} v_j \bigg| & \ge |m_{\semicircle}|^2 C_1 \big( 2 U_2 + |m_{\semicircle}| \big) N^{- \varepsilon / 20} + \displaystyle\frac{2 |v_i|^2}{|m_{\semicircle}|} \\
& \quad + 2 |m_{\semicircle}|^2 \widetilde{C} U_2 (\log N)^{2 \xi} \Big( N^{- \varepsilon / 10} + \displaystyle\frac{1}{\sqrt{N \eta}} \Big) \Bigg] \le \exp \big( - \widetilde{\nu} (\log N)^{\xi} \big) + P_i,
\end{aligned}
\end{flalign}
\noindent for any fixed $i \in \mathcal{T}$. Applying \eqref{linearvnotdeviantsmall1} to all $i \in \mathcal{T}$ and applying a union estimate, we obtain that
\begin{flalign}
\label{linearvnotdeviantsmall2}
\begin{aligned}
\mathbb{P} \Bigg[ \Big\| \big( \Id - m_{\semicircle}^2 \widetilde{\textbf{S}} \big) \widetilde{\textbf{v}} \Big\|_{\infty} & \ge |m_{\semicircle}|^2 C_1 \big( 2 U_2 + |m_{\semicircle}| \big) N^{- \varepsilon / 20} + \displaystyle\frac{2 |v_i|^2}{|m_{\semicircle}|} \\
& \quad + (\log N)^{2 \xi} \Big( N^{- \varepsilon / 10 } + \displaystyle\frac{1}{\sqrt{N \eta}} \Big) \Bigg] \le N \exp \big( - \widetilde{\nu} (\log N)^{\xi} \big) + \displaystyle\sum_{i = 1}^N P_i,
\end{aligned}
\end{flalign}
\noindent where $\widetilde{\textbf{v}} = (v_i)_{i \in \mathcal{T}}$ $|\mathcal{T}|$-dimensional vector, and $\widetilde{\textbf{S}} = \{ s_{jk} \}_{j, k \in \mathcal{T}}$ is a $|\mathcal{T}| \times |\mathcal{T}|$ matrix.
Now, from \eqref{mtestimate1} of Lemma \ref{testimate} below, we deduce the existence of a constant $\widehat{C} > 0$ (only dependent on $\kappa$) such that $\big\| (\Id - m_{\semicircle}^2 \widetilde{\textbf{S}} )^{-1} \big\| < \widehat{C} \log N$. Inserting this into \eqref{linearvnotdeviantsmall2} yields
\begin{flalign}
\label{linearvnotdeviantsmall3}
\begin{aligned}
\mathbb{P} \Bigg[ \displaystyle\max_{j \in \mathcal{T}} |v_j| & \ge |m_{\semicircle}|^2 C_1 \big( 2 U_2 + |m_{\semicircle}| \big) \widehat{C} \log N + 2 |m_{\semicircle}|^2 \widehat{C} \widetilde{C} U_2 (\log N)^{2 \xi + 1} \Big( N^{- \varepsilon / 10} + \displaystyle\frac{1}{\sqrt{N \eta}} \Big) \\
& \qquad + 2 |m_{\semicircle}|^{-1} \widehat{C} \big( \log N \big) \displaystyle\max_{j \in \mathcal{T}} |v_j|^2 \Bigg] \le N \exp \big( - \widetilde{\nu} (\log N)^{\xi} \big) + \displaystyle\sum_{i = 1}^N P_i,
\end{aligned}
\end{flalign}
Now, by \eqref{giismall1notdeviant1}, we have that
\begin{flalign*}
\Big| 1 - 2 |m_{\semicircle}|^{-1} \widehat{C} \big( \log N \big) |v_i| \Big| \ge \displaystyle\frac{1}{2},
\end{flalign*}
\noindent for each $i \in \mathcal{T}$ (after restricting to the event $\bigcap_{i \in \mathcal{T}} \overline{\Omega_i}$), assuming that $N$ is sufficiently large (in a way that only depends on $U_1$ and $U_2$).
Inserting this into \eqref{linearvnotdeviantsmall3} yields the existence of $C, \nu > 0$ (only dependent on $\kappa$, $U_2$, $C_1$, and $C_2$) satisfying \eqref{linearvnotdeviantsmall4}; this implies the proposition.
\end{proof}
The previous proposition estimates the diagonal terms $\big| G_{ii} \big|$. We must also estimate the off-diagonal terms $\big| G_{ij} \big|$: this is given by the following proposition.
\begin{prop}
\label{omegaijnotdeviant}
Adopt the notation of Proposition \ref{incrementnotdeviant}. There exist constants $C, \nu > 0$ (only dependent on $\kappa$, $C_1$, $C_2$, and $U_2$) such that
\begin{flalign}
\label{linearvnotdeviantsmall3gij}
\begin{aligned}
\mathbb{P}_{\textbf{\emph{H}} (\textbf{\emph{L}})} \Bigg[ \displaystyle\max_{\substack{i \in \mathcal{T} \\ 1 \le j \le N \\ i \ne j}} \big| G_{ij} \big| & \ge C (\log N)^{3 \xi} \bigg( N^{- \varepsilon / 10} + \sqrt{ \displaystyle\frac{1}{N \eta} } \bigg) \Bigg] \le \exp \big( - \nu (\log N)^{\xi} \big) + N \displaystyle\sum_{i = 1}^N P_i,
\end{aligned}
\end{flalign}
\noindent for any $2 \le \xi \le \log \log N$ and all sufficiently large $N$ (in comparison to $\kappa$, $C_1$, $C_2$, $U_1$, and $U_2$).
\end{prop}
\begin{proof}
We will establish this corollary through \eqref{suma2} from Lemma \ref{largeprobability2}. In particular, fix $i \in \mathcal{T}$ and $j \in [1, N]$; applying that lemma with $R_j = G_{kj}^{(i)}$, $X_j = a_{ij}$, and $N^{\delta} = N^{- \varepsilon / 10} = q^{-1}$ (and using the fact that $\big| \mathbb{E} [a_{ij}] \big| < 2 C_2 N^{-1 - \varepsilon / 10}$ and the estimate \eqref{qhijsmallmoments}) yields
\begin{flalign}
\label{linearvnotdeviantsmall3gij1}
\begin{aligned}
\mathbb{P}_{\textbf{H} (\textbf{\emph{L}})} \Bigg[ \bigg| \displaystyle\sum_{\substack{k \in [1, N] \\ k \ne i}} a_{ik} G_{kj}^{(i)} \bigg| \ge (\log N)^{\xi} \bigg( (2 C_2 + 1) N^{- \varepsilon / 10} \displaystyle\max_{\substack{1 \le k \le N \\ k \ne i}} \big| G_{jk}^{(i)} \big| + & \Big( \displaystyle\sum_{\substack{k \in [1, N] \\ k \ne i}} \big| G_{jk}^{(i)} \big|^2 \Big)^{1 / 2} \bigg) \Bigg] \\
& \le \exp \big( - \widetilde{\nu} (\log N)^{\xi} \big),
\end{aligned}
\end{flalign}
\noindent where $\widetilde{\nu}$ is the constant $\nu (C_1, 2 C_2)$ from Lemma \ref{largeprobability2}. In what follows, we restrict to the event $\bigcap_{i = 1}^N \overline{\Omega_i}$, where $\overline{\Omega_i}$ was defined in \eqref{notdeviantevent}. Applying Ward's identity \eqref{sumgij}, we find that
\begin{flalign}
\label{gjksumnotdeviant}
\displaystyle\frac{1}{N} \displaystyle\sum_{\substack{k \in [1, N] \\ k \ne i}} \big| G_{jk}^{(i)} \big|^2 = \displaystyle\frac{\Im G_{jj}^{(i)}}{N \eta} \le \displaystyle\frac{2 U_2}{N \eta},
\end{flalign}
\noindent where we used \eqref{notdeviantgjj1} to establish the third estimate above.
Inserting \eqref{notdeviantgjk3} and \eqref{gjksumnotdeviant} into \eqref{linearvnotdeviantsmall3gij1}, and also using the facts that $C_2 > 1$ and $\mathbb{P} \big[ \overline{\Omega_i} \big] \le P_i$, yields
\begin{flalign}
\label{linearvnotdeviantsmall3gij2}
\mathbb{P}_{\textbf{H} (\textbf{L})} \Bigg[ \bigg| \displaystyle\sum_{\substack{k \in [1, N] \\ k \ne i}} a_{ik} G_{kj}^{(i)} \bigg| \ge (\log N)^{\xi} \bigg( 6 C_2 U_2 N^{- \varepsilon / 10} + \sqrt{ \displaystyle\frac{2 U_2}{N \eta} } \bigg) \Bigg] \le \exp \big( - \widetilde{\nu} (\log N)^{\xi} \big) + P_i.
\end{flalign}
\noindent Applying \eqref{notdeviantgij1} in \eqref{linearvnotdeviantsmall3gij2}, the fact that $h_{ij}$ has the same distribution as $a_{ij}$ (since $i$ is typical), and \eqref{gij} yields
\begin{flalign}
\label{linearvnotdeviantsmall3gij3}
\mathbb{P}_{\textbf{H} (\textbf{L})} \Bigg[ |G_{ij}| \ge 2 U_2 (\log N)^{\xi} \bigg( 6 C_2 U_2 N^{- \varepsilon / 10} + \sqrt{ \displaystyle\frac{2U_2}{N \eta} } \bigg) \Bigg] \le \exp \big( - \widetilde{\nu} (\log N)^{\xi} \big) + P_i.
\end{flalign}
\noindent Now, the existence of $C, \nu > 0$ satisfying \eqref{linearvnotdeviantsmall3gij} follows from applying \eqref{linearvnotdeviantsmall3gij3} and a union estimate over all $i \in \mathcal{T}$ and $j \in [1, N]$.
\end{proof}
Above, we used the following lemma, which bounds the (deterministic) norm $\| \big( \Id - m_{\semicircle}^2 \widetilde{\textbf{S}} \big)^{-1} \|$; its proof is very similar to that of Lemma 2.15 in \cite{URM} and is thus omitted.
\begin{lem}
\label{testimate}
Fix some $\kappa \in (0, 1)$, let $z \in \mathbb{H}$ satifsy $\Re z \in (\kappa - 2, 2 - \kappa)$, let $M \in \mathbb{Z}_{> 0}$, and recall the definition of $m_{\semicircle} = m_{\semicircle} (z)$ from \eqref{mquadratic}. Let $\textbf{\emph{T}} = \{ t_{ij} \}$ be some $M \times M$ symmetric matrix with positive entries satisfying $\sum_{j = 1}^N t_{ij} \le 1$; assume that there exist $\widetilde{c}, \widetilde{C} > 0$ such that $\widetilde{c} < M t_{ij} < \widetilde{C}$ for each $i, j$. Then, there exists some constant $C > 0$ (only dependent on $\kappa$, $\widetilde{c}$, and $\widetilde{C}$) such that
\begin{flalign}
\label{mtestimate1}
\left\| \big( \Id - m_{\semicircle}^2 \textbf{\emph{T}} \big)^{-1} \right\|_{\infty} \le C \log M.
\end{flalign}
\end{lem}
\section{The Multiscale Argument for Deviant Indices}
\label{LawSmallDeviant}
The following result is an analog of Proposition \ref{incrementnotdeviant} and Proposition \ref{omegaijnotdeviant} that now addresses the resolvent entries indexed by deviant integers. Observe here that the sum on the right side of \eqref{psi} has significantly more terms than does the sum on the right side of \eqref{probabilitynotdeviantevent}. Further observe that, as in Remark \ref{u1u2c}, the constants $C$ and $\nu$ below do not depend on the parameter $U$; however, the smallest value of $N$ for which \eqref{linearvdeviantsmall4} holds might depend on $U$.
\begin{prop}
\label{incrementdeviant}
Fix $\kappa > 0$ and $U > 1$, and let $E \in [\kappa - 2, 2 - \kappa]$; also, let $\eta \in \mathbb{R}_{> 0}$ such that $N \eta > (\log N)^{8 \log \log N}$, and denote $z = E + \textbf{\emph{i}} \eta$. Fix a positive integer $N$, and let $\textbf{\emph{H}}$ be an $N \times N$ generalized Wigner matrix. Fix an admissible $N \times N$ $AB$ label $\textbf{\emph{L}} \in \mathcal{A}$. Recall the definitions of $v_i$ from \eqref{vj}; and of $\Omega_{C; \xi}^{(c)} (i, j)$ and $\Omega_C (i, j)$ from \eqref{omegaij}. Denote $r = \log \log N$ and
\begin{flalign}
\label{psi}
\begin{aligned}
\psi_i & = \displaystyle\sum_{\substack{\mathcal{S} \subset [1, N] \\ |\mathcal{S}| \le 2 r}} \displaystyle\sum_{\substack{j, k \in [1, N] \\ j, k \notin \mathcal{S}}} \mathbb{P}_{\textbf{\emph{H}}^{(\mathcal{S})} (\textbf{\emph{L}}^{(\mathcal{S})})} \Bigg[ \Omega_U \bigg( j, k; \textbf{\emph{H}}^{(\mathcal{S})}; E + \textbf{\emph{i}} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big) \bigg) \Bigg] \\
& \qquad + \displaystyle\sum_{\substack{\mathcal{S} \subset [1, N] \\ |\mathcal{S}| \le 2 r}} \displaystyle\sum_{\substack{j \in [1, N] \\ j \notin \mathcal{S} \\ j \in \mathcal{T}_{\textbf{\emph{L}}}}} \mathbb{P}_{\textbf{\emph{H}}^{(\mathcal{S})} (\textbf{\emph{L}}^{(\mathcal{S})})} \Bigg[ \Omega_{U; \xi}^{(1 / 20)} \bigg( j, j; \textbf{\emph{H}}^{(\mathcal{S})}; E + \textbf{\emph{i}} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big) \bigg) \Bigg].
\end{aligned}
\end{flalign}
\noindent Then, there exist constants $\nu, C > 0$ (only dependent on $\kappa$, $\varepsilon$, $C_1$, and $C_2$) such that
\begin{flalign}
\label{linearvdeviantsmall4}
\begin{aligned}
\mathbb{P}_{\textbf{\emph{H}} (\textbf{\emph{L}})} \Bigg[ \displaystyle\max_{1 \le i, j \le N} |G_{ij}| & \ge C \Bigg] \le \exp \big( - \nu (\log N)^{\xi} \big) + 3 N \displaystyle\sum_{i = 1}^N \psi_i,
\end{aligned}
\end{flalign}
\noindent for any $2 \le \xi \le \log \log N$ and sufficiently large $N$ (in comparison to $\kappa$, $\varepsilon$, $C_1$, $C_2$, and $U$).
\end{prop}
\begin{proof}
In what follows, let us fix deviant indices $i, j \in [1, N]$, and let us restrict to the event
\begin{flalign}
\label{deviantevent}
\begin{aligned}
\overline{\Psi_i} & = \bigcap_{\substack{\mathcal{S} \subset [1, N] \\ |\mathcal{S}| \le 2 r}} \bigcap_{\substack{j, k \in [1, N] \\ j, k \notin \mathcal{S}}} \overline{\Omega}_U \bigg( j, k; \textbf{H}^{(\mathcal{S})}; E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big) \bigg) \\
& \qquad \cap \bigcap_{\substack{\mathcal{S} \subset [1, N] \\ |\mathcal{S}| \le 2 r}} \bigcap_{\substack{j \in [1, N] \\ j \notin \mathcal{S} \\ j \in \mathcal{T}_{\textbf{L}} }} \overline{\Omega}_{U; \xi}^{(1 / 20)} \bigg( j, j; \textbf{H}^{(\mathcal{S})}; E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big) \bigg).
\end{aligned}
\end{flalign}
\noindent By a union estimate, we find that $\mathbb{P}_{\textbf{H} (\textbf{L})} \big[ \Psi_i \big] \le \psi_i$, where $\Psi_i$ is the complement of $\overline{\Psi_i}$.
We would like to estimate $\big| G_{ij} (E + \textbf{i} \eta) \big|$. By conjugating $\textbf{L}$ (and $\textbf{H}$) by a permutation matrix if necessary, we may assume that $i = 1$ and that $j \in \{ 1, 2 \}$ (depending on whether $i = j$ or $i \ne j$). Recalling the definitions from the beginning of Section \ref{IndicesGraph}, let $\mathscr{S} \subset [1, N]$ denote the union of $\{ i, j \}$ and the set of indices connected to either $i$ or $j$; we may assume that $\mathscr{S} = \{ 1, 2, \ldots , k \}$ for some $k < 2 r$, since $\textbf{L}$ is admissible.
We must consider two possibilities, when $i = j$ or when $i \ne j$. First assume that $i$ and $j$ are distinct, so that $j = 2$. Denote $\textbf{H} = \left[\begin{smallmatrix} \textbf{A} & \textbf{B} \\ \textbf{C} & \textbf{D} \end{smallmatrix} \right]$, where $\textbf{A}$ is the top-left $k \times k$ submatrix of $\textbf{H}$. Then, by the Schur complement identity \eqref{blockinverse}, we deduce that the $(i, j)$ entry of $(\textbf{H} - z)^{-1}$ is equal to the $(i, j)$ entry of the $k \times k$ matrix $\big( \textbf{A} - z - \textbf{B} ( \textbf{D} - z )^{-1} \textbf{C} \big)^{-1}$.
Denoting $\textbf{Y} = \textbf{B} ( \textbf{D} - z )^{-1} \textbf{C}$ and setting $\textbf{Y} = \{ y_{ij} \} = \{ y_{ij} (z) \}$, we have that
\begin{flalign}
\label{yij}
y_{ij} & = \displaystyle\sum_{i', j' \in [1, N] \setminus \mathscr{S}} a_{ii'} G_{i' j'}^{(\mathscr{S})} a_{j' j}.
\end{flalign}
\noindent Here, we are using the fact that the $(i, i')$ entry (and $(j, j')$ entry) of $\textbf{H} (\textbf{L})$ are equal to $a_{ii'}$ (and $a_{jj'}$), which follows since $i$ and $i'$ are unlinked (and as are $j$ and $j'$).
Since we are restricting to the event $\overline{\Psi_i}$, we have that
\begin{flalign}
\label{deviantgjk1}
\Bigg| G_{i' j'}^{(\mathscr{S})} \bigg( E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big) \bigg) \Bigg| \le U,
\end{flalign}
\noindent for all $i', j' \in [1, N] \setminus \mathscr{S}$. Thus, through a similar way as in the derivation of \eqref{notdeviantgjk3} from \eqref{notdeviantgjk1}, we find that
\begin{flalign}
\label{deviantgjk3}
\big| G_{i' j'}^{(\mathscr{S})} ( E + \textbf{i} \eta ) \big| \le 2 U,
\end{flalign}
\noindent for all $i', j' \in [1, N] \setminus \mathscr{S}$, assuming that $N > 10$.
Using \eqref{deviantgjk3}, we apply \eqref{abij2} with $X_{i'} = a_{ii'}$; $Y_{j'} = a_{j' j}$; $N^{-\delta} = N^{- \varepsilon / 10} = q^{-1}$; and $R_{i' j'} = G_{i' j'}^{(\mathscr{S})}$. Due to the estimates $\mathbb{E} \big[ | a_{i' j'} | \big] \le 2 C_2 N^{-1 - \varepsilon / 10}$ and \eqref{qhijsmallmoments}, this yields
\begin{flalign}
\label{deviantestimatey1}
\begin{aligned}
\mathbb{P} \Bigg[ \bigg| \displaystyle\sum_{i', j' \in [1, N] \setminus \mathscr{S}} a_{ii'} G_{i' j'}^{(\mathscr{S})} a_{j' j} \bigg| \ge & (\log N)^{2 \xi} \bigg( (20 C_2^2 + 2) N^{-\varepsilon / 10} \displaystyle\max_{\substack{1 \le i', j' \le N \\ i', j' \notin \mathscr{S}}} \big| G_{i' j'}^{(\mathscr{S})} \big| \\
& \qquad + \Big( \displaystyle\frac{1}{N^2} \displaystyle\sum_{\substack{1 \le i', j' \le N \\ i', j' \notin \mathscr{S}}} \big| G_{i' j'}^{(\mathscr{S})} \big|^2 \Big)^{1 / 2} \bigg) \Bigg] \le \exp \big( - \widetilde{\nu} (\log N)^{\xi} \big).
\end{aligned}
\end{flalign}
\noindent where we have denoted $\widetilde{\nu}$ as the constant $\nu (C_1, 2C_2)$ from Lemma \ref{largeprobability2}.
Inserting \eqref{deviantgjk3} and \eqref{yij} into \eqref{deviantestimatey1} (and using the fact that $C_2 > 1$) yields
\begin{flalign}
\label{deviantestimatey2}
\mathbb{P} \Bigg[ \big| y_{ij} \big| \ge (\log N)^{2 \xi} \bigg( 44 C_2^2 U N^{-\varepsilon / 10} + & \Big( \displaystyle\frac{1}{N^2} \displaystyle\sum_{\substack{1 \le i', j' \le N \\ i', j' \notin \mathscr{S}}} \big| G_{i' j'}^{(\mathscr{S})} \big|^2 \Big)^{1 / 2} \bigg) \Bigg] \le \exp \big( - \widetilde{\nu} (\log N)^{\xi} \big) + \mathbb{P} \big[ \Psi_i \big].
\end{flalign}
\noindent Applying Ward's identity \eqref{sumgij} and \eqref{deviantgjk3}, we find that
\begin{flalign}
\label{sumgijdeviant}
\left| \displaystyle\frac{1}{N^2} \displaystyle\sum_{\substack{1 \le i', j' \le N \\ i', j' \notin \mathscr{S}}} \big| G_{i' j'}^{(\mathscr{S})} \big|^2 \right| \ge \displaystyle\frac{1}{N^2 \eta} \displaystyle\sum_{i' \in [1, N] \setminus \mathscr{S}} \big| \Im G_{i' j'}^{(\mathscr{S})} \big| \le \displaystyle\frac{2 U}{N \eta}.
\end{flalign}
Inserting \eqref{sumgijdeviant} into \eqref{deviantestimatey2} yields
\begin{flalign}
\label{deviantestimatey3}
\mathbb{P} \Bigg[ \big| y_{ij} \big| \ge (\log N)^{2 \xi} \bigg( 44 C_2^2 U N^{-\varepsilon / 10} + \sqrt{\displaystyle\frac{2 U}{N \eta}} \bigg) \Bigg] \le \exp \big( - \widetilde{\nu} (\log N)^{\xi} \big) + \mathbb{P} \big[ \Psi_i \big].
\end{flalign}
\noindent The bound \eqref{deviantestimatey3} shows that $|y_{ij}|$ is small if $i \ne j$.
Now, assume that $i = j$. Then, using \eqref{yij} and the fact that $\sum_{j = 1}^N s_{ij} = 1 + t_i$, we find that
\begin{flalign}
\label{yii}
\begin{aligned}
y_{ii} & = \displaystyle\sum_{i', j' \notin \mathscr{S}} a_{ii'} G_{i' j'}^{(\mathscr{S})} a_{j' i} \\
& = m_{\semicircle} + \displaystyle\sum_{\substack{i', j' \notin \mathscr{S} \\ i' \ne j'}} a_{ii'} G_{i' j'}^{(\mathscr{S})} a_{j' i} + \displaystyle\sum_{j \notin S} \big( |a_{ij}|^2 - s_{ij} \big) G_{jj}^{(\mathscr{S})} + \displaystyle\sum_{j \notin \mathscr{S}} s_{ij} \big( G_{jj}^{(\mathscr{S})} - m_{\semicircle} \big) - m_{\semicircle} \left( \displaystyle\sum_{j \in \mathscr{S}} s_{ij} - t_i \right).
\end{aligned}
\end{flalign}
\noindent Now, since $\big| \mathscr{S} \big| \le 2 r$, $s_{ij} \le C_1 N^{-1}$, and $|t_i| < C_1 N^{-\varepsilon}$, we have that
\begin{flalign}
\label{sijs}
\left| m_{\semicircle} \displaystyle\sum_{j \in \mathscr{S}} s_{ij} \right| + \big| m_{\semicircle} t_i \big| \le 3 r C_1 |m_{\semicircle}| (N^{-1} + N^{-\varepsilon}).
\end{flalign}
\noindent Furthermore, since we are restricting to the event $\overline{\Psi_i}$, we have that
\begin{flalign}
\label{deviantgjj2}
\Bigg| G_{j' j'}^{(\mathscr{S})} \bigg( E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big) \bigg) - m_{\semicircle} \Bigg| \le U (\log N)^{3 \xi} \left( N^{- \varepsilon / 20} + \displaystyle\frac{1}{\sqrt{N \eta}} \right),
\end{flalign}
\noindent if $j' \in \mathcal{T}_{\textbf{L}}$ is typical. Following the derivation of \eqref{giismall1notdeviant1} from \eqref{giigjjsmall1notdeviant1}, we obtain that
\begin{flalign}
\label{deviantgjj3}
\big| G_{j' j'}^{(\mathscr{S})} ( E + \textbf{i} \eta ) - m_{\semicircle} \big| \le U (\log N)^{3 \xi} \left( N^{- \varepsilon / 20} + \displaystyle\frac{1}{\sqrt{N \eta}} \right) + \displaystyle\frac{2 U}{(\log N)^2},
\end{flalign}
\noindent if $j' \in \mathcal{T}_{\textbf{L}}$ is typical.
Furthermore, if $j' \in \mathcal{D}_{\textbf{L}}$ is deviant, then we have that $\big| G_{j' j'}^{(\mathscr{S})} - m_{\semicircle} \big| \le 2 U + |m_{\semicircle}|$, in view of \eqref{deviantgjk3}. Thus, since $s_{ij} \le C_1 N^{-1}$ and $\mathcal{D}_{\textbf{L}} \le N^{1 - \varepsilon / 20}$ (since $\textbf{L}$ is admissible), it follows that
\begin{flalign}
\label{sijgjjm}
\begin{aligned}
\left| \displaystyle\sum_{j \notin \mathscr{S}} s_{ij} \big( G_{jj}^{(\mathscr{S})} - m_{\semicircle} \big) \right| & \le \left| \displaystyle\sum_{\substack{j \notin \mathscr{S} \\ j \in \mathcal{T}_{\textbf{L}}}} s_{ij} \big( G_{jj}^{(\mathscr{S})} - m_{\semicircle} \big) \right| + \left| \displaystyle\sum_{\substack{j \notin \mathscr{S} \\ j \in \mathcal{D}_{\textbf{L}}}} s_{ij} \big( G_{jj}^{(\mathscr{S})} - m_{\semicircle} \big) \right| \\
& \le C_1 U (\log N)^{\xi} \left( N^{-c \varepsilon} + \displaystyle\frac{1}{\sqrt{N \eta}} \right) + \displaystyle\frac{2 C_1 U}{(\log N)^2} + C_1 \big( 2 U + |m_{\semicircle}| \big) N^{-\varepsilon / 20}.
\end{aligned}
\end{flalign}
\noindent To estimate the remaining terms in \eqref{yii}, we apply Lemma \ref{largeprobability2}. Specifically, applying \eqref{bij2} with $X_{i'} = a_{ii'}$; $B_{i' j'} = G_{i' j'}^{(\mathscr{S})}$; and $N^{-\delta} = N^{-\varepsilon / 10} = q^{-1}$, we obtain that
\begin{flalign}
\label{deviantestimateyii1}
\begin{aligned}
\mathbb{P} \Bigg[ \bigg| \displaystyle\sum_{\substack{i', j' \notin \mathscr{S} \\ i' \ne j'}} a_{ii'} G_{i' j'}^{(\mathscr{S})} a_{j' i} \bigg| \ge & (\log N)^{2 \xi} \bigg( (20 C_1^2 + 1) N^{-\varepsilon / 10} \displaystyle\max_{\substack{1 \le i', j' \le N \\ i', j' \notin \mathscr{S}}} \big| G_{i' j'}^{(\mathscr{S})} \big| \\
& \qquad + \Big( \displaystyle\frac{1}{N^2} \displaystyle\sum_{\substack{1 \le i', j' \le N \\ i', j' \notin \mathscr{S}}} \big| G_{i' j'}^{(\mathscr{S})} \big|^2 \Big)^{1 / 2} \bigg) \Bigg] \le \exp \big( - \widetilde{\nu} (\log N)^{\xi} \big).
\end{aligned}
\end{flalign}
\noindent Similar to \eqref{deviantestimatey3}, it quickly follows that
\begin{flalign}
\label{deviantestimateyii3}
\mathbb{P} \Bigg[ \bigg| \displaystyle\sum_{\substack{i', j' \notin \mathscr{S} \\ i' \ne j'}} a_{ii'} G_{i' j'}^{(\mathscr{S})} a_{j' i} \bigg| \ge (\log N)^{2 \xi} \bigg( 21 C_2^2 N^{-\varepsilon / 10} U + \sqrt{\displaystyle\frac{2 U}{N \eta}} \bigg) \Bigg] \le \exp \big( - \widetilde{\nu} (\log N)^{\xi} \big) + \mathbb{P} \big[ \Psi_i \big].
\end{flalign}
\noindent Applying \eqref{bi2} with $X_j = a_{ij}$; $s_j = s_{ij}$; $B_j = G_{jj}^{(\mathscr{S})}$; and $N^{-\delta} = N^{-\varepsilon / 10} = q^{-1}$, we obtain that
\begin{flalign}
\label{deviantestimateyii4}
\mathbb{P} \Bigg[ \bigg| \displaystyle\sum_{\substack{1 \le j \le N \\ j \notin \mathscr{S}}} \big( | a_{ij} |^2 - s_{ij} \big) G_{jj}^{(\mathscr{S})} \bigg| \ge & (\log N)^{\xi} (20 C_2^2 + 1) N^{-\varepsilon / 10} \displaystyle\max_{\substack{1 \le j \le N \\ j \notin \mathscr{S}}} \big| G_{jj}^{(\mathscr{S})} \big| \Bigg] \le \exp \big( - \widetilde{\nu} (\log N)^{\xi} \big).
\end{flalign}
\noindent Inserting \eqref{deviantgjk3} into \eqref{deviantestimateyii4} yields
\begin{flalign}
\label{deviantestimateyii5}
\mathbb{P} \Bigg[ \bigg| \displaystyle\sum_{\substack{1 \le j \le N \\ j \notin \mathscr{S}}} \big( | a_{ij} |^2 - s_{ij} \big) G_{jj}^{(\mathscr{S})} \bigg| \ge 21 C_2^2 (\log N)^{\xi} U N^{-\varepsilon / 10} \Bigg] \le \exp \big( - \widetilde{\nu} (\log N)^{\xi} \big) + \mathbb{P} \big[ \Psi_i \big].
\end{flalign}
\noindent Combining \eqref{yii}, \eqref{sijs}, \eqref{sijgjjm}, \eqref{deviantestimateyii3}, \eqref{deviantestimateyii5}, and a union estimate yields
\begin{flalign}
\label{yijdeviantsmall}
\mathbb{P} \left[ |y_{ii} - m_{\semicircle}| > \displaystyle\frac{1}{\log N} \right] \le 2 \exp \big( - \widetilde{\nu} (\log N)^{\xi} \big) + 3 \mathbb{P} \big[ \Psi_i \big],
\end{flalign}
\noindent assuming that $N$ is sufficiently large in comparison to $C_1$, $C_2$, $U$, and $\varepsilon^{-1}$.
Applying \eqref{deviantestimatey3}, \eqref{yijdeviantsmall}, and a union estimate yields
\begin{flalign}
\label{deviantestimateyentries}
\mathbb{P} \left[ \displaystyle\max_{1 \le i, j \le k} \big| y_{ij} - \textbf{1}_{i = j} m_{\semicircle} \big| \le \displaystyle\frac{1}{\log N} \right] \le 2 N^2 \exp \big( - \widetilde{\nu} ( \log N )^{\xi} \big) + 3 N \displaystyle\sum_{i = 1}^N \mathbb{P} \big[ \Psi_i \big],
\end{flalign}
\noindent assuming that $N$ is sufficiently large in comparison to $C_1$, $C_2$, $U$, and $\varepsilon^{-1}$.
From \eqref{deviantestimateyentries}, it follows that
\begin{flalign}
\label{deviantestimatey}
\mathbb{P} \left[ \| \textbf{Y} - m_{\semicircle} \Id \|_{\infty} \le \displaystyle\frac{2 \log \log N }{\log N} \right] \le 2 N^2 \exp \big( - \widetilde{\nu} ( \log N )^{\xi} \big) + 3 N \displaystyle\sum_{i = 1}^N \mathbb{P} \big[ \Psi_i \big],
\end{flalign}
\noindent where we have used the fact that $k \le 2 r = 2 \log \log N$.
Now, since $\Im z \in [\kappa - 2, 2 - \kappa]$, there exists some constant $c > 0$ (only dependent on $\kappa$) such that $\big| \Im (m_{\semicircle} + z) \big|> c$. Therefore, since $\textbf{A}$ is Hermitian, there exists some $\widetilde{C} > 0$ (only dependent on $\kappa$) such that $\big\| \big( \textbf{A} - (m_{\semicircle} + z) \Id \big)^{-1} \big\| < \widetilde{C}$. This implies by \eqref{deviantestimatey} that
\begin{flalign*}
\mathbb{P} \bigg[ \Big\| \big( \textbf{A} - z \Id - \textbf{Y} \big)^{-1} \Big\| \ge 2 \widetilde{C} \bigg] \le 2 N^2 \exp \big( - \widetilde{\nu} ( \log N )^{\xi} \big) + 3 N \displaystyle\sum_{i = 1}^N \mathbb{P} \big[ \Psi_i \big],
\end{flalign*}
\noindent for sufficiently large $N$ (in comparison to $C_1$, $C_2$, $U$, $\varepsilon^{-1}$, and $\kappa$), from which it follows that
\begin{flalign}
\label{gijdeviantestimate}
\mathbb{P} \Big[ \big| G_{ij} \big| \ge 2 \widetilde{C} \Big] \le 2 N^2 \exp \big( - \widetilde{\nu} ( \log N )^{\xi} \big) + 3 N \displaystyle\sum_{i = 1}^N \mathbb{P} \big[ \Psi_i \big],
\end{flalign}
\noindent where we used the facts that $G_{ij}$ is the $(i, j)$-entry of $\big( \textbf{A} - z \Id - \textbf{Y} \big)^{-1}$ and that both $\textbf{A}$ and $\textbf{Y}$ are symmetric. Now the existence of constants $C, \nu > 0$ satisfying \eqref{linearvdeviantsmall4} follows from \eqref{gijdeviantestimate}.
\end{proof}
\section{Proof of Theorem \ref{localmomentsdeltalabel}}
\label{ProofLocalLaw}
The purpose of this section is to establish Theorem \ref{localmomentsdeltalabel}, which will essentially follow from repeated application of Proposition \ref{incrementnotdeviant}, Proposition \ref{omegaijnotdeviant}, and Proposition \ref{incrementdeviant}. However, before doing so, we must first choose the constants from those propositions in such a way that those results can be applied simultaneously; to that end, we introduce the following notation.
\begin{itemize}
\item{Denote by $\gamma_1$ the constant $C$ from Proposition \ref{locallargeeta}; it only depends on $C_1$ and $C_2$.}
\item{Denote by $\mu_1$ the constant $\nu$ from Proposition \ref{locallargeeta}; it also only depends on $C_1$ and $C_2$.}
\item{Recall that Proposition \ref{locallargeeta} holds when $N$ is sufficiently large in comparison to $\varepsilon$; let $\Phi_1$ (dependent on only $\varepsilon$) be such that it holds whenever $N > \Phi_1$.}
\item{Denote by $\gamma_2$ the constant $C$ from Proposition \ref{incrementdeviant}; it only depends on $\varepsilon$, $\kappa$, $C_1$, and $C_2$.}
\item{Denote by $\mu_2$ the constant $\nu$ from Proposition \ref{incrementdeviant}; it only depends on $\varepsilon$, $\kappa$, $C_1$, and $C_2$.}
\item{Apply Proposition \ref{incrementnotdeviant} and Proposition \ref{omegaijnotdeviant} with $U_2 = \max \{ 1, \gamma_2 \}$ and $U_1 > 1$ arbitrary. Denote by $\gamma_3$ the resulting constant $C$ from Proposition \ref{incrementnotdeviant}, and denote by $\gamma_4$ the resulting constant from Proposition \ref{omegaijnotdeviant}. Recall that $\gamma_3$ and $\gamma_4$ only depend on $\varepsilon$, $\kappa$, $C_1$, $C_2$, and $U_2$; in particular, since $U_2$ only depends on the first four parameters, $\gamma_3$ and $\gamma_4$ only depend on $\varepsilon$, $\kappa$, $C_1$, and $C_2$.}
\item{Denote by $\mu_3$ and $\mu_4$ denote the constants $\nu$ from Proposition \ref{incrementnotdeviant} and Proposition \ref{omegaijnotdeviant}, respectively; they only depend on $\varepsilon$, $\kappa$, $C_1$, and $C_2$.}
\item{Let $\gamma = \max \{ 1, \gamma_1, \gamma_3, \gamma_4 \}$ and $\widetilde{\gamma} = \max \{ 1, \gamma_2\}$; furthermore, let $\widehat{\gamma} = \max \{ \gamma, \widetilde{\gamma} \}$. These parameters only depend on $\varepsilon$, $\kappa$, $C_1$, and $C_2$.}
\item{Let $\nu = \min \{ \mu_1, \mu_2, \mu_3, \mu_4 \}$; it only depends on $\varepsilon$, $\kappa$, $C_1$, and $C_2$. }
\item{Apply Proposition \ref{incrementnotdeviant} and Proposition \ref{omegaijnotdeviant} with $U_1 = \gamma$ and $U_2 = \widetilde{\gamma}$. Recall that both of these propositions hold when $N$ is sufficiently large in comparison to $\varepsilon$, $\kappa$, $C_1$, $C_2$, $U_1 = \gamma$, and $U_2 = \widetilde{\gamma}$. Let $\Phi_2$ and $\Phi_3$ (only dependent on $\varepsilon$, $\kappa$, $C_1$, and $C_2$, since $\gamma$ and $\widetilde{\gamma}$ are determined from those four parameters) be such that Proposition \ref{incrementnotdeviant} and Proposition \ref{omegaijnotdeviant} hold whenever $N > \Phi_2$ and $N > \Phi_3$, respectively. }
\item{Apply Proposition \ref{incrementdeviant} with $U = \widehat{\gamma}$. Recall that this proposition holds when $N$ is sufficiently large in comparison to $\varepsilon$, $\kappa$, $C_1$, $C_2$, and $\widehat{\gamma}$. Let $\Phi_4$ (dependent only on $\varepsilon$, $\kappa$, $C_1$, and $C_2$) be such this propostion holds for all $N > \Phi_4$.}
\item{Denote $\Phi = \max \{ \Phi_1, \Phi_2, \Phi_3, \Phi_4 \}$; it only depends on $\varepsilon$, $\kappa$, $C_1$, and $C_2$.}
\item{Set $r = \log \log N$.}
\end{itemize}
Now, select $N$ to be sufficiently large (in comparison to $\varepsilon$, $\kappa$, $C_1$, and $C_2$) such that
\begin{flalign}
\label{largen}
\left( 1 + \displaystyle\frac{1}{(\log N)^2} \right)^{(\log N)^4} > \widehat{\gamma} N; \qquad N - 2 r (\log N)^4 > \displaystyle\frac{N}{2} > \Phi.
\end{flalign}
Let us apply Proposition \ref{incrementnotdeviant} and Proposition \ref{omegaijnotdeviant} with $U_1 = \gamma$ and $U_2 = \widetilde{\gamma}$, and then apply Proposition \ref{incrementdeviant} with $U = \widehat{\gamma}$. From a union estimate, we obtain that
\begin{flalign}
\label{omega1}
& \mathbb{P}_{\textbf{H} (\textbf{L})} \Big[ \bigcap_{i \in \mathcal{T}_{\textbf{L}}} \bigcap_{j = 1}^N \Omega_{\gamma; \xi}^{(1 / 20)} \big( i, j; \textbf{H}, E + \textbf{i} \eta \big) \Big] \le 2 \exp \big( - \nu (\log N)^{\xi} \big) + (N + 1) \displaystyle\sum_{i = 1}^N P_i, \\
\label{omega2}
& \mathbb{P}_{\textbf{H} (\textbf{L})} \Big[ \bigcap_{1 \le i, j \le N} \Omega_{\widetilde{\gamma}} \big( i, j; \textbf{H}, E + \textbf{i} \eta \big) \Big] \le \exp \big( - \nu (\log N)^{\xi} \big) + 3 N \displaystyle\sum_{i = 1}^N \psi_i,
\end{flalign}
\noindent where $P_i$ and $\psi_i$ were defined in \eqref{probabilitynotdeviantevent} and \eqref{psi}, respectively.
Now, observe that
\begin{flalign}
\label{psi1}
\begin{aligned}
P_i, \psi_i & \le N^2 \binom{N}{2 r} \displaystyle\max_{\substack{\mathcal{S} \subset [1, N] \\ |\mathcal{S}| \le 2 r}} \displaystyle\max_{\substack{j, k \in [1, N] \\ j, k \notin \mathcal{S}}} \mathbb{P}_{\textbf{H}^{(\mathcal{S})} (\textbf{L}^{(\mathcal{S})})} \Bigg[ \Omega_{\widetilde{\gamma}} \bigg( j, k; \textbf{H}^{(\mathcal{S})}; E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big) \bigg) \Bigg] \\
& \qquad + N \binom{N}{2 r} \displaystyle\max_{\substack{\mathcal{S} \subset [1, N] \\ |\mathcal{S}| \le 2 r}} \displaystyle\max_{\substack{j \in [1, N] \\ j \notin \mathcal{S} \\ j \in \mathcal{T}_{\textbf{L}}}} \mathbb{P}_{\textbf{H}^{(\mathcal{S})} (\textbf{L}^{(\mathcal{S})})} \Bigg[ \Omega_{\gamma; \xi}^{(1 / 20)} \bigg( j, j; \textbf{H}^{(\mathcal{S})}; E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big) \bigg) \Bigg].
\end{aligned}
\end{flalign}
\noindent Inserting \eqref{psi1} into \eqref{omega1} and \eqref{omega2} yields
\begin{flalign}
\label{omega3}
\begin{aligned}
& \mathbb{P}_{\textbf{H} (\textbf{L})} \Big[ \bigcap_{i \in \mathcal{T}_{\textbf{L}}} \bigcap_{j = 1}^N \Omega_{\gamma; \xi}^{(1 / 20)} \big( i, j; \textbf{H}, E + \textbf{i} \eta \big) \Big] - 2 \exp \big( - \nu (\log N)^{\xi} \big)\\
& \qquad \le N^{6r} \displaystyle\max_{\substack{\mathcal{S} \subset [1, N] \\ |\mathcal{S}| \le 2 r}} \displaystyle\max_{\substack{j, k \in [1, N] \\ j, k \notin \mathcal{S}}} \mathbb{P}_{\textbf{H}^{(\mathcal{S})} (\textbf{L}^{(\mathcal{S})})} \Bigg[ \Omega_{\widetilde{\gamma}} \bigg( j, k; \textbf{H}^{(\mathcal{S})}; E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big) \bigg) \Bigg] \\
& \qquad \qquad + N^{6r} \displaystyle\max_{\substack{\mathcal{S} \subset [1, N] \\ |\mathcal{S}| \le 2 r}} \displaystyle\max_{\substack{j \in [1, N] \\ j \notin \mathcal{S} \\ j \in \mathcal{T}_{\textbf{L}}}} \mathbb{P}_{\textbf{H}^{(\mathcal{S})} (\textbf{L}^{(\mathcal{S})})} \Bigg[ \Omega_{\gamma; \xi}^{(1 / 20)} \bigg( j, j; \textbf{H}^{(\mathcal{S})}; E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big) \bigg) \Bigg],
\end{aligned}
\end{flalign}
\noindent and
\begin{flalign}
\label{omega4}
\begin{aligned}
& \mathbb{P}_{\textbf{H} (\textbf{L})} \Big[ \bigcap_{1 \le i, j \le N} \Omega_{\widetilde{\gamma}} \big( i, j; \textbf{H}, E + \textbf{i} \eta \big) \Big] - 2 \exp \big( - \nu (\log N)^{\xi} \big) \\
& \qquad \le N^{6 r} \displaystyle\max_{\substack{\mathcal{S} \subset [1, N] \\ |\mathcal{S}| \le 2 r}} \displaystyle\max_{\substack{j, k \in [1, N] \\ j, k \notin \mathcal{S}}} \mathbb{P}_{\textbf{H}^{(\mathcal{S})} (\textbf{L}^{(\mathcal{S})})} \Bigg[ \Omega_{\widetilde{\gamma}} \bigg( j, k; \textbf{H}^{(\mathcal{S})}; E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big) \bigg) \Bigg] \\
& \qquad \qquad + N^{6 r} \displaystyle\max_{\substack{\mathcal{S} \subset [1, N] \\ |\mathcal{S}| \le 2 r}} \displaystyle\max_{\substack{j \in [1, N] \\ j \notin \mathcal{S} \\ j \in \mathcal{T}_{\textbf{L}}}} \mathbb{P}_{\textbf{H}^{(\mathcal{S})} (\textbf{L}^{(\mathcal{S})})} \Bigg[ \Omega_{\gamma; \xi}^{(1 / 20)} \bigg( j, j; \textbf{H}^{(\mathcal{S})}; E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big) \bigg) \Bigg].
\end{aligned}
\end{flalign}
Now, fix an integer $k \ge 0$. Let us apply \eqref{omega1} and \eqref{omega2} again, but with $\textbf{H}$ replaced by $\textbf{H}^{(\mathcal{S})}$, and apply a union estimate over all $\mathcal{S} \subset [1, N]$ satisfying $\big| \mathcal{S} \big| < 2 k r$. We obtain that
\begin{flalign}
\label{omega5}
\begin{aligned}
& \displaystyle\max_{\substack{\mathcal{S} \subset [1, N] \\ |\mathcal{S}| \le 2 k r}} \displaystyle\max_{\substack{j, k \in [1, N] \\ j, k \notin \mathcal{S}}} \mathbb{P}_{\textbf{H}^{(\mathcal{S})} (\textbf{L}^{(\mathcal{S})})} \Bigg[ \Omega_{\gamma; \xi}^{(1 / 20)} \bigg( j, k; \textbf{H}^{(\mathcal{S})}; E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big)^k \bigg) \Bigg] \\
& \quad \le N^{12 (k + 1) r} \displaystyle\max_{\substack{\mathcal{S} \subset [1, N] \\ |\mathcal{S}| \le 2 (k + 1) r}} \displaystyle\max_{\substack{j, k \in [1, N] \\ j, k \notin \mathcal{S}}} \mathbb{P}_{\textbf{H}^{(\mathcal{S})} (\textbf{L}^{(\mathcal{S})})} \Bigg[ \Omega_{\widetilde{\gamma}} \bigg( j, k; \textbf{H}^{(\mathcal{S})}; E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big)^{k + 1} \bigg) \Bigg] \\
& \quad \quad + N^{12 (k + 1) r} \displaystyle\max_{\substack{\mathcal{S} \subset [1, N] \\ |\mathcal{S}| \le 2 (k + 1) r}} \displaystyle\max_{\substack{j \in [1, N] \\ j \notin \mathcal{S} \\ j \in \mathcal{T}_{\textbf{L}}}} \mathbb{P}_{\textbf{H}^{(\mathcal{S})} (\textbf{L}^{(\mathcal{S})})} \Bigg[ \Omega_{\gamma; \xi}^{(1 / 20)} \bigg( j, j; \textbf{H}^{(\mathcal{S})}; E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big)^{k + 1} \bigg) \Bigg] \\
& \quad \quad + 3 N^{12 k r} \exp \big( - \nu (\log N)^{\xi} \big),
\end{aligned}
\end{flalign}
\noindent and
\begin{flalign}
\label{omega6}
\begin{aligned}
& \displaystyle\max_{\substack{\mathcal{S} \subset [1, N] \\ |\mathcal{S}| \le 2 k r}} \displaystyle\max_{\substack{j, k \in [1, N] \\ j, k \notin \mathcal{S}}} \mathbb{P}_{\textbf{H}^{(\mathcal{S})} (\textbf{L}^{(\mathcal{S})})} \Bigg[ \Omega_{\widetilde{\gamma}} \bigg( j, k; \textbf{H}^{(\mathcal{S})}; E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big)^k \bigg) \Bigg] \\
& \quad \le N^{12 (k + 1) r} \displaystyle\max_{\substack{\mathcal{S} \subset [1, N] \\ |\mathcal{S}| \le 2 (k + 1) r}} \displaystyle\max_{\substack{j, k \in [1, N] \\ j, k \notin \mathcal{S}}} \mathbb{P}_{\textbf{H}^{(\mathcal{S})} (\textbf{L}^{(\mathcal{S})})} \Bigg[ \Omega_{\widetilde{\gamma}} \bigg( j, k; \textbf{H}^{(\mathcal{S})}; E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big)^{k + 1} \bigg) \Bigg] \\
& \quad \quad + N^{12 (k + 1) r} \displaystyle\max_{\substack{\mathcal{S} \subset [1, N] \\ |\mathcal{S}| \le 2 (k + 1) r}} \displaystyle\max_{\substack{j \in [1, N] \\ j \notin \mathcal{S} \\ j \in \mathcal{T}_{\textbf{L}}}} \mathbb{P}_{\textbf{H}^{(\mathcal{S})} (\textbf{L}^{(\mathcal{S})})} \Bigg[ \Omega_{\gamma; \xi}^{(1 / 20)} \bigg( j, j; \textbf{H}^{(\mathcal{S})}; E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big)^{k + 1} \bigg) \Bigg] \\
& \quad \quad + 3 N^{12 k r} \exp \big( - \nu (\log N)^{\xi} \big),
\end{aligned}
\end{flalign}
\noindent Now let $\zeta = \lceil (\log N)^4 \rceil$, and repeatedly apply \eqref{omega5} and \eqref{omega6} for all $k \in [0, \zeta - 1]$. We obtain that
\begin{flalign}
\label{omega7}
\begin{aligned}
& \mathbb{P}_{\textbf{H} (\textbf{L})} \Big[ \bigcap_{i \in \mathcal{T}_{\textbf{L}}} \bigcap_{j = 1}^N \Omega_{\gamma; \xi}^{(1 / 20)} \big( i, j; \textbf{H}, E + \textbf{i} \eta \big) \Big] + \mathbb{P}_{\textbf{H} (\textbf{L})} \Big[ \bigcap_{1 \le i, j \le N} \Omega_{\widetilde{\gamma}} \big( i, j; \textbf{H}, E + \textbf{i} \eta \big) \Big] \\
& \quad \le 2^{\zeta} N^{12 \zeta^2 r } \displaystyle\max_{\substack{\mathcal{S} \subset [1, N] \\ |\mathcal{S}| \le 2 \zeta r}} \displaystyle\max_{\substack{j, k \in [1, N] \\ j, k \notin \mathcal{S}}} \mathbb{P}_{\textbf{H}^{(\mathcal{S})} (\textbf{L}^{(\mathcal{S})})} \Bigg[ \Omega_{\widetilde{\gamma}} \bigg( j, k; \textbf{H}^{(\mathcal{S})}; E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big)^{\zeta} \bigg) \Bigg] \\
& \quad \quad + 2^{\zeta} N^{12 \zeta^2 r} \displaystyle\max_{\substack{\mathcal{S} \subset [1, N] \\ |\mathcal{S}| \le 2 \zeta r}} \displaystyle\max_{\substack{j \in [1, N] \\ j \notin \mathcal{S} \\ j \in \mathcal{T}_{\textbf{L}}}} \mathbb{P}_{\textbf{H}^{(\mathcal{S})} (\textbf{L}^{(\mathcal{S})})} \Bigg[ \Omega_{\gamma; \xi}^{(1 / 20)} \bigg( j, j; \textbf{H}^{(\mathcal{S})}; E + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big)^{\zeta} \bigg) \Bigg] \\
& \quad \quad + 6^{\zeta} N^{12 \zeta^2 r} \exp \big( - \nu (\log (N / 2) )^{\xi} \big),
\end{aligned}
\end{flalign}
\noindent for all $2 \le \xi \le \log \log N$.
Now, observe that $\eta \big( 1 + (\log N)^{-2} \big)^{\zeta} > \gamma$, due to the first estimate in \eqref{largen} and the fact that $\eta > N^{-1}$. This, and the second estimate of \eqref{largen} shows that we can apply Proposition \ref{locallargeeta} to obtain
\begin{flalign}
\label{omega8}
\begin{aligned}
\displaystyle\max_{\substack{\mathcal{S} \subset [1, N] \\ |\mathcal{S}| \le 2 \zeta r}} \displaystyle\max_{\substack{j, k \in [1, N] \\ j, k \notin \mathcal{S}}} \mathbb{P}_{\textbf{H}^{(\mathcal{S})} (\textbf{L}^{(\mathcal{S})})} \Bigg[ \Omega_{\widetilde{\gamma}} \bigg( j, k; \textbf{H}^{(\mathcal{S})}; E & + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log (N / 2))^2} \Big)^{\zeta} \bigg) \Bigg] \\
& \le N^{2 \zeta r + 2} \exp \big( - \nu (\log (N / 2) )^{\xi} \big)
\end{aligned}
\end{flalign}
\noindent and
\begin{flalign}
\label{omega9}
\begin{aligned}
\displaystyle\max_{\substack{\mathcal{S} \subset [1, N] \\ |\mathcal{S}| \le 2 \zeta r}} \displaystyle\max_{\substack{j \in [1, N] \\ j \notin \mathcal{S} \\ j \in \mathcal{T}_{\textbf{L}}}} \mathbb{P}_{\textbf{H}^{(\mathcal{S})} (\textbf{L}^{(\mathcal{S})})} \Bigg[ \Omega_{\gamma; \xi}^{(1 / 20)} \bigg( j, j; \textbf{H}^{(\mathcal{S})}; E & + \textbf{i} \eta \Big( 1 + \displaystyle\frac{1}{(\log N)^2} \Big)^{\zeta} \bigg) \Bigg] \\
& \le N^{2 \zeta r + 2} \exp \big( - \nu (\log (N / 2) )^{\xi} \big),
\end{aligned}
\end{flalign}
\noindent for all $2 \le \xi \le \log \log N$; here, we have applied a union estimate to bound the maximum.
Now \eqref{typicalomega} (with $C = \gamma$, $c = 1 / 20$, and $\xi = 10$, for $N$ sufficiently large) and \eqref{deviantomega} (with $C = \widetilde{\gamma}$ and $\xi = 10$, for $N$ sufficiently large) follow from inserting \eqref{omega8} and \eqref{omega9} into \eqref{omega7} and taking $\xi = 10$.
\section{Universality of Local Statistics}
\label{LocalH}
The goal of this section is to use the local semicircle law Theorem \ref{localmoments} to establish the bulk universality results Theorem \ref{gapsfunctions} and Theorem \ref{bulkfunctions} for local eigenvalue statistics of (heavy-tailed) generalized Wigner matrices. This will comprise the latter two parts of the three-step strategy.
Recall that the first of those is to apply a Dyson Brownian motion (or, in our case, a matrix Ornstein-Uhlenbeck process) to the original random matrix $\textbf{H}$, thereby forming matrix $\textbf{H}_t$, and then show that the local statistics of $\textbf{H}_t$ converge to those of $\textbf{GOE}_N$. The second is to show that the local statistics of $\textbf{H}_t$ are very similar to those of $\textbf{H}$.
These two steps have been implemented many times in the random matrix literature \cite{BESRRG, FEUGM, URMCE, SSGESEE, UM, URMLRF, LRFULSRM, URMFTD, BUGM, SSSG, BUSM, CLSM, FEUM, ULES}. In particular, our exposition will closely follow that of \cite{URMCE, SSSG, BUSM}. Therefore, we will only outline the proofs, explaining the differences where they arise.
\subsection{Bulk and Gap Universality Under Gaussian Perturbations}
\label{PerturbedUniversality}
We begin with the first step, that is, we apply an Ornstein-Uhlenbeck process to $\textbf{H}$ and show that the local statistics of the resulting matrix converge to those of the GOE in the large $N$ limit; see Theorem \ref{universalityperturbation3}. This will largely use the results of the recent works of Landon-Yau \cite{CLSM} and Landon-Sosoe-Yau \cite{FEUM}, which establish very quick convergence of bulk local statistics under Dyson Brownian motion.
The discussion in this section will be very similar to that in Section 3 of \cite{BUSM} and Section 4.2 of \cite{SSSG}, so we omit most proofs and refer to those papers for a more thorough explanation.
To proceed, we require the following definition of \cite{CLSM}, which defines a class of initial data for which it is possible to show quick convergence of Dyson Brownian motion.
\begin{definition}[{\cite[Definition 2.1]{CLSM}}]
Fix some $E_0 \in \mathbb{R}$, and let $\delta$ be a positive real number. For each positive real number $N$, let $r = r_N$ and $R = R_N$ be two parameters satisfying $N^{\delta - 1} \le r \le N^{-\delta}$ and $N^{\delta} r \le R \le N^{-\delta}$.
We call a diagonal $N \times N$ matrix $\textbf{D} = \textbf{D}_N = \{ V_1, V_2, \ldots , V_N \}$ \emph{$(r, R)$-regular with respect to $E_0$} if there exist constants $c, C > 0$ (independent of $N$) such that the estimates
\begin{flalign*}
c \le \Im m_{\textbf{D}} (E + \textbf{i} \eta) \le C; \qquad |V_i| \le N^C,
\end{flalign*}
\noindent both hold for all $E \in (E - R, E + R)$ and $r \le \eta \le 10$, where $m_{\textbf{D}} (z)$ denotes the Stieltjes transform of $\textbf{D}$ for all $z \in \mathbb{H}$, as defined in \eqref{mn}. We call an arbitrary symmetric matrix $\textbf{M}$ \emph{$(r, R)$-regular with respect to $E_0$} if $\textbf{D} (M)$ is $(r, R)$-regular with respect to $E_0$, where $\textbf{D} (M)$ denotes a diagonal matrix whose entries are the eigenvalues of $\textbf{M}$.
\end{definition}
The results of \cite{CLSM, FEUM} essentially state that, if we start with a $(r, R)$-regular diagonal matrix and then add an independent small Gaussian component of order greater than $r$ but less than $R$, then the local statistics of the result will asymptotically coincide with those of the GOE. To state this more precisely, we must introduce the free convolution \cite{FCSD} of a probability distribution with the semicircle law.
To that end, fix $N \in \mathbb{Z}_{> 0}$ and a symmetric $N \times N$ matrix $\textbf{A}$. For each $s \ge 0$, define $\textbf{A}^{(s)} = \textbf{A} + s^{1 / 2} \textbf{GOE}_N$. Further denote by $m^{(s)} (z) = m_{\textbf{A}^{(s)}} (z)$ the Stieltjes transform \eqref{mn} of the empirical spectral density of $\textbf{A}^{(s)}$, which we denote by $\rho^{(s)} (x) = \pi^{-1} \lim_{\eta \rightarrow 0} \Im m^{(s)} (E + \textbf{i} \eta)$.
For each $i \in [1, N]$, let $\gamma_i$ and $\gamma_i^{(s)}$ denote the \emph{classical eigenvalue locations} of the distributions $\rho_{\semicircle}$ and $\rho^{(s)}$, respectively, defined by the equations
\begin{flalign*}
\displaystyle\int_{-\infty}^{\gamma_i} \rho_{\semicircle} (x) dx = \displaystyle\frac{i}{N}; \qquad \displaystyle\int_{-\infty}^{\gamma_i^{(s)}} \rho^{(s)} (x) dx = \displaystyle\frac{i}{N}.
\end{flalign*}
The following two theorems establish the universality of gap statistics and correlation functions of the random matrix $\textbf{M}^{(s)}$, assuming that $\textbf{M}$ is regular.
\begin{prop}[{\cite[Theorem 2.5]{CLSM}}]
\label{universalityperturbation1}
Let $N$ be a positive integer, and let $r = r_N$ and $R = R_N$ be positive real parameters dependent on $N$. Fix a real number $\kappa > 0$, and let $\textbf{\emph{M}}$ be a real, symmetric $N \times N$ matrix. Assume that $\textbf{\emph{M}}$ is $(r, R)$-regular with respect to some fixed $E \in (\kappa - 2, 2 - \kappa)$.
Fix $\delta > 0$ (independent of $N$), and assume that there exists some $s > 0$ satisfying $N^{\delta} r < s < N^{-\delta} R$. Let $i \in [1, N]$ be an integer satisfying $\gamma_i^{(s)} \in [E - G / 2, E + G / 2]$.
Fix a positive integer $k$. Then, there exists a sufficiently small real number $c = c_{\delta; k} > 0$ such that the following holds. For any compactly supported smooth function $F \in \mathcal{C}_0^{\infty} (\mathbb{R}^k)$ and any positive integers $i_1 < i_2 < \cdots < i_k < N^c$, we have (for sufficiently large $N$) that
\begin{flalign}
\label{fmt}
\begin{aligned}
\bigg| & \mathbb{E}_{\textbf{\emph{M}}_t} \Big[ F \big( N \rho^{(s)} (\gamma_i^{(s)}) (\lambda_i - \lambda_{i + i_1}), N \rho^{(s)} (\gamma_i^{(s)}) (\lambda_i - \lambda_{i + i_2}), \ldots , N \rho_i^{(s)} (\gamma_i^{(s)}) (\lambda_i - \lambda_{i + i_n} )\big) \Big] \\
& - \mathbb{E}_{\textbf{\emph{GOE}}_N} \Big[ F \big( N \rho_{\semicircle} (\gamma_i) (\lambda_i - \lambda_{i + i_1}), N \rho_{\semicircle} (\gamma_i) (\lambda_{i} - \lambda_{i + 1}), \ldots , N \rho_{\semicircle} (\gamma_i) (\lambda_{i + n - 1} - \lambda_{i + n} ) \big) \Big] \bigg| < N^{-c}.
\end{aligned}
\end{flalign}
\end{prop}
\begin{prop}[{\cite[Theorem 2.2]{FEUM}}]
\label{universalityperturbation2}
Adopt the notation of Proposition \ref{universalityperturbation1}, and fix a positive integer $k$. Then, there exists a sufficiently small real number $c = c_{\delta; k} > 0$ such that the following holds. For any $F \in \mathcal{C}_0^{\infty} (\mathbb{R}^k)$, we have (for sufficiently large $N$) that
\begin{flalign}
\label{pmt}
\begin{aligned}
& \Bigg| \displaystyle\int_{\mathbb{R}^k} F (a_1, a_2, \ldots , a_k) p_{\textbf{\emph{M}}^{(s)}}^{(k)} \left( E + \displaystyle\frac{a_1}{N \rho^{(s)} (E)}, E + \displaystyle\frac{a_2}{N \rho^{(s)} (E)}, \ldots , E + \displaystyle\frac{a_k}{N \rho^{(s)} (E)} \right) \displaystyle\prod_{j = 1}^k d a_j \\
& - \displaystyle\int_{\mathbb{R}^k} F (a_1, a_2, \ldots , a_k) p_{\textbf{\emph{GOE}}_N}^{(k)} \left( E + \displaystyle\frac{a_1}{N \rho_{\semicircle} (E)}, E + \displaystyle\frac{a_2}{N \rho_{\semicircle} (E)}, \ldots , E + \displaystyle\frac{a_k}{N \rho_{\semicircle} (E)} \right) \displaystyle\prod_{j = 1}^k d a_j \Bigg| < N^{-c}.
\end{aligned}
\end{flalign}
\end{prop}
These two propositions can be applied to deduce universality of a matrix $\textbf{H}_t$, defined from the original generalized Wigner matrix $\textbf{H} = \{ h_{ij} \}$, as follows. For each $1 \le i, j \le N$, let $B_{ij} (s)$ be a Brownian motion so that $B_{ij} (s) = B_{ji} (s)$ and the $\big\{ B_{ij} (s) \big\}$ are mutually independent (and also independent from $\textbf{H}$) for $1 \le i \le j \le N$. Denote by $h_{ij} (s)$ the solution to the Ornstein-Uhlenbeck equation
\begin{flalign}
\label{hijt}
d h_{ij} (s) = N^{-1 / 2} d B_{ij} (s) - (2 N s_{ij})^{-1} h_{ij} (s) ds,
\end{flalign}
\noindent and define the $N \times N$ random real symmetric matrix $\textbf{H}_t = \big\{ h_{ij} (s) \big\}$.
Using Proposition \ref{universalityperturbation1} and Proposition \ref{universalityperturbation2}, one can deduce the following result.
\begin{prop}
\label{universalityperturbation3}
Fix constants $\kappa > 0$; $0 < \delta < \varepsilon < 1 / 2$; $0 < c_1 < 1 < C_1$; and $C_2 > 1$. Let $\big\{ \textbf{\emph{H}} = \textbf{\emph{H}}_N \big\}_{N \ge 1}$ be a family of generalized Wigner matrices as in Definition \ref{momentassumption}. Let $\lambda_1, \lambda_2, \ldots , \lambda_N$ denote the eigenvalues of $\textbf{\emph{H}}$, and denote $t = t_N = t^{\delta - 1}$. Define $\textbf{\emph{H}}_t$ as above.
Fix a positive integer $k$. Then, there exists a sufficiently small real number $c = c_{\delta; k} > 0$ such that the following holds. For any compactly supported smooth function $F \in \mathcal{C}_0^{\infty} (\mathbb{R}^k)$ and any positive integers $i_1 < i_2 < \cdots < i_k < N^c$, we have (for sufficiently large $N$) that
\begin{flalign}
\label{htf}
\begin{aligned}
\bigg| \mathbb{E}_{\textbf{\emph{H}}_t} \Big[ & F \big( N (\lambda_i - \lambda_{i + i_1}), N (\lambda_i - \lambda_{i + i_2}), \ldots , N (\lambda_i - \lambda_{i + i_n} )\big) \Big] \\
& - \mathbb{E}_{\textbf{\emph{GOE}}_N} \Big[ F \big( N (\lambda_i - \lambda_{i + i_1}), N (\lambda_{i} - \lambda_{i + 1}), \ldots , N (\lambda_{i + n - 1} - \lambda_{i + n} ) \big) \Big] \bigg| < N^{-c}.
\end{aligned}
\end{flalign}
Furthermore, we have (for sufficiently large $N$) that
\begin{flalign}
\label{pht}
\begin{aligned}
\Bigg| & \displaystyle\int_{\mathbb{R}^k} F (a_1, a_2, \ldots , a_k) p_{\textbf{\emph{H}}_t}^{(k)} \left( E + \displaystyle\frac{a_1}{N \rho_{\semicircle} (E)}, E + \displaystyle\frac{a_2}{N \rho_{\semicircle} (E)}, \ldots , E + \displaystyle\frac{a_k}{N \rho_{\semicircle} (E)} \right) \displaystyle\prod_{j = 1}^k d a_j \\
& - \displaystyle\int_{\mathbb{R}^k} F (a_1, a_2, \ldots , a_k) p_{\textbf{\emph{GOE}}_N}^{(k)} \left( E + \displaystyle\frac{a_1}{N \rho_{\semicircle} (E)}, E + \displaystyle\frac{a_2}{N \rho_{\semicircle} (E)}, \ldots , E + \displaystyle\frac{a_k}{N \rho_{\semicircle} (E)} \right) \displaystyle\prod_{j = 1}^k d a_j \Bigg| < N^{-c}.
\end{aligned}
\end{flalign}
\end{prop}
Given the local semicircle law Theorem \ref{localmoments} and the universality statements Proposition \ref{universalityperturbation1} and Proposition \ref{universalityperturbation2} above, the proof of this proposition is very similar to that of Theorem 3.1 in \cite{BUSM} and Proposition 4.9 in \cite{SSSG}; therefore, it is omitted. However, let us briefly explain the idea of the proof, referring to the references \cite{BUSM, SSSG} for the remaining details.
First observe that $\textbf{H}_t$ is formed by $\textbf{H}$ from applying an Ornstein-Uhlenbeck process for time $t = N^{\delta - 1}$, while Proposition \ref{universalityperturbation1} and Proposition \ref{universalityperturbation2} are stated for matrices of the form $\textbf{M} + s^{1 / 2} \textbf{GOE}_N$. It happens that $\textbf{H}_t$ is also of the latter form (see, for example equation (2.17) in \cite{BUSM}), with $\textbf{M} = \textbf{H}_t^{(1)} = \{ h_{ij; t}^{(1)} \}$ defined by
\begin{flalign*}
h_{ij; t}^{(1)} = e^{- t / 2 N s_{ij}} h_{ij} + N^{-1 / 2} B_{ij} (s) \sqrt{N s_{ij} \big( 1 - e^{t / N s_{ij}} \big) - r \left( \displaystyle\frac{1 + \textbf{1}_{i = j}}{2} \right) \big( 1 - e^{-t / r} \big) },
\end{flalign*}
\noindent where $r = N \min_{1 \le i, j \le N} s_{ij} \in (c_1, C_1)$. In particular, the law of $\textbf{H}_t$ is that of
\begin{flalign}
\label{sh1}
\textbf{H}_t^{(1)} + s^{1 / 2} \textbf{GOE}_N, \qquad \text{where $s = \displaystyle\sqrt{r(1 - e^{-t / r})}{2}$,}
\end{flalign}
\noindent and $\textbf{GOE}_N$ is chosen to be independent from $\textbf{H}_t^{(1)}$; observe that $s$ is of order $t = N^{\delta - 1}$.
It can be quickly verified that $\textbf{H}_t^{(1)}$ is also a generalized Wigner matrix in the sense of Definition \ref{momentassumption}, meaning by Theorem \ref{localmoments} that it satisfies a local semicircle law on some event $\Omega$ that has probability at least $1 - C N^{-c \log \log N}$, for some constants $c, C > 0$. One can then condition on the matrix $\textbf{H}_t^{(1)}$ and apply Proposition \ref{universalityperturbation1} and Proposition \ref{universalityperturbation2} with $\textbf{A} = \textbf{H}_t^{(1)}$ and $s$ as in \eqref{sh1}, to deduce that \eqref{fmt} and \eqref{pmt} hold for $\textbf{A}^{(s)} = \textbf{H}_t$.
The remaining difference between \eqref{fmt} and \eqref{htf} and between \eqref{pmt} and \eqref{pht} is in the scaling. Specifically, one must approximate the factors of $\rho^{(s)} \big( \gamma_j^{(s)} \big)$ by $\rho_{\semicircle} (\gamma_i)$ in \eqref{fmt} and the factors of $\rho^{(s)} (E)$ by $\rho_{\semicircle} (E)$ in \eqref{pmt}. This approximation can be justified using the local semicircle law Theorem \ref{localmoments}; this can be done in a very similar way to what was explained in Lemma 3.3 and Lemma 3.4 of \cite{BUSM} and Lemma 4.12 of \cite{SSSG} and thus we omit further details.
This provides an outline of the proof of Proposition \ref{universalityperturbation3} assuming Theorem \ref{localmoments}, Proposition \ref{universalityperturbation1}, and Proposition \ref{universalityperturbation2}; we again refer to Section 3 of \cite{BUSM} and Section 4.2 of \cite{SSSG} for a more comprehensive exposition.
\subsection{Continuity Estimates}
\label{HHt}
Recall that one of our goals is to establish Theorem \ref{bulkfunctions}, which states that the correlation functions of $\textbf{H}$ are universal; in view of Proposition \ref{universalityperturbation3} it suffices to show that the correlation functions of the perturbed matrix $\textbf{H}_t$ (from \eqref{hijt}) equal those of $\textbf{H}$ in the large $N$ limit. The following lemma, which first appeared in some weaker form as Lemma 6.4 in \cite{BUGM} (but was later \cite{DRM, BUSM} altered, with very little modification in the proof, to essentially as below) provides a sufficient condition for when the correlation functions of two generalized Wigner matrices asymptotically coincide.
\begin{lem}[{\cite[Theorem 15.3]{DRM}, \cite[Theorem 6.4]{BUGM}, \cite[Theorem 5.3]{BUSM}} ]
\label{hhtcorrelations}
Fix $\kappa > 0$, $\varepsilon > 0$, $0 < c_1 < 1 < C_1$, and $C_2 > 1$. Let $\{ \textbf{\emph{A}} = \textbf{\emph{A}}_N \}_{N \in \mathbb{Z}_{\ge 1}}$ and $\{ \textbf{\emph{B}} = \textbf{\emph{B}}_N \}_{N \in \mathbb{Z}_{\ge 1}}$ be two families of generalized Wigner matrices. Further fix an arbitrary $k \in \mathbb{Z}_{\ge 1}$ and an arbitrary real $\omega \in (0, 1]$ (bounded away from $0$ independently of $N$). Also fix positive integers $r_1, r_2, \ldots , r_k$, and let $\{ z_1^{(j)}, z_2^{(j)}, \ldots , z_{r_j}^{(j)} \}_{1 \le j \le k} \subset \mathbb{H}$ be families of complex numbers such that $\Re z_i^{(j)} \in [\kappa - 2, 2 - \kappa]$ and $\Im z_i^{(j)} \in [N^{-\omega - 1}, N^{-1}]$ for each $i, j$.
For each $z \in \mathbb{H}$, denote $\textbf{\emph{G}}^{(\textbf{\emph{A}})} (z) = (\textbf{\emph{A}} - z)^{-1}$ and define $\textbf{\emph{G}}^{(\textbf{\emph{B}})} (z)$ similarly. Assume that there exist constants $\widetilde{c} = \widetilde{c}_{k; r_1, r_2, \ldots , r_k} > 0$ and $\widetilde{C} = \widetilde{C}_{k; r_1, r_2, \ldots , r_k} > 0$ such that the following holds whenever $\omega < \widetilde{c}$. For any compactly supported smooth function $\Lambda \in \mathcal{C}_0^{\infty} (\mathbb{C}^k)$ satisfying
\begin{flalign}
\label{lambdaderivatives}
\displaystyle\max_{1 \le |\alpha| \le 4} \displaystyle\sup_{|y_j| \le N^{\beta}} \big| \partial^{\alpha} \Lambda (y_1, y_2, \ldots , y_k) \big| \le N^{\widetilde{C} \beta}; \qquad \displaystyle\max_{1 \le |\alpha| \le 4} \displaystyle\sup_{|y_j| \le N^2 } \big| \partial^{\alpha} \Lambda (y_1, y_2, \ldots , y_k) \big| \le N^{\widetilde{C}},
\end{flalign}
\noindent for each $\beta > 0$, we have (for sufficiently large $N$) that
\begin{flalign}
\label{ablambda}
\begin{aligned}
\Bigg| \mathbb{E} & \bigg[ \Lambda \Big( N^{-r_1} \Tr \displaystyle\prod_{j = 1}^{r_1} \textbf{\emph{G}}^{(\textbf{\emph{A}})} \big( z_j^{(1)} \big), N^{-r_2} \Tr \displaystyle\prod_{j = 1}^{r_2} \textbf{\emph{G}}^{(\textbf{\emph{A}})} \big( z_j^{(2)} \big), \ldots , N^{-r_k} \Tr \displaystyle\prod_{j = 1}^{r_k} \textbf{\emph{G}}^{(\textbf{\emph{A}})} \big( z_j^{(k)} \big) \Big) \bigg] \\
& - \mathbb{E} \bigg[ \Lambda \Big( N^{-r_1} \Tr \displaystyle\prod_{j = 1}^{r_1} \textbf{\emph{G}}^{(\textbf{\emph{B}})} \big( z_j^{(1)} \big), N^{-r_2} \Tr \displaystyle\prod_{j = 1}^{r_2} \textbf{\emph{G}}^{(\textbf{\emph{B}})} \big( z_j^{(2)} \big), \ldots , N^{-r_k} \Tr \displaystyle\prod_{j = 1}^{r_k} \textbf{\emph{G}}^{(\textbf{\emph{B}})} \big( z_j^{(k)} \big) \Big) \bigg] \Bigg| < N^{-\widetilde{c}}.
\end{aligned}
\end{flalign}
\noindent Then, there exists some $c = c_k > 0$ such that for any $F \in \mathcal{C}_0^{\infty} (\mathbb{R}^k)$, we have (for sufficiently large $N$) that
\begin{flalign*}
\Bigg| \displaystyle\int_{\mathbb{R}^n} F (a_1, a_2, \ldots , a_N) & \bigg( p_{\textbf{\emph{A}}}^{(k)} \Big( E + \displaystyle\frac{a_1}{N}, E + \displaystyle\frac{a_2}{N}, \ldots , E + \displaystyle\frac{a_N}{N} \Big) \\
& - p_{\textbf{\emph{B}}}^{(k)} \Big( E + \displaystyle\frac{a_1}{N}, E + \displaystyle\frac{a_2}{N}, \ldots , E + \displaystyle\frac{a_N}{N} \Big) \bigg) \displaystyle\prod_{i = 1}^N d a_i \Bigg| < N^{- c}.
\end{flalign*}
\end{lem}
\noindent In view of Lemma \ref{hhtcorrelations}, we would like to establish the following proposition.
\begin{prop}
\label{functionghht}
Adopt the notation of Theorem \ref{hhtcorrelations}, and let $\{ \textbf{\emph{H}} = \textbf{\emph{H}}_N \}_{N \in \mathbb{Z}_{\ge 0}}$ denote a family of generalized Wigner matrices. If $\omega$ is sufficiently small (in a way that only depends on $\widetilde{C}$ and $\varepsilon$), then there exists some $\delta > 0$ (independent of $N$) such that \eqref{ablambda} holds with $\textbf{\emph{A}} = \textbf{\emph{H}}$ and $\textbf{\emph{B}} = \textbf{\emph{H}}_t$. Here, we have set $t = N^{\delta - 1}$ and recalled the definition of $\textbf{\emph{H}}_t$ from \eqref{hijt}.
\end{prop}
We will establish Proposition \ref{functionghht} later, in Section \ref{UniversalityMatrix}.
In the context of less singular Wigner matrix (whose entry laws have at least three moments, for example), proofs of Proposition \ref{functionghht} have appeared in several previous works; for instance, see Lemma 5.2 of \cite{BUSM}. The recent proofs of such results are based on a certain \emph{continuity estimate} that originally appeared as Lemma A.1 of \cite{EMFLQUE}.
Unfortunately, that lemma assumes that $\mathbb{E} \big[ |h_{ij} \sqrt{N} |^3 \big] < \infty$, which might be false in our setting. Thus we require a modification of that result, which is given below as Lemma \ref{fsmall}.
To state this lemma, we require some additional notation. Fix some positive integer $N$. For each pair of integers $1 \le a, b \le N$, let $\textbf{X}_{ab}$ denote the $N \times N$ matrix whose entries are all equal to $0$, except for the $(a, b)$ and $(b, a)$ entries which are equal to $1$.
Furthermore, fix some $N \times N$ real symmetric matrix $\textbf{M} = \{ m_{ij} \}$. For any $\theta \in [0, 1]$ and integers $1 \le a, b \le N$, let $\Theta_{ab} \textbf{M} = \{ \widetilde{m_{ij}} \}$ denote the $N \times N$ symmetric matrix whose entries $\widetilde{m_{ij}} = m_{ij}$ if $(i, j) \notin \{ (a, b), (b, a) \}$ and $\widetilde{m_{ij}} = \theta m_{ij}$ otherwise. When $\theta = 0$, we denote $\Theta_{ab} \textbf{M} = \textbf{Z}_{ab} \textbf{M}$.
Moreover, for any smooth function $F$ (from the set of $N \times N$ real symmetric matrices to $\mathbb{C}$), let $\partial_{ij} F$ denote the partial derivative of $F$ in the $X_{ij}$-coordinate. Specifically, we set $\partial_{ij} F (\textbf{M}) = \lim_{y \rightarrow 0} y^{-1} \big( F(\textbf{M} + y \textbf{X}_{ij}) - F(\textbf{M}) \big)$, if it exists.
Now we have the following estimate.
\begin{lem}
\label{fsmall}
Let $\textbf{\emph{H}} = \{ h_{ij} \}$ be an $N \times N$ generalized Wigner matrix in the sense of Definition \ref{momentassumption}. For any $t \ge 0$, define $\textbf{\emph{H}}_t = \{ h_{ij} (t) \}$ as in \eqref{hijt}. Let $F$ be a smooth function. Then,
\begin{flalign}
\label{fsmall1}
\Big| \mathbb{E} \big[ F (\textbf{\emph{H}}_t) - F (\textbf{\emph{H}}_0) \big] \Big| \le t N \Xi,
\end{flalign}
\noindent where
\begin{flalign}
\label{fsmall2}
\begin{aligned}
\Xi & = \displaystyle\max_{1 \le i, j \le N} \Bigg| s_{ij}^{-1} \mathbb{E} \bigg[ \textbf{\emph{1}}_{|h_{ij} (t)| \ge N^{-\varepsilon / 10}} \big| h_{ij} (t) \big| \displaystyle\sup_{0 \le \theta \le 1} \big| \partial_{ij} F (\Theta_{ij} \textbf{\emph{H}}_t) \big| \bigg] \\
& \qquad \qquad + s_{ij}^{-1} \mathbb{E} \bigg[ \textbf{\emph{1}}_{|h_{ij} (t)| \ge N^{-\varepsilon / 10}} \big| h_{ij} (t) \big|^2 \displaystyle\sup_{0 \le \theta \le 1} \big| \partial_{ij}^2 F (\Theta_{ij} \textbf{\emph{H}}_t) \big| \bigg] \\
& \qquad \qquad + \mathbb{E} \bigg[\textbf{\emph{1}}_{|h_{ij} (t)| \ge N^{-\varepsilon / 10}} \displaystyle\sup_{0 \le \theta \le 1} \big| \partial_{ij}^2 F (\Theta_{ij} \textbf{\emph{H}}_t) \big| \bigg] \\
& \qquad \qquad + s_{ij}^{-1} \mathbb{E} \bigg[ \textbf{\emph{1}}_{|h_{ij} (t)| < N^{-\varepsilon / 10}} \big| h_{ij} (t) \big|^3 \displaystyle\sup_{0 \le \theta \le 1} \big| \partial_{ij}^3 F (\Theta_{ij} \textbf{\emph{H}}_t) \big| \bigg] \\
& \qquad \qquad + \mathbb{E} \bigg[ \textbf{\emph{1}}_{|h_{ij} (t)| < N^{-\varepsilon / 10}} \big| h_{ij} (t) \big| \displaystyle\sup_{0 \le \theta \le 1} \big| \partial_{ij}^3 F (\Theta_{ij} \textbf{\emph{H}}_t) \big| \bigg] \Bigg|.
\end{aligned}
\end{flalign}
\end{lem}
\begin{proof}
Applying It\^{o}'s Lemma in the definition \ref{hijt} of the $h_{ij} (t)$ yields
\begin{flalign}
\label{fsmall3}
\partial_t \mathbb{E} \big[ F (\textbf{H}_t) \big]= \displaystyle\frac{1}{2N} \displaystyle\sum_{1 \le i \le j \le N} \Big( \mathbb{E} \big[ \partial_{ij}^2 F (\textbf{H}_t) \big] - s_{ij}^{-1} \mathbb{E} \big[ h_{ij} (t) \partial_{ij} F (\textbf{H}_t) \big] \Big).
\end{flalign}
\noindent We would now like to Taylor expand both $\partial_{ij}^2 F (\textbf{H}_t)$ and $h_{ij} (t) \partial_{ij} F (\textbf{H}_t)$ in a neighborhood of $h_{ij} (t) = 0$. However, there is the issue that $h_{ij} (t)$ might not be small.
To resolve that, we observe
\begin{flalign}
\label{hfhderivative}
\begin{aligned}
\Big| h_{ij} (t) & \partial_{ij} F (\textbf{H}_t) - h_{ij} (t) \partial_{ij} F (\textbf{Z}_{ij} \textbf{H}_t) - h_{ij} (t)^2 \partial_{ij}^2 F (\textbf{Z}_{ij} \textbf{H}_t) \Big| \\
& \le \textbf{1}_{|h_{ij} (t)| \ge N^{-\varepsilon / 10}} \bigg( 2 \big| h_{ij} (t) \big| \displaystyle\sup_{0 \le \theta \le 1} \big| \partial_{ij} F (\Theta_{ij} \textbf{H}_t) \big| + \big| h_{ij} (t) \big|^2 \displaystyle\sup_{0 \le \theta \le 1} \big| \partial_{ij}^2 F (\Theta_{ij} \textbf{H}_t) \big| \bigg) \\
& \quad + \textbf{1}_{|h_{ij} (t)| < N^{-\varepsilon / 10}} \big| h_{ij} (t) \big|^3 \displaystyle\sup_{0 \le \theta \le 1} \big| \partial_{ij}^3 F (\Theta_{ij} \textbf{H}_t) \big|.
\end{aligned}
\end{flalign}
\noindent Similarly, we find that
\begin{flalign}
\label{2fhderivative}
\begin{aligned}
\Big| \partial_{ij}^2 F (\textbf{H}_t) - \partial_{ij}^2 F (\textbf{Z}_{ij} \textbf{H}_t) \Big| & \le 2 \textbf{1}_{|h_{ij} (t)| \ge N^{-\varepsilon / 10}} \displaystyle\sup_{0 \le \theta \le 1} \big| \partial_{ij}^2 F (\Theta_{ij} \textbf{H}_t) \big| \\
& \quad + \textbf{1}_{|h_{ij} (t)| < N^{-\varepsilon / 10}} \big| h_{ij} (t) \big| \displaystyle\sup_{0 \le \theta \le 1} \big| \partial_{ij}^3 F (\Theta_{ij} \textbf{H}_t) \big|.
\end{aligned}
\end{flalign}
\noindent Now, since the $h_{ij} (t)$ are centered, $\mathbb{E} \big[ |h_{ij} (t)|^2 \big] = \mathbb{E} \big[ |h_{ij}|^2 \big] = s_{ij}$, and $h_{ij} (t)$ is independent from $\textbf{Z}_{ij} \textbf{H}_t$, we find that
\begin{flalign}
\label{hijfexpectation}
s_{ij}^{-1} \mathbb{E} \big[ h_{ij} (t) \partial_{ij} F (\textbf{Z}_{ij} \textbf{H}_t) + h_{ij} (t)^2 \partial_{ij}^2 F (\textbf{Z}_{ij} \textbf{H}_t) \big] = \mathbb{E} \big[ \partial_{ij}^2 F (\textbf{Z}_{ij} \textbf{H}_t) \big].
\end{flalign}
\noindent The claimed estimate \eqref{fsmall1} now follows from combining \eqref{fsmall3}, \eqref{hfhderivative}, \eqref{2fhderivative}, and \eqref{hijfexpectation}, and summing over all $1 \le i \le j \le N$.
\end{proof}
To use Lemma \ref{fsmall}, we must estimate certain derivatives of $F$. The following lemma does this in the case when $F = m_{\textbf{H}} (z)$ is the Stieltjes transform of $\textbf{H}$, which will be useful for us later in Section \ref{UniversalityMatrix}.
\begin{lem}
\label{derivativesm}
Fix constants $\kappa > 0$, $\varepsilon > 0$, $0 < c_1 < 1 < C_1$, and $C_2 > 1$. Let $\textbf{\emph{H}}$ be an $N \times N$ generalized Wigner matrix in the sense of Definition \ref{momentassumption}. Then, there exist constants $0 < c < C$ such that
\begin{flalign}
\label{entriesthetag}
\mathbb{P} \left[ \displaystyle\sup_{z \in \mathscr{D}_{\kappa; N}} \displaystyle\max_{1 \le i, j \le N} \displaystyle\max_{1 \le a, b \le N} \displaystyle\sup_{0 \le \theta \le 1} \big| (\Theta_{ab} \textbf{\emph{H}} - z)_{ij}^{-1} \big| > C \right] \le C N^{-c \log \log N},
\end{flalign}
\noindent where we recall the definition of $\mathscr{D}_{\kappa; N}$ from below \eqref{dkappanr}. Furthermore, fix a real number $0 \le \omega < 1$ and set $r = r_N = r_{N; \omega} = N^{-1 - \omega}$. Then, we have that
\begin{flalign}
\label{entriesthetagomega}
\mathbb{P} \left[ \displaystyle\sup_{z \in \mathscr{D}_{\kappa; N; r}} \displaystyle\max_{1 \le i, j \le N} \displaystyle\max_{1 \le a, b \le N} \displaystyle\sup_{0 \le \theta \le 1} \big| (\Theta_{ab} \textbf{\emph{H}} - z)_{ij}^{-1} \big| > C N^{\omega} (\log N)^{C \log \log N} \right] \le C N^{-c \log \log N}.
\end{flalign}
\noindent Moreover, for any integer $1 \le k \le 4$, we have that
\begin{flalign}
\label{entriesthetagomegaderivative}
\mathbb{P} \left[ \displaystyle\sup_{z \in \mathscr{D}_{\kappa; N; r}} \displaystyle\max_{\substack{1 \le i, j \le N \\ 1 \le a, b \le N}} \displaystyle\sup_{0 \le \theta \le 1} \Big| N^{-1} \partial_{ij}^{(k)} \big( \Tr (\Theta_{ab} \textbf{\emph{H}} - z )^{-1} \big) \Big| > C N^{(k + 1) \omega} (\log N)^{C \log \log N} \right] \le C N^{- c \log \log N}.
\end{flalign}
\end{lem}
\begin{proof}
We begin with the proof of \eqref{entriesthetag}. First observe that, since $\Var h_{ij} < C_1 N^{-1}$, each $\Theta_{ab} \textbf{H}$ is a generalized Wigner matrix in the sense of Definition \ref{momentassumption} (perhaps with a slightly larger value of $C_1$). Hence, we can apply Theorem \ref{gijestimate} to deduce the existence positive constants $\widetilde{c}$ and $\widetilde{C}$ such that
\begin{flalign*}
\mathbb{P} \left[ \displaystyle\sup_{z \in \mathscr{D}_{\kappa; N}} \displaystyle\max_{1 \le i, j \le N} \big| (\Theta_{ab} \textbf{H} - z)_{ij}^{-1} \big| > \widetilde{C} \right] < \widetilde{C} N^{-\widetilde{c} \log \log N},
\end{flalign*}
\noindent for each fixed $\theta \in [0, 1]$ and fixed $1 \le a, b \le N$. Now, let $\mathbb{M}_N = \big\{ \theta \in [0, 1] : N^8 \theta \in \mathbb{Z} \big\}$. From a union estimate, it follows that
\begin{flalign}
\label{entriesthetag2}
\mathbb{P} \left[ \displaystyle\sup_{z \in \mathscr{D}_{\kappa; N}} \displaystyle\max_{1 \le i, j \le N} \displaystyle\max_{1 \le a, b \le N} \displaystyle\sup_{\theta \in \mathbb{M}_N} \big| (\Theta_{ab} \textbf{H} - z)_{ij}^{-1} \big| > \widetilde{C} \right] < \widetilde{C} N^{10 - \widetilde{c} \log \log N}.
\end{flalign}
Now the existence of constants $0 < c < C$ satisfying \eqref{entriesthetag} follows from \eqref{entriesthetag2} and the fact that $\big| ( \Theta_{ab} \textbf{H} - z )_{ij}^{-1} - ( \Theta_{ab} \textbf{H} - z' )_{ij}^{-1} \big| < 1$ if $\Im z, \Im z' > N^{-2}$ and $|z - z'| > N^{-7}$; the latter estimate holds due to the resolvent identity \eqref{resolvent} and the deterministic estimate \eqref{gijeta}.
The estimate \eqref{entriesthetagomega} follows directly from \eqref{entriesthetag} and the fact that $\Gamma (E + \textbf{i} \eta / R) \le R \Gamma (E + \textbf{i} \eta)$, for any real number $R$, where $\Gamma (z) = \Gamma_{\textbf{M}} (z) = \max_{1 \le i, j \le N} \max \big\{ 1, \big| (\textbf{M} - z)_{ij}^{-1} \big| \big\} $ for any $N \times N$ deterministic matrix $\textbf{M}$; the latter estimate appears as Lemma 2.1 of \cite{LSLRRG}.
To derive \eqref{entriesthetagomegaderivative}, one uses \eqref{entriesthetagomega} and the fact that
\begin{flalign}
\label{derivativeijabhz}
\partial_{ij}^{(k)} \Tr (\Theta_{ab} \textbf{H} - z)^{-1} = (-1)^k k! \Tr \Big( \big( (\Theta_{ab} \textbf{H} - z)^{-1} \textbf{X}_{ij} \big)^k (\Theta_{ab} \textbf{H} - z)^{-1} \Big).
\end{flalign}
\noindent In particular, since $\textbf{X}_{ij}$ only has two nonzero entries (both of which are equal to $1$), this trace is a sum of at most $2^k N$ terms that are each products of at most $k + 1$ entries of $(\Theta_{ab} \textbf{H} - z)^{-1}$. Each of these entries is bounded by $C N^{\omega} (\log N)^{C \log \log N}$ with very high probability in view of \eqref{entriesthetagomega}, from which we deduce \eqref{entriesthetagomegaderivative} (after incrementing $C$ if necessary).
\end{proof}
\subsection{Comparing \texorpdfstring{$\textbf{H}$}{} and \texorpdfstring{$\textbf{H}_t$}{}}
\label{UniversalityMatrix}
We now use the estimates from Section \ref{HHt} to establish Proposition \ref{functionghht}.
\begin{proof}[Proof of Proposition \ref{functionghht}]
To ease notation, we assume that $k = 1$ and $r_1 = 1$; the proof in the more general case is very similar.
For any symmetric matrix $\textbf{M}$, denote $\widetilde{\Lambda} (\textbf{M}) = \Lambda \big( N^{-1} \Tr \textbf{G}^{(\textbf{M})} (z_1^{(1)}) \big)$. To establish the proposition, we would like to apply Lemma \ref{fsmall} with $F = \widetilde{\Lambda}$; this requires estimates on the derivatives of $\widetilde{\Lambda}$. To that end, observe that
\begin{flalign}
\label{derivativelambdaij}
\partial_{ij}^{(k)} \widetilde{\Lambda} (\textbf{M}) = \displaystyle\sum_{j = 0}^k \binom{k}{j} \Big( \partial^{(j)} \Lambda (\textbf{M}) \Big) \partial_{ij}^{(k - j)} \big( N^{-1} \Tr (\textbf{M} - z)^{-1} \big).
\end{flalign}
We will obtain two types of estimates on these derivatives of $\widetilde{\Lambda}$, a deterministic bound and a singificantly improved high-probability estimate. Let us begin with the deterministic estimate. To that end, observe that in view of \eqref{derivativeijabhz}, \eqref{gijeta}, the fact that $\omega < 1$, and the fact that $k \le 4$, we deterministically have that
\begin{flalign}
\label{derivativelambdaij1}
\Big| \partial_{ij}^{(k - m)} \big( N^{-1} \Tr (\textbf{M} - z)^{-1} \big) \Big| < 1000 N^{20}.
\end{flalign}
\noindent Furthermore, the second estimate in \eqref{lambdaderivatives} and the fact that $N^{-1} \Tr \textbf{G}^{(\textbf{H})} (z_1^{(1)}) < N^2$ (again due to \eqref{gijeta}) together yield that $\big| \partial^{(k)} \Lambda \big( N^{-1} \Tr (\Theta_{ij} \textbf{M} - z_1^{(1)} )^{-1}\big) \big| < N^{\widetilde{C}}$. Combining with \eqref{derivativelambdaij} and \eqref{derivativelambdaij1} yields the deterministic estimate
\begin{flalign}
\label{derivativelambdaij2}
\Big| \partial_{ij}^{(k)} \widetilde{\Lambda} (\textbf{M}) \Big| < 10000 N^{\widetilde{C} + 20}.
\end{flalign}
Now let us obtain a very high probability estimate on the right side of \eqref{derivativelambdaij} in the case when $\textbf{M} = \textbf{H}_t$ (which we recall is a generalized Wigner matrix). In view of \eqref{entriesthetagomegaderivative} and the fact that $k \le 4$, we can estimate
\begin{flalign}
\label{derivativelambdaij3}
\begin{aligned}
\mathbb{P} & \left[ \displaystyle\sup_{z \in \mathscr{D}_{\kappa; N; r}} \displaystyle\max_{\substack{1 \le i, j \le N \\ 1 \le a, b \le N}} \displaystyle\sup_{0 \le \theta \le 1} \Big| \partial_{ij}^{(k - j)} \big( N^{-1} \Tr (\Theta_{ab} \textbf{H}_t - z)^{-1} \big) \Big| > \overline{C} N^{5 \omega} (\log N)^{\overline{C} \log \log N} \right] \\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad < \overline{C} N^{-\overline{c} \log \log N},
\end{aligned}
\end{flalign}
\noindent for some constants $\overline{c}$ and $\overline{C}$; above, $r = r_{N; \omega} = N^{-\omega - 1}$.
Furthermore, in view of the first estimate in \eqref{lambdaderivatives}, the fact that $\omega$ is bounded away from $0$, and the estimate \eqref{entriesthetagomega}, we deduce that
\begin{flalign}
\label{derivativelambdaij4}
\begin{aligned}
\mathbb{P} & \left[ \displaystyle\sup_{z \in \mathscr{D}_{\kappa; N; r}} \displaystyle\max_{\substack{1 \le i, j \le N \\ 1 \le a, b \le N}} \displaystyle\sup_{0 \le \theta \le 1} \Big| \partial^{(j)} \Lambda \big( N^{-1} \Tr (\Theta_{ij} \textbf{H}_t - z_1^{(1)} )^{-1} \big) \Big| > \overline{C} N^{2 \widetilde{C} \omega} \right] <\overline{C} N^{-\overline{c} \log \log N},
\end{aligned}
\end{flalign}
\noindent for sufficiently large $N$. Therefore, it follows from \eqref{derivativelambdaij}, \eqref{derivativelambdaij3}, and \eqref{derivativelambdaij4} that there exist positive constants $\widehat{c}, \widehat{C}$ such that if we denote the event
\begin{flalign}
\label{evente}
\mathcal{E} = \mathcal{E}_{\widehat{C}} \left\{ \displaystyle\sup_{z \in \mathscr{D}_{\kappa; N; r}} \displaystyle\sup_{0 \le \theta \le 1} \displaystyle\max_{\substack{1 \le i, j \le N \\ 1 \le a, b \le N}} \big| \partial_{ij}^{(k)} \widetilde{\Lambda} (\Theta_{ab} \textbf{H}) \big| > \widehat{C} N^{\widehat{C} \omega} \right\},
\end{flalign}
\noindent then we have that
\begin{flalign}
\label{derivativelambdaij5}
\mathbb{P} [\mathcal{E}] < \widehat{C} N^{-\widehat{c} \log \log N}.
\end{flalign}
Using \eqref{derivativelambdaij2} and \eqref{derivativelambdaij5}, we can estimate the value of $\Xi$ \eqref{fsmall2} from Lemma \ref{fsmall}. In particular, we claim that there exists a constant $C$ such that $\Xi \le C N^{C \omega - \varepsilon / 20}$.
To establish this, we recall from \eqref{fsmall2} that $\Xi$ is the sum of five terms; each one will be bounded by some quantity of the form $C N^{C \omega - \varepsilon / 20}$. We will only explicitly verify this for the second and fourth term; the remaining three terms can be addressed similarly.
We begin with the second term. For fixed $i, j$, it is equal to
\begin{flalign}
\label{derivativelambdaij6}
\begin{aligned}
& s_{ij}^{-1} \mathbb{E} \bigg[ \textbf{1}_{|h_{ij} (t)| \ge N^{-\varepsilon / 10}} \big| h_{ij} (t) \big|^2 \displaystyle\sup_{0 \le \theta \le 1} \big| \partial_{ij}^2 F (\Theta_{ij} \textbf{H}_t) \big| \bigg] \\
& \qquad = s_{ij}^{-1} \mathbb{E} \bigg[ \textbf{1}_{\mathcal{E}} \textbf{1}_{|h_{ij} (t)| \ge N^{-\varepsilon / 10}} \big| h_{ij} (t) \big|^2 \displaystyle\sup_{0 \le \theta \le 1} \big| \partial_{ij}^2 F (\Theta_{ij} \textbf{H}_t) \big| \bigg] \\
& \qquad \qquad + s_{ij}^{-1} \mathbb{E} \bigg[ \textbf{1}_{\overline{\mathcal{E}}} \textbf{1}_{|h_{ij} (t)| \ge N^{-\varepsilon / 10}} \big| h_{ij} (t) \big|^2 \displaystyle\sup_{0 \le \theta \le 1} \big| \partial_{ij}^2 F (\Theta_{ij} \textbf{H}_t) \big| \bigg].
\end{aligned}
\end{flalign}
\noindent where $\overline{\mathcal{E}}$ denotes the complement of the event $\mathcal{E}$. The first summand on the right side of \eqref{derivativelambdaij6} can be bounded by $N^{C - \widehat{c} \log \log N}$ (for some constant $C > 0$), due to the deterministic estimate \eqref{derivativelambdaij2}, the probability estimate \eqref{derivativelambdaij5}, the fact that $|s_{ij}| > c_1 N^{-1}$, and the boundedness of the second moment of $|h_{ij} \sqrt{N}|$.
The second summand can be bounded as
\begin{flalign*}
s_{ij}^{-1} \mathbb{E} \bigg[ \textbf{1}_{\overline{\mathcal{E}}} \textbf{1}_{|h_{ij} (t)| \ge N^{-\varepsilon / 10}} \big| h_{ij} (t) \big|^2 \displaystyle\sup_{0 \le \theta \le 1} \big| \partial_{ij}^2 F (\Theta_{ij} \textbf{H}_t) \big| \bigg] & \le c_1^{-1} \widehat{C} N^{1 + \widehat{C} \omega} \mathbb{E} \big[ \textbf{1}_{|h_{ij} (t)| \ge N^{-\varepsilon / 10}} \big| h_{ij} (t) \big|^2 \big] \\
& \le c_1^{-1} C_2 \widehat{C} N^{\widehat{C} \omega - \varepsilon / 20}.
\end{flalign*}
\noindent Here, we used the fact that $s_{ij} \ge c_1 N^{-1}$ and the definition \eqref{evente} of the event $\mathcal{E}$ in the first estimate; in the second estimate, we used assumption \ref{moments} and the fact that
\begin{flalign*}
\mathbb{E} \big[ \textbf{1}_{|h_{ij} (t)| \ge N^{-\varepsilon / 10}} \big| h_{ij} (t) \big|^2 \big] \le N^{\varepsilon^2 / 10} \mathbb{E} \big[ |h_{ij} (t)|^{2 + \varepsilon} \big] \le C_2 N^{- 1 - \varepsilon / 20}.
\end{flalign*}
\noindent Hence, it follows that the left side of \eqref{derivativelambdaij6} is bounded by $N^{C - \widehat{c} \log \log N} + c_1^{-1} C_2 \widehat{C} N^{\widehat{C} \omega - \varepsilon / 20} = \mathcal{O} (N^{\widehat{C} \omega - \varepsilon / 20})$.
The fourth term in the definition \eqref{fsmall2} of $\Xi$ can be estimated similarly. Specifically,
\begin{flalign}
\label{derivativelambdaij8}
\begin{aligned}
& s_{ij}^{-1} \mathbb{E} \bigg[ \textbf{1}_{|h_{ij} (t)| < N^{-\varepsilon / 10}} \big| h_{ij} (t) \big|^3 \displaystyle\sup_{0 \le \theta \le 1} \big| \partial_{ij}^3 F (\Theta_{ij} \textbf{H}_t) \big| \bigg] \\
& \qquad = s_{ij}^{-1} \mathbb{E} \bigg[ \textbf{1}_{\mathcal{E}} \textbf{1}_{|h_{ij} (t)| < N^{-\varepsilon / 10}} \big| h_{ij} (t) \big|^3 \displaystyle\sup_{0 \le \theta \le 1} \big| \partial_{ij}^3 F (\Theta_{ij} \textbf{H}_t) \big| \bigg] \\
& \qquad \qquad + s_{ij}^{-1} \mathbb{E} \bigg[ \textbf{1}_{\mathcal{E}} \textbf{1}_{|h_{ij} (t)| < N^{-\varepsilon / 10}} \big| h_{ij} (t) \big|^3 \displaystyle\sup_{0 \le \theta \le 1} \big| \partial_{ij}^3 F (\Theta_{ij} \textbf{H}_t) \big| \bigg].
\end{aligned}
\end{flalign}
Again, the first term on the right side of \eqref{derivativelambdaij8} can be bounded by $N^{C - \widehat{c} \log \log N}$ due to \eqref{derivativelambdaij2} and \eqref{derivativelambdaij5}. To estimate the second term, we observe that
\begin{flalign}
\label{derivativelambdaij9}
\begin{aligned}
s_{ij}^{-1} \mathbb{E} \bigg[ \textbf{1}_{\overline{\mathcal{E}}} \textbf{1}_{|h_{ij} (t)| \ge N^{-\varepsilon / 10}} \big| h_{ij} (t) \big|^3 \displaystyle\sup_{0 \le \theta \le 1} \big| \partial_{ij}^3 F (\Theta_{ij} \textbf{H}_t) \big| \bigg] & \le c_1^{-1} \widehat{C} N^{1 + \widehat{C} \omega} \mathbb{E} \big[ \textbf{1}_{|h_{ij} (t)| < N^{-\varepsilon / 10}} \big| h_{ij} (t) \big|^3 \big] \\
& \le c_1^{-1} C_2 \widehat{C} N^{\widehat{C} \omega - \varepsilon / 20},
\end{aligned}
\end{flalign}
\noindent where in the second estimate we used the fact that
\begin{flalign*}
\mathbb{E} \big[ \textbf{1}_{|h_{ij} (t)| < N^{-\varepsilon / 10}} \big| h_{ij} (t) \big|^3 \big] < N^{(\varepsilon / 10)(\varepsilon - 1)} \mathbb{E} \big[ |h_{ij} (t)|^{2 + \varepsilon} \big] < N^{- 1 - \varepsilon / 20}.
\end{flalign*}
Thus, we can bound the right side of \eqref{derivativelambdaij9} by $N^{C - \widehat{c} \log \log N} + c_1^{-1} C_2 \widehat{C} N^{\widehat{C} \omega - \varepsilon / 20} = \mathcal{O} (N^{\widehat{C} \omega - \varepsilon / 20})$; this estimates the fourth term in \eqref{fsmall2}.
As mentioned previously, the other three terms in the definition \eqref{fsmall2} of $\Xi$ can be bounded similarly. It follows that there exists a constant $C > 0$ such that $\Xi < C N^{C \omega - \varepsilon / 20}$. Recalling that $t = N^{\delta - 1}$ and inserting the result into \eqref{fsmall1} yields that the left side of \eqref{ablambda} is bounded by $C N^{C \omega - \varepsilon / 20 + \delta}$. Setting $\omega$ and $\delta$ sufficiently small so that $C \omega + \delta < \varepsilon / 40$ then yields \eqref{ablambda} with $\widetilde{c} = \varepsilon / 40$; this confirms the proposition.
\end{proof}
\begin{proof}[Proof of Theorem \ref{bulkfunctions}]
This follows from Proposition \ref{universalityperturbation3}, Lemma \ref{hhtcorrelations}, and Proposition \ref{functionghht}.
\end{proof}
This establishes Theorem \ref{bulkfunctions}. We can also prove Theorem \ref{gapsfunctions} but, given what we have already done, this is very similar to what was already explained in several previous works; see, for example, Section 4 and Lemma 5.1 of \cite{BUSM} or Section 4.1.2 of \cite{SSSG}.
The main difference between what should be done in our setting and what was done in their setting is that we must use the less restrictive continuity estimate Lemma \ref{fsmall} as opposed to Lemma 4.5 of \cite{SSSG} or Lemma 4.3 of \cite{BUSM} (which both require that $\mathbb{E} \big[ |h_{ij} \sqrt{N}|^3 \big]$ is bounded). Usage of Lemma \ref{fsmall} has already been explained in the proof of Proposition \ref{functionghht} above and, since all other parts of the proof of Theorem \ref{gapsfunctions} are essentially the same as those in \cite{SSSG, BUSM}, we omit further details.
| {
"timestamp": "2016-12-02T02:09:25",
"yymm": "1612",
"arxiv_id": "1612.00421",
"language": "en",
"url": "https://arxiv.org/abs/1612.00421",
"abstract": "In this paper we consider $N \\times N$ real generalized Wigner matrices whose entries are only assumed to have finite $(2 + \\varepsilon)$-th moment for some fixed, but arbitrarily small, $\\varepsilon > 0$. We show that the Stieltjes transforms $m_N (z)$ of these matrices satisfy a weak local semicircle law on the nearly smallest possible scale, when $\\eta = \\Im (z)$ is almost of order $N^{-1}$. As a consequence, we establish bulk universality for local spectral statistics of these matrices at fixed energy levels, both in terms of eigenvalue gap distributions and correlation functions, meaning that these statistics converge to those of the Gaussian Orthogonal Ensemble (GOE) in the large $N$ limit.",
"subjects": "Probability (math.PR); Mathematical Physics (math-ph)",
"title": "Bulk Universality for Generalized Wigner Matrices With Few Moments",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754456551737,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.7075103943107596
} |
https://arxiv.org/abs/1502.01786 | Complete graph immersions in dense graphs | In this article we consider the relationship between vertex coloring and the immersion order. Specifically, a conjecture proposed by Abu-Khzam and Langston in 2003, which says that the complete graph with $t$ vertices can be immersed in any $t$-chromatic graph, is studied.First, we present a general result about immersions and prove that the conjecture holds for graphs whose complement does not contain any induced cycle of length four and also for graphs having the property that every set of five vertices induces a subgraph with at least six edges.Then, we study the class of all graphs with independence number less than three, which are graphs of interest for Hadwiger's Conjecture. We study such graphs for the immersion-analog. If Abu-Khzam and Langston's conjecture is true for this class of graphs, then an easy argument shows that every graph of independence number less than $3$ contains $K_{\left\lceil\frac{n}{2}\right\rceil}$ as an immersion. We show that the converse is also true. That is, if every graph with independence number less than $3$ contains an immersion of $K_{\left\lceil\frac{n}{2}\right\rceil}$, then Abu-Khzam and Langston's conjecture is true for this class of graphs. Furthermore, we show that every graph of independence number less than $3$ has an immersion of $K_{\left\lceil\frac{n}{3}\right\rceil}$. | \section{Introduction}
Vertex coloring has been a very important topic in graph theory. The usual goal, and the one considered here, is to color every vertex of a graph such that adjacent vertices get different colors. The \textit{chromatic number} of a graph $G$, denoted $\chi(G)$, is the minimum number of colors required to color its vertices. If $\chi(G)=t$, then we say that $G$ is \textit{$t$-chromatic}.
It has been suspected for a long time that if a graph cannot be colored with $t-1$ colors, then it has to somehow contain the complete graph $K_t$ with $t$ vertices. At some point in the 40's, Haj\'os \cite{HajosConjecture} conjectured that the relation of containment was the topological order. This conjecture is true for $t \leq 4$ \cite{DiracHajosConjecture}, but false for $t \geq 7$ \cite{CatlinHajosConjecture}. It remains open for $t \in \{5,6\}$. In 1943 Hadwiger \cite{HadwigerConjecture} suggested that the containment had to be the minor order, i.e. he conjectured that every $t$-chromatic graph contains $K_t$ as a minor. It was shown that Hadwiger's conjecture holds for $t=5$ \cite{WagnerHadwigerConjecture} and $t=6$ \cite{RSTHadwigerConjecture}. But it remains uncertain whether or not the conjecture is true for $t \geq 7$.
In this article we study a different order, the immersion order, which is defined by lifts of edges. A \textit{lift} of two (adjacent) edges $uv$ and $vw$, with $u \neq w$ and $uw \notin E(G)$, consists of deleting $uv$ and $vw$, and adding the edge $uw$. And a graph $H$ is \textit{immersed} in a graph $G$ if $H$ can be obtained from $G$ by performing lifts of edges and deleting vertices and/or edges. We denote this by $H \preccurlyeq_i G$. We also say that $G$ contains an \textit{immersion} of $H$. This definition is equivalent \cite{GraphColImm} to the existence of an injective function $\phi:V(H)\rightarrow V(G)$ such that:
\begin{enumerate}
\item For every $uv \in E(H)$, there is a path in $G$, denoted $P_{uv}$, which connects $\phi(u)$ and $\phi(v)$.
\item The paths $\{P_{uv} : uv \in E(H)\}$ are pairwise edge-disjoint.
\end{enumerate}
If the paths $P_{uv}$ are internally disjoint from $\phi(V(H))$, then we say that the immersion is \textit{strong}. We call the vertices in $\phi(V(H))$ the \textit{corner vertices} of the immersion.
Clearly topological containment implies immersion containment (strong immersion containment, actually). However, the minor order and the immersion order are not comparable. The immersion order, although initially much less studied than the minor and topological orders, has received a large amount of attention recently \cite{Booth1999344, Fellows1988727, Fellows:1992:WTA:131829.131839, Fellows1994769, ForbKuratImm, Langston1998191, StructureNotFixedImm}. In fact, Robertson and Seymour extended their proof of Wagner's famous conjecture \cite{GraphMinorTheorem}, to prove that the immersion order is a well-quasi-order \cite{GraphMinorTheoremImmersion}.
In analogy to Hadwiger and Haj\'os' conjectures, Lescure and Meyniel \cite{Lescure1988325} conjectured the following.
\begin{conjj}
If $\chi(G) \geq t$, then $G$ contains a strong immersion of $K_t$.
\end{conjj}
Independently, Abu-Khzam and Langston \cite{GraphColImm} proposed a weaker statement.
\begin{conj}[Abu-Khzam and Langston] \label{conj1}
If $\chi(G) \geq t$, then $K_t$ is immersed in $G$.
\end{conj}
Since Haj\'os' conjecture holds for $t \leq 4$, Abu-Khzam and Langston's conjecture is true for $t \leq 4$, as topological order is just a particular case of immersion order.
Each graph $G$ with $\chi(G)=t$ must contain a \textit{$t$-critical} subgraph, i.e., a graph $\widetilde{G}$ such that $\chi(\widetilde{G})=t$ and $\chi(H)<t$ for every proper subgraph $H$ of $\widetilde{G}$. Furthermore, it is easy to see that every $t$-critical graph must have minimum degree at least $t-1$. Using this fact, DeVos, Kawarabayashi, Mohar and Okamura \cite{ImmSmall} resolved Abu-Khzam and Langston's conjecture for small values of $t$.
\begin{teo}[\cite{ImmSmall}] \label{ALchico}
Let $f(k)$ be the smallest integer such that every graph of minimum degree at least $f(k)$ contains an immersion of $K_k$. Then $f(k)=k-1$ for $k \in \{5,6,7\}$.
\end{teo}
For $k \geq 8$, however, $f(k) \geq k$ \cite{NoImm, MinDegImm}, i.e. $\delta(G) \geq k-1$ does not guarantee an immersion of $K_k$ in $G$.
Theorem~\ref{ALchico} solves Abu-Khzam and Langston's conjecture for very small values of $t$. We are interested here in the other end of the spectrum, where $t$ is close to the number of vertices. So we restrict our attention to classes of graphs which are quite dense. We already know some properties about dense graphs, such that if a graph has $2cn^2$ edges, then it contains a strong immersion of the complete graph on at least $c^2n$ vertices \cite{MinDegImm}.
One very special case of dense graphs are the complete multipartite graphs. We prove the following result.
\begin{teo} \label{multi}
Let $G$ be a complete multipartite graph of $k\geq2$ classes with $s$ vertices each. Then $G$ has a strong immersion of $H$, where,
$$H =
\begin{cases}
K_{(k-1)s+1} & \text{ if } s \text{ is even}\\
K_{(k-1)s} & \text{ if } s\neq1 \text{ and } s \text{ is odd} \\
K_k & \text{ if } s=1
\end{cases}
$$
\end{teo}
We will call a graph \textit{$(k,s)$-dense} if every set of $k$ vertices induces a subgraph with at least $s$ edges. We prove the following two results.
\begin{teo} \label{Teo56Denso}
Every (5,6)-dense graph $G$ contains a strong immersion of $K_{\chi(G)}$.
\end{teo}
\begin{teo} \label{TeoC4Inducido}
Any graph $G$ whose complement has no induced cycle of length four contains a strong immersion of $K_{\chi(G)}$.
\end{teo}
Finally, we focus on the study of a special class of graphs, the graphs $G$ that have no independent set of size three, or equivalently, whose independence number $\alpha(G)$ is at most $2$. This class of graphs has been extensively studied in an attempt to solve Hadwiger's conjecture (see \cite{ASpecialCaseHadwiger, HadwigerSeagullPacking, PackingSeagulls, OnASpecialCaseHadwiger}). It is for this reason that we are interested in Abu-Khzam and Langston's conjecture restricted to these graphs. Abu-Khzam and Langston's conjecture restricted to that class reads as follows.
\begin{conj} \label{conj2}
Any graph $G$ with $\alpha(G) \leq 2$ contains an immersion of $K_{\chi(G)}$.
\end{conj}
If $\alpha(G) \leq 2$, then in any vertex coloring of $G$, every color class, being an independent set, has at most two vertices, which implies that $\chi(G) \geq \frac{n}{2}$. Abu-Khzam and Langston's conjecture would thus imply that $G$ must contain an immersion of $K_{\lceil \frac{n}{2} \rceil}$. The latter gives rise to a new conjecture.
\begin{conj} \label{conj3}
Any graph $G$ with $\alpha(G) \leq 2$ contains an immersion of $K_{\lceil \frac{n}{2} \rceil}$.
\end{conj}
We just saw that Conjecture~\ref{conj2} implies Conjecture~\ref{conj3}. However, the two conjectures are actually equivalent. Following ideas from \cite{OnASpecialCaseHadwiger} we show the next result.
\begin{teo} \label{TeoConjeturasEquivalentes}
Conjectures~\ref{conj2} and~\ref{conj3} are equivalent.
\end{teo}
A weaker version of Conjecture~\ref{conj3} is shown, namely the following result.
\begin{teo} \label{TeoEneTercios}
If $G$ is a graph with $\alpha(G) \leq 2$, then $G$ contains a strong immersion of $K_{\lceil\frac{n}{3}\rceil}$.
\end{teo}
An analogous result was shown by Chudnovsky \cite{HadwigerSeagullPacking}, namely that if $G$ is a graph with $n$ vertices and no independent set of size three, then $G$ contains a complete minor of size $\lceil\frac{n}{3}\rceil$. The technique used there is a nice use of induced paths of length two. Here we present a different technique.
In this article every graph is simple, without loops and parallel edges, unless stated otherwise.
This work is organized as follows. In Section~\ref{seccionDefiniciones} we present a quick review of some definitions and properties about vertex coloring that will be used through the text. In Section~\ref{seccionResultadosGenerales} we immerse a large complete graph into a multipartite complete graph (see Theorem~\ref{multi}), and also prove Theorems~\ref{Teo56Denso} and~\ref{TeoC4Inducido}. And in Section~\ref{SeccionAlpha} we prove Theorems~\ref{TeoConjeturasEquivalentes} and~\ref{TeoEneTercios}, and show a series of properties that a counterexample of Conjecture~\ref{conj2} with minimum number of vertices should satisfy.
\section{Vertex coloring} \label{seccionDefiniciones}
Given a vertex coloring $c:V(G) \rightarrow \{1,...,k\}$, we denote $c_i = \{u: c(u)=i\}$ and $c_{ij}$ the subgraph induced by the set of vertices $\{u: c(u) \in \{i, j\}\}$.
We call a path in $c_{ij}$ a \textit{chain}, and for each $u \in V(c_{ij})$, we denote $c_{ij}(u)$ the connected component of $c_{ij}$ that contains $u$.
If $\{i,j\} \neq \{k,l\}$, then $c_{ij}$ and $c_{kl}$ are edge-disjoint graphs.
This observation is particularly important to find immersions in graphs, considering the second definition of immersion. For this reason, the use of chains will be very helpful.
Let $c:V(G) \rightarrow \{1,...,k\}$ be a vertex coloring of $G$ and let $i \in \{1,...,k\}$. We say that $u \in V(G)$ is a \textit{dominating vertex} for color $i$, if $c(u)=i$ and if for each color $j \neq i$, there is a vertex $v$ such that $c(v)=j$ and $uv \in E(G)$.
If $c:V(G) \rightarrow \{1,...,\chi(G)\}$ is a coloring of $G$ with minimum number of colors, then it is easy to check that every $i \in \{1,...,\chi(G)\}$ has a dominating vertex.
\section{Complete Graph Immersions} \label{seccionResultadosGenerales}
Let us see first, that in a complete multipartite graph we can find an immersion of a complete graph of relatively large size. That is, let us prove Theorem~\ref{multi}.
\begin{proof}[Proof of Theorem~\ref{multi}]
The $s=1$ case is trivial, so we can assume $s>1$. We choose the vertices of $ k-1$ classes as corner vertices (in the case that $s$ is even, we will add an additional corner vertex later), and the vertices of the remaining class, let us call it $U$, will be used for the edge-disjoint paths. The paths between two vertices from different classes already exist (they are the edges between them), so we only need to worry about those vertices that are in the same class. We know that $\chi'(K_s)= s-1$ if $s$ is even, and $\chi'(K_s) = s$ if $s$ is odd (\cite[p.133]{Soifer}).
For each class of $s$ corner vertices, consider a $\chi'(K_s)$-edge-coloring of the edges that are missing (all of them). As $|U| \geq \chi'(K_s)$, we can assign each of the used colors on the edges of $K_s$ to some vertex in $U$. Say vertex $u_i \in U$ gets color $i$. Then, for two corner vertices $v$ and $w$ in the same class, we assign $P_{vw} = vu_iw$ where $vw$ is colored with color $i$.
Observe that these paths are edge-disjoint. Indeed, if two paths $P_{vw}$ and $P_{xy}$ share an edge, then $vw$ would have to be adjacent to $xy$. In addition, we would have $P_{vw} = vu_iw$, $P_{xy}=xu_iy$ for some $i \leq \chi'(G)$. That is, both $vw$ and $xy$ would have assigned color $i$, which is a contradiction.
Note that if $s$ is even, then in $U$ there is a vertex that is not being used in the edge-disjoint paths, so we can add it as a corner vertex of the immersion, as it is adjacent to all other corner vertices. Thus, we find the desired immersion, which is strong because no corner vertex is used as an internal vertex of some path.
\end{proof}
\begin{obs}
Actually, a more general result follows directly from the proof of the theorem. If $G$ is a complete multipartite graph with $k\geq2$ classes of sizes $s_1, s_2, \ldots, s_k$, with $s_k \geq s_i$, for $i \leq k-1$, then $G$ contains a strong immersion of $K_{s_1+s_2+\ldots+s_{k-1}}$.
\end{obs}
We now prove Theorem~\ref{Teo56Denso}.
\begin{proof}[Proof of Theorem~\ref{Teo56Denso}]
Let us suppose first that $G$ has fewer than five vertices. The cases $\chi(G) \in \{1,2\}$ are trivial. If $\chi(G)=3$, $G$ must contain a triangle, so $K_3 \subseteq G$. And if $\chi(G)=4$, it is easy to check that the only option is $G=K_4$. So, we can assume $|V(G)|\geq5$.
Let $c$ be a coloring of $V(G)$ with minimum number of colors and let $k=\chi(G)$. Note that $|c_i| \leq 3$, for $1 \leq i \leq k$, since there cannot be independent sets of size four. This, because if there were any, then, adding any other vertex, we would have a set of five vertices inducing less than six edges.
Observe that if $c_i = \{u,x\}$ and $c_j = \{v,y\}$ are such that $c_{ij}$ is not connected, then the vertices in $c_i \cup c_j$ are adjacent to all other vertices. Indeed, if $c_{ij}$ is not connected, it has exactly two edges. Then, any other vertex must be adjacent to $u$, $v$, $x$ and $y$, because of the ($5$,$6$)-density of $G$. By symmetry, there are two cases.
\begin{itemize}
\item $uv, xy \in E(c_{ij})$, in which case every vertex in $c_i \cup c_j$ is a dominating vertex for its color.
\item $uv, vx \in E(c_{ij})$, in which case $v$ has to be the dominating vertex for color $j$, and both $u$ and $x$ are dominating vertices for color $i$.
\end{itemize}
We choose a dominating vertex $u_t$ of each color $t$ as the corner vertices of the immersion with the extra requirement that if $i \neq j$ and $|c_i|=|c_j|=2$ with $c_{ij}$ disconnected, then we choose a pair of adjacent dominating vertices as corner vertices. Note that this choice is possible because of the above observation. Let $i,j$ be any two colors and we will show that $u_i,u_j$ are connected by a chain.
\begin{itemize}
\item If $|c_i|=1$, then $u_iu_j \in E(G)$, as $u_j$ is dominating. The edge $u_iu_j$ is the chain we want.
\item If $|c_i|=2, |c_j|=3$, then $u_iu_j \in E(G)$, due to the $(5,6)$-density of the graph.
\item If $|c_i|=3, |c_j|=3$, then considering $c_i$ plus $u_j$ and a vertex in $c_j \setminus \{u_j\}$, it holds that the induced subgraph must necesarily be a complete bipartite graph, because of the $(5,6)$-density of $G$. Then, $u_iu_j \in E(G)$.
\item If $|c_i|=2, |c_j|=2$, there are two cases. If $c_{ij}$ is connected, we can always find a chain between $u_i$ and $u_j$. And if $c_{ij}$ is not connected, then by the choice of $u_i,u_j$, it holds that $u_iu_j \in E(G)$.
\end{itemize}
By symmetry, the above are all possible cases, and so, between each pair of corner vertices there is a chain that connects them, and therefore, we have found an immersion of $K_{\chi(G)}$. None of the chains we described uses another corner vertex as an internal vertex, so the immersion is strong
\end{proof}
Let us prove now Theorem~\ref{TeoC4Inducido}.
\begin{proof}[Proof of Theorem~\ref{TeoC4Inducido}]
Let $c$ be a vertex coloring of $G$ with minimum number of colors, and choose a dominating vertex of each color as the set of corner vertices. Consider two corner vertices, $u$ and $v$, with $c(u)=i$, $c(v)=j$ and let us see that there is a chain that joins them (so we ensure that paths will be edge-disjoint).
If $uv \in E(G)$, then the edge $uv$ is the chain we want. If $uv \notin E(G)$, there are vertices $x \in c_j, y \in c_i$, such that $ux, vy \in E(G)$, because $u$ and $v$ are dominating. Also, as $C_4$ is not an induced subgraph of $\overline{G}$, necessarily $xy \in E(G)$. Thus $uxyv$ is the chain we want. Then we have an immersion of $K_{\chi(G)}$, which is strong since the paths being chains, they do not use another corner vertex as an internal vertex.
\end{proof}
\begin{obs}
At first, the condition that there are no induced cycles of length four in the complement of the graph might seem too restrictive, however, unlike in $(5,6)$-dense graphs, color classes can be arbitrarily big. Indeed, consider the graph obtained from $K_{2,n-2}$ by adding the edge between the two vertices in the class of size two. This graph has no induced cycle of length four in the complement, but any coloring with minimum number of colors contains a class of size $n-2$.
\end{obs}
\section{Graphs with small independence number} \label{SeccionAlpha}
Here we study the class of graphs that have no independent set of size three. It is easy to check that the non-neighbourhood of any vertex of a graph $G$ with $\alpha(G) \leq 2$ induces a complete graph.
We shall now see that if we replaced $K_{\lceil \frac{n}{2} \rceil}$ with $K_{\lceil \frac{n}{3} \rceil}$ in Conjecture~\ref{conj3}, then the statement is true, as claimed in Theorem~\ref{TeoEneTercios}. Moreover, either $G$ contains $K_{\lceil \frac{n}{3} \rceil}$ as a subgraph, or any set of $\lceil \frac{n}{3} \rceil$ vertices can be a set of corner vertices. Also, the immersion is strong.
From now on we will use the following notation: $$\overline{N}(v) = V(G) \setminus \left(N(v) \cup \{v\}\right).$$
\begin{proof}[Proof of Theorem~\ref{TeoEneTercios}]
Let us define, for a vertex $v \in V(G)$ and a set $U \subseteq V(G)$, $$N_U(v)=N(v) \cap U$$ $$\overline{N}_U(v)=\overline{N}(v) \cap U.$$
If there was a vertex $v$ with $d(v)< \lfloor\frac{2n}{3}\rfloor$, then the non-neighborhood of $v$ would have size at least $\lceil\frac{n}{3}\rceil$, and as it induces a complete graph, we would have $G$ containing $K_{\lceil\frac{n}{3}\rceil}$ as a subgraph. Therefor, we can assume $\delta(G) \geq \lfloor \frac{2n}{3} \rfloor$.
We will find an immersion of $K_{\left\lceil\frac{n}{3}\right\rceil}$ in $G$. We partition $V(G)$ into two disjoint sets $U$ and $W$, such that $|U| = \lceil\frac{n}{3}\rceil$ and $|W| = \lfloor\frac{2n}{3}\rfloor$. The vertices from the set $U$ will be the corner vertices and we denote $P_{uv}$ the path between $u$ and $v$ in the immersion, which will be constructed as follows. We arrange the pairs $\{u,v\}$ with $u,v \in U$ arbitrarily and we assign the paths of the immersion in the following way. If $uv \in E(G)$, then $P_{uv} = uv$. If $uv \notin E(G)$, then we will choose a vertex $z \in N_W(u) \cap N_W(v)$ such that $z$ has not been used at some $P_{ux}$, with $x \in U$ or some $P_{vx}$, with $x \in U$, and we will assign $P_{uv} = uzv$. Note that given the latter condition, the paths will be edge-disjoint. Furthermore, no corner vertex is used as an internal vertex of a path, so the immersion is indeed strong. Let us see that this assignment is possible (we only need to verify this for the case $uv \notin E(G)$).
Let $uv \not\in E(G)$. We must prove that $u$ and $v$ have enough common vertices in $W$. That is to say, we need to prove the following.
$$|N_W(u)\cap N_W(v)| \geq |\overline{N}_U(u)| + |\overline{N}_U(v)| - 1$$
The term $-1$ is there because the non-existing edge $uv$ is being counted twice.
\begin{eqnarray*}
|\overline{N}_U(u)| + |\overline{N}_U(v)| - 1 & = & |U| - |\{u\}| - |N_U(u)| + |U| - |\{v\}| - |N_U(v)| - 1\\
& = & \left\lceil\frac{n}{3}\right\rceil - 1 - |N_U(u)| + \left\lceil\frac{n}{3}\right\rceil - 1 - |N_U(v)| - 1\\
& = & 2\left\lceil\frac{n}{3}\right\rceil - 3 - \left( |N(u)| - |N_W(u)| + |N(v)| - |N_W(v)| \right) \\
& = & 2\left\lceil\frac{n}{3}\right\rceil - 3 + |N_W(u)| + |N_W(v)| - \left( |N(u)| + |N(v)| \right) \\
& \leq & 2\left\lceil\frac{n}{3}\right\rceil - 3 + |N_W(u)| + |N_W(v)| - \left( \left\lfloor\frac{2n}{3}\right\rfloor + \left\lfloor\frac{2n}{3}\right\rfloor \right) \\
& = & 2\left\lceil\frac{n}{3}\right\rceil - 3 + |N_W(u)\cup N_W(v)| + |N_W(u) \cap N_W(v)| - 2\left\lfloor\frac{2n}{3}\right\rfloor \\
\end{eqnarray*}
Since $uv \notin E(G)$ and $\alpha(G)\leq2$, we have that for each $w \in W, uw \in E(G)$ or $vw \in E(G)$. This implies that $N_W(u) \cup N_W(v) = W$. Then,
\begin{eqnarray*}
|\overline{N}_U(u)| + |\overline{N}_U(v)| - 1 & \leq & 2\left\lceil\frac{n}{3}\right\rceil - 3 + |W| + |N_W(u) \cap N_W(v)| - 2\left\lfloor\frac{2n}{3}\right\rfloor \\
& = & 2\left\lceil\frac{n}{3}\right\rceil - 3 + \left\lfloor\frac{2n}{3}\right\rfloor + |N_W(u) \cap N_W(v)| - 2\left\lfloor\frac{2n}{3}\right\rfloor \\
& = & 2\left\lceil\frac{n}{3}\right\rceil - 3 - \left\lfloor\frac{2n}{3}\right\rfloor + |N_W(u) \cap N_W(v)| \\
\end{eqnarray*}
So, we only need to prove that $2\left\lceil\frac{n}{3}\right\rceil - 3 - \left\lfloor\frac{2n}{3}\right\rfloor \leq 0$.
\begin{eqnarray*}
2\left\lceil\frac{n}{3}\right\rceil - 3 - \left\lfloor\frac{2n}{3}\right\rfloor & \leq & 2\left\lfloor\frac{n}{3}+1\right\rfloor - 3 - \left\lfloor\frac{2n}{3}\right\rfloor \\
& = & 2\left\lfloor\frac{n}{3}\right\rfloor + 2 - 3 - \left\lfloor\frac{2n}{3}\right\rfloor \\
& \leq & \left\lfloor\frac{2n}{3}\right\rfloor - 1 - \left\lfloor\frac{2n}{3}\right\rfloor \\
& = & - 1 \\
& \leq & 0
\end{eqnarray*}
That is, less vertices are needed than those that are available, to construct the edge-disjoint paths of the immersion. Therefore, there exists $z \in N_W(u) \cap N_W(v)$ that has not been used in other paths $P_{ux}$ or $P_{vx}$, and then we can assign $P_{uv} = uzv$. Thus, we have obtained a strong immersion of $K_{\lceil\frac{n}{3}\rceil}$ in $G$.
\end{proof}
\subsection{Equivalence of Conjectures \ref{conj2} and \ref{conj3}} \label{SeccionEquivalenciaConjeturas}
The proof of Theorem~\ref{TeoConjeturasEquivalentes} is strongly inspired from \cite{OnASpecialCaseHadwiger}. We will need to use some preliminary results. Suppose Conjecture~\ref{conj2} fails, and let $G$ be a counterexample that minimizes the number of vertices. Observe that the number of vertices is upper bounded by the product of the independence number and the chromatic number. So we have the following inequality:
\begin{equation} \label{eq2}
|V(G)|\leq2\chi(G).
\end{equation}
We will prove some properties that $G$ satisfies.
\begin{defn}
A graph $G$ is \textit{$k$-color-critical} if $\chi(G)=k$ and $\chi(G-v)<k$, for each $v \in V(G)$.
\end{defn}
\begin{lema} \label{critic}
$G$ is $\chi(G)$-color-critical.
\begin{proof}
Indeed, if there is a vertex $v \in V(G)$, such that $\chi(G-v)=\chi(G)$, then as $G-v$ has less vertices than $G$ and $\alpha(G-v) \leq 2$, we would have that, $$K_{\chi(G)} = K_{\chi(G-v)} \preccurlyeq_i G-v \preccurlyeq_i G,$$ which contradicts the fact that $G$ is a counterexample for Conjecture~\ref{conj2}.
\end{proof}
\end{lema}
\begin{lema} \label{compCon}
$\overline{G}$ is connected.
\begin{proof}
If not, $G$ consists of two disjoint subgraphs $G_1$ and $G_2$, such that for all $u \in V(G_1)$ and for all $v \in V(G_2)$, $uv \in E(G)$. Then, as both $G_1$ and $G_2$ have less vertices than $G$, it holds that $K_{\chi(G_1)} \preccurlyeq_i G_1$ and $K_{\chi(G_2)} \preccurlyeq_i G_2$, and then, $$K_{\chi(G)} = K_{\chi(G_1)+\chi(G_2)} \preccurlyeq_i G,$$ which leads to a contradiction.
\end{proof}
\end{lema}
For the next property, we will use the next result.
\begin{teo}[\cite{CriticalGraphsConnectedComplements}] \label{teoUtil}
If $x$ is any vertex of a $k$-color-critical graph $G$ such that $\overline{G}$ is connected, then $G-x$ has a $(k-1)$-coloring in which every color class contains at least $2$ vertices.
\end{teo}
\begin{lema} \label{numvert}
$|V(G)|=2\chi(G)-1$.
\begin{proof}
By Lemmas \ref{critic}, \ref{compCon} and Theorem \ref{teoUtil} we know that $G-v$ has a $(\chi(G)-1)$-coloring such that each color class contains at least two vertices. Since $\alpha(G) \leq 2$, each color class in that coloring has size exactly two. So, $|V(G)|=2\chi(G)-1$.
\end{proof}
\end{lema}
We are now able to prove Theorem~\ref{TeoConjeturasEquivalentes}.
\begin{proof}[Proof of Theorem~\ref{TeoConjeturasEquivalentes}]
By Lemma~\ref{numvert}, we have that $\left \lceil \frac{|V(G)|}{2} \right \rceil = \frac{|V(G)|+1}{2} = \chi(G)$. Then, $K_{\lceil\frac{n}{2}\rceil} \not\preccurlyeq_i G$ and therefore, $G$ is also a counterexample for Conjecture~\ref{conj3}.
\end{proof}
Observe that $G$ turns out to be a counterexample with minimum number of vertices for Conjecture~\ref{conj3} as well. Indeed, let $H$ be a counterexample of Conjecture~\ref{conj3} such that $|V(H)|<|V(G)|$. Then, $$K_{\frac{|V(H)|}{2}} \not\preccurlyeq_i H.$$ And as $|V(H)|\leq 2 \chi(H)$, $$K_{\chi(H)} \not\preccurlyeq_i H.$$ So, $H$ is a counterexample of Conjecture~\ref{conj2} and $|V(H)|<|V(G)|$, which is a contradiction.
\subsection{Properties of a minimum counterexample of Conjecture~\ref{conj2}} \label{SeccionContraejemploMinimo}
In this subsection we will prove a series of properties that a counterexample of Conjecture~\ref{conj2} with minimum number of vertices satisfies besides those mentioned by Lemmas~\ref{critic},~\ref{compCon} and~\ref{numvert}. The next result enumerates them.
\begin{teo}
Let $G$ be a counterexample to Conjecture~\ref{conj2} which minimizes the number of vertices. Then the following hold:
\begin{enumerate}
\item $G$ is a counterexample to Conjecture~\ref{conj2} which minimizes the chromatic number. \label{MinCromatico}
\item For every $v \in V(G)$, $\overline{G}-v$ has a perfect matching. \label{match}
\item For every pair of nonadjacent vertices $x$, $y$ of $G$, $|N(x) \cap N(y)| \leq \frac{n-1}{2}$. \label{cod}
\item $\omega(G) \geq \frac{n+1}{4}$. \label{MinOmega}
\item $G$ is connected. \label{MinConnected}
\item $\delta(G) \geq \lceil \frac{n}{2} \rceil$. \label{gradmin}
\item $G$ is Hamiltonian. \label{ham}
\item For every $v \in V(G)$, $G-v$ has a perfect matching. \label{MinMatching}
\item For every $x, y \in V(G)$, it holds that $d(x,y) \leq 2$. \label{dist}
\item $\chi(G) \geq 8$. \label{MinNumeroCromatico8}
\end{enumerate}
Suppose now that $G$, among all counterexamples of Conjecture~\ref{conj2} minimizing the number of vertices, is one that minimizes the number of edges. Then the next additional property hold:
\newcounter{i}
\setcounter{i}{\value{enumi}}
\begin{enumerate}
\setcounter{enumi}{\value{i}}
\item For every edge $e \in E(G)$, it holds that $\alpha(G-e)>\alpha(G)$. \label{LemaAlphaCritic}
\end{enumerate}
\begin{proof}
\begin{enumerate}
\item[\ref{MinCromatico}.]
Let $\widetilde{G}$ be any counterexample to Conjecture~\ref{conj2} with minimum chromatic number. Then,
$$2\chi(G)-1 = |V(G)| \leq |V(\widetilde{G})| \leq 2\chi(\widetilde{G}) \leq 2\chi(G).$$
Therefore, $\chi(G)=\chi(\widetilde{G})$.
\item[\ref{match}.]
We know by Theorem~\ref{teoUtil} that $G-v$ has a ($\chi(G)-1$)-coloring, in which every color class has exactly two vertices. This corresponds to a perfect matching in $\overline{G}-v$.
\item[\ref{cod}.]
If this is not so, let $x$ and $y$ be two vertices such that $xy \notin E(G)$ and $|N(x) \cap N(y)| \geq \frac{n-1}{2}+1$. As $G$ is a minimal counterexample for Conjecture~\ref{conj3}, we know that $$K_{\frac{n-1}{2}} = K_{\lceil\frac{n-2}{2}\rceil} \preccurlyeq_i G-\{x,y\}.$$
Let $U$ be the set of corner vertices of such an immersion and let $W=V(G-\{x,y\})\setminus U$. As $\alpha(G)\leq2$ and $xy \notin E(G)$, we have that for every $u \in U$, $ux \in E(G)$ or $uy \in E(G)$. Without loss of generality, assume that $x$ is adjacent to at least half of the vertices in $U$ (and that $x$ has more neighbors than $y$ in $U$). Note that every non-neighbor of $x$ has to be adjacent to $y$.
Let us see that $x$ is connected to every vertex $u$ in $U$, by edge-disjoint paths $P_{xu}$. If $xu \in E(G)$, then $P_{xu}=xu$. If $xu \notin E(G)$, then $P_{xu}=xzyu$, with $z \in N_W(x)\cap N_W(y)$. Observe that for this to work, it needs to hold that $|N_W(x) \cap N_W(y)| \geq |\overline{N}_U(x)|$.
We know that $|\overline{N}_U(x)| \leq \frac{n-1}{4}$, so, $$|\overline{N}_U(x)| = \left\lfloor\frac{n-1}{4}\right\rfloor-i \text{ with } i \in \left\{0,...,\left\lfloor\frac{n-1}{4}\right\rfloor\right\}.$$ Besides, $$|N_W(x) \cap N_W(y)| = |N(x) \cap N(y)| - |N_U(x) \cap N_U(y)| \geq |N(x) \cap N(y)|-(2i+1).$$ The last inequality is obtained by assuming that $|N_U(x)| \geq |N_U(y)|$, so the number of neighbors that $x$ and $y$ share in $U$ is bounded. Indeed, $N_U(y) = \overline{N}_U(x) \cup (N_U(x) \cap N_U(y))$ and as we assumed $|N_U(y)| \leq |N_U(x)|$, we have that $$|\overline{N}_U(x)| + |N_U(x) \cap N_U(y)| \leq |N_U(x)|.$$ Then,
\begin{eqnarray*}
|N_U(x) \cap N_U(y)| & \leq & |N_U(x)| - |\overline{N}_U(x)| \\
& = & \left(\left\lceil\frac{n-1}{4}\right\rceil + i\right) - \left(\left\lfloor\frac{n-1}{4}\right\rfloor - i\right) \\
& \leq & 2i+1.
\end{eqnarray*}
And as $|N(x) \cap N(y)| \geq \frac{n-1}{2}+1$, we have that $$|\overline{N}_U(x)| = \left\lfloor\frac{n-1}{4}\right\rfloor-i \leq |N(x) \cap N(y)|-(2i+1) \leq |N_W(x) \cap N_W(y)|.$$
It is important to notice that the paths $P_{xu}$, from $x$ to $u \in U$, do not interfere with the already existing paths between corner vertices in $U$. This is so, because the new paths only use edges which are incident to $x$ and $y$. Therefore, we get an immersion of $K_{\left\lceil\frac{n-2}{2}\right\rceil+1} = K_{\left\lceil\frac{n}{2}\right\rceil}$ in $G$, which is a contradiction.
\item[\ref{MinOmega}.]
Let $x$, $y$ be any two vertices such that $xy \notin E(G)$ and divide the rest of the vertices into $A = N(x) \setminus N(y)$, $B = N(x) \cap N(y)$ and $C = N(y) \setminus N(x)$. Observe that both $A$ and $C$ induce a complete graph because $\alpha(G)\leq2$. Besides, by property~\ref{cod}, it holds that $|B| \leq \frac{n-1}{2}$. Therefore, at least one of the other two sets, say $A$, satisfies that $|A \cup \{x\}| \geq \frac{n+1}{4}$. And as $\omega(G) \geq |A \cup \{x\}|$, we are done.
\item[\ref{MinConnected}.]
Indeed, if not so, $G$ would have at least two connected components. In fact, since $\alpha(G) \leq 2$, it would have exactly two connected components and every component would be a complete graph. Then, $K_{\chi(G)} \subseteq G$, which contradicts that $G$ is a counterexample for Conjecture~\ref{conj2}.
\item[\ref{gradmin}.]
Observe first that it is straightforward to prove that $\delta(G) \geq \lfloor\frac{n}{2}\rfloor$, since the non-neighborhood of any vertex induces a complete graph. Indeed, if $\delta(G) < \lfloor\frac{n}{2}\rfloor$, $K_{\lceil \frac{n}{2} \rceil}$ would be a subgraph of $G$, a contradiction.
So suppose that $\delta(G) = \lfloor \frac{n}{2} \rfloor$ and let $v$ be such that $d(v)=\delta(G)$. Divide $V(G)-{v}$ into the neighbors and the non-neighbors of $v$, $N(v)$ and $\overline{N}(v)$ respectively. We know, by property~\ref{match}, that $\overline{G}-v$ has a perfect matching. And given that $\overline{N}(v)$ induces a complete graph, every vertex in $\overline{N}(v)$ is matched with a vertex in $N(v)$. Besides, as $|N(v)|=\lfloor \frac{n}{2} \rfloor$, then $|\overline{N}(v)|=\lfloor \frac{n}{2} \rfloor$. This matching represents a coloring of $G$, in which all color classes have exactly two vertices. We claim that $K_{\chi(G)} \preccurlyeq_i G$, where the corner vertices are $\{v\} \cup N(v)$.
Indeed, $vu \in E(G)$, for every $u \in N(v)$. Then, we can assign $P_{vu}=vu$. Consider now $u, w \in N(v)$. If $uw \in E(G)$, then $P_{uw}=uw$. If $uw \notin E(G)$, then, as $\alpha(G) \leq 2$, it holds that $ux_w, wx_u \in E(G)$, where $x_w, x_u$ are the vertices that are matched with $w$ and with $u$, respectively. Also, $x_ux_w \in E(G)$, since $x_u, x_w \in \overline{N}(v)$, which is a complete graph. Therefore, we can assign $P_{uw}=ux_wx_uw$. The paths are edge-disjoint, because by seeing the matching in $\overline{G}-v$ as a coloring in $G-v$, the chosen paths are precisely chains between corner vertices of different colors.
\item[\ref{ham}.]
It follows from property~\ref{gradmin} along with Dirac's Theorem for Hamiltonian graphs \cite{DiracTheo}.
\item[\ref{MinMatching}.]
It is implied by property~\ref{ham} and Lemma~\ref{numvert}.
\item[\ref{dist}.]
There are two cases. If $x,y \in E(G)$, then $d(x,y)=1$. If $x,y \notin E(G)$, the by property~\ref{gradmin}, we know that both $x$ and $y$ have at least $\left\lceil\frac{n}{2}\right\rceil$ neighbors into a set of $n-2$ vertices ($V(G)\setminus\{x,y\}$). That means they have at least one common neighbor, so $d(x,y)=2$.
\item[\ref{MinNumeroCromatico8}.]
It follows directly from Theorem~\ref{ALchico}.
\item[\ref{LemaAlphaCritic}.]
If there were an edge $e \in E(G)$, such that $\alpha(G-e) \leq \alpha(G) = 2$, then, $$K_{\chi(G-e)} \preccurlyeq_i G-e \preccurlyeq_i G.$$ So, $\chi(G-e) \leq \chi(G)-1$. Therefore, $G-e$ has $|V(G)|=2\chi(G)-1$ vertices and can be colored with $\chi(G)-1$ colors. Necessarily one color class has at least 3 vertices, which is a contradiction.
\end{enumerate}
\end{proof}
\end{teo}
\section{Conclusion}
The question of whether Abu-Khzam and Langston's conjecture is true still remains open, even in the special case of $\alpha(G) \leq 2$. A possible way would be to continue studying a counterexample of Conjecture~\ref{conj2} minimizing the number of vertices. More structural properties can be found in \cite{Memoria}.
After seeing the proofs of Theorems~\ref{Teo56Denso} and~\ref{TeoC4Inducido} it is tempting to try to look for an immersion of a complete graph with a vertex of every color as the set of corner vertices and chains as paths between them. However there are examples of graphs with colorings in which it is impossible to find this type of immersion (the reader is referred also to \cite{Memoria}).
\section{Acknowledgments}
The author would like to thank Maya Stein for her tremendous help and support during the creation of this article, and also the two anonymous referees for their very valuable comments.
\bibliographystyle{plain}
| {
"timestamp": "2017-01-03T02:02:42",
"yymm": "1502",
"arxiv_id": "1502.01786",
"language": "en",
"url": "https://arxiv.org/abs/1502.01786",
"abstract": "In this article we consider the relationship between vertex coloring and the immersion order. Specifically, a conjecture proposed by Abu-Khzam and Langston in 2003, which says that the complete graph with $t$ vertices can be immersed in any $t$-chromatic graph, is studied.First, we present a general result about immersions and prove that the conjecture holds for graphs whose complement does not contain any induced cycle of length four and also for graphs having the property that every set of five vertices induces a subgraph with at least six edges.Then, we study the class of all graphs with independence number less than three, which are graphs of interest for Hadwiger's Conjecture. We study such graphs for the immersion-analog. If Abu-Khzam and Langston's conjecture is true for this class of graphs, then an easy argument shows that every graph of independence number less than $3$ contains $K_{\\left\\lceil\\frac{n}{2}\\right\\rceil}$ as an immersion. We show that the converse is also true. That is, if every graph with independence number less than $3$ contains an immersion of $K_{\\left\\lceil\\frac{n}{2}\\right\\rceil}$, then Abu-Khzam and Langston's conjecture is true for this class of graphs. Furthermore, we show that every graph of independence number less than $3$ has an immersion of $K_{\\left\\lceil\\frac{n}{3}\\right\\rceil}$.",
"subjects": "Combinatorics (math.CO)",
"title": "Complete graph immersions in dense graphs",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754452025766,
"lm_q2_score": 0.7185943925708562,
"lm_q1q2_score": 0.7075103939855258
} |
https://arxiv.org/abs/1807.11465 | Edge Coloring Signed Graphs | We define a method for edge coloring signed graphs and what it means for such a coloring to be proper. Our method has many desirable properties: it specializes to the usual notion of edge coloring when the signed graph is all-negative, it has a natural definition in terms of vertex coloring of a line graph, and the minimum number of colors required for a proper coloring of a signed simple graph is bounded above by {\Delta} + 1 in parallel with Vizing's Theorem. In fact, Vizing's Theorem is a special case of the more difficult theorem concerning signed graphs. | \section{Introduction}
A \textit{signed graph} is a graph in which each edge is assigned either a positive or negative sign. Signed graphs were invented by Harary in 1953 in order to help study a question in social psychology \cite{harary} and since then have proved to be a natural generalization of ordinary graphs in many ways. Typically the signed graph version of a graph theoretic structure is similar to the unsigned version, but more complicated due to the presence of the edge signs. Various phenomena that are unseen in the world of ordinary graphs will manifest themselves in the signed world, leading to interesting insights into both ordinary graphs and signed graphs. For example, Zaslavsky \cite{zaslav2} discovered a theory of signed vertex coloring complete with deletion/contraction recurrence and chromatic polynomials that specializes to ordinary graphs when the signed graph is all positive. Zaslavsky's coloring construction makes use of ``signed colors", which are a new twist that must be introduced to define vertex coloring of a signed graph in an interesting fashion. In this paper we will use these same signed colors to construct a theory of signed edge coloring that has many nice properties, including compatibility with Zaslavsky's signed vertex coloring.
It is a well-known theorem of Vizing that the number of colors needed to properly edge color an ordinary simple graph is either $\Delta$ or $\Delta+1$ \cite{vz}. Thus there are two kinds of graphs---those that can be $\Delta$-colored (class $1$), and those that require one additional color (class $2$). In what follows, we will prove that every signed simple graph can be edge colored with $\Delta$ or $\Delta+1$ colors (Theorem \ref{goodvizing}), rendering ordinary Vizing's Theorem a special case of our new theorem concerning signed graphs. Interestingly, it is possible to change the class of a signed graph by modifying its signature, so that two signed graphs on the same underlying graph may fall into distinct sides of the dichotomy.
To prove Signed Vizing's Theorem we will employ the signed graph analogues of several techniques that are commonly used when studying Vizing's theorem, such as Kempe chains and the ``fan" used to recolor edges locally. Signed Kempe chains are particularly interesting, as they exhibit certain properties that are unseen in ordinary Kempe chains.
It is reasonable to expect nuances or exceptions to arise when generalizing an ordinary graph theory concept to signed graphs. For example, in ordinary graph theory the well-known Brooks' Theorem states that the number of colors needed to vertex color a connected graph is bounded above by $\Delta$ with two exceptions---complete graphs and odd cycles require one additional color. The signed version of Brooks' Theorem is nearly identical, but it turns out that negative cycles of even length, in addition to positive complete graphs and positive even cycles, are a third exception \cite{maca}. Remarkably, no additional exceptions arise in the signed version of Vizing's theorem, and the upper bound for the number of colors is $\Delta+1$ for both ordinary and signed graphs alike.
\section{Graphs and Signed Graphs}
\subsection{Graphs}
We write $\Gamma$ for a \emph{graph} and we write $V(\Gamma)$ and $E(\Gamma)$ for its vertex and edge sets respectively. Throughout, we will assume that every graph edge has two distinct endpoints, and that no two edges have the same pair of endpoints. In other words, we assume that all graphs are \emph{simple}. Often we write $e{:}vw$ for an edge $e$ with endpoints $v$ and $w$. If vertices $v$ and $w$ are connected by an edge we say that they are \emph{adjacent} or that they are \emph{neighbors}.
An \emph{incidence} of $\Gamma$ is a pair $(v,e)$ such that vertex $v$ is an endpoint of edge $e$. The set of all incidences of $\Gamma$ is written $I(\Gamma)$. If we write $(v,vw)$ it is understood that we are referring to the incidence between $v$ and edge $e{:}vw$.
A \emph{circle} is a connected $2$-regular subgraph. A \emph{path} is a sequence of adjacent vertices and connecting edges that never repeats an edge or a vertex. A \emph{trail} has the same definition as a path, except that a trail may repeat vertices (but not edges). Thus, every path is a trail, but not every trail is a path. For trails and paths, we call $v_0$ and $v_n$ the \emph{endpoints}, while the other vertices are \emph{interior vertices}. In a path the endpoints and interior vertices are distinct, but in a trail there may be interior vertices that are also endpoints. Often, we specify a trail by listing its vertices in order inside of parenthesis. For a trail $T$ with endpoints $t_0$ and $t_n$ we write $T=(t_0,...,t_n)$.
A \emph{matching} $M$ in $\Gamma$ is a collection of edges of $\Gamma$ such that no two edges of $M$ share an endpoint. An \emph{independent set} $J$ in $\Gamma$ is a collection of vertices such that no edge has both endpoints in $J$.
\subsection{Signed Graphs}
A \emph{signed graph} is a pair $\Sigma = (\Gamma,\sigma)$, where $\Gamma$ is a graph and $\sigma : E(\Gamma) \rightarrow \{+,-\}$ is the \emph{signature}. We write $|\Sigma| = \Gamma$ for the \emph{underlying graph} of $\Sigma$, the unsigned graph obtained by forgetting all of the signs.
A circle in $\Sigma$ is \emph{positive} if the product of its edge signs is positive, and \emph{negative} otherwise. A subgraph of $\Sigma$ is \emph{balanced} if each of its circles is positive, and \emph{unbalanced} otherwise. A subgraph of $\Sigma$ is \emph{antibalanced} if each of its circles is either positive and even in length or negative and odd in length.
We write $-\Sigma$ for the \emph{negation} of $\Sigma$, defined by $-\Sigma = (|\Sigma|, -\sigma)$. It is easy (even easier, once we define switching) to check that $\Sigma$ is balanced if and only if $-\Sigma$ is antibalanced.
\emph{Switching} $\Sigma$ by $v \in V(\Sigma)$ means negating the sign of every edge that has $v$ as an endpoint. Switching $\Sigma$ by $X \subseteq V(\Sigma)$ means switching each $v \in X$ in turn. If $\Sigma'$ is obtained from $\Sigma$ by switching, we say they are \emph{switching equivalent}, written $\Sigma' \sim \Sigma$. It is straightforward to check that switching equivalence is indeed an equivalence relation. The \emph{switching class} of $\Sigma$ is the equivalence class of $\Sigma$ under this equivalence relation and is denoted by $[\Sigma]$.
A fundamental theorem concerning switching (as found in \cite{zaslav3}) is that $\Sigma' \sim \Sigma$ if and only if $|\Sigma'|=|\Sigma|$ and $\Sigma'$ and $\Sigma$ have the same balanced circles. Thus, any property of signed graphs that depends only on the signs of the circles is invariant for all graphs contained in $[\Sigma]$. As we will see, signed edge coloring using a certain number of colors is an example of this phenomenon.
\section{Edge Colorings}
In this section we will give a natural definition for edge coloring a signed graph. Recall that an \emph{edge coloring} of an ordinary graph $\Gamma$ is an assignment of colors (typically elements of $\{1,\ldots,n\}$) to its edges. Such a coloring is \emph{proper} if no two adjacent edges receive the same color. Our definition is similar, but we define edge coloring in terms of incidences (rather than edges themselves) in order to incorporate edge signs.
To edge color a signed graph, we need a more sophisticated set of colors than $\{1,\ldots,n\}$. Let $M_n=\{0,\pm 1,\ldots,\pm k\}$ if $n=2k+1$, and $M_n=\{\pm 1,\ldots,\pm k\}$ if $n=2k$. The $M_n$ are called \emph{signed color sets} and they contain \emph{signed colors}. The colors $+a$ and $-a$ have the same \emph{magnitude}, but are \emph{opposite}. These are the same signed color sets used in both \cite{zaslav2} and \cite{maca} to study signed vertex coloring.
\begin{defn}An \emph{$n$-edge coloring} (or more briefly, an \emph{$n$-coloring}) $\gamma$ of $\Sigma$ is an assignment of colors from $M_n$ to each vertex-edge incidence of $\Sigma$ subject to the condition that $\gamma(v,e) = -\sigma(e) \gamma(w,e)$ for each edge $e{:}vw$.
An $n$-coloring is \emph{proper} if for any two incidences $(v,e)$ and $(v,f)$ involving the same vertex, $\gamma(v,e) \neq \gamma(v,f)$.
\end{defn}
Intuitively, negative edges act like unsigned edges since they have the same color at both incidences. However, positive edges act differently and instead have opposite colors at their incidences. See Figure \ref{fig1} for an example of a proper coloring.
Since both incidences of a negative edge $e{:}vw$ receive the same color, we will sometimes write $\gamma(e) = \gamma(v,e) = \gamma(w,e)$. When such notation is used it is understood that the edge in question is negatively signed. We cannot afford this luxury when it comes to positive edges.
The color $0$ is its own opposite, so when the color $0$ is available we are allowed to color both positive and negative edges with $0$ at both of their incidences. The subsequent theory often becomes simpler when the color $0$ is not available, so it is sometimes convenient to discuss \emph{zero-free} colorings---those which omit $0$ and hence have an even number of colors available.
\begin{figure}[h!]
\centering
\includegraphics[scale=.07]{image1}
\caption[A signed graph and a proper $3$-coloring.]{Here we see a signed graph and a proper $3$-coloring. Throughout, we indicate negative edges with dashed lines and positive edges with solid lines.}
\label{fig1}
\end{figure}
\begin{defn}Let $\gamma$ be an $n$-coloring of $\Sigma$. If there exists an edge $e$ such that $\gamma(v,e) = a$, then we say the color $a$ is \emph{present} at $v$. Otherwise, $a$ is \emph{absent} at $v$.
\end{defn}
To rephrase the definition of a proper $n$-coloring, $\gamma$ is proper if and only if each color from $M_n$ is present at each vertex at most once.
\subsection{Basic Coloring Properties}
We begin by reiterating that negative edges receive the same color at both of their incidences. Hence, colored negative edges are essentially the same as colored unsigned edges. We use this to record the following lemma.
\begin{lem}\label{specialize} If every edge of $\Sigma$ is negative, then there is a one-to-one correspondence between (proper) $n$-colorings of $\Sigma$ and (proper) $n$-colorings (in the usual unsigned sense) of $|\Sigma|$.
\end{lem}
The fact that signed edge coloring specializes to ordinary edge coloring when $\Sigma$ is all negative (as opposed to all positive, as one might expect) is a consequence of the definition of the signed line graph, as we will see in later sections.
Another basic feature of signed edge coloring is its compatibility with switching.
\begin{lem}\label{switchin} Suppose $\gamma$ is a proper $n$-coloring of $\Sigma=(\Gamma,\sigma)$ and suppose $\Sigma'=(\Gamma,\sigma')$ is obtained from $\Sigma$ by switching a vertex set $X$. Define a new coloring $\gamma \,'$ which is obtained from $\gamma$ by negating all colors on all incidences involving vertices from $X$. Then, $\gamma \,'$ is a proper $n$-coloring of $\Sigma'$.
\begin{proof} We will describe the case where we switch a single vertex, since all larger cases can be considered one vertex at a time. Consider a vertex $v \in X$. Since $\gamma$ is an edge coloring, $\gamma(v,e)=-\sigma(e)\gamma(w,e)$ for any edge $e{:}vw$ incident with $v$. Switching $v$ changes the signs of all edges adjacent to $v$. Hence, we have $\gamma \,'(v,e)=-\sigma'(e)\gamma \,'(w,e)$ and thus $\gamma \,'$ is an edge coloring. We further see that $\gamma\, '$ is proper, since $\gamma(v,e)\neq \gamma(v,f)$ implies that $\gamma \,'(v,e) \neq \gamma \,'(v,f)$.
\end{proof}
\end{lem}
Thus, if we obtain a proper $n$-coloring of $\Sigma$, we automatically obtain a proper $n$-coloring for every member of $[\Sigma]$. Lemma \ref{switchin} is illustrated in Figure \ref{fig2}.
\begin{figure}[h!]
\centering
\includegraphics[scale=.05]{image2}
\caption[Applying a switching function to an edge colored signed graph.]{On the left-hand graph, we have the same proper coloring as seen in Figure 1. On the right-hand graph, we have switched the top-right vertex and negated all colors incident with this vertex to obtain a proper coloring of the switched graph.}
\label{fig2}
\end{figure}
\subsection{Magnitude Subgraphs}
In this section we study the properties of subgraphs whose edges are colored with a single magnitude. Let $\Sigma$ be a signed graph and let $\gamma$ be a proper $n$-coloring of $\Sigma$. We write $\Sigma_{a}[\gamma]$ for the set of edges of $\Sigma$ that are colored using $\pm a$ with respect to $\gamma$. If there is only one coloring that we have in mind, we write $\Sigma_a$. We call $\Sigma_a$ the \emph{$a$-graph} of $\Sigma$ with respect to $\gamma$.
We first observe that $\Sigma_a$ has maximum degree $2$, since at most $a$ and $-a$ are present at each vertex of $\Sigma_a$. Thus, each component of $\Sigma_a$ is either a path or a circle. When $a=0$ the maximum degree is $1$, and hence $\Sigma_0$ is a matching. We now describe which kinds of paths and cycles can possibly appear in $\Sigma_a$ when $a\neq 0$.
\begin{lem} Every signed path can be properly edge colored with $\pm a$ (where $a \neq 0$). Furthermore, every signed path has exactly two different $\pm a$ colorings.
\begin{proof} Switch so that the path is all negative. Then, color the edges of the path so that they alternate between $-a$ and $a$. Finally, switch back to the original signature of the path, negating colors at the switched vertices as in Lemma \ref{switchin}. Clearly there are exactly two possibilities for each path---simply negate all colors to change between the two possible $\pm a$ colorings.
\end{proof}
\end{lem}
\begin{lem}A signed circle $C$ can be properly colored with $\pm a$ ($a \neq 0$) if and only if $C$ is positive. Furthermore, every positive circle has exactly $two$ $\pm a$ colorings.
\begin{proof} First, suppose $C$ can be properly colored with $\pm a$. Choose an edge $e{:}vw$ of $C$ and switch so that $C{\setminus}e$ is an all-positive path from $v$ to $w$. Since $C{\setminus}e$ is all-positive, up to choice of names $a$ is present at $v$ in $C{\setminus}e$ and $-a$ is present at $w$ in $C{\setminus}e$. Hence, $e$ must be positive, or else a coloring using $\pm a$ would be impossible.
Conversely, any positive circle can be colored with $\pm a$. Simply switch so that the circle is all positive and color incidences alternating between $a$ and $-a$ around the circle.
Finally, we note that given any $2$-coloring of a positive circle, we can obtain another $2$-coloring by negating all of the colors. Clearly, every positive circle has just two possible $2$-colorings. A $2$-colored positive circle is shown in Figure \ref{fig3}.
\end{proof}
\end{lem}
\begin{figure}[h!]
\centering
\includegraphics[scale=.069]{image3}
\caption{A positive circle and one of its two possible $2$-colorings. \label{fig3}}
\end{figure}
Therefore in a proper edge coloring, $\Sigma_a$ consists of paths and positive circles so that $\Sigma_a$ is balanced. Thus, a proper coloring corresponds to a partition of the edges of $\Sigma$ into balanced subgraphs of maximum degree $2$ and a single matching (if the color $0$ is available). This is not a one-to-one correspondence, as each connected component in such a partition may be colored in one of two ways (except for edges that are colored $0$).
We see the phenomenon mentioned above in ordinary graphs as well. Suppose $c$ is an ordinary edge coloring of $\Gamma$ (using $M_n$ for the color set). In this case, $\Sigma_a$ ($a \neq 0$) is a bipartite subgraph of maximum degree $2$ (or a matching if $a=0$).
When $\Sigma$ is all negative, the balanced subgraphs are precisely the bipartite subgraphs. Hence, the partition into bipartite subgraphs induced by an ordinary edge coloring is a special case of the balanced partition induced by a signed edge coloring.
\section{Vizing's Theorem for Signed Graphs}
For an ordinary graph $\Gamma$, we write $\chi'(\Gamma)$ for the \emph{chromatic index} of $\Gamma$---the minimum number of colors used in any proper edge coloring of $\Gamma$. It is a classic result of Vizing that every simple ordinary graph $\Gamma$ satisfies $\Delta(\Gamma) \leq \chi'(\Gamma) \leq \Delta(\Gamma)+1$, where $\Delta(\Gamma)$ is the maximum degree of a vertex in $\Gamma$ \cite{vz}.
For a signed graph $\Sigma$ we borrow the existing notation and write $\chi'(\Sigma)$ for its \emph{chromatic index}, which we define to be the smallest $n$ such that $\Sigma$ has a proper edge coloring using colors from $M_n$. As an easy application of Lemma \ref{specialize}, we see that $\chi'(|\Sigma|)=\chi'(\Sigma)$ if $\Sigma$ is all negative. Because of this, our use of the usual notation is justified.
The lower bound for $\chi'(\Sigma)$ is $\Delta(\Sigma)$ since there are $\Delta(\Sigma)$ different incidences to be colored at a vertex of maximum degree. Our goal is now to prove an upper bound of $\Delta(\Sigma)+1$, but we first require some new machinery. We will define a signed version of a Kempe chain, which is a tool that is frequently used in the study of ordinary edge coloring.
\subsection{Signed Kempe Chains}
The main difference between ordinary and signed Kempe chains is that the signed version is a trail (which may intersect itself at a vertex), and not a path.
We begin with a convenient notational definition. If $T = (v_0,\ldots,v_m)$ is a signed trail we write $t_k$ for the number of positive edges that appear on $T$ between $v_0$ and $v_k$.
\begin{defn}\label{kempe}Suppose $\gamma$ is a proper $n$-coloring of $\Sigma$. If $a$ is absent at a vertex $v_0$ and $b$ is present at $v_0$, we define the \emph{$a/b$-chain at $v_0$} to be the maximal trail $T=(v_0,\ldots,v_m)$ starting at $v_0$ with the properties:
\begin{enumerate}
\item The edge magnitudes alternate between $|a|$ and $|b|$ along $T$ (starting with $|b|$).
\item $\{\gamma(v_i, v_{i-1}v_i), \gamma(v_i, v_iv_{i+1})\} = \{ (-1)^{t_i} a, (-1)^{t_i} b \}$ for all $i \neq 0,m$.
\end{enumerate}
\end{defn}
We write $K_{a,b}(v_0, v_m)$ for the $a/b$-chain at $v_0$. In the following lemma we note that signed Kempe chains specialize to ordinary Kempe chains when $\Sigma$ is all negative.
\begin{lem}If $\Sigma$ is all negative, $K_{a,b}(v_0, v_m)$ is a path.
\begin{proof} Since all edges are negative, the second property in the definition of the signed Kempe chain becomes $\{\gamma(v_i, v_{i-1}v_i), \gamma(v_i, v_iv_{i+1})\} = \{a , b \}$ for all $i \neq 0,m$. Thus, the signed Kempe chain has maximum degree $2$ and hence does not intersect itself. It must be a path.
\end{proof}
\end{lem}
While an ordinary Kempe chain does not intersect itself, a signed Kempe chain may intersect itself. For example, let $(v_0,\ldots,v_m)$ be the signed $a/b$-chain at $v_0$. Suppose at vertex $v_j$ we have $\gamma(v_j, v_{j-1}v_j) = a$ and $\gamma(v_j, v_jv_{j+1}) = b$. Then, vertex $v_j$ may appear once again in the Kempe chain (say, $v_j = v_s$, $s > j$), as long as $\gamma(v_s, v_{s-1}v_s) = -a$ and $\gamma(v_s, v_s v_{s+1}) = -b$ (or vice versa). There is at most one self intersection at each vertex, since the only available colors are $\pm a$ and $\pm b$. A self-intersecting Kempe chain is shown in Figure \ref{fig4}. The following lemma shows that in the event of a self-intersection, the path between $v_j$ and $v_s$ must have a certain sign.
\begin{lem} Suppose $K_{a,b}(v_0, v_m)$ has a vertex $v_j = v_s$ that appears twice $(s > j)$. Let $Q$ be the subtrail $(v_j,\ldots,v_s)$. Then, $Q$ has an odd number of positive edges.
\begin{proof} If $Q$ has an even number of positive edges, $\{\gamma(v_j, v_{j-1}v_j), \gamma(v_j, v_jv_{j+1})\} = \{\gamma(v_s, v_{s-1}v_s), \gamma(v_s, v_sv_{s+1})\}$, which contradicts the definition of a proper coloring.
\end{proof}
\end{lem}
\begin{figure}[h!]
\centering
\includegraphics[scale=.05]{image4}
\caption[Two examples of $a/b$-chains.]{On the left, an $a/b$-chain that does not intersect itself. On the right, an $a/b$-chain that does intersect itself. Parenthetical colors indicate their absence at the specified vertex.}
\label{fig4}
\end{figure}
The main purpose of signed Kempe chains is that we can use them to change the colors present at a given vertex.
When $a,b\neq0$, we modify $K_{a,b}(v_0, v_m)$ by performing the $a/b$-\emph{swap} at $v_0$, which is the act of interchanging $a$ with $b$ and $-a$ with $-b$ for all incidences in $K_{a,b}(v_0, v_m)$. Thus, the $a/b$ swap changes the color present at $v_0$ from $b$ to $a$, and also changes the color present at $v_m$.
Intuitively, we imagine that changing $\gamma(v_0, v_0v_1)$ from $b$ to $a$ creates a sequence of subsequent changes that must be made in order to preserve the propriety of the coloring. This sequence of changes propagates along $K_{a,b}(v_0,v_m)$ and ends when $K_{a,b}(v_0,v_m)$ ends, allowing us to maintain a proper coloring. We now describe in detail what happens when performing a swap.
\begin{lem}\label{swap1}Consider $K_{a,b}(v_0, v_m)$ where $a \neq 0$ and $b \neq 0$. Performing the $a/b$-swap at $v_0$ does not change the present or absent colors at any vertex except for $v_0$ and $v_m$. It changes the present colors at $v_0$ and $v_m$, interchanging $a$ with $b$ at $v_0$ and $(-1)^{t_m}a$ with $(-1)^{t_m}b$ at $v_m$.
\end{lem}
We now discuss what happens when a signed Kempe chain involving the color $0$ intersects itself. Since $+0 = -0$, such a chain has maximum degree $3$ and hence does not behave the same as a zero-free chain.
\begin{lem} Suppose $K_{a,0}(v_0,v_m)$ has its first self-intersection at $v_j=v_s$. Then, $v_s = v_m$. In other words, the Kempe chain must terminate at its first self-intersection.
\begin{proof} Suppose without loss of generality that $\gamma(v_j, v_{j-1}v_j) = a$ and $\gamma(v_j,v_jv_{j+1}) = 0$. Then the only possibility for $\gamma(v_s, v_{s-1}v_s)$ is $-a$. However, the chain cannot continue past $v_s$ since all three available colors are already present at $v_s$.
\end{proof}
\end{lem}
Finally, we make note of what happens if we swap a self-intersecting Kempe chain that involves the color $0$.
\begin{lem}Consider $K_{a,0}(v_0,v_m)$ where $a \neq 0$. If $K_{a,0}(v_0,v_m)$ intersects itself at $v_j = v_m$ where $j < m$, then performing the $a/0$-swap at $v_0$ creates a single impropriety at $v_m$. Namely, either $\gamma(v_m, v_{m-1}v_m)=\gamma(v_j, v_{j-1}v_j) = 0$ or $\gamma(v_m, v_{m-1}v_m)=\gamma(v_j, v_{j}v_{j+1}) = 0$.
\begin{proof} Since $a, -a$, and $0$ are present at $v_m$ before the swap, then $0$, $0$, and either $a$ or $-a$ are present at $v_m$ after the swap. No two incidences involving the same vertex may be colored $0$ in a proper coloring.
\end{proof}
\end{lem}
Because swapping a self-intersecting chain involving $0$ creates an improper coloring, we will consider only zero-free Kempe chains in subsequent arguments and deal with $0$ a different way. The above lemma is illustrated in Figure \ref{fig5}.
\begin{figure}[h!]
\centering
\includegraphics[scale=.081]{image5}
\caption[Swapping an $a/0$-chain.]{On the top, an $a/0$-chain that terminates when it intersects itself. On the bottom, the result of swapping the chain. Swapping the chain ruins the propriety of the coloring.}
\label{fig5}
\end{figure}
\subsection{Signed Vizing's Theorem}
We are now ready to prove the signed generalization of Vizing's theorem. We will first prove the zero-free version of the theorem (Theorem \ref{vizing}) using a method that involves Kempe chains. The zero-free version gives the desired upper bound of $\Delta(\Sigma)+1$ when $\Delta(\Sigma)$ is odd, and a weaker upper bound of $\Delta(\Sigma)+2$ when $\Delta(\Sigma)$ is even. This occurs because it is impossible to use $\Delta(\Sigma)+1$ colors when $\Delta(\Sigma)$ is even and the color $0$ is unavailable. After proving Theorem \ref{vizing}, we incorporate the color $0$ using a method that does not involve Kempe chains, bringing the upper bound down to $\Delta(\Sigma)+1$ when $\Delta(\Sigma)$ is even.
The statement of our main theorem is as follows.
\begin{thm}[Signed Vizing's Theorem]\label{goodvizing} For a signed simple graph $\Sigma$, $\Delta(\Sigma) \leq \chi'(\Sigma) \leq \Delta(\Sigma)+1$.
\end{thm}
We first define a \emph{fan}, which is a device that allows us to manipulate the colors locally at a given vertex without affecting the propriety of a coloring.
\begin{defn} Let $\Sigma$ be a signed graph and let $e{:}uv_0$ (called the \emph{initial edge}) be an edge of $\Sigma$. Let $\gamma_0$ be a proper edge coloring (called the \emph{initial coloring}) of $\Sigma {\setminus} e$. Assume by switching that all edges incident with $u$ are negative. Let $v=v_0,\ldots,v_s$ be a maximal sequence of neighbors of $u$ such that $\gamma_0(v_i, uv_i)$ is absent at $v_{i-1}$ for all $1 \leq i \leq s$. The \emph{fan} at $u$ is the subgraph induced by all of the $uv_i$ edges. The edges of the fan are written $e_i:=uv_i$. We say that $u$ is the \emph{hinge} of the fan.
\end{defn}
The purpose of a fan is to allow us to interchange the colors on the edges adjacent to $u$. The colorings we obtain by interchanging colors are called \emph{shifted colorings} and are defined as follows.
\begin{defn} Let $F$ be a fan with initial coloring $\gamma_0$ and edges $e_0,\ldots,e_s$ (where $e_0{:}uv_0$ is the uncolored initial edge). We define a sequence of \emph{shifted colorings}, $\gamma=\gamma_1,\ldots,\gamma_s$, such that:
\begin{enumerate}
\item The edge $e_i$ is not colored in $\gamma_i$.
\item $\gamma_i(u,e_j)=\gamma_i(v_j,e_j) = \gamma_0(v_{j+1},e_{j+1})$ for $j \in \{0,\ldots,i-1\}$.
\item $\gamma_i=\gamma_0$ otherwise.
\end{enumerate}
\end{defn}
The reader is encouraged to envision $v_0$ as being at the bottom, $v_s$ as being at the top, and the $\gamma_i$ as being obtained by shifting the colors of the edges $e_1,\ldots,e_i$ downwards, leaving $e_i$ uncolored. By design, each of the $\gamma_i$ is proper and uses the same set of colors as $\gamma_0$. Also notice that the colors present or absent at $u$ are exactly the same in all of the $\gamma_i$.
Under certain conditions, we will be able to use a fan in conjunction with Kempe chains to extend a proper coloring of $\Sigma{\setminus}e_0$ to a proper coloring of $\Sigma$ using the same set of colors. The following lemma forms the bulk of the proof of Theorem \ref{vizing}, and hence is a substantial portion of the proof of Theorem \ref{goodvizing}.
\begin{lem}\label{fan} Let $\Sigma$ be a signed graph and let $e_0{:}uv_0$ be one of its edges. Let $\gamma_0$ be a proper zero-free edge coloring of $\Sigma{\setminus}e_0$ using $n$ colors, and suppose that there is at least one color absent at $u$ and at each neighbor of $u$. Furthermore, suppose colors of the same magnitude are absent at $u$ and $v_0$. Then, there exists an $n$-coloring of $\Sigma$.
\begin{proof}
First, assume by switching that all edges incident with $u$ are negative. Let $a$ be a color absent at $u$, and suppose a color of the same magnitude is absent at $v_0$. If $a$ is also absent at $v_0$, we extend the coloring by setting $\gamma_0(e_0)=a$ and we are done. Otherwise, assume $-a$ is absent at $v_0$. If $-a$ is absent at $u$, then we can also extend the coloring, and so we assume $-a$ is present at $u$. So, both $u$ and $v_0$ have degree $1$ in $(\Sigma{\setminus}e)_a$.
Now we build a fan $F$ with hinge $u$ and initial edge $e_0$, and with shifted colorings $\gamma_0,\ldots,\gamma_s$. Let $e_0,\ldots,e_s$ be the edges of $F$. Since $-a$ is present at $u$ and absent at $v_0$, we choose $e_1$ to be the edge adjacent to $u$ with color $-a$.
Next, let $b$ be a signed color that is absent at $v_s$ with respect to $\gamma_0$. Then, $b$ is absent at $v_s$ with respect to all of the $\gamma_i$. If $b$ is absent at $u$, then we can simply extend $\gamma_s$ by setting $\gamma_s(e_s) = b$. So, we assume that $b$ is present at $u$. Since $F$ is maximal (by definition), there must be some $1 \leq j \leq s-1$ such that $\gamma_0(e_j) = b$. See Figure \ref{fig6} for an illustration of $F$.
\begin{figure}[h!]
\centering
\includegraphics[scale=.050]{image6}
\caption[A depiction of $\gamma_0$ and $F$.]{A depiction of $\gamma_0$ and $F$. The triple marking on the initial edge $e_0$ indicates that it is currently uncolored.\label{fig6}}
\end{figure}
Now we consider the final shifted coloring $\gamma_s$. If $a$ is absent at $v_s$ with respect to $\gamma_s$, we simply extend $\gamma_s$ by coloring $\gamma_s(e_s)=a$. Thus, we assume that $a$ is present at $v_s$. Since $b$ is absent at $v_s$, there is an $a/b$-chain starting at $v_s$, which we denote by $T$. We write the vertices of $T$, in order, as $v_s=t_0,t_1,\ldots,t_r$, so that $T=(v_s,t_1,...,t_r)$. If we swap the $a/b$-chain at $v_s$ we can color both incidences of $e_s$ with the color $a$, unless performing the $a/b$-swap at $v_s$ makes it so that $a$ is present at $u$. By Lemma \ref{swap1}, this can only happen if $u=t_r$ (i.e., $T$ ends at $u$), which we now assume. So, to use our Kempe chain notation, $T = K_{a,b}(v_s,u)$ with respect to $\gamma_s$. It is worth noting that while $T$ ends at $u$, it is also possible for $T$ to pass through $u$ once before ending at $u$. If $T$ does pass through $u$ before ending at $u$, it must pass through $u$ at consecutive edges whose colors are $-a$ and $-b$ (or vice versa). Thus, there may be $1$, $2$, or $3$ edges of $T$ that are contained in $F$.
Once again consider $\gamma_s$. Since performing the $a/b$-swap at $v_s$ makes the color $a$ present at $u$, the last incidence of $T$ must be colored $b$ . Thus, the last edge of $T$ is $uv_{j-1}$, which is the edge of $F$ that is colored $b$ with respect to $\gamma_s$. We now break the proof into several cases, depending on the nature of the edges in the intersection of $T$ and $F$. Let $X$ be the set of edges in the intersection of $T$ and $F$.
Case 1: The edge $e_{j-1}$ (colored $b$) is above all other edges of $X$ with respect to $\gamma_s$. By above, we mean that the index $j-1$ is greater than the index of all other edges in $X$. Shift from $\gamma_s$ to $\gamma_{j-1}$, so that $e_{j-1}$ is uncolored and $b$ is absent at $v_{j-1}$. The key is that performing this shift only disturbs $T$ at its last edge $e_{j-1}$, since all other edges in $X$ are below $e_{j-1}$ and hence do not have their colors changed when shifting from $\gamma_s$ to $\gamma_{j-1}$. Now, we perform the $a/b$-swap at $v_{j-1}$, which propagates backwards along the trail $(v_{j-1}=t_{r-1},t_{r-2},\ldots,t_1,t_0=v_s)$ and terminates at $v_s$. This changes the color present at $v_s$ from $a$ to $b$ and also changes the color absent at $v_{j-1}$ from $b$ to $a$ without changing any other present or absent colors (by Lemma \ref{swap1}). Thus, we are now free to color both incidences of $e_{j-1}$ with the color $a$, completing this case.
We pause to note two things. First, if $b=-a$ then the only edge in $X$ is $e_{j-1}$. Thus we have disposed of the case where $b=-a$ in the previous paragraph. In what follows, we will assume $b \neq -a$. Second, the cases where $|X|=1$ and $|X|=2$ are proved in the previous paragraph. Indeed, if $|X|=1$ then $T$ must end at $e_{j-1}$, and if $|X|=2$ then $X$ must contain only $e_0$ (which is colored $-a$ with respect to $v_s$) and $e_{j-1}$. To see this, suppose that $|X|=2$ and the edges of $X$ are $e_k$ (colored $-b$) and $e_{j-1}$ (colored $b$). Then either $T=(t_0,\ldots, v_k, u,\ldots u)$ or $T=(t_0,\ldots, u,v_k,\ldots u)$. In the first case, the incidence after $(u, uv_k)$ along $T$ must be colored $-a$, since $(u,uv_k)$ is colored $-b$. In the second case, the incidence before $(u, uv_k)$ must be colored $-a$. Thus, in both cases the edge $e_0$ (which is colored $-a$ and is contained in $F$) must be contained in $T$. This contradicts the fact that $|X|=2$. Thus, if $|X|=2$ it must contain $e_{j-1}$ and $e_0$ and hence Case 1 applies.
We point out that if $\Sigma$ is all negative then our work so far essentially implies ordinary Vizing's Theorem. One can prove ordinary Vizing's Theorem using the techniques described above, but the only possible case is where $|X|=1$. Case 1 is illustrated in Figure \ref{fig7}.
\begin{figure}[h!]
\centering
\includegraphics[scale=.056]{image7}
\caption[An illustration of Case 1 in Theorem \ref{goodvizing}.]{An illustration of Case 1. On the left, we see $\gamma_s$ along with $T$, which ends at $u$ and intersects the fan at three edges. The $b$-colored edge is above the other two. On the right, we have shifted up to $\gamma_{j-1}$. Performing this shift does not break the Kempe chain exept at its final edge (the $b$-colored edge). On the right-hand graph, performing the $a/b$-swap at $v_{j-1}$ allows us to color $e_{j-1}$ with color $a$. Once again, the triple marking indicates that the given edge is uncolored.}
\label{fig7}
\end{figure}
Case 2: The edge $e_{j-1}$ is not above all other edges of $X$ with respect to $\gamma_s$. Due to the previous discussion, this implies that $|X|=3$. Let the edges in $X$ be $e_k$ (colored $-b$), $e_0$ (colored $-a$), and $e_{j-1}$ (colored $b$). Since $\gamma_s(e_0)=-a$, we see that $e_k$ is above all the other edges. In other words, $k > j-1 >0$.
To prove Case 1 above, we relied on the fact that shifting to $\gamma_{j-1}$ disturbed $T$ only at its final edge. In Case 2 we can no longer shift to $\gamma_{j-1}$ and swap the $a/b$-chain backwards along $T$, since shifting to $\gamma_{j-1}$ will change the color of $e_k$ and will hence break $T$ at an edge other than its final edge.
We break Case 2 into two subcases, depending on whether $T$ passes through $e_k$ or $e_0$ first. In the first subcase, we will have $T=(v_s,\ldots,v_k,u,v_0,\ldots,v_{j-1},u)$, and in the second we will have $T=(v_s,\ldots v_0,u,v_k,\ldots,v_{j-1},u)$. The second case is easy to take care of. If $T=(v_s,\ldots v_0,u,v_k,\ldots,v_{j-1},u)$, then we shift to coloring $\gamma_{j-1}$, leaving $e_{j-1}$ uncolored. Notice that $-b$ is absent at $v_k$ with respect to $\gamma_{j-1}$ (since $k >j-1$). We now perform the $a/b$-swap at $v_{j-1}$, which travels along the trail $(v_{j-1},\ldots,v_k)$ and terminates at $v_k$ (because $-b$ is absent at $v_k$). We can now color $v_{j-1}$ with the color $a$ and we are done.
Now we consider the case where $T=(v_s,\ldots,v_k,u,v_0,\ldots,v_{j-1},u)$. We note that the argument given in the previous paragraph will not work in this case. To see why, consider what happens if we shift to $\gamma_{j-1}$ and swap the $a/b$-chain at $v_{j-1}$. Since we have shifted to $\gamma_{j-1}$, the edge $uv_{k+1}$ is now colored $-b$. Thus, the swap will travel along the trail $T'=(v_{j-1},\ldots, v_0, u, v_{k+1},\ldots)$. We have not specified a second endpoint for $T'$ for good reason---we simply do not know where $T'$ will end. In fact, it is possible that the last two vertices of $T'$ are $v_j$ and $u$ so that performing the swap makes $a$ present at $u$.
Thus, to tackle the case where $T=(v_s,\ldots,v_k,u,v_0,\ldots,v_{j-1},u)$, we shall instead do this: first, shift to the coloring $\gamma_k$, leaving $e_k$ uncolored and $-b$ absent at $v_k$. Perform the $-a/-b$-swap at $v_k$, which travels along the trail $(v_k,\ldots,v_s)$. This leaves $-a$ absent at $v_k$, but this is still not quite what we want. We need to force $a$ to be absent at $v_k$ rather than $-a$. We will call the coloring that we have obtained $\gamma_k \,'$.
Next, we consider the $-a/a$-chain at $v_k$ with respect to $\gamma_k \,'$, denoted by $A$. One of two things may happen. First, $A$ may terminate at a vertex other than $u$. In this case, perform the $-a/a$-swap at at $v_k$ and extend by coloring $e_k$ with color $a$. The other possibility is that $A$ terminates at $u$. If it does, then $A=(v_k,...,v_0,u)$. Notice that when we modified $\gamma_k$ to obtain $\gamma_k \,'$, we interchanged the color present at $v_k$ from $-a$ to $-b$ and the color present at $v_s$ from $b$ to $a$, but we did not change the present or absent colors at any of the other $v_i$. Thus, with respect to $\gamma_k \,'$, we can still shift the colors on all edges below $e_k$ upwards, leaving $e_0$ uncolored. It is also important here that $e_{k-1}$ is not colored $-a$ with respect to $\gamma_k \,'$, or else shifting the colors on the edges upwards would make $-a$ present at $v_k$. Fortunately it is impossible that $e_{k-1}$ is colored $-a$, because $e_0$ is colored $-a$ and $0<j-1<k$, so that $k \geq 2$.
Once we have shifted the colors upwards on the edges below $e_k$ with respect to $\gamma_{k}\,'$, we simply perform the $-a/a$-swap at $v_0$. This swap travels along the trail $(v_0,\ldots,v_k)$ (i.e., backwards along $A$), and terminates at $v_k$. We now color $e_0$ with the color $a$, and we are done. The part of Case 2 where $j-1 < k$ and where $T=(v_s,\ldots,v_k,u,v_0,\ldots,v_{j-1},u)$ is illustrated in Figures \ref{fig8} and \ref{fig9}.
\end{proof}
\end{lem}
We will now prove the zero-free version of signed Vizing's Theorem. We reiterate that the upper bound of $\Delta(\Sigma)+2$ appears due to the fact that a zero-free coloring always uses an even number of colors.
\begin{thm}[Zero-free Signed Vizing's Theorem]\label{vizing}Let $\Sigma$ be a signed simple graph. Then $\Delta(\Sigma) \leq \chi'(\Sigma) \leq \Delta(\Sigma)+1$ if $\Delta(\Sigma)$ is odd, and $\Delta(\Sigma) \leq \chi'(\Sigma) \leq \Delta(\Sigma)+2$ if $\Delta(\Sigma)$ is even.
\begin{proof} The proof is by induction on the number of edges, with the result being clear for signed graphs on $0$, $1$, or $2$ edges. Suppose $\Sigma$ has $n \geq 3$ edges. Choose an edge $e$ of $\Sigma$ and consider $\Sigma{\setminus}e$. If $\Delta(\Sigma) = \Delta(\Sigma{\setminus}e)$, then we obtain a coloring of $\Sigma{\setminus}e$ with either $\Delta(\Sigma)+1$ or $\Delta(\Sigma)+2$ colors (depending on the parity of $\Delta(\Sigma)$) by induction. If $\Delta(\Sigma) -1 = \Delta(\Sigma{\setminus}e)$, then we obtain a coloring of $\Sigma{\setminus}e$ with either $\Delta(\Sigma)$ or $\Delta(\Sigma)+1$ colors. Clearly a zero-free coloring with $\Delta(\Sigma)$ colors can be transformed into a coloring with $\Delta(\Sigma)+2$ colors (just add two more colors to the color set), so in either case we have a zero-free coloring of $\Sigma{\setminus}e$ such that each vertex has at least one absent color. We call this coloring $\gamma_0$.
\begin{figure}[h!]
\centering
\includegraphics[scale=.056]{image8}
\caption[An illustration of Case 2 in Theorem \ref{goodvizing}---part 1.]{On the left we have $\gamma_s$ along with $T=(v_s,\ldots,v_k,u,v_0,\ldots,v_{j-1},u)$, which intersects the fan at three edges, $e_k$, $e_0$, and $e_{j-1}$ (in order). On the right, we have shifted up to coloring $\gamma_k$ and then swapped the $-a/-b$-chain at $v_k$ to obtain the new coloring $\gamma_k\,'$. From the picture on the right, we proceed by attempting to swap the $-a/a$-chain at $v_k$. The details of this second swap are illustrated in Figure \ref{fig9}. \label{fig8}}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[scale=.056]{image9}
\caption[An illustration of Case 2 in Theorem \ref{goodvizing}---part 2.]{A continuation of Figure \ref{fig8}. On the left, we see the modified coloring $\gamma_k\,'$ along with the $-a/a$-chain at $v_k$ that ends at $u$ via edge $e_0$. On the right, we have shifted the colors on the edges below $e_k$ upwards leaving $e_0$ uncolored. We are now free to swap $-a/a$ at $v_0$, which will terminate at $v_k$. This allows us to color $e_0$ with the color $a$. \label{fig9} }
\end{figure}
Start a fan $F$ at $e{:}uv_0$ using $\gamma_0$ as the initial coloring. Write $\gamma_0,\ldots,\gamma_s$ for the shifted colorings. If colors of the same magnitude are absent at both $u$ and $v_0$ with respect to $\gamma_0$, then we can apply Lemma \ref{fan} to $F$ and obtain the desired coloring of $\Sigma$. Therefore, we assume that colors of the same magnitude are not absent at $u$ and $v_0$.
Let $b$ be absent at $v_s$ with respect to $\gamma_0$. Shift to coloring $\gamma_s$. In a similar fashion to the proof of Lemma \ref{fan}, we have an $a/b$-chain $T$ starting at $v_s$ that must end at a $b$-colored edge of $F$. Let $e_k=uv_k$ be the first edge where $T$ intersects $F$, so that either $T=(v_s,\ldots,v_k,u,\ldots,u)$, or $T=(v_s,\ldots u, v_k,\ldots, u)$, or simply $T=(v_s,\ldots,v_k, u)$. In the first two cases we are saying that $T$ passes through $e_k$ before leaving and returning to $u$ (in two different ways), and in the third case we are saying that $T$ passes through $e_k$ and then immediately ends. Notice that the third case occurs if and only if $\gamma_s(e_k)=b$.
If $\gamma_s(e_k)=b$, shift to $\gamma_k$ and perform the $a/b$-swap at $v_k$, which propagates along the trail $(v_k,...,v_s)$ and terminates at $v_s$. We can then color $e_k$ with the color $a$, finishing this case.
If $\gamma_s(e_k) = -a$, then $-a$ is absent at $v_k$ with respect to $\gamma_k$. Thus, in this case we may begin a new fan
$F'$ with initial edge $e_k{:}uv_k$ and initial coloring $\gamma_k$. The key is that colors of the same magnitude absent at $u$ and $v_k$ with respect to $\gamma_k$. We then apply Lemma \ref{fan} to complete this case.
If $\gamma_s(e_k) = -b$, we consider two different cases. First, if $T=(v_s,\ldots,v_k,u,\ldots,u)$, then we shift to $\gamma_k$ and perform the $-a/-b$-swap at $v_k$, which travels along the trail $(v_k,\ldots v_s)$. The edge $e_k$ is now uncolored with colors of the same magnitude absent at its endpoints, and we apply Lemma \ref{fan}.
In the second case, suppose that $T=(v_s,\ldots,u,v_k,\ldots,u)$. Since $e_k$ is colored $-b$ and $T$ passes through $u$ first and $v_k$ second, we see that no edge of $F$ is colored $-a$. If an edge of $F$ were colored $-a$, then $T$ would have to pass through it first in order to get to $e_k$, but $e_k$ is the first edge where $T$ and $F$ intersect. Thus, $T$ intersects $F$ at two edges, namely $e_k$ and $e_j$ (which is the final edge of $T$ and is colored $b$). Moreover, $T=(v_s,\ldots, u,v_k, \ldots, v_j , u)$. So, if $k >j$, we shift to $\gamma_j$ (which leaves $e_j$ uncolored and $-b$ absent at $v_k$) and perform the $a/b$-swap at $v_j$, which moves along the trail $(v_j,...,v_k)$ and terminates at $v_k$. We can then color $e_j$ with $a$ and we have finished. If $k <j$ then we shift to coloring $\gamma_k$, and perform the $-a/-b$-swap at $v_k$, which travels along $(v_k,...,v_j)$ and terminates at $v_j$. Then, we have $e_k$ uncolored with colors of the same magnitude absent at its endpoints. We apply Lemma \ref{fan} to extend the coloring.
In any case, we are able to color all of $\Sigma$, proving the theorem.
\end{proof}
\end{thm}
We will now present another theorem that allows us to deal with the color $0$. This theorem will be pivotal in the proof of Theorem \ref{goodvizing}. Given a signed graph $\Sigma$, we write $M(\Sigma)$ for the subgraph induced by all vertices of maximum degree.
\begin{thm}\label{indeplem} Suppose $\Sigma$ is a signed graph with $\Delta(\Sigma)$ even. If $M(\Sigma)$ is an independent set, then $\Sigma$ admits an edge coloring with $\Delta(\Sigma)$ colors.
\begin{proof} We proceed by induction on the number of vertices contained in $M(\Sigma)$. For a base case, suppose that $|M(\Sigma)|=1$. Delete an edge $e$ such that $e$ is incident with the vertex of maximum degree $u$. Then, $\Delta(\Sigma{\setminus}e)=\Delta(\Sigma)-1$ and hence there is a $\Delta(\Sigma)$-coloring of $\Sigma{\setminus}e$ by Theorem \ref{vizing}. We shall call this coloring $\gamma$.
Now, since $u$ is maximum degree, no neighbor of $u$ is maximum degree by assumption. Therefore, there is at least one absent color with respect to $\gamma$ at $u$ and at all neighbors of $u$. Thus, we are able to start a fan $F$ with initial edge $e=e_0$ and with $u$ as the hinge. From this point on, the proof is identical to that of Theorem \ref{vizing}---if colors of the same magnitude are absent at $u$ and $v_0$ then we apply Lemma \ref{fan}, and if not, we follow the process described in Theorem \ref{vizing} to extend the coloring.
Now we proceed by induction, supposing that $|M(\Sigma)| \geq 2$ and that the statement is true for all graphs with smaller sets of maximum degree vertices. Once again, let $e$ be an edge incident with a vertex of maximum degree $u$. By induction we obtain a $\Delta(\Sigma)$-coloring of $\Sigma{\setminus}e$. We can once again build a fan with hinge $u$ and initial edge $e=e_0$ in the same way as described above, mimicing the proof of Theorem \ref{vizing}.
\end{proof}
\end{thm}
Finally, we show how the bound of $\Delta(\Sigma)+2$ can be lowered in the case where $\Delta(\Sigma)$ is even. This will complete the proof of Theorem \ref{goodvizing}.
\begin{thm} \label{vizingzero}Suppose $\Sigma$ is a signed graph with $\Delta(\Sigma)$ even. Then $\chi'(\Sigma) \leq \Delta(\Sigma)+1$.
\begin{proof}Suppose that $\Delta(\Sigma)$ is even. Remove a maximal matching $N$ from $M(\Sigma)$. If $\Delta(\Sigma{\setminus}N) = \Delta(\Sigma)-1$, then Theorem \ref{vizing} gives a $\Delta(\Sigma)$-coloring of $\Sigma{\setminus}N$. If $\Delta(\Sigma{\setminus}N) = \Delta(\Sigma)$, then Theorem \ref{indeplem} gives a $\Delta(\Sigma)$-coloring of $\Sigma{\setminus}N$, since removing a maximal matching $N$ from $M(\Sigma)$ that does not cover all of $M(\Sigma)$ will leave $M(\Sigma{\setminus}N)$ as an independent set.
In either case, we have obtained a zero-free coloring of $\Sigma{\setminus}N$ using $\Delta(\Sigma)$ colors. We are now free to color every edge of $N$ with the color $0$, resulting in a proper coloring of $\Sigma$ using the prescribed number of colors.
\end{proof}
\end{thm}
In parallel with the ordinary Vizing's Theorem, the signed version of Vizing's Theorem partitions signed graphs into two classes. Indeed, we say that a signed graph $\Sigma$ is \emph{class $1$} if it admits a coloring that achieves the lower bound of $\Delta(\Sigma)$ colors. Otherwise, a signed graph is called \emph{class $2$}.
It is possible to produce two different signed graphs on the same underlying graph, one of which is class $1$ and the other class $2$---for example, positive and negative circles. However, every graph in $[\Sigma]$ has the same class as $\Sigma$. We define the \emph{class ratio} of an unsigned graph $\Gamma$ to be the number of signatures on $\Gamma$ such that the resulting signed graph is $\Delta$-colorable, divided by $2^m$, the number of possible signatures on $\Gamma$. The class ratio is denoted by $\mathcal{C}(\Gamma)$. Thus for example $\mathcal{C}(C_n)=1/2$, as only balanced circles are $2$-colorable. Since each switching class on $\Gamma$ contains the same number of signatures, $\mathcal{C}(\Gamma)$ can also be computed by counting the ratio of $\Delta$-colorable switching classes.
\subsection{Snarks}
In ordinary edge coloring, a \emph{snark} is a connected isthmus-free $3$-regular graph that does not admit an edge coloring with $3$ colors. Snarks are known for being quite hard to find, with relatively few examples known. A \emph{signed snark} is a connected isthmus-free $3$-regular signed graph that is not $3$-colorable. By Theorem \ref{goodvizing}, all signed snarks are $4$-colorable. Every unsigned snark corresponds to a signed snark on the same graph---simply sign all the edges as negative.
A natural question one might ask is the following: are the any connected isthmus-free $3$-regular graphs $\Gamma$ such that every signed graph on $\Gamma$ is a signed snark? We answer this question negatively.
\begin{lem}Let $\Gamma$ be an ordinary connected isthmus-free $3$-regular graph. There exists a signature $\sigma$ on $\Gamma$ such that $\Sigma=(\Gamma, \sigma)$ is $3$-colorable. In other words, $\mathcal{C}(\Gamma)>0$.
\begin{proof} We apply Petersen's theorem---every connected $3$-regular isthmus-free graph has a perfect matching. Let $M$ be a perfect matching in $\Gamma$. Then $\Gamma {\setminus} M$ is a $2$-regular graph---it is a union of circles. We choose $\sigma$ such that $\Gamma {\setminus}M$ is balanced, and we choose the signature of $M$ arbitrarily. Thus, we can color $\Sigma{\setminus}M$ with $\pm a$ and color $M$ with $0$, obtaining a $3$-coloring of $\Sigma$.
\end{proof}
\end{lem}
The opposite question to that posed above is also interesting: are there any connected $3$-regular isthmus-free graphs such that $\mathcal{C}(\Gamma)=1$?
Before giving an example, we introduce a helpful concept. The \emph{frustration index} of a signed graph $\Sigma$ is the minimum number of negative edges that occur over all signed graphs in $[\Sigma]$. Equivalently, the frustration index is the minimum number of edges that must be deleted to obtain a balanced signed graph. The \emph{maximum frustration} of a graph $\Gamma$ is the maximum frustration index over all possible signatures.
\begin{exm} Every signature of $K_{3,3}$ is $3$-colorable. In other words, $\mathcal{C}(K_{3,3})=1$.
\begin{proof} It sufficies to explain that any signature on $K_{3,3}$ contains a positive $6$-circle. The complement of the $6$-circle is a matching, so we can color the $6$-circle with $\pm a$ and the matching with $0$.
Indeed, let $\Sigma=(K_{3,3}, \sigma)$. It is known (see \cite{bowlin}) that the maximum frustration of $K_{3,3}$ is $2$. Thus, we assume that $\Sigma$ has $2$ or less negative edges. If $\Sigma$ has $0$ or $1$ negative edges, simply choose a matching that contains them. The complement of this matching is a balanced $6$-circle. If $\Sigma$ has $2$ negative edges they must be non-adjacent, since if they are adjacent we can switch to a signature with $1$ negative edge. There is a perfect matching containing any two non-adjacent edges of $K_{3,3}$.
\end{proof}
\end{exm}
In fact there is nothing particularly special about $K_{3,3}$ here---if $\Gamma$ has maximum frustration $2$ and if there is a perfect matching covering any two given edges, then every signature of $\Gamma$ is $\Delta$-colorable. Thus $\mathcal{C}(K_4)=1$, for example.
We close this section by giving a bound on the class ratio for Hamiltonian connected $3$-regular isthmus-free graphs.
\begin{exm} If $\Gamma$ is a Hamiltonian connected $3$-regular isthmus-free graph, then $\mathcal{C}(\Gamma) \geq 1/2$.
\begin{proof} Let $H$ be a Hamilton circle in $\Gamma$. Then the complement of $H$ is a perfect matching. We note that $H$ is positive in precisely half of the possible signatues on $\Gamma$. So, $H$ can be $2$-colored in precisely half the signatures. Thus, $\Gamma$ is $3$-colorable in at least half of its possible signatures.
\end{proof}
\end{exm}
The above argument can be generalized. If $\Gamma$ contains a $2$-regular spanning subgraph $K$ with $k$ components, then $K$ is balanced in exactly $1/2^k$ of the possible signatures. Thus in this case $\mathcal{C}(\Gamma) \geq 1/2^k$.
\subsection{Class Ratio of Complete Graphs}
It is desirable to calculate $\mathcal{C}$ for certain classes of well known graphs. Here we briefly mention one of the simplest possibilies, $\mathcal{C}(K_n)$. We have already seen that $\mathcal{C}(K_2)=1$, $\mathcal{C}(K_3)=1/2$, and $\mathcal{C}(K_4)=1$. Based on this evidence one might guess that $\mathcal{C}(K_n) = 1$ if $n$ is even, and $\mathcal{C}(K_n)=1/2$ if $n$ is odd. However, in reality the situation is not quite this simple.
First, we note that if $n$ is odd then $\mathcal{C}(K_n)\leq1/2$. This is because any $\Delta$-coloring of $K_n$ (where $n$ is odd) is a decomposition of $K_n$ into $\Delta/2$ balanced $2$-regular spanning subgraphs. If the signature has an odd number of negative edges, then some circle in the decomposition must contain an odd number of negative edges and hence be unbalanced. Exactly half of all possible signatures have an odd number of negative edges.
In order to prove that $\mathcal{C}(K_n)\geq1/2$ when $n$ is odd it would suffice to show that any signature with an even number of negative edges can be $\Delta$-colored. However, this is not true---consider $K_5$ with an all-negative signature. This signature has $10$ negative edges, but a decomposition of $K_5$ into two balanced spanning subgraphs must be a decomposition into two circles of length $5$. Neither of these circles will be positive as they each contain $5$ negative edges. Thus, it is in fact true that $\mathcal{C}(K_5) < 1/2$. We leave it as an open problem to determine a precise formula for $\mathcal{C}(K_n)$.
\section{Line Graphs}
In this section we show that every signed edge coloring can be realized as a vertex coloring of a signed line graph. This is a desirable property for signed edge coloring to possess, since unsigned edge coloring posseses the very same property. We recall that a \emph{vertex coloring} of $\Gamma$ is an assignment of a color to each of the vertices of $\Gamma$. Such a coloring is \emph{proper} if no two adjacent vertices have the same color.
To vertex color a signed graph, we assign a color from the set $M_n$ to each of its vertices. We employ the definition of propriety discovered by Zaslavsky \cite{zaslav2}---a \emph{proper signed vertex coloring} has the requirement that positive edges do not have the same color at their endpoints, and negative edges do not have colors with the same magnitude and opposite sign at their endpoints.
One of the nice features of this definition is that it extends to switching classes in a natural way. If $c$ is a proper vertex coloring of $\Sigma$ and $\Sigma \sim \Sigma'$ via switching vertex set $X$, then we can obtain a proper coloring $c'$ of $\Sigma'$ by simply negating $c(x)$ for all $x \in X$. In this way $c$ generates a proper vertex coloring for each member of $[\Sigma]$.
\subsection{Bidirected Graphs}
The easiest way to define the line graph of a signed graph is through the use of bidirected graphs. A \emph{bidirected graph} is a pair $(\Gamma, \tau)$, where $\Gamma$ is a graph and $\tau:I(\Gamma)\rightarrow \{+,-\}$ is a \emph{bidirection}. When $\tau(v,e)=+$ we imagine an arrow drawn on $e$ that points into $v$, and when $\tau(v,e)=-$ we imagine an arrow drawn on $e$ that points away from $v$. An edge $e$ is \emph{extraverted} if both of its $\tau$ values are $+$, \emph{introverted} if both of its $\tau$ values are $-$, and \emph{coherent} otherwise.
A negation of the $\tau$ values for a certain edge $e$ is a \emph{reorientation} of $e$. Thus a reorientation of an extraverted edge is an introverted edge and vice versa, while a reorientation of a coherent edge remains coherent. Reorientation is an equivalence relation on bidirected graphs, and hence there is a partition of the set of bidirected graphs into \emph{reorientation classes}. The reorientation class of a given bidirected graph $B$ is denoted by $\vec{B}$.
Bidirected graphs can be thought of as orientations of signed graphs. For a given bidirected graph $B=(\Gamma, \tau)$, there is a natural associated signed graph $\Sigma_B = (\Gamma, \sigma_{\tau})$, obtained by setting $\sigma_{\tau}(e) = -\tau(v,e)\tau(w,e)$ for $e{:}{vw}$. We say that $B$ is an \emph{orientation} of $\Sigma_B$. In other words, positive edges correspond to coherent edges, and negative edges correspond to extraverted and introverted edges.
A signed graph with $m$ edges has $2^m$ possible orientations, and each of these orientations can be obtained from any other by reorientation of the appropriate edges. Thus, if $\Sigma$ has a single orientation $B$, then $\vec{B}$ contains exactly the $2^m$ orientations of $\Sigma$. See Figure \ref{fig10} for a picture of a signed graph and an orientation.
In light of the fact that bidirected graphs are orientations of signed graphs, we may use the terminology of signed graphs to refer to bidirected graphs when it is not confusing to do so. For example, a subgraph of a bidirected graph is \emph{balanced} if its correspoding signed subgraph is balanced.
A vertex $v$ of a bidirected graph can be \emph{switched} by negating all $\tau$ values involving $v$. It is easy to check that bidirected switching is compatible with signed switching as far as orientations go---switching both $B$ and $\Sigma_B$ at $v$ will result in $B'$ and $\Sigma_B'$ such that $B'$ is an orientation of $\Sigma_B'$.
\begin{figure}[h!]
\centering
\includegraphics[scale=.064]{image10}
\caption[A signed graph $\Sigma$ and one of its orientations $B$.]{On the left, a signed graph $\Sigma$. On the right, $B$, one of its $32$ possible orientations. \label{fig10} }
\end{figure}
\subsection{Coloring Bidirected Graphs}
The definition of edge coloring for a signed graph cooperates nicely with bidirected graphs. We define an edge coloring of a bidirected graph in the following way. Notice that our definition is in terms of edges rather than in terms of incidences.
\begin{defn}An $n$-edge coloring (or more briefly, an $n$-coloring) $\gamma$ of a bidirected graph $B$ is an assignment of colors from $M_n$ to each edge of $B$. Such a coloring is \emph{proper} if $\tau(v,e)\gamma(e)\neq \tau(v,f) \gamma(f)$ for all edges $e$ and $f$ that are adjacent at vertex $v$.
\end{defn}
Thus in a proper coloring if the $\tau$ values are equal the edges may not have the same color, while if they are not equal the edges may not have opposite colors.
The purpose of this definition is to enable us to view an edge coloring of $\Sigma$ as an edge coloring of one of its orientations.
\begin{lem}\label{orientcolor}Suppose $\Sigma$ is a signed graph and $B$ is one of its orientations. Suppose $\gamma$ is an edge coloring of $\Sigma$. Then there is a unique edge coloring $\gamma_B$ of $B$ such that $\tau(v,e)\gamma_B(e)=\gamma(v,e)$ for all $v$ and incident $e$.
\begin{proof} We describe how to define $\gamma_B$ for each edge of $B$. Take any edge $e$ and let $v$ be one of its endpoints. We set $\gamma_B(e)=\tau(v,e)\gamma(v,e)$. Thus, $\tau(v,e)\gamma_B(e)=\gamma(v,e)$. We now must check the other endpoint of $e$. Let $w$ be the other endpoint of $e$. Then $\tau(w,e)\gamma_B(e) = -\sigma(e)\tau(v,e)\gamma_B(e)=-\sigma(e)\gamma(v,e)=\gamma(w,e)$.
\end{proof}
\end{lem}
Furthermore, if we are given an edge coloring $\gamma_B$ of $B$, we can uniquely recover the edge coloring $\gamma$ of $\Sigma$ by setting $\gamma(v,e)=\tau(v,e)\gamma_B(e)$.
The following lemma shows that a coloring of one orientation automatically generates a coloring of every other possible orientation.
\begin{lem}\label{orientcolor2} Suppose $\gamma_B$ is an edge coloring of $B$. Let $B'$ be a reorientation of $B$. Then there is a unique edge coloring $\gamma_{B'}$ such that $\gamma_{B'}(e) \tau'(v,e) = \gamma_B(e) \tau(v,e)$ for all incidences $(v,e)$.
\begin{proof} Suppose $e$ is an edge of $B$, and suppose that $e$ is one of the edges that gets reoriented when passing from $B$ to $B'$. In this case we define $\gamma_{B'}(e):= -\gamma_B(e)$. If $e$ is not reoriented, we set $\gamma_{B'}(e) = \gamma_B(e)$. Either way, $\gamma_{B'}(e) \tau'(v,e) = \gamma_B(e) \tau(v,e)$.
\end{proof}
\end{lem}
Suppose $\Sigma$ is a signed graph with two different orientations $B$ and $B'$. Let $\gamma$ be an edge coloring of $\Sigma$, and let $\gamma_{B}$ and $\gamma_{B'}$ be the corresponding edge colorings of $B$ and $B'$ in the sense of Lemma \ref{orientcolor}. Then, Lemma \ref{orientcolor2} tells us that we can obtain $\gamma_{B}$ from $\gamma_{B'}$ by negating the colors on the edges that must be reoriented to change from $B$ to $B'$.
Thus, $\gamma$ uniquely determines $\gamma_{B}$ for every orientation $B$ of $\Sigma$, and conversely $\gamma_{B}$ determines all other $\gamma_{B'}$, each of which determine $\gamma$.
The following figure illustrates Lemma \ref{orientcolor} and Lemma \ref{orientcolor2}.
\begin{center}
\begin{figure}[h!]\label{fig11}
\centering
\includegraphics[scale=.056]{image11}
\caption[$\Sigma$, two of its orientations, and two edge colorings.]{From left to right; $\Sigma$ along with an edge coloring, one of its orientations $B$ with the corresponding bidirected edge coloring, and a second orientation $B'$ with its corresponding edge coloring. We can pass from $B$ to $B'$ by negating colors on the edges that have been reoriented.}
\end{figure}
\end{center}
\subsection{Defining Line Graphs}
First, we present the definition of the signed line graph, as originally studied by Zaslavsky \cite{zaslav4}. Recall that the \emph{line graph} of an ordinary graph $\Gamma$ is the graph $L(\Gamma)$ whose vertices are the edges of $\Gamma$, and two vertices of $L(\Gamma)$ are adjacent in $L(\Gamma)$ if and only the corresponding edges are adjacent in $\Gamma$. Thus, $L(\Gamma)$ is the graph of edge adjacency for $\Gamma$. If $e$ is an edge of $\Gamma$, we write $\ell_e$ to represent the corresponding vertex in $L(\Gamma)$.
\begin{defn} The \emph{line graph} of a bidirected graph $B=(\Gamma,\tau)$ is $L(B):=(L(|B|),\vec{\tau})$, where $\vec{\tau}(\ell_e,\ell_e\ell_f):=\tau(v,e)$ (where $v$ is the common vertex of edges $e$ and $f$ in $B$).
\end{defn}
Thus for example the line graph of an all-extraverted $B$ is itself all extraverted. The main purpose of the bidirected line graph is to act as a tool that enables us to define the line graph of a signed graph. Before proceeding with the definition, we point out that the line graph of $\Sigma$ turns out to be a switching class of signed graphs, rather than a single signed graph.
\begin{defn}[Signed Line Graph]\label{slg} The \emph{line graph} of a signed graph $\Sigma$ is obtained by the following procedure:
\begin{enumerate}
\item Choose any orientation $B$ of $\Sigma$.
\item Find $L(B)$, the bidirected line graph of $B$.
\item Find the signed graph corresponding to $L(B)$. Denote this by $\Sigma_{L(B)}$.
\item The line graph of $\Sigma$ is defined as the switching class of $\Sigma_{L(B)}$. We write $\Lambda(\Sigma) = [\Sigma_{L(B)}]$.
\end{enumerate}
\end{defn}
It is important to notice that the above definition does not depend on the choice of $B$. Indeed, reorienting an edge $e$ of $B$ will have the effect of switching the vertex $\ell_e$ in $L(B)$. Since $\Lambda(\Sigma)$ is a switching class, reorienting $e$ does not change the line graph. An illustration of Definition \ref{slg} is given in Figure \ref{fig12}.
Not only is the line graph of $\Sigma$ a switching class, but every signed graph switching equivalent to $\Sigma$ has the same line graph as $\Sigma$.
\begin{lem} If $\Sigma \sim \Sigma'$, then $\Sigma$ and $\Sigma'$ have the same line graph.
\begin{proof} We switch a single vertex $v$ of $\Sigma$ and observe what effect it has on the line graph. Let $e$ and $f$ be two edges that are adjacent at $v$. Indeed, switching $v$ has the effect of negating all $\tau$ values at $v$ for any orientation $B$ of $\Sigma$. Thus, in $L(B)$, the edge $\ell_e\ell_f$ gets reoriented. This reorientation has no effect on $\Lambda(\Sigma)$, and hence does not change the line graph.
\end{proof}
\end{lem}
\begin{center}
\begin{figure}[h!]
\centering
\includegraphics[scale=.063]{image12}
\caption[A signed graph $\Sigma$, one of its orientations $B$, the bidirected line graph $L(B)$ of $B$, and the signed graph corresponding to $L(B)$, $\Sigma_{L(B)}$.]{A signed graph $\Sigma$, one of its orientations $B$, the bidirected line graph $L(B)$ of $B$, and the signed graph corresponding to $L(B)$, $\Sigma_{L(B)}$. The line graph of $\Sigma$ is $\Lambda(\Sigma)=[\Sigma_{L(B)}]$, the switching class of $\Sigma_{L(B)}$. A reorientation of one of the edges of $B$ will switch the corresponding vertex in $L(B)$, which switches the corresponding vertex in $\Sigma_{L(B)}$.\label{fig12}}
\end{figure}
\end{center}
\subsection{Edge Coloring in Terms of Vertex Coloring}
We will now study how one can interpret an edge coloring of $\Sigma$ in terms of a vertex coloring of the line graph of $\Sigma$. It turns out that an edge coloring of $\Sigma$ corresponds with a vertex coloring (in Zaslavsky's sense) of $[-\Sigma_{L(B)}]$, the negative of the line graph of $\Sigma$.
\begin{thm}\label{lineg} There is a bijection between (proper) edge colorings of $\Sigma$ and (proper) vertex colorings (in Zaslavsky's sense) of $[-\Sigma_{L(B)}]=-\Lambda(\Sigma)$, the negative of the line graph of $\Sigma$.
\begin{proof} Let $\gamma$ be a (proper) edge coloring of $\Sigma$. We will describe how to define a (proper) vertex coloring $c$ of $-\Lambda(\Sigma)$ in terms of $\gamma$.
Choose any orientation $B$ of $\Sigma$, using $\gamma$ to induce the unique edge coloring $\gamma_B$ of $B$. Since the edges of $B$ are the vertices of $L(B)$, we can think of $\gamma_B$ as a vertex coloring of $L(B)$. In turn, we think of $\gamma_B$ as a vertex coloring of $-\Lambda(\Sigma)$. Thus, we have a bijection between (not necessarily proper) edge colorings of $\Sigma$ and vertex colorings of $-\Lambda(\Sigma)$.
We now wish to prove that $\gamma$ is a proper edge coloring of $\Sigma$ if and only if $\gamma_B$ is a proper vertex coloring of $-\Lambda(\Sigma)$. Indeed, let edges $e$ and $f$ be adjacent at vertex $v$ in $\Sigma$. Since $\gamma$ is proper, $\gamma(v,e)\neq \gamma(v,f)$. Equivalently, in $B$, $\gamma_B(e) \tau(v,e) \neq \gamma_B(f) \tau(v,f)$. Thus, equivalently in $L(B)$, the previous rule becomes $\gamma_B(\ell_e) \tau(\ell_e, \ell_e\ell_f) \neq \gamma_B(\ell_f) \tau(\ell_f, \ell_e\ell_f)$. Thus, when passing to $-\Lambda(\Sigma)$, we have $\gamma_B(\ell_e) \neq \sigma(\ell_e \ell_f) \gamma_B(\ell_f)$, which is precisely the definition of a proper vertex coloring.
\end{proof}
\end{thm}
\begin{center}
\begin{figure}[h!]
\centering
\includegraphics[scale=.063]{image13}
\caption[A proper edge coloring of $\Sigma$ and the corresponding vertex coloring of its line graph.]{A proper edge coloring of $\Sigma$, the corresponding proper edge coloring of $B$, the corresponding vertex coloring of $L(B)$, and the corresponding proper vertex coloring of $-\Sigma_{L(B)}$. \label{fig13}}
\end{figure}
\end{center}
Figure \ref{fig13} shows a signed graph along with an edge coloring and the corresponding vertex coloring of the negative of its line graph. The intermediate bidirected graph steps are shown as well.
Theorem \ref{lineg} specializes to ordinary graphs nicely when $\Sigma$ is all-negative. Indeed, if $\Sigma$ is all-negative we can choose $B$ so that it is all-extraverted. Then $L(B)$ is also all-extraverted, so that $-\Sigma_{L(B)}$ is all-positive. Thus, edge colorings of an all-negative $\Sigma$ correspond to vertex colorings of an all-positive $-\Sigma_{L(B)}$. This is what we expect, since edge colorings of an all-negative $\Sigma$ correspond to ordinary edge colorings, and vertex colorings of an all-positive $-\Sigma_{L(B)}$ correspond to ordinary vertex colorings.
\section{Additional Topics}
\subsection{Reversibility and The Linear Arboricity Conjecture}
The \emph{linear arboricity} $\la(\Gamma)$ of an ordinary simple graph is the minimum number of linear forests (acyclic subgraphs of maximum degree $2$) into which its edges can be partitioned. The linear arboricity of $\Gamma$ is at least $\lceil \Delta / 2 \rceil$ since each linear forest in such a partition uses at most two edges incident with a vertex of maximum degree. In 1981, Akiyama et al.\ conjectured \cite{lineararb} that the linear arboricity of a graph is bounded above by $\lceil (\Delta+1) / 2 \rceil$, a conjecture that remains unresolved at present. Since $\lceil \Delta/2 \rceil$ and $\lceil (\Delta +1) /2 \rceil$ are either equal or consecutive integers, the linear arboricity conjecture states that $\la(\Gamma)$ is either $\lceil \Delta/2 \rceil$ or $\lceil (\Delta +1 )/2 \rceil$---a dichotomous statement that is reminiscent of Vizing's Theorem. As we are about to see, the problem of computing Linear Arboricity can be naturally phrased in terms of signed graph edge coloring.
Suppose $\Sigma$ is a signed graph, and $\gamma$ is a proper edge coloring. We say that an edge $e$ is \emph{reversible with respect to $\gamma$} if $e$ lies in a path component of $\Sigma_{\gamma(e)}$. If $e$ is reversible with respect to $\gamma$, we can negate the sign of $e$ and easily find another proper coloring $\gamma \,'$ such that the magnitudes of the colors in $\gamma$ are the same as those in $\gamma \,'$. This occurs because we can always color any bidirected path with only two colors.
We say that the proper coloring $\gamma$ is \emph{completely reversible} if every edge of $\Sigma$ is reversible with respect to $\gamma$. The following two lemmas should now be evident.
\begin{lem} Let $\Sigma$ be a signed graph. A proper edge coloring $\gamma$ is completely reversible if and only if every component of every magnitude graph is a path.
\end{lem}
\begin{lem} If $\Sigma=(\Gamma,\sigma)$ admits a completely reversible $n$-coloring, then so does every signed graph with underlying graph $\Gamma$.
\end{lem}
Thus, in a completely reversible coloring each $\Sigma_a$ is a linear forest, and conversely any partition of the edges of $\Sigma$ into linear forests yields a completely reversible coloring. We note that if $a=0$ this linear forest is a matching (not an arbitrary linear forest), so we look only at zero-free colorings. We write $\chi'_R(\Sigma)$ for the minimum number of colors needed in a completely reversible zero-free proper coloring of $\Sigma$. We can now phrase the linear arboricity conjecture in terms of edge coloring.
\begin{cnj}[Linear Arboricity Conjecture] For any simple signed graph $\Sigma$, $\Delta(\Sigma) \leq \chi'_R(\Sigma) \leq \Delta(\Sigma) +2$.
\end{cnj}
This is equivalent to the linear arboricity conjecture since each $\Sigma_a$ requires two colors but is one linear forest.
\subsection{A Change of Convention}
If we tweak our definition of signed edge coloring slightly we end up with $\Sigma_a$ graphs that differ from those we see normally. In fact, the change that we make to the definition will turn the $\Sigma_a$ into arbitrary antibalanced subgraphs, which links edge coloring to a problem that has already been studied---balanced decomposition.
The change is this: when we define a proper edge coloring, we insist that $\gamma(v,e) \neq -\gamma(v,f)$ for all edges $e$ and $f$ adjacent at vertex $v$. This is the negative of the usual definition, so let us call such a coloring \emph{antiproper}. In the language of orientations, an antiproper coloring is one that satisfies $\gamma(e) \tau(v,e) \neq - \gamma(f) \tau(v,f)$.
We first note that the Vizing-style lower bound of $\Delta(\Sigma)$ does not hold for an antiproper coloring, since every incidence at a given vertex may be colored the same. However, despite this, the magnitude graphs $\Sigma_a$ still do have some nice structure.
\begin{lem} Let $\gamma$ be an antiproper edge coloring of $\Sigma$. Then each $\Sigma_a$ graph is antibalanced. Furthermore, any antibalanced graph can be antiproperly colored with $2$ colors ($\pm a$).
\begin{proof} Let $C$ be a circle in $\Sigma_a$ and suppose $C$ has $m$ edges. Switch $\Sigma$ so that $C$ consists of a negative path of length $m-1$, and one additional edge $e$ that is either positive or negative depending on the sign of $C$. Since $\gamma$ is antiproper, every edge in the path $C{\setminus}e$ must have both its incidences colored (without loss of generality) $a$. Thus, $e$ must be negative as well, or else we do not have an antiproper coloring. Hence, $C$ switches to all negative and so $\Sigma_a$ is antibalanced.
To prove the other statement, let $A$ be an arbitrary antibalanced signed graph. Switch $A$ to all negative and color every incidence as $a$. This gives a $2$-coloring (not a $1$-coloring, since we must use $M_2=\{-a,a\}$).
\end{proof}
\end{lem}
Thus, an antiproper coloring of $\Sigma$ is equivalent to a decomposition of $\Sigma$ into antibalanced subgraphs. These antibalanced subgraphs are arbitrary (except in the case of $\Sigma_0$, which must be a matching). In turn, a decomposition of $\Sigma$ into antibalanced subgraphs corresponds to a decomposition of $-\Sigma$ into balanced subgraphs.
In \cite{zaslav1}, Zaslavsky studies the \emph{balanced decomposition number} of $\Sigma$---the smallest number of balanced sets into which its edges can be partitioned, denoted by $\delta_0(\Sigma)$. The balanced decomposition number is a parameter that encapsulates how far a given signed graph is from being balanced---the higher the balanced decomposition number, the ``less balanced'' $\Sigma$ is.
The balanced decomposition number is a generalization of the \emph{biparticity} $\beta_0$ of an unsigned graph---the fewest number of bipartite sets into which the edges can be partitioned. In particular, $\beta_0(\Gamma) = \delta_0(-\Gamma)$ (here $-\Gamma$ means an all negatively signed $\Gamma$), since the balanced subgraphs in an all negative graph are precisely the bipartite subgraphs. Biparticity is known to be connected to the chromatic number $\chi(\Gamma)$ by the formula
$$\beta_0(\Gamma) = \lceil \log_2(\chi(\Gamma)) \rceil,$$
discovered independently by Harary-Hsu-Miller \cite{bipar1} and Matula \cite{bipar2}.
A similar theorem for $\delta_0$ was given by Zaslavsky in \cite{zaslav1}.
\begin{thm}[Zaslavsky's Balanced Decomposition Theorem] \label{zaslav2} If $\Sigma$ has at least one edge, $\delta_0(\Sigma)=\lceil \log_2( \chi^*(-\Sigma))\rceil$, where $\chi^*(\Sigma)$ is the zero-free vertex chromatic number of $\Sigma$.
\end{thm}
Let us write $\chi_A'$ for the minimum number of colors needed in any zero-free antiproper coloring. Then Zaslavsky's Theorem immediately gives $\chi_A'(\Sigma) = 2\delta_0(-\Sigma)=2\lceil \log_2( \chi^*(\Sigma))\rceil$. The reason for the multiplication by $2$ is the fact that in any antibalanced decomposition, each antibalanced set requires two colors.
We close this section by offering an interesting interpretation of antiproper colorings in terms of a line graph. In contrast with proper colorings, we do not have to negate the line graph to obtain the correspondence. The proof is evident from our discussion of signed line graphs.
\begin{thm} Antiproper edge colorings of $\Sigma$ correspond to proper vertex colorings of the line graph $\Lambda(\Sigma)$.
\end{thm}
\subsection{Total Coloring} In this section we will discuss how to define total coloring for a signed graph. There are a couple of ways that we can do this, both of which are interesting in their own right.
Recall that a \emph{total coloring} of an ordinary graph $\Gamma$ is an assignment of colors to its vertices and edges such that no two adjacent vertices, adjacent edges, or incident vertices and edges share a color. In other words, a total coloring is simultaneously a proper vertex coloring and a proper edge coloring, and the interaction between them is that incident vertices and edges also do not share a color.
We would like to make a similar definition for signed graphs, and ideally, our definition should specialize to the ordinary definition in some way. However, there is a subtlety that we must deal with---signed vertex colorings correspond to ordinary vertex colorings when $\Sigma$ is balanced, but signed edge colorings correspond to ordinary edge colorings when $\Sigma$ is antibalanced. Thus, we are compelled to make the following definition:
\begin{defn} A \emph{signed total coloring} $\mu$ of $\Sigma$ is an assignment of colors from $M_n$ to the vertices and incidences of $\Sigma$ such that:
\begin{enumerate}
\item $\mu$ restricted to $V(\Sigma)$ is a proper vertex coloring of $-\Sigma$.
\item $\mu$ restricted to $I(\Sigma)$ is a proper edge coloring.
\item $\mu(v) \neq \mu(v,e)$ for all incident vertices and edges $v$ and $e$.
\end{enumerate}
\end{defn}
Thus, a signed total coloring of an all-negative $\Sigma$ corresponds to a total coloring of $|\Sigma|$, which is what we desire. Since signed vertex and edge coloring are both compatible with switching, signed total coloring is too.
\begin{lem} Let $\mu$ be a signed total coloring of $\Sigma$. Let $\Sigma' \sim \Sigma$ via switching function $\eta$. Then $\mu'$ is a total coloring of $\Sigma'$, where $\mu'$ is obtained from $\mu$ by negating the colors on all vertices and incidences that were switched via $\eta$.
\begin{proof} This follows immediately from the compatibility of vertex and edge coloring with switching.
\end{proof}
\end{lem}
Let $\chi''(\Sigma)$ be the total chromatic number---the fewest number of colors used in any signed total coloring of $\Sigma$. It is easy to notice that $\chi''(\Sigma) \geq \Delta(\Sigma) +1$, as a maximum degree vertex requires $\Delta$ different colors on its incident edges and one additional color for itself.
It is natural to look for a Vizing-style upper bound for $\chi''$, and indeed for ordinary graphs it has been long conjectured that the upper bound is $\Delta +2$, although no proof has been found. This is known as the \emph{total coloring conjecture}, first posed by Behzad. Interestingly, many people attribute this conjecture to Vizing, but according to Shahmohamad \cite{shah}, Behzad is indeed the sole author of the conjecture. The upper-bound of $\Delta+2$ is known to hold for some specific classes of ordinary graphs, such as $r$-partite graphs \cite{yap}, $3$-regular graphs \cite{rosen}, and most planar graphs.
Based on our results concerning signed Vizing's theorem, we conjecture the following.
\begin{cnj}[Signed Total Coloring Conjecture] Any simple signed graph $\Sigma$ admits a total coloring with $\Delta(\Sigma)+2$ colors.
\end{cnj}
It is left as an open problem to prove various special cases of this conjecture, such as $r$-partite graphs and $3$-regular graphs.
Let us revisit the definition of total coloring. In the definition, we insisted on having a proper vertex coloring of $-\Sigma$ so that total coloring would specialize for antibalanced graphs. Instead, let us do the following:
\begin{defn} A \emph{twisted signed total coloring} $\mu$ of $\Sigma$ is an assignment of colors from $M_n$ to the vertices and incidences of $\Sigma$ such that:
\begin{enumerate}
\item $\mu$ restricted to $V(\Sigma)$ is a proper vertex coloring of $\Sigma$.
\item $\mu$ restricted to $I(\Sigma)$ is a proper edge coloring.
\item $\mu(v) \neq \mu(v,e)$ for all incident vertices and edges $v$ and $e$.
\end{enumerate}
\end{defn}
The difference in definitions is only a single negative sign, but something interesting happens. If $\Sigma$ is all-negative, a twisted total coloring corresponds to an ordinary edge coloring of $|\Sigma|$. However, if $\Sigma$ is all-positive, a twisted total coloring corresponds to an ordinary vertex coloring of $|\Sigma|$. Thus, a twisted total coloring specializes in two different ways, but not necessarily at the same time. In fact, both ways may happen at the same time.
\begin{lem} A twisted total coloring of $\Sigma$ corresponds to a total coloring of $|\Sigma|$ if and only if $\Sigma$ is both balanced and antibalanced (i.e., if and only if $\Sigma$ is balanced and bipartite).
\begin{proof} We require $\Sigma$ to switch to both all-positive and all-negative.
\end{proof}
\end{lem}
| {
"timestamp": "2018-12-05T02:16:09",
"yymm": "1807",
"arxiv_id": "1807.11465",
"language": "en",
"url": "https://arxiv.org/abs/1807.11465",
"abstract": "We define a method for edge coloring signed graphs and what it means for such a coloring to be proper. Our method has many desirable properties: it specializes to the usual notion of edge coloring when the signed graph is all-negative, it has a natural definition in terms of vertex coloring of a line graph, and the minimum number of colors required for a proper coloring of a signed simple graph is bounded above by {\\Delta} + 1 in parallel with Vizing's Theorem. In fact, Vizing's Theorem is a special case of the more difficult theorem concerning signed graphs.",
"subjects": "Combinatorics (math.CO)",
"title": "Edge Coloring Signed Graphs",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754452025766,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.7075103939855257
} |
https://arxiv.org/abs/1802.02072 | Local Energy Optimality of Periodic Sets | We study the local optimality of periodic point sets in $\mathbb{R}^n$ for energy minimization in the Gaussian core model, that is, for radial pair potential functions $f_c(r)=e^{-c r}$ with $c>0$. By considering suitable parameter spaces for $m$-periodic sets, we can locally rigorously analyze the energy of point sets, within the family of periodic sets having the same point density. We derive a characterization of periodic point sets being $f_c$-critical for all $c$ in terms of weighted spherical $2$-designs contained in the set. Especially for $2$-periodic sets like the family $\mathsf{D}^+_n$ we obtain expressions for the hessian of the energy function, allowing to certify $f_c$-optimality in certain cases. For odd integers $n\geq 9$ we can hereby in particular show that $\mathsf{D}^+_n$ is locally $f_c$-optimal among periodic sets for all sufficiently large~$c$. | \section{Introduction}
Point configurations which minimize energy for a given pair
potential function occur in diverse branches of mathematics and its applications.
There are various numerical approaches to find locally stable configurations.
However, in general, proving optimality of a point configuration
appears hardly possible, except maybe for some very special sets.
In \cite{MR2257398} Cohn and Kumar introduced the notion of a {\em universally
optimal point configuration}, that is, a set of points in a given space,
which minimizes energy for all completely monotonic potential functions.
There exist several fascinating examples among spherical point sets.
However, considering infinite point sets in Euclidean spaces is more
difficult. Even a proper definition of potential energy bears subtle convergence problems.
For \textit{periodic sets} such problems can be avoided,
so that these point configurations are the ones usually considered in
the Euclidean setting.
When working with local variations of periodic sets it is convenient to
work with a parameter space up to translations and orthogonal
transformations, as introduced in \cite{MR2466406}.
With it, a larger experimental study of energy minima
among periodic sets in low dimensions ($n\leq 9$)
was undertaken
in the {\em Gaussian core model}, that is,
for potential functions $f_c(x) := e^{ - c \, x}$, with $c > 0$
(see \cite{PhysRev09}).
These experiments support a conjecture of Cohn and Kumar that the
hexagonal lattice $\mathsf{A}_2$ in dimension~$2$ and the root lattice $\mathsf{E}_8$
in dimension~$8$ are universally optimal among periodic sets in their dimension.
Somewhat surprising, the numerical experiments also suggest that the
root lattice $\mathsf{D}_4$ in dimension $4$ is universally optimal.
Since proving global optimality seemed out of reach,
we considered a kind of local universal optimality among periodic sets
in \cite{MR2889159}.
We showed that lattices whose shells are spherical $4$-designs and which
are locally optimal among lattices can not locally be improved to another
periodic set with lower energy.
By a result due to Sarnak and Str\"ombergsson \cite{MR2221138}, this implies
local universal optimality among periodic sets
for the lattices $\mathsf{A}_2$, $\mathsf{D}_4$ and $\mathsf{E}_8$, as well as for the
exceptional Leech lattice $\Lambda_{24}$.
A corresponding result for the ``sphere packing case'' $c\to \infty$
is shown in~\cite{MR3074813}.
In all other dimensions the situation is much less clear.
In dimension $3$, for instance, there is a small intervall
for $c$ with a {\em phase transition}, for which periodic point-configurations
seem not to minimize energy at all.
For all larger~$c$ the fcc-lattice (also known as $\mathsf{D}_3$) and for all smaller~$c$ the
bcc-lattice (also known as $\mathsf{D}^\ast_3$) appear to be energy minimizers.
Similarly, there appear to be no universal optima in dimensions $5$, $6$ and~$7$.
Contrary to a conjecture of Torquato and
Stillinger from 2008~\cite{PhysRev08}, there even
seem to be various non-lattice configurations which
minimize energies in each of these dimensions.
Quite surprising, the situation appears to be very different in dimension~$9$:
According to our numerical experiments
it is possible that there exists a universally optimal $2$-periodic
(non-lattice) set in dimension~$9$. This set, known as $\mathsf{D}^+_9$,
is a union of two translates of the root lattice $\mathsf{D}_9$.
From the viewpoint of energy minimization, respectively our numerical experiments,
$\mathsf{D}^+_9$ seems almost of a similar nature as the
exceptional lattice structures $\mathsf{E}_8$ and $\Lambda_{24}$.
However, its shells are only spherical $3$-designs (and not
$4$-designs), which makes a major difference for our proofs.
The purpose of this paper is to shed more light onto
the energy minimizing properties of $\mathsf{D}^+_9$
and similar periodic non-lattice sets that might exist in other dimensions.
Here, we in particular derive criteria for $f_c$-critical
periodic point sets (Theorem~\ref{th:periodic-critical}) and we show that
$\mathsf{D}^+_9$ is locally $f_c$-optimal for all sufficiently
large~$c$ (Theorem~\ref{thm:final}).
\medskip
Our paper is organized as follows: In Section~\ref{sec:pps} we collect
some necessary preliminary remarks on periodic sets, in particular
about their representations, their symmetries
and attached average theta series.
In Section~\ref{sec:two} we define the $f$-potential energy of a
periodic set and show how it can be expanded in the neighborhood of
a given $m$-periodic representation. Section~\ref{sec:critical} gives necessary and
sufficient conditions for a periodic set to be an $f_c$-critical
configuration for all $c>0$.
We provide a simplification for the expression of energy for
the special case of $2$-periodic sets in Section~\ref{sec:2periodic}.
This can in particular be applied
to the sets $\mathsf{D}^+_n$, which we describe in more
detail in Section~\ref{sec:four}.
In Section~\ref{sec:hessian} we obtain all necessary ingredients
to show that $\mathsf{D}^+_n$ for odd $n\geq 9$ is locally
$f_c$-optimal for all sufficiently large $c$.
In our concluding Section~\ref{sec:conclusion} we also explain how this
result could possibly be extended, to prove at least locally a kind of
universal optimality of the set~$\mathsf{D}^+_9$.
\section{Preliminaries on periodic sets}\label{sec:pps}
We record in this section some preliminary remarks about periodic
sets. These may be of interest in their own, but will in particular
be useful in subsequent computations.
The first of these remarks is about minimal representations of periodic sets.
\begin{defn}\label{period}
A \emph{periodic set} in $\mathbb{R}^n$ is a closed discrete subset $\Lambda$ of $\mathbb{R}^n$ which is invariant under translations by all the vectors of a full dimensional lattice $L$ in $\mathbb{R}^n$, that is
\begin{equation}\label{eqn:translat}
\Lambda + L =\Lambda.
\end{equation}
A lattice for which \eqref{eqn:translat} holds is called a \emph{period lattice for $\Lambda$}.
\end{defn}
If \eqref{eqn:translat} holds, then the quotient $\Lambda/L$ is discrete and compact, hence finite. From this we can derive an alternative definition of a periodic set in $\mathbb{R}^n$, as a set of points which can be written as a union of finitely many cosets of a full-rank lattice $L$, {\it i.e. }
\begin{equation} \label{eqn:periodic}
\Lambda = \bigcup_{i=1}^m\left( t_i + L\right)
\end{equation}
for some vectors $t_1, \dots, t_m$ in $\mathbb{R}^n$, which we assume to be pairwise incongruent $\mod L$. In that case we say that $\Lambda$ is $m$-periodic.
Note that closedness is necessary in Definition~\ref{period}, as shown
by the counterexample
$\Lambda=\bigcup_{n \in
\mathbb{N}^*}\left(\frac{1}{n}+\mathbb{Z}\right)$
which is invariant under translations by $\mathbb{Z}$ but not of the
form~\eqref{eqn:periodic} for any $m$.
\subsection*{Representations}
We call the set of data, {\it i.e. } a lattice $L$ together with a collection
$\mathbf{t}=\left(t_1,\ldots,t_m\right)$ of translational vectors, a
\textit{representation} of $\Lambda$, which we write
$\left(L,\mathbf{t}\right)$ for short.
A given periodic set $\Lambda$ admits infinitely many period lattices and representations, in which the number $m=\left\vert\Lambda\slash L\right \vert$ varies. For instance one can replace $L$ by any of its sublattice $L'$ and obtain a representation as a union of $m\left[L :L' \right]$ translates of $L'$, as in the example in Figure~\ref{fig:fig1}, where the same set is represented as a $4$ and $8$-periodic set.
\begin{figure}[h]\label{fig:fig1}
\centering
\includegraphics[scale=0.1]{figure13.pdf} \hspace{1cm} \includegraphics[scale=0.1]{figure23.pdf}
\caption{}
\end{figure}
However, the set of period lattices, which is partially ordered by inclusion, admits a maximum $L_{\max}$, which we call the \emph{maximal period lattice of $\Lambda$} (see Proposition \ref{lmax} below), corresponding to an essentially unique \emph{minimal} representation of $\Lambda$ ({\it i.e. } with a minimal number of cosets).
Note also that the \emph{point density} $p\delta(\Lambda)=\dfrac{m}{\sqrt{\det L}}$ of a periodic set $\Lambda$, which counts the "number of points per unit volume of space", does not depend on the choice of a representation. When studying properties which are invariant by scaling, we restrict to periodic sets with fixed point density.
We will be interested in quantities, such as energy, which depend only on the pairwise differences of elements of $\Lambda$ (see Definition \ref{pnrj} below). For any $x$ in $\Lambda$, we define the \textit{difference set $\Lambda_x$ of $x$} as the translate of $\Lambda$ by the vector $-x$:
\begin{equation}
\Lambda_x \coloneqq\Lambda-x=\set{y-x \mid y \in \Lambda}.
\end{equation}
Two points $x$ and $y$ in $\Lambda$ have the same difference set if and only if $\Lambda$ is invariant under the translation by $x-y$. This is the case in particular if $x$ and $y$ are congruent modulo a period lattice of $\Lambda$.
The following proposition shows that the number $m(\Lambda)$ of distinct difference sets $\Lambda_x$ as $x$ runs through $\Lambda$ is equal to the minimal number of cosets needed to represent $\Lambda$ as a periodic set, {\it i.e. } the cardinality of the quotient of $\Lambda$ by its maximal period lattice:
\begin{proposition}\label{lmax} Let $\Lambda$ be a periodic set in $\mathbb{R}^n$, and let $m(\Lambda)$ be the number of distinct difference sets $\Lambda_x$ as $x$ runs through $\Lambda$. Then the following holds:
\begin{enumerate}
\item\label{uno} For every period lattice $L$ of $\Lambda$ one has
\begin{equation*}
\left\vert\Lambda\slash L\right \vert \geq m(\Lambda)
\end{equation*}
with equality if and only if $L$ is maximal with respect to inclusion among period lattices of~$\Lambda$.
\item\label{dos} There exists a unique period lattice $L_{\max}$ containing all period lattices of $\Lambda$, defined as
$$ L_{\max}=\left\lbrace v \in \mathbb{R}^n \mid v + \Lambda = \Lambda \right\rbrace .$$
We call it the \emph{maximal period lattice} of $\Lambda$.
It corresponds to an essentially unique minimal representation of
$\Lambda$ as a union of
$m(\Lambda) = \left \vert\Lambda\slash L_{\max}\right \vert $
translates of $L_{\max}$ (up to the choice of representatives modulo $L_{\max}$ and reordering).
\item \label{tres} For $x$ and $y$ in $\Lambda$ one has
$$\Lambda_x = \Lambda_y \Leftrightarrow x \equiv y \!\!\mod L_{\max}.$$
\end{enumerate}
\end{proposition}
\begin{proof}
\ref{uno}. As already noticed, two elements of $\Lambda$ which are congruent modulo a period lattice $L$ have the same difference set, so that $m(\Lambda)$ is at most $\left\vert\Lambda\slash L\right \vert$. If $L$ is not maximal, then there exists a period lattice $L'$ containing $L$ with finite index and we have
$$ \left\vert\Lambda\slash L\right \vert = \left[L':L\right]\left\vert\Lambda\slash L'\right \vert \geq \left[L':L\right] m(\Lambda) > m(\Lambda).$$
Conversely, if $\left\vert\Lambda\slash L\right \vert > m(\Lambda)$, then there are at least two elements $x$ and $y$ in $\Lambda$ which are not congruent modulo $L$ and have the same difference sets. Then $\Lambda-x=\Lambda-y$, so that $\Lambda +(x-y)=\Lambda=\Lambda+(y-x)$ and more generally, $\Lambda$ is stable under translation by any vector in $\mathbb{Z}(x-y)$. The group $L'\coloneqq L +\mathbb{Z}(x-y)$ is discrete (it is contained in a translate of the discrete set $\Lambda$) hence a full dimensional lattice in $\mathbb{R}^n$ strictly containing $L$, and since $\Lambda + L' =\Lambda$, it is indeed a period lattice of $\Lambda$.
\ref{dos}. Starting from any period lattice $L$, we can enlarge it using the construction described above as long as $\left\vert\Lambda\slash L\right \vert > m(\Lambda)$. The process ends up with a maximal period lattice. Since the sum $L+L'$ of two period lattices $L$ and $L'$ for $\Lambda$ is again a period lattice containing $L$ and $L'$, we see that such a maximal period lattice is unique, and contains all period lattices. It is also clear from its construction that it consists precisely of the vectors $v$ in the ambient space such that $v+\Lambda=\Lambda$.
\ref{tres}. This follows since $\Lambda_x = \Lambda_y \Leftrightarrow \Lambda-x = \Lambda-y \Leftrightarrow (x-y)+\Lambda=\Lambda$.
\end{proof}
For a given representation $\Lambda = \bigcup_{i=1}^m\left(t_i + L\right)$ of a periodic set $\Lambda$, the set "$\Lambda - \Lambda$" of pairwise differences of elements of $\Lambda$ can be described as
$$\Lambda - \Lambda= \bigcup_{1 \leq i \leq m}\Lambda_{t_i}.$$
As an ordinary set, it does not depend on the choice of a representation $\left(L,\mathbf{t}\right)$, but it does as a "multiset", since the difference of two elements of $\Lambda$ may occur in several difference sets $\Lambda_{t_i}$. Moreover, the number of difference sets to which a given element of $\Lambda - \Lambda$ belongs depends on the representation chosen.
To eliminate this dependency, we define a weight function $\nu$ on $\Lambda - \Lambda$, setting
\begin{equation}
\nu(w)=\frac{1}{m}\left| \left\lbrace i \mid w \in \Lambda_{t_i}\right\rbrace\right|.
\end{equation}
This definition is independent of the choice of a representation of $\Lambda$, namely one has
\begin{equation}\label{poids}
\nu(w)=\frac{1}{m(\Lambda)}\left| \left\lbrace x \in \mathcal{R}\mid w
\in \Lambda_x\right\rbrace\right|
,
\end{equation}
where $\mathcal{R}$ is a set of representatives of $\Lambda \!\!\mod L_{\max}$.
Note also that $\nu(w)=1$ if and only if $w \in L_{\max}$. Indeed,
$w$ has weight $1$ if and only if it belongs to all difference sets
$\Lambda_x$: It is clearly the case if $w \in L_{\max}$, and
conversely, if $w$ belongs to
$\displaystyle \bigcap_{x \in \mathcal{R}}\Lambda_x$,
then there exists a permutation $\sigma$ of $\mathcal{R}$ such that
$$\forall x \in \mathcal{R}, \ w +x \equiv \sigma(x) \!\!\mod L_{\max}$$
which implies that $w+\Lambda=\Lambda$, so that $w \in L_{\max}$.
Note also, in the same spirit, the following two observations:
\begin{itemize}
\item if $m(\Lambda)=1$, {\it i.e. } if $\Lambda$ is a translate of a lattice, then one has $\nu(w)= 1$ for all $w \in \Lambda - \Lambda=\Lambda$.
\item if $m(\Lambda)=2$, then one has $\nu(w)= 1$ or $\frac{1}{2}$
according to $w$ belonging to the maximal period lattice of $\Lambda$ or not.
\end{itemize}
\subsection*{Symmetries}
We continue this preliminary section with some considerations on
automorphisms.
To a lattice $L$ in $\mathbb{R}^n$ one associates the group $\operatorname{Aut} L$ of its
\emph{linear} automorphisms defined as
\begin{equation}
\operatorname{Aut} L=\left\lbrace f \in O(\mathbb{R}^n) \mid f(L) =L\right\rbrace.
\end{equation}
For a more general periodic set $\Lambda$, the natural group of
transformations to consider is the group
$\operatorname{Isom}\Lambda$ of \emph{affine isometries} preserving it.
If $f$ is such an affine isometry, then its associated orthogonal
automorphism $\bar{f}$, defined by the property that
$\bar{f}(x-y)=f(x)-f(y)$ for all $x$ and $y$ in $\mathbb{R}^n$, stabilizes the
maximal period lattice $L_{\max}$. Indeed, for every $\ell \in L_{\max}$, one has
$$\bar{f}(\ell)+\Lambda=\bar{f}(\ell)+f(\Lambda)=f(\ell+\Lambda)=f(\Lambda)=\Lambda,$$
whence $\bar{f}(\ell) \in L_{\max}$, by the very definition of $L_{\max}$.
We denote by $\operatorname{Aut} \Lambda$ the image of $\operatorname{Isom} \Lambda$ in $\operatorname{Aut} L_{\max}$, {\it i.e. } the subgroup of $\operatorname{Aut} L_{\max}$ consisting of all maps
$\bar{f}$ as $f$ runs through $\operatorname{Isom} \Lambda$, and call it the
\emph{group of orthogonal automorphisms of $\Lambda$}.
Two affine isometries of $\Lambda$ with the same associated
orthogonal automorphism $\bar{f}$ differ by a translation by a vector in
$L_{\max}$. Therefore, we get the following short exact sequence
\begin{equation}
\begin{array}{rcrlcl}
1 \longrightarrow & L_{\max} \longrightarrow & \operatorname{Isom} \Lambda \longrightarrow &\operatorname{Aut} \Lambda \longrightarrow &1 \\
& & f \longmapsto & \bar{f} &
\end{array}
\end{equation}
which is no split in general (it is split for instance when $\Lambda$ is a lattice). Disregarding translations by $L_{\max}$, the main object of interest is thus the group $\operatorname{Aut} \Lambda$ of orthogonal automorphisms which we now characterize:
\begin{lemma}\label{auto}
Let $\Lambda= \bigcup_{i=1}^{m} t_i + L_{\max}$ be an $m$-periodic set in $\mathbb{R}^n$ given by
a minimal representation. Let $\operatorname{Isom} \Lambda$ be the group of its
affine isometries and
$\operatorname{Aut} \Lambda=\left\lbrace \bar{f} \mid f \in \operatorname{Isom} \Lambda
\right\rbrace \subseteq \operatorname{Aut} L_{\max}$
be the group of its orthogonal automorphisms. Then:
\begin{enumerate}
\item\label{perm} For every $f \in \operatorname{Isom} \Lambda$ there exists a unique
permutation
$\sigma \in \mathfrak{S}_m$ such that
\begin{equation*}
f(t_i) \equiv t_{\sigma(i)} \!\!\mod L_{\max} \text{ for all } i \in \left\lbrace 1, \dots, m
\right\rbrace.
\end{equation*}
\item \label{aut} An element $\varphi \in \operatorname{Aut} L_{\max}$ belongs to $\operatorname{Aut} \Lambda$ if and only if
\begin{equation}\label{orthaut}
\exists \sigma \in \mathfrak{S}_m \text{ s.t. } \varphi(t_i - t_1) \equiv t_{\sigma(i)}-t_{\sigma(1)} \!\!\mod L_{\max} \text{ for all } i \in \left\lbrace 1, \dots, m
\right\rbrace
\end{equation}
in which case it is associated to the affine isometry $x \mapsto \varphi(x-t_1)+t_{\sigma(1)}$.
\end{enumerate}
\end{lemma}
\begin{proof}
If $f \in \operatorname{Isom} \Lambda$, then for each $i \in \left\lbrace 1, \dots, m
\right\rbrace$ there exists an index
$\sigma(i)$ such that $f(t_i) \in t_{\sigma(i)} + L_{\max}$, and $\sigma$ is a
permutation since
$$f(t_i) \equiv f(t_j) \!\!\mod L_{\max}\Leftrightarrow \bar{f}(t_i-t_j) \in
L_{\max} \Leftrightarrow t_i-t_j \in L_{\max} \Leftrightarrow i=j.$$
This proves~\ref{perm} as well as the congruence $\bar{f}(t_i - t_1) \equiv t_{\sigma(i)}-t_{\sigma(1)}
\!\!\mod L_{\max}$ for all $i\in \left\lbrace 1, \dots, m
\right\rbrace$. Conversely, if $\varphi \in \operatorname{Aut}
L_{\max}$ satisfies \eqref{orthaut} for some permutation $\sigma$, then the map $f_{\varphi}(x):= \varphi
(x-t_1)+t_{\sigma(1)}$ is in $\operatorname{Isom} \Lambda$ and
$\bar{f_{\varphi}}=\varphi$, which establishes \ref{aut}.
\end{proof}
Note that for each $\varphi$ in $\operatorname{Aut} \Lambda$, the associated permutation
$\sigma$ is unique, as a consequence of the maximality of $L_{\max}$:
If $\sigma$ and $\gamma$ are two permutations of $\Lambda \slash L_{\max}$
such that $\varphi(t_i - t_1) \equiv t_{\sigma(i)}-t_{\sigma(1)} \equiv
t_{\gamma(i)}-t_{\gamma(1)}\mod L_{\max}$ for all $i$, then
$t_{\sigma(i)}-t_{\gamma(i)} \equiv t_{\sigma(1)}-t_{\gamma(1)}=: u\mod L_{\max}$
for all $i$, whence $t_{\sigma(i)}\equiv u+t_{\gamma(i)}$, which implies that $u \in
L_{\max}$, so that $\sigma=\gamma$.
Also, the elements of $\operatorname{Aut} \Lambda$ stabilize the set $\Lambda - \Lambda= \bigcup_{1 \leq i \leq m}\Lambda_{t_i}$. More precisely, for each $\varphi \in \operatorname{Aut} \Lambda$ one has $\varphi (\Lambda_{t_i})=\Lambda_{t_{\sigma(i)}}$ where $\sigma$ is the permutation of $\Lambda \slash L_{\max}$ canonically associated to $\varphi$. This last property makes this group the right object to consider in the sequel.
\begin{rem} \label{rem:auto-discuss}
For a given periodic set
$\Lambda = \bigcup_{i=1}^m\left(t_i + L_{\max}\right)$,
we can often assume
without loss of generality that $t_1=0$ (it amounts to translate
$\Lambda$ by a fixed vector). In such a situation, $\operatorname{Aut} \Lambda$
contains, with index at most $m$, the subgroup $$\operatorname{Aut}_{0} \Lambda =
\left\lbrace \varphi \in \operatorname{Aut} L_{\max} \mid \varphi(\Lambda)=\Lambda
\right\rbrace.$$ This corresponds to permutations $\sigma$ fixing $1$
in \eqref{orthaut} and could be a natural choice for an alternative
definition of the group of automorphisms of $\Lambda$. Nevertheless,
it would introduce a somewhat unnecessary dissymmetry between the
$t_i$'s, and would lead to disregard some automorphisms which are
natural to consider.
\end{rem}
For example, for a $2$-periodic set
$$\Lambda=L_{\max} \cup \left( v+L_{\max}\right) \ , \ 2v\notin L_{\max},$$
we have $-\id \in \operatorname{Aut} \Lambda \setminus \operatorname{Aut}_{0} \Lambda$ and $\left[ \operatorname{Aut} \Lambda : \operatorname{Aut}_0\Lambda\right]=2$.
At the other end, if $\Lambda$ is a $3$-periodic set of the form
$$\Lambda=L_{\max} \cup \left(v+L_{\max}\right)\cup\left( -v+L_{\max}\right) \ , \ 2v\notin L_{\max},$$
then one checks that $\operatorname{Aut} \Lambda = \operatorname{Aut}_0\Lambda$.
\subsection*{Review on theta series and modular forms}
For some estimates needed in Section~\ref{sec:plc} we use
certain theta series and their properties, which we review here.
To start with, we state a rather general result about the modularity of theta series with spherical coefficients attached to a rational periodic set.
If $L$ is a lattice in $\mathbb{R}^n$ and $\rho$ is any vector in
$\mathbb{R}^n$, one defines, for $z$ in the upper half-plane $\mathbb H =\left\lbrace z \in
\mathbb{C} \mid \Im z >0 \right\rbrace$
\begin{equation}\label{theta}
\theta_{\rho+L}(z) =\sum\limits_{x \in \rho + L} e\left(\dfrac{\Vert
x \Vert^2 z}{2}\right)
\end{equation}
where $e(z)=e^{2\pi i z}$. When $\rho=0$, this reduces to the standard theta series
of the lattice $L$.
As in the lattice case, one can introduce \emph{spherical
coefficients} in the previous definition, namely, if $P$ is a harmonic polynomial, one
defines
\begin{equation}\label{thetaspher}
\theta_{\rho+L,P}(z) =\sum\limits_{x \in \rho + L} P(x)e\left(\dfrac{\Vert
x \Vert^2 z}{2}\right).
\end{equation}
From this, and following \cite{MR688626}, we define the \emph{average theta series} with spherical coefficients $P$ of a periodic set $\Lambda= \bigcup_{i=1}^m\left( t_i + L\right)$ as
\begin{align*}
\theta_{\Lambda,P}(\tau)&=\dfrac{1}{m}\sum\limits_{1\leq i,j\leq m} \theta_{t_i-t_j+L,P}(\tau)\\
&=\theta_{L,P}(\tau) + \dfrac{2}{m}\sum\limits_{1\leq i < j\leq m} \theta_{t_i-t_j+L,P}(\tau).
\end{align*}
Both, \eqref{theta} and \eqref{thetaspher}, satisfy transformation formulas under
$\SL(2,\mathbb{Z})$, from which one deduces,
under suitable assumptions on $L$ and $\rho$, that $\theta_{\rho+L,P}(z)$ (resp. $\theta_{\Lambda,P}$) is a \emph{modular form} for some modular group and character (see Proposition \ref{mod} below).
Let $L$ be an even integral lattice, {\it i.e. } $x \cdot x$ is even for all $x \in L$.
The level of $L$ is the smallest integer $N$ such that $\sqrt{N}L^{*}$ is even integral (this implies in particular that $NL^{*} \subseteq L$).
\begin{proposition}\label{mod}
Let $L$ be an even integral lattice of dimension $n$ and level
$N$. Then, for any $\rho \in L^{*}$, and any spherical
harmonic polynomial $P$ of degree $k$,
the theta series $\theta_{\rho+L,P}(z)$ is a modular form of weight $k+\dfrac{n}{2}$ for the principal congruence group $$\Gamma (4N)=\left\lbrace \tau =\begin{pmatrix}
a & b \\
c & d
\end{pmatrix} \in \SL(2,\mathbb{Z}) \mid \tau \equiv \begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix}\mod 4N\right\rbrace $$ and the character $$\vartheta\left(\tau \right) = \left(\dfrac{2c}{d}\right)^n
.$$
Moreover, if $k>0$, then $\theta_{\rho+L,P}(z)$ is a cusp form.
\end{proposition}
\begin{proof}
This is essentially \cite[Corollary 10.7]{MR1474964}, up to
reformulation: setting $L=g \mathbb{Z}^n$ for some $g \in \GL(n,\mathbb{R})$,
$A=g^tg$,
and $h=Ng^{-1}(\rho)$, the condition $\rho \in L^{*}$ is equivalent to
$Ah \equiv 0 \mod N$ (the condition defining the set $\mathcal H$ in
\cite[Corollary 10.7]{MR1474964}) and $\theta_{\rho+L,P}(z)$ coincides
with the congruence theta series $\Theta(z;h)$ in the above reference,
whence the conclusion follows.
\end{proof}
\section{Energy of periodic sets}\label{sec:two}
We recall in this section some basic facts about the energy of a periodic set and its local study, which were established in \cite{MR2889159}.
Following Cohn and Kumar \cite{MR2257398} we define the energy of a periodic set with respect to a non negative potential function as follows:
\begin{defn}\label{pnrj}
Let $\Lambda$ be a periodic set with maximal period lattice $L_{\max}$, and $f$ a non-negative potential function. We set
\begin{equation}\label{eqn:nrj}
E(f,\Lambda)=\frac{1}{m(\Lambda)}\sum_{x \in \mathcal{R}}\sum_{\substack{u\in \Lambda_x\\ u \neq 0}}f(\norm{u}^2)
\end{equation}
where $\mathcal R$ is a set of representatives of $\Lambda$ modulo $L_{\max}$.
\end{defn}
This sum may diverge, in which case the energy is \emph{infinite}.
Note that if $\Lambda$ is given by an $m$-periodic representation $\Lambda = \bigcup_{i=1}^m x_i + L$, non necessarily minimal, then one has
$$E(f,\Lambda)=\frac{1}{m}\sum_{i=1}^m\sum_{\substack{u \in \Lambda _{x_i}\\ u \neq 0}} f(\norm{u}^2)=\frac{1}{m}\sum_{1 \leq i, j \leq m}\sum_{\substack{w \in L \\ w+x_j-x_i \neq 0}} f(\norm{w+x_j-x_i}^2)$$
in accordance with the definition used in \cite{MR2257398}.
This "non intrinsic" formulation
$\frac{1}{m}\sum_{i=1}^m\sum_{\substack{u \in \Lambda _{t_i}\\ u \neq
0}} f(\norm{u}^2)$ is often better suited for explicit computations because it allows to use representations of periodic sets that are not assumed to be minimal.
We want to expand the $f$-energy in a neighbourhood of a given $m$-periodic set $$\Lambda_0=\bigcup_{i=1}^m t^0_i+L_0.$$
Note that the question of periodic sets with minimal $f$-energy
(with $f$ being monotone decreasing) only makes sense if we restrict to
periodic sets with a fixed point density.
Otherwise, the energy can be
made arbitrary small by scaling.
So we restrict to $m$-periodic sets $\Lambda$ with fixed point density, {\it i.e. } of the form
$$\Lambda =\displaystyle\bigcup_{i=1}^m (t^0_i+t_i)+gL_0$$
with $g \in \SL(n,\mathbb{R})$.
As in \cite[\S 3]{MR2889159}, we set
$g^{t}g=A_0^{t}A_0\exp\left(A^{-1}_0HA_0\right)$
where $L_0=A_0\mathbb{Z}^n$ and $H$ is a trace zero symmetric matrix. Then the evaluation of the energy $E(f,\Lambda)$ as $\Lambda$ varies in a neighbourhood of the initial periodic set $\Lambda_0$ reduces to the study of the quantity
\begin{equation}\label{nrg3}
E_f(H,\mathbf{t}) \coloneqq\frac{1}{m} \; \sum_{1\leq i,j\leq m} \; \sum_{0\neq w \in t^0_i-t^0_j+L_0} f(\exp(H)\br{w+t_i-t_j})
\end{equation}
for small enough $\mathbf{t}\in \mathbb{R}^{mn}$ and $H \in\mathcal T:=\set{Q \in {\mathcal S}^{n} \mid \Tr(Q)=0}$, where ${\mathcal S}^{n}$ stands for the space of $n\times n$ real symmetric matrices (see \cite[\S 3]{MR2889159} for details).
Using the Taylor expansion of the matrix exponential we write
\begin{equation*}
\exp(H)[w+t_i-t_{j}]={\|w\|^2}+\mathcal{L}\pr{H,\mathbf{t}}+\mathcal{S}\pr{H,\mathbf{t}}+ o (\|\pr{H,\mathbf{t}}\|^2)
\end{equation*}
where
\begin{equation*}
\mathcal{L}\pr{H,\mathbf{t}}=H[w]+ 2w^{t}(t_i-t_{j})
\end{equation*}
and
\begin{equation*}
\mathcal{S}\pr{H,\mathbf{t}}=\|t_i-t_{j}\|^2+2w^{t}H(t_i-t_{j})+\frac{1}{2}H^2\br{w}.
\end{equation*}
In particular, if $f(r)=e^{-cr}$ we get
\begin{equation*}
e^{-c\exp(H)[w+t_i-t_{j}]}=e^{-c\|w\|^2}\left(1-c\left(\mathcal{L} +\mathcal{S} \right)+\frac{c^2}{2}\mathcal{L}^2\right)+ o (\|\pr{H,\mathbf{t}}\|^2)
\end{equation*}
and hence the following expressions for the gradient
\begin{equation*}
\grad = -\frac{c}{m}
\sum_{1\leq i,j\leq m}\sum_{0\neq w \in t^0_i-t^0_j+L_0}
\left(
H[w]+ 2w^{t}(t_i-t_j)
\right) e^{-c \|w\|^2}
\end{equation*}
and the Hessian
\begin{eqnarray*}
\hess&=&\frac{c}{m}
\sum_{1\leq i,j\leq m}\sum_{0\neq w \in t^0_i-t^0_j+L_0}
\left( \left\lbrace \frac{c}{2}H[w]^2-\frac{1}{2}H^2\br{w}\right\rbrace
\right.
\\&+&
\left.
\left\lbrace 2c \left( w^{t}(t_i-t_j)\right)^2 - \|t_i-t_j\|^2-2w^{t}H(t_i-t_j)+2cw^{t}(t_i-t_j)H[w]\right\rbrace
\right) e^{-c \|w\|^2}
.
\end{eqnarray*}
\section{Critical Points} \label{sec:critical}
A periodic set is said to be $f$-critical if it is a critical point for the energy $E_f$. We will be especially interested in $f_c$-critical periodic sets, where~$f_c(x)=e^{-cx}$ with~$c>0$, since these functions generate the space of completely monotonic functions (see \cite[Theorem 12b, p. 161]{MR0005923}).
We want to give a necessary and sufficient criterion for
a periodic set $\Lambda$ in $\mathbb{R}^n$ to be $f_c$-critical for all $c>0$. Using the formulas of the previous section this amounts to show that the gradient vanishes for all choices of~$c>0$.
Collecting the terms in the sum above with the same value $e^{-c\|w\|^2}$, we obtain the following:
\begin{lemma}\label{lem:sums}
A periodic set~$\Lambda$ in $\mathbb{R}^n$ is $f_c$-critical for all~$c>0$ if and only if the terms
$$
\sum_{1\leq i,j\leq m}
\; \sum_{ w \in t^0_i-t^0_j+L_0 , \; \|w\| = r} \;
H[w]+ 2w^{t}(t_i-t_j)
$$
vanish for any representation $\Lambda=\bigcup_{i=1}^m t^0_i+L_0$
and any choice of $r>0$ and $(H,\mathbf{t})$.
\end{lemma}
\begin{proof} According to the previous section, the gradient of $E_{f_c}$can be written as
\begin{equation}
\grad = -\frac{c}{m}
\sum_{r>0} \left[\sum_{1\leq i,j\leq m}\;\sum_{ w \in t^0_i-t^0_j+L_0, \|w\|=r}
\left(
H[w]+ 2w^{t}(t_i-t_j)
\right)\right] e^{-c r^2}
\end{equation}
Suppose there is a representation of~$\Lambda$
and a minimal $r>0$ for which the sum between brackets does
not vanish for some choice of $(H,\mathbf{t})$.
Then for sufficiently large~$c$ the gradient is
essentially given by the corresponding term
(in front of $e^{-c r^2}$). So the gradient does not vanish as well.
If on the other hand the gradient vanishes for all $c$,
we find that the corresponding sums of the proposition
all have to vanish.
\end{proof}
We want to state a necessary and sufficient condition for the vanishing
of all the sums of the previous propositions in terms of \emph{weighted spherical designs}.
For a periodic set $\Lambda$, $x\in \Lambda$ and $r >0$ we define
$$
\Lambda_x(r) \; = \; \{ y-x \mid \| y-x \| = r, \; y \in \Lambda \}
$$
and we set $\Lambda(r) = \bigcup_{x\in \Lambda} \Lambda_x(r)$.
A \emph{weighted spherical $t$-design} is a pair $(X,\nu)$ of a
finite set $X$ contained in a sphere of radius~$r$
and a weight function $\nu$ on $X$ such that
\begin{equation}
\dfrac{1}{\left| S \right|}\int_S f(x) dx = \dfrac{1}{\left| X \right|}\sum_{x \in X} \nu(x) f(x)
\end{equation}
for all polynomials $f(x)=f(x_1, \dots,x_n)$ of degree at most $t$. This is a special case of a \emph{cubature formula} on the sphere, studied {\it e.g. } by Goethals and Seidel in \cite{MR661779}, and reduces to the classical notion of spherical $t$-design when the weight function is equal to $1$.
Note that, for $t=1$, this simply means that the weighted sum $\sum_{x \in X} \nu(x) x$ is $0$. When all weights are $1$, this reduces to the condition
$$\sum_{x \in X} x = 0$$
which we refer to in the sequel as $X$ being a \emph{balanced} set. One may think of forces acting on the origin that balance each other.
Finally, we mention the following useful characterization of the $2$-design property, which will be used throughout the rest of the paper:
\begin{lemma}[\cite{MR964837}, Theorem 4.3]\label{NS}\quad
A weighted set $(X,\nu)$ on a sphere of radius $r$ in $\mathbb{R}^n$ is a weighted spherical $2$-design if and only if
$$\sum_{x \in X} \nu(x) x =0 \text{ and } \sum_{x \in X} \nu(x) x x^t=c\id_n$$ for some constant $c$.
\end{lemma}
\begin{theorem} \label{th:periodic-critical}
A periodic set~$\Lambda$ in $\mathbb{R}^n$ is $f_c$-critical for any~$c>0$ if and only if
\begin{enumerate}
\item
All non-empty shells $\Lambda_x(r)$ for $x\in \Lambda$ and $r>0$ are balanced.
\item
All non-empty shells $\Lambda(r)$ for $r>0$ are weighted spherical $2$-designs with respect to the weight $\nu(w)=\frac{1}{m(\Lambda)}\left| \left\lbrace x \in \Lambda \!\!\mod L_{\max} \mid w \in \Lambda_x\right\rbrace\right|$.
\end{enumerate}
\end{theorem}
Note that the statement of the theorem, in contrast to the one of Lemma~\ref{lem:sums},
is independent of the possible representations of~$\Lambda$.
\begin{proof}
First observe that the sums of Lemma~\ref{lem:sums}
split
for any representation $\Lambda=\bigcup_{i=1}^m t^0_i+L_0$
into two parts: one depending on $H$ only and one depending on $\mathbf{t}$ only.
The part depending on $\mathbf{t}$ is (up to a factor of $2$) equal to
$$
\sum_{1\leq i,j\leq m}
\; \sum_{ w \in t^0_i-t^0_j+L_0, \; \|w\| = r} \;
w^t (t_i - t_j)
.
$$
We can rearrange the sum, collecting terms that occur with a fixed $t_k$,
either for $k=i$ as $u=w \in (t^0_i +L_0) - (t^0_j+L_0)$
or for $k=j$ as $u=-w \in (t^0_j +L_0) - (t^0_i+L_0)$ and get
$$
-2\sum_{1\leq k\leq m}
\left(
\; \sum_{ u \in \Lambda_{t^0_k}(r)} \;
u^t
\right)t_k.
$$
So this sum vanishes for all choices of $\mathbf{t}$ if and only if
the coefficients of each $t_k$ vanish.
This is precisely the case if and only if $\Lambda_x(r)$ is balanced
for every $x\in \Lambda$. This implies that $\Lambda(r)$ itself is a
weighted balanced set (weighted spherical $1$-design) since
\begin{equation*}
\sum_{x \in \mathcal R} \sum_{u\in \Lambda_x(r)}u=\sum_{u\in
\Lambda(r)}\nu(u) u,
\end{equation*}
with $\mathcal{R}$ being a set of representatives of $\Lambda \!\!\mod L_0$.
The part depending on $H$ can be rewritten as
\begin{align*}
\sum_{1\leq i,j\leq m}
\; \sum_{ w \in t^0_i-t^0_j+L_0 , \; \|w\| = r} \;
H[w]
&\;= \;
\left\langle
H ,
\sum_{1\leq i,j\leq m}
\; \sum_{ w \in t^0_i-t^0_j+L_0 , \; \|w\| = r} \;
w w^t
\right\rangle\\
&\;= \;\left\langle
H ,
\sum_{1\leq j \leq m}\;\sum_{w \in \Lambda_{t^0_j}(r)}
\;
w w^t
\right\rangle\\
&\;= \;m\left\langle
H ,
\sum_{w \in \Lambda(r)}
\; \nu(w)
w w^t
\right\rangle
.
\end{align*}
It vanishes for all choices of trace zero symmetric matrice $H$ if and only if the sum of rank-$1$ forms (matrices) $\nu(w)w w^t$ is a (positive) multiple of the identity, namely
\begin{equation}\label{2des}
\sum_{w \in \Lambda(r)} \nu(w) w w^t = c_r \id_n
\end{equation}
with
$$c_r=\dfrac{r^2\sum_{w \in \Lambda(r)} \nu(w)}{n},$$
where the value of the constant $c_r$ is obtained by taking the trace of \eqref{2des}.
Combined with the first part of the theorem which insures that $\Lambda(r)$ is already a weighted spherical $1$-design, this last condition is equivalent to $\Lambda(r)$ being a weighted spherical $2$-design, due to Lemma \ref{NS}.
\end{proof}
\section{Expressing energy of $2$-periodic sets} \label{sec:2periodic}
In order to deal with the energy of $\mathsf{D}_n^+$ and more general
for other $2$-periodic sets, a reordering of contributing terms
will be very helpful.
Let $\Lambda$ be a periodic set. Without loss of generality, we
can assume that $\Lambda$ contains $0$ (it amounts to translate $\Lambda$ by a
well-chosen vector). Note that this is equivalent to the property that $\Lambda$
contains its maximal period lattice $L_{\max}$. If we assume moreover that
$m(\Lambda)=2$, then we have
$\Lambda=
L_{\max}
\cup \left(v + L_{\max}\right)$ for any $v \in \Lambda \setminus L_{\max}$ and
$$\Lambda_x=\begin{cases}
\Lambda \text{ if } x \in L_{\max}\\
-\Lambda \text{ otherwise.}
\end{cases}$$
In particular, $\Lambda -\Lambda =\Lambda \cup \left( -\Lambda\right)=L_{\max}
\cup \left(v + L_{\max}\right)\cup \left(-v + L_{\max}\right)$. The next lemma clarifies the consequences of these properties on a non-minimal representation of $\Lambda$.
\begin{lemma}\label{lem:perm}
Let $\Lambda =\displaystyle\bigcup_{i=1}^m t_i+L \subset \mathbb{R} ^n$ be a periodic set
\textbf{containing} $\mathbf{0}$. Suppose $m(\Lambda)=2$. Then there is a partition of $I=\left\lbrace 1, \ldots , m \right\rbrace$ into two equipotent subsets $J$ and $J'$ and a map $\sigma : I \times I \rightarrow I$ such that
\[
\displaystyle
\forall i \in \set{1,\dots,m} \quad t_i-t_k \equiv
\left\{
\begin{array}{ll}
\phantom{-}t_{\sigma(i,k)} \!\!\mod L \; \mbox{ if} \; k\in J\\
-t_{\sigma(i,k)} \!\!\mod L \; \mbox{ if} \; k\in J'
\end{array}
\right.
.
\]
Moreover, for any fixed $i$ or $k$ in $J$
(resp. in $J'$), the maps $\sigma(i,\cdot)$
and $\sigma(\cdot,k)$ bijectively map $J$ onto
$J$ and $J'$ onto $J'$ (resp $J$ onto $J'$ and $J'$ onto $J$).
\end{lemma}
\begin{proof}
If $m(\Lambda)=2$ then the maximal period lattice $L_{\max}$ of $\Lambda$ contains $L$ with index $\frac{m}{2}$ and, as mentioned above, if $v$ is any element in $\Lambda \setminus L_{\max}$, one has
$$\Lambda =\bigcup_{i=1}^m t_i+L = L_{\max} \cup \left(v + L_{\max}\right).$$ Consequently, $t_i \in L_{\max}$ for exactly one half of the indices $ i\in I$ and $\Lambda-t_i = \Lambda$ or $-\Lambda$ according as $t_i $ belongs to $L_{\max}$ or $v + L_{\max}$. Setting $J=\left\lbrace i \in I \mid t_i \in L_{\max}\right\rbrace$ and $J'=\left\lbrace i \in I \mid t_i \in v + L_{\max}\right\rbrace$ one can construct the map $\sigma$ as follows:
\begin{itemize}
\item If $k \in J$, that is to say if $t_k \in L_{\max}$, then $\Lambda-t_k = \Lambda=\bigcup_{i=1}^m t_i+L$ so that for all $i \in I$ there is a well-defined index $\sigma(i,k) \in I$ such that $t_i-t_k \equiv t_{\sigma(i,k)} \!\!\mod L$. On the other hand, since $\Lambda-t_k = \Lambda= L_{\max} \cup \left(v + L_{\max}\right)$, we infer that $t_i-t_k$ belongs to $ L_{\max}$ or $v + L_{\max}$, depending on whether $t_i$ is in $ L_{\max}$ ($\Leftrightarrow i\in J$) or $t_i$ is in $v + L_{\max}$ ($\Leftrightarrow i\in J'$), which means that $\sigma(\cdot,k)$ maps $J$ to $J$ and $J'$ to $J'$. The injectivity of $\sigma(\cdot,k)$ is straightforward, as the $t_i$'s are noncongruent $\mod L$.
\item If $k \in J'$, then $t_k \in v + L_{\max}$ and $\Lambda-t_k = -\Lambda$ so that for all $i \in I$ there is a well-defined index $\sigma(i,k) \in I$ such that $t_i-t_k \equiv -t_{\sigma(i,k)} \!\!\mod L$. Now, since $\Lambda-t_k = -\Lambda=- L_{\max} \cup -\left(v + L_{\max}\right)$ we have this time that $t_i-t_k$ belongs to $- L_{\max}$ or $-\left(v + L_{\max}\right)$ according to $i$ being in $J'$ or $J$, which means that $\sigma(i,k)$ belongs to $J$ if $i\in J'$ and to $J'$ if $i\in J$. Again, the injectivity of $\sigma(\cdot,k)$ is clear.
\end{itemize}
It remains to prove that, for fixed $i$, the map $\sigma(i,\cdot)$ also satisfies the required properties, which proceeds by an easy case by case verification, as above.
\end{proof}
Using the results of Section \ref{sec:two}, we know that in a suitable
neighborhood of our given set $\Lambda_0=\bigcup_{i=1}^m (t^0_i +
L_0)$, the $f$-energy varies according to
\begin{equation}\label{eqn:local_energy}
E_f(H,t) \coloneqq\frac{1}{m}
\; \sum_{1\leq i,j\leq m} \; \sum_{0\neq w \in t^0_i-t^0_j+L_0} f(\exp(H)[w+t_i-t_j])
\end{equation}
In what follows we will extensively use the following reordering of contributions:
\begin{lemma} \label{lem:ReorderingLemma}
Suppose $\Lambda_0=\bigcup_{i=1}^m (t^0_i + L_0) = L_{\max} \cup (v+ L_{\max})$
with $v \in\mathbb{R}^n$ and lattice $L_0 \subseteq L_{\max}\subset \mathbb{R}^n$ is a $2$-periodic set, and that
$t^0_i\in L_{\max}$ for $i\in J=\{1,\ldots,\frac{m}{2}\}$, $t^0_i \in v + L_{\max}$ for $i\in J'=\{\frac{m}{2}+1,\ldots, m\}$.
Then
\begin{eqnarray*}
E_f(H,t)
& = &
\frac{2}{m^2}\left[
\left(
\sum_{0\not= w \in L_{\max}} \; \sum_{i=1}^m \; \sum_{k\in J} \;
f\left( \exp(H) [ w+t_i-t_{\sigma(i,k)}]\right)
\right)
\right. \\
& & \qquad +
\left(
\sum_{0\not= w \in -(v+ L_{\max})} \; \sum_{i\in J} \; \sum_{k\in J'} \;
f\left( \exp(H) [ w+t_i-t_{\sigma(i,k)}]\right)
\right) \\
& & \qquad +
\left.
\left(
\sum_{0\not= w \in (v+ L_{\max})} \; \sum_{i\in J'} \; \sum_{k\in J'} \;
f\left( \exp(H) [ w+t_i-t_{\sigma(i,k)}]\right)
\right)
\right]
,
\end{eqnarray*}
where $\sigma(i,k)$ is defined as in Lemma~\ref{lem:perm}, that is
\[
\displaystyle
t^0_i-t^0_k \equiv_{\!\!\mod L_0}
\left\{
\begin{array}{ll}
\phantom{-}t^0_{\sigma(i,k)} \; \mbox{if} \; k\in J\\
-t^0_{\sigma(i,k)} \; \mbox{if} \; k\in J'
\end{array}
\right.
.
\]
\end{lemma}
\begin{proof}
For the local expression of energy,
we start with the expression~\eqref{eqn:local_energy} for $E_f(H,t)$ and
split the sum over $i,j\in\{1,\ldots, m\}=J\cup J'$ into four parts
1A, 2A, 1B, 2B according to $j\in J$ or $j\in J'$ (cases with 1 or 2)
and $i\in J$ or $i \in J'$ (cases with~A or~B):
\[
\frac{1}{m}
\left[
\underbrace{\left( \sum_{(i,j)\in J\times J} (\ast) \right)}_{\mbox{part 1A}}
+
\underbrace{\left( \sum_{(i,j)\in J\times J'} (\ast) \right)}_{\mbox{part 2A}}
+
\underbrace{\left( \sum_{(i,j)\in J'\times J} (\ast) \right)}_{\mbox{part 1B}}
+
\underbrace{\left( \sum_{(i,j)\in J'\times J'} (\ast) \right)}_{\mbox{part 2B}}
\right]
,
\]
where $(\ast)$ is a placeholder
for $\displaystyle\sum_{0\neq w \in t^0_i-t^0_j+L_0} f(\exp(H)[w+t_i-t_j])$.
\bigskip
{\bf Part 1A:}
First we reorder terms by substituting $j$ with $\sigma(i,k)$.
Here we use that $j=\sigma(i,k)$ is a bijection of $J$ for fixed $i$, mapping index~$k$ to~$j$.
So Part~1A is equal to
\[
\sum_{i \in J} \; \sum_{k \in J} \; \sum_{0\neq w \in t^0_i-t^0_{\sigma(i,k)}+L_0} f(\exp(H)[w+t_i-t_{\sigma(i,k)}])
.
\]
The translate $t^0_i-t^0_{\sigma(i,k)}+L_0$ can be written as $t^0_{\ell}+L_0$ with
$\ell =\sigma(i,j)=\sigma(i,\sigma(i,k))\in J$ depending on~$i$ and~$k$.
So we get for Part~1A:
\[
\sum_{i \in J} \; \sum_{k \in J} \; \sum_{0\neq w \in t^0_{\ell}+L_0} f(\exp(H)[w+t_i-t_{\sigma(i,k)}])
\]
with the vectors $w \in t^0_{\ell}+L_0$ for $\ell \in J$
running through all non-zero elements of the lattice~$ L_{\max}$.
Therefore a shift of the $w$ by any vectors of $ L_{\max}$ does not effect the outcome
for Part~1A. For every $j\in J$ we may shift by $-t^0_j$ and get the same value
as for Part~1A also in
\[
\sum_{i \in J} \; \sum_{k \in J} \; \sum_{0\neq w \in t^0_{\ell} -t^0_j +L_0} f( \cdots )
=
\sum_{i \in J} \; \sum_{k \in J} \; \sum_{0\neq w \in t^0_{\sigma(l,j)}+L_0} f( \cdots )
.
\]
Here and in the following $f( \cdots )$ abbreviates $f(\exp(H)[w+t_i-t_{\sigma(i,k)}])$.
Since $$\bigcup_{j\in J} t^0_{\sigma(l,j)}+L_0 = L_{\max}$$ for every fixed~$\ell$, we can take an average over all $j\in J$
and get for Part~1A:
\[
\frac{1}{|J|}
\sum_{i \in J} \;
\sum_{k \in J} \;
\sum_{0\neq w \in L_{\max}}
f( \cdots )
=
\frac{1}{|J|}
\sum_{0\neq w \in L_{\max}} \;
\sum_{i \in J} \;
\sum_{k \in J}
f( \cdots )
\]
\bigskip
{\bf Part 2A:}
First we reorder terms again, by substituting $j$ with $\sigma(i,k)$.
Here we use that $j=\sigma(i,k)$ is a bijection of $J'$ for fixed $i$, mapping index~$k$ to~$j$.
So Part~2A is equal to
\[
\sum_{i \in J} \; \sum_{k \in J'} \; \sum_{0\neq w \in t^0_i-t^0_{\sigma(i,k)}+L_0} f(\exp(H)[w+t_i-t_{\sigma(i,k)}])
.
\]
The translate $t^0_i-t^0_{\sigma(i,k)}+L_0$ can be written as $-t^0_{\ell}+L_0$ with
$\ell =\sigma(i,j)=\sigma(i,\sigma(i,k))\in J'$ depending on~$i$ and~$k$.
So Part~2A can be written as:
\[
\sum_{i \in J} \; \sum_{k \in J'} \; \sum_{0\neq w \in -t^0_{\ell}+L_0} f(\exp(H)[w+t_i-t_{\sigma(i,k)}])
\]
with the vectors $w \in -t^0_{\ell}+L_0$ for $\ell \in J'$
running through all non-zero elements of the lattice translate~$-( v+L_{\max})$.
A shift of the $w$ by any vectors of $ L_{\max}$ does not effect the outcome
for Part~2A. So for every $j\in J$ we may shift by $t^0_j$ and get the same value
as for Part~2A also in
\[
\sum_{i \in J} \; \sum_{k \in J} \; \sum_{0\neq w \in t^0_j -t^0_{\ell} +L_0} f( \cdots )
=
\sum_{i \in J} \; \sum_{k \in J} \; \sum_{0\neq w \in -t^0_{\sigma(j,l)}+L_0} f( \cdots )
.
\]
Here, $\sigma(j,l)\in J'$ since $j\in J$ and $\ell \in J'$,
and $f( \cdots )$ abbreviates $f(\exp(H)[w+t_i-t_{\sigma(i,k)}])$ again.
Since $\bigcup_{j\in J} -t^0_{\sigma(j,l)}+L_0 = -( v+L_{\max})$ for every fixed~$\ell$, we can take an average over all $j\in J$
and get for Part~2A:
\[
\frac{1}{|J|}
\sum_{i \in J} \;
\sum_{k \in J'} \;
\sum_{0\neq w \in -( v+L_{\max})}
f( \cdots )
=
\frac{1}{|J|}
\sum_{0\neq w \in -( v+L_{\max})} \;
\sum_{i \in J} \;
\sum_{k \in J'}
f( \cdots )
\]
\bigskip
{\bf Part 1B:}
We start by substituting $j$ with $\sigma(i,k)$ again, where
$j=\sigma(i,k)$ is a bijection from $J'$ to $J$ for fixed $i$, mapping index~$k$ to~$j$.
So Part~1B is equal to
\[
\sum_{i \in J'} \; \sum_{k \in J'} \; \sum_{0\neq w \in t^0_i-t^0_{\sigma(i,k)}+L_0} f(\exp(H)[w+t_i-t_{\sigma(i,k)}])
.
\]
The translate $t^0_i-t^0_{\sigma(i,k)}+L_0$ can be written as $t^0_{\ell}+L_0$ with
$\ell =\sigma(i,j)=\sigma(i,\sigma(i,k))\in J'$ depending on~$i$ and~$k$.
So Part~1B can be written as:
\[
\sum_{i \in J'} \; \sum_{k \in J'} \; \sum_{0\neq w \in t^0_{\ell}+L_0} f(\exp(H)[w+t_i-t_{\sigma(i,k)}])
\]
with the vectors $w \in t^0_{\ell}+L_0$ for $\ell \in J'$
running through all non-zero elements of the lattice translate~$ v+L_{\max}$.
Again, a shift of the $w$ by any vectors of $ L_{\max}$ does not effect the outcome
for Part~1B. So for every $j\in J$ we may shift by $-t^0_j$ and get the same value
as for Part~1B also in
\[
\sum_{i \in J'} \; \sum_{k \in J'} \; \sum_{0\neq w \in t^0_{\ell} -t^0_j +L_0} f( \cdots )
=
\sum_{i \in J'} \; \sum_{k \in J'} \; \sum_{0\neq w \in t^0_{\sigma(l,j)}+L_0} f( \cdots )
.
\]
Here, $\sigma(l,j)\in J'$ since $\ell \in J'$ and $j\in J$,
and $f( \cdots )$ abbreviates $f(\exp(H)[w+t_i-t_{\sigma(i,k)}])$ again.
Since $\bigcup_{j\in J} t^0_{\sigma(l,j)}+L_0 = ( v+L_{\max})$ for every fixed~$\ell$, we can take an average over all $j\in J$
and get for Part~1B:
\[
\frac{1}{|J|}
\sum_{i \in J'} \;
\sum_{k \in J'} \;
\sum_{0\neq w \in ( v+L_{\max})}
f( \cdots )
=
\frac{1}{|J|}
\sum_{0\neq w \in ( v+L_{\max})} \;
\sum_{i \in J'} \;
\sum_{k \in J'}
f( \cdots )
\]
\bigskip
{\bf Part 2B:}
We reorder terms by substituting $j$ with $\sigma(i,k)$
where $j=\sigma(i,k)$ is a bijection from $J$ to $J'$ for fixed $i$,
mapping index~$k$ to~$j$.
So Part~2B is equal to
\[
\sum_{i \in J'} \; \sum_{k \in J} \; \sum_{0\neq w \in t^0_i-t^0_{\sigma(i,k)}+L_0} f(\exp(H)[w+t_i-t_{\sigma(i,k)}])
.
\]
The translate $t^0_i-t^0_{\sigma(i,k)}+L_0$ can be written as $-t^0_{\ell}+L_0$ with
$\ell =\sigma(i,j)=\sigma(i,\sigma(i,k))\in J$ depending on~$i$ and~$k$.
So we get for Part~2B:
\[
\sum_{i \in J'} \; \sum_{k \in J} \; \sum_{0\neq w \in -t^0_{\ell}+L_0} f(\exp(H)[w+t_i-t_{\sigma(i,k)}])
\]
with the vectors $w \in -t^0_{\ell}+L_0$ for $\ell \in J$
running through all non-zero elements of the lattice~$ L_{\max}=- L_{\max}$.
A shift of the $w$ by any vectors of $ L_{\max}$ does not effect the outcome
for Part~2B. In particular, for every $j\in J$ we may shift by $t^0_j$ and get the same value
as for Part~2B also in
\[
\sum_{i \in J'} \; \sum_{k \in J} \; \sum_{0\neq w \in t^0_j -t^0_{\ell} +L_0} f( \cdots )
=
\sum_{i \in J'} \; \sum_{k \in J} \; \sum_{0\neq w \in t^0_{\sigma(j,l)}+L_0} f( \cdots )
.
\]
Here, $\sigma(j,l)\in J$ since $j,\ell \in J$
and $f( \cdots )$ abbreviates $f(\exp(H)[w+t_i-t_{\sigma(i,k)}])$.
Since $\bigcup_{j\in J} t^0_{\sigma(j,l)}+L_0 = L_{\max}$ for every fixed~$\ell$, we can take an average over all $j\in J$
and get for Part~2B:
\[
\frac{1}{|J|}
\sum_{i \in J'} \;
\sum_{k \in J} \;
\sum_{0\neq w \in L_{\max}}
f( \cdots )
=
\frac{1}{|J|}
\sum_{0\neq w \in L_{\max}} \;
\sum_{i \in J'} \;
\sum_{k \in J}
f( \cdots )
\]
\bigskip
{\bf Summing all up:}
Finally, we can combine Parts~1A and~2B to get:
\[
\frac{1}{|J|}
\sum_{0\not= w \in L_{\max}} \; \sum_{i \in J\cup J'} \; \sum_{k\in J} \;
f\left( \exp(H) [ w+t_i-t_{\sigma(i,k)}] \right)
\]
Altogether, with Parts 2A and 1B and with the observation $|J|=\frac{m}{2}$,
we get the asserted formula for $E_f(H,t)$.
\end{proof}
\section{The $\mathsf{D}_n^+$ example}\label{sec:four}
For $n\geq 1$ the lattice $\mathsf{D}_n$ consists of all
integral vectors with an even coordinate sum:
$$
\mathsf{D}_n = \left\{ x\in \mathbb{Z}^n \; : \; x_1 + \ldots + x_n \; \mbox{ even } \right\}
$$
The lattice is sometimes also referred to as the {\em checkerboard lattice}.
It gives one of the two families of irreducible {\em root lattices} which exist in every dimension, the other one being~$\mathsf{A}_n$.
The set $\mathsf{D}^+_n$ is defined as the $2$-periodic set
$$
\mathsf{D}^+_n \; = \; \mathsf{D}_n \cup \left( \frac{\mathbb{1}}{2} + \mathsf{D}_n \right)
,
$$
where $\mathbb{1}$ stands for the all-one vector $(1,\ldots , 1)^t $
It is easy to show that $\mathsf{D}^+_n$ is a lattice if and only if $n$ is even,
as the vector $2 \frac{\mathbb{1}}{2}=\mathbb{1}$ is an element of $\mathsf{D}_n$ only
if $n$ is even.
For $n=8$, $\mathsf{D}^+_n$ is equal to the famous root lattice
$\mathsf{E}_8$, with a lot of remarkable properties, not only for
energy minimization (see {\it e.g. } \cite{cs1999}).
For $n=9$, $\mathsf{D}^+_n$ is a $2$-periodic non-lattice set
sharing several of the remarkable properties of~$\mathsf{E}_8$.
It is for instance also a conjectured optimal sphere packing in its dimension,
although as such it is not unique, but part of an infinite family of
``fluid diamond packings'' in dimension~$9$.
Besides its putative optimality for the more general energy
minimization problem (see \cite{PhysRev09}),
$\mathsf{D}^+_9$ has for instance also been found to give the best known set
for the quantization problem, being in particular better than any
lattice in dimension~$9$ (see \cite{ae1998}).
In the following we collect some of the properties of
$\mathsf{D}^+_n$, which are needed in later sections.
We start with its symmetries.
The finite orthogonal group preserving $\mathsf{D}_n$ contains
the {\em hyperoctahedral group},
which is isomorphic to $S_n\rtimes (\mathbb{Z}/2\mathbb{Z})^{n}$,
since every coordinate permutation and every sign flip leaves the parity of
the coordinate sum unchanged.
Only for $n=4$ there exists an additional threefold symmetry
(see {\it e.g. } \cite[Section 4.3]{MR1957723}).
The group $\operatorname{Aut}_0 \mathsf{D}^+_n= \left\lbrace \varphi \in \operatorname{Aut} \mathsf{D}_n \mid \varphi(\mathsf{D}^+_n)=\mathsf{D}^+_n \right\rbrace$, contains all the
coordinate permutations and every even number of sign flips,
so it is a group isomorphic to $S_n\rtimes (\mathbb{Z}/2\mathbb{Z})^{n-1}$.
This is precisely the {\em Weyl group} $W(\mathsf{D}_n)$ of the $\mathsf{D}_n$ {\em root system}
(the minimal vectors of $\mathsf{D}_n$).
For even $n$, this gives all automorphisms of the lattice
$\mathsf{D}^+_n$
(see {\it loc. cit. }), {\it i.e. } we have $\operatorname{Aut} \mathsf{D}^+_n=\operatorname{Aut}_0 \mathsf{D}^+_n=W(\mathsf{D}_n)$.
For odd $n$, the maximal period lattice of $\mathsf{D}^+_n$ is $\mathsf{D}_n$, and it
follows from the discussion in Remark~\ref{rem:auto-discuss} that $\operatorname{Aut}_0
\mathsf{D}^+_n=W(\mathsf{D}_n)$ has index $2$ in $\operatorname{Aut} \mathsf{D}^+_n=W(\mathsf{D}_n) \cup -W(\mathsf{D}_n)$. The
orthogonal automorphisms of $\mathsf{D}^+_n$ coming from $W(\mathsf{D}_n)$ correspond to
affine isometries fixing $0$ and $\frac{\mathbb{1}}{2}$
modulo $L_{\max}=\mathsf{D}_n$ while those from $ -W(\mathsf{D}_n)$ correspond to affine isometries exchanging $0$ and $\frac{\mathbb{1}}{2}$. In particular, all non-empty shells $\Lambda_x(r)$ of $\mathsf{D}_n^+$ are fixed by $W(\mathsf{D}_n)$. We will take advantage of this invariance property in the sequel, using classical results about the invariant theory of the Weyl group $W(\mathsf{D}_n)$.
\begin{proposition}
Every non-empty shell $\Lambda(r)$ and $\Lambda_x(r)$ of $\Lambda=\mathsf{D}_n^+$
forms a spherical $3$-design.
\end{proposition}
\begin{proof}
For a finite set $X$ on a sphere of radius~$r$ being a spherical $3$-design is
equivalent to
$$
\sum_{x\in X} ( y^t x )^2 = c \cdot ( y^t y )
\quad \mbox {and} \quad
\sum_{x\in X} ( y^t x )^3 = 0
$$
for some constant $c$ and any $y \in \mathbb{R}^n$.
The first property is actually that of a $2$-design.
It is satisfied for any set $X$ which is invariant under a group that
acts irreducibly on $\mathbb{R}^n$ (see \cite[Theorem 3.6.6.]{MR1957723} where the synonymous expression "strongly eutactic configuration" is used ) .
The second property is satisfied, since the Weyl group of the root system~$\mathsf{D}_n$
has no non-zero invariant homogeneous polynomials of degree~$3$
(see \cite[\S 3.7, Table 1]{MR1066460}).
\end{proof}
\begin{rem}
For half-integral $r$ the shells $\Lambda_x(r)$ are not centrally symmetric and therefore the $2$-design property does not immediately imply the $3$-design property.
\end{rem}
As a consequence of the preceding proposition, $\mathsf{D}_n^+$
satisfies the properties of Theorem~\ref{th:periodic-critical}: the shells
$\Lambda_x(r)$ are balanced and $\Lambda(r)$ is a spherical $2$-design for all~$r$.
Consequently, $\mathsf{D}_n^+$ is $f_c$-critical for any~$c>0$. On the other hand, the shells
are not $4$-designs in general, as can be checked numerically for small $r$. If they
were, then the study of the Hessian in the following section
would be significantly simpler, in the spirit of what
was done in~\cite{MR2889159}.
\section{The Hessian of $2$-periodic sets
and in particular of $\mathsf{D}_n^+$} \label{sec:hessian}
For $f(r)=e^{-cr}$, we consider the Hessian of $E_{f_c}(H,\mathbf{t})$ at a $2$-periodic set $L_{\max}\cup(v+L_{\max})$ given by an
$m$-periodic representation $\bigcup_{i=1}^m (t^0_i + L_0)$. We will then use the obtained expression for the Hessian to analyze whether or not $\mathsf{D}_n^+$ is a local minimum among $m$-periodic sets.
According to Section \ref{sec:two} this Hessian is equal to
\begin{equation}\label{eq:splithess}
\frac{c}{m}
\sum_{r>0} \left[I(r)+II(r)+III(r)\right] e^{-c r^2}
\end{equation}
where
\begin{align*}
I(r)&=\sum_{1\leq i,j\leq m}\;\sum_{w \in t^0_i-t^0_j+L_0, \|w \|=r}2c \left( w^{t}(t_i-t_j)\right)^2 - \| t_i-t_j\|^2\\
II(r)&=\sum_{1\leq i,j\leq m}\;\sum_{w \in t^0_i-t^0_j+L_0, \|w \|=r}- 2w^{t}H(t_i-t_j)+2cw^{t}(t_i-t_j)H[w]\\
III(r)&=\sum_{1\leq i,j\leq m}\;\sum_{w \in t^0_i-t^0_j+L_0, \|w
\|=r}\frac{c}{2}H[w]^2-\frac{1}{2}H^2\br{w}
\end{align*}
In this decomposition we distinguish three types of terms: purely translational terms ($I(r)$), mixed terms ($II(r)$) and purely lattice changing terms ($III(r)$). Note that we can reorder individually each of these three terms according to Lemma \ref{lem:ReorderingLemma}. In particular we will use that
\begin{eqnarray*}
I(r)
& = &
\frac{2}{m}\left[
\left(
\sum_{\substack{0\not= w \in L_{\max}\\ \|w\| = r}} \; \sum_{i=1}^m \; \sum_{k\in J} \;
2c \left( w^{t}(t_i-t_{\sigma(i,k)})\right)^2 - \| t_i-t_{\sigma(i,k)}\|^2
\right)
\right. \\
& & \qquad +
\left(
\sum_{\substack{0\not= w \in -(v+ L_{\max})\\ \|w\| = r}} \; \sum_{i\in J} \; \sum_{k\in J'} \;
2c \left( w^{t}(t_i-t_{\sigma(i,k)})\right)^2 - \| t_i-t_{\sigma(i,k)}\|^2
\right) \\
& & \qquad +
\left.
\left(
\sum_{\substack{0\not= w \in (v+ L_{\max})\\ \|w\| = r}} \; \sum_{i\in J'} \; \sum_{k\in J'} \;
2c \left( w^{t}(t_i-t_{\sigma(i,k)})\right)^2 - \| t_i-t_{\sigma(i,k)}\|^2
\right)
\right],
\end{eqnarray*}
where we assume that $t^0_i\in L_{\max}$ for $i\in J=\{1,\ldots,\frac{m}{2}\}$, and $t^0_i \in v + L_{\max}$ for $i\in J'=\{\frac{m}{2}+1,\ldots, m\}$, which finally simplifies to
\begin{equation}
\begin{aligned}
I(r)
&=
\frac{2}{m}\left[
\left(
\sum_{\substack{0\not= w \in L_{\max}\\ \|w\| = r}} \; \sum_{i=1}^m \; \sum_{k\in J} \;
2c \left( w^{t}(t_i-t_{\sigma(i,k)})\right)^2 - \| t_i-t_{\sigma(i,k)}\|^2
\right)
\right. \\
&\qquad +
\left.
\left(
\sum_{\substack{0\not= w \in (v+ L_{\max})\\ \|w\| = r}} \; \sum_{i=1}^m \; \sum_{k\in J'} \;
2c \left( w^{t}(t_i-t_{\sigma(i,k)})\right)^2 - \| t_i-t_{\sigma(i,k)}\|^2
\right)
\right]
\end{aligned}
\end{equation}
since the inner sums are invariant towards negation of~$w$.
\subsection{Purely translational terms for $\mathsf{D}_n^+$}\label{ptt}
This formula simplifies for $\Lambda=\mathsf{D}^+_n$ with odd $n$ since the elements of a given non-²empty shell $\Lambda(r)$ are either all contained in $\mathsf{D}_n$
or in $\pm \frac{\mathbb{1}}{2} + \mathsf{D}_n$, depending on wether $r$ is integral or half-integral.
This gives us two cases to consider:
In one case, assuming $\Lambda(r)\subset \mathsf{D}_n$ we get, for fixed $r>0$,
\begin{equation} \label{eqn:CaseI_TranslationalPart}
I(r)=\frac{2}{m}\sum_{w\in \mathsf{D}_n, \|w\|=r} \;
\sum_{i=1}^m \;
\sum_{k=1}^{m/2} \;
2c
\left( w^{t}(t_i-t_{\sigma{(i,k)}})\right)^2
- \| t_i-t_{\sigma{(i,k)}}\|^2
\end{equation}
and in the other case,
\begin{equation} \label{eqn:CaseI_TranslationalPartbis}
I(r)=\frac{2}{m}\sum_{w\in \left(\frac{\mathbb{1}}{2} + \mathsf{D}_n\right), \|w\|=r} \;
\sum_{i=1}^m \;
\sum_{k=m/2+1}^{m} \;
2c
\left( w^{t}(t_i-t_{\sigma{(i,k)}})\right)^2
- \| t_i-t_{\sigma{(i,k)}}\|^2
\end{equation}
In both cases, we can use the relation
\begin{align}\label{eqn:TraceRewriteTranslationalTerms}
\left( w^{t}(t_i-t_{\sigma(i,k)})\right)^2
& = w^t (t_i-t_{\sigma(i,k)}) (t_i-t_{\sigma(i,k)})^t w\\
\notag & =\Tr \left( (t_i-t_{\sigma(i,k)}) (t_i-t_{\sigma(i,k)})^t (w w^t) \right).
\end{align}
to simplify the part of the sum involving $w$.
Using the linearity of the trace we get in the first case, that is if $\Lambda(r)\subset \mathsf{D}_n$,
\begin{equation}\label{tr1}
\begin{split}
&\sum_{w\in \mathsf{D}_n, \|w\|=r} \;\sum_{i=1}^{m} \;
\sum_{k=1}^{m/2} \;
2c
\left( w^{t}(t_i-t_{\sigma{(i,k)}})\right)^2\\
=&\quad 2c
\Tr
\left(
\left(
\sum_{i=1}^m \;
\sum_{k=1}^{m/2} \;
(t_i-t_{\sigma{(i,k)}}) (t_i-t_{\sigma{(i,k)}})^t
\right)
\sum_{w\in \mathsf{D}_n, \|w\|=r} \;
(w w^t)
\right)
\end{split}
\end{equation}
and in the second case,
\begin{equation}\label{tr2}
\begin{split}
&\sum_{w\in \left( \frac{\mathbb{1}}{2} + \mathsf{D}_n\right), \|w\|=r} \;\sum_{i=1}^{m} \;
\sum_{k=m/2+1}^{m} \;
2c
\left( w^{t}(t_i-t_{\sigma{(i,k)}})\right)^2\\
=&\quad 2c
\Tr
\left(
\left(
\sum_{i=1}^m \;
\sum_{k=m/2+1}^{m} \;
(t_i-t_{\sigma{(i,k)}}) (t_i-t_{\sigma{(i,k)}})^t
\right)
\sum_{w\in \left( \frac{\mathbb{1}}{2} + \mathsf{D}_n\right), \|w\|=r} \;
(w w^t)
\right).
\end{split}
\end{equation}
Using the $2$-design property of the shell $\Lambda(r)$
(see \eqref{2des}), and noticing that a typical element $w$ of $\mathsf{D}^+_n$ has weight
$$\nu(w)=\begin{cases}1 \text{ if } w \in \mathsf{D}_n\\ \frac{1}{2} \text{ if }w \in \frac{\mathbb{1}}{2} + \mathsf{D}_n \end{cases}$$
we may substitute
$\displaystyle\sum_{w\in \mathsf{D}_n, \|w\|=r} \; (w w^t)$
and $\displaystyle\sum_{w\in \left( \frac{\mathbb{1}}{2} + \mathsf{D}_n\right), \|w\|=r} \;
(w w^t)$ by
$\frac{r^2|\Lambda(r)|}{n} \id_n$. Therefore, formula \eqref{tr1} and \eqref{tr2} simplify respectively to
$$
\frac{2c r^2|\Lambda(r)|}{n}
\sum_{i=1}^m \;
\sum_{k=1}^{m/2} \;
\| t_i-t_{\sigma{(i,k)}}) \|^2
\text{ and }
\frac{2c r^2|\Lambda(r)|}{n}
\sum_{i=1}^m \;
\sum_{k=m/2+1}^{m} \;
\| t_i-t_{\sigma{(i,k)}} \|^2.
$$
We finally get
$$
I(r)=\frac{2}{m}\left( \frac{2c r^2}{n}-1 \right) |\Lambda(r)|
\sum_{i=1}^m \;
\sum_{k=1}^{m/2} \;
\| t_i-t_{\sigma{(i,k)}} \|^2
$$
in the first case ($\Lambda(r)\subset \mathsf{D}_n$) and
$$
I(r)=\frac{2}{m}\left( \frac{2c r^2}{n}-1 \right) |\Lambda(r)|
\sum_{i=1}^m \;
\sum_{k=m/2+1}^{m} \;
\| t_i-t_{\sigma{(i,k)}} \|^2
$$
in the second case.
In both cases, this is nonnegative for all $c \geq \frac{n}{2r^2}$.
As $r^2\geq 2$ for $\mathsf{D}^+_n$ with $n\geq 8$, we overall find
for $\Lambda=\mathsf{D}^+_n$
that the purely translational terms are nonnegative
for all $c \geq \frac{n}{4}$ and $n\geq 8$.
\subsection{Mixed terms}\label{mt}
There are two different mixed terms in $II(r)$:
The first one is the sum over terms $w^{t}H(t_i-t_j)$
and the second one is the sum over terms $w^{t}(t_i-t_j)H[w]$.
The first sum evaluates to~$0$ for balanced configurations
as it can be reordered as follows:
$$
\sum_{i,j}\sum_{w \in t^0_i-t^0_j+L_0 , \|w\|=r} w^{t}H(t_i-t_j)
=
\Tr \left( H \cdot ( \sum_{i,j}\sum_{w \in t^0_i-t^0_j+L_0 , \|w\|=r} (t_i-t_j)w^t ) \right)
$$
with
$$
\sum_{i,j}\sum_{w \in t^0_i-t^0_j+L_0 , \|w\|=r} (t_i-t_j)w^t
\;
=
\;
\sum_{1\leq k\leq m}
t_k
\left(
\; \sum_{u \in t^0_k-t^0_{\ell}+L_0 \, \atop \mbox{\footnotesize for some } \, 1\leq l \leq m , \; \|u\| = r} \;
u
\right)
$$
as seen in the proof of Theorem~\ref{th:periodic-critical}.
Thus for balanced shells $\Lambda(r)$ this part of the Hessian vanishes.
\bigskip
For the second sum of mixed terms over a fixed shell we get:
\begin{eqnarray*}
& &
\sum_{i,j}\sum_{w \in t^0_i-t^0_j+L_0 , \|w\|=r}
w^{t}(t_i-t_j)H[w]
\\
& = &
\sum_{i,j}\sum_{w \in t^0_i-t^0_j+L_0 , \|w\|=r}
H[w] w^{t} t_i
-
\sum_{i,j}\sum_{w \in t^0_i-t^0_j+L_0 , \|w\|=r}
H[w] w^{t} t_j
\\
& = &
\sum_{i}
\left( \sum_j \sum_{w \in t^0_i-t^0_j+L_0 , \|w\|=r} H[w] w^{t} \right) t_i
-
\sum_{j}
\left( \sum_i \sum_{w \in t^0_i-t^0_j+L_0 , \|w\|=r} H[w] w^{t} \right) t_j
\\
& = &
\sum_{i}
\left( \sum_{w\in -\Lambda_{t_i^0}, \|w\|=r} H[w] w^{t} \right) t_i
-
\sum_{j}
\left( \sum_{w\in \Lambda_{t_j^0}, \|w\|=r} H[w] w^{t} \right) t_j
\\
& = &
-2 \sum_{i}
\left( \sum_{w\in \Lambda_{t_i^0}, \|w\|=r} H[w] w^{t} \right) t_i
\end{eqnarray*}
Here the inner sum is a homogeneous degree~$3$ polynomial in $w$ evaluated
on the shell $\Lambda_{t_i^0}(r)$. Since these shells are $3$-designs
for $\Lambda=\mathsf{D}_n^+$, the inner sum vanishes for all shells of
$\mathsf{D}_n^+$. Indeed, any degree $3$ homogeneous polynomial $P(w)$
decomposes uniquely as a sum $P(w)=F(w)+\|w\|^2 G(w)$ where $F(w)$ is
a harmonic degree $3$ polynomial and $G(w)$ is a linear
form. Consequently the sum $\sum_{w \in X} P(w)$, where $X$ is any
spherical $3$-design contained in a sphere of radius~$r$, reduces to
$$\sum_{w \in X} P(w)=\sum_{w \in X} F(w)+r^2\sum_{w \in X} G(w)$$
and both the sums $\sum_{w \in X} F(w)$ and $\sum_{w \in X} G(w)$
vanish from the $3$-design property.
\subsection{Purely lattice changing terms in the case of $\mathsf{D}_n^+$}
\label{sec:plc}
It remains to look at the sum $III(r)$, which we can also write as
\begin{equation}\label{eq:h1}
III(r)=m\sum_{w \in \Lambda_r} \nu(w)\left(
\frac{c}{2}H[w]^2-\frac{1}{2}H^2\br{w} \right)
.
\end{equation}
This sum corresponds to an effect coming from local changes of
the underlying lattice~$L_0$, respectively of $\mathsf{D}_n$ in case of
$\mathsf{D}_n^+$.
The sum of the terms $H^2[w]$ over any given shell simplifies to
\begin{equation}\label{eq:h2}
m\sum_{w\in \Lambda(r)}\nu(w)H^2\br{w}
\; = \; m\,\nu_r\dfrac{r^2 |\Lambda(r)|}{n}\Tr H^2
\end{equation}
because of the weighted-$2$-design property, where
$\nu_r=\dfrac{\sum_{w \in \Lambda(r)} \nu(w)}{|\Lambda(r)|}$ is the
average weight on~$\Lambda(r)$. In the case of $\mathsf{D}_n^+$, the
weight is constant ($1$ or $\frac{1}{2}$) on each $\Lambda(r)$, so that \eqref{eq:h1} simplifies to
\begin{equation}\label{eq:h3}
m \, \nu_r \sum_{w \in \Lambda(r)} \left(
\frac{c}{2}H[w]^2-\frac{1}{2}H^2\br{w} \right)
\; = \; m \, \nu_r
\sum_{w \in \Lambda(r)} \left( \frac{c}{2}H[w]^2\right)
-m \, \nu_r\dfrac{r^2 |\Lambda(r)|}{n}\Tr H^2.
\end{equation}
For the terms involving $(H[w])^2$, we note that, for any positive $r$, the
polynomial $\sum_{w \in \Lambda(r)}H[w]^2$
is a quadratic $G$-invariant polynomial
in $H$, where $G=\operatorname{Aut}_0(\mathsf{D}_n^+)=W(\mathsf{D}_n)$. We will make use of the following
classical result about the polynomial invariants of $G$.
\begin{lemma} \label{lem:quadratic_poly_on_sd}
Let $n\geq 5$.
Then any homogeneous quadratic polynomial
on the space ${\mathcal S}^{n}$ of symmetric $n\times n$ matrices $H=(h_{ij})$,
which is invariant under the Weyl group
of $\mathsf{D}_n$
(acting on ${\mathcal S}^{n}$ by $H\mapsto M^t H M$ by the $n\times n$ permutation matrices and
diagonal matrices~$M$ having an even number of $-1$s and $1$s otherwise on the diagonal),
is a linear combination of the
three quadratic polynomials
\[
(\Tr H)^2 = \sum_{i,j=1}^{n}h_{jj}h_{ii}, \quad
\Tr (H^2) = \sum_{i,j=1}^{n}h^2_{ij} \quad
\mbox{and} \quad
\sum_{i<j} h^2_{ij}
.
\]
\end{lemma}
\begin{proof}
Since we are not aware of a pinpoint reference for this statement, we give a short argument here for the convenience of the reader. The homogeneous quadratic polynomials on ${\mathcal S}^{n}$
have seven types of monomials
(where different indices are actually chosen to be different):
\[
h_{ii}^2, \quad
h_{ii} h_{jj}, \quad
h_{ii} h_{ij}, \quad
h_{ii} h_{jk}, \quad
h_{ij}^2, \quad
h_{ij} h_{ik}, \quad
h_{ij} h_{kl}
\]
Note that there are less of these monomials for $n=2,3$.
From the invariance towards permutation matrices we can conclude
that coefficients in front of any given type of monomials have to be the same.
From the invariance towards
diagonal matrices with an even number of $-1$s (and $1$s otherwise)
we then deduce that only monomials of the three types
$h_{ii}^2$, $h_{ii}h_{jj}$ and $h_{ij}^2$ are invariant under the Weyl group of $\mathsf{D}_n$.
Among the others, some monomials are mapped to their negatives.
The only exception is the case $n=4$, where also the set of monomials of the type
$h_{ij} h_{kl}$ is invariant under the action of the group.
\end{proof}
Note that the lemma and its proof can be adapted to the description of the space of quadratic $G$-invariant differential operators on functions with matrix argument. In particular, this space has dimension $3$. Using the local system of coordinates $h_{ij}, \, 1 \leq i \leq j \leq n$, of ${\mathcal S}^{n}$ and denoting by $\partial_{ij}$ the partial derivative with respect to $h_{ij}$, a spanning system is given by
\begin{align*}
\delta_{1}&=\dfrac{1}{n(n-1)}\sum_{i<j}\partial_{ii}\partial_{jj},\\
\delta_{2}&=\dfrac{1}{2n}\sum_{i}\partial^2_{ii}-\dfrac{1}{n(n-1)}\sum_{i<j}\partial_{ii}\partial_{jj}\quad\text{ and } \\
\delta_{3}&=-\dfrac{1}{n}\sum_{i}\partial^2_{ii}+\dfrac{2}{n(n-1)}
\sum_{i<j}\partial_{ii}\partial_{jj}+\dfrac{1}{n(n-1)}\sum_{i<j}\partial_{ij}^2.
\end{align*}
This particular basis satisfies the relations $$\delta_i(F_j)=\delta_{ij},\quad 1 \leq i,j \leq 3$$ for $$F_1(H)=(\Tr
H )^2=\left(\sum_{i=1}^n h_{ii}\right)^2,$$ $$F_2(H)=\Tr H^2=\sum_{i=1}^n h_{ii}^2+2\displaystyle\sum_{1\leq i<j\leq n}h_{ij}^2$$ and $$F_3(H)=\displaystyle\sum_{1 \leq i<j\leq n} h_{ij}^2.$$
For any positive $r$, the polynomial $\sum_{w \in \Lambda(r)}H[w]^2$
is a quadratic $G$-invariant polynomial in $H$. As such, it is a linear combination
\begin{equation}\label{abc}
\sum_{w \in \Lambda(r)}H[w]^2=\alpha_r F_1(H)
+\beta_r F_2(H) +\gamma_r F_3(H)
\end{equation}
for some constants $\alpha_r$, $\beta_r$ and $\gamma_r$ to be computed.
To compute the constants $\alpha_r$, $\beta_r$ and $\gamma_r$ in \eqref{abc}, it suffices to evaluate $\delta_i(\sum_{w \in \Lambda(r)}H[w]^2)$ for $1\leq i \leq 3$: setting $Z_r =\sum_{w \in \Lambda(r)}\left(\sum_{i=1}^n w_i^4\right)$, one has
\begin{align*}
\alpha _r &=\delta_1(\sum_{w \in \Lambda(r)}H[w]^2)=\dfrac{1}{n(n-1)}\left(r^4\vert \Lambda(r)\vert - Z_r\right)\\
\beta_r &=\delta_2(\sum_{w \in \Lambda(r)}H[w]^2)=\dfrac{1}{n-1}Z_r -\dfrac{1}{n(n-1)}r^4\vert\Lambda(r)\vert\\
\gamma_r &=\delta_3(\sum_{w \in \Lambda(r)}H[w]^2)=-2\dfrac{n+2}{n(n-1)}Z_r+\dfrac{6}{n(n-1)}r^4\vert\Lambda(r)\vert.
\end{align*}
\bigskip
We are now in the position to estimate $III(r)$.
Recall that we restrict to $H$ with $\Tr H =0$, in which case $F_1(H)=0$. Using the above formulas, the relation $\Tr H^2= \sum_i h_{ii}^2 + 2\sum_{i<j}h_{ij}^2$ and formula \eqref{eq:h2}, we get
\begin{align*}
\dfrac{1}{m \, \nu_r}III(r)&=\left(c\beta_r+\dfrac{c\gamma_r}{2}-\dfrac{r^2}{n}\vert\Lambda(r)\vert\right)\sum_{i<j}h_{ij}^2 + \dfrac{1}{2}\left(c\beta_r-\dfrac{r^2}{n}\vert\Lambda(r)\vert\right)\sum_i h_{ii}^2\\
&=\left(\dfrac{2c}{n(n-1)}\left(r^4\vert \Lambda(r)\vert - Z_r\right)-\dfrac{r^2}{n}\vert\Lambda(r)\vert\right)\sum_{i<j}h_{ij}^2\\& \qquad + \dfrac{1}{2}\left(\dfrac{c}{n-1}\left(Z_r-\dfrac{r^4}{n}\vert \Lambda(r)\vert\right)-\dfrac{r^2}{n}\vert\Lambda(r)\vert\right) \sum_i h_{ii}^2.
\end{align*}
In order that $III(r)$ be positive, it is enough that the coefficients of $
\sum_{i<j}h_{ij}^2 $ and $\sum_i h_{ii}^2 $ are positive. This is of course impossible
for small $c$, but as we show below, it is achievable for big enough $c$. To see this, we
introduce the polynomial $$P(x)=\sum_{i=1}^n x_i^4
-\dfrac{3}{n+2}\left(\sum_{i=1}^n x_i^2\right)^2$$ which is readily seen to be
harmonic. As a consequence of Proposition \ref{mod}, the average theta series $f(\tau)=\theta_{\Lambda,P}(\tau)$ is a
cusp modular form of weight $k=\dfrac{n}{2}+4$, and its Fourier coefficients $a_r(f)$ are "small", in a sense to be
made more precise.
Finally, from the relation
$Z_r=a_r(f)+\dfrac{3}{n+2}r^4\vert \Lambda(r)\vert$, we can rewrite the coefficients of
$ \sum_{i<j}h_{ij}^2$ and $\sum_i h_{ii}^2$ in the expression for
$\dfrac{1}{m\, \nu_r}III(r)$ as
\begin{equation}\label{coeff1}
\dfrac{2c}{n(n+2)}r^4\vert\Lambda(r)\vert\left[\left(
1-\dfrac{n+2}{2cr^2}\right)-\dfrac{n+2}{n-1}\,\dfrac{a_r(f)}{r^4\vert\Lambda(r)\vert}\right]
\end{equation}
and
\begin{equation}\label{coeff2}
\dfrac{c}{n(n+2)}r^4\vert\Lambda(r)\vert\left[\left(
1-\dfrac{n+2}{2cr^2}\right)+\dfrac{n(n+2)}{2(n-1)}\,\dfrac{a_r(f)}{r^4\vert\Lambda(r)\vert}\right].
\end{equation}
Note that if all shells $\Lambda(r)$ were spherical $4$-designs, then
$a_r(f)$ would be zero, and the above coefficients would be positive
for any $c>\dfrac{n+2}{4}$. As mentioned before, not all shells of
$\mathsf{D}_n^{+}$ do have the $4$-design property. We can nevertheless
obtain the same conclusion, using some classical estimates on the
growth of the coefficients of cusp forms:
\begin{lemma}\label{cusp}
For any $r>0$ such that the shell $\Lambda(r)$ of $\mathsf{D}_n^+$ is non-empty one has
$$\dfrac{a_r(f)}{r^4\vert\Lambda(r)\vert}=\mathcal{O}\left(r^{-\frac{n}{2}+2}\right).$$
\end{lemma}
\begin{proof}
Using elementary bounds on the size of coefficients of cusp forms (see {\it e.g. }
\cite[(5.7)]{MR1474964}) we see that
$$a_r(f)=\mathcal{O}\left(r^{\frac{n}{2}+4}\right).$$
As for the size of $\Lambda(r)$ we can use classical estimates on the number of
representations by quadratic forms (see {\it e.g. } \cite[chapter 11 ]{MR1474964}) . For
shells $\Lambda(r)$ which are contained in $\mathsf{D}_n^{+}$, corresponding to $r$ such
that $r^2$ is integral, one can apply Corollary 11.3 of \cite{MR1474964} to
conclude that $\vert\Lambda(r)\vert \asymp r^{n-2}$.
For shells contained in
$(-\frac{\mathbb{1}}{2} + \mathsf{D}_n) \cup
(\frac{\mathbb{1}}{2} + \mathsf{D}_n)$
the same argument
applies since these shells are indeed shells of the lattice $\mathsf{D}_n^{*}$.
Altogether, we obtain the desired estimate for the quotient
$\dfrac{a_r(f)}{r^4\vert\Lambda(r)\vert}$.
\end{proof}
\section{Concluding remarks} \label{sec:conclusion}
\begin{theorem} \label{thm:final}
Let $n$ be an odd integer $\geq 9$. Then there exists a constant $c_n$ such that $\mathsf{D}_n^{+}$ is locally $f_c$-optimal for any $c > c_n$.
\end{theorem}
\begin{proof}
This is mainly the collection of facts proven before: we know from Section~\ref{ptt}
that the purely translational part of the hessian is $>0$ as
soon as $c >\frac{n}{4}$
and that the mixed terms vanish (Section~\ref{mt}). As for the pure lattice changes, the sign of their contribution is governed by that of \eqref{coeff1} and \eqref{coeff2}, which is positive if $c$
is big enough, thanks to Lemma \ref{cusp}.
\end{proof}
For $n=9$, the result of Theorem~\ref{thm:final} is of course not fully
satisfactory as one would expect local $f_c$-optimality to hold for
any $c>0$, in accordance with the conjecture and experimental results
about $\mathsf{D}_9^{+}$ mentioned at the beginning of this paper. A strategy
to get such a universal local optimality result --- which we used in
\cite{MR2889159} for the lattices $\mathsf{A}_2$, $\mathsf{D}_4$
an $\mathsf{E}_8$ --- is roughly speaking as follows:
First one proves local extremality for all $c$ bigger than an
\emph{explicit} $c_0$ (as small as possible, but certainly not $0$!),
and then, if $c_0$ is small enough, one can use self-duality together
with the Poisson summation formula to switch from "big $c$'' to "small
$c$" (see \cite{MR2889159} for details). In our situation here, there are
two difficulties in applying this strategy. First, as explained in
\cite{MR3289409}, there is no good notion of duality, let alone
self-duality, and the Poisson summation formula for general periodic
sets. This first obstruction seems unavoidable, and incidentally one
does not expect universal local optimality of $\mathsf{D}_n^{+}$ for general
$n\geq 8$. But fortunately
the $2$-periodic set $\mathsf{D}_n^{+}$ (with $n$ odd)
is precisely one instance of a non-lattice configuration
for which a formal self-duality holds together with a Poisson formula
(see~\cite{PhysRev09}).
So it is not hopeless to overcome this first obstruction in this
particular case.
The second impediment, not theoretical in nature but really critical
in practice, is the need for an \emph{explicit} threshold $c_0$. To
this end, one needs an effective version of Lemma \ref{cusp}, {\it i.e. }
effective bounds for the coefficients of the cusp form involved, in
the spirit of \cite{MR2854563} for instance.
But those seem to be quite difficult in our case,
given that the cusp form $\theta_{\Lambda,P}(\tau)$ has half-integral weight.
Here, further research appears to be necessary.
\section*{Acknowledgments}
Both authors were supported
by the Erwin-Schr\"odinger-Institute (ESI) during a stay in fall 2014 for
the program on Minimal Energy Point Sets, Lattices and Designs.
The second author gratefully acknowledges support by
DFG grant SCHU 1503/7-1.
The authors like to thank Jeremy Rouse,
Frieder Ladisch and Robert Sch\"uler for several valuable
remarks.
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| {
"timestamp": "2018-02-07T02:11:55",
"yymm": "1802",
"arxiv_id": "1802.02072",
"language": "en",
"url": "https://arxiv.org/abs/1802.02072",
"abstract": "We study the local optimality of periodic point sets in $\\mathbb{R}^n$ for energy minimization in the Gaussian core model, that is, for radial pair potential functions $f_c(r)=e^{-c r}$ with $c>0$. By considering suitable parameter spaces for $m$-periodic sets, we can locally rigorously analyze the energy of point sets, within the family of periodic sets having the same point density. We derive a characterization of periodic point sets being $f_c$-critical for all $c$ in terms of weighted spherical $2$-designs contained in the set. Especially for $2$-periodic sets like the family $\\mathsf{D}^+_n$ we obtain expressions for the hessian of the energy function, allowing to certify $f_c$-optimality in certain cases. For odd integers $n\\geq 9$ we can hereby in particular show that $\\mathsf{D}^+_n$ is locally $f_c$-optimal among periodic sets for all sufficiently large~$c$.",
"subjects": "Metric Geometry (math.MG); Mathematical Physics (math-ph); Number Theory (math.NT)",
"title": "Local Energy Optimality of Periodic Sets",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754447499796,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.7075103936602921
} |
https://arxiv.org/abs/2107.14713 | Crowns in linear $3$-graphs | A \textit{linear $3$-graph}, $H = (V, E)$, is a set, $V$, of vertices together with a set, $E$, of $3$-element subsets of $V$, called edges, so that any two distinct edges intersect in at most one vertex. The linear Turán number, ${\rm ex}(n,F)$, is the maximum number of edges in a linear $3$-graph $H$ with $n$ vertices containing no copy of $F$.We focus here on the \textit{crown}, $C$, which consists of three pairwise disjoint edges (jewels) and a fourth edge (base) which intersects all of the jewels. Our main result is that every linear $3$-graph with minimum degree at least $4$ contains a crown. This is not true if $4$ is replaced by $3$. In fact the known bounds of the Turán number are \[ 6 \left\lfloor{\frac{n - 3}{4}}\right\rfloor \leq {\rm ex}(n, C) \leq 2n, \] and in the construction providing the lower bound all but three vertices have degree $3$. We conjecture that ${\rm ex}(n, C) \sim \frac{3n}{2}$ but even if this were known it would not imply our main result.Our second result is a step towards a possible proof of ${\rm ex}(n,C) \leq \frac{3n}{2}$ (i.e., determining it within a constant error). We show that a minimal counterexample to this statement must contain certain configurations with $9$ edges and we conjecture that all of them lead to contradiction. |
\section{Introduction}
A \textit{$3$-graph}, $H = (V,E)$, is a set, $V$, whose elements are called points or vertices together with a set, $E$, of $3$-element subsets of $V$ called edges. If not clear from the context, we use the notation $V(H)$ and $E(H)$ for $V$ and $E$ respectively. We restrict ourselves to the important family of \textit{linear} $3$-graphs where \textit{any two distinct edges intersect in at most one vertex}. In the remainder of this paper we use the term $3$-graph for linear $3$-graph.
The number of edges containing a point $v \in V(H)$ is the \textit{degree} of $v$ and is denoted by $d(v)$ or $d_H(v)$. We denote by $\delta (H)$ the minimum degree of $H$. Similar notations are used for graphs ($2$-uniform linear hypergraphs). We use $[k]$ to denote $\{1, \ldots, k\}$.
Let $F$ be a fixed $3$-graph. A $3$-graph, $H$, is called \textit{$F$-free} if $H$ has no subgraph isomorphic to $F$. The \textit{(linear) Tur\'an number of $F$}, ${\rm ex}(n,F)$, is the maximum number of edges in an $F$-free $3$-graph on $n$ vertices.
The behavior of ${\rm ex}(n,F)$ is interesting even if $F$ has three or four edges. A famous theorem of Ruzsa and Szemer\'edi \cite{RSZ} is that ${\rm ex}(n,T) = o(n^2)$ if $T$ is the \textit{triangle}. For the \textit{Pasch configuration}, $P$, ${\rm ex}(n,P) = \frac{n (n-1)}{6}$ for infinitely many $n$ since there are $P$-free Steiner triple systems (see \cite{CR}). For the \textit{fan}, $F$, we have ${\rm ex}(n,F) = \frac{n^{2}}{9}$ if $n$ is divisible by $3$ (see \cite{FGY}). Figure \ref{fig-ex} shows these $3$-graphs (drawn with the convention that edges are represented as straight line segments).
\begin{figure}[H]
\centering
\includegraphics[width=0.6\linewidth]{figures/triangle-pasch-fan}
\caption{Triangle, Pasch configuration, and Fan}
\label{fig-ex}
\end{figure}
This paper is related to the Tur\'an number of the \textit{crown}, $C$ (Figure \ref{fig-crown}).
\begin{figure}[H]
\centering
\includegraphics[width=0.25\linewidth]{figures/crown}
\caption{The crown}
\label{fig-crown}
\end{figure}
It seems that $C$ deserves a descriptive name, the list of small configurations in \cite{CR} refers to it as $C_{13}$. We call the horizontal edge of the crown the \textit{base} and the vertical edges \textit{jewels}. The crown is the smallest $3$-tree with unknown Tur\'an number, the following bounds are from \cite{GYRS}:
\begin{equation} \label{crownbound}
6 \floor{\frac{n - 3}{4}} \leq {\rm ex}(n, C) \leq 2n.
\end{equation}
The construction for the lower bound in (\ref{crownbound}) (for the case $n \equiv 3 \pmod 4$) is the following. Choose three vertices $\{ a, b, c \}$, and define edges
\[ (a,x_i,y_i), \ (a,z_i,w_i), \ (b,x_i,w_i), \ (b,y_i,z_i), \ (c,x_i,z_i), \ (c,y_i,w_i) \]
where $i = 1, 2, \ldots, \floor{ (n - 3) / 4}$ and $x_i$, $y_i$, $z_i$, and $w_i$ are distinct vertices.
In this construction (overlay of Fano planes) all but three vertices have degree $3$. This poses the question whether raising the minimum degree of a $3$-graph $H$ from $3$ to $4$ ensures a crown. Our main result is an affirmative answer.
\begin{theorem} \label{main-theorem}
Every $3$-graph with minimum degree $\delta(H) \geq 4$ contains a crown.
\end{theorem}
It is worth noting that even if ${\rm ex}(n, C) \leq \frac{3n}{2}$ were known, Theorem \ref{main-theorem} would not follow since minimum degree $4$ ensures only $\frac{4n}{3} < \frac{3n}{2}$ edges.
For $e = (a, b, c) \in E(H)$, let $D(e)$ denote the degree vector $\degvec{ d(a), d(b), d(c) }$ with coordinates in non-increasing order. We define a partial order on these vectors by considering $D(e) \geq D(f)$ if at all positions the coordinate of $e$ is larger than or equal to that of $f$.
One tool in proving the upper bound ${\rm ex}(n,C) \leq 2n$ in \cite{GYRS} was showing that in a crown-free $3$-graph, the set of $11$ edges incident to an edge, $e$, with $D(e) = \degvec{5,5,3}$ form two possible $3$-graphs. To prove Theorem \ref{main-theorem} we need a similar result for the case $D(e) = \degvec{4,4,4}$. Lemma \ref{triple-four-lemma} in Section \ref{sectlinkgraph} proves that we have $5$ possible $3$-graphs in this case.
Our second result is a ``reduction theorem'' showing that the (almost) sharp upper bound ${\rm ex}(n,C) \leq \frac{3n}{2}$ would follow if edges with $D(e) \geq \degvec{4,4,3}$ or $D(e) \geq \degvec{5,4,2}$ are not present. It is worth noting that $D(e) \geq \degvec{6,4,2}$ is not possible in a crown-free $3$-graph (an easy exercise).
\begin{theorem} \label{sec-theorem}
Assume that a crown-free $3$-graph, $H$, with $n$ vertices has no edge $e \in E(H)$ with $D(e) \geq \degvec{4,4,3}$ or $D(e) \geq \degvec{5,4,2}$. Then $\abs{E(H)} \leq \frac{3n}{2}$.
\end{theorem}
A \textit{critical configuration} in a crown-free $3$-graph is defined by the $9$ edges incident to an edge $e$ with $D(e) = \degvec{4,4,3}$ or with $D(e) = \degvec{5,4,2}$. An immediate corollary of Theorem \ref{sec-theorem} is the following:
\begin{corollary} \label{cor}
If a crown-free $3$-graph $H$ contains no critical configuration then $\abs{E(H)} \leq \frac{3n}{2}$.
\end{corollary}
Corollary \ref{cor} may lead to a proof of ${\rm ex}(n,C) \leq \frac{3n}{2}$, since minimal counterexamples (with $n$ as small as possible) probably cannot contain critical configurations.
\begin{conjecture} \label{conj}
Minimal counterexamples to ${\rm ex}(n,C) \leq \frac{3n}{2}$ cannot contain critical configurations.
\end{conjecture}
We prove Conjecture \ref{conj} for one particular critical configuration in Section \ref{concluding-section}. The method seems to work for all others but new ideas are needed to achieve a reasonably short proof this way.
In Sections \ref{sectlinkgraph} and \ref{secgoodquint} we define our tools. In Sections \ref{sectproof1} and \ref{sectproof2} we prove Theorems \ref{main-theorem} and \ref{sec-theorem}.
\section{Link graphs of edges with $D(e) = \degvec{4, 4, 4}$}
\label{sectlinkgraph}
\begin{definition}[Link graph of an edge] \label{linkgraph}
Assume that $H$ is a $3$-graph and $e = (a,b,c) \in E(H)$. The \textit{link graph}, $G(e)$, is the graph whose edges are the pairs $(x,y)$ for which there exists $(x,y,z) \in E(H)$ with $z \in \{a,b,c\}$. The set of vertices of $G(e)$ is defined as the subset of $V(H)$ covered by the edges of $G(e)$.
\end{definition}
Note that Definition \ref{linkgraph} provides a proper $3$-coloring of the edges of $G(e)$ with colors $a$, $b$, and $c$. We denote by $\varphi(x,y)$ the color of the edge $(x,y)$ in this coloring. Edges with colors $a$, $b$, and $c$ will be labelled $\alpha$, $\beta$, and $\gamma$, respectively and are colored red, blue, and green in colored figures. Observe that a crown with base edge $e$ exists in $H$ if and only if $G(e)$ has three pairwise disjoint edges with different colors, which we call a \textit{rainbow matching}.
\begin{lemma} \label{triple-four-lemma}
If a crown-free $3$-graph $H$ has an edge $e = (a, b, c)$ such that $D(e) = \degvec{4, 4, 4}$, then $G(e)$ is isomorphic (up to permutation of colors) to one of the following five graphs (see Figure \ref{fig-link-graphs}).
\end{lemma}
\begin{figure}[H]
\centering
\includegraphics[width=0.75\linewidth]{figures/link-graphs-2}
\caption{Link graphs $G_1, \ldots, G_5$}
\label{fig-link-graphs}
\end{figure}
\begin{proof}
For $i \in [3]$, let $M_i$ denote the vertex set of the matching of color $a$, $b$, and $c$ in $G(e)$, respectively. Observe that for all $i, j \in [3]$, where $i \neq j$, $M_i$ must intersect all the three edges in $M_j$. Otherwise, there exists an edge $f \in M_j$ not intersecting $M_i$ and an edge $g \in M_k$, for $k \neq i \text{ or } j$, such that $g \cap f = \emptyset$. Then $f$, $g$, and some edge in $M_i$ is a rainbow matching.
First, we show that $\abs{M_1 \cap M_2} > 3$. It follows from the previous observation that $\abs{M_1 \cap M_2} \geq 3$. Assume for contradiction that $\abs{M_1 \cap M_2} = 3$. Then $S = M_1 \cap M_2$ cannot contain an edge from $M_1 \cup M_2$, otherwise we get a contradiction with our observation. Thus, $M_1$ and $M_2$ are matchings from $S$ to $M_1 \setminus S$ and from $S$ to $M_2 \setminus S$, respectively. If $e \in M_3$, then $e \in S$ since otherwise $e$ would be part of a rainbow matching. However, it is impossible for there to be three disjoint edges in $S$ since $\abs{S} = 3$ by assumption.
Now, suppose $\abs{M_1 \cap M_2} = 4$. Then, in $S$ there is one edge
of $M_1$ and one edge of $M_2$. If the two edges are disjoint, we have two disjoint $\alpha$-$\beta$ paths both with four vertices. One has two $\alpha$-edges and one $\beta$-edge, and the other has the opposite. To avoid a rainbow matching, any $\gamma$-edge must intersect every edge in one of the paths. There are thus four possible locations for a $\gamma$-edge. Choose any three of them gives $G_2$. On the other hand, if the two edges intersect, we will get a contradiction. In this case, the edges of $M_1 \cup M_2$ form two disjoint, alternating $\alpha$-$\beta$ paths with three and five vertices, respectively. Following the paths, label the three vertices $v_1$ through $v_3$, and the five vertices $w_1$ through $w_5$. The only possible location for a $\gamma$-edge that does not intersect $v_2$ and would not form a rainbow matching is $(w_2, w_4)$. However, then at least two $\gamma$-edges must intersect $v_2$, a contradiction.
We show that the next case implies that $G(e)$ is isomorphic (up to permutation of colors) to one of the $G_i$'s.
Assume $\abs{M_1 \cap M_2} = 5$. Then either $M_1 \cup M_2$ is an alternating $\alpha$-$\beta$ path on seven vertices, or it is a disjoint alternating $\alpha$-$\beta$ four-cycle and $\alpha$-$\beta$ path on three vertices.
In the former case, label the vertices $v_1$ through $v_7$ along the path. Then the possible $\gamma$-edges that don't create a rainbow matching are $(v_1, v_3)$, $(v_1, v_6)$, $(v_2, v_4)$, $(v_2, v_6)$, $(v_2, v_7)$, $(v_4, v_6)$, and $(v_5, v_7)$. Apart from the symmetry (reflection of a point of the path through $v_4$), the non-intersecting triples of these edges are
\[ \{ (v_1, v_3), (v_2, v_4), (v_5,v_7) \}, \]
\[ \{ (v_1, v_3), (v_2, v_7), (v_4,v_6) \}, \ \{ (v_1, v_3), (v_2, v_6), (v_5,v_7)\}. \]
The first triple gives $G_3$, and the last two triples give $G_4$.
In the latter case, label the vertices $v_1$ through $v_3$ along the path and $w_1$ through $w_4$ along the cycle. If there is a $\gamma$-edge containing a vertex, $u \notin M_1 \cup M_2$, then that edge must be $(u, v_2)$ or part of a rainbow matching. Thus, there is at most one such edge. The vertex $v_2$ may also be in a $\gamma$-edge with $w_i$ for $i \in [4]$; however, it may only be in one so we only consider the edge $(v_2, w_1)$. The other possible $\gamma$-edges are $(v_1, v_3)$, $(w_1, w_3)$, and $(w_2, w_4)$. Thus, apart from the symmetry (choice of diagonal in the cycle), the possible non-intersecting triples from these edges are
\[ \{ (v_1, v_3), (w_1, w_3), (w_2, w_4) \}, \ \{ (v_1, v_3), (u, v_2), (w_1, w_3) \}, \]
\[ \{ (v_1, v_3), (v_2, w_1), (w_2, w_4) \}, \ \{ (u, v_2), (w_1, w_3), (w_2, w_4) \}. \]
These triples give $G_1$, $G_2$, $G_3$, and $G_5$, respectively.
Lastly, suppose $\abs{M_1 \cap M_2} = 6$. Then $M_1 \cup M_2$ is an alternating $\alpha$-$\beta$ six-cycle. Any $\gamma$-edges
intersecting the cycle in at most one vertex or along a long diagonal are in rainbow matchings. However, at most two $\gamma$-edges can be short diagonals without intersecting, which is a contradiction thus concludes the proof.
\end{proof}
\section{Good quintuple lemma}
\label{secgoodquint}
As shown in Section \ref{sectlinkgraph}, it is easy to recognize a crown with base edge $e$: We have to find a rainbow matching in $G(e)$. To recognize other crowns related to $G(e)$, we introduce the following definition:
\begin{definition}[Good quintuple]
A quintuple $Q=\{ x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \}$ of vertices of $G(e)$ is \textit{good} if
\begin{itemize}
\item $x_{1} x_{2}$, $x_{2} x_{3}$, and $x_{4} x_{5}$ are edges of $G(e)$
\item $\varphi(x_{1} x_{2}) = \varphi(x_{4} x_{5})$
\end{itemize}
\end{definition}
\begin{remark}\label{quintremark}
The ordering of the vertices in $Q=\{ x_1,x_2,x_3,x_4,x_5 \}$ is important. Assume that $Q$ is a good quintuple. Then the quintuple $\{ x_1,x_2,x_3,x_5,x_4 \}$ is still good. However, observe that $\{ x_2,x_1,x_3,x_4,x_5 \}$ is good if and only if $(x_1,x_3)$ is an edge in $G(e)$. On the other hand, $\{ x_3,x_2,x_1,x_4,x_5 \}$ is never good.
\end{remark}
\begin{remark} \label{allbutv7}
Observe (see Figure \ref{fig-link-graphs}) that apart from $v_7 \in V(G_5)$, every vertex in each $G_i$ is the first vertex of some good quintuple.
\end{remark}
\begin{lemma}[Good quintuple lemma] \label{quintuple-lemma}
Assume $H$ is a crown-free $3$-graph and $Q = \{ x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \}$ is a good quintuple in $G(e)$ for some $e = (a, b, c) \in E(H)$. Then there is no edge $f \in E(H)$ such that $f \cap e = \emptyset$ and that $f \cap Q = \{x_1\}$.
\end{lemma}
\begin{proof}
Without loss of generality, $Q$ defines the edges $(x_1,x_2,a)$, $(x_2,x_3,b)$, and $(x_4,x_5,a)$ in $H$. Assume towards contradiction that edge $f = (p,q,x_1)$ where (from the assumptions) $p, q \notin Q \cup \{ a,b,c \}$.
Observe that
\[(p,q,x_1), (x_2,x_3,b), (x_4,x_5,a)\]
are pairwise disjoint edges and $(x_1,x_2,a)$ intersects all of them, thus we have a crown (with base $(x_1,x_2,a)$), a contradiction.
\end{proof}
\section{Proof of Theorem \ref{main-theorem}}
\label{sectproof1}
Suppose that Theorem \ref{main-theorem} is not true, there exists a crown-free $3$-graph $H$ with $\delta(H) \geq 4$. Select an arbitrary edge $e = (a,b,c) \in E(H)$ and let $H'$ be the $3$-graph obtained from $H$ by removing edges intersecting $e$ until $D(e) = \degvec{4,4,4}$ in $H'$. Then Lemma \ref{triple-four-lemma} can be applied to $H'$ and we get that $G_i \subseteq G(e)$ for some $i \in [5]$. Further, note that every vertex $v$ in $G_i$ has degree at most three in $H'$, thus we can select $f_v \in E(H)$ such that $v \in f_v$ and $f_v \cap e = \emptyset$. Selecting $v \neq v_7$, there exists a good quintuple $Q$ with first vertex $v$ in $G_i$ (see Remark \ref{allbutv7}). We shall get a contradiction from Lemma \ref{quintuple-lemma}, finding a good quintuple $Q$ satisfying $f_v \cap Q = \{v\}$. This is obvious if $f_v \cap V(G_i) = \{v\}$, therefore in the subsequent cases we may assume that $f_v = (v,p,q)$ where $v,p \in V(G_i)$.
\begin{itemize}
\item $G(e) = G_1$. Set $v=v_1$ and from the symmetry of $G_1$ we may assume that $f_{v_1} = (v_1, v_5, q)$ (where $q \notin V(G_1)$). Then $Q = \{v_1,v_2,v_3,v_6,v_7\}$ is a good quintuple.
\item $G(e) = G_2$. Set $v=v_1$ and (apart from symmetry) we have to consider either $f_{v_1} = (v_1,v_5,q)$ (where $q=v_8$ is possible) or $f_{v_1} = (v_1,p,q)$ where $p \in \{v_7,v_8\}$ and $q \notin V(G_2)$. In the former case $Q = \{v_1,v_2,v_3,v_6,v_7\}$ and in the latter $Q = \{v_1,v_3,v_4,v_5,v_6\}$ is a good quintuple.
\item $G(e) = G_3$. Set $v=v_2$ and (up to symmetry) we have to consider either $f_{v_2} = (v_2, v_6, q)$ (where $q = v_4$ is possible) or $f_{v_2} = (v_2, v_4, q)$ (where $q\not \in G(e)$). In both cases $Q = \{v_2, v_5, v_7, v_1, v_3 \}$ is a good quintuple.
\item $G(e) = G_4$. Set $v=v_1$. We have to consider three cases: either $f_{v_1} = (v_1, v_4, q)$ (where $q = v_6$ is possible), $f_{v_1} = (v_1, v_6, q)$ (where $q \not = v_4$), and $f_{v_1} = (v_1, v_5, q)$. In the first two cases $Q = \{v_1, v_3, v_2, v_5, v_7 \}$ is a good quintuple, and in the last case $ Q = \{v_1, v_2, v_3, v_6, v_7 \}$ is a good quintuple.
\item $G(e) = G_5$. Set $v=v_5$. Assume first that $f = f_{v_5} = \{v_5, v_6, v_8\}$. In this case $d_H(a) = d_H(b) = d_H(c) = 4$, i.e. $H=H'$ since otherwise we have an edge $g$ intersecting $e$ and intersecting $V(G_5)$ in at most one point (in one of $v_5,v_6,v_8$). Then $g$ would be a jewel in a crown with base $e = (a,b,c)$ leading to contradiction. Since $d_{H'}(v_5) \geq 3$, there exists $f' = f'_{v_5} = (v_5,p,q) \in E(H)$ different from $f_{v_5}$ and from $(a,v_5,v_7)$. Since $f' \cap e = \emptyset$ (from $H=H'$), we can select $f'$ instead of $f$. Up to symmetry, we may assume that $p = v_6$ and $q = v_1$. Then $Q = \{v_5, v_7, v_8, v_2, v_3 \}$ is a good quintuple.
\end{itemize}
Since all cases ended by finding $v \in V(G_i)$, $f_v \in E(H)$, and a good quintuple $Q$ such that $f_v \cap e = \emptyset$ and $f_v \cap Q = \{v\}$, we get a contradiction from Lemma \ref{quintuple-lemma}, concluding the proof. \qed
\section{Proof of Theorem \ref{sec-theorem}}
\label{sectproof2}
Suppose that Theorem \ref{sec-theorem} is not true: Let $H$ be a minimal counterexample, that is, a crown-free $3$-graph satisfying the conditions in Theorem \ref{sec-theorem} and $\abs{E(H)} > 3n/2$, with $n$ as small as possible. If there is a vertex $v \in V(H)$ with degree $d(v) \leq 1$, we remove $v$ together with the one possible edge containing $v$ and get a smaller counterexample, contradicting the minimality of $H$. Thus $d(v) \geq 2$ holds for all $v \in V(H)$.
We define the partition $V(H) = Y \cup Z$ where $Z$ contains the vertices of degree at most three and $Y$ is the set of remaining vertices (of degree at least four).
A \textit{special vertex} is a vertex $v$ with $d_{H}(v) = 2$, such that for the two edges $(a_1, a_2, v)$ and $(b_1, b_2, v)$ containing $v$ we have
\[ d_{H}(a_1) = d_{H}(a_2) = d_{H}(b_1) = d_{H}(b_2) = 4. \]
Partition $Z$ into three parts as follows: Let $Z_1$ be the set of vertices in $Z$ with $d_H(v) = 3$, $Z_2$ be the set of non-special vertices in $Z$ with $d_H(v) = 2$, and $Z_3$ be the set of special vertices. Let $E_1$ denote the set of edges in $H$ intersecting $Z$ in at least two vertices and set $E_2 = E(H) \setminus E_1$. Since there is no edge $(a, b, c)$ in $E(H)$ with $\degvec{a, b, c} \geq \degvec{4, 4, 4}$, all edges in $E_2$ intersect $Z$ in exactly one vertex. See Figure \ref{fig:prop51fig} illustrating the definitions, where the numbers indicate degrees and edges of $E_2$ are dotted.
\input{figures/prop51figure}
\begin{proposition} \label{prop}
For all $v \in Z_1$, we have $d_{E_2}(v) = 0$. For all $v \in Z_2$, we have $d_{E_2}(v) \leq 1$. Moreover, $\abs{Z_3} \leq \abs{Y}$.
\end{proposition}
\begin{proof}
Since $H$ has no edge $(a, b, c) \in E(H)$ with $\degvec{a, b, c} \geq \degvec{4, 4, 3}$, for all $v \in Z_1$ we have $d_{E_2}(v) = 0$, proving the first statement.
Assume that for some vertex $v \in Z_2$, $d_{E_2}(v) = 2$. Since H has no edge with $\degvec{a, b, c} \geq \degvec{5, 4, 2}$, the two edges in $E_2$ containing $v$ have all other vertices of degree $4$. But then $v \in Z_3$, contradicting the assumption and proving the second statement.
To prove the third statement, let $Y_1$ denote the set of vertices with degree $4$ in $Y$. Let $G$ be the graph with vertex set $Y_1$ and let $(y,y')$ be an edge of $G$ for $y,y' \in Y_1$ if and only if there exists $z \in Z_3$ such that $(y,y',z) \in E_2$. Then
\[ 2 \abs{Z_3} = \abs{E(G)} = \frac{1}{2} \sum_{y \in Y_1} d_{G}(y) \leq \frac{1}{2} \sum_{y \in Y_1} d_{H}(y) = 2 \abs{Y_1} \leq 2 \abs{Y}, \]
implying $\abs{Z_3} \leq \abs{Y}$ as required.
\end{proof}
Note that $\sum_{v \in Z} d_{E_1} (v)$ is at least a double-count of $\abs{E_1}$ from the definition of $E_1$. Therefore, using that $d_{E_1} (v) = 2 - d_{E_2} (v)$ for $v \in Z_2$ and that $d_{E_1} (v) = 0$ for $v \in Z_3$, we have
\[ \abs{E_1} \leq \frac{1}{2} \left( \sum_{v \in Z_1} d_{E_1} (v) + \sum_{v \in Z_2} ( 2 - d_{E_2} (v) ) \right). \]
From the first statement of Proposition \ref{prop}, we have
\[ \abs{E_2} = \sum_{v \in Z_2} d_{E_2} (v) + \sum_{v \in Z_3} d_{E_2} (v). \]
Therefore,
\begin{align*}
\abs{E(H)} & = \abs{E_{1}} + \abs{E_{2}} \\
& \leq \frac{1}{2} \left( \sum_{v \in Z_1} d_{E_1}(v) + \sum_{v \in Z_2} (2 - d_{E_2}(v)) \right) + \sum_{v \in Z_2} d_{E_2}(v) + \sum_{v \in Z_3} d_{E_2}(v) \\
& = \frac{1}{2} \left( \sum_{v \in Z_1} d_{E_1}(v) + 2 \abs{Z_2} + \sum_{v \in Z_2} d_{E_2}(v) \right) + \sum_{v \in Z_3} d_{E_2}(v) \\
& \leq \frac{1}{2} \left( 3 \abs{Z_{1}} + 3 \abs{Z_{2}} \right) + 2 \abs{Z_{3}} \numberthis \label{first-ineq} \\
& \leq \frac{1}{2} \left( 3 \abs{Z_{1}} + 3 \abs{Z_{2}} \right) + \abs{Z_{3}} + \abs{Y} \numberthis \label{second-ineq} \\
& = \abs{Z_{1}} + \abs{Z_{2}} + \abs{Z_{3}} + \abs{Y} + \frac{1}{2} (\abs{Z_{1}} + \abs{Z_{2}}) \\
& \leq n + \frac{n}{2} = \frac{3n}{2}
\end{align*}
where the inequality \eqref{first-ineq} follows from $d_{E_2}(v) \leq 1$ for $v \in Z_2$, and the inequality \eqref{second-ineq} follows from $\abs{Z_3} \leq \abs{Y}$ (second and third statements of Proposition \ref{prop}). We conclude that $\abs{E(H)} \leq \frac{3n}{2}$ contradicting the assumption that $H$ is a counterexample. \qed
\section{Excluding critical configurations}
\label{concluding-section}
Here we prove Conjecture \ref{conj} for one particular critical configuration: an edge $e=(a,b,c)$ with $D(e) = \degvec{4,4,3}$ with link graph $G_6$ (see Figure \ref{fig-G6}). Note that although $D(e) < \degvec{4,4,4}$, $G_6$ cannot be obtained from any $G_i$ with $i \in [5]$ by deleting an edge, since those graphs (see Figure \ref{fig-link-graphs}) do not contain two vertex-disjoint four-cycles.
\begin{figure}[H]
\centering
\includegraphics[width=0.6\linewidth]{figures/G6}
\caption{Link graph $G_6$}
\label{fig-G6}
\end{figure}
\begin{proof}
Let $H$ be a minimal counterexample with $n$ vertices containing an edge $e=(a,b,c)$ with $D(e) = \degvec{4,4,3}$ and with link graph $G_6$.
Set $X = V(G_6) \cup \{a,b,c\}$. We prove that $X$ is incident to at most $\frac{3 \abs{X}}{2} = \frac{3 \times 11}{2} = 16.5$ edges of $H$. This will conclude the proof because after removing $X$ and its incident edges from $H$, we could have a smaller counterexample.
In fact, we claim more: Apart from the extensions of the $8$ edges of $G_6$ and the edge $e$, $X$ can be incident only to those four possible edges of $H$ that contain a diagonal of the two four-cycles of $G_6$. Indeed, suppose that $f \in E(H)$ is not among these $14$ possibilities:
\begin{enumerate}
\item $a$ or $b$ is in $f$, without loss of generality, $a \in f$. Note that $f$ intersect $V(G_6)$ in at most one point and that point need to be one of the endpoints of the $\beta$-edge of the first component of $G_6$. Let $g$ be the edge containing $c$ and the $\gamma$-edge of the first component of $G_6$ which does not intersect $f$. Let $h$ be an edge containing $b$ and a $\beta$-edge of the second component of $G_6$. Then $f,g,h$ are the jewels of a crown with base $e$, contradiction.
\item $c$ is in $f$. Note that $f$ intersect $V(G_6)$ in at most one point and that point need to be on the second component of $G_6$. Let $g$ be the edge containing $b$ and the $\beta$-edge of the second component of $G_6$ which does not intersect $f$. Let $h$ be the edge containing $a$ and the $\alpha$-edge of the first component of $G_6$. Then $f,g,h$ are the jewels of a crown with base $e$, contradiction.
\item $f$ does not intersect $e$ and $x_1=V(G_6)\cap f$ is in the first component of $G_6$. We can select an $\alpha$-$\gamma$ path (or a $\beta$-$\gamma$ path) $x_1,x_2,x_3$ in the first component of $G_6$ and a $\beta$ (or an $\alpha$) edge $(x_4,x_5)$ in the second component of $G_6$ such that this edge does not contain the possible intersection point of $f$ with the second component. Applying Lemma \ref{quintuple-lemma} with $Q=\{x_1,x_2,x_3,x_4,x_5\}$ leads to a contradiction.
\item $f$ does not intersect $e$ and $x_1=V(G_6)\cap f$ is in the second component of $G_6$. We can select an $\alpha$-$\beta$ path $x_1,x_2,x_3$ in the second component of $G_6$ and a $\gamma$-edge $(x_4,x_5)$ in the first component of $G_6$ such that this edge does not contain the possible intersection point of $f$ with the first component. Again, applying Lemma \ref{quintuple-lemma} with $Q=\{x_1,x_2,x_3,x_4,x_5\}$ leads to a contradiction.
\end{enumerate}
\end{proof}
| {
"timestamp": "2021-08-02T02:20:57",
"yymm": "2107",
"arxiv_id": "2107.14713",
"language": "en",
"url": "https://arxiv.org/abs/2107.14713",
"abstract": "A \\textit{linear $3$-graph}, $H = (V, E)$, is a set, $V$, of vertices together with a set, $E$, of $3$-element subsets of $V$, called edges, so that any two distinct edges intersect in at most one vertex. The linear Turán number, ${\\rm ex}(n,F)$, is the maximum number of edges in a linear $3$-graph $H$ with $n$ vertices containing no copy of $F$.We focus here on the \\textit{crown}, $C$, which consists of three pairwise disjoint edges (jewels) and a fourth edge (base) which intersects all of the jewels. Our main result is that every linear $3$-graph with minimum degree at least $4$ contains a crown. This is not true if $4$ is replaced by $3$. In fact the known bounds of the Turán number are \\[ 6 \\left\\lfloor{\\frac{n - 3}{4}}\\right\\rfloor \\leq {\\rm ex}(n, C) \\leq 2n, \\] and in the construction providing the lower bound all but three vertices have degree $3$. We conjecture that ${\\rm ex}(n, C) \\sim \\frac{3n}{2}$ but even if this were known it would not imply our main result.Our second result is a step towards a possible proof of ${\\rm ex}(n,C) \\leq \\frac{3n}{2}$ (i.e., determining it within a constant error). We show that a minimal counterexample to this statement must contain certain configurations with $9$ edges and we conjecture that all of them lead to contradiction.",
"subjects": "Combinatorics (math.CO)",
"title": "Crowns in linear $3$-graphs",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754447499795,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.707510393660292
} |
https://arxiv.org/abs/2109.03527 | Matrix functions via linear systems built from continued fractions | A widely used approach to compute the action $f(A)v$ of a matrix function $f(A)$ on a vector $v$ is to use a rational approximation $r$ for $f$ and compute $r(A)v$ instead. If $r$ is not computed adaptively as in rational Krylov methods, this is usually done using the partial fraction expansion of $r$ and solving linear systems with matrices $A- \tau I$ for the various poles $\tau$ of $r$. Here we investigate an alternative approach for the case that a continued fraction representation for the rational function is known rather than a partial fraction expansion. This is typically the case, for example, for Padé approximations. From the continued fraction, we first construct a matrix pencil from which we then obtain what we call the CF-matrix (continued fraction matrix), a block tridiagonal matrix whose blocks consist of polynomials of $A$ with degree bounded by 1 for many continued fractions. We show that one can evaluate $r(A)v$ by solving a single linear system with the CF-matrix and present a number of first theoretical results as a basis for an analysis of future, specific solution methods for the large linear system. While the CF-matrix approach is of principal interest on its own as a new way to compute $f(A)v$, it can in particular be beneficial when a partial fraction expansion is not known beforehand and computing its parameters is ill-conditioned. We report some numerical experiments which show that with standard preconditioners we can achieve fast convergence in the iterative solution of the large linear system. | \section{Introduction}
If $A\in \complns^{m \times m}$ and $f: D \subset \complns \to \complns$, the matrix function $f(A)$ is defined provided $f$ is $n(\lambda)-1$ times differentiable at all eigenvalues $\lambda$ of $A$ with $n(\lambda)$ being their multiplicity in the minimal polynomial of $A$. For diagonalizable $A$,
i.e., $A = W\Lambda W^{-1}$ with $\Lambda = \operatorname{diag}\opfences{\lambda_1,\ldots,\lambda_n}$ (implying $n(\lambda_i) =1$ for all $i$), one has
\[
f(A) = W \operatorname{diag}\opfences{f(\lambda_1),\ldots,f(\lambda_n)} W^{-1},
\]
and if $A$ is not diagonalizable, a similar representation for $f(A)$ can be given involving the Jordan blocks of $A$. We refer the reader to \cite{Higham} for the above facts as well as for many further results on properties and computational methods for matrix functions.
Even when $A$ is sparse, $f(A)$ is typically a full matrix. This implies that if $A$ is large and sparse, we are practically bound to compute the action $f(A)v$ of $f(A)$ on a vector $v$ rather than computing the full matrix $f(A)$, and it is this situation that we consider in this paper.
Most of the existing methods fall into one of the two following categories:
$f(A)v$ is computed approximately as the action of a matrix polynomial $p(A)$ on $v$, or it is approximated as the action of a rational matrix function $r(A)$ on $v$. The polynomial $p$ or the rational function $r$ can either be obtained by using \emph{a priori} information about the matrix $A$ like information about its spectrum or its numerical range, or it can be constructed adaptively in a process that accumulates increasingly accurate information on the spectral properties of $A$, depending on the vector $v$.
For example, if $A$ is Hermitian and if we know an interval $[a,b]$ containing
the spectrum of $A$, we can use Chebyshev best polynomial or rational
approximations for $f$ on $[a,b]$ or approximations to these best approximations
like truncated Chebyshev series approximations, see, e.g., \cite{Higham,Gallopoulos,Kenney,Moler,Trefethen}. On the other hand, Arnoldi approximations for $f(A)v$ take their values
from the Krylov subspace spanned by $A$ and $v$ and as such adaptively produce
polynomial approximations, depending not only on $A$ but also on $v$. The same
holds true for the rational approximations obtained in rational Krylov subspace
methods. We refer to \cite{FrSi06} or \cite{Higham} for an
overview of Arnoldi-type methods and to the review \cite{Guettel} for
rational Krylov methods.
In polynomial methods, the computational cost is determined by matrix-vector
products with $A$, whereas in rational approximation methods the cost typically
resides in solving linear systems with matrices $A - \tau I$, where $\tau$ is a
pole. Of course, solving linear systems is usually much more expensive than a
matrix-vector multiplication, but the additional cost when using rational
approximations is often more than balanced for by the fact that we get much better approximation properties.
In this paper, we study the situation where a rational approximation $r$ to $f$ is given via a continued fraction expansion. This is naturally the case, for example, for Pad\'e approximations. It is well known that one can evaluate a continued fraction
solving a linear system with a tridiagonal matrix, see, e.g., \cite{Haydock75,Ozaki07}. In a similar fashion,
we use in this paper the continued fraction (``CF'') to construct a block tridiagonal ``CF-matrix'' of size $m(n+1)$, where $n$ is the degree of the
continued fraction, such that we can retrieve $r(A)v$ as the first $m$
components of a linear system involving the CF-matrix. We then investigate
spectral properties of the CF-matrix. In this manner, we contribute to establishing and analyzing a new way
to approximate $f(A)v$ using a rational function for which we do not have to know or compute its poles.
Our investigations are meant to provide first results which lay the ground for being able to develop particularly efficient solution methods for the CF-matrix in the future like, for example, adequately preconditioned Krylov subspace methods or multigrid approaches. The numerical examples in this paper illustrate first steps into this direction.
The paper is organized as follows: We recall the most important definitions and properties of continuous fractions in \cref{sec:continued_fractions}.
In \cref{sec:5d_matrix} we then develop in detail how to construct the CF-matrix from which we get $f(A)v$ as a part of the solution for a particular right-hand side. \Cref{sec:simple} discusses special but important cases in which the CF-matrix takes a particularly appealing form in the sense that no higher powers of $A$ appear in its blocks.
This allows us to fully characterize the partial fraction expansion form of the rational function via the Weierstrass canonical form of certain matrix pencils and to establish relations between
a block Gauss-Seidel and a block Jacobi iteration on the CF-matrix with Gauss-Seidel and Jacobi performed on the systems $A-\tau_iI$, $\tau_i$ being the poles. \Cref{sec:numerics} presents a few numerical examples comparing the convergence of preconditioned GMRES \cite{Saad_GMRES} on the CF-matrix and on the systems resulting from the partial fraction expansion.
\section{Continued fractions and rational functions} \label{sec:continued_fractions}
We start by reviewing some properties of continued fractions that can be found in standard literature, e.g., \cite{Cuyt}.
Given a formal continued fraction as
\begin{equation*}
g = b_0 + \contf{i=1}{}{c_i}{b_i} = b_0 + \cfrac{c_1}{b_1 + \cfrac{c_2}{b_2 + \dots}}, \qquad b_i, c_i \in \complns
\end{equation*}
its $n$-th \emph{approximant} (sometimes also called \emph{convergent}) is defined by
\begin{equation*} \label{eq:approx:def}
g_n = b_0 + \contf{i=1}{n}{c_i}{b_i} = b_0 + \cfrac{c_1}{b_1 + \cfrac{c_2}{b_2 + \quad\raisebox{-3mm}{$\ddots$}\quad\raisebox{-6mm}{$+\cfrac{c_n}{b_n}$}}}
\end{equation*} with corresponding \emph{tail}
\begin{equation*} \label{eq:tail_def}
t_{n} = \contf{i=n+1}{\infty}{c_i}{b_i}.
\end{equation*}
In this definition, we always assume that either $c_i \neq 0$ for all $i$ or that if $c_n = 0$ for some $n$, then $b_i = 1$ and $c_i = 0$ for all $i \geq n$. In the latter case, we say that the continued fraction is
{\em finite} and also write \[
g = b_0 + \contf{i=1}{n}{c_i}{b_i} = g_n.
\]
With these assumptions, considering the usual extension of complex
arithmetic to $\complns \cup \{\infty\}$, we see that each approximant $g_n$ to a formal continued fraction is defined with
value in $\complns \cup \{\infty\}$, and we say that $g$ exists with value $\lim_{n \to \infty} g_n \in \complns \cup \{\infty\}$ if this limit exists.
The approximant $g_n$ of a continued fraction can be expressed as a simple fraction $g_n = \frac{p_n}{q_n}$ with $p_n$, $q_n$ determined by the recursion
\begin{equation}
\label{eq:cf_recursion}
\begin{bmatrix} p_{-1} \\ q_{-1} \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix},\quad
\begin{bmatrix} p_{0} \\ q_{0} \end{bmatrix} = \begin{bmatrix} b_0 \\ 1 \end{bmatrix}, \quad
\begin{bmatrix} p_n \\ q_n \end{bmatrix} = b_n \begin{bmatrix} p_{n-1} \\ q_{n-1} \end{bmatrix} + c_n \begin{bmatrix} p_{n-2} \\ q_{n-2} \end{bmatrix} \enspace \text{for } n \geq 1.
\end{equation}
The tails obey the simple ``backward'' recursion
\begin{equation} \label{eq:tail_rec}
t_n = \frac{c_{n+1}}{b_{n+1}+t_{n+1}} \enspace \text{\ for\ } n=0,1,\ldots,
\end{equation}
with $b_0 + t_0 = g$.
\begin{remark} \label{rem:conf_rec_zero}
The recursion \cref{eq:cf_recursion} allows that for a given $n$ either $p_n$ or $q_n$ can be zero (which then gives $g_n = 0$ or
$g_n = \infty$), but that $p_n$ and $q_n$ cannot vanish at the same time. Indeed, if this were the case, the recursion shows that the vectors $\begin{bsmallmatrix} p_i\\ q_i\end{bsmallmatrix}$ end up being all collinear for $i=n-1, n-2, \ldots, -1$, an obvious contradiction to the definition of these vectors for $i=0$ and $i=-1$.
\end{remark}
Note that the value of a continued fraction does not change if we expand each inner fraction with arbitrary factors $d_i\neq 0$ in the sense that
\begin{equation}
\label{eq:expanding_cf}
g = b_0 + \contf{i=1}{}{c_i}{b_i} = b_0 + \contf{i=1}{}{d_{i-1}d_i c_i}{d_i b_i}, \quad d_0 = 1.
\end{equation}
Continued fractions can be extended to also represent functions of a (complex) variable $z$. This is usually achieved by using polynomials as partial numerators and denominators,
\begin{equation*}
g(z) = b_0(z) + \contf{i=1}{}{c_i(z)}{b_i(z)} \text{\ with\ } c_i(z), b_i(z) \text{\ polynomials in\ } z.
\end{equation*}
The approximants $g_n(z)$ are then rational functions, see \cref{eq:cf_recursion}. For instance, continued fractions of the form
\begin{equation}
\label{eq:C_frac}
g(z) = b_0 + \contf{i=1}{\infty}{c_i z}{1}
\end{equation}
are called \emph{regular C-fractions}.
It is known that for every Stieltjes function $f(z)$ (see \cite{Henrici} for a definition), there exists a regular C-fraction such that the approximants yield a \emph{descending staircase} of Padé approximations for $f(z)$, i.e.,
\begin{equation*}
g_n(z) = r_{\ceil{\frac{n}{2}}, \floor{\frac{n}{2}}}(z).
\end{equation*}
Recall that the Padé approximation $r_{k,l}(z) = \frac{p_{k,l}(z)}{q_{k,l}(z)}$ is the rational function with numerator degree $k$ and denominator degree $l$ for which at least the first $k+l$ terms of the Taylor expansion at 0 agree with the approximated function $f(z)$,
\begin{equation*}
f(z) - \frac{p_{k,l}(z)}{q_{k,l}(z)} = \mathcal{O}\opfences{z^{k+l+1}}.
\end{equation*}
We refer to \cite{Baker} as a reference for the theory of Pad\'e approximations and to \cite{Baker, Cuyt} for the many connections
which exist between Pad\'e approximations and continued fractions.
C-fractions can be \emph{contracted}, see \cite{Cuyt}, which gives the new continued fraction
\begin{equation}
\label{eq:C_frac_contracted}
\begin{gathered}
\widetilde{g}(z) = b_0 + \frac{c_1 z}{1 + c_2 z + \contf{i=2}{\infty}{-c_{2i-2}c_{2i-1} z^2}{1 + (c_{2i}+c_{2i-1})z}},\\
\widetilde{g}_{n}(z) = r_{n,n}(z) = g_{2n}(z),
\end{gathered}
\end{equation}
in which the $n$-th approximant corresponds to the $2n$-th approximant of the original C-fraction. We can thus represent the diagonal $(n,n)$-Pad\'e approximation of a Stieltjes function as a contracted C-fraction of the form \cref{eq:C_frac_contracted}.
More generally, for each $j\in \mathbb{N}$ the Padé approximations $r_{j+\ceil{\frac{n}{2}}, \floor{\frac{n}{2}}}(z)$ can be obtained as the approximants of a continued fraction of the form
\begin{equation*}
r_{j,0}(z) + z^j\contf{i=1}{\infty}{c_i z}{1}.
\end{equation*}
For Padé approximations with $k<l$, we note that the reciprocal rational function $r_{k,l}(z)^{-1} = \frac{q_{k,l}(z)}{p_{k,l}(z)}$ is the Padé approximation of $f(z)^{-1}$ with switched degrees. As the inverse of a continued fraction is again a continued fraction
\begin{equation*}
g = b_0 + \contf{i=1}{}{c_i}{b_i} \quad\implies\quad g^{-1} = 0 + \contf{i=0}{}{c_i}{b_i}, \quad c_0=1,
\end{equation*}
one can construct Padé approximants with $k<l$ by considering $f(z)^{-1}$.
Every rational function can be written as a finite continued fraction. One way to obtain such a finite continued fraction is by using the Euclidean algorithm with polynomial long division to find the greatest common divisor of the polynomials $p(z)$ and $q(z)$, see, e.g., \cite{Knopfmacher},
\begin{equation*}
\frac{p(z)}{q(z)} = b_0(z) + \frac{a(z)}{q(z)} = b_0(z) + \frac{1}{\frac{q(z)}{a(z)}}.
\end{equation*}
Here, the degree of the remainder $a(z)$ is less than the degree of $q(z)$, and one recurses with the new rational function $\frac{q(z)}{a(z)}$ until the remainder is zero. We write the resulting continued fraction as
\begin{equation*}
\frac{p(z)}{q(z)} = b_0(z) + \contf{i=1}{n}{1}{b_i(z)}.
\end{equation*}
Note that the continued fractions obtained in this manner are not C-fractions:
The polynomials are in the partial denominators instead of the partial numerators. In particular, even if one starts at a Padé approximation, the process just described yields a continued fraction whose approximants are not necessarily Padé approximations of lower degree.
\section{CF-matrices} \label{sec:5d_matrix}
In this section, we show how the inverse of the approximant $g_n$ of any (formal) continued fraction $g$ can be obtained as the $(1,1)$ entry of the inverse of a tridiagonal matrix $T_n$ built from the partial numerators and denominators. By extension, we show the connection between the action $r(A)v$ of a rational matrix function $r(A)$ on a vector $v$ and the solution of a single, block tridiagonal linear system.
\subsection{Continued fractions and tridiagonal matrices}
\begin{theorem} \label{thm:trid}
Let the continued fraction $g = b_0 + \contf{i=1}{}{c_i}{b_i}$ be given and $g_n$ denote its $n$-th approximant with $g_n \neq 0$. If the entries of the tridiagonal matrix
\begin{equation*}
T_n = \begin{bmatrix}
\beta_{0} & \gamma_{1} \\
\alpha_{1} & \beta_{1} & \ddots \\
& \ddots & \ddots & \ddots \\
& & \ddots & \beta_{n-1} & \gamma_{n} \\
& & & \alpha_{n} & \beta_{n}
\end{bmatrix}
\in \complns^{(n+1)\times(n+1)}
\end{equation*}
fulfill
\begin{align*}
\beta_{i} = b_i,\quad i=0,\ldots,n, \\
-\alpha_i \gamma_i = c_i,\quad i=1,\ldots,n,
\end{align*}
then the matrix $T_n$ is nonsingular and
\begin{equation*}
g_n^{-1} = (T_n^{-1})_{1,1}.
\end{equation*}
\end{theorem}
\begin{proof}
$(T_n^{-1})_{1,1}$ is the first entry of the solution $x$ of the linear system $T_n x = e_1$ with $e_1$ being the first unit vector. Using Cramer's rule we have
\begin{equation} \label{Cramer:eq}
(T_n^{-1})_{1,1} = \transpose{e}_1 T_n^{-1} e_1 = \frac{\transpose{e}_1\operatorname{adj}(T_n)e_1}{\det(T_n)} = \frac{\det(T_n')}{\det(T_n)},
\end{equation}
where $T_n'$ is obtained from $T_n$ by replacing $\beta_0$ with $1$ and $\alpha_1$ with 0.
Let $d_i = \det(T_i)$ denote the $i$-th principal minor of $T_n$. Then we have the recurrence relation
\[
d_{-1} = 1,\quad d_0 = \beta_0, \quad
d_i = \beta_i d_{i-1} - \alpha_i \gamma_i d_{i-2}, \quad \mbox{for } i=1, \dots, n.
\]
Due to $\beta_i = b_i$ and $\alpha_i\gamma_i = -c_i$, this is exactly the recursion for the numerators $p_i$ from \cref{eq:cf_recursion}, i.e., $d_i = p_i$. As we assumed $g_n \neq 0$, we have $d_n = p_n \neq 0$ and thus $T_n$ is nonsingular. In a similar manner, we obtain that the principal minors $d'_i$ of $T'$ satisfy the same recursion as $q_i$ from \cref{eq:cf_recursion}. Thus, for $i=n$, we obtain from \cref{Cramer:eq} that
\[
(T_n^{-1})_{1,1} = \frac{q_n}{p_n} = g_n^{-1}. \qedhere
\]
\end{proof}
\begin{remark} If $T_n$ is singular, we still have $p_n = d_n$, but now with value $0$. \cref{rem:conf_rec_zero} shows that in this case $q_n = \det(T_n') \neq 0$. Thus \cref{thm:trid} also holds in the case $g_n = 0$, where $T_n$ is singular, if we interpret $(T_n^{-1})_{1,1}$ as $\infty$. \end{remark}
\begin{corollary} \label{cor:freedom}
\Cref{thm:trid} still holds if $T_n$ is multiplied with any two nonsingular matrices from the left and the right as long as $e_1$ is a right and left eigenvector, respectively, of these matrices with reciprocal eigenvalue:
\begin{equation*}
\left.\begin{aligned}
H_\ell^{-1} e_1 &= \lambda e_1 \\
\transpose{e}_1 H_r^{-1} &= \lambda^{-1} \transpose{e}_1
\end{aligned}\right\}
\implies g_n^{-1} = \transpose{e}_1 T_n^{-1} e_1 = \transpose{e}_1 (H_\ell T_n H_r)^{-1} e_1.
\end{equation*}
In particular, using diagonal matrices $D=\operatorname{diag}\opfences{1, d_1, \dots, d_n}$ for $H_\ell$ and $H_r$ is equivalent to expanding the continued fraction as in \cref{eq:expanding_cf}.
\end{corollary}
\subsection{Extension to matrix functions}
We now consider (formal) continued fractions as functions by assuming that the coefficients of the continued fraction $g(z)$ are polynomials in $z\in\complns$ of maximum degree $\ell$,
\begin{equation} \label{eq:poly_conf_def}
g(z) = b_0(z) + \contf{i=1}{}{c_i(z)}{b_i(z)},\text{\ where\ } b_i(z) = \sum_{j=0}^\ell b_i^{(j)}z^j \text{\ and\ }
c_i(z) = \sum_{j=0}^\ell c_i^{(j)}z^j.
\end{equation}
Define the tridiagonal matrices
\[
T_n^{(j)} = \begin{bmatrix}
\beta_{0}^{(j)} & \gamma_{1}^{(j)} \\
\alpha_{1}^{(j)} & \beta_{1}^{(j)} & \ddots \\
& \ddots & \ddots & \ddots \\
& & \ddots & \beta_{n-1}^{(j)} & \gamma_{n}^{(j)} \\
& & & \alpha_{n}^{(j)} & \beta_{n}^{(j)}
\end{bmatrix}, \enspace j=0,\ldots,\ell,
\]
and
\begin{equation} \label{eq:5d_sum}
T_n(z) = \sum_{j=0}^{\ell} T_n^{(j)} z^j = \begin{bmatrix}
\beta_{0}(z) & \gamma_{1}(z) \\
\alpha_{1}(z) & \beta_{1}(z) & \ddots \\
& \ddots & \ddots & \ddots \\
& & \ddots & \beta_{n-1}(z) & \gamma_{n}(z) \\
& & & \alpha_{n}(z) & \beta_{n}(z)
\end{bmatrix},
\end{equation}
where
$\alpha_i(z) = \sum_{j=0}^{\ell} \alpha_i^{(j)}z^j$, $\beta_i(z) = \sum_{j=0}^{\ell} \beta_i^{(j)}z^j$,
$\gamma_i(z) = \sum_{j=0}^{\ell} \gamma_i^{(j)}z^j$.
Then for all $z$ for which $c_i(z) \neq 0$ for $i=1,\ldots,n$, \cref{thm:trid} shows that we have $g_n(z)^{-1} = (T_n(z)^{-1})_{1,1}$ provided
\begin{equation} \label{eq:polynomial_identities}
\left\{
\begin{array}{rl}
\beta_i(z) = b_i(z), & i=0,\ldots,n, \\
-\alpha_i(z)\gamma_i(z) = c_i(z), & i=1,\ldots,n.
\end{array}
\right.
\end{equation}
By continuity, \cref{eq:polynomial_identities} also implies $g_n(z)^{-1} = (T_{n}(z)^{-1})_{1,1}$ for those $z$ which are a zero of one of the $c_i$.
Note that \cref{eq:polynomial_identities} means that $\beta_i^{(j)} = b_i^{(j)}$ for the coefficients of the polynomial $\beta_i$, but that we have freedom in choosing the $\alpha_i^{(j)}$ and $\gamma_i^{(j)}$ in the sense that only the product of the two polynomials $\alpha_i$ and $\gamma_j$ with these coefficients is prescribed. This means that for the coefficients we in general have $-\alpha_i^{(j)}\gamma_i^{(j)} \neq c_i^{(j)} $.
\begin{remark}
Consider the special case
\begin{equation*}
T_n(z) = T_n^{(0)} - zI.
\end{equation*}
From \cref{thm:trid}, we know that the continued fraction $g_n$ constructed from the entries of $T_n(z)$ fulfills
\begin{equation*}
g_n(z)^{-1} = \frac{q_n(z)}{p_n(z)} =
\frac{q_n(z)}{\det(T_n^{(0)} - zI)}.
\end{equation*}
Hence, the eigenvalues of a tridiagonal matrix are the zeros of the continued fraction constructed from its entries, a known fact that dates back at least to Rutishauer; see \cite[Anhang~§1]{Rutishauser}.
\end{remark}
Let us now consider a matrix $A\in\complns^{m \times m}$ instead of $z\in\complns$ and assume that the function $g(z)$, given by a continued fraction, is defined on the spectrum of $A$ in the sense of \cite{Higham}.
Any approximant $g_n(z)$ of $g(z)$ is a rational function of $z$ and as such defined on the spectrum of $A$ as long as no eigenvalue of $A$ is a pole of $g_n(z)$.
The value of a matrix function is independent
from the way we represent the function, see \cite{Higham}, so that with a slight abuse of notation we can write
\begin{equation*}
g_n(A) = b_0(A) + \contf{i=1}{n}{c_i(A)}{b_i(A)},
\end{equation*}
where each denominator is to be understood as a matrix inversion. On the other hand, let us define the matrix $T_n(A)$ as
\begin{equation} \label{eq:5d_sumA}
T_n(A) = \begin{bmatrix}
\beta_{0}(A) & \gamma_{1}(A) \\
\alpha_{1}(A) & \beta_{1}(A) & \ddots \\
& \ddots & \ddots & \ddots \\
& & \ddots & \beta_{n-1}(A) & \gamma_{n}(A) \\
& & & \alpha_{n}(A) & \beta_{n}(A)
\end{bmatrix}
= \sum_{j=0}^{\ell} T_n^{(j)} \otimes A^j,
\end{equation}
where $\otimes$ denotes the Kronecker product. Note that $T_n(A)\in\complns^{(n+1)m \times (n+1)m}$ is a \emph{block} tridiagonal matrix.
\begin{definition}
The matrix $T_n(A)\in\complns^{(n+1)m \times (n+1)m}$ from \cref{eq:5d_sumA}, constructed from the approximant of a continued fraction with polynomial partial numerators and denominators, is called \emph{CF-matrix}. (CF stands for ``continued fraction''.)
\end{definition}
Due to its construction, we already know that $\beta_i(A) = b_i(A)$ and $-\alpha_i(A)\gamma_i(A)=c_i(A)$ and so one might wonder whether---by analogy with \cref{thm:trid}---the block $(T_n(A)^{-1})_{1,1}$ yields the matrix function $g_n(A)^{-1}$ of the continued fraction $g_n(z)^{-1}$. To show that this is indeed true, let us first introduce the UDL decomposition for block tridiagonal matrices.
\begin{lemma}
\label{lem:UDL}
Let $T\in\complns^{(n+1)m \times (n+1)m}$ be a block tridiagonal matrix and denote its blocks by $T_{i,j} \in\complns^{m \times m}$. If $T$ is nonsingular and if all matrices $\Sigma_i$ defined below are nonsingular, too, the following decomposition exists:
\[
T = UDL,
\]
where $D = \operatorname{diag}\opfences{\Sigma_0, \dots, \Sigma_n}$ and
\begin{equation*}
U = \renewcommand*{\arraystretch}{2} \begin{bmatrix}
I & T_{0,1} \Sigma_1^{-1} \\
& \ddots & \ddots \\
& & I & T_{n-1,n} \Sigma_n^{-1} \\
& & & I
\end{bmatrix}, \enspace
L = \renewcommand*{\arraystretch}{2} \begin{bmatrix}
I \\
\Sigma_1^{-1} T_{1,0} & I \\
& \ddots & \ddots \\
& & \Sigma_n^{-1} T_{n,n-1} & I
\end{bmatrix} .
\end{equation*}
Herein, the $\Sigma_i$ are the block Schur complements described by the backward recursion
\begin{equation*} \label{eq:Schur_compl_rec}
\Sigma_n = T_{n,n}, \enspace \Sigma_i = T_{i,i} - T_{i,i+1}\Sigma_{i+1}^{-1}T_{i+1,i}, \enspace i= n-1,\ldots,1.
\end{equation*}
\end{lemma}
\begin{proof}
Straightforward algebra.
\end{proof}
For a CF-matrix $T_n(A)$, the Schur complements---if they exist---are rational matrix functions of $A$. Since these commute with polynomials in $A$, we see that
\begin{equation*}
\Sigma_i = \beta_i(A) - \gamma_{i+1}(A)\Sigma_{i+1}^{-1}\alpha_{i+1}(A) = b_i(A) + c_{i+1}(A)\Sigma_{i+1}^{-1}.
\end{equation*}
Having thus expressed the recursions for the Schur complements in terms of the $b_i$ and $c_i$, we see from \cref{eq:tail_rec} that we actually have
\[
\Sigma_i = b_i(A) + t_i(A),
\]
where $t_i(z)$ is the tail of the (finite) continued fraction $g_n$. In particular, $\Sigma_0 = g_n(A)$.
The following theorem is now the matrix analog of \cref{thm:trid}.
\begin{theorem}
\label{thm:5d_linear}
Let $g(z)$ be a continued fraction with polynomial partial numerators and denominators of the form \cref{eq:poly_conf_def} and let its $n$-th approximant $g_n(z)$ be the inverse of the rational function $r(z) = g_n(z)^{-1}$. Then
\begin{equation} \label{eq:rational_func_TA}
r(A)v = (\transpose{e}_1\otimes I) T_n(A)^{-1} (e_1\otimes v),
\end{equation}
i.e., the action of the rational function on a vector $v\in\complns^m$ can be computed by solving a linear system with the CF-matrix $T_n(A)$.
\end{theorem}
\begin{proof}
We first assume that for the given matrix $A$ the UDL decomposition $T_n(A)^{-1} = L^{-1}D^{-1}U^{-1}$ of \cref{lem:UDL} exists, i.e., that all Schur complements $\Sigma_i$ are nonsingular. Then, using $(e_1\otimes v) = (e_1\otimes I)v$,
\begin{equation*}
(\transpose{e}_1\otimes I) T_n(A)^{-1} (e_1\otimes v) = (\transpose{e}_1\otimes I) L^{-1}D^{-1}U^{-1} (e_1\otimes I) v.
\end{equation*}
Analogously to \cref{cor:freedom}, since $(\transpose{e}_1\otimes I) L^{-1} = \transpose{e}_1\otimes I$, $U^{-1} (e_1\otimes I) = e_1\otimes I$, we can simplify this to
\begin{equation*}
(\transpose{e}_1\otimes I) L^{-1} D^{-1} U^{-1} (e_1\otimes I) =
(\transpose{e}_1\otimes I) D^{-1} (e_1\otimes I) = \Sigma_0^{-1}.
\end{equation*}
By construction, $\Sigma_0 = g_n(A) = r(A)^{-1}$, thus giving \cref{eq:rational_func_TA}.
For a general matrix $A$, the theorem now follows by a continuity argument: The Schur complements $\Sigma_i$ are
rational matrix functions $s_i(A)$, with $s_i$ defined by the entries of $T_n(z)$, evaluated at the matrix $A$. The
Schur complements are thus nonsingular if $s_i(\lambda) \neq 0$ for all $\lambda \in \operatorname{spec}(A)$ and
$i=0,\ldots,n$. Let $A = VJV^{-1}$ be the Jordan canonical form of $A$, take $D_\varepsilon = (1+\varepsilon) I$ with
$\varepsilon >0$ and consider $A_\varepsilon = VD_\varepsilon JV^{-1}$. Then $\operatorname{spec}(A_\varepsilon) = \{(1+\epsilon)\lambda,\; \lambda \in \operatorname{spec}(A)\}$. The set of all zeros of all the
rational functions $s_i(z)$ is finite and so is $\operatorname{spec}(A)$. This implies that for $\varepsilon>0$ sufficiently small the
set $\operatorname{spec}(A_\varepsilon)$ is disjoint from the set of all zeros of all $s_i$. For such $\varepsilon$, from what we have
already shown, we know that \cref{eq:rational_func_TA} holds for $A_\varepsilon$, and letting $\varepsilon \to 0$ shows that \cref{eq:rational_func_TA} also holds for $A$.
\end{proof}
\begin{remark}
If the approximants $g_n$ of $g(z)$ yield Padé approximations for each $n$, we can increase the degree of the Pad\'e approximation by simply appending $k$ block rows and columns to $T_n(A)$ which yields $T_{n+k}(A)$.
\end{remark}
\subsection{Eigendecomposition}
Assume that $A \in \complns^{m \times m}$ is diagonalizable with the eigendecomposition
\begin{equation*}
AW = W\Lambda,
\end{equation*}
where the columns $w_i$ of $W$ are the eigenvectors and the diagonal elements $\lambda_{i}$ of the diagonal matrix $\Lambda$ the corresponding eigenvalues.
Then it is easy to find the eigendecomposition of the general CF-matrix $T_n(A)$: For any vector $v \in \complns^{n+1}$ we have
\begin{align*}
\MoveEqLeft T_n(A) (v \otimes w_i) \\
&= \sum_{j=0}^{\ell} (T_n^{(j)} \otimes A^j)(v \otimes w_i) = \sum_{j=0}^{\ell} T_n^{(j)} v \otimes A^j w_i = \left(\sum_{j=0}^{\ell} \lambda_i^j T_n^{(j)} v \right) \otimes w_i \\
&= \left( T_n(\lambda_i) v \right) \otimes w_i.
\end{align*}
Thus, for each $i$, if $v$ is an eigenvector of $T_n(\lambda_i)$, then $v \otimes w_i$ is an eigenvector of $T_n(A)$. As a consequence, if each of the matrices $T_n(\lambda_i)$ is diagonalizable with eigenvectors $v_{k,i}$ and eigenvalues $\mu_{k,i}$, then the $(n+1)m$ vectors $v_{k,i} \otimes w_i$ represent a full system of eigenvectors for $T_n(A)$ with eigenvalues $\mu_{k,i}$.
\section{Special CF-matrices}\label{sec:simple}
Up until now, the polynomials that appear in the continued fraction were allowed to be of arbitrarily high degree, and thus, expressing $T_n(A)$ as in \cref{eq:5d_sum}, might involve many terms. In many cases, however, we only need the first two terms.
In this section, we now assume that $T_n(z) = T_n^{(0)} - zT_n^{(1)}$, such that
\begin{equation*}
T_n(A) = T_n^{(0)}\otimes I - T_n^{(1)}\otimes A
\end{equation*}
with tridiagonal matrices $T_n^{(0)}, T_n^{(1)} \in \complns^{(n+1)\times(n+1)}$. We take a closer look at the linear system
\begin{equation}
\label{eq:5d_linearsystem}
T_n(A)x = e_1 \otimes v,
\end{equation}
the solution of which gives $g_n(A)^{-1}v$ according to \cref{thm:5d_linear}.
\subsection{Construction}
To illustrate the connection between the matrices $T_n^{(0)}$, $T_n^{(1)}$ and the partial numerators and denominators of the underlying continued fraction, we discuss three special cases.
\begin{example}[Regular C-fractions]
In the approximant of a regular C-fraction
\begin{equation*}
g_n(z) = b_0 + \contf{i=1}{n}{c_iz}{1},
\end{equation*}
the partial denominators are all 1. For the numerators we use $-c_iz = (-1)\cdot(c_iz)$ and obtain
\begin{equation*}
T_n^{(0)} = \begin{bmatrix}
b_0 & 1 \\
& 1 & \ddots \\
& &\ddots & 1 \\
& & & 1
\end{bmatrix}, \quad
T_n^{(1)} = \begin{bmatrix}
0 & \\
c_1 & \ddots& \\
& \ddots&\ddots & \\
& & c_n & 0
\end{bmatrix}.
\end{equation*}
Here, we chose the subdiagonal to contain the coefficients $c_i$, but we could as well choose the superdiagonal, i.e., we could take the pair $(T_n^{(0)})^{\mathsf{T}}$, $(T_n^{(1)})^{\mathsf{T}}$ instead of $T_n^{(0)}$, $T_n^{(1)}$.
\end{example}
\begin{example}[Contracted regular C-fractions]\label{ex:contracted}
Assume we contract the regular C-fraction before constructing the CF-matrix. Recall that by \cref{eq:C_frac_contracted} the approximants of the contracted partial fraction are given by
\begin{equation*}
\widetilde{g}_{n}(z) = b_0 + \frac{c_1 z}{1 + c_2 z + \contf{i=2}{n}{-c_{2i-2}c_{2i-1} z^2}{1 + (c_{2i}+c_{2i-1})z}}.
\end{equation*}
In this continued fraction, monomials of degree 2 appear in the partial numerators.
Factorizing $c_{2i-2}c_{2i-1} z^2 = (c_{2i-2}z)(c_{2i-1} z)$, we see that a CF-matrix can be constructed as $T_n(z) = T_n^{(0)} - z T_n^{(1)} $ with
\begin{align*}
T_n^{(0)} &= \begin{bmatrix}
b_0 & -1 \\
& 1 &\\
& & \ddots & \\
& & & 1
\end{bmatrix},\\
T_n^{(1)} &= (-1)\cdot\begin{bmatrix}
0 & 0\\
c_1 & c_2 & c_2 \\
& c_3 & c_3+c_4 & c_4 \\
& & c_5 & c_5+c_6 & \ddots \\
& & & \ddots & \ddots & c_{2n-2} \\
& & & & c_{2n-1}& c_{2n-1}+c_{2n}
\end{bmatrix}.
\end{align*}
Since the approximant $\widetilde{g}_{n}(z)$ of the contracted C-fraction is the same rational function as $g_{2n}(z)$ of the original C-fraction,
using the contracted form for the same diagonal Padé approximation reduces the size of the CF-matrix from $2n+1$ to $n+1$.
\end{example}
\begin{example}[Continued fractions via polynomial long division]
Let us consider an approximant of a continued fraction obtained by repeated polynomial long division
\begin{equation*}
g_n(z) = b_0(z) + \contf{i=1}{n}{1}{b_i(z)},
\end{equation*}
and assume that all quotients have degree at most 1, i.e., $b_i(z) = b_i^{(0)} - b_i^{(1)}z$. Then a possible construction for $T_n^{(0)}, T_n^{(1)}$ is
\begin{equation*}
T_n^{(0)} = \begin{bmatrix}
b_0^{(0)} & -1 \\
1 & \ddots&\ddots \\
&\ddots &\ddots & -1 \\
& & 1 & b_n^{(0)}
\end{bmatrix}, \quad
T_n^{(1)} = \begin{bmatrix}
b_0^{(1)} & \\
& \ddots& \\
& & b_n^{(1)}
\end{bmatrix}.
\end{equation*}
In this case, it is possible to transform $T_n^{(0)}$ and $T_n^{(1)}$ along the lines of
\cref{cor:freedom} such that they are symmetric. For instance, with the unitary diagonal matrices
\begin{align*}
D_\mathrm{L} &= \operatorname{diag}\opfences{(-1)^{\floor{0/2}}, \dots, (-1)^{\floor{n/2}}},\\
D_\mathrm{R} &= \operatorname{diag}\opfences{(-1)^{\floor{(0+1)/2}}, \dots, (-1)^{\floor{(n+1)/2}}}
\end{align*}
we obtain
\begin{align*}
D_\mathrm{L} T_n^{(0)} D_\mathrm{R} &= \begin{bmatrix}
(-1)^0 b_0^{(0)}& 1 \\
1 &(-1)^1 b_1^{(0)} &\ddots \\
&\ddots &\ddots & 1 \\
& & 1 & (-1)^n b_n^{(0)}
\end{bmatrix},\\
D_\mathrm{L} T_n^{(1)} D_\mathrm{R} &= \operatorname{diag}\opfences{(-1)^0 b_0^{(1)}, \dots, (-1)^n b_n^{(1)}}
\end{align*}
or, alternatively,
\begin{align*}
D_\mathrm{R} T_n^{(0)} D_\mathrm{L} &= \begin{bmatrix}
(-1)^0 b_0^{(0)}& -1 \\
-1 &(-1)^1 b_1^{(0)} &\ddots \\
&\ddots &\ddots & -1 \\
& & -1 & (-1)^n b_n^{(0)}
\end{bmatrix},\\
D_\mathrm{R} T_n^{(1)} D_\mathrm{L} & = D_\mathrm{L} T_n^{(1)} D_\mathrm{R}.
\end{align*}
\end{example}
Let us remark that these examples illustrate that the matrix $T_n^{(1)}$ is often singular.
\subsection{Weierstrass canonical form and partial fraction expansion}
If we have $\det(T_n(z)) \not\equiv 0$, i.e., the determinant does not vanish identically as a function of $z$, then the \emph{Weierstrass canonical form}\footnote{The Weierstrass canonical form is a special case of the \emph{Kronecker canonical form} for regular pencils.} exists, see \cite{Gantmacher}.
That is, there exist nonsingular matrices $U,V \in \complns^{(n+1) \times (n+1)}$ such that
\begin{equation*}
U(T_n^{(0)} - zT_n^{(1)})V =
\begin{bmatrix}
J^{(0)} - zI_{n^{(0)}} \\
& I_{n^{(1)}} - zJ^{(1)}
\end{bmatrix},
\end{equation*}
with the Jordan matrices
\begin{equation} \label{eq:Kronecker_form}
\left\{ \begin{array}{rcl}
J^{(0)} &=& \bigoplus_{j=1}^{k_0} J(\tau_j,n_j^{(0)}), \enspace \tau_j \in \complns, \\
J^{(1)} &=& \bigoplus_{j=1}^{k_1} J(0,n_j^{(1)}),
\end{array} \right.
\end{equation}
where $n^{(i)} = \sum_{j=1}^{k_i} n_j^{(i)}$, $n^{(0)}+n^{(1)} = n+1$
and $J(\mu,m)$ denotes a Jordan block of size $m$ given as
\[
J(\mu,m) = \mu I_m + S_{m}, \enspace S_{m} = \begin{bmatrix} 0 & 1 & 0 & \cdots &0 \\
\vdots & \ddots & \ddots & \ddots & 0 \\
\vdots& & \ddots & \ddots & 0 \\
\vdots & & & 0 & 1 \\
0 & \cdots & \cdots & 0 & 0
\end{bmatrix}, \enspace I_{m}, S_{m} \in \complns^{m \times m}.
\]
The $\tau_i$ are the (generalized) eigenvalues of the pencil $T_n^{(0)}-zT_n^{(1)}$. With the Weierstrass canonical form, we are able to describe the partial fraction expansion of $g_n(z)^{-1}$ via the pencil $T_n^{(0)}-zT_n^{(1)}$ as we show in the following theorem. In particular, the eigenvalues of $T_n^{(0)}-zT_n^{(1)}$ are the poles of $g_n(z)^{-1}$.
\begin{theorem} \label{thm:5d_pfe}
Assume that $\det(T_n(z)) \not\equiv 0$. Let $U,V$ and $J(\tau_j,n_j^{(0)})$, $J(0, n_j^{(1)})$
be the matrices and parameters of the Weierstrass canonical form \cref{eq:Kronecker_form}. Let $u = Ue_1$ and $\transpose{v}= \transpose{e}_1V$, and let $u^{(j)}$ and $(v^{(j)})^\mathsf{T}$ denote the blocks of $u$ and $\transpose{v}$ corresponding to block $j$ of the Weierstrass canonical form.
Then
\begin{equation*}
\transpose{e}_1 T_n(z)^{-1} e_1 = \sum_{j=1}^{k_0}\sum_{i=0}^{n_j^{(0)}-1} \frac{-\omega_{j,i}}{(z-\tau_j)^{i+1}} + \sum_{j=1}^{k_1}\sum_{i=0}^{n_j^{(1)}-1} z^{i} \sigma_{j,i}
\end{equation*}
where
\begin{align*}
\omega_{j,i} &= (v^{(j)})^\mathsf{T} \big(S_{n_j^{(0)}}\big)^i u^{(j)},\\
\sigma_{j,i} &= (v^{(k_0+j)})^\mathsf{T} \big(S_{n_j^{(1)}}\big)^i u^{(k_0+j)}.
\end{align*}
\end{theorem}
\begin{proof} We have
\begin{equation} \label{eq:Tninv11}
e_1^\mathsf{T} T_n(z)^{-1}e_1 = \underbrace{e_1^\mathsf{T} V}_{=v^\mathsf{T}} \left((J^{(0)} - zI)^{-1} \oplus (-zJ^{(1)} + I)^{-1} \right) \underbrace{Ue_1}_{=u}.
\end{equation}
Since for any Jordan block $J(\mu,m)$ we have
\[
J(\mu,m)^{-1} = \sum_{i=0}^{m-1} \frac{(-1)^i}{\mu^{i+1}} \big(S_{m}\big)^i = \sum_{i=0}^{m-1} \frac{-1}{(-\mu)^{i+1}} \big(S_{m}\big)^i,
\]
this gives
\begin{equation*}
(J^{(0)} - zI)^{-1} = \bigoplus_{j=1}^{k_0} \sum_{i=0}^{n_j^{(0)}-1} \frac{-1}{(z-\tau_j)^{i+1}} \big(S_{n_j^{(0)}}\big)^i
\end{equation*}
and similarly
\begin{equation*}
(-z J^{(1)} + I)^{-1} = -z^{-1}(J^{(1)} - z^{-1}I)^{-1} = \bigoplus_{j=1}^{k_1} \sum_{i=0}^{n_j^{(1)}-1} z^{i} \big(S_{n_j^{(1)}}\big)^i.
\end{equation*}
Inserting the last two equalities into \cref{eq:Tninv11} gives
\begin{align*}
\transpose{e}_1 T_n(z)^{-1} e_1 =& \sum_{j=1}^{k_0}\sum_{i=0}^{n_j^{(0)}-1} \frac{-1}{(z-\tau_j)^{i+1}} \underbrace{(v^{(j)})^\mathsf{T} \big(S_{n_j^{(0)}}\big)^i u^{(j)}}_{=\omega_{j,i}} \\
{}& + \sum_{j=1}^{k_1}\sum_{i=0}^{n_j^{(1)}-1} z^{i} \underbrace{(v^{(k_0+j)})^\mathsf{T} \big(S_{n_j^{(1)}}\big)^i u^{(k_0+j)}}_{=\sigma_{j,i}}. \qedhere
\end{align*}
\end{proof}
We note that a similar result was given in \cite{Berljafa} under the additional assumption that $T_n^{(1)}$ is nonsingular if there are higher-order poles.
\begin{remark}
The coefficients $\omega_{j,i}$ and $\sigma_{j,i}$ are essentially determined by the vectors $u = Ue_1$ and $\transpose{v}= \transpose{e}_1V$. By choosing vectors other than $\transpose{e}_1$ and $e_1$, one would still obtain a rational function with the same poles and same multiplicities. Moreover, the polynomial part would have degree at most $\max_j n_j^{(1)}-1$.
\end{remark}
\subsection{Generalized Sylvester equation}
There is an interesting connection of the linear system $T_n(A)x = e_1 \otimes v$ to Sylvester-type matrix equations which we shortly discuss here. Let us denote by $\operatorname{vec}\opfences{}$ the operator which maps a matrix to the vector obtained by stacking its columns from left to right. Then,
see for example \cite{vanLoan,vanLoan2}, for compatible matrices $A,X$ and $B$, we have
\begin{equation*}
\operatorname{vec}\opfences{AXB} = (\transpose{B}\otimes A)\operatorname{vec}\opfences{X}.
\end{equation*}
Because of this, the {\em Sylvester equation}
\begin{equation*}
AX + XB = C .
\end{equation*}
is equivalent to the linear system
\begin{equation*}
(I \otimes A + \transpose{B} \otimes I)\operatorname{vec}\opfences{X} = \operatorname{vec}\opfences{C}.
\end{equation*}
This is the basis for the following corollary.
\begin{corollary} \label{cor:matrix_equation}
The linear system \cref{eq:5d_linearsystem} with $T_n(A) = T_n^{(0)}\otimes I - T_n^{(1)}\otimes A$ is equivalent to the matrix equation
\begin{equation} \label{eq:Sylvester_type}
X(T_n^{(0)})^{\mathsf{T}} - AX(T_n^{(1)})^{\mathsf{T}} = v \transpose{e}_1 \quad \mbox{where } x = \operatorname{vec}\opfences{X}.
\end{equation}
\end{corollary}
Solution methods for Sylvester equations have been an active research area in recent years. See, e.g., \cite{Simoncini16}, for a review. Consequently, it might be worthwhile to adapt such methods to \cref{eq:Sylvester_type}---an idea that, however, we do not develop any further here.
\subsection{Block Jacobi and block Gauß-Seidel}
If one wants to compute $r(A)v$, one might use its partial fraction expansion
(see \cref{thm:5d_pfe}) to get
\begin{equation}\label{eq:pfe}
r(A)v = -\sum_{j=1}^{k_0}\sum_{i=0}^{n_j^{(0)}-1} \omega_{j,i}(A-\tau_jI)^{-(i+1)}v + \sum_{j=1}^{k_1}\sum_{i=0}^{n_j^{(1)}-1} \sigma_{j,i} A^{i}v.
\end{equation}
This involves solving the shifted systems
\begin{equation*}
(A-\tau_jI)^{i+1}x = v.
\end{equation*}
We will now show that for the CF-system~\cref{eq:5d_linearsystem}, block iterative methods like block
Jacobi and block Gauß-Seidel with a specific choice for the blocks can be expected to expose the
same convergence properties as their nonblock versions on the shifted systems arising from the partial fraction expansion.
As a preparation, we introduce two theorems about the eigenvalues of pencils.
We use the standard terminology for eigenvalues of a pencil as, e.g., in \cite{Stewart}: The spectrum $\operatorname{spec}(T-z\tilde{T})$ of a pencil $T-z\tilde{T}$ are all $\lambda \in \complns$ for which there exists a nonzero eigenvector $x$ such that $Tx - \lambda \tilde{T}x = 0$. In addition,
the pencil has an eigenvalue at infinity if $\tilde{T}$ is singular.
\begin{theorem}\label{thm:ews_pencils_diag}
Let the Weierstrass canonical form of the regular pencil $T^{(0)} -z T^{(1)} \in \complns^{(n+1)\times(n+1)}$ be diagonal, i.e., $k_0 = n^{(0)}$, $k_1 = n^{(1)}$ and
\begin{equation*}
U(T^{(0)} - zT^{(1)})V =
\underbrace{\begin{bmatrix}
D\\
& I_{n^{(1)}}
\end{bmatrix}}_{K^{(0)}} - z
\underbrace{\begin{bmatrix}
I_{n^{(0)}} \\
& 0
\end{bmatrix}}_{K^{(1)}}
\enspace \mbox{ with } D = \operatorname{diag}\opfences{\tau_1,\ldots,\tau_{k_0}}.
\end{equation*}
Furthermore, define the following matrices:
\begin{align*}
T &= T^{(0)} \otimes I - T^{(1)} \otimes A, \\
\widetilde{T} &= T^{(0)} \otimes I - T^{(1)} \otimes \widetilde{A}, \enspace \mbox{ with } A, \widetilde{A} \in \complns^{m\times m}.
\end{align*}
Then the following holds:
\begin{enumerate}[(i)
\item $\operatorname{spec}(T-z\tilde{T}) = \begin{cases}
\bigcup_{i=1}^{k_0} \operatorname{spec}\big((\tau_i I - A) - z(\tau_i I - \tilde{A})\big) & \text{if } n^{(1)} = 0,\\
\bigcup_{i=1}^{k_0} \operatorname{spec}\big((\tau_i I - A) - z(\tau_i I - \tilde{A})\big) \cup \{1\} & \text{if } n^{(1)} > 0.
\end{cases}$
\item The pencil $T-z\tilde{T}$ has an eigenvalue at infinity if and only if one of the pencils $(\tau_\alpha I - A) - z(\tau_\alpha I - \tilde{A})$ has an eigenvalue at infinity.
\item If $w$ is an eigenvector of the pencil $(\tau_\alpha I - A) - z(\tau_\alpha I - \tilde{A})$ with eigenvalue $\lambda$, then there exists a vector $v\neq 0$ such that $v\otimes w$ is an eigenvector of $T-z\tilde{T}$ with eigenvalue $\lambda$.
\end{enumerate}
\end{theorem}
\begin{proof}
Multiplying the eigen equation $Tx = \lambda \tilde{T}x $ with $U \otimes I$ from the left and putting $y = (V^{-1} \otimes I)x$ gives the equivalent equation
\begin{equation} \label{eq:eig_for_y}
\big(K^{(0)} \otimes I - K^{(1)} \otimes A\big) y = \lambda \big(K^{(0)} \otimes I - K^{(1)} \otimes \tilde{A}\big) y .
\end{equation}
The matrices $K^{(0)} \otimes I - K^{(1)} \otimes A$ and $K^{(0)} \otimes I - K^{(1)} \otimes \tilde{A}$ are both block diagonal with $n+1$ diagonal blocks of size $m$. We use the index $\alpha \in\{1,\ldots,n+1\}$ to denote such a block.
Now assume that \cref{eq:eig_for_y} holds for some $y \neq 0$ and let $\alpha$ be a block for which $y_\alpha \neq 0$. Then block $\alpha$ in \cref{eq:eig_for_y} reads
\begin{align*}
(\tau_\alpha I - A) y_\alpha &= \lambda(\tau_\alpha I - \tilde{A})y_\alpha &&\mbox{if } \alpha \leq n^{(0)}, \\
Iy_\alpha &= \lambda Iy_\alpha &&\mbox{if } n^{(0)} < \alpha \leq n+1.
\end{align*}
This proves that the set to the left is contained in the set to the right in (i).
%
For the opposite inclusion, assume that for some $\alpha \in
\{1, \ldots, n^{(0)}\}$ we have $(\tau_\alpha I - A) w = \lambda(\tau_\alpha I - \tilde{A})w$ for some vector $w \neq 0$. Then the vector $y$ with block components
\[
y_\beta = \left\{ \begin{array}{rl} w & \mbox{if } \beta = \alpha, \\
0 & \mbox{otherwise} \end{array} \right .
\]
is nonzero and satisfies \cref{eq:eig_for_y}. This, in passing, proves (iii) because $y=e_\alpha \otimes w$ with $e_\alpha$ being a unit vector gives $x=(Ve_\alpha)\otimes w = v\otimes w$ as an eigenvector of $T-z\tilde{T}$. In case $n^{(0)} < n+1$, for $\alpha \in \{n^{(0)}+1,\ldots,n+1\}$ the same construction with $w$ being just any nonzero vector from $\complns^m$ gives an eigenvector $y$ with eigenvalue 1 in \cref{eq:eig_for_y} which concludes the proof of (i).
Finally for (ii), $\infty$ being an eigenvalue of the pencil $T-z\tilde{T}$ is equivalent to $\tilde{T}$ being singular. Then $K^{(0)}\otimes I - K^{(1)} \otimes \tilde{A}$ is singular which means that for some $\alpha \in \{1, \ldots, n^{(0)}\}$ the diagonal block $\tau_\alpha I - \tilde{A}$ is singular. This in turn implies that $\infty$ is an eigenvalue of the pencil $(\tau_\alpha I - A) - z(\tau_\alpha I - \tilde{A})$.
\end{proof}
\begin{remark}\label{re:diagonal_simple_pfe}
If and only if the pencil $T_n^{(0)} - zT_n^{(1)}$ of the CF-matrix $T_n$ has a diagonal Weierstrass form, the corresponding partial fraction expansion has only simple poles and its polynomial part is constant, since it then simplifies to
\begin{equation*}
\transpose{e}_1 T_n(z)^{-1} e_1 = \sum_{j=1}^{k_0} \frac{\omega_{j,0}}{\tau_j-z} + \sum_{j=1}^{k_1} \sigma_{j,0}.
\end{equation*}
\end{remark}
\begin{theorem}\label{thm:ews_pencil_gen}
Let the Weierstrass canonical form of the pencil $T^{(0)} - zT^{(1)}$ in \cref{thm:ews_pencils_diag} not be diagonal, i.e.,
\begin{equation*}
U(T^{(0)} - zT^{(1)})V =
\begin{bmatrix}
J^{(0)}\\
& I_{n^{(1)}}
\end{bmatrix} - z
\begin{bmatrix}
I_{n^{(0)}} \\
& J^{(1)}
\end{bmatrix}.
\end{equation*}
Then (i) of \cref{thm:ews_pencils_diag} still holds,
\begin{equation*}
\operatorname{spec}(T-z\tilde{T}) = \begin{cases}
\bigcup_{i=1}^{k_0} \operatorname{spec}\big((\tau_i I - A) - z(\tau_i I - \tilde{A})\big) & \text{if } n^{(1)} = 0,\\
\bigcup_{i=1}^{k_0} \operatorname{spec}\big((\tau_i I - A) - z(\tau_i I - \tilde{A})\big) \cup \{1\} & \text{if } n^{(1)} > 0.
\end{cases}
\end{equation*}
\end{theorem}
\begin{proof}
To circumvent the nontrivial Jordan blocks in $J^{(0)}$, $J^{(1)}$, we perturb the pencil $T^{(0)} - zT^{(1)}$ using the pencil $E - zF$ defined such that
\begin{alignat*}{2}
UEV &= \begin{bmatrix}
\varepsilon E^{(0)}\\
& 0
\end{bmatrix}, \quad &E^{(0)} = \delta \cdot &\operatorname{diag}\opfences{2^{-1}, \dots, 2^{-n^{(0)}}},\\
UFV &= \begin{bmatrix}
0\\
& \varepsilon F^{(1)}
\end{bmatrix}, \quad &F^{(1)} = &\operatorname{diag}\opfences{2^{-1}, \dots, 2^{-n^{(1)}}}.
\end{alignat*}
We now choose $\delta$ such that for $ 0 < \varepsilon \leq 1$
the diagonal entries of $J^{(0)} + \varepsilon E^{(0)}$ are pairwise distinct. One such choice is
\begin{equation*}
\delta = \begin{cases}
1 & \text{if } k_0 = 1,\\
\min_{i\neq j} \abs{\tau_i-\tau_j} & \text{if } k_0 > 1.
\end{cases}
\end{equation*}
As a consequence, the perturbed pencil $(T^{(0)}+E) - z(T^{(1)}+F)$ is diagonalizable, i.e., there exist matrices $M^{(0)}$ and $M^{(1)}$ such that
\begin{align*}
M^{(0)} (J^{(0)} + \varepsilon E^{(0)}) (M^{(0)})^{-1} &= \operatorname{diag}\opfences{J^{(0)}_{1,1} + \varepsilon E^{(0)}_{1,1}, \dots, J^{(0)}_{n^{(0)},n^{(0)}} + \varepsilon E^{(0)}_{n^{(0)},n^{(0)}}}, \\
M^{(1)}(J^{(1)} + \varepsilon F^{(1)}) (M^{(1)})^{-1} &= \operatorname{diag}\opfences{\varepsilon F^{(1)}_{1,1}, \dots, \varepsilon F^{(1)}_{n^{(1)},n^{(1)}}}.
\end{align*}
We now consider the perturbation $\Tpert - z\widetilde{\Tpert}$ of the pencil $T - z\widetilde{T}$ given by
\begin{gather*}
\Tpert = T + (E\otimes I - F\otimes A) = (T^{(0)}+E) \otimes I - (T^{(1)}+F) \otimes A, \\
\widetilde{\Tpert} = \widetilde{T} + (E\otimes I - F\otimes \widetilde{A}) = (T^{(0)}+E) \otimes I - (T^{(1)}+F) \otimes \widetilde{A}.
\end{gather*}
Clearly
\begin{align*}
\lim_{\varepsilon \to 0} ((T^{(0)}+E) - z(T^{(1)}+F)) &= T^{(0)} - zT^{(1)}, \\
\lim_{\varepsilon \to 0} (\Tpert - z\widetilde{\Tpert}) &= T - z\widetilde{T}.
\end{align*}
Let $\mu_j$ be the eigenvalues of $\Tpert - z\widetilde{\Tpert}$. Then, see \cite[Theorem~2.1]{Stewart} for example, the eigenvalues $\lambda_j$ of $T - z\widetilde{T}$ can be ordered such that
\begin{equation*}
\lim_{\varepsilon \to 0} \mu_j = \lambda_j.
\end{equation*}
By construction, we can block-diagonalize the pencil $\Tpert - z\widetilde{\Tpert}$ by multiplying with $\left(\begin{bmatrix} M^{(0)} & 0\\0 & M^{(1)} \end{bmatrix} U\right)\otimes I$ from the left and with $\left(V\begin{bmatrix} (M^{(0)})^{-1} & 0\\0 & (M^{(1)})^{-1} \end{bmatrix}\right)\otimes I$ from the right.
When applied to the eigen equation, we get
\begin{align*}
((J^{(0)}_{\alpha,\alpha} + \varepsilon E^{(0)}_{\alpha,\alpha}) I - A) y_\alpha &= \mu((J^{(0)}_{\alpha,\alpha} + \varepsilon E^{(0)}_{\alpha,\alpha}) I - \tilde{A})y_\alpha &&\mbox{if } \alpha \leq n^{(0)}, \\
(I-\varepsilon F^{(1)}_{\alpha,\alpha} A) y_\alpha &= \mu (I-\varepsilon F^{(1)}_{\alpha,\alpha} \widetilde{A}) y_\alpha &&\mbox{if } \alpha > n^{(0)}.
\end{align*}
By analogy with \cref{thm:ews_pencils_diag}, we have
\begin{align*}
\operatorname{spec}(\Tpert-z\tilde{\Tpert}) =&
\bigcup_{i=1}^{n^{(0)}} \operatorname{spec}\left(((J^{(0)}_{i,i}+\varepsilon E_{i,i}^{(0)}) I - A) - z((J^{(0)}_{i,i}+\varepsilon E_{i,i}^{(0)}) I - \tilde{A})\right) \\
&\cup
\bigcup_{i=1}^{n^{(1)}} \operatorname{spec}\left((I - \varepsilon F_{i,i}^{(1)} A) - z(I - \varepsilon F_{i,i}^{(1)}\tilde{A})\right).
\end{align*}
In the limit $\varepsilon \to 0$, we retrieve the eigenvalues of the pencils $(J^{(0)}_{\alpha,\alpha} I - A) - z(J^{(0)}_{\alpha,\alpha} I - \widetilde{A})$ for $\alpha \leq n^{(0)}$ and the eigenvalue 1 for $\alpha > n^{(0)}$. Note that there are only $k_0 < n^{(0)}$ different values $\tau_j$ for $J^{(0)}_{\alpha,\alpha}$ because the Weierstrass form is not diagonal. Thus, we obtain the pencils $(\tau_j I - A) - z(\tau_j I - \widetilde{A})$ with $j \leq k_0$.
\end{proof}
Let us now use the above results for an analysis of splitting-based iterative methods for the system \cref{eq:5d_linearsystem}.
We recall that splitting methods for the system $Ax=b$ can be written as
\begin{equation}
\label{eq:smoother}
x_{i+1} = x_i + \widetilde{A}^{-1} (b-Ax_i)
\end{equation}
where $\widetilde{A}$ is an easily invertible approximation to $A$. For example, $\widetilde{A}$ may be the diagonal of $A$---giving the Jacobi iteration---or the lower triangular part---giving the Gauß-Seidel iteration.
In a splitting method, the errors $\epsilon_i = x-x_i$ satisfy
\begin{equation*}
\epsilon_{i+1} = (I - \widetilde{A}^{-1}A) \epsilon_i
\end{equation*}
with the error propagator $M = I - \widetilde{A}^{-1}A$.
We now consider a special block version of the general splitting approach for the system $T_n(A) x = e_1\otimes v$ where
$\widetilde{T_n(A)}$ has the form
\[
\widetilde{T_n(A)} = T_n(\widetilde{A}) = T_n^{(0)}\otimes I - T_n^{(1)}\otimes \widetilde{A}.
\]
Depending on the choice for $\widetilde{A}$, the matrix $T_n(\widetilde{A})$ can be interpreted as resulting from a {\em block} splitting of $T_n(A)$. To see this, we first observe that for a Kronecker product of two matrices $B\in \complns^{k \times k}$ and $C\in \complns^{\ell \times \ell}$, there exists a permutation matrix $P$ such that
\begin{equation*}
P(B\otimes C)\transpose{P} = C \otimes B.
\end{equation*}
For more details, see, e.g., \cite[eq.~(1.3.5)]{vanLoan}. Thus,
we have
\begin{equation*}
P\widetilde{T_n(A)} \transpose{P} = I \otimes T_n^{(0)} - \widetilde{A} \otimes T_n^{(1)},
\end{equation*}
in which the nonzero blocks, which themselves are at most tridiagonal, are determined by the sparsity of $\widetilde{A}$ (and $I$). Therefore, any $\widetilde{A}$ resulting from a splitting for $A$ induces a block splitting $ PT_n(\widetilde{A}) \transpose{P}$ for $P T_n(A) \transpose{P}$.
For instance, if we take $\widetilde{A}$ to be the diagonal of $A$, the matrix $PT_n(\widetilde{A}) \transpose{P}$ is block diagonal and it represents the approximation to $P T_n(A) \transpose{P}$ corresponding to the block Jacobi splitting. Similarly, if $\widetilde{A}$ is the lower triangular part of $A$, we retrieve the block Gauß-Seidel splitting for the (permuted) matrix $T_n(A)$.
In the next theorem, we relate the spectrum of the error propagator $M_{T_n} = I - T_n(\widetilde{A})^{-1}T_n(A)$ for the CF-matrix to the spectra of error propagators $I-(\widetilde{A}-\tau I)^{-1}(A-\tau I)$ where the shifts $\tau$ are the poles of the rational function represented by the continued fraction.
\begin{theorem}\label{thm:smoother_gen}
Given $\widetilde{A}$ as an approximation for $A$, assume that for any $\tau \in \complns$ we take
\[
\widetilde{A-\tau I} = \widetilde{A}-\tau I.
\]
Let $M(\tau) = I - (\widetilde{A-\tau I})^{-1} (A-\tau I)$ and $M_{T_n} = I - T_n(\widetilde{A})^{-1}T_n(A) $ be the error propagators for the shifted matrices $A-\tau I$ and the CF-matrix $T_n(A)$, respectively.
Then
\begin{equation*}
\operatorname{spec}(M_{T_n}) = \begin{cases}
\bigcup_{i=1}^{k_0} \operatorname{spec}(M(\tau_i)) & \text{if } n^{(1)} = 0,\\
\bigcup_{i=1}^{k_0} \operatorname{spec}(M(\tau_i)) \cup \{0\} & \text{if } n^{(1)} > 0,
\end{cases}
\end{equation*}
where $\tau_1,\ldots,\tau_{k_0}$ are the eigenvalues of $T_n(A)$ from the Weierstrass canonical form; see \cref{eq:Kronecker_form}.
\end{theorem}
\begin{proof}
The eigenvalues of
\begin{equation*}
M_{T_n} = I - T_n(\widetilde{A})^{-1}T_n(A)
\end{equation*}
are obviously $1-\mu$ where $\mu$ are the eigenvalues of $T_n(\widetilde{A})^{-1}T_n(A)$. They can be interpreted as the eigenvalues of the pencil $T_n(A) - zT_n(\widetilde{A})$. \Cref{thm:ews_pencil_gen} tells us that if $n^{(1)} = 0$ they are given by the eigenvalues of the pencils $(\tau_i I - A) - z(\tau_i I - \widetilde{A})$.
Similarly, the eigenvalues of
\begin{equation*}
M(\tau_i) = I - (\widetilde{A}-\tau_iI)^{-1} (A-\tau_iI)
\end{equation*}
are given by $1-\mu$ where $\mu$ are the eigenvalues of the pencil $(A-\tau_i I) - z(\widetilde{A}-\tau_i I)$ which are those of the pencil $(\tau_i I - A) - z(\tau_i I - \widetilde{A})$.
In the case $n^{(1)} > 0$, the same argument as above can be made but in addition $\mu=1$ is possible.
\end{proof}
\Cref{thm:smoother_gen} tells us that the convergence speed for the block splitting method for $T_n(A)$, measured by the largest eigenvalue of the error propagator, is the slowest of all convergence speeds for the splitting based methods $(A-\tau_iI)$.
If the partial fraction expansion of the rational function has only simple poles and the polynomial part is constant, we can relate the eigenvectors of the error propagators according to the following corollary.
\begin{corollary}\label{cor:smoother_diag}
Let the pencil $T_n^{(0)} - zT_n^{(1)}$ have a diagonal Weierstrass canonical form. Then in addition to \cref{thm:smoother_gen},
\begin{equation*}
\quad M(\tau_i)w = \lambda w \quad\implies\quad \exists v\neq 0: M_{T_n} (v\otimes w) = \lambda (v\otimes w).
\end{equation*}
\end{corollary}
\begin{proof}
By analogy with the proof for \cref{thm:smoother_gen}, the identities in the error propagators do not influence the eigenvectors. What remains are the eigenvectors of the pencils $(A-\tau_i I) - z(\widetilde{A}-\tau_i I)$ which are also the eigenvectors of the pencils $(\tau_i I - A) - z(\tau_i I - \widetilde{A})$ on the left side and the eigenvectors of the pencil $T_n(A) - zT_n(\widetilde{A})$ on the right side. Part (iii) in \cref{thm:ews_pencils_diag} proves the rest.
\end{proof}
We conclude this section by remarking that each diagonal block of the (permuted) CF-matrix is tridiagonal. So their LU-factorizations can be obtained at low cost implying that the cost for performing one block Jacobi or block Gauss-Seidel iteration for the CF-matrix becomes comparable to the accumulated cost for (nonblock) Jacobi or Gauss-Seidel for all shifted systems $A-\tau I$. Our analysis shows that when working with these block iterations ``stand-alone'', we cannot expect to top the approach where we perform the corresponding nonblock iteration on all shifted systems from the partial fraction. We anticipate, however, that our analysis might be helpful when devising a smoother for a multigrid method (see, e.g., \cite{Trottenberg}) for the CF system in future work.
\section{Numerical examples}\label{sec:numerics}
To illustrate the potential of the CF-matrix approach, we now present numerical examples. Emphasis is placed on how the preconditioned CF-matrix compares with the approach of the partial fraction expansion (see \cref{eq:pfe}). All calculations were done in MATLAB R2021a \cite{matlab}.
\begin{example}[Exponential function]
\label{ex:exp}
First, consider the exponential function
\begin{equation*}
f(A)v = \exp(-A)v.
\end{equation*}
According to \cite[eq.~(11.1.3)]{Cuyt}, a regular C-fraction for the inverse of the function, $f(z)^{-1} = \exp(z)$, is given by
\begin{equation*}
b_0 = 1,\quad c_1 = 1,\quad c_i = \begin{cases}
\frac{-1}{2(i-1)} & \text{if } i>1 \text{ is even,}\\
\frac{1}{2i} & \text{if } i>1 \text{ is odd.}
\end{cases}
\end{equation*}
We construct the CF-matrix from the contracted continued fraction as illustrated in \cref{ex:contracted} for $n=20$ and apply (full) GMRES \cite{Saad_GMRES} to the resulting system $T_{20}(A)x = e_1\otimes v$ without preconditioning and with ILU(0) \cite{vanderVorst77,Saad} preconditioning.
For comparison, we also calculate the poles $\tau_j$ of the partial fraction expansion by solving the eigenvalue problem of the pencil $T_{20}^{(0)} - z T_{20}^{(1)}$.\footnote{Note that the Padé approximations in this case have simple poles only. Thus, the Weierstrass canonical form is diagonal.} and apply GMRES to the resulting systems $(A-\tau_j I) x_j = v$. We report, for each iteration, the largest relative residual across all shifts, denoted as PFE in our figures.
For $A$, we use two matrices. First, we take the discrete 2D Laplace operator on a square grid with Dirichlet boundary conditions, i.e.,
\begin{equation*}
A = A_0 \otimes I + I \otimes A_0 \enspace \text{ with } A_0 = \begin{bmatrix}
2 & -1\\
-1 & 2 & \ddots\\
& \ddots & \ddots & -1 \\
& & -1 & 2
\end{bmatrix} \in \complns^{100\times 100},
\end{equation*}
which results in $m=100^2$. As a less conventional matrix, we second consider a random sparse nonsingular M-matrix by creating a Z-matrix $B$ via \texttt{sprand} in MATLAB and shifting it by its spectral radius plus $0.1$, i.e.\ $A = B+ (\rho(B)+0.01)I$.
To make it comparable to the 2D Laplace example, we use the same size $m=100^2$ and a similar density of $5 \cdot 10^{-4}$.
The vector $v$ is chosen as a random vector via the function \texttt{rand}. The convergence behavior for both matrices is plotted in \cref{fig:exp}.
\begin{figure}
\centering
\subfloat[2D Laplace]{
\begin{tikzpicture}
\begin{semilogyaxis}[
width=0.5\textwidth,
xlabel={iteration},
ylabel={relative residual},
legend pos=south east,
ymin=1e-14, ymax=10,
]
\addplot+[mark repeat=2,] table {plots/exp/laplace_wo.txt};
\addlegendentry{CF-matrix}
\addplot table {plots/exp/laplace_ILU.txt};
\addlegendentry{ILU(0)}
\pgfplotsset{cycle list shift=1}
\addplot+[green] table {plots/exp/laplace_PFE.txt};
\addlegendentry{PFE}
\legend{}
\end{semilogyaxis}
\end{tikzpicture}
}
\subfloat[random sparse M-matrix]{
\begin{tikzpicture}
\begin{semilogyaxis}[
width=0.5\textwidth,
xlabel={iteration},
yticklabels={,,},
legend pos=south east,
ymin=1e-14, ymax=10,
]
\addplot+[mark repeat=2,] table {plots/exp/sprand_wo.txt};
\addlegendentry{CF-matrix}
\addplot table {plots/exp/sprand_ILU.txt};
\addlegendentry{ILU(0)}
\pgfplotsset{cycle list shift=1}
\addplot+[green] table {plots/exp/sprand_PFE.txt};
\addlegendentry{PFE}
\end{semilogyaxis}
\end{tikzpicture}
}
\caption{Convergence for $\exp(-A)$ via GMRES}
\label{fig:exp}
\end{figure}
Note that the poles $\tau_j$ in this example are complex numbers which necessitates the use of complex arithmetic for the shifted systems $(A-\tau_j I) x_j = v$ even though $A$ and $v$ are real. Since, on the other side, the coefficients of the continued fraction are real, complex arithmetic is not required when working with the CF-matrix. In this sense, we can say that the roughly 50\%--80\% increase in iterations for the ILU(0) preconditioned CF-matrix is approximately compensated for by the fact that we avoid complex arithmetic.
\end{example}
\begin{example}[Inverse square root]
We now consider
\begin{equation*}
f(A)v = A^{-1/2} v.
\end{equation*}
The inverse function $f(z)^{-1} = \sqrt{z}$ is not differentiable at 0 which is why we use the Padé approximations for the function $\hat{f}(z) = \sqrt{z+1}$. A C-fraction for $\hat{f}(z)$ under the condition that $\abs{\operatorname{Arg}(z+1)} < \pi$ is given by \cite[eq.~(11.7.1)]{Cuyt}
\begin{equation*}
b_0 = 1,\quad c_1 = \frac{1}{2},\quad c_i = \frac{1}{4} \mbox{ for } i > 1.
\end{equation*}
To obtain a pencil for $f(z)$, we exploit $\hat{f}(z-1)=f(z)$ which leads to
\begin{equation*}
T_{n}^{(0)} - (z-1)T_{n}^{(1)} = (T_n^{(0)} + T_n^{(1)}) - zT_n^{(1)}.
\end{equation*}
Thus, we consider the modified CF-matrix
\begin{equation*}
T_{20}(A) = (T_{20}^{(0)} + T_{20}^{(1)})\otimes I - T_{20}^{(1)} \otimes A.
\end{equation*}
We repeat the procedure of the previous example\footnote{The Padé approximations for this example have again only simple poles.} and plot the results in \cref{fig:invsqrt}. For the 2D Laplace matrix, we see that the ILU(0) preconditioned CF-matrix approach results in much faster convergence than when using the partial fraction expansion, and for the random spare matrix example, the situation is similar, although significantly less pronounced.
\begin{figure}
\centering
\subfloat[2D Laplace]{
\begin{tikzpicture}
\begin{semilogyaxis}[
width=0.5\textwidth,
xlabel={iteration},
ylabel={relative residual},
legend pos=south west,
ymin=1e-14, ymax=10,
mark repeat=4,
]
\addplot table {plots/invsqrt/laplace_wo.txt};
\addlegendentry{CF-matrix}
\addplot table {plots/invsqrt/laplace_ILU.txt};
\addlegendentry{ILU(0)}
\pgfplotsset{cycle list shift=1}
\addplot+[green] table {plots/invsqrt/laplace_PFE.txt};
\addlegendentry{PFE}
\end{semilogyaxis}
\end{tikzpicture}
}
\subfloat[random sparse M-matrix]{
\begin{tikzpicture}
\begin{semilogyaxis}[
width=0.5\textwidth,
xlabel={iteration},
yticklabels={,,},
legend pos=south east,
ymin=1e-14, ymax=10,
mark repeat=4,
]
\addplot table {plots/invsqrt/sprand_wo.txt};
\addlegendentry{CF-matrix}
\addplot table {plots/invsqrt/sprand_ILU.txt};
\addlegendentry{ILU(0)}
\pgfplotsset{cycle list shift=1}
\addplot+[green] table {plots/invsqrt/sprand_PFE.txt};
\addlegendentry{PFE}
\legend{}
\end{semilogyaxis}
\end{tikzpicture}
}
\caption{Convergence for $A^{-1/2}$ via GMRES}
\label{fig:invsqrt}
\end{figure}
Until now, we only compared the relative residuals of the linear system corresponding to the Pad\'e approximation to the inverse square root. Due
\begin{equation*}
(A^2)^{-1/2} = A^{-1},
\end{equation*}
we can determine the {\em error} for the approximation of the inverse square root if we start with a matrix $A$ but approximate $(A^2)^{-1/2}v$ which we compare with the solution of the linear system $Ax = v$.
We use the 2D Laplace operator again. To prevent the condition number of $A^2$ from being too large, we add a shift of $0.1$ to the matrix $A$, $A \to A+0.01I$. The result is shown in \cref{fig:invsqrt_error}. The plateau to be observed for $n=20$ at $10^{-4}$ for ILU(0)-preconditioned GMRES for the CF-matrix reflects the accuracy of the Pad\`e approximation of degree $n=20$ to the exact inverse square root.
When increasing the degree of the diagonal Padé approximation $n$, we expect this accuracy to increase, as well. Indeed, for $n=40$, the plateau starts to form at around $10^{-7}$ only (right part of \cref{fig:invsqrt_error}).
\begin{figure}
\centering
\subfloat[$n=20$]{
\begin{tikzpicture}
\begin{semilogyaxis}[
width=0.5\textwidth,
xlabel={iteration},
ylabel={relative error},
legend pos=south west,
ymin=1e-8, ymax=10,
mark repeat=2,
]
\addplot table {plots/invsqrt/error_20_wo.txt};
\addlegendentry{CF-matrix}
\addplot table {plots/invsqrt/error_20_ILU.txt};
\addlegendentry{ILU(0)}
\pgfplotsset{cycle list shift=1}
\addplot+[green] table {plots/invsqrt/error_20_PFE.txt};
\addlegendentry{PFE}
\end{semilogyaxis}
\end{tikzpicture}
}
\subfloat[$n=40$]{
\begin{tikzpicture}
\begin{semilogyaxis}[
width=0.5\textwidth,
xlabel={iteration},
yticklabels={,,},
legend pos=south east,
ymin=1e-8, ymax=10,
mark repeat=2,
]
\addplot table {plots/invsqrt/error_40_wo.txt};
\addlegendentry{CF-matrix}
\addplot table {plots/invsqrt/error_40_ILU.txt};
\addlegendentry{ILU(0)}
\pgfplotsset{cycle list shift=1}
\addplot+[green] table {plots/invsqrt/error_40_PFE.txt};
\addlegendentry{PFE}
\legend{}
\end{semilogyaxis}
\end{tikzpicture}
}
\caption{Comparison of the errors for $(A^2)^{-1/2}$ via GMRES}
\label{fig:invsqrt_error}
\end{figure}
\end{example}
\section{Conclusion and Outlook}
Starting from the observation that the approximants of a continued fraction can be expressed as the (1,1) element of the inverse of a corresponding tridiagonal matrix, we showed that for a rational function $r$ we can describe $r(A)v$ as the solution of a linear system whose coefficient matrix, the CF-matrix, is block tridiagonal.
What we need is a continued fraction describing $r(z)$. For some continued fractions like regular C-fractions, the resulting CF-matrix contains no higher powers of $A$ and is thus readily available.
For C-fractions, their contractions and similar continued fractions,
the Kronecker structure of the CF-matrix involves a matrix pencil with tridiagonal matrices. We showed how the partial fraction expansion of $r(z)$ is related to the Weierstrass canonical form of this pencil. We established a connection to Sylvester-type matrix equations and proved that if $r(z)$ has simple poles only and a certain block structure is used, splitting methods like block Jacobi and block Gauß-Seidel on the CF-matrix exhibit the same convergence properties as their nonblock version on the shifted systems $(A-\tau_i I)$ of the partial fraction expansion. In our numerical experiments, we
showed results that used (preconditioned) GMRES rather than (block) Jacobi or Gauß-Seidel and obtained fast convergence using ILU(0) as a preconditioner.
An immediate application of the CF-matrix approach is in situations where a continued fraction is available but the computation of its partial fraction expansion is ill-conditioned and thus error-prone. In addition, we expect that the investigations presented here lay the ground for the development of further approaches. Our analysis of the connection between Jacobi and Gauß-Seidel on the CF-matrix with these methods on the shifted matrices can be regarded as the first step towards an understanding of smoothers to be used in a multigrid approach on the CF-matrix. The question of how to develop appropriate coarsening strategies is open and has not been addressed in this paper.
| {
"timestamp": "2021-09-09T02:16:52",
"yymm": "2109",
"arxiv_id": "2109.03527",
"language": "en",
"url": "https://arxiv.org/abs/2109.03527",
"abstract": "A widely used approach to compute the action $f(A)v$ of a matrix function $f(A)$ on a vector $v$ is to use a rational approximation $r$ for $f$ and compute $r(A)v$ instead. If $r$ is not computed adaptively as in rational Krylov methods, this is usually done using the partial fraction expansion of $r$ and solving linear systems with matrices $A- \\tau I$ for the various poles $\\tau$ of $r$. Here we investigate an alternative approach for the case that a continued fraction representation for the rational function is known rather than a partial fraction expansion. This is typically the case, for example, for Padé approximations. From the continued fraction, we first construct a matrix pencil from which we then obtain what we call the CF-matrix (continued fraction matrix), a block tridiagonal matrix whose blocks consist of polynomials of $A$ with degree bounded by 1 for many continued fractions. We show that one can evaluate $r(A)v$ by solving a single linear system with the CF-matrix and present a number of first theoretical results as a basis for an analysis of future, specific solution methods for the large linear system. While the CF-matrix approach is of principal interest on its own as a new way to compute $f(A)v$, it can in particular be beneficial when a partial fraction expansion is not known beforehand and computing its parameters is ill-conditioned. We report some numerical experiments which show that with standard preconditioners we can achieve fast convergence in the iterative solution of the large linear system.",
"subjects": "Numerical Analysis (math.NA)",
"title": "Matrix functions via linear systems built from continued fractions",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754442973824,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.7075103933350583
} |
https://arxiv.org/abs/2008.08270 | On the collapsing of homogeneous bundles in arbitrary characteristic | We study the geometry of equivariant, proper maps from homogeneous bundles $G\times_P V$ over flag varieties $G/P$ to representations of $G$, called collapsing maps. Kempf showed that, provided the bundle is completely reducible, the image $G\cdot V$ of a collapsing map has rational singularities in characteristic zero. We extend this result to positive characteristic and show that for the analogous bundles the saturation $G\cdot V$ is strongly $F$-regular if its coordinate ring has a good filtration. We further show that in this case the images of collapsing maps of homogeneous bundles restricted to Schubert varieties are $F$-rational in positive characteristic, and have rational singularities in characteristic zero. We provide results on the singularities and defining equations of saturations $G\cdot X$ for $P$-stable closed subvarieties $X\subset V$. We give criteria for the existence of good filtrations for the coordinate ring of $G\cdot X$. Our results give a uniform, characteristic-free approach for the study of the geometry of a number of important varieties: multicones over Schubert varieties, determinantal varieties in the space of matrices, symmetric matrices, skew-symmetric matrices, and certain matrix Schubert varieties therein, representation varieties of radical square zero algebras (e.g. varieties of complexes), subspace varieties, higher rank varieties, etc. | \section{Introduction}
Let $G$ be a connected reductive group over an algebraically closed field $\Bbbk$. Consider a parabolic subgroup $P$ of $G$, and let $W$ be a $G$-module and $V\subset W$ a $P$-stable submodule. The saturation $G\cdot V \subset W$ is the image of the homogeneous vector bundle $G \times_P V$ under the proper \lq\lq collapsing map" \, $G \times_P V \to W$\, induced by the action of $G$ on $W$.
Many remarkable varieties can be realized through such collapsing of bundles for various choices of $G$, $P$, $W$, $V$ (cf.\ Section \ref{sec:applications}; for more such examples, see \cite{weymanbook}). Generally, the study of their geometry has been undertaken on case-by-case basis. An exception is the seminal work \cite{Kempf76}, where it is shown that in characteristic zero $G\cdot V$ has rational singularities whenever the unipotent radical $U(P)$ of $P$ acts trivially on $V$ (see also \cite{Kempf86}). Further, in this case the singularities of $G\cdot X$ are shown to be well-behaved for a closed $P$-stable subvariety $X \subset V$ \cite[Proposition 1 and Theorem 3]{Kempf76}.
In this paper, we generalize and extend the scope of Kempf's results along several directions. In particular, we give characteristic-free strengthenings of the statements above, under the presence of good filtrations as initiated by Donkin \cite{donkin}, \cite{donkinconj}. We say that a $G$-variety $Z$ is good, if $\Bbbk[Z]$ has a good filtration (see Section \ref{sec:goodfil}). We point out to the reader that all good-related properties hold automatically when $\cha \Bbbk = 0$, and our results below are new in this case as well (with the exception of Theorem \ref{thm:introgood}).
Let $B\subset P$ a Borel subgroup of $G$ and $T\subset B$ a maximal torus. We denote the set of dominant weights of $G$ by $X(T)_+$. For $\lambda \in X(T)_+$ we let $\Delta_G(\lambda)$ denote the corresponding Weyl module (see Section \ref{sec:cohomo}). We consider the Levi decomposition $P= L \ltimes U(P)$ with $L$ reductive. Pick any $\lambda_1, \lambda_2,\dots, \lambda_n \in X(T)_+$, and for the rest of the introduction fix
\begin{equation}\label{eq:mainsetup}
W= \bigoplus_{i=1}^{n} \Delta_{G}(\lambda_i) \quad \mbox{and} \quad V=\bigoplus_{i=1}^{n} \Delta_{L}(\lambda_i).
\end{equation}
We have a natural inclusion $V\subseteq W^{U(P)}$, with equality if $\cha \Bbbk = 0$ (when the bundle is completely reducible \cite{Kempf76}). While the examples in Section \ref{sec:applications} fit into the setup (\ref{eq:mainsetup}), we note that in Section \ref{sec:mainresults} we develop the results in a more general setting (see (\ref{eq:hsplit})).
\begin{theorem}\label{thm:introsat}
Let $X \subset V$ be an $L$-submodule such that $G\cdot X$ is good. Then $G\cdot X$ is strongly $F$-regular when $\cha \Bbbk >0$ (resp.\ is of strongly $F$-regular type when $\cha \Bbbk =0$).
\end{theorem}
This illustrates that good filtrations are responsible for the geometric behavior of saturations in positive characteristic, a phenomenon that is apparent in invariant theory as well \cite{hashiinv}, \cite{hashiunip}. Example \ref{ex:counter} demonstrates that this assumption cannot be dropped.
The following is our main criterion for the existence of good filtrations (for the definition of good pairs, see Section \ref{sec:goodfil}).
\begin{theorem}\label{thm:introgood}
Assume that $W$ is good, and that $(V,X)$ is a good pair for some closed $L$-variety $X\subset V$. Then $(\,W\,, \,\, G\cdot X)$ is a good pair of $G$-varieties.
\end{theorem}
In particular, this implies that $G\cdot V$ is good whenever $\cha \Bbbk > \max \{ \dim \Delta_{G}(\lambda_i) \,| \, 1\leq i \leq n\}$. However, in concrete situations the bound on $\cha \Bbbk$ can be further improved significantly (cf. Sections \ref{subsec:det}, \ref{sec:radsquare}). See Theorem \ref{thm:good} for other criteria in this direction.
We extend the collapsing method to various relative settings, thus greatly increasing its versatility. These include restrictions to Schubert varieties or multiplicity-free subvarieties of flag varieties (for the latter, see Corollary \ref{cor:multfree}). Below $\mathcal{W}$ denotes the Weyl group of $G$.
\begin{theorem}\label{thm:intromain}
Consider a closed $L$-variety $X\subset V$ and assume that $G\cdot X$ is good. For any $w\in \mathcal{W}$, we have:
\begin{enumerate}
\item $\overline{BwX}$ is normal if and only if $X$ is so.
\item If $\cha \Bbbk = 0$, then $\overline{BwX}$ has rational singularities if and only if so does $X$.
\item If $\cha \Bbbk> 0$ and $X$ is an $L$-submodule of $V$, then $\overline{BwX}$ is $F$-rational.
\end{enumerate}
\end{theorem}
Note that when $w$ is the longest element in $\mathcal{W}$, we have $\overline{BwX}=G\cdot X$.
\smallskip
Frequently (e.g.\ when $G\cdot X$ is a spherical variety), the varieties $\overline{BwX}$ are orbit closures under the action of the Borel subgroup $B$ (see Section \ref{sec:applications}). The singularities of such varieties have been investigated mostly in the spherical case (e.g. \cite{projnormschub}, \cite{brispher}, \cite{brithom}), but they are not well understood \cite[Comments 4.4.4]{perrin}. Theorem \ref{thm:intromain} is one of the first of its kind at this level of generality, applicable equally in non-spherical situations as well.
When $P$ is itself a Borel subgroup, we sharpen some results on singularities (see Corollary \ref{cor:borel}), extending the case of multicones over Schubert varieties \cite{multicone}, \cite{hashischub}.
Next, we provide a relative result on the defining ideals of saturations $G\cdot X$. For this, we introduce the notion of good generators of an ideal, see Definition \ref{def:goodgen}.
\begin{theorem}\label{thm:introdefi}
Let $(V,X)$ be a good pair with $G\cdot V$ good, and denote by $I_X \subset \Bbbk[V]$ the defining ideal of $X \subset V$. Let $M$ be the span of a set of good generators of $I_X$ and take a basis $\mathcal{P}'$ of the $G$-module $H^0(G/P, \mathcal{V}(M)) \, \subset \Bbbk[G\cdot V]$. Consider:
\begin{enumerate}
\item A set of generators $\mathcal{P}_{G\cdot V}$ of the defining ideal $I_{G\cdot V} \subset \Bbbk[W]$ of $G\cdot V$;
\item A lift $\tilde{\mathcal{P}'} \subset \Bbbk[W]$ of the set $\mathcal{P}' \subset \, \Bbbk[W]/I_{G\cdot V}$.
\end{enumerate}
Then the defining ideal of $G \cdot X$ in $\Bbbk[W]$ is generated by the set $\mathcal{P}_{G\cdot V} \, \cup \, \tilde{\mathcal{P}'}$.
\end{theorem}
In Theorem \ref{thm:defi} we give a version of the above that yields good defining equations, which we use to readily find (good) defining equations for the examples in Sections \ref{subsec:det} and \ref{sec:radsquare}.
Saturations of the type $G\cdot V$ appear in various forms throughout the existing literature, and a range of techniques have been developed to better understand their geometry. Applying the results above in the special case of radical square zero algebras (see Section \ref{sec:radsquare}), we simultaneously sharpen and generalize the main results in \cite{Kempfcomp}, \cite{DecoStrick}, \cite{Strick1}, \cite{brioncomp},
\cite{Strick2}, \cite{MT1}, \cite{MT2} that concern the singularities and defining equations of the Buchsbaum--Eisenbud varieties of complexes as well as varieties of complexes of other type. In addition, we obtain that certain $B$-orbit closures in varieties of complexes are $F$-rational when $\cha \Bbbk >0$ (resp.\ have rational singularities when $\cha \Bbbk=0$).
Our results provide a general method for the investigation of the geometry of parabolically induced orbit closures in a representation $W$ of a reductive group $G$. Namely, for any choice of a parabolic $P \subset G$, we can take the representation $V$ of the smaller reductive group $L$ as in (\ref{eq:mainsetup}) with trivial $U(P)$-action; choosing an $L$-orbit closure $X=\overline{Lx}$ (for any $x\in V$), saturation gives a $G$-orbit closure $G\cdot X = \overline{Gx} \subset G\cdot V \subset W$. By considering all such possible choices, we obtain a large set of $G$-orbit closures in $W$ whose singularities and defining equations are inherited from the smaller ones according to the results above.
The Cohen--Macaulay property for collapsing of bundles in positive characteristic is a consequence of the study of their $F$-singularities. This relies on techniques from tight closure theory that was developed by Hochster and Huneke \cite{hh1}, \cite{hh}. In Section \ref{subsec:ample}, we translate this property into Griffiths-type vanishing results for the cohomology of such bundles on Schubert varieties in positive characteristic, extending the classical Kodaira-type vanishing results for line bundles \cite{schubsplit}, \cite{schubfreg}, \cite{smith2}.
\subsection*{Acknowledgments}
The author would like to express his gratitude to Ryan Kinser for his valuable comments and suggestions on this work.
\section{Preliminaries}
We work over an algebraically closed field $\Bbbk$ of arbitrary characteristic (see Remark \ref{rem:field}).
An action of an algebraic group $G$ on an algebraic variety $X$ is always assumed to be algebraic, so that the map
$G \times X \to X$ is a morphism of algebraic varieties. We call a (possibly infinite-dimensional) vector space $V$ a rational $G$-module, if $V$ is equipped with a linear action of $G$, such that every $v\in V$ is contained in a finite-dimensional $G$-stable subspace on which $G$ acts algebraically. All modules considered are assumed to be rational of countable dimension.
Unless otherwise stated, throughout a ring or algebra is commutative, finitely generated over $\Bbbk$ with a multiplicative identity.
\subsection{Reductive groups}
Let $G$ be a connected reductive group over $\Bbbk$, $B$ a Borel subgroup and $U$ its unipotent radical. We fix a maximal torus $T \subset B$, and denote by $X(T)$ its group of characters. We denote by $\langle \cdot, \cdot \rangle$ the standard pairing between $X(T)$ and the group of cocharacters. Let $\Phi \subset X(T)$ denote the set of roots and $\Phi_+ \subset \Phi$ the set of positive roots with respect to the choice of $B$. We denote by $\rho$ the half sum of all the positive roots. The set of simple roots in $\Phi_+$ is denoted by $S$. We let $\mathcal{W}=N(T)/T$ be the Weyl group of $G$, and $w_0 \in W$ its longest element.
For $I\subset S$, consider the standard parabolic subgroup $P:=P_I \subset G$. We have a Levi decomposition $P_I= L_I \ltimes U_I$, where $U_I$ is the unipotent radical of $P$ and $L:=L_I$ is reductive. Let $\mathcal{W}_I$ be the subgroup generated by the reflections $s_\alpha$ with $\alpha \in I$, and $w_I$ the longest element in $\mathcal{W}_I$. We choose the set $\mathcal{W}^I$ of representatives of the cosets of $\mathcal{W}/\mathcal{W}_I$ as
\begin{equation}\label{eq:weylrep}
\mathcal{W}^I=\{w\in \mathcal{W} \, | \, w(\alpha) \in \Phi_+, \mbox{ for all } \alpha \in I\}.
\end{equation}
We have the Bruhat decomposition of $G$ into $B\times P$-orbits (see \cite[Section II.13]{jantzen}):
\[G \, = \, \bigcup_{w \in \mathcal{W}^I} BwP.\]
For $w \in \mathcal{W}^I$, we put $U(w):=U \cap w U^- w^{-1}$, where $U^-$ is the opposite unipotent radical. The multiplication map induces an isomorphism of $U(w)$-varieties (see \cite[Section II.13.8]{jantzen})
\begin{equation}\label{eq:opencell}
U(w) \times P \xrightarrow{\cong} BwP, \qquad (u,p) \, \mapsto \, u w p.
\end{equation}
We denote by $X(w)_P$ the Schubert variety that is the image of $\overline{BwP}$ under the locally trivial projection $G\to G/P$. For $P=B$, we write $X(w):=X(w)_B$.
\subsection{Cohomology of homogeneous bundles}\label{sec:cohomo}
For any representation $M$ of $P$, we denote by $\mathcal{V}(M)$ the sheaf of sections of the homogeneous vector bundle $G\times_P M$. For $\lambda \in X(T)$, we put $\mathcal{L}(\lambda):=\mathcal{V}( \Bbbk_{-\lambda})$, where $\Bbbk_{-\lambda}$ is the $1$-dimensional representation of $B$.
A weight $\lambda \in X(T)$ is dominant if $\langle \lambda , \alpha^\vee \rangle \geq 0$, for all simple roots $\alpha \in S$. The set of dominant weights is denoted by $X(T)_+$. For $\lambda \in X(T)_+$, we call the space of sections
\[\nabla_G(\lambda):=H^0(G/B, \mathcal{L}(\lambda)),\]
a \emph{dual Weyl module}. It has lowest weight $-\lambda$ and highest weight $-w_0 \cdot \lambda$. The module $\Delta_G(\lambda) = \nabla_G(\lambda)^*$ is called a \emph{Weyl module}, that has a non-zero highest weight vector of weight $\lambda$, and this generates $\Delta_G(\lambda)$ as a $G$-module. It is known that $\nabla_G(\lambda)$ has a unique simple submodule, of highest weight $-w_0 \cdot \lambda$.
When $\cha \Bbbk = p >0$ and $e\geq 1$ is an integer such that $(p^e-1)\rho$ is a weight of $G$, we denote by $\operatorname{St}_e = \nabla_G((p^e-1)\rho)$ the $e^{\operatorname{th}}$ Steinberg module, and put $\operatorname{St}:=\operatorname{St}_1$. The assumption is superficial as we can always replace $G$ by $\operatorname{rad} G \times \tilde{G}$, where $\operatorname{rad} G$ denotes the radical of $G$ and $\tilde{G}$ the universal cover of $[G,G]$, and $(p^e-1)\rho$ is a weight of $\operatorname{rad} G \times \tilde{G}$ for all $e\geq 1$.
Let $P=P_I$ be a parabolic subgroup. For $\lambda \in X(T)_+$, put $\mathcal{V}(\lambda):= \mathcal{V}(\nabla_L(\lambda))$ (here $U_I$ acts trivially on $\nabla_L(\lambda)$). The quotient map $\pi: G/B \to G/P$ induces a quasi-isomorphism
\begin{equation}\label{eq:pushfiber}
\mathbf{R}\pi_* \mathcal{L}(\lambda) \cong \mathcal{V}(\lambda).
\end{equation}
By abuse of notation, we use the same notation for the respective bundles on Schubert varieties that are obtained by restriction. We record the following result.
\begin{lemma}\label{lem:surj}
Let $\lambda \in X(T)_+$, and $w \in \mathcal{W}^I$. For all $i\geq 0$ we have $H^i(X(w)_P, \mathcal{V}(\lambda)) \cong H^i(X(w\cdot w_I), \, \mathcal{L}(\lambda))$, and the map induced by restriction is surjective:
\[H^i(G/P, \, \mathcal{V}(\lambda)) \to H^i(X(w)_P, \mathcal{V}(\lambda)).\]
Moreover, $H^i(G/P, \, \mathcal{V}(\lambda))=0$ for $i>0$.
\end{lemma}
\begin{proof}
We have $\pi^{-1} (X(w)_P) = X(ww_I)$ \cite[Section 13.8]{jantzen} and a Cartesian square
\[
\xymatrix@R-0.8pc{
G/B \ar[r]^{\pi} & G/P \\
X(ww_I) \ar[u] \ar[r]^{\,\,\pi^w} & X(w) \ar[u]
}\]
where the vertical maps are inclusions. As $\pi$ is proper and flat, by a base change argument (see \cite[Corollary 12.9]{hartshorne}) and (\ref{eq:pushfiber}) we get $\mathbf{R}\pi_*^w \, \mathcal{L}(\lambda) \cong \mathcal{V}(\lambda)$. This shows that $H^i(X(w)_P, \mathcal{V}(\lambda)) \cong H^i(X(w w_I), \mathcal{L}(\lambda))$ for all $i\geq 0$. The rest of the claims now follows from the diagram above using \cite[Theorem 2]{projnormschub}.
\end{proof}
\subsection{Classes of singularities}\label{sec:fsing}
When $\cha \Bbbk = p > 0$, for a $\Bbbk$-space $V$ and $e\in \mathbb{Z}_{\geq 0}$ we denote by $V^{(e)}$ the abelian group $V$ with the new $\Bbbk$-space structure $c \cdot v := c^{1/p^e} \cdot v$. When $V$ is a module over an algebraic group $G$, then $V^{(e)}$ also has a $G$-module structure \cite[Section I.9.10]{jantzen}. If $A$ is a $\Bbbk$-algebra, then so is $A^{(e)}$ by using the same multiplicative structure.
We call a domain $A$ \textit{strongly} $F$\textit{-regular} if for every non-zero $c\in A$ there exists $e>0$ such that the $A^{(e)}$-map $cF^{e}: A^{(e)} \to A$ given by $x \mapsto c x^{p^e}$ is $A^{(e)}$-split.
As we do not need it for our purposes, we refer the reader to \cite{hh} for the definition of $F$-\textit{rational} rings (see (\ref{eq:f-sing}) below for some of its important properties).
When $\cha \Bbbk =0$, an algebraic variety $X$ has \emph{rational singularities}, if for some (hence, any) resolution of singularities $f: Z\to X$ (i.e.\ $Z$ is smooth, and $f$ proper and birational), the natural map $\mathcal{O}_X \to \mathbf{R}f_* \mathcal{O}_Z$ is a (quasi-)isomorphism. Further, we say a ring $A$ is of \textit{strongly} $F$\textit{-regular type} if there exist some subring $R$ of $\Bbbk$ which is of finite type over $\mathbb{Z}$, and some $R$-algebra $A_R$ which is flat of finite type over $R$, such that $A_R \otimes_R \Bbbk \cong A$ and for the closed points $\mathfrak{m}$ in a dense open subset of $\operatorname{Spec} R$, the ring $A_R \otimes_R R/\mathfrak{m}$ is strongly $F$-regular.
An affine variety $X$ is $F$-rational (resp.\ strongly $F$-regular or of strongly $F$-regular type) if $\Bbbk[X]$ is so. We have the following implications (where CM stands for Cohen--Macaulay):
\begin{equation}\label{eq:f-sing}
\begin{aligned}
\cha \Bbbk = 0: & \mbox{ regular} \Rightarrow \mbox{strongly }F\mbox{-regular type} \! \Rightarrow \mbox{rational sing.} \Rightarrow \mbox{normal, CM}; \\
\cha \Bbbk > 0: & \mbox{ regular} \Rightarrow \, \mbox{strongly }F\mbox{-regular} \,\, \Longrightarrow \, F\mbox{-rational} \Rightarrow \mbox{normal, CM.}
\end{aligned}
\end{equation}
Furthermore, $F$-rationality implies pseudo-rationality \cite{smith} and rational singularities in positive characteristic as defined in \cite{kovacs2}. When $\cha \Bbbk = 0$, a ring has log terminal singularities if and only if it
is of strongly $F$-regular type and $\mathbb{Q}$-Gorenstein (see \cite{hawa}).
Now let $A$ be a $G$-algebra and $\cha \Bbbk = p >0$. We can assume that $(p^e-1)\rho$ is a weight of $G$ for $e\geq 1$ (otherwise replace $G$ by $\operatorname{rad} G \times \tilde{G}$). Following \cite[Section 4]{hashiunip}, we say that $A$ is $G$-$F$-\textit{pure} if there exists some $e \geq 1$ such that the map $\operatorname{id} \otimes F^e : \operatorname{St}_e \otimes A^{(e)} \to \operatorname{St}_e \otimes A$ splits as a $(G, A^{(e)})$-linear map.
Now we study the coordinate ring of $\overline{BwB} \subset G$, where $w\in \mathcal{W}$. Consider the rational $B\times T$-subalgebra of $\Bbbk[\overline{BwB}]^{U}$ consisting of dominant $T$-weight spaces
\[\Bbbk[\overline{BwB}]^{U}_+ := \bigoplus_{\lambda \in X(T)_+} \Bbbk[\overline{BwB}]^{U}_{\lambda}.\]
Consider the section ring $C(X(w)):= \bigoplus_{\lambda \in X(T)_+} \! H^0(X(w), \mathcal{L}(\lambda))$.
\begin{lemma}\label{lem:cone}
For any $w\in \mathcal{W}$, we have an isomorphism of $B\times T$-algebras
\[\Bbbk[\overline{BwB}]^{U}_+ \cong C(X(w)).\]
As a consequence, the algebra $\Bbbk[\overline{BwB}]^{U}_+$ is finitely generated, and strongly $F$-regular when $\cha \Bbbk > 0$ (resp.\ of strongly $F$-regular type when $\cha \Bbbk =0$).
\end{lemma}
\begin{proof}
Let $\Gamma := - X(T)_+$ and consider the semigroup ring $\Bbbk[\Gamma]$, which is naturally a subalgebra of $\Bbbk[T]$. We have an isomorphism of $B\times T$-algebras
\[(\Bbbk[\overline{BwB}]^U \otimes \Bbbk[\Gamma])^T \cong C(X(w)).\]
On the other hand, we have
\[(\Bbbk[\overline{BwB}]^U \otimes \Bbbk[\Gamma])^T=(\Bbbk[\overline{BwB}]^U_+ \otimes \Bbbk[\Gamma])^T =(\Bbbk[\overline{BwB}]^U_+ \otimes \Bbbk[T])^T \cong \Bbbk[\overline{BwB}]^U_+,\]
where the second equality follows from the decomposition $\Bbbk[T] \cong \bigoplus_{\lambda \in X(T)} \Bbbk_{\lambda}$ as $T$-modules, and the last isomorphism from Lemma \ref{lem:transfer}.
We now show that $C(X(w))$ is finitely generated. By \cite[Theorem 16.2]{grossbook}, $\Bbbk[G]^U$ is finitely generated, and therefore so is $C(X(w_0))=(\Bbbk[G]^U\otimes \Bbbk[\Gamma])^T$. By Lemma \ref{lem:surj}, we see that the map $C(X(w_0)) \to C(X(w))$ induced by restriction is onto, hence $C(X(w))$ is finitely generated (alternatively, this follows also from \cite[Theorem 2]{projnormschub}).
Let $\cha \Bbbk > 0$. The (not necessarily noetherian) algebra $\bigoplus_{\lambda \in X(T)} H^0(X(w), \mathcal{L}(\lambda))$ is quasi-$F$-regular, by \cite[Theorem 2.6 (4)]{hashisurjgraded} and the global $F$-regularity of Schubert varieties in the sense of \cite{smith2}, see \cite{schubfreg}, \cite{hashischub}. Therefore, the algebra $C(X(w))$ is also quasi-$F$-regular by \cite[Lemma 2.4]{hashisurjgraded}. The latter is finitely generated, so strongly $F$-regular (see \cite[Section 2.1]{hashisurjgraded}).
For ring $R$, consider the $R$-algebra $C(X(w)_{R})= \bigoplus_{\lambda \in X(T)_+} \! H^0(X(w)_{R}, \mathcal{L}(\lambda)_{R})$. We have $C(X(w)_{\Bbbk'}) = C(X(w)_{\mathbb{Z}}) \otimes_\mathbb{Z} k'$ (see \cite[Section II.14.15]{jantzen}), for any field $\Bbbk'$, and $C(X(w)_{\mathbb{Z}})$ is flat and finitely generated over $\mathbb{Z}$ (e.g.\ from \cite[Sections II.14.1 and II.14.21]{jantzen}). By \cite[Theorem 5.5]{hh}, $C(X(w)_{\Bbbk'})$ is strongly $F$-regular for a perfect field $\Bbbk'\subset \Bbbk$. This shows that when $\cha \Bbbk = 0$, $C(X(w))$ is of strongly $F$-regular type.
\end{proof}
For the remainder of the subsection, we assume that $\cha \Bbbk >0$.
\begin{lemma}\label{lem:cartanalg}
Let $\Gamma \subset X(T)_+$ be a finitely generated semigroup, and $A= \bigoplus_{\lambda \in \Gamma} A_\lambda$ a $\Gamma$-graded integral domain with a $G$-action such that $A_\lambda\cong \nabla_G(\lambda)$. Then $A$ is $G$-$F$-pure.
\end{lemma}
\begin{proof}
The proof follows closely that of \cite[Lemma 3]{hashiunipshort}. We can assume that $G=\tilde{G} \times \operatorname{rad} G$. Further, we can assume that the product $\nabla_G(\lambda) \otimes \nabla_G(\mu) \to \nabla_G(\lambda+\mu)$ in $A$ is given by multiplication of sections of the corresponding line bundles on $G/B$, as seen in the proof of \cite[Lemma 5.6]{hashisurjgraded}. We denote by $\phi$ the composition of $G$-maps
\[\phi \colon \operatorname{St} \otimes A \twoheadrightarrow \bigoplus_{\lambda \in \Gamma} \operatorname{St} \otimes \nabla_G(p\lambda) \twoheadrightarrow \bigoplus_{\lambda \in \Gamma} \nabla_G(p(\lambda+\rho)-\rho) \xrightarrow{\cong} \operatorname{St} \otimes A^{(1)},\]
where the first map is given by projection, the second by multiplication (see \cite[Theorem 1]{projnormschub}), and the third by the inverse of $G$-isomorphism $\operatorname{St} \otimes \nabla_G(\lambda)^{(1)} \xrightarrow{\cong} \nabla_G(p(\lambda+\rho) - \rho)$ induced also by multiplication of sections (see \cite[Theorem 2.5]{andersen}). Then $\phi$ gives the required splitting, since it is $A^{(1)}$-linear. The latter can be checked on the graded components, where it follows from the commutative diagram (with the obvious maps induced by multiplication):
\[\xymatrix@R-0.3pc@C-0.3pc{
\operatorname{St} \otimes \nabla_G(p\lambda) \otimes \nabla_G(\mu)^{(1)} \ar[r] \ar[d] & \nabla_G(p(\lambda+\rho)-\rho) \otimes \nabla_G(\mu)^{(1)} \ar[r]^{\cong} \ar[d] & \operatorname{St} \otimes \nabla_G(\lambda)^{(1)} \otimes \nabla_G(\mu)^{(1)} \ar[d] \\
\operatorname{St} \otimes \nabla_G(p(\lambda+\mu)) \ar[r] \quad & \quad \nabla_G(p(\lambda+\mu+\rho)-\rho) \ar[r]^{\cong} \quad & \quad \operatorname{St} \otimes \nabla_G(\lambda+\mu)^{(1)}
}\]
\end{proof}
When $\Gamma$ is saturated, the algebra $A$ as above is strongly $F$-regular \cite[Lemma 5.6]{hashisurjgraded}.
\begin{corollary}\label{cor:lpure}
The algebra $\Bbbk[G]^{U_I \times U}$ is strongly $F$-regular and $L$-$F$-pure.
\end{corollary}
\begin{proof}
The algebra $A=\Bbbk[G]^{U_I \times U}$ has an $L\times T$-action so that we have a decomposition $A=\bigoplus_{\lambda \in \Gamma} \nabla_L(w_I w_0\lambda)$ as $L$-modules (e.g.\ see \cite[Theorem 3]{donkinunip}). Clearly, the set $\{w_I w_0 \lambda \}_{\lambda \in X(T)_+}$ forms a saturated subsemigroup in the semigroup of dominant weights of $L$. Hence, the claims follow by \cite[Lemma 5.6]{hashisurjgraded} and Lemma \ref{lem:cartanalg}, respectively.
\end{proof}
\subsection{Good filtrations}\label{sec:goodfil}
Take a (possibly infinite-dimensional) $G$-module $V$. Following Donkin \cite{donkin}, an ascending exhaustive filtration
\[0 = V_0 \subset V_1 \subset V_2 \subset \dots\]
of $G$-submodules of $V$ is a \textit{good filtration} (resp.\ Weyl filtration) of $V$, if each $V_i/V_{i-1}$ is isomorphic to a dual Weyl module (resp.\ to a Weyl module). If $V$ has both good and Weyl filtrations, then we call $V$ \textit{tilting}.
Now let $w\in \mathcal{W}$. We say that a $B$-module $V$ has a $w$-\textit{excellent filtration}, if it has a $B$-module filtration with successive quotients isomorphic to some $H^0(X(w), \mathcal{L}(\lambda))$, with $\lambda \in X(T)_+$. This is a special type of excellent filtration, as defined in \cite[Definition 2.3.6]{vanderkallen}. Note that a good filtration of a $G$-module is a $w_0$-excellent filtration.
A finite-dimensional $G$-module $W$ \textit{good} if $\Sym_d W^*$ has a good filtration for all $d\geq 0$. In particular, in this case $W$ must have a Weyl filtration. Similarly, we call an affine $G$-variety (resp.\ $B$-variety) $X$ good (resp $w$-excellent) if $\Bbbk[X]$ has a good (resp.\ $w$-excellent) filtration.
If $X\subset Y$ is a closed $G$-stable subvariety, then we say that $(Y,X)$ is a \textit{good pair} whenever $Y$ is good and the defining ideal $I_X \subset \Bbbk[Y]$ has a good filtration (see \cite[Section 1.3]{donkinconj}). In this case $X$ is automatically good.
If $\cha \Bbbk = 0$, then all (pairs of) affine $G$-varieties are good. An important feature of good filtrations is the following result of Donkin \cite{donkin} and Mathieu \cite[Theorem 1]{mathieu}.
\begin{proposition}\label{prop:mathieu}
If $M$ and $N$ are $G$-modules with good filtrations, then $M\otimes_\Bbbk N$ has a good filtration. In particular, if $X$ and $Y$ are good affine $G$-varieties, then so is $X\times Y$.
\end{proposition}
We list some cases that imply the existence of good filtrations (see \cite[Section 4]{andjan}).
\begin{lemma}\label{lem:good}
Let $V,W$ be finite-dimensional $G$-modules.
\begin{enumerate}
\item If $\langle \chi+ \rho , \alpha^\vee \rangle \leq \cha \Bbbk$ for all weights $\chi$ of $V$ and all $\alpha \in \Phi_+$, then $V$ has a good filtration.
\item If $V$ has a good filtration and $\cha \Bbbk >i$, then $\bigwedge^i V$ and $\Sym_i V$ have good filtrations.
\item If $\bigwedge V$ and $ \bigwedge W$ have good filtrations, then $V\otimes W$ is good.
\item $\bigwedge V$ has a good filtration if and only if so does $\bigwedge V^*$ (i.e.\ $\bigwedge V$ is tilting).
\end{enumerate}
\end{lemma}
We further need some basic results.
\begin{lemma}\label{lem:gr}
Let $f: M\to N$ be a $G$-module map. If $M$ has a good filtration and the induced map $M^{U} \to N^{U}$ is onto, then $N$ and $\ker f$ have good filtrations and $f$ is onto.
\end{lemma}
\begin{proof}
Put $I=\im f$ and $K = \ker f$. Fix any $\lambda \in X(T)_+$. Since $M$ has a good filtration, we have an exact sequence (see \cite[Proposition II.4.16]{jantzen})
\[0 \to \operatorname{Hom}_G(\Delta_G(\lambda), K) \to \operatorname{Hom}_G(\Delta_G(\lambda), M) \to \operatorname{Hom}_G(\Delta_G(\lambda), I) \to \Ext^1_G(\Delta_G(\lambda), K) \to 0.\]
The assumption gives an exact sequence
\[0 \to K^{U} \to M^{U} \to I^{U} \to 0.\]
Taking $\lambda$-weights above we obtain that $\Ext^1_G(\Delta_G(\lambda), K)=0$ (see \cite[Lemma II.2.13]{jantzen}). Since $\lambda \in X(T)_+$ was arbitrary, this shows that $K$ has a good filtration (see \cite[Proposition II.4.16]{jantzen}), and hence so does $I$. Let $C=\coker f$ and consider an exact sequence $0 \to I \to N \to C \to 0$. Since $I$ has a good filtration, we see as above that the induced sequence $0 \to I^{U} \to N^{U} \to C^{U} \to 0$ is also exact. By assumption $C^{U} = 0$, hence $C=0$.
\end{proof}
\begin{corollary}\label{cor:goodpair}
Let $Y$ be a good affine $G$-variety and $X\subset Y$ a closed $G$-stable subvariety. Then $(Y,X)$ is a good pair if and only if the map $\Bbbk[Y]^{U} \to \Bbbk[X]^{U}$ is surjective.
\end{corollary}
\begin{proof}
If $\Bbbk[Y]^{U} \to \Bbbk[X]^{U}$ is surjective, then it follows from Lemma \ref{lem:gr} that $(Y,X)$ is a good pair. The converse follows from \cite[Proposition 1.4 and Proposition 2]{donkinunip}.
\end{proof}
We introduce a notion for generators of ideals, that is again relevant only in positive characteristic.
\begin{defn}\label{def:goodgen}
Let $Y$ be a good affine $G$-variety and $X\subset Y$ a closed $G$-stable subvariety with defining ideal $I_X \subset \Bbbk[Y]$. We say that a finite set of equations $\mathcal{P} \subset I_X$ are good defining equations (resp.\ good generators) of $X$ (resp.\ of $I_X$) if the following hold for $M_{\mathcal{P}} := \operatorname{span}_{\Bbbk} \mathcal{P} \, \subset I_X$:
\begin{enumerate}
\item $M_{\mathcal{P}}$ is a $G$-module with a good filtration;
\item The multiplication map $m_{\mathcal{P}}: k[Y] \otimes M_{\mathcal{P}} \to I_X$ induces a surjective map on $U$-invariants $(k[Y] \otimes M_{\mathcal{P}})^{U} \to I_X^{U}$.
\end{enumerate}
\end{defn}
Let us record some useful results regarding this notion. We continue with the notation in Definition \ref{def:goodgen}.
\begin{lemma}\label{lem:exist}
There exist good defining equations for $X \subset Y$ if and only if $(Y,X)$ is a good pair.
\end{lemma}
\begin{proof}
Assume that $(Y,X)$ is a good pair. By \cite[Theorem 16.2]{grossbook}, $\Bbbk[Y]^{U}$ is noetherian, hence $I_X^{U}$ is finitely generated. Choose a finite set of generators. Taking a good filtration of $I_X$, there exists a finite dimensional piece $M$ that contains these generators. We can pick $\mathcal{P}$ to be a basis of $M$.
Conversely, let $\mathcal{P} \subset I_X$ be a set of good generators. By Proposition \ref{prop:mathieu}, the domain of the multiplication map $m_{\mathcal{P}}$ has a good filtration. By Lemma \ref{lem:gr}, we obtain that $m_{\mathcal{P}}$ is surjective, and $I_X$ has a good filtration.
\end{proof}
The proof above shows assumption (2) in Definition \ref{def:goodgen} can be replaced with the equivalent assumption that $\mathcal{P}$ generates $I_X$ and $\ker m_{\mathcal{P}}$ has a good filtration. In particular, the notion does not depend on the choice of the Borel subgroup (see \cite[Remark II.4.16 (2)]{jantzen}). We record another convenient fact.
\begin{lemma}\label{lem:compint}
Assume that $Y$ is good and let $M \subset I_X $ be $G$-module such that a basis $\mathcal{P}$ of $M$ generates $I_X$ and forms a regular sequence in $\Bbbk[Y]$. Assume that $\bigwedge M$ has a good filtration. Then $\mathcal{P}$ are good defining equations of $X \subset Y$.
\end{lemma}
\begin{proof}
This follows readily by considering the Koszul resolution, and using \cite[Proposition 3.2.4]{donkin} together with Proposition \ref{prop:mathieu} repeatedly.
\end{proof}
Although we do not need it in this article, the assumption on $\bigwedge M$ in the lemma above can be weakened by requiring only that the good filtration dimension of $\bigwedge^i M$ is at most $i-1$, for all $i\geq 1$ (see \cite[Section 1.3]{donkinconj}).
\subsection{Deformation of algebras}\label{sec:gropop}
We recall a filtration of algebras considered in \cite{popov} and \cite{gross}. There exists a homomorphism $h: X(T) \to \mathbb{Z}$ satisfying the following properties:
\begin{enumerate}
\item $h(\lambda)$ is a non-negative integer for all $\lambda\in X(T)_+$;
\item if $\chi',\chi \in X(T)$ with $\chi' > \chi$, then $h(\chi')> h(\chi)$.
\end{enumerate}
For a commutative $G$-algebra $A$ over $k$, we define the $\mathbb{Z}_{\geq 0}$-filtration
\[ F^i A := \{ a \in A \, | \, h(\chi) \leq i \mbox{ for all } T\mbox{-weights } \chi \mbox{ of } \operatorname{span}_\Bbbk G \cdot a \}.\]
Denote by $\gr A$ the associated graded algebra. Then there is an injective map of $G$-algebras
\begin{equation}\label{eq:grA}
\gr A \hookrightarrow (A^{U^-} \otimes_\Bbbk \Bbbk[G/U])^T,
\end{equation}
which is onto if and only if $A$ has a good filtration \cite[Theorem 16]{gross}.
Consider $L$ a linear algebraic group, and $H\subset L$ a closed subgroup. Let $N:=N_L(H)$ be the normalizer of $H$ in $L$. Let $R$ be an $L$-algebra. The group $N$ acts naturally on $R^H$ and on $H$-invariants $\Bbbk[L]^H=\Bbbk[L/H]$ (by right multiplication). The following is a consequence of \cite[Theorem 4]{popov} (see also \cite[Theorem 9.1]{grossbook}).
\begin{lemma}\label{lem:transfer}
There is an isomorphism of $N$-algebras $R^H \cong (R \otimes_\Bbbk \Bbbk[L/H])^L$.
\end{lemma}
\section{Main results}\label{sec:mainresults}
In this section we develop our general results on collapsing of bundles. We work over an algebraically closed field $\Bbbk$ of arbitrary characteristic (see Remark \ref{rem:field}). In the special case when $\cha \Bbbk = 0$ and the Schubert variety considered is the flag variety itself, the general framework agrees with that of completely reducible bundles as in \cite{Kempf76}.
We fix the notation that is used throughout the section. Consider a parabolic subgroup $P\subset G$. Without loss of generality, we assume that $P$ is standard corresponding to a set of simple roots $I\subset S$. Let $U_I$ be the unipotent radical of $P$. Let $P= L \ltimes U_I$ be the Levi decomposition, with $L:=L_I$ reductive. We denote by $P^-$ the opposite parabolic subgroup, having decomposition $P^-= L \ltimes U_I^-$.
Let $W$ be a finite-dimensional $G$-module. We introduce the map of $L$-modules
\begin{equation}\label{eq:hsplit}
\psi \colon W^{U_I} \longrightarrow \left((W^*)^{U^-_I}\right)^*,
\end{equation}
which is the dual of the composition $(W^*)^{U^-_I} \hookrightarrow W^* \twoheadrightarrow (W^{U_I})^*$.
\smallskip
Throughout we take an $L$-submodule $V\subset W^{U_I}$ such that the map $\left.\psi \right|_{V} : V \to ((W^*)^{U^-_I})^*$ is injective. The following shows that tracking the map $\left.\psi \right|_{V}$ is relevant only when $\cha \Bbbk > 0$.
\begin{lemma}\label{lem:iso} In either of the following cases, $\left.\psi \right|_{V}$ is an isomorphism:
\begin{itemize}
\item[(a)] $W$ is a semi-simple $G$-module and $V=W^{U_I}$.
\item[(b)] $W= \bigoplus_{i=1}^{n} \Delta_{G}(\lambda_i)$ for some $\lambda_i \in X(T)_+$, and $V \subset W^{U_I}$ is $V=\bigoplus_{i=1}^{n} \Delta_{L}(\lambda_i)$.
\end{itemize}
\end{lemma}
\begin{proof}
For part (a), we can assume that $W$ is a simple $G$-module. Both $W^{U_I}$ and $((W^*)^{U^-_I})^*$ are simple $L$-modules \cite[Proposition II.2.11]{jantzen}, and $\psi$ gives a non-trivial map between their respective highest weight vectors. Therefore, $\psi$ is an isomorphism.
For part (b), we can assume that $W = \Delta_G(\lambda)$ is a Weyl module. The restriction map $\nabla_G(\lambda) \to \nabla_L(\lambda)$ induced by $P/B \subset G/B$ is surjective (see Lemma \ref{lem:surj}). Therefore, the $L$-submodule of $W$ generated by its highest weight vector (of weight $\lambda$) is $V\cong \Delta_L(\lambda)$. On the other hand, we have $((W^*)^{U^-_I})^* \cong \Delta_L(\lambda)$ as $L$-modules (see \cite[Section 1.2]{donkinunip}), generated as an $L$-module by the highest weight vector. Since on the weight space of $\lambda$ the map $\left.\psi \right|_{V}$ is easily seen to be non-zero, it is also surjective, hence an isomorphism.
\end{proof}
\smallskip
Let $X$ be a closed $L$-stable subvariety of $V$. As $U_I$ acts on $V$ trivially, $X$ is $P$-stable closed subvariety of $W$. We have the following proper collapsing map
\begin{equation}\label{eq:collapse}
q \colon G\times_P X \longrightarrow W,
\end{equation}
with $\operatorname{im} q = G\cdot X$ a closed subvariety of $W$. Let $\pi : G\times_P X \to G/P$ be the bundle map. For any closed subset $Y \subset G/P$, the subvariety $q(\pi^{-1}(Y)) \subset W$ is closed. In the case when $Y=X(w)_P$ is a Schubert variety, then $q(\pi^{-1} (Y)) = \overline{BwX}$ is a $B$-stable subvariety in $X$.
\begin{proposition}\label{prop:invariantalg}
For any $w\in \mathcal{W}^I$, the restriction map $\Bbbk[\overline{BwX}] \to \Bbbk[wX]$ induces an isomorphism of algebras
\[\Bbbk[\overline{BwX}]^{U(w)} \, \xrightarrow{\,\,\cong\,\,} \, \Bbbk[wX].\]
Thus, the algebra $\Bbbk[X]$ is a direct summand of $\Bbbk[\overline{BwX}]$ as a $\Bbbk[X]$-module via $\Bbbk[X]\cong \Bbbk[wX]$.
\end{proposition}
\begin{proof}
The inclusions $wX \subset \overline{BwX} \subset W$ give rise to a commutative diagram
\[
\xymatrix@R-0.5pc{
\Bbbk[\overline{BwX}]^{U(w)} \ar[r]^{\quad f} & \Bbbk[wX] \\
\Bbbk[W]^{U(w)} \ar[u] \ar[r]^{\,\,g} & \Bbbk[wV] \ar@{->>}[u]
}\]
To show that $f$ is onto, it is enough to show that $g$ is so. For this, we show that the map $(W^*)^{U(w)} \to w \cdot V^*$ is onto. As $w^{-1} \cdot (W^*)^{U(w)}=(W^*)^{w^{-1} U w \,\cap\, U^-}$ and $w^{-1} U w \, \cap \, U^- \subset U_I^-$ (cf. (\ref{eq:weylrep})), this follows since the $L$-module map $\left.\psi \right|_{V}^*: (W^*)^{U^-_I} \to V^*$ is onto.
The morphism (\ref{eq:collapse}) induces an injective map of algebras
\[\Bbbk[\overline{BwX}] \hookrightarrow (\Bbbk[\overline{BwP}] \otimes \Bbbk[X])^P.\]
The multiplication map (\ref{eq:opencell}) gives an open immersion into $\overline{BwP}$, inducing an injective map $\Bbbk[\overline{BwP}]^{U(w)} \hookrightarrow \Bbbk[wP]$. The previous maps give
\[\Bbbk[\overline{BwX}]^{U(w)} \hookrightarrow (\Bbbk[\overline{BwP}]^{U(w)} \!\! \otimes \Bbbk[X])^P \hookrightarrow (\Bbbk[wP] \otimes \Bbbk[X])^P \cong \Bbbk[wX],\]
thus proving the injectivity of $f$.
\end{proof}
\begin{remark}\label{rem:unipara}
Putting $w=w_0 w_I^{-1}$ in Proposition \ref{prop:invariantalg}, and twisting by $w$ we obtain an isomorphism of $L$-algebras $\Bbbk[G\cdot X]^{U_I^{-}} \xrightarrow{\cong} \Bbbk[X]$.
\end{remark}
\subsection{Good saturations}\label{subsec:good} The following is our main tool for inducing the property of being good via saturations.
\begin{theorem}\label{thm:good}
\begin{itemize}
\item[(a)] The $G$-variety $G\cdot X$ is good if and only if the $L$-variety $X$ is good and the induced map $k[W] \to q_* \mathcal{O}_{G\times_P X}$ is onto.
\item[(b)] Assume that $(V,X)$ is a good pair of $L$-varieties and $\left.\psi \right|_{V}$ is a split map of $L$-modules. If there is a good closed $G$-subvariety $Z \subset W$ with $G\cdot X \subset Z$, then $(\,Z\,, \,\, G\cdot X)$ is a good pair.
\item[(c)] Let $Y\subset V$ be a closed $L$-stable subvariety such that $(Y,X)$ is a good pair and $G\cdot Y$ is good. Then $(G\cdot Y\, , \, G\cdot X)$ is a good pair.
\end{itemize}
\end{theorem}
\begin{proof}
Assume that $G\cdot X$ is good. By Remark \ref{rem:unipara} and \cite{donkinunip}, we obtain that $X$ is good. From the proof of Proposition \ref{prop:invariantalg} we have $\Bbbk[G \cdot X]^{U^-_I} \xrightarrow{\,\cong \,} (q_* \mathcal{O}_{G\times_P X})^{U^-_I}$, which yields isomorphisms
\begin{equation}\label{eq:unip}
\Bbbk[G \cdot X]^{U^-} \xrightarrow{\,\,\cong \,\,} (q_* \mathcal{O}_{G\times_P X})^{U^-} \xrightarrow{\,\,\cong \,\,} \Bbbk[X]^{U_L^-}.
\end{equation}
Therefore, the map $\Bbbk[G \cdot X] \to q_* \mathcal{O}_{G\times_P X}$ is onto by Lemma \ref{lem:gr}.
Now assume that $X$ is good and $\Bbbk[G \cdot X] \to q_* \mathcal{O}_{G\times_P X}$ is onto (hence, an isomorphism). By \cite[Theorem 3]{donkinunip}, Proposition \ref{prop:mathieu} and \cite[Proposition 1.2e (iii)]{donkinconj} the $G$-module $q_* \mathcal{O}_{G\times_P X}= (\Bbbk[G/U_I] \otimes \Bbbk[X])^L$ has a good filtration, thus $G\cdot X$ is good.
For part (b), by Corollary \ref{cor:goodpair} the claim is equivalent to the map $\Bbbk[Z]^{U^-} \to \Bbbk[G\cdot X]^{U^-}$ being onto. By Proposition \ref{prop:invariantalg}, it is enough to show that the map $\Bbbk[W]^{U^-} \to \Bbbk[X]^{U_L^-}$ is onto. By Corollary \ref{cor:goodpair}, the map $\Bbbk[V]^{U_L^-} \to \Bbbk[X]^{U_L^-}$ is onto. Hence, the claim follows if we show that the map $\Bbbk[W]^{U^-} \to \Bbbk[V]^{U_L^-}$ is onto. For this, we prove that the restriction of the latter map to the subalgebra $(\Sym ((W^*)^{U_I^-}))^{U_L^-}$ is already onto.
Since the $L$-map $\left.\psi \right|_{V}$ is split, then so is $\Sym ((W^*)^{U_I^-}) \to \Sym(V^*)$. Therefore, taking $U_L^-$-invariants yields a surjective map.
Now we consider part (c). By Corollary \ref{cor:goodpair} it is enough to see that the morphism $\Bbbk[G\cdot Y]^{U^-} \to \Bbbk[G\cdot X]^{U^-}$ is surjective. By Proposition \ref{prop:invariantalg}, this is equivalent to showing that $\Bbbk[Y]^{U^-_L} \to \Bbbk[X]^{U^-_L}$ is onto. This follows again by Corollary \ref{cor:goodpair}.
\end{proof}
\begin{remark}\label{rem:xi}
Assume $V$ is good and put $\eta= \mathcal{V}(V^*)$ and $\xi = \mathcal{V}(W^*)/\eta$. Then:
\begin{itemize}
\item[(a)] $G\cdot V$ is good if and only if $H^{i}(G/P, \, \bigwedge^i \xi)=0$, for all $i>0$, by Theorem \ref{thm:good} (a), \cite[Theorem 5.1.2]{weymanbook} and Remark \ref{rem:normalize} below.
\item[(b)] Assume further that $W$ has Weyl filtration and the $L$-map $\left.\psi \right|_{V}$ is a split. Then using Lemma \ref{lem:gr} we see as in the proof above that the induced map $W^* \to H^0(G/P, \, \eta)$ is onto. Hence, by Theorem \ref{thm:good} (a), $G\cdot V$ is good if and only if the algebra $q_* \mathcal{O}_{G\times_P V} \cong H^0(G/P, \, \Sym \eta)$ is generated by $H^0(G/P, \, \eta)$.
\end{itemize}
\end{remark}
\begin{corollary}\label{cor:largep}
If $\cha \Bbbk > \dim W$ and $ \cha \Bbbk\geq \langle \chi+ \rho , \alpha^\vee \rangle$ for all weights $\chi$ of $W$ and all $\alpha \in \Phi_+$, then $(\,W\,, \, G\cdot V\,)$ is a good pair.
\end{corollary}
\begin{proof}
By Lemma \ref{lem:good} parts (1)--(3), we see that both $V$ and $W$ are good. By \cite[Section 5.6]{jantzen}, both $V$ and $W$ are semi-simple, therefore $\left.\psi \right|_{V}$ is split injective (see Lemma \ref{lem:iso}). The conclusion now follows from Theorem \ref{thm:good} (b).
\end{proof}
If $W$ is as in (\ref{eq:mainsetup}), then putting $X=V$ and $Z=W$ in Theorem \ref{thm:good} (b), we see that $(W, \,G\cdot V)$ is a good pair whenever $\cha \Bbbk > \max \{ \dim \Delta_{G}(\lambda_i) \,| \, 1\leq i \leq n\}$ by Proposition \ref{prop:mathieu} and Lemma \ref{lem:good}. In particular, $G\cdot V$ is then good as claimed in the Introduction.
\subsection{Singularities via Schubert collapsing}\label{subsec:sing}
Now we turn to Theorems \ref{thm:introsat} and \ref{thm:intromain}. The following result describes the behavior of singularities under collapsing, and it strengthens \cite[Proposition 1 and Theorem 3]{Kempf76} when $w=w_0 w_I^{-1}$ (i.e.\ when $\overline{BwX} = G\cdot X$) in the characteristic zero case as well.
\begin{theorem}\label{thm:main}
Assume that $G\cdot X$ is good. For $w\in \mathcal{W}^I$, the $B$-variety $\overline{BwX}$ is $ww_I$-excellent. Furthermore, the following statements hold:
\begin{enumerate}
\item The map $\mathcal{O}_{\overline{BwX}} \, \xrightarrow{\,\cong\,} \, \mathbf{R} q_*\mathcal{O}_{\overline{BwP}\times_P X}$ is an isomorphism.
\item $\overline{BwX}$ is normal if and only if $X$ is so.
\item If $\cha \Bbbk = 0$, then $\overline{BwX}$ has rational singularities if and only if so does $X$.
\item If $X$ is an $L$-submodule of $V$, then $G\cdot X$ is strongly $F$-regular (resp.\ of strongly $F$-regular type) when $\cha \Bbbk > 0$ (resp.\ when $\cha \Bbbk =0$), and $\overline{BwX}$ is $F$-rational when $\cha \Bbbk> 0$.
\end{enumerate}
\end{theorem}
\begin{proof}
For part (1), observe that by (\ref{eq:unip}) a good filtration of $\Bbbk[X]$ has composition factors $\Delta_L(\lambda)$ with such that $\lambda\in X(T)_+$. By Lemma \ref{lem:surj}, we obtain by induction on filtration that $\mathbf{R}^i q_*\mathcal{O}_{\overline{BwP}\times_P X} = 0$, for all $i>0$. The map $\mathcal{O}_{\overline{BwX}} \to q_*\mathcal{O}_{\overline{BwP}\times_P X}$ is an isomorphism, since the composition $\Bbbk[W] \to q_*\mathcal{O}_{G\times_P X} \to q_*\mathcal{O}_{\overline{BwP}\times_P X}$ is surjective by Theorem \ref{thm:good} (a) and Lemma \ref{lem:surj}.
For part (2), if $\overline{BwX}$ is normal, then by Proposition \ref{prop:invariantalg} so is $X$. Conversely, if $X$ is normal, then so is $\overline{BwX}$ by the normality of $X(w)_P$ \cite{projnormschub} and $\mathcal{O}_{\overline{BwX}} \cong q_*\mathcal{O}_{\overline{BwP}\times_P X}$.
Next, we prove the statements regarding $\overline{BwX}$ in part (3) and (4). If $\overline{BwX}$ has rational singularities, then due to the direct summand property in Proposition \ref{prop:invariantalg} so does $X$ according to \cite[Th\'eor\`eme]{boutot}.
Consider the filtration $F^i \Bbbk[X]$ as in Section \ref{sec:gropop}. This gives an exhaustive filtration on $A:=\Bbbk[\overline{BwX}]$ by $F^i A:= (\Bbbk[\overline{BwP}] \otimes F^i \Bbbk[X])^P$. The associated graded is
\[\gr A = (\Bbbk[\overline{BwP}]^{U_I} \otimes \gr \Bbbk[X])^L \,\stackrel{(\ref{eq:grA})}{\cong}\, (\Bbbk[\overline{BwP}]^{U_I} \otimes (\Bbbk[L/U_L]\otimes \Bbbk[X]^{U^-_L})^T)^L \cong\]
\[\cong ((\Bbbk[\overline{BwP}]^{U_I} \otimes \Bbbk[L/U_L])^L\otimes \Bbbk[X]^{U^-_L})^T \cong (\Bbbk[\overline{BwP}]^U\otimes \Bbbk[X]^{U^-_L})^T = (\Bbbk[\overline{Bww_IB}]^U_+\otimes \Bbbk[X]^{U^-_L})^T, \]
where the last equality is a consequence of $\overline{BwP}=\overline{Bww_I B}$ and (\ref{eq:unip}), and the isomorphism before it follows from Lemma \ref{lem:transfer}.
Now assume that $X$ has rational singularities when $\cha \Bbbk = 0$ (resp.\ $X$ is an $L$-module when $\cha \Bbbk >0$). By \cite[Theorem 6]{popov} (resp.\ by \cite[Corollary 4.14]{hashiunip}), $\Bbbk[X]^{U^-_L}$ has rational singularities (resp.\ is strongly $F$-regular). By Lemma \ref{lem:cone} and (\ref{eq:f-sing}), $\Bbbk[\overline{Bww_IB}]^U_+$ has rational singularities (resp.\ is strongly $F$-regular). Hence, $\gr A$ has rational singularities (resp.\ is strongly $F$-regular) by \cite{boutot} (resp.\ \cite[Theorem 5.5]{hh}). As in \cite[Section 5]{popov}, the algebra $\gr A$ is a flat deformation of $A$. Therefore, $A$ has rational singularities by \cite{elkik} (resp.\ is $F$-rational by (\ref{eq:f-sing}) and \cite[Theorem 4.2]{hh}).
Now we show that $G\cdot X$ is strongly $F$-regular in part (4). Let $G' = \tilde{G} \times Z$, with $\tilde{G}$ a covering of $[G,G]$ and $Z \subset T$ a torus so that $G$ is a quotient of $G'$. We can view $W$ as a $G'$-representation. Since $T \subset L$, we have $G\cdot X = \tilde{G} \cdot X$. Moreover, we can lift $P$ to a parabolic $P'$ of $\tilde{G}$ with unipotent radical $U'_I$ and Levi subgroup $L'$. We have $W^{U'_I} = W^{U_I}$ and $(W^*)^{U'^-_I} = (W^*)^{U^-_I}$. Furthermore, $G\cdot X$ (resp.\ $X$) is $G$-good (resp.\ $L$-good) if and only if it is $\tilde{G}$-good (resp.\ $L'$-good) \cite[Section 3]{donkin}. This shows that we can assume that $G$ is simply connected and semisimple.
Assume that $\cha \Bbbk > 0$. Since $X$ and $G$ are good, using \cite[Theorem 3]{donkinunip} and Proposition \ref{prop:mathieu} we have
\[q_* \mathcal{O}_{G\times_P X}= (\Bbbk[G/U_I] \otimes \Bbbk[X])^L = \left((\Bbbk[G/U_I] \otimes \Bbbk[X])^{U_L}\right)^T.\]
As $T$ is linearly reductive, by \cite[Theorem 5.5]{hh} the claim follows once we show that $R:=(\Bbbk[G/U_I] \otimes \Bbbk[X])^{U_L}$ is strongly $F$-regular. Since $\Bbbk[X]$ and $\Bbbk[G]$ are factorial rings (see \cite{popovufd}), so is $R$ and $\Bbbk[G]^{U \times U_I}$ (see \cite[Theorem 3.17]{popvin}). In particular, since $\Bbbk[G]^{U \times U_I}$ is Cohen--Macaulay by Corollary \ref{cor:lpure} and (\ref{eq:f-sing}), it is Gorenstein \cite{murthy}.
We have an action of $G$ on $R$ induced from its left action on $\Bbbk[G]$. We have an isomorphism $R \cong (\Bbbk[L/U_L] \otimes \Bbbk[G/U_I] \otimes \Bbbk[X])^L$, which is easily seen to be $G$-equivariant. The algebra $\Bbbk[L/U_L] \otimes \Bbbk[G/U_I] \otimes \Bbbk[X]$ has a good filtration as a $G\times L$-module, as seen using \cite[Theorem 3]{donkinunip} and Proposition \ref{prop:mathieu}. By \cite[Proposition 1.2e (iii)]{donkinconj}, we obtain that $R$ has a good filtration as a $G$-module. We consider the invariant ring $R^{U}$. By Corollary \ref{cor:lpure}, \cite[Theorem 5.2]{hashisurjgraded} and \cite[Theorem 4.4 and Lemma 4.7]{hashiunip}, the $\mathbb{Z}_{\geq 0}$-graded ring $\Bbbk[G]^{U \times U_I} \otimes \Bbbk[X]$ is Gorenstein, strongly $F$-regular, and $L$-$F$-pure. Then \cite[Corollary 4.13]{hashiunip} implies that $R^{U}$ is strongly $F$-regular. Using the filtration in Section \ref{sec:gropop}, this implies that $R$ is $F$-rational by (\ref{eq:f-sing}) and \cite[Theorem 4.2]{hh} (see also \cite[Corollary 3.9]{hashiunip}). Since $R$ is factorial and Cohen--Macaulay, it is also Gorenstein \cite{murthy}. This shows that $R$ is strongly $F$-regular (see \cite[Corollary 4.7]{hh} or \cite{hh2}).
Now let $\cha \Bbbk = 0$. We can choose a suitable large set of primes $S$ such that for $D=\mathbb{Z}[S^{-1}]$ we have: the map $G\times W \to W$ (resp.\ $G\times X \to G\cdot X$) is defined over $D$;
$G\cdot X = (G_D\cdot X_{D}) \times_{\operatorname{Spec} D} \operatorname{Spec}(\Bbbk)$; the affine scheme $(G\cdot X)_{D}=G_D\cdot X_{D}$ is flat over $D$; both $W_{\overline{\mathbb{F}}_p}$ and $(G\cdot X)_{\overline{\mathbb{F}}_p}$ are good for $p\notin S$ (see Corollary \ref{cor:largep}); $W_{\overline{\mathbb{F}}_p}$ (resp.\ $X_{\overline{\mathbb{F}}_p}$) is a semi-simple $G_{\overline{\mathbb{F}}_p}$-module (resp.\ $L_{\overline{\mathbb{F}}_p}$-module) (see \cite[Section II.5.6]{jantzen}). For such $p\notin S$, for $V=X_{\overline{\mathbb{F}}_p}$ the map $\left.\psi \right|_{V}$ in (\ref{eq:hsplit}) is injective (see Lemma \ref{lem:iso}). By the previous paragraph and \cite[Theorem 5.5]{hh}, we obtain that $(G \cdot X)_{\mathbb{F}_p}$ is strongly $F$-regular. Hence, $G\cdot X$ is of strongly $F$-regular type.
\end{proof}
\begin{remark}\label{rem:field}
As seen in the proof above, the assumption on the field to be algebraically closed is not essential. The claims about rational singularities and strongly $F$-regular type (resp.\ $F$-rational singularities) hold over any field, e.g.\ by \cite{boutot} (resp.\ proof of \cite[Lemma 1.4]{smith}), as do claims (1) and (2). The claim on strong $F$-regularity holds for any $F$-finite (e.g.\ perfect) field \cite[Theorem 5.5]{hh}.
\end{remark}
\begin{remark}\label{rem:normalize}
Even if $X$ is good, it may happen that $G\cdot X$ is not, as can be seen in Example \ref{ex:counter}. Nevertheless, we still have $\mathbf{R}^i q_*\mathcal{O}_{\overline{BwP}\times_P X}=0$ for $i>0$. Further if $X$ is good, normal, and $q: \overline{BwP} \times_P X \to \overline{BwX}$ is birational (or, more generally, the generic fiber of $q$ is connected and $q$ is separable, as in \cite[Theorem 2.1 (a)]{lorwey}),
then the results in Theorem \ref{thm:main} carry over if we replace the variety $\overline{BwX}$ in each statement (besides part (2)) with its normalization, which is then in turn a $ww_I$-excellent variety.
\end{remark}
We further note that if one knows a good filtration of $\Bbbk[X]$ explicitly, then by Theorem \ref{thm:main} one obtains readily a corresponding $ww_I$-excellent filtration for $\Bbbk[\overline{BwX}]$. It is then possible to compute the ($T$-equivariant) Hilbert function for $\Bbbk[\overline{BwX}]$ using Lemma \ref{lem:surj} and the Demazure character formula (e.g.\ \cite[Corollary 3.3.11]{brionkumar}).
By Proposition \ref{prop:invariantalg} and \cite[Theorem 5.5]{hh} if $\overline{BwX}$ is strongly $F$-regular (when $\cha \Bbbk > 0$), then $X$ must also be strongly $F$-regular. In the case of a Borel subgroup, we can strengthen Theorem \ref{thm:main} by giving the following converse to this statement.
\begin{corollary}\label{cor:borel}
Assume that $P=B$ is a Borel subgroup and $W$ has a Weyl filtration. Then $G\cdot X$ is good. Moreover, for $w\in \mathcal{W}$, the variety $\overline{BwX}$ is strongly $F$-regular (resp.\ of strongly $F$-regular type) when $\cha \Bbbk > 0$ (resp.\ when $\cha \Bbbk =0$) if and only if so is $X$.
\end{corollary}
\begin{proof}
We can assume that $P=B$. Since $T$ is linearly reductive, $(V,X)$ is a good pair. By Theorem \ref{thm:good} (c), in order to show that $G\cdot X$ is good it is enough to show that $G\cdot V$ is so. For this, we use Theorem \ref{thm:good}(a). Since $V\subset W^U$, we have a $T$-decomposition $V=\bigoplus_{i=1}^n \Bbbk_{\lambda_i}$, where $\lambda_i \in X(T)_+$. The section ring
\[q_* \mathcal{O}_{G\times_B V} = \bigoplus_{(m_i)\in \mathbb{N}^n} H^0(\mathcal{L}(\sum_{i=1}^n m_i\lambda_i))\]
is generated in the components of the unit tuples, i.e.\ by the sum $\bigoplus_{i=1}^{n} \nabla_{G}(\lambda_i)$, as it follows from \cite{projnormschub} (see also \cite{multicone}). By Remark \ref{rem:xi} (b), $G\cdot V$ is good.
Assume that $X$ is strongly $F$-regular. Note that both $\Bbbk[\overline{BwB}]^U_+$ and $\Bbbk[X]$ are $X(T)_+$-graded algebras, so also $\mathbb{Z}_{\geq 0}$-graded, using for instance the map $h$ in Section \ref{sec:gropop}. Then the algebra $q_* \mathcal{O}_{\overline{BwB}\times_B X} = (\Bbbk[\overline{BwB}]^U \otimes \Bbbk[X])^T=(\Bbbk[\overline{BwB}]^U_+ \otimes \Bbbk[X])^T$ is strongly $F$-regular, as it follows by combining Lemma \ref{lem:cone}, \cite[Theorem 5.2]{hashisurjgraded} and \cite[Theorem 5.5]{hh}. Since $G\cdot X$ is good, the conclusion follows from Theorem \ref{thm:main} (1).
Now let $\cha \Bbbk = 0$. Assume $X$ is of strongly $F$-regular type, and consider a finitely generated $\mathbb{Z}$-algebra $R \subset k$ as in the definition in Section \ref{sec:fsing} (enlarging, if necessary, so that the action of $T_R$ is well-defined). Let $(\overline{BwX})_R = \operatorname{Spec}((C(X(w)_R) \otimes R[X_R])^{T_R})$. As in the proof of Lemma \ref{lem:cone}, $(\overline{BwX})_R$ is flat of finite type over $R$, and $(C(X(w)_R) \otimes R[X_R])^{T_R} \otimes_R \Bbbk' \cong (C(X(w)_{\Bbbk'}) \otimes \Bbbk'[X_{\Bbbk'}])^{T_{\Bbbk'}}$, for any field $\Bbbk'$ over $R$ (see \cite[Section I.2.11]{jantzen}). By Theorem \ref{thm:main} (1), we have $(\overline{BwX})_R \times_{\operatorname{Spec}(R)} \operatorname{Spec}(\Bbbk) \cong \overline{BwX}$. When $\Bbbk'$ is a residue field of $R$, it is finite, in which case $C(X(w)_{\Bbbk'})$ is strongly $F$-regular, as seen in the proof of Lemma \ref{lem:cone}. As in the previous paragraph, we conclude that $(\overline{BwX})_{R/\mathfrak{m}}$ is strongly $F$-regular for maximal ideals $\mathfrak{m}$ in a dense open subset of $\operatorname{Spec}(R)$.
Finally, if $\overline{BwX}$ is of strongly $F$-regular type, using Proposition \ref{prop:invariantalg} we see by an argument similar to the above that $X$ is also of strongly $F$-regular type.
\end{proof}
Further, we provide a result that can lead to more general varieties outside the equivariant setting. Following \cite{brionmultfree}, we call a closed subvariety $Y\subset G/P$ multiplicity-free if it is rationally equivalent to a multiplicity-free linear combination of Schubert cycles.
\begin{corollary}\label{cor:multfree}
Let $Y$ be a multiplicity-free subvariety of $G/P$, and assume that $G\cdot X$ is good. Then $\mathcal{O}_{q(\pi^{-1}(Y))} \, \xrightarrow{\,\cong\,} \, \mathbf{R} q_*\mathcal{O}_{\pi^{-1}(Y)}$ is an isomorphism. Moreover, if $X$ is normal (resp.\ has rational singularities when $\cha \Bbbk = 0$), then $q(\pi^{-1}(Y))$ is normal (resp.\ has rational singularities).
\end{corollary}
\begin{proof}
The proof of the isomorphism $\mathcal{O}_{q(\pi^{-1}(Y))} \, \xrightarrow{\,\cong\,} \, \mathbf{R} q_*\mathcal{O}_{\pi^{-1}(Y)}$ follows as in Theorem \ref{thm:main} (a) using \cite[Theorem 0.1]{brionmultfree} and Lemma \ref{lem:surj}. The claim on normality follows from this, as $Y$ itself is normal \cite[Theorem 0.1]{brionmultfree}. Moreover, $Y$ has rational singularities when $\cha \Bbbk = 0$ \cite[Theorem 0.1 and Remark 3.3]{brionmultfree}, hence we conclude that so does $q(\pi^{-1}(Y))$ by \cite[Theorem 1]{kovacs}.
\end{proof}
\subsection{Defining equations of saturations}\label{subsec:def} In this section we give a result on the defining equations of $G\cdot X$ in $W$. Assume that $G\cdot V$ is good. Let $M \subset \Bbbk[V]$ be an $L$-stable module with a good filtration. We can associate to it a $G$-module $M' \subset \Bbbk[G\cdot V]$ in the following way. Consider the inclusion of sheaves $\mathcal{V}(M) \subset \mathcal{V}(\Sym V^*)$ on $G/P$. Then we put $M' = H^0(G/P, \mathcal{V}(M))$. As in the proof of Theorem \ref{thm:main} (1), we see that $M'$ has a good filtration as a $G$-module. Note that $M'$ contains $\operatorname{span}_{\Bbbk} G\cdot M$ via the inclusion given by Remark \ref{rem:unipara}, and this containment is an equality when $M'$ is a semi-simple $G$-module.
\begin{theorem}\label{thm:defi}
Let $(V,X)$ be a good pair with $G\cdot V$ good, and denote by $I_X \subset \Bbbk[V]$ the defining ideal of $X \subset V$. Let $M$ be the span of a set of good generators of $I_X$ and take a basis $\mathcal{P}'$ of the $G$-module $M'\subset \Bbbk[G\cdot V]$ associated to $M$ as above. Consider the following:
\begin{enumerate}
\item A set of generators $\mathcal{P}_{G\cdot V}$ of the defining ideal $I_{G\cdot V} \subset \Bbbk[W]$ of $G\cdot V$;
\item A lift $\tilde{\mathcal{P}'} \subset \Bbbk[W]$ of the set $\mathcal{P}' \subset \, \Bbbk[W]/I_{G\cdot V}$.
\end{enumerate}
Then the defining ideal of $G \cdot X$ in $\Bbbk[W]$ is generated by $\mathcal{P}:= \, \mathcal{P}_{G\cdot V} \, \cup \, \tilde{\mathcal{P}'}$.
Furthermore, assume that $(W, \, G\cdot V)$ is a good pair. If either $M'$ is a tilting module, or there are no dominant weights $\lambda >\mu$ such that $(M')^{U}_\lambda \neq 0 \neq (I_{G\cdot V})^{U}_\mu$ , then the lift $\tilde{\mathcal{P}'}$ can be chosen such that $\operatorname{span}_{\Bbbk} \tilde{\mathcal{P}'} \, \subset \Bbbk[W]$ is $G$-stable; with such lift, if $\mathcal{P}_{G\cdot V}$ are good generators of $I_{G\cdot V}$ then $\mathcal{P}$ is a set of good defining equations of $G\cdot X \, \subset W$.
\end{theorem}
\begin{proof}
Let $J \subset \Bbbk[G\cdot V]$ denote the defining ideal of $G\cdot X$ in $G\cdot V$. We have an exact sequence
\[0 \to J \to \Bbbk[G\cdot V] \to\Bbbk[G\cdot X] \to 0.\]
By Remark \ref{rem:unipara}, taking $U^-_I$-invariants in the sequence above we get that $J^{U^-_I} \cong I_X$. Furthermore, by construction $M'\subset J$ and $M\subset M'^{U^-_I}$. Consider the multiplication map
\[m_{\mathcal{P}'}: \Bbbk[G\cdot V] \otimes M' \to J.\]
By Lemma \ref{lem:gr} and Proposition \ref{prop:mathieu}, to see that $m_{\mathcal{P}'}$ is surjective, it is enough to show that the induced map on $U^-$-invariants is so. This is a consequence of the fact that the following composition of maps is surjective by the assumption on good generators of $I_X$:
\begin{equation}\label{eq:ontoinv}
(\Bbbk[V] \otimes M)^{U_L^-} \hookrightarrow (\Bbbk[G\cdot V]^{U^-_I} \otimes M'^{U^-_I})^{U_L^-} \hookrightarrow (\Bbbk[G\cdot V] \otimes M')^{U^-} \to J^{U^-} = I_X^{U^-_L}.
\end{equation}
As $\mathcal{P}'$ generates $J = I_{G\cdot X}/ I_{G\cdot V}$, it is clear that $\mathcal{P}$ generates $I_{G\cdot X}$.
Let $N$ be the $G$-submodule $N \subset I_{G\cdot X}$ corresponding to $M' \subset J$. We have an exact sequence
\[0 \to I_{G\cdot V} \to N \to M'\to 0.\]
To show that $\tilde{\mathcal{P}'}$ can be chosen in the required way, we show that the sequence splits as $\Ext^1_G(M', I_{G\cdot V}) = 0$. When $M'$ is tilting, this is a consequence of \cite[Proposition II.4.13]{jantzen}, as $I_{G\cdot V}$ has a good filtration and $M'$ has a Weyl filtration. The other case is a consequence of \cite[Proposition 2]{friedlander}.
By the splitting above, we have $M_{\tilde{\mathcal{P}'}} = \operatorname{span} \tilde{\mathcal{P}'} \cong M' $ as $G$-modules. It has a good filtration, as the module $M_{\mathcal{P}_{G\cdot V}}$, since $\mathcal{P}_{G\cdot V}$ is a set of good generators. Therefore, $M_{\mathcal{P}} = M_{\tilde{\mathcal{P}'}} \oplus M_{\mathcal{P}_{G\cdot V}}$ has a good filtration \cite[Corollary 3.2.5]{donkin}. Consider the commutative diagram
\[\xymatrix@R-0.4pc@C-0.3pc{
0 \ar[r] & (k[W] \otimes M_{\mathcal{P}_{G\cdot V}})^{U} \ar[r]\ar[d] & (k[W] \otimes M_{\mathcal{P}})
^{U} \ar[r]\ar[d] & (k[W] \otimes M_{\tilde{\mathcal{P}'}})^{U} \ar[r]\ar[d] & 0 \\
0 \ar[r] & \quad (I_{G\cdot V})^{U} \quad \ar[r] & \quad (I_{G\cdot X})^{U} \quad \ar[r] & \quad J^{U} \quad \ar[r] & 0
}\]
Due to the respective modules having good filtrations by Proposition \ref{prop:mathieu}, the rows of the diagrams are exact \cite[Proposition 1.4 and Proposition 2]{donkinunip}. Since $\mathcal{P}_{G\cdot V}$ is a set of good generators, the first vertical map is onto. We are left to show that the third vertical map is onto, or equivalently, that the following composition is surjective (see comment after Lemma \ref{lem:exist}):
\[ (\Bbbk[W] \otimes M_{\tilde{\mathcal{P}'}})^{U^-} \to (\Bbbk[G\cdot V] \otimes M')^{U^-} \to J^{U^-}.\]
The first map is onto since $M_{\tilde{\mathcal{P}'}} \xrightarrow{\cong} M'$ and $(W, \, G\cdot V)$ is a good pair. The second map is onto as seen in (\ref{eq:ontoinv}). Thus, $\mathcal{P}$ is a good generating set of $I_{G\cdot X}$.
\end{proof}
\begin{remark}
With the assumptions above, one can similarly give defining equations of $\overline{BwX}$, provided we have defining equations of $\overline{BwV}$ in $\Bbbk[G\cdot V]$.
\end{remark}
When $G\cdot V$ is good, by Theorem \ref{thm:main} one can in principle apply \cite[Theorem 5.1.3]{weymanbook} to obtain a (minimal) set of generators $\mathcal{P}_{G\cdot V}$ (as seen in Remark \ref{rem:xi}), or even its minimal free resolution. We note that the minimal free resolution of $G\cdot V$ given by \textit{loc.\ cit.}\ has length equal to $\codim_{G\cdot V} W$, since $G\cdot V$ is Cohen--Macaulay (\ref{eq:f-sing}). For variations of this technique, see for example \cite[Section 6]{weymanbook} or \cite[Proposition 4.4]{kinlor}.
\section{Special cases and applications}\label{sec:applications}
This section is devoted to demonstrate the strength of our results through some important applications, both classical and new. The examples in the next three subsections fit into the situation described in the Introduction (\ref{eq:mainsetup}).
\subsection{Varieties of determinantal type}\label{subsec:det}
Let $m\geq n\geq 0$, and consider the case when $W$ is the space of $m\times n$ matrices, $n\times n$ skew-symmetric matrices, or $n\times n$ symmetric matrices -- the latter can be also identified with the 2nd divided power of $\Bbbk^n$. Then we choose $G$ to be $\operatorname{GL}(m)\times \operatorname{GL}(n)$, $\operatorname{GL}(n)$ or $\operatorname{GL}(n)$, and $W=\Delta_G(\lambda)$ to be $\Bbbk^m \otimes \Bbbk^n$, $\bigwedge^2 \Bbbk^n$, or $\Delta_G(2\omega_1)$, respectively.
For $0\leq r \leq n$, we put $L$ to be $\operatorname{GL}(r)\times \operatorname{GL}(r)$, $\operatorname{GL}(r)$ or $\operatorname{GL}(r)$, respectively (and $V=\Delta_L(\lambda)$). Then $G\cdot V$ is precisely the closed subvariety in $W$ of matrices of rank at most $r$ (see \cite[Section 6]{weymanbook}).
The variety $W$ (resp.\ $V$) is good in arbitrary characteristic (see Lemma \ref{lem:good} and \cite{boffi}). Thus, by Theorem \ref{thm:introgood} (with $X=V$) the $G$-variety $G\cdot V$ is good as well. Therefore, by Theorem \ref{thm:main} $G\cdot V$ is strongly $F$-regular when $\cha \Bbbk >0$ (resp.\ is of strongly $F$-regular type when $\cha \Bbbk = 0$) and $\overline{BwV}$ is $F$-rational (resp.\ has rational singularities if $\cha \Bbbk = 0 $). This yields all $G$-orbit closures $G\cdot V$ and many $B$-orbit closures $\overline{BwV}$ in $W$.
For $G$-orbit closures in the case of symmetric matrices, this answers \cite[Question 5.10]{katzmir}. For $G$-orbit closures in $\Bbbk^m \otimes \Bbbk^n$ and $\bigwedge^2 \Bbbk^n$, we recover the results \cite{hh2}, \cite[Theorem 1.3]{baetica} (see also \cite[Chapter 7]{baeticabook}).
The $B$-orbit closures are called matrix Schubert varieties in the literature. As far as we are aware, in this case the results are new even in characteristic $0$, except in the space of $m\times n$ matrices, when it is known that all matrix Schubert varieties are strongly $F$-regular, as this can be reduced to the corresponding statement on Schubert varieties \cite{schubfreg} (see Corollary \ref{cor:borel}) by an identification as done in \cite{fulton}.
Let us show that the $(r+1)\times (r+1)$ minors of a generic symmetric matrix give good defining equations for the space of symmetric matrices of rank $\leq r$ in $W$ using Theorem \ref{thm:defi} (the other cases are analogous and slightly easier). We work on downwards induction on $r$, the case $r=n$ being trivial. Let $V$ be the space of $r\times r$ symmetric matrices as above, and consider $X\subset V$ the matrices of rank $< r$. Clearly, the symmetric determinant is a good defining equation for $X\subset V$ (e.g.\ Lemma \ref{lem:compint}). The associated $G$-module $M'\subset \Bbbk[G\cdot V]$ in Theorem \ref{thm:defi} is $M'=\nabla_G(2 \omega_r)$, and it is easy to see that it satisfies the condition that there are no dominant weights $\mu < 2 \omega_r$ with $(I_{G\cdot V})^{U}_\mu \neq 0$. The lift $\tilde{\mathcal{P}}'$ can be chosen to be the $r \times r$ minors of a generic symmetric matrix, while $\mathcal{P}_{G\cdot V}$ are the $(r+1)\times (r+1)$ minors, by the induction hypothesis. By Theorem \ref{thm:defi}, we conclude that $\tilde{\mathcal{P}}'$ is a good set of defining equations for $G\cdot X$ in $W$.
\subsection{Varieties of complexes on arbitrary quivers}\label{sec:radsquare}
The geometry of the Buchsbaum--Eisenbud varieties of complexes has been investigated thoroughly in a number of articles. In \cite{Kempfcomp} it has been shown that these varieties have rational singularities in characteristic zero, based on the method in \cite{Kempf76}. A characteristic-free approach has been pursued in \cite{DecoStrick} using Hodge algebras, where defining equations are provided as well. In characteristic zero, this result has been proved also in \cite{brioncomp} by showing that their algebra of covariants is a polynomial ring.
Frobenius splitting methods have been applied in \cite{MT2}. One can realize such varieties as certain open subsets in Schubert varieties \cite{zele}, \cite{LM98}. Similar varieties have been studied in \cite{Strick1}, \cite{Strick2}, \cite{MT1} for other special quivers. These varieties can be considered for any quiver, and are particular cases of certain rank varieties of radical square zero algebras, as in explained in \cite{kinlor}. In \emph{ibid.}, it is shown that in characteristic zero all such varieties have rational singularities, and defining equations are provided. We explain now how to extend such results to arbitrary characteristic, as announced in Remark 4.16 of \emph{ibid.} Additionally, we obtain analogous results for $B$-varieties.
We follow closely the notation established in \cite{kinlor}. Consider the (associative, non-commutative) radical square zero algebra $A=\Bbbk Q/\Bbbk Q_{\geq 2}$, with $Q$ an arbitrary finite quiver with the set of vertices $Q_0$ and arrows $Q_1$. For a dimension vector $\mathbf{d}: Q_0 \to \mathbb{Z}_{\geq 0}$, we consider the representation space
\[
\rep_Q(\mathbf{d}) = \prod_{\ensuremath{\alpha} \in Q_1} \operatorname{Hom}_\Bbbk(\Bbbk^{\mathbf{d}(t\ensuremath{\alpha})}, \Bbbk^{\mathbf{d}(h\ensuremath{\alpha})}) = \bigoplus_{\ensuremath{\alpha} \in Q_1} (\Bbbk^{\mathbf{d}(t\ensuremath{\alpha})})^* \otimes \Bbbk^{\mathbf{d}(h\ensuremath{\alpha})},
\]
and within the representation variety of $A$
\[\rep_A(\mathbf{d}) = \{ M \in \rep_Q(\mathbf{d}) \, \mid \, M_{\beta}\circ M_{\ensuremath{\alpha}}=0, \mbox{ for all } \ensuremath{\alpha},\beta\in Q_1 \mbox{ with } h\ensuremath{\alpha} = t\beta\},\]
which has a natural action of the reductive group $\operatorname{GL}(\mathbf{d}) = \prod_{x \in Q_0} \operatorname{GL}(\mathbf{d}(x))$. For $x\in Q_0$ and $M\in \rep_Q(\mathbf{d})$, we put
\[h_x(M)=\bigoplus_{h\ensuremath{\alpha} =x} M_{\ensuremath{\alpha}} \colon \, \bigoplus_{h\ensuremath{\alpha} =x} M_{t\ensuremath{\alpha}} \to M_x.\]
For a dimension vector $\mathbf{r} \leq \mathbf{d}$, we denote by $C_\mathbf{r}$ the closure of the set of representations $M \in \rep_A(\mathbf{d})$ such that $\rank h_x(M)=\mathbf{r}(x)$, for all $x\in Q_0$. Let $\mathbf{s}=\mathbf{d} - \mathbf{r}$. By \cite[Theorem 3.19]{kinlor} the variety $C_\mathbf{r}$ is irreducible, and it is non-empty if and only if
\begin{equation}\label{eq:nonzero}
\sum_{h\ensuremath{\alpha} = x} \mathbf{s}(t\ensuremath{\alpha}) \, \geq \mathbf{r}(x), \mbox{ for all } x\in Q_0.
\end{equation}
Furthermore, each irreducible component of $\rep_A(\mathbf{d})$ is of the form $C_\mathbf{r}$, for some $\mathbf{r}\leq \mathbf{d}$.
Now fix $\mathbf{r} \leq \mathbf{d}$ as in (\ref{eq:nonzero}). With the notation from Section \ref{sec:mainresults}, we let $W=\rep_Q(\mathbf{d})$, $V=\bigoplus_{\ensuremath{\alpha} \in Q_1} (\Bbbk^{\mathbf{s}(t\ensuremath{\alpha})})^* \otimes \Bbbk^{\mathbf{r}(h\ensuremath{\alpha})}$, $G=\operatorname{GL}(\mathbf{d})$, $L= \prod_{x \in Q_0} (\operatorname{GL}(\mathbf{s}(x)) \times \operatorname{GL}(\mathbf{r}(x)))$ . It is implicit from the proof of \cite[Theorem 3.19]{kinlor} that $C_\mathbf{r} = G\cdot V$ (in fact, the collapsing map $q: G\times_P V \to C_\mathbf{r}$ is a resolution of singularities). The variety $W$ (resp.\ $V$) is good in arbitrary characteristic by Lemma \ref{lem:good} and Proposition \ref{prop:mathieu}. Thus, by Theorem \ref{thm:introgood} the $G$-variety $C_\mathbf{r}$ is good and Theorem \ref{thm:main} implies the following result.
\begin{corollary}\label{cor:nodefreg}
The rank variety $C_\mathbf{r}$ is strongly $F$-regular when $\cha \Bbbk >0$ (resp.\ of strongly $F$-regular type when $\cha \Bbbk =0$).
\end{corollary}
Moreover, the varieties $\overline{BwV} \subset C_\mathbf{r}$ are $F$-rational when $\cha \Bbbk > 0$ (resp.\ have rational singularities when $\cha \Bbbk = 0$). Note that the Buchsbaum--Eisenbud varieties of complexes are spherical (e.g.\ \cite{brioncomp}), therefore such varieties are always $B$-orbit closures in this case as there are only finitely many $B$-orbits \cite{brisph}, \cite{vinsph}. We leave the details of the combinatorial characterization of such $B$-orbit closures to the interested reader.
In \cite[Corollary 4.13]{kinlor}, explicit defining equations are provided for all $C_\mathbf{r}$ when $\cha \Bbbk = 0$.
We give a self-contained argument to show that, in the case when $Q$ has no loops, these equations are also defining equations when $\cha \Bbbk > 0$.
For $\ensuremath{\alpha} \in Q_1$, we let $X_{\ensuremath{\alpha}}$ be the $\mathbf{d}(t\ensuremath{\alpha})\times \mathbf{d}(h\ensuremath{\alpha})$ generic matrix of variables. We identify the coordinate ring $\Bbbk[\rep_{\Bbbk Q}(\mathbf{d})]$ with a polynomial ring in the entries of the matrices $\{X_\ensuremath{\alpha}\}_{\ensuremath{\alpha}\in Q_1}$. For $x \in Q_0$, we write $H_x$ (resp.\ $T_x$) for the $\mathbf{d}(x)\times \left(\displaystyle\sum_{h\ensuremath{\alpha} = x} \mathbf{d}(t\ensuremath{\alpha})\right)$ matrix (resp.\ $\left(\displaystyle\sum_{t\ensuremath{\alpha} = x} \mathbf{d}(h\ensuremath{\alpha})\right)\times \mathbf{d}(x)$ matrix) obtained by placing the matrices $X_\ensuremath{\alpha}$ with $h\ensuremath{\alpha} = x$ next to (resp.\ with $t\ensuremath{\alpha} = x$ on top of) each other.
\begin{corollary}\label{cor:nodedefi}
Assume $Q$ has no loops, and let $C_\mathbf{r} \subset \rep_A(\mathbf{d})$ be non-empty. The following set of polynomials in $\Bbbk[\rep_{\Bbbk Q}(\mathbf{d})]$ form a good set of generators for the prime ideal of $C_\mathbf{r}$, as $x$ runs through all the vertices in $Q_0$:
\begin{enumerate}
\item The $(\mathbf{r}(x)+1)\times (\mathbf{r}(x)+1)$ minors of $H_x$;
\item The $(\mathbf{s}(x)+1) \times (\mathbf{s}(x)+1)$ minors of $T_x$;
\item The entries of \, $T_x \cdot H_x$;
\end{enumerate}
\end{corollary}
\begin{proof}
We work by splitting nodes one at a time, analogously to \cite[Corollary 4.13]{kinlor}. We note that in Theorem \ref{thm:defi}, the module $M'$ is tilting in this case (see Lemma \ref{lem:good} (4)). To conclude using Theorem \ref{thm:defi} as in \cite[Corollary 4.13]{kinlor}, we are reduced to show that the equations (1)--(3) with $x=2$ are good defining equations of $C_\mathbf{r}$ for the following quiver (compare with \cite[Proposition 4.4]{kinlor})
\[\xymatrix{ 1 \ar[r]^a & 2 \ar[r]^b & 3 } \]
As in the case of determinantal varieties in Section \ref{subsec:det}, we can further reduce using Theorem \ref{thm:defi} (applied at vertices $1$ and $3$) to the case $\mathbf{r}=(0,d_1,d_3)$ (when we have $d_2 \geq d_1 + d_3$). In such case only the equations of type (3) appear, and they form a regular sequence. Using the Jacobian criterion, one readily obtains that the ideal generated by these polynomials is radical. Moreover, by Lemma \ref{lem:compint} they give good defining equations for $C_\mathbf{r} \subset \rep_{\Bbbk Q}(\mathbf{d})$, thus yielding the conclusion.
\end{proof}
The article \cite{kinlor} further demonstrates the usefulness of working in the relative situation $X\subset V$. By splitting nodes one at a time, the method is applied to a large number of other quiver varieties in characteristic zero. The main obstruction to extending such results to positive characteristics readily is that so far the good property of the corresponding $L$-variety $X$ has been studied only in a handful of cases (e.g.\ \cite{donkinconj}).
\subsection{Further examples}
When $G=\operatorname{GL}(n)$, $L=\operatorname{GL}(r)$ (with $r\leq n$), $W=\Delta_G(\lambda)$ and $V=\Delta_L(\lambda)$, the variety $G\cdot V$ is called higher rank variety \cite[Section 7]{weymanbook}. Thus, Theorem \ref{thm:main} generalizes Proposition 7.1.2 in \emph{loc.\ cit.}\ to characteristics that are not \lq\lq too small", and further gives new results for the varieties $\overline{BwV}$. We note that the result does not hold in arbitrary characteristic, as the following example shows.
\begin{example}\label{ex:counter} Let $G=\operatorname{GL}(3)$, $W=\bigwedge^3 \Bbbk^6$, $V=\bigwedge^3 \Bbbk^5$ with $\cha \Bbbk = 2$. Then $V$ is a good variety, but $G\cdot V$ is not normal, as shown by Weyman \cite[Proposition 7.3.10]{weymanbook}. Using Theorems \ref{thm:good} and \ref{thm:main} we see that $W$ is not good (nor is the hypersurface given by the discriminant of degree $4$), a fact further observed in \cite[Example 3.3]{kallen}. Nevertheless, by Remark \ref{rem:normalize} the normalization of $G\cdot V$ is strongly $F$-regular.
We can extrapolate this to higher dimensions as follows. Set $X:=G\cdot V$ from above. Let $n\geq 6$, and consider inclusions $\bigwedge^3 \Bbbk^5 \subset \bigwedge^3 \Bbbk^6 \subset \bigwedge^3 \Bbbk^n$. Then the saturation $Y:=\operatorname{GL}(n) \cdot \bigwedge^3 \Bbbk^5 \, \subset \, \bigwedge^3 \Bbbk^n$ is the same as the saturation $\operatorname{GL}(n) \cdot X \, \subset \operatorname{GL}(n)\cdot \bigwedge^3 \Bbbk^6 \, \subset \bigwedge^3 \Bbbk^n$. We have seen that $X$ is not strongly $F$-regular, hence neither is $Y$ by Proposition \ref{prop:invariantalg} and \cite[Theorem 5.5]{hh}, but the normalization of $Y$ is again strongly $F$-regular by Remark \ref{rem:normalize}. In particular, $\bigwedge^3 \Bbbk^n$ is not good by Theorems \ref{thm:good} and \ref{thm:main}.
\end{example}
Other examples of saturations $G\cdot V$ (and $\overline{BwV}$) where our results can be readily applied include varieties considered in \cite[Section 2]{Kempf76}, \cite{multicone}, \cite{samwey}, \cite{marcus}, \cite{landwey2}, and the subspace varieties in \cite{landwey1} (including the relative setting for secant varieties, as in \cite[Proposition 5.1]{landwey1}), thus strengthening the corresponding results therein.
As explained in the Introduction, the results can be effectively used in the study of the geometry of orbit closures for any representation $W$ (as in (\ref{eq:mainsetup})) of a reductive group. Since such problems have been pursued intensively in numerous articles for various special representations, it would be difficult to list them all in relation with our results.
We simply direct the reader to \cite{weymanbook} and the references therein for a large collection of such examples.
\subsection{Vanishing results for bundles on Schubert varieties}\label{subsec:ample}
First, we record the following positive characteristic version of the Grauert--Riemenschneider theorem for collapsing of bundles (cf. \cite[Section 3]{Kempf76}). Such results are of interest (see \cite[Theorem 1.3.14]{brionkumar}), as in general they do not hold in positive characteristic. We continue with the notation from Section \ref{sec:mainresults}. We denote by $\omega_Y$ the canonical sheaf of a Cohen--Macaulay variety $Y$ and put $\eta=\mathcal{V}(V^*)$ as in Remark \ref{rem:xi}.
\begin{proposition}\label{prop:duality}
Take $w\in \mathcal{W}^I$ and put $c=\dim X(w)+\dim V - \dim \overline{BwV}$. If $G\cdot V$ is good then $\mathbf{R}^c q_* \, \omega_{\overline{BwP}\times_P V} \cong \omega_{\overline{BwV}}$ and
\[H^i(X(w)_P, \,\, \Sym_d \eta \otimes \det \eta \otimes \omega_{X(w)_P}\,)=0 \mbox{ for all } i\neq c, d\geq 0.\]
\end{proposition}
\begin{proof}
Put $Y=\overline{BwP}\times_P V$ and $Z=\overline{BwV}$. By Theorem \ref{thm:main} we have $\mathbf{R} q_* \mathcal{O}_Y \cong \mathcal{O}_Z$. and $Z$ is Cohen--Macaulay (\ref{eq:f-sing}). As $q^{!} \omega_Z \cong \omega_Y[c]$, we obtain using Grothendieck duality \cite[Theorem III.11.1]{hartshorne2}
\[\mathbf{R} q_* \omega_Y \cong \mathbf{R} q_* \mathcal{H}\!om_{\mathcal{O}_Y} (\mathcal{O}_Y, \omega_Y) \cong \mathcal{H}\!om_{\mathcal{O}_Z} (\mathcal{O}_Z, \omega_Z[-c])\cong \omega_Z[-c].\]
The conclusion follows by the adjunction formula \cite[Proposition II.8.20]{hartshorne}.
\end{proof}
\begin{remark}\label{rem:bott}
When $X(w)=G/P$ and $\cha \Bbbk = 0$, the bundle $\Sym_d \eta \otimes \det \eta \otimes \omega_{X(w)_P}$ is semi-simple. Thus, using the Borel--Weil--Bott theorem (see \cite[Section 4]{weymanbook} and \cite[Corollary 5.5]{jantzen}) and Serre duality \cite[Corollary 7.7]{hartshorne}, in this case we can deduce from Proposition \ref{prop:duality} that the $L$-dominant weights that appear in $\Sym V \otimes \det V$ are either singular or lie in a single Bott chamber (giving cohomology in degree $\dim G\cdot V - \dim V$).
\end{remark}
If we only assume that $V$ is good, one can give an analogous result to Proposition \ref{prop:duality} using normalization as in Remark \ref{rem:normalize}. Along these lines, we give the following version of Griffiths' vanishing theorem \cite{griff} for Schubert varieties in positive characteristic.
\begin{corollary}\label{cor:griff}
Assume $V$ is a good and let $\lambda \in X(T)_+$ with $\langle \lambda , \alpha_i^\vee \rangle = 0$ if and only if $i\in I$ (i.e.\ $\mathcal{L}(\lambda)$ is ample on $G/P$). Then
\[H^i(X(w)_P, \,\, \Sym_d \eta \otimes \det \eta \otimes \mathcal{L}(\lambda) \otimes \omega_{X(w)_P}\,)=0 \mbox{ for all } i>0, d\geq 0, w\in \mathcal{W}^I.\]
\end{corollary}
\begin{proof}
We put $W'= \Delta_G(\lambda)\oplus W$, $V' = \Bbbk_{\lambda} \oplus V$ and consider $q: G \times_P V' \to G\cdot V'$. To conclude by Proposition \ref{prop:duality} in combination with Remark \ref{rem:normalize}, it is enough to show that $q$ is an isomorphism on the open $G\times_P ((\Bbbk_{\lambda} \setminus\{0\}) \times V)$ (so $q$ is birational). It is known that the map $q_1 : G\times_P \Bbbk_{\lambda} \to G\cdot \Bbbk_{\lambda}$ is an isomorphism on the open $G\times_P (\Bbbk_{\lambda} \setminus\{0\})$ (e.g.\ \cite[Exercise 5.8]{weymanbook}). Further, we have an isomorphism $G\times_P (\Bbbk_{\lambda} \times W) \cong (G\times_P \Bbbk_{\lambda}) \times W$ given by $(g,l,w) \mapsto (g,l,gw)$. Composing the latter map with $q_1$ we obtain the result.
\end{proof}
Note that when $\eta$ is a line bundle and $d=0$ (or when $V=0$), the result amounts to the classical Kodaira-type vanishing property for Schubert varieties that can be realized as a consequence of Frobenius splitting or global $F$-regularity \cite{schubsplit}, \cite{schubfreg}, \cite{smith2}.
\bibliographystyle{alpha}
| {
"timestamp": "2021-10-06T02:19:00",
"yymm": "2008",
"arxiv_id": "2008.08270",
"language": "en",
"url": "https://arxiv.org/abs/2008.08270",
"abstract": "We study the geometry of equivariant, proper maps from homogeneous bundles $G\\times_P V$ over flag varieties $G/P$ to representations of $G$, called collapsing maps. Kempf showed that, provided the bundle is completely reducible, the image $G\\cdot V$ of a collapsing map has rational singularities in characteristic zero. We extend this result to positive characteristic and show that for the analogous bundles the saturation $G\\cdot V$ is strongly $F$-regular if its coordinate ring has a good filtration. We further show that in this case the images of collapsing maps of homogeneous bundles restricted to Schubert varieties are $F$-rational in positive characteristic, and have rational singularities in characteristic zero. We provide results on the singularities and defining equations of saturations $G\\cdot X$ for $P$-stable closed subvarieties $X\\subset V$. We give criteria for the existence of good filtrations for the coordinate ring of $G\\cdot X$. Our results give a uniform, characteristic-free approach for the study of the geometry of a number of important varieties: multicones over Schubert varieties, determinantal varieties in the space of matrices, symmetric matrices, skew-symmetric matrices, and certain matrix Schubert varieties therein, representation varieties of radical square zero algebras (e.g. varieties of complexes), subspace varieties, higher rank varieties, etc.",
"subjects": "Algebraic Geometry (math.AG); Commutative Algebra (math.AC); Representation Theory (math.RT)",
"title": "On the collapsing of homogeneous bundles in arbitrary characteristic",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754501811437,
"lm_q2_score": 0.718594386544335,
"lm_q1q2_score": 0.7075103916295314
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https://arxiv.org/abs/1803.03511 | Divisibility of L-Polynomials for a Family of Artin-Schreier Curves | In this paper we consider the curves $C_k^{(p,a)} : y^p-y=x^{p^k+1}+ax$ defined over $\mathbb F_p$ and give a positive answer to a conjecture about a divisibility condition on $L$-polynomials of the curves $C_k^{(p,a)}$. Our proof involves finding an exact formula for the number of $\mathbb F_{p^n}$-rational points on $C_k^{(p,a)}$ for all $n$, and uses a result we proved elsewhere about the number of rational points on supersingular curves. | \section*{Divisibility of L-Polynomials for a Family of Artin-Schreier Curves}
{\large
Gary McGuire\footnote{email gary.mcguire@ucd.ie, Research supported by Science Foundation Ireland Grant 13/IA/1914} and
Emrah Sercan Y{\i}lmaz \footnote {Research supported by Science Foundation Ireland Grant 13/IA/1914}
\\
School of Mathematics and Statistics\\
University College Dublin\\
Ireland}
\end{center}
\subsection*{Abstract}
In this paper we consider the curves $C_k^{(p,a)} : y^p-y=x^{p^k+1}+ax$ defined
over $\mathbb F_p$ and give a positive answer to a conjecture about a divisibility condition on $L$-polynomials of the curves $C_k^{(p,a)}$.
Our proof involves finding an exact formula for the number of $\F_{p^n}$-rational points
on $C_k^{(p,a)}$ for all $n$, and uses a result we proved
elsewhere about the number of rational points on supersingular curves.
\section{Introduction}
Let $p$ be a prime and let $q=p^r$ where $r$ is a positive integer.
Let $\mathbb{F}_q$ be the finite field with $q$ elements.
Let $X$ be a projective smooth absolutely irreducible curve of genus $g$
defined over $\mathbb{F}_q$.
The $L$-polynomial of the curve $X$ over $\mathbb F_{q}$ is defined by
$$L_{X/\mathbb{F}_q}(T)=L_X(T)=\exp\left( \sum_{n=1}^\infty ( \#X(\mathbb F_{q^n}) - q^n - 1 )\frac{T^n}{n} \right).$$
where $\#X(\mathbb F_{q^n})$ denotes the number of $\mathbb F_{q^n}$-rational points of $X$.
It is well known that $L_X(T)$ is a polynomial of degree $2g$ with integer coefficients, so we write it as
\begin{equation} \label{L-poly}
L_X(T)= \sum_{i=0}^{2g} c_i T^i, \ c_i \in \mathbb Z.
\end{equation}
It is also well known that $c_0=1$ and $c_{2g}=q^g$.
We wish to consider the question of divisibility of L-polynomials.
In previous papers \cite{chapman:AM}, \cite{chapman:AMR},
we have studied conditions on the curves under which the L-polynomial of
one curve divides the L-polynomial of another curve.
A theorem of Tate gives an answer in terms of Jacobians.
We refer the reader to these papers for a longer discussion of this topic.
Artin-Schreier curves are degree $p$ coverings of the projective line,
and are cyclic extensions of degree $p$ of the rational function field.
It can be shown that all Artin-Schreier curves have an equation of the
form $y^p-y=f(x)$.
Let $k$ be a positive integer. In this paper we will study
the family of Artin-Schreier curves $$C_k^{(p,a)} : y^p-y=x^{p^k+1}+ax$$
where $a\in \F_p$,
which are defined over $\mathbb F_p$
and have genus $p^k(p-1)/2$.
We will prove the following conjecture, which is stated in \cite{conj4}.
\begin{conj}\label{conj}
Let $k$ and $m$ be positive integers. Then the $L$-polynomial of $C_{km}^{(p,a)}$ is divisible by the $L$-polynomial
of $C_k^{(p,a)}$.
\end{conj}
The L-polynomials in the conjecture are over $\F_p$.
This conjecture was proved for $p=2$ in \cite{conj4}, so we will assume that $p$ is odd for this paper.
In Section \ref{a} we explain why we can assume $a=1$ without loss of generality.
We prove this conjecture by finding an exact expression for the number
of $\mathbb F_{p^n}$-rational points on $C_k^{(p)}=C_k^{(p,1)}$,
for any $n$, see Section \ref{sectck}.
This is done by first
finding an exact expression for the number
of $\mathbb F_{p^n}$-rational points on related curves
$B_k^{(p)}$, see Section \ref{sectbk}.
Sections \ref{sectb0} and \ref{sectc0} deal with the curves $B_0^{(p)}$
and $C_0^{(p)}$ respectively, which need separate consideration.
Section \ref{proofconj} gives the proof of Conjecture \ref{conj}.
In Section \ref{Bkmaps} we consider the corresponding divisibility
result for the $B_k$ family.
Section \ref{opp} contains some results on the opposite problem to the conjecture;
we prove that if $k$ does not divide $\ell$ then
the L-polynomial of $C_k^{(p)}$ does not divide the L-polynomial of $C_\ell^{(p)}$.
We will usually drop the superscript in $C_k^{(p)}=C_k^{(p,1)}$, and write $C_k$.
The trace map is always the absolute trace, unless otherwise stated.
Throughout the paper $\left(\frac{\cdot }{p}\right)$ denotes the Legendre symbol.
Finally, the L-polynomials of a more general class of curves
than $C_k$ were found in \cite{Betal}, however they were L-polynomials
over an extension of $\F_p$, and not L-polynomials over $\F_p$, which is the
subject of this paper.
\section{Background}
In this section we will give some basic facts that we will use.
Some of this requires that $p$ is odd, some of it does not,
but we remind that reader that we are going to assume $p$ is
odd for this paper.
\subsection{More on Curves}\label{morec}
Let $p$ be a prime and let $q=p^r$ where $r$ is a positive integer.
Let $X$ be a projective smooth absolutely irreducible curve of genus $g$
defined over $\mathbb{F}_q$. Let $\eta_1,\cdots,\eta_{2g}$ be the roots of the
reciprocal of the $L$-polynomial of $X$ over
$\mathbb F_{q}$ (sometimes called the Weil numbers of $X$, or
Frobenius eigenvalues). Then, for any $n\geq 1$,
the number of rational points of $X$ over $\mathbb F_{q^{n}}$ is
given by
\begin{equation}\label{eqn-sum of roots}
\#X(\mathbb F_{q^{n}})=(q^{n}+1)- \sum\limits_{i=1}^{2g}\eta_i^n.
\end{equation}
The Riemann Hypothesis for curves over finite fields
states that $|\eta_i|=\sqrt{q}$ for all $i=1,\ldots,2g$.
It follows immediately from this property and \eqref{eqn-sum of roots}
that
\begin{equation}
|\#X(\mathbb F_{q^n})-(q^n+1)|\leq 2g\sqrt{q^n}
\end{equation}
which is the Hasse-Weil bound.
We call $X(\mathbb F_{q})$ \emph{maximal} if $\eta_i=-\sqrt{q}$ for all $i=1,\cdots,2g$, so the Hasse-Weil upper bound is met. Equivalently, $X(\mathbb F_{q})$ is maximal
if and only if $L_X(T)=(1+\sqrt{q} T)^{2g}$.
We call $X(\mathbb F_{q})$ \emph{minimal} if $\eta_i=\sqrt{q}$ for all $i=1,\cdots,2g$,
so the Hasse-Weil lower bound is met.
Equivalently, $X(\mathbb F_{q})$ is minimal
if and only if $L_X(T)=(1-\sqrt{q} T)^{2g}$.
Note that if $X(\mathbb F_{q})$ is minimal or maximal then $q$ must be a square
(i.e.\ $r$ must be even).
The following properties follow immediately.
\begin{prop} \label{minimal-prop}
\begin{enumerate}
\item If $X(\mathbb F_{q})$ is maximal then $X(\mathbb F_{q^{n}})$ is minimal for even $n$ and maximal for odd $n$.
\item If $X(\mathbb F_{q})$ is minimal then $X(\mathbb F_{q^{n}})$ is minimal for all $n$.
\end{enumerate}
\end{prop}
We also record another Proposition here.
\begin{prop}\label{pureimag}
If $X$ is a curve defined over $\F_q$ and $X(\mathbb F_{q^{2n}})$ is maximal,
then $\#X(\mathbb F_{q^n})=q^n+1$ and the L-polynomial of $X$ over $\F_{q^n}$
is $(1+q^{n}t^2)^{g}$.
\end{prop}
\begin{proof}
Let $\eta_1,\cdots,\eta_{2g}$ be the Weil numbers of $X$ over $\F_q$.
Then $\eta_j^{2n}=-\sqrt{q^{2n}}=-q^n$ for all $j$ because $X(\mathbb F_{q^{2n}})$ is maximal.
But then $\eta_j^{n}=\pm i \sqrt{q^n}$ for all $j$, which implies that
\[
\sum\limits_{j=1}^{2g}\eta_j^n =(q^{n}+1)-\#X(\mathbb F_{q^{n}})
\]
is a purely imaginary complex number and also an integer.
This number is therefore 0.
\end{proof}
\subsection{Supersingular Curves}\label{supsing}
A curve $X$ of genus $g$ defined over $\mathbb F_q$
($q=p^r$) is \emph{supersingular} if any of the following
equivalent properties hold.
\begin{enumerate}
\item All Weil numbers of $X$ have the form $\eta_i = \sqrt{q}\cdot \zeta_i$ where $\zeta_i$ is a root of unity.
\item The Newton polygon of $X$ is a straight line of slope $1/2$.
\item The Jacobian of $X$ is geometrically isogenous to $E^g$ where
$E$ is a supersingular elliptic curve.
\item If $X$ has $L$-polynomial
$L_X(T)=1+\sum\limits_{i=1}^{2g} c_iT^i$
then
$$ord_p(c_i)\geq \frac{ir}{2}, \ \mbox{for all $i=1,\ldots
,2g$.}$$
\end{enumerate}
By the first property, a supersingular curve defined over $\mathbb F_q$ becomes minimal over some finite extension of $\mathbb F_q$.
Conversely, any minimal or maximal curve is supersingular.
\subsection{Quadratic forms}\label{QF}
We now recall the basic theory of quadratic forms over $\mathbb{F}_{q}$, where $q$ is odd.
Let $K=\mathbb{F}_{q^n}$, and
let $Q:K\longrightarrow \mathbb{F}_{q}$ be a quadratic form.
The polarization of $Q$ is the symplectic bilinear form $B$ defined by $B(x,y)=Q(x+y)-Q(x)-Q(y)$. By definition the radical of $B$ (denoted $W$) is $
W =\{ x\in K : B(x,y)=0 \text{ for all $y\in K$}\}$. The rank of $B$ is defined to be $n-\dim(W)$. The rank of $Q$ is defined to be the rank of $B$.
The following result is well known, see Chapter $6$ of \cite{lidl} for example.
\bigskip
\begin{prop}\label{counts}
Continue the above notation. Let $N=|\{x\in K : Q(x)=0\}|$, and let $w=\dim(W)$. If $Q$ has odd rank then $N=q^{n-1}$; if $Q$ has even rank then $N=q^{n-1}\pm (q-1)q^{(n-2+w)/2}$.
\end{prop}
In this paper we will be concerned with quadratic forms of the type
$Q(x)=\Tr(f(x))$ where $f(x)$ has the form $\sum a_{ij} x^{p^i+p^j}$.
If $N$ is the number of $x\in \mathbb{F}_{p^n}$ with $\Tr(f(x))=0$, then
because elements of trace 0 have the form $y^p-y$, finding $N$ is equivalent
to finding the exact number of $\mathbb{F}_{p^n}$-rational points on
the curve $C: y^p-y=f(x)$.
Indeed,
\begin{equation}\label{quadpts}
\#C(\mathbb F_{p^n})=pN+1.
\end{equation}
\subsection{Discrete Fourier Transform}
In this section we recall the statement of the Discrete Fourier Transform
and its inverse.
\begin{prop}[Inverse Discrete Fourier Transform]
Let $N$ be a positive integer and let $w_N$ be a primitive $N$-th root of unity
over any field where $N$ is invertible. If $$F_n=\sum_{j=0}^{N-1}f_jw_N^{-jn}$$ for $n=0,1\cdots, N-1$ then we have $$f_n=\frac1{N}\sum_{j=0}^{N-1}F_jw_N^{jn}$$ for $n=0,1\cdots, N-1$.
\end{prop}
\subsection{Relations on the Number of Rational Points}
In this section we state a theorem which allows us to find the number of
$\F_{p^n}$-rational points of a supersingular curve by finding the the number of
$\F_{p^m}$-rational points only for the divisors $m$ of $s$, where the Weil numbers
are $\sqrt{p}$ times an $s$-th root of unity. Note that $s$ is even because
equality holds in the Hasse--Weil bound over $\F_{p^s}$.
\begin{thm}[\cite{MY2}]\label{reduction-thm}
Let $p$ be an odd prime.
Let $X$ be a supersingular curve of genus $g$ defined over $\mathbb F_p$
whose Weil numbers are $\sqrt{p}$ times an $s$-th root of unity.
Let $n$ be a positive integer, let $\gcd (n,s)=m$ and write $n=mt$. Then we have\bigskip \\
$-p^{-n/2}[\#X(\F_{p^n})-(p^n+1)]=$
$$\begin{cases}
-p^{-m/2}[\#X(\F_{p^m})-(p^m+1)] &\text{if } m \text{ is even},\\
-p^{-m/2}[\#X(F_{q^m})-(q^m+1)]&\text{if } m \text{ is odd and } p\mid t,\\
-p^{-m/2}[\#X(\F_{p^m})-(p^m+1)]\left(\frac{(-1)^{(t-1)/2}t}{p}\right)&\text{if }
m \text{ is odd and } p\nmid t,
\end{cases}$$ where $\left(\frac{.}{p}\right)$ is the Legendre symbol.
\end{thm}
\subsection{A Divisibilty Theorem}
The following theorem is well-known.
\begin{thm}\label{chapman:KleimanSerre}
(Kleiman--Serre)
If there is a surjective morphism of curves
$C \longrightarrow D$ that is defined over $\mathbb{F}_q$
then $\mathrm{L}_{D}(T)$ divides $\mathrm{L}_C(T)$.
\end{thm}
This theorem is sometimes used to show divisibility.
The $p=2$ case of Conjecture \ref{conj} was proved in \cite{conj4}
by finding a map $C_{km}^{(2)} \longrightarrow C_{k}^{(2)}$.
However, there are
cases where there is no map of curves and yet there is divisibility of
L-polynomials.
We suspect that $C_k^{(p)} $ and $C_{2k}^{(p)} $ is such a case, see Theorem \ref{divisibility-2k}.
We are unable to find a map $C_{2k}^{(p)} \longrightarrow C_{k}^{(p)}$
when $p>2$.
\section{The Curve $B_0: y^p-y=x^2$ over $\mathbb F_p$}\label{sectb0}
From now on in this paper we will assume that $p$ is an odd prime.
Given a bilinear form $B$ we define
$$W^{(n)}:=\{x\in \mathbb F_{p^n} \: | \: B(x,y)=0 \text{ for all } y \in \mathbb F_{p^n} \}.$$
In this section we will give the exact number of $\mathbb{F}_{p^n}$-rational points on
$B_0:y^p-y=x^{2}$ for all positive integers $n$.
Note that $B_0$ has genus $(p-1)/2$.
\begin{lemma}\label{b01}
The number of $\mathbb F_p$-rational points of $B_0$ is $p+1$.
\end{lemma}
\begin{proof}
Since $x^2=0$ if and only if $x=0$ and since $y^p-y=0$ for all $y \in \mathbb F_p$, we have that the number of $\mathbb F_p$-rational points of $B_0$
(including $\infty$) is $p+1$.
\end{proof}
\begin{lemma}\label{radical-lemma}
Let $n$ be a positive integer. The radical of the quadratic form $Q_0(x)=\Tr(x^2)$ is $\{0\} $where $\Tr: \mathbb F_{p^n} \to \mathbb F_p$ is the trace map.
\end{lemma}
\begin{proof}
We have $$B_0(x,y):=Q_0(x+y)-Q_0(x)-Q_0(y)=\Tr(2xy)$$ and
$W^{(n)}=\{0\}$
because $\Tr (xy)$ is a non-degenerate bilinear form.
\end{proof}
\begin{lemma}\label{b02}
The number of $\mathbb F_{p^2}$-rational points of $B_0$ is $$\begin{cases}
p^2+1-(p-1)p &\text{ if } p \equiv 1 \mod 4,\\
p^2+1+(p-1)p &\text{ if } p \equiv 3 \mod 4.\\
\end{cases}$$
\end{lemma}
\begin{proof}
Since $2-\dim(W^{(2)})=2-0=2$ is even by Lemma \ref{radical-lemma},
the $N$ in Proposition \ref{counts} is equal to $p \pm (p-1)$.
By \eqref{quadpts} we get
\[
\#B_0(\mathbb F_{p^2})=pN+1=p^2+1\pm p(p-1).
\]
Because the genus of $B_0$ is $(p-1)/2$ we have $2g\sqrt{p^2}=p(p-1)$ and so
the curve $B_0$ is maximal or minimal over $\mathbb F_{p^2}$
because the Hasse-Weil bound is met.
Let $\Tr:\mathbb F_{p^2}\to \mathbb F_p$ be the trace map. Then $$Tr(x^2)=x^2+x^{2p}=x^2(x^{2p-2}+1).$$
We know that $B_0$ is maximal or minimal over $\mathbb F_{p^2}$. If it is minimal (resp. maximal), then $$|\{ x \in \mathbb F_{q^2} \: | \: Tr(x^2)=0\}|=1 \text{ (resp. $2p-1$)}.$$
In other words, the degree of the greatest common divisor $(x^{2p}+x^2, x^{p^2}-x)$ is $$\begin{cases}
1 &\text{ if } B_0 \text{ is minimal over } \mathbb F_{q^2},\\
2p-1 &\text{ if } B_0 \text{ is maximal over } \mathbb F_{q^2}\\
\end{cases}$$ or the degree of the greatest common divisor $(x^{2p-2}+1, x^{p^2-1}-1)$ is $$\begin{cases}
0 &\text{ if } B_0 \text{ is minimal over } \mathbb F_{q^2},\\
2p-2 &\text{ if } B_0 \text{ is maximal over } \mathbb F_{q^2}.\\
\end{cases}$$
Assume $p \equiv 1 \mod 4$. Then $(p+1)/2$ is a positive odd integer and $$x^{p^2-1}-1=(x^{2p-2})^{(p+1)/2}-1\equiv (-1)^{(p+1)/2}-1=-2 \mod (x^{2p-2}+1)$$ which implies that $x^{2p-2}+1$ does not divide $x^{p^2-1}-1$. Therefore, $$(x^{2p-2}+1, x^{p^2-1}-1)=1.$$
Assume $p \equiv 3 \mod 4$. Then we have that $(p+1)/4$ is a positive integer and $$x^{p^2-1}-1=(x^{4p-4})^{(p+1)/4}-1$$ is divisible by $x^{4p-4}-1$ which equals to $(x^{2p-2}+1)(x^{2p-2}-1)$. Hence $x^{p^2-1}-1$ is divisible by $x^{2p-2}+1$. Therefore, $$(x^{2p-2}+1, x^{p^2-1}-1)=x^{2p-2}+1.$$
\end{proof}
\begin{thm}\label{thm-B0}
Let $p\equiv 1 \mod 4$ and $n\ge 1$ be an integer. Then
$$-p^{-n/2}\left[\#B_0(\mathbb F_{p^n})-(p^n+1)\right]=\begin{cases} 0 &\text{ if $n$ is odd},\\p-1 &\text{ if $n$ is even}. \end{cases}$$ Let $p\equiv 3 \mod 4$ and $n\ge 1$ be an integer. Then
$$-p^{-n/2}\left[\#B_0(\mathbb F_{p^n})-(p^n+1)\right]=\begin{cases} 0 &\text{ if } (4,n)=1,\\-(p-1) &\text{ if } (4,n)=2,\\p-1 &\text{ if } (4,n)=4. \end{cases}$$
\end{thm}
\begin{proof}
It follows by Lemma \ref{b01}, \ref{b02} and Theorem \ref{reduction-thm}.
\end{proof}
\section{The Curve $C_0: y^p-y=x^2+x$ over $\mathbb F_p$}\label{sectc0}
In this section we will give the exact number of $\mathbb{F}_{p^n}$-rational points on $C_0: y^p-y=x^2+x$ for all positive integer $n$.
Let $n \ge 1$ be a positive integer. The map $(x, y)\to (x-2^{-1},y)$ is a one-to-one map over $\mathbb F_{p^n}^2$. Let $\Tr:\mathbb F_{p^n}\to \mathbb F_p$ be the trace map. Since $$\Tr\left( (x-2^{-1})^2+x\right)=\Tr\left( x^2+4^{-1}\right)=\Tr(x^2)+n4^{-1},$$ we can use the information on the curve $B_0$.
\begin{lemma}\label{C0p}
Let $n$ be a positive integer. The number of $\mathbb F_{p^{pn}}$-rational points of $C_0$ equals the number of $\mathbb F_{p^{pn}}$-rational points of $B_0$.
\end{lemma}
\begin{proof}
Let $\Tr:\mathbb F_{p^{pn}}\to \mathbb F_p$ be the trace map. Since $p\cdot n$ is divisible by $p$, we have $$Tr(x^2+x)=Tr(x^2).$$ Hence we have the result.
\end{proof}
\begin{lemma}\label{C0B0np1}
Let $n$ be a positive integer with $(n,p)=1$. If $\#B_0(\mathbb F_{p^n})-(p^n+1)\not= 0$
then
$$-(p-1)\bigg(\#C_0(\mathbb F_{p^n})-(p^n+1)\bigg)=
\bigg( \#B_0(\mathbb F_{p^n})-(p^n+1)\bigg).$$
\end{lemma}
\begin{proof}
The proof of this lemma is exactly the same as that of Lemma \ref{evenspread}.
\end{proof}
\begin{lemma}\label{C01}
The number $\#C_0(\mathbb F_p)$ is $2p+1$.
\end{lemma}
\begin{proof}
We have $y^p-y=0$ for all $y \in \mathbb F_p$. Also
$x^2+x=x(x+1)=0$ if and only if $x=0$ or $x=-1$.
Therefore, $\#C_0(\mathbb F_p)=2\cdot p+1$.
\end{proof}
\begin{lemma}\label{C0np1}
$C_0(\mathbb F_{p^{2p}})$ is minimal if $p\equiv 1 \mod 4$ and maximal $p\equiv 3 \mod 4$.
\end{lemma}
\begin{proof}
By Lemma \ref{C0p} we know that $\#C_0(\mathbb F_{p^{2p}})=\#B_0(\mathbb F_{p^{2p}})$. Hence it follows by Theorem \ref{thm-B0}.
\end{proof}
We put all these results together in the final Theorem of this section.
\begin{thm}
Let $p\equiv 1 \mod 4$ and $n\ge 1$ be an integer. Then we have that$$-p^{-n/2}\left[\#C_0(\mathbb F_{p^n})-(p^n+1)\right]=\begin{cases} -\left(\frac{n}{p}\right) \sqrt{p}&\text{ if } (n,2p)=1,\\ -1 &\text{ if } (n,2p)=2,\\ 0 &\text{ if } (n,2p)=p,\\ p-1 &\text{ if } (n,2p)=2p.\end{cases}$$
Let $p\equiv 3 \mod 4$ and $n\ge 1$ be an integer. Then we have that $$-p^{-n/2}\left[\#C_0(\mathbb F_{p^n})-(p^n+1)\right]=\begin{cases} -\left(\frac{(-1)^{(n-1)/2}n}{p}\right) \sqrt{p}&\text{ if } (n,4p)=1,\\ 1 &\text{ if } (n,4p)=2,\\ -1 &\text{ if } (n,4p)=4,\\ 0 &\text{ if } (n,4p)=p,\\ -(p-1) &\text{ if } (n,4p)=2p,\\p-1 &\text{ if } (n,4p)=4p.\end{cases}$$
\end{thm}
\begin{proof}
It follows by Lemmas \ref{C0p}, \ref{C0B0np1}, \ref{C01}, \ref{C0np1} and Theorems \ref{thm-B0} and \ref{reduction-thm}.
\end{proof}
\section{The Curve $B_k: y^p-y=x^{p^k+1}$ over $\mathbb F_p$}\label{sectbk}
In this section we will give the exact number of $\mathbb{F}_{p^n}$-rational points on
$B_k^{(p)}=B_k: y^p-y=x^{p^k+1}$ for all positive integers $k$ and $n$.
\begin{lemma}
Let $d\mid k$. The number of $\mathbb F_{p^d}$-rational points of $B_k$ is
equal to the number of $\mathbb F_{p^d}$-rational points of $B_0$.
\end{lemma}
\begin{proof}
Since $x^{p^k+1}=x^{p^k}\cdot x=x\cdot x=x^2$ in $\mathbb F_{p^d}$ for all $d\mid k$, the result is immediate.
\end{proof}
\begin{lemma}
Let $d\mid k$ with $2d \nmid k$. The number of $\mathbb F_{p^{2d}}$-rational points of $B_k$ is $(p^{2d}+1)-(p-1)p^{d}$.
\end{lemma}
\begin{proof}
Since $d\mid k$ and $2d\nmid k$, we have $e:= k/d$ is odd. Define
$\Tr_n: \mathbb F_{p^n} \to \F_p$ be the trace map.
We have \begin{align*}
\Tr_{2d}(x^{p^k+1})&=\Tr_d(x^{p^k+1}+x^{p^{k+d}+p^d})\\ &=\Tr_d(x^{p^k+1}+x^{p^{ed+d}+p^d})\\&=\Tr_d(x^{p^{ed}+1}+x^{p^{d(e+1)}+p^d})\\&=\Tr_d(x^{p^d+1}+x^{p^{d(e+1)}+p^d})\\&=\Tr_d(x^{p^d+1}+x^{1+p^d})\\&=\Tr_d(2x^{p^d+1}).
\end{align*}
Since $x\to x^{p^d+1}$ is $p^d+1$-to-$1$ map from $\mathbb F_{p^{2d}}^\times$ to $\mathbb F_{p^d}^\times$ and since $\Tr_d(x)$ is a linear map from $\mathbb F_{p^{d}}$ to $\mathbb F_p$, we have that the number of $\mathbb F_{p^{2d}}$-rational points of $B_k$ is $$1+p(1+(p^d+1)(p^{d-1}-1))=(p^{2d}+1)-(p-1)p^{d}.$$
\end{proof}
\begin{lemma}
The curve $B_k$ is minimal over $\mathbb F_{p^{4k}}$.
\end{lemma}
\begin{proof}
Define $Q_k(x)=\Tr(x^{p^k+1})$ where $\Tr: \mathbb F_{p^n} \to \mathbb F_p$ is the trace map ($n=4k$).
We have $$B(x,y):=Q(x+y)-Q(x)-Q(y)=\Tr(x^{p^k}y+xy^{p^k})=\Tr(y^{p^{k}}(x^{p^{2k}}+x))$$ and $$W^{(n)}:=\{x\in \mathbb F_{p^n} \: | \: B(x,y)=0 \text{ for all } y \in \mathbb F_{p^n} \}=\{x\in \mathbb F_{p^n} \: | \:x^{p^{2k}}+x=0 \}.$$
So $W^{(4k)} \subseteq \F_{p^{4k}}$ and so the rank of $Q_k$ is
$n-\dim W^{(n)}=4k-2k$ which is even, and so the $N$ in Proposition \ref{counts} is equal to $p^{n-1} \pm (p-1)p^{3k-1}$.
By \eqref{quadpts} we get
\[
\#B_0(\mathbb F_{p^{4k}})=pN+1=p^n+1\pm (p-1)p^{3k}.
\]
The genus of $B_k$ is $p^k(p-1)/2$ so $2g\sqrt{p^{4k}}=p^{3k}(p-1)$, and so
the curve $B_k$ is maximal or minimal over $\mathbb F_{p^{4k}}$
because the Hasse-Weil bound is met.
If the curve $B_k$ is maximal over $\mathbb F_{p^{4k}}$, then $\#B_k(\mathbb F_{p^{2k}})$ has to be $p^{2k}+1$ by Proposition \ref{pureimag}.
However, $W^{(2k)}=\{0\}$ and so $2k-\dim W^{(2k)}=2k-0=2k$ is even, which means that
$\#B_k(\mathbb F_{p^{2k}})$ cannot be $p^{2k}+1$ by Proposition \ref{counts}.
Hence the curve $B_k$ is minimal over $\mathbb F_{p^{4k}}$.
\end{proof}
\begin{cor}\label{bkperiod}
We have $\zeta^{4k}=1$ for all $\zeta$ where $\sqrt{q} \zeta$ is a
Weil number of $B_k$.
\end{cor}
\begin{proof}
We have shown that $B_k$ is minimal over $\mathbb F_{p^{4k}}$,
and it follows from Sections \ref{morec} and \ref{supsing}.
\end{proof}
\begin{lemma}
Let $d\mid 4k$ with $d \nmid 2k$. The number of $\mathbb F_{p^{d}}$-rational points of $B_k$ is $(p^d+1)-(p-1)p^{3d/4}$.
\end{lemma}
\begin{proof}
Since $d\mid 4k$ and $d\nmid 2k$, we have $e:= 4k/d$ is odd and $d=4f$ for some integer $f$. Define $\Tr_n: \mathbb F_{p^n} \to F_p$ be the trace map.
We have \begin{align*}
\Tr_{d}(x^{p^k+1})&=\Tr_{f}(x^{p^k+1}+x^{p^{k+f}+p^f}+x^{p^{k+2f}+p^{2f}}+x^{p^{k+3f}+p^{3f}})\\ &=\Tr_{f}(x^{p^{ef}+1}+x^{p^{f(e+1)}+p^f}+x^{p^{f(e+2)}+p^{2f}}+x^{p^{f(e+3)}+p^{3f}})\\&=\Tr_{f}(x^{p^f+1}+x^{p^{2f}+p^f}+x^{p^{3f}+p^{2f}}++x^{p^{4f}+p^{3f}})\\&=\Tr_d(x^{p^f+1}).
\end{align*}
Since $4f=d \mid d$, the curve $B_f$ is minimal over $\mathbb F_{p^d}$ and hence The number of $\mathbb F_{p^{d}}$-rational points of $B_k$ is $$(p^d+1)-(p-1)p^{3d/4}.$$
\end{proof}
\begin{cor}\label{cor-bk}
If $d \mid k$ and $d$ is odd, then $$\#B_k(\mathbb F_{p^d})=p^d+1.$$ If $d \mid k$ and $d$ is even, then $$\#B_k(\mathbb F_{p^d})=\begin{cases}(p^d+1)+(p-1)p^{d/2} &\text{ if } 2 \mid\mid d \text{ and } p\equiv 3 \mod 4, \\
(p^d+1)-(p-1)p^{d/2} &\text{ if } 4 \mid d \text{ or } p\equiv 1 \mod 4.\end{cases}$$ If $d\nmid k$ and $\frac d2\mid k$, then $$\#B_k(\mathbb F_{p^d})=p^d+1-(p-1)p^{d/2}.$$ If $d\nmid 2k$ and $d \mid 4k$, then $$\#B_k(\mathbb F_{p^d})=(p^d+1)-(p-1)p^{3d/4}.$$
\end{cor}
We put all these results together.
\begin{thm}
Let $n\ge 1$ be an integer and let $d=(n,4k)$. Then we have that$$-p^{-n/2}\left[\#B_k(\mathbb F_{p^n})-(p^n+1)\right]=\begin{cases} 0&\text{ if } d\mid k \text{ and $d$ is odd},\\ (-1)^{n(p-1)/4}(p-1) &\text{ if } d\mid k \text{ and $d$ is even},\\ -(p-1)&\text{ if } d\nmid k \text{ and } \frac d2\mid k,\\ (p-1)p^{(k,n)} &\text{ if } d\nmid 2k \text{ and } d\mid 4k.\end{cases}$$
\end{thm}
\begin{proof}
It follows by Corollary \ref{cor-bk} and Theorem \ref{reduction-thm}.
\end{proof}
\section{The Curve $C_k: y^p-y=x^{p^k+1}+x$ over $\mathbb F_p$}\label{sectck}
In this section we will give the exact number of $\mathbb{F}_{p^n}$-rational points on
$C_k: y^p-y=x^{p^k+1}+x$ for all positive integers $k$ and $n$.
Let $(x, y)\to (x-2^{-1},y)$ is a one-to-one map over $\mathbb F_{p^n}^2$. Since
\begin{align}
\Tr\left( (x-2^{-1})^{p^k+1}+(x-2^{-1})\right)&=
\Tr\left( x^{p^k+1}-2^{-1}x^{p^k}+2^{-1}x-4^{-1}\right)\\
&=\Tr(x^{p^k+1})-n4^{-1}, \label{onetoone}
\end{align}
we can use the information on the curve $B_k$.
\begin{lemma}\label{isooverp}
If $p|n$, the number of $\mathbb F_{p^{n}}$-rational points of $C_k$ equals the
number of $\mathbb F_{p^{n}}$-rational points of $B_k$.
\end{lemma}
\begin{proof}
Since $n$ is divisible by $p$,
by \eqref{onetoone}
we have $$\mid \{ x\in \mathbb F_{p^{n}} : Tr(x^{p^k+1}+x)=0 \} \mid =
\mid \{ x\in \mathbb F_{p^{n}} : Tr(x^{p^k+1})=0 \}.$$ Hence we have the result.
\end{proof}
\begin{cor}\label{ckperiod1}
If $p | k$ we have $\zeta^{4k}=1$ for all $\zeta$ where $\sqrt{q} \zeta$ is a
Weil number of $C_k$.
\end{cor}
\begin{proof}
Follows from Lemma \ref{isooverp} and Corollary \ref{bkperiod}.
\end{proof}
{\bf Remark}.
It follows from Lemma \ref{isooverp} that $C_k$ and $B_k$ have the same
L-polynomial when considered as curves defined over $\F_{p^p}$.
They do not have the same L-polynomial when considered as curves defined over $\F_{p}$,
as the results in this paper show (see Lemma \ref{evenspread} below).
This means that the $p$-th powers of the Weil numbers of $B_k$ and $C_k$
(considered as curves defined over $\F_{p}$) are equal,
but the Weil numbers themselves are not the same.
For example, the L-polynomial of $B_2^{(3)}$ is
\[
19683T^{18} + 6561T^{16} - 486T^{10} - 162T^8 + 3T^2 + 1
\]
and the L-polynomial of $C_2^{(3)}$ is
\[
19683T^{18} - 19683T^{17} + 6561T^{16} + 243T^{10} - 243T^9 + 81T^8 + 3T^2 - 3T+ 1.
\]
For both of these, the polynomial whose roots are the cubes of the roots is
\[
1 + 27T^2 - 1062882T^8 - 28697814T^{10} + 282429536481T^{16} +
7625597484987T^{18}
\]
which is the L-polynomial of both $B_2$ and $C_2$ considered as curves defined over $\F_{3^3}$.
\begin{lemma}
Let $d\mid k$. The number of $\mathbb F_{p^d}$-rational points of $C_k$ is the number of $\mathbb F_{p^d}$-rational points of $C_0$.
\end{lemma}
\begin{proof}
Since $$x^{p^k+1}+x=x^{p^k}\cdot x+x=x\cdot x+x=x^2+x$$ in $\mathbb F_{p^d}$ for all $d\mid k$, the result is immediate.
\end{proof}
\begin{lemma}\label{evenspread}
Let $n$ be a positive integer with $(n,p)=1$. If $\#B_k(\mathbb F_{p^n})-(p^n+1)\not= 0$
then
$$-(p-1)\bigg(\#C_k(\mathbb F_{p^n})-(p^n+1)\bigg)=
\bigg( \#B_k(\mathbb F_{p^n})-(p^n+1)\bigg).$$
\end{lemma}
\begin{proof}
Let $b:=n4^{-1}\ne 0$, let
\[
N_0=| \{ x \in \F_{p^n} : \Tr(x^{p^k+1})=0 \} |
\]
and let
\[
N_1=| \{ x \in \F_{p^n} : \Tr(x^{p^k+1})=b \} |.
\]
By \eqref{onetoone} and also \eqref{quadpts}
we get $\#C_k(\mathbb F_{p^n}) = pN_1+1$.
So
\begin{equation}\label{ckn1}
\#C_k(\mathbb F_{p^n})-(p^n+1)=pN_1-p^n.
\end{equation}
The nonzero values of the quadratic form $\Tr(x^{p^k+1})$ are evenly distributed over the nonzero elements of $\F_p$ by \cite[Theorem 6.26]{lidl},
so \[ N_0+(p-1)N_1=p^n. \]
Substituting for $N_1$ into \eqref{ckn1} gives
\[
\#C_k(\mathbb F_{p^n})-(p^n+1)=\frac{p^{n+1}-pN_0}{p-1}-p^n
\]
or
\[
(p-1)\bigg( \#C_k(\mathbb F_{p^n})-(p^n+1)\bigg)
=p^{n}-pN_0.
\]
Finally, by \eqref{quadpts} again we note that
\[
\#B_k(\mathbb F_{p^n})-(p^n+1)=pN_0-p^n.
\]
\end{proof}
\begin{cor}\label{ckperiod2}
If $p \nmid k$ we have $\zeta^{4kp}=1$ for all $\zeta$ where $\sqrt{q} \zeta$ is a
Weil number of $C_k$.
\end{cor}
\begin{proof}
The previous lemma shows that that $C_k$ is minimal over $\mathbb F_{p^{4kp}}$,
and it follows from Sections \ref{morec} and \ref{supsing}.
\end{proof}
\begin{cor}\label{cor-Ck}
Let $k$ be a positive integer. Define
$$l=\begin{cases}
k &\text{if }p\mid k,\\
kp &\text{if }p\nmid k.
\end{cases}$$ \\ If $d \mid l$ and $d$ is odd and relatively prime to $p$, then
$$\#C(\mathbb F_{p^d})=p^d+1+\left(\frac{(-1)^{(d-1)/2}d}{p}\right)p^{(d+1)/2}.$$
If $d \mid l $ and $d$ is odd and divisible by $p$, then $$\#C_k(\mathbb F_{p^d})=0.$$ If $d \mid l $ and $d$ is even and relatively prime to $p$, then $$\#C_k(\mathbb F_{p^d})=\begin{cases}(p^d+1)-p^{d/2} &\text{ if } 2 \mid\mid d \text{ and } p\equiv 3 \mod 4, \\
(p^d+1)+p^{d/2} &\text{ if } 4 \mid d \text{ or } p\equiv 1 \mod 4.\end{cases}$$ If $d \mid l $ and $d$ is even and divisible by $p$, then $$\#C_k(\mathbb F_{p^d})=\begin{cases}(p^d+1)+(p-1)p^{d/2} &\text{ if } 2 \mid\mid d \text{ and } p\equiv 3 \mod 4, \\
(p^d+1)-(p-1)p^{d/2} &\text{ if } 4 \mid d \text{ or } p\equiv 1 \mod 4.\end{cases}$$ If $d\nmid l$ and $\frac{d}{2} \mid l$ and $d$ is relatively prime to $p$, then $$C_k(\mathbb F_{p^d})=(p^d+1)+p^{d/2}.$$ If $d\nmid l$ and $\frac{d}{2} \mid l$ and $d$ is relatively prime to $p$, then $$C_k(\mathbb F_{p^d})=(p^d+1)-(p-1)p^{d/2}.$$ If $d\nmid 2l$, $d \mid 4l$ and $d$ is relatively prime to $p$, then $$\#C_k(\mathbb F_{p^d})=(p^d+1)+p^{3d/4}.$$ If $d\nmid 2l$, $d \mid 4l$ and $d$ is divisible by $p$, then $$\#C_k(\mathbb F_{p^d})=(p^d+1)-(p-1)p^{3d/4}.$$
\end{cor}
\begin{thm}\label{finalck}
Let $k$ be a positive integer. Define
$$l=\begin{cases}
k &\text{if }p\mid k,\\
kp &\text{if }p\nmid k.
\end{cases}$$
Let $n\ge 1$ be an integer with $d=(n,4l)$. Then we have
$$-p^{-n/2}[\#C_k(\mathbb F_{p^n})-(p^n+1)]=\begin{cases}
-\left(\frac{(-1)^{(n-1)/2}n}{p}\right) \sqrt{p}&\text{ if } d \mid l \text{ and $n$ is odd and $p\nmid n$} ,\\ 0 &\text{ if } d \mid l \text{ and $n$ is odd and $p\mid n$},\\ -(-1)^{n(p-1)/4} &\text{ if } d \mid l \text{ and $n$ is even and $p\nmid n$},\\
(-1)^{n(p-1)/4}(p-1) &\text{ if } d \mid l \text{ and $n$ is even and $p\mid n$},\\ -1 &\text{ if } \text{ $d\nmid l$ and $\frac{d}{2} \mid l$ and $p\nmid n$},\\ p-1 &\text{ if } \text{ $d\nmid l$ and $\frac{d}{2} \mid l$ and $p\mid n$}, \\-p^{(k,n)} &\text{ if } d \nmid 2l \text{ and } d \mid 4l \text{ and } p \nmid n,\\(p-1)p^{(k,n)} &\text{ if } d \nmid 2l \text{ and } d \mid 4l \text{ and } p \mid n.
\end{cases}$$
\end{thm}
\begin{proof}
It follows by Corollary \ref{cor-Ck} and Theorem \ref{reduction-thm}.
\end{proof}
\section{Divisibility Property of the Curves $C_k$}\label{proofconj}
In this section we will prove Conjecture \ref{conj}.
The proof will be broken into a few parts.
The first part is to show that the L-polynomial of $C_k$
divides the L-polynomial of $C_{2k}$.
The next part is to show that the L-polynomial of $C_k$
divides the L-polynomial of $C_{tk}$ where $t$ is odd.
Finally, these results are combined to prove the conjecture.
\begin{lemma}\label{divisibility-lemma-un} Let $k$ be a positive integer. Define
$$s=\begin{cases}
8k &\text{if }p\mid k,\\
8kp &\text{if }p\nmid k.
\end{cases}$$ \\
For $n\geq 1$ define
$$U_n=-p^{-n/2}[\#C_{2k}(\mathbb F_{p^n})-\#C_{k}(\mathbb F_{p^n})]$$and
write $U_n$ as a linear combination
of the $s$-th roots of unity as
$$U_n=\sum_{j=0}^{s-1}u_jw_{s}^{-jn}.$$
Then we have $$u_n \ge 0$$ for all $n \in \{ 0,1,\cdots, s-1\}$.
\end{lemma}
\begin{proof} Let $U_0=U_s$.
Write $k=2^vt$ where $v$ is a positive integer and $t$ is an odd integer.
By Theorem \ref{finalck} we have $$U_n=\begin{cases}
0 &\text{if } 2^{v+1} \nmid n, \\
-1+(-1)^{n(p-1)/4} &\text{if } 2^{v+1} \mid \mid n \text{ and } p \nmid n,\\
(p-1)(1-(-1)^{n(p-1)/4})) &\text{if } 2^{v+1} \mid \mid n \text{ and } p \mid n,\\
p^{(k,n)}-1 &\text{if } 2^{v+2} \mid \mid n \text{ and } p \nmid n,\\
-(p-1)(p^{(k,n)}-1) &\text{if } 2^{v+2} \mid \mid n \text{ and } p \mid n,\\
-(p^{(2k,n)}-p^{(k,n)}) &\text{if } 2^{v+3} \mid n \text{ and } p \nmid n,\\
(p-1)(p^{(2k,n)}-p^{(k,n)}) &\text{if } 2^{v+3} \mid n \text{ and } p \mid n.
\end{cases}$$
If $k$ is divisible by $p$, by using Inverse Discrete Fourier Transform we have
\begin{align*}
u_{n}&=\frac{1}{8k}\sum_{j=0}^{8k-1}U_jw_{8k}^{jn}\\
&=\frac{1}{8k}\sum_{j=0}^{4t-1}U_{2^{v+1}j}w_{2t}^{jn} \quad \textrm{because
$U_n=0$ if $2^{v+1} \nmid n$}\\
&\ge \frac{1}{8k}\left(U_0- \sum_{j=1}^{4t-1}|U_{2^{v+2}j}| \right) \quad \textrm{by the
triangle inequality}\\
&\ge \frac{1}{8k}\left[(p-1)(p^{2k}-p^{k})-(4t-1)(p-1)p^{k}\right] \quad \textrm{because
$U_s=U_0=(p-1)(p^{2k}-p^k)$}\\
& \qquad \qquad \qquad \qquad \qquad \qquad\qquad \qquad \qquad \qquad \textrm{ and all others are $\le p^k(p-1)$} \\
&= \frac{1}{8k}(p-1)p^{k}(p^k-4t)\\&\ge \frac{1}{8k}(p-1)p^{k}(p^k-4k)\\ &\ge 0.
\end{align*}
If $k$ and $n$ are not divisible by $p$, by using the
Inverse Discrete Fourier Transform we have
\begin{align*}
u_{n}&=\frac{1}{8kp}\sum_{j=0}^{8kp-1}U_jw_{8kp}^{jn}\\
&=\frac{1}{8kp}\sum_{j=0}^{4tp-1}U_{2^{v+1}j}w_{2tp}^{jn} \\
&\geq \frac{1}{8kp}\left( U_0+\sum_{j=1}^{p-1}U_{2^{v+3}tj}w_p^j-\sum_{j=0,2t\nmid j}^{2tp-1}|U_{2^{v+2}j}|- \sum_{j=0}^{2tp-1}|U_{2^{v+1}(2j+1)}|\right).
\end{align*}
We have $$|U_{2^{v+1}(2j+1)}| \le \begin{cases}
2 &\text{ if } p\nmid (2j+1), \\
2(p-1) &\text{ if } p\mid (2j+1). \\
\end{cases}$$
Since there are $2t$ (resp. $2t(p-1)$) integers which is (resp. not) divisible by $p$ between $0$ and $2tp-1$, we have $$\sum_{j=0}^{2tp-1}|U_{2^{v+1}(2j+1)}|\le 2t\cdot 2(p-1)+2t(p-1)\cdot 2=8t(p-1).$$
Therefore, we have\begin{align*}
u_n &\ge \frac{1}{8kp}\left[(p-1)(p^{2k}-p^{k})+(p^{2k}-p^{k})-(2tp-p)(p-1)p^{k}-8t(p-1)\right] \\
&\ge \frac{1}{8k}p^{k}(p^k-(p-1)(2t-1))-\frac{p-1}{p}\\&> -1.
\end{align*}
Since $u_n$ is an integer, we have $u_n\ge 0$.\\
Assume $k$ is not divisible by $p$ and $n$ is divisible by $p$ and write $n=mp$.
We will show that $u_n=0$.
By the Inverse Discrete Fourier Transform we have
\begin{align*}
u_{n}&=\frac{1}{8kp}\sum_{j=0}^{8kp-1}U_jw_{8kp}^{jn}\\
&=\frac{1}{8kp}\sum_{j=0}^{4tp-1}U_{2^{v+1}j}w_{4t}^{jm}\\
&=\frac{1}{8kp}\sum_{j=0}^{4t-1}\left[
\left(\sum_{i=0}^{p-1}U_{2^{v+1}(4ti+j)}\right)w_{2t}^{jm}\right].
\end{align*}
Since $(4t,p)=1$, for any integer $j$ we have
$$\{ 4ti+j \mod p \: |\: 0 \le i\le p-1\}=\{i \mod p\: | \: 0 \le i\le p-1 \}$$
and so exactly one of the $4ti+j$ is divisible by $p$.
Therefore, for each $j$,
\[
\sum_{i=0}^{p-1}U_{2^{v+2}(2ti+j)}=0
\]
because if $j$ is odd then one term is $(p-1)(1-(-1)^{n(p-1)/4})$ and
the other $p-1$ terms are $-1+(-1)^{n(p-1)/4}$, if $2\mid \mid j$ then one term is $-(p-1)(p^{(k,n)}-1)$ and
the other $p-1$ terms are $p^{(k,n)}-1$,
and if $4 \mid j$ is even then one term is $(p-1)(p^{(2k,n)}-p^{(k,n)})$ and
the other $p-1$ terms are $-(p^{(2k,n)}-p^{(k,n)})$.
\end{proof}
We write $L(C_k)$ for $L_{C_k}$.
\begin{cor}\label{divisibility-2k}
Let $k$ be a positive integer. Then $$L(C_k) \mid L(C_{2k}).$$
\end{cor}
\begin{proof}
Lemma \ref{divisibility-lemma-un} shows that the multiplicity of each root of $L(C_k)$ is smaller than or equal to
its multiplicity as a root of $L(C_{2k})$.
\end{proof}
\begin{lemma}\label{divisiblity-odd}
Let $k$ be an integer and $t$ be an odd integer. Then $$L(C_k)\mid L(C_{kt}).$$
\end{lemma}
\begin{proof}
Let $X_k: y^p-y=x^{p^k+1}-4^{-1}$ over $\mathbb F_{p}$ and check that
\[
(x,y) \mapsto \left(x-\frac12, y-\frac12\sum_{i=0}^{k-1}x^{p^{i}}\right)
\]
is a map $X_k \longrightarrow C_k$.
The map is defined everywhere and is invertible, so $C_k$ is isomorphic to $X_k$, and hence
$L(C_k)=L(X_k).$ Therefore, it is enough to show that $L(X_k) \mid L(X_{kt})$.
Since $t$ is odd, $p^k+1$ divides $p^{kt}+1$ and therefore there is a map
of curves $X_{kt} \longrightarrow X_k$ given by
$(x,y)\to (x^{(p^{kt}+1)/(p^k+1)},y)$. Hence
$L(X_k) \mid L(X_{kt})$ by Theorem \ref{chapman:KleimanSerre}.
\end{proof}
\begin{thm}\label{wholeconj}
Let $k$ and $m$ be positive integers. Then $$L(C_k) \mid L(C_{km}).$$
\end{thm}
\begin{proof}
If $m=1$, then the result is trivial. Assume $m\ge 2$ and write $m=2^st$ where $t$ is odd. Since $t$ is odd, by Lemma \ref{divisiblity-odd} we have $$L(C_k)\mid L(C_{kt})$$ and by Corollary \ref{divisibility-2k} we have $$L(C_{2^{i-1}kt})\mid L(C_{2^{i}kt})$$ for all $i\in\{1,\cdots,s\}$. Hence $$L(C_k) \mid L(C_{km}).$$
\end{proof}
\section{Remark on the Divisibility Property of the Curves $B_k$}\label{Bkmaps}
The divisibility property of the curves $B_k$ can be proved in the same way as for $C_k$.
For odd $t$, we have a natural map from $B_{tk}$ to $B_{k}$ which sends $(x,y)$ to
$(x^{(p^{kt}+1)/(p^k+1)},y)$.
We are unable to find a map from $B_{2k}$ to $B_k$, so we use a similar argument.
For $n\ge 1$ we can define
$$U_n=-p^{-n/2}[\#B_{2k}(\mathbb F_{p^n})-\#B_{k}(\mathbb F_{p^n})]=\begin{cases}
0 &\text{if } 2^{v+1} \nmid n, \\
(p-1)(1-(-1)^{n(p-1)/4})) &\text{if } 2^{v+1} \mid \mid n \text{ and } p \mid n,\\
-(p-1)(p^{(k,n)}-1) &\text{if } 2^{v+2} \mid \mid n \text{ and } p \mid n,\\
(p-1)(p^{(2k,n)}-p^{(k,n)}) &\text{if } 2^{v+3} \mid n \text{ and } p \mid n.
\end{cases}$$ and can similarly show that $u_j\ge 0$ for all $j\in\{0,\cdots,4k-1\}$ where
$$U_n=\sum_{j=0}^{4k-1}u_jw_{4k}^{-jn}.$$
We write $L(B_k)$ for $L_{B_k}$.
\begin{thm}
Let $k$ and $m$ be positive integers. Then $$L(B_k) \mid L(B_{km}).$$
\end{thm}
The proof is similar to the proof of Theorem \ref{wholeconj}.
\section{Opposite Direction}\label{opp}
In this section, we will prove that the opposite directions of the divisibility theorems for $B_k$ and $C_k$ are also valid.
To be precise, we have shown that if $k$ divides $\ell$ then the L-polynomial
of $C_k$ (or $B_k$) divides the L-polynomial of $C_\ell$ (or $B_\ell$).
We now prove that if $k$ does not divide $\ell$ then the L-polynomials do not divide.
Let $X$ be a supersingular curve defined over $\F_q$.
The smallest positive integer $s=s_X$ such that $\zeta_i^{s}=1$ for all
$i=1,\ldots ,2g$ will be called the \emph{period} of $X$.
The period depends on $q$, in the sense that $X(\F_{q^n})$
may have a different period to $X(\F_{q})$.
\begin{prop}\label{divisiblity-period}
Let $C$ and $D$ be supersingular curves over $\mathbb F_q$. If $L(C)$ divides $L(D)$, then $s_C$ divides $s_D$.
\end{prop}
\begin{proof}
Since $L(C)$ divides $L(D)$, the roots of $L(C)$ are also roots of $L(D)$. Therefore, any Weil number of $C$ is also a Weil number of $D$. Let $\sqrt{q}\zeta$ be a Weil number for $C$. Since $\zeta^{s_D}=1$, the order of $\zeta$ divides $s_D$. Since this happens for all Weil numbers of $C$, $s_C$ divides $s_D$.
\end{proof}
\begin{thm}\label{Bknotdiv}
Let $k$ and $\ell $ be positive integers such that $k$ does not divide $\ell$.
Then $L(B_k)$ does not divide $L(B_{\ell })$.
\end{thm}
\begin{proof}
By Corollary \ref{bkperiod}
the period of $B_k$ is $4k$ and the period of $B_\ell$ is $4\ell$. Since $4k$ does not divide $4\ell$, we have $L(B_k)$ does not divide $L(B_\ell)$ by Proposition \ref{divisiblity-period}.
\end{proof}
\begin{cor}
Let $k$ and $\ell $ be positive integers such that $k<\ell $ and $k$ does not divide $\ell$.
Then there is no map from $B_\ell$ to $B_k$.
\end{cor}
\begin{proof}
By the Kleiman-Serre theorem (Theorem \ref{chapman:KleimanSerre}) and Theorem \ref{Bknotdiv}.
\end{proof}
Now we turn to $C_k$.
\begin{lemma}\label{Cp-nmid-Cl}
Let $\ell$ be a positive integer coprime to $p$. Then $L(C_p)$ does not divide $L(C_{\ell})$.
\end{lemma}
\begin{proof}
We will check the multiplicities of $T-\sqrt{p}$ in $L_{C_p}(T)$ and $L_{C_\ell}(T)$.
Using the Inverse Fourier Transform, the multiplicity of $T-\sqrt{p}$ in $L_{C_p}(T)$ is $$\frac{1}{4p}\sum_{j=1}^{4p}(p^{-j/2}[\#C_p(\mathbb F_{p^j})-(p^j+1)])\ge\frac{1}{4p} \left[(p-1)p^p-(4p-1)p\right] >0$$ by Theorem \ref{finalck} and triangle inequality.
Using the Inverse Fourier Transform, the multiplicity of $T-\sqrt{p}$ in $L_{C_\ell}(T)$ is $$\frac{1}{4p\ell}\sum_{j=1}^{4p\ell}(p^{-j/2}[\#C_\ell(\mathbb F_{p^j})-(p^j+1)])$$
which is 0 because $$|\{k \in \mathbb Z \ | \ (4p\ell,k)=d \ \text{ and } \ 1 \le j \le 4p\ell \}|=(p-1)\cdot |\{k \in \mathbb Z \ | \ (4p\ell,k)=dp \ \text{ and } \ 1 \le j \le 4p\ell \}|$$ for any $d \mid l$ and by Theorem \ref{finalck}.
\end{proof}
\begin{thm}\label{Cknotdiv}
Let $k$ and $\ell $ be positive integers such that $k$ does not divide $\ell$.
Then $L(C_k)$ does not divide $L(C_{\ell })$.
\end{thm}
\begin{proof}
We use Corollaries $\ref{ckperiod1}$ and $\ref{ckperiod2}$ which give the
period of $C_k$.
Case I: If $p \mid k,l$, then the period of $C_k$ is $4k$ and the period of $C_\ell$ is $4\ell$. Since $4k \nmid 4\ell$, $L(C_k)$ does not divide $L(C_\ell)$ by Proposition \ref{divisiblity-period}.
Case II: If $p \nmid k$and $p\mid \ell$, then the period of $C_k$ is $4kp$ and the period of $C_\ell$ is $4\ell$. Since $4kp \nmid 4\ell$, $L(C_k)$ does not divide $L(C_\ell)$ by Proposition \ref{divisiblity-period}.
Case III: If $p \nmid k$and $p\nmid \ell$, then the period of $C_k$ is $4kp$ and the period of $C_\ell$ is $4\ell p$. Since $4kp \nmid 4\ell p$, $L(C_k)$ does not divide $L(C_\ell)$ by Proposition \ref{divisiblity-period}.
Case IV - A: If $p \mid k$, $p\nmid \ell$ and $(k/p) \nmid \ell$, then the period of $C_k$ is $4k$ and the period of $C_\ell$ is $4\ell p$. Since $4k \nmid 4\ell p$, $L(C_k)$ does not divide $L(C_\ell)$ by Proposition \ref{divisiblity-period}.
Case IV - B: If $p \mid k$, $p\nmid \ell$ and $(k/p) \mid \ell$, then the period of $C_k$ is $4k$ and the period of $C_\ell$ is $4\ell p$. Since $4k \mid 4\ell p$, we cannot use the Proposition \ref{divisiblity-period}.
Since $p\mid k$, we have $L(C_p)$ divides $L(C_k)$. If $L(C_k)$ divides $L(C_\ell)$, then $L(C_p)$ divides $L(C_\ell)$ which gives a contradiction by Lemma \ref{Cp-nmid-Cl}.
\end{proof}
\begin{cor}
Let $k$ and $\ell $ be positive integers such that $k<\ell $ and $k$ does not divide $\ell$.
Then there is no map from $C_\ell$ to $C_k$.
\end{cor}
\begin{proof}
By the Kleiman-Serre theorem (Theorem \ref{chapman:KleimanSerre}) and Theorem \ref{Cknotdiv}.
\end{proof}
We close this section by remarking again that we do not know if there
is a rational map from $B_{2k}$ to $B_k$ or from $C_{2k}$ to $C_k$.
\section{Remark on the the Curves $y^p-y=x^{p^k+1}+ax$ over $\mathbb F_p$}\label{a}
We remark that the number of $\mathbb{F}_{p^n}$-rational points on
$C_{k,a}: y^p-y=x^{p^k+1}+ax$ is equal to the number of $\mathbb{F}_{p^n}$-rational points on
$C_k$, where $a\in \mathbb F_p^\times$.
All the proofs in this paper go through, with minor changes.
There is the same relationship to $B_k$, the
map in \eqref{onetoone} must be changed to the map $(x, y)\mapsto (x-a2^{-1},y)$
and all proofs go through.
The curve $X_k$ in the proof of Lemma \ref{divisiblity-odd}
must be replaced by $y^p-y=x^{p^k+1}-a4^{-1}$.
Therefore, the divisibility property (and its opposite direction) also holds for these curves.
| {
"timestamp": "2018-05-28T02:02:43",
"yymm": "1803",
"arxiv_id": "1803.03511",
"language": "en",
"url": "https://arxiv.org/abs/1803.03511",
"abstract": "In this paper we consider the curves $C_k^{(p,a)} : y^p-y=x^{p^k+1}+ax$ defined over $\\mathbb F_p$ and give a positive answer to a conjecture about a divisibility condition on $L$-polynomials of the curves $C_k^{(p,a)}$. Our proof involves finding an exact formula for the number of $\\mathbb F_{p^n}$-rational points on $C_k^{(p,a)}$ for all $n$, and uses a result we proved elsewhere about the number of rational points on supersingular curves.",
"subjects": "Algebraic Geometry (math.AG)",
"title": "Divisibility of L-Polynomials for a Family of Artin-Schreier Curves",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754492759499,
"lm_q2_score": 0.718594386544335,
"lm_q1q2_score": 0.7075103909790643
} |
https://arxiv.org/abs/2007.06694 | Pansu pullback and exterior differentiation for Sobolev maps on Carnot groups | We show that in an $m$-step Carnot group, a probability measure with finite $m^{th}$ moment has a well-defined Buser-Karcher center-of-mass, which is a polynomial in the moments of the measure, with respect to exponential coordinates. Using this, we improve the main technical result of our previous paper concerning Sobolev mappings between Carnot groups; as a consequence, a number of rigidity and structural results from recent papers hold under weaker assumptions on the Sobolev exponent. We also give applications to quasiregular mappings, extending earlier work in the $2$-step case to general Carnot groups. | \section{Introduction}
This is part of a series of papers on geometric mapping theory in Carnot groups, in which we establish regularity, rigidity, and partial rigidity results for bilipschitz, quasiconformal, or more generally, Sobolev mappings, between Carnot groups \cite{KMX1,KMX2,kmx_rumin,kmx_iwasawa}. In \cite{KMX1} we showed that Reshetnyak's theorem on pullbacks of differential forms has a partial generalization to mappings between Carnot groups (see also \cite{dairbekov_morphism_property_bounded_distortion,vodopyanov_bounded_distortion,vodopyanov_foundations}).
Our aim here is to strengthen the pullback theorem from \cite{KMX1} by relaxing the assumptions on the Sobolev exponent. This yields new applications to quasiregular mappings, in addition to stronger versions of results from \cite{KMX1,KMX2,kmx_rumin,kmx_iwasawa}.
We expect further applications to geometric mapping theory in Carnot groups, in particular to understanding the threshold between flexibility and rigidity.
We refer the interested reader to \cite{KMX1} for more background.
Before stating our results, we briefly recall some facts and notation; see Section~\ref{sec_prelim} for more detail.
Let $G$ be a Carnot group with Lie algebra $\mathfrak{g}$, grading $\mathfrak{g}=\oplus_{j\geq 1}^sV_j$, and dilation group $\{\delta_r:G\rightarrow G\}_{r\in (0,\infty)}$. The exponential map $\exp:\mathfrak{g}\rightarrow G$ is a diffeomorphism, with inverse $\log:G\rightarrow \mathfrak{g}$. Without explicit mention, in what follows all Carnot groups will be equipped with Haar measure and a Carnot-Caratheodory metric denoted generically by $d_{CC}$. If $f:G\supset U\rightarrow G'$ is a Sobolev mapping between Carnot groups, where $U$ is open, then $f$ has a well-defined approximate Pansu differential $D_Pf(x):G\rightarrow G'$ for a.e. $x\in U$, which is a graded group homomorphism (Theorem~\ref{th:Lp*_pansu_differentiability_new});
By abuse of notation, we also denote the associated homomorphism of graded Lie algebras by $D_Pf(x):\mathfrak{g}\rightarrow \mathfrak{g}'$; furthermore, for the sake of brevity we will often shorten ``approximate Pansu differential'' to ``Pansu differential''.
If $\omega$ is a differential form defined on the range of $f$, then the Pansu pullback $f_P^*\omega$ is given by $f_P^*\omega(x)=(D_Pf(x))^*\omega(f(x))$ for a.e. $x\in U$.
Let $G$ be a Carnot group with Lie algebra $\mathfrak{g}$. For every $x\in G$ we let $\log_x:G\rightarrow \mathfrak{g}$ be the logarithm map ``centered at $x$'', i.e. $\log_x(y):=\log(x^{-1}y)$. We recall \cite{karcher_buser_almost_flat_manifolds,KMX1} that if $\nu$ is a compactly supported probability measure in $G$, then $\nu$ has a well-defined Buser-Karcher center of mass $\operatorname{com}_\nu$, which is characterized as the unique point $x\in G$ such that $\nu$ is ``balanced'' with respect to logarithmic coordinates centered at $x$: $\int_G\log_x\,d\nu=0$. Our first result is a generalization of this center of mass to the case of measures with noncompact support.
\medskip
\begin{theorem}[Theorem~\ref{le:C_nu_diffeomorphism}]~
\label{thm_com_finite_moment_intro}
Suppose $G$ is an $m$-step Carnot group, and $\nu$ is a probability measure on $G$ with finite $m^{th}$-moment, i.e. for some $x\in G$ we have
$$
\int_Gd_{CC}^m(x,y)\,d\nu(y) <\infty\,.
$$
Then for every $x\in G$ the map $\log_x$ is integrable w.r.t. $\nu$, and there is a unique point $\operatorname{com}_\nu\in G$ such that $\int_G\log_{\operatorname{com}_\nu}\,d\nu=0$. Moreover, $\log(\operatorname{com}_\nu)$ is a polynomial in the polynomial moments of the pushforward measure $(\log_{x_0})_*\nu$, for any $x_0\in G$.
\end{theorem}
\begin{remark}
With minor modifications, the same proof works for general simply connected nilpotent groups.
\end{remark}
Applying Theorem~\ref{thm_com_finite_moment_intro} in a standard way, one may define a mollification process for $L^m_{\operatorname{loc}}$-mappings into an $m$-step Carnot group, which yields a family of smooth approximations.
For a Sobolev mapping $f$ with mollification $f_\rho$, our main result relates the ordinary pullback $f_\rho^*\omega$ of a differential form $\omega$ with the Pansu pullback $f_P^*\omega$, as defined above. To state the result, we require the notion of the weight $\operatorname{wt}(\alpha)$ of a differential form $\alpha$; this is defined using the decomposition of $\Lambda^*\mathfrak{g}$ with respect to the diagonalizable action of the Carnot scaling, see Subsection~\ref{subsec_differential_forms_on_carnot_groups}.
\begin{theorem}[Approximation theorem] \label{thm_approximation_theorem_intro}~
Let $G$, $G'$ be Carnot groups, and $f: U \to G'$ be a map in $W^{1,p}_{loc}(U, G')$, where $U\subset G$ is open. Suppose:
\begin{itemize}
\item $\eta\in \Omega^k(G)$, $\omega\in\Omega^\ell(G')$ are differential forms, where $k+\ell=N:=\dim G$.
\item $\eta$ is left-invariant.
\item $\omega$ is continuous and bounded.
\item $\operatorname{wt}(\omega)+\operatorname{wt}(\eta)\leq -\nu$, where $\nu$ is the homogeneous dimension of $G$.
\item $p\geq -\operatorname{wt}(\omega)$.
\item $\frac1p\leq \frac1m+\frac{1}{\nu}$, where $G'$ has step $m$.
\end{itemize}
Then
\begin{equation*}
f_\rho^*\omega \wedge \eta \to f_P^* \omega \wedge \eta \quad \text{in $L^s_{\operatorname{loc}}(U)$ with $s = \frac{p}{-\operatorname{wt}(\omega)}$,}
\end{equation*}
where $f_\rho$ is the mollification of $f$ at scale $\rho$, see Section~\ref{subsec_mollifying_maps_between_carnot_groups}. In particular, when $\omega\in\Omega^N(G')$ and $\operatorname{wt}(\omega)\leq -\nu$, then
\begin{equation*}
f_\rho^*\omega \to f_P^* \omega \quad
\text{in $L^{\frac{p}{\nu}}_{\operatorname{loc}}(U)$.}
\end{equation*}
\end{theorem}
\bigskip
We refer the reader to Section~\ref{sec_pansu_pullback_mollification} for more refined statements.
Although the overall outline of the proof of Theorem~\ref{thm_approximation_theorem_intro} is the same as for \cite[Theorem 1.18]{KMX1}, the fact that $p\leq \nu$ creates several complications: a Sobolev mapping $f\in W^{1,p}_{\operatorname{loc}}(U,G')$ as in the theorem need not be either (classically) Pansu differentiable almost everywhere or continuous; in particular, the argument cannot be localized in the target.
\bigskip
As immediate consequences of Theorem~\ref{thm_approximation_theorem_intro}, the rigidity and partial rigidity results from \cite{KMX1} hold under weaker assumptions on the exponent. For instance:
\begin{enumerate}
\item Let $\{G_i\}_{1 \le i \le n}$, $\{G'_j\}_{1 \le j \le n'}$ be collections on Carnot groups where each $G_i, G'_j$ is nonabelian and does not admit
a nontrivial decomposition as a product of Carnot groups. Let $G = \prod_i G_i$, $G' = \prod_j G'_j$. Set
$ K_i : = \{ k \in \{1, \ldots, n\} : G_k \simeq G_i \}$
and assume that
\begin{equation} \label{eq:p_rigidity_intro}
p \ge \max\{ \nu_i - 1: |K_i| \ge 2 \}
\end{equation}
where $\nu_i$ denotes the homogeneous dimension of $G_i$.
Assume that $f: G \supset U \to G'$ is a $W^{1,p}_{\rm loc}$ mapping,
$U = \prod_i U_i$ is a product of connected open sets $U_i \subset G_i$,
and the (approximate) Pansu differential $D_P f(x)$ is an isomorphism for a.e.\ $x \in G$.
Then $f$ coincides almost everywhere with a product mapping, modulo a permutation of the factors,
see Theorem~\ref{th:product_rigidity} below. In \cite[Theorem 1.1]{KMX1} the result was proved under the stronger hypothesis
$p > \sum_i \nu_i$.
If each $G_i$ is either a higher Heisenberg group $\H_{m_i}$ (with $m_i \ge 2$) or any complex Heisenberg group $\H^\mathbb{C}_{m_i}$ (with $m_i \ge 1$)
then the condition
\eqref{eq:p_rigidity_intro} can be improved to $p \ge 2$,
see Corollary~\ref{co:product_rigidity} below.
%
\item If $\H^\mathbb{C}_m$ is the complexification of the $m^{th}$ Heisenberg group $\H_m$, $U \subset \H^\mathbb{C}_m$ is open and connected
and $f: U\rightarrow\H^\mathbb{C}_m$ is a $W^{1,2m+1}_{\operatorname{loc}}$-mapping such that the (approximate) Pansu differential $D_Pf(x)$ is an isomorphism for a.e. $x$, then $f$ coincides almost everywhere with a holomorphic or antiholomorphic mapping, cf. \cite[Theorem 1.6]{KMX1}
for the same result under the stronger condition $p > 4m +2$.
\end{enumerate}
\bigskip
Another application of Theorem~\ref{thm_approximation_theorem_intro} is to quasiregular mappings between Carnot groups, addressing questions originating in \cite{rickman_quasiregular_mappings,heinonen_holopainen}. We recall that a fundamental step in Reshetnyak's approach to quasiregular mappings in $\mathbb{R}^n$ is showing that the composition of an $n$-harmonic function with a quasiregular mapping is a solution to a quasilinear elliptic PDE; this ``morphism property'' depends crucially on the fact that pullback commutes with exterior differentiation \cite{reshetnyak_space_mappings_bounded_distortion}. Using Theorem~\ref{thm_approximation_theorem_intro}, we are able to extend earlier work of \cite{vodopyanov_foundations} (see also \cite{heinonen_holopainen}), so as to generalize a portion of Reshetnyak's theory to all Carnot groups. In particular, if a Carnot group $G$ has homogeneous dimension $\nu$ and $f:G\supset U\rightarrow G$ is a quasiregular mapping, then (see Section~\ref{sec_quasiregular_mappings} for more details):
\begin{itemize}
\item The ``morphism'' property, which was first shown by Reshetnyak in the $\mathbb{R}^n$ case, holds for locally Lipschitz $\nu$-harmonic functions: if $u:G\rightarrow \mathbb{R}$ is a locally Lipschitz $\nu$-harmonic function then the composition $u\circ f$ is $\mathcal{A}$-harmonic.
\item $f\in W^{1,\nu'}$ for some $\nu'>\nu$.
\item $f$ is H\"older continuous, Pansu differentiable almost everywhere, and maps null sets to null sets.
\end{itemize}
\bigskip\bigskip
We conclude with some open questions.
\begin{question}
What is the exponent threshold for rigidity/flexibility in the results mentioned above?
\end{question}
For instance, suppose $f:\H\times\H\rightarrow\H\times\H$ is a $W^{1,p}$-mapping whose Pansu differential is an isomorphism almost everywhere. For which $p$ must $f$ agree with a product mapping almost everywhere? Are there counterexamples when $p=1$? Optimal Sobolev exponents were obtained for an analogous product rigidity question in the Euclidean setting in \cite{kmsx_infinitesimally_split_globally_split,kmsx_counterexample}.
\medskip
We recall that a Carnot group $G$ is rigid in the sense of Ottazzi-Warhurst if for any connected open subset $U\subset G$, the family of smooth contact embeddings $U\rightarrow G$ is finite dimensional. We conjectured \cite[Conjecture 1.10]{KMX1} that quasiconformal homeomorphisms of rigid Carnot groups are smooth. One may ask if there is a rigidity/flexibility threshold for these groups.
\begin{question}
Let $f:G\supset U\rightarrow G$ be a $W^{1,p}$-mapping, where $U$ is an open subset of an Ottazzi-Warhurst rigid Carnot group, and $D_Pf(x)$
is an isomorphism for a.e. $x\in U$ (recall that $D_Pf(x)$ denotes the approximate Pansu differential). For which $p$ can we conclude that $f$ is smooth? What if $f$ is a homeomorphism?\footnote{Ottazi-Warhurst
showed that $C^2$ quasiconformal homeomorphisms of rigid Carnot groups are smooth.
Recently Jonas Lelmi improved this result, replacing the $C^2$ regularity assumption with $C^1$ (or even Euclidean bilipschitz);
the same result was shown
by Alex Austin for the $(2,3,5)$ distribution \cite{lelmi,austin_235}.}
\end{question}
Motivated by \cite{iwaniec_martin_quasiregular_even_dimensions,iwaniec_p_harmonic_tensor_quasiregular_mappings} one may ask about minimal regular requirements for quasiregular mappings.
\begin{question}
Suppose $f:G\supset U\rightarrow G$ is a weakly quasiregular mapping, i.e. $f\in W^{1,p}$ and for some $C$ we have $|D_hf|^\nu\leq C\det D_Pf$ almost everywhere. For which $p<\nu$ can we conclude that $f\in W^{1,\nu}$?
\end{question}
\bigskip
\subsection*{Organization of the paper}~
We review some background material on Carnot groups and Sobolev mappings in Section~\ref{sec_prelim}. Section~\ref{sec_center_of_mass_mollification} establishes existence and estimates for the center of mass for measures which satisfy a moment condition, and establishes bounds for the associated mollification procedure. The proof of the main approximation theorem and some applications to the exterior derivative are proven in Section~\ref{sec_pansu_pullback_mollification}. Section~\ref{sec_quasiregular_mappings} gives applications to quasiregular mappings. For the convenience of the reader, we have included proofs of some background results in the appendices. In Appendix~\ref{se:W1p_differentiability}, we give a new direct proof of the $L^p$ Pansu differentiability of Sobolev mappings; see Subsection~\ref{subsec_sobolev} and Appendix~\ref{se:W1p_differentiability} for a comparison with the original proof by Vodopyanov. In Appendix~\ref{se:app_compact} we prove the compact Sobolev embedding, and in Appendix~\ref{se:sobolev} we discuss Sobolev spaces defined using weak upper gradients, collecting some results from the literature, and comparing with with distributional approach of Reshetnyak and Vodopyanov.
\section{Preliminaries}
\label{sec_prelim}
\subsection{Carnot groups}~
In this subsection we recall some standard facts about Lie groups, in particular nilpotent Lie groups and Carnot groups, and prove a simple estimate for the
nonlinear term in the Baker-Campbell-Hausdorff (BCH) formula. This will be useful to define the center of mass for probability measures which do not necessarily have compact support, but only satisfy bounds on certain moments.
Our main interest is in Carnot groups and the reader may focus on this case. Since the construction of the center of mass extends
to connected, simply connected nilpotent groups without additional effort, we include a short discussion of nilpotent groups as well. Since the facts mentioned below will be standard for most geometers, our discussion is more calibrated for analysts.
\medskip
\medskip
Let $G$ be a Lie group of dimension $N$. In this paper we will only consider connected, simply connected
Lie groups.
We give the tangent space $T_e G$ at the identity the structure of a Lie algebra $\mathfrak{g}$ in the usual way: each tangent vector $X \in T_e G$
can be extended to a left-invariant vectorfield $\tilde X$ through push-forward by left translation $\ell_a (g) = a g$, i.e $\tilde X(a) f = X (f \circ \ell_a)$.
For two left-invariant vector fields $\tilde X$ and $\tilde Y$ one easily sees that the commutator $[\tilde X, \tilde Y] = \tilde X \tilde Y - \tilde Y \tilde X$ is
a left-invariant vectorfield. We define the Lie bracket on $T_e G$ by $[X,Y] = [\tilde X, \tilde Y](e)$. In the following we do not distinguish between
$X$ and $\tilde X$. Similarly we identify left-invariant differential $k$-forms on $G$ with $\Lambda^k \mathfrak{g}$.
The descending series of the Lie algebra is defined by $\mathfrak{g}_1 = \mathfrak{g}$ and $\mathfrak{g}_{i+1} = [\mathfrak{g}, \mathfrak{g}_i]$ where the right hand side denotes the linear space generated by of all brackets of the form $[X, Y]$
with $X \in \mathfrak{g}$, $Y \in \mathfrak{g}_i$.
We say that $G$ is a nilpotent group of step $m$ if $[\mathfrak{g}]_m \ne \{0\}$ and $[\mathfrak{g}]_{m+1} = \{0\}$.
We say that $G$ is a Carnot group of step $m$ if, in addition, $\mathfrak{g}$ is graded, i.e.
if we are given a direct sum decomposition $\mathfrak{g} = \oplus_{j=1}^m V_j$ (as a vectorspace) with $V_{j+1} = [V_1, V_j]$ for $1 \le j \le m-1$.
In the general nilpotent case it will be convenient to introduce subspaces $W_1, \ldots, W_m$ such that $\mathfrak{g}_{i} = W_i \oplus \mathfrak{g}_{i+1}$.
There is no canonical choice of the spaces $W_i$ (except for $W_m$), but for our analysis any choice will do (see also
Remark~\ref{re:norms_and_complement} below).
In the Carnot and general cases, respectively, we have
\begin{equation} \label{eq:bracket_grading}
[V_i, V_j] \subset \mathfrak{g}_{i+j} \quad \text{and} \quad [W_i, W_j] \subset \oplus_{k=i+j}^{m} W_k.
\end{equation}
By uniqueness of solutions of ordinary differential equations the integral curve $\gamma_X: \mathbb{R} \to G$ of a left invariant vectorfield $X$ with $\gamma_X(0) = e$ is a subgroup.
We define the exponential map $\exp: \mathfrak{g} \to G$ by $\exp X(e) = \gamma_X(1)$. Thus $\exp: \mathfrak{g} \to G$ is smooth.
By the Baker-Campbell-Hausdorff (BCH) Theorem for sufficiently small
$X$ and $Y$ we have $\exp X \exp Y = \exp (X + Y + P(X,Y))$ where $P(X,Y)$ is a series in iterated Lie brackets of $X$ and $Y$, see, e.g.
\cite[eqn. (2), p.\ 12]{corwin_greenleaf_book} or
\cite[Thm. 4.29]{Michor} .
For a nilpotent Lie group of step $m$
the Lie brackets of order $m+1$ and higher vanish. Then the expression $P(X,Y)$ is a polynomial,
the exponential map is a diffeomorphism and the BCH formula holds for all $X$ and $Y$ \cite[Thm. 1.21]{corwin_greenleaf_book}. We often write $\log = \exp^{-1}$,
and denote the induced group action on $\mathfrak{g}$ by
$$
X \ast Y :=\log(\exp X\exp Y)= X + Y + P(X,Y)\,.
$$
One can use $\log: G \to \mathfrak{g}$ as a global chart for $G$ with the group operation given by $\ast$, but we will usually not do this.
We denote by $\pi_i$ the projection $\mathfrak{g} \to V_i$ (or $\mathfrak{g} \to W_i$ for nilpotent groups). It follows from \eqref{eq:bracket_grading}
that $\pi_i [X,Y]$ depends only on $\pi_1(X), \ldots, \pi_{i-1}(X)$
and $\pi_1(Y), \ldots, \pi_{i-1}(Y)$. Thus the differential of $P$ with respect to the first or second variable is block lower triangular with
respect to the decompositions $\mathfrak{g} = \oplus_{i=1}^m V_i$ or $\mathfrak{g} = \oplus_{i=1}^m W_i$, with zero entries on the block diagonal.
It follows that the Lebesgue measure $\mathcal L^N$ on $\mathfrak{g}$ is invariant under the left and right group operation $\ast$.
Thus the push-forward measure $\exp_* \mathcal L^N$ is the biinvariant Haar measure on $G$ (up to a factor).
The horizontal bundle $\mathcal H \subset TG$ is the span of the left-invariant vectorfields $X$ which satisfy $X(e) \in V_1$ (or $X(e) \in W_1$ in the nilpotent case).
We fix a scalar product on $\mathfrak{g}$. This induces a left-invariant metric on $G$ by left-translation. The Carnot-Carath\'eodory distance on $G$ as the shortest length of horizontal curves, i.e.
\begin{equation}
d_{CC}(x, y) = \inf \{ \int_a^b |\gamma'(t)| \, dt : \text{ $\gamma: [a,b] \to G$ rectifiable, $\gamma'(t) \in \mathcal H$} \}.
\end{equation}
Push-forward by left translation preserves the horizontal bundle. Thus the left translation of a horizontal curve is horizontal
and the metric $d_{CC}$ is left-invariant. By Chow's theorem every two points in $G$ can be connected by a horizontal curve
of finite length so that $d_{CC}(x,y) < \infty$ for all $x, y \in G$. Moreover $d_{CC}$ induces the usual manifold topology on $G$
\cite[Thm 2.1.2 and Thm 2.1.3 ]{montgomery_book}.
\bigskip
We now focus on Carnot groups.
We define a one parameter group of dilations $\delta_r: \mathfrak{g} \to \mathfrak{g}$ by $\delta_r X = r^j X$ for $X \in V_j$ and linear extension. Then $\delta_r [X, Y] = [\delta_r X, \delta_r Y]$
so that $\delta_r$ is a Lie algebra homomorphism. Since $P(X,Y)$ is a sum of iterated Lie brackets it follows that $\delta_r (X \ast Y) =( \delta_r X) \ast (\delta_r Y)$.
Thus $\exp \circ \delta_r \circ \exp^{-1}: G \to G$ is a group homomorphism which we also denote by $\delta_r$.
Then $\delta_r (\ell_a x) = \ell_{\delta_r a} \delta_r x$. Since $\delta_r$ as a map on $\mathfrak{g}$ preserves $V_1$ and is scaling by $r$ on $V_1$ it follows that
$\delta_r$ maps horizontal curves to horizontal curves and
\begin{equation}
d_{CC}(\delta_rx, \delta_r y) = r d_{CC}(x,y).
\end{equation}
Since $d_{CC}$ is also left-invariant, the bi-invariant measure of a ball $B(x,r)$ in the $d_{CC}$ metric is given by
\begin{align}
& \, \mu(B(x,r)) = \mu(B(e,r)) = \mu(\delta_r B(e,1)) \\
=& \, \mathcal L^N(\delta_r \log B(e,1)) = r^\nu \mu(B(e,1))
\end{align}
where
\begin{equation}
\nu := \sum_{j=1}^m j \dim \mathfrak{g}_j \quad \text{is the homogeneous dimension of $G$.}
\end{equation}
We define a Euclidean norm $| \cdot|_e$ on $V_j$ by restriction of the scalar product on $\mathfrak{g}$ to $V_j$.
Recall that $\pi_j$ denotes the projection from $\mathfrak{g}$ to $V_j$.
To reflect the action of $\delta_r$ on $\mathfrak{g}$ we introduce the `homogeneous norm'
\begin{equation} \label{eq:homogeneous_norm}
|X| := \left( \sum_{i=1}^m | \pi_i X|_e^{2m!/i}\right)^{1/ 2m!}. \quad
\end{equation}
Then
\begin{equation} \label{eq:scaling_homogeneous_norm}
|\delta_r X| = r |X|.
\end{equation}
Note that $| X|$ is comparable to $\sum_{j=1}^m |\pi_j X|_e^{1/j}$ and that $| \cdot|$ does not satisfy the triangle inequality
but only the weaker estimate $| X + Y| \le C |X| + C |Y|$.
It follows from the ball-box theorem, see e.g. \cite[Theorem 2.4.2]{montgomery_book}, that there exists constant $C_1$ and $C_2$ such that
\begin{equation} \label{eq:ball_box}
C_1 d_{CC}(e, \exp X) \le |X| \le C_2 d_{CC}(e, \exp X) \quad \forall X \in \mathfrak{g}.
\end{equation}
In fact, in Carnot groups the ball-box theorem follows immediately from the seemingly weaker statement that
the Riemannian distance $d$ and $d_{CC}$ induce the same topology on $G$. Indeed, together with the fact
that $\exp$ is a homeomorphism from $\mathbb{R}^n$ to $G$ equipped with $d$ this implies that the set
$S := \{ X \in \mathfrak{g} : d_{CC}(e, \exp X) = 1\}$ is compact. Thus $|X|$ attains its minimum and maximum on $S$ and
the inequality \eqref{eq:ball_box} follows since all terms scale by $r$ if we replace $X$ by $\delta_r X$.
Note also that
\begin{equation} \label{eq:norm_euclidean}
C^{-1} \sum_{j=1}^m |\pi_j X|_e^2 \le |X|_e^2 \le C \sum_{j=1}^m |\pi_j X|_e^2 \quad \forall X \in \mathfrak{g}
\end{equation}
since all norms on a finite-dimensional vector space are equivalent.
In fact we can take $C=1$ if we choose a scalar product on $\mathfrak{g}$ such that the subspaces $V_j$
are orthogonal.
One of our main goals is to construct a center of mass for probability measures $\nu$ on $G$
which is invariant under left-translation and group homomorphisms. Equivalently, we want to construct
a center of mass for probability measures on $\mathfrak{g}$ which is compatible with the group action $\ast$.
Since we want to allow measures which do not have compact support but only satisfy suitable moment bounds
we need good control of the nonlinear term $P(X,Y)$ in the BCH formula in terms of $|X|$ and $|Y|$.
To write the estimate we use the following notation for a multiindex
$J = {j_1, \ldots, j_k}$ with $k \ge 1$ and $j_i \in \mathbb{N} \setminus \{0\}$. We set
$\# J= k$ and $| J| = \sum_{i=1}^k j_i$.
\begin{proposition} \label{pr:bounds_P}
Let $E_1, \ldots, E_N$ be a basis of $\mathfrak{g}$. There exist multilinear forms $M_I$ and $L_J$ such that
\begin{align} \label{eq:decomposition_P}
& \qquad P(X,Y) = \\
& \sum_{i=1}^N \sum_ { \underset{ j \ge 1, k \ge 1}{j, k, j+k \le m} }
\sum_{ \underset{ \# I =j, \# J = k}{I,J} }
M^i_I(\pi_{i_1} X, \ldots, \pi_{i_j} X) \, L^i_J(\pi_{i_1} Y, \ldots, \pi_{i_k} Y) \, E_i
\nonumber
\end{align}
and
\begin{equation} \label{eq:bound_P_multilinear}
M^i_I(\pi_{i_1} X, \ldots, \pi_{i_j} X) \le C |X|^{| I |}, \quad
L^i_J(\pi_{i_1} Y, \ldots, \pi_{i_k} Y) \le C |Y|^{| J |}.
\end{equation}
In particular
\begin{equation} \label{eq:bound_P}
|P(X,Y)|_e \le C(R) (1 + |Y|^{m-1}) \quad \text{for all $X$ with $|X| \le R$}
\end{equation}
and the derivatives of $P$ with respect to the first variable satisfy
\begin{align} \label{eq:bound_derivatives_P}
& |D^k_1 P(X,Y) (\dot X, \ldots, \dot X)| \le C(R) (1 + |Y|^{m-1}) \\
& \quad \text{for all $X, \dot X$ with $|X| \le R$ and $|\dot X| \le 1$.} \nonumber
\end{align}
for $1 \le k \le m-1$ and $D^m_1 P = 0$.
Moreover
\begin{equation} \label{eq:bound_P_homogeneous_norm}
| [X, Y]| \le C (|X| + |Y|) \quad \text{and} \quad |P(X,Y)| \le C (|X| + |Y|).
\end{equation}
\end{proposition}
\begin{proof} Since $X = \sum_i \pi_i X$ and $Y = \sum_i \pi_i Y$ and since $P(X,Y)$ is a multilinear expression
in $X$ and $Y$ it is clear that $P$ can be expanded into sums of products of multilinear terms as in \eqref{eq:decomposition_P}. To show
\eqref{eq:decomposition_P}
we only have to show that the terms corresponding to $ | I | + |J | > m$ vanish.
This follows immediately from the fact that $P(X,Y)$ is a sum of iterated Lie brackets and
the first inclusion in \eqref{eq:bracket_grading}
Since $M^i_I$ is a multilinear form it follows that
$M^i_I(\pi_{i_1} X, \ldots, \pi_{i_j} X) \le C \prod_{k=1}^j |\pi_{i_k} X|_e$.
Now by the definition of the homogeneous norm we have $ |\pi_{i_k} X|_e \le |X|^{i_k}$.
This implies the estimate for $M^i_I$ and the same argument applies to $L^i_J$.
The estimate \eqref{eq:bound_P} is an immediate consequence of
\eqref{eq:decomposition_P}, \eqref{eq:bound_P_multilinear}, the fact that
only terms with $|J| \le m-1$ appear in \eqref{eq:decomposition_P}
and the estimate $a^k \le 1 + a^{m-1}$ for $1 \le k \le m-1$.
Since the terms $M_I^i$ are multilinear, their derivatives are uniformly bounded for $|X| \le R$
and thus \eqref{eq:bound_derivatives_P} follows in the same way.
The second estimate in \eqref{eq:bound_P_homogeneous_norm} follows from the first since $P(X,Y)$ is a linear combination of
iterated Lie brackets. To show the first inequality, assume first that $X \in V_j$, $Y \in V_k$. Then $[X, Y] \in V_{j+k}$.
Thus
$$|[X, Y]| = |[X,Y]|_e^{\frac1{k+j}} \le C |X|_e^{\frac1{k+j}} \, |Y|_e^{\frac1{k+j}} \le C |X|^{\frac{j}{k+j}} \, |Y|^{\frac{k}{k+j}}$$
and the estimate follows by Young's inequality. For general $X, Y$ the estimate follows by bilinearity of the Lie bracket.
\end{proof}
\begin{remark} \label{re:BCH_bounds_nilpotent}
The estimates \eqref{eq:bound_P_multilinear}, \eqref{eq:bound_P} and \eqref{eq:bound_derivatives_P}
also hold for nilpotent groups if $\pi_i$ denotes the projection
to the spaces $W_i$ and the homogeneous norm is defined with this definition of $\pi_i$.
Indeed, we can use the second inclusion in \eqref{eq:bracket_grading} to see that also in the nilpotent
case the sum in \eqref{eq:decomposition_P} only contains terms
with $|I| + |J| \le m$. The rest of the argument is the same.
Instead of \eqref{eq:bound_P_homogeneous_norm}
we have the slightly weaker estimates
\begin{equation} \label{eq:bound_P_nilpotent_homogeneous_norm}
| [X, Y]| \le C (1 + |X| + |Y|) \quad \text{and} \quad |P(X,Y)| \le C ( 1+ |X| + |Y|).
\end{equation}
Again the second estimate follows from the first. For the first estimate we first consider $X \in W_j$, $Y \in W_k$.
Then $[X, Y] \in \oplus_{i=j+k}^m W_i$
and thus by Young's inequality and the previous estimate for
$|[X,Y]|_e^{\frac1{j+k}}$
$$ | [X, Y] | \le \sum_{i=j+k}^m |\pi_i [X,Y]|_e^{\frac1i} \le C(1 + |[X,Y]|_e^{\frac1{j+k}}) \le C (1 + |X| + |Y|).$$
For general $X$, $Y$ the estimate follows by bilinearity.
\end{remark}
\bigskip\bigskip
\begin{remark} \label{re:norms_and_complement}
Note that in the nilpotent case the homogeneous norm does not just depend on the group and the
scalar product on $\mathfrak{g}$, but also on the choice of the complementing spaces $W_1, \ldots, W_m$.
Different choices lead, however, to norms which are essentially equivalent in the following sense. Let $\tilde W_i$ be different spaces with
$\mathfrak{g}_i = \tilde W_i \oplus \mathfrak{g}_{i+1}$, let $\tilde \pi_i$ be the corresponding projections
and let $|\cdot |_{\sim}$ be the corresponding homogeneous norm.
Then there exists a constant $C$ such that
\begin{equation} \label{eq:equivalent_norms_nilpotent}
|X|_{\sim} \le C( |X|^{\frac1m} + |X| ) \quad \text{and} \quad |X| \le C( |X|_{\sim}^{\frac1m} + |X|_{\sim} ).
\end{equation}
It suffices to prove the first inequality, the second follows by reversing the roles of $W_i$ and $\tilde W_i$.
We have $\tilde \pi_i |_{\mathfrak{g}_{i+1}} = 0$. Since $\pi_{i+1} X, \ldots, \pi_{m} X \in \mathfrak{g}_{i+1}$ there exist linear maps
$L_i : \oplus_{k=1}^i W_k \to \tilde W_i$
such that
$ \tilde \pi_i X = L_i(\pi_1 X, \ldots, \pi_i X)$.
Thus
$$ | \tilde \pi_i X|_e \le C \sum_{k=1}^i |\pi_k X|_e \le C \sum_{k=1}^i |X|^k$$
and
hence
$$ |X|_\sim \le C \sum_{i=1}^m |\tilde \pi_i X|_e^{\frac1i} \le C \sum_{i=1}^m |X|^{\frac1i}.$$
From this the assertion easily follows by Young's inequality.
\end{remark}
\bigskip\bigskip
\subsection{Differential forms on Carnot groups}\label{subsec_differential_forms_on_carnot_groups}
Let $G$ be a Carnot group with graded Lie algebra $\mathfrak{g}=\oplus_i V_i$. The grading defines a simultaneous eigenspace decomposition for the dilations $\{\delta_r\}_{r\in (0,\infty)}$. Therefore the action of $\{\delta_r\}_{r\in (0,\infty)}$ on $\Lambda^k\mathfrak{g}$ also has a simultaneous eigenspace decomposition
\begin{equation}
\label{eqn_weight_eigenspace_decomposition}
\Lambda^k\mathfrak{g}=\oplus_w \Lambda^{k,w}\mathfrak{g}
\end{equation}
where $\delta_r$ acts on $\Lambda^{k,w}\mathfrak{g}$ by scalar multiplication by $r^w$. In particular, for any $\alpha\in\Lambda^k\mathfrak{g}$, we have a canonical decomposition
\begin{equation}
\label{eqn_weight_decomposition}
\alpha=\sum_w\alpha_w
\end{equation}
where $(\delta_r)_*\alpha_w=r^w\alpha_w$ for every $w$. Concretely, if $X_1,\ldots,X_N$ is a graded basis of $\mathfrak{g}$, and $\th_1,\ldots,\th_N$ is the dual basis, then the actions of $\delta_r$ on $\mathfrak{g}$ and $\mathfrak{g}^*$ are diagonal with respect to these bases, and the action on $\Lambda^k\mathfrak{g}$ is diagonal with respect to the basis given by exterior powers of the $\th_i$s.
\begin{definition}
An element $\alpha\in \Lambda^k(\mathfrak{g})$ is {\bf homogeneous with weight $w$}
if $\alpha\in\Lambda^{k,w}\mathfrak{g}$; it has {\bf weight $\leq w$} if $\alpha\in \Lambda^{k,\leq w}$ where
\begin{equation}
\label{eqn_def_wt_leq_w}
\Lambda^{k,\leq w}:=\oplus_{\bar w\leq w}\Lambda^{k,\bar w}\,.
\end{equation}
If $U\subset G$ is open, then a $k$-form $\alpha\in \Omega^k(U)$ is {\bf homogeneous of weight $w$} or has {\bf weight $\leq w$} if $\omega(x)\in \Lambda^{k,w}$ or $\omega(x)\in \Lambda^{k,\leq w}$, for every $x\in U$, respectively. We let $\Omega^{k,w}(U)$ and $\Omega^{k,\leq w}$ denote the homogeneous forms of weight $w$ and the forms of weight $\leq w$, respectively, so $\Omega^k(U)=\oplus_w\Omega^{k,w}(U)$. Note that $0\in\Lambda^k(\mathfrak{g})$ has weight $w$ for every $w\in \mathbb{R}$.
\end{definition}
\begin{lemma}
\label{lem_weight_facts}
\mbox{}
\begin{enumerate}
\item If $\alpha_i\in\Omega^{k_i,w_i}$ for $1\leq i\leq 2$, then $\alpha_1\wedge \alpha_2\in\Omega^{k_1+k_2,w_1+w_2}$.
\item $\th_{i_1}\wedge\ldots\wedge\th_{i_k}$ is homogeneous of weight $\sum_j\operatorname{weight}(\th_{i_j})$. In particular, such wedge products give a basis for the left invariant
$k$-forms.
\item \label{it:weight_facts_pullback}
If $\beta\in \Omega^{k,w}(G')$ and
$\Phi:G\rightarrow G'$ is a graded group homomorphism, then $\Phi^*\beta=\Phi_P^*\beta$ belongs to $\Omega^{k,w}(G)$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) is immediate from the definitions, and (2) follows from (1).
(3). Since $\Phi$ is a graded group homomorphism, the Pansu derivative
and the ordinary derivative coincide. Therefore
$$
(\delta_r)_*\Phi_P^*\beta= (\delta_{r^{-1}})^*\Phi^*\beta
=\Phi^*(\delta_{r^{-1}})^*\beta=\Phi^*(\delta_r)_*\beta
=r^{w_\beta}\Phi_P^*\beta\,.
$$
\end{proof}
\bigskip\bigskip
\subsection{Sobolev spaces on Carnot groups} \label{subsec_sobolev}
In this subsection we discuss $L^p$ and Sobolev spaces for maps between Carnot
groups; in addition to the definitions, we cover two key properties needed for the proof of the approximation theorem -- the Poincar\'e-Sobolev
inequality and almost everywhere Pansu differentiability (in an $L^p$ sense).
In the literature there are different approaches to Sobolev mappings between Carnot groups -- some are based on distribution derivatives, and others on (weak) upper gradients (see \cite{vodopyanov_monotone_1996,vodopyanov_bounded_distortion,HKST}). In this subsection we use the distributional definition of Sobolev mappings, and cover the upper gradient version in Appendix~\ref{se:sobolev}.
In fact, the two definitions are equivalent in our setting (see Appendix~\ref{se:sobolev}), so one could work equally well work with either.
We treat the case where the domain is an open set in a Carnot group; however most statements and proofs apply without modification to equiregular subriemannian manifolds satisfying a suitable Poincar\'e inequality.
In this subsection we let $G$ denote a Carnot group with graded Lie algebra $\mathfrak{g} = \oplus_{i=1}^m V_i$. Let $X_1, \ldots, X_K$ be an orthonormal basis of the first layer $V_1$.
As usual we identify the vectors $X_i \in V_1$ with left-invariant vectorfields on $G$. Then $X_1(p), \ldots, X_K(p)$ is a
basis of the horizontal subspace at $p$.
\begin{definition} \label{de:sobolev_scalar_new} Let $U \subset G$ be open. We say that $u: U \to \mathbb{R}$ is in the Sobolev space $W^{1,p}(U)$
if $u \in L^p(U)$ and
if the distributional derivatives $X_1 u, \ldots X_K u$ are in $L^p(U)$, i.e., if there exist $g_i \in L^p(U)$ such that
$$ \int_U u \, X_i \varphi \, d\mu = - \int_U g_i \, \varphi \, d\mu \quad \text{for all $\varphi \in C_c^\infty(U)$.}
$$
We say that $u \in W^{1,p}_{loc}(U)$ if $u \in W^{1,p}(V)$ for every open set $V$ whose closure is compact and contained in $U$.
\end{definition}
\bigskip
We write $X_i u$ for the weak derivatives $g_i$ and we define
\begin{equation}
D_h u = (X_1 u, \ldots, X_K u), \quad |D_h u| := \left( \sum_{i=1}^K |X_i f|^2\right)^{1/2}.
\end{equation}
We recall some basic properties of Sobolev functions.
\begin{proposition} \label{pr:sobolev_basic} Let $U \subset G$ be open. Then the following assertions hold.
\begin{enumerate}
\item \label{it:basic_density} $C^\infty(U)$ is dense in $W^{1,p}(U)$;
\item \label{it:basic_chain} if $u\in W^{1,p}(U)$ and $\psi: \mathbb{R} \to \mathbb{R}$ is $C^1$ with bounded derivative then $\psi \circ u - \psi(0) \in W^{1,p}(U)$ and
the weak derivatives satisfy the chain rule;
\item \label{it:basic_absolute} if $u \in W^{1,p}(U)$ then $|u| \in W^{1,p}(U)$ and the weak derivatives satisfy $X_i |u| = \pm X_i u$ a.e. in the set $\{ \pm u > 0\}$
while $X_i |u| = 0$ a.e.\ in the set $\{u = 0\}$;
\item \label{it:basic_min} if $u,v \in W^{1,p}(U)$ then $\min(u,v) \in W^{1,p}(U)$ and $|D_h \min(u,v)| \le \max(|D_h u|, |D_h v|)$ a.e.;
\item \label{it:basic_inf} if $u_k \in W^{1,p}(U)$ for $k \in \mathbb{N}$ and there exist functions $g, h \in L^p(U)$ such that $ |D_h u_k| \le g$ a.e. and
$u_k \ge h$ a.e., for all $k \in \mathbb{N}$ then $\underline u := \inf_k u_k \in W^{1,p}(U)$ and $|D_h \underline u| \le g$ a.e.
\end{enumerate}
\end{proposition}
\begin{proof} For assertion~\eqref{it:basic_density} see
Friedrichs \cite{friedrichs_1944} or Thm.\ 1.13 and Thm.\ A.2 in
\cite{garofalo_nhieu_1996}. Assertions~\eqref{it:basic_chain} and \eqref{it:basic_absolute} follow from \eqref{it:basic_density}
in the same way as in the Euclidean case (see, for example \cite[Sec. 7.4]{gilbarg_trudinger} for the Euclidean setting). Indeed, for \eqref{it:basic_absolute}
one applies \eqref{it:basic_chain} with $\psi_\varepsilon(t) = \sqrt{t^2 + \varepsilon^2} - \varepsilon$ and takes the limit $\varepsilon \to 0$.
Assertion~\eqref{it:basic_min} follows from \eqref{it:basic_absolute} since $\min(u,v) = \frac12( u+v) - \frac12 |u-v|$. To prove
assertion~\eqref{it:basic_inf} set $w_k = \inf_{j \le k} u_j$. It follows from \eqref{it:basic_min} that $w_k \in W^{1,p}(U)$ and $|D_h w_k| \le g$.
Moreover $k \mapsto w_k$ is non-increasing. Since $w_k \ge h$ and $h \in L^p(U)$, the monotone convergence theorem implies that
$w_k \to \underline u$ in $L^p(U)$. Moreover a subsequence of the weak derivatives $X_i w_k$ converges weakly in $L^p(U)$ to a limit
$h_i$ (for $p=1$ use the Dunford-Pettis theorem). Thus $\underline u$ is weakly differentiable with weak derivatives $h_i$. By weak lower semicontinuity
of the norm we deduce that $(\sum_i h_i^2)^{1/2} \le |g|$.
\end{proof}
\bigskip
We now consider spaces of $L^p$ functions and Sobolev functions with values in a metric space. We will later only consider a Carnot group $G'$
with the Carnot-Caratheodory metric as the target space, but we state the results for general targets to emphasize that they do
not use the structure of a Carnot group.
The following definition is due to Reshetnyak \cite{reshetnyak_1997} for open subsets of $\mathbb{R}^n$ or a Riemannian manifold
and has been extended
by Vodopyanov \cite[Proposition 3, p.\ 674] {vodopyanov_bounded_distortion} to the setting of Carnot groups.
\begin{definition} \label{de:sobolev_carnot_new}
Let $(X', d')$ be a complete separable metric space and let $U \subset G$ be open.
\begin{enumerate}
\item We say that a map $f: U \to X'$ is in $L^p(U,X')$ if $f$ is measurable and if there exist an $a \in X'$ such that
the map $x \mapsto d(f(x), a)$ is in $L^p(U)$.
\item We say that $f \in L^p(U,X')$ is in
the Sobolev space $W^{1,p}(U;X')$
if for all $z \in X'$ the functions $u_z(\cdot) := d'(f(\cdot), z) - d'(a,z)$ are in $W^{1,p}(U)$
and if there exists a function $g \in L^p(U)$ such that
\begin{equation} \label{eq:bound_D_h_metric} |D_h u_z| \le g
\end{equation}
almost everywhere.
\end{enumerate}
The spaces $L^p_{loc}(U;X')$ and $W^{1,p}_{loc}(X')$ are defined as usual.
\end{definition}
Note that by the triangle inequality the map $x \mapsto d'(f(x), z)$ is in $L^p_{loc}(U)$ for all $z \in X'$
if it is in $L^p_{loc}(U)$ for one $z \in X'$; if $\mu(U) < \infty$ then the same assertion holds for $L^p(U)$. Note however that the assertion fails for $L^p(U)$ when $\mu(U)=\infty$.
Definition~\ref{de:sobolev_carnot_new} imposes estimates on the weak derivatives of $f$ composed with the distance functions $d(z, \cdot)$. These imply similar estimates on the composition with general Lipschitz functions from $G'$ to a finite-dimensional linear space:
\begin{proposition} \label{pr:composition_by_lip}
Let $(X', d')$ be a complete separable metric space and let $U \subset G$ be open.
Let $f \in W^{1,p}(U; G')$ and let $a \in G'$ be such that $x \mapsto d'(a, f(x))$ is in $L^p(U)$.
Let $Y$ be a finite-dimensional $\mathbb{R}$-vector space and $v : G' \to Y$ be Lipschitz. Then $v \circ f - v(a)
\in W^{1,p}(U;Y)$.
\end{proposition}
\begin{proof}
It suffices to show the assertion for $Y= \mathbb{R}$. So let $u :X' \to \mathbb{R}$ be $L$-Lipschitz and let $D \subset X'$ be a countable dense set.
Then $v \circ f$ in $L^p(U)$ and
$$ v(z) = \inf_{z' \in D} v(z') + d'(z', z).$$
Thus
the assertion follows from the definition of $W^{1,p}(U;X')$ and Proposition~\ref{pr:sobolev_basic}~\eqref{it:basic_inf}.
\end{proof}
\bigskip
We will use the following version of the Lebesgue point theorem. Here and elsewhere in the paper we use the standard notation $\Xint-$ to denote an average.
\begin{lemma} \label{le:lebesgue_point_new}
Let $f \in L^p_{loc}(U, X')$. Then for a.e. $x \in U$ we have
\begin{equation} \label{eq:p_lebesgue_point_new}
\lim_{r \to 0}
\Xint-_{B(x,r)} [ d'(f(y), f(x))]^p \, d\mu(y) = 0.
\end{equation}
\end{lemma}
\begin{proof}This follows easily by applying the usual Lebesgue point theorem to the scalar functions
$u_z(y) = d'(f(y), z)$ where $z$ runs through a countable dense subset of $X'$.\end{proof}
To recall the Poincar\'e-Sobolev inequality for metric-space-valued maps we define the $L^p$-oscillation as follows.
\begin{definition} Let $X'$ be a metric space. Let $A \subset G$ be a measurable set and let $f \in L^p(A,X')$.
The $L^p$ oscillation on $A$ is defined by
\begin{equation}
\operatorname{osc}_p(f, A) := \inf_{a \in X'} \left( \int_{A} {d'}^p(f(x), a) d\mu(x) \right)^{1/p}.
\end{equation}
\end{definition}
\bigskip
There is a general strategy for deducing the Poincar\'e-Sobolev inequality for metric-space-valued maps
from the Poincar\'e-Sobolev inequality for scalar valued functions. It is based on the derivation of a pointwise estimate for a.e.\ pair of points
and a chaining argument, see Theorem 9.1.15 in \cite{HKST} for an implementation of this approach in the context of the upper gradient definition.
Since we are only interested in Carnot groups as targets we use a more pedestrian approach.
Recall that a metric space is doubling
if every ball of radius $r >0$ can be covered by a fixed number $M$
of balls of radius $\frac{r}{2}$. Carnot groups are doubling since by compactness $B(e,1)$ can be covered by $M$ balls
of radius $\frac12$. By translation and scaling every ball of radius $r$ can be covered by $M$ balls of radius $\frac{r}{2}$.
\begin{theorem} Let $G$ be a Carnot group of homogeneous dimension $\nu$ and let $X'$ be a complete, separable metric space which is doubling. Let $1 \le p < \nu$ and define $p^*$ by
$$\frac{1}{p*} = \frac1p - \frac1\nu.$$ There exists a constant $C = C(G,p)$ with the following property.
Let $B(x,r)$ be a ball in $G$, let
$f \in W^{1,p}(B(x,r),X')$ and let $g \in L^p(B(x,r))$ be the function in Definition~\ref{de:sobolev_carnot_new}. Then
\begin{equation} \label{eq:sobolev_poincare_new}
\operatorname{osc}_{p^*}(f, B(x,r)) \leq C \|g\|_{L^p(B(x,r))}
\end{equation}
and
\begin{equation} \label{eq:p_poincare_new}
\operatorname{osc}_{p}( f, B(x,r)) \le C r \|g\|_{L^p(B(x,r))}.
\end{equation}
\end{theorem}
\begin{proof} If suffices to prove the first estimate, since the second follows from the first by H\"older's inequality.
For the Poincar\'e-Sobolev inequality for scalar functions see \cite[Corollary 1.6.]{garofalo_nhieu_1996} or \cite[Theorem 2.1]{lu_poincare_1994}.
By scaling and translation in $G$ it suffices to show the Poincar\'e inequality for $X'$-valued maps
for the set $B = B(e,1)$ and we may assume that the Haar measure $\mu$ is normalized so that
$\mu(B) = 1$.
Let $f \in W^{1,p}(B, X')$ and for $z \in X'$ define $u_z(x) = d'(z, f(x))$.
By Definition~\ref{de:sobolev_carnot_new} and the Poincar\'e inequality for
scalar functions we see that $u_z \in L^{p*}(B)$ for all $z \in D$. Thus $ L:= \operatorname{osc}_{p*}( f, B) = \inf_z \|u_z\|_{p^*,B} < \infty$.
Let $\bar a \in X'$ be such that the infimum is achieved.
Then
$$ \mu \{ x \in B: f(x) \in B(\bar a, 2L) \} \ge (1- 2^{-p^*}) \mu(B) \ge \frac12.$$
Since $X'$ is doubling there exist $M^2$ balls of radius $\frac{L}{2}$ which cover $B(\bar a, 2L)$. Thus there exist $z \in X'$ such that
$$ \mu(E) \ge \frac12 M^{-2} \quad \text{where $E= f^{-1}(B(z, \frac{L}{2}))$}.$$
By the Sobolev-Poincar\'e inequality for scalar-valued
functions and the triangle inequality we have
\begin{equation} \label{eq:poincare_double} \int_B \int_B |u_z(x) - u_z(y) |^{p^*} \, d\mu(x) \, d\mu(y)
\le C \|g \|_{p, B}^{p^*} \,.
\end{equation}
Let $$v(x) = \max( u_z(x) - \frac{L}{2}, 0).$$ For $y \in E$ we have $u_z(y) \le \frac{L}{2}$ and hence $v(x) \le |u_z(x) - u_z(y)|$.
Restricting the outer integral on the left hand side of \eqref{eq:poincare_double} to the set $E$ we get
$$ \int_B v^{p^*} \, d\mu \le 2 C M^2 \|g \|_{p, B}^{p^*}.$$
Thus $\| v \|_{p^*, B} \le C(M, p) \| g\|_{p, B}$.
By the definition of the oscillation and the definition of $v$ we have
$$ \| v \|_{p^*,B} \ge \| u_z\|_{p^*, B} - \frac{L}{2} \ge \frac{L}{2}.$$
Hence $ \operatorname{osc}_{p^*}(f, B) = L \le 2 C(M,p) \| g\|_{p, B}$.
\end{proof}
\bigskip\bigskip
We finally discuss differentiability results. It is well known that locally Lipschitz maps from $\mathbb{R}^n$ to $\mathbb{R}^m$
are differentiable a.e. In fact maps
in $f \in W^{1,p}_{loc}(\mathbb{R}^n; \mathbb{R}^m)$ are differentiable a.e.\ if $p> n$. For $1 \le p < n$ the map $f$ is differentiable in an $W^{1,p}$ sense.
For maps between Carnot groups, Pansu \cite{pansu} showed that Lipschitz maps (with respect to the Carnot-Caratheodory metrics) on open sets
are a.e. differentiable in the following sense, now known as Pansu differentiability. For a.e. $x \in U$
there exist a graded group homomorphism $\Phi: G \to G'$ such that
the rescaled maps
\begin{equation} \label{eq:pansu_rescaling_new}
f_{x,r} := \delta_{r^{-1}} \circ \ell_{f(x)^{-1}} \circ f \circ \ell_x \circ \delta_r.
\end{equation}
converge locally uniformly to as $r\to 0$.
The corresponding Pansu differentiability results for Sobolev maps between Carnot groups
have been obtained by Vodopyanov \cite[Theorems 1 and 2, Corollaries 1 and 2]{vodopyanov_differentiability_2003}, see also \cite{vodopyanov_carnot_manifolds} for extensions to
maps between Carnot manifolds.
We will use the following result.
\begin{theorem}[$L^{p*}$ Pansu differentiability a.e., \cite{vodopyanov_differentiability_2003}, Corollary 2]
\label{th:Lp*_pansu_differentiability_new}
Let $ U \subset G$ be open, let $1 \le p < \nu$
and define $p^*$ by $\frac1{p^*} = \frac1p - \frac1\nu$.
Let $f \in W^{1,p}(U;G')$. For $x \in U$ consider the rescaled maps
$$ f_{x,r} = \delta_{r^{-1}} \circ \ell_{f(x)^{-1}} \circ f \circ \ell_x \circ \delta_r.$$
Then, for a.e. $x \in U$, there exists a group homomorphism $\Phi: G \to G'$ such that
\begin{equation} \label{eq:Lpstart_pansu_differentiability_new}
f_{x,r} \to \Phi \quad \text{in $L^{p^*}_{loc}(G;G')$ as $r \to 0$.}
\end{equation}
\end{theorem}
\begin{remark} \label{re:g_equal_norm_DPf}
It follows easily from Theorem~\ref{th:pansu_diff_Lpstar} that for all $z \in G'$
the functions $u_z:= d'(z, f(\cdot))$ satisfy
\begin{equation} \label{eq:optimal_g}
\text{$|D_h u_z| \le |D_P f|$ a.e., where}
\end{equation}
\begin{equation}
|D_P f(x)| = \max \{ |D_P f(x) X|_{V'_1} : \, X \in V_1, \, |X|_{V_1} \le 1 \}.
\end{equation}
Here $| \cdot |_{V_1}$ and $| \cdot |_{V'_1}$ denote the norms induced by the scalar product on the first layer of $\mathfrak{g}$ and $\mathfrak{g}'$,
respectively. Thus the condition \eqref{eq:bound_D_h_metric}
in Definition~\ref{de:sobolev_carnot_new} holds with $g = |D_P f|$.
The short proof of \eqref{eq:optimal_g}can be found in Appendix~\ref{se:W1p_differentiability}.
\end{remark}
\bigskip
\begin{remark} \label{re:weak_derivative_abel}
Let $\pi_{G'} : G' \to G'/[G',G']$ denote the abelianization map. Since $\pi_{G'}$ is globally Lipschitz,
the map $\pi_{G'} \circ f$ is in $W^{1,p}(U, G'/[G',G'])$. It easily follows from
Theorem~\ref{th:Lp*_pansu_differentiability_new} that the weak (or distributional) derivative of
$\pi_{G'} \circ f$ in direction of a horizontal vectorfield $X$ satisfies
\begin{equation} \label{eq:weak_derivative_abel}
X (\pi_{G'} \circ f)(x) = D_P f(x) X \quad \text{for a.e. $x$,}
\end{equation}
see Remark~\ref{rem_step_2_of_proof} below.
In \eqref{eq:weak_derivative_abel} we have identified the abelian group $G'/[G',G']$ with the first layer $V'_1$ of $\mathfrak{g}'$.
\end{remark}
\bigskip
Vodopyanov's proof in \cite{vodopyanov_differentiability_2003} combines work from a series of earlier papers \cite{vodopyanov_ukhlov_approximate_differentiable,vodopyanov_monotone_1996,vodopyanov_bounded_distortion,vodopyanov_P_differentiability}. His argument is based on Lipschitz approximation on sets of almost full measure,
an extension of Pansu's result to Lipschitz maps defined on sets $E \subset G$
which are not open and a careful estimate of the remainder terms at points in $E$ of density one.
In Appendix~\ref{se:W1p_differentiability} we give a direct alternative proof of $L^p$ differentiability which
is based on blow-up, the Poincar\'e-Sobolev inequality, the compact Sobolev embedding (which is an immediate consequence of the Poincar\'e-Sobolev inequality)
and the following observation: if $F : G \to G'$ is a Lipschitz map with $F(e) = e$ and the abelianization $\pi_{G'} \circ F$ is affine
(i.e. has constant weak horizontal derivatives) then $F$ is a graded group homomorphism.
\section{Center of mass and mollification}
\label{sec_center_of_mass_mollification}
In this section we first define a center of mass for measures on a Carnot group which satisfy a moment condition. We then use this to define a mollification procedure for maps with finite $L^m$-oscillation taking values in an $m$-step Carnot group.
We will be using the notation and results from Section~\ref{sec_prelim}, in particular the `homogeneous norm' $|\cdot|$ on $\mathfrak{g}$ defined in (\ref{eq:homogeneous_norm}), and the Euclidean norm $|\cdot|_e$.
\subsection{Center of mass in Carnot groups}
\label{subsec_center_of_mass}
Let $G$ be a $m$-step Carnot group (for an extension to connected, simple connected nilpotent Lie groups see
Remark~\ref{re:com_nilpotent} and Remark~\ref{re:naturality_com_nilpotent} below).
Let $\nu$ be a Borel probability measure on $G$. We say that $\nu$ has finite $p$-th moment if
\begin{equation} \label{eq:p_th_moment_nu}
\int_G d^p_{CC}(e, y) \, d\nu(y) < \infty.
\end{equation}
In view of \eqref{eq:ball_box} this is equivalent to
\begin{equation}
\label{eq:p_th_moment_nu_bis}
\int_\mathfrak{g} |Y|^p \, d(\log_*\nu)(Y) < \infty.
\end{equation}
In this subsection we define, for probability measures with finite $m$-th moment, a center of mass
which is compatible with left translation and group homomorphisms.
For probability measures $\nu$ with compact support our notion of center of mass agrees with the one by Buser and Karcher \cite[Example 8.1.8] {karcher_buser_almost_flat_manifolds}.
Their proof of the existence of the center of mass is different.
They use the bi-invariant flat connection $D$ such that left-invariant vector fields are $D$-parallel, and base their proof on some estimates for the convexity radius of $D$ with respect to some auxiliary left invariant Riemannian metric.
Here we argue directly on the Lie algebra
and also show that there is an explicit recursive formula for (the logarithm of) the center of mass
and that the logarithm of the center of mass is a polynomial in certain polynomial moments of
$\log_* \nu$.
The extension from compactly supported measures to measures with finite $m$-th moment
will be crucial in the next subsection where we use the center of mass to define a group compatible
mollification for (Sobolev) functions which may be unbounded.
We define
\begin{equation} \label{eq:log_x}
\log_x = \log \circ \ell_{x^{-1}}
\end{equation}
\begin{theorem} \label{le:C_nu_diffeomorphism}
Let $G$ be an $m$-step Carnot group and let $\nu$ be a Borel probability measure on $G$
with finite $m$-th moment. Then $\log_x$ is $\nu$ integrable and the map $C_\nu: G \to \mathfrak{g}$ defined
by
\begin{equation}
C_\nu(x) := \int_G \log_x \, d\nu
\end{equation}
is a diffeomorphism. Moreover $C_\nu \circ \exp: \mathfrak{g} \to \mathfrak{g}$ is a polynomial of degree not larger than $m-1$.
For any $Z\in \mathfrak{g}$, the equation $C_\nu(\exp X) = Z$ can be solved recursively
and $\log (C_\nu)^{-1}(Z)$ is a polynomial in $Z$ and certain polynomial moments of $\log_* \nu$.
In particular there exist a $\mathfrak{g}$-valued polynomial $Q$
with $Q(0, \ldots, 0) = 0$ and $\mathfrak{g}$-valued
multilinear forms $L_1, \ldots L_K: \mathfrak{g} \to \mathfrak{g}$ such that
\begin{equation} \label{eq:formula_com1}
\log (C_\nu)^{-1}(0) = Q(A_1, \ldots A_K),
\end{equation}
where
\begin{equation} \label{eq:formula_com2}
A_i = \int_{\mathfrak{g}} L_i(Y, \ldots, Y) \, d(\log_*\nu)(Y).
\end{equation}
Moreover
\begin{equation} \label{eq:bound_Qi}
|L_i(Y, \ldots, Y)|_e \le C_i (1 + |Y|^m) \quad \text{for $1 \le i \le K$.}
\end{equation}
\end{theorem}
We call
\begin{equation} \label{eq:define_com}
\operatorname{com}_\nu := (C_\nu)^{-1}(0)
\end{equation}
the center of mass of $\nu$.
\bigskip\bigskip
\begin{remark}
The proof shows the condition that $\nu$ has finite $m$-th moment can be slightly weakened. It suffices to assume that
\begin{equation} \label{eq:weakened_moment_condition}
\int_{\mathfrak{g}} (|Y|^{m-1} + |Y|_e ) \, d\log_* \nu < \infty.
\end{equation}
The reason is that elements of $V_m$ do not appear in the BCH term $P(X,Y)$ and that $P(X,Y)$
is polynomial in $Y$ of degree
not exceeding $m-1$, see also \eqref{eq:bound_P}. Recall from \eqref{eq:homogeneous_norm} that the homogeneous norm $|X|$ is equivalent to $\sum_{i=1}^m |\pi_j X|_e^{\frac1j}$ where $\pi_j : \mathfrak{g} \to V_j = \mathfrak{g}_j$
is the projection to the $j$-th layer of the algebra. Thus condition \eqref{eq:weakened_moment_condition} is equivalent to the condition that
$|Y|^{m-1} + |\pi_m Y|_e$ is $\log_* \nu$ integrable.
\end{remark}
\bigskip\bigskip
\begin{proof}[Proof of Theorem~\ref{le:C_nu_diffeomorphism}]
It is easier to work on the algebra $\mathfrak{g}$ rather than the group $G$.
We thus define
$$ \tilde C_\nu(X) = C_\nu(\exp X).$$
Then, using the BCH formula, we get
\begin{equation}
\label{eqn_c_nu_tilde_formula}
\begin{aligned}
\tilde C_\nu(X) = & \, \int_G \log( \exp(-X) y) \, d\nu(y) \\
= & \, \int_{\mathfrak{g}} \log( \exp(-X) \exp Y) \, d\log_*\nu(Y) \\
= & \, - X + \int_{\mathfrak{g}} ( Y + P(-X,Y)) \, d\log_*\nu(Y).
\end{aligned}
\end{equation}
The integrand is a polynomial of degree at most $m-1$ in $X$.
By the definition of the homogeneous norm we have $|Y|_e \le C( |Y| + |Y|^m)$.
Moreover by \eqref{eq:bound_P} we have $|P(X,Y)| \le C(X) ( 1 + |Y|^{m-1})$.
Since $\nu$ is a probability measure with finite $m$-th moment the integral in (\ref{eqn_c_nu_tilde_formula}) exists.
The bounds \eqref{eq:bound_derivatives_P} on the derivatives imply that differentiation with respect
to $X$ and integration commute. Thus $\tilde C_\nu$ is a polynomial of degree at most $m-1$.
We now show that for every $Z\in \mathfrak{g}$, the equation
\begin{equation} \label{eq:equation_inverse_Cnu} \tilde C_\nu(X) = Z
\end{equation}
has a unique solution which is a polynomial in $Z$, and moreover depends only on certain polynomial moments of $\log_* \nu$ of degree at most $m-1$.
Recall that $\pi_i$ denotes the projection from $\mathfrak{g} = \oplus_{i=j}^m V_j$ to $V_i$.
Define functions $P^i$ by $P^i(X,Y) = \pi_i P(X,Y)$ and set $X^i = \pi_i X$, $Y^i = \pi_i Y$.
Applying $\pi_i$ to \eqref{eq:equation_inverse_Cnu} we get a system of $m$ equations, namely
\begin{equation} \label{eq:inverse_C_nu}
- X^i + \int_{\mathfrak{g}} Y^i + P^i(-X, Y) d\log_*\nu(Y) = Z^i \quad \text{for $i \in \{1, \ldots, m\}$.}
\end{equation}
Since $P$ consists of iterated commutators one easily sees that $P_i(X,Y)$ depends on $X$
only through $(X^1, \ldots, X^{i-1})$.
Thus the system can be solved recursively starting with
$$ X^1 = -Z^1 + \int_{\mathfrak{g}} Y^1 \, d\log_*\nu(Y).$$
Moreover the solution is polynomial in $Z$ and in particular smooth.
We finally discuss the dependence of the solution on $\log_* \nu$. Since $[V_i, V_j] \subset V_{i+j}$ we see as
in Lemma~\ref{pr:bounds_P} that
$P^i(X,Y)$ can be written as
\begin{align} \label{eq:decomp_Pi}
& \, \quad P^i(X,Y) \\
= & \sum_{p=1}^{d_i} \, \sum_{j=2}^{i} \sum_{k=1}^{j-1} \sum_{ \sum_{\ell = 1}^j i_\ell = i}
M^p_{i_1 i_2 \ldots i_k}(X^{i_1}, \ldots X^{i_k}) \, \,
L^p_{i_{k+1} \ldots i_j}(Y^{i_{k+1}}, \ldots Y^{i_j}) E_p \nonumber
\end{align}
where $E_1, \ldots, E_{d_i}$ is a basis of $V_i$ and where
$M_{I}^p$ and $L_{I}^p$ are multilinear forms.
Let $\bar X = \log \operatorname{com}_\nu$. Then it follows from \eqref{eq:inverse_C_nu}
that $\bar X$ can be recursively computed as
\begin{equation} \label{eq:formula_com}
\bar X^i = \int_{\mathfrak{g}} \Big( Y^i + P^i(-\bar X, Y) \Big) \, d\log_*\nu(Y) \quad \text{for $i \in \{1, \ldots, m\}$, }
\end{equation}
starting with $\bar X^1 = \int_{\mathfrak{g}} Y^1 \, d\log_*\nu(Y)$.
Hence $\bar X$ is a polynomial expression in
$$ \bar Y^i = \int_{\mathfrak{g}} Y^i \, d\log_*\nu(Y)$$
and
the polynomial moments
$$ A^i_{i_{k+1}, \ldots, i_j} := \int_{\mathfrak{g}}
L^i_{i_{k+1} \ldots i_j}(Y^{i_{k+1}}, \ldots Y^{i_j}) \, d\log_*\nu(Y).
$$
Since $s := \sum_{\ell = k+1}^j i_\ell \le i-k \le m-1$ we have by multilinearity
\begin{align} \label{eq:bound_Li_com}
& \, | L^i_{i_{k+1} \ldots i_j}(Y^{i_{k+1}}, \ldots Y^{i_j}) |_e
\le C \prod_{\ell = k+1}^j |Y^{i_\ell}|_e \\
\le & \, C \prod_{\ell = k+1}^j |Y^{i_\ell}|^{i_\ell}
\le C |Y|^s \le C (1 + |Y|^m). \nonumber
\end{align}
Moreover $|Y^i|_e \le |Y|^i \le 1 + |Y|^m$.
\end{proof}
\bigskip\bigskip
\begin{remark} \label{re:com_nilpotent} The conclusion of Theorem~\ref{le:C_nu_diffeomorphism} continues to hold if we consider a nilpotent group instead of a Carnot group, and use (\ref{eq:p_th_moment_nu_bis}) rather than (\ref{eq:p_th_moment_nu}) to define the $p$-th moment, where the homogenous norm is defined as
in
Remark~\ref{re:BCH_bounds_nilpotent} using a (noncanonical) decomposition $\mathfrak{g} = \oplus W_j$ with $ \mathfrak{g}_j= W_j \oplus \mathfrak{g}_{j+1}$ and denoting by $\pi_j$ the projection from $\mathfrak{g}$ to $W_j$.
Indeed, by Remark~\ref{re:BCH_bounds_nilpotent} the bounds in Lemma~\ref{pr:bounds_P} also hold in the nilpotent case. Thus $C_\nu \circ \exp$ is well-defined
and a polynomial of degree at most $m-1$. In view of Remark \ref{re:norms_and_complement} the condition that the probability measure $\log_* \nu$ has finite $m$-th moment
is independent of the choice of the auxiliary spaces $W_i$ since different choices lead to homogeneous norms $|\cdot|_\sim$ and $|\cdot|$
which satisfy $|X|_\sim \le C( 1 + |X|^m)$ and $|X| \le C(1 + |X|_{\sim}^m)$.
To see that the equation \eqref{eq:equation_inverse_Cnu} can be solved recursively, consider the
projections $\pi_j : \mathfrak{g} \to W_j$ and $\tilde \pi_j = \sum_{i=1}^j \pi_i$.
Since $[W_j, W_k] \subset \oplus_{i=j+k}^m W_i$, the expression $\tilde \pi_j [X,Y]$ depends only on $\tilde \pi_{j-1}(X)$ and $ \tilde \pi_{j-1}(Y)$
and thus $\tilde \pi_j P(X,Y)$ depends only on $\tilde \pi_{j-1}(X)$ and $ \tilde \pi_{j-1}(Y)$. Hence
\eqref{eq:equation_inverse_Cnu} can be again solved recursively by successively applying the projections $\tilde \pi_1, \ldots, \tilde \pi_m = \operatorname{id}$.
The projection $\tilde \pi_i P(X,Y)$ can again be expressed as a sum of products of multilinear terms in $X$ and $Y$.
The only difference is that the condition $ \sum_{\ell=1}^j = i$ is replaced by $ \sum_{\ell=1}^j \le i$.
Nonetheless the bound \eqref{eq:bound_Li_com} still holds and this implies
\eqref{eq:bound_Qi}.
We note in passing that one can show the recursive solvability of \eqref{eq:equation_inverse_Cnu} without introducing
the spaces $W_i$, by considering the abstract projections $\bar \pi_1, \ldots, \bar \pi_m = \operatorname{id}$ given by
$\bar \pi_i: \mathfrak{g} \to \mathfrak{g} / \mathfrak{g}_{i+1}$.
\end{remark}
\bigskip
\begin{remark} \label{re:com_two_step} For a step-$2$ group $C_\nu \circ \exp$ is an affine function and thus
\begin{equation} \label{eq:mollification_2step}
\log \operatorname{com}_\nu = \int_{\mathfrak{g}} Y \, d\log_* \nu(Y) \quad \text{for step-$2$ groups.}
\end{equation}
\end{remark}
\bigskip\bigskip
We now show that the center of mass defined by $\operatorname{com}_\nu = C_\nu^{-1}(0)$ commutes with left translations,
inversion and group homomorphisms.
\begin{lemma} \label{le:center_behaves_natural}
Let $G$ and $G'$ be Carnot groups of step $m$ and $m'$, respectively. Let $\Phi: G \to G'$ be a group homomorphism (with derivative $D\Phi: \mathfrak{g} \to \mathfrak{g}'$), let $I: G \to G$ be the inversion map given by $I(x) = x^{-1}$, and $\nu$ be a Borel probability measure on $G$.
If $\nu$ has finite $m$-th moment then $(\ell_z)_* \nu$ and $I_* \nu$ are Borel probability measures on $G$ with finite $m$-th moment and
\begin{eqnarray}
\operatorname{com}_{(\ell_z)_* \nu} &=& \ell_z(\operatorname{com}_\nu), \label{eq:com_translation}\\
\operatorname{com}_{I_* \nu} &=& I(\operatorname{com}_\nu). \label{eq:com_inversion}
\end{eqnarray}
In particular, if $\nu$ is reflection symmetric, i.e., if $I_* \nu = \nu$, then
\begin{equation} \label{eq:com_inversion_invariant}
\operatorname{com}_{(\ell_z)_*\nu} = z.
\end{equation}
If $\nu$ has finite $p$-th moment for $p\geq \max(m,m')$, then $\Phi_* \nu$ is a Borel probability measure on $G'$ with finite $p$-th moment, and
\begin{equation}
\operatorname{com}_{\Phi_* \nu} = \Phi(\operatorname{com}_\nu)\,. \label{eq:com_Phi}
\end{equation}
\end{lemma}
\begin{remark} \label{re:naturality_com_nilpotent} The assertion and the proof immediately extend to nilpotent groups. To bound the $m$-th moment of the measures
$I_* \nu$ and $(\ell_x)_*\nu$ one uses \eqref{eq:bound_P_nilpotent_homogeneous_norm} instead of
\eqref{eq:bound_P_homogeneous_norm}.
\end{remark}
\begin{proof}
Assume that $\nu$ has finite $p$-th moment. We have $|\log \circ I(y)| = |- \log y| = |\log y|$. Thus $I_* \nu$ has finite $p$-th moment.
Similarly $\log \circ \ell_{x}(y) = \log x + \log y + P(\log x, \log y)$. By
\eqref{eq:bound_P_homogeneous_norm} we have $|P(\log x, \log y)| \le C(|\log x| + |\log y|)$. Thus $(\ell_x)_*\nu$ has finite $m$-th moment.
To control the moment of $\Phi_* \nu$ we first note that $\Phi$ preserves the one-parameter subgroups and thus
$ \log_{G'} \circ \Phi \circ \exp_G = D \Phi$.
It follows that
$ (\log_{G'})_* \Phi_* \nu = (D\Phi)_* (\log_{G})_*\nu$.
It thus suffices to show that
\begin{equation} \label{eq:bound_DPhi_natural}
|D \Phi (X)| \le C (|X|^\frac1m + |X|).
\end{equation}
To show this, observe that in a Carnot algebra the elements $\mathfrak{g}_j$ of the descending series are given by
$\mathfrak{g}_j = \oplus_{i=j}^m V_i$. Since $D\Phi$ is a Lie algebra homomorphism we have $D\Phi(\mathfrak{g}_j) \subset \mathfrak{g}'_j$.
Note also that $D\Phi$ is linear and hence bounded with respect to the Euclidean norms. Thus we have
for $X \in V_j$
$$ |D \Phi (X)| \le C \sum_{i=j}^m |\pi_i D\Phi (X)|_e^{\frac1i}
\le C \sum_{i=j}^m |X|_e^{\frac1i} = C \sum_{i=j}^m |X|^\frac{j}{i}.$$
By Young's inequality we have $|X|^\frac{j}{i} \le C (|X|^\frac1m + |X|)$ whenever $i \ge j$.
By linearity we get \eqref{eq:bound_DPhi_natural} for all $X$.
To prove \eqref{eq:com_translation}--\eqref{eq:com_Phi}
it suffices to verify the corresponding
transformation rules for $C_\nu$. We have
$$ \log[ (\Phi(x))^{-1} \Phi(y) ]= \log[ \Phi(x^{-1} y)] = D\Phi(\log(x^{-1}y))$$
and thus
$$ C_{\Phi_* \nu}(\Phi(x)) = D\Phi (C_{\nu}(x)).$$
Setting $x = \operatorname{com}_\nu$ we get $C_{\Phi_* \nu}(\Phi(\operatorname{com}_\nu)) = 0$ and thus \eqref{eq:com_Phi}.
Similarly the relation
$$ \log [I(x^{-1} I(y)] = \log [ I(x^{-1}y) ] = - \log(x^{-1}y)$$
gives \eqref{eq:com_inversion} while the identity
$$ (\ell_z x)^{-1} (\ell_z y) = x^{-1} z^{-1} z y = x^{-1} y$$
gives \eqref{eq:com_translation}.
Now assume that $\nu$ is reflection symmetric. Then \eqref{eq:com_inversion} implies that
$\operatorname{com}_{\nu} = e$. In combination with
\eqref{eq:com_translation} we obtain \eqref{eq:com_inversion_invariant}.
\end{proof}
\bigskip\bigskip
\subsection{Mollifying maps between Carnot groups}
\label{subsec_mollifying_maps_between_carnot_groups}
In this subsection we define a mollification procedure for $L^p_{\operatorname{loc}}$-mappings into a Carnot group. Traditional mollification of mappings into a linear target is based on averaging; since Carnot groups are not linear spaces, we replace averaging with the center of mass from Subsection~\ref{subsec_center_of_mass}.
Let $\sigma_1$ be a smooth probability measure on a Carnot group $G$ with $\operatorname{spt}(\sigma_1)
\subset B(e,1)$. Thus $\sigma_1 = \alpha \mu$ where $\mu$ is the biinvariant measure on $G$
and $\alpha \in C_c^\infty(B(e,1))$. We also assume that $\sigma_1$ is symmetric under inversion:
$I_*\sigma_1=\sigma_1$, where $I(x)=x^{-1}$.
For $x\in G$, $\rho\in (0,\infty)$, let $\sigma_\rho$, $\sigma_x$, and $\sigma_{x,\rho}$
be the pushforwards of $\sigma_1$ under the the corresponding Carnot scaling
and left translation:
\begin{equation} \label{eq:transported_measures}
\sigma_\rho=(\delta_\rho)_*\sigma_1\,,\quad \sigma_x=(\ell_x)_*\sigma_1\,,\quad
\sigma_{x,\rho}=(\ell_x\circ\delta_\rho)_*\sigma_1=(\ell_x)_*(\sigma_\rho)\,.
\end{equation}
Let $G$ be any Carnot group and let $G'$ be an $m$-step Carnot group.
Recall that $f:G\rightarrow G'$ is in $L^m_{loc}(G,G')$ if $f$ is measurable
and
\begin{equation} \label{eq:f_in_Lm_loc}
\hbox{$ y \mapsto d_{CC, G'} (f(y), e) $
belongs to $L^m_{loc}(G)$}.
\end{equation}
In particular every continuous map belongs to $L^m_{loc}(G, G')$.
We will see shortly that $f \in L^m_{loc}(G,G')$ implies that the push-forward measure
$f_* \sigma_z$ has finite $m$-th moment.
We may then define a mollified map $f_1:G\rightarrow G'$ by
\begin{equation} \label{eq:mollification}
f_1(x)=\operatorname{com}(f_*(\sigma_{x}))\,,
\end{equation}
and maps $f_\rho:G\rightarrow G'$ by
\begin{equation} \label{eq:mollification_scaled}
f_\rho=\delta_\rho\circ(\delta_{\rho^{-1}}\circ f\circ \delta_\rho)_1\circ \delta_{\rho^{-1}}\,.
\end{equation}
Recall that for $p \in [1, \infty)$ the $L^p$ oscillation on a set $A$ is defined by
\begin{equation} \label{eq:osc_Lp}
\operatorname{osc}_p(f,A) := \inf_{a \in G'} \left( \int_A d_{CC, G'}^p(f(y), a) \, \mu(dy) \right)^{1/p}
\end{equation}
where $\mu$ is the biinvariant measure on $G$.
\begin{lemma}
\label{le:moll_prop_Lm}
Let $G$ be a Carnot group, let $G'$ be an $m$-step Carnot group, let $ p \in [m, \infty)$ and $f \in L^p_{loc}(G,G')$.
As above let $\sigma_1 = \alpha \mu$ with $\alpha \in C_c^\infty(B(e,1))$ and $I_* \sigma_1 = \sigma_1$. Define $\sigma_z$
by \eqref{eq:transported_measures} and $f_1$ and $f_\rho$ by \eqref{eq:mollification} and
\eqref{eq:mollification_scaled}.
Then:
\begin{enumerate}
\item For all $z \in G$ the measures $f_* \sigma_z$ have finite $p$-th moment.
\item For all $\rho\in(0,\infty)$,
$$
\delta_{\rho^{-1}}\circ f_\rho\circ\delta_\rho
=(\delta_{\rho^{-1}}\circ f\circ \delta_\rho)_1\,.
$$
\item For all $a \in G$ and $b \in G'$
\begin{equation*} \label{eq:smoothing_commutes_translation}
(\ell_b \circ f \circ \ell_a)_1 = \ell_b \circ f_1 \circ \ell_a.
\end{equation*}
\item For all $\rho > 0$
$$ (\delta_\rho \circ f)_1 = \delta_\rho \circ f_1.$$
\item \label{it:bound_oscillation_f1}
Assume that for some $x_0 \in G$ and some $a \in G'$
$$ \int_{B(x_0,1)} d^m_{CC, G'}(f(x), a) \, d\mu \le R^m.$$
Then
$$ d_{CC, G'}(f_1(x_0), a) \le C R \quad \text{where $C= C(G,G', \sigma_1)$.}
$$
\item \label{it:convergence_f_rho}
We have
$$ f_\rho \to f \quad \text{a.e. and in $L^p_{loc}(G)$.} $$
If $f$ is continuous, then $f_\rho \to f$ locally uniformly.
\item \label{it:bound_derivatives_f1}
If $\operatorname{osc}_{m}(f, B(x_0,1)) \le R$
then the (Riemannian) norms of the derivatives of
$f_1$ are
controlled at $x_0$, i.e. $\|D^i( f_1)(x_0)\|<C=C(i,R, G, G', \sigma_1)$ and
$\| D^i\big((\delta_{R^{-1}} \circ f_1)(x_0)\big)\| \le C(i,G,G', \sigma_1)$.
\end{enumerate}
\end{lemma}
In assertion \eqref{it:bound_derivatives_f1}
the 'Riemannian' derivatives are computed with respect to the charts $\varphi= \log_{G} \circ \ell_{x_0^{-1}}: G \to \mathfrak{g}$
and $\psi = \log_{G'} \circ \ell_{f_1(x_0)^{-1}}: G' \to \mathfrak{g}'$, i.e. we estimate the derivatives of the map
$$ \psi \circ f_1 \circ \varphi^{-1} = \log \circ \ell_{f_1(x_0)^{-1}} \circ f_1 \circ \ell_{x_0} \circ \exp : \mathfrak{g} \to \mathfrak{g}'$$
at $0$.
\bigskip
\begin{remark} One can consider more general domains. First, if $U \subset G$ is open, then it
follows from the proof that the results extend to maps in $L^m_{loc}(U;G)$, whenever the expressions
make sense. In particular we need that $\operatorname{spt} \sigma_z \subset U$. Taking $G = \mathbb{R}^N$ we in particular obtain a smoothing operation
for maps $U \subset \mathbb{R}^N \to G'$. If $U$ is an open subset in a metric measure space $X$ we can abstractly define the mollification $f_1(z)$
using a general family of compactly supported Borel probability measures $\sigma_z$.
In this case there is no notion of left translation to define $\sigma_z$, but the measures
$\sigma_z$ should be in a suitable sense concentrated near $z$. Then (1) still holds and it is easy to prove counterpart of (5). Moreover the
proof of (6) shows that $z \mapsto f_1(z)$ is locally Lipschitz (with bounds on the local Lipschitz constant in terms of the $L^m$-oscillation of $f$),
provided that the measures are such that for each function $h \in L^1_{loc}(X)$ the map $z \mapsto \int_X h \sigma_z$ is Lipschitz.
\end{remark}
\bigskip
\begin{proof}[Proof of Lemma~\ref{le:moll_prop_Lm}]
We will sometimes denote the Carnot-Caratheodory distance $d_{CC,G'}$ generically by $d$ for brevity.
(1). We have
$$
\int_{G'} d_{CC, G'}^p(y',e) \, df_* \sigma_z(y') \\
= \int_G d_{CC, G'}^p(f(y),e)\, d\sigma_z(y).
$$
Now $\operatorname{spt} \sigma_z = z \operatorname{spt} \sigma$ is compact. Thus the right hand side is finite
since by assumption $y \mapsto d_{CC, G'}^p(f(y), e)$ is integrable over compact sets.
By \eqref{eq:ball_box} this implies that $f_*\sigma_z$ has finite $p$-th moment.
(2). This is immediate from the definition.
(3). We have $(\ell_a)_* \sigma_z =(\ell_a)_* ((\ell_z)_* \sigma) = \sigma_{az}$ and thus
$$(f \circ \ell_a)_1(z) = \operatorname{com}_{(f \circ \ell_a)_*\sigma_z} = \operatorname{com}_{f_*\sigma_{az}} = f_1 (az) = (f_1 \circ \ell_a)(z).
$$
The identity $(\ell_b \circ f)_1 = \ell_b \circ f_1$ follows from \eqref{eq:com_translation}.
(4). This follows from \eqref{eq:com_Phi}.
\eqref{it:bound_oscillation_f1}.
Since mollification commutes with pre- and postcomposition by left translation we may assume that $a=e$
and $x_0 = e$.
Since mollification commutes with dilation we may also assume $R=1$.
By \eqref{eq:formula_com1} and \eqref{eq:formula_com2}
$$ \log f_1 = \log \operatorname{com} f_* \sigma$$
is a polynomial of the polynomial moments
$$ A_i= \int_{\mathfrak{g}'} L_i(Y, \ldots, Y) \, d\log_* f_*\sigma_{1}(Y).$$
Here and in the following we just write $\log$ instead of $\log_{G'}$ for ease of notation.
It just suffices to prove bounds on the $A_i$ (which only depend on $G$, $G'$ and $\sigma$).
Now $\sigma = \alpha \mu$ where $\mu$ is the biinvariant measure on $G$ and $\alpha \in C_c^\infty(B(e,1))$
and thus
$$ A_i = \int_{G} L_i( \log f(z), \ldots, \log f(z)) \, \alpha(z) \, \mu(dz).$$
Now by
\eqref{eq:bound_Qi} we have
$ |L_i(Y, \ldots, Y)| \le C (1 + |Y|^m)$.
By \eqref{eq:ball_box} we have $|\log f(z)| \le C d_{CC, G'}(f(z), e)$.
Hence
$$ |A_i| \le C \int_{B(e,1)} (1 + d_{CC, G'}^m(f(z), e)) \, \| \alpha\|_\infty \, \mu(dz) \le C,$$
as desired.
\eqref{it:convergence_f_rho}. Taking $a= f(x_0)$ in assertion \eqref{it:bound_oscillation_f1}
and unwinding definitions we see that
\begin{equation}
\label{eqn_unwinding_5}
d^m(f_\rho(x_0), f(x_0)) \le C \Xint-_{B(x_0, \rho)} d^m( f(x), f(x_0)) \, d\mu(x).
\end{equation}
Thus at every Lebesgue point $x_0$ we have $f_\rho(x_0) \to f(x_0)$, and by Lemma~\ref{le:lebesgue_point_new} we therefore have $f_\rho\rightarrow f$ almost everywhere.
If $f$ is continuous, then it is uniformly continuous on compact sets and thus $f_\rho \to f$ locally uniformly.
To get convergence in $L^p_{\operatorname{loc}}(G)$ for $f \in L^p_{\operatorname{loc}}(G)$ with $p\ge m$, it suffices to show that the restriction of $y \mapsto d^p_{CC,G'}(f_\rho(y), f(y))$
to any ball $B(e,R)$ is equi-integrable for $0 < \rho \le 1$, see Proposition~\ref{pr:dominated_equiintegrable} below.
To that end, we first observe that (\ref{eqn_unwinding_5}) gives, by Jensen's inequality and the triangle inequality:
\begin{equation}
\label{eqn_d_m_jensen_triangle}
\begin{aligned}
d^p(f_\rho(y),f(y))&=\left(d^m(f_\rho(y),f(y)) \right)^{\frac{p}{m}}\\
&\leq \left(C \Xint-_{B(y, \rho)} d^m( f(x), f(y)) \, d\mu(x) \right)^{\frac{p}{m}}\\
&\leq C_p\Xint-_{B(y,\rho)}d^p(f(x),f(y))\,d\mu(x)\\
&\leq C_p\Xint-_{B(y,\rho)}\left(d^p(f(x),f(e))+d^p(f(y),e)\right)\,d\mu(x)\\
&=C_p\left(\Xint-_{B(y,\rho)}h(x)\,d\mu(x) + h(y) \right)
\end{aligned}
\end{equation}
where $h(x) := d^p_{CC, G'}(f(x), e)$ and $C_p$ denotes a generic constant depending on $p$. The right hand side of \eqref{eqn_d_m_jensen_triangle} can be written as $C_p( h \ast \varphi_\rho + h)(y)$ with
$ (f\ast g)(y) := \int_G f(x) g(x^{-1} y) \, \mu(dx)$ and
$\varphi_\rho(z)= \rho^{-\nu} 1_{B(0,1)}(\delta_{\rho^{-1}} z)$.
Since $h \in L^1_{\rm loc}(G)$, the mollifications $h \ast \varphi_\rho$ converge to $h$ in $L^1_{loc}(G)$
as $\rho \to 0$. Thus the right hand side of \eqref{eqn_d_m_jensen_triangle} is equi-integrable on each ball $B(e,R)$ for $0 < \rho \le 1$.
\eqref{it:bound_derivatives_f1}. It suffices to prove the estimate for the derivatives of $\delta_{R^{-1}} \circ f_1$
at $x_0$. Then the other estimate follows from the chain rule since $\delta_R$ is smooth. In view of assertion
(4) we may in addition assume $R=1$.
In view of assertion (3) we may assume without loss of generality that $x_0 = e$ and $f_{1}(x_0) = e$.
Thus we have to estimate the derivatives of the map
$$g = \log_{G'} \circ f_1 \circ \exp_G.$$
We begin with the following observation. Let $h \in L^1_{loc}(G)$ and define
$$ \bar h(x) := \int_G h \, \sigma_x.$$
Then $\bar h$ is smooth and the derivatives are uniformly controlled. In particular
\begin{equation} \label{eq:smoothness_in_sigma_z}
\sup_{1 \le j \le k} |D^k (\bar h \circ \exp)(0)| \le C(G, k, \| \alpha\|_{C_k}, \| g\|_{L^1(B(0,1))}).
\end{equation}
Indeed, using the definition of $\sigma_x$ and the fact that $\log_*\mu = \mathcal L^N$ we get
\begin{align*} & \,
(\bar h \circ \exp)(X) = \int_G h(y) \alpha((\exp X)^{-1} y) \, d \mu(y) \\
= & \, \int_{\log B(e,1)} (h \circ \exp)(Y) (\alpha \circ \exp)( (-X) \ast Y) \, d\mathcal L^N(Y).
\end{align*}
Here $(-X) \ast Y = -X + Y + P(-X,Y)$ is the induced group operation on the Lie algebra.
Since $\alpha$ has compact support in $B(e,1)$ and the group operation $\ast$ is continuous, it follows
that $Y \mapsto (\alpha \circ \exp) ((-X) \ast Y)$ is supported in a fixed compact subset of $\log B(e,1)$ for all sufficiently small $X$.
Hence differentiation and integration commute and the assertion follows since $\| h \circ \exp\|_{L^1(\log(B(e,1)))} = \| h \|_{L^1(B(e,1))}$.
By \eqref{eq:formula_com1} and \eqref{eq:formula_com2} the quantity
$ \log f_1 (x) = \log \operatorname{com}_{f_*(\sigma_x)}$
is a polynomial in the polynomial
moments
\begin{align*} & \, A_i(x) = \int_{g'} L_i(Y, \ldots, Y) \, d\log_* f_*(\sigma_{x})(Y)
= \, \int_G L_i (\log f, \ldots, \log f) \, d\sigma_x,
\end{align*}
and by \eqref{eq:bound_Qi} we have
$$ |L_i(Y, \ldots, Y)| \le C (1 + |Y|^m).$$
By \eqref{eq:ball_box} we have $|Y| \le C d_{CC, G'}(\exp Y)$.
Thus the function $h$ defined by $h(y) := L_i (\log f(y), \ldots, \log f(y))$ is in $L^1_{loc}(G)$.
Hence by \eqref{eq:smoothness_in_sigma_z}, the map $x \mapsto A_i(x)$ is smooth with uniform bounds in terms of
$G$, $\alpha$ and $\| d_{CC, G'}^m(f( \cdot), e)\|_{L^1}$. Since $\log f_1$ is a polynomial (depending on $G'$) in the $A_i$ it is
also smooth and the derivatives are controlled in terms of the same quantities and $G'$.
It only remains to show that $\| d_{CC, G'}^m(f( \cdot), e)\|_{L^1}$ is controlled by a constant, taking into account the normalisations
$R=1$ and $f(e) =e$.
By assumption there exists an $a \in G'$ such that
$$ \int_{B(e,1)} d_{CC, G'}^m(f(x), a) \, \mu(dx) \le 1.$$
Thus assertion \eqref{it:bound_oscillation_f1} yields
$$ d_{CC, G'}(f_1(e), a) \le C(G, G', \sigma_{1}).$$
Since $f_1(e) = e$ it follows from the triangle inequality that
$$ \int_{B(e,1)} d_{CC, G'}^m(f(x), e) \, \mu(dx) \le (1 + C(G, G', \sigma))^m.$$
This concludes the proof of assertion
\eqref{it:bound_derivatives_f1}.
\end{proof}
For the proof of assertion \eqref{it:convergence_f_rho} we used the following standard extension of the dominated convergence
theorem. Let $(A, \mathcal A, \mu)$ be a measure space with $\mu(A) < \infty$. We say that a family
of integrable functions $f_\alpha: E \to \mathbb{R}$ is equi-integrable if for every $\varepsilon >0$ there exists a $\delta > 0$
such that $\mu(E) < \delta$ implies $\int_E |f_\alpha| \, d\mu < \varepsilon$ for all $k$. Clearly every finite family of integrable
functions is equi-integrable and thus every $L^1$-convergent sequence of functions is equi-integrable.
\begin{proposition} \label{pr:dominated_equiintegrable} Let $(A, \mathcal A, \mu)$ be a measure space with $\mu(A) < \infty$.
Assume that $f_k \to f$ a.e. in $A$ and that for some $s \in [1, \infty)$ the family $\{|f_k|^s\}$ is equi-integrable. Then
$f_k \to f$ in $L^s(A)$.
\end{proposition}
\begin{proof} An equi-integrable sequence is in particular bounded in $L^1(A)$. Thus by Fatou's lemma $f \in L^s(A)$.
Since $|f_k - f|^s \le 2^s (|f_k|^s + |f|^s)$ and since the right hand side is integrable it suffices to consider the case $s=1$,
$f=0$, $f_k \ge 0$. Pick $\rho>0$. Let $F_{k, \rho} : \{ x \in A : f_k > \rho \}$. Then $\mu(F_{k, \rho}) \to 0$ as $k \to \infty$, so by equi-integrability of $\{f_k\}$ we have
$$ \limsup_{k \to \infty} \int_A f_k \le \rho \mu(A) + \limsup_{k \to \infty} \int_{F_{k, \rho}} f_k \le \rho \mu(A)\,.$$
Since $\rho >0$ was arbitrary, we get $\| f_k \|_{L^1(A)} \to 0$.
\end{proof}
\bigskip\bigskip
\begin{lemma} \label{le:moll_prop_Lm_bis}
Let $G$ be a Carnot group and let $G'$ be an $m$-step Carnot group.
\begin{enumerate}
\item If $f:G\rightarrow G'$ is a group homomorphism, then $f_1=f$.
\item
If $\{f_k:G\rightarrow G'\}$ is a sequence of continuous maps,
and $f_k\rightarrow f_\infty$ in $L^m_{loc}(G,G')$, i.e.
$d_{CC,G'}(f_k, f_\infty) \to 0$
in $L^m_{loc}(G)$,
then the sequence of
mollified maps $\{(f_k)_1\}$ converges in $C^j_{loc}$ (with respect to the Riemannian structure)
to $(f_\infty)_1$, for all $j$.
\end{enumerate}
\end{lemma}
\begin{proof} (1). This follows directly from \eqref{eq:com_Phi} and
\eqref{eq:com_inversion_invariant}.
(2). The main point is to show that
\begin{equation} \label{eq:pointwise_convergence_fk1}
(f_k)_1(x_0) \to (f_\infty)_1(x_0) \quad \forall x_0 \in G.
\end{equation}
Then $C^i_{loc}$ convergence will follow from the uniform bounds in
Lemma \ref{le:moll_prop_Lm}.
Since mollification commutes with pre- and postcomposition by left-translation we may assume that
$x_0 = e$ and $(f_\infty)(e) = e$.
To prove the pointwise convergence $(f_k)_1(e) \to e$
we use the following fact.
Suppose that
$$ \text{$\varphi: G' \to \mathbb{R}$
is continuous and $\varphi(y) \le C d^m_{CC, G'}(y, e)$.}$$
Then
\begin{equation} \label{eq:dominated_convergence_continuous}
\varphi \circ f_k \to \varphi \circ f_\infty \quad \text{in $L^1_{loc}(G)$.}
\end{equation}
This follows easily from Proposition~\ref{pr:dominated_equiintegrable} by first passing to an a.e.
converging sequence and then using uniqueness of the limit.
Now recall that $\log (f_k)_{1}(e)$
is a polynomial $P$ in the polynomial moments
$$ A_i^k:= \int_{\mathfrak{g}'} L_i(Y, \ldots, Y) (\log \circ f_k)_*\sigma_1(Y) = \int_{G} Q_i(\log f_k, \ldots, \log f_k) \, \sigma_1.$$
It thus suffices to show that $\lim_{k \to \infty} A^k_i = A^\infty_i$.
In view of (\ref{eq:bound_Qi}), this follows from
\eqref{eq:dominated_convergence_continuous}
applied to the function $\varphi(y) = L_i( \log y, \ldots, \log y)$ since $\sigma_1 = \alpha \mu$ and
$\alpha$ is bounded and compactly supported.
\end{proof}
\section{Pansu pullback and mollification}
\label{sec_pansu_pullback_mollification}
We now consider the behavior of pulling back using
a mollified map between Carnot groups $G$ and $G'$.
For an open set $U \subset G$ define
$$ U_\rho := \{ x \in U : \operatorname{dist}(x, G \setminus U) > \rho\}.$$
Assertion \eqref{it:pullback_pre_pansu} of the following lemma
provides a key connection between convergence of
the mollified pullback $f_\rho^*\alpha$ and Pansu differentiability.
\begin{lemma} \label{lem_moll_calc_forms}
Let $U \subset G$ be open and let $f \in L^m_{loc}(U, G')$.
Suppose that $\alpha \in \Omega^{k,w_\alpha}(G')$ and $\gamma \in \Omega^{N-k,w_\gamma}(G)$ are left-invariant forms.
In particular, if $k=N$ then $\gamma$ is a constant zero-form, i.e. a constant function and $w_\gamma=0$.
Then
\begin{enumerate}
\item \label{it:pullback_alpha_rho} For every $x\in U_\rho$,
$$
(f_\rho^*\alpha \wedge \gamma)(x)=\rho^{-(\nu + w_\alpha+w_\gamma)}(h_1^*\alpha \wedge \gamma)(\delta_{\rho^{-1}}(x))\,,
$$
where $h=\delta_{\rho^{-1}}\circ f\circ \delta_\rho$.
\item \label{it:pullback_pre_pansu} For every $x \in U_\rho$,
$$
(f_\rho^*\alpha \wedge \gamma)(x)=\rho^{-(\nu + w_\alpha+w_\gamma)}
\big((\delta_{\rho^{-1}}\circ f_x\circ \delta_\rho)_1^* \alpha \wedge \gamma\big)(e)\,,
$$
where $f_x=\ell_{f(x)^{-1}}\circ f\circ \ell_x$.
\item \label{it:bound_pullback} If $x \in U_\rho$ and $\operatorname{osc} _m (f,B(x,\rho)) \le C \rho^{1 + \frac{\nu}{m}}$, then
\begin{equation*}
\|(f_\rho^*\alpha) \wedge \gamma)(x) \| \leq
C'\, C^{-w_\alpha}\rho^{-( \nu + w_\alpha+w_\gamma)}
\| \alpha\| \, \| \gamma\|.
\end{equation*}
\end{enumerate}
\end{lemma}
\begin{proof} The proof of the first two assertions is exactly the same
as the proof of the corresponding assertions in Lemma 6.4 in \cite{KMX1}.
We include the short calculation for the convenience of the reader.
\eqref{it:pullback_alpha_rho}.
Note that $\{ z : B(z, 1) \subset \delta_{\rho^{-1}} U\} = \delta_{\rho^{-1}} U_\rho$
and thus
$$h: \delta_{\rho^{-1}} U \to G', \quad h_1: \delta_{\rho^{-1}} U_\delta \to G'. $$
For $x \in U_\rho$ we have
\begin{align*}
(f_\rho^*(\alpha) \wedge \gamma)(x)
=& \big( (\delta_\rho \circ h_1 \circ \delta_{\rho^{-1}})^* \alpha \wedge \gamma\big)(x)\\
=& (\delta_{\rho^{-1}}^* h_1^* \delta_\rho^* \alpha \, \wedge\, \delta_{\rho^{-1}}^* \delta_\rho^* \gamma)(x)\\
=& \rho^{-(w_\alpha + w_\gamma)} (\delta_{\rho^{-1}}^* h_1^* \alpha \wedge \delta_{\rho^{-1}}^* \gamma)(x)\\
=& \rho^{-(w_\alpha + w_\gamma)} \big(\delta_{\rho^{-1}}^* (h_1^* \alpha \wedge \gamma) \big)(x)\\
=& \rho^{-(\nu + w_\alpha + w_\gamma)} (h_1^* \alpha \wedge \gamma)(\delta_{\rho^{-1}} x).
\end{align*}
In the last step we used that $h_1^* \alpha \wedge \gamma$ is a multiple of the volume form, which has weight $-\nu$.
\bigskip
\eqref{it:pullback_pre_pansu}.
With $h$ as in \eqref{it:pullback_alpha_rho} we get
\begin{align*}
h=&\delta_{\rho^{-1}}\circ f \circ \delta_\rho\\
=&(\delta_{\rho^{-1}}\circ \ell_{f(x)}\circ \delta_\rho)\circ
\delta_{\rho^{-1}}\circ \ell_{f(x)^{-1}}\circ f\circ \ell_x\circ\delta_\rho
\circ(\delta_{\rho^{-1}}\circ \ell_{x^{-1}}\circ \delta_\rho)\\
=&\ell_{\delta_{\rho^{-1}}f(x)}
\circ \delta_{\rho^{-1}}\circ f_x\circ \delta_\rho
\circ \ell_{\delta_{\rho^{-1}}x^{-1}}
\end{align*}
and so
$$
h_1=\ell_{\delta_{\rho^{-1}}f(x)}
\circ (\delta_{\rho^{-1}}\circ f_x\circ \delta_\rho)_1
\circ \ell_{\delta_{\rho^{-1}}x^{-1}}\,.
$$
Since $\alpha$ and $\gamma$ are left invariant we have for $x \in U_\rho$
\begin{align} \label{eqn_h1_al}
&( h_1^*\alpha \wedge \gamma)
(\delta_{\rho^{-1}}(x) ) \notag \\
=&\ell_{\delta_{\rho^{-1}}x^{-1}}^*[(\delta_{\rho^{-1}}\circ f_x\circ \delta_\rho)_1^*\alpha
\wedge \gamma] (\delta_{\rho^{-1}}(x)) \\
=& [ (\delta_{\rho^{-1}}\circ f_x\circ \delta_\rho)_1^*\alpha
\wedge \gamma](e). \notag
\end{align}
Combining \eqref{it:pullback_alpha_rho}
with (\ref{eqn_h1_al}) gives (2).
\eqref{it:bound_pullback}. Note that our assumptions imply
that $\operatorname{osc}_m (h,B(\delta_{\rho^{-1}} x,1)) \le C$.
Thus Lemma~\ref{le:moll_prop_Lm}~\eqref{it:bound_derivatives_f1}
implies that
$$ \| D( \delta_{C^{-1}} \circ h_1)(\delta_{\rho^{-1}} x) \| \le C'.$$
Using assertion~\eqref{it:pullback_alpha_rho} we get
\begin{align*}
& \| (f_\rho^* \alpha \wedge \gamma)(x) \| \\
= & \| \rho^{-(\nu + w_\alpha + w_\gamma)}
\big( \big( \delta_C \circ (\delta_{C^{-1}} \circ h_1) \big)^*\alpha
\wedge \gamma \big)(\delta_{\rho^{-1}}(x)) \| \\
= & C^{-w_\alpha} \| \rho^{-(\nu + w_\alpha + w_\gamma)}
\big( (\delta_{C^{-1}} \circ h_1)^*\alpha
\wedge \gamma \big)(\delta_{\rho^{-1}}(x)) \| \\
\le & C' \, C^{-w_\alpha} \rho^{-(\nu + w_\alpha + w_\gamma)} \| \alpha\| \, \| \gamma\|.
\end{align*}
\end{proof}
\begin{theorem}[Approximation theorem] \label{th:weight_controlled_pullback}
Let $G$ be a Carnot group of topological dimension $N$ and homogeneous
dimension $\nu$ and let $G'$ be an $m$-step Carnot group. Let $U \subset G$ and $U' \subset G'$ be open. Suppose that $\omega \in \Omega^{k,w_\omega}(U')$ has continuous and bounded coefficients
and $\gamma \in \Omega^{N-k,w_\gamma}(G)$ is a left-invariant form
such that
\begin{equation} \label{eq:condition_minimal_weight} w_\omega+w_\gamma \le - \nu.
\end{equation}
Assume that $p \ge - w_\omega$ and $\frac{1}{p} \le \frac1m + \frac{1}{\nu}$.
Let $f: U \to U'$ be a map in $W^{1,p}_{loc}(U, G')$.
Let $\overline \omega$ denote the extension of $\omega$ to $G' \setminus U'$ by zero.
Then
\begin{equation} \label{eq:L1_convergence_pullback}
f_\rho^*\overline{\omega} \wedge \gamma \to f_P^* \omega \wedge \gamma \quad \text{in $L^s_{\operatorname{loc}}(U)$ with $s = \frac{p}{-w_\omega}$.}
\end{equation}
Equivalently, we have convergence of weight $w$ components
$$
(f_\rho^*\overline\omega)_w\rightarrow (f_P^*\omega)_w
$$
for $w\geq w_\omega$, see Remark~\ref{rem_w_component_convergence} below.
In particular we have
\begin{equation} \label{eq:L1_convergence_pullback_top_degree}
f_\rho^* \overline{\omega} \to f_P^* \omega \quad
\text{in $L^{\frac{p}{\nu}}_{\operatorname{loc}}(U)$ \quad
if $\omega \in \Omega^N(U')$}
\end{equation}
\end{theorem}
\bigskip\bigskip
\begin{remark} The mollifications $f_\rho$ may take values outside $U'$. This is why we need to extend $\omega$ outside $U'$ to define the pull-back by $f_\rho$.
The proof shows that convergence in \eqref{eq:L1_convergence_pullback} does not depend on which extension we choose. More precisely, if $ \widetilde \omega$
is any extension of $\omega$ which is everywhere defined, bounded, measurable and satisfies $\widetilde{\omega}(x)\in\Lambda^{k,w_\omega}$ at each point,
then
\begin{equation} \label{eq:L1_convergence_pullback_other_ext}
f_\rho^*\widetilde{\omega} \wedge \gamma \to f_P^* \omega \wedge \gamma \quad \text{in $L^s_{\operatorname{loc}}(U)$ with $s = \frac{p}{-w_\omega}$.}
\end{equation}
\end{remark}
\bigskip\bigskip
\begin{remark}
\label{rem_w_component_convergence}
The convergence in \eqref{eq:L1_convergence_pullback} in connection with the condition \eqref{eq:condition_minimal_weight} is equivalent to convergence of weight $w$ components
$$
(f_\rho^*\overline \omega)_w\rightarrow (f_P^*\omega)_w
$$
for $w\geq w_\omega$. To see this, note that for $\omega$ fixed and $w\geq w_\omega$, we may choose a basis $\{\gamma_i\}$ of the space of left-invariant forms $\gamma\in\Omega^{N-k,-\nu-w}(G)$, and this is dual via the wedge product to a basis $\{\alpha_{w,i}\}$ for the left invariant forms in $\Omega^{k,w}(G)$. Thus \eqref{eq:L1_convergence_pullback} applied to each $\gamma_i$ yields convergence
$$
(f_\rho^*\overline \omega)_{w,i}\rightarrow (f_P^*\omega)_{w,i}
$$
in $L^s_{\operatorname{loc}}$ where the notation $(\beta)_{w,i}$ for a form $\beta$ is defined by $\beta_w=\sum_i(\beta)_{w,i}\alpha_{w,i}$. In particular, if $G=G'$ and the weight of $\omega$ is minimal among nonzero forms of degree $k$,
then all components converge and thus $f_\rho^*\overline \omega \to f_P^*\omega$ in $L^s_{\operatorname{loc}}(U)$.
\end{remark}
\bigskip\bigskip
\begin{remark} \label{re:natural_exponent_p}
We now comment on the assumptions on the exponent $p$. The obvious estimate for the pullback is
$$
|f_P^*\omega|(x)\leq C|D_Pf(x)|^{-w_\omega}|\omega|(f(x))\,.
$$
Therefore, in general, one would expect $p\geq -w_\omega$ to be the optimal lower bound on the Sobolev exponent. However if $w_\omega + w_\gamma < - \nu$ then some improvement is possible, see Corollary~\ref{co:improved_exponents_approximation} below.
In the abelian case we have $w_\omega = -k$ and it is known that the condition $p \ge k$ is necessary to have $L^1_{\operatorname{loc}}$ convergence of $f_\rho^* \omega$.
Typical counterexamples are given by suitable $0$-homogeneous functions. For example, if $G=G'= \mathbb{R}^N$ and $\omega =dy_1 \wedge \ldots \wedge dy_N$
one can take $f = \frac{x}{|x|}$. Then $f^*\omega = 0$, $f \in W^{1,p}(U; \mathbb{R}^N)$ for all $p < N$, but it is easily seen, e.g. by a degree argument, that $f_\rho^*\omega$ weak$*$ converges to the Dirac mass
$\mu(B(0,1)) \delta_0$ as $\rho\rightarrow 0$, where we identify top degree forms with measures. We do not know the optimal exponent $p$ for which
the conclusion $f_\rho^*\omega \wedge \gamma \to f_P^*\omega \wedge \gamma$ in $L^1_{\operatorname{loc}}(U)$ holds.
\end{remark}
\bigskip
\begin{proof}[Proof of Theorem~\ref{th:weight_controlled_pullback}]
We will prove the result using the dominated convergence theorem.
In brief, this is implemented as follows. Pointwise convergence almost everywhere follows from the formula
in Lemma~\ref{lem_moll_calc_forms}~\eqref{it:pullback_pre_pansu},
Pansu differentiability a.e. (in an $L^m$ sense) and the fact the
mollification improves $L^m$-convergence to $C^1$-convergence.
The majorant is obtained from the estimate in Lemma~\ref{lem_moll_calc_forms}~\eqref{it:bound_pullback}
and the Sobolev-Poincar\'e inequality which provides
a uniform estimate of the $L^m$ oscillation in terms of the maximal function of the $p$-th power of the (horizontal) derivative.
We begin with some preparations.
Since we only want to prove convergence in $L^s_{\operatorname{loc}}$ we may assume that $f \in W^{1,p}(U;G')$. By linearity it suffices to verify the theorem for forms $\omega = a \alpha$ where $\alpha$ is a left-invariant form with $w_\alpha + w_\gamma \le - \nu$ and $a$ is a continuous and bounded function. We denote by $\overline a$ the extension of $a$ by zero to $G' \setminus U'$.
Set $w_\alpha = \operatorname{wt}(\alpha)$ and $w_\gamma = \operatorname{wt}(\gamma)$.
Fix a compact set $ K \subset U$.
We next show pointwise convergence a.e. in $K$.
Recall that $U_\rho := \{ x \in U : \operatorname{dist}(x, G \setminus U) > \rho\}.$
For $\rho >0$ small enough we have $K \subset U_\rho$.
By Lemma~\ref{lem_moll_calc_forms}~\eqref{it:pullback_pre_pansu}
we have for $x \in K$
\begin{align} \label{eq:pullback_pointwise_convergence}
&(f_\rho^*( \overline \omega) \wedge \gamma)(x)\\
=&(\overline a\circ f_\rho)(x) \, (f_\rho^*\alpha \wedge \gamma)(x) \nonumber \\
=&( \overline a\circ f_\rho)(x) \, \rho^{-(\nu + w_\alpha+w_\gamma)}
\big( (\delta_{\rho^{-1}}\circ f_x\circ \delta_\rho)_1^*\alpha \wedge \gamma \big)(e)\,.
\nonumber
\end{align}
By Theorem~\ref{th:Lp*_pansu_differentiability_new} and the condition $\frac1p \le \frac1m + \frac1\nu$
we have for a.e.\ $x \in K$
the convergence
$\delta_{\rho^{-1}}\circ f_x\circ \delta_\rho \stackrel{L^m_{loc}}{\longrightarrow}
D_P f$.
(Recall that we are using the notation $D_Pf(x)$ to denote a graded Lie algebra homomorphism $\mathfrak{g}\rightarrow \mathfrak{g}'$ and a homomorphism of Carnot groups $G\rightarrow G'$, depending on the context.)
By Lemma~\ref{le:moll_prop_Lm_bis} we get
$D(\delta_{\rho^{-1}}\circ f_x\circ \delta_\rho)_1(e)\rightarrow D_Pf(x)$
as $\rho\rightarrow 0$.
Moreover by Lemma~\ref{le:moll_prop_Lm}~\eqref{it:convergence_f_rho} we have
$f_\rho(x) \to f(x)$ almost everywhere.
Let $N \subset K$ be a null set such that for all $x \in K \setminus N$ we have $f_\rho(x) \to f(x)$ and
$D(\delta_{\rho^{-1}}\circ f_x\circ \delta_\rho)_1(e)\rightarrow D_Pf(x)$.
Since $U'$ is open, for each $x\in K\setminus N$ there exist a $\rho_0(x) > 0$ such that $f_\rho(x) \in U'$ for
all $\rho < \rho_0(x)$. Since $\overline a$ is continuous in $U'$ (and agrees there with $a$) it follows that
$\overline a \circ f_\rho(x) \to a \circ f(x)$ for all $x \in K\setminus N$. Note that this convergence is independent of how we extend
$a$ outside $U'$.
Now if $w_\alpha+w_\gamma= - \nu $, then
\begin{align} \label{eq:pointwise_convergence_pull_back}
(f_\rho^* \overline \omega \wedge \gamma) (x) \rightarrow & (a\circ f)(x) \, ((D_Pf(x))^* \alpha)(x) \wedge \gamma\\
=&(f_P^*\omega \wedge \gamma)(x) \nonumber
\end{align}
so we have pointwise convergence in this case. If $w_\alpha+w_\gamma< - \nu$, then
$(f_\rho^*\overline \omega \wedge \gamma) (x) \rightarrow 0$ as $\rho\rightarrow 0$, while
$$
(f_P^*\omega \wedge \gamma)(x) =(a\circ f)(x)
\, \, ((D_P f)(x)^* \alpha)(x) \wedge \gamma\,.
$$
Now by Lemma~\ref{lem_weight_facts}~\eqref{it:weight_facts_pullback}
we deduce that $((D_P f)(x)^* \alpha)(x) \wedge \gamma$ is a form of weight strictly
less than $-\nu$ and hence zero. Thus if $w_\alpha + w_\gamma < - \nu$ we have $(f_P^*\omega \wedge \gamma)(x) = 0$.
Hence we have shown that $(f_\rho^* \overline\omega \wedge \gamma) (x) \rightarrow (f_P^*\omega \wedge \gamma)(x)$
for a.e. $x \in K$.
By Proposition~\ref{pr:dominated_equiintegrable}, it remains only to show that $|f_\rho^*\overline \omega \wedge \eta|^s$ is equi-integrable for $s = \frac{p}{-\omega_\alpha}$.
If $m \ge 2$ define $q > 1$ by $\frac1q = \frac1m + \frac1\nu$ (if $m=1$, i.e., if $G$ is abelian, take $q=1$; then
\eqref{eq:osc_bound_max_function} below follows directly from the Poincar\'e inequality).
Set $\psi = | D_h f|^{q}$.
Then $\psi \in L^{\frac{p}{q}}(U)$.
By the Sobolev-Poincar\'e inequality (\ref{eq:sobolev_poincare_new}) we have for $x \in K$
$$
\rho^{-\frac{\nu}{m}} \operatorname{osc}_m (f,B(x,\rho)) \le C \rho \left( \rho^{-{\nu}} \int_{B(x,\rho)} \psi \right)^{\frac1q} = C \rho \, \psi_\rho^{\frac1q}(x)
$$
where
$$ \psi_\rho: = \psi \ast \rho^{-\nu} 1_{B(0, \rho)}$$
and
$$ (f\ast g)(x) := \int_G f(x y^{-1}) g(y) \, \mu(dy) = \int_G f(y) g(y^{-1} x) \, \mu(dy).$$
Since $\psi \in L^{\frac{p}{q}}_{\operatorname{loc}}(G)$ we have
\begin{equation}
\label{eqn_psi_rho_converges_l_p_q}
\psi_{\rho} \to \psi \quad \text{in $L^{\frac{p}{q}}(K)$}
\end{equation}
as $\rho\rightarrow 0$.
Moreover
\begin{equation} \label{eq:osc_bound_max_function}
\operatorname{osc}_m (f,B(x,\rho)) \le C \rho^{1 + \frac{\nu}{m}} \psi^{\frac1q}_\rho(x).
\end{equation}
Now let $s =\frac{p}{ -w_\alpha}$. Then by
Lemma~\ref{lem_moll_calc_forms}~\eqref{it:bound_pullback}
\begin{align} \label{eq:pullback_equiintegrability}
|f_\rho^* \overline\omega \wedge \eta|^s(x) \le & \, \|\bar a\|_\infty^s \, |(f_\rho^*\alpha)(x) \wedge \eta|^s \\
\le & \, C \psi_{\rho}^{\frac{- s w_\alpha}{q}}(x) \rho^{-s(\nu+ w_\alpha + w_\gamma)} \,
\| \alpha\|^s \, \| \gamma\|^s \, \|\bar a\|_\infty^s \nonumber \\
\le & \, C \rho^{-s(\nu+ w_\alpha + w_\gamma)} \psi_\rho^{\frac{p}{q}}. \nonumber
\end{align}
In view of \eqref{eqn_psi_rho_converges_l_p_q} the family $\psi_\rho^{\frac{p}{q}}$ is equi-integrable, so \eqref{eq:pullback_equiintegrability} gives the desired equi-integrability of $|f_\rho^* \overline\omega \wedge \eta|^s$. Note also that the argument used only the fact that the extension $\bar a$ is bounded.
\end{proof}
\bigskip
The argument above shows easily that we have better convergence results if $\operatorname{wt}(\omega) + \operatorname{wt}(\gamma) < - \nu$. We summarize these as follows.
\begin{corollary} \label{co:improved_exponents_approximation} With the notation and assumptions of Theorem~\ref{th:weight_controlled_pullback} the following refinements of \eqref{eq:L1_convergence_pullback} hold.
\begin{enumerate}
\item If $\frac1p \le \frac1m + \frac1\nu$, $p \ge -w_\omega$ and $w_\omega+w_\gamma < - \nu$ then
\begin{equation}
\rho^{\nu + w_\omega + w_\gamma} f_\rho^*\overline \omega \wedge \gamma\to 0 \quad \text{in $L^s_{\operatorname{loc}}(U)$ with $s = \frac{p}{-w_\omega}$.}
\end{equation}
\item
Set $\beta = -\frac{p}{\nu}(w_\omega+w_\gamma+\nu)$. If $\beta < -w_\omega$ then
\begin{equation}
f_\rho^*\overline\omega\wedge\gamma \to 0 \quad \text{in $L^{\frac{p}{-w_\omega-\beta}}_{\operatorname{loc}}(U)$.} \end{equation}
If $\beta \ge -w_\omega$ then
\begin{equation}
f_\rho^*\overline\omega\wedge\gamma \to 0 \quad \text{locally uniformly}. \end{equation}
\end{enumerate}
\end{corollary}
\bigskip\bigskip
\begin{proof} The first assertion follows directly from the proof of Theorem~\ref{th:weight_controlled_pullback}.
Indeed, \eqref{eq:pullback_pointwise_convergence}, Pansu differentiability a.e., and
the estimates for the mollification
imply that $\rho^{\nu + w_{\omega}+w_{\gamma}} f_\rho^*\overline\omega\wedge\gamma \to (a \circ f) f_P^* \alpha \wedge \gamma$ a.e.
Moreover $f_P^* \alpha \wedge \gamma =0$ since forms of weight strictly less than $-\nu$ must vanish.
Regarding equi-integrability, \eqref{eq:pullback_equiintegrability} yields
$$| \rho^{\nu + w_{\omega}+w_{\gamma}
} f_\rho^*\overline\omega|^{\frac{p}{-\operatorname{wt}(\omega)}}
\le C |\psi_\rho|^{\frac{p}{q}} \to C |\psi|^{\frac{p}{q}}$$
in $L^1_{\operatorname{loc}}(U)$. Hence the assertion follows from Proposition~\ref{pr:dominated_equiintegrable}.
To prove the second assertion, we note that
Lemma~\ref{lem_moll_calc_forms}~\eqref{it:bound_pullback} yields
\begin{align} \label{eq:pullback_equiintegrability_refined}
|f_\rho^* \overline\omega \wedge \gamma|(x)
\le &
\, C
\Big( \rho^{-(1+ \frac{\nu}{m})}
\operatorname{osc}_m (f,B(x,\rho) ) \Big)^{-w_\omega}
\rho^{ e}
\end{align}
where where $C$ is a constant independent of $x$ and $\rho$, and
$$ e= -(w_\omega + w_\gamma+ \nu).$$
By \eqref{eq:osc_bound_max_function} and the Poincar\'e-Sobolev inequality we have
\begin{align}
\operatorname{osc}_m(f, B(x,\rho)) \le & \, C \rho^{1 + \frac{\nu}m} \psi_\rho^{\frac1q}(x), \label{eq:osc1}\\
\operatorname{osc}_m(f, B(x,\rho)) \le & \, C \rho^{1 + \frac{\nu}m- \frac\nu{p}} \label{eq:osc2}
\|D_h f\|_{L^p(B(x,\rho)}.
\end{align}
Recall that $\beta = ep/ \nu$.
If $\beta < -w_\omega$ then we take \eqref{eq:osc1} to the power $-w_\omega-\beta$ and
\eqref{eq:osc2} to the power $\beta$ to get
\begin{align} \label{eq:pullback_equiintegrability_refined}
|f_\rho^*\overline\omega \wedge \gamma|(x)
\le &
\, C
\psi_\rho(x)^{\frac{-w_\omega - \beta}{q}} \, \|D_h f\|_{L^p(B(x,\rho))}^\beta. \,
\end{align}
Since $\|D_h f\|_{L^p(B(x,\rho))} \to 0$ as $\rho \to 0$, locally uniformly in $x$, and
$\psi_\rho^{\frac1q}$ converges in $L^p_{\operatorname{loc}}(U)$
it follows that
$ |f_\rho^*\overline \omega \wedge \gamma| \to 0$
in $L_{\operatorname{loc}}^{ \frac{p}{-w_\omega-\beta}}(U)$.
If $\beta \ge -w_\omega$ we take \eqref{eq:osc2} to the power $-w_\omega$ and get (for $\rho \le 1$)
the estimate
$ |f_\rho^* \overline\omega \wedge \gamma|(x)
\le
\, C
\|D_h f\|_{L^p(B(x,\rho))}^{-w_\omega}$. The assertion follows since the right hand side converges locally uniformly to zero.
\end{proof}
\bigskip
\bigskip
\bigskip
\bigskip
\bigskip
Next we apply the approximation theorem to show that for certain components the Pansu pullback of differential forms commutes with exterior differentiation.
Note that in general the Pansu pullback does not commute with exterior differentiation (see \cite{KMX1}).
\begin{theorem} \label{th:pansupullback_exterior_d}~
Let $G$ be a Carnot group of topological dimension $N$ and homogeneous dimension $\nu$, let $G'$ be a $m$-step Carnot group, and
$$
f:G\supset U\rightarrow U'\subset G'
$$
be a $W^{1,p}_{\operatorname{loc}}$-mapping between open subsets.
Suppose that $\alpha \in \Omega^{k,w_\alpha}(G')$ has continuous and bounded coefficients such that the weak exterior differential $d\alpha$
also has continuous and bounded coefficients. Let $\beta \in \Omega^{N-k-1,w_\beta}(G)$ be a closed left-invariant form. Assume that
\begin{equation}
w_\alpha+w_\beta = - \nu + 1.
\end{equation}
Then the following assertions hold.
\begin{enumerate}
\item If $\alpha$ is weakly closed, $p \ge -w_\alpha$ and $\frac1p \le \frac1m + \frac1\nu$
then $f_P^*(\alpha) \wedge \beta$ is weakly closed, i.e.
\begin{equation}
\int_{G} f_P^*(\alpha) \wedge \beta \wedge d\varphi = 0 \quad \text{for all $\varphi \in C^\infty_c(U)$.}
\end{equation}
\item Assume $\operatorname{wt}(d\alpha)<w_\alpha$, and that $d\alpha = \sum_{s\leq w< w_\alpha}\omega^{(w)}$ is the weight decomposition of $d\alpha$. Assume that $p\geq -s$ and $\frac1p \le \frac1m + \frac1\nu$.
Then
\begin{equation}
d( f_P^*\alpha \wedge \beta) = f_P^*(d\alpha) \wedge \beta \quad \text{in the sense of distributions,}
\end{equation}
i.e.
\begin{equation} \label{eq:distributional_exterior_derivative}
(-1)^N
\int_{U} f_P^*\alpha \wedge \beta \wedge d\varphi = \int_{U} f_P^*(d\alpha) \wedge \beta \, \varphi \quad
\forall \varphi \in C_c^\infty(U).
\end{equation}
\end{enumerate}
\end{theorem}
\begin{remark} \label{re:codegree_coweight_1}
If $G = G'$, $k = N-1$ and $w_\alpha = - \nu +1$ we have $d\alpha \in \Omega^N(G')$ and hence $\operatorname{wt}(d\alpha) = -\nu$. Thus for $p \ge \nu$
we can take $\beta \equiv 1$ and we get
\begin{equation} \label{eq:pullback_adjoint}
d f_P^*\alpha = f_P^*(d\alpha) \quad \text{if $G=G'$, $k=N-1$, $\operatorname{wt}(\alpha) = - \nu +1$}
\end{equation}
in the sense of distributions. For $2$-step groups this was first shown by Vodopyanov, see \cite{vodopyanov_foundations}.
\end{remark}
\begin{remark} If we use Corollary~\ref{co:improved_exponents_approximation} then the condition on the exponent
in the second assertion can be slightly improved if $s > - w_\alpha + 1$. In that case we can replace the condition
$ p \ge s$ by
$$ p \ge s-\frac{p}\nu (s + w_\beta + \nu) = s- \frac{p}{\nu}(s+1-w_\alpha),$$
or, equivalently,
\begin{equation}
\frac{s}{p} \le 1+ \frac1\nu(s+1-w_\alpha).
\end{equation}
\end{remark}
\bigskip\bigskip
\begin{proof}[Proof of Theorem~\ref{th:pansupullback_exterior_d}] Since $\beta$ is closed we have $d (\varphi \beta) = d\varphi \wedge \beta$, and hence $\operatorname{wt}(d(\phi\beta))\leq \operatorname{wt}\beta-1$.
Using that the (weak) exterior derivative commutes with pullback by smooth functions we get
\begin{equation}
\label{eqn_smooth_pullback}
\int_G f_\rho^*\alpha \wedge \beta \wedge d\varphi = (-1)^{N} \int_G f_\rho^*(d\alpha)
\wedge \beta \, \varphi \quad
\forall \varphi \in C_c^\infty(U).
\end{equation}
Hence both assertions follow by applying Theorem~\ref{th:weight_controlled_pullback} to both sides of \eqref{eqn_smooth_pullback}; on the right hand side the theorem is applied to each component of the weight decomposition of $d\alpha$ separately. Note that the condition $\operatorname{wt}(d\alpha)<w_\alpha$
ensures that Theorem~\ref{th:weight_controlled_pullback} can be applied.
\end{proof}
\section{Quasiregular mappings}
\label{sec_quasiregular_mappings}
In this section we review some results from \cite{vodopyanov_foundations} which were stated only for $2$-step Carnot groups, but which extend immediately to general Carnot groups using the Approximation Theorem \ref{th:weight_controlled_pullback}.
In this section we fix a Carnot group $G$ of homogeneous dimension $\nu$, and an open subset $U\subset G$.
\begin{definition}[\cite{vodopyanov_foundations}]
A mapping $f:G\supset U\rightarrow G$ is {\bf quasiregular} (has {\bf bounded distortion}) if $f\in W^{1,\nu}_{\operatorname{loc}}$ and there is a constant $C$ such that $|D_hf|^\nu\leq C\det D_Pf$ almost everywhere.
\end{definition}
We now fix a quasiregular mapping $f:G\supset U\rightarrow G$.
Following Reshetnyak \cite{reshetnyak_space_mappings_bounded_distortion,heinonen_holopainen,vodopyanov_foundations}, we exploit the pullbacks of $\nu$-harmonic functions to control quasiregular mappings.
\begin{theorem}
\label{thm_morphism_property}
If $u:G\rightarrow \mathbb{R}$ is a Lipschitz $\nu$-harmonic function, then the composition $u\circ f$ is $\mathcal{A}$-harmonic. See \cite[Sec. 2]{heinonen_holopainen}, \cite[Subsec. 4.3]{vodopyanov_foundations} for the definition and basic properties of $\mathcal{A}$-harmonic functions.
\end{theorem}
Note that if $f$ takes values in an open subset $U'\subset G$, then the theorem holds when $u$ is locally Lipschitz, see below.
\begin{proof}
In the $2$-step case, the proof is contained in \cite{vodopyanov_foundations}. This extends to general Carnot groups using the Approximation Theorem.
We give an outline of the steps, to facilitate reading of \cite{vodopyanov_foundations}:
\begin{itemize}
\item By Remark~\ref{re:codegree_coweight_1},
if $\omega$ is a smooth differential form on $G$ with codegree and coweight $1$, and both $\omega$ and $d\omega$ are bounded, then
\begin{equation}
\label{eqn_d_commutes_with_pullback_on_codegree_1_coweight_1}
df_P^*\omega=f_P^*d\omega
\end{equation}
distributionally.
\item If $\Sigma\subset G$ is a Borel null set, then the (approximate) Pansu differential $D_Pf(x)$ satisfies $\det D_Pf(x) = 0$ for a.e. $x\in f^{-1}(\Sigma)$ \cite{vodopyanov_P_differentiability}; hence by the bounded distortion assumption in fact $D_Pf(x)=0$
for a.e.\ $x\in f^{-1}(\Sigma)$. If $\omega$ is a measurable differential form on $G$, and we define $f_P^*\omega(x)$ to be zero whenever $D_Pf(x)=0$, then the Pansu pullback $f_P^*\omega$ is well-defined almost everywhere.
\item By an approximation argument (\ref{eqn_d_commutes_with_pullback_on_codegree_1_coweight_1}) remains true if $\omega\,,d\omega\in L^\infty$, see \cite[Corollaries 2.15, 2.18]{vodopyanov_foundations}.
\item It follows from Proposition~\ref{pr:composition_by_lip} that the composition $v:=u\circ f$ belongs to $W^{1,\nu}_{\operatorname{loc}}(U)$.
\item To see that $v$ is $\mathcal{A}$-harmonic, it suffices to show that its horizontal differential $d_hv$ satisfies the distributional equation $\delta(\mathcal{A} d_hv)=0$ (cf. \cite[(2.11)]{heinonen_holopainen}). This is equivalent to the vanishing of the distributional exterior derivative of $\star_\mathcal{A} d_hv$, see \cite[Section 3, Theorem 3.14]{heinonen_holopainen}. Since $\star_\mathcal{A} d_hv=f_P^*(\star d_hu)$, this follows from (\ref{eqn_d_commutes_with_pullback_on_codegree_1_coweight_1}).
\end{itemize}
\end{proof}
\bigskip\bigskip
The composition $u$ of the abelianization map $G\rightarrow G/[G,G]$ with a coordinate function is Lipschitz and $\nu$-harmonic. Hence by Theorem~\ref{thm_morphism_property} the composition $u\circ f$ is $\mathcal{A}$-harmonic. Following \cite{reshetnyak_space_mappings_bounded_distortion,BI83,heinonen_holopainen,vodopyanov_foundations}, by applying the Caccioppoli inequality for $\mathcal{A}$-harmonic functions and the Poincare inequality one obtains a number of results, including:
\begin{itemize}
\item $f\in W^{1,\nu'}$ for some $\nu'>\nu$.
\item $f$ is H\"older continuous, (classically) Pansu differentiable almost everywhere, and maps null sets to null sets.
\item A suitable change of variables formula holds for $f$.
\end{itemize}
Since $f$ is continuous, the proof of Theorem~\ref{thm_morphism_property} may be localized in the target:
\begin{corollary}
Suppose the image of $f$ is contained in an open subset $U'\subset G$, and $u:U'\rightarrow\mathbb{R}$ is a locally Lipschitz $\nu$-harmonic function. Then $u\circ f$ is $\mathcal{A}$-harmonic.
\end{corollary}
If there exists for some $r>0$ a locally Lipschitz $\nu$-harmonic function $u:B(e,r)\setminus\{e\}\rightarrow (0,\infty)$ such that $\lim_{x\rightarrow e}u(x)\rightarrow\infty$, then the method of Reshetnyak could be applied to show that $f$ is open and discrete, which would have a number of further consequences, see \cite{vodopyanov_foundations} (Theorem 4.11 and the ensuing discussion). Unfortunately, the existence of such $\nu$-harmonic functions remains an open problem.
\section{Product rigidity}
In this section we show how the results in \cite{KMX1} on product rigiditiy can be improved by using the improved version
of the Pullback Theorem, Theorem~\ref{th:pansupullback_exterior_d} and a better choice of forms to be pulled back.
\begin{theorem}[Product rigidity] \label{th:product_rigidity}
Let $\{G_i\}_{1 \le i \le n}$, $\{G'_j\}_{1 \le j \le n'}$ be collections of Carnot groups where each $G_i, G'_j$ is nonabelian and does not admit
a nontrivial decomposition of Carnot groups. Let $G = \prod_i G_i$, $G' = \prod_j G'_j$. Set
$$ K_i : = \{ k \in \{1, \ldots, n\} : G_k \simeq G_i \}$$
and if $|K_i|\geq 2$ for some $i$, assume that
\begin{equation} \label{eq:p_rigidity}
p \ge \max\{ \nu_i - 1: |K_i| \ge 2 \}
\end{equation}
where $\nu_i$ denotes the homogeneous dimension of $G_i$.
Suppose that $f: G \supset U \to G'$ is a $W^{1,p}_{\operatorname{loc}}$-mapping, $U = \prod_i U_i$ is a product of open connected sets $U_i \subset G_i$,
and the (approximate) Pansu differential $D_P f(x)$ is an
isomorphism for a.e.\ $x\in U$. Then $f$ is a product of mappings, i.e. $n=n'$
and for some permutation $\sigma:\{1, \ldots, n\} \to \{1, \ldots, n\}$ there are mappings
$\{f_{\sigma(i)}:U_i \to G'_{\sigma(i)}\}_{1 \le i \le n}$
such that
\begin{equation} \label{eq:product_mapping}
f(x_1, \ldots, x_n) = \left(f_1(x_{\sigma^{-1}(1)}), \ldots, f_n(x_{\sigma^{-1}(n)}) \right)
\end{equation}
for a.e. $(x_1,\ldots,x_n)\in\prod_iU_i$.
\end{theorem}
For many groups the condition \eqref{eq:p_rigidity} on $p$ can be improved.
\begin{corollary} \label{co:product_rigidity}
Let $G_i, G'_j, U_i$ be as in Theorem~\ref{th:product_rigidity}, assume that $G_i$ is a group of step $m_i$ and set
$$ \bar m = \max \{ m_i : |K_i| \ge 2\}, \quad \bar \nu = \sum_{i : |K_i| \ge 2} \nu_i.$$
Assume further for all $i$ with $|K_i| \ge 2$ the Lie algebra $\mathfrak{g}_i/ \oplus_{j=3}^{m_i} V_j$ is not a free Lie algebra.
Then the conclusions of Theorem~\ref{th:product_rigidity} hold provided \eqref{eq:p_rigidity}
is replaced by the weaker conditions
\begin{equation} p \ge 2 \quad \text{and} \quad \frac1p \le \frac1{\bar m} + \frac{1}{\bar \nu}.
\end{equation}
\end{corollary}
For example, the conclusion of Theorem~\ref{th:product_rigidity} holds for $p=2$ if all $G_i$ are isomorphic to a higher Heisenberg group $\H_{d_i}$ with $d_i \ge 2$ or to a complex Heisenberg group $\H^\mathbb{C}_{d_j}$ with $d_j \ge 1$. On the other hand, the assumptions are not satisfied if some $G_i$ is a copy of the first Heisenberg group $\H_1$ which is a free Carnot group of step $2$.
\medskip
We use the following result from \cite[Prop. 2.5]{Xie_Pacific2013}, see also \cite{KMX1}.
\begin{lemma}
\label{lem_isom_is_product}
Suppose $\mathfrak{g}=\oplus_{i\in I}\mathfrak{g}_i$, $\mathfrak{g}'=\oplus_{j\in I'}\mathfrak{g}_j'$ where every $\mathfrak{g}_i$, and $\mathfrak{g}'_j$ is nonabelian and does not admit a nontrivial decomposition as a direct sum of graded ideals.
Then any graded isomorphism $\phi:\mathfrak{g}\rightarrow \mathfrak{g}'$ is a product of graded isomorphisms, i.e. there is a bijection $\sigma:I\rightarrow I'$ and for every $i\in I$ there exists a graded isomorphism $\phi_i:\mathfrak{g}_i\rightarrow \mathfrak{g}'_{\sigma(i)}$ such that for all $i\in I$ we have $\pi_{\sigma(i)}\circ\phi=\phi_i\circ\pi_i$.
\end{lemma}
We also use the following Fubini-type property of Sobolev maps.
\begin{lemma} \label{le:fubini_sobolev}
Let $G_1$, $G_2$, and $G'$ be Carnot groups, let $U_i \subset G_i$ be open sets, let $\imath_i$ be the injections
$G_i \to G_1 \times G_2$ and let $\pi_i: G_1 \times G_2 \to G_i$ be the corresponding projections.
Let $f$ be the representative of a map in $W^{1,p}(U_1 \times U_2; G')$, let $D_P f$ be a representative
of the
(approximate) Pansu differential.
Then the following assertions hold.
\begin{enumerate}
\item For a.e.\ $a \in U_1$ the map $f_a: U_2 \to G'$ defined by $f_a(y) = f(a,y)$ is in the Sobolev space $W^{1,p}(U_2;G')$
and the
Pansu differential $D_P f_a$ satisfies
\begin{equation} \label{eq:fubini_chain}
D_P f_a = D_P f(a, \cdot) \circ \imath_2 \qquad \text{$\mu_{G_2}$-a.e.\ in $U_2$.}
\end{equation}
\item If, in addition, $U_2$ is connected and $D_P f \circ \imath_2 = 0$ a.e.\ then there exists a function $\bar f: U_1 \to G'$
such that $f = \bar f \circ \pi_1$ a.e.
\item If $G'=G_1\times G_2$, $U_i$ is connected, and for a.e. $x\in U$ we have
$$
\pi_1\circ D_Pf(x)\circ\imath_2=0\,,\quad\text{and} \quad \pi_2\circ D_Pf(x)\circ\imath_1=0\,,
$$
then there exist mappings $\bar f_i:U_i\rightarrow G_i$ such that $f(x_1,x_2)=(\bar f_1(x_1),\bar f_2(x_2))$ for a.e. $(x_1,x_2)\in U_1\times U_2$.
\end{enumerate}
\end{lemma}
\begin{proof}
To prove (1), first note that for $G' = \mathbb{R}^N$ the Pansu differential (viewed as map on the Lie algebra) is given by $D_P f = D_h f \circ \Pi_1$
a.e., where $D_h f$ is the weak horizontal differential and $\Pi_1$ is the projection to the horizontal subspace. Thus, for $G' = \mathbb{R}^N$, assertion (1)
follows directly from the definition of the weak horizontal derivative and Fubini's theorem.
For a general Carnot group $G'$ there exists a $p$-integrable function $g:U_1 \times U_2 \to \mathbb{R}$ such that
$|D_h d(z, f(\cdot))| \le g$ a.e.
Applying the result for real-valued maps to the maps $(x,y) \mapsto d(z, f(x,y))$ for all $z$ in a countable
dense subset $D$ of $G'$ we easily conclude that $|D_h d(z, f_a(\cdot)| \le g_a$ and hence $f_a \in W^{1,p}(U_2, G')$ for a.e. $a \in U_1$.
Let $\Pi_{G'}$ denote the abelianization map. Then we can apply the result for $\mathbb{R}^N$-valued maps to $\Pi_{G'} \circ f$
and we get, for a.e $a \in U_1$,
$$ \Pi_{G'} \circ D_P f_a = D_P (\Pi_{G'} \circ f_a) = \big(D_P (\Pi_{G'} \circ f)(a, \cdot) \big) \circ \imath_2 = \Pi_{G'} \circ (D_P f \circ \imath_2)(a, \cdot)
$$
$\mu_{G_2}$-a.e.\ in $U_2$.
Now if $\Phi, \Psi: G_2 \to G'$ are graded group homomorphism such that $\Pi_{G'} \circ \Phi = \Pi_{G'} \circ \Psi$ then $\Phi = \Psi$.
Hence \eqref{eq:fubini_chain} holds.
Assertion (2) is an immediate consequence of assertion (1).
Assertion (3) follows by applying assertion (2) to the compositions $\pi_i\circ f$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{th:product_rigidity}]
The result was established in \cite{KMX1} under the stronger condition $p > \nu(\Pi_i G_i) = \sum_i \nu_i$. We first briefly
recall the argument in \cite{KMX1} and indicate the strategy to obtain the improved condition
\eqref{eq:p_rigidity}.
First, Lemma~\ref{lem_isom_is_product} implies that we may assume without loss of generality
that
$n=n'$ and $\mathfrak{g}_i=\mathfrak{g}_i'$ for all $i\in I$, and so there
is a measurable function $\sigma:U\rightarrow \operatorname{Perm}(\{1, \ldots, n\})$ such that $D_Pf(x)(\mathfrak{g}_i)=\mathfrak{g}_{\sigma(x)(i)}$ for a.e. $x\in U$.
Moreover $\sigma(i) = i$ if $|K_i| = 1$. Hence it follows by applying Lemma~\ref{le:fubini_sobolev}(3) repeatedly that we may assume without loss of generality that
\begin{equation} \label{eq:no_trivial_factors}
|K_i| \ge 2 \quad \forall i=1, \ldots, n.
\end{equation}
The main point is to show that there exists a constant permutation $\bar \sigma$ such that $\sigma = \bar \sigma$ a.e.
Then, using again Lemma~\ref{le:fubini_sobolev}, we see $f$ has the desired product structure.
To prove that $\sigma$ is constant
a.e.\ we argue as follows. We choose closed left-invariant forms $\alpha$
such that the pullback $f_P^*\alpha$ can 'detect' the permutation $\sigma$. Then we use the Pullback Theorem to deduce that
for suitable left-invariant forms $\beta$, we have
$d f_P^*\alpha \wedge \beta = 0$ in the sense of distributions
and conclude that $\sigma$ is constant. In \cite{KMX1} we use for $\alpha$ the
volume forms $\omega_i$ of the factors $G_i$. Then Lemma~\ref{lem_isom_is_product} implies that
$f_P^*\omega_i = \sum_{j \in K_i} a_j \omega_j$
where the $a_j$ are integrable functions. Moreover, for each
$x$, exactly one of the functions $a_j$ is different from zero, namely $a_{\sigma^{-1}(i)}$
Let $j' \in \{1, \ldots, n\}$ and $l \ne j'$. For $X \in V_1 \cap \mathfrak{g}_{l}$ we apply the Pullback theorem with the closed codegree $N_{j'} + 1$
test forms
$\beta =i_X \omega_l \wedge \bigwedge_{i' \notin\{j',l\}} \omega_{i'}$ and easily
conclude that $a_{j'}(x_1, \ldots, x_n)$ depends only on $x_{j'}$. Then one easily deduces that $\sigma = \bar \sigma$ almost everywhere.
The requirement $p > \nu(\Pi_i G_i)$ in \cite{KMX1} comes from the hypotheses of the Pullback Theorem \cite[Theorem 4.5]{KMX1}. Using the improved Pullback Theorem,
Theorem~\ref{th:pansupullback_exterior_d}, we see that this argument works
if $p \ge \max_i \nu_i$.
The second condition in Theorem~\ref{th:pansupullback_exterior_d}, namely $\frac1p \le \frac1m + \frac1\nu$ is then automatically
satisfied since
\begin{equation} \label{eq:bound_m}
m \le \max_i \nu_i - 1.
\end{equation}
To get the improved condition
\eqref{eq:p_rigidity} we will apply the pullback theorem not to the volume forms of each factor, but to a codegree one form on each factor,
with weight $-\nu_i +1$.
To set the notation, we assign to a form $ \alpha \in \Omega^*(G_i)$ the form $\pi_i^* \alpha \in \Omega^*(G)$ where $\pi_i: G \to G_i$ is the projection map.
Note that $\pi_i^* \alpha$ is closed if and only if $\alpha$ is closed. We will usually write $\alpha$ also for the form $\pi_i^* \alpha$ if no confusion can occur.
Similar we identify a vector field $X \in TG_i$ with a vectorfield in $TG$ through the push-forward by the canonical injection $G_i \to G$.
For $i\in \{1, \ldots, n\}$ let $\operatorname{vol}_{G_i}$ denote the volume form in $G_i$ and let $Y \in V_1(G_i) \setminus \{0\}$.
By Cartan's formula and the biinvariance of $\operatorname{vol}_{G_i}$ the $N_i-1$ form
$i_Y \operatorname{vol}_{G_i}$ is closed. Let $\alpha_i = i_Y \operatorname{vol}_{G_i}$.
Then the $N_i-1$ form $\alpha_i$ is left-invariant, closed and has weight $-\nu_i +1$.
In view of \eqref{eq:no_trivial_factors} we have $p \ge \nu_{i}-1$.
For $j \in K_i$ let $X_{j,k}$, $k=1, \ldots \dim V_1(G_{i})$ be a basis of $V_1(G_{j})$.
Then $ i_{X_{j,k}}\operatorname{vol}_{G_j}$, $k=1, \ldots \dim V_1(G_{i})$ is a basis of the left-invariant forms
on $G_{j}$ with degree $N_i-1$ and weight $-\nu_i +1$.
Since pullback by a graded isomorphism preserves degree and weight we have
$$f_P^*\alpha_{i} = \sum_{j \in K_i} \sum_{k=1}^{\dim V_1 (G_{i})} a_{j,k} \, i_{X_{j,k}} \operatorname{vol}_{G_j}$$
with $a_{j,k} \in L^1_{\operatorname{loc}}(U)$.
Set
$$ a_j :=(a_{j,1}, \ldots, a_{j, \dim V_1(G_{i})})$$
and
$$ E_j = \{ x \in U : \sigma^{-1}(x) (i) = j \}.$$
Then $U \setminus \bigcup_{j \in K_i} E_j$ is a null set. Since $D_P f(x)$ is a graded automorphism for a.e.\ $x \in U$, we have,
for all $j \in K_i$,
\begin{equation} \label{eq:disjoint_pullback}
a_j \ne 0 \quad \text{a.e.\ in $E_j$}, \qquad a_j = 0 \quad \text{a.e.\ in $U \setminus E_j$}.
\end{equation}
We next show that
\begin{equation} \label{eq:product_distributional_derivative}
Z a_{j'} = 0 \quad \text{in distributions for all $ j' \in K_i, \, Z \in \oplus_{l \ne j'} \mathfrak{g}_{l}$.}
\end{equation}
To prove \eqref{eq:product_distributional_derivative},
let $\theta_{j',k'}$ be basis of left-invariant one-forms which vanish on
$\oplus_{l = 2}^s V_l(G_{j'})$
which is dual to the basis $X_{j',k}$ of $V_{1}(\mathfrak{g}_{j'})$
, i.e. $\theta_{j',k'}(X_{j',k}) = \delta_{k k'}$.
Note that the forms $\theta_{j',k'}$ are closed.
For $l \in \{1, \ldots, n\} \setminus \{j'\}$, and $X \in V_1(G_{l})$ consider the closed form
$$ \beta = \theta_{j',k'} \wedge i_{X} \operatorname{vol}_{G_{l}} \wedge (\Lambda_{i'\neq j',l} \operatorname{vol}_{G_{i'}}).
$$
Then, for a.e.\ $x \in U$,
$$ (D_Pf)^*(x) \alpha_i \wedge \beta = \pm a_{j',k'} \, i_X \operatorname{vol}_G.$$
In view of \eqref{eq:bound_m}
and the assumption $p \ge \nu_{i} - 1$
(recall that we may assume \eqref{eq:no_trivial_factors})
we get from
the Pullback Theorem,
Theorem~\ref{th:pansupullback_exterior_d},
$$
0 = \int_U f_P^*\alpha \wedge \beta \wedge d\varphi =\pm \int_U a_{j',k'} \, \, X\varphi \, \, \operatorname{vol}_G
$$
for all $\varphi \in C_c^\infty(U)$.
Since $V_1\cap \mathfrak{g}_{l}$ generates $\mathfrak{g}_{l}$ as a Lie algebra, we
see that \eqref{eq:product_distributional_derivative} holds.
It follows from \eqref{eq:product_distributional_derivative} that $a_j(x) = a_j(x_j)$. Thus \eqref{eq:disjoint_pullback} implies that
$\chi_{E_j}(x) = \chi_{E_j}(x_j)$ for all $j \in K_i$.
Since $\sum_{j \in K_i} \chi_{E_j} = 1$ a.e.\ there exists one $j_0$ such that
$\chi_{E_{j_0}} = 1$ almost everywhere. Thus $\sigma^{-1}(i) = j_0$ almost everywhere.
Summarizing, we have shown that for all $i$ the function $\sigma^{-1}(x) (i)$ is constant almost everywhere.
Hence $\sigma$ is constant almost everywhere.
\end{proof}
\bigskip
The proof of Corollary~\ref{co:product_rigidity} is very similar to the proof of Theorem~\ref{th:product_rigidity}.
The main change is that instead of the closed codegree one forms $i_{Y} \operatorname{vol}_{G_i}$ of weight $-\nu_i + 1$ we pull back certain closed
two-forms of weight $-2$ in $G_i$. To identify suitable two-forms we use the setting in \cite{kmx_rumin}.
For a Carnot algebra $\mathfrak{g} = \oplus_{i=1}^s V_i$, let $\Lambda^{1}_v\mathfrak{g}$ denote the space of one-forms which vanish on
the first layer $V_1$
and let
$I^*\mathfrak{g} \subset \Lambda^*\mathfrak{g}$ be the differential ideal generated by $\Lambda^{1}_v\mathfrak{g}$.
Thus
\begin{equation}
I^*\mathfrak{g} = \operatorname{span} \{ \alpha \wedge \tau + \beta \wedge d\eta : \alpha, \beta \in \Lambda^*\mathfrak{g}, \tau, \eta \in \Lambda^{1}_v\mathfrak{g} \}.
\end{equation}
The set of $k$-forms in $I^*\mathfrak{g}$ is denoted by $I^k\mathfrak{g} = I^*\mathfrak{g} \cap \Lambda^k\mathfrak{g}$.
We define $J^*\mathfrak{g}$ to be the annihilator $\operatorname{Ann}(I^*\mathfrak{g})$ of $I^*\mathfrak{g}$, i.e.,
\begin{equation}
J^*\mathfrak{g} = \{ \alpha \in \Lambda^*\mathfrak{g} : \alpha \wedge \beta = 0 \quad \forall \beta \in I^*\mathfrak{g} \}.
\end{equation}
We will use the following facts about $I^*\mathfrak{g}$ and $J^*\mathfrak{g}$ which easily follow from exterior algebra and the
formula for $d\alpha$. For the convenience of the reader we include a proof after the proof of
Corollary~\ref{co:product_rigidity}.
\begin{proposition} \label{pr:IJ} Let $\mathfrak{g} = \oplus_{i=1}^s V_i$ be a Carnot algebra of dimension $N$, homogeneous dimension $\nu$
and step $s \ge 2$. Let $G$ be the corresponding Carnot group.
Then the following assertions hold.
\begin{enumerate}
\item \label{it:jnmk}
For all $0\leq k\leq N$ we have
$$
J^{N-k}\mathfrak{g}=\operatorname{Ann}(I^k\mathfrak{g})\cap \Lambda^{N-k}\mathfrak{g}
=\{\alpha\in \Lambda^{N-k}\mathfrak{g}\mid \alpha\wedge\beta=0\,,\; \forall\beta\in I^k\mathfrak{g}\}.
$$
\item \label{it:IJ1} If $I^k\mathfrak{g} = \Lambda^k\mathfrak{g}$ then $J^k\mathfrak{g} = \{0\}$. If $I^k\mathfrak{g} \ne \Lambda^k\mathfrak{g}$ then the wedge product induces a nondegenerate pairing
$$ \Lambda^k\mathfrak{g}/ I^k\mathfrak{g} \times J^{N-k}\mathfrak{g} \overset{\wedge}{\longrightarrow} \Lambda^N \mathfrak{g} \simeq \mathbb{R}.$$
In particular, $\dim \Lambda^k\mathfrak{g}/ I^k\mathfrak{g} = \dim J^{N-k}\mathfrak{g}$
and for each basis $\{ \tilde \alpha_i\} $ of $ \Lambda^k\mathfrak{g}/ I^k\mathfrak{g}$ there exists a dual basis $ \{ \gamma_j\}$
of $J^{N-k}\mathfrak{g}$ such that $\tilde \alpha_i \wedge \gamma_j = \delta_{ij} \operatorname{vol}_G$.
\item \label{it:IJ2} $J^*\mathfrak{g}$ is a differential ideal, i.e. $\alpha \in J^k\mathfrak{g} \Longrightarrow d\alpha \in J^{k+1}\mathfrak{g}$.
\item \label{it:IJ3} If $\mathfrak{g}'$ is another Carnot algebra and $\Phi: \mathfrak{g} \to \mathfrak{g}'$ is a graded isomorphism then $\Phi^*(I^k\mathfrak{g}') = I^k\mathfrak{g}$
and $\Phi^*$ induces an isomorphism from $\Lambda^k \mathfrak{g}'/ I^k\mathfrak{g}'$ to $\Lambda^k\mathfrak{g}/ I^k\mathfrak{g}$.
\item \label{it:IJ4} If $\gamma \in J^{k}\mathfrak{g} \setminus \{0\}$ then $\gamma$ is homogeneous with coweight equal to its codegree, i.e. $\operatorname{wt}(\alpha)=N-k-\nu$.
\item \label{it:IJ5} If $\gamma \in J^k\mathfrak{g}$ then $d\gamma = 0$.
\end{enumerate}
\end{proposition}
The main new ingredient in the proof of Corollary~\ref{co:product_rigidity} is the following simple observation.
For a Carnot algebra $\mathfrak{g} = \oplus_{i=1}^s V_i$ with $s \ge 3$ let $\pi_{1,2}$ denote the projection
$\mathfrak{g} \to V_1 \oplus V_2$. We define $ \tilde \mathfrak{g} := \mathfrak{g}/ \oplus_{j=3}^s V_j$ as the algebra $V_1 \oplus V_2$ with
bracket $[X,Y]_\sim = \pi_{1,2}[X,Y]$. Then $\tilde \mathfrak{g}$ is a Carnot algebra. If $s=2$, we set $\tilde \mathfrak{g} = \mathfrak{g}$.
\begin{proposition} \label{pr:good_two_forms}
If $\mathfrak{g}/ \oplus_{j=3}^s V_j$ is not a free Carnot algebra then
$I^2(\mathfrak{g}) \ne \Lambda^2(\mathfrak{g})$. Moreover
\begin{equation} \label{eq:I2_horizontal}
\Lambda^2\mathfrak{g}/ I^2\mathfrak{g} \simeq (\Lambda^1_h \mathfrak{g} \wedge \Lambda^1_h \mathfrak{g} )/ (I^2\mathfrak{g} \cap (\Lambda^1_h \mathfrak{g} \wedge \Lambda^1_h \mathfrak{g} ))
\end{equation}
where $\Lambda^1_h\mathfrak{g}$ denotes the space of horizontal one-forms, i.e. one-forms which vanish on $\oplus_{i=2}^s V_i$.
\end{proposition}
\begin{proof} Since $\Lambda^1\mathfrak{g} = \Lambda^1_h \mathfrak{g} \oplus \Lambda^1_v\mathfrak{g}$ we have
$$ \Lambda^2 \mathfrak{g} = (\Lambda^1_h \mathfrak{g} \wedge \Lambda^1_h \mathfrak{g}) \oplus (\Lambda^1_v \mathfrak{g} \wedge \Lambda^1_h \mathfrak{g}) \oplus
(\Lambda^1_v \mathfrak{g} \wedge \Lambda^1_v \mathfrak{g} ).$$
Since the second and third summand on the right hand side are contained in $I^2\mathfrak{g}$ we get \eqref{eq:I2_horizontal}.
Now assume that $\mathfrak{g}/ \oplus_{j=3}^s V_j = V_1 \oplus V_2$ is not a free Carnot algebra. Let $\{X_i\}_{1 \le i \le \dim V_1}$
be a basis of $V_1$. Then there exist coefficients
$\{a_{i,j}\}_{1 \le i < j \le \dim V_1}$ which do not all vanish such that
$$ \sum_{1 \le i < j \le \dim V_1} a_{i,j} [X_i, X_j]_\sim = 0.$$
Since the $a_{i,j}$ do not all vanish there exists $\gamma \in \Lambda^1_h \wedge \Lambda^1_h$ such that
$$ \gamma(\sum_{i < j} a_{i,j} X_i \wedge X_j) \ne 0.$$
We claim that $\gamma \notin I^2\mathfrak{g}$. Otherwise there exist $\alpha \in \Lambda^1$ and $\tau, \eta \in \Lambda^1_v$ such that
\begin{align*}
0 \ne \,( \alpha \wedge \tau + d\eta)\left(\sum_{i < j} a_{i,j} X_i \wedge X_j\right)
= - \eta\left( \sum_{i < j} a_{i,j} [X_i, X_j]_\sim \right) = 0.
\end{align*}
This contradiction concludes the proof.
\end{proof}
\bigskip
\begin{proof}[Proof of Corollary~\ref{co:product_rigidity}]
Again we may assume without loss of generality that $|K_i| \ge 2$ for all $i$, that $n=n'$ and that $\mathfrak{g}'_i = \mathfrak{g}_i$.
Then $G'$ is a step $\bar m$ group.
By Propositions~\ref{pr:IJ} and~\ref{pr:good_two_forms} there exist horizontal two-forms $\alpha_{j,k} \in \Lambda_h^1 \wedge \Lambda_h^1$
such that $\alpha_{j,k} + I^2\mathfrak{g}_j$ is a basis of $\Lambda^2\mathfrak{g}_j/ I^2\mathfrak{g}_j$ and there exist dual bases
$\gamma_{j,k'}$ of $J^{N_j-2}\mathfrak{g}_j$ such that
\begin{equation} \alpha_{j,k} \wedge \gamma_{j,k'} = \delta_{kk'} \operatorname{vol}_{G_j}.
\end{equation}
Note also that forms in $ \Lambda_h^1 \wedge \Lambda_h^1$ are closed since forms in $\Lambda_h^1$ are closed.
Now we can proceed as in the proof of Theorem~\ref{th:product_rigidity}.
Let $i \in \{1, \ldots, n\}$. Then, for a.e.\ $x \in U$ we have
$(D_P f)^*(x) \alpha_{i,1} \in \Lambda^1_h(\mathfrak{g}_{\sigma^{-1}(i)}) \wedge \Lambda^1_h(\mathfrak{g}_{\sigma^{-1}(i)})$.
Thus
$$
f_P^*\alpha_{i,1}
= \sum_{j \in K_i} \sum_k (a_{j,k} \alpha_{j,k} + \beta_k) \, $$
with $a_{j,k} \in L^1_{\operatorname{loc}}(U)$ and $\beta_k \in L^1_{\operatorname{loc}}(U; I^2\mathfrak{g}_j)$.
Set
$$ a_j :=(a_{j,1}, \ldots, a_{j, \dim( \Lambda^2(\mathfrak{g}_j)/ I^2\mathfrak{g}_j)})$$
and
$$ E_j = \{ x \in U : (\sigma(x))^{-1} (i) = j \}.$$
Then $U \setminus \bigcup_{j \in K_i} E_j$ is a null set.
By Proposition~\ref{pr:IJ}~\eqref{it:IJ3} we have, for a.e. $x \in U$, $(D_P f)^*(x) \alpha_{i,1} \notin I^2\mathfrak{g}_{\sigma^{-1}(i)}.$
Thus,
for all $j \in K_i$,
\begin{equation} \label{eq:disjoint_pullback_bis}
a_j \ne 0 \quad \text{a.e.\ in $E_j$}, \qquad a_j = 0 \quad \text{a.e.\ in $U \setminus E_j$}.
\end{equation}
We next show that
\begin{equation} \label{eq:product_distributional_derivative_bis}
Z a_j = 0 \quad \text{in distributions for all $j\in K_i, \, Z \in \oplus_{j' \ne j} \mathfrak{g}_{j'}$.}
\end{equation}
To prove \eqref{eq:product_distributional_derivative_bis}, let
$j' \in K_i$, $l \in \{1, \ldots, n\} \setminus \{j'\}$, $X \in V_1(G_{l})$, and consider the closed form
$$ \beta = \gamma_{j',k'} \wedge i_{X} \operatorname{vol}_{G_{l}} \wedge \Lambda_{i'\neq j',l} \operatorname{vol}_{G_{i'}}.
$$
Then $\beta$ is a closed form of degree $N-3$ and weight $-\nu + 3$, where $N$ and $\nu$ are the topological
and homogeneous dimension of $G$, respectively.
Moreover
$$ \alpha_{j,k} \wedge \beta = \pm \delta_{jj'} \, \delta_{kk'} \, i_X \operatorname{vol}_G.$$
Finally,
$\alpha_{i,1}$ is a closed left-invariant two-form of weight $-2$. Thus the conditions on $p$ in Corollary~\ref{co:product_rigidity}
allow us to apply the
Pullback Theorem,
Theorem~\ref{th:pansupullback_exterior_d}, and we get
$$
0 = \int_U f_P^*\alpha_{i,1} \wedge \beta \wedge d\varphi =\pm \int_U a_{j',k'} \, \, X\varphi\, \, \operatorname{vol}_{G}.
$$
for all $\varphi \in C_c^\infty(U)$, all $j' \in K_i$, and all $k'$.
As in the proof Theorem~\ref{th:product_rigidity} we conclude that $a_{j,k}(x_1, \ldots, x_n)$ depends only on $x_j$
and that $\sigma$ is constant almost everywhere.
\end{proof}
\medskip
Looking back, we see that the arguments in the proofs of Theorem~\ref{th:product_rigidity} and
Corollary~\ref{co:product_rigidity} are exactly analogous. The only difference is that we use different
forms in the ideals $I^*$ and $J^*$ as forms to be pulled back and as test forms.
Indeed, note that $I^1\mathfrak{g}_i = \Lambda^1_v\mathfrak{g}_i$
and thus $\Lambda^1\mathfrak{g}_i/I^1\mathfrak{g}_i \simeq \Lambda^1_h\mathfrak{g}_i$ and $J^1\mathfrak{g}_i = \{ i_X \operatorname{vol}_{G_i} : X \in \Lambda^1_h\mathfrak{g}_i\}$.
Thus in the proof of Theorem~\ref{th:product_rigidity} we pull back a form in $J^1\mathfrak{g}_i$ and use test forms of the type
$\Lambda^1\mathfrak{g}_{j'}/I^1\mathfrak{g}_{j'} \wedge i_X \operatorname{vol}_{G_l} \wedge \Lambda_{i' \ne \{j',l\}} \operatorname{vol}_{G_{i'}}$,
while for the proof
of Corollary~\ref{co:product_rigidity} we pull back forms in $\Lambda^2\mathfrak{g}_i/I^2\mathfrak{g}_i$ and use test forms of the type
$J^2\mathfrak{g}_{j'} \wedge i_X \operatorname{vol}_{G_l} \wedge \Lambda_{i' \ne \{j',l\}} \operatorname{vol}_{G_{i'}}$.
\bigskip
We finally provide a proof of Proposition~\ref{pr:IJ} for the convenience of the reader.
\begin{proof}[Proof of Proposition~\ref{pr:IJ}]
\eqref{it:jnmk}
Since $I^k\mathfrak{g}\subset I^*\mathfrak{g}$ we have $\operatorname{Ann}(I^k\mathfrak{g})\cap \Lambda^{N-k}\mathfrak{g}\supseteq \operatorname{Ann}(I^*\mathfrak{g})\cap \Lambda^{N-k}\mathfrak{g}$. To establish the opposite inclusion, choose $\alpha\in \operatorname{Ann}(I^k\mathfrak{g})\cap \Lambda^{N-k}\mathfrak{g}$, and $\beta\in I^j\mathfrak{g}$ for some $0\leq j\leq N$.
If $k<j\leq N$ we have $\alpha\wedge\beta=0$ since $\deg\alpha+\deg\beta>N$. If $0\leq j\leq k$ and $\gamma\in \Lambda^{k-j}\mathfrak{g}$ we have $
\beta\wedge\gamma\in I^k\mathfrak{g}$ since $I^*\mathfrak{g}$ is an ideal, so
\begin{equation}
\label{eqn_al_be_ga}
(\alpha\wedge\beta)\wedge\gamma=\alpha\wedge(\beta\wedge\gamma)=0
\end{equation}
because $\alpha\in\operatorname{Ann}(I^k\mathfrak{g})$ by assumption. The pairing $\Lambda^{N-(k-j)}\mathfrak{g}\times\Lambda^{k-j}\mathfrak{g}\rightarrow \Lambda^N\mathfrak{g}$ is nondegenerate and $\gamma\in \Lambda^{k-j}\mathfrak{g}$ was arbitrary, so \eqref{eqn_al_be_ga} implies that $\alpha\wedge\beta=0$. Since $\beta \in \Lambda^j\mathfrak{g}$ was arbitrary, we have $\alpha\in \operatorname{Ann}(I^j\mathfrak{g})$. Since $j$ was arbitrary we get $\alpha\in \cap_{0\leq j\leq N}\operatorname{Ann}(I^j\mathfrak{g})=\operatorname{Ann}(I^*\mathfrak{g})$.
\eqref{it:IJ1}
Since the wedge product $\Lambda^k\mathfrak{g}\times\Lambda^{N-k}\mathfrak{g}\rightarrow \Lambda^N\mathfrak{g}\simeq \mathbb{R}$ is a nondegenerate pairing, \eqref{it:IJ1} follows from \eqref{it:jnmk} and the fact that for any nondegenerate pairing $E\times E'\stackrel{b}{\rightarrow}\mathbb{R}$ of finite dimensional vector spaces and any subspace $W\subset E$, there is a nondegenerate pairing
$(E/W)\times W^\perp\rightarrow \mathbb{R}$ induced by $b$, where $W^\perp:=\{e'\in E'\mid b(e,e')=0\,,\;\forall e\in W\}$.
\eqref{it:IJ2} This follows from the fact that $I$ is a differential ideal and the graded Leibniz rule.
\eqref{it:IJ3} This follows from the facts that $\Phi^*(\Lambda^1_v\mathfrak{g}') = \Lambda^1_v\mathfrak{g}$ and that $d$ commutes
with pullback by $\Phi$.
\eqref{it:IJ4} Let $\{X_{i,j}\}$, $i = 1, \ldots, s$, $j = 1, \ldots, \dim V_i$ be a graded basis of $\mathfrak{g}$, i.e., $X_{i,j} \in V_i$.
Let $\theta_{i,j}$ be the dual basis of one-forms. Then the forms $\theta_{i,j}$ are homogeneous with $\operatorname{wt}(\theta_{i,j}) = -i$.
Moreover
\begin{align*} \Lambda^1_h\mathfrak{g} = & \, \operatorname{span} \{ \theta_{1,j} : 1 \le j \le \dim V_1\}, \\
\Lambda^1_v\mathfrak{g} = & \, \operatorname{span} \{ \theta_{i,j} : i \ge 2, 1 \le j \le \dim V_i \}.
\end{align*}
Set $\tau = \Lambda_{i \ge 2, 1 \le j \le \dim V_i} \th_{i,j}$.
It is easy to see that for $k \ge N - \dim V_1$
every $\gamma \in J^k\mathfrak{g}$ is of the form
$$ \gamma = \alpha \wedge \tau, \quad \text{with $\alpha \in \Lambda^{k - (N- \dim V_1)} (\Lambda_h^1\mathfrak{g})$,}$$
and $J^k\mathfrak{g} = \{0\}$ if $k < N- \dim V_1$. Thus every non-zero element of $J^k\mathfrak{g}$ is a homogeneous form
with weight $-\nu + (N-k)$.
\eqref{it:IJ5} Let $\gamma \in J^k\mathfrak{g}$ and assume that $d\gamma \ne 0$. By properties \eqref{it:IJ2} and \eqref{it:IJ4} the form $\gamma$ is homogeneous and
\begin{equation} \label{eq:weights_dJk} \operatorname{wt}(d\gamma) = \operatorname{wt}(\gamma) - 1.
\end{equation}
On the other hand for $\gamma \in \Lambda^k\mathfrak{g}$ we have
$$ d\gamma(X_0, \ldots, X_k) = \sum_{0 \le i < j \le k} \gamma([X_i, X_j], X_0, \ldots, \hat X_i, \ldots, \hat X_j, \ldots, X_k) $$
where $\hat X_i$ denotes that $X_i$ is omitted (see, for example, \cite[Lemma 14.14]{Michor}).
If $\gamma$ is homogeneous and $d\gamma \ne 0$ it follows that $d\gamma$ is homogeneous and
$\operatorname{wt}(d\gamma) = \operatorname{wt}(\gamma)$. This contradicts \eqref{eq:weights_dJk}. Hence $d\gamma = 0$.
\end{proof}
\section{Complexified Carnot algebras}
In \cite{KMX1} it was shown that under suitable conditions nondegenerate Sobolev maps of a complexified Carnot group are
automatically holomorphic or antiholomorphic. In this section show these results with improved conditions on the Sobolev exponent.
We first recall the setting in \cite{KMX1} to which we refer for further details.
Let $H$ be a Carnot group of topological dimension $N$ and homogeneous dimension $\nu$. Let $\mathfrak{h}$ be the corresponding Carnot algebra.
Let $\mathfrak{g}$ denote the complexified Carnot algebra, i.e. $\mathfrak{g} = \mathfrak{h}^\mathbb{C}$
equipped with the grading $\mathfrak{g}=\oplus_jV_j^\mathbb{C}$.
The corresponding Carnot group $G$ has topological dimension
$2N$ and homogeneous dimension $2\nu$.
We now denote by $J$ the almost complex structure on $G$
coming from $\mathfrak{g}$; it follows from the Baker-Campbell-Hausdorff formula that $J$ is integrable, i.e.\ $(G,J)$ is a complex manifold, and the group operations are holomorphic.
Also, complex conjugation $\mathfrak{g}\rightarrow \mathfrak{g}$ is induced by a unique graded automorphism $G\rightarrow G$, since $G$ is simply-connected.
\begin{theorem} \label{th:holomorphic} Let $U \subset G$ be a connected open subset, let $p \ge \nu$, let $f \in W^{1,p}_{\operatorname{loc}}(U, G)$ and assume that
(approximate) Pansu differential $D_P f (x)$ is either a $J$-linear
isomorphism or a $J$-antilinear graded isomorphism for a.e.\
$x \in U$. Then $f$ is holomorphic or antiholomorphic (with respect to the complex structure $J$).
\end{theorem}
\begin{corollary} \label{co:holomorphic} Let $U \subset G$ be open, let $p \ge \nu$, let $f \in W^{1,p}_{\operatorname{loc}}(U, G)$.
Suppose that any graded isomorphism $\mathfrak{g} \to \mathfrak{g}$ is either $J$-linear or $J$-antilinear and that $D_P f(x)$ is an isomorphism
for a.e.\ $x \in U$. Then $f$ is holomorphic or antiholomorphic (with respect to the complex structure $J$).
\end{corollary}
The condition that any graded isomorphism $\mathfrak{g} \to \mathfrak{g}$ is either $J$-linear or $J$-antilinear is in particular satisfied
for the complexified Heisenberg algebras $\mathfrak{h}^\mathbb{C}_m$, see \cite[Section 6]{reimann_ricci}.
\begin{proof}[Proof of Theorem~\ref{th:holomorphic}] The result is proved in \cite{KMX1} under the stron\-ger condition
$p > \text{homogeneous dimension of $G$} = 2 \nu$.
The key step in the proof is to show that $D_P f$ cannot switch
between a $J$-linear and a $J$-antilinear map.
To show this we use the Pullback Theorem to prove that the the pullback of
the top degree holomorphic form cannot oscillate between a holomorphic and an anti-holomorphic form.
Since the top degree holomorphic form has weight $-\nu$,
the improved version of the Pullback Theorem, Theorem~\ref{th:pansupullback_exterior_d},
gives this result already under the weaker condition $p \ge \nu$.
Note that $G$ is a group of step $m$ with $m < \nu$ so the condition $\frac1p \le \frac1m + \frac1{\bar \nu}$, where $\bar \nu = 2 \nu$ is the
homogeneous dimension of $G = H^\mathbb{C}$, is automatically satisfied.
Thus, under the assumption $p \ge \nu$, we still may asssume that $D_P f$ is $J$-linear a.e.\ (the case
that $D_P f$ is $J$-antilinear a.e.\ being analogous). Let $\pi_{G} : G \to G/[G,G]$ denote the abelianization map.
By Remark~\ref{re:weak_derivative_abel}, the map $\pi_{G} \circ f$ belongs to $W^{1,p}_{\operatorname{loc}}$ and
for each horizontal vectorfield $X$ the weak derivative satisfies $X (\pi_{G} \circ f)(x) = D_P f(x) X$
for a.e.\ $x \in U$. Since $D_P f(x)$ is $J$-linear, the horizontal anticonformal derivatives
$\bar Z (\pi_{G} \circ f)$ vanish. It follows easily that $\pi_{G} \circ f$ is holomorphic (see, e.g., \cite{KMX1}).
In particular $\pi_{G} \circ f$ is smooth and hence $|D_P f(x)| := \max \{ |D_P f(x) X|_{\mathfrak{g}} : |X|_{\mathfrak{g}} \le1, X \in V_1\}$
is locally bounded. By \eqref{eq:optimal_g}, it follows that $f \in W^{1, \infty}_{\operatorname{loc}}(U;G)$.
Thus the assertion follows from the result in \cite{KMX1} for $p > 2 \nu$.
\end{proof}
\bigskip
\begin{proof}[Proof of Corollary~\ref{co:holomorphic}] This follows immediately from Theorem~\ref{th:holomorphic}.
\end{proof}
| {
"timestamp": "2021-12-06T02:21:14",
"yymm": "2007",
"arxiv_id": "2007.06694",
"language": "en",
"url": "https://arxiv.org/abs/2007.06694",
"abstract": "We show that in an $m$-step Carnot group, a probability measure with finite $m^{th}$ moment has a well-defined Buser-Karcher center-of-mass, which is a polynomial in the moments of the measure, with respect to exponential coordinates. Using this, we improve the main technical result of our previous paper concerning Sobolev mappings between Carnot groups; as a consequence, a number of rigidity and structural results from recent papers hold under weaker assumptions on the Sobolev exponent. We also give applications to quasiregular mappings, extending earlier work in the $2$-step case to general Carnot groups.",
"subjects": "Differential Geometry (math.DG)",
"title": "Pansu pullback and exterior differentiation for Sobolev maps on Carnot groups",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754488233528,
"lm_q2_score": 0.718594386544335,
"lm_q1q2_score": 0.7075103906538306
} |
https://arxiv.org/abs/0712.3411 | The Parabolic Two-Phase Membrane Problem: Regularity in Higher Dimensions | For the parabolic obstacle-problem-like equation $$\Delta u - \partial_t u = \lambda_+ \chi_{\{u>0\}} - \lambda_- \chi_{\{u<0\}} ,$$ where $\lambda_+$ and $\lambda_-$ are positive Lipschitz functions, we prove in arbitrary finite dimension that the free boundary $\partial\{u>0\} \cup\partial\{u<0\}$ is in a neighborhood of each ``branch point'' the union of two Lipschitz graphs that are continuously differentiable with respect to the space variables. The result extends the elliptic paper \cite{imrn} to the parabolic case. The result is optimal in the sense that the graphs are in general not better than Lipschitz, as shown by a counter-example. | \section{Introduction}
\subsection{Background and main result}
In this paper we study the regularity of the
parabolic obstacle-problem-like equation
\begin{equation}\label{obst}
\Delta u - \partial_t u = \lambda_+ \chi_{\{u>0\}} \> - \> \lambda_-
\chi_{\{u<0\}}\; \qquad \hbox{in } (0,T)\times \Omega ,
\end{equation}
where $T<+\infty, \lambda_+> 0, \lambda_->0$ are Lipschitz functions and $\Omega \subset {\bf R}^n$ is a
given domain. The problem arises as limiting case in the model
of temperature control through the interior described
in \cite[2.3.2]{duvaut} as $h_1,h_2\to 0$.\\
We are interested in the regularity of the free boundary $\partial\{ u>0\} \cup\partial\{ u<0\}$.
As the one-phase case (i.e. the case of a non-negative or non-positive solution)
is covered by classical results, and
regularity of the set $\{ u=0\} \cap \{ \nabla u\ne 0\}$ can be obtained via
the implicit function theorem (see Section \ref{non} for higher regularity),
the research focusses on the study of $\partial\{ u>0\}
\cap \partial\{ u<0\} \cap \{ \nabla u=0\}$.\\
In the stationary case --- the two-phase membrane problem ---
the authors proved (\cite{advances} and \cite{imrn}) that
the free boundary $\partial\{ u>0\} \cup\partial\{ u<0\}$ is in a neighborhood of each branch point,
i.e. a point in the set $\Omega \cap \partial\{ u>0\}
\cap \partial\{ u<0\} \cap \{ \nabla u=0\}$, the
union of (at most) two $C^1$-graphs. Note that the definition of
``branch point'' does not necessarily imply a bifurcation
as that in Figure \ref{branchfig}.
\begin{figure}
\begin{center}
\input{suw_par2.pstex_t}
\end{center}
\caption{Example of a Stationary Branch Point}\label{branchfig}
\end{figure}
\newline
We formulate the main result in this paper.
\begin{theorem}\label{main}
Suppose that
$$0<\lambda_{\rm min}\le \inf_{Q_1(0)} \min(\lambda_+,\lambda_-),
\qquad \sup_{Q_1(0)}\max(|\nabla \lambda_+|,|\nabla
\lambda_-|, |\partial_t \lambda_+|,|\partial_t \lambda_-|)<+\infty$$ and that $u$ is a weak solution of
$$\Delta u - \partial_t u = \lambda_+ \chi_{\{u>0\}} \> - \>
\lambda_- \chi_{\{u<0\}}\hbox{ in } Q_1(0)\; ;$$
here $Q_1(0)$ is the parabolic cylinder $(-1,1)\times B_1(0)$.
\newline
Then there are constants $\sigma>0$ and $r_0>0$ such that
\begin{equation}\label{cond}
u(0)=0\> ,\> |\nabla u(0)|\le \sigma\> ,\> \hbox{\rm pardist}(0,\{ u>0\})\le \sigma
\> \hbox{ and }\> \hbox{\rm pardist}(0,\{ u<0\})\le \sigma\end{equation}
imply
$\partial \{ u>0\}\cap Q_{r_0}(0)$ and $\partial
\{ u<0\}\cap Q_{r_0}(0)$ being graphs of
Lipschitz functions (in some space direction) that are continuously differentiable with respect to the space variables.
The constants $\sigma, r_0$, the Lipschitz norms and the modulus of continuity of the spatial normal vectors to these surfaces
depend only on $\inf_{Q_1(0)} \min(\lambda_+,\lambda_-)$, the Lipschitz norms of $\lambda_\pm,$
the supremum norm of $u$ and the space dimension $n$.
\newline
Moreover the regularity above is optimal in the sense that the
graphs are in general not better than Lipschitz.
\end{theorem}
\begin{corollary}
Suppose that
$$0<\lambda_{\rm min}\le \inf_{Q_1(0)} \min(\lambda_+,\lambda_-),
\qquad \sup_{Q_1(0)}\max(|\nabla \lambda_+|,|\nabla
\lambda_-|, |\partial_t \lambda_+|,|\partial_t \lambda_-|)<+\infty$$ and that $u$ is a weak solution of
$$\Delta u -\partial_t u = \lambda_+ \chi_{\{u>0\}} \> - \>
\lambda_- \chi_{\{u<0\}}\hbox{ in } Q_1(0)\; .$$
Then there is a constant $r_0>0$ such that
if the origin is a branch point, then
$\partial \{ u>0\}\cap Q_{r_0}(0)$ and $\partial
\{ u<0\}\cap Q_{r_0}(0)$ are graphs of
Lipschitz functions (in some space direction)
that are continuously differentiable with respect to the space variables.
The constant $r_0$, the Lipschitz norms and the modulus of continuity of the spatial normal vectors to these surfaces
depend only on $\inf_{Q_1(0)} \min(\lambda_+,\lambda_-)$, the Lipschitz norms of $\lambda_\pm,$
the supremum norm of $u$ and the space dimension $n$.
\end{corollary}
As to the proof we extend the method of \cite{imrn} to the parabolic case.
There is however a difficulty as the time derivative $\partial_t u$
is in general not continuous, so that it is not possible to apply
directly the comparison principle. We deal with that problem
by a {\em two-stage} proof of directional monotonicity.
\section{Notation}
Throughout this article ${\bf R}^n$ will be equipped with the Euclidean
inner product $x\cdot y$ and the induced norm $\vert x \vert\> ,\>
B_r(x^0)$ will denote the open $n$-dimensional ball of center
$x^0\> ,$ radius $r$ and volume $r^n\> \omega_n\> ,
\> B'_r(0)$ the open $n-1$-dimensional ball of center $0$ and radius
$r\> ,$
and $e_i$ the $i$-th unit vector in ${\bf R}^n\> .$
We define $Q_r(t^0,x^0) := (t^0-r^2 , t^0+ r^2)\times B_r(x^0)$ to be
the cylinder
of radius $r$ and height $2r^2$,
$Q^-_r(t^0,x^0) := (t^0-r^2 , t^0)\times B_r(x^0)$ its ``negative part''
and $Q^+_r(t^0,x^0) := (t^0, t^0+r^2)\times B_r(x^0)$ its ``positive part''.
When omitted, $x^0$ (or $(t^0,x^0)$, respectively) is assumed to be the
origin.
Moreover let $\partial_{\rm par} Q_r(t^0,x^0) :=
(t^0-r^2 , t^0+ r^2)\times \partial B_r(x^0)\cup \{ t^0-r^2 \}\times B_r(x^0)$
denote the parabolic boundary of $Q_r(t^0,x^0)$.
Let us also introduce the parabolic distance
$\hbox{\rm pardist}((t,x),A) := \inf_{(s,y)\in A}
\sqrt{\vert x-y\vert^2 + \vert t-s\vert}\> .$
Given a set $A\subset {\bf R}^{n+1}\> ,$ we denote its interior by $A^\circ$
and its characteristic function by $\chi_A\> .$
By $\nabla u$ we mean the gradient with respect to the space
variables.
In the text we use the $n$-dimensional Lebesgue-measure
${\mathcal L}^n$ and
the $m$-dimensional Hausdorff measure
${\mathcal H}^m$.
Finally, ${\bf C}^{\beta,\mu}:={\bf H}^{\mu,\beta}$
denotes the parabolic
H\"older-space as defined in
\cite{lady}.
\section{A supremum-mean-value estimate}
In this section we show that at branch points the time derivative
$\partial_t u$, in general a discontinuous function, satisfies a $\sup$-mean-value estimate.
\begin{lemma}\label{mean}
Let $Q^-_{2r}(t^0,x^0)\subset (0,T)\times \Omega$ and let $\lambda_+,\lambda_-$ be non-negative and Lipschitz continuous with respect to the time variable. Then each solution $u$ of (\ref{obst})
satisfies
$$\sup_{Q_1} \vert \partial_t u_{r_k}\vert=\sup_{Q^-_r(t^0,x^0)} |\partial_t u| \le C \left(r^2+\left(r^{-n-2}\int_{Q^-_{2r}(t^0,x^0)}
|\partial_t u|^2\right)^{1\over 2}\right)\; .$$
\end{lemma}
\proof
Using the scaling invariance of the equation
with respect to the scaling
$$u_r(t,x)=r^{-2} u(t^0+r^2t,x^0+rx)$$
we may assume that $r=1/2, t^0=0$ and $x^0=0$.\\
Let $H(t,x,z) = \lambda_+(t,x)\chi_{\{ z>0\}} - \lambda_-(t,x) \chi_{\{ z<0\}}$.
For $$v(t,x) := \partial_t^\tau u(t,x):= {u(t+\tau,x)-u(t)\over \tau}$$
and $\eta\in L^2((-1,1);W^{1,2}(B_1))$ such that $\eta=0$ on
$(-1,0)\times \partial B_1$,
we calculate
\begin{equation}\label{mean1}
\begin{array}{l}
\int_{-1}^{s} \int_{B_1}(\eta \partial_t v + \nabla v \cdot \nabla \eta)\\
= - \int_{-1}^{s} \int_{B_1}\eta \partial_t^\tau H(t,x,u(t,x))\; , \; s\in (-1,0)
\; .
\end{array}
\end{equation}
Here
$$\partial_t^\tau H(t,x,u(t,x))
= \lambda_+(t^0+r^2t,x^0+rx) \partial_t^\tau \chi_{\{ u>0\}}
- \lambda_-(t^0+r^2t,x^0+rx) \partial_t^\tau \chi_{\{ u<0\}}$$ $$
+ \chi_{\{ u(t^0+r^2(t+\tau),x^0+rx)>0\}} \partial_t^\tau\lambda_+
- \chi_{\{ u(t^0+r^2(t+\tau),x^0+rx)<0\}} \partial_t^\tau\lambda_-\; .$$
Testing with
$\eta(t,x) := \zeta^2(x)\phi^2(t) \max(v(t,x)-k,0)$ where $k\ge 0$,
$\zeta\in C^{0,1}_0(B_1)$ and $\phi \in C^{0,1}(-1,1)$ such that
$\phi(t)\in [0,1]$ and
$$\phi(t) := \left\{\begin{array}{l}
1, t\ge -1/2\\
0, t\le -1\> ,\end{array}\right.$$
and observing that
$$\max(v(t,x)-k,0) \partial_t^\tau H(t,x,u(t,x))\ge -C_1 r^2 \max(v(t,x)-k,0)$$
we obtain
\begin{equation}\label{supmean}
\sup_{-1 < s < 0}
\int_{B_1} \phi^2(s) \zeta^2 \max(v(s,\cdot)-k,0)^2
+ \int_{-1}^{s} \int_{B_1} \phi^2 \zeta^2 |\nabla \max(v-k,0)|^2
\end{equation}
$$\le C_2 \int_{-1}^{s} \int_{B_1} [
\max(v-k,0)^2 (\phi^2 |\nabla \zeta|^2 | +
\phi |\partial_t \phi| \zeta^2)+ r^2 \phi^2\zeta^2\max(v-k,0)]\; .$$
From the proof of \cite[Theorem 4.7]{lieberman} we infer that
\begin{equation}\label{time_1}\sup_{Q_{1/2}^-} v
\le C_3 \left(r^2+\left(\int_{Q_1^-} v^2\right)^{1\over 2}\right)\; .\end{equation}
Testing with
$\eta(t,x) := \zeta^2(x)\phi^2(t) \max(-v(t,x)-k,0)$ where $k\ge 0$,
we obtain in a similar way that
\begin{equation}\label{time_2}
\sup_{Q_{1/2}^-} (-v)
\le C_3 \left(r^2+\int_{Q_1^-} v^2\right)^{1\over 2}\; .
\end{equation}
Letting $\tau\to 0$ and scaling back we obtain the statement.
\qed
\section{Non-degeneracy and regularity of the solution}
\begin{lemma}[Non-Degeneracy]
\label{ndeg}
For every $Q_{2r}(t^0,x^0)\subset (0,T)\times \Omega$ the following holds:
\[ \textrm{1) If } (t^0,x^0)\in \partial \{ u>0\}, \textrm{ then }\sup_{Q_r^-(t^0,x^0)} u\; \ge \;
{1\over {8n}} \inf_{Q_{r}(t^0,x^0)}\lambda_+ \> r^2\; .\]
\[ \textrm{2) If } (t^0,x^0)\in \partial \{ u<0\}, \textrm{ then } \inf_{Q_r^-(t^0,x^0)} u\; \le \;
-{1\over {8n}}\inf_{Q_{r}(t^0,x^0)}\lambda_- \> r^2\; .
\]
\end{lemma}
\proof
We choose a sequence
$\{u>0\}\ni (t^m,x^m) \to (t^0,x^0)$ as $m\to \infty\> .$
Supposing that $\sup_{Q_r^-(t^m,x^m)} u \> \le \>
{1\over {8n}} \inf_{Q_{r}(t^0,x^0)}\lambda_+ \> r^2\> ,$ the comparison principle
yields that $u(t,x)\le v(t,x) := ({t^m-t\over 2}+{1\over {8n}}{\vert x-x_m\vert}^2)\>\inf_{Q_{r}(t^0,x^0)}\lambda_+
$ in $Q_r^-(t^m,x^m)\> ,$ a contradiction to the fact that $u(t^m,x^m)>0\> .$\\
The estimate for $\inf_{Q_r^-(t^0,x^0)} u$ is obtained the same
way, replacing $u$ by $-u$ and $\lambda_+$ by $\lambda_-\> .$ \qed
\begin{lemma}\label{bounded}
Let $\lambda_+,\lambda_-\in C^{0,1}_{\rm loc}((0,T)\times \Omega)$. Then each solution $u$ of (\ref{obst})
satisfies the following:\\
1) $\partial_t u\in L^\infty_{\rm loc}((0,T)\times \Omega)$.\\
2) $\partial_t\nabla u\in L^2_{\rm loc}((0,T)\times \Omega)$.
\end{lemma}
\proof
1) follows from Lemma \ref{mean}.\\
2) follows from (\ref{supmean}) with $k=0$ and from the analogous estimate for $\max(-v,0)$.
\qed
\begin{corollary}\label{ndeg2}
\label{ndeg2}
For every $Q_{2r}(t^0,x^0)\subset (0,T)\times \Omega$,
there exists a constant $c_0>0$ depending only on $n$ and
$\Vert\partial_t u\Vert_{L^\infty(Q_r(t^0,x^0))}$
such that
\[ u \ge 0 \textrm{ in } Q_r^-(t^0,x^0) \textrm{ implies } u\ge 0 \textrm{ in } Q_{c_0r}(t^0,x^0)\; ,\textrm{ and}\]
\[ u \le 0 \textrm{ in } Q_r^-(t^0,x^0) \textrm{ implies } u\le 0 \textrm{ in } Q_{c_0r}(t^0,x^0)\; .\]
\end{corollary}
\proof
Suppose towards a contradiction that
$u(t^1,x^1)<0$ for some $(t^1,x^1)\in Q_{c_0r}^+(t^0,x^0)$. Then there is a point
$(t^2,x^2)\in \partial\{ u<0\}\cap \overline{Q_{c_0r}^+(t^0,x^0)}$. Applying Lemma \ref{ndeg}
at $(t^2,x^2)$ with respect to the cylinder $Q_{(1-c_0)r}(t^2,x^2)$ yields a contradiction to Lemma \ref{bounded} 1) provided
that $c_0$ has been chosen small enough.\\
The second estimate is proved in the same fashion.\qed
\begin{proposition}\label{regular}
Let $\lambda_+,\lambda_-\in C^{0,1}_{\rm loc}((0,T)\times \Omega)$. Then each solution $u$ of (\ref{obst})
satisfies $\nabla u\in {\bf C}^{1/2,1}_{\rm loc}((0,T)\times \Omega)$, that is, the gradient is Lipschitz continuous with respect to the space variables and H\"older continuous with exponent $1/2$ with respect to the time variable.
\end{proposition}
\proof
Let us first show that for any $e\in \partial B_1$,
$(\Delta-\partial_t)(\max(\partial_e u,0))\ge -C$ and
$(\Delta-\partial_t)(\max(-\partial_e u,0))\ge -C$ in $\Omega$.
We give a formal proof that can be made rigorous translating
everything into a weak formulation. In $\{ \partial_e u > 0\}$,
$$(\Delta-\partial_t)(\partial_e u)
$$ $$= {\partial_e u \over {|\nabla u|}} (\lambda_+ {\mathcal H}^{n-1}\lfloor (\{ \nabla u\ne 0\} \cap \partial\{ u>0\})
+ \lambda_- {\mathcal H}^{n-1}\lfloor (\{ \nabla u\ne 0\} \cap \partial\{ u<0\}))$$
$$+ \partial_e \lambda_+ \chi_{\{ u>0\}} - \partial_e \lambda_- \chi_{\{ u<0\}}\;
\ge \; -C\; .$$
As $\partial_e u$ is continuous,
we obtain $(\Delta-\partial_t)(\max(\partial_e u,0))\ge -C$.\\
Considering $-e$ instead of $e$ we obtain also
$(\Delta-\partial_t)(\max(-\partial_e u,0))\ge -C$.
But then the ``almost monotonicity formula'' Theorem I of \cite{edquist} applies
and we proceed as follows:
at each point $(t^0,x^0)\in \{ u\ne 0\}\cap \{ \nabla u=0\}$,
we obtain from the almost monotonicity formula that
$\nabla \partial_e u$ is
bounded at $(t^0,x^0)$ by a locally uniform constant.
\\
At each point $(t^0,x^0)\in \{ u\ne 0\}\cap \{ \nabla u\ne 0\}$,
we obtain in a similar way that
{\em for every $e\bot \nabla u(t^0,x^0)$},
$|\nabla \partial_e u(t^0,x^0)|$ is
bounded by a locally uniform constant.
Let $e_1 = {\nabla u(t^0,x^0)\over {|\nabla u(t^0,x^0)|}}$.
Then $-\partial_{11} u(t^0,x^0) = -\lambda_+ \chi_{\{ u(t^0,x^0)>0\}}
+ \lambda_- \chi_{\{ u(t^0,x^0)<0\}} - \partial_t u(t^0,x^0) + \sum_{j=2}^n \partial_{jj} u(t^0,x^0)$
is by Lemma \ref{bounded} bounded by a locally uniform constant.
\qed
\begin{corollary}\label{density}
${\mathcal L}^{n+1}(\partial \{ u > 0\}\cup \partial \{ u< 0\})=0$
\end{corollary}
\proof First, we obtain from Lemma \ref{ndeg}, Lemma \ref{bounded}
and Proposition \ref{regular} that there exists a locally uniform
constant $c>0$ such that for $Q_{2r}(s,y)\subset (0,T)\times \Omega$,
$$\frac{{\mathcal L}^{n+1}(Q_r(s,y)\cap \{ u>0\})}{{\mathcal L}^{n+1}(Q_r)} \ge c >0\textrm{ if }(s,y)\in \partial \{ u> 0\}$$
$$\textrm{and }\frac{{\mathcal L}^{n+1}(Q_r(s,y)\cap \{ u<0\})}{{\mathcal L}^{n+1}(Q_r)} \ge c >0\textrm{ if }(s,y)\in \partial \{ u< 0\}\; .$$
Since $\chi_{\{ u>0\}} * \chi_{Q_r}/{\mathcal L}^{n+1}(Q_r)\to \chi_{\{ u>0\}}$ in $L^1_{\rm loc}((0,T)\times \Omega)$ as $r\to 0$
and the analogous fact holds for $\chi_{\{ u<0\}}$, we obtain
that $\chi_{\{ u>0\}}\ge c >0$ ${\mathcal L}^{n+1}$-a.e. on $\partial \{ u> 0\}$
and $\chi_{\{ u<0\}}\ge c>0$ ${\mathcal L}^{n+1}$-a.e. on $\partial \{ u< 0\}$. Thus
${\mathcal L}^{n+1}(\partial \{ u > 0\}\cup \partial \{ u < 0\})=0$.
\qed
\section{Vanishing time derivative}
As a corollary of Lemma \ref{mean} we obtain now that
at points at which the blow-up limit depends only
on the space variables, the time derivative
$\partial_t u$ -- in general a discontinuous function --
attains the limit $0$.
\begin{corollary}\label{zerolim}
Let $Q_{2r}(t^0,x^0)\subset (0,T)\times \Omega$
and suppose that for a sequence of solutions $u_k$ in $(0,T)\times \Omega$
$$u_{r_k}(t,x)={r_k}^{-2} u_k(t^k+r_k^2t,x^k+r_kx)\to u_0(x)
\textrm{ in } L^1_{\rm loc}({\bf R}^{n+1})\textrm{ as }r_k\to 0\; .$$
Then
$$\sup_{Q_{r_k}(t^k,x^k)} |\partial_t u_k| \to 0$$
as $r_k\to 0$.
\end{corollary}
\proof
The statement follows from Lemma \ref{mean}
and the fact that $\partial_t u_{r_k}$ converges to $0$ in $L^2_{\rm loc}({\bf R}^{n+1})$
as $r_k\to 0$.
The $L^2$-convergence in turn may be shown as follows:
as $\partial_t u_k$ is by Lemma \ref{mean} bounded in $L^\infty(Q_{r}(t^0,x^0))$, it is sufficient to prove
a.e. convergence.
For $(s,y)\in \{ u_0 =0\}^0$ we obtain from Lemma \ref{ndeg} that
$u_{r_k}=0$ in $Q_\delta(s,y)$ for some $\delta>0$ and large $k$.
For $(s,y)\in \{ u_0 > 0\}\cup \{ u_0 < 0\}$, $u_{r_k}$ converges in $C^1(Q_\delta(s,y))$
for some $\delta>0$ as $k\to\infty$.
Moreover we know from Corollary \ref{density} that ${\mathcal L}^{n+1}(\partial \{ u_0 > 0\}\cup \partial \{ u_0< 0\})=0$.
It follows that $\partial_t u_{r_k}$ converges ${\mathcal L}^{n+1}$-a.e. to $\partial_t u_0$.
\qed
\section{Directional monotonicity}\label{dirmon}
In a first stage, we show that
if the solution is close to the one-dimensional solution
\begin{equation}\label{one-dim-sol}
h(x) :={\lambda_+(0)\over 2} \max(x_1,0)^2\> -{\lambda_-(0)\over 2}
\min(x_1,0)^2 \ .
\end{equation}
then it is increasing in a cone of {\em spatial} directions.
Later on we will extend the result to a cone of {\em tempo-spatial} directions.
\begin{proposition}\label{directional-monoton}
Let $0<\lambda_{\rm min}\le \inf_{Q_1(0)}
\min(\lambda_+,\lambda_-)$, $h$ as in (\ref{one-dim-sol}), and let
$\varepsilon\in (0,1)$. Then each solution $u$ of (\ref{obst}) in
$Q_1(0)$ such that
$$\hbox{\rm dist}_{L^\infty((-1,1);W^{1,\infty}(B_1))}(u,h)
\le \delta := {\lambda_{\rm min}\varepsilon\over {48 n}}$$ and
$$\sup_{Q_1(0)}\max(|\nabla \lambda_+|,|\nabla \lambda_-|)\le
\delta$$ satisfies $\varepsilon^{-1}
\partial_e u - |u| \ge 0$ in $Q_{1/2}(0)$ for every $e\in \partial
B_1(0)$ such that $e_1\ge \varepsilon$;
here $e_1$ denotes the first component of the vector $e$.
\end{proposition}
\proof
First note that $\varepsilon^{-1} \partial_e h - |h|\ge 0$ in $Q_2(0)$.
It follows that
\begin{equation}\label{approx}
\varepsilon^{-1} \partial_e u - |u|\ge -3\delta \varepsilon^{-1} \textrm{ in } Q_1(0)
\end{equation}
provided that $\hbox{\rm dist}_{L^\infty((-1,1);W^{1,\infty}(B_1))}(u,h) \le\delta$. Suppose now
towards a contradiction that the statement is not true. Then there
exist $\lambda_+,\lambda_-\in (\lambda_{\rm min},+\infty), (t^*,x^*) \in
Q_{1/2}(0), e^*, $ and a solution $u$ of (\ref{obst}) in $Q_1(0)$
such that $\hbox{\rm dist}_{L^\infty((-1,1);W^{1,\infty}(B_1))}(u,h) \le\delta$,
$$\sup_{Q_1(0)}\max(|\nabla \lambda_+|,|\nabla \lambda_-|)\le
\delta,$$ $e^*_1 \ge \varepsilon$ and
$\varepsilon^{-1}\partial_{e^*} u(t^*,x^*) - |u(t^*,x^*)|<0.$ For the
positive constant $c$ to be defined later the functions $v :=
\varepsilon^{-1}\partial_{e^*} u - |u|$ and $w:=
\varepsilon^{-1}\partial_{e^*} u - |u|+c |x-x^*|^2 - c (t-t^*)$ satisfy then
the
following: in the set $D := Q_1(0)\cap \{ v<0\}\cap\{ t<t^*\}$,
\[ \Delta w \> -\> \partial_t w\le 2nc + c -{\lambda_+} \chi_{\{ u>0\}}-{\lambda_-} \chi_{\{ u<0\}}\]\[
+\varepsilon^{-1} (\lambda_+ +\lambda_-) \nu_x\cdot e^* {\mathcal
H}^{n-1}\lfloor (\{ u=0\}\cap \{ \nabla u\ne 0\}) \]\[+\varepsilon^{-1}
( \chi_{\{ u>0\}}\partial_{e^*} \lambda_+ - \chi_{\{
u<0\}}\partial_{e^*} \lambda_-)
\]
where $\nu_x = {\nabla u \over {|\nabla u|}}$. As
\[\nu_x\cdot e^*< 0 \> \hbox{ on }\> \{ u=0\}\cap \{ v<0\} = \{ u=0\}\cap \{
\varepsilon^{-1}\partial_{e^*} u <0\}\; ,\] we obtain by the definition of $\delta$ that
$w$ is supercaloric in $D$ provided that $c$ has been chosen
accordingly, say $c:= \lambda_{\rm min}/(4n)$. It follows that the
negative infimum of $w$ is attained on
\[ \partial_{\rm par} D \subset (\partial_{\rm par} Q_1(0)\cap \{ t\le t^*\}) \cup
\left( Q_1(0)\cap \partial \{ v<0\}\right)
\; .\]
Consequently it is attained on $\{ t\le t^*\}\cap \partial_{\rm par} Q_1(0)$,
say at the point $(\bar t,\bar x) \in \{ t \le t^*\}\cap \partial_{\rm par} Q_1(0)$.
Since $\hbox{\rm pardist}((\bar t,\bar x),(t^*,x^*))\ge 1/2$, we obtain that
\[ \varepsilon^{-1}\partial_{e^*} u(\bar t,\bar x) - |u(\bar t,\bar x)|=
v(\bar t,\bar x) = w(\bar t,\bar x) - c |x^* - \bar x|^2 + c (\bar t-t^*)
\]\[ < -c/4=-\lambda_{\rm min}/(16n)\; .\] But this contradicts
(\ref{approx}) in view of $\delta = {\lambda_{\rm
min}\varepsilon\over {48 n}}$.\qed
\section{The set of non-vanishing gradient}\label{nonvanish}
In the sequel we are going to need higher regularity of
the level set $\{ u=0\}\cap \{\nabla u\ne 0\}$.
Higher regularity can be obtained in a standard way using the
von Mises transform:
\begin{lemma}\label{cinf}
The set $\{ u=0\}\cap \{\nabla u\ne 0\}$ is locally in $(0,T)\times \Omega$ a $C^1$-surface
and $\partial_t u$ is continuous on that surface.
\end{lemma}
\proof
Let $(t^0,x^0)\in \{ u=0\}\cap \{\nabla u\ne 0\}$.
We may assume that $\nabla u(t^0,x^0)=\partial_1 u(t^0,x^0)$ and
that in $Q_\delta(t^0,x^0)$,
$u$ is strictly increasing in the $x_1$-direction and
$\{ u=0\}$ is the graph
of a function, say $x_1=g(t,x')$ for $(t,x)\in Q_\delta(t^0,x^0)$,
where $g\in C^0((t^0-\delta^2,t^0+\delta^2);C^1(B'_\delta(x^0)))$.
It is sufficient to prove that
$g\in C^1(Q_{\delta/2}(t^0,x^0))$.
To do so, we use von Mises variables, i.e.
$$y=u(t,x_1,x') \textrm{ and } x_1=v(t,y,x')\; .$$
A calculation assures that
$$\left(\frac{-1-|\nabla' v|^2}{(\partial_y v)^3} \partial_{yy} v\right)
-\frac{\Delta' v}{\partial_y v} + 2 \frac{\nabla' v \cdot \nabla' \partial_y v}{(\partial_y v)^2}
+ \frac{\partial_t v}{\partial_y v} $$ $$= \left\{\begin{array}{l}
\lambda_+(t,v(t,y,x'),x') , \> y>0\\
-\lambda_-(t,v(t,y,x'),x') , \> y<0\end{array}\right. \; .$$
Thus
$$\partial_t v - a_{ij}(\nabla v) \partial_{ij} v =
f(t,y,x')\partial_y v:=
\left\{\begin{array}{l}
-\lambda_+(t,v(t,y,x'),x') \partial_y v, \> y>0\\
\lambda_-(t,v(t,y,x'),x') \partial_y v, \> y<0\> .\end{array}\right.$$
Provided that $\delta$ has been chosen small enough,
$|\nabla' v|\le 1/2$, $0< \partial_y v \le 1/2$ and
the above equation is uniformly parabolic.
Moreover,
$$\partial_t \partial^h_t v - a_{ij}(\nabla v) \partial_{ij} \partial^h_t v
- \frac{\partial a_{ij}(z_h)}
{ \partial p_k}\partial_{ij} v(t+h,y,x') \partial_k \partial^h_t v$$
$$=
f(t,y,x')\partial_y \partial^h_t v + \partial_y v(t+h,y,x')\partial^h_t f(t,y,x')$$
where $z_h=\theta(t,y,x') \nabla v(t+h,y,x') + (1-\theta(t,y,x')) \nabla v(t,y,x')$
and $\theta(t,y,x')\in [0,1]$.
Calculating
$$\partial_1 v = 1/\partial_1 u, \partial_t v = -\partial_t u/\partial_1 u, \partial_i v = -\partial_i u/\partial_1 u
\textrm{ for } 2\le i \le n,$$
$$\partial_{ij} v + \partial_{i1} v \partial_j u + \partial_{1j}v\partial_i u +\partial_1 v \partial_{ij} u+\partial_{11}v\partial_i u\partial_j u=0
\textrm{ for } 2\le i,j \le n
$$
$$\partial_{i1} v \partial_1 u + \partial_{11} v \partial_1 u \partial_i u + \partial_1 v \partial_{i1} u =0,
\partial_{11} v = -\partial_{11} u/(\partial_1 u)^3$$
shows that all spatial second derivatives of $v$
and $\partial_t v$ are bounded.
Thus $f(t,y,x'), \partial^h_t f(t,y,x')$ and $\frac{\partial a_{ij}(z_h)}
{ \partial p_k}\partial_{ij} v(t+h,y,x')$
are bounded uniformly in $h$, and we obtain from \cite{krylov}
that $\partial^h_t v$ is uniformly H\"older continuous with respect to $h$
and that $\partial_t u$ is H\"older continuous in $Q_{\delta/2}(t^0,x^0)$.
\qed
\section{Global solutions}
In this section we extend our characterization of elliptic global solutions
\cite[Theorem 4.3]{global} to the parabolic case.
We are going to need the following version of the Caffarelli-Kenig
monotonicity formula of \cite{caffarellikenig}:
\begin{theorem}\label{mono}
Let
$$\Phi(r,w) := {1\over {r^4}} I(r,\max(w,0))I(r,\max(-w,0))$$
where
$$I(r,v) := \int_{-r^2}^0 \int_{{\bf R}^n} |\nabla v|^2 G(t,x)$$
and $G$ is the backwards heat kernel
$$G(t,x) = (4\pi (-t))^{n/2} \exp({|x|^2\over {4t}})\; .$$
If $\max(w,0)$ and $\max(-w,0)$ are continuous subcaloric functions, then
$r\mapsto \Phi(r,w)$ is non-decreasing, and
$\Phi(\sigma,w)=\Phi(\rho,w)$ for some $0<\rho<\sigma$ implies that
either
\\
(A) $\nabla \max(w,0)=0$ in $-\sigma^2<t<0$ or $\nabla \max(-w,0)=0$ in $-\sigma^2<t<0$.\\
or
\\
(B) $\max(w,0)(\partial_t - \Delta)\max(w,0)=0$
and $\max(-w,0)(\partial_t - \Delta)\max(-w,0)=0$
in $-\sigma^2<t<0$
in the sense of measures.
\end{theorem}
\proof
For $v := \max(w,0)$ (or $v := \max(-w,0)$, respectively) we calculate
$$I(r,v)=
- {1\over 2} \int_{-r^2}^0 \int_{{\bf R}^n} G(t,x)(\partial_t - \Delta)v^2
+ \int_{-r^2}^0 \int_{{\bf R}^n} G(t,x)v(\partial_t - \Delta)v\; ,$$
$$I'(r,v)\ge 2r\int_{{\bf R}^n} |\nabla v|^2 G(-r^2,x)\; .$$
In what follows we assume that $I(r,v)\ne 0$.
It follows that
$$\frac{I'(r,v)}{I(r,v)} \ge 4r\frac{\int_{{\bf R}^n} |\nabla v(-r^2,x)|^2 G(-r^2,x)}{\int_{{\bf R}^n} v^2(-r^2,x)G(-r^2,x)}\; .$$
In the case $I'(r,v)\ne 0$ the inequality is strict unless $\int_{-r^2}^0 \int_{{\bf R}^n}v(\partial_t - \Delta)v=0$.
Consequently $\Phi(r,w)=0$, or else
$$\frac{\Phi'(r,w)}{\Phi(r,w)} \ge {4\over r}
\Bigg[-1\;+ \; r^2 \frac{\int_{{\bf R}^n} |\nabla \max(w,0)|^2 G(-r^2,x)}{\int_{{\bf R}^n} \max(w,0)^2G(-r^2,x)}$$
$$+ r^2 \frac{\int_{{\bf R}^n} |\nabla \max(-w,0)|^2 G(-r^2,x)}{\int_{{\bf R}^n} \max(-w,0)^2G(-r^2,x)}\Bigg]\; ,$$
where the inequality is strict unless both $\int_{-r^2}^0 \int_{{\bf R}^n}\max(w,0)(\partial_t - \Delta)\max(w,0)=0$
and $\int_{-r^2}^0 \int_{{\bf R}^n}\max(-w,0)(\partial_t - \Delta)\max(-w,0)=0$.
Moreover, by \cite[Corollary 2.4.6]{caffarellikenig}, the right-hand side is non-negative.
\qed
\begin{lemma}\label{forward}
Let $v^1,v^2$ be solutions of (\ref{obst}) in ${\bf R}^{n+1}$ with
such that
$v^1=v^2$ in $\{ t<0\}$ and $v^1,v^2$ have polynomial growth with respect to the space variables.
Then $v^1=v^2$ in ${\bf R}^{n+1}$.
\end{lemma}
\proof
Multiplying the difference of the two equations by $(v^1-v^2)W$ where $W(t,x)=G(t-T,x)$
and
integrating, we obtain for each $0<T<+\infty$, $0<S<T$ and $H$ defined in Lemma \ref{mean} that $0\; =$
$$ \int_0^S \int_{{\bf R}^n} W [|\nabla (v^1-v^2)|^2 \> + \> (H(v^1)-H(v^2))(v^1-v^2)]
\; - \; {1\over 2}\int_0^S \int_{{\bf R}^n} (v^1-v^2)^2 \partial_t W$$ $$
+ \; {1\over 2} \int_{{\bf R}^n} W(S) (v^1(S)-v^2(S))^2
\; + \; \int_0^S \int_{{\bf R}^n} (v^1-v^2) \nabla W \cdot \nabla (v^1-v^2)
$$
$$\ge {1\over 2} \int_{{\bf R}^n} W(S) (v^1(S)-v^2(S))^2\; + \; {1\over 2}\int_0^S \int_{{\bf R}^n} (v^1-v^2)^2 [-\partial_t W-\Delta W]$$
$$= \; {1\over 2} \int_{{\bf R}^n} W(S) (v^1(S)-v^2(S))^2\; .$$\qed
\begin{lemma}\label{backself}
Assume that $w$ is a backward self-similar solution with constant coefficients $\lambda_+,\lambda_-$,
i.e.
$$ w(\theta^2 t,\theta x)= \theta^2 w(t,x) \textrm{ for all }
\theta\ge 0, t<0 \textrm{ and } x\in {\bf R}^n\; .$$
Then $\nabla w=0$ on $\{ w=0\}$.
\end{lemma}
\proof
First, the self-similarity implies that
\begin{equation}\label{homog}
\partial_e w(\lambda^2t,\lambda x)=\lambda \partial_e w(t,x)
\textrm{ for all }e\in \partial B_1, \lambda \ge 0, t<0 \textrm{ and }x\in {\bf R}^n.
\end{equation}
Consequently the function $r\mapsto \Phi(r, \partial_e w)$ of
the monotonicity formula Theorem \ref{mono}
is constant in $(0,+\infty)$, implying by Theorem \ref{mono}
that
either
\\
(A) $\nabla \max(\partial_e w,0)=0$ in $\{ t<0\}$ or $\nabla \max(-\partial_e w,0)=0$ in $\{ t<0\}$.
\\
or\\
(B) $\max(\partial_e w,0)(\partial_t - \Delta)\max(\partial_e w,0)=0$ in $\{ t<0\}$
and $\max(-\partial_e w,0)(\partial_t - \Delta)\max(-\partial_e w,0)=0$ in $\{ t<0\}$
in the sense of measures.\\
Suppose now towards a contradiction that
there is a point $(t^1,x^1)\in \{ t<0\} \cap \{ w=0\} \cap \{ \nabla w \ne 0\}$
and denote $\nu={\nabla w\over {|\nabla w|}}$,
$\nu^0= {\nabla w(t^1,x^1)\over {|\nabla w(t^1,x^1)|}}$ and let $Q_\kappa(t^1,x^1)$ such
that $\partial_{\nu^0} w >0$ in $Q_\kappa(t^1,x^1)$ and
$\{ w=0\}\cap Q_\kappa(t^1,x^1)$ is a $C^1$-surface.
In the case $\nu^0\cdot e\ne 0$,
$$ |(\partial_t - \Delta)\partial_e w|(Q_\kappa(t^1,x^1))
=
|\lambda_++\lambda_-|
\int_{t^1-\kappa^2}^{t^1+\kappa^2}
\int_{B_\kappa(x^1)\cap \{ w(t)=0\} } |e\cdot \nu| \> d {\mathcal H}^{n-1}\> dt
\ne 0\; .$$
Thus (A) holds.
From (\ref{homog}) we infer that $\partial_e w\ge 0$ in $\{ t<0\}$ if $e\cdot \nu^0>0$ and
$\partial_e w\le 0$ in $\{ t<0\}$ if $e\cdot \nu^0<0$.
Hence $\partial_e w= 0$ in $\{ t<0\}$ for all $e\bot \nu^0$.
As in \cite[p. 844]{cps} we may
write
$$ w(t,x)= -t f({x_n\over{\sqrt{-t}}})$$
and calculate the $2$-parameter family of solutions
of the ODE which $f(\xi)=w(-1,\xi)$ satisfies in $(0,+\infty)$,
$$f(\xi)=\lambda_++C_1 (\xi^2-2)$$ $$+\; C_2\left( -2\xi e^{\xi^2/4} + (\xi^2-2) \int_0^{\xi} e^{s^2/4}\> ds\right)\; \textrm{ in } \{ f>0\}$$
and
$$f(\xi)=-\lambda_-+C_3 (\xi^2-2)$$ $$+\; C_4\left( -2\xi e^{\xi^2/4} + (\xi^2-2) \int_0^{\xi} e^{s^2/4}\> ds\right)\; \textrm{ in } \{ f<0\}\; .$$
As $w$ has polynomial growth towards infinity
we conclude that
$0=C_2=C_4$ and that
$$f(\xi)=\lambda_++C_1 (\xi^2-2) \textrm{ in } \{ f>0\}$$
and
$$f(\xi)=-\lambda_-+C_3(\xi^2-2) \textrm{ in } \{ f<0\}\, .$$
If $f(a)=0$ and $f'(a)\ne 0$ for some $a\in {\bf R}$ then
$C_1=C_3=-\lambda_+/(a^2-2)=\lambda_-/(a^2-2)$, a contradiction.
Therefore $f(a)=0$ implies $f'(a)=0$ and $a=0$.
It follows that
$\nabla w=0$ on $\{ w=0\}$.\qed
\begin{theorem}\label{global-solution}
Let $w$ be a global solution
with constant coefficients $\lambda_+,\lambda_-$
such that $\partial_t w$ and $D^2 w$ are bounded,
and suppose that
the origin (in time-space) is a branch point of $w$.
Then after rotation
$$w(t,x)=w^*(t,x) := \lambda_+ {\max(x_n,0)^2/2} - \lambda_- {\max(-x_n,0)^2/2}
\textrm{ for } (t,x)\in {\bf R}^{n+1} \; .$$
\end{theorem}
\noindent {\sl Proof.}\\
{\bf Step 1:} Let us first assume that $w$ is a backward self-similar solution.
By Lemma \ref{backself} $\nabla w=0$ on $w=0$.
But then $z_1:= \max(w,0)$ and $z_2:= \max(-w,0)$ are in $\{ t \le 0\}$ non-negative backward self-similar solutions. Concerning those, it has been
shown in \cite[Lemma 6.3]{cps} and \cite[Theorem 8.1]{cps} that
either $z_j$ is a half-plane solution of the form
$z_j(t,x) = \lambda_\pm/2 \max(x\cdot e,0)^2$ for some
$e\in \partial B_1$, or
$z_j(t,x)=- a_0 t + \sum_{i=1}^n a_i x_i^2$ with non-negative
constants $a_i, 0\le i\le n$.
In the latter case the symmetry of $z_j$ implies that
$z_k=0$ in $\{ t<0\}$ for $k\ne j$, and by Corollary
\ref{ndeg2} the origin cannot be a branch point.
\\
It follows that after rotation
$$w(t,x)= w^*(t,x)\textrm{ for } t<0 \; .$$
\\
{\bf Step 2:}
In the case of a general solution $w$ as in the statement of our theorem,
we consider the blow-up up $w_0$ of $w$ at the origin and the blow-down
$w_\infty$. By the non-degeneracy Lemma \ref{ndeg} and
\cite[Theorem 4.1]{siam},
both $w_0$ and $w_\infty$ satisfy
the assumptions of Step 1.
Thus both $w_0$ and $w_\infty$ are after rotation
of the form
$\lambda_+ {\max(x_n,0)^2/2} - \lambda_- {\max(-x_n,0)^2/2}$ for $t<0$,
and the monotonicity formula
\cite{siam} implies
that $w$ is backward self-similar. But then it follows from Step 1
that
after rotation
$$w(t,x)=w^*(t,x)
\textrm{ for } t<0 \; .$$
Last, we apply Lemma \ref{forward} to obtain the same
for $t\ge 0$.
\qed
\section{Uniform closeness to $h$}
We are now ready to prove uniform closeness of the scaled solution
to the global solution $h$ of (\ref{one-dim-sol}), assuming that we are in the setting
of Theorem \ref{main}.
\begin{lemma}\label{uniform-closeness}
Let $u$ be a solution of (\ref{obst}) in $Q_1(0)$. Then, given
$\delta >0$, there are constants $r_\delta
>0,\sigma_\delta>0$ (depending only on $\inf_{Q_1(0)} \min(\lambda_+,\lambda_-)$,
the Lipschitz norms of $\lambda_\pm,$ the supremum norm of $u$ and
the space dimension $n$) such that the following holds:
\newline
If $r\in (0,r_\delta]\> , \> u(s,y)=0\> ,\> |\nabla u(s,y)|\leq \sigma_\delta r$,
$\hbox{\rm pardist}((s,y),\{ u>0\})\le \sigma_\delta r$ and $\hbox{\rm pardist}((s,y),\{ u<0\})\le \sigma_\delta r$
for some $(s,y) \in
Q_{1/2}(0)$ then in
$Q_{r}(s,y)$,
the solution
$u(s+\cdot,y+\cdot )$ is $\delta r^2$-close to a rotated version $\tilde h$ of the
one-dimensional solution $h$ defined in (\ref{one-dim-sol}), more precisely
$$
r^{-2}\sup\limits_{Q_{r}(0)}|u(s+\cdot,y+\cdot)-\tilde h|+
r^{-1}\sup\limits_{Q_{r}(0)}|\nabla u(s+\cdot,y+\cdot)-\nabla {\tilde h}|+
\sup\limits_{Q_{r}(0)}|\partial_t u(s+\cdot,y+\cdot)|$$
$$\leq \delta.
$$
\end{lemma}
\proof Suppose towards a contradiction that the statement of the lemma fails. Then for some $\delta
>0$ there exist $\sigma_j\to 0, r_j \to 0$, $(s^j,y^j) \to (s^0,y^0)\in \overline{Q_{1/2}}$, a sequence $u_j$
of solutions such that
$(s^j,y^j)\in Q_{1/2}(0)$,
$u_j(s^j,y^j)=0$,
$|\nabla u_j(s^j,y^j)|\leq \sigma_j r_j$,
$\>\hbox{\rm pardist}((s^j,y^j),\{ u_j>0\})\le \sigma_jr_j$, $\>\hbox{\rm pardist}((s^j,y^j),\{ u_j<0\})\le \sigma_jr_j$
and
$$
r^{-2}_{j}\sup\limits_{Q_1(0)}| u_j(s^j+r_j^2 \cdot,y^j+r_j\cdot)-\tilde h
(r_j\cdot)| \> + \> r^{-1}_{j }\sup\limits_{Q_1(0)}| \nabla
u_j(s^j+r_j^2 \cdot,y^j+r_j\cdot)-\nabla \tilde h (r_j\cdot)|
$$
$$ + \> \sup\limits_{Q_1(0)}| \partial_t u_j(s^j+r_j^2 \cdot,y^j+r_j\cdot)|
> \delta
$$
for all possible rotations $\tilde h$ of $h$.
\\
We may define
$$
U_j(x):=\frac{u_j(r_j^2 t + s^j, r_j x + y^j)}{r_j^{2}}
$$
and arrive at
\begin{equation}\label{hdiff}
\Vert U_j-\tilde h\Vert_{W^{1,\infty}(Q_{1})} > \delta ,
\end{equation}
for all possible rotations $\tilde h$ of $h$.
\newline
Observe that $U_j$ is a solution of (\ref{obst}) in $Q_1$ with
respect to the scaled coefficients $\lambda_+(r_j^2 t+s^j,r_j x + y^j)$ and
$\lambda_-(r_j^2 t+s^j, r_j x + y^j)$. Since $U_j(0)=0$, $\>|\nabla U_j(0)|\leq
\sigma_j$,$\>\hbox{\rm pardist}(0,\{ U_j>0\})\le \sigma_j$,$\>\hbox{\rm pardist}(0,\{ U_j<0\})\le \sigma_j$ and the derivatives $D^2 U_j, \partial_t U_j$ are uniformly
bounded, we obtain by standard compactness arguments a global limit
solution $U_0$ of (\ref{obst}) in ${\bf R}^n$ with respect to
$\lambda_+(s^0,y^0)$ and $\lambda_-(s^0,y^0)$ which
satisfies $0\in \partial\{ U_0>0\}\cap \partial\{ U_0<0\}\cap \{ \nabla U_0=0\}$.
By Theorem \ref{global-solution},
$U_0=\tilde h$ where $\tilde h$ is a rotated version of $h$.
Thus $U_j$ and $\nabla U_j$ converge in $Q_1$ uniformly to $\tilde h$
and $\nabla \tilde h$, respectively, and by Corollary \ref{zerolim}
$\partial_t U_j \to 0$ in $L^\infty(Q_1)$ as $j\to\infty$.
We obtain a contradiction to (\ref{hdiff}).\qed
\section{Continuity of the time derivative}
Assuming once more that we are in the setting of Theorem \ref{main},
we show in the present section that the time derivative
of the solution is {\em continuous} in a suitable neighborhood of the origin.
\begin{proposition}\label{time-indep}
Let $u$ be a solution of (\ref{obst}) in $Q_1$.
Then
there are positive constants $\tilde r$ and $\tilde \sigma$
(depending on $\inf_{Q_{1}} \min(\lambda_+,\lambda_-)$,
the Lipschitz norms of $\lambda_\pm,$ the supremum norm of $u$ and
the space dimension $n$)
such that
the following holds.
If $u(0)=0\> ,\> |\nabla u(0)|\leq \tilde \sigma\tilde r$,
$\hbox{\rm pardist}(0,\{ u>0\})\le \tilde \sigma\tilde r$ and $\hbox{\rm pardist}(0,\{ u<0\})\le \tilde \sigma\tilde r$
then
each blow-up limit at a point
$(t^1,x^1)\in Q_{\tilde r}\cap \{ u=0\} \cap \{ \nabla u=0\}$ is time-independent.
\end{proposition}
\proof
Let us consider $(t^1,x^1)\in \{ u=0\} \cap \{ \nabla u=0\}$.
As the statement of the Proposition is by Theorem \ref{global-solution} true
when $(t^1,x^1)$ is a branch point, we may assume that $u\ge 0$
in some neighborhood of $(t^1,x^1)$.
From Lemma \ref{uniform-closeness} (with $\delta := \inf_{Q_1} \min(\lambda_+,\lambda_-)/(96n)$) and Proposition \ref{directional-monoton}
we know that $u$ is non-decreasing, say in the direction $e$
for every $e$ close to $x_n$ in $Q_{\tilde r}$
and that $|\partial_t u|\le \inf_{Q_{\tilde r}} \min(\lambda_+,\lambda_-)/2$
in $Q_{\tilde r}$.
\\
From \cite[Theorem 4.1]{siam} we infer now that
each blow-up limit $z$ at $(t^1,x^1)$ is a non-negative backward self-similar solution.
Concerning those, it has been
shown in \cite[Lemma 6.3]{cps} and \cite[Theorem 8.1]{cps} that
either $z$ is a half-plane solution of the form
$z(t,x) = \lambda_+(t^1,x^1)/2 \max(x\cdot e,0)^2$ for some
$e\in \partial B_1$, or
$z(t,x)=- a_0 t + \sum_{i=1}^n a_i x_i^2$ with non-negative
constants $a_i, 0\le i\le n$ and $a_0\le \lambda_+(t^1,x^1)/2$.
In the latter case the symmetry of $z$
contradicts the fact that $z$ is non-decreasing in every
direction $e$ as above. Consequently $\partial_t z=0$ in $\{ t<0\}$,
and Lemma \ref{forward} and Corollary \ref{zerolim}
imply that
$\partial_t u(t^1,x^1)=0$.
\qed
\begin{corollary}\label{conti}
Let $u$ be a solution of (\ref{obst}) in $Q_1$.
Then
there are positive constants $\tilde r$ and $\tilde \sigma$
(depending on $\inf_{Q_{1}} \min(\lambda_+,\lambda_-)$,
the Lipschitz norms of $\lambda_\pm,$ the supremum norm of $u$ and
the space dimension $n$)
such that
the following holds.
If $u(0)=0\> ,\> |\nabla u(0)|\leq \tilde \sigma\tilde r$,
$\hbox{\rm pardist}(0,\{ u>0\})\le \tilde \sigma\tilde r$ and $\hbox{\rm pardist}(0,\{ u<0\})\le \tilde \sigma\tilde r$
then
$\partial_t u$ is continuous
in $Q_{\tilde r}$.
\end{corollary}
\proof
The corollary follows immediately from Lemma \ref{cinf}, Proposition \ref{time-indep}
and Corollary \ref{zerolim}.\qed
\section{Directional Monotonicity II}
It is now possible to extend the directional monotonicity result
of Section \ref{dirmon} to a directional monotonicity result
with respect to time-space variables.
\begin{proposition}\label{directional-monoton-par}
Let $0<\lambda_{\rm min}\le \inf_{Q_1(0)}
\min(1,\lambda_+,\lambda_-)$, $h$ as in (\ref{one-dim-sol}), let
$\varepsilon\in (0,1)$
and let $\tilde r$ and $\tilde \sigma$ be the constants of Corollary \ref{conti}. Then each solution $u$ of (\ref{obst}) in
$Q_1(0)$ such that
$$\hbox{\rm dist}_{W^{1,\infty}(Q_1(0))}(u,h)
\le \delta := {\lambda_{\rm min}\varepsilon\over {48 n}}\tilde r^2 \tilde \sigma^2$$ and
$$\sup_{Q_1(0)}\max(|\nabla \lambda_+|,|\partial_t \lambda_+|,|\nabla \lambda_-|,|\partial_t \lambda_-|)\le
\delta$$ satisfies $\varepsilon^{-1}\alpha \partial_t u \> + \> \varepsilon^{-1}
\partial_e u - |u| \ge 0$ in $Q_{1/2}(0)$ for every $\alpha \in [-1,1]$ and every $e\in \partial
B_1(0)$ such that $e_1\ge \varepsilon$;
here $e_1$ denotes the first component of the vector $e$.
\end{proposition}
\proof
First note that $Q_1\cap\{ u=0\}$
is by the assumptions contained in the strip
$|x_1| < \tilde \sigma \tilde r/2$, implying by
Corollary \ref{conti} and Lemma \ref{cinf}
that $\partial_t u$ is continuous in $Q_1$.
We know that $\varepsilon^{-1}\alpha\partial_t h \> +\> \varepsilon^{-1} \partial_e h - |h|\ge 0$.
It follows that
\begin{equation}\label{approx_p}
\varepsilon^{-1} \alpha\partial_t u \> +\> \varepsilon^{-1} \partial_e u - |u|\ge -3\delta \varepsilon^{-1}
\end{equation}
provided that $\hbox{\rm dist}_{W^{1,\infty}(Q_1(0))}(u,h) \le\delta$. Suppose now
towards a contradiction that the statement is not true. Then there
exist $\lambda_+,\lambda_-\in (\lambda_{\rm min},+\infty), (t^*,x^*) \in
Q_{1/2}(0), \alpha^*,e^*, $ and a solution $u$ of (\ref{obst}) in $Q_1(0)$
such that $\hbox{\rm dist}_{W^{1,\infty}(Q_1(0))}(u,h) \le\delta$,
$$\sup_{Q_1(0)}\max(|\nabla \lambda_+|,|\partial_t \lambda_+|,|\nabla \lambda_-|,|\partial_t \lambda_-|)\le
\delta,$$ $|\alpha^*|\le 1,e^*_1 \ge \varepsilon$ and
$\varepsilon^{-1}\alpha^* \partial_t u(t^*,x^*) \> +\>\varepsilon^{-1}\partial_{e^*} u(t^*,x^*) - |u(t^*,x^*)|<0.$ For the
positive constant $c$ to be defined later the functions $v :=
\varepsilon^{-1}\alpha^* \partial_t u \> +\>\varepsilon^{-1}\partial_{e^*} u - |u|$ and $w:=
\varepsilon^{-1}\alpha^* \partial_t u \> +\>\varepsilon^{-1}\partial_{e^*} u - |u|+c |x-x^*|^2 - c (t-t^*)$ satisfy then
by the definition of $\delta$ the
following: in the set $D := Q_1(0)\cap \{ v<0\}\cap\{ t<t^*\}$,
\[ \Delta w \> -\> \partial_t w\le 2nc + c -{\lambda_+} \chi_{\{ u>0\}}-{\lambda_-} \chi_{\{ u<0\}}\]\[
+\varepsilon^{-1} (\lambda_+ +\lambda_-) \nu_x\cdot e^* {\mathcal
H}^{n}\lfloor (\{ u=0\}\cap \{ \nabla u\ne 0\}) \]\[
+\varepsilon^{-1} (\lambda_+ +\lambda_-) \nu_t \alpha^* {\mathcal
H}^{n}\lfloor (\{ u=0\}\cap \{ \nabla u\ne 0\}) \]\[+\varepsilon^{-1}
( \chi_{\{ u>0\}}(\alpha^* \partial_t \> + \> \partial_{e^*}) \lambda_+ - \chi_{\{
u<0\}}(\alpha^* \partial_t \> + \>\partial_{e^*}) \lambda_-)
\]
where $\nu = {(\partial_t u,\nabla u) \over {|(\partial_t u,\nabla u)|}}$. As
\[\nu\cdot (\alpha^*,e^*)\le 0 \> \hbox{ on }\> \{ u=0\}\cap \{ v<0\} = \{ u=0\}\cap \{
\varepsilon^{-1}\alpha^* \partial_t u \> +\>\varepsilon^{-1}\partial_{e^*} u <0\}\; ,\] we obtain by the definition of $\delta$ that
$w$ is supercaloric in $D$ provided that $c$ has been chosen
accordingly, say $c:= \lambda_{\rm min}/(4n)$. It follows that the
negative infimum of $w$ is attained on
\[ \partial_{\rm par} D \subset (\partial_{\rm par} Q_1(0)\cap \{ t<t^*\})\cup
\left( Q_1(0)\cap \partial \{ v<0\}\right)
\; .\]
Consequently it is attained on $\{ t<t^*\}\cap \partial_{\rm par} Q_1(0)$,
say at the point $(\bar t,\bar x) \in \{ t<t^*\}\cap \partial_{\rm par} Q_1(0)$.
Since $\hbox{\rm pardist}((\bar t,\bar x),(t^*,x^*))\ge 1/2$, we obtain that
\[ \varepsilon^{-1}\alpha^* \partial_t u(\bar t,\bar x) \> +\>\varepsilon^{-1}\partial_{e^*} u(\bar t,\bar x) - |u(\bar t,\bar x)|\]\[ =
v(\bar t,\bar x) = w(\bar t,\bar x) - c |x^* - \bar x|^2 + c (\bar t-t^*)
< -c/4=-\lambda_{\rm min}/(16n)\; .\] But this contradicts
(\ref{approx_p}) in view of $\delta = {\lambda_{\rm
min}\varepsilon\over {48 n}}\tilde r^2\tilde \sigma^2$.\qed
\section{Proof of the main theorem}
The theorem is proven in several simple steps, using mainly
Proposition \ref{directional-monoton-par}, and Lemma
\ref{uniform-closeness}. Note that the proof can be simplified substantially
in the case that we are dealing not with a whole class of solutions
but a single solution.
\newline
{\bf Part I:} In this first part we prove uniform Lipschitz regularity
and continuous differentiability with respect to the space variables.
\newline
{\bf Step 1 (Directional monotonicity):} Given $\varepsilon >0$, there are
$\sigma_\varepsilon>0$ and $r_\varepsilon >0$ (depending only on the parameters of the
statement) such that $2\alpha \varepsilon^{-1}r^2_\varepsilon\partial_t u+2\varepsilon^{-1}r_\varepsilon\partial_e u
-|u|\geq 0$ in $Q_{r_\varepsilon/2}(y)$ for every $\alpha \in [-1,1]$. The inequality holds for every $(s,y)\in
Q_{1/2}(0)$ satisfying
$u(s,y)=0$,$\>|\nabla u(s,y)|\leq \sigma_\varepsilon r_\varepsilon$,$\>\hbox{\rm pardist}((s,y),\{ u>0\})\le \sigma_\varepsilon r_\varepsilon$ and
$\hbox{\rm pardist}((s,y),\{ u<0\})\le \sigma_\varepsilon r_\varepsilon$,
for some unit vector
$\nu_\varepsilon(s,y)$ and for every $e \in \partial B_1$ satisfying $e\cdot \nu_\varepsilon(s,y)
\ge {\varepsilon \over 2}$. In particular, for $\varepsilon=1$, the solution $u$ is
by condition (\ref{cond}) with $\sigma=\sigma_1 r_1$
non-decreasing in
$Q_{r_1/2}(0)$ in direction $(r_1,e)$ for every $e \in \partial B_1(0)$ such that $e \cdot
\nu_\varepsilon(0) \geq {1\over 2}$.
\newline
{\it Proof:} By Lemma \ref{uniform-closeness} there are
$\sigma_\varepsilon>0$ and $r_\varepsilon>0$ as above such that the scaled function
$u_{r_\varepsilon}(t,x) = u(s+r_\varepsilon^2 t,y+r_\varepsilon x)/r_\varepsilon^2$ is
$\delta:=\varepsilon \frac{\lambda_{\rm min}}{64n}\tilde r^2\tilde \sigma^2$-close in
$C^1(Q_1(0))$ to a rotated version $\tilde h$ of $h$ in $Q_1$.
Let $\nu_\varepsilon(s,y)$ be the accordingly rotated version of the unit
vector $e_1$.
Since
$u_{r_\varepsilon}$ solves (\ref{obst}) with respect to
$\lambda_+(r_\varepsilon^2 \cdot + s, r_\varepsilon \cdot +y)$ and $\lambda_-(r_\varepsilon^2 \cdot + s,r_\varepsilon
\cdot +y)$, and since $\max(|\nabla (\lambda_+(r_\varepsilon^2 \cdot + s,r_\varepsilon \cdot
+ y))|,|\nabla (\lambda_-(r_\varepsilon^2 \cdot + s,r_\varepsilon \cdot+y))|,|\partial_t (\lambda_+(r_\varepsilon^2 \cdot + s,r_\varepsilon \cdot
+ y))|,|\partial_t (\lambda_-(r_\varepsilon^2 \cdot + s,r_\varepsilon \cdot
+ y))|)\le C_1
r_\varepsilon $, we may choose $r_\varepsilon < \delta/C_1$ in order
to apply
Proposition \ref{directional-monoton-par} to
$u_{r_\varepsilon}$ in $Q_1$ and to conclude that
$2\alpha \varepsilon^{-1}\partial_t u_{r_\varepsilon}+2\varepsilon^{-1}\partial_e u_{r_\varepsilon} - |u_{r_\varepsilon}|\ge 0$ in $Q_{1/2}(0)$ for
every $\alpha \in [-1,1]$ and every $e\in \partial B_1(0)$ such that $e\cdot \nu_\varepsilon(s,y) \ge
\varepsilon/2$. Scaling back we obtain the statement of Step 1.
\newline
{\bf Step 2 (Lipschitz continuity):} $\partial \{u>0\}\cap
Q_{r_1/2}(0)$ and $\partial \{u<0\}\cap Q_{r_1/2}(0)$ are Lipschitz
graphs in the direction of $(0,\nu_\varepsilon(0))$ with spatial Lipschitz norms less than
$1$ and temporal Lipschitz norms less than $r_1^{-1}$. Moreover, for each $\varepsilon\in (0,1)$ and $(s,y)\in
\{ u=0\} \cap Q_{1/2}$ satisfying $|\nabla u(s,y)|\leq \sigma_\varepsilon r_\varepsilon$,
$\hbox{\rm pardist}((s,y),\{ u>0\})\le \sigma_\varepsilon r_\varepsilon$ and
$\hbox{\rm pardist}((s,y),\{ u<0\})\le \sigma_\varepsilon r_\varepsilon$,
the free boundaries $\partial
\{ u>0\} \cap Q_{r_{\varepsilon /2}}(s,y)$ and $\partial \{ u<0\} \cap
Q_{r_{\varepsilon /2}}(s,y)$ are Lipschitz graphs (in the direction of
$\nu_\varepsilon(s,y)$) with spatial Lipschitz norms not greater than $\varepsilon$.
\newline
{\it Proof:} This follows from the monotonicity obtained in Step 1.
\newline
{\bf Step 3 (Existence of a spatial tangent plane at points
$(s,y) \in \partial\{ u>0\}\cap \partial \{ u<0\}\cap Q_{1/2}(0)$ satisfying $|\nabla u(s,y)|=0$):}
The Lipschitz graphs of Step 2 are both
differentiable with respect to the space variables
at the point $(s,y)$, and the two spatial tangent planes
at $(s,y)$ coincide.
\newline
{\it Proof:} This follows from Step 2 by letting $\varepsilon$ tend
to zero.
\newline
{\bf Step 4 (One-phase points are regular):} If $(s,y) \in Q_{r_1/2}(0)$
is a free boundary point and the solution $u$ is non-negative or
non-positive in $Q_\delta(s,y)$, then the free boundary is the graph
of a $C^{1,\alpha}$-function in $Q_{c_1\delta}(s,y)$, where $c_1$ and
the $C^{1,\alpha}$-norm depend only on the parameters in the
statement. Consequently, in $Q_{r_1/2}(0)$, there exist no singular
one-phase free boundary points.
\newline
{\it Proof:} By Step 2, the sets
$\{ u>0\} \cap Q_{r_1/2}(0)$ and $
\{ u<0\} \cap Q_{r_1/2}(0)$ are sub/supergraphs of
Lipschitz continuous functions.
Therefore $\{ u=0\}\cap Q_\delta(s,y)$
satisfies the thickness condition required for
\cite[Theorem 15.1]{cps}
and the statement follows.
\newline
{\bf Step 5 (Existence of space normals in $Q_{r_1/2}(0)$):}
$\partial
\{ u>0\} \cap Q_{r_1/2}(0)$ and $\partial
\{ u<0\} \cap Q_{r_1/2}(0)$ are graphs
of Lipschitz continuous functions which are differentiable
with respect to the space variables.
\newline
{\it Proof:} Let $(s,y) \in Q_{r_1/2}(0)$ be a free boundary point.
We have to prove existence of a tangent plane at $(s,y)$.\newline
First, if $(s,y)$ is a one-phase point,
i.e. if the solution $u$ is non-negative or non-positive in
$Q_\delta(s,y)$, then the statement holds at $(s,y)$ by the result of Step 4.
Second, if $|\nabla u(s,y)|\ne 0$, the statement holds by Lemma \ref{cinf}.
Last, if $|\nabla u(s,y)|=0$ and $(s,y)$ is the limit point
of both phases $\{ u>0\}$ and $\{ u<0\}$, then
Step 3 applies.
\newline
{\bf Step 6 (Equicontinuity of the space normals):} It remains to prove
that the space normals are equicontinuous on $Q_{r_1/2}(0)\cap \partial \{
u>0\}$ and on $Q_{r_1/2}(0)\cap \partial \{ u<0\}$ for $u$ in the
class of solutions specified in the statement of the main theorem.
\newline\noindent
{\it Proof:} By Step 2 we know already that the spatial Lipschitz norms of
$\partial \{ u>0\} \cap Q_{r_1/2}(0)$ and $\partial \{ u<0\} \cap
Q_{r_1/2}(0)$ are less than $1$. We prove that the space normals are
equicontinuous on $Q_{r_1/2}(0)\cap\partial \{ u>0\}$.
\newline
We may assume that $\nu(0)$ points in the direction of the $x_1$-axis and
that $x_1=f(t,x_2,\dots,x_n)$ is the representation of $\partial
\{u>0\}\cap Q_{r_1/2}(0)$. Besides we have $|\nabla f(t,x')|<1$ for
$(t,x)=(t,x_1,x') \in
\partial \{u>0\}\cap Q_{r_1/2}(0)$.
We claim that for $\varepsilon >0$ there is $\delta_\varepsilon >0$
depending only on the parameters in the statement such that for any
pair of free boundary points $(s^1,y^1),(s^2,y^2) \in
\partial \{ u>0\} \cap Q_{r_1/2}(0)$,
\begin{equation}\label{continuity-of-normal}
\hbox{\rm pardist}((s^1,y^1),(s^2,y^2)) \leq \delta_\varepsilon \quad \Rightarrow \quad |\nu(s^1,y^1)
-\nu(s^2,y^2)|\leq 2\varepsilon .
\end{equation}
In what follows let $\rho_\varepsilon := \sigma_\varepsilon r_\varepsilon/2 \le r_1/2$.
\newline
Suppose first that $u$ is non-negative in $Q_{\rho_\varepsilon}(s^1,y^1)$. Here we
may as in Step 4 apply \cite[Theorem 15.1]{cps} to the scaled
function $w(t,x) := u(s^1+\rho_\varepsilon^2 t, y^1+\rho_\varepsilon x)/{\rho^2_\varepsilon}$; since
the $C^{1,\alpha}$-norm of the free boundary normal of $w$ is on $Q_{c_2}\cap \partial\{ w>0\}$
bounded by a constant $C_3$, where $c_2>0$ and $C_3<+\infty$ depend only on the parameters in the
statement, we may choose
$$\delta_\varepsilon := \min({\varepsilon^{1 \over \alpha}\over {C_3^{1 \over \alpha}}},c_2)\rho_\varepsilon$$
to obtain (\ref{continuity-of-normal}).
\newline
Next, suppose that
$u$ changes its sign at $Q_{\rho_\varepsilon}(s^1,y^1)$.
If there is a point
$(s,y)\in
Q_{\rho_\varepsilon}(s^1,y^1)\cap \partial\{ u>0\}$ such that
$|\nabla u(s,y)|\leq\rho_\varepsilon$ then we are in the situation of Step 1. By Step 2 the
free boundary $\partial \{ u>0\} \cap Q_{r_{\varepsilon}/2}(s,y)$ is
Lipschitz with spatial Lipschitz norm not greater than ${\varepsilon}$.
Hence (\ref{continuity-of-normal})
follows in this case with $\delta_\varepsilon := r_{\varepsilon}/2$.
\newline
Last, if $|\nabla u(s,y)|\geq \rho_\varepsilon$
for all points $(s,y)\in Q_{\rho_\varepsilon}(s^1,y^1)\cap \partial\{ u>0\}$,
we proceed as follows:
from the equation $u(t,f(t,x'),x')=0$ we infer that
$\nabla' u+\partial_1 u\> \nabla'f=0$
on $\partial \{u>0\}\cap
Q_{r_1/2}(0)$. Hence we obtain
$$
|\nabla' f(s^1,(y^1)')-\nabla' f(s^2,(y^2)')|
= \left\vert {\nabla' u(s^1,y^1)\over {\partial_1 u(s^1,y^1)}}-
{\nabla' u(s^2,y^2)\over {\partial_1 u(s^2,y^2)}}\right\vert$$
$$\le {|\nabla' u(s^2,y^2)-\nabla' u(s^1,y^1)|\over {|\partial_1 u(s^1,y^1)|}}$$
$$+ \> \left|{\nabla' u(s^2,y^2)\over {\partial_1 u(s^2,y^2)}}\right|
{|\partial_1 u(s^2,y^2)-\partial_1 u(s^1,y^1)|\over {|\partial_1 u(s^1,y^1)|}}$$
$$ \leq 4M\rho_\varepsilon^{-1}\hbox{\rm pardist}((s^1,y^1),(s^2,y^2))\; ,
$$
where $M=\Vert \nabla u \Vert_{C^{1/2,1}(Q_{1/2}(0))}$.
In particular we may choose
$$\delta_\varepsilon := {\varepsilon \over {4M}}\rho_\varepsilon$$
to arrive at (\ref{continuity-of-normal}).\\[.3cm]
{\bf Part II:} We conclude the proof of the main theorem by pointing out a counter-example to $C^1$-regularity.
\newline
\begin{figure}
\begin{center}
\input{suw_par.pstex_t}
\end{center}
\caption{A counter-example to $C^1$-regularity}\label{counter}
\end{figure}
Consider the one-phase counter-example $u:[-r^2,r^2]\times [0,r]\to [0,+\infty)$
from \cite[p. 376]{asu} satisfying the following:
$u(t,0)=0$ for $-r^2\le t \le r^2$,
and the free boundary touches the lateral boundary at the origin
in a non-tangential way.
Thus we may reflect $u$ to a solution
$$v(t,x) :=\left\{ \begin{array}{l}
u(t,x),x\ge 0\\
-u(t,-x),x<0\end{array}\right.$$
and obtain that $v$ is a solution of our two-phase problem
(\ref{obst}) in $Q_r$ for $\lambda_+=\lambda_-=1$.
As the free boundary $\partial\{ v>0\}$ is only Lipschitz at the origin,
we conclude that differentiability with respect to the time variable is
in general not true.\qed
\bibliographystyle{plain}
| {
"timestamp": "2007-12-20T14:32:08",
"yymm": "0712",
"arxiv_id": "0712.3411",
"language": "en",
"url": "https://arxiv.org/abs/0712.3411",
"abstract": "For the parabolic obstacle-problem-like equation $$\\Delta u - \\partial_t u = \\lambda_+ \\chi_{\\{u>0\\}} - \\lambda_- \\chi_{\\{u<0\\}} ,$$ where $\\lambda_+$ and $\\lambda_-$ are positive Lipschitz functions, we prove in arbitrary finite dimension that the free boundary $\\partial\\{u>0\\} \\cup\\partial\\{u<0\\}$ is in a neighborhood of each ``branch point'' the union of two Lipschitz graphs that are continuously differentiable with respect to the space variables. The result extends the elliptic paper \\cite{imrn} to the parabolic case. The result is optimal in the sense that the graphs are in general not better than Lipschitz, as shown by a counter-example.",
"subjects": "Analysis of PDEs (math.AP)",
"title": "The Parabolic Two-Phase Membrane Problem: Regularity in Higher Dimensions",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754488233528,
"lm_q2_score": 0.718594386544335,
"lm_q1q2_score": 0.7075103906538306
} |
https://arxiv.org/abs/math/0609494 | A pinching theorem for the first eigenvalue of the laplacian on hypersurface of the euclidean space | In this paper, we give pinching Theorems for the first nonzero eigenvalue $\lambda$ of the Laplacian on the compact hypersurfaces of the Euclidean space. Indeed, we prove that if the volume of $M$ is 1 then, for any $\epsilon>0$, there exists a constant $C\_{\epsilon}$ depending on the dimension $n$ of $M$ and the $L\_{\infty}$-norm of the mean curvature $H$, so that if the $L\_{2p}$-norm $\|H\|\_{2p}$ ($p\geq 2$) of $H$ satisfies $n\|H\|\_{2p}-C\_{\epsilon}<\lambda$, then the Hausdorff-distance between $M$ and a round sphere of radius $(n/\lambda)^{1/2}$ is smaller than $\epsilon$. Furthermore, we prove that if $C$ is a small enough constant depending on $n$ and the $L\_{\infty}$-norm of the second fundamental form, then the pinching condition $n\|H\|\_{2p}-C<\la$ implies that $M$ is diffeomorphic to an $n$-dimensional sphere. | \section{Introduction and preliminaries}
\ \
Let $(M^n,g)$ be a compact, connected and oriented $n$-dimensional Riemannian
ma\-nifold without boundary isometrically immersed by $\phi$ into the
$n+1$-dimensional euclidean space $(\mathbb{R}^{n+1}, can)$
(i.e. $\phi^{\star}can=g$). A well known inequality due to Reilly
(\cite{rei}) gives an extrinsic upper bound for the first nonzero eigenvalue $\lambda_1(M)$ of the Laplacian of $(M^n,g)$ in terms of the square of the length of the mean curvature. Indeed, we have
\begin{align}\label{a}\lambda_1(M)\leq\frac{n}{V(M)}\insm |H|^2\vol\end{align}
\ndt where $\vol$ and $V(M)$ denote respectively the Riemannian volume
element and the volume of $(M^n,g)$. Moreover the equality holds if and only
if $(M^n,g)$ is a geodesic hypersphere of $\mathbb{R}^{n+1}$.
By using H\"older inequality, we obtain some other similar estimates
for the $L_{2p}$-norm ($p\geq 1$) with $H$ denoted by $\|H\|_{2p}^2$
\begin{align}\label{b}\lambda_1(M)\leq\frac{n}{V(M)^{1/p}}\|H\|_{2p}^2,\end{align}
\ndt and as for the inequality (\ref{a}), the equality case is characterized
by the geodesic hyperspheres of $\mathbb{R}^{n+1}$.
A first natural question is to know if there exists a pinching result as
the one we state now: does a constant
$C$ depending on minimum geometric invariants exist so that if we have the
pinching condition
\saut
\begin{tabular}{cc}$(P_C)$
&\hspace{3cm} $\displaystyle\frac{n}{V(M)^{1/p}}\|H\|_{2p}^2-C<\lambda_1(M)$\end{tabular}
\saut
\ndt then $M$ is close to a sphere in a certain sense?
Such questions are known for the intrinsic lower bound of
Lichnerowicz-Obata (\cite{lic}) of $\lambda_1(M)$ in terms of the lower bound of the
Ricci curvature (see \cite{cro}, \cite{ili}, \cite{pet}).
Other pinching results have been proved for Riemannian manifolds
with positive Ricci curvature, with a pinching condition on the $n+1$-st eigenvalue
(\cite{pet}), the diameter (\cite{esc}, \cite{ili}, \cite{wu}), the volume
or the radius (see for instance \cite{col1} and \cite{col2}).
For instance,
S. Ilias proved in \cite{ili} that there exists $\varepsilon$ depending on
$n$ and an upper bound of the sectional curvature so that if the Ricci
curvature $Ric$ of $M$ satisfies $Ric\geq n-1$ and
$\lambda_1(M)\leq\lambda_1(\mathbb{S}^n)+\varepsilon$, then $M$ is homeomorphic to
$\mathbb{S}^n$.
In this article, we investigate the case of hypersurfaces where, as
far as we know, very little is known about pinching and stability results
(see however \cite{shixu1}, \cite{shixu2}).
More precisely, in our paper, the hypothesis made in \cite{ili} that $M$ has a positive Ricci curvature is replaced
by the fact that $M$ is isometrically immersed as a hypersurface in
$\mathbb{R}^{n+1}$, and the bound on the sectional curvature by an $L^{\infty}$-bound on the mean curvature or on the second fundamental form. Note
that we do not know if such bounds are sharp, or if a bound on the
$L^q$-norm (for some $q$) of the mean curvature would be enough.
\medskip
We get the following results
\begin{theo}\label{hausdist} Let $(M^n,g)$ be a compact, connected and
oriented $n$-dimensional Riemannian ma\-nifold without boundary
isometrically immersed by $\phi$ in $\mathbb{R}^{n+1}$. Assume that $V(M)=1$ and let $x_0$ be the center of mass of
$M$. Then for any $p\geq 2$ and for any $\varepsilon>0$, there
exists a constant $C_{\varepsilon}$ depending only on $n$, $\varepsilon>0$ and
on the $L_{\infty}$-norm of $H$ so that if
\saut
\begin{tabular}{cc}$(P_{C_{\varepsilon}})$
&\hspace{3cm} $n\|H\|_{2p}^2-C_{\varepsilon}<\lambda_1(M)$\end{tabular}
\saut
\ndt then the Hausdorff-distance $d_H$ of $M$ to the sphere
$S\left(x_0,\sqrt{\frac{n}{\lambda_1(M)}}\right)$ of center $x_0$ and radius
$\sqrt{\frac{n}{\lambda_1(M)}}$ satisfies $d_H\left(\phi(M),S\left(x_0,\sqrt{\frac{n}{\lambda_1(M)}}\right)\right)<\varepsilon$.
\end{theo}
We recall that the Hausdorff-distance between two compact subsets $A$ and
$B$ of a metric space is given by
$$d_H(A,B)=\inf\{\eta| V_{\eta}(A)\supset B\; \text{and}\; V_{\eta}(B)\supset
A\}$$
\ndt where for any subset $A$, $V_{\eta}(A)$ is the tubular neighborhood of
$A$ defined by $V_{\eta}(A)= \{x|dist(x, A)<\eta\}$.
\saut
\ndt {\bf Remark} We will see in the proof that $C_{\varepsilon}(n,\|H\|_{\infty})\rightarrow
0$ when $\|H\|_{\infty}\rightarrow\infty$ or $\varepsilon\rightarrow 0$.
\saut
In fact the previous Theorem is a consequence of the above definition and
the following Theorem
\begin{theo}\label{maintheo} Let $(M^n,g)$ be a compact, connected and
oriented $n$-dimensional Riemannian ma\-nifold without boundary
isometrically immersed by $\phi$ in $\mathbb{R}^{n+1}$. Assume that $V(M)=1$ and
let $x_0$ be the center of mass of $M$. Then for any $p\geq 2$ and for any $\varepsilon>0$,
there exists a constant $C_{\varepsilon}$ depending only on
$n$, $\varepsilon>0$ and on the $L_{\infty}$-norm of $H$ so that if
\saut
\begin{tabular}{cc}$(P_{C_{\varepsilon}})$
&\hspace{3cm} $n\|H\|_{2p}^2-C_{\varepsilon}<\lambda_1(M)$\end{tabular}
\saut
\ndt then
\begin{enumerate} \item $\phi(M)\subset
B\left(x_0,\sqrt{\frac{n}{\la}}+\varepsilon\right)\backslash
B\left(x_0,\sqrt{\frac{n}{\la}}-\varepsilon\right)$.
\item $\forall x\in S\left(x_0,\sqrt{\frac{n}{\la}}\right), B(x,\varepsilon)\cap
\phi(M)\neq\O$.
\end{enumerate}
\end{theo}
In the following Theorem, if the pinching is strong enough, with a
control on $n$ and the
$L_{\infty}$-norm of the second fundamental form, we obtain
that $M$ is diffeomorphic to a sphere and even almost isometric with
a round sphere in a sense we will make precise.
\begin{theo}\label{diffeo} Let $(M^n,g)$ be a compact, connected and
oriented $n$-dimensional Riemannian ma\-nifold ($n\geq 2$) without boundary
isometrically immersed by $\phi$ in $\mathbb{R}^{n+1}$. Assume that
$V(M)=1$. Then for any $p\geq 2$, there exists a constant $C$ depending
only on $n$ and the $L_{\infty}$-norm of the second fundamental form $B$ so that if
\saut
\begin{tabular}{cc}$(P_{C})$
&\hspace{3cm} $n\|H\|_{2p}^2-C <\lambda_1(M)$\end{tabular}
\saut
\ndt Then $M$ is diffeomorphic to $\mathbb{S}^n$.
More precisely, there exists a diffeomorphism $F$ from $M$ into the sphere
$\mathbb{S}^n\left(\sqrt{\crit}\right)$ of radius $\sqrt{\crit}$ which is a quasi-isometry.
Namely, for any $\theta$, $0<\theta<1$, there
exists a constant $C$ depending only on $n$, the $L_{\infty}$-norm of $B$
and $\theta$, so that the pinching condition $(P_{C})$ implies
$$\left||dF_x(u)|^2-1\right|\leq \theta$$
\ndt for any $x\in M$ and $u\in T_x M$ so that $|u|=1$.
\end{theo}
Now we will give some preliminaries for the proof of these
Theorems. Throughout the paper, we consider a compact, connected and
oriented $n$-dimensional Riemannian ma\-nifold $(M^n,g)$ without boundary
isometrically immersed by $\phi$ into $(\mathbb{R}^{n+1}, can)$
(i.e. $\phi^{\star}can=g$). Let $\nu$ be the outward normal vector field. Then the second fundamental form of the immersion will be defined by $B(X,Y)=\scal{\nabla^0_X \nu}{Y}$,
where $\nabla^0$ and $\langle \ \ ,\ \ \rangle$ are respectively the Riemannian
connection and the inner product of $\mathbb{R}^{n+1}$. Moreover the mean curvature $H$ will be given by $H=(1/n)trace(B)$.
Now let $\partial_i$ be an orthonormal frame of $\mathbb{R}^{n+1}$ and let $x_i : \mathbb{R}^{n+1}\rightarrow \mathbb{R}$ be the associated component functions. Putting $X_i= x_i\circ\phi$, a straightforward calculation shows us that
$$B\otimes\nu=-\sum_{i\leq n+1}\nabla dX_i\otimes\partial_i$$
\ndt and
$$nH\nu=\sum_{i\leq n+1}\Delta X_i\partial_i$$
\ndt where $\nabla$ and $\Delta$ denote respectively the Riemannian
connection and the Laplace-Beltrami operator of $(M^n,g)$. On the other hand,
we have the well known formula
\begin{align}\label{hsiung}\frac{1}{2}\Delta |X|^2=nH\scal{\nu}{X}-n\end{align}
\ndt where $X$ is the position vector given by $X=\sum_{i\leq n+1}X_i\partial_i$.
We recall that to prove the Reilly inequality, we use the functions $X_i$
as test functions (cf \cite{rei}). Indeed, doing a translation if
necessary, we can assume that $\insm X_i\vol=0$ for all $i\leq n+1$ and we
can apply the variational characterization of $\lambda_1(M)$ to $X_i$. If the equality
holds in (\ref{a}) or (\ref{b}), then the functions are nothing but
eigenfunctions of $\lambda_1(M)$ and from the Takahashi's Theorem (\cite{tak}) $M$
is immersed isometrically in $\mathbb{R}^{n+1}$ as a geodesic sphere of radius $\sqrt{\frac{n}{\la}}$.
Throughout the paper we use some notations. From now on, the inner product
and the norm induced by $g$ and $can$ on a tensor $T$ will be denoted
respectively by $\langle \; ,\;\rangle$ and $|\;|^2$, and the $L_p$-norm will be given by
$$\|T\|_p=\left(\insm|T|^p\vol\right)^{1/p}$$
\ndt and
$$\|T\|_{\infty}=\sup_M |T|$$
We end these preliminaries by a convenient result
\begin{lem}\label{pratic} Let $(M^n,g)$ be a compact, connected and
oriented $n$-dimensional Riemannian ma\-nifold ($n\geq 2$) without boundary
isometrically immersed by $\phi$ in $\mathbb{R}^{n+1}$. Assume that
$V(M)=1$. Then there exist constants $c_n$ and $d_n$ depending only on $n$ so
that for any $p\geq 2$, if $(P_C)$ is true with $C< c_n$ then
\begin{align}\label{titi}\frac{n}{\lambda_1(M)}\leq d_n\end{align}
\end{lem}
\saut
\begin{proof} We recall the standard Sobolev inequality (cf \cite{hofspr},
\cite{hofspr'}, \cite{xu} and p 216 in \cite{burzal}). If $f$ is a smooth function and $f\geq 0$, then
\begin{align}\label{sobol}\left(\insm
f^{\frac{n}{n-1}}\vol\right)^{1-(1/n)}\leq
K(n)\insm\left(|df|+|H|f\right)\vol\end{align}
\ndt where $K(n)$ is a constant depending on $n$ and the volume of the unit
ball in $\mathbb{R}^n$. Taking $f=1$ on $M$, and using the fact that $V(M)=1$, we
deduce that
\begin{align*}\|H\|_{2p}\geq\frac{1}{K(n)}\end{align*}
\ndt and if $(P_C)$ is satisfied and $C\leq\frac{n}{2K(n)^2}=c_n$, then
\begin{align*}\frac{n}{\lambda_1(M)}\leq\frac{1}{n\|H\|_{2p}^2-C}\leq 2K(n)^2=d_n\end{align*}
\end{proof}
\saut
Throughout the paper, we will assume that $V(M)=1$ and $\insm X_i\vol=0$
for all $i\leq n+1$. The last assertion implies that the center of mass of
$M$ is the origin of $\mathbb{R}^{n+1}$.
\section{An $L^2$-approach of the problem}
\ \
A first step in the proof of the Theorem \ref{maintheo} is to
prove that if the pinching condition $(P_C)$ is satisfied, then $M$ is close
to a sphere in an $L^2$-sense.
In the following Lemma, we prove that the $L^2$-norm of the position vector
is close to $\sqrt\frac{n}{\la}$.
\begin{lem}\label{vectpos} If we have the pinching condition $(P_C)$ with $C<c_n$, then
$$\frac{n\lambda_1(M)}{(C+\lambda_1(M))^2}\leq\|X\|_2^2\leq \frac{n}{\lambda_1(M)}\leq d_n$$
\end{lem}
\begin{proof} Since $\insm X_i\vol=0$, we can apply the variational
characterization of the eigenvalues to obtain
\begin{align*}\lambda_1(M)\insm\sum_{i\leq
n+1}|X_i|^2\vol\leq\insm\sum_{i\leq n+1}|dX_i|^2\vol=n\end{align*}
\ndt which gives the inequality of the right-hand side
Let us prove now the inequality of the left-hand side.
\begin{align*}\lambda_1(M)\insm|X|^2\vol&\leq\frac{\left(\insm\displaystyle\sum_{i\leq n+1}|dX_i|^2\vol\right)^4}{\left(\insm\displaystyle\sum_{i\leq n+1}|dX_i|^2\vol\right)^3}=\frac{\left(\insm\displaystyle\sum_{i\leq n+1}(\Delta X_i) X_i\vol\right)^4}{n^3}\displaybreak[2]\\
&\leq\frac{\left(\insm\displaystyle \sum_{i\leq n+1}(\Delta
X_i)^2\vol\right)^2\left(\insm|X|^2\vol\right)^2}{n^3}\displaybreak[2]\\
&= n\left(\insm
H^2\vol\right)^2\left(\insm|X|^2\vol\right)^2\end{align*}
\ndt then using again the H\"older inequality, we get
\begin{align*}\lambda_1(M)&\leq\frac{1}{n}\left(n\|H\|_{2p}^2\right)^2\insm|X|^2\vol\leq\frac{(C+\lambda_1(M))^2}{n}\insm|X|^2\vol\end{align*}
\ndt This completes the proof. \end{proof}
\saut
From now on, we will denote by $X^{T}$ the orthogonal tangential
projection on $M$. In fact, at $x\in M$, $X^{T}$ is nothing but the
vector of $T_{x}M$ defined by $X^{T}=\displaystyle\sum_{1\leq i\leq
n}\scal{X}{\ei}\ei$ where $(\ei)_{1\leq i\leq n}$ is an orthonormal basis of $T_{x}M$.
In the following Lemma, we will show that the condition $(P_C)$ implies
that the $L^2$-norm of $X^{T}$ of $X$ on $M$ is close to $0$.
\begin{lem}\label{l2xt} If we have the pinching condition $(P_C)$, then
$$\|X^{T}\|_2^2\leq A(n)C$$
\end{lem}
\begin{proof} From the lemma \ref{vectpos} and the relation (\ref{hsiung}), we have
\begin{align*}\lambda_1(M)\insm|X|^2\vol&\leq n=n\left(\insm H\scal{X}{\nu}\vol\right)^2\displaybreak[2]\\
&\leq\left(\insm |H||\scal{X}{\nu}|\vol\right)^2\leqn\|H\|_{2p}^2\left(\insm
|\scal{X}{\nu}|^{\frac{2p}{2p-1}}\vol\right)^{\frac{2p-1}{p}}\displaybreak[2]\\
&\leqn\|H\|_{2p}^2\left(\insm
|\scal{X}{\nu}|^2\vol\right)=n\|H\|_{2p}^2\insm|X|^2\vol\end{align*}
\ndt Then we deduce that
\begin{align*}n\|H\|_{2p}^2\|X^{T}\|_2^2&=n\|H\|_{2p}^2\left(\insm\left(|X|^2-|\scal{X}{\nu}|^2\right)\vol\right)\displaybreak[2]\\
&\leq (n\|H\|_{2p}^2-\lambda_1(M))\|X\|_2^2\leq d_n C\end{align*}
\ndt where in the last inequality we have used the pinching condition and
the Lemma \ref{vectpos}.\end{proof}
\saut
Now, we will show that the condition $(P_C)$ implies that
the component functions are almost eigenfunctions in an $L^2$-sense. For
this, let us consider the vector field $Y$ on $M$ defined by
$$Y=\sum_{i\leq n+1}\left(\Delta X_i-\lambda_1(M) X_i\right)\partial_i=nH\nu-\lambda_1(M) X$$
\begin{lem}\label{Y} If $(P_C)$ is satisfied, then
$$\|Y\|_2^2\leq nC$$
\end{lem}
\saut
\begin{proof} We have
\begin{align*}\insm |Y|^2\vol=\insm\left(n^2H^2-2n\lambda_1(M) H\scal{\nu}{X}+\lambda_1(M)^2|X|^2\right)\vol\end{align*}
\ndt Now by integrating the relation (\ref{hsiung}) we deduce that
$$\insm H\scal{\nu}{X}\vol=1$$
\ndt Furthermore, since $\insm X_i\vol=0$, we can apply the variational
characterization of the eigenvalues to obtain
\begin{align*}\lambda_1(M)\insm|X|^2\vol=\lambda_1(M)\insm\sum_{i\leq n+1}|X_i|^2\vol\leq\insm\sum_{i\leq n+1}|dX_i|^2\vol=n\end{align*}
\ndt Then
\begin{align*}\insm |Y|^2\vol&\leq n^2\insm|H|^2\vol-n\lambda_1(M) \leq n \left(n\|H\|_{2p}^2-\lambda_1(M)\right)\leq nC\end{align*}
\ndt where in this last inequality we have used the H\"older inequality. \end{proof}
\saut
To prove Assertion 1 of Theorem \ref{maintheo}, we will show that
$\left\| |X|-\left(\frac{n}{\la}\right)^{1/2}\right\|_{\infty}\leq\varepsilon$. For this
we need to have an $L^2$-upper bound on the function
$\varphi=|X|\left(|X|-\left(\frac{n}{\la}\right)^{1/2}\right)^2$.
Before giving such estimate, we will introduce the vector field $Z$ on $M$
defined by
$$Z=\left(\frac{n}{\la}\right)^{1/2}|X|^{1/2}H\nu-\frac{X}{|X|^{1/2}}$$
\ndt We have
\begin{lem}\label{Z} If $(P_C)$ is satisfied with $C<c_n$, then
$$\|Z\|_2^2\leq B(n)C$$
\end{lem}
\begin{proof} We have
\begin{align*}\|Z\|_2^2&=\left\|\left(\frac{n}{\la}\right)^{1/2}|X|^{1/2}H\nu-\frac{X}{|X|^{1/2}}\right\|_2^2\displaybreak[2]\\
&=\insm\left(\frac{n}{\la} |X|H^2-2\left(\frac{n}{\la}\right)^{1/2}H\scal{\nu}{X}+|X|\right)\vol\\
&\leq\frac{n}{\la}\left(\insm|X|^2\vol\right)^{1/2}\left(\insm H^4\vol\right)^{1/2}-2\left(\frac{n}{\la}\right)^{1/2}+\left(\insm|X|^2\vol\right)^{1/2}\end{align*}
\ndt Note that we have used the relation (\ref{hsiung}). Finally for $p\geq 2$,
we get
\begin{align*}\|Z\|_2^2&\leq\left(\insm|X|^2\vol\right)^{1/2}\left(\frac{n}{\la}\|H\|_{2p}^2+1\right)-2\left(\frac{n}{\la}\right)^{1/2}\displaybreak[2]\\
&\leq\left(\frac{n}{\la}\right)^{1/2} \left(\frac{C}{\lambda_1(M)}+2\right)-2\left(\frac{n}{\la}\right)^{1/2}
\\
&=\left(\frac{n}{\la}\right)^{1/2}\frac{C}{\lambda_1(M)}\leq\frac{d_n^{3/2}}{n}C\end{align*}
\ndt This concludes the proof of the Lemma. \end{proof}
\saut
\ndt Now we give an $L^2$-upper bound of $\varphi$
\saut
\begin{lem}\label{norml2} Let $p\geq 2$ and $C\leq c_n$. If we have the pinching condition $(P_C)$, then
$$\|\varphi\|_2\leq D(n)\|\varphi\|_{\infty}^{3/4}C^{1/4}$$
\end{lem}
\saut
\begin{proof} We have
$$\|\varphi\|_2=\left(\insm\varphi^{3/2}\varphi^{1/2}\vol\right)^{1/2}\leq\|\varphi\|_{\infty}^{3/4}\|\varphi^{1/2}\|_1^{1/2}$$
\ndt and noting that
$$|X|\left(|X|-\left(\frac{n}{\la}\right)^{1/2}\right)^2=\left|
|X|^{1/2}X-\left(\frac{n}{\la}\right)^{1/2}\frac{X}{|X|^{1/2}}\right|^2$$
\ndt we get
\begin{align}\label{ineq}\insm\varphi^{1/2}\vol&=\left\||X|^{1/2}X-\left(\frac{n}{\la}\right)^{1/2}\frac{X}{|X|^{1/2}}\right\|_1\notag\\
&=\left\|-\frac{|X|^{1/2}}{\lambda_1(M)}Y+\frac{n}{\la}|X|^{1/2}H\nu-\left(\frac{n}{\la}\right)^{1/2}\frac{X}{|X|^{1/2}}\right\|_1\notag\displaybreak[2]\\
&\leq\left\|\frac{|X|^{1/2}}{\lambda_1(M)}Y\right\|_1+\left(\frac{n}{\la}\right)^{1/2}\left\|Z\right\|_1\end{align}
\ndt From Lemmas \ref{Y} and \ref{pratic} we get
\begin{align*}\left\|\frac{|X|^{1/2}}{\lambda_1(M)}Y\right\|_1&\leq\frac{1}{\lambda_1(M)}\left(\insm
|X|\vol\right)^{1/2}\|Y\|_2\\
&\leq \frac{1}{\lambda_1(M)}\left(\insm
|X|^2\vol\right)^{1/4}\|Y\|_2\leq\frac{d_n^{3/4}}{n^{1/2}}C^{1/2}\end{align*}
\ndt Moreover, using Lemmas \ref{Z} and \ref{pratic} again it is easy to see that the last term of (\ref{ineq}) is bounded
by $d_n^{1/2}B(n)^{1/2}C^{1/2}$. Then $\|\varphi^{1/2}\|_1^{1/2}\leq D(n)C^{1/4}$.
\end{proof}
\section{Proof of Theorem \ref{maintheo}}
\ \
The proof of Theorem \ref{maintheo} is immediate from the two following
technical Lemmas which we state below.
\begin{lem}\label{normi} For $p\geq 2$ and for any $\eta>0$, there exists
$K_{\eta}(n, \|H\|_{\infty})\leq c_n$ so that if $(P_{K_{\eta}})$ is true, then
$\|\varphi\|_{\infty}\leq\eta$. Moreover, $K_{\eta}\rightarrow 0$ when
$\|H\|_{\infty}\rightarrow\infty$ or $\eta\rightarrow 0$.
\end{lem}
\saut
\ndt and
\begin{lem}\label{contactpoint} Let $x_0$ be a point of the sphere $S(O,R)$
of $\mathbb{R}^{n+1}$ with the center at the origin and of radius $R$. Assume that $x_0=Re$ where $e\in\mathbb{S}^n$. Now let $(M^n,g)$ be a compact oriented
$n$-dimensional Riemannian manifold without boundary isometrically
immersed by $\phi$ in $\mathbb{R}^{n+1}$ so that
$\phi(M)\subset\left(B(O,R+\eta)\backslash B(O,R-\eta)\right)\backslash
B(x_0,\rho)$ with $\rho=4(2n-1)\eta$ and suppose that there exists
a point $p\in M$ so that $\scal{X}{e}>0$. Then there exists $y_0\in M$
so that the mean curvature $H(y_0)$ at $y_0$ satisfies $|H(y_0)|\geq \frac{1}{4n\eta}$.
\end{lem}
Now, let us see how to use these Lemmas to prove Theorem \ref{maintheo}.
\saut
\begin{proofmaintheo} Let $\varepsilon>0$ and let us consider the function
$f(t)=t\left(t-\left(\frac{n}{\la}\right)^{1/2}\right)^2$. Let us put
\begin{align*}\eta(\varepsilon)&=\min\left(\left(\frac{1}{\|H\|_{\infty}}-\varepsilon\right)\varepsilon^2,\left(\frac{1}{\|H\|_{\infty}}+\varepsilon\right)\varepsilon^2,\frac{1}{27\|H\|_{\infty}^3}\right)\\
&\leq\min\left(f\left(\left(\frac{n}{\la}\right)^{1/2}-\varepsilon\right),f\left(\left(\frac{n}{\la}\right)^{1/2}+\varepsilon\right),\frac{1}{27\|H\|_{\infty}^3}\right)\end{align*}
\ndt Then, as $\eta(\varepsilon)>0$ and from Lemma \ref{normi}, it follows that if the pinching condition
$(P_{K_{\eta(\varepsilon)}})$ is satisfied with $K_{\eta(\varepsilon)}\leq c_n$, then for any $x\in M$, we have
\begin{align}\label{major}f(|X|)\leq\eta(\varepsilon)\end{align}
\ndt Now to prove Theorem \ref{maintheo}, it is sufficient to assume
$\varepsilon<\frac{2}{3\|H\|_{\infty}}$. Let us show that either
\begin{align}\label{etude}\left(\frac{n}{\la}\right)^{1/2}-\varepsilon\leq |X|\leq
\left(\frac{n}{\la}\right)^{1/2}+\varepsilon\;\;\;\;\text{or}\;\;\;\;
|X|<\frac{1}{3}\left(\frac{n}{\la}\right)^{1/2}\end{align}
\ndt By studying the function $f$, it is easy to see that $f$ has a
unique local maximum in $\frac{1}{3}\left(\frac{n}{\la}\right)^{1/2}$ and from the
definition of $\eta(\varepsilon)$ we have $\eta(\varepsilon)<\frac{4}{27}\frac{1}{\|H\|_{\infty}^3}\leq\frac{4}{27}\left(\frac{n}{\la}\right)^{3/2}=f\left(\frac{1}{3}\left(\frac{n}{\la}\right)^{1/2}\right)$.
Now since $\varepsilon<\frac{2}{3\|H\|_{\infty}}$, we have
$\varepsilon<\frac{2}{3}\left(\frac{n}{\la}\right)^{1/2}$, and
$\frac{1}{3}\left(\frac{n}{\la}\right)^{1/2}<\left(\frac{n}{\la}\right)^{1/2}-\varepsilon$.
This and (\ref{major}) yield (\ref{etude}).
Now, from Lemma
\ref{vectpos} we deduce that there exists a point $y_0\in M$ so that
$|X(y_0)|\geq\frac{n^{1/2}\lambda_1(M)^{1/2}}{(K_{\eta(\varepsilon)}+\lambda_1(M))}$ and since
$K_{\eta(\varepsilon)}\leq c_n=\frac{n}{d_n}\leq \lambda_1(M)\leq 2\lambda_1(M)$ (see the
proof of the Lemma \ref{pratic}), we obtain $|X(y_0)|\geq\frac{1}{3}\left(\frac{n}{\la}\right)^{1/2}$.
\ndt By the connectedness of $M$, it follows that $\left(\frac{n}{\la}\right)^{1/2}-\varepsilon\leq|X|\leq
\left(\frac{n}{\la}\right)^{1/2}+\varepsilon$ for any point of $M$ and
Assertion 1 of Theorem \ref{maintheo} is shown for the condition $(P_{K_{\eta(\varepsilon)}})$.
In order to prove the second assertion, let us consider the pinching condition
$(P_{C_{\varepsilon}})$ with
$C_{\varepsilon}=K_{\eta\left(\frac{\varepsilon}{4(2n-1)}\right)}$. Then Assertion 1
is still valid. Let $x=\left(\frac{n}{\la}\right)^{1/2}e\in
S\left(O,\sqrt{\frac{n}{\la}}\right)$, with $e\in\mathbb{S}^n$ and suppose that
$B(x,\varepsilon)\cap M=\O$. Since $\insm X_i\vol=0$ for any
$i\leq n+1$, there exists a point $p\in M$ so that $\scal{X}{e}>0$ and we
can apply Lemma \ref{contactpoint}. Therefore there is a point $y_0\in
M$ so that $H(y_0)\geq\frac{2n-1}{n\varepsilon}>\|H\|_{\infty}$ since we have
assumed $\varepsilon<\frac{2}{3\|H\|_{\infty}}\leq\frac{2n-1}{2n\|H\|_{\infty}}$. Then we
obtain a contradiction which implies $B(x,\varepsilon)\cap M\neq\O$ and
Assertion 2 is satisfied. Furthermore,
$C_{\varepsilon}\rightarrow 0$ when $\|H\|_{\infty}\rightarrow\infty$ or
$\varepsilon\rightarrow 0$. \end{proofmaintheo}
\section{Proof of Theorem \ref{diffeo}}
\ \
From Theorem \ref{maintheo}, we know that for any $\varepsilon>0$, there
exists $C_{\varepsilon}$ depending only on $n$ and $\|H\|_{\infty}$ so that if $(P_{C_{\varepsilon}})$ is true then
$$\left| |X|_x-\sqrt{\crit}\right|\leq\varepsilon$$
\ndt for any $x\in M$. Now, since $\sqrt{n}\|H\|_{\infty}\leq\|B\|_{\infty}$, it is easy to see from the previous
proofs that we can assume that $C_{\varepsilon}$ is depending only on $n$ and
$\|B\|_{\infty}$.
The proof of Theorem \ref{diffeo} is a consequence of the following
Lemma on the $L_{\infty}$-norm of $\psi=|X^{T}|$
\begin{lem}\label{normpsi} For $p\geq 2$ and for any $\eta>0$, there exists
$K_{\eta}(n,\|B\|_{\infty})$ so that if $(P_{K_{\eta}})$ is true, then
$\|\psi\|_{\infty}\leq\eta$. Moreover, $K_{\eta}\rightarrow 0$ when
$\|B\|_{\infty}\rightarrow\infty$ or $\eta\rightarrow 0$.
\end{lem}
\ndt This Lemma will be proved in the Section 5.
\saut
\begin{prooftheodiffeo} Let
$\varepsilon<\frac{1}{2}\sqrt{\frac{n}{\|B\|_{\infty}}}\leq\sqrt{\crit}$. From the choice of
$\varepsilon$, we deduce that the condition $(P_{C_{\varepsilon}})$ implies that
$|X_x|$ is nonzero for any $x\in M$ (see the proof of Theorem \ref{maintheo}) and we can consider the differential application
\begin{center}\begin{tabular}{ccccc }$F$ & $:$ & $M$ & $\longrightarrow$ &
$S\left(O,\sqrt{\crit}\right)$\\
\ \ &\ \ & $x$ & $\longmapsto$ &
$\sqrt{\frac{n}{\la}}\frac{X_x}{|X_x|}$\end{tabular}\end{center}
\ndt We will prove that $F$ is a quasi isometry. Indeed, for any
$0<\theta<1$, we can choose a
constant $\varepsilon(n,\|B\|_{\infty}, \theta)$ so that for any $x\in M$ and any unit
vector $u\in T_x M$, the pinching condition $(P_{C_{\varepsilon(n,\|B\|_{\infty},\theta)}})$ implies
$$\left||dF_x(u)|^2-1\right|\leq\theta$$
\ndt For this, let us compute $dF_x(u)$. We have
\begin{align*}dF_x(u)&=\sqrt{\crit}\nabla^0_u\left(\frac{X}{|X|}\right)\Big{|}_x=\sqrt{\crit} u\left(\frac{1}{|X|}\right)X+\sqrt{\crit}\frac{1}{|X|}\nabla^0_u X\displaybreak[2]\\
&=-\frac{1}{2}\sqrt{\crit}\frac{1}{|X|^3}u(|X|^2)X+\sqrt{\crit}\frac{1}{|X|}u\\
&=-\sqrt{\crit}\frac{1}{|X|^3}\scal{u}{X}X+\sqrt{\crit}\frac{1}{|X|}u\displaybreak[2]\\
&=\sqrt{\crit}\frac{1}{|X|}\left(-\frac{\scal{u}{X}}{|X|^2}X+u\right)\end{align*}
\ndt By a straightforward computation, we obtain
\begin{align}\label{quasisom}\left||dF_x(u)|^2-1\right|&=\left|\frac{n}{\la}\frac{1}{|X|^2}\left(1-\frac{\scal{u}{X}^2}{|X|^2}\right)-1\right|\notag\\
&\leq\left|\frac{n}{\la}\frac{1}{|X|^2}-1\right|+\frac{n}{\la}\frac{1}{|X|^4}\scal{u}{X}^2\end{align}
\ndt Now
\begin{align*}\left|\frac{n}{\la}\frac{1}{|X|^2}-1\right|=\frac{1}{|X|^2}\left|\frac{n}{\la}-|X|^2\right|&\leq\varepsilon\frac{\left|\sqrt{\crit}+|X|\right|}{|X|^2}\leq\varepsilon\frac{2\sqrt{\crit}+\varepsilon}{\left(\sqrt{\crit}-\varepsilon\right)^2}\end{align*}
\ndt Let us recall that $\frac{n}{d_n}\leq \lambda_1(M)\leq \|B\|_{\infty}^2$ (see (\ref{titi}) for the first
inequality). Since we assume $\varepsilon<\frac{1}{2}\sqrt{\frac{n}{\|B\|_{\infty}}}$, the right-hand side is bounded above by a constant depending only
on $n$ and $\|B\|_{\infty}$ and we have
\begin{align}\label{pinc1}\left|\frac{n}{\la}\frac{1}{|X|^2}-1\right|\leq\varepsilon\gamma(n,\|B\|_{\infty})\end{align}
\ndt On the other hand, since $C_{\varepsilon}(n,\|B\|_{\infty})\rightarrow 0$ when
$\varepsilon\rightarrow 0$, there exists $\varepsilon(n,\|B\|_{\infty},\eta)$ so that
$C_{\varepsilon_{(n,\|B\|_{\infty},\eta)}}\leq K_{\eta}(n,\|B\|_{\infty})$ (where $K_{\eta}$ is
the constant of the Lemma) and then by Lemma
\ref{normpsi}, $\|\psi\|_{\infty}^2\leq\eta^2$.
Thus, there exists a constant $\delta$ depending only on $n$ and
$\|B\|_{\infty}$ so that
\begin{align}\label{pinc2}\frac{n}{\la}\frac{1}{|X|^4}\scal{u}{X}^2\leq\frac{n}{\la}\frac{1}{|X|^4}\|\psi\|_{\infty}^2\leq\eta^2\delta(n,\|B\|_{\infty})\end{align}
\ndt then from (\ref{quasisom}), (\ref{pinc1}) and (\ref{pinc2}) we deduce that the
condition $(P_{C_{\varepsilon(n,\|B\|_{\infty},\eta)}})$ implies
$$\left||dF_x(u)|^2-1\right|\leq\varepsilon\gamma(n,\|B\|_{\infty})+\eta^2\delta(n,\|B\|_{\infty})$$
\ndt Now let us choose
$\eta=\left(\frac{\theta}{2\delta}\right)^{1/2}$. Then we can assume that
$\varepsilon(n,\|B\|_{\infty},\eta)$ is small enough in order to have
$\varepsilon(n,\|B\|_{\infty},\eta)\gamma(n\|B\|_{\infty})\leq\frac{\theta}{2}$. In this case we have
$$\left||dF_x(u)|^2-1\right|\leq \theta$$
\ndt Now let us fix $\theta$, $0<\theta<1$. It follows that $F$ is a local
diffeomorphism from $M$ to $S\left(O,\sqrt{\crit}\right)$. Since
$S\left(O,\sqrt{\crit}\right)$ is simply connected for $n\geq 2$, $F$ is a diffeomorphism. \end{prooftheodiffeo}
\section{Proof of the technical Lemmas }
\ \
The proofs of Lemmas \ref{normi} and \ref{normpsi} are providing from a
result stated in the following Proposition using a Nirenberg-Moser type of proof.
\begin{pro}\label{nirmos} Let $(M^n,g)$ be a compact, connected and oriented $n$-dimensional Riemannian
ma\-nifold without boundary isometrically immersed into the
$n+1$-dimensional euclidean space $(\mathbb{R}^{n+1}, can)$. Let $\xi$ be a
nonnegative continuous function so that $\xi^k$ is smooth for $k\geq
2$. Let $0\leq r<s\leq 2$ so that
$$\frac{1}{2}\Delta\xi^2
\xi^{2k-2}\leq\delta\omega+(A_1+kA_2)\xi^{2k-r}+(B_1+kB_2)\xi^{2k-s}$$
\ndt where $\delta\omega$ is the codifferential of a $1$-form and $A_1,A_2,B_1,B_2$ are nonnegative constants. Then for any
$\eta>0$, there exists a constant $L(n,A_1,A_2,B_1,B_2,\|H\|_{\infty},\eta)$ depending only on $n$,
$A_1$, $A_2$, $B_1$, $B_2$, $\|H\|_{\infty}$ and $\eta$ so that if $\|\xi\|_{\infty}>\eta$ then
$$\|\xi\|_{\infty}\leq L(n,A_1, A_2, B_1, B_2, \|H\|_{\infty},\eta)\|\xi\|_2$$
\ndt Moreover, $L$ is bounded when $\eta\rightarrow\infty$, and if $B_1>0$, $L\rightarrow\infty$ when $\|H\|_{\infty}\rightarrow\infty$ or
$\eta\rightarrow 0$.
\end{pro}
\saut
This Proposition will be proved at the end of the paper.
\saut
Before giving the proofs of Lemmas \ref{normi} and \ref{normpsi}, we
will show that under the pinching condition $(P_C)$ with $C$ small enough,
the $L_{\infty}$-norm of $X$ is bounded by a constant depending only on $n$ and $\|H\|_{\infty}$.
\begin{lem}\label{ninfx} If we have the pinching condition $(P_C)$ with
$C<c_n$, then there exists $E(n,\|H\|_{\infty})$ depending only on $n$ and $\|H\|_{\infty}$ so
that $\|X\|_{\infty}\leq E(n,\|H\|_{\infty})$.
\end{lem}
\begin{proof} From the relation (\ref{hsiung}), we have
$$\frac{1}{2}\Delta |X|^2 |X|^{2k-2}\leq n\|H\|_{\infty} |X|^{2k-1}$$
\ndt Then applying Proposition \ref{nirmos} to the function $\xi=|X|$
with $r=0$ and $s=1$, we obtain that if $\|X\|_{\infty} > E$, then there
exists a constant $L(n,\|H\|_{\infty},E)$ depending only on $n$, $\|H\|_{\infty}$ and $E$
so that
$$\|X\|_{\infty}\leq L(n,\|H\|_{\infty},E)\|X\|_2$$
\ndt and under the pinching condition $(P_C)$ with $C<c_n$ we have from
Lemma \ref{vectpos}
$$\|X\|_{\infty}\leq L(n,\|H\|_{\infty},E)d_n^{1/2}$$
\ndt Now since $L$ is bounded when
$E\rightarrow\infty$, we can choose $E=E(n,\|H\|_{\infty})$ great enough so that
$$L(n,\|H\|_{\infty},E)d_n^{1/2}<E$$
\ndt In this case, we have $\|X\|_{\infty}\leq E(n,\|H\|_{\infty})$.
\end{proof}
\saut
\begin{proofnormi} First we compute the Laplacian of the square of
$\varphi^2$. We have
\begin{align*}
\Delta\varphi^2&=\Delta\left(|X|^4-2\left(\frac{n}{\la}\right)^{1/2}|X|^3+\frac{n}{\la}
|X|^2\right)\\
&=-2|X|^2|d|X|^2|^2+2|X|^2\Delta|X|^2\\
&-2\left(\frac{n}{\la}\right)^{1/2}\left(-\frac{3}{4}|X|^{-1}|d|X|^2|^2+\frac{3}{2}|X|\Delta|X|^2\right)+\frac{n}{\la}\Delta|X|^2\end{align*}
\ndt Now by a direct computation one gets $|d|X|^2|^2\leq 4|X|^2$. Moreover
by the relation (\ref{hsiung}) we have $|\Delta|X|^2|\leq 2n\|H\|_{\infty} |X|+n$. Then
applying Lemmas \ref{pratic} and \ref{ninfx} we get
$$\Delta\varphi^2\leq \alpha(n,\|H\|_{\infty})$$
\ndt and
$$\frac{1}{2}\Delta\varphi^2\varphi^{2k-2}\leq
\alpha(n,\|H\|_{\infty})\varphi^{2k-2}$$
\ndt Now, we apply Proposition \ref{nirmos} with $r=0$ and $s=2$. Then
if $\|\varphi\|_{\infty}>\eta$, there exists a constant $L(n,\|H\|_{\infty})$ depending only on
$n$ and $\|H\|_{\infty}$ so that
$$\|\varphi\|_{\infty}\leq L\|\varphi\|_2$$
\ndt From Lemma \ref{norml2}, if $C\leq c_n$ and
$(P_C)$ is true, we have $\|\varphi\|_2\leq D(n)\|\varphi\|_{\infty}^{3/4}C^{1/4}$. Therefore
$$\|\varphi\|_{\infty}\leq (L D)^4C$$
\ndt Consequently, if we choose $C=K_{\eta}=\inf\left(\frac{\eta}{(LD)^4},c_n\right)$, then we obtain that $\|\varphi\|_{\infty}\leq\eta$.
\end{proofnormi}
\saut
\begin{proofnormpsi} First we will prove that for any $C<c_n$, if $(P_C)$ is
true, then
\begin{align}\label{deltapsi}\frac{1}{2}(\Delta\psi^2)\psi^{2k-2}\leq\delta\omega+\left( \alpha_1 (n,\|B\|_{\infty})+k\alpha_2 (n,\|B\|_{\infty}) \right) \psi^{2k-2}\end{align}
\ndt where $\delta\omega$ is the codifferential of a $1$-form $\omega$.
First observe that the gradient $\nabla^M |X|^2$ of $|X|^2$ satisfies $\nabla^M |X|^2=2X^{T}$. Then by the Bochner formula we get
\begin{align*}\frac{1}{2}\Delta|X^{T}|^2&=\frac{1}{4}\scal{\Delta d|X|^2}{d|X|^2}-\frac{1}{4}|\nabla d|X|^2|^2-\frac{1}{4}\text{Ric\ } (\nabla^M |X|^2,\nabla^M |X|^2)\notag\\
&\leq \frac{1}{4}\scal{d\Delta |X|^2}{d|X|^2}-\frac{1}{4}\text{Ric\ } (\nabla^M |X|^2,\nabla^M |X|^2)\end{align*}
\ndt and by the Gauss formula we obtain
\begin{align*}\frac{1}{2}\Delta|X^{T}|^2&\leq \frac{1}{4}\scal{d\Delta |X|^2}{d|X|^2}-\frac{1}{4}nH\scal{B\nabla^M |X|^2}{\nabla^M |X|^2}+\frac{1}{4}|B\nabla^M |X|^2|^2\\
&=\frac{1}{4}\scal{d\Delta |X|^2}{d|X|^2}-nH\scal{BX^{T}}{X^{T}}+|BX^{T}|^2
\end{align*}
\ndt By Lemma \ref{ninfx} we know that $\|X\|_{\infty}\leq E(n,\|B\|_{\infty})$ (the
dependance in $\|H\|_{\infty}$ can be replaced by $\|B\|_{\infty}$). Then it follows that
\begin{align}\label{estipsi}\frac{1}{2}(\Delta\psi^2)\psi^{2k-2}\leq \frac{1}{4}\scal{d\Delta |X|^2}{d|X|^2}\psi^{2k-2}+\alpha'(n,\|B\|_{\infty})\psi^{2k-2}\end{align}
\ndt Now, let us compute the term $\scal{d\Delta |X|^2}{d|X|^2}\psi^{2k-2}$. We
have
\begin{align*}\scal{d\Delta |X|^2}{d|X|^2}\psi^{2k-2}&=\delta\omega+(\Delta |X|^2)^2\psi^{2k-2}-(2k-2)\Delta |X|^2\scal{d|X|^2}{d\psi}\psi^{2k-3}\\
&=\delta\omega+(\Delta |X|^2)^2\psi^{2k-2}-2(2k-2)\Delta |X|^2\scal{X^{T}}{\nabla^M \psi}\psi^{2k-3}\end{align*}
\ndt where $\omega=-\Delta |X|^2 \psi^{2k-2} d|X|^2$. Now,
$$\ei(\psi)=\frac{\ei|X^{T}|^2}{2|X^{T}|}=\frac{\ei|X|^2-\ei\scal{X}{\nu}^2}{2|X^{T}|}=\frac{\scal{\ei}{X}-B_{ij}\scal{X}{e_j}\scal{X}{\nu}}{|X^{T}|}$$
\ndt Then
\begin{align*}\scal{d\Delta |X|^2}{d|X|^2}\psi^{2k-2}&=\delta\omega+(\Delta |X|^2)^2\psi^{2k-2}-2(2k-2)\Delta |X|^2 |X^{T}|\psi^{2k-3}\\
&\hspace{4cm}+2(2k-2)\Delta |X|^2\frac{\scal{BX^{T}}{X^{T}}}{|X^{T}|}\scal{X}{\nu}\psi^{2k-3}\\
&\leq \delta\omega+(\Delta |X|^2)^2\psi^{2k-2}+2(2k-2)|\Delta |X|^2|\psi^{2k-2}\\
&\hspace{4cm}+2(2k-2)|\Delta |X|^2||B||X|\psi^{2k-2}\end{align*}
\ndt Now by relation (\ref{hsiung}) and Lemma \ref{ninfx} we have
$$\scal{d\Delta |X|^2}{d|X|^2}\psi^{2k-2}\leq \delta\omega+\left( \alpha_1''(n,\|B\|_{\infty})+k\alpha_2''(n,\|B\|_{\infty})\right)\psi^{2k-2}$$
\ndt Inserting this in (\ref{estipsi}), we obtain the desired
inequality (\ref{deltapsi}).
Now applying again Proposition \ref{nirmos}, we get that there exists
$L(n,\|B\|_{\infty},\eta)$ so that if $\|\psi\|_{\infty}>\eta$ then
$$\|\psi\|_{\infty}\leq L \|\psi\|_2$$
\ndt From the Lemma \ref{l2xt} we deduce that if the pinching condition
$(P_C)$ holds then $\|\psi\|_2\leq A(n)^{1/2}C^{1/2}$. Then taking
$C=K_{\eta}=\inf\left(\frac{\eta}{L A^{1/2}},c_n\right)$, then
$\|\psi\|_{\infty}\leq\eta$.
\end{proofnormpsi}
\saut
\begin{proofcontactpoint} The idea of the proof consists in
foliating the region $B(O,R+\eta) \backslash B(O,R-\eta)$ with
hypersurfaces of large mean curvature and to show that one of these
hypersurfaces is tangent to $\phi(M)$. This will imply that $\phi(M)$
has a large mean curvature at the contact point.
Consider $\mathbb{S}^{n-1}\subset\mathbb{R}^n$ and
$\mathbb{R}^{n+1}=\mathbb{R}^n\times\mathbb{R} e$. Let $a$, $L>l>0$ and
\begin{center}\begin{tabular}{ccccc}$\Phi_{L,l,a}$ & $:$ &
$\mathbb{S}^{n-1}\times\mathbb{S}^1$ & $\longrightarrow$ & $\mathbb{R}^{n+1}$\\
\ \ &\ \ & $(\xi,\theta)$ & $\longmapsto$ &
$L\xi-l\cos\theta\xi+l\sin\theta e+ae$\end{tabular}\end{center}
\ndt Then $\Phi_{L,l,a}$ is a family of embeddings from
$\mathbb{S}^{n-1}\times\mathbb{S}^1$ in $\mathbb{R}^{n+1}$. If we orient the family
of hypersurfaces $\Phi_{L,l,a}(\mathbb{S}^{n-1}\times\mathbb{S}^1)$ by the unit outward
normal vector field, a straightforward computation shows
that the mean curvature $H(\theta)$ depends only on $\theta$ and we
have
\begin{align}\label{meancurv}H(\theta)=\frac{1}{n}\left(\frac{1}{l}-\frac{(n-1)\cos\theta}{L-l\cos\theta}\right)\geq\frac{1}{n}\left(\frac{1}{l}-\frac{n-1}{L-l}\right)\end{align}
\ndt Now, let us consider the hypotheses of the Lemma and for
$t_0=2\arcsin\left(\frac{\rho}{2R}\right)\leq t\leq\frac{\pi}{2}$, put
$L=R\sin t$, $l=2\eta$ and $a=R\cos t$. Then $L>l$ and we can consider for $t_0\leq
t\leq\frac{\pi}{2}$ the family ${\cal M}_{R,\eta, t}$ of hypersurfaces
defined by ${\cal M}_{R,\eta, t}=\Phi_{R\sin t, 2\eta, R\cos
t}(\mathbb{S}^{n-1}\times\mathbb{S}^1)$.
From the relation (\ref{meancurv}), the mean curvature $H_{R,\eta, t}$ of
${\cal M}_{R,\eta, t}$ satisfies
\begin{align*}H_{R,\eta,
t}&\geq\frac{1}{n}\left(\frac{1}{2\eta}-\frac{n-1}{R\sin t
-2\eta}\right)\geq\frac{1}{n}\left(\frac{1}{2\eta}-\frac{n-1}{R\sin t_0
-2\eta}\right)\displaybreak[2]\\
&\geq\frac{1}{n}\left(\frac{1}{2\eta}-\frac{n-1}{R\sin(t_0/2)
-2\eta}\right)=\frac{1}{n}\left(\frac{1}{2\eta}-\frac{n-1}{\frac{\rho}{2}
-2\eta}\right)=\frac{1}{4n\eta}\displaybreak[2]\end{align*}
\ndt where we have used in this last equality the fact that
$\rho=4(2n-1)\eta$.
Since there exists a point $p\in M$ so that $\scal{X(p)}{e}>0$, we can find
$t\in [t_0, \pi/2]$ and a point $y_0\in M$ which is a contact point with
${\cal M}_{R,\eta, t}$. Therefore $|H(y_0)|\geq \frac{1}{4n\eta}$.
\includegraphics[height=10cm]{dessin}
\vspace{-8,9cm}
\hspace{0,5cm} {\scriptsize ${\cal M}_{R,\eta, t_0}\cap F$ } \hspace{5,8cm}
{\scriptsize $\rho$} \hspace{3cm} {\scriptsize ${\cal M}_{R,\eta, t_0}\cap F$ }
\vspace{0,7cm}
\hspace{7cm}{\scriptsize $x_0$}
\vspace{0,18cm}
\hspace{7,5cm}{\scriptsize $\eta$}
\vspace{0,2cm}
\hspace{0,5cm} {\scriptsize ${\cal M}_{R,\eta, t}\cap F$ } \hspace{2cm}
{\scriptsize $2\eta$} \hspace{8cm} {\scriptsize ${\cal M}_{R,\eta, t}\cap F$ }
\vspace{0,5cm}
{\scriptsize $M\cap F$}
\vspace{-0,4cm}
\hspace{2,7cm} {\scriptsize $y_0$}
\vspace{0,76cm}
\hspace{6,3cm} {\scriptsize $t_0$}
\vspace{-0,3cm}
\hspace{2,7cm} {\scriptsize $2\eta$}
\vspace{0,23cm}
\hspace{4,1cm} {\scriptsize $R$} \hspace{2,8cm} {\scriptsize $e$}
\vspace{0,2cm}
\hspace{6,1cm} {\scriptsize $t$}
\vspace{0,05cm}
\hspace{8,4cm} {\scriptsize $\xi$}
\vspace{0,1cm}
\hspace{7cm} {\scriptsize $O$}
{\scriptsize $F$ is the vector space spanned by $e$ and $\xi$}
\end{proofcontactpoint}
\saut
\saut
\begin{proofnirmos} Integrating by parts we have
\begin{align*}\insm\frac{1}{2}\Delta\xi^2\xi^{2k-2}\vol&=\frac{1}{2}\insm\scal{d\xi^2}{d\xi^{2k-2}}\vol=2\left(\frac{k-1}{k^2}\right)\insm|d\xi^k|^2\vol\\
&\leq (A_1+kA_2)\insm\xi^{2k-r}\vol+(B_1+kB_2)\insm\xi^{2k-s}\vol\end{align*}
\ndt Now, given a smooth function $f$ and applying the Sobolev inequality
(\ref{sobol}) to $f^2$, we get
\begin{align*}\left(\insm f^{\frac{2n}{n-1}}\vol\right)^{1-(1/n)}&\leq K(n)\insm\left(2|f||df|+|H|f^2\right)\vol\\
&\leq 2K(n)\left(\insm f^2\vol\right)^{1/2}\left(\insm|df|^2\vol\right)^{1/2}+K(n)\|H\|_{\infty}\insm f^2\vol\displaybreak[2]\\
&=K(n)\left(\insm f^2\vol\right)^{1/2}\left(2\left(\insm|df|^2\vol\right)^{1/2}+\|H\|_{\infty}\left(\insm f^2\vol\right)^{1/2}\right)\end{align*}
\ndt where in the second inequality, we have used the H\"older
inequality. Using it again, by assuming that $V(M)=1$, we have
$$\left(\insm f^2\vol\right)^{1/2}\leq\left(\insm f^{\frac{2n}{n-1}}\vol\right)^{\frac{n-1}{2n}}$$
\ndt And finally, we obtain
$$\|f\|_{\frac{2n}{n-1}}\leq K(n)\left(2\|df\|_2+\|H\|_{\infty}\|f\|_2\right)$$
\ndt For $k\geq 2$, $\xi^k$ is smooth and we apply the above inequality
to $f=\xi^k$. Then we get
\begin{align*}\|\xi\|_{\frac{2kn}{n-1}}^k&\leq K(n)\left[2\left(\insm|d\xi^k|^2\vol\right)^{1/2}+\|H\|_{\infty}\left(\insm\xi^{2k}\vol\right)^{1/2}\right]\\
&\leq
K(n)\left[2\left(\frac{k^2}{2(k-1)}\right)^{1/2}\left((A_1+kA_2)\insm\xi^{2k-r}\vol+(B_1+kB_2)\insm\xi^{2k-s}\vol\right)^{1/2}\right.\displaybreak[2]\\
&\left.\hspace{9cm}+\|H\|_{\infty}\left(\insm\xi^{2k}\vol\right)^{1/2}\right]\\
&\leq K(n)\left[2\left(\frac{k^2}{2(k-1)}\right)^{1/2}
\left((A_1+kA_2)\|\xi\|_{\infty}^{2-r}+(B_1+kB_2)\|\xi\|_{\infty}^{2-s}\right)^{1/2}\|\xi\|_{2k-2}^{k-1}\right.
\\
&\hspace{9cm}\Biggl.+\|H\|_{\infty}\|\xi\|_{\infty}\|\xi\|_{2k-2}^{k-1}\ \ \Biggr]\displaybreak[2]\\
&\leq
K(n)\left[2\left(\frac{k^2}{2(k-1)}\right)^{1/2}\left(\frac{A_1+kA_2}{\|\xi\|_{\infty}^r}+\frac{B_1+kB_2}{\|\xi\|_{\infty}^s}\right)^{1/2}\right.
\Biggl.+\|H\|_{\infty}\Biggr]\|\xi\|_{\infty}\|\xi\|_{2k-2}^{k-1}\displaybreak[2]\\
&\leq K(n)\left[2\left(\frac{k^2}{2(k-1)}\right)^{1/2}
\left(\frac{A_1^{1/2}+k^{1/2}A_2^{1/2}}{\|\xi\|_{\infty}^{r/2}}+\frac{B_1^{1/2}+k^{1/2}B_2^{1/2}}{\|\xi\|_{\infty}^{s/2}}\right)\right.\\
&\hspace{9cm}\Biggl.+\|H\|_{\infty}\Biggr]\|\xi\|_{\infty}\|\xi\|_{2k-2}^{k-1}
\end{align*}
\ndt Now if we assume that $\|\xi\|_{\infty}>\eta$, the last inequality becomes
\begin{align*}\|\xi\|_{\frac{2kn}{n-1}}^{k}&\leq
K(n)\left[2\left(\frac{k^2}{2(k-1)}\right)^{1/2}\left(\frac{A_1^{1/2}+k^{1/2}A_2^{1/2}}{\eta^{r/2}}+\frac{B_1^{1/2}+k^{1/2}B_2^{1/2}}{\eta^{s/2}}\right)\right.\\
&\Biggl.\hspace{9cm}+\|H\|_{\infty}\Biggr]\|\xi\|_{\infty}\|\xi\|_{2k-2}^{k-1}\displaybreak[2]\\
&=\left[(K_1+k^{1/2}K_2)\left(\frac{k^2}{k-1}\right)^{1/2}+K'\right]\|\xi\|_{\infty}\|\xi\|_{2k-2}^{k-1}\end{align*}
\ndt Now let $q=\frac{n}{n-1}>1$ and for $i\geq 0$ let $k=q^i+1\geq 2$ . Then
\begin{align*}\|\xi\|_{2(q^{i+1}+q)}&\leq\left(\left(K_1+(q^i+1)^{1/2}K_2\right)\left(\frac{q^i+1}{q^{i/2}}\right)+K''\right)^{\frac{1}{q^i+1}}\|\xi\|_{\infty}^{\frac{1}{q^i+1}}\|\xi\|_{2q^i}^{1-\frac{1}{q^i+1}}\\
&\leq\left(\tilde{K} q^{i}\right)^{\frac{1}{q^i+1}}\|\xi\|_{\infty}^{\frac{1}{q^i+1}}\|\xi\|_{2q^i}^{1-\frac{1}{q^i+1}}\end{align*}
\ndt where $\tilde{K}=2K_1+2^{3/2}K_2+K'$. We see that $\tilde{K}$ has a finite limit when $\eta\rightarrow\infty$ and if $B_1 >0$, $\tilde{K}\rightarrow\infty$ when
$\|H\|_{\infty}\rightarrow\infty$ or $\eta\rightarrow 0$. Moreover the H\"older inequality gives
$$\|\xi\|_{2q^{i+1}}\leq\|\xi\|_{2(q^{i+1}+q)}$$
\ndt which implies
$$\|\xi\|_{2q^{i+1}}\leq\left(\tilde{K} q^{i}\right)^{\frac{1}{q^i+1}}\|\xi\|_{\infty}^{\frac{1}{q^i+1}}\|\xi\|_{2q^i}^{1-\frac{1}{q^i+1}}$$
\ndt Now, by iterating from $0$ to $i$, we get
\begin{align*}\|\xi\|_{2q^{i+1}}&\leq \tilde{K}^{\left(1-\prod_{k=i-j}^i\left(1-\frac{1}{q^k+1}\right)\right)}q^{\sum_{k=i-j}^i\frac{k}{q^k+1}}\|\xi\|_{\infty}^{\left(1-\prod_{k=i-j}^i\left(1-\frac{1}{q^k+1}\right)\right)}\|\xi\|_{2q^{i-j}}^{\prod_{k=i-j}^i\left(1-\frac{1}{q^k+1}\right)}\\
&\leq \tilde{K}^{\left(1-\prod_{k=0}^i\left(1-\frac{1}{q^k+1}\right)\right)} q^{\sum_{k=0}^i\frac{k}{q^k+1}}\|\xi\|_{\infty}^{\left(1-\prod_{k=0}^i\left(1-\frac{1}{q^k+1}\right)\right)}\|\xi\|_{2}^{\prod_{k=0}^i\left(1-\frac{1}{q^k+1}\right)}\end{align*}
\ndt Let $\alpha=\sum_{k=0}^{\infty}\frac{k}{q^k+1}$ and $\beta=\prod_{k=0}^{\infty}\left(1-\frac{1}{q^k+1}\right)=\prod_{k=0}^{\infty}\left(\frac{1}{1+(1/q)^k}\right)$. Then
$$\|\xi\|_{\infty}\leq
\tilde{K}^{1-\beta}q^{\alpha}\|\xi\|_{\infty}^{(1-\beta)}\|\xi\|_2^{\beta}$$
\ndt and finally
$$\|\xi\|_{\infty}\leq L\|\xi\|_2$$
\ndt where $L=\tilde{K}^{\frac{1-\beta}{\beta}}q^{\alpha/\beta}$ is a constant
depending only on $n$, $A_1$, $A_2$, $B_1$, $B_2$, $\|H\|_{\infty}$ and $\eta$. From classical methods we show
that $\beta\in [e^{-n},e^{-n/2}]$. In particular, $0<\beta<1$ and we deduce
that $L$ is bounded when $\eta\rightarrow\infty$ and $L\rightarrow\infty$ when $\|H\|_{\infty}\rightarrow\infty$ or $\eta\rightarrow
0$ with $B_1>0$.
\end{proofnirmos}
\saut
\ndt {\bf Remark} In \cite{shixu1} and \cite{shixu2} Shihohama and Xu have
proved that if $(M^n,g)$ is a compact $n$-dimensional Riemannian ma\-nifold
without boundary isometrically immersed in $\mathbb{R}^{n+1}$ and if
$\insm\left(|B|^2-n|H|^2\right)<D_n$ where $D_n$ is a constant depending on
$n$, then all Betti numbers are zero. For $n=2$, $D_2=4\pi$, and it
follows that if
$$\insm|B|^2\vol-4\pi<\lambda_1(M) V(M)$$
\ndt then we deduce from the Reilly inequality $\lambda_1(M) V(M)\leq 2\insm
H^2\vol$ that $\insm(|B|^2-2|H|^2)\vol<4\pi$ and by the result of Shihohama
and Xu $M$ is diffeomorphic to $\mathbb{S}^2$.
| {
"timestamp": "2006-09-18T14:12:43",
"yymm": "0609",
"arxiv_id": "math/0609494",
"language": "en",
"url": "https://arxiv.org/abs/math/0609494",
"abstract": "In this paper, we give pinching Theorems for the first nonzero eigenvalue $\\lambda$ of the Laplacian on the compact hypersurfaces of the Euclidean space. Indeed, we prove that if the volume of $M$ is 1 then, for any $\\epsilon>0$, there exists a constant $C\\_{\\epsilon}$ depending on the dimension $n$ of $M$ and the $L\\_{\\infty}$-norm of the mean curvature $H$, so that if the $L\\_{2p}$-norm $\\|H\\|\\_{2p}$ ($p\\geq 2$) of $H$ satisfies $n\\|H\\|\\_{2p}-C\\_{\\epsilon}<\\lambda$, then the Hausdorff-distance between $M$ and a round sphere of radius $(n/\\lambda)^{1/2}$ is smaller than $\\epsilon$. Furthermore, we prove that if $C$ is a small enough constant depending on $n$ and the $L\\_{\\infty}$-norm of the second fundamental form, then the pinching condition $n\\|H\\|\\_{2p}-C<\\la$ implies that $M$ is diffeomorphic to an $n$-dimensional sphere.",
"subjects": "Differential Geometry (math.DG)",
"title": "A pinching theorem for the first eigenvalue of the laplacian on hypersurface of the euclidean space",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754474655619,
"lm_q2_score": 0.718594386544335,
"lm_q1q2_score": 0.7075103896781296
} |
https://arxiv.org/abs/1711.11399 | A note on power generalized extreme value distribution and its properties | Similar to the generalized extreme value (GEV) family, the generalized extreme value distributions under power normalization are introduced by Roudsari (1999) and Barakat et al. (2013). In this article, we study the asymptotic behavior of GEV laws under power normalization and derive expressions for the kth moments, entropy, ordering in dispersion, rare event estimation and application of real data set. We also show that, under some conditions, the Shannon entropy and variance of GEV families are ordered. | \section{Introduction}
Let $X_1, X_2,\ldots, X_n$ is a sequence of independent and identically distributed (iid) random variables (rvs) with distribution function (df) $F.$ If, for some non-degenerate df $G,$ a df $F$ belongs to the max domain of attraction of $G$ under linear normalization and it denotes by $F\in\mathcal{D}_\ell(G),$ then for some norming constants $a_n>0$ and $b_n\in\mathbb R$
\begin{equation}
\lim_{n\to\infty}\Pr \left(\bigvee_{i=1}^{n}X_i\leq a_nx+b_n\right)=\lim_{n\to\infty}F^n \left(a_nx+b_n\right)=G(x).\label{Introduction_e1}
\end{equation}
Limit df $G$ satisfying (\ref{Introduction_e1}) are the well known generalized extreme value (GEV) distribution, namely,
\begin{equation}\label{gev} G_{\tilde{\xi}}(x;\tilde{\mu},\tilde{\sigma})=\exp\left(-\left(1+\frac{\tilde{\xi}}{\tilde{\sigma}} (x-\tilde{\mu})\right)^{-1/\tilde{\xi}}_+\right),\end{equation}
where, $\tilde{\xi}\in\mathbb R/\{0\},$ $\tilde{\mu}\in\mathbb R$ and $\tilde{\sigma}>0.$ The subset of the GEV family with $\tilde{\xi}=0$ is interpreted as the limit of (\ref{gev}) as $\tilde{\xi}\to 0,$ leading to the Gumbel family with df \[G(x;\tilde{\mu},\tilde{\sigma})=\exp\left(-\exp\left(-\frac{x-\tilde{\mu}}{\tilde{\sigma}}\right)\right),\;\;x\in\mathbb R.\]
Criteria for $F\in \mathcal{D}_\ell(G)$ are described, for example, in the books of Embrechts et al. (1997) and de Haan and Ferreira (2006). Coles (2001) is good reference to the application of GEV distribution.
Pancheva (1984) studies limit laws of partial maxima of iid rvs under power normalization. Namely, $K$ is called p-max stable law and $F$ belongs to the p-max domain of attraction of $K$ under power normalization and denote it by $F\in \mathcal{D}_p(K),$ if for some $\delta_n>0,$ $\beta_n>0$
\begin{eqnarray}\label{pmax_lim}
\lim_{n\to\infty}\Pr\left(\left(\dfrac{\abs {\bigvee_{i=1}^{n}X_i}}{\delta_n}\right)^{1/\beta_n}\text{sign}\left(\bigvee_{i=1}^{n}X_i\right)\leq x\right)=K(x).
\end{eqnarray}
The limit laws $K$ satisfying (\ref{pmax_lim}) are the six p-max stable laws which we represent them in appendix \ref{pmax}.
Mohan and Ravi (1993) show that if a df $F\in \mathcal{D}_\ell (G)$ then there always exists a p-max stable law $K$ such that $F \in \mathcal{D}_p (K)$ and the converse need not be true always. They also investigate, the p-max stable laws attract more dfs to their max domains than the $\ell-$max stable laws. See also Christoph and Falk (1996) and Falk et al. (2004) for properties of dfs to belong to the p-max domain of attraction.
Roudsari (1999) demonstrates that the six p-max stable laws can be represented as two families. We call them log-GEV distribution in positive support and negative log-GEV distribution in negative support.
Suppose a positive rv $X^+$ is said to have the log-GEV with location, scale and shape parameters $\mu\in\mathbb R,$ $\sigma>0$ and $\xi\in\mathbb R /\{0\}$ if its df is given by
\begin{eqnarray}\label{df1} L_{1,\xi}(x;\mu,\sigma)=\left\lbrace
\begin{array}{l l} \exp\left(-\left(1+\frac{\xi}{\sigma}\log(e^{-\mu} x)\right)_+^{-1/\xi}\right), &\text{for, } \xi\neq 0;\\ \exp\left(-\left(xe^{-\mu}\right)^{-\frac{1}{\sigma}}_+\right), & \text{for, }\xi=0,\\
\end{array}\right.
\end{eqnarray}
where, $y_+=\max(0,y),$ and define a negative rv $X^-$ with df of negative log-GEV, if its df is given by,
\begin{eqnarray}\label{df2} L_{2,\xi}(x;\mu,\sigma)=\left\lbrace \begin{array}{ll} \exp\left(-\left(1-\frac{\xi}{\sigma}\log(e^{-\mu}\abs{x})\right)_+^{-1/\xi}\right), &\text{for } \xi\neq 0;\\ \exp\left(-(\abs{x}e^{-\mu})^{\frac{1}{\sigma}}_+\right),& \text{for }\xi= 0.\\
\end{array}\right.
\end{eqnarray}
The summarization these two families as a single one is easier to implement.
In other words, the unification of the log-GEV and negative log-GEV families into single family and it is called the power generalized extreme value (PGEV) family.
Suppose a rv $X$ is said to have the PGEV distribution with three parameters $\mu\in\mathbb R,$ $\sigma>0$ and $\xi\in\mathbb R/\{0\}$ if its df is given by
\begin{eqnarray}\label{df3}
L_{\xi}(x;\mu,\sigma)=\exp\left(-\left(1+\frac{\xi}{\sigma}\log(e^{-\mu}\abs{x})\text{sign}(x)\right)_+^{-1/\xi}\right),
\end{eqnarray}
The limit of (\ref{df3}) as $\xi\to 0,$ tending to the GEV distribution with $\tilde{\sigma}=e^{\mu}$ and $\tilde{\xi}=\sigma\,\text{sign}(x).$ The df of PGEV for $\xi=0$ is well known in (\ref{gev}).
Barakat et al. (2013) study the statistical inference about the PGEV.
In appendix \ref{fig}, illustrate the density functions and confidence interval for quantile estimator of PGEV family and gives the figures \ref{fig2} and \ref{fig3} of standardized density function of $l_{\xi}$ for different values of $\xi.$
In this article, we obtain some mathematical properties of PGEV family and discuss maximum likelihood estimation of parameters and estimate the rare event by using the Bayesian method. We also, show that the PGEV has big variance and entropy in the class of extreme value distributions. The article is outlined as follows. In section 2, we first study the asymptotic behavior of generalized extreme value distributions under power normalization and we also, derive expressions of $k$th moments, the Shannon entropy and ordering in dispersion of GEV families. Maximum likelihood estimation, Bayesian modeling and illustrates the importance of the PGEV through the analysis of real data set are addressed in section 3. We provide the some calculating, plots and tables in appendices.
Throughout the manuscript $\gamma=-\int_{0}^{\infty}\log x \,e^{-x}dx$ denotes the Euler constant with value $0.577\ldots$ and $\Gamma^{(k)}(\cdot)$ is $k$th derivative of gamma function. The inverse function of $h(\cdot)$ denoted by $h^{\leftarrow} (\cdot)$ and $\nabla_x h(x)$ is derivative of $h$ with respect to $x.$ Also, we employ the notation, $\Phi_\alpha(x)=e^{-x^{-\alpha}},$ $x>0$ is the distribution of Fr\'{e}chet and $\Psi_\alpha(x)=e^{-\abs{x}^{\alpha}},$ $x<0$ is the distribution of Weibull with parameter $\alpha$. For right extremity of $F,$ we shall denote by $r(F)=\sup\{x:F(x)<1\}\leq\infty,$ and survival function is $\bar{F}(\cdot)=1-F(\cdot).$
\section{Distribution properties}
\subsection{Limiting distributions}
Throughout we consider measurable real valued function $U:\mathbb R^+\to\mathbb R^+$ is regularly varying function with index $\rho$ if
$$\lim_{t\to\infty}\frac{U(tx)}{U(t)}=x^{\rho},\;\;\text{for }x>0.$$
We write $U\in RV_{\rho}$ and we call $\rho$ the exponent of variation.
The regular varying function plays an important role in the asymptotic analysis of various problems. It is well known, following de Haan and Ferreira (2006) that a necessary and sufficient condition for the existence of constants $a_n=a(n)$ and $b_n=\frac{1}{\bar{F}(n)}$ such that (\ref{Introduction_e1}) is equivalent
$$\lim_{t\to r(F)}\frac{\bar{F}(t+u(t)x)}{\bar{F}(t)}=(1+\xi x)_+^{-1/\xi},$$
where, $u(t)=a(1/\bar{F}(t)).$
In this section, we establish the regular variation of the dfs belongs to the p-max domain of attraction of the log-GEV and negative log-GEV laws.
The following result reveals that the upper tail behavior of $F$ might determine whether $F\in D_p(L_{i,\xi}),$ $i=1,2.$ We first state and prove a lemma of independent interest which will be used subsequently.
\begin{lem}\label{lem}
If $\mathcal{L}$ is slowly varying as represented (\ref{rep}), then
\begin{eqnarray}
\mathcal{L}(e^y)=c(e^y)\exp\Big\{\int_{e^{x_0}}^{e^y}\frac{u^*(t)}{t\log t}dt\Big\},
\end{eqnarray}
where, $c(e^t)\to c$ and $u^*(t)=\log t\,u(t)\to 0$ as $t\to\infty.$
\end{lem}
\begin{proof}
From (\ref{rep}), taking $x=e^y,$ we have
\begin{eqnarray}
\mathcal{L}(e^y)&=&c(e^y)\exp\Big\{\int_{x_0}^{y}u(e^{t'})dt'\Big\},\;\;\;(\text{ where, } t=e^{t'}).\nonumber
\end{eqnarray}
Setting, $u^*(e^{t'})=t'\; u(e^{t'})\to 0,$ for lager $t',$ then
\begin{eqnarray}
\mathcal{L}(e^y)&=&c(e^y)\exp\Big\{\int_{x_0}^{y}\frac{u^*(e^{t'})}{t'}dt'\Big\},\nonumber\\
&=&c(e^y)\exp\Big\{\int_{e^{x_0}}^{e^y}\frac{u^*(t)}{t\log t}dt\Big\},\nonumber
\end{eqnarray}
where, $t'=\log t.$
\end{proof}
Now we obtain necessary and sufficient conditions for a df $F$ belongs to the p-max domain of attraction of log-GEV and negative log-GEV stable laws. The next theorems examines the properties of regularly varying function for standardized these families.
\begin{thm}\label{thm1}
A df $F\in\mathcal{D}_p(L_{1,\xi}),$\\
(i) $r(F)=\infty,$ and $\xi>0$ if and only if
\begin{eqnarray}
\lim_{t\to \infty}\dfrac{\bar{F}(x^{t\xi}e^t)}{\bar{F}(e^{t})}=(\log (x^{\xi}e))^{-\frac{1}{\xi}},\label{thm1_e1}
\end{eqnarray}
(ii) $0<r(F)<\infty,$ and $\xi<0$ if and only if
\begin{eqnarray}
\lim_{t\to \infty}\dfrac{\bar{F}(r(F)e^{-t/\log(x^{\xi}e)})}{\bar{F}(r(F)e^{-t})}=(\log (x^{\xi}e))^{-\frac{1}{\xi}}.\label{thm1_e2}
\end{eqnarray}
\end{thm}
\begin{proof} (i) For $\xi> 0$ and $r(F)=\infty,$ we have $L_{1,\xi}(x) =K_{1,\alpha}(x^{\alpha^{-1}}e)$ for $\alpha=\frac{1}{\xi}.$ By Theorem \ref{RV1}-(a), is then equivalent to $\bar{F}(\exp(.))\in RV_{-\alpha}.$ Taking $z=\log (x^{\frac{1}{\alpha}}e)$ and from Theorem \ref{thm_von}-(1), setting $\frac{u(t)}{t\log t}=\frac{f(t)}{\bar{F}(t)}-\frac{\alpha}{t\log t}\to 0,$ for large $t$ and from Lemma \ref{lem},
\begin{eqnarray}
\dfrac{\mathcal{L}(e^{tz})}{\mathcal{L}(e^{t})}&=&\dfrac{c(e^{tz})\bar{F}(e^{t})}{c(e^{t})\bar{F} (e^{tz})}z^{-\alpha},\nonumber
\end{eqnarray}
Taking limit both side as $t\to\infty$ which is (\ref{thm1_e1}). If (\ref{thm1_e1}) holds, choose $d_n =\log F^{\leftarrow} (1-1/n),$ then $1/\bar{F} (e^{d_n} )=n$ (see, Mohan and Ravi 1993) and then,
\[\lim_{n\to\infty}\dfrac{\bar{F}(x^{\frac{d_n}{\alpha}}e^{d_n})}{\bar{F}(e^{d_n})}=\lim_{n\to\infty}n\bar{F}(x^{\frac{d_n}{\alpha}}e^{d_n})=(\log (x^{\frac{1}{\alpha}}e))^{-\alpha},\]
whence, from (\ref{pmax_lim}), $F\in\mathcal{D}(L_{1,\xi}),$ for $\xi=\alpha^{-1}.$
(ii) Now, we have $L_{1,\xi}(x) =K_{2,\alpha}( x^{\frac{1}{\alpha}}e),$ for $\alpha=-\frac{1}{\xi},$ $\xi< 0$ and $0<r(F)<\infty.$ By Theorem \ref{RV2}-(a), $\bar{F}(r(F)\exp(-1/(.)))$ is regularly varying with exponent $(-\alpha).$ From Theorem \ref{thm_von}-(2), we choose $\frac{u(t)}{t\log (r(F)/t)}=\frac{f(t)}{\bar{F}(t)}-\frac{\alpha}{t\log (r(F)/t)}\to 0,$ for $t\to \infty$ and from Lemma \ref{lem},
\begin{eqnarray}
\dfrac{\mathcal{L}(r(F)e^{-t/z)})}{\mathcal{L}(r(F)e^{-t})}
&=&\dfrac{c(r(F)e^{-t/z})\bar{F}(r(F)e^{-t})}{c(r(F)e^{-t})\bar{F} (r(F)e^{-t/z})}z^{\alpha}.\nonumber
\end{eqnarray}
where, $z=\log (x^{-\frac{1}{\alpha}}e).$
Taking limit both side as $t\to \infty$ which is (\ref{thm1_e2}). Conversely, if (\ref{thm1_e2}) holds, setting $d_n =-\log\frac{r(F)}{F^{\leftarrow} (1-1/n)},$ then $1/\bar{F} (r(F)e^{-d_n})=n,$ (see, Mohan and Ravi 1993) and then,
\[\lim_{n\to\infty} \frac{\bar{F} (r(F)e^{-d_n/x})}{\bar{F} (r(F)e^{-d_n})} =\lim_{n\to\infty} n\bar{F} (r(F)e^{-d_n/x})=\left(\log (x^{-\frac{1}{\alpha}}e)\right)^{\alpha}.\]
whence again, from (\ref{pmax_lim}), $F\in\mathcal{D}(L_{1,\xi}),$
\end{proof}
\begin{thm}\label{thm3}
A df $F\in\mathcal{D}_p(L_{2,\xi}),$\\
(i) $r(F)=0,$ and $\xi>0$ if and only if
\begin{eqnarray}
\lim_{t\to\infty}\dfrac{\bar{F}(-(-x)^{\xi t}e^{-t}))}{\bar{F}(-e^{-t})}=(-\log ((-x)^{\xi}e))^{-\frac{1}{\xi}}.\label{thm3_e1}
\end{eqnarray}
(ii) $r(F)<0,$ and $\xi<0$ if and only if
\begin{eqnarray}
\lim_{t\to\infty}\dfrac{\bar{F}(r(F)e^{t/\log ((-x)^\xi e)})}{\bar{F}(r(F)e^{t})}=(-\log ((-x)^{\xi}e))^{-\frac{1}{\xi}};\label{thm4_e1}
\end{eqnarray}
\end{thm}
\begin{proof}
(i) We have $L_{2,\xi}(x) =K_{4,\alpha}(-(-x)^{\frac{1}{\alpha}}e)$ for $\alpha=\frac{1}{\xi}.$ Suppose, $\xi> 0$ and $r(F)=0,$ by Theorem \ref{RV1}-(b), $\bar{F}(-\exp(-(.)))\in RV_{-\alpha}.$ Putting $z=\log ((-x)^{-\frac{1}{\alpha}}e)$ and from Theorem \ref{thm_von}-(4), $\frac{u(t)}{t\log(-t)}=\frac{f(t)}{\bar{F}(t)}-\frac{\alpha}{t\log(-t)}\to 0,$ for $t\to\infty$ and from Lemma \ref{lem},
\begin{eqnarray}
\dfrac{\mathcal{L}(-e^{-tz})}{\mathcal{L}(-e^{-t})}=\dfrac{c(-e^{-tz})\bar{F}(-e^{-t})}{c(-e^{-t})\bar{F} (-e^{-tz})}z^{-\alpha},\nonumber
\end{eqnarray}
Taking limit both side as $t\to\infty$ and hence (\ref{thm3_e1}). Now, if (\ref{thm3_e1}) holds, define $d_n =-\log (-F^{\leftarrow} (1-1/n)),$ then $1/\bar{F} (-e^{-d_n} )=n$ (see, Mohan and Ravi 1993) and,
\[\lim_{n\to\infty}\dfrac{\bar{F}(-(-x)^{\frac{d_n}{\alpha}}e^{d_n})}{\bar{F}(-e^{-d_n})}=\lim_{n\to\infty}n\bar{F}((-x)^{-\frac{d_n}{\alpha}}e^{-d_n})=(\log ((-x)^{-\frac{1}{\alpha}}e))^{-\alpha}.\]
From (\ref{pmax_lim}), this implies that $F\in\mathcal{D}(L_{2,\xi}),$ for $\xi=\alpha^{-1}.$
(ii) Suppose, $\xi< 0$ and $r(F)<0,$ we have $L_{\xi}(x) =K_{5,\alpha}(-(-x)^{\frac{1}{\alpha}}e)$ for $\alpha=-\frac{1}{\xi}.$ By Theorem \ref{RV2}-(b), is then equivalent to $\bar{F}(r(F)\exp(1/(\cdot)))\in RV_{-\alpha}.$ From Theorem \ref{thm_von}-(5), we choose $\frac{u(t)}{t\log(r(F)/t)}=\frac{f(t)}{\bar{F}(t)}-\frac{\alpha}{t\log (r(F)/t)}\to 0,$ for $t\to \infty$ and from Lemma \ref{lem},
\begin{eqnarray}
\dfrac{\mathcal{L}(r(F)e^{t/z)})}{\mathcal{L}(r(F)e^{t})}
=\dfrac{c(r(F)e^{t/z})\bar{F}(r(F)e^{t})}{c(r(F)e^{t})\bar{F} (r(F)e^{t/z})}z^{\alpha}.\nonumber
\end{eqnarray}
where, $z=\log ((-x)^{-\frac{1}{\alpha}}e).$
Taking limit both side as $t\to r(F)$ which is (\ref{thm1_e1}). If (\ref{thm1_e1}) holds, setting $d_n =\log\frac{r(F)}{F^{\leftarrow} (1-1/n)},$ then $1/\bar{F} (r(F)e^{d_n} )=n$ (see, Mohan and Ravi 1993) and then,
\[\lim_{n\to\infty} \frac{\bar{F} (r(F)e^{d_n/x})}{\bar{F} (r(F)e^{d_n})} =\lim_{n\to\infty} n\bar{F} (r(F)e^{d_n/x})=\left(\log((-x)^{\frac{1}{\alpha}}e)\right)^{\alpha}.\]
whence, from (\ref{pmax_lim}), $F\in\mathcal{D}(L_{2,\xi}),$ for $\xi=-\alpha^{-1}.$
\end{proof}
\begin{rem} In case of $\xi=0$ (eq. \ref{gev}) for $L_{\tilde{\xi}}(x)=\Phi_1(x)$ is proved in Theorem \ref{Ap_thm1} and for $L_{\tilde{\xi}}(x)=\Psi_1(x)$ presented Theorem \ref{Ap_thm2}.
\end{rem}
\subsection{Moments}
Some of the most important features and characteristics of a distribution can be studied through moments. The $k$th moments of PGEV are derived in the following theorems. In our proofs of $k$th moments of PGEV, the moment generating function (MGF) of Weibull with positive support plays and important role. Note that the MGF corresponding to a standard Weibull rv of $Y$ with positive support specified as
\begin{eqnarray}\label{mgf1}
M_Y(t;\alpha)=\alpha\int_{0}^{\infty}x^{\alpha-1}\exp\left(-tx-x^{\alpha}\right)dx.\label{mgf_weibull}
\end{eqnarray}
Cheng et al. (2004) derived the moment generating function (MGF) of $Y,$ when the parameter $\alpha$ takes integer values. Nadarajah and Kotz (2007) show that a closed form expression for MGF of $Y,$ for all rational values of shape parameter. Since, we assume $\alpha=p/q,$ where $p\geq 1$ and $q\geq 1,$ are coprime integers, the integral in (\ref{mgf_weibull}) can be provided that
\begin{eqnarray}\label{mgf_1}
M_Y(t;\alpha)=\left\lbrace
\begin{array}{ll}
\alpha\sum\limits_{j=0}^{q-1}\frac{(-1)^j}{j! t^{\alpha+\alpha j}}\Gamma(\alpha+\alpha j) \left[{_{p+1}\mathcal{F}_q}(1,\Delta(p,j\alpha+j);\Delta(q,1+j);(-1)^q z)\right],&\\
\text{if }0<\alpha<1;&\\
&\\
\sum\limits_{j=0}^{p-1}\frac{(-t)^j}{j!}\Gamma\left(1+\frac{j}{\alpha}\right)\left[_{q+1}\mathcal{F}_p\left(1,\Delta\left(q,1+\frac{\alpha}{j}\right);\Delta(p,1+j);\frac{(-1)^p}{z}\right)\right],&\\
\text{ if }\alpha>1,&\\
\end{array}\right.
\end{eqnarray}
where, $z=p^p/(t^p q^q)$ and $\Delta(c,d)=\{d/c,(d+1)/c,\cdots,(c+d-1)/c\}$ and ${_p\mathcal{F}_q}$ is the generalized hyper geometric function defined by $${_p\mathcal{F}_q}(a_1,\cdots,a_p;b_1,\cdots,b_q;x)=\sum_{k=0}^{\infty}\frac{(a_1)_k(a_2)_k\cdots (a_p)_k}{(b_1)_k(b_2)_k\cdots (b_q)_k}\frac{x^k}{k!}$$
where, $(\upsilon)_k=\upsilon(\upsilon+1)\cdots(\upsilon+k-1).$
In particular value $\alpha=1$ simple integration of (\ref{mgf_weibull}) gives,
\begin{eqnarray}\label{mgf_2}
M_Y(t;1)=\frac{1}{1+t}.
\end{eqnarray}
In the case $\alpha=2$ the MGF becomes
\begin{eqnarray}\label{mgf3}
M_Y(t;2)=1-\frac{t\sqrt{\pi}}{2}\exp\left(\frac{t^2}{4}\right)\erf\left(\frac{t}{2}\right),
\end{eqnarray}
where, the complementary error function defined by $\erf(x)=1-\frac{2}{\sqrt{\pi}}\int_{0}^{x}\exp(-t^2)dt.$
The generalized hypergeometric function is widely available in many scientific software packages, such as R and Matlab.
The following results show that, the proofs of the $k$th moments of PGEV involve the application of MGF of standard Weibull distribution function.
\begin{thm}
Let $Y$ is a rv with standard Weibull df and $X$ is a rv with PGEV in (\ref{df3}). For $k>0,$
\begin{enumerate}
\item[(i)] $X^+$ is positive support and $\xi<0$
\begin{eqnarray}
E(X^+)^k=e^{k\left(\mu-\frac{\sigma}{\xi}\right)}M_Y\left(\frac{k\sigma}{\abs{\xi}},\frac{1}{\abs{\xi}}\right).\nonumber
\end{eqnarray}
where, $M_Y(\cdot)$ defined in (\ref{mgf_1}).
\item[(ii)] $X^-$ is negative support and $\xi>0$
\begin{eqnarray}
E\abs{X^-}^k=e^{k\left(\mu+\frac{\sigma}{\xi}\right)}M_{Y^{-1}}\left(\frac{k\sigma}{\xi},\frac{1}{\xi}\right).\nonumber
\end{eqnarray}
\end{enumerate}
\end{thm}
\begin{proof} Suppose $\abs{Z}=\left(\abs{X}e^{-\mu}\right)^{\frac{1}{\sigma}}$ is a standardizing rv with df in (\ref{df3}) for $A=\{z:1+\text{sign}(z)\xi\log \abs{z}>0\}.$ We write
\begin{equation}
E\abs{Z}^k=\int_{A} \abs{z}^{k-1}(1+\text{sign}(Z)\xi\log \abs{z})^{-1-1/\xi}e^{-(1+\text{sign}(Z)\xi\log \abs{z})^{-1/\xi}}dz,
\end{equation}
We have
\begin{equation}\label{moment}
E\abs{Z}^k=\frac{1}{\xi}\int_{A} e^{\frac{k}{\xi}(y-1)\text{sign}(Z)-y^{-\frac{1}{\xi}}}y^{-1-\frac{1}{\xi}}dy,\;\text{ where, } y=1+\xi\,\text{sign}(Z)\log\abs{z}.
\end{equation}
(i) Let $Z^+$ is a rv with positive support. From (\ref{moment}), the $k$th moment does not exist for $\xi>0.$ For $\xi<0,$ we have
\begin{eqnarray}
E(Z^+)^k&=&-\frac{1}{\xi}\int_{0}^{\infty} e^{\frac{k}{\xi}(y-1)-y^{-\frac{1}{\xi}}}y^{-1-\frac{1}{\xi}}dy,\nonumber\\
&=&e^{-\frac{k}{\xi}}M_Y\left(\frac{k}{\abs{\xi}},\frac{1}{\abs{\xi}}\right).\nonumber
\end{eqnarray}
where, $Y$ is a positive rv with standard Weibull distribution and
$M_Y(\cdot)$ defined in (\ref{mgf_1}).
The $k$th moment of $X^+$ can be obtained as
\begin{eqnarray}
E(X^+)^k=e^{k\left(\mu-\frac{\sigma}{\xi}\right)}M_Y\left(\frac{k\sigma}{\abs{\xi}},\frac{1}{\abs{\xi}}\right).\nonumber
\end{eqnarray}
(ii) Similarly, let $Z^-$ is a rv with neagitve support. From (\ref{moment}), the $k$th moment does not exist, for $\xi<0.$ For $\xi>0$ we get
\begin{eqnarray}
E\abs{Z^-}^k&=&\frac{1}{\xi}\int_{0}^{\infty} e^{\frac{k}{\xi}(1-y)-y^{-\frac{1}{\xi}}}y^{-1-\frac{1}{\xi}}dy,\nonumber\\
&=&e^{\frac{k}{\xi}}M_{Y^{-1}}\left(\frac{k}{\xi},\frac{1}{\xi}\right).\nonumber
\end{eqnarray}
The $k$th moment of $X^-$ can be obtained as
\begin{eqnarray}
E\abs{X^-}^k=e^{k\left(\mu+\frac{\sigma}{\xi}\right)}M_{Y^{-1}}\left(\frac{k\sigma}{\xi},\frac{1}{\xi}\right).\nonumber
\end{eqnarray}
\end{proof}
\begin{rem}
The $k$th moment of rvs $X^+$ with PGEV for $\xi>0$ and the $k$th moment of rvs $X^-$ with PGEV $\xi<0$ do not exist.
\end{rem}
The $k$th central moments of $X$ are easily obtained from the ordinary moments by
\begin{eqnarray}\label{cm}
E(X-E(X))^k=\sum_{j=0}^{k}{{k}\choose{j}}(-1)^j(E(X))^jE(X^{k-j}).
\end{eqnarray}
From (\ref{cm}) and $k=2,$ the variances of PGEV listed in Appendix \ref{tab1}.
\subsection{Entropy.}
An entropy of rv $X$ is a measure of variation of the uncertainty. Shannon entropy is defined by
\begin{eqnarray}\label{Shannon_entropy}
H(X)=-\int_A \log f(x) f(x)dx,
\end{eqnarray}
where, $A=\{x: f(x)>0\}.$
Here, the Shannon entropy of GEV family is well known as
\begin{equation}
H(X)=\log\tilde{\sigma}+(\tilde{\xi}+1)\gamma+1.
\end{equation}
The Shannon entropy of six type of p-max stable laws are evaluated by Ravi and Saeb (2012).
Now, we illustrate the Shannon entropy of PGEV family.
\begin{thm}\label{thm_ent}
If $X$ is a rv with df PGEV for $\xi< 0,$ then the Shannon entropy of $X$ is given by
\begin{eqnarray}
H(X)=
\mu+\log\sigma+(1+\xi)\gamma+\frac{\sigma}{\xi}E(\text{sign}(X))\left[\Gamma(1-\xi)-1\right]+1.
\end{eqnarray}
\end{thm}
\begin{proof}
Let $Z$ is a standardized rv with df PGEV ($\xi<0$) in (\ref{pdf_1}), the Shannon entropy is given by
\begin{eqnarray}
H(Z)&=&E(\log \abs{Z})+E\left(\log (1+\text{sign}(Z)\xi\log \abs{Z})^{1+1/\xi}+(1+\text{sign}(Z)\xi\log \abs{Z})^{-1/\xi}\right),\nonumber\\
&=&E_1+E_2.\label{ZEnt}
\end{eqnarray}
Putting $Y=(1+\text{sign}(Z)\xi\log \abs{Z})^{-1/\xi},$ and $Y$ has standard exponential distribution.
\begin{eqnarray}
E_1=\xi^{-1}E(\text{sign}(Z))E(Y^{-\xi}-1))
=\frac{1}{\xi}E(\text{sign}(Z))\left[\Gamma(1-\xi)-1\right].\label{E1}
\end{eqnarray}
Next,
\begin{eqnarray}
E_2=-(1+\xi)E_Y(\log(Y))+E_Y(Y)
=(1+\xi)\gamma+1,\label{E2}
\end{eqnarray}
From (\ref{E1}), (\ref{E2}) we get
\begin{eqnarray}
H(Z)=
(1+\xi)\gamma+\frac{1}{\xi}E(\text{sign}(Z))\left[\Gamma(1-\xi)-1\right]+1.
\end{eqnarray}
From lemma 1.3, Ravi and Saeb (2012), If $X=\abs{Z}^{\sigma} e^{\mu}$ then
\begin{eqnarray}
H(X)&=&\mu+\log\sigma+\left(\sigma-1\right)E\log \abs{Z}+H(Z), \nonumber\\
&=&\mu+\log\sigma+(1+\xi)\gamma+\frac{\sigma}{\xi}E(\text{sign}(X))\left[\Gamma(1-\xi)-1\right]+1.\nonumber
\end{eqnarray}
\end{proof}
\begin{rem}\label{rem2}
Note that, the Shannon entropy of the PGEV distribution for $\xi>0$ does not exist.
\end{rem}
Suppose $Y$ is a rv with df $F_Y$ and $X=h(Y)$ with df $F_X$ where $h$ is a continuous function. The entropy ordering $H(Y)<H(X),$ will be denoted as $F_Y\stackrel{E}{<}F_X$ or $Y\stackrel{E}{<}X.$ In general case, the following lemma finds a direct relationship for entropy.
\begin{lem}\label{lem2}
If $E_X\left(\log\abs{\nabla_X h^{\leftarrow}(X)}\right)<0$ then $Y\stackrel{E}{<}X.$
\end{lem}
\begin{proof}
We write,
\[F_X(x)=\Pr(h(Y)\leq x)=\Pr(Y\leq h^{\leftarrow}(x))=F_Y(h^{\leftarrow}(x)),\]
with respective density function
\[f_X(x)=f_Y(h^{\leftarrow}(x))\abs{\nabla_x(h^{\leftarrow}(x))}.\]
From definition of entropy we have
\begin{eqnarray}\label{lem1}
H(X)&=&-\int_\mathbb R f_Y(h^{\leftarrow}(x))\log\left(f_Y(h^{\leftarrow}(x)\right) d(h^{\leftarrow}(x))-\int_{\mathbb R}f_X(x)\log\abs{\nabla_x h^{\leftarrow}(x)}dx,\nonumber\\
&=&H(Y)-E_X(\log\abs{\nabla_X(h^{\leftarrow}(X))}).\label{entropy}
\end{eqnarray}
Noting that, if $E_X\left(\log\abs{\nabla_X h^{\leftarrow}(X)}\right)<0$ then $Y\stackrel{E}{<}X.$
\end{proof}
The following theorem investigates the entropy ordering in GEV families with $\xi<0.$
\begin{thm}\label{entropy_ord} Suppose $Y$ has GEV family.
If $X=\text{sign(X)}\exp(\abs{Y})$ is a rv with PGEV $(\xi<0)$ then $Y\stackrel{E}{<}X.$
\end{thm}
\begin{proof}
(i) Let $X^{+}$ is a positive rv and $h(x)=\exp(x).$ From Lemma \ref{lem2}, it is enough to show that, $E_X(\log(X))>0.$ Use the proof of Theorem \ref{thm_ent}, we have
$E_X(\log(X))=\mu+\frac{\sigma}{\xi}(\Gamma(1+\abs{\xi})-1).$ Since, the Shannon entropy of PGEV for $\xi<0$ exists, $\Gamma(1+\abs{\xi})>0$ for all $\xi<0,$ and $Y\stackrel{E}{<}X^{+}$ holds.
(ii) Similarly, define $h(x)=-\exp(-x)$ and $X^{-}$ is a negative rv. From Lemma \ref{lem2} and Theorem \ref{thm_ent}, $E_X\left(\log\abs{X}\right)=\mu-\frac{\sigma}{\xi}(\Gamma(1-\xi)-1)>0$ for all $\xi<0$ and hence the proof.
\end{proof}
\subsection{Dispersion ordering.}
Lewis and Thompson (1981) have defined the concept of \textquotedblleft ordering in dispersion\textquotedblright. Two distributions $F_X$ and $F_Y$ are said to be ordered in dispersion, denoted by $F_Y\stackrel{disp}{<}F_X$ if and only if
\[F_Y^{\leftarrow}(u)-F_Y^{\leftarrow}(v)\leq F_X^{\leftarrow}(u)-F_X^{\leftarrow}(v),\;\;\text{ for all }0<v<u<1.\]
It is easily seen by putting $u=F_Y(y)$ and $v=F_Y(x)$ where $y\leq x$ that $F_Y\stackrel{disp}{<}F_X$ if and only if
\begin{eqnarray}\label{dis_1}
F_X^{\leftarrow}(F_Y(x))+x\;\text{ is nondecreasing in }x,
\end{eqnarray}
then, $F_Y$ is said to be tail-ordered with respect to $F_X$ $(F_Y \stackrel{tail}{<} F_X).$ Thus we see that dispersive ordering is the same as tail-ordering. Oja (1981) shows that the dispersion ordering implies both variance ordering and entropy ordering $(\stackrel{EV}{<}).$ In other word, $F_Y\stackrel{disp}{<}F_X$ is a sufficient condition for $Y\stackrel{EV}{<}X$ (variance and entropy order similarly). Entropy ordering of distributions within many parametric families are studied in Ebrahimi et al. (1999).
Let $L(x_p)=1-p,$ where, $L(\cdot)$ is the distribution (\ref{df3}) so that
\begin{eqnarray}\label{quantile_e1}
x_p=\text{sign}(X)\exp\left(\frac{\sigma}{\xi}\text{sign}(X)\left(y_p^{-\xi}-1\right)+\mu\right);
\end{eqnarray}
We also well known the quantile for $\xi=0$ in (\ref{gev}) we get
\begin{eqnarray}
x_p=\frac{\tilde{\sigma}}{\tilde{\xi}}(y_p^{-\tilde{\xi}}-1)+\tilde{\mu},\label{quantile_e2}
\end{eqnarray}
where $y_p =-\log(1-p).$
The following corollary investigates the dispersion ordering in the GEV families.
\begin{coll} Suppose $X$ and $Y$ are rvs to correspond PGEV and GEV families. Let $X^+$ is a positive support, from (\ref{dis_1}) and (\ref{quantile_e1}) we have
\begin{eqnarray}\label{dispresion}
L^{\leftarrow}(G(x))=\exp(x)+x,\nonumber
\end{eqnarray}
is a nondecreasing function for all $x$ in support of GEV, then, $Y\stackrel{disp}{<}X^+.$ On the other hand, the result from Oja (1981) and Theorem \ref{entropy_ord}, the entropy of GEV and PGEV families are ordered for $\xi<0,$ we conclude that the variances are also ordered in $\xi,$
so, $Y\stackrel{EV}{<}X^+$ for $\xi<0.$
Similarly, if $X^-$ is a rv with negative support, from Theorem \ref{entropy_ord}, $Y\stackrel{EV}{<}X^-$ for $\xi<0$ and hence the proof.
\end{coll}
\section{Methods of Estimations}
\subsection{Maximum Likelihood Estimation.} The method of maximum likelihood estimation (MLE) using Newton-Raphson iteration to maximize the likelihood function of GEV, as recommended by Prescott and Walden (1980).
The log-likelihood function for $(\mu,\sigma,\xi)$ based on PGEV family, given by
\begin{eqnarray}\label{mle2}
\ell(x;\mu,\sigma,\xi)&=&-k\log \sigma-\sum_{i=1}^{k}\log \abs{x_i}-\left(1+\frac{1}{\xi}\right)\sum_{i=1}^{k}\log\left(1+\xi\,\text{sign}(x)\left(\frac{\log \abs{x_i}-\mu}{\sigma}\right)\right)\nonumber\\
&&-\sum_{i=1}^{k}\left(1+\xi\,\text{sign}(x)\left(\frac{\log\abs{ x_i}-\mu}{\sigma}\right)\right)^{-1/\xi};
\end{eqnarray}
For determining the MLEs of the parameters $\mu,$ $\sigma$ and $\xi,$ we can use the same procedure as for the GEV law. Since, there is no analytical solution, but for any given dataset the maximization is straightforward using standard numerical optimization algorithms. Jenkinson (1969), Prescott and Walden (1980) show that the elements of the Fisher information matrix for GEV distribution$(\xi\neq 0).$
Since the $\log \abs{x}$ is free from of parameters,
the Fisher information matrix for PGEV is similar the Fisher information matrix for GEV law. Since, the Shannon entropy is equivalent to the negative log-likelihood function and from remark \ref{rem2} the MLEs exists for $\xi<1.$
Smith (1985) has investigated the classical regularity conditions for the asymptotic properties of MLEs are not satisfied but he shows that, when $\xi>-0.5$ the MLEs have usual asymptotic properties. For $\xi=-0.5$ the MLEs are asymptotically efficient and normally distributed, but with a different rate of convergence. We remark that results of Smith applies also to the three parameters. The MLEs may nonregular for $\xi< -0.5$ and $\xi\geq 1$, but Bayesian techniques offer an alternative that is often preferable.
\subsection{Bayesian Estimation.} Let $\bm{\theta}$ is a vector of the model parameters in a space $\Theta$ and $\pi(\bm{\theta})$ denote the density of the prior distribution for $\bm{\theta}.$ The posterior density of $\bm{\theta}$ is given by
\begin{eqnarray}
\pi(\bm{\theta}|\bm{x})=\frac{\pi(\bm{\theta})\exp(\ell (\bm{x};\bm{\theta}))}{\int_{\Theta} \pi(\bm{\theta})\exp(\ell (\bm{x};\bm{\theta}))d\bm{\theta}}\propto \pi(\bm{\theta})\exp(\ell (\bm{x};\bm{\theta})).\nonumber
\end{eqnarray}
where, $\ell(\cdot)$ is log-likelihood function. However, computing posterior inference directly is difficult. To bypass this problem we can use simulation bases techniques such as Markov Chain Monte Carlo (MCMC).
The Markov Chain is generated using standard Metropolis (Hastings, 1970) within Gibbs (Geman and Geman, 1984) methods. Setting $\bm{\theta}=(\mu,\eta, \xi)$ where, $\eta=\log\sigma$ is easier to work.
We might choose a prior density function
\[\pi(\bm{\theta})=\pi_\mu(\mu)\pi_\eta(\eta)\pi_\xi(\xi),\]
where the marginal priors, $\pi_\mu(\cdot), \pi_\eta(\cdot)$ and $\pi_\xi(\cdot),$ are normal density function with mean zero and variances, $v_\mu,\;v_\eta$ and $v_\xi$ respectively.
These are independent normal priors with large variances. The variances are chosen large enough to make the distributions almost flat and therefore should correspond to prior ignorance. The choice of normal densities is arbitrary. The proposed value $\bm{\theta}^*$ at point $i$ is $\bm{\theta}^* = \bm{\theta}^{(i)} + \bm{\epsilon}.$ The
$\bm{\epsilon}=(\epsilon_\mu, \epsilon_\eta, \epsilon_\xi)$ are normally distributed variables, with zero means and variances $w_\mu,$ $w_\eta$ and $w_\xi$ respectively.
Now we specify an arbitrary probability rule $q(\bm{\theta}_{i+1}|\bm{\theta}_i)$ for iterative simulation of successive values. The distribution $q$ is called the proposal distribution. Possibilities include $(\bm{\theta}_{i+1}| \bm{\theta}_i)$ is Normal density with mean $\bm{\theta}_i$ and variance one. Then $q(\bm{\theta}_i| \bm{\theta}^*) =\tilde{f}(\bm{\theta}^*-\bm{\theta}_i),$ where $\tilde{f}(\cdot)$ is
the density function of $\bm{\epsilon}.$ Since the distribution of $\bm{\epsilon}$ is symmetric about zero $q(\bm{\theta}_i|\bm{\theta^*})=q(\bm{\theta^*}|\bm{\theta}_i).$
The acceptance probability
\begin{eqnarray}\label{Omega}
\Omega_i=\min\left\{1,\frac{\exp(\ell(x;\bm{\theta}^*))\pi(\bm{\theta}^*)}{\exp(\ell(x;\bm{\theta}_i))\pi(\bm{\theta}_i)}\right\},
\end{eqnarray}
was suggested by Hastings (1970). Here we accepted the proposed value $\bm{\theta}^*$ with probability $\Omega_i.$
We note that, the variance of $\bm{\epsilon}$ affects the acceptance
probability, if the variance is too low most proposals will be accepted, resulting in very slow convergence, and if it is too high very few will be accepted and the moves in the chain will often be large. Appendix \ref{Ape1} gives the details of the required algorithm.
Here we find few papers linking the Bayesian method and extreme value analysis. Smith and Naylor (1987) who compare Bayesian and maximum likelihood estimators for the Weibull distribution. Coles and Powell (1996) and Coles and Tawn (1996) for a
detailed review of Bayesian methods in extreme value modelling.
Stephenson and Tawn (2004) perform inference using reversible jump MCMC techniques for extremal types.
\subsection{Prediction.} We are interested
in the outcome $y$ of the future experiment
Within the Bayesian framework, the predictive distribution function is argued by Aitchison and Dunsmore (1975). In particular, since
the objective of an extreme value analysis is usually an estimate of the
probability of future events reaching extreme levels, expression through
predictive distributions is natural. Let $Y$ is a rv with annual maximum distribution over a
future period of years and $\bm{x}$ represents historical observations. The predictive distribution function is defined as
\begin{eqnarray}\label{pred}
\Pr(Y< y|\bm{x})&=&\int_\Theta \Pr(Y<y|\bm{\theta})\pi(\bm{\theta}|\bm{x})d\bm{\theta},\nonumber\\
&\simeq&\frac{1}{n}\sum_{i=1}^{n}\Pr(Y < y|\bm{\theta}_i),\nonumber
\end{eqnarray}
where $\bm{\theta}_i$ is the output from the $i^{th}$ iteration of a sample of size $n$ from the Gibbs
sampler of posterior distribution of $\bm{\theta}.$ Estimates of extreme quantiles
of the annual maximum distribution are then obtained by solving the equation
\begin{eqnarray}\label{return}
\frac{1}{n}\sum_{i=1}^{n}\Pr(Y < x_p|\bm{\theta}_i)=1-p,
\end{eqnarray}
for $x_p$ with various values of $m$ where, $m=1/p$ is defined as return period.
\subsection{Real Data Analysis.}
In this section we shall use the PGEV model to a real data set. This analysis is based on the annual maximum yearly rainfall data of station Eudunda, Australia (Latitude 34.18S; Longitude 139.08E and
Elevation 415 m) which collected during 1881-2015. Annual maxima, corresponding to the year from 1881, were found from the 135 years worth of data and are plotted in Fig \ref{Data}.
We assume that the pattern of variation has stayed constant over the observation period, so we model the data as independent observations from the GEV families.
Here, maximization of GEV and PGEV log-likelihood functions using the "Nelder-Mead" algorithms. All the computations were done using R programming language.
In what follows we shall apply formal goodness of fit tests in order to verify which distribution fits better to these data. We apply the Cram\'{e}r-von Mises ($C$) and Anderson-Darling ($A$) test statistics. The test statistics $C$ and $A$ are described in detail in Chen and Balakrishnan (1995). In general, the smaller values of statistics $C$ and $A$, the better fit to data. Additionally, from the critical values of statistics $C$ and $A$ given in Chen and Balakrishnan (1995), it is possible to calculate the p-values corresponding to each test statistics. The null hypothesis is $H_0:\{X_1,\ldots,X_n\}$ comes from GEV/PGEV families. To test $H_0,$ we can proceed as appendix \ref{Ape}.
The values of statistics $C$ and $A$ (p-values between parentheses) for all models are given in Table \ref{Table1}.
From this table we conclude that does not evidence to reject the null hypothesis for GEV/PGEV distributions
Table \ref{Table2} lists the MLE method of the parameters estimation and standard errors in parentheses. Since the values of standard errors in PGEV model are lower than other laws, we suggesting that the PGEV model is best fit model for these data.
Within the Bayesian model with non-informative prior distributions, the algorithm in \ref{Ape1} was applied to annual maxima dataset. Initializing the MCMC algorithm with maximum likelihood estimates as our initial vector, $\theta_0 = (4.3614, 02853, -02386)$ should produce a chain with small burn-in period. After some pilot runs, a Markov chain of $1000$
iterations was then generated with good mixing properties (Figure \ref{graph_bayes}). The burn-in period was taken to be the first $400$ iterations which the stochastic variations in the chain seem reasonably homogeneous. If we accept this, after deleting the
first $600$ simulations, the remaining $400$ simulated values can be treated as
dependent realizations whose marginal distribution is the target posterior. The sample means (and standard error) of each marginal component
of the chain are
\[\hat{\mu}=4.3615\;(0.0265)\;\;\hat{\sigma}= 0.2848\;(0.0144)\;\;\hat{\xi}= -0.2411\;(0.0340).\]
Finally, using eq. \ref{return}, a plot of the predictive distribution of a future annual maximum is shown in Fig. \ref{return_graph} on the usual return period scale. Table.\ref{Table3} shows the predictive return levels $x_p$ for $m$ years where, $m=\frac{1}{p}$ is return period. For example, the corresponding estimate for the 4 years return level is $x_{0.75}= 106.59.$
| {
"timestamp": "2017-12-01T02:09:14",
"yymm": "1711",
"arxiv_id": "1711.11399",
"language": "en",
"url": "https://arxiv.org/abs/1711.11399",
"abstract": "Similar to the generalized extreme value (GEV) family, the generalized extreme value distributions under power normalization are introduced by Roudsari (1999) and Barakat et al. (2013). In this article, we study the asymptotic behavior of GEV laws under power normalization and derive expressions for the kth moments, entropy, ordering in dispersion, rare event estimation and application of real data set. We also show that, under some conditions, the Shannon entropy and variance of GEV families are ordered.",
"subjects": "Applications (stat.AP)",
"title": "A note on power generalized extreme value distribution and its properties",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754474655619,
"lm_q2_score": 0.718594386544335,
"lm_q1q2_score": 0.7075103896781296
} |
https://arxiv.org/abs/1603.07364 | Special divisors on marked chains of cycles | We completely describe all Brill-Noether loci on metric graphs consisting of a chain of g cycles with arbitrary edge lengths, generalizing work of Cools, Draisma, Payne, and Robeva. The structure of these loci is determined by displacement tableaux on rectangular partitions, which we define. More generally, we fix a marked point on the rightmost cycle, and completely analyze the loci of divisor classes with specified ramification at the marked point, classifying them using displacement tableaux. Our results give a tropical proof of the generalized Brill-Noether theorem for general marked curves, and serve as a foundation for the analysis of general algebraic curves of fixed gonality. | \section{Introduction}
The tropical proof of the Brill-Noether theorem \cite{cdpr} gave a parameterization of the sets $W^r_d(\Gamma)$ of special divisors on a metric graph $\Gamma$ composed of a chain of cycles with generic edge lengths. The purpose of this paper to to generalize this analysis, in two ways. We use these generalizations to give a tropical proof of the generalized Brill-Noether theorem (for curves with a marked point), and also as a foundation for our paper \cite{pfl2}, which gives applications to the geometry of general curves of fixed gonality.
The first generalization is that we consider \textit{arbitrary} chains of cycles, with any edge lengths. We show that the behavior of the loci of special divisor classes on $\Gamma$ depends only on a sequence of numbers $\underline{m} = (m_2,m_3,\cdots,m_g)$, easily computed from the edge lengths, called the \emph{torsion profile} of $\Gamma$ (Definition \ref{def:tp}). The torsion profile determines whether the chain $\Gamma$ is Brill-Noether general in the sense of \cite{cdpr}. The metric graph $\Gamma$ is called \emph{Brill-Noether general} if $\dim W^r_d(\Gamma) = g-(r+1)(g-d+r)$ whenever this number is nonnegative, and $W^r_d(\Gamma)$ is empty otherwise. The original genericity condition of \cite{cdpr} is equivalent to saying that $m_i = 0$ or $m_i > 2g-2$.
\begin{thm} \label{thm_genericity1}
A chain of cycles $\Gamma$ is Brill-Noether general if and only if for each $i \in \{2,3,\cdots,g-1\}$, either $m_i = 0$ or $$m_i > \min( i, g+1-i).$$
\end{thm}
The second direction in which we generalize \cite{cdpr} is by considering special divisors with prescribed ramification at a marked point $w \in \Gamma$, which we always take to be on the rightmost cycle of the chain. In algebraic geometry, imposing ramification conditions amounts to considering only those linear series that correspond to maps to projective space in which the marked point is inflected in a particular way (see Definition \ref{def:wrad}). Although this generalization is interesting in its own right, it is in fact a necessary ingredient in our arguments even in the ordinary case. We will define, for any metric graph $\Gamma$ with a marked point $w$, generalized Brill-Noether loci $W^\lambda(\Gamma,w) \subseteq \Pic^0(\Gamma)$, where $\lambda$ is a partition.
Here by a \emph{partition} we mean a finite, non-increasing sequence of nonnegative integers, where two such sequences are considered the same if one is obtained by adding a sequence of $0$s to the end of the other. We will also identify partitions with their Young diagrams (in French notation), which we in turn regard as subsets of $\textbf{Z}^2_{>0}$, according to the following convention.
\begin{conv} \label{conv_youngdiagram}
Any partition $\lambda = (\lambda_0,\lambda_1,\cdots,\lambda_r)$ (where $\lambda_0 \geq \lambda_1 \geq \cdots \geq \lambda_r$) will be identified with the subset $$\{(x,y) \in \textbf{Z}^2_{>0}:\ 1 \leq x \leq \lambda_{y-1},\ 1 \leq y \leq r+1 \},$$ which we will refer to as the Young diagram of $\lambda$. We denote the number of elements in this set (i.e. the sum of the elements $\lambda_i$) by $|\lambda|$.
\end{conv}
The link between the generalized Brill-Noether loci $W^\lambda(\Gamma,w)$ and the ordinary Brill-Noether loci is that $W^r_d(\Gamma)$ is isomorphic to $W^\lambda(\Gamma,w)$, where $\lambda$ is a \textit{rectangular} partition of height $r+1$ and width $g-d+r$, and $w$ is any marked point. The isomorphism $W^\lambda(\Gamma,w) \rightarrow W^r_d(\Gamma)$ is given by $[D] \mapsto [D + d \cdot w]$.
\begin{center}$\begin{array}{m{7cm} m{5cm}}
\begin{tikzpicture}[scale=0.5]
\draw (-0.5,2) node[left] {$\lambda = $};
\draw (0,0) rectangle (6,4);
\foreach \i in {1,2,...,5} \draw (\i,0) -- (\i,4);
\foreach \j in {1,2,3} \draw (0,\j) -- (6,\j);
\draw[snake=brace,thick] (6,-0.25) -- (0,-0.25) node[midway,below] {$g-d+r$};
\draw[snake=brace,thick] (6.25,4) -- (6.25,0) node[midway,right] {$r+1$};
\end{tikzpicture}
&
$W^r_d(\Gamma) \cong W^\lambda(\Gamma,w)$
\end{array}$\end{center}
The analog of the Brill-Noether number is $g - |\lambda|$. Note in particular that when $\lambda$ is the rectangle above, this is the usual Brill-Noether number $g - (r+1)(g-d+r)$. We call a \emph{marked} metric graph $(\Gamma,w)$ \emph{Brill-Noether general} if for all partitions $\lambda$, $\dim W^\lambda(\Gamma,w) = g - |\lambda|$ if this number is nonnegative, and $W^\lambda(\Gamma,w)$ is empty otherwise. We obtain the following marked-point version of Theorem \ref{thm_genericity1}.
\begin{thm} \label{thm_genericity2}
Let $(\Gamma,w)$ be a marked chain of cycles with $w$ on the rightmost cycle and torsion profile $\underline{m} = (m_2,\cdots,m_g)$. Then $(\Gamma,w)$ is Brill-Noether general if and only if for each $i \in \{2,3,\cdots,g\}$, either $m_i = 0$ or $m_i > i$.
\end{thm}
Our main result is an explicit parameterization of the loci $W^\lambda(\Gamma,w)$, when $\Gamma$ is any chain of cycles of genus $g$ (with arbitrary edge lengths), $w$ is a marked point on the rightmost cycle, and $\lambda$ is any partition. This parameterization is based on combinatorial objects called \emph{$\underline{m}$-displacement tableaux} (Definition \ref{def:dt}), which are Young tableaux on $\lambda$ with certain constraints depending on the torsion profile $\underline{m}$. We write $t \vdash_{\underline{m}} \lambda$ to indicate that $t$ is an $\underline{m}$-displacement tableau on the partition $\lambda$. Each such tableau $t$ defines a locus $\textbf{T}(t) \subseteq \Pic^0(\Gamma)$, homeomorphic to a torus of dimension equal to $g$ minus the number of symbols appearing in $t$ (Definition \ref{def:Tt}). In particular, $\dim \textbf{T}(t) \geq g - |\lambda|$, with equality if and only if $\lambda$ has no repeated symbols.
\begin{thm} \label{t_tori}
For any chain $\Gamma$ of torsion profile $\underline{m}$ and any partition $\lambda$, $$W^\lambda(\Gamma,w) = \bigcup_{t \vdash_{\underline{m}} \lambda} \textbf{T}(t).$$
\end{thm}
A consequence of this (Corollary \ref{c_dp}) is that we can compute the dimensions of the loci $W^\lambda(\Gamma,w)$ by determining the minimum number of distinct symbols in an $\underline{m}$-displacement tableau on $\lambda$. Together with semicontinuity results for tropicalization of algebraic curves (summarized in Section \ref{sec_trop}), we can deduce upper bounds on the dimensions of Brill-Noether varieties of (marked or unmarked) algebraic curves. Most importantly, choosing \textit{special} edge lengths in a chain of cycles opens new applications of the theory of linear series on metric graphs, as in our \cite{pfl2}. We also obtain a tropical proof of the generalized Brill-Noether theorem for algebraic curves with one marked point, originally proved by Eisenbud and Harris \cite[Theorem 4.5]{eh}.
\begin{thm}[Generalized Brill-Noether theorem]
\label{thm_gbn}
Let $(C,p)$ be a general marked algebraic curve of genus $g$ over an algebraically closed field of any characteristic. Let $r,d$ be positive integers such that $g-d+r \geq 0$, and let $\alpha = (\alpha_0,\alpha_1,\cdots,\alpha_r)$ be a nondecreasing sequence of nonnegative integers. Then the variety $W^{r,\alpha}_d(C,p) \subseteq \Pic^d(C)$ of line bundles of rank at least $r$ and ramification at least $\alpha$ at the point $p$ (see Definition \ref{def:wrad}) is nonempty if and only if the adjusted Brill-Noether number
$$
\rho(g,d,r,\alpha) = g - (r+1)(g-d+r) - \sum_{i=0}^r \alpha_i
$$
is nonnegative. If nonempty, this locus has dimension exactly $\rho(g,d,r,\alpha)$.
\end{thm}
This theorem will follow from Corollary \ref{cor_gbn}. We now define and give the geometric interpretation of generalized Brill-Noether loci, in both the algebraic and tropical contexts.
\subsection{Brill-Noether loci of marked algebraic curves} Let $C$ be a smooth projective algebraic curve of genus $g$, and let $d,r$ be nonnegative integers with $r \geq d-g$. Brill-Noether theory concerns the geometry of the schemes $W^r_d(C)$ parameterizing degree $d$ line bundles $L$ on $C$ such that $h^0(L) \geq r+1$. The reason that we require $r \geq d-g$ is that the Riemann-Roch formula guarantees that $h^0(L) \geq (d-g)+1$, regardless of the line bundle. If a marked point $p \in C$ is chosen, one can also impose ramification data at $p$. The \emph{vanishing orders} of a line bundle $L$ of rank $r$ (that is, $h^0(L) = r+1)$ at the point $p$ are the integers $a_0 < a_1 < \cdots < a_r$ such that there exists a section of $L$ vanishing to that order at $p$. Alternatively, one can define $a_i = \max \{ n:\ h^0( L (-n\cdot p) ) \geq r+1-i$. The \emph{ramification orders} are the nondecreasing sequence of integers $\alpha_0 \leq \alpha_1 \leq \cdots \leq \alpha_r$ defined by $\alpha_i = a_i - i$. We therefore make the following definition.
\begin{defn} \label{def:wrad}
Let $(C,p)$ be a smooth projective curve with a marked point. The variety of line bundles of rank at least $r$ with ramification at least $\alpha$ at $p$ is
$$
W^{r,\alpha}_d(C,p) = \{ L \in \Pic^d(C):\ h^0(L( - (\alpha_i+i) p ) ) \geq r+1 - i\ \mbox{for }i = 0,1,\cdots,r\}.
$$
\end{defn}
To the data $g,r,d,\alpha$, we associate a partition $(\lambda_0,\cdots,\lambda_r)$ as follows (see Figure \ref{fig:wrad_lambda} for a visual illustration).
$$
\lambda_i = (g-d+r) + \alpha_{r-i}
$$
\begin{figure}
\begin{tikzpicture}[scale=0.5]
\draw (0,0) rectangle (6,4);
\foreach \i in {1,2,...,5} \draw (\i,0) -- (\i,4);
\foreach \j in {1,2,3} \draw (0,\j) -- (6,\j);
\draw[snake=brace,thick] (6,-0.25) -- (0,-0.25) node[midway,below] {$g-d+r$};
\draw[snake=brace,thick] (-0.25,0) -- (-0.25,4) node[midway,left] {$r+1$};
\begin{scope}[xshift=0.4cm]
\draw (6,0) rectangle (9,1); \draw (7,0) -- (7,3); \draw (8,0) -- (8,1);
\draw (6,1) rectangle (8,2);
\draw (6,2) rectangle (8,3);
\draw (6,3) rectangle (6,4);
\draw (9,0) node[above right] {$\alpha_r$};
\draw (8,1) node[above right] {$\vdots$};
\draw (8,2) node[above right] {$\alpha_1$};
\draw (6,3) node[above right] {$\alpha_0$};
\end{scope}
\end{tikzpicture}
\caption{The partition $\lambda$ associated to the data $g,d,r,\alpha$. In this example, $r=3,\ d=g-3$, and $\alpha = (0,2,2,3)$.} \label{fig:wrad_lambda}
\end{figure}
An alternative description that we find useful is that $\lambda$ is the minimal partition whose Young diagram (regarded as a subset of $\textbf{Z}^2_{>0}$) contains all of the points in
$$\{ (g-d+r + \alpha_i, r+1-i):\ i =0,1,\cdots,r\} \cap \textbf{Z}^2_{>0}.$$
Observe that many such varieties $W^{r,\alpha}_d(C,p)$ are automatically isomorphic: if $d$ is increased by $1$, as are all of the integers $\alpha_i$ (that is, if a \emph{base point} is added at $p$), the result is an isomorphic variety, now in $\Pic^{d+1}(C)$, obtained by applying the map $L \mapsto L(p)$ to the original. Therefore there is substantial redundancy in studying all possible data $g,r,d,\alpha$. However, note that any two sets of data $(g,r,d,\alpha)$ that are isomorphic for the reason described give the same partition $\lambda$ (the difference is simply where the gap is drawn in Figure \ref{fig:wrad_lambda}). Indeed, we can identify both with a single subscheme of $\Pic^0(C)$, defined as follows.
\begin{defn} \label{def:wl_alg}
The \emph{Brill-Noether locus} corresponding to the partition $\lambda$ and the marked curve $(C,p)$ is
\begin{align*}
W^\lambda(C,p) =& \{ L \in \Pic^0(C):\ h^0( L(d' \cdot p) ) \geq r'+1 \\&\mbox{ whenever } (g-d'+r',r'+1) \in \lambda\}.
\end{align*}
\end{defn}
\begin{rem}
If $h^0(L(d \cdot p)) \geq r+1$, then it follows automatically that $h^0(L((d+1) \cdot p)) \geq r+1$ and $h^0(L((d-1) \cdot p)) \geq r$. In other words, once $(x,y) \in \lambda$, there are no additional restrictions imposed by $(x-1,y)$ or $(x,y-1)$. This explains why there is no loss of generality in requiring $\lambda$ to be a partition, i.e. to be closed leftward and downward. Also, the Riemann-Roch theorem guarantees that if $r<0$ or $g-d+r \leq 0$, then the condition $h^0(L(d\cdot p)) \geq r+1$ holds for \emph{all} line bundles $L \in \Pic^0(C)$, so there is no loss of generality in only allowing $\lambda$ to have points with both coordinates positive.
\end{rem}
Comparing Definition \ref{def:wrad} and Definition \ref{def:wl_alg}, it follows that the map $L \mapsto L(-d\cdot p )$ gives an isomorphism $W^{r,\alpha}_d(C,p) \rightarrow W^\lambda(C,p)$, where $\lambda$ is the partition associated to $g,r,d,\alpha$ as described above. Note in particular that when $\lambda$ is a rectangle (i.e. the minimal partition containing $(g-d+r,r+1)$), we have $W^\lambda(C,p) \cong W^r_d(C)$ (regardless of the marked point $p$). Therefore it suffices to study the geometry of $W^\lambda(C,p)$. We hope that the results of this paper will convince the reader that indexing Brill-Noether varieties (with one marked point) by the partition $\lambda$ is the most natural way to study their geometry.
The scheme structure on $W^\lambda(C,p)$ is defined as an intersection of degeneracy loci of certain maps of vector bundles over $\Pic^0(C)$. We summarize this description in Section \ref{sec_ag}, where we also deduce from intersection theory that $\dim W^\lambda(C,p) \geq g - |\lambda|$, with equality if and only if the class of $W^\lambda(C,p)$ (either in the Chow group or in singular cohomology) is
\begin{equation} \label{eq_class}
[W^\lambda(C,p)] = \left. \Theta^{|\lambda|} \middle/ \prod_{(x,y) \in \lambda} \hook(x,y). \right.
\end{equation}
In formula \eqref{eq_class}, $\hook(x,y)$ denotes the hook length of the box $(x,y)$ in the Young diagram of $\lambda$, i.e. the total number of boxes in $\lambda$ that are either of the form $(x',y)$ with $x' \geq x$ or $(x,y')$ with $y' \geq y$. In particular, since $\deg \Theta^g = g!$, this formula shows in the case $g = |\lambda|$ that if $W^\lambda(C,p)$ is $0$-dimensional, then its cardinality is equal, by the hook length formula \cite{frame}, to the number of standard Young tableaux on $\lambda$. Observe in particular that equation \eqref{eq_class} does not depend on $g$ at all, but only on $\lambda$; we offer this as evidence that $\lambda$ is the most natural combinatorial object with which to parameterize Brill-Noether varieties of marked curves.
Theorem \ref{thm_gbn} can therefore be restated to say that, for a general point $(C,p)$ in $\mathcal{M}_{g,1}$ over an algebraically closed field, the dimension of $W^\lambda(C,p)$ is always equal to $g - |\lambda|$ (or $W^\lambda(C,p)$ is empty if $|\lambda| > g$), and formula \eqref{eq_class} gives the class of this locus. This will be proved as Corollary \ref{cor_gbn}.
\subsection{Chains of cycles}
If $\Gamma$ is a metric graph, $w \in \Gamma$ is a point, and $\lambda$ is a partition, then we define a generalized Brill-Noether locus $W^\lambda(\Gamma,w) \subseteq \Pic^0(\Gamma)$ in the same manner as for marked algebraic curves. Here $r(\cdot)$ denotes the Baker-Norine rank of a divisor (see \cite{mz} or \cite{gk}).
\begin{align*}
W^\lambda(\Gamma,w) =& \{ [D] \in \Pic^0(\Gamma):\ r(D + d'\cdot p) \geq r' \\& \mbox{ whenever } (g-d'+r',r'+1) \in \lambda \}
\end{align*}
We will consider a metric graph $\Gamma$ composed of a chain of cycles as shown in Figure \ref{fig_chain}. We require that $v_i$ and $w_i$ are \emph{distinct} points for each $i$. The point $w_g$ will serve as the marked point. We have drawn the chain with bridges from $w_i$ to $v_{i+1}$ to make some of our notation simpler, but all of our results are unaffected if the bridges have length $0$.
\begin{figure}
\begin{center}
\begin{tikzpicture}
\draw (0,0) circle[radius=1];
\coordinate (w1) at ({cos(70)},{sin(70)});
\coordinate (v1) at ({cos(110)},{sin(110)});
\draw (3,0) circle[radius=1];
\coordinate (w2) at ({3+cos(70)},{sin(70)});
\coordinate (v2) at ({3+cos(110)},{sin(110)});
\draw (7,0) circle[radius=1];
\coordinate (wg) at ({7+cos(70)},{sin(70)});
\coordinate (vg) at ({7+cos(110)},{sin(110)});
\draw[fill=black] (w1) circle[radius=0.05];
\draw[fill=black] (v1) circle[radius=0.05];
\draw[fill=black] (w2) circle[radius=0.05];
\draw[fill=black] (v2) circle[radius=0.05];
\draw[fill=black] (wg) circle[radius=0.05];
\draw[fill=black] (vg) circle[radius=0.05];
\draw[rounded corners = 2ex] (w1) -- (1,1.25) -- (2,1.25) -- (v2);
\draw[rounded corners = 2ex] (w2) -- (4,1.25) -- (4.25,1.25);
\draw[rounded corners = 2ex] (5.75,1.25) -- (6,1.25) -- (vg);
\draw (5,1.25) node {$\cdots$};
\draw (5,0) node {$\cdots$};
\draw (v1) node[above] {$v_1$};
\draw (w1) node[above] {$w_1$};
\draw (v2) node[above] {$v_2$};
\draw (w2) node[above] {$w_2$};
\draw (vg) node[above] {$v_g$};
\draw (wg) node[above] {$w_g$};
\end{tikzpicture}
\end{center}
\caption{The chain of cycles $\Gamma$.}\label{fig_chain}
\end{figure}
We will show that all of the information about the Brill-Noether loci of $(\Gamma,w_g)$ is encoded in a sequence $\underline{m} = (m_2, \cdots, m_g)$ of $g-1$ nonnegative integers, called the \emph{torsion profile} of $\Gamma$. The integer $m_i$ will be called the \emph{$i$th torsion order} and is defined as follows.
\begin{defn} \label{def:tp}
Let $\ell_i$ denote the length of the $i$th cycle, and let $\ell(v_i w_i)$ denote the length of the clockwise edge from $v_i$ to $w_i$. If $\ell(v_i w_i)$ is an irrational multiple of $\ell_i$, then the $i$th torsion order $m_i$ is $0$. Otherwise, $m_i$ is the minimum positive integer such that $m_i \cdot \ell(v_i w_i)$ is an integer multiple of $\ell_i$.
\end{defn}
An equivalent description (which explains the name) is that $m_i$ identifies precisely which multiples of the divisor $w_i - v_i$ are linearly equivalent to $0$. Note that $m_i \neq 1$, since we prohibit $v_i = w_i$.
We summarize this notation in the following situation, for easy reference.
\begin{sit} \label{sit_comb}
Let $\Gamma$ be a chain of cycles with points $v_i,w_i$ as shown in Figure \ref{fig_chain}, and let $m_i$ (for $i=1,2,\cdots,g$) denote the $i$th torsion order of this chain. The \emph{torsion profile} of $\Gamma$ is denoted by $\underline{m} = (m_2,\cdots,m_g)$.
\end{sit}
Although we have defined $m_i$ for all $i \in \{1,2,\cdots,g\}$, note that we do not include the first torsion order $m_1$ in the torsion profile $\underline{m}$. This is because the value of $m_1$ is determined only by the position of $v_1$ on the first cycle, which in immaterial to the properties of the marked metric graph $(\Gamma,w_g)$.
The Jacobian $\Pic^0(\Gamma)$ is naturally isomorphic to the product of all $g$ cycles of $\Gamma$ (see Lemma \ref{l_stdform} for an explicit isomorphism that we will be the basis of our constructions). Our main result on the structure of $W^\lambda(\Gamma,w_g)$, Theorem \ref{t_tori}, states that $W^\lambda(\Gamma,w_g)$ is always equal to a union of sub-tori, each given by fixing a point on some of the cycles but not others. These tori are in bijection with $\underline{m}$-displacement tableaux, which depend on the torsion profile.
The proof of Theorem \ref{t_tori} is largely similar to the method in \cite{cdpr}: special divisors are constructed one cycle at a time, by placing a single chip on each cycle. The choice of where to place the chip of cycle $i$ is determined in \cite{cdpr} by the data of a \emph{lingering lattice path}, which can in turn be specified by a Young tableau on $\lambda$. We give a slightly different construction of divisors from Young tableaux. Unlike in \cite{cdpr}, the divisors we produce are not necessary $v_1$-reduced divisors. However, the dependence of the chip positions on the tableau is more transparent. The idea underlying our construction is that we construct, inductively, divisors on a \emph{sequence of chains} $\Gamma_1,\cdots, \Gamma_g = \Gamma$ of genera $1,2,\cdots, g$, each obtained by taking only the first $i$ cycles of $\Gamma$. Each divisor (when twisted by a multiple of $w_i$ until it has degree $0$) lies in a Brill-Noether locus $W^{\lambda_i}(\Gamma_i,w_i)$, where the partitions $\lambda_i$ form an increasing sequence wherein the box $(x,y)$ is added to the partition $\lambda_i$ if and only if $(x,y)$ has label $i$ in the chosen tableau. Even in the case where we ultimately care only about the loci $W^r_d(\Gamma)$, the more general loci $W^\lambda(\Gamma,w_g)$ are essential, as they are used to record the intermediate steps in this construction. In this way, our method provides a geometric interpretation of the lingering lattice paths of \cite{cdpr}: a specific lingering lattice path records the precise way in which the ramification conditions grow as cycles are attached to $\Gamma$.
\begin{rem}
There are two contexts in algebraic geometry very similar to divisor theory on chains of cycles: limit linear series on chains of elliptic curves and limit linear series (as defined for curves of pseudo-compact type in \cite{oss14dim}) on stable curves formed as a chain of pairs of rational curves joined at two points. In both cases, we suspect that a similar combinatorial analysis should yield a description of the dimension of these spaces of limit linear series, and a procedure to enumerate their irreducible components. Osserman discusses the link between the latter context and that of tropical chains of cycles in \cite{oss14dim}.
\end{rem}
\subsection{Tropicalization}
The interplay between algebraic and tropical geometry will take place in the following situation.
\begin{sit} \label{sit_main}
In addition to the notation of Situation \ref{sit_comb}, let $K$ be a complete valued field with infinite residue field, and let $C$ be a smooth projective curve over $K$. Assume that $C$ has totally split reduction and minimal skeleton isometric to $\Gamma$, and that $w_g$ is rational over the value group of $K$. Let $p \in C$ be a $K$-point specializing to $w_g \in \Gamma$.
\end{sit}
We show in Section \ref{sec_trop} that Theorem \ref{thm_genericity2} implies the following result on algebraic curves.
\begin{thm} \label{thm_introalg}
In Situation \ref{sit_main}, if the torsion profile $\underline{m}$ of $\Gamma$ satisfies $m_i = 0$ or $m_i > i$ for all $i$, then $(C,p)$ is Brill-Noether general (as a marked algebraic curve).
\end{thm}
The generalized Brill-Noether Theorem \ref{thm_gbn} will follow, via Corollary \ref{cor_gbn}.
\subsection{Organization of the paper}
We describe displacement tableaux in Section \ref{sec_dt}. The core of the paper is Section \ref{sec_comb}, in which we parameterize the Brill-Noether loci on marked chains of cycles and prove our main result, Theorem \ref{t_tori}, which establishes a correspondence between the components of $W^\lambda(\Gamma,w_g)$ and displacement tableaux. We also deduce Theorems \ref{thm_genericity1} and \ref{thm_genericity2}. Section \ref{sec_ag} summarizes the basic facts about the scheme structure of $W^\lambda(C,p)$ in the algebraic case, and its analysis via intersection theory. Section \ref{sec_trop} proves two semicontinuity properties needed to derive results in algebraic geometry from our tropical results, which we apply to prove Theorem \ref{thm_introalg} and complete our tropical proof of the generalized Brill-Noether Theorem \ref{thm_gbn}. Finally, Section \ref{sec_qu} states some further questions suggested by our work.
\subsection*{Acknowledgements}
The key ideas for this paper came about through conversations with Melody Chan, David Jensen, and Sam Payne. The author is also grateful to Dhruv Ranganathan and Nicola Tarasca for several helpful conversations.
\section{Displacement tableaux} \label{sec_dt}
The description in \cite{cdpr} of special divisors on generic chains gave a bijection between irreducible components of Brill-Noether loci and standard Young tableaux. Our description requires the following generalization of standard tableaux, which incorporates the possibility of special torsion profiles by allowing some symbols from the alphabet to occur more than once.
\begin{defn} \label{def:dt}
Let $\lambda$ be a partition, and let $\underline{m} = (m_2,\cdots,m_g)$ be a $(g-1)$-tuple of nonnegative integers. An \emph{$\underline{m}$-displacement tableux} on $\lambda$ is a function $t:\ \lambda \rightarrow \{1,2,\cdots,g\}$ satisfying the following properties.
\begin{enumerate}
\item $t$ is strictly increasing in any given row or column of $\lambda$.
\item For any two distinct boxes $(x,y),(x',y')$ in $\lambda$ such that $t(x,y) = t(x',y')$, the following congruence holds. $$x-y \equiv x'-y' \pmod{m_{t(x,y)}}$$
\end{enumerate}
We write $t \vdash_{\underline{m}} \lambda$ to denote that $t$ is an $\underline{m}$-displacement tableau on $\lambda$.
\end{defn}
Observe that if $t$ satisfies property (1), then it is impossible for two different boxes in $t$ to have the label $1$. Therefore property (2) will never refer to the (unspecified) value of $m_1$.
\begin{eg}
The tableau $$\young(6,3456,1234)$$ is a $(0,3,3,0,2)$-displacement tableau. The only repeated symbols are $3$, $4$, and $6$, each of which occurs twice, in two places separated by a lattice distances $3$, $3$, and $4$, respectively. Since $m_3 \mid 3$, $m_4 \mid 3$, and $m_6 \mid 4$, this is a $(0,3,3,0,2)$-displacement tableau.
\end{eg}
\begin{eg}
Let $\underline{0}$ denote the $(g-1)$-tuple $(0,0,\cdots,0)$. Then a $\underline{0}$-displacement tableau is the same thing as a standard Young tableau with alphabet $\{1, 2, \cdots, g\}$ (note that the alphabet may have more elements than $\lambda$ has boxes, so not all symbols from the alphabet need to appear).
\end{eg}
The following notation will be convenient.
\begin{notn}
Let $\lambda$ be a partition (identified, as usual, with a subset of $\textbf{Z}_{>0} \times \textbf{Z}_{>0}$). Let $\overline{\lambda}$ denote the union of $\lambda$ with all points $(x,y) \in \textbf{Z} \times \textbf{Z}$ such that either $x \leq 0$ or $y \leq 0$.
\end{notn}
\begin{rem}
In some sense, $\overline{\lambda}$ is the more natural set in our application, since the Riemann-Roch formula guarantees that if either coordinate of the point $(g-d+r,r+1)$ is nonpositive, then \emph{all} degree $d$ divisors have rank at least $r$.
\end{rem}
A displacement tableau $t$ may be interpreted as assembly instructions for the partition $\lambda$: boxes are added in $g$ successive steps to produce a sequence of partitions. The permissible boxes to add at any given step are governed by the notion of displacement of a partition, which we now define (see Lemma \ref{lem_disp} for the origin of displacement in our application).
\begin{defn}
Let $\lambda$ be a partition, and let $S$ be a set of integers. The \emph{upward displacement} of $\lambda$ by $S$ is the partition
$$\disp^+(\lambda,S) = \lambda \cup L,$$
where $L$ is the set of boxes $(x,y) \not\in \lambda$ such that $x-y \in S$, $(x-1,y) \in \overline{\lambda}$, and $(x,y-1) \in \overline{\lambda}$. The elements of $L$ are called the \emph{loose boxes of $\lambda$ with respect to $S$}.
\end{defn}
The displacement process is easy to visualize: each element of $z \in S$ causes a slope-$1$ line through $(x,y) = (z,0)$ to be drawn through the Young diagram. Wherever these lines meet an inward corner, they turn that corner outwards, adding a box to the partition. Here, we regard the four corners of the box $(x,y)$ as being the points $(x,y),(x-1,y),(x-1,y-1)$, and $(x,y-1)$ in the plane.
\begin{eg}
Let $\lambda = (7,6,5,1)$, and let $S = 1 + 3 \textbf{Z}$. Then $\disp^+(\lambda,S) = (8,6,5,2)$, as illustrated in the figure below.
\begin{center}
\begin{tikzpicture}[scale=0.4]
\draw (0,4) -- (0,0) -- (7,0);
\draw (0,4) -- (1,4) -- (1,0);
\draw (0,3) -- (5,3) -- (5,0);
\draw (2,3) -- (2,0);
\draw (3,3) -- (3,0);
\draw (4,3) -- (4,0);
\draw (0,2) -- (6,2) -- (6,0);
\draw (0,1) -- (7,1) -- (7,0);
\draw[style=ultra thick, <->] (0,6) -- (0,4) -- (1,4) -- (1,3) -- (5,3) -- (5,2) -- (6,2) -- (6,1) -- (7,1) -- (7,0) -- (9,0);
\draw[style=dotted,thick,->] (-1,4) -- (2,7);
\draw[style=dotted,thick,->] (-1,1) -- (5,7);
\draw[style=dotted,thick,->] (0,-1) -- (8,7);
\draw[style=dotted,thick,->] (3,-1) -- (11,7);
\draw[style=dotted,thick,->] (6,-1) -- (11,4);
\draw[style=dotted,thick,->] (9,-1) -- (11,1);
\draw[pattern=north west lines,pattern color=gray] (1,3) rectangle (2,4);
\draw[pattern=north west lines,pattern color=gray] (7,0) rectangle (8,1);
\end{tikzpicture}
\end{center}
\end{eg}
\begin{lemma} \label{l_dispmono}
If $\lambda,\lambda'$ are two partitions such that $\lambda \subseteq \lambda'$ and $S$ is any set of integers, then $\disp^+(\lambda,S) \subseteq \disp^+(\lambda',S)$.
\end{lemma}
\begin{proof}
Suppose that $(x,y)$ is a box in $\disp^+(\lambda,S)$ that is not contained in $\lambda'$. Then certainly it is not contained in $\lambda$, so it was loose in $\lambda$. Therefore $(x-1,y)$ and $(x,y-1)$ both lie in $\overline{\lambda}$, and hence in $\overline{\lambda'}$. Thus $(x,y)$ was also loose in $\lambda'$, so it lies in $\disp^+(\lambda',S)$. Therefore every box in $\disp^+(\lambda,S)$ also lies in $\disp^+(\lambda',S)$.
\end{proof}
Displacement of partitions and displacement tableaux are related by the following combinatorial fact.
\begin{cor} \label{c_tabpart}
Let $t$ be an $\underline{m}$-displacement tableau on a partition $\lambda$, and let $S_1,S_2,\cdots,S_g$ be sets of integers such that for all boxes $(x,y) \in \lambda$, $x-y \in S_{t(x,y)}$. Define a sequence of partitions $\lambda_0,\lambda_1,\cdots,\lambda_g$ inductively as follows.
\begin{align*}
\lambda'_0 &= \emptyset\\
\lambda'_{i+1} &= \disp^+(\lambda'_i, S_{i+1})
\end{align*}
Then $\lambda \subseteq \lambda'_g$.
\end{cor}
\begin{proof}
Let $\lambda_i = t^{-1}(\{1,2,\cdots,i\})$ be the sub-partition of $\lambda$ given by boxes with labels less than or equal to $i$. By definition of displacement tableaux, all boxes with label $i+1$ are loose in $\lambda_i$ with respect to the set $S_i$. Therefore $\lambda_{i+1} \subseteq \disp^+(\lambda_i,S_{i+1})$. It follows from Lemma \ref{l_dispmono} and induction on $i$ that $\lambda_i \subseteq \lambda'_i$ for all $i$.
\end{proof}
\section{Combinatorics of $W^\lambda(\Gamma,w_g)$} \label{sec_comb}
Let $(\Gamma,w_g)$ be as in Situation \ref{sit_comb}. The language of displacement tableaux allows a succinct and explicit parameterization (Theorem \ref{t_tori}) of all Brill-Noether loci $W^\lambda(\Gamma,w_g)$. We will use the following notation to conveniently denote points of the cycles of $\Gamma$.
\begin{defn}
In Situation \ref{sit_comb}, for any $\xi \in \textbf{R}$ let $\la{\xi}_i$ denote the point on the $i$th cycle that is located $\xi \cdot \ell(v_iw_i)$ units clockwise from $w_i$, where $\ell(v_iw_i)$ denotes the length of the clockwise edge from $v_i$ to $w_i$.
\end{defn}
\begin{rem} \label{r_lanot}
In this notation,
\begin{enumerate}
\item The points $v_i$ and $w_i$ are equal to $\la{-1}_i$ and $\la{0}_i$, respectively;
\item If $n_1,n_2$ are \emph{integers}, then $\la{n_1}_i = \la{n_2}_i$ if and only if $n_1\equiv n_2 \pmod{m_i}$, where $m_i$ denotes the torsion order on the $i$th cycle.
\end{enumerate}
\end{rem}
The following lemma gives a convenient bijection between $\Pic^d(\Gamma)$ and the product of the $g$ cycles of $\Gamma$.
\begin{lemma} \label{l_stdform}
Let $D$ be any divisor of degree $d$ on $\Gamma$. Then $D$ is linearly equivalent to a unique divisor of the form $$\sum_{i=1}^g \la{\xi_i}_i + (d-g)\cdot w_g.$$
\end{lemma}
\begin{proof} Assume without loss of generality that $d = g$. So we must show that every degree $g$ divisor is linearly equivalent to a unique sum of $g$ points $\la{\xi_i}_i$, with exactly one point on each cycle.
\textit{Existence of $\xi_i$:} We may assume that the support of $D$ is contained only on the cycles, since any point on a bridge is linearly equivalent to every other point on the the bridge. Next, by adding an integer linear combination of the divisors $w_i - v_{i+1}$ (each of which is linearly equivalent to $0$), we may assume that the portion of $D$ supported on the $i$th cycle has degree $1$. By the Riemann-Roch formula, any degree $1$ divisor on a genus $1$ metric graph is linearly equivalent to an effective degree $1$ divisor. Therefore we can replace the part of $D$ supported on cycle $i$ by a linearly equivalent divisor of the form $\la{\xi_i}_i$ for some $\xi_i$.
\textit{Uniqueness of $\la{\xi_i}_i$:} Suppose that $D = \sum_{i=1}^g \la{\xi_i}_i$ and $D' = \sum_{i=1}^g \la{\xi_i'}_i$ are two linearly equivalent degree $g$ divisors, each consisting of exactly one chip on each cycle. Then $D-D'$ is the principle divisor of a rational function $f$ on $\Gamma$. Since the degree of the part of $D-D'$ to the left of $v_i$ or the right of $w_i$ is $0$, the slope of $f$ is $0$ on all bridges. Therefore, by restricting $f$ to the $i$th cycle, we obtain a rational function on the cycle whose principle divisor is equal to $\la{\xi_i}_i - \la{\xi_i'}_i$. But two points on a cycle are linearly equivalent as divisors if and only if they are the same point: otherwise the rational function whose principle divisor is their difference would have slope $s$ on one interval, and slope $s+1$ on a complementary interval of the cycle, which is impossible since both these slopes would be nonnegative or both nonpositive, with at least one nonzero.
\end{proof}
\begin{rem}
The values $\xi_i$ are easy to compute, given the divisor $D$. Define, for any point $p \in \Gamma$ and index $i$, the number $\widetilde{\xi}_i(p)$ to be $-1$ if $p$ lies to the left of $v_i$, $0$ if $p$ lies to the right of $w_i$, and otherwise to be $\zeta$ such that $p = \la{\zeta}_i$. The function $\widetilde{\xi}_i$ extends to all divisors by linearity. Then the desired values $\xi_i$ are
$$\xi_i(D) = (i-1) + \widetilde{\xi}_i(D).$$
The value $\xi_i$ is well-defined modulo the length of the the $i$th cycle. Note that $\xi_i(D)$ is not a linear function of the divisor $D$. Instead, it has the feature that $\xi_i(K_\Gamma -D) = -\xi_i(D)$, since $\xi_i(K_{\Gamma}) = 2(i-1)$.
\end{rem}
Lemma \ref{l_stdform} identifies $\Pic^d(\Gamma)$ with the product of the $g$ cycles. Every displacement tableau $t \vdash_{\underline{m}} \lambda$ defines a sub-torus in $\Pic^0(\Gamma)$ as follows.
\begin{defn} \label{def:Tt}
Use the notation of Situation \ref{sit_comb}. Let $t$ be an $\underline{m}$-displacement tableau on a partition $\lambda$. Denote by $\textbf{T}(t)$ the set of divisor classes of the form $$\displaystyle \sum_{i=1}^g \la{\xi_i}_i - g \cdot w_g,$$ where $\xi_1,\cdots,\xi_g$ are real numbers such that $$\xi_{t(x,y)} \equiv x-y \mod{m_{t(x,y)}}$$ for all $(x,y) \in \lambda$.
\end{defn}
Observe that while this definition makes use of the first torsion order $m_1$ of the chain in question, although this number is not included in the torsion profile $\underline{m}$. This apparent discrepancy is harmless since the value of $m_1$ is immaterial to the definition of an $\underline{m}$-displacement tableau.
\begin{rem}
If $t^\ast$ is the dual tableau of $t$ (obtained by switching the axes), then $\textbf{T}(t^\ast)$ is the Serre dual of $\textbf{T}(t)$, twisted by $-(2g-2)w_g$ (i.e. the set of classes $[K_\Gamma - D - (2g-2)w_g]$ such that $[D] \in \textbf{T}(t)$).
\end{rem}
The definition of $\underline{m}$-displacement tableaux ensures that $\textbf{T}(t)$ is well-defined and non-empty, since no two boxes of $t$ will impose conflicting conditions on $\xi_i$ (due to the second part of Remark \ref{r_lanot}).
The main result of this paper, Theorem \ref{t_tori} from the introduction, asserts that
$$W^\lambda(\Gamma,w_g) = \bigcup_{t \vdash_{\underline{m}} \lambda} \textbf{T}(t).$$
We prove Theorem \ref{t_tori} in Section \ref{ss_pftori}. The key ingredient is Lemma \ref{lem_disp}, which illuminates the link between displacement of partitions and Brill-Noether loci.
\subsection{Applications and examples of Theorem \ref{t_tori}}
For now, we will assume the result of Theorem \ref{t_tori}, demonstrate how to use it in several examples, and deduce some consequences.
\begin{cor} \label{c_dp}
The dimension of the largest component of $W^\lambda(\Gamma,w_g)$ is equal to the maximum number of omitted symbols in an $\underline{m}$-displacement tableau on $\lambda$. This is equal to $g - |\lambda|$ if and only if it is impossible for an $\underline{m}$-displacement tableau on $\lambda$ to have any repeated symbols.
\end{cor}
\begin{proof}
Each $\textbf{T}(t)$ is homeomorphic to a torus, whose dimension is equal to the number of symbols $i \in \{1,2,\cdots,g\}$ that do not occur in $t$. This dimension is equal to $g - |\lambda|$ if and only if each symbol in $t$ occurs exactly once.
\end{proof}
\begin{cor} \label{c_gencrit}
The metric graph $\Gamma$ is Brill-Noether general (as a metric graph without a marked point, i.e. in the original sense of \cite{cdpr}) if and only if every $\underline{m}$-displacement tableau on a \emph{rectangular} partition has all entries distinct. The marked metric graph $(\Gamma,w_g)$ is Brill-Noether general (as a marked metric graph) if and only if every $\underline{m}$-displacement tableau on \emph{any} partition has all entries distinct.
\end{cor}
\begin{eg}
Suppose that $g=4$ and consider the locus $W^1_3(\Gamma)$. This is in bijection with $W^\lambda(\Gamma,w_g)$, where $\lambda = (2,2)$. From the theory of algebraic curves, we expect this to be finite set with two elements, unless $\Gamma$ is special. Indeed, there are two standard Young tableaux on $\lambda$, which correspond to the following two divisor classes of degree $3$.
\begin{align*}
\young(34,12) \hspace{1cm} & \begin{tikzpicture}[thick,scale=0.6, every node/.style={scale=0.6}]
\draw (0,0) circle[radius = 1];
\draw[rounded corners = 1ex] ({0+cos(70)},{sin(70)}) -- (1.000000,1.25) -- (1.500000,1.25);
\draw (3,0) circle[radius = 1];
\draw[rounded corners = 1ex] ({3+cos(110)},{sin(110)}) -- (2.000000,1.25) -- (1.500000,1.25);
\draw[rounded corners = 1ex] ({3+cos(70)},{sin(70)}) -- (4.000000,1.25) -- (4.500000,1.25);
\draw (6,0) circle[radius = 1];
\draw[rounded corners = 1ex] ({6+cos(110)},{sin(110)}) -- (5.000000,1.25) -- (4.500000,1.25);
\draw[rounded corners = 1ex] ({6+cos(70)},{sin(70)}) -- (7.000000,1.25) -- (7.500000,1.25);
\draw (9,0) circle[radius = 1];
\draw[rounded corners = 1ex] ({9+cos(110)},{sin(110)}) -- (8.000000,1.25) -- (7.500000,1.25);
\draw[fill=black] ({0+cos(70)},{sin(70)}) circle[radius=0.1];
\draw[fill=black] ({6+cos(110)},{sin(110)}) circle[radius=0.1];
\draw[fill=black] ({3+cos(30)},{sin(30)}) circle[radius=0.1];
\end{tikzpicture}
\\
&\la{0}_1 + \la{1}_2 + \la{-1}_3 + \la{0}_4 - w_g\\
\young(24,13) \hspace{1cm} &\begin{tikzpicture}[thick,scale=0.6, every node/.style={scale=0.6}]
\draw (0,0) circle[radius = 1];
\draw[rounded corners = 1ex] ({0+cos(70)},{sin(70)}) -- (1.000000,1.25) -- (1.500000,1.25);
\draw (3,0) circle[radius = 1];
\draw[rounded corners = 1ex] ({3+cos(110)},{sin(110)}) -- (2.000000,1.25) -- (1.500000,1.25);
\draw[rounded corners = 1ex] ({3+cos(70)},{sin(70)}) -- (4.000000,1.25) -- (4.500000,1.25);
\draw (6,0) circle[radius = 1];
\draw[rounded corners = 1ex] ({6+cos(110)},{sin(110)}) -- (5.000000,1.25) -- (4.500000,1.25);
\draw[rounded corners = 1ex] ({6+cos(70)},{sin(70)}) -- (7.000000,1.25) -- (7.500000,1.25);
\draw (9,0) circle[radius = 1];
\draw[rounded corners = 1ex] ({9+cos(110)},{sin(110)}) -- (8.000000,1.25) -- (7.500000,1.25);
\draw[fill=black] ({0+cos(70)},{sin(70)}) circle[radius=0.1];
\draw[fill=black] ({3+cos(110)},{sin(110)}) circle[radius=0.1];
\draw[fill=black] ({6+cos(30)},{sin(30)}) circle[radius=0.1];
\end{tikzpicture}\\
& \la{0}_1 + \la{-1}_2 + \la{1}_3 + \la{0}_4 - w_g\\
\end{align*}
If $\underline{m} = \underline{0}$ (i.e. $\Gamma$ is general), then these are the only points of $W^1_3(\Gamma)$. However, there are two situations in which additional points exist: if $m_2 = 2$ or if $m_3 = 2$. For example, if $m_2 = 2$, there are two additional one-dimensional tori in $W^1_3(\Gamma)$, coming from the two $(0,2,0,0)$-displacement tableaux below. The second loop is drawn differently to show that $m_2 = 2$, and $\Asterisk$s denote either variables or points which are free to vary (giving the degree of freedom of the torus).
\begin{align*}
\young(24,12) \hspace{1cm} & \begin{tikzpicture}[thick,scale=0.600000, every node/.style={scale=0.600000}]
\draw (0,0) circle[radius = 1];
\draw[rounded corners = 1ex] (0.342020,0.939693) -- (0.444626,1.221600) -- (1.700000,0.000000) -- (2.000000,0.000000);
\draw (3,0) circle[radius = 1];
\draw[rounded corners = 1ex] (4.000000,0.000000) -- (4.300000,0.000000) -- (5.555374,1.221600) -- (5.657980,0.939693);
\draw (6,0) circle[radius = 1];
\draw[rounded corners = 1ex] (6.342020,0.939693) -- (6.444626,1.221600) -- (8.555374,1.221600) -- (8.657980,0.939693);
\draw (9,0) circle[radius = 1];
\draw[fill=black] (0.342020,0.939693) circle[radius=0.1];
\draw[fill=black] (2.000000,-0.000000) circle[radius=0.1];
\draw (6.342020,-0.939693) node {$\Asterisk$};
\end{tikzpicture}\\
& \la{0}_1 + \la{1}_2 + \la{\Asterisk}_3 + \la{0}_4 - w_g\\
\\
\young(23,12) \hspace{1cm} & \begin{tikzpicture}[thick,scale=0.600000, every node/.style={scale=0.600000}]
\draw (0,0) circle[radius = 1];
\draw[rounded corners = 1ex] (0.342020,0.939693) -- (0.444626,1.221600) -- (1.700000,0.000000) -- (2.000000,0.000000);
\draw (3,0) circle[radius = 1];
\draw[rounded corners = 1ex] (4.000000,0.000000) -- (4.300000,0.000000) -- (5.555374,1.221600) -- (5.657980,0.939693);
\draw (6,0) circle[radius = 1];
\draw[rounded corners = 1ex] (6.342020,0.939693) -- (6.444626,1.221600) -- (8.555374,1.221600) -- (8.657980,0.939693);
\draw (9,0) circle[radius = 1];
\draw[fill=black] (6.342020,0.939693) circle[radius=0.1];
\draw[fill=black] (0.342020,0.939693) circle[radius=0.1];
\draw[fill=black] (2.000000,-0.000000) circle[radius=0.1];
\draw[fill=red] (9.342020,0.939693) circle[radius=0.1];
\draw (9.342020,0.939693) node[above] {$-1$};
\draw (8.133975,-0.500000) node {$\Asterisk$};
\end{tikzpicture}\\
& \la{0}_1 + \la{1}_2 + \la{0}_3 + \la{\Asterisk}_4 - w_g\\
\end{align*}
\end{eg}
\begin{eg} (\textit{Hyperelliptic chains}) The locus $W^1_2(\Gamma)$ is nonempty if and only if there exists an $\underline{m}$-displacement tableau on $\lambda = (g-1,g-1)$. The only possible displacement tableau on this partition is the following.
$$
t = \begin{array}{|c|c|c|c|c|}\hline
2 & 3 & 4 & \cdots & g\\\hline
1 & 2 & 3 & \cdots & g-1\\\hline
\end{array}
$$
This is an $\underline{m}$-displacement tableau if and only if $m_2 = m_3 = \cdots = m_{g-1} = 2$. Therefore a chain has a degree $2$ divisor class of rank $1$ if and only if all of these torsion orders are $2$. Theorem \ref{t_tori} also shows that this divisor class is unique: it must be the divisor class of $\la{0}_1 + \sum_{i=2}^g \la{i-3}_i - (g-2) w_g$.
\end{eg}
We close this subsection by showing how Theorem \ref{t_tori} implies Theorems \ref{thm_genericity1} and \ref{thm_genericity2}, which give necessary and sufficient criteria for a chain of cycles to be Brill-Noether general (in the sense of \cite{cdpr}), and for a marked chain of cycles (with marked point on the rightmost cycle) to be Brill-Noether general (as a marked metric graph).
\begin{proof}[Proof of Theorem \ref{thm_genericity1}]
First, we show that the condition that $m_i = 0$ or $m_i > \min(i,g+1-i)$ is \textit{necessary} for Brill-Noether generality. Suppose that $i$ is an index such that $m_i \neq 0$, $m_i \leq g+1-i$, and $m_i \leq i$. Let $\lambda = (m_i,m_i)$ and define a function $t: \lambda \rightarrow \textbf{Z}$ by
\begin{align*}
t(x,1) &= x-m_i + i,\\
t(x,2) &= x+i-1.
\end{align*}
The given inequalities guarantee that $1 \leq t(x,y) \leq g$, and $t$ has exactly one repeated value: $t(m_i,1) = t(1,2) = i$. Therefore $t$ is an $\underline{m}$-displacement tableau on a rectangular partition with a repeated entry. It follows from Corollary \ref{c_gencrit} that $\Gamma$ is not Brill-Noether general.
Now we show that the condition is \textit{sufficient}. Suppose that $\Gamma$ is \textit{not} Brill-Noether general; we will show that there exists some $i$ such that $0 < m_i \leq \min(i,g+1-i)$. By corollary \ref{c_gencrit}, there exists a rectangular partition $\lambda$ and a displacement tableau $t \vdash_{\underline{m}} \lambda$ with a repeated symbol. Select such a rectangular partition $\lambda$ and tableau $t$ so that $|\lambda|$ is as small as possible. Suppose that $(x,y),(x',y')$ are two distinct boxes containing the index $i$ in $t$. Assume without loss of generality that $x < x'$ and $y > y'$. Then in fact $(x,y)$ is the upper-left corner of $\lambda$, and $(x',y')$ is the lower-right corner, since otherwise one can restrict $t$ to the rectangle with corners $(x,y),(x,y'),(x',y'),(x',y)$ and regard this as an $\underline{m}$-displacement tableau on a smaller rectangle, contradicting the minimality of $|\lambda|$. It follows that $x=1,\ y'=1$, and the \emph{only} repetition of symbols in $t$ is $t(1,y) = t(x',1)$. Let this common label be $i$. Then the boxes $(1,y-1),\ (1,y-2),\ \cdots (1,1), (2,1), \cdots, (x'-1,1)$ ($x'+y-3$ boxes total) all have distinct labels strictly less than $i$, so $x' + y - 3 \leq i-1$. Since $t$ is an $\underline{m}$-displacement tableau, $(y-1) - (1-x')$ is divisible by $m_i$; since $(y-1)-(1-x') = x' + y - 2$ is positive, it follows that $m_i > 0$ and $m_i \leq x' + y - 2 \leq i$. Similarly, all the boxes $(2,y),(3,y),\cdots,(x',y),(x',y-1),\cdots,(x',1)$ have distinct labels greater than $i$, and it follows that $x' + y - 3 \leq g-i$, hence $m_i \leq g+1-i$. Therefore $0 < m_i \leq \min(i,g+1-i)$, as desired.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm_genericity2}]
First, suppose that there exists an index $i$ such that $0 < m_i \leq i$. Let $\lambda$ be the partition $(m_i,1)$, and define a function $t$ in exactly the same way as in the proof of Theorem \ref{thm_genericity1} (but restricted to the smaller, non-rectangular partition). Then $t$ is an $\underline{m}$-displacement tableau with the symbol $i$ repeated, so $(\Gamma,w_g)$ is not Brill-Noether general.
Conversely, suppose that $(\Gamma,w_g)$ is not Brill-Noether general. We will show that there exists some $i \in \{2,3\cdots,g\}$ such that $m_i \leq i$. There exists some partition $\lambda$ with a $\underline{m}$-displacement tableau $t$ that repeats some symbol $i$. Suppose that $i = t(x,y) = t(x',y')$, and assume without loss of generality that $x < x'$ and $y > y'$. Then the boxes $(x,y-1), (x,y-2), \cdots, (x,y'), (x+1,y'), \cdots, (x'-1,y')$ constitute $(x-y) - (x'-y') - 1$ boxes with \emph{distinct} labels in $t$. Hence $i \geq (x-y) - (x'-y') > 0$. Since $m_i$ divides $(x-y) - (x'-y')$, it follows that $m_i \leq i$.
\end{proof}
\subsection{Displacement of Weierstrass partitions}
We now begin the process of proving Theorem \ref{t_tori}. We begin by linking the notion of displacement of partitions to divisors on metric graphs.
\begin{defn}
Let $D$ be a degree $0$ divisor on a marked metric graph $(\Gamma,w)$. Define
$$ \overline{\lambda}_{\Gamma,w}(D) = \{ (g-d+r,\ r+1):\ d,r \in \textbf{Z},\ r(D + d\cdot w) \geq r \}.$$
If $D$ is a divisor of \emph{any} degree, define $\overline{\lambda}_{\Gamma,w}(D)$ to be $\overline{\lambda}_{\Gamma,w}(D-\deg D \cdot w)$. The \emph{Weierstrass partition} of $D$ at $w$ is the set $$
\lambda_{\Gamma,w}(D) = \overline{\lambda}_{\Gamma,w}(D) \cap \textbf{Z}_{>0}^2.
$$
\end{defn}
\begin{rem}
The word \emph{Weierstrass} is chosen due to the following analogy with \emph{Weierstrass semigroups} on algebraic curve: if $(C,p)$ is a marked algebraic curve, $D$ is a divisor, and $\lambda_{C,p}(D)$ is defined in the same manner as above, then $\lambda_{C,p}(0)$ (the Weierstrass partition of divisor $0$) encodes the data of the Weierstrass semigroup of the point $p$.
\end{rem}
This set $\lambda_{\Gamma,w}(D)$ is (the boxes in the Young diagram of) a partition since $r(D-w) \leq r(D)$ and $r(D+w) \leq r(D) + 1$. The Riemann-Roch formula implies that $\overline{\lambda}_{\Gamma,w}$ always contains all points $(x,y)$ with $x\leq 0$ or $y\leq 0$, so no information is lost in considering only the (finite) set $\lambda_{\Gamma,w}(D)$. The definition of $W^\lambda(\Gamma,w)$ implies that for any partition $\lambda$,
$$[D] \in W^\lambda(\Gamma,w) \mbox{ if and only if } \lambda \subseteq \lambda_{\Gamma,w}(D).$$
The purpose of this subsection and the next is to explain how to compute the Weierstrass partition of any divisor on a chain of cycles as in Situation \ref{sit_comb}. We begin in a more general situation. Let $A$ be any metric graph, and $u \in A$ a point. Let $C$ be a cycle, and $v,w$ be two distinct points on $C$. Denote by $\Gamma$ the metric graph obtained by attaching $C$ to $A$ by an edge connecting $u$ to $v$ (see Figure \ref{fig_disp}). We will explain how the Weierstrass partition of a divisor $D$ at $w$ on $\Gamma$ is related to the Weierstrass partition of the restriction of $D$ to the metric graph $A$ at the marked point $u$. The relationship is surprisingly simple, and it is the origin of our choice of the word displacement: when the marked graph $(A,u)$ is ``displaced'' to the marked graph $(\Gamma,w)$, the Weierstrass partition is displaced upward (in the sense of the $\disp^+$ operation) in a manner than depends on the torsion order of the divisor $w-v$ (which in turn depends only on the edge lengths of $C$).
\begin{figure}
\begin{tikzpicture}
\draw (-3,0) node {$\Gamma = $};
\draw[fill, color = lightgray] (0,0) circle[radius=2cm];
\draw (0,0) node {$A$};
\draw[fill] ({1+cos(70)},{sin(70)}) circle[radius=0.05cm] node[above] {$u$};
\draw[fill] ({4+cos(110)},{sin(110)}) circle[radius=0.05cm] node[above] {$v$};
\draw (4,0) circle[radius=1cm];
\draw[fill] ({4+cos(70)},{sin(70)}) circle[radius=0.05cm] node[above] {$w$};
\draw[rounded corners = 2ex] ({1+cos(70)},{sin(70)}) -- (2,1.25) -- (3,1.25) -- ({4+cos(110)},{sin(110)});
\draw[decorate,decoration={brace,amplitude=0.25cm}] (5,-1) -- (3,-1);
\draw (4,-1.25) node[below] {$C$};
\end{tikzpicture}
\caption{A marked point being displaced across a single cycle.}\label{fig_disp}
\end{figure}
We will denote the points of $C$ as follows, parallel to our notation when describing the chain of cycles. Here $\ell(vw)$ denotes the length of the clockwise edge from $v$ to $w$. $$\la{\xi}= \textrm{the point on $C$ located $\xi\cdot \ell(vw)$ units clockwise from $w$.}$$ Let $m$ denote the nonnegative generator of $\{n \in \textbf{Z}:\ \la{n} = \la{0}\}$, i.e. the torsion order of the divisor $w-v$.
We summarize in the following lemma a few useful facts about metric graphs of this form.
\begin{lemma} \label{l_bridge}
Let $\Gamma$ be the metric graph described above.
\begin{enumerate}
\item Let $D_A,D_A'$ be two divisors on $A$, and let $D_C,D_C'$ be two divisors on $C$. Then $D_A + D_C$ is linearly equivalent to $D_A' + D_C'$ if and only if both
\begin{align*}
D_A + \deg(D_C) \cdot u &\sim D_A' + \deg(D_C')\cdot u \textrm{ as divisors on $A$, and}\\
D_C + \deg(D_A) \cdot v &\sim D_C' + \deg(D_A') \cdot v \textrm{ as divisors on $C$.}
\end{align*}
\item Let $\ell(C)$ denote the length of the cycle $C$. Define a function $\sigma: \Div(C) \rightarrow \textbf{R}/\left( \ell(C)\cdot \textbf{Z} \right)$ by $\la{\xi} \mapsto (\xi+\ell(C) \cdot \textbf{Z})$, extended by linearity. Two divisors on $C$ are linearly equivalent if and only if they have the same degree and the same image under $\sigma$.
\item Let $D_A,D_C$ be divisors supported on $A$ and $C$ respectively. The linear series $|D_A+D_C|_\Gamma$ is nonempty if and only if there exists an integer $n$ such that both $|D_A+n\cdot u|_A$ and $|D_C - n \cdot v|_C$ are nonempty. Here the subscript in the notation $|\cdot|_X$ indicates the metric graph on which the complete linear series is to be formed.
\end{enumerate}
\end{lemma}
\begin{proof}
The first part follows from the observation that a divisor $D$ is principal on $\Gamma$ if and only if the restriction of $D$ to $A$ differs from a principal divisor by a multiple of $u$ and the restriction of $D$ to $C$ differs from a principal divisor by a multiple of $v$. The second part is standard. The third part is a corollary of the first; we can take $D_A'$ and $D_C'$ to be the restrictions to $A$ and $C$ of any element of the complete linear series of $D_A + D_C$, after first moving all chips on the edge $uv$ to one end or the other.
\end{proof}
The following lemma is the key to all of our analysis of chains of cycles.
\begin{lemma} \label{lem_disp}
Let $D$ be any divisor of degree $g-1$ on $A$, where $g$ is the genus of $\Gamma$, and $p$ any point on $C$. Then
$$\lambda_{\Gamma,w}(D+p) =
\begin{cases}
\disp^+(\lambda_{A,u}(D), z+m\textbf{Z}) & \mbox{if $p=\la{z}$ for some integer $z$,}\\
\lambda_{A,u}(D) & \mbox{otherwise.}
\end{cases}$$
\end{lemma}
\begin{proof}
Throughout this proof, we will use the following vocabulary to make several statements more succinct: we call a divisor $D$ on a marked metric graph $(\Gamma,w)$ a \emph{weak $g^r_d$ for $(\Gamma,w)$} if $r(D + (d-\deg D) w) \geq r$. In other words, a weak $g^r_d$ differs from a $g^r_d$ by a multiple of the marked point\footnote{In the context of algebraic curves, ``a $g^r_d$'' usually means a \emph{linear series}, i.e. a choice of both a degree $d$ divisor class and a $r+1$-dimensional space of sections of the associated line bundle. In our context, there is nothing analogous to a vector space of sections, so we will use the phrase to refer only to the divisor class.}.
Fix positive integers $x,y$, and define integers $r = y-1$ and $d = y-x+g-1$. Then $(x,y) \in \lambda_{\Gamma,w}(D+p)$ if and only if $D+p$ is a weak $g^r_d$ for $(\Gamma,w)$. We will establish the following two claims.
\begin{enumerate}
\item If $p \neq \la{x-y}$, then $D+p$ is a weak $g^r_d$ for $\Gamma$ if and only if $D$ is a weak $g^r_{d-1}$ for $A$.
\item If $p = \la{x-y}$, then $D+p$ is a weak $g^r_d$ for $\Gamma$ if and only if $D$ is both a weak $g^r_d$ and a weak $g^{r-1}_{d-2}$ for $A$.
\end{enumerate}
Translated into the language of Weierstrass partitions, these claims assert that $(x,y) \in \lambda_{\Gamma,w}(D+p)$ if and only if either:
\begin{enumerate}
\item[(1')] $p \neq \la{x-y}$ and $(x,y) \in \lambda_{A,u}(D)$, or
\item[(2')] $p = \la{x-y}$ and both $(x,y-1)$ and $(x-1,y)$ lie in $\overline{\lambda}_{A,u}(D)$.
\end{enumerate}
We now show that these two claims imply the lemma. Consider first the case where $p = \la{z}$ for some integer $z$. Then the set of all integers $z'$ such that $\la{z'} = \la{z}$ is the arithmetic progression $z + m \textbf{Z}$, hence $p = \la{x-y}$ if any only if $x-y \equiv z \pmod{m}$. Therefore, claims (1) and (2) amount, in this case, to saying that $(x,y) \in \lambda_{\Gamma,w}(D+p)$ if and only if $(x,y) \in \disp^+(\lambda_{A,u}(D),z+m \textbf{Z})$. Consider next the case where $p$ is not equal to $\la{z}$ for any integer $z$. Then $p \neq \la{x-y}$ for all choices of $(x,y)$, so claim (1) shows that $\lambda_{\Gamma,w}(D+p) = \lambda_{A,u}(D)$ in this case. Therefore the two claims will imply the lemma.
\textit{Proof of claims (1) and (2)}. Let $\xi$ be a real number such that $p = \la{\xi}$. Observe that the divisor $D+p+(d-g)w$ is linearly equivalent to $D + (d-g)v + \la{\xi+d-g}$ by Lemma \ref{l_bridge}, which is equal to $D + (d-g)v + \la{\xi+y-x-1}$. Let $D'$ denote this degree $d$ divisor.
By definition, $D+p$ is a weak $g^r_d$ for $(\Gamma,w)$ if and only if $|D' - E| \neq \emptyset$ for all effective divisors $E$ of degree $r$. Replacing $E$ by a linearly equivalent divisor if necessary, it suffices to consider only effective divisors $E$ with at most one chip on $C$, and no chips on the interior of the edge from $u$ to $v$.
\textit{Case 1: $E$ has one chip on $C$ and $r-1$ chips on $A$.} Let the chip on $C$ be placed at the point $\la{\delta}$, and let $E_A = E-\la{\delta}$ be the part supported on $A$. By Lemma \ref{l_bridge}, $|D'-E| \neq \emptyset$ if and only if there exists an integer $n$ such that $|D - E_A + n\cdot u|_A \neq \emptyset$ and $|(d-g-n)\cdot v + \la{\xi+y-x-1}-\la{\delta}|_C \neq \emptyset$. By Lemma \ref{l_bridge} and the fact that $v = \la{-1}$, the second condition is equivalent to $$| (d-g-n+1)\cdot v - \la{\delta - \xi -y + x}| \neq \emptyset \textrm{ on $C$.}$$
This holds either if $d-g-n+1 \geq 2$ or if $d-g-n+1 = 1$ and $\la{\delta - \xi - y + x} = v$. Therefore, as long as $\la{\delta}$ is not equal to one specific point (namely, $\la{\xi+y-x-1}$), the maximum value of $n$ satisfying the second condition of Lemma \ref{l_bridge}(3) is $n = d-g-1$. Hence assuming that $\la{\delta}$ is not equal to this point, $|D'-E| \neq \emptyset$ on $\Gamma$ if and only if $|D + (d-g-1)u - E_A| \neq \emptyset$. This holds for \emph{every} choice of degree-$(r-1)$ effective $E_A$ on $A$ if and only if $D$ is a weak $g^{r-1}_{d-2}$ for $A$.
We conclude that $|D+p-E| \neq \emptyset$ for \emph{all} degree $r$ effective $E$ with exactly one chip on $C$ if and only if $D$ is a weak $g^{r-1}_{d-2}$ on $A$.
\textit{Case 2: $E$ is supported on $A$. }Again, we analyze the maximum value of $n$ such that when $n \cdot v$ is subtracted from the part of $D'-E$ supported on $C$, the result has nonempty linear series. The part of $D'-E$ supported on $C$ consists of $(d-g)v + \la{\xi+y-x-1}$, hence there are two cases: if $\la{\xi+y-x-1} = v$, then the maximum $n$ is $d-g+1$, and otherwise it is $d-g$. The hypothesis of the first case is equivalent to $p = \la{x-y}$. Define $$\epsilon = \begin{cases} 1 & \mbox{if } p = \la{x-y}\\ 0 & \mbox{otherwise.}\end{cases}$$
Lemma \ref{l_bridge} now says that $|D'-E| \neq \emptyset$ on $\Gamma$ if and only if $|D+(d~-~g~+~\epsilon)u - E| \neq \emptyset$ on $A$. This holds for every effective degree $r$ divisor on $A$ if and only if $D$ is a weak $g^r_{d-1+\epsilon}$ for $A$.
\textit{End of the proof:} combining these two cases, we see that $D+p$ is a weak $g^r_d$ for $\Gamma$ if and only if $D$ is both a weak $g^{r-1}_{d-2}$ and a weak $g^r_{d-1+\epsilon}$ for $A$. If $p = \la{x-y}$, i.e. $\epsilon = 1$, this is precisely claim (2). If $p \neq \la{x-y}$, i.e. $\epsilon = 0$, then this is equivalent to saying only that $D$ is a weak $g^r_{d-1}$, since a weak $g^r_{d-1}$ is automatically also a weak $g^{r-1}_{d-2}$; this gives claim (1) and completes the proof of the lemma.
\end{proof}
\subsection{Proof of Theorem \ref{t_tori}} \label{ss_pftori}
We now return Situation \ref{sit_comb}, where $\Gamma$ is a chain of cycles with torsion profile $\underline{m}$. By induction on the genus (where the base case may be taken to be a genus $0$ metric graph consisting of a single vertex), Lemma \ref{lem_disp} immediately gives the a description, in terms of displacement, of the Weierstrass partition of a divisor of the form $D = \sum_{i=1}^g \la{\xi_i}_i$, as follows.
\begin{cor} \label{c_computewp}
Let $D$ be the degree $g$ divisor $\sum_{i=1}^g \la{\xi}_i$ on the marked graph $(\Gamma,w_g)$ of Situation \ref{sit_comb}. For $i=1,2,\cdots,g$, let $S_i = \{z \in \textbf{Z}:\ \la{\xi_i}_i = \la{z}_i \}$. This is either empty, a set with one element (in which case $m_i =0$), or a congruence class modulo $m_i$. Define partitions $\lambda_0, \lambda_1,\cdots,\lambda_g$ as follows.
\begin{align*}
\lambda_0 &= \emptyset\\
\lambda_{i+1} &= \disp^+(\lambda_i,S_{i+1})
\end{align*}
Then the Weierstrass partition of $D$ on $(\Gamma,w_g$) is $\lambda_g$.
\end{cor}
\begin{lemma} \label{l_tori1}
If $t \vdash_{\underline{m}} \lambda$, then $\textbf{T}(t) \subseteq W^\lambda(\Gamma,w_g)$.
\end{lemma}
\begin{proof}
Suppose that $[D] \in \textbf{T}(t)$. Define sets $S_1,S_2, \cdots,S_g$ and partitions $\lambda_0,\lambda_1,\cdots,\lambda_g$ as in corollary \ref{c_computewp}; by that corollary, $\lambda_g$ is the Weierstrass partition of $D$ for $(\Gamma,w_g)$. It suffices to prove that $\lambda \subseteq \lambda_g$. By the definition of $\textbf{T}(t)$, the sets $S_i$ satisfy the condition that $x-y \in S_{t(x,y)}$ for all $(x,y) \in \lambda$. Therefore $S_i$ also meet the hypothesis of Corollary \ref{c_tabpart}, from which it follows that $\lambda \subseteq \lambda_g$, as desired.
\end{proof}
\begin{lemma} \label{l_tori2}
If $[D] \in W^\lambda(\Gamma,w_g)$, then there exists an $\underline{m}$-displacement tableau $t$ such that $[D] \in \textbf{T}(t)$.
\end{lemma}
\begin{proof}
By Lemma \ref{l_stdform}, we may assume without loss of generality that $D+g\cdot w_g$ consists of exactly one chip on each cycle of $\Gamma$. Let the position of the chip on the $i$th cycle be $\la{\xi_i}_i$.
For each $i \in \{1,2,\cdots,g\}$, denote by $\Gamma_i$ the first $i$ cycles of the chain $\Gamma$ (with the bridges between them), and denote by $D_i$ the restriction of $D+g\cdot w_g$ to $\Gamma_i$. Let $\lambda_i$ be the Weierstrass partition of $D_i$ for the marked metric graph $(\Gamma_i,w_i)$. Then by Lemma \ref{lem_disp}, these partitions are related by
$$\lambda_{i+1} = \disp^+(\lambda_i,S_{i+1}),$$
where $S_i = \{z \in \textbf{Z}:\ \la{\xi_i}_i = \la{z}_i \}$. In particular, all of the boxes $(x,y)$ in $\lambda_{i+1} \backslash \lambda_i$ have values of $x-y$ that are congruent modulo $m_i$.
Define a function $t:\ \lambda \rightarrow \{1,2,\cdots,g\}$ by
$$t(x,y) = \min \{i:\ (x,y) \in \lambda_i \}.$$
The previous paragraph shows that $t$ is in fact an $\underline{m}$-displacement tableau. Each $(x,y) \in \lambda$ is a loose box of $\lambda_{t(x,y)-1}$ with respect to the set $S_{t(x,y)}$, so in particular $x-y \in S_{t(x,y)}$, i.e. $\la{\xi_{t(x,y)}}_{t(x,y)} = \la{x-y}_{t(x,y)}$. Therefore $[D] \in \textbf{T}(t)$.
\end{proof}
Theorem \ref{t_tori} follows directly from Lemmas \ref{l_tori1} and \ref{l_tori2}.
\section{Algebraic geometry of $W^\lambda(C,p)$} \label{sec_ag}
Throughout this section, let $C$ denote a smooth projective curve over a field $K$, and $p \in C(K)$ a $K$-point. We will define the structure of $W^\lambda(C,p)$ as a subscheme of $\Pic^0(C)$, and compute its expected dimension and class via intersection theory.
We first review the scheme structure of $W^r_d(C)$, following \cite[IV \S 3]{acgh}. If an effective divisor $E$ of degree at least $2g-d-1$ is fixed, then the vector spaces $H^0( L(E) )$ (where $L$ varies over all degree $d$ line bundles) all have the same dimension $\deg E + d-g + 1$, and form the fibers of a vector bundle $\mathcal{M}$ over $\Pic^d(C)$. A second vector bundle $\mathcal{P}$ may be formed, whose fiber over $[L] \in \Pic^d(C)$ is $H^0( L(E) / L)$. There is an obvious map $\mathcal{M} \rightarrow \mathcal{P}$ of vector bundles. Observe that for any particular line bundle $L$ of degree $d$, $h^0(L) \geq r+1$ if and only if $\dim \ker \left( H^0(L(E)) \rightarrow H^0(L(E)/L)\right) \geq (r+1)$. Therefore $W^r_d(C)$ is defined to be the subscheme of $\Pic^d(C)$ whose defining equations are, locally, the $N \times N$ minors of a trivialization of the bundle map $\mathcal{M} \rightarrow \mathcal{P}$, where $N = (\deg E + d - g - r+1)$. The details of the construction of the vector bundles $\mathcal{M},\mathcal{P}$, as well as the fact that this scheme does not depend on any of the choices made, can be found in \cite{acgh}.
Each scheme $W^r_d(C)$ may be regarded as a subscheme of $\Pic^0(C)$, via the isomorphism $L \mapsto L(-d \cdot p)$.
\begin{defn} \label{d_wlfirst}
The scheme $W^\lambda(C,p)$ is the intersection of all the schemes $W^r_d(C)$ such that $(g-d+r,r+1) \in \lambda$, where these are regarded as subschemes of $\Pic^0(C)$.
\end{defn}
Definition \ref{d_wlfirst} is the most natural definition, but it is difficult to use since the various schemes being intersected are not transverse. We therefore give an alternative characterization, to which we can apply intersection theory.
First, since we wish to describe a locus in $\Pic^0(C)$ rather than $\Pic^d(C)$, we may replace $L$ by $L(d \cdot p)$ in the discussion above.
Next, suppose that in the construction of the schemes $W^r_d(C)$ we choose the divisor $E$ to be $(2g-1-d) \cdot p$. Then the same vector bundle $\mathcal{M}$ can be used in the construction of every locus $W^r_d(C)$; its fiber over $[L]$ will be $H^0(L((2g-1)p))$. The bundle $\mathcal{P}$ used in the definition of $W^r_d(C)$ has fibers naturally identified with $H^0( L((2g-1) \cdot p) / L(d \cdot p))$ (note that we now write $L(d \cdot p)$ since $L$ is a degree $0$ line bundle, and $L(d \cdot p)$ is the line bundle which is required to have $r+1$ sections). Write this bundle as $\mathcal{P}_d$. These vector bundles, for various choices of $d$, fit into a flag of quotients.
$$
\mathcal{P} = \mathcal{P}_{-1} \twoheadrightarrow \mathcal{P}_{0}
\twoheadrightarrow \mathcal{P}_{1}
\twoheadrightarrow \cdots
\twoheadrightarrow \mathcal{P}_{2g-1} = 0
$$
In this notation, we can alternatively characterize $W^\lambda(C,p)$ as the following degeneracy locus.
\begin{align*}
W^\lambda(C,p) =& \left\{[L] \in \Pic^0(C):\ \textrm{rank}( \mathcal{M}_{[L]} \rightarrow \left(\mathcal{P}_d)_{[L]} \right) \leq g-r-1\right. \\
& \left.\mbox{ whenever } (r+1,g-d+r) \in \lambda \right\}
\end{align*}
Degeneracy loci of this form are analyzed in \cite{fulton}. The expected codimension of this degeneracy locus is equal to $|\lambda|$, and the class of the locus, in case it has the correct dimension, can be computed from the Chern classes of $\mathcal{M}$ and $\mathcal{P}_d$. By \cite[VII \S 4]{acgh} (where the same bundles are used to analyze $W^r_d(C)$), $\mathcal{P}_i$ has trivial Chern classes, and $\mathcal{M}$ has Chern classes given by
$$
c_i(-\mathcal{M}) = \Theta^i / i!.
$$
By \cite[Theorem 10.1]{fulton}, this degeneracy locus supports a (Chow or singular cohomology) class given by the following determinant, with this class being equal to the class of the degeneracy locus in case the dimensions match. Here $\lambda^*_1$ denotes the largest element of the dual partition, i.e. the number of nonzero elements of $\lambda$.
\begin{align*}
\Omega^\lambda(C) &= \det \left( c_{\lambda_i-i+j} ( - \mathcal{M} ) \right)_{1 \leq i,j \leq \lambda^*_1}\\
&= \det \left( \Theta^{\lambda_i - i + j} / (\lambda_i-i+j)! \right)_{1 \leq i,j \leq \lambda^*_1}\\
&= \Theta^{|\lambda|} \cdot \det \left( 1 / (\lambda_i - i + j)! \right)_{1 \leq i,j \leq \lambda^*_1}
\end{align*}
In the last line, the number $1 / n!$ should be interpreted to be $0$ when $n < 0$. By Aitken's determinantal formula \cite{aitken}, this determinant is equal to the number of standard Young tableaux on $\lambda$ divided by $|\lambda|!$; by the hook length formula \cite{frame} we can express this as the product of the reciprocals of the hook lengths of $\lambda$.
We therefore deduce the following fact.
\begin{prop} \label{prop_lowerbound}
If $g \geq |\lambda|$, then the Brill-Noether locus $W^\lambda(C,p)$ is nonempty of dimension at least $g - |\lambda|$. If the dimension of $W^\lambda(C,p)$ is exactly $g - |\lambda|$, then its (Chow or singular cohomology) class is equal to
$$
\left. \Theta^{|\lambda|} \middle/ \prod_{(x,y) \in \lambda} \hook(x,y) \right. .
$$
\end{prop}
\section{Tropicalization} \label{sec_trop}
This section is concerned with the marked curve $(C,p)$ from Situation \ref{sit_main}. We will prove Theorem \ref{thm_introalg}, which is a corollary of a pair of ``semicontinuity'' results.
The retraction from the Berkovich analytic space $C^{\textrm{an}}$ to its skeleton extends by linearity to divisors and induces a map
$$
\Trop:\ \Pic(C) \rightarrow \Pic(\Gamma).
$$
When $K$ is the field of fractions of a discrete valuation ring, this map is identical to the tropicalization map of \cite{baker08}. This map has two important properties, which can be interpreted as two forms of semicontinuity for $\Trop$.
\begin{prop}\ \label{prop_semi}
\begin{enumerate}
\item The image of $W^\lambda(C,p)$ under $\Trop$ lies in the locus $W^\lambda(\Gamma,w_g)$.
\item $\dim W^\lambda(\Gamma,w_g) \geq \dim W^\lambda(C,p)$.
\end{enumerate}
\end{prop}
\begin{proof}
The definition of $\Trop$ implies that for any divisor $D$ on $C$ and any integer $n$, $\Trop(D + n \cdot p) = \Trop(D) + n w_g$. Therefore it suffices to show that $\Trop(W^r_d(C)) \subseteq W^r_d(\Gamma)$ for all integers $r,d$. This is precisely the specialization inequality of \cite{baker08}; although that paper works in the discrete valuation case, the proof applies without modification to the general case. This establishes part (1).
The tropical Jacobian $\Pic^0(\Gamma)$ is isomorphic to the skeleton $\Sigma(\Pic^0(C)^{\textrm{an}})$ \cite{br}, and this isomorphism is compatible with the tropicalization map. Therefore Gubler's results on tropicalization of subvarieties of abelian varieties apply: by \cite[Theorem 6.9]{gubler}, a variety of pure dimension $d$ in $\Pic^0(C)$ has image, under the retraction to the skeleton $\Sigma(\Pic^0(C))$, of pure dimension $d$. Combining this with part (1), it follows that the image $\Trop(W^\lambda(C,p)) \subseteq \Gamma$ contains a tropical variety of dimension equal to the dimension of $W^\lambda(C,p)$, and part (2) follows.
\end{proof}
\begin{cor}[Theorem \ref{thm_introalg}]
In Situation \ref{sit_main}, if for each $i$ either $m_i = 0$ or $m_i > i$, then for every partition $\lambda$, $W^\lambda(C,p)$ has the expected dimension.
\end{cor}
\begin{proof}
Theorem \ref{thm_genericity2} and Proposition \ref{prop_semi}.
\end{proof}
\begin{cor} \label{cor_gbn}
Working over any algebraically closed field, if $(C,p)$ is a general marked curve, then for every partition $\lambda$, $W^\lambda(C,p)$ has the expected dimension, and Chow class given by Formula \eqref{eq_class}.
\end{cor}
\begin{proof}
Theorem \ref{thm_introalg} and deformation theory (cf. \cite[Appendix]{baker08}) show the existence of Brill-Noether general marked curves over any complete valued field with infinite residue field. The existence of Brill-Noether general marked curves over an arbitrary algebraically closed field follows from the fact that the coarse moduli space $M_{g,1}$ of marked smooth curves is a scheme of finite type over $\textrm{Spec} \textbf{Z}$, and the locus of Brill-Noether general points is Zariski open and surjects onto $\textrm{Spec} \textbf{Z}$.
Proposition \ref{prop_lowerbound} shows that the Chow class is given by formula \eqref{eq_class}.
\end{proof}
The generalized Brill-Noether theorem (Theorem \ref{thm_gbn}) follows from Corollary \ref{cor_gbn}.
\section{Further questions for arbitrary metric graphs} \label{sec_qu}
The detailed description, for marked chains of cycles $(\Gamma,w_g)$, of the Brill-Noether loci $W^\lambda(\Gamma,w_g)$, makes it possible to see an intriguing analogy with the theory of algebraic curves; we believe that extending this analogy to arbitrary metric graphs might yield further insight. In this section we formulate this analogy and the questions arising from it.
For a marked algebraic curve $(C,p)$, classical intersection theory, as discussed in Section \ref{sec_ag}, guarantees not only that $W^{\lambda}(C,p)$ is nonempty (when $g-|\lambda| \geq 0$), but that all of its irreducible components are at least $(g-|\lambda|)$-dimensional and that it supports a particular intersection class, namely $\Theta_C^{|\lambda|} / \prod_{(x,y) \in \lambda} \hook(x,y)$. Furthermore, this is precisely the class of $W^\lambda(C,p)$ when its dimension is exactly $g - |\lambda|$.
Even though there is currently no equivalent in tropical intersection theory of the machinery available to study degeneracy loci in algebraic geometry (as in \cite{fulton}), the same facts outlined in the previous paragraph hold in the context of chains of cycles, when interpreted in a suitable way. Indeed, observe that if $(\Gamma,w_g)$ is a marked chain of cycles as in Situation \ref{sit_comb}, and $\lambda$ is a partition, then regardless of the torsion profile there is always a locus given by those tableaux that have no repeating symbols, namely
$$
\bigcup_{t \vdash_{\underline{0}} \lambda} \textbf{T}(t) \subseteq W^\lambda(\Gamma,w_g).
$$
The parameterization provided by Defintion \ref{def:Tt} shows that this locus varies continuously as the edge lengths of $\Gamma$ vary. In this section we will refer to this subset as the \emph{stable locus}. It consists of a union of $(g-|\lambda|)$-dimensional tori. Furthermore, observe that any point of $W^\lambda(\Gamma,w_g)$ that is not in the stable locus necessarily sits in a torus $\textbf{T}(t)$ coming from a tableau $t$ with a repeated symbol, i.e. a torus of dimension strictly greater than $g-|\lambda|$. Thus the locus $W^\lambda(\Gamma,w_g)$ is equal to the stable locus if and only if it is exactly $(g-|\lambda|)$-dimensional.
To complete the analogy with algebraic curves, we point out that, in a suitable sense, the stable locus has intersection class $\Theta_\Gamma^{|\lambda|} / \prod_{(x,y) \in \lambda} \hook(x,y)$. We must define what we mean by these terms.
For any metric graph $\Gamma$ of genus $g$, there is a tropical subvariety $\Theta_\Gamma \subset \Pic^{g-1}$, called the tropical theta divisor, which can be identified with $W^{0}_{g-1}(\Gamma)$; see \cite{mz}. As usual, we suppose that we have chosen a marked point $w \in \Gamma$, and regard $\Theta_\Gamma$ as a subset of $\Pic^0(\Gamma)$. When $\Gamma$ is a chain of cycles, Theorem \ref{t_tori} shows that $\Theta_\Gamma$ is a union of $(g-1)$-dimensional tori (the Young diagram associated to $r=0,\ d=g-1$ is a single box), each obtained by fixing one coordinate $\xi_i$ to be equal to $0$ (in the notation of Lemma \ref{l_stdform}). If $d$ is a positive integer, and we choose $d$ general translates of $\Theta_\Gamma$, we can easily describe the intersection of these translates: it is a union of $g (g-1) \cdots (g-d+1)$ sub-tori, each giving by fixing $d$ coordinates $\xi_i$. More precisely, for each choice of $d$ coordinates, the intersection includes $d!$ tori given by fixing these $d$ coordinates.
Now, observe that the stable locus of $W^\lambda(\Gamma,w_g)$ has a similar description: for each standard Young tableaux on $\lambda$ and choice of $|\lambda|$ symbols from the set $\{1,2,\cdots, g\}$, there is one torus in the stable locus, given by fixing the $|\lambda|$ coordinates chosen. By the hook-length formula \cite{frame}, the number of tori in the stable locus fixing a given set of $|\lambda|$ coordinates is
$$
|\lambda| ! / \prod_{(x,y) \in \lambda} \hook(x,y).
$$
Provided that any subtorus of $\Pic^0(\Gamma)$ is regarded as ``equivalent'' to any translate of itself, it follows that the stable locus of $W^\lambda(\Gamma,w_g)$ may be regarded as being equivalent to $1 / \prod_{(x,y) \in \lambda} \hook(x,y)$ times the locus given by intersecting $|\lambda|$ general translates of $\Theta_\Gamma$. This precisely mirrors the formula for the expected class of $W^\lambda(C,p)$ in the algebraic case.
To formulate a more general statement for arbitrary marked metric graphs $(\Gamma,w)$, we require a suitable equivalence relation on tropical subvarieties of $\Pic^0(\Gamma)$. One candidate is the notion of rational equivalence described in \cite{ahr}. Whatever choice is made, it should be compatible with tropical intersection theory, e.g. as developed in \cite{ar}.
To extend the analogy outlined above to arbitrary marked metric graphs, several questions present themselves.
\begin{qu}
If $(\Gamma,w)$ is any marked metric graph, is the local dimension of $W^\lambda(\Gamma,w)$ at least $g- |\lambda|$ at every point?
\end{qu}
Note that the semicontinuity results of Section \ref{sec_trop} imply that the local dimension of $W^\lambda(\Gamma,w)$ is at least $g-|\lambda|$ at every point that lifts to a point of $W^\lambda(C,p)$ on an algebraic marked curve. This does not rule out the existence of extra components of smaller dimension.
\begin{qu}
If $(\Gamma,w)$ is any marked metric graph, does the locus $W^\lambda(\Gamma,w)$ always support a tropical subvariety equivalent (in a suitable sense) to $\Theta_\Gamma^{|\lambda|} / \prod_{(x,y) \in \lambda} \hook(x,y)$? Is there a ``canonical'' such locus that varies continuously as the edge lengths vary (analogous to the ``stable'' locus defined above for chains of cycles)?
\end{qu}
\begin{qu}
If $\Gamma$ is a metric graph, $w$ is a marked point, and $\lambda$ is a partition such that $W^\lambda(\Gamma,w)$ has pure dimension $g - |\lambda|$, must $W^\lambda(\Gamma,w)$ be equivalent (in a suitable sense) to $\Theta_\Gamma^{|\lambda|} / \prod_{(x,y) \in \lambda} \hook(x,y)$?
\end{qu}
| {
"timestamp": "2017-01-19T02:01:05",
"yymm": "1603",
"arxiv_id": "1603.07364",
"language": "en",
"url": "https://arxiv.org/abs/1603.07364",
"abstract": "We completely describe all Brill-Noether loci on metric graphs consisting of a chain of g cycles with arbitrary edge lengths, generalizing work of Cools, Draisma, Payne, and Robeva. The structure of these loci is determined by displacement tableaux on rectangular partitions, which we define. More generally, we fix a marked point on the rightmost cycle, and completely analyze the loci of divisor classes with specified ramification at the marked point, classifying them using displacement tableaux. Our results give a tropical proof of the generalized Brill-Noether theorem for general marked curves, and serve as a foundation for the analysis of general algebraic curves of fixed gonality.",
"subjects": "Combinatorics (math.CO); Algebraic Geometry (math.AG)",
"title": "Special divisors on marked chains of cycles",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754474655618,
"lm_q2_score": 0.718594386544335,
"lm_q1q2_score": 0.7075103896781295
} |
https://arxiv.org/abs/1410.6723 | The FedEx problem | The original shipping strategy of FedEx is to fly all packages to a hub location during the afternoon and evening, sort them there, and then fly them to their destinations during the night for delivery the next day. This leads to interesting mathematical questions: Given a population represented by points in Euclidean space or on a sphere, what is the location of the point of the hub that minimizes the total distance to all the points? Is such a point unique? Then using census data from 2000 we examine how close the FedEx hub in Memphis is to the hub for the U.S. population. | \section*{The Euclidean FedEx Problem}
The Euclidean FedEx problem in one dimension has a well-known solution: a hub for the population is any median. Here is a quick proof. For a population located at the points $x_1\leq x_2 \leq \cdots \leq x_k$, a hub must be between $x_1$ and $x_k$, for otherwise all the distances to the $x_i$ could be decreased by moving the potential hub toward the points. As far as $x_1$ and $x_k$ are concerned, any point between them is equally optimal because the sum of the distances to those two points is constant. Thus, we can eliminate $x_1$ and $x_k$ from the population and reduce the problem to a smaller population. We continue to eliminate the endpoints until there is either one or two left. If there is one point left (corresponding to $k$ odd), then that point is the unique hub, and if there are two points left (corresponding to $k$ even), then any point between them is a hub. The results in either case are medians for the population locations.
In higher dimensions the story starts with Fermat, who generally receives credit for first posing the following problem, which we have updated and renamed for the twenty-first century.
\begin{fredex*}
Fred has been married and divorced three times. He has three children, one from each marriage, and the children live with Fred's ex-wives. Each weekend Fred visits one of his children in a regular rotation. Where should he live in order to minimize the distance he travels?
Let the locations of the residences be $a, b, c \in \mathbb{R}^2$. We are assuming that they are near enough so that we do not have to take into account the curvature of the Earth and that the metric is Euclidean distance. The mathematical problem is to find the point $h$ in $\mathbb{R}^2$ that minimizes
\[ f(x)= \Vert x-a\Vert + \Vert x-b\Vert + \Vert x-c\Vert. \]
\end{fredex*}
Torricelli gave the first solution to Fermat's problem and eventually offered several different proofs.
The solution, to be discussed in more detail following the proof of Theorem 1, has two cases. First, if the angles of the triangle $abc$ are less than $2\pi/3$, then $h$ is the point within the triangle such that the lines from $h$ to the vertices form three equal angles of size $2 \pi/3$. Second, if some angle is larger than $2 \pi/3$, then $h$ is the vertex of that angle. Note that the solution can be constructed by compass and straightedge.
\begin{figure}
\centerline {
\includegraphics[width=5in]{morrison1}
}
\vspace{-30mm}
\caption{Solution of the FredEx problem with $h$ inside the triangle.}
\end{figure}
Since then the problem of minimizing total or average distance has arisen repeatedly and in different contexts so that several different names are attached. In addition to being called the Fermat-Torricelli problem, it is known as the Weber problem---named for the economist Alfred Weber who was interested in the problem in connection with the location of industries. In the sub-area of operations research known as location science, the problem is called the \emph{median problem} or the \emph{single facility location problem}. Statisticians may refer to the minimal point as the spatial, multivariate, or multidimensional median. For a survey of the problem, its generalizations, and its history see the article by
Wesolowsky \cite{ Wesolowsky93} with its extensive bibliography or the papers in the collection edited by Drezner and Hamacher \cite{DreznerHamacher02}.
For the general FedEx problem in $n$-dimensional Euclidean space, we consider a population of size $k$ located at the points $x_1,x_2,\ldots,x_k \in \mathbb{R}^n$, which are not necessarily distinct. The function to be minimized is the sum of the distances from the $x_i$ to a variable point $x \in \mathbb{R}^n$
\[ f(x) = \sum_i \|x-x_i\|. \]
Just as in one dimension where the hub is located between the extremes, a hub in higher dimensions should be located ``between'' the points, which means, as we will show in Theorem 1, that the hub lies in the convex hull of the points (i.e., the smallest convex set containing them). Recall that a subset of $\mathbb{R}^n$ is convex if, for any two points in the set, the points on the line segment between them are also in the set. The convex hull of $x_1,x_2,\ldots,x_k$ can be seen as the image of the compact set $\{(\alpha_1,\ldots,\alpha_k) | \alpha_i \geq 0, \, \sum \alpha_i =1\}$ under the continuous map sending $(\alpha_1,\ldots,\alpha_k)$ to $\sum \alpha_i x_i$. In particular, the convex hull is compact.
The function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is continuous everywhere and differentiable at all $x \in \mathbb{R}^n$ except the points $x_i$. At a point $x$ not equal to any of the $x_i$ the derivative $Df(x)$ is the linear map from $\mathbb{R}^n$ to $\mathbb{R}$ defined by
\begin{equation} \label{Df} Df(x)(v)=
\sum_i \left\langle \frac{x-x_i}{\|x-x_i\|}, v \right\rangle
\end{equation}
We now have the ingredients to prove the existence and uniqueness of a hub for a finite set of non-collinear points in $\mathbb{R}^n$. Although this result has probably been proved again and again, we cannot find it in the literature clearly stated with a complete proof. A brief article by Haldane \cite{Haldane48} contains the result for $\mathbb{R}^2$ without mention of the convex hull.
\begin{theorem}[Existence and Uniqueness of a Hub] \label{exuniq}
For any non-collinear points $x_1,x_2,\ldots,x_k$ in $\mathbb{R}^n$ there is a unique hub contained in the convex hull of the points.
\end{theorem}
\begin{proof}
Let $Z$ be the convex hull of $x_1,x_2,\ldots,x_k$ and let $x$ be a point not in $Z$. There is a separating hyperplane $H$ between $x$ and $Z$. (See, for example, the Basic Separation Theorem in \cite[p. 158]{PSU93}). Let $n$ be the unit normal to $H$ pointing toward $x$. Thus, $\langle x,n \rangle > \langle z,n \rangle$ for all $z \in Z$. In particular, $\langle x,n \rangle > \langle x_i,n \rangle $ for $i=1,\ldots,n$. Then it follows that $Df(x)(n) >0$, and so $f$ decreases from $x$ in the direction $-n$. More precisely, for some $\epsilon > 0$, $f(x-\epsilon n) < f(x)$. Therefore, a minimum of $f | Z$, which exists because $f$ is continuous and $Z$ is compact, will actually be a global minimum of $f$.
Next we show that the function $f$ is strictly convex. This means that for all $x, y \in \mathbb{R}^n$ and any $t$ in the open interval $(0,1)$,
\[ f(t x + (1-t) y) < t f(x) + (1-t)f(y) .\] We have
\begin{align*}
f(t x + (1-t) y) &= \sum_i \| tx + (1-t)y-x_i \| \\
& =\sum_i \| tx -tx_i +(1-t)y - (1-t)x_i \| \\
&= \sum_i \| t(x-x_i) +(1-t)(y-x_i) \| \\
& \leq \sum_i( \|t(x-x_i) \| +\|(1-t)(y-x_i) \|) \\
&=t \sum_i \|x-x_i \| + (1-t)\sum_i \|y-x_i \| \\
&= t f(x) + (1-t)f(y).
\end{align*}
The triangle inequality in the fourth line will be strict, unless $t(x-x_i)$ and $(1-t)(y-x_i)$ are linearly dependent, which is equivalent to $x,y$ and $x_i$ being collinear. Therefore, if the $x_i$ are not collinear, then there are no $x$ and $y$ simultaneously collinear with all the $x_i$, and it follows that the inequality is strict. Thus, $f$ is strictly convex.
Finally, if there were two distinct minima for $f$, say $x$ and $y$, then $f(x)=f(y)$ and \[f(tx +(1-t)y) < tf(x)+(1-t)f(y) = f(x)\] for all $t \in (0,1)$, contradicting the fact that $x$ is a minimum. \end{proof}
The hub $h$ of the points $x_1,x_2,\ldots,x_k$ must be a critical point of $f$, so that either $h=x_i$ for some $i$, or $Df(h)=0$, which means
\[ \sum_i\frac{h-x_i}{\|h-x_i\|} = 0. \]
Now we return to the case of three points in $\mathbb{R}^2$, which we can assume are not collinear. If the hub is one of the points, then it must be the vertex opposite the longest side of the triangle formed by the three points, because $f(x_i)$ is the sum of the lengths of the two sides meeting at $x_i$, and this is minimal when the two sides are the shorter ones. If the hub is not one of the $x_i$, then it lies within the triangle. Let $u_i$ be the unit vector pointing from $h$ to $x_i$. Then $u_1+u_2+u_3=0$. By rotating the coordinate system we may assume that $u_1=(1,0)$. Then $u_2=(a,b)$ and $u_3=(-1-a,-b)$ with $\|u_2 \|^2=a^2+b^2=1$ and $\|u_3\|^2 = (-1-a)^2+b^2=1$. The last two equations imply that $a=-1/2$ and $b= \pm \sqrt{3}/2$. Thus, the angle between any pair of the vectors $u_1,u_2,u_3$ is $2\pi/3$.
If one of the angles of the triangle is $2\pi/3$ or greater, then at any other point of the closed triangle, the vectors from it to the other two vertices span an angle greater than $2\pi/3$. Such a point cannot be a hub, and so the hub must be the vertex of the large angle (see Figure 2).
\begin{figure}
\centerline {
\includegraphics[width=5in]{morrison2}
}
\vspace{-4cm}
\caption{The solution of the FredEx problem is $h=a$. An interior point $x$ cannot be the hub because $\measuredangle bxc > \measuredangle bac > 2\pi/3.$ }
\end{figure}
If, on the other hand, all the angles are less than $2\pi/3$, then the following continuity argument, illustrated in Figure 3, shows that the hub is an interior point.Let $L_1$, $L_2$, and $L_3$ be three rays issuing from a point $P$ and making equal angles of $2\pi/3$ between them. Place the triangle with one vertex at $P$, one vertex on $L_1$, and the third vertex in the region between $L_1$ and $L_3$. Now continuously slide the triangle so that the vertex originally at $P$ moves out along $L_2$, and the vertex on $L_1$ stays on $L_1$ and moves toward $P$. When the third vertex crosses $L_3$ the vertices of the triangle are on the three rays, and so the point $P$ satisfies the property that the unit vectors from $P$ toward the vertices of the triangle sum to $0$. Therefore, $P$ is the hub and is an interior point of the triangle. For more on this classical problem we recommend \cite{Honsberger73} for an extended discussion and \cite{Hajja94} for another proof using calculus.
\begin{figure}
\centerline{
\includegraphics[width=5in]{morrison3}
}
\vspace{-2cm}
\caption{Beginning with one side on $L_1$ the triangle is moved until the third vertex meets $L_3$. The hub is at $P$.}
\end{figure}
There is also a nice description of the hub for four distinct points in $\mathbb{R}^2$. We assume that the four points are not on a line for otherwise we are in the one-dimensional setting. Again there are two cases; the convex hull is either a quadrilateral or a triangle. In the first case the hub is the intersection of the two diagonals. In the second case one of the four points is in the triangular convex hull of the other three, and that point is the hub.
To prove this result, first consider the case in which the four points are the vertices of a convex quadrilateral and labeled so that the sequence $x_1,x_2,x_3,x_4$ makes a circuit of the quadrilateral. Let $h$ be the intersection of the diagonals; one diagonal is the line between $x_1$ and $x_3$ and the other is the line between $x_2$ and $x_4$. Let $u_i$ be the unit vector pointing from $h$ to $x_i$. Therefore, $u_1=-u_3$ and $u_2=-u_4$, and so $u_1 +u_2 +u_3+u_4=0$, from which it follows that $h$ is the hub.
For the second case suppose that $x_4$ is in the triangular convex hull of the other three points. Let $h$ be the hub. Then either $Df(h)=0$ or $h$ is one of the $x_i$. Suppose $Df(h)=0$. Then the unit vectors $u_i$ pointing from $h$ toward the $x_i$ satisfy $u_1 +u_2 +u_3+u_4=0$. We may assume that they are ordered so that they run counter-clockwise around the unit circle. Therefore the points $0, u_1, u_1+u_2, u_1+u_2+u_3$ form a quadrilateral with sides of equal length, and such a quadrilateral must be a rhombus. Thus, $u_1=-u_3$ and $u_2=-u_4$. Then $x_1$ and $x_3$ are on the line through $h$ with direction vector $u_1$, and $x_2$ and $x_4$ are on the line through $h$ with direction vector $u_2$. This means that the $x_i$ are actually the vertices of a convex quadrilateral with $h$ as the intersection of the diagonals. This contradicts our assumption that the convex hull is a triangle. It follows that $h$ must be one of the $x_i$. Finally, we claim that if $x_4 \neq x_i$, then $f(x_4) < f(x_i)$. Consider $i=1$, since the other two are similar. The inequality $f(x_4) < f(x_1)$ is equivalent to
\[ \|x_4-x_2\| +\|x_4-x_3\| < \| x_1-x_2 \| + \|x_1-x_3\| ,\]
which is clear from the picture in Figure 4 (proof left to the reader). Therefore, the hub is $x_4$, the point that is in the convex hull of the other three points. Note that $x_4$ may be on the boundary of the triangle.
\begin{figure}
\vspace{-2cm}
\centerline {
\includegraphics[width=6in]{morrison4}
}
\vspace{-4cm}
\caption{Proof that $f(x_4)<f(x_1)$.}
\end{figure}
The ``Varignon frame'' is a mechanical device invented by Pierre Varignon (1654--1722) for finding the hub of points $x_1,\ldots,x_k$ in $\mathbb{R}^2$. On a flat piece of wood mark the locations of the points and drill a hole at each point. For each hole take a piece of string and attach a weight to one end--all weights the same. Put the string through the hole with the weight below the board and tie all of the loose ends together. Hold the board level and above the ground so that the weights can hang freely. The knot tying all the strings together will move to the location of the hub.
\section*{The U.S. Population Hub}
Let's return to the original FedEx Problem of where to establish a shipping hub. Every ten years the U.S. Census Bureau calculates a point called the ``center of population.'' Could this point be the hub we are looking for? According to the current Bureau website \cite{Census01} the center of population is
\begin{quote}
the point at which an imaginary, flat, weightless, and rigid map of the United States would balance perfectly if weights of identical value were placed on it so that each weight represented the location of one person on the date of the census.
\end{quote}
This clearly is not what we want in a hub. The U.S. is large enough so that the curvature matters in measuring distance, and even if we could treat the U.S. area as flat, this population center is the center of gravity for the population distribution, and, thus in effect it is the point minimizing the average squared distances to the population, or, equivalently, the point minimizing the aggregate squared distance to all the people in the country. However, it is an often held misconception that the center of gravity or centroid minimizes the average distance. Even the Census Bureau suffered from this confusion as witnessed by this passage from its Bulletin of the 1920 census (quoted in \cite[p.34]{Eells30}):
\begin{quote}
If all the people in the United States were to be assembled at one place, the center of population would be the point which they could reach with the minimum aggregate travel, assuming that they all travelled in direct lines from their residence to the meeting place.
\end{quote}
After correspondence in 1926 between the Census Bureau, the American Statistical Association, and a group of interested people, the Census Bureau fixed the problem by taking out references to the minimum aggregate travel \cite{Editor30}.
Although the population center of the Census Bureau is not the hub we are looking for, it is interesting nevertheless to understand how it is calculated. This description comes from \cite{Census01} and is the method used for the censuses from 1950 to the most recent one in 2000.
The center of population is given as a pair of numbers $(\bar{\phi},\bar{\lambda})$ representing the center's latitude and the longitude.
The latitude $\bar{\phi}$ is simply the average latitude of the population:
\[ \bar{\phi} = \frac{1}{k} \sum_i \phi_i . \]
However, the longitude $\bar{\lambda}$ is \emph{not} the average longitude. Instead it is defined by
\[ \bar{\lambda}=\frac{\sum_i\lambda_i \cos \phi_i}{\sum_i \cos \phi_i} . \]
To make some sense of this, notice that the distance from the point with latitude $\phi$ and longitude $\lambda$ to the Greenwich meridian (longitude zero) along the latitude line is $\lambda \cos \phi$. Therefore, the distance to the Greenwich meridian (along lines of constant latitude) averaged over the entire population is $(1/k) \sum_i \lambda_i \cos \phi_i$. This needs to be converted to a longitude value. If we use the average cosine of the latitude of the population in order to convert, then we get the formula of the Census Bureau:
\[\bar{\lambda} = \frac{\frac{1}{k}\sum_i \lambda_i \cos \phi_i }{\frac{1}{k}\sum_i \cos \phi_i}=
\frac{\sum_i\lambda_i \cos \phi_i}{\sum_i \cos \phi_i} . \]
The calculation of $\bar{\phi}$ and $\bar{\lambda}$ gives $(\bar{\phi},\bar{\lambda})=(37.7^\circ,91.8^\circ)$, which is a point in Phelps County, Missouri, but there are some difficulties with this definition of the population center. Here is one example. Suppose the population in question consisted of two individuals, one located at $(\phi_1,\lambda_1)=(35^\circ,120^\circ)$, which is near San Luis Obispo, California, and one located at $(\phi_2,\lambda_2)=(35^\circ,80^\circ)$ near Charlotte, North Carolina. It is easy to check that $\bar{\phi}=35^\circ$ and $\bar{\lambda}=100^\circ$, which is the point midway between on the same line of latitude. However, it is difficult to imagine any reasonable definition of population center for this case that would give a point different from the midpoint of the great circle between the two locations, which is $(36.7^\circ, 100^\circ)$. In this case there is quite a discrepancy between the two answers. Both points are in Oklahoma, but they are 116 miles apart.
If the Census Bureau insists that the population center be the center of gravity, then it could be done more accurately by treating the area of the United States as a region on a spherical shell with a unit of mass for each person.
On the other hand the Bureau could return to its definition of 1920 in which the population center is the point of minimum aggregate travel but calculate it correctly. While it may have been a daunting task back then, it is now a calculation that can be done easily with the data provided by the Census Bureau.
\subsection*{Calculating the hub}
On its web pages \cite{PopCenters00} the Census Bureau provides 2000 census data for the 65,443 census tracts in the 50 states. Each line of the comma-delimited text file contains six numbers. The first three numbers identify the state, county, and census tract. The last three numbers give the population of the tract and the latitude and longitude in degrees of the population center of the tract. Note that the longitude is negative because of the convention that east is the positive direction from Greenwich. Here are a few lines from that file with the commas removed for readability.
\vspace{2mm}
\texttt{
\begin{tabular}{llllll}
State&County&Tract&Pop.&Latitude&Longitude\\
\hline
06&077&005404&6511&+37.732419&-121.425296\\
06&077&005500&6876&+37.71513&-121.322774\\
06&079&010000&6803&+35.701192&-120.801674\\
06&079&010100&8787&+35.634833&-120.69533\\
06&079&010201&4687&+35.642344&-120.655632\\
06&079&010202&4180&+35.606719&-120.650357\\
06&079&010203&8069&+35.615338&-120.670017
\end{tabular} }
\vspace{3mm}
We wrote a program to use this data to compute the function that is the aggregate great circle distance from the entire population to the point with latitude $\phi$ and longitude $\lambda$. Then we minimized this function on a grid with one degree increments. Each evaluation of the function on a grid point took about two seconds, and so in several minutes all points with any chance of being optimal could be checked. We found the minimum to be $(39^\circ,87^\circ)$, a location in Greene County, Indiana, about 70 miles southwest of Indianapolis. From this point the average distance to any person in the country is 795 miles, while the average distance to Memphis is 843 miles. It is interesting to note that as FedEx has grown it has established some secondary hubs, one of them being in Indianapolis. Furthermore, the optimal location is only about 85 miles northwest of Louisville where UPS has established its main hub.
The optimal location is 275 miles from the Census Bureau's population center $(37.7^\circ,91.8^\circ)$ located in Phelps County, Missouri and 315 miles from Memphis $(35^\circ,90^\circ)$.
\begin{figure}
\vspace{.3in}
\centerline {
\includegraphics[width=5in]{morrison5}
}
\caption{Location of the 2000 Census Bureau population center (C), the 2000 U.S. population hub (H), and the FedEx Memphis hub (M). The median center is very close to H.}
\end{figure}
The Census Bureau also computes a point called the ``median center of population'' by finding the median latitude and median longitude for the U.S. population; one half of the population lives north of the median latitude and one half of the population lives east of the median longitude. In the plane, this point minimizes the expected distance using the 1-norm to define distance between $x, y$ by $\| x - y \| = |x_1-y_1| + |x_2-y_2|$, and so the Census Bureau is essentially using the distance defined by the sum of differences in latitude and longitude. In general the optimal points depend on the metric used, but in this case the median center of population when rounded to the nearest degree is the same as the hub we found. (Their precise answer is $(38.75644^\circ, 86.93074^\circ)$. See \cite{Census01}.) Refer to Figure 5 for a map showing these locations.
\section*{Generalizations and Extensions}
\subsection*{More than four points in the plane}
With five or more points in the plane there is no simple description of the hub. The difficulty stems from the fact that the set $\{(u_1,\ldots,u_k) \, | \, \|u_i\|=1,\sum u_i =0\} \subset (\mathbb{R}^2)^k$ has dimension $k-2$. The one-dimensional orthogonal group $O(2)$ acts on the set by rotations; the quotient space is the space of geometrically distinct configurations and has dimension $k-3$. For $k=3$ there is a unique configuration up to the action of $O(2)$, and for $k=4$ there is a one-dimensional set of geometrically distinct configurations parameterized by the smallest angle between two of the $u_i$. For $k \geq 5$ the dimension of the set of geometrically distinct configurations is two or more, and there are too many possibilities. Figure 6 is a typical example with $k=6$, showing the unit vectors and the associated hexagon that has no obvious symmetry. The hexagon can be deformed continuously into other hexagons with three degrees of freedom. (Challenge to the reader: identify the degrees of freedom.)
\begin{figure}
\vspace{-.5in}
\centerline {
\includegraphics[width=4.5in]{morrison6}
}
\vspace{-3cm}
\caption{A configuration of six unit vectors with zero sum.}
\end{figure}
\subsection*{From points to probability measures}
A finite number of points $x_1,x_2,\ldots,x_k$ in $\mathbb{R}^n$ can be regarded as a discrete probability measure with each point having probability $1/k$. It is natural to define a hub for a probability measure $\rho$ as a point having minimal expected distance to points distributed according to $\rho$. Then Theorem 1 has the following generalization and can be proved in essentially in the same way. (An even more general result can be found in \cite{MilasevicDucharme87}.)
\begin{theorem}
Let $\rho$ be a probability measure on $\mathbb{R}^n$ with compact support not lying on a line. Then $\rho$ has a unique hub, and this hub is contained in the convex hull of the support.
\end{theorem}
With this result we know that reasonable sets in $\mathbb{R}^n$ have unique hubs. For example, let $X$ be a bounded subset of $\mathbb{R}^n$ having positive Lebesgue measure $\lambda(X)$ and define $\rho$ to be normalized Lebesgue measure restricted to $X$. That is, for a measurable subset $E$
\[\rho (E)=\lambda(E \cap X)/\lambda(X).\]
There are only a few regions for which the hub can be exactly determined. For the interior of a rhombus or an ellipse the hub is the center, but for a region as simple as the interior of an isosceles triangle there does not appear to be an exact formula for the hub.
The theorem also applies to bounded curves in the plane for which $\rho$ is normalized arc length of the curve. Again there are very few curves for which the hub can be described exactly.
\subsection*{Hubs on a sphere}
Essential to the proof of uniqueness of a hub for points in $\mathbb{R}^n$ is that the function to be minimized is strictly convex, a property that depends strongly on the Euclidean structure of $\mathbb{R}^n$. Interesting and challenging questions immediately arise in other metric spaces such as the sphere. Given points $x_1,x_2,\ldots,x_k$ on the sphere,
a hub is a point $h$ minimizing the function $f(x)= \sum_i d(x_i,x)$, where $d(x,y)$ is the great circle distance between $x$ and $y$. Hubs \emph{exist} because the function to be minimized is continuous and the sphere is compact, but \emph{uniqueness} may fail. An easy example is the case of two antipodal points, say the north and south poles; in this case any point on the equator is a hub.
A subset of the sphere is spherically convex if it contains the geodesics between any two points in the subset, and the (spherical) convex hull of a subset is the smallest convex set containing the subset. Aly, Kay, and Litwhiler \cite{AlyKayLitwhiler79} prove that if $x_1,\ldots,x_k$ lie in an open hemisphere, then the hub (or hubs) must be in the convex hull of the points. One might conjecture in that case that the hub is unique, but even for three points that is not always true. For example, three points equally spaced on the same latitude just above the equator have the property that the points themselves are minima and they are the only minima. A complete description of the minima for three points was given by Cockayne in 1972 \cite{Cockayne72}. It would be interesting to find reasonable assumptions that guarantee unique hubs for $k$ points, as well as to prove existence and uniqueness results for more general probability measures on $S^2$ and on higher dimensional spheres.
\subsection*{Multiple hubs}
As FedEx and other package shipping companies have grown, they have established additional hubs so that packages from Boston to New York, for example, are shipped through an intermediate location on the East Coast rather than through Memphis. Deciding where to put a second or third hub is a ``multiple facility location'' problem in operations research, and there is ongoing interest in such problems. The two hub problem for finite sets in Euclidean space is the following. Given $x_1,x_2,\ldots,x_k$ in $\mathbb{R}^n$ and two hubs $u, v \in \mathbb{R}^n$ a package shipped from $x_i$ to $x_j$ goes via the hub that results in the shorter total distance traveled by the package.
One can prove the existence of minimizing pairs using continuity and compactness arguments, but uniqueness may not hold in the generality of the one hub case because the function $f$ is no longer convex. Explicitly finding solutions, however, appears to be virtually impossible. Consider, for example, the two hub problem for the uniform distribution on $[0,1]$. The function to be minimized is
\[ f(u,v)= \int_0^1 \int_0^1 \min(|x-u|+|y-u|,|x-v|+|y-v|)\, dx\,dy .\]
Numerical optimization gives the optimal locations as approximately $0.29$ and $0.71$. For these hubs the average distance a package travels is $0.39$, a significant decrease from the one hub average, which is $1/2$, and not too more than the average distance between any two points, which is $1/3$.
| {
"timestamp": "2014-10-27T01:12:27",
"yymm": "1410",
"arxiv_id": "1410.6723",
"language": "en",
"url": "https://arxiv.org/abs/1410.6723",
"abstract": "The original shipping strategy of FedEx is to fly all packages to a hub location during the afternoon and evening, sort them there, and then fly them to their destinations during the night for delivery the next day. This leads to interesting mathematical questions: Given a population represented by points in Euclidean space or on a sphere, what is the location of the point of the hub that minimizes the total distance to all the points? Is such a point unique? Then using census data from 2000 we examine how close the FedEx hub in Memphis is to the hub for the U.S. population.",
"subjects": "History and Overview (math.HO); Metric Geometry (math.MG); Probability (math.PR)",
"title": "The FedEx problem",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754463340692,
"lm_q2_score": 0.718594386544335,
"lm_q1q2_score": 0.7075103888650454
} |
https://arxiv.org/abs/1602.04055 | On the Higher Dimensional Quasi-Power Theorem and a Berry-Esseen Inequality | Hwang's quasi-power theorem asserts that a sequence of random variables whose moment generating functions are approximately given by powers of some analytic function is asymptotically normally distributed. This theorem is generalised to higher dimensional random variables. To obtain this result, a higher dimensional analogue of the Berry-Esseen inequality is proved, generalising a two-dimensional version of Sadikova. | \section{Introduction}
Asymptotic normality is a frequently occurring phenomenon in combinatorics, the
classical central limit theorem being the very first example. The first step in
the proof is the observation that the
moment generating function of the sum of $n$ identically independently distributed
random variables is the $n$-th power of the moment generating function of the
distribution underlying the summands. As similar moment generating functions
occur in many examples in combinatorics, a general theorem to prove asymptotic
normality is desirable. Such a theorem was proved by
Hwang~\cite{Hwang:1998}, usually called the ``quasi-power theorem''.
\begin{theoremnonumber}[Hwang~\cite{Hwang:1998}]
Let $\{\Omega_n\}_{n\ge 1}$ be a sequence of integral random
variables. Suppose that the moment generating function satisfies the asymptotic expression
\begin{equation}\label{eq:moments-1d}
M_n(s):=\mathbb{E}(e^{\Omega_ns})=e^{W_n(s)}(1+O(\kappa_n^{-1})),
\end{equation}
the $O$-term being uniform for $\abs{s}\le \tau$, $s\in\mathbb{C}$,
$\tau>0$, where
\begin{enumerate}
\item $W_n(s)=u(s)\phi_{n}+v(s)$, with $u(s)$ and $v(s)$
analytic for $\abs{s}\le \tau$ and independent of $n$; and $u''(0)\neq 0$;
\item $\lim_{n\to\infty}\phi_{n}=\infty$;
\item $\lim_{n\to\infty}\kappa_n=\infty$.
\end{enumerate}
Then the distribution of $\Omega_n$ is asymptotically normal, i.e.,
\begin{equation*}
\sup_{x\in\mathbb{R}}\left\vert\P\left(\frac{\Omega_n- u'(0)\phi_{n}}{\sqrt{u''(0)\phi_{n}}} <
x\right)-
\Phi(x)\right\vert=O\left(\frac{1}{\sqrt{\phi_{n}}}+\frac{1}{\kappa_n}\right),
\end{equation*}
where $\Phi$ denotes the standard normal distribution
\begin{equation*}
\Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}\exp\left(-\frac12
y^2\right)\,dy.
\end{equation*}
\end{theoremnonumber}
See Hwang's article~\cite{Hwang:1998} as well as
Flajolet-Sedgewick~\cite[Sec.~IX.5]{Flajolet-Sedgewick:ta:analy} for many
applications of this theorem.
A generalisation of the quasi-power theorem to dimension~$2$ has been provided
in \cite{Heuberger:2007:quasi-power}. It has been used in \cite{Heuberger-Prodinger:2006:analy-alter},
\cite{Heuberger-Prodinger:2007:hammin-weigh},
\cite{Eagle-Gao-Omar-Panario:2008:distr:short},
\cite{Heuberger-Kropf-Wagner:2014:combin-charac} and
\cite{Kropf:2015:varian-and}. In~\cite[Thm.~2.22]{Drmota:2009:random},
an $m$-dimensional version of the quasi-power theorem is stated
without speed of convergence. Also
in~\cite{Bender-Richmond:1983:centr}, such an $m$-dimensional theorem
without speed of convergence is proved. There, several multidimensional applications
are given, too.
In contrast to many results about the speed of convergence in classical probability
theory (see, e.g.,~\cite{Gut:2005:probab}), the sequence of
random variables is not assumed to be independent. The only assumption is that the
moment generating function behaves asymptotically like a large
power. This mirrors the fact that the moment generating function of
the sum of independent, identically distributed random variables is exactly a
large power. The advantage is that the asymptotic expression~\eqref{eq:moments-1d} arises naturally in combinatorics by
using techniques such as singularity analysis or saddle point
approximation (see \cite{Flajolet-Sedgewick:ta:analy}).
The purpose of this article is to generalise the quasi-power theorem
including the speed of convergence to
arbitrary dimension $m$. We first state this main result in
Theorem~\ref{th:quasi-power-dD} in this section. In
Section~\ref{sec:Berry--Esseen}, a new Berry--Esseen inequality
(Theorem~\ref{theorem:Berry-Esseen-dimension-m}) is presented, which we
use to prove the $m$-dimensional quasi-power theorem. We give sketches of the
proofs of these two theorems in
Section~\ref{sec:sketch-proofs}. All details of these proofs can be
found in the full version of this extended abstract. In
Section~\ref{sec:exampl-mult-centr}, we give some applications of the
multidimensional quasi-power theorem.
We use the following conventions: vectors are denoted by
boldface letters such as $\mathbf{s}$, their components are then denoted by
regular letters with indices such as $s_j$. For a vector $\mathbf{s}$, $\|\mathbf{s}\|$ denotes the
maximum norm $\max\{\abs{s_j}\}$.
All implicit constants of
$O$-terms may depend on the dimension $m$ as well as on $\tau$ which is introduced in
Theorem~\ref{th:quasi-power-dD}.
Our first main result is the following $m$-dimensional version of Hwang's theorem.
\begin{theorem}\label{th:quasi-power-dD}
Let $\{\boldsymbol{\Omega}_n\}_{n\ge 1}$ be a sequence of $m$-dimensional real random
vectors. Suppose that the moment generating function satisfies the asymptotic expression
\begin{equation}\label{eq:moment-asymp}
M_n(\mathbf{s}):=\mathbb{E}(e^{\langle \boldsymbol{\Omega}_n,\mathbf{s}\rangle})=e^{W_n(\mathbf{s})}(1+O(\kappa_n^{-1})),
\end{equation}
the $O$-term being uniform for $\norm{\mathbf{s}}\le \tau$, $\mathbf{s}\in\mathbb{C}^m$,
$\tau>0$, where
\begin{enumerate}
\item $W_n(\mathbf{s})=u(\mathbf{s})\phi_{n}+v(\mathbf{s})$, with $u(\mathbf{s})$ and $v(\mathbf{s})$
analytic for $\norm{\mathbf{s}}\le \tau$ and independent of $n$; and the Hessian
$H_u(\boldsymbol{0})$ of $u$ at the origin is non-singular;
\item $\lim_{n\to\infty}\phi_{n}=\infty$;
\item $\lim_{n\to\infty}\kappa_n=\infty$.
\end{enumerate}
Then, the distribution of $\boldsymbol{\Omega}_n$ is
asymptotically normal with speed of convergence $O(\phi_n^{-1/2})$, i.e.,
\begin{equation}\label{eq:quasi-power-result}
\sup_{\mathbf{x}\in\mathbb{R}^{m}}\left\vert\P\left(\frac{\boldsymbol{\Omega}_n-\grad u (\boldsymbol{0})\phi_{n}}{\sqrt{\phi_{n}}} \le
\mathbf{x}\right)-
\Phi_{H_u(\boldsymbol{0})}(\mathbf{x})\right\vert=O\left(\frac{1}{\sqrt{\phi_{n}}}\right),
\end{equation}
where $\Phi_{\Sigma}$ denotes the distribution function
of the non-degenerate $m$-dimensional normal distribution
with mean $\boldsymbol{0}$ and variance-covariance
matrix $\Sigma$, i.e.,
\begin{equation*}
\Phi_\Sigma(\mathbf{x})=\frac{1}{(2\pi)^{m/2}\sqrt{\det \Sigma}}\int_{\mathbf{y}\le \mathbf{x}}\exp\left(-\frac12
\mathbf{y}^\top \Sigma^{-1} \mathbf{y} \right)\,d\mathbf{y},
\end{equation*}
where $\mathbf{y}\le \mathbf{x}$ means $y_\ell\le x_\ell$ for $1\le \ell\le m$.
If $H_{u}(\boldsymbol{0})$ is singular, the random variables
\begin{equation*}
\frac{\boldsymbol{\Omega}_{n}-\grad u(\boldsymbol{0})\phi_{n}}{\sqrt{\phi_{n}}}
\end{equation*}
converge in distribution to a
degenerate normal distribution with mean $\boldsymbol{0}$ and variance-covariance
matrix $H_{u}(\boldsymbol{0})$.
\end{theorem}
Note that in the case of the singular $H_{u}(\boldsymbol{0})$, a uniform speed of
convergence cannot be guaranteed.
To see this,
consider the (constant) sequence of random variables $\Omega_{n}$
which takes values $\pm1$ each with probability $1/2$. Then the
moment generating function is $(e^{t}+e^{-t})/2$,
which is of the form \eqref{eq:moment-asymp} with $\phi_{n}=n$,
$u(s)=0$, $v(s)=\log (e^{t}+e^{-t})/2$ and $\kappa_{n}$ arbitrary. However, the distribution function of $\Omega_{n}/\sqrt{n}$ is
given by
\begin{equation*}
\mathbb{P}\biggl(\frac{\Omega_{n}}{\sqrt{n}}\le x\biggr)=
\begin{cases}
0& \text{if }x<-1/\sqrt{n},\\
1/2& \text{if }-1/\sqrt{n}\le x<1/\sqrt{n},\\
1& \text{if }1/\sqrt{n}\le x,
\end{cases}
\end{equation*}
which does not converge uniformly.
In contrast to the original quasi-power theorem, the error term in our result does not
contain the summand $O(1/\kappa_n)$. In fact, this summand could also be
omitted in the original proof of the quasi-power theorem by using a better
estimate for the error $E_{n}(\mathbf{s})=M_n(\mathbf{s})e^{-W_{n}(\mathbf{s})}-1$.
The proof of Theorem~\ref{th:quasi-power-dD} relies on an
$m$-dimensional Berry--Esseen inequality
(Theorem~\ref{theorem:Berry-Esseen-dimension-m}). It is a generalisation of
Sadikova's result~\cite{Sadikova:1966:esseen, Sadikova:1966:esseen:englisch} in
dimension $2$. The main challenge is to provide a version which leads to
bounded integrands around the origin, but still allows to use excellent bounds for the
tails of the characteristic functions. To achieve this, linear combinations
involving all partitions of the set $\{1,\ldots, m\}$ are used.
Note that there are several generalisations of the one-dimensional
Berry--Esseen inequality \cite{Berry:1941:gauss,Esseen:1945:fourier}
to arbitrary dimension, see, e.g., Gamkrelidze~\cite{Gamkrelidze:1977,
Gamkrelidze:1977:englisch} and Prakasa
Rao~\cite{Rao:2002:anoth-esseen}. However, using these results would lead to
the less precise error term in~\eqref{eq:quasi-power-result}, see the end of
Section~\ref{sec:Berry--Esseen} for more details. For that reason we generalise Sadikova's
result, which was already successfully used by the first author
in~\cite{Heuberger:2007:quasi-power} to prove a $2$-dimensional
quasi-power theorem. Also note
that our theorem can deal with discrete random variables, in contrast
to \cite{Roussas:2001:esseen}, where density functions are considered.
For the sake of completeness, we also state the following result
about the moments of $\boldsymbol{\Omega}_{n}$.
\begin{proposition}\label{proposition:moments}
The cross-moments of $\boldsymbol{\Omega}_{n}$ satisfy
\begin{equation*}
\frac{1}{\prod_{\ell=1}^{m}k_{\ell}!}\mathbb E\Big(\prod_{\ell=1}^{m}\Omega_{n,\ell}^{k_{\ell}}\Big)=p_{\mathbf{k}}(\phi_{n})+O\Big(\kappa_{n}^{-1}\phi_{n}^{k_{1}+\cdots+k_{m}}\Big),
\end{equation*}
for $k_{\ell}$ nonnegative integers, where $p_{\mathbf{k}}$ is a polynomial of degree
$\sum_{\ell=1}^{m}k_{\ell}$ defined by
\begin{equation*}
p_{\mathbf{k}}(X)=[s_{1}^{k_{1}}\cdots s_{m}^{k_{m}}]e^{u(\mathbf{s})X+v(\mathbf{s})}.
\end{equation*}
In particular, the mean and the variance-covariance matrix are
\begin{align*}
\mathbb E(\boldsymbol{\Omega}_{n})&=\grad u(\boldsymbol{0})\phi_{n}+\grad
v(\boldsymbol{0})+O(\kappa_{n}^{-1}),\\
\Cov(\boldsymbol{\Omega}_{n})&=H_{u}(\boldsymbol{0})\phi_{n}+H_{v}(\boldsymbol{0})+O(\kappa_{n}^{-1}),
\end{align*}
respectively.
\end{proposition}
\section{A Berry--Esseen Inequality}\label{sec:Berry--Esseen}
This section is devoted to a generalisation of Sadikova's Berry--Esseen
inequality~\cite{Sadikova:1966:esseen, Sadikova:1966:esseen:englisch} in
dimension 2 to dimension $m$. Before stating the theorem, we introduce our notation.
Let $L=\{1,\ldots, m\}$. For $K\subseteq L$, we write
$\mathbf{s}_K=(s_k)_{k\in K}$ for the projection of $\mathbf{s}\in\mathbb{C}^L$ to $\mathbb{C}^K$.
For $J\subseteq K\subseteq L$, let
$\chi_{J,K}\colon \mathbb{C}^{J}\to\mathbb{C}^{K}$, $(s_{j})_{j\in J}\mapsto
(s_{k}\iverson{k\in J})_{k\in K}$ be an injection from $\mathbb{C}^{J}$ into
$\mathbb{C}^{K}$. Similarly, let $\psi_{J,K}\colon \mathbb{C}^{K}\to\mathbb{C}^{K}$,
$(s_{k})_{k\in K}\mapsto (s_{k}\iverson{k\in J})_{k\in K}$ be the
projection which sets all coordinates corresponding to $K\setminus J$ to
$0$.
We denote the
set of all partitions of $K$ by $\Pi_K$. We consider a partition as a set
$\alpha=\{J_{1},\ldots,J_{k}\}$. Thus $\abs{\alpha}$ denotes the number of parts of the partition
$\alpha$. Furthermore, $J\in\alpha$ means that $J$ is a part of the
partition $\alpha$.
Now, we can define an operator which we later use to state our Berry--Esseen
inequality. The motivation behind this definition is explained at the end of
this section.
\begin{definition}\label{definition:Lambda-K}
Let $K\subseteq L$ and $h\colon \mathbb{C}^K\to \mathbb{C}$.
We define the non-linear operator
\begin{equation*}
\Lambda_K(h):=\sum_{\alpha\in\Pi_K}\mu_\alpha \prod_{J\in
\alpha}h\circ \psi_{J, K}
\end{equation*}
where
\begin{equation*}
\mu_\alpha = (-1)^{\abs{\alpha}-1}(\abs{\alpha}-1)!\,.
\end{equation*}
We denote $\Lambda_{L}$ briefly by $\Lambda$.
\end{definition}
For any random variable $\mathbf{Z}$, we denote its cumulative distribution function
by $F_\mathbf{Z}$ and its
characteristic function by $\varphi_\mathbf{Z}$.
With these definitions, we are able to state our second main result, an
$m$-dimensional version of the Berry--Esseen inequality.
\begin{theorem}\label{theorem:Berry-Esseen-dimension-m}
Let $m\ge 1$ and $\mathbf{X}$ and $\mathbf{Y}$ be $m$-dimensional random variables. Assume
that $F_\mathbf{Y}$ is differentiable.
Let
\begin{align*}
A_j&=\sup_{\mathbf{y} \in\mathbb{R}^m}\frac{\partial F_\mathbf{Y}(\mathbf{y})}{\partial y_j},\\
B_j&=\sum_{k=1}^{j} \stirlingpartition{j}{k} k!\ ,\\
C_1&=\sqrt[3]{\frac{32}{\pi\bigl(1-\bigl(\frac{3}{4}\bigr)^{1/m}\bigr)}},\\
C_2&=\frac{12}{\pi}
\end{align*}
for $1\le
j\le m$ where $\stirlingpartition{j}{k}$ denotes a Stirling partition number
(Stirling number of the second kind).
Let $T>0$ be fixed. Then
\begin{equation}\label{eq:Berry-Esseen}
\begin{aligned}
\sup_{\mathbf{z}\in\mathbb{R}^m}\abs{F_{\mathbf{X}}(\mathbf{z})-F_{\mathbf{Y}}(\mathbf{z})}&\le
\frac{2}{(2\pi)^m} \int_{\norm{\mathbf{t}}\le
T}\abs[\Big]{\frac{\Lambda(\varphi_{\mathbf{X}})(\mathbf{t})-\Lambda(\varphi_{\mathbf{Y}})(\mathbf{t})}{\prod_{\ell\in
L} t_\ell}}\,d\mathbf{t} \\
&\qquad+ 2\sum_{\emptyset\neq J\subsetneq
L}B_{m-\abs{J}}\sup_{\mathbf{z}_J\in\mathbb{R}^J}\abs[\big]{F_{\mathbf{X}_{J}}(\mathbf{z}_J)-F_{\mathbf{Y}_{J}}(\mathbf{z}_J)}
\\
&\qquad +\frac{2\sum_{j=1}^m A_j}{T}(C_1+C_2).
\end{aligned}
\end{equation}
Existence of $\mathbb{E}(\mathbf{X})$ and $\mathbb{E}(\mathbf{Y})$ is sufficient for the finiteness
of the integral in \eqref{eq:Berry-Esseen}.
\end{theorem}
Let us give two remarks on the distribution functions occurring in this theorem: The distribution function $F_\mathbf{Y}$ is non-decreasing in every
variable, thus $A_j>0$ for all $j$. Furthermore, our general notations imply that $F_{\mathbf{X}_J}$ is a marginal
distribution of $\mathbf{X}$.
The numbers $B_j$ are known as ``Fubini numbers'' or ``ordered Bell numbers''.
They form the sequence \href{http://oeis.org/A000670}{A000670} in \cite{OEIS:2015}.
Recursive application of \eqref{eq:Berry-Esseen} leads to the following
corollary, where we no longer explicitly state the constants depending on the
dimension.
\begin{corollary}\label{corollary:Berry-Esseen}
Let $m\ge 1$ and $\mathbf{X}$ and $\mathbf{Y}$ be $m$-dimensional random
variables. Assume
that $F_\mathbf{Y}$ is differentiable and
let
\begin{equation*}
A_j=\sup_{\mathbf{y} \in\mathbb{R}^m}\frac{\partial F_\mathbf{Y}(\mathbf{y})}{\partial y_j}, \qquad 1\le
j\le m.
\end{equation*}
Then
\begin{multline}\label{eq:Berry-Esseen-recursive}
\sup_{\mathbf{z}\in\mathbb{R}^m}\abs{F_{\mathbf{X}}(\mathbf{z})-F_{\mathbf{Y}}(\mathbf{z})}\\=
O\biggl(\sum_{\emptyset \neq K\subseteq L}\int_{\norm{\mathbf{t}_K}\le
T}\abs[\Big]{\frac{\Lambda_K(\varphi_{\mathbf{X}}\circ\chi_{K, L})(\mathbf{t}_K)-\Lambda_K(\varphi_{\mathbf{Y}}\circ\chi_{K, L})(\mathbf{t}_K)}{\prod_{k\in
K} t_k}}\,d\mathbf{t}_K + \frac{\sum_{j=1}^m A_j}{T}\biggr)
\end{multline}
where the $O$-constants only depend on the dimension $m$.
Existence of $\mathbb{E}(\mathbf{X})$ and $\mathbb{E}(\mathbf{Y})$ is sufficient for the finiteness
of the integrals in \eqref{eq:Berry-Esseen-recursive}.
\end{corollary}
In order to explain the choice of the operator $\Lambda$, we first state it in
dimension $2$:
\begin{equation}\label{eq:Sadikova-simple}
\Lambda(h)(s_1, s_2) = h(s_1, s_2) - h(s_1, 0)h(0, s_2).
\end{equation}
This coincides with Sadikova's definition. This also shows that our
operator is non-linear as, e.g., $\Lambda(s_{1}+s_{2})(s_{1},s_{2})\neq\Lambda(s_{1})(s_{1},s_{2})+\Lambda(s_{2})(s_{1},s_{2})$.
In Theorem~\ref{theorem:Berry-Esseen-dimension-m}, we apply $\Lambda$ to
characteristic functions; so we may restrict our attention to functions $h$
with $h(\boldsymbol{0})=1$. From~\eqref{eq:Sadikova-simple}, we see that
$\Lambda(h)(s_1, 0) = \Lambda(h)(0, s_2)=0$, so that $\Lambda(h)(s_1,
s_2)/(s_1s_2)$ is bounded around the origin. This is essential for the boundedness
of the integral in Theorem~\ref{theorem:Berry-Esseen-dimension-m}. In general,
this property will be guaranteed by our particular choice of coefficients. It
is no coincidence that for $\alpha\in \Pi_L$, the coefficient $\mu_\alpha$
equals the value $\mu(\alpha, \{L\})$ of the Möbius function in the lattice of partitions:
Weisner's theorem (see Stanley~\cite[Corollary~3.9.3]{Stanley:2012:enumer_1}) is crucial in the proof that $\Lambda(h)(\mathbf{s})/(s_1\cdots
s_m)$ is bounded around the origin.
The second property is that our proof of the quasi-power theorem needs
estimates for the tails of the integral in Theorem~\ref{theorem:Berry-Esseen-dimension-m}. These estimates have to be
exponentially small in every variable, which means that every variable has to
occur in every summand. This is trivially fulfilled as every summand
in the definition of $\Lambda$ is
formulated in terms of a partition.
Note that Gamkrelidze~\cite{Gamkrelidze:1977:englisch} (and also
Prakasa Rao~\cite{Rao:2002:anoth-esseen}) use a linear operator $L$ mapping $h$ to
\begin{equation}\label{eq:gamkrelidze}
(s_1, s_2) \mapsto h(s_1, s_2) - h(s_1, 0) - h(0, s_2).
\end{equation}
When taking the difference of two characteristic functions, we may assume that
$h(0, 0)=0$ so that the first crucial property as defined above still
holds. However, the tails are no longer exponentially small in every
variable: The last summand $h(0,s_{2})$ in \eqref{eq:gamkrelidze} is not exponentially small in
$s_{1}$ because it is independent of $s_{1}$ and nonzero in
general. However, the first two summands are exponentially small in
$s_{1}$ by our assumption~\eqref{eq:moment-asymp}.
For that reason, using the Berry--Esseen inequality by
Gamkrelidze~\cite{Gamkrelidze:1977:englisch} to prove a quasi-power theorem leads to a less precise error term
$O(\phi_{n}^{-1/2}\log^{m-1}\phi_n)$
in~\eqref{eq:quasi-power-result}. It can be shown that the less precise
error term necessarily appears when using Gamkrelidze's result by considering
the example of $\boldsymbol{\Omega}_n$ being the $2$-dimensional vector consisting of a normal distribution
with mean $-1$ and variance $n$ and a normal distribution with mean $0$ and
variance $n$. This is a consequence of the linearity of the
operator $L$ in Gamkrelidze's result.
\section{Examples of Multidimensional Central Limit
Theorems}\label{sec:exampl-mult-centr}
In this section, we give two examples from combinatorics where we can
apply Theorem~\ref{th:quasi-power-dD}. Asymptotic normality was
already shown in earlier publications
\cite{Drmota:1997:system-funct-equat,Bender-Richmond:1983:centr}, but
we additionally provide an estimate for the
speed of convergence.
\subsection{Context-Free Languages}
Consider the following example of a context-free grammar $G$ with
non-terminal symbols $S$ and $T$, terminal symbols $\{a,b,c\}$,
starting symbol $S$ and the rules
\begin{equation*}
P=\{S\to aSbS,\, S\to bT,\, T\to bS,\, T\to cT,\, T\to a\}.
\end{equation*}
The corresponding context-free language $L(G)$ consists of all words which
can be generated starting with $S$ using the rules in $P$ to replace
all non-terminal symbols. For example, $abcabababba\in L(G)$ because it can be derived as
\begin{equation*}
S\to aSbS
\to abTbaSbS
\to abcTbabTbbT
\to abcabababba.
\end{equation*}
Let $\P(\boldsymbol{\Omega}_{n}=\mathbf{x})$ be the probability that a word of length $n$
in $L(G)$ consists of $x_{1}$ and $x_{2}$ terminal symbols
$a$ and $b$, respectively. Thus there are $n-x_{1}-x_{2}$ terminal
symbols $c$. For simplicity, this random variable is only
$2$-dimensional. But it can be easily
extended to higher dimensions.
Following
Drmota~\cite[Sec.~3.2]{Drmota:1997:system-funct-equat}, we obtain that the
moment generating function is
\begin{equation*}
\mathbb{E}(e^{\langle
\boldsymbol{\Omega}_n,\mathbf{s}\rangle})=\frac{y_{n}(e^{\mathbf{s}})}{y_{n}(\boldsymbol{1})}
\end{equation*}
with $y_{n}(\boldsymbol{z})$ defined
in \cite{Drmota:1997:system-funct-equat}. Using
\cite[Equ.~(4.9)]{Drmota:1997:system-funct-equat}, this moment
generating function has an asymptotic expansion as in \eqref{eq:moment-asymp} with $\phi_{n}=n$. Thus
$\boldsymbol{\Omega}_{n}$ is asymptotically normally distributed after
standardisation (as was shown in~\cite{Drmota:1997:system-funct-equat})
and additionally the speed of convergence is $O(n^{-1/2})$.
Other context-free languages can be analysed in the same way, either
by directly using the results in \cite{Drmota:1997:system-funct-equat} (if the
underlying system is strongly connected) or by similar methods. This
has applications, for example, in genetics (see~\cite{Poznanovic-Heitsch:2014:asymp-rna}).
\subsection{Dissections of Labelled Convex Polygons}
Let $S_{1}\mathbin{\mathaccent\cdot\cup}\cdots\mathbin{\mathaccent\cdot\cup} S_{t+1}=\{3,4,\ldots\}$ be a
partition. We dissect a labelled convex $n$-gon into smaller convex polygons by
choosing some non-intersecting diagonals. Each small polygon should be a
$k$-gon with $k\not\in S_{t+1}$. Define $a_{n}(\mathbf{r})$ to be the number
of dissections of an $n$-gon such that it consists of exactly $r_{i}$ small
polygons whose number of vertices is in $S_{i}$, for $i=1$, \dots,
$t$. For convenience, we use $a_{2}(\mathbf{r})=[\mathbf{r}=\boldsymbol{0}]$. Asymptotic normality was proved in
\cite[Sec.~3]{Bender-Richmond:1983:centr}, see also
\cite[Ex.~7.1]{Bender:1974:asymp-method-enumer} for a one-dimensional
version. We additionally provide an estimate for the speed of convergence.
Let
\begin{equation*}
f(z,\mathbf{x})=\sum_{\substack{n\geq2\\ \mathbf{r}\geq 0}}a_{n}(\mathbf{r})\mathbf{x}^{\mathbf{r}}z^{n-1}.
\end{equation*}
Then choosing a $k$-gon with $k\in S_{1}\mathbin{\mathaccent\cdot\cup}\cdots\mathbin{\mathaccent\cdot\cup} S_{t}$
and gluing dissected polygons to $k-1$ of its sides translates into
the equation
\begin{equation*}
f=z+\sum_{i=1}^{t}x_{i}\sum_{k\in S_{i}}f^{k-1}.
\end{equation*}
Following \cite{Bender:1974:asymp-method-enumer}, this equation can be used to obtain an
asymptotic expression for the moment generating function as in
\eqref{eq:moment-asymp} with $\phi_{n}=n$. The asymptotic normal
distribution follows after suitable standardisation with speed of
convergence $O(n^{-1/2})$.
\section{Sketch of the Proofs}\label{sec:sketch-proofs}
We now sketch the main ideas of the proofs of
Theorems~\ref{theorem:Berry-Esseen-dimension-m}
and~\ref{th:quasi-power-dD}. All details can be found in the full
version of this extended abstract.
\begin{sketch}[Sketch of the proof of Theorem~\ref{theorem:Berry-Esseen-dimension-m}]
As in
\cite{Sadikova:1966:esseen:englisch,Gamkrelidze:1977:englisch,Rao:2002:anoth-esseen}, our
proof of the Berry--Esseen inequality proceeds via adding a continuous random
variable $\mathbf{Q}$ to
our random variables $\mathbf{X}$ and $\mathbf{Y}$. The
characteristic function of $\mathbf{Q}$ vanishes outside $[-T, T]^m$. The error resulting from replacing the difference of the distribution functions $\abs{F_{\mathbf{X}}-F_{\mathbf{Y}}}$ by $\abs{F_{\mathbf{X}+\mathbf{Q}} - F_{\mathbf{Y}+\mathbf{Q}}}$ can be estimated by the final summand
in~\eqref{eq:Berry-Esseen}. In principle, L\'evy's theorem then allows to
bound the difference of the distribution functions by the difference of the
characteristic functions.
Instead of only using the difference of the
characteristic functions, we use the difference $\abs{\Lambda(\varphi_{\mathbf{X}})-\Lambda(\varphi_{\mathbf{Y}})}$, which
ensures boundedness of the integral
in~\eqref{eq:Berry-Esseen} at least if the first moments exist. However, we have to compensate $\Lambda$ by the
sum over the differences of the marginal distribution functions,
which yields the second summand in~\eqref{eq:Berry-Esseen}.
\end{sketch}
\begin{sketch}[Sketch of the proof of Theorem~\ref{th:quasi-power-dD}]
First, the characteristic
function of the standardised random variable $\mathbf{X}=(\boldsymbol{\Omega}_{n}-\grad u(\boldsymbol{0})\phi_{n})/\sqrt{\phi_{n}}$ is
\begin{equation*}
\varphi_{\mathbf{X}}(\mathbf{s})=\exp\Bigl(-\frac12 \mathbf{s}^\top \Sigma \mathbf{s} + O\Bigl(\frac{\norm{\mathbf{s}}^3+\norm{\mathbf{s}}}{\sqrt{\phi_{n}}}\Bigr)\Bigr)
\end{equation*}
for $\norm{\mathbf{s}}<\tau\sqrt{\phi_{n}}/2$. Thus, we obtain convergence
in distribution as stated in the theorem.
To obtain a bound for the speed of convergence, we use the Berry--Esseen
inequality given in
Theorem~\ref{theorem:Berry-Esseen-dimension-m} for $\mathbf{Y}$ an
$m$-dimensional normal distribution. We bound the
difference of $\Lambda$ evaluated at the characteristic function of
$\mathbf{X}$ and the one of the
normal distribution by the exponentially decreasing function
\begin{equation*}
\abs{\Lambda(\varphi_\mathbf{X})(\mathbf{s})-\Lambda(\varphi_\mathbf{Y})(\mathbf{s})}\le
\exp\Bigl(-\frac{\sigma}{4}\norm{\mathbf{s}}^2 + O(\norm{\mathbf{s}})\Bigr)O\Bigl(\frac{\norm{\mathbf{s}}^3+\norm{\mathbf{s}}}{\sqrt{\phi_{n}}}\Bigr)
\end{equation*}
for suitable $\mathbf{s}$ where $\sigma$ is the smallest eigenvalue of $\Sigma$.
We then estimate the integral in
\eqref{eq:Berry-Esseen}.
For
the variables in a neighbourhood of zero, we get rid of the denominator by
Taylor expansion using the zero of $\Lambda(\varphi_{\mathbf{X}})-\Lambda(\varphi_{\mathbf{Y}})$ at $\boldsymbol{0}$. The error term of the Taylor expansion can be
estimated by the difference of the characteristic
functions using Cauchy's formula. The
exponentially small tails are used to bound the contribution of the
large variables in the integral in~\eqref{eq:Berry-Esseen}.
The second summand in
\eqref{eq:Berry-Esseen} can be estimated inductively.
\end{sketch}
\bibliographystyle{amsplain}
| {
"timestamp": "2016-05-09T02:06:59",
"yymm": "1602",
"arxiv_id": "1602.04055",
"language": "en",
"url": "https://arxiv.org/abs/1602.04055",
"abstract": "Hwang's quasi-power theorem asserts that a sequence of random variables whose moment generating functions are approximately given by powers of some analytic function is asymptotically normally distributed. This theorem is generalised to higher dimensional random variables. To obtain this result, a higher dimensional analogue of the Berry-Esseen inequality is proved, generalising a two-dimensional version of Sadikova.",
"subjects": "Probability (math.PR); Combinatorics (math.CO)",
"title": "On the Higher Dimensional Quasi-Power Theorem and a Berry-Esseen Inequality",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754461077707,
"lm_q2_score": 0.718594386544335,
"lm_q1q2_score": 0.7075103887024285
} |
https://arxiv.org/abs/1601.06287 | Geometric properties of a sine function extendable to arbitrary normed planes | In this paper we study a metric generalization of the sine function which can be extended to arbitrary normed planes. We derive its main properties and give also some characterizations of Radon planes. Furthermore, we prove that the existence of an angular measure which is "well-behaving" with respect to the sine is only possible in the Euclidean plane, and we also define some new constants that estimate how non-Radon or non-Euclidean a normed plane can be. Sine preserving self-mappings are studied, and a complete description of the linear ones is given. In the last section we exhibit a version of the Law of Sines for Radon planes. | \section{Introduction} \label{secintro}
The notion of angle plays an important role in the geometry of (Minkowski or) normed planes. In \cite{brass} Brass introduced the notion of angular measure in normed planes and used such a measure to study packings of the unit circle. Later this concept was more explored. In \cite{duvelmeyer1} D\"{u}velmeyer characterized the Euclidean plane as the only plane where Busemann and Glogovskii angular bisectors can be defined by using an angular measure. Fankh\"{a}nel (see \cite{fankhanel1} and \cite{fankhanel2}) studied particular types of angular measures which carry some properties concerning orthogonality types, and he found some interesting characterizations of the Euclidean plane. Moreover, in \cite{martiniantinorms} Martini and Swanepoel briefly discussed angular measures which are proportional to arcs and to areas of sectors of the unit circle.
Having all this in mind, our purpose is to develop some kind of trigonometry in an arbitrary normed plane, not using any angular measure, but considering a suitable metric extension of the sine function to general normed planes, which was introduced by Szostok \cite{szostok} when studying functional equations. We claim that this generalized sine function has some interesting geometric properties, which we explore throughout this paper.
The idea of studying trigonometry in normed spaces is not new: functions which somehow play the role of cosine and sine functions in normed spaces were studied by authors like Finsler \cite{finsler}, Busemann \cite{busemann}, and Thompson \cite{thompson} (see Chapter 8 there). The last two references provide definitions for sine functions in terms or area and volume. We show that when dealing with the planar case, our definition based on a metric point of view can be, in some sense, given in terms of Euclidean area (Proposition \ref{prop2}). As observed by Thompson (\cite{thompson}, Section 8.5), Minkowskian trigonometric functions can also be related to solutions of second-order linear differential equations of the type $x'' + f(t)x = 0$ (known as Hill equations).
In Section \ref{secsine} we define the generalized sine function and present some of its basic properties. Section \ref{secradonsine} is devoted to characterizations of Radon planes using the sine function (in particular, a new proof of D\"{u}velmeyer's characterization of Radon planes via Busemann and Glogovskii angular bisectors presented in \cite{duvelmeyer} is given). In Section \ref{secangle} we prove that if a Radon plane is endowed with an angular measure which provides equal or supplementary measures for angles with the same sine, then this plane has to be Euclidean. In Section \ref{secconstants} sine based constants are defined and used to estimate how non-Radon or non-Euclidean a normed plane can be (extremal cases are also considered). In Section \ref{secconformal} we study sine preserving self-mappings, which we call \textit{sine conformal}. Last, a Law of Sines for Radon planes is given in Section \ref{lawofsines}.
As usual, $(V,||\cdot||)$ denotes a \emph{normed plane}, also called \emph{Minkowski plane}, i.e., an affine plane whose norm $\|\cdot\|$ is determined by a \emph{unit ball} which is an arbitrary
compact, convex figure centered at the origin; the boundary
$S = \{x \in V : \|x\| = 1\}$ of this figure is called \emph{unit circle} of $(V, \|\cdot\|)$.
For such planes we consider three orthogonality types. Given two vectors $x,y \in V$, we say that \\
$\bullet$ $x$ is \textit{Birkhoff orthogonal} to $y$ (denoted by $x \dashv_B y$) when $||x + ty|| \geq ||x||$ for every $t \in \mathbb{R}$,\\
$\bullet$ $x$ is \textit{isosceles orthogonal} to $y$ (denoted by $x \dashv_I y$) when $||x+y|| = ||x-y||$, and \\
$\bullet$ $x$ is \textit{Roberts orthogonal} to $y$ (denoted by $x \dashv_R y$) when $||x + ty|| = ||x - ty||$ for all $t \in \mathbb{R}$. \\
Also, we denote by $[ab]$, $\left.[ab\right>$, and $\left<ab\right>$ the \emph{segment} from $a$ to $b$, the
\emph{half-line} with origin in $a$ and passing through $b$, and the \emph{line} spanned by $a$ and $b$, respectively.
Good introductions to the geometry of Minkowski planes and spaces are given in the book \cite{thompson}, by the surveys \cite{martini1} and \cite{martini2}, and in the papers \cite{martiniantinorms} and \cite{alonso}.
\section{The generalized sine function} \label{secsine}
When dealing with inner product planes $(V,\left<,\right>)$, the (fairly known) sine between two non-zero vectors $x,y \in V$ is defined by \\
\[ s(x,y) = \sqrt{1 - \frac{\left<x,y\right>^2}{||x||_E^2||y||_E^2}}, \]\\
where the norm comes from the inner product ($||x||_E = \sqrt{\left<x,x\right>}$). Since we are defining the sine function only for non-zero vectors, and since it is clear that $s(\alpha x,y) = s(x,\beta y) = s(x,y)$ for any non-zero real numbers $\alpha$ and $\beta$, we may, for the sake of simplicity, restrict the definition of the sine function to unit vectors. For extending this definition properly to general normed planes, the only suitable tool
is the concept of distances given by the norm. Fortunately, we may describe the sine function of an Euclidean plane easily in terms of distances.
\begin{lemma}\label{lemma1} Let $(V,\left<,\right>)$ be an inner product plane, and denote by $S_E$ the unit circle of the norm derived from the inner product
$\left<,\right>$. If $x, y \in S_E$, then $s(x,y)$ is the distance (in the usual norm) from the origin to the line $l:t\mapsto x + ty$. In other words, $s(x,y) = \inf_{t\in\mathbb{R}}||x+ty||_E$.
\end{lemma}
\noindent\textbf{Proof.} From the standard theory of inner product planes it is clear that this infimum is attained for some $t_0 \in\mathbb{R}$ satisfying $\left<x+t_0y,y\right> = 0$. Hence $t_0 = -\left<x,y\right>$. Now
we have \\
\[ \inf_{t\in\mathbb{R}}||x+ty||_E = ||x+t_0y||_E = \sqrt{\left<x-\left<x,y\right>y,x-\left<x,y\right>y\right>} = \sqrt{1-\left<x,y\right>^2}, \]\\
and this is what we need.
\begin{flushright} $\square$ \end{flushright}
Now we are ready for extending the definition of the sine function in a natural way to general normed planes. \\
\begin{defi}\label{defi1}\normalfont Let $(V,||\cdot||)$ be a normed plane. We define the \textit{sine function} $s:S\times S \rightarrow \mathbb{R}$ by \\
\[s(x,y) = \inf_{t\in\mathbb{R}}||x+ty||. \]\\
In other words, the sine between $x,y\in S$ is the distance from the origin to the line $l:t\mapsto x +ty$.
\end{defi}
\begin{remark}\label{remark2}\normalfont Despite the fact that we defined the sine function only for unit vectors, we will sometimes abuse of the notation and, for any non-zero $x, y \in V$, denote by $s(x,y)$ the sine between the vectors in the respective directions. In other words, sometimes it is more suitable to think about the sine function as a function defined for directions in $V$.
\end{remark}
The next step is to investigate which properties of the sine function still hold in general normed planes.
\begin{lemma}\label{lemma2} In any normed plane $(V,||\cdot||)$ we have $0 \leq s(x,y) \leq 1$ for every $x,y \in S$. Moreover, $s(x,y) = 0$ if and only if $x = \pm y$, and $s(x,y) = 1$ if and only if $x \dashv_B y$.
\end{lemma}
\noindent\textbf{Proof.} The inequality $||x+ty|| \geq 0$ is immediate, and the other one comes from the equality $||x + ty|| = 1$ for $t = 0$. If $x \neq\pm y$, then the line $l:t\mapsto x+ty$ does not passes through the origin, and so $s(x,y) > 0$. The other direction is obvious. Now we look at the last bi-implication. We have $s(x,y) = 1$ if and only if $||x+ty|| \geq 1 = ||x||$ for every $t \in \mathbb{R}$, i.e., if and only if $x \dashv_B y$. This finishes the proof.
\begin{flushright} $\square$ \end{flushright}
\begin{remark}\label{remark1}\normalfont Notice that what we have done so far clearly holds for normed spaces of dimensions larger than $2$. In general, the sine function is not symmetric. More precisely, in the planar case symmetry of the sine function characterizes Radon planes, and for higher dimensions it characterizes inner product spaces. This will be discussed later.
\end{remark}
Geometrically we can characterize the sine $s(x,y)$ for linearly independent $x,y \in S$ as follows: the line $l_1:t\mapsto x+ty$ divides the plane into two half-planes. Denote by $H$ the (open) one which does not contain the origin $o$. Hence, a line $l_2$ parallel to $y$ supports $S\cap H$ at a point $p$, say (see Figure \ref{fig60}). Therefore, if the segment $[op]$ intersects $l_1$ at the point $q$, then $||q|| = s(x,y)$.
\begin{figure}[h]
\centering
\includegraphics{fig60.eps}
\caption{$||q|| = s(x,y)$}
\label{fig60}
\end{figure}
This follows since distances to lines in direction $y$ are attained by any segments in a direction $z$ with $z \dashv_B y$. In \cite{martini1} it is proved that any normed plane has a \emph{pair of conjugate directions}, i.e., a pair $x,y \in S$ such that $x \dashv_B y$ and $y \dashv_B x$. We will study now some kind of triangle trigonometry for normed planes. Using conjugate diameters, we may also derive some sort of polar coordinates. This is the subject of the next proposition.
\begin{prop}\label{prop3} Let $(V,||\cdot||)$ be a normed plane and assume that $x,y\in S$ are such that $x \dashv_B y$. If $\Delta\mathbf{abc}$ is a triangle such that the segment $[ba]$ is in the direction of $x$ and the segment $[bc]$ is in the direction $y$, then \\
\[ s(c-a,c-b) = \frac{||b-a||}{||c-a||}. \]\\
In particular, if $x,y \in S$ are conjugate diameters, then for any vector $z$, which can be written as $z = \alpha x + \beta y$ for non-negative $a,b \in \mathbb{R}$, we have \\
\[ z = ||z||\left(s(z,y)x + s(z,x)y\right). \]\\
These may be interpreted as polar coordinates for normed planes.
\end{prop}
\noindent\textbf{Proof.} Obviously, we may assume that $a$ is the origin $o$ and that $||a-c|| = 1$. Hence we have to prove that $s(c,c-b) = ||b||$. But this is easy since $s(c,c-b)$ is the distance from the line $\left<cb\right>$ to the origin, and because $x \dashv_B y$, this distance is attained precisely at $b$ (see Figure \ref{fig61}). \\
Assume now that $z = \alpha x + \beta y$ with $\alpha,\beta > 0$, and consider the triangle $\Delta\mathbf{(\alpha x)oz}$ (see Figure \ref{fig62}). By the previous arguments we have $s(z,y) = \frac{||\alpha x||}{||z||} = \frac{\alpha}{||z||}$ and $s(z,x) = \frac{||\beta y||}{||z||} = \frac{\beta}{||z||}$. If $\alpha = 0$ or $\beta = 0$, the result is immediate.
\begin{flushright} $\square$ \end{flushright}
\begin{figure}[t]
\centering
\includegraphics{fig61.eps}
\caption{$s(c,c-b) = ||b||$}
\label{fig61}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics{fig62.eps}
\caption{Polar coordinates ($z\in S$)}
\label{fig62}
\end{figure}
\begin{remark}\label{remark3}\normalfont We can clearly work also with vectors which are not in the quadrant $\mathrm{conv}(\left.[ox\right>\cup\left.[oy\right>)$, by using absolute values. For example, if $z = \alpha x + \beta y$ is a point of $S$ with $\alpha < 0$, then $s(z,x) = |\alpha|$.
\end{remark}
We notice here that triangles which have two sides lying in conjugate directions behave, in the sense of the sine of the corresponding (ordered) angle, like rectangular triangles in the Euclidean plane. Due to this, it would be natural do define a cosine function $c:S\times S \rightarrow\mathbb{R}$ by setting $c(z,x) = s(z,y)$, where $y\in S$ is a conjugate direction of $x$. The problem here is that we can have directions with no conjugate, and if there is a conjugate, it might not be unique. This is the main reason why we work with only one trigonometric function. If we would work with an inner product plane, then we would clearly have $s(z,x)^2 + s(z,y)^2 = 1$ for any orthogonal pair $x,y \in S$ and every $z \in S$. This is not true for an arbitrary normed plane, and thus the ``distortion" in this equality may provide a way to measure how far the plane is from being Euclidean. This is made in Section \ref{secconstants}, but we can already estimate the range of $s(z,x)^2 + s(z,y)^2$.
\begin{lemma}\label{lemma7} If $x,y \in S$ are conjugate directions, then $\frac{1}{2} \leq s(z,x)^2 + s(z,y)^2 \leq 2$ for all $z \in S$. On the left side, equality holds if and only if $z = \frac{x+y}{2}$ or $z = \frac{y-x}{2}$ (in particular, if one of the segments $[xy]$ or $[y(-x)]$ is contained in the unit circle), and on the right side equality holds if and only if $z = x+y$ or $z = y-x$ (in particular, if $[y(x+y)]$ and $[(x+y)x]$ or $[y(y-x)]$ and $[(y-x)(-x)]$ are contained in $S$).
\end{lemma}
\noindent\textbf{Proof.} It is clear that $s(z,x)^2 + s(z,y)^2 \leq 2$, and that equality holds if and only if $s(z,x) = s(z,y) = 1$. Since a unit vector $z$ of the half-circle $S_1$ from $x$ to $-x$ passing through $y$ can be written as $z = s(z,y)x + s(z,x)y$ or $z = s(z,y)(-x)+s(z,x)y$, it follows that $s(z,x)^2 + s(z,y)^2 = 2$ if and only if $x+y \in S$ or $y-x \in S$. By convexity, if a point $z = \alpha x+\beta y$ belongs to the unit circle, then the intersection of the ray $\left.[oz\right>$ with the segment $[xy]$ is a point $z_0 = \alpha_0x + \beta_0y$ for which $|\alpha_0| \leq |\alpha|$ and $|\beta_0| \leq |\beta|$ hold. Hence, the minimum value for $s(z,x)^2 + s(z,y)^2$ is attained if and only if one of the segments $[xy]$ or $[(-x)y]$ is contained in the unit circle. In this case the value $\frac{1}{2}$ is easily achieved, since $\frac{1}{2} = \min\{\alpha^2+\beta^2:\alpha + \beta = 1\}$. Figure \ref{fig63} illustrates the situation.
\begin{flushright} $\square$ \end{flushright}
\begin{figure}[h]
\centering
\includegraphics{fig63.eps}
\caption{Lemma \ref{lemma7}}
\label{fig63}
\end{figure}
Despite the fact that we will discuss the continuity of the sine function later, we will finish this section by using the geometric interpretation to prove that, similarly as in the Euclidean case, the generalized sine function of an arbitrary normed plane attains every value between $0$ and $1$.
\begin{lemma}\label{lemma5} Let $(V,||\cdot||)$ be a normed plane with associated sine function $s:S\times S\rightarrow\mathbb{R}$. Then for every $0 \leq \varepsilon \leq 1$ there exist $x,y \in S$ such that $s(x,y) = \varepsilon$.
\end{lemma}
\noindent\textbf{Proof.} Let $x, y \in S$ be conjugate directions. Let $p \in [ox]$ be such that $||p|| = \varepsilon$, and let $l$ be the line parallel to $y$ and passing through $p$. Hence, if $z$ is any point in $S\cap l$, we have immediately $s(z,y) = \varepsilon$.
\begin{flushright} $\square$ \end{flushright}
\section{Characterizing Radon planes} \label{secradonsine}
Throughout this section we follow the approach given by Martini and Swanepoel in \cite{martiniantinorms}. Thus, $(V,||\cdot||)$ is a Minkowski plane endowed with a non-degenerate symplectic bilinear form $[\cdot,\cdot]:V\times V \rightarrow \mathbb{R}$. We define the \textit{antinorm} of $V$ to be
\[ ||x||_a = \sup_{y\in S}|[x,y]|, \] \\
for every $x\in V$. Moreover, the supremum is attained for some $z \in S$ if and only if $z \dashv_B x$.
\begin{lemma}\label{lemma3} Let $(V,||\cdot||)$ be a normed plane with antinorm $||\cdot||_a$. Then the following statements are equivalent: \\
\noindent\textbf{(a)} \textit{The antinorm is a multiple of the norm}.\\
\noindent\textbf{(b)} \textit{Birkhoff orthogonality is a symmetric relation}.\\
\end{lemma}
For a proof we refer the reader to \cite{martiniantinorms}.
\begin{flushright} $\square$ \end{flushright}
It is well known that a normed plane $(V,||\cdot||)$ which satisfies either \textbf{(a)} or \textbf{(b)} is called a \textit{Radon plane}, see again \cite{martiniantinorms}. \\
In order to characterize Radon planes using only the generalized sine function, we will rewrite it in terms of the antinorm. This rewriting has also some other interesting consequences which show that, in some aspects,
the sine function behaves for Radon planes as it does in the Euclidean plane.
\begin{prop}\label{prop2} Let $(V,||\cdot||)$ be a normed plane endowed with a nondegenerate symplectic bilinear form $[\cdot,\cdot]$ and with associated antinorm $||\cdot||_a$. Then, for any $x,y \in S$, we have\\
\begin{align}\label{eqsine} s(x,y) = \frac{|[x,y]|}{||y||_a}. \end{align}
\end{prop}
\noindent\textbf{Proof.} If $x = \pm y$, the equality is immediate. If not, then every point of $S\setminus \{y,-y\}$ can be written as $\frac{x+ty}{||x+ty||}$ for some $t \in \mathbb{R}$. Hence\\
\[||y||_a = \sup_{z\in S}|[x,y]| = \sup_{t\in\mathbb{R}}\left|\left[\frac{x+ty}{||x+ty||},y\right]\right| = \sup_{t\in\mathbb{R}}\frac{|[x,y]|}{||x+ty||} = \frac{|[x,y]|}{\inf_{t\in\mathbb{R}}||x+ty||} = \frac{|[x,y]|}{s(x,y)}, \]\\
where one should be aware of the fact that in the second equality the supremum is not attained for $z = \pm y$.
\begin{flushright} $\square$ \end{flushright}
\begin{remark}\label{remark5} \normalfont Notice that if we consider the sine function defined for directions instead of for unit vectors, then formula (\ref{eqsine}) becomes \\
\begin{align}\label{eqsine2} s(x,y) = \frac{|[x,y]|}{||y||_a||x||}. \end{align}
Hence, if we denote by $s_a$ the sine function associated to the antinorm $||\cdot||_a$, we immediately have $s_a(x,y) = s(y,x)$ for any directions $x,y \in V$.
\end{remark}
\begin{coro}\label{coro1} The generalized sine function is continuous in any normed plane.
\end{coro}
This comes straightforwardly from formula (\ref{eqsine}). \\
\begin{flushright} $\square$ \end{flushright}
\begin{prop}\label{prop2} A normed plane $(V,||\cdot||)$ is Radon if and only if its associated sine function is symmetric.
\end{prop}
\noindent\textbf{Proof.} If the norm is Radon, then we may write $||\cdot||_a = \lambda ||\cdot||$ for some $\lambda >0$. Therefore, for any $x,y \in S$ we have \\
\[s(x,y) = \frac{|[x,y]|}{||y||_a} = \frac{|[x,y]|}{\lambda} = \frac{|[y,x]|}{||x||_a} = s(y,x) \,.
\] \\
For the converse we just check whether symmetry of the sine function implies symmetry of Birkhoff orthogonality. In fact, if $x \dashv_B y$, then $s(x,y) = 1$, and thus $s(y,x) = 1$. But this implies $y \dashv_B x$.
\begin{flushright} $\square$ \end{flushright}
\begin{remark}\label{remark6}\normalfont In dimensions $\geq 3$, the symmetry of the generalized sine function implies that the space is an inner product space. In fact, the above shows that the symmetry of the sine function is equivalent to the symmetry of Birkhoff orthogonality, and so the statement follows. For characterizations of inner product spaces via the generalized sine function we refer to \cite{szostok}.
\end{remark}
The next corollary is a kind of Minkowskian analogue of the Euclidean statement ``in a triangle, equal angles yield equal sides".
\begin{coro}\label{coro3} Let $\Delta\mathbf{abc}$ be a triangle in a normed plane $(V,||\cdot||)$. We have $s\left(b-a,c-b\right) = s\left(c-a,c-b\right)$ if and only if $||b-a|| = ||c-a||$.
\end{coro}
\noindent\textbf{Proof.} From the formula (\ref{eqsine2}) we have \\
\[
s\left(b-a,c-b\right) = \frac{|[b-a,c-b]|}{||b-a||.||c-b||_a} \ \mathrm{and} \]\\
\[
s\left(c-a,c-b\right) = \frac{|[c-a,c-b]|}{||c-a||.||c-b||_a}\,.
\] \\
Thus, by $|[b-a,c-b]| = |[c-a,c-b]|$ the desired follows.
\begin{flushright} $\square$ \end{flushright}
The corollary above allows us to characterize isosceles orthogonality (defined in Section \ref{secintro}) via the sine function.
\begin{lemma}\label{lemma8} Let $x,y \in V$ be linearly independent non-zero vectors. Then the following statements are equivalent: \\
\normalfont\noindent\textbf{(i)} $x \dashv_I y$, \\
\noindent\textbf{(ii)} $s(x+y,y) = s(x-y,y)$, \emph{and} \\
\noindent\textbf{(iii)} $s(x+y,x) = s(x-y,x)$.
\end{lemma}
\noindent\textbf{Proof.} By Corollary \ref{coro3}, the proof is immediate.
\begin{flushright} $\square$ \end{flushright}
We continue with deriving some characterizations of Radon planes via ``Euclidean properties'' of the sine function. For the first one we note that we may characterize Radon planes by changing the orientation of the angle in Corollary \ref{coro3}. Of course, this is related to the symmetry of the sine function in these planes.
\begin{lemma}\label{lemma4} In any plane which is not Radon there exists a triangle $\Delta\mathbf{abc}$ for which $||b-a|| = ||c-a||$, but $s\left(c-b,b-a\right) \neq s\left(c-b,c-a\right)$.
\end{lemma}
\noindent\textbf{Proof.} We just have to choose $x, y \in S$ such that $||x||_a \neq ||y||_a$ and consider the triangle $\Delta\mathbf{oxy}$. We have $||x|| = ||y|| = 1$, and it is easy to see that $s\left(y-x,x\right) \neq s\left(x-y,y\right)$.
\begin{flushright} $\square$ \end{flushright}
Let $a,b,c \in V$ be three non-collinear points. The \textit{angle} $\mbox{$\,<$\hspace{-5pt}\footnotesize )$\,$}\mathbf{abc}$ is the convex hull of the union of the half-lines $\left.[b(a-b)\right>$ and $\left.[b(c-b)\right>$. Having this in mind, we may consider two types of angular bisectors in normed planes:\\
$\bullet$ The \textit{Busemann angular bisector} of $\mbox{$\,<$\hspace{-5pt}\footnotesize )$\,$}\mathbf{abc}$ is the half-line with origin $b$ in the direction $\frac{a-b}{||a-b||} + \frac{c-b}{||c-b||}$. \\
$\bullet$ The \textit{Glogovskii angular bisector} of $\mbox{$\,<$\hspace{-5pt}\footnotesize )$\,$}\mathbf{abc}$ is the set of the points $p \in \mbox{$\,<$\hspace{-5pt}\footnotesize )$\,$}\mathbf{abc}$ equidistant to $\left.[b(a-b)\right>$ and $\left.[b(c-b)\right>$.\\
In \cite{duvelmeyer} D\"{u}velmeyer characterized Radon planes as the only Minkowski planes for which the Glogovskii angular bisector and the Busemann angular bisector coincide for any angle;
see also \cite{martiniantinorms} for a discussion of angular bisectors. We can easily obtain this result by characterizing these angular bisectors in terms of the sine function. This is our next aim.
\begin{prop}\label{prop6} Let $x,y \in S$ be unit vectors which form an angle $\mbox{$\,<$\hspace{-5pt}\footnotesize )$\,$}\mathbf{xoy}$, and let $z \in S$ be such that the half-line $\left.[oz\right>$ lies in the interior of this angle. Then \\
\noindent\textbf{(a)} $\left.[oz\right>$ is the Glogovskii angular bisector of $\mbox{$\,<$\hspace{-5pt}\footnotesize )$\,$}\mathbf{xoy}$ if and only if $s(z,x) = s(z,y)$, and \\
\noindent\textbf{(b)} $\left.[oz\right>$ is the Busemann angular bisector of $\mbox{$\,<$\hspace{-5pt}\footnotesize )$\,$}\mathbf{xoy}$ if and only if $s(x,z) = s(y,z)$.\\
As a consequence we have that the Busemann angular bisector in the norm coincides with the Glogovskii angular bisector in the antinorm (and vice versa). Also, it follows that a normed plane is Radon if and only if these bisectors coincide for any angle.
\end{prop}
\noindent\textbf{Proof.} For \textbf{(a)}, consider a point $p$ within $\mbox{$\,<$\hspace{-5pt}\footnotesize )$\,$}\mathbf{xoy}$ attaining the distance to the sides of the angle, respectively, at $a \in \left.[ox\right>$ and $b\in\left.[oy\right>$. Then we just have to apply Proposition \ref{prop3} to the triangles $\Delta\mathbf{aop}$ and $\Delta\mathbf{bop}$. \\
We now come to \textbf{(b)}. It is known that the Busemann angular bisector of $\mbox{$\,<$\hspace{-5pt}\footnotesize )$\,$}\mathbf{xoy}$ is the half-line $\left.[o(x+y)\right>$. Hence the result follows immediately from formula (\ref{eqsine}).\\
The affirmation that Busemann angular bisectors in the norm coincide with Glogovskii angular bisectors in the antinorm (and vice versa) comes now from Remark \ref{remark5}. By Proposition \ref{prop2} it follows that these types of bisectors coincide in any Radon plane. For the converse, assume that the bisectors coincide, and let $x,y \in S$ be in distinct directions. After changing some sign, if necessary, we may consider that the Busemann angular bisector of $\mbox{$\,<$\hspace{-5pt}\footnotesize )$\,$}\mathbf{xoy}$ is the half-line $\left.[oz\right>$, where $z = \frac{x+y}{||x+y||}$. Since this is also the Glogovskii angular bisector, we have from \textbf{(a)} that $s(z,x) = s(z,y)$, and this gives immediately $||x||_a = ||y||_a$. It follows that the antinorm is a multiple of the norm, and therefore the plane is Radon.
\begin{flushright} $\square$ \end{flushright}
In the Euclidean plane the area of a parallelogram can be calculated as $A = \alpha\beta\sin(\theta)$, where $\alpha$ and $\beta$ are the respective lengths of two consecutive sides and $\theta$ is the angle between these sides. Using a fixed non-degenerate symplectic bilinear form $[\cdot,\cdot]$, the area of the parallelogram with consecutive sides given by $x$ and $y$ is $|[x,y]|$. It turns out that we may characterize Radon planes as being the ones for which this formula still holds.
\begin{prop}\label{prop4} A normed plane $(V,||\cdot||)$ is Radon if and only if there exists a number $\lambda >0$ for which the area of any (ordered) parallelogram $\mathbf{abcd}$ is given by $\lambda||a-b|| \cdot ||a-d||s(v,w)$, where $v$ and $w$ are the unit vectors in the directions $b-a$ and $d-a$, respectively. In this case, rescaling the symplectic bilinear form in such a way that the antinorm coincides with the norm, we will have $\lambda = 1$.
\end{prop}
\noindent\textbf{Proof.} Assume first that $(V,||\cdot||)$ is Radon, and let $\lambda > 0$ be the number such that $||\cdot||_a = \lambda ||\cdot||$. Then, denoting by $A$ the area of the parallelogram $\mathbf{abcd}$, we have \\
\[ A = |[b-a,d-a]| = ||a-b||.||a-d||.|[v,w]| = \lambda||a-b||.||a-d||\frac{|[v,w]|}{||w||_a} = \lambda||a-b||.||a-d||s(v,w). \]\\
For the converse, let $x,y \in S$ be linearly independent unit vectors. Then, considering the parallelogram $\mathbf{oxy(x+y)}$, we have \\
\[ |[x,y]| = \lambda s(x,y) = \lambda\frac{|[x,y]|}{||y||_a} \]\\
for some constant $\lambda >0$. Therefore, $||y||_a = \lambda$. It follows that $||\cdot||_a = \lambda||\cdot||$, and thus the norm is Radon. The remaining part is straightforward.
\begin{flushright} $\square$ \end{flushright}
\section{Angular Measures} \label{secangle}
We follow D\"{u}velmeyer (see \cite{duvelmeyer1}) to introduce angular measures for normed planes in the axiomatic way. The main goal of this section is to prove that the existence of such a measure which is ``coherent'' with the sine function is only possible in the Euclidean plane.
\begin{defi}\label{defi3}\normalfont An \textit{angular measure} in a normed plane $(V,||\cdot||)$ is a Borel measure $\mu$ on the unit circle $S$ satisfying \\
\noindent\textbf{(i)} $\mu(S) = 2\pi$,\\
\noindent\textbf{(ii)} for any Borel set $A \subseteq S$ the equality $\mu(-A) = \mu(A)$ holds,\\
\noindent\textbf{(iii)} for each $v \in S$ we have $\mu(\{v\}) = 0$, and\\
\noindent\textbf{(iv)} any nondegenerate arc of the unit circle has positive measure.\\
\end{defi}
We may define the measure of an angle to be the measure of the arc of $S$ determined by its translate to the origin. In view of this, we may define the measure $\mu(x,y)$ of the angle $\mbox{$\,<$\hspace{-5pt}\footnotesize )$\,$}\mathbf{xoy}$ between two vectors $x,y \in V\setminus\{0\}$ to be the measure of the smallest arc of the unit circle connecting $\frac{x}{||x||}$ and $\frac{y}{||y||}$. Extending our definition so that opposite half-lines with the same origin also form an angle, it clearly follows from \textbf{(i)}, \textbf{(ii)}, and \textbf{(iv)} that $\mu(x,y) = \pi$ if and only if $\frac{x}{||x||} = -\frac{y}{||y||}$. Standard measure theory gives
\begin{lemma}\label{lemma10} Let $\lambda(S)$ be the length, in the norm, of the unit circle $S$ and consider the arclength parametrization $p:\left[0,\frac{\lambda(S)}{2}\right] \rightarrow S$ of one of the arcs from $x_0$ to $-x_0$, where $x_0 \in S$ is any fixed vector. Then the mapping $t \mapsto \mu(x_0,p(t))$ is continuous.
\end{lemma}
This can be used to derive
\begin{teo}\label{teo3} Let $(V,||\cdot||)$ be a Radon plane. If there exists an angular measure $\mu$ on the unit circle $S$ such that $s(x,y) = s(v,w)$ if and only if $\mu(x,y) = \mu(v,w)$ or $\mu(x,y) + \mu(v,w) = \pi$, then $V$ is the Euclidean plane and $\mu$ is the standard angular measure.
\end{teo}
\noindent\textbf{Proof.} We prove that if such a measure exists, then isosceles orthogonality implies Birkhoff orthogonality. This implication characterizes inner product planes (see \cite{amir}). The first step is to prove that given non-zero vectors $x,y \in V$, we have $\mu(x,y) = \pi/2$ if and only if $x \dashv_B y$. Let us begin by fixing $x,y \in S$ such that $x \dashv_B y$ (and, consequently, $y \dashv_B x$, since we are working with Radon planes). Notice that by the additivity of $\mu$ it follows that $\mu(x,y) + \mu(-x,y) = \pi$. Suppose that $\mu(x,y) > \pi/2$ (the opposite case is analogous). Then, by Lemma \ref{lemma10}, we may choose a point $z$ belonging to the (open) smallest arc from $x$ to $y$ such that $\mu(x,z) = \pi/2$. Assume that the line parallel to $x$ and passing through $z$ intersects the (closed) smallest arc from $y$ to $(-x)$ in a point $z_0$. Thus, the geometric characterization of the sine function given in Section \ref{secsine} shows that $s(-x,z_0) = s(x,z)$, and hence $\mu(-x,z_0) = \mu(x,z) = \pi/2$. But this is a contradiction, since the union of the smallest arcs from $x$ to $z$ and from $-x$ to $z_0$ is properly contained in an arc joining $x$ to $-x$ (see Figure \ref{fig67}). \\
Now assume that the unit vectors $x$ and $y$ are not Birkhoff orthogonal. Choose $z \in S$ such that $x \dashv_B z$ and assume, changing signs if necessary, that $\mbox{$\,<$\hspace{-5pt}\footnotesize )$\,$}\mathbf{xoy} \subseteq \mbox{$\,<$\hspace{-5pt}\footnotesize )$\,$}\mathbf{xoz}$. Since the inclusion is obviously proper, it follows that $\mu(x,y) < \mu(x,z) = \pi/2$. \\
We now prove that if $x,y \in V$ are non-zero vectors which are isosceles orthogonal, then $\mu(x,y) = \pi/2$. By Lemma \ref{lemma8}, for $x \dashv_I y$ we have $s(x+y,x) = s(y-x,x)$ and $s(x+y,y) = s(y-x,y)$. Notice that $\mu(x,y) + \mu(v,w) = \pi$ if and only if $\mu(x,y) = \mu(-v,w)$. Hence, by the hypothesis and since the vectors $\frac{x+y}{||x+y||}$ and $\frac{y-x}{||y-x||}$ must lie, respectively, in the smallest arcs from $\frac{x}{||x||}$ to $\frac{y}{||y||}$ and from $\frac{y}{||y||}$ to $\frac{-x}{||x||}$, it follows that $\mu(x+y,x) = \mu(y-x,-x)$ and $\mu(x+y,y) = \mu(y-x,x)$. The union of these four respective arcs is an arc which connects $\frac{x}{||x||}$ and $\frac{-x}{||x||}$ (the one which contains $\frac{y}{||y||}$, to be exact). Hence, additivity of $\mu$ gives $\mu(x+y,x) + \mu(x+y,y) = \mu(y-x,y) + \mu(y-x,-x) = \pi/2$. Finally, the equality $\mu(x,y) = \mu(x,x+y) + \mu(x+y,y)$ yields the desired. \\
Therefore, given non-zero vectors $x,y \in V$ such that $x \dashv_I y$, we have $\mu(x,y) = \pi/2$, and this implies $x \dashv_B y$. This shows that $(V,||\cdot||)$ is Euclidean. Moreover, the measure $\mu$ agrees with the Euclidean sine. It follows that $\mu$ is the standard Euclidean measure of angles.
\begin{figure}[t]
\centering
\includegraphics{fig67.eps}
\caption{$\mu(x,z)=\mu(-x,z_0)$}
\label{fig67}
\end{figure}
\begin{flushright} $\square$ \end{flushright}
\section{Trigonometric Constants and Extremal Values} \label{secconstants}
This section is devoted to the study of some constants defined via the sine function, and also to characterizations of some types of normed planes for which those constants attain extremal values. We first want to estimate how ``non-Euclidean'' a normed plane $(V,||\cdot||)$ can be. In view of Proposition \ref{prop3} we can do this by measuring how far triangles with two sides in conjugate directions can be from being Pythagorean. For any conjugate directions $x,y \in S$ we define \\
\[ c_E(x,y) := \sup_{z\in S}\left(s(z,x)^2+s(z,y)^2\right) - \inf_{z\in S}\left(s(z,x)^2 + s(z,y)^2\right). \] \\
It is clear that this number is $0$ for any pair of conjugate directions in an inner product plane. The next lemma states that we only need one pair of ``well-behaving'' conjugate directions to characterize such a plane.
\begin{lemma}\label{lemma6} There exists a pair $x,y \in S$ of conjugate directions such that $c_E(x,y) = 0$ if and only if $(V,||\cdot||)$ is an inner product plane.
\end{lemma}
\noindent\textbf{Proof.} If we have conjugate directions $x,y \in S$ such that $c_E(x,y) = 0$, then $s(z,x)^2 + s(z,y)^2 = 1$ for every $z \in S$. Then we simply define an inner product $\left<\cdot,\cdot\right>:V\times V \rightarrow \mathbb{R}$ by setting $\left<x,y\right> =0$ and $\left<x,x\right>=\left<y,y\right>=1$, and we denote by $||\cdot||_E$ the norm given by it. If $z \in S$, we may write $z = \alpha x + \beta y$ for some $\alpha,\beta\in\mathbb{R}$. By Proposition \ref{prop3} we may assume that $|\alpha| = s(z,y)$ and $|\beta| = s(z,x)$. Hence $\alpha^2 + \beta^2 = 1$. Since $||z||_E$ clearly equals $\alpha^2 + \beta^2$, it follows that $||\cdot|| = ||\cdot||_E$, as we wished. The converse is obvious.
\begin{flushright} $\square$ \end{flushright}
It is natural to ask now what the largest ``distortion" among all pairs of conjugate diameters is. We define the constant
\[ c_E\left(||\cdot||\right) := \sup\{c_E(x,y):x\dashv_B y \ \mathrm{and} \ y\dashv_B x\}. \]\\
For this constant we have the following
\begin{prop}\label{prop5} For any normed plane $(V,||\cdot||)$ we have $0 \leq c_E\left(||\cdot||\right) \leq \frac{3}{2}$. Equality on the left side holds if and only if the norm is derived from an inner product, and equality on the right side holds if and only if the unit circle is an affine regular hexagon.
\end{prop}
\noindent\textbf{Proof.} The left side is immediate due to Lemma \ref{lemma6}. For the right side, notice first that Lemma \ref{lemma7} yields the inequality immediately. By compactness the supremum is in fact a maximum, and hence we have the following: if equality holds, then we have conjugate directions $x,y \in S$ such that $\sup_{z\in S}\left(s(z,x)^2 + s(z,y)^2\right) = 2$ and $\inf_{z\in S}\left(s(z,x)^2+ s(z,y)^2\right) = \frac{1}{2}$. Using again compactness and Lemma \ref{lemma7}, we see that this is only possible if the unit circle is the affine regular hexagon with vertices $\pm x$, $\pm y$ and $\pm (x+y)$, or $\pm (x-y)$.
\begin{flushright} $\square$ \end{flushright}
Recall that Proposition \ref{prop2} states that a Minkowski plane is Radon if and only if its associated sine function is symmetric. Thus, it is natural to estimate how far a norm is from being Radon by calculating the differences between $s(x,y)$ and $s(y,x)$ for pairs $x,y\in S$. We define the respective constant $c_R(||\cdot||)$ by \\
\[ c_R(||\cdot||) = \sup_{x,y\in S}|s(x,y) - s(y,x)|. \]
In view of Remark \ref{remark5} one can instantly check that $c_R(||\cdot||_a) = c_R(||\cdot||)$. In some sense, according to $c_R$ a norm and its antinorm are ``equally non-Radon" . The next theorem characterizes the rectilinear planes as the ``most non-Radon" planes.
\begin{teo}\label{teo1} For any normed plane $(V,||\cdot||)$ we have
\[ c_R(||\cdot||) \leq \frac{1}{2}. \] \\
\noindent Equality holds if and only if the plane is rectilinear.
\end{teo}
\noindent\textbf{Proof.} Let $x,y \in S$ be vectors with $x \neq \pm y$ and assume, without loss of generality, that $s(x,y) \geq s(y,x)$. Denote by $H$ the (open) half-plane determined by the line parallel to $y$ and passing through the origin which contains also $x$, and let $w$ be a point at which a line of direction $y$ supports $S\cap H$. We will denote this supporting line by $l_1$. Then the ray $\left.[ow\right>$ intersects the line $l_2: t \mapsto x + ty$ at the point $q = s(x,y)w$. Let $z$ be the intersection of the ray $\left.[ox\right>$ with the line $l_1$ and assume that $||x-z|| = \beta$ (notice that $\beta = 0$ if and only if $x \dashv_B y$). We may calculate $s(x,y)$ in terms of $\beta$: from the triangles $\Delta\mathbf{owz}$ and $\Delta\mathbf{oqx}$ we have $\frac{||x||}{||x-z||} = \frac{||q||}{||w||}$ (see Figure \ref{fig59}). This gives immediately $s(x,y) = \frac{1}{1+\beta}$.\\
If we denote by $p$ the point where the line $\left<(-x)y\right>$ intersects $l_1$, then it follows that the portion of $S$ from $x$ to $y$ (the one which does not contain $-x$) is contained in $\mathrm{conv}\{y,p,z,o\}$. In particular, this means that the rays of the quadrant $\mathrm{conv}(\left.[oy\right>\cup\left.[ox\right>)$ intersect the unit circle before they intersect $[yp]\cup[pz]$. This follows by convexity and from the fact that $l_1$ supports $S$ (see Figure \ref{teo1}).
\begin{figure}[h]
\centering
\includegraphics{fig59.eps}
\label{fig59}
\caption{Proof of Theorem \ref{teo1}}
\end{figure}
We will use this ``location" of that portion of the unit circle to estimate $||y + tx||$ whenever $t \geq 0$. It will be necessary to consider two cases, but first we need to find the intersection $a$ of $\left.[op\right>$ with the line $l_3:t \mapsto y+tx$. If $l_1$ and $l_3$ intersect at $b$ then, by using homothety between triangles, we have \\
\[ \frac{||x+z||}{||y-b||} = \frac{||p-z||}{||p-b||} = \frac{||z||}{||a-b||} \,.
\] \\
Hence $\frac{2+\beta}{1+\beta} = \frac{1+\beta}{||a-b||}$. Now put $||y-a|| = 1+\beta - ||a-b|| = \frac{1+\beta}{2+\beta}$. Therefore, $a = y + \frac{1+\beta}{2+\beta}x$. Denote by $\lambda_t$ the non-negative number such that $\lambda_t(y+tx)$ is the point where the ray $\left.[o(y+tx)\right>$ intersects the unit circle $S$. Then $||y + tx|| = \lambda_t^{-1}$. If $0 \leq t \leq \frac{1+\beta}{2+\beta}$, a simple calculation shows that the ray $\left[o(y+tx)\right>$ intersects the segment $[yp]$ at the point $\frac{1}{1-t}(y+tx)$. Since the ray intersects the unit circle before it intersects the segment, it follows that $\lambda_t \leq \frac{1}{1-t}$. Hence \\
\[ ||y + tx|| = \lambda_t^{-1} \geq 1-t \geq 1-\frac{1+\beta}{2+\beta} = \frac{1}{2+\beta}. \]\\
If $t \geq \frac{1+\beta}{2+\beta}$, then the ray $\left[o(y+tx)\right>$ intersects the segment $[pz)$ at $\frac{1+\beta}{t}(y+tx)$. Thus $\lambda_t \leq \frac{1+\beta}{t}$, and therefore \\
\[ ||y+tx|| = \lambda_t^{-1} \geq \frac{t}{1+\beta} \geq \frac{1}{2+\beta}. \] \\
We may repeat exactly the same argument to show that this estimate still holds if $t < 0$. Then \\
\[
|s(x,y) - s(y,x)| = s(x,y) - s(y,x) \leq \frac{1}{1+\beta} - \frac{1}{2+\beta} = \frac{1}{(1+\beta)(2+\beta)} \leq \frac{1}{2} \,,
\] \\
and the desired inequality follows. \\
Notice that if we have equality, then by continuity of the sine function and compactness of $S\times S$ there exists a pair $x, y \in S$ satisfying $s(x,y) - s(y,x) = \frac{1}{2}$. In this case, we have necessarily $\beta = 0$, and therefore $x \dashv_B y$. Also we must have $\lambda_t = 2$, and this happens only when $t = \pm \frac{1}{2}$. If $t = \frac{1}{2}$, then by $\left|\left|y + \frac{1}{2}x\right|\right| = \frac{1}{2}$ it follows that $x + 2y \in S$. Therefore, since $-x$, $y$ and $x+2y$ are collinear points in $S$, we have that $[(-x)(x+2y)]$ is a segment of the unit circle. This, together with central symmetry and the orthogonality $x \dashv_B y$, yields immediately that $S$ is the parallelogram whose vertices are the points $\pm x$ and $\pm (x+2y)$. If $t = -\frac{1}{2}$, we repeat the argument, but will obtain the parallelogram with vertices $\pm x$ and $\pm (-x+2y)$. \\
It remains to prove that the supremum is attained for any rectilinear plane, but this is obvious. If $S$ is the parallelogram with vertices $\pm v$ and $\pm w$, we just have to consider the vectors $v$ and $\frac{v+w}{2}$.
\hfill $\Box$
\begin{flushright} $\square$ \end{flushright}
It is known that the affine regular $(4n+2)$-gons, $n \in \mathbb{N}$, are Radon curves, and that this is never true for the affine regular $(4n)$-gons (this was noticed by Heil in \cite{heil}). We now calculate the distortion $c_R$ for norms whose unit circle is a regular $(4n)$-gon.
\begin{teo}\label{teo2} Let $||\cdot||_{4n}$ denote the norm whose unit circle is given by an affine regular $(4n)$-gon, $n\in\mathbb{N}$. Then we have \\
\[ c_R\left(||\cdot||_{4n}\right) = \left(\sin\frac{\pi}{4n}\right)^ 2. \]
\end{teo}
\noindent\textbf{Proof.} It is clear that the sine function is invariant under linear transformations in the following sense: if $T:V\rightarrow V$ is a linear transformation and $s$ is the sine function associated to a norm whose unit circle is $S$ then, if we denote by $s_T$ the sine function induced by the norm with unit circle $T(S)$, we will have $s(x,y) = s_T(Tx,Ty)$. Consequently, the constant $c_R$ remains the same if we modify the unit circle by an affine transform, and therefore we may use the standard Euclidean $(4n)$-gons to perform the calculations. \\
For simplicity, denote by $||\cdot||$ the norm whose unit circle S is a regular $(4n)$-gon. It is known that the anticircle $S_a$ of $S$ is a homothet of the polygon whose vertices are the midpoints of the sides of $S$. In particular, we may rescale $[\cdot,\cdot]$ so that $S_a$ is precisely this polygon. Our first task is to determine, in this case, the minimum and maximum values of $||x||_a$ as $x$ ranges through $S$. First, from $S_a \subseteq S$ it follows that $||x||_a \geq 1$ for every $x \in S$. Since equality holds whenever $x$ is the midpoint of some side of $S$, it follows that $\min_{w\in S}||w||_a = 1$. For the maximum, assume that the ray $\left.[ox\right>$ intersects the segment joining the respective midpoints $m_1$ and $m_2$ of consecutive sides $[a_1a_2]$ and $[a_2a_3]$ of $S$. Since $[m_1m_2] \subseteq S_a,$ we have that this intersection occurs at $x_0 = \frac{x}{||x||_a}$. Thus, the value of $||x||_a$ is the ratio between the Euclidean lengths of the segments $[ox]$ and $[ox_0]$ (see Figure \ref{fig66}).
\begin{figure}[h]
\centering
\includegraphics{fig66.eps}
\label{fig66}
\caption{Estimating $||x||_a$}
\end{figure}
At this point, our problem becomes a problem of planar Euclidean geometry: we have to find the maximum ratio between lengths of $[ox]$ and $[ox_0]$ as $x$ ranges from $a_2$ to $m_2$ (evidently, we have
correspondingly equal values if $x$ varies from $m_1$ to $a_2$). Let $\theta$ be the standard Euclidean measure of the angle $\mbox{$\,<$\hspace{-5pt}\footnotesize )$\,$}\mathbf{xom_2}$ and assume, without loss of generality, that $||a_2||_E = 1$, where $||\cdot||_E$ is the Euclidean norm. Then basic trigonometry gives \\
\[ \frac{||x||_E}{||x_0||_E} = 1 + \tan\theta\tan\left(\frac{\pi}{4n}\right). \] \\
Hence the maximum value for $\frac{||x||_E}{||x_0||_E}$ is $1 + \left(\tan\frac{\pi}{4n}\right)^2 = \left(\cos\frac{\pi}{4n}\right)^{-2}$. It follows that $\max_{z\in S}||z||_a = \left(\cos\frac{\pi}{4n}\right)^{-2}$.\\
Now, given $x,y \in S$, we may assume that $s(x,y) \geq s(y,x)$, which is equivalent to $||y||_a \leq ||x||_a$. Due to this we have \\
\[s(x,y) - s(y,x) = s(x,y)\left(1 - \frac{s(y,x)}{s(x,y)}\right) \leq 1 - \frac{||y||_a}{||x||_a} \leq 1 - \frac{\min_{w\in S}||w||_a}{\max_{z\in S}||z||_a} = 1 - \left(\cos\frac{\pi}{4n}\right)^2 = \left(\sin\frac{\pi}{4n}\right)^2 \,.
\]\\
To check that this value is optimal, let $a_1, a_2, ..., a_{4n}$ be the vertices of $S$. Denote by $m_j$ the midpoint of the side $[a_ja_{j+1}]$ (identifying $a_1$ with $a_{4n+1}$). Since $S$ is a regular $(4n)$-gon, the vector $m_n$ is parallel to $[a_{4n}a_1]$, and hence $a_1 \dashv_B m_n$. This gives $|[a_1,m_n]| = ||m_n||_a = 1$ (recall that we rescaled $[\cdot,\cdot]$ in such a way that the unit anticircle is the polygon whose vertices are $m_1, m_2,...m_{4n}$). Moreover, it is clear from the previous calculations that $||a_1||_a = \left(\cos\frac{\pi}{4n}\right)^{-2}$. It follows that $|s(a_1,m_n)-s(m_n,a_1)| = \left(\sin\frac{\pi}{4n}\right)^2$, as desired.
\begin{flushright} $\square$ \end{flushright}
We should note that a geometric constant which can be written in terms of the sine function has been studied previously. It quantifies the difference between isosceles orthogonality and Birkhoff orthogonality and it is defined as: \\
\[ D(X) := \inf\left\{\inf_{\lambda\in \mathbb{R}}||x+\lambda y|| : x,y \in S, \ x \dashv_I y\right\}. \] \\
Clearly, $D(X) = \inf\left\{s(x,y):x,y \in S, \ x\dashv_I y\right\}$. This constant was defined in \cite{wu}. Another reference is \cite{alonso}.
\section{Sine Conformal Mappings} \label{secconformal}
Conformal mappings are usually defined to be mappings which preserve measures of angles.
For that reason we will call sine preserving self-mappings of $V$ sine conformal. More precisely, we fix this by
\begin{defi}\label{defi2}\normalfont Let $(V,||\cdot||)$ be a normed plane with associated sine function (in the sense of Remark \ref{remark2}) $s: V\setminus\{0\}\times V\setminus\{0\} \rightarrow \mathbb{R}$. A mapping $f:V\rightarrow V$ for which $f(x) \neq 0$ if $x \neq 0$ is said to be \textit{sine conformal} if $s(f(x),f(y)) = s(x,y)$ for every $x,y \in V\setminus\{0\}$.
\end{defi}
Notice that sine conformal mappings, in particular, preserve Birkhoff orthogonality (see Lemma \ref{lemma2}). It is known that any linear map which preserves Birkhoff orthogonality is a scalar multiple of an isometry (this was established in \cite{blanco}). Using this, we may describe the sine conformal mappings which are linear.
\begin{prop}\label{prop7} A linear map $f:V \rightarrow V$ is sine conformal if and only if it is a scalar multiple of an isometry.
\end{prop}
\noindent\textbf{Proof.} We just have to prove that a scalar multiple of a linear isometry is sine conformal, since the other direction of the bi-implication comes immediately from the previous comments. Let $f = kL$, where $k \neq 0$ and $L: V\rightarrow V$ is a linear isometry. Fix any linearly independent $x,y \in V$ and let $z \in S$ be such that $z \dashv_B y$. Assume that we may write $z = \alpha x + \beta y$. Hence \\
\[
s(x,y) = \frac{|[x,y]|}{||y||_a||x||} =\frac{|[x,y]|}{|[y,z]| \cdot ||x||} = \frac{1}{|\alpha| \cdot ||x||}\,.
\] \\
On the other hand, since $f$ obviously preserves Birkhoff orthogonality, we have that $||f(y)||_a = \left|\left[f(y),\frac{f(z)}{k}\right]\right|$. Hence \\
\[
s(f(x),f(y)) =\frac{|[f(x),f(y)]|}{||f(y)||_a||f(x)||} = \frac{|[f(x),f(y)]|}{|[f(y),f(z)]| \cdot ||x||} = \frac{1}{|\alpha| \cdot ||x||},
\]\\
showing that $f$ is sine conformal.\\
\begin{flushright} $\square$ \end{flushright}
In our next lemma we give a first result concerning the existence of a non-trivial sine conformal mapping. As a consequence, we will derive a characterization of inner product planes among Radon planes.
A linear map whose eigenvalues are $1$ and $-1$ we call a \textit{reflection}. For the following notice that Roberts orthogonality (defined in Section \ref{secintro}) in the norm is equivalent to Roberts orthogonality in the antinorm. In fact, assume that $x \dashv_R y$. In this case, fixing any $\alpha \in\mathbb{R}$, we have\\
\[ \left|\left[x+\alpha y, \frac{x - ty}{||x - ty||} \right]\right| = \left|\left[x-\alpha y, \frac{x + ty}{||x + ty||} \right]\right| \]\\
for every $t \in \mathbb{R}$. It follows that the continuous maps $z \mapsto |[x+\alpha y,z]|$ and $z \mapsto |[x - \alpha y,z]|$ have the same range when $z$ varies through $S$. Thus, $||x+\alpha y||_a = ||x - \alpha y||_a$.
\begin{lemma}\label{lemma9} Let $f:V \rightarrow V$ be a reflection whose eigenvectors are $x$ and $y$. Then $f$ is sine conformal if and only if we have that $x$ and $y$ are Roberts orthogonal.
\end{lemma}
\noindent\textbf{Proof.} We must have $f(x) = x$ and $f(y) = -y$ or $f(x) = -x$ and $f(y) = y$. In both cases, formula (\ref{eqsine2}) gives that $s(f(v),f(w)) = s(v,w)$ for every $v,w \in V\setminus\{0\}$ if and only if \\
\[ ||\alpha x + \beta y||_a||\nu x + \xi y|| = ||\alpha x - \beta y||_a||\nu x - \xi y||
\] \\
for any $\alpha,\beta,\nu,\xi \in \mathbb{R}$. Hence $f$ is sine conformal if and only if $x$ and $y$ are Roberts orthogonal in both, the norm and the antinorm. But this is equivalent to $x \dashv_R y$.
\begin{flushright} $\square$ \end{flushright}
\begin{coro}\label{coro4} Let $(V,||\cdot||)$ be a Radon plane. Then $V$ is an inner product plane if and only if for every conjugate pair $\{x,y\} \subseteq S$ the reflection $f:V\rightarrow V$ given by $f(x) = x$ and $f(y) = -y$ is sine conformal.
\end{coro}
\noindent\textbf{Proof.} Radon planes are precisely the planes with the property that for every $x \in S$ there exists a $y \in S$ such that $x \dashv_B y$ and $y \dashv_B x$. Hence the hypothesis that the reflection given by $f(x) = x$ and $f(y) = -y$ is sine conformal whenever $x$ and $y$ are conjugate gives that for every $x \in S$ there exists some $y \in S$ for which $x \dashv_R y$. This characterizes inner product planes (see \cite{alonso}). The converse is obvious.
\begin{flushright} $\square$ \end{flushright}
\section{The Law of Sines} \label{lawofsines}
In this final part we present a few results related to the Law of Sines in Radon planes.
\begin{teo}[Law of Sines]\label{teolaw} Let $(V,||\cdot||)$ be a Radon plane, and let $x,y,z \in S$ be non-collinear. Then in the triangle $\Delta\mathbf{xyz}$ we have\\
\begin{align}\label{sinelaw} \frac{||x-y||}{s(x-z,y-z)} = \frac{||y-z||}{s(x-y,x-z)} = \frac{||x-z||}{s(y-z,x-y)}. \end{align}\\
If this ratio equals $2$ for any triple of distinct unit vectors, then $V$ is an inner product plane. \\
\end{teo}
\noindent\textbf{Proof.} The equality follows immediately from (3.2). Assume now that the ratio is $2$ for any $x,y,z \in S$. If $x,y \in S$, then, looking to the triangle $\Delta\mathbf{xy(-x)}$, we have \\
\[ \frac{||2x||}{s(x+y,x-y)} = 2, \]\\
and hence $s(x+y,x-y) = 1$. It follows that $(x+y) \dashv_B (x-y)$ whenever $x,y \in S$. This property characterizes inner product planes (see \cite{alonso}, the comment below Theorem 4.20).
\begin{flushright} $\square$ \end{flushright}
\begin{remark}\normalfont Notice that to characterize inner product planes it is enough to demand only that the ratio (\ref{sinelaw}) equals $2$ for any triangle inscribed in the unit circle having a diameter as one of its sides. In other words, this means that ``any inscribed angle which is opened to a diameter is a right angle". For a related classification of triangles in normed planes (having then obtuse, right, and acute ones) we refer to the paper \cite{AMS}.
\end{remark}
The question that arises is whether or not for any Radon plane we have a triangle inscribed in the unit circle for which the ratio expressed in Theorem \ref{teolaw} equals $2$. To answer this question we enunciate (a part of) the main theorem of \cite{benitez}.
\begin{teo}\label{teo4} Let $(V,||\cdot||)$ be a normed space. For any $x,y \in V\setminus\{0\}$ there exists a unique number $\alpha = \alpha(x,y) > 0$ such that $(x+\alpha y) \dashv_B (x - \alpha y)$. Moreover, the function which associates each pair $(x,y) \in V\setminus\{0\}\times V\setminus\{0\}$ to the number $\alpha(x,y) \in \mathbb{R}$ is continuous.
\end{teo}
\begin{flushright} $\square$ \end{flushright}
\begin{prop}\label{prop8} In any Radon plane, there exist vectors $x,y \in S$ with different directions such that $(x+y) \dashv_B (x-y)$.
\end{prop}
\noindent\textbf{Proof.} Fix arbitrary $v,w \in S$ with $v \neq \pm w$ and let $\alpha:V\setminus\{0\}\times V\setminus\{0\} \rightarrow \mathbb{R}$ be as in Theorem \ref{teo4}. Define $f:[0,1] \rightarrow \mathbb{R}$ by \\
\[ f(\lambda) = \alpha\left(\frac{(1-\lambda)v+\lambda w}{||(1-\lambda)v + \lambda w||}, \frac{(1-\lambda)w+\lambda (-v)}{||(1-\lambda)w + \lambda (-v)||}\right). \]\\
Notice that $f$ is continuous. Moreover, since the plane is Radon, we have $f(0) = \alpha(v,w) = \alpha(w,-v)^{-1} = f(1)^{-1}$. It follows from the Intermediate Value Theorem that there exists some $\lambda_0 \in [0,1]$ such that $f(\lambda_0) = 1$. Setting $x = \frac{(1-\lambda_0)v + \lambda_0w}{||(1-\lambda_0)v + \lambda_0w||}$ and $y = \frac{(1-\lambda_0)w + \lambda_0(-v)}{||(1-\lambda_0)w + \lambda_0(-v)||}$, we have the desired. \\
\begin{flushright} $\square$ \end{flushright}
\begin{coro} In any Radon plane there exists a triangle inscribed in the unit circle for which the ratio (\ref{sinelaw}) attains the ``Euclidean value'' $2$.
\end{coro}
\noindent\textbf{Proof.} Let $x,y \in S$ be as in Proposition \ref{prop8} and consider the triangle $\Delta\mathbf{xy(-x)}$. The statement follows immediately.
\begin{flushright} $\square$ \end{flushright}
\begin{remark}\label{remark7}\normalfont Inspired by formula (\ref{eqsine2}), one can easily verify a weaker Law of Sines for non-Radon planes. Given a triangle $\Delta\mathbf{abc}$ in such a plane, we have the equality \\
\[ \frac{||c-a||}{s(b-a,c-b)} = \frac{||b-a||}{s(c-a,c-b)}, \] \\
and the two analogous equalities (obtained by interchanging suitably the vertices). Only in a Radon plane the three equalities yield the same value for any given triangle. Moreover, notice that Proposition \ref{prop3} and Corollary \ref{coro3} can be seen as special cases of this law.
\end{remark}
| {
"timestamp": "2016-01-26T02:05:40",
"yymm": "1601",
"arxiv_id": "1601.06287",
"language": "en",
"url": "https://arxiv.org/abs/1601.06287",
"abstract": "In this paper we study a metric generalization of the sine function which can be extended to arbitrary normed planes. We derive its main properties and give also some characterizations of Radon planes. Furthermore, we prove that the existence of an angular measure which is \"well-behaving\" with respect to the sine is only possible in the Euclidean plane, and we also define some new constants that estimate how non-Radon or non-Euclidean a normed plane can be. Sine preserving self-mappings are studied, and a complete description of the linear ones is given. In the last section we exhibit a version of the Law of Sines for Radon planes.",
"subjects": "Metric Geometry (math.MG)",
"title": "Geometric properties of a sine function extendable to arbitrary normed planes",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754452025766,
"lm_q2_score": 0.718594386544335,
"lm_q1q2_score": 0.7075103880519611
} |
https://arxiv.org/abs/2108.01538 | Geometry of Linear Convolutional Networks | We study the family of functions that are represented by a linear convolutional neural network (LCN). These functions form a semi-algebraic subset of the set of linear maps from input space to output space. In contrast, the families of functions represented by fully-connected linear networks form algebraic sets. We observe that the functions represented by LCNs can be identified with polynomials that admit certain factorizations, and we use this perspective to describe the impact of the network's architecture on the geometry of the resulting function space. We further study the optimization of an objective function over an LCN, analyzing critical points in function space and in parameter space, and describing dynamical invariants for gradient descent. Overall, our theory predicts that the optimized parameters of an LCN will often correspond to repeated filters across layers, or filters that can be decomposed as repeated filters. We also conduct numerical and symbolic experiments that illustrate our results and present an in-depth analysis of the landscape for small architectures. | \section{Introduction}
A neural network is a parameterized family of functions.
The class of representable functions, also known as the \emph{neuromanifold} or \emph{function space}, is determined by the architecture of the network. Optimizing an objective function over such a parametrized set usually depends on both the set and the parametrization.
The objective function is typically convex in function space but non-convex in parameter space. This observation has guided numerous recent works on parameter optimization in neural networks; see, {\em e.g.},
\cite{NEURIPS2018_5a4be1fa,
du2018gradient,
pmlr-v97-allen-zhu19a,
pmlr-v119-dukler20a}.
However, the geometry of function space is not well understood in general.
Fully-connected networks, also known as dense networks, have been studied in significantly more detail than convolutional networks, which are the topic of this article. Whereas weight matrices in fully-connected networks have independent and unconstrained entries, weight matrices in convolutional networks are sparse and have repeated entries.
Neural networks with linear activation functions, called linear networks, represent linear functions as compositions of linear functions.
An important motivation for studying such networks is to better understand the effects of the parameterization when optimizing an objective function.
We will see that it also can help us better understand the effects that the geometry of function space has on the optimization problem.
Although neural networks are usually presented in a parameterized form, it is in principle possible to characterize the representable functions implicitly as the solutions to certain constraints.
Such descriptions have been particularly fruitful in algebraic statistics, {\em e.g.}, \cite{sullivant2018algebraic,SeigalMontufar,722fd1ac5816467ebadbd10f15ff2a74,10.1007/978-3-030-43120-4_29,ccelik2020wasserstein}.
In the case of fully-connected linear networks, the function space is characterized by rank constraints depending on the layer widths. A study of critical points and local optimizers in parameter and function space for fully-connected linear networks has been carried out in \cite{geometryLinearNets}.
Here we follow a similar perspective,
analyzing the function space and the optimization of parameters for
\emph{linear convolutional networks} (LCNs).
\paragraph{Main results}
In \Cref{section:LCNs} we introduce LCNs and present our general setup.
Each layer of an LCN is represented by a convolutional matrix or, more generally, a convolutional tensor.
We observe that composing convolutional layers corresponds to polynomial multiplication.
In \Cref{sec:discriminants} we recall some general properties of discriminants and multiplicity root patterns for polynomials, which we use in later sections.
In \Cref{sec:geometry} we show that the function space of an LCN ({\em i.e.}, the set of linear maps that can be represented by the network) is a semi-algebraic set, whose natural ambient space is the vector space of convolutional tensors of a certain format.
In contrast to fully-connected linear networks, whose function space is cut out by polynomial equations only, the function space of LCNs is determined by polynomial equations and inequalities.
For one-dimensional convolutions with stride one, the function space is a full-dimensional subset of its natural ambient space and its boundary is contained in the discriminant hypersurface of univariate polynomials of a certain degree. The function space is equal to its natural ambient space if and only if at most one of the filters has even size.
The latter case is interesting since optimizing a convex loss in function space reduces to a convex optimization problem on a vector space.
For higher-dimensional convolutions or larger strides, the function space is typically a lower-dimensional subset of its natural ambient space.
In \Cref{sec:optimization-function} we study the optimization problem for a general loss and one-dimensional convolutions with stride one.
Different LCNs can have the same function space, but different parameterizations can influence the optimization behavior.
Hence, we distinguish \emph{pure critical points} given by the optimization on function space from \emph{spurious critical points} induced only by the parameterization.
We show that the critical points in parameter space are either global minima or correspond to polynomials with repeated roots. The latter actually correspond to critical points of the loss in function space restricted to subsets of polynomials with particular real root multiplicity patterns that we characterize combinatorially depending on the LCN architecture.
We further describe the gradient flow dynamics in parameter space and provide invariants that allow us to formulate it as a local Riemannian gradient in function space.
In \Cref{sec:squareloss} we take a closer look at the the square loss.
We provide upper bounds on the number of critical points in function space in terms of \emph{Euclidean distance degrees}, and observe that for certain normalizations of the training data, the square loss becomes the \emph{Bombieri norm}, which is expected to have fewer critical points.
In \Cref{sec:experiments} we conduct numerical experiments illustrating our theoretical results.
These demonstrate how the type of training data, the geometry of function space, and the parametrization lead to different types of critical points and solutions found using gradient descent.
\paragraph{Overview of previous works on linear networks}
\emph{Loss surface.}
The properties of the optimization problem in linear networks have been studied in \cite{BALDI198953,NIPS1988_123,10.1109/72.392248}.
For the square loss in fully-connected linear networks there is a unique local and global minimum, up to equivalence, and all other critical points are saddles.
A proof of this statement for the deep case was given in \cite{NIPS2016_6112}.
For deep residual linear networks, \cite{DBLP:conf/iclr/HardtM17} showed that the square loss has no critical points other than global minima.
Further, \cite{DBLP:journals/corr/LuK17} and \cite{47812} argue that depth alone does not create bad local minima, although it induces a non-convex loss surface.
The analytic form of the critical points for the square loss in fully-connected linear networks was discussed in \cite{zhou2018critical}.
For arbitrary differentiable convex losses and fully-connected linear networks having layers at least as wide as the input or output layers, \cite{pmlr-v80-laurent18a} showed that all local minima are global.
A geometric analysis was given in \cite{geometryLinearNets}, showing that the absence of non-global local minima in linear networks is expected for arbitrary smooth convex losses only if the architecture can express all linear maps (as in \cite{pmlr-v80-laurent18a}) or for the quadratic loss and arbitrary architectures (as in \cite{NIPS2016_6112}).
Taking an algebraic standpoint, \cite{9397294} studied fully-connected linear networks and upper bounds on the number of critical points of the square loss with weight norm regularization.
The above works focus mostly on fully-connected networks.
A paper dating back to 1989~\cite{NIPS1988_123} asked whether non-global local minima for the square loss exist in the case non-fully-connected ({\em i.e.}, locally-connected) multi-layer linear networks, and this seems to have remained an open question.
We show that LCNs can indeed have non-trivial local minima in function space and in parameter space.
\emph{Gradient dynamics.}
An overview of non-convex optimization in low-rank matrix factorization (two-layer linear networks) was given in \cite{8811622}.
Exact solutions to the nonlinear gradient dynamics for the square loss in fully-connected deep linear networks with orthogonal input data have been obtained in \cite{DBLP:journals/corr/SaxeMG13}.
Also for the square loss in fully-connected linear networks, \cite{arora2018a} showed linear convergence to a global minimum when the hidden layers are at least as wide as the input or output layers, weight matrices at initialization are approximately balanced, and the initial loss is less than for any rank deficient solution.
The work \cite{pmlr-v80-arora18a} shows that
depth amounts to a pre-conditioner
which may accelerate convergence.
A detailed analysis of gradient flow optimization and convergence for the square loss in fully-connected linear networks was presented in \cite{DBLP:journals/corr/abs-1910-05505}. Using a balancedness condition they obtain a Riemannian gradient in function space and show that the flow always converges to a critical point, which for almost all initializations is the global optimum in function space restricted to the set of rank-$k$ matrices for some $k$.
We characterize gradient flow invariants in LCNs and obtain corresponding descriptions in function space. The critical points can also be characterized in function space, in terms of real root multiplicity patterns rather than rank constraints.
\emph{Linear convolutional networks and implicit bias of gradient descent.}
LCNs have been~studied in \cite{dft} for the case of 1D convolutions with circulant weight matrices (with stride~1) having full-width filters and a single output. Such networks can express any linear map.
That work focuses on binary classification with linearly separable data and asks which of the infinitely many linear classifiers with zero loss are obtained by gradient descent.
For the exponential loss, it is shown that for fully-connected networks gradient descent converges to the hard margin support vector machine linear classifier independently of the depth of the network (as was previously observed for the direct parameterization in \cite{JMLR:v19:18-188}), whereas for convolutional networks it is biased towards linear classifiers that are sparse in frequency domain depending on the depth.
In contrast, we consider LCNs with arbitrary filter sizes (and also provide first geometric insights for LCNs with larger strides or higher-dimensional convolutions).
As we will see, the filter sizes play an important role.
Our results suggest that gradient descent in LCNs will be biased towards solutions with repeated filters (or repeated factors of filters), even when these are not global optimizers of the training objective.
\paragraph{Notation}
Throughout the paper, we use zero-based indices for our vectors and matrices.
For $n\in\mathbb{Z}_{\ge 0}$, we write $[n]=\{0,\ldots, n-1\}$.
We write $\mathbb{R}[\mathsf{x},\mathsf{y}]_{d}$ (resp. $\mathbb{C}[\mathsf{x},\mathsf{y}]_{d}$) for the space of bivariate homogeneous polynomials of degree $d$ with real (resp. complex) coefficients.
\section{Linear convolutional networks}
\label{section:LCNs}
A \emph{feedforward neural network} is a family of functions $f: \mathbb{R}^{d_0} \rightarrow \mathbb{R}^{d_L}$ that are compositions of linear and nonlinear maps:
\begin{equation}\label{eq:feedforward}
f(x) = (\alpha_L \circ \rho \circ \alpha_{L-1} \circ \rho \circ \ldots \rho \circ \alpha_1)(x).
\end{equation}
Here $\alpha_l: \mathbb{R}^{d_{l-1}} \rightarrow \mathbb{R}^{d_{l}}$ is an affine map and each $\rho: \mathbb{R}^{d_*} \rightarrow \mathbb{R}^{d_*}$ is a (typically nonlinear) \emph{activation map} that acts on each element of a vector. The composition of consecutive affine and activation maps is called a \emph{layer} of the network.
The coefficients of the affine functions $\alpha_i$ are called the \emph{weights} and \emph{biases} of the network and serve as trainable parameters.
A \emph{linear network} is a feedforward network where the activation function $\rho$ is the identity. In this setting, the end-to-end map $f$ in~\cref{eq:feedforward} is affine, however the parameterization of the representable functions provided by the network's weights is nonlinear. Thus, linear networks preserve some of the nonlinear aspects of general feedforward networks while being theoretically more tractable.
A \emph{linear convolutional network (LCN)} is a linear network with the additional constraint that the affine functions in~\cref{eq:feedforward} have a convolutional structure. The convolutional structure can be seen as constraining general affine maps to satisfy certain linear equations, corresponding to conditions of \emph{restricted connectivity} (some weight coefficients must be zero) and \emph{weight sharing} (some weight coefficients must be repeated). Although an LCN function $f$ can also be seen as a particular fully-connected linear network, the family of all representable functions using an LCN architecture is a strict subset of the family of representable functions using a fully-connected architecture.
In particular, the study of LCNs cannot be reduced to the study of general linear networks. We will point out several important differences between the geometry and optimization properties of LCNs and those of fully-connected networks.
In this paper, we consider convolutional networks with a single filter per layer and no biases. For most of our discussion, we focus on convolutions applied to one-dimensional signals, which means that the input $x$ of the network function is a vector (we discuss the case where $x$ is an image or higher-dimensional in \Cref{sec:higher_dimension}, but we defer general definitions to that section). In this setting, a \emph{convolution} is a map $\alpha_l: \mathbb{R}^{d_{l-1}} \rightarrow \mathbb{R}^{d_{l}}$ of the form
\begin{equation}
(\alpha_l(x))_i = \sum_{j \in [k_l]} w_{l,j} \cdot x_{is_l+j},
\label{eq:convolutionDefinition}
\end{equation}
where $w_l = (w_{l,0},\ldots,w_{l,k_l-1}) \in \mathbb{R}^{k_l}$ is a \emph{filter} of \emph{width} $k_l$ and $s_l \in \mathbb{Z}_{>0}$ is the $\emph{stride}$ associated with the convolution.\footnote{The expression in~\cref{eq:convolutionDefinition} is actually the \emph{cross-correlation} between $x$ and $w_l$ or equivalently the convolution of $x$ with the \emph{adjoint filter} $w^*_l = (w_{l,k_l-1},\ldots,w_{l,0})$. We refer to it simply as a ``convolution'' following common machine learning terminology.
Exchanging all filters with their adjoints would have no effect on our discussion.}
We are assuming ``no padding'' in the way the boundaries of the domain are handled.
This tacitly assumes that input dimension and stride fit with the output dimension.
We compare this approach with using ``circular padding'' in \Cref{subsec:toep_vs_circulant} below.
The linear map $\alpha_l$ in~\cref{eq:convolutionDefinition} can be represented as a generalized Toeplitz matrix (generalized in the sense that standard Toeplitz matrices would correspond to stride $s_l=1$). For example,
for stride $s_l = 2$, filter size $k_l=3$, and input size $d_{l-1}=7$, the matrix takes the form
\begin{equation}
W_l =
\begin{bmatrix}
w_{l,0}&w_{l,1}&w_{l,2}&&&\\
&&w_{l,0}&w_{l,1}&w_{l,2}&\\
&&&&w_{l,0}&w_{l,1}&w_{l,2}\\
\end{bmatrix}.
\label{eq:toeplitz}
\end{equation}
We will call generalized Toeplitz matrices \emph{convolutional matrices}.
For convolutional matrices as in~\cref{eq:toeplitz}, the input and output sizes $d_{l-1}$, $d_{l}$, filter size $k_l$, and stride $s_l$ are related by
\begin{equation}
\label{eq:dimensionConstraintsToeplitz}
d_{l} = \frac{d_{l-1} - k_l}{s_l} + 1.
\end{equation}
An $L$-layer LCN function is given by $f(x) = \overline W x$, where $\overline W \in \mathbb{R}^{d_L \times d_0}$ can be written as $\overline{W} = W_LW_{L-1}\cdots W_{2}W_{1}$ and each $W_l \in \R^{d_{l} \times d_{l-1}}$ is a convolutional matrix.
In this work, we study the \emph{function space}, {\em i.e.}, the set of linear maps that can be represented by an LCN as the parameters vary:
\begin{equation*}
\sM_{\bm{d},\bm{k},\bm{s}} = \Big\{ \overline{W} \in \R^{d_L \times d_0} \colon \overline{W} = \prod_{l=1}^L W_l, \; W_l \in \R^{d_{l} \times d_{l-1}} \text{ convolutional} \Big\}.
\label{eq:functionSpace}
\end{equation*}
Here $\bm d = (d_0, \ldots, d_L)$ denotes the dimensions, $\bm k = (k_1, \ldots, k_L)$ the filter sizes, and
$\bm s = (s_1, \ldots, s_L)$ the strides of the $L$ layers.
We refer to the triple $(\bm d, \bm k, \bm s)$ as the \emph{architecture} of the LCN.
We tacitly assume in the following
that the dimensions $\boldsymbol{d}$ are compatible by satisfying~\cref{eq:dimensionConstraintsToeplitz}.
We also note that by \cref{eq:dimensionConstraintsToeplitz} the function space $\sM_{\bm{d},\bm{k},\bm{s}}$ is fully determined by fixing $d_L$, $\bm k$, and $\bm s$.
As such, we may omit any of the indices $\boldsymbol{d}$, $\boldsymbol{k}$ and $\bm s$ when clear from the context.
The function space $\sM_{\bm{d},\bm{k},\bm{s}}$ is the image of the \textit{parameterization map}
\begin{equation} \label{eqn:Wbar}
\mu_{\bm{d},\bm{k},\bm{s}}\colon \mathbb{R}^{k_1}\times \cdots\times \mathbb{R}^{k_L} \longrightarrow \sM_{\bm{d},\bm{k},\bm{s}} ;\quad (w_1,\ldots, w_L) \longmapsto \overline{W},
\end{equation}
that takes $L$ filters to the product of their associated convolutional matrices.
\begin{remark} Although we use $\sM_{\bm{d},\bm{k},\bm{s}}$ to denote the function space of LCNs, the sequence of dimensions $\bm d$ does not play an essential role. More precisely,
since the product of convolutional matrices is again a convolutional matrix (see \Cref{prop:nonzerodiagsExtension}),
we can also define the parameterization map~\cref{eqn:Wbar} at the level of filters $
\mu_{\bm{k},\bm{s}}\colon \mathbb{R}^{k_1}\times \cdots\times \mathbb{R}^{k_L} \to \mathbb{R}^{k} ;\; (w_1,\ldots, w_L) \mapsto \overline w,
$
where $\overline{w}$ is the filter of the product convolutional matrix.
\end{remark}
\subsection{Compositions of convolutional matrices}
It is well known that the composition of convolution operations is again a convolution. More precisely, for two convolutional maps $\alpha_1: \mathbb{R}^{d_0} \rightarrow \mathbb{R}^{d_1}$ and $\alpha_2: \mathbb{R}^{d_1} \rightarrow \mathbb{R}^{d_2}$ with filters $w_1$ and $w_2$ respectively and satisfying~\eqref{eq:dimensionConstraintsToeplitz}, the composition is given by
\begin{align}
\label{eq:convolutionalMatricesComposition}
\begin{split}
(\alpha_2 \circ \alpha_1) (x)_i &= \sum_{j \in [k_2]} w_{2,j} \cdot \alpha_1(x)_{is_2+j} = \sum_{j \in [k_2]} w_{2,j} \cdot \sum_{\ell \in [k_1]} w_{1,\ell} \cdot x_{(is_2 + j)s_1+\ell}\\
&= \sum_{j \in [k_2]} \sum_{\ell \in [k_1]} w_{2,j} w_{1,\ell} \cdot x_{is_2 s_1 + (js_1 + \ell)} = \sum_{m \in [(k_2-1)s_1 + k_1]} u_{m} \cdot x_{is_2s_1 + m}.
\end{split}
\end{align}
Here the resulting filter $u$ has size $(k_2-1)s_1 + k_1$ and has entries
\begin{align}
\label{eq:filterProduct}
u_m = \sum_{\substack{j \in [k_2], \,\ \ell \in [k_1]\\ js_1 + \ell = m}} w_{2,j} \, w_{1, \ell}.
\end{align}
\begin{proposition}
\label{prop:nonzerodiagsExtension}
The composition of $L$ convolutions with filter sizes ${\bm k} = (k_1,\ldots,k_L)$ and strides ${\bm s} = (s_1,\ldots, s_L)$ is a convolution with
filter size $k = k_1 + \sum_{l=2}^L (k_l-1) \prod_{m=1}^{l-1} s_m$ and stride $s = \prod_{l=1}^L s_l$.
If all strides $s_i$ are equal to $\tilde s$, then
$k=d_0 - (d_L-1)s$ and $s = \tilde s^L$.
\end{proposition}
\begin{proof}
The expression in~\cref{eq:convolutionalMatricesComposition} shows that the composition of two convolutions with filter sizes $k_1, k_2$ and strides $s_1, s_2$ is again a convolution, with filter size $k=k_1 + (k_2-1)s_1$ and stride $s_2 s_1$. If the strides $s_1,s_2$ are equal to $\tilde s$,
then by~\cref{eq:dimensionConstraintsToeplitz}, we have $k_1 = d_0 - (d_1-1) \tilde s$ and $k_2 = d_1 - (d_2-1) \tilde s$, which yields $k = k_1 + (k_2-1) \tilde s = d_0 - (d_2-1)\tilde s^2$.
The statement then follows by induction, whereby at the $l$-th iteration we multiply the convolution represented up to layer $l-1$ with the $l$-th layer convolution.
\end{proof}
\noindent By \Cref{prop:nonzerodiagsExtension}, $\sM_{\bm d, \bm k, \bm s}$ is contained in the vector space $\sM_{(d_0,d_L),k,s}$ of convolutional $d_L\times d_0$ matrices with filter size $k$ and stride $s$. A natural question is whether this containment is strict.
\begin{definition}%
\label{def:filling}\rm
An architecture $(\bm d, \bm k, \bm s)$ is \emph{filling} if $\sM_{\bm d, \bm k, \bm s} = \sM_{(d_0,d_L),k,s}$ holds, where $k$ and $s$ are as in \Cref{prop:nonzerodiagsExtension}.
\end{definition}
Filling LCN architectures might be desirable since in function space training using a convex loss is simply
convex optimization in a vector space.
We emphasize that the vector space $\sM_{(d_0,d_L),k,s}$ is properly contained in $\mathbb{R}^{d_L \times d_0}$, since convolutional matrices have restricted patterns of non-zero entries as well as rows with repeated entries.
For non-filling architectures, we will see in \Cref{sec:geometry} that $\sM_{\bm d, \bm k, \bm s}$ has additional constraints in the form of polynomial equalities and inequalities.
%
Below we give examples of filling and non-filling LCN architectures.
For fully-connected linear networks, the only constraints on the space of representable matrices are rank constraints (vanishing of certain minors, which are polynomial equations). For LCNs, the matrix sizes are decreasing by \cref{eq:dimensionConstraintsToeplitz}, and such rank constraints do not occur.
\begin{example}\label{example:minimal}\rm
Consider the LCN architecture with $\bm d = (3,2,1)$, $\bm k = (2,2)$, $\bm s = (1,1)$. The corresponding function space $\sM \subset \mathbb{R}^{1 \times 3}$ consists of matrices
\begin{equation*}
\overline{W} = W_2W_1 = \begin{bmatrix} c&d \end{bmatrix} \begin{bmatrix} a&b&0\\0&a&b \end{bmatrix} = \begin{bmatrix} ac & ad + bc & bd \end{bmatrix} =: \begin{bmatrix} A & B & C \end{bmatrix} .
\end{equation*}
We can see that this architecture is not filling, {\em i.e.}, that the containment $\sM \subsetneq \mathbb{R}^{1 \times 3}$ is strict. Indeed, the entries in $\overline{W}$ are given by the coefficients $(A,B,C)$ of quadratic polynomials that factor into two real linear forms:
\begin{equation*}
(a \mathsf{x} + b)(c \mathsf{x} + d) = ac\mathsf{x}^2 + (ad + bc) \mathsf{x} + bd = A\mathsf{x}^2 + B \mathsf{x} + C.
\end{equation*}
These correspond to quadratic equations with one or more real roots (or linear equations with $A=0$).
Hence the set of representable coefficients $(A,B,C)$ is characterized by the discriminant condition $B^2 - 4AC \ge 0$.
See \Cref{fig:zwei} (left).
\end{example}
\begin{example} \label{example:larger}\rm
Consider the LCN with $\bm d = (5,3,2)$, $\bm k = (3,2)$, $\bm s = (1,1)$. Then
\begin{equation*}
\overline{W} = W_2W_1 =
\begin{bmatrix} d&e&0\\
0&d&e\end{bmatrix}
\begin{bmatrix}
a&b&c&0&0\\
0&a&b&c&0\\
0&0&a&b&c
\end{bmatrix}
=
\begin{bmatrix}
A&B&C&D&0\\
0&A&B&C&D
\end{bmatrix},
\end{equation*}
where $(A,B,C,D) = (ad, \, bd + ae, \, cd + be, \, ce)$.
The set $\sM$ is the set of all possible $\overline{W}$ where its entries are related by the above equation as $(a,b,c,d,e)$ vary over $\R^5$.
Rephrasing this question as which real cubic polynomials factor into a real quadratic and linear factor, {\em i.e.},
\begin{equation*}
(a\mathsf{x}^2 + b\mathsf{x} + c)(d\mathsf{x} + e) = ad\mathsf{x}^3 + (bd+ae)\mathsf{x}^2 + (cd + be)\mathsf{x} + ce = A\mathsf{x}^3 + B\mathsf{x}^2 + C\mathsf{x} + D,
\end{equation*}
one can immediately draw on the fundamental theorem of algebra to conclude that every choice of $(A,B,C,D)$ has a corresponding choice of $(a,b,c,d,e)$. From this, we note that $\sM$ is filling, so $\sM_{(5,3,2),(3,2),(1,1)} = \sM_{(5,2),4,1}$.
\end{example}
\subsection{Polynomial multiplication}
The previous examples illustrate a useful identification between convolutional matrices and polynomials. More precisely, we associate a filter $w \in \mathbb{R}^k$ with the polynomial
\begin{equation}\label{eq:polynomial_identification}
\pi(w) = w_0 \mathsf{x}^{k-1} + w_1 \mathsf{x}^{k-2}\mathsf{y} + \cdots + w_{k-2}\mathsf{x}\mathsf{y}^{k-2} + w_{k-1}\mathsf{y}^{k-1} \in \mathbb{R}[\mathsf{x},\mathsf{y}]_{k-1}.
\end{equation}
This is a formal polynomial, where the monomials $\mathsf{x}^i\mathsf{y}^j$ are used to index the weights of the filter. To better handle situations where leading coefficients vanish, it is convenient to use homogeneous bivariate polynomials instead of univariate affine polynomials as in the two previous examples. Recall that a root of a homogeneous polynomial $p(\mathsf{x},\mathsf{y})$ is an equivalence class of pairs $(r_\mathsf{x}, r_\mathsf{y}) \in \mathbb{C}^2 \setminus \{ (0,0) \}$ such that $p(r_\mathsf{x}, r_\mathsf{y}) = 0$, up to complex scalar factors ({\em i.e.}, a point in $\mathbb C\mathbb P^1$).
With some abuse of notation, we will also use $\pi$ to map a convolutional matrix $W$ with filter $w$ to the polynomial in~\cref{eq:polynomial_identification}.
\begin{proposition}\label{prop:deep_poly_multiplication}
Consider convolutional matrices $W_1,\ldots, W_L$ with compatible sizes and stride one.
Then the convolutional matrix $\overline W = W_L \cdots W_1$ satisfies $\pi(\overline W) =\pi(W_L) \cdots \pi(W_1)$.
\end{proposition}
\begin{proof}
For the product of two convolutional matrices, the statement follows from \cref{eq:filterProduct}.
The general case follows by induction.
\end{proof}
\begin{corollary}\label{cor:polynomial_space_identification}
Let $(\boldsymbol d, \boldsymbol k, \boldsymbol s)$ be an LCN architecture of depth $L$ with $\boldsymbol{s} = (1, \ldots, 1)$.
Then $\pi$ identifies $\mathcal{M}_{\boldsymbol d, \boldsymbol k, \boldsymbol s}$ with the set of polynomials that factor into polynomials of degrees $k_1 - 1, \ldots, k_L-1$, that is, $\pi(\mathcal{M}_{\boldsymbol d, \boldsymbol k, \boldsymbol s}) = \left\lbrace p = q_L \cdots q_1 \colon q_i \in \mathbb{R}[\mathsf{x},\mathsf{y}]_{k_i-1} \right\rbrace \subseteq \mathbb{R}[\mathsf{x},\mathsf{y}]_{k-1}$ where $k = \sum_{i=1}^L (k_i -1) + 1$ is the size of the end-to-end filter.
\end{corollary}
Since the ordering of the factors does not affect the product polynomial, the function space of an LCN with stride-one 1D convolutional layers is the same for any permutation of the layers.
This also holds for higher-dimensional convolutions, as we will see in \Cref{sec:higher_dimension}.
\begin{remark}
\label{rem:polynomialCorrespondenceHigherStride}
\Cref{prop:deep_poly_multiplication} can be generalized to deal with arbitrary strides $\boldsymbol s$.
For example, for two convolutional matrices $W_2,W_1$ with strides $s_2,s_1$, the product $W_2 W_1$ can be identified with
the polynomial $p = \pi(W_2 W_1) \in \mathbb{R}[\mathsf{x},\mathsf{y}]$ that factors as $p = q_2q_1$, where $q_1 = \pi(W_1)$ as in~\cref{eq:polynomial_identification}, while $q_2$ is defined as follows:
if $w$ denotes the filter of $W_2$, we see from \cref{eq:filterProduct} that
\[q_2 = w_0 \mathsf{x}^{s_1(k_2-1)} + w_1 \mathsf{x}^{s_1(k_2-2)}\mathsf{y}^{s_1} + \cdots + w_{k_2-2} \mathsf{x}^{s_1} \mathsf{y}^{s_1(k_2-2)} + w_{k_2-1}\mathsf{y}^{s_1(k_2-1)}.
\]
\end{remark}
\subsection{Toeplitz vs.\ circulant matrices}
\label{subsec:toep_vs_circulant}
Discrete convolutions over finite-dimensional data can be defined in slightly different ways depending on how they deal with the boundary of the input. Until now, we have been discussing a ``no padding'' setting where the convolution is not applied across the boundary, and the dimension of the output is smaller than the dimension of the input (see \cref{eq:toeplitz}). An alternative approach would be to consider \emph{cyclic convolutions}, where the input space $\mathbb{R}^d$ is viewed as representing ``cyclic signals'' $\mathbb{Z}/d \rightarrow \mathbb{R}$. In this section we point out that the two approaches are effectively equivalent in terms of the function space.
Let $W_1,\ldots, W_L$ be the weight matrices of an LCN. These matrices are generalized Toeplitz matrices and must have compatible sizes \cref{eq:dimensionConstraintsToeplitz}.
If $W_l$ has size $d_{l} \times d_{l-1}$ and corresponding filter $w_l$ of size $k_l$, then we construct a \emph{generalized circulant} matrix of size $d_{0} \times d_{0}$ as follows.
If the stride is $s_l = 1$, the circulant matrix is
\[
\begin{bmatrix} w_{l,0} & w_{l,1} & \ldots & w_{l, d_0-1} \\
w_{l, d_0-1} & w_{l,0} & \ldots & w_{l, d_0-2} \\
\vdots &\vdots&\ddots& \vdots \\
w_{l,1} & w_{l,2} &\ldots & w_{l,0} \\
\end{bmatrix} ,
\]
where $w_{l,i} = 0$ for $i \geq k_l$.
For Toeplitz matrices with larger strides $s_l$, we can similarly define a $d_0 \times d_0$ generalized circulant matrix of stride $s_l$, where each row is obtained by shifting the previous row by $s_l$ steps.
The generalized Toeplitz matrix $W_l$ is the top-left $d_l \times d_{l-1}$ block of its associated generalized circulant matrix.
Under this identification, the multiplication of generalized Toeplitz matrices with compatible sizes is the multiplication of the corresponding generalized circulant matrices.
In particular, the resulting filter $\overline{w}$ of the Toeplitz matrix product $\overline{W}$ also results from the product of the circulant matrices. Therefore both formulations are equivalent.
\begin{example} \rm
For the Toeplitz matrices in \Cref{example:minimal},
the corresponding product of circulant matrices~is
\begin{equation*}
\begin{bmatrix}
c&d&0\\
0&c&d\\
d&0&c\\
\end{bmatrix}
\begin{bmatrix}
a&b&0\\
0&a&b\\
b&0&a\\
\end{bmatrix}
= \begin{bmatrix}
ac&ad+bc&bd\\
bd&ac&ad+bc\\
ad+bc&bd&ac\\
\end{bmatrix}.
\end{equation*}
\end{example}
\section{Discriminants and multiple roots}
\label{sec:discriminants}
In this section, we review some properties of the structure of real polynomials with repeated roots, recalling the definitions of \emph{discriminants}, \emph{multiple root loci}, and \emph{real root multiplicity patterns} (\emph{rrmp} for short).
We will use these notions in the next sections to study the function space of LCNs and to investigate the critical points of the associated loss functions.
A reader who is more interested in the applications of these concepts to LCNs might skip this section on a first read and return to it later to better understand the structure of repeated roots in polynomials.
The \emph{discriminant} of a polynomial $p=a_n\mathsf{x}^n+a_{n-1}\mathsf{x}^{n-1}\mathsf{y} +\cdots+a_1 \mathsf{x}\mathsf{y}^{n-1} +a_0\mathsf{y}^n \in \mathbb{R}[\mathsf{x},\mathsf{y}]$ is a polynomial in the coefficients $a_n,\ldots, a_0$ that vanishes if and only if $p$ has a (complex) double root \cite[Chapter~4]{10.5555/1197095}.
The iterated singular loci of the discriminant hypersurface are the \emph{coincident/multiple root loci}.
\begin{definition} \rm
For a partition $\lambda = (\lambda_1, \ldots, \lambda_r)$ of $k-1$,
the \emph{multiple root locus} $\Delta_{\lambda} \subset \mathbb{R}[\mathsf{x},\mathsf{y}]_{k-1}$ is the set of homogeneous polynomials that have $r$ (not necessarily distinct) complex roots with multiplicities $\lambda_1, \ldots, \lambda_r$. That is, $\Delta_\lambda$ is the set of polynomials $P \in \mathbb{R}[\mathsf{x},\mathsf{y}]_{k-1}$ that can be written as $P=Q_{1}^{\lambda_1} \cdots Q_r^{\lambda_r}$, where each $Q_i \in \mathbb{C}[\mathsf{x},\mathsf{y}]_1$ is a complex linear form.
\end{definition}
For instance, the discriminant hypersurface is $\Delta_{(2,1,\ldots,1)}$.
The singular locus of a multiple root locus is a union of certain higher-order multiple root loci; see \cite{kurmann2012some} for a combinatorial description.
\footnote{For two partitions $\lambda$ and $\mu$ of $k-1$, $\Delta_{\mu}$ is contained in $\Delta_{\lambda}$ if and only if $\mu$ is a coarsening of $\lambda$. In \cite[Proposition 2.1(i)]{kurmann2012some} it is shown that $\Delta_{\mu}$ is contained in the singular locus of $\Delta_{\lambda}$ if and only if either there are at least two ways of obtaining $\lambda$ as a splitting of $\mu$ or there is a splitting such that one of the elements $\mu_i$ in $\mu$ is split into subsets which do not all have the same size.
An illustrative example is given in~\cite[Example 2.4]{kurmann2012some}.}
We are also interested in distinguishing between real and complex roots. This more fine-grained information is captured by the \emph{real root multiplicity pattern}.
\begin{definition} \rm
A homogeneous polynomial $P \in\mathbb{R}[\mathsf{x},\mathsf{y}]_{k-1}$ has \emph{real root multiplicity pattern}, short \emph{rrmp}, $(\rho \mid \gamma) = (\rho_1, \ldots, \rho_r \mid \gamma_1, \ldots, \gamma_c)$ if it can be written as \begin{equation}
\label{eq:realFactorization}
P = p_1^{\rho_1} \cdots p_r^{\rho_r} q_1^{\gamma_1} \cdots q_c^{\gamma_c}, \end{equation}
where $p_i \in \mathbb{R}[\mathsf{x},\mathsf{y}]_1$ and $q_j \in \mathbb{R}[\mathsf{x},\mathsf{y}]_2$ are irreducible and pairwise linearly independent.
\end{definition}
Recall that each factor $q_j$ has two complex conjugate roots.
Polynomials with rrmp\, $(\rho \mid \gamma)$ are smooth points on the multiple root locus $\Delta_{\lambda_{\rho|\gamma}}$,
where $\lambda_{\rho|\gamma} = (\rho_1, \ldots, \rho_r, \gamma_1, \gamma_1, \ldots, \gamma_c, \gamma_c)$.
\begin{example}\rm \label{ex:rrmps}
We describe the possible rrmp\, of a quadratic, cubic, and quartic polynomial.
The discriminant polynomial
$\Delta$ of $a\mathsf{x}^2+b\mathsf{x}\mathsf{y}+c\mathsf{y}^2$ distinguishes three rrmp's:
\vspace{.2cm}
\begin{center}
\small
\begin{tabular}{ll}
{$11|0$}& 2 distinct real roots: $\Delta = b^2-4ac > 0$. \\
{$2|0$}& 1 double real root: $\Delta =0$. \\
{$0|1$}& no real roots: $\Delta<0$.
\end{tabular}
\end{center}
\vspace{.2cm}
\noindent
For cubic polynomials $a\mathsf{x}^3+b\mathsf{x}^2\mathsf{y}+c\mathsf{x}\mathsf{y}^2+d\mathsf{y}^3$ with real coefficients, there are four rrmp's shown below, discussed in \cite{ARNON198837,BlinnHowto,Gonzalez2015RootCO}.
\vspace{.2cm}
\begin{center}
\small
\begin{tabular}{ll}
{$111|0$} & 3 distinct real roots: $\Delta = b^2 c^2 - 4 a c^3 -4 b^3 d - 27 a^2 d^2 + 18 a b c d > 0$. \\
{$12|0$} & 1 single real root and 1 double real root: $\Delta = 0$.\\
{$3|0$} & 1 triple real root: $\delta_1 = 3ac-b^2 = 0$ and $\delta_2 = 9ad-bc= 0$ and $\delta_3 = 3bd-c^2= 0$. \\
{$1|1$} & 1 real root and 2 complex conjugate roots: $\Delta < 0$.
\end{tabular}
\end{center}
\vspace{.2cm}
\noindent
Following \cite{ARNON198837}, a polynomial $f_n=\mathsf{x}^n+a_1\mathsf{x}^{n-1}\mathsf{y}+a_2\mathsf{x}^{n-2}\mathsf{y}^2+ \cdots +a_n\mathsf{y}^n$ can be translated to a polynomial $g_n=f_n(\mathsf{x}-\nicefrac{a_1}{n}\mathsf{y},\mathsf{y}) = \mathsf{x}^n+b_2\mathsf{x}^{n-2}\mathsf{y}^2+b_3\mathsf{x}^{n-3}\mathsf{y}^3+\cdots+b_n\mathsf{y}^n$. The two polynomials $f_n$ and $g_n$ have the same rrmp.
For a quartic polynomial $g_4=\mathsf{x}^4+p\mathsf{x}^2\mathsf{y}^2+q\mathsf{x}\mathsf{y}^3+r\mathsf{y}^4$, letting
\begin{equation*}
\begin{split}
\delta(p,q,r) =& 256 r^3 - 128p^2r^2 + 144 pq^2r +16p^4r -27q^4 -4p^3q^2\\
\delta'(p,q,r) =& 8pr - 9q^2 -2p^3,
\end{split}
\end{equation*}
where $\delta$ corresponds to the discriminant, the different rrmp's are as follows:
\begin{center}
\begin{tabular}{ll}
{$1111|0$} & 4 distinct real roots: $\delta>0$ and $\delta'>0$ and $p<0$. \\
{$112|0$} & 2 distinct real roots and 1 double real root: $\delta=0$ and $\delta'>0$ and $p<0$. \\
{$22|0$} & 2 real double roots: $\delta=0$ and $\delta'=0$ and $p<0$ and $q=0$. \\
{$13|0$} & 1 single real root and 1 triple real root: $\delta=0$ and $\delta'=0$ and $p<0$ and $q\neq0$. \\
{$4|0$} & 1 quadruple real root: $\delta=0$ and $\delta'=0$ and $p=0$. \\
%
{$11|1$} & 2 distinct real roots and 2 distinct complex conjugate roots: $\delta<0$. \\
{$2|1$} & 1 double real root and 2 distinct complex conjugate roots: $\delta=0$ and $\delta'<0$. \\
%
{$0|2$} & 2 complex conjugate double roots: $\delta=0$ and $\delta'=0$ and $p>0$. \\
{$0|11$} & 4 distinct complex roots: $\delta>0$ and ($\delta'\leq 0$ or $p>0$).
\end{tabular}
\end{center}
\end{example}
\section{Function space of linear convolutional networks}
\label{sec:geometry}
We show that the function space of an LCN is a semi-algebraic set,
discuss its dimension, and whether it is filling (\Cref{def:filling}).
\subsection{Convolutional matrices of stride one}
We first focus on one-dimensional convolutions and stride one.
Recall that a semi-algebraic set is a solution set to a finite set of polynomial equations and polynomial inequalities, or a finite union of such sets.
\begin{theorem}%
\label{thm:fillingCircular}
Let $(\bm d, \bm k, \bm s)$ be an LCN architecture with $\bm s= (1,\ldots,1)$ and depth $L$.
Then $\sM_{\bm d, \bm k, \bm s}$ is a full-dimensional semi-algebraic subset of $\sM_{(d_0, d_L), k, s}$, where
$k= \sum_{i=1}^L k_i - L + 1$ and $s=1$.
In particular, $\sM_{\bm d, \bm k, \bm s}$ is cut out from $\sM_{(d_0, d_L), k, s}$ by inequalities only.
Moreover, the architecture is filling if and only if at most one of the filter sizes $k_1,\ldots, k_L$ is even.
\end{theorem}
This result explains the difference between \Cref{example:minimal,example:larger}.
To prove it, we use \Cref{cor:polynomial_space_identification}:
We identify $\sM_{(d_0, d_L), k, s}$ with the space $\mathbb{R}[\mathsf{x},\mathsf{y}]_{k-1}$ of polynomials of degree $k-1$,
and the subset $\sM_{\bm d, \bm k, \bm s}$ with the polynomials that factor as a product of polynomials of degrees $k_1-1,\ldots, k_{L}-1$.
Under this identification, we have the following.
\begin{lemma}
\label{lem:realRoots}
Let $\bm s= (1,\ldots,1)$. A polynomial $p \in \mathbb{R}[\mathsf{x},\mathsf{y}]_{k-1}$
corresponds to a convolutional matrix in $\sM_{\bm d, \bm k, \bm s}$ if and only if $p$ has at least $e := |\{k_i \colon k_i \mbox{ is even}\}|$ real roots (counted with multiplicity), {\em i.e.}, if the rrmp $(\rho \, | \, \gamma)$ of $p$ satisfies $\sum_{i=1}^r \rho_i \ge e$.
\end{lemma}
\begin{proof} By \Cref{cor:polynomial_space_identification}, it suffices to show that a polynomial $p$ of degree $ k-1$ has at least $e$ real roots if and only if $p$ is a product of polynomials of degrees $k_1-1,\ldots, k_{L}-1$.
One direction is immediate. A factorization of $p$ of the desired type has $e$ odd-degree factors, and any polynomial $p$ with $e$ odd-degree factors has at least $e$ real roots.
For the other direction,
let $e'$ denote the number of real roots of $p$.
The irreducible factors of $p$ are $e'$ linear terms and $(k-1 - e')/2$ quadratic terms.
If $e' \geq e$, these factors can be multiplied to build a set of factors $q_1, \ldots, q_L$ of degrees $k_1-1,\ldots,k_L-1$ as follows. We first assign one of the $e'$ linear terms to each of the $e$ factors $q_i$ of odd degree and then we use the remaining $(e'-e) + (k-1 - e')/2$ irreducible factors of $p$ to fill up the factors $q_1, \ldots, q_L$ until their degrees are indeed $k_1-1,\ldots,k_L-1$.
Note that this procedure works as the degrees of the irreducible factors of $p$ and the degrees of $q_1, \ldots, q_L$ both sum up to the same value~$k-1$.
\end{proof}
\begin{example} \rm
\label{ex:quartic}
When $\bm s = (1,\ldots,1)$ and $\bm k = (k_1,\ldots,k_L)$ are chosen such that the resulting filter size is $k = 5$, by \Cref{lem:realRoots} the matrices in $\sM_{\bm d, \bm k, \bm s}$ correspond to quartic polynomials with a particular real root structure, specified by the individual filter sizes $k_i$:
\begin{enumerate}[leftmargin=*]
\item If $L=2$ and $(k_1,k_2)=(3,3)$, then every polynomial of degree four is in $\pi(\sM_{\bm d, \bm k, \bm s})$.
\item \label{ex:quartic2} If $L = 2$ and $(k_1, k_2) = (4,2)$, then a quartic polynomial $p$ corresponds to a matrix in the function space if and only if it has two or four real roots.
Hence, the complement of the function space corresponds to all quartics with no real roots.
\item \label{ex:quartic3} If $L = 3$ and $(k_1, k_2, k_3) = (3,2,2)$, this architecture has the same function space as the previous one. However, the parameterization of the function space is different: Here $\pi(\sM_{\bm d, \bm k, \bm s})$ arises by taking the product of two linear and one quadratic factor, whereas in the previous architecture one considers the product of a linear and a cubic factor.
\item \label{ex:quartic4} If $L=4$ and $(k_1, k_2, k_3, k_4) = (2,2,2,2)$, a polynomial of degree four is in $\pi(\sM_{\bm d, \bm k, \bm s})$ if and only if all its roots are real.
\end{enumerate}
\end{example}
\begin{proof}[Proof of \Cref{thm:fillingCircular}]
We first consider the case where all filter sizes are odd.
Using \Cref{lem:realRoots}, we see that $e=0$, so every polynomial of degree $k-1$ corresponds to a matrix in the function space $\sM_{\bm d, \bm k, \bm s}$.
If exactly one filter size is even, then $k-1$ is odd.
Hence, every polynomial $p$ of degree $k-1$ has at least one real root.
In \Cref{lem:realRoots} we have $e=1$, which shows that $p \in \pi(\sM_{\bm d, \bm k, \bm s})$.
We conclude that the architecture is filling if at most one filter size is~even.
Finally, assume that two or more filter sizes are even. In this case, we consider a polynomial $p$ of degree $k-1$ that has zero (if $k-1$ is even) or one (if $k-1$ is odd) real root. Then $p$ does not belong to $\pi(\sM_{\bm d, \bm k, \bm s})$ by \Cref{lem:realRoots}. This shows that the architecture is not filling.
That the function space is semi-algebraic follows from the fact that it has a polynomial parametrization (by Tarski-Seidenberg).
Finally, we argue that the set of polynomials representable by a given LCN is full dimensional in $\mathbb{R}[\mathsf{x},\mathsf{y}]_{k-1}$.
For this, it is enough to notice that small perturbations of the coefficients of a generic polynomial will not affect its root structure.
We will discuss this in more detail in the following statements.
\end{proof}
When the architecture of an LCN is not filling, it is interesting to study the boundary of the function space.
\begin{proposition} \label{prop:boundaryCondition}
Let $(\bm d, \bm k, \bm s)$ be an LCN architecture with $e := |\{k_i \colon k_i \text{ is even}\}| \geq 2$ and $\bm s= (1,\ldots,1)$.
In the Euclidean topology on $\sM_{(d_0,d_L), k, s}$, the function space $\sM_{\bm d, \bm k, \bm s}$ is closed and its boundary consists of all convolutional matrices that correspond to polynomials with rrmp\, $(\rho \mid \gamma)$ satisfying
$\sum_{i=1}^r \rho_i \geq e$ and $|\{\rho_i \colon \rho_i \mbox{ is odd}\}| \leq e-2$.
\end{proposition}
\begin{proof}
A sequence of polynomials, each having at least $e$ real roots counted with multiplicity, will also have at least $e$ real roots in the limit.
Hence, by \Cref{lem:realRoots}, the function space is closed.
Using the identification $\pi$ with polynomials, the boundary of $\sM_{\bm d, \bm k, \bm s}$ consists of all polynomials in $\sM_{\bm d, \bm k, \bm s}$ that are limits of sequences of polynomials in its complement.
\\ \indent
Given a sequence of polynomials $P^{(j)}$ of degree $(k-1) = \sum_{i=1}^L (k_i-1)$ in the complement of $\sM_{\bm d, \bm k, \bm s}$ that converges to a polynomial in $\sM_{\bm d, \bm k, \bm s}$, we may assume (by restricting to a sub-sequence) that all $P^{(j)}$ have the same rrmp\, $(\rho \mid \gamma)$.
Since $e$ and $(k-1)$ have the same parity and $P^{(j)} \notin \sM_{\bm d, \bm k, \bm s}$, we have $\sum_{i=1}^r \rho_i \leq e-2$.
In particular, the number of odd $\rho_i$ is at most $e-2$.
In the limit of the $P^{(j)}$, pairs of complex conjugate roots might become real double roots (which does not affect the number of odd $\rho_i$) or distinct real roots might become equal (which cannot increase the number of odd $\rho_i$).
Hence, the limit of the $P^{(j)}$ has rrmp\, $(\rho' \mid \gamma')$ with at most $e-2$ odd $\rho'_i$ and $\sum \rho'_i \geq e$ (since we assumed the limit to be in $\sM_{\bm d, \bm k, \bm s}$).
\\\indent
Conversely, given a polynomial $P$ with rrmp\, $(\rho \mid \gamma)$ satisfying $|\{\rho_i \colon \rho_i \mbox{ is odd}\}| \leq e-2$ and $\sum_{i=1}^r \rho_i \geq e$, we write $\rho_i = 2\alpha_i+\beta_i$, where $\beta_i \in \{0,1\}$.
We consider its factorization as in \cref{eq:realFactorization} and perturb each factor $p_i^{\rho_i}$ to a polynomial with $\alpha_i$ pairs of complex conjugate roots, plus one real root if $\beta_i=1$.
That way, $P$ gets perturbed to a polynomial $\tilde P$ with exactly $\sum \beta_i$ real roots.
We have that $\sum \beta_i = |\{\rho_i \colon \rho_i \mbox{ is odd}\}| \leq e-2$, so $\tilde P \notin \sM_{\bm d, \bm k, \bm s}$.
\end{proof}
\begin{corollary}
Let $(\bm d, \bm k, \bm s)$ be an LCN architecture with $\bm s= (1,\ldots,1)$ and two or more even filter sizes.
The Zariski closure of the Euclidean boundary of the function space $\sM_{\bm d, \bm k, \bm s}$ is the discriminant hypersurface $\pi^{-1}(\Delta_{(2,1,\ldots,1)})$.
\end{corollary}
\begin{proof}
Let $e := |\{k_i \colon k_i \text{ is even}\}|$.
Since the rrmp\, $(\rho\mid\gamma)$ of any polynomial on the Euclidean boundary of $\sM_{\bm d, \bm k, \bm s}$ satisfies by \Cref{prop:boundaryCondition} that at least one of the $\rho_i$ is larger than $1$, the boundary is contained in the discriminant hypersurface.
Moreover, any polynomial with one real double root, $e-2$ pairwise distinct real single roots, and all other roots non-real with multiplicity one, is both on the boundary of $\sM_{\bm d, \bm k, \bm s}$ and a smooth point of the discriminant.
\end{proof}
\begin{example}\rm \label{ex:boundaries}
The following table lists all non-filling architectures with stride one and where the end-to-end filter has size $3,4,5$ (up to permutations of the filters):
\begin{center}
\small
\begin{tabular}{c|ccc}
$\bm k$ &function space $\mathcal M_{\bm k}$ & complement & Euclidean boundary \\ \hline
$(2,2)$ & $11|0$, $2|0$ & $0|1$ & $2|0$ \\
$(2,2,2)$ & $111|0$, $12|0$, $3|0$ & $1|1$ & $12|0$, $3|0$ \\
$(3,\!2,\!2)$ \!or\! $(4,\!2)$ & $1111|0$, $112|0$, $22|0$, $13|0$, $4|0$, $11|1$, $2|1$ & $0|2$, $0|11$ & $2|1$, $22|0$, $4|0$ \\
$(2,2,2,2)$ & $1111|0$, $112|0$, $22|0$, $13|0$, $4|0$ & $11|1$, $2|1$, $0|2$, $0|11$ & $112|0$, $13|0$, $22|0$, $4|0$
\\
\hline
\end{tabular}
\end{center}
\vspace{.35cm}
\noindent
For each architecture $\bm k$, the function space $\mathcal{M}_{\bm k}$ consists of all polynomials with one of the rrmp's listed above.
We described the possible rrmp's for quadratic, cubic, and quartic polynomials in \Cref{ex:rrmps}.
Similarly, the complement and boundary of $\mathcal{M}_{\bm k}$ are as shown above.
\end{example}
When exactly two filter sizes are even, the non-filling geometry of $\sM_{\bm d, \bm k, \bm s}$ has a particularly simple description.
\begin{proposition}
\label{prop:twoConvexCones}
Let $(\bm d, \bm k, \bm s)$ be an LCN architecture with $\bm s= (1,\ldots,1)$ and exactly two even filter sizes.
The complement of $\pi(\sM_{\bm d, \bm k, \bm s})$ in $\mathbb{R}[\mathsf{x},\mathsf{y}]_{k-1}$ is a union of two open convex cones, each consisting of all positive or respectively negative polynomials.
\end{proposition}
\begin{proof}
If exactly two filters have even size, then $k-1$ is even, and a polynomial of degree $k-1$ is in $\pi(\sM_{\bm d, \bm k, \bm s})$ if and only if it has at least two real roots (by \Cref{lem:realRoots}).
Hence, the complement of $\pi(\sM_{\bm d, \bm k, \bm s})$ consists of all polynomials of even degree $k-1$ without real~roots.
\end{proof}
\noindent This behaviour is exhibited in \Cref{example:minimal}, illustrated on the left of \Cref{fig:zwei}, and in \Cref{ex:quartic} cases 2 and 3 (see also \Cref{ex:boundaries} first and third rows).
In general, if strictly more than two filter sizes are even, the complement of $\pi(\sM_{\bm d, \bm k, \bm s})$ is \emph{not} a union of two convex cones, as the following example demonstrates.
\begin{example}\rm
Consider the case $|\{k_i \colon k_i \mbox{ is even}\}| = 3$ with stride one.
A polynomial of odd degree $k-1$ is in $\pi(\sM_{\bm d, \bm k, \bm s})$ if and only if $p$ has at least three real roots (by \Cref{lem:realRoots}).
Hence, the complement $\mathbb{R}[\mathsf{x},\mathsf{y}]_{k-1} \setminus \pi(\sM_{\bm d, \bm k, \bm s})$ is the set of all polynomials of degree $k-1$ with exactly one real root.
Even when restricting ourselves to degree-$(k-1)$ polynomials with one real root such that the remaining degree-$(k-2)$ factor is positive, this set does not form a convex cone.
For instance, if $k-1 = 3$, consider the polynomials $(\mathsf{x}-\mathsf{y})\mathsf{x}(\mathsf{x}+\mathsf{y})+2\mathsf{y}^3$ and $(\mathsf{x}-\mathsf{y})\mathsf{x}(\mathsf{x}+\mathsf{y})-2\mathsf{y}^3$:
Both have exactly one real root such that the remaining quadratic factor is positive.
However, their sum clearly has three real roots.
\end{example}
\subsection{Generalization to larger strides}
\label{sec:largerStrides}
In this section, we consider one-dimensional convolutions that may have stride larger than one. In this setting, the identification with polynomials holds as described in Remark~\ref{rem:polynomialCorrespondenceHigherStride}.
Specifically, we write
\begin{equation*}
\pi_s: \mathbb{R}^k \longrightarrow \mathbb{R}[\mathsf{x},\mathsf{y}]_{(k-1)s}, \quad w \longmapsto w_{0} \mathsf{x}^{(k-1)s} + w_1 \mathsf{x}^{(k-2)s}\mathsf{y}^s + \cdots + w_{k-2} \mathsf{x}^s\mathsf{y}^{(k-2)s} + w_{k-1} \mathsf{y}^{(k-1)s}.
\end{equation*}
This generalizes the map $\pi = \pi_1$ defined in~\cref{eq:polynomial_identification}.
If $W_2$ and $W_1$ are convolutional matrices with strides $s_2$ and $s_1$, then $\pi(W_2 W_1) = \pi_{s_1}(W_2) \pi(W_1)$.
Note that $\pi_{s_1}$ is applied to $W_2$, while $s_2$ does not have any effect on the filter of $W_2 W_1$ (only on its stride which is given by $s_2 s_1$).
Based on this fact, we deduce the following two results.
\begin{lemma}
\label{lem:specialCases}
\begin{enumerate}[leftmargin=*]
\item[a)] If $\bm s \!=\! (s_1,\ldots, s_L)$ and $\bm s' \!=\! (s_1,\ldots, s_{L-1},1)$, then
$\pi(\mathcal M_{\bm d, \bm k, \bm s}) = \pi(\mathcal M_{\bm d, \bm k, \bm s'})$.
\item[b)] If $\bm k = (k_1, \ldots, k_{L-1}, 1)$ and $\bm k' = (k_1, \ldots, k_{L-1})$, $\bm d' = (d_0, \ldots, d_{L-1})$, $\bm s' = (s_1, \ldots, s_{L-1})$,
then $\pi(\mathcal M_{\bm d, \bm k, \bm s}) = \pi(\mathcal M_{\bm d', \bm k', \bm s'})$.
\end{enumerate}
\end{lemma}
\begin{proof}
This follows from the fact that $\pi(W_L \cdots W_1) = \pi_s(W_L) \pi_1(W_{L-1}\cdots W_1)$, where $s = s_{L-1} \ldots s_1$ is the stride of $W_{L-1}\cdots W_1$. Part a) holds since this fact is independent of $s_L$. Part~b) holds since $k_L=1$ implies that $\pi_s(W_L)$ is a degree-0 polynomial, which does not play a role in the factorization of the end-to-end filter.
\end{proof}
\begin{corollary}
Let $s_{1} = \cdots = s_{L-1} = 1$ and $s_L$ arbitrary.
Then the LCN architecture
$(\bm d, \bm k, \bm s)$ satisfies the properties in \Cref{thm:fillingCircular} (with $s=s_L$) and \Cref{prop:boundaryCondition,prop:twoConvexCones}.
\end{corollary}
The corollary is immediately implied by \Cref{lem:specialCases}~a).
This covers the case where only the last layer has an arbitrary stride.
Similarly, \Cref{lem:specialCases}~b) says that if the last filter size is one, we can understand the function space from the smaller architecture that only consists of the first $L-1$ layers.
For the remaining cases, we obtain the following result.
\begin{proposition}
\label{prop:nonfillingstride}
If $k_L > 1$ and $s_i > 1$ for some $i \leq L-1 $, the LCN architecture is non-filling.
In fact, the function space is a lower-dimensional semi-algebraic subset of~$\sM_{(d_0,d_L),k,s}$.
\end{proposition}
\begin{proof}
Consider the last product $W_L \cdot W'$ where $W' = W_{L-1} \cdots W_1$.
We write $k' = k_1 + \sum_{l=2}^{L-1} (k_l-1)\prod_{m=1}^{l-1} s_m $ for the filter size of $W'$ and $s' = s_1 \cdots s_{L-1} > 1$ for its stride.
Even if $W'$ is an arbitrary convolutional matrix with filter size and stride $(k',s')$ ({\em i.e.}, even if the LCN architecture from the first $L-1$ layers is filling),
the product $W_L \cdot W'$ is \emph{not an arbitrary} convolutional matrix with filter size $k = k'+(k_L-1)s'$ and stride $s = s's_L$.
Otherwise, every polynomial $p \in \mathbb{R}[\mathsf{x},\mathsf{y}]_{k-1}$ could be factored into a real polynomial $p'$ of degree $k'-1$ in $\mathsf{x},\mathsf{y}$ and a real polynomial $p_L$ of degree $k_L-1$ in $\mathsf{x}^{s'},\mathsf{y}^{s'}$, by \Cref{rem:polynomialCorrespondenceHigherStride}.
Modding out global scaling, we may assume that the polynomial $p$ and its two factors $p'$ and $p_L$ are monic, so their remaining degrees of freedom ({\em i.e.}, the number of their remaining coefficients) are $k-1$ for $p$, $k'-1$ for $p'$, and $k_L-1$ for $p_L$.
Since $k_L > 1$ and $s' > 1$, we have that $(k'-1)+(k_L-1) < k'+(k_L-1)s'-1 = k-1$,
so the monic polynomials $p'$ and $p_L$ do not have enough degrees of freedom for their product to yield all monic polynomials $p$.
This dimension count shows that $\dim (\mathcal M_{\bm d, \bm k, \bm s}) \leq k'+k_L -1 < k = \dim(\sM_{(d_0,d_L),k,s}). $
\end{proof}
For the non-filling architectures in \cref{prop:nonfillingstride},
$\sM_{\bm d, \bm k, \bm s}$ has a strictly smaller dimension than
$\sM_{(d_L, d_0), k,s}$ and is thus cut out by polynomial equations and inequalities.
This is in contrast to the non-filling architectures appearing for stride one in \Cref{thm:fillingCircular}: in that case,
$\sM_{\bm d, \bm k, \bm s}$ has the same dimension as
$\sM_{(d_0,d_L),k,s}$ and is cut out only by polynomial inequalities.
\begin{example}
\rm
Consider the LCN with $\bm d = (5,2,1)$, $\bm k =(3,2)$, $\bm s =(2,1)$.
Note $s_2=1$ is inconsequential by \Cref{lem:specialCases} a).
This setting is similar to \Cref{example:larger} where we considered $\bm s = (1,1)$. In that case, the architecture was filling, but this is no longer the case for stride size $2$ as we will see.
In the current example, the function space $\sM\in \mathbb{R}^{1\times 5}$ consists of matrices
\begin{align}
\label{eq:strideTwo}
\overline{W} = W_2 W_1 = \begin{bmatrix}
d & e
\end{bmatrix} \begin{bmatrix}
a & b & c & 0 & 0 \\
0 & 0 & a & b & c
\end{bmatrix}
= \begin{bmatrix}
ad & bd & ae+cd & be & ce
\end{bmatrix}.
\end{align}
We see that the coefficients in~\cref{eq:strideTwo} are given by
\begin{equation}\label{eq:poly_stride}
\pi(\overline W) = (d \mathsf{x}^2 + e \mathsf{y}^2)(a \mathsf{x}^2+b\mathsf{x}\mathsf{y}+c\mathsf{y}^2).
\end{equation}
The architecture is filling if and only if every real quartic polynomial
$A\mathsf{x}^4 + B\mathsf{x}^3\mathsf{y} + C\mathsf{x}^2\mathsf{y}^2 + D\mathsf{x}\mathsf{y}^3 + E\mathsf{y}^4$
can be factored into two real quadratic polynomials as above.
However, it is easy to verify that the coefficients of the polynomial in~\cref{eq:poly_stride}
satisfy the equation $AD^2+B^2E = BCD$.
In fact, we can fully characterize the function space as follows:
\begin{align}
\label{eq:strideTwoFunctionSpace}
\mathcal{M} = \{ (A,B,C,D,E) \in \mathbb{R}^5 \colon AD^2+B^2E = BCD \text{ and } C^2 \geq 4AE \}.
\end{align}
Indeed, the coefficients in \cref{eq:strideTwo} satisfy $(ae+cd)^2-4adce = (ae-cd)^2 \geq 0$.
For the other containment, we consider a tuple $(A, \ldots, E)$ in the right-hand side of \cref{eq:strideTwoFunctionSpace} and explicitly find a factorization as in~\cref{eq:poly_stride}.
For the case $E=D=0$, set $a=A$, $b=B$, $c=C$, $d=1$, $e=0$.
For the case $E=0, D\neq0$, set $a= C/D$, $b=1$, $c=0$, $d=B$, $e=D$, which works since $AD=BC$.
It remains to show the case $E \neq 0$. We may assume without loss of generality that $E=1$ and we set $c=1, e=1, b = D$. To determine $a$ and $d$, we distinguish
four cases:
\begin{itemize}
\item $D \neq 0, B \neq 0$: Set $d = B/D$ and $a = AD/B$.
\item $D \neq 0, B = 0$: Set $d = 0$ and $a = C$.
\item $D = 0, A \neq 0$:
At least one of the two values $\frac{C}{2} \pm \sqrt{\frac{C^2}{4}-A}$ is non-zero since $A \neq 0$. Set $d$ to be such a non-zero value and $a = A/d$.
\item $D = 0, A=0$: Set $a = C$ and $d=0$.
\end{itemize}
Using the equation and inequality in the right-hand side of \cref{eq:strideTwoFunctionSpace}, we verify that these four cases provide factorizations as in \cref{eq:poly_stride}.
\end{example}
\subsection{Higher dimensional convolutions}
\label{sec:higher_dimension}
We briefly discuss convolutions over inputs having a $D$-dimensional structure, $D>1$ ({\em e.g.}, raster images). The output of each layer is now a tensor of order $D$. Thus, we identify $\mathbb{R}^{d_l} \cong \mathbb{R}^{\Lambda_l}$ with $\Lambda_l = [d^1_l] \times \cdots \times [d^D_l]$. Each filter is also a tensor $w_l \in \mathbb{R}^{\lambda_l}$ with $\lambda_l = [k^1_l]\times \cdots \times[k^D_l]$. The convolution of $x \in \mathbb{R}^{\Lambda_{l-1}}$ with $w_l \in \mathbb{R}^{\lambda_l}$ is
\begin{equation}\label{eq:D_conv}
(\alpha_l(x))_i = \sum_{j\in \lambda_l} w_{l,j} \cdot x_{is_l+j}, \quad i \in \Lambda_l,
\end{equation}
where $s_l \in \mathbb{Z}_{>0}$ is the stride of the convolution (cf.\ \cref{eq:convolutionDefinition} for 1D convolutions). The map~\eqref{eq:D_conv} is linear in $x$ and can be represented as a \emph{convolutional tensor} $T_l \in \mathbb{R}^{\Lambda_{l} \times \Lambda_{l-1}}$ of order $2D$. The dimensions of the tensor $T_l$ satisfy~\eqref{eq:dimensionConstraintsToeplitz} along each dimension: $d_l^{h} = \frac{d_{l-1}^{h} - k_l^{h}}{s_l} + 1$.
\begin{example} \label{ex:tensorConv} \rm
Let $D=2$ and let us consider a filter $w$ of size $2 \times 2$ with stride one.
The convolution of an input $ x= \left[\begin{smallmatrix}* & * \\ * \circ& * \circ\\ \circ& \circ\end{smallmatrix}\right]$
of size $3 \times 2$ with the filter $w$
yields an output of size $2 \times 1$:
Its first resp.\ last entry is the inner product of $w$ with the first resp.\ last four entries of $x$ (marked by $*$ resp. $\circ$).
The associated convolutional tensor $T$ has size $2 \times 1 \times 3 \times 2$ and its entries (shown in two slices) are as follows:
\begin{align*}
T_{00::} = \begin{bmatrix}
w_{00} & w_{01} \\ w_{10} & w_{11} \\ 0 & 0
\end{bmatrix}
\quad \text{ and }
\quad
T_{10::} = \begin{bmatrix}
0 & 0 \\ w_{00} & w_{01} \\ w_{10} & w_{11}
\end{bmatrix}.
\end{align*}
\end{example}
In the following, we assume that all strides are one. Similar to the one-dimensional case, we can represent convolutions using polynomials. If $w \in \mathbb{R}^{\lambda_l}$, we define
\begin{equation}\label{eq:poly_tensor}
\pi(w) = \sum_{i \in \lambda_l} w_{i_1, \ldots, i_D} \, \mathsf{x}_1^{i_1} {\mathsf{y}_1^{k_l^1-1-i_1}} \cdots \mathsf{x}_D^{i_D} {\mathsf{y}_D^{k_l^D-1-i_D}} \,\, \in \,\, \mathbb{R}[\mathsf{x}_1,{\mathsf{y}_1},\ldots,\mathsf{x}_D, {\mathsf{y}_D}],
\end{equation}
where the weights of the filter are the coefficients of a multivariate polynomial {that is multi-homogeneous, that is, it has degree $k_l^j-1$ in each pair $\mathsf{x}_j,\mathsf{y}_j$.}
{For instance, in \Cref{ex:tensorConv} we have
$\pi(w) = w_{00} {\mathsf{y}_1\mathsf{y}_2} +w_{10} \mathsf{x}_1 {\mathsf{y}_2} + w_{01} {\mathsf{y}_1} \mathsf{x}_2 + w_{11} \mathsf{x}_1\mathsf{x}_2$.}
If $T$ is a convolutional tensor with filter $w$, we also write $\pi(T)$ for the polynomial in~\eqref{eq:poly_tensor}.
The following is a direct generalization of~\Cref{prop:deep_poly_multiplication}.
\begin{proposition} Consider a collection of convolutional tensors $T_1,\ldots, T_L$ with compatible sizes and stride one.
The tensor $\overline T = T_L \circ \cdots \circ T_1$ corresponding to the composition of the associated linear maps~\eqref{eq:D_conv} is a convolutional tensor and satisfies $\pi(\overline T) =\pi(T_L) \cdots \pi(T_1)$.
\end{proposition}
\begin{proof} The proof follows from the composition of two maps as in~\eqref{eq:D_conv} with stride one:
\begin{align*}
\begin{split}
(\alpha_2 \circ \alpha_1) (x)_i &= \sum_{j \in \lambda_2} w_{2,j} \cdot \alpha_1(x)_{i+j} = \sum_{j \in \lambda_2} w_{2,j} \cdot \sum_{\ell \in \lambda_1} w_{1,\ell} x_{(i + j)+\ell}\\
&= \sum_{j \in \lambda_2} \sum_{\ell \in \lambda_1} w_{2,j} w_{1,\ell} x_{i + (j + \ell)} = \sum_{m \in \mu} u_{m} \cdot x_{i + m},
\end{split}
\end{align*}
where $\mu = [k_1^1+k_2^1-1] \times \cdots \times [k_1^D + k_2^D -1]$ and $u \in \mathbb{R}^\mu$ is the filter with entries
$
u_m = \sum_{j \in \lambda_1, \ell \in \lambda_2, j + \ell = m} w_{2,j} w_{1, \ell}.
$
From this expression it follows that $\pi(u) = \pi(w_2) \pi(w_1)$.
\end{proof}
\begin{corollary}
\label{prop:tensorFunctionSpace}
Given an LCN architecture with $L$ layers, inputs of format $[d_0^1]\times \cdots[d_0^D]$, and stride one, we write $(k_i^1, \ldots, k_i^D)$ for the filter size in the $i$-th layer.
The function space can be identified with the family of polynomials $P \in \mathbb{R}[\mathsf{x}_1,{\mathsf{y}_1},\ldots,\mathsf{x}_D, {\mathsf{y}_D}]$ that can be factored as
$P = Q_L \cdots Q_1$,
{where $Q_i$ is homogeneous of degree $k_i^j-1$ in the variables $\mathsf{x}_j,\mathsf{y}_j$.}
\end{corollary}
The natural ambient space of the LCN function space is the set of polynomials $P\in \mathbb{R}[\mathsf{x}_1,{\mathsf{y}_1},\ldots,\mathsf{x}_D, {\mathsf{y}_D}]$
that are homogeneous of degree $k^j-1$ in $\mathsf{x}_j,\mathsf{y}_j$
where $k^j:= (k_1^j-1) + \cdots + (k_L^j-1)+1$.
We denote this vector space by
$\mathcal{M}_{(d_0,d_L),k,s}$, where
$d_0 = (d^1_0,\ldots, d^D_0)$, $ d_L = (d^1_L,\ldots, d^D_L)$, $k=(k^1,\ldots, k^D)$, $s =(1,\ldots, 1)$.
Using~\Cref{prop:tensorFunctionSpace}, we show that for higher-dimensional convolutions and at least two layers (and excluding trivial cases where filters have size one) the LCN function space is always a lower-dimensional semi-algebraic subset of $\mathcal{M}_{(d_0,d_L),k,s}$ and in particular is not filling.
\begin{corollary}
Consider an LCN architecture with stride one, $D > 1$, and $L > 1$, where no layer has trivial filter size $(1, \ldots, 1)$.
If there are distinct $j_1, j_2 \in \{ 1, \ldots, D \}$ such that $k^{j_1} > 1$ and $k^{j_2} > 1$, the architecture is not filling. In fact, the function space is a lower-dimensional semi-algebraic subset of $\mathcal{M}_{(d_0,d_L),k,s}$.
\end{corollary}
\begin{proof}
The statement is a consequence of the fact that multivariate polynomials are generically irreducible.
Since at least two layers have non-trivial filter sizes, the function space is contained in the set of polynomials that are reducible over $\mathbb{C}$, which is Zariski closed and thus lower-dimensional in the ambient space $\mathcal{M}_{(d_0,d_L),k,s}$.
As before, the fact that the function space is semi-algebraic follows from Tarski-Seidenberg.
For completeness, we give a formal argument showing that a generic polynomial $P$ in the ambient space $\mathcal{M}_{(d_0,d_L),k,s}$ is irreducible.
We may assume without loss of generality that $k^j > 1$ for all $1 \le j \le n$ and that $k^j = 1$ for all $n < j \le D$.
The latter means that the variables $\mathsf{x}_j, \mathsf{y}_j$ for $n<j\leq D$ do not appear in the polynomial $P$; see \cref{eq:poly_tensor}.
Hence, the zero locus of $P$ is a hypersurface in $(\mathbb{P}^1)^n$ of multidegree $(k^1-1, \ldots, k^n-1)$.
Given the assumptions of the architecture, note that $n \ge 2$.
To proceed, we consider the Segre-Veronese embedding $(\mathbb{P}^1)^n \hookrightarrow \mathbb{P}^{N-1}$ that maps $(\mathsf{x}_j:\mathsf{y}_j)_{j=1}^n$ to the $N$-tuple of all monomials that are homogeneous of degree $k^j-1$ in each pair $\mathsf{x}_j,\mathsf{y}_j$.
Under this map, the polynomial $P$ becomes linear in the new coordinates of $\mathbb{P}^{N-1}$.
In turn, the zero locus of $P$ is a hyperplane section of $(\mathbb{P}^1)^n$ embedded in $\mathbb{P}^{N-1}$.
Now, Bertini's Theorem \cite[Ch. II Thm. 8.18 + Ch. III Rem. 7.9.1]{hartshorne} states that, for smooth complex projective varieties of dimension at least two (in our case, $(\mathbb{P}^1)^n$ with $n\geq2$), generic hyperplane sections are irreducible.
This implies that a generic polynomial $P$ in $\mathcal{M}_{(d_0,d_L),k,s}$ is irreducible over $\mathbb{C}$.
\end{proof}
\begin{example}\rm
Consider an LCN architecture with $3 \times 3$ matrices as inputs ({\em i.e.}, $D=2$ and $d_0 = (3,3)$), $L=2$ layers, filter sizes $(k_1^1, k_1^2) = (2,2) = (k_2^1, k_2^2)$, and strides one.
The two convolutions are maps $\mathbb{R}^{3 \times 3} \to \mathbb{R}^{2 \times 2} \to \mathbb{R}^{1 \times 1}$.
Their composition is a convolution with filter size $3 \times 3$.
Hence, the natural ambient space of the LCN function space is the 9-dimensional vector space of polynomials in two {pairs of} variables such that the degree in each {pair is two},
and the function space is the subset of polynomials that factor into two {polynomials that are linear in each pair}.
The function space has codimension two in the ambient space.
A \texttt{Macaulay2}~\cite{m2} computation reveals that its Zariski closure has degree 10 and that its vanishing ideal is minimally generated by 39 homogeneous polynomials of degree~6.
\end{example}
\section{Optimization}
\label{sec:optimization-function}
We study the optimization of an objective function, or \emph{loss}, using LCNs with 1D convolutions and stride one.
A {loss} $\L(\theta)$ is a function of the filters $\theta=(w_1,\ldots,w_L)$ that can be written as a composition $\L = \ell \circ \mu$, where $\mu = \mu_{\bm{d},\bm{k},\bm{s}}$ is the polynomial map described in~\cref{eqn:Wbar} and $\ell$ is a smooth function in matrix space (so $\L$ only depends on the end-to-end convolutional matrix $\overline W = \mu(\theta)$).
This includes not only the square loss discussed further in \Cref{sec:squareloss}, but also classification losses (such as cross-entropy) or any other smooth function that depends only on the output of the LCN. A tuple of filters $\theta=(w_1,\ldots,w_L)$ is said to be \emph{critical} for the loss $\L$ if all partial derivatives of $\L$ vanish at $\theta$.
In~\Cref{sec:critPts} we analyze critical points of the loss function, explaining their relation to polynomials with repeated roots. In \Cref{sec:training-dynamics}, we discuss the training dynamics and invariants in parameter space for LCNs, which allow us to formulate local Riemannian gradient dynamics in function space.
\subsection{Critical points in parameter and function space}
\label{sec:critPts}
In this section, we discuss how critical points of an LCN
correspond to polynomials that have repeated roots.
We provide the proofs of the following proposition and theorems at the end of this section.
\begin{proposition}
\label{prop:critical-points}
Let $\L = \ell \circ \mu$ be a loss function for an LCN with stride one, where $\ell$ is a smooth and convex function on the space of convolutional matrices. If $\theta = (w_1,\ldots,w_L)$ is critical for $\L$, then one of the following holds:
\begin{enumerate}
\item $\theta$ is a \emph{global minimum} for $\L$, or
\item for some $i \neq j$, the homogeneous polynomials $\Pi(w_i)$ and $\Pi(w_j)$ have a common factor.
\end{enumerate}
\end{proposition}
Polynomials with multiple roots form in fact the boundaries of non-filling architectures (as shown in \Cref{prop:boundaryCondition}), however we will see that they arise as critical points more generally: \emph{Every} critical point $\theta$ for the loss of an LCN corresponds to a critical point $\mu(\theta)$ of $\ell$ restricted to one of the multiple root loci $\Delta_\lambda$, {\em i.e.}, $\mu(\theta) \in \mathrm{Crit}(\ell |_{\Pi^{-1}(\Delta_{\lambda})} )$, {even if $\mu(\theta)$ does not necessarily lie on the boundary of the function space.}
In other words, $\mu(\theta)$ is a smooth point of $\Pi^{-1}(\Delta_\lambda)$ and the differential $d \ell$ at $\mu(\theta)$ vanishes on the tangent space $T_{\mu(\theta)}\Pi^{-1}(\Delta_\lambda)$.
\begin{theorem}
\label{thm:criticalPointsAreOnMultipleRootLoci}
Let $\L = \ell \circ \mu$ be a loss function for an LCN with stride one. Let $\theta = (w_1,\ldots,w_L) \in \mathrm{Crit}(\L)$ and let $\overline w = \mu(\theta) \in \mathbb{R}^{k}$ be the end-to-end filter. Let $\lambda = (\lambda_1, \ldots, \lambda_r)$ be the partition of $k-1$ corresponding to the root structure of $\Pi(\overline w)$. Then $\overline w \in \mathrm{Crit}(\ell |_{\Pi^{-1}(\Delta_{\lambda})} )$.
\end{theorem}
To reverse \Cref{thm:criticalPointsAreOnMultipleRootLoci}, we characterize which critical points on the multiple root loci come from critical points of a given LCN.
\begin{theorem}
\label{thm:MultipleRootLocigivecriticalpoints}
Let $\mathcal L = \ell \circ \mu$ be a loss function for an LCN with stride one.
Let $\lambda$ be a partition of $k-1$ and let $\overline w \in \mathrm{Crit}(\ell |_{\Pi^{-1}(\Delta_{\lambda})} )$. If there exists $\theta = (w_1,\ldots,w_L)$ such that $\mu(\theta) = \overline w$ and each polynomial $\Pi(w_1),\ldots,\Pi(w_L)$ has no repeated roots, then $\theta \in \mathrm{Crit}(\L)$.
\end{theorem}
Moreover, if the loss $\ell$ in function space is sufficiently generic ({\em e.g.}, if the training data is sufficiently generic), then we do not expect there to be any further critical points of the LCN other than the ones described in \Cref{thm:MultipleRootLocigivecriticalpoints}.
We provide a formal argument for this assertion when $\ell=\ell_{X,Y}$ is the square loss (defined in \cref{eq:objective}) with generic training data.
\begin{theorem}
\label{thm:noDoubleRootsInFilters}
Let $X\in\mathbb{R}^{d_0\times N}$ and $Y\in \mathbb{R}^{d_L\times N}$ be generic data matrices with $N \geq d_0$ and let $\mathcal L = \ell_{X,Y} \circ \mu$ be the square loss for an LCN with stride one.
Then every $\theta = (w_1,\ldots,w_L) \in \mathrm{Crit}(\L)$ satisfies that each polynomial $\Pi(w_1),\ldots,\Pi(w_L)$ has no repeated roots.
\end{theorem}
To summarize, a critical point $\overline{w}$ of a generic loss $\ell$ on a multiple root locus $\Pi^{-1}(\Delta_\lambda)$ comes from a critical point of the LCN if and only if
$\Pi(\overline{w})$ can be factored according to the LCN architecture such that no factor has a double root.
Whether a polynomial with rrmp\, $(\rho\mid\gamma)$ can be factored according to the LCN architecture without double roots in any of the factors depends on a purely combinatorial property of $(\rho\mid\gamma)$ and the partition $(k_1-1, \ldots, k_L-1)$ of $k-1$ that is given by the architecture.
Indeed, the factorization of the polynomial is equivalent to the placing of $\rho_i$ balls of size $1$ and color $i$ (for $1 \leq i \leq r$) and $\gamma_j$ balls of size $2$ and color $r+j$ (for $1 \leq j \leq c$) into $L$ bins of sizes $k_1-1, \ldots, k_L-1$ such that no bin contains two balls of the same color. If this placement of balls into bins is possible, we say that $(\rho\mid\gamma)$ is \emph{compatible} with the LCN architecture.
All in all, the previous \Cref{thm:criticalPointsAreOnMultipleRootLoci,thm:MultipleRootLocigivecriticalpoints,thm:noDoubleRootsInFilters} show that the critical points of an LCN with a generic loss $\ell$ correspond exactly to the critical points on multiple root loci with an rrmp\, that is compatible with the architecture:
\begin{corollary}
\label{cor:rrmp}
Let $X\in\mathbb{R}^{d_0\times N}$ and $Y\in \mathbb{R}^{d_L\times N}$ be generic data matrices with $N \geq d_0$ and let $\mathcal L = \ell_{X,Y} \circ \mu$ be the square loss for an LCN with stride one.
Let $(\rho \mid \gamma)$ be such that $\lambda = \lambda_{\rho|\gamma} $ is a partition of $k-1$.
If $(\rho \mid \gamma)$ is compatible with the LCN architecture, then for every $\overline w \in \mathrm{Crit}(\ell |_{\Pi^{-1}(\Delta_{\lambda})} )$ with rrmp\, $(\rho \mid \gamma)$ there exists some $\theta \in \mu^{-1}(\overline{w}) \cap \mathrm{Crit}(\L)$.
Otherwise, every $\overline w \in \mathrm{Crit}(\ell |_{\Pi^{-1}(\Delta_{\lambda})} )$ with rrmp\, $(\rho \mid \gamma)$ satisfies $\mu^{-1}(\overline{w}) \cap \mathrm{Crit}(\L) = \emptyset$.
\end{corollary}
\begin{example}
\label{ex:partitionVsArchitecture} \rm
We compare the LCN architectures with $\bm k = (3,2,2)$ and $\bm k' = (4,2)$. We discussed in \Cref{ex:quartic} that the function space associated with these architectures is the same. The polynomial $\Pi(\overline{w}) = p_1 p_2^3$, where $p_i\in \mathbb{R}[\mathsf{x},\mathsf{y}]_1$, has rrmp\, $(13 \mid 0)$.
We can find a factorization compatible with the architecture $\bm k = (3,2,2)$ where each polynomial associated with a filter does not have any repeated roots: $\Pi(\overline{w}) = (p_1 p_2) \cdot p_2 \cdot p_2$.
In contrast, for $\bm k' = (4,2)$, any factorization of $\Pi(\overline w)$ according to the architecture will be such that the cubic polynomial associated with the filter of size $4$ will have at least one pair of repeated roots.
For both architectures, we now list all compatible $(\rho \mid \gamma)$ by providing a feasible factorization; the remaining ones are marked with a dash:
\noindent
\begin{tabular}{c|ccccccccc}
$\rho|\gamma$ & $1111|0$ & $112|0$ & $22|0$ & $13|0$ & $4|0$ & $11|1$ & $2|1$ & $0|2$ & $0|11$ \\ \hline
\small $(3,2,2)$ & \small $p_1p_2 {\cdot} p_3 {\cdot} p_4$ & \small $p_1p_2 {\cdot} p_3 {\cdot} p_3$ & \small $p_1p_2 {\cdot} p_1 {\cdot} p_2$ & \small $p_1p_2 {\cdot} p_2 {\cdot} p_2$ & $-$ & \small $q_1 {\cdot} p_1 {\cdot} p_2$ & \small $q_1 {\cdot} p_1 {\cdot} p_1$ & $-$ & $-$ \\
\small $(4,2)$ & \small $p_1p_2p_3{\cdot}p_4$ & \small $p_1p_2p_3{\cdot}p_3$ & $-$
& $-$ & $-$ & \small $p_1q_1{\cdot}p_2$ & \small $p_1q_1{\cdot}p_1$ & $-$ & $-$
\end{tabular}
\end{example}
To prove the theorems above, we make use of the composition rule of differentials:
\begin{equation*}
d \L = d \ell \circ d \mu_{\bm{d},\bm{k},\bm{s}}.
\end{equation*}
Even if the parameterization map $\mu$ is surjective ({\em i.e.}, the function space $\mathcal{M}=\mathcal M_{\bm d, \bm k, \bm s}$ is filling), the differential $d \mu$ (equivalently, the Jacobian matrix of $\mu$) might not have maximal rank everywhere. This implies that the parameterization may give rise to critical points for $\L$ in addition to the global minimum even if the function space $\mathcal M$ is a vector space and $\ell$ is convex.
Following~\cite{geometryLinearNets}, we call this kind of critical points \emph{spurious}.
In contrast, a critical point $\theta$ for $\L$ is $\emph{pure}$ if $\mu(\theta)$ is a critical point of the loss in function space, {\em i.e.}, of $\ell |_{\mathcal{M}}$.
Given an end-to-end filter $\overline w \in \mathbb{R}^k$ of an LCN, we describe the set of parameters $\theta$ such that $\mu(\theta) = \overline{w}$, {\em i.e.}, the fiber of the parameterization.
We say that two parameters $\theta = (w_1,\ldots, w_L)$ and $\theta'=(w_1',\ldots,w_L')$ are equivalent up to scalar factors if there exists $(\kappa_1,\ldots,\kappa_L) \in (\mathbb{R}^*)^L$ with $\prod_{i=1}^L \kappa_i = 1$ such that $w_i = \kappa_i w_i'$.
\begin{lemma} \label{lem:fiber}
Let $\overline w \in \mathbb{R}^k \setminus\lbrace 0 \rbrace$ be an end-to-end filter of an LCN with stride one. Then the fiber $\mu^{-1}(\overline w)$ consists of a finite set of scaling equivalence classes. Moreover, either all points in the same equivalence class are critical or none are.
\end{lemma}
\begin{proof} The fact that $\mu^{-1}(\overline w)$ consists of a finite set of scaling equivalence classes follows from the uniqueness of the decomposition of $\pi(\overline w) \in \mathbb{R}[\mathsf{x},\mathsf{y}]$ into irreducible factors: Each equivalence class corresponds to a way of aggregating factors of $\pi(\overline w)$ into polynomials of degrees $k_1-1,\ldots,k_L-1$, where $(k_1,\ldots,k_L)$ are the filter widths of the LCN. For the second claim, we use the fact that if $\theta$ and $\theta'$ are equivalent under rescaling, then the image of the differential $d\mu$ at these points is the same.
This will follow from our characterization of the image of $d \mu$ in~\eqref{eq:image_differential_ideal}. Hence, $d\mathcal L(\theta) = d \ell(\overline w) \circ d \mu(\theta) = 0$ if and only if \mbox{$d\mathcal L(\theta') = d \ell(\overline w) \circ d \mu(\theta') = 0$.}
\end{proof}
\begin{example}\rm
We discuss the parameters corresponding to particular points in the function space of the LCN with $\bm k = (2,2)$ from \Cref{example:minimal}.
Using the identification $\pi$ with polynomials,
a point in the interior of the function space is a quadratic polynomial with two distinct real roots: $P = p_1p_2$, where $p_1 = a\mathsf{x}+b\mathsf{y}$ and $p_2= c\mathsf{x}+d\mathsf{y}$ are not scalar multiples of each other.
The fiber $\mu^{-1}(P)$ of the parameterization has four connected one-dimensional components:
$\{ (\kappa p_1, \frac{1}{\kappa} p_2) : \kappa > 0 \}$,
$\{ (\kappa p_1, \frac{1}{\kappa} p_2) : \kappa < 0 \}$,
$\{ (\kappa p_2, \frac{1}{\kappa} p_1) : \kappa > 0 \}$, and
$\{ (\kappa p_2, \frac{1}{\kappa} p_1) : \kappa < 0 \}$.
\\\indent
The points on the boundary of the function space are exactly those $P=p_1p_2$ where $p_1$ and $p_2$ are linearly dependent. In other words, the parameters mapping to boundary points are those which satisfy $bc - ad = 0$.
We will see in \Cref{prop:rank_differential} that these are exactly the parameters where the Jacobian of the parameterization $\mu$ drops rank.
\\\indent
Hence, a non-zero point on the boundary of the function space is of the form $P = \pm p^2$ for some linear term $p \neq 0$.
Thus, the fiber $\mu^{-1}(P)$ has two connected one-dimensional components:
$\{ (\pm \kappa p, \frac{ 1}{\kappa} p) : \kappa > 0 \}$ and
$\{ (\pm \kappa p, \frac{1}{\kappa} p) : \kappa < 0 \}$.
Finally, the fiber $\mu^{-1}(0)$ over the origin is the union of the two planes
$\{ (p, 0) : p \in \mathbb{R}[\mathsf{x},\mathsf{y}]_1 \}$ and $\{ (0, p) : p \in \mathbb{R}[\mathsf{x},\mathsf{y}]_1 \}$.
\end{example}
\begin{remark}
\label{rem:fiberstructure}
In the case of fully-connected linear networks, spurious critical points are always \emph{saddles}. More precisely, for a fully-connected network, a loss function $\L = \ell \circ \mu$ where $\ell$ is a smooth convex function has non-global minima if and only if the restriction of $\ell$ on the function space has non-global minima~\cite[Proposition~10]{geometryLinearNets}.
In Example~\ref{ex:badMinimum}, we will see that this is \emph{not} true for LCNs, and it is possible for the loss function $\L$ to have non-global minima even if the function space is filling.
The intuitive geometric reason for this different behavior between fully-connected and convolutional linear networks lies in the very different structure of the fibers.
For fully-connected networks, as shown in~\cite[Proposition~9]{geometryLinearNets},
it is possible to perturb a spurious critical point within the fiber to obtain another point that is not a critical point for the loss.
This in turn implies that the original spurious critical point could not have been a minimum for the loss. In the case of LCNs, however, this argument does not apply, by \Cref{lem:fiber}.
Thus, for LCNs, it is possible for spurious critical points to be (non-global) local minima.
\end{remark}
\begin{example}
\label{ex:badMinimum}\rm
Let us consider an LCN with filter sizes $\bm k = (2,3)$. This is a filling architecture.
In terms of polynomials, its function space is the set of all cubic polynomials.
The two layers of the LCN factor such a polynomial into a linear and a quadratic term.
We fix the target polynomial $u = (\mathsf{x}+\mathsf{y})(\mathsf{x}^2 + 1/10 \, \mathsf{y}^2)$.
In the space of polynomials $\mathbb{R}[\mathsf{x},\mathsf{y}]_{3}$, we use the loss $\ell(p)$ that is the squared sum of the coefficients of $p-u$.
We will see that $\L = \ell \circ \mu$ has a non-global minimum, although $\ell$ clearly only has one local and global minimum.
\\\indent
Almost every cubic polynomial $p$ can be factored as $p = q_1 q_2$ where $q_1 = \mathsf{x}+a\mathsf{y}$ and $q_2 = b\mathsf{x}^2+c\mathsf{x}\mathsf{y}+d\mathsf{y}^2$.
We assume that $q_1$ is monic ({\em i.e.}, the coefficient for $\mathsf{x}$ is $1$) since there is otherwise a scaling ambiguity among $q_1$ and $q_2$ which does not affect the product $q_1q_2$ nor the property of critical points of $\L$. Our objective function $\L$ thus becomes
\begin{align*}
\L: \mathbb{R}^4 \longrightarrow \mathbb{R};\quad
(a,b,c,d) \longmapsto (b-1)^2 + (ab+c-1)^2 + (ac+d-\nicefrac{1}{10})^2 + (ad-\nicefrac{1}{10})^2.
\end{align*}
A direct computation reveals that $\L$ has 10 critical points over the complex numbers:
Three of them (1 real, 2 complex) correspond to factorizations of the target polynomial $u$, so that $\mathcal L=0$.
The other seven (3 real, 4 complex) are spurious critical points and yield factorizations of seven different cubic polynomials.
One of the latter three real spurious critical points is
\begin{align*}
a \approx 0.0578445483987, b \approx 1.0000187825172, c \approx 0.941829719725, d \approx 0.0511336556138.
\end{align*}
The Hessian of $\L$ at this point is positive definite, showing that it is indeed a local minimum.
\end{example}
We now investigate the existence of spurious critical points for LCNs by studying the rank of the differential of the parameterization map $\mu$.
\begin{proposition}\label{prop:rank_differential} Let $\mu$ be the parameterization map of an LCN with stride one. Given filter weights $\theta = (w_1,\ldots,w_L)$, we write $\overline w = \mu(\theta) \in \mathbb{R}^k$ for the end-to-end filter.
Using the identification $\Pi$, the image of the differential $d \mu$ at the point $\theta$ is the set of polynomials
\begin{equation}\label{eq:image_differential_ideal}
H(\theta) := \left\{h \, g \,\colon h \in \mathbb{R}[\mathsf{x},\mathsf{y}]_{k-1-\deg(g)}\right\} \subset \mathbb{R}[\mathsf{x},\mathsf{y}]_{k-1},
\end{equation}
where $g = {\rm gcd}\left(\frac{\Pi(\overline w)}{\Pi(w_1)},\ldots,\frac{\Pi(\overline w)}{\Pi(w_L)}\right) \in \mathbb{R}[\mathsf{x},\mathsf{y}].$
In particular, $d\mu (\theta)$ is surjective if and only if ${\rm gcd}(\Pi(w_i),\Pi(w_j)) = 1$ for every pair $i,j \in \{1,\ldots,L\}$ with $i \ne j$.
\end{proposition}
To prove this result, we make use of the following homogeneous version of Bezout's identity.
\begin{lemma}\label{lemma:ideal_gcd} Let $p_1,\ldots,p_n$ be homogeneous polynomials in $\mathbb{R}[\mathsf{x},\mathsf{y}]$ of arbitrary degree and let
$g = \mathrm{gcd}(p_1,\ldots,p_n)$ and $l={\rm lcm}(p_1,\ldots,p_n)$. For any $d \ge \deg(l)-1$ the following sets of homogeneous polynomials in $\mathbb{R}[\mathsf{x},\mathsf{y}]_d$ are equal:
\[
\begin{aligned}
I_d := \{\alpha_1 p_1 + \cdots + \alpha_n p_n \colon \alpha_i \in \mathbb{R}[\mathsf{x},\mathsf{y}]_{d - \deg(p_i)} \}, \quad J_d := \{h g \colon h \in \mathbb{R}[\mathsf{x},\mathsf{y}]_{d-\deg(g)}\}.\\
\end{aligned}
\]
\end{lemma}
\begin{proof}
If $g = l$ ({\em i.e.} \, $p_1=\cdots=p_n$), the assertion is clear. Hence, we assume that $\deg(g) < \deg(l)$.
We always have that $I_d \subset J_d$, since any algebraic combination of $p_1, \ldots, p_n$ is a multiple of $g$. For the converse, we observe that it is enough to prove that $J_{\deg(l)-1} \subset I_{\deg(l)-1}$ (if $hg$ belongs to the ideal generated by $p_1,\ldots,p_n$, then so do $\mathsf{x} hg$ and $\mathsf{y} hg$).
Moreover, it is sufficient to assume
that the linear form $\mathsf{y}$ is not a factor of $l$ (otherwise, we apply a general invertible linear transformation to the input variables, {\em e.g.}\, $\mathsf{x}'=\mathsf{x}, \mathsf{y}' = \mathsf{y} + t\mathsf{x}$ for appropriate $t$).
We can thus safely \emph{dehomogenize} and obtain univariate polynomials $\tilde p_i = p_i(\mathsf{x},1)$, $\tilde g = g(\mathsf{x},1)$, $\tilde l = l(\mathsf{x},1)$ in $\mathbb{R}[\mathsf{x}]$. Note that these polynomials have the same degrees as their homogeneous counterparts and also that $\tilde g = \mathrm{gcd}(\tilde p_1,\ldots,\tilde p_n)$ and $\tilde l = {\rm lcm}(\tilde p_1,\ldots,\tilde p_n)$.
By Bezout's identity, we can write $\tilde \beta_1 \tilde p_1 + \cdots + \tilde \beta_n \tilde p_n = \tilde g$ for some $\tilde \beta_i \in \mathbb{R}[\mathsf{x}]$.
Thus, for any $\tilde h \in \mathbb{R}[\mathsf{x}]$, we get a decomposition
$\tilde \alpha_1 \tilde p_1 + \cdots + \tilde \alpha_n \tilde p_n = \tilde h \tilde g$.
We now claim that we can assume $\deg(\tilde \alpha_i) < \deg(\tilde l) - \deg(\tilde p_i)$ whenever $\deg(\tilde h \tilde g) \le \deg(\tilde l)-1$.
Indeed, if this is not the case, then we write $\tilde \alpha_i = \tilde q_i \tilde p_i^* + \tilde r_i$ with $\tilde p_i^* = \tilde l / \tilde p_i$ and $\deg(\tilde r_i) < \deg(\tilde p_i^*) = \deg(\tilde l) - \deg(\tilde p_i)$, and we obtain $(\sum_{i=1}^n \tilde q_i) \tilde l + \tilde r_1 \tilde p_1 + \cdots + r_n \tilde p_n = \tilde h \tilde g$.
Since $\deg(\tilde h \tilde g) < \deg (\tilde l)$, we deduce that $\sum_{i=1}^n \tilde q_i = 0$, which yields a decomposition with the desired degree bounds.
We can now \emph{homogenize} this relation so that the total degree is $\deg(l) - 1 = \deg(\tilde l)-1$ to obtain $\alpha_1 p_1 + \cdots +\alpha_n p_n = hg$, where
\[
\begin{aligned}
&\alpha_i(\mathsf{x},\mathsf{y}) = \mathsf{y}^{\deg(\tilde l) - 1 -\deg(\tilde p_i)}\tilde \alpha_i\left(\frac{\mathsf{x}}{\mathsf{y}}\right), &&h(\mathsf{x}, \mathsf{y}) = \mathsf{y}^{\deg(\tilde l) - 1 -\deg(\tilde g)}\tilde h\left(\frac{\mathsf{x}}{\mathsf{y}}\right),\\
&p_i(\mathsf{x},\mathsf{y})=\mathsf{y}^{\deg(\tilde p_i)}\tilde p_i\left(\frac{\mathsf{x}}{\mathsf{y}}\right), &&g(\mathsf{x},\mathsf{y})=\mathsf{y}^{\deg(\tilde g)}\tilde g\left(\frac{\mathsf{x}}{\mathsf{y}}\right).
\end{aligned}
\]
Note that $p_i$ and $g$ are the original homogeneous polynomials from the statement. Since $\tilde h$ can be chosen arbitrarily in $\mathbb{R}[\mathsf{x}]_{\le \deg(\tilde l) - 1 - \deg(\tilde g)}$, we have that $h(\mathsf{x},\mathsf{y})$ can be an arbitrary polynomial in $\mathbb{R}[\mathsf{x},\mathsf{y}]_{\deg(l) - 1 - \deg(g)}$. We thus conclude that $J_{\deg(l)-1} \subset I_{\deg(l)-1}$.
\end{proof}
\begin{proof}[Proof of \Cref{prop:rank_differential}]
For notational simplicity, we omit writing the map $\Pi$ and identify $\mathbb{R}^k$ with $\mathbb{R}[\mathsf{x},\mathsf{y}]_{k-1}$. The differential of the parameterization can be written as
\begin{equation}\label{eq:differential_general}
d \mu(\theta)\colon \mathbb{R}^{k_1 + \cdots + k_L}
\longrightarrow \mathbb{R}^{k}; \quad
(\dot w_1, \ldots, \dot w_L)
\longmapsto \dot w_1\frac{\overline w}{w_1} + \cdots + \dot w_L\frac{\overline w}{w_L},
\end{equation}
where the product means polynomial multiplication and each $\dot w_i$ is simply a symbol for the coordinates of the tangent vector corresponding to the variable $w_i$. Since $\overline w$ is a common multiple of $\frac{\overline w}{w_1},\ldots, \frac{\overline w}{w_L}$, using~\Cref{lemma:ideal_gcd} we deduce that the image of~\eqref{eq:differential_general} is equal to $H(\theta)$ in~\eqref{eq:image_differential_ideal}. This shows that the differential is surjective if and only if $\mathrm{gcd}\left(\frac{\overline w}{w_1}, \ldots, \frac{\overline w}{w_L}\right)=1$.
The latter is equivalent to ${\rm gcd}(w_i,w_j) = 1$ for every pair $i \neq j$. Indeed, any common factor of $w_i, w_j$ is a factor of each $\frac{\overline w}{w_m}$; conversely, if $q$ is an irreducible common factor of $\frac{\overline w}{w_1}, \ldots, \frac{\overline w}{w_L}$, then $q | w_i$ for some $i \in \{1,\ldots,L\}$, and from $q | \frac{\overline w}{w_i}$ we deduce $q | w_j$ for some $j \ne i$.
\end{proof}
We are finally ready to prove the main theorems of this section.
As in the proof of \Cref{prop:rank_differential}, we omit writing the map $\pi$ and identify filters with polynomials.
\begin{proof} [Proof of \Cref{prop:critical-points}]
Let $\overline W = \mu(w_1,\ldots,w_L)$. If $(d \ell \circ d \mu)(\theta) = 0$ then either $d \ell (\overline{W}) = 0$ or $d \mu(\theta)$ must not have full rank. The assumption that $\ell$ is convex means that if $d\ell(\overline W) = 0$ then $\overline W$ is a global minimum for $\ell$ and thus $\theta$ is a global minimum for $\L$. Alternatively, if $d \mu(\theta)$ does not have full rank, then by \cref{prop:rank_differential} the second condition holds.
\end{proof}
\begin{proof}[Proof of \Cref{thm:criticalPointsAreOnMultipleRootLoci}]
If $\lambda = (1,\ldots,1)$, then $\overline w$ has $k-1$ distinct roots. By \Cref{prop:rank_differential}, the differential $d\mu(\theta)$ has full rank, {\em i.e.}, $\theta$ is a pure critical point. This in turn implies that $\overline w$ is a critical point for $\ell$ in $\Delta_{\lambda} = \mathbb{R}[x,y]_{k-1}$.
We next assume that $\lambda = (\lambda_1,\ldots,\lambda_r)$ is such that $\lambda_1 > 1$. Let $\overline w = q_1^{\lambda_1} \cdots q_r^{\lambda_r}$ be the decomposition of $\overline w$ into pairwise distinct complex linear forms $q_i \in \mathbb{C}[\mathsf{x}, \mathsf{y}]_1$. Note that if $\overline w = \prod_{i=1}^r q_i^{\lambda_i}$ is real, then so is $\prod_{i=1}^r q_i^{\lambda_i - 1}$. We claim that the (real) tangent space of $\Delta_{\lambda}$ at the smooth point $\overline w$ is given by
\begin{equation}\label{eq:tangent_multiple_root_locus}
T_{\overline w} \Delta_{\lambda} = \left\{h \prod_{i=1}^r q_i^{\lambda_i - 1} \,\, \colon \,\, h \in \mathbb{R}[\mathsf{x},\mathsf{y}]_r\right\} \subset \mathbb{R}[\mathsf{x},\mathsf{y}]_{k-1}.
\end{equation}
Indeed, let us assume without loss of generality that $q_{i_1},\ldots,q_{i_{r'}}$ are real linear forms, while $q_{j_1}, q_{j_1+1}, \ldots,\allowbreak q_{j_{r''}}, q_{j_{r''}+1}$ are such that each $(q_{j_l}, q_{j_l+1})$ is a conjugate pair of complex linear forms, with $r'+2r'' = r$ (in particular we must also have $\lambda_{j_l} = \lambda_{j_l+1}$ for all $l=1,\ldots,r''$). We write $t_{j_l} = q_{j_l} q_{j_l+1}$ for the irreducible real quadratic form that is the product of the pair $q_{j_l}, q_{j_l+1}$.
In a neighborhood of $\overline w$, the multiple root locus $\Delta_{\lambda}$ is the image of
\begin{equation}\label{eq:parameterization_multi_root_locus}
(q_{i_1},\ldots,q_{i_{r'}}, t_{j_1},\ldots,t_{j_{r''}}) \mapsto \prod_{m=1}^{r'} q_{i_m}^{\lambda_{i_m}} \prod_{l=1}^{r''} t_{j_l}^{\lambda_{j_l}},
\end{equation}
as $q_{i_m} \in \mathbb{R}[\mathsf{x},\mathsf{y}]_1$ and $t_{j_l} \in \mathbb{R}[\mathsf{x},\mathsf{y}]_2$ vary. Arguing as in the proof of \Cref{prop:rank_differential} (based on \Cref{lemma:ideal_gcd}), we have the image of the differential of~\eqref{eq:parameterization_multi_root_locus} is the set of homogeneous polynomials of degree $k-1$ that are multiples of $g' = \mathrm{gcd}\left(\frac{\overline w}{q_{i_1}}, \ldots, \frac{\overline w}{q_{i_{r'}}},\frac{\overline w}{t_{j_1}},\ldots,\frac{\overline w}{t_{j_{r''}}} \right)$. We finally observe that $g' = \prod_{i=1}^r q_i^{\lambda_i - 1}$.
Having shown~\eqref{eq:tangent_multiple_root_locus}, we recall that the image of the differential of the parameterization $\mu$ is $\mathrm{Im}(d\mu(\theta)) = \left\{h g \,\colon h \in \mathbb{R}[\mathsf{x},\mathsf{y}]_{k-1-\deg(g)}\right\} \subset \mathbb{R}[\mathsf{x},\mathsf{y}]_{k-1}$, where $g$ is as in~\Cref{prop:rank_differential}. To conclude the proof, we note that $T_{\overline w} \Delta_{\lambda} \subset \mathrm{Im}(d\mu(\theta))$ since $g \, | \, \prod_{i=1}^r q_i^{\lambda_i - 1}$. Indeed, the linear form $q_i$ can appear in the factorization of $g$ with multiplicity at most $\lambda_i - 1$, for if $q_i | w_j$ then $q_i^{\lambda_i}\not|\frac{\overline w}{w_j}$.
The fact that $T_{\overline w} \Delta_{\lambda} \subset \mathrm{Im}(d\mu(\theta))$ implies the statement of the theorem since $0=d\mathcal L(\theta) = d \ell(\overline w) \circ d \mu(\theta)$ implies $d \ell(\overline w)|_{T_{\overline w} \Delta_{\lambda}} = 0$, {\em i.e.}, $\overline w$ is a critical point of $\ell|_{\Delta_{\lambda}}$.
\end{proof}
\begin{proof}[Proof of \Cref{thm:MultipleRootLocigivecriticalpoints}]
Below we show the following claim for $\overline w$ being a smooth point of $\Delta_{\lambda}$:
For every $\theta \in \mu^{-1}(\overline w)$, it holds $\mathrm{Im}(d\mu(\theta)) = T_{w}(\Delta_\lambda)$ if and only if each $w_i$ has no repeated roots. Hence, under the assumptions in the theorem, $d \mathcal L(\theta) = d \ell(\overline w) \circ d \mu(\theta) = d \ell(\overline w)|_{T_{\overline w} \Delta_{\lambda}} = 0$.
To prove our claim, we use the fact that $T_{\overline w}(\Delta_\lambda) = \{h \prod_{i=1}^r q_i^{\lambda_i - 1} \,\, \colon \,\, h \in \mathbb{R}[\mathsf{x},\mathsf{y}]_r\}$ as in \cref{eq:tangent_multiple_root_locus} if $\overline w = \prod_{i=1}^r q_i^{\lambda_i}$ and that $\mathrm{Im}(d\mu(\theta)) = \left\{h \, g \,\colon h \in \mathbb{R}[\mathsf{x},\mathsf{y}]_{k-1-\deg(g)}\right\}$, where $g$ is the gcd defined in \Cref{prop:rank_differential}.
We observe that $w_i$ has no repeated roots if and only if $\prod_{i=1}^r q_i^{\lambda_i - 1} | (\nicefrac{\overline{w}}{w_i})$.
This happens for every $i=1, \ldots, L$ if and only if
$\prod_{i=1}^r q_i^{\lambda_i - 1} | g$.
Since $g|\prod_{i=1}^r q_i^{\lambda_i - 1}$ always holds, as seen in the proof of \Cref{thm:criticalPointsAreOnMultipleRootLoci}, the latter is equivalent to $\mathrm{Im}(d\mu(\theta)) = T_{\overline w}(\Delta_\lambda)$.
\end{proof}
\begin{proof}[Proof of \Cref{thm:noDoubleRootsInFilters}]
As we will see in \cref{eq:objective,eq:squareLossFilter,eq:filterLoss}, if $XX^\top$ has full rank, any square loss $\ell_{X,Y}(\overline w)$ is equivalent up to a constant additive term to a function of the form $\ell_{\Sigma,u}(\overline w) = \|u - \overline w\|^2_{\Sigma}$, where $u$ is a target filter and $\langle \cdot, \cdot \rangle_{\Sigma}$ is a non-degenerate inner product on $\mathbb{R}^{k}$.
{Here $k$ is the size of the filters $u$ and $\overline{w}$.}
We now show that for \emph{arbitrary} inner products $\langle \cdot, \cdot \rangle_{\Sigma}$ and \emph{generic} target filters $u$, any critical point $\theta = (w_1,\ldots,w_L)$ of $\mathcal L_{\Sigma,u} = \ell_{\Sigma,u} \circ \mu$ is such that each polynomial $w_1,\ldots,w_L$ has no repeated roots.\footnote{In the setting of~\Cref{ssec:squareloss}, we need to observe that the inner product only depends on the input data matrix $X \in \mathbb{R}^{d_0 \times N}$, and that as the output data matrix $Y \in \mathbb{R}^{d_L \times N}$ varies the target matrix $U = YX^\top(XX^\top)^{-1}$ can be an arbitrary matrix of size $d_L \times d_0$ due to the assumption $N \geq d_0$. In particular, for any non-degenerate inner product $\langle \cdot, \cdot \rangle_{\Sigma}$, all possible target filters $u$ arise.} Fixing the inner product arbitrarily, we consider for all $i=1,\ldots,L$ the set
\[
Z_i = \{(\theta, \overline w, u) \, \colon \, \theta \in \mathrm{Crit}(\mathcal L_{\Sigma,u}), \, \mu(\theta) = \overline w, \,\, \theta=(w_1,\ldots,w_L),\,\, w_i \text{ has a repeated root}\}.
\]
Note that this is an algebraic set. We claim that $\dim({\rm pr}_3(Z_i)) \le k-1$ holds for all $i=1,\ldots,L$, where ${\rm pr}_3$ denotes the projection on the third factor. This implies the statement of the theorem since it shows that the set of target filters $u$ such that any of the corresponding critical points has a filter with repeated roots has positive codimension in $\mathbb{R}^k$. To prove our claim on the dimension of $Z_i$, we further consider the sets
\[
Z_{i,\lambda} = \{(\theta, \overline w, u) \, \colon (\theta, \overline w, u) \in Z_i, \,\, \overline w \in \mathrm{Reg}( \Delta_{\lambda} )\},
\]
where $\lambda$ is a partition of $k-1$ and $\mathrm{Reg}( \Delta_{\lambda})$ denotes the locus of smooth points on $\Delta_\lambda$.
Clearly, $Z_i = \bigcup_{\lambda} Z_{i,\lambda}$.
As shown in the proof of~\Cref{thm:MultipleRootLocigivecriticalpoints}, if $\overline w$ is a smooth point of $\Delta_{\lambda}$ and $w_i$ has a repeated root, then $T_{\overline w} \Delta_{\lambda} \subsetneq {\rm Im}(d\mu(\theta))$.
The fiber $\mu^{-1}(\overline{w})$ consists of finitely many scaling equivalence classes (see \Cref{lem:fiber}) and all $\theta$ in the same equivalence class yield the same ${\rm Im}(d\mu(\theta))$.
We note that $\theta\in\mathrm{Crit}(\mathcal L_{\Sigma,u})$ means equivalently that $u - \overline w \perp_{\Sigma}{\rm Im}(d\mu(\theta))$, or in other words $u\in\overline{w}+{\rm Im}(d\mu(\theta))^{\perp_{\Sigma}}$.
We write $N_{\Sigma,\overline{w}}$ for the finite union of the affine spaces $\overline{w}+{\rm Im}(d\mu(\theta))^{\perp_{\Sigma}}$ where $\theta=(w_1,\ldots,w_L)\in\mu^{-1}(\overline{w})$ and some $w_i$ has a repeated root.
Hence, $u\in{\rm pr}_3(Z_{i,\lambda})$ if and only if there is some smooth point $\overline{w}$ on $\Delta_\lambda$ such that $u\in N_{\Sigma,\overline{w}}$.
Since $\dim(N_{\Sigma,\overline{w}})<\dim((T_{\overline w} \Delta_{\lambda})^{\perp_{\Sigma}})=k-\dim(\Delta_\lambda)$, we see that $\dim({\rm pr}_3(Z_{i,\lambda}))\leq \dim(\Delta_\lambda)+\max_{\overline{w}\in\mathrm{Reg}(\Delta_\lambda)}\dim(N_{\Sigma,\overline{w}})<k$.
This concludes the proof.
\end{proof}
\subsection{Training dynamics}
\label{sec:training-dynamics}
In this section, we discuss some matters related to the training dynamics of linear convolutional networks. We begin by proving the existence of dynamical invariants in parameter space. Similar results have been shown for linear networks~\cite{DBLP:journals/corr/abs-1910-05505} and ReLU networks~\cite{williams2019gradient}.
We restrict ourselves to networks with stride one.
\begin{proposition}\label{prop:invariants} Let $\L= \ell \circ \mu$ be a loss function for an LCN with stride one, where $\ell$ is a smooth function on the space of convolutional matrices. Let $\theta(t)$ be an integral curve in parameter space for the negative gradient field of $\L$, {\em i.e.}, $\dot
\theta(t) = - \nabla (\L(\theta(t)))$. If we write $\theta(t) = (w_1(t),\ldots,w_L(t))$, where $w_i$ are filter vectors, then the quantities $\delta_{ij}(t) = \|w_{i}(t)\|^2 - \|w_j(t)\|^2$ for $i,j \in \{1,\ldots,L\}$ remain constant for all $t \in \mathbb{R}$.
\end{proposition}
\begin{proof} For any $\kappa \in \mathbb{R} \setminus \{0\}$, let $\nu_{ij}(\theta,\kappa) = (w_1,\ldots,\kappa w_i, \ldots, \nicefrac{1}{\kappa} \, w_{j},\ldots, w_L)$. Because the map $\mu$ is multilinear in the filters, we have that $\L(\nu_{ij}(\theta,\kappa))$ does not depend on $\kappa$. This implies that $\frac{d}{d\kappa}\L(\nu_{ij}(\theta,\kappa)) \equiv 0$ and in particular
\[
0=\left[\frac{d}{d\kappa}\L(\nu_{ij}(\theta,\kappa))\right]_{\kappa=1} = \frac{d}{d\theta}\L(\theta) \cdot \left[\frac{d}{d\kappa}\nu_{ij}(\theta,\kappa)\right]_{\kappa=1} =\frac{d}{d\theta}\L(\theta) \cdot [0,\ldots,w_i, \ldots, -w_j,\ldots 0]^\top.
\]
We now observe that
\[
\frac{d}{dt}\delta_{ij}(t) = \frac{d}{d\theta} \delta_{ij} \cdot \frac{d}{dt} \theta(t) = -2 \, [0,\ldots,w_i,\ldots,-w_j,\ldots,0] \cdot \frac{d}{d\theta} \L(\theta) = 0.
\]
This shows that $\delta_{ij}$ remains constant for all $t$.
\end{proof}
As discussed in \Cref{lem:fiber}, each connected component of the fiber $\mu^{-1}(p)$ of a polynomial $p$ under the paramaterization map $\mu$ consists of the different rescalings of a fixed factorization of $p$.
\Cref{prop:invariants} can be used to eliminate the scaling ambiguity, assuming that the parameters (or the invariants) are known at initialization.
\begin{corollary}\label{cor:invariants_system}
Let $\mu$ be the parameterization map of an LCN with stride one.
Given filters $\theta = (w_1,\ldots,w_L)$, the set of parameter values $\theta'$ such that $\mu(\theta) = \mu(\theta')$ and $\delta_{i, i+1} = \delta'_{i,i+1}$ for all $i = 1, \ldots, L-1$ is finite, where $\delta_{ij}$ and $\delta'_{ij}$ are the invariants associated with $\theta$ and $\theta'$, respectively, as defined in \Cref{prop:invariants}.
\end{corollary}
\begin{proof} Let $q_i = \pi(w_i)$ be the polynomials associated with the filters $w_i$, and let $p = q_L \cdots q_1 = \pi(\mu(\theta))$.
As in Lemma~\ref{lem:fiber}, the factors $q_i$ are determined by $p$ up to a finite number of root permutations and also up to rescalings:
$
q_i' = \kappa_i q_i,$ where $\prod_{i=1}^L \kappa_i = 1.
$
It remains to show that there are only finitely many choices of the scale factors $\kappa_i$ such that
$\delta_{i+1,i} = \|\kappa_{i+1} q_{i+1}\|^2 - \|\kappa_i q_{i}\|^2$ for $i=1,\ldots,L-1$.
We can recover the possible $\kappa_i$ by solving a simple polynomial system. Setting $\beta_i = \|\kappa_i q_i\|^2 = \kappa_i^2 \| q_i \|^2$ and $\delta_i = \delta_{i+1,i}$, we have that
\begin{equation*}
\beta_{i+1} - \beta_i = \delta_i, \,\, (i=1,\ldots,L-1), \quad
\prod_{i=1}^L \beta_i = H,
\end{equation*}
where $H = \prod_{i=1}^L \|q_i\|^2$.
In particular, all $\beta_i$ are determined by $\beta_1$: $\beta_{i+1} = \beta_1 + \delta_1 + \cdots + \delta_i$ for all $i=1,\ldots,L-1$.
Hence, we obtain one polynomial equation of degree $L$ in $\beta_1$, namely
\begin{equation}\label{eq:1eq_1var}
\beta_1 \cdot (\beta_1 + \delta_1) \cdots (\beta_1 + \delta_1 + \cdots + \delta_{L-1}) - H = 0.
\end{equation}
There are finitely many real solutions for $\beta_1$ in~\cref{eq:1eq_1var} such that $\beta_1, \ldots,\beta_1 + \delta_1 + \cdots +\delta_{L-1}$ are all positive. Each such solution gives finitely many choices for the $\kappa_i$.
\end{proof}
The previous statement can be used to study the impact of the initialization (within the fiber) on the training dynamics. More precisely, we can compute the Riemannian metric that locally characterizes the gradient dynamics in function space. In general if $\L = \ell \circ \mu$ and $\theta(t)$ is an
integral curve for the negative gradient field of $\L$, then setting $\overline W(t) = \mu(\theta(t))$
we have
\begin{equation}\label{eq:function_dynamics}
\begin{aligned}
\frac{d }{dt}\overline W(t)
&= - K(\theta(t)) \,\cdot \, \nabla \ell (\overline W(t)),
\end{aligned}
\end{equation}
where $K(\theta) = \operatorname{Jac}_{\mu}(\theta)\operatorname{Jac}_{\mu}(\theta)^\top$ and $\operatorname{Jac}_{\mu}(\theta)$ is the Jacobian matrix of $\mu$ at $\theta$.
The matrix $K(\theta)$ corresponds to the \emph{neural tangent kernel} \cite{NEURIPS2018_5a4be1fa} popular in the analysis of neural networks and can be viewed as a ``pre-conditioning'' matrix that modifies the gradient of $\ell$ in function space. If $K(\theta)$ could be written as a function of $\overline W = \mu(\theta)$, then we could view~\cref{eq:function_dynamics} as a Riemannian gradient flow for the metric tensor $K^{-1}$ (this assumes $K$ is not singular, which is generically the case; note that $K$ is always positive semi-definite). In general, however, the matrix $K(\theta)$ depends on the parameter $\theta$, and different parameters within the fiber of $\overline W$ will yield different matrices. In particular, the trajectories in function space are susceptible to the relative scalings of the filters at initialization. Using the dynamical invariants, however, we can bypass this problem by restricting ourselves to the parameters that are dynamically reachable, defining a local metric in function space associated with the initialization.
We show this idea with a simple example.
\begin{example} \rm
Let us consider the factorization of a cubic polynomial into a linear and a quadratic factor. We write $C = A \cdot B$, where
\[
C = c_3 \mathsf{x}^3 + c_2 \mathsf{x}^2\mathsf{y} + c_1 \mathsf{x}\mathsf{y}^2 + c_0\mathsf{y}^3,\quad
A = a_2 \mathsf{x}^2 + a_1 \mathsf{x}\mathsf{y} + a_0\mathsf{y}^2,\quad
B = b_1 \mathsf{x} + b_0\mathsf{y}
\]
The Jacobian matrix of the parameterization map $\mu:(a_2,a_1,a_0,b_1,b_0) \mapsto (c_3,c_2,c_1,c_0) = (a_2 b_1, a_2 b_0 + a_1 b_1, a_1 b_0 + a_0 b_1, a_0b_0)$ is
\begin{equation*}
\operatorname{Jac}_{\mu}(a_2,a_1,a_0,b_1,b_0) =
\begin{bmatrix}
b_1 & 0 & 0 & a_2 & 0\\
b_0 & b_1 & 0 & a_1 & a_2\\
0 & b_0 & b_1 & a_0 & a_1\\
0 & 0 & b_0 & 0 & a_0\\
\end{bmatrix} \in \mathbb{R}^{4 \times 5}.
\end{equation*}
The matrix $K(\theta) = \operatorname{Jac}_{\mu} \operatorname{Jac}_{\mu}^\top \in \mathbb{R}^{4 \times 4}$ can be written as $K(\theta)=K_B+K_A$, where
\begin{equation*}
K_B =
\begin{bmatrix}
b_{1}^{2} & b_{0} b_{1} & 0 & 0 \\
b_{0} b_{1} & b_{0}^{2} + b_{1}^{2} & b_{0} b_{1} & 0 \\
0 & b_{0} b_{1} & b_{0}^{2} + b_{1}^{2} & b_{0} b_{1} \\
0 & 0 & b_{0} b_{1} & b_{0}^{2}
\end{bmatrix},\;\;
K_A =
\begin{bmatrix}
a_{2}^{2} & a_{1} a_{2} & a_{0} a_{2} & 0 \\
a_{1} a_{2} & a_{1}^{2} + a_{2}^{2} & a_{0} a_{1} + a_{1} a_{2} & a_{0} a_{2} \\
a_{0} a_{2} & a_{0} a_{1} + a_{1} a_{2} & a_{0}^{2} + a_{1}^{2} & a_{0} a_{1} \\
0 & a_{0} a_{2} & a_{0} a_{1} & a_{0}^{2}
\end{bmatrix}.
\end{equation*}
This matrix clearly depends on the coefficients of $A$ and $B$; however, we can recover it given only $C$ and the value of $\delta = \|A\|^2 - \|B\|^2$, which is fixed from the initialization. To resolve the scale ambiguity, we replace $A,B$ with $\gamma A, 1/\gamma B$ with $\gamma \in \mathbb{R}\setminus \{0\}$ (here $A$ and $B$ can be fixed arbitrarily so that $A \cdot B = C$). We see that $K(\theta)$ becomes a one-dimensional (quadratic) family of matrices $K(\gamma) = (1/\gamma)^2 K_B + \gamma^2 K_A$. Using $\|\gamma A\|^2 - \|\nicefrac{1}{\gamma} \, B\|^2 = \delta$ and writing $\beta = \|\nicefrac{1}{\gamma} \, B\|^2$ and $H = \|A\|^2\|B\|^2$ we deduce that
\[
\beta^2 + \delta \beta - H = 0, \quad \beta > 0, \beta+\delta > 0, H > 0,
\]
which is the same as~\cref{eq:1eq_1var}.
This yields $\beta = \frac{-\delta + \sqrt{\delta^2 + 4H}}{2}$ and $\gamma^2 = \frac{2 \|B\|^2}{-\delta + \sqrt{\delta^2 + 4H}}$.
\end{example}
\section{Optimization of the square loss}
\label{sec:squareloss}
In this section, we investigate the minimization of quadratic losses and the Euclidean distance degrees for LCNs.
\subsection{The square loss}
\label{ssec:squareloss}
Given data $X\in\mathbb{R}^{d_0\times N}$ and $Y\in \mathbb{R}^{d_L\times N}$ (where $N$ is the number of data samples), and assuming that
$XX^\top$ has full rank,
we write the quadratic loss as:
\begin{equation}
\begin{aligned}
\ell(W) =\ell_{X,Y}(W) &= \|W X - Y\|^2 =
\langle Y, Y \rangle - 2 \langle WX,Y \rangle + \langle WX, WX \rangle \\
&= const. + 2 \langle W (XX^\top), YX^\top(XX^\top)^{-1} \rangle + \langle WX, WX \rangle\\
&= const. + 2\langle W, U \rangle_{XX^\top} + \langle W, W \rangle_{XX^\top} \\
&= const. + \|W - U\|^2_{XX^\top},
\end{aligned}
\label{eq:objective}
\end{equation}
where $U$ is the unconstrained optimizer of this quadratic problem, over the set of all matrices:
\begin{equation*}
U = \operatorname{argmin} \limits_{V\in\mathbb{R}^{d_L\times d_0}}\|V X - Y\|^2 = YX^\top (XX^\top)^{-1}.
\end{equation*}
As discussed in the previous sections, an LCN corresponds to a semi-algebraic set $\mathcal{M}$ of convolutional matrices $\overline{W}$.
By~\cref{eq:objective}, the optimization of the square error $\ell(\overline W)$ over the function space $\mathcal{M}$ corresponds to a constrained linear regression problem, which is equivalent to solving
$$
\min_{\overline W\in \mathcal{M}} \|U-\overline W\|_{XX^\top}^2.
\label{eq:optimization_model}
$$
If $U\in\mathcal{M}$, then the solution is $U$.
If $U\not\in \mathcal{M}$, we can optimize the objective over the boundary of the function space. We also may assume that the matrix $U$ is convolutional of the same format as the matrices in $\mathcal{M}$, for if not, we can consider the orthogonal projection (with respect to $\| \cdot \|_{XX^\top}$) to the appropriate linear space of convolutional matrices.
\begin{remark}
\label{rem:populationLoss}
The same expressions as above hold for population losses, by replacing $\frac{1}{N} XX^\top = \frac{1}{N}\sum_{i=1}^N x_i x_i^\top$ with $\mathbb E_{x \sim \mathcal D} [xx^\top]$ where $\mathcal D$ is the distribution over the input data.
\end{remark}
If we assume that $U$ is a convolutional matrix with filter $u$, then we can also write the square loss in terms of the filter $\overline w$ of $\overline{W}$:
\begin{equation}
\label{eq:squareLossFilter}
\begin{aligned}
\ell( \overline w) = \|\overline{W} - U\|_{XX^\top}^2
&= \mathrm{tr} [(\overline W - U) XX^\top (\overline W - U)^\top]\\
&= \sum_{m=0}^{d_L-1} [(\overline W - U)_{m,:} XX^\top (\overline W - U)_{m,:}^\top],
\end{aligned}
\end{equation}
where we write $M_{m,:}$ for the $m$-th row of a given matrix $M$.
We denote by $k$ and $s$ the filter size and stride of the convolutional matrices in the function space $\mathcal{M}$.
Since the $m$-th row of $\overline{W}-U$ is obtained by shifting $\overline{w}-u$ by $sm$ positions to the right, we see that
\begin{equation}
\label{eq:filterLoss}
\begin{aligned}
\ell(\overline{w})= (\overline w - u)^\top \tau({XX^\top}) (\overline w - u),
\end{aligned}
\end{equation}
where $\tau: \mathbb{R}^{d_0 \times d_0} \rightarrow \mathbb{R}^{k \times k}$ is an operator that sums all shifts of rows and columns of a
matrix:
\begin{equation}
\label{eq:tauoperator}
\tau(M)_{ij} = \sum_{m=0}^{d_L-1} M_{i+sm,j+sm}.
\end{equation}
\begin{remark}\label{rmk:circulant_tau}
If we consider circulant matrices instead of convolutional matrices, then all matrices appearing in~\cref{eq:squareLossFilter} would be square matrices of size $d_0 \times d_0$ and the trace would sum from $m=0$ to $d_0-1$.
The sum in~\cref{eq:tauoperator} would change accordingly and would be cyclic.
\end{remark}
\begin{lemma}\label{lem:euclidean_dist}
Let $XX^\top$ be a multiple of the identity matrix. For any stride $s$, the square loss $\ell$ on the filters is equivalent to the Euclidean distance between $\overline w$ and $u$, {\em i.e.},
\mbox{$
\ell(\overline w) \propto \|\overline w-u\|^2.
$}
\end{lemma}
\begin{proof}
If $XX^\top$ is a multiple of the identity,
then $\tau(XX^\top) = d_L XX^\top$ for convolutional matrices
resp. $\tau(XX^\top) = d_0 XX^\top$ for circulant matrices.
Now the claim follows from \cref{eq:filterLoss}.
\end{proof}
\begin{remark}
\label{rem:diagonalCirculant}
The lemma above holds both for convolutional and circulant matrices.
If we restrict ourselves to the circulant setting, then the statement holds more generally if $XX^\top$ is an arbitrary diagonal matrix as long as $s$ and $d_0$ are coprime. This follows from \Cref{rmk:circulant_tau}. Indeed,
if $s$ and $d_0$ are coprime, $\tau(XX^\top)_{ii}$ sums over all entries of the diagonal matrix $XX^\top$ and so $\tau(XX^\top)$ is a multiple of the identity matrix.
\end{remark}
We can apply \Cref{lem:euclidean_dist} also for population losses (see \Cref{rem:populationLoss}) where the data distribution $\mathcal D$ is such that the covariance matrix $\mathbb E_{x \sim \mathcal D}[xx^\top]$ is a multiple of the identity ({\em e.g.}, Gaussian data), and conclude that using the square loss on the LCN is equivalent to using a standard Euclidean distance at the level of filters. If we make use of circulant convolutions and restrict ourselves to a stride that is coprime to $d_0$, then this is also the case if the covariance matrix is simply diagonal.
Next we give a detailed example illustrating the critical points of the quadratic loss for an LCN and how these depend on the particular training data. In the next subsection we will discuss the number of critical points in more generality.
\begin{example}\rm
\label{ex:tinyexample}
We return to \Cref{example:minimal} to consider minimizing the square loss over the set of functions $\mathcal{M} = \{ W = (A,B,C) \colon B^2 \geq 4AC \}$ represented by the two layer LCN with architecture specified by $\bm d = (3,2,1)$, $\bm k = (2,2)$, $\bm s = (1,1)$.
Given data matrices $X,Y$, we are minimizing an objective $\ell(W)=\|WX-Y\|^2$ which is convex over the space of all matrices $W\in\mathbb{R}^{1\times 3}$.
Hence, if the unconstrained minimizer is outside $\mathcal{M}$ and $XX^\top$ is full rank, then the set of minimizers over $\mathcal{M}$ will be at the boundary of $\mathcal{M}$.
The situation is illustrated in \Cref{fig:zwei}.
The boundary $\partial \mathcal{M} = \{W=(A,B,C) \colon B^2 = 4AC\}$ can be parametrized by $(A,C) \mapsto W = (A,\pm 2\sqrt{AC},C)$ with $(A,C)$ either both non-negative or both non-positive.
In the following we consider the case $A,C\geq0$ (the case $A,C\leq 0$ is similar).
The gradient of the objective function with respect to $A,C$ is
\begin{align*}
\nabla_{A,C} \ell(A,C) =&
(-2YX^\top + 2WXX^\top) \nabla_{A,C}W^\top .
\end{align*}
The first order criticality condition $\nabla_{A,C}\ell(A,C)=0$ means that
\begin{align}
- YX^\top + W XX^\top
= \lambda WJ,
\label{eq:ex1storder}
\end{align}
where we introduce the matrix $J=\left[\begin{smallmatrix} 0&0&1\\
0&-1/2&0\\
1&0&0 \end{smallmatrix}\right]$, so that $W J = [C, \mp\sqrt{AC}, A]$ is a basis of the left null space of $\nabla_{A,C}W^\top$.
We can solve \cref{eq:ex1storder} by writing
\begin{equation}
W= YX^\top (XX^\top - \lambda J)^{-1},
\label{eq:tiny-critical}
\end{equation}
and plugging this into the defining equation of the boundary, $B^2=4AC$, which gives us an equation in $\lambda$.
As discussed in~\cite[Section~4.4]{1027497}, computing a square distance from a point to a quadric surface results in a degree $6$ polynomial in $\lambda$.
Since our boundary equation is homogeneous, we can clear the determinant in Cramer's rule for the matrix inverse and use simply the adjugate matrix, which results in a degree $4$ polynomial in $\lambda$:
\begin{equation}
(YX^\top \operatorname{adj}(XX^\top-\lambda J)_{:,2})^2 - 4(YX^\top \operatorname{adj}(XX^\top-\lambda J)_{:,1})(YX^\top \operatorname{adj}(XX^\top-\lambda J)_{:,3}).
\label{eq:ex-lambda}
\end{equation}
Solving this, we typically obtain $4$ values of $\lambda$, which when inserted in \cref{eq:tiny-critical} give us the critical points in the boundary of the function space.
\begin{figure}
\centering
\begin{tikzpicture}
\node at (0,0) {\includegraphics[width=6cm]{M144118-foo3.png}};
\node at (7.5,0) {
\scalebox{.8}{
\begin{tikzpicture}[x=1cm,y=1cm]
\definecolor{col1}{rgb}{0, 0.4470, 0.7410}
\definecolor{col2}{rgb}{0.8500, 0.3250, 0.0980}
\definecolor{col3}{rgb}{0.8361, 0.6246, 0.1125}
\definecolor{col4}{rgb}{0.4940, 0.1840, 0.5560}
\definecolor{col5}{rgb}{0.4660, 0.6740, 0.1880}
\hypersetup{linkcolor=col5}
\node at (0,0) {\includegraphics[clip=true, trim=0cm 0cm 0cm 1cm, width=10cm]{M144118-discriminant_tiny_example_slice.pdf}};
\node at (3.5,2.5) {\textcolor{col1}{$u_2^2 - 4 u_1 u_3$
}};
\node at (3.2,1) {\textcolor{col2}{$u_2^2 - u_1 u_3$}};
\node at (.3,3) {\textcolor{col3}{$u_2^2$}};
\node at (-2.7,0.2) {\textcolor{col4}{$(u_1-u_3)^2$}};
\node at (1.3,1) {\textcolor{col5}{\cref{eq:discriminant_long_tiny_example}}};
\node
fill=white, fill opacity=1, text opacity=1, inner sep = 1] (N1) at (3.8,.125) {2 local min and 2 saddles};
\node
fill=white, fill opacity=1, text opacity=1, inner sep = 1] (N2) at (3.8,-.4375) {1 min and 1 saddle};
\node
fill=white, fill opacity=1, text opacity=1, inner sep = 1] (N3) at (3.8,-1) {2 local min};
\draw (.4,.125) -- (N1.west);
\draw (1.25,-.4375) -- (N2.west);
\draw (2.4,-1) -- (N3.west);
\end{tikzpicture}
}
};
\end{tikzpicture}
\caption{
Left:
Boundary of the function space in $\mathbb{R}^{1\times 3}$ from \Cref{example:minimal}.
The complement of the function space is inside the elliptic double cone with axis $(1,0,1)$.
The boundary is colored by values of the square loss for random data $X,Y$ with $U=YX^\top(XX^\top)^{-1}$ shown as a red dot.
For this data there are two local minima and two saddles shown as green dots.
Right: Discrimination of datasets with $XX^\top=I$ in terms of $U=(u_1,u_2,u_3)=YX^\top$ by the types of critical points of the square loss as discussed in \Cref{ex:tinyexample}.
Shown is the slice $u_3=-u_1+1$.
For points with $u_2^2-4u_1u_3<0$ outside the function space,
we have cases with two local minima and two saddles, one minimum and one saddle, and two local minima.
}
\label{fig:zwei}
\label{fig:discriminant_tiny_example}
\end{figure}
Now we take a closer look at the special case with $XX^\top=I$. In this case, the unconstrained optimal solution is $U = (u_1, u_2, u_3)= YX^\top$ and \cref{eq:ex-lambda} simplifies to
\begin{multline}
(u_2^2-u_1u_3)\lambda^4+(-u_1^2-4u_1u_3-u_3^2)\lambda^3+(-4u_1^2-2u_2^2-5u_1u_3-4u_3^2)\lambda^2\\
+(-4u_1^2-4u_1u_3-4u_3^2)\lambda + (u_2^2-4u_1u_3).
\label{eq:ex-lambda1}
\end{multline}
We can classify the points $(u_1,u_2,u_3)$ for which the problem has different numbers of solutions.
When $u_2^2-u_1u_3\neq 0$, \cref{eq:ex-lambda1} is a degree 4 polynomial in $\lambda$ which generally has 4 or 2 real solutions (the global minimum corresponds to a real root).
The cases are separated by the discriminant of \cref{eq:ex-lambda1} with factors
$u_2^2$, $(u_1-u_3)^2$, $(u_1+u_3)^2$,
and
\begin{multline}
32u_1^6 + 435u_1^4u_2^2 + 384u_1^2u_2^4+256u_2^6 - 240u_1^5u_3 - 960u_1^3u_2^2u_3 - 960u_1u_2^4u_3 + 696u_1^4u_3^2\\
+ 1098u_1^2u_2^2u_3^2 + 384u_2^4u_3^2 - 980u_1^3u_3^3 - 960u_1u_2^2u_3^3 + 696u_1^2u_3^4 + 435u_2^2u_3^4 - 240u_1u_3^5 + 32u_3^6.
\label{eq:discriminant_long_tiny_example}
\end{multline}
We visualize the situation in Figure~\ref{fig:discriminant_tiny_example}.
The figure shows the slice $u_3 = -u_1+1$.
Since the discriminant is homogeneous, each point in the slice is representative of a line through the origin in $\mathbb{R}^3$. The zeros of the factor $(u_1+u_3)^2$ are not visible over this slice.
We see that the discriminant separates $12$ generic cases outside of the function space ({\em i.e.}, inside the blue ellipse, where $u_2^2 - 4u_1u_3<0$).
By evaluating individual instances in each of these cases,
we find that the points inside the green curvy diamond (caustic) have two local minima and two saddles,
the points between the green curvy diamond and the red circle have one minimum and one saddle,
and the points between the red circle and the blue ellipse have two local minima (whereby we observed that only one of them satisfies $A,C\geq 0$).
\end{example}
\subsection{Euclidean distance degrees}
\label{sec:EDdegree}
We now discuss the number of critical points of the optimization problem in function space using the square loss~\cref{eq:objective}.
This problem is given by
\begin{align*}
\min_{W \in \mathcal{M}} \| U-W \|^2_{XX^\top},
\end{align*}
where $\mathcal{M}$ is the function space of an LCN and $U$ and $X$ are fixed data matrices.
We focus on one-dimensional convolutions with stride one.
We first assume that $XX^\top$ is the identity matrix ({\em i.e.}, we are interested in minimizing the standard Euclidean distance of the data $U$ to the function space $\mathcal{M}$) and comment on other choices of $X$ later.
By \Cref{thm:criticalPointsAreOnMultipleRootLoci}, each critical point in the parameter space of the LCN corresponds in function space to a critical point of the Euclidean distance from the data $U$ to a multiple root locus $\Delta_\lambda$.
For almost every choice of $U$ the number of \emph{complex} critical points of minimizing the Euclidean distance from $U$ to $\Delta_\lambda$ is the same.
This number is known as the \emph{Euclidean distance degree} (\emph{ED degree} for short) of $\Delta_\lambda$~\cite{draisma2016euclidean}.
Hence, if we assume that the data $U$ is generic, \Cref{cor:rrmp} shows that
the number of $\mu(\theta)$, where $\theta$ is a critical point in the parameter space of the given LCN, is upper bounded by the sum of the ED degrees of all multiple root loci $\Delta_\lambda \subset \mathbb{R}[\mathsf{x},\mathsf{y}]_{k-1}$, where $k$ is the filter size in the function space $\mathcal{M}$ and $\lambda=\lambda_{\rho|\gamma}$ comes from an rrmp\, $(\rho \mid \gamma)$ that is compatible with the LCN architecture.
\begin{example}
\label{ex:upperEDdegreeBound}
For generic data $U$, \Cref{ex:partitionVsArchitecture} shows the following:
The number of $\mu(\theta)$, where $\theta$ is a critical point of an LCN with architecture $\bm k = (3,2,2)$ resp.\ $\bm {k'} = (4,2)$ is at most the sum of the ED degrees of
$\Delta_{(1,1,1,1)}, \Delta_{(1,1,2)},\Delta_{(2,2)}, \Delta_{(1,3)}$ resp.\ $\Delta_{(1,1,1,1)}, \Delta_{(1,1,2)}$.
\end{example}
The state-of-the-art on ED degrees of multiple root loci is presented in \cite{lee2016duality}.
For instance, the ED degree of the discriminant hypersurface of degree-$(k-1)$ polynomials is $3(k-1)-2$~\cite[Example 5.12]{draisma2016euclidean}.
More generally, the ED degree of the multiple root locus $\Delta_{(\alpha, 1, \ldots, 1)}$ (with $\alpha >1$ and $k-1-\alpha$ ones) is $(2\alpha-1)(k-1)-2(\alpha-1)^2$~\cite[Theorem 5.1]{lee2016duality}.
Many more ED degrees are computed in \cite[Table 1]{lee2016duality}, but a closed formula for general partitions is not known.
\begin{example}\rm
\label{ex:EDdegreesGeneric}
$(k=5)$
The discriminant of quartic polynomials has ED degree $10$.
Its iterated singular loci are the following
multiple root loci: $\Delta_{(2,2)}$ with ED degree $13$, $\Delta_{(1,3)}$ with ED degree $12$, and $\Delta_{(4)}$ with ED degree $10$.
Hence, the upper bounds on the number of $\mu(\theta)$ for critical points $\theta$ given in \Cref{ex:upperEDdegreeBound} are $1+10+13+12=36$ resp.\ $1+10=11$.
\end{example}
\begin{remark}\label{rmk:bombieri}
The ED degrees discussed in this section are often referred to as \emph{generic ED degrees} in the algebraic literature.
The \emph{special ED degree} refers to the following slightly different norm to be optimized: for two polynomials $P = \sum_{i=0}^{k-1} w_i \mathsf{x}^i$ and $Q = \sum_{i=0}^{k-1} u_i \mathsf{x}^i$,
\begin{equation}\label{eq:bombieri}
\langle P, Q \rangle_{B}:=\sum_{i=0}^{k-1} \frac{i!(k-i-1)!}{(k-1)!} u_i w_i.
\end{equation}
The letter ``$B$'' stands for \emph{Bombieri norm}. For example, if $P$ and $Q$ are quadratic polynomials, then $\langle P,Q \rangle_B = u_2 w_2 + \frac{1}{2}u_1 w_1 + u_0 w_0$.
The Euclidean inner product that arises for LCNs if $XX^\top$ is the identity is instead simply $\langle P,Q \rangle_2 = u_2 w_2 + u_1 w_1 + u_0 w_0$. This slight difference has an impact on the ED degrees. Indeed, the ED degree of the discriminant hypersurface $\{(w_0,w_1,w_2) \colon w_1^2 = w_0 w_2\}$ is $2$ for the Bombieri norm, and $4$ for the standard Euclidean norm (see Example~\ref{ex:tinyexample} for a computation of the four critical points).
More generally, if we consider the multiple root loci for polynomials of degree $k-1$, then the special ED degree of the discriminant hypersurface is $k-1$~\cite[Corollary 8.7]{draisma2016euclidean} (whereas the generic ED degree is $3(k-1) - 2$, as noted above). The special ED degree of the multiple root locus $\Delta_{(\alpha, 1, \ldots, 1)}$ (with $\alpha >1$ and $k-1-\alpha$ ones) is $k-1$, independently of $\alpha$~\cite[Theorem 5.1]{lee2016duality}.
\end{remark}
\begin{example} \label{ex:specialEDdegree} \rm $(k=5)$
The generic ED degrees of the multiple root loci of quartic polynomials are listed in Example~\ref{ex:EDdegreesGeneric}.
The special ED degrees are smaller:
for $\Delta_{(1,1,2)}$ it is $4$,
for $\Delta_{(2,2)}$ it is $7$,
for $\Delta_{(1,3)}$ it is $4$,
and for $\Delta_{(4)}$ it is $4$.
\end{example}
\begin{remark}
The above discussion shows that the Bombieri norm can be thought of as a ``special'' norm that leads to lower ED degrees and thus makes the optimization landscape simpler. Perhaps counterintuitively, the standard Euclidean distance is instead \emph{generic}, in the sense that almost all pairings $\langle\cdot,\cdot\rangle$ will lead to the same ED degrees for multiple-root loci.
In other words, for \emph{almost all} data matrices the ED degrees of the multiple root loci would be the same as assuming $XX^\top$ is the identity.
\end{remark}
A natural question in this setting is if there exists a ``special'' data distribution that induces the Bombieri norm~\cref{eq:bombieri} in polynomial space. For circulant convolutions this is not possible. Indeed, for any matrix $M$, the matrix $\tau(M)$ when $d_0 = d_L$ will always have constant diagonal entries by the same reasoning as in Remark~\ref{rem:diagonalCirculant}.
In the convolutional (Toeplitz) case, however, such special data distributions can exist, as the following simple example demonstrates.
\begin{example}\rm
We consider a single convolutional layer with $k=3$, $d_0 = 4$ and $d_1 = 2$. If the data covariance matrix is such that
\[
XX^\top = \left[\begin{smallmatrix}
3/4 & & & \\
& 1/4 & & \\
& & 1/4 & \\
& & & 3/4\\
\end{smallmatrix}\right],
\]
then the induced metric in the space of quadratic polynomials is indeed the Bombieri norm $\langle P,Q \rangle_B = u_2 w_2 + \frac{1}{2} u_1 w_1 + u_0 w_0$. This example suggests that non-standard normalizations of the data might lead to more favorable optimization landscapes for convolutional networks.
\end{example}
\begin{example}\rm
We return to Example~\ref{ex:tinyexample}, an LCN with $\mathcal{M} = \{W=(A,B,C)\colon B^2\geq4AC\}$, and investigate the structure of critical points of the square loss over the boundary $\partial \mathcal{M} = \{W=(A,B,C)\colon B^2=4AC\}$ when $XX^\top$ is compatible with the Bombieri norm,
\begin{equation}
XX^\top = \left[\begin{smallmatrix}
1& & \\
& 1/2& \\
& & 1
\end{smallmatrix}\right].
\label{eq:tiny-example-bombieri-norm}
\end{equation}
A geometric interpretation of this setting is that the balls $\{W \colon \|W-U\|_{XX^\top}\leq c\}$ are ellipsoids with the same $1\times 2\times 1$ aspect ratio as the boundary of the function space.
As before, we write $U=(u_1,u_2,u_3) = YX^\top (XX^\top)^{-1}$ for the unconstrained least squares solution.
With $XX^\top$ taking the form \cref{eq:tiny-example-bombieri-norm}, equation \cref{eq:ex-lambda} for the case $A,C\geq0$ becomes
\begin{multline}
(-1) (\lambda + 1) (\lambda + 1) ((u_1 u_3 - 4 u_2^2)\lambda^2 + (u_1^2 + 8 u_2^2 + u_3^2)\lambda + (u_1 u_3 - 4 u_2^2)).
\label{eq:ex-lambda-Bombieri}
\end{multline}
This quartic polynomial in $\lambda$ has a repeated root $\lambda = -1$ for any values of $U$ and hence its discriminant vanishes identically.
The distinction in terms of $U$ is determined by the last factor alone.
We discuss the zeros of the individual factors in turn.
\begin{itemize}[leftmargin=*]
\item
Consider the factor $(\lambda+1)$ of \cref{eq:ex-lambda-Bombieri}.
The root $\lambda=-1$ plugged into \cref{eq:ex1storder}
gives equation $W(XX^\top+J)=YX^\top$ for $W$.
Since the matrix $XX^\top+J = \left[\begin{smallmatrix}
1 &0 &1\\
0&0&0 \\
1&0&1
\end{smallmatrix}\right]$ is not of full rank, for generic $YX^\top$ there is no solution and this case can be ignored.
Solutions only exist in the non-generic case that $YX^\top = c \begin{bmatrix} 1& 0& 1\end{bmatrix}$, namely $\{W = (A, B, C) \colon A+C = c\}$, which is a hyperplane.
The intersection of this hyperplane with the boundary of the function space gives us the set of critical points $\{(A,B,C)\colon A+C=c, B^2=4AC\}$. For the assumed case $A,C\geq0$, this is empty over the reals for $c<0$, a single point for $c=0$, and an ellipse for $c>0$. All critical points have the same Bombieri distance $c$ from $U=\begin{bmatrix} c& 0& c\end{bmatrix}$.
\item
Consider now the last factor of \cref{eq:ex-lambda-Bombieri}, which is quadratic in $\lambda$.
Its discriminant has factors $(u_1 + u_3)^2$ and $u_1^2 - 2 u_1 u_3 + 16 u_2^2 + u_3^2$.
The former has zero set $\{u_1=-u_3\}$,
which also appeared in the case $XX^\top=I$ and is inside the function space.
The latter has real zero set $\{u_2=0, u_1=u_3\}$, which is the center line of our boundary cone $\partial \mathcal{M}$ and can be interpreted as a degeneration of the caustic that appeared in the case $XX^\top=I$. It also corresponds to the non-generic case discussed above.
\end{itemize}
Hence, when $XX^\top$ takes the form \cref{eq:tiny-example-bombieri-norm} compatible with the Bombieri norm, there is a single generic case for the datasets.
We find that generic datasets $U=(u_1,u_2,u_3)$ outside the function space have two critical points over the boundary of the function space: 1 global minimum and 1 saddle.
This is in contrast with the more complicated case distinction in Example~\ref{ex:tinyexample} and Figure~\ref{fig:discriminant_tiny_example} when $XX^\top=I$, where some generic datasets led to 4 critical points.
On the other hand, for the Bombieri norm there are special datasets, with $\{u_2=0, u_1=u_3 >0\}$, with infinitely many critical points, namely entire ellipses of global minima.
\end{example}
\section{Examples and numerical evaluation}
\label{sec:experiments}
We present computational experiments on LCNs with 1D convolutions with stride one.
In \cref{prop:critical-points} we saw that critical parameters are global minimizers of the objective or are contained in the discriminant hypersurface. In fact, in \Cref{thm:criticalPointsAreOnMultipleRootLoci} we observed that any critical parameter corresponds to a critical point over the locus with the same root structure.
We discuss the relevant root loci and evaluate their appearance in gradient optimization using different LCN architectures.
As observed in the previous sections, the order of the filters does not affect the function space nor the gradient dynamics. For this reason, we consider architectures with filter widths $k_1 \ge \cdots \ge k_L$.
We perform our experiments with the five non-filling architectures from \Cref{ex:boundaries}.
The left hand side of \Cref{fig:zwei} visualizes the function space $\mathcal M_{(2,2)}$.
The function spaces representing cubic and quartic polynomials are depicted in \Cref{fig:cubic_quartic_discriminants}:
On the left, $\mathcal M_{(2,2,2)}$ is the ``inside'' of the ``cuspidal wedge.''
On the right,
the complement of $\mathcal M_{(3,2,2)} = \mathcal M_{(4,2)}$ are the points above the blue ``bowl.''
These points correspond to one of the two convex cones described in \cref{prop:twoConvexCones}.
The other cone is not visible since the figure fixes the highest degree coefficient to $1$.
Moreover, $\mathcal M_{(2,2,2,2)}$
is the set inside the triangular region.
\begin{figure}[htb]
\centering
\begin{tabular}{cc}
Cubic & Quartic\\
\begin{tikzpicture}
\node at (0,0) {\includegraphics[width=5cm,clip=true,trim=0cm 0cm 0cm 1cm]{M144118-Cubic_discriminant3.pdf}};
\node at (-.75,2) {\textcolor{black}{$\Delta=0$}};
\node at (0.2,-1.5) {\textcolor{red}{\small $\delta_1=\delta_2=\delta_3=0$}};
\node at (2.2,0.78) {{\small $\mathcal{M}_{(2,2,2)}$}};
\end{tikzpicture}
&
\begin{tikzpicture}
\node at (0,0) {\includegraphics[width=5cm,clip=true,trim=0cm 0cm 0cm 1cm]{M144118-Quartic_discriminant3.pdf}};
\node at (1.1,0.1) {\textcolor{black}{$\delta=0$}};
\node at (.8,-1) {\textcolor{red}{$\delta'=0$}};
\node[fill=white, fill opacity=1, text opacity=1, inner sep = 0]
at (-2,1.2) {{\small $\mathcal{M}_{(3,2,2)}$}};
\node[fill=white, fill opacity=1, text opacity=1, inner sep = 0]
(M2222) at (-2,-1.5) {{\small $\mathcal{M}_{(2,2,2,2)}$}};
\draw (-1,0) -- (M2222.north);
\end{tikzpicture}
\end{tabular}
\caption{
Left: The surface $\Delta=0$ of coefficients $(a,b,c,d)$, $a\neq0$, for which a cubic polynomial $a\mathsf{x}^3+b \mathsf{x}^2\mathsf{y}+c\mathsf{x}\mathsf{y}^2+d\mathsf{y}^3$ has a double root. The coefficients with triple roots are shown in red.
The coefficients with $\Delta>0$
(the interior of $\mathcal{M}_{(2,2,2)}$)
are on the side with smaller values of $c$.
%
Right: The surfaces $\delta=0$ and $\delta'=0$ (as in \Cref{ex:rrmps}) of coefficients $(p,q,r)$, that separate the real root multiplicities of a depressed quartic polynomial $\mathsf{x}^4+p\mathsf{x}^2\mathsf{y}^2+q\mathsf{x}\mathsf{y}^3+r\mathsf{y}^4$.
The positive values of $\delta$ and $\delta'$ are on the sides that do not contain the labels.
The complement of $\mathcal{M}_{(3,2,2)}$ is above the blue bowl.
The interior of $\mathcal{M}_{(2,2,2,2)}$ is within the triangular blue region.
}
\label{fig:cubic_quartic_discriminants}
\end{figure}
\subsection{Critical points reached by gradient descent}
For each of the architectures listed above, we generated $10,000$ random data sets $X,Y$ each consisting of $10$ i.i.d.\ input-output pairs from a standard normal distribution, with inputs in $\mathbb{R}^{d_0}$ and outputs in $\mathbb{R}$. Since $d_L=1$, we have from \Cref{prop:nonzerodiagsExtension} that $d_0 = k$.
For each dataset and architecture, we initialized the parameters at random from a standard normal distribution and ran gradient descent optimization of the square loss $\L$
with fixed step size $0.01$ and stopping criterion $\|\nabla \L\|^2 \leq 10^{-14}$.
Instances which did not reach the stopping criterion within a maximum of $15,000$ iterations were discarded.
Since the final coefficients are only approximately critical points, they rarely correspond to polynomials with exactly repeated roots. To assess the rrmp's of solutions, we compute the roots of all factor filters and consider a root $r$ as real if $|r-\operatorname{real}(r)| \leq \text{\texttt{tol}} \cdot |r|$, and consider two roots $r,r'$ as same if $|r-r'| \leq \text{\texttt{tol}}\cdot \max\{|r|,|r'|\}$ with $\text{\texttt{tol}} = 10^{-4}$. We use a similar procedure for the initializations and for the target.
This classification scheme is relatively robust, but is not exact.
The results are shown in Table~\ref{table:solution_types}.
\begin{itemize}[leftmargin=*]
\item
For the architecture $\bm k =(2,2)$, about 66\% of the datasets correspond to targets with rrmp\, {$11|0$} inside the function space, and gradient descent finds a solution with nearly zero training error.
About 34\% of the datasets correspond to targets with rrmp\, {$0|1$} outside the function space and gradient descent finds solutions with rrmp\, {$2|0$} at its boundary.
\item
For $\bm k=(2,2,2)$,
about 25\% of the datasets have rrmp\, {$111|0$} inside the function space, and for most of these, gradient descent finds a solution of the same rrmp\, with nearly zero loss.
About 75\% of the datasets have rrmp\, {$1|1$} outside the function space. Of these, about 70\% led to a solution with rrmp\, {$12|0$}, {\em i.e.}, a smooth point of the discriminant hypersurface, and about 30\% led to a solution with rrmp\, {$3|0$}, {\em i.e.}, a point on the red curve in Figure~\ref{fig:cubic_quartic_discriminants}.
\item
For the architectures $\bm k = (4,2)$ or $(3,2,2)$, which have the same function space, the different parametrizations lead to different optimization behaviors.
First, the initialization procedure leads to different distributions in function space,
with $\bm k = (4,2)$ being more frequently initialized with rrmp\, {$11|1$}
than $\bm k = (3,2,2)$.
For targets with rrmp\, {$0|11$}, outside the function space,
the solutions often correspond to zeros of the discriminant and are critical points in parameter space but they do not lie on the Euclidean boundary of $\mathcal{M}_{\bm k}$.
Hence, they are spurious critical points.
The effect is more pronounced in the deeper architecture, which has a more intricate parametrization. The optimization landscapes are illustrated in Figure~\ref{fig:landscapes}.
For targets with rrmp\, {$11|1$}, inside the function space, a good fraction of the optimization instances with $\bm k = (4,2)$ converged to sub-optimal solutions with rrmp\, {$112|0$} or {$2|1$}, and for $\bm k = (3,2,2)$ an even larger fraction converged to sub-optimal solutions with rrmp\, {$112|0$}, {$22|0$}, {$13|0$}, or {$2|1$}.
This is in line with our theoretical discussion, by which $\bm k = (3,2,2)$, but not $\bm k = (4,2)$, can have critical points in parameter space with rrmp's {$22|0$} and {$13|0$} for generic data (see \Cref{cor:rrmp} and \Cref{ex:partitionVsArchitecture}).
\item For $\bm k = (2,2,2,2)$, in contrast to the previous architectures, a fraction of the targets outside the function space ({\em i.e.}, with rrmp\, {$11|1$} or {$0|11$}) led to a solution with rrmp\, {$4|0$}, {\em i.e.}, the codimension 3 point at the origin in Figure~\ref{fig:cubic_quartic_discriminants}, which was predicted by \Cref{cor:rrmp}.
\end{itemize}
\begin{table}
\caption{Rrmp's of solutions returned by gradient descent optimization of the square loss
in parameter space for random datasets and initializations.
The table omits a small fraction of targets and initializations numerically classified in rrmp's of positive codimension. }
\label{table:solution_types}
\centering
\small
\begin{tabular}{ll}
$\mathcal{M}_{(2,2)}$&
\begin{tabular}[t]{l S[table-format=2.2] | l S[table-format=3.3] l}
& & \multicolumn{3}{c}{initialization {$11|0$} \;\; 100\% } \\
target & \% & solution & \%& mean loss\\
\hline
{$11|0$\phantom{11}} & 65.8 & {$11|0$} & 100 & 1.15e-15\\
\hline
{$0|1$} & 34.2 & {$2|0$} & 100 & 0.185 \\
\bottomrule
\end{tabular}\\
$\mathcal{M}_{(2,2,2)}$&
\begin{tabular}[t]{l S[table-format=2.2] | l S[table-format=3.3] l}
&&\multicolumn{3}{c}{initialization {$111|0$} \;\; 100\% }\\
target & \% & solution & \%& mean loss\\
\hline
{$111|0$\phantom{1}}& 24.9 & {$111|0$} & 99.7 & 2.32e-15\\
&& {$12|0$} & 0.051 & 0.945\\
&& {$3|0$} & 0.205 & 1.19\\
\hline
{$1|1$} & 75 & {$12|0$} & 69.8 & 0.224 \\
&& {$3|0$} & 30.2 & 0.709 \\
\bottomrule
\end{tabular}\\
$\mathcal{M}_{(4,2)}$&
\begin{tabular}[t]{l S[table-format=2.2] | l S[table-format=3.3] l|l S[table-format=3.3] l}
&&\multicolumn{3}{c|}{initialization {$1111|0$}\;\; 24.6\%} & \multicolumn{3}{c}{initialization {$11|1$}\;\; 75.3\%}\\
target & \% & solution & \%& mean loss & solution & \%& mean loss\\
\hline
{$1111|0$}& 5.28 & {$1111|0$} & 100 & 3.04e-15 & {$1111|0$} & 100 & 3.1e-15 \\
\hline
{$11|1$} & 72.6 & {$112|0$} & 15.5 & 0.228 & {$112|0$} & 12.4 & 0.193 \\
&& {$11|1$} & 83.2 & 1.94e-15 & {$11|1$} & 86.7 & 1.84e-15 \\
&& {$2|1$} & 1.36 & 0.54 & {$2|1$} & 0.886 & 0.397 \\
\hline
{$0|11$} & 22.1 & {$112|0$} & 7.85 & 0.347 & {$112|0$} & 4.81 & 0.35 \\
&& {$2|1$} & 92.2 & 0.231 & {$2|1$} & 95.2 & 0.229 \\
\bottomrule
\end{tabular}\\
$\mathcal{M}_{(3,2,2)}$&
\begin{tabular}[t]{l S[table-format=2.2] | l S[table-format=3.3] l| l S[table-format=3.3] l}
&& \multicolumn{3}{c|}{initialization {$1111|0$}\;\; 65.1\%} & \multicolumn{3}{c}{initialization {$11|1$}\;\; 34.9\%} \\
target & \% & solution & \%& mean loss & solution & \%& mean loss\\
\hline
{$1111|0$}& 4.82 & {$1111|0$} & 99.6 & 4.68e-15 & {$1111|0$} & 100 & 4e-15\\
&& {$13|0$} & 0.429 & 0.71 &&&\\
\hline
{$11|1$}& 72.9 & {$112|0$} & 27.1 & 0.221 & {$112|0$} & 21.8 & 0.207\\
&& {$22|0$} & 1.28 & 0.992 & {$22|0$} & 0.663& 0.879\\
&& {$13|0$} & 25.8 & 0.798 & {$13|0$} & 15.6 & 0.71\\
&& {$11|1$} & 45.5 & 1.78e-15 & {$11|1$} & 61.7 & 1.7e-15\\
&& {$2|1$} & 0.381 & 0.446 & {$2|1$} & 0.306& 0.418 \\
\hline
{$0|11$}& 22.3 & {$112|0$} & 11.2 & 0.374 & {$112|0$} & 9.09 & 0.365 \\
&& {$22|0$} & 25.5 & 0.855 & {$22|0$} & 17.5 & 0.882 \\
&& {$13|0$} & 7.1 & 0.895 & {$13|0$} & 4.21 & 0.937 \\
&& {$2|1$} & 56.2 & 0.224 & {$2|1$} & 69.2 & 0.208 \\
\bottomrule
\end{tabular}\\
$\mathcal{M}_{(2,2,2,2)}$&
\begin{tabular}[t]{l S[table-format=2.2] | l S[table-format=3.3] l}
&& \multicolumn{3}{c}{initialization {$1111|0$}\;\; 99.8\%}\\
target & \% & solution & \%& mean loss \\
\hline
{$1111|0$}& 4.79 & {$1111|0$} & 99.7 & 5.41e-15 \\
&& {$13|0$} & 0.274 & 0.297 \\
\hline
{$11|1$}& 72.7 & {$112|0$} & 33.3 & 0.259 \\
&& {$22|0$} & 2.24 & 0.908 \\
&& {$13|0$} & 52.1 & 0.774 \\
&& {$4|0$} & 12.4 & 1.53 \\
\hline
{$0|11$}& 22.5 & {$112|0$} & 13.2 & 0.411 \\
&& {$22|0$} & 40.7 & 0.779 \\
&& {$13|0$} & 16.1 & 1.02 \\
&& {$4|0$} & 30 & 1.48 \\
\bottomrule
\end{tabular}
\end{tabular}
\end{table}
\begin{figure}
\centering
\setlength\tabcolsep{0pt}
\begin{tabular}{ccccc}
$\mathcal{M}_{(2,2)}$&$\mathcal{M}_{(2,2,2)}$&$\mathcal{M}_{(4,2)}$&
$\mathcal{M}_{(3,2,2)}$&$\mathcal{M}_{(2,2,2,2)}$\\
\includegraphics[clip=true,trim=0cm 3.5cm 1cm 3.6cm,width=.2\textwidth]{M144118-Loss_Net2__2rep4.pdf}&
\includegraphics[clip=true,trim=0cm 3.5cm 1cm 3.6cm,width=.2\textwidth]{M144118-Loss_Net2__2__2rep3.pdf}&
\includegraphics[clip=true,trim=0cm 3.5cm 1cm 3.6cm,width=.2\textwidth]{M144118-Loss_Net2__4rep5.pdf}&
\includegraphics[clip=true,trim=0cm 3.5cm 1cm 3.6cm,width=.2\textwidth]{M144118-Loss_Net3__2__2rep1.pdf}&
\includegraphics[clip=true,trim=0cm 3.5cm 1cm 3.6cm,width=.2\textwidth]{M144118-Loss_Net2__2__2__2rep3.pdf}
\end{tabular}
\caption{The square loss $(w_1,\ldots, w_L) \mapsto \|U - \mu(w_1,\ldots, w_L)\|^2_{XX^\top}$, $U = YX^\top (XX^\top)^{-1}$, for random data $X,Y$, plotted in log-scale over random 2D affine subsets of parameter values.
The bottom shows the discriminant at the corresponding filters, $(w_1,\ldots, w_L)\mapsto |\Delta(\pi(\mu(w_1,\ldots, w_L)))|$.
As can be seen, deeper networks tend to have a more intricate loss surface and critical parameters often correspond to functions where the discriminant vanishes.
}
\label{fig:landscapes}
\end{figure}
\subsection{Euclidean vs.\ Bombieri norm}
In this experiment we generated 500 random targets $U$ for each architecture.
For each of them, we ran gradient descent with 50 random initializations for the loss with Euclidean and Bombieri norms.
For each target we recorded the list of converged solutions, using the settings of the previous experiment.
To determine the number of distinct solutions we considered two filters $w_i$ and $w_{i'}$ as equal if on each entry
$|w_{ik}-w_{i'k}| \leq \text{\texttt{tol}}\cdot \max\{\max\{|w_{jk}|\colon j\},1\}$ with $\text{\texttt{tol}}=10^{-4}$.
The results are shown in Table~\ref{table:compare_metric}.
In line with our theoretical discussion in \Cref{sec:EDdegree} we observed fewer distinct solutions for the Bombieri norm than for the generic norm.
\begin{table}
\caption{Percentages of targets for which gradient descent with different initializations converged to a given number of distinct solutions.
For the Bombieri norm $B$ we usually observe a smaller number of distinct solutions compared with the generic Euclidean norm $I$. }
\label{table:compare_metric}
\centering
\small
\begin{tabular}{c c| S[table-format=3.1] S[table-format=2.2] S[table-format=2.2] S[table-format=1.3]}
&& \multicolumn{4}{c}{number of distinct solutions}\\
$\bm k$ & metric & 1 & 2 & 3 & 4\\
\hline
$(2,2)$ & $I$ & 95.8 & 4.2 & \\
& $B$ & 100 & & \\
\hline
$(2,2,2)$ & $I$ & 84.7 & 14.5 & 0.84 \\
& $B$ & 95.6 & 1.1 & 3.29 \\
\hline
$(4,2)$ & $I$ & 66.3 & 33.7 & \\
& $B$ & 73.8 & 26.2 & \\
\hline
$(3,2,2)$ & $I$ & 22.5 & 58.8 & 17.7 & 1.01 \\
& $B$ & 32 & 48.3 & 19.5 & 0.209\\
\hline
$(2,2,2,2)$ & $I$ & 70.7 & 26.5 & 2.61 & 0.217 \\
& $B$ & 84.8 & 8.21 & 6.97 & \\
\hline
\end{tabular}
\end{table}
\subsection{Symbolic computations}
We illustrate several of our results with a final example. We consider linearly generated data $\mathcal D_N = \{(x_i,y_i) \in \mathbb{R}^5 \times \mathbb{R} \colon i=1,\ldots,N, y_i = u^\top x_i\}$ where
$u=[2,0,5,0,2]^\top \in \mathbb{R}^5$
is a ``target filter'' and $x_i$ is sampled from a standard Gaussian distribution. We study the square loss regression problem for different 1D LCN architectures with compatible dimensions, as described in~\Cref{sec:squareloss}. In light of~\Cref{lem:euclidean_dist}, if the data is normalized so that the empirical covariance $XX^\top$ is the identity, or if $N \rightarrow \infty$, then this corresponds to an objective function $\mathcal L(\theta) = \ell(\mu(\theta)) =\|\mu(\theta) - u\|^2$, where $\overline w = \mu(\theta)$ is the end-to-end filter and $\|\cdot \|$ is the standard Euclidean norm.
The possible LCN architectures with stride one, $d_0=5, d_L=1$ and {non-trivial filter sizes $1 < k_i < 5$ are} $\bm k = (4,2)$, $(3,3)$, $(3,2,2)$, $(2,2,2,2)$ (not considering permutations). By \Cref{thm:fillingCircular}, only the architecture $\bm k = (3,3)$ is filling. The target filter corresponds to the polynomial
$\pi(u) = 2\mathsf{x}^4 + 5\mathsf{x}^2\mathsf{y}^2 + 2\mathsf{y}^4 = (\mathsf{x}^2+2\mathsf{y}^2)(2\mathsf{x}^2+\mathsf{y}^2)$,
which does not have any real roots and is thus not contained in
$\mathcal M_{(4,2)}$, $\mathcal M_{(3,2,2)}$, nor $\mathcal M_{(2,2,2,2)}$.
Following our results in \Cref{sec:critPts}, for all architectures listed above, any critical point $\theta \in \mathrm{Crit}(\mathcal L)$ with $\mu(\theta) \ne u$ corresponds to polynomials with repeated roots.
More precisely, if $\theta$ is a critical point for $\mathcal L$, then $\mu(\theta)$ is a critical point for $\ell|_{\Delta_{\lambda}}$ where $\ell(\overline w) = \|\overline w - u\|^2$ and $\Delta_{\lambda}$ is the multiple root locus where $\lambda$ describes the {(complex)} root pattern of $\mu(\theta)$. Based on this, we can compute the critical points for $\ell|_{\Delta_{\lambda}}$ for any $\lambda$ by solving algebraic systems.
Recall that the number of complex critical points was predicted by the ED degrees in \Cref{ex:EDdegreesGeneric}.
\begin{itemize}[leftmargin=*]
\item $\lambda = (2,1,1)$:
out of the $10$ complex critical points on $\Delta_{\lambda}$, four are real and in fact rational, namely
$[\frac{1}{5},\pm \frac{9}{5},\frac{16}{5},\pm \frac{9}{5},\frac{1}{5}]^\top {\equiv \frac{1}{5}(\mathsf{x}\pm \mathsf{y})^2(\mathsf{x}^2 \pm 7\mathsf{x}\mathsf{y} +\mathsf{y}^2)}$, \,\, $ [0,0,5,0,2]^\top {\equiv \mathsf{y}^2(5\mathsf{x}^2+2\mathsf{y}^2)}$, and $[2,0,5,0,0]^\top \equiv \allowbreak \mathsf{x}^2(5\mathsf{y}^2+2\mathsf{x}^2)$.
\item $\lambda = (3,1)$:
out of the $12$ complex critical points on $\Delta_{\lambda}$, four are real and none are rational.
\item $\lambda = (2,2)$: out of the $13$ complex critical points on $\Delta_{\lambda}$, five are real and three are rational:
$[-1,0,2,0,-1]^\top \allowbreak \equiv \allowbreak -(\mathsf{x}-\mathsf{y})^2(\mathsf{x}+\mathsf{y})^2$,
$[0,0,5,0,0]^\top \equiv 5 \mathsf{x}^2\mathsf{y}^2$,
$[\frac{7}{3},0,\frac{14}{3},0,\frac{7}{3}]^\top \equiv \frac{7}{3} (\mathsf{x}^2+\mathsf{y}^2)^2$.
\item $\lambda = (4)$: out of the $10$ complex critical points on $\Delta_{\lambda}$, four are real and also rational: $[0,0,0,0,2]^\top \equiv 2\mathsf{y}^4$, $[2,0,0,0,0]^\top \equiv 2\mathsf{x}^4$, $[\frac{17}{35},\pm \frac{68}{35},\frac{102}{35},\pm \frac{68}{35},\frac{17}{35}]^\top \equiv \frac{17}{35}(\mathsf{x}\pm \mathsf{y})^4$.
\end{itemize}
For any choice of quartic architecture, all critical points are mapped to {either $u$ or} one of the $17$ real critical points listed above.
We have verified numerically that gradient descent indeed always converges to one of these critical points. In fact, it always seems to converge to one of the rational critical points. For example, for $\bm k= (4,2)$, gradient descent converges to either
$[2,0,5,0,0]^\top$ or $[0,0,5,0,2]^\top$
which arise with $\lambda = (2,1,1)$.
Note that in this case we do not expect critical points with $\lambda = (3,1), (2,2), (4)$ due to~\Cref{ex:partitionVsArchitecture,cor:rrmp}.
Finally, we remark that the previous list only describes the critical points in function space but not the actual parameters.
However, as discussed in~\Cref{sec:training-dynamics}, it is possible recover these parameters up to a finite ambiguity given the initialization of gradient descent. For example, assume that $\bm k = (4,2)$ and we initialize gradient descent at $\theta(0) = (w_1(0), w_2(0)) = ([1,6,11,6],[4,1])$, {\em i.e.}, a factorization of the polynomial $(\mathsf{x}+\mathsf{y})(\mathsf{x}+2\mathsf{y})(\mathsf{x}+3\mathsf{y}) \cdot {(4\mathsf{x}+\mathsf{y})}$.
Then, following the approach outlined in \Cref{cor:invariants_system}, we can derive that if the gradient descent path $\theta(t)$ converges to $\mu(\theta)=[2,0,5,0,0]^\top$ in function space, then in parameter space $\theta(t)$ converges to one of $(w_1,w_2) = (1/\kappa[2,0,5,0]^\top, \kappa [1,0]^\top)$
with $\kappa = \pm \sqrt{\frac{1}{2} \left(\sqrt{31445} - {177}\right)} \approx \pm 0.4045867$.
Numerically, we verify that gradient descent converges to these filters with $\kappa > 0$.
\section{Conclusion}
We presented a semi-algebraic description of the families of functions that can be represented by linear convolutional networks. Using the fact that compositions of convolutions correspond to polynomial multiplication, we showed that the function space can be described as sets of polynomials that admit certain factorizations. We then investigated the optimization of an objective function over such sets and described the critical points in function space and in parameter space. Unlike fully-connected linear networks, convolutional linear networks can have non-global local optimizers in function space as well as spurious local optimizers in parameter space. In more detail, our analysis showed that all critical points that are not global minima correspond to polynomials with repeated roots, according to multiplicity patterns that depend on the network's architecture.
Possible extensions of this work might include a more detailed study of the function space for linear convolutions in higher dimensions or with larger strides (which we discussed briefly in Sections~\ref{sec:higher_dimension} and \ref{sec:largerStrides}) and, naturally, carrying out a similar analysis for \emph{non-linear} convolutional networks. In particular, one might first consider the special case of polynomial activation functions, since in this setting the function space is still a semi-algebraic set.
\section*{Acknowledgment}
Kathl\'en Kohn was partially supported by the Knut and Alice Wallenberg Foundation within their WASP
(Wallenberg AI, Autonomous Systems and Software Program) AI/Math initiative.
Guido Mont\'ufar and Thomas Merkh have been supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant n\textsuperscript{o}~757983). While at New York University, Matthew Trager was supported in part by Samsung Electronics.
\bibliographystyle{alpha}
| {
"timestamp": "2022-06-09T02:19:13",
"yymm": "2108",
"arxiv_id": "2108.01538",
"language": "en",
"url": "https://arxiv.org/abs/2108.01538",
"abstract": "We study the family of functions that are represented by a linear convolutional neural network (LCN). These functions form a semi-algebraic subset of the set of linear maps from input space to output space. In contrast, the families of functions represented by fully-connected linear networks form algebraic sets. We observe that the functions represented by LCNs can be identified with polynomials that admit certain factorizations, and we use this perspective to describe the impact of the network's architecture on the geometry of the resulting function space. We further study the optimization of an objective function over an LCN, analyzing critical points in function space and in parameter space, and describing dynamical invariants for gradient descent. Overall, our theory predicts that the optimized parameters of an LCN will often correspond to repeated filters across layers, or filters that can be decomposed as repeated filters. We also conduct numerical and symbolic experiments that illustrate our results and present an in-depth analysis of the landscape for small architectures.",
"subjects": "Machine Learning (cs.LG); Algebraic Geometry (math.AG)",
"title": "Geometry of Linear Convolutional Networks",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754442973824,
"lm_q2_score": 0.7185943865443352,
"lm_q1q2_score": 0.7075103874014937
} |
https://arxiv.org/abs/1508.00956 | Asymptotic formula on average path length of fractal networks modelled on Sierpinski Gasket | In this paper, we introduce a new method to construct evolving networks based on the construction of the Sierpinski gasket. Using self-similarity and renewal theorem, we obtain the asymptotic formula for average path length of our evolving networks. | \section{Introduction}
The Sierpinski gasket described in 1915 by W.~Sierpi\'{n}ski is a classical
fractal. Suppose $K$ is the solid regular triangle with vertexes
a_{1}=(0,0),$ $a_{2}=(1,0),$ $a_{3}=(1/2,\sqrt{3}/2).$ Let
T_{i}(x)=x/2+a_{i}/2$ be the contracting similitude for $i=1,2,3.$ Then
T_{i}:K\rightarrow K\ $and the Sierpinski gasket $E$ is the self-similar
set, which is the unique invariant set \cite{H} of IFS $\{T_{1},T_{2},T_{3
\},$ satisfyin
\begin{equation*}
E=\cup _{i=1}^{3}T_{i}(E).
\end{equation*
The Sierpinski gasket is important for the study of fractals, e.g., the Sierpinski gasket is a typical example of post-critically finite self-similar fractals on which the Dirichlet forms and Laplacians can be constructed by Kigami \cite{Ki1,Ki2},
see also Strichartz \cite{St}.
\begin{figure}[tbph]
\centering\includegraphics[width=0.65\textwidth]{3.eps} \vspace{-0.35cm}
\caption{The first two constructions of Sierpinski gasket}
\end{figure}
For the word $\sigma =i_{1}\cdots i_{k}$ with letters in $\{1,2,3\}$, i.e.,
every letter $i_{t}\in \{1,2,3\}$ for all $t\leq k,$ we denote by $|\sigma
|(=k)$ the length of word $\sigma .$ Given words $\sigma =i_{1}\cdots i_{k}$
and $\tau =j_{1}\cdots j_{n},$ we call $\sigma $ a prefix of $\tau $ and
denote by $\tau \prec \sigma ,$ if $k<n$ and $i_{1}\cdots i_{k}=j_{1}\cdots
j_{k}.$ We also write $\tau \preceq \sigma $ if $\tau =\sigma $ or $\tau
\prec \sigma .$ When $\tau \prec \sigma $ with $|\tau |=|\sigma |-1,$ we say
that $\tau $ is the father of $\sigma $ and $\sigma $ is a child of $\tau .$
Given $\sigma =i_{1}\cdots i_{k},$ we write $T_{\sigma }=T_{i_{1}}\circ
\cdots \circ T_{i_{k}}\text{ and }K_{\sigma }=T_{\sigma }(K)$ which is a
solid regular triangle with side length $2^{-|\sigma |}.$ For notational
convenience, we write $K_{\emptyset }=K$ with empty word $\emptyset .$ We
also denote $|\emptyset |=0.$ If $\tau \prec \sigma ,$ then $K_{\sigma
}\subset K_{\tau }.$ For solid triangle $K_{\sigma }$ with word $\sigma ,$
we denote by $\partial K_{\sigma }$ its boundary consisting of $3$ sides,
where every side is a line segment with side length $2^{-|\sigma |}.$
Complex networks arise from natural and social phenomena, such as the
Internet, the collaborations in research, and the social relationships.
These networks have in common two structural characteristics: the
small-world effect and the scale-freeness (\emph{power-law} degree
distribution), as indicated, respectively, in the seminal papers by Watts
and Strogatz \cite{WS98} and by Barab\'{a}si and Albert \cite{BA99}. In fact
complex networks also exhibit \emph{self-similarity} as demonstrated by
Song, Havlin and Makse \cite{SHM05} and fractals possess the feature of
\emph{power law} in terms of their fractal dimension (e.g. see \cit
{Falconer03}). Recently self-similar fractals are used to model evolving
networks, for example, in a series of papers, Zhang et al.\ \cit
{ZZFGZ07,ZZZCG08,GWZZW09} use the Sierpinski gasket to construct evolving
networks. There are also some complex networks modelled on self-similar
fractals, for example, Liu and Kong \cite{LK10} and Chen et al.\ \cite{CFW12}
study Koch networks, Zhang et al.\ \cite{ZZCYG07} investigate the
networks constructed from Vicsek fractals. See also Dai and Liu
\cite{D}, Sun et al. \cite{S} and Zhou et al. \cite{Zhou}.
\medskip
In the paper, we introduce a new method to construct evolving networks
modelled on Sierpinski gasket and study the asymptotic formula for average
path length.
Since $E$ is connected, we can construct the network from geometry as
follows.
Fix an integer $t,$ we consider a network $G_{t}$ with vertex set
V_{t}=\{\sigma :0\leq |\sigma |\leq t\}$ where $\#V_{t}=1+3+...+3^{t}=\frac{
}{2}(3^{t+1}-1).$ For the edge set of $G_{t},$ there is a unique edge
between distinct words $\sigma $ and $\tau $ (denoted by $\tau \sim \sigma
) if and only if
\begin{equation}
\partial K_{\sigma }\cap \partial K_{\tau }\neq \varnothing . \label{vvv}
\end{equation
We can illustrate the geodesic paths in Figure 2 for $t=3$.
We have $233\sim 32\sim 312$ since $\partial K_{233}\cap \partial
K_{32}=\{C\}$ and $\partial K_{32}\cap \partial K_{312}=\{F\}.$ We also get
another geodesic path from $233\ $to $312:$ $233\sim 3\sim 312$ since
\partial K_{233}\cap \partial K_{3}=\{C\}$ and $\partial K_{3}\cap \partial
K_{312}=[G,F],$ the line segment between $G$ and $F.$ We also have some
geodesic paths from $21\ $to $312:$ $21\sim 1\sim 311\sim 312,\ 21\sim
\emptyset \sim 3\sim 312$ and $21\sim 2\sim 32\sim 312.$ Also we have
132\sim 1\sim \emptyset $ but $132\not\sim \emptyset ,$ then the geodesic
distance between $132$ and $\emptyset $ is $2.$
\begin{figure}[tbph]
\centering\includegraphics[width=0.9\textwidth]{1.eps} \vspace{-0.35cm}
\caption{{}}
\end{figure}
In Figure 2, $312\not\sim \emptyset $, $231\not\sim \emptyset $ and $132\not\sim
\emptyset$. In fact, by observation we have
\begin{claim}
\label{C:two}Suppose $\sigma \prec \tau $ with $\tau =\sigma \beta .$ Then
\sigma \sim \tau $ if and only if there are at most two letters in $\beta .$
In particular, if $\sigma \prec \tau $ and $|\tau |-|\sigma |\leq 2,$ then
\sigma \sim \tau .$
\end{claim}
For example, $123123$ and $1231$ are neighbors, but $123123$ and $123$ are
not.
For every $t,$ we denote $d_{t}(\sigma ,\tau )$ the geodesic distance on
V_{t}.$ Le
\begin{equation*}
\bar{D}(t)=\frac{\sum\nolimits_{\sigma \neq \tau \in V_{t}}d_{t}(\sigma
,\tau )}{\#V_{t}(\#V_{t}-1)/2}
\end{equation*
be the average path length of the complex network $V_{t}.$
We can state our main result as follows.
\begin{theorem}
We have the asymptotic formula
\begin{equation}
\lim_{t\rightarrow \infty }\frac{\bar{D}(t)}{t}=\frac{4}{9}. \label{a0}
\end{equation}
\end{theorem}
\begin{remark}
\label{R:6} Since $t\propto \ln (\#V_{t})$, Theorem $1$ implies that the
evolving networks $G_{t}$ have small average path length, namely $\bar{D
(t)\propto \ln (\#V_{t})$.
\end{remark}
The paper is organized as follows. In Section 2 we give notations and sketch of proof
for Theorem 1,consisting of four steps. In sections 3-6, we will provide details for
the four steps respectively. Our main techniques come from the
self-similarity and the renewal theorem.
\section{Sketch of proof for Theorem 1}
We will illustrate our following four steps needed to prove Theorem 1.
\textbf{Step 1.} We calculate the geodesic distance between a word and the
empty word.
Given a small solid triangle $\Delta ,$ we can find a maximal solid triangle
$\Delta ^{\prime }$ which contains $\Delta $ and their boundaries are
touching. Translating into the language of words, for a given word $\sigma
\neq \emptyset ,$ we can find a unique shortest word $f(\sigma )$ such that
f(\sigma )\prec \sigma $ and $f(\sigma )\sim \sigma .$ For a word $\sigma
=\tau _{2}\tau _{1},$ where $\tau _{1}$ is the maximal suffix with at
most two letters appearing, using Claim \ref{C:two} we have $f(\sigma )=\tau _{2}.$
Iterating $f$ again and again, we obtain a sequence $\sigma \sim f(\sigma
)\sim \cdots \sim f^{n-1}(\sigma )\sim f^{n}(\sigma )=\emptyset .$ Let
\begin{equation*}
\omega (\sigma )=\min \{n:f^{n}(\sigma )=\emptyset \}.
\end{equation*
In particular, we define $\omega (\emptyset )=0.$ For $\sigma
=112113112312=(112)(11311)(23)(12),$ we have $f(\sigma )=(112)(11311)(23),$
f^{2}(\sigma )=(112)(11311)$, $f^{3}(\sigma )=(112)\ $and $f^{4}(\sigma
)=\emptyset .$ Then $\omega (\sigma )=4.$ We will prove in Section 3
\begin{proposition}
For $\sigma \in V_{t},$ we have $d_{t}(\sigma ,\emptyset )=\omega (\sigma ).$
\end{proposition}
In fact, this proposition shows that $d_{t}(\sigma ,\emptyset )$ is
independent of the choice of $t$ whenever $t\geq |\sigma |.$ In this case, we
also write $d(\sigma ,\emptyset ).$ Write
\begin{equation*}
L(\tau )=d(\tau ,\emptyset )-1\text{ for }\tau \neq \emptyset \text{ and
L(\emptyset )=0.
\end{equation*
Then $L(\tau )$ is independent of $t,$ in fact, $L(\tau )$ is the minimal
number of moves for $K_{\tau }$ to touch the boundary of $K.$
\medskip
\textbf{Step 2.} Given $m\geq 1,$ we consider the average geodesic
distance between the empty word and word of length $m$ and set
\begin{equation*}
\,\bar{\alpha}_{m}=\frac{\sum\nolimits_{|\sigma |=m}d(\sigma ,\emptyset )}
\#\{\sigma :|\sigma |=m\}}-1=\frac{\sum\nolimits_{|\sigma |=m}L(\sigma )}
\#\{\sigma :|\sigma |=m\}},
\end{equation*
and $\bar{\alpha}_{0}=0,$ we will obtain the limit property of $\bar{\alpha
_{m}/m$ as $m\rightarrow \infty$ in Section 4.
In fact, by the \textbf{Jordan curve theorem},\textbf{\ }we can obtain
\begin{equation}
L(\tau )+L(\sigma )\leq L(\tau \sigma )\leq L(\tau )+L(\sigma )+1.
\end{equation
From (2.1), we can verify $\{\bar{\alpha}_{m}\}_{m}$ is
superadditive which implies
\begin{proposition}
$\lim\limits_{m\rightarrow \infty }\bar{\alpha}_{m}/m$ $=\sup (\bar{\alpha
_{m}/m)<\infty .$
\end{proposition}
Denote
\begin{equation*}
\alpha ^{\ast }=\sup (\bar{\alpha}_{m}/m)=\lim\limits_{m\rightarrow \infty
\bar{\alpha}_{m}/m.
\end{equation*}
\medskip
\textbf{Step 3.} We obtain the asymptotic formula of $\bar{D}(t)$ in
terms of $\alpha ^{\ast }.$
Using the similarity of Sierpinski gasket, e.g., for $i=1,2,3,
\begin{equation*}
d(i\sigma ,i\tau )=d(\sigma ,\tau ),
\end{equation*
and for $i\neq j,
\begin{equation*}
L(\sigma )+L(\tau )\leq d(i\sigma ,j\tau )\leq (L(\sigma )+1)+(L(\tau )+1)+1,
\end{equation*
we will prove the following in Section 5
\begin{proposition}
\label{P:asym}$\lim\limits_{t\rightarrow \infty }\frac{\bar{D}(t)}{t
=2\alpha ^{\ast }.$
\end{proposition}
\begin{figure}[tbph]
\centering\includegraphics[width=0.5\textwidth]{x.eps} \vspace{-0.35cm}
\caption{The typical geodesic path between $\Delta _{1}$ and $\Delta _{3}$}
\end{figure}
As illustrated in Figure 3, Proposition \ref{P:asym} shows that the \emph{typical}
geodesic path is the geodesic path between $\Delta _{1}$ and $\Delta
_{3}$ whose first letters of codings are different. On the other hand, for
example the geodesic path between $\Delta _{1}$ and $\Delta _{2}$ with the same first letter
will give negligible contribution to
\bar{D}(t).$ Using $L(\sigma )+L(\tau )\leq d(i\sigma ,j\tau )\leq L(\sigma
)+L(\tau )+3$ with $i\neq j,$ we obtain that $d(i\sigma ,j\tau )\approx
L(\sigma )+L(\tau )$, ignoring the terms like $d(i\sigma ,i\tau ),$ we have
\begin{equation*}
\frac{\bar{D}(t)}{t}\approx 2\cdot \frac{1}{t}\cdot \frac
\sum\nolimits_{|\tau |\leq t-1}L(\tau )}{\#\{\tau :|\tau |\leq t-1\}},
\end{equation*
where $\frac{\sum\nolimits_{|\tau |\leq t-1}L(\tau )}{\#\{\tau :|\tau |\leq
t-1\}}$ is the average value of $L(\tau ).$ Using Stolz theorem, we have
\begin{eqnarray}
\lim\limits_{t\rightarrow \infty }\frac{2\sum\nolimits_{|\tau |\leq
t-1}L(\tau )}{t\cdot \#\{\tau :|\tau |\leq t-1\}} &=&\lim\limits_{
\rightarrow \infty }\frac{2\sum\nolimits_{|\tau |=t-1}L(\tau )}{3^{t-1}t
\frac{3^{t}}{6}-\frac{1}{2}} \notag \\
&=&\lim\limits_{t\rightarrow \infty }\frac{2}{t}\cdot \frac
\sum\nolimits_{|\tau |=t-1}L(\tau )}{\#\{\tau :|\tau |=t-1\}} \label{mmm} \\
&=&\lim\limits_{t\rightarrow \infty }\frac{2\bar{\alpha}_{t-1}}{t}=2\alpha
^{\ast }. \notag
\end{eqnarray}
\bigskip
\textbf{Step 4.} Using the renewal theorem,
we will prove in Section 6
\begin{proposition}
$\alpha ^{\ast }=2/9.$
\end{proposition}
By programming, we have
\begin{equation*}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
$t=$ & $300$ & $400$ & $500$ & $600$ & $700$ & $800$ \\ \hline
$\bar{\alpha}_{t}/t=$ & $0.2207\cdots $ & $0.2211\cdots $ & $0.2213\cdots $
& $0.2214\cdots $ & $0.2215\cdots $ & $0.2216\cdots $ \\ \hline
\end{tabular
\end{equation*
which is in line with $\alpha ^{\ast }=2/9=0.2222\cdots.$
In fact, suppose $\Sigma =\{\cdots x_{2}x_{1}:$ $x_{i}=1,2$ or $3$ for all
t\}$ is composed of infinite words with letters in $\{1,2,3\}.$ Then we have
a natural mass distribution $\mu $ on $\Sigma $ such that for any word
\sigma $\ of length $k,$
\begin{equation*}
\mu (\{\cdots x_{k}\cdots x_{1}:x_{k}\cdots x_{1}=\sigma \})=1/3^{k}.
\end{equation*
For any word $\sigma ,$ let $\#(\sigma )$ denote the cardinality of letters
appearing in word $\sigma .$ For $\mu $-almost all $x,$ let
\begin{equation*}
S(\cdots x_{p}x_{p-1}\cdots x_{1})=p\text{ if }\#(x_{p-1}\cdots x_{1}1)=
\text{ and }\#(x_{p}x_{p-1}\cdots x_{1}1)=3.
\end{equation*
Then $\mathbb{E}(S)=\sum_{k=2}^{\infty }k\cdot \mu
\{x:S(x)=k\}=\sum_{k=2}^{\infty }k\cdot (2^{k}-2)/3^{k}=9/2<\infty .$
For $\mu $-almost all $x=\cdots x_{2}x_{1},$ suppose $x_{0}=1$ and $p_{0}=0$
and there is an infinite sequence $\{p_{n}\}_{n\geq 0}$ of integers such that
$p_{n+1}>p_{n}$ for all $n$ and
\begin{equation*}
x1=\cdots x_{p_{3}}\cdots x_{p_{2}}\cdots x_{p_{1}}\cdots x_{1}x_{0}
\end{equation*
satisfying $\#(x_{p_{n}}\cdots x_{p_{n-1}})=3$ and $\#(x_{(p_{n}-1)}\cdots
x_{p_{n-1}})=2$ for all $n.$ We then let
\begin{equation*}
S_{i}(x)=p_{i}-p_{i-1}\text{ for any }i\geq 1.
\end{equation*
Since $x_{p_{n}}$ is uniquely determined by word $x_{(p_{n}-1)}\cdots
x_{p_{n-1}},$ and $\mu $ is symmetric for letters in $\{1,2,3\},$ we find
that $\{S_{i}\}_{i}$ is a sequence of positive independent identically
distributed random variables with $S_{1}=S.$ For example, for $x=\cdots
321223121,$ then $x1=\cdots 3(21)(223)(1211)$ and $S_{1}(x)=S(x)=4,$
S_{2}(x)=3$, $S_{3}(x)=2,\cdots .$
Let $J_{n}=S_{1}+\cdots +S_{n}$ and $Y_{t}=\sup \{n:J_{n}\leq t\}.$ Then
J_{n}=p_{n}$ and $Y_{t}=\max \{n: p_{n}\leq t\}.$ By the elementary renewal theorem, we
have
\begin{equation*}
\frac{\mathbb{E}(Y_{t})}{t}\rightarrow \frac{1}{\mathbb{E}(S)}=\frac{2}{9
=0.222\cdots .
\end{equation*
Using the following estimates (Lemma \ref{l: lim} in Section 6)
\begin{eqnarray*}
\liminf\limits_{t\rightarrow \infty }\frac{\sum\nolimits_{|\sigma
|=t}d(\sigma ,\emptyset )}{t3^{t}} &\geq &\liminf\limits_{t\rightarrow
\infty }\frac{\sum\nolimits_{k=2}^{t-1}\mathbb{E}(Y_{t-k})\frac{2^{k}-2}
3^{k}}}{t}, \\
\limsup_{t\rightarrow \infty }\frac{\sum\nolimits_{|\sigma |=t}d(\sigma
,\emptyset )}{t3^{t}} &\leq &\limsup_{t\rightarrow \infty }\frac
\sum\nolimits_{k=2}^{t-1}\mathbb{E}(Y_{t-k})\frac{2^{k}-2}{3^{k}}}{t},
\end{eqnarray*
we can prove that $\alpha ^{\ast }=\lim\limits_{t\rightarrow \infty }\frac
\sum\nolimits_{|\sigma |=t}d(\sigma ,\emptyset )}{t3^{t}}=\lim\limits_{
\rightarrow \infty }\frac{\mathbb{E}(Y_{t})}{t}=2/9.$
\medskip Theorem 1 follows from Propositions 3 and 4.
\bigskip
\section{Basic formulas on geodesic distance}
\subsection{\textbf{Criteria of }neighbor}
$\ $
Given distinct words $\sigma $ and $\tau $ with $|\sigma |,|\tau |\leq t,$
we give the following \textbf{criteria} to test whether they are neighbors
or not. At first, we delete the common prefix of $\sigma $ and $\tau ,$ say
\sigma =\beta \sigma ^{\prime }$ and $\tau =\beta \tau ^{\prime }$ where the
first letters of $\sigma ^{\prime }$ and $\tau ^{\prime }$ are different. We
can distinguish two cases: \newline
\textbf{Case 1}. If one of $\sigma ^{\prime }$ and $\tau ^{\prime }$ is the
empty word, say $\sigma ^{\prime }=\emptyset ,$ then $\sigma $ and $\tau
\textbf{\ }are not neighbors if and only if every letter $i\in \{1,2,3\}$
appears in the word $\tau ^{\prime }$. \newline
\textbf{Case 2}. If neither $\sigma ^{\prime }$ nor $\tau ^{\prime }$ is the
empty word, say $i,j$ the first letters of $\sigma ^{\prime }$ and $\tau
^{\prime }$ respectively, then $\sigma $ and $\tau $\textbf{\ }are neighbors
if and only if
\begin{equation*}
\sigma ^{\prime }=i[j]^{k}\text{ and }\tau ^{\prime }=j[i]^{k^{\prime }
\text{ with }k,k^{\prime }\geq 0.
\end{equation*
We say that $\sigma $ and $\tau $ are neighbors of type $1$ or $2$ according
to Case 1 or 2 respectively.
\subsection{Estimates on distance}
$\ $
The first lemma show the \emph{self-similarity} of geodesic distance.
\begin{lemma}
\label{l:3}If $\sigma =i\sigma ^{\prime }$ and $\tau =i\tau ^{\prime }$ with
$i\in \{1,2,3\},$ then
\begin{equation*}
d_{t}(\sigma ,\tau )=d_{t}(i\sigma ^{\prime },i\tau ^{\prime
})=d_{t-1}(\sigma ^{\prime },\tau ^{\prime }).
\end{equation*
Consequently, given any word $i_{1}\cdots i_{k},
\begin{equation*}
d_{t}(i_{1}\cdots i_{k}\sigma ^{\prime },i_{1}\cdots i_{k}\tau ^{\prime
})=d_{t-k}(\sigma ^{\prime },\tau ^{\prime })\text{ for all }\sigma ^{\prime
},\tau ^{\prime }.
\end{equation*}
\end{lemma}
\begin{proof}
Fix the letter $i,$ we define $g:V_{t}\rightarrow V_{t}$ by
\begin{equation*}
g(\beta )=\left\{
\begin{array}{ll}
\beta & \text{if }i\preceq \beta ,\text{ } \\
i & \text{otherwise}
\end{array
\right.
\end{equation*
If we give a shortest sequence $(\sigma =)\sigma ^{1}\sim \sigma ^{2}\sim
\cdots \sim \sigma ^{k}(=\tau )$ in $G_{t}$, then $\sigma =g(\sigma
^{1})\simeq g(\sigma ^{2})\simeq \cdots \simeq g(\sigma ^{k})=\tau $ is also
a sequence and all word $\{g(\sigma ^{i})\}_{i=1}^{k}$ have the same first
letter $i.$ Deleting the first letter $i$, we get a ($\simeq $)-sequence
from $\sigma ^{\prime }$ to $\tau ^{\prime }$ in $G_{t-1}.$ Hence
d_{t}(\sigma ,\tau )\geq d_{t-1}(\sigma ^{\prime },\tau ^{\prime }).$
On the other hand, for given ($\sim $)-sequence from $\sigma ^{\prime }$ to
\tau ^{\prime }$ in $G_{t-1},$ by adding the first letter $i,$ we obtain a (
\sim $)-sequence $\sigma $ to $\tau $ in $G_{t}$ which implies $d_{t}(\sigma
,\tau )\leq d_{t-1}(\sigma ^{\prime },\tau ^{\prime }).$ The lemma follows.
\end{proof}
Given $t,$ let $L_{t}(\sigma )=d_{t}(\sigma ,\emptyset )-1$ for $\sigma \neq
\emptyset .$ The second lemma shows that $L_{t}(\sigma )$ is independent of
t$ whenever $t\geq |\sigma |.$ We can write $L(\sigma ).$
\begin{lemma}
\label{l: 1}For $\sigma ,\tau \in V_{k},$ $d_{t}(\sigma ,\tau )=d_{k}(\sigma
,\tau ).$ As a result, $L_{t}(\sigma )=L_{|\sigma |}(\sigma ).$
\end{lemma}
\begin{proof}
Fix $k\leq t.$ Let $h:V_{t}\rightarrow V_{k}$ be defined by
\begin{equation*}
h(\beta )=\left\{
\begin{array}{ll}
\beta & \text{if }|\beta |\leq k,\text{ } \\
i_{1}\cdots i_{k} & \text{if }|\beta |>k\text{ and }\beta =i_{1}\cdots
i_{k}\cdots i_{|\beta |}
\end{array
\right.
\end{equation*
Given $\sigma ,\tau \in V_{k},$ if we give a shortest sequence $(\sigma
=)\sigma ^{1}\sim \sigma ^{2}\sim \cdots \sim \sigma ^{k}(=\tau )$ in $G_{t}
, by criteria of neighbor we obtain that $\sigma =h(\sigma ^{1})\simeq
h(\sigma ^{2})\simeq \cdots \simeq h(\sigma ^{k})=\tau $ is also a ($\simeq
) $-sequence and all words in $V_{k}.$ Therefore, $d_{t}(\sigma ,\tau )\geq
d_{k}(\sigma ,\tau ).$ On the other hand, any ($\sim $)-sequence from
\sigma $ to $\tau $ in $G_{k}$ is also a ($\sim $)-sequence in $G_{k},$ that
means $d_{t}(\sigma ,\tau )\leq d_{k}(\sigma ,\tau ).$ The lemma follows.
\end{proof}
By the \emph{Jordan curve theorem}, when a point\ in the \emph{interior}
moves to the \emph{exterior}, it must touch the \emph{boundary}. Therefore,
we have the following
\begin{lemma}
\label{l:double}For any word $\tau \sigma $ with $\tau ,\sigma \neq
\emptyset ,$ we have
\begin{equation*}
L(\tau )+L(\sigma )\leq L(\tau \sigma )\leq L(\tau )+L(\sigma )+1.
\end{equation*}
\end{lemma}
\begin{proof}
In fact, using triangle inequality and Lemmas \ref{l:3} and \ref{l: 1}, we hav
\begin{eqnarray*}
L(\tau \sigma )=d_{t}(\tau \sigma ,\emptyset )-1 &\leq &d_{t}(\tau \sigma
,\tau )+d_{t}(\tau ,\emptyset )-1 \\
&=&d_{t-|\tau |}(\sigma ,\emptyset )+d_{t}(\tau ,\emptyset )-1 \\
&=&L(\sigma )+L(\tau )+1.
\end{eqnarray*}
On the other hand, using the self-similarity in Lemma \ref{l:3}, the minimal
number of moves for $K_{\tau \sigma }$ to touch the boundary of $K_{\tau }$
is $L(\sigma ),$ and $L(\tau )$ is the minimal number of moves for $K_{\tau
} $ to touch the boundary of $K,$ by the Jordan curve theorem, there are at
least $L(\sigma )+L(\tau )$ moves for $K_{\tau \sigma }$ to touch the
boundary of $K,$ that means $L(\tau )+L(\sigma )\leq L(\tau \sigma ).$
\end{proof}
\subsection{Proof of Proposition 1}
$\ $
Suppose $\omega $ and $f$ are defined as in Section 2. By the definition of
f,$ if $\tau ^{\prime }\preceq \sigma ^{\prime },$ then we have $f(\tau
^{\prime })\preceq f(\sigma ^{\prime }).$ Therefore we have
\begin{claim}
\label{C:2}If $f(\sigma )\preceq f(\tau )\preceq \sigma ,$ then
f^{k}(\sigma )\preceq f^{k}(\tau )\preceq f^{k-1}(\sigma )$ for all $k\geq
0. $ As a result, $|\omega (\sigma )-\omega (\tau )|\leq 1.$
\end{claim}
\begin{example}
\label{E:1}Let $i=1$, $j=3$ and $\sigma =\beta 321211$ for some word $\beta
, $ we have $f(\sigma i)=\beta 3$ and $f(\sigma j)=\beta 3212.$ Then
\begin{equation*}
f(\sigma i)\preceq f(\sigma j)\preceq \sigma i,
\end{equation*
and thus $|\omega (\sigma i)-\omega (\sigma j)|\leq 1.$ For $\beta =132$ and
$\beta ^{\prime }=\beta 1211$ with $\beta \sim \beta ^{\prime }$, we have
f(\beta )\preceq f(\beta ^{\prime })\preceq \beta ,$ then $|\omega (\beta
)-\omega (\beta ^{\prime })|\leq 1.$
\end{example}
\begin{lemma}
\label{l:omega}If $\sigma \sim \tau ,$ then $\omega (\sigma )\geq \omega
(\tau )-1.$
\end{lemma}
\begin{proof}
By the criteria of neighbor, without loss of generality, we only need to
deal with three cases: (1) $\sigma \prec \tau ;$ (2) $\sigma i[j]^{p}\ $and
\sigma j[i]^{q}$ with $i\neq j$ and $p,q\geq 1;$ (3) $\sigma i\ $and $\sigma
j[i]^{q}$ with $i\neq j$ and $q\geq 0.$ For case (2), it is clear that
\begin{equation*}
\omega (\sigma i[j]^{p})=\omega (\sigma j[i]^{q})=\omega (\sigma ij).
\end{equation*
For cases (1) and (3), we only use Claim \ref{C:2} as in Example
\ref{E:1} above.
\end{proof}
\begin{proof}[Proof of Proposition 1]
As shown in Section 2, we can find the following path from $\sigma $ to the
empty word $\emptyset :
\begin{equation*}
\sigma \sim f(\sigma )\sim f^{2}(\sigma )\sim \cdots \sim f^{\omega (\sigma
)}(\sigma _{k})=\emptyset .
\end{equation*
That means $d_{t}(\sigma ,\emptyset )\leq \omega (\sigma ).$
It suffices to show $d_{t}(\sigma ,\emptyset )\geq \omega (\sigma ).$
Suppose on the contrary, if we give a sequence
\begin{equation*}
\sigma =\sigma _{0}\sim \sigma _{1}\sim \sigma _{2}\sim \cdots \sim \sigma
_{k}=\emptyset \text{ with }k\leq \omega (\sigma )-1,
\end{equation*
then $\omega (\sigma _{i+1})\geq \omega (\sigma _{i})-1$ for all $i$ by
Lemma \ref{l:omega}. Therefore, $0=\omega (\emptyset )\geq \omega (\sigma
)-k>0$ which is impossible. That means $d_{t}(\sigma ,\emptyset )=\omega
(\sigma ).$
\end{proof}
\bigskip
\section{Average geodesic distance to empty word}
We first recall some notations. For every $t,$ we denote $d_{k}(\sigma ,\tau )$ the
geodesic distance on $V_{k}.$ Given $k\geq 0,\,$let $L_{k}(\emptyset )=0$
and
\begin{equation*}
L_{k}(\sigma )=d_{k}(\sigma ,\emptyset )-1\ \text{for }\sigma \in V_{k}\text{
with }\sigma \neq \emptyset .
\end{equation*}
As shown in (\ref{mmm}), for average geodesic distance $\frac
\sum\nolimits_{|\tau |\leq t-1}L(\tau )}{\#\{\tau :|\tau |\leq t-1\}}+1$ to
the empty word, when we estimate $\frac{1}{t}\frac{\sum\nolimits_{|\tau |\leq
t-1}L(\tau )}{\#\{\tau :|\tau |\leq t-1\}},$ it is important for us to
estimate $\bar{\alpha}_{k}/k,$ where
\begin{equation*}
\bar{\alpha}_{k}=\frac{\sum_{|\sigma |=k}L_{k}(\sigma )}{\#\{\sigma :|\sigma
|=k\}}\text{ for }k\geq 0.
\end{equation*
\begin{lemma}
For any $k_{1},k_{2}\geq 1,$ we have
\begin{equation}
\bar{\alpha}_{k_{1}}+\bar{\alpha}_{k_{2}}\leq \bar{\alpha}_{k_{1}+k_{2}}\leq
\bar{\alpha}_{k_{1}}+\bar{\alpha}_{k_{2}}+1. \label{hha}
\end{equation
In particular, $\{\bar{\alpha}_{k}\}_{k}$ is non-decreasing, i.e.,
\begin{equation}
\bar{\alpha}_{k+1}\geq \bar{\alpha}_{k}\text{ for all }k. \label{mono}
\end{equation
As a result, for any positive integers $q$ and $k$ we obtain tha
\begin{equation}
\bar{\alpha}_{k}\geq \bar{\alpha}_{q}\big[\frac{k}{q}\big]\geq \frac{\bar{\alpha}_{q
}{q}(k-q+1), \label{lower}
\end{equation}
\end{lemma}
\begin{proof}
We obtain that
\begin{equation*}
\bar{\alpha}_{k_{1}+k_{2}}=\frac{\sum\nolimits_{|\tau |=k_{1}}\sum_{|\sigma
|=k_{2}}L_{k_{1}+k_{2}}(\tau \sigma )}{\#\{\tau :|\tau |=k_{1}\}\cdot
\#\{\sigma :|\sigma |=k_{2}\}}.
\end{equation*
If $|\tau |=k_{1}$ and $|\sigma |=k_{2},$ using Lemma \ref{l:double}, we hav
\begin{equation}
L_{k_{1}}(\tau )+L_{k_{2}}(\sigma )\leq L_{k_{1}+k_{2}}(\tau \sigma )\leq
L_{k_{1}}(\tau )+L_{k_{2}}(\sigma )+1, \label{oki}
\end{equation
which implies
\begin{eqnarray*}
\bar{\alpha}_{k_{1}+k_{2}} &\geq &\frac{\sum\nolimits_{|\tau
|=k_{1}}\sum_{|\sigma |=k_{2}}\left( L_{k_{1}}(\tau )+L_{k_{2}}(\sigma
)\right) }{\#\{\tau :|\tau |=k_{1}\}\cdot \#\{\sigma :|\sigma |=k_{2}\}}
\bar{\alpha}_{k_{1}}+\bar{\alpha}_{k_{2}}, \\
\bar{\alpha}_{k_{1}+k_{2}} &\leq &\frac{\sum\nolimits_{|\tau
|=k_{1}}\sum_{|\sigma |=k_{2}}\left( L_{k_{1}}(\tau )+L_{k_{2}}(\sigma
)+1\right) }{\#\{\tau :|\tau |=k_{1}\}\cdot \#\{\sigma :|\sigma |=k_{2}\}}
\bar{\alpha}_{k_{1}}+\bar{\alpha}_{k_{2}}+1,
\end{eqnarray*
then (\ref{hha}) follows. In particular, we have $\bar{\alpha}_{k+1}\geq
\bar{\alpha}_{k}+\bar{\alpha}_{1}\geq \bar{\alpha}_{k}.$
Using (\ref{hha}) repeatedly, we have $\bar{\alpha}_{qm}\geq \bar{\alpha
_{q(m-1)}+\bar{\alpha}_{q}\geq \cdots \geq m\bar{\alpha}_{q}.$ It follows
from (\ref{mono}) tha
\begin{equation*}
\bar{\alpha}_{qm+(q-1)}\geq \bar{\alpha}_{qm+(q-2)}\geq \cdots \geq \bar
\alpha}_{qm+1}\geq \bar{\alpha}_{qm}\geq m\bar{\alpha}_{q}
\end{equation*
which implies $\bar{\alpha}_{k}\geq \bar{\alpha}_{q}[\frac{k}{q}]\geq \frac
\bar{\alpha}_{q}}{q}(k-q+1).$
\end{proof}
\begin{proof}[Proof of Proposition 2]
Since $\{\bar{\alpha}_{m}\}_{m}$ is superadditive, i.e., $\bar{\alpha
_{k_{1}+k_{2}}\geq \bar{\alpha}_{k_{1}}+\bar{\alpha}_{k_{2}}$, by Fekete's
superadditive lemma (\cite{Fe}), the limit $\lim\limits_{m\rightarrow \infty
}\frac{\bar{\alpha}_{m}}{m}$ exists and is equal to $\sup_{m}\frac{\bar
\alpha}_{m}}{m}$. We shall verify that $\lim\limits_{m\rightarrow \infty
\frac{\bar{\alpha}_{m}}{m}<+\infty .$
Fix an integer $q.$ For any $p=0,1,\cdots ,(q-1),$ using (\ref{hha}) and
\ref{mono}), we have $\bar{\alpha}_{qk+p}\leq \bar{\alpha}_{q(k+1)}\leq (k+1
\bar{\alpha}_{q}+k.$ Letting $k\rightarrow \infty ,$ we have
\limsup\limits_{m\rightarrow \infty }\frac{\bar{\alpha}_{m}}{m}\leq \frac
\bar{\alpha}_{q}}{q}+\frac{1}{q}.$
\end{proof}
Set $\alpha ^{\ast }=\lim_{m\rightarrow \infty }\frac{\bar{\alpha}_{m}}{m
=\sup_{m}\frac{\bar{\alpha}_{m}}{m}.$
\medskip
\section{Asymptotic formula}
Now we will investigate
\begin{equation*}
\kappa _{t}=\frac{\sum\nolimits_{\sigma \in V_{t}}L(\sigma )}{\#V_{t}}=\frac
\sum\nolimits_{k=0}^{t}\sum\nolimits_{|\sigma |=k}L_{k}(\sigma )}
\sum\nolimits_{k=0}^{t}\#\{\sigma :|\sigma |=k\}}=\frac{\su
\nolimits_{k=0}^{t}\bar{\alpha}_{k}3^{k}}{\sum\nolimits_{k=0}^{t}3^{k}},
\end{equation*
where we let $L(\emptyset )=\bar{\alpha}_{0}=\frac{\bar{\alpha}_{0}}{0}=0.$
At first, we hav
\begin{equation}
\kappa _{t}\leq \Big(\sup_{m\geq 1}\frac{\bar{\alpha}_{m}}{m}\Big)\frac
\sum\nolimits_{k=0}^{t}k3^{k}}{\sum\nolimits_{k=0}^{t}3^{k}}=\alpha ^{\ast
\frac{\sum\nolimits_{k=0}^{t}k3^{k}}{\sum\nolimits_{k=0}^{t}3^{k}}\leq
(\alpha ^{\ast }\chi (t))t, \label{k-upper}
\end{equation
where
\begin{equation}
\chi (t)=\frac{\sum\nolimits_{k=0}^{t}k3^{k}}{t\sum\nolimits_{k=0}^{t}3^{k}}
\frac{(3t-\frac{3}{2})+\frac{3}{2}\frac{1}{3^{t}}}{(3t)(1-\frac{1}{3^{t+1}})
\leq 1, \label{less}
\end{equation
since $(3t-\frac{3}{2})+\frac{3}{2}\frac{1}{3^{t}}-(3t)(1-\frac{1}{3^{t+1}})
\frac{1}{2}\left( 2t-3^{t+1}+3\right) \frac{1}{3^{t}}<0$ for any $t\geq 1.$
We also hav
\begin{equation}
\lim_{t\rightarrow \infty }\chi (t)=1. \label{equal}
\end{equation}
On the other hand, using $\bar{\alpha}_{k}\geq \frac{\bar{\alpha}_{q}}{q
(k-q+1)$ in (\ref{lower}), for any $t$ we have
\begin{equation*}
\kappa _{t}\geq \frac{\sum\nolimits_{k=0}^{t}\frac{\bar{\alpha}_{q}(k-q+1)}{
}3^{k}}{\sum\nolimits_{k=0}^{t}3^{k}}\geq \frac{\bar{\alpha}_{q}}{q}\left(
\frac{\sum\nolimits_{k=0}^{t}k\cdot 3^{k}}{\sum\nolimits_{k=0}^{t}3^{k}
-q+1\right) ,
\end{equation*
that is
\begin{equation}
\kappa _{t}\geq \frac{\bar{\alpha}_{q}}{q}(\chi (t)\cdot t-q+1).
\label{k-lower}
\end{equation}
Denot
\[
\begin{split}
\pi _{t} =\sum\limits_{\sigma ,\tau \in V_{t}}d_{t}(\sigma ,\tau ),\quad \;
&\mu _{t}=\sum\limits_{i}\sum\limits_{i\preceq \sigma ,i\preceq \tau
}d_{t}(\sigma ,\tau ), \\
\lambda _{t} =\sum\limits_{\sigma \in V_{t}}d_{t}(\sigma ,\emptyset )
\quad \quad &\nu _{t}=\sum\limits_{i\neq j}\sum\limits_{i\preceq \sigma
,j\preceq \tau }d_{t}(\sigma ,\tau ).
\end{split
\]
Then
\begin{equation*}
\pi _{t}=\mu _{t}+\lambda _{t}+\nu _{t}.
\end{equation*
By Lemma \ref{l:3}, we have
\begin{equation*}
\mu _{t}=3\sum\nolimits_{\sigma ^{\prime },\tau ^{\prime }\in
V_{t-1}}d_{t-1}(\sigma ^{\prime },\tau ^{\prime })=3\pi _{t-1},
\end{equation*
and thu
\begin{equation}
\pi _{t}=3\pi _{t-1}+\lambda _{t}+\nu _{t}. \label{imp3}
\end{equation}
\textbf{(1) The estimate of }$\lambda _{t}:$ Using (\ref{k-upper})-(\re
{less}), we hav
\begin{eqnarray}
\lambda _{t} &=&\left( \frac{\sum\nolimits_{\sigma \in V_{t}}d_{t}(\sigma
,\emptyset )}{\#V_{t}}\right) \#V_{t} \notag \\
&=&\left( \frac{\sum\nolimits_{\sigma \in V_{t}}L_{t}(\sigma ,\emptyset )}
\#V_{t}}+\frac{\sum\nolimits_{\sigma \neq \emptyset }1}{\#V_{t}}\right)
\#V_{t} \label{lambda=} \\
&=&\kappa _{t}\#V_{t}+(\#V_{t}-1)\leq \alpha ^{\ast }t(\#V_{t})+(\#V_{t}-1).
\notag
\end{eqnarray}
\textbf{(2) The estimate of} $\nu _{t}:$ For $\sigma =i\sigma ^{\prime },$
\tau =j\tau ^{\prime }$ with $i\neq j,$ by Lemma \ref{l:3} we have
\begin{equation}
d_{t}(\sigma ,\tau )\leq d_{t}(i\sigma ^{\prime },i)+d_{t}(j\tau ^{\prime
},j)+d_{t}(i,j)=d_{t-1}(\sigma ^{\prime },\emptyset )+d_{t-1}(\tau ^{\prime
},\emptyset )+1. \label{new}
\end{equation
Notice that
\begin{equation}
d_{t-1}(\sigma ^{\prime },\emptyset )\leq L(\sigma ^{\prime })+1\text{ for
all }\sigma ^{\prime }\in V_{t-1}, \label{new1}
\end{equation
since $d_{t-1}(\sigma ^{\prime },\emptyset )=L(\sigma ^{\prime })+1$ for
\sigma ^{\prime }\neq \emptyset $ and (\ref{new1}) is also true for $\sigma
^{\prime }=\emptyset .$
Using (\ref{k-upper})-(\ref{less}) and (\ref{new})-(\ref{new1}), we have the
following upper bound of $\nu _{t}.
\begin{eqnarray}
\nu _{t} &=&\sum\nolimits_{i\neq j}\sum\nolimits_{i\preceq \sigma ,j\preceq
\tau }d_{t}(\sigma ,\tau ) \notag \\
&\leq &C_{3}^{2}\sum\nolimits_{\sigma ^{\prime },\tau ^{\prime }\in
V_{t-1}}(d_{t-1}(\sigma ^{\prime },\emptyset )+d_{t-1}(\tau ^{\prime
},\emptyset )+1) \notag \\
&\leq &3(\#V_{t-1})^{2}+6(\#V_{t-1})^{2}\frac{\sum\nolimits_{\sigma ^{\prime
}\in V_{t-1}}(L(\sigma ^{\prime })+1)}{\#V_{t-1}} \label{nu-upper} \\
&\leq &3(\#V_{t-1})^{2}+6(\#V_{t-1})^{2}\left( \kappa _{t-1}+1\right) \notag
\\
&\leq &9(\#V_{t-1})^{2}+6(\#V_{t-1})^{2}\cdot \alpha ^{\ast }\cdot (t-1).
\notag
\end{eqnarray}
On the other hand, for $i\neq j,$ using the Jordan curve theorem, we have
\begin{equation*}
d_{t}(i\sigma ^{\prime },j\tau ^{\prime })\geq L_{t-1}(\sigma ^{\prime
})+L_{t-1}(\tau ^{\prime }).
\end{equation*
Then we obtain the following lower bound of $\nu _{t}.$
\begin{eqnarray}
\nu _{t} &=&\sum\nolimits_{i\neq j}\sum\nolimits_{i\preceq \sigma ,j\preceq
\tau }d_{t}(\sigma ,\tau ) \notag \\
&\geq &C_{3}^{2}\sum\nolimits_{\sigma ^{\prime },\tau ^{\prime }\in
V_{t-1}}(L_{t-1}(\sigma ^{\prime })+L_{t-1}(\tau ^{\prime })) \notag \\
&\geq &6\frac{\sum\nolimits_{\sigma ^{\prime }\in V_{t-1}}L_{t-1}(\sigma
^{\prime })}{(\#V_{t-1})}(\#V_{t-1})^{2} \label{nu-lower} \\
&\geq &6\kappa _{t-1}(\#V_{t-1})^{2} \notag \\
&\geq &6\left( \frac{\kappa _{t-1}}{t-1}\right) (\#V_{t-1})^{2}(t-1), \notag
\end{eqnarray
where
\begin{equation}
\frac{\kappa _{t-1}}{t-1}\rightarrow \alpha ^{\ast }\text{ as }t\rightarrow
\infty \label{text}
\end{equation
since
\begin{equation*}
\frac{\kappa _{t-1}}{t-1}=\frac{\sum\nolimits_{k=0}^{t-1}\left( \frac{\bar
\alpha}_{k}}{k}\right) k3^{k}}{(t-1)\sum\nolimits_{k=0}^{t-1}3^{k}}
\end{equation*
with $\lim\limits_{k\rightarrow \infty }\frac{\bar{\alpha}_{k}}{k}=\alpha
^{\ast }$ and $\lim\limits_{t\rightarrow \infty }\frac{\su
\nolimits_{k=0}^{t-1}k3^{k}}{(t-1)\sum\nolimits_{k=0}^{t-1}3^{k}
=\lim\limits_{t\rightarrow \infty }\chi (t-1)=1.$
\begin{proof}[Proof of Proposition 3]
\
\textbf{(i)} \textbf{Upper bound of} $\pi _{t}:$ Using (\ref{lambda=}) and
\ref{nu-upper})$,$ we hav
\begin{equation}
\lambda _{t}+\nu _{t}\leq \psi (t)+6(\#V_{t-1})^{2}\alpha ^{\ast }(t-1),
\label{nu_lambda}
\end{equation
where $\psi (t)=\alpha ^{\ast }t(\#V_{t})+(\#V_{t}-1)+9(\#V_{t-1})^{2}.$
Fix an integer $q.$ Using (\ref{imp3}) and (\ref{nu_lambda}) again and
again, for $t>q$ we hav
\[
\begin{split}
\pi _{t} &\leq 3\pi _{t-1}+\psi (t)+6\alpha ^{\ast }(t-1)(\#V_{t-1})^{2} \\
&\leq 3^{2}\pi _{t-2}+\left( \psi (t)+3\psi (t-1)\right) +6\alpha ^{\ast
}\left( (t-1)(\#V_{t-1})^{2}+18\alpha ^{\ast }(t-2)(\#V_{t-2})^{2}\right) \\
&\leq \cdots \leq 3^{t-q}\pi _{q}+\sum\limits_{k=q}^{t-1}3^{t-k-1}\psi
(k+1)+6\alpha ^{\ast }\sum\limits_{k=q}^{t-1}3^{t-k-1}k(\#V_{k})^{2}.
\end{split}
\]
We can check that
\begin{equation*}
\frac{3^{t-q}\pi _{q}+\sum\nolimits_{k=q}^{t-1}3^{t-k-1}\psi (k+1)}
t(\#V_{t})^{2}}\rightarrow 0\text{ as }t\rightarrow \infty .
\end{equation*
In fact, we only need to estimate \newline
(i) $\frac{\sum\nolimits_{k=q}^{t-1}3^{t-k-1}(k+1)(\#V_{k+1})}{t(\#V_{t})^{2
}=\frac{\sum\nolimits_{k=q+1}^{t}3^{t-k}k\frac{3^{k+1}-1}{2}}{t(\frac
3^{t+1}-1}{2})^{2}}\leq \frac{3}{2}\frac{\frac{t(t+1)}{2}3^{t}}{t(\frac
3^{t+1}-1}{2})^{2}}\rightarrow 0$ as $t\rightarrow \infty ;$ \newline
(ii) $\frac{\sum\nolimits_{k=q}^{t-1}3^{t-k-1}(\#V_{k+1}-1)}{t(\#V_{t})^{2}
\leq \frac{\sum\nolimits_{k=q}^{t}3^{t-k-1}(\#V_{k})^{2}}{t(\#V_{t})^{2}
\leq \frac{1}{4}\frac{3^{t}\sum\nolimits_{k=0}^{t}3^{k+1}}{t(\frac{3^{t+1}-
}{2})^{2}}\rightarrow 0$ as $t\rightarrow \infty .$
Therefore we obtain tha
\[
\begin{split}
\limsup_{t\rightarrow \infty }\frac{\bar{D}(t)}{t} &=\limsup_{t\rightarrow
\infty }\frac{\pi _{t}}{t(\#V_{t}-1)\#V_{t}/2} \\
&\leq 6\alpha ^{\ast }\lim_{t\rightarrow \infty }\frac{\su
\nolimits_{k=q}^{t-1}3^{t-k-1}k(\#V_{k})^{2}}{t(\#V_{t}-1)\#V_{t}/2} \\
&=12\alpha ^{\ast }\lim_{t\rightarrow \infty }\frac{\su
\nolimits_{k=q}^{t-1}3^{t-k-1}k(\#V_{k})^{2}}{t(\#V_{t})^{2}}.
\end{split
\]
Using Stolz theorem, we hav
\[
\begin{split}
\lim_{t\rightarrow \infty }\frac{\sum\nolimits_{k=q}^{t-1}3^{t-k-1}k
\#V_{k})^{2}}{t(\#V_{t})^{2}} &=\frac{1}{3}\lim_{t\rightarrow \infty }\frac
\sum\nolimits_{k=q}^{t-1}k(\#V_{k})^{2}/3^{k}}{t(V_{t})^{2}/3^{t}} \\
&=\frac{1}{3}\lim_{t\rightarrow \infty }\frac{(t-1)(\#V_{t-1})^{2}/3^{t-1}}
t(\#V_{t})^{2}/3^{t}-(t-1)(\#V_{t-1})^{2}/3^{t-1}} \\
&=\lim_{t\rightarrow \infty }\frac{(t-1)(3^{t}-1)^{2}}{6(t\cdot
3^{2t})+3^{2t+1}-6\cdot 3^{t}-2t+3} \\
&=\frac{1}{6}.
\end{split
\]
That means
\begin{equation}
\limsup_{t\rightarrow \infty }\frac{\bar{D}(t)}{t}\leq 2\alpha ^{\ast }.
\label{Dlimsup}
\end{equation}
\textbf{(ii) Lower bound of} $\pi _{t}:$ By (\ref{text}), suppose there
exists an integer $k_{0}$ such that $\frac{\kappa _{t-1}}{t-1}\geq (\alpha
^{\ast }-\varepsilon )$ for all $t\geq k_{0}.$ Using (\ref{imp3}) and (\re
{nu-lower}) we hav
\begin{eqnarray*}
\pi _{t} &\geq &3\pi _{t-1}+\nu _{t} \\
&\geq &3\pi _{t-1}+6(\alpha ^{\ast }-\varepsilon )(\#V_{t-1})^{2}(t-1) \\
&\geq &3^{2}\pi _{t-2}+6(\alpha ^{\ast }-\varepsilon )\left(
(\#V_{t-1})^{2}(t-1)+3(\#V_{t-2})^{2}(t-2)\right) \\
&\geq &\cdots \geq 6(\alpha ^{\ast }-\varepsilon
)\sum\nolimits_{k=k_{0}}^{t-1}3^{t-1-k}k(\#V_{k})^{2}.
\end{eqnarray*
In the same way as above, we obtain that
\begin{equation}
\liminf_{t\rightarrow \infty }\frac{\bar{D}(t)}{t}\geq 2\alpha ^{\ast }.
\label{Dliminf}
\end{equation}
It follows from (\ref{Dlimsup}) and (\ref{Dliminf}) tha
\begin{equation*}
\lim_{t\rightarrow \infty }\frac{\bar{D}(t)}{t}=2\alpha ^{\ast }.
\end{equation*}
\end{proof}
\medskip
\section{Determination of $\alpha^*$}
\subsection{Normal decomposition}
$\ $
Given a word $\sigma ,$ let $C(\sigma )$ be the set of letters appearing in
\sigma ,$ $\#\sigma $ the cardinality of $C(\sigma ),$ and $\sigma |_{-1}$
the last letter of $\sigma .$
For $\sigma =222113112312$, $\omega (\sigma )=4,$ we can obtain a
decomposition
\begin{equation*}
\sigma =(222)(11311)(23)(12)=\tau _{1}\tau _{2}\tau _{3}\tau _{4}
\end{equation*
such that $\tau _{2},\tau _{3},\tau _{4}$ contains $2$ letters and $3\tau
_{4},$ $1\tau _{3},$ $2\tau _{2}$ contains $3$ letters, where $3,$ $1$ and
2 $ are the last letter of $\tau _{3},\tau _{2}$ and $\tau _{1}$
respectively. In the same way, for $\sigma $ with $\omega (\sigma )=l>1,$ we
have the decomposition
\begin{equation}
\sigma =\tau _{1}\tau _{2}\cdots \tau _{l-1}\tau _{l}\text{ with }\omega
(\sigma )=l>1 \label{t-1}
\end{equation
satisfyin
\begin{equation}\label{tt}
\begin{split}
&\#\tau _{1} \leq 2\text{ and }\#\tau _{i}=2\text{ for }i\geq 2, \\
&|\tau _{1}| \geq 1\text{ and }|\tau _{i}|\geq 2\text{ for }i\geq 2, \\
&\{\tau _{i}|_{-1}\} =\{1,2,3\}\backslash C(\tau _{i+1}),
\end{split
\end{equation}
where the last one means the tail of $\tau _{i}$ with $i<l$ is \emph
uniquely determined} by $\tau _{i+1}.$ For $\sigma $ with $\omega (\sigma
)=1,$ we have $\#\sigma \leq 2,$ we give the decomposition
\begin{equation}
\sigma =\tau _{1}\text{ with }\omega (\sigma )=1 \label{t+1}
\end{equation
and (\ref{tt}) also holds. We call the decomposition (\ref{t-1}) or (\re
{t+1}) the \textbf{normal decomposition} if (\ref{tt}) holds. For $k\geq 3,$
let
\begin{eqnarray*}
T(k) &=&\#\{|\tau |=k\text{ with letters in }\{1,2\}:\#\tau =2\}, \\
h(k) &=&\#\{|\tau |=k\text{ with letters in }\{1,2,3\}:\tau |_{-1}=1\text{
and }\#\tau \leq 2\}, \\
M(k) &=&\#\{|\tau |=k\text{ with letters in }\{1,2\}:\tau |_{-1}=1\text{ and
}\#\tau =2\}, \\
e(k) &=&\#\{|\tau |=k\text{ with letters in }\{1,2,3\}:\#\tau \leq 2\}.
\end{eqnarray*
Then
\begin{eqnarray*}
T(k) &=&2^{k}-2,\text{ \quad }h(k)=2^{k}-1, \\
M(k) &=&2^{k-1}-1,\text{ }e(k)=3\cdot 2^{k}-3=3h(k).
\end{eqnarray*
Notice that $e(k)=\#\{|\tau |=k:\omega (\tau )=1\}$ an
\begin{equation}
2M(k)=T(k). \label{4}
\end{equation}
Given $k_{1}+\cdots +k_{l}=t$ with $k_{1}\geq 1$ and $k_{2},\cdots
,k_{l}\geq 2,$ consider
\begin{equation*}
W_{k_{1}\cdots k_{l}}=\#\{|\sigma |=t:\sigma =\tau _{1}\tau _{2}\cdots \tau
_{l-1}\tau _{l}\text{ are normal with }|\tau _{i}|=k_{i}\text{ for all }i\}.
\end{equation*}
\begin{lemma}
If $l\geq 3,$ then
\begin{equation}
W_{k_{1}\cdots k_{l}}=h(k_{1})\left[ (C_{2}^{1}M(k_{2}))\cdots
((C_{2}^{1}M(k_{l-1}))\right] (C_{3}^{2}T(k_{l})).\text{ } \label{hei}
\end{equation
For $l=2,$ we have $W_{k_{1}k_{2}}=h(k_{1})(C_{3}^{2}T(k_{l})).$
\end{lemma}
\begin{proof}
Fix $l\geq 3.$ At first we can choose two distinct letters $i_{1}<i_{2}$
from $\{1,2,3\}$ such that $C(\tau _{l})=\{i_{1},i_{2}\},$ then the number
of choices for $\tau _{l}$ is $C_{3}^{2}T(k_{l}).$ When $\tau _{l}$ is
given, the tail of $\tau _{l-1},$ say $1,$ is uniquely determined by $\tau
_{l},$ then the number of choices for $\tau _{l-1}$ is $C_{2}^{1}M(k_{l-1})
. Again and again, when $\tau _{2}$ is given$,$ then the tail of $\tau _{1}$
is uniquely determined and number of choices for $\tau _{1}$ is $h(k_{1}).$
Then (\ref{hei}) follows.
\end{proof}
Then we have
\begin{equation*}
\sum\limits_{l\geq 1}\sum\limits_{\substack{ k_{1}\geq 1,k_{2},\cdots
k_{l}\geq 3 \\ k_{1}+\cdots +k_{l}=t}}W_{k_{1}\cdots
k_{l}}=e(t)+\#\{|\sigma |=t:\omega (\sigma )\geq 2\}=3^{t},
\end{equation*
and
\begin{equation*}
\sum\limits_{|\sigma |=t}\omega (\sigma )=e(t)+\sum\limits_{l\geq 2}\Bigg(
l\cdot \sum\limits_{\substack{ k_{1}\geq 1,\text{ }k_{2},\cdots ,k_{l}\geq 2
\\ k_{1}+\cdots +k_{l}=t}}W_{k_{1}\cdots k_{l}}\Bigg) .
\end{equation*}
\begin{lemma}
If $l\geq 2,$ the
\begin{equation}
W_{k_{1}\cdots k_{l-1}k_{l}}=T(k_{l})W_{k_{1}\cdots k_{l-1}}\text{.}
\label{test}
\end{equation}
\end{lemma}
\begin{proof}
If $l\geq 3,$ using (\ref{4}) and (\ref{hei}), we obtain (\ref{test}). If
l=2,$ we have
\begin{equation}
W_{k_{1}k_{2}}=h(k_{1})(C_{3}^{2}T(k_{2}))=T(k_{2})(3h(k_{1}))=T(k_{2})(e(k_{1}))=T(k_{2})W_{k_{1}}
\label{666}
\end{equation
since $3h(k)=e(k).$
\end{proof}
\subsection{Proof of Proposition$\ $4}
\
For $x=\cdots x_{2}x_{1}\in \Sigma ,$ let $x|_{-k}=x_{k}x_{k-1}\cdots x_{1}.$
Given $k_{1}\geq 1$ and $k_{2},\cdots ,k_{q}\geq 2,$ consider
\[
\begin{split}
A_{k_{1}\cdots k_{q}}=\{x\in \Sigma : \;&(x1)|_{-(k_{1}+\cdots +k_{q})}=\tau
_{1}\tau _{2}\cdots \tau _{q-1}\tau _{q}\text{ is a } \\
&\text{normal decomposition with }|\tau _{i}|=k_{i}\text{ for all }i\}.
\end{split
\]
Since $1$ is the tail of $\tau _{q}$, we have
\begin{equation}
\mu (A_{k_{1}\cdots k_{q}})=\frac{W_{k_{1}\cdots k_{q}}/3}{3^{k_{1}+\cdots
+k_{q}-1}}=\frac{W_{k_{1}\cdots k_{q}}}{3^{k_{1}+\cdots +k_{q}}}.
\label{haha}
\end{equation}
Suppose $x1=\cdots x_{p_{n+1}}\cdots x_{p_{n}}\cdots x_{p_{1}}\cdots 1\ $and
$Y_{t}(x)=n,$ then
\begin{equation*}
(x1)|_{-t}=(x_{t}\cdots x_{p_{n}})(x_{(p_{n}-1)}\cdots x_{p_{n-1}})\cdots
(x_{p_{2}}\cdots x_{p_{1}})(x_{(p_{1}-1)}\cdots 1)
\end{equation*
with $p_{n+1}>t\geq p_{n}$ and
\begin{equation}
x\in A_{k_{1}k_{2}\cdots k_{n+1}}\text{ with } k_1+\cdots +k_{n+1}=t \text{ and } Y_t(x) =n \label{haha1}
\end{equation
where $k_{1}=t-p_{n}+1 (\geq 1)$ and $k_{i}=p_{n-i+2}-p_{n-i+1} (\geq 2)$ for $i\geq 2.$
\begin{lemma}
\label{l: lim}Suppose $J_{n}=S_{1}+S_{2}+\cdots +S_{n}$ and $Y_{t}=\sup
\{n:J_{n}\leq t\}\ $are defined in Section 2. Then
\begin{eqnarray*}
\liminf\limits_{t\rightarrow \infty }\frac{\sum\nolimits_{|\sigma
|=t}d(\sigma ,\emptyset )}{t3^{t}} &\geq &\liminf\limits_{t\rightarrow
\infty }\frac{\sum\nolimits_{k=2}^{t-1}\mathbb{E}(Y_{t-k})\frac{2^{k}-2}
3^{k}}}{t}, \\
\limsup_{t\rightarrow \infty }\frac{\sum\nolimits_{|\sigma |=t}d(\sigma
,\emptyset )}{t3^{t}} &\leq &\limsup_{t\rightarrow \infty }\frac
\sum\nolimits_{k=2}^{t-1}\mathbb{E}(Y_{t-k})\frac{2^{k}-2}{3^{k}}}{t}.
\end{eqnarray*}
\end{lemma}
\begin{proof}
For $Y_{t^{\prime }}=\sup \{n:J_{n}\leq t^{\prime }\}$, using (\ref{haha1})
we hav
\begin{equation*}
\mathbb{E}(Y_{t^{\prime }})=\sum_{q}\sum\limits_{\substack{ k_{1}+\cdots
+k_{q}=t^{\prime } \\ k_{1}\geq 1,k_{2},\cdots ,k_{q}\geq 2}}(q-1)\mu
(A_{k_{1}\cdots k_{q}}).
\end{equation*
Using (\ref{test}) and (\ref{haha}), we obtain that
\begin{eqnarray*}
&&\frac{\sum\nolimits_{|\sigma |=t}\omega (\sigma )}{3^{t}} \\
&=&\frac{e(t)}{3^{t}}+\sum\limits_{k=2}^{t-1}\sum\limits_{l\geq 2}\Bigg(
l\cdot \frac{T(k)}{3^{k}}\sum\limits_{\substack{ k_{1}+\cdots +k_{l-1}=t-k
\\ k_{1}\geq 1,k_{2},\cdots ,k_{l-1}\geq 2}}\mu (A_{k_{1}\cdots
k_{l-1}})\Bigg)
\end{eqnarray*
where $\frac{e(t)}{3^{t}}\rightarrow 0$. We notice tha
\begin{eqnarray*}
&&\sum\limits_{k=2}^{t-1}\sum\limits_{l\geq 2}\sum\limits_{\substack{
k_{1}+\cdots +k_{l-1}=t-k \\ k_{1}\geq 1,k_{2},\cdots ,k_{l-1}\geq 2}}l\cdot
\frac{T(k)}{3^{k}}\mu (A_{k_{1}\cdots k_{l-1}}) \\
&\leq &2\sum\limits_{k\geq 2}\frac{T(k)}{3^{k}}+\sum\limits_{k=2}^{t-1}\su
\limits_{l\geq 2}\sum\limits_{\substack{ k_{1}+\cdots +k_{l-1}=t-k \\
k_{1}\geq 1,k_{2},\cdots ,k_{l-1}\geq 2}}(l-2)\frac{T(k)}{3^{k}}\mu
(A_{k_{1}\cdots k_{l-1}}) \\
&\leq &2+\sum\limits_{k=2}^{t-1}\mathbb{E}(Y_{t-k})\frac{T(k)}{3^{k}}.
\end{eqnarray*
On the other hand, we hav
\begin{eqnarray*}
&&\sum\limits_{k=2}^{t-1}\sum\limits_{l\geq 2}\sum\limits_{\substack{
k_{1}+\cdots +k_{l-1}=t-k \\ k_{1}\geq 1,k_{2},\cdots ,k_{l-1}\geq 2}}l\cdot
\frac{T(k)}{3^{k}}\mu (A_{k_{1}\cdots k_{l-1}}) \\
&\geq &\sum\limits_{k=2}^{t-1}\sum\limits_{l\geq 2}\sum\limits_{\substack{
k_{1}+\cdots +k_{l-1}=t-k \\ k_{1}\geq 1,k_{2},\cdots ,k_{l-1}\geq 2}}(l-2
\frac{T(k)}{3^{k}}\mu (A_{k_{1}\cdots k_{l-1}}) \\
&\geq &\sum\limits_{k=2}^{t-1}\mathbb{E}(Y_{t-k})\frac{T(k)}{3^{k}}.
\end{eqnarray*
Notice that $\frac{e(t)}{3^{t}}\rightarrow 0,$ then the lemma follows.
\end{proof}
\begin{proof}[Proof of Proposition 4]
Notice that $\frac{\mathbb{E}(Y_{i})}{i}\rightarrow \frac{2}{9}$ as
i\rightarrow \infty $ and
\begin{equation*}
\frac{\sum\nolimits_{k=2}^{t-1}\left( \mathbb{(}t-k\mathbb{)\cdot }\frac
2^{k}-2}{3^{k}}\right) }{t}\rightarrow 1\text{ as }t\rightarrow \infty ,
\end{equation*
using Lemma \ref{l: lim} we obtain tha
\begin{eqnarray*}
\alpha ^{\ast }=\lim_{t\rightarrow \infty }\frac{\sum\nolimits_{|\sigma
|=t}\omega (\sigma )}{t3^{t}} &=&\lim_{t\rightarrow \infty }\frac
\sum\nolimits_{k=2}^{t-1}\mathbb{E}(Y_{t-k})\frac{2^{k}-2}{3^{k}}}{t} \\
&=&\lim_{t\rightarrow \infty }\frac{\sum\nolimits_{k=2}^{t-1}\frac{\mathbb{E
(Y_{t-k})}{t-k}(t-k)\frac{2^{k}-2}{3^{k}}}{t} \\
&=&\lim_{i\rightarrow \infty }\frac{\mathbb{E}(Y_{i})}{i}=\frac{2}{9}.
\end{eqnarray*
\bigskip
\end{proof}
| {
"timestamp": "2015-08-06T02:04:12",
"yymm": "1508",
"arxiv_id": "1508.00956",
"language": "en",
"url": "https://arxiv.org/abs/1508.00956",
"abstract": "In this paper, we introduce a new method to construct evolving networks based on the construction of the Sierpinski gasket. Using self-similarity and renewal theorem, we obtain the asymptotic formula for average path length of our evolving networks.",
"subjects": "Metric Geometry (math.MG)",
"title": "Asymptotic formula on average path length of fractal networks modelled on Sierpinski Gasket",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754501811437,
"lm_q2_score": 0.7185943805178139,
"lm_q1q2_score": 0.7075103856959667
} |
https://arxiv.org/abs/1909.13406 | Embedding Dimension Phenomena in Intersection Complete Codes | Two tantalizing invariants of a combinatorial code $\mathcal C\subseteq 2^{[n]}$ are cdim$(\mathcal C)$ and odim$(\mathcal C)$, the smallest dimension in which $\mathcal C$ can be realized by convex closed or open sets, respectively. Cruz, Giusti, Itskov, and Kronholm showed that for intersection complete codes $\mathcal C$ with $m+1$ maximal codewords, odim$(\mathcal C)$ and cdim$(\mathcal C)$ are both bounded above by $\max\{2,m\}$. Results of Lienkaemper, Shiu, and Woodstock imply that odim and cdim may differ, even for intersection complete codes. We add to the literature on open and closed embedding dimensions of intersection complete codes with the following results:(*) If $\mathcal C$ is a simplicial complex, then cdim$(\mathcal{C}) = \mbox{odim}(\mathcal C)$,(*) If $\mathcal C$ is intersection complete, then cdim$(\mathcal C)\le \mbox{odim}(\mathcal C)$,(*) If $\mathcal C\subseteq 2^{[n]}$ is intersection complete with $n\ge 2$, then cdim$(\mathcal C) \le \min \{2d+1, n-1\}$, where $d$ is the dimension of the simplicial complex of $\mathcal C$, and(*) For each simplicial complex $\Delta\subseteq 2^{[n]}$ with $m\ge 2$ facets, the code $\mathcal S_\Delta := (\Delta \ast (n+1)) \cup \{[n]\}\subseteq 2^{[n+1]}$ is intersection complete, has $m+1$ maximal codewords, and satisfies odim$(\mathcal S_\Delta)=m$. In particular, for each $n\ge 3$ there exists an intersection complete code $\mathcal C\subseteq 2^{[n]}$ with odim$(\mathcal C) = \binom{n-1}{\lfloor (n-1)/2\rfloor}$.A key tool in our work is the study of sunflowers: arrangements of convex open sets in which the sets simultaneously meet in a central region, and nowhere else. We use Tverberg's theorem to study the structure of "$k$-flexible" sunflowers, and consequently obtain new lower bounds on odim$(\mathcal C)$ for intersection complete codes $\mathcal C$. | \section{Introduction}\label{sec:intro}
In \cite{neuralring13}, Curto, Itskov, Veliz-Cuba and Youngs introduced \emph{convex codes} to mathematically model stimulus reconstruction from neural data, particularly in the context of hippocampal place cells. Classifying and understanding convex codes has been an active area of recent mathematical research, bringing together tools and perspectives from topology \cite{local15, connected}, algebra \cite{grobner, polarization, factorcomplex}, and discrete geometry \cite{undecidability, polarcomplex,sunflowers,CUR, obstructions}. A complete classification of convex codes is far out of reach for the moment, but progress can yield new techniques for analyzing neural data, as well as a deeper understanding of the mathematical theory of convex sets. In this paper, we give new bounds on the open and closed embedding dimensions of intersection complete codes, including families of examples where these bounds are tight. In particular, we provide infinite families of intersection complete codes for which open embedding dimension grows exponentially in the number of neurons, while closed embedding dimension grows only linearly.
Before stating our results we recall some definitions and frame our main questions of study. A convex code (see Definition \ref{def:convexcode} below) is a special case of a \emph{combinatorial code}, which is a collection of subsets of $[n]:= \{1,\ldots, n\}$. Due to the biological motivation behind our work, we think of the elements of $[n]$ as \emph{neurons}, and each element of a code as recording a set of neurons which fired together in a small window of time.
The elements of a code are called \emph{codewords}, and for concision we often omit braces and commas when writing codewords. For example, we may write 124 instead of $\{1,2,4\}$. The \emph{weight} of a codeword is simply the number of neurons it contains. We will often think of a code as a partially ordered set under containment---for example, we may speak of maximal codewords, which are not properly contained in any other codeword. When writing down a specific code, we will bold the maximal codewords.
Codes can arise abstractly when one wishes to describe how a certain collection of sets covers a space, as follows. Let $X$ be a set, and $\U = \{U_1,\ldots, U_n\}$ a collection of subsets of $X$. One may form the \emph{code of $\U$ in $X$}, a combinatorial code whose codewords describe how the $U_i$ intersect and cover one another: \[
\code(\U, X) := \bigg\{\sigma\subseteq [n]\ \bigg|\ \bigcap_{i\in\sigma} U_i \setminus \bigcup_{j\in[n]\setminus\sigma} U_j\neq \emptyset\bigg\}.
\]
The region $\bigcap_{i\in\sigma} U_i \setminus \bigcup_{j\in[n]\setminus\sigma} U_j$ is called the \emph{atom} of $\sigma$, and denoted $\mathcal A_\U^\sigma$. The space $X$ is called the \emph{ambient space} or \emph{stimulus space}, and the $U_i$ are called \emph{receptive fields} or \emph{firing regions}. Note that the receptive fields are indexed by neurons. If $\mathcal C = \code(\U, X)$, then the collection $\U$ is called a \emph{realization} of $\mathcal C$ in $X$. For concision, we will write $U_\sigma$ for $\bigcap_{i\in\sigma} U_i$, and adopt the convention that $U_\emptyset = X$.
Unless otherwise specified, throughout this paper the ambient space will be $\mathbb R^d$, and the $U_i$ will be (possibly empty) convex sets that are either all open, or all closed. We will write $\code(\U)$ instead of $\code(\U, \mathbb R^d)$ when the ambient dimension is clear. We will also adopt the usual convention in the study of convex codes that $\emptyset$ is contained in all codes, i.e. that there is always a point in the ambient space not covered by any $U_i$.
\begin{definition}\label{def:convexcode}
A code $\mathcal C\subseteq 2^{[n]}$ is called an \emph{open convex} code if it has a realization consisting of convex open sets in $\mathbb R^d$. Similarly, $\mathcal C$ is called \emph{closed convex} if it has a realization consisting of closed convex sets in $\mathbb R^d$.
\end{definition}
\begin{example}\label{ex:realization}
The figure below shows a realization in $\mathbb R^2$ of the (open/closed) convex code $\mathcal C = \{\mathbf{123}, 12, 23, 2, 3, \emptyset\}$. The atom $\mathcal A_\U^{23}$ is highlighted in grey.
\[
\includegraphics[width=22em]{example_realization.pdf}
\]
\end{example}
In the neuroscientific context mentioned above, open convex codes are of greater interest than closed convex codes, since receptive fields have been experimentally observed to be full-dimensional (see \cite[Figure 1]{local15} for example). However, we will also study closed convex codes in this paper to build on the work of \cite{openclosed}, and to contrast their behavior to that of open convex codes. Moreover, it is of broad mathematical interest to develop our understanding of closed convex sets and their intersection patterns, since they are ubiquitous in fields such as optimization and discrete geometry.
The study of convex codes asks two main questions. First, given a code $\mathcal C\subseteq 2^{[n]}$, \textbf{when can we find a (closed or open) convex realization of $\mathbf{\mathcal C}$?} Second, if we can find a realization, \textbf{what is the smallest dimension in which we can do so?} Formally, we wish to investigate the open and closed embedding dimensions of combinatorial codes, described below.
\begin{definition}\label{def:odim}\label{def:cdim}
Let $\mathcal C\subseteq 2^{[n]}$ be a code. The \emph{open embedding dimension} of $\mathcal C$, denoted $\odim(\mathcal C)$, is the smallest $d$ so that $\mathcal C$ has a realization in $\mathbb R^d$ consisting of convex open sets, or $\infty$ if no open realization exists. Similarly, the \emph{closed embedding dimension}, denoted $\cdim(\mathcal C)$, is the smallest dimension in which $\mathcal C$ has a closed convex realization, or $\infty$ if none exist.
\end{definition}
Note that the realization in Example \ref{ex:realization} is not minimal with respect to dimension, since we could flatten the $U_i$ into (closed or open) intervals to obtain a realization in $\mathbb R^1$. Thus $\odim(\mathcal C) = \cdim(\mathcal C) = 1$ for $\mathcal C = \{\mathbf{123}, 12, 23, 2, 3, \emptyset\}$.
In this paper we will study codes that are \emph{intersection complete}: the intersection of any two codewords is again a codeword. It was shown in \cite{openclosed} that such codes are always convex, and their open and closed embedding dimensions are bounded above by $\max\{2, k-1\}$, where $k$ is the number of maximal codewords. Our work provides new upper and lower bounds on open and closed embedding dimension, as well as examples where these bounds are tight. Except where stated otherwise, every code we work with in this paper is intersection complete.
A special case of intersection complete codes is that of a simplicial complex. For simplicial complexes, open and closed embedding dimensions are equal. Although this result is well known among the neural codes community, we are not aware of any written proofs. We provide one below.
\begin{theorem}\label{thm:complexes}
Let $\mathcal C\subseteq 2^{[n]}$ be a simplicial complex. Then $\cdim(\mathcal C) = \odim(\mathcal C)$.
\end{theorem}
\begin{proof}
In Theorem \ref{thm:cdimleodim}, we will show that $\cdim(\mathcal C)\le \odim(\mathcal C)$. Thus we just need to prove that $\odim(\mathcal C)\le\cdim(\mathcal C)$. Let $\V = \{V_1,\ldots, V_n\}$ be a closed realization of $\mathcal C$ in $\mathbb R^{\cdim(\mathcal C)}$. By intersecting all the $V_i$ with a sufficiently large closed ball, we may assume that they are bounded, and hence compact. For each nonempty codeword $c\in\mathcal C$, choose a point $p_c\in \mathcal A_\V^c$. By compactness, each $p_c$ has positive distance to any set $V_i$ that does not contain it. Likewise, any disjoint $V_\sigma$ and $V_\tau$ have positive distance between one another. Thus we may choose $\varepsilon$ such that replacing the $V_i$ by their Minkowski sums with a $\varepsilon$-ball neither causes any $V_i$ to cover some $p_c$ it did not before, nor causes disjoint $V_\sigma$ and $V_\tau$ to intersect. This creates a collection of convex open sets whose code contains all the codewords of $\mathcal C$, and no new maximal codewords. Since $\mathcal C$ is a simplicial complex, this is exactly a convex open realization of $\mathcal C$.
\end{proof}
\begin{example}
Consider the code $\mathcal C = \{\mathbf{123},\mathbf{34}, 12, 13, 23, 1,2,3,4,\emptyset\}$, and note that $\mathcal C$ is a simplicial complex. The figure below shows a realization of $\mathcal C$ in $\mathbb R^2$ with closed convex sets, as well as possible choices of points $p_c$ for $c\in \mathcal C$ as used in the proof above. The righthand side shows the open realization given in the proof above, which results from adding a small $\varepsilon$-ball to each $V_i$.
\[
\includegraphics[width=34em]{2_closed_to_open.pdf}
\]
\end{example}
Can the above techniques be extended to realizations of codes that are not simplicial complexes? The answer in general is no, even for intersection complete codes, a fact which was first observed implicitly in the results of \cite{sunflowers, obstructions}. Corollary \ref{cor:exponential} will yield a plethora of examples of intersection complete codes on $n$ neurons that are closed convex in $\mathbb R^{n-1}$, but not open convex in $\mathbb R^{n-1}$. For such codes, adding an $\varepsilon$-ball to sets in a closed realization in $\mathbb R^{n-1}$ will always fail to produce an open realization.
The following theorems are the main contributions of this work, and give us a handle on how open and closed dimension behave for intersection complete codes.
\begin{theorem}\label{thm:cdimleodim}
Let $\mathcal C\subseteq 2^{[n]}$ be an intersection complete code. Then $\cdim(\mathcal C)\le \odim(\mathcal C)$.
\end{theorem}
It is known that this inequality may be strict for intersection complete codes $\mathcal C\subseteq 2^{[n]}$, as mentioned above. In fact, the gap may be quite large: Theorem \ref{thm:cdimlinear} implies that $\cdim(\mathcal C)\le n-1$, but Corollary \ref{cor:exponential} says that $\odim(\mathcal C)$ \textbf{may be exponential in $n$}.
\begin{theorem}\label{thm:cdimlinear}
Let $\mathcal C\subseteq 2^{[n]}$ be an intersection complete code, and $d$ be one less than the weight of the largest codeword in $\mathcal C$ (i.e. $d=\dim(\Delta(\mathcal C))$). Then $\cdim(\mathcal C)\le \min \{2d+1, n-1\}$.
\end{theorem}
This bound is known to be tight. For every $d\ge 0$, \cite{2dplus1} describes a $d$-dimensional simplicial complex on $n$ vertices whose closed embedding dimension is exactly $2d+1$ (which, in the family given, is the same as $\min\{2d+1, n-1\}$). Interestingly, the bound above does not hold for $\odim(\mathcal C)$. Theorem \ref{thm:SDelta} below gives us a way to construct numerous examples of intersection complete codes for which $\odim(\mathcal C) \gg \min \{2d+1, n-1\}$.
\begin{definition}\label{def:SDelta}
Let $\Delta\subseteq 2^{[n]}$ be a simplicial complex. Define $\S_\Delta\subseteq 2^{[n+1]}$ to be the code \[
\S_\Delta := \left(\Delta \ast (n+1)\right)\cup\{[n]\},
\]
where $\Delta\ast (n+1)$ denotes the cone over $\Delta$ with apex $n+1$.
\end{definition}
\begin{theorem}\label{thm:SDelta}
Let $\Delta\subseteq 2^{[n]}$ be a simplicial complex with $m\ge 2$ facets. Then $\S_\Delta$ is an intersection complete code with $m+1$ maximal codewords, and $\odim(\S_\Delta) = m$.
\end{theorem}
A key tool in proving Theorem \ref{thm:SDelta} is an application of a ``Sunflower Theorem" that we proved in \cite{sunflowers}. In this paper, we will generalize this theorem to ``$k$-flexible" sunflowers of convex open sets, defined formally below. These are collections of convex open sets which have a common intersection, but which do not overlap with degree more than $k$ outside of this common intersection.
\begin{definition}\label{def:flexiblesunflower}
Let $\U = \{U_1,\ldots, U_n\}$ be a collection of convex sets in $\mathbb R^d$ and let $\mathcal C = \code(\U)$. The collection $\U$ is called a \emph{$k$-flexible sunflower} if $[n]\in\mathcal C$, and all other codewords have weight at most $k$. The $U_i$ are called \emph{petals} and $U_{[n]}$ is called the \emph{center} of $\U$. \end{definition}
The following theorem tells us that if a $k$-flexible sunflower $\U$ in $\mathbb R^d$ has ``enough petals," then sampling a point from each petal and taking the convex hull always yields a point in the center of $\U$. Our proof of this theorem is given in Section \ref{sec:flexible} and relies on an application of Tverberg's theorem.
By considering a set of line segments in $\mathbb R^2$ which meet at a point, one can see that this result does not hold for closed convex sets.
\begin{theorem}\label{thm:flexible}
Let $\U = \{U_1,\ldots, U_n\}$ be an open $k$-flexible sunflower in $\mathbb R^d$. Suppose that $n\ge dk+1$, and for each $i\in[n]$ let $p_i\in U_i$. Then $\conv\{p_1,\ldots, p_n\}$ contains a point in the center of $\U$. Moreover, if $d\ge 2$ this result may fail when $n < dk+1$.
\end{theorem}
\begin{example}
Consider the $2$-flexible sunflower $\{U_1,U_2,U_3,U_4,U_5\}$ in $\mathbb R^2$ below. The center of this sunflower is the unit square highlighted in gray. Note that $d=2$, $k=2$ and $n=5$. Thus $n \ge dk+1$, and so Theorem \ref{thm:flexible} applies. Indeed, any choice of $p_1\in U_1,\ldots, p_5\in U_5$ has the property that $\conv\{p_1,p_2,p_3,p_4,p_5\}$ intersects the center of the sunflower. One choice of such points is shown below.
\[
\includegraphics[width=18em]{3_flexible_example.pdf}
\]
One last observation worth making about the above figure is that deleting $U_5$ yields a $2$-flexible sunflower in $\mathbb R^2$ for which the conclusion of Theorem \ref{thm:flexible} does not hold: the set $\conv\{p_1, p_2, p_3,p_4\}$ does not intersect the center of $\{U_1, U_2, U_3, U_4\}$.
\end{example}
In Section \ref{sec:background} we will recall some relevant background material. The subsequent sections are devoted to proving the theorems stated above, with one self-contained section per theorem. An exception to this is Section \ref{sec:sunflowercodeversion}, which should be read accompanying Section \ref{sec:SDelta} since it provides some important supporting results.
Section \ref{sec:Tn} describes a new family $\mathcal T_n$ of intersection complete codes, and initiates the study of their open embedding dimensions. The codes $\mathcal T_n$ are related to sunflowers, but the theorems that we prove regarding sunflowers are not sufficient to precisely determine $\odim(\mathcal T_n)$.
Section \ref{sec:pcode} provides a unifying capstone to our results. We contextualize our new bounds and examples by examining a partially ordered set $\mathbf{P}_\Code$ consisting of all neural codes, which was first introduced in \cite{morphisms}. We show that some of our bounds on open embedding dimension can be proven combinatorially using this partial order. We also generalize Definition \ref{def:SDelta}, and apply Theorem \ref{thm:flexible} to prove a generalization of Theorem \ref{thm:SDelta}, viewing these results through the lens of $\mathbf{P}_\Code$.
\section{Background and Preliminaries}\label{sec:background}
Throughout this paper we will assume familiarity with standard concepts in topology and convex geometry; for example the interior, closure, and boundary of a set in $\mathbb R^d$, convex hulls, hyperplanes, and halfspaces. Recall that each hyperplane $H$ in $\mathbb R^d$ may be given an orientation, so that we can speak of the (open) halfspaces $H^>$ and $H^<$ consisting of points lying on the positive and negative sides of $H$, respectively. We will also use $H^\ge$ and $H^\le$ to denote the (closed) non-negative and non-positive respective halfspaces associated to $H$. For any convex set $U\subseteq \mathbb R^d$, and any boundary point $p$ of $U$, one can find a \emph{supporting hyperplane} through $p$: an oriented hyperplane $H$ containing $p$ with $U\subseteq H^\ge$.
Below, we provide additional background on convex codes, simplicial complexes, and polytopes.
\subsection{Convex Codes}
In Section \ref{sec:intro} we gave a brief overview of the theory of convex neural codes. We will need one additional concept related to neural codes, described below.
\begin{definition}\label{def:trunk}
Let $\mathcal C\subseteq 2^{[n]}$ be a code, and let $\sigma\subseteq [n]$. The \emph{trunk} of $\sigma$ in $\mathcal C$ is \[
\Tk_\mathcal C(\sigma) := \{c\in \mathcal C\mid \sigma\subseteq c\}.
\]
A subset of $\mathcal C$ is called a trunk if it is empty, or equal to $\Tk_\mathcal C(\sigma)$ for some $\sigma\subseteq [n]$. When $\sigma = \{i\}$ we will call $\Tk_\mathcal C(\sigma)$ a \emph{simple} trunk, and denote it $\Tk_\mathcal C(i)$.
\end{definition}
We introduced trunks in \cite{morphisms} and used them (and a consequent notion of morphism) to define a convenient partial order on neural codes, in which convex codes form a down-set. We will make use of this partial order to contextualize our results in Section \ref{sec:pcode}.
It is worth briefly justifying our requirement that realizations consist of all closed or all open sets. As mentioned in Section \ref{sec:intro}, openness is a natural requirement from the perspective of neuroscience, in which receptive fields are full-dimensional and do not terminate in sharp boundaries. From a mathematical perspective, requiring closed or open sets is also natural, so that we may think of the receptive fields as a collection of closed or open sets covering some topological subspace of $\mathbb R^d$. A further reason to place topological constraints on the sets in our realizations is the following: in \cite{allcodesconvex}, it was shown that every code has a realization consisting of convex sets (possibly neither open nor closed). Thus topological constraints are imperative to make the overall question of classifying convex codes meaningful.
\subsection{Simplicial Complexes}
For our purposes, an \emph{(abstract) simplicial complex} is just a code that is closed under taking subsets (i.e. a subset of a codeword is again a codeword). If $\Delta\subseteq 2^{[n]}$ is a simplicial complex, the maximal codewords may be called \emph{facets}, the codewords called \emph{faces}, and elements of $[n]$ called \emph{vertices}. Observe that every simplicial complex is uniquely specified by its facets together with the vertex set $[n]$. In contrast to the usual theory of simplicial complexes, we allow the case in which $i$ is a vertex but $\{i\}\notin \mathcal C$.
The \emph{dimension} of a simplicial complex $\Delta$, denoted $\dim(\Delta)$, is one less than the size of the largest face in $\Delta$.
If $\Delta\subseteq 2^{[n]}$ is a simplicial complex, and $m > n$, the \emph{cone over $\Delta$ with apex $m$} is the simplicial complex \[
\Delta \ast m := \{\sigma \subseteq [m] \mid \sigma\setminus \{m\} \in \Delta\}.
\]
That is, $\Delta\ast m$ is the simpicial complex whose facets are the facets of $\Delta$ with $m$ added to them.
Finally, for any code $\mathcal C\subseteq 2^{[n]}$, the \emph{simplicial complex of $\mathcal C$}, denoted $\Delta(\mathcal C)$, is the smallest simplicial complex containing $\mathcal C$.
\subsection{Polytopes and Polytopal Complexes}
A \emph{polytope} is the convex hull of a finite set of points in $\mathbb R^d$, or equivalently a bounded intersection of finitely many closed halfspaces. The \emph{dimension} of a polytope is the dimension of its affine hull. The \emph{(proper) faces} of a $d$-dimensional polytope in $\mathbb R^d$ are its intersections with supporting hyperplanes; faces consisting of a single point are called \emph{vertices}, and maximal faces are called \emph{facets}. We will also consider the empty set to be a proper face of any polytope $P$, and its associated supporting hyperplane to be any hyperplane that does not intersect $P$.
One can partially order the faces of a polytope by inclusion to form its \emph{face poset}. Two polytopes are called \emph{combinatorially equivalent} if their face posets are isomorphic. Every polytope $P\subseteq \mathbb R^d$ admits a \emph{dual polytope} $P^*\subseteq \mathbb R^d$, which has the property that the face poset of $P^*$ is isomorphic to the dual of the face poset of $P$ (i.e. one obtains the face poset of $P^*$ by turning the face poset of $P$ upside down).
A polytope is called \emph{$d$-neighborly} if the convex hull of any $d$ of its vertices is a face. Conveniently, $d$-neighborly polytopes with an arbitrarily large number of vertices can always be found in $\mathbb R^{2d}$ (e.g. the cyclic polytope; see \cite[Corollary 0.8]{ziegler-polytopesbook}).
A \emph{polytopal complex} in $\mathbb R^d$ is a finite set of polytopes $\P$ with the properties that (i) if $P\in \P$, then any face of $P$ is also in $\P$, and (ii) the intersection of two polytopes $P_1,P_2\in \P$ is a face of both $P_1$ and $P_2$. Polytopes in $\P$ are called \emph{faces}.
Each polytopal complex $\mathcal P$ has a \emph{face poset}, consisting of all faces in $\P$ partially ordered by containment. Two polytopal complexes are called \emph{combinatorially equivalent} if their face posets are isomorphic. Maximal faces in $\P$ are called \emph{facets}, and if all facets have the same dimension then $\P$ is called \emph{pure}. Finally, we say that a polytopal complex $\P$ in $\mathbb R^d$ is \emph{full-dimensional} if it has a facet of dimension $d$.
Given a $d$-dimensional polytope $P\subseteq \mathbb R^d$ and a facet $F$ of $P$, one can form a pure, full-dimensional polytopal complex in $\mathbb R^{d-1}$ called the \emph{Schlegel diagram of $P$ based at $F$}. Roughly, one does this by ``looking through" the facet $F$ to project all other faces of $P$ into $\mathbb R^{d-1}$. The key fact about Schlegel diagrams that we will need is the following: as a polytopal complex, the Schlegel diagram is combinatorially equivalent to the complex of all proper faces of $P$, but with $F$ removed.
For further background on polytopes and polyhedral complexes, we refer the reader to \cite[Chapter 5]{ziegler-polytopesbook}.
\section{Closed Embedding Dimension is Bounded by Open Embedding Dimension}\label{sec:cdimleodim}
To begin our investigation, we recall a useful characterization of intersection complete codes in terms of their realizations. This fact has been observed before in various forms, for example \cite[Theorem 1.9]{signatures}.
\begin{proposition}\label{prop:singlecover}
A code $\mathcal C\subseteq 2^{[n]}$ is intersection complete if and only if the following holds: for all $\sigma\subseteq \Delta(\mathcal C)\setminus\mathcal C$ and all (possibly non-convex) realizations $\U = \{U_1,\ldots U_n\}$ of $\mathcal C$ there is some $i\in [n]\setminus \sigma$ with $U_\sigma\subseteq U_i$.
\end{proposition}
\begin{proof}
First suppose that $\mathcal C$ is intersection complete, and has a realization $\U = \{U_1,\ldots,U_n\}$. Let $\sigma\in \Delta(\mathcal C)\setminus \mathcal C$ and define $c_0 = \bigcap_{c\in\Tk_\mathcal C(\sigma)} c$. The trunk $\Tk_\mathcal C(\sigma)$ is nonempty since $\sigma\in \Delta(\mathcal C)$, and $c_0\in \mathcal C$ since $\mathcal C$ is intersection complete. Moreover, $\sigma$ is a proper subset of $c_0$ since $\sigma\notin\mathcal C$. Thus we may choose $i\in c_0\setminus \sigma$.
We claim that $U_\sigma\subseteq U_i$. Indeed, since $c_0$ is the unique minimal element of $\Tk_\mathcal C(\sigma)$, every codeword containing $\sigma$ also contains $i$. This implies that $U_\sigma\subseteq U_i$.
For the converse, we prove the contrapositive. Suppose that $\mathcal C$ is not intersection complete, so there exist $c_1$ and $c_2$ in $\mathcal C$ such that $c_1\cap c_2\notin \mathcal C$. Define $\sigma = c_1\cap c_2$ and note that $\sigma\in \Delta(\mathcal C)\setminus \mathcal C$. Then choose any (possibly non-convex) realization $\U = \{U_1,\ldots, U_n\}$ of $\mathcal C$, and let $i\in [n]\setminus \sigma$. Observe that $i$ is contained in at most one of $c_1$ and $c_2$. Since $U_\sigma$ contains $U_{c_1}$ and $U_{c_2}$, it follows that there is a point in $U_\sigma$ that is not contained in $U_i$. This proves the result. \end{proof}
In addition to Proposition \ref{prop:singlecover}, we will need the following ``trimming" operation, which was also employed in \cite{sparse}.
\begin{definition}\label{def:trim}
Let $U\subseteq \mathbb R^d$ be any set and $\varepsilon>0$. The \emph{trim} of $U$ by $\varepsilon$ is the set \[
\trim(U,\varepsilon) := \{p\in U\mid B_\varepsilon(p)\subseteq U\},
\]
where $B_\varepsilon(p)$ is the closed ball of radius $\varepsilon$ centered at $p$.
\end{definition}
\begin{proposition}\label{prop:trimconvex}
If $U\subseteq \mathbb R^d$ is convex and open, then $\trim(U,\varepsilon)$ is convex and open for any $\varepsilon >0$. Moreover, $\cl(\trim(U,\varepsilon))\subseteq U$.
\end{proposition}
\begin{proof}
Let $p\in \trim(U,\varepsilon)$. Since $B_\varepsilon(p)$ is a closed subset of $U$ and $U$ is open and convex, there exists $\delta > 0$ such that $B_{\varepsilon+\delta}(p) \subseteq U$. This implies that the open ball of radius $\delta$ centered at $p$ is contained in $\trim(U,\varepsilon)$. Thus $p$ is an interior point of $\trim(U, \varepsilon)$, so $\trim(U,\varepsilon)$ is open.
Next let $p$ and $q$ be points in $\trim(U,\varepsilon)$. By convexity of $U$, the Minkowski sum $C = \overline{pq} + B_\varepsilon(0)$ is contained in $U$. For any $r$ on $\overline{pq}$, this implies that $B_\varepsilon(r) \subseteq C\subseteq U$. Thus $r$ lies in $\trim(U,\varepsilon)$, proving that $\trim(U,\varepsilon)$ is convex.
For the final statement, observe that no boundary point of $U$ is a boundary point of $\trim(U,\varepsilon)$. Thus all boundary points of $\trim(U,\varepsilon)$ lie in $U$, and the closure $\cl(\trim(U,\varepsilon))$ must be a subset of $U$.
\end{proof}
\begin{proposition}\label{prop:trimcommutes}
Let $U$ and $V$ be sets in $\mathbb R^d$. Then $\trim(U\cap V,\varepsilon) = \trim(U,\varepsilon)\cap \trim(V,\varepsilon)$. If $U\subseteq V$, then $\trim(U,\varepsilon)\subseteq \trim(V,\varepsilon)$.
\end{proposition}
\begin{proof}
The first statement follows from the fact that $B_\varepsilon(p)$ is contained in both $U$ and $V$ if and only if it is contained in their intersection. The second statement is immediate from Definition \ref{def:trim}.
\end{proof}
A notion of non-degeneracy for realizations was introduced in \cite{openclosed}. Intuitively, non-degeneracy requires that the different regions in the realization do not get too close to one another, unless they intersect. The formal definition is given below.
\begin{definition}[\cite{openclosed}]\label{def:nondegen}
A collection $\U = \{U_1,\ldots, U_n\}$ of convex sets in $\mathbb R^d$ is called \emph{non-degenerate} if the following two conditions hold:\begin{itemize}
\item[(i)] For all $\sigma\in \code(\U)$, the atom $\mathcal A_\U^\sigma$ is top dimensional (i.e. its intersection with any open set is either empty, or has nonempty interior).
\item[(ii)] For all nonempty $\sigma\subseteq [n]$, we have $\bigcap_{i\in\sigma} \partial U_i \subseteq \partial U_\sigma$.
\end{itemize}
\end{definition}
When $\U$ is a collection of convex open sets, \cite{openclosed} proved that (ii) implies (i). We will show that trimming an open realization of an intersection complete code $\mathcal C$ by a sufficiently small $\varepsilon$ yields a non-degenerate realization of $\mathcal C$.
\begin{lemma}\label{lem:trimrealization}
Let $\mathcal C\subseteq 2^{[n]}$ be an intersection complete code, and let $\U = \{U_1,\ldots, U_n\}$ be a convex open realization of $\mathcal C$. Then there exists $\varepsilon >0$ such that the sets $V_i = \trim(U_i,\varepsilon)$ form a non-degenerate convex open realization of $\mathcal C$.\end{lemma}
\begin{proof}
For each codeword $c\in\mathcal C$, choose a point $p_c\in \mathcal A_\U^c$. Observe that we may choose $\varepsilon$ small enough that $B_\varepsilon(p_c)\subseteq U_c$ for all $c\in\mathcal C$. We claim that this suffices.
Note that by choice of $\varepsilon$, $p_c\in V_c$ for all $c\in\mathcal C$. In particular, if $U_\sigma$ is nonempty then so is $V_\sigma$.
To prove that $\mathcal C = \code(\{V_1,\ldots, V_n\})$, we must show for all nonempty $\sigma\subseteq [n]$ that $U_\sigma$ is covered by $\{U_i\mid i\in[n]\setminus \sigma\}$ if and only if $V_\sigma$ is covered by $\{V_i\mid i\in [n]\setminus \sigma\}$. Suppose first that $U_\sigma$ is covered by $\{U_i\mid i\in[n]\setminus \sigma\}$. By Proposition \ref{prop:singlecover}, there exists some $i\in[n]\setminus \sigma$ with $U_\sigma\subseteq U_i$. But since trimming commutes with intersections and preserves containment, we see that $V_\sigma\subseteq V_i$, so $V_\sigma$ is covered as desired.
For the converse, we prove the contrapositive. Suppose that $U_\sigma$ is not covered by $\{U_i\mid i\in[n]\setminus \sigma\}$. Then $\sigma\in \mathcal C$ and we may and consider the point $p_\sigma$. By choice of $\varepsilon$, $p_\sigma$ is in $V_\sigma$ but not any $U_i$ with $i\in[n]\setminus \sigma$. Since $V_i\subseteq U_i$, this implies that $p_\sigma$ is not covered by $\{V_i\mid i\in[n]\setminus \sigma\}$. This proves that $\mathcal C = \code(\{V_1,\ldots, V_n\})$.
To see that the $V_i$ form a non-degenerate realization, we must check (ii) of Definition \ref{def:nondegen}. For any nonempty $\sigma\subseteq [n]$, let $p$ be a point in $\bigcap_{i\in\sigma} \partial V_i$. Observe that since the closure of any $V_i$ is contained in $U_i$, the point $p$ lies in $U_\sigma$. Since $U_\sigma$ is nonempty, we may choose a point $q\in V_\sigma$, and consider the line segment $\overline{pq}$. Since $p$ is a boundary point of all $V_i$ with $i\in\sigma$, the line segment $\overline{pq}$ is contained in $V_i$ except for the point $p$. But this implies that all points on the line segment except $p$ lie in $V_\sigma$. Thus $p$ is a boundary point of $V_\sigma$ and the result follows.\end{proof}
\begin{example}\label{ex:trim}
Below we show the construction used in Lemma \ref{lem:trimrealization} and Theorem \ref{thm:cdimleodim} for two realizations of intersection complete codes.
The first is the code $\{\mathbf{123}, 1,2,3,\emptyset\}$. In this case the $U_i$ already formed a non-degenerate realization, but trimming them slightly does not cause any issues.
\[
\includegraphics[width=25em]{4a_trimming_example_1.pdf}
\]
The figure below shows a degenerate realization of the code $\{\mathbf{13},\mathbf{14},\mathbf{23},\mathbf{24}, 1,2,3,4,\emptyset\}$. This realization is degenerate since $U_1$ and $U_2$ are disjoint but share boundary points, and similarly for $U_3$ and $U_4$. On the lefthand side, we have labeled the regions corresponding to maximal codewords.
\[
\includegraphics[width=21em]{4b_trimming_example_2.pdf}
\]
\end{example}
The importance of non-degeneracy is the following: when $\U$ is a non-degenerate collection of convex open sets, taking the closures of these sets does not change the code of the collection (see \cite{openclosed}, Theorem 2.12). With this, we are ready to prove Theorem \ref{thm:cdimleodim}.
\begin{restatetheorem}{\ref{thm:cdimleodim}}
Let $\mathcal C\subseteq 2^{[n]}$ be an intersection complete code. Then $\cdim(\mathcal C)\le \odim(\mathcal C)$.
\end{restatetheorem}
\begin{proof}
Let $\U = \{U_1,\ldots, U_n\}$ be an open realization of $\mathcal C$ in $\mathbb R^{\odim(\mathcal C)}$. By Lemma \ref{lem:trimrealization}, we may assume that $\U$ is non-degenerate by possibly trimming the sets in the realization. By \cite[Theorem 2.10]{openclosed}, the realization consisting of closures of the $U_i$ is a closed convex realization of $\mathcal C$. Thus $\cdim(\mathcal C)\le \odim(\mathcal C)$.
\end{proof}
\begin{example}\label{ex:trimfails}
Trimming a realization may fail when a code is not intersection complete. The following shows a realization of the code $\{\mathbf{123}, 12, 13, \emptyset\}$ with labeled atoms, and a trimming of that realization. One can observe that no matter how small we choose $\varepsilon$, trimming this realization always yields an arrangement in which part of $V_1$ is not covered by $V_2$ and $V_3$.
\[
\includegraphics[width=30em]{5_failtrim.pdf}
\]
Of course, we could have drawn a better realization of this code. However, we are not always so lucky. There are examples of open convex codes where trimming will fail for \emph{any} convex open realization---see for example \cite[Section 2.3]{openclosed}, which describes a code for which every convex open realization is degenerate.
\end{example}
\section{Closed Embedding Dimension is Bounded by $\min\{2d+1, n-1\}$}\label{sec:cdimlinear}
Throughout this section, let us fix a (possibly not intersection complete) code $\mathcal C\subseteq 2^{[n]}$, and let $d = \dim(\Delta(\mathcal C))$. We will attempt to build a realization of $\mathcal C$ using closed convex sets satisfying the bound of Theorem \ref{thm:cdimlinear}. As we will prove in Lemma \ref{lem:itworks}, this construction will succeed if and only if $\mathcal C$ is intersection complete. This result echoes \cite[Lemma 5.9]{openclosed}, but our approach allows us stronger control over the dimension of the ambient space. Our approach is inspired by the construction described in \cite[Theorem 3.1]{tancer}.
Throughout this section we will refer to the \emph{intersection completion} of $\mathcal C$, which is the code containing all intersections of codewords in $\mathcal C$. Note that $\mathcal C$ is intersection complete if and only if it is equal to its intersection completion. To begin building our attempted realization, we need to introduce several objects.
\begin{lemma}\label{lem:polycomplex}
Let $m = \min \{2d+1, n-1\}$. There exists a pure, full-dimensional polytopal complex $\P$ in $\mathbb R^m$ with facets $\{P_1,\ldots, P_n\}$ such that any $d+1$ facets of $\P$ meet in a unique nonempty face of $\P$. In particular, $\code(\{P_1,\ldots, P_n\})$ contains all $\sigma\subseteq [n]$ with $|\sigma| \le d+1$.
\end{lemma}
\begin{proof}
First, recall that there exists a $(d+1)$-neighborly polytope in $\mathbb R^{m+1}$ with $n+1$ vertices. When $m=2d+1$, one example is the cyclic polytope, and when $m=n-1$ the $n$-simplex suffices. Let $P\subseteq \mathbb R^{m+1}$ be a polytope dual to a $(d+1)$-neighborly polytope with $n+1$ vertices. Let $F_1,\ldots, F_n, F_{n+1}$ be the facets of $P$, and observe that any $d+1$ facets of $P$ meet in a unique face of $P$. Consider the Schlegel diagram of $P$ in $\mathbb R^m$ based at the facet $F_{n+1}$. For $1\le i\le n$, define $P_i$ to be the image of $F_i$ in the Schlegel diagram. We claim that the complex $\P$ with facets $\{P_1,\ldots, P_n\}$ is the desired polytopal complex.
Each $P_i$ is full-dimensional since each $F_i$ has dimension $m$. Furthermore, if $\sigma\subseteq [n]$ and $|\sigma|\le d+1$, then (by $(d+1)$-neighborliness of $P$) the facets $P_i$ with $i\in\sigma$ meet at a unique face in this complex. A point in the relative interior of this face will not lie in any $P_j$ with $j\notin\sigma$, and so $\sigma\in \code(\P)$. This proves the result.
\end{proof}
For the remainder of this section, let us fix a polytopal complex $\P$ with facets $\{P_1,\ldots, P_n\}$ as given by Lemma \ref{lem:polycomplex}. So far we have a fixed code $\mathcal C$, and a fixed complex $\P$. We begin to relate these two objects to one another below.
\begin{definition}\label{def:psigma}\label{def:Vi}
For each nonempty $\sigma\subseteq [n]$, define the following:
\begin{itemize}
\item Let $P_\sigma$ denote the face $\bigcap_{i\in \sigma} P_i$ of $\P$.
\item When $|\sigma|\le d+1$, let $p_\sigma$ be a relative interior point of $P_\sigma$.
\end{itemize}
For each $i\in[n]$, define $V_i := \conv\{p_c\mid c\in \Tk_\mathcal C(i)\}$.
\end{definition}
These objects are illustrated in Example \ref{ex:2d+1} below. Since the various $P_\sigma$ with $|\sigma|\le d+1$ are distinct faces of $\P$, $p_\sigma \in P_i$ if and only if $i\in\sigma$. Observe also that $V_i\subseteq P_i$ for all $i$, and as a consequence $V_\sigma\subseteq P_\sigma$ for all nonempty $\sigma\subseteq [n]$. The following lemmas build the connection between the sets $V_i$ and the structure of our code $\mathcal C$.
\begin{lemma}\label{lem:Hsupport}
Let $\sigma\subseteq [n]$ with $|\sigma|\ge 2$, and let $i\in\sigma$. Let $H$ be a supporting hyperplane for the proper face $P_\sigma$ of $P_i$. Then $V_i\cap H = \conv\{p_c\mid c\in\Tk_\mathcal C(\sigma)\}$.
\end{lemma}
\begin{proof}
Consider the points $\{p_c\mid c\in \Tk_\mathcal C(i)\}$, the convex hull of which is equal to $V_i$ by definition. Since $V_i\subseteq P_i$, we see that $V_i\subseteq H^\ge$. Thus $V_i\cap H$ is the convex hull of all points in $\{p_c\mid c\in \Tk_\mathcal C(i)\}$ which lie in $H$. If $c\in\Tk_\mathcal C(i)$ but $\sigma\not\subseteq c$, then we may choose $j\in \sigma\setminus c$, noting that $p_c\notin P_j$. In particular, $p_c\in P_i$ but $p_c\notin P_\sigma$. Thus $p_c$ lies in $H^>$ when $\sigma\not\subseteq c$. On the other hand, if $\sigma\subseteq p_c$ then $p_c\in P_\sigma\subseteq H$. Thus $V_i\cap H$ is the convex hull of $\{p_c\mid c\in \Tk_\mathcal C(\sigma)\}$ as desired.
\end{proof}
\begin{lemma}\label{lem:trunkconv}
Let $\sigma\subseteq [n]$ be nonempty. Then $V_\sigma = \conv\{p_c\mid c\in\Tk_\mathcal C(\sigma)\}$.
\end{lemma}
\begin{proof}
Let $C = \conv\{p_c\mid c\in\Tk_\mathcal C(\sigma)\}$. Then $C\subseteq V_\sigma$ since each $p_c$ with $c\in\Tk_\mathcal C(\sigma)$ lies in $V_j$ for all $j\in\sigma$. For the reverse inclusion, we consider two cases. If $\sigma= \{i\}$ then $C = V_i$ and the result is immediate. Otherwise, $|\sigma|\ge 2$ and we may choose $i\in\sigma$ and $H$ a supporting hyperplane for the face $P_\sigma$ of $P_i$. Observe that $V_\sigma\subseteq V_i\cap P_\sigma \subseteq V_i\cap H$, and by Lemma \ref{lem:Hsupport} $V_i\cap H = C$, proving the result.
\end{proof}
\begin{lemma}\label{lem:faces}
Let $\sigma$ and $\tau$ be nonempty subsets of $[n]$. Then $V_\sigma$ is a face of $V_\tau$ if and only if $\Tk_\mathcal C(\sigma)\subseteq \Tk_\mathcal C(\tau)$.
\end{lemma}
\begin{proof}
First suppose that $\Tk_\mathcal C(\sigma)\subseteq \Tk_\mathcal C(\tau)$. This implies that every codeword that contains $\sigma$ also contains $\tau$, and so $\Tk_\mathcal C(\sigma) = \Tk_\mathcal C(\sigma\cup \tau)$. Lemma \ref{lem:trunkconv} then implies that $V_\sigma = V_{\sigma\cup\tau}$, and so it suffices to prove that $V_{\sigma\cup\tau}$ is a face of $V_\tau$. Equivalently, we may reduce to the case in which $\tau\subseteq \sigma$. It will suffice to prove that $V_\sigma$ is a face of all $V_i$ with $i\in\tau$. If $\sigma = \{i\}$ then $\tau= \{i\}$ and the result is immediate. Otherwise, $|\sigma|\ge 2$, and for any $i\in\tau$ we may choose a hyperplane $H$ supporting the face $P_\sigma$ of $P_i$. Lemma \ref{lem:Hsupport} implies that $H\cap V_i = \conv\{p_c\mid c\in\Tk_\mathcal C(\sigma)\}$, and Lemma \ref{lem:trunkconv} implies that this is $V_\sigma$. Thus $V_i\cap H = V_\sigma$ and $V_\sigma$ is a face of $V_i$ for all $i\in\tau$ as desired.
For the converse, we argue by contrapositive. If $\Tk_\mathcal C(\sigma)\not\subseteq \Tk_\mathcal C(\tau)$ then there exists $c\in\mathcal C$ with $\sigma\subseteq c$ but $\tau\not\subseteq c$. Consider the point $p_c$. Since $\tau\not\subseteq c$, there exists $i\in \tau\setminus c$, and we see that $p_c\notin P_i$. But $V_\tau\subseteq V_i\subseteq P_i$, so $p_c\notin V_\tau$. On the other hand, $p_c\in V_\sigma$, so $V_\sigma$ is not contained in $V_\tau$, proving the result.
\end{proof}
\begin{lemma}\label{lem:properstuff}
Let $\sigma\subseteq [n]$ be nonempty. Then $\sigma$ lies in the intersection completion of $\mathcal C$ if and only if the following holds: $\Tk_\mathcal C(\sigma)$ is nonempty and properly contains $\Tk_\mathcal C(\sigma\cup\{i\})$ for all $i\in[n]\setminus \sigma$.
\end{lemma}
\begin{proof}
If $\sigma$ is an intersection of codewords in $\mathcal C$, then there must be a codeword containing $\sigma$, and thus $\Tk_\mathcal C(\sigma)$ is nonempty. If there exists $i\in[n]\setminus \sigma$ such that $\Tk_\mathcal C(\sigma) = \Tk_\mathcal C(\sigma\cup\{i\})$, then every codeword of $\mathcal C$ containing $\sigma$ also contains $i$. This is a contradiction, since $\sigma$ is the intersection of all codewords in $\mathcal C$ that contain it.
For the converse we consider two cases. If $\sigma = [n]$ and $\Tk_\mathcal C(\sigma)$ is nonempty then $[n]\in \mathcal C$ and the result follows. Otherwise $\sigma$ is a proper subset of $[n]$. Since $\Tk_\mathcal C(\sigma)$ is nonempty and properly contains $\Tk_\mathcal C(\sigma\cup\{i\})$ for all $i\in[n]\setminus \sigma$, for every $i\in[n]\setminus \sigma$ we may choose a codeword $c_i$ with $\sigma\subseteq c_i$ and $i\notin c_i$. The intersection of all such $c_i$ is $\sigma$, proving the result.
\end{proof}
\begin{lemma}\label{lem:itworks}The set $\V = \{V_1,\ldots, V_n\}$ is a closed realization of the intersection completion of $\mathcal C$. In particular, $\V$ is a realization of $\mathcal C$ if and only if $\mathcal C$ is intersection complete.
\end{lemma}
\begin{proof}
Let $\widehat{\mathcal C}$ denote the intersection completion of $\mathcal C$. We argue for each nonempty $\sigma\subseteq[n]$ that $\sigma\in \widehat\mathcal C$ if and only if $\sigma\in \code(\V)$. By Lemma \ref{lem:properstuff} it suffices to argue that $\sigma\in\code(\V)$ if and only if $\Tk_\mathcal C(\sigma)$ is nonempty and $\Tk_\mathcal C(\sigma\cup\{i\})$ is a proper subset of $\Tk_\mathcal C(\sigma)$ for all $i\in[n]\setminus\sigma$. By Lemma \ref{lem:faces}, this condition is equivalent to the requirement that $V_\sigma$ is nonempty, and $V_{\sigma\cup\{i\}}$ is a proper face of $V_\sigma$ for all $i\in[n]\setminus \sigma$. This is in turn equivalent to the statement that $V_\sigma$ is nonempty and not covered by $\{V_i\mid i\in[n]\setminus\sigma\}$, which happens if and only if $\sigma\in \code\{\V\}$, proving the result.
\end{proof}
\begin{example}\label{ex:2d+1} To make the construction in Lemma \ref{lem:itworks} concrete, we give an example for the intersection complete code $\mathcal C = \{\mathbf{123}, 12, 1,2,3,\emptyset\}$. We choose $\P$ in $\mathbb R^2$ with facets $P_1, P_2, P_3$ which are triangles meeting at a common vertex. This is shown below, and the various $p_\sigma$ are represented by dots. The sets $V_1$ and $V_2$ are triangles, and $V_3$ is the line segment from $p_{3}$ to $p_{123}$.\[
\includegraphics[width=18em]{6_complex_example.pdf}
\]
\end{example}
\begin{restatetheorem}{\ref{thm:cdimlinear}}
Let $\mathcal C\subseteq 2^{[n]}$ be an intersection complete code, and $d=\dim(\Delta(\mathcal C))$. Then $\cdim(\mathcal C)\le \min \{2d+1, n-1\}$.
\end{restatetheorem}
\begin{proof}
In this section we have chosen a polytopal complex $\P$ in $\mathbb R^{\min\{2d+1,n-1\}}$, and used it to construct a collection $\V = \{V_1,\ldots, V_n\}$ of closed convex sets. Lemma \ref{lem:itworks} says that $\V$ realizes $\mathcal C$ if and only if $\mathcal C$ is intersection complete. This proves the result.
\end{proof}
In Section \ref{sec:SDelta}, we will see that this bound on closed embedding dimension may fail dramatically for open embedding dimension. Before proving this, we use Section \ref{sec:sunflowercodeversion} to recall a theorem from \cite{sunflowers}, and show that it is equivalent to a statement about the open embedding dimension of a family of intersection complete codes.
\section{A Code Version of the Sunflower Theorem}\label{sec:sunflowercodeversion}
In this section we recall a result regarding sunflowers of convex open sets. In Section \ref{sec:SDelta}, we will use this result to build a family of intersection complete codes with large open embedding dimension.
\begin{definition}\label{def:sunflower}
Let $\U = \{U_1,\ldots, U_n\}$ be a collection of convex sets in $\mathbb R^d$ and let $\mathcal C = \code(\U)$. The collection $\U$ is called a \emph{sunflower} if $[n]\in\mathcal C$, and $\mathcal C\setminus \{[n]\}$ contains $\emptyset$ and codewords of weight at most 1. That is, a sunflower is just a $1$-flexible sunflower. As in Definition \ref{def:flexiblesunflower}, we will call the $U_i$ \emph{petals} and $U_{[n]}$ will be called the \emph{center} of $\U$. \end{definition}
\begin{theorem}[Sunflower Theorem, \cite{sunflowers}]\label{thm:sunflower}
Let $d\ge 1$, let $\U = \{U_1,\ldots, U_{d+1}\}$ be a convex open sunflower in $\mathbb R^d$, and for each $i\in[d+1]$ choose a point $p_i\in U_i$. Then $\conv\{p_1,\ldots, p_{d+1}\}$ contains a point in the center of $\U$.
\end{theorem}
Note that the result above fails when we consider a sunflower with $d$ petals in $\mathbb R^d$. In particular, one may take an infinite rectangular cylinder about each coordinate axis to form a sunflower whose center is a hypercube at the origin. In this situation, choosing the $p_i$ with sufficiently large positive coordinates yields points in each petal whose convex hull does not touch the center of the sunflower.
The sunflower theorem may be restated purely in the language of convex codes. We do this below in order to simplify our discussion in the following section, and also to foreshadow our applications of Theorem \ref{thm:flexible} in Section \ref{subsec:SCoverD}.
\begin{definition}\label{def:Sn}
For $n\ge 1$, define $\S_n\subseteq 2^{[n+1]}$ to be the code consisting of the following codewords: $[n]$, all singleton sets, all pairs $\{i, n+1\}$ for $1\le i \le n$, and the empty set.
\end{definition}
Note that $\S_n$ is an intersection complete code. The sunflower theorem can be restated as follows:
\begin{theorem}[Sunflower Theorem, Code Version]\label{thm:sunflowercodeversion}
For all $n\ge 1$, $\odim(\S_n) = n$.
\end{theorem}
\begin{proof}
When $n=1$, we have $\S_n = \{12, 1, 2, \emptyset\}$, which can be realized by two overlapping intervals in $\mathbb R^1$. For $n\ge 2$, $\S_n$ has $n+1$ maximal codewords, and so by \cite[Theorem 1.2]{openclosed} $\S_n$ has an open realization in $\mathbb R^n$. We will show that it does not have an open realization in $\mathbb R^{n-1}$. Suppose for contradiction that there exists an open realization $\U = \{U_1,\ldots, U_{n+1}\}$ of $\S_n$ in $\mathbb R^{n-1}$. Observe that $\{U_1,\ldots, U_n\}$ is a sunflower, and $U_{n+1}$ intersects $U_i$ for all $i\in[n]$. Thus for each $i\in[n]$ we may choose $p_i\in U_i\cap U_{n+1}$. The convex hull $\conv\{p_1,\ldots, p_n\}$ is contained in $U_{n+1}$, but by Theorem \ref{thm:sunflower} this convex hull also meets $U_{[n]}$. Thus $U_{n+1}\cap U_{[n]}$ is nonempty. Since $[n+1]$ is not a codeword in $\S_n$ this is a contradiction.
\end{proof}
\begin{example}
Let us look at the first few $\S_n$:\begin{align*}
\S_1 = & \{\mathbf{12}, 1, 2, \emptyset\},\\
\S_2 = &\{\mathbf{12},\mathbf{13},\mathbf{23}, 1, 2, 3, \emptyset\},\\
\S_3 = &\{\mathbf{123}, \mathbf{14},\mathbf{24},\mathbf{34}, 1, 2, 3, 4, \emptyset\}.
\end{align*} These have realizations in $\mathbb R^1,\mathbb R^2$, and $\mathbb R^3$ respectively, illustrated below. Theorem \ref{thm:sunflowercodeversion} says that these realizations are minimal in dimension.
\[
\includegraphics[width=31em]{7_S123.pdf}
\]
\end{example}
In the following section we build on the family $\S_n$ to construct a family of intersection complete codes on $n$ neurons whose open embedding dimension is $\binom{n-1}{\lfloor (n-1)/2\rfloor}$.
\section{A Family of Codes with Large Open Embedding Dimension}\label{sec:SDelta}
In this section, we will associate to every simplicial complex $\Delta\subseteq 2^{[n]}$ an intersection complete code $\S_\Delta\subseteq 2^{[n+1]}$. As long as $\Delta$ has at least two facets, the open embedding dimension of $\S_\Delta$ is exactly the number of facets in $\Delta$.
\begin{restatedefinition}{\ref{def:SDelta}}
Let $\Delta\subseteq 2^{[n]}$ be a simplicial complex. Define $\S_\Delta\subseteq 2^{[n+1]}$ to be the code \[
\S_\Delta := \left(\Delta \ast (n+1)\right)\cup\{[n]\},
\]
where $\Delta\ast (n+1)$ denotes the cone over $\Delta$ with apex $n+1$.
\end{restatedefinition}
We start with some straightforward structural observations about the code $\S_\Delta$.
\begin{proposition}\label{prop:SDelta}
$\S_\Delta$ is intersection complete. If $\Delta\subsetneq 2^{[n]}$ and has $m$ facets, then $\S_\Delta$ has $m+1$ maximal codewords. In particular, $\odim(\S_\Delta)\le \max\{2, m\}$.
\end{proposition}
\begin{proof}
First note that $\S_\Delta$ is a simplicial complex, plus the codeword $[n]$. Adding a single codeword to a simplicial complex always yields an intersection complete code, so $\S_\Delta$ is intersection complete.
Let $F_1,\ldots, F_{m}$ be the facets of $\Delta$. Observe that the maximal codewords of $\S_\Delta$ are either facets of $\Delta\ast (n+1)$, or equal to $[n]$. The facets of $\Delta\ast(n+1)$ are just $F_i\cup \{n+1\}$ for $i\in[m]$. Since $\Delta\subsetneq 2^{[n]}$, $[n]$ is also a maximal codeword of $\S_\Delta$, so $\S_\Delta$ has $m+1$ maximal codewords in total. The bound $\odim(\S_\Delta)\le \max\{2, m\}$ then follows immediately from \cite[Theorem 1.2]{openclosed}.
\end{proof}
\begin{restatetheorem}{\ref{thm:SDelta}}
Let $\Delta\subseteq 2^{[n]}$ be a simplicial complex with $m\ge 2$ facets. Then $\S_\Delta$ (as given by Definition \ref{def:SDelta}) is an intersection complete code with $m+1$ maximal codewords, and $\odim(\S_\Delta) = m$.
\end{restatetheorem}
\begin{proof} By Proposition \ref{prop:SDelta} we know that $\S_\Delta$ is intersection complete, has $m+1$ maximal codewords, and $\odim(\S_\Delta) \le m$. Thus it suffices to show that $\S_\Delta$ does not have an open realization in $\mathbb R^{m-1}$. Suppose for contradiction that we had such a realization $\U = \{U_1,\ldots, U_{n+1}\}$.
Label the facets of $\Delta$ as $F_1,\ldots, F_{m}$, and for each $i\in[m]$ define $V_i = U_{F_i}$. Lastly, define $V_{m+1} = U_{n+1}$. Now observe that the pairwise intersection of any two distinct $V_i$ with $i\in[m]$ is $U_{[n]}$, so $\{V_1,\ldots, V_{m}\}$ is a sunflower. Note that $V_{m+1}$ intersects each petal of this sunflower since $F_i\cup\{n+1\}$ is a codeword of $\S_\Delta$ for all $i\in[m]$. However, $V_{m+1}$ does not intersect $U_{[n]}$. In particular, $\{V_1,\ldots, V_{m+1}\}$ is a convex open realization of $\S_{m}$ in $\mathbb R^{m-1}$. This contradicts Theorem \ref{thm:sunflowercodeversion}, and so $\odim(\S_\Delta)$ must be equal to $m$ as desired.
\end{proof}
\begin{corollary}\label{cor:choice}
For any $n\ge 2$ and $1\le m\le \binom{n-1}{\lfloor (n-1)/2\rfloor}$, there exists an intersection complete code on $n$ neurons with $m+1$ maximal codewords, and open embedding dimension equal to $m$.
\end{corollary}
\begin{proof}
For $m = 1$, the code $\{\mathbf{1}, \emptyset\}$ suffices. For $m \ge 2$ we apply Theorem \ref{thm:SDelta}. Among all $\binom{n-1}{\lfloor (n-1)/2\rfloor}$ subsets of $[n-1]$ with size $\lfloor (n-1)/2 \rfloor$, we may select $m$. Letting $\Delta$ be the simplicial complex with these subsets as its facets, we see that $\S_\Delta$ is the desired code.
\end{proof}
\begin{corollary}\label{cor:exponential}
There is a family of codes $\mathcal E_n\subseteq 2^{[n]}$ such that $\odim(\mathcal E_n)$ grows exponentially in $n$.
\end{corollary}
\begin{proof}
By Corollary \ref{cor:choice}, we may choose $\mathcal E_n$ so that $\odim(\mathcal E_n) = \binom{n-1}{\lfloor (n-1)/2\rfloor}$. But $\binom{n-1}{\lfloor (n-1)/2\rfloor} \ge \frac{2^{n-1}}{n}$, which grows exponentially in $n$.\end{proof}
Qualitatively, these results are very surprising. The codes $\S_\Delta$ are ``almost" simplicial complexes (we have added the single codeword $[n]$ to a simplicial complex), but their open embedding dimensions grow exponentially faster than that of any simplicial complex. Strikingly, these codes provide the first example of codes whose embedding dimension (open or closed) is larger than $n-1$.
\begin{remark}
From the perspective of the neuroscience which motivates the study of convex codes, Corollary \ref{cor:exponential} has the following interpretation: theoretically, $n$ neurons may ``recognize" dimensions that are exponentially large in $n$. Whether such a phenomenon ever occurs in experimental data could be an interesting avenue of investigation.
\end{remark}
\section{Flexible Sunflowers}\label{sec:flexible}
In this section our goal is to investigate $k$-flexible sunflowers of convex open sets. These are a generalization of sunflowers in which we allow petals to overlap outside the center of the sunflower, but no more than $k$ at a time. For sunflowers, we saw in Theorem \ref{thm:sunflower} that sampling a point in each petal and taking the convex hull always yielded a point in the center of the sunflower if we had enough petals relative to our ambient dimension. We will see that the same holds for $k$-flexible sunflowers, and the minimum number of petals needed is proportional to $k$, as well as the ambient dimension. Qualitatively, the more flexibility we allow in a sunflower, the larger the number of petals we need to sample in order to guarantee that the convex hull of the sampled points intersects the center of the sunflower.
To begin, let us recall the definition of a $k$-flexible sunflower.
\begin{restatedefinition}{\ref{def:flexiblesunflower}}
Let $\U = \{U_1,\ldots, U_n\}$ be a collection of convex sets in $\mathbb R^d$ and let $\mathcal C = \code(\U)$. The collection $\U$ is called a \emph{$k$-flexible sunflower} if $[n]\in\mathcal C$, and all other codewords have weight at most $k$. The $U_i$ are called \emph{petals} and $U_{[n]}$ is called the \emph{center} of $\U$. \end{restatedefinition}
We start with a family of examples. For each $d\ge 2$ and $k \ge 1$, Proposition \ref{prop:fails} describes a $k$-flexible $\U$ sunflower in $\mathbb R^d$ with $dk$ petals in which we can sample points from each petal whose convex hull does not contain a point in the center of $\U$.
\begin{proposition}\label{prop:fails}
For all $d\ge 2$ and $k\ge1$, there exists a $k$-flexible sunflower $\U =\{U_1,\ldots, U_n\}$ in $\mathbb R^d$ with $n=dk$, and points $p_1,\ldots, p_n$ with $p_i\in U_i$, such that $\conv\{p_1,\ldots, p_n\}$ does not contain a point in the center of $\U$.
\end{proposition}
\begin{proof}
For $k=1$, we begin with an open unit hypercube in $\mathbb R^d$ centered at the origin, and let $U_i$ be the Minkowski sum of this hypercube with a line segment from the origin to a large positive multiple of $e_i$. We can see that the $U_i$ form a $d$-petal sunflower, and our desired $p_i$ are just the large multiples of $e_i$.
For $k\ge 2$, we can take the sunflower described above and duplicate each of the $d$ petals $k$ times. This creates a $k$-flexible sunflower, and the same sampling of points (with each duplicated $k$ times) satisfies the proposition.
\end{proof}
\begin{remark}
One might argue that the construction above is unsatisfying. Should we not stipulate that petals diverge in different directions, or at least are distinct? It turns out we can address these concerns. Start with the usual coordinate-direction sunflower whose center is a unit hypercube, as described above. If $k=1$ we are done. Otherwise, choose a cyclic permutation $\sigma$ of $[d]$, for example $i\mapsto i+1$ mod $d$. Then, we can duplicate each petal in our coordinate-direction sunflower $k$ times, but when duplicating the $i$-th petal we ``skew" it slightly in the direction of $-e_{\sigma(i)}$. If each duplicated petal is skewed a different amount, our petals will diverge from one another. As long as we skew a small enough amount, this yields a $k$-flexible sunflower from which we can sample the desired $p_i$.
This construction is illustrated below for $k=3$ and $d=2$:\[
\includegraphics[width=11em]{8_flexible_R2.pdf}
\]
\end{remark}
We now turn our attention to proving Theorem \ref{thm:flexible}. We will see that some technical lemmas regarding $k$-flexible sunflowers together with Tverberg's theorem do the trick. We start by showing that the center of every open $k$-flexible sunflower admits a set of supporting halfspaces that cut away a dense subset of the boundary points, and any one of which contains all but at most $k$ of the petals.
\begin{definition}\label{def:wellsupported}
Let $\U=\{U_1,\ldots, U_n\}$ be a $k$-flexible sunflower in $\mathbb R^d$ with center $U$. A point $b\in\partial U$ is called \emph{well supported} if it is not in the boundary of $U_i\cap \partial U$ (considered as a subset of the topological space $\partial U$) for any $i\in[n]$.
\end{definition}
\begin{lemma}\label{lem:densesupport}
Let $\U=\{U_1,\ldots, U_n\}$ be a $k$-flexible sunflower in $\mathbb R^d$ with center $U$. The set of well supported points is dense in $\partial U$.
\end{lemma}
\begin{proof}
Consider the sets $U_i\cap \partial U$ in $\partial U$. For each of these sets, the set of non-boundary points in $\partial U$ is dense and open when considered as a subset of $\partial U$. The set of well supported points is just the intersection of non-boundary points of $U_i\cap \partial U$ in $\partial U$for all $i$, and a finite intersection of dense open sets is again open and dense. Thus the well supported points are dense in $\partial U$.
\end{proof}
\begin{lemma}\label{lem:goodhalfspaces}
Let $\U=\{U_1,\ldots, U_n\}$ be a $k$-flexible sunflower in $\mathbb R^d$ with center $U$, and let $b\in \partial U$ be well supported. Let $H_b$ be a supporting halfspace for $U$ at $b$, and let $\sigma = \{i\in[n]\mid b\notin U_i\}$. Then $U_i\subseteq H_b^>$ for all $i\in \sigma$.
\end{lemma}
\begin{proof}
Suppose not, so that there exists $i\in\sigma$ for which $U_i$ is not contained in $H_b^>$. Since $U_i$ is open, we may assume that there exists a point $p\in U_i$ strictly on the negative side of $H_b$. Then choose any point $q\in U$, and consider the line segment $\overline{qb}$. All points on this line segment other than $b$ lie in $U$. For each $r\in \overline{qb}$ with $r\neq b$, note that the line segment $\overline{pr}$ is contained in $U_i$ and intersects $\partial U$ since it begins in the interior of $U$ and ends outside of $U$. The set of these intersection points forms a subset of $U_i\cap \partial U$ whose closure contains $b$. This is illustrated in the figure below, with the points in $U_i\cap \partial U$ converging to $b$ shown in the bold curved line segment.
\[
\includegraphics[width=12em]{9_well_supported.pdf}
\]
But since $b\notin U_i$, this implies that $b$ is a boundary point of $U_i\cap \partial U$ in $\partial U$. This contradicts the fact that $b$ is well supported, proving the result.
\end{proof}
\begin{lemma}\label{lem:densecut}
Let $U\subseteq \mathbb R^d$ be a convex open set. Let $B$ be a dense subset of the boundary of $U$, and for each $b\in B$ let $H_b$ be a supporting hyperplane to $U$ at $b$. Then $\bigcap_{b\in B} H_b^>$ is contained in $\overline U$.
\end{lemma}
\begin{proof}
Consider any point $p\notin \overline U$. Since $p$ lies a positive distance away from $U$, the intersection of $\interior(\conv(\{p\} \cup U))$ with $\partial U$ is a relatively open subset of $\partial U$, and thus contains some $b\in B$. Since $\interior(\conv(\{p\}\cup U))$ is open, the line segment $\overline{pb}$ can be extended so that it ends at a point $q\in U$, as shown in the following figure.\[
\includegraphics[width=18em]{10_separating.pdf}
\]Now, consider the supporting hyperplane $H_b$. We have $U\subseteq H_b^>$. In particular, $H_b^>$ contains $q$ but not $b$. Since $b$ lies between $q$ and $p$, we see that $H_b^>$ does not contain $p$. Thus $p\notin \bigcap_{b\in B} H_b^>$ and the lemma follows.
\end{proof}
Finally, we recall Tverberg's theorem. After stating this theorem, we are ready to prove Theorem \ref{thm:flexible}.
\begin{theorem}[Tverberg's theorem] \label{thm:tverberg}
Let $d\ge1$, $r\ge 2$, and $n = (d+1)(r-1)+1$. For any set of points $P = \{p_1,\ldots, p_n\}$ in $\mathbb R^d$, there is a partition of $P$ into $r$ parts $P_1,\ldots, P_r$ such that $\bigcap_{i=1}^r \conv\{P_i\} \neq \emptyset$.
\end{theorem}
\begin{restatetheorem}{\ref{thm:flexible}}
Let $\U = \{U_1,\ldots, U_n\}$ be an open $k$-flexible sunflower in $\mathbb R^d$. Suppose that $n\ge dk+1$, and for each $i\in[n]$ let $p_i\in U_i$. Then $\conv\{p_1,\ldots, p_n\}$ contains a point in the center of $\U$. Moreover, if $d\ge 2$ this result may fail when $n < dk+1$.
\end{restatetheorem}
\begin{proof}
It suffices to prove the first statement for $n=dk+1$. Let $U$ denote the center of $\U$. Suppose for contradiction that the theorem does not hold, so that $\conv\{p_1,\ldots, p_n\}$ does not contain a point in $U$. Since the $U_i$ are open, we may move each $p_i$ a fixed distance $\varepsilon$ away from a chosen point $U$, and choose a separating hyperplane $H$ between $\conv\{p_1,\ldots, p_n\}$ and $U$ such that $H$ does not contain any boundary point of $U$. Moreover, we can replace each $p_i$ by the intersection of the line segment $\overline{p_ip}$ with $H$, so that all $p_i$ lie inside $H$.
Now, $H$ has dimension $d-1$, so we may apply Tverberg's theorem to our points $p_i$ with $r=k+1$. We obtain a partition $P_1,\ldots, P_{k+1}$ such that $\bigcap_{i=1}^{k+1} \conv\{P_i\} \neq \emptyset$. Choose any point $p$ lying in this intersection, and observe that $p\in H$.
Let $B$ be the set of well supported points in $\partial U$, and choose supporting halfspaces $\{H_b\mid b\in B\}$ as per Lemma \ref{lem:goodhalfspaces}. Since $b\notin U$, it lies in at most $k$ petals of $\U$. Therefore by Lemma \ref{lem:goodhalfspaces}, each $H_b^>$ contains all $p_j$ except for at most $k$. In particular, there must be some $P_i$ such that $H_b^>$ contains all points in $P_i$, and hence also their convex hull. Thus $p\in H_b^>$ for all $b\in B$. But by Lemma \ref{lem:densecut}, this implies that $p\in \overline U$. Since $p\in H$ and $H$ was constructed not to contain $U$ or any of its boundary points, this is a contradiction.
To prove the second part of the theorem, recall that Proposition \ref{prop:fails} shows that when $d\ge 2$ and $n = dk$, we can choose a $k$-flexible sunflower $\U$ in $\mathbb R^d$ and points in each petal whose convex hull does not intersect the center of $\U$. This proves the result.
\end{proof}
\begin{remark}
Note that when $k=1$, Theorem \ref{thm:flexible} is the same as Theorem \ref{thm:sunflower} (the usual Sunflower Theorem), and the application of Tverberg's theorem in the proof above reduces to an application of Radon's Theorem. Thus the fact that Theorem \ref{thm:flexible} generalizes Theorem \ref{thm:sunflower} is directly analogous to the fact that Tverberg's theorem generalizes Radon's.
\end{remark}
\begin{remark}
In terms of neuroscientific motivation, flexible sunflowers are natural to investigate. Allowing some codewords beyond singletons, but of a fixed weight, accounts for some tolerance to error in data gathering and also captures a wider range of possibilities. We hope that flexible sunflowers may yield meaningful bounds on dimensions in experimental data.
\end{remark}
Theorem \ref{thm:flexible} has implications regarding the open embedding dimensions of intersection complete codes, which we will illustrate in Section \ref{subsec:SCoverD}, in particular by generalizing the families $\S_n$ and $\S_\Delta$ that were defined in Sections \ref{sec:sunflowercodeversion} and \ref{sec:SDelta} respectively.
We conclude with a corollary which examines the extremal case in which we have a $k$-flexible sunflower $\U$ with $n=dk$ petals for which Theorem \ref{thm:flexible} fails. In this case Theorem \ref{thm:flexible} implies $\code(\U)$ must contain at least one codeword of weight $k$, but we can actually say something slightly stronger:
\begin{corollary}\label{cor:manywords}
Let $\U = \{U_1,\ldots, U_n\}$ be an open $k$-flexible sunflower in $\mathbb R^d$. Suppose that $n = dk$, and there exist points $p_1,\ldots, p_n$ such that $p_i\in U_i$ and $\conv\{p_1,\ldots, p_n\}$ does not contain a point in the center of $\U$. Then $\code(\U)$ contains at least $d$ distinct codewords of weight $k$.
\end{corollary}
\begin{proof}
We work by induction on $k$. When $k=1$ the result is clear since if there are fewer than $d$ codewords of weight $k$ in $\code(\U)$ then some $U_i$ is equal to the center of $\U$, and so some $p_i$ lies in the center of $\U$, a contradiction. For $k\ge 2$, suppose for contradiction that $\code(\U)$ contains fewer than $d$ codewords of weight $k$. For each of these codewords $c$, select some petal $U_i$ with $i\in c$. Deleting these $U_i$ yields a $(k-1)$-flexible sunflower, and since we have deleted fewer than $d$ petals our new $(k-1)$-flexible sunflower has more than $d(k-1)$ petals. But the same choice of $p_i$ yields a collection of points whose convex hull does not contain a point in the center of this $(k-1)$-flexible sunflower, contradicting Theorem \ref{thm:flexible}.
\end{proof}
\section{Tangled Sunflowers}\label{sec:Tn}
For $n\ge 1$ we construct an intersection complete code $\mathcal T_n\subseteq 2^{[2n]}$, and investigate $t_n:= \odim(\mathcal T_n)$. We use Theorem \ref{thm:sunflower} to prove the following: for $d\ge 1$ there exists $n$ such that $t_n = d$. Thus for every $d\ge 1$, one of the $\mathcal T_n$ codes describes an arrangement of convex open sets which can be achieved in $\mathbb R^d$ but not a smaller dimension. Beyond this statement and some basic bounds, however, determining the behavior of $t_n$ remains an open problem, ripe for future investigation.
\begin{definition}\label{def:Tn}
Let $n\ge 1$. Define $\mathcal T_n\subseteq 2^{[2n]}$ to be the code consisting of the following codewords:\begin{itemize}
\item[(i)] $\{2k-1, 2k\}$ for $k=1,2,\ldots, n$,
\item[(ii)] $\{1, 3,5,\ldots, 2n-1\}$ and $\{2,4,6,\ldots, 2n\}$,
\item[(iii)] all singletons, and
\item[(iv)] the empty set.
\end{itemize}
For each $n$ define $t_n := \odim( \mathcal T_n)$.
\end{definition}
Observe that codewords of type (i) and (ii) are the maximal codewords in $\mathcal T_n$ for $n\ge 2$; in particular $\mathcal T_n$ has $n+2$ maximal codewords. Furthermore observe that $\mathcal T_n$ is intersection complete, and hence convex. Thus $t_n$ is finite for all $n$.
Moreover, note that the odd-numbered sets in any realization of $\mathcal T_n$ form an $n$-petal sunflower, as do the even-numbered sets. These two sunflowers are ``tangled," in that their petals are matched and overlap one another.
\begin{example}\label{ex:S1-S4}
The first four $\mathcal T_n$ are given below: \begin{align*}
\mathcal T_1 &= \{\mathbf{12},1, 2, \emptyset\},\\
\mathcal T_2 &= \{\mathbf{13}, \mathbf{24}, \mathbf{12}, \mathbf{34}, 1, 2, 3, 4, \emptyset\},\\
\mathcal T_3 &= \{\mathbf{135}, \mathbf{246}, \mathbf{12}, \mathbf{34}, \mathbf{56}, 1, 2, 3, 4, 5, 6, \emptyset\}. \\
\mathcal T_4 &= \{\mathbf{1357}, \mathbf{2468}, \mathbf{12}, \mathbf{34}, \mathbf{56}, \mathbf{78}, 1, 2, 3, 4, 5, 6,7, 8, \emptyset\}.
\end{align*}
These have convex realizations in $\mathbb R^1, \mathbb R^2$, $\mathbb R^3$ and $\mathbb R^3$ respectively.\[
\includegraphics[width=32em]{T1-T4.pdf}
\]
\end{example}
We will see that in fact each of the realizations in Example \ref{ex:S1-S4} is minimal with respect to dimension. That is, $t_1 = 1, t_2 =2, t_3 = t_4=3$. To build towards this result, we first prove some general results about the minimal embedding dimensions $\{t_n\mid n\ge 1\}$.
\begin{proposition}\label{prop:sublinear}
For all $n\ge 1$, $t_n\le t_{n+1}\le t_n+1$. That is, the sequence $\{t_n\mid n\ge 1\}$ is weakly increasing and changes by at most 1 at each step.
\end{proposition}
\begin{proof}
The inequality $t_n\le t_{n+1}$ follows from the fact that a realization of $\mathcal T_n$ can be obtained from a realization of $\mathcal T_{n+1}$ by simply deleting $U_{2n+1}$ and $U_{2n+2}$. To prove the inequality $t_{n+1}\le t_n + 1$ we argue that if $\mathcal T_n$ is convex in $\mathbb R^d$, then $\mathcal T_{n+1}$ is convex in $\mathbb R^{d+1}$.
Since $\mathcal T_n$ is intersection complete, we may apply Lemma \ref{lem:trimrealization} to obtain a realization $\U = \{U_1,\ldots, U_{2n}\}$ of $\mathcal T_n$ in $\mathbb R^d$ in which disjoint $U_\sigma$ have positive distance between them. We will use this to create a realization of $\mathcal T_{n+1}$ in $\mathbb R^{d+1}$. To start, identify $\mathbb R^d$ with the subspace of $\mathbb R^{d+1}$ in which $x_{d+1} = 0$, and define $W_1 = U_1\cap U_3\cap\cdots\cap U_{2n-1}$ and $W_2 = U_2\cap U_4\cap\cdots \cap U_{2n}$. We may assume that the origin lies in $W_1$. Now choose a vector $w\in W_2$ and a small positive $\varepsilon$, and define a collection $\V = \{V_1, V_2,\ldots, V_{2n+2}\}$ as follows:\[
V_i = \begin{cases}
\{v + \gamma e_{d+1} \mid v\in U_i \text{ and } 0 < \gamma < \varepsilon\} & \text{for } i = 1, 3, \ldots, 2n-1,\\
\{v + \gamma (e_{d+1}-w) \mid v\in U_i \text{ and } 0 < \gamma < \varepsilon\} & \text{for } i = 2, 4, \ldots, 2n,\\
\{v + \gamma e_{d+1} \mid v\in W_1\text{ and } \gamma >0\}& \text{for } i=2n+1,\\
\{v + \gamma (e_{d+1}-w) \mid v\in W_2\text{ and } \gamma >0\}& \text{for } i=2n+2.
\end{cases}
\]
This construction is shown below when $d=2$. The set $V_{2n+1}$ is a vertical cylinder over $W_1$, and the set $V_{2n+2}$ is the skewed cylinder over $W_2$. The remaining $V_i$ are $\varepsilon$-thick cylinders over the corresponding $U_i$, with even $V_i$ skewed at the same angle as $V_{2n+2}$. The origin is represented by the black dot. \[
\includegraphics[width=24em]{Tn_skew.pdf}
\]
We claim that the collection $\V$ is a realization of $\mathcal T_{n+1}$. First, observe that all $V_i$ are open and convex in $\mathbb R^{d+1}$. To see that they form a realization of $\mathcal T_{n+1}$, we must check that the odd $V_i$ and even $V_i$ both form sunflowers, and that only the appropriate petals intersect one another.
For the odd $V_i$, note that $\{V_1,V_3,\ldots, V_{2n-1}\}$ is a sunflower since the odd $U_i$ form a sunflower. Adding $V_{2n+1}$ to this collection preserves the sunflower property since $V_{2n+1}$ is simply the product of $W_1$ with an open ray. Similar logic holds for the even $V_i$: we see that $\{V_2,V_4,\ldots, V_{2n}\}$ forms a sunflower, and the additional petal $V_{2n+2}$ only overlaps any other petal in the region $\{v + \gamma (e_{d+1}-w) \mid v\in W_2\text{ and } 0<\gamma< \varepsilon\}$, which is the intersection of all the petals.
To see that the petals overlap in the correct manner, first note that $V_{2i-1}\cap V_{2i}$ is nonempty for $i=1,\ldots, n$ since the same holds for $U_{2i-1}\cap U_{2i}$. For $V_{2n+1}\cap V_{2n+2}$, simply note that $e_{d+1}\in V_{2n+1}\cap V_{2n+2}$ so the intersection is nonempty. Thus we have at least the appropriate overlapping between the petals of our two sunflowers, and it remains to show that no additional overlap has been introduced.
For this it suffices to argue that for all $j< k$ with different parity, the sets $V_j$ and $V_k$ are disjoint unless $j=2i-1$ and $k=2i$. We know that this property holds for the $U_i$, and since we chose a nondegenerate realization we know that disjoint $U_i$ have positive distance between them. Except for $V_{2n+1}$ and $V_{2n+2}$, all the $V_i$ are simply a slightly thickened $U_i$, possibly with a small skew by the vector $w$. By choosing $\varepsilon$ small enough, we can assume that the skew does not overcome the distance between disjoint $U_i$, so the $V_i$ satisfy the same disjointness for $i=1,2,\ldots, 2n$. This leaves the case of $V_{2n+1}$ and $V_{2n+2}$. For these, observe that all $V_i$ with $i\le 2n$ contain only points whose $(d+1)$-st coordinate is between 0 and $\varepsilon$. As discussed previously, the only points in $V_{2n+1}$ and $V_{2n+2}$ whose $(d+1)$-st coordinate satisfies these constraints are those in the center of the respective sunflowers. Thus neither of these sets overlap any petals they should not, and we have indeed formed a realization of $\mathcal T_{n+1}$ in $\mathbb R^{d+1}$. This proves the result.
\end{proof}
\begin{theorem}\label{thm:unbounded}
For all $n$, $t_n \ge \lceil n/2 \rceil$. In particular, the sequence $\{t_n\mid n\ge 1\}$ is unbounded.
\end{theorem}
\begin{proof}
Let $d = \lceil n/2 \rceil-1$. We must show that $\mathcal T_n$ does not have a realization in $\mathbb R^d$. Suppose for contradiction that such a realization existed, consisting of sets $\{U_1,\ldots, U_{2n}\}$. Define $V_1 = U_1\cap U_3\cap \cdots\cap U_{2n-1}$ and $V_2 = U_2\cap U_4\cap\cdots\cap U_{2n}$. Observe that $V_1$ and $V_2$ are disjoint, and let $H$ be a hyperplane separating $V_1$ and $V_2$.
Choose $p_1\in V_1$ and $p_2\in V_2$, and for $k\in[n]$ choose a point $q_k\in U_{2k-1}\cap U_{2k}$ (this intersection is nonempty since $\{2k-1, 2k\}$ is a codeword in $\mathcal T_n$). Now, for $k\in[n]$ consider the line segments $L_k = \overline{p_1q_k}$ and $M_k = \overline{q_k p_2}$. The union $L_k\cup M_k$ forms a path that begins on one side of $H$ and ends on the other, so for all $k$ either $L_k$ or $M_k$ contains a point in $H$ (and possibly both do). By choice of $d$ and pigeonhole principle, either at least $d+1$ of the line segments $\{L_k\}$ contain a point in $H$, or at least $d+1$ of the line segments $\{M_k\}$ contain a point in $H$.
Without loss of generality, we may assume that at least $d+1$ of the $\{L_k\}$ contain a point $p_k$ in $H$. The convex hull of these $p_k$ lies in $H$, and therefore does not intersect the center $V_1$ of the sunflower $\{U_1,U_3,\ldots, U_{2n-1}\}$. But $L_k\subseteq U_{2k-1}$, so each $p_k$ lies in the petal $U_{2k-1}$. Since there are at least $d+1$ points $p_k$, Theorem \ref{thm:sunflower} implies that their convex hull must intersect $V_1$, a contradiction.\end{proof}
\begin{corollary}
The sequence $\{t_n\mid n\ge 1\}$ takes on all positive integer values.
\end{corollary}
\begin{proof}
We know that $t_1 = 1$. Theorem \ref{thm:unbounded} implies that the sequence is unbounded, and Proposition \ref{prop:sublinear} tells us that it increases by at most 1 at each step. Thus it must achieve every positive integer value.
\end{proof}
In the remainder of this section, we determine $t_n$ for all $n\le 5$. The arguments used below are concrete, but seem difficult to generalize.
\begin{proposition}\label{prop:T3}
The code $\mathcal T_3$ does not have a realization in $\mathbb R^2$, but does have a realization in $\mathbb R^3$.
\end{proposition}
\begin{proof}
A realization of $\mathcal T_3$ in $\mathbb R^3$ is given in Example \ref{ex:S1-S4}. Thus we just have to argue that $\mathcal T_3$ does not have a convex realization in $\mathbb R^2$. Suppose for contradiction that $\{U_1, U_2, U_3, U_4, U_5, U_6\}$ is a realization of $\mathcal T_3$ in $\mathbb R^2$. Choose points $q_1\in U_1\cap U_2, q_2\in U_3\cap U_4$, and $q_3\in U_5\cap U_6$. Note that $\{U_1,U_3,U_5\}$ and $\{U_2, U_4,U_6\}$ are both sunflowers and that $\{q_1,q_2,q_3\}$ is a set containing one point from each petal for both of these sunflowers. By Theorem \ref{thm:sunflower} the triangle $\conv\{q_1,q_2,q_3\}$ contains a point $p_1\in U_1\cap U_3\cap U_5$ and $p_2\in U_2\cap U_4\cap U_6$. Since all the $U_i$ are open sets, we may assume that $\{p_1,p_2, q_1,q_2,q_3\}$ is in general position. The set of points $\{p_1, q_1,q_2,q_3\}$ can be visualized as follows:\[
\includegraphics[width=8em]{triangle.pdf}
\]
Now, $p_2$ falls in the interior of one of the three triangular regions surrounding $p_1$. Suppose that $p_2$ lies in the interior of $\conv\{p_1, q_1, q_2\}$ (i.e. the top right triangle above). Then consider the line segment $L = \overline{p_2 q_3}$, observing that $L$ is contained in $U_6$. The line segment $L$ must cross either the line segment $\overline{p_1q_1}\subseteq U_1$ or $\overline{p_1q_2}\subseteq U_3$. In the former case we see that $U_6\cap U_1 \neq \emptyset$, and in the latter $U_6\cap U_3 \neq\emptyset$. But there is no codeword in $\mathcal T_3$ containing $\{1,6\}$ or $\{3,6\}$, so both of these situations lead to a contradiction. Thus $\mathcal T_3$ is not convex in $\mathbb R^2$.
\end{proof}
The lemma below will allow us to prove that $t_5 \ge 4$ by showing that if $\mathcal T_5$ has a realization in $\mathbb R^3$, then $\mathcal T_3$ has a realization in $\mathbb R^2$, contradicting Proposition \ref{prop:T3}.
\begin{lemma}\label{lem:5points}
Given five points in $\mathbb R^3$ in general position, there exists a plane $H$ containing three of the points and with the remaining two points on opposite sides of $H$.
\end{lemma}
\begin{proof}
Up to affine transformation we may assume that our set of points is $\{0, e_1, e_2, e_3, p\}$ where $p$ is a point none of whose coordinates are zero. We consider two cases. First suppose that one of the coordinates of $p$ is negative. By permuting our coordinates we can assume this is the last coordinate. Then choose $H = \spann\{e_1,e_2\}$. This contains the three points $0, e_1$, and $e_2$. Moreover since $e_3$ has positive last coordinate and $p$ has negative last coordinate, they lie on opposite sides of $H$ and the lemma follows.
Otherwise every coordinate of $p$ is positive. In this case, write $p=(x,y,z)$ and choose $H = \spann\{e_3, p\}$. Observe that $H$ contains the three points $0, e_3,$ and $p$, and that $v = (y,-x,0)$ is a normal vector to $H$. We see that $v\cdot e_1 >0$ and $v\cdot e_2< 0$, so the remaining two points $e_1$ and $e_2$ lie on opposite sides of $H$. This proves the result.
\end{proof}
\begin{proposition}\label{prop:S5}
The code $\mathcal T_5$ does not have a realization in $\mathbb R^3$.
\end{proposition}
\begin{proof}
Suppose for contradiction that we have a realization $\U = \{U_1, U_2,\ldots, U_{10}\}$ of $\mathcal T_5$ in $\mathbb R^3$. For $i=1,\ldots, 5$, choose a point $p_i$ in the open set $U_{2i-1}\cap U_{2i}$, such that all $p_i$ are in general position. Applying Lemma \ref{lem:5points} to these five points, we obtain a hyperplane $H$ with contains three of them, and with the remaining two on opposite sides. By permuting the labels on our realization of $\mathcal T_5$, we may assume that $p_1, p_2,$ and $p_3$ all lie in $H$.
Now, consider the two tetrahedra $\Delta_1 = \conv\{p_1,p_2,p_3,p_4\}$ and $\Delta_2 = \conv\{p_1,p_2,p_3,p_5\}$. The vertices of these tetrahedra belong to distinct petals of the sunflowers $\{U_1, U_3,\ldots, U_9\}$ and $\{U_2, U_4, \ldots,U_{10}\}$, so by Theorem \ref{thm:sunflower} each of these tetrahedra contain a point in the center of both of these sunflowers. Since the tetrahedra lie on opposite sides of $H$, each of the centers of these two sunflowers contains a point on each side of $H$. But the center of a sunflower is convex, and so $H$ itself must contain a point in the center of each of the two sunflowers.
With this observation, consider the set $\V = \{V_1,\ldots, V_6\}$ where $V_i = U_i\cap H$. Since $H\cong \mathbb R^2$, we can regard this set as a convex realization of a code in $\mathbb R^2$. We claim that in fact this code is $\mathcal T_3$. To verify this, it suffices to show that (i) $\{V_1, V_3, V_5\}$ and $\{V_2, V_4, V_6\}$ are both sunflowers and (ii) that $V_1\cap V_2$, $V_3\cap V_4$, and $V_5\cap V_6$ are nonempty, and that (iii) no other petals overlap.
Condition (i) follows from the fact that the $V_i$ are subsets of the $U_i$ and that the sunflowers making up the realization of $\mathcal T_5$ both have centers that intersect $H$. Condition (ii) follows by considering the points $p_1, p_2$, and $p_3$, which all lie in the desired respective intersections.
Condition (iii) is a consequence of the fact that the the petals of the $U_i$ sunflowers overlap appropriately.
However, this is a contradiction: $\mathcal T_3$ is not convex in $\mathbb R^2$ by Proposition \ref{prop:T3}. Thus $\mathcal T_5$ cannot be convex in $\mathbb R^3$.
\end{proof}
\begin{corollary}\label{cor:smallvalues}
The sequence $t_n$ begins as follows:\[\begin{tabular}{|c||c|c|c|c|c|}
\hline n & 1 & 2 & 3 & 4 & 5\\\hline
$t_n$ & 1 & 2 & 3 & 3 & 4\\\hline
\end{tabular}\]
\end{corollary}
\begin{proof}
Clearly $t_1 = 1$ since $\mathcal T_1$ is convex in $\mathbb R^1$ but has more than one codeword, so is not convex in $\mathbb R^0$. The code $\mathcal T_2$ has a realization in $\mathbb R^2$ as given in Example \ref{ex:S1-S4}, but has no realization in $\mathbb R^1$ since any realization contains a non-crossing loop. Thus $t_2 = 2$.
Note that $t_3 \le 3$ and $t_4\le 3$ by Example \ref{ex:S1-S4}, and both bounds are tight by Proposition \ref{prop:T3} and monotonicity of the $t_n$. By Proposition \ref{prop:S5} we know that $t_5\ge 4$, and simultaneously Proposition \ref{prop:sublinear} implies that $t_5 \le t_4+1 = 4$. This proves the result.
\end{proof}
The proofs presented in Propositions \ref{prop:T3} and \ref{prop:S5} are both somewhat ad hoc and do not seem ripe for generalization. Determining $t_n$ for $n\ge 6$ remains an open problem, perhaps of significant difficulty.
\section{Contextualizing Our Results via Code Minors}\label{sec:pcode}
In \cite{morphisms}, we introduced a notion of morphism for neural codes. Morphisms have a strong relationship to convexity, and provide a useful framework in which to state and compare results about convex neural codes. Let us begin by stating some of the basic definitions and results regarding morphisms.
\begin{definition}\label{def:morphism}
Let $\mathcal C$ and $\mathcal D$ be codes. A function $f:\mathcal C\to\mathcal D$ is called a \emph{morphism} if the preimage of any trunk in $\mathcal D$ under $f$ is a trunk in $\mathcal C$.
\end{definition}
Morphisms make the class of all combinatorial codes into a category $\mathbf{Code}$. Suppose that $\mathcal C$ is open or closed convex in $\mathbb R^d$. Theorem 1.3 of \cite{morphisms} states that the same is true of its image under any morphism, and Proposition 4.3 of \cite{morphisms} states that the same is true of any trunk in $\mathcal C$. An analogous result holds for intersection completeness: the image or trunk of an intersection complete code is again intersection complete. In particular, convexity and intersection completeness are isomorphism invariants in the category $\mathbf{Code}$. This motivates the following notion of minors for codes.
\begin{definition}
We say that a code $\mathcal C$ is a \emph{minor} of a code $\mathcal D$ if there exists a sequence $\mathcal D = \mathcal C_0, \mathcal C_1,\cdots, \mathcal C_{k-1}, \mathcal C_k = \mathcal C$ of codes such that for all $1\le i \le k$, either there is a surjective morphism $\mathcal C_{i-1}\to \mathcal C_i$, or $\mathcal C_i$ is a trunk in $\mathcal C_{i-1}$. The relation ``$\mathcal C$ is a minor of $\mathcal D$" forms a partial order on isomorphism classes of codes. We denote the resulting partially ordered set by $\mathbf{P}_\Code$.
\end{definition}
In \cite{morphisms} we studied the partial order $\mathbf{P}_\Code$. We did not use the term ``minor" in \cite{morphisms}, but we will employ it here for convenience. The aforementioned Theorem 1.3 and Proposition 4.3 of \cite{morphisms} tell us that (open/closed) convex codes in $\mathbb R^d$ form a down-set in $\mathbf{P}_\Code$.
Implicit in these results is the following interesting fact, observed by Caitlin Lienkaemper: if $\mathcal C\subseteq 2^{[n]}$ is a code with a realization $\U = \{U_1,\ldots, U_n\}$ in a space $X$, then there are bijections between the following sets:
\[\left\{\parbox{7em}{\begin{center}Images of $\mathcal C$ under morphisms\end{center}}\right\} \quad\longleftrightarrow \quad\left\{\parbox{9em}{\begin{center}Codes that can be\\realized in $X$ using sets of the form $U_\sigma$.\end{center}}\right\}\]\[
\left\{\parbox{7em}{\begin{center}Codes below $\mathcal C$\\ in $\mathbf{P}_\Code$\end{center}}\right\} \quad\longleftrightarrow \quad\left\{\parbox{9em}{\begin{center}Codes that can be\\realized in spaces $U_\tau$ using sets of the form $U_\sigma\subseteq U_\tau$.\end{center}}\right\}
\]
The following summarizes the relationship between $\mathbf{P}_\Code$, open/closed convexity, and intersection completeness. For details on the proposition below, see \cite{morphisms}.
\begin{proposition}
The following properties are \emph{minor-closed} (that is, if $\mathcal C$ has one of the properties below, then so does every $\mathcal D\le \mathcal C$):\begin{itemize}
\item Open convexity in $\mathbb R^d$,
\item Closed convexity in $\mathbb R^d$,
\item Non-degenerate open/closed convexity in $\mathbb R^d$ (see Definition \ref{def:nondegen}),
\item Intersection completeness.
\end{itemize}
\end{proposition}
The final bullet point above implies that intersection completeness is also an isomorphism invariant, and that restricting $\mathbf{P}_\Code$ to only intersection complete codes amounts to restricting to a downward-closed set in $\mathbf{P}_\Code$. Throughout the rest of this section, we will examine exclusively nonempty intersection complete codes, with the partial order inherited from $\mathbf{P}_\Code$. We will focus on the open embedding dimensions of these codes.
Although the $\mathbf{P}_\Code$ framework allows for codes which do not include the empty set, every nonempty intersection complete code contains the empty set up to isomorphism\footnote{Simply delete all neurons which lie in the unique minimal element of the code to obtain an isomorphism class representative containing $\emptyset$.} so for intersection complete codes, the structure of $\mathbf{P}_\Code$ does not conflict with our convention that the empty set lies in every code.
One can visualize the partial order on intersection complete codes as stratified by open embedding dimension into different ``layers," as shown in the figure below.\[
\includegraphics[width=30em]{pcode.pdf}
\]
Note that $\{\emptyset\}$ is the only nonempty intersection complete code whose open embedding dimension is zero. For any $d\ge1$, however, there are infinitely many nonempty intersection complete codes with open embedding dimension $d$.
The figure above is slightly misleading: each ``layer" of codes with open embedding dimension $d$ is not finitely thick. Indeed, each layer may contain chains that are infinitely long (``tall"), and antichains that are infinitely large (``wide").
Our main goal in the rest of this section will be to understand where the codes we have constructed in this paper sit inside this partial order. In Section \ref{subsec:SCoverD}, we will also provide some more general examples of codes in $\mathbf{P}_\Code$ using Theorem \ref{thm:flexible}.
We will make heavy use of the following definition and proposition, which give a combinatorial description of all morphisms. For details, see \cite[Section 2]{morphisms}.
\begin{definition}\label{def:determined}
Let $\mathcal C\subseteq 2^{[n]}$ be a code, and for $i\in[m]$ let $T_i\subseteq \mathcal C$ be a trunk. The \emph{morphism determined by the trunks $\{T_1,\ldots T_m\}$} is the map $f:\mathcal C\to 2^{[m]}$ given by $f(c) = \{i\in[m] \mid c\in T_i\}$.
\end{definition}
\begin{proposition}\label{prop:determined}
The map described in Definition \ref{def:determined} is a morphism from $\mathcal C$ to $2^{[m]}$. Moreover, every morphism arises in this way. Formally, for codes $\mathcal C\subseteq 2^{[n]}$ and $\mathcal D\subseteq 2^{[m]}$, and any morphism $f:\mathcal C\to \mathcal D$, $f$ is the morphism determined by the trunks $\{T_i:= f^{-1}(\Tk_\mathcal D(i))\mid i\in[m]\}$ (restricted to the range $\mathcal D$). Equivalently, for all $c\in \mathcal C$,\[
f(c) = \{i\in[m]\mid c\in f^{-1}(\Tk_\mathcal D(i))\}.
\]
\end{proposition}
\subsection{The codes $\S_n, \S_\Delta,$ and $\mathcal T_n$ in $\mathbf{P}_\Code$}
Let us begin by establishing a relationship between codes of the type $\S_n$ and the type $\S_\Delta$. Recall that $\S_n$ is a special case of $\S_\Delta$---in particular, $\S_n = \S_\Delta$ where $\Delta$ is $n$ points. More generally, we have the following:
\begin{proposition}\label{prop:SDeltaSn} Let $\Delta\subsetneq 2^{[n]}$ be a simplicial complex with $m$ facets. Then there exists a surjective morphism $\S_\Delta\to \S_{m}$. In particular, $\S_{m} \le \S_\Delta$.
\end{proposition}
\begin{proof}
Let $F_1,\ldots, F_m$ be the facets of $\Delta$. For $i\in[m]$ define $T_i = \Tk_{\S_\Delta}(F_i)$, and define $T_{m+1} = \Tk_{\S_\Delta}(m+1)$. We claim that $\S_m$ is the image of $\S_{\Delta}$ under the morphism $f$ determined by the trunks $\{T_1,T_2,\ldots, T_{m+1}\}$. Recall from Definition \ref{def:SDelta} that the codewords of $S_\Delta$ are:\begin{itemize}
\item $\sigma$ for $\sigma\in\Delta$,
\item $\sigma \cup \{m+1\}$ for $\sigma\in \Delta$, and
\item $[n]$.
\end{itemize}
The images of these codewords under $f$ are as follows:\begin{itemize}
\item $f(\sigma)$ is equal to $\emptyset$ if $\sigma$ is not a facet of $\Delta$, and equal to $\{i\}$ if $\sigma = F_i$,
\item $f(\sigma\cup\{m+1\})$ is equal to $\{m+1\}$ if $\sigma$ is not a facet of $\Delta$, and equal to $\{i, m+1\}$ if $\sigma = F_i$, and
\item $f([n]) = [m]$ since $[n]$ contains all facets of $\Delta$, but does not contain $m+1$.
\end{itemize}
But comparing these to Definition \ref{def:Sn}, we see that these are exactly the codewords of $\S_m$, proving the result.
\end{proof}
\begin{remark}\label{rem:Sm} One way to think of Proposition \ref{prop:SDeltaSn} is as follows. The set \[\{\S_\Delta\mid \Delta\text{ is a simplicial complex with $m$ facets}\}\] inherits a partial order from $\mathbf{P}_\Code$, and with this inherited order $\S_m$ is the unique minimal element of the set. Theorem \ref{thm:SDelta} says that for $m\ge 2$ all of these live in the ``layer" of codes with open embedding dimension $m$. We can visualize this situation as follows. \[
\includegraphics[width=18em]{SDeltaSn.pdf}
\]\end{remark}
It is also worth noting the following, regarding the codes $\mathcal T_n$ described in Section \ref{sec:Tn}.
\begin{proposition}\label{prop:Tnchain}
For any $n\ge 1$, $\mathcal T_n\le \mathcal T_{n+1}$. In particular, the codes $\{\mathcal T_n\mid n\ge 1\}$ form a chain in $\mathbf{P}_\Code$.
\end{proposition}
\begin{proof}
Given a set of neurons $\sigma\subseteq [n]$, one can form a natural ``restriction" of a code $\mathcal C\subseteq 2^{[n]}$ by mapping $c\mapsto c\cap \sigma$. This restriction is a morphism (see \cite[Section 2]{morphisms}). In the case of the codes $\mathcal T_n\subseteq 2^{[2n]}$, one can note that $\mathcal T_n$ is the image of $\mathcal T_{n+1}$ under the restriction map with $\sigma = [2n]\subseteq [2n+2]$. This surjective morphism from $\mathcal T_{n+1}$ to $\mathcal T_n$ implies that $\mathcal T_n\le \mathcal T_{n+1}$ as desired.
\end{proof}
\subsection{Generalizing $\S_n$ and $\S_\Delta$ using Theorem \ref{thm:flexible}}\label{subsec:SCoverD}
We begin with a definition generalizing that of $\S_\Delta$. \begin{definition}\label{def:SCoverD}
Let $\mathcal D\subseteq \mathcal C\subseteq 2^{[n]}$ be intersection complete codes. We define \[
\S_{\mathcal C/\mathcal D} := \mathcal C \cup \{[n]\}\cup \{d\cup\{n+1\}\mid d\in \mathcal D\} \subseteq 2^{[n+1]}.
\]
Note that choosing $\mathcal D = \{\text{minimal nonempty codewords in $\mathcal C$}\}$ always satisfies the above conditions. In this case, we will let $\S_{\mathcal C/\text{min}}$ denote $\S_{\mathcal C/\mathcal D}$.
\end{definition}
Qualitatively, $\S_{\mathcal C/\mathcal D}$ is the result of forming a flexible sunflower using the codewords in $\mathcal C$, and then ``gluing" the petals of that sunflower to a new set $U_{n+1}$ along codewords in $\mathcal D$. Observe that $\S_\Delta$ of Definition \ref{def:SDelta} is equal to $\S_{\Delta/\Delta}$ in this notation. Also, if $\mathcal C = \{\{1\},\{2\},\ldots, \{n\}, \emptyset\}$, then we see $\S_n$ of Definition \ref{def:Sn} is equal to $\S_{\mathcal C/\text{min}}$.
\begin{proposition}
Let $\mathcal D\subseteq \mathcal C\subseteq 2^{[n]}$ be intersection complete codes. The code $\S_{\mathcal C/\mathcal D}$ is intersection complete. If $\mathcal D$ has $m$ maximal codewords and does not contain $[n]$, then $\S_{\mathcal C/\mathcal D}$ has $m+1$ maximal codewords. In particular, $\odim(\S_{\mathcal C/\mathcal D}) \le \max\{2,m\}$.
\end{proposition}
\begin{proof}
Codewords in $\S_{\mathcal C/\mathcal D}$ come in three types: codewords from $\mathcal C$, the codeword $[n]$, and those of the form $d\cup\{n+1\}$ where $d\in\mathcal D$. Since $\mathcal C$ and $\mathcal D$ are intersection complete, the intersection of two codewords of the same type always yields another codeword of that type (and hence lying in $\S_{\mathcal C/\mathcal D}$). This leaves the intersections of codewords of different types. The intersection of a codeword in $\mathcal C$ with $[n]$ is simply the same codeword in $\mathcal C$. The intersection of $d\cup\{n+1\}$ with $[n]$ is just $d$, which lies in $\S_{\mathcal C/\mathcal D}$ since $\mathcal D\subseteq \mathcal C$. Finally, the intersection of $c\in\mathcal C$ with $d\cup\{n+1\}$ is $c\cap d$, which lies in $\mathcal C$ since since $\mathcal C$ is intersection complete.
For the second part of the statement, note that if $d$ is a maximal codeword of $\mathcal D$, then $d\cup\{n+1\}$ is a maximal codeword of $\S_{\mathcal C/\mathcal D}$. Since $[n]\notin\mathcal D$, the codeword $[n]$ is also a maximal codeword of $\S_{\mathcal C/\mathcal D}$, yielding $m+1$ total maximal codewords. The bound on $\odim(\S_{\mathcal C/\mathcal D})$ follows immediately from \cite[Theorem 1.2]{openclosed}.
\end{proof}
The following proposition provides a generalization of Theorem \ref{thm:sunflowercodeversion} to the codes $\S_{\mathcal C/\text{min}}$.
\begin{proposition}\label{prop:SCovern}
Let $\mathcal C\subseteq 2^{[n]}$ be an intersection complete code which contains every singleton set. Then \[\odim(\S_{\mathcal C/\text{min}}) \ge \left\lceil \frac{n}{\dim(\Delta(\mathcal C))+1}\right\rceil.\]
\end{proposition}
\begin{proof}
We start with a degenerate case: if $n=1$, then $\mathcal C = \{\emptyset, 1\}$ and $\S_{\mathcal C/\text{min}} = \{12, 1, 2, \emptyset\}$. In this case $\odim(\S_{\mathcal C/\text{min}}) = 1$, while $n = 1$ and $\dim(\Delta(\mathcal C)) + 1 = 1$. We see that the bound given above is satisfied as desired.
Otherwise, $n\ge 2$. In this case, let $\{U_1,\ldots, U_{m+1}\}$ be an open convex realization of $\S_{\mathcal C/\text{min}}$ in $\mathbb R^d$. Since the minimal nonempty codewords of $\mathcal C$ are all singletons, the code $\S_{\mathcal C/\text{min}}$ consists of codewords from $\mathcal C$, the codeword $[n]$, codewords of the form $\{i, n+1\}$ where $i\in[n]$, and lastly the codeword $\{n+1\}$. Since $[n]$ is a codeword, the sets $\{U_1,\ldots, U_n\}$ all meet in a central point. In particular, $\{U_1,\ldots, U_n\}$ is a $k$-flexible sunflower, where $k$ is the largest weight of a codeword in $\mathcal C$ other than possibly $[n]$. In particular $k \le \dim(\Delta(\mathcal C))+1$, with equality if $[n]\notin \mathcal C$.
But consider the set $U_{n+1}$. This set does not meet $U_{[n]}$ since $[n+1]$ is not a codeword of $\S_{\mathcal C/\text{min}}$. However, it does touch each $U_i$ since $\{i, n+1\}$ is a codeword. If we choose $p_i\in U_i\cap U_{n+1}$, then the convex hull of $\{p_1,\ldots, p_n\}$ is contained in $U_{n+1}$ and therefore does not contain a point in the center of $\{U_1,\ldots, U_n\}$. By Theorem \ref{thm:flexible}, such a sampling of $p_i$ cannot be chosen if $n\ge dk+1$. Therefore we must have $n\le dk$. Rearranging, this implies $d\ge \lceil n/k\rceil$. Using the inequality $k\le \dim(\Delta(\mathcal C))+1$ yields the result.
\end{proof}
The added assumption in Proposition \ref{prop:SCovern} that $\mathcal C$ contains all singletons is not too restrictive, since adding singletons to an intersection complete code always maintains intersection completeness.
Continuing our pattern of generalizations, the proposition below is analogous to Theorem \ref{thm:SDelta} and its second part generalizes Proposition \ref{prop:SDeltaSn}.
\begin{proposition}\label{prop:SCoverDSE}
Let $\mathcal D\subseteq \mathcal C\subseteq 2^{[n]}$ be intersection complete codes. Let $m\ge 2$ be the number of maximal codewords in $\mathcal D$, and let $k$ be the largest number of maximal codewords in $\mathcal D$ whose union lies in $\Delta(\mathcal C)$. Then there exists an intersection complete code $\mathcal E\subseteq 2^{[m]}$ containing all singleton sets such that (i) $k=\dim(\Delta(\mathcal E))+1$, and (ii) there exists a surjective morphism $\S_{\mathcal C/\mathcal D}\to \S_{\mathcal E/\text{min}}$. In particular, $\S_{\mathcal E/\text{min}} \le \S_{\mathcal C/\mathcal D}$ and $m\ge \odim(\S_{\mathcal C/\mathcal D}) \ge \left\lceil \frac{m}{k}\right\rceil$.
\end{proposition}\begin{proof}
We will mirror the proof of Proposition \ref{prop:SDeltaSn}. Let $F_1,\ldots, F_m$ be the maximal codewords of $\mathcal D$. For $i\in[m]$ define $T_i = \Tk_{\S_{\mathcal C/\mathcal D}}(F_i)$, and define $T_{m+1} = \Tk_{\S_{\mathcal C/\mathcal D}}(n+1)$. Let us consider the image of $\S_{\mathcal C/\mathcal D}$ under the morphism $f$ determined by $\{T_1,\ldots, T_{m+1}\}$. Recall from Definition \ref{def:SCoverD} that the codewords of $\S_{\mathcal C/\mathcal D}$ come in the following types:\begin{itemize}
\item $c$ for $c\in \mathcal C$,
\item $d\cup \{n+1\}$ for $d\in \mathcal D$, and
\item $[n]$.
\end{itemize} The images of these codewords under $f$ are as follows:\begin{itemize}
\item $f(c)$ is equal to $\{i\in[m]\mid c \text{ contains $F_i$}\}$,
\item $f(d\cup \{n+1\})$ is equal to $\{m+1\}$ if $d$ is not equal to some $F_i$, and is equal to $\{i,m+1\}$ if $d = F_i$,
\item $f([n]) = [m]$ since $[n]$ contains all maximal codewords in $\mathcal D$, but not $n+1$.
\end{itemize}
Let $\mathcal E\subseteq 2^{[m]}$ be the collection of codewords in the first bullet above, i.e. $\mathcal E$ is the image of $\mathcal C$ under $f$. Since the image of an intersection complete code is again intersection complete, we see that $\mathcal E$ is intersection complete. Moreover, $\mathcal E$ contains every singleton set since $f(F_i) = \{i\}$.
The image of $\S_{\mathcal C/\mathcal D}$ under $f$ therefore contains codewords in $\mathcal E$, codewords of the form $\{i, m+1\}$ for all $i\in[m]$, the codeword $\{m+1\}$, and $[m]$. But these are exactly the codewords of $\S_{\mathcal E/\text{min}}$. Thus $\S_{\mathcal E/\text{min}}$ is the image of $\S_{\mathcal C/\mathcal D}$ under $f$.
To prove the result, it remains to show that $k = \dim(\Delta(\mathcal E)) + 1$. The codewords in $\mathcal E$ are of the form $f(c) = \{i\in[m]\mid F_i\subseteq c\}$. Thus a codeword in $\mathcal E$ corresponds to a collection of maximal codewords in $\mathcal D$ all of which are contained in some $c\in \mathcal C$. A codeword in $\mathcal E$ with largest weight thus corresponds to a largest possible collection of maximal codewords in $\mathcal D$ whose union is contained in $\Delta(\mathcal C)$. The largest such collection has size $k$ by definition, so any largest codeword in $\mathcal E$ has weight $k$, proving the result.
\end{proof}
\begin{remark}Generalizing Remark \ref{rem:Sm} from the last section, we see that among all codes of the form $\S_{\mathcal C/\mathcal D}$ with parameters $m$ and $k$ as described in Proposition \ref{prop:SCoverDSE}, the minimal elements are always of the form $\S_{\mathcal E/\text{min}}$ where $\mathcal E\subseteq 2^{[m]}$ contains all singletons, and $k =\dim(\Delta(\mathcal E))+1$. The following diagram shows this:\[
\includegraphics[width=30em]{SCoverD.pdf}
\]
\end{remark}
These results use Theorem \ref{thm:flexible} to provide a more complete picture of the open embedding dimensions of intersection complete codes. There is still much to be done, however. As one example, the bound $m\ge \odim(\S_{\mathcal C/\mathcal D}) \ge \left\lceil \frac{m}{k}\right\rceil$ of Proposition \ref{prop:SCoverDSE} leaves quite a large gap for $k\ge 2$. Sharpening this bound based on the combinatorial structure of $\mathcal C$ and $\mathcal D$ would be a natural task of interest.
\section{Conclusion}\label{sec:conclusion}
We have seen a number of phenomena arise in the closed and open embedding dimensions of intersection complete codes. Some of these, like Theorems \ref{thm:complexes}, \ref{thm:cdimleodim} and \ref{thm:cdimlinear}, gave us improved control over the embedding dimensions. Others, like Theorem \ref{thm:SDelta}, showed that embedding dimension may be difficult to control. With Theorem \ref{thm:flexible}, we developed new tools to understand open embedding dimension using $k$-flexible sunflowers, but the picture is still far from complete.
One direction for future work would be to search for analogous phenomena among codes that are not intersection complete. One could start with the following.
\begin{question}\label{q:odimlecdim}
Does there exist a code $\mathcal C$ with $\odim(\mathcal C)<\cdim(\mathcal C)<\infty$?
\end{question}
Theorem \ref{thm:cdimleodim} tells us that such a code cannot be intersection complete. There are examples due to \cite{5neurons, openclosed} of codes with $\odim(\mathcal C) < \cdim(\mathcal C) = \infty$, in which a minimum distance via compactness argument is used to prove $\cdim(\mathcal C) = \infty$. A similar approach, paired with a classic convexity theorem that depends on dimension such as Radon's Theorem, could yield a positive answer to Question \ref{q:odimlecdim}, and also possibly Question \ref{q:cdimexp} below.
\begin{question}\label{q:cdimexp}
Little is known about whether closed embedding dimension can be large relative to the number of neurons, $n$. A few open areas to investigate are the following, in increasing order of difficulty:\begin{itemize}
\item Does there exist a code $\mathcal C\subseteq 2^{[n]}$ for which $\cdim(\mathcal C)$ is finite, but larger than $n-1$?
\item Does there exist a family of codes $\{\mathcal C_n\subseteq 2^{[n]}\mid n\ge 1\}$ such that $\cdim(\mathcal C_n)$ grows faster than any linear function of $n$?
\item Does there exist a family of codes $\{\mathcal C_n\subseteq 2^{[n]}\mid n\ge 1\}$ such that $\cdim(\mathcal C_n)$ grows faster than any polynomial function of $n$?
\end{itemize}
Note that Theorem \ref{thm:cdimlinear} tells us that if such codes exist, they cannot be intersection complete. The construction of $\S_\Delta$ in Section \ref{sec:SDelta} provided affirmative answers to the $\odim$ versions of the above questions.
\end{question}
Theorem \ref{thm:flexible} provides a characterization of the ``worst case" for $k$-flexible sunflowers in $\mathbb R^d$: we can have up to $dk$ petals from which we can sample points whose convex hull does not intersect the center. But among these ``hard to sample" $k$-flexible sunflowers, how many codewords of weight $k$ are actually necessary? Presumably there should be a difference between containing a few codewords of weight $k$ and containing many codewords of weight $k$. The following question asks this formally.
\begin{question}
Let $k\ge 1$, $d\ge 2$ and $n\le dk$. Among all $k$-flexible sunflowers $\U$ in $\mathbb R^d$ with $n$ petals that do not satisfy Theorem \ref{thm:flexible} (i.e. for which we can choose points from each petal whose convex hull does not contain a point in the center of $\U$), what is the smallest number of codewords of weight $k$ in $\code(\U)$?
\end{question}
Regarding the tangled sunflower codes $\mathcal T_n$ of Section \ref{sec:Tn}, there is much to be done. A good first step would be to improve the embedding dimension bounds that we currently have, or, more ambitiously, find an exact characterization of the embedding dimension.
\begin{question}
Does there exist an explicit characterization of the open embedding dimensions $t_n$ described in Definition \ref{def:Tn}? Can we improve the bounds of $\lceil n/2\rceil \le t_n \le n$?
\end{question}
One might also consider codes that describe more than two sunflowers whose petals are ``tangled" (i.e. incident) in some way. This would be a significantly more complicated problem, but perhaps of some interest. Another generalization would be to consider a notion of tangled flexible sunflowers. This would be even more challenging to investigate, but would perhaps be more relevant to applications in experimental data.
\begin{question}
In Section \ref{sec:pcode} we contextualized our results via a partial order on codes, denoted $\mathbf{P}_\Code$. In this partial order, both $\odim$ and $\cdim$ are monotone functions. In \cite{morphisms} we showed that a code is intersection complete if and only if it lies below a simplicial complex in $\mathbf{P}_\Code$. An interesting question is thus the following: do the simplicial complexes lying above an intersection complete code $\mathcal C$ in $\mathbf{P}_\Code$ determine $\odim(\mathcal C)$? That is, among the simplicial complexes lying above $\mathcal C$ in $\mathbf{P}_\Code$, does one have minimal embedding dimension equal to $\odim(\mathcal C)$?
\end{question}
A positive answer to the above question would reduce the problem of determining open embedding dimension for intersection complete codes to the problem of determining open embedding dimension for simplicial complexes, which is very closely tied to the well-studied problem of determining when a complex is $d$-representable, as described in \cite[Section 1.2]{tancer}.
Note that the answer to this question cannot be positive when we replace $\odim$ with $\cdim$. Open and closed embedding dimension for simplicial complexes are always the same, but the code $\S_3$ already shows that closed dimension and open dimension are different for intersection complete codes. Thus the simplicial complexes lying over a code are not enough information to determine its closed embedding dimension.
In Section \ref{sec:SDelta}, we showed that $\odim(\S_\Delta)$ was equal to the number of facets in $\Delta$ by showing that any realization of $\S_\Delta$ gave rise to a realization of the code $\S_m$ described in Section \ref{sec:sunflowercodeversion}. Equivalently, we found a surjective morphism from $\S_\Delta$ to $\S_m$. This technique could be generalized to analyze arbitrary codes as follows. Given a code $\mathcal C$, look for the largest $m$ so that there is a surjective morphism $\mathcal C\to \S_m$. This largest $m$ then provides a lower bound on the open embedding dimension of $\mathcal C$.
Existing techniques for providing lower bounds on $\odim(\mathcal C)$ rely on homological information obtained from $\Delta(\mathcal C)$ (see for example \cite{leray}). In contrast, the sunflower approach is completely agnostic to homology of $\Delta(\mathcal C)$. Whether this approach could be useful in analyzing experimental data may be an interesting open question.
\begin{question}
Among experimental data, is it ever the case that (fractional) sunflower results such as Theorem \ref{thm:sunflower} and Theorem \ref{thm:flexible} yield a stronger lower bound on $\odim(\mathcal C)$ than other techniques such as the Leray dimension described in \cite{leray}?
\end{question}
\section*{Acknowledgements}
We would like to thank Florian Frick for raising the question of whether there exist open convex codes $\mathcal C\subseteq 2^{[n]}$ with $\odim(\mathcal C) > n-1$, and for interesting discussions on this question. Isabella Novik provided detailed feedback on initial drafts of this paper, as well as helpful discussions on its mathematical content.
\bibliographystyle{plain}
| {
"timestamp": "2019-10-01T02:21:22",
"yymm": "1909",
"arxiv_id": "1909.13406",
"language": "en",
"url": "https://arxiv.org/abs/1909.13406",
"abstract": "Two tantalizing invariants of a combinatorial code $\\mathcal C\\subseteq 2^{[n]}$ are cdim$(\\mathcal C)$ and odim$(\\mathcal C)$, the smallest dimension in which $\\mathcal C$ can be realized by convex closed or open sets, respectively. Cruz, Giusti, Itskov, and Kronholm showed that for intersection complete codes $\\mathcal C$ with $m+1$ maximal codewords, odim$(\\mathcal C)$ and cdim$(\\mathcal C)$ are both bounded above by $\\max\\{2,m\\}$. Results of Lienkaemper, Shiu, and Woodstock imply that odim and cdim may differ, even for intersection complete codes. We add to the literature on open and closed embedding dimensions of intersection complete codes with the following results:(*) If $\\mathcal C$ is a simplicial complex, then cdim$(\\mathcal{C}) = \\mbox{odim}(\\mathcal C)$,(*) If $\\mathcal C$ is intersection complete, then cdim$(\\mathcal C)\\le \\mbox{odim}(\\mathcal C)$,(*) If $\\mathcal C\\subseteq 2^{[n]}$ is intersection complete with $n\\ge 2$, then cdim$(\\mathcal C) \\le \\min \\{2d+1, n-1\\}$, where $d$ is the dimension of the simplicial complex of $\\mathcal C$, and(*) For each simplicial complex $\\Delta\\subseteq 2^{[n]}$ with $m\\ge 2$ facets, the code $\\mathcal S_\\Delta := (\\Delta \\ast (n+1)) \\cup \\{[n]\\}\\subseteq 2^{[n+1]}$ is intersection complete, has $m+1$ maximal codewords, and satisfies odim$(\\mathcal S_\\Delta)=m$. In particular, for each $n\\ge 3$ there exists an intersection complete code $\\mathcal C\\subseteq 2^{[n]}$ with odim$(\\mathcal C) = \\binom{n-1}{\\lfloor (n-1)/2\\rfloor}$.A key tool in our work is the study of sunflowers: arrangements of convex open sets in which the sets simultaneously meet in a central region, and nowhere else. We use Tverberg's theorem to study the structure of \"$k$-flexible\" sunflowers, and consequently obtain new lower bounds on odim$(\\mathcal C)$ for intersection complete codes $\\mathcal C$.",
"subjects": "Combinatorics (math.CO)",
"title": "Embedding Dimension Phenomena in Intersection Complete Codes",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754492759499,
"lm_q2_score": 0.7185943805178139,
"lm_q1q2_score": 0.7075103850454995
} |
https://arxiv.org/abs/1509.05660 | Closest multiplication tables of groups | Suppose that all groups of order $n$ are defined on the same set $G$ of cardinality $n$, and let the \emph{distance} of two groups of order $n$ be the number of pairs $(a,b)\in G\times G$ where the two group operations differ. Given a group $G(\circ)$ of order $n$, we find all groups of order $n$, up to isomorphism, that are closest to $G(\circ)$. | \section{Introduction}
Let $G$ be a finite set of cardinality $n$, and let $\circ$, $\ast$, $\gc$, $\bullet$ be group operations defined on $G$. For groups $G(\circ)$, $G(\ast)$, let
\begin{align*}
\mathrm{diff}(\circ,\ast) &= \{(a,b)\in G\times G;\;a\circ b\ne a\ast b\},\\
\mathrm{dist}(\circ,\ast) &= |\mathrm{diff}(\circ,\ast)|,
\end{align*}
and call $\mathrm{dist}(\circ,\ast)$ the \emph{(Hamming) distance of groups $G(\circ)$, $G(\ast)$}.
In a research programme spanning two decades, Ale\v{s} Dr\'apal showed that there is a strong relationship between algebraic properties of groups and their distances, as will become apparent from many of his results we quote below.
In this paper we solve the following problem: \emph{Given a group $G(\circ)$, determine all multiplication tables of groups $G(\ast)$
(up to isomorphism) that are as close to the multiplication table of $G(\circ)$ as possible.}
More formally, let
\begin{align*}
\cst(\circ) &= \min\{\mathrm{dist}(\circ,\ast);\;G(\circ)\ne G(\ast)\},\\
\cnb(\circ) &= \{G(\ast);\;\mathrm{dist}(\circ,\ast) = \cst(\circ)\}.
\end{align*}
Our task is then to find $\cst(\circ)$ and to construct one group $G(\ast)$ of minimum distance from $G(\circ)$ for every isomorphism class of groups intersecting $\cnb(\circ)$.\
In particular, we determine the minimal distance
\begin{align*}
\cst(n) = \min\{\cst(\circ);\;G(\circ)\text{ is a group of order }n\}
\end{align*}
and all pairs of groups $G(\circ)$, $G(\ast)$ (up to isomorphism) of order $n$ satisfying $\mathrm{dist}(\circ,\ast)=\cst(n)$.
\subsection{The context}
Let
\begin{align*}
\cst_{\cong}(\circ) &= \min\{\mathrm{dist}(\circ,\ast);\;G(\circ)\cong G(\ast)\ne G(\circ)\},\\
\cst_{\not\cong}(\circ) &= \min\{\mathrm{dist}(\circ,\ast);\;G(\circ)\not\cong G(\ast)\},
\end{align*}
where the second quantity is set to $\infty$ if all groups of order $n$ are isomorphic. Obviously, we have $\cst(\circ) = \min\{\cst_{\cong}(\circ),\,\cst_{\not\cong}(\circ)\}$.
An important threshold for $\cst(\circ)$ is obtained by considering pairs of groups isomorphic via a transposition. Note that if $f=(a,b)$ is an isomorphism between $G(\circ)$ and $G(\ast)$ then $\mathrm{diff}(\circ,\ast)$ is a subset of the rows and columns indexed by $a$, $b$, and of the ``diagonal'' entries $(x,y)$ with $x\circ y\in\{a,b\}$. This means that $\delta(n)$ will not exceed $6n$. More precisely:
As in \cite{DrEJC}, for a nontrivial commutative group $O$ of odd order, let $D(O)$ be the generalized dihedral group defined on $O\times C_2$ by
\begin{displaymath}
(a,0)(b,h) = (ab,h),\quad (a,1)(b,h) = (ab^{-1},1+h).
\end{displaymath}
Then let
\begin{equation}\label{Eq:Delta0}
\cst_0(\circ) = \left\{\begin{array}{ll}
6n-18,&\text{if $n$ is odd},\\
6n-20,&\text{if $G(\circ)\cong D(O)$ for some $O$},\\
6n-24,&\text{otherwise}.
\end{array}\right.
\end{equation}
The main results of \cite{DrEJC} can be summarized as follows:
\begin{theorem}[Dr\'apal]\label{Th:DrEJC} Let $|G|=n$ and let $G(\circ)$, $G(\ast)$ be groups defined on $G$. If $\mathrm{dist}(\circ,\ast)<n^2/9$ then $G(\circ)$ and $G(\ast)$ are isomorphic. If $n\ge 5$ then $\mathrm{dist}(\circ,\ast)\ge\cst_0(\circ)$ whenever $G(\ast)$ is isomorphic to $G(\circ)$ via a transposition, and $\mathrm{dist}(\circ,\ast) = \cst_0(\circ)$ for some $G(\ast)$ isomorphic to $G(\circ)$ via a transposition. Consequently, if $n\ge 51$ then $\cst(\circ) = \cst_0(\circ) = \cst_{\cong}(\circ) < \cst_{\not\cong}(\circ)$.
\end{theorem}
Moreover, \cite[Proposition 5.8]{DrEJC} describes in detail the transpositions that achieve the distance $\cst_0(\circ)$. Hence our problem has already been solved in all but finitely many cases. Here is an overview of other known results concerning distances of groups:
To determine $\cst_{\not\cong}(\circ)$ appears to be a very difficult problem. We already know from \tref{Th:DrEJC} that $\cst_{\not\cong}(\circ)\ge n^2/9$ whenever $n\ge 5$. When $G(\circ)$ is a $2$-group then $\cst_{\not\cong}(\circ)\ge n^2/4$ by \cite{Dr2}. Examples of non-isomorphic $2$-groups at \emph{quarter distance}, that is, with $\mathrm{dist}(\circ,\ast) = n^2/4$, can be found in \cite{DrConstr1} and \cite{DrConstr2}. In \cite{DrProc}, Dr\'apal constructed a family of $p$-groups for every prime $p>2$ with the property $\cst_{\not\cong}(\circ) = (n^2/4)(1-1/p^2)$. In particular, there is a $3$-group satisfying $\cst_{\not\cong}(\circ) = 2n^2/9$ (see also Construction 2 in Subsection \ref{Ss:Other}). Ivanyos {\em et al.} \cite{ILY} showed, after this paper had been submitted, that $\cst_{\not\cong}(\circ)\ge 2n^2/9$ always holds.
Let $\mathcal G(n)$ be a graph whose vertices are the isomorphism classes of groups of order $n$, and in which two vertices, possibly the same, form an edge if and only if they contain representatives at distance $\cst(n)$.
When $n$ is a power of two, let $\mathcal G'(n)$ be a graph on the same vertices as $\mathcal G(n)$ in which two vertices, possibly the same, form an edge if an only if they contain representatives at distance $n^2/4$ obtained by one of the two constructions of Dr\'apal \cite{DrConstr1} that we recall in Subsection \ref{Ss:Quarter}. When $n\in\{8,16\}$, it turns out that $\cst(n)=n^2/4$, so $\mathcal G'(n)$ is a subgraph of $\mathcal G(n)$.
By \cite{DrDiscr}, $\cst(\circ)\ge n^2/4$ for any $2$-group $G(\circ)$ of order $n\le 16$. In \cite{VoMS, VoProc}, the first author determined the connected graph $\mathcal G(8)$ with $\cst(n)=8^2/4=16$ (we checked that $\mathcal G'(8) = \mathcal G(8)$), calculated $\cst(\circ)$ for cyclic groups $G(\circ)$ of order less than $13$, proved that $\cst(\circ)=6n-18$ whenever $G(\circ)$ is a group of prime order $n>7$, and constructed a class of groups with $\cst(\circ)<\cst_0(\circ)$, of which the largest member has order $21$. (As we are going to show, $n=21$ happens to be the largest order for which $\cst(\circ)<\cst_0(\circ)$ can occur.)
B\'alek \cite{Ba} computed the subgraph $\mathcal G'(16)$ (excluding the diagonal entries) of $\mathcal G(16)$. Since $\mathcal G'(16)$ turns out to be connected, it follows that $\cst(\circ) = n^2/4$ for every group $G(\circ)$ of order $n=16$. A more direct argument establishing the connectedness of $\mathcal G(16)$ can be found in \cite{DrZh}. Our computational results show that $\mathcal G'(16) = \mathcal G(16)$. The two constructions of Subsection \ref{Ss:Quarter} can therefore be seen as canonical for $n\in\{8,16\}$.
Groups at quarter distance received attention even for orders $n=2^k>16$, although then $\cst(n)<n^2/4$ so $\mathcal G'(n)$ is no longer a subgraph of $\mathcal G(n)$. In \cite{Zh}, Zhukavets calculated $\mathcal G'(32)$ and $\mathcal G'(64)$; the first graph is connected while the second one has two connected components.
The quarter distance is of interest outside the variety of groups, too. In \cite{DrVo}, Dr\'apal and the first author generalized the constructions of \cite{DrConstr1} for \emph{Moufang loops}, that is, loops satisfying the identity $x(y(xz)) = ((xy)x)z$. The first author went on to construct a large family of Moufang loops of order $64$ \cite{VoEJC}, starting with the well-known Moufang loops $M_{2n}(G,2)$ of Chein \cite[pp. 35--38]{Chein} and using the constructions of \cite{DrVo}. Nagy and the first author eventually proved in \cite{NaVo} that the family of \cite{VoEJC} actually contains all Moufang loops of order $64$ up to isomorphism.
Distances of infinite groups are somewhat trivial, as it was shown in \cite{DrEJC} that if $G(\circ)$ is a group of infinite cardinality $\kappa$ then $\cst_{\cong}(\circ) = \cst_{\not\cong}(\circ) = \kappa$.
\subsection{The content}
For the convenience of the reader, the main result is stated at the outset in Section \ref{Sc:Main}.
For two subsets $\mathcal A$, $\mathcal B$ of groups defined on $G$, let
\begin{displaymath}
\mathrm{dist}(\mathcal A,\mathcal B) = \min\{\mathrm{dist}(\circ,\ast);\;G(\circ)\in \mathcal A,\,G(\ast)\in\mathcal B,\,G(\circ)\ne G(\ast)\}.
\end{displaymath}
Denote by $[\circ]$ the class of all groups defined on $G$ and isomorphic to $G(\circ)$. In Section \ref{Sc:Aut}, we recall that $\mathrm{dist}([\circ],[\ast])= \mathrm{dist}([\circ],\ast)$. Consequently, the values of $\cst(\circ)$, $\cst_{\cong}(\circ)$ and $\cst_{\not\cong}(\circ)$ depend only on the isomorphism type of $G(\circ)$. If $n\ge 5$, \lref{Lm:Neutral} allows us to assume that closest groups have the same neutral element. \lref{Lm:Aut} shows how automorphism groups of $G(\circ)$, $G(\ast)$ come into play to speed up the calculation of $\mathrm{dist}([\circ],[\ast])$.
In Section \ref{Sc:Tools} we introduce, following Dr\'apal, these concepts and parameters:
\begin{align} \label{Eq:Tools}
\begin{split}
\textstyle \mathrm{diff}_a(\circ,\ast) &=
\{(a,b);\;b\in G,\,a\circ b\ne a\ast b\}|,\quad
\textstyle \mathrm{dist}_a(\circ,\ast) = |\mathrm{diff}_a(\circ,\ast)|,\\
m(\circ,\ast) &=
\textstyle
\min\{\mathrm{dist}_a(\circ,\ast);\;a\in G,\,\mathrm{dist}_a(\circ,\ast)>0\},\\
H(\circ,\ast) &=
\textstyle
\{a\in G;\;\mathrm{dist}_a(\circ,\ast)=0\},\quad h(\circ,\ast)=|H(\circ,\ast)|,\\
K(\circ,\ast) &=
\textstyle
\{a\in G;\;\mathrm{dist}_a(\circ,\ast)<n/3\},\quad k(\circ,\ast)=|K(\circ,\ast)|.
\end{split}
\end{align}
When $\circ$, $\ast$ are fixed, we drop the operations from the names of the parameters and write $\mathrm{dist}_a$, $m$, $H$, and so on.
Among other results, we recall in Section \ref{Sc:Tools} that $a\circ b\ne a\ast b$ implies $\mathrm{dist}_a+\mathrm{dist}_b + \mathrm{dist}_{a\circ b}\ge n$; the set $H$ is either empty or it is a subgroup of both $G(\circ)$ and $G(\ast)$; if $|k|>3n/4$ then $\mathrm{dist}(\circ,\ast)>\cst_0(\circ)$; $m\ge 2$ if $n$ is even and $m\ge 3$ if $n$ is odd. We also study $\mathrm{dist}_a$ when the orders of $a$ in $G(\circ)$ and $G(\ast)$ disagree.
Building on these results, in Section \ref{Sc:Inequalities} we develop a series of inequalities relating $n$, $h$, $k$, $m$ and, consequently, we find only a few (less than hundred) quadruples $(n,h,k,m)$ in the range $22<n<51$ that can possibly yield $\mathrm{dist}(\circ,\ast)\le \cst_0(\circ)$. This will already imply that $\mathrm{dist}(\circ,\ast)<\cst_0(\circ)$ cannot hold for $n\ge 43$, improving upon the bound $n\ge 51$ of \tref{Th:DrEJC}.
In Section \ref{Sc:m2}, we first show that the case $m=2$ can be reduced to the study of distances of the cyclic group $C_n$ from a group possessing an element of order $n/2$, a case that is not difficult to handle computationally. We can proceed similarly when $n$ is a prime, independently verifying the results of \cite{VoMS, VoProc}.
The general algorithm for finding $\mathrm{dist}([\circ],[\ast])$ is given in Section \ref{Sc:Algorithm}. The algorithm is sufficiently fast to deal with all orders $n\le 22$ and also all cases when $h>1$, leaving us with only $20$ quadruples $(n,h,k,m)$, which require a very delicate analysis.
In Section \ref{Sc:Graph} we study the question: \emph{Given an edge-colored graph on $v$ vertices such that no color is used more than $m$ times and no vertex is adjacent to more than two edges of the same color, how many edges must the graph have to guarantee a rainbow $i$-matching?} A partial answer can be found in \pref{Pr:Graph}.
Returning to the problem of group distances, in Section \ref{Sc:ApplyGraph} we study the set $\{(a,b)\in\mathrm{diff}(\circ,\ast);\;a\in K,b\not\in K,a\circ b\not\in K\}$ and similar sets which give rise to edge-colored graphs. The main idea of Section \ref{Sc:ApplyGraph} is to exhibit a large enough rainbow matching in a certain graph to push the distance over the threshold $\cst_0(\circ)$.
Only $7$ quadruples $(n,h,k,m)$ remain after this analysis, all with $n\le 28$. These are disposed of in Section \ref{Sc:Stubborn}, using a series of increasingly more specialized lemmas.
Finally, in Section \ref{Sc:Constructions} we present several constructions that produce all pairs $G(\circ)$, $G(\ast)$ with $\mathrm{dist}(\circ,\ast) = \cst(\circ)<\cst_0(\circ)$. These are the constructions alluded to in \tref{Th:Main}, the main result.
\section{Main result}\label{Sc:Main}
\begin{theorem}\label{Th:Main}
Let $G$ be a set of size $n\ge 4$. Let $G(\circ)$ be a group defined on $G$, $\cst(\circ) = \min\{\mathrm{dist}(\circ,\ast);\;G(\ast)\text{ is a group different from }G(\circ)\}$, $\cnb(\circ) = \{G(\ast);\;\mathrm{dist}(\circ,\ast)=\cst(\circ)\}$, and let $\cst_0(\circ)$ be defined as in \eqref{Eq:Delta0}.
Then the value of $\cst(\circ)$ and one representative from $\cnb(\circ)$
for every isomorphism type of groups present in $\cnb(\circ)$ can be
found as follows:
\begin{enumerate}
\item[$\bullet$] If $n\not\in\{4,6,7,8,9,10,12,14,15,16,18,21\}$ then
$\cst(\circ)=\cst_0(\circ)$, all groups in $\cnb(\circ)$ are isomorphic to
$G(\circ)$, and there is a transposition $f$ of $G$ such that
$f:G(\circ)\to G(\ast)$ is an isomorphism and $G(\ast)\in \cnb(\circ)$.
\item[$\bullet$] Otherwise the value of $\cst(\circ)$ and the
isomorphism types of groups in $\cnb(\circ)$ can be found in Table
\ref{Tb:Main}. When $n$ is a power of two and also in the case $\mathrm{dist}(C_3\times S_3,C_3\times S_3)$, the representatives of
$\cnb(\circ)$ can be obtained by the constructions
of Subsection $\ref{Ss:Quarter}$. When $n$ is not a power of two, the
representatives of $\cnb(\circ)$ can be obtained by one of the three
types of constructions of Subsection $\ref{Ss:Other}$, as indicated by
the superscript in the table.
\end{enumerate}
In particular,
\begin{enumerate}
\item[$\bullet$] $\cst(\circ)<\cst_0(\circ)$ if and only if $G(\circ)$ is one of the following groups: $C_6$, $C_{10}$, $C_{14}$, $C_{21}$, a group of order $12$ except for $A_4$, a group of order $7$, $8$, $9$, $15$, $16$ or $18$.
\item[$\bullet$] $\cnb(\circ)$ contains groups of more than one isomorphism type if and only if $G(\circ)$ is one of the following groups: $C_9$, $D_{10}$, a group of order $8$, a group of order $16$, $D_{18}$, $C_{18}$, $C_6\times C_3$.
\item[$\bullet$] $\cnb(\circ)$ contains no groups isomorphic to $G(\circ)$ if and only if $G(\circ)$ is one of the following groups: $C_4$, $(C_2)^2$, $S_3$, $Q_8$, $(C_2)^3$, $(C_3)^2$, $(C_2)^4$, $(C_3)^2\rtimes C_2$.
\end{enumerate}
\end{theorem}
\begin{scriptsize}
\begin{table}
\caption{Distances of isomorphism classes of groups for all orders $n$ where at least one group $G(\circ)$ satisfies $\cst(\circ)<\cst_0(\circ)$. A group of order $n$ labeled by $i$ is the $i$th group of order $n$ as listed in \texttt{GAP}. The row labels are structural descriptions of the groups with the usual conventions. The distance $\mathrm{dist}([\circ],[\ast])$ between the $i$th group $G(\circ)$ and the $j$th group $G(\ast)$ of order $n$ can be found in row $i$ and column $j$ of the table for $n$. This value is underlined if it is less than $\cst_0(\circ)$ (this has the potential to break the diagonal symmetry of the tables but actually never does), it is in bold face if it equals $\cst(n)$, and it is replaced with ``?'' if it was not calculated exactly but exceeds $\cst_0(\circ)$. The superscript points to a construction in Subsection \ref{Ss:Other} that achieves the distance.}\label{Tb:Main}
\begin{displaymath}
\addtolength{\arraycolsep}{-1.5pt}
\begin{array}{r|c|c}
n=4
&1&2\\
\hline\vline depth 0pt height 10pt width 0pt
C_4=1&7&\bf{4}\\
(C_2)^2=2&\bf{4}&16\\
\end{array}
\addtolength{\arraycolsep}{1.5pt}
\quad
\addtolength{\arraycolsep}{-1.5pt}
\begin{array}{r|c|c}
n=6
&1&2\\
\hline\vline depth 0pt height 10pt width 0pt
S_3=1&16&12^{1}\\
C_6=2&12&\best{8}^2\\
\end{array}
\addtolength{\arraycolsep}{1.5pt}
\quad
\addtolength{\arraycolsep}{-1.5pt}
\begin{array}{r|c|c}
n=9
&1&2\\
\hline\vline depth 0pt height 10pt width 0pt
C_9=1&\best{18}^3&\best{18}^2\\
(C_3)^2=2&\best{18}^2&36\\
\end{array}
\addtolength{\arraycolsep}{1.5pt}
\end{displaymath}
\begin{displaymath}
\addtolength{\arraycolsep}{-1.5pt}
\begin{array}{r|c}
n=7
&1\\
\hline\vline depth 0pt height 10pt width 0pt
C_7=1&\best{18}^3\\
\end{array}
\addtolength{\arraycolsep}{1.5pt}
\quad\quad
\addtolength{\arraycolsep}{-1.5pt}
\begin{array}{r|c}
n=15
&1\\
\hline\vline depth 0pt height 10pt width 0pt
C_{15}=1&\best{50}^2\\
\end{array}
\end{displaymath}
\begin{displaymath}
\addtolength{\arraycolsep}{-1.5pt}
\begin{array}{r|c|c}
n=10
&1&2\\
\hline\vline depth 0pt height 10pt width 0pt
D_{10}=1&40&40^1\\
C_{10}=2&40&\best{24}^2\\
\end{array}
\addtolength{\arraycolsep}{1.5pt}
\quad
\addtolength{\arraycolsep}{-1.5pt}
\begin{array}{r|c|c}
n=14
&1&2\\
\hline\vline depth 0pt height 10pt width 0pt
D_{14}=1&64&84\\
C_{14}=2&84&\best{48}^2\\
\end{array}
\addtolength{\arraycolsep}{1.5pt}
\quad
\addtolength{\arraycolsep}{-1.5pt}
\begin{array}{r|c|c}
n=21
&1&2\\
\hline\vline depth 0pt height 10pt width 0pt
C_7\rtimes C_3=1&108&?\\
C_{21}=2&?&\best{98}^2\\
\end{array}
\addtolength{\arraycolsep}{1.5pt}
\end{displaymath}
\begin{displaymath}
\addtolength{\arraycolsep}{-1.5pt}
\begin{array}{r|c|c|c|c|c}
n=8
&1&2&3&4&5\\
\hline\vline depth 0pt height 10pt width 0pt
C_8=1 &\best{16}&\best{16}&24&24&28\\
C_4\times C_2=2 &\best{16}&\best{16}&\best{16}&\best{16}&\best{16}\\
D_8=3 &24&\best{16}&\best{16}&\best{16}&\best{16}\\
Q_8=4 &24&\best{16}&\best{16}&24&24\\
(C_2)^3=5 &28&\best{16}&\best{16}&24&24\\
\end{array}
\addtolength{\arraycolsep}{1.5pt}
\end{displaymath}
\begin{displaymath}
\addtolength{\arraycolsep}{-1.5pt}
\begin{array}{r|c|c|c|c|c}
n=12
&1&2&3&4&5\\
\hline\vline depth 0pt height 10pt width 0pt
\mathrm{Dic}_3=1 &\best{32}^2&48&82&\good{36}&60\\
C_{12}=2 &48&\best{32}^2&70&60&\good{36}\\
A_4=3 &82&70&48&72&60\\
D_{12}=4 &\good{36}&60&72&\best{32}^2&48\\
C_6\times C_2=5 &60&\good{36}&60&48&\best{32}^2\\
\end{array}
\addtolength{\arraycolsep}{1.5pt}
\quad\quad
\addtolength{\arraycolsep}{-1.5pt}
\begin{array}{r|c|c|c|c|c}
n=18
&1&2&3&4&5\\
\hline\vline depth 0pt height 10pt width 0pt
D_{18}=1 & \best{72}^3&144&144&\best{72}^2&180\\
C_{18}=2 &144& \best{72}^3&138&180&\best{72}^2\\
C_3\times S_3=3 &144&138& \good{81}&108&108\\
(C_3)^2\rtimes C_2=4 & \best{72}^2&180&108& 88&144\\
C_6\times C_3=5 &180& \best{72}^2&108&144& \best{72}^2\\
\end{array}
\addtolength{\arraycolsep}{1.5pt}
\end{displaymath}
\begin{displaymath}
\addtolength{\arraycolsep}{-1.5pt}
\begin{array}{r|c|c|c|c|c|c|c|c|c|c|c|c|c|c}
n=16
&1&2&3&4&5&6&7&8&9&10&11&12&13&14\\
\hline\vline depth 0pt height 10pt width 0pt
C_{16}=1 & \best{64}& \best{64}&112&112& \best{64}& 96&112&112&112&112&136&136&128&148\\
(C_4)^2=2 & \best{64}& \best{64}& \best{64}& \best{64}& \best{64}& 88&128&112&112& \best{64}& 96& 96& 96&112\\
\text{rank\ 2\ }(C_4\times C_2)\rtimes C_2=3 &112& \best{64}& \best{64}& \best{64}& 88& \best{64}& 96& \best{64}& 96& \best{64}& \best{64}& 96& 96& 96\\
C_4\rtimes C_4=4&112& \best{64}& \best{64}& \best{64}& 88& \best{64}& 96& 96& \best{64}& \best{64}& \best{64}& \best{64}& 96&112\\
C_8\times C_2=5 & \best{64}& \best{64}& 88& 88& \best{64}& \best{64}& 96& 96& 96& \best{64}& 96& 96& 96&112\\
C_8\rtimes C_2=6 & 96& 88& \best{64}& \best{64}& \best{64}& \best{64}& 96& 96& 96& 88& 96& 96& \best{64}&128\\
D_{16}=7 &112&128& 96& 96& 96& 96& \best{64}& \best{64}& \best{64}&112& \best{64}&112& 96&112\\
QD_{16}=8 &112&112& \best{64}& 96& 96& 96& \best{64}& \best{64}& \best{64}&112& 96& 96& \best{64}&128\\
Q_{16}=9 &112&112& 96& \best{64}& 96& 96& \best{64}& \best{64}& \best{64}&112& 96& \best{64}& 96&136\\
C_4\times (C_2)^2=10&112& \best{64}& \best{64}& \best{64}& \best{64}& 88&112&112&112& \best{64}& \best{64}& \best{64}& \best{64}& \best{64}\\
C_2\times D_8=11&136& 96& \best{64}& \best{64}& 96& 96& \best{64}& 96& 96& \best{64}& \best{64}& \best{64}& \best{64}& \best{64}\\
C_2\times Q_8=12&136& 96& 96& \best{64}& 96& 96&112& 96& \best{64}& \best{64}& \best{64}& \best{64}& \best{64}& 96\\
\text{rank\ 3\ }(C_4\times C_2)\rtimes C_2 = 13&128& 96& 96& 96& 96& \best{64}& 96& \best{64}& 96& \best{64}& \best{64}& \best{64}& \best{64}& 88\\
(C_2)^4=14&148&112& 96&112&112&128&112&128&136& \best{64}& \best{64}& 96& 88& 72
\end{array}
\addtolength{\arraycolsep}{1.5pt}
\end{displaymath}
\end{table}
\end{scriptsize}
\subsection{Additional results}
The values $\cst_{\cong}(C_n)$ for $4\le n\le 22$ are as follows:
\begin{displaymath}
\begin{array}{|c|ccccccccccccccccccc|}
\hline
n&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19&20&21&22\\
\cst_{\cong}(C_n)&7&12&8&18&16&18&24&48&32&60&48&50&64&84&72&96&96&98&108\\
\hline
\end{array}
\end{displaymath}
The distances for $n\in\{20,22\}$ are as follows, with the same notational conventions as in Table \ref{Tb:Main}:
\begin{displaymath}
\begin{array}{r|c|c|c|c|c}
n=20
&1&2&3&4&5\\
\hline\vline depth 0pt height 10pt width 0pt
\mathrm{Dic}_5=1 & \mathbf{96}& ?& ?&100&?\\
C_{20}=2 & ?& \mathbf{96}& ?& ?&100\\
C_5\rtimes C_4=3 & ?& ?&\mathbf{96}& ?&?\\
D_{20}=4 &100& ?& ?&\mathbf{96} &160\\
C_{10}\times C_2=5 & ?&100& ?& 160&\mathbf{96}\\
\end{array}
\addtolength{\arraycolsep}{1.5pt}
\quad\quad
\addtolength{\arraycolsep}{-1.5pt}
\begin{array}{r|c|c}
n=22
&1&2\\
\hline\vline depth 0pt height 10pt width 0pt
D_{22}&112&?\\
C_{22}&?&\mathbf{108}\\
\end{array}
\quad\quad
\end{displaymath}
\section{Distances of isomorphism classes}\label{Sc:Aut}
For a group $G(\circ)$ and a bijection $f:G\to G$ there is a unique group $G(\ast)$ such that $f:G(\circ)\to G(\ast)$ is an isomorphism, namely $a\ast b = f(f^{-1}(a)\circ f^{-1}(b))$. We denote this operation $\ast$ by $\circ_f$.
\begin{lemma}\label{Lm:IsoMove}
Let $G(\circ)$, $G(\ast)$ be groups and $f:G\to G$ a bijection. Then $\mathrm{dist}_a(\circ,\ast) = \mathrm{dist}_{f(a)}(\circ_f,\ast_f)$ for every $a\in G$. In particular, $\mathrm{dist}(\circ,\ast) = \mathrm{dist}(\circ_f,\ast_f)$.
\end{lemma}
\begin{proof}
Fix $a\in G$. The cardinalities of the sets of elements $b\in G$ satisfying any of the following conditions are the same:
\begin{align*}
a\circ b&\ne a\ast b,\\
f^{-1}(f(a))\circ b&\ne f^{-1}(f(a))\ast b,\\
f^{-1}(f(a))\circ f^{-1}(b)&\ne f^{-1}(f(a))\ast f^{-1}(b),\\
f(f^{-1}(f(a))\circ f^{-1}(b))&\ne f(f^{-1}(f(a))\ast f^{-1}(b)),\\
f(a)\circ_f b&\ne f(a)\ast_f b.
\end{align*}
\end{proof}
\begin{proposition}\label{Pr:IsoDist}
Let $G(\circ)$, $G(\ast)$ be groups. Then $\mathrm{dist}([\circ],[\ast]) = \mathrm{dist}([\circ],\ast)$. Moreover, if $G(\circ)\cong G(\ast)$ then $\cst(\circ) = \cst(\ast)$, $\cst_{\cong}(\circ) = \cst_{\cong}(\ast)$ and $\cst_{\not\cong}(\circ) = \cst_{\not\cong}(\ast)$.
\end{proposition}
\begin{proof}
Let $f$, $g:G\to G$ be bijections for which $\mathrm{dist}([\circ],[\ast]) = \mathrm{dist}(\circ_f,\ast_g)$. Then, by \lref{Lm:IsoMove}, $\mathrm{dist}([\circ],[\ast]) = \mathrm{dist}(\circ_f,\ast_g) = \mathrm{dist}((\circ_f)_{g^{-1}},\ast)\ge\mathrm{dist}([\circ],\ast)$. The other inequality is obvious.
Now assume that $\ast = \circ_f$ for some bijection $f:G\to G$, and let $G(\gc)$ be such that $\cst(\circ) = \mathrm{dist}(\circ,\gc)$. Then $\cst(\ast)\le \mathrm{dist}(\ast,\gc_f) = \mathrm{dist}(\circ_f,\gc_f) = \mathrm{dist}(\circ,\gc)=\cst(\circ)$, the other inequality follows by symmetry, so $\cst(\circ)=\cst(\ast)$. The equalities $\cst_{\cong}(\circ) = \cst_{\cong}(\ast)$ and $\cst_{\not\cong}(\circ) = \cst_{\not\cong}(\ast)$ are proved similarly.
\end{proof}
To determine $\mathrm{dist}([\circ],[\ast])$ it therefore suffices to find the minimal value of $\mathrm{dist}(\circ_f,\ast)$, where $f:G\to G$ is a bijection.
Let us denote the neutral element of $G(\circ)$ by $1(\circ)$, and the inverse of $a$ in $G(\circ)$ by $a^\circ$.
\begin{lemma}\label{Lm:Neutral}
Assume that $G(\circ)$, $G(\ast)$ have the same neutral element $1(\circ)=1(\ast)$, and let $f:G\to G$ be a bijection such that $\mathrm{dist}([\circ],[\ast]) = \mathrm{dist}(\circ_f,\ast)$. Then either $f(1(\circ)) = 1(\circ)$, or else $\circ_f =\ast_\ell$ for some transposition $\ell$ and $\mathrm{dist}([\circ],[\ast]) = \mathrm{dist}(\ast_\ell,\ast)$.
\end{lemma}
\begin{proof}
Let $G(\gc) = G(\circ_f)$, so $\mathrm{dist}([\circ],[\ast]) = \mathrm{dist}(\gc,\ast)$. Since $f:G(\circ)\to G(\gc)$ is an isomorphism, we have $1(\gc) = f(1(\circ))$. If $1(\gc)=1(\circ)$ we are done, so assume that $1(\gc) = f(1(\circ))\ne 1(\circ)$. Let $g = \ell\circ f$ be the composition of $f$ with the
transposition $\ell$ of $1(\circ)$ and $1(\gc)$, and let $G(\bullet) = G(\circ_g)$. We claim that $\mathrm{dist}(\bullet,\ast) < \mathrm{dist}(\gc,\ast)$.
Recall that $1(\circ)=1(\ast)$, and consider the set $E=\{(a,b)\in G\times G;\;\{a,b\}\cap\{1(\gc),1(\ast)\}\ne\emptyset\}$. We first show that $G(\gc)$ and $G(\ast)$ disagree on every entry of $E$. Indeed, if $a=1(\gc)$ and $b\in G$ then $a\gc b = 1(\gc)\gc b = b = 1(\ast)\ast b\ne 1(\gc)\ast b = a\ast b$, if $a=1(\ast)$ then $a\gc b = 1(\ast)\gc b \ne 1(\gc)\gc b = b = 1(\ast)\ast b = a\ast b$, and similarly if $b\in\{1(\gc),1(\ast)\}$. On the other hand, we claim that $G(\bullet)$ and $G(\ast)$ agree on the row of $E$ indexed by $1(\ast)$, and on the column of $E$ indexed by $1(\ast)$. Indeed, we have $g^{-1}(1(\ast)) = f^{-1}(1(\gc)) = 1(\circ)$, and hence $1(\ast)\bullet b = g(g^{-1}(1(\ast))\circ g^{-1}(b)) = g(1(\circ)\circ g^{-1}(b)) = g(g^{-1}(b)) = b = 1(\ast)\ast b$, and, similarly, $b\bullet 1(\ast) = b\ast 1(\ast)$. Hence $|E\cap\mathrm{diff}(\gc,\ast)| - |E\cap\mathrm{diff}(\bullet,\ast)|\ge 2n-1$.
Since the operation $\bullet = \circ_g$ is obtained from $\gc = \circ_f$ by applying the transposition $\ell$, the two operations agree outside of $E$, except possibly on the two ``diagonals''
\begin{displaymath}
F=\{(a,b)\in G\times G;\;a\gc b = 1(\ast)\text{ or }a\gc b = 1(\gc)\}.
\end{displaymath}
Recall that $1(\ast)\bullet b = 1(\ast)\ast b$ for every $b\in G$, in particular for the two values of $b$ with $(1(\ast),b)\in F$. Thus, in the worst case, $|F\cap\mathrm{diff}(\gc,\ast)| - |F\cap\mathrm{diff}(\bullet,\ast)| \ge 0 - (|F|-2) = 2-2n$. We conclude that $\mathrm{dist}(\bullet,\ast)<\mathrm{dist}(\gc,\ast)$.
This means that $\mathrm{dist}(\bullet,\ast)=0$ and thus $\bullet = \ast$. Since $\bullet = \circ_g = (\circ_f)_\ell$, we see that $\circ_f = \ast_\ell$.
\end{proof}
While calculating $\mathrm{dist}([\circ],[\ast])$, we can certainly assume that $1(\circ) = 1(\ast) = 1$. Lemma \ref{Lm:Neutral} therefore allows us to consider only mappings $f$ fixing the element $1$, or to conclude that $\mathrm{dist}([\circ],[\ast]) = \mathrm{dist}(\ast_\ell,\ast)$ for some transposition $\ell$, a case fully resolved by Theorem \ref{Th:DrEJC} as long as $n\ge 5$. This speeds up the search slightly. A much larger improvement is achieved by looking at the automorphism groups of $G(\circ)$ and $G(\ast)$. Denote by $\mathrm{Aut}(\circ)$ the automorphism group of $G(\circ)$.
\begin{lemma}\label{Lm:Aut}
Let $G(\circ)$, $G(\ast)$ be groups, $f:G\to G$ a bijection, and $g\in\mathrm{Aut}(\circ)$, $\ell\in\mathrm{Aut}(\ast)$. Then $\mathrm{dist}(\circ_f,\ast) = \mathrm{dist}(\circ_{\ell fg},\ast)$.
\end{lemma}
\begin{proof}
Note that $\circ_g = \circ$ and $\ast_\ell=\ast$. Using these facts and \lref{Lm:IsoMove}, we have $\mathrm{dist}(\circ_f,\ast) = \mathrm{dist}((\circ_g)_f,\ast) = \mathrm{dist}(\circ_{fg},\ast) = \mathrm{dist}((\circ_{fg})_\ell,\ast_\ell) = \mathrm{dist}(\circ_{\ell fg},\ast_\ell)$.
\end{proof}
\section{Structural tools}\label{Sc:Tools}
Recall the parameters \eqref{Eq:Tools}. The results \ref{Lm:Triple}--\ref{Lm:ConstantOnCosets} and \ref{Lm:HalfH}--\ref{Cr:BigK} are taken from \cite{DrEJC} and \cite{DrSzeged}, or are immediate corollaries of results therein. We do not hesitate to include short proofs here, and we refer the reader to \cite{DrEJC} and \cite{DrSzeged} for the longer, omitted proofs.
\begin{lemma}\label{Lm:Triple}
If $a\circ b\ne a\ast b$ then $\mathrm{dist}_a+\mathrm{dist}_b+\mathrm{dist}_{a\circ b}\ge n$.
\end{lemma}
\begin{proof}
Let $c\in G$ and suppose that $b\circ c = b\ast c$ and $(a\circ b)\circ c = (a\circ b)\ast c$. Then $a\circ (b\circ c) = (a\circ b)\circ c = (a\circ b)\ast c\ne (a\ast b)\ast c = a\ast(b\ast c) = a\ast(b\circ c)$.
\end{proof}
\begin{lemma}\label{Lm:HSubgroup}
Let $H=H(\circ,\ast)$. Then either $H=\emptyset$ or else $H\le G(\circ)$
and $H\le G(\ast)$.
\end{lemma}
\begin{proof}
Assume that $a$, $b\in H$. Then for every $c\in G$ we have $(a\circ b)\circ c = a\circ (b\circ c) = a\circ (b\ast c) = a\ast (b\ast c) = (a\ast b)\ast c = (a\circ b)\ast c$, so $a\circ b\in H$.
\end{proof}
We remark that, as per the previous section, we can always assume that $1\in
H$, so the case when $H=\emptyset$ will not arise in our work.
\begin{lemma}\label{Lm:ConstantOnCosets}
Suppose that $H\ne\emptyset$. If $b\in H\circ a$ then $\mathrm{dist}_a = \mathrm{dist}_b$.
\end{lemma}
\begin{proof}
Let $b=c\circ a$ for $c\in H$. Let $d\in G$ and suppose that $b\circ d=b\ast d$. Then $c\ast(a\circ d) = c\circ (a\circ d) = (c\circ a)\circ d = (c\circ a)\ast d = (c\ast a)\ast d = c\ast(a\ast d)$, and thus $a\circ d = a\ast d$. This shows that $\mathrm{dist}_a\le \mathrm{dist}_b$, and the other inequality follows from $a\in H\circ b$.
\end{proof}
\begin{lemma}\label{Lm:TwoCosets}
If $a\circ b\ne a\ast b$ then $H\circ b\ne H\circ(a\circ b)$.
\end{lemma}
\begin{proof}
If $H\circ b = H\circ(a\circ b)$ then $a = a\circ b\circ b^\circ\in H$, contradicting $\mathrm{dist}_a>0$.
\end{proof}
\begin{lemma}\label{Lm:HalfH}
If $h(\circ,\ast)=n/2$ then $\mathrm{dist}(\circ,\ast)\ge n^2/4$.
\end{lemma}
\begin{lemma}\label{Lm:HDividesK}
If $h>0$ then $h$ divides $k$.
\end{lemma}
\begin{proof}
Since the function $\mathrm{dist}:G\to \mathbb N$, $a\mapsto \mathrm{dist}_a$ takes on different values in $K$ and $G\setminus K$, \lref{Lm:ConstantOnCosets} implies that $K$ is a union of (right) cosets of $H$.
\end{proof}
\begin{proposition}\label{Pr:3n4}
If $k(\circ,\ast)>3n/4$ then there is an isomorphism $f:G(\circ)\to G(\ast)$
fixing all elements of $K(\circ,\ast)$.
\end{proposition}
The following example shows that \pref{Pr:3n4} is best possible. Let
$\circ,\ast$ be defined as follows, where differences are shaded.
\[
\begin{array}{c|cccccccc}
\circ&1&2&3&4&5&6&7&8\\
\hline
1&1&2&3&4&5&6&7&8\\
2&2&1&4&3&6&5&8&7\\
3&3&4&\mk1&\mk2&\mk8&\mk7&6&5\\
4&4&3&\mk2&\mk1&\mk7&\mk8&5&6\\
5&5&6&\mk8&\mk7&3&4&2&1\\
6&6&5&\mk7&\mk8&4&3&1&2\\
7&7&8&6&5&2&1&\mk3&\mk4\\
8&8&7&5&6&1&2&\mk4&\mk3\\
\end{array}
\qquad
\begin{array}{c|cccccccc}
\ast&1&2&3&4&5&6&7&8\\
\hline
1&1&2&3&4&5&6&7&8\\
2&2&1&4&3&6&5&8&7\\
3&3&4&\mk2&\mk1&\mk7&\mk8&6&5\\
4&4&3&\mk1&\mk2&\mk8&\mk7&5&6\\
5&5&6&\mk7&\mk8&3&4&2&1\\
6&6&5&\mk8&\mk7&4&3&1&2\\
7&7&8&6&5&2&1&\mk4&\mk3\\
8&8&7&5&6&1&2&\mk3&\mk4\\
\end{array}
\]
In this example, $k=6=3n/4$, but the groups are not isomorphic;
$G(\circ)\cong C_4\times C_2$ and $G(\ast)\cong C_8$.
\begin{proposition}\label{Pr:ManyFixedPoints}
Assume that $n\ge 12$, and let $f:G(\circ)\to G(\ast)$ be a non-identity isomorphism with more than $2n/3$ fixed points. Then $\mathrm{dist}(\circ,\ast)\ge \cst_0(\circ)$.
\end{proposition}
\begin{corollary}\label{Cr:BigK}
Assume that $n\ge 12$. If $k(\circ,\ast)>3n/4$ then $G(\circ)\cong G(\ast)$ and $\mathrm{dist}(\circ,\ast)\ge\cst_0(\circ)$.
\end{corollary}
In our search for closest groups $G(\ast)$ to $G(\circ)$, we can therefore assume that $k\le3n/4$ when $n\ge 12$.
Denote by $L_a(\circ)$ the left translation by $a$ in $G(\circ)$, that is, $L_a(\circ)(b) = a\circ b$. Let $\beta_a(\circ,\ast) = (L_a(\circ))^{-1}L_a(\ast)$. Then $\beta_a(\circ,\ast)(b) = b$ if and only if $a\circ b=a\ast b$, and thus $\mathrm{dist}_a(\circ,\ast)$ is the number of points moved by $\beta_a(\circ,\ast)$.
\begin{lemma}\label{Lm:MinimalM}
Assume that $\mathrm{dist}_a = \mathrm{dist}_a(\circ,\ast)>0$. Then $\mathrm{dist}_a \ge 2$. If $\beta_a(\circ,\ast)$ is an even permutation then $\mathrm{dist}_a\ge 3$. In particular, if $n$ is odd then $\mathrm{dist}_a\ge 3$.
\end{lemma}
\begin{proof}
The case $\mathrm{dist}_a=1$ is impossible since $\beta_a$ cannot move precisely $1$ point. When $\beta_a$ is even, it is not a transposition, and hence it moves at least $3$ points. When $n$ is odd, the left translations $L_a(\circ)$, $L_a(\ast)$ are products of cycles of odd length, hence $\beta_a$ is an even permutation.
\end{proof}
Finally, we investigate $\mathrm{dist}_a(\circ,\ast)$ depending on whether $a$ has the same order in $G(\circ)$ and $G(\ast)$. Denote by $|a|_\circ$ the order of $a$ in $G(\circ)$. If $|a|_\circ = |a|_\ast$, we say that $a$ is \emph{order matched}, otherwise it is \emph{order mismatched}.
\begin{lemma}\label{Lm:OrderMismatched}
Assume that $\sigma=|a|_\circ > |a|_\ast = \tau$. Then $\mathrm{dist}_a(\circ,\ast)\ge (n/\sigma)\lceil \sigma/\tau\rceil \ge n/\tau$.
\end{lemma}
\begin{proof}
The left translation $L_a(\circ)$ is a product of $n/\sigma$ disjoint cycles of length $\sigma$, and $L_a(\ast)$ is a product of $n/\tau$ disjoint cycles of length $\tau<\sigma$. Consider a cycle $(b_0,\dots,b_{\sigma-1})$ of $L_a(\circ)$. By definition then, $a\circ b_i = b_{i+1\mod \sigma}$. Let us focus on $b_0$. Without loss of generality, there is a cycle $(c_0,\dots,c_{\tau-1})$ of $L_a(\ast)$ such that $b_0=c_0$. Let $i$ be the least integer with $1\le i\le \tau$ such that $b_i\ne c_{i\mod \tau}$. (Such an $i$ exists, since $c_{\tau\mod \tau} = c_0=b_0\ne b_\tau$.) Then $a\circ c_{i-1} = a\circ b_{i-1} = b_i\ne c_{i\mod \tau} = a\ast c_{i-1} = a\ast b_{i-1}$.
Hence, corresponding to the segment $b_0$, $\dots$, $b_\tau$, we found a
difference $a\circ b_j\ne a\ast b_j$ with $0\le j\le \tau-1$. Repeating this
argument shows that there must be $\lceil \sigma/\tau\rceil$ differences within
each of the $n/\sigma$ cycles of $L_a(\circ)$.
\end{proof}
By \tref{Th:DrEJC}, $\cst_{\cong}(\circ)<\cst_{\not\cong}(\circ)$ when $n\ge 51$. We can reach the same conclusion for some smaller orders $n$, too:
\begin{lemma}\label{Lm:2p}
Let $n=2p$ for a prime $p\ge 11$. Let $G(\circ)$ be a group of order $n$. Then $\cst_{\cong}(\circ)<\cst_{\not\cong}(\circ)$.
\end{lemma}
\begin{proof}
Up to isomorphism, there are only two groups of order $2p$, the cyclic group $C_{2p}=G(\circ)$ and the dihedral group $D_{2p}=G(\ast)$. There is a unique involution in $C_{2p}$ and there are $p$ involutions in $D_{2p}$. Hence at least $p-1$ involutions are order mismatched. By \lref{Lm:OrderMismatched}, $d_a(\circ,\ast)\ge 2p/2=p$ for every order mismatched involution $a$. We therefore have $\mathrm{dist}(\circ,\ast)\ge (p-1)p$. On the other hand, $\cst(C_{2p})\le 12p-24$ and $\cst(D_{2p})\le 12p-20$ by \tref{Th:DrEJC}. The inequality $(p-1)p>12p-20$ holds for every $p\ge 13$.
It remains to discuss the case $p=11$. If at least one element $a$ in the cyclic subgroup $C_p$ of $D_{2p}$ satisfies $\mathrm{dist}_a>0$ (hence $\mathrm{dist}_a\ge 2$), then the same inequality holds for every nonidentity element of $C_p$, by \lref{Lm:HSubgroup}, and thus $\mathrm{dist}(\circ,\ast)\ge (p-1)p+2(p-1)>12p-20$. Otherwise, $C_p=H$, and $\mathrm{dist}(\circ,\ast)\ge 2p^2>12p-20$ by \lref{Lm:HalfH}.
\end{proof}
\begin{lemma}\label{Lm:OrderMatched} If $\mathrm{dist}_a>0$ and $a$ is order matched then $\mathrm{dist}_a\ge 3$.
\end{lemma}
\begin{proof}
The two left translations $L_a(\circ)$ and $L_a(\ast)$ have the same cycle structure, thus $\beta_a(\circ,\ast)$ is an even permutation, and we are done by \lref{Lm:MinimalM}.
\end{proof}
We can now narrow down possible isomorphism types of $G(\circ)$ and
$G(\ast)$ when $m=2$.
\begin{proposition}\label{Pr:m2}
Assume that $\mathrm{dist}_a(\circ,\ast)=2$. Then, without loss of generality, $|a|_\circ = n$ and $|a|_\ast = n/2$.
\end{proposition}
\begin{proof}
Since $\mathrm{dist}_a=2$, $a$ must be order mismatched, by
\lref{Lm:OrderMatched}. Let $\sigma = |a|_\circ$ and
$\tau=|a|_\ast$. Without loss of generality, $\sigma>\tau$. Then, by
\lref{Lm:OrderMismatched},
$2=\mathrm{dist}_a\ge(n/\sigma)\lceil\sigma/\tau\rceil$. As $\sigma>\tau$, we
must have $n/\sigma=1$ and $\lceil \sigma/\tau\rceil = 2$, hence
$n=\sigma$, $\lceil n/\tau\rceil =2$, and because $\tau$ divides $n$,
it follows that $\tau=n/2$.
\end{proof}
For a group $G(\circ)$ and integer $\ell\ge 1$, let $o_\ell(\circ)$ be the number of elements of order $\ell$ in $G(\circ)$. Motivated by \pref{Pr:m2}, we let
\begin{displaymath}
\omega(\circ,\ast) = \min\{o_n(\circ),o_{n/2}(\ast)\} + \min\{o_{n/2}(\circ),o_n(\ast)\}.
\end{displaymath}
Let $\varphi$ denote Euler's totient function.
\begin{lemma}\label{Lm:LimitOnm2}
For groups $G(\circ)$, $G(\ast)$ of even order $n$, there are at most $h(\circ,\ast)+2\varphi(n/2)$ rows $a\in G$ with $\mathrm{dist}_a<3$.
\end{lemma}
\begin{proof}
Consider $a\not\in H$. If $a$ is order matched, then $\mathrm{dist}_a\ge 3$ by
\lref{Lm:OrderMatched}. If $a$ is order mismatched and $\mathrm{dist}_a=2$, we
must have $\{|a|_\circ,|a|_\ast\} = \{n,n/2\}$, by
\lref{Lm:OrderMismatched}. The number of elements $a$ with
$\{|a|_\circ,|a|_\ast\}=\{n,n/2\}$ cannot exceed $\omega(\circ,\ast)$. Thus
it suffices to show that $\omega(\circ,\ast)\le2\varphi(n/2)$.
Suppose $G(\circ)$ is not cyclic. Then
$\omega(\circ,\ast)=\min\{0,o_{n/2}(\ast)\}+\min\{o_{n/2}(\circ),o_n(\ast)\}
\le o_n(\ast)\le\varphi(n)\le2\varphi(n/2)$. A similar argument works if
$G(\ast)$ is not cyclic, so we may as well assume that both
$G(\circ)$ and $G(\ast)$ are cyclic. In that case
$\omega(\circ,\ast)=2\min\{o_{n/2}(\circ),o_n(\ast)\}
=2\min\{\varphi(n/2),\varphi(n)\}
=2\varphi(n/2)$.
\end{proof}
\section{Inequalities}\label{Sc:Inequalities}
We now start the search for closest multiplication tables of groups.
Let $G(\circ)$, $G(\ast)$ be two groups of order $n$, and let $h=h(\circ,\ast)$, $k=k(\circ,\ast)$, $m=m(\circ,\ast)$. Keeping our goal in mind, we can make the following assumptions on $n$, $h$, $k$ and $m$:
\begin{enumerate}
\item[-] $23\le n\le 50$ (the case $n\ge 51$ is covered by
\tref{Th:DrEJC}, the case $n\le 22$ will be addressed later),
\item[-] $1\le h<n$ and $h$ divides $n$ (we can assume $1\le h$ by
\lref{Lm:Neutral}, $h<n$ to avoid $G(\circ)=G(\ast)$, and $h$ divides
$n$ by \lref{Lm:HSubgroup}),
\item[-] $k\le 3n/4$ and $h$ divides $k$ (by
\cref{Cr:BigK} and \lref{Lm:HDividesK}),
\item[-] $m\ge 2$ when $n$ is even and $m\ge 3$ when $n$ is odd (by
\lref{Lm:MinimalM}). By the definition of $k$, we also know $m<n/3$
if $h<k$, whereas $n/3\le m\le n$ if $h=k$.
\end{enumerate}
We will consider quadruples $(n,h,k,m)$ satisfying the above
conditions. We are interested only in such quadruples for which
$\mathrm{dist}(\circ,\ast)\le\cst(\circ)$ occurs. Since we do not want to assume
(yet) anything about the isomorphism type of $G(\circ)$, we set
\begin{displaymath}
\cst_0(n) = \left\{\begin{array}{ll}
6n-18,&\text{ when $n$ is odd},\\
6n-20,&\text{ when $n\equiv2\mod4$},\\
6n-24,&\text{ when $n\equiv0\mod4$},
\end{array}\right.
\end{displaymath}
and we keep only those quadruples for which it is possible that
$\mathrm{dist}(\circ,\ast)\le \cst_0(n)$. We will eliminate most quadruples by a
series of inequalities.
We start with a fundamental inequality based on both $H$ and
$K$. Every element of $G\setminus K$ satisfies
$\mathrm{dist}_a\ge\lceil n/3\rceil$, and $H\subseteq K$, thus
\begin{equation}\label{Eq:I4}
\mathrm{dist}(\circ,\ast)\ge (n-k)\lceil n/3\rceil + (k-h)m.
\end{equation}
There are $309$ quadruples $[n,h,k,m]$ that satisfy this constraint.
We will gradually whittle these away until none remain (at the end of Section \ref{Sc:Stubborn}).
Let $a$ be such that $\mathrm{dist}_a=m$. By \lref{Lm:Triple}, there is
$b$ such that $\mathrm{dist}_a+\mathrm{dist}_b+\mathrm{dist}_{a\circ b}\ge n$. Hence
$\mathrm{dist}_b+\mathrm{dist}_{a\circ b}\ge n-m$, and we conclude that there exists $c$
such that $\mathrm{dist}_c\ge \lceil (n-m)/2\rceil$. Then by
\lref{Lm:ConstantOnCosets}, there are (at least) $h$ elements $c$ with
$\mathrm{dist}_c\ge \lceil (n-m)/2\rceil$, all in $G\setminus H$. The
remaining $n-2h\ge 0$ elements of $G\setminus H$ satisfy $\mathrm{dist}_a\ge m$,
and we have
\begin{equation}\label{Eq:I2}
\mathrm{dist}(\circ,\ast) \ge h\Big\lceil\frac{n-m}{2}\Big\rceil + (n-2h)m.
\end{equation}
(282 quadruples remain.)
By \lref{Lm:HalfH},
\begin{equation}\label{Eq:I3}
\text{if $h=n/2$ then }\mathrm{dist}(\circ,\ast)\ge n^2/4.
\end{equation}
(207 quadruples remain, all with $m < n/3$ and $h<k$.)
Let again $a\circ b\ne a\ast b$, and assume $\mathrm{dist}_a=m$. Then $\mathrm{dist}_b + \mathrm{dist}_{a\circ b}\ge n-m$. By \lref{Lm:TwoCosets}, the cosets $H\circ b$ and $H\circ (a\circ b)$ are distinct. Since $\mathrm{dist}_c$ is constant within every right coset of $H$ by \lref{Lm:ConstantOnCosets}, there are $2h$ elements with average value of $\mathrm{dist}_c$ at least $(n-m)/2$. On one of these $2$ cosets, $\mathrm{dist}_c\ge (n-m)/2$, which puts this coset into $G\setminus K$, as $(n-m)/2 > n/3$ (using $m<n/3$). If we temporarily assume that $n-k<2h$, the second coset cannot be located in $G\setminus K$, so we have
\begin{equation}\label{Eq:I5}
\text{if $n-k<2h$ then }\mathrm{dist}(\circ,\ast)\ge h(n-m) + (n-k-h)\lceil n/3\rceil + (k-2h)m.
\end{equation}
($188$ quadruples remain, all with $n-k\ge 2h$.)
Returning to the two cosets with average value of $\mathrm{dist}_c$ at least $(n-m)/2$, even if both are located within $G\setminus K$, we at least have
\begin{equation}\label{Eq:I6}
\mathrm{dist}(\circ,\ast)\ge h(n-m) + (n-k-2h)\lceil n/3\rceil + (k-h)m.
\end{equation}
($99$ quadruples remain.)
In the previous inequality, we have used $\mathrm{dist}_a>m$ on $n-k$ rows. If $m=2$, there are at most $h+2\varphi(n/2)$ rows with $\mathrm{dist}_a=2$, by \lref{Lm:LimitOnm2}, so there are at least $n - (h+2\varphi(n/2)) - (n-k) = k-h-2\varphi(n/2)$ rows where we used $\mathrm{dist}_a=2$ in \eqref{Eq:I6} but could have used $\mathrm{dist}_a\ge 3$. This number of rows might be negative, but we certainly have
\begin{equation}\label{Eq:I7}
\text{if $m=2$ then }
\mathrm{dist}(\circ,\ast)\ge h(n-m) + (n-k-2h)\lceil n/3\rceil + (k-h)m + k-h-2\varphi(n/2).
\end{equation}
($89$ quadruples remain.)
Finally, we eliminate the case $n=32$:
\begin{lemma}[\cite{DrDiscr}, Lemma 4.4]\label{Lm:ManyFixedPoints2} Let $G(\circ)$, $G(\ast)$ be isomorphic $2$-groups of order $n$ satisfying $\mathrm{dist}(\circ,\ast)<n^2/4$. Then there exists a bijection $f:G\to G$ with at least $(n/4)(3+1/\sqrt{3})$ fixed points and such that $\ast = \circ_f$.
\end{lemma}
\begin{corollary}\label{Cr:32}
Let $G(\circ)$ be a group of order $32$. Then $\cst_{\not\cong}(\circ) > \cst_{\cong}(\circ) = \cst_0(\circ) = 168$, and there is a transposition $g:G\to G$ such that $\cst(\circ) = \mathrm{dist}(\circ,\circ_g)$.
\end{corollary}
\begin{proof}
Let $n=32$. Recalling the results from the Introduction, we know that
$\cst_{\not\cong}(\circ) \ge n^2/4 > \cst_0(\circ) = 6\cdot 32-24 =
168$. Let $G(\ast)\cong G(\circ)$ be such that
$\cst(\circ)=\mathrm{dist}(\circ,\ast)$. Since $\cst(\circ)<n^2/4$,
\lref{Lm:ManyFixedPoints2} yields a bijection $f:G\to G$ with at least
$(n/4)(3+1/\sqrt{3}) > 2n/3$ fixed points. By
\pref{Pr:ManyFixedPoints}, $\mathrm{dist}(\circ,\ast)\ge\cst_0(\circ)$. We are done
by Theorem \ref{Th:DrEJC}.
\end{proof}
The remaining $82$ quadruples $(n,h,k,m)$ are as follows (quadruples with the same $n$, $h$, $m$ are grouped):
\begin{equation}
\begin{array}{lll}
( 23, 1, \{13,14,15,16,17\}, 3 ), &( 23, 1, \{16,17\}, 4 ), &( 24, 1, \{14,15,16,17,18\}, 2 ),\\
( 24, 1, \{15,16,17,18\}, 3 ), &( 24, 1, 18, 4 ), &( 24, 2, \{14,16,18\}, 2 ),\\
( 24, 2, \{16,18\}, 3 ), &( 24, 2, 18, 4 ), &( 24, 3, \{15,18\}, 2 ),\\
( 24, 3, 18, 3 ), &( 24, 3, 18, 4 ), &( 24, 4, 16, 2 ),\\
( 24, 4, 16, 3 ), &( 25, 1, \{16,17,18\}, 3 ), &( 26, 1, \{15,16,17,18,19\}, 2 ),\\
( 26, 1, \{17,18,19\}, 3 ), &( 26, 2, \{16,18\}, 2 ), &( 26, 2, 18, 3 ),\\
( 27, 1, \{17,18,19,20\}, 3 ), &( 27, 1, 20, 4 ), &( 27, 3, 18, 3 ),\\
( 28, 1, \{19,20,21\}, 2 ), &(28, 1, \{20,21\}, 3 ), &( 28, 2, 20, 2 ),\\
( 28, 2, 20, 3 ), &( 28, 4, 20, 2 ), &( 29, 1, \{20,21\}, 3 ),\\
( 30, 1, \{19,20,21,22\}, 2 ), &( 30, 1, \{21,22\}, 3 ), &( 30, 2, \{20,22\}, 2 ),\\
( 30, 2, 22, 3 ), &( 30, 3, 21, 2 ), &( 31, 1, \{22,23\}, 3 ),\\
( 33, 1, 24, 3 ), &( 34, 1, \{23,24,25\}, 2 ), &( 34, 2, 24, 2 ),\\
( 35, 1, 26, 3 ), &( 36, 1, 27, 2 ), &( 38, 1, \{27,28\}, 2 ),\\
( 38, 2, 28, 2 ), &( 42, 1, 31, 2 ).&
\end{array}\label{Eq:Quadruples}
\end{equation}
\section{Special row differences}\label{Sc:m2}
\subsection{The case $m=2$}\label{Ss:m2}
In this subsection we describe an algorithm that determines all pairs of groups $G(\circ)$, $G(\ast)$ with $m(\circ,\ast)=2$.
By \pref{Pr:m2}, we can assume that $G(\ast)$ is a fixed cyclic group of even order $n$, and there is $a\in G$ such that $|a|_\ast = n$, $|a|_\circ = n/2$.
The automorphism group $\mathrm{Aut}(C_n)$ acts transitively on the generators of $C_n$. Thus, if $b$ is a generator of $G(\ast)$, there is $f\in\mathrm{Aut}(\ast)$ such that $f(a)=b$. By \lref{Lm:IsoMove}, we then have $\mathrm{dist}_a(\circ,\ast) = \mathrm{dist}_{f(a)}(\circ_f,\ast_f) = \mathrm{dist}_b(\circ_f,\ast)$ and $\mathrm{dist}(\circ,\ast) = \mathrm{dist}(\circ_f,\ast)$. We can therefore assume without loss of generality that $a$ is a fixed generator of $G(\ast)$.
The input of the algorithm is a cyclic group $G(\ast)=C_n$ and its generator $a$. To obtain $\mathrm{dist}_a(\ast,\circ)=2$, we must modify the row $a$ of $G(\ast)$ in two places; say there are $v\ne w$ such that $a\circ b = a\ast b$ except for $a\circ v = a\ast w$, $a\circ w = a\ast v$. Since $a\circ b$ is now determined for every $b\in G$, we can see if $|a|_\circ = n/2$, as desired. If not, we choose different $v$, $w$.
Assume now that the locations $v$, $w$ of differences in row $a$ were chosen so that $|a|_\circ=n/2$. Let $A$ be the subgroup generated by $a$ in $G(\circ)$, and let $b$ be any element of $G\setminus A$. Denote by $a^i$ the $i$th power of $a$ in $G(\circ)$. Since $G= A\cup (A\circ b) = A\cup (b\circ A)$, we must have $b\circ a = a^\alpha\circ b$ for some $1\le \alpha<n/2$, and $b\circ b = a^\beta$ for some $0\le \beta<n/2$. Once the parameters $\alpha$, $\beta$ are chosen, the operation $\circ$ is determined, namely:
\begin{align*}
a^i\circ a^j &= a^{i+j},\\
a^i\circ (a^j\circ b) &= a^{i+j}\circ b,\\
(a^i\circ b)\circ a^j &= a^i\circ (b\circ a^j) = a^i\circ (a^{j\alpha}\circ b) = a^{i+j\alpha}\circ b,\\
(a^i\circ b)\circ (a^j\circ b) &= a^i\circ (b\circ a^j)\circ b = a^{i+j\alpha}\circ b\circ b = a^{i+j\alpha+\beta},
\end{align*}
for $0\le i$, $j<n/2$. We do not claim that this operation defines a group, only that there is no alternative way to define $\circ$ that does produce a group (as it happens, the smallest distance is achieved when $\circ$ does define a group).
It therefore suffices to consider all choices of $v$, $w$, $\alpha$, $\beta$ and find the resulting groups closest to $G(\ast)$.
Both authors independently ran this algorithm and discovered that in all cases the nearest group $G(\circ)$ was isomorphic to $C_{n/2}\times C_2$ and satisfied
\[
\mathrm{dist}(\circ,\ast)=
\begin{cases}
n^2/4&\text{when }n\equiv0\mod4,\\
n^2/4-1&\text{when }n\equiv2\mod4.\\
\end{cases}
\]
Since $n^2/4-1>\cst_0(n)$ when $n>20$, the quadruples of
\eqref{Eq:Quadruples} with $m=2$ can therefore be eliminated. (43
quadruples remain.)
\subsection{Some cyclic cases}\label{Ss:Cyclic}
Among the remaining orders $n$ of \eqref{Eq:Quadruples}, if $n$ belongs to $\{23$, $29$, $31$, $33$, $35\}$, the only group of order $n$ is the cyclic group $C_n$. For these orders, the search therefore amounts to determination of $\mathrm{dist}([C_n],[C_n])$, a difficult task in general.
Let $G(\circ)$ be a cyclic group of order $n$. For any group $G(\ast)$, define
\begin{displaymath}
m' = m'(\circ,\ast) = \textstyle\min\{\mathrm{dist}_a(\circ,\ast);\;|a|_\circ = n\}.
\end{displaymath}
Recall that $C_n$ has $\varphi(n)$ generators. Since $m'$ might be bigger than $m$, we can refine \eqref{Eq:I6} as follows,
\begin{equation}\label{Eq:Ig}
\mathrm{dist}(\circ,\ast)\ge h(n-m) + (n-k-2h)\lceil n/3\rceil + (\varphi(n) - (n-k))m' + (n-\varphi(n)-h)m,
\end{equation}
where we first count elements in the two cosets of $H$, then all remaining elements of $G\setminus K$, then all remaining generators, and then the remaining elements in $G\setminus H$, if any.
To eliminate all remaining quadruples with $n\in\{29,31,33,35\}$ (resp.\ $n=23$), it suffices to set $m'=4$ (resp.\ $m'=5$) in \eqref{Eq:Ig}.
We are therefore interested in the following algorithm, with parameter
$d$: Given $G(\circ)\cong C_n$, find $G(\circ)\cong C_n$ closest to
$G(\ast)$ that has $\mathrm{dist}_a(\circ,\ast)=d$ for some generator $a$ of
$G(\circ)$.
The idea is similar to Subsection \ref{Ss:m2}, but we reverse the
roles of the groups $G(\circ)$ and $G(\ast)$. Let $a\in G$ be such that
$|a|_{\ast}=\ell$. We wish to have $|a|_{\circ}=n$ and
$\mathrm{dist}_a(\circ,\ast)=m'$. By \lref{Lm:OrderMismatched}, we can assume
that $n/\ell\le d$ (since $|a|_{\circ}=n$), that is, $\ell\ge n/d$.
Let us fix $a\in G$ with the above properties. We now need to make $d$
changes to row $a$ of $G(\ast)$, focusing on only those changes that
result in $|a|_{\circ}=n$. Once such a change is made, the group
$G(\circ)$ is determined.
\begin{remark}
When $n$ is a prime, the search can be sped up by taking advantage of
the automorphism group of $C_n$ (since all nonidentity elements are
generators), and by analyzing which permutations of $\mathrm{diff}_a(\circ,\ast)$
result in $|a|_{\circ}=n$. See \cite{VoMS} or \cite{VoProc} for details.
We did not employ these improvements here in order to keep the code
simpler.
\end{remark}
For every quadruple $(n,h,k,m)$ of \eqref{Eq:Quadruples} with
$n\in\{23,29,31,33,35\}$, the algorithm (with $d=3$ if
$n\in\{29,31,33,35\}$ and with $d\in\{3,4\}$ if $n=23$) returns
minimal distance at least as big as $\cst_0(n)$. (30 quadruples
remain.)
\section{General algorithm for $\mathrm{dist}([\circ],[\ast])$}\label{Sc:Algorithm}
Here is an algorithm that finds $d=\mathrm{dist}([\circ],[\ast])$. By \pref{Pr:IsoDist}, we have $d=\mathrm{dist}([\circ],\ast) = \min\{\mathrm{dist}(\circ_f,\ast);\;f:G\to G$ is a bijection, $G(\circ_f)\ne G(\ast)\}$.
\bigskip
\hrule
\bigskip
When $n<5$ a brute force algorithm is sufficient. Let us therefore assume that $n\ge 5$ and, by Lemma \ref{Lm:Neutral}, that $f(1)=1$ and thus $1\in H$.
Either $H=1$ or there exists a prime $p$ and a subgroup $\overline{H}\le H$ of $G(\ast)$ of order $p$. The main cycle of the algorithm proceeds over all subgroups $\overline{H}\le G(\ast)$ of prime order $p$ or $p=1$, with $|\overline{H}|$ in descending order. From now on we will write $H$ instead of $\overline{H}$, since the fact that $H$ might be larger is irrelevant in the search.
Assume that $\mathrm{dist}_{min}$ is the smallest distance found by the algorithm so far, and let $H\le G(\ast)$, $|H|=p$ be given. We need to consider all bijections $f:G\to G$ such that $G(\gc) = G(\circ_{f^{-1}})$ and $G(\circ)$ agree on at least $H$. The inverse $f^{-1}$, rather than $f$, is used for notational convenience, and we then have $f(a\gc b) = f(a)\circ f(b)$.
The algorithm is a depth-first search on all partially defined $1$-to-$1$ maps $f:G\to G$, where the maps are lexicographically ordered as follows: Let $\mathrm{Dom}(f)$ denote the domain of $f$, and let $G=\{1,\dots,n\}$. Let $f$, $g:G\to G$ be two partially defined maps. Then we say that $g<f$ if and only if there exists $i\in\mathrm{Dom}(f)$ such that (a) for every $j\le i$, if $j\in\mathrm{Dom}(f)$ then $j\in\mathrm{Dom}(g)$, (b) for every $j<i$, if $j\in\mathrm{Dom}(f)$ then $g(j)=f(j)$, (c) $g(i)<f(i)$.
The search starts as follows: Let $x$ be a generator of $H$. Then $f(x)$ is an element of order $p$ in $G(\circ)$, because we demand that $x\in H(\gc,\ast) = H$ and that $f:G(\gc)\to G(\circ)$ is an isomorphism. The second cycle of the algorithm is therefore over all elements $y=f(x)$ such that $|y|_\circ=p$.
Once $f(x)$ is known, we can extend $f$ onto $H$. Indeed, we have $f(x\ast x) = f(x\gc x)$ by our assumption that $H=H(\gc,\ast)$, and $f(x\gc x) = f(x)\circ f(x)$ because $f:G(\gc)\to G(\circ)$ is a homomorphism. Similarly for higher powers of $x$.
To extend the domain of $f$ further, we systematically choose $b\not\in\mathrm{Dom}(f)$, $c\not\in\mathrm{Im}(f)$, and declare $f(b)=c$. Once again, we can now extend $f$ onto the coset $H\ast b$, as for $y\in H$ we must have $f(y\ast b) = f(y\gc b) = f(y)\circ f(b)$.
Anytime we extend the domain of $f$ by another coset of $H$, we can calculate the guaranteed distance between the partially defined group $G(\gc)$ and the group $G(\ast)$ by counting only those pairs $(a,b)$ that satisfy: $a\in\mathrm{Dom}(f)$, $b\in\mathrm{Dom}(f)$, $a\gc b\in\mathrm{Dom}(f)$ and $f(a\gc b)\ne f(a)\circ f(b)$. If this distance exceeds $\mathrm{dist}_{min}$, we terminate this branch of the depth-first search.
Whenever we extend the domain of $f$ by another coset, we consider the automorphisms $g\in\mathrm{Aut}(\circ)$ and $\ell\in \mathrm{Aut}(\ast)$. By \lref{Lm:Aut}, $\mathrm{dist}(\circ_{\ell fg},\ast) = \mathrm{dist}(\circ_{f},\ast)$. It is also easy to see that $H(\circ_{\ell fg}, \ast) = H(\circ_f,\ast)$. Therefore, if $\ell fg<f$, we have seen $\ell fg$ before $f$ (in this cycle with the same $H$), $f$ cannot do better than $\ell fg$ as far as distance is concerned, so we terminate the branch.
If $\mathrm{Dom}(f)=G$ anytime in the search, we calculate the full distance $\mathrm{dist}(\gc,\ast)$ and compare it to $\mathrm{dist}_{min}$.
\bigskip
\hrule
\bigskip
The following improvements make the algorithm faster.
\begin{enumerate}
\item[-] the distance $\mathrm{dist}(\gc,\ast)$ is calculated incrementally, in every step considering only rows, columns and values from the coset of $H$ on which $f$ has just been defined,
\item[-] the comparison of $\ell fg$ to $f$ is costly, and it is better to stop using it in the search from a certain (heuristically determined) depth in the search,
\item[-] assuming that the algorithm has gone through all values of $p>1$ and is now in the cycle $p=1$, the guaranteed distance can be calculated with a bonus. Namely, since we have $H=1$ at this stage, we can assume that every row not in the domain of $f$ contains $2$ (resp. $3$) differences when $n$ is even (resp. odd), by \lref{Lm:MinimalM}.
\end{enumerate}
The algorithm is sufficiently fast to deal with all orders $n\le 22$, albeit in some cases we merely verified that $\mathrm{dist}([\circ],[\ast])$ exceeds $\cst(\circ)$, without actually determining $\mathrm{dist}([\circ],[\ast])$. The case $n=22$ alone took more than a week of computing time. It was therefore of some importance that we could assume $G(\circ)\cong G(\ast)$ when $n=22$, by \lref{Lm:2p}.
The results of the search for $n\le 22$ are summarized in Theorem \ref{Th:Main}.
The algorithm can also be used to eliminate all remaining cases of \eqref{Eq:Quadruples} with $h>1$; we simply do not run the algorithm with any values $p$ less than $h$. This leaves us with the following twenty quadruples $(n,h,k,m)$:
\begin{equation}\label{Eq:Quadruples2}
\begin{array}{lll}
( 24, 1, \{15,16,17,18\}, 3 ),& (24, 1, 18, 4), &(25, 1, \{16,17,18\}, 3)\\
( 26, 1, \{17,18,19\}, 3 ),&( 27, 1, \{17,18,19,20\}, 3 ), & ( 27, 1, 20, 4 )\\
( 28, 1, \{20,21\}, 3 ),&( 30, 1, \{21,22\}, 3 ).&
\end{array}
\end{equation}
We eliminate them in Section \ref{Sc:Stubborn}, but first we need to introduce results on rainbow matchings in edge-colored graphs.
\section{Rainbow matchings and the graph $\Gamma_U$}\label{Sc:Graph}
Call an edge-colored graph \emph{restricted} if it has at most $3$
edges of any given color, and if at most two edges of the same color
are incident at any vertex. Recall that a \emph{rainbow
$\ell$-matching} in an edge-colored graph is a set of $\ell$
disjoint edges colored by distinct colors. For $v>1$ and $\ell>0$,
define $\mu_\ell(v)$ to be the minimum number of edges a
restricted graph on $v$ vertices must have in order to guarantee a
rainbow $\ell$-matching. If there exists a coloring of the
complete graph on $v$ vertices that yields a restricted graph without
a rainbow $\ell$-matching, then we define $\mu_\ell(v)=\binom{v}{2}+1$.
\begin{proposition}\label{Pr:Graph} We have $\mu_1(v)=1$ for every $v\ge 2$, $\mu_2(v)=7$ if $4\le v\le 6$, $\mu_2(v)=v$ if $v\ge 7$, $\mu_3(6)=13$, $\mu_3(7)=15$, $\mu_3(8)=15$, $\mu_3(9)=16$ and $\mu_3(10)=18$.
\end{proposition}
We now describe the algorithm used to
establish Proposition~\ref{Pr:Graph}. The aim was to find the
greatest number of edges that a restricted graph on $v$ vertices can
have without containing a rainbow $\ell$-matching. We began with an
empty graph on $v$ vertices, and added the edges one color at a
time. We will refer to the process of adding all the edges of a
particular color as a {\em stage}. In each stage, we read in each of
the graphs from the previous stage, one at a time, added edges of
the new color in all possible ways, and output any graph which was
not isomorphic (by an isomorphism that respects the edge coloring,
but is allowed to permute colors) to a graph we had already seen. The
isomorphism testing was accomplished by \texttt{nauty} \cite{nauty}.
After a graph was read in stage $c$, we found all rainbow
$(\ell-1)$-matchings in it. Any edge disjoint from any such matching is
unavailable to be colored $c$. Typically this rule leaves very few
edges still available. We also sped up the search by making several
other assumptions. Firstly, since all isolated vertices are isomorphic,
vertex $j+1$ would not be connected to its first edge before vertex $j$
was. Secondly, for $c>1$ we insisted that there were not more edges of
color $c$ than there were of color $c-1$. Thirdly, we assumed that
there was at most one color which occurs on only one edge. This last
assumption is justified because if two colors each only occurred on
one edge then we could replace those two colors by a single color.
The result would still be a restricted graph, and would not have a
rainbow $\ell$-matching unless the original graph did.
As a partial validation of our computations, it is easy to confirm by
hand that the values quoted in Proposition~\ref{Pr:Graph} are lower
bounds on $\mu_\ell(v)$. First note that we can prevent a rainbow
$\ell$-matching by having no $\ell$-matchings at all. This can be achieved
by having a set of $\ell-1$ vertices that cover all edges, in which case
we can have up to ${\ell-1\choose 2}+(\ell-1)(v-\ell+1)=(\ell-1)(v-\ell/2)$ edges.
Thus $\mu_\ell(v)\ge 1+(\ell-1)(v-\ell/2)$ whenever $v\ge \ell-1$. This
elementary lower bound is actually achieved for $\mu_1(v),\,v\ge1$;
$\mu_2(v),\,v\ge7$; and $\mu_3(v),\,v\in\{9,10\}$. To give a lower
bound for the other values quoted in Proposition~\ref{Pr:Graph}, we
display in \fref{f:rnbw} graphs with (a) 4 vertices, 6 edges and no
rainbow $2$-matching, (b)
7 vertices, 14 edges and no rainbow $3$-matching. Edge colors are
indicated by the different styles of lines. By deleting either
of the degree 2 vertices from (b) we obtain a graph with
6 vertices, 12 edges and no rainbow $3$-matching. These examples show
that $\mu_2(v)\ge7$ for $v\ge4$, $\mu_3(6)\ge13$
and $\mu_3(8)\ge\mu_3(7)\ge15$.
\begin{figure}[htb]
\[(a)\;\includegraphics[scale=0.6]{rnbwA.eps}\qquad
(b
\;\includegraphics[scale=0.6]{rnbwC.eps}
\]
\caption{\label{f:rnbw}Restricted graphs giving lower bounds for Proposition~\ref{Pr:Graph}.}
\end{figure}
The statement in \pref{Pr:Graph} that $\mu_2(v)=v$ for $v\ge7$ is
easily seen. We have already argued that $\mu_2(v)\ge v$. Suppose we
have a restricted graph with $v\ge7$ vertices and $v$ edges and
no rainbow $2$-matching. Any graph with $v>3$ vertices and $v$ edges
has a $2$-matching; in our case both edges must have
the same color $c$. Every edge of color different from $c$ must join
the two edges of the $2$-matching, and there are only 4 possible
places to put such an edge. There may be a third edge of color $c$, but
that is all. Thus our graph has at most $7$ edges. The case $v=e=7$ can
be handled by more detailed case analysis, or ruled out by our computer
programs.
Let us now return to the problem of distances of groups. The following subsets of $\mathrm{diff}(\circ,\ast)$ will play an important role in the analysis of the cases \eqref{Eq:Quadruples2}. Let
\begin{equation}\label{e:defRSTU}
\begin{split}
R = R(\circ,\ast)&= \{(a,a)\in \mathrm{diff}(\circ,\ast);\;a\in K\},\quad r=r(\circ,\ast)=|R|,\\
S = S(\circ,\ast)&= \{(a,b)\in \mathrm{diff}(\circ,\ast);\;a\in K,\,b\in K,\,a\ne b\},\quad s=s(\circ,\ast)=|S|,\\
T = T(\circ,\ast)&= \{(a,b)\in \mathrm{diff}(\circ,\ast);\;a\in K,\,a\circ b\in K\},\quad t=t(\circ,\ast)=|T|,\\
U' = U'(\circ,\ast)&= \{(a,b)\in \mathrm{diff}(\circ,\ast);\;a\in K,\,a\circ b\not\in K,\,b\not\in K\}.
\end{split}
\end{equation}
Note that, $R$, $S$, $T$, $U'$ are disjoint and $R\cup S\cup T\cup U'
= \mathrm{diff}(\circ,\ast)\cap (K\times G)$, a set that contains
at least $m\ge3$ elements in every row indexed by $K\setminus H$. Let $U$
be any minimal subset of $U'$ subject to the condition that $R\cup
S\cup T\cup U$ contains at least $3$ elements within each row indexed
by $K\setminus H$. Let $u=u(\circ,\ast)=|U|$. We have
\begin{equation}\label{Eq:rstu}
r+s+t+u\ge 3(k-h).
\end{equation}
Note that if $(a,b)\in S(\circ,\ast)$, then we must have $a\circ b\not\in K$
(and $a\ast b\not\in K$), since otherwise $\mathrm{dist}_a+\mathrm{dist}_b+\mathrm{dist}_{a\circ b} < n$
(and $\mathrm{dist}_a+\mathrm{dist}_b+\mathrm{dist}_{a\ast b}< n$), a contradiction of
\lref{Lm:Triple}. Similarly, if $(a,b)\in T(\circ,\ast)$ then $b\not\in
K$.
Define a multigraph $\Gamma'_U$ on vertices $V=G\setminus K$ by
declaring $\{x,y\}\subseteq V$ to be an edge if and only if $x\ne y$
and $\{x,y\}=\{b,a\circ b\}$ for some $(a,b)\in U$. Such an edge
$\{x,y\} = \{b,a\circ b\}$ will be colored $a$.
If $\{x,y\}=\{b,a\circ b\} = \{d,c\circ d\}$ is an edge of $\Gamma'_U$ for
some $(a,b)$, $(c,d)\in U$, one of the following situations occurs. If
$b=d$ then $a\circ b=c\circ b$, $a=c$, and $(a,c)=(b,d)$. Otherwise
$b=c\circ d$, $d=a\circ b$, $a\circ c\circ d=d$, and $c=a^\circ$. Therefore
$\Gamma'_U$ has at most two edges between any two given vertices. If
two distinct edges colored $a$ are incident to a vertex of
$\Gamma'_U$, they are of the form $\{b,a\circ b\}$, $\{c,a\circ c\}$ for
some $b\ne c$. Then, without loss of generality, we have $b=a\circ
c$. This means that no more than two distinct edges colored $a$ are
incident to a vertex of $\Gamma'_U$.
Let $\Gamma_U$ be the simple subgraph of $\Gamma'_U$ obtained by
suppressing any multiple edges. By construction, $\Gamma_U$ is a
restricted graph on $n-k$ vertices. Moreover, any edge of $\Gamma_U$
colored $a$ stems from some element $(a,b)\in U$. Later we will use
\eqref{Eq:rstu} to find a lower bound for $u$. In creating $\Gamma_U$
from $\Gamma'_U$, there are at least $\lceil u/2\rceil$ edges that
remain. Having built a restricted graph with at least a certain number
of edges, we will be in a position to employ \pref{Pr:Graph}.
\section{Eliminating cases with a rainbow $3$-matching in $\Gamma_U$}\label{Sc:ApplyGraph}
For the rest of this section, fix $G(\circ)$, $G(\ast)$, assume that $m(\circ,\ast)\ge 3$, let $q=\lceil n/3\rceil$, and let
\begin{displaymath}
\pi = \mathrm{dist}(\circ,\ast) - \big((k-h)m + (n-k)q\big)
\end{displaymath}
be the number of differences above those guaranteed by the fundamental inequality \eqref{Eq:I4}. We will refer to $\pi$ as the \emph{profit}. If we wish to indicate the profit obtained in particular rows $r_1$, $\dots$, $r_\ell$, we use the notation $\pi(r_1,\dots,r_\ell)$.
We present a series of lemmas that eliminate most quadruples of
\eqref{Eq:Quadruples2}. While attempting to eliminate a quadruple
$(n,h,k,m)$ from \eqref{Eq:Quadruples2}, we proceed as follows: We use
Lemmas \ref{Lm:R}, \ref{Lm:LargeR} and, if $n=2p$, also \lref{Lm:2pR},
to obtain an upper bound on $r$, with default bound $r\le k-h$. Lemmas
\ref{Lm:RowS} and \ref{Lm:7S} yield an upper bound on $s$, with
default bound $s\le (k-1)(k-h)$. The dual Lemmas \ref{Lm:RowT} and
\ref{Lm:7T} yield an upper bound on $t$, with default bound
$t\le(n-k)(k-h)$. Then \eqref{Eq:rstu} provides a lower bound for $u$.
Recall that there are $n-k$ vertices and at least
$\lceil u/2\rceil$ edges in $\Gamma_U$. We then use \pref{Pr:Graph}
to determine the maximal $\ell$ such that
$\lceil u/2\rceil\ge\mu_\ell(n-k)$. Finally, we apply
\lref{Lm:Matching}, and if this yields a sufficient profit then
$(n,k,h,m)$ is eliminated.
The challenge is not to count profit on the same row more than once. We often use the following \emph{disjunction tricks} to make sure that this does not happen. If $(a,a)\in R$ then we have $2\mathrm{dist}_a+\mathrm{dist}_{a\circ a}\ge n$ (by Lemma \ref{Lm:Triple} that we are going to use without reference) and $2\mathrm{dist}_a+\mathrm{dist}_{a\ast a}\ge n$. Thus $\pi(a\circ a)$, $\pi(a\ast a)\ge n-(q-1)$ and we are free to choose one of the two distinct rows $a\circ a$, $a\ast a$ of $G\setminus K$. If $(a,b)\in S$ then $\mathrm{dist}_a+\mathrm{dist}_b+\mathrm{dist}_{a\circ b}\ge n$ and $\mathrm{dist}_a+\mathrm{dist}_b+\mathrm{dist}_{a\ast b}\ge n$. Since $a$, $b\in K$, we must have $a\ast b\in G\setminus K$, too, $\pi(a,b,a\circ b)$, $\pi(a,b,a\ast b)\ge n-(2m+q)$, and we are free to choose one of the two distinct rows $a\circ b$, $a\ast b$ of $G\setminus K$. Finally, if $(a,b)\in T$, then again $\mathrm{dist}_a+\mathrm{dist}_b+\mathrm{dist}_{a\circ b}\ge n$, $\mathrm{dist}_a+\mathrm{dist}_b+\mathrm{dist}_{a\ast b}\ge n$, we have $a\circ b\in K$, but we might have $a\ast b\in G\setminus K$. It is therefore better to consider the element $c = a^{\ast}*(a\circ b)$ and the triple $(a,c,a\ast c)$ with respect to $G(\ast)$. Indeed, $a\in K$, $a\ast c = a\circ b\in K$, $b\ne c$ (since $a\circ b\ne a\ast b$), thus $a\circ c\ne a\circ b = a\ast a^\ast\gb(a\circ b) = a\ast c$, $c\not\in K$, and $(a,c)\in T(\circ,\ast)$. We then have $\pi(a,b,a\circ b)$, $\pi(a,c,a\ast c)\ge n-2m-q$ and we are free to choose one of the two alternatives.
\begin{lemma}\label{Lm:R}
Suppose that $(a_1,a_1)$, $\dots$, $(a_\ell,a_\ell)\in R$ are
distinct. Then $\pi\ge\ell(n-2q-m+1)$ provided that for $1\le
i\le\ell$ there is $\gc_i\in\{\circ,\ast\}$ such that $a_1\gc_1 a_1$,
$\dots$, $a_\ell\gc_\ell a_\ell$ are distinct. In particular, this
condition is always satisfied if $n$ is odd or if $\ell=2$.
\end{lemma}
\begin{proof}
For any $a$ with $(a,a)\in R$ we have
$\mathrm{dist}_a+\mathrm{dist}_a+\mathrm{dist}_{a\circ a}\ge n$ by \lref{Lm:Triple}.
Since $a\in K$, it follows that
$\mathrm{dist}_a+\mathrm{dist}_{a\circ a}\ge n-\mathrm{dist}_a\ge n-q+1$.
Since \eqref{Eq:I4} guaranteed
only $m+q$ differences on the two rows $a$, $a\circ a$, the profit on
these two rows is at least $n-q+1-(m+q) = n-2q-m+1$.
A similar argument applies to the pair of rows $a$ and $a\ast a$.
When $a_1\gc_1 a_1$, $\dots$, $a_\ell\gc_\ell a_\ell$ are distinct, we
immediately obtain $\pi\ge \ell(n-2q-m+1)$ as $a_i\gc_i a_i\in
G\setminus K$ and $a_i\in K$ for all $i$. In particular, if $n$ is odd
we can choose $\gc_i=\circ$ for all $i$, since the squaring map is a
permutation in groups of odd order.
The case $\ell=2$ is resolved by a disjunction trick, using
$a_2\circ a_2$ or $a_2\ast a_2$.
\end{proof}
\begin{lemma}\label{Lm:LargeR}
Suppose that $r\ge 4$. Then $\pi\ge \min\{2(n-q-2m)$, $3(n-2q-m+1)\}$.
\end{lemma}
\begin{proof}
First suppose that there are $(a,a)$, $(b,b)\in R$ such that $M=\{a\circ a$, $a\ast a$, $b\circ b$, $b\ast b\}$ satisfies $|M|\ge 3$. Pick any $c$ such that $a\ne c\ne b$ and $(c,c)\in R$, which is possible since $r\ge 3$. If $c\circ c\not\in M$ then $|\{a\gc a$, $b\bullet b$, $c\circ c\}|\ge 3$ for some $\gc$, $\bullet\in\{\circ,\ast\}$, and \lref{Lm:R} implies $\pi\ge 3(n-2q-m+1)$. Let us therefore assume without loss of generality that $c\circ c=a\circ a$. Note that we then have $c\circ c\ne a\ast a$. If $c\circ c=b\circ b$ then $c\circ c\ne b\ast b$ and also $b\ast b\ne a\ast a$ (else $a\circ a=c\circ c=b\circ b$, $b\ast b=a\ast a$, $|M|<3$), so $a\ast a$, $c\circ c$, $b\ast b$ are distinct, and we are done by \lref{Lm:R}. If $c\circ c =b\ast b$ then $c\circ c\ne b\circ b$ and $b\circ b\ne a\ast a$ (else $b\circ b=a\ast a$, $b\ast b = c\circ c = a\circ a$, $|M|<3$), so $a\ast a$, $c\circ c$, $b\circ b$ are distinct, and we are done by \lref{Lm:R}. Thus we can assume $b\circ b\ne c\circ c\ne b\ast b$. Since either $a\ast a\ne b\circ b$ or $a\ast a\ne b\ast b$, the elements $c\circ c$, $a\ast a$, $b\gc b$ are distinct for some $\gc\in\{\circ,\ast\}$, and we finish with \lref{Lm:R} again.
We can therefore suppose that there are $x$, $y\in G$ such that $\{a\circ a,a\ast a\}=\{x,y\}$ for every $(a,a)\in R$. Let $\rho = \min\{\mathrm{dist}_x,\mathrm{dist}_y\}$. Then for every $(a,a)\in R$ we have $\mathrm{dist}_a\ge (n-\rho)/2$, because $\mathrm{dist}_a+\mathrm{dist}_a+\mathrm{dist}_{a\gc a}\ge n$ for $\gc\in\{\circ,\ast\}$, and $\mathrm{dist}_{a\gc a}\le \rho$ for some $\gc\in\{\circ,\ast\}$. The profit on the rows $\{a;\;(a,a)\in R\}\cup\{x,y\}$ is therefore at least $r((n-\rho)/2-m) + 2(\rho-q)$. If $(n-\rho)/2-m\ge 0$, the assumption $r\ge 4$ yields profit at least $2(n-q-2m)$. Suppose that $(n-\rho)/2-m<0$. Then $\rho>n-2m$, so $\mathrm{dist}_{a\circ a}$, $\mathrm{dist}_{a\ast a}>n-2m$ for every $(a,a)\in R$. Let $(a,a)$, $(b,b)\in R$ be distinct. Then there is $\gc\in\{\circ,\ast\}$ such that $a\circ a$, $b\gc b$ are distinct, and the profit on these rows is at least $2(n-2m-q+1)$.
\end{proof}
\begin{lemma}\label{Lm:2pR}
Suppose that $n=2p$ for some prime $p$. Then $\pi\ge \lceil r/2\rceil(n-2q-m+1)$.
\end{lemma}
\begin{proof}
The only groups of order $2p$ are the cyclic group $C_{2p}$ and the
dihedral group $D_{2p}$. In these groups, for every $a\ne 1$ there are
at most two elements $b$ such that $a=b^2$. Hence there are at least
$\ell = \lceil r/2\rceil$ distinct elements $(a_1,a_1)$, $\dots$,
$(a_\ell,a_\ell)\in R$ with $a_1\circ a_1$, $\dots$, $a_\ell\circ a_\ell$
distinct. We are done by \lref{Lm:R}.
\end{proof}
Let us now establish several results concerning an upper bound on $s$.
\begin{lemma}\label{Lm:RowS}
Let $a\in K$ and let $b_1$, $\dots$, $b_\ell\in K$ be distinct.
Suppose that either $(a,b_1)$, $\dots$, $(a,b_\ell)\in S$, or $(b_1,a)$, $\dots$, $(b_\ell,a)\in S$. Then $\pi\ge \ell(n-2q-m+1)+q-m-1$.
\end{lemma}
\begin{proof}
Assume that $(a,b_1)$, $\dots$, $(a,b_\ell)\in S$, with the transposed situation being similar. By \lref{Lm:Triple}, for every $i$ we have $\mathrm{dist}_{b_i}+\mathrm{dist}_{a\circ b_i} \ge n-\mathrm{dist}_a\ge n-q+1$. Since $(a,b_i)\in S$, we have $a\ne b_i$ for every $1\le i\le\ell$. Hence
the elements $a$, $b_1$, $\dots$, $b_\ell$, $a\circ b_1$, $\dots$, $a\circ b_\ell$ are distinct, with $a\circ b_i\not\in K$. The profit on $a$, $b_1$, $a\circ b_1$ is at least $n-(2m+q)$, while the profit on each of the $\ell-1$ pairs of rows $b_i$, $a\circ b_i$ for $i>1$ is at least $n-q+1-(m+q)$.
\end{proof}
\begin{lemma}\label{Lm:SquareS}
If there are $(a,b)$, $(c,d)\in S$ such that $|\{a,b,c,d\}|=4$ then
$\pi\ge 2(n-q-2m)$.
\end{lemma}
\begin{proof}
If $a\circ b\ne c\circ d$ then the profit at the distinct rows $a$, $b$,
$c$, $d$, $a\circ b$, $c\circ d$ is at least $2(n-q-2m)$, by
\lref{Lm:Triple}. Otherwise use a disjunction trick and $c\ast d$ instead of $c\circ d$.
\end{proof}
\begin{lemma}\label{Lm:7S}
If $s\ge 7$ then $\pi\ge 2(n-q-2m)$.
\end{lemma}
\begin{proof}
If there are three elements of $S$ in the same row or in the same
column, \lref{Lm:RowS} implies $\pi\ge 3(n-2q-m+1)+(q-m-1)\ge
2(n-q-2m)$. Suppose that no three elements of $S$ are in the same row
or in the same column.
Define a multigraph $\Gamma_S$ on $K$ where $\{x,y\}$ is an edge if
and only if $(x,y)\in S$ or $(y,x)\in S$. Then $\Gamma_S$ has $s$
edges, there are no more than two edges between any two vertices of
$S$, and we claim that $\Gamma_S$ has a $2$-matching.
Suppose that $\Gamma_S$ has a vertex $x$ with two distinct neighbours
$y$ and $z$. By our assumptions on $S$, there are at most $4$ edges
incident with $x$. Also, there are at most $2$ edges between $y$ and
$z$. Therefore if $s\ge7$ then there is an edge disjoint from either
$\{x,y\}$ or $\{x,z\}$, yielding the required $2$-matching.
Alternatively, if no such $x$ exists then edges are disjoint unless
they join the same pair of vertices, and it is trivial to find a
$2$-matching.
Any $2$-matching in $\Gamma_S$ yields $\pi\ge 2(n-q-2m)$ by
\lref{Lm:SquareS}.
\end{proof}
We are now going to establish results for $t$ dual to Lemmas
\ref{Lm:RowS}--\ref{Lm:7S}.
\begin{lemma}\label{Lm:RowT}
Let $a\in K$ and let $b_1$, $\dots$, $b_\ell\not\in K$ be
distinct. Suppose that either $(a,b_1)$, $\dots$, $(a,b_\ell)\in T$,
or that $(a_1,b_1)$, $\dots$, $(a_\ell,b_\ell)\in T$ for some $a_1$,
$\dots$, $a_\ell\in K$ such that $a_i\circ b_i=a$. Then $\pi\ge
\ell(n-2q-m+1)+q-m-1$.
\end{lemma}
\begin{proof}
Let $(a,b_1)$, $\dots$, $(a,b_\ell)\in T$. By \lref{Lm:Triple}, for every $i$ we have $\mathrm{dist}_{b_i}+\mathrm{dist}_{a\circ b_i} \ge n-\mathrm{dist}_a\ge n-q+1$. We cannot have $a=a\circ b_i$ for some $i$, else $b_i=1$, $(a,b_i)\not\in\mathrm{diff}(\circ,\ast)$, so $(a,b_i)\not\in T$. Hence the elements $a$, $b_1$, $\dots$, $b_\ell$, $a\circ b_1$, $\dots$, $a\circ b_\ell$ are distinct, with $a\circ b_i\in K$. The profit on $a$, $b_1$, $a\circ b_1$ is at least $n-(2m+q)$, while the profit on each of the $\ell-1$ pairs of rows $b_i$, $a\circ b_i$ for $i>1$ is at least $n-q+1-(m+q)$.
Now assume that $(a_i,b_i)\in T$, $a_i\circ b_i=a$ for some $a_i\in K$, $1\le i\le \ell$. By \lref{Lm:Triple}, for every $i$ we have $\mathrm{dist}_{a_i}+\mathrm{dist}_{b_i} \ge n-\mathrm{dist}_a\ge n-q+1$. We cannot have $a=a_i$ for some $i$, else $a=a_i\circ b_i=a\circ b_i$, $b_i=1$, $(a_i,b_i)\not\in T$. Hence the elements $a$, $a_1$, $\dots$, $a_\ell$, $b_1$, $\dots$, $b_\ell$ are distinct. The profit on $a_1$, $b_1$, $a=a_1\circ b_1$ is at least $n-(2m+q)$, while the profit on each of the $\ell-1$ pairs of rows $a_i$, $b_i$ for $i>1$ is at least $n-q+1-(m+q)$.
\end{proof}
\begin{lemma}\label{Lm:SquareT}
If there are $(a,b)$, $(c,d)\in T$ such that $|\{a,c,a\circ b,c\circ d\}|=4$ then $\pi\ge 2(n-q-2m)$.
\end{lemma}
\begin{proof}
If $b\ne d$ then $|\{a,b,c,d,a\circ b,c\circ d\}|=6$ and we are done by \lref{Lm:Triple}. So let us assume that $b=d$. We can apply a disjunction trick and consider $e=c^\ast\gb(c\circ b)\in G\setminus K$, obtaining $e\ne b$, $\pi(c,e,c\ast e)\ge n-(2m+q)$. By our assumption, $\{a,a\circ b\}\cap\{c,c\ast e\}=\emptyset$. We therefore have additional profit of at least $n-(2m+q)$ on the rows $a$, $b$, $a\circ b$.
\end{proof}
\begin{lemma}\label{Lm:7T}
If $t\ge 7$ then $\pi\ge 2(n-q-2m)$.
\end{lemma}
\begin{proof}
If there are three elements of $T$ in the same row or with the same
product, \lref{Lm:RowT} implies $\pi\ge 3(n-2q-m+1)+(q-m-1)\ge
2(n-q-2m)$. Suppose that no three elements of $T$ are in the same row
or have the same product.
Define a multigraph $\Gamma_T$ on $K$ where $\{x,y\}$ is an edge if
and only if there is $z$ such that either $(x,z)\in T$ and $x\circ z=y$,
or $(y,z)\in T$ and $y\circ z=x$. Then $\Gamma_T$ has $t$ edges and
there are no more than two edges between any two vertices of
$T$. Arguing as in the proof of \lref{Lm:7S}, we can show that
$\Gamma_T$ has a $2$-matching.
Hence there are $(a,b)$, $(c,d)\in T$ such that $|\{a,c,a\circ b,c\circ d\}|=4$, and we are done by \lref{Lm:SquareT}.
\end{proof}
Finally, we return to the graph $\Gamma_U$ based on the set $U$.
\begin{lemma}\label{Lm:Matching}
If $\Gamma_U$ has a rainbow $\ell$-matching then $\pi\ge\ell(n-2q-m)$.
\end{lemma}
\begin{proof}
The existence of a rainbow $\ell$-matching in $\Gamma_U$ is equivalent to the existence of $\ell$ pairwise disjoint sets $\{a_i,b_i,a\circ b_i\}$, where $(a_i,b_i)\in U$, so $a_i\in K$, $b_i$, $a\circ b_i\in G\setminus K$. The rest follows from \lref{Lm:Triple}.
\end{proof}
To illustrate the procedure outlined at the beginning of this section, let us eliminate $(n,h,k,m)=(24,1,16,3)$. Since $\cst_0(24) = 120$, $q=\lceil n/3\rceil=8$, and $(n-k)q+(k-h)m=109$, we need a profit of at least $12$. \lref{Lm:R} with $r=2$ (thus $\ell=2$) yields precisely $\pi\ge 12$. We can therefore assume $r\le 1$, which \lref{Lm:LargeR} cannot improve. \lref{Lm:RowS} yields a sufficient $\pi\ge 16$ with $\ell=2$ (but $\ell=1$ does not suffice), so $s\le1(k-h)=15$. Since \lref{Lm:7S} yields $\pi\ge 20$, we can improve the bound to $s\le 6$. Similarly, \lref{Lm:RowT} with $\ell=2$ yields $t\le15$, which \lref{Lm:7T} improves with $\pi\ge 20$ to $t\le 6$. Then \eqref{Eq:rstu} allows us to assume that $u\ge3(k-h)-1-6-6=32$, and thus that $\Gamma_U$ has at least $\lceil 32/2\rceil = 16$ edges. Since $\mu_3(n-k)=\mu_3(8)=15$ by \pref{Pr:Graph}, $\Gamma_U$ contains a rainbow $3$-matching. Then $\pi\ge 3(n-2q-m)=15>12$ by \lref{Lm:Matching}, which is what we need, and $(24,1,16,3)$ is eliminated.
A straightforward calculation shows that the only remaining cases of \eqref{Eq:Quadruples} are
\begin{equation}\label{Eq:Quadruples3}
(24,1,\{17,18\},3),\quad(25,1,\{17,18\},3),\quad (26,1,19,3),\quad (27,1,\{19,20\},3).
\end{equation}
For these surviving cases the above procedure at least yields upper bounds on $r$, $s$, $t$ and a lower bound on $u$ as follows:
\begin{align*}
&(24,1,17,3):\quad r\le 3,\,s\le 6,\,t\le 6,\,u\ge 33,\\
&(24,1,18,3):\quad r\le 17,\,s\le 34,\,t\le 34,\,u\ge 0,\\
&(25,1,17,3):\quad r\le 2,\,s\le 6,\,t\le 6,\,u\ge 34,\\
&(25,1,18,3):\quad r\le 3,\,s\le 6,\,t\le 6,\,u\ge 36,\\
&(26,1,19,3):\quad r\le 6,\,s\le 6,\,t\le 6,\,u\ge 36,\\
&(27,1,19,3):\quad r\le 2,\,s\le 6,\,t\le 6,\,u\ge 40,\\
&(27,1,20,3):\quad r\le 3,\,s\le 38,\,t\le 38,\,u\ge 0.
\end{align*}
\section{Stubborn cases}\label{Sc:Stubborn}
It is easy to check that the profit obtained from a rainbow $3$-matching in $U$ is not sufficient to eliminate any of the cases \eqref{Eq:Quadruples3}. We will need more delicate profits, for instance obtained from a rainbow $2$-matching in $U$ and an element $(a,b)\in S$ such that $a$, $b$, $a\circ b$ are disjoint from the vertices and colors of the rainbow $2$-matching. We start with two dual lemmas that in certain circumstances provide upper bounds on $s$ and $t$.
\begin{lemma}\label{Lm:s3}
If $s\ge 3$ and $q\ge m+1$ then $\pi\ge 2n-3q-3m+1$.
\end{lemma}
\begin{proof}
If there are $(a,b)$, $(c,d)\in S$ with $|\{a,b,c,d\}|=4$, we are done by \lref{Lm:SquareS} and $q\ge m+1$. Otherwise there are $(a,b)$, $(c,d)\in S$ with $|\{a,b,c,d\}|=3$. If either $a=c$ and $b\ne d$, or $a\ne c$ and $b=d$, then $\pi\ge 2n-3q-3m+1$ by \lref{Lm:RowS} with $\ell=2$. The cases when $a=d$ or $b=c$ yield the same profit by an argument similar to \lref{Lm:RowS}. We cannot have $a=b$ or $c=d$ by the definition of $S$.
\end{proof}
\begin{lemma}\label{Lm:t3}
If $t\ge 3$ and $q\ge m+1$ then $\pi\ge 2n-3q-3m+1$.
\end{lemma}
\begin{proof}
If there are $(a,b)$, $(c,d)\in T$ with $|\{a,c,a\circ b, c\circ d\}|=4$, we are done by \lref{Lm:SquareT} and $q\ge m+1$. Otherwise there are $(a,b)$, $(c,d)\in T$ with $|\{a,c,a\circ b,c\circ d\}|=3$. The cases when $a=c$ or $a\circ b=c\circ d$ are handled by \lref{Lm:RowT}.
If either $a=c\circ d$ or $c=a\circ b$, we can assume without loss of generality that $a = c\circ d$. If $b\ne d$ then $a$, $b$, $c$, $d$, $a\circ b$ are distinct, and the profit on the rows $a$, $b$, $a\circ b$ is at least $n-(2m+q)$. Since $\mathrm{dist}_c+\mathrm{dist}_d\ge n-\mathrm{dist}_{c\circ d}\ge n-q+1$, the profit on the rows $c$, $d$ is at least $n-2q-m+1$, and the total profit is at least $2n-3q-3m+1$.
Finally suppose that $a=c\circ d$, $b=d$, and the elements $a$, $b$, $c$, $a\circ b$ are distinct. Using a disjunction trick for $(a,b)$, let us consider $(a,e=a^\ast\gb(a\circ b))\in T$ and $(c,b=d)\in T$, focusing on the rows $a$, $e$, $a\ast e = a\circ b$, $c$, $b=d$, $c\circ d$, which are distinct, except that $a=c\circ d$. We finish as above.
\end{proof}
\begin{lemma}\label{Lm:U}
We have $u\le (n-k)(n-k-1)$.
\end{lemma}
\begin{proof}
An element $(c,d)\in U$ determines the ordered pair $(d,c\circ d)\in (G\setminus K)\times (G\setminus K)$ with $d\ne c\circ d$ (since $c\ne 1$) and vice versa.
\end{proof}
We now elaborate on the idea of rainbow matchings in $U$ disjoint from
elements of $R$, $S$ and/or $T$.
For $(a,b)\in R\cup S\cup T$, let $U\hbox{$\setminus\!\!\setminus$} (a,b)=\{(c,d)\in U;\;\{c,d,c\circ d\}\cap \{a,b,a\circ b\}=\emptyset\}$. For $(a,b)$, $(c,d)\in R\cup S\cup T$, let $U\hbox{$\setminus\!\!\setminus$}(a,b)(c,d) = \{(e,f)\in U;\;\{a,b,a\circ b,c,d,c\circ d\}\cap \{e,f,e\circ f\}=\emptyset\}$.
\begin{lemma}\label{Lm:Uwr}
For $(a,b)\in R\cup S\cup T$, we have
\begin{displaymath}
|U\hbox{$\setminus\!\!\setminus$}(a,b)|\ge
\left\{\begin{array}{ll}
u - (2n-2k+1),&\text{ if }(a,b)\in R,\\
u - (2n-2k+4),&\text{ if }(a,b)\in S\cup T.
\end{array}\right.
\end{displaymath}
\end{lemma}
\begin{proof}
Assume that $(a,b)\in S$. Then an element $(c,d)\in U$ does not belong to $U\hbox{$\setminus\!\!\setminus$}(a,b)$ if and only if one of the following occurs: $c=a$, $c=b$, $d=a\circ b$, $c\circ d=a\circ b$. Now, $c=a$ can occur for at most $2$ elements of $U$, by the definition of $U$, given that row $a$ contains $(a,b)\in S$. We have $c=b$ at most $3$ times. We have $d=a\circ b$ at most $n-k$ times, because the column $a\circ b$ contains at most $n-k$ values from $G\setminus K$. Finally, $c\circ d=a\circ b$ occurs at most another $n-k-1$ times, because the value $a\circ b$ can occur at most once in every column of $G\setminus K$, and we have already accounted for all elements of $U$ in column $a\circ b$. The result for $(a,b)\in S$ follows.
Assume that $(a,b)\in T$. Then an element $(c,d)\in U$ does not belong to $U\hbox{$\setminus\!\!\setminus$}(a,b)$ if and only if one of the following occurs: $c=a$, $c=a\circ b$, $d=b$, $c\circ d=b$. The rest is analogous to the case $(a,b)\in S$.
Assume that $(a,b)=(a,a)\in R$. Then an element $(c,d)\in U$ does not belong to $U\hbox{$\setminus\!\!\setminus$}(a,b)$ if and only if one of the following occurs: $c=a$, $d=a\circ a$, $c\circ d = a\circ a$. The rest is analogous to the case $(a,b)\in S$.
\end{proof}
\begin{lemma}\label{Lm:Uwr2}
If $(a,b)$, $(c,d)\in S\cup T$ then $|U\hbox{$\setminus\!\!\setminus$}(a,b)(c,d)| \ge u - (4n-4k+8)$. If $(a,b)$, $(c,d)\in S$ and $|\{a,b,c,d\}|=3$ then $|U\hbox{$\setminus\!\!\setminus$}(a,b)(c,d)| \ge u - (4n-4k+5)$.
\end{lemma}
\begin{proof}
For $(a,b)$, $(c,d)\in S\cup T$, apply a variation of \lref{Lm:Uwr} twice. The worst case estimate $|U\hbox{$\setminus\!\!\setminus$}(a,b)(c,d)| \ge u - (4n-4k+8)$ is obtained when $|\{a,b,a\circ b,c,d,c\circ d\}|=6$.
Suppose that $(a,b)$, $(c,d)\in S$ and $|\{a,b,c,d\}|=3$. An element
$(e,f)\in U$ does not belong to $U\hbox{$\setminus\!\!\setminus$}(a,b)(c,d)$ if and only if one
of the following occurs: $e\in\{a,b,c,d\}$, $f\in\{a\circ b,c\circ d\}$,
or $e\circ f\in\{a\circ b,c\circ d\}$. Since
$|\{a,b,c,d\}|=3$, we can assume without loss of generality that
either $a=c$, $b$, $d$ are distinct, or $a=d$, $b$, $c$ are
distinct. (Note that $a=b$ is impossible since $(a,b)\in S$.) If
$a=c$, $b$, $d$ are distinct, then $e=a$ occurs at most once (since
$(a,b)$, $(c,d)\in S$), $e=b$ at most $3$ times, and $e=d$ at most $3$
times. If $a=d$, $b$, $c$ are distinct, then $e=a$ occurs at most
twice, $e=b$ at most $3$ times, and $e=c$ at most twice. Hence in both
cases, $e\in\{a,b,c,d\}$ occurs for at most $7$ elements $(e,f)\in U$.
As before, we eliminate up to $2(n-k)$ elements $(e,f)\in U$ with
$f\in\{a\circ b,c\circ d\}$, and a further $2(n-k-1)$ with
$e\circ f\in\{a\circ b,c\circ d\}$.
\end{proof}
Note that in all cases \eqref{Eq:Quadruples3} we have $k>2n/3$. The following lemma will therefore apply to these cases.
\begin{lemma}\label{Lm:rs0}
Assume that $n\ge 12$ and $k>2n/3$. Then $r+s>0$ or $G(\circ)$, $G(\ast)$ are isomorphic via a transposition.
\end{lemma}
\begin{proof}
Assume that $r+s=0$. The proof of \cite[Proposition 3.1]{DrEJC} (our \pref{Pr:3n4}) goes through with $k>2n/3$ (rather than $k>3n/4$), except for part (iv), as explicitly noted already by Dr\'apal in \cite{DrEJC}. With our assumption $r+s=0$, we can replace the proof of (iv) with the following: Let $g\in G$. Then there are $a$, $b\in K$ such that $g=a\circ b$, since $k>n/2$. Assume $g=a_i\circ b_i$ for some $a_i$, $b_i\in K$, $1\le i\le 2$. If $a_1\ast b_1\ne a_2\ast b_2$ then there is $i$ such that $a_i\circ b_i\ne a_i\ast b_i$, and for this $i$ we have $(a_i,b_i)\in R\cup S$, a contradiction. Thus $a_1\ast b_1 = a_2\ast b_2$.
We can now conclude from \cite[Proposition 3.1]{DrEJC} that there is an isomorphism $f:G(\circ)\to G(\ast)$ such that $f(a)=a$ for every $a\in K$. Then by \cite[Proposition 6.1]{DrEJC}, $\mathrm{dist}(\circ,\ast)\ge \cst_0(\circ)$, and if equality holds, $f$ must be a transposition.
\end{proof}
The following example shows that \lref{Lm:rs0} is best possible. Let $\circ,\ast$
be defined by
\[
\begin{array}{c|ccccccccc}
\circ& 1& 2& 3& 4& 5& 6& 7& 8& 9\\
\hline
1& 1& 2& 3& 4& 5& 6& 7& 8& 9\\
2& 2& 3& 1& 5& 6& 4& 8& 9& 7\\
3& 3& 1& 2& 6& 4& 5& 9& 7& 8\\
4& 4& 5& 6& 7& 8& 9& \mk1& \mk2& \mk3\\
5& 5& 6& 4& 8& 9& 7& \mk2& \mk3& \mk1\\
6& 6& 4& 5& 9& 7& 8& \mk3& \mk1& \mk2\\
7& 7& 8& 9& \mk1& \mk2& \mk3& \mk4& \mk5& \mk6\\
8& 8& 9& 7& \mk2& \mk3& \mk1& \mk5& \mk6& \mk4\\
9& 9& 7& 8& \mk3& \mk1& \mk2& \mk6& \mk4& \mk5
\end{array}
\qquad
\begin{array}{c|ccccccccc}
\ast& 1& 2& 3& 4& 5& 6& 7& 8& 9\\
\hline
1& 1& 2& 3& 4& 5& 6& 7& 8& 9\\
2& 2& 3& 1& 5& 6& 4& 8& 9& 7\\
3& 3& 1& 2& 6& 4& 5& 9& 7& 8\\
4& 4& 5& 6& 7& 8& 9& \mk2& \mk3& \mk1\\
5& 5& 6& 4& 8& 9& 7& \mk3& \mk1& \mk2\\
6& 6& 4& 5& 9& 7& 8& \mk1& \mk2& \mk3\\
7& 7& 8& 9& \mk2& \mk3& \mk1& \mk5& \mk6& \mk4\\
8& 8& 9& 7& \mk3& \mk1& \mk2& \mk6& \mk4& \mk5\\
9& 9& 7& 8& \mk1& \mk2& \mk3& \mk4& \mk5& \mk6
\end{array}
\]
where the differences are shaded. Then $k=2n/3$ and yet the groups
are not isomorphic; $G(\circ)\cong (C_3)^2$ and $G(\ast)\cong C_9$.
By taking direct products of these two groups with other groups we
can make arbitrarily large non-isomorphic pairs where $k=2n/3$
and $r=s=0$.
\begin{lemma}\label{Lm:Profit1}
Suppose that $k=n-q+2$ and $x$, $y\in G\setminus K$, $x\ne y$. Then
there is $(v,w)\in \mathrm{diff}(\circ,\ast)$ such that $\{v,w,v\circ w\}\cap
\{x,y\}=\emptyset$, $v\in G\setminus K$, and either $w\in K$ or $v\circ
w\in K$.
\end{lemma}
\begin{proof}
The set $L=G\setminus(K\cup\{x,y\})$ is not closed under $\circ$ since
it does not contain $1$, so there are $v$, $w\in L$ such that $v\circ
w\not\in L$. If $v\circ w\in K$, we are done. Otherwise $v\circ w\in
\{x,y\}$, and we can assume without loss of generality that $v\circ w=x$.
Since $v\in G\setminus K$, $\mathrm{dist}_v\ge q = n - k + 2$, but
$|(G\setminus K)\cup\{v^\circ\ga x,v^\circ\ga y\}|\le n-k+1$ (as $v^\circ\ga
x = v^\circ \circ v\circ w = w\in G\setminus K$), so there is $z\in K$ with
$(v,z)\in\mathrm{diff}(\circ,\ast)$, and $v\circ z\not\in\{x,y\}$. Then $\{v,z,v\circ
z\}\cap \{x,y\}=\emptyset$, $v\in G\setminus K$, $z\in K$, and $(v,z)$
does the job.
\end{proof}
We now eliminate all the quadruples of \eqref{Eq:Quadruples3}, sorting them according to the difference $n-k$.
\emph{Case $(n,h,k,m) = (25,1,17,3)$.} To eliminate this case, we need a profit of at least $\cst_0(n)-(n-k)q -(k-h)m + 1 = 13$, and we can assume $r\le 2$, $s\le 6$, $u\ge 34$. If $s>0$ and $(a,b)\in S$ then $|U\hbox{$\setminus\!\!\setminus$}(a,b)|\ge u - (2n-2k+4) \ge 14$ by \lref{Lm:Uwr}, so there is $(c,d)\in U$ such that $a$, $b$, $a\circ b$, $c$, $d$, $c\circ d$ are distinct, yielding the profit of at least $(n-q-2m) + (n-2q-m)=14>13$.
We can therefore assume that $s=0$ and $u\ge 40$. By \lref{Lm:rs0}, $r>0$ and there is $(a,a)\in R$. Then $|U\hbox{$\setminus\!\!\setminus$}(a,a)|\ge u-(2n-2k+1) \ge 23$ by \lref{Lm:Uwr}. Since $\mu_2(n-k)=\mu_2(8)=8\le \lceil 23/2\rceil$, there is a rainbow $2$-matching in $U$ disjoint from $\{a,a\circ a\}$, and we obtain a sufficient profit of at least $(n-2q-m+1) + 2(n-2q-m) = 13$.
\emph{Case $(n,h,k,m) = (27,1,19,3)$.} We need a profit of at least $19$, and we can assume $r\le 2$, $s\le 6$, $u\ge 40$. If $s>0$ and $(a,b)\in S$ then $|U\hbox{$\setminus\!\!\setminus$}(a,b)|\ge u-(2n-2k+4)\ge 20$ by \lref{Lm:Uwr}, $\mu_2(n-k)=\mu_2(8)=8\le \lceil 20/2\rceil$, so there is a rainbow $2$-matching disjoint from $\{a,b,a\circ b\}$, yielding a sufficient profit of $(n-q-2m)+2(n-2q-m) = 24$. We can therefore assume that $s=0$ and $u\ge 46$. By \lref{Lm:rs0}, $r>0$ and there is $(a,a)\in R$. Then $|U\hbox{$\setminus\!\!\setminus$}(a,a)|\ge u-(2n-2k+1)\ge 29$ by \lref{Lm:Uwr}. Since $\mu_2(n-k)=8\le \lceil 29/2\rceil$, there is a rainbow $2$-matching disjoint from $\{a,a\circ a\}$, and we obtain a sufficient profit of at least $(n-2q-m+1)+2(n-2q-m) = 19$.
\emph{Case $(n,h,k,m) = (24,1,17,3)$.} We need a profit of at least $17$, and we can assume $r\le 3$, $s\le 6$, $t\le 6$, $u\ge 33$. If $s>0$ and $(a,b)\in S$ then $|U\hbox{$\setminus\!\!\setminus$}(a,b)|\ge 15$ by \lref{Lm:Uwr}, $\mu_2(n-k) = \mu_2(7) = 7 \le \lceil 15/2\rceil$, so there is a rainbow $2$-matching in $U$ disjoint from $\{a,b,a\circ b\}$, for a sufficient profit of at least $(n-q-2m)+2(n-2q-m)=20$. Similarly if $t>0$. We can therefore assume that $s=0$, $t=0$ and $u\ge 45$. There is $(a,a)\in R$ by \lref{Lm:rs0}, $|U\hbox{$\setminus\!\!\setminus$}(a,a)| \ge 30$ by \lref{Lm:Uwr}, $\mu_3(n-k) = \mu_3(7) = 15 = \lceil 30/2\rceil$, so there is a rainbow $3$-matching in $U$ disjoint from $\{a,a\circ a\}$, giving a sufficient profit of at least $(n-2q-m+1)+3(n-2q-m) = 21$.
\emph{Case $(n,h,k,m) = (26,1,19,3)$.} We need a profit of at least $20$, and we can assume $r\le 6$, $s\le 6$, $t\le 6$, $u\ge 36$. If $s>0$ and $(a,b)\in S$ then $|U\hbox{$\setminus\!\!\setminus$}(a,b)|\ge 18$ by \lref{Lm:Uwr}, $\mu_2(n-k) = \mu_2(7) = 7 \le \lceil 18/2\rceil$, so there is a rainbow $2$-matching in $U$ disjoint from $\{a,b,a\circ b\}$, for a sufficient profit of at least $(n-q-2m)+2(n-2q-m)=21$. Similarly if $t>0$. If $s=0=t$ then $u\ge 52$, a contradiction of \lref{Lm:U}, which yields $u\le 42$.
\emph{Case $(n,h,k,m) = (25,1,18,3)$.} We need a profit of at least $19$, and we can assume $r\le 3$, $s\le 6$, $t\le 6$, $u\ge 36$. Suppose that $s\ge 3$. If there are $(a,b)$, $(c,d)\in S$ such that $|\{a,b,c,d\}|=4$ then \lref{Lm:SquareS} yields a sufficient profit of at least $2(n-q-2m)=20$. Otherwise, as in the proof of \lref{Lm:s3}, there are $(a,b)$, $(c,d)\in S$ such that $|\{a,b,c,d\}|=3$ and $\pi(a,b,c,d,a\circ b,c\circ d)\ge 2n-3q-3m+1 = 15$. Moreover, \lref{Lm:Uwr2} implies that $|U\hbox{$\setminus\!\!\setminus$}(a,b)(c,d)|\ge 3$, so there is $(e,f)\in U$ such that $\{e,f,e\circ f\}\cap \{a,b,c,d,a\circ b,c\circ d\}=\emptyset$. Since $\pi(e,f,e\circ f)\ge n-2q-m = 4$, we have $\pi\ge 15+4=19$, as desired. We can therefore assume that $s\le 2$ and $u\ge 40$. Using \lref{Lm:Uwr2} once more, we may now deduce that $t\le2$. Hence $u\ge 44$, contradicting $u\le 42$ from \lref{Lm:U}.
\emph{Case $(n,h,k,m) = (27,1,20,3)$.} We need a profit of at least $25$, and we can assume $r\le 3$. Suppose that $s\ge 7$. Then by \lref{Lm:7S}, there are $(a,b)$, $(c,d)\in S$ such that $\pi(a,b,c,d,a\circ b,c\circ d)\ge 2(n-q-2m) = 24$. Using $(x,y) = (a\circ b,c\circ d)$ in \lref{Lm:Profit1}, we obtain $(v,w)\in\mathrm{diff}(\circ,\ast)$ such that $\{v,w,v\circ w\}\cap \{x,y\}=\emptyset$, $v\in G\setminus H$, and either $w\in K$ or $v\circ w\in K$. We have not yet used any of the rows $v$, $w$, $v\circ w$ that happen to be in $G\setminus K$ in our calculation of the profit. We have therefore counted at most $q + q + (q-1) = 3q-1$ differences on the rows $v$, $w$, $v\circ w$ so far, however, we have $\mathrm{dist}_v+\mathrm{dist}_w+\mathrm{dist}_{v\circ w}\ge n =3q$ because $(v,w)\in\mathrm{diff}(\circ,\ast)$. We can now increase the profit of $24$ by $1$, and we are done. Similarly, if $t\ge 7$, there are $(a,b)$, $(c,d)\in T$ such that $\pi(a,b,c,d,a\circ b,c\circ d)\ge24$ by \lref{Lm:RowT}, and we can apply \lref{Lm:Profit1} with $(x,y)=(b,d)$ to increase the profit by $1$. We can therefore assume $s\le 6$, $t\le 6$ and $u\ge 42$.
If $s\ge 3$, there are $(a,b)$, $(c,d)\in S$ with $\pi(a,b,c,d,a\circ b,c\circ d)\ge 2n-3q-3m+1 = 19$ by \lref{Lm:s3}, $|U\hbox{$\setminus\!\!\setminus$}(a,b)(c,d)|\ge 6$ by \lref{Lm:Uwr2}, $(e,f)\in U$ with $\{e,f,e\circ f\}\cap \{a,b,c,d,a\circ b,c\circ d\}=\emptyset$, and $\pi(e,f,e\circ f)\ge n-2q-m = 6$, for a sufficient profit of $19+6=25$. We can therefore assume $s\le 2$ and $u\ge 46$, contradicting $u\le 42$ from \lref{Lm:U}.
\emph{Case $(n,h,k,m) = (24,1,18,3)$}. We need a profit of at least $22$.
Define $\lambda$ to be the maximum integer for which there exist distinct
$x,y\in G$ such that $\mathrm{dist}_x\ge\mathrm{dist}_y\ge\lambda$. Suppose that $\lambda\ge17$.
By \lref{Lm:Profit1} there is $(v,w)\in\mathrm{diff}(\circ,\ast)$ with
$\{v,w,v\circ w\}\cap\{x,y\}=\emptyset$ and $|K\cap\{w,v\circ w\}|\ge1$
so $\pi(v,w,v\circ w,x,y)\ge n-2q-m+2(\lambda-q)\ge23$. Thus we may assume that
$\lambda\le16$.
Let $\Omega$ be a maximal subset of $R\cup S\cup T$ under the constraint that
there should be a maximum of 3 elements of $\Omega$ within any row. Let
$\Sigma$ be the sum over $\Omega$ of $\mathrm{dist}_a+\mathrm{dist}_b-2m$ for elements
$(a,b)\in R\cup S$, and $\mathrm{dist}_a+\mathrm{dist}_{a\circ b}-2m$ for $(a,b)\in T$.
We claim that $\Sigma\ge|\Omega|(n-2m-\lambda)$. Each $(a,b)\in R\cup S$
satisfies
$\mathrm{dist}_a+\mathrm{dist}_b\ge n-\min\{\mathrm{dist}_{a\circ b},\mathrm{dist}_{a\ast b}\}\ge n-\lambda$.
So it suffices to show that each $(a,b)\in T$ satisfies
$\mathrm{dist}_a+\mathrm{dist}_{a\circ b}\ge n-\lambda$. Since $(a,b)\in T$, we have
$\mathrm{dist}_a+\mathrm{dist}_{a\circ b}\ge n-\mathrm{dist}_b$. By a disjunction trick, $(a,c)\in T$ where $c=a^\ast\gb(a\circ b)$, so
$\mathrm{dist}_a+\mathrm{dist}_{a\circ b}\ge n-\mathrm{dist}_c$.
Since $b$, $c$ are distinct elements of $G\setminus K$, we have
$\lambda\le\min\{\mathrm{dist}_b,\mathrm{dist}_c\}$, from which the claim follows.
Next we claim that $\Sigma\le8(37-2\lambda)$. Consider $a\in K\setminus H$.
By construction, $a$ is a row coordinate for at most 3 cells in $\Omega$.
By \lref{Lm:RowS}, there are at most 2 cells in $S$ for which $a$ is
the column coordinate, otherwise we realize a sufficient profit
of $3(n-2q-m+1)+q-m+1=24$. Similarly, using \lref{Lm:RowT}, there
are at most 2 cells $(c,d)$ in $T$ for which $a=c\circ d$. It is also
possible that $a$ is the column coordinate for a single cell in $R$.
It follows that $\Sigma\le 8\Sigma'$, where $\Sigma'$ is the sum over
$a\in K\setminus H$ of $\mathrm{dist}_a-m$. As the profit from $K\cup\{x,y\}$
is at least $\Sigma'+2(\lambda-q)$ we are done unless
$\Sigma'\le 21+2q-2\lambda=37-2\lambda$. This proves the claim.
Combining the previous two claims we find that
$|\Omega|\le8(37-2\lambda)/(n-2m-\lambda)=16+8/(18-\lambda)\le20$, since
$\lambda\le16$. As $\Omega\cup U$ contains three differences in every row
indexed by $K\setminus H$, it follows that $u\ge3(k-h)-|\Omega|\ge31$.
This contradicts \lref{Lm:U}, finishing the last case.
\section{Constructions}\label{Sc:Constructions}
We have now established all distances mentioned in Theorem \ref{Th:Main}. It remains to present the constructions that realize the minimal distances $\cst(\circ)=\mathrm{dist}(\circ,\ast)$ in situations when $\cst(\circ)<\cst_0(\circ)$.
\subsection{Cyclic and dihedral constructions}\label{Ss:Quarter}
The following two constructions \eqref{Eq:CyclicConstruction} and \eqref{Eq:DihedralConstruction} were introduced in \cite{DrConstr1}. Given a certain group $G(\circ)$ of even order $n$, they produce a group $G(\ast)$ at distance $n^2/4$ from $G(\circ)$.
Recall the graphs $\mathcal G(n)$ and $\mathcal G'(n)$ from the Introduction. It turns out that whenever two groups $G(\circ)$, $G(\ast)$ of order $n=8$ or $n=16$ are at distance $n^2/4$, there is a group $G(\gc)$ obtained from $G(\circ)$ by one of the two constructions and such that $G(\ast)\cong G(\gc)$. This follows from the fact that the graph $\mathcal G(8)$ (calculated in \cite{VoMS} and independently here) coincides with $\mathcal G'(8)$, and from the fact that the graph $\mathcal G(16)$ (calculated here for the first time) coincides with $\mathcal G'(16)$ (calculated by B\'alek \cite{Ba} and independently here).
For a fixed positive integer $m$ and the set $M = \{-m+1$, $-m+2$,
$\dots$, $m-1$, $m\}$, define $\sigma:\mathbb{Z}\to \{-1$, $0$, $1\}$ by
\begin{equation*}
\sigma(i) = \left\{\begin{array}{ll}
1,&i>m,\\
0,&i\in M,\\
-1,&i<1-m.
\end{array}\right.
\end{equation*}
\emph{The cyclic construction.}
Let $G(\circ)$ be a group of order $n$, $S\unlhd G$, $G/S=\langle \alpha \rangle$ a cyclic group of order $2m$ and $1\ne h\in S\cap Z(G)$. Then $G(\circ)$ is the disjoint union $\bigcup_{i\in M}\alpha^i$, and we can define a new multiplication $\ast$ on $G$ by
\begin{equation}\label{Eq:CyclicConstruction}
x\ast y = x\circ y\circ h^{\sigma(i+j)},
\end{equation}
where $x\in\alpha^i$, $y\in\alpha^j$, and $i$, $j\in M$. Then $G(\ast)$ is a group and $\mathrm{dist}(\circ,\ast)=n^2/4$.
\emph{The dihedral construction.}
Let $G(\circ)$ be a group of order $n$, $S\unlhd G$, $G/S$ a dihedral group of order $4m$ (where we allow $m=1$), and $\beta$, $\gamma$ involutions of $G/S$ such that $\alpha=\beta\gamma$ is of order $2m$. Let $G_0=\bigcup_{i\in M}\alpha^i$ and $G_1=G\setminus G_0$. Let $1\ne h\in S\cap Z(G_0)$ be such that $hxh=x$ for some (and hence every) $x\in G_1$. Then there are $e\in\beta$ and $f\in\gamma$ so that $G$ is the disjoint union $\bigcup_{i\in M}(\alpha^i\cup e\alpha^i)$ or $\bigcup_{j\in M}(\alpha^j\cup \alpha^jf)$, and we can define a new multiplication $*$ on $G$ by
\begin{equation}\label{Eq:DihedralConstruction}
x\ast y=x\circ y\circ h^{(-1)^r\sigma(i+j)},
\end{equation}
where $x\in\alpha^i\cup e\alpha^i$, $y\in(\alpha^j\cup\alpha^jf)\cap G_r$, $i$, $j\in M$, and $r\in\{0,\,1\}$. Then $G(\ast)$ is a group and $\mathrm{dist}(\circ,\ast)=n^2/4$.
\subsection{Other constructions}\label{Ss:Other}
The following three constructions furnish the distances of
\tref{Th:Main} with $\mathrm{dist}(\circ,\ast)<\cst_0(\circ)$ and $n\ne 2^k$.
\emph{Construction 1.} Suppose $n\equiv 2\bmod4$ and $n\ge6$. Let $O$ be an abelian group of order $n/2$. We have two groups defined on the set $O\times C_2$, namely $D(O)$ and the usual direct product on $O\times C_2$. The distance between these two groups is $n(n-2)/2$. When $n\in\{6,10\}$, this is $\cst(D_n)$ so $\cnb(D_n)$ contains a group isomorphic to $C_{n/2}\times C_2\cong C_n$ (although $\cnb(D_{10})$ also contains a group isomorphic to $D_{10}$, because $n(n-2)/2=6n-20=\delta_0(D_{10})$).
\def\odot} \def\grtwo{\circledast{\odot} \def\grtwo{\circledast}
\emph{Construction 2.} We construct two abelian group operations $\odot} \def\grtwo{\circledast$, $\grtwo$ on the set $C_a\times C_b$ where $a$ is odd.
\[
(s,t)\odot} \def\grtwo{\circledast(u,v)=
\begin{cases}
(s+u,t+v+1)&\text{if $s+u\ge a$}\\
(s+u,t+v)&\text{otherwise}.
\end{cases}
\]
Clearly $\odot} \def\grtwo{\circledast$ is isomorphic to $C_{ab}$ by the map $(s,t)\mapsto s+at$.
To form $\grtwo$ we take the usual group on $C_a\times C_b$ and apply the isomorphism
\[
(s,t)\mapsto\begin{cases}
(s,t+1)&\text{if $s\ge\frac12(a+1)$}\\
(s,t)&\text{otherwise}.
\end{cases}
\]
It is routine to check that $d(\odot} \def\grtwo{\circledast,\grtwo)=n^2(1-a^{-2})/4$. In particular, $d(\odot} \def\grtwo{\circledast,\grtwo)=2n^2/9$ when $a=3$, the nearest (proportional) distance between non-isomorphic groups \cite{ILY}. Note that $2n^2/9<\delta_0(n)$ for $n\le 21$, and indeed $\cst(C_{3b})=2n^2/9$ for $2\le b\le 7$. The above construction proves this for $b\in\{2,4,5,7\}$.
Construction 2 shows directly that the following achieve $2n^2/9$:
\begin{align*}
&\mathrm{dist}(C_6,C_6),
\mathrm{dist}(C_9,C_3^2),
\mathrm{dist}(C_{12},C_{12}),
\mathrm{dist}(C_{15},C_{15}),\\
\mathrm{dist}(C_{18},C_6\times C_3),
\mathrm{dist}(C_{21},C_{21}).
\end{align*}
Taking appropriate extensions of the example that realises $\mathrm{dist}(C_6, C_6)$,
we can show that $2n^2/9$ is also achieved in these cases:
\begin{align*}
&\mathrm{dist}(C_6\times C_2, C_6\times C_2),
\mathrm{dist}(C_6\times C_3, C_6\times C_3),
\mathrm{dist}(D_{12},D_{12}),\mathrm{dist}(\mathrm{Dic}_{12},\mathrm{Dic}_{12}).
\end{align*}
Similarly, $\mathrm{dist}(D_{18},(C_3)^2\rtimes C_2)$ is achieved by an extension
of the example that yields $\mathrm{dist}(C_9,(C_3)^2)$.
The above is a complete catalogue of cases where
two groups are at distance precisely $2n^2/9$, except for the ad hoc
constructions for $\mathrm{dist}(C_9,C_9)$, $\mathrm{dist}(C_{18},C_{18})$
and $\mathrm{dist}(D_{18},D_{18})$ below.
Construction 2 can also be used directly to realize $\mathrm{dist}(C_{10},C_{10})$ and
$\mathrm{dist}(C_{14},C_{14})$.
\emph{Construction 3. (Ad hoc)}
$\mathrm{dist}(C_7,C_7)$: The distance between $C_7=\{0,\dots,6\}$ and its $(12)(56)$
isomorph is $18$.
$\mathrm{dist}(C_9,C_9)$: The distance between $C_9=\{0,\dots,8\}$ and its $(36)(47)(58)$
isomorph is $2n^2/9=18$.
Appropriate extensions of this last example realize
both $\mathrm{dist}(C_{18},C_{18})$ and $\mathrm{dist}(D_{18},D_{18})$.
\begin{remark} The computer calculations used in this paper were as follows: The graphs $\mathcal G'(8)$ and $\mathcal G'(16)$ were calculated by the first author using the \texttt{GAP} \cite{GAP} package \texttt{LOOPS} \cite{LOOPS} and modified code from \cite{VoEJC}. The inequalities of Section \ref{Sc:Inequalities} were independently verified by both authors, resulting in the list \eqref{Eq:Quadruples}. The algorithm for $m=2$ of Subsection \ref{Ss:m2} was implemented by both authors independently, and so was the algorithm for distances of cyclic groups of Subsection \ref{Ss:Cyclic}. The general algorithm for $\mathrm{dist}([\circ],[\ast])$ was run by the second author for all $n\le 22$ (which took several months on a single processor computer), and by the first author for $n\le 15$. Both authors verified the values $\mu_3(6)$--$\mu_3(10)$ of Proposition \ref{Pr:Graph} with independent programs. Finally, the upper bounds on $r$, $s$, $t$ and lower bounds on $u$ of Section \ref{Sc:ApplyGraph} were also performed independently by the two authors.
\end{remark}
| {
"timestamp": "2015-09-21T02:12:17",
"yymm": "1509",
"arxiv_id": "1509.05660",
"language": "en",
"url": "https://arxiv.org/abs/1509.05660",
"abstract": "Suppose that all groups of order $n$ are defined on the same set $G$ of cardinality $n$, and let the \\emph{distance} of two groups of order $n$ be the number of pairs $(a,b)\\in G\\times G$ where the two group operations differ. Given a group $G(\\circ)$ of order $n$, we find all groups of order $n$, up to isomorphism, that are closest to $G(\\circ)$.",
"subjects": "Group Theory (math.GR)",
"title": "Closest multiplication tables of groups",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754492759499,
"lm_q2_score": 0.7185943805178139,
"lm_q1q2_score": 0.7075103850454995
} |
https://arxiv.org/abs/1701.06029 | Hamiltonicity in locally finite graphs: two extensions and a counterexample | We state a sufficient condition for the square of a locally finite graph to contain a Hamilton circle, extending a result of Harary and Schwenk about finite graphs.We also give an alternative proof of an extension to locally finite graphs of the result of Chartrand and Harary that a finite graph not containing $K^4$ or $K_{2,3}$ as a minor is Hamiltonian if and only if it is $2$-connected. We show furthermore that, if a Hamilton circle exists in such a graph, then it is unique and spanned by the $2$-contractible edges.The third result of this paper is a construction of a graph which answers positively the question of Mohar whether regular infinite graphs with a unique Hamilton circle exist. | \section{Introduction}
Results about Hamilton cycles in finite graphs can be extended to locally finite graphs in the following way.
For a locally finite connected graph $G$ we consider its Freudenthal compactification $|G|$~\cite{diestel_buch, diestel_arx}. This is a topological space obtained by taking~$G$, seen as a $1$-complex, and adding certain points to it.
These additional point are the \textit{ends} of $G$, which are the equivalence classes of the rays of $G$ under the relation of being inseparable by finitely many vertices.
Extending the notion of cycles, we define \textit{circles}~\cite{inf-cyc-1, inf-cyc-2} in $|G|$ as homeomorphic images of the unit circle $S^1 \subseteq \mathbb{R}^2$ in $|G|$, and we call them \textit{Hamilton circles} of $G$ if they contain all vertices of $G$.
As a consequence of being a closed subspace of $|G|$, Hamilton circles also contain all ends of $G$.
Following this notion we call $G$ \textit{Hamiltonian} if there is a Hamilton circle in $|G|$.
One of the first and probably one of the deepest results about Hamilton circles was Georgakopoulos's extension of Fleischner's theorem to locally finite graphs.
\begin{theorem}\label{fin_fleisch}\cite{fleisch}
The square of any finite $2$-connected graph is Hamiltonian.
\end{theorem}
\begin{theorem}\label{inf-fleisch}\cite[Thm.\ 3]{agelos}
The square of any locally finite $2$-connected graph is Hamiltonian.
\end{theorem}
\noindent Following this breakthrough, more Hamiltonicity theorems have been extended to locally finite graphs in this way \cite{brewster-funk, bruhn-HC, agelos, Ha_Leh_Po, heuer_ObSu, heuer_Asra, lehner-HC}.
The purpose of this paper is to extend two more Hamiltonicity results about finite graphs to locally finite ones and to construct a graph which shows that another result does not extend.
The first result we consider is a corollary of the following theorem of Harary and Schwenk.
A \textit{caterpillar} is a tree such that after deleting its leaves only a path is left.
Let $S(K_{1, 3})$ denote the graph obtained by taking the star with three leaves, $K_{1, 3}$, and subdividing each edge once.
\begin{theorem}\label{fin_cater}\cite[Thm.\ 1]{cat_HC}
Let $T$ be a finite tree with at least three vertices. Then the following statements are equivalent:
\begin{enumerate}[\normalfont(i)]
\item $T^2$ is Hamiltonian.
\item $T$ does not contain $S(K_{1,3})$ as a subgraph.
\item $T$ is a caterpillar.
\end{enumerate}
\end{theorem}
Theorem~\ref{fin_cater} has the following obvious corollary.
\begin{corollary}\label{fin_cater_impl}~\cite{cat_HC}
The square of any finite graph $G$ on at least three vertices such that $G$ contains a spanning caterpillar is Hamiltonian.
\end{corollary}
While the proof of Corollary~\ref{fin_cater_impl} is immediate, the proof of the following extension of it, which is the first result of this paper, needs more work.
We call the closure $\overline{H}$ in $|G|$ of a subgraph $H$ of $G$ a \textit{standard subspace} of $|G|$.
Extending the notion of trees, we define \textit{topological trees} as topologically connected standard subspaces not containing any circles.
As an analogue of a path, we define an \textit{arc} as a homeomorphic image of the unit interval $[0,1] \subseteq \mathbb{R}$ in $|G|$.
Note that for standard subspaces being topologically connected is equivalent to being arc-connected by Lemma~\ref{arc_conn}.
For our extension we adapt the notion of a caterpillar to the space $|G|$ and work with \textit{topological caterpillars}, which are topological trees $\overline{T}$ such that $\overline{T-L}$ is an arc, where $T$ is a forest in $G$ and $L$ denotes the set of vertices of degree $1$ in~$T$.
\begin{theorem}\label{top_catp_HC}
The square of any locally finite connected graph $G$ on at least three vertices such that $|G|$ contains a spanning topological caterpillar is Hamiltonian.
\end{theorem}
The other two results of this paper concern the uniqueness of Hamilton circles.
The first is about finite \textit{outerplanar graphs}.
These are finite graphs that can be embedded in the plane so that all vertices lie on the boundary of a common face.
Clearly, finite outerplanar graphs have a Hamilton cycle if and only if they are $2$-connected.
In a $2$-connected graph call an edge $2$-\textit{contractible} if its contraction leaves the graph $2$-connected.
It is also easy to see that any finite $2$-connected outerplanar graph has a unique Hamilton cycle.
This cycle consists precisely of the $2$-contractible edges of the graph (except for the $K^3$), as pointed out by Sys\l{}o~\cite{syslo}.
We summarise this with the following proposition.
\begin{proposition}\label{summary}
\begin{enumerate}[\normalfont(i)]
\item A finite outerplanar graph is Hamiltonian if and only if it is $2$-connected.
\item \cite[Thm.~6]{syslo} Finite $2$-connected outerplanar graphs have a unique Hamilton cycle, which consists precisely of the $2$-contractible edges unless the graph is isomorphic to a $K^3$.
\end{enumerate}
\end{proposition}
Finite outerplanar graphs can also be characterised by forbidden minors, which was done by Chartrand and Harary.
\begin{theorem}\label{outerplanar_count_char}\cite[Thm.\ 1]{char_har}
A finite graph is outerplanar if and only if it contains neither a $K^4$ nor a $K_{2, 3}$ as a minor.\footnote{\label{stronger_K_4}Actually these statements can be strengthened a little bit by replacing the part about not containing a $K^4$ as a minor by not containing it as a subgraph.
This follows from Lemma~\ref{K^4_minor_subgr}.}
\end{theorem}
In the light of Theorem~\ref{outerplanar_count_char} we first prove the following extension of statement~(i) of Proposition~\ref{summary} to locally finite graphs.
\begin{theorem}\label{HC_K_4-K_2,3}
Let $G$ be a locally finite connected graph.
Then the following statements are equivalent:
\begin{enumerate}[\normalfont(i)]
\item $G$ is $2$-connected and contains neither $K^4$ nor $K_{2,3}$ as a minor.$^{\ref{stronger_K_4}}$
\item $|G|$ has a Hamilton circle $C$ and there exists an embedding of $|G|$ into a closed disk such that $C$ is mapped onto the boundary of the disk.
\end{enumerate}
Furthermore, if statements (i) and (ii) hold, then $|G|$ has a unique Hamilton circle.
\end{theorem}
\noindent From this we then obtain the following corollary, which extends statement (ii) of Proposition~\ref{summary}.
\begin{corollary}\label{Cor_contr}
Let $G$ be a locally finite $2$-connected graph not containing $K^4$ or $K_{2,3}$ as a minor, and not isomorphic to $K^3$.
Then the edges contained in the Hamilton circle of $|G|$ are precisely the $2$-contractible edges of $G$.
\end{corollary}
We should note here that parts of Theorem~\ref{HC_K_4-K_2,3} and Corollary~\ref{Cor_contr} are already known.
Chan~\cite[Thm.~20 with Thm.~27]{chan} proved that a locally finite $2$-connected graph not isomorphic to $K^3$ and not containing $K^4$ or $K_{2,3}$ as a minor has a Hamilton circle that consists precisely of the $2$-contractible edges of the graph.
He deduces this from other general results about $2$-contractible edges in locally finite $2$-connected graphs.
In our proof, however, we directly construct the Hamilton circle and show its uniqueness without working with $2$-contractible edges.
Afterwards, we deduce Corollary~\ref{Cor_contr}.
Our third result is related to the following conjecture Sheehan made for finite graphs.
\begin{conjecture}\label{sheehan}\cite{sheehan}
There is no finite $r$-regular graph with a unique Hamilton cycle for any $r > 2$.
\end{conjecture}
\noindent This conjecture is still open, but some partial results have been proved \cite{haxell, thomason, thomassen}.
For $r=3$ the statement of the conjecture was first verified by C.~A.~B.~Smith.
This was noted in an article of Tutte~\cite{tutte} where the statement for $r=3$ was published for the first time.
For infinite graphs Conjecture~\ref{sheehan} is not true in this formulation.
It fails already with $r=3$.
To see this consider the graph depicted in Figure~\ref{double_ladder}, called the \textit{double ladder}.
\begin{figure}[htbp]
\centering
\includegraphics{double_ladder.pdf}
\caption{The double ladder}
\label{double_ladder}
\end{figure}
\noindent It is easy to check that the double ladder has a unique Hamilton circle, but all vertices have degree $3$.
Mohar has modified the statement of the conjecture and raised the following question.
To state them we need to define two terms.
We define the \textit{vertex-} or \textit{edge-degree} of an end $\omega$ to be the supremum of the number of vertex- or edge-disjoint rays in $\omega$, respectively.
In particular, ends of a graph $G$ can have infinite degree, even if $G$ is locally finite.
\vspace{6pt}
\begin{question}\label{Q2}~\cite{mohar}
Does an infinite graph exist that has a unique Hamilton circle and degree $r > 2$ at every vertex as well as vertex-degree $r$ at every end?
\end{question}
Our result shows that, in contrast to Conjecture~\ref{sheehan} and its known cases, there are infinite graphs having the same degree at every vertex and end while being Hamiltonian in a unique way.
\begin{theorem}\label{Q_mohar_Yes}
There exists an infinite connected graph $G$ with a unique Hamilton circle that has degree $3$ at every vertex and vertex- as well as edge-degree $3$ at every end.
\end{theorem}
\noindent So with Theorem~\ref{Q_mohar_Yes} we answer Question~\ref{Q2} positively and, therefore, disprove the modified version of Conjecture~\ref{sheehan} for infinite graphs in the way Mohar suggested by considering degrees of both, vertices and ends.
The rest of this paper is structured as follows.
In Section~2 we establish all necessary notation and terminology for this the paper.
We also list some lemmas that will serve as auxiliary tools for the proofs of the main theorems.
Section~3 is dedicated to Theorem~\ref{top_catp_HC} where at the beginning of that section we discuss how one can sensibly extend Corollary~\ref{fin_cater_impl} and which problems arise when we try to extend Theorem~\ref{fin_cater} in a similar way.
In Section~4 we present a proof of Theorem~\ref{HC_K_4-K_2,3}.
Afterwards we describe how a different proof of this theorem works which copies the ideas of a proof of statement~(i) of Proposition~\ref{summary}.
We conclude this section by comparing the two proofs.
The last section, Section~5, contains the construction of a graph witnessing Theorem~\ref{Q_mohar_Yes}.
\section{Preliminaries}
When we mention a graph in this paper we always mean an undirected and simple graph.
For basic facts and notation about finite as well as infinite graphs we refer the reader to \cite{diestel_buch}.
For a broader survey about locally finite graphs and a topological approach to them see \cite{diestel_arx}.
Now we list important notions and concepts that we shall need in this paper followed by useful statements about them.
In a graph $G$ with a vertex $v$ we denote by $\delta(v)$ the set of edges incident with $v$ in $G$.
Similarly, for a subgraph $H$ of $G$ or just its vertex set we denote by $\delta(H)$ the set of edges that have only one endvertex in $H$.
Although formally different, we will not always distinguish between a cut $\delta(H)$ and the partition $(V(H), V(G) \setminus V(H))$ it is induced by.
For two vertices $v, w \in V(G)$ let $d_G(v,w)$ denote the distance between $v$ and $w$ in $G$.
We call a finite graph \textit{outerplanar} if it can be embedded in the plane such that all vertices lie on the boundary of a common face.
For a graph $G$ and an integer $k \geq 2$ we define the $k$\textit{-th power} of $G$ as the graph obtained by taking $G$ and adding additional edges $vw$ for any two vertices $v, w \in V(G)$ such that $1 < d_G(v, w) \leq k$.
A tree is called a \textit{caterpillar} if after the deletion of its leaves only a path is left.
We denote by $S(K_{1, 3})$ the graph obtained by taking the star with three leaves $K_{1, 3}$ and subdividing each edge once.
We call a graph \textit{locally finite} if each vertex has finite degree.
A one-way infinite path in a graph $G$ is called a \textit{ray} of $G$, while we call a two-way infinite path in $G$ a \textit{double ray} of $G$.
Every ray contains a unique vertex that has degree $1$ it.
We call this vertex the \textit{start vertex} of the ray.
An equivalence relation can be defined on the set of rays of a graph $G$ by saying that two rays are equivalent if and only if they cannot be separated by finitely many vertices in $G$.
The equivalence classes of this relation are called the \textit{ends} of $G$.
We denote the set of all ends of a graph $G$ by $\Omega(G)$.
The union of a ray $R$ with infinitely many disjoint paths $P_i$ for $i \in \mathbb{N}$ each having precisely one endvertex on $R$ is called a \textit{comb}.
We call the endvertices of the paths $P_i$ that do not lie on $R$ and those vertices $v$ for which there is a $j \in \mathbb{N}$ such that $v = P_j$ the \textit{teeth} of the comb.
The following lemma is a basic tool for infinite graphs.
Especially for locally finite graphs it helps us to get a comb whose teeth lie in a previously fixed infinite set of vertex.
\begin{lemma}\label{star-comb}\cite[Prop.\ 8.2.2]{diestel_buch}
Let $U$ be an infinite set of vertices in a connected graph $G$.
Then $G$ contains either a comb with all teeth in $U$ or a subdivision of an infinite star with all leaves in $U$.
\end{lemma}
For a locally finite and connected graph $G$ we can endow $G$ together with its ends with a topology that yields the space $|G|$.
A precise definition of $|G|$ can be found in \cite[Ch.\ 8.5]{diestel_buch}.
Let us point out here that a ray of $G$ converges in $|G|$ to the end of $G$ it is contained in.
Another way of describing $|G|$ is to endow $G$ with the topology of a $1$-complex and then forming the Freudenthal compactification \cite{freud-equi}.
For a point set $X$ in $|G|$, we denote its closure in $|G|$ by $\overline{X}$.
We shall often write $\overline{M}$ for some $M$ that is a set of edges or a subgraph of $G$.
In this case we implicitly assume to first identify $M$ with the set of points in $|G|$ which corresponds to the edges and vertices that are contained in $M$.
We call a subspace $Z$ of $|G|$ \textit{standard} if $Z = \overline{H}$ for a subgraph $H$ of $G$.
A \textit{circle} in $|G|$ is the image of a homeomorphism having the unit circle $S^1$ in $\mathbb{R}^2$ as domain and mapping into $|G|$.
Note that all finite cycles of a locally finite connected graph $G$ correspond to circles in $|G|$, but there might also be infinite subgraphs $H$ of $G$ such that $\overline{H}$ is a circle in $|G|$.
Similar to finite graphs we call a locally finite connected graph $G$ \textit{Hamiltonian} if there exists a circle in $|G|$ which contains all vertices of $G$.
Such circles are called \textit{Hamilton circles} of $G$.
We call the image of a homeomorphism with the closed real unit interval $[0, 1]$ as domain and mapping into $|G|$ an \textit{arc} in $|G|$.
Given an arc $\alpha$ in $|G|$, we call a point $x$ of $|G|$ an \textit{endpoint} of $\alpha$ if $0$ or $1$ is mapped to $x$ by the homeomorphism defining~$\alpha$.
If the endpoint of an arc corresponds to a vertex of the graph, we also call the endpoint an \textit{endvertex} of the arc.
Similarly as for paths, we call an arc an $x$--$y$ arc if $x$ and $y$ are the endpoints of the arc.
Possibly the simplest example of a nontrivial arc is a ray together with the end it converges to.
However, the structure of arcs is more complicated in general and they might contain up to $2^{\aleph_0}$ many ends.
We call a subspace $X$ of $|G|$ \textit{arc-connected} if for any two points $x$ and $y$ of $X$ there is an $x$--$y$ arc in $X$.
Using the notions of circles and arc-connectedness we now extend trees in a similar topological way.
We call an arc-connected standard subspace of $|G|$ a \textit{topological tree} if it does not contain any circle.
Note that, similar as for finite trees, for any two points $x, y$ of a topological tree there is a unique $x$--$y$ arc in that topological tree.
Generalizing the definition of caterpillars, we call a topological tree $\overline{T}$ in $|G|$ a \textit{topological caterpillar} if $\overline{T-L}$ is an arc, where $T$ is a forest in $G$ and $L$ denotes the set of all leaves of $T$, i.e., vertices of degree $1$ in~$T$.
Now let $\omega$ be an end of a locally finite connected graph $G$.
We define the \textit{vertex-} or \textit{edge-degree of $\omega$ in $G$} as the supremum of the number of vertex- or edge-disjoint rays in $G$, respectively, which are contained in $\omega$.
By this definition ends may have infinite vertex- or edge-degree.
Similarly, we define the \textit{vertex-} or \textit{edge-degree of $\omega$ in a standard subspace $X$ of $|G|$} as the supremum of vertex- or edge-disjoint arcs in $X$, respectively, that have $\omega$ as an endpoint.
We should mention here that the supremum is actually an attained maximum in both definitions.
Furthermore, when we consider the whole space $|G|$ as a standard subspace of itself, the vertex-degree in $G$ of any end $\omega$ of $G$ coincides with the vertex-degree in $|G|$ of $\omega$.
The same holds for the edge-degree.
The proofs of these statements are nontrivial and since it is enough for us to work with the supremum, we will not go into detail here.
We make one last definition with respect to end degrees which allows us to distinguish the parity of degrees of ends when they are infinite.
The idea of this definition is due to Bruhn and Stein~\cite{circle}.
We call the vertex- or edge-degree of an end~$\omega$ of $G$ in a standard subspace $X$ of $|G|$ \textit{even} if there is a finite set $S \subseteq V(G)$ such that for every finite set $S' \subseteq V(G)$ with $S \subseteq S'$ the maximum number of vertex- or edge-disjoint arcs in $X$, respectively, with $\omega$ as endpoint and some $s \in S'$ is even. Otherwise, we call the vertex- or edge-degree of $\omega$ in~$X$, respectively,~\textit{odd}.
Next we collect some useful statements about the space $|G|$ for a locally finite connected graph $G$.
\begin{proposition}\label{compact}\cite[Prop.\ 8.5.1]{diestel_buch}
If $G$ is a locally finite connected graph, then $|G|$ is a compact Hausdorff space.
\end{proposition}
Having Proposition~\ref{compact} in mind the following basic lemma helps us to work with continuous maps and to verify homeomorphisms, for example when considering circles or arcs.
\begin{lemma}\label{invers_cont}
Let $X$ be a compact space, $Y$ be a Hausdorff space and $f: X \longrightarrow Y$ be a continuous injection. Then $f^{-1}$ is continuous too.
\end{lemma}
The following lemma tells us an important combinatorial property of arcs.
To state the lemma more easily, let $\mathring{F}$ denote the set of inner points of edges $e \in F$ in $|G|$ for an edge set $F \subseteq E(G)$.
\begin{lemma}\label{jumping-arc}\cite[Lemma 8.5.3]{diestel_buch}
Let $G$ be a locally finite connected graph and \linebreak ${F \subseteq E(G)}$ be a cut with sides $V_1$ and $V_2$.
\textnormal{
\begin{enumerate}[\normalfont(i)]
\item \textit{If $F$ is finite, then $\overline{V_1} \cap \overline{V_2} = \emptyset$, and there is no arc in $|G| \setminus \mathring{F}$ with one endpoint in $V_1$ and the other in $V_2$.}
\item \textit{If $F$ is infinite, then $\overline{V_1} \cap \overline{V_2} \neq \emptyset$, and there may be such an arc.}
\end{enumerate}
}
\end{lemma}
The next lemma ensures that connectedness and arc-connectedness are equivalent for the spaces we are mostly interested in, namely standard subspaces, which are closed by definition.
\begin{lemma}\label{arc_conn}\cite[Thm.\ 2.6]{path-cyc-tree}
If $G$ is a locally finite connected graph, then every closed topologically connected subset of $|G|$ is arc-connected.
\end{lemma}
We continue in the spirit of Lemma~\ref{jumping-arc} by characterising important topological properties of the space $|G|$ in terms of combinatorial ones.
The following lemma deals with arc-connected subspaces.
It will be convenient for us to use this in a proof later on.
\begin{lemma}\label{top_conn}\cite[Lemma 8.5.5]{diestel_buch}
If $G$ is a locally finite connected graph, then a standard subspace of $|G|$ is topologically connected (equivalently: arc-connected) if and only if it contains an edge from every finite cut of $G$ of which it meets both sides.
\end{lemma}
The next theorem is actually part of a bigger one containing more equivalent statements.
Since we shall need only one equivalence, we reduced it to the following formulation.
For us it will be helpful to check or at least bound the degree of an end in a standard subspace just by looking at finite cuts instead of dealing with the homeomorphisms that actually define the relevant arcs.
\begin{theorem}\label{cycspace}\cite[Thm.\ 2.5]{diestel_arx}
Let $G$ be a locally finite connected graph. Then the following are equivalent for $D \subseteq E(G)$:
\begin{enumerate}[\normalfont(i)]
\item $D$ meets every finite cut in an even number of edges.
\item Every vertex of $G$ has even degree in $\overline{D}$ and every end of $G$ has even edge-degree in $\overline{D}$.
\end{enumerate}
\end{theorem}
The following lemma gives us a nice combinatorial description of circles and will be especially useful in combination with Theorem~\ref{cycspace} and Lemma~\ref{top_conn}.
\begin{lemma}\label{circ}\cite[Prop.\ 3]{circle}
Let $C$ be a subgraph of a locally finite connected graph~$G$. Then $\overline{C}$ is a circle if and only if $\overline{C}$ is topologically connected, every vertex in $\overline{C}$ has degree $2$ in $\overline{C}$ and every end of $G$ contained in $\overline{C}$ has edge-degree $2$ in $\overline{C}$.
\end{lemma}
A basic fact about finite Hamiltonian graphs is that they are always $2$-connected.
For locally finite connected graphs this is also a well-known fact, although it has not separately been published.
Since we shall need this fact later and can easily deduce it from the lemmas above, we include a proof here.
\begin{corollary}\label{2-con}
Every locally finite connected Hamiltonian graph is $2$-connected.
\end{corollary}
\begin{proof}
Let $G$ be a locally finite connected Hamiltonian graph and suppose for a contradiction that it is not $2$-connected.
Fix a subgraph $C$ of $G$ whose closure $\overline{C}$ is a Hamilton circle of $G$ and a cut vertex $v$ of $G$.
Let $K_1$ and $K_2$ be two different components of $G-v$.
By Theorem~\ref{cycspace} the circle $\overline{C}$ uses evenly many edges of each of the finite cuts $\delta(K_1)$ and $\delta(K_2)$.
Since $\overline{C}$ is a Hamilton circle and, therefore, topologically connected, we also get that it uses at least two edges of each of these cuts by Lemma~\ref{top_conn}.
This implies that $v$ has degree at least $4$ in $C$, which contradicts Lemma~\ref{circ}.
\end{proof}
\section{Topological caterpillars}
In this section we close a gap with respect to the general question of when the $k$-th power of a graph has a Hamilton circle.
Let us begin by summarizing the results in this field.
We start with finite graphs.
The first result to mention is the famous theorem of Fleischner, Theorem~\ref{fin_fleisch}, which deals with $2$-connected graphs.
For higher powers of graphs the following theorem captures the whole situation.
\begin{theorem}\label{fin_3-con}\cite{karaganis, sekanina}
The cube of any finite connected graph on at least three vertices is Hamiltonian.
\end{theorem}
These theorems leave the question whether and when one can weaken the assumption of being $2$-connected and still maintain the property of being Hamiltonian.
Theorem~\ref{fin_cater} gives an answer to this question.
Now let us turn our attention towards locally finite infinite graphs.
As mentioned in the introduction, Georgakopoulos has completely generalized Theorem~\ref{fin_fleisch} to locally finite graphs by proving Theorem~\ref{inf-fleisch}.
Furthermore, he also gave a complete generalization of Theorem~\ref{fin_3-con} to locally finite graphs with the following theorem.
\begin{theorem}\label{inf-3-con}\cite[Thm.\ 5]{agelos}
The cube of any locally finite connected graph on at least three vertices is Hamiltonian.
\end{theorem}
What is left and what we do in the rest of this section is to prove lemmas about locally finite graphs covering implications similar to those in Theorem~\ref{fin_cater}, and mainly Theorem~\ref{top_catp_HC}, which extends Corollary~\ref{fin_cater_impl} to locally finite graphs.
Let us first consider a naive way of extending Theorem~\ref{fin_cater} and Corollary~\ref{fin_cater_impl} to locally finite graphs.
Since we consider spanning caterpillars for Corollary~\ref{fin_cater_impl}, we need a definition of these objects in infinite graphs that allows them to contain infinitely many vertices.
So let us modify the definition of caterpillars as follows:
A locally finite tree is called a \textit{caterpillar} if after deleting its leaves only a finite path, a ray or a double ray is left.
Using this definition Theorem~\ref{fin_cater} remains true for locally finite infinite trees $T$ and Hamilton circles in $|T^2|$.
The same proof as the one Harary and Schwenk~\cite[Thm.\ 1]{cat_HC} gave for Theorem~\ref{fin_cater} in finite graphs can be used to show this.
Corollary~\ref{fin_cater_impl} remains also true for locally finite graphs using this adapted definition of caterpillars.
Its proof, however, is no trivial deduction anymore.
The problem is that for a spanning tree $T$ of a locally finite connected graph $G$ the topological spaces $|T^2|$ and $|G^2|$ might differ not only in inner points of edges but also in ends.
More precisely, there might be two equivalent rays in $G^2$ that belong to different ends of $T^2$.
So the Hamiltonicity of $T^2$ does not directly imply the one of $G^2$.
However, for $T$ being a spanning caterpillar of $G$, this problem can only occur when $T$ contains a double ray such that all subrays belong to the same end of $G$.
Then the same construction as in the proof for the implication from~(iii) to (i) of Theorem~\ref{fin_cater} can be used to build a spanning double ray in $T^2$ which is also a Hamilton circle in $|G^2|$.
The idea for the construction which is used for this implication is covered in Lemma~\ref{order}.
The downside of this naive extension is the following.
For a locally finite infinite graph the assumption of having a spanning caterpillar is quite restrictive.
Such graphs can especially have at most two ends since having three ends would imply that the spanning caterpillar must contain three disjoint rays.
This, however, is impossible because it would force the caterpillar to contain a $S(K_{1, 3})$.
For this reason we have defined a topological version of caterpillars, namely topological caterpillars.
Their definition allows graphs with arbitrary many ends to have a spanning topological caterpillar.
Furthermore, it yields with Theorem~\ref{top_catp_HC} a more relevant extension of Corollary~\ref{fin_cater_impl} to locally finite graphs.
We briefly recall the definition of topological caterpillars.
Let $G$ be a locally finite connected graph.
A topological tree $\overline{T}$ in $|G|$ is a topological caterpillar if $\overline{T-L}$ is an arc, where $T$ is a forest in $G$ and $L$ denotes the set of all leaves of $T$, i.e., vertices of degree $1$ in $T$.
The following basic lemma about topological caterpillars is easy to show and so we omit its proof.
It is an analogue of the equivalence of the statements~(ii) and~(iii) of Theorem~\ref{fin_cater} for topological caterpillars.
\begin{lemma}\label{_inf_forbd_sub}
Let $G$ be a locally finite connected graph.
A topological tree $\overline{T}$ in $|G|$ is a topological caterpillar if and only if $T$ does not contain $S(K_{1,3})$ as a subgraph and all ends of $G$ have vertex-degree in $\overline{T}$ at most $2$.
\end{lemma}
Before we completely turn towards the preparation of the proof of Theorem~\ref{top_catp_HC} let us consider statement~(i) of Theorem~\ref{fin_cater} again.
A complete extension of Theorem~\ref{fin_cater} to locally finite graphs using topological caterpillars seems impossible because of statement~(i).
To see this we should first make precise what the adapted version of statement~(i) most possibly should be.
In order to state it let $G$ denote a locally finite connected graph and let $\overline{T}$ be a topological tree in $|G|$.
Now the formulation of the adapted statement should be as follows:
\vspace{6pt}
\textnormal{
\begin{enumerate}[\normalfont(i*)]
\item \textit{In the subspace $\overline{T^2}$ of $|G^2|$ is a circle containing all vertices of $T$.}
\end{enumerate}
}
\vspace{6pt}
\noindent This statement does not hold if $T$ has more than one graph theoretical component.
Therefore, it cannot be equivalent to $\overline{T}$ being a topological caterpillar in $|G|$, which is the adapted version of statement~(iii) of Theorem~\ref{fin_cater} for locally finite graphs.
Note that any two vertices of $T$ lie in the same graph theoretical component of $T$ if and only if they lie in the same graph theoretical component of~$T^2$.
Hence, we can deduce that statement~(i*) fails if $T$ has more than one graph theoretical component from the following claim.
\begin{claim}
Let $G$ be a locally finite connected graph and let $\overline{T}$ be a topological tree in~$|G|$.
Then there is no circle in the subspace $\overline{T^2}$ of $|G^2|$ that contains vertices from different graph theoretical components of $T^2$.
\end{claim}
\begin{proof}
We begin with a basic observation.
The inclusion map from $G$ into $G^2$ induces an embedding from $|G|$ into $|G^2|$ in a canonical way.
Moreover, all ends of $G^2$ are contained in the image of this embedding.
To see this note that any two non-equivalent rays in $G$ stay non-equivalent in $G^2$ since $G$ is locally finite.
Furthermore, by applying Lemma~\ref{star-comb} it is easy to see that every end in $G^2$ contains a ray that is also a ray of $G$.
This already yields an injection from $|G|$ to $|G^2|$ whose image contains all of $\Omega(G^2)$.
Verifying the continuity of this map and its inverse is immediate.
Now let us suppose for a contradiction that there is a circle $C$ in $\overline{T^2}$ containing vertices $v, v'$ from two different graph theoretical components $K, K'$ of $T^2$.
Say $v \in V(K)$ and $v' \in V(K')$.
Let $A_1$ and $A_2$ denote the two $v'$--$v$ arcs on $C$.
Since $A_1$ and $A_2$ are disjoint except from their endpoints, they have to enter $K$ via different ends $\omega^2_1$ and $\omega^2_2$ of $G^2$ that are contained in $\overline{K} \subseteq |G^2|$.
Say $\omega^2_1 \in A_1$ and $\omega^2_2 \in A_2$.
By the observation above $\omega^2_1$ and $\omega^2_2$ correspond to two different ends $\omega_1$ and $\omega_2$ of~$G$.
Only one of them, say~$\omega_1$, lies on the unique $v'$--$v$ arc that is contained in the topological tree $\overline{T}$.
Now we modify $A_2$ by replacing each edge $uw$ of $A_2$ which is not in $E(T)$ by a $u$--$w$ path of length $2$ that lies in $T$.
By Lemma~\ref{top_conn} this yields an arc-connected subspace of $\overline{T}$ that contains $v$ and $v'$.
By our observation above the unique $v'$--$v$ arc in this subspace must contain the end $\omega_2$.
This, however, is a contradiction since we have found two different $v'$--$v$ arcs in $\overline{T}$.
\end{proof}
Now we start preparing the proof of Theorem~\ref{top_catp_HC}.
For this we define a certain partition of the vertex set of a topological caterpillar.
Additionally, we define a linear order of these partition classes.
Let $G$ be a locally finite connected graph and $\overline{T}$ a topological caterpillar in~$|G|$.
Furthermore, let $L$ denote the set of leaves of $T$.
By definition, $\overline{T-L}$ is an arc, call it $A$.
This arc induces a linear order $<_A$ of the vertices of $V(T)-L$.
For consecutive vertices $v, w \in V(T)-L$ with $v <_A w$ we now define the set
\[{P_w := \lbrace w \rbrace \cup (N_T(v) \cap L)} \]
(cf.~Figure~\ref{partition}).
If $A$ has a maximal element $m$ with respect to $<_A$, we define an additional set $P^+ = N_T(m) \cap L$.
Should $A$ have a minimal element $s$ with respect to $<_A$, we define another additional set $P^- = \lbrace s \rbrace$.
The sets $P_w$, possibly together with $P^+$ and $P^-$, form a partition $\mathcal{P}_T$ of $V(T)$.
For any $v \in V(T)$ we denote the corresponding partition class containing $v$ by $V_v$.
Next we use the linear order $<_A$ to define a linear order $<_T$ on $\mathcal{P}_T$.
For any two vertices ${v, w \in V(T)-L}$ with $v <_A w$ set $V_v <_T V_w$.
If $P^+$ (resp.~$P^-$) exists, set $P_v <_T P^+$ (resp.~${P^- <_T P_v}$) for every ${v \in V(T)-L}$.
Finally we define for two vertices~${v, w \in V(T)}$ with $V_v \leq_T V_w$ the set
\[I_{vw} := \bigcup \lbrace V_u \; ; \; V_v \leq_T V_u \leq_T V_w \rbrace. \]
\begin{figure}[htbp]
\centering
\includegraphics{partition.pdf}
\caption{The partition classes $P_w$.}
\label{partition}
\end{figure}
The following basic lemma lists important properties of the partition $\mathcal{P}_T$ together with its order $<_T$.
The proof of this lemma is immediate from the definitions of $\mathcal{P}_T$ and $<_T$.
Especially for Lemma~\ref{decomp} and in the proof of Theorem~\ref{top_catp_HC} the listed properties will be applied intensively.
Furthermore, the proof that statement~(iii) of Theorem~\ref{fin_cater} implies statement~(i) of Theorem~\ref{fin_cater} follows easily from this lemma.
\begin{lemma}\label{order}
Let $\overline{T}$ be a topological caterpillar in $|G|$ for a locally finite connected graph~$G$. Then the partition $\mathcal{P}_T$ of $V(T)$ has the following properties:
\begin{enumerate}[\normalfont(i)]
\item Any two different vertices belonging to the same partition class of $\mathcal{P}_T$ have distance~$2$ from each other in $T$.
\item For consecutive partition classes $Q$ and $R$ with $Q <_T R$, there is a unique vertex in $Q$ that has distance $1$ in $T$ to every vertex of $R$.
For $Q \neq P^-$, this vertex is the one of $Q$ that is not a leaf of $T$.
\end{enumerate}
\end{lemma}
\begin{proof}
\end{proof}
Referring to statement~(ii) of Lemma~\ref{order}, let us call the vertex in a partition class~$Q \in \mathcal{P}_T$ that is not a leaf of $T$ the \textit{jumping vertex} of $Q$.
We still need a bit of notation and preparation work before we can prove the main theorem of this section.
Now let $\overline{T}$ denote a topological caterpillar with only one graph-theoretical component.
Let $(\mathcal{X}_1, \mathcal{X}_2)$ be a bipartition of the partition classes $V_v$ such that consecutive classes with respect to $\leq_T$ lie not both in $\mathcal{X}_1$, or in $\mathcal{X}_2$.
Furthermore, let $v, w \in V(T)$ be two vertices, say with $V_v \leq_T V_w$, whose distance is even in $T$.
We define a $(v, w)$ \textit{square string} $S$ in $T^2$ to be a path in $T^2$ with the following properties:
\begin{enumerate}
\item $S$ uses only vertices of partitions that lie in the bipartition class $\mathcal{X}_i$ in which $V_v$ and $V_w$ lie.
\item $S$ contains all vertices of partition classes $V_u \in \mathcal{X}_i$ for $V_v <_T V_u <_T V_w$.
\item $S$ contains only $v$ and $w$ from $V_v$ and $V_w$, respectively.
\end{enumerate}
Similarly, we define $(v, w]$, $[v, w)$ and $[v, w]$ square strings in $T^2$, but with the difference in (3) that they shall also contain all vertices of $V_w$, $V_v$ and $V_v \cup V_w$, respectively.
We call the first two types of square strings \textit{left open} and the latter ones \textit{left closed}.
The notion of being \textit{right open} and \textit{right closed} is analogously defined.
From the properties of $\mathcal{P}_T$ listed in Lemma~\ref{order}, it is immediate how to construct square strings.
The next lemma gives us two possibilities to cover the vertex set of a graph-theoretical component of a topological caterpillar $\overline{T}$ that contains a double ray.
Each cover will consist of two, possibly infinite, paths of $T^2$.
Later on we will use these covers to connect all graph-theoretical components of $\overline{T}$ in a certain way such that a Hamilton circle of $G^2$ is formed.
\begin{lemma}\label{decomp}
Let $G$ be a locally finite connected graph and let $\overline{T}$ be a topological caterpillar in $|G|$.
Suppose $T$ has only one graph-theoretical component and contains a double ray.
Furthermore, let $v$ and $w$ be vertices of $T$ with $V_v \leq_T V_w$.
\begin{enumerate}[\normalfont(i)]
\item If $d_T(v, w)$ is even, then in $T^2$ there exist a $v$--$w$ path $P$, a double ray $D$ and two rays $R_v$ and $R_w$ with the following properties:
\begin{itemize}
\item $P$ and $D$ are disjoint as well as $R_v$ and $R_w$.
\item $V(T) = V(P) \cup V(D) = V(R_v) \cup V(R_w)$.
\item $v$ and $w$ are the start vertices of $R_v$ and $R_w$, respectively.
\item $R_v \cap V_x = \emptyset$ for every $V_x >_T V_w$.
\item $R_w \cap V_y = \emptyset$ for every $V_y <_T V_v$.
\end{itemize}
\vspace{5 pt}
\item If $d_T(v, w)$ is odd, then in $T^2$ there exist rays $R_v, R_w, R'_v, R'_w$ with the following properties:
\begin{itemize}
\item $R_v$ and $R_w$ are disjoint as well as $R'_v$ and $R'_w$.
\item $V(T) = V(R_v) \cup V(R_w) = V(R'_v) \cup V(R'_w)$.
\item $v$ is the start vertex of $R_v$ and $R'_v$ while $w$ is the one of $R_w$ and $R'_w$.
\item $R_v \cap V_x = R'_w \cap V_x = \emptyset$ for every $V_x >_T V_w$.
\item $R_w \cap V_y = R'_v \cap V_y = \emptyset$ for every $V_y <_T V_v$.
\end{itemize}
\end{enumerate}
\end{lemma}
\begin{proof}
We sketch the proof of statement (i).
As $v$--$w$ path $P$ we take a square string $S_{vw}$ in $T^2$ with $v$ and $w$ as endvertices.
Depending whether $v$ is a jumping vertex or not we take a left open or closed square string, respectively.
Depending on $w$ we take a right closed or open square string if $w$ is a jumping vertex or not, respectively.
Since $d_T(v, w)$ is even, we can find such square strings.
To construct the double ray $D$ start with a $(v^-, w^-]$ square string in $T^2$ where $v^-$ and $w^-$ denote the jumping vertices in the partition classes proceeding $V_v$ and $V_w$, respectively.
Using the properties~(i) and (ii) of the partition $\mathcal{P}_T$ mentioned in Lemma~\ref{order}, the $(v^-, w^-]$ square string can be extend to a desired double ray $D$ containing all vertices of $T$ that do not lie in $S_{vw}$ (cf.~Figure~\ref{caterpillar_pattern}).
To define $R_v$ we start with a square string $S_v$ having $v$ as one endvertex.
For the definition of $S_v$ we distinguish four cases.
If $v$ and $w$ are jumping vertices, we set $S_v$ as a path obtained by taking a $(v, w]$ square string and deleting $w$ from it.
If $v$ is not a jumping vertex, but~$w$ is one, take a $[v, w]$ square string, delete $w$ from it and set the remaining path as~$S_v$.
In the case that $v$ is a jumping vertex, but $w$ is none, $S_v$ is defined as a path obtained from a $(v, w)$ square string from which we delete $w$.
In the case that neither $v$ nor $w$ is a jumping vertex, we take a $[v, w)$ square string, delete $w$ from it and set the remaining path as $S_v$.
Next we extend $S_v$ using a square string to a path with $v$ as one endvertex containing all vertices in partition classes $V_u$ with $V_v <_T V_u <_T V_w$.
We extend the remaining path to a ray that contains also all vertices in partition classes $V_u$ with $V_u \leq_T V_v$, but none from partition classes $V_x$ for $V_x >_T V_w$.
The desired second ray $R_w$ can now easily be build in $T^2 -R_v$.
The rays for statement (ii) are defined in a very similar way (cf.~Figure~\ref{caterpillar_pattern}).
Therefore, we omit their definitions here.
\end{proof}
\begin{figure}[htbp]
\centering
\includegraphics{caterpillar_pattern_1_new.pdf} \vspace{10pt}
\includegraphics{caterpillar_pattern_4_new.pdf} \vspace{10pt}
\includegraphics{caterpillar_pattern_2_new.pdf} \vspace{10pt}
\includegraphics{caterpillar_pattern_3_new.pdf}
\caption{Examples for covering the vertices of a caterpillar as in Lemma~\ref{decomp}.}
\label{caterpillar_pattern}
\end{figure}
The following lemma is essential for connecting the parts of the vertex covers of two different graph-theoretical components of $\overline{T}$.
Especially, here we make use of the structure of $|G|$ instead of arguing only inside of $\overline{T}$ or $\overline{T^2}$.
This allows us to build a Hamilton circle using square strings and to ``jump over" an end to avoid producing an edge-degree bigger than $2$ at that end.
\begin{lemma}\label{shortcut}
Let $\overline{T}$ be a spanning topological caterpillar of a locally finite connected graph $G$ and let $v, w \in V(G)$ where $V_v \leq_T V_w$.
Then for any two vertices $x, y$ with ${V_v <_T V_x <_T V_w}$ and ${V_v <_T V_y <_T V_w}$ there exists a finite $x$--$y$ path in $G[I_{vw}]$.
\end{lemma}
\begin{proof}
Let the vertices $v, w, x$ and $y$ be as in the statement of the lemma and, as before, let $L$ denote the set of leaves of $T$.
Now suppose for a contradiction that there is no finite $x$--$y$ path in $G[I_{vw}]$.
Then we can find an empty cut $D$ of $G[I_{vw}]$ with sides $M$ and $N$ such that $x$ and $y$ lie on different sides of it.
Since $\overline{T \cap G[I_{vw}]}$ contains an $x$--$y$ arc, there must exist an end $\omega \in \overline{M} \cap \overline{N} \cap \overline{T-L}$.
Let us show next that there exists an open set $O$ in $|G|$ that contains $\omega$ and, additionally, every vertex in $O$ is an element of $I_{vw}$.
To see this we first pick a set $O_A \subseteq \overline{T-L}$ so that it is open in the subspace $\overline{T-L}$, topologically connected and contains $\omega$, but its closure does not contain the jumping vertices of $V_v$ and~$V_w$.
Now let $O'$ be an open set in $|G|$ witnessing that $O_A$ is open in $\overline{T-L}$.
We prove that $O'$ contains only finitely many vertices of $V(G) \setminus I_{vw}$.
Suppose for a contradiction that this is not the case.
Then we would find an infinite sequence $(z_n)_{n \in \mathbb{N}}$ of different vertices in $O' \setminus I_{vw}$ that must converge to some point $p \in |G|$ by the compactness of $|G|$.
Since~$\overline{T}$ is a spanning topological caterpillar of $G$, it contains all the vertices~$z_n$.
Using that $G$ is locally finite, we get that the jumping vertices of the sets $V_{z_n}$ also form a sequence that converges to $p$.
So we can deduce that ${p \in \overline{T-L}}$, because $\overline{T-L}$ is a closed subspace containing all jumping vertices.
Hence, ${p \in \overline{O'} \cap (\overline{T-L}) = \overline{O_A}}$.
This is a contradiction to our choice of $O_A$ ensuring $p \notin \overline{O_A}$.
Hence, $O'$ contains only finitely many vertices of $V(G) \setminus I_{vw}$, say $v_1, \ldots, v_n$ for some $n \in \mathbb{N}$.
Before we define our desired set $O$ using $O'$, note that ${O_v := |G| \setminus \lbrace v \rbrace}$ defines an open set in $|G|$ for every vertex $v \in V(G)$.
Therefore, ${O := O' \cap \bigcap^n_{i=1} O_{v_i}}$ is an open set in $|G|$ containing no vertex of $V(G) \setminus I_{vw}$.
Inside $O$ we can find a basic open set $B$ around $\omega$, which contains a graph-theoretical connected subgraph with all vertices of $B$.
Now $B$ contains vertices of $M$ and $N$ as well as a finite path between them, which must then also exist in $G[I_{vw}]$.
Such a path would have to cross $D$ contradicting the assumption that $D$ is an empty cut in $G[I_{vw}]$.
\end{proof}
To figure out which parts of the vertex covers of which graph-theoretical components of~$\overline{T}$ we can connect such that afterwards we are still able to extend this construction to a Hamilton circle of $G$, we shall use the next lemma.
For the formulation of the lemma, we use the notion of \textit{splits}.
Let $G$ be a multigraph and $v \in V(G)$.
Furthermore, let $E_1, E_2 \subseteq \delta(v)$ such that $E_1 \cup E_2 = \delta(v)$ but $E_1 \cap E_2 = \emptyset$ where $E_i \neq \emptyset$ for $i \in \lbrace 1, 2 \rbrace$.
Now we call a multigraph $G'$ a $v$\textit{-split} of $G$ if
\[{V(G') = V(G) \setminus \lbrace v \rbrace \cup \lbrace v_1, v_2 \rbrace}\]
with ${v_1, v_2 \notin V(G)}$ and
\[{E(G') = E(G-v) \cup \lbrace v_1w \; ; \; wv \in E_1 \rbrace \cup \lbrace v_2u \; ; \; uv \in E_2 \rbrace}.\]
We call the vertices $v_1$ and $v_2$ \textit{replacement vertices} of $v$.
\begin{lemma}\label{euler_split}
Let $G$ be a finite Eulerian multigraph and $v$ be a vertex of degree $4$ in~$G$.
Then there exist two $v$-splits $G_1$ and $G_2$ of $G$ both of which are also Eulerian.
\end{lemma}
\begin{proof}
There are ${\frac{1}{2} } \cdot {\binom {4} {2}} = 3$ possible non-isomorphic $v$-splits of $G$ such that $v_1$ and $v_2$ have degree $2$ in the $v$-split.
Assume that one of them, call it $G'$, is not Eulerian.
This can only be the case if $G'$ is not connected.
Let $(A, B)$ be an empty cut of $G'$.
Note that $G-v$ has precisely two components $C_1$ and $C_2$ since $G$ is Eulerian and $v$ has degree $4$ in $G$.
So $C_1$ and $C_2$ must lie in different sides of $(A, B)$, say $C_1 \subseteq A$.
Since $G$ was connected, we get that $v_1$ and $v_2$ lie in different sides of the cut $(A, B)$, say $v_1 \in A$.
Therefore, ${A = C_1 \cup \lbrace v_1 \rbrace}$ and ${B = C_2 \cup \lbrace v_2 \rbrace}$.
If ${\delta(v) = \lbrace vw_1, vw_2, vw_3, vw_4 \rbrace}$ and ${\lbrace v_1w_1, v_1w_2 \rbrace , \lbrace v_2w_3, v_2w_4 \rbrace \subseteq E(G')}$, set $G_1$ and $G_2$ as $v$-splits of $G$ such that the inclusions $\lbrace v_1w_1, v_1w_3 \rbrace , \lbrace v_2w_2, v_2w_4 \rbrace \subseteq E(G_1)$ and $\lbrace v_1w_1, v_1w_4 \rbrace , \lbrace v_2w_2, v_2w_3 \rbrace \subseteq E(G_2)$ hold.
Now $G_1$ and $G_2$ are Eulerian, because every vertex has even degree in each of those multigraphs and both multigraphs are connected.
To see the latter statement, note that any empty cut $(X, Y)$ of $G_i$ for $i \in \lbrace 1, 2 \rbrace$ would need to have $C_1$ and $C_2$ on different sides.
If also $v_1$ and $v_2$ are on different sides, we would have $(A, B) = (X, Y)$, which does not define an empty cut of $G_i$ by definition of $G_i$.
However, $v_1$ and $v_2$ cannot lie on the same side of the cut $(X, Y)$.
This is because otherwise the cut $(X, Y)$ would induce an empty cut in $G$ after identifying $v_1$ and $v_2$ in $G_i$.
Since $G$ is Eulerian and therefore especially connected, we would have a contradiction.
\end{proof}
Now we have all tools together to prove Theorem~\ref{top_catp_HC}.
Before we start the proof, let us recall the statement of the theorem.
\setcounter{section}{1}
\setcounter{theorem}{4}
\begin{theorem}
The square of any locally finite connected graph $G$ on at least three vertices such that $|G|$ contains a spanning topological caterpillar is Hamiltonian.
\end{theorem}
\setcounter{theorem}{7}
\setcounter{section}{3}
\begin{proof}
Let $G$ be a graph as in the statement of the theorem and let $\overline{T}$ be a spanning topological caterpillar of $G$.
We may assume by Corollary~\ref{fin_cater_impl} that $G$ has infinitely many vertices.
Now let us fix an enumeration of the vertices, which is possible since every locally finite connected graph is countable.
We inductively build a Hamilton circle of $G^2$ in at most $\omega$ many steps.
We ensure that in each step $i \in \mathbb{N}$ we have two disjoint arcs $\overline{A^i}$ and $\overline{B^i}$ in $|G^2|$ whose endpoints are vertices of subgraphs $A^i$ and $B^i$ of $G^2$, respectively.
Let $a^i_{\ell}$ and $a^i_{r}$ (resp.\ $b^i_{\ell}$ and $b^i_{r}$) denote the endvertices of $\overline{A^i}$ (resp.\ $\overline{B^i}$) such that $V_{a^i_{\ell}} \leq_T V_{a^i_{r}}$ (resp.\ $V_{b^i_{\ell}} \leq_T V_{b^i_{r}}$).
For the construction we further ensure the following properties in each step $i \in \mathbb{N}$:
\begin{enumerate}
\item The vertices $a^i_{r}$ and $b^i_{r}$ are the jumping vertices of $V_{a^i_{r}}$ and $V_{b^i_{r}}$, respectively.
\item The partition sets $V_{a^i_{\ell}}$ and $V_{b^i_{\ell}}$ as well as $V_{a^i_{r}}$ and $V_{b^i_{r}}$ are consecutive with respect to $\leq_T$.
\item If $V_v \cap V(A^i \cup B^i) \neq \emptyset$ holds for any vertex $v \in V(G)$, then $V_v \subseteq V(A^i \cup B^i)$.
\item If for any vertex $v \in V(G)$ there are vertices $u, w \in V(G)$ such that ${V_u, V_w \subseteq V(A^i \cup B^i)}$ and $V_u \leq_T V_v \leq_T V_w$, then $V_v \subseteq V(A^i \cup B^i)$ is true.
\item $A^i \cap A^{i+1} = A^i$ and $B^i \cap B^{i+1} = B^i$, but $V(A^{i+1} \cup B^{i+1})$ contains the least vertex with respect to the fixed vertex enumeration that was not already contained in $V(A^i \cup B^i)$.
\end{enumerate}
We start the construction by picking two adjacent vertices $t$ and $t'$ in $T$ that are no leaves in $T$.
Then $V_t$ and $V_t'$ are consecutive with respect to $\leq_T$.
Note that $G^2[V_t]$ and $G^2[V_{t'}]$ are cliques by property~(i) of the partition $\mathcal{P}_T$ mentioned in Lemma~\ref{order}.
We set $A^1$ to be a Hamilton path of $G^2[V_t]$ with endvertex $t$ and $B^1$ to be one of $G^2[V_{t'}]$ with endvertex $t'$.
This completes the first step of the construction.
Suppose we have already constructed $A^n$ and $B^n$.
Let $v \in V(G)$ be the least vertex with respect to the fixed vertex enumeration that is not already contained in $V(A^n \cup B^n)$.
We know by our construction that either $V_v <_T V_x$ or $V_v >_T V_x$ for every vertex $x \in V(A^n \cup B^n)$.
Consider the second case, since the argument for the first works analogously.
Let $v' \in V(G)$ be a vertex such that $V_{v'}$ is the predecessor of $V_{v}$ with respect to $\leq_T$.
Further, let $w \in V(G)$ be a vertex such that $V_w >_T V_{a^n_{r}}, V_{b^n_{r}}$ and $V_w$ is the successor of either $V_{a^n_{r}}$ or $V_{b^n_{r}}$, say $V_{b^n_{r}}$.
By Lemma~\ref{shortcut} there exists a $v'$--$w$ path $P$ in $G[I_{b^n_r, v}]$.
We may assume that $E(P) \setminus E(T)$ does not contain an edge whose endvertices lie in the same graph-theoretical component of~$T$.
Furthermore, we may assume that every graph-theoretical component of $T$ is incident with at most two edges of~$E(P) \setminus E(T)$.
Otherwise we could modify the path $P$ using edges of $E(T)$ to meet these conditions.
Next we inductively define a finite sequence of finite Eulerian auxiliary multigraphs $H_1, \ldots, H_k$ where $H_k$ is a cycle for some $k \in \mathbb{N}$.
Every vertex in each of these multigraphs will have either degree $2$ or degree $4$.
Furthermore, we shall obtain $H_{i+1}$ from $H_i$ as a $h$-split for some vertex $h \in V(H_i)$ of degree $4$ until we end up with a multigraph $H_k$ that is a cycle.
As $V(H_1)$ take the set of all graph-theoretical components $T_1, \ldots, T_n$ of $T$ that are incident with an edge of $E(P) \setminus E(T)$.
Two vertices $T_i$ and $T_j$ are adjacent if either there is an edge in $E(P) \setminus E(T)$ whose endpoints lie in $T_i$ and $T_j$ or there is a $t_i$--$t_j$ arc $\overline{A}$ in $\overline{T}$ for a subgraph $A$ of $T$ and vertices $t_i \in V(T_i)$ and $t_j \in V(T_j)$ such that no endvertex of any edge of $E(P) \setminus E(T)$ lies in $V(A) \cup N_T(A)$.
Since $\overline{T}$ is a spanning topological caterpillar, the multigraph $H_1$ is connected.
By definition of~$P$, the multigraph $H_1$ is also Eulerian where all vertices have either degree $2$ or~$4$.
Now suppose we have already constructed $H_i$ and there exists a vertex ${h \in V(H_i)}$ with degree $4$ in $H_i$.
Since $H_i$ is obtained from $H_1$ via repeated splitting operations, we know that $h$ is incident with two edges $d, e$ in $H_i$ that correspond to edges~${d_P, e_P}$, respectively, of ${E(P) \setminus E(T)}$.
Furthermore, $h$ is incident with two edges $f, g$ that correspond to arcs $\overline{A_f}$ and $\overline{A_g}$, respectively, of $\overline{T}$ for subgraphs $A_f$ and $A_g$ of $T$ such that neither $V(A_f) \cup N_T(A_f)$ nor $V(A_g) \cup N_T(A_g)$ contain an endvertex of an edge of ${E(P) \setminus E(T)}$.
Let $T_j$ be the graph-theoretical component of $T$ in which each of $d_P$ and $e_P$ has an endvertex, say $w_d$ and $w_e$, respectively.
Here we consider two cases:
\begin{case}
The distance in $T_j$ between $w_d$ and $w_e$ is even.
\end{case}
In this case we define $H_{i+1}$ as a Eulerian $h$-split of $H_i$ such that one of the following two options holds for the edge $d_{i+1}$ in $H_{i+1}$ corresponding to $d$.
The first option is that $d_{i+1}$ is adjacent to the edge in $H_{i+1}$ corresponding to $e$.
The second options is that $d_{i+1}$ is adjacent to the edge in $H_{i+1}$ corresponding to either $f$ or $g$ with the property that the path in $T_j$ connecting~$w_d$ and $A_f$ (resp.~ $A_g$) does not contain $w_e$.
This is possible since two of the three possible non-isomorphic $v$-splits of $H_i$ are Eulerian by Lemma~\ref{euler_split}.
\begin{case}
The distance in $T_j$ between $w_d$ and $w_e$ is odd.
\end{case}
Here we set $H_{i+1}$ as a Eulerian $h$-split of $H_i$ such that the edge in $H_{i+1}$ corresponding to $d$ is not adjacent to the one corresponding to $e$.
As in the first case, this is possible because two of the three possible non-isomorphic $h$-splits of $H_i$ are Eulerian by Lemma~\ref{euler_split}.
This completes the definition of the sequence of auxiliary multigraphs.
\\
Now we use the last auxiliary multigraph $H_k$ of the sequence to define the arcs $\overline{A^{n+1}}$ and $\overline{B^{n+1}}$.
Note that $P$ is a $w$--$v'$ path in $G[I_{b^n_r, v}]$ where $v'$ and $w$ lie in the same graph-theoretical components $T_{v'}$ and $T_w$ of $T$ as $v$ and $b^n_r$, respectively.
Since we may assume that $E(P) \setminus E(T) \neq \emptyset$ holds, let $e \in E(P) \setminus E(T)$ denote the edge which contains one endvertex $w_e$ in $T_w$.
Then either the distance between $w_e$ and $a^n_r$ or between $w_e$ and $b^n_r$ is even, say the latter one holds.
Now we first extend $B^n$ via a $(b^n_r, w_e]$ square string in $T^2$ and $A^n$ by a $(a^n_r, w_e^+]$ square string in~$T^2$ where $V_{w_e^+}$ is the successor of $V_{w_e}$ with respect to $\leq_T$ and $w_e^+$ is the jumping vertex of~$V_{w_e^+}$.
Then we extend $A^n$ further using a ray to contain all vertices of partition classes $V_x$ with $V_x >_T V_{w_e^+}$ for $x \in T_w$.
This is possible due to the properties~(i) and (ii) of the partition $\mathcal{P}_T$ mentioned in Lemma~\ref{order}.
Next let $P_1$ and $P_2$ be the two edge-disjoint $T_{v'}$--$T_w$ paths in $H_k$.
Since every edge of $E(P) \setminus E(T)$ corresponds to an edge of $H_k$, we get that $e$ corresponds either to $P_1$ or $P_2$, say to the former one.
Therefore, we will use $P_1$ to obtain arcs to extend~$B^n$ and $P_2$ for arcs extending $A^n$.
Now we make use of the definition of $H_k$ via splittings.
For any vertex $T_j$ of $H_1$ of degree $4$ we have performed a $T_j$-split.
We did this in such a way that the partition of the edges incident with $T_j$ into pairs of edges incident with a replacement vertex of $T_j$ corresponds to a cover of $V(T_j)$ via two, possibly infinite, paths as in Lemma~\ref{decomp}.
So for every vertex of $H_1$ of degree $4$ we take such a cover.
For every graph-theoretical component $T_m$ of $T$ such that there exist two consecutive edges $T_iT_j$ and $T_jT_{\ell}$ of $P_1$ or $P_2$ that do not correspond to edges of $E(P) \setminus E(T)$ and $V_{t_i} <_T V_{t_m} <_T V_{t_j}$ or $V_{t_j} <_T V_{t_m} <_T V_{t_{\ell}}$ holds for every choice of $t_i \in T_i$, $t_j \in T_j$, $t_{\ell} \in T_{\ell}$ and $t_m \in T_m$, we take a spanning double ray of $T^2_m$.
We can find such spanning double rays by using again the properties~(i) and (ii) of the partition $\mathcal{P}_T$ mentioned in Lemma~\ref{order}.
Since $H_k = P_1 \cup P_2$ is a cycle, we can use these covers and double rays to extend $\overline{A^n}$ and $\overline{B^n}$ to be disjoint arcs~$\alpha^n$ and $\beta^n$ with endvertices on $T_{v'}$.
With the same construction that we have used for extending $A^n$ and $B^n$ on $T_w$, we can extend $\alpha^n$ and $\beta^n$ to have endvertices~$v'_j$ and $v_j$ which are the jumping vertices of $V_{v'}$ and $V_{v}$, respectively.
Additionally, we incorporate that these extensions contain all vertices of partition classes $V_y$ for $y \in T_{v'}$ and $V_y \leq V_v$.
Then we take these arcs as $\overline{A^{n+1}}$ and $\overline{B^{n+1}}$ where $A^{n+1}$ and $B^{n+1}$ are the corresponding subgraphs of $G^2$ whose closures give the arcs.
By setting $a^{n+1}_r$ and $b^{n+1}_r$ to be $v'_j$ and $v_j$, depending on which of the two arcs $\overline{A^{n+1}}$ or $\overline{B^{n+1}}$ ends in these vertices, we have guaranteed all properties from $(1)$ to $(5)$ for the construction.
Now the properties $(3) - (5)$ yield not only that $\overline{A}$ and $\overline{B}$ are disjoint arcs for ${A = \bigcup_{i \in \mathbb{N}} A^i}$ and ${B = \bigcup_{i \in \mathbb{N}} B^i}$, but also that $V(G) = V(A \cup B)$.
If there exists neither a maximal nor minimal partition class with respect to $\leq_T$, the union~$\overline{A \cup B}$ forms a Hamilton circle of $G^2$ by Lemma~\ref{circ}.
Should there exist a maximal partition class, say $V_{a^n_r}$ for some $n \in \mathbb{N}$ with jumping vertex $a^n_r$, the vertex $a^n_r$ will also be an endvertex of~$\overline{A}$.
In this case we connect the endvertices $a^n_r$ and $b^n_r$ of $\overline{A}$ and~$\overline{B}$ via an edge.
Such an edge exists since $V_{a^n_r}$ and $V_{b^n_r}$ are consecutive with respect to $\leq_T$ by property $(2)$ and $a^n_r$ as well as $b^n_r$ are jumping vertices by property $(1)$.
Analogously, we add an edge if there exists a minimal partition class.
Therefore, we can always obtain the desired Hamilton circle of $G^2$.
\end{proof}
\section{Graphs without $K^4$ or $K_{2,3}$ as minor}
We begin this section with a small observation which allows us to strengthen Theorem~\ref{HC_K_4-K_2,3} a bit by forbidding subgraphs isomorphic to a $K^4$ instead of minors.
\begin{lemma}\label{K^4_minor_subgr}
For graphs without $K_{2, 3}$ as a minor it is equivalent to contain a $K^4$ as a minor or as a subgraph.
\end{lemma}
\begin{proof}
One implication is clear.
So suppose for a contradiction we have a graph without a $K_{2, 3}$ as a minor that does not contain $K^4$ as a subgraph but as a subdivision.
Note that containing a $K^4$ as a subdivision is equivalent to containing a $K^4$ as a minor since $K^4$ is cubic.
Consider a subdivided $K^4$ where at least one edge $e$ of the $K^4$ corresponds to a path $P_e$ in the subdivision whose length is at least two.
Let $v$ be an interior vertex of $P_e$ and $a, b$ be the endvertices of $P_e$.
Let the other two branch vertices of the subdivision of $K^4$ be called $c$ and $d$.
Now we take $\lbrace a, b, c, d, v \rbrace$ as branch vertex set of a subdivision of $K_{2,3}$.
The vertices $a$ and $b$ can be joined to $c$ and $d$ by internally disjoint paths using the ones of the subdivision of $K^4$ except the path $P_e$.
Furthermore, the vertex $v$ can be joined to $a$ and $b$ using the paths $vP_ea$ and $vP_eb$.
So we can find a subdivision of $K_{2,3}$ in the whole graph, which contradicts our assumption.
\end{proof}
Before we start with the proof of Theorem~\ref{HC_K_4-K_2,3} we need to prepare two structural lemmas.
The first one will be very convenient for controlling end degrees because it bounds the size of certain separators.
\begin{lemma}\label{struct_1}
Let $G$ be a $2$-connected graph without $K_{2,3}$ as a minor and let $K_0$ be a connected subgraph of $G$.
Then $|N(K_1)| = 2$ holds for every component $K_1$ of $G-(K_0 \cup N(K_0))$.
\end{lemma}
\begin{proof}
Let $K_0$, $G$ and $K_1$ be defined as in the statement of the lemma.
Since $G$ is $2$-connected, we know that $|N(K_1)| \geq 2$ holds.
Now suppose for a contradiction that $N(K_1) \subseteq N(K_0)$ contains three vertices, say $u, v$ and $w$.
Pick neighbours $u_i$,~$v_i$ and $w_i$ of $u, v$ and $w$, respectively, in $K_i$ for $i \in \lbrace 0, 1 \rbrace$.
Furthermore, take a finite tree $T_i$ in $K_i$ whose leaves are precisely $u_i$, $v_i$ and $w_i$ for $i \in \lbrace 0,1 \rbrace$.
This is possible because $K_0$ and $K_1$ are connected.
Now we have a contradiction since the graph $H$ with $V(H) = \lbrace u, v, w \rbrace \cup V(T_0) \cup V(T_1)$ and $E(H) = \bigcup^1_{i=0} (\lbrace uu_i, vv_i, ww_i \rbrace \cup E(T_i))$ forms a subdivision of $K_{2,3}$.
\end{proof}
Let $G$ be a connected graph and $H$ be a connected subgraph of $G$.
We define the operation of \textit{contracting} $H$ \textit{in} $G$ as taking the minor of $G$ which is attained by contracting in $G$ all edges of $H$.
Now let $K$ be any subgraph of $G$.
We denote by $G_K$ the following minor of $G$:
First contract in $G$ each subgraph that corresponds to a component of~$G-K$.
Then delete all multiple edges.
Obviously $G_K$ is connected if $G$ was connected.
We can push this observation a bit further towards $2$-connectedness with the following lemma.
\begin{lemma}\label{fin_2_con}
Let $K$ be a connected subgraph with at least three vertices of a \linebreak $2$-connected graph $G$.
Then $G_K$ is $2$-connected.
\end{lemma}
\begin{proof}
Suppose for a contradiction that $G_K$ is not $2$-connected for some $G$ and $K$ as in the statement of the lemma.
Since $K$ has at least three vertices, we obtain that $G_K$ has at least three vertices too.
So there exists a cut vertex $v$ in $G_K$.
If $v$ is also a vertex of $G$ and, therefore, does not correspond to a contracted component of $G-K$, then $v$ would also be a cut vertex of $G$.
This contradicts the assumption that $G$ is $2$-connected.
Otherwise $v$ corresponds to a contracted component of $G-K$.
Note that two vertices of $G_K$ both of which correspond to contracted components of $G-K$ are never adjacent by definition of $G_K$.
However, $v$ being a cut vertex in $G_K$ must have at least one neighbour in each component of $G_K-v$.
So in particular we get that $v$ separates two vertices, say $x$ and $y$, of $G_K$ that do not correspond to contracted components of $G-K$.
This yields a contradiction because $K$ is connected and, therefore, contains an $x$--$y$ path.
This path still exists in $G_K$ and contradicts the statement that $v$ separates $x$ and $y$ in~$G_K$.
\end{proof}
We shall need another lemma for the proof Theorem~\ref{HC_K_4-K_2,3}.
In that proof we shall construct an embedding of an infinite graph into a fixed closed disk $D$ by first embedding a finite subgraph into $D$.
Then we extend this embedding stepwise to bigger finite subgraphs so that eventually we define an embedding of the whole graph into $D$.
The following lemma will allow us to redraw newly embedded edges as straight lines in each step while keeping the embedding of every edge that was already embedded as a straight line.
Additionally, we will be able to keep the embedding of those edges that are mapped into the boundary of the disk.
\begin{lemma}\label{straight_embed}
Let $G$ be a finite $2$-connected outerplanar graph and $C$ be its Hamilton cycle.
Furthermore, let $\sigma: G \longrightarrow D$ be an embedding of $G$ into a fixed closed disk $D$ such that $C$ is mapped onto the boundary $\partial D$ of $D$.
Then there is an embedding $\sigma^*: G \longrightarrow D$ such that
\textnormal{
\begin{enumerate}[\normalfont(i)]
\item \textit{$\sigma^*(e)$ is a straight line for every $e \in E(G) \setminus E(C)$.}
\item \textit{$\sigma^*(e) = \sigma(e)$ if $e \in E(C)$ or $\sigma(e)$ is a straight line.}
\end{enumerate}
}\end{lemma}
\begin{proof}
We prove the statement by induction on $\ell := |E(G) \setminus E(C)|$.
For $\ell = 0$ we can choose the given embedding $\sigma$ as our desired embedding $\sigma^*$.
Now let $\ell \geq 1$ and suppose $\sigma$ does not already fulfill all properties of $\sigma^*$.
Then there exists an edge $e \in E(G) \setminus E(C)$ such that $\sigma(e)$ is not a straight line.
Hence, $G-e$ is still a $2$-connected outerplanar graph that contains $C$ as its Hamilton cycle.
Also $\sigma \upharpoonright _{G-e}$ is an embedding of $G-e$ into $D$ such that $C$ is mapped onto $\partial D$.
So by the induction hypothesis we get an embedding $\tilde{\sigma}^*$ satisfying~(i)~and~(ii) with respect to $\sigma \upharpoonright _{G-e}$.
Now let $e = uv$ and suppose for a contradiction that we cannot additionally embed $e$ as a straight line between $u$ and $v$.
Then there exists an edge $xy \in E(G-e) \setminus E(C)$ such that $\tilde{\sigma}^*(xy)$ is crossed by the straight line between $u$ and $v$.
Because $\tilde{\sigma}^*(xy)$ is a straight line between $x$ and $y$ by property~(ii), we know that the vertices $u, v, x$ and $y$ are pairwise distinct.
This, however, is a contradiction to $G$ being outerplanar since the cycle $C$ together with the edges $uv$ and $xy$ witness the existence of a $K^4$ minor in $G$ with $u, v, x$ and $y$ as branch sets.
So we can extend $\tilde{\sigma}^*$ by embedding $e = uv$ as a straight line between $u$ and $v$, which yields our desired embedding of $G$ into $D$.
\end{proof}
With the lemmas above we are now prepared to prove Theorem~\ref{HC_K_4-K_2,3}.
We recall the formulation of the theorem.
\setcounter{section}{1}
\setcounter{theorem}{7}
\begin{theorem}
Let $G$ be a locally finite connected graph.
Then the following statements are equivalent:
\begin{enumerate}[\normalfont(i)]
\item $G$ is $2$-connected and contains neither $K^4$ nor $K_{2,3}$ as a minor.
\item $|G|$ has a Hamilton circle $C$ and there exists an embedding of $|G|$ into a closed disk such that $C$ is mapped onto the boundary of the disk.
\end{enumerate}
Furthermore, if statements (i) and (ii) hold, then $|G|$ has a unique Hamilton circle.
\end{theorem}
\setcounter{theorem}{4}
\setcounter{section}{4}
\begin{proof}
First we show that $(ii)$ implies $(i)$.
Since $G$ is Hamiltonian, we know by Corollary~\ref{2-con} that $G$ is $2$-connected.
Suppose for a contradiction that $G$ contains $K^4$ or $K_{2,3}$ as a minor.
Then $G$ has a finite subgraph $H$ which already has $K^4$ or $K_{2,3}$ as a minor.
Now take any finite connected subgraph $K_0$ of $G$ which contains~$H$ and set~${K = G[V(K_0) \cup N(K_0)]}$.
Next let us take an embedding of $|G|$ as in statement~$(ii)$ of this theorem.
It is easy to see using Lemma~\ref{struct_1} that our fixed embedding of $|G|$ induces an embedding of $G_K$ into a closed disk such that all vertices of~$G_K$ lie on the boundary of the disk.
This implies that $G_K$ is outerplanar.
So~$G_K$ can neither contain $K^4$ nor $K_{2,3}$ as a minor by Theorem~\ref{outerplanar_count_char}, which contradicts that $H$ is a subgraph of $G_K$.
Now let us assume $(i)$ to prove the remaining implication.
We set $K_0$ as an arbitrary connected subgraph of $G$ with at least three vertices.
Next we define~${K_{i+1} = G[V(K_i) \cup N(K_i)]}$ for every ${i \geq 0}$.
Inside $G$ we define the vertex sets ${L_{i} = \lbrace v \in V(K_i) \; ; \; N(v) \subseteq V(K_i) \rbrace}$ for every $i \geq 1$.
Let then ${\tilde{K}_{i+1} = G_{K_{i+1}} - L_i}$ for every ${i \geq 1}$.
By Lemma~\ref{fin_2_con} we know that $G_{K_i}$ is $2$-connected for each ${i \geq 0}$.
Furthermore, $G_{K_i}$ contains neither $K^4$ nor $K_{2,3}$ as a minor for every ${i \geq 0}$ since it would also be a minor of $G$ contradicting our assumption.
So each $G_{K_i}$ is outerplanar by Theorem~\ref{outerplanar_count_char}.
Using statement~(ii) of Proposition~\ref{summary} we obtain that each $G_{K_i}$ has a unique Hamilton cycle $C_i$ and that there is an embedding $\sigma_i$ of $G_{K_i}$ into a fixed closed disk $D$ such that $C_i$ is mapped onto the boundary~$\partial D$ of $D$.
Set ${E_i = E(C_i) \cap E(K_i)}$ for every ${i \geq 1}$.
Next we define an embedding of $G$ into $D$ and extend it to the desired embedding of $|G|$.
We start by taking $\sigma_1$.
Note again that $G_{K_1}$ is a finite $2$-connected outerplanar graph by Lemma~\ref{fin_2_con}.
Furthermore, $\sigma_1(C_1) = \partial D$ .
So we can use Lemma~\ref{straight_embed} to obtain an embedding $\sigma^*_1: G_{K_1} \longrightarrow D$ as in the statement of that lemma.
Because of Lemma~\ref{struct_1} we can extend $\sigma^*_1 \upharpoonright _{K_1}$ using $\sigma_2 \upharpoonright _{\tilde{K}_2}$, maybe after rescaling the latter embedding, to obtain an embedding $\varphi_2: G_{K_2} \longrightarrow D$ such that $\varphi_2(C_2) = \partial D$.
We apply again Lemma~\ref{straight_embed} with $\varphi_2$, which yields an embedding $\sigma^*_2: G_{K_2} \longrightarrow D$ as in the statement of that lemma.
Note that this construction ensures $\sigma^*_2 \upharpoonright _{K_1} = \sigma^*_1 \upharpoonright _{K_1}$.
Proceeding in the same way, we get an embedding $\sigma^*:G \longrightarrow D$ by setting $\sigma^* := \bigcup_{i \in \mathbb{N}} \sigma^*_i \upharpoonright _{K_i}$.
The use of Lemma~\ref{straight_embed} in the construction of $\sigma^*$ ensures that all edges are embedded as straight lines unless they are contained in any $E_i$.
However, all edges in the sets $E_i$, and therefore also all vertices of $G$, are embedded into $\partial D$.
Furthermore, we may assure that $\sigma^*$ has the following property:
\begin{table}[h]
\centering
\begin{tabular}{cl}
\begin{minipage}{0.9\textwidth}
\textit{Let $(M_i)_{i \geq 1}$ be any infinite sequence of components $M_i$ of ${G - K_i}$ where ${M_{i+1} \subseteq M_i}$. Also, let $\lbrace u_i, w_i \rbrace$ be the neighbourhood of $M_i$ in $G$.
Then the sequences $(\sigma^*(u_i))_{i \geq 1}$ and $(\sigma^*(w_i))_{i \geq 1}$ converge to a common point on $\partial D$.}
\end{minipage} & $(\ast)$
\end{tabular}
\end{table}
It remains to extend this embedding $\sigma^*$ to an embedding $\overline{\sigma}^*$ of all of $|G|$ into $D$.
First we shall extend the domain of $\sigma^*$ to all of $|G|$.
For this we need to prove the following claim.
\begin{claim}
For every end $\omega$ of $G$ there exists an infinite sequence $(M_i)_{i \geq 1}$ of components $M_i$ of ${G - K_i}$ with ${M_{i+1} \subseteq M_i}$ such that $\bigcap_{i \geq 1}\overline{M_i} = \lbrace \omega \rbrace$.
\end{claim}
Since $K_i$ is finite, there exists a unique component of ${G - K_i}$ in which all $\omega$-rays have a tail.
Set this component as $M_i$.
It follows from the definition that $\omega$ lies in~$\overline{M_i}$.
Furthermore, we get that $\bigcap_{i \geq 1}\overline{M_i}$ does neither contain any vertex nor an inner point of any edge.
So suppose for a contradiction that $\bigcap_{i \geq 1}\overline{M_i}$ contains another end~$\omega' \neq \omega$.
We know there exists a finite set $S$ of vertices such that all tails of $\omega$-rays lie in a different component of $G-S$ than all tails of $\omega'$-rays.
By definition of the graphs~$K_i$ we can find an index $j$ such that $S \subseteq V(K_j)$.
So $\omega$ lies in $\overline{M_j}$ and $\omega'$ in~$\overline{M'_j}$ where~$M'_j$ is the component of $G-K_j$ in which all tails of $\omega'$-rays lie.
Since $G$ is locally finite, the cut $E(M_j, K_j)$ is finite.
Using Lemma~\ref{jumping-arc} we obtain that $\overline{M_j} \cap \overline{M'_j} = \emptyset$.
Therefore, $\omega' \notin \overline{M_j} \supseteq \bigcap_{i \geq 1}\overline{M_i}$.
This contradiction completes the proof of the claim.
\newline
Now let us define the map $\overline{\sigma}^*$.
For every vertex or inner point of an edge $x$, we set~$\overline{\sigma}^*(x) = \sigma^*(x)$.
For an end $\omega$ let $(M_i)_{i \geq 1}$ be the sequence of components $M_i$ of~${G - K_i}$ given by Claim~1 and $\lbrace u_i, w_i \rbrace$ be the neighbourhood of $M_i$ in $G$.
Using property~$(\ast)$ we know that $(\sigma^*(u_i))_{i \geq 1}$ and $(\sigma^*(w_i))_{i \geq 1}$ converge to a common point~$p_{\omega}$ on $\partial D$.
We use this to set $\overline{\sigma}^*(\omega) = p_{\omega}$.
This completes the definition of~$\overline{\sigma}^*$.
Next we prove the continuity of $\overline{\sigma}^*$.
For every vertex or inner point of an edge $x$, it is easy to see that an open set around $\overline{\sigma}^*(x)$ in $D$ contains $\overline{\sigma}^*(U)$ for some open set~$U$ around $x$ in $|G|$.
This holds because $G$ is locally finite and so it follows from the definition of $\overline{\sigma}^*$ using the embeddings $\sigma^*_i$.
Let us check continuity for ends.
Consider an open set $O$ around $\overline{\sigma}^*(\omega)$ in $D$, where $\omega$ is an end of $G$.
Let $B_{\varepsilon}(\overline{\sigma}^*(\omega))$ denote the restriction to $D$ of an open ball around $\overline{\sigma}^*(\omega)$ with radius $\varepsilon > 0$.
Then $B_{\varepsilon}(\overline{\sigma}^*(\omega))$ is an open set and, for sufficiently small $\varepsilon$, contained in~$O$.
We fix such an $\varepsilon$ for the rest of this proof.
Let $(M_i)_{i \geq 1}$ be a sequence as in Claim~1 for $\omega$ and $\lbrace u_i, w_i \rbrace$ be the neighbourhood of $M_i$ in $G$.
By property $(\ast)$ and the definition of $\overline{\sigma}^*$, we get that $(\sigma^*(u_i))_{i \geq 1}$ and $(\sigma^*(w_i))_{i \geq 1}$ converge to $\overline{\sigma}^*(\omega)$ on $\partial D$.
So there exists a $j \in \mathbb{N}$ such that $B_{\varepsilon}(\overline{\sigma}^*(\omega))$ contains $\sigma^*(u_i)$ and $\sigma^*(w_i)$ for every $i \geq j$.
By the definitions of $\overline{\sigma}^*$ and $\sigma^*$ using the embeddings $\sigma^*_i$, it follows that $\overline{\sigma}^*(\overline{M_j}) \subsetneqq B_{\varepsilon}(\overline{\sigma}^*(\omega)) \subseteq O$.
At this point we use the property of $\sigma^*$ that every edge of $G$ is embedded as a straight line unless it is embedded into $\partial D$.
Hence, if $vw \in E(G)$ and $\overline{\sigma}^*(v), \overline{\sigma}^*(w) \in B_{\varepsilon}(\overline{\sigma}^*(\omega))$, then $\overline{\sigma}^*(vw)$ is also contained in $B_{\varepsilon}(\overline{\sigma}^*(\omega))$ by the convexity of the ball.
Since $\overline{M_j}$ together with the inner points of the edges of $E(M_j, K_j)$ is a basic open set in~$|G|$ containing $\omega$ whose image under~$\overline{\sigma}^*$ is contained in $O$, continuity holds for ends too.
The next step is to check that $\overline{\sigma}^*$ is injective.
If $x$ and $y$ are each either a vertex or an inner point of an edge, then they already lie in some $K_j$.
By the definition of $\overline{\sigma}^*$ we get that $\overline{\sigma}^*(x) = \overline{\sigma}^*(y)$ if and only if there exists a $j \in \mathbb{N}$ such that $x$ and $y$ are mapped to the same point by the embedding of $K_j$ defined by $\bigcup^j_{i=1} \sigma^*_{i} \upharpoonright _{K_{i}}$.
So $x$ and $y$ need to be equal.
For an and $\omega$ of $G$, let $(M_i)_{i \geq 1}$ be a sequence of components of $G-K_i$ such that $\bigcap_{i \geq 1}\overline{M_i} = \lbrace \omega \rbrace$, which exists by Claim~1.
Let $\lbrace u_i, w_i \rbrace$ be the neighbourhood of $M_i$ in $G$.
Since $G$ is locally finite, there exists an integer $j$ such that $y$ lies in $K_j$ if it is a vertex or an inner point of an edge, or $y$ lies in $\overline{M'_j}$ for some component $M'_j \neq M_j$ of $G-K_j$ if $y$ is an end of $G$ that is different from $\omega$.
By the definition of $\overline{\sigma}^*$ and property $(\ast)$ we get that the arc on $\partial D$ between $\sigma^*(u_j)$ and $\sigma^*(w_j)$ into which the vertices of $M_j$ are mapped contains also $\overline{\sigma}^*(\omega)$ but not $y$.
Hence, $\overline{\sigma}^*(\omega) \neq \overline{\sigma}^*(y)$ if $\omega \neq y$.
This shows the injectivity of the map $\overline{\sigma}^*$.
To see that the inverse function of ${\overline{\sigma}^*}$ is continuous, note that $|G|$ is compact by Proposition~\ref{compact} and~$D$ is Hausdorff.
So Lemma~\ref{invers_cont} immediately implies that the inverse function of ${\overline{\sigma}^*}$ is continuous.
This completes the proof that $\overline{\sigma}$ is an embedding.
It remains to show the existence of a unique Hamilton circle of $G$ that is mapped onto $\partial D$ by $\overline{\sigma}$.
For this we first prove that $\partial D \subseteq \mathrm{Im}(\overline{\sigma})$.
This then implies that the inverse function of ${\overline{\sigma}^*}$ restricted to $\partial D$ is a homeomorphism defining a Hamilton circle of $G$ since it contains all vertices of $G$.
We begin by proving the following claim.
\begin{claim}
For every infinite sequence $(M_i)_{i \geq 1}$ of components $M_i$ of ${G - K_i}$ with ${M_{i+1} \subseteq M_i}$ there exists an end $\omega$ of $G$ such that $\bigcap_{i \geq 1}\overline{M_i} = \lbrace \omega \rbrace$.
\end{claim}
Let $(M_i)_{i \geq 1}$ be any sequence as in the statement of the claim.
Since for every vertex $v$ there exists a $j \in \mathbb{N}$ such that $v \in K_j$, we get that $\bigcap_{i \geq 1}\overline{M_i}$ is either empty or contains ends of $G$.
Using that each $M_i$ is connected and that $M_{i+1} \subseteq M_i$, we can find a ray $R$ such that every $M_i$ contains a tail of $R$.
Therefore, $\bigcap_{i \geq 1}\overline{M_i}$ contains the end in which $R$ lies.
The argument that $\bigcap_{i \geq 1}\overline{M_i}$ contains at most one end is the same as in the proof of Claim~1.
This completes the proof of Claim~2.
\newline
Suppose a point $p \in \partial D$ does not already lie in $\mathrm{Im}(\sigma^*)$.
Then it does not lie in $\mathrm{Im}(\sigma^*_i \upharpoonright _{K_i})$ for any $i \geq 1$.
So there exists an infinite sequence $(M_i)_{i \geq 1}$ of components $M_i$ of ${G - K_i}$ with ${M_{i+1} \subseteq M_i}$ such that $p$ lies in the arc~$A_i$ of $\partial D$ between $\sigma^*(u_i)$ and $\sigma^*(w_i)$ into which the vertices of $M_i$ are mapped, where $\lbrace u_i, w_i \rbrace$ denotes the neighbourhood of $M_i$ in $G$.
Using Claim~2 we obtain that there exists an end $\omega$ of $G$ such that $\bigcap_{i \geq 1}\overline{M_i} = \lbrace \omega \rbrace$.
By property $(\ast)$ of the map~$\sigma^*$ the sequences $(\sigma^*(u_i))_{i \geq 1}$ and $(\sigma^*(w_i))_{i \geq 1}$ converge to a common point on~$\partial D$.
This point must be $p$ since the arcs $A_i$ are nested.
Now the definition of $\overline{\sigma}^*$ tells us that~$\overline{\sigma}^*(\omega) = p$.
Hence $\partial D \subseteq \mathrm{Im}(\overline{\sigma}^*)$ and $G$ is Hamiltonian.
We finish the proof by showing the uniqueness of the Hamilton circle of $G$.
Suppose for a contradiction that $G$ has two subgraphs $C_1$ and $C_2$ yielding different Hamilton circles $\overline{C_1}$ and $\overline{C_2}$.
Then there must be an edge $e \in E(C_1) \setminus E(C_2)$.
Let $j \in \mathbb{N}$ be chosen such that $e \in E(K_j)$.
By Lemma~\ref{struct_1} we obtain that $G_{K_j}[E(C_1) \cap E(G_{K_j})]$ and $G_{K_j}[E(C_2) \cap E(G_{K_j})]$ are two Hamilton cycles of $G_{K_j}$ differing in the edge $e$.
Note that $G_{K_j}$ is a finite $2$-connected outerplanar graph.
The argument for this is the same as for $G_K$ in the proof that $(ii)$ implies $(i)$.
This yields a contradiction since $G_{K_j}$ has a unique Hamilton cycle by statement~(ii) of Proposition~\ref{summary}.
\end{proof}
Next we deduce Corollary~\ref{Cor_contr}.
Let us recall its statement first.
\setcounter{section}{1}
\setcounter{theorem}{8}
\begin{corollary}
The edges contained in the Hamilton circle of a locally finite \linebreak $2$-connected graph not containing $K^4$ or $K_{2,3}$ as a minor are precisely the \linebreak $2$-contractible edges of the graph unless the graph is isomorphic to a $K^3$.
\end{corollary}
\setcounter{theorem}{4}
\setcounter{section}{4}
\begin{proof}
Let $G$ be a locally finite $2$-connected graph not isomorphic to a $K^3$ and not containing $K^4$ or $K_{2,3}$ as a minor.
Further, let $C$ be the subgraph of $G$ such that $\overline{C}$ is the Hamilton circle of $G$.
First we show that each edge $e \in E(C)$ is a $2$-contractible edge.
Note for this that the closure of the subgraph of $G/e$ formed by the edge set $E(C) \setminus \lbrace e \rbrace$ is a Hamilton circle in $|G/e|$.
Hence, $G/e$ is $2$-connected by Corollary~\ref{2-con}.
It remains to verify that no edge of $E(G) \setminus E(C)$ is $2$-contractible.
For this we consider any edge $e = uv \in E(G) \setminus E(C)$.
Let $K$ be a finite connected induced subgraph of $G$ containing at least four vertices as well as $N(u) \cup N(v)$, which is a finite set since $G$ is locally finite.
Then we know by Lemma~\ref{fin_2_con} and by using the locally finiteness of $G$ again that $G_K$ is a finite $2$-connected graph not containing $K^4$ or $K_{2,3}$ as a minor.
So by Theorem~\ref{outerplanar_count_char} and Proposition~\ref{summary} we get that $G_K$ has a unique Hamilton cycle consisting precisely of its $2$-contractible edges.
However, as we have seen in the proof of Theorem~\ref{HC_K_4-K_2,3}, $G_K[E(C) \cap E(G_K)]$ is the unique Hamilton cycle of $G_K$ and does not contain $e$.
Since $G_K$ is outerplanar, we get that the vertex of $G_K/e$ corresponding to the edge $e$ is a cut vertex in $G_K/e$.
By our choice of $K$ containing $N(u) \cup N(v)$, we get that the vertex in $G/e$ corresponding to the edge $e$ is a cut vertex of $G/e$ too.
So $e$ is not $2$-contractible.
\end{proof}
The question arises whether one could prove the more complicated part of Theorem~\ref{HC_K_4-K_2,3}, the implication $(i) \Longrightarrow (ii)$, by mimicking a proof for finite graphs.
To see the positive answer for this question, let us summarize the proof for finite graphs except the part about the uniqueness.
By Theorem~\ref{outerplanar_count_char} every finite graph without $K^4$ or $K_{2,3}$ as a minor can be embedded into the plane such that all vertices lie on a common face boundary.
Since every face of an embedded $2$-connected graph is bounded by a cycle, we obtain the desired Hamilton cycle.
So for our purpose we would first need to prove a version of Theorem~\ref{outerplanar_count_char} for~$|G|$ where $G$ is a locally finite connected graph.
This can similarly be done in the way we have defined the embedding for the Hamilton circle in Theorem~\ref{HC_K_4-K_2,3} by decomposing the graph into finite parts using Lemma~\ref{struct_1}.
Since none of these parts contains a $K^4$ or a $K_{2,3}$ as a minor, we can fix appropriate embeddings of them and stick them together.
However, in order to obtain an embedding of $|G|$ we have to be careful.
We also need to ensure that the embeddings of finite parts that converge to an end in $|G|$ also converge to a point in the plane where we can map the corresponding end to.
The second ingredient of the proof is the following lemma pointed out by Bruhn and Stein, but which is a corollary of a stronger and more general result of Richter and Thomassen~\cite[Prop.\ 3]{richter}.
\begin{lemma}\label{circ_boundary}\cite[Cor.\ 21]{bruhn_stein}
Let $G$ be a locally finite $2$-connected graph with an embedding ${\varphi : |G| \longrightarrow S^2}$.
Then the face boundaries of $\varphi(|G|)$ are circles of $|G|$.
\end{lemma}
\noindent These observations show that the proof idea for finite graphs is still applicable for locally finite graphs.
Let us compare the proof for the implication $(i) \Longrightarrow (ii)$ of Theorem~\ref{HC_K_4-K_2,3} that we sketched right above, with the one we outlined completely.
The two proofs share a big similarity.
Both need to show first that $|G|$ can be embedded into the plane such that all vertices lie on a common face boundary if $G$ is a connected or $2$-connected, respectively, locally finite graph without $K^4$ or $K_{2,3}$ as a minor.
At this point the proof we outlined completely already incorporates further properties into the embedding without too much additional effort.
Especially, we use the $2$-connectedness of the graph there by finding suitable finite $2$-connected contraction minors.
Then we apply Proposition~\ref{summary} for these.
The embeddings we obtain for the contraction minors allow us to define an embedding of $|G|$ into a fixed closed disk.
Furthermore, this embedding of $|G|$ has the additional property that its restriction onto the boundary of the disk directly witnesses the existence of a Hamilton circle.
The second proof, however, takes a step backward and argues more general.
There the $2$-connectedness of $G$ is used to apply Lemma~\ref{circ_boundary}, which, as noted before, is a corollary of a more general result of Richter and Thomassen~\cite[Prop.\ 3]{richter}.
At this point we forget about the special embedding of $|G|$ into the plane that we had to construct before.
We continue the argument with an arbitrary one given that $G$ is a $2$-connected locally finite graph.
So for the purpose of proving the implication $(i) \Longrightarrow (ii)$ of Theorem~\ref{HC_K_4-K_2,3}, the outlined proof is more straightforward and self-contained.
\section{A cubic infinite graph with a unique Hamilton circle}
This section is dedicated to Theorem~\ref{Q_mohar_Yes}.
We shall construct an infinite graph with a unique Hamilton circle where all vertices in the graph have degree $3$.
Furthermore, all ends of that graph have vertex-degree $3$ as well as edge-degree $3$.
The main ingredient in our construction is the finite graph $T$ depicted in Figure~\ref{Tutte_HCs}.
This graph has three distinguished vertices of degree $1$, which we denote by $u$, $l$ and $r$ as in Figure~\ref{Tutte_HCs}.
For us, the important feature of $T$ is that we know where all \textit{Hamilton paths}, i.e., spanning paths, of $T-u$ and $T-r$ proceed.
Tutte~\cite{tutte} came up with the graph $T$ to construct a counterexample to Tait's conjecture~\cite{tait}, which said that every $3$-connected cubic planar graph is Hamiltonian.
The crucial observation of Tutte in~\cite{tutte} was that $T-u$ does not contain a Hamilton path.
We shall use this observation as well, but we need more facts about $T$, which are covered in the following lemma.
The proof is straightforward, but involves several cases that need to be distinguished.
\begin{lemma}\label{HCs_in_T}
There is no Hamilton path in $T-u$, but there are precisely two in $T-r$ (see Figure~\ref{Tutte_HCs}).
\end{lemma}
\begin{figure}[htbp]
\centering
\includegraphics{Tutte_fragment_HCs.pdf}
\hspace{20pt}
\includegraphics{Tutte_fragment_HCs_gruen.pdf}
\hspace{20pt}
\includegraphics{Tutte_fragment_HCs_rot.pdf}
\caption{The fat edges in the most left picture are in every Hamilton path of $T-r$. The fat edges in the other two pictures mark the two Hamilton paths of $T-r$.}
\label{Tutte_HCs}
\end{figure}
\begin{proof}
As mentioned already by Tutte~\cite{tutte}, the graph $T-u$ does not have a Hamilton path.
It remains to show that $T-r$ has precisely two Hamilton paths.
For this we need to check several cases, but afterwards we can precisely state the Hamilton paths.
For convenience, we label each edge with a number as depicted in Figure~\ref{Tutte_edges} and refer to the edges just by their labels for the rest of the proof.
\begin{figure}[htbp]
\centering
\includegraphics[width=55mm]{Tutte_fragment_all_edges.pdf}
\caption{Our fixed labelling of the relevant edges of $T$.}
\label{Tutte_edges}
\end{figure}
Obviously, the edges incident with $\ell$ and $u$ would need to be in every Hamilton path of $T-r$ since these vertices have degree $1$.
Furthermore, the edges $2$ and $3$ need to be in every Hamilton path of $T-r$ since the vertex incident with $2$ and $3$ has degree $2$ in $T-r$.
\begin{claim}
The edge $4$ needs to be in every Hamilton path of $T-r$.
\end{claim}
Suppose for a contradiction that there is a Hamilton path $P$ in $T-r$ that does not use $4$.
Then it needs to contain $1$.
Since it also contains $2$, we know $5 \notin E(P)$.
This implies further that $7,8 \in E(P)$.
We can use $4 \notin E(P)$ also to deduce that $6, 10 \in E(P)$ holds.
Now we get $11 \notin E(P)$ since $6, 7 \in E(P)$.
This implies $20, 21 \in E(P)$.
But now $14 \notin E(P)$ holds because $10, 20 \in E(P)$.
From this we get then $16, 18 \in E(P)$.
So $19$ cannot be contained in $P$, which implies $13, 17 \in E(P)$.
Now we arrived at a contradiction since the edges incident with $l$ and $u$ together with the edges of the set $\lbrace 1, 2, 3, 13,17, 16, 18 \rbrace$ form a $\ell$-$u$ path in $T-r$ that is contained in $P$ and needs therefore to be equal to $P$.
Then, however, $P$ would not be a Hamilton path $T-r$.
This completes the proof of Claim~3
\newline
We immediately get from Claim~3 that $5$ needs to be in every Hamilton path of $T-r$ and since $8$ and $9$ can not both be contained in any Hamilton path of $T-r$, because they would close a cycle together with $5, 2$ and $3$, we also know that $12$ needs to be in every Hamilton path of $T-r$.
\begin{claim}
The edges $14$ and $16$ lie in every Hamilton path of $T-r$.
\end{claim}
Suppose for a contradiction that the claim is not true.
Then there is a Hamilton path $P$ of $T-r$ containing $18$.
So $P$ cannot contain $19$, which implies $13, 17 \in E(P)$.
Since $3, 13 \in E(P)$, we obtain $9 \notin E(P)$, from which we follow that $8 \in P$ holds.
Furthermore, $15$ cannot be contained in $P$, because then the edges $15, 17, 13, 3, 2, 5, 8, 12$ would form a cycle in $P$.
Therefore, $16$ is an edge of $P$.
From $5, 8 \in E(P)$ we can deduce that $7 \notin E(P)$ holds.
So $6$ and $11$ are edges of $P$, which that implies $10 \notin E(P)$.
Then $14, 20 \in E(P)$ needs to be true.
Now, however, we have a contradiction, because $P$ would have a vertex incident with three vertices, namely $14, 16$ and $18$.
This completes the proof of Claim~4
\newline
It follows from Claim~4 that $19$ is contained in every Hamilton path of $T-r$.
We continue with another claim.
\begin{claim}
The edges $6$ and $20$ lie in every Hamilton path of $T-r$.
\end{claim}
Suppose for a contradiction that the claim is not true.
Then there is a Hamilton path $P$ of $T-r$ containing $10$.
This immediately implies that $6 \notin E(P)$, yielding $7, 11 \in E(P)$, and $20 \notin E(P)$, yielding $21 \in E(P)$.
We note that $8$ cannot be an edge of $P$ since $P$ would then contain a cycle spanned by the edge set $\lbrace 8, 7, 11, 21, 12 \rbrace$.
Therefore, $9 \in E(P)$ must hold.
Here we arrive at a contradiction, since $P$ now contains a cycle spanned by the edge set $\lbrace 9, 3, 2, 5, 7, 11, 21, 12 \rbrace$.
This completes the proof of Claim~5
\newline
Using all the observations we have made so far, we can now show that $T-r$ has precisely two Hamilton paths and state them by looking at the edge $11$.
Assume that $11$ is contained in a Hamilton path $P_1$ of $T-r$.
Then $7, 21 \notin E(P_1)$ follows, because $6, 20 \in E(P_1)$ holds by Claim~5.
Since we could deduce from Claim~3 that $5, 12 \in E(P_1)$ holds, we get furthermore $8, 15 \in E(P_1)$.
This now yields $9, 17 \notin E(P_1)$ and, therefore, $13 \in E(P_1)$.
As we can see, the assumption that $11$ is contained in a Hamilton path $P_1$ of $T-r$ is true.
Also, $P_1$ is uniquely determined with respect to this property and consists of the fat edges in the most right picture of Figure~\ref{Tutte_HCs}.
Next assume that there is a Hamilton path $P_2$ of $T-r$ that does not contain the edge $11$.
Then $7$ and $21$ have to be edges of $P_2$.
Using again that $5, 12 \in E(P_2)$ holds, we deduce $8, 15 \notin E(P_2)$.
Then, however, we get $9, 17 \in E(P)$ and have already uniquely determined $P_2$, which corresponds to the fat edges in the middle picture of Figure~\ref{Tutte_HCs}.
\end{proof}
Using Lemma~\ref{HCs_in_T} we shall now prove Theorem~\ref{Q_mohar_Yes} by constructing a prescribed graph.
During the construction we shall often refer to certain distinguished vertices of $T$ that are named as depicted in Figure~\ref{Tutte_HCs}.
Let us recall the statement of the theorem.
\setcounter{section}{1}
\setcounter{theorem}{10}
\begin{theorem}
There exists an infinite connected graph $G$ with a unique Hamilton circle that has degree $3$ at every vertex and vertex- as well as edge-degree $3$ at every end.
\end{theorem}
\setcounter{theorem}{1}
\setcounter{section}{5}
\begin{proof}
We construct a sequence of graphs $(G_n)_{n \in \mathbb{N}}$ inductively and obtain the desired one $G$ as a limit of the sequence.
We start with $G_0 = T^1_0 = T$.
Now suppose we have already constructed $G_n$ for $n \geq 0$.
Furthermore, let ${\lbrace T^i_n \; ; \; 1 \leq i \leq 2^n \rbrace}$ be a specified set of disjoint subgraphs of $G_n$ each of which each is isomorphic to $T$.
We define $G_{n+1}$ as follows.
Take $G_n$ and two copies $T_c$ and $T_v$~of $T$ for each $T^i_n \subseteq G_n$.
Then identify for every $i$ the vertices of $T_c$ that correspond to $u$, $\ell$ and $r$, respectively, with the vertices of the related $T^i_n \subseteq G_n$ corresponding to $\ell$, $s$ and~$t$, respectively.
Also identify for every $i$ the vertices of $T_v$ corresponding to $u$,~$\ell$ and~$r$, respectively, with the ones of the related $T^i_n \subseteq G_n$ corresponding to~$w$,~$x$ and~$y$, respectively.
Finally, delete in each $T^i_n \subseteq G_n$ the vertices corresponding to~$c$ and~$v$, see Figure~\ref{Tutte_insert_constr}.
This completes the definition of~$G_{n+1}$.
It remains to fix the set of $2^{n+1}$ many disjoint copies of $T$ that occur as disjoint subgraphs in $G_{n+1}$.
For this we take the set of all copies $T_c$ and $T_v$ of $T$ that we have inserted in the subgraphs $T^i_n$ of $G_n$.
\begin{figure}[htbp]
\centering
\includegraphics{Tutte_fragment_insert.pdf}
\hspace{20pt}
\includegraphics{Tutte_constr.pdf}
\caption{A sketch of the construction of $G_1$. The fat black, grey and dashed edges incident with the grey vertices in the right picture correspond to the ones in the left picture.}
\label{Tutte_insert_constr}
\end{figure}
Using the graphs $G_n$ we define a graph $\hat{G}$ as a limit of them.
We set
\[\hat{G} = G[\hat{E}] \; \textnormal{ where } \; \hat{E} = \left \lbrace e \in \bigcup_{n \in \mathbb{N}} E(G_n) \; ; \; \exists N \in \mathbb{N} : e \in \bigcap_{n \geq N} E(G_n) \right \rbrace .\]
Note that an edge $e \in E(G_n)$ is an element of $\hat{E}$ if and only if it was not deleted during the construction of $G_{n+1}$ as an edge incident with one of the vertices that correspond to $c$ or $v$ in $T^i_n$ for some $i$.
Finally, we define $G$ as the graph obtained from $\hat{G}$ by identifying the three vertices that correspond to $u$, $\ell$ and $r$ of $T^1_0$.
Next let us verify that every vertex of $G$ has degree $3$ and that every end of $G$ has vertex- as well as edge-degree $3$ in $G$.
Since every vertex of $T$ except $u$, $\ell$ and $r$ has degree $3$, the construction ensures that every vertex of $G$ has degree $3$ too.
In order to analyse the end degrees, we have to make some observations first.
The edges of $G$ that are adjacent to vertices corresponding to $u$, $\ell$ and $r$ of any $T^i_n$ define a cut $E(A^i_n, B^i_n)$ of $G$.
Note that for any finite cut of a graph all rays in one end of the graph have tails that lie completely on one side of the cut.
Therefore, the construction of $G$ ensures that for every end $\omega$ of $G$ there exists a function $f: \mathbb{N} \longrightarrow \mathbb{N}$ with $f(n) \in \lbrace 1, \ldots, 2^n \rbrace$ such that all rays in $\omega$ have tails in $B^{f(n)}_n$ for each $n \in \mathbb{N}$ and $B^{f(n)}_n \supseteq B^{f(n+1)}_{n+1}$ with $\bigcap_{n \in \mathbb{N}} B^{f(n)}_n = \emptyset$.
Using that $|E(A^i_n, B^i_n)| = 3$ for every $n$ and $i$, this implies that every end of $G$ has edge-degree at most $3$.
Since there are three disjoint paths from $\lbrace u, \ell, r \rbrace$ to $\lbrace s, \ell, t \rbrace$ as well as to $\lbrace x, w, y \rbrace$ in~$T$, we can also easily construct three disjoint rays along the cuts $E(A^i_n, B^i_n)$ that belong to an arbitrary chosen end of $G$.
So every end of $G$ has vertex-degree $3$.
In total this yields that every end of $G$ has vertex- as well as edge-degree $3$ in $G$.
It remains to prove that $G$ has precisely one Hamilton circle.
We begin by stating the edge set of the subgraph $C$ defining the Hamilton circle $\overline{C}$ of $G$.
Let $E(C)$ consist of those edges of $E(G) \cap T^i_n$ for every $n$ and $i$ that correspond to the fat edges of $T$ in the most right picture of Figure~\ref{Tutte_HCs}.
Now consider any finite cut $D$ of~$G$.
The construction of $G$ yields that there exists an $N \in \mathbb{N}$ such that $D$ is already a cut of the graph obtained from $G_n$ by identifying the vertices corresponding to $u$, $\ell$ and $r$ of $T^1_0 \subseteq G_n$ for all~$n \geq N$.
Using this observation we can easily see that every vertex of $G$ has degree $2$ in $\overline{C}$.
We also obtain that every finite cut is met at least twice, but always in an even number of edges of $C$.
By Lemma~\ref{top_conn} we get that $\overline{C}$ is topologically and also arc-connected.
Therefore, every end of $G$ has edge-degree at least $1$ and at most~$3$ in~$\overline{C}$.
Together with Theorem~\ref{cycspace} this implies that every end of $G$ has edge-degree~$2$ in~$\overline{C}$.
Hence, Lemma~\ref{circ} tells us that $\overline{C}$ is a circle, which is Hamiltonian since it contains all vertices of $G$.
We finish the proof by showing that $\overline{C}$ is the unique Hamilton circle of $G$.
Since any Hamilton circle $\overline{H}$ of $G$ meets each cut $E(A^i_n, B^i_n)$ precisely twice, $\overline{H}$ induces a path through $T$ that contains all vertices of $T$ except one out of the set $\lbrace u, \ell, r \rbrace$.
By Lemma~\ref{HCs_in_T} we know that such paths must contain the edge adjacent to $u$.
Let us consider any $T^i_n$ in $G_n$.
Now let $T^j_{n+1}$ be the copy of $T$ whose vertices of degree~$1$ we have identified with the vertices corresponding to the neighbours of $c$ in $T^i_n$ during the construction of $G_{n+1}$.
The way we have identified the vertices implies that the path induced by $\overline{H}$ through $T^i_n$ must also use the edge adjacent to $\ell$ since the induced path in $T^j_{n+1}$ must use the edge adjacent to $u$.
With a similar argument we obtain that the induced path inside $T^i_n$ must use the edge corresponding to $vw$.
We know from Lemma~\ref{HCs_in_T} that there is a unique Hamilton path in $T-r$ that uses the edges~$\ell c$ and $vw$, namely the one corresponding to the fat edges in the most right picture of Figure~\ref{Tutte_HCs}.
So the edges which must be contained in every Hamilton circle are precisely those of $C$.
\end{proof}
\begin{remark}
\textnormal{After reading a preprint of this paper Max Pitz~\cite{pitz} carried further some ideas of this paper.
Also using the graph $T$, he recently constructed a two-ended cubic graph with a unique Hamilton circle where both ends have vertex- as well as edge-degree $3$.
He further proved that every one-ended Hamiltonian cubic graph whose end has edge-degree $3$ (or vertex-degree $3$) admits a second Hamilton circle.}
\end{remark}
\section*{Acknowledgement}
I would like to thank Tim R\"uhmann for reading an early draft of this paper and giving helpful comments.
| {
"timestamp": "2018-12-06T02:05:58",
"yymm": "1701",
"arxiv_id": "1701.06029",
"language": "en",
"url": "https://arxiv.org/abs/1701.06029",
"abstract": "We state a sufficient condition for the square of a locally finite graph to contain a Hamilton circle, extending a result of Harary and Schwenk about finite graphs.We also give an alternative proof of an extension to locally finite graphs of the result of Chartrand and Harary that a finite graph not containing $K^4$ or $K_{2,3}$ as a minor is Hamiltonian if and only if it is $2$-connected. We show furthermore that, if a Hamilton circle exists in such a graph, then it is unique and spanned by the $2$-contractible edges.The third result of this paper is a construction of a graph which answers positively the question of Mohar whether regular infinite graphs with a unique Hamilton circle exist.",
"subjects": "Combinatorics (math.CO)",
"title": "Hamiltonicity in locally finite graphs: two extensions and a counterexample",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754492759499,
"lm_q2_score": 0.7185943805178139,
"lm_q1q2_score": 0.7075103850454995
} |
https://arxiv.org/abs/2010.07505 | Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies | We calculate the Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies of the Taft algebra $T_p$ for any integer $p>2$ which is a nonquasi-triangular Hopf algebra. We show that the bracket is indeed zero on Hopf algebra cohomology of $T_p$, as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi-triangular algebra. Also, we find a general formula for the bracket on Hopf algebra cohomology of any Hopf algebra with bijective antipode on the bar resolution that is reminiscent of Gerstenhaber's original formula for Hochschild cohomology. | \section{Introduction}\label{sec:introduction}
Gerstenhaber brackets were originally defined on Hochschild cohomology by M. Gerstenhaber himself \cite[Section 1.1]{G}. In 2002, A. Farinati and A. Solotar showed that for any Hopf algebra $A$, Hopf algebra cohomology H$^*(A):=\Ext^*_A(k,k)$ is a Gerstenhaber algebra \cite{FAR}. Hence, we can define a Gerstenhaber bracket on Hopf algebra cohomology. In the same year, R. Taillefer used a different approach and found a bracket on Hopf algebra cohomology \cite{Tail} which is equivalent to the bracket constructed by A. Farinati and A. Solotar. The category of $A$-modules and the category of $A^e$-modules are examples of strong exact monoidal categories. In 2016, Reiner Hermann \cite[Theorem 6.3.12, Corollary 6.3.15]{Hermann} proved that if the strong exact monoidal category is lax braided, then the bracket is constantly zero. Therefore, the Gerstenhaber bracket on the Hopf algebra cohomology of a quasi-triangular Hopf algebra is trivial. However, we do not know the bracket structure for a nonquasi-triangular Hopf algebra. Taft algebras are nice examples of nonquasi-triangular Hopf algebras. In this paper, we show that the Gerstenhaber bracket on the Hochschild cohomology of a Taft Algebra is nontrivial. However, the bracket structure on Hopf algebra cohomology of a Taft algebra is constantly zero. Also, we take the Gerstenhaber bracket formula on Hochschild comology and find a general formula for Gerstenhaber bracket on Hopf algebra cohomology.
We start by giving some basic definitions and some tools to calculate the bracket on Hochschild cohomology in Section 2. Then, we compute the Gerstenhaber bracket on the Hochschild cohomology of $A=k[x]/(x^p)$ where the field $k$ has characteristic 0 and the integer $p>2$ in Section 3. We use the technique introduced by C. Negron and S. Witherspoon \cite{NW} and note that they computed the bracket on Hochschild cohomology of $A$ for the case that $k$ has positive characteristic $p$ \cite[Section 5]{NW}.
In Section 4, we compute the Gerstenhaber bracket for the Taft algebra $T_p$ which is a nonquasi-triangular Hopf algebra. We use a similar technique as in \cite{NW} to calculate the bracket on Hochschild cohomology of $T_p$. It is also known that the Hopf algebra cohomology of any Hopf algebra with a bijective antipode can be embedded in the Hochschild cohomology of the algebra \cite[Theorem 9.4.5 and Corollary 9.4.7]{HH}. Since all finite dimensional Hopf algebras (also most of known infinite dimensional Hopf algebras) have bijective antipode, we can embed the Hopf algebra cohomology of $T_p$ into the Hochschild cohomology of $T_p$. Then, we use this explicit embedding and find the bracket on the Hopf algebra cohomology of $T_p$. As a result of our calculation, we obtain that the bracket on Hopf algebra cohomology of $T_p$ is also trivial.
In the last section, we derive a general expression for the bracket on Hopf algebra cohomology of any Hopf algebra $A$ with bijective antipode. We first consider a specific resolution that agrees with the bar resolution of $A$ and find a bracket formula for it. Then, we use the composition of various isomorphisms and an embedding from Hopf algebra cohomology into Hochschild cohomology in order to discover the bracket formula on Hopf algebra cohomology.
\section{Gerstenhaber Bracket on Hochschild Cohomology}
Let $k$ be a field, $A$ be a $k$-algebra, and $A^e=A\ot_k A^{op}$ where $A^{op}$ is the opposite algebra with reverse multiplication. For simplicity, we write $\ot$ instead of $\ot_k$. The following resolution $B(A)$ is a free resolution of the $A^e$-module $A$, called the \textit{bar resolution},
\begin{equation}\label{bar complex}
B(A):\cdots \stackrel{d_3}{\longrightarrow} A^{\otimes 4} \stackrel{d_2}{\longrightarrow} A^{\otimes 3} \stackrel{d_1}{\longrightarrow} A^{\otimes 2} \stackrel{\pi}{\longrightarrow} A \longrightarrow 0,
\end{equation}
where
$$d_n(a_0\ot a_1\ot \cdots \ot a_{n+1})=\sum_{i=0}^{n}(-1)^ia_0\ot a_1\ot \cdots\ot a_ia_{i+1}\ot \cdots \ot a_{n+1}$$
and $\pi$ is multiplication.
Consider the following complex that is derived by applying Hom$_{A^e}(-,A)$ to the bar resolution $B(A)$
\begin{equation}\label{hom bar}
0 {\longrightarrow} \text{Hom}_{A^e}(A^{\otimes 2},A) \stackrel{d^{*}_1}{\longrightarrow} \text{Hom}_{A^e}(A^{\otimes 3},A) \stackrel{d^{*}_2}{\longrightarrow} \text{Hom}_{A^e}(A^{\otimes 4},A) \stackrel{d^{*}_3}{\longrightarrow} \cdots
\end{equation}
where $d_n^{*}(f)=fd_n$. The \textit{Hochshild cohomology} of the algebra $A$ is the cohomology of the cochain complex (\ref{bar complex}), i.e. $$\text{HH}^{*}(A,A)=\bigoplus_{n\geq 0}\Ext_{A^e}^n(A,A).$$
We also define the \textit{Hopf algebra cohomology} of the Hopf algebra $A$ over the field $k$ as $$\textup{H}^{*}(A,k)=\bigoplus_{n\geq 0}\textup{Ext}^n_A(k,k)$$ under the cup product.
Let $f\in \text{Hom}_k(A^{\otimes m},A)$ and $g\in \text{Hom}_k(A^{\otimes n},A)$. The Hochschild cohomology of $A$ is an algebra with the following cup product and the Gerstenhaber bracket structures. The \textit{cup product} $f\smile g\in$Hom$_k(A^{\otimes(m+n)},A)$ is defined by
$$(f\smile g)(a_1\otimes \cdots \otimes a_{m+n}):=(-1)^{mn}f(a_1\otimes \cdots \otimes a_m)g(a_{m+1}\otimes\cdots a_{m+n})$$
for all $a_1,\cdots, a_{m+n}\in A$, and the \textit{Gerstenhaber bracket} $[f,g]$ is an element of\\
Hom$_k(A^{\otimes(m+n-1)},A)$ given by
$$[f, g] := f\circ g-(-1)^{(m-1)(n-1)}g\circ f$$
where the circle product $f\circ g$ is
\begin{align*}
&(f\circ g)(a_1\otimes \cdots \otimes a_{m+n-1}):=\\
&\sum_{i=1}^{m}(-1)^{(n-1)(i-1)}f(a_1\otimes\cdots a_{i-1}\otimes g(a_i\otimes \cdots a_{i+n-1})\otimes a_{i+n}\otimes \cdots \otimes a_{m+n-1})
\end{align*}
for all $a_1,\cdots, a_{m+n-1}\in A$. We note that these definitions directly come from the bar resolution.
There is an identity between cup product and bracket \cite[Section 1]{G}:
\begin{equation}\label{cupbrac}
[f^*\smile g^*,h^*]=[f^*,h^*]\smile g^*+(-1)^{|f^*|(|h^*|-1)}f^*\smile [g^*,h^*],
\end{equation}
where $f^*, g^*,$ and $h^*$ are the images (in Hochschild cohomology) of the cocyles $f,g$, and $h$, respectively.
Computing the bracket on the bar resolution is not an ideal method. Instead, we can use another resolution, $\mathbb{A}\stackrel{\mu}\rightarrow A$, satisfying the following hypotheses \cite[(3.1) and Lemma 3.4.1]{NW}:
(a) $\mathbb{A}$ admits an embedding $\iota:\mathbb{A}\to B(A)$ of complexes of $A$-bimodules for which
the following diagram commutes
\begin{center}
\begin{tikzcd}[column sep=small]
\mathbb{A} \arrow{r}{\iota} \arrow{rd}
& B(A) \arrow{d} \\
& A
\end{tikzcd}
\end{center}
(b) The embedding $\iota$ admits a section $\pi:B\to \mathbb{A}$, i.e.\ an $A^e$-chain map $\pi$ with
$\pi\iota= id_{\mathbb{A}}$.
(c) There is a diagonal map that satisfies $\Delta_{\mathbb{A}}^{(2)}=(\pi\ot_A\pi\ot_A\pi)\Delta^{(2)}_{B(A)}\iota$ where $\Delta^{(2)}=(id\ot \Delta)\Delta$.
We give the following theorem which is the combination of \cite[Theorem 3.2.5]{NW} and \cite[Lemma 3.4.1]{NW} that allows us to use a different resolution for the bracket calculation.
\begin{theo}\label{theo brac}
Suppose $\mathbb{A}\stackrel{\mu}\rightarrow A$ is a projective $A$-bimodule resolution of $A$ that satisfies the hypotheses (a)-(c). Let $\phi:\mathbb{A}\otimes_A \mathbb{A}\to \mathbb{A}$ be any contracting homotopy for the chain map $F_\mathbb{A}:\mathbb{A}\ot_A \mathbb{A}\to \mathbb{A}$ defined by $F_\mathbb{A}:=(\mu\otimes_A id_\mathbb{A}- id_\mathbb{A}\otimes_A \mu)$, i.e.
\begin{equation}\label{org phi}
d(\phi):=d_\mathbb{A}\phi+\phi d_{ \mathbb{A}\otimes_A \mathbb{A}}=F_\mathbb{A}.
\end{equation}
Then for cocycles $f$ and $g$ in
Hom$_{A^e}(\mathbb{A},A)$, the bracket given by
\begin{equation}\label{brac}
[f, g]_\phi = f\circ_\phi g-(-1)^{(|f|-1)(|g|-1)}g\circ_\phi f
\end{equation}
where the circle product is
\begin{equation} \label{circ}
f\circ_\phi g=f\phi(id_\mathbb{A}\otimes_A g \otimes_A id_\mathbb{A})\Delta^{(2)}
\end{equation}
agrees with the Gerstenhaber bracket on cohomology.
\end{theo}
In general, it is not easy to calculate the map $\phi$ by the formula \eqref{org phi}. We use alternative way to find $\phi$.
Let $h$ be any $k$-linear contracting homotopy for the identity map on the extended complex $\mathbb{A}\to A\to 0$ where $\mathbb{A}$ is free. A contracting homotopy $\phi_i:(\mathbb{A}\otimes_A\mathbb{A})_i\longrightarrow\mathbb{A}_{i+1}$ in Theorem \ref{theo brac} is constructed by the following formula \cite[Lemma 3.3.1]{NW}:
\begin{equation}\label{mapphi}
\phi_i=h_i((F_\mathbb{A})_i-\phi_{i-1}d_{(\mathbb{A}\otimes_A\mathbb{A})_i}).
\end{equation}
\section{Bracket on Hochschild cohomology of $A=k[x]/(x^p)$}
Let $A=k[x]/(x^p)$ where $k$ is a field of characteristic 0 and $p>2$ is an integer. We compute the Lie bracket on Hochschild cohomology of $A$ by Theorem \ref{theo brac}. We work on a smaller resolution of $A$ than the bar resolution of $A$. Consider the following $A^e$-module resolution of $A$:
\begin{equation}\label{res A}
\mathbb{A}:\cdots \stackrel{v.}{\longrightarrow} A^{e}\stackrel{u.}{\longrightarrow} A^{e} \stackrel{v.}{\longrightarrow} A^{e} \stackrel{u.}{\longrightarrow} A^{e} \stackrel{\pi}{\longrightarrow} A \longrightarrow 0,
\end{equation}
where $u=x\otimes 1-1\otimes x$, $v=x^{p-1}\otimes 1+x^{p-2}\ot x+\cdots+x\otimes x^{p-2}+1\otimes x^{p-1}$, and $\pi$ is the multiplication.
The bracket on $A$ where $k$ is a field with positive characteristic, is calculated by C. Negron and S. Witherspoon \cite[Section 5]{NW}. We adopt the contracting homotopy $h$ for the identity map from that calculation and obtain a new map $h$ for our setup. Let $\xi_i$ be the element $1\otimes1$ of $\mathbb{A}_i$. The following maps $h_n:\mathbb{A}_n\longrightarrow \mathbb{A}_{n+1}$ form a contracting homotopy for identity map, as we can see by direct calculation:
\begin{align}\label{hn}
\begin{split}
h_{-1}(x^i)&=\xi_0x^i,\\
h_{0}(x^i\xi_0x^j)&=\sum_{l=0}^{i-1}x^l\xi_1x^{i+j-1-l},\\
h_{1}(x^i\xi_1x^j)&=\delta_{i,p-1}x^j\xi_2,\\
h_{2n}(x^i\xi_{2n}x^j)&=-\sum_{l=0}^{j-1}x^{i+j-1-l}\xi_{2n+1}x^l \text{ $(n\geq 2)$ ,}\\
h_{2n+1}(x^i\xi_{2n+1}x^j)&=\delta_{j,p-1}x^i\xi_{2n+2} \text{ $(n\geq 2)$ .}
\end{split}
\end{align}
Then, we take $\phi_{-1}=0$ and construct the following $A^e$-linear maps $\phi_i:(\mathbb{A}\otimes_A\mathbb{A})_i\longrightarrow\mathbb{A}_{i+1}$ for degree 1 and 2 by (\ref{mapphi}):
\begin{align}\label{phin}
\begin{split}
&\phi_0(\xi_0\otimes_A x^i\xi_0)=\sum_{l=0}^{i-1}x^l\xi_1x^{i-1-l},\\
&\phi_1(\xi_1\otimes_A x^i\xi_0)=-\delta_{i,p-1}\xi_2,\\
&\phi_1(\xi_0\otimes_A x^i\xi_1)=\delta_{i,p-1}\xi_2.
\end{split}
\end{align}
Lastly, we form the following diagonal map $\Delta:\mathbb{A}\longrightarrow\mathbb{A}\otimes_A\mathbb{A}$:
\begin{align}\label{Deltan}
\begin{split}
&\Delta_0(\xi_0) = \xi_0\otimes_A\xi_0,\\
&\Delta_1(\xi_1) =\xi_1\otimes_A \xi_0 + \xi_0 \otimes_A \xi_1,\\
&\Delta_{2n}(\xi_{2n}) = \sum_{i=0}^{n}\xi_{2i} \otimes_A\xi_{2n-2i} +\sum_{i=0}^{n-1}\sum_{a+b+c=p-2}^{}x^a\xi_{2i+1} \otimes_A x^b\xi_{2n-2i-1}x^c, \text{ for }n\geq 1\\
&\Delta_{2n+1}(\xi_{2n+1}) = \sum_{i=0}^{2n+1}\xi_i \otimes_A \xi_{2n+1-i}, \text{ for }n\geq 1.
\end{split}
\end{align}
It can be seen that the map $\Delta$ is a chain map lifting the canonical isomorphism $A\stackrel{\sim}\rightarrow A\ot_A A$ by direct calculation.
Now, we are ready to calculate the brackets on cohomology in low degrees. By applying Hom$_{A^e}(-,A)$
to $\mathbb{A}$, we see that the differentials are all 0 in odd degrees and $(px^{p-1})\cdot$ in even degrees. In each degree, the term in the Hom complex is the free $A$-module Hom$_{A^e}(A^e,A)\cong A$. Moreover, since $p$ is not divisible by the characteristic of $k$, we deduce HH$^0(A)\cong A, \text{HH}^{2i+1}(A)\cong (x),\text{ and }\text{HH}^{2i}(A)\cong A/(x^{p-1})$ \cite[Section 1.1]{HH}.
Let $x^j\xi^*_i \in$ Hom$_{A^e}(A^e,A)$ denote the function that takes $\xi_i$ to $x^j$. Since the characteristic of $k$ does not divide $p$, the Hochschild cohomology as an $A$-algebra is generated by $\xi_1^{*}$ and $\xi_2^{*}$ \cite[Example 2.2.2]{HH}. We only calculate the brackets of the elements of degrees 1 and 2 which can be extended to higher degrees by the formula (\ref{cupbrac}). Hence, we have the following calculations:
The bracket of the elements of degrees 1 and 1:\\
\begin{align*}
&(x^i\xi_1^{*}\circ_{\phi}x^j\xi_1^{*})(\xi_1)\\
&=x^i\xi_1^{*}\phi(1\otimes_A x^j\xi_1^{*}\otimes_A1)\Delta^{(2)}(\xi_1)\\
&=x^i\xi_1^{*}\phi(1\otimes_Ax^j\xi_1^{*}\otimes_A1)(\xi_1\otimes_A\xi_0\otimes_A\xi_0+\xi_0\otimes_A\xi_1\otimes_A\xi_0+\xi_0\otimes_A\xi_0\otimes\xi_1)\\
&=x^i\xi_1^{*}\phi(\xi_0\otimes_A x^j\xi_0)\\
&=x^i\xi_1^{*}(\xi_1x^{j-1}+x\xi_1x^{j-2}+\cdots+x^{j-1}\xi_1)\\
&=jx^{i+j-1}
\end{align*}
and by symmetry
$(x^j\xi_1^{*}\circ_{\phi}x^i\xi_1^{*})(\xi_1)=ix^{i+j-1}$. Therefore, we have
$$[x^i\xi_1^{*},x^j\xi_1^{*}]=(j-i)x^{i+j-1}\xi_1^{*}.$$
The bracket of the elements of degrees 1 and 2:\\
\begin{align*}
&(x^i\xi_1^{*}\circ_{\phi}x^j\xi_2^{*})(\xi_2)\\
&=x^i\xi_1^{*}\phi(1\otimes_A x^j\xi_2^{*}\otimes_A1)\Delta^{(2)}(\xi_2)\\
&=x^i\xi_1^{*}\phi(1\otimes_A x^j\xi_2^{*}\otimes_A1)(\xi_0\otimes_A\xi_0\otimes_A\xi_2+\xi_0\otimes_A\xi_2\otimes_A\xi_0+\xi_2\otimes_A \xi_0\otimes_A\xi_0\\
&+\xi_0\otimes_A\sum_{\substack{a+b+c \\ =p-2}}(x^a\xi_1\ot_A x^b\xi_1x^c)+\sum_{\substack{a+b+c \\ =p-2}}x^a\xi_1\ot_A x^b(\xi_0\ot_A \xi_1+\xi_1\ot_A x_0)x^c)\\
&=x^i\xi_1^{*}\phi(\xi_0\otimes_A x^j\xi_0)=x^i\xi_1^{*}(\xi_1x^{j-1}+x\xi_1x^{j-2}+\cdots+x^{j-1}\xi_1)=jx^{i+j-1}.
\end{align*}
The circle product in the reverse order is
\begin{align*}
&(x^j\xi_2^{*}\circ_{\phi}x^{p-1}\xi_1^{*})(\xi_2)\\
&=x^j\xi_2^{*}\phi(1\otimes_A x^{p-1}\xi_1^{*}\otimes_A1)\Delta^{(2)}(\xi_2)\\
&=x^j\xi_2^{*}\phi(1\otimes_A x^{p-1}\xi_1^{*}\otimes_A1)(\xi_0\otimes_A\xi_0\otimes_A\xi_2+\xi_0\otimes_A\xi_2\otimes_A\xi_0+\xi_2\otimes_A \xi_0\otimes_A\xi_0\\
&+\xi_0\otimes_A\sum_{\substack{a+b+c \\ =p-2}}(x^a\xi_1\ot_A x^b\xi_1x^c)+\sum_{\substack{a+b+c \\ =p-2}}x^a\xi_1\ot_A x^b(\xi_0\ot_A \xi_1+\xi_1\ot_A x_0)x^c)\\
&=x^j\xi_2^{*}\phi(\sum_{\substack{a+b+c \\ =p-2}}(\xi_0\otimes_A x^{a+b+i}\xi_1x^c-x^a\xi_1\ot_A x^{b+i}\xi_0x^c))\\
&=x^j\xi_2^{*}(\sum_{\substack{a+b+c \\ =p-2}}(\delta_{a+b+i,p-1}\xi_2x^c+x^a\delta_{b+i,p-1}\xi_2x^c))\\
&=x^j\xi_2^{*}((p-i)\xi_2x^{i-1}+\sum_{\substack{a+c \\ =i-1}}x^a\xi_2x^c)\\
&=(p-i)x^{i+j-1}+\sum_{\substack{a+c \\ =i-1}}x^{a+c+j}=(p-i)x^{i+j-1}+ix^{i+j-1}=px^{i+j-1}.
\end{align*}
Therefore, we obtain
$$[x^i\xi_1^{*},x^j\xi_2^{*}]=(j-p)x^{i+j-1}\xi^{*}_2.$$
Lastly, the bracket of the elements of degrees 2 and 2:\\
\begin{align*}
(x^i\xi_2^{*}\circ_{\phi}x^j\xi_2^{*})(\xi_3)&=x^i\xi_2^{*}\phi(1\otimes_A x^j\xi_2^{*}\otimes_A1)\Delta^{(2)}(\xi_3)\\
&=x^i\xi_2^{*}\phi(\xi_1\otimes_A x^j\xi_0+\xi_0\otimes_A x^j\xi_1)=x^i\xi_2^{*}(0)=0
\end{align*}
and by symmetry
$(x^j\xi_2^{*}\circ_{\phi}x^i\xi_2^{*})(\xi_3)=0$. Therefore, we have
$$[(x^i\xi_2^{*},x^j\xi_2^{*})]=0.$$
As a consequence, the brackets for the elements of degrees 1 and 2 are
\begin{align*}
[(x^i\xi_1^{*},x^j\xi_1^{*})]&=(j-i)x^{i+j-1}\xi_1^{*},\\
[(x^i\xi_1^{*},x^j\xi_2^{*})]&=(j-p)x^{i+j-1}\xi^{*}_2,\\
[(x^i\xi_2^{*},x^j\xi_2^{*})]&=0.
\end{align*}
Brackets in higher degrees can be determined from these and the identity (\ref{cupbrac})
since the Hochschild cohomology is generated as an $A$-algebra under the cup product in degrees 1
and 2.
L. Grimley, V. C. Nguyen, and S. Witherspoon \cite{GNW} calculated Gerstenhaber brackets on Hochschild cohomology of a twisted tensor product of algebras. S. Sanchez-Flores \cite{SF} also calculated the bracket on group algebras of a cyclic group over a field of positive characteristic which is isomorphic to $A=k[x]/(x^p)$. C. Negron and S. Witherspoon \cite{NW} calculated the bracket on group algebras of a cyclic group over a field of positive characteristic as well with the same $h,\phi$, and $\Delta$ maps. Our calculation agrees with those except slightly different $[(x^i\xi_1^{*},x^j\xi_2^{*})]$.
\section{Bracket on Hopf algebra cohomology of a Taft algebra}
The Taft algebra $T_p$ with $p>2$ is a $k$-algebra generated by $g$ and $x$ satisfying the relations : $g^p = 1, x^p = 0,\text{ and }xg =\omega gx$ where $\omega$ is a primitive $p$-th root of unity. It is a Hopf algebra with the structure:
\begin{itemize}
\item $\Delta(g)=g\otimes g$, $\Delta(x)=1\otimes x+x\otimes g$
\item $\varepsilon(g)=1,\varepsilon(x)=0$
\item $S(g)=g^{-1},S(x)=-xg^{-1}.$
\end{itemize}
Note that as an algebra, $T_p$ is a skew group algebra $A\rtimes kG$ where $A=k[x]/(x^p)$ and $G=<g\mid g^p=1>$. The action of $G$ on $A$ is given by $^gx=\omega x$.
In this section, our main goal is to calculate the bracket on Hochschild cohomology of $T_p$ with the same technique in Section 3 and find the bracket on Hopf algebra cohomology of $T_p$ by using the embedding of $\text{H}^*(T_p,k)$ into $\text{HH}^*(T_p,T_p)$.
We first find the bracket on Hochschild cohomology of $T_p$. Let $\mathcal{D}$ be the skew group algebra $A^e\rtimes G$ where the action of $G$ on $A^e$ is diagonal, i.e. $^g(a\ot b)=(^ga)\ot (^gb)$. Then, there is the following isomorphism \cite[Section 2]{BW}
$$\mathcal{D}=A^e\rtimes G\cong \underset{g\in G}{\bigoplus}Ag\ot Ag^{-1}\subset T_p^e.$$
Hence $\mathcal{D}$ is isomorphic to a subalgebra of $T_p^e$ via $a_1\ot a_2\ot g\mapsto a_1g\ot (^{g^{-1}}a_2g^{-1})$. Moreover, $A$ is a $\mathcal{D}$-module under the following left and right action \cite[Section 4]{BW}:
$$(a_1g\ot a_2g^{-1})a_3=a_1ga_3a_2g^{-1}=a_1(^g(a_3a_2))$$
$$a_3(a_1g\ot a_2g^{-1})=a_2g^{-1}a_3a_1g=a_2(^{g^{-1}}(a_3a_1)).$$
Remember the resolution \eqref{res A}
\begin{equation*}
\mathbb{A}:\cdots \stackrel{v.}{\longrightarrow} A^{e}\stackrel{u.}{\longrightarrow} A^{e} \stackrel{v.}{\longrightarrow} A^{e} \stackrel{u.}{\longrightarrow} A^{e} \stackrel{\pi}{\longrightarrow} A \longrightarrow 0.
\end{equation*}
This is also a $\mathcal{D}$-projective resolution of A and the action of $G$ on $A^e$ is given by
\begin{itemize}
\item $g\cdot(a_1\ot a_2)=(^ga_1)\ot(^ga_2)$ in even degrees,
\item $g\cdot(a_1\ot a_2)=\omega (^ga_1)\ot(^ga_2)$ in odd degrees.
\end{itemize}
From the resolution $\mathbb{A}$, we construct the following $T_p^{e}$ resolution of $T_p$:
\begin{equation} \label{resol}
T_p^{e}\ot_{\mathcal{D}}\mathbb{A}:\cdots {\longrightarrow} T_p^{e}\ot_{\mathcal{D}}{A^{e}}{\longrightarrow} T_p^{e}\ot_{\mathcal{D}}{A^{e}} {\longrightarrow} T_p^{e}\ot_{\mathcal{D}}{A^{e}}{\longrightarrow} T_p^{e}\ot_{\mathcal{D}}{A} \longrightarrow 0.
\end{equation}
It is known that, $ T_p\cong T_p^{e}\ot_{\mathcal{D}}{A} $ as $T_p$-bimodules via the map sending $x^i\ot g^k$ to $(1\ot g^k)\ot_{\mathcal{D}}x^i$ \cite[Section 3.5]{HH}. Then we have $ A\ot T_p\cong T_p^{e}\ot_{\mathcal{D}}{A^e} $ with the $T_p$-bimodule isomorphism given by
\begin{equation}\label{iso Tp}
\kappa(x^i\ot(x^j\ot g^k))=(1\ot g^k)\ot_{\mathcal{D}}(x^i\ot x^j).
\end{equation}
Then, we obtain the following resolution $\tilde{\mathbb{A}}$ which is isomorphic to the resolution (\ref{resol}), i.e.
\begin{equation}\label{resol2}
\tilde{\mathbb{A}}:\cdots\stackrel{\tilde{u}.}{\longrightarrow} A\ot T_p\stackrel{\tilde{v}.} {\longrightarrow}A\ot T_p\stackrel{\tilde{u}.}{\longrightarrow} A\ot T_p \stackrel{\tilde{\pi}.}{\longrightarrow}T_p\longrightarrow 0
\end{equation}
where $\tilde{v}=v\ot id_{kG},\tilde{u}=u\ot id_{kG}$, and $\tilde{\pi}=\pi\ot id_{kG}$.
The following lemma gives us a contracting homotopy for the identity map on the resolution $\tilde{\mathbb{A}}$.
\begin{lemma}\label{tildehn}
Let $h_n$ be a contracting homotopy in \eqref{hn}. Then $\tilde{h}_n=h_n\otimes1_{kG}$ forms a contracting homotopy for the identity map on $\tilde{\mathbb{A}}$.
\end{lemma}
\begin{proof}
For $n\geq 0$, the domain of $h_n\ot 1_{kG}$ is $A\otimes A\otimes kG$ which is $A\otimes T_p$ as a vector space. Moreover, by definition of contracting homotopy, $h_n$ satisfy
$$h_{i-1}d_i+d_{i+1}h_{i}=id_{\mathbb{A}_i}.$$
Then,
\begin{align*}
\tilde{h}_{i-1}\tilde{d}_i+\tilde{d}_{i+1}\tilde{h}_{i}&=(h_{i-1}\ot id_{kG})(d_i\ot id_{kG})+(d_{i+1}\ot id_{kG})(h_{i}\ot id_{kG})\\
&=(h_{i-1}d_i\ot id_{kG})+(d_{i+1}h_{i}\ot id_{kG})=(h_{i-1}d_i+d_{i+1}h_{i})\ot id_{kG}\\
&=id_{\mathbb{A}_i}\ot id_{kG}=id_{\tilde{\mathbb{A}}_i}
\end{align*}
and that implies $\tilde{h}_n$ is a contracting homotopy for $\tilde{\mathbb{A}}$. The proof is similar for $n=-1$.
\end{proof}
We abbreviate $a_1\ot a_2\ot g\in A\ot T_p$ by $a_1\ot a_2g$. By the Lemma \ref{tildehn}, we obtain
\begin{align*}
\tilde{h}_{-1}(x^i g)&=\xi_0x^i g,\\
\tilde{h}_{0}(x^i\xi_0x^j g)&=\sum_{l=0}^{i-1}x^l\xi_1x^{i+j-1-l} g,\\
\tilde{h}_{1}(x^i\xi_1x^j g)&=\delta_{i,p-1}x^j\xi_2 g,\\
\tilde{h}_{2n}(x^i\xi_{2n}x^jg)&=-\sum_{l=0}^{j-1}x^{i+j-1-l}\xi_{2n+1}x^lg,\\
\tilde{h}_{2n+1}(x^i\xi_{2n+1}x^jg)&=\delta_{j,p-1}x^i\xi_{2n+2} g.
\end{align*}
We need a lemma to have the linear maps $\tilde{\phi}_i:(\tilde{\mathbb{A}}\otimes_{T_p}\tilde{\mathbb{A}})_i\longrightarrow \tilde{\mathbb{A}}_{i+1}$. However, we first mention that there is an isomorphism from $(A\ot T_p)\ot_{T_p}(A\ot T_p)$ to $(A\ot A)\ot_A(A\ot A)\ot kG$ as $T_p^e$-modules given by
\begin{equation}\label{iso ATp}
\psi((x^{i_1}\ot x^{j_1}g^{k_1})\ot_{T_p}(x^{i_2}\ot x^{j_2}g^{k_2}))=\omega^{k_1(i_2+j_2)}(x^{i_1}\ot x^{j_1})\ot_A(x^{i_2}\ot x^{j_2}) g^{(k_1+k_2)}.
\end{equation}
\begin{lemma}\label{lemma tildephi}
Let $F_\mathbb{A}=(\pi\otimes_A id_\mathbb{A}- id_\mathbb{A}\otimes_A \pi)$ be the chain map for the resolution $\mathbb{A}$ in \eqref{res A} which is used for calculation of $\phi$ in \eqref{phin}. Then $F_{\tilde{\mathbb{A}}}:\tilde{\mathbb{A}}\ot_{T_p}\tilde{\mathbb{A}}\to \tilde{\mathbb{A}}$ defined by $(\tilde{\pi}\otimes_{T_p} id_{\tilde{\mathbb{A}}}- id_{\tilde{\mathbb{A}}}\otimes_{T_p} \tilde{\pi})$ is exactly $(F_\mathbb{A}\ot id_{kG})\psi$. Moreover $\tilde{\phi}:=(\phi\ot id_{kG})\psi$ is a contracting homotopy for $F_{\tilde{\mathbb{A}}}$.
\end{lemma}
\begin{proof}
Let $(x^{i_1}\ot x^{j_1}g^{k_1})\ot_{T_p}(x^{i_2}\ot x^{j_2}g^{k_2})\in (A\ot T_p)\ot_{T_p}(A\ot T_p)$. Note that $F_{\tilde{\mathbb{A}}}$ is zero if degrees of $(x^{i_1}\ot x^{j_1}g^{k_1})$ and $(x^{i_2}\ot x^{j_2}g^{k_2})$ are both nonzero since $\tilde{\pi}$ is only defined on degree zero. Also remember that $\tilde{\pi}=\pi\ot id_{kG}$ for the resolution $\tilde{\mathbb{A}}$.
We check the case that the degree of $(x^{i_1}\ot x^{j_1}g^{k_1})$ is zero and the degree of $(x^{i_2}\ot x^{j_2}g^{k_2})$ is nonzero. By using definition of $F_{\tilde{\mathbb{A}}}$, we obtain
\begin{align*}
F_{\tilde{\mathbb{A}}}((x^{i_1}\ot x^{j_1}g^{k_1})\ot_{T_p}(x^{i_2}\ot x^{j_2}g^{k_2}))&=(x^{i_1+j_1}g^{k_1})\ot_{T_p}(x^{i_2}\ot x^{j_2}g^{k_2})\\
&=\omega^{k_1(i_2+j_2)}x^{i_1+i_2+j_1}\ot x^{i_2}g^{k_1+k_2}.
\end{align*}
On the other hand, we also have
\begin{align*}
&(F_\mathbb{A}\ot id_{kG})\psi((x^{i_1}\ot x^{j_1}g^{k_1})\ot_{T_p}(x^{i_2}\ot x^{j_2}g^{k_2}))\\
&=(F_\mathbb{A}\ot id_{kG})(\omega^{k_1(i_2+j_2)}(x^{i_1}\ot x^{j_1})\ot_A (x^{i_2}\ot x^{j_2})g^{k_1+k_2})\\
&=\omega^{k_1(i_2+j_2)}x^{i_1+i_2+j_1}\ot x^{i_2}g^{k_1+k_2}.
\end{align*}
The proof for other cases are similar. Hence $F_{\tilde{\mathbb{A}}}$ and $(F_\mathbb{A}\ot id_{kG})\psi$ are identical.
In order to prove $\tilde{\phi}:=(\phi \ot id_{kG})\psi$ is a contracting homotopy for $F_{\tilde{\mathbb{A}}}$, we need to show that $$\tilde{d}_{\tilde{\mathbb{A}}}\tilde{\phi}+\tilde{\phi}\tilde{d}_{\tilde{\mathbb{A}}\ot_{T_p}\tilde{\mathbb{A}}}= F_{\tilde{\mathbb{A}}}.$$
It is clear that
\begin{equation}\label{d phi}
\tilde{d}_{\tilde{\mathbb{A}}}\tilde{\phi}=(d_{\mathbb{A}}\ot id_{kG})(\phi \ot id_{kG})\psi=(d_{\mathbb{A}}\phi \ot id_{kG})\psi.
\end{equation}
We now claim that
\begin{equation}\label{d_AotA}
\psi\tilde{d}_{\tilde{\mathbb{A}}\ot_{T_p}\tilde{\mathbb{A}}}=(d_{\mathbb{A}\ot_A \mathbb{A}}\ot id_{kG})\psi.
\end{equation}
By definition $$\tilde{d}_{\tilde{\mathbb{A}}\ot_{T_p}\tilde{\mathbb{A}}}=\tilde{d}_{\tilde{\mathbb{A}}}\ot_{T_p}id_{T_p}+(-1)^{*} id_{T_p} \ot_{T_p} \tilde{d}_{\tilde{\mathbb{A}}}$$ where $*$ is the degree of the element in left $A\ot T_p$. Moreover, $(A\ot T_p)\ot_{T_p}(A\ot T_p)$ is generated by $\xi_m1_G\ot_{T_p} x^i\xi_n1_G$ as $T_p$-bimodule. Without loss of generality, assume $m$ and $n$ are odd. Then we have the following calculation:
\begin{flalign*}
&\begin{aligned}
&\psi \tilde{d}_{\tilde{\mathbb{A}}\ot_{T_p}\tilde{\mathbb{A}}}(\xi_m1_G\ot_{T_p} x^i\xi_n1_G)\\
&=\psi ((x\xi_m1_G-\xi_mx1_G)\ot_{T_p}x^{i}\xi_n1_G- \xi_m1_G\ot_{T_p} (x^{i+1}\xi_n1_G-x^i\xi_n x1_G))\\
&=(x\xi_m-\xi_mx)\ot_Ax^{i}\xi_n1_G-\xi_m\ot_A (x^{i+1}\xi_n-x^i\xi_n x)1_G
\end{aligned}&&
\end{flalign*}
and
\begin{flalign*}
&\begin{aligned}
&(d_{\mathbb{A}\ot_A \mathbb{A}}\ot id_{kG})\psi(\xi_m1_G\ot_{T_p} x^i\xi_n1_G)\\
&=(d_{\mathbb{A}\ot_A \mathbb{A}}\ot id_{kG})(\xi_m\ot_A x^i\xi_n1_G)\\
&=(x\xi_m-\xi_mx)\ot_Ax^{i}\xi_n1_G-\xi_m\ot_A (x^{i+1}\xi_n-x^i\xi_n x)1_G.
\end{aligned}&&
\end{flalign*}
The calculation is similar for the other cases of $m$ and $n$. Therefore,
\begin{equation}\label{phi d}
\tilde{\phi}\tilde{d}_{\tilde{\mathbb{A}}\ot_{T_p}\tilde{\mathbb{A}}}=(\phi \ot id_{kG})\psi\tilde{d}_{\tilde{\mathbb{A}}\ot_{T_p}\tilde{\mathbb{A}}}=(\phi \ot id_{kG})(d_{\mathbb{A}\ot_A\mathbb{A}} \ot id_{kG})\psi=(\phi d_{\mathbb{A}\ot_A\mathbb{A}} \ot id_{kG})\psi.
\end{equation}
By combining \eqref{d phi} and \eqref{phi d}, we obtain
\begin{equation*}
\tilde{d}_{\tilde{\mathbb{A}}}\tilde{\phi}+\tilde{\phi}\tilde{d}_{\tilde{\mathbb{A}}\ot_{T_p}\tilde{\mathbb{A}}}=((d_\mathbb{A}\phi+\phi d_{ \mathbb{A}\otimes_A \mathbb{A}})\ot id_{kG})\psi=(F_\mathbb{A}\ot id_{kG})\psi= F_{\tilde{\mathbb{A}}}
\end{equation*}
whence $\tilde{\phi}=(\phi\ot id_{kG})\psi$ is a contracting homotopy for $F_{\tilde{\mathbb{A}}}$.
\end{proof}
We use the Lemma \ref{lemma tildephi} and find the following $T_p^e$-linear maps $\tilde{\phi}_i:(\tilde{\mathbb{A}}\otimes_{T_p}\tilde{\mathbb{A}})_i\longrightarrow \tilde{\mathbb{A}}_{i+1}$:
\begin{align*}
&\tilde{\phi}_0(\xi_01_G\ot_{T_p} x^i\xi_01_G)=\sum_{l=0}^{i-1}x^l\xi_1x^{i-1-l}1_G,\\
&\tilde{\phi}_1(\xi_11_G\ot_{T_p} x^i\xi_01_G)=-\delta_{i,p-1}\xi_21_G,\\
&\tilde{\phi}_1(\xi_01_G\ot_{T_p} x^i\xi_11_G)=\delta_{i,p-1}\xi_21_G.
\end{align*}
Next, we give a lemma to find the the diagonal map.
\begin{lemma}\label{lemma diagonal}
The map $\tilde{\Delta}:=\psi^{-1}(\Delta\ot id_{kG})$ is a diagonal map on $\tilde{\mathbb{A}}$ where $\Delta$ is in \eqref{Deltan}.
\end{lemma}
\begin{proof}
We need to check that $\tilde{\Delta}$ is a chain map. The following equations are straightforward by considering the fact that $\Delta$ is a chain map and \eqref{d_AotA}:
\begin{align*}
\tilde{d}_{\tilde{\mathbb{A}}\ot_{T_p}\tilde{\mathbb{A}}}\tilde{\Delta}&=\tilde{d}_{\tilde{\mathbb{A}}\ot_{T_p}\tilde{\mathbb{A}}}\psi^{-1}(\Delta\ot id_{kG})=\psi^{-1}(d_{\mathbb{A}\ot_A \mathbb{A}}\ot id_{kG})(\Delta\ot id_{kG})\\
&=\psi^{-1}(d_{\mathbb{A}\ot_A \mathbb{A}}\Delta\ot id_{kG})=\psi^{-1}(\Delta d_{\mathbb{A}}\ot id_{kG})=\psi^{-1}(\Delta\ot id_{kG})( d_{\mathbb{A}}\ot id_{kG})\\
&=\tilde{\Delta}\tilde{d}_{\tilde{\mathbb{A}}}.
\end{align*}
\end{proof}
Lemma \ref{lemma diagonal} allows us to compute the $T_p$-linear map $\tilde{\Delta}:\tilde{\mathbb{A}}_{i+1}\longrightarrow (\tilde{\mathbb{A}}\otimes_{T_p}\tilde{\mathbb{A}})_i$ as follows:
\begin{align*}
\tilde{\Delta}_0(\xi_0 1_G) &= \xi_01_G\ot_{T_p}\xi_01_G,\\
\tilde{\Delta}_1(\xi_11_G) &=\xi_11_G\ot_{T_p}\xi_0 1_G + \xi_01_G \ot_{T_p} \xi_11_G,\\
\tilde{\Delta}_{2n}(\xi_{2n}1_G) &= \sum_{i=0}^{n}\xi_{2i} 1_G\ot_{T_p}\xi_{2n-2i} 1_G \\
&+\sum_{i=0}^{n-1}\sum_{\substack{a+b+c \\ =p-2}}^{}x^a\xi_{2i+1} 1_G\ot_{T_p}x^b\xi_{2n-2i-1}x^c1_G, \text{ for }n\geq 1\\
\tilde{\Delta}_{2n+1}(\xi_{2n+1} 1_G) &= \sum_{i=0}^{2n+1}\xi_i 1_G\ot_{T_p} \xi_{2n+1-i} 1_G, \text{ for }n\geq 1.
\end{align*}
Before computing the bracket on Hochschild cohomology of $T_p$, we need to find a basis of $\text{Hom}_{T_p^e}(\tilde{\mathbb{A}}, T_p)$. In particular, we must find a basis of $\text{Hom}_{T_p^e}(A\ot T_p, T_p)$ as it is an invariant in each degree.
It is known that $$\HH^*(T_p):=\text{Ext}^*_{T_p^{e}}(T_p,T_p)\cong\text{Ext}^*_{\mathcal{D}}(A,T_p)\cong \text{Ext}^*_{A^{e}}(A,T_p)^G.$$
The Eckmann-Shapiro Lemma (Lemma \ref{Eck}) and (\ref{iso Tp}) imply the first isomorphism and see \cite[Theorem 3.6.2]{HH} for the second isomorphism.
Consider the following resolution
\begin{equation}\label{2nd}
\text{Hom}_{A^e}(\mathbb{A},T_p)^G:0{\longrightarrow} \text{Hom}_{A^e}(A^e,T_p)^G{\longrightarrow} \text{Hom}_{A^e}(A^e,T_p)^G {\longrightarrow} \cdots
\end{equation}
where the action of $G$ on $\text{Hom}_{A^e}(A^e,T_p)^G$ is defined by
\begin{equation}\label{action on T_p}
g\cdot f(a_1\ot a_2)={^g}f(^{g^{-1}}(a_1\ot a_2)).
\end{equation}
This resolution is clearly isomorphic to
\begin{equation}\label{1st}
0{\longrightarrow} T_p^{G}{\longrightarrow} T_p^{G} {\longrightarrow} T_p^{G} \longrightarrow \cdots
\end{equation}
with the correspondence
\begin{equation}\label{corres}
f_t\mapsto t \text{ where } f_t(\xi_*)=t\text{ for all } t\in T_p.
\end{equation}
We claim that $\text{Hom}_{T_p^e}(A\ot T_p, T_p)\cong T_p^G$. Suppose $x^ig^j \in T_p^{G}$. Then, we have $f_{x^ig^j}\in \text{Hom}_{A^e}(A^e,T_p)^G$ defined by $f_{x^ig^j}(x^k\ot x^l):=x^{k+l+i}g^j$ where $x^*\in A$. Now observe that, $f_{x^ig^j}\in \text{Hom}_{A^e}(A^e,T_p)^G$ is a $\mathcal{D}$-module homomorphism since
\begin{align*}
f_{x^ig^j}((x^k\xi_* x^l g)(a_1\ot a_2))&=f_{x^ig^j}((x^k\xi_* x^l 1_G)g(a_1\ot a_2))=(x^k\xi_* x^l 1_G)f_{x^ig^j}(g(a_1\ot a_2))\\
&=(x^k\xi_*x^l 1_G)gf_{x^ig^j}(a_1\ot a_2)=(x^k\xi_* x^l g)f_{x^ig^j}(a_1\ot a_2)
\end{align*}
where $x^k\xi_* x^l g\in \mathcal{D}, a_1\ot a_2\in A^e$. Moreover, if $f\in \text{Hom}_{\mathcal{D}}(A^e,T_p)$, then $f$ is $G$-invariant as
$$g\cdot f(a_1\ot a_2)={^g}f(^{g^{-1}}(a_1\ot a_2))={^{(gg^{-1})}}f(a_1\ot a_2)=f(a_1\ot a_2)$$
where $g\in G, a_1\ot a_2\in A^e$. Hence, the isomorphism
from $\text{Hom} _{A^e}(A^e,T_p)^G$ to $\text{Hom}_{\mathcal{D}}(A^e,T_p)$ is the identity, so that $f_{x^ig^j}$ is also in $\text{Hom}_{\mathcal{D}}(A^e,T_p)$. We next use the Eckmann-Shapiro lemma (Lemma \ref{Eck}) which implies that $\text{Ext}_{\mathcal{D}}^{*}(A,T_p)\cong \text{Ext}_{T_p^{e}}^{*}(T_p^{e}\ot_\mathcal{D}A,T_p)$ and the isomorphism is given by
\begin{align*}
\sigma(f_{x^ig^j})(x^mg^s\ot x^ng^r\ot_{\mathcal{D}}x^k\ot x^l)&=x^mg^s\ot x^ng^rf_{x^ig^j}(x^k\ot x^l)=x^mg^s\ot x^ng^r(x^{k+l+i}g^j)\\
&=(x^mg^s)(x^{k+l+i}g^j)(x^ng^r)\\
&=((x^m(^{g^s}x^{k+l+i}))g^{s+j})(x^ng^r)\\
&=\omega^{s(k+l+i)}(x^{m+k+l+i}g^{s+j})(x^ng^r)\\
&=\omega^{s(k+l+i)}(x^{m+k+l+i}(^{g^{s+j}}x^n))g^{j+s+r}\\
&=\omega^{s(k+l+i+n)+jn}x^{i+k+l+m+n}g^{j+s+r}.
\end{align*}
Hence, $\sigma(f_{x^ig^j})$ is in $\text{Hom}_{T_p^{e}}(T_p^{e}\ot_{\mathcal{D}} A^e,T_p)$. Lastly, recall that $T_p^{e}\ot_{\mathcal{D}}A^e\cong A\ot T_p$ via $\kappa$ (\ref{iso Tp}); so that,
\begin{align*}
\kappa^*(\sigma(f_{x^ig^j}))(x^k\ot x^lg^r)=\sigma(f_{x^ig^j})((1_{T_p}\ot \xi_*g^r)\ot_{\mathcal{D}}x^k\ot x^l)=x^{i+k+l}g^{j+r}
\end{align*}
which implies $\kappa^{*}(\sigma(f_{x^ig^j}))\in \text{Hom}_{T_p^e}(A\ot T_p, T_p)$. For simplicity, we define $\tilde{f}_{x^ig^j}:=\kappa^{*}(\sigma(f_{x^ig^j}))$.
The action of $G$ on $T_p$ given by \eqref{action on T_p} and \eqref{corres} depends on degree. Since $T_p^{G}$ is spanned by $\{1,g,\cdots,g^{p-1}\}$ in even degrees and $\{x,xg,\cdots, xg^{p-1}\}$ in odd degrees \cite[Section 8.2]{Ng}, we have $\{\tilde{f}_1,\tilde{f}_g,\cdots, \tilde{f}_{g^{p-1}}\}$ in even degrees and $\{\tilde{f}_x,\tilde{f}_{xg},\cdots,\tilde{f}_{xg^{p-1}}\}$ in the odd degrees as a basis of $\text{Hom}_{T_p^e}(A\ot T_p, T_p)$.
We only calculate the bracket in degree 1 and 2 as before so we can extend it to higher degrees by the relation between cup product and the bracket. Since $A\ot T_p\cong A^e\ot kG$ as vector spaces, $\xi_i 1_G$ generates $A\ot T_p$ as a $T_p$-bimodule. Through the calculation, $id$ represents $id_{A\ot T_p}$ and $\ot$ represents $\ot_{T_p}$.
The circle product of two elements in degree one is
\begin{align*}
(\tilde{f}_{xg^i}\circ_{\tilde{\phi}}\tilde{f}_{xg^j})(\xi_11_G)&=\tilde{f}_{xg^i}\tilde{\phi}(id\otimes\tilde{f}_{xg^j}\ot id)\tilde{\Delta}^{(2)}(\xi_1 1_G)\\
&=\tilde{f}_{xg^i}\tilde{\phi}(id\ot \tilde{f}_{xg^j}\ot id)(\xi_01_G\ot\xi_01_G\ot\xi_11_G+\xi_01_G\ot\xi_11_G\ot\xi_01_G\\
&+\xi_11_G\ot\xi_01_G\ot\xi_01_G)\\
&=\tilde{f}_{xg^i}\tilde{\phi}(\xi_01_G\ot x\xi_0g^j)=\tilde{f}_{xg^i}(\xi_1 g^j)=xg^{i+j}.
\end{align*}
Because of the symmetry, $(\tilde{f}_{xg^j}\circ_{\tilde{\phi}}\tilde{f}_{xg^i})(\xi_11_G)=xg^{i+j}$. Therefore $$[\tilde{f}_{xg^i},\tilde{f}_{xg^j}](\xi_11_G)=xg^{i+j}-(-1)^{0} xg^{i+j}=0.$$
The circle product of the elements of degrees 1 and 2:
\begin{align*}
(\tilde{f}_{xg^i}\circ_{\tilde{\phi}}\tilde{f}_{g^j})(\xi_21_G)&=\tilde{f}_{xg^i}\tilde{\phi}(id\ot\tilde{f}_{g^j}\ot id)\tilde{\Delta}^{(2)}(\xi_2 1_G)=\tilde{f}_{xg^i}\tilde{\phi}(id\ot \tilde{f}_{g^j}\ot id)\\
&(\xi_01_G\ot\xi_01_G\ot\xi_21_G+\xi_01_G\ot\xi_21_G\ot\xi_01_G\\
&+\xi_01_G\ot\sum_{\substack{a+b+c \\ =p-2}}(x^a\xi_11_G\ot x^b\xi_1x^c1_G)+\xi_21_G\ot\xi_01_G\ot\xi_01_G\\
&+\sum_{\substack{a+b+c \\ =p-2}}(x^a\xi_11_G\ot (x^b\xi_01_G\ot \xi_1x^c1_G+x^b\xi_11_G\ot\xi_0x^c1_G)))\\
&=\tilde{f}_{xg^i}\tilde{\phi}(\xi_01_G\ot\xi_0g^j)=0.
\end{align*}
And the circle product on the reverse order:
\begin{align*}
(\tilde{f}_{g^j}\circ_{\tilde{\phi}}\tilde{f}_{xg^i})(\xi_21_G)&=\tilde{f}_{g^j}\tilde{\phi}(id\ot\tilde{f}_{xg^i}\ot id)\tilde{\Delta}^{(2)}(\xi_2 1_G)=\tilde{f}_{g^j}\tilde{\phi}(id\ot \tilde{f}_{xg^i}\ot id)\\
&(\xi_01_G\ot\xi_01_G\ot\xi_21_G+\xi_01_G\ot\xi_21_G\ot\xi_01_G\\
&+\xi_01_G\ot\sum_{\substack{a+b+c \\ =p-2}}(x^a\xi_11_G\ot x^b\xi_1x^c1_G)+\xi_21_G\ot\xi_01_G\ot\xi_01_G\\
&+\sum_{\substack{a+b+c \\ =p-2}}(x^a\xi_11_G\ot (x^b\xi_01_G\ot \xi_1x^c1_G+x^b\xi_11_G\ot \xi_0x^c1_G)))\\
&=\tilde{f}_{g^j}\tilde{\phi}(\sum_{\substack{a+b+c \\ =p-2}}\omega^{i(b+c)}\xi_01_G\ot x^{a+b+1}\xi_1x^cg^i+\omega^{ic}x^a\xi_11_G\ot x^{b+1}\xi_0x^cg^i)\\
&=\tilde{f}_{g^j}(\sum_{\substack{a+b+c \\ =p-2}}\omega^{i(b+c)}\delta_{a+b+1,p-1}x^c\xi_2g^i-\omega^{ic}\delta_{b+1,p-1}x^{a+c}\xi_2 g^i)\\
&=\tilde{f}_{g^j}(\sum_{b=0}^{p-2}\omega^{ib}\xi_2g^i)-\tilde{f}_{g^j}(\xi_2g^i)\\
&=\left\{\begin{array}{lr}
(p-2)g^j, & \text{for } i=0\\
-(\omega^{-i}+1)g^{i+j}, & \text{for } i\neq 0
\end{array}\right..
\end{align*}
Therefore, we obtain
$$[\tilde{f}_{xg^i},\tilde{f}_{g^j}]=\left\{\begin{array}{lr}
-(p-2)g^j, & \text{for } i=0\\
(\omega^{-i}+1)g^{i+j}, & \text{for } i\neq 0
\end{array}\right..$$
Lastly, the bracket of the elements of degrees 2 and 2:\\
\begin{align*}
(\tilde{f}_{g^i}\circ_{\tilde{\phi}}\tilde{f}_{g^j})(\xi_31_G)&=\tilde{f}_{g^i}\tilde{\phi}(id\ot\tilde{f}_{g^j}\ot id)\tilde{\Delta}^{(2)}(\xi_31_G)=\tilde{f}_{g^i}\tilde{\phi}(id\ot\tilde{f}_{g^j}\ot id)\\
&(\xi_01_G\ot\xi_01_G\ot\xi_31_G+\xi_01_G\ot\xi_11_G\ot\xi_21_G+\xi_01_G\ot\xi_21_G\ot\xi_11_G\\
&+\xi_01_G\ot\xi_31_G\ot\xi_01_G+\xi_11_G\ot\xi_21_G\ot\xi_01_G+\xi_11_G\ot\xi_01_G\ot\xi_21_G\\
&+\xi_21_G\ot\xi_11_G\ot\xi_01_G+\xi_21_G\ot\xi_01_G\ot\xi_11_G+\xi_31_G\ot\xi_01_G\ot\xi_01_G)\\
&=\tilde{f}_{g^i}\tilde{\phi}(\xi_01_G\ot\xi_1g^j+\xi_11_G\ot\xi_0g^j)=0
\end{align*}
and by symmetry
$(\tilde{f}_{g^j}\circ_{\tilde{\phi}}\tilde{f}_{g^i})(\xi_31_G)=0$. Therefore, we have
$[\tilde{f}_{g^i},\tilde{f}_{g^j}]=0$.
As a consequence, the bracket for the elements of degree 1 and 2 are
$$[\tilde{f}_{xg^i},\tilde{f}_{xg^j}]=0,[\tilde{f}_{xg^i},\tilde{f}_{g^j}]=\left\{\begin{array}{lr}
-(p-2)g^j, & \text{for } i=0\\
(\omega^{-i}+1)g^{i+j}, & \text{for } i\neq 0
\end{array}\right.,[\tilde{f}_{g^i},\tilde{f}_{g^j}]=0.$$
By the identity (\ref{cupbrac}), brackets in higher degrees can be determined, since the
Hochschild cohomology is generated as an algebra under cup product in degrees 1
and 2.
Hopf algebra cohomology of $T_p$ and Hochschild cohomology of $T_p$ were calculated before by V. C. Nguyen \cite[Section 8]{Ng} as the Hopf algebra cohomology
\[
\text{H}^n(T_p,k)=
\begin{cases}
k & \text{if $n$ is even,} \\
0 & \text{if $n$ is odd,}
\end{cases}
\]
and the Hochschild cohomology
\[
\text{HH}^n(T_p,k)=
\begin{cases}
k & \text{if $n$ is even,} \\
Span_k\{x\} & \text{if $n$ is odd.}
\end{cases}
\]
It is known that for any Hopf algebra with bijective antipode, the Hopf algebra cohomology can be embedded into the Hochschild cohomology \cite[Theorem 9.4.5 and Corollary 9.4.7]{HH}. Since any finite dimensional Hopf algebra has a bijective antipode, the Taft algebra $T_p$ is also a Hopf algebra with a bijective antipode. The embedding of $\text{H}^n(T_p,k)$ into $\text{HH}^n(T_p,T_p)$ turns out to be the map that is identity in even degrees and zero on odd degrees. Then, the corresponding bracket in Hopf algebra cohomology is
$$[\tilde{f}_{g^i},\tilde{f}_{g^j}]=0,$$
so that, the bracket on Hopf algebra cohomology for the elements of all degrees is 0 by the identity (\ref{cupbrac}).
This is the first example of the Gerstenhaber bracket on the Hopf algebra cohomology of a nonquasi-triangular Hopf algebra and our calculation shows that the bracket on Hopf algebra cohomology of a Taft algebra is zero as it is on the Hopf algebra cohomology of any quasi-triangular algebra. A natural question that arises whether the bracket structure on the Hopf algebra cohomology is always trivial. In the next section, we explore a general expression for the bracket on the Hopf algebra cohomology that may help us to approach this question with a more theoretical perspective in the future researches.
\section{Gerstenhaber bracket for Hopf algebras}
In this section, we want to explore an expression for Gerstenhaber bracket on a Hopf algebra $A$ with a bijective antipode $S$.
We give the following lemma which helps us to define the Gerstenhaber bracket on an equivalent resolution to the bar resolution of $A$ as an $A$-bimodule.
\begin{lemma}
Let $A$ be a Hopf algebra with bijective antipode. Let $P_{\bullet}$ be the bar resolution of $k$ as a left $A$-module:
\begin{equation*}
P_{\bullet}:\cdots \stackrel{d_3}{\longrightarrow} A^{\otimes 3} \stackrel{d_2}{\longrightarrow} A^{\otimes 2} \stackrel{d_1}{\longrightarrow} A \stackrel{\varepsilon}\longrightarrow k\longrightarrow 0,
\end{equation*}
with differentials $$d_n(a_0\otimes a_1\otimes\cdots \otimes a_{n})=\sum_{i=0}^{n-1}(-1)^ia_0\otimes a_1\otimes \cdots \otimes a_ia_{i+1}\otimes\cdots \otimes a_n+(-1)^n\varepsilon(a_n)a_0\ot \cdots\ot a_{n-1}$$
Then $X_{\bullet}=A^e\ot_AP_{\bullet}$ is equivalent to the bar resolution of $A$ as an $A$-bimodule.
\end{lemma}
\begin{proof}
Since $S$ is bijective \cite[Lemma 9.2.9]{HH}, $A^e$ is projective as a right $A$-module. Also there is an $A^e$-module isomorphism $\rho:A\to A^e\ot_Ak$ defined by $\rho(a)=a\ot 1\ot 1$ for all $a\in A$ \cite[Lemma 9.4.2]{HH}.
For each $n$, define $\theta_n:X_n\to A^{\ot (n+2)}$ by
$$\theta_n((a\ot b)\ot_A(1\ot c^1\ot c^2\ot \cdots \ot c^n))=\sum a\ot c_1^1\ot c_1^2\ot\cdots\ot c_1^n\ot S(c_2^1c_2^2\cdots c_2^n)b$$
for all $a,b,c^1,\cdots c^n\in A.$
Now, we show that $\theta$ is a chain map:
\begin{flalign*}
\theta&_{n-1}d_n((a\ot b)\ot_A(1\ot c^1\ot c^2\ot \cdots \ot c^n))\\
=&\theta_{n-1}((a\ot b)\ot_A(c^1\ot c^2\ot \cdots \ot c^n)\\
&+\sum_{i=1}^{n-1}(-1)^i(a\ot b)\ot_A(1\ot c^1\ot c^2\ot \cdots\ot c^ic^{i+1}\ot\cdots \ot c^n)\\
&+(-1)^n(a\ot b)\ot_A(\varepsilon(c^n)\ot c^1\ot c^2\ot \cdots \ot c^{n-1}))\\
=&\theta_{n-1}(\sum (ac_1^1\ot S(c_2^1) b)\ot_A(1\ot c^2\ot \cdots \ot c^n)\\
&+\sum_{i=1}^{n-1}(-1)^i(a\ot b)\ot_A(1\ot c^1\ot c^2\ot \cdots\ot c^ic^{i+1}\ot\cdots \ot c^n)\\
&+(-1)^n(\varepsilon(c^n)a\ot b)\ot_A(1 \ot c^1\ot c^2\ot \cdots \ot c^{n-1}))\\
=&\sum ac_1^1\ot c_1^2\ot \cdots \ot c_1^n\ot S(c_2^2\cdots c_2^n) S(c_2^1)b\\
&+\sum_{i=1}^{n-1}(-1)^i\sum a\ot c_1^1\ot \cdots \ot c_1^ic_1^{i+1}\ot \cdots \ot c_1^n\ot S(c_2^1\cdots c_2^n)b\\
&+\sum (-1)^na\ot c_1^1\ot \cdots \ot c_1^{n-1}\ot \varepsilon(c^n)S(c_2^1\cdots c_2^{n-1})b
\end{flalign*}
and
\begin{align*}
d_n&\theta_n((a\ot b)\ot_A(1\ot c^1\ot c^2\ot \cdots \ot c^n))\\
=&d_n(\sum a\ot c_1^1\ot c_1^2\ot\cdots\ot c_1^n\ot S(c_2^1c_2^2\cdots c_2^n)b)\\
=&\sum ac_1^1\ot c_1^2\ot \cdots \ot c_1^n\ot S(c_2^1c_2^2\cdots c_2^n)b\\
&+\sum\sum_{i=1}^{n-1}(-1)^i a\ot c_1^1\ot \cdots \ot c_1^ic_1^{i+1}\ot \cdots \ot c_1^n\ot S(c_2^1\cdots c_2^n)b\\
&+\sum (-1)^na\ot c_1^1\ot \cdots \ot c_1^{n-1}\ot c_1^nS(c_2^1\cdots c_2^{n})b.
\end{align*}
Since $S$ is an algebra anti-homomorphism that is convolution inverse to the identity map,
$$\sum c_1^nS(c_2^1\cdots c_2^n)=\sum c_1^nS(c_2^n)S(c_2^{n-1})\cdots S(c_2^1)=\sum \varepsilon(c^n)S(c_2^1\cdots c_2^{n-1})$$
and
$$S(c_2^2\cdots c_2^{n})S(c_2^1)=S(c_2^1c_2^2\cdots c_2^{n})$$
so that the two expressions are equal which follows $\theta$ is a chain map.
Lastly, one can see that the $A^e$-module homomorphism $$\psi_n(a\ot c^1\ot c^2\ot \cdots c^n\ot b)=\sum (a\ot c_2^1c_2^2\cdots c_2^nb)\ot_A (1\ot c_1^1\ot c_1^2\ot \cdots \ot c_1^n)$$
is the inverse of $\theta_n$ by using the property that $S$ is an algebra anti-homomorphism that is convolution inverse to the identity map.
\end{proof}
Let $f_x\in$Hom$_{A^e}(X_m,A)$ and $g_x\in$Hom$_{A^e}(X_n,A)$. Then we define the $X$-bracket $[f_x,g_x]_X\in$Hom$_{A^e}(X_{m+n-1},A)$ to be a composition $X\stackrel{\theta}\longrightarrow B(A)\xrightarrow{[\psi^{*}f_x.\psi^{*}g_x]} A$; so that, we have $$[f_x,g_x]_X=[\psi^{*}f_x,\psi^{*}g_x]\theta=(\psi^{*}f_x\circ \psi^{*}g_x)\theta-(-1)^{(m-1)(n-1)}(\psi^{*}g_x\circ \psi^{*}f_x)\theta$$
where
\begin{align*}
(\psi&^{*}f_x\circ \psi^{*}g_x)\theta_{m+n-1}((a\ot b)\ot_A 1\ot c^1\otimes \cdots \otimes c^{m+n-1})\\
=&(\psi^{*}f_x\circ \psi^{*}g_x)(\sum a\ot c_1^1\ot c_1^2\ot \cdots \ot c_1^{m+n-1}\ot S(c_2^1c_2^2\cdots c_2^{m+n-1})b)\\
=&\sum \sum_{i=1}^{m}(-1)^{(n-1)(i-1)}f_x\psi_m(a\otimes c_1^1\otimes\cdots \ot c_1^{i-1}\otimes g_x\psi_n(1\ot c_1^{i}\otimes \cdots \otimes c_1^{i+n-1}\ot 1)\\
&\otimes c_1^{i+n}\otimes \cdots \otimes c_1^{m+n-1}\ot S(c_2^1c_2^2\cdots c_2^{m+n-1})b)\\
=&\sum \sum_{i=1}^{m}(-1)^{(n-1)(i-1)}f_x\psi_m(a\otimes c_1^1\otimes\cdots \ot c_1^{i-1}\\
& \otimes\sum g_x(1\ot c_{2}^{i}c_2^{i+1}\cdots c_{2}^{i+n-1}\ot_A 1\ot c_{1}^{i}\ot c_{1}^{i+1}\ot \cdots \ot c_{1}^{i+n-1} )\\
&\otimes c_1^{i+n}\otimes \cdots \otimes c_1^{m+n-1}\ot S(c_2^1\cdots c_2^{i-1}c_3^{i}\cdots c_3^{i+n-1}c_2^{i+n} \cdots c_2^{m+n-1})b)\\
=&\sum \sum_{i=1}^{m}(-1)^{(n-1)(i-1)}f_x(a\otimes c_{2}^{1}c_{2}^{2}\cdots c_{2}^{i-1}c_{2}^{*} c_{2}^{i+n}\cdots c_{2}^{m+n-1}S(c_3^1c_3^2\cdots c_3^{m+n-1})b\\
& \ot_A 1\ot c_{1}^{1}\ot c_{1}^{2}\ot \cdots \ot c_{1}^{i-1}\ot c_{1}^{*}\ot c_{1}^{i+n}\ot \cdots \ot c_{1}^{m+n-1})\\
\end{align*}
where
\begin{align*}
\Delta(c)&=\sum c_1\ot c_2, \Delta^{(2)}(c)=\sum c_1\ot c_2\ot c_3,\Delta(c^*)=\sum c_{1}^*\ot c_{2}^*\text{ and }\\
c^*&=\sum g_x(1\ot c_{2}^{i}c_2^{i+1}\cdots c_{2}^{i+n-1}\ot_A 1\ot c_{1}^{i}\ot c_{1}^{i+1}\ot \cdots \ot c_{1}^{i+n-1} ).
\end{align*}
This is the general expression of the Gerstenhaber bracket on Hochschild cohomology of $A$. Next, we start with the following theorem \cite[Theorem 9.4.5]{HH} to construct an embedding from H$^*(A,k)$ into HH$^*(A)$.
\begin{theo}
Let $A$ be a Hopf algebra over k with bijective antipode. Then $$\HH^*(A)\cong \coh^*(A,A^{ad}).$$
\end{theo}
In this theorem $A^{ad}$ is an $A$-module $A$ under left adjoint action, given by \\
$a\cdot b=\sum a_1bS(a_2)$ for all $a,b\in A$. To find explicit isomorphism between HH$^*(A)$ and H$^*(A,A^{ad})$, we give the Eckmann-Shapiro lemma.
\begin{lemma}[Eckmann-Shapiro]\label{Eck}
Let $A$ be a ring and let $B$ be a subring of $A$ such that A is projective as a right $B$-module. Let $M$ be an $A$-module and $N$ be a $B$-module. Then $$\emph{Ext}_B^n(N,M)\cong \emph{Ext}_A^n(A\ot_BN,M). $$
\end{lemma}
\begin{proof}
Let $P_{\bullet}\to N$ be a $B$ projective resolution of $N$. Then $A\ot_BP_n$ is projective as A-module so that $A\ot_BP_{\bullet}\to A\ot_BN$ is a projective resolution of $A\ot_BN$ as an $A$-module. Let $$\sigma: \text{Hom}_B(P_n,M)\to \text{Hom}_A(A\ot_BP_n,M) \text{ defined by } \sigma(f)(a\ot_B p)=af(p),$$
$$\tau: \text{Hom}_A(A\ot_BP_n,M)\to \text{Hom}_B(P_n,M) \text{ defined by } \tau(g)(p)=g(1\ot_B p)$$
where $a\in A,p\in P_n,f\in \text{Hom}_B(P_n,M), g\in \text{Hom}_A(A\ot_BP_n,M).$ Since $\sigma$ and $\tau$ are inverse of each other and they are homomorphisms, $\text{Hom}_A(A\ot_BP_n,M)\cong \text{Hom}_B(P_n,M)$.
\end{proof}
If we replace $A$ with $A^e$, $B$ with $A$ and take $M=A,N=k$ in the Eckmann-Shapiro lemma, we have the isomorphism Ext$_{A^e}^n(A^e\ot_Ak,A)\cong$Ext$_A^n(k,A^{ad})$. We also know that $A\cong A^e\ot_A k$ \cite[Lemma 9.4.2]{HH} and the isomorphism is given by $\rho(a)=a\ot 1\ot 1$ for all $a\in A$. Therefore Ext$_{A^e}^n(A,A)\cong$Ext$_{A^e}^n(A^e\ot_Ak,A)\cong$Ext$_A^n(k,A^{ad})$.
We already have the Gerstenhaber bracket $[,]_X$ on Ext$_{A^e}^n(A^e\ot_Ak,A)$. Hence we can use the isomorphisms $\sigma$ and $\tau$ in Eckmann-Shapiro Lemma and find the bracket expression on H$^*(A,A^{ad})$. Now let $\tilde{f}\in$Hom$_{A}(P_m,A^{ad})$ and $\tilde{g}\in$Hom$_{A}(P_n,A^{ad})$. Then $[\tilde{f},\tilde{g}]_{P}\in$Hom$_{A}(P_{m+n-1},A^{ad})$ and we have
\begin{align*}
[\tilde{f},\tilde{g}]_P&=\tau[\sigma(\tilde{f}),\sigma(\tilde{g})]_X\\
&=\tau((\psi^*(\sigma(\tilde{f}))\circ\psi^*(\sigma(\tilde{g})))\theta)-(-1)^{(m-1)(n-1)}\tau((\psi^*(\sigma(\tilde{g}))\circ \psi^*(\sigma(\tilde{f})))\theta).
\end{align*}
For simplification we define $$\tilde{f}\circ_{P}\tilde{g}:=\tau((\psi^*(\sigma(\tilde{f}))\circ\psi^*(\sigma(\tilde{g})))\theta).$$
Then by using previous circle product formula we obtain:
\begin{align*}
\tilde{f}&\circ_{P}\tilde{g}(1\ot c^1\ot c^2\ot\cdots\ot c^{m+n-1})\\
=&\tau((\psi^*(\sigma(\tilde{f}))\circ\psi^*(\sigma(\tilde{g})))\theta)(1\ot c^1\ot c^2\ot\cdots\ot c^{m+n-1})\\
=&(\psi^*(\sigma(\tilde{f}))\circ\psi^*(\sigma(\tilde{g})))\theta((1\ot 1) \ot_A 1\ot c^1\ot c^2\ot\cdots\ot c^{m+n-1})\\
=&\sum \sum_{i=1}^{m}(-1)^{(n-1)(i-1)}\sigma(\tilde{f})(1\otimes c_{2}^{1}c_{2}^{2}\cdots c_{2}^{i-1}c_{2}^{*} c_{2}^{i+n}\cdots c_{2}^{m+n-1}S(c_3^1c_3^2\cdots c_3^{m+n-1})\\
&\ot_A 1\ot c_{1}^{1}\ot c_{1}^{2}\ot \cdots \ot c_{1}^{i-1}\ot c_{1}^{*}\ot c_{1}^{i+n}\ot \cdots \ot c_{1}^{m+n-1})\\
=&\sum \sum_{i=1}^{m}(-1)^{(n-1)(i-1)}\tilde{f}(1\ot c_{1}^{1}\ot c_{1}^{2}\ot \cdots \ot c_{1}^{i-1}\ot c_{1}^{*}\ot c_{1}^{i+n}\ot \cdots \ot c_{1}^{m+n-1})\\
&c_{2}^{1}c_{2}^{2}\cdots c_{2}^{i-1}c_{2}^{*} c_{2}^{i+n}\cdots c_{2}^{m+n-1}S(c_3^1c_3^2\cdots c_3^{m+n-1}))\\
\end{align*}
with $\Delta(c^*)=\sum c_{1}^*\ot c_{2}^*$ and
\begin{align*}
c^*&=\sum \sigma(\tilde{g})(1\ot c_{2}^{i}c_2^{i+1}\cdots c_{2}^{i+n-1}\ot_A 1\ot c_{1}^{i}\ot c_{1}^{i+1}\ot \cdots \ot c_{1}^{i+n-1} )\\
&=\sum (1\ot c_{2}^{i}c_2^{i+1}\cdots c_{2}^{i+n-1})\tilde{g}(1\ot c_{1}^{i}\ot c_{1}^{i+1}\ot \cdots \ot c_{1}^{i+n-1})\\
&=\sum\tilde{g}(1\ot c_{1}^i\ot c_{1}^{i+1}\ot \cdots \ot c_{1}^{i+n-1})c_{2}^{i}c_2^{i+1}\cdots c_{2}^{i+n-1}.
\end{align*}
We now have the Lie bracket $[,]_P$ on H$^*(A,A^{ad})$. Next, we embed H$^*(A,k)$ into H$^*(A,A^{ad})$ \cite[Corollary 9.4.7]{HH} via the unit map $$\eta_*:\text{Hom}_A(P_{\bullet},k)\to \text{Hom}_A(P_{\bullet},A^{ad}).$$ Let $f\in \text{Hom}_A(P_m,k)$ and $g\in \text{Hom}_A(P_n,k)$. Then by using counit map $$\varepsilon_*:\text{Hom}_A(P_{\bullet},A) \to \text{Hom}_A(P_{\bullet},k),$$ $\eta_*$ and bracket on H$^*(A,A^{ad})$, we derive the formula for $[f,g]\in \text{Hom}_A(P_{m+n-1},k)$:
$$[f,g]=\varepsilon_*[\eta_*(f),\eta_*(g)]_P=\varepsilon_*(\eta_*(f)\circ_P\eta_*(g))-(-1)^{(m-1)(n-1)}\varepsilon_*(\eta_*(g)\circ_P\eta_*(f))$$
where
\begin{align*}
\varepsilon_*&((\eta_*(f)\circ_P\eta_*(g))(1\ot c^1\ot c^2\ot\cdots\ot c^{m+n-1}))\\
=&\varepsilon(\sum \sum_{i=1}^{m}(-1)^{(n-1)(i-1)}\eta(f(1\ot c_{1}^{1}\ot c_{1}^{2}\ot \cdots \ot c_{1}^{i-1}\ot c_{1}^{*}\ot c_{1}^{i+n}\ot \cdots \ot c_{1}^{m+n-1}))\\
&c_{2}^{1}c_{2}^{2}\cdots c_{2}^{i-1}c_{2}^{*} c_{2}^{i+n}\cdots c_{2}^{m+n-1}S(c_3^1c_3^2\cdots c_3^{m+n-1}))
\end{align*}
with
\begin{align*}
\Delta(c^*)&=\sum c_{1}^*\ot c_{2}^* \text{ and }\\
c^*&=\sum\eta(g(1\ot c_{1}^i\ot c_{1}^{i+1}\ot \cdots \ot c_{1}^{i+n-1}))c_{2}^{i}c_2^{i+1}\cdots c_{2}^{i+n-1}.
\end{align*}
Therefore, the last formula is a general expression of the Gerstenhaber bracket on a Hopf algebra cohomology which is indeed inherited from the formula of the bracket on Hochschild cohomology.
\section*{Acknowledgement}
The author would like to thank S. Witherspoon for her precious time, suggestions and support.
| {
"timestamp": "2020-10-16T02:09:01",
"yymm": "2010",
"arxiv_id": "2010.07505",
"language": "en",
"url": "https://arxiv.org/abs/2010.07505",
"abstract": "We calculate the Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies of the Taft algebra $T_p$ for any integer $p>2$ which is a nonquasi-triangular Hopf algebra. We show that the bracket is indeed zero on Hopf algebra cohomology of $T_p$, as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi-triangular algebra. Also, we find a general formula for the bracket on Hopf algebra cohomology of any Hopf algebra with bijective antipode on the bar resolution that is reminiscent of Gerstenhaber's original formula for Hochschild cohomology.",
"subjects": "Quantum Algebra (math.QA); K-Theory and Homology (math.KT); Rings and Algebras (math.RA); Representation Theory (math.RT)",
"title": "Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754483707558,
"lm_q2_score": 0.7185943805178139,
"lm_q1q2_score": 0.7075103843950321
} |
https://arxiv.org/abs/math/9808053 | Computations of the Yamabe invariant | For a compact connected manifold M of dimension n greater than 3 and with no metric of positive scalar curvature, we prove that the Yamabe invariant is unchanged under surgery on spheres of dimension different from 1, n-2 and n-1. We use this result to give new computations of the invariant in dimension four and display new examples of 4-manifolds which do not admit Einstein metrics. | \section{Introduction}
Consider a smooth compact connected manifold $M^n$ of dimension $n$.
It is well known that set of the critical points of the functional
$${\cal S} (g) =
\frac{\int\limits_M s_g \ dvol_g}{(Vol_g (M))^{\frac{n-2}{n}}}$$
\noindent
(on the space ${\cal M}$ of all Riemannian metrics on $M$)
is the set of Einstein metrics on $M$. But it is also well known that this
functional is never bounded below or above and hence it is not possible
to prove existence of critical points by minimizing or maximizing the
functional.
An alternative strategy to find critical points is a minimax argument,
the first ideas of which were introduced by
Yamabe in \cite{Yamabe}. First we have to restrict
the functional to some fixed conformal class; the functional is bounded
below in this subspace and we can define the Yamabe constant of the
conformal class ${\cal C}$ by
$$Y(M, {\cal C})= \inf\limits_{g\in {\cal C}} {\cal S} (g).$$
Moreover, the infimum is always achieved by a smooth metric, which
therefore has constant scalar curvature.
Yamabe claimed to prove this result in \cite{Yamabe}, but his proof
contained an error. This was recognized by Trudinger (see \cite{Trudinger}),
and the problem was finally solved in several steps by Trudinger
\cite{Trudinger},
Aubin \cite{Aubin} and Schoen \cite{Schoen2}.
\vspace{.3cm}
We now define the {\it Yamabe invariant} of $M$ by
$$Y(M)=\sup\limits_{\cal{C} }Y(M,\cal{C})$$
\noindent
(where the supremum is taken over the family of all conformal classes of
metrics on $M$).
This invariant was introduced by O. Kobayashi
in \cite{Kobayashi} and it is also
frequently called the {\it sigma constant}
of $M$ \cite{Schoen3}.
Note that the Yamabe invariant of $M$
is an invariant of the smooth structure of $M$.
\vspace{.3cm}
We will be concerned in this paper with manifolds for which the Yamabe
invariant is non-positive.
This is equivalent to say that the manifold does not admit any metric
of strictly positive scalar curvature. In this case, there is always a unique
metric of constant scalar curvature and unit volume
in each conformal class of metrics on $M$, and the Yamabe invariant is the
supremum of the scalar curvature over the space of metrics of constant scalar
curvature and unit volume. Furthermore,
if the Yamabe invariant is realized by such a metric, then
the metric is a critical point of the functional ${\cal S}$ and it
is therefore an Einstein metric.
This is the minimax approach to find Einstein metrics
we mentioned in the beginning of this introduction.
For the proofs of these last statements, and additional
references, see \cite[Section 1]{Schoen3}.
\vspace{.3cm}
Recall also that, still assuming that $M$ does not admit any metric of
positive scalar curvature,
$$Y(M)=-\left( \inf\limits_{\cal M} {\int}_M |s_g|^{n/2} \ dvol_g
\right)^{2/n},$$
\noindent
where ${\cal M}$ is, as before, the space of all Riemannian metrics on
$M$ (see \cite{Anderson2, LeBrun}). Hence all our results
regarding the Yamabe invariant of a manifold can also be
reinterpreted in terms of the infimum
over ${\cal M}$ of the
$L^{n/2}$ norm of the scalar curvature.
\vspace{.3cm}
We will prove,
\begin{theorem}: Suppose $M$ is a connected smooth compact manifold
of dimension $n\geq 4$ with $Y(M)\leq 0$.
Let $\widehat{M}$ be a manifold obtained
from $M$ by performing surgeries on spheres of dimension different from
1, n-1 and n-2. Then $Y(\widehat{M} )=Y(M)$.
\end{theorem}
The proof of Theorem 1 will be based on the results of \cite{Petean},
where it is proved that surgery in codimension greater than 2 does not
decrease the Yamabe invariant (assuming that the invariant is
non-positive).
\vspace{.3cm}
Some computations of the invariant have been carried out in low dimensions.
In dimension four, LeBrun computed the Yamabe invariant of all compact complex
surfaces of K\"{a}hler type which do not admit metrics of
positive scalar curvature \cite{LeBrun}.
We will use Theorem 1 and the computations in \cite{LeBrun} to give new
exact computations of the Yamabe invariant for
four-dimensional manifolds.
Note that the computation of the invariant in dimension two follows from
the Gauss-Bonnet formula and in dimension three, in the non-positive case,
would follow from Anderson's
program for the hyperbolization conjecture \cite{Anderson}.
We will also show that it follows from Theorem 1 that
the Yamabe invariant of a manifold of dimension greater than four is
unchanged under
connecting sum with the product of two spheres (always assuming that the
invariant is non-positive).
\vspace{.3cm}
Finally, we will use these results to display new examples of
compact four-dimensional manifolds which
do not admit Einstein metrics. For instance, we
will prove:
\begin{theorem}: The connected sum of a compact complex hyperbolic
four-manifold
with any number of copies of $S^1 \times S^3$ does not admit any Einstein
metric.
\end{theorem}
Note that similar results, in the case of {\it real} hyperbolic
four-dimensional manifolds,
have been proved by Sambusetti in \cite{Sambusetti}.
Numerous other examples will be given in the last section of this paper.
\section{Computations of the invariant}
We will now prove Theorem 1. We will use a ``double surgery'' argument
and the following result, proved in \cite[Theorem 1]{Petean}:
\begin{proposition}:
If $\widehat{M}$ is obtained from $M$ by performing surgery
in codimension at least 3 and $Y(M)\leq 0$, then $Y(\widehat{M})\geq Y(M)$.
\end{proposition}
Since we know that, under certain restrictions in codimension,
surgery can only increase the Yamabe invariant, what we will do
is to try to kill the surgery we realized by another surgery
that also verifies the restriction in codimension.
\vspace{.2cm}
There is a canonical way to undo a surgery done on a manifold.
Let ${\cal S}^k$ be a $k$-dimensional sphere embedded in the
manifold $M$ with trivialized normal bundle.
Let $\widehat{M}$
be the manifold obtained by doing surgery on ${\cal S}^k$
(of course,
$\widehat{M}$ will depend on the homotopy class of the trivialization of the
normal bundle).
The boundary of a tubular neighbourhood $U$ of ${\cal S}^k$ is diffeomorphic
to $S^k \times S^{n-k-1}$ and $\widehat{M}$ is obtained by gluing
$D^{k+1} \times S^{n-k-1}$ to $M-U$ along their boundaries. Doing
surgery
on the (n-k-1)-{\it belt sphere}, $\{ 0\} \times S^{n-k-1} \subset
D^{k+1} \times S^{n-k-1}$, in $\widehat{M}$, obviously produces a manifold
diffeomorphic to $M$.
\vspace{.2cm}
Assume now that $k\leq n-3$ and $n-k-1\leq n-3$. Then it follows from
Theorem 2 and the previous observations
that $Y(M)\leq Y(\widehat{M})\leq Y(M)$. Therefore
$Y(M)=Y(\widehat{M})$. Note that $\widehat{M}$ does not admit any
metric of positive scalar curvature because if that were the case
then $M$ would also admit such a metric
(since the family of manifolds which admit metrics of positive
scalar curvature is closed under surgery on codimension greater
than two from the results of Gromov-Lawson
and Schoen-Yau; see \cite{Gromov, Schoen}).
\vspace{.2cm}
The previous condition on $k$ can also be written as $2\leq k\leq n-3$.
To conclude the proof of Theorem 1 we therefore need only to deal with the case
$k=0$.
There is another canonical way to undo surgery in the 0-dimensional case.
Doing 0-dimensional surgery on a connected manifold $M$
means that we take out 2 disjoint balls in $M$
and we glue the boundaries to the boundary of
$S^{n-1}\times [0,1]$. Pick any point
$q\in S^{n-1}$ and consider the loop obtained by joinning the endpoints of
$\{ q\} \times [0,1]$ with a smoothly embedded
curve inside of $M$. The result will be a smoothly embedded circle in
the manifold obtained by doing the 0-dimensional surgery on $M$.
Doing surgery on such
a circle will undo the 0-dimensional surgery. Another way to see this is
the following: the manifold obtained by doing 0-dimensional surgery on $M$ is
$M\#(S^1 \times S^{n-1})$, and doing surgery on the circle of
$S^1 \times S^{n-1}$ we get $S^n$. Hence the result of the two surgeries is
$M\#S^n =M$.
When $n\geq 4$,
this proves that the Yamabe invariant is also unchanged under 0-dimensional
surgeries and, therefore, we have finished the proof of Theorem 1.
{\hfill$\Box$\medskip}
\vspace{.4cm}
{\it Remark}: It is
also clear that the Yamabe invariant is unchanged under
surgery on an (n-1)-sphere
$ S$ unless $S$ separates $M$.
\vspace{.3cm}
Let $\widehat{M}= M\# (S^k \times S^{n-k})$, where $M$ is a smooth connected
compact manifold of dimension $n\geq 5$. Doing surgery on any of the spheres
$S^k$ or $S^{n-k}$ in $\widehat{M}$ produces a manifold diffeomorphic to $M$.
And, of course, either $k$ or $n-k$ is different from 1 and n-2
(note also that $M$ admits a metric of positive scalar
curvature if and only if $\widehat{M}$ does). Therefore,
it follows from Theorem 1 that,
\begin{corollary}: Let $M$ be any
compact smooth connected manifold $M$ of dimension $n\geq 5$. Assume that
the Yamabe invariant of $M$ is non-positive.
Then $Y(M)=Y\left( M\#\ (S^k \times S^{n-k})\right) $ for all $0\leq k\leq n$.
\end{corollary}
The previous result is definitely not true in the 4-dimensional
case. Recall that C.T.C. Wall proved in \cite{Wall} that if two compact
simply connected four
manifolds $M^1$ and $M^2$ have isomorphic intersection
forms, then for $l$ big enough the
connected sum of $M^1$ with $l$ copies of $S^2 \times S^2$ is diffeomorphic
to the connected sum of $M^2$ with $l$ copies of $S^2 \times S^2$.
Suppose that we have a pair $M^1$, $M^2$ of 4-manifolds
with isomorphic intersection forms such that
$Y(M^1 )<0$ while $Y(M^2 )>0$.
Then it follows from the previous comments (and the fact that the family of
manifolds admitting positive scalar curvature metrics is closed under
connected sums \cite{Gromov, Schoen}), that for $l$ big enough,
$Y(M^1 \# l(S^2 \times S^2 ))>0$.
For example, we can take $M^1$ to be a smooth hypersurface of
odd degree $d\geq 5$ in ${\bf CP}^3$. C. LeBrun proved
in \cite{LeBrun2} that $Y(M^1 )<0$, while it is well known, from
Freedman's classification of compact simply connected topological
four-manifolds, that $M^1$ is
homeomorphic to a connected sum of copies of ${\bf CP}^2$ (some of these
copies taken with the reversed orientation); and such a connected sum
has positive Yamabe invariant (since the Fubini-Study metric in
${\bf CP}^2$ has positive scalar curvature).
\vspace{.3cm}
Note that given a
4-manifold $M$ and any ``trivial $S^1$'' in $M$ (i.e. a
small, simple loop in a neighbourhood of a point), doing surgery on such
a circle produces $M\# (S^2 \times S^2)$. Hence we can see from the
previous observation that Theorem 1 is
close to optimal:
\begin{proposition}: There are examples of 4-manifolds $M$, with negative
Yamabe invariant, on which one can strictly increase the Yamabe invariant
by doing surgery on dimension 1. It follows that there are also examples
where one can strictly decrease the invariant by doing surgery in dimension 2.
\end{proposition}
We can now give some new exact computations of the
invariant in dimension 4. Using Seiberg-Witten techniques, C. LeBrun
proved in \cite{LeBrun2} that if $X$ is a minimal compact complex
surface of general type and $M$ is obtained by
blowing up $X$ any number of times, then
$$Y(M)=Y(X)=-4\pi\sqrt{2c_1^2 (X)}.$$
Note also that doing 0-dimensional surgery on a connected four-manifold
$M$ produces a manifold diffeomorphic to $M\# (S^1 \times S^3 )$. Therefore,
it follows from Theorem 1 that,
\begin{proposition}: Let $M$ be a compact four-dimensional
manifold with non-positive
Yamabe invariant. Then $Y(M)=Y(M\# (S^1 \times S^3 ))$.
\end{proposition}
Using the results of LeBrun mentioned previously we get new exact
computations of the invariant:
\begin{proposition}: Let $X$ be a minimal compact complex surface
of general type. Let $M$ be a manifold obtained by taking the connected sum
of any number of copies of $S^1 \times S^3$ with
the blow-up of $X$ at any number of points. Then
$Y(M)=-4\pi\sqrt{2c_1^2 (X)}$.
\end{proposition}
\section{Manifolds without Einstein metrics}
Finally, the results from the previous section produce
numerous new examples of compact
four-manifolds
which do not admit any Einstein metric. Let $M$ be obtained by blowing up
$l$ times the minimal compact complex surface of general type $X$.
Let $N=N(k,l)$ be the manifold obtained as the
connected sum of $M$ with $k$ copies of $S^1 \times S^3$. Suppose that
$N$ admits an Einstein metric $g$. Then it
follows from the Gauss-Bonnet formula
that
$$2\chi (N)\pm 3\tau (N)=\frac{1}{4{\pi}^2}
\int\limits_N ( 2|W_{\pm}|^2 +(s_g^2 /24)) \
dvol_g.$$
But,
$$\frac{1}{4{\pi}^2}
\int\limits_N ( 2|W_{\pm}|^2 +(s_g^2 /24))\
dvol_g \ \ \geq \ \
\frac{1}{4{\pi}^2}
\int\limits_N
\frac{s_g^2}{24}\
dvol_g \ \
\geq$$
$$\geq \ \ \frac{1}{96{\pi}^2}(Y(N))^2 \ \ =\ \ \frac{1}{3}c_1^2 (X).$$
\vspace{.2cm}
\noindent
where the last inequality follows because the Yamabe invariant of $N$ is
the negative of
the square root of the infimum over the
space of all Riemannian metrics on $N$ of the
$L^2$ norm of the scalar curvature (we have already
mentioned this result in the introduction)
and the final equality is precisely Corollary 2.
\vspace{.3cm}
Hence, we have seen that if $N$ admits an Einstein metric then
$$2\chi (N)-3|\tau (N)|\geq (1/3) c_1^2 (X).$$
This implies in many cases that
such an Einstein metric cannot exist.
For instance,
let us now prove Theorem 2. Let $M$ be a (compact)
complex
hyperbolic surface. In this case, one knows that
$\chi (M)=3\tau (M)>0$; and if $N$ is obtained as the
connected sum of $M$ and $k$ copies of $S^1 \times S^3$, the above
inequality reads:
$3\tau (M) -4k \geq 3\tau (M)$. Therefore, $N$ cannot admite any Einstein
metric.
\vspace{.2cm}
{\it Remark}: Note that the Hitchin-Thorpe inequality only assures that
an Einstein metric does not exist on $M\# k(S^1 \times S^3 )$ when $k\geq (3/4)
\tau (M)$.
\vspace{.3cm}
For a more general case,
let $X$ be a minimal compact complex surface of general type
which has non-positive signature.
As before, let $M$ be the blow up of $X$ at $l$ points and
let $N$ be the connected sum of $M$ and $k$ copies of $S^1 \times S^3$.
Note that
$M$ and $N$ also have
non-positive signature. If $N$ admits an Einstein metric we
get from the previous computations that $c_1^2 (X)-l -4k\geq (1/3) c_1^2 (X)$.
Therefore,
\begin{theorem}: If $k> (1/6)c_1^2 (X) -l/4$,
then $N=N(k,l)$ does not
admit an Einstein metric.
\end{theorem}
{\it Remark}: In this case the Hitchin-Thorpe inequality only assures the
non-existence of an Einstein metric when $k\geq (1/4)c_1^2 (X) -l/4$.
\vspace{.3cm}
{\bf Acknowledgements}: The author would like to thank Claude LeBrun
for his support and for numerous observations related to this work. He would
also like to thank
Vyacheslav Krushkal for many useful discussions, and the directors
and staff of the Max-Planck Institut for their hospitality during
the preparation of this work.
| {
"timestamp": "1998-08-11T19:42:31",
"yymm": "9808",
"arxiv_id": "math/9808053",
"language": "en",
"url": "https://arxiv.org/abs/math/9808053",
"abstract": "For a compact connected manifold M of dimension n greater than 3 and with no metric of positive scalar curvature, we prove that the Yamabe invariant is unchanged under surgery on spheres of dimension different from 1, n-2 and n-1. We use this result to give new computations of the invariant in dimension four and display new examples of 4-manifolds which do not admit Einstein metrics.",
"subjects": "Differential Geometry (math.DG)",
"title": "Computations of the Yamabe invariant",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754465603679,
"lm_q2_score": 0.7185943805178139,
"lm_q1q2_score": 0.7075103830940975
} |
https://arxiv.org/abs/1604.01727 | One-dimensional parametric determining form for the two-dimensional Navier-Stokes equations | The evolution of a determining form for the 2D Navier-Stokes equations (NSE), which is an ODE on a space of trajectories is completely described. It is proved that at every stage of its evolution, the solution is a convex combination of the initial trajectory and the fixed steady state, with a dynamical convexity parameter $\theta$, which will be called the characteristic determining parameter. That is, we show a remarkable separation of variables formula for the solution of the determining form. Moreover, for a given initial trajectory, the dynamics of the infinite-dimensional determining form are equivalent to those of the characteristic determining parameter $\theta$ which is governed by a one-dimensional ODE. %for the parameter specifying the position on the line segment. This one-dimensional ODE is used to show that if the solution to the determining form converges to the fixed state it does so no faster than $\mathcal{O}(\tau^{-1/2})$, otherwise it converges to a projection of some other trajectory in the global attractor of the NSE, but no faster than $\mathcal{O}(\tau^{-1})$, as $\tau \to \infty$, where $\tau$ is the evolutionary variable in determining form. The one-dimensional ODE also exploited in computations which suggest that the one-sided convergence rate estimates are in fact achieved. The ODE is then modified to accelerate the convergence to an exponential rate. Remarkably, it is shown that the zeros of the scalar function that governs the dynamics of $\theta$, which are called characteristic determining values, identify in a unique fashion the trajectories in the global attractor of the 2D NSE. Furthermore, the one-dimensional characteristic determining form enables us to find unanticipated geometric features of the global attractor, a subject of future research. | \section{Introduction}\label{sec1}
Many dissipative infinite-dimensional dynamical systems have been reduced
to finite-dimensional ordinary differential equations (ODE) by the restriction to
inertial manifolds \cite{CFNTbook, CFNT,inertial1,FSTi}. The list includes
the Kuramoto-Sivashinsky, complex Ginzburg-Landau, and certain reaction diffusion equations \cite{M-PS,T97},
just to name a few. Whether the 2D Navier-Stokes equations (NSE) enjoys a
finite-dimensional reduction via an inertial manifold
has remained an open question since the
mid-1980's. Recently however, it has been shown that the global attractor ${\mathcal A}$ of the 2D NSE
can in fact be captured by an ODE $dv/d\tau=F(v)$ in a Banach space of trajectories. This is an ODE in the
true sense that $F$ is a globally Lipschitz map. This ODE has its own
evolution variable denoted by $\tau$, which is distinct from that in the NSE, which we denote by $s$.
There have been two different constructions of such an ODE, referred to as a {\it determining form}
due its connection to the notion of determining modes, nodes, volume elements, etc. \cite{CJTi1,FP,FTfinelt,FT1, FTi,HTi,JT93}.
In the first approach, in \cite{FJKrT1}, trajectories in the global attractor of the NSE,
are precisely traveling waves in the variables $\tau$ and $s$, while in the second, in \cite{FJKrT2},
they are precisely steady states of the determining form.
This paper focuses on the latter type
of determining form. We show in section \ref{sec3} that its evolution always proceeds along
a segment in $X$ the phase space of trajectories (see \eqref{X}), connecting the initial trajectory $v_0$, and $Ju^*$, where
$u^*$ is a fixed steady state $u^*$ of the NSE, and $J$ is a finite-rank projector
(see \eqref{ape1}, \eqref{ape2} below).
This leads to a separation of variables formula for the solution of the determining form, specifically, we show in section \ref{sec3}, that the solution is a convex combination $v(\tau,s)=\theta(\tau) v_0(s)+(1-\theta(\tau))Ju^*$, $s \in {\Bbb R}$, $\tau \ge 0$. We will refer to the convexity parameter
as the {\it characteristic determining parameter}. Moreover, if
$v(\cdot) \neq Ju(\cdot)$ for any trajectory $u(\cdot)\subset \mathcal{ A } }\def\BBB{ \mathcal{ B } }\def\CCC{ \mathcal{ C } $ and there is no
trajectory in ${\mathcal A}$ strictly between $v_0$ and $Ju^*$, then $v(\tau) \to Ju^*$, as $\tau \to \infty$, no faster than $\OOO(\tau^{-1/2})$. Otherwise $v(\tau) \to J\tilde{u}$ no faster than $\OOO(\tau^{-1})$, where $\tilde{u}(\cdot) \subset {\mathcal A}\setminus \{u^*\}$ is on the
closed segment connecting $Ju^*$ with $v_0$. For a given $v_0$, the evolution of the determining form is equivalent to the dynamics of a one-dimensional ODE in the characteristic determining parameter $\theta$,
$d \theta/d\tau=\Phi( \theta;v_0,u^*)$, which we refer to as the {\it characteristic parametric determining form}, where $\Phi$ is a Lipschitz function in $\theta \in [0,1]$.
We show in section \ref{sec4} that the function $\Phi$ is readily modified to accelerate the convergence to an exponential rate.
This surprising reduction to a one-dimensional ODE may have far reaching consequences.
It enables us to find unanticipated geometric features of the global attractor, and develop
alternative computational approaches to finding this set, a subject of ongoing research; in particular bifurcation analysis of the 2D NSE. Remarkably, the zeros of the real function $\Phi$, which we refer to as {\it characteristic determining values}, identify the trajectories on the global attractor of the 2D NSE in a unique fashion.
We demonstrate the utility of
this reduction by finding a limit cycle to roughly machine accuracy with just eight steps
of the secant method.
In section \ref{sec6} we exploit this scalar ODE in computations which suggest that the
one-sided estimates on the two rates of convergence are in fact achieved.
A significant advantage of this approach
is that it avoids the padding
of a time interval to account for compounding relaxation times.
In contrast, the direct numerical simulation of the determining form $dv/d\tau=F(v)$ would involve the sequential evaluation of a map $W(v)$ (see Theorem \ref{mainthm}).
The image of $W$ is the unique bounded solution $w(s)$ of a system similar to the 2D NSE (see \eqref{weqn}) but
driven by $v(\tau,s)$, where $\tau$ is fixed, and $-\infty < s < \infty$.
On a computer, trajectories over all time
must be truncated to, say $0 \le s \le s_2$. Given
$v(\tau,s)$ for $s \in [0,s_2]$, the image $W(v)$ can be effectively
approximated over subinterval $s_1 \le s \le s_2$, after a short relaxation time $s_1 > 0$,
by solving this driven NSE system, with initial condition $w(0)=0$. Thus in order to make $N$ sequential
evaluations of $F$, one would need to pad the initial trajectory $v(0,s)$, i.e., specify
$s$ over the interval $0 \le s \le Ns_1+s_2$. In the case of the one-dimensional ODE, however,
$W(v)$ is always evaluated at a convex combination $v=\theta v_0+(1-\theta)Ju^*$, so the
relaxation times are not compounded.
Notably, the 2D NSE considered in the paper is only a prototype example of a dissipative evolutionary equation; the results presented in this
work can be equally extended to other dissipative evolution equations
(cf. \cite{JMST,JST1,JST2}).
\section{Background and preliminaries} \label{sec2}
The two-dimensional incompressible Navier-Stokes equations (NSE)
\begin{equation}\label{NSES}
\begin{aligned}
&\frac{\partial u}{\partial s} -
\nu \Delta u + (u\cdot\nabla)u + \nabla p = g \\
&\text {div} u = 0 \\
& \int_{\Omega} u\, dx =0 \;,\qquad \int_{\Omega} g\, dx =0 \\
& u(0,x) = u_0(x),
\end{aligned}
\end{equation}
subject to periodic boundary conditions with the basic periodic-domain $\Omega =[0,L]^2$, can be written as
an evolution equation in a Hilbert space $H$ (cf. \cite{CF88,Robinson,T97})
\begin{equation}\label{NSE}
\begin{aligned}
&\frac{d}{ds}u(s) + \nu Au(s) + B(u(s),u(s)) = f, \,\, \text{for}\, s >0,\\
& u(0)=u_0.
\end{aligned}
\end{equation}
Here $H$ is the closure of $\mathcal{V}$ in $(L^2_{\rm{per}}(\Omega))^2$, where
$$
\mathcal{V}=\{\varphi:\, \varphi\,\, \text{is}\,\, \mathbb{R}^2-{\hbox{valued trigonometric polynomials}}, \,\, \nabla \cdot \varphi = 0, \, \text{ and} \, \int_{\Omega} \varphi(x)\, dx =0 \}\;.
$$
The Stokes operator $A$, the bilinear operator $B$, and force $f$ are defined as
\begin{equation}
\label{nabla1}
A=-{\mathcal P}\Delta=-\Delta \;, \quad B(u,v)={\mathcal P}\left( (u \cdot \nabla) v \right)\;,\quad f={\mathcal P} g\;,
\end{equation}
where ${\mathcal P}$ is the Helmholtz-Leray orthogonal projector from $(L^2(\Omega))^2$ onto $H$.
We denote $|\cdot|=\|\cdot\|_{L^2}$ and $\|u\|=|A^{1/2}u|$, with
fractional powers of $A$ defined by $A^\alpha \varphi_j=\lambda_j^\alpha \varphi_j$, $\alpha \in {\Bbb R}$, where
$\{\varphi_j\}$ is an othornormal basis for $H$ consisting of eigenfunctions of $A$ corresponding
to eigenvalues $\{\lambda_j\}$ satisfying
$$
0<\lambda _1=\left(\frac{2\pi}{ L}\right)^2
\leq \lambda _2\leq\lambda_3 \le \cdots \;.
$$
It is well known that for a time independent forcing term $f\in H$, the two-dimensional NSE \eqref{NSE} is globally well-posed, and that it has a global attractor
\begin{equation}\label{attractordef}
\mathcal{A}=\{u_0\in H: \, \exists \, \text{ a solution}\, u(s,u_0)\, \text{of \eqref{NSE}} \, \, \forall \ s\in\mathbb{R},\,\, \sup_{s\in {\Bbb R}} \|u(s)\|<\infty\}\;.
\end{equation}
Moreover, it is also well known that
\begin{equation}\label{Grashof}
\mathcal{A} \subset \{u\in D(A^{1/2}): \|u\|\leq G\nu\kappa_0\}
\end{equation}
where $\kappa_0=\lambda_1^{1/2}=2\pi/L$ and $G=|f|/(\nu^2\lambda_1)$ is the Grashof
number, a dimensionless parameter which plays the role of the Reynolds number in turbulent flows \cite{CF88,FMRT,Hale,Robinson,T97}.
It is also proved in \cite{FJLRYZ} that
$$
|Au| \le c\nu\kappa_0^2G^3\;,\quad |A \frac{du}{ds}| \le c\nu^2\kappa_0^4G^7 \quad \forall \ u \in \mathcal{ A } }\def\BBB{ \mathcal{ B } }\def\CCC{ \mathcal{ C } \;.
$$
For a subset $ \mathcal{ M } }\def\NNN{ \mathcal{ N } }\def\OOO{ \mathcal{ O } \subset (L^2_{\text{per}}(\Omega))^2$, we denote $\dot \mathcal{ M } }\def\NNN{ \mathcal{ N } }\def\OOO{ \mathcal{ O } =\{\varphi \in \mathcal{ M } }\def\NNN{ \mathcal{ N } }\def\OOO{ \mathcal{ O } : \int_\Omega \varphi(x) \ dx =0\}$.
Let $J=J_h$ be a finite-rank linear interpolant operator,
$$
J:(\dot H_{\text{per}}^2(\Omega))^2 \to (\dot H_{\text{per}}^1(\Omega))^2\;,
$$ which approximates the identity in the
sense that for every $w \in (\dot {H}^2(\Omega))^2$
\begin{align}
|Jw-w| &\le c_1h|\nabla w|+c_2h^2|\Delta w| \label{ape1} \\
|\nabla(Jw-w)|&\le \tilde{c}_1|\nabla w|+\tilde{c}_2h|\Delta w| \label{ape2}\;,
\end{align}
where $h$ represents the spatial resolution of the interpolant operator $J$. Observe that the above approximate identity inequalities imply that $J:(\dot H^2)^2 \to(\dot H^1)^2$ is a bounded linear operator. Indeed, we have from \eqref{ape1} that for every $w \in (\dot{H}^2)^2$
$$
|Jw| \le |w| + c_1h|\nabla w| + c_2h^2|\Delta w| \sim \|w\|_{H^2}\;,
$$
while from \eqref{ape2}
$$
|\nabla Jw| \le |\nabla w| + \tilde{c}_1|\nabla w| + \tilde{c}_2h|\Delta w| \sim \|w\|_{H^2}\;.
$$
Adding these two inequalities implies
$$
\|Jw\|_{\dot H^1} \le c_J \|w\|_{\dot H^2} \;, \quad \text{for every} \ w \in (\dot{H}^2)^2\;.
$$
Such an interpolant operator could be defined in terms of Fourier modes, nodal values, volume elements,
or finite elements (see, e.g., \cite{AOT,CJTi1,FP,FTfinelt,FTi,HTi, JT93} and references therein).
Let $X$ denote the Banach space $C_b^1(\mathbb{R},J(\dot{H}^2_{\rm{per}}(\Omega))^2)$. We endow $X$ with the norm
\begin{equation}\label{X}
\|v\|_X=\|v\|_{X^0}+\sup_{s\in {\Bbb R}}\frac{ \|v'(s)\|}{\nu^2\kappa_0^3}\;,
\quad \text{where}\quad
\|v\|_{X^0}=\sup_{s \in {\Bbb R}} \frac{\|v(s)\|}{\nu\kappa_0} \;.
\end{equation}
Now consider for any $v\in X$
the associated ``feedback control" system
\begin{equation} \label{weqn}
\frac{dw}{ds}+Aw+B(w,w)=f-\mu\nu\kappa_0^2 (J_hw-v)\;, \quad w \in H\;.
\end{equation}
The following is proved in \cite{FJKrT2}.
\begin{theorem}\label{mainthm} \hfill\break
\begin{enumerate}
\item Let $\rho > 0$ and $\mu\gtrsim \rho^2$, $h \lesssim 1/\sqrt{\mu}$. Then for every $v \in \BBB_X^\rho(0)$ equation \eqref{weqn} has a unique solution $w \in Y=C_b^1({\Bbb R},D(A))$, defining a globally Lipschitz map
$W:\BBB_X^\rho(0)\to Y$, by $W(v)=w$,
\item $\| J u\|_X \le R \sim G^7$ for all $u(\cdot) \subset \mathcal{ A } }\def\BBB{ \mathcal{ B } }\def\CCC{ \mathcal{ C } $,
\item let $v \in \BBB^{4R}_X(0)$, then $JW(v)=v$ if and only if $W(v)(\cdot) \subset \mathcal{ A } }\def\BBB{ \mathcal{ B } }\def\CCC{ \mathcal{ C } $.
\end{enumerate}
\end{theorem}
Let $u^*$ be a steady state of the NSE. In \cite{FJKrT2} the term {\it determining form}
was introduced for the following evolution equation
\begin{equation}\label{detform}
\frac{dv}{d\tau}=F(v)=-\|v-JW(v)\|_{X^0} ^2\ (v-Ju^*)\;, \quad v \in \BBB_{X}^{3R}(Ju^*)\subset X \;,
\end{equation}
which is an ordinary differential equation in the true sense, i.e. $F$ is globally Lipschitz in
$\BBB_{X}^{3R}(Ju^*)$, which is a positively invariant set for \eqref{detform} (here $\mu$, $h$ and
$W$ are determined by $\rho=4R$).
Moreover, by (ii),(iii) the
set of steady states of \eqref{detform} is precisely
$\{Ju(\cdot)$, where $u(\cdot) \subset \mathcal{ A } }\def\BBB{ \mathcal{ B } }\def\CCC{ \mathcal{ C } \}$. In fact, it was shown in \cite{FJKrT2} (and we will repeat this proof later)
that every solution of \eqref{detform} approaches
a steady state, as $\tau\to \infty$.
Thus, trajectories in the global attractor of the NSE
are readily recognized/realized/identified by the long-time behavior of the determining form.
Note that at each ``evolutionary time" instant $\tau$
we may write $v(\tau)=v(\tau,s)$, $-\infty < s < \infty$, just as for the NSE written in the form \eqref{NSE},
at each instant $s$ one may write $u(s)=u(s,x)$ to denote the dependence on the suppressed spatial variable $x$.
This type of determining form and its properties have also been established for
the subcritical surface quasigeostrophic equation (SQG), a damped, driven nonlinear Schr\"odinger equation (NLS), and a damped, driven Korteveg-de Vries equation (KdV)
\cite{JMST,JST1,JST2}. The analysis used to prove analogs of Theorem \ref{mainthm} for these systems differ from that for the NSE.
The approach for the SQG involves the Littlewood-Paley decomposition and Di Giorgi techniques in order to obtain $L^p$ estimates for \eqref{weqn} over the full subcritical range.
For the weakly dissipative NLS and KdV the analysis uses of certain compound functionals
resulting in different spaces $X$ and $Y$ in each case. An earlier type of determining form in which trajectories in
the global attractor the 2D NSE were identified with traveling wave solutions $v(\tau,s)=v(0,\tau+s)$ was developed in \cite {FJKrT1}. In this paper we focus on the dynamics of the determining form in \eqref{detform}, describing completely the basin of attraction for each steady state, and distinguishing between lower bounds on the rate of convergence toward $Ju^*$ and the rate toward any other trajectory in $ \mathcal{ A } }\def\BBB{ \mathcal{ B } }\def\CCC{ \mathcal{ C } $.
\section{One-dimensional ODE determining the global dynamics of the NSE}\label{sec3}
In this section we investigate the dynamical behavior of the determining form \eqref{detform}. Remarkably, the global dynamics of the NSE, i.e., the trajectories on the global attractor of the NSE,
are precisely the liftings of the steady state solutions of a one-dimensional ODE \eqref{titi6}, below,
which will be referred to as the ``characteristic parametric
determining form".
The determining form \eqref{detform} is equivalent to
\begin{equation} \label{titi1}
\begin{split}
\frac{d}{d\tau}(v - Ju^*) & = -\| v - JW(v) \| _{X^0} ^2 (v - Ju^*) \;,\\
v(0) &= v_0 \in \BBB_X ^{3R}(Ju^*) \;,
\end{split}
\end{equation}
whose solution can be written as
\begin{equation} \label{titi2}
v(\tau) - Ju^ * = \theta (\tau) (v_0 - Ju^*) ,
\end{equation}
where
\begin{equation} \label{titi3}
\begin{split}
\theta (\tau) &= \exp \left( - \int _0 ^ \tau \| v(\sigma) - J W(v(\sigma)) \| _{X^0} ^ 2 \, d\sigma \right) \\
\theta(0) &= 1, \text{ and } \theta(\tau) \in [0,1].
\end{split}
\end{equation}
Observe that
\begin{equation} \label{titi4}
\frac{d\theta}{d\tau} = - \theta \| v(\tau) - JW(v(\tau)) \| _{X^0} ^2 .
\end{equation}
Thanks to \eqref{titi2} we have
\begin{equation} \label{titi5}
v(\tau) = \theta(\tau) v_0 + (1-\theta (\tau)) Ju^*
\end{equation}
and combined with \eqref{titi4} we have a one-dimensional ODE for the characteristic determining parameter $\theta$:
\begin{equation} \label{titi6}
\begin{split}
\frac{d\theta}{d\tau} &= -\theta \| \theta v_0 + (1-\theta) Ju^* - JW(\theta v_0 + (1-\theta)J(u^*)) \| _{X^0} ^2 =: \Phi(\theta;v_0,u^*) \\
\theta(0) &= 1.
\end{split}
\end{equation}
We term equation \eqref{titi6} the characteristic parametric determining form.
Observe that \eqref{titi5} states that the solution of the determining form \eqref{detform} is a convex combination of the given initial data $v_0$ and the steady state projection $Ju^*$, where the convexity parameter is the characteristic determining parameter.
Moreover, the evolution of \eqref{detform} is equivalent to the evolution of the one parameter ODE \eqref{titi6}, the characteristic parametric determining form. Note also that \eqref{titi5} gives a separation of variables, i.e.
\begin{equation*}
\begin{split}
v(\tau,s) &= \theta(\tau)v_0(s) - (1-\theta(\tau)) Ju^* \\
\text{for} \quad \tau\ &\geq 0 , ~ s\in \RR, \text{ where } v_0 \in \BBB_X ^{3R}(Ju^*).
\end{split}
\end{equation*}
\subsection{The behavior of the $\theta(\tau)$: Lower bounds on convergence rate}
\begin{theorem}\label{titithm}
The limit $\lim _{t\rightarrow \infty} \theta(\tau) = \bar{\theta}$ exists and
\begin{enumerate
\item $\bar{\theta} = 0$ if and only if
\[ \int _0 ^ \infty \| v(\sigma ) - J W(v(\sigma)) \| _{X^0} ^ 2 \, d\sigma = \infty \quad \text{and} \quad
v(\tau) \to \bar{v} = Ju^*\;, \text{ as } \tau \to \infty\;.\]
\item $\bar{\theta} \in (0,1]$ if and only if
\[ \int _0 ^ \infty \| v(\sigma ) - J W(v(\sigma)) \| _{X^0} ^ 2 \, d\sigma < \infty\;. \]
Moreover, in this case $\bar v=\bar\theta v_0+(1-\bar \theta)Ju^*$ satisfies $W(\bar v)(\cdot) \subset {\mathcal A}$.
\item $\Phi(\bar\theta;v_0,u^*)=0$ if and only if $\bar\theta$ satisfies (i) or (ii); where
$\Phi(\bar\theta;v_0,u^*)$ is given in \eqref{titi6}. The zeros, $\bar \theta$, of
$\Phi(\theta;v_0,u^*)$ are called characteristic determining values.
\item If $\bar{\theta} = 0$ then
\begin{equation}\label{thalf}
\theta(\tau) \geq \frac{1}{(1+2\tilde{c}^2\| v_0 - Ju^*\|_{X} ^2 \tau) ^{1/2}}\;, \quad \tau\geq 0\;,
\end{equation}
\begin{equation}\label{thalfX}
\| v(\tau) - Ju^* \| _X \geq \frac{\|v_0 - Ju^* \|_{X}}{(1+2\tilde{c}^2\|v_0 - Ju^* \| ^2 _{X} \tau ) ^{1/2}}\;, \quad \tau \geq 0
\end{equation}
and
\begin{equation}\label{thalfXX}
\| v(\tau) - Ju^* \| _{X^0} \geq \frac{\|v_0 - Ju^* \|_{X^0}}{(1+2\tilde{c}^2\|v_0 - Ju^* \| ^2 _{X} \tau ) ^{1/2}}\;, \quad \tau \geq 0.
\end{equation}
where $\tilde{c}$ is a constant depending only on $c_J$ and the Lipschitz constant of $W$.
\item If $\bar{\theta} \in (0,1]$ then $\bar{v} = v_0 \bar{\theta} + (1-\bar{\theta} )Ju^*$ and
\begin{equation} \label{whole}
\theta(\tau) - \bar{\theta} \geq \frac{1-\bar{\theta}}{1+\tilde{c}^2\|v_0 - Ju^* \| ^2 _{X} \tau }\;,
\quad \tau \geq 0,
\end{equation}
\begin{equation}\label{wholeX}
\| v(\tau) - \bar{v} \| _X \geq \frac{(1-\bar{\theta})\|v_0 - Ju^*\|_{X}}{1+\tilde{c}^2\|v_0 - Ju^* \| ^2 _{X} \tau }\;, \quad \tau \geq 0
\end{equation}
and
\begin{equation}\label{wholeXX}
\| v(\tau) - \bar{v} \| _{X^0} \geq \frac{(1-\bar{\theta})\|v_0 - Ju^*\|_{X^0}}{1+\tilde{c}^2\|v_0 - Ju^* \| ^2 _{X} \tau }\;, \quad \tau \geq 0.
\end{equation}
\end{enumerate}
\end{theorem}
As a consequence of the above theorem we have the following corollary which states that the
trajectories in the global attractor of the NSE are determined in a unique fashion by the characteristic determining values, i.e., zeros of a
real-valued function of a real variable, the vector field of the parametric determining form, $\Phi(\theta;v_0,u^*)$.
\begin{corollary} Let $v_0 \in \BBB^{3R}_X(Ju^*)$, and let $\bar\theta$ satisfy
$\Phi(\bar\theta;v_0,u^*)=0$. \hfill\break Then $w(\cdot)=W(v_0 \bar{\theta} + (1-\bar{\theta} )Ju^*)(\cdot) \subset {\mathcal A}$. Conversely, for every $u(\cdot) \subset {\mathcal A}$, there exists $v_0 \in \BBB_X^{3R}(Ju^*)$ and $\bar\theta \in [0,1]$ such that $\Phi(\bar\theta, v_0,u^*)=0$ and \hfill\break
$u(\cdot)=W(v_0 \bar{\theta} + (1-\bar{\theta} )Ju^*)(\cdot)$.
\end{corollary}
The proof of the first part of the corollary follows immediately from part (iii) of Theorem \ref{titithm} and part (iii) of Theorem \ref{mainthm}. To prove the second part of the corollary
we take $v_0=Ju$ where $u(\cdot) \subset \mathcal{ A } }\def\BBB{ \mathcal{ B } }\def\CCC{ \mathcal{ C } $ and $\bar\theta=1$, because in this case $v_0$ is a steady state of the determining form \eqref{titi1}.
\begin{remark} In fact for a given $u(\cdot)\subset \mathcal{ A } }\def\BBB{ \mathcal{ B } }\def\CCC{ \mathcal{ C } $ there might be infinitely many
$v_0$ satisfying the second part of the proposition. They must have the form
$$
v_0(\cdot)=\alpha Ju(\cdot) + (1-\alpha)Ju^* \quad \text{for some} \quad
\alpha \ge 1\;, \quad \text{with} \quad v_0 \in \BBB_X^{3R}(Ju^*)\;.
$$
In the above proof
we have shown the existence of at least one such $v_0$, which is the trivial choice.
However, in the computational section, below, we take a nontrivial choice for $v_0$ to demonstrate the validity of the analytical results.
\end{remark}
Next we give the proof of Theorem \ref{titithm}.
\begin{proof}
The existence of the limit $\bar\theta$ follows from the fact that $\theta(t) \in [0,1]$ is monotonic and non-increasing.
Part (i) is immediate.
To prove part (ii) observe that since $\bar\theta > 0$, we have
\begin{align}\label{finiteint}
\int_0^ \infty \| v(\sigma)-JW(v(\sigma))\|_{X^0}^2 \ d\sigma < \infty \;.
\end{align}
As a result
of \eqref{titi5} we conclude that
$$
v(\tau)=\theta(\tau)v_0 + (1-\theta(\tau))Ju^* \to \bar v=\bar\theta v_0 + (1-\bar\theta)Ju^*\;,
$$
as $\tau \to \infty$.
Since $J$ and $W$ are continuous maps we have $JW(v(\tau)) \to JW(\bar v)$, as $\tau \to \infty$.
Thus, the integrand in \eqref{finiteint}, $\| v(\sigma)-JW(v(\sigma))\|_{X^0}^2$, converges to $\| \bar v(\sigma)-JW(\bar v(\sigma))\|_{X^0}^2$, as
$\sigma \to \infty$. Since the integral is finite we have
$$
\| v(\sigma)-JW(v(\sigma))\|_{X^0}^2 \to 0\;,
\quad \text{as} \quad \sigma \to \infty\;.
$$
As a consequence, we have $\bar v-JW(\bar v)= 0$. Thanks to part (iii)
of Theorem \ref{mainthm}, $W(\bar v)(\cdot) \subset {\mathcal A}$.
To prove (iv) observe that $Ju^* = JW(Ju^*)$, therefore
\begin{equation} \label{bigguy}
\begin{aligned}
\| v - JW(v) \| _{X^0} &= \| v - Ju^* + JW(Ju^*) - JW(v) \| _{X^0} \\
& \le \|v-Ju^*\|_{X^0}+ \|JW(Ju^*)-JW(v)\|_{X^0} \\
(J:\dot H^2 \to \dot H^1 \text { bounded}) \qquad & \le \|v-Ju^*\|_{X^0}+c_J c \sup_{s \in {\Bbb R}}|A(W(Ju^*)-W(v)| \\
& \le \|v-Ju^*\|_{X^0}+ c_J c \|W(Ju^*)-W(v)\|_Y \\
(W: X \to Y \text{ Lipschitz}) \qquad & \le \|v-Ju^*\|_{X^0}+ c_J c L_{\text{Lip}}\|Ju^*-v\|_X \\
(\|\cdot\|_X \text{ stronger than } \|\cdot \|_{X^0}) \qquad &\leq \tilde{c} \| v - Ju^* \|_{X} \\
\text{by \eqref{titi5}}\qquad & =\tilde{c}\theta\|v_0-Ju^*\|_{X}
\end{aligned}
\end{equation}
As a consequence of \eqref{bigguy} and \eqref{titi4} we have
\[
\frac{d\theta}{d\tau} \geq -\theta ^3 \tilde{c}^2\| v_0 - Ju^* \| _{X} ^2 ; \quad \theta(0) = 1.
\]
We integrate to obtain \eqref{thalf}, then use \eqref{titi5} to get \eqref{thalfX} and \eqref{thalfXX}.
To prove (v) suppose $\bar{\theta}\in (0,1]$, and $\bar{v} = \bar{\theta}v_0 + (1-\bar{\theta })Ju^*$. Note that by part (ii) $\bar{v}=JW(\bar{v})$, so that $\| \bar{v} - JW(\bar{v}) \| _{X^0} = 0$. In view of this and $\eqref{titi4}$, we have
\begin{align*}
\frac{d}{d\tau}(\theta - \bar{\theta}) &= -\theta \| v - JW(v) \| _{X^0} ^2 \\
&= -\theta \left[ \| v - JW(v) \| _{X^0} - \| \bar{v} - JW(\bar{v} ) \| _{X^0} \right] ^2\;,
\end{align*}
while proceeding as in \eqref{bigguy}, we have
\begin{align*}
\left \vert \| v - JW(v) \| _{X^0} - \| \bar{v} - JW(\bar{v}) \| _{X^0} \right \vert &\leq \| (v - \bar{v}) - (JW(v) - JW(\bar{v}) ) \| _{X^0} \\
&\leq \tilde{c}\| v - \bar{v} \| _{X} \\
&\leq \tilde{c} (\theta - \bar{\theta} ) \| v_0 - Ju^* \| _{X} \;.
\end{align*}
Since $\theta(\tau) \geq \bar{\theta}$ and $\theta(\tau) \in (0,1]$,
\begin{align*}
\frac{d}{d\tau} (\theta-\bar{\theta}) &\geq -\theta (\theta - \bar{\theta} ) ^2 \tilde{c}^2 \| v_0 -Ju^* \| _{X} ^2 \\
&\geq -(\theta - \bar{\theta})^2 \tilde{c}^2\| v_0 - Ju^* \| _{X} ^2\;,
\end{align*}
with $\theta(0) = 1$. Once again, integrating yields \eqref{whole}, and applying \eqref{titi5}
gives \eqref{wholeX} and \eqref{wholeXX}.
\end{proof}
\section{Accelerating the convergence rate} \label{sec4}
Following the proof of Theorem \ref{mainthm} in \cite{FJKrT2} one can modify
the determining form slightly and prove similar statements for
the ODE (a modified determining form)
\begin{equation}\label{detform3}
\frac{dv}{d\tau}=-\|v-JW(v)\|_{X^0}\ (v-Ju^*)\;, \quad v(0)=v_0 \;
\end{equation}
The solution to \eqref{detform3} can be expressed through the convexity parameter
$$
\tilde\theta(\tau)=\exp\left(\int_0^\tau -\|v(s)-JW(v(s))\|_{X^0} \ ds \right) $$
satisfying
$$
\frac{d \tilde\theta}{d\tau}=-\tilde\theta \|v(s)-JW(v(s))\|_{X^0} \;.
$$
Due to the reduced power on the norm, one can then follow the proof of Theorem \ref{titithm}, to derive faster, but still algebraic, lower bounds on the rates of convergence in analogs of \eqref{thalf}-\eqref{wholeX}.
For yet another parametric ODE
\begin{align} \label{etaeqn}
\frac{d\eta}{d\tau} = -\|v(\tau)-JW(v(\tau))\|_{X^0}\;, \quad \eta(0)=1\;,
\end{align}
where $v(\tau)=\eta(\tau)v_0+(1-\eta(\tau))Ju^*$, one can again follow the proof of Theorem \ref{titithm} to show an exponential lower bound on the rate of convergence to $\bar \eta$
\begin{align}\label{exprate}
\eta(\tau)-\bar \eta \ge (1-\bar \eta) e^{-c\tau}
\end{align}
for both cases $\bar \eta=0$ and $\bar \eta \in (0,1)$.
The ``determining form" in the space of trajectories associated with \eqref{etaeqn} is
\begin{equation}\label{detform4}
\frac{dv}{d\tau}=-\|v-JW(v)\|_{X^0}\ (v_0-Ju^*)\;, \quad v(0)=v_0 \;
\end{equation}
which is, strictly speaking, not an ODE, due to the involvement of the initial condition $v_0$ in the vector field. The solution to \eqref{detform4}, however, follows the same trajectory as that for
\eqref{detform3} and \eqref{detform}, only in each case the parametrization is different.
\section{Further properties of solutions of the determining form} \label{sec5}
\begin{proposition}\label{shifty}
Suppose $\tilde{v} \in \BBB _X ^{3R} (Ju^*)$. Denote by $\tilde{v}_\sigma (s) = \tilde{v}(s+\sigma)$, then
\begin{align}\label{shifteqn}
W(\tilde{v} _\sigma) (s) = W(\tilde{v})(\sigma + s)\;.
\end{align}
\end{proposition}
\begin{proof}
Let $w=W(\tilde{v})$ be the unique bounded solution of the equation
\begin{align}\label{E1}
\frac{d}{ds} w + \nu A w + B(w,w) = f - \mu (Jw-\tilde{v}) \;.
\end{align}
Notice that $\|\tilde{v}_\sigma\|_X=\|\tilde{v}\|_X < 3R$.
Since \eqref{E1} is autonomous it is clear that if $w(s)$ corresponds to $\tilde{v}(s)$ then $w(s+\sigma)$ is the unique bounded solution of \eqref{weqn} corresponding to $v(s)=\tilde{v} (s+\sigma) = \tilde{v}_\sigma (s)$, and so \eqref{shifteqn} follows.
\end{proof}
\begin{corollary}\label{noway}
If $\tilde{v} \in \BBB _X ^\rho (Ju^*)$ is periodic with respect to $s$ with period $P > 0$, then $W(\tilde{v})$ is also periodic with period $P$. In particular, if $\tilde v$ is independent of $s$, then $w=W(\tilde v)$ is the
unique steady state of \eqref{E1}.
\end{corollary}
\begin{proof}
By the assumption we have $\tilde{v}(s+P) = \tilde{v}_P (s) = \tilde{v}(s)$. That is $\tilde{v}_P = \tilde{v}$. Hence, by Proposition \ref{shifty} $W(\tilde{v}_P)(s) = W(\tilde{v})(s+P)=W(\tilde{v})(s)$. Thus, $W(\tilde{v})$ is periodic with period $P$. If $\tilde v (s)=\tilde v ^*=\text{constant}$, then $\tilde v(s+P)=\tilde v^*$ for every
$P \in \RR$. Thus the unique bounded solution of \eqref{E1} satisfies $w(s+P)=w(s)$
for all $s \in \RR$, and all $P \in \RR$, which implies $w=\text{constant}$ and as a result $W(\tilde v)$ is
a steady state of \eqref{E1}.
\end{proof}
\begin{remark} For an alternative proof of Corollary \ref{noway} in the case where
$\tilde{v} ^*$ is independent of $s$, consider the steady state version of \eqref{weqn}.
As in the case of the NSE one can show that
\eqref{weqn} with $\tilde{v}^*$ in the right-hand side has a steady state solution.
Since it is bounded with respect to $s$, it is the only solution (one can also show the uniqueness
directly). Thus the steady state solution $w^*$ is the only bounded solution and hence $w^*=W(\tilde{v}^*)$.
\end{remark}
\begin{proposition}
Let $P^* >0$, and assume that there is no solution of the NSE (or the underlying equation), which is periodic with period $P^*$. Then all the periodic initial data $v_0\in \BBB _X ^{3R}(Ju^*)$ with period $P^*$ converge to $Ju^*$. Moreover, if $v_0$ is independent of $s$, then $v(t,s)$ is also independent of $s$, for all $t >0$, and the image of its limit $W(\bar v)$, as $t \to \infty$, is a steady state of the NSE.
\end{proposition}
\begin{proof}
Thanks to \eqref{titi5} we have $v(\tau) = \theta(\tau) v_0 + (1-\theta(\tau))Ju^*$. Since $v_0(s+P^*) = v_0(s)$ for all $s\in \RR$, then if $\theta(\tau) \rightarrow \bar{\theta} >0$, as $\tau\rightarrow \infty$, we have
$\bar v=\bar \theta v_0+(1-\bar \theta)Ju^*$ satisfying
$\bar{v} (s + P^*) = \bar{v}(s)$, where $W(\bar v)$ is a trajectory in $ \mathcal{ A } }\def\BBB{ \mathcal{ B } }\def\CCC{ \mathcal{ C } $, which is periodic with period $P^*$, a contradiction. Hence $\bar{\theta} = 0$. The statement regarding initial data which are independent of $s$ follows from \eqref{titi2}.
\end{proof}
\begin{proposition}
The determining form \eqref{detform} does not have any traveling wave solutions.
\end{proposition}
\begin{proof}
This follows from the fact that each solution of \eqref{detform} converges to a steady state of \eqref{detform}.
\end{proof}
\section{Computational evidence that rate estimates are achieved} \label{sec6}
Here we carry out simulations of \eqref{titi6}, the characteristic parametric determining form, to demonstrate that the lower bounds on the rates of convergence
in Theorem \ref{titithm} are achieved for a particular forcing term of the NSE \eqref{NSE}. The use of either \eqref{detform} or \eqref{titi6} for the expressed purpose of locating the attractor is a subject of future work.
Note that the only connection \eqref{titi6} has to the NSE is
through the map $W$. While the rigorous construction of $W$ in \cite{FJKrT2} involves solving \eqref{weqn} with initial data
$w(s_0)=0$, and taking $s_0\to -\infty$, we can effectively approximate $w(s)$ for $0 < s_1 \le s \le s_2$ by taking
$w(0)=0$ and solving forward in time, i.e., taking $s_1$ sufficiently large. The final time $s_2$ is chosen so as to compute the sup norm of what will be a periodic function of $s$. Just as sufficiently large $\mu$, and correspondingly small $h$ guarantee
that the map $W$ is well-defined, so for similarly chosen $\mu$, $h$ will $|w(s)-u(s)| \to 0$, as $s\to \infty$, at an exponential rate, if we take $v=Ju$ as is done in data assimilation \cite{AOT}
(see also the computational study \cite{GOT}). Using Theorem \ref{mainthm} (iii), we empirically determine that $s_1=1.0$ is a sufficient relaxation time by recovering a particular periodic solution $\tilde{u}(\cdot) \subset \mathcal{ A } }\def\BBB{ \mathcal{ B } }\def\CCC{ \mathcal{ C } $ (see Figure 1). The interpolant $J$ is the projection onto the Fourier modes $(i,j)$, $|i|,|j| \le 5$, and the relaxation parameter $\mu=150$.
We consider two cases for $v_0$: one where we know $\theta \to \bar\theta > 0$,
and another where we can expect $\theta \to 0$. For the former we take for $v_0= J(2\tilde{u}-u^*)$, a point on the ray from
$Ju^*$ through $J\tilde{u}(s_1)$, beyond $J\tilde{u}(s_1)$ (double the distance). In the latter case, $v_0=J\tilde{u}+\delta$, where
$$
\hat{\delta}_{1,1}= \hat{\delta}_{-i,-j} =0.5 \;,\quad
\text{and} \quad \hat{\delta}_{i,j} =0 \quad \text{for} \quad i,j \neq \pm(1,1)\;.
$$
Several techniques are used for computational efficiency. The NSE and \eqref{weqn} are
solved in vorticity form. The steady state $u^*$ and force $f$ are defined through the curl;
$$
\omega^*=\nabla \times u^*, \ \phi=\nabla \times f, \text{ where }
\hat{\omega}^*_{3,4}=24+36i, \ \hat{\omega}^*_{5,0}=60+84i,
\text{ and } \hat{\phi}_{i,j}=25 \hat{\omega}^*_{i,j}\;,
$$
with complex conjugate values at wave vectors $(-3,-4)$ and $(-5,0)$.
Otherwise all Fourier coefficients are 0.
Rather than compute
the supremum norm at each step in $\tau$, this norm is sampled
at 150 values of $\theta \in [0,1]$ (see Figure \ref{fig2}). This approach avoids solving over an extended interval
in $s$ which would be needed in a direct approach due to compounding relaxation times.
For each sample point $\theta$, the solution $w(s)$, $s\in [1.0,1.5]$ is computed by solving (in curl form)
a fully dealiased, $256\times256$ mode pseudospectral discretization of \eqref{weqn} with $\nu=\kappa_0=1$, and $v=\theta v_0 + (1-\theta)J(u^*)$. The time-stepper to solve \eqref{weqn} is a third order Adams-Bashforth scheme, with $\Delta s=5\times 10^{-5}$,
which respects a CFL condition (see \cite{olson2008determining} for more details). The supremum norm in \eqref{titi6} is concurrently computed along with $w$.
Using this approximation of the supremum norm dependence on $\theta$ in \eqref{titi6} results in a piecewise linear
ODE which is solved exactly over each subinterval in $\tau$.
Higher resolution sampling for $\theta \sim 10^{-4}$ is needed to produce smooth convergence for large $\tau$ in Figure \ref{fig3}, demonstrating that the lower bounds
on the rates in Theorem \ref{titithm} are achieved.
We follow a similar procedure for \eqref{etaeqn} to make Figure \ref{fig4}
which indicates that $\eta$ converges at an exponential rate, regardless of whether
$v \to Ju^*$ or $v \to J\tilde{u}$, though in the case of the former the rate still appears to be slower. We also demonstrate in Table \ref{secant} the advantage of having a real-valued function of
a real variable, i.e., the function $\Phi$ in \eqref{titi6}, identify trajectories in $ \mathcal{ A } }\def\BBB{ \mathcal{ B } }\def\CCC{ \mathcal{ C } $ by applying the secant method to converge
in $\tilde{u}$ in just 8 steps.
\begin{figure}[h]
\centerline{\includegraphics[scale=.47]{1_alt2.eps}
\includegraphics[scale=.47]{2.eps}}
\caption{(a) $\tilde{u}(\cdot)\subset \mathcal{ A } }\def\BBB{ \mathcal{ B } }\def\CCC{ \mathcal{ C } $
\quad (b) Relaxation of solution of \eqref{weqn} to $\tilde{u}(\cdot)$.}
\label{fig1}
\end{figure}
\begin{figure}[h]
\centerline{\includegraphics[scale=.25]{5_x_noy.eps}
\includegraphics[scale=.25]{9_nox_noy.eps}}
\caption{Sampling of supremum norm: (a) $v_0=J\tilde{u}+\delta $, where we expect $\theta \to 0$,
(b) $v_0= J(2\tilde{u}-u^*)$, where we know $\theta \to \bar\theta =.5$}
\label{fig2}
\end{figure}
\begin{figure}[h]
\psfrag{g}{a}
\centerline{\includegraphics[scale=.47]{7.eps}
\includegraphics[scale=.47]{12.eps}}
\caption{Convergence for \eqref{titi4}: (a) $v_0=J\tilde{u}+\delta$, (b) $v_0=J(2\tilde{u}-u^*)$}
\label{fig3}
\end{figure}
\begin{figure}[h]
\psfrag{g}{a}
\centerline{\includegraphics[scale=.65]{14.eps}
\includegraphics[scale=.65]{13.eps}}
\caption{Convergence for \eqref{etaeqn}: (a) $v_0=J\tilde{u}+\delta$, (b) $v_0=J(2\tilde{u}-u^*)$}
\label{fig4}
\end{figure}
\begin{table}
\begin{center}
\begin{tabular}{|c|c||c|c|c|c|} \hline
$i$& $\eta_i$ & $\|v_i-JW(v_i)\|_{X^0}$ \\ \hline \hline
0 & 1.00000000000000 & 13.4013441411815 \\ \hline
1 & 0.980000000000000 & 12.8017448867074 \\ \hline
2 & 0.552989966509007 & 1.09311945051355 \\ \hline
3 & 0.513124230285987 & 0.255930982604035 \\ \hline
4 & 0.500937157201634 & 1.792679690457618E-002 \\ \hline
5 & 0.500019210387369 & 3.669263629982810E-004 \\ \hline
6 & 0.500000029216005 & 5.580203907596127E-007 \\ \hline
7 & 0.500000000000913 & 1.745075917466212E-011 \\ \hline
8 & 0.500000000000000 & 2.455150177114436E-014 \\ \hline
\end{tabular}
\end{center}
\caption{Secant method, $v_0=J(2\tilde{u}-u^*)$, $v_i=\eta_i v_0 + (1-
\eta_i)Ju^*$}
\label{secant}
\end{table}
\section{Final remarks} \label{sec7}
The steady state solutions $\bar \theta$ of \eqref{titi6}, i.e., characteristic determining values which are the zeros of the function
$\Phi$ in \eqref{titi6}, capture all the trajectories on the global
attractor $ \mathcal{ A } }\def\BBB{ \mathcal{ B } }\def\CCC{ \mathcal{ C } $ of the NSE through $W(\bar \theta v_0+(1-\bar \theta )Ju^*)$ as $v_0$ varies
throughout $\BBB^{3R}_X(Ju^*)$. We do not know if $W$ is differentiable
so using the Newton-Raphson method to find the zeros of $\Phi(\theta;v_0,u^*)$ may not work.
However, since $W$ is Lipschitz one can (and we do here) find the zeros
by the secant method. In particular, every steady state of the NSE is realized as
the image $W(\bar \theta v_0+(1-\bar \theta )Ju^*)$, where $v_0$ is independent of $s$. It is thus possible, and perhaps beneficial to study bifurcations of steady states through \eqref{titi6}. Moreover, if $v_0$ is periodic (with positive minimal period), then
$W(\bar \theta v_0+(1-\bar \theta )Ju^*)$ is either periodic (with the same minimal period) or the steady state $u^*$. If, on the other hand, $v_0$ is independent of $s \in {\Bbb R}$,
then $W(\bar \theta v_0+(1-\bar \theta )Ju^*)$ is some steady state of the 2D NSE. Rigorous lower bounds suggest the rate of convergence toward $Ju^*$ is slower than toward any other steady
state of the determining form. It remains to study the complete basin of attraction for the determining form of the exceptional state
$Ju^*$.
\section*{Acknowledgements}
The work of C.F. was partially supported by National Science Foundation (NSF) grant
DMS-1109784 and Office of Naval Research (ONR) grant N00014-15-1-2333, that of M.S.J. by NSF grant DMS-1418911 and the Leverhulme Trust grant VP1-2015-036, and D.L. by NSF grant DMS-1418911. The work of E.S.T. was supported in part by the ONR grant N00014-15-1-2333 and the NSF grants DMS-1109640 and DMS-1109645.
\bibliographystyle{plain}
| {
"timestamp": "2016-04-14T02:07:58",
"yymm": "1604",
"arxiv_id": "1604.01727",
"language": "en",
"url": "https://arxiv.org/abs/1604.01727",
"abstract": "The evolution of a determining form for the 2D Navier-Stokes equations (NSE), which is an ODE on a space of trajectories is completely described. It is proved that at every stage of its evolution, the solution is a convex combination of the initial trajectory and the fixed steady state, with a dynamical convexity parameter $\\theta$, which will be called the characteristic determining parameter. That is, we show a remarkable separation of variables formula for the solution of the determining form. Moreover, for a given initial trajectory, the dynamics of the infinite-dimensional determining form are equivalent to those of the characteristic determining parameter $\\theta$ which is governed by a one-dimensional ODE. %for the parameter specifying the position on the line segment. This one-dimensional ODE is used to show that if the solution to the determining form converges to the fixed state it does so no faster than $\\mathcal{O}(\\tau^{-1/2})$, otherwise it converges to a projection of some other trajectory in the global attractor of the NSE, but no faster than $\\mathcal{O}(\\tau^{-1})$, as $\\tau \\to \\infty$, where $\\tau$ is the evolutionary variable in determining form. The one-dimensional ODE also exploited in computations which suggest that the one-sided convergence rate estimates are in fact achieved. The ODE is then modified to accelerate the convergence to an exponential rate. Remarkably, it is shown that the zeros of the scalar function that governs the dynamics of $\\theta$, which are called characteristic determining values, identify in a unique fashion the trajectories in the global attractor of the 2D NSE. Furthermore, the one-dimensional characteristic determining form enables us to find unanticipated geometric features of the global attractor, a subject of future research.",
"subjects": "Dynamical Systems (math.DS)",
"title": "One-dimensional parametric determining form for the two-dimensional Navier-Stokes equations",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9845754447499796,
"lm_q2_score": 0.7185943805178139,
"lm_q1q2_score": 0.7075103817931627
} |
https://arxiv.org/abs/1511.02723 | Variance inequalities for quadratic forms with applications | We obtain variance inequalities for quadratic forms of weakly dependent random variables with bounded fourth moments. We also discuss two application. Namely, we use these inequalities for deriving the limiting spectral distribution of a random matrix and estimating the long-run variance of a stationary time series. | \section{Introduction}
Moment inequalities for quadratic forms constitute a powerful tool in time series analysis and the random matrix theory. In particular, they are used in the study of consistency and optimality properties of spectral density estimates (e.g., see Section V.4 in Hannan \cite{H}) as well as provide low-level conditions under which the limiting spectral distribution of a random matrix can be derived (e.g., see Chapter 19 in Pastur and Shcherbina \cite{PS}).
When random variables $\{X_{i}\}_{i=1}^n$ are independent, moment inequalities for quadratic forms $\sum_{i,j=1}^n a_{ij}X_iX_j$ are well-studied (e.g., see Lemma B.26 in Bai and Silverstein \cite{BS} and Chen \cite{C}). In the time series context, similar inequalities were obtained by many authors in connection with spectral density estimation and long-run variance estimation (e.g., see Chapter VI in White \cite{W}, Sections 6 and 7 in Wu and Xiao \cite{WX} and the references therein). In particular, high-order moment inequalities for causal time series were obtained by Wu and Xiao \cite{WX}.
In the present paper we study variance inequalities for quadratic forms $\sum_{i,j=1}^n a_{ij}X_iX_j$ of weakly dependent random variables $\{X_i\}_{i=1}^n$ with bounded fourth moments.
Our assumptions deal with covariances of $X_i$'s products up to the fourth order only and are closely related to the classical fourth-order cumulant condition for a stationary time series (see Theorem V.4 in Hannan \cite{H} and Assumption A in Andrews \cite{A}). These assumptions can be easily verified under standard weak dependence conditions (e.g., strong mixing). We also demonstrate how our results can be applied in the random matrix theory and time series analysis.
The paper is structured as follows. Main results are given in Section 2. Section 3 is devoted to applications.
Section 4 contains all proofs.
\section{Main results}
Let $\{X_k\}_{k=1}^\infty $ be a sequence of centred random variables and let $\{\varphi_k\}_{k=1}^\infty$ be a non-increasing sequence of non-negative numbers such that, for all $i<j<k<l$,
\begin{equation}\label{vphi}|\cov(X_i,X_jX_kX_l)|\leqslant \varphi_{j-i},\quad |\cov(X_iX_jX_k,X_l)|\leqslant \varphi_{l-k},
\end{equation}
\begin{equation}\label{vphi1}|\cov(X_iX_j,X_kX_l)|\leqslant \varphi_{k-j},\quad\text{and}\quad |\cov(X_i,X_{j})|\leqslant \varphi_{j-i}.
\end{equation}
Assumptions of this kind go back to R\'ev\'esz \cite{R} and studied by Koml\'os and R\'ev\'esz \cite{KR}, Gaposhkin \cite{G}, and Longecker and Serfling \cite{LS} (see also M\'oricz \cite{M}, Section 4.3 and 4.4 in Doukhan et al. (2007)).
For simplicity, we will further assume that $\e X_k^2\leqslant 1$, $k\geqslant 1$.
Define $\X_p=(X_1,\ldots,X_p)$ for $p\geqslant 1$, $\Phi_0=\sup\{\e X_1^4,\e X_2^4,\ldots \}$, and $ \Phi_1=\sum_{k=1}^\infty k\varphi_k.$
\begin{theorem}\label{t1}
There is a universal constant $C>0$ such that, for any $a\in\bR^p$ and all $p\times p$ matrices $A$ with zero diagonal, \[\e(\X_p^\top a)^4\leqslant C(\Phi_0+\Phi_1)(a^\top a)^2\;\;\text{and}\;\; \var(\X_p^\top A\X_p)\leqslant C(\Phi_0+\Phi_1 )\tr(A A^\top).\]
\end{theorem}
A version of the first inequality in Theorem \ref{t1} is proved by Koml\'os and R\'ev\'esz \cite{KR}, Gaposhkin \cite{G}, and Longecker and Serfling \cite{LS}. The second inequality is new.
Let now $\phi_k,$ $k\geqslant 1,$ satisfy
\begin{equation}\label{phi}
\cov(X_i^2,X_j^2)\leqslant \phi_{j-i} \quad\text{for all }i<j.
\end{equation}
Define $\Phi_2=\sum_{k=1}^\infty \phi_k$.
\begin{theorem}\label{t2} There is a universal constant $C>0$ such that, for all $p\times p$ matrices $A$,
\[\var(\X_p^\top A\X_p)\leqslant C(\Phi_0+\Phi_1+ \Phi_2)\tr(A A^\top).\]
\end{theorem}
Let us give two examples of $\{X_k\}_{k=1}^\infty$ that satisfy \eqref{vphi}, \eqref{vphi1}, and \eqref{phi}. \\
\noindent {\bf Example 1.} Let $\{X_k\}_{k=1}^\infty$ be a martingale difference sequence with bounded 4th moments. Then \eqref{vphi} and \eqref{vphi1} hold for $\varphi_k=0,$ $k\geqslant 1,$ and $\Phi_1=0$. However, in general, there are no such $\phi_k,$ $k\geqslant 1$, that \eqref{phi} holds and $\Phi_2<\infty$. This explains why we introduce two sets of coefficients $\{\varphi_k\}_{k=1}^\infty$ and $\{\phi_k\}_{k=1}^\infty$. If, in addition, $\{X_k^2-\e X_k^2\}_{k=1}^\infty$ is a martingale difference sequence, then, of course, $\phi_k=0$, $k\geqslant 1,$ and $\Phi_2=0.$
\noindent {\bf Example 2.} Let $\{X_k\}_{k=1}^\infty $ be strongly mixing random variables with mixing coefficients $(\alpha_k)_{k=1}^\infty$, zero mean, and bounded moments of order $4\delta$ for some $\delta>1$. Then \eqref{vphi}--\eqref{phi} hold for $\varphi_k=\phi_k=C\alpha_k^{(\delta-1)/\delta}$ and large enough $C>0$ (e.g., see Corollary A.2 in Hall and Heyde \cite{HH}). One can give similar bounds for other weak dependence conditions.
\begin{remark} We believe that higher order moment inequalities for quadratic forms $\X_p^\top A\X_p$ can be derived under similar conditions on $\cov(X_{i_1}\ldots X_{i_k},X_{i_{k+1}}\ldots X_{i_p})$ for $i_1< \ldots<i_p$, $k=1,\ldots,p-1$, and $p>4$. However, the proofs are quite technical even in the case of the second-order inequalities and we leave this question for future research.
\end{remark}
Consider the special case when $X_k,$ $k\geqslant 1$, are centred orthonormal random variables. In this case, \eqref{vphi} and \eqref{vphi1} reduce to
\begin{equation*}|\e X_iX_jX_kX_l|\leqslant\min\{ \varphi_{j-i},\varphi_{k-j},\varphi_{l-k}\},\quad i<j<k<l.
\end{equation*}
Let $\y_p=(Y_1,\ldots,Y_p)$, where each $Y_j$ can be written as $\sum_{k=1}^\infty a_{k}X_k$ in $L_2$ for some $a_{k}\in\bR,$ $k\geqslant 1,$ with $\sum_{k=1}^\infty a_k^2<\infty.$
\begin{corollary}\label{t4} Let $\Sigma_p=\e\y_p\y_p^\top$. Then there is $C>0$ such that, for any $p\times p$ matrix $A$, \[\var(\y_p^\top A\y_p)\leqslant C(\Phi_0+\Phi_1+\Phi_2)\tr(\Sigma_p A\Sigma_p A^\top).\]
\end{corollary}
If $\{X_k\}_{k=1}^\infty$ are independent standard normal variables and $A$ is a $p\times p$ symmetric matrix, then $\var(\y_p^\top A\y_p)=2\tr((\Sigma_p A)^2)$
(e.g., see Lemma 2.3 in Magnus \cite{JM}). Thus, Corollary \ref{t4} delivers an optimal estimate of the variance up to the factor $C(\Phi_0+\Phi_1+\Phi_2).$
\section{Applications}
In this section we discuss two applications of the obtained inequalities.
Our first application will be in the random matrix theory.
For each $p,n\geqslant 1$, let $\bY_{pn}$ be a $p\times n$ random matrix whose columns are independent copies
of $\y_p$, where $\y_p$ is given either in Corollary \ref{t4}, or $\y_p=\X_p$ for $\X_p$ from Theorem \ref{t2}.
\begin{theorem}\label{rm}
Let $\Phi_0,\Phi_1,\Phi_2<\infty$. If the following conditions hold
\begin{itemize}
\item[$(1)$] $p=p(n)$ is such that $p/n\to c$ for some $c>0,$
\item[$(2)$] the spectral norm of $\Sigma_{p}=\e \y_p\y_p^\top $ is bounded over $p$,
\item[$(3)$] the empirical spectral distribution of $\Sigma_p$'s eigenvalues has a weak limit $P(d\lambda)$,
\end{itemize}
then, with probability one, the empirical spectral distribution of $n^{-1}\bY_{pn}\bY_{pn}^\top$'s eigenvalues has a weak limit whose Stieltjes transform $m = m(z)$ satisfies
\[m(z)=\int_0^\infty\frac{P(d\lambda)}{\lambda(1-c-czm(z))-z},\quad z\in\mathbb C,\quad {\rm\,Im} (z)>0.\]
\end{theorem}
The next application concerns long-run variance estimation. First, let us recall the generic form of the central limit theorem for a weakly stationary time series $(X_t)_{t=-\infty}^\infty$:
\[\sqrt{n}(\overline{X}_n-\mu)\stackrel{d}\to\mathcal N(0,\sigma^2),\quad n\to\infty,\]
where $\overline{X}_n=n^{-1}\sum_{t=1}^n X_t$, $\e X_t=\mu$, and $\sigma^2$ is the long-run variance of $(X_t)_{t=-\infty}^\infty$, i.e.
\[\sigma^2=\sum_{j=-\infty}^\infty\cov(X_t,X_{t+j}).\]
This theorem can be proved under different weak dependence assumptions (e.g., see the books of Doukhan et al. \cite{D} and Bulinski and Shashkin \cite{BS}). In statistical applications, this theorem takes the form
\[\frac{\sqrt{n}(\overline{X}_n-\mu)}{\wh\sigma_n}\stackrel{d}\to\mathcal N(0,1),\quad n\to\infty,\]
where $\wh\sigma^2_n$ is a consistent estimator of $\sigma^2.$ Recall also that $\sigma^2$ can be written as $\sigma^2=2\pi f(0),$ where $f=f(x),$ $x\in[-\pi,\pi),$ is the spectral density of $(X_t)_{t=-\infty}^\infty$. Therefore, long-run variance estimation is closely related to spectral density estimation.
A number of papers is devoted to the study of consistency and optimality properties of long-run variance estimators (e.g., see Andrews \cite{A}, Hansen \cite{H1}, de Jong and Davidson \cite{JD}, and Jansson \cite{J} among others). When $\mu=0$, a typical estimator has the form
\begin{equation}\label{sig}
\wh\sigma^2_n=\frac{1}{n}\sum_{s,t=1}^n K\bigg(\frac{|s-t|}{m}\bigg)X_sX_t,
\end{equation}
where $K=K(x),$ $x\geqslant 0,$ is a kernel function. Standard assumptions on $K=K(x)$ include
$(a)$ $K(0)=1$, $K$ is continuous at $x=0$, and $\sup_{x\geqslant 0}|K(x)|<\infty$,
$(b)$ $\int_{0}^\infty \bar{K}^2(x)\,dx<\infty$ for $\bar{K}(x)=\sup_{y\geqslant x}|K(y)|,$
$(c)$ $k_q=\lim_{x\to 0+}x^{-q}(K(x)-1)$ exists for some $q> 0$.
Assumptions $(a)$--$(b)$ are inspired by Assumption 3 of Jansson \cite{J}. However, $(b)$ is weaker than Assumption 3(ii) in \cite{J}, where the integrability of $\bar{K}$ is assumed.
To our knowledge, the weakest alternative to $(b)$ considered in the literature
is the integrability of $K^2$. However, as discussed in Jansson \cite{J}, many previous results (Andrews \cite{A}, Hansen \cite{H1}, among others) are incorrect as they stated and need stronger alternatives to the integrability of $K^2$. Assumption $(c)$ is classical and goes back to Parzen \cite{Pa} (see also Andrews \cite{A}).
Let further $(X_t)_{t=-\infty}^\infty$ be a centred weakly stationary time series that satisfies conditions from Section 2 (in particular, $\e X_t^2\leqslant 1$). Our first result is the consistency of $\wh\sigma^2_n$.
\begin{theorem}\label{lv} Let $K=K(x)$ satisfy $(a)$--$(b)$. If $\Phi_0,\Phi_1,\Phi_2$ are finite, then
$\wh\sigma^2_n\to \sigma^2$ in mean square as $m,n\to\infty$ and $m=o(n)$.
\end{theorem}
The dependence $m=o(n)$ is optimal. It can be seen by taking a Gaussian white noise $(X_t)_{t=-\infty}^\infty$ and showing that $\var(\wh\sigma^2)\not\to 0$ when $m/n\not \to0$ due to the variance formula for Gaussian quadratic forms given in the end of Section 2. Andrews \cite{A} following Hannan \cite{H} proved consistency of $\wh\sigma^2_n$ under the cumulant condition
\begin{equation}\label{cum}
\sum_{j,k,l=1}^\infty \sup_{t\geqslant 1}|\kappa(X_t,X_{t+j},X_{t+k},X_{t+l})|<\infty.
\end{equation}
Here $\kappa(X_i,X_{j},X_{k},X_{l})$ is the fourth-order cumulant that is equal to
\[\e X_iX_{j}X_{k}X_{l}-\e X_iX_{j} \e X_{k}X_{l} -
\e X_iX_{k}\e X_{j}X_{l}- \e X_iX_{l}\e X_{k}\e X_{j}\]
when each $X_t$ has zero mean. By Lemma 1 of Andrews \cite{A}, \eqref{cum} holds when $(X_t)_{t=-\infty}^\infty$ is a strongly mixing sequence with mixing coefficients $(\alpha_k)_{k=1}^\infty$ satisfying \[\sum_{k=1}^\infty k^2 \alpha_k^{(\delta-1)/\delta}<\infty\] and bounded moments of order $4\delta$ for some $\delta>1$. By Example 2, our Theorem \ref{lv} is applicable whenever $\sum_{k=1}^\infty k \alpha_k^{(\delta-1)/\delta}<\infty$.
The cumulant condition allows to calculate the limit of the mean squared error (MSE) of $\wh\sigma_n^2$ explicitly. We can not do it under our assumptions. However, we can give an upper bound for MSE which is very similar to the exact limit (see Proposition 1 in Andrews \cite{A}).
\begin{theorem}\label{lv1} Under conditions of Theorem \ref{lv}, let $(c)$ hold for some $q>0$ and
\[\text{the series }\Gamma_q=\sum_{j=1}^\infty j^q\cov(X_t,X_{t+j})\text{ converges absolutely.}\]Then there is an absolute constant $C>0$ such that, as $m,n\to\infty,$
\[\e|\wh\sigma^2_n-\sigma^2|^2\leqslant C(\Phi_0+\Phi_1+\Phi_2)\frac{m}{n}\int_{0}^\infty \bar{K}^2(x)\,dx+\frac{4(k_q\Gamma_q)^2}{m^{2q}}+o(m^{-2q})+O(n^{-1}).\]
\end{theorem}
\section{Proofs}
Below we assume that $\Phi_0,\Phi_1,\Phi_2$ are finite otherwise all bounds become trivial.
\noindent {\bf Proof of Theorem \ref{t1}.} To prove the first inequality, we reproduce the proof given in Gaposhkin \cite{G} with the only difference that we derive explicit constants in his inequality.
Note first that, as $\e X_i=0$ for all $i\geqslant 1$, it follows from \eqref{vphi} that
\begin{equation}
\label{vphi2}
|\e X_iX_jX_kX_l|\leqslant\min\{\varphi_{j-i},\varphi_{l-k}\},\quad i<j<k<l.
\end{equation}
Write $a=(a_1,\ldots,a_p)$. Using Lemma 1 in Moricz \cite{M} with $p=2$ and $r=4$, we get \[|(\X_p^\top a)^4-24 T|\leqslant C_0(S^4+|\X_p^\top a|^3S),\]
where $C_0>0$ is a universal constant,
\[T=\sum_{i<j<k<l}a_ia_ja_ka_lX_iX_jX_kX_l,\quad S=\Big(\sum_{i=1}^pa_i^2X_i^2\Big)^{1/2},\]
hereinafter $i,j,k,l$ are any numbers in $\{1,\ldots,p\}$.
By H\"older's inequality,
\begin{align*}
\e(\X_p^\top a)^4\leqslant 24 \e T+C_0\big(\e S^4 +(\e S^4)^{1/4}(\e |\X_p^\top a|^4)^{3/4}).
\end{align*}
By \eqref{vphi2},
\[
|\e T|\leqslant\sum_{ i<j<k<l}|a_ia_ja_ka_l| \min\{\varphi_{j-i},\varphi_{l-k}\}\leqslant \frac{1}{4}\sum_{ i<j<k<l}(a_i^2+a_j^2)(a_k^2+a_l^2) \min\{\varphi_{j-i},\varphi_{l-k}\}. \]
We estimate only the term
\[J=\sum_{ i<j<k<l}a_i^2a_k^2\min\{\varphi_{j-i},\varphi_{l-k}\}. \]
Other terms with $a_j^2a_k^2$, $a_i^2a_l^2$, and $a_j^2a_l^2$ instead of $a_i^2a_k^2$ can be estimated similarly. We have
\[J\leqslant \sum_{ i<k}a_i^2a_k^2\sum_{q,r=1}^\infty \min\{\varphi_{q},\varphi_{r}\}\]
and
\begin{equation}
\label{qr}
\sum_{q,r=1}^\infty \min\{\varphi_{q},\varphi_{r}\}\leqslant
\sum_{q=1}^\infty \Big( q\varphi_q+\sum_{r=q+1}^\infty \varphi_r \Big)=\Phi_1+\sum_{r=2}^\infty\sum_{q=1}^{r-1}\varphi_r\leqslant 2\Phi_1.
\end{equation}
As a result, \[J\leqslant 2\Phi_1\sum_{ i<k}a_i^2a_k^2\leqslant \Phi_1\Big(\sum_{ i=1}^pa_i^2\Big)^2\quad\text{and}\quad |\e T|\leqslant \Phi_1\|a\|^4.\]
Let us also note that
\[\e S^4=\sum_{i,j=1}^pa_i^2 a_j^2\e X_i^2X_j^2\leqslant \Phi_0\|a\|^4.\]
Combining the above estimates, we infer that
\[\e(\X_p^\top a)^4\leqslant (24+C_0) (\Phi_0+\Phi_1)\|a\|^4 +C_0[(\Phi_0+\Phi_1)\|a\|^4]^{1/4}[\e(\X_p^\top a)^4]^{3/4}.\]
Put $R=[\e(\X_p^\top a)^4/(\Phi_0+\Phi_1)]^{1/4}/\|a\|$. Then $R^4\leqslant 24+C_0+C_0R^3.$ Therefore, $R\leqslant R_0$, where $R_0>0$ is the largest positive root of the equation $x^4=24+C_0+C_0x^3.$ Finally, we obtain
\[
\e|\X_p^\top a|^4\leqslant R_0^4(\Phi_0+\Phi_1)\|a\|^4.
\]
We now verify the second inequality. First, note that, for $i<j,$
\[|\cov( X_i,X_j)|=|\e X_iX_j|\leqslant \sqrt{\e X_i^2\e X_j^2}\leqslant 1\]
In addition, for $i<j<k<l,$ \begin{equation}\label{fi}
|\cov(X_iX_j,X_kX_l)|\leqslant 2\min\{\varphi_{j-i},\varphi_{k-j},\varphi_{l-k}\}\quad\text{and}\quad I\leqslant 2\min\{\varphi_{j-i},\varphi_{l-k}\},
\end{equation}
where $I$ is equal to $|\cov(X_iX_k,X_jX_l)|$ or $|\cov(X_iX_l,X_jX_k)|$. Indeed, by \eqref{vphi1} and \eqref{vphi2},
\begin{align*}|\cov(X_iX_j,X_kX_l)|&\leqslant\min\{\varphi_{k-j},|\e X_iX_jX_kX_l|+| \e X_iX_j\e X_kX_l|\}\\
&\leqslant \min\{\varphi_{k-j},2\min\{\varphi_{j-i},\varphi_{l-k}\}\}\\
&\leqslant 2\min\{\varphi_{j-i},\varphi_{k-j},\varphi_{l-k}\},
\end{align*}
and, by the monotonicity of $\varphi_k$,
\begin{align*}
|\cov(X_iX_k,X_jX_l)|&\leqslant |\e X_iX_k\e X_jX_l|+|\e X_iX_j X_kX_l |\\
&\leqslant\min\{\varphi_{k-i},\varphi_{l-j}\}+\min\{\varphi_{j-i},\varphi_{l-k}\}\\
&\leqslant 2\min\{\varphi_{j-i},\varphi_{l-k}\}.
\end{align*}
A similar bound holds for $\cov(X_iX_l,X_jX_k).$
Let $A=(a_{ij})_{i,j=1}^p$ and $a_{ii}=0$, $1\leqslant i\leqslant p$. Set $B=(A^\top+A)/2$. Then $\X_p^\top A\X_p=\X_p^\top B\X_p$ and
\begin{equation}\label{BB}
\tr(BB^\top)=\sum_{i,j=1}^p \Big(\frac{a_{ij}+a_{ji}}2\Big)^2\leqslant
\sum_{i,j=1}^p \frac{a_{ij}^2+a_{ji}^2}2=\sum_{i,j=1}^p a_{ij}^2=\tr(AA^\top).
\end{equation}
Hence, we may assume w.l.o.g. that $A=A^\top.$ Then
\begin{align*}
\var(&\X_p^\top A\X_p)=4
\var\Big(\sum_{i=1}^{p-1} X_i \sum_{k=i+1}^p a_{ik} X_k\Big)=4\sum_{i=1}^{p-1} \var\Big( X_i\sum_{k=i+1}^p a_{ik} X_k\Big)+\\
&+8\sum_{i<j}\cov\Big( X_i\sum_{k=i+1}^p a_{ik} X_k,X_j\sum_{k=j+1}^p a_{jk} X_k\Big)=4I_1+8I_2+8I_3+8I_4,
\end{align*}
where
\begin{align*}
I_1&=\sum_{i=1}^{p-1} \var\Big( X_i\sum_{k=i+1}^p a_{ik} X_k\Big),\\
I_2&=\sum_{i<j}\cov\Big( X_i \sum_{k=i+1}^{j-1} a_{ik} X_k, X_j \sum_{k=j+1}^p a_{jk} X_k\Big),\\
I_3&=\sum_{i<j}a_{ij}\cov\Big( X_i X_j,X_j \sum_{k=j+1}^p a_{jk} X_k\Big),\\
I_4&=\sum_{i<j}\cov\Big( X_i\sum_{k=j+1}^p a_{ik} X_k,X_j\sum_{k=j+1}^p a_{jk} X_k\Big),
\end{align*}
and the sums over the empty set are zeros.
\\
{\it Control of $I_1$.} By the Cauchy-Schwartz inequality and the first inequality in Theorem \ref{t1},
\begin{align*}
I_1\leqslant &
\sum_{i=1}^{p-1} \sqrt{\e X_i^4}\Big(\e\Big|\sum_{k=i+1}^p a_{ik} X_k\Big|^4\Big)^{1/2}\\
\leqslant&
C(\Phi_0+\Phi_1) \sum_{i=1}^{p-1} \sum_{k=i+1}^p a_{ik}^2=C(\Phi_0+\Phi_1)\frac{\tr(A^2)}2
\end{align*}
\\
{\it Control of $I_2$.} By the Cauchy inequality and \eqref{fi},
\begin{align*} I_2\leqslant&\sum_{i<k<j<l}|a_{ik}a_{jl}|\,|\cov(X_iX_k,X_jX_l)|\\
\leqslant& 2\sum_{i<k<j<l}|a_{ik}a_{jl}|\min\{\varphi_{k-i},\varphi_{j-k},\varphi_{l-j}\}\\
\leqslant & I_5+I_6 ,\end{align*}
where
\[ I_5=\sum_{i<k<j<l}
a_{ik}^2\min\{\varphi_{j-k},\varphi_{l-j}\},\quad
I_6=\sum_{i<k<j<l}
a_{jl}^2\min\{\varphi_{k-i},\varphi_{j-k}\}.
\]
Additionally, by \eqref{qr},
\begin{align*}
I_5\leqslant&\sum_{i<k}a_{ik}^2\sum_{q,r=1}^\infty \min\{\varphi_q,\varphi_r\}\leqslant \frac{\tr(A^2)}2\,(2\Phi_1)= \tr(A^2)\Phi_1.
\end{align*}
We similarly derive that $I_6\leqslant\tr(A^2)\Phi_1.$ Hence, $I_2\leqslant 2\tr(A^2)\Phi_1 $.
\\
{\it Control of $I_3$.} By the Cauchy-Schwartz inequality and the first inequality in Theorem \ref{t1},
\begin{align*}
I_3=&\sum_{j=2}^{p-1}\cov\Big(X_j\sum_{i=1}^{j-1} a_{ij} X_i,X_j\sum_{k=j+1}^p a_{jk} X_k\Big)\\
&\leqslant \sum_{j=2}^{p-1}\Big(\e X_j^2\Big|\sum_{i=1}^{j-1} a_{ij} X_i\Big|^{2}\Big)^{1/2}\Big(\e X_j^2\Big|\sum_{k=j+1}^p a_{jk} X_k\Big|^2\Big)^{1/2}\\
&\leqslant\sum_{j=2}^{p-1}\sqrt{\e X_j^4}\bigg[\e\Big(\sum_{i=1}^{j-1} a_{ij} X_i\Big)^4\e\Big(\sum_{k=j+1}^p a_{jk} X_k\Big)^4\bigg]^{1/4}\\&\leqslant\sqrt{C(\Phi_0+\Phi_1)}(I_7+I_8)/2,
\end{align*}
where
\[I_7=\sum_{j=2}^{p-1}\bigg[\e\Big(\sum_{i=1}^{j-1} a_{ij} X_i\Big)^4\bigg]^{1/2},\quad I_8=\sum_{j=2}^{p-1}\bigg[\e\Big(\sum_{k=j+1}^p a_{jk} X_k\Big)^4\bigg]^{1/2}.\]
By the first inequality in Theorem \ref{t1},
\[I_7\leqslant K\sum_{j=2}^{p-1}\sum_{i=1}^{j-1} a_{ij}^2\leqslant\frac{K\tr(A^2)}{2},\quad
I_8\leqslant K\sum_{j=2}^{p-1}\sum_{k=j+1}^p a_{jk}^2\leqslant\frac{K\tr(A^2)}{2},\]
where $K=\sqrt{C(\Phi_0+\Phi_1)}.$ As a result, $I_3\leqslant C(\Phi_0+\Phi_1)\tr(A^2)/2$.
\\
{\it Control of $I_4$.} We have $I_4=I_9+I_{10}+I_{11}$, where
\[I_9=\sum_{i<j<k}\cov(a_{ik }X_i X_k,a_{jk} X_jX_k),\;\; I_{10}=
\sum_{i<j<k<l}a_{ik}a_{jl}\cov( X_iX_k,X_{j}X_l),\]\[
I_{11}=\sum_{i<j<k<l}a_{il}a_{jk}\cov( X_iX_l,X_{j}X_k).\]
By the first inequality in Theorem \ref{t1},
\[I_9=\frac{1}{2}\sum_{k=3}^p \var\Big(X_k\sum_{i=1}^{k-1}a_{ik}X_i\Big)-
\frac{1}{2}\sum_{k=3}^p \sum_{i=1}^{k-1}\var( a_{ik}X_iX_k) \leqslant \]
\[\leqslant\frac{1}{2} \sum_{k=3}^p\bigg[ \e X_k^4\e\Big(\sum_{i=1}^{k-1}a_{ik}X_i\Big)^4\bigg]^{1/2}\leqslant C(\Phi_0+\Phi_1)
\sum_{k=3}^p \sum_{i=1}^{k-1}\frac{a_{ik}^2}2\leqslant C(\Phi_0+\Phi_1)\,\frac{\tr(A^2)}{4}.
\]
Let us now estimate $I_{10}$ and $I_{11}$. By \eqref{fi},
\begin{align*}
I_{10}\leqslant 2\sum_{i<j<k<l}|a_{ik}a_{jl}| \min\{\varphi_{j-i},\varphi_{l-k}\}\quad\text{and}\quad
I_{11}\leqslant 2\sum_{i<j<k<l}|a_{il}a_{jk}| \min\{\varphi_{j-i},\varphi_{l-k}\}.
\end{align*}
By the Cauchy inequality, $I_{10}\leqslant I_{12}+I_{13}$ and $I_{11}\leqslant I_{14}+I_{15}$ with
\[I_{12}=\sum_{i<j<k<l}a_{ik}^2 \min\{\varphi_{j-i},\varphi_{l-k}\},\quad I_{13}=\sum_{i<j<k<l}a_{jl}^2 \min\{\varphi_{j-i},\varphi_{l-k}\},\]
\[I_{14}=\sum_{i<j<k<l}a_{il}^2 \min\{\varphi_{j-i},\varphi_{l-k}\},\quad I_{15}=\sum_{i<j<k<l}a_{jk}^2 \min\{\varphi_{j-i},\varphi_{l-k}\},\]
As previously, we have
\begin{align*}
I_{12}&\leqslant\sum_{i<k}a_{ik}^2\sum_{q,r=1}^\infty \min\{\varphi_q,\varphi_r\}\leqslant \tr(A^2)\Phi_1.
\end{align*}
By the same arguments, $I_{13}$, $I_{14},$ and $I_{15}$ can be bounded from above by $\tr(A^2)\Phi_1$.
Thus, we conclude that $I_{10}+I_{11}\leqslant 4\tr(A^2)\Phi_1$.
Combining all above estimates, we get
$\var(\X_p^\top A\X_p)\leqslant C(\Phi_0+\Phi_1)\tr(A^2)$ for a universal constant $C>0.$
Q.e.d.
\\
\noindent{\bf Proof of Theorem \ref{t2}.}
Let $A=(a_{ij})_{i,j=1}^p$ and $D$ be the $p\times p$ diagonal matrix with diagonal entries $a_{11},\ldots,a_{pp}$.
By Theorem \ref{t1},
\[\var(\X_p^\top (A-D)\X_p) \leqslant C(\Phi_0+\Phi_1)\tr((A-D)(A-D)^\top).\]
In addition, $\var(\X_p^\top A\X_p)\leqslant 2\var(\X_p^\top D\X_p)+2\var(\X_p^\top (A-D)\X_p) .$
Noting that \[\tr (AA^\top)=\tr((A-D)(A-D)^\top)+\tr(D^2),\] we only need to bound $\var(\X_p^\top D\X_p)$ from above by $\tr(D^2)$ up to a constant factor. Write $D=D_1-D_2$, where $D_i$ are diagonal matrices with non-negative diagonal entries and $\tr(D^2)=\tr(D_1^2)+\tr(D_2^2)$. By the Cauchy inequality, $\var(\X_p^\top D\X_p)\leqslant 2\sum_{i=1}^2\var(\X_p^\top D_i\X_p).$ Hence,
we may assume w.l.o.g. that diagonal elements of $D$ are non-negative.
We see that
\begin{align*}
\var(\X_p^\top D\X_p )=\var\Big(\sum_{i=1}^pa_{ii}X_i^2 \Big)&=\sum_{i=1}^p a_{ii}^2\var( X_i^2)+\sum_{i\neq j}a_{ii}a_{jj}\cov( X_i^2, X_j^2)\\
&\leqslant \Phi_0\sum_{i=1}^n a_{ii}^2+\sum_{i\neq j}a_{ii}a_{jj}\phi_{|i-j|}
\end{align*}
and, as a result,
\begin{align*}
\var(\X_p^\top D\X_p )\leqslant&\Phi_0\tr(D^2)+\sum_{i\neq j}\frac{a_{ii}^2+a_{jj}^2}2 \phi_{|i-j|}\\
\leqslant&\Phi_0\tr(D^2)+\sum_{i=1}^p a_{ii}^2\sum_{j: j\neq i}\phi_{|i-j|}\\
\leqslant& 2\tr(D^2)\Big(\Phi_0+\sum_{k=1}^\infty \phi_k\Big)= 2(\Phi_0+\Phi_2)\tr (D^2).
\end{align*}
Combining the above bounds, we get the desired inequality. Q.e.d.\\
{\bf Proof of Corollary \ref{t4}.}
By the definition of $\y_p$, $\Gamma_n \X_n\to \y_p $ in probability and in mean square as $n\to\infty$ for some $p\times n$ matrices $\Gamma_n$ and $\X_n=(X_1,\ldots,X_n)$. Since $X_k, k\geqslant 1,$ are orthonormal, we have $\Gamma_n\Gamma_n^\top=\e (\Gamma_n \X_n)(\Gamma_n \X_n)^\top\to \e \y_p\y_p^\top=\Sigma_p$, \[\X_n^\top(\Gamma_n^\top A \Gamma_n)\X_n=(\Gamma_n \X_n)^\top A( \Gamma_n \X_n)\to \y_p^\top A\y_p\quad\text{in probability},\]
and \[\e \X_n^\top( \Gamma_n^\top A\Gamma_n) \X_n=\tr(\Gamma_n^\top A \Gamma_n)=\tr( \Gamma_n\Gamma_n^\top A)\to \tr(\Sigma_pA)=\e \y_p^\top A\y_p\] as $n\to\infty.$ We need the following version of Fatou's lemma:
\begin{center}
if $\xi_n\to \xi$ in probability, then $\e |\xi|\leqslant \varliminf\limits_{n\to\infty}\e |\xi_n|.$
\end{center}By this lemma and Theorem \ref{t2},
\begin{align*}
\e|\y_p^\top A\y_p-\tr(\Sigma_p A)|^2\leqslant & \varliminf\limits_{n\to\infty} \e|\X_n^\top( \Gamma_n^\top A\Gamma_n) \X_n-\tr(\Gamma_n^\top A\Gamma_n)|^2\\
\leqslant&\varliminf\limits_{n\to\infty} C(\Phi_0+\Phi_1+\Phi_2) \tr(\Gamma_n^\top A\Gamma_n\Gamma_n^\top A^\top \Gamma_n)
\end{align*}
Note that $\tr(\Gamma_n^\top A\Gamma_n\Gamma_n^\top A^\top \Gamma_n)=\tr(\Gamma_n\Gamma_n^\top A\Gamma_n\Gamma_n^\top A^\top)\to\tr(\Sigma_pA\Sigma_pA^\top ).$ Q.e.d.
\noindent{\bf Proof of Theorem \ref{rm}.} Denote the spectral norm of a matrix $A$ by $\|A\|$. Recall that $\|A\|=\sqrt{\|AA^\top\|}=\sqrt{\|A^\top A\|}$. In addition, let $A^{1/2}$ be the principal square root of a square positive semi-definite matrix $A$. By Theorem 1.1 in Bai and Zhou \cite{BZ}, we will prove the theorem by checking that $\var(\y_p^\top A_p\y_p)=o(p^2)$ as $p\to\infty$ for any sequence $(A_p)_{p=1}^\infty$ with $\|A_p\|=O(1)$, where $A_p$ is a $p\times p$ matrix.
First, let $\y_p$ be as in Corollary \ref{t4}. Then \[\tr(\Sigma_pA_p\Sigma_pA_p^\top)=\tr(\Sigma_p^{1/2}A_p\Sigma_pA_p^\top \Sigma_p^{1/2})=\tr(Q \Sigma_pQ^\top)\]
with $Q=\Sigma_p^{1/2}A_p$. If $I_p$ is the $p\times p$ identity matrix, then $\|\Sigma_p\|I_p-\Sigma_p$ is positive semi-definite and, as a result, $Q(\|\Sigma_p\|I_p-\Sigma_p) Q^\top$ is positive semi-definite for any $Q$. Hence,
\[\tr(Q^\top \Sigma_pQ)\leqslant\|\Sigma_p\|\tr(QQ^\top )=\|\Sigma_p\|\tr(Q^\top Q)=\|\Sigma_p\|\tr(A_p^\top \Sigma_p A_p)\leqslant \|\Sigma_p\|^2\tr(A_p^\top A_p)
\]
and $\tr(A_p^\top A_p)\leqslant \|A_p\|^2p$. Therefore, by Corollary \ref{t4}, \[\var(\y_p^\top A_p\y_p)\leqslant C(\Phi_0+\Phi_1+\Phi_2)\|A_p\|^2\|\Sigma_p\|^2p=o(p^2)\]
whenever $\|A_p\|=O(1).$ The case $\y_p=\X_p$ with $\X_p$ as in Theorem \ref{t2} can be considered along the same lines due to the inequality $\tr(A_pA_p^\top)\leqslant \|A_p\|^2p$. Q.e.d.
\noindent {\bf Proof of Theorem \ref{lv}.} Since $\Phi_1<\infty$, we have \begin{equation}
\label{covc} \sum_{j=1}^\infty j|C(j)|<\infty\end{equation} and $\sigma^2$ is well-defined, where $C(j)=\cov(X_t,X_{t+j}),$ $j\in\mathbb Z$. We also have the bias-variance decomposition $\e (\wh\sigma^2-\sigma^2)^2=\var(\wh\sigma^2)+(\e\wh\sigma^2-\sigma^2)^2.$
First, let us estimate the bias term $\e\wh\sigma^2-\sigma^2.$ Using $K(0)=1,$ we get
\begin{align*}
\e\wh\sigma^2-\sigma^2=&\sum_{j=-n}^n \bigg(1-\frac{|j|}{n}\bigg)K\Big(\frac{|j|}m\Big)C(j)-\sum_{j=-\infty}^\infty C(j)\\
=&2\sum_{j=1}^n \bigg(K\Big(\frac{j}m\Big)-1\bigg)C(j)-\frac{2}n\sum_{j=1}^nK\Big(\frac{j}m\Big)jC(j)-2\sum_{j>n} C(j)
\end{align*}
Now, setting $M=\sup_{x\geqslant 0}|K(x)|,$
\[\Big|\sum_{j=1}^n K(j/m)jC(j)\Big|\leqslant M\sum_{j=1}^\infty j|C(j)|=O(1).\]
Additionally,
\[\Big|\sum_{j>n} C(j)\Big|\leqslant \frac{1}{n}\sum_{j>n} j|C(j)|=o(1/n). \]
Combining these relations yields
\begin{equation}
\label{bias}\e\wh\sigma^2-\sigma^2=2\sum_{j=1}^n (K(j/m)-1)C(j) +O(1/n)=o(1),
\end{equation}
where the last equality follows from $(a)$ and \eqref{covc}.
Now, consider the variance term $\var(\wh\sigma^2)$. By Theorem \ref{t2},
\[\var(\wh\sigma^2)\leqslant \frac{C_0}{n^2}\sum_{s,t=1}^n K^2\bigg(\frac{|s-t|}m\bigg), \]
where $C_0=C(\Phi_0+\Phi_1+\Phi_2)$ with $C$ given in Theorem \ref{t2}. Using that $\bar K(x)=\sup_{y\geqslant x}|K(y)|$ is a non-decreasing function in $L_2(\bR)$, we derive
\[\frac{1}{mn}\sum_{s,t=1}^n K^2\bigg(\frac{|s-t|}m\bigg)\leqslant\frac{1}{m}+ \frac{2}{m}\sum_{j=1}^n \bar K^2\Big(\frac{j}m\Big)\leqslant \frac{1}{m}+2\int_0^\infty \bar K^2(x)\,dx.\]
As a result,
\begin{equation}\label{var}
\var(\wh\sigma^2)\leqslant \frac{C_0}{n}+\frac{2C_0m}{n}\int_0^\infty \bar K^2(x)\,dx
\end{equation}
and $\var(\wh\sigma^2)=o(1)$ whenever $m,n\to\infty$ and $m/n\to0.$
Combining the above bounds for the bias and the variance, we finish the proof. Q.e.d.
\noindent {\bf Proof of Theorem \ref{lv1}.} The proof follows the same line as the proof of Theorem \ref{lv}. We only need to note the following. If $(c)$ holds for some $q>0$. Then, by $(c)$ and the boundedness of $K$, $x^{-q}(K(x)-1)$ is bounded on $\bR_+$. Therefore, by $(a)$, $(c)$, and the absolute convergence of $\sum_{j\geqslant 1}j^qC(j)$,
\[m^q\sum_{j=1}^n (K(j/m)-1)C(j)=\sum_{j=1}^n \frac{K(j/m)-1}{(j/m)^q}\,j^qC(j)=k_q\sum_{j=1}^\infty j^qC(j)+o(1).\]
By \eqref{bias} and \eqref{var}, the latter yields the desired bound. Q.e.d.
| {
"timestamp": "2015-11-10T02:21:57",
"yymm": "1511",
"arxiv_id": "1511.02723",
"language": "en",
"url": "https://arxiv.org/abs/1511.02723",
"abstract": "We obtain variance inequalities for quadratic forms of weakly dependent random variables with bounded fourth moments. We also discuss two application. Namely, we use these inequalities for deriving the limiting spectral distribution of a random matrix and estimating the long-run variance of a stationary time series.",
"subjects": "Probability (math.PR)",
"title": "Variance inequalities for quadratic forms with applications",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9597620585273153,
"lm_q2_score": 0.7371581568543043,
"lm_q1q2_score": 0.7074964300826887
} |
https://arxiv.org/abs/0706.4314 | Domain growth morphology in curvature driven two dimensional coarsening | We study the distribution of domain areas, areas enclosed by domain boundaries (''hulls''), and perimeters for curvature-driven two-dimensional coarsening, employing a combination of exact analysis and numerical studies, for various initial conditions. We show that the number of hulls per unit area, $n_h(A,t) dA$, with enclosed area in the interval $(A,A+dA)$, is described, for a disordered initial condition, by the scaling function $n_h(A,t) = 2c_h/(A + \lambda_h t)^2$, where $c_h=1/8\pi\sqrt{3} \approx 0.023$ is a universal constant and $\lambda_h$ is a material parameter. For a critical initial condition, the same form is obtained, with the same $\lambda_h$ but with $c_h$ replaced by $c_h/2$. For the distribution of domain areas, we argue that the corresponding scaling function has, for random initial conditions, the form $n_d(A,t) = 2c_d (\lambda_d t)^{\tau'-2}/(A + \lambda_d t)^{\tau'}$, where $c_d=c_h + {\cal O}(c_h^2)$, $\lambda_d=\lambda_h + {\cal O}(c_h)$, and $\tau' = 187/91 \approx 2.055$. For critical initial conditions, one replaces $c_d$ by $c_d/2$ (possibly with corrections of ${\cal O}(c_h^2)$) and the exponent is $\tau = 379/187 \approx 2.027$. These results are extended to describe the number density of the length of hulls and domain walls surrounding connected clusters of aligned spins. These predictions are supported by extensive numerical simulations. We also study numerically the geometric properties of the boundaries and areas. | \section{Introduction}
In this work we obtain some exact and new results for the coarsening
dynamics~\cite{BrayReview} of a non-conserved scalar field in two
dimensions, demonstrating, {\em en passant}, the validity of the {\em
dynamical scaling hypothesis} for this system. We study the morphology
of the domain structure, which is illustrated in
Fig.~\ref{fig:snapshots0}
for the coarsening
of the two-dimensional Ising model ($2d$IM) on a square lattice
quenched from an equilibrium state at $T_0>T_c$.
A {\it domain} is a region of connected aligned spins. Each domain has
one external perimeter which is called the {\it hull}. The {\it hull
enclosed area} is the total area within this perimeter, {\it i.e.} the
domain area plus the area of any internal sub-domain. The {\it domain
perimeter} is the total length of the interface between the chosen
domain and the neighbouring ones -- including the hull and internal
borders. See
Fig.~\ref{fig:sketch-domains-hulls} for a sketch explaining these
definitions.
The paper deals primarily with the distributions of two characteristic
areas, the domain area and the hull enclosed area, and their
associated lengths, the domain wall perimeter and the hull length.
Naively, one may imagine that coarsening is basically due to the
coalescence of small domains that form larger ones. However, in two
dimensional curvature driven coarsening coalescence processes are
quite unimportant as shown by the Allen-Cahn result. All the
interfaces move with a local velocity that is proportional to the
local curvature and points in the direction of decreasing the
curvature; therefore, interfaces tend to disappear independently of
one another. This is the reason why we first focused on the statistics
of hull enclosed areas, quantities that depend on the motion of a
single and connected interface, and not on the statistics of the more
natural domain areas. Next we expressed the statistics of the domain
areas in terms of the simpler and more clear statistics of hull
enclosed areas.
Hull enclosed and domain areas have distributions that, at late times
after the quench into the ordered phase, exhibit, according to the
scaling hypothesis, the scaling form $n(A,t) = t^{-2} f(A/t)$, where
$n(A,t)dA$ is the number of hulls (domains) per unit area with area in
the range $(A, A+dA)$. The argument of the scaling function arises
from the fact that the characteristic length scale is known to grow as
$t^{1/2}$, so the characteristic area (of hulls and domains) grows as
$t$. The scaling function $f(x)$ will be different for domains and
hulls. The prefactor $t^{-2}$ follows from the fact that there is of
order one hull (or domain) per scale area.
Our analytical result is an elegant application of the Gauss-Bonnet
theorem and, it should be emphasized, its simplicity relies on the
dimensionality of the system being 2. Indeed, in three dimensions, the time
variation of the hull enclosed volume depends on a characteristic
size of the domain. A similar dependence on the dimension
is also observed in the von Neumann's law for cellular systems, whose
simple form, independent of any linear size of the system, is also
only observed in two dimensions \cite{Glazier, Mac}.
In this paper we derive these scaling forms from first principles (i.e.
without recourse to the scaling hypothesis), and determine explicitly
the scaling functions. Some of our results have appeared earlier
in Letter form \cite{us}.
Hulls and domain boundary lengths are themselves distributed
quantities related in a non-trivial manner to their corresponding
areas. In this paper we examine the geometry of these
structures and we derive the number density of hull and domain
wall lengths showing that these distributions also satisfy
scaling.
The organisation of the paper is as follows. In
Sect.~\ref{sec:equilibrium} we recall known results about the
equilibrium distribution of hull enclosed and domain areas at critical
percolation and critical Ising initial conditions. We also summarize known
results about the equilibrium distribution of domain walls and hulls
and their geometrical relation to their associated areas. In
Sect.~\ref{sec:generic} we derive some generic results that stem from
a number of sum rules and the use of the scaling hypothesis. In
Sect.~\ref{sec:analytic} we explain the analytic derivation of the
time-dependent hull enclosed and domain area distributions. These
arguments do not rely on any scaling hypothesis but rather demonstrate its
validity. In Sect.~\ref{sec:numeric} we show our numerical results for
the statistics of areas in the $2d$IM evolving with Monte Carlo
dynamics. Section~\ref{sec:domain-walls} is devoted to the analysis,
both analytical and numerical, of the geometry of hulls and domain
walls during the dynamics and their relation to their corresponding
areas. Finally, in the Conclusion we discuss future studies along
these lines. We also add two appendices in which we describe the
algorithm used to identify and count the hull enclosed areas, and for
the sake of comparison we present the distribution of domain lengths
in the one-dimensional Ising model.
\begin{figure}[h]
\includegraphics[width=4.cm]{other_1a.eps}
\includegraphics[width=4.cm]{other_1b.eps}
\caption{Snapshots of the $2d$ Ising model at time
$t=32$ MCs after a quench from infinite temperature, $T_0\to\infty$,
to the working temperature $T=1.5\simeq 0.66T_c$. In the left panel we
show the raw data, where the domain structure as well as the thermal
fluctuations within the domains are visible. In the right panel
we show the boundaries between regions of opposite sign in the configuration
to the left, using a variant of the algorithm in
Ref.~\cite{Derrida} to make the domain structure clearer.}
\label{fig:snapshots0}
\end{figure}
\section{Equilibrium distributions}
\label{sec:equilibrium}
In this Section we summarize known results on the {\it equilibrium} hull
enclosed and domain area distributions as well as the number density
of perimeters, both at critical percolation and critical Ising conditions in
two-dimensions, see Fig.~\ref{fig:sketch-domains-hulls}.
These will act as initial conditions for the coarsening dynamics.
\subsection{Area distributions}
A {\it hull enclosed area} is defined as the full interior of a domain
boundary -- irrespective of there being other interfaces and thus
regions of the opposite phase within. The equilibrium hull enclosed
area distributions
at percolating criticality and Ising criticality, have been computed by Cardy and
Ziff~\cite{Cardy} in two dimensions:
\begin{eqnarray}
n_h(A,0) \sim \left\{
\begin{tabular}{ll}
$2c_h/A^2 \;$ , & \qquad \mbox{critical percolation}, \\
$c_h/A^2 \;$ , & \qquad \mbox{critical Ising}.
\end{tabular}
\right.
\label{eq:nhpercolation}
\end{eqnarray}
These results are valid for $A_0 \ll A \ll L^2$, with $A_0$ a
microscopic area and $L^2$ the system size. Note also that we are
taking an extra factor 2 arising from the fact that there are
two types of hull enclosed areas, corresponding to the two phases,
while the Cardy-Ziff results accounts only for clusters of occupied
sites (and not clusters of unoccupied sites).
$n_h(A,0)\,dA$ is the
number density of hulls per unit area with enclosed area in the
interval $(A,A+dA)$ (we keep the notation to be used later and set
$t=0$). The adimensional constant $c_h$ is a universal quantity that
takes a very small value: $c_h = 1/(8\pi\sqrt{3})\approx
0.022972$. The smallness of $c_h$ plays an important role in the
analysis of Sect.~IV.
\begin{figure}[h]
\begin{center}
\psfrag{r1}{$R_1$}
\psfrag{r2}{$R_2$}
\includegraphics[width=3cm]{other_2.eps}
\end{center}
\caption{(Colour online.) A sketch of a configuration with two
concentric and circular interfaces with radius $R_1$ and $R_2$ is
shown to illustrate the definition of hull enclosed and domain areas
as well as hull and domain-wall perimeters. This configuration has
two hull enclosed areas $A_h^{(1)}=\pi R_1^2$ and $A_h^{(2)}=\pi
R_2^2$ and two domains with areas $A_d^{(1)}=\pi (R_1^2-R_2^2)$ and
$A_d^{(2)}=\pi R_2^2$. There are two hulls with length
$p_h^{(1)}=2\pi R_1$ and $p_h^{(2)}=2\pi R_2$, and two domain walls
with length $p_d^{(1)}=2\pi (R_1+R_2)$ and $p_d^{(2)}=2\pi R_2$.
}
\label{fig:sketch-domains-hulls}
\end{figure}
The distribution of {\it domain} areas -- we recall that domains are
clusters of connected aligned spins -- at critical percolation is
given by~\cite{Stauffer}
\begin{equation}
n_d(A,0) \sim \frac{2c_d A_0^{\tau'-2}}{A^{\tau'}}
\; ,
\qquad \mbox{with}\ \ \tau' = \frac{187}{91} \approx 2.055
\; .
\label{eq:domequilTinfty}
\end{equation}
Of course, the constants $c_d$ and $A_0$ are not separately defined by
this relationship, only the ratio $c_d A_0^{\tau'-2}$. In practice
it is convenient to choose $c_d$ to be the value appearing in the
domain area distribution at general times -- see Eq.\ (\ref{eq:guess2-bis}).
The quantity $A_0$ can be interpreted as a microscopic area, and will
be introduced at various points as a small-area cut-off. The quantity
$A_0^{\tau'-2}$ in Eq. (\ref{eq:domequilTinfty}) sets the units in such
a way that $[n_d]=A_0^{-2}$. This result is also valid in the limit
$A_0\ll A \ll L^2$.
In equilibrium at $T_c$, Stella and Vanderzande~\cite{Stella} computed
the number density of domains with area $A$
\begin{equation}
n_d(A,0) \sim \frac{c_d A_0^{\tau-2}}{A^\tau}
\; , \qquad \mbox{with} \ \ \tau=\frac{379}{187} \approx 2.027
\label{eq:Janke}
\end{equation}
in the large $A$ limit.
Janke and Schakel~\cite{Janke} confirmed this claim numerically
finding $\tau\approx 2.0269$. Motivated by the Cardy-Ziff result for
hull enclosed areas, we conjecture that the prefactor $c_d$ in
(\ref{eq:Janke}) is the same $c_d$ (up to terms of order $c_h^2$)
as that appearing in the prefactor $2c_d$ for critical percolation.
We discuss this point in detail at the beginning of section IV.B, and
we check this hypothesis numerically in Sect.~\ref{sec:numeric}.
We find it useful to include the small-area cut-off, $A_0$, in these number
densities, transforming the denominators to $(A+A_0)^2$
or $(A+A_0)^{\tau,\tau'}$ for hull enclosed and domain areas,
respectively.
\subsection{Perimeter distributions}
One can equally study the length of {\it hulls} (external perimeters) and
{\it domain walls} (including external and internal boundaries).
The number density $n_h(p,0)$ of hulls with length $p$, defined as
the number of spins on the hull, at the critical point
of the $2d$IM was computed by Vanderzande and Stella~\cite{Vander}
\begin{equation}
n_h(p,0) \sim \frac{c_{p_h} \; p_0^{\zeta_h-3}}{(p+p_0)^{\zeta_h}} \; ,
\;\; \mbox{with} \;\; \zeta_h=\frac{27}{11}
\approx 2.454
\; .
\end{equation}
The value of the constant $c_{p_h}$ was not estimated. As far as we
know there is no analytic prediction for the number density of domain
walls at critical Ising conditions that should have the same
functional form though with a possibly different exponent $\zeta_d$
and a different constant $c_{p_d}$. $p_0$ is a microscopic length that
we define as $p_0^2=A_0$.
For critical percolation, the number density of hulls with given perimeter
was obtained by Saleur and Duplantier~\cite{Saleur-Duplantier}
\begin{equation}
n_h(p,0) \sim
\frac{c'_{p_h} {p'_0}^{\zeta'_h-3}}{(p+p_0)^{\zeta'_h}} \; ,
\;\; \mbox{with} \;\;
\zeta'_h=\frac{15}{7}
\approx 2.143 \; ,
\label{perimeters}
\end{equation}
compatible with the numerical study~\cite{Ziff-perc}.
Again, the numerical value of the constant $c'_{p_h}$ is not
known. As far as we know, the analog of (\ref{perimeters}) for domain
walls at critical percolation is not known, though we expect the same
functional form with different constant $c'_{p_d}$ and exponent $\zeta'_{d}$.
In Sect.~\ref{sec:domain-walls} we show numerical results for the
equilibrium perimeter length number densities of critical Ising
and infinite temperature configurations.
\subsection{Fractal properties}
\label{sec:Cambier}
Several authors studied the fractal properties of areas and perimeters
in critical Ising and critical percolation equilibrium conditions
using different analytic methods that include conformal invariance and
renormalization group and Coulomb gas
techniques~\cite{Stella-Vanderzande,Vander,Stella,Saleur-Duplantier,Duplantier-Saleur}.
Many numerical studies~\cite{Janke,Ziff-perc,Aharony,Voss,Grassberger}
confirmed and complemented the results in these analytic works.
These works focused on the fractal dimensions of the domain area, $D_d$,
and of the hull length, $D_h$. In two-dimensions
these exponents are linked to the distribution
exponents $\tau$ and $\zeta_h$ as~\cite{Stauffer}
\begin{equation}
D_d = \frac{2}{\tau-1} \; , \qquad
D_h = \frac{2}{\zeta_h-1} \; ,
\end{equation}
and similarly for the primed quantities.
We concentrate on the
fractal properties of geometric structures by
comparing the area of the clusters to their associated
perimeter. This approach was used by
Cambier and Nauenberg~\cite{Cambier} who studied domain
walls in the $2d$IM -- with internal and external border -- at
equilibrium below, but near $T_c$,
and found
\begin{equation}
A \sim p^{1.43}\;.
\label{eq:equil-Ap}
\end{equation}
The proportionality constant is not given. In
Sect.~\ref{sec:domain-walls} we revisit the geometric
properties of the clusters at $T_c$ and the
corresponding one at very high temperature as well as their zero
temperature evolution.
\section{Some general results}
\label{sec:generic}
A number of general properties of hull enclosed and domain areas as
well as hull and domain wall perimeters can be easily derived just by
using the scaling hypothesis and two sum rules. We summarize them
here.
\subsection{Scaling}
\label{sec:scaling}
At long times, and irrespective of the initial condition, the total
number of domain and hull enclosed areas per unit area,
$N_{d,h}(t) = \int_0^\infty dA \; n_{d,h}(A,t)$, should
scale as $R^{-d}(t)$ in $d$ dimensions, with $R(t)$ a
characteristic length scale usually associated to
the `typical'
domain radius. For pure ferromagnetic coarsening,
$R(t) \sim (\lambda_d t)^{1/2}$ from which it follows that
$N_{d,h}(t) \sim t^{-1}$ in $d=2$. Since characteristic areas scale as
$R^2(t) \sim t$, the scaling hypothesis implies that the domain and hull
enclosed area distributions have the forms $n_{d,h}(A,t) =
t^{-2}f_{d,h}(A/t)$.
In Sect.~IV we present arguments that the initial distributions
of hull enclosed and domain areas determine the forms of these distributions
at late times. In particular, the initial forms are modified in a
rather simple way at later times, such that the scaling forms are already
suggested by the initial conditions. In the following we outline the
consequences of this line of argument and defer detailed analysis to
Sect.~IV.
Let us first discuss domains with critical Ising initial conditions.
Retaining the initial form~(\ref{eq:Janke}), but including an extra
time-dependent factor to satisfy the desired scaling
form at large times, the expression for $n_d(A,t)$ is,
\begin{equation}
n_d(A,t) \sim
\frac{c_d \; [\lambda_d(t+t_0)]^{\tau-2}}{[A+\lambda_d (t+t_0)]^\tau}
\label{eq:guess1}
\end{equation}
at long times. Here, $t_0$ is a microscopic time, and $\lambda_d$ is a
phenomenological parameter with the dimensions of a diffusion constant.
Setting $t=0$ one recovers
Eq.~(\ref{eq:Janke}) provided $A_0 = \lambda_d t_0$ and $A\gg
A_0$. Indeed, $t_0$ is defined through this requirement while $A_0$ is
the by now usual microscopic area. For large $t$, $t_0$ can be neglected
and the conventional scaling form is recovered.
For infinite temperature initial conditions we propose
\begin{equation}
n_d(A,t) \sim
\frac{2c_d \; [\lambda_d(t+t_0)]^{\tau'-2}}{[A+\lambda_d (t+t_0)]^{\tau'}}
\label{eq:guess2-bis}
\end{equation}
that also satisfies the scaling form
asymptotically.
One can easily check that the obvious generalization of
Eq.~(\ref{eq:nhpercolation}) including time
\begin{equation}
n_h(A,t) \sim \frac{(2) c_h}{(A+\lambda_h t)^2}
\label{eq:guess3}
\end{equation}
also has the desired scaling form. The constant $\lambda_h$ is another
phenomenological parameter. Its value will turn out to be very close to
$\lambda_d$. The factor 2 in the brackets in Eq.~(\ref{eq:guess3})
takes into account the two types of initial condition.
We remark that in Eq.~(\ref{eq:guess3}) we do not explicitly include
a short-time cut-off $t_0$, while it is necessary in Eqs.~(\ref{eq:guess1})
and (\ref{eq:guess2-bis}) to connect smoothly to the initial condition.
Scaling can also be used to predict the time-dependence of the
number density of hull and domain wall lengths. It yields
\begin{equation}
n_{d,h}(p,t) \sim
\frac{c_{p_d,p_h}[(\lambda_{d,h} t)^{1/2}+p_0]^{\zeta_{d,h}-3}}
{[p+(\lambda_{d,h} t)^{1/2}+p_0]^{\zeta_{d,h}}} \; .
\label{eq:scaling-np}
\end{equation}
with the constant and exponent values depending on the
initial condition and whether we are studying hulls or domain
walls. This form is based on the assumption that the characteristic
perimeters evolve in time as $(\lambda_{d,h} t)^{1/2}$.
\subsection{Sum rules}
We now present two exact sum rules which provide useful input for the
analysis of $n_{d,h}(A,t)$. These sum rules apply at all times $t\geq 0$,
for any initial condition, and for any working temperature.
The first sum rule follows from the fact that the total domain area,
per unit area of the system, is unity since each space point (or
lattice site) belongs to one and only one domain. This gives
\begin{equation}
\int_0^\infty dA \; A \; n_d(A,t) = 1
\; .
\label{eq:totalarea}
\end{equation}
The second sum rule follows from the fact that the total number of hull
enclosed areas,
$N_h(t)$, is equal to the total number of domains, $N_d(t)$, since each
domain can be associated with a unique hull, namely the hull that forms
its external boundary. This yields
\begin{eqnarray}
N_d(t) &\equiv&
\int_0^\infty dA \; n_d(A,t)
\nonumber\\
&=&\int_0^\infty dA \; n_h(A,t)
\equiv N_h(t)
\; .
\label{eq:hulls-equal-domains}
\end{eqnarray}
Equations~(\ref{eq:nhpercolation})-(\ref{eq:domequilTinfty}) and their
generalization describing the time-dependence, have been shown to hold
for large areas ($A\gg A_0$) only and the number density can take a
different form at small values of $A$. From the constraints
(\ref{eq:totalarea}) and (\ref{eq:hulls-equal-domains}), using
Eqs.~(\ref{eq:guess1}), (\ref{eq:guess2-bis}) and (\ref{eq:guess3}),
we shall derive {\it approximate} relations between the constants
$c_d$, $c_h$, $\lambda_d$, $\lambda_h$, $\tau$ and $\tau'$ expected to
hold at any working temperature $T$. These relations are exact to
first order in the small quantity $c_h$.
For the number densities of hulls or domain walls with given perimeter
we have only one sum rule. The total number density must equal the number
density of domains and hull enclosed areas:
\begin{equation}
N_{p_d,p_h}(t) \equiv
\int_0^\infty dp \; n_{d,h}(p,t) = N_d(t) =N_h(t)
\; .
\end{equation}
\subsubsection{Critical Ising initial conditions.}$\;$
The constraint on the total area (\ref{eq:totalarea})
using Eq.~(\ref{eq:guess1}) for the number density of domain areas
yields
\begin{equation}
c_d = (\tau-2)(\tau-1) \approx 0.02745
\label{eq:relation1}
\end{equation}
where the numerical value was obtained for $\tau=379/187$.
If one takes into account that the minimal area is $A_0$ (and not $0$)
the result is slightly different and it varies, though very weakly,
with $t$. Indeed, the value of $c_d$ decreases from $c_d=(\tau-2)(\tau-1)
2^{\tau-1}/\tau$ at $t=0$ to $c_d=(\tau-2)(\tau-1)$ at $t\gg t_0 =
A_0/\lambda_d$. Inserting $\tau=379/187$ gives
\begin{equation}
0.02745
\leq c_d \leq 0.02760,
\end{equation}
a rather narrow interval.
Note that $c_d$ computed using (\ref{eq:relation1}) is quite close to the
analytical result for $c_h$, namely $c_h^{\rm an} = 1/8\pi\sqrt{3}
\sim 0.02297$. Indeed, $c_d - c_h^{\rm an} \approx 0.00448$
(which is order $c^{\rm an}_h/5$)
where we used the minimum value of $c_d$ and the analytical value of $c_h$.
We now consider the sum rule~(\ref{eq:hulls-equal-domains}), at times
$t=0$ and $t \gg A_0/\lambda_h$, distinguishing the integral over the
full interval $[0,\infty)$ from the one that takes into account the
finite minimal area $[A_0,\infty)$. We denote by $A_1$ the lower
limit of the integration interval. We also include a small area
cut-off $A_0$ in the hull enclosed area number density, $n_h(A,0) \sim
c_h/(A+A_0)^2$.
At time $t=0$ we find
\begin{eqnarray}
c_d &=&
c_h\; (\tau-1) \;
\frac{(A_1+A_0)^{\tau-2}}{A_0^{\tau-2}}
\; .
\label{eq:relation1a}
\end{eqnarray}
Using now the expressions for $c_d$ derived above
we relate $c_h$ to $\tau$:
\begin{eqnarray}
c_h &=&
\left\{
\begin{array}{lcll}
(\tau-2) & \approx & 0.02674
\; , \qquad &A_1=0
\; ,
\\
\frac{2 (\tau-2)}{\tau} & \approx & 0.02639
\; , \qquad & A_1= A_0
\; .
\end{array}
\right.
\end{eqnarray}
These values are quite close to the analytic one
$c^{\rm an}_h \approx 0.02297$. The relative difference is
$(c_h-c_h^{\rm an})/c_h^{an} \approx 0.164$ ($A_1=0$) and
$(c_h-c_h^{\rm an})/c_h^{\rm an} \approx 0.149$ ($A_1=A_0$).
Note that using the sum rules we over-estimate the value of $c_h$.
We can then expect to have over-estimated the value of $c_d$ too.
This remark will be important when comparing to numerical data.
Finally, we can evaluate condition (\ref{eq:hulls-equal-domains}) at
late times, $t \gg A_0/\lambda_h$, using the conjecture
(\ref{eq:guess1}) for $n_d$ and the result $n_h(A,t) \sim
c_h/(A+\lambda_h t)^2$ deduced from scaling in Sect.~\ref{sec:scaling}
and to be shown analytically in Sect.~\ref{sec:hulls}. In this case we find,
independently of $A_1$:
\begin{equation}
\frac{\lambda_d}{\lambda_h} = \frac{c_d}{c_h} \frac1{(\tau-1)}
\; .
\label{eq:relation2}
\end{equation}
This equation can be used to relate the two factors $\lambda_d$ and
$\lambda_h$. Indeed,
replacing $c_d$ and $c_h$ by their expressions as functions of
$\tau$ one finds
\begin{eqnarray}
\frac{\lambda_d}{\lambda_h}
&=&
\left\{
\begin{array}{lll}
1 \;, \qquad & A_1=0
\; ,
\\
2^{\tau-2} \approx 1.019 \;, \qquad & A_1=A_0
\; .
\end{array}
\right.
\end{eqnarray}
If, instead, we use $c_d$ as derived above and the analytic $c_h$,
$c_h^{\rm an}=0.022972$ we
find
\begin{eqnarray}
1.164 \leq \frac{\lambda_d}{\lambda_h} \leq 1.170
\; .
\label{eq:rel3}
\end{eqnarray}
Since we derived Eq.~(\ref{eq:relation2}) using the sum rules, it seems
more appropriate to use the values of $c_d$ and $c_h$ obtained
from the same relations. The values of $\lambda_d$ and $\lambda_h$ are
then very close and consistent with the relation that we shall find in
the next Section using the approximate equation for the evolution of
domain areas [while Eq.~(\ref{eq:rel3}) yields a too large value
for $\lambda_d$].
The condition $N_{p_h}(t)=N_{p_d}(t)=N_h(t)=N_d(t)$ implies
\begin{eqnarray}
c_{p_h} &=& -c_h \, (1-\zeta)
\; ,
\nonumber\\
c_{p_d} &=& c_d \; \frac{1-\zeta}{1-\tau}
\; ,
\end{eqnarray}
for critical Ising initial conditions.
\subsubsection{Infinite-temperature initial conditions.}$\;$
The infinite-temperature initial conditions turn out to be, after just a few
time steps, equivalent to critical percolation ones -- see the numerical
evidence in Sect.~\ref{sec:infTinit}. The sum rules yield, in this case,
\begin{eqnarray}
c_d &=& \frac{(\tau'-2) (\tau'-1)}{2} \approx 0.029\; ,
\\
c_h &=& \frac{c_d}{(\tau'-1)} \approx 0.027 \; ,
\\
\frac{\lambda_d}{\lambda_h} &=& \frac{c_d}{c_h \, (\tau'-1)}
=1
\; ,
\end{eqnarray}
where, for simplicity, we present results obtained with
$A_1=0$ only.
The conditions $N_{p_h}(t)=N_{p_d}(t)=N_h(t)=N_d(t)$ imply
\begin{equation}
c'_{p_h} = -2c_h \, (1-\zeta'_h)
\; ,
\qquad
c'_{p_d} = 2c_d \; \frac{1-\zeta'_d}{1-\tau'}
\; ,
\end{equation}
for critical percolation.
\section{Statistics of areas: analytic results}
\label{sec:analytic}
Our analytic results are obtained using a continuum description of
domain growth in which the non conserved order parameter is a scalar field,
$\phi(\vec x,t)$, defined on a $d$-dimensional space. For a
review in this problem, see \cite{BrayReview}.
Its evolution is determined by the time-dependent
Ginzburg-Landau equation or model A dynamics:
\begin{eqnarray}
\gamma \frac{\partial \phi(\vec x,t)}{\partial t} = \nabla^2 \phi(\vec x,t)
- \frac{\delta V(\phi)}{\delta \phi(\vec x,t)} + \xi(\vec x,t)
\; .
\label{eq:langevin}
\end{eqnarray}
The potential $V$ is a symmetric double well, with $V(\phi\to\pm
\infty) = \infty$ and two minima at $\pm\phi_0$. $\xi$ is a Gaussian
distributed random scalar field with zero mean and correlation
\begin{equation}
\langle \, \xi(\vec x,t) \xi(\vec x', t') \, \rangle
= 2 k_{\scriptscriptstyle\rm B} T \gamma \; \delta^d(\vec x-\vec x') \,
\delta (t-t')
\; .
\end{equation}
This white noise introduces thermal agitation. $T$ is the temperature
of the thermal bath, $k_{\scriptscriptstyle\rm B}$ is the Boltzmann
constant and $\gamma$ is
the friction coefficient. From now on we set the units in such a way
that $k_{\scriptscriptstyle\rm B}=\gamma=1$. The
low-temperature ordering dynamics from a disordered initial condition
corresponds to the growth of ordered domains of the two equilibrium states,
$\phi(\vec x,t) =\pm \phi_0$, separated by
interfaces. Using the evolution equation (\ref{eq:langevin}) at
{\it zero temperature}, Allen and Cahn showed that in any dimension $d$
the velocity, $v$, of each element of a domain boundary is
proportional to the local interfacial mean curvature,
$\kappa$~\cite{AC,BrayReview},
\begin{equation}
v = - \frac{\lambda_h}{2\pi} \; \kappa
\; .
\label{allen-cahn}
\end{equation}
$\lambda_h$ is a material constant with the dimensions of a
diffusion constant, and the factor $1/2\pi$ is for later convenience.
The velocity is normal to the interface and
points in the direction of reducing the curvature.
The dynamics is then purely curvature driven at zero temperature.
Temperature fluctuations have a two-fold
effect. On the one hand they generate {\it equilibrium} thermal
domains that are not related to the coarsening process. On the other
hand they {\it roughen} the domain walls thus opposing the curvature driven
growth and slowing it down. Then equation (\ref{allen-cahn})
is no longer valid. However, it has been conjectured and verified
numerically that, at least for averaged dynamic quantities
well-described by the scaling hypothesis, all temperature
effects are captured by introducing a $T$ dependent $\lambda_h$
parameter. We shall use this working hypothesis in the
analytic part of our paper and we shall put it to the
test numerically.
The number density of hull enclosed or domain areas at time $t$ as a
function of their initial distribution is
\begin{equation}
n(A,t) = \int_0^\infty dA_i \; \delta(A-A(t,A_i)) \, n(A_i,t_i)
\;
\label{eq:integration}
\end{equation}
with $A_i$ the initial area and $n(A_i,t_i)$ their number
distribution at the initial time $t_i$. $A(t,A_i)$ is the
hull enclosed/domain area, at time $t$, having started from
an area $A_i$ at time $t_i$.
\subsection{Hull enclosed areas}
\label{sec:hulls}
In two dimensions~\cite{Comment} we can immediately deduce the
time-dependence of the area contained within any finite hull by
integrating the velocity around the hull:
\begin{equation}
\frac{dA}{dt} = \oint v\,dl = -\frac{\lambda_h}{2\pi} \oint
\kappa\,dl = -\lambda_h
\end{equation}
In the second equality we used the zero-temperature
Allen-Cahn Eq.\ (\ref{allen-cahn}),
and in the final one we used the Gauss-Bonnet theorem.
Integrating over time, with initial time $t_i$, we find
\begin{equation}
A_h(t,A_i) = A_i-\lambda_h (t-t_i)
\; .
\end{equation}
Therefore
\begin{eqnarray}
n_h(A,t) &=&
\int_0^\infty dA_i \; \delta(A-A_i+\lambda_h (t-t_i)) \, n_h(A_i,t_i)
\nonumber\\
&=& n_h(A+\lambda_h(t-t_i),t_i)
\; .
\label{eq:advection}
\end{eqnarray}
In deriving this result we have implicitly assumed that a single domain
cannot split into two, and that two domains cannot coalesce. A little
thought shows that neither process is possible for two-dimensional
curvature-driven growth since both processes require that two parts
of a single domain boundary (for splitting) or parts of two different
domain boundaries (for coalescence) come together and touch. But it is
clear that the curvature driven dynamics always acts to prevent this
happening, since the velocities of the domain boundaries at the incipient
contact point are in opposite directions.
The initial distributions, $n_h(A_i,t_i)$, are given by the Cardy-Ziff
results displayed in Eq.\ (\ref{eq:nhpercolation}) -- assuming, in the case
of a quench from infinite temperature, that the system rapidly sets into
the critical percolation condition (see Sect.~\ref{sec:infTinit}).
For $t\gg t_i$ one immediately recovers the results in~\cite{us}:
\begin{eqnarray}
n_h(A,t) & = & \frac{2c_h}{(A+\lambda_h t)^2}
\; , \qquad T_0\to\infty
\; ,
\label{eq:analytic-nh-Tinf}
\\
n_h(A,t) & = & \frac{c_h}{(A+\lambda_h t)^2}
\; , \qquad T_0=T_c
\; ,
\label{eq:analytic-nh-Tc}
\end{eqnarray}
in the limit $A_0 \ll A \ll L^2$, i.e.\ for hull enclosed areas much
larger than microscopic areas but much smaller than the area of the
system.
Equations~(\ref{eq:analytic-nh-Tinf}) and (\ref{eq:analytic-nh-Tc})
have the expected scaling forms $n_h(A,t) = t^{-2} f(A/t)$
corresponding to a system with characteristic area proportional to $t$
or characteristic length scale $R(t) \propto t^{1/2}$, which is the
known result if scaling is {\em assumed} \cite{BrayReview}. Here,
however, we do not {\em assume} scaling -- rather, it emerges from the
calculation. Furthermore, the conventional scaling phenomenology is
restricted to the `scaling limit': $A \to \infty$, $t \to \infty$ with
$A/t$ fixed. Equations~(\ref{eq:analytic-nh-Tinf}) and
(\ref{eq:analytic-nh-Tc}), by contrast, are valid whenever $t$ is
sufficiently large and $A \gg A_0$. This follows from the fact that,
for large $t$, the forms (\ref{eq:analytic-nh-Tinf}) and
(\ref{eq:analytic-nh-Tc}) probe, for any $A \gg A_0$, the tail (i.e.\
the large-$A$ regime) of the Cardy-Ziff results, which is just the
regime in which the latter is valid. The restriction $A \gg A_0$ is
needed to justify the use of Eq.\ (\ref{allen-cahn}), which breaks
down when the reciprocal of the curvature becomes comparable with the
width of a domain wall.
The averaged area enclosed by a hull is then given by
\begin{eqnarray}
\langle A\rangle(t) &=& \frac{\int dA' A' n_h(A',t)}{\int dA'
n_h(A',t)} \\ &\propto& \lambda_h t
\end{eqnarray}
with a time-independent prefactor that behaves as $(A_0^2 \ln L^2)$
for large system sizes. The reason for the divergent prefactor in the
infinite size limit is that a site can belong to several hulls.
\subsection{Domains}
\label{sec:domains}
For the domains we need to write an evolution equation and derive, at
least approximately, the area at time $t$, $A_d(t,A_i)$, of a domain
with initial area $A_i$. We shall show that the time-dependent number
density of domain areas is indeed given by our guess,
Eqs.~(\ref{eq:guess1}) and (\ref{eq:guess2-bis}), for the two classes
of initial conditions.
Our strategy is to exploit the smallness of the parameter $c_h \approx
0.023$. Although $c_h$ is a constant, we can exploit a formal
expansion in $c_h$ in the following sense. Since the total number of
hulls per unit area is proportional to $c_h$, the number of interior
hulls within a given hull is also proportional to $c_h$, and so on.
This means that, in dealing with domains we need consider only the
first generation of interior hulls, since the number of ``hulls within
hulls'' is smaller by a factor $c_h$. With this approach, only one
approximation -- a kind of mean-field one on the number of
first-generation hulls within a parent hull (see below) -- is
necessary.
The same line of reasoning shows that, in a hypothetical theory in
which $c_h$ can be treated as variable, the distinction between hulls
and domains will disappear in the limit $c_h \to 0$. In this limit,
therefore, the exponents $\tau$ and $\tau'$ must both approach the
value 2, i.e.\ we can formally write $\tau = 2 +O(c_h)$ and $\tau' = 2
+ O(c_h)$. Furthermore, due to the factor 2 that appears in
(\ref{eq:analytic-nh-Tinf}) but not in (\ref{eq:analytic-nh-Tc}),
the ratio $(\tau'-2)/(\tau-2)$ must approach the value 2
in the limit $c_h \to 0$. The actual value of this ratio is $187/91 =
\tau'= 2.055$, not very far from 2. Indeed the difference is of order
$c_h$ as expected.
We can use the same line of argument to discuss $c_d$, $c_h$, $\lambda_d$
and $\lambda_h$. Since in the (hypothetical) limit $c_h \to 0$, hulls
and domains become identical, it follows that in this limit one must have
$c_d \to c_h$ and $\lambda_d \to \lambda_h$, i.e.\ $c_d = c_h + O(c_h^2)$,
and $\lambda_d = \lambda_h[1 + O(c_h)]$. All of these results are
consistent with the relations (\ref{eq:relation1}) and
(\ref{eq:relation2}) derived from the sum rules (\ref{eq:totalarea})
and (\ref{eq:hulls-equal-domains}).
\subsubsection{The evolution of domain areas.}$\;$
Take a hull with enclosed area $A_h$ at time $t$. This hull is also the
external border of a domain, which may itself contain one or more `first
level' sub-domains whose external borders form the internal border (which
may be disconnected) of the original domain. These external borders of
the first level sub-domains are themselves `first generation' hulls lying
within the parent hull. These interior hulls can themselves have
interfaces in their bulk separating domains of the reversed phase (higher
generation hulls), see Fig.~\ref{fig:snapshot-Ising} where we show a
sketch with this structure.
\begin{figure}[h]
\psfrag{r1}{$R_1$}
\psfrag{r2}{$R_2$}
\psfrag{r3}{$R_3$}
\psfrag{r4}{$R_4$}
\begin{center}
\includegraphics[width=4cm]{other_3.eps}
\end{center}
\caption{(Colour online.) Sketch of a configuration with four circular
hulls and domains. The parent hull has radius $R_1$. There are two
first generation hulls with radius $R_2$ and $R_3$ and one second
generation hull with radius $R_4$. $\nu=2$ in this example. The
interior border of the external domain is disconnected and has two
components.}
\label{fig:snapshot-Ising}
\end{figure}
Let us call $\nu(t)$ the number of first-generation hulls within the
parent one. It is clear that $\nu(t)$ is semi-positive definite,
monotonically decreasing as a function of time and reaching zero at a
given instant $t_{max}$, when all interior hulls disappear and
$A_d=A_h$ thereafter. One can estimate $t_{max}$ from
$0=A^{int}_h(t_{max})=A^{int}_h(t_i)-\lambda_h (t_{max}-t_i)$, which
yields $t_{max} -t_i= A^{int}_h(t_i)/\lambda_h$ where the index $int$
indicates that we are studying here the first generation hull with
maximal initial area (all others having already
disappeared). It is clear that $t_{max}-t_i$ is smaller but of the order
of $A_h(t_i)/\lambda_h$ where we replaced $A_h^{int}(t_i)$ by the initial
area of the parent hull:
\begin{equation}
(t_{max}-t_i) \stackrel{<}{\sim} \frac{A_h(t_i)}{\lambda_h}
\; .
\label{eq:tmax}
\end{equation}
We wish to write a differential equation for the time-evolution
of the parent domain area. It is clear that, at first order in $dt$:
\begin{equation}
A_d(t+dt) = A_d(t) - \lambda_h dt + \nu(t) \lambda_h dt
\end{equation}
where the second term in the right-hand-side represents the
loss in area due to the inward motion of the external domain-wall
while the last term is the gain in area due to the outward
motion of the first-generation internal domain-walls. This gives
\begin{equation}
\frac{dA_d(t)}{dt} = - \lambda_h \; [1- \nu(t)]
\; .
\label{eq:diff-eq-domains}
\end{equation}
Differently from hull enclosed areas, that always decrease in size
as time passes, domain can either diminish ($\nu = 0$), increase
($\nu>1$) or conserve ($\nu=1$) their area in time.
\subsubsection{The number of first generation interior hulls.}$\;$
\label{subsubsec:nu}
We cannot, of course, know the exact number of first generation
hulls falling within a selected hull with enclosed area $A_h$.
We can, however, estimate it with an
upper bound obtained by counting all interior hulls
and averaging over all parent hulls
using $n_h(A,t)$ derived in Sect.~\ref{sec:hulls}.
Thus, we expect
\begin{eqnarray}
\qquad\qquad\nu(t) &<& \langle \, \nu(t) \, \rangle_{A_h(t)}
\nonumber\\
\langle \, \nu(t) \, \rangle_{A_h(t)}
&\sim&
A_h(t) \int_0^{A_h(t)} dA \; n_h(A,t)
\nonumber\\
&=&
\frac{c_h A^2_h(t)[\lambda_h (t-t_i)+A_0]^{-1}}
{[A_h(t) + \lambda_h (t-t_i) +A_0]}
\; ,
\label{eq:numedia}
\end{eqnarray}
where we include a small area cut-off, $A_0$, in the denominator
of $n_h$ and, for concreteness, we use the hull enclosed
area distribution for critical Ising initial conditions.
This equation can be further simplified if one uses
that at time $t$ the hull enclosed area we are interested in is given by
\begin{equation}
A_h(t) = A_h(t_i)-\lambda_h (t-t_i)
\; .
\end{equation}
(We call here $A_h(t_i)$ the initial area of the hull.) Then
\begin{eqnarray}
\langle \, \nu(t) \, \rangle_{A_h(t)} =
\frac{c_h [A_h(t_i)-\lambda_h (t-t_i)]^2}{[\lambda_h (t-t_i) +A_0]\;
[A_h(t_i) +A_0]}
\; .
\label{eq:nuaveraged}
\end{eqnarray}
Note that, although we over-counted the interior hulls by including
second-generation, third-generation, etc.\ hulls, the number of these
is of order $c_h^2$, $c_h^3,\ldots$ respectively, so this treatment is
exact to leading order in $c_h$ except for the replacement of $\nu(t)$
by its average over all first-generation hulls of the same area.
The most interesting cases are such that $A_h(t_i) \gg A_0$, otherwise
the hull and domain areas are just identical or very similar. In these
cases $\langle \, \nu(t_i) \, \rangle_{A_h(t_i)} \sim c_h A_h(t_i)/A_0$.
Expression (\ref{eq:nuaveraged}) has the following limiting values
\begin{eqnarray*}
\langle \, \nu(t) \, \rangle_{A_h(t)} \sim
\left\{
\begin{array}{ll}
\displaystyle
\frac{c_h\ A_h(t_i)}{\lambda_h (t-t_i)+A_0},
& A_h(t_i) \gg \lambda_h (t-t_i)
\; ,
\\ & \\ & \\
a c_h,
& A_h(t_i) \sim \lambda_h (t-t_i)
\; ,
\end{array}
\right.
\label{eq:limits}
\end{eqnarray*}
we used $A_h(t_i) \gg A_0$ in the last case, and $a$ is
a numerical constant of the order of $A_h(t_i)$.
The result is a very small quantity, of the order of $c_h$, in both
cases. The remaining mathematical possibility, $A_h(t_i) < \lambda_h
(t-t_i)$ is not realized because $A_h(t)$ cannot be negative.
While $\nu(t)$ vanishes at $t_{max}$, see
Eq.~(\ref{eq:tmax}), $\langle \, \nu(t) \, \rangle_{A_h(t)}$ is different
from zero at all times. Thus,
Eq.~(\ref{eq:nuaveraged}) cannot be used beyond the limit $t_{max}$
when all internal hulls have already disappeared and it is no longer
correct to replace $\nu(t)$ by $\langle \, \nu(t) \,
\rangle_{A_h(t)}$.
The analysis of infinite temperature initial conditions is identical
to the one above with $c_h$ replaced by $2c_h$. Thus, $\langle \, \nu(t) \,
\rangle_{A_h(t)}$ is expected to take twice the value it takes for
critical Ising initial configurations.
We have checked the accuracy of this approximation numerically by
counting the number of first generation internal hulls falling within
each parent hull at different times.
Figure~\ref{fig:nu} shows the results for the zero
temperature evolution of the $2d$IM starting from $T_0\to\infty$ and
$T_0=T_c$ initial conditions. While at very short times one sees
deviations between the numerical data and analytic prediction, the
agreement between the two becomes very satisfactory for times of the
order of $t=64$ MCs and longer, as shown in the figure.
\begin{figure}[h]
\centerline{
\psfrag{x}{\large\hspace{-10mm} $A_h(t)/\lambda_h t$}
\psfrag{y}{\large $\nu$}
\psfrag{a}{$T_0=T_c$}
\psfrag{b}{$T_0=\infty$}
\includegraphics[width=8cm]{other_4.eps}
}
\caption{(Colour online.) Comparison between $\nu(t)$ and $\langle \, \nu(t) \,
\rangle_{A_h(t)}$ for the $T=0$ evolution of the $2d$IM with
$T_0\to\infty$ and $T_0=T_c$ initial conditions. The measuring time is
$t=64$ MCs. The curves are given by Eq.~(\ref{eq:numedia}) in the
limit $t\gg t_i$ and $A_h(t) \gg A_0$, leading to the functional form
$\langle \, \nu(t) \, \rangle_{A_h(t)} = c_h x^2/(1+x) \sim c_h x$
when $x\gg 1$, with $x=A_h(t)/\lambda_h t$ and $\lambda_h=2.1$, see
Sect.~\ref{sec:numeric} for an explanation of the choice of this
value.}
\label{fig:nu}
\end{figure}
\subsubsection{The instantaneous domain area.}$\;$
If we now replace $\nu(t)$ by $\langle \, \nu(t) \, \rangle_{A_h(t)}$
given in Eq.~(\ref{eq:nuaveraged}),
it is quite simple to integrate the differential
equation~(\ref{eq:diff-eq-domains}). One finds
\begin{eqnarray*}
A_d(t) &=& A_d(t_i) - \lambda_h (1+2c_h) (t-t_i)
\nonumber\\
&&
+ \,
\frac{c_h A_0^2}{2[A_h(t_i)+A_0]}
\left\{\left[1+\frac{\lambda_h (t-t_i)}{A_0}\right]^2-1 \right\}
\nonumber\\
&&
+ \,
c_h [A_h(t_i)+A_0] \ln \left[1 +\frac{\lambda_h (t-t_i)}{A_0} \right]
\; .
\end{eqnarray*}
Setting $t=t_i$ one recovers $A_d(t)=A_d(t_i)$ as required.
In the natural cases in which $A_h(t_i) \gg A_0$ and
for long times such that $\lambda_h(t-t_i) \gg A_0$
this expression can be rewritten as
\begin{eqnarray}
A_d(t) &=&
A_d(t_i) - \lambda_h
\left[1+2c_h-\frac{c_h}{2} \frac{\lambda_h (t-t_i)}{A_h(t_i)} \right]
(t-t_i)
\nonumber\\
&&
+ \,
c_h A_h(t_i) \ln \left[1+\frac{\lambda_h (t-t_i)}{A_0} \right]
\; .
\label{Ad(t)}
\end{eqnarray}
The factor in the second term
\begin{eqnarray}
\lambda_d(t) \equiv
\lambda_h \left[1+2c_h-\frac{c_h}{2} \frac{\lambda_h (t-t_i)}{A_h(t_i)}\right]
\;
\end{eqnarray}
is a very weakly time-dependent function. Since
$t$ can take values between the initial time, $t=t_i$, and the maximum
time before the first generation hull itself disappears,
$t_{max}=t_i+A_h(t_i)/\lambda_h$, $\lambda_d(t)$ varies within the interval:
\begin{eqnarray}
\lambda_h \left(1+\frac{3c_h}{2} \right) \leq \lambda_d(t) \leq
\lambda_h (1 + 2c_h)
\; .
\end{eqnarray}
These bounds are indeed very close. As expected from the analysis
of the sum rules, see Sect.~\ref{sec:generic},
$\lambda_d$ takes a slightly higher value than
$\lambda_h$; it equals
$\lambda_h$ plus a small correction of order $c_h$ (in practice, $1.035
\, \lambda_h \leq \lambda_d \leq 1.046 \, \lambda_h$ using the analytic
value for $c_h$).
The coefficient in front of the logarithm, $c_h A_h(t_i)$, is ${\cal
O}(c_h)$. The sum rules, imply $c_h =c_d +{\cal O}(c_h^2)$.
Neglecting the higher order correction we can then replace $c_h$ by
$c_d$. The same applies to $A_h(t_i)$, which equals $A_d(t_i)$ plus a
term ${\cal O}(c_h)$ that we can equally neglect. Thus $A_h(t_i)
\approx A_d(t_i) \equiv A_i$. In this way we obtain
\begin{displaymath}
A_d(t,A_i) \simeq A_i - \lambda_d (t-t_i) + c_d A_i
\;
\ln \left[1+\frac{\lambda_h(t-t_i)}{A_0} \right]
\; .
\label{eq:expectation}
\end{displaymath}
Inserting this result into Eq.~(\ref{eq:integration}),
including the microscopic area $A_0$ as a small-area cut-off in the
denominator of Eq.~(\ref{eq:nhpercolation}), then gives
\begin{eqnarray}
n_d(A,t) & \simeq & c_d A_0^{\tau-2}
\left\{1+ c_d
\ln\left[ 1 + \frac{\lambda_d(t-t_i)}{A_0}\right] \right\}^{\tau-1}
\nonumber \\
& & \times [A + \lambda_d (t-t_i) + A_0]^{-\tau}
\label{eq:replace}
\end{eqnarray}
where we have replaced $\lambda_h$ by $\lambda_d$ inside the logarithm,
which is correct to leading order in $c_h$.
Using the fact that $c_d$ is very small and of the order of
$(\tau-2)(\tau-1) = (\tau-2) + {\cal O}(c_h^2)$, as implied by the sum rules,
we can now exponentiate, correct to leading order in $c_d$, the logarithm
in the curly brackets to obtain
\begin{equation}
n_d(A,t) \simeq \frac{c_d \, [A_0 + \lambda_d(t-t_i)]^{\tau-2}}
{[A + A_0 + \lambda_d(t-t_i)]^\tau}
\; .
\end{equation}
Finally we set the initial time, $t_i$, to zero and write the
microscopic area, $A_0$, as $\lambda_d t_0$ to obtain the expected form
(\ref{eq:guess1}),
\begin{equation}
n_d(A,t) \simeq
\frac{c_d \; [\lambda_d(t+t_0)]^{\tau-2}}{[A+\lambda_d (t+t_0)]^\tau}
\; ,
\label{guess1confirmed}
\end{equation}
for the time-dependent number density of domain areas.
The same sequence of steps for infinite-temperature initial
conditions leads to the same form but with $c_d$ replaced by
$2c_d$ and $\tau$ replaced by $\tau'$. The effects of temperature
are expected to appear only through the parameters $\lambda_d$ and
$\lambda_h$ once thermal fluctuations are extracted from the analysis.
The averaged domain area is then given by
\begin{eqnarray}
\langle A\rangle(t)
&=&
\frac{\int dA' A' n_d(A',t)}{\int dA' n_d(A',t)} = \frac{1}{N_d(t)}
\\
&\propto& \lambda_d t
\; .
\end{eqnarray}
\section{Statistics of areas: numerical tests}
\label{sec:numeric}
To test our analytic results we carried out numerical simulations on the
$2d$ square-lattice Ising model ($2d$IM) with periodic boundary
conditions using a heat-bath algorithm with random sequential
updates. All data have been obtained using systems with size
$L^2=10^3\times 10^3$ and $2\times 10^3$ runs using independent
initial conditions.
Domain areas are identified with the Hoshen-Kopelman
algorithm~\cite{Hoshen} while hull-enclosed ones are measured by
performing a directed walk along the interfaces, in analogy with the
algorithm in~\cite{Aharony}. A detailed description of our
algorithm is given in Appendix~\ref{app:algorithm}.
The equilibrium critical Ising initial conditions have
one spanning cluster (since the system is at the limit
of the percolation threshold), that grows during evolution. No other
spanning cluster is later formed. By contrast, equilibrium infinite
temperature initial conditions are below the critical random percolation
point in $d=2$ but often after 2 MC steps two spanning
clusters appear that also grow during evolution. After 20 MCs roughly
50\% of the spins lie typically on these clusters. This
implies that we need to simulate a large number of independent samples
to obtain a good statistics.
It is important to note that the dynamics of the discrete model
includes processes that are not taken into account in the continuous
model, as given in Eq.~(\ref{allen-cahn}), for which we derived our
analytical results. Some of these processes are the fission of a big
domain into two smaller ones (that usually occurs by cutting a thin
bottle neck that joined them), or the coalescence of two domains to
form a bigger one. However, we shall prove that these processes are not
important and the dynamics of the discrete model is well described by
the analytic results.
\subsection{Initial conditions.}
\label{sec:infTinit}
We used three types of initial conditions: equilibrium at infinite
temperature, $T_0\to\infty$; equilibrium at the critical point,
$T_0=T_c$; equilibrium within the high temperature phase,
$T_0=T_c+\Delta T$ with $\Delta T>0$.
\begin{figure}[h]
\centerline{
\psfrag{x}{\large\hspace{-5mm} $A$}
\psfrag{y}{\large\hspace{-5mm} $n_h(A)$}
\includegraphics[width=8cm]{other_5a.eps}
}
\centerline{
\psfrag{x}{\large\hspace{-5mm} $A$}
\psfrag{y}{\large\hspace{-5mm} $n_d(A)$}
\includegraphics[width=8cm]{other_5b.eps}
}
\caption{(Colour online.) Early evolution of the infinite temperature
initial condition. Top panel: in a few MC steps the hull enclosed area
distribution reaches the one of critical percolation
(\ref{eq:nhpercolation}), whose slope, -2, is shown by the straight
line~\cite{Cardy}. Bottom panel: the domain area distribution. The
straight line is the power-law decay $A^{-2.055}$~\cite{Vander}.}
\label{fig:evolinitcondT05fromTinfhull}
\end{figure}
We mimicked an instantaneous quench from $T_0\to\infty$ by using
random initial states with spins pointing up or down with probability
$1/2$. Assigning site occupation to up spins and vacant sites to down
spins the infinite temperature initial condition can be interpreted as
a percolation problem at $p=0.5$ and thus below the random percolation
transition $p_c=0.5927$ in a square bi-dimensional lattice. Even if
initially away from criticality, in a few MC steps the hull enclosed
area distribution becomes the one in Eq.~(\ref{eq:nhpercolation}), as
shown in the top panel in
Fig.~\ref{fig:evolinitcondT05fromTinfhull}. The initial
distribution lacks large areas, there being almost none with $A>10^3$,
and the tail of $n_h$ falls off too quickly well below the critical
percolation curve. In a few time steps large structures appear and the
tail of the distribution approaches the expected form at critical
percolation. Simultaneously, the weight at small areas diminishes and
the curve progressively gets flatter. This effect can also be seen in
Fig.~\ref{fig:pdf-perim-equil-Tinf} where we display data for
perimeter lengths. In the bottom panel in
Fig.~\ref{fig:evolinitcondT05fromTinfhull}
we display our numerical results for the domain area distribution,
which are compatible with the form (\ref{eq:domequilTinfty}). It is
intuitively clear why this must be so. If the system is
coarse-grained on the domain typical scale, $R(t)$, it will look
completely disordered. When $R(t)$ is large compared to the lattice
spacing, the disorder will be that of continuum percolation, for which
the critical density is one half by symmetry in two
dimensions~\cite{continuous-perc,continuous-perc2}.
It follows that the coarsening
system will be asymptotically at percolative criticality, i.e.\ the
dynamics self-tunes the system to percolative criticality in two
dimensions [provided $R(t)$ remains much smaller than the system
size]. The data show that, as far as the hull and domain area
distributions are concerned, this only takes a few Monte Carlo steps
in practice. During these few steps many small domains coalesce to
form larger ones meaning that the dynamics is dominated by processes
that are not taken into account by Eq.~(\ref{allen-cahn}).
This argument also shows that the domain distribution in the scaling limit
indeed has the predicted $A^{-\tau'}$ tail. It is interesting to
remark that the system approaches the percolative critical state not
by increasing $p$ (indeed, the magnetization during the coarsening
process initially remains close to zero), but by decreasing the value
of $p_c$, from 0.5927 to 0.5, as the correlation between spins
increases.
We can look at this from another perspective in the context of the
continuum model. Consider a random field $\phi(\vec x)$, symetrically
distributed with respect to zero, with bounded variance and two-point
correlator $C(r)=\langle\, \phi(\vec x)\phi(\vec x + \vec r) \,
\rangle$ with $r=|\vec r|$. The zero contour lines of this field can
be imagined to divide the plane into regions of black and white with
each contour line forming a boundary between black and white
regions. Provided that $C(r)$ falls off faster than $r^{-3/4}$ for
large $r$, this problem is known to belong to the standard percolation
universality class~\cite{continuous-perc2}. If we
now identify $\phi(\vec x)$ with the order parameter field when
well-defined domain walls (the zero contour lines) have formed, we see
that the resulting domain structure corresponds to critical
percolation.
In the plots we use a double logarithmic scale that serves as a first
check of the power-law decay of the probability distributions but it
is not accurate enough to examine the value of the constants $c_h$ and
$c_d$. We delay the presentation of a very precise test of these
parameters to Sect.~\ref{subsec:coarseningT0} where we analyse the
time-evolution of the distribution functions.
We obtained the initial states for the coarsening dynamics at the critical
temperature, $T_0=T_c$, and at $T_0=T_c+\Delta T$, after running $10^3$
Swendsen-Wang algorithm steps. We checked that the systems are well
equilibrated after these runs. The distribution of hull enclosed and domain
areas at $T_c$ are consistent with the analytic forms -- not shown.
\begin{figure}[h]
\begin{center}
\psfrag{x}{\large $A$}
\psfrag{y}{\large\hspace{-5mm} $n_h(A,t)$}
\includegraphics[width=8cm]{other_6.eps}
\end{center}
\caption{(Colour online.) Equilibrium number density of hulls per unit
area for the high temperature phase of the $2d$IM. The distributions
seem power law for small areas, with a temperature dependent exponent
that approaches 2 when $T\to T_c$. Indeed, data for $T=2.3\simeq 1.01
\; T_c$ is almost coincident with Eq.~(\ref{eq:analytic-nh-Tc}).}
\label{fig:tmaiortc}
\end{figure}
\subsection{Coarsening at zero temperature}
\label{subsec:coarseningT0}
\subsubsection{Hull enclosed areas.}
In Fig.~\ref{fig:hulls} we show the time-dependent hull enclosed area
distribution in double logarithmic scale, at seven different times,
following a quench from $T_0\to\infty$. The figure shows a strong
time dependence at small areas and a very weak one on the tail, which
is clearly very close to a power law. The curves at small areas move
downwards and the breaking point from the asymptotic power law decay
moves towards larger values of $A$ for increasing $t$.
\begin{figure}[h]
\begin{center}
\psfrag{x}{\large $A$}
\psfrag{y}{\large $n_h(A,t)$}
\includegraphics[width=8cm]{other_7.eps}
\end{center}
\caption{(Colour online.) Number density of hull enclosed areas per
unit system area for the zero-temperature dynamics of the $2d$IM at
seven times evolving from an infinite temperature initial
condition. The lines represent Eq.~(\ref{eq:analytic-nh-Tinf})
with $c_h=1/8\pi\sqrt{3}$.}
\label{fig:hulls}
\end{figure}
\begin{figure}[h]
\begin{center}
\psfrag{x}{\large $A/\lambda_ht$}
\psfrag{y}{\large $(\lambda_ht)^2 \, n_h(A,t)$}
\includegraphics[width=8cm]{other_8.eps}
\end{center}
\caption{(Colour online.)
Number density of hull enclosed areas per unit system area for the
zero temperature dynamics of the $2d$IM
evolving from an infinite temperature initial condition. The full
line is the prediction (\ref{eq:analytic-nh-Tinf}) with $c_h=1/8\pi\sqrt{3}$
and $\lambda_h=2.1$.}
\label{fig:scaling-hulls-prl}
\end{figure}
\begin{figure}[h]
\begin{center}
\psfrag{x}{\large $A/\lambda_ht$}
\psfrag{y}{\large $(\lambda_ht)^2 \, n_h(A,t)$}
\psfrag{1000}{1000}
\psfrag{500}{500}
\psfrag{200}{200}
\psfrag{L=100}{$L=100$}
\includegraphics[width=8cm]{other_9.eps}
\end{center}
\caption{(Colour online.)
Finite size effects at $t=16$ MCs;
four linear sizes of the sample are used and indicated by the
data-points. The value of $A/t$ at which the data
separate from the master curve grows very fast with $L$ with
an exponent close to 2.}
\label{fig:finite-size}
\end{figure}
In Fig.~\ref{fig:critic-init-cond} (bottom) we zoom on the small area
region ($A<10^3$) where the time-dependence is clearer and we scale
the data by plotting $(\lambda_h t)^2 n_h(A,t)$ against $A/\lambda_h
t$ with $\lambda_h=2.1$. We tried other time-dependent factors but
$\lambda_h t$ with this particular value of $\lambda_h$ is the one
yielding the best collapse of data at small areas, $A_0 \ll A <
\lambda_h t$. For $A$ larger than the `typical' value $\lambda_h t$
the time and $\lambda_h$ dependence becomes less and less important.
In Fig.~\ref{fig:scaling-hulls-prl} we show the data in their full
range of variation in log-log form to test the prediction $n_h(A,t)
\propto A^{-2}$ for large $A$. The data are in remarkably good
agreement with the prediction~(\ref{eq:analytic-nh-Tinf}) -- shown as
a continuous curve in the figure -- over the whole range of $A$ and
$t$. The downward deviations from the scaling curve are due to
finite-size effects. The latter are shown in more detail in
Fig.~\ref{fig:finite-size}, where we display the $t=16$ MCs results
for several linear sizes. Finite size effects appear only when the
weight of the distribution has fallen by many orders of magnitude (7
for a system with $L=10^3$) and are thus quite irrelevant. In the
tail of the probability distribution function (pdf) the numerical
error is smaller than the size of the data points. The nearly perfect
agreement between the analytical theory and the data is all the more
impressive given that the curvature-driven growth underlying the
prediction (\ref{eq:analytic-nh-Tinf}) only holds in a statistical
sense for the lattice Ising model~\cite{Anisotropy}. Indeed, even at
small values of $A/\lambda_h t$, where the lattice and continuous
descriptions are expected to differ most, the difference is only a few
percent, as we shall show below.
It is clear that the evolution of the hull-enclosed area distribution
follows the same `advection law' (\ref{eq:advection}), with the same
value of $\lambda_h$, for other initial conditions. The evolution from
a critical temperature initial condition is shown in the top panel of
Fig.~\ref{fig:critic-init-cond}. A fit of the data at small areas
yields the value of the parameter $\lambda_h$ that, consistently with
the analytic prediction, takes the same value $\lambda_h=2.1$. In the
bottom panel, we compare the time-dependent hull enclosed area
distributions for the initial conditions $T_0\to\infty$ and $T_0=T_c$
and we zoom on the behaviour of $n_h(A,t)$ at small areas,
$A/\lambda_h t \leq 10$. The two solid lines correspond to the
numerator in $n_h$ being equal to $2c_h$ for infinite temperature
initial conditions and $c_h$ for critical Ising initial
conditions. The difference between the numerical data for the two
initial states is clear and it goes in the direction of the analytic
prediction (a factor 2 difference in the constant). Finally, while
the log-log plot in Fig.~\ref{fig:scaling-hulls-prl} suggests that the
data are compatible with $c_h \approx 0.023$ this way of presenting
the data is not precise enough to let us quantify the accuracy with
which we match the analytic prediction. We test the numerical values
of the constant $c_h$ in detail in Sect.~\ref{sec:numeric} where the
numerical error is also estimated.
\begin{figure}[h]
\begin{center}
\psfrag{x}{\large $A/\lambda_ht$}
\psfrag{y}{\large $(\lambda_h t)^2 \, n_h(A,t)$}
\includegraphics[width=8cm]{other_10a.eps}
\includegraphics[width=8cm]{other_10b.eps}
\end{center}
\caption{(Colour online.) Number density of hulls per unit area for
the zero-temperature $2d$IM evolving from critical initial conditions.
The initial states are obtained after running $10^3$ Swendsen-Wang
algorithm steps. Top panel: the full (red) line is
Eq.~(\ref{eq:analytic-nh-Tc}) with $\lambda_h = 2.1$ which again
yields the best fit of the data at small areas. For comparison we
include with a dotted (blue) line the analytic prediction for an
infinite temperature initial condition,
i.e. Eq.~(\ref{eq:analytic-nh-Tinf}). Bottom panel: more details on
the influence of the initial conditions. The two data sets correspond
to configurations taken at several times after a quench from
$T_0\to\infty$ and $T_0=T_c$. The solid lines are the analytic
predictions (\ref{eq:analytic-nh-Tinf}), blue line, and
(\ref{eq:analytic-nh-Tc}), red line.}
\label{fig:critic-init-cond}
\end{figure}
Moreover, Eq.~(\ref{eq:analytic-nh-Tinf}) applies to {\it any}
$T_0>T_c$ equilibrium initial condition asymptotically. Equilibrium
initial conditions at different $T_0>T_c$ show only a different
transient behaviour: the closer they are from $T_c$, the longer it
takes to reach the asymptotic law, Eq.~(\ref{eq:analytic-nh-Tinf}).
Equilibrium initial distributions, for $T_c<T_0<\infty$, are shown in
Fig.~\ref{fig:tmaiortc}, while Fig.~\ref{fig:intermTihulls} shows an
example of their subsequent evolution. In the latter, both analytic
predictions, for $T_0=T_c$ and $T_0=\infty$, are shown as solid lines
along with data for increasing times after a zero-temperature quench
from $T_0=2.5$. In the first steps, the curve follows the one for
critical initial conditions at small $A/\lambda_ht$ and then departs
to reach the one for infinite temperature initial conditions at large
$A/\lambda_ht$. At longer times, the deviation from the critical
initial condition line occurs at a smaller value of $A/\lambda_ht$.
Initially the system has a finite, though relatively small,
correlation length $\xi(T)$. Thermal fluctuations with linear size of
the order of $\xi$ and also significantly larger than $\xi$ exist (see
the discussion on the effect of thermal fluctuations in
Sect.~\ref{sec:effectoftemperature}). Notice that $\xi(T)$ does not
correspond exactly to the size of geometric domains: thermal
fluctuations are not perfectly described by domains of aligned spins,
since not all of them are correlated. At any given temperature above
$T_c$, fluctuations smaller than $\xi(T)$ have the same statistics
than those occurring at $T_c$ and are thus described by
Eq.~(\ref{eq:analytic-nh-Tc}), while domains larger than $\xi(T)$ are
not made of correlated spins and thus are described by the infinite
temperature distribution, Eq.~(\ref{eq:analytic-nh-Tinf}). As time
increases, the system loses memory of the finite-size fluctuations and
the asymptotic state does not differ from the infinite temperature
one. Only when fluctuations exist over all spatial scales does the
asymptotic state differ.
This behaviour can be interpreted as follows. At
fixed $A/t$, shorter times correspond to small areas while longer
times are related to larger areas. Very small areas correspond to
short linear sizes, of the order of the domains in the initial
configurations, and thus reminiscent of critical ones. Instead large
areas correspond to long linear sizes that are much longer than the
correlation length and closer to the ones reached from the infinite
temperature initial condition.
\begin{figure}[h]
\begin{center}
\psfrag{x}{\large\hspace{-5mm} $A/\lambda_ht$}
\psfrag{y}{\large\hspace{-1cm} $(\lambda_ht)^2 \, n_h(A,t)$}
\psfrag{a}{$T_0=T_c$ [Eq.~(\ref{eq:analytic-nh-Tc})]}
\psfrag{b}{$T_0=\infty$ [Eq.~(\ref{eq:analytic-nh-Tinf})]}
\includegraphics[width=8cm]{other_11.eps}
\end{center}
\caption{(Colour online.) Effect of the initial condition temperature
on the hull enclosed area distribution. The continuous lines are the analytic
results for equilibrium initial conditions at $T_0=\infty$ and
$T_0=T_c$, bottom (red) and top (blue) lines, respectively. In
between, we present numerical data for two different times given
in the key after the
quench from an initial state equilibrated at $T_0=2.5$.}
\label{fig:intermTihulls}
\end{figure}
\subsubsection{Domain areas.}
We now study the domain areas and perform the same tests
as above though focusing on the analytic predictions
(\ref{eq:guess1}) and (\ref{eq:guess2-bis}).
In Fig.~\ref{fig:domains} we display the number density of domain
areas in the scaled form for two initial conditions, $T_0\to\infty$
and $T_0=T_c$, after removing any spanning domain from the statistics.
For comparison, in Fig.~\ref{fig:domains-complete}, the same distributions
with the spanning domains are shown. As done for the hull enclosed areas
we fit the parameter $\lambda_d$ by analysing the behaviour at small areas,
$A<\lambda_d t$, and we find, once again that $\lambda_d=2.1$ yields
the best collapse of data (see the discussion in
Sect.~\ref{sec:domains}). We use the extrapolated value $c_d=0.025$
obtained with the numerical analysis described in full detail in
Sect.~\ref{subsub:constants}. Note that we expect the difference between
$c_d$ and $c_h$ to be of order $c_h^2$, and thus rather hard to observe
numerically.
\begin{figure}[h]
\psfrag{x}{\large\hspace{-5mm} $A/\lambda_dt$}
\psfrag{y}{\large\hspace{-1cm} $(\lambda_dt)^2 \, n_d(A,t)$}
\includegraphics[width=225pt]{other_12a.eps}
\includegraphics[width=225pt]{other_12b.eps}
\caption{(Colour online.) Number density of domains per unit area
for the zero-temperature $2d$IM evolving from $T_0\to\infty$ (top) and
$T_0=T_c$ (bottom) initial conditions. In both figures the spanning
clusters have been extracted from the analysis (compare with
Fig.~\ref{fig:domains-complete} where we include them).
The full (red) line represents Eq.~(\ref{guess1confirmed}), with
$c_d=0.025$ and $\tau'=2.055$, $c_d\to c_d/2$ and $\tau=2.027$ (bottom panel),
and $\lambda_d=2.1$ in both cases.}
\label{fig:domains}
\end{figure}
\begin{figure}[h]
\psfrag{x}{\large\hspace{-5mm} $A/\lambda_dt$}
\psfrag{y}{\large\hspace{-1cm} $(\lambda_dt)^2 \, n_d(A,t)$}
\includegraphics[width=225pt]{other_13a.eps}
\includegraphics[width=225pt]{other_13b.eps}
\caption{(Colour online.) The same as Fig.~\ref{fig:domains}, but with
the spanning domains included in the statistics. Notice that due to
the finiteness of the lattice, the actual area of these spanning
domains are smaller than the value they would have on an infinite
system, generating the overshoot of the distribution for values of $A$
close to $L^2$. The larger is the system, the more to the right these
peaks are localized.}
\label{fig:domains-complete}
\end{figure}
Both sets of figures, \ref{fig:domains} and
\ref{fig:domains-complete}, exhibit finite size effects in the tail of
the distributions, where the number of domain areas has already
decreased by several orders of magnitude. As for the hull enclosed
areas, the point where these finite size effects cause the deviation
from the collapsed curve moves towards the right as the system size
increases, becoming less and less relevant. In
Fig.~\ref{fig:domains}, large domain areas (violating the limiting
condition $A\ll L^2$) that would nonetheless be accounted for in an
infinite system are here removed since they span the system in one of
the directions, leading to the downward bending of the
distribution. On the other hand, in Fig.~\ref{fig:domains-complete},
when counting these domains, they are chopped by the system
boundaries, thus contributing to the distribution in a region shifted
to the left, accounting for the bumps seen in the figure.
\subsubsection{Study of the constants $c_h$ and $c_d$.}$\;$
\label{subsub:constants}
In order to improve the data analysis we followed the procedure used
by Cardy and Ziff, who studied the `finite area' scaling of the
cumulative distribution between $A$ and $2A$~\cite{Cardy}. The method
is as follows. For hull enclosed and domain areas the total number of
areas between $A$ and $2A$ is $N_{h,d}(A,2A)= \int_{A}^{2A} dA' \;
n_{h,d}(A')$. Using the analytic prediction for $n_h$ one finds $2A
N_{h}(A,2A)\sim c_h$ for large areas at critical Ising conditions and
without the factor two at critical percolation. Following Ref.~\cite{Cardy},
we assume that there are power-law finite area corrections and add
a term like $a A^{-b}$ to the above expressions. From this relation one
extracts the value of $c_h$. Similarly, for domains one can use
$(1-\tau) N_{d}(A,2A)/[(2A)^{1-\tau}-A^{1-\tau}]= c_d + a A^{-b}$
at critical Ising initial and its modified form at critical percolation
initial conditions.
\begin{figure}[h]
\centerline{
\psfrag{x}{\large\hspace{-5mm} $A^{-0.875}$}
\psfrag{y}{\large $c_h$}
\includegraphics[width=8cm]{other_14a.eps}
}
\centerline{
\psfrag{x}{\large\hspace{-7mm} {$ {(A/\lambda_dt)}^{-0.875}$} }
\psfrag{y}{\large $c_h$}
\includegraphics[width=8cm]{other_14b.eps}
}
\caption{(Colour online.) Study of $c_h$ using finite area scaling
implemented as in~\cite{Cardy}. The numerical data correspond to
$2AN_h(A,2A)$ in the equilibrium case and the left-hand-side of
Eq.~(\ref{eq:dyn-cd}) in the dynamic case. Top panel: equilibrium
distribution of hull-enclosed areas at criticality. The upper
horizontal line is the predictions for $c_h$ stemming from the use of
the sum rule. The lower horizontal line is the analytic prediction.
The intercept of the inclined straight line with $x=0$ is the
numerical test of the analytic prediction. Bottom panel: study of $c_h$ during
coarsening after a quench from criticality to zero temperature. Note
that the prediction from the fit is slightly higher than the analytic
value, see the discussion in the text.}
\label{fig:chcd-improved}
\end{figure}
In the dynamic case we have
\begin{eqnarray}
N_{h,d}(A,2A;t) & \equiv & N_{h,d}(A,t) - N_{h,d}(2A,t) \nonumber \\
& = & \int_{A}^{2A} dA' \; n_{h,d}(A',t)\; .
\end{eqnarray}
Using the predictions for $n_{h,d}$ we find
\begin{equation}
N_h(A,2A;t) = \frac{2c_h A}{(A+\lambda_h t)(2A+\lambda_h t)}
\end{equation}
and
\begin{eqnarray}
N_d(A,2A;t) & = & \frac{2c_d (\lambda_d t)^{\tau'-2}}{1-\tau'}\;
[(2A+\lambda_d t)^{1-\tau'} \nonumber \\
&& \hspace*{0.5cm} - (A+\lambda_d t)^{1-\tau'}]
\end{eqnarray}
for $T_0\to\infty$ and without the factor two and with the exponent
$\tau'$ replaced by $\tau$ for $T_0=T_c$. To extract the values of the
constants $c_h$ and $c_d$, we rewrite these forms as
\begin{eqnarray}
&& \hspace*{-1.5cm} (2A)^{-1}(A+\lambda_h t)(2A+\lambda_h t)N_h(A,2A;t)
\nonumber \\
&& \hspace*{2cm}= c_h +a\left(\frac{A}{\lambda_h t}\right)^{-b}
\label{eq:dyn-ch}
\end{eqnarray}
and
\begin{eqnarray}
&& \hspace*{-1.5cm} \frac{(\lambda_d t)^{2-\tau'} (1-\tau') N_d(A,2A;t)}
{2[(2A+\lambda_d t)^{1-\tau'}-(A+\lambda_d t)^{1-\tau'}]} \nonumber \\
&& \hspace*{2cm} = c_d + a \left(\frac{A}{\lambda_d t}\right)^{-b}
\label{eq:dyn-cd}
\end{eqnarray}
and similarly for $T_0=T_c$. $a$ is a constant that takes different
values for different times.
In Figs.~\ref{fig:chcd-improved}-\ref{fig:chcd-improved3}
we show the outcome of this analysis. We use the same scale
on the vertical axis in all plots to compare the accuracy
of the results.
\begin{figure}[h]
\centerline{
\psfrag{x}{\large\hspace{-7mm} {$ {(A/\lambda_dt)}^{-0.875}$} }
\psfrag{y}{\large $c_h$}
\includegraphics[width=8cm]{other_15.eps}
}
\caption{(Colour online.) Study of $c_h$ during coarsening after a
quench from $T_0\to\infty$ to $T=0$ using finite area scaling
implemented as in~\cite{Cardy}. Note that the prediction from the fit
weakly depends on time.}
\label{fig:chcd-improved2}
\end{figure}
In Fig.~\ref{fig:chcd-improved} we show results for the hull enclosed
areas. In the top panel we study equilibrium data at $T_c$. The
extrapolation of the numerical data approaches the analytic prediction
for $c_h$ within a 2\% accuracy. We use the value of the exponent
$b$ proposed by Cardy and Ziff~\cite{Cardy}. The estimation from the
approximate use of the sum rules is well above the analytic prediction
and numerical value. In the bottom panel we test the value of $c_h$ in our
dynamic prediction by studying data at three instants, $t=16, \, 64,
\, 128$ MCs, after a critical Ising initial configuration. In this
analysis we use the same exponent, $b=0.875$, as in the study of the
equilibrium data and we fit the other parameters, $c_h$ and $a$.
The fits of the time-dependent data to straight lines extrapolate to
the {\it same} value that is, however, of the order of 5\% off the
analytic prediction. Note the non-monotonic character of the
time-dependence in the slopes $a$. One has to keep in mind that
the extrapolated value is very sensitive to the fit, in particular, to
how many data points are considered.
In the analysis of the infinite temperature initial conditions we are
forced to use dynamic results to reach, first the critical percolation
situation, and next follow the coarsening evolution. In
Fig.~\ref{fig:chcd-improved2} we display this type of data for the
instants given in the key. The prediction from the fit is slightly
different from the analytic result at short times but it approaches the
analytic value, shown with a horizontal line,
at sufficiently long times (look at the $t=128$ MCs
results). In conclusion we find
\begin{equation}
c_h \sim 0.0229 \; \pm \; 0.0015
\; .
\end{equation}
\begin{figure}[h]
\centerline{
\psfrag{x}{\large\hspace{-7mm} {$ {(A/\lambda_dt)}^{-0.875}$} }
\psfrag{y}{\large $c_d$}
\includegraphics[width=8cm]{other_16.eps}
}
\caption{(Colour online.) Study of $c_d$ using finite area scaling
implemented as in~\cite{Cardy}. $c_d$ extracted from time-dependent
pdfs evolving at $T=0$ after a quench from criticality. The numerical
value of $c_d$ is roughly the same for all times.}
\label{fig:chcd-improved3}
\end{figure}
Next we study the constant $c_d$. First we use critical temperature
initial conditions using the dynamic results to avoid the ambiguity
introduced by the units restoring constant $A_0$ in equilibrium.
In short we find
\begin{equation}
c_d \sim 0.0251 \; \pm \; 0.0015
\; .
\end{equation}
Note that this value is slightly higher than $c_h$, in accord with the
prediction from the sum rules and the analytic argument, and the
difference between the two ($c_d-c_h\sim 0.002$) is slightly smaller
than the one that is obtained from the sum rules.
\subsection{Effect of the working temperature}
\label{sec:effectoftemperature}
Up to now we have considered quenches to a zero working temperature.
In this Section we investigate the effect of having a finite
temperature on the dynamics.
The arguments in Sect.~\ref{sec:analytic} rely on the $T=0$ Allen-Cahn
equation~(\ref{allen-cahn}). Temperature fluctuations have a two-fold
effect. On the one hand they generate {\it equilibrium} thermal
domains that are not related to the coarsening process. On the other
hand they roughen the domain walls thus opposing the curvature driven
growth and slowing it down.
Renormalization group treatments
of domain growth dynamics~\cite{Humayun} have led to the idea that
a $T=0$ fix point controls the domain growth for all $T<T_c$, i.e. that
thermal fluctuations are irrelevant to the asymptotic dynamics of the
ordering system, their contribution being limited primarily to the
renormalization of temperature-dependent prefactors.
\begin{figure}[h]
\begin{center}
\psfrag{x}{\large $r$ }
\psfrag{y}{\large\hspace{-7mm} $C(r,t)$}
\includegraphics[width=8cm]{other_17a.eps}
\includegraphics[width=8cm]{other_17b.eps}
\end{center}
\caption{(Colour online.) Top panel: spatial decay of the equal-time
correlation, Eq.~ (\ref{eq:Crt}), at fixed $T=0$ and several different
times $t$. Bottom panel: the $T$ dependence of the parameters
$\lambda_{d,h}$. Two sets of data points are extracted from the
analysis of the correlations shown in the top panel. The data named
half-value are obtained from $C(r,t)=1/2$ and the data named collapse
from collapsing the curves on the range $r\gg a$. The two other sets
are obtained from fitting $n_{d,h}(A,t)$ with $\lambda_{d,h}(T)$ as
free parameters. }
\label{fig:Tdep-lambda}
\end{figure}
For the distribution of domain areas and hull enclosed
areas, one may expect that
once equilibrium thermal domains are
subtracted -- hulls and domains associated to the coarsening
process are correctly identified -- the full temperature dependence
enters only through the values of $\lambda_h$ and $\lambda_d$, which
set the time scale.
The first step then is to identify the temperature dependence
of the parameter $\lambda_d$. The simplest and most
direct way to do this is to use the scaling hypothesis and
analyze the behaviour of the spatial correlation
\begin{eqnarray}
C(r,t) & \equiv & \frac{1}{N} \sum_{i=1}^N
\langle \, s_i(t) s_j(t) \, \rangle|_{|\vec r_i - \vec r_j|=r} \nonumber \\
& \sim & m^2(T) \; f\left(\frac{r}{R(t)}\right)
\; ,
\label{eq:Crt}
\end{eqnarray}
where $m(T)$ is the equilibrium magnetization density and
$a \ll r \ll L$ and $t_0 \ll t$. Using $R(t) \sim
[\lambda_d(T) t]^{1/2}$, the $T$-dependence of
$\lambda_d$ can be estimated either by collapsing all
curves or by studying the value of $r$ at which
$C(r,t)=1/2$. The resulting $\lambda_d(T)$ obtained using these
two prescriptions is shown in
Fig.~\ref{fig:Tdep-lambda}. $\lambda_d(T)$ is a monotonically decreasing
function of temperature, starting at $\lambda_d(T=0)=2.1$ and
falling-off to zero at $T_c$. These results are consistent with the
evaluation of $\lambda_{d,h}(T)$ from the analysis of $n_{d,h}(A,t)$, see
below, and it is at variance with what was previously presented
in Refs.~\cite{Kaski}.
Assuming that $\lambda_d$ vanishes at $T_c$ one can derive the way in
which it does with a simple argument~\cite{lacasse}. We require that the coarsening
law for coarsening below $T_c$, namely $R(t) \sim [\lambda(T)
t]^{1/2}$, match critical coarsening at $T_c$, {\it viz.} $R(t) \sim
t^{1/z}$ with $z$ the dynamic exponent, for $T\to T_c$. Near (but just
below) $T_c$ the coarsening length grows as $\xi^{-a}(T) t^{1/2}$ as
long as $R(t) \gg \xi(T)$ with $\xi(T)$ the equilibrium correlation
length. For $R(t)$ comparable with $\xi(T)$, this has to be modified
by a function of $R(t)/\xi$ and, since $R(t) \sim t^{1/z}$ at $T_c$,
we can write
\begin{equation}
R(t)
\sim \xi^{-a}(T) t^{1/2} f\left(\frac{t}{\xi^z(T)}\right)
\; .
\end{equation}
In the limit $\xi(T) \to \infty$, the $\xi$ -
dependence must drop out.
In order to cancel the time
dependence at large times, one needs $f(x) \sim x^{1/2}$ for
$x \to \infty$. This yields $R(t) \sim
t^{1/z}$, which fixes the exponent `$a$' as $a=(2-z)/2$, giving
$\lambda(T) \sim \xi^{-2a}(T) \sim (T_c-T)^{\nu(z-2)}$.
Inserting the exact value $\nu=1$ and the numerical value
$z=2.15(2)$~\cite{Leung} implies
\begin{equation}
\lambda_d(T) \sim (T_c-T)^{0.15} \; .
\end{equation}
Note that we are
matching two nonequilibrium growth laws -- the one below $T_c$ and the
one at $T_c$ -- not an equilibrium and a nonequlibrium one.
The data in Fig.~\ref{fig:Tdep-lambda} are still
far from the critical region where this small power-law decay
should show up.
\begin{figure}
\centerline{
\psfrag{x}{\large {$A$} }
\psfrag{y}{\large\hspace{-5mm} $n_h(A)$}
\includegraphics[width=8cm]{other_18.eps}
}
\caption{(Colour online.) The number density of hull enclosed areas after
$t=128$ MCs at the working temperatures
$T=0.5,\; 1.5, \; 2$ and $T=T_c$.}
\label{fig:three-T}
\end{figure}
\begin{figure}
\begin{center}
\psfrag{x}{\large {$A$} }
\psfrag{y}{\large\hspace{-5mm} $n_h(A)$}
\includegraphics[width=8cm]{other_19a.eps}
\psfrag{x}{\large\hspace{-5mm} $A/\lambda_ht$}
\psfrag{y}{\large\hspace{-1cm} $(\lambda_ht)^2 \, n_h(A,t)$}
\includegraphics[width=8cm]{other_19b.eps}
\end{center}
\caption{(Colour online.) The number density of hulls for $T=1.5$
after different times (top) and the scaling of these data
points (bottom)}
\label{fig:oneT-severaltimes}
\end{figure}
Finite working temperatures also affect the distribution of domain
areas. In Fig.~\ref{fig:three-T} the raw data at $t=128$ MCs is
shown for four working temperatures. Upward deviations with respect to
the result of zero working temperature are prominent in the small
areas region of the figure, and increase with temperature.
In Fig.~\ref{fig:oneT-severaltimes}~(top) we display the raw data at
the working temperature $T=1.5$, for several times. Notice that
although the curves move downwards, the small areas region becomes
time independent. This region also fails to collapse (bottom) with the
proposed scaling using the temperature dependent values of
$\lambda_d(T)$. The reason is that the distribution counts thermal
equilibrium domains, that is to say fluctuations that are present in
an equilibrated sample at the working temperature, but are not due to
the coarsening process. Thus, these fluctuations should be identified
and eliminated from the statistics. We tried to apply the method
introduced by Derrida~\cite{Derrida}, and extended by Hinrichsen and
Antoni~\cite{Antoni}, to eliminate thermal domains, but the results
were not satisfactory, as not all of them could be eliminated. Thus,
instead of removing each thermal domain, we tried to directly remove
their contribution to the distributions by simulating samples in
equilibrium at the working temperature, starting with a fully
magnetized state, and computing the number density of thermal domain
areas. These data are shown with green data points in
Fig.~\ref{fig:oneT-severaltimes} and ~\ref{fig:Tdep-dist}.
Surprisingly enough, thermal fluctuations generate areas that are
larger than one would have naively expected. Equilibrium arguments
suggest that the averaged area of thermally generated domains scale as
$A_T \sim \xi^2(T)$ with $\xi(T)/p_0\sim f^- (1-T/T_c)^{-\nu}$,
$\nu=1$ and $f^{-}=0.18$~\cite{Fisher}. This estimate yields, for
example, $A_T\sim 4A_0$ at $T=1.5$. In equilibrium at this temperature
the average size of the domains found numerically is $\langle A_T
\rangle\sim 1.5 A_0$. However, the probability distribution of thermal
areas has a non-negligible weight -- as compared to the one of coarsening
domains -- that goes well beyond this value. For example, in
Figs.~\ref{fig:oneT-severaltimes} and \ref{fig:Tdep-dist}
we see that the crossover between the thermal area distribution
and the coarsening area distribution occurs at $A\sim 10 A_0$.
In Fig.~\ref{fig:Tdep-dist} we also present data for
the dynamic distribution at three different times, and
compared with the analytic prediction using $\lambda_d(T)$ estimated
from the analysis of the global spatial correlation, see
Fig.~\ref{fig:Tdep-lambda}. We conclude that the agreement between
analytic prediction and numerical results is very good in the region
in which the thermal domains are subdominant, {\it i.e.} when the blue
dynamic curves deviate from the green equilibrium one, indeed the
regime in which the analytic calculation is expected to apply.
\begin{figure}[h]
\centerline{
\psfrag{x}{\large {$A$} }
\psfrag{y}{\large\hspace{-5mm} $n_d(A)$}
\includegraphics[width=8cm]{other_20.eps}
}
\caption{(Colour online.) The contribution of
`thermal domains' obtained by simulating an equilibrated sample at the
working temperature $T=1.5$, along with the evolution of the
distribution of domain sizes after a quench to the same temperature.}
\label{fig:Tdep-dist}
\end{figure}
\begin{figure}[h]
\centerline{
\psfrag{x}{\large {$A$} }
\psfrag{y}{\large\hspace{-5mm} $n_d(A)$}
\includegraphics[width=8cm]{other_21.eps}
}
\caption{(Colour online.) Zoom on the number density of domain areas
at $t=128$ MCs at three working temperatures given in the key. The
black lines are the equilibrium distributions at $T=1.5$ and $T=2$
and the other lines (pink, blue and green) represent our analytic
prediction for the coarsening areas.}
\label{fig:Tdep-dist2}
\end{figure}
One can also use the results in Fig.~\ref{fig:Tdep-dist2} to estimate
the value of $\lambda_d(T)$. Indeed, a fit of the numerical data for
areas larger than the value at which the equilibrium thermal
contribution (green points) deviates from the dynamic one, yields the
values of $\lambda_d(T)$ [and $\lambda_h(T)$] shown in
Fig.~\ref{fig:Tdep-lambda}. This analysis allows us to extract
independent predictions for $\lambda_d(T)$ and $\lambda_h(T)$. We find
that the qualitative $T$-dependence is the same. As regards the
absolute values, the numerical data yield $\lambda_h(T) \leq
\lambda_d(T)$ on the whole range. Note that the sum rules suggested
$\lambda_d=\lambda_h$ and the analytic prediction $\lambda_d=\lambda_h
+ {\cal O}(c_h)$.
\section{Statistics of perimeters and fractal properties}
\label{sec:domain-walls}
The analytic argument described in Sect.~\ref{sec:analytic} can be
extended to study the distribution of domain wall lengths or
perimeters. In this Section we present the analytic prediction for this
function together with numeric results that confirm it. We study two types
of domain boundaries: those associated to the hulls and those associated to
the domains that is to say that
include external and internal perimeters. In
the simulations we define the length of the
boundary as the number of broken bonds.
\subsection{Initial conditions}
\subsubsection{Equilibrium at $T_0=T_c$}
In equilibrium we find numerically that the domain areas and their
corresponding boundaries are related by (see Fig.~\ref{fig:new1}
where the scatter plots have been averaged to make the trend clearer)
\begin{eqnarray*}
A_h &\sim& c^{(i)}_h \; p^{\alpha^{(i)}_h}, \;\; \mbox{with} \;\; \alpha^{(i)}_h \sim 1.47 \pm 0.1,
\\
\\
A_d &\sim&
\left\{
\begin{array}{lll}
c^{(i>)}_d \; p^{\alpha^{(i>)}_d},
\; & \alpha^{(i>)}_d \sim
1.14\pm 0.1 & {\mbox{for}} \;\; p \stackrel{>}{\sim} 50
\; ,
\\
\\
c^{(i<)}_d \; p^{\alpha^{(i<)}_d},
\; & \alpha^{(i<)}_d \sim
1.47\pm 0.1 & {\mbox{for}} \;\; p \stackrel{<}{\sim} 50
\; ,
\end{array}
\right.
\label{eq:initial-powers-domains-Tc}
\end{eqnarray*}
in the whole range of variation. Note that the longest lengths,
$p\approx 10^3-10^4$ may be affected by finite size effects given that
the linear size of the simulating box is $L=10^3$. The spanning
clusters are not counted (note that their perimeters would be severely
under estimated due to the periodic boundary conditions).
The exponent $\alpha^{(i<)}_d \sim 1.47\pm 0.1$ is consistent with the result in~\cite{Cambier}
mentioned in Sect.~\ref{sec:Cambier}. The constants take the values
$c^{(i)}_h = 0.15$, $c^{(i<)}_d = 0.15$ and $c^{(i>)}_d = 0.70$.
The difference between the small and large $p$ regimes in the relation
between areas and perimeters for the domains is due to the existance of
holes in the large structures. The small domains and hulls are just the same
objects because the former do not have holes within.
\begin{figure}[h]
\centerline{
\psfrag{x}{\large $p$}
\psfrag{y}{\large $A$}
\includegraphics[width=8cm]{other_22.eps}
}
\caption{(Colour online.) Relation between areas and perimeters in equilibrium at $T_0=T_c$.}
\label{fig:new1}
\end{figure}
\begin{figure}[h]
\centerline{
\psfrag{x}{\large $p$ }
\psfrag{y}{\large\hspace{-5mm} $n_{h,d}(p)$}
\includegraphics[width=8cm]{other_23a.eps}
}
\caption{(Colour online.) Distribution of domain and hull lengths in equilibrium at $T_0=T_c$.}
\label{fig:new2}
\end{figure}
Numerically, we find that
the number densities of hull and domain lengths
at critical Ising conditions are (see Fig.~\ref{fig:new2})
\begin{eqnarray*}
n_h(p,0) &\sim& p^{-\zeta_h}\;\; \mbox{with} \;\; \zeta_h \sim 2.48\pm 0.05.
\\
\\
n_d(p,0) &\sim& \left\{
\begin{array}{lll}
p^{-\zeta^>_d},
\; & \zeta^>_d \sim
2.17\pm 0.05 & {\mbox{for}} \;\; p \stackrel{>}{\sim} 50
\; ,
\\ \\
p^{-\zeta^<_d},
\; & \zeta^<_d \sim
2.48\pm 0.05 & {\mbox{for}} \;\; p \stackrel{<}{\sim} 50
\; ,
\end{array}
\right.
\label{eq:initial-powers-domains-Tc}
\end{eqnarray*}
The value of $\zeta_h$ is to be
compared to the analytic result $\zeta_h =27/11\approx
2.454$~\cite{Vander}. It is interesting to notice that
the distribution of domain lengths is not a single power law
in constrast to the distribution of domain areas.
\subsubsection{Equilibrium at $T_0=\infty$}
After a few time-steps evolving at $T=0$ from the
infinite temperature initial condition, we reach critical
percolation conditions.
In Fig.~\ref{fig:new3}, we show the area perimeter relation
for hulls, in equilibrium at $T_0 \to \infty$ and after a few
time steps. The analysis of this figure and the corresponding
one for domains yields
\begin{eqnarray*}
A_h &\sim& c^{(i')}_h \; p^{\alpha^{(i')}_h}, \;\; \mbox{with} \;\; \alpha^{(i')}_h \sim 1.12 \pm 0.1
\\
\\
A_d &\sim&
\left\{
\begin{array}{lll}
c^{(i'>)}_d \; p^{\alpha^{(i'>)}_d},
\; & \alpha^{(i'>)}_d \sim
1.01\pm 0.1 & {\mbox{for}} \;\; p \stackrel{>}{\sim} 50
\; ,
\\ \\
c^{(i'<)}_d \; p^{\alpha^{(i'<)}_d},
\; & \alpha^{(i'<)}_d & {\mbox{for}} \;\; p \stackrel{<}{\sim} 50
\; ,
\end{array}
\right.
\label{eq:initial-powers-domains-Tc}
\end{eqnarray*}
The constants take the values
$c^{(i')}_h = 0.96$, and $c^{(i'>)}_d = 1.50$.
The exponent $\alpha^{(i'<)}_d$ cannot be determined numerically since
critical percolation is not accessible exactly.
\begin{figure}[h]
\centerline{
\psfrag{x}{\large $p$ }
\psfrag{y}{\large $A_h$}
\includegraphics[width=8cm]{other_23b.eps}
}
\caption{(Colour online.) Relation between areas and perimeters in equilibrium at $T_0\to\infty$
and after a few time steps when critical percolation is approximatively reached.}
\label{fig:new3}
\end{figure}
As we can see in Fig.~\ref{fig:pdf-perim-equil-Tinf}, the
initial weight of the number density at large values of the
perimeter is lower than expected at critical percolation. In a few
time-steps long perimeters develop and the weight reaches
the asymptotic power law at large values of $p$ while it
looses weight at small values of $p$. This effect is the same
as the one observed in the study of the initial and early times
number density of areas, see the discussion in Sect.~\ref{sec:infTinit}
and Fig.~\ref{fig:evolinitcondT05fromTinfhull}. For the hull and domain
length distributions one finds
\begin{eqnarray*}
n_h(p,0) &\sim& p^{-\zeta^{'}_h} \;\; \mbox{with} \;\; \zeta^{'}_h \sim 2.12\pm 0.05
\\
\\
n_d(p,0) &\sim& \left\{
\begin{array}{lll}
p^{-\zeta^{'>}_d},
\; & \zeta^{'>}_d \sim
2.01\pm 0.05 & {\mbox{for}} \;\; p \stackrel{>}{\sim} 50
\; ,
\\ \\
p^{-\zeta^<_d},
\; & \zeta^<_d & {\mbox{for}} \;\; p \stackrel{<}{\sim} 50
\; ,
\end{array}
\right.
\label{eq:initial-powers-domains-Tc}
\end{eqnarray*}
The analytical result for the hull exponent in critical percolation is
$\zeta'_h=15/7 \approx 2.14$~\cite{Saleur-Duplantier}.
\begin{figure}
\centerline{
\psfrag{x}{\large $p$ }
\psfrag{y}{\large\hspace{-5mm} $n_h(p)$}
\includegraphics[width=8cm]{other_24.eps}
}
\caption{(Colour online.) The number density of hull lengths
in equilibrium at $T_0\to\infty$ labelled as $t=0$ and
the same quantity evaluated at two later times after evolution
at zero working temperature. The lines are
$n_h \sim p^{-2.05}$; see the text for a discussion.}
\label{fig:pdf-perim-equil-Tinf}
\end{figure}
\subsubsection{General comments on both initial cases}
It is interesting to note that the exponents characterising the number
density of perimeter lengths at the two initial conditions are
significantly different. They are approximately equal to 2.5 at $T_c$ and 2
at $T_0\to\infty$. This is to be contrasted with the behaviour of the
area number densities for which the exponents were identical for hull
enclosed areas and very close indeed for domains.
The exponents $\alpha$ and $\zeta$
are linked by the fact that
each hull-enclosed area or domain area is in one-to-one relation
to its own boundary. Thus, $n_h(A,0) dA = n_h(p,0) dp$
and one finds
\begin{equation}
n_h(p,0) \sim \frac{c_h \alpha^{(i)}_h}{c^{(i)}_h}
\; p^{-1-\alpha^{(i)}_h} \; ,
\end{equation}
which implies
\begin{equation}
\zeta_h = 1+\alpha_h^{(i)} \; ,
\label{eq:zetah}
\end{equation}
These conditions are also satisfied for the primed ($T_0 \rightarrow \infty$) quantities.
Within our numerical accuracy these relations are respected,
for instance
\begin{eqnarray}
\begin{array}{lll}
\zeta'_h \sim 2.12 \; , & \qquad {\alpha'}_h^{(i)} \sim 1.12 \;,
& \qquad T_0\to\infty
\; ,
\nonumber\\
\zeta_h\sim 2.48 \; , & \qquad \alpha^{(i)}_h \sim 1.47 \;, & \qquad T_0=T_c
\;
\end{array}
\end{eqnarray}
Similarly, for domain areas and domain boundaries one obtains
\begin{equation}
n_d(p,0) \sim \frac{c_d\alpha^{(i)}_d}{{c_d^{(i)}}^{\tau-1}} \; p^{-1-(\tau-1)
\alpha^{(i)}_d} \; ,
\end{equation}
therefore,
\begin{equation}
\zeta_d = 1+(\tau-1) \alpha_d^{(i)} \; ,
\label{eq:zetad}
\end{equation}
These relations are satisfied for both $\alpha^{(i>)}_d$
and $\alpha^{(i<)}_d$
as well as for primed ($T_0 \rightarrow \infty$) quantities.
They are respected by our measures.
The main sources of error in the determination of the exponents and
the constants in the study of the initial conditions are the following:
(i) statistical errors, although we have a rather good sampling;
(ii) the choice of the large area-perimeter limit that is not perturbed
by finite size effects, and (iii) the fact that the $T_0\to\infty$ initial
condition is not exactly at critical percolation. We estimated the magnitude
of the error to be $\pm 0.1$ in the $\alpha$ exponents, and $\pm 0.05$ in the
$\zeta$ exponents, which correspond to, roughly, less than $10\%$ in
both cases. Within this level of accuracy, the relations between
exponents (\ref{eq:zetad}) and (\ref{eq:zetah}) are satisfied.
\subsection{Time evolution at zero temperature}
\subsubsection{Hulls}
After a quench from $T_0=T_c$, the hull enclosed areas and their corresponding
perimeters, during coarsening at zero temperature, obey the scaling
relations (see Fig.~\ref{fig:new4-5})
\begin{figure}[h]
\begin{center}
{
\psfrag{x}{\large $p$ }
\psfrag{y}{\large $A_h$}
\includegraphics[width=8cm]{other_25a.eps}
\psfrag{x}{\large\hspace{-10mm} $p/\sqrt{\lambda_h t}$ }
\psfrag{y}{\large\hspace{-5mm} $A_h/\lambda_h t$}
\includegraphics[width=8cm]{other_25b.eps}
}
\end{center}
\caption{(Colour online.) Time evolution of the hull enclosed area
vs. perimeter relation for $T_0=T_c$ and different
times indicated in the legend.}
\label{fig:new4-5}
\end{figure}
\begin{figure}[h]
\begin{center}
{
\psfrag{x}{\large\hspace{-10mm} $p/\sqrt{\lambda_h t}$ }
\psfrag{y}{\large\hspace{-5mm} $A_h/\lambda_h t$}
\psfrag{zuluzulu}{$T_0 = T_c \;\;\;\;$}
\psfrag{wuluzulu}{$T_0 = \infty$}
\includegraphics[width=8cm]{other_25c.eps}
}
\end{center}
\caption{(Colour online). Hull enclosed area vs. perimeter relation
at time $t=32$ MCs from two different initial conditions.}
\label{fig:new6}
\end{figure}
\begin{equation}
\frac{A}{\lambda_h t}
\sim \eta_h
\left(\frac{p}{\sqrt{\lambda_h t}}\right)^{\alpha_h},
\label{eq:t-dep-ApTc}
\end{equation}
with
\begin{equation}
\left.
\begin{array}{l}
\alpha^>_h \sim 1.37 \pm 0.2
\\
\eta_h^> \sim 0.26
\end{array}
\right\}
\;\; \mbox{for} \;\;
\frac{A}{\lambda_h t} \; \stackrel{>}{\sim} 50 \; ,
\end{equation}
and
\begin{equation}
\left.
\begin{array}{l}
\alpha^<_h \sim 1.83 \pm 0.2
\\
\eta_h^< \sim 0.06
\end{array}
\right\}
\;\; \mbox{for} \;\;
\frac{A}{\lambda_h t} \; \stackrel{<}{\sim} 10 \; ,
\end{equation}
and, after a quench from $T_0\to\infty$,
\begin{equation}
\frac{A}{\lambda_h t}
\sim \eta'_h
\left(\frac{p}{\sqrt{\lambda_h t}}\right)^{\alpha'_h},
\label{eq:t-dep-ApTinfty}
\end{equation}
with
\begin{equation}
\left.
\begin{array}{l}
{\alpha'}^>_h \sim 1.12 \pm 0.2
\\
{\eta'}_h^> \sim 0.38
\end{array}
\right\}
\;\; \mbox{for} \;\;
\frac{A}{\lambda_h t} \; \stackrel{>}{\sim} 50 \; ,
\end{equation}
and
\begin{equation}
\left.
\begin{array}{l}
{\alpha'}^<_h \sim 1.83 \pm 0.2
\\
{\eta'}_h^< \sim 0.057
\end{array}
\right\}
\;\; \mbox{for} \;\;
\frac{A}{\lambda_h t} \; \stackrel{<}{\sim} 10 \; .
\end{equation}
We note that the relation between area and perimeter exhibits two
distinct regimes. During the coarsening process a characteristic scale
$A^*(t) \sim \lambda_h t$ develops such that domains with area $A>A^*$
have the same exponent as in the initial condition (structures that
are highly ramified with $\alpha$ smaller than two) and domains with
$A<A^*$ are regular ($\alpha \sim 2$) (as shown in Fig.~\ref{fig:new6} the
structure of these small domains does not depend on the initial condition).
This phenomenon is reminiscent
of an {\it unroughening transition} occurring at a velocity
$\lambda_h$. The same features were observed by Grest and
Srolovitz~\cite{Grest} and Fialkowski and Holyst~\cite{Holyst}
in the study of the {\it domain} fractal dimension
during coarsening.
The hull structures of any size do not have holes, therefore
the crossover we see is of pure dynamical origin, contrasting
the idea presented in~\cite{Jacobs}, where the crossover in the domains
where explained by only geometrical reasons.
Note that we estimated the error in the exponents $\alpha$ to be $\pm
0.2$ and thus more important than in the analysis of the initial
conditions. The reason is that the crossover from the small area to
the large area regime is not sufficiently sharp and the choice of the
fitting interval introduces an additional source of error. Indeed,
note that in Eqs.~(\ref{eq:t-dep-ApTc}) and (\ref{eq:t-dep-ApTinfty})
we did not use the intermediate regime $10 \leq \frac{A}{\lambda_h t} \leq 50$
to fit the power laws.
In analogy with the derivation in Sect.~\ref{sec:analytic} for the
time-dependent number density of domain areas, the time-dependent
number densities of hull and domain wall lengths are given by
\begin{equation}
n_{h,d}(p,t) = \int dp_i \; \delta(p-p(t,p_i)) \, n_{h,d}(p_i,t_i)
\end{equation}
with $n_{h,d}(p_i,t_i)$ the initial condition and $p(t,p_i)$ the
perimeter length of a boundary at time $t$ that had initial length
$p_i$ at time $t_i$.
Let us here discuss the hull lengths. In this case one can simply use
the exact number density of hull enclosed areas, $n_h(A,t) \sim
c_h/(A+\lambda_h t)^2$ for, say, $T_0=T_c$ and
Eq.~(\ref{eq:t-dep-ApTc}) to relate time-dependent areas to their
perimeters on the two regimes of large and small areas.
After a little algebra one derives
\begin{equation}
(\lambda_h t)^{3/2}
\;
n_h(p,t) \sim
\frac{
\alpha^<_h \eta^<_h c_h \left(
\frac{p}{\sqrt{\lambda_h t}}\right)^{\alpha^<_h-1}}
{\left[
1+\eta^<_h \left(\frac{p}{\sqrt{\lambda_h t}}\right)^{\alpha^<_h}
\right]^{2}
}
\label{eq:analytic-np-small}
\end{equation}
for small areas, $A/\lambda_h t<10$, and
\begin{equation}
(\lambda_h t)^{3/2}
\;
n_h(p,t) \sim
\frac{
\alpha^>_h \eta^>_h c_h \left(
\frac{p}{\sqrt{\lambda_h t}}\right)^{\alpha^>_h-1}}
{\left[
1+\eta^>_h \left(\frac{p}{\sqrt{\lambda_h t}}\right)^{\alpha^>_h}
\right]^{2}
}
\label{eq:analytic-np-large}
\end{equation}
for large areas $A/\lambda_h t>50$.
Note that these expressions satisfy scaling -- see Eq.~(\ref{eq:scaling-np}).
Interestingly, the scaling
function, $f_<(x)=x^{\alpha_h^<-1}/(1+\eta_h^< x^{\alpha_h^<})^2$ with
$x=p/\sqrt{\lambda_h t}$ reaches a maximum at
\begin{equation}
x_{max} = \left(\frac{\alpha_h^<-1}{\eta^<_h (\alpha_h^<+1)}\right)
^{1/\alpha_h^<}
\label{eq:maximum}
\end{equation}
and then falls-off to zero as another power-law. There is then a
maximum at a finite and positive value of $p$ as long as
$\alpha_h^<>1$, that is to say, in the regime of not too large
areas. The numeric evaluation of the right-hand-side yields $x_{max} =
p_{max}/(\sqrt{\lambda_h t}) \sim 3$ which is in the range of validity
of the scaling function $f_<$. The behaviour of the time-dependent
perimeter number density for long perimeters is controlled by
Eq.~(\ref{eq:analytic-np-large}) that falls-off as a power law $f_>(x)
\sim x^{-(1+\alpha_h^>)}$. Although the function $f_>$ also has
a maximum, this one falls out of its range of validity.
Above we used the critical Ising parameters. The results
after a quench from $T_0\to \infty$ follow the same functional
form with the corresponding primed values of $\alpha$ and $\eta$
and $c_h \to 2c_h$.
The power law describing the tail of the number density of long
perimeters is the same as the one characterising the initial
distribution, since $\alpha_h^>=\alpha_h^{(i)}$ and then
$1+\alpha_h^>=\zeta_h$. Therefore, the decay of the time-dependent
number density at long perimeters after a quench from $T_0=T_c$ and
$T_0\to\infty$ are distinguishably different with $\zeta_h\approx 2.5$
and $\zeta'_h\approx 2$. This is to be contrasted with the small
difference in the area number densities that fall with two power laws
that are so close (powers of 2 and 2.05) that are impossible to
distinguish numerically.
In Fig.~\ref{fig:perimeters-hulls}, top and bottom, we display the
time-dependent perimeter number densities for a system evolving at
zero temperature after a quench from $T_0=T_c$ and $T_0\to\infty$,
respectively. Notice that the perimeter length definition we are
using on the lattice can only take even values and thus when
constructing the histogram we have to take into account the extra
factor of $2$ in the binning.
\begin{figure}[h]
\begin{center}
\psfrag{x}{\large $p$ }
\psfrag{y}{\large\hspace{-5mm} $n_h(p)$}
\includegraphics[width=8cm]{other_26a.eps}
\includegraphics[width=8cm]{other_26b.eps}
\end{center}
\caption{(Colour online.) The time-dependent number density of
perimeters evolving at $T=0$ from an initial condition at $T_0=T_c$
(top) and $T_0\to\infty$ (bottom). Note that the time-dependence is
visible in the whole range of values of $p$ (while in the area number
densities the large area tails were very weakly dependent on time, see
Fig.~\ref{fig:hulls}).}
\label{fig:perimeters-hulls}
\end{figure}
\begin{figure}[h]
\begin{center}
\psfrag{y}{\large\hspace{-10mm} ${(\lambda_h t)}^{3/2} \, n_h(p)$ }
\psfrag{x}{\large\hspace{-5mm} $p/ \sqrt{\lambda_h t}$}
\includegraphics[width=8cm]{other_27a.eps}
\includegraphics[width=8cm]{other_27b.eps}
\end{center}
\caption{(Colour online.) Scaling of the time-dependent number density
of hull lengths evolving at $T=0$ from an initial condition at
$T_0=T_c$ (top) and $T_0\to\infty$ (bottom). The solid black lines
represent the theoretical prediction valid for $A/\lambda_h t<10$
and for $A/\lambda_h t>50$. The
agreement between theory and numerical data is again very
impressive. The small grey line in the top plot represents the slope in
the bottom plot, showing that in contrast with domain size distribution,
perimeter distribution is very sensitive to initial conditions.
The isolated data points that lie above the scaling
function correspond to reversed, isolated spins within a
bulk of the opposite sign that give rise to a perimeter length $p=4$
(four broken bonds). The area number densities also showed this
anomalous behaviour for $A=1$.}
\label{fig:perimeters-hulls-scaling}
\end{figure}
In Fig.~\ref{fig:perimeters-hulls-scaling} we display the scaling plot
of the number density of hull lengths and we compare it to the
analytic prediction~(\ref{eq:analytic-np-small}) and
(\ref{eq:analytic-np-large}). The data are in remarkably good
agreement with the analytic prediction; the lines represent the
theoretical functional forms for long and short lengths, and describe
very well the two limiting wings of the number density. The maximum
is located at a value that is in agreement with the prediction,
Eq.~(\ref{eq:maximum}).
\subsubsection{Domains}
We studied the relation between domain areas and their corresponding
perimeters during coarsening at zero temperature finding that the
scaling forms
\begin{equation}
\frac{A}{\lambda_d t}
\sim \eta_d
\left(\frac{p}{\sqrt{\lambda_d t}}\right)^{\alpha_d},
\label{eq:t-dep-ApTc-domains}\\
\end{equation}
with
\begin{equation}
\left.
\begin{array}{l}
\alpha^>_d \sim 1.16
\\
\eta_d^> \sim 0.63
\end{array}
\right\}
\;\; \mbox{for} \;\;
\frac{A}{\lambda_d t} \; \stackrel{>}{\sim} 50 \; ,
\end{equation}
and
\begin{equation}
\left.
\begin{array}{l}
\alpha^<_d \sim 1.83
\\
\eta_d^< \sim 0.057
\end{array}
\right\}
\;\; \mbox{for} \;\;
\frac{A}{\lambda_d t} \; \stackrel{<}{\sim} 10 \; ,
\end{equation}
after a quench from $T_0=T_c$ and
\begin{equation}
\frac{A}{\lambda_d t}
\sim \eta'_d
\left(\frac{p}{\sqrt{\lambda_d t}}\right)^{\alpha'_d},
\label{eq:t-dep-ApTinf-domains}
\end{equation}
with
\begin{equation}
\left.
\begin{array}{l}
{\alpha'_d}^> \sim 1.01
\\
{\eta'_d}^> \sim 0.52
\end{array}
\right\}
\;\; \mbox{for} \;\;
\frac{A}{\lambda_d t} \; \stackrel{>}{\sim} 50 \; ,
\end{equation}
and
\begin{equation}
\left.
\begin{array}{l}
{\alpha'_d}^< \sim 1.83
\\
{\eta'_d}^< \sim 0.06
\end{array}
\right\}
\;\; \mbox{for} \;\;
\frac{A}{\lambda_d t} \; \stackrel{<}{\sim} 10 \; ,
\end{equation}
after a quench from $T_0\to\infty$.
\begin{figure}[h]
\begin{center}
\psfrag{x}{\large\hspace{-10mm} $p_d/\sqrt{\lambda_d t}$ }
\psfrag{y}{\large\hspace{-5mm} $A_d/\lambda_d t$}
\includegraphics[width=8cm]{other_28a.eps}
\includegraphics[width=8cm]{other_28b.eps}
\end{center}
\caption{(Colour online.) The time-dependent relation between the area and the
perimeter of a cluster of aligned spins (domain) evolving at $T=0$
after a quench from $T_0=T_c$ (top) and
$T_0\to\infty$ (bottom). In both cases $\lambda_d=2.1$.}
\label{fig:t-dep-Ap1}
\end{figure}
These results are shown in Fig.~\ref{fig:t-dep-Ap1}. Note
that the large area results match the behaviour of the initial conditions
in both cases, and small domains are much more compact that the initial ones.
With the same line of argument exposed above we can analyse the statistics of
the domain walls, that is to say, including external and internal
perimeters. One finds basically the same results as for the hulls;
for critical Ising initial conditions:
\begin{equation}
(\lambda_d t)^{3/2} \; n_d(p,t)
\sim
\frac{\alpha_d^< \eta_d^< c_d
\left( \frac{p}{\sqrt{\lambda_d t}}\right)^{\alpha_d^<-1}
}
{
\left[
1+ \eta_d^< \left( \frac{p}{\sqrt{\lambda_d t}}\right)^{\alpha_d^<}
\right]^{\tau}}
\end{equation}
for small areas and its obvious modification for large areas. For
$T_0\to\infty$ one replaces $\eta_d$ and $\alpha_d$ by the primed
quantities and $c_d \to 2c_d$. The scaling analysis of the number
density of domain wall lengths is displayed in
Fig.~\ref{fig:perimeters-domain-scaling} for both initial conditions.
Once again we find a very good agreement between the analytic predictions
and the numerical data.
\begin{figure}[h]
\begin{center}
\psfrag{y}{\large\hspace{-10mm} ${(\lambda_d t)}^{3/2} \, n_d(p)$ }
\psfrag{x}{\large\hspace{-5mm} $p/ \sqrt{\lambda_d t}$}
\includegraphics[width=8cm]{other_29a.eps}
\includegraphics[width=8cm]{other_29b.eps}
\end{center}
\caption{(Colour online.) (Colour online.) Scaling of the time-dependent number density
of domain wall lengths evolving at $T=0$ from an initial condition at
$T_0=T_c$ (top) and $T_0\to\infty$ (bottom). The solid black lines
represent the theoretical prediction valid for $A/\lambda_h t<10$
and for $A/\lambda_h t>50$. The
origin of the isolated data points is the same as in
Fig.~\ref{fig:perimeters-hulls-scaling}.}
\label{fig:perimeters-domain-scaling}
\end{figure}
\subsection{Finite temperature evolution}
Once we analysed the statistics of perimeters in the
zero temperature dynamics we focus on the effects of a
finite working temperature. We briefly list
the results below without presenting the
data.
\subsubsection{The area-perimeter relations}
For large areas we find the same exponent as for zero
temperature coarsening that is also the initial condition exponent
($T_0\to\infty$ or $T_0=T_c$). This is reasonable since the large
structures are still `unaware' of the coarsening process and thus
retain the form they had in the initial configuration. For small
areas, instead, we see domain walls roughening due to thermal agitation
but it is hard to extract the value of the exponent $\alpha^<$
with sufficient accuracy.
\subsubsection{Perimeter number densities}
The scaling of the perimeter number densities and the functional form
for the scaling function predicted analytically describe the numerical
data with high precision once the values of the exponents $\alpha$,
the prefactors $\eta$, and the parameter $\lambda$ are modified to
take into account thermal agitation (for absolute area values larger
than $A\sim 10 A_0$, this value limits the range of areas where the
effect of thermal fluctuations is larger than the ones of coarsening).
The analytic prediction is very accurate in the region of
small coarsening domains,
$A/\lambda_{d,h} t < 10$ and $A>10A_0$ where the maximum is located,
and in the region of large coarsening domains, $A/\lambda_{d,h} t > 10$ and
$A>10A_0$, for both domains and hulls and the two initial conditions.
\section{Conclusions}
In this paper we studied the statistics and geometry of hull enclosed
and domain areas and interfaces during the non-equilibrium dynamics of
curvature driven pure coarsening in two dimensions. The analytical
part of our work relies on the Allen-Cahn equation derived from the
continuous Ginzburg-Landau field-theory in two-dimensions while the
numerical part of it dealts with Monte Carlo simulations of the
$2d$IM. Our main results are:
\newline
(i) We proved scaling of the various number densities studied.
\newline
(ii) We derived the exact number density of hull enclosed areas
and hull lengths; we obtained approximate expressions for the
number density of domain areas and domain wall lengths.
\newline
(iii) The geometrical properties and distribution of the time
dependent large structures (by large we mean much larger than the
average ones) are the ones of critical continuous percolation (for
all initial conditions equilibrated at $T_0>T_c$) and critical
Ising (for $T_0=T_c$). The long interfaces retain the fractal
geometry imposed by the equilibrium initial condition and the scaling
function of all number densities decay as power laws.
\newline
(iv) Instead, small structures progressively become regular
and the area-perimeter relation is $A\sim p^2$.
\newline
(v) We took into account the effects of a finite working temperature
by correctly eliminating purely thermal fluctuations and thus
correctly identifying the coarsening structures. The temperature
effect thus amounts to introducing the temperature dependence in the
prefactor in the growth law, $R(t) \sim [\lambda(T) t]^{1/2}$.
$\lambda(T)$ is a monotonically decreasing function of $T$ that
vanishes at $T_c$.
It is important to stress that our analytic results rely on the use of
the Allen-Cahn result for the velocity of an almost flat interface.
Thus, they would be expected to hold only in a statistical sense and
for large structures in the lattice model. Surprisingly, we
found with numerical simulations that the number density area
distributions in the $2d$IM match the analytic predictions for very
small structures, and even after a few MC steps evolution of a
critical Ising initial condition for which rather rough interfaces
exist.
Using the Allen-Cahn result and a variety of numerical measurements we
verified the well-known result for pure coarsening with non-conserved
order parameter: there is a characteristic growing length that
increases in time as $t^{1/2}$. The mesoscopic analysis presented
here allows us to demonstrate that the reason for the growth of the
characteristic length is the disappearance of small structures.
Our analytic results hold only in two dimensions. In Appendix
\ref{sec:oned} we summarized the behaviour of the distribution of
domain lengths in the Ising chain. As expected from scaling arguments,
this quantity scales with the typical domain length, $R(t) \sim
t^{1/2}$ but the form of the scaling function is very different from
the one in two dimensions: the pdf vanishes at zero scaling argument
($x=0$), it then increases linearly to reach a maximum and then
falls-off to zero exponentially. On the other hand, we cannot extend
the analytic argument to dimensions higher than two since the hull
enclosed volumes no longer decrease with time in a manner that does
not depend on their own volume~\cite{Comment}.
This paper and \cite{us} open the way to a number of related
studies. For instance, it would be interesting to extend the analysis
presented here to dynamic clusters of correlated spins (droplets),
that are known to describe thermal fluctuations close to the
transition (see ~\cite{coniglio} and ~\cite{Sator} for a review). These droplets are smaller
than the geometric clusters in which they are embedded because some of
the neighboring parallel spins are discarded (by a temperature
dependent criteria) for not being correlated (remember that even at
infinite temperature, where all spins are uncorrelated, there are
domains of parallel spins).
Two-dimensional coarsening with conserved order parameter is another
problem that deserves a careful study along these lines.
In~\cite{us} we include a preliminary analysis of the hull and domain
structure in the finite temperature dynamics of the bi-dimensional
random bond ferromagnetic Ising model after a quench from infinite
temperature. In such a disordered case a finite working temperature
is necessary to help the interfaces depin from pinning centers in the
quenched disordered potential through thermal activation. We first
computed the typical domain radius that scales the time-dependent
spatial correlation, $C(r,t) \sim f(r/R(t))$, in the scaling
regime. Due to the presence of quenched disorder $R(t)$ strongly
depends on temperature and the strength of randomness and it is slower
than the simple square root behaviour of the pure Ising case. We then
showed numerically that the number density of hull enclosed and domain
areas scale as $R^4(t) n_{h,d}(A,t) \sim g(A/R^2(t))$ for areas
satisfying $10^{-1} \stackrel{<}{\sim} A/R^2(t)$ and that are smaller
than the cut-off set by finite size effects. The effect of a
non-trivial typical radius $R(t)$ determined by the quenched disorder
can be tested in the intermediate regime, say $10^{-1}
\stackrel{<}{\sim} A/R^2(t)\stackrel{<}{\sim} 10^1$, where the quality
of the scaling plot is excellent. The scaling function $g(x)$ does
not depend on the disorder strength satisfying the hyper-scaling
hypothesis~\cite{BrayReview}. For smaller areas, say $A/R^2(t)
\stackrel{<}{\sim} 10^{-1}$, the contribution of thermal domains with
domain walls roughened by disorder is important. We shall give more
details on the domain morphology of the quenched disordered coarsening
problem in a separate publication~\cite{us-EPL}.
These results give an idea of the richness and complexity of
coarsening phenomena even in the absence of quenched randomness. We
expect them to be of help in understanding the fluctuating dynamics of
even more complex situations, like spin-glasses and
glassy problems~\cite{Chamon}, in
which the mere existence of a domain growth of two competing
equilibrium phases is not even established.
| {
"timestamp": "2007-11-08T11:55:55",
"yymm": "0706",
"arxiv_id": "0706.4314",
"language": "en",
"url": "https://arxiv.org/abs/0706.4314",
"abstract": "We study the distribution of domain areas, areas enclosed by domain boundaries (''hulls''), and perimeters for curvature-driven two-dimensional coarsening, employing a combination of exact analysis and numerical studies, for various initial conditions. We show that the number of hulls per unit area, $n_h(A,t) dA$, with enclosed area in the interval $(A,A+dA)$, is described, for a disordered initial condition, by the scaling function $n_h(A,t) = 2c_h/(A + \\lambda_h t)^2$, where $c_h=1/8\\pi\\sqrt{3} \\approx 0.023$ is a universal constant and $\\lambda_h$ is a material parameter. For a critical initial condition, the same form is obtained, with the same $\\lambda_h$ but with $c_h$ replaced by $c_h/2$. For the distribution of domain areas, we argue that the corresponding scaling function has, for random initial conditions, the form $n_d(A,t) = 2c_d (\\lambda_d t)^{\\tau'-2}/(A + \\lambda_d t)^{\\tau'}$, where $c_d=c_h + {\\cal O}(c_h^2)$, $\\lambda_d=\\lambda_h + {\\cal O}(c_h)$, and $\\tau' = 187/91 \\approx 2.055$. For critical initial conditions, one replaces $c_d$ by $c_d/2$ (possibly with corrections of ${\\cal O}(c_h^2)$) and the exponent is $\\tau = 379/187 \\approx 2.027$. These results are extended to describe the number density of the length of hulls and domain walls surrounding connected clusters of aligned spins. These predictions are supported by extensive numerical simulations. We also study numerically the geometric properties of the boundaries and areas.",
"subjects": "Statistical Mechanics (cond-mat.stat-mech)",
"title": "Domain growth morphology in curvature driven two dimensional coarsening",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9597620550745211,
"lm_q2_score": 0.7371581510799252,
"lm_q1q2_score": 0.7074964219954033
} |
https://arxiv.org/abs/2103.10476 | Smooth Aggregation for Difficult Stretched Mesh and Coefficient Variation Problems | Four adaptations of the smoothed aggregation algebraic multigrid (SA-AMG) method are proposed with an eye towards improving the convergence and robustness of the solver in situations when the discretization matrix contains many weak connections. These weak connections can cause higher than expected levels of fill-in within the coarse discretization matrices and can also give rise to sub-optimal smoothing within the prolongator smoothing phase. These smoothing drawbacks are due to the relatively small size of some diagonal entries within the filtered matrix that one obtains after dropping the weak connections. The new algorithms consider modifications to the Jacobi-like step that defines the prolongator smoother, modifications to the filtered matrix, and also direct modifications to the resulting grid transfer operators. Numerical results are given illustrating the potential benefits of the proposed adaptations. |
\section{Smoothed Aggregation Variations} \label{sec:sa variations}
We now outline four algorithm variations to address some of the difficulties associated with weak connections
and small aggregates.
\subsection{A 1-norm diagonal approximation}\label{sec:1norm diag approx}
As noted, the $\ensuremath{\bar{D}}\xspace^{-1}$ within the Jacobi iteration can be viewed as a diagonal approximation to
$\bar{A}^{-1}$. There are, however, alternatives, and one natural possibility is to replace $\ensuremath{\bar{D}}\xspace$ by
the diagonal matrix $\widetilde{D}$ where
$\widetilde{D}_{ii}$ is the sum of the absolute values of nonzeros
in the $i^{th}$ row of $\bar{A}$.
That is,
\begin{equation} \label{eq:absRowSum}
\widetilde{D}_{ii} = \sum_{j} | \bar{A}_{ij} | ~~,
\end{equation}
which is the $1-$norm of this $i^{th}$ row.
One could argue that this $1-$norm
choice for $\widetilde{D}_{ii}$ better captures the scaling of the entire row and that an iterative procedure based on
this will be less sensitive to the diagonal dominance properties of the matrix. This approximation
is not new, and is related to a variant of the well-known pressure-correction
algorithm, SIMPLE, that is referred to as SIMPLEC~\cite{doi:10.1080/01495728408961817}. As we will see, this relatively straight-forward change can have a
significant convergence effect.
Notice that
the entries of this $\widetilde{D}$ matrix are larger (assuming more than one nonzero per row) than those of $\ensuremath{\bar{D}}\xspace$, and so the entries of $\widetilde{D}^{-1} \bar{A}$ are smaller in magnitude than those of $\ensuremath{\bar{D}}\xspace^{-1}
\bar{A}$. Overall, however, it is the entries of $\frac{4}{3 \bar{\lambda}_m} \ensuremath{\bar{D}}\xspace^{-1} \bar{A}$ and
$\frac{4}{3 \widetilde{\lambda}_m} \widetilde{D}^{-1} \bar{A}$ that appear within the Jacobi iteration, and these will be
comparable when $\bar{A}$ is diagonally dominant, as $\widetilde{\lambda}_m $ will be smaller than $\bar{\lambda}_m $.
For example, when every row of $\bar{A}$ is given by the stencil
$$
\begin{matrix}
& -1 & \cr
-1 & ~~4 & -1 \cr
& -1 &
\end{matrix} ~~,
$$
then it is easy to see that
$\frac{1}{\bar{\lambda}_m} \ensuremath{\bar{D}}\xspace^{-1} \bar{A} = \frac{1}{\widetilde{\lambda}_m} \widetilde{D}^{-1} \bar{A}$.
This case corresponds to a constant coefficient periodic Poisson problem where
all diagonal entries of $\ensuremath{\bar{D}}\xspace$ are 4, all diagonal entries of $\widetilde{D}$ are 8,
$\bar{\lambda}_m = 2$, and $\widetilde{\lambda}_m = 1$. Clearly, this exact equality will hold whenever
$\bar{A}$ is defined by any circulant matrix.
Thus, the proposed $1-$norm diagonal modification will exactly reproduce the behavior
of the traditional smoothed aggregation method in this case. We normally expect to see similar SA-AMG convergence
behavior with either $\ensuremath{\bar{D}}\xspace$ or $\widetilde{D}$ when $\bar{A}$ is diagonally dominant.
For more general problems, however, we will see that the behavior can be significantly different.
One interesting consequence of this $\widetilde{D}$ definition is that the magnitude of $\widetilde{\lambda}_m $,
the maximum eigenvalue associated with $\widetilde{D}^{-1} \bar{A}$, is always bounded by 1,
again as a consequence of the Gershgorin circle theorem. This implies that one can
omit the eigenvalue calculation (which is problematic for the wind
simulation in \S~\ref{sec:exawind}) and consider using
$1$ as an estimate for $\widetilde{\lambda}_m $.
\subsection{A Safe Guarded Diagonal Approximation}\label{sec:safeguard}
While this $\widetilde{D}_{ii}$ choice works well in practice, we propose one additional modification that is
motivated by cases when the diagonal entry within a row of $\bar{A}$ is much larger than the
sum of the magnitudes of the row's off-diagonal entries.
While this cannot happen for rows where the sum of
the entries is zero,
it
might occur for matrix rows associated with or near Dirichlet boundary conditions.
To better understand this large diagonal scenario, consider the extreme case
when $\bar{A}$ is in fact a diagonal matrix. Then, $\widetilde{D}^{-1} \bar{A} = I $, $\widetilde{\lambda}_m$ is 1,
and the prolongator smoothing step \ref{eq:prolongator_smoothing} becomes
$$
P = \left (I - \frac{4}{3\lambda_m}\widetilde{D}^{-1} \bar{A} \right) = \left (I - \frac{4}{3} I \right) \ensuremath{{P}^{(t)}}\xspace = -\frac{1}{3} \ensuremath{{P}^{(t)}}\xspace .
$$
If instead the damping parameter had been defined as $\omega = 1 / ( \widetilde{\lambda}_m)$, which is
not an unreasonable choice, then $P$ would be identically zero.
Essentially, the Jacobi iteration converges too rapidly to the $P = 0$ solution when the diagonal is
much larger than the sum of the magnitudes of the row's off-diagonals. Instead of only reducing high frequencies,
the whole spectrum is reduced, which is undesirable. If only a subset
of rows have a very large diagonal, it would be better
to use the corresponding row of $\ensuremath{{P}^{(t)}}\xspace$ or to somehow limit the damping. While this could be
accomplished by setting $\omega$ to a suitably small value, this would have the unintended
consequence of limiting the smoothing effect for \textit{all} rows, even those that are not so diagonally
dominant. Instead, we prefer modifying or boosting the value of $\widetilde{D}_{ii}$ for any row
where the modified Jacobi step significantly reduces low frequencies. This effectively corresponds
to augmenting the definition of $\widetilde{D}_{ii}$ with a safeguard. To do this, we must have a
criterion to detect the low frequency reduction.
To understand a possible remedy for low frequency reduction,
consider the effect of the prolongator smoother step on just one row of the matrix. Specifically, assume that
vertex $i$ defines the center or root point of the $k^{th}$ aggregate. That is, $i \in {\cal A}_k $ and
$j \in {\cal A}_k $ for any $j$ such that $\bar{A}_{ij} \neq 0$ . If we additionally take $\widetilde{\lambda}_m = 1$,
then the $i^{th}$ row of the resulting prolongator is given by
$$
P_{ij} = 1 - \frac{4}{3} \frac{s_i}{\widetilde{D}_{ii}},
$$
where $s_i$ denotes the sum of $\bar{A}$'s entries in the $i^{th}$ row.
Note that if $s_i=0$ and $\widetilde{D}_{ii} \neq 0$, then $P_{ij} = 1$. One can argue that it is natural for a prolongator
basis function to have a value of $1$ at the aggregate's central node and to then decay smoothly toward
zero. This is analogous to the use of injection to interpolate points that are co-located on the fine
and coarse grids. Here, a coarse unknown associated with an aggregate is viewed as being co-located
with the root node. As the basis function should be largest at the root node, it should not have a negative
value or be too small at this root vertex.
In other words, we seek to enforce the condition that
\[
P_{ij} \ge \sigma,
\]
where $i$ is again the root node of the $j^{th}$ aggregate
and $\sigma$ is the minimum acceptable value for the prolongator basis function at the root vertex.
This implies that $\widetilde{D}_{ii}$ must be chosen
such that
\[
1 - \frac{4}{3} \frac{s_i}{\widetilde{D}_{ii}} \ge \sigma,
\]
or
\begin{equation} \label{eq:not too small}
\frac{4 s_i}{3 (1 - \sigma)} \le \widetilde{D}_{ii} .
\end{equation}
Notice that when $s_i = 0$ and $\sigma < 1$, any non-negative $\widetilde{D}_{ii}$ will satisfy this condition.
Thus, \eqref{eq:absRowSum} can be used to define $\widetilde{D}_{ii}$.
When $s_i$ is not zero, an initial value computed via \eqref{eq:absRowSum} is checked to see
that it satisfies \eqref{eq:not too small}. If this second condition is not satisfied, \eqref{eq:absRowSum}
is discarded and instead the smallest $\widetilde{D}_{ii}$ satisfying \eqref{eq:not too small} is used to
define $\widetilde{D}_{ii}$.
While the precise choice of $\sigma$ is not obvious, we have found experimentally that
$\sigma = 1/3$ works well.
For this choice of $\sigma$, \eqref{eq:not too small} becomes
\begin{equation} \label{eq:not 2 small}
2 s_i \le \widetilde{D}_{ii}
\end{equation}
Thus, we first compute $\widetilde{D}_{ii}$ using \eqref{eq:absRowSum}. For $\widetilde{D}_{ii}$ that are identically zero
(due to an entire zero row of $\bar{A}$) we set $\widetilde{D}_{ii} = 1$, and for those remaining rows violating \eqref{eq:not 2 small}
we set $\widetilde{D}_{ii} = 2 s_i$.
While the arguments were motivated by considering $i$ to be a root node of an aggregate, we apply this
criteria for all rows in the matrix, regardless as to whether or not the node is a root node. At non-root nodes, the
associated $P_{ij}$ can be smaller than $\sigma$ due to the fact $s_i$ does not reflect the $\bar{A} \ensuremath{{P}^{(t)}}\xspace$
product.
\subsection{Alternative lumping strategies} \label{sec:alt lumping}
As already noted,
near null space vectors should be accurately represented in the range space of the interpolation operator.
If the null space of $A$
coincides with that of $\bar{A}$ and is also contained within the range space of $\ensuremath{{P}^{(t)}}\xspace$,
then it
will additionally be contained within the range space of the smoothed prolongator.
To force the
two null spaces to coincide, $\bar{A}$'s entries must be modified to account for dropped nonzeros.
This is traditionally accomplished by only changing the diagonal.
As discussed, any dropped off-diagonal $A_{ij}$ is simply added
to $A_{ii}$ to define $\bar{A}_{ii}$. Unfortunately, this simple process can drastically alter the properties
of the resulting filtered matrix. In extreme cases, it is possible that $A_{ii}$ and $\bar{A}_{ii}$
have opposite signs.
More generally, the diagonal dominance properties of a row might change. That is,
we could have
$$
{\cal D}(\bar{A}_i) \gg {\cal D}(A_i)
~~\mbox{where}~~
{\cal D}(A_i) = \frac{\sum_{j\ne i} |A_{ij}|}{|A_{ii}|} .
$$
Here, $A_i$ refers to the $i^{th}$ row of $A$ and
${\cal D}()$ measures the magnitude of off-diagonal entries relative to the diagonal.
Notice that ${\cal D}(A_i) = 1$ when all off-diagonals are negative and the sum of $A_i$'s entries are zero.
In this case, any reasonable lumping strategy will result in ${\cal D}(\bar{A}_i) = 1$.
However, this will not be the case when there are both positive and negative off-diagonals or when $A_i$'s entries
do not sum to zero (e.g., at a boundary).
For the most part, small values of ${\cal D}()$ are preferred by a damped Jacobi iterative method
(i.e., the prolongator smoothing step) as
its convergence rate is generally more rapid for matrices with small off-diagonal entries relative to
the diagonal entries. This suggests that it might be effective to consider a
strategy that restricts or limits the lumping of terms to the diagonal to
avoid significant growth in the resulting ${\cal D}(\bar{A}_i)$. There are many possible such strategies.
We now describe a scheme that is primarily oriented toward scalar diffusion-type PDEs,
and so may not be appropriate for other operators.
For scalar diffusion-like PDEs, one can argue that positive off-diagonal entries are somewhat irregular.
To see this, consider the simplified prolongator smoother step $ u = (I - \ensuremath{\bar{D}}\xspace^{-1} \bar{A}$)v .
When all off-diagonals are negative and when the constant
is in the null space of $\bar{A}$, then each $u_i$ is just a weighted average of the $v_i$'s within
its immediate neighborhood.
In this way, the relaxation process mimics a diffusion process associated with a heat equation.
When some off-diagonal entries are instead positive, then some weights will be negative, which no longer
resembles a diffusion process. This can be made more rigorous by consider the relationship
between Jacobi iterations and time marching for ordinary differential equations~\cite{OlScTu10}.
Thus, a possible lumping algorithm might make decisions based on the sign of matrix entries.
Our overall lumping strategy considers modifying or perturbing the retained positive off-diagonals,
the diagonal, or the retained negative off-diagonals {\it in this order of preference}
when the perturbation is negative. The general aim is to
enforce ${\cal D}(\bar{A}_i) \le \tau {\cal D}(A_i)$ where $\tau$ is a user-supplied growth factor.
\begin{algorithm} \label{algo:spreadLumping}
\begin{tabbing}
\hskip .6in \= \hskip .3in \= \hskip .3in \= \hskip .3in \= \kill \\ \linefill \\
$\bar{A}_{i} = $\textsf{Lump\_AvoidSmallDiag}( $A_i, {\cal R}_i, \tau$)\\[2pt]
~~Input: \\
~~~~~${A}_i$ \> $i^{th}$ row of matrix with entries to be dropped \\
~~~~~${\cal R}_i$ \> set of column indices in $i^{th}$ row to be removed\\
~~~~~$\tau$ \> tolerance indicating that ${\cal D}(\bar{A}_i)$ should not exceed $\tau {\cal D}(A_i) $ \\
~~Output: \\
~~~~~$\bar{A}_{i} $ \> matrix row where $\bar{A}_{ij} = 0 ~\mbox{for}~ j \in {\cal R}_i$ and
$\bar{A}_i v = A v $ where $v$ is a constant vector
\\[-4pt] \linefill \\[5pt]
~1. Let $r_i \leftarrow \sum_{k \in {\cal R}_i } A_{ik} $ \\[2pt]
~2. {\bf if} $r_i > 0 ~{\bf then}~ \bar{A}_{ii} \leftarrow A_{ii} + r_i$ ~~~\codeComment{decreases ${\cal D}(\bar{A}_i)$}\\
~3. {\bf else \{} \\
~4. \> Let ${\cal K}_i^+ \leftarrow \{ k ~|~~ A_{ik} > 0 ~\land~ k \neq i ~\land~ k \notin {\cal R}_i \} $ \\[2pt]
~5. \> Let ${\cal K}_i^- \leftarrow \{ k ~|~~ A_{ik} < 0 ~\land~ k \neq i ~\land~ k \notin {\cal R}_i \} $ \\[2pt]
~6. \> Let $\kappa_i^+ \leftarrow \sum_{k \in {\cal K}_i^+} A_{ik} $;~~~$\kappa_i^- \leftarrow \sum_{k \in {\cal K}_i^-} A_{ik}$;\\[2pt]
~7. \> {\bf if} $|r_i| \le \kappa_i^+$ {\bf then}~ ${\bar{A}}_{ij} \leftarrow {A}_{ij} (1 + \delta_{i})$ for $j \in {\cal K}_i^+$ where $ \delta_i \leftarrow r_i / \kappa^+ $\\
~8. \> {\bf else \{} \\
~9. \> \> ${\bar{A}}_{ij} \leftarrow 0$ for $j \in {\cal K}_i^+$ \codeComment{zero out the ${\cal K}_i^+$ by distributing a} \\[3pt]
10. \> \> $\hat{r}_i \leftarrow r_i + \kappa_i^+ $ ~~~~~~~~\codeComment{portion of $r_i$ ($= \kappa_i^+$) to them} \\
11. \> \> {\bf if} ${\cal K}_i^- == \emptyset$ {\bf then} redistribute to ${\cal K}^+$ if possible or if not \\
12. \>\> ~~~~~~~~~~~~~~~~~~~~~~~~possible do not modify row $i$ and return \\
13. \> \> {\bf else \{} \\
14. \> \> \> find largest positive $r_i^* < \min(d_{ii},|\hat{r}_i|)$ such that ${\cal D}(\bar{A}_{i}) \le \tau {\cal D}(A_i) $\\
15. \> \> \> define $\bar{A}_{i}$ such that its only nonzero values are \\
16. \> \> \> ~~~~~~~~ $\bar{A}_{ii} \leftarrow {A}_{ii} - r_i^* $\\
17. \> \> \> ~~~~~~~~ $\bar{A}_{ij} \leftarrow {A}_{ij} (1 + \delta_{i})$ for $i \in {\cal K}_i^-$ \\
18. \> \> \> ~~ where $ \delta_i \leftarrow (\hat{r}_i + r_i^* )/ \kappa^- $ \\
19. \> \> {\bf \}} \\
20. \> {\bf \}} \\
21. {\bf \}} \\
\end{tabbing}
\caption{Alternative lumping procedure for prolongator smoothing step}
\end{algorithm}
A detailed algorithm description is given in Figure~\ref{algo:spreadLumping}. The algorithm is
supplied a set of indices ${\cal R}_i$ denoting the nonzero columns that should be {\it removed}
from the $i^{th}$ row. The sum of these entries $r_i$ must be then distributed to the {\it kept} entries of $\bar{A}_i$.
If this sum is positive, then only the diagonal is modified in \textsf{Line 2}
as this lowers ${\cal D}(\bar{A}_i)$. If instead $r_i$ is negative, then more care is necessary.
We first split the set of kept
indices into two subsets ${\cal K}^+$ and ${\cal K}^-$ corresponding to
entries that have positive $A_{ij}$ values or negative $A_{ij}$ values, respectively.
The sum of the nonzero values associated with these two sets is denoted by ${\kappa_i}^+$ and ${\kappa_i}^-$,
respectively. \textsf{Line 7} corresponds to the case when all of the lumping can be distributed
to the positive kept entries without creating any new negative entries. If this is impossible,
we distribute a portion of $r_i$ equal to $-\kappa_i^+$ to the ${\cal K}^+$, effectively zeroing them out.
That is, we prefer not creating new negative entries as this might fundamentally change the equation's character.
For the remaining $\hat{r}_i = r_i + \kappa_i^+$,
we seek in \textsf{Line 14} the largest magnitude perturbation,
$|r^*|$, to the diagonal
that does not violate the ${\cal D}()$ growth restriction.
The remaining $\hat{r}_i - r_i^*$ is then distributed proportionally to the negative kept off-diagonals.
In many cases,
$r_i^* = \hat{r}_i$, so $\hat{r}_i$ is lumped entirely to the diagonal. When this is not true, $r_i^*$ is generally given by
$$
r_i^* = \frac{\hat{r}_i + \kappa_i^- + \tau {\cal D}(A_i) A_{ii} }{1-\tau {\cal D}(A_i)} .
$$
This is obtained by some algebraic manipulations after first setting ${\cal D}(\bar{A}_i) = \tau {\cal D}(A_i)$ and
recognizing that the sum of the absolute values of $\bar{A}_i$'s off-diagonals is $(-\kappa_i^-) (1 + \delta)$ while its
diagonal is $A_{ii} + r_i^*$.
However, safe-guards must be added for situations where there is no suitable value of $r_i^*$
satisfying the ${\cal D}()$ growth restriction. This might occur if the set ${\cal K}^-$ is empty.
If it is instead possible to satisfy the ${\cal D}()$ growth restriction by further lumping to ${\cal K}^+$,
then this is done even though these off-diagonals now become negative. Otherwise, if there are no kept
off-diagonal entries in the row,
we skip the perturbation entirely, no longer preserving the row sums in these problematic
rows.
\subsection{Prolongator Constraints} \label{sec:constraints}
From a geometric multigrid perspective, one can argue that the entries of $P$ should lie
between $0$ and $1$ inclusive. For example, consider the vector $u = P e^{(j)}$ where $e^{(j)} $ is a
canonical basis vector with only one nonzero entry, the $j^{th}$ element, that is set to one. It is clear
that all entries of $u$ should be positive and not greater than one for any
sort of geometric interpolation scheme (as opposed to an extrapolation scheme). For smoothed aggregation,
this connection is a little less apparent. Obviously, the tentative prolongator $\ensuremath{{P}^{(t)}}\xspace$
satisfies these constraints when the null space of $A$ is given by the vector of all ones.
As the objective of the prolongator smoother step is to produce low energy grid transfer basis functions,
it can also be argued that these smoothed basis functions should decay smoothly from the peak
value to zero. Thus, these basis functions should not be negative anywhere.
Though somewhat less obvious, the peak will typically be either one or less than one.
Specifically, smoothed prolongator basis functions and tentative prolongator basis functions
will coincide for vertices that are not part of the aggregate boundary. This is a property of the
null space of $\bar{A}$ being a constant function in the case of
the Laplace operator and was discussed for ideal aggregates in \S~\ref{sec:safeguard}.
The remaining nonzero basis function entries will typically
be less than one due to the energy minimization and smooth decay properties just mentioned.
When instead prolongator values do not lie between $0$ and $1$ inclusive, it is often
an indicator that the smoothing of some basis functions is sub-optimal.
In these cases, one can consider enforcing a condition that all $\bar{P}_{ij}$ lie
between $0$ and $1$. Specifically, one can construct a minimization problem for the $i^{th}$
row of $\bar{P}$
$$
\left\{
\begin{aligned}
&\bar{P}_i = \argmin_{\widetilde{P}_i} \; || \widetilde{P}_i - P_i ||_2 \\
&\;\,~~~~~~\text{\small subject to}\hspace{1em}
\widetilde{P}_{ij} = 0 ~\mbox{if}~ P_{ij} = 0,~~~
\widetilde{P}_{ij} \ge 0,~~~ \widetilde{P}_{ij} \le 1,~~~
\widetilde{P}_i v = P_i v
\end{aligned}
\right .
$$
where $v$ is the vector of all ones.
That is, find a new prolongator row that is closest to $P_i$ and satisfies the two bound constraints, has
a sparsity pattern not extending beyond $P_i$'s pattern, and where the sum of $P_i$'s entries and $\bar{P}_i$'s entries are identical.
If the row sum of $P_i$ is negative, then there is no feasible solution to this minimization problem.
If the $P_i$ row sum is zero, then the only solution is that $\bar{P}_i$ is identically
zero. There is also no feasible solution when the row sum is greater than the number
of nonzeros in the sparsity pattern of $P_i$. When the minimization problem has a
feasible solution, it can be obtained by the procedure described in
Algorithm~\ref{alg:post process}. The algorithm itself is relatively
straightforward. On each pass of the while loop, we take the worst
constraint violators on both ends (both negative entries and entries which are
greater than one), pin them to their constraint values, and then split the combined change in values
among all remaining non-violating entries. This will then be repeated
until no constraint violating entries remain.
When the constraints cannot be satisfied within the $i^{th}$ row, then we simple take $\bar{P}_{i} = \ensuremath{{P}^{(t)}}\xspace_{i}$.
\begin{algorithm}
\begin{algorithmic}
\caption{$\bar{P}_i$=\textsf{Constrain\_One\_P\_Row}$(P_i)$ \label{alg:post process}}
\State Let $\bar{P}_{i} = P_{i} $
\State Let $w = \{ k ~|~~ \bar{P}_{ik} \neq 0 \} $ \;
\State \While{$(\min_k \bar{P}_{ik} < 0 )~ \land ~(\max_k \bar{P}_{ik} > 1 )$}{
\State $\hat{k} \gets \argmin_{k} \bar{P}_{ik} $\;
\State $\tilde{k} \gets \argmax_{k} \bar{P}_{ik}$\;
\State $\delta \gets 0$\;
\State \lIf{$\bar{P}_{i\hat{k}} ~<~ 0 $} { $\delta \gets \delta +\bar{P}_{i\hat{k} }$; ~~$\bar{P}_{i\hat{k} } \gets 0$; ~~$w \gets w \setminus \hat{k}$}\;
\State \lIf{$\bar{P}_{i\tilde{k}} ~>~ 1 $} { $\delta \gets \delta +\bar{P}_{i\tilde{k}} - 1$; ~~$\bar{P}_{i\tilde{k}} \gets 1$; ~~$w \gets w \setminus \tilde{k}$}\;
\State $\bar{P}_{ik} \gets \bar{P}_{ik} + \delta/|w| $ ~for~ $k \in w$\;
}
\end{algorithmic}
\end{algorithm}
\subsection{Further sparsification}\label{sec:root_node}
To understand a possible further sparsification of the smoothed aggregation prolongator operator,
we first review the aggregation process within smoothed aggregation.
As noted, smoothed aggregation applies an algorithm to the graph of $\bar{A}$ to construct aggregates
${\cal A}_j$ such that each fine mesh vertex belongs to only one aggregate. The basic idea is that a root node
is first chosen and then an initial aggregate is defined as the root node and all of its strong neighbors. Each root
node is chosen among vertices that have not yet been aggregated and are not adjacent (via strong connections) to
any existing already
aggregated vertex. This aggregation procedure is repeated until it is no longer possible to find such a root
node as all unassigned vertices are adjacent to assigned vertices. At this juncture, some heuristics are needed
to assign these remaining unassigned vertices by either creating new aggregates or enlarging existing aggregates.
This implies that most aggregates are
composed of a central root node and its strong neighbors, thus the shape of the aggregates is primarily governed
by the strong neighbors of the root node. A typical aggregate will have a diameter of length 3. Figure~\ref{fig:hotdog example}
illustrates two aggregate scenarios on a regular mesh. In the leftmost image, all connections are strong
and the corresponding aggregates happen to be perfect squares. In the rightmost image,
only the vertical connections are strong with the exception of a few horizontal edges that in
this contrived example never coincide with edges emanating from a root node. Once again, the aggregates are
perfect, consisting of 3 points aligned in the vertical direction. That is, the aggregate shapes are
determined by the root nodes, which only have strong vertical connections. The main issue is that the horizontal
connections shown in the rightmost image will lead to nonzero fill-in within the coarse level discretization
matrix due to the prolongator smoothing step.
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.3]{figs/boxes.jpg} ~~~~~~~~~~~~
\includegraphics[scale=0.3]{figs/hotdogs.jpg}
\caption{Two aggregation examples (aggregates indicated by blue enclosures) on a 2D structured mesh. Each red arrow indicate a strong $\bar{A}_{ij}$ connection for vertex $i$ at the arrow base and vertex $j$ is the closest node in the direction pointed to by the arrow.}
\label{fig:hotdog example}
\end{figure}
Specifically, additional nonzeros connections arise between coarse vertices associated with non-neighboring aggregates (distance two aggregates) in the horizontal direction.
In particular, it is easy to show that for nontrivial $\bar{A}$ matrices a nonzero $(i,j)$ entry
occurs in the coarse matrix discretization whenever $(\ensuremath{{P}^{(t)}}\xspace_{.,i})^{T} \bar{A}^T A \bar{A} (\ensuremath{{P}^{(t)}}\xspace)_{.,j}$ is
nonzero where $\ensuremath{{P}^{(t)}}\xspace_{.,j}$ refers to the $j^{th}$ column of $\ensuremath{{P}^{(t)}}\xspace$. This will certainly occur if there
is a distance 3 path in the filtered graph (along the red arrows in Figure~\ref{fig:hotdog example}).
More generally, fill-in between distant aggregates can occur due to a conflict between the
characterization of strong connections between the root node and the non-root nodes. Specifically, the root node indicates that connections to certain neighboring aggregates
are weak
while some member of the root node's aggregate
has a strong connection to this very same aggregate.
To reduce fill-in, we can look for conflicts and re-label some conflicting connections.
As our conflict characterization is based on aggregate choices, it is most practical to only
consider non-root strong connections for re-labeling
as these do not alter the definition of the already-chosen aggregates.
This re-labeling corresponds to a further sparsification of $\bar{A}$ that is performed immediately
preceding the prolongator smoother step. This sparsification occurs aggregate-by-aggregate. First,
the weak connections of the root node are examined to determine the neighboring aggregates (in the graph
of $A$) associated with these weak connections. If these weak-neighbor aggregates have no strong connections
to the root node, then these aggregates are put into a set {\sf CandidatesForPruning}.
Second, we examine all the non-root vertices in the aggregate looking for
strong connections to any vertex within each {\sf CandidatesForPruning} aggregate. If there is just one strong connection to a particular aggregate in {\sf CandidatesForPruning}, this strong connection is re-labeled as weak and dropped from the $\bar{A}$ that will be used in the prolongator smoothing step. The diagonal entry is modified
to reflect the dropped entry following the usual method.
Thus, strong connections to each aggregate in {\sf CandidatesForPruning} are retained if there
are multiple strong connections but dropped if there is only one strong connection.
This leads to a further sparsification of $\bar{A}$ that may now be non-symmetric even if
$\bar{A}$ is symmetric.
To remedy this, all $(j,i)$ entries are dropped if the associated $(i,j)$ entry was dropped in the re-labeling phase.
This restores the symmetry in the resulting filtered
matrix, denoted as $\widetilde{A}$, that is then used in the prolongator smoothing step. As the
final prolongator is now sparser, we can expect that the amount of fill-in will be reduced on coarse
level matrices. However, this may also cause the convergence rate to be somewhat slower.
\section{Rand Cube Input}\label{appendix:randcube}
The results from \S~\ref{sec:results_randcube}, were generated from
meshes using the following Pamgen \cite{Pamgen} template. The mesh is
uniform in $x$ and $y$ and stretched with $100:1$ in the
$z$-direction. The nodes are each given a random perturbation up to
$20\%$ of the distance to the neighboring node in each direction.
The random number generator was seeded with 50 different seeds in order to
generate the meshes considered.
\begin{verbatim}
mesh
brick
zmin = 0.0
xmin = 0.0
ymin = 0.0
numz 1
zblock 1 1.0 interval 60
numx 1
xblock 1 1.0 interval 60
numy 1
yblock 1 1.0 interval 60
end
set assign
sideset, ilo, 1
sideset, jlo, 2
sideset, klo, 3
sideset, ihi, 4
sideset, jhi, 5
sideset, khi, 6
end
user defined geometry transformation
'
outxcoord = (inxcoord + 0.2*drand()/60)*1.;
outycoord = (inycoord + 0.2*drand()/60)*1.0;
outzcoord = (inzcoord + 0.2*drand()/60)*100.0;
'
end
end
\end{verbatim}
\section{Triaxially Stretched Cube Input}\label{appendix:stretchcube}
The results from \S~\ref{sec:results_stretchcube}, were generated from
meshes using the following Pamgen \cite{Pamgen} template. The
elements linear vary in size in each dimension, depending on the
parameters \textsf{KX}, \textsf{KY}, and \textsf{KZ}, which must be
substituted into the input deck below (quantities inside braces are replaced).
\begin{verbatim}
mesh
brick
zmin = 0.0
xmin = 0.0
ymin = 0.0
numx 1
xblock 1 {3.0*(_KX_+1)}, first size .1, last size {_KX_/10}
numy 1
yblock 1 {3.0*(_KY_+1)}, first size .1, last size {_KY_/10}
numz 1
zblock 1 {3.0*(_KZ_+1)}, first size .1, last size {_KZ_/10}
end
set assign
sideset, ilo, 1
sideset, jlo, 2
sideset, klo, 3
sideset, ihi, 4
sideset, jhi, 5
sideset, khi, 6
end
end
\end{verbatim}
\section{Conclusion} \label{sec:conclusion}
In this paper we have presented four new algorithmic variants to SA-AMG that focus on improving the smoothed prolongator
grid transfer, especially for
problems with many weak connections. Such systems commonly arise in practice, and can lead to poor SA-AMG performance,
which can manifest as high operator complexity, increased iteration counts, and even failure to converge (due to iteration
matrices with negative eigenvalues).
Whereas other SA-AMG research has focused on developing new strength-of-connection measures to mitigate these issues, we
have assumed a standard scalar strength measure is utilized, and in this paper present algorithms aimed at improving the
final smoothed prolongator. These algorithms are algebraic in nature and build naturally on the existing SA-AMG machinery.
We have demonstrated the efficacy of these new algorithms on a suite of problems that are challenging for standard SA-AMG to solve:
manufactured scalar Poisson problems with severe variable mesh stretching and lack of diagonal dominance, a standard oil reservoir benchmark, and
linear systems arising from a low-Mach CFD application.
The main take-away is that the four new variants generally yield improvements over standard SA-AMG. The \textsf{1Norm}\xspace and \textsf{OffLmp}\xspace
variations very rarely take more iterations that traditional SA-AMG. While there are cases where the convergence behavior
is similar to that of SA-AMG, there are other cases where \textsf{1Norm}\xspace and \textsf{OffLmp}\xspace are significantly faster and more robust
than SA-AMG. The results with \textsf{Cnstrnt}\xspace are a bit mixed. Sometimes it helps dramatically but other times it is not so
robust. The \textsf{Sprsfy}\xspace results do help a modest amount with the multigrid operator complexity, but in most cases
convergence does suffer. However, we do note that \textsf{Sprsfy}\xspace was robust on the random cube problem, which was not true
for SA-AMG.
Exploring the use of these algorithms in the context of a new weak-connection threshold approach will be the subject of a forthcoming
paper, where \textsf{Sprsfy}\xspace's ability to reduce multigrid operator complexity is more significant.
Another potential topic for future research is the adaptation of one or more of these algorithms to systems of PDEs.
\subsection{Prolongator Smoothing and Small Diagonal Entries }\label{sec:motivation}
We now motivate \S\ref{sec:sa variations} by considering some possible prolongator smoothing shortcomings.
The prolongator smoothing step
corresponds to one iteration of the damped Jacobi method applied to the matrix equation
\begin{equation} \label{eq:energy minimization}
\bar{A} P = 0,
\end{equation}
starting with an initial guess of $P=\ensuremath{{P}^{(t)}}\xspace$ and $\bar{A}$ given by \eqref{eq:filteredA}.
Here, we now drop the subscript $\ell$ to simplify the notation for the remainder of the paper.
Of course, $P = 0$ is a trivial solution
as is $P = C$ when $\bar{A}$'s null space is given by the space of constant vectors. Here, $C$ is a
$n \times m$ dense matrix with
$c_{ik} = c_{q k}$ for all $i$ and $q$ such that $ 1 \le i,q \le n $ and each $k$ such that $1 \le k \le m$.
While repeated Jacobi iterations
converge to something uninteresting,
one Jacobi step extends the nonzero support of the interpolation basis functions by one in the graph of $\bar{A}$.
When suitably damped, it also reduces the energy of these basis functions in the norm defined by
$||p||_{\bar{A}} = \sqrt{p^T \bar{A} p}$ when $\bar{A}$ is symmetric positive definite.
Most AMG convergence theories rely on some bound for the energy of the interpolation basis functions.
$\ensuremath{\bar{D}}\xspace^{-1}$ can be viewed as an inexpensive approximation to
$\bar{A}^{-1}$ that will reduce high frequencies in $\ensuremath{{P}^{(t)}}\xspace$'s basis functions when used
within a Jacobi step.
Undamped Jacobi is guaranteed to converge when $\bar{A}$ is strictly diagonally dominant, i.e., the magnitude of the diagonal
entry in every row is greater than the sum of the absolute value of all other nonzeros in that row.
When $\bar{A}$ is symmetric positive definite but not strictly diagonally dominant, damping may
be needed to ensure that the spectral radius
of $I - \omega \ensuremath{\bar{D}}\xspace^{-1} \bar{A} $ is less than one. For smoothed aggregation the damping parameter is
typically chosen as $\omega = 4 / ( 3 \bar{\lambda}_m)$ and is based on a Chebyshev minimization principle~\cite{Br1997}.
For strictly diagonally dominant matrices, it is easy to see that $\bar{\lambda}_m < 2$ using the
Gershgorin circle theorem. When the matrix is not strictly diagonally dominant,
it is possible that $\bar{\lambda}_m \gg 2$ or even undefined if some $\ensuremath{\bar{D}}\xspace_{ii} = 0$.
Unfortunately, $\bar{\lambda}_m $ might be large even if only one row violates the strictly diagonally
dominant condition. Obviously, the Jacobi step has little effect when $\omega$ is small,
which occurs for large $\bar{\lambda}_m $.
Some discretization schemes can produce diagonally dominant matrices\footnote{Diagonally dominant
matrices are defined by replacing the {\it greater than the sum} condition in the strictly diagonally
dominant definition by a condition {\it greater than or equal to the sum}.}
where ${\lambda}_m \le 2$.
However, matrices arising from finite element discretization are generally not diagonally dominant,
though often ${\lambda}_m$ is still not much larger than 2. Unfortunately,
dropping might lead to a
$\bar{\lambda}_m$ that is much larger than ${\lambda}_m$. In fact, a $\ensuremath{\bar{D}}\xspace_{ii}$ might even
become zero or negative.
For example, a nodal linear finite
element discretization of a Poisson operator on a mesh where all elements are
$1 \times 1 \times h_z$ hexahedrons produces matrices with 27-point
interior stencils. After scaling by $36 h_z$, these
identical stencils are given by
$32+64 h_z^2$ (diagonal entry),
$-1 - 2 h_z^2$ (8 neighbors on cell corners),
$-4- 2 h_z^2$ (8 neighbors sharing a $x-z$ or $y-z$ face),
$16 h_z^2 - 16$ (2 neighbors sharing a $z$ edge),
$2 - 8 h_z^2$ (4 neighbors sharing a $x-y$ face),
and $8-8 h_z^2$ (4 neighbors sharing a $x$ or $y$ edge).
When $h_z = \sqrt{7}/2$, these stencils take on the values
$144 , -4.5 , -7.5 , 12 , -12, ~\mbox{and}~ -6$.
For $h_z > \sqrt{7}/2$, the positive off-diagonal entry becomes the largest in magnitude value. Thus, any magnitude-based
dropping criteria that employs a threshold that happens to retain the largest entry but drops all others will
result in a negative
$\bar{A}_{ii}$. This follows from the fact that $\bar{A}$'s rows all sum to zero.
Further, small diagonal entries can be even more problematic for non-symmetric systems.
While there may not be a damping parameter to guarantee Jacobi convergence
for non-symmetric matrices, it is still effective for matrices that are {\it close} to
symmetric positive definite. In a wind simulation that will be
described in \S~\ref{sec:exawind},
the discretization matrix includes a symmetric sub-block of only slightly smaller dimension than the
entire matrix. However, a few of the non-symmetric rows/columns have small diagonal entries,
so that the non-symmetric part of $\bar{A}$ is emphasized in the matrix $\ensuremath{\bar{D}}\xspace^{-1} \bar{A}$.
We have observed that this causes
severe eigenvalue convergence problems and very poor $\bar{\lambda}_m$ estimates
coming from the typical power method used to provide these estimates.
\section{Introduction} \label{sec: intro}
The smoothed aggregation algebraic multigrid (SA-AMG) algorithm was originally proposed over twenty years ago as an effective and scalable
solution strategy for linear systems arising from discretized elliptic partial differential equations (PDEs)~\cite{VaMaBr96,VaBrMa01}.
The basic form of the original algorithm has been employed without major mathematical modification within numerous applications to tackle a wide
range of non-trivial problems. While generally successful, convergence difficulties can arise for some complex applications, even for
matrices coming from elliptic PDEs.
Multigrid methods, including SA-AMG, are based on the idea that simple relaxation methods such as Jacobi generally smooth high frequency errors
and that these smoothed errors can then be accurately represented and more efficiently reduced by relaxation iterations on a coarser grid
representation of the linear system. This paper considers modifications to the SA-AMG method, targeting some potentially vulnerable
components that tend to be more fragile when irregular or anisotropic coarsening is needed. While simple relaxation methods
generally smooth errors, they do not necessarily do so in a uniform or isotropic fashion. That is, errors after relaxation may be
much smoother in certain directions than in others. In these cases, algebraic multigrid (AMG) coarsening
must only occur in directions where errors are {\it algebraically smooth}
(i.e., directions where the relaxation method significantly damps some local error components).
These algebraically smooth errors are not necessarily geometrically smooth, but they do need to be well
represented within the range of the coarse grid interpolation.
Irregular coarsening occurs when the strength-of-connection phase of the standard SA-AMG algorithm labels many nonzeros as {\it weak}.
The presence of many weak connections accentuates some problematic facets of the
hierarchy construction process. Specifically,
SA-AMG constructs a graph based on only the strong matrix connections
(as determined by a strength-of-connection algorithm), which is then coarsened to generate coarse discretization operators.
In general, sparser graphs (i.e., graphs with few connections) lead to less coarsening, larger coarse matrices,
denser coarse matrices, and more total AMG levels. The overall effect
is a potentially significant cost increase in both the
setup and apply phases. Further, the presence of weak connections requires that a special filtered discretization
matrix be devised in order to define the grid transfers. This filtering step can be somewhat fragile, and if not done properly
can lead to sub-optimal grid transfer operators. All of this is exacerbated by limitations in the standard strength-of-connection
algorithms. The original strength-of-connection idea dates back to the 1980s~\cite{BrMcRu84,RuSt85} and is
motivated by M-matrix assumptions. While computationally inexpensive, it is easy to construct examples
(see \S~\ref{sec:motivation}) where common strength criteria give a poor indication of
the
directions that correspond to algebraically smooth errors. To avoid the adverse convergence effects of coarsening in
directions that are not aligned with algebraically smooth errors, often a somewhat large threshold or cutoff value is chosen to
encourage the labeling of many connections as weak, and thus coarsen slowly.
While there are some interesting alternatives to the classical coarsening
approaches~\cite{Livne04,OBroeker_2003a,BrZi2005,BrBrMaMaMc06,BrBrKaLi15,BrannickF10,OlScTu10},
that might reduce some of these ill effects,
most of these alternatives are expensive and not fully robust. For this reason, these
alternatives have not been generally adopted and are not considered further in this paper.
Four algorithmic adaptations are proposed to address potential SA-AMG deficiencies that tend to arise in the presence of many
weak connections. The new algorithms are algebraic and do not require any additional information or intervention from the application.
Furthermore, these adaptations fit naturally into the existing SA-AMG setup workflow.
The first idea considers an alternative diagonal approximation to the matrix inverse used within the Jacobi
{\em prolongator smoothing} step.
Relatively small diagonal entries can occur when formulating the filtered discretization operators. Unfortunately, these
small entries are highly problematic for the Jacobi step.
Effectively, we propose an alternative Jacobi-like step that uses the inverse
of a diagonal matrix based on the 1-norm of individual rows of the filtered matrix.
The second algorithm addresses how the filtered matrix is defined. Normally, the filtered diagonal entries
are modified to reflect weak off-diagonal entries that are dropped from the original matrix. To prevent potentially small
diagonal entries, we propose that dropped entries are accounted for by modifying off-diagonal nonzeros in some circumstances.
The third variant introduces a set of constraints that the prolongator must normally satisfy
(e.g., all entries lie between zero and one). The algorithm attempts to reformulate any SA-AMG prolongator row that violates
the constraints by finding a suitable nearby row.
The fourth and final algorithm introduces a second dropping or filtering stage
that is performed after the standard SA-AMG coarsening algorithm. That is, the second filtering stage
does not alter the coarsening process, but indirectly yields a sparser grid transfer operator
by effectively sparsifying the matrix used within the prolongator smoothing step. Here, the aim is
to reduce the cost of the solver
while maintaining SA-AMG's convergence properties.
In some cases, these four modifications do not
significantly alter cost or convergence behavior (when compared to traditional SA-AMG). Under certain conditions, however,
modified SA-AMG performs much better than the traditional version, especially for problems where
irregular or anisotropic coarsening is needed.
The paper is structured as follows. In \S\ref{sec: saamg} we give an overview of the SA-AMG method. In
\S\S\ref{sec:1norm diag approx}--\ref{sec:root_node},
we present the new algorithmic variants for
how SA-AMG generates the grid transfer.
In \S\ref{sec:results_randcube} and \S\ref{sec:results_stretchcube} we present results for Poisson and
reaction-diffusion tests problem on a regular domain. In \S\ref{sec:results_sp10}, we explore the
effectiveness of the new algorithms on systems arising from a well-known reservoir model. In \S\ref{sec:exawind}, we detail experiments
on systems arising in a low-Mach computational fluid dynamics wind turbine simulation.
\subsection{Smoothed Aggregation} \label{sec: saamg}
\input{background}
\input{dbar}
\section{Numerical Results} \label{sec:results}
Four sample problems are presented to examine the behavior of the four
proposed SA-AMG algorithm variations. The first two correspond to
fairly academic cubes with perturbations or stretched mesh
spacing. The other two are more realistic. One is the SPE10 benchmark
problem~\cite{spe10} from
the Society of Petroleum Engineers, which has highly heterogeneous material jumps. The second comes from a wind turbine
simulation where the underlying mesh has problematic aspect ratios within some parts of the domain.
\subsection{Randomly Perturbed Cube}\label{sec:results_randcube}
The first test problem considers a Poisson
equation
\begin{equation} \label{LAPLACE_EQN}
\begin{aligned}
-\nabla^2 u(x) &= f(x) \hskip .4in &x \mbox{~in~} \Omega, \\
u(x) &= u_D(x) &x \mbox{~on~} \partial \Omega,
\end{aligned}
\end{equation}
where the domain is defined by $\Omega=[0,1]\times[0,1]\times[0,100]$, the forcing function
$f(x)$ is identically zero,
and $u_D(x) = 1 + x_1 + x_2 + x_3 + x_1 x_2+ x_1 x_3+ x_2 x_3 + x_1 x_2 x_3$.
We discretize \eqref{LAPLACE_EQN} using linear hexahedral finite elements on
a tensor product mesh using 60 elements in each coordinate direction.
The element sizes are given by a perturbation of a uniform spacing. Specifically,
along each coordinate direction mesh points are randomly perturbed by up to $20\%$
from the uniform spacing location (details given in Appendix~\ref{appendix:randcube}).
This leads to a mesh where the spacing in the first two coordinate directions is
generally finer than that in the third dimension.
For our experiments, we generated 50 different test meshes using
different random seeds.
The smoothed aggregation multigrid solver in MueLu \cite{MueLu} is used as a preconditioner
for the conjugate gradient (CG) method to solve the 50 linear systems. It is well-known
that linear hexahedral elements on highly stretched meshes give rise to
matrix coefficients that are problematic from a strength-of-connection perspective.
That is, the magnitude of the matrix entries are not well correlated with
mesh stretching. For this reason, an alternative matrix termed a distance Laplacian,
$L^{(d)}$, is used in this experiment for the aggregation/coarsening phase of the algorithm. Specifically,
strong connections satisfy $ |L^{(d)}_{ij}| \geq \theta \sqrt{L^{(d)}_{ii} L^{(d)}_{jj}} $.
The matrix $L^{(d)}$ has the same nonzero pattern as the discretization matrix $A$. The $L^{(d)}_{ij}$
off-diagonal values, however, are defined as the reciprocal of the negative distance between the
$i^{th}$ and $j^{th}$ coordinate, requiring coordinates be supplied to the solver.
The diagonal is then chosen so that sum of all entries within each
row is identically zero.
Here, $\theta$ is taken as $0.025$, which was determined
experimentally to produce desirable aggregates that are primarily oriented along the first two coordinate
directions. The smoothed
aggregation hierarchy was generated so that the resulting discretization
matrix is coarsened until only 1,000 or fewer unknowns remain, at which point a
direct solver is applied. Two sweeps of Chebyshev pre- and post-smoothing are applied on all
other levels. The Chebyshev eigenvalue interval is given by $[\lambda^*/10,\lambda^*]$ where
$\lambda^*$ is an estimate of the maximum eigenvalue of $A$ obtained by $10$ sweeps
of the power method. The conjugate gradient
iteration is terminated when the residual is reduced by a factor of $10^{-10}$.
We now consider all 16 combinations of enabling and disabling the four adaptations:
the \textsf{1Norm}\xspace diagonal modification of \S\S~\ref{sec:1norm diag approx}--\ref{sec:safeguard},
the \textsf{OffLmp}\xspace lumping modification of \S~\ref{sec:alt lumping},
the \textsf{Cnstrnt}\xspace constraint enforcement of \S~\ref{sec:constraints}, and
the \textsf{Sprsfy}\xspace sparsification of \S~\ref{sec:root_node}.
Disabling all four options corresponds
to traditional smoothed aggregation.
In all experiments involving \textsf{OffLmp}\xspace, we choose $\tau = 1.1$ as the maximum allowable ${\cal D}()$ growth.
Results can be found in Table~\ref{tbl:randcube}.
\begin{table}[htb]\label{tbl:randcube}
\begin{center}
\include{randcube_results_paper}
\end{center}
\caption{
CG iterations and operator complexities using all combinations of
enabling and disabling \textsf{1Norm}\xspace, \textsf{Sprsfy}\xspace, \textsf{OffLmp}\xspace and \textsf{Cnstrnt}\xspace.
Each algorithm denoted in \textcolor{red}{red} had 16 solution
failures due to negative eigenvalues. These failures are ignored for
iteration counts and operator complexities.}
\end{table}
We note first that traditional SA-AMG (all options off) fails in 16 of the 50
test cases.
These failures arise from a negative eigenvalue estimate for $\bar{\lambda}_m$
in \eqref{eq:prolongator_smoothing} due to the $\bar{A}$ matrix, which
has poor diagonal dominance properties. These negative eigenvalues
lead to catastrophic failures within the solver. Here, operator complexity
is defined as the ratio of the number of nonzeros within all the hierarchy
discretization matrices divided by the number of nonzeros for the
finest level matrix.
Three algorithms used by themselves (or in combination) --- \textsf{1Norm}\xspace,
\textsf{Sprsfy}\xspace and \textsf{OffLmp}\xspace --- prevent all negative eigenvalues in
the test problem. In the case of \textsf{1Norm}\xspace or \textsf{OffLmp}\xspace, the
diagonal entries are less sensitive to the dropping schemes that lead
to small values in $\ensuremath{\bar{D}}\xspace$. The \textsf{Sprsfy}\xspace algorithm,
indirectly removes some problematic entries from $\bar{A}$ which
would otherwise lead to negative eigenvalues. The \textsf{Cnstrnt}\xspace algorithm has no
effect on reducing the number of failures.
Second we note that \textsf{OffLmp}\xspace and \textsf{1Norm}\xspace each modestly reduce the
iteration counts whenever they are used (either individually or in
combination), while \textsf{Sprsfy}\xspace tends to slightly increase the
iteration counts on average. The \textsf{Cnstrnt}\xspace algorithm yields a
substantial decrease in iterations, though it needs to be used in
combination with at least one other option to avoid failures.
For example, the combination of \textsf{Cnstrnt}\xspace and \textsf{OffLmp}\xspace reduced the iteration
count from $26.1$ with $32\%$ failures to $16.0$ with no failures. This corresponds to
an average iteration reduction of $60\%$ (not counting the failures).
Finally, we note \textsf{Sprsfy}\xspace consistently leads to a small reduction
in operator complexity, 1.28 to 1.25, whenever it is used.
\subsection{Triaxially Stretched Cube}\label{sec:results_stretchcube}
We take $(k_x, k_y, k_z)\in \{1,5,10\}^3$, restricted to $k_x
\leq k_y \leq k_z$, yielding 10 different configurations. For each
$(k_x,k_y,k_z)$ we define a domain $\Omega = [0,3k_x+3] \times [0,3k_y+3] \times [0,3k_z+3]$.
On $\Omega$, we solve the reaction-diffusion equation
\begin{equation} \label{REACTDIFF_EQN}
\begin{aligned}
\sigma^{-1} u(x) -\nabla^2 u(x) &= f(x) \hskip .4in &x \mbox{~in~} \Omega, \\
u(x) &= u_D(x) &x \mbox{~on~} \partial \Omega,
\end{aligned}
\end{equation}
where the forcing function
$f(x)$ is identically zero,
$u_D(x) = 1 + x_1 + x_2 + x_3 + x_1 x_2+ x_1 x_3+ x_2 x_3 + x_1 x_2
x_3$
and the reaction term, $\sigma$ is chosen to be $10,10^2,10^3,10^4,$ or $10^5$.
Notice that the nonzero $\sigma$ implies that the matrix has a positive row sum.
We discretize \eqref{REACTDIFF_EQN} using linear hexahedral finite elements on
a tensor product mesh using 60 elements in each coordinate
direction. The element sizes linearly vary from $0.1$ to $k_x/10$ in
the $x$-dimension (and similarly in the $y$ and $z$ dimensions).
Details are given in Appendix~\ref{appendix:stretchcube}.
Following \S~\ref{sec:results_randcube}, CG with a $10^{-10}$ tolerance is used with a MueLu
preconditioner to solve the linear system. Again, the distance
Laplacian criterion is used for the aggregation/coarsening phase of
the algorithm with $\theta=0.025$. Two sweeps of Chebyshev smoothing
is applied (with identical parameters to
\S~\ref{sec:results_randcube}) to all levels except the coarsest
level, where a direct solver is used.
First we consider the traditional SA-AMG algorithm and \textsf{1Norm}\xspace
modifications. Figure~\ref{fig:stretchcube:dmod} presents a probability
histogram of the number of iterations taken by traditional SA-AMG minus
the number of iterations taken with the \textsf{1Norm}\xspace diagonal modification. We note
that in all fifty cases, the \textsf{1Norm}\xspace modification takes no more than the number of
iterations taken by traditional SA-AMG. While 20 of the \textsf{1Norm}\xspace
modification runs are within three iterations of traditional SA-AMG,
four runs have a difference of 15 or more iterations, which is a substantial
savings. Similarly, Figure~\ref{fig:stretchcube:spreadlump} shows the same
results for the \textsf{OffLmp}\xspace lumping modification, with 19 of the
runs withing three iterations of traditional SA-AMG, and four runs
with a difference of 15 or more iterations. We do not show results
for the \textsf{Cnstrnt}\xspace constraint enforcement, but note that 40 of those runs are within three iterations
of traditional SA-AMG while no runs had an iteration difference of 15
or more. More problematically, two of the 50 runs fail due to
negative eigenvalues. We also do not show results for \textsf{Sprsfy}\xspace
sparsification, as in three cases it increases the iteration count substantially. Thus we
cannot recommend using \textsf{Cnstrnt}\xspace or
\textsf{Sprsfy}\xspace on this problem, unless they are paired with
either one of the modifications which can overcome their deficiencies.
\begin{figure}[htb]
\begin{center}
\subfigure[Traditional SA-AMG (Trad) minus $\textsf{1Norm}\xspace$ diagonal modification.]{
\includegraphics[scale=0.5]{figs/iteration_difference_paperTrad_1Norm.png}
\label{fig:stretchcube:dmod}
}
\subfigure[Traditional SA-AMG (Trad) minus $\textsf{OffLmp}\xspace$ lumping modification.]{
\includegraphics[scale=0.5]{figs/iteration_difference_paperTrad_OffLmp.png}
\label{fig:stretchcube:spreadlump}
}
\end{center}
\caption{Probability histogram of the difference in iteration counts
for MueLu-preconditioned CG on \eqref{REACTDIFF_EQN} between
traditional SA-AMG and one of the proposed algorithms, taken over
ten mesh configurations $(k_x, k_y, k_z) \in \{1,5,10\}^3$,
restricted to $k_x \leq k_y \leq k_z$,
and five reaction rates $\sigma=1e1,1e2,1e3,1e4,1e5$.}\label{fig:stretchcube}
\end{figure}
\subsection{Subsurface SPE10 Problem}\label{sec:results_sp10}
This model comes from a dataset associated with the $10^{th}$ SPE Comparative Solution Project (SPE10)~\cite{spe10}. It describes a reservoir
simulation that models flow through porous media to predict well production from hydrocarbon deposits.
The discrete matrix problem was formed using the open source toolbox MRST~\cite{lie_2019}. The underlying PDE
equations are defined by Darcy's law for a single fluid along with external influences such as wells
and are discretized using a two-point flux approximation method.
The porosity is shown in figure~\ref{fig:porosity} and leads to large permeability variations that range up to 12 orders of
magnitude.
\begin{figure}
\centering
\includegraphics[scale=0.5]{figs/spe10Porosity.png}
\caption{Porosity of SPE10 benchmark problem \label{fig:porosity}}
\end{figure}
The large permeability variation requires that the multigrid solver coarsen irregularly.
For these experiments, the same Chebyshev smoother choices are used as with the first example.
A direct solver is also used on the coarsest grid. A standard smoothed aggregation strength-of-connection criteria is
employed using the matrix coefficients (i.e., not the distance Laplacian) and three values of
$\theta$ are considered. The number of multigrid levels is fixed at 5 where the finest level discretization
matrix is $660K \times 660K $ and the coarsest level matrix dimensions is always less than $14K$
for the largest $\theta$ value and is much smaller for the other two $\theta$.
The conjugate gradient
iteration is terminated when the residual is reduced by a factor of $10^{-8}$.
Table~\ref{results:spe10} illustrates the results for different combinations of our proposed
variants. All combinations that did not use the \textsf{1Norm}\xspace variant failed
and so these are not shown.
\begin{table}
\begin{center}
\begin{tabular} {lll||cc|cc|cc}
\toprule
& & & \multicolumn{2}{c|}{$\theta = .02$} &\multicolumn{2}{c|}{$\theta = .05$} &\multicolumn{2}{c}{$\theta = .1$} \\
\multicolumn{3}{c||}{Algorithm Choice} & AMG & its. & AMG & its. & AMG & its. \\
& & & complex. & & complex. & & complex. & \\
\midrule
& & & 1.85 & 24 & 2.00 & 26 & 2.60 & 32 \\
& & \textsf{Cnstrnt}\xspace & 1.85 & 24 & 2.00 & 26 & 2.60 & 28 \\
\textsf{OffLmp}\xspace & & & 1.85 & 23 & 2.00 & 20 & 2.60 & 15 \\
\textsf{OffLmp}\xspace & & \textsf{Cnstrnt}\xspace & 1.85 & 24 & 2.00 & 20 & 2.60 & 15 \\
& \textsf{Sprsfy}\xspace & & 1.77 & 53 & 1.85 & 35 & 2.12 & 26 \\
& \textsf{Sprsfy}\xspace & \textsf{Cnstrnt}\xspace & 1.77 & 49 & 1.85 & 35 & 2.12 & 26 \\
\textsf{OffLmp}\xspace & \textsf{Sprsfy}\xspace & & 1.77 & 48 & 1.85 & 33 & 2.11 & 27 \\
\textsf{OffLmp}\xspace & \textsf{Sprsfy}\xspace & \textsf{Cnstrnt}\xspace & 1.77 & 47 & 1.85 & 35 & 2.12 & 27 \\
\bottomrule
\end{tabular}
\end{center}
\caption{SPE10 results for combinations of different AMG variants (all results use \textsf{1Norm}\xspace).\label{results:spe10}}
\end{table}
For the smallest value of $\theta$, the \textsf{Cnstrnt}\xspace and \textsf{OffLmp}\xspace procedures do not have too significant
an effect on convergence. The \textsf{Sprsfy}\xspace approach does reduce the operator complexity, but
this comes at a fairly significant increase in iteration count (approximately double). For large values of $\theta$,
however, we see that some of the different algorithm choices do have a more pronounced effect. In particular, the
best iteration counts employ \textsf{OffLmp}\xspace for $\theta=.1$, which are about half those of the other methods
when \textsf{Sprsfy}\xspace is not used.
The \textsf{Sprsfy}\xspace algorithm more significantly improves the AMG operator complexity when $\theta=.1$ and also improves
the iteration counts in the case that \textsf{OffLmp}\xspace is not used, though the iterations are worse when both
\textsf{OffLmp}\xspace and \textsf{Sprsfy}\xspace are employed. The \textsf{Cnstrnt}\xspace option has only a modest effect, giving some
reduction in iterations for a couple of cases (e.g. when the only other employed variant was \textsf{1Norm}\xspace for $\theta=.1$).
The main point, however, is that the traditional SA-AMG algorithm failed to converge on this problem
and so the diagonal $\textsf{1Norm}\xspace$ modification is critical in getting the solver to converge.
The additional algorithm modifications can then provide some additional benefit, though not always.
As the choice of $\theta$ is unknown, it is reassuring that with \textsf{OffLmp}\xspace the number of
iterations decrease as the cost/AMG operator complexity increases (associated with an increasing value of $\theta$).
\subsection{Wind Turbine}\label{sec:exawind}
We now consider the effects of the various solver options within a wind turbine simulation run in the low Mach computational
fluid dynamics (CFD) code NaluWind~\cite{naluwind}. Figure~\ref{fig:nrel5mw} depicts a 5-megawatt wind turbine
that is composed of three turbine blades and a hub.
\begin{figure}
\includegraphics[scale=0.15]{figs/5mw_bmesh_front.png}
\includegraphics[scale=0.15]{figs/5mw_bmesh_side.png}
\caption{Front and side views of NREL 5 megawatt turbine mesh, showing blades and hub.}
\label{fig:nrel5mw}
\end{figure}
The blades and background are meshed separately and coupled via constraints. NaluWind has two main techniques for addressing these constraints.
The first ``coupled" technique can be viewed as an alternating Schwarz approach, in which each mesh has equally valid solutions.
The corresponding linear systems have equations corresponding to constraints, and it is these equations which cause difficulties for the linear solver.
In the second ``decoupled" approach, the constraints are eliminated from the corresponding linear systems, which can be solved independently.
The simulation itself is time-dependent and consists of two main physics solve phases: momentum and pressure.
While the momentum linear system is amenable to GMRES preconditioned with symmetric Gauss-Seidel, the pressure system
requires a more robust, scalable solver. The current solver of choice is
GMRES preconditioned by smoothed-aggregation multigrid, with a convergence criteria requiring a relative residual reduction of $10^{-5}$.
For this problem 5 AMG levels are used. The finest level matrix system has 23 million DOFs. The number of DOFs in
the coarsest level AMG matrices varies, but is typically 9k--15k unknowns. One pre- and one post-smoothing sweep using a degree 2 Chebyshev smoother is employed on all
levels with the exception of the coarsest level where a direct solver is used.
A standard SA-AMG strength-of-connection criteria is employed using the matrix coefficients (i.e., not the distance Laplacian) for
a fixed $\theta$ threshold of $.02$.
We will consider the effect of the proposed SA-AMG options on iteration counts over 10 time steps in the simulation.
The left side of
Figure~\ref{fig:nrel5mw-absrowsum} compares linear iteration totals for standard SA-AMG versus SA-AMG using various options
for NaluWind run in decoupled mode. The best improvement in iteration counts comes with either the \textsf{1Norm}\xspace or the \textsf{OffLmp}\xspace
options. The \textsf{Cnstrnt}\xspace option is only shown in one case as it generally produced very little improvement.
The \textsf{1Norm}\xspace option could not be improved further by using it in conjunction with any of the other algorithms, though it is worth
noting that \textsf{OffLmp}\xspace and \textsf{Cnstrnt}\xspace did not further degrade performance. As expected,
\textsf{Sprsfy}\xspace generally increased the iteration count and unfortunately provides only a modest gain in AMG operator
complexity from about 1.66 to about 1.61. When used with the traditional method by itself (shown in the plot) or in
conjunction with \textsf{1Norm}\xspace (not shown), it generally increases the average iteration count by about 1 iteration.
Finally, when not using the \textsf{1Norm}\xspace option, it is possible to improve the \textsf{Sprsfy}\xspace iteration counts and make it
fairly competitive with the best runs by applying both \textsf{OffLmp}\xspace and \textsf{Cnstrnt}\xspace.
Most of the improvements are due to the \textsf{OffLmp}\xspace (on average a 2.575 iteration improvement) while the
\textsf{Cnstrnt}\xspace gains are more modest (on average an additional .75 iteration improvement).
\begin{figure}[tbph]
\centering
\includegraphics[scale=0.163]{figs/decoupledSubset.jpg}
~~~~~~ \includegraphics[scale=0.45]{figs/5MW-coupled-uncoupled.png}
\label{fig:nrel5mw-absrowsum}
\caption{Comparison of effect of prolongator options on linear solver iterations over 10 timesteps (4 linear solvers per timestep).
In the left plot,
all combinations of the four algorithm variants were tested, but only the most effective strategies are shown.
``Traditional" denotes standard SA-AMG.
``DMod" is diagonal modification, ``AltLmp" is alternative lumping, and``Sprsify" is prolongator
sparsification.
In the right plot, only the effect of "DMod" is shown versus standard SA-AMG for an ExaWind 5MW linear
system formulated either either as a coupled system (containing constraints) or an uncoupled system.
}
\end{figure}
As the \textsf{1Norm}\xspace option alone provides the biggest improvement for this problem,
the right side highlights the impact of only the \textsf{1Norm}\xspace option (all others are turned off) for the coupled formulation. For the sake of
comparison, the same information is repeated for the decoupled formulation. Here, one can see that the coupled mode leads to generally harder
linear systems. For both coupled and decoupled, there is a nice reduction in the number of iterations using the \textsf{1Norm}\xspace option. Further,
the iteration count is generally less erratic as well with the \textsf{1Norm}\xspace option. Figure~\ref{fig:exawind} considers a robustness
study for one representative linear system within a McAlister fixed wing simulation using the coupled formulation. In this study,
iteration counts are shown as a function of a varying drop threshold $\theta$ parameter. Without the \textsf{1Norm}\xspace option, we again see
erratic behavior including 3 threshold choices where
the solver does not converge. However, the iteration counts vary smoothly without any failures when the \textsf{1Norm}\xspace option is used.
\begin{figure}[h!tb]
\includegraphics[height=0.2\textwidth,width=2.7in]{figs/mcalister_mesh_wing.png}\\
\vskip -1.6in
{\color{white} .}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\includegraphics[height=0.27\textwidth]{figs/robust2ness.jpg}
\caption{Left: Sample mesh sections for \textsc{5MegaWatt} turbine. Middle: GMRES/AMG iterations for time sequence on 5MegaWatt
simulation (coupled formulation being more challenging).
Right: GMRES/AMG iterations for user-supplied strong/weak threshold for a \textsc{McAlister} fixed wing configuration.}
\label{fig:exawind}
\end{figure}
That is, one benefit of \textsf{1Norm}\xspace appears to be enhanced robustness.
| {
"timestamp": "2021-03-22T01:02:08",
"yymm": "2103",
"arxiv_id": "2103.10476",
"language": "en",
"url": "https://arxiv.org/abs/2103.10476",
"abstract": "Four adaptations of the smoothed aggregation algebraic multigrid (SA-AMG) method are proposed with an eye towards improving the convergence and robustness of the solver in situations when the discretization matrix contains many weak connections. These weak connections can cause higher than expected levels of fill-in within the coarse discretization matrices and can also give rise to sub-optimal smoothing within the prolongator smoothing phase. These smoothing drawbacks are due to the relatively small size of some diagonal entries within the filtered matrix that one obtains after dropping the weak connections. The new algorithms consider modifications to the Jacobi-like step that defines the prolongator smoother, modifications to the filtered matrix, and also direct modifications to the resulting grid transfer operators. Numerical results are given illustrating the potential benefits of the proposed adaptations.",
"subjects": "Numerical Analysis (math.NA)",
"title": "Smooth Aggregation for Difficult Stretched Mesh and Coefficient Variation Problems",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9407897492587141,
"lm_q2_score": 0.7520125848754471,
"lm_q1q2_score": 0.7074857311643693
} |
https://arxiv.org/abs/1510.07281 | Eigenvalue inequalities on Riemannian manifolds with a lower Ricci curvature bound | We revisit classical eigenvalue inequalities due to Buser, Cheng, and Gromov on closed Riemannian manifolds, and prove the versions of these results for the Dirichlet and Neumann boundary value problems. Eigenvalue multiplicity bounds and related open problems are also discussed. | \section{Statements and discussion}
\label{results}
\subsection{Notation and preliminaries}
Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold with or without boundary, and let
$$
0=\lambda_0(g)<\lambda_1(g)\leqslant\lambda_2(g)\leqslant\ldots\leqslant\lambda_k(g)\leqslant\ldots
$$
be the eigenvalues of the Laplace-Beltrami operator on $M$. If the boundary $\partial M$ is non-empty we assume for now that the Neumann boundary conditions are imposed. Later we also consider the Dirichlet eigenvalue problem; its eigenvalues are denoted by $\nu_k(g)$. Recall that the eigenvalue counting function $N_g(\lambda)$ is defined for any $\lambda> 0$ as the number of Laplace eigenvalues, counted with multiplicity, that are strictly less than $\lambda$. By the celebrated Weyl's law the counting function satisfies the following asymptotics:
\begin{equation}
\label{Weyl}
N_g(\lambda)\sim\frac{\omega_n}{(2\pi)^n}\mathit{Vol}_g(M)\lambda^{n/2}\qquad\text{as}\quad\lambda\to+\infty,
\end{equation}
where $\mathit{Vol}_g(M)$ is the volume of $M$ and $\omega_n$ is the volume of a unit ball in the Euclidean space~$\mathbb R^n$, see~\cite{SV} for the refined asymptotics and other developments in the subject. By $m_k(g)$ we denote the multiplicity of the $k$th eigenvalue $\lambda_k(g)$. Clearly, we have $m_k(g)=N_g(\lambda_k+0)-N_g(\lambda_k)$, and hence, $m_k(g)=o(\lambda_k^{n/2})$ as $k\to +\infty$.
The purpose of this paper is three-fold. First, we revisit classical lower bounds for Laplace eigenvalues on closed Riemannian manifolds due to Gromov and Buser, and give an alternative unified approach to these statements. It avoids delicate isoperimetric arguments used in the original proofs, and uses only the Neumann-Poincar\'e inequality and geometric estimates for the cardinality of certain coverings. The advantage of our argument is that it carries over directly to the boundary value problems for geodesically convex domains, and yields rather explicit eigenvalue bounds, which appear to be new. Next, we turn our attention to the eigenvalue upper bounds originally obtained by Cheng and Buser on closed manifolds. Some of their versions for boundary value problems also appear to be missing in the literature, and we fill this gap by presenting such results. Finally, we discuss eigenvalue multiplicity bounds on Riemannian manifolds, showing, for example, that for geodesically convex compact domains in complete manifolds of non-negative Ricci curvature the multiplicities $m_k(g)$ of Neumann eigenvalues are bounded in terms of the dimension and the index $k$ only. We end Section~\ref{results} with a few related open problems. Section~\ref{prem} contains the necessary background material, and the proofs of lower and upper eigenvalue bounds appear in Sections~\ref{proofs:lower} and~\ref{proofs:upper} respectively.
\subsection{Lower eigenvalue bounds: Gromov and Buser revisited}
\label{section:lower}
Let $M$ be a closed manifold of non-negative Ricci curvature. A classical result by Li and Yau~\cite{LY80} says that the first Laplace eigenvalue $\lambda_1(g)$ of $M$ satisfies the inequality $\lambda_1(g)\geqslant \pi^2/(4d^2)$, where $d$ is the diameter of $M$. Later it has been improved by Zhong and Yang~\cite{ZY84} to the estimate $\lambda_1(g)\geqslant \pi^2/d^2$. For more general closed manifolds the following inequalities for all Laplace eigenvalues hold.
\begin{theorem}
\label{gromov}
Let $(M,g)$ be a closed Riemannian manifold whose Ricci curvature satisfies the bound $\mathit{Ricci}\geqslant -(n-1)\kappa$, where $\kappa\geqslant 0$. Then there exist constants $C_i$, $i=1,\ldots,3$, depending on the dimension $n$ of $M$ only, such that:
\begin{equation}
\label{leb:gro}
\lambda_k(g)\geqslant C_1^{1+d\sqrt{\kappa}}d^{-2}k^{2/n}\qquad\text{for any}\quad k\geqslant 1,
\end{equation}
and
\begin{equation}
\label{leb:bu}
\lambda_k(g)\geqslant C_2\mathit{Vol}_g(M)^{-2/n}k^{2/n} \qquad\text{for any}\quad k\geqslant 3C_3\mathit{Vol}_g(M)\max\{\kappa^{n/2},\inj^{-n}\},
\end{equation}
where $d$ and $\inj$ are the diameter and the injectivity radius of $M$ respectively.
\end{theorem}
Inequality~\eqref{leb:gro} is due to Gromov~\cite[Appendix~C]{Gro}. Motivated by Weyl's law~\eqref{Weyl}, he also poses a question whether there is an asymptotically sharp lower bound in terms of volume. This question has been answered by Buser who proved inequality~\eqref{leb:bu}, which however has been stated in~\cite[Theorem~6.2]{Bu82} in a slightly different form. Note that the hypothesis on the index $k$ in~\eqref{leb:bu} is necessary: for any given integer $k$ no geometry-free lower bound for the renormalised eigenvalue $\lambda_k(g)\mathit{Vol}_g(M)^{2/n}$ can hold. Indeed, the standard examples of manifolds with long necks (the connected sums of the so-called Cheeger dumbbells) show that there are sequences of metrics whose $k$th renormalised eigenvalues converge to zero. Moreover, as the examples of long thin flat tori show, the appearance of the injectivity radius in the hypothesis on the index $k$ also can not be easily removed. A number of related eigenvalue bounds have also been obtained by Li and Yau~\cite{LY80}, and Donnelly and Li~\cite{DL82}.
Both arguments by Gromov and Buser use methods based on isoperimetric inequalities: in the former case it is the circle of ideas around Levy's isoperimetric inequality, and in the latter -- the estimate for the Cheeger constant. The lower eigenvalue bounds above can be re-written in the form of upper bounds for the counting function $N_g(\lambda)$. In particular, Gromov's bound~\eqref{leb:gro} is equivalent to the inequality
\begin{equation}
\label{leb:gro1}
N_g(\lambda)\leqslant \max\{C_4^{1+d\sqrt{\kappa}}d^n{\lambda}^{n/2}, 1\}\qquad\text{for any}\quad \lambda \geqslant 0,
\end{equation}
and Buser's inequality~\eqref{leb:bu}
is a consequence of
\begin{equation}
\label{improved}
\quad N_g(\lambda)\leqslant C_3\mathit{Vol}_g(M)({\lambda}^{n/2}+\kappa^{n/2}+\inj^{-n}) \qquad\text{for any}\quad \lambda \geqslant 0.
\end{equation}
In Section~\ref{proofs:lower} we give a rather direct argument for the inequalities~\eqref{leb:gro1} and~\eqref{improved} that allows to bound the values of the counting function via the cardinality of an appropriate covering by metric balls, and avoids using isoperimetric inequalities as in~\cite{Bu82,Gro}. We also obtain versions of these inequalities for boundary value problems, which we discuss now.
Suppose that $(M,g)$ is a complete Riemannian manifold whose Ricci curvature is bounded below $\mathit{Ricci}\geqslant -(n-1)\kappa$, where $\kappa\geqslant 0$, and $\Omega\subset M$ is a compact domain with a Lipschitz boundary.
Recall that a domain with a smooth boundary is said to satisfy the {\it interior rolling $\delta$-ball condition} if for any $x\in\partial\Omega$ there exists a ball $B$ of radius $\delta$ contained in $\Omega$ that touches the boundary $\partial\Omega$ at the point $x$, that is $B\subset\Omega$ and $B\cap\partial\Omega=\{x\}$.
We define the {\it maximal radius} $\rad(\Omega)$ of an interior rolling ball as the supremum of $\delta>0$ such that $\Omega$ satisfies the interior rolling $\delta$-ball condition; equivalently, it can be defined as
$$
\rad(\Omega)=\inf_{x\in\partial\Omega}\sup\{r>0:\text{ there exists } B(z,r)\subset\Omega\text{ tangent to }\partial\Omega\text{ at }x\}.
$$
Finally, below by the injectivity radius $\inj(\Omega)$ of a domain $\Omega\subset M$ we mean the infimum of the injectivity radii $\inj(p)$ of the ambient manifold $M$ as the point $p$ ranges over $\Omega$.
The following theorem is a version of the Gromov and Buser eigenvalue bounds for the Neumann eigenvalue problem. For the convenience of future references we state it in the form of upper bounds for the counting function.
\begin{theorem}
\label{neumann:gromov}
Let $(M,g)$ be a complete Riemannian manifold whose Ricci curvature is bounded below $\mathit{Ricci}\geqslant -(n-1)\kappa$, where $\kappa\geqslant 0$, and $\Omega\subset M$ be a geodesically convex precompact domain with Lipschitz boundary. Then the counting function for the Neumann eigenvalue problem on $\Omega$ satisfies inequality~\eqref{leb:gro1}, and hence the eigenvalues satisfy inequality~\eqref{leb:gro}, with $d=d(\Omega)$ being the diameter of the domain. In addition, if the boundary of $\Omega$ is smooth, then the counting function $N_g(\lambda)$ also satisfies the inequality
\begin{equation}
\label{neumann:volume}
N_g(\lambda)\leqslant C_{5}\mathit{Vol}_g(\Omega)(\lambda^{n/2}+\kappa^{n/2}+\inj(\Omega)^{-n}+\rad(\Omega)^{-n}) \qquad\text{for any}\quad \lambda \geqslant 0,
\end{equation}
where $\rad(\Omega)$ is the maximal radius of an interior rolling ball, $\inj(\Omega)$ is the infimum of the injectivity radii over $\Omega$, and the constant $C_5$ depends on the dimension of $M$ only.
\end{theorem}
To our knowledge, the equivalent estimates \eqref{leb:gro} and \eqref{leb:gro1} have not been available in the literature for domains with Neumann boundary conditions under such general assumptions. Previously, Li and Yau~\cite[Theorem 5.3]{LY86} showed that when $M$ has non-negative Ricci curvature and the second fundamental form of $\partial\Omega$ is non-negative definite, the Neumann eigenvalues $\lambda_k(g)$ satisfy the inequalities $\lambda_k(g)\geqslant C\cdot d^{-2}k^{2/n}$, with the constant $C$ depending only on the dimension. It is likely that the method in~\cite{LY86} can be also used to get eigenvalue lower bounds under more general hypotheses, but probably with a more implicit dependence on the diameter and the lower Ricci curvature bound, cf.~\cite{DL82,Wang97}.
The convexity hypothesis on a domain $\Omega\subset M$ in estimates~\eqref{leb:gro} and~\eqref{leb:gro1} can not be easily removed. Indeed, consider a Euclidean domain obtained from a disjoint union of small balls connected by even tinier passages. As the size of the balls tends to zero and their number increases, so that the domain remains to be contained in a ball of fixed radius, the number of eigenvalues close to zero tends to infinity, while the diameter remains bounded.
To our knowledge, inequality~\eqref{neumann:volume} for the counting function is new even if $M$ is a Euclidean space. Examples, obtained by smoothing long thin rectangles, show that it fails to hold if the interior rolling ball radius $\rad(\Omega)$ is removed on the right-hand side. However, if the manifold $M$ has a finite volume, then $\rad(\Omega)$ can be dispensed at the price of replacing the volume $\mathit{Vol}(\Omega)$ by the total volume $\mathit{Vol}(M)$, see Remark~\ref{finite}.
Finally, by the variational principle the Neumann eigenvalues are always not greater than the corresponding Dirichlet eigenvalues, and therefore inequalities~\eqref{leb:gro1} and~\eqref{neumann:volume} hold also for the Dirichlet counting function under the assumption that $\Omega$ is geodesically convex. A different upper bound for the Dirichlet counting function can be found in~\cite{CY}; it is a generalization of the classical results by Berezin~\cite{Be} and Li and Yau~\cite{LY83} to the setting of eigenvalue problems on Riemannian manifolds. The bound does not assume that $\Omega$ is convex, but involves a less explicit geometric quantity, which could be expressed in terms of the mean curvatures of $\Omega$ with respect to isometric embeddings of $M$ into a Euclidean space. It is worth mentioning that upper bounds on the eigenvalue counting function are important in applications, such as image processing and machine learning~\cite{JMS}.
\subsection{Upper eigenvalue bounds: extensions of Cheng and Buser}
Now we discuss the upper eigenvalue bounds on Riemannian manifolds with a lower Ricci curvature bound. We start with recalling classical results due to Cheng~\cite{Che75} and Buser~\cite{Bu79} for the closed eigenvalue problem.
\begin{theorem}
\label{ueb:closed}
Let $(M,g)$ be a closed Riemannian manifold whose Ricci curvature satisfies the bound $\mathit{Ricci}\geqslant -(n-1)\kappa$, where $\kappa\geqslant 0$. Then there exist constants $C_6$ and $C_7$ depending on the dimension $n$ of $M$ only, such that:
\begin{equation}
\label{cheng}
\lambda_k(g)\leqslant \frac{(n-1)^2}{4}\kappa+C_6(k/d)^2\qquad\text{for any}\quad k\geqslant 1,
\end{equation}
and
\begin{equation}
\label{buser79}
\lambda_k(g)\leqslant \frac{(n-1)^2}{4}\kappa+C_{7}(k/\mathit{Vol}_g(M))^{2/n}\qquad\text{for any}\quad k\geqslant 1,
\end{equation}
where $d=d(M)$ is the diameter of $M$.
\end{theorem}
When a manifold $(M,g)$ has non-negative Ricci curvature, Cheng proves the version of inequality~\eqref{cheng} with an explicit constant:
$$
\lambda_k(g) \leqslant \frac{4k^2 j^2_{\frac{n}{2}-1}}{d^2} < \frac{2k^2 n(n+4)}{d^2},
$$
where $j_{\frac{n}{2}-1}$ is the first zero of the Bessel function $J_{\frac{n}{2}-1}$. The striking difference about the eigenvalue inequalities in Theorem~\ref{ueb:closed} is that the power of $k$ in the second is optimal in the sense of Weyl's law, while in the first it is not. Nevertheless, as can be seen from the examples of thin flat tori, the quadratic growth in Cheng's inequality~\eqref{cheng} can not be improved. Note that inequalities similar to~\eqref{buser79} have been also obtained by Li and Yau~\cite{LY80} under somewhat stronger hypotheses.
Buser's inequality~\eqref{buser79} has been generalised by Colbois and Maerten~\cite{CM08} to the Neumann eigenvalues for compact domains in complete manifolds with a lower Ricci curvature bound. More precisely, they show that there exist constants $C_{8}$ and $C_{9}$ depending on the dimension only such that for any compact domain $\Omega\subset M$ with a Lipschitz boundary the Neumann eigenvalues of $\Omega$ satisfy
\begin{equation}
\label{cm}
\lambda_k(g)\leqslant C_{8}\kappa+C_{9}(k/\mathit{Vol}_g(\Omega))^{2/n}\qquad\text{for any}\quad k\geqslant 1.
\end{equation}
To complete the picture of eigenvalue upper bounds for the Neumann problem, in Section~\ref{proofs:upper} we prove the following version of Cheng's inequality~\eqref{cheng}.
\begin{theorem}
\label{ueb:np}
Let $(M,g)$ be a complete Riemannian manifold whose Ricci curvature is bounded below $\mathit{Ricci}\geqslant -(n-1)\kappa$, where $\kappa\geqslant 0$, and $\Omega\subset M$ be a geodesically convex precompact domain with Lipschitz boundary. Then there exists a constant $C_{10}$ depending on dimension $n$ of $M$ only such that the following inequality for the Neumann eigenvalues of $\Omega$ holds:
\begin{equation}
\label{ourcheng}
\lambda_k(g)\leqslant C_{10}(\kappa+(k/d)^2)\qquad\text{for any}\quad k\geqslant 1,
\end{equation}
where $d=d(\Omega)$ is the diameter of $\Omega$.
\end{theorem}
In the case when $\Omega$ is a convex Euclidean domain, inequality~\eqref{ourcheng} has been obtained in~\cite{Kr}. As the following example shows the convexity hypothesis on a domain $\Omega$ in the theorem above can not be easily removed. First, note that when a domain $\Omega$ is non-convex, its diameter can be also measured using the so-called intrinsic distance on $\Omega$. Recall that it is defined as the infimum of the lengths of paths that lie in $\Omega$ and join two given points.
\begin{example}
\label{counter}
For a given real number $R>0$ consider a surface of revolution
$$
\Sigma_R=\{(x,y,z)\in\mathbb R^3: y^2+z^2=e^{-2xR}/R^2, x\in [0,1]\}.
$$
As is shown in~\cite[Lemma~5.1]{FT00}, the first non-zero Neumann eigenvalue of $\Sigma_R$ satisfies the inequality $\lambda_1(\Sigma_R)\geqslant R^2/8$. Hence, for the first eigenvalue of the product $\Sigma_R\times [0,\delta]$ we have
$$
\lambda_1(\Sigma_R\times [0,\delta])\geqslant R^2/8\qquad\text{when }0<\delta\leqslant \sqrt{8}\pi R^{-1}.
$$
Now for a sufficiently small $\delta>0$ consider a Euclidean domain in $\mathbb R^3$
$$
\Omega_R(\delta)=\{\exp_p(tv): p\in\Sigma_R, v\text{ is a unit outward normal vector}, ~t\in[0,\delta]\},
$$
where $\exp$ denotes the exponentional map in $\mathbb R^3$. Clearly, it is quasi-isometric to the Riemannian product $\Sigma_R\times [0,\delta]$, and the quasi-isometry constant converges to $1$ as $\delta\to 0+$. Thus, for any sequence $R_\ell\to +\infty$ we may choose a sequence $\delta_\ell\to 0+$ such that the first eigenvalues of the domains $\Omega_\ell=\Omega_{R_\ell}(\delta_\ell)$ satisfy the inequality $\lambda_1(\Omega_\ell)\geqslant R_\ell^2/16$. Note that the extrinsic diameter of $\Sigma_R$, and hence of any domain containing it, is always greater than $1$. In particular, the extrinsic diameters of $\Omega_\ell$ are bounded away from zero, and we obtain a counterexample to inequality~\eqref{ourcheng} for non-convex Euclidean domains in $\mathbb R^3$, independently of whether the notion of extrinsic or intrinsic diameter is used. It is straightforward to construct other examples of Euclidean domains in $\mathbb R^n$, where $n\geqslant 3$, with similar properties. As was mentioned to us by A. Savo~\cite{S}, there are also examples of non-convex planar domains for which Cheng's upper bound~\eqref{ourcheng} fails. All these examples are closely related to the concentration of measure phenomenon for large eigenvalues, see~\cite{CS11} for details.
\end{example}
Now we state the version of Theorem~\ref{ueb:closed} for the Dirichlet eigenvalue problem, which to our knowledge, appears to be missing in the literature. It involves the maximal radius $\rad(\Omega)$ of an interior rolling ball, and holds for domains with smooth boundary that are not necessarily convex.
\begin{theorem}
\label{ueb:dp}
Let $(M,g)$ be a complete Riemannian manifold whose Ricci curvature is bounded below $\mathit{Ricci}\geqslant -(n-1)\kappa$, where $\kappa\geqslant 0$, and $\Omega\subset M$ be a precompact domain with smooth boundary. Then there exist constants $C_i$, where $i=11,\dots, 14$ depending on the dimension only such that the Dirichlet eigenvalues $\nu_k(\Omega)$ satisfy the following inequalities:
\begin{equation}
\label{cheng:dp}
\nu_k(\Omega)\leqslant C_{11}(\kappa+\rad^{-2})+C_{12}((k+1)/\bar d)^2\qquad\text{for any}\quad k\geqslant 0,
\end{equation}
and
\begin{equation}
\label{buser:dp}
\nu_k(\Omega)\leqslant C_{13}(\kappa+\rad^{-2})+C_{14}((k+1)/\mathit{Vol}(\Omega))^{2/n}\qquad\text{for any}\quad k\geqslant 0,
\end{equation}
where $\rad=\rad(\Omega)$ is the maximal radius of an interior rolling ball, and $\bar d=\bar d(\Omega)$ is the intrinsic diameter of $\Omega$.
\end{theorem}
Since the extrinsic diameter $d(\Omega)$ is not greater than the intrinsic diameter $\bar d(\Omega)$, we conclude that estimate~\eqref{cheng:dp} holds also for the former in the place of the latter. The examples obtained by rounding long thin rectangles in the Euclidean plane show that the inequalities in the theorem above fail to hold even for convex domains if the quantity $\rad(\Omega)$ on the right-hand side is removed. If a domain $\Omega$ has corners, and thus $\rad(\Omega)=0$, Theorem \ref{ueb:dp} can be applied to any smooth domain contained inside $\Omega$, yielding upper bounds on $\nu_k(\Omega)$ using the domain monotonicity.
It is important to mention that the upper bounds for the Dirichlet eigenvalues in Theorem~\ref{ueb:dp} are also upper bounds for the Neumann eigenvalues. In particular, inequality~\eqref{cheng:dp} for the Neumann eigenvalues can be viewed as a version of~\eqref{ourcheng} for non-convex domains; due to Example~\ref{counter} the quantity $\rad(\Omega)$ is necessary. On the other hand, inequality~\eqref{buser:dp} does not give anything new for the Neumann problem, since a stronger inequality~\eqref{cm} due to Colbois and Maerten holds.
The proofs of Theorems~\ref{ueb:np} and~\ref{ueb:dp} follow the original strategy, used by Cheng and Buser, and are based on versions of volume comparison theorems. They appear in Section~\ref{proofs:upper}.
\subsection{Multiplicity bounds and related open problems}
Recall that a classical result due to Cheng~\cite{Che76} says that the multiplicities $m_k(g)$ of the Laplace eigenvalues $\lambda_k(g)$ on a closed Riemannian surface are bounded in terms of the index $k$ and the topology of the surface. The estimate obtained by Cheng has been further improved by Besson~\cite{Be80} and Nadirashvili~\cite{Na87}, and since then
related questions have been studied extensively in the literature, see~\cite{CC,CdV86,HMN,KKP,GK} and references therein for further details. Note that even the fact that eigenvalue multiplicities on Riemannian surfaces of fixed topology are bounded is by no means trivial, and due to the results of Colin de Verdi\`ere~\cite{CdV}, fails in higher dimensions. More precisely, in dimension $n\geqslant 3$ for any closed manifold $M$ any finite part of the spectrum can be prescribed by choosing an appropriate Riemmannian metric.
The purpose of the remaining part of the section is to discuss multiplicity bounds for Laplace eigenvalues in terms of geometric quantities, which seem to have been unnoticed in the literature. Recall that by the definition of the counting function, the multiplicity $m_k(g)$ of the Laplace eigenvalue $\lambda_k(g)$ satisfies the inequality $m_k(g)\leqslant N_g(\lambda_k+0)$. Thus, the combination of upper bounds for the counting function and the upper bounds for the Laplace eigenvalues yields the desired bounds for the multiplicities. For the convenience of references we state them below in the form of corollaries, considering the cases of the closed, Neumann, and Dirichlet eigenvalue problems consecutively. The first statement follows by combination of Theorem~\ref{gromov}, or rather inequalities~\eqref{leb:gro1} and~\eqref{improved}, with Theorem~\ref{ueb:closed}.
\begin{cor}
\label{mbc}
Let $(M,g)$ be a closed Riemannian manifold whose Ricci curvature satisfies the bound $\mathit{Ricci}\geqslant -(n-1)\kappa$, where $\kappa\geqslant 0$. Then there exist constants $C_{15}$ and $C_{16}$ depending on the dimension $n$ of $M$ only, such that the multiplicities $m_k(g)$ of the Laplace eigenvalues $\lambda_k(g)$ satisfy the inequalities
\begin{equation}
\label{mbc:diameter}
m_k(g)\leqslant C_{15}^{1+d\sqrt{\kappa}}(d\sqrt{\kappa}+k^n)\qquad\text{for any }k\geqslant 1,
\end{equation}
and
\begin{equation}
\label{mbc:volume}
m_k(g)\leqslant C_{16}(k+\mathit{Vol}_g(M)(\kappa^{n/2}+\inj^{-n}))\qquad\text{for any }k\geqslant 1,
\end{equation}
where $d$ and $\inj$ are the diameter and the injectivity radius of $M$ respectively.
\end{cor}
As a direct consequence of inequality~\eqref{mbc:diameter}, we see that for manifolds of non-negative Ricci curvature the multiplicities $m_k(g)$ are bounded in terms of the index $k$ and the dimension only. In this statement the hypothesis $\kappa=0$ can not be replaced by a weaker assumption $\kappa>0$, that is by a negative lower Ricci curvature bound. Indeed, this follows from the prescription results~\cite{CdV} together with the fact that the multiplicities $m_k(g)$ are invariant under scaling of a metric $g$. In a similar vein, Lohkamp~\cite{Lohk} shows that any finite part of spectrum can be prescribed by choosing an appropriate Riemannian metric whose volume can be normalised $\mathit{Vol}_g(M)=1$ and the Ricci curvature can be made negative and arbitrarily large in absolute value. This result indicates that the presence of the scale-invariant quantity $\mathit{Vol}(M)\kappa^{n/2}$ in inequality~\eqref{mbc:volume} is rather natural, and one may ask the following question.
\begin{open}
\label{quest1}
Apart from the index $k$ and the dimension, can the multiplicity $m_k(g)$ of a Laplace eigenvalue $\lambda_k(g)$ on a closed manifold $M$ be controlled by the volume and the lower Ricci curvature bound only?
\end{open}
The inequalities in Corollary~\ref{mbc} have two notable differences. First, the second inequality~\eqref{mbc:volume} is geometry free for a sufficiently large index $k$ in the sense that the second term is dominated by the first one. Second, it is linear in $k$, while the growth in $k$ in inequality~\eqref{mbc:diameter} has order $n$. Concerning the growth of multiplicities in the index $k$, recall that by the result of H\"ormander~\cite{Ho} the sharp remainder estimate in Weyl's law~\eqref{Weyl} is $O(\lambda^{(n-1)/2})$, and hence, for any given metric $g$ the quantity $m_k(g)k^{(1-n)/n}$ is bounded as $k\to+\infty$. In other words, for a sufficiently large $k$ the multiplicity $m_k(g)$ can not be greater than $C(g)\cdot k^{1-1/n}$, where $C(g)$ is a constant depending on a metric $g$. Though the dependence on the index $k$ in bound~\eqref{mbc:volume} might be satisfactory when the dimension $n$ is large, we ask the following question.
\begin{open}
In inequality~\eqref{mbc:volume} is the linear growth in $k$ the best possible? Can it be replaced by $ k^{1-1/n}$, where $n$ is the dimension of $M$?
\end{open}
Now we state a version of Corollary~\ref{mbc} for the Neumann eigenvalue problem. It is a consequence of Theorems~\ref{neumann:gromov} and~\ref{ueb:np}, and inequality~\eqref{cm}.
\begin{cor}
\label{mbn}
Let $(M,g)$ be a complete Riemannian manifold whose Ricci curvature is bounded below $\mathit{Ricci}\geqslant -(n-1)\kappa$, where $\kappa\geqslant 0$, and $\Omega\subset M$ be a geodesically convex precompact domain with Lipschitz boundary. Then the multiplicities $m_k(g)$ of the Neumann eigenvalue problem on $\Omega$ satisfy inequality~\eqref{mbc:diameter}, with $d=d(\Omega)$ being the diameter of the domain. In addition, if the boundary of $\Omega$ is smooth, then the multiplicities $m_k(g)$ also satisfy the inequality
\begin{equation}
\label{mbn:volume}
m_k(g)\leqslant C_{17}(k+\mathit{Vol}_g(\Omega)(\kappa^{n/2}+\inj^{-n}+\rad^{-n}))\qquad\text{for any }k\geqslant 1,
\end{equation}
where $\rad(\Omega)$ is the maximal radius of an interior rolling ball, $\inj(\Omega)$ is the infimum of the injectivity radii over $\Omega$, and the constant $C_{17}$ depends on the dimension of $M$ only.
\end{cor}
Following the discussion above for the eigenvalue problem on a closed manifold, we see that the multiplicities $m_k(g)$ of the Neumann eigenvalues of any geodesically convex domain $\Omega$ in the manifold of non-negative Ricci curvature are bounded in terms of the index $k$ and the dimension $n$ only. This statement, and hence also inequality~\eqref{mbc:diameter}, is false without the convexity assumption: indeed, by~\cite{CdV} in dimension $n\geqslant 3$ one can construct Euclidean domains with arbitrary high multiplicities of Neumann eigenvalues.
We end this section with a discussion of the multiplicity bounds for the Dirichlet eigenvalue problem. The following statement is a consequence of Theorems~\ref{neumann:gromov} and~\ref{ueb:dp}.
\begin{cor}
\label{mbd}
Let $(M,g)$ be a complete Riemannian manifold whose Ricci curvature is bounded below $\mathit{Ricci}\geqslant -(n-1)\kappa$, where $\kappa\geqslant 0$, and $\Omega\subset M$ be a geodesically convex precompact domain with smooth boundary. Then there exist constants $C_{18}$ and $C_{19}$ depending on the dimension only such that the multiplicities $m_k(g)$ of the Dirichlet eigenvalues $\nu_k(g)$ satisfy the following inequalities:
\begin{equation}
\label{mbd:diameter}
m_k(g)\leqslant C_{18}^{1+d\sqrt{\kappa}}((d\sqrt{\kappa})^n+(d/\rad)^n+k^n)
\end{equation}
and
\begin{equation}
\label{mbd:volume}
m_k(g)\leqslant C_{19}(k+1+\mathit{Vol}(\Omega)(\kappa^{n/2}+\inj^{-n}+\rad^{-n})),
\end{equation}
where $\rad=\rad(\Omega)$ is the maximal radius of an interior rolling ball, and $d=d(\Omega)$ is the diameter of $\Omega$.
\end{cor}
Note that all multiplicity bounds in the corollaries above are in fact bounds for the sums $\sum_{i\leqslant k} m_i(g)$, and in particular, may not reflect the actual behaviour of the individual multiplicities. It is plausible that in particular instances the multiplicities satisfy better bounds. For example, considering inequality~\eqref{mbd:diameter} for Euclidean domains, one can ask whether the remaining dependence on geometry is actually necessary.
\begin{open}
\label{open:1}
Does there exist a constant $C(n,k)$ depending on the dimension $n\geqslant 3$ and the index $k \geqslant 1$, such that the multiplicity of the $k$-th Dirichlet eigenvalue of a Euclidean domain $\Omega\subset\mathbb R^n$ is bounded above by $C(n,k)$?
\end{open}
Clearly, $C(n,0)=1$ for all $n$, and by the results of~\cite{Na87}, see also~\cite{HMN, KKP, Berd}, one can take $C(2,k)=2k+1$ for $k\geqslant 1$. To our knowledge, the question above is open even for convex domains, where we have a positive answer for the Neumann problem, see the first statement in Corollary~\ref{mbn}. If instead of Euclidean domains we consider arbitrary Riemannian manifolds with boundary, then the answer to Open Problem \ref{open:1} is negative.
Indeed, by~\cite{CC, CdV} for any integers $n\geqslant 2$, $k\geqslant 1$, and $N>0$ there exists a closed manifold $M$ of dimension $n$, such that $m_{k}(M)>N$. Then, for a sufficiently small $\epsilon>0$ the multiplicity of the $k$-th Dirichlet eigenvalue of the cylinder $M \times [-\epsilon, \epsilon]$, equipped with the product metric, also satisfies $m_{k}(M)>N$.
Note also that the methods used in~\cite{CC, CdV} to construct closed surfaces with Laplace eigenvalues of high multiplicity can be generalized directly to surfaces with Neumann boundary conditions. However, the approach does not extend in a straightforward way to the case of the Dirichlet boundary conditions. It would be interesting to know whether for any $k\geqslant 1$ there exists a surface with boundary whose $k$th Dirichlet eigenvalue has an arbitrary large multiplicity.
In higher dimensions the Dirichlet eigenvalues also behave differently: they satisfy the so-called universal inequalities, and hence, there is no analogue of the eigenvalue prescription results~\cite{CdV} for this problem. Nevertheless, it is still possible that the multiplicities can be prescribed; we state this question in the form of the following problem.
\begin{open}
Let $M$ be a manifold with boundary of dimension $n \geqslant 3$. For given integers $k\geqslant 1$ and $N\geqslant 1$ does there exist a Riemannian metric on $M$ such that the multiplicity of the $k$-th Dirichlet eigenvalue is equal to $N$?
\end{open}
We conclude with a few remarks on multiplicity bounds similar to inequality~\eqref{mbd:volume}. Recall that for a convex Euclidean domain $\Omega\subset\mathbb R^n$ it takes the form
$$
m_k(g)\leqslant C_{19}(k+1+\mathit{Vol}(\Omega)/\rad^n).
$$
For arbitrary precompact Euclidean domains one can also bound the multiplicity in terms of volume and inradius; the latter quantity is defined as the maximal radius of an inscribed ball
$$
\rho(\Omega)=\sup\{r: B(x,r)\subset \Omega~\text{for some $x\in\Omega$}\}.
$$
In more detail, by the result of Li and Yau~\cite{LY83} the Dirichlet counting function of an arbitrary domain $\Omega\subset\mathbb R^n$ satisfies the inequality $N(\lambda)\leqslant C_{20}\mathit{Vol}(\Omega)\lambda^{n/2}$, where $C_{20}$ is a constant depending only on the dimension. Combining this inequality with the upper bound due to Cheng and Yang~\cite[Proposition 3.1]{CY07}:
$$
\nu_k(\Omega)\leqslant \frac{n+3}{n}\nu_0(\Omega)(k+1)^{2/n}\qquad\text{for any }k\geqslant n,
$$
we obtain
\begin{multline*}
m_k(g)\leqslant C_{21}\mathit{Vol}(\Omega)\nu_0(\Omega)^{n/2}(k+1)\leqslant C_{21}\mathit{Vol}(\Omega)\nu_0(B({\rho}))^{n/2}(k+1)\\ \leqslant C_{22}\left(\mathit{Vol}(\Omega)/\rho(\Omega)^n\right)(k+1),
\end{multline*}
where $B({\rho})$ is an inscribed ball of radius $\rho=\rho(\Omega)$, and in the second inequality we used the domain monotonicity. Note that a similar multiplicity bound for Neumann eigenvalues does not hold if $n\geqslant 3 $, as one can prescribe any finite part of the Neumann spectrum while keeping the volume and the inradius of a domain bounded. The last statement can be deduced by inspecting the arguments in~\cite[pp. 610-611]{CdV}.
\section{Poincar\'e inequality and coverings by metric balls}
\label{prem}
\subsection{{Poincar\'e inequality}}
A key ingredient in our approach to the lower eigenvalue bounds by Gromov and Buser is the following version of the Neumann-Poincar\'e inequality.
\begin{prop}
\label{np:intro}
Let $(M,g)$ be a complete Riemannian manifold whose Ricci curvature is bounded below, $\mathit{Ricci}\geqslant -(n-1)\kappa$, where $\kappa\geqslant 0$ and $n$ is the dimension of $M$. Then, for any $p\geqslant 1$, there exists a constant $C_N=C_N(n,p)$ that depends on the dimension $n$ and $p$ only, such that for any smooth function $u$ on $M$ the following inequality holds:
$$
\int_{B_R}\abs{u-u_R}^pd\mathit{Vol}\leqslant C_NR^p{e^{(n-1)R\sqrt{\kappa}}}\int_{B_{2R}}\abs{\nabla u}^pd\mathit{Vol},
$$
where $B_R$ and $B_{2R}$ are concentric metric balls in $M$ of radii $R$ and $2R$ respectively, and $u_R$ is the mean-value of $u$ on $B_R$, i.e. $u_R=\mathit{Vol}(B_R)^{-1}\int_{B_R}u\,d\!\mathit{Vol}$.
\end{prop}
The statement above is folkloric; related results can be found in~\cite[section 5]{Bu82} and~\cite{SC02}. We extend the above inequality to the case of convex domains in Riemannian manifolds.
\begin{prop}
\label{np4np}
Let $(M,g)$ be a complete Riemannian manifold whose Ricci curvature is bounded below, $\mathit{Ricci}\geqslant -(n-1)\kappa$, where $\kappa\geqslant 0$ and $n$ is the dimension of $M$. Then for any $p\geqslant 1$ there exists a constant $C_N=C_N(n,p)$ that depends on the dimension $n$ and $p$ only such that for any geodesically convex domain $\Omega\subset M$ and for any smooth function $u$ on $\Omega$ the following inequality holds
\begin{equation}\label{poindomain}
\int_{B_R\cap\Omega}\abs{u-u_R}^pd\mathit{Vol}\leqslant C_NR^p{e^{(n-1)R\sqrt{\kappa}}}\int_{B_{2R}\cap\Omega}\abs{\nabla u}^pd\mathit{Vol},
\end{equation}
where $B_R$ and $B_{2R}$ are concentric metric balls in $M$ of radii $R$ and $2R$ respectively, and $u_R$ is the mean-value of $u$ on $B_R\cap\Omega$, i.e. $u_R=\mathit{Vol}(B_R\cap\Omega)^{-1}\int_{B_R\cap\Omega}ud\mathit{Vol}$.
\end{prop}
The inequality in Proposition \ref{np4np} (with a slightly different constant in the exponent) can be obtained by building on the argument used in~\cite[Chap.~5]{SC02}. Below we give a shorter proof, avoiding technicalities by using the so-called segment inequality due to Cheeger and Colding~\cite{CC96}. Before stating it we introduce the following notation: we set
$$
C(n,\kappa, R)=2R\sup_{0<s/2\leqslant t\leqslant s}\frac{\mathit{Vol}(\partial B_\kappa(s))}{\mathit{Vol}(\partial B_\kappa(t))},
$$
where $R>0$ and $\partial B_\kappa(r)$ is a sphere of radius $r$ in an $n$-dimensional simply connected space of constant sectional curvature $-\kappa$. Note that for $\kappa\geqslant 0$ the ratio of volumes above is not greater than $(s/t)^{n-1}e^{(n-1)s\sqrt{\kappa}}$, and we obtain
\begin{equation}
\label{rem}
C(n,\kappa,R)\leqslant 2^nRe^{(n-1)R\sqrt{k}}
\end{equation}
The following proposition is a reformulation of~~\cite[Theorem 2.11]{CC96}.
\begin{prop}[The segment inequality]
\label{segmentinq}
Let $(M,g)$ be a complete Riemannian manifold whose Ricci curvature is bounded below, $\mathit{Ricci}\geqslant -(n-1)\kappa$, where $\kappa\geqslant 0$ and $n$ is the dimension of $M$. Let $B_R$ be a metric ball, $A$ and $B$ be open subsets in $B_R$, and $W\subset M$ be an open subset that contains the convex hull of the union $A\cup B$. Then for any nonnegative integrable function $F$ on $W$ the following inequality holds:
\begin{equation}
\label{cheeger-colding}
\int_{A\times B}\int_0^{d(x,y)}F(\gamma_{x,y}(s))ds\, dx \,dy\leqslant C(n,\kappa,R) (\mathit{Vol}(A)+\mathit{Vol}(B))\int_{W} F(z) dz,
\end{equation}
where $\gamma_{x,y}:[0,d(x,y)]\to M$ is a shortest geodesic joining $x$ and $y$, and the first integral on the left hand-side is taken over the subset of $A\times B$ formed by the pairs $(x,y)$ of points that can be joined by such a unique geodesic.
\end{prop}
\begin{proof}[Proof of Proposition \ref{np4np}]
For arbitrary open subsets $A$ and $B$ consider the set of pairs $(x,y)\in A\times B$ such that the points $x$ and $y$ can be joined by a unique shortest geodesic $\gamma_{x,y}$. By standard results in Riemannian geometry, see~\cite{Cha}, its complement in $A\times B$ has zero measure, and abusing the notation, we also denote it below by $A\times B$.
It is not hard to see that for any $x\in M$ the inequality
$$
\abs{u-u_R}^p(x)\leqslant\mathit{Vol}(B_R\cap\Omega)^{-1}\int_{B_R\cap\Omega}\abs{u(x)-u(y)}^pdy.
$$
holds, where $u_R=\mathit{Vol}(B_R\cap\Omega)^{-1}\int_{B_R\cap\Omega}u$. Indeed, for $p=1$ it is straightforward, and for $p>1$ it can be obtained from the former case by using the H\"older inequality. Integrating it over $B_R\cap\Omega$, we obtain
\begin{multline*}
\int_{B_R\cap\Omega}\abs{u(x)-u_R}^pdx\leqslant\mathit{Vol}(B_R\cap\Omega)^{-1}\int_{B_R\cap\Omega}\int_{B_R\cap\Omega}\abs{u(x)-u(y)}^pdxdy\\
\leqslant\mathit{Vol}(B_R\cap\Omega)^{-1}\int_{(B_R\cap\Omega)\times (B_R\cap\Omega)}\left(\int_0^{d(x,y)}\abs{\nabla u(\gamma_{x,y}(s))}ds\right)^pdxdy\\
\leqslant (2R)^{p-1}\mathit{Vol}(B_R\cap\Omega)^{-1}\int_{(B_R\cap\Omega)\times(B_R\cap\Omega)}\int_0^{d(x,y)}\abs{\nabla u(\gamma_{x,y}(s))}^pdsdxdy,
\end{multline*}
where in the last inequality we used the H\"older inequality and the relation $\dist(x,y)\leqslant 2R$. Since $\Omega$ is convex, the convex hull of $B_R\cap\Omega$ lies in $B_{2R}\cap\Omega$. Thus, applying Proposition~\ref{segmentinq} with $A=B=B_R\cap\Omega$ and $W=B_{2R}\cap\Omega$, and using inequality~\eqref{rem}, we obtain
$$
\int_{(B_R\cap\Omega)\times (B_R\cap\Omega)}\int_{0}^{d(x,y)}\abs{\nabla u(\gamma_{x,y}(s))}^p ds\,dx\, dy\leqslant 2^{n+1}e^{(n-1)R\sqrt{\kappa}}R\mathit{Vol}(B_R\cap\Omega)\int_{B_{2R}\cap\Omega}\abs{\nabla u(z)}^pdz.
$$
Combining the last two inequalities, we arrive at the Poincare inequality \eqref{poindomain}.
\end{proof}
\subsection{{Coverings by metric balls}: closed manifolds}
We proceed with the estimates for the cardinality and multiplicity of certain coverings. The following lemma is by now a standard application of the Gromov-Bishop volume comparison theorem. We outline its proof for the sake of completeness.
\begin{lemma}
\label{cp:l1}
Let $(M,g)$ be a closed Riemannian manifold whose Ricci curvature satisfies the bound $\mathit{Ricci}\geqslant -(n-1)\kappa$, where $\kappa\geqslant 0$. Let $(B_i)$ be a covering of $M$ by balls $B_i=B(x_i,\rho)$ such that the balls $B(x_i,\rho/2)$ are disjoint. Then:
\begin{itemize}
\item[(i)] for any $0<\rho\leqslant 2d$ the cardinality of the family $(B_i)$ is not greater than $2^ne^{(n-1)d\sqrt{\kappa}}(d/\rho)^n$, where $d$ is the diameter of $M$;
\item[(ii)] for any $\rho>0$ and for any $x\in M$ the number of balls from $(B(x_i,2\rho))$ that contain $x$ is not greater than ${12}^ne^{6(n-1)\rho\sqrt{\kappa}}$.
\end{itemize}
\end{lemma}
\begin{proof}
First, by the relative volume comparison theorem, see~\cite{Cha}, it is straightforward to show that the volumes of concentric metric balls of radii $0<r\leqslant R$ satisfy the relation
\begin{equation}
\label{vc}
\mathit{Vol}(B_R)\leqslant e^{(n-1)R\sqrt{\kappa}}(R/r)^n\mathit{Vol}(B_r).
\end{equation}
Now to prove~$(i)$ note that $m=\card(B_i)$ satisfies the following relations
$$
m\cdot\inf_i\mathit{Vol}(B(x_i,\rho/2))\leqslant\sum_i\mathit{Vol}(B(x_i,\rho/2))\leqslant\mathit{Vol}(M).
$$
Let $x_{i_0}$ be a point at which the infimum in the left hand-side above is achieved. Then, for any $0<\rho\leqslant 2d$ we obtain
$$
m\leqslant\mathit{Vol}(B(x_{i_0},d))/\mathit{Vol}(B(x_{i_0},\rho/2))\leqslant 2^ne^{(n-1)d\sqrt{\kappa}}(d/\rho)^n,
$$
where in the last inequality we used~\eqref{vc}.
To prove the statement~$(ii)$ we re-denote by $x_{i_0}$ the point at which the infimum $\inf\mathit{Vol}(B(x_i,\rho/2))$ is achieved while $i$ ranges over all indices such that the balls $B(x_i,2\rho)$ contain $x$. Note that if $x\in B(x_i,2\rho)$, then $B(x_i,2\rho)\subset B(x_{i_0},6\rho)$. Thus, for any $\rho>0$ we obtain that
$$
\mult_x(B_i)\leqslant\mathit{Vol}(B(x_{i_0},6\rho))/\mathit{Vol}(B(x_{i_0},\rho/2))\leqslant {12}^ne^{6(n-1)\rho\sqrt{\kappa}},
$$
where in the last inequality we again used~\eqref{vc}.
\end{proof}
For a proof of the Buser inequality in Theorem~\ref{gromov} we also need the following supplement to Lemma~\ref{cp:l1}.
\begin{lemma}
\label{cp:l2}
Under the hypotheses of Lemma~\ref{cp:l1}, the cardinality of the family $(B(x_i,\rho))$
is not greater than $c_1\mathit{Vol}(M)(\min\{\rho,\inj\})^{-n}$, where $\inj$ is the injectivity radius of $M$, and $c_1$ is a constant that depends on the dimension $n$ only.
\end{lemma}
\begin{proof}
As in the proof of Lemma~\ref{cp:l1}, we see that
$$
m=\card(B_i)\leqslant\mathit{Vol}(M)/\mathit{Vol}(B(x_{i_0},\rho/2))
$$
for some point $x_{i_0}$. Recall that by~\cite[Prop.~14]{Cro80} the volume of a geodesic ball satisfies the inequality
$$
c_2\rho^n\leqslant\mathit{Vol}(B(x,\rho/2))\qquad\text{for any}\quad\rho\leqslant\inj,
$$
where $c_2$ is a constant that depends on $n$ only. For $\rho\geqslant\inj$, we clearly have
$$
c_2\inj^n\leqslant\mathit{Vol}(B(x,\inj/2))\leqslant\mathit{Vol}(B(x,\rho/2)).
$$
Combining these inequalities with the bound for the cardinality $m$ above, we complete the proof of the lemma.
\end{proof}
\subsection{Coverings by metric balls: domains}
Now we discuss versions of the above statements for coverings of domains in Riemannian manifolds.
\begin{lemma}
\label{np:l}
Let $(M,g)$ be a complete Riemannian manifold whose Ricci curvature is bounded below, $\mathit{Ricci}\geqslant -(n-1)\kappa$, where $\kappa\geqslant 0$ and $n$ is the dimension of $M$. Let $\Omega\subset M$ be a precompact domain, and $(B_i)$ be its covering by balls $B_i=B(x_i,\rho)$ such that $x_i\in\Omega$ and the balls $B(x_i,\rho/2)$ are disjoint. Then:
\begin{itemize}
\item[(i)] if $\Omega$ is convex, the conclusions of Lemma~\ref{cp:l1} hold, where $d=d(\Omega)$ is the (extrinsic) diameter of $\Omega$;
\item[(ii)] if $\Omega$ has a smooth boundary, the cardinality of the covering $(B_i)$ is not greater than $c_3\mathit{Vol}(\Omega)(\min\{\rho,\inj,\rad\})^{-n}$, where $c_3$ is a constant that depends on $n$ only, $\inj=\inj(\Omega)$ is the injectivity radius of $\Omega$, and $\rad=\rad(\Omega)$ is the maximal radius of an inscribed rolling ball;
\item[(iii)] if $M$ has finite volume, then the cardinality of the covering $(B_i)$ is not greater than $c_4\mathit{Vol}(M)(\min\{\rho,\inj\})^{-n}$.
\end{itemize}
\end{lemma}
In the sequel we use the following folkloric version of Gromov-Bishop relative volume comparison theorem, see~\cite[p.524]{Gro}; we outline its proof for the sake of completeness.
\begin{lemma}
\label{GB}
Let $(M,g)$ be a complete Riemannian manifold whose Ricci curvature is bounded below, $\mathit{Ricci}\geqslant -(n-1)\kappa$, where $\kappa\geqslant 0$, and $\Omega\subset M$ be a precompact domain that is star-shaped with respect to a point $x\in\bar\Omega$. Then the quotient $\mathit{Vol}(B(x,r)\cap\Omega)/\mathit{Vol}(B_\kappa(r))$, where $B_\kappa(r)$ is a ball in the space of constant curvature $(-\kappa)$, is a non-increasing function in $r>0$. In particular, for any $0<r\leqslant R$ we have
$$
\mathit{Vol}(B(x,R)\cap\Omega)\leqslant e^{(n-1)R\sqrt{\kappa}}(R/r)^n\mathit{Vol}(B(x,r)\cap\Omega).
$$
\end{lemma}
\begin{proof}
For a given subset $S$ of a unit sphere $S^{n-1}\subset\mathbb R^n$ denote by $C_S$ the cone $\{\exp_x(t\xi): t>0, \xi\in S\}$. The standard proof of the Gromov-Bishop comparison theorem, see~\cite[p.134-135]{Cha}, shows that the quotient $\mathit{Vol}(B(x,r)\cap C_S)/\mathit{Vol}(B_\kappa(r))$ is a non-increasing function in $r>0$. For a given $0<r\leqslant R$ define $S$ as the set formed by $\xi\in S^{n-1}$ such that $\exp_x(r\xi)\in\Omega$. Since $\Omega$ is star-shaped with respect to $x$, we conclude that:
\begin{itemize}
\item[{\it (a)}] $B(x,r)\cap C_S\subset B(x,r)\cap\Omega$,
\item[{\it (b)}] $(B(x,R)\backslash B(x,r))\cap\Omega\subset(B(x,R)\backslash B(x,r))\cap C_S$.
\end{itemize}
By relation $(a)$ the quantity
$$
h:=\mathit{Vol}(B(x,r)\cap\Omega)-\mathit{Vol}(B(x,r)\cap C_S)
$$
is non-negative, and by~$(b)$, we obtain
$$
\mathit{Vol}(B(x,R)\cap\Omega)-h\leqslant\mathit{Vol}(B(x,R)\cap C_S).
$$
Finally, using the Gromov-Bishop theorem for the intersections of balls with cones, we obtain
\begin{multline*}
\mathit{Vol}(B(x,R)\cap\Omega)/\mathit{Vol}(B(x,r)\cap\Omega)\leqslant (\mathit{Vol}(B(x,R)\cap\Omega)-h)/(\mathit{Vol}(B(x,r)\cap\Omega)-h)\\ \leqslant \mathit{Vol}(B(x,R)\cap C_S)/\mathit{Vol}(B(x,r)\cap C_S)\leqslant \mathit{Vol}(B_\kappa(R))/\mathit{Vol}(B_\kappa(r)).
\end{multline*}
The last statement of the lemma follows from the standard formula for the volume $\mathit{Vol}(B_k(r))$, see~\cite{Cha}, which leads to the estimate for the quotient $\mathit{Vol}(B_\kappa(R))/\mathit{Vol}(B_\kappa(r))$.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{np:l}]
Following the argument in the proof of Lemma~\ref{cp:l1}, we see that
\begin{equation}
\label{lemma:aux1}
m=\card(B_i)\leqslant\mathit{Vol}(\Omega)/\mathit{Vol}(B(x_{i_0},\rho/2)\cap\Omega)
\end{equation}
for some point $x_{i_0}\in\Omega$. If $d=d(\Omega)$ is the diameter of $\Omega$, then $\Omega$ lies in the ball $B(x_{i_0},d)$, and by Lemma~\ref{GB}, we obtain
$$
m\leqslant\mathit{Vol}(B(x_{i_0},d)\cap\Omega)/\mathit{Vol}(B(x_{i_0},\rho/2)\cap\Omega)\leqslant 2^ne^{(n-1)d\sqrt{\kappa}}(d/\rho)^n.
$$
The estimate for the multiplicity of the covering $(B_i)$ is the same as in the proof of Lemma~\ref{cp:l1}. We proceed with the statement~(ii): by relation~\eqref{lemma:aux1} for a proof it is sufficient to show that
\begin{equation}
\label{lemma:aux2}
c_5\rho^n\leqslant\mathit{Vol}(B(x,\rho/2)\cap\Omega)\qquad\text{for any}\quad\rho\leqslant\min\{\inj,\rad\}.
\end{equation}
To see that the above relation holds note that for any ball $B(x,r)$, where $x\in\Omega$ and $r< 2\rad(\Omega)$, there exists a point $\tilde x\in B(x,r)$ such that
$$
\dist(x,\tilde x)< r/2\qquad\text{and}\qquad B(\tilde x,r/2)\subset B(x,r)\cap\Omega.
$$
Indeed, the statement is clear if $B(x,r/2)\subset\Omega$. If $B(x,r/2)$ does not lie entirely in $\Omega$, then since $B(\tilde x,r/2)\subset\Omega$, one can take an inscribed ball that touches the boundary $\partial\Omega$ at a point $p$ where the minimum of the distance $\dist(q,x)$, while $q$ ranges over $\partial\Omega$, is achieved. It is straightforward to see that the point $x$ belongs to the shortest geodesic arc joining $\tilde x$ and $p$, which meets the boundary $\partial\Omega$ at the point $p$ orthogonally. In particular, the ball $B(\tilde x,r/2)$ is also contained in the ball $B(x,r)$. Thus, we conclude that under our hypotheses
$$
\mathit{Vol}(B(\tilde x,\rho/4))\leqslant\mathit{Vol}(B(x,\rho/2)\cap\Omega),
$$
and by~\cite[Prop.~14]{Cro80} the quantity on the left-hand side is at least $c_5\rho^n$ when $\rho/2<\inj(\Omega)$. Combining the last statement with the hypothesis $\rho/2<\rad(\Omega)$, we prove relation~\eqref{lemma:aux2}.
Under the hypotheses of the last statement of the lemma we may bound the cardinality $m$ of the covering by $\mathit{Vol}(M)/\mathit{Vol}(B(x_{i_0},\rho/2))$, and then appeal directly to Croke's result~\cite[Prop.~14]{Cro80} in the same fashion as in the proof of Lemma~\ref{cp:l2}.
\end{proof}
\section{Lower eigenvalue bounds}
\label{proofs:lower}
\subsection{Proof of Theorem~\ref{gromov}: Gromov's inequalities}
We prove estimate~\eqref{leb:gro1} for the counting function $N_g(\lambda)$.
{For a given real number $\lambda>0$ denote by $E(\lambda)$ the sum of all eigenspaces that correspond to the eigenvalues $\lambda_k(g)<\lambda$. Recall that by the variational principle, for any $0\ne\varphi\in E(\lambda)$ we have
\begin{equation}
\label{rayleigh}
\int_M\abs{\nabla\varphi}^2d\mathit{Vol}<\lambda\int_M\varphi^2d\mathit{Vol}.
\end{equation}
For a given $\rho>0$ consider a covering of $M$ by balls $B_i=B(x_i,\rho)$ such that the balls $B(x_i,\rho/2)$ are disjoint. It can be obtained by choosing the collection of balls $B(x_i,\rho/2)$ to be a maximal collection of disjoint balls. Given such a covering $(B_i)$ we define the map
$$
\Phi_\lambda:E(\lambda)\to\mathbb R^m,\qquad u\mapsto\mathit{Vol}(B_i)^{-1}\int_{B_i}u\, d\mathit{Vol},
$$
where $m$ stands for the cardinality of $(B_i)$, and $i=1,\ldots, m$, cf.~\cite{ML}. We claim that there exists a constant $c_6$ depending on the dimension $n$ only such that if $\lambda^{-1}\geqslant c_6\rho^{2}e^{7(n-1)\rho\sqrt{\kappa}}$, then the map $\Phi_\lambda$ is injective. To see this we define $c_6={12}^{n}C_N$, where $C_N=C_N(n,2)$ is the Poincar\'e constant from Proposition~\ref{np:intro}, and argue by assuming the contrary. Suppose that $\varphi\ne 0$ belongs to the kernel of $\Phi_\lambda$. Then we obtain
\begin{multline*}
\int_M\varphi^2d\mathit{Vol}\leqslant\sum_i\int_{B_i}\varphi^2d\mathit{Vol}\leqslant C_N\rho^2e^{(n-1)\rho\sqrt{\kappa}}\sum_i\int_{2B_i}\abs{\nabla\varphi}^2d\mathit{Vol}\\
\leqslant c_6\rho^2e^{7(n-1)\rho\sqrt{\kappa}}\int_M\abs{\nabla\varphi}^2d\mathit{Vol},
\end{multline*}
where we used Proposition~\ref{np:intro} in the second inequality and Lemma~\ref{cp:l1} in the last. Now combining these relations with~\eqref{rayleigh}, we conclude that $\lambda^{-1}<c_6\rho^{2}e^{7(n-1)\rho\sqrt{\kappa}}$, and arrive at a contradiction. Thus, for a sufficiently small $\rho$ the map $\Phi_\lambda$ is injective, and the value $N_g(\lambda)$ is not greater than the cardinality $m$ of a covering $(B_i)$.}
For a given $\lambda>0$ we set
$$
\rho_0=(c_6\lambda e^{14(n-1)d\sqrt{\kappa}})^{-1/2},
$$
where $d$ is the diameter of $M$. When $\rho_0\leqslant 2d$, it is straightforward to check that the relation $\lambda^{-1}\geqslant c_6\rho_0^{2}e^{7(n-1)\rho_0\sqrt{\kappa}}$ holds, and by Lemma~\ref{cp:l1} we obtain
$$
N_g(\lambda)\leqslant m\leqslant 2^ne^{(n-1)d\sqrt{\kappa}}(d/\rho_0)^n\leqslant C_4^{1+d\sqrt{\kappa}}d^n\lambda^{n/2}.
$$
To treat the case $\rho_0>2d$, note that there is only one covering with balls of radius $\rho\geqslant 2d$ that satisfies our hypotheses, and it consists of only one ball. In particular, if $\rho_0>2d$, then the covering under the consideration coincides with the one for $\rho_*=2d$ for which $\Phi_\lambda$ is also injective. Indeed, by the definition of $\rho_0$, it is straightforward to see that the relation $\rho_0\geqslant 2d$ implies that $\lambda^{-1}\geqslant c_6\rho_*^{2}e^{7(n-1)\rho_*\sqrt{\kappa}}$. Since such a covering consists of only one ball, we conclude that in this case $N_g(\lambda)$ is not greater than $1$. Combining these two cases, we finish the proof of the theorem.\qed
\begin{remark}
The idea to use the bounds for the first eigenvalue on small sets to get estimates for higher eigenvalues is not new; see, for example, the already mentioned papers by Cheng~\cite{Che75}, Gromov~\cite[Appendix~C]{Gro}, and Li and Yau~\cite{LY80}. A similar strategy has been used in~\cite{ML} in the context of the multiplicity bounds for Laplace eigenvalues. Note that the eigenvalue multiplicity bound for closed manifolds of non-negative Ricci curvature obtained in~\cite{ML}, see formula~$(6)$ in~\cite[Theorem~3.1]{ML}, is a partial case of~\eqref{mbc:diameter}, which is a consequence of the results of Cheng and Gromov cited above.
\end{remark}
\subsection{Proof of Theorem~\ref{gromov}: Buser's inequalities}
Consider a covering of $M$ by balls $B_i=B(x_i,\rho)$ such that $B(x_i,\rho/2)$ form a maximal family of disjoint balls. As is shown above, if a real number $\lambda>0$ satisfies the inequality $\lambda^{-1}\geqslant c_6\rho^{2}e^{7(n-1)\rho\sqrt{\kappa}}$, where $c_6$ depends on $n$ only, then $N_g(\lambda)$ is not greater than the cardinality $m=\card(B_i)$. In this case by Lemma~\ref{cp:l2} we have
\begin{equation}
\label{eq:count}
N_g(\lambda)\leqslant m\leqslant c_1\mathit{Vol}(M)(\rho^{-n}+\inj^{-n}).
\end{equation}
The hypothesis on $\rho$ is clearly satisfied, if $\lambda^{-1}\geqslant 2c_6\rho^2$ {and} $2\geqslant e^{7(n-1)\rho\sqrt{\kappa}}$. Thus, choosing $\rho=\rho_0$ as the minimum of the values $(2c_6\lambda)^{-1/2}$ and $\ln 2(7(n-1)\sqrt{\kappa})^{-1}$, by relation~\eqref{eq:count} we obtain
$$
N_g(\lambda)\leqslant C_3\mathit{Vol}(M)(\lambda^{n/2}+\kappa^{n/2}+\inj^{-n}),
$$
where the value of the constant $C_3$ depends on $c_1$, $c_6$, and the dimension $n$.
\qed
\subsection{Proof of Theorem~\ref{neumann:gromov}: Gromov's inequalities for domains}
The proof of estimate \eqref{leb:gro1} for convex domains follows a line of argument similar to the one in the proof of Theorem~\ref{gromov}; it uses the Neumann-Poincar\'e inequality in Proposition~\ref{np4np} and Lemma~\ref{np:l}.
More precisely, for $\lambda>0$ denote by $E(\lambda)$ the sum of all eigenspaces that correspond to the Neumann eigenvalues $\lambda_k(g)<\lambda$. Let $(B_i)$ be a covering of $\Omega$ by balls $B_i=B(x_i,\rho)$ such that $x_i\in\Omega$ and the smaller balls $B(x_i,\rho/2)$ are disjoint. We claim that if $\lambda^{-1}\geqslant c_6\rho^{2}e^{7(n-1)\rho\sqrt{\kappa}}$, then the map
$$
\Phi_\lambda:E(\lambda)\to\mathbb R^m,\qquad u\mapsto\mathit{Vol}(B_i\cap\Omega)^{-1}\int_{B_i\cap\Omega}ud\mathit{Vol},
$$
is injective, and the value $N_g(\lambda)$ is not greater than $m=\card(B_i)$. Indeed, suppose that a function $\varphi\ne 0$ belongs to the kernel of $\Phi_\lambda$. Then, setting $c_6={12}^nC_N$ with $C_N=C_N(n,2)$ being the constant from Proposition~\ref{np4np}, we obtain
\begin{multline*}
\int_\Omega\varphi^2d\mathit{Vol}\leqslant\sum_i\int_{B_i\cap\Omega}\varphi^2d\mathit{Vol}\leqslant C_N\rho^2e^{(n-1)\rho\sqrt{\kappa}}\sum_i\int_{2B_i\cap\Omega}\abs{\nabla\varphi}^2d\mathit{Vol}\\
\leqslant c_6\rho^2e^{7(n-1)\rho\sqrt{\kappa}}\int_\Omega\abs{\nabla\varphi}^2d\mathit{Vol},
\end{multline*}
where in the last relation we used Lemma~\ref{np:l}. Now we arrive at a contradiction in the same fashion as above.
For a given $\lambda>0$ we set
$$
\rho_0=(c_6\lambda e^{14(n-1)d\sqrt{\kappa}})^{-1/2},
$$
where $d$ is the diameter of $\Omega$. When $\rho_0\leqslant 2d$, it is straightforward to check that the above hypothesis on $\lambda$ holds, and the value $N_g(\lambda)$ is bounded by the cardinality of the
covering $B(x_i,\rho_0)$ such that $x_i\in\Omega$ and the balls $B(x_i,\rho_0/2)$ are disjoint. Then, by Lemma~\ref{np:l} we obtain
$$
N_g(\lambda)\leqslant m\leqslant 2^ne^{(n-1)d\sqrt{\kappa}}(d/\rho_0)^n\leqslant C_{4}^{1+d\sqrt{\kappa}}d^n\lambda^{n/2}.
$$
The case $\rho_0>2d$ is treated in the fashion similar to the one in the proof of Theorem~\ref{gromov}.
\qed
\subsection{Proof of Theorem~\ref{neumann:gromov}: Buser's inequalities for domains}
Let $(B_i)$ be a covering of $\Omega$ by balls $B_i=B(x_i,\rho)$ such that $x_i\in\Omega$ and the smaller balls $B(x_i,\rho/2)$ form a maximal family of disjoint balls. As is shown in the proof of Gromov's inequalities for Neumann eigenvalues, if $\lambda^{-1}\geqslant c_6\rho^{2}e^{7(n-1)\rho\sqrt{\kappa}}$, then the value $N_g(\lambda)$ is not greater than $m=\card(B_i)$. Now by Lemma~\ref{np:l} we have
$$
N_g(\lambda)\leqslant m\leqslant c_3\mathit{Vol}(\Omega)(\rho^{-n}+\inj^{-n}+\rad^{-n}).
$$
Choosing $\rho=\rho_0$ as the minimum of the values $(2c_6\lambda)^{-1/2}$ and $\ln 2(7(n-1)\sqrt{\kappa})^{-1}$, we obtain the desired bound on the counting function.
\qed
\begin{remark}
\label{finite}
When a manifold $M$ has a finite volume, the argument above yields the estimate
$$
N_\Omega(\lambda)\leqslant C\cdot\mathit{Vol}_g(M)(\lambda^{n/2}+\kappa^{n/2}+\inj(\Omega)^{-n}) \qquad\text{for any}\quad \lambda \geqslant 0,
$$
for the Neumann eigenvalues counting function of any compact geodesically convex domain $\Omega\subset M$. Indeed, this is a consequence of the following estimate for the cardinality $m$ of the covering $(B_i)$ with the properties described above:
$$
m=\card(B_i)\leqslant c_4\mathit{Vol}(M)(\rho^{-n}+\inj^{-n}),
$$
see Lemma~\ref{np:l}.
\end{remark}
\section{Upper eigenvalue bounds}
\label{proofs:upper}
\subsection{Proof of Theorem~\ref{ueb:np}}
Since $\Omega\subset M$ is geodesically convex, by Lemma~\ref{GB} for any $x\in\bar\Omega$ and any $0<r\leqslant 1/\sqrt{\kappa}$ we have
\begin{equation}
\label{gb}
\mathit{Vol}(B(x,r)\cap\Omega)/\mathit{Vol}(B(x,r/2)\cap\Omega)\leqslant 2^ne^{n-1}.
\end{equation}
For a given integer $k>0$ let $\rho(k)$ be the supremum of all $r>0$ such that there exists $k$ points $x_1,\ldots, x_k\in\bar\Omega$ with $\dist(x_i,x_j)>r$ for all $i\ne j$. We consider the following cases.
\smallskip
\noindent
{\it Case 1: $\rho(k)\geqslant 1/\sqrt{\kappa}$.} Then for any $r<1/\sqrt{\kappa}$ there exist points $x_1,\ldots,x_k\in\bar\Omega$ such that the balls $B(x_i,r/2)$ are disjoint. Consider the plateau functions $u_i$ supported in $B(x_i,r/2)$ such that $u_i\equiv 1$ on $B(x_i,r/4)$ and $\abs{\nabla u_i}\leqslant 4/r$. Their restrictions to $\Omega$ can be used as test-functions for the Neumann eigenvalue $\lambda_k(g)$, and by the variational principle we obtain
\begin{multline*}
\lambda_k(g)\leqslant\max_i\left(\int_\Omega\abs{\nabla u_i}^2d\mathit{Vol}\right)/\left(\int_\Omega u_i^2d\mathit{Vol}\right)\\
\leqslant 16r^{-2}\max_i\mathit{Vol}(B(x_i,r/2)\cap\Omega)/\mathit{Vol}(B(x_i,r/4)\cap\Omega)\leqslant 2^{n+4}e^{n-1}r^{-2}.
\end{multline*}
Taking the limit as $r\to 1/\sqrt{\kappa}$, we see that $\lambda_k(g)\leqslant C_{10}\kappa$.
\smallskip
\noindent
{\it Case 2: $\rho(k)<1/\sqrt{\kappa}$.} Following the argument above, we see that $\lambda_k(g)\leqslant C_{10}r^{-2}$ for any $0<r<\rho(k)$, and tending $r\to\rho(k)$, we obtain that $\lambda_k(g)\leqslant C_{10}\rho(k)^{-2}$. Now we claim that $\rho(k)\geqslant d/k$. Indeed, to see this we note that the closure a convex domain $\Omega$ contains a geodesic arc whose length equals the diameter $d$. Breaking it into sub-arcs of the length $d/k$, we conclude that $\rho(k)\geqslant (d/k)$, and hence, $\lambda_k(g)\leqslant C_{10}(k/d)^2$. Taking into account both cases we finish the proof of the theorem.
\qed
\subsection{Proof of Theorem \ref{ueb:dp}: Cheng's inequalities for the Dirichlet problem}
Below we give an argument based on Cheng's comparison theorem for the principal Dirichlet eigenvalue~\cite{Che75}; however, one can also argue as in the proof of Theorem~\ref{ueb:np} using the volume comparison theorem and constructing test-functions explicitly.
Denote by $\overline{\dist}(x,y)$ the intrinsic distance on $\Omega$, that is the infimum of lengths of paths in $\Omega$ joining the points $x$ and $y$. Note that the closure of $\Omega$ contains a continuous path $\gamma$ whose length equals the intrinsic diameter $\bar d=\bar d(\Omega)$; its existence follows from the Arzela-Ascoli theorem, see~\cite{BBI} for details. Breaking it into sub-arcs of the length $\bar d/(k+1)$, we find $(k+1)$ points $x_i$ on $\gamma$, where $i=0,\ldots, k$, such that
$$
\overline{\dist}(x_i,x_j)\geqslant2r:=\overline{d}/(k+1)\qquad\text{for any}\quad i\ne j.
$$
In particular, we see that the sets $D(x_i,r)=\{y\in \Omega: \overline{\dist}(x_i,y)< r\}$ are disjoint. Since the extrinsic distance is not greater than the intrinsic distance, we also conclude that each $D(x_i,r)$ lies in $B(x_i,r)\cap\Omega$.
Denote by $k_0$ the integer $\lfloor\bar d/(4\rad(\Omega))\rfloor $, the greatest integer that is at most $\bar d/(4\rad(\Omega))$. Following the argument in the proof of Lemma~\ref{np:l}, it is straightforward to see that for any $k\geqslant k_0$ and any $0\leqslant i\leqslant k$ there exists $\tilde x_i\in B(x_i,r)\cap\Omega$ such that
\begin{equation}
\label{dp:aux3}
\overline\dist(\tilde x_i,x_i)=\dist(\tilde x_i,x_i)\leqslant r/2\quad\text{and}\quad B(\tilde x_i,r/2)\subset B(x_i,r)\cap\Omega.
\end{equation}
Here in the first relation we used the fact that the point $\tilde x_i$ can be chosen such that $\tilde x_i$ and $x_i$ lie on a extrinsically shortest geodesic arc that is contained in $\Omega$. Since any extrinsically shortest path joining points in the ball $B(\tilde x_i,r/4)$ lies in the ball $B(\tilde x_i,r/2)\subset\Omega$, we conclude that the intrinsic and extrinsic distances coincide on $B(\tilde x_i,r/4)$. Using the first relation in~\eqref{dp:aux3}, it is then straightforward to see that the ball $B(\tilde x_i,r/4)$ lies in $D(x_i,r)$. In particular, the balls $B(\tilde x_i,r/4)$ are disjoint, and by the domain monotonicity principle and Cheng's comparison for the principal eigenvalue, we obtain
$$
\nu_k(\Omega)\leqslant\max_i\nu_0(B(\tilde x_i,r/4))\leqslant\nu_0(B_\kappa(r/4)),
$$
where $B_\kappa(r/4)$ is a ball of radius $r/4$ in the simply connected space of constant sectional curvature $(-\kappa)$. As is shown by Cheng~\cite{Che75}, there is a constant $c_7$ depending on the dimension only such that
$$
\nu_0(B_\kappa(r/4))\leqslant c_7(\kappa+r^{-2}).
$$
From the consideration above we conclude that for any integer $k\geqslant k_0$ the Dirichlet eigenvalue $\nu_k(\Omega)$ satisfies the inequality
\begin{equation}
\label{aux:k}
\nu_k(\Omega)\leqslant c_7\kappa+c_8((k+1)/\overline{d})^2.
\end{equation}
If $k_0=0$, then the statement is proved. If $k_0\geqslant 1$, then we can estimate the eigenvalue $\nu_{k_0}(\Omega)$ in the following fashion
\begin{equation}
\label{aux:k_0}
\nu_{k_0}(\Omega)\leqslant c_7(\kappa+r_0^{-2})\leqslant c_7(\kappa+\rad^{-2}),
\end{equation}
where we used that
$$
2r_0:=\bar d/(k_0+1)\geqslant 2\rad(\Omega).
$$
Finally, combining relations~\eqref{aux:k} and~\eqref{aux:k_0}, for any $k\geqslant 0$ we obtain
$$
\nu_k(\Omega)\leqslant\max\{\nu_{k_0}(\Omega),\nu_k(\Omega)\}\leqslant C_{11}(\kappa+\rad^{-2})+C_{12}((k+1)/\overline{d})^2,
$$
which is the desired inequality~\eqref{cheng:dp}.
\qed
\subsection{Proof of Theorem \ref{ueb:dp}: Buser's inequalities for the Dirichlet problem}
We start with recalling that by the Bishop volume comparison theorem for any $0<r\leqslant 1/\sqrt{\kappa}$ the volume of a metric ball $B(x,r)$ in $M$ satisfies the inequality
\begin{equation}
\label{bishop}
\mathit{Vol}(B(x,r))\leqslant n\omega_n\int_0^r t^{n-1}e^{(n-1)t\sqrt{\kappa}}dt\leqslant \omega_ne^{n-1}r^n,
\end{equation}
where $\omega_n$ is the volume of a unit ball in the Euclidean space $\mathbb R^n$, see~\cite{Cha}.
For a given integer $k>0$ let $\rho(k+1)$ be the supremum of all $r>0$ such that there exist $(k+1)$ points $x_0,\ldots,x_k\in\Omega$ with $\dist(x_i,x_j)>r$ for any $i\ne j$. Following the argument in the proof of Theorem~\ref{ueb:np}, we consider the two cases below.
\smallskip
\noindent
{\it Case~1: $\rho(k+1)\geqslant 1/\sqrt{\kappa}$.} For every $r<1/\sqrt{\kappa}$ there exist points $x_0,\ldots, x_k$ such that the balls $B(x_i,r/2)$ are disjoint. When $r\leqslant \rad$, then repeating the argument in the proof of Lemma~\ref{np:l}, we find points $\tilde x_i\in B(x_i,r/2)$ such that
$$
B(\tilde x_i,r/4)\subset B(x_i,r/2)\cap\Omega.
$$
Now by the domain monotonicity and Cheng's comparison for the zero Dirichlet eigenvalue, we have
\begin{equation}
\label{dp:aux}
\nu_k(\Omega)\leqslant\max_i\nu_0(B(\tilde x_i,r/4))\leqslant c_7(\kappa+r^{-2}).
\end{equation}
Taking the limit as $r\to\min\{\rad,1/\sqrt{\kappa}\}$, we obtain that $\nu_k(\Omega)$ is not greater than $c_9(\kappa+\rad^{-2})$.
\smallskip
\noindent
{\it Case~2: $\rho(k+1)<1/\sqrt{\kappa}$.} Following the line of an argument in Case~1, we see that for any $r<\min\{\rad,\rho(k+1)\}$ relation~\eqref{dp:aux} holds. Tending $r\to\min\{\rad,\rho(k+1)\}$, we obtain
\begin{equation}
\label{dp:aux2}
\nu_k(\Omega)\leqslant c_{7}(\kappa+\rad^{-2}+\rho(k+1)^{-2}).
\end{equation}
Now we estimate the value $\rho(k+1)$. For any given $s$ such that $\rho(k+1)<s<1/\sqrt{\kappa}$ let $m$ be the maximal number of points $y_1,\ldots,y_m\in\Omega$ such that $\dist(y_i,y_j)>s$ for any $i\ne j$. In particular, the balls $B(y_i,s)$, where $i=1,\ldots,m$, cover the domain $\Omega$. By the definition of $\rho(k+1)$ we also conclude that $m\leqslant k$. Thus, by inequality~\eqref{bishop}, we obtain
$$
\mathit{Vol}(\Omega)\leqslant\sum\mathit{Vol}(B(y_i,s))\leqslant m\omega_n e^{n-1}s^n\leqslant c_{10}ks^n.
$$
Letting $s$ tend to $\rho(k+1)$, we further obtain
$$
\rho(k+1)^{-2}\leqslant (c_{10})^{2/n}(k/\mathit{Vol}(\Omega))^{2/n}.
$$
Combining the last relation with inequality~\eqref{dp:aux2}, we get
$$
\nu_k(\Omega)\leqslant c_{7}(\kappa+\rad^{-2})+c_{11}(k/\mathit{Vol}(\Omega))^{2/n}.
$$
Taking into account both cases, we finish the proof of the theorem.
\qed
\subsection*{Acknowledgements}
The authors are grateful to A. Savo for proposing the idea of Example~\ref{counter}, as well as to B. Colbois and L. Polterovich for useful discussions. The project has originated out of a number of discussions the authors had while GK and AH were visiting the Centre de recherche math{\'e}matiques (CRM) in Montreal. During the work on the paper AH has been supported by the CRM-ISM postdoctoral fellowship and the postdoctoral programme at the Max-Planck Institute for mathematics in Bonn. The support and hospitality of both institutes is gratefully acknowledged. The research of IP was partially supported by NSERC, FRQNT and Canada Research Chairs program.
| {
"timestamp": "2016-05-17T02:15:45",
"yymm": "1510",
"arxiv_id": "1510.07281",
"language": "en",
"url": "https://arxiv.org/abs/1510.07281",
"abstract": "We revisit classical eigenvalue inequalities due to Buser, Cheng, and Gromov on closed Riemannian manifolds, and prove the versions of these results for the Dirichlet and Neumann boundary value problems. Eigenvalue multiplicity bounds and related open problems are also discussed.",
"subjects": "Differential Geometry (math.DG); Spectral Theory (math.SP)",
"title": "Eigenvalue inequalities on Riemannian manifolds with a lower Ricci curvature bound",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9888419671077918,
"lm_q2_score": 0.7154240079185319,
"lm_q1q2_score": 0.7074412833063015
} |
https://arxiv.org/abs/0706.2421 | A Simple Method Which Generates Infinitely Many Congruence Identities | A simple method called symbolic representation for piecewise linear functions on the real line is introduced and used to compute the numbers of periodic points of all periods for some such functions. Since, for every positive integer m, the number of periodic points of minimal period m must be divisible by m, we obtain infinitely many congruence identities. | \section{Introduction}
Let $\phi(m)$ be an integer-valued function defined on the set of all positive integers. If $m=p_1^{k_1}p_2^{k_2} \cdots p_r^{k_r}$, where the $p_i$'s are distinct prime numbers, $r$ and the $k_i$'s are positive integers, we define $\Phi_1(1, \phi)=\phi(1)$ and $\Phi_1(m, \phi) =$
$$
\phi(m)-\sum_{i=1}^r \phi(\frac m{p_i})+\sum_{i_1<i_2} \phi(\frac m{p_{i_1}p_{i_2}})
- \sum_{i_1<i_2<i_3} \phi(\frac m{p_{i_1}p_{i_2}p_{i_3}}) + \cdots
+ (-1)^r \phi(\frac m{p_1p_2 \cdots p_r}),
$$
\noindent
where the summation $\sum_{i_1<i_2< \cdots < i_j}$ is taken over all integers $i_1, i_2, \cdots, i_j$ with $1 \le i_1 < i_2 <$ $\cdots < i_j \le r$.
If $m = 2^{k_0}p_1^{k_1}p_2^{k_2} \cdots p_r^{k_r}$, where the $p_i$'s are distinct odd prime numbers, and $k_0 \ge 0, r\ge
1$, and the $k_i$'s $\ge 1$ are integers, we define, similarly, $\Phi_2(m, \phi) =$
$$
\phi(m)-\sum_{i=1}^r \phi(\frac m{p_i})+\sum_{i_1<i_2} \phi(\frac m{p_{i_1}p_{i_2}})
- \sum_{i_1<i_2<i_3} \phi(\frac m{p_{i_1}p_{i_2}p_{i_3}}) + \cdots
+ (-1)^r \phi(\frac m{p_1p_2 \cdots p_r}),
$$
\noindent
If $m = 2^k$, where $k \ge 0$ is an integer, we define $\Phi_2(m, \phi) = \phi(m) - 1$.
\noindent
If, for some integer $n \ge 2$, we have $\phi(m) = n^m$ for all positive integer $m$, then we denote $\Phi_i(m, \phi)$ by $\Phi_i(m, n)$, $i = 1, 2$ to emphasize the role of this integer $n$.
On the other hand, let $S$ be a subset of the real numbers and let $f$ be a function from $S$ into itself. For every positive integer $n$, we let $f^n$ denote the $n^{\text{th}}$ iterate of $f$: $f^1 = f$ and $f^n = f \circ f^{n-1}$ for $n \ge 2$. For every $x_0 \in S$, we call the set $\{ f^k(x_0) : k \ge 0 \}$ the orbit of $x_0$ under $f$. If $x_0$ satisfies $f^m(x_0) = x_0$ for some positive integer $m$, then we call $x_0$ a periodic point of $f$ and call the smallest such positive integer $m$ the minimal period of $x_0$ and of the orbit of $x_0$ (under $f$). Note that, if $x_0$ is a periodic point of $f$ with minimal period $m$, then, for every integer $1 \le k \le m$, $f^k(x_0)$ is also a periodic point of $f$ with minimal period $m$ and they are all distinct. So, every periodic orbit of $f$ with minimal period $m$ consists of exactly $m$ distinct points. Since it is obvious that distinct periodic orbits of $f$ are pairwise disjoint, the number (if finite) of distinct periodic points of $f$ with minimal period $m$ is divisible by $m$ and the quotient equals the number of distinct periodic orbits of $f$ with minimal period $m$. This observation, together with a standard inclusion-exclusion argument, gives the following well-known result.
\noindent
{\bf Theorem 1.}
{\it Let $S$ be a subset of the real numbers and let $f : S \rightarrow S$ be a mapping with the property that, for every positive integer $m$, the equation $f^m(x) = x$ \, (or $-x$, respectively) has only finitely many distinct solutions. Let $\phi(m)$ (or $\psi(m)$, respectively) denote the number of these solutions. Then, for every positive integer $m$, the following hold:
\begin{itemize}
\item[(i)]
The number of periodic points of $f$ with minimal period $m$ is $\Phi_1(m, \phi)$. So $\Phi_1(m, \phi) \equiv 0 \, (\text{mod} \,\,\, m)$.
\item[(ii)]
If $0 \in S$ and $f$ is odd, then the number of symmetric periodic points (i.e., periodic points whose orbit are symmetric with respect to the origin) of $f$ with minimal period $2m$ is $\Phi_2(m, \psi)$. Thus, $\Phi_2(m, \psi) \equiv 0 \, (\text{mod} \,\,\, 2m)$.
\end{itemize}}
Successful applications of the above theorem depend of course on a knowledge of the function $\phi$ or $\psi$. For example, if we let $S$ denote the set of all real numbers and, for every integer $n \ge 2$ and every odd integer $t = 2k+1 > 1$, let $$f_n(x) = a_n \cdot \Pi_{j=1}^n (x-j)$$ and let $$g_t(x) = b_t \cdot x \cdot \Pi_{j=1}^k (x^2-j^2),$$ where $a_n$ and $b_t$ are fixed sufficiently large positive numbers depending only on $n$ and $t$, respectively. Then it is easy to see that, for every positive integer $m$, the equation $f_n^m(x) = x \,\,\, (g_t^m(x) = -x$, resp.) has exactly $n^m \,\,\, (t^m$, resp.) distinct solutions in $S$. Therefore, if $\phi(m,n) = n^m$ and $\psi(m,t) = t^m$, then we have as a consequence of Theorem 1 the following well-known congruence identities which include Fermat's Little Theorem as a special case.
\noindent
{\bf Corollary 2.} {\it (i) Let $m \ge 1$ and $n \ge 2$ be integers. Then $\Phi_1(m, n) \equiv 0 \,\,(\text{mod} \,\,\, m)$.
\qquad\qquad\,\, (ii) Let $m \ge 1$ be an integer and let $n > 1$ be an odd integer. Then $\Phi_2(m, n) \equiv 0 \,\,(\text{mod} \,\,\,2m)$.}
In this note, we indicate that the method introduced in {\bf{\cite{du1}}} can also be used to recursively define infinitely many $\phi$ and $\psi$ and thus produce infinitely many families of congruence identities related to Theorem 1. In Section 2, we will review this method, and to illustrate it we will prove the following result in Section 3.
\noindent
{\bf Theorem 3.}
{\it For every positive integer $n \ge 3$, let $\phi_n$ be the integer-valued function on the set of all positive integers defined recursively by letting $\phi_n(m) = 2^m - 1$ for all $1 \le m \le n-1$ and $$\phi_n(n+k) = \sum_{j=1}^{n-1} \phi_n(n+k-j), \,\,\, \text{for all} \,\,\, k \ge 0.$$ Then, for every positive integer $m$, $\Phi_1(m, \phi_n) \equiv 0 \,\,\, (\text{mod} \,\,\, m)$. Furthermore, $$\lim_{m \to \infty} [\log \Phi_1(m, \phi_n)]/m = \lim_{m \to \infty} [\log \phi_n(m)]/m = \log \alpha_n,$$ where $\alpha_n$ is the (unique) positive (and the largest in absolute value) zero of the polynomial $$x^{n-1} - \sum_{k=0}^{n-2} x^k.$$}
Note that in the above theorem these numbers $\phi_n(m)$, $m \ge 1$ are generalized Fibonacci numbers {\bf{\cite{ga,mi}}} and when $n = 3$, these numbers $\phi_3(m)$, $m \ge 1$, are the well-known Lucas numbers: $1, 3, 4, 7, 11, 18, 29, \cdots$.
Just for comparison, we also include the following two results which can be verified numerically. The rigorous proofs of these two results which are similar to that of Theorem 3 below can be found in ${\bf [1}$, Theorem 2 ${\bf ]}$ and ${\bf [2}$, Theorem 3 ${\bf ]}$, respectively.
\noindent
{\bf Theorem 4.}
{\it For every positive integer $n \ge 2$, let sequences $$<b_{k,1,j,n}>, <b_{k,2,j,n}>, 1\le j \le n,$$ be defined recursively as follows:
$$\begin{cases}
b_{1,1,j,n} = 0, \,\,\, 1 \le j \le n, \\
b_{2,1,j,n} = 1, \,\,\, 1 \le j \le n, \\
b_{1,2,j,n} = b_{2,2,j,n} = 0, \,\,\, 1 \le j \le n-1, \\
b_{1,2,n,n} = b_{2,2,n,n} = 1. \\
\end{cases}
$$ For $i = 1$ or 2, and $k \ge 1$,
$$\begin{cases}
b_{k+2,i,j,n} = b_{k,i,1,n} + b_{k,i,j+1,n}, \,\,\, 1 \le j \le n-1, \\
b_{k+2,i,n,n} = b_{k,i,1,n} + b_{k+1,i,n,n}. \\
\end{cases}
$$ Let $b_{k,1,j,n} = 0$ for all $-2n+3 \le k \le 0$ and $1 \le j \le n$, and for all positive integers $m$, let $$\phi_n(m) = b_{m,2,n,n} + 2 \cdot \sum_{j=1}^n b_{m+2-2j,1,j,n}.$$ Then, for every positive integer $m$, $\Phi_1(m, \phi_n) \equiv 0 \,\,\, (\text{mod} \,\,\, m)$. Furthermore, $$\lim_{m \to \infty} [\log \Phi_1(m, \phi_n)]/m = \lim_{m \to \infty} [\log \phi_n(m)]/m = \log \beta_n,$$ where $\beta_n$ is the (unique) positive (and the largest in absolute value) zero of the polynomial $x^{2n+1} - 2x^{2n-1} - 1$.}
\noindent
{\bf Remark 1.}
For all positive integers $m$ and $n$, let $$A_{m,n} = \Phi_1(2m-1, \phi_n)/(2m-1),$$ where $\phi_n$ is defined as in Theorem 3 for $n = 1$ and as in Theorem 4 for $n \ge 2$. Table 1 lists the first 31 values of $A_{m,n}$, for $1 \le n \le 6$. It seems that $A_{m,n} = 2^{m-n-1}$ for $n+1 \le m \le 3n+2$ and $A_{m,n} > 2^{m-n-1}$ for $m > 3n + 2$. If, for all positive integers $m$ and $n$, we define sequences $<B_{m,n,k}>$ by letting $$B_{m,n,1} = A_{m+3n+2,n} - 2A_{m+3n+1,n} \qquad\quad\,\,$$ and $$B_{m,n,k} = B_{m+2n+1,n,k-1} - B_{m+2n+1,n+1,k-1}$$ for $k > 1$, then more extensive numerical computations seem to show that, for all positive integers $k$, we have
\noindent
(i) \,\, $B_{1,n,k} = 2$ \,\,\,\,\,\, for all \,\, $n \ge 1$, \\
(ii) \, $B_{2,n,k} = 4k$ \,\, for all \,\, $n \ge 1$, \\
(iii) $B_{3,n,k}$ is a constant depending only on $k$, and \\
(iv) for all \,\, $1 \le m \le 2n+1$, \,\, $B_{m,n,k} = B_{m,j,k}$ \,\, for all \,\, $j \ge n \ge 1$. \\
\noindent
{\bf Theorem 5.}
{\it Fix any integer $n \ge 2$. For all integers $i$, $j$, and $k$ with $i = 1, 2$, $1 \le |j| \le n$, and $k \ge 1$, we define $c_{k,i,j,n}$ recursively as follows: $$c_{1,1,n,n} = 1 \,\,\, \text{and} \,\,\, c_{1,1,j,n} = 0 \,\,\, \text{for} \,\,\, j \ne n,$$ $$c_{1,2,1,n} = 1 \,\,\, \text{and} \,\,\, c_{1,2,j,n} = 0 \,\,\, \text{for} \,\,\, j \ne 1,$$ For $i = 1, 2$, and $k \ge 1$,
$$
\begin{cases}
c_{k+1,i,1,n} &= c_{k,i,1,n} + c_{k,i,-n,n} + c_{k,i,n,n}, \\
c_{k+1,i,j,n} &= c_{k,i,j-1,n} + c_{k,i,n,n} \,\,\, \text{for all} \,\,\, 2\le j \le n, \\
c_{k+1,i,-1,n} &= c_{k,i,-1,n} + c_{k,i,-n,n} + c_{k,i,n,n}, \\
c_{k+1,i,-j,n} &= c_{k,i,-j+1,n} + c_{k,i,-n,n} \,\,\, \text{for all} \,\,\, 2 \le j \le n. \\
\end{cases}
$$ Let $c_{k,1,j,n} = 0$ for all integers $k$, $j$ with $4-n \le k \le 0$ and $1 \le |j| \le n$, and, for all positive integers $m$, let $$ \phi_n(m) = 2 \sum_{k=1}^{n-1} c_{m+2-k,1,n+1-k,n} + 2c_{m+1,2,1,n} - 1\,\,\,\,$$ and $$\psi_n(m) = 2 \sum_{k=1}^{n-1} c_{m+2-k,1,k-n-1,n} + 2c_{m+1,2,-1,n} + 1.$$ Then, for every positive integer $m$, $$\Phi_1(m, \phi_n) \equiv 0 \,\,\, (\text{mod} \,\,\, m) \quad \text{and} \quad \Phi_2(m, \psi_n) \equiv 0 \,\,\, (\text{mod} \,\,\, 2m).$$ Furthermore, $$\lim_{m \to \infty} [\log \Phi_1(m, \phi_n)]/m = \lim_{m \to \infty} [\log \phi_n(m)]/m = \lim_{m \to \infty} [\log \psi_n(m)]/m$$ $$\qquad\qquad\qquad= \lim_{m \to \infty} [\log \Phi_2(m, \psi_n)]/m = \log \gamma_n.$$ where $\gamma_n$ is the (unique) positive (and the largest in absolute value) zero of the polynomial $x^n - 2x^{n-1} - 1$.}
\noindent
{\bf Remark 2.}
For all positive integers $m \ge 1$ and $n \ge 2$, let $$D_{m,n} = \Phi_2(m, \psi_n)/(2m),$$ where the $\psi_n$'s are defined as in the above theorem. Table 2 lists the first 25 values of $D_{m,n}$ for $2 \le n \le 6$. It seems that $D_{m,n} = 2^{m-n}$ for $n \le m \le 3n$, and $D_{m,n} > 2^{m-n}$ for $m > 3n$. If, for all integers $m \ge 1$ and $n \ge 2$, we define the sequences $<E_{m,n,k}>$ by letting $$E_{m,n,1} = D_{m+3n,n} - 2D_{m+3n-1,n}\qquad\,\,$$ and $$E_{m,n,k} = E_{m+2n,n,k-1} - E_{m+2n,n+1,k-1}$$ for $k > 1$, then more extensive computations seem to show that, for all positive integers $k$, we have
\noindent
(i) \,\, $E_{1,n,k} = 2$ \,\,\,\,\,\, for all \,\, $n \ge 2$, \\
(ii) \, $E_{2,n,k} = 4k$ \,\, for all \,\, $n \ge 2$, \\
(iii) $E_{3,n,k}$ and $E_{4,n,k}$ are constants depending only on $k$, and \\
(iv) for all \,\, $1 \le m \le 2n$, \,\, $E_{m,n,k} = E_{m,j,k}$ \,\, for all \,\, $j \ge n \ge 2$. \\
\noindent
See Tables 1 and 2 below.
\pagebreak
$$\text{Table 1}$$
\bigskip
\noindent
$m$ \qquad\qquad $A_{m,1}$ \qquad\qquad $A_{m,2}$ \qquad\qquad $A_{m,3}$ \qquad\qquad $A_{m,4}$ \qquad\qquad $A_{m,5}$ \qquad\qquad $A_{m,6}$
\noindent
1 \quad \qquad\qquad 1 \qquad\qquad\qquad 1 \quad\qquad\qquad 1 \quad\qquad\qquad 1 \quad\quad\qquad\qquad 1 \quad\qquad\qquad 1 \\
2 \quad\qquad\qquad 1 \qquad\qquad\qquad 1 \quad\qquad\qquad 1 \quad\qquad\qquad 1 \quad\quad\qquad\qquad 1 \quad\qquad\qquad 1 \\
3 \quad\qquad\qquad 1 \qquad\qquad\qquad 0 \quad\qquad\qquad 0 \quad\qquad\qquad 0 \quad\quad\qquad\qquad 0 \quad\qquad\qquad 0 \\
4 \quad\qquad\qquad 1 \qquad\qquad\qquad 1 \quad\qquad\qquad 1 \quad\qquad\qquad 1 \quad\quad\qquad\qquad 1 \quad\qquad\qquad 1 \\
5 \quad\qquad\qquad 2 \qquad\qquad\qquad 1 \quad\qquad\qquad 0 \quad\qquad\qquad 0 \quad\quad\qquad\qquad 0 \quad\qquad\qquad 0 \\
6 \quad\qquad\qquad 2 \qquad\qquad\qquad 2 \quad\qquad\qquad 2 \quad\qquad\qquad 2 \quad\quad\qquad\qquad 2 \quad\qquad\qquad 2 \\
7 \quad\qquad\qquad 4 \qquad\qquad\qquad 2 \quad\qquad\qquad 1 \quad\qquad\qquad 0 \quad\quad\qquad\qquad 0 \quad\qquad\qquad 0 \\
8 \quad\qquad\qquad 5 \qquad\qquad\qquad 3 \quad\qquad\qquad 3 \quad\qquad\qquad 3 \quad\quad\qquad\qquad 3 \quad\qquad\qquad 3 \\
9 \quad\qquad\qquad 8 \qquad\qquad\qquad 4 \quad\qquad\qquad 2 \quad\qquad\qquad 1 \quad\quad\qquad\qquad 0 \quad\qquad\qquad 0 \\
10 \qquad\quad\quad\,\, 11 \qquad\qquad\qquad\, 6 \quad\quad\qquad 6 \quad\qquad\qquad 6 \quad\quad\qquad\qquad 6 \quad\qquad\qquad 6 \\
11 \qquad\quad\quad\,\, 18 \qquad\qquad\qquad\, 8 \quad\quad\qquad 4 \quad\qquad\qquad 2 \quad\quad\qquad\qquad 1 \quad\qquad\qquad 0 \\
12 \qquad\quad\quad\,\, 25 \qquad\qquad\qquad 11 \quad\quad\qquad 9 \quad\qquad\qquad 9 \quad\quad\qquad\qquad 9 \,\,\qquad\qquad\,\, 9 \\
13 \qquad\quad\quad\,\, 40 \qquad\qquad\qquad 16 \quad\quad\qquad 8 \quad\qquad\qquad 4 \quad\quad\qquad\qquad 2 \,\,\qquad\qquad\,\, 1 \\
14 \qquad\quad\quad\,\, 58 \qquad\qquad\qquad 23 \quad\quad\qquad 18 \qquad\qquad 18 \quad\quad\qquad\qquad 18 \qquad\qquad 18 \\
15 \qquad\quad\quad\,\, 90 \qquad\qquad\qquad 32 \quad\quad\qquad 16 \quad\qquad\qquad 8 \quad\quad\qquad\qquad 4 \qquad\qquad\,\, 2 \\
16 \qquad\quad\quad 135 \qquad\qquad\qquad 46 \quad\quad\qquad 32 \quad\qquad\qquad 30 \quad\qquad\qquad 30 \qquad\qquad 30 \\
17 \qquad\quad\quad 210 \qquad\qquad\qquad 66 \quad\quad\qquad 32 \quad\qquad\qquad 16 \quad\qquad\qquad 8 \qquad\qquad\,\,\,\, 4 \\
18 \qquad\quad\quad 316 \qquad\qquad\qquad 94 \quad\quad\qquad 61 \quad\qquad\qquad 56 \quad\qquad\qquad 56 \qquad\qquad 56 \\
19 \qquad\quad\quad 492 \quad\qquad\qquad\,\, 136 \quad\quad\qquad 64 \quad\qquad\qquad 32 \quad\qquad\qquad 16 \qquad\qquad\,\, 8 \\
20 \qquad\quad\quad 750 \quad\qquad\qquad\,\, 195 \quad\qquad\,\,\, 115 \quad\qquad\qquad 101 \quad\qquad\qquad 99 \qquad\qquad 99 \\
21 \,\,\,\quad\quad\quad 1164 \quad\qquad\qquad\,\, 282 \quad\quad\qquad 128 \quad\qquad\qquad 64 \quad\qquad\qquad 32 \qquad\qquad 16 \\
22 \,\,\quad\quad\quad 1791 \quad\qquad\qquad\,\, 408 \quad\quad\qquad 224 \qquad\qquad\,\, 191 \quad\qquad\qquad 186 \quad\qquad\,\, 186 \\
23 \,\,\quad\quad\quad 2786 \quad\qquad\qquad\,\, 592 \quad\quad\qquad 258 \qquad\qquad\,\, 128 \quad\qquad\qquad\,\, 64 \qquad\qquad 32 \\
24 \,\,\quad\quad\quad 4305 \quad\qquad\qquad\,\, 856 \quad\quad\qquad 431 \qquad\qquad\,\, 351 \quad\qquad\qquad 337 \quad\qquad\,\, 335 \\
25 \,\,\quad\quad\quad 6710 \quad\qquad\qquad 1248 \qquad\qquad\,\, 520 \qquad\qquad 256 \quad\qquad\qquad 128 \qquad\qquad 64 \\
26 \quad\quad\quad 10420 \,\,\,\qquad\qquad\,\, 1814 \quad\qquad\,\,\,\, 850 \qquad\qquad\,\, 668 \qquad\qquad\,\, 635 \qquad\qquad 630 \\
27 \quad\quad\quad 16264 \quad\qquad\qquad 2646 \qquad\qquad 1050 \qquad\qquad\,\, 512 \qquad\qquad\,\, 256 \qquad\qquad 128 \\
28 \quad\quad\quad 25350 \quad\qquad\qquad 3858 \qquad\qquad 1673 \qquad\qquad 1257 \qquad\qquad 1177 \qquad\qquad 1163 \\
29 \quad\quad\quad 39650 \quad\qquad\qquad 5644 \qquad\qquad 2128 \qquad\qquad 1026 \qquad\qquad\,\, 512 \qquad\qquad 256 \\
30 \quad\quad\quad 61967 \quad\qquad\qquad 8246 \qquad\qquad 3328 \qquad\qquad 2402 \qquad\qquad 2220 \qquad\qquad 2187 \\
31 \quad\quad\quad 97108 \qquad\qquad\,\, 12088 \qquad\qquad 4320 \qquad\qquad 2056 \qquad\qquad 1024 \qquad\qquad 512 \\
\pagebreak
$$\text{Table 2}$$
\bigskip
\noindent
$m$ \quad\quad\qquad\quad $D_{m,2}$ \qquad\qquad\qquad\,\, $D_{m,3}$ \qquad\qquad\qquad $D_{m,4}$ \quad\qquad\qquad $D_{m,5}$ \quad\qquad\qquad $D_{m,6}$
\noindent
1 \quad\qquad\qquad\qquad\,\,\,\, 0 \quad\qquad\qquad\qquad\,\,\,\,\,\, 0 \qquad\qquad\qquad\qquad 0 \qquad\quad\quad\quad\,\, 0 \quad\quad\qquad\qquad 0 \\
2 \quad\qquad\qquad\qquad\,\,\,\, 1 \quad\qquad\qquad\qquad\,\,\,\,\,\, 0 \qquad\qquad\qquad\qquad 0 \qquad\quad\quad\quad\,\, 0 \quad\quad\qquad\qquad 0 \\
3 \quad\qquad\qquad\qquad\,\,\,\, 2 \quad\qquad\qquad\qquad\,\,\,\,\,\, 1 \qquad\qquad\qquad\qquad 0 \qquad\quad\quad\quad\,\, 0 \quad\quad\qquad\qquad 0 \\
4 \quad\qquad\qquad\qquad\,\,\,\, 4 \quad\qquad\qquad\qquad\,\,\,\,\,\, 2 \qquad\qquad\qquad\qquad 1 \qquad\quad\quad\quad\,\, 0 \quad\quad\qquad\qquad 0 \\
5 \quad\qquad\qquad\qquad\,\,\,\, 8 \quad\qquad\qquad\qquad\,\,\,\,\,\, 4 \qquad\qquad\qquad\qquad 2 \qquad\quad\quad\quad\,\, 1 \quad\quad\qquad\qquad 0 \\
6 \quad\qquad\qquad\qquad\, 16 \,\,\,\qquad\qquad\qquad\quad\,\,\, 8 \qquad\qquad\qquad\qquad 4 \qquad\quad\qquad\quad\,\, 2 \quad\quad\quad\qquad 1 \\
7 \quad\qquad\qquad\qquad\, 34 \quad\quad\qquad\qquad\quad\,\,\, 16 \qquad\qquad\qquad\qquad 8 \qquad\quad\qquad\quad\,\, 4 \quad\quad\quad\qquad 2 \\
8 \quad\qquad\qquad\qquad\,\, 72 \quad\qquad\qquad\qquad\,\, 32 \quad\qquad\qquad\qquad\,\, 16 \qquad\quad\qquad\quad\,\, 8 \quad\quad\quad\qquad 4 \\
9 \quad\qquad\qquad\qquad 154 \quad\qquad\qquad\qquad\,\, 64 \quad\qquad\qquad\qquad\,\, 32 \qquad\quad\qquad\quad 16 \quad\quad\quad\qquad 8 \\
10 \qquad\quad\quad\qquad\,\, 336 \quad\qquad\qquad\qquad 130 \quad\qquad\qquad\qquad\,\, 64 \qquad\quad\qquad\quad 32 \quad\quad\quad\qquad 16 \\
11 \qquad\quad\quad\qquad\,\, 738 \qquad\qquad\qquad\quad 264 \quad\qquad\qquad\qquad 128 \quad\qquad\qquad\,\,\,\,\, 64 \quad\quad\quad\qquad 32 \\
12 \qquad\quad\quad\qquad 1632 \qquad\qquad\qquad\quad 538 \quad\qquad\qquad\qquad 256 \quad\qquad\qquad\,\,\, 128 \quad\quad\quad\qquad 64 \\
13 \qquad\quad\quad\qquad 3640 \qquad\qquad\qquad\,\, 1104 \quad\qquad\qquad\qquad 514 \quad\qquad\qquad\,\,\, 256 \quad\quad\quad\quad 128 \\
14 \qquad\quad\quad\qquad 8160 \qquad\qquad\qquad\,\, 2272 \quad\quad\qquad\qquad\, 1032 \qquad\qquad\qquad 512 \quad\quad\quad\quad 256 \\
15 \qquad\quad\quad\quad\,\, 18384 \qquad\qquad\qquad\,\, 4692 \quad\quad\qquad\qquad\, 2074 \quad\qquad\qquad\,\, 1024 \quad\quad\quad\quad 512 \\
16 \qquad\quad\quad\quad\, 41616 \qquad\qquad\qquad\,\, 9730 \qquad\qquad\qquad\,\, 4176 \quad\qquad\qquad\,\, 2050 \quad\quad\qquad 1024 \\
17 \qquad\quad\quad\,\,\quad 94560 \qquad\qquad\qquad 20236 \qquad\qquad\qquad\,\, 8416 \quad\qquad\qquad\,\, 4104 \quad\quad\qquad 2048 \\
18 \qquad\quad\quad\quad 215600 \qquad\qquad\qquad 42208 \qquad\qquad\qquad 16980 \quad\qquad\qquad\,\, 8218 \quad\quad\qquad 4096 \\
19 \qquad\quad\quad\,\,\,\,\, 493122 \qquad\qquad\qquad 88288 \qquad\qquad\qquad 34304 \quad\qquad\qquad 16464 \quad\quad\qquad 8194 \\
20 \qquad\quad\quad\,\, 1130976 \qquad\qquad\quad\,\, 185126 \qquad\qquad\qquad 69376 \quad\qquad\qquad 32992 \quad\quad\qquad 16392 \\
21 \qquad\quad\quad\,\, 2600388 \quad\qquad\qquad\,\, 389072 \quad\qquad\qquad\,\, 140458 \quad\qquad\qquad 66132 \quad\quad\qquad 32794 \\
22 \qquad\quad\quad\,\, 5992560 \quad\qquad\qquad\,\, 819458 \quad\qquad\qquad\,\, 284684 \qquad\qquad\,\, 132608 \quad\quad\qquad 65616 \\
23 \qquad\quad\quad 13838306 \quad\qquad\qquad 1729296 \quad\qquad\qquad\,\, 577592 \qquad\qquad\,\, 265984 \quad\quad\qquad 131296 \\
24 \qquad\quad\quad 32016576 \quad\qquad\qquad 3655936 \quad\qquad\qquad 1173040 \qquad\qquad\,\, 533672 \quad\quad\qquad 262740 \\
25 \qquad\quad\quad 74203112 \quad\qquad\qquad 7742124 \quad\qquad\qquad 2384678 \qquad\qquad 1071104 \quad\quad\qquad 525824 \\
\section{Symbolic representation for continuous piecewise linear functions}
In this section, we review the method introduced in {\bf{\cite{du1}}}. Throughout this section, let $g$ be a continuous piecewise linear function from the interval $[c, d]$ into itself. We call the set $\{(x_i, y_i) : i = 1,2, \cdots, k \}$ a set of nodes for (the graph of) $y = g(x)$ if the following three conditions hold:
\begin{itemize}
\item[(1)]
$k \ge 2$,
\item[(2)]
$x_1 = c$, $x_k = d$, $x_1 < x_2 < \cdots < x_k$, and
\item[(3)]
$g$ is linear on $[x_i, x_{i+1}]$ for all $1 \le i \le k-1$ and $y_i = g(x_i)$ for all $1 \le i \le k$.
\end{itemize}
\noindent
For any such set, we will use its $y$-coordinates $y_1, y_2, \cdots, y_k$ to represent its graph and call $y_1y_2 \cdots y_k$ (in that order) a (symbolic) representation for (the graph of) $y = g(x)$. For $1 \le i < j \le k$, we call $y_iy_{i+1} \cdots y_j$ the representation for $y = g(x)$ on $[x_i, x_j]$ obtained by restricting $y_1y_2 \cdots y_k$ to $[x_i, x_j]$. For convenience, we will also call every $y_i$ in $y_1y_2 \cdots y_k$ a node. If $y_i = y_{i+1}$ for some $i$ (i.e., $g$ is constant on $[x_i, x_{i+1}]$), we will simply write $$y_1 \cdots y_iy_{i+2} \cdots y_k$$ instead of
$$y_1 \cdots y_iy_{i+1}y_{i+2} \cdots y_k.$$ That is, we will delete $y_{i+1}$ from the (symbolic) representation $y_1y_2 \cdots y_k$. Therefore, every two consecutive nodes in a (symbolic) representation are distinct. Note that a continuous piecewise linear function obviously has more than one (symbolic) representation. However, as we will soon see that there is no need to worry about that.
Now assume that $\{ (x_i, y_i) : i = 1, 2, \cdots, k \}$ is a set of nodes for $y = g(x)$ and $a_1a_2 \cdots a_r$ is a representation for $y = g(x)$ with $$\{ a_1, a_2, \cdots, a_r \} \subset \{ y_1, y_2, \cdots, y_k \}$$ and $a_i \ne a_{i+1}$ for all $1 \le i \le r-1$. If $$\{ y_1, y_2, \cdots, y_k \} \subset \{ x_1, x_2, \cdots, x_k \},$$ then there is an easy way to obtain a representation for $y = g^2(x)$ from the one $a_1a_2 \cdots a_r$ for $y = g(x)$. The procedure is as follows. First, for any two distinct real numbers $u$ and $v$, let $[u : v]$ denote the closed interval with endpoints $u$ and $v$. Then let $b_{i,1}b_{i,2} \cdots b_{i,t_i}$ be the representation for $y = g(x)$ on $[a_i:a_{i+1}]$ which is obtained by restricting $a_1a_2 \cdots a_r$ to $[a_i:a_{i+1}]$. We use the following notation to indicate this fact: $$a_ia_{i+1} \rightarrow b_{i,1}b_{i,2} \cdots b_{i,t_i} \,\,\, (\text{under} \,\,\, g) \,\,\, \text{if} \,\,\, a_i < a_{i+1},$$ or $$a_ia_{i+1} \rightarrow b_{i,t_i} \cdots b_{i,2}b_{i,1} \,\,\, (\text{under} \,\,\, g) \,\,\, \text{if} \,\,\, a_i > a_{i+1}.$$ The above representation on $[a_i:a_{i+1}]$ exists since $$\{ a_1, a_2, \cdots, a_r \} \subset \{ x_1, x_2, \cdots, x_k \}.$$ Finally, if $a_i < a_{i+1}$, let $z_{i,j} = b_{i,j}$ for all $1 \le j \le t_i$. If $a_i > a_{i+1}$, let $z_{i,j} = b_{i,t_i+1-j}$ for all $1 \le j \le t_i$. Let $$Z = z_{1,1} \cdots z_{1,t_1}z_{2,2} \cdots z_{2,t_2} \cdots z_{r,2} \cdots z_{r,t_r}.$$ (Note that $z_{i,t_i} = z_{i+1,1}$ \, for all \, $1 \le i \le r-1$). Then it is easy to see that $Z$ is a representation for $y = g^2(x)$. It is also obvious that the above procedure can be applied to the representation $Z$ for $y = g^2(x)$ to obtain one for $y = g^3(x)$, and so on.
\section{Proof of Theorem 3}
In this section we fix an integer $n \ge 3$ and let $f_n(x)$ be the continuous function from the interval $[1, n]$ onto itself defined by $$f_n(x) = x + 1 \,\,\, \text{for} \,\,\, 1 \le x \le n-1 \qquad\qquad\qquad\qquad\,\,\,\,$$ and $$f_n(x) = -(n-1)x + n^2 - n + 1 \,\,\, \text{for} \,\,\, n-1 \le x \le n.$$ Using the notations introduced in Section 2, we have the following result.
\noindent
{\bf Lemma 6.}
{\it Under $f_n$, we have
$$
\begin{cases}
k(k+1) &\rightarrow (k+1)(k+2), \,\,\, 2 \le k \le n-2, \,\,\, \text{if} \,\,\, n > 3, \\
(k+1)k &\rightarrow (k+2)(k+1), \,\,\, 2 \le k \le n-2, \,\,\, \text{if} \,\,\, n > 3. \\
(n-1)n &\rightarrow n1, \quad n(n-1) \rightarrow 1n, \\
n1 &\rightarrow 1n(n-1) \cdots 432, \quad 1n \rightarrow 234 \cdots (n-1)n1. \\
\end{cases}
$$}
In the following when we say the representation for $y = f_n^k(x)$, we mean the representation obtained, following the procedure as described in Section 2, by applying Lemma 6 to the representation $234 \cdots (n-1)n1$ for $y = f_n(x)$ successively until we get to the one for $y = f_n^k(x)$.
For every positive integer $k$ and all integers $i$, $j$, with $1 \le i, j \le n-1$, let $a_{k,i,j,n}$ denote the number of $uv$'s and $vu$'s in the representation for $y = f_n^k(x)$ whose corresponding $x$-coordinates are in the interval $[i, i+1]$, where $uv = 1n$ if $j = 1$, and $uv = j(j+1)$ if $2 \le j \le n-1$. It is obvious that $$a_{1,i,i+1,n} = 1 \,\,\, \text{for all} \,\,\, 1 \le i \le n-2, \,\quad\quad$$ $$a_{1,n-1,1,n} = 1, \,\, \text{and} \,\,\, a_{1,i,j,n} = 0 \,\,\, \text{elsewhere}.$$ From the above lemma, we find that these sequences $<a_{k,i,j,n}>$ can be computed recursively.
\noindent
{\bf Lemma 7.}
{\it For every positive integer $k$ and all integers $i$ with $1 \le i \le n-1$, we have
$$
\begin{cases}
a_{k+1,i,1,n} & = a_{k,i,1,n} + a_{k,i,n-1,n}, \\
a_{k+1,i,2,n} & = a_{k,i,1,n}, \\
a_{k+1,i,j,n} & = a_{k,i,1,n} + a_{k,i,j-1,n}, \,\,\, 3 \le j \le n-1 \,\,\, \text{if} \,\,\, n > 3. \\
\end{cases}
$$}
It then follows from the above lemma that the sequences $<a_{k,i,j,n}>$ can all be computed from the sequences $<a_{k,n-1,j,n}>$.
\noindent
{\bf Lemma 8.}
{\it For every positive integer $k$ and all integers $j$ with $1 \le j \le n-1$, we have $$a_{k,n-1,j,n} = a_{k+i,n-1-i,j,n}, \,\,\, 1 \le i \le n-2.$$}
For every positive integer $k$, let $$c_{k,n} = \sum_{i=1}^{n-1} a_{k,i,1,n} + \sum_{i=2}^{n-1} a_{k,i,i,n}.$$ Then it is easy to see that $c_{k,n}$ is exactly the number of distinct solutions of the equation $f_n^k(x) = x$ in the interval $[1, n]$. From the above lemma, we also have, for all $k \ge 1$, the identities: $$c_{k,n} = \sum_{i=0}^{n-2} a_{k-i,n-1,1,n} + \sum_{i=0}^{n-3} a_{k-i,n-1,n-1-i,n}$$ provided that $a_{m,n-1,j,n} = 0$ for all $m \le 0$ and $j > 0$. Since, for every positive integer $k$, $$a_{k,n-1,1,n} = a_{k-1,n-1,1,n} + a_{k-1,n-1,n-1,n} = a_{k-1,n-1,1,n} + a_{k-2,n-1,1,n} + a_{k-2,n-1,n-2,n}$$ $$= a_{k-1,n-1,1,n} + a_{k-2,n-1,1,n} + a_{k-3,n-1,1,n} + a_{k-3,n-1,n-3,n} = \cdots = \sum_{i=1}^{n-1} a_{k-i,n-1,1,n}$$ and $$c_{k,n} = \sum_{i=0}^{n-2} a_{k-i,n-1,1,n} + \sum_{i=0}^{n-3} a_{k-i,n-1,n-1-i,n}$$ $$= a_{k,n-1,1,n} + a_{k-1,n-1,1,n} + \sum_{i=2}^{n-2} a_{k-i,n-1,1,n} + a_{k-1,n-1,1,n} + a_{k-1,n-1,n-2,n} + \sum_{i=1}^{n-3} a_{k-i,n-1,n-1-i,n}$$ $$= a_{k,n-1,1,n} + 2a_{k-1,n-1,1,n} + \sum_{i=2}^{n-2} a_{k-i,n-1,1,n} + 2a_{k-1,n-1,n-2,n} + \sum_{i=2}^{n-3} a_{k-i,n-1,n-1-i,n} = \cdots$$ $$= \sum_{i=0}^{n-2} (i+1)a_{k-i,n-1,1,n}$$ provided that $a_{m,n-1,1,n} = 0$ if $m \le 0$, we obtain that $c_{k,n} = 2^k - 1$ for all $1 \le k \le n-1$ and $$c_{k,n} = \sum_{i=1}^{n-1} c_{k-i,n} \,\,\, \text{for all integers} \,\,\, k \ge n.$$ If, for every positive integer $m$, we let $\phi_n(m) = c_{m,n}$, then, by Theorem 1, we have $\Phi_1(m, \phi_n) \equiv 0$ (mod $m$). The proof of the other statement of Theorem 3 is easy and omitted (see {\bf{\cite{ga}}} and {\bf{\cite{mi}}}). This completes the proof of Theorem 3.
| {
"timestamp": "2007-06-16T17:36:19",
"yymm": "0706",
"arxiv_id": "0706.2421",
"language": "en",
"url": "https://arxiv.org/abs/0706.2421",
"abstract": "A simple method called symbolic representation for piecewise linear functions on the real line is introduced and used to compute the numbers of periodic points of all periods for some such functions. Since, for every positive integer m, the number of periodic points of minimal period m must be divisible by m, we obtain infinitely many congruence identities.",
"subjects": "Number Theory (math.NT); Dynamical Systems (math.DS)",
"title": "A Simple Method Which Generates Infinitely Many Congruence Identities",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.988841966450142,
"lm_q2_score": 0.7154240079185319,
"lm_q1q2_score": 0.707441282835803
} |
https://arxiv.org/abs/2005.03251 | Structured inversion of the Bernstein-Vandermonde Matrix | Bernstein polynomials, long a staple of approximation theory and computational geometry, have also increasingly become of interest in finite element methods. Many fundamental problems in interpolation and approximation give rise to interesting linear algebra questions. When attempting to find a polynomial approximation of boundary or initial data, one encounters the Bernstein-Vandermonde matrix, which is found to be highly ill-conditioned. Previously, we used the relationship between monomial Bezout matrices and the inverse of Hankel matrices to obtain a decomposition of the inverse of the Bernstein mass matrix in terms of Hankel, Toeplitz, and diagonal matrices. In this paper, we use properties of the Bernstein-Bezout matrix to factor the inverse of the Bernstein-Vandermonde matrix into a difference of products of Hankel, Toeplitz, and diagonal matrices. We also use a nonstandard matrix norm to study the conditioning of the Bernstein-Vandermonde matrix, showing that the conditioning in this case is better than in the standard 2-norm. Additionally, we use properties of multivariate Bernstein polynomials to derive a block $LU$ decomposition of the Bernstein-Vandermonde matrix corresponding to equispaced nodes on the $d$-simplex. | \section{Introduction}
Given data $\{f_j\}_{j=0}^n$ and distinct nodes $\{x_j\}_{j=0}^n$, the interpolation problem consists of finding a polynomial $p$ of degree $n$ that satisfies
\begin{equation}
p(x_j)=f_j
\end{equation}
for each $0\leq j\leq n$. If a basis for the space of polynomials of degree at most $n$ is chosen, then the interpolation problem can be expressed as a system of linear equations, where the coefficient matrix is a Vandermonde-like matrix \cite{gautschi1999orthogonal}. Recently, Bernstein polynomials have been considered as a tool for high-order approximation of partial differential equations via the finite element method \cite{ainsworth2011bernstein,duffy1982quadrature}, and the interpolant is often used as a polynomial approximation of initial or boundary data. The corresponding Bernstein--Vandermonde matrix is found to be highly ill-conditioned \cite{davis1975condition}, but the structure of the matrix has led to fast algorithms that avoid some of the issues that arise from the ill-conditioning. For example, Marco and Mart\'{i}nez \cite{marco2007bidiagonal} used the fact that the Bernstein--Vandermonde matrix is strictly totally positive \cite{gasca1992positivity} to obtain a bidiagonal factorization of the inverse, and Ainsworth and Sanchez \cite{ainsworth2016newton} adapted the standard divided difference algorithm for the monomial Vandermonde matrix \cite{burden2012difference} to the Bernstein basis.
In \cite{allenkirby2020mass}, we used the relationship between monomial B\'{e}zout matrices and the inverse of Hankel matrices described by Heinig and Rost \cite{heinig1984algebraic} to obtain a decomposition of the inverse of the Bernstein mass matrix in terms of Hankel, Toeplitz, and diagonal matrices. In this paper, we generalize an argument made by Kaplan \cite{kaplan2006bezout} to obtain a decomposition of the inverse of Vandermonde-like matrices in terms of their transpose, a diagonal matrix, and the corresponding B\'{e}zout matrix. When applied to the Bernstein basis, this gives a decomposition of the inverse of the Bernstein--Vandermonde matrix in terms of Hankel, Toeplitz, and diagonal matrices, which in turn leads to a fast algorithm for solving the interpolation problem. Additionally, we use the nonstandard matrix norm defined in \cite{allenkirby2020mass} to give an explanation for the relatively good performance of the Bernstein--Vandermonde matrix despite its massive condition number.
Bernstein polynomials also extend naturally to give a basis for multivariate polynomials of total degree $n$. Properties of Bernstein polynomials lead to special recursive blockwise-structure for finite element matrices \cite{kirby2017fast,kirby2012fast}. In this paper, we use this structure to obtain a block $LU$ decomposition of the Bernstein--Vandermonde matrix associated to equispaced nodes on the $d$-simplex.
\section{Inverse Formulas}
\label{sec:inverse}
\subsection{General Formulas}
\label{ssec:general}
For integers $m,n\geq 0$, let $\{ b^n_j(x) \}_{j=0}^n$ be a basis for the space of univariate polynomials of degree at most $n$, and let $\mathbf{x}\in\mathbb{R}^{m+1}$ with $\mathbf{x}_i<\mathbf{x}_{i+1}$ for each $0\leq i<m$, where boldface is used to distinguish the vector of nodes $\mathbf{x}$ from the indeterminate $x$ (a similar convention will be used to distinguish a polynomial $v$ from its vector of coefficients $\mathbf{v}$). The Vandermonde matrix $V^n(\mathbf{x})$ associated to $\{ b^n_j(x) \}_{j=0}^n$ and $\mathbf{x}$ is the $(m+1)\times(n+1)$ matrix given by
\begin{equation}
\label{eq:vandermonde}
V^n_{ij}(\mathbf{x})=b^n_j(\mathbf{x}_i).
\end{equation}
We consider the case $m=n$ so that $V^n(\mathbf{x})$ is invertible.
If $v$ and $w$ are polynomials of degree at most $n+1$ expressed in the basis $\{b^{n+1}_j(x)\}_{j=0}^{n+1}$, then the B\'{e}zout matrix generated by $v$ and $w$, denoted $\Bez(v,w)$, is the $(n+1)\times (n+1)$ matrix whose entries satisfy
\begin{equation}
\label{eq:bezout}
\frac{v(s)w(t)-v(t)w(s)}{s-t} = \sum_{i,j=0}^n \Bez_{ij}(v,w)b^n_i(s)b^n_j(t).
\end{equation}
In \cite{kaplan2006bezout}, it was shown that if the monomial basis is used, then the inverse of a B\'{e}zout matrix is a Hankel matrix. As a consequence of the arguments involved, one obtains a formula for the inverse of the monomial Vandermonde matrix in terms of its transpose, a diagonal matrix, and a particular B\'{e}zout matrix. Since none of the arguments require anything specific to the monomial basis, we can generalize the arguments to any basis, which we summarize here.
For $t\in\mathbb{R}$, let $\mathbf{b}^n(t)$ denote the column vector $(b^n_0(t),\dots,b^n_n(t))^T$. As a consequence of (\ref{eq:bezout}), if $s\neq t$, then
\begin{align*}
(\mathbf{b}^n(s))^T\Bez(v,w)\mathbf{b}^n(t)
&= \sum_{i,j=0}^n \Bez_{ij}(v,w)b^n_i(s)b^n_j(t) \\
&= \frac{v(s)w(t)-v(t)w(s)}{s-t}.
\end{align*}
This implies that
\begin{align*}
(\mathbf{b}^n(t))^T\Bez(v,w)\mathbf{b}^n(t)
&= \lim_{\varepsilon\rightarrow 0} \left[ (\mathbf{b}^n(t))^T\Bez(v,w)\mathbf{b}^n(t+\varepsilon) \right] \\
&= \lim_{\varepsilon\rightarrow 0} \frac{v(t+\varepsilon)w(t)-v(t)w(t+\varepsilon)}{\varepsilon} \\
&= v'(t)w(t)-v(t)w'(t).
\end{align*}
Therefore, if $v$ has simple zeros at $\mathbf{x}_i$ and $w(\mathbf{x}_i)\neq 0$ for each $0\leq i\leq n$, then
\begin{equation}
V^n(\mathbf{x})\Bez(v,w)\left(V^n(\mathbf{x})\right)^T = \diag(v'(\mathbf{x}_j)w(\mathbf{x}_j))_{j=0}^n,
\end{equation}
and so we have the following:
\begin{theorem}
\label{thm:bezout}
The inverse of $V^n(\mathbf{x})$ is given by
\begin{equation}
\left(V^n(\mathbf{x})\right)^{-1} = \Bez(v,w)\left(V^n(\mathbf{x})\right)^T\diag\left(\frac{1}{v'(\mathbf{x}_j)w(\mathbf{x}_j)}\right)_{j=0}^n,
\end{equation}
where $v$ and $w$ are any polynomials of degree $n+1$ satisfying $v(\mathbf{x}_i)=0$, $v'(\mathbf{x}_i)\neq 0$, and $w(\mathbf{x}_i)\neq 0$ for each $0\leq i\leq n$.
\end{theorem}
\begin{corollary}
\label{cor:bezout}
The inverse of $V^n(\mathbf{x})$ is given by
\begin{equation}
\left(V^n(\mathbf{x})\right)^{-1} = \Bez(v,1)\left(V^n(\mathbf{x})\right)^T\diag\left(\frac{1}{v'(\mathbf{x}_j)}\right)_{j=0}^n,
\end{equation}
where $v$ is any polynomial of degree $n+1$ that has simple zeros at $\mathbf{x}_i$ for each $0\leq i\leq n$.
\end{corollary}
\subsection{Bernstein--Vandermonde}
\label{ssec:BernsteinVandermonde}
We now focus on the case where $0\leq \mathbf{x}_i\leq 1$ for each $0\leq i\leq n$ and the basis consists of the polynomials $\{B^n_j(x)\}_{j=0}^n$, where
\begin{equation}
\label{eq:bernstein}
B^n_j(x) = \binom{n}{j}x^j(1-x)^{n-j}
\end{equation}
are the Bernstein polynomials of degree $n$. In order to apply Theorem~\ref{thm:bezout} or Corollary~\ref{cor:bezout}, we need a method for computing the entries of the Bernstein--B\'{e}zout matrix; in fact, they satisfy a recurrence relation involving the Bernstein coefficients of the polynomials.
\begin{theorem}
\label{thm:recursion}
If $v(t)=\sum_{i=0}^{n+1} \mathbf{v}_iB^{n+1}_i(t)$ and $w(t)=\sum_{i=0}^{n+1} \mathbf{w}_iB^{n+1}_i(t)$ are polynomials of degree at most $n+1$, then the entries $b_{ij}$ of the Bernstein--B\'{e}zout matrix $\Bez(v,w)$ generated by $v$ and $w$ satisfy
\begin{equation}
\label{eq:recursion}
b_{ij} = \frac{1}{(i+1)(n-j+1)}\left[j(n-i)b_{i+1,j-1}+(n+1)^2\left( \mathbf{v}_{i+1}\mathbf{w}_j-\mathbf{v}_j\mathbf{w}_{i+1}\right)\right].
\end{equation}
\end{theorem}
\begin{proof}
By (\ref{eq:bernstein}), we have that
\begin{equation}
tB^n_j(t) = \frac{j+1}{n+1}B^{n+1}_{j+1}(t)
\end{equation}
and
\begin{equation}
B^n_j(t) = \frac{n-j+1}{n+1}B^{n+1}_j(t)+\frac{j+1}{n+1}B^{n+1}_{j+1}(t).
\end{equation}
Therefore,
\begin{align*}
s\sum_{i,j=0}^n b_{ij}B^n_i(s)B^n_j(t) &= \sum_{i,j=0}^n \frac{i+1}{n+1} b_{ij}B^{n+1}_{i+1}(s)B^n_j(t) \\
&= \sum_{i,j=0}^n \frac{(i+1)(n-j+1)}{(n+1)^2}b_{ij} B^{n+1}_{i+1}(s)B^{n+1}_j(t) \\
&\qquad +\sum_{i,j=0}^n \frac{(i+1)(j+1)}{(n+1)^2}b_{ij} B^{n+1}_{i+1}(s)B^{n+1}_{j+1}(t).
\end{align*}
Similarly, we have that
\begin{align*}
t\sum_{i,j=0}^n b_{ij}B^n_i(s)B^n_j(t) &= \sum_{i,j=0}^n \frac{j+1}{n+1}b_{ij}B^n_i(s)B^{n+1}_{j+1}(t) \\
&= \sum_{i,j=0}^n \frac{(n-i+1)(j+1)}{(n+1)^2} b_{ij} B^{n+1}_i(s)B^{n+1}_{j+1}(t) \\
&\qquad + \sum_{i,j=0}^n \frac{(i+1)(j+1)}{(n+1)^2} b_{ij} B^{n+1}_{i+1}(s)B^{n+1}_{j+1}(t).
\end{align*}
This implies that
\begin{align*}
(s-t)\sum_{i,j=0}^n b_{ij} B^n_i(s)B^n_j(t) &= \sum_{i,j=0}^n \frac{(i+1)(n-j+1)}{(n+1)^2}b_{ij} B^{n+1}_{i+1}(s)B^{n+1}_j(t) \\
&\quad -\sum_{i,j=0}^n \frac{(n-i+1)(j+1)}{(n+1)^2}b_{ij}B^{n+1}_i(s)B^{n+1}_{j+1}(t),
\end{align*}
and so
\begin{align*}
(s-t)\sum_{i,j=0}^n b_{ij}B^n_i(s)B^n_j(t) &= \sum_{i,j=0}^{n+1}\frac{i(n-j+1)}{(n+1)^2}b_{i-1,j}B^{n+1}_i(s)B^{n+1}_j(t) \\
&\quad -\sum_{i,j=0}^{n+1}\frac{j(n-i+1)}{(n+1)^2}b_{i,j-1}B^{n+1}_i(s)B^{n+1}_j(t).
\end{align*}
On the other hand,
\begin{equation}
v(s)w(t)-v(t)w(s) = \sum_{i,j=0}^{n+1} (\mathbf{v}_i\mathbf{w}_j-\mathbf{v}_j\mathbf{w}_i)B^{n+1}_i(s)B^{n+1}_j(t),
\end{equation}
and so the result follows by comparing coefficients in (\ref{eq:bezout}).
\end{proof}
We can form the first column and last row of $\Bez(v,w)$ by using (\ref{eq:recursion}) with $j=0$ and $i=n$. The rest of the columns can then be built by applying the recurrence relation. Therefore,
\begin{corollary}
\label{cor:bezoutcost}
$\Bez(v,w)$ can be constructed in $\mathcal{O}(n^2)$ operations.
\end{corollary}
By repeatedly applying (\ref{eq:recursion}), we obtain a closed form for the entries of the Bernstein--B\'{e}zout matrix.
\begin{corollary}
\label{cor:bernbezout}
If $v(t)=\sum_{i=0}^{n+1} \mathbf{v}_iB^{n+1}_i(t)$ and $w(t)=\sum_{i=0}^{n+1} \mathbf{w}_iB^{n+1}_i(t)$ are polynomials of degree at most $n+1$, then the entries $b_{ij}$ of the Bernstein--B\'{e}zout matrix $\Bez(v,w)$ generated by $v$ and $w$ are given by
\begin{equation}
\label{eq:bernbezout}
b_{ij} = \frac{1}{\binom{n}{i}\binom{n}{j}}\sum_{k=0}^{m_{ij}} \binom{n+1}{i+k+1}\binom{n+1}{j-k}\left( \mathbf{v}_{i+k+1}\mathbf{w}_{j-k}-\mathbf{v}_{j-k}\mathbf{w}_{i+k+1}\right),
\end{equation}
where $m_{ij} = \min\{j,n-i\}$.
\end{corollary}
Constructing $\Bez(v,w)$ by using (\ref{eq:bernbezout}) would require $\mathcal{O}(n^3)$ operations, and so Corollary~\ref{cor:bezoutcost} implies that (\ref{eq:bernbezout}) is not optimal for building $\Bez(v,w)$; however, since $\binom{n}{i}=0$ whenever $i>n$, the upper limit of the sum in (\ref{eq:bernbezout}) can be replaced with $k=n$, and so we recognize that $\Bez(v,w)$ is given by a difference of matrix products.
\begin{corollary}
\label{cor:factorization}
Let $v(t) = \sum_{i=0}^{n+1} \mathbf{v}_iB^{n+1}_i(t)$ and $w(t)=\sum_{i=0}^{n+1}\mathbf{w}_iB^{n+1}_i(t)$ be polynomials of degree at most $n+1$. Define the Toeplitz matrices $T^{v,n}$ and $T^{w,n}$ by
\begin{equation*}
T^{v,n}_{ij} = \binom{n+1}{j-i}\mathbf{v}_{j-i}\qquad \text{and}\qquad T^{w,n}_{ij} = \binom{n+1}{j-i}\mathbf{w}_{j-i},
\end{equation*}
define the Hankel matrices $H^{v,n}$ and $H^{w,n}$ by
\begin{equation*}
H^{v,n}_{ij} = \binom{n+1}{i+j+1}\mathbf{v}_{i+j+1}\qquad \text{and}\qquad H^{w,n}_{ij} = \binom{n+1}{i+j+1}\mathbf{w}_{i+j+1},
\end{equation*}
and let $\Delta^n=\diag\left(\binom{n}{j}\right)_{j=0}^n$. Then the Bernstein--B\'{e}zout matrix $\Bez(v,w)$ generated by $v$ and $w$ is given by
\begin{equation}
\label{eq:bezoutinverse}
\Bez(v,w) = \left(\Delta^n\right)^{-1}\left[ H^{v,n}T^{w,n}-H^{w,n}T^{v,n}\right]\left(\Delta^n\right)^{-1}.
\end{equation}
\end{corollary}
\begin{remark}
\label{rmk:connection}
In \cite{allenkirby2020mass}, it was shown that the inverse of the degree $n$ Bernstein mass matrix $M^n$ has the decomposition
\begin{equation}
\label{eq:massinverse}
\left(M^n\right)^{-1} = \left(\Delta^n\right)^{-1}\left[ \widetilde{T}^nH^n-T^n\widetilde{H}^n\right] \left(\Delta^n\right)^{-1},
\end{equation}
where $\widetilde{H}^n$ and $H^n$ are Hankel matrices, $\widetilde{T}^n$ and $T^n$ are Toeplitz matrices, and $\Delta^n = \diag\left(\binom{n}{j}\right)_{j=0}^n$. This decomposition was obtained from the monomial B\'{e}zout matrix corresponding to polynomials whose coefficients are contained in the last column of $\left(M^n\right)^{-1}$. It is interesting to note the similarities between (\ref{eq:bezoutinverse}) and (\ref{eq:massinverse}) despite the different matrices and bases being considered.
\end{remark}
Given a polynomial $v(t)=\sum_{k=0}^{n+1} \mathbf{v}_k B^{n+1}_k(t)$ of degree at most $n+1$, we can express $v$ in the monomial basis as $v(t)=\sum_{k=0}^{n+1} \widetilde{\mathbf{v}}_kt^k$. It was shown in \cite{farouki2000legendre} that
\begin{equation}
\label{eq:montobern}
\mathbf{v}_k = \frac{1}{\binom{n+1}{k}} \sum_{\ell=0}^k \binom{n-\ell+1}{k-\ell} \widetilde{\mathbf{v}}_{\ell}.
\end{equation}
This implies that
\begin{align*}
H^{v,n}_{ij} = \binom{n+1}{i+j+1}\mathbf{v}_{i+j+1} &= \sum_{k=0}^{i+j+1} \binom{n-k+1}{i+j-k+1}\widetilde{\mathbf{v}}_k \\
&= \sum_{k=0}^{i+j+1} \binom{n-k+1}{n-i-j}\widetilde{\mathbf{v}}_k.
\end{align*}
Similarly,
\begin{equation*}
T^{v,n}_{ij} = \sum_{k=0}^{j-i}\binom{n-k+1}{j-i-k}\widetilde{\mathbf{v}}_ k.
\end{equation*}
By Vieta's formula (see, for example, \cite{rotman2015algebra}),
\begin{equation}
\label{eq:vieta}
\prod_{i=0}^n (x-\mathbf{x}_i) = \sum_{k=0}^n (-1)^{n-k+1}\sigma_{n-k+1}(\mathbf{x})x^k,
\end{equation}
where
\begin{equation}
\sigma_k(\mathbf{x}) = \begin{cases} 1, & \text{if}\ k=0; \\ \sum_{0\leq i_0<\dots<i_{k-1}\leq n} \mathbf{x}_{i_0}\cdots \mathbf{x}_{i_{k-1}}, & \text{otherwise}; \end{cases}
\end{equation}
is the $k^{\text{th}}$ elementary symmetric function in the $n+1$ variables $\mathbf{x}_0,\dots,\mathbf{x}_n$. In addition, since
\begin{equation}
1 = \sum_{i=0}^{n+1} B^{n+1}_i(x)
\end{equation}
and
\begin{equation}
\frac{d}{dx}\left[ \prod_{i=0}^n (x-\mathbf{x}_i)\right]_{x=x_j} = \prod_{i\in\{0,\dots,n\}\setminus\{j\}} (\mathbf{x}_j-\mathbf{x}_i),
\end{equation}
we can combine the previous discussion with Corollary~\ref{cor:bezout} and Corollary~\ref{cor:factorization} to obtain a decomposition of the inverse of the Bernstein--Vandermonde matrix.
\begin{theorem}
\label{thm:factoredinverse}
Define the Hankel matrices $H^n$ and $\widetilde{H}^n(\mathbf{x})$ by
\begin{equation*}
H^n_{ij} = \binom{n+1}{i+j+1}\qquad \text{and}\qquad \widetilde{H}^n_{ij}(\mathbf{x}) = \sum_{k=0}^{i+j+1} (-1)^{n-k+1}\binom{n-k+1}{n-i-j}\sigma_{n-k+1}(\mathbf{x}),
\end{equation*}
define the Toeplitz matrices $T^n$ and $\widetilde{T}^n(\mathbf{x})$ by
\begin{equation*}
T^n_{ij} = \binom{n+1}{j-i}\qquad \text{and}\qquad \widetilde{T}^n_{ij}(\mathbf{x}) = \sum_{k=0}^{j-i} (-1)^{n-k+1} \binom{n-k+1}{j-i-k}\sigma_{n-k+1}(\mathbf{x}),
\end{equation*}
and define the diagonal matrices $D^n(\mathbf{x})$ and $\Delta^n$ by
\begin{equation*}
D^n(\mathbf{x}) = \diag\left( \prod_{i\in\{0,\dots,n\}\setminus\{j\}}(\mathbf{x}_j-\mathbf{x}_i)\right)_{j=0}^n\qquad \text{and}\qquad \Delta^n = \diag\left( \binom{n}{j}\right)_{j=0}^n.
\end{equation*}
In addition, let $\widetilde{V}^n(\mathbf{x})$ be the scaled Bernstein--Vandermonde matrix
\begin{equation*}
\widetilde{V}^n_{ij}(\mathbf{x}) = \mathbf{x}_i^j(1-\mathbf{x}_i)^{n-j}.
\end{equation*}
Then
\begin{equation}
\left(V^n(\mathbf{x})\right)^{-1} = \left(\Delta^n\right)^{-1}\left[ \widetilde{H}^n(\mathbf{x})T^n-H^n\widetilde{T}^n(\mathbf{x})\right]\left(\widetilde{V}^n(\mathbf{x})\right)^T\left(D^n(\mathbf{x})\right)^{-1}.
\end{equation}
\end{theorem}
\subsection{Equispaced Nodes}
\label{ssec:equispaced}
In Section~\ref{sec:dimensions}, we will derive a block $LU$ decomposition for the Bernstein--Vandermonde matrix associated to equispaced nodes on the $d$-simplex. This decomposition leads to a recursive, block-structured algorithm for the corresponding interpolation problem. The base for this algorithm is the one-dimensional Bernstein--Vandermonde matrix associated to equispaced nodes, and so we now briefly focus on the case where $\mathbf{x}_i=i/n$ for each $0\leq i\leq n$. For these nodes, we observe that (\ref{eq:vieta}) becomes
\begin{align*}
\prod_{i=0}^n (x-i/n) = \frac{1}{n^{n+1}}\prod_{i=0}^{n+1} (nx-i) &= \frac{1}{n^{n+1}}\sum_{k=0}^{n+1} s(n+1,k)(nx)^k \\
&= \sum_{k=0}^{n+1} \frac{s(n+1,k)}{n^{n-k+1}}x^k,
\end{align*}
where $s(n,k)$ are the (signed) Stirling numbers of the first kind \cite{stegun1965handbook}. In addition, we have that
\begin{equation}
\prod_{i\in\{0,\dots,n\}\setminus\{j\}}(j/n-i/n) = (-1)^{n-j} \frac{j!(n-j)!}{n^n}.
\end{equation}
\begin{corollary}
\label{cor:equispacedinverse}
Define the Hankel matrices $H^n$ and $\widetilde{H}^n$ by
\begin{equation*}
H^n_{ij} = \binom{n+1}{i+j+1}\qquad \text{and}\qquad \widetilde{H}^n_{ij} = \sum_{k=0}^{i+j+1}\binom{n-k+1}{n-i-j}\frac{s(n+1,k)}{n^{n-k+1}},
\end{equation*}
define the Toeplitz matrices $T^n$ and $\widetilde{T}^n$ by
\begin{equation*}
T^n_{ij} = \binom{n+1}{j-i}\qquad \text{and}\qquad \widetilde{T}^n_{ij} = \sum_{k=0}^{j-i} \binom{n-k+1}{j-i-k}\frac{s(n+1,k)}{n^{n-k+1}},
\end{equation*}
and define the diagonal matrices $D^n$ and $\Delta^n$ by
\begin{equation*}
D^n = \diag\left( (-1)^{n-j}\left[j!(n-j)!\right]\right)_{j=0}^n\qquad \text{and}\qquad \Delta^n = \diag\left( \binom{n}{j}\right)_{j=0}^n.
\end{equation*}
In addition, let $\widetilde{V}^n$ be the scaled Bernstein--Vandermonde matrix
\begin{equation*}
\widetilde{V}^n_{ij} = i^j(n-i)^{n-j}.
\end{equation*}
Then the inverse of $V^n=V^n(0,1/n,\dots,1)$ is given by
\begin{equation}
\left(V^n\right)^{-1} = \left(\Delta^n\right)^{-1}\left[ \widetilde{H}^nT^n-H^n\widetilde{T}^n\right]\left(\widetilde{V}^n\right)^T\left(D^n\right)^{-1}.
\end{equation}
\end{corollary}
\section{Applying the inverse}
\label{sec:apply}
Now, we describe several approaches to applying $\left(V^n(\mathbf{x})\right)^{-1}$ to a vector.
\paragraph{$LU$ factorization} We can decompose $V^n(\mathbf{x})$ as
\begin{equation}
V^n(\mathbf{x}) = L^n(\mathbf{x})U^n(\mathbf{x}),
\end{equation}
where $L^n(\mathbf{x})$ and $U^n(\mathbf{x})$ are lower and upper triangular matrices, respectively. Widely available in libraries, computing $L^n(\mathbf{x})$ and $U^n(\mathbf{x})$ requires $\mathcal{O}(n^3)$ operations, and each of the subsequent triangular solves require $\mathcal{O}(n^2)$ operations to perform.
\paragraph{Exact inverse} In light of Corollary~\ref{cor:bezout}, we can directly form $\left(V^n(\mathbf{x})\right)^{-1}$. By Theorem~\ref{thm:recursion}, we can form the inverse in $\mathcal{O}(n^2)$ operations. The inverse can then be applied to any vector in $\mathcal{O}(n^2)$ operations using the standard algorithm.
\paragraph{DFT-based application} By Theorem~\ref{thm:factoredinverse}, we can express the inverse of $V^n(\mathbf{x})$ in terms of its (scaled) transpose and Hankel, Toeplitz, and diagonal matrices. The matrices $H^n(\mathbf{x})$ and $T^n(\mathbf{x})$ can be formed in $\mathcal{O}(n)$ operations, but the matrices $\widetilde{H}^n(\mathbf{x})$ and $\widetilde{T}^n(\mathbf{x})$ require $\mathcal{O}(n^2)$ operations to form, since each entry involves a sum. Even though the Hankel and Toeplitz matrices can be applied to a vector in $\mathcal{O}(n\log n)$ operations via circulant embedding and a couple of FFT/iFFT \cite{vogel2002fft}, the scaled transpose still requires $\mathcal{O}(n^2)$ operations to apply to a vector.
\paragraph{Newton algorithm} In \cite{ainsworth2016newton}, Ainsworth and Sanchez gave a fast algorithm for solving $V^n(\mathbf{x})\mathbf{c}=\mathbf{b}$ via a recursion relation for the B\'{e}zier control points of the Newton form of the Lagrangian interpolant. The numerical results given suggest that the algorithm is stable even for high polynomial degree, and so we include it for comparison.
\section{Conditioning and accuracy}
\label{sec:cond}
In \cite{allenkirby2020mass}, we gave a partial explanation of the relatively high accuracy, compared to its large 2-norm condition number, of working with the Bernstein mass matrix. We will use similar reasoning to investigate the performance of the Bernstein--Vandermonde matrix.
Recall that for an integer $n\geq 0$, the Bernstein mass matrix $M^n$ is the $(n+1)\times(n+1)$ matrix given by
\begin{equation}
\label{eq:mass}
M^n_{ij} = \int_0^1 B^n_i(x)B^n_j(x)dx.
\end{equation}
The mass matrix (for Bernstein or any other family of polynomials) plays an important role in connecting the $L^2$ topology on the finite-dimensional space to linear algebra. To see this, we first define mappings connecting polynomials of degree $n$ to $\mathbb{R}^{n+1}$. Given any $\mathbf{c} \in \mathbb{R}^{n+1}$, we let $\pi(\mathbf{c})$ be the polynomial expressed in the Bernstein basis with coefficients contained in $\mathbf{c}$:
\begin{equation}
\pi(\mathbf{c})(x) = \sum_{i=0}^n \mathbf{c}_i B^n_i(x).
\end{equation}
We let $\mathbf{\Pi}$ be the inverse of this mapping, sending any polynomial of degree at most $n$ to the vector of $n+1$ coefficients with respect to the Bernstein basis.
Now, let $p(x)$ and $q(x)$ be polynomials of degree $n$ with expansion coefficients $\mathbf{\Pi}(p) = \mathbf{p}$ and $\mathbf{\Pi}(q) = \mathbf{q}$. Then the $L^2$ inner product of $p$ and $q$ is given by the $M^n$-weighted inner product of $\mathbf{p}$ and $\mathbf{q}$, for
\begin{equation}
\label{eq:innprod}
\int_0^1 p(x) q(x) dx
=
\sum_{i,j=0}^n \mathbf{p}_i \mathbf{q}_j \int_0^1 B^n_i(x) B^n_j(x) dx
= \mathbf{p}^T M^n \mathbf{q}.
\end{equation}
Similarly, if
\begin{equation}
\| \mathbf{p} \|_{M^n} = \sqrt{ \mathbf{p}^T M^n \mathbf{p} }
\end{equation}
is the $M^n$-weighted vector norm, then we have for $p = \pi(\mathbf{p})$,
\begin{equation}
\label{eq:topology}
\| p \|_{L^2} = \| \mathbf{p} \|_{M^n}.
\end{equation}
We can interpret the Bernstein--Vandermonde matrix $V^n(\mathbf{x})$ as an operator mapping polynomials of degree at most $n$ to $\mathbb{R}^{n+1}$ via $p\mapsto V^n(\mathbf{x})\mathbf{\Pi}(p)$. Therefore, it makes sense to measure $p$ in the $L^2$ norm and $V^n(\mathbf{x})\mathbf{\Pi}(p)$ in the 2-norm. By (\ref{eq:topology}), we can also consider $\mathbf{\Pi}(p)$ in the $M^n$-norm, which leads us to define the operator norm
\begin{equation}
\label{eq:Mto2}
\| A \|_{M^n\rightarrow 2} = \max_{\mathbf{y}\neq 0} \frac{ \| A\mathbf{y} \|_2 }{\| \mathbf{y} \|_{M^n}},
\end{equation}
and going in the opposite direction,
\begin{equation}
\label{eq:2toM}
\| A \|_{2\rightarrow M^n} = \max_{\mathbf{y}\neq 0} \frac{ \| A\mathbf{y} \|_{M^n}}{\| \mathbf{y} \|_2}.
\end{equation}
These two norms naturally combine to define a new condition number
\begin{equation}
\label{eq:kappaMto2}
\kappa_{M^n\rightarrow 2}(A) = \| A \|_{M^n\rightarrow 2} \left\| A^{-1} \right\|_{2\rightarrow M^n}.
\end{equation}
Since $M^n$ is symmetric and positive definite, it has a well-defined positive square root via the spectral decomposition.
\begin{lemma}
\label{lem:kapparelation}
\begin{equation}
\kappa_{M^n\rightarrow 2}\left(V^n(\mathbf{x})\right) = \kappa_2 \left( V^n(\mathbf{x})\left(M^n\right)^{-1/2}\right).
\end{equation}
\end{lemma}
\begin{proof}
Observe that if $\mathbf{y}\neq 0$, then
\begin{align*}
\frac{\left\| \left(V^n(\mathbf{x})\right)^{-1} \mathbf{y} \right\|^2_{M^n}}{ \| \mathbf{y} \|^2_2} &= \frac{ \mathbf{y}^T\left(V^n(\mathbf{x})\right)^{-T} M^n \left(V^n(\mathbf{x})\right)^{-1}\mathbf{y}}{\| \mathbf{y} \|^2_2} \\
&= \frac{ \left( \left(M^n\right)^{1/2} \left(V^n(\mathbf{x})\right)^{-1} \mathbf{y}\right)^T\left( \left(M^n\right)^{1/2} \left(V^n(\mathbf{x})\right)^{-1} \mathbf{y}\right)}{\| \mathbf{y} \|^2_2} \\
&= \frac{ \left\| \left(M^n\right)^{1/2}\left( V^n(\mathbf{x})\right)^{-1} \mathbf{y}\right\|^2_2}{ \| \mathbf{y} \|^2_2 },
\end{align*}
and so
\begin{equation}
\left\| \left(V^n(\mathbf{x})\right)^{-1}\right\|_{2\rightarrow M^n} = \left\| \left(M^n\right)^{1/2}\left(V^n(\mathbf{x})\right)^{-1} \right\|_2.
\end{equation}
Let $\mathbf{z} = \left(M^n\right)^{1/2} \mathbf{y}$. Then $\mathbf{z}\neq 0$ and
\begin{align*}
\frac{ \left\| V^n(\mathbf{x})\mathbf{y}\right\|^2_2}{\left\| \mathbf{y}\right\|^2_{M^n}} &= \frac{ \mathbf{y}^T\left(V^n(\mathbf{x})\right)^TV^n(\mathbf{x})\mathbf{y}}{ \mathbf{y}^T M^n \mathbf{y} } \\
&= \frac{ \left( V^n(\mathbf{x})\left(M^n\right)^{-1/2}\mathbf{z}\right)^T\left( V^n(\mathbf{x})\left(M^n\right)^{-1/2}\mathbf{z}\right)}{\mathbf{z}^T\mathbf{z}} \\
&= \frac{ \left\| V^n(\mathbf{x})\left(M^n\right)^{-1/2}\mathbf{z}\right\|^2_2}{ \| \mathbf{z} \|^2_2},
\end{align*}
and so
\begin{equation}
\left\| V^n(\mathbf{x})\right\|_{M^n\rightarrow 2} = \left\| V^n(\mathbf{x})\left(M^n\right)^{-1/2}\right\|_2.
\end{equation}
It follows that
\begin{align*}
\kappa_{M^n\rightarrow 2}\left(V^n(\mathbf{x})\right) &= \left\| V^n(\mathbf{x})\right\|_{M^n\rightarrow 2}\left\| \left(V^n(\mathbf{x})\right)^{-1}\right\|_{2\rightarrow M^n} \\
&= \left\| V^n(\mathbf{x})\left(M^n\right)^{-1/2}\right\|_2 \left\| \left(M^n\right)^{1/2}\left(V^n(\mathbf{x})\right)^{-1} \right\|_2 \\
&= \left\| V^n(\mathbf{x})\left(M^n\right)^{-1/2}\right\|_2 \left\| \left(V^n(\mathbf{x})\left(M^n\right)^{-1/2}\right)^{-1}\right\|_2 \\
&= \kappa_2\left( V^n(\mathbf{x})\left(M^n\right)^{-1/2}\right).
\end{align*}
\end{proof}
To analyze $\kappa_{2}\left(V^n(\mathbf{x})\left(M^n\right)^{-1/2}\right)$, we will need reference to the Legendre polynomials, mapped from their typical home on $[-1,1]$ to $[0,1]$. For $0\leq j\leq n$, let $L^j(x)$ denote the Legendre polynomial of degree $n$ over $[0,1]$, scaled so that $L^n(1)=1$ and
\begin{equation}
\label{eq:LegendreL2norm}
\| L^j \|^2_{L^2} = \frac{1}{2j+1}.
\end{equation}
The Legendre--Vandermonde matrix is the $(n+1)\times (n+1)$ matrix $\widehat{V}^n(\mathbf{x})$ given by
\begin{equation}
\label{eq:LegendreVandermonde}
\widehat{V}^n_{ij}(\mathbf{x}) = L^j(\mathbf{x}_i).
\end{equation}
Given any polynomial $p$ of degree at most $n$, let $\mathbf{\Theta}(p)$ denote the vector of $n+1$ coefficients with respect to the Legendre basis. Define $T^n$ to be the $(n+1)\times (n+1)$ matrix satisfying $T^n\mathbf{\Theta}(p)= \mathbf{\Pi}(p)$ for all polynomials $p$ of degree at most $n$. In particular, we have the relationship
\begin{equation}
\label{eq:Vandermondeconversion}
\widehat{V}^n(\mathbf{x}) = V^n(\mathbf{x})T^n.
\end{equation}
Let $\{\lambda^n_j\}_{j=0}^n$ be the set of eigenvalues of $M^n$. In \cite{allenkirby2020mass}, it was shown that $M^n$ admits the spectral decomposition
\begin{equation}
\label{eq:spectral}
M^n = Q^n\Lambda^n\left(Q^n\right)^T,
\end{equation}
where
\begin{equation}
Q^n = T^n\diag\left( \sqrt{(2j+1)\lambda^n_j}\right)_{j=0}^n
\end{equation}
is an orthogonal matrix and $\Lambda^n = \diag\left(\lambda^n_j\right)_{j=0}^n$ contains the eigenvalues. This decomposition combined with Lemma~\ref{lem:kapparelation} and the transformation given in (\ref{eq:Vandermondeconversion}) suggest a relationship between the $M^n\rightarrow 2$ condition number of $V^n(x)$ and the $2$-norm condition number of $\widehat{V}^n(\mathbf{x})$ scaled by a diagonal matrix.
\begin{lemma}
\label{lem:kappa2Legendre}
\begin{equation}
\label{eq:kappa2Legendre}
\kappa_{M^n\rightarrow 2}\left(V^n(\mathbf{x})\right) = \kappa_2\left( \widehat{V}^n(\mathbf{x})\diag\left( \sqrt{2j+1}\right)_{j=0}^n\right).
\end{equation}
\end{lemma}
\begin{proof}
We have that
\begin{align*}
V^n(\mathbf{x})\left(M^n\right)^{-1/2} &= V^n(\mathbf{x})Q^n\left(\Lambda^n\right)^{-1/2}\left(Q^n\right)^T \\
&= V^n(\mathbf{x})T^n\diag\left(\sqrt{(2j+1)\lambda^n_j}\right)_{j=0}^n\diag\left(\frac{1}{\sqrt{\lambda^n_j}}\right)_{j=0}^n\left(Q^n\right)^T \\
&= \widehat{V}^n(\mathbf{x})\diag\left(\sqrt{2j+1}\right)_{j=0}^n\left(Q^n\right)^T.
\end{align*}
Since $Q^n$ is orthogonal, the result follows from Lemma~\ref{lem:kapparelation}.
\end{proof}
In \cite{gautschi1983condition}, Gautschi computed the condition number in the Frobenius norm $\| \cdot \|_F$ for Vandermonde matrices associated with families of orthonormal polynomials. Since the diagonal matrix in Lemma~\ref{lem:kappa2Legendre} contains the reciprocal of the $L^2$ norms of the Legendre polynomials, we can adapt his arguments to bound $\left(\widehat{V}^n(\mathbf{x})\diag\left( \sqrt{2j+1}\right)_{j=0}^n\right)^{-1}$ in the Frobenius norm. In addition, the Legendre polynomials have the special property that $|L^j(x)|\leq 1$ for all $0\leq x\leq 1$, and so we can bound the Frobenius norm of $\widehat{V}^n(\mathbf{x})\diag\left( \sqrt{2j+1}\right)_{j=0}^n$.
For each integer $0\leq j\leq n$, let $\ell^{j,n}$ be the $j^{\text{th}}$ Lagrange polynomial with respect to $\mathbf{x}$; that is,
\begin{equation}
\ell^{j,n}(x) = \prod_{i\in\{0,\dots,n\}\setminus\{j\}} \frac{x-\mathbf{x}_i}{\mathbf{x}_j-\mathbf{x}_i}.
\end{equation}
Define $\mathbf{w}^n\in\mathbb{R}^{n+1}$ by
\begin{equation}
\mathbf{w}^n_j = \| \ell^{j,n} \|_{L^2}.
\end{equation}
\begin{theorem}
\label{thm:kappaMto2}
\begin{equation}
\kappa_{M^n\rightarrow 2}(V^n(\mathbf{x})) \leq (n+1)^{3/2}\| \mathbf{w}^n \|_2.
\end{equation}
\end{theorem}
\begin{proof}
Since $|L^j(x)|\leq 1$ for all $0\leq x\leq 1$, we have that
\begin{align*}
\left\| \widehat{V}^n(\mathbf{x})\diag\left(\sqrt{2j+1}\right)_{j=0}^n\right\|_F &= \left( \sum_{i,j=0}^n (2j+1)\left(L^j(\mathbf{x}_i)\right)^2\right)^{1/2} \\
&\leq \left( \sum_{i,j=0}^n (2j+1) \right)^{1/2} \\
&= (n+1)^{3/2}.
\end{align*}
Since
\begin{equation}
\ell^{j,n}(\mathbf{x}_i) = \begin{cases} 1, & \text{if}\ i=j; \\ 0, & \text{if}\ i\neq j; \end{cases}
\end{equation}
we have that $\left(\widehat{V}^n(\mathbf{x})\right)^{-1}_{ij} = \mathbf{\Theta}\left(\ell^{j,n}\right)_i$. Therefore, by (\ref{eq:LegendreL2norm}), we have that
\begin{align*}
\| \mathbf{w}^n \|_2 &= \left( \sum_{j=0}^n \left(\mathbf{w}^n_j\right)^2\right)^{1/2} \\
&= \left(\sum_{j=0}^n \int_0^1 \left(\ell^{j,n}(x)\right)^2dx\right)^{1/2} \\
&= \left(\sum_{j=0}^n \int_0^1 \sum_{i,k=0}^n \mathbf{\Theta}(\ell^{j,n})_i\mathbf{\Theta}(\ell^{j,n})_kL^i(x)L^k(x)dx\right)^{1/2} \\
&= \left(\sum_{j=0}^n \sum_{i,k=0}^n \mathbf{\Theta}(\ell^{j,n})_i\mathbf{\Theta}(\ell^{j,n})_k \int_0^1 L^i(x)L^k(x)dx\right)^{1/2} \\
&= \left(\sum_{i,j=0}^n \mathbf{\Theta}(\ell^{j,n})^2_i \int_0^1 (L^i(x))^2dx\right)^{1/2} \\
&= \left(\sum_{i,j=0}^n \frac{\mathbf{\Theta}(\ell^{j,n})^2_i}{2i+1}\right)^{1/2} \\
&= \left\| \diag\left( \frac{1}{\sqrt{2i+1}}\right)_{i=0}^n \left(\widehat{V}^n(\mathbf{x})\right)^{-1}\right\|_F \\
&= \left\| \left(\widehat{V}^n(\mathbf{x})\diag\left(\sqrt{2j+1}\right)_{j=0}^n\right)^{-1}\right\|_F.
\end{align*}
Since the condition number in the 2 norm is bounded above by the condition number in the Frobenius norm, the result follows from Lemma~\ref{lem:kappa2Legendre}.
\end{proof}
\begin{figure}
\begin{center}
\begin{tikzpicture}
\begin{semilogyaxis}[legend style={at={(0.05,0.95)}, anchor=north west}, xlabel=$n$, ylabel=Condition Numbers, xmin=1, xmax=20, ymin=0]
\addplot [mark=square] table [x=n, y=k2LD, col sep=comma] {conditioning.dat};
\addplot [mark=triangle] table [x=n, y=ub, col sep=comma] {conditioning.dat};
\addplot [mark=o] table [x=n, y=k2V, col sep=comma] {conditioning.dat};
\legend{$\kappa_{M^n\rightarrow 2}(V^n(\mathbf{x}))$, Estimate, $\kappa_2(V^n(\mathbf{x}))$}
\end{semilogyaxis}
\end{tikzpicture}
\end{center}
\caption{Comparsion of the condition number of $V^n$ in the $M^n\rightarrow 2$ norm and the $2$ norm for $1\leq n\leq 20$, where $V^n$ is the matrix described in Subsection~\ref{ssec:equispaced}. We also include the estimate given in Theorem~\ref{thm:kappaMto2}. We use Lemma~\ref{lem:kappa2Legendre} to compute $\kappa_{M^n\rightarrow 2}(V^n(\mathbf{x}))$.}
\label{fig:kappa}
\end{figure}
\section{Higher dimensions}
\label{sec:dimensions}
Now, in the case of equispaced nodes on the $d$-simplex, we can use block-recursive structure as discussed in \cite{kirby2012fast} to give low-complexity algorithms for the inversion of simplicial Bernstein--Vandermonde matrices. This gives, at least for the equispaced lattice, an alternate algorithm for solving simplicial Bernstein interpolation problems to the one worked out in \cite{ainsworth2011bernstein}.
\subsection{Notation and Preliminaries}
\label{ssec:notation}
For an integer $d\geq 1$, let $S_d$ be a nondegenerate simplex in $\mathbb{R}^d$. Let $\{\mathbf{v}_i\}_{i=0}^d\subset \mathbb{R}^d$ be the vertices of $S_d$, and let $\{\mathbf{b}_i\}_{i=0}^d$ denote the barycentric coordinates of $S_d$. Each $\mathbf{b}_i$ is an affine map from $\mathbb{R}^d$ to $\mathbb{R}$ such that
\begin{equation}
\mathbf{b}_i(\mathbf{v}_j) = \begin{cases} 1, & \text{if}\ i=j; \\ 0, & \text{if}\ i\neq j; \end{cases}
\end{equation}
for each vertex $\mathbf{v}_j$. Each $\mathbf{b}_i$ is nonnegative on $S_d$, and
\begin{equation}
\sum_{i=0}^d \mathbf{b}_i = 1.
\end{equation}
A multiindex $\mathbf{\beta}$ of length $d+1$ is a $(d+1)$-tuple of nonnegative integers, written
\begin{equation}
\mathbf{\beta} = (\mathbf{\beta}_0,\dots,\mathbf{\beta}_d).
\end{equation}
The order of $\mathbf{\beta}$, denoted $|\mathbf{\beta}|$, is given by
\begin{equation}
|\mathbf{\beta}| = \sum_{i=0}^d \mathbf{\beta}_i.
\end{equation}
The factorial $\mathbf{\beta}!$ of a multiindex $\mathbf{\beta}$ is defined by
\begin{equation}
\mathbf{\beta}! = \prod_{i=0}^d \mathbf{\beta}_i.
\end{equation}
For a multiindex $\mathbf{\beta}$ of length $d+1$, denote by $\mathbf{\beta}'$ the multiindex of length $d$ given by
\begin{equation}
\mathbf{\beta}' = (\mathbf{\beta}_1, \dots, \mathbf{\beta}_d).
\end{equation}
Given a nonnegative integer $b$ and a multiindex $\mathbf{\beta}' = (\mathbf{\beta}_1,\dots, \mathbf{\beta}_d)$ of length $d$, define a new multiindex $b\vdash\mathbf{\beta}$ of length $d+1$ by
\begin{equation}
b\vdash\mathbf{\beta}' = (b,\mathbf{\beta}_1,\dots,\mathbf{\beta}_d).
\end{equation}
In particular,
\begin{equation}
\mathbf{\beta} = \mathbf{\beta}_0\vdash\mathbf{\beta}'.
\end{equation}
Multiindices have a natural partial ordering given by
\begin{equation}
\label{eq:ordering}
\mathbf{\beta} \leq \widetilde{\mathbf{\beta}}\qquad \text{if and only if}\qquad \mathbf{\beta}_i\leq \widetilde{\mathbf{\beta}}_i\quad \text{for all}\quad 0\leq i\leq d.
\end{equation}
The Bernstein polynomials of degree $n$ on the $d$-simplex $S_d$ are defined by
\begin{equation}
B^n_{\mathbf{\beta}} = \frac{n!}{\mathbf{\beta}!} \prod_{i=0}^d \mathbf{b}_i^{\mathbf{\beta}_i}.
\end{equation}
The complete set of Bernstein polynomials $\{B^n_{\mathbf{\beta}}\}_{|\mathbf{\beta}|=n}$ form a basis for polynomials in $d$ variables of complete degree at most $n$.
If $n_0\leq n$, then any polynomial expressed in the basis $\{B^{n_0}_{\mathbf{\beta}}\}_{|\mathbf{\beta}|=n_0}$ can also be expressed in the basis $\{B^n_{\mathbf{\beta}}\}_{|\mathbf{\beta}|=n}$. We denote by $E^{d,n_0,n}$ the $\binom{n+d}{d}\times\binom{n_0+d}{d}$ matrix that maps the coefficients of the degree $n_0$ representation to the coefficients of the degree $n$ representation. The matrix $E^{d,n_0,n}$ is sparse and can be applied matrix-free \cite{kirby2017fast}, if desired.
For a nonnegative integer $m$ and a set of distinct nodes $\{\mathbf{x}^m_{\mathbf{\alpha}}\}_{|\mathbf{\alpha}|=m}\subset S_d$, define the Bernstein--Vandermonde matrix $V^{d,m,n}$ to be the $\binom{m+d}{d}\times\binom{n+d}{d}$ matrix given by
\begin{equation}
V^{d,m,n}_{\mathbf{\alpha}\mathbf{\beta}} = B^n_{\mathbf{\beta}}(\mathbf{x}^m_{\mathbf{\alpha}})
\end{equation}
\begin{theorem}
\label{thm:elev}
Let $d,m,n_0$, and $n$ be nonnegative integers with $d\geq 1$ and $n_0\leq n$, and let $\{\mathbf{x}^m_{\mathbf{\alpha}}\}_{|\mathbf{\alpha}|=m}$ be distinct nodes contained in the $d$-simplex $S_d$. Then
\begin{equation}
\label{eq:elev}
V^{d,m,n}E^{d,n_0,n}=V^{d,m,n_0}.
\end{equation}
\end{theorem}
\begin{proof}
Given a polynomial $p\in\Span\{B^{n_0}_{\mathbf{\beta}}\}_{|\mathbf{\beta}|=n_0}$, the matrix $V^{d,m,n_0}$ evaluates $p$ at each of the nodes $\mathbf{x}^m_{\mathbf{\alpha}}$. On the other hand, the matrix $V^{d,m,n}E^{d,n_0,n}$ evaluates the degree $n$ representation of $p$ at each of the nodes $\mathbf{x}^m_{\mathbf{\alpha}}$. Since we are evaluating at the same nodes and the polynomial has not been modified, both evaluations must give the same result.
\end{proof}
The partial ordering (\ref{eq:ordering}) implies a natural way to order the entries of $V^{d,m,n}$ and also imposes a block structure on $V^{d,m,n}$ by dividing the matrix into sections where $\mathbf{\alpha}_0$ and $\mathbf{\beta}_0$ are constant. For integers $0\leq \mathbf{\alpha}_0\leq m$ and $0\leq \mathbf{\beta}_0\leq n$, let $V^{d,m,n}_{\mathbf{\alpha}_0\mathbf{\beta}_0}$ denote the $\binom{m-\mathbf{\alpha}_0+d}{d}\times\binom{n-\mathbf{\beta}_0+d}{d}$ submatrix of $V^{d,m,n}$ whose entries satisfy
\begin{equation}
\left(V^{d,m,n}_{\mathbf{\alpha}_0\mathbf{\beta}_0}\right)_{\mathbf{\alpha}'\mathbf{\beta}'} = V^{d,m,n}_{\left(\mathbf{\alpha}_0\vdash\mathbf{\alpha}'\right)\left(\mathbf{\beta}_0\vdash\mathbf{\beta}'\right)}.
\end{equation}
\subsection{Degree Reduction}
\label{ssec:reduction}
We now consider the case of equispaced nodes on the $d$-simplex $S^d$. For this case, the entries of the Bernstein--Vandermonde matrix are given by
\begin{equation}
\label{eq:entries}
V^{d,m,n}_{\mathbf{\alpha}\mathbf{\beta}} = \frac{n!}{\mathbf{\beta}!}\frac{1}{m^n} \prod_{i=0}^d \mathbf{\alpha}_i^{\mathbf{\beta}_i}.
\end{equation}
This representation of the entries allows us to represent the submatrix $V^{d,m,n}_{\mathbf{\alpha}_0\mathbf{\beta}_0}$ in terms of a lower-dimensional Bernstein--Vandermonde matrix.
\begin{theorem}
\label{thm:reduction}
Let $d>1$ and $m,n\geq 0$ be integers, and let $\mathbf{\alpha}_0,\mathbf{\beta}_0$ be integers with $0\leq\mathbf{\alpha}_0\leq m$ and $0\leq\mathbf{\beta}_0\leq n$. Then
\begin{equation}
\label{eq:reduction}
V^{d,m,n}_{\mathbf{\alpha}_0\mathbf{\beta}_0} = m_{\mathbf{\alpha}_0\mathbf{\beta}_0}V^{d-1,m-\mathbf{\alpha}_0,n-\beta_0},
\end{equation}
where
\begin{equation}
m_{\mathbf{\alpha}_0\mathbf{\beta}_0} = \binom{n}{\mathbf{\beta}_0}(\mathbf{\alpha}_0/m)^{\mathbf{\beta}_0}(1-\mathbf{\alpha}_0/m)^{n-\mathbf{\beta}_0}.
\end{equation}
\end{theorem}
\begin{proof}
If $\mathbf{\alpha}_0=m$ and $\mathbf{\beta}_0\neq n$, then both sides equal zero and we are done. Otherwise, by (\ref{eq:entries}), we have that
\begin{equation}
V^{d-1,m-\mathbf{\alpha}_0,n-\mathbf{\beta}_0}_{\mathbf{\alpha}'\mathbf{\beta}'} = \frac{(n-\mathbf{\beta}_0)!}{\mathbf{\beta}'!}\frac{1}{(m-\mathbf{\alpha}_0)^{n-\mathbf{\beta}_0}}\prod_{i=1}^d \mathbf{\alpha}_i^{\mathbf{\beta}_i}.
\end{equation}
On the other hand,
\begin{align*}
\left(V^{d,m,n}_{\mathbf{\alpha}_0\mathbf{\beta}_0}\right)_{\mathbf{\alpha}'\mathbf{\beta}'} &= V^{d,m,n}_{\mathbf{\alpha}\mathbf{\beta}} \\
&= \frac{n!}{\mathbf{\beta}!}\frac{1}{m^n} \prod_{i=0}^d \mathbf{\alpha}_i^{\mathbf{\beta}_i} \\
&= \frac{n!}{\mathbf{\beta}_0!(n-\mathbf{\beta}_0)!}\frac{(n-\mathbf{\beta}_0)!}{\mathbf{\beta}'!} \frac{\mathbf{\alpha}_0^{\mathbf{\beta}_0}}{m^{\mathbf{\beta}_0}} \frac{(m-\mathbf{\alpha}_0)^{n-\mathbf{\beta}_0}}{m^{n-\mathbf{\beta}_0}} \frac{1}{(m-\mathbf{\alpha}_0)^{n-\mathbf{\beta}_0}}\prod_{i=1}^d \mathbf{\alpha}_i^{\mathbf{\beta}_i},
\end{align*}
where we have separated terms containing $\mathbf{\alpha}_0$ and $\mathbf{\beta}_0$ and multiplied and divided by $(n-\mathbf{\beta}_0)!$ and $(m-\mathbf{\alpha}_0)^{n-\mathbf{\beta}_0}$. Since
\begin{equation}
\frac{n!}{\mathbf{\beta}_0!(n-\mathbf{\beta}_0)!}\frac{\mathbf{\alpha}_0^{\mathbf{\beta}_0}}{m^{\mathbf{\beta}_0}}\frac{(m-\alpha_0)^{n-\mathbf{\beta}_0}}{m^{n-\mathbf{\beta}_0}} = \binom{n}{\mathbf{\beta}_0}(\mathbf{\alpha}_0/m)^{\mathbf{\beta}_0}(1-\mathbf{\alpha}_0/m)^{n-\mathbf{\beta}_0},
\end{equation}
we have the desired result.
\end{proof}
The fact that the blocks are multiples of lower-dimensional matrices is analogous to what has been observed for other matrices related to Bernstein polynomials. For example, the Bernstein--Vandermonde matrix associated with the zeroes of Legendre polynomials has a similar structure \cite{kirby2012fast}, and so does the Bernstein mass matrix \cite{kirby2017fast}.
We recognize that $m_{\mathbf{\alpha}_0\mathbf{\beta}_0}$ describes a univariate Bernstein polynomial of degree $n$ being evaluated at equispaced points. Therefore, if $V^n$ is the one-dimensional Bernstein--Vandermonde matrix associated with equispaced nodes as described in Subsection~\ref{ssec:equispaced}, then we have that $V^{d,n,n}$ admits the block structure
\begin{equation}
\label{eq:block}
V^{d,n,n} =
\begin{pmatrix}
V^n_{00}V^{d-1,n,n} & V^n_{01}V^{d-1,n,n-1} & \cdots & V^n_{0n}V^{d-1,n,0} \\
V^n_{10}V^{d-1,n-1,n} & V^n_{11}V^{d-1,n-1,n-1} & \cdots & V^n_{1n}V^{d-1,n-1,0} \\
\vdots & \vdots & \ddots & \vdots \\
V^n_{n0}V^{d-1,0,n} & V^n_{n1}V^{d-1,0,n-1} & \cdots & V^n_{nn}V^{d-1,0,0}
\end{pmatrix}.
\end{equation}
We can use this block structure and Theorem~\ref{thm:elev} to perform block Gaussian elimination on $V^{d,n,n}$. This method was used in \cite{kirby2017fast} to obtain the block $LU$ decomposition of the Bernstein mass matrix. In the same way, we have the following:
\begin{theorem}
\label{thm:LU}
Let $V^n$ be the one-dimensional Bernstein--Vandermonde matrix associated with equispaced nodes as described in Subsection~\ref{ssec:equispaced}. Suppose $V^n=L^nU^n$ is the $LU$ decomposition of $V^n$. Then
\begin{equation}
V^{d,n,n}=L^{d,n}U^{d,n},
\end{equation}
where $L^{d,n}$ is the block lower triangular matrix with blocks given by
\begin{equation}
L^{d,n}_{\mathbf{\alpha}_0\mathbf{\beta}_0} = L^n_{\mathbf{\alpha}_0\mathbf{\beta}_0}V^{d-1,n-\mathbf{\alpha}_0,n-\mathbf{\beta}_0}
\end{equation}
and $U^{d,n}$ is the block upper triangular matrix with blocks given by
\begin{equation}
U^{d,n}_{\mathbf{\alpha}_0\mathbf{\beta}_0} = U^n_{\mathbf{\alpha}_0\mathbf{\beta}_0}E^{d-1,n-\mathbf{\beta}_0,n-\mathbf{\alpha}_0}.
\end{equation}
\end{theorem}
\begin{proof}
By Theorem~\ref{thm:elev}, if $0\leq \mathbf{\alpha}_0,\mathbf{\beta}_0\leq n$ and $\bm{\gamma}_0\leq \mathbf{\beta}_0$, then
\begin{align*}
L^{d,n}_{\mathbf{\alpha}_0\bm{\gamma}_0}U^{d,n}_{\bm{\gamma}_0\mathbf{\beta}_0} &= L^n_{\mathbf{\alpha}_0\bm{\gamma}_0}U^n_{\bm{\gamma}_0\mathbf{\beta}_0}V^{d-1,n-\mathbf{\alpha}_0,n-\bm{\gamma}_0}E^{d-1,n-\mathbf{\beta}_0,n-\bm{\gamma}_0} \\
&= L^n_{\mathbf{\alpha}_0\bm{\gamma}_0}U^n_{\bm{\gamma}_0\mathbf{\beta}_0} V^{d-1,n-\mathbf{\alpha}_0,n-\mathbf{\beta}_0}.
\end{align*}
Therefore,
\begin{align*}
\left(L^{d,n}U^{d,n}\right)_{\mathbf{\alpha}_0\mathbf{\beta}_0} &= \sum_{\bm{\gamma}_0=0}^{\mathbf{\beta}_0} L^{d,n}_{\mathbf{\alpha}_0\bm{\gamma}_0}U^{d,n}_{\bm{\gamma}_0\mathbf{\beta}_0} \\
&= \left( \sum_{\bm{\gamma}_0=0}^{\mathbf{\beta}_0} L^n_{\mathbf{\alpha}_0\bm{\gamma}_0}U^n_{\bm{\gamma}_0\mathbf{\beta}_0}\right) V^{d-1,n-\mathbf{\alpha}_0,n-\mathbf{\beta}_0} \\
&= V^n_{\mathbf{\alpha}_0\mathbf{\beta}_0}V^{d-1,n-\mathbf{\alpha}_0,n-\mathbf{\beta}_0} \\
&= V^{d,n,n}_{\mathbf{\alpha}_0\mathbf{\beta}_0}.
\end{align*}
\end{proof}
\section{Numerical results}
Now, we consider the accuracy of the methods described above on several problems. For all of the problems, we chose random solution vectors, computed the right-hand side by matrix multiplication, and then attempted to recover the solution. The numerical results are run in double precision arithmetic on a 2014 Macbook Air running macOS 10.13 and using Python 2.7.13. Cholesky factorization and FFTs are performed using the \texttt{numpy} (v1.12.0) function calls. Also, because our code is a mix of pure Python and low-level compiled libraries, timings are not terribly informative. Consequently, we focus on assessing the stability and accuracy of our methods. If future work leads to more stable fast algorithms, greater care will be afforded to tuning our implementations for performance.
In Figure~\ref{fig:equispacedinfo}, we consider the case of equispaced nodes. When considering the Euclidean norm of the error (Figure~\ref{subfig:equispaced2err}), the $LU$ factorization and the Newton algorithm have the best perfomance, followed by the DFT-based application and multiplying by the inverse. We observe that the $LU$ and Newton methods have comparable performance, as do the DFT-based algorithm and multiplying by the inverse. This is in contrast to what was observed in \cite{allenkirby2020mass}, where a similar DFT-based algorithm quickly became unstable. The same behavior can be observed when comparing the Euclidean norm of the residual (Figure~\ref{subfig:equispaced2res}), although some separation does occur between the Newton method and the $LU$ factorization. Since the solution vector can be viewed as the Bernstein coefficients of the interpolation polynomial, we also measure the $L^2$ difference between the exact and computed solutions (Figure~\ref{subfig:equispacedMerr}); equivalently, we measure the relative $M^n$ error, where $M^n$ is the Bernstein mass matrix given in (\ref{eq:mass}). All four solution methods have very similar behavior in the $M^n$ norm as they do when considering the Euclidean norm of the residual.
To ensure that the behavior of the solution methods does not depend on the choice of nodes, we also considered the case where the nodes $\mathbf{x}_j$ are randomly selected from $[j/(n+1),(j+1)/(n+1))$ for each $0\leq j\leq n$ (Figure~\ref{fig:randominfo}); however, there was no significant difference in the quantities measured between this case and the equispaced case.
We also used the block $LU$ decomposition given in Theorem~\ref{thm:LU} combined with the one-dimensional $LU$ algorithm to solve the interpolation problem for equispaced nodes on the $d$-simplex for $d=2$ and $d=3$ (Figure~\ref{fig:blockinfo}). Due to the recursive nature of the algorithm, the quantities measured are very similar to the ones observed for equispaced nodes.
\begin{figure}
\centering
\begin{subfigure}{0.475\linewidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{semilogyaxis}[legend style={at={(0.05,0.95)}, anchor=north west}, xlabel=$n$, ylabel=$\| \mathbf{c}-\widehat{\mathbf{c}} \|_2 / \| \mathbf{c} \|_2$, xmin=1, xmax=20, ymin=1e-18]
\addplot [mark=square] table [x=n, y=BezoutL2err, col sep=comma] {equispaced_info.dat};
\addplot [mark=triangle] table [x=n, y=DFTL2err, col sep=comma] {equispaced_info.dat};
\addplot [mark=diamond] table [x=n, y=LUL2err, col sep=comma] {equispaced_info.dat};
\addplot[mark=o] table [x=n, y=NewtonL2err, col sep=comma]{equispaced_info.dat};
\legend{B\'{e}zout, DFT, $LU$, Newton}
\end{semilogyaxis}
\end{tikzpicture}
\caption{Error in the 2-norm.}
\label{subfig:equispaced2err}
\end{subfigure}
\hfill
\begin{subfigure}{0.475\linewidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{semilogyaxis}[legend style={at={(0.05,0.95)}, anchor=north west}, xlabel=$n$, ylabel=$\| \mathbf{c}-\widehat{\mathbf{c}} \|_{M^n} / \| \mathbf{c} \|_{M^n}$, xmin=1, xmax=20, ymin=1e-18]
\addplot [mark=square] table [x=n, y=BezoutMerr, col sep=comma] {equispaced_info.dat};
\addplot [mark=triangle] table [x=n, y=DFTMerr, col sep=comma] {equispaced_info.dat};
\addplot [mark=diamond] table [x=n, y=LUMerr, col sep=comma] {equispaced_info.dat};
\addplot[mark=o] table [x=n, y=NewtonMerr, col sep=comma]{equispaced_info.dat};
\legend{B\'{e}zout, DFT, $LU$, Newton}
\end{semilogyaxis}
\end{tikzpicture}
\caption{Error in the $M^n$ norm.}
\label{subfig:equispacedMerr}
\end{subfigure}
\vskip\baselineskip
\begin{subfigure}{\linewidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{semilogyaxis}[legend style={at={(0.05,0.95)}, anchor=north west}, xlabel=$n$, ylabel=$\| V^n\widehat{\mathbf{c}}-\mathbf{b} \|_2$, xmin=1, xmax=20, ymin=1e-18]
\addplot [mark=square] table [x=n, y=Bezoutres, col sep=comma] {equispaced_info.dat};
\addplot [mark=triangle] table [x=n, y=DFTres, col sep=comma] {equispaced_info.dat};
\addplot [mark=diamond] table [x=n, y=LUres, col sep=comma] {equispaced_info.dat};
\addplot[mark=o] table [x=n, y=Newtonres, col sep=comma]{equispaced_info.dat};
\legend{B\'{e}zout, DFT, $LU$, Newton}
\end{semilogyaxis}
\end{tikzpicture}
\caption{Residual in the 2-norm.}
\label{subfig:equispaced2res}
\end{subfigure}
\caption{Error/residual in using the methods described in Section~\ref{sec:apply} to solve $V^n\mathbf{c}=\mathbf{b}$ for $1\leq n\leq 20$, where $V^n$ is the matrix described in Subsection~\ref{ssec:equispaced} and $\mathbf{b}$ is a random vector in $[-1,1]^{n+1}$. B\'{e}zout refers to Corollary~\ref{cor:bezout}, DFT refers to Corollary~\ref{cor:equispacedinverse}, $LU$ refers to $LU$ decomposition of $V^n$, and Newton refers to the Ainsworth--Sanchez algorithm. We use $\widehat{\mathbf{c}}$ to denote the computed solution.}
\label{fig:equispacedinfo}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}{0.475\linewidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{semilogyaxis}[legend style={at={(0.05,0.95)}, anchor=north west}, xlabel=$n$, ylabel=$\| \mathbf{c}-\widehat{\mathbf{c}} \|_2 / \| \mathbf{c} \|_2$, xmin=1, xmax=20, ymin=1e-18]
\addplot [mark=square] table [x=n, y=BezoutL2err, col sep=comma] {random_info.dat};
\addplot [mark=triangle] table [x=n, y=DFTL2err, col sep=comma] {random_info.dat};
\addplot [mark=diamond] table [x=n, y=LUL2err, col sep=comma] {random_info.dat};
\addplot[mark=o] table [x=n, y=NewtonL2err, col sep=comma]{random_info.dat};
\legend{B\'{e}zout, DFT, $LU$, Newton}
\end{semilogyaxis}
\end{tikzpicture}
\caption{Error in the 2-norm.}
\label{subfig:random2err}
\end{subfigure}
\hfill
\begin{subfigure}{0.475\linewidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{semilogyaxis}[legend style={at={(0.05,0.95)}, anchor=north west}, xlabel=$n$, ylabel=$\| \mathbf{c}-\widehat{\mathbf{c}} \|_{M^n} / \| \mathbf{c} \|_{M^n}$, xmin=1, xmax=20, ymin=1e-18]
\addplot [mark=square] table [x=n, y=BezoutMerr, col sep=comma] {random_info.dat};
\addplot [mark=triangle] table [x=n, y=DFTMerr, col sep=comma] {random_info.dat};
\addplot [mark=diamond] table [x=n, y=LUMerr, col sep=comma] {random_info.dat};
\addplot[mark=o] table [x=n, y=NewtonMerr, col sep=comma]{random_info.dat};
\legend{B\'{e}zout, DFT, $LU$, Newton}
\end{semilogyaxis}
\end{tikzpicture}
\caption{Error in the $M^n$ norm.}
\label{subfig:randomMerr}
\end{subfigure}
\vskip\baselineskip
\begin{subfigure}{\linewidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{semilogyaxis}[legend style={at={(0.05,0.95)}, anchor=north west}, xlabel=$n$, ylabel=$\| V^n(\mathbf{x})\widehat{\mathbf{c}}-\mathbf{b} \|_2$, xmin=1, xmax=20, ymin=1e-18]
\addplot [mark=square] table [x=n, y=Bezoutres, col sep=comma] {random_info.dat};
\addplot [mark=triangle] table [x=n, y=DFTres, col sep=comma] {random_info.dat};
\addplot [mark=diamond] table [x=n, y=LUres, col sep=comma] {random_info.dat};
\addplot[mark=o] table [x=n, y=Newtonres, col sep=comma]{random_info.dat};
\legend{B\'{e}zout, DFT, $LU$, Newton}
\end{semilogyaxis}
\end{tikzpicture}
\caption{Residual in the 2-norm.}
\label{subfig:random2res}
\end{subfigure}
\caption{Error/residual in using the methods described in Section~\ref{sec:apply} to solve $V^n(\mathbf{x})\mathbf{c}=\mathbf{b}$ for $1\leq n\leq 20$, where $V^n(\mathbf{x})$ is the Bernstein--Vandermonde matrix associated to $\mathbf{x}$, the nodes $\mathbf{x}_j$ are randomly selected from $[j/(n+1),(j+1)/(n+1))$ for $0\leq j\leq n$, and $\mathbf{b}$ is a random vector in $[-1,1]^{n+1}$. B\'{e}zout refers to Corollary~\ref{cor:bezout}, DFT refers to Theorem~\ref{thm:factoredinverse}, $LU$ refers to $LU$ decomposition of $V^n$, and Newton refers to the Ainsworth--Sanchez algorithm. We use $\widehat{\mathbf{c}}$ to denote the computed solution.}
\label{fig:randominfo}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}{0.475\linewidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{semilogyaxis}[legend style={at={(0.05,0.95)}, anchor=north west}, xlabel=$n$, ylabel=$\| \mathbf{c}-\widehat{\mathbf{c}} \|_2 / \| \mathbf{c} \|_2$, xmin=1, xmax=20, ymin=1e-18]
\addplot [mark=square] table [x=n, y=2dL2err, col sep=comma] {blockLU_info.dat};
\addplot [mark=triangle] table [x=n, y=3dL2err, col sep=comma] {blockLU_info.dat};
\legend{$d=2$, $d=3$}
\end{semilogyaxis}
\end{tikzpicture}
\caption{Error in the 2-norm.}
\label{subfig:block2err}
\end{subfigure}
\hfill
\begin{subfigure}{0.475\linewidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{semilogyaxis}[legend style={at={(0.05,0.95)}, anchor=north west}, xlabel=$n$, ylabel=$\| \mathbf{c}-\widehat{\mathbf{c}} \|_{M^{d,n}} / \| \mathbf{c} \|_{M^{d,n}}$, xmin=1, xmax=20, ymin=1e-18]
\addplot [mark=square] table [x=n, y=2dMerr, col sep=comma] {blockLU_info.dat};
\addplot [mark=triangle] table [x=n, y=3dMerr, col sep=comma] {blockLU_info.dat};
\legend{$d=2$, $d=3$}
\end{semilogyaxis}
\end{tikzpicture}
\caption{Error in the $M^{d,n}$ norm.}
\label{subfig:blockMerr}
\end{subfigure}
\vskip\baselineskip
\begin{subfigure}{\linewidth}
\centering
\begin{tikzpicture}[scale=0.7]
\begin{semilogyaxis}[legend style={at={(0.05,0.95)}, anchor=north west}, xlabel=$n$, ylabel=$\| V^{d,n,n}\widehat{\mathbf{c}}-\mathbf{b} \|_2$, xmin=1, xmax=20, ymin=1e-18]
\addplot [mark=square] table [x=n, y=2dres, col sep=comma] {blockLU_info.dat};
\addplot [mark=triangle] table [x=n, y=3dres, col sep=comma] {blockLU_info.dat};
\legend{$d=2$, $d=3$}
\end{semilogyaxis}
\end{tikzpicture}
\caption{Residual in the 2-norm.}
\label{subfig:block2res}
\end{subfigure}
\caption{Error/residual in using the block $LU$ decomposition given in Theorem~\ref{thm:LU} to solve $V^{d,n,n}\mathbf{c}=\mathbf{b}$ for $1\leq n\leq 20$ and $d=2,3$, where $V^{d,n,n}$ is the Bernstein--Vandermonde matrix given in (\ref{eq:entries}) and $\mathbf{b}$ is a random vector in $[-1,1]^{\binom{n+d}{d}}$. We use $\widehat{\mathbf{c}}$ to denote the computed solution.}
\label{fig:blockinfo}
\end{figure}
\section{Conclusion}
We have studied several algorithms for the inversion of the univariate Bernstein--Vandermonde matrix. These algorithms, while less stable than algorithms discovered previously, provide insight into the structure of the Bernstein--Vandermonde matrix and are remarkably similar to algorithms derived for the Bernstein mass matrix. In addition, we have used a block $LU$ decomposition of the Bernstein--Vandermonde matrix corresponding to equispaced nodes on the $d$-simplex to give a recursive, block-structured algorithm with comparable accuracy to the one-dimensional algorithm. Moreover, we have given a new perspective on the conditioning of the Bernstein--Vandermonde matrix, indicating that the interpolation problem is better-conditioned with respect to the $L^2$ norm than the Euclidean norm. In the future, we hope to expand this perspective to other polynomial problems and continue the development of fast and accurate methods for problems involving Bernstein polynomials.
\clearpage
\bibliographystyle{plain}
| {
"timestamp": "2020-05-08T02:06:00",
"yymm": "2005",
"arxiv_id": "2005.03251",
"language": "en",
"url": "https://arxiv.org/abs/2005.03251",
"abstract": "Bernstein polynomials, long a staple of approximation theory and computational geometry, have also increasingly become of interest in finite element methods. Many fundamental problems in interpolation and approximation give rise to interesting linear algebra questions. When attempting to find a polynomial approximation of boundary or initial data, one encounters the Bernstein-Vandermonde matrix, which is found to be highly ill-conditioned. Previously, we used the relationship between monomial Bezout matrices and the inverse of Hankel matrices to obtain a decomposition of the inverse of the Bernstein mass matrix in terms of Hankel, Toeplitz, and diagonal matrices. In this paper, we use properties of the Bernstein-Bezout matrix to factor the inverse of the Bernstein-Vandermonde matrix into a difference of products of Hankel, Toeplitz, and diagonal matrices. We also use a nonstandard matrix norm to study the conditioning of the Bernstein-Vandermonde matrix, showing that the conditioning in this case is better than in the standard 2-norm. Additionally, we use properties of multivariate Bernstein polynomials to derive a block $LU$ decomposition of the Bernstein-Vandermonde matrix corresponding to equispaced nodes on the $d$-simplex.",
"subjects": "Numerical Analysis (math.NA)",
"title": "Structured inversion of the Bernstein-Vandermonde Matrix",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9888419674366167,
"lm_q2_score": 0.7154240018510026,
"lm_q1q2_score": 0.7074412775417231
} |
https://arxiv.org/abs/1605.02281 | Optimal quantizers for a nonuniform distribution on a Sierpinski carpet | The purpose of quantization for a probability distribution is to estimate the probability by a discrete probability with finite support. In this paper, a nonuniform probability measure $P$ on $\mathbb R^2$ which has support the Sierpiński carpet generated by a set of four contractive similarity mappings with equal similarity ratios has been considered. For this probability measure, the optimal sets of $n$-means and the $n$th quantization errors are investigated for all $n\geq 2$. | \section{Introduction}
Quantization is a destructive process. Its purpose is to reduce the cardinality of the representation space, in particular when the input data is real-valued. It is a fundamental problem in signal processing, data compression and information theory. We refer to \cite{GG, GN, Z} for surveys on the subject and comprehensive lists of references to the literature, see also \cite{AW, GKL, GL1, GL2}. Let $\D R^d$ denote the $d$-dimensional Euclidean space, $\|\cdot\|$ denote the Euclidean norm on $\D R^d$ for any $d\geq 1$, and $n\in \D N$. Then the $n$th \textit{quantization
error} for a Borel probability measure $P$ on $\D R^d$ is defined by
\begin{equation*} \label{eq0} V_n:=V_n(P)=\inf \Big\{\int \min_{a\in\alpha} \|x-a\|^2 dP(x) : \alpha \subset \mathbb R^d, \text{ card}(\alpha) \leq n \Big\},\end{equation*}
where the infimum is taken over all subsets $\alpha$ of $\mathbb R^d$ with card$(\alpha)\leq n$. If $\int \| x\|^2 dP(x)<\infty$ then there is some set $\alpha$ for
which the infimum is achieved (see \cite{AW, GKL, GL1, GL2}). Such a set $\alpha$ for which the infimum occurs and contains no more than $n$ points is called an \tit{optimal set of $n$-means}, or \tit{optimal set of $n$-quantizers}. The collection of all optimal sets of $n$-means for a probability measure $P$ is denoted by $\C C_n:=\C C_n(P)$. It is known that for a continuous probability measure an optimal set of $n$-means always has exactly $n$-elements (see \cite{GL2}).
Given a finite subset $\alpha\sci \D R^d$, the Voronoi region generated by $a\in \alpha$ is defined by
\[M(a|\alpha)=\set{x \in \D R^d : \|x-a\|=\min_{b \in \alpha}\|x-b\|}\]
i.e., the Voronoi region generated by $a\in \alpha$ is the set of all points in $\D R^d$ which are closest to $a \in \alpha$, and the set $\set{M(a|\alpha) : a \in \alpha}$ is called the \tit{Voronoi diagram} or \tit{Voronoi tessellation} of $\D R^d$ with respect to $\alpha$. A Borel measurable partition $\set{A_a : a \in \alpha}$ of $\D R^d$ is called a \tit{Voronoi partition} of $\D R^d$ with respect to $\alpha$ (and $P$) if
$A_a \sci M(a|\alpha) \te{ ($P$-a.e.) for every $a \in \alpha$}.$ Given a Voronoi tessellation $\set{M_i}_{i=1}^k$ generated by a set of points $\set{z_i}_{i=1}^k$ (called \tit{sites} or \tit{generators}), the mass centroid $c_i$ of $M_i$ with respect to the probability measure $P$ is given by
\begin{align*}
c_i=\frac{1}{P(M_i)}\int_{M_i} x dP=\frac{\int_{M_i} x dP}{\int_{M_i} dP}.
\end{align*}
The Voronoi tessellation is called the \tit{centroidal Voronoi tessellation} (CVT) if $z_i=c_i$ for $i=1, 2, \cdots, k$, that is, if the generators are also the centroids of the corresponding Voronoi regions.
Let us now state the following proposition (see \cite{GG, GL2}):
\begin{prop} \label{prop10}
Let $\alpha$ be an optimal set of $n$-means and $a\in \alpha$. Then,
$(i)$ $P(M(a|\alpha))>0$, $(ii)$ $ P(\partial M(a|\alpha))=0$, $(iii)$ $a=E(X : X \in M(a|\alpha))$, and $(iv)$ $P$-almost surely the set $\set{M(a|\alpha) : a \in \alpha}$ forms a Voronoi partition of $\D R^d$.
\end{prop}
Let $\alpha$ be an optimal set of $n$-means and $a \in \alpha$, then by Proposition~\ref{prop10}, we have
\begin{align*}
a=\frac{1}{P(M(a|\alpha))}\int_{M(a|\alpha)} x dP=\frac{\int_{M(a|\alpha)} x dP}{\int_{M(a|\alpha)} dP},
\end{align*}
which implies that $a$ is the centroid of the Voronoi region $M(a|\alpha)$ associated with the probability measure $P$ (see also \cite{DFG, R1}).
A transformation $f: X \to X$ on a metric space $(X, d)$ is called \tit{contractive} or a \tit{contraction mapping} if there is a constant $0<c<1$ such that $d(f(x), f(y))\leq c d(x, y)$ for all $x, y \in X$. On the other hand, $f$ is called a \tit{similarity mapping} or a \tit{similitude} if there exists a constant $s>0$ such that $d(f(x), f(y))=s d(x, y)$ for all $x, y\in X$. Here $s$ is called the similarity ratio of the similarity mapping $f$.
Let $C$ be the Cantor set generated by the two contractive similarity mappings $S_1$ and $S_2$ on $\D R$ given by $S_1(x)=r_1 x$ and $ S_2 (x)=r_2 x +(1-r_2)$ where $0<r_1, r_2<1$ and $r_1+r_2<\frac 12$. Let $P=p_1 P\circ S_1^{-1}+p_2 P\circ S_2^{-1}$, where $P\circ S_i^{-1}$ denotes the image measure of $P$ with respect to
$S_i$ for $i=1, 2$ and $(p_1, p_2)$ is a probability vector with $0<p_1, p_2<1$. Then, $P$ is a singular continuous probability measure on $\D R$ with support the Cantor set $C$ (see \cite{H}). For $r_1=r_2=\frac 13$ and $p_1=p_2=\frac 12$, Graf and Luschgy gave a closed formula to determine the optimal sets of $n$-means for the probability distribution $P$ for any $n\geq 2$ (see \cite{GL3}). For $r_1=\frac 14$, $r_2=\frac 12$, $p_1=\frac 14$ and $p_2=\frac 34$, L. Roychowdhury gave an induction formula to determine the optimal sets of $n$-means and the $n$th quantization error for the probability distribution $P$ for any $n\geq 2 $ (see \cite{R2}). Let us now consider the Sierpi\'nski carpet which is generated by the four contractive similarity mappings $S_1, S_2, S_3$ and $S_4$ on $\D R^2$ such that $S_1(x_1, x_2)=\frac 13(x_1, x_2)$, $S_2(x_1, x_2)=\frac 13(x_1, x_2) + (\frac 23, 0)$, $S_3(x_1, x_2)=\frac 13(x_1, x_2) +(0, \frac 23)$, and $S_4(x_1, x_2)=\frac 13(x_1, x_2)+(\frac 23, \frac 23)$ for all $(x_1, x_2) \in \D R^2$. If $P$ is a Borel probability measure on $\D R^2$ such that $P=\frac 1 4P\circ S_1^{-1}+\frac 1 4P\circ S_2^{-1}+\frac 1 4P\circ S_3^{-1}+\frac 1 4P\circ S_4^{-1}$, then $P$ has support the Sierpi\'nski carpet. For this probability measure, C\"omez and Roychowdhury gave closed formulas to determine the optimal sets of $n$-means and the $n$th quantization error for any $n\geq 2$ (see \cite{CR}).
In this paper, we have considered the probability distribution $P$ given by $P=\frac 1 8P\circ S_1^{-1}+\frac 1 8P\circ S_2^{-1}+\frac 38P\circ S_3^{-1}+\frac 38P\circ S_4^{-1}$ which has support the Sierpi\'nski carpet generated by the four contractive similarity mappings given by $S_1(x_1, x_2)=\frac 13(x_1, x_2)$, $S_2(x_1, x_2)=\frac 13(x_1, x_2) + (\frac 23, 0)$, $S_3(x_1, x_2)=\frac 13(x_1, x_2) +(0, \frac 23)$, and $S_4(x_1, x_2)=\frac 13(x_1, x_2)+(\frac 23, \frac 23)$ for all $(x_1, x_2) \in \D R^2$. The probability distribution $P$ considered in this paper is called `nonhomogeneous' to mean that the probabilities associated with the mappings $S_1, S_2, S_3$ and $S_4$ are not equal. For this probability distribution in Proposition~\ref{prop1}, Proposition~\ref{prop2} and Proposition~\ref{prop3}, first we have determined the optimal sets of $n$-means and the $n$th quantization errors for $n=2, 3, \te{ and } 4$. Then, in Theorem~\ref{Th1} we state and prove an induction formula to determine the optimal sets of $n$-means for all $n\geq 2$. We also give some figures to illustrate the location of the optimal points (see Figure~\ref{Fig1}). In addition, running the induction formula in computer algorithm, we obtain some results and observations about the optimal sets of $n$-means which are given in Section~4; a tree diagram of the optimal sets of $n$-means for a certain range of $n$ is also given (see Figure~\ref{Fig2}).
\section{Preliminaries}
In this section, we give the basic definitions and lemmas that will be instrumental in our analysis. For $k\geq 1$, by a word $\omega$ of length $k$ over the alphabet $I:=\set{1, 2,3, 4}$ it is meant that $\omega:=\omega_1\omega_2\cdots \omega_k$, i.e., $\omega$ is a finite sequence of symbols over the alphabet $I$. Here $k$ is called the length of the word $\omega$. If $k=0$, i.e., if $\omega$ is a word of length zero, we call it the empty word and is denoted by $\es$. Length of a word $\omega$ is denoted by $|\omega|$. $I^\ast$ denotes the set of all words over the alphabet $I$ including the empty word $\es$. By $\omega\tau:=\omega_1\cdots \omega_k\tau_1\cdots \tau_\ell$ it is meant that the word obtained from the concatenations of the words $\omega:=\omega_1\omega_2\cdots \omega_k$ and $\tau:=\tau_1\tau_2\cdots\tau_\ell$ for $k,\ell\geq 0$. The maps $S_i :\D R^2 \to \D R^2,\ 1\leq i \leq 4, $ will be the generating maps of the Sierpi\'nski carpet defined as before.
For $\omega=\omega_1\omega_2 \cdots\omega_k \in I^k$, set $S_\omega=S_{\omega_1}\circ \cdots \circ S_{\omega_k}$
and $J_\omega=S_{\omega}([0, 1]\times [0, 1])$. For the empty word $\emptyset $, by $S_{\emptyset}$ we mean the identity mapping on $\D R^2$, and write $J=J_{\emptyset}=S_{\emptyset}([0,1]\times [0, 1])=[0, 1]\times [0, 1]$. The sets $\{J_\omega : \omega \in \{1, 2, 3, 4 \}^k \}$ are just the $4^k$ squares in the $k$th level in the construction of the Sierpi\'nski carpet. The squares $J_{\omega 1}$, $J_{\omega 2}$, $J_{\omega 3}$ and $J_{\omega 4}$ into which $J_\omega$ is split up at the $(k+1)$th level are called the basic squares of $J_\omega$. The set $S=\cap_{k \in \D N} \cup_{\omega \in \{1, 2, 3, 4 \}^k} J_\omega$ is the Sierpi\'nski carpet and equals the support of the probability measure $P$ given by $P =\frac 1 8 P \circ S_1^{-1} + \frac 1 8 P\circ S_2^{-1} +\frac 38 P\circ S_3^{-1}+\frac 38 P\circ S_4^{-1}$. Set $s_1=s_2=s_3=s_4=\frac 13$, $p_1=p_2=\frac 18$ and $p_3=p_4=\frac 38$, and for $\omega=\omega_1 \omega_2 \cdots \omega_k \in I^k$, write
$c(\omega):=\te{card}(\{i : \omega_i=3 \te{ or } 4, \, 1\leq i\leq k\})$, where $\te{card}(A)$ of a set $A$ represents the number of elements in the set $A$. Then, for $\omega=\omega_1\omega_2 \cdots\omega_k \in I^k$, $k\geq 1$, we have
\[s_\omega=\frac 1 {3^k} \te{ and } p_\omega=p_{\omega_1}p_{\omega_2}\cdots p_{\omega_k}=\frac{3^{c(\omega)}}{8^k}.\]
Let us now give the following lemma.
\begin{lemma} \label{lemma1} Let $f: \D R \to \D R^+$ be Borel measurable and $k\in \D N$. Then,
\[\int f \,dP=\sum_{\omega \in I^k} p_\omega\int f\circ S_\omega \,dP.\]
\end{lemma}
\begin{proof}
We know $P =p_1 P \circ S_1^{-1} + p_2 P\circ S_2^{-1} +p_3 P\circ S_3^{-1}+p_4 P\circ S_4^{-1}$, and so by induction $P=\sum_{\omega \in I^k} p_\omega P\circ S_\omega^{-1}$, and thus the lemma is yielded.
\end{proof}
Let $S_{(i1)}, \, S_{(i2)}$ be the horizontal and vertical components of the transformation $S_i$ for $i=1, 2, 3, 4$. Then for any $(x_1, x_2) \in \D R^2$ we have
$S_{(11)}(x_1) =\frac 1 3 x_1$, $ S_{(12)}(x_2)=\frac 1 3 x_2$, $S_{(21)}(x_1)=\frac 1 3 x_1 +\frac 23$, $S_{(22)}(x_2)=\frac 1 3 x_2$, $S_{(31)}(x_1)=\frac 1 3 x_1$, $S_{(32)}(x_2)= \frac 1 3 x_2+ \frac 2 3$, and $S_{(41)}(x_1)=\frac 1 3 x_1 +\frac 23$, $S_{(42)}(x_2)= \frac 1 3 x_2+ \frac 2 3$. Let $X=(X_1, X_2)$ be a bivariate random variable with distribution $P$. Let $P_1, P_2$ be the marginal distributions of $P$, i.e., $P_1(A)=P(A\times \D R)$ for all $A \in \F B$, and $P_2(B)=P(\D R \times B)$ for all $B \in \F B$. Here $\F B$ is the Borel $\sigma$-algebra on $\D R$. Then $X_1$ has distribution $P_1$ and $X_2$ has distribution $P_2$.
Let us now state the following lemma. The proof is similar to Lemma~2.2 in \cite{CR}.
\begin{lemma} \label{lemma2222} Let $P_1$ and $P_2$ be the marginal distributions of the probability measure $P$. Then,
\begin{itemize}
\item[]
$P_1 =\frac 1{8} P_1 \circ S_{(11)}^{-1} + \frac 1{8} P_1\circ S_{(21)}^{-1} +\frac {3}{8} P_1\circ S_{(31)}^{-1}+\frac {3}{8} P_1\circ S_{(41)}^{-1}$ and
\item[]
$P_2 =\frac 1 {8} P_2 \circ S_{(12)}^{-1} + \frac 1{8} P_2\circ S_{(22)}^{-1} +\frac {3}{8} P_2\circ S_{(32)}^{-1}+\frac 38 P_2\circ S_{(42)}^{-1}$.
\end{itemize}
\end{lemma}
Let us now give the following lemma.
\begin{lemma} \label{lemma333} Let $E(X)$ and $V(X)$ denote the the expectation and the variance of the random variable $X$. Then, \[E(X)=(E(X_1), \, E(X_2))=(\frac 12, \frac 34) \te{ and } V:=V(X)=E\|X-(\frac 12, \frac 34)\|^2=\frac 7{32}.\]
\end{lemma}
\begin{proof} We have
\begin{align*}
&E(X_1)=\int x \, dP_1=\frac 1 8 \int x\, dP_1 \circ S_{(11)}^{-1} + \frac 1 8 \int x\, dP_1\circ S_{(21)}^{-1}+ \frac 38 \int x\, dP_1\circ S_{(31)}^{-1} +\frac 38 \int x\, dP_1\circ S_{(41)}^{-1}\\
&=\frac 1 8\int \frac 13 x\, dP_1 + \frac 18 \int (\frac 1 3 x+\frac 23)\, dP_1 +\frac 38 \int \frac 13 x\, dP_1 + \frac 38\int (\frac 1 3 x+\frac 23)\, dP_1,
\end{align*}
which after simplification yields $E(X_1)=\frac 12$, and similarly $E(X_2)=\frac 34$. Now,
\begin{align*}
&E(X_1^2)=\int x^2 \, dP_1\\
&=\frac 1 8 \int x^2\, dP_1 \circ S_{(11)}^{-1} + \frac 1 8 \int x^2\, dP_1\circ S_{(21)}^{-1} +\frac 3 8 \int x^2\, dP_1\circ S_{(31)}^{-1} +\frac 38 \int x^2\, dP_1\circ S_{(41)}^{-1}\\
&=\frac 1 8 \int (\frac 13 x)^2\, dP_1 + \frac 1 8\int (\frac 1 3 x+\frac 23)^2\, dP_1 +\frac 38 \int (\frac 13 x)^2\, dP_1 + \frac 38\int (\frac 1 3 x+\frac 23)^2\, dP_1\\
&=\frac 1 2 \int \frac 19 x^2\, dP_1 + \frac 12\int (\frac 1 9 x^2+ \frac 4 9 x + \frac 49)\, dP_1\\
&=\frac 1 {18} E(X_1^2)+\frac 1 {18} E(X_1^2)+\frac 4{18} E(X_1) +\frac {4}{18}\\
&=\frac 1 9 E(X_1^2)+\frac 1 3.
\end{align*}
This implies $E(X_1^2)=\frac 38$. Similarly, we can show $E(X_2^2)=\frac {21}{32}$. Thus,
$V(X_1)=E(X_1^2)-(E(X_1))^2=\frac 38 - \frac 14=\frac 18,$
and similarly $V(X_2)=\frac 3{32}$. Hence,
\begin{align*} & E\|X-(\frac 12, \frac 34)\|^2=E(X_1-\frac 12)^2 +E(X_2-\frac 34)^2=V(X_1)+V(X_2)=\frac 7{32}.
\end{align*}
Thus, the proof of the lemma follows.
\end{proof}
Let us now give the following note.
\begin{note} \label{note1} From Lemma~\ref{lemma333} it follows that the optimal set of one-mean is the expected value and the corresponding quantization error is the variance $V$ of the random variable $X$. For words $\beta, \gg, \cdots, \delta$ in $I^\ast$, by $a(\beta, \gg, \cdots, \delta)$ we mean the conditional expectation of the random variable $X$ given $J_\beta\uu J_\gg \uu\cdots \uu J_\delta,$ i.e.,
\begin{equation} \label{eq0}a(\beta, \gg, \cdots, \delta)=E(X|X\in J_\beta \uu J_\gg \uu \cdots \uu J_\delta)=\frac{1}{P(J_\beta\uu \cdots \uu J_\delta)}\int_{J_\beta\uu \cdots \uu J_\delta} x dP.
\end{equation}
For $\omega \in I^k$, $k\geq 1$, since $a(\omega)=E(X : X \in J_\omega)$, using Lemma~\ref{lemma1}, we have
\begin{align*}
&a(\omega)=\frac{1}{P(J_\omega)} \int_{J_\omega} x \,dP(x)=\int_{J_\omega} x\, dP\circ S_\omega^{-1}(x)=\int S_\omega(x)\, dP(x)=E(S_\omega(X))=S_\omega(\frac 12, \frac 34).
\end{align*} For any $(a, b) \in \D R^2$, $E\|X-(a, b)\|^2=V+\|(\frac 12, \frac 34)-(a, b)\|^2.$
In fact, for any $\omega \in I^k$, $k\geq 1$, we have
$\int_{J_\omega}\|x-(a, b)\|^2 dP= p_\omega \int\|(x_1, x_2) -(a, b)\|^2 dP\circ S_\omega^{-1},$
which implies
\begin{equation} \label{eq1}
\int_{J_\omega}\|x-(a, b)\|^2 dP=p_\omega \Big(s_\omega^2V+\|a(\omega)-(a, b)\|^2\Big).
\end{equation}
The expressions \eqref{eq0} and \eqref{eq1} are useful to obtain the optimal sets and the corresponding quantization errors with respect to the probability distribution $P$.
\end{note}
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\caption{Configuration of the points in an optimal set of $n$-means for $1\leq n\leq 16$.} \label{Fig1}
\end{figure}
\section{Optimal sets of $n$-means for all $n\geq 2$}
In this section we determine the optimal sets of $n$-means for all $n\geq 2$. First, prove the following proposition.
\begin{prop}\label{prop1} The set $\alpha=\set{a(1, 3), a(2, 4)}$, where $a(1, 3)=(\frac{1}{6}, \frac{3}{4})$ and $a(2,4)=(\frac{5}{6}, \frac{3}{4})$, is an optimal set of two-means with quantization error $V_2=\frac{31}{288}=0.107639$.
\end{prop}
\begin{proof}
With respect to the vertical line passing through the centroid $(\frac 12, \frac 34)$, the Sierpi\'nski carpet has the maximum symmetry, i.e., with respect to the line $x_1=\frac 12$ the Sierpi\'nski carpet is geometrically symmetric. Also,
observe that, if the two basic rectangles of similar geometrical shape lie in the opposite sides of the line $x_1=\frac 12$, and are equidistant from the line $x_1=\frac 12$, then they have the same probability; hence, they are symmetric with respect to the probability distribution $P$ as well. Due to this, among all the pairs of two points which have the boundaries of the Voronoi regions oblique lines passing through the point $(\frac 12, \frac 34)$, the two points which have the boundary of the Voronoi regions the line $x_1=\frac 12$ will give the smallest distortion error. Again, we know that the two points which give the smallest distortion error are the centroids of their own Voronoi regions. Let $(a_1, b_1)$ and $(a_2, b_2)$ be the centroids of the left half and the right half of the Sierpi\'nski carpet with respect to the line $x_1=\frac 12$ respectively. Then using \eqref{eq0}, we have
\begin{align*}
(a_1, b_1)=E(X : X\in J_1\uu J_3)=\frac{1}{P(J_1)+P(J_3)}\Big(P(J_1)\mathop{\int}\limits_{J_1}x dP+P(J_3)\mathop{\int}\limits_{J_2}x dP\Big)=(\frac{1}{6},\frac{3}{4})
\end{align*}
and
\begin{align*}
(a_2, b_2)=E(X : X\in J_2\uu J_4)=\frac{1}{P(J_2)+P(J_4)}\Big(P(J_2)\mathop{\int}\limits_{J_2}x dP+P(J_4)\mathop{\int}\limits_{J_4}x dP\Big)=(\frac{5}{6},\frac{3}{4}).
\end{align*}
Write $\alpha:=\set{(\frac{1}{6},\frac{3}{4}), (\frac{5}{6},\frac{3}{4})}$. Then, the distortion error is obtained as
\[\int\min_{c\in \alpha}\|x-c\|^2 dP=\mathop{\int}\limits_{J_1\uu J_3} \|x-(\frac{1}{6},\frac{3}{4})\|^2 dP+\mathop{\int}\limits_{J_2\uu J_4} \|x-(\frac{5}{6},\frac{3}{4})\|^2 dP=\frac{31}{288}=0.107639.\]
Since $V_2$ is the quantization error for two-means, we have $0.107639\geq V_2$. Suppose that the points in an optimal set of two-means lie on a vertical line. Then, we can assume that $\beta=\set{(p, a), (p, b)}$ is an optimal set of two-means with $a\leq b$. Then, by the properties of centroids we have
\[(p, a) P(M((p, a)|\beta))+(p, b) P(M((p, b)|\beta))=(\frac 12, \frac 34),\]
which implies $p P(M((p, a)|\beta))+p P(M((p, b)|\beta))=\frac 12$ and $a P(M((p, a)|\beta))+b P(M((p, b)|\beta))=\frac 34$. Thus, we see that $p=\frac 12 $, and the two points $(p, a)$ and $(p, b)$ lie on the opposite sides of the point $(\frac 12, \frac 34)$. Since the optimal points are the centroids of their own Voronoi regions, we have $0\leq a\leq \frac 34\leq b\leq 1$. Then, notice that $J_{3 3} \uu J_{3 4}\uu J_{43}\uu J_{44} \sci M((\frac 12, b)|\beta)$ and $J_1\uu J_2\sci M((\frac 12, a)|\beta)$. Suppose that $a\geq \frac 5{12}$. Then as $a(33,34, 43, 44)=E(X : X\in J_{3 3} \uu J_{3 4}\uu J_{43}\uu J_{44})=(\frac{1}{2},\frac{35}{36})$, we have
\[\int\min_{c\in \alpha}\|x-c\|^2 dP\geq \mathop{\int}\limits_{J_1\uu J_2} \|x-(\frac{1}{2},\frac 5{12})\|^2 dP+\mathop{\int}\limits_{J_{3 3} \uu J_{3 4}\uu J_{43}\uu J_{44}} \|x-(\frac{1}{2},\frac{35}{36})\|^2 dP=\frac{515}{4608}=0.111762,\]
which is a contradiction, as $0.111762>0.107639\geq V_2$ and $\alpha$ is an optimal set of two-means. Thus, we can assume that $a<\frac 5{12}$. Since $a<\frac 5{12}$ and $b\leq 1$, we have $\frac 12(a+b)\leq \frac 12(\frac 5{12}+1)=\frac {17}{24}$, which yields that $B \sci M((\frac 12, b)|\alpha)$ where $B=J_{3 3} \uu J_{3 4}\uu J_{43}\uu J_{44}\uu J_{313}\uu J_{314}\uu J_{323}\uu J_{324}\uu J_{413}\uu J_{414}\uu J_{423}\uu J_{424}$. Using \eqref{eq0}, we have $E(X : X\in B)=(\frac{1}{2},\frac{503}{540})$ which implies that $b\leq \frac{503}{540}$. Now if $a\geq \frac 13$, we have
\[\int\min_{c\in \alpha}\|x-c\|^2 dP\geq \mathop{\int}\limits_{J_1\uu J_2} \|x-(\frac{1}{2},\frac 13)\|^2 dP+\mathop{\int}\limits_{B} \|x-(\frac{1}{2}, \frac{503}{540})\|^2 dP=\frac{106847}{829440}=0.128818>V_2,\]
which is a contradiction. So, we can assume that $a<\frac 13$. Then, $J_1\uu J_1\sci M((\frac 12, a)|\alpha)$ and $J_3\uu J_4\sci M((\frac 12, b)|\alpha)$, and so
$(\frac 12, a)=E(X : X\in J_1\uu J_2)=(\frac{1}{2},\frac{1}{4})$ and $(\frac 12, b)=E(X : X \in J_3\uu J_4)=(\frac{1}{2},\frac{11}{12})$, and
\[\int\min_{c\in \alpha}\|x-c\|^2 dP= \mathop{\int}\limits_{J_1\uu J_2} \|x-(\frac{1}{2},\frac 14)\|^2 dP+\mathop{\int}\limits_{J_3\uu J_4} \|x-(\frac{1}{2},\frac{11}{12})\|^2 dP=\frac{13}{96}=0.135417>V_2,\]
which leads to another contradiction. Therefore, we can assume that the points in an optimal set of two-means can not lie on a vertical line. Hence, $\alpha=\set{(\frac{1}{6},\frac{3}{4}), (\frac{5}{6},\frac{3}{4})}$ forms an optimal set of two-means with quantization error $V_2=\frac{31}{288}=0.107639$.
\end{proof}
\begin{remark}
The set $\alpha$ in Proposition~\ref{prop1} forms a unique optimal set of two-means.
\end{remark}
\begin{prop}\label{prop2}
The set $\alpha=\set{a(1, 2), a(3), a(4)}$, where $a(1, 2)=E(X : X \in J_1\uu J_2)=(\frac{1}{2},\frac{1}{4})$, $a(3)=E(X : X\in J_3)=(\frac{1}{6},\frac{11}{12})$ and $a(4)=E(X : X \in J_4)=(\frac{5}{6},\frac{11}{12})$, forms an optimal set of three-means with quantization error $V_3=\frac{5}{96}=0.0520833$.
\end{prop}
\begin{proof}
Let us first consider the three-point set $\beta$ given by $\beta=\set{a(1, 2), a(3), a(4)} $. Then, the distortion error is obtained as
\begin{align*}
&\int\min_{c\in \alpha}\|x-c\|^2 dP\\
&=\int_{J_1\uu J_2} \|x-a(1, 2)\|^2 dP+ \int_{J_3} \|x-a(3)\|^2 dP+\int_{J_4} \|x-a(4)\|^2 dP=0.0520833.
\end{align*}
Since $V_3$ is the quantization error for an optimal set of three-means, we have $0.0520833\geq V_3$. Let $\alpha:=\set{(a_i, b_i) : 1\leq i\leq 3}$ be an optimal set of three-means. Since the optimal points are the centroids of their own Voronoi regions, we have $\alpha\sci [0, 1]\times [0, 1]$. Then, by the definition of centroid, we have
\[\sum_{(a_i, b_i)\in \alpha} (a_i, b_i) P(M((a_i, b_i)|\alpha)) =(\frac 12, \frac 34),\]
which implies $\sum_{(a_i, b_i)\in \alpha} a_i P(M((a_i, b_i)|\alpha)) =\frac 12$ and $\sum_{(a_i, b_i)\in \alpha} b_i P(M((a_i, b_i)|\alpha)) =\frac 34$. Thus, we conclude that all the optimal points can not lie in one side of the vertical line $x_1=\frac 12$ or in one side of the horizontal line $x_2=\frac 34$. Without any loss of generality, due to symmetry we can assume that one of the optimal points, say $(a_1, b_1)$, lies on the vertical line $x_1=\frac 12$, i.e., $a_1=\frac 12$, and the optimal points $(a_2, b_2)$ and $(a_3, b_3)$ lie on a horizontal line and are equidistant from the vertical line $x_1=\frac 12$. Further, due to symmetry we can assume that $(a_2, b_2)$ and $(a_3, b_3)$ lie on the vertical lines $x_1=\frac 16$ and $x_1=\frac 56$ respectively, i.e., $a_2=\frac 16$ and $a_3=\frac 56$.
Suppose that $(\frac 12, b_1)$ lies on or above the horizontal line $x_2=\frac 34$, and so $(\frac 16, b_2)$ and $(\frac 56 , b_3)$ lie on or below the line $x_2=\frac 34$. Then, if $\frac 23\leq b_2, b_3\leq \frac 34$, we have
\begin{align*}
&\int\min_{c\in \alpha}\|x-c\|^2 dP\geq 2\mathop{ \int}\limits_{J_1\uu J_{31}\uu J_{33}} \min_{\frac{2}{3}\leq b\leq \frac{3}{4}}\|x-(\frac 16, b)\|^2 dP=0.0820313>V_3,
\end{align*}
which is a contradiction. If $\frac 12\leq b_2, b_3\leq \frac 23$,
\begin{align*}
&\int\min_{c\in \alpha}\|x-c\|^2 dP\\
&\geq 2\Big(\mathop{ \int}\limits_{J_1\uu J_{31}\uu J_{321}} \min_{\frac{1}{2}\leq b\leq \frac{2}{3}}\|x-(\frac 16, b)\|^2 dP+\mathop{\int}\limits_{J_{33}}\|x-(\frac 16, \frac 23)\|^2 dP+\mathop{ \int}\limits_{J_{342}\uu J_{344}} \min_{\frac{3}{4}\leq b\leq 1}\|x-(\frac 12, b)\|^2 dP\Big)\\
&=2 \Big(\frac{6521}{442368}+\frac{281}{18432}+\frac{277}{110592}\Big)=0.0649821>V_3,
\end{align*}
which leads to a contradiction.
If $\frac 13\leq b_2, b_3\leq \frac 12$, then
\begin{align*}
&\int\min_{c\in \alpha}\|x-c\|^2 dP\\
&\geq 2\Big(\mathop{ \int}\limits_{J_{31}\uu J_{321}\uu J_{331}} \|x-(\frac 16, \frac 12)\|^2 dP+\mathop{\int}\limits_{J_1}\|x-(\frac 16, \frac 13)\|^2 dP+\mathop{ \int}\limits_{J_{34}\uu J_{334}} \min_{\frac{3}{4}\leq b\leq 1}\|x-(\frac 12, b)\|^2 dP\Big)\\
&=2 \Big(\frac{811}{110592}+\frac{1}{256}+\frac{78373}{4866048}\Big)=0.0546912>V_3,
\end{align*}
which gives a contradiction. If $0\leq b_2, b_3\leq \frac 13$, then
\begin{align*}
&\int\min_{c\in \alpha}\|x-c\|^2 dP\geq 2\Big(\mathop{ \int}\limits_{J_1} \|x-a(1)\|^2 dP+\mathop{ \int}\limits_{J_{33}\uu J_{34}} \min_{\frac{3}{4}\leq b\leq 1}\|x-(\frac 12, b)\|^2 dP\Big)\\
&=2 \Big(\frac{7}{2304}+\frac{109}{3072}\Big)=0.0770399>V_3
\end{align*}
which leads to another contradiction. Therefore, we can assume that $(\frac 12, b_1)$ lies on or below the horizontal line $x_2=\frac 34$, and $(\frac 16, b_2)$ and $(\frac 56 , b_3)$ lie on or above the line $x_2=\frac 34$. Notice that for any position of $(\frac 12, b_1)$ on or below the line $x_2=\frac 34$, always $J_{31}\uu J_{33}\uu J_{34} \sci M((\frac 16, b_2)|\alpha)$ which implies that $b_2\leq \frac{79}{84}$. Similarly, $b_3\leq \frac{79}{84}$. Suppose that $\frac 12\leq b_1\leq \frac 34$. Then, writing $A=J_{133}\uu J_{321}\uu J_{324}$ and $B=J_{11}\uu J_{12}\uu J_{14}\uu J_{132}$, we have
\begin{align*}
&\int\min_{c\in \alpha}\|x-c\|^2 dP\\
&\geq 2\Big(\mathop{ \int}\limits_{J_{31}\uu J_{33}\uu J_{34} \uu J_{323}}\min_{\frac{3}{4}\leq b\leq \frac{79}{84}} \|x-(\frac 16, b)\|^2 dP+\mathop{ \int}\limits_{A} \|x-(\frac 16, \frac 34)\|^2 dP+\mathop{ \int}\limits_{B} \|x-(\frac 12, \frac 12)\|^2 dP\Big)\\
&=2 \Big(\frac{588517}{78299136}+\frac{5347}{1327104}+\frac{6601}{442368}\Big)=0.0529346>V_3,
\end{align*}
which is a contradiction. So, we can assume that $b_1< \frac 12$. Suppose that $\frac 13\leq b_1< \frac 12$. Then, as $\frac 34\leq b_2\leq \frac{79}{84}$, we see that $J_{31}\uu J_{33}\uu J_{34} \uu J_{321}\uu J_{323}\uu J_{324}\sci M((\frac 16, b_2)|\alpha)$.
Then, writing $A_1:=J_{31}\uu J_{33}\uu J_{34} \uu J_{321}\uu J_{323}\uu J_{324}$ and $A_2:=J_{322}\uu J_{1331}\uu J_{1333}\uu J_{1334}\uu J_{13323}\uu J_{13324}$ and $A_3:=J_{11}\uu J_{12}\uu J_{14}\uu J_{131}\uu J_{132}\uu J_{134}\uu J_{13322}$, we have
\begin{align*}
&\int\min_{c\in \alpha}\|x-c\|^2 dP\\
&\geq 2\Big(\mathop{ \int}\limits_{A_1}\min_{\frac{3}{4}\leq b\leq \frac{79}{84}} \|x-(\frac 16, b)\|^2 dP+\mathop{ \int}\limits_{A_2} \|x-(\frac 16, \frac 34)\|^2 dP+\mathop{ \int}\limits_{A_3} \|x-(\frac 12, \frac 13)\|^2 dP\Big)\\
&=2 \Big(\frac{242191}{27869184}+\frac{4135547}{1146617856}+\frac{31584803}{2293235712}\Big)=0.0521401>V_3,
\end{align*}
which gives a contradiction. So, we can assume that $b_1\leq \frac 13$. Then, notice that $J_{11}\uu J_{12}\uu J_{132}\uu J_{141}\uu J_{142}\uu J_{144}\uu J_{21}\uu J_{22}\uu J_{241}\uu J_{231}\uu J_{232}\uu J_{233}\sci M((\frac 12, b_1)|\alpha)$ which implies that $b_1\geq \frac{13}{68}$. Thus, we have $\frac{13}{68}\leq b_1\leq \frac 13$. Suppose that $\frac 34\leq b_2, b_3\leq \frac 56$. Then,
\begin{align*}
&\int\min_{c\in \alpha}\|x-c\|^2 dP\\
&\geq 2\Big(\mathop{ \int}\limits_{J_{3}}\min_{\frac{3}{4}\leq b\leq \frac 56} \|x-(\frac 16, b)\|^2 dP+\mathop{ \int}\limits_{J_{11}\uu J_{12}\uu J_{14}\uu J_{131}\uu J_{132}} \min_{\frac{13}{68}\leq b\leq \frac{1}{3}} \|x-(\frac 12, b)\|^2 dP\\
&+\mathop{ \int}\limits_{J_{1331}\uu J_{1333}\uu J_{1334}} \|x-(\frac 16, \frac 34)\|^2 dP+\mathop{ \int}\limits_{J_{134}} \|x-(\frac 12, \frac 13)\|^2 dP\Big)\\
&=2 \Big(\frac{3}{256}+ \frac{147359}{15261696}+\frac{32969}{10616832}+ \frac{3881}{1327104}\Big)=0.054808>V_3,
\end{align*}
which leads to a contradiction. So, we can assume that $\frac 56<b_2, b_3\leq 1$. Then, we have $J_1\uu J_2\sci M((\frac 12, b_1)|\alpha)$, $J_3\sci M((\frac 16, b_2)|\alpha)$ and $J_4\sci M((\frac 56, b_3)|\alpha)$ which yield that $(\frac 12, b_1)=a(1, 2)$, $(\frac 16, b_2)=a(3)$ and $(\frac 56, b_3)=a(4)$, and the quantization error is $V_3=\frac{5}{96}=0.0520833$. Thus, the proof of the proposition is complete.
\end{proof}
\begin{prop} \label{prop3} The set $\alpha=\set{a(1), a(2), a(3), a(4)}$ forms an optimal set of four-means with quantization error $V_4=\frac{7}{288}=0.0243056$.
\end{prop}
\begin{proof}
Let us consider the four-point set $\beta$ given by $\beta:=\set{a(1), a(2), a(3), a(4)}$. Then, the distortion error is given by
\[\int\min_{c\in \beta}\|x-c\|^2 dP=\sum_{i=1}^4 \int_{J_i}\|x-a(i)\|^2 dP=\frac{7}{288}=0.0243056.\]
Since, $V_4$ is the quantization error for four-means, we have $0.0243056\geq V_4$. As the optimal points are the centroids of their own Voronoi regions, $\alpha \sci J$. Let $\alpha$ be an optimal set of $n$-means for $n=4$.
By the definition of centroid, we know
\begin{equation} \label{eq100}
\sum_{(a, b) \in \alpha} (a, b) P(M((a, b)|\alpha))=(\frac 12, \frac 34).
\end{equation}
If all the points of $\alpha$ are below the line $x_2=\frac 34$, i.e., if $b<\frac 34$ for all $(a, b)\in \alpha$, then by \eqref{eq100}, we see that $\frac 34=\sum_{(a, b) \in \alpha} b P(M((a, b)|\alpha))<\sum_{(a, b) \in \alpha} \frac 34 P(M((a, b)|\alpha))=\frac 34$, which is a contradiction. Similarly, it follows that if all the points of $\alpha$ are above the line $x_2=\frac 34$, or left of the line $x_1=\frac 12$, or right of the line $x_1=\frac 12$, a contradiction will arise.
Suppose that all the points of $\alpha$ are on the line $x_2=\frac 34 $. Then, for $(x_1, x_2) \in\uu_{i, j=3}^4 J_{ij}$, we have $\min_{c\in \alpha}\|(x_1,x_2)-c\|\geq \frac{5}{36}$, and for $(x_1, x_2) \in\uu_{i, j=1}^2 J_{ij}$, we have $\min_{c\in \alpha}\|(x_1,x_2)-c\|\geq \frac{23}{36}$, which implies that
\begin{align*}
&\int \min_{c\in \alpha}\|x-c\|^2 dP\geq 4 \mathop{\int}\limits_{J_{33}}\min_{c\in \alpha}\|(x_1,x_2)-c\|^2dP+4 \mathop{\int}\limits_{J_{11}}\min_{c\in \alpha}\|(x_1,x_2)-c\|^2dP\\
&=4 \Big(\frac 5{36}\Big)^2 P(J_{33})+4 \Big(\frac {23}{36}\Big)^2 P(J_{11})=\frac{377}{10368}=0.0363619>V_4,
\end{align*}
which is a contradiction. Thus, we see that all the points of $\alpha$ can not lie on $x_2=\frac34$. Similarly, all the points of $\alpha$ can not lie on $x_1=\frac 12$. Recall that the Sierpi\'nski carpet has maximum symmetry with respect to the line $x_1=\frac 12$. As all the points of $\alpha$ can not lie on the line $x_1=\frac 12$, due to symmetry we can assume that the points of $\alpha$ lie either on the three lines $x_1=\frac 16$, $x_1=\frac 56$ and $x_1=\frac 12$, or on the two lines $x_1=\frac 16$ and $x_1=\frac 56$.
Suppose $\alpha$ contains points from the line $x_1=\frac 12$.
As $\alpha$ can not contain all the points from $x_1=\frac 12$, we can assume that $\alpha$ contains two points, say $(\frac 12, b_1)$ and $(\frac 12, b_2)$ with $b_1<b_2$, from the line $x_1=\frac 12$ which are in the opposite sides of the centroid $(\frac 12, \frac 34)$, and the other two points, say $(\frac 16, a_1)$ and $(\frac 56, a_2)$, from the lines $x_1=\frac 16$ and $x_1=\frac 56$. Then, if $\alpha$ does not contain any point from $J_3\uu J_4$, we have
\begin{align*}
&\int \min_{c\in \alpha}\|x-c\|^2 dP\geq 2 \mathop{\int}\limits_{J_{31}\uu J_{33}}\|x-(\frac 16, \frac 23)\|^2dP=\frac{25}{768}=0.0325521>V_4,
\end{align*}
which leads to a contradiction. So, we can assume that $(\frac 16, a_1)\in J_3$ and $(\frac 56, a_2)\in J_4$. Suppose $\frac 23 \leq a_1, a_2\leq \frac 56$. Then, notice that $J_{31}\uu J_{33}\uu J_{321}\uu J_{323} \sci M((\frac 16, a_1)|\alpha)$ and similar is the expression for the point $(\frac 56, a_2)$. Further, notice that $J_{11}\uu J_{12}\uu J_{14}\uu J_{21}\uu J_{22}\uu J_{23} \sci M((\frac 12, \frac 13)|\alpha)$. Therefore, under the assumption $\frac 23 \leq a_1, a_2\leq \frac 56$, writing $A_1:=J_{31}\uu J_{33}\uu J_{321}\uu J_{323}$ and $A_2:=J_{11}\uu J_{12}\uu J_{14}$, we have the distortion error as
\begin{align*}
&\int \min_{c\in \alpha}\|x-c\|^2 dP\geq 2 \Big(\mathop{\int}\limits_{A_1}\min_{\frac 23\leq b\leq \frac 56}\|x-(\frac 16, b)\|^2dP+\mathop{\int}\limits_{A_2}\min_{0\leq b\leq \frac 34}\|x-(\frac 12, b)\|^2dP\Big)\\
&=2 \Big(\frac{2051}{331776}+\frac{2021}{276480}\Big)=0.0269833>V_4,
\end{align*}
which leads to a contradiction. So, we can assume that $\frac 56<a_1, a_2\leq 1$. Then, we see that $J_1\uu J_2\sci M((\frac 12, b_1)|\alpha)$ for $b_1=\frac 12$, and so the distortion error is
\[\int \min_{c\in \alpha}\|x-c\|^2 dP\geq 2 \int_{J_1} \min_{0\leq b\leq \frac 34}\|x-(\frac 12, b)\|^2dP=\frac{13}{384}=0.0338542>V_4\]
which is a contradiction. All these contradictions arise due to our assumption that $\alpha$ contains points from the line $x_1=\frac 12$. So, we can assume that $\alpha$ can not contain any point from the line $x_1=\frac 12$, i.e., we can assume that $\alpha$ contains two points from the line $x_1=\frac 16$ and two points from the line $x_1=\frac 56$. Thus, we can take $\alpha:=\set{(\frac 16, a_1), (\frac 16, b_1), (\frac 56, a_2), (\frac 56, b_2)}$ where $a_1\leq \frac 34\leq b_1$ and $a_2\leq \frac 34\leq b_2$. Notice that the Voronoi region of $(\frac 16, a_1)$ contains $J_1$ and the Voronoi region of $(\frac 56, a_2)$ contains $J_2$. If the Voronoi region of $(\frac 16, a_1)$ contains points from $J_3$, we must have $\frac 12(a_1+b_1)\geq \frac 23$ which yields $a_1\geq \frac 43-b_1\geq \frac 43 -\frac 34=\frac 7{12}$, and similarly if the Voronoi region of $(\frac 56, a_2)$ contains points from $J_4$, we must have $a_2\geq \frac 7{12}$. But, then
\begin{align*}
&\int \min_{c\in \alpha}\|x-c\|^2 dP\geq 2\mathop{\int}\limits_{J_{1}}\|x-(\frac 16, \frac 7{12})\|^2dP+ 2\mathop{\int}\limits_{J_{33}\uu J_{34}}\|x-a(33, 34)\|^2dP\\
&=\frac{65}{1536}=0.0423177>V_4,
\end{align*}
which is a contradiction. So, we can assume that the Voronoi regions of $(\frac 16, a_1)$ and $(\frac 56, a_2)$ do not contain any point from $J_3\uu J_4$. Thus, we have
$(\frac 16, a_1)=a(1)=(\frac{1}{6},\frac{1}{4})$, $(\frac 56, a_2)=a(2)=(\frac 56, \frac 14)$, $(\frac 16, b_1)=a(3)=(\frac 1 6, \frac {11}{12})$, and $(\frac 56, b_2)=a(4)=(\frac 56, \frac {11}{12})$, and the quantization error is $V_4=\frac{7}{288}=0.0243056$. Thus, the proof of the proposition is complete.
\end{proof}
\begin{note}\label{note1}
Let $\alpha$ be an optimal set of $n$-means for some $n\geq 2$. Then, for $a\in \alpha$, we have $a=a(\omega)$, $a=a(\go1, \go3)$, or $a=a(\go2, \go4)$ for some $\omega \in I^\ast$. Moreover, if $a\in \alpha$, then $P$-almost surely $M(a|\alpha)=J_\omega$ if $a=a(\omega)$, $M(a|\alpha)=J_{\go1}\uu J_{\go3}$ if $a=a(\go1, \go3)$, and $M(a|\alpha)=J_{\go2}\uu J_{\go4}$ if $a=a(\go2, \go4)$. For $\omega \in I^\ast$, $(i=1$ and $j=3)$, $(i=2$ and $j=4)$, or $(i=1, j=2)$ write
\begin{align}\label{eq2}
E(\omega):=\mathop{\int}\limits_{J_\omega}& \|x-a(\omega)\|^2 dP, \te{ and } E(\omega i, \omega j):=\mathop{\int}\limits_{J_{\omega i}\uu J_{\omega j}}\|x-a(\omega i, \omega j)\|^2 dP.
\end{align}
\end{note}
Let us now give the following lemma.
\begin{lemma} \label{lemma10}
For any $\omega \in I^\ast$, let $E(\omega)$, $E(\go1, \go3)$, $E(\go2, \go4)$, and $E(\go1, \go2)$ be defined by \eqref{eq2}. Then, $E(\omega 1, \go3)=E(\go2, \go4)=\frac{31}{126} E(\omega)$, $E(\go1, \omega 2)=\frac{13}{84}E(\omega)$, $E(\go1)=E(\go2)=\frac 1{72} E(\omega)$, and $E(\go3)=E(\go4)=\frac 1{24} E(\omega)$.
\end{lemma}
\begin{proof} By \eqref{eq1}, we have
\begin{align*}
&E(\go1, \go3)=\mathop{\int}\limits_{J_{\go1}\uu J_{\go3}}\|x-a(\go1, \go3)\|^2 dP=\mathop{\int}\limits_{J_{\go1}}\|x-a(\go1, \go3)\|^2 dP+\mathop{\int}\limits_{J_{\go3}}\|x-a(\go1, \go3)\|^2 dP\\
&=p_{\go1} (s_{\go1}^2V+\|a(\go1)-a(\go1, \go3)\|^2)+p_{\go3} (s_{\go3}^2V+\|a(\go3)-a(\go1, \go3)\|^2).
\end{align*}
Notice that
\[a(\go1, \go3)=\frac{1}{p_{\go1}+p_{\go3}}\Big (p_{\go1}S_{\go1}(\frac12, \frac 34)+p_{\go3}S_{\go3}(\frac12, \frac 34)\Big)=\frac{1}{\frac 18+\frac 38}\Big(\frac 18 S_{\go1}(\frac12, \frac 34)+\frac 38S_{\go3}(\frac12, \frac 34)\Big),\]
which implies $a(\go1, \go3)=\frac 14 S_{\go1}(\frac12, \frac 34)+\frac 34 S_{\go3}(\frac12, \frac 34)$. Thus, we have
\begin{align*}
&\|a(\go1)-a(\go1, \go3)\|^2=\|S_{\go1}(\frac 12, \frac 34)-\frac 14 S_{\go1}(\frac12, \frac 34)-\frac 34 S_{\go3}(\frac12, \frac 34)\|^2=\frac 9{16} s_\omega^2 \|(0, \frac 23)\|^2=\frac 14 s_\omega^2,
\end{align*}
and similarly, $\|a(\go3)-a(\go1, \go3)\|^2=\frac 1{16} s_\omega^2\|(0, \frac 23)\|^2 =\frac 1{36} s_\omega^2$. Thus, we obtain,
\begin{align*}
&E(\go1, \go3)=p_{\go1} (s_{\go1}^2V+\frac 1{4} s_\omega^2)+p_{\go3} (s_{\go3}^2V+\frac 1{36} s_\omega^2)=p_\omega s_\omega^2 V(p_1s_1^2+p_3s_3^2)+p_\omega s_\omega^2(\frac 14 p_1+\frac 1{36}p_3)\\
&= p_\omega s_\omega^2 V(\frac 1{18}+\frac1{24}\frac 1 V)=\frac {31}{126} E(\omega),
\end{align*}
and similarly, we can prove the rest of the lemma. Thus, the proof of the lemma is complete.
\end{proof}
\begin{remark}
From the above lemma it follows that $E(\go1, \go3)=E(\go2, \go4)>E(\go1, \go2)>E(\go3)=E(\go4)>E(\go1)=E(\go2)$.
\end{remark}
The following lemma gives some important properties about the distortion error.
\begin{lemma} \label{lemma11}
Let $\omega, \tau \in I^\ast$. Then
$(i)$ $E(\omega)> E(\tau)$ if and only if $E(\go1, \go3)+E(\go2, \go4)+E(\tau)< E(\omega)+E(\gt1, \gt3)+E(\gt2, \gt4)$;
$(ii)$ $E(\omega)> E(\tau 1, \tau 3)(=E(\gt2, \gt4))$ if and only if $E(\go1, \go3)+E(\go2, \omega 4)+E(\tau 1, \tau 3)+E(\tau 2, \tau 4)< E(\omega)+E(\tau 1, \gt2)+E(\tau 3)+E(\gt4)$;
$(iii)$ $E(\go1, \go3)(=E(\omega 2, \omega 4))> E(\tau 1, \tau 3)(=E(\gt2, \gt4))$ if and only if $E(\go1, \go2)+E(\go3)+E(\omega 4)+E(\tau 1, \tau 3)+E(\tau 2, \gt4)< E(\go1, \go3)+E(\go2, \go4)+E(\tau 1, \gt2)+E(\tau 3)+E(\gt4)$;
$(iv)$ $E(\go1, \go3)(=E(\omega 2, \omega 4))> E(\tau)$ if and only if $E(\go1, \go2)+E(\go3)+E(\omega 4)+E(\tau)< E(\go1, \go3)+E(\go2, \go4)+E(\tau 1, \gt3)+E(\tau 2, \tau 4)$;
$(v)$ $E(\go1, \go2)> E(\tau)$ if and only if $E(\go1)+E(\go2)+E(\tau)< E(\go1, \go2)+E(\gt1, \gt3)+E(\gt2, \gt4)$;
$(vi)$ $E(\go1, \go2)> E(\tau 1, \tau 3)(=E(\gt2, \gt4))$ if and only if $E(\go1)+E(\go2)+E(\tau 1, \tau 3)+E(\tau 2, \tau 4)< E(\go1, \go2)+E(\tau 1, \gt2)+E(\tau 3)+E(\gt4)$;
$(vii)$ $E(\go1, \go2)> E(\tau 1, \tau 2)$ if and only if $E(\go1)+E(\go2)+E(\tau 1, \tau 2)< E(\go1, \go2)+E(\tau 1)+E(\gt2)$;
$(viii)$ $E(\omega)> E(\gt1,\gt2)$ if and only if $E(\go1, \go3)+E(\go2, \go4)+E(\gt1, \tau 2)< E(\omega)+E(\gt1)+E(\gt2)$.
\end{lemma}
\begin{proof} Let us first prove $(iii)$. Using Lemma~\ref{lemma10}, we see that
\begin{align*}
LHS&=E(\go1, \go2)+E(\go3)+E(\omega 4)+E(\tau 1, \tau 3)+E(\tau 2, \gt4)=\frac {5}{21} E(\omega)+\frac{31}{63}E(\tau),\\
RHS&= E(\go1, \go3)+E(\go2, \go4)+E(\tau 1, \gt2)+E(\tau 3)+E(\gt4)=\frac{31}{63} E(\omega)+\frac {5}{21} E(\tau).
\end{align*}
Thus, $LHS< RHS$ if and only if $\frac {5}{21} E(\omega)+\frac{31}{63}E(\tau)< \frac{31}{63} E(\omega)+\frac {5}{21} E(\tau)$, which yields $E(\omega)>E(\tau)$, i.e., $E(\go1, \go3)>E(\gt1, \gt3)$. Thus $(iii)$ is proved. The other parts of the lemma can similarly be proved. Thus, the lemma follows.
\end{proof}
\begin{figure}\label{Fig3}
\centerline{\includegraphics[width=7.5 in, height=8 in]{Figure.pdf}}
\caption{Tree diagram of the optimal sets from $\alpha_8$ to $\alpha_{21}$.} \label{Fig2}
\end{figure}
In the following theorem, we give the induction formula to determine the optimal sets of $n$-means for any $n\geq 2$.
\begin{theorem} \label{Th1} For any $n\geq 2$, let $\alpha_n:=\set{a(i) : 1\leq i\leq n}$ be an optimal set of $n$-means, i.e., $\alpha_n \in\C C_n:= \mathcal{C}_n(P)$. For $\omega \in I^\ast$, let $E(\omega)$, $E(\go1, \go3)$ and $E(\omega 2, \go4)$ be defined by \eqref{eq2}. Set
\[\tilde E(a(i)):=\left\{\begin{array} {ll}
E(\omega) \te{ if } a(i)=a(\omega) \te{ for some } \omega \in I^\ast, \\
E(\omega k, \omega \ell) \te{ if } a(i)=a(\omega k, \omega \ell) \te { for some } \omega \in I^\ast,
\end{array} \right.
\]
where $(k=1, \ell=3)$, or $(k=2, \ell=4)$, or $(k=1, \ell=2)$,
and $W(\alpha_n):=\set{a(j) : a(j) \in \alpha_n \te{ and } \tilde E(a(j))\geq \tilde E(a(i)) \te{ for all } 1\leq i\leq n}$. Take any $a(j) \in W(\alpha_n)$, and write
\[\alpha_{n+1}(a(j)):=\left\{\begin{array}{ll}
(\alpha_n\setminus \set{a(j)})\uu \set{a(\go1, \go3), a(\go2, \go4)} \te{ if } a(j)=a(\omega), &\\
(\alpha_n \setminus \set{a(\go1, \go3), a(\go2, \go4)})\uu \set{a(\omega 1, \go2), a(\omega 3), a(\omega 4)} & \\
\qquad \qquad \te{ if } a(j)=a(\omega 1, \omega 3) \te{ or } a(\go2, \omega 4), &\\
(\alpha_n\setminus \set{a(j)})\uu \set{a(\go1), a(\go2)} \te{ if } a(j)=a(\go1, \go2),
\end{array}\right.
\]
Then $\alpha_{n+1}(a(j))$ is an optimal set of $(n+1)$-means, and the number
of such sets is given by
\[\te{card}\Big(\UU_{\alpha_n \in \C{C}_n}\{\alpha_{n+1}(a(j)) : a(j) \in W(\alpha_n)\}\Big).\]
\end{theorem}
\begin{proof}
By Proposition~\ref{prop1}, Proposition~\ref{prop2} and Proposition~\ref{prop3}, we know that the optimal sets of two-, three-, and four-means are respectively $\{a(1,3), a(2, 4)\}$,
$\{a(1, 2), a(3), a(4)\}$, and $\set{a(1), a(2), a(3), a(4)}$. Notice that by Lemma~\ref{lemma10}, we know $E(1, 3)\geq E(2, 4)$, and $E(1, 2)\geq E(3)=E(4)$. Thus, the lemma is true for $n=2$ and $n=3$.
For any $n\geq 3$, let us now assume that $\alpha_n$ is an optimal
set of $n$-means. Let $\alpha_n:=\set{a(i) : 1\leq i\leq n}$. Let $\tilde E(a(i))$ and $W(\alpha_n)$ be defined as in the hypothesis. If $a(j) \not \in W(\alpha_n)$, i.e., if $a(j) \in \alpha_n\setminus W(\alpha_n)$, then by Lemma~\ref{lemma11}, the error
\[\sum_{a(i)\in (\alpha_n\setminus \set{a(j)})} \tilde E(a(i))+E(\go1, \go3)+E(\go2, \go4) \te{ if } a(j)=a(\omega),\]
\[\sum_{a(i)\in (\alpha_n\setminus \set{a(\go1, \go3), \, a(\go2, \go4)})} \tilde E(a(i))+E(\go1, \go2)+E(\go3)+E(\go4) \te{ if } a(j)=a(\go1, \go3) \te{ or } a(\omega 2, \omega 4),\]
\[\sum_{a(i)\in (\alpha_n\setminus \set{a(j)})} \tilde E(a(i))+E(\go1)+E(\go2) \te{ if } a(j)=a(\go1, \go2),\]
obtained in this case is strictly greater than the corresponding error obtained in the case when $a(j)\in W(\alpha_n)$. Hence for any $a(j) \in W(\alpha_n)$, the set $\alpha_{n+1}(a(j))$, where
\[\alpha_{n+1}(a(j)):=\left\{\begin{array}{ll}
(\alpha_n\setminus \set{a(j)})\uu \set{a(\go1, \go3), a(\go2, \go4)} \te{ if } a(j)=a(\omega), &\\
(\alpha_n \setminus \set{a(\go1, \go3), a(\go2, \go4)})\uu \set{a(\omega 1, \go2), a(\omega 3), a(\omega 4)} & \\
\qquad \qquad \te{ if } a(j)=a(\omega 1, \omega 3) \te{ or } a(\go2, \omega 4), &\\
(\alpha_n\setminus \set{a(j)})\uu \set{a(\go1), a(\go2)} \te{ if } a(j)=a(\go1, \go2),
\end{array}\right.
\] is an optimal set of $(n+1)$-means, and the number
of such sets is
\[\te{card}\Big(\UU_{\alpha_n \in \C{C}_n}\{\alpha_{n+1}(a(j)) : a(j) \in W(\alpha_n)\}\Big).\]
Thus the proof of the theorem is complete (also see Note~\ref{note10}).
\end{proof}
\begin{remark}
Once an optimal set of $n$-means is known, by using \eqref{eq1}, the corresponding quantization error can easily be calculated.
\end{remark}
\begin{remark}
By Theorem~\ref{Th1}, we note that to obtain an optimal set of $(n+1)$-means one needs to know an optimal set of $n$-means. We conjecture that unlike the homogeneous probability distribution, i.e., when the probability measures on the basic rectangles at each level of the Sierpi\'nski carpet construction are equal, for the nonhomogeneous probability distribution considered in this paper, to obtain the optimal sets of $n$-means a closed formula can not be obtained.
\end{remark}
Running the induction formula given by Theorem~\ref{Th1} in computer algorithm, we obtain some results and observations about the optimal sets of $n$-means, which are given in the following section.
\section{Some results and observations}
First, we explain about some notations that we are going to use in this section. Recall that the optimal set of one-mean consists of the expected value of the random variable $X$, and the corresponding quantization error is its variance. Let $\alpha_n$ be an optimal set of $n$-means, i.e., $\alpha_n \in \C C_n$, and then for any $a\in \alpha_n$, we have $a=a(\omega)$, or $a=a(\omega i, \omega j)$ for some $\omega \in I^\ast$, where $(i=1, j=3)$, $(i=2, j=4)$, or $(i=1, j=2)$. For $\omega=\omega_1\omega_2\cdots\omega_k\in I^k$, $k\geq 1$, in the sequel, we will identify the elements $a(\omega)$ and $a(\omega i, \omega j)$ by the set $\set{\set{\omega_1, \omega_2, \cdots, \omega_k}}$ and $\set{\set{\omega_1, \omega_2, \cdots, \omega_k, i}, \set{\omega_1, \omega_2, \cdots, \omega_k, j}}$ respectively. Thus, we can write
\begin{align*} &\alpha_2= \set{\set{\set{1}, \set{3}}, \set{\set{2} ,\set{4}}}, \, \alpha_3=\set{\set{\set{1}, \set {2}}, \set{\set{3}}, \set{\set{4}}}, \\
&\alpha_4=\set{\set{\set{1}}, \set{\set{2}}, \set{\set{3}}, \set{\set{4}}},
\end{align*}
and so on. For any $n\geq 2$, if $\te{card}(\C C_n)=k$, we write
\[\C C_n=\left\{\begin{array}{ccc}
\set{\alpha_{n, 1}, \alpha_{n, 2}, \cdots, \alpha_{n, k}} & \te{ if } k\geq 2,\\
\set{\alpha_{n}} & \te{ if } k=1.
\end{array}\right.
\]
If $\te{card}(\C C_n)=k$ and $\te{card}(\C C_{n+1})=m$, then either $1\leq k\leq m$, or $1\leq m\leq k$ (see Table~\ref{tab1}). Moreover, by Theorem~\ref{Th1}, an optimal set at stage $n$ can contribute multiple distinct optimal sets at stage $n+1$, and multiple distinct optimal sets at stage $n$ can contribute one common optimal set at stage $n+1$; for example from Table~\ref{tab1}, one can see that the number of $\alpha_{21}=8$, the number of $\alpha_{22}=28$, the number of $\alpha_{23}=56$, the number of $\alpha_{24}=70$, and the number of $\alpha_{25}=56$.
By $\alpha_{n, i} \rightarrow \alpha_{n+1, j}$, it is meant that the optimal set $\alpha_{n+1, j}$ at stage $n+1$ is obtained from the optimal set $\alpha_{n, i}$ at stage $n$, similar is the meaning for the notations $\alpha_n\rightarrow \alpha_{n+1, j}$, or $\alpha_{n, i} \rightarrow \alpha_{n+1}$, for example from Figure~\ref{Fig3}:
\begin{align*} &\left\{\alpha _{16}\to \alpha _{17,1},\alpha _{16}\to \alpha _{17,2},\alpha _{16}\to \alpha _{17,3},\alpha _{16}\to \alpha _{17,4}\right\};\\
&\{\left\{\alpha _{17,1}\to \alpha _{18,1},\alpha _{17,1}\to \alpha _{18,2},\alpha _{17,1}\to \alpha _{18,4}\right\}, \left\{\alpha _{17,2}\to \alpha _{18,1},\alpha _{17,2}\to \alpha _{18,3},\alpha _{17,2}\to \alpha _{18,5}\right\}, \\
& \left\{\alpha _{17,3}\to \alpha _{18,2},\alpha _{17,3}\to \alpha _{18,3},\alpha _{17,3}\to \alpha _{18,6}\right\},\left\{\alpha _{17,4}\to \alpha _{18,4},\alpha _{17,4}\to \alpha _{18,5},\alpha _{17,4}\to \alpha _{18,6}\right\}\};\\
& \{\left\{\alpha _{18,1}\to \alpha _{19,1},\alpha _{18,1}\to \alpha _{19,2}\right\},\left\{\alpha _{18,2}\to \alpha _{19,1},\alpha _{18,2}\to \alpha _{19,3}\right\}, \left\{\alpha _{18,3}\to \alpha _{19,1},\alpha _{18,3}\to \alpha _{19,4}\right\}, \\
& \left\{\alpha _{18,4}\to \alpha _{19,2},\alpha _{18,4}\to \alpha _{19,3}\right\}, \left.\left\{\alpha _{18,5}\to \alpha _{19,2},\alpha _{18,5}\to \alpha _{19,4}\right\},\left\{\alpha _{18,6}\to \alpha _{19,3},\alpha _{18,6}\to \alpha _{19,4}\right\}\right\};\\
& \left\{\alpha _{19,1}\to \alpha _{20},\alpha _{19,2}\to \alpha _{20},\alpha _{19,3}\to \alpha _{20},\alpha _{19,4}\to \alpha _{20}\right\}.
\end{align*}
Moreover, one can see that
\begin{align*}
\alpha_8&=\Big\{\{\{1,1\},\{1,3\}\},\{\{1,2\},\{1,4\}\},\{\{2,1\},\{2,3\}\},\{\{2,2\},\{2,4\}\},\{\{3,1\},\{3,3\}\},\\
& \{\{3,2\},\{3,4\}\},\{\{4,1\},\{4,3\}\},\{\{4,2\},\{4,4\}\Big\} \te{ with } V _8= \frac{31}{2592}=0.0119599;\\
\alpha_{9,1}&= \Big \{\{\{3,3\}\},\{\{3,4\}\},\{\{1,1\},\{1,3\}\},\{\{1,2\},\{1,4\}\},\{\{2,1\},\{2,3\}\},\{\{2,2\},\{2,4\}\},\\
& \{\{3,1\},\{3,2\}\},\{\{4,1\},\{4,3\}\},\{\{4,2\},\{4,4\}\}\Big \},\\
\alpha_{9,2}&= \Big \{\{\{4,3\}\},\{\{4,4\}\},\{\{1,1\},\{1,3\}\},\{\{1,2\},\{1,4\}\},\{\{2,1\},\{2,3\}\},\{\{2,2\},\{2,4\}\},\\
& \{\{3,1\},\{3,3\}\}, \{\{3,2\},\{3,4\}\},\{\{4,1\},\{4,2\}\}\Big \} \te{ with } V_9= \frac{25}{2592}=0.00964506;\\
\alpha_{10} &=\Big\{\{\{3,3\}\},\{\{3,4\}\},\{\{4,3\}\},\{\{4,4\}\},\{\{1,1\},\{1,3\}\},\{\{1,2\},\{1,4\}\},\{\{2,1\},\{2,3\}\},\\
& \{\{2,2\},\{2,4\}\},\{\{3,1\},\{3,2\}\},\{\{4,1\},\{4,2\}\}\Big \} \te{ with } V_{10}=\frac{19}{2592}=0.00733025,\end{align*}
and so on.
\begin{table}
\begin{center}
\begin{tabular}{ |c|c||c|c|| c|c||c|c|c||c|c||c|c}
\hline
$n$ & $\te{card}(\C C_n) $ & $n$ & $\te{card}(\C C_n) $ & $n$ & $\te{card}(\C C_n) $ & $n$ & $\te{card}(\C C_n)$ & $n$ & $\te{card}(\C C_n)$ & $n$ & $\te{card}(\C C_n)$\\
\hline
5 & 2 & 18 & 6 & 31& 4 & 44 & 70& 57 & 8& 70 & 6 \\6 & 1 & 19 & 4 & 32 & 1 & 45& 56 & 58 & 28& 71 & 4\\7 & 2 & 20 & 1 & 33& 4& 46 & 28 & 59 & 56& 72 & 1\\8 & 1 & 21 & 8 & 34 & 6& 47& 8 & 60&70 & 73 & 24\\9 & 2 & 22& 28 & 35 & 4 & 48 & 1 & 61 & 56 & 74 & 276 \\10 & 1 & 23 & 56 & 36 & 1 & 49 & 8& 62 & 28& 75 & 2024 \\11 & 2 & 24 & 70 & 37 & 4& 50 & 28& 63 & 8 & 76 & 10626 \\12 & 1 & 25 & 56 & 38 & 6 & 51 & 56 & 64& 1& 77 & 42504\\13 & 2 & 26 & 28 & 39& 4 & 52 &70 & 65 & 4& 78 & 134596 \\14 & 1 & 27& 8 & 40 & 1& 53 & 56& 66 & 6 & 79 & 346104\\15 & 2 & 28& 1 & 41 &8 & 54 & 28& 67 & 4 & 80 & 735471 \\16 & 1 &29 & 4 & 42 & 28 & 55 &8& 68 &1& 81 & 1307504\\17 & 4 &30 & 6 & 43& 56 & 56 & 1 & 69 & 4 & 82 & 1961256\\
\hline
\end{tabular}
\end{center}
\
\caption{Number of $\alpha_n$ in the range $5\leq n\leq 82$.}
\label{tab1}
\end{table}
\begin{note} \label{note10} Notice that there is only one optimal set of $n$-means for $n=72$. By the notations used in Theorem~\ref{Th1}, we can write $\alpha_{72}=\set{a(i) : 1\leq i\leq 72}$. Then,
\begin{align*}W(\alpha_{72})&= \set{\set{\set{1, 3, 3}}, \set{\set{1, 3, 4}}, \set{\set{1, 4, 3}}, \set{\set{1, 4, 4}}, \set{\set{2, 3, 3}}, \set{\set{2,
3, 4}}, \set{\set{2, 4, 3}}, \\
& \set{\set{2, 4, 4}}, \set{\set{3, 1, 3}}, \set{\set{3, 1, 4}}, \set{\set{3,
2, 3}}, \set{\set{3, 2, 4}}, \set{\set{3, 3, 1}}, \set{\set{3, 3, 2}}, \\
& \set{\set{3, 4, 1}}, \set{\set{3, 4,
2}}, \set{\set{4, 1, 3}}, \set{\set{4, 1, 4}}, \set{\set{4, 2, 3}}, \set{\set{4, 2, 4}}, \set{\set{4, 3,
1}}, \set{\set{4, 3, 2}}, \\
& \set{\set{4, 4, 1}}, \set{\set{4, 4, 2}}}.
\end{align*}
Since $\te{card}(W(\alpha_{72}))=24$, by the theorem, we have
$\te{card}(\C C_{73})= \binom{24}{1}=24, \, \te{card}(\C C_{74})= \binom{24}{2}=276, \, \te{card}(\C C_{75})= \binom{24}{3}=2024, \, \te{card}(\C C_{76})= \binom{24}{4}=10626$, etc., for details see Table~\ref{tab1}.
\end{note}
Let us now conclude the paper with the following remark:
\begin {remark}
Consider a set of four contractive affine transformations $S_{(i, j)}$ on $\D R^2$, such that
$S_{(1,1)}(x_1, x_2)=(\frac 1 4 x_1, \frac 1 4 x_2)$, $S_{(2,1)}(x_1, x_2)=(\frac 12 x_1+\frac 12, \frac 1 4 x_2)$, $S_{(1,2)}(x_1, x_2)=(\frac 14 x_1, \frac 12 x_2+\frac 12)$, and $S_{(2,2)}(x_1, x_2)=(\frac 12 x_1+\frac 12, \frac 12 x_2+\frac 12)$ for all $(x_1, x_2) \in \D R^2$. Let $S$ be the limit set of these contractive mappings. Then, $S$ is called the Sierpi\'nski carpet generated by $S_{(i, j)}$ for all $1\leq i, j\leq 2$.
Let $P$ be the Borel probability measure on $\D R^2$ such that $P=\frac{1}{16} P\circ S_{(1,1)}^{-1}+\frac 3{16} P\circ S_{(2,1)}^{-1} +\frac 3{16} P\circ S_{(1,2)}^{-1} +\frac {9}{16} P\circ S_{(2,2)}^{-1}$. Then, $P$ has support the Siepi\'nski carpet $S$. For this probability measure, the optimal sets of $n$-means and the $n$th quantization error are not known yet for all $n\geq 2$.
\end{remark}
| {
"timestamp": "2016-05-10T02:09:34",
"yymm": "1605",
"arxiv_id": "1605.02281",
"language": "ceb",
"url": "https://arxiv.org/abs/1605.02281",
"abstract": "The purpose of quantization for a probability distribution is to estimate the probability by a discrete probability with finite support. In this paper, a nonuniform probability measure $P$ on $\\mathbb R^2$ which has support the Sierpiński carpet generated by a set of four contractive similarity mappings with equal similarity ratios has been considered. For this probability measure, the optimal sets of $n$-means and the $n$th quantization errors are investigated for all $n\\geq 2$.",
"subjects": "Information Theory (cs.IT)",
"title": "Optimal quantizers for a nonuniform distribution on a Sierpinski carpet",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9888419671077918,
"lm_q2_score": 0.7154240018510026,
"lm_q1q2_score": 0.707441277306474
} |
https://arxiv.org/abs/1511.08405 | Gains and Losses are Fundamentally Different in Regret Minimization: The Sparse Case | We demonstrate that, in the classical non-stochastic regret minimization problem with $d$ decisions, gains and losses to be respectively maximized or minimized are fundamentally different. Indeed, by considering the additional sparsity assumption (at each stage, at most $s$ decisions incur a nonzero outcome), we derive optimal regret bounds of different orders. Specifically, with gains, we obtain an optimal regret guarantee after $T$ stages of order $\sqrt{T\log s}$, so the classical dependency in the dimension is replaced by the sparsity size. With losses, we provide matching upper and lower bounds of order $\sqrt{Ts\log(d)/d}$, which is decreasing in $d$. Eventually, we also study the bandit setting, and obtain an upper bound of order $\sqrt{Ts\log (d/s)}$ when outcomes are losses. This bound is proven to be optimal up to the logarithmic factor $\sqrt{\log(d/s)}$. | \section{Introduction}
We consider the classical problem of regret
minimization \cite{hannan1957approximation} that has been well
developed during the last decade
\cite{cesa2006prediction,rakhlinlecture,bubeck2011introduction,shalev2011online,hazan201210,bubeck2012regret}. We
recall that in this sequential decision problem, a decision maker (or
agent, player, algorithm, strategy, policy, depending on the context) chooses at each stage a decision in a finite set
(that we write as $[d]:=\{1,\ldots,d\}$) and obtains as an
\emph{outcome} a real number in $[0,1]$. We specifically chose the
word \emph{outcome}, as opposed to \emph{gain} or \emph{loss}, as our
results show that there exists a fundamental discrepancy between these
two concepts.
The criterion used to evaluate the policy of the decision maker is
the \emph{regret}, i.e., the difference between the
cumulative performance of the best stationary policy (that always
picks a given action $i \in [d]$) and the cumulative performance of
the policy of the decision maker.
We focus here on the \emph{non-stochastic} framework, where no
assumption (apart from boundedness) is made on the sequence of
possible outcomes. In particular, they are not i.i.d.\ and we can even
assume, as usual, that they depend on the past choices of the
decision maker. This broad setup, sometimes referred to as
\emph{individual sequences} (since a policy must be good against \emph{any}
sequence of possible outcomes) incorporates prediction with
expert advice \cite{cesa2006prediction}, data with time-evolving
laws, etc. Perhaps the most fundamental results in this setup are the upper bound of order $\sqrt{T\log d}$
achieved by the Exponential Weight Algorithm
\cite{littlestone1994weighted,vovk1990aggregating,cesa1997analysis,auer2002adaptive}
and the asymptotic lower bound of the same order \cite{cesa1997use}. This
general bound is the same whether outcomes are gains in $[0,1]$ (in
which case, the objective is to maximize the cumulative sum of gains) or
losses in $[0,1]$ (where the decision maker aims at minimizing the
cumulative sum). Indeed, a loss $\ell$ can easily be turned into gain
$g$ by defining $g:=1 -\ell$, the regret being invariant under this
transformation.
This idea does not apply anymore with
structural assumption. For instance, consider the framework where the outcomes
are limited to \emph{$s$-sparse vectors}, i.e. vectors that have at most $s$ nonzero coordinates.
The coordinates which are nonzero may
change arbitrarily over time.
In this framework, the aforementioned
transformation does not preserve the sparsity assumption. Indeed, if $(\ell_1,\ldots,
\ell_d)$ is a $s$-sparse loss vector, the corresponding gain vector $(1-\ell_1,\dots,1-\ell_d)$
may even have full support.
Consequently, results for loss vectors do not apply directly to sparse gains, and
vice versa. It turns out that both setups are fundamentally
different.
The sparsity assumption is actually quite natural in learning and have also
received some attention in online learning
\cite{gerchinovitz2013sparsity,carpentier2012bandit,abbasi2012online,djolonga2013high}. In the case of gains, it reflects the fact
that the problem has some hidden structure and that many
options are irrelevant. For instance, in the canonical
click-through-rate example, a website displays an ad and gets rewarded
if the user clicks on it; we can safely assume that there are only a
small number of ads on which a user would click.
The sparse scenario can also be seen through the scope of prediction
with experts. Given a finite set of expert, we call the \emph{winner of a
stage} the expert with the highest revenue (or the smallest loss);
ties are broken arbitrarily. And the objective would be to win as many
stages as possible. The $s$-sparse setting would represent the case
where $s$ experts are designated as winners (or, non-loser) at each stage.
In the case of losses, the sparsity assumption is motivated by
situations where rare failures might happen at each stage, and the decision maker wants to avoid them.
For instance, in network routing problems, it could be assumes that only a small number of paths would
lose packets as a result of a single, rare, server failure. Or a learner could have access to a finite number of
classification algorithms that perform ideally most of the time; unfortunately, some of them makes mistakes on some examples and the learner would like to prevent that. The general setup is therefore a number of algorithms/experts/actions that
mostly perform well (i.e., find the
correct path, classify correctly, optimize correctly some target function, etc.); however, at each time instance, there are rare
mistakes/accidents and the objective would be to find the action/algorithm
that has the smallest number (or probability in the stochastic case) of failures.
\subsection{Summary of Results}
We investigate regret minimization scenarios both when outcomes are
gains on the one hand, and losses on the other hand. We recall that our objectives are to prove that
they are fundamentally different by exhibiting rates of convergence of different order.
When outcomes are gains, we construct an algorithm based on the Online
Mirror Descent family
\cite{shalev2007online,shalev2011online,bubeck2011introduction}. By
choosing a regularizer based on the $\ell^p$ norm, and then tuning the
parameter $p$ as a function of $s$, we get in
Theorem~\ref{thm:full-info-gains-m-2} a regret bound of order $\sqrt{T\log s}$,
which has the interesting property of being independent of the number
of decisions $d$. This bound is trivially optimal, up to the constant.
If outcomes are losses instead of gains, although the previous
analysis remains valid, a much better bound can be obtained. We build
upon a regret bound for the Exponential Weight Algorithm
\cite{littlestone1994weighted,freund1997decision} and we manage to get
in Theorem~\ref{thm:upper bound-losses-full-info} a regret bound of
order $\sqrt{\frac{Ts\log d}{d}}$,
which is \emph{decreasing} in $d$, for a given
$s$. A nontrivial matching lower bound is established in
Theorem~\ref{thm:lower bound-losses}.
Both of these algorithms need to be tuned as a function of $s$. In
Theorem~\ref{thm:adative-losses} and Theorem~\ref{thm:adaptive-gains},
we construct algorithms which essentially achieve the same regret
bounds without prior knowledge of $s$, by adapting over time to the
sparsity level of past outcome vectors, using an adapted version of the doubling trick.
Finally, we investigate the bandit setting, where the only
feedback available to the decision maker is the outcome of his
decisions (and, not
the outcome of all possible decisions). In the case of losses
we obtain in Theorem~\ref{thm:bandit-upper bound-losses} an
upper bound of order $\sqrt{Ts\log (d/s)}$,
using the Greedy Online Mirror Descent family of algorithms
\cite{audibert2009minimax,audibert2013regret,bubeck2011introduction}.
This bound is proven to be optimal up to a logarithmic factor, as
Theorem~\ref{thm:bandit-losses-lowerbound} establishes a lower bound
of order $\sqrt{Ts}$.
The rates of convergence achieved by our algorithms are summarized in Figure~\ref{fig:tableau}.
\renewcommand{\arraystretch}{2}
\begin{figure}
\centering
\begin{tabular}{r|c|c|c|c|}
\cline{2-5}
&\multicolumn{2}{|c|}{Full information}&\multicolumn{2}{|c|}{Bandit}\\
\cline{2-5}
&Gains&Losses&Gains&Losses\\
\hline
\multicolumn{1}{|r|}{Upper bound}&\multirow{2}{*}{$\sqrt{T\log s}$}&\multirow{2}{*}{$\sqrt{Ts\frac{\log d}{d}}$}&$\sqrt{Td}$&$\sqrt{Ts\log \frac{d}{s}}$\\
\cline{1-1}\cline{4-5}
\multicolumn{1}{|r|}{Lower bound}&&&$\sqrt{Ts}$&$\sqrt{Ts}$\\
\hline
\end{tabular}
\caption{Summary of upper and lower bounds.}
\label{fig:tableau}
\end{figure}
\subsection{General Model and Notation}
\label{sec:framework-notation}
We recall the classical non-stochastic regret minimization problem.
At each time instance $t \geq 1$, the decision maker chooses a decision $d_t$
in the finite set $[d]=\{1,\ldots,d\}$, possibly at random,
accordingly to $x_t \in \Delta_d$, where
\[ \Delta_d= \left\{ x=(x^{(1)},\ldots,x^{(d)})\in \mathbb{R}^d_+\,\middle|\,\sum_{i=1}^dx^{(i)}=1 \right\} \]
is the the set of probability distributions over $[d]$.
Nature then reveals an outcome
vector $\omega_t \in [0,1]^d$ and the decision maker receives
$\omega_t^{(d_t)} \in [0,1]$. As outcomes are bounded, we can easily
replace $\omega_t^{(d_t)}$ by its expectation that we denote by
$\langle \omega_t,x_t\rangle$. Indeed, Hoeffding-Azuma concentration inequality will imply that all the results we will state in expectation hold with high probability.
Given a time horizon $T \geqslant 1$, the objective of the decision
maker is to minimize his regret, whose definition depends on whether
outcomes are \emph{gains} or \emph{losses}. In the case of gains
(resp. losses), the notation $\omega_t$ is then changed to $g_t$
(resp. $\ell_t$) and the regret is:
\[ R_T=\max_{i \in [d]}\sum_{t=1}^Tg_t^{(i)} -\sum_{t=1}^T \langle g_t, x_t\rangle\quad \left( \text{resp.}\ R_T=\sum_{t=1}^T \langle \ell_t,x_t\rangle-\min_{i \in [d]}\sum_{t=1}^T\ell_t^{(i)} \right). \]
In both cases, the well-known Exponential Weight
Algorithm guarantees a bound on the
regret of order $\sqrt{T\log d}$. Moreover, this bound cannot be
improved in general as it matches a lower bound.
\bigskip
We shall consider an additional structural
assumption on the outcomes, namely that $\omega_t$ is $s$-sparse in
the sense that $\|\omega_t\|_0 \leq s$, i.e., the number of nonzero components
of $\omega_t$ is less than $s$, where $s$ is a fixed known
parameter. The set of components which are nonzero is not fixed nor known, and
may change arbitrarily over time.
We aim at proving that it is then possible to drastically improve the
previously mentioned guarantee of order $\sqrt{T\log d}$ and that
losses and gains are two fundamentally different settings with minimax
regrets of different orders.
\section{When Outcomes are Gains to be Maximized}
\subsection{Online Mirror Descent Algorithms}
\label{sec:online-mirr-desc}
We quickly present the general Online Mirror Descent algorithm \cite{shalev2011online,bubeck2011introduction,bubeck2012regret,kwon2014continuous} and
state the regret bound it incurs; it will be used as a key element in
Theorem~\ref{thm:full-info-gains-m-2}.
A convex function $h:\mathbb{R}^d\to \mathbb{R}\cup\{+\infty\}$ is called a \emph{regularizer} on $\Delta_d$ if $h$ is strictly convex
and continuous on its domain $\Delta_d$, and $h(x)=+\infty$ outside $\Delta_d$. Denote $\delta_h=\max_{\Delta_d}h-\min_{\Delta_d}h$ and $h^*:\mathbb{R}^d\to \mathbb{R}^d$ the Legendre-Fenchel
transform of $h$:
\[ h^*(y)=\sup_{x\in \mathbb{R}^d}\left\{ \left< y , x \right>
-h(x) \right\},\quad y\in \mathbb{R}^d, \]
which is differentiable since $h$ is strictly convex. For all $y\in
\mathbb{R}^d$, it holds that $\nabla h^*(y)\in \Delta_d$.
Let $\eta \in \mathbb{R}$ be a parameter to be tuned. The Online Mirror Descent Algorithm
associated with regularizer $h$ and parameter $\eta$ is defined by:
\[ x_t=\nabla h^*\left( \eta\sum_{k=1}^{t-1}\omega_k \right),\quad t \geq 1, \]
where $\omega_t \in [0,1]^d$ denote the vector of outcomes and $x_t$ the probability distribution chosen at stage $t$. The specific choice $h(x)=\sum_{i=1}^dx^{(i)}\log x^{(i)}$ for $x=(x^{(1)},\ldots,x^{(d)})\in \Delta_d$ (and
$h(x)=+\infty$ otherwise) gives the celebrated Exponential Weight
Algorithm, which can we written explicitly, component by component:
\[ x_t^{(i)}=\frac{\exp \left( \eta\sum_{k=1}^{t-1}\omega_k^{(i)} \right)}{\sum_{j=1}^d\exp \left( \eta\sum_{k=1}^{t-1}\omega_k^{(j)} \right) },\quad
t \geq 1,\ i\in [d]. \]
The following general regret guarantee for strongly convex regularizers
is expressed in terms of the dual norm $\left\| \,\cdot\, \right\|_*$ of $\left\| \,\cdot\,
\right\|$.
\begin{theorem}[\cite{shalev2011online} Th.\ 2.21; \cite{bubeck2012regret} Th.\ 5.6; \cite{kwon2014continuous} Th.\ 5.1]
\label{thm:mirror-descent}
Let $K>0$ and assume $h$ to be $K$-strongly convex with respect to a norm $\left\|
\,\cdot\, \right\|$. Then, for all sequence of outcome vectors
$(\omega_t)_{t\geq 1}$ in $\mathbb{R}^d$, the Online Mirror Descent
strategy associated with $h$ and $\eta$ (with $\eta >0$ in cases of gains and $\eta <0$ in cases of losses) guarantees, for $T\geq 1$, the following
regret bound:
\[ R_T\leq \frac{\delta_h}{|\eta|}+\frac{|\eta|}{2K}\sum_{t=1}^T\left\| \omega_t \right\|_*^2. \]
\end{theorem}
\subsection{Upper Bound on the Regret}
\label{sec:gains}
We first assume $s\geq 2$. Let $p\in (1,2]$ and define the following regularizer:
\[ h_p(x)= \begin{cases}
\frac{1}{2}\left\| x \right\|_p^2&\text{if $x\in \Delta_d$}\\
+\infty&\text{otherwise}.
\end{cases}
\]
One can easily check that $h_p$ is indeed a regularizer on $\Delta_d$
and that $\delta_{h_p}\leqslant 1/2$. Moreover, it is
$(p-1)$-strongly convex with respect to $\left\|\,\cdot\,
\right\|_p$ (see \cite[Lemma
5.7]{bubeck2011introduction} or \cite[Lemma 9]{kakade2012regularization}).
We can now state our first result, the general upper bound on regret when outcomes are $s$-sparse gains.
\begin{theorem}
\label{thm:full-info-gains-m-2}
Let $\eta>0$ and $s\geq 3$. Against all sequence of
$s$-sparse gain vectors $g_t$, i.e., $g_t \in [0,1]^d$ and $\|g_t\|_0\leq s$, the Online Mirror Descent algorithm associated
with regularizer $h_p$ and parameter $\eta$ guarantees:
\[ R_T\leq \frac{1}{2\eta}+\frac{\eta Ts^{2/q}}{2(p-1)}, \]
where $1/p+1/q=1$. In particular, the choices $\eta=\sqrt{(p-1)/Ts^{2/q}}$ and $p=1+(2\log s-1)^{-1}$ give:
\[ R_T\leq \sqrt{2eT\log s}. \]
\end{theorem}
\begin{proof}
$h_p$ being $(p-1)$-strongly convex with respect to
$\left\| \,\cdot\, \right\|_p$, and $\left\| \,\cdot\, \right\|_q$
being the dual norm of $\left\| \,\cdot\, \right\|_p$, Theorem~\ref{thm:mirror-descent} gives:
\[ R_T\leq \frac{\delta_{h_p}}{\eta}+\frac{\eta}{2(p-1)}\sum_{t=1}^T\left\| g_t \right\|_q^2. \]
For each $t\geq 1$, the norm of $g_t$ can be bounded as follows:
\[ \left\| g_t \right\|_q^2=\left( \sum_{i=1}^d\left| g_t^{(i)}\right|^q \right)^{2/q}\leq \left( \sum_{\text{$s$ terms}}^{}\left| g_t^{(i)} \right|^q \right)^{2/q}\leq s^{2/q}, \]
which yields
\[ R_T\leq \frac{1}{2\eta}+\frac{\eta Ts^{2/q}}{2(p-1)}. \]
We can now balance both terms by choosing $\eta=\sqrt{(p-1)/(Ts^{2/q})}$ and get:
\[ R_T\leq \sqrt{\frac{Ts^{2/q}}{p-1}}. \]
Finally, since $s\geq 3$, we have $2\log s>1$ and we set $p=1+(2\log s-1)^{-1}\in(1,2]$, which gives:
\[ \frac{1}{q}=1-\frac{1}{p}=\frac{p-1}{p}=\frac{(2\log s-1)^{-1}}{1+(2\log s-1)^{-1}}=\frac{1}{2\log s}, \]
and thus:
\[ R_T\leq \sqrt{\frac{Ts^{2/q}}{p-1}}=\sqrt{2T\log s\,e^{2\log s/q}}=\sqrt{2e\, T\log s}. \]
\end{proof}
We emphasize the fact that we obtain, up to a multiplicative constant,
the exact same rate as when the decision maker only has a set of $s$ decisions.
In the case $s=1,2$, we can easily derive a bound of respectively $\sqrt{T}$ and $\sqrt{2T}$
using the same regularizer with $p=2$.
\subsection{Matching Lower Bound}
\label{sec:fullinfo-gains-lowerbound}
For $s \in [d]$ and $T\geq 1$, we denote $v_T^{g,s,d}$
the minimax regret of the $T$-stage decision problem with
outcome vectors restricted to $s$-sparse gains:
\[ v_T^{g,s,d}=\min_{\text{strat.}}\max_{ (g_t)_t} R_T \]
where the minimum is taken over all possible policies of the decision maker,
and the maximum over all sequences of $s$-sparse gains vectors.
To establish a lower bound in the present setting, we can assume that only the first $s$
coordinates of $g_t$ might be positive (for all $t\geqslant 1$) and even that the decision
maker is aware of that. Therefore he has no interest in assigning
positive probabilities to any decision but the first $s$ ones. That
setup, which is simpler for the decision maker than the original one,
is obviously equivalent to the basic regret minimization problem with
only $s$ decisions. Therefore, the classical lower bound
\cite[Theorem 3.2.3]{cesa1997use} holds and we obtain the following.
\begin{theorem}
\label{thm:lowerbound-full-info-gains}
\[ \liminf_{\substack{s\to +\infty\\d\geq s}}\liminf_{T\to +\infty}\frac{v_T^{g,s,d}}{\sqrt{T \log s}}\geq \frac{\sqrt{2}}{2}. \]
\end{theorem}
The same lower bound, up to the multiplicative constant actually holds non asymptotically, see \cite[Theorem~3.6]{cesa2006prediction}.
\medskip
An immediate consequence of Theorem \ref{thm:lowerbound-full-info-gains} is that the regret bound
derived in Theorem~\ref{thm:full-info-gains-m-2} is asymptotically minimax
optimal, up to a multiplicative constant.
\section{When Outcomes are Losses to be Minimized}
\label{sec:losses}
\subsection{Upper Bound on the Regret}
We now consider the case of losses, and the regularizer shall no longer depend on $s$ (as with gains), as we will always use the Exponential Weight Algorithm. Instead, it is the parameter $\eta$ that will be tuned as a function of $s$.
\begin{theorem}
\label{thm:upper bound-losses-full-info}
Let $s\geq 1$. For all sequence of $s$-sparse loss vectors $(\ell_t)_{t\geq 1}$, i.e., $\ell_t \in [0,1]^d$ and $\|\ell_t\|_0\leq s$, the Exponential Weight Algorithm with parameter
$-\eta$ where $\eta:=\log \left( 1+\sqrt{2d\log d/sT} \right)>0$ guarantees, for $T\geq 1$:
\[ R_T\leq \sqrt{\frac{2sT\log d}{d}}+\log d. \]
\end{theorem}
We build upon the following regret bound for losses
which is written in terms of the performance of the best action.
\begin{theorem}[\cite{littlestone1994weighted}; \cite{cesa2006prediction} Th\ 2.4]
\label{thm:upper-bound-small-losses}
Let $\eta>0$. For all sequence of loss vectors $(\ell_t)_{t\geq 1}$ in $[0,1]^d$, the Exponential Weight Algorithm with parameter $-\eta$ guarantees, for all $T\geq 1$:
\[ R_T\leq \frac{\log d}{1-e^{-\eta}}+\left( \frac{\eta}{1-e^{-\eta}}-1 \right)L_T^*\enspace, \]
where $\displaystyle L_T^*=\min_{i \in [d]}\sum_{t=1}^T\ell_t^{(i)}$ is the loss of the best stationary decision.
\end{theorem}
\begin{proof}[Proof of Theorem~\ref{thm:upper bound-losses-full-info}]
Let $T\geq 1$ and $ L_T^*=\min_{i \in [d]}\sum_{t=1}^T\ell_t^{(i)}$
be the loss of the best stationary policy.
First note that since the loss vectors $\ell_t$ are $s$-sparse, we have $s\geq \sum_{i=1}^d \ell_t^{(i)}$. By summing over $1\leqslant t\leqslant T$:
\[ sT\geq
\sum_{t=1}^T\sum_{i=1}^d\ell_t^{(i)}=\sum_{i=1}^d\left( \sum_{t=1}^{T}\ell_t^{(i)}\right) \geq
d\left( \min_{i \in [d]}\sum_{t=1}^T \ell_t^{(i)}\right) =dL_T^*, \]
and therefore, we have $L_T^*\leq Ts/d$.
Then, by using the inequality $\eta\leq (e^\eta-e^{-\eta})/2$, the bound from Theorem~\ref{thm:upper-bound-small-losses} becomes:
\[ R_T\leq \frac{\log d}{1-e^{-\eta}}+\left( \frac{e^\eta-e^{-\eta}}{2(1-e^{-\eta})}-1 \right)L_T^*\enspace. \]
The factor of $L_T^*$ in the second term can be transformed as follows:
\[ \frac{e^\eta-e^{-\eta}}{2(1-e^{-\eta})}-1=\frac{(1+e^{-\eta})(e^{\eta}-e^{-\eta})}{2(1-e^{-2\eta})}-1=\frac{(1+e^{-\eta})e^\eta}{2}-1=\frac{e^\eta-1}{2}\enspace, \]
and therefore the bound on the regret becomes:
\[ R_T\leq \frac{\log d}{1-e^{-\eta}}+\frac{e^\eta-1}{2}L_T^*\leq \frac{\log d}{1-e^{-\eta}}+\frac{(e^{\eta}-1)Ts}{2d}\enspace, \]
where we have been able to use the upper-bound on $L_T^*$ since $\frac{e^\eta-1}{2}\geq 0$. Along with the choice $\eta=\log (1+\sqrt{2d\log d/Ts} )$ and standard computations, this yields:\[ R_T\leq \sqrt{\frac{2Ts\log d}{d}}+\log d\enspace. \]
\end{proof}
Interestingly, the bound from Theorem \ref{thm:upper
bound-losses-full-info} shows that $\sqrt{2sT\log d/d}$, the
dominating term of the regret bound, is \emph{decreasing} when the
number of decisions $d$ increases. This is due to the sparsity
assumptions (as the regret increases with $s$, the maximal number of
decision with positive losses). Indeed, when $s$ is fixed and $d$
increases, more and more decisions are optimal at each stage, a
proportion $1-s/d$ to be precise. As a consequence, it becomes
\emph{easier} to find an optimal decisions when $d$ increases. However,
this intuition will turn out not to be valid in the bandit framework.
On the other hand, if the proportion $s/d$ of positive losses remains constant then the regret bound achieved is of the same order as in the usual case.
\subsection{Matching Lower Bound}
\label{sec:losses-1}
When outcomes are losses, the argument from
Section~\ref{sec:fullinfo-gains-lowerbound} does not allow to derive a
lower bound. Indeed, if we assume that only the first $s$ coordinates
of the loss vectors $\ell_t$ can be positive, and that the decision
maker knows it, then he just has to take at each stage the decision
$d_t=d$ which incurs a loss of 0. As a consequence, he trivially has a
regret $R_T=0$. Choosing at random, but once and for all, a fixed
subset of $s$ coordinates does not provide any interesting lower bound
either. Instead, the key idea of the following result is to choose at
random and at each stage the $s$ coordinates associated to positive losses.
And we therefore use the following classical probabilistic
argument. Assume that we have found a probability distribution on
$(\ell_t)_t$ such that the expected regret can be bounded from below
by a quantity which does not depend on the strategy of the decision
maker. This would imply that for any algorithm, there exists a sequence of $(\ell_t)_t$
such that the regret is greater than the same quantity.
In the following statement, $v_T^{\ell,s,d}$ stands for the minimax
regret in the case where outcomes are losses.
\begin{theorem}
\label{thm:lower bound-losses}
For all $s\geq 1$,
\[ \liminf_{d\to +\infty}\liminf_{T \to +\infty}\frac{v_T^{\ell,s,d}}{\sqrt{T\frac{s}{d}\log d}}\geq \frac{\sqrt{2}}{2}. \]
\end{theorem}
The main consequences of this theorem are that the algorithm described
in Theorem~\ref{thm:upper bound-losses-full-info} is asymptotically minimax
optimal (up to a multiplicative constant) and that gains and losses are fundamentally different from the point of view of regret minimization.
\begin{proof}
We first define the sequence of loss vectors $\ell_t$ ($t\geq 1$) i.i.d. as follows. Firs, we draw a set $I_t\subset [d]$ of cardinal $s$ uniformly among the $\binom{d}{s}$ possibilities. Then, if $i\in I_t$ set $\ell_t^{(i)}=1$ with probability $1/2$ and $\ell_t^{(i)}=0$ with probability $1/2$, independently for each component. If $i\not \in I_t$, we set $\ell_t^{(i)}=0$.
As a consequence, we always have that $\ell_t$ is $s$-sparse. Moreover, for each $t\geq 1$ and each coordinate $i\in [d]$, $\ell_t^{(i)}$ satisfies:
\[ \mathbb{P}\left[ \ell_t^{(i)}=1 \right]=\frac{s}{2d}\quad \text{and}\quad \mathbb{P}\left[ \ell_t^{(i)}=0 \right]=1-\frac{s}{2d}\enspace, \]
thus $\mathbb{E}\left[ \ell_t^{(i)} \right]=s/2d$.
Therefore we obtain that for any algorithm $(x_t)_{t\geq 1}$,
$\mathbb{E}\left[ \langle \ell_t , x_t \rangle
\right]=s/2d$. This yields that
\begin{align*}
\mathbb{E}\left[ \frac{R_T}{\sqrt{T}} \right]&=\mathbb{E}\left[ \frac{1}{\sqrt{T}}\left( \sum_{t=1}^T \langle \ell_t , x_t \rangle -\min_{i \in [d]}\sum_{t=1}^T\ell_t^{(i)}\right) \right] \\
&=\mathbb{E}\left[ \max_{i \in [d]}\frac{1}{\sqrt{T}}\sum_{t=1}^T\left(\frac{s}{2d}- \ell_t^{(i)} \right) \right]\\
&=\mathbb{E}\left[ \max_{i \in [d]}\frac{1}{\sqrt{T}}\sum_{t=1}^TX_t^{(i)} \right],
\end{align*}
where $t\geq 1$, we have defined the random vector $X_t$ by $X_t^{(i)}=s/2d-\ell_t^{(i)}$ for all $i\in [d]$. For $t\geq 1$, the $X_t$ are i.i.d. zero-mean random vectors with values in $[-1,1]^d$. We can therefore apply the comparison Lemma~\ref{lm:1} to get:
\[ \liminf_{T\to +\infty}\mathbb{E}\left[ \frac{R_T}{\sqrt{T}} \right]=\liminf_{T\to +\infty}\mathbb{E}\left[ \max_{i \in [d]}\frac{1}{\sqrt{T}}\sum_{t=1}^TX_t^{(i)} \right]\geq \mathbb{E}\left[ \max_{i \in [d]}Z^{(i)} \right]\enspace, \]
where $Z \sim \mathcal N(0,\Sigma)$ with $\Sigma=(\operatorname{cov}(X_1^{(i)},X_1^{(j)}))_{i,j}$.
We now make appeal to Slepian's lemma, recalled in
Proposition~\ref{prop:slepian} below. Therefore, we introduce the
Gaussian vector $W \sim \mathcal N( 0, \tilde{\Sigma})$ where
\[ \tilde{\Sigma}=\operatorname{diag}\left(
\operatorname{Var}X_1^{(1)},\dots,\operatorname{Var}X_1^{(1)}
\right). \]
As a consequence, the first two hypotheses of Proposition~\ref{prop:slepian} from the definitions of $Z$ and $W$. Let $i\neq j$, then
\begin{align*}
\mathbb{E}\left[ Z^{(i)}Z^{(j)} \right]&=\operatorname{cov}(Z^{(i)},Z^{(j)})=\operatorname{cov}(\ell_1^{(i)},\ell_1^{(j)})=\mathbb{E}\left[ \ell_1^{(i)}\ell_1^{(j)} \right]-\mathbb{E}\left[ \ell_1^{(i)} \right]\mathbb{E}\left[ \ell_1^{(j)} \right].
\end{align*}
By definition of $\ell_1$, $\ell_1^{(i)}\ell_1^{(j)}=1$ if and only if $\ell_1^{(i)}=\ell_1^{(j)}=1$ and $\ell_1^{(i)}\ell_1^{(j)}=0$ otherwise. Therefore, using the random subset $I_1$ that appears in the definition of $\ell_1$:
\begin{align*}
\mathbb{E}\left[ Z^{(i)}Z^{(j)} \right]&=\mathbb{P}\left[ \ell_1^{(i)}=\ell_1^{(j)}=1 \right]-\left( \frac{s}{2d} \right)^2 \\
&=\mathbb{P}\left[ \ell_1^{(i)}=\ell_1^{(j)}=1\,\middle|\, \left\{ i,j \right\}\subset I_1 \right] \mathbb{P}\left[ \left\{ i,j \right\}\subset I_1 \right]-\left( \frac{s}{2d} \right)^2 \\
&=\frac{1}{4}\cdot \frac{\binom{d-2}{s-2}}{\binom{d}{s}}-\left( \frac{s}{2d} \right)^2\\
&=\frac{1}{4}\left( \frac{s(s-1)}{d(d-1)}-\frac{s^2}{d^2} \right)\leq 0,\\
\end{align*}
and since $\mathbb{E}\left[ W^{(i)}W^{(i)} \right]=0$, the third hypothesis of Slepian's lemma is also satisfied. It yields that, for all $\theta\in \mathbb{R}$:
\begin{align*}
\mathbb{P}\left[ \max_{i \in [d]}Z^{(i)}\leq \theta \right]&=\mathbb{P}\left[ Z^{(1)}\leq \theta,\dots, Z^{(d)}\leq \theta\right]\\
&\leq \mathbb{P}\left[ W^{(1)}\leq \theta,\dots, W^{(d)} \leq \theta \right]=\mathbb{P}\left[ \max_{i \in [d]}W^{(i)}\leq \theta \right].
\end{align*}
This inequality between two cumulative distribution functions implies, the reverse inequality on expectations:
\[ \mathbb{E}\left[ \max_{i \in [d]}Z^{(i)} \right]\geq \mathbb{E}\left[ \max_{i \in [d]}W^{(i)}\right]\enspace. \]
The components of the Gaussian vector $W$ being independent, and of variance $\operatorname{Var}\ell_1^{(1)}$, we have
\[ \mathbb{E}\left[ \max_{i \in [d]} W^{(i)} \right]= \kappa_d\sqrt{\operatorname{Var}\ell_1^{(1)}}=\kappa_d\sqrt{\frac{s}{2d}\left( 1-\frac{s}{2d} \right) }\geq \kappa_d\sqrt{\frac{s}{4d}}\enspace, \]
where $\kappa_d$ is the expectation of the maximum of $d$ Gaussian variables. Combining everything gives:
\[ \liminf_{T\to +\infty}\frac{v_T^{\ell,s,d}}{\sqrt{T}}\geq \liminf_{T\to +\infty}\mathbb{E}\left[ \frac{R_T}{\sqrt{T}} \right]\geq \mathbb{E}\left[ \max_{i \in [d]} Z^{(i)} \right]\geq \mathbb{E}\left[ \max_{i \in [d]} W^{(i)} \right]\geq \kappa_d\sqrt{\frac{s}{4d}}\enspace. \]
And for large $d$, since $\kappa_d$ is equivalent to $\sqrt{2\log d}$, see e.g., \cite{galambos1980asymptotic}
\[ \liminf_{d\to +\infty}\liminf_{T\to +\infty}\frac{v_T^{\ell,s,d}}{\sqrt{T\frac{s}{d}\log d}}\geq \frac{\sqrt{2}}{2}\enspace. \]
\end{proof}
\begin{proposition}[Slepian's lemma \cite{slepian1962one}]
\label{prop:slepian}
Let $Z=(Z^{(1)},\dots,Z^{(d)})$ and $W=(W^{(1)},\dots,W^{(d)})$ be Gaussian random vectors in $\mathbb{R}^d$ satisfying:
\begin{enumerate}[(i)]
\item $\mathbb{E}\left[ Z \right]=\mathbb{E}\left[ W \right]=0$;
\item $\mathbb{E}\left[ (Z^{(i)})^2 \right]=\mathbb{E}\left[ (W^{(i)})^2 \right]$ for $i \in [d]$;
\item $\mathbb{E}\left[ Z^{(i)}Z^{(j)} \right]\leq \mathbb{E}\left[W^{(i)}W^{(j)}\right]$ for $i\neq j \in [d]$.
\end{enumerate}
Then, for all real numbers $\theta_1,\dots,\theta_d$, we have:
\[ \mathbb{P}\left[ Z^{(1)}\leq \theta_1,\dots,Z^{(d)}\leq \theta_d \right]\leq \mathbb{P}\left[ W^{(1)}\leq \theta_1,\dots,W^{(d)}\leq \theta_d \right]\enspace. \]
\end{proposition}
The following lemma is an extension of e.g.\ \cite[Lemma
A.11]{cesa2006prediction} to random vectors with correlated components.
\begin{lemma}[Comparison lemma]
\label{lm:1}
For $t \geq 1$, let $(X_t)_{t\geqslant 1}$ be i.i.d. zero-mean random vectors in $[-1,1]^d$, $\Sigma$ be the covariance matrix of $X_t$ and $Z \sim \mathcal N(0,\Sigma)$. Then,
\[ \liminf_{T\to +\infty}\mathbb{E}\left[ \max_{i \in [d]}\frac{1}{\sqrt{T}}\sum_{t=1}^TX_t^{(i)} \right]\geq \mathbb{E}\left[ \max_{i \in [d]} Z^{(i)} \right]. \]
\end{lemma}
\begin{proof}
Denote
\[ Y_T=\max_{i\in [d]}\frac{1}{\sqrt{T}}\sum_{t=1}^TX_t^{(i)}. \]
Let $A\leqslant 0$ and consider the function $\phi_A:\mathbb{R}\to \mathbb{R}$ defined by $\phi_A(x)=\max_{}(x,A)$.
\begin{align*}
\mathbb{E}\left[ Y_T \right]&=\mathbb{E}\left[ Y_T\cdot \mathbbm{1}_{\left\{ Y_T\geqslant A \right\} } \right]+\mathbb{E}\left[ Y_T\cdot \mathbbm{1}_{\left\{ Y_T<A \right\} } \right]\\
&=\mathbb{E}\left[ \phi_A(Y_T)\cdot \mathbbm{1}_{\left\{ Y_T\geqslant A \right\} } \right]+\mathbb{E}\left[ Y_T\cdot \mathbbm{1}_{\left\{ Y_T<A \right\} } \right]\\
&=\mathbb{E}\left[ \phi_A(Y_T) \right]-\mathbb{E}\left[ \phi_A(Y_T)\cdot \mathbbm{1}_{\left\{ Y_T<A \right\} } \right]+\mathbb{E}\left[ Y_T\cdot \mathbbm{1}_{\left\{ Y_T<A \right\} } \right]\\
&=\mathbb{E}\left[ \phi_A(Y_T) \right]-\mathbb{E}\left[ (A-Y_T)\cdot \mathbbm{1}_{\left\{ A-Y_T>0 \right\} } \right].
\end{align*}
Let us estimate the second term. Denote $Z_T=(A-Y_T)\cdot \mathbbm{1}_{A-Y_T>0}$. We clearly have, for all $u>0$, $\mathbb{P}\left[ Z_T>u \right]=\mathbb{P}\left[ A-Y_T>u \right]$. And $Z_T$ being nonnegative, we can write:
\begin{align*}
0&\leqslant \mathbb{E}\left[ (A-Y_T)\cdot \mathbbm{1}_{\left\{ A-Y_T \right\}>0 } \right]=\mathbb{E}\left[ Z_T \right]\\
&=\int_0^{+\infty}\mathbb{P}\left[ Z_T>u \right]\,\mathrm{d}u\\
&=\int_0^{+\infty}\mathbb{P}\left[ A-Y_T>u \right]\,\mathrm{d}u \\
&=\int_{-A}^{+\infty}\mathbb{P}\left[ Y_T<-u \right]\,\mathrm{d}u\\
&=\int_{-A}^{+\infty}\mathbb{P}\left[ \max_{i\in [d]}\frac{1}{\sqrt{T}}\sum_{t=1}^TX_t^{(i)}<u \right]\,\mathrm{d}u\\
&\leqslant \int_{-A}^{+\infty}\mathbb{P}\left[ \sum_{t=1}^TX_t^{(1)}<u\sqrt{T} \right]\,\mathrm{d}u.
\end{align*}
For $u>0$, using Hoeffding's inequality together with the assumptions $\mathbb{E}\left[ X_t^{(1)} \right]=0$ and $X_t^{(1)}\in [-1,1]$, we can bound the last integrand:
\[ \mathbb{P}\left[ \sum_{t=1}^TX_t^{(1)}<u\sqrt{T} \right]\leqslant e^{-u^2/2}, \]
Which gives:
\[ 0\leqslant \mathbb{E}\left[ (A-Y_T)\cdot \mathbbm{1}_{\left\{ A-Y_T \right\}>0 } \right]\leqslant \int_{-A}^{+\infty}e^{-u^2/2}\,\mathrm{d}u\leqslant \frac{e^{-A^2/2}}{-A}. \]
Therefore:
\[ \mathbb{E}\left[ Y_T \right]\geqslant \mathbb{E}\left[ \phi_A(Y_T) \right]+\frac{e^{-A^2/2}}{A}. \]
We now take the liminf on both sides as $t\to +\infty$. The left-hand side is the quantity that appears in the statement. We now focus on the second term of the right-hand side.
The central limit theorem gives the following convergence in distribution:
\[ \frac{1}{\sqrt{T}}\sum_{t=1}^TX_t \xrightarrow[T \rightarrow +\infty]{\mathcal L}X. \]
The application $(x^{(1)},\dots,x^{(d)})\longmapsto \max_{i\in [d]}x^{(i)}$ being continuous, we can apply the continuous mapping theorem:
\[ Y_T=\max_{i\in [d]}\frac{1}{\sqrt{T}}\sum_{t=1}^TX_t^{(i)} \xrightarrow[n \rightarrow +\infty]{\mathcal L}\max_{i\in [d]}X^{(i)}. \]
This convergence in distribution allows the use of the portmanteau lemma: $\phi_A$ being lower semi-continuous and bounded from below, we have:
\[ \liminf_{t\to +\infty}\mathbb{E}\left[ \phi_A(Y_T) \right]\geqslant \mathbb{E}\left[ \phi_A\left( \max_{i\in [d]}X^{(i)} \right) \right], \]
and thus:
\[ \liminf_{t\to +\infty}\mathbb{E}\left[ Y_T \right]\geqslant \mathbb{E}\left[ \phi_A\left( \max_{i\in [d]}X^{(i)}\right) \right]+\frac{e^{-A^2/2}}{A}. \]
We would now like to take the limit as $A\to -\infty$. By definition of $\phi_A$, for $A\leqslant 0$, we have the following domination:
\[ \left| \phi_A\left( \max_{i\in [d]}X^{(i)} \right) \right| \leqslant \left| \max_{i\in [d]}X^{(i)} \right|\leqslant \max_{i\in [d]}\left| X^{(i)} \right| \leqslant \sum_{i=1}^d\left| X^{(i)} \right|, \]
where each $X^{(i)}$ is $L^1$ since it is a normal random variable. We can therefore apply the dominated convergence theorem as $A\to -\infty$:
\[ \mathbb{E}\left[ \phi_A\left( \max_{i\in [d]}X^{(i)} \right) \right] \xrightarrow[A \rightarrow -\infty]{}\mathbb{E}\left[ \max_{i\in [d]}X^{(i)} \right], \]
and eventually, we get the stated result:
\[ \liminf_{t\to +\infty}\mathbb{E}\left[ Y_T \right]\geqslant \mathbb{E}\left[ \max_{i\in [d]}X^{(i)} \right]. \]
\end{proof}
\section{When the sparsity level $s$ is unknown}
\label{sec:when-sparsity-level}
We now longer assume in this section that the decision maker have the
knowledge of the sparsity level $s$. We modify our algorithms to be
adaptive over the sparsity level of the observed gain/loss vectors, following the same ideas behind the classical doubling trick (yet it cannot be directly applied here). The
algorithms are proved to essentially achieve the same regret bounds as in the case
where $s$ is known.
Specifically, let $T\geqslant 1$ be the number of rounds and $s^*$ the
highest sparsity level of the gain/loss vectors chosen by Nature up to time $T$. In the following, we
construct algorithms which achieve regret bounds of order $\sqrt{T\log
s^*}$ and $\sqrt{T\frac{s^*\log d}{d}}$ for gains and losses
respectively, without prior knowledge of $s^*$.
\subsection{For Losses}
\label{sec:doubling-trick-like}
Let $(\ell_t)_{t\geqslant 1}$ be the sequence of loss vectors in $[0,1]^d$ chosen
by Nature, and $T\geqslant 1$ the number of rounds. We denote
$s^*=\max_{1\leqslant t\leqslant T}\left\| \ell_t \right\|_0$ the higher
sparsity level of the loss vectors up to time $T$. The goal is to
construct an algorithm which achieves a regret bound of order
$\sqrt{\frac{Ts^*\log d}{d}}$ without any prior knowledge about the
sparsity level of the loss vectors.
The time instances
$\left\{ 1,\dots,T \right\}$ will be divided into several time
intervals. On each of those, the previous loss vectors will be left aside,
and a new instance of the Exponential Weight Algorithm with a specific
parameter will be run. Let $M=\lceil \log_2s^* \rceil$ and
$\tau(0)=0$. Then, for $1\leqslant m<M$ we define
\[ \tau(m)=\min_{}\left\{ 1\leqslant t\leqslant T\,\middle|\, \left\| \ell_t \right\|_0>2^m \right\}\quad \text{and}\quad \tau(M)=T. \]
In other words, $\tau(m)$ is the first time instance at which the
sparsity level of the loss vector execeeds
$2^m$. $(\tau(m))_{1\leqslant m\leqslant M}$ is thus a nondecreasing
sequence. We can then define the time intervals $I(m)$ as follows. For $1\leqslant m\leqslant M$, let
\[ I(m)= \begin{cases}
\left\{ \tau(m-1)+1,\dots,\tau(m) \right\}&\text{if }
\tau(m-1)<\tau(m)\\
\varnothing&\text{if }\tau(m-1)=\tau(m).
\end{cases} . \]
The sets $(I(m))_{1\leqslant m\leqslant M}$ clearly is a partition
of $\left\{ 1,\dots,T \right\}$ (some of the intervals may be empty).
For $1\leqslant t\leqslant T$, we define $m_t=\min_{}\left\{
m\geqslant 1\,\middle|\,\tau(m)\geqslant t \right\}$ which implies
$t\in I(m_t)$. In other words, $m_t$ is the index of the only interval $t$
belongs to.
Let $C>0$ be a constant to be chosen later and for $1\leqslant
m\leqslant M$, let
\[ \eta(m)=\log \left( 1+C\sqrt{\frac{d\log d}{2^mT}} \right) \]
be the parameter of the Exponential Weight Algorithm to
be used on interval $I(m)$.
In this section, $h$ will be entropic regularizer on the simplex
$h(x)=\sum_{i=1}^dx^{(i)}\log x^{(i)}$, so that $y\longmapsto \nabla
h^*(y)$ is the \emph{logit map} used in the Exponential
Weight Algorithm. We can then define the played actions to be:
\[ x_t=\nabla h^*\left( -\eta(m_t)\sum_{\substack{t'<t\\t'\in I(m_t)}}^{}\ell_{t'} \right),\quad t=1,\dots,T. \]
\begin{algorithm}
\SetAlgoLined
\SetKwInOut{Input}{input}
\Input{$T\geqslant 1$, $d\geqslant 1$ integers, and $C>0$.}
$\eta\leftarrow \log (1+C\sqrt{d\log d/2T})$\;
$m\leftarrow 1$\;
\For{$i\leftarrow 1$ \KwTo $d$}{
$w^{(i)}\leftarrow 1/d$\;}
\For{$t\leftarrow 1$ \KwTo $T$}{
draw and play decision $i$ with probability
$w^{(i)}/\sum_{j=1}^{d}w^{(j)}$\;
observe loss vector $\ell_t$\;
\eIf{$\left\| \ell_t \right\|_0\leqslant 2^m$}{
\For{$i\leftarrow 1$ \KwTo $d$}{
$w^{(i)}\leftarrow w^{(i)}e^{-\eta\ell_t^{(i)}}$\;}
}{
$m\leftarrow \lceil \log_2\left\| \ell_t \right\|_0 \rceil$\;
$\eta\leftarrow \log (1+C\sqrt{d\log d/2^mT})$\;
\For{$i\leftarrow 1$ \KwTo $d$}{
$w^{(i)}\leftarrow 1/d$\;}
}
}
\caption{For losses in full information without prior knowledge about
sparsity}
\end{algorithm}
\begin{theorem}
\label{thm:adative-losses}
The above algorithm with $C=2^{3/4}(\sqrt{2}+1)^{1/2}$ guarantees
\[ R_T\leqslant 4\sqrt{\frac{Ts^*\log d}{d}}+\frac{\lceil \log s^* \rceil\log d}{2}+5s^*\sqrt{\frac{\log d}{dT}}. \]
\end{theorem}
\begin{proof}
Let $1\leqslant m\leqslant M$. On time interval $I(m)$, the
Exponential Weight Algorithm is run with parameter $\eta(m)$ against
loss vectors in $[0,1]^d$. Therefore, the following regret bound
derived in the proof of Theorem~\ref{thm:upper bound-losses-full-info}
applies:
\begin{align*}
R(m):=&\sum_{t\in I(m)}^{}\left< \ell_t , x_t \right> -\min_{i\in
[d]}\sum_{t\in I(m)}^{}\ell_t^{(i)}\\
&\leqslant \frac{\log d}{1-e^{-\eta(m)}}+\frac{e^{\eta(m)}-1}{2}\min_{i\in
[d]}\sum_{t\in
I(m)}^{}\ell_t^{(i)}\\
&=\frac{1}{C}\sqrt{\frac{2^mT\log d}{d}}+\frac{\log d}{C}+\frac{C}{2}\sqrt{\frac{d\log d}{2^mT}}\cdot \min_{i\in
[d]}\sum_{t\in
I(m)}^{}\ell_t^{(i)}.
\end{align*}
We now bound the ``best loss'' quantity from above, using the fact
that $\ell_t$ is $2^m$-sparse for $t\in
I(m)\setminus \left\{ \tau(m) \right\}$ and that $\ell_{\tau(m)}$ is $s^*$-sparse:
\begin{align*}
\sum_{i=1}^d \sum_{t\in I(m)}^{}\ell_t^{(i)}&=\sum_{t\in I(m)}^{}
\sum_{i=1}^d\ell_t^{(i)} =\sum_{\substack{t<\tau(m)\\t\in I(m)}}^{} \sum_{i=1}^d\ell_t^{(i)}
+\sum_{i=1}^d \ell_{\tau(m)}^{(i)}\\
&\leqslant (\tau(m)-\tau(m-1))2^m+s^*,
\end{align*}
which implies:
\[ \min_{i\in [d]}\sum_{t\in I(m)}^{}\ell_t^{(i)}\leqslant \frac{(\tau(m)-\tau(m-1))2^m+s^*}{d}. \]
Therefore, the regret on interval $I(m)$, which we will denote $R(m)$,
is bounded by:
\begin{align*}
R(m)&:=\sum_{t\in I(m)}^{}\left< \ell_t , x_t \right> -\min_{i\in
[d]}\sum_{t\in I(m)}^{}\ell_t^{(i)}\\
&\leqslant \frac{1}{C}\sqrt{\frac{2^mT\log
d}{d}}+\frac{\log d}{C}+
\frac{C}{2}\sqrt{\frac{2^m\log d}{dT}}\left(
\tau(m)-\tau(m-1) \right)+\frac{C}{2}\sqrt{\frac{\log d}{2^mdT}}s^*\\
&\leqslant \frac{1}{C}\sqrt{\frac{2^mT\log
d}{d}}+\frac{\log d}{C}+
\frac{C}{2}\sqrt{\frac{2s^*\log d}{dT}}\left(
\tau(m)-\tau(m-1) \right)+\frac{C}{2}\sqrt{\frac{\log d}{2^mdT}}s^*,
\end{align*}
where we used $2^m\leqslant 2^M=2^{\lceil \log_2s^* \rceil}\leqslant
2^{\log_2s^*+1}=2s^*$ for the third term of the last line.
We now turn the whole regret $R_T$ from $1$ to $T$. Since $(I(m))_{1\leqslant
m\leqslant M}$ is a partition of $\left\{ 1,\dots,T \right\}$, we obtain
\begin{align*}
R_T&=\sum_{t=1}^T\left< \ell_t , x_t \right> -\min_{i\in
[d]}\sum_{t=1}^T\ell_t^{(i)}\\
&\leqslant \sum_{m=1}^M \sum_{t\in I(m)}^{}\left< \ell_t , x_t \right> -\sum_{m=1}^M\min_{i\in [d]}\sum_{t\in I(m)}^{}\ell_t^{(i)}\\
&=\sum_{m=1}^MR(m)\\
&\leqslant \frac{1}{C}\sqrt{\frac{T\log d}{d}}\sum_{m=1}^M\sqrt{2^m}+C\sqrt{\frac{s^*T\log d}{2d}}+\frac{M\log d}{C}+\frac{C}{2}\sqrt{\frac{\log d}{dT}}s^*\sum_{m=1}^M2^{-m/2}.
\end{align*}
The sum in the first term above can be bounded as follows
\[ \sum_{m=1}^M\sqrt{2^m}\leqslant \sum_{m=1}^M\sqrt{2}^m=\sqrt{2}\frac{\sqrt{2}^M-1}{\sqrt{2}-1}\leqslant \sqrt{2}\frac{\sqrt{2^{\log_2s^*+1}}}{\sqrt{2}-1}=2\frac{\sqrt{s^*}}{\sqrt{2}-1}=2(\sqrt{2}+1)\sqrt{s^*}, \]
whereas the sum in the last term can be bounded by $\sqrt{2}+1$.
Eventually, the choice $C=2^{3/4}(\sqrt{2}+1)^{1/2}$ give:
\[ R_T\leqslant 2^{5/4}(\sqrt{2}+1)^{1/2}\sqrt{\frac{Ts^*\log
d}{d}}+\frac{\lceil \log s^* \rceil \log d
}{2^{3/4}(\sqrt{2}+1)^{1/2}}+2^{1/4}(\sqrt{2}+1)^{3/2}s^*\sqrt{\frac{\log d}{dT}}, \]
and the statement follows from numerical computation of the constant factors.
\end{proof}
\subsection{For Gains}
\label{sec:gains-1}
The construction is similar to the case of losses, but the time
intervals are slightly different.
Let $(g_t)_{t\geqslant 1}$ be the sequence of gain vectors in
$[0,1]^d$ chosen by Nature.
We assume $s^*\geqslant 2$ and set $M=\lceil \log_2\log_2s^* \rceil$
and $\tau(0)=0$. For $1\leqslant m\leqslant M$ we define
\[ \tau(m)=\min_{}\left\{ 1\leqslant t\leqslant T\,\middle|\,\left\| g_t \right\|_0>2^{2^m} \right\}\quad \text{and}\quad \tau(M)=T. \]
We now define the time intervals $I(m)$. For $1\leqslant m\leqslant M$,
\[ I(m)= \begin{cases}
\left\{ \tau(m-1)+1,\dots,\tau(m) \right\}&\text{if }
\tau(m-1)<\tau(m)\\
\varnothing&\text{if }\tau(m-1)=\tau(m).
\end{cases} \]
Therefore, for $1\leqslant m\leqslant M$ and $t<\tau(m)$, we have $\left\| g_t \right\|_0\leqslant 2^{2^{m}}$.
For $1\leqslant t\leqslant T$, we denote $m_t=\min_{}\left\{ m\geqslant 1\,\middle|\,\tau(m)\geqslant t \right\}$.
Let $C>0$ be a constant to be chosen later and for $1\leqslant
m\leqslant M$, let
\begin{align*}
p(m)&=1+\frac{1}{\log 2\cdot 2^{m+1}-1},\\
q(m)&=\left( 1-\frac{1}{p(m)}\right)^{-1},\\
\eta(m)&=C\sqrt{\frac{p(m)-1}{T2^{2^{m+1}/q(m)}}}.
\end{align*}
As in Section~\ref{sec:gains}, for $p\in (1,2]$, we denote $h_p$ the regularizer on the simplex defined by:
\[ h_p(x)= \begin{cases}
\frac{1}{2}\left\| x \right\|_p^2&\text{if $x\in \Delta_d$}\\
+\infty&\text{otherwise}.
\end{cases}
\]
The algorithm is then defined by:
\[ x_t=\nabla h_{p(m_t)}^*\left( \eta(m_t)\sum_{\substack{t'<t\\t'\in I(m_t)}}^{}g_{t'} \right),\quad t=1,\dots,T. \]
\begin{algorithm}
\SetAlgoLined
\SetKwInOut{Input}{input}
\Input{$T\geqslant 1$, $d\geqslant 1$ integers, and $C>0$.}
$p\leftarrow 1+(4\log 2-1)^{-1}$\;
$q\leftarrow (1-1/p)^{-1}$\;
$\eta\leftarrow C\sqrt{(p-1)/2^{4/q}T}$\;
$m\leftarrow 1$\;
$y\leftarrow (0,\dots,0)\in \mathbb{R}^d$\;
\For{$t\leftarrow 1$ \KwTo $T$}{
draw and play decision $i\sim \nabla h^*_p(\eta\cdot y)$\;
observe gain vector $g_t$\;
\eIf{$\left\| g_t \right\|_0\leqslant 2^{2^{m}}$}{
$y\leftarrow y+g_t$\;
}{
$m\leftarrow \lceil \log_2\log_2\left\| g_t \right\|_0 \rceil$\;
$p\leftarrow 1+(\log 2\cdot 2^{m+1}-1)^{-1}$\;
$q\leftarrow (1-1/p)^{-1}$\;
$\eta\leftarrow C\sqrt{(p-1)/2^{2^{m+1}/q}T}$\;
$y\leftarrow (0,\dots,0)$\;}}
\caption{For gains in full information without prior knowledge about sparsity.}
\end{algorithm}
\begin{theorem}
\label{thm:adaptive-gains}
The above algorithm with $C=(e\sqrt{2}(\sqrt{2}+1))^{1/2}$ guarantees
\[ R_T\leqslant 7\sqrt{T\log s^*}+\frac{4s^*}{\sqrt{T}}. \]
\end{theorem}
\begin{proof}
Let $1\leqslant m\leqslant M$. On time interval $I(m)$, the algorithm boils down to an Online Mirror
Descent algorithm with regularizer $h_{p(m)}$ and parameter
$\eta(m)$. Therefore, using Theorem~\ref{thm:mirror-descent}, the regret on this interval is bounded as follows.
\begin{align*}
R(m)&:=\max_{i\in [d]}\sum_{t\in I(m)}^{}g_t^{(i)}-\sum_{t\in
I(m)}^{}\left< g_t , x_t \right>\\
&\leqslant \frac{1}{2\eta(m)}+\frac{\eta(m)}{2(p(m)-1)}\sum_{t\in
I(m)}^{}\left\| g_t \right\|_{q(m)}^2\\
&=\frac{1}{2\eta(m)}+\frac{\eta(m)}{2(p(m)-1)}\left( \sum_{\substack{t\in I(m)\\t<\tau(m)}}^{}\left\| g_t \right\|_{q(m)}^2+\left\| g_{\tau(m)} \right\|_{q(m)}^2 \right).
\end{align*}
$g_t$ being $2^{2^m}$-sparse for $t<\tau(m)$ and $g_{\tau(m)}$ being
$s^*$-sparse, the $q(m)$-norms can therefore bounded from above as
follows:
\[ \left\| g_t \right\|_{q(m)}^2\leqslant 2^{2^{m+1}/q(m)}\quad \text{and}\quad \left\| g_{\tau(m)} \right\|_{q(m)}^2\leqslant (s^*)^{2/q(m)}. \]
The bound on $R(m)$ then becomes
\begin{align*}
R(m)&\leqslant
\frac{1}{2\eta(m)}+\frac{\eta(m)(\tau(m)-\tau(m-1))2^{2^{m+1}/q(m)}}{2(p(m)-1)}+\frac{\eta(m)(s^*)^{2/q(m)}}{2(p(m)-1)}\\
&=\frac{1}{2C}\sqrt{Te(\log 2\cdot
2^{m+1}-1)}+\frac{C}{2}\sqrt{\frac{e(\log 2\cdot
2^{m+1}-1)}{T}}(\tau(m)-\tau(m-1))\\
&\quad \quad \quad \quad \quad +\frac{C}{2}(s^*)^{1/(\log 2\cdot 2^m)}\sqrt{\frac{e(\log 2\cdot 2^{m+1}-1)}{T}}\\
&\leqslant \frac{1}{2C}\sqrt{Te\log 2\cdot 2^{m+1}}+C\sqrt{\frac{e\log
s^*}{T}}\left( \tau(m)-\tau(m-1) \right)\\
&\quad \quad \quad \quad \quad +\frac{C}{2}s^*\sqrt{\frac{e\log 2\cdot 2^{m+1}}{T}},
\end{align*}
where for the second term of the last expression we used:
\begin{align*}
\log 2\cdot 2^{m+1}-1&\leqslant \log 2\cdot 2^{M+1}=\log 2\cdot \exp \left(
\log 2\left( \lceil \log_2\log_2s^* \rceil
+1\right) \right) \\
&\leqslant \log 2\cdot \exp \left( \log 2\left( \log_2\log_2s^*+2
\right) \right)\\
&=\log 2\cdot e^{2\log 2}\exp \left( \log 2\cdot \log_2\log_2s^*
\right)\\
&= 4\log 2\cdot \exp \left( \log \log_2s^* \right)\\
&=4\log 2\cdot \log_2s^* \\
&=4\log s^*.
\end{align*}
Then, the whole regret $R_T$ is bounded by the sum of the regrets on
each interval:
\begin{align*}
R_T&\leqslant \sum_{m=1}^MR(m)\leqslant \frac{1}{2C}\sqrt{Te\log
2}\sum_{m=1}^M\sqrt{2^{m+1}}+C\sqrt{\frac{e\log
s^*}{T}}\sum_{m=1}^M(\tau(m)-\tau(m-1))\\
&\quad \quad \quad \quad \quad +\frac{Cs^*}{2}\sqrt{\frac{e\log 2}{T}}\sum_{m=1}^M2^{-(m+1)/2}.
\end{align*}
The second sum is equal to $T$ and the third sum is bounded from above
by $(\sqrt{2}+1)/\sqrt{2}$. Let us bound the first sum from above:
\begin{align*}
\sqrt{\log 2}\sum_{m=1}^M\sqrt{2^{m+1}}&=2\sqrt{\log
2}\frac{2^{M/2}-1}{\sqrt{2}-1}\\
&\leqslant 2(\sqrt{2}+1)\sqrt{\log 2}\cdot \exp \left(
\frac{\log 2}{2}\left(
\log_2\log_2s^*+1 \right)
\right) \\
&=2(\sqrt{2}+1)\sqrt{\log 2}\cdot \sqrt{2e^{\log \log_2s^*}}\\
&=2\sqrt{2}(\sqrt{2}+1)\sqrt{\log 2 \log_2s^*}\\
&=2\sqrt{2}(\sqrt{2}+1)\sqrt{\log s^*}.
\end{align*}
Therefore,
\[ R_T\leqslant \frac{\sqrt{2}(\sqrt{2}+1)}{C}\sqrt{Te\log s^*}+C\sqrt{Te\log s^*}+\frac{C(\sqrt{2}+1)s^*}{2}\sqrt{\frac{e\log 2}{2T}}. \]
Choosing $C=(e\sqrt{2}(\sqrt{2}+1))^{1/2}$ balance the first two term
and gives:
\begin{align*}
R_T&\leqslant 2(e\sqrt{2}(\sqrt{2}+1))^{1/2}\sqrt{T\log s^*}+2^{-5/4}e\sqrt{\log 2}(\sqrt{2}+1)^{3/2}\frac{s^*}{\sqrt{T}}\\
&\leqslant 7\sqrt{T\log s^*}+\frac{4s^*}{\sqrt{T}}.
\end{align*}
\[ \]
\end{proof}
\section{The Bandit Setting}
\label{sec:bandit}
We now turn to the bandit framework (see for instance \cite{bubeck2012regret}
for a recent survey).
Recall that the minimax regret \cite{audibert2009minimax} in the basic bandit framework
(without sparsity) is of order $\sqrt{Td}$.
In the case of losses, we manage to take advantage of the sparsity
assumption and obtain in Theorem~\ref{thm:bandit-upper bound-losses} an upper bound of order $\sqrt{Ts\log
\frac{d}{s}}$, and an lower bound of order
$\sqrt{Ts}$ in Theorem~\ref{thm:bandit-losses-lowerbound}. This establishes the order of the minimax regret up to a
logarithmic factor. In the case of gains however, the same upper bound
and lower bound techniques do not seem to work; this difficulty is
discussed below in remark \ref{RM:upper_bound_bandit}.
For simplicity, we shall assume that the
sequence of outcome vectors $(\omega_t)_{t\geq 1}$ is chosen before stage 1 by the
environment, which is called \emph{oblivious} in that case. We refer to \cite[Section 3]{bubeck2012regret} for a detailed discussion
on the difference between oblivious and non-oblivious opponent, and
between regret and pseudo-regret.
As before, at stage $t$, the
decision maker chooses $x_t \in
\Delta_d$ and draws decision $d_t \in [d]$ according to $x_t$. The main difference with the previous framework is that the decision maker only observes
his own outcome $\omega_t^{d_t}$ before choosing the next decision $d_{t+1}$.
\subsection{Upper Bounds on the Regret with Sparse Losses}
\label{sec:greedy-online-mirror}
We shall focus in this section on $s$-sparse losses. The algorithm we consider belongs to the family of Greedy Online Mirror
Descent. We follow \cite[Section 5]{bubeck2012regret} and refer to it
for the detailed and rigorous construction. Let $F_q(x)$ be
the Legendre function associated with potential $\psi(x)=(-x)^{-q}$
($q>1$), i.e.,
\[ F_q(x)=-\frac{q}{q-1}\sum_{i=1}^d(x^i)^{1-1/q}. \]
The algorithm, which depends on a parameter $\eta>0$ to be fixed later, is defined as follows. Set
$x_1=(\frac{1}{d},\dots,\frac{1}{d}) \in \Delta_d$. For all $t\geq 1$,
we define the estimator $\hat{\ell}_t$ of $\ell_t$ as usual:
\[ \hat{\ell}_t^{(i)}=\mathbbm{1}_{\left\{ d_t=i\right\} }\frac{\ell_t^{(i)}}{x_t^{(i)}},\quad i \in [d], \]
which is then used to compute
\[ z_{t+1}=\nabla F_q^*(\nabla F_q(x_t)-\eta \hat{\ell}_t)\quad \text{and}\quad x_{t+1}=\operatorname{argmin}_{x\in \Delta_d}D_{F_q}(x,z_{t+1}), \]
where $D_{F_q}: \bar{\mathcal D}\times \mathcal D\to \mathbb{R}$ is the
Bregman divergence associated with $F_q$:
\[ D_{F_q}(x',x)=F_q(x')-F_q(x)-\left< \nabla F_q(x) , x'-x \right>. \]
\begin{theorem}
\label{thm:bandit-upper bound-losses}
Let $\eta>0$ and $q>1$. For all sequence of $s$-sparse loss vectors,
the above strategy with parameter $\eta$ guarantees, for $T\geq
1$:
\[ R_T\leq q\left( \frac{d^{1/q}}{\eta(q-1)}+\frac{\eta Ts^{1-1/q}}{2} \right). \]
In particular, if $d/s\geq e^2$, the choices
\[ \eta=\sqrt{\frac{2d^{1/q}}{(q-1)Ts^{1-1/q}}}\quad \text{and}\quad
q=\log (d/s) \]
the following regret bound:
\[ R_T\leq 2\sqrt{e}\sqrt{Ts\log \frac{d}{s}}. \]
\end{theorem}
\begin{proof}
\cite[Theorem 5.10]{bubeck2012regret} gives:
\[ R_T\leq \frac{\max_{x\in \Delta_d}F(x)-F(x_1)}{\eta}+\frac{\eta}{2}\sum_{t=1}^T\sum_{i=1}^{d}\mathbb{E}\left[ \frac{(\hat{\ell}_t^{(i)})^2}{(\psi^{-1})'(x_t^{(i)})} \right], \]
with $(\psi^{-1})'(x)=(q\, x^{1+1/q})^{-1}$.
Let us bound the first term.
\[ \frac{1}{\eta}\max_{x\in \Delta_d}F_q(x)-F_q(x_1)\leq \frac{1}{\eta}\frac{q}{q-1}\left(0+d\left( 1/d \right)^{1-1/q} \right)=\frac{qd^{1/q}}{\eta(q-1)}. \]
We turn to the second term. Let $1\leq t\leq T$.
\begin{align*}
\sum_{i=1}^{d}\mathbb{E}\left[
\frac{(\hat{\ell}_t^{(i)})^2}{(\psi^{-1})'(x_t^{(i)})}
\right]&=q\sum_{i=1}^d\mathbb{E}\left[
(\hat{\ell}_t^{(i)})^2(x_t^{(i)})^{1+1/q}
\right]\\
&=q\sum_{i=1}^d\mathbb{E}\left[ \mathbb{E}\left[
\mathbbm{1}_{\left\{ d_t=i \right\}
}\frac{(\ell_t^{(i)})^2}{(x_t^i)^2}(x_t^i)^{1+1/q}\middle|x_t\right]
\right]\\
&=q\sum_{i=1}^d\mathbb{E}\left[ (\ell_t^{(i)})^2(x_k^{(i)})^{1/q}
\right]\\
& = q\, \mathbb{E}\left[ \sum_{\text{$s$ terms}}^{}(\ell_t^{(i)})^2(x_t^{(i)})^{1/q}\right] \\
&\leq qs(1/s)^{1/q}=qs^{1-1/q},
\end{align*}
where we used the assumption that $\ell_t$ has at most $s$ nonzero
components, and the fact that $x_t\in \Delta_d$. The first regret
bound is thus proven.
By choosing $\eta=\sqrt{\frac{2s^{1-1/q}}{(q-1)Td^{1/q}}}$, we balance both terms and get:
\[ R_T\leq 2q\sqrt{\frac{Td^{1/q}s^{1-1/q}}{2(q-1)}}=\sqrt{2q}\sqrt{Ts\left(\frac{d}{s}\right)^{1/q}\left( \frac{q}{q-1} \right) }\enspace. \]
If $d/s\geq e^2$ and $q=\log (d/s)$, then $q/(q-1)\leq 2$
and finally:
\[ R_T\leq 2\sqrt{e}\sqrt{Ts\log \frac{d}{s}}. \]
\end{proof}
\begin{remark}\label{RM:upper_bound_bandit}
The previous analysis cannot be carried in the case of gains because
the bound from \cite[Theorem 5.10]{bubeck2012regret} that we use above only holds for
nonnegative losses (and its proof strongly relies on this
assumption). We are unaware of techniques which could provide a similar
bound in the case of nonnegative gains.
\end{remark}
\subsection{Matching Lower Bound}
The following theorem establishes that the bound from
Theorem~\ref{thm:bandit-upper bound-losses}
is optimal up to a
logarithmic factor.
We denote $\hat{v}_T^{\ell,s,d}$ the minimax regret
in the bandit setting with losses.
\begin{theorem}
\label{thm:bandit-losses-lowerbound}
For all $d\geqslant 2$, $s\in [d]$ and $T\geqslant d^2/4s$, the
following lower bound holds:
\[ \hat{v}_T^{\ell,s,d}\geqslant \frac{1}{32}\sqrt{Ts}. \]
\end{theorem}
The intuition behind the proof is the following. Let us consider the case where $s=1$ and assume that $\omega_t$ is a unit vector $e_{i_t} = (\mathbbm{1}\{j=i_t\})_j$ where $ \mathbb{P}(i_t=i)\simeq (1-\varepsilon)/d$ for all $i \in [d]$, except one fixed coordinate $i^*$ where $ \mathbb{P}(i_t=i^*)\simeq 1/d+\varepsilon$.
Since $1/d$ goes to 0 as $d$ increases, the Kullback-Leibler divergence between two Bernoulli of parameters $(1-\varepsilon)/d$ and $1/d+\varepsilon$ is of order $d\varepsilon^2$. As a consequence, it would require approximately $1/d\varepsilon^2$ samples to distinguish between the two. The standard argument that one of the coordinates has not been chosen more than $T/d$ times, yields that one should take $1/d\varepsilon^2 \simeq T/d$ so that the regret is of order $T\varepsilon$. This provides a lower bound of order $\sqrt{T}$. Similar arguments with $s>1$ give a lower bound of order $\sqrt{sT}$.
We emphasize that one cannot simply assume that the $s$
components with positive losses are chosen at the beginning once for
all, and apply standard lower bound techniques. Indeed, with
this additional information, the decision maker just has to choose, at
each stage, a decision associated with a zero loss. His regret
would then be uniformly bounded (or even possibly equal to zero).
\subsection{Proof of Theorem~\ref{thm:bandit-losses-lowerbound}}
Let $d\geqslant 1$, $1\leqslant s\leqslant d$, $T\geqslant 1$, and $\varepsilon\in (0,s/2d)$. Denote $\mathfrak{P}_s([d])$
the set of subsets of $[d]$ of cardinality $s$, $\delta_{ij}$ the
Kronecker symbol, and $B(1,p)$ the Bernoulli distribution of parameter
$p\in [0,1]$. If $P,Q$ are two probability distributions on the same
set, $D_{}\left( P \,\middle|\!\middle|\, Q \right)$ will denote the
relative entropy of $P$ and $Q$.
\subsubsection{Random $s$-sparse loss vectors $\ell_t$ and $\ell'_t$}
\label{sec:random-loss-vectors}
For $t\geqslant 1$, define the random $s$-sparse loss vectors $(\ell_t)_{t\geqslant 1}$ as follows.
Draw $Z$ uniformly from $[d]$. We will denote $\mathbb{P}_i\left[ \,\cdot\,
\right]=\mathbb{P}\left[ \,\cdot\,\,\middle|\,Z=i \right]$ and
$\mathbb{E}_i\left[ \,\cdot\, \right]=\mathbb{E}\left[
\,\cdot\,\,\middle|\,Z=i \right]$.
Knowing $Z=i$, the random vectors $\ell_t$ are
i.i.d and defined as follows. Draw $I_t$ uniformly from $\mathfrak{P}_s([d])$.
If $j\in I_t$, define $\ell_t^{(j)}$ such that:
\[ \mathbb{P}_i\left[ \ell_t^{(j)}=1 \right]=1-\mathbb{P}_i\left[
\ell_t^{(j)}=0 \right]=\frac{1}{2}-\frac{\varepsilon
d}{s}\delta_{ij}. \]
If $j\not \in I_t$, set $\ell_t^{(j)}=0$. Therefore, one can check that for each component $j\in [d]$ and
all $t\geqslant 1$,
\[ \mathbb{E}_i\left[ \ell_t^{(j)} \right]=\frac{s}{2d}-\varepsilon \delta_{ij}. \]
For $t\geqslant 1$, define the i.i.d. random $s$-sparse loss vectors $(\ell'_t)_{t\geqslant 1}$ as follows.
Draw $I'_t$ uniformly from $\mathfrak{P}_s([d])$. Then if $j\in I'_t$, set $(\ell'_t)^{(j)}$ such that:
\[ \mathbb{P}\left[ (\ell'_t)^{(j)}=1 \right]=\mathbb{P}\left[ (\ell'_t)^{(j)}=0 \right]=1/2. \]
And if $j\not \in I'_t$, set $(\ell'_t)^{(j)}=0$. Therefore, one can check that for each component $j\in [d]$ and
all $t\geqslant 1$,
\[ \mathbb{E}_i\left[ (\ell'_t)^{(j)} \right]=\frac{s}{2d}. \]
By construction, $\ell_t$ and $\ell_t'$ are indeed random $s$-sparse loss vectors.
\subsubsection{A deterministic strategy $\sigma$ for the player}
\label{sec:determ-strat-sigma}
We assume given a deterministic strategy
$\sigma=(\sigma_t)_{t\geqslant 1}$ for the player:
\[ \sigma_t:([d]\times [0,1])^{t-1}\longrightarrow [d]. \]
Therefore,
\[ d_t=\sigma_t(d_1,\omega_1^{(d_1)},\dots,d_{t-1},\omega_{t-1}^{(d_{t-1})}), \]
where $d_t$ denotes the decision chosen by the strategy at stage $t$ and
$\omega_t$ the outcome vector of stage $t$. But since $d_t$ is determined by
previous decisions and outcomes, we can consider that $\sigma_t$ only
depends on the received outcomes:
\[ \sigma_t:[0,1]^{t-1}\longrightarrow [d], \]
\[ d_t=\sigma_t(\omega_1^{(d_1)},\dots,\omega_{t-1}^{(d_{t-1})}). \]
We define $d_t$ and $d'_t$ to be the (random) decisions played by
deterministic strategy $\sigma$ against the random loss vectors $(\ell_t)_{t\geqslant 1}$ and
$(\ell'_t)_{t\geqslant 1}$ respectively:
\begin{align*}
d_t&=\sigma_t(\ell_1^{(d_1)},\dots,\ell_{t-1}^{(d_{t-1})}),\\
d'_t&=\sigma_t((\ell'_1)^{(d'_1)},\dots,(\ell'_{t-1})^{(d'_{t-1})}).
\end{align*}
For $t\geqslant 1$ and $i\in [d]$, define $A_t^{(i)}$ to be the set of sequences of
outcomes in $\left\{ 0,1 \right\}$ of the first $t-1$ stages for
which strategy $\sigma$ plays decision $i$ at stage $t$:
\[ A_t^{(i)}=\left\{ (u_1,\dots,u_{t-1})\in \left\{ 0,1 \right\}^{t-1}\,\middle|\,\sigma_t(u_1,\dots,u_{t-1})=i \right\}, \]
and $B_t^{(i)}$ the complement:
\[ B_t^{(i)}=\left\{0,1\right\}^{t-1}\setminus A_t^{(i)}. \]
Note that for a given $t\geqslant 1$, $(A_t^{(i)})_{i\in [d]}$ is a
partition of $\left\{ 0,1 \right\}^{t-1}$ (with possibly some empty sets).
For $i\in [d]$, define $\tau_i(T)$ (resp. $\tau'_i(T)$) to be the number of
times decision $i$ is played by strategy $\sigma$ against loss vectors $(\ell_t)_{t\geqslant 1}$
(resp. against $(\ell'_t)_{t\geqslant 1}$) between stages $1$ and $T$:
\[ \tau_i(T)=\sum_{t=1}^T\mathbbm{1}_{\left\{ d_t=i \right\} }\quad \text{and}\quad \tau'_i(T)=\sum_{t=1}^T\mathbbm{1}_{\left\{ d'_t=i \right\} }. \]
\subsubsection{The probability distributions $\mathbb Q$ and $\mathbb Q_i$
($i\in [d]$) on binary sequences}
\label{sec:prob-distr-mathbb}
We consider binary sequences $\vec{u}=(u_1,\dots,u_T)\in \left\{ 0,1 \right\}^T$.
We define $\mathbb Q$ and $\mathbb Q_i$ ($i\in [d]$) to be probability
distributions on $\left\{ 0,1 \right\}^T$ as follows:
\begin{align*}
\mathbb Q_i\left[ \vec{u} \right]&=\mathbb{P}_i\left[
\ell_1^{(d_1)}=u_1,\dots,\ell_T^{(d_T)}=u_T
\right],\\
\mathbb Q\left[ \vec{u} \right]&=\mathbb{P}\left[ (\ell'_1)^{(d'_1)}=u_1,\dots,(\ell'_T)^{(d'_T)}=u_T \right].
\end{align*}
Fix $(u_1,\dots,u_{t-1})\in \left\{ 0,1 \right\}^t$. The applications
\[ u_t\longmapsto \mathbb Q\left[ u_t\,\middle|\,u_1,\dots,u_{t-1} \right]\quad \text{and}\quad u_t\longmapsto \mathbb Q_i\left[ u_t\,\middle|\,u_1,\dots,u_{t-1} \right], \]
are probability distributions on $\left\{ 0,1 \right\}$, which we now
aim at identifying. The first one is Bernoulli of
parameter $s/2d$. Indeed,
\begin{align*}
\mathbb Q\left[ 1\,\middle|\,u_1,\dots,u_{t-1} \right]&=\mathbb{P}\left[ (\ell'_t)^{(d'_t)}=1\,\middle|\,(\ell'_1)^{(d'_1)}=u_1,\dots,(\ell'_{t-1})^{(d'_{t-1})}=u_{t-1} \right]\\
&=\mathbb{P}\left[ (\ell'_t)^{(d'_t)}=1 \right]\\
&=\mathbb{P}\left[ d'_t\in I'_t \right]\mathbb{P}\left[ (\ell'_t)^{(d_t)}=1\,\middle|\,d'_t\in I'_t \right]\\
&=\frac{s}{d}\times \frac{1}{2}\\
&=\frac{s}{2d},
\end{align*}
where we used the independence of the random vectors $(\ell'_t)_{t\geqslant 1}$ for
the second inequality. We now turn to the second distribution, which
depends on $(u_1,\dots,u_{t-1})$. If $(u_1,\dots,u_{t-1})\in A_t^{(i)}$,
it is a Bernoulli of parameter $s/2d-\varepsilon$:
\begin{align*}
\mathbb Q_i\left[ 1\,\middle|\,u_1,\dots,u_{t-1} \right]&=\mathbb{P}_i\left[ \ell_t^{(d_t)}=1\,\middle|\,\ell_1^{(d_1)}=u_1,\dots,\ell_{t-1}^{(d_{t-1})}=u_{t-1}\right]\\
&=\mathbb{P}_i\left[ \ell_t^{(i)}=1\,\middle|\,\ell_1^{(d_1)}=u_1,\dots,\ell_{t-1}^{(d_{t-1})}=u_{t-1} \right]\\
&=\mathbb{P}_i\left[ \ell_t^{(i)}=1 \right]\\
&=\mathbb{P}_i\left[ i\in I_t \right]\mathbb{P}_i\left[ \ell_t^{(i)}=1\,\middle|\,i\in I_t \right]\\
&=\frac{s}{d}\times \left( \frac{1}{2}-\frac{\varepsilon d}{s} \right)\\
&=\frac{s}{2d}-\varepsilon.
\end{align*}
where for the third inequality, we used the assumption that the
random vectors $(\ell_t)_{t\geqslant 1}$ are independent under $\mathbb P_i$, i.e. knowing
$Z=i$. On the other hand, if $(u_1,\dots,u_{t-1})\in B_t^{(i)}$, we can
prove similarly that the distribution is a Bernoulli of parameter $s/2d$.
\subsubsection{Computation the relative entropy of $\mathbb Q_i$ and
$\mathbb Q$}
\label{sec:comp-relat-entr}
We apply iteratively the chain rule to the relative entropy of $\mathbb
Q[\vec{u}]$ and $\mathbb Q_i[\vec{u}]$. Using the short-hand $\mathbb D_i[\,\cdot\,]:=D_{}\left( \mathbb Q[\,\cdot\,] \,\middle|\!\middle|\, \mathbb Q_i[\,\cdot\,] \right)$,
\begin{align*}
D_{}\left( \mathbb Q\left[\vec{u}\right] \,\middle|\!\middle|\,
\mathbb Q_i\left[\vec{u}\right] \right)&=\mathbb{D}_i[\vec{u}]\\
&=\mathbb D_i\left[u_1\right]+\mathbb D_i\left[u_2,\dots,u_T \,\middle|\, u_1\right]\\
&=\mathbb D_i\left[u_1\right]+\mathbb D_i\left[u_2\,\middle|\,u_1\right]+\mathbb D_i\left[u_3,\dots,u_T\,\middle|\,u_1,u_2\right]\\
&=\sum_{t=1}^T\mathbb D_i\left[u_t\,\middle|\,u_1,\dots,u_{t-1}\right].
\end{align*}
We now use the definition of the conditional relative entropy, and
make the previously discussed Bernoulli distributions appear. For $1\leqslant t\leqslant T$,
\begin{align*}
\mathbb D_i&\left[ u_t\,\middle|\,u_1,\dots , u_{t-1}
\right]=\sum_{u_1,\dots,u_{t-1}}^{}\mathbb Q\left[
u_1,\dots,u_{t-1} \right]\\
&\qquad \qquad \qquad \qquad \qquad \qquad \times \sum_{u_t}^{}\mathbb Q\left[ u_t\,\middle|\,u_1,\dots,u_{t-1} \right]\log \frac{\mathbb Q\left[ u_t\,\middle|\,u_1,\dots,u_{t-1} \right] }{\mathbb Q_i\left[ u_t\,\middle|\,u_1,\dots,u_{t-1} \right] }\\
&\quad \quad \quad =\frac{1}{2^{t-1}}\sum_{u_1,\dots,u_{t-1}}^{}\sum_{u_t}^{}\mathbb Q\left[ u_t\,\middle|\,u_1,\dots,u_{t-1} \right]\log \frac{\mathbb Q\left[ u_t\,\middle|\,u_1,\dots,u_{t-1} \right] }{\mathbb Q_i\left[ u_t\,\middle|\,u_1,\dots,u_{t-1} \right] }\\
&\quad \quad \quad =\frac{1}{2^{t-1}}\sum_{(u_1,\dots,u_{t-1})\in A_t^{(i)}}^{}D_{}\left( B\left(1,\frac{s}{2d}\right) \,\middle|\!\middle|\, B\left(1,\frac{s}{2d}-\varepsilon\right) \right)\\
&\quad \quad\quad \quad \quad \quad \quad +\frac{1}{2^{t-1}}\sum_{(u_1,\dots,u_{t-1})\in B_t^{(i)}}^{}D_{}\left( B\left(1,\frac{s}{2d}\right) \,\middle|\!\middle|\, B\left(1,\frac{s}{2d}\right) \right)\\
&\quad \quad \quad =\frac{1}{2^{t-1}}\sum_{(u_1,\dots,u_{t-1})\in A_t^{(i)}}^{} \mathbb B\left( \frac{s}{2d},\varepsilon \right),
\end{align*}
where we used the short-hand $\mathbb B\left( \frac{s}{2d},\varepsilon \right):= D\left( B\left(1,\frac{s}{2d}\right) \,\middle|\!\middle|\, B\left(1,\frac{s}{2d}-\varepsilon\right)\right)$.
Eventually:
\[
D_{}\left( \mathbb Q[\vec{u}] \,\middle|\!\middle|\, \mathbb
Q_i[\vec{u}] \right) =\mathbb B\left(\frac{m}{2d},\varepsilon\right)\sum_{t=1}^T\frac{\left| A_t^{(i)} \right|}{2^{t-1}}. \]
\subsubsection{Upper bound on
$\frac{1}{d}\sum_{i=1}^{d}\mathbb{E}_i\left[ \tau_i(T) \right]$ using
Pinsker's inequality}
\label{sec:upper-bound-frac1ds}
In this step, we will make use of Pinsker's inequality to make the
relative entropy appear.
\begin{proposition}[Pinsker's inequality]
Let $X$ be a finite set, and $P,Q$ probability distributions on $X$. Then,
\[ \frac{1}{2}\sum_{x\in X}^{}\left| P(x)-Q(x) \right| \leqslant \sqrt{\frac{1}{2}D_{}\left( P \,\middle|\!\middle|\, Q \right)}. \]
Immediate consequence:
\[ \sum_{\substack{x\in X\\P(x)>Q(x)}}^{}\left( P(x)-Q(x) \right)\leqslant \sqrt{\frac{1}{2}D_{}\left( P \,\middle|\!\middle|\, Q \right)}. \]
\end{proposition}
Let $i\in [d]$. If $(u_1,\dots,u_T)\in \left\{ 0,1 \right\}^T$ is
given, since the decisions $d_t$ and $d'_t$ are determined by the previous
losses $\ell_t^{(d_t)}$ and $(\ell'_t)^{(d'_t)}$ respectively, we have in particular:
\[ \mathbb{E}_i\left[ \tau_i(T)\,\middle|\,\ell_1^{(d_1)}=u_1,\dots,\ell_T^{(d_T)}=u_T \right]=\mathbb{E}\left[ \tau_i'(T)\,\middle|\,(\ell'_1)^{(d'_1)}=u_1,\dots,(\ell'_T)^{(d'_T)} =u_T\right]. \]
Therefore,
\begin{align*}
\mathbb{E}_i\left[ \tau_i(T) \right]-\mathbb{E}\left[ \tau'_i(T)
\right]&=\sum_{\vec{u}}^{}\mathbb Q_i[\vec{u}]\cdot
\mathbb{E}_i\left[ \tau_i(T)\,\middle|\,\forall t,\
\ell_t^{(d_t)}=u_t \right]\\
&\quad \quad\quad -\sum_{\vec{u}}^{}\mathbb Q[\vec{u}]\cdot \mathbb{E}\left[ \tau'_i(T)\,\middle|\,\forall t,\ (\ell'_t)^{d'_t}=u_t \right]\\
&=\sum_{\vec{u}}^{}\left( \mathbb Q_i[\vec{u}]-\mathbb Q[\vec{u}] \right)\mathbb{E}_i\left[ \tau_i(T)\,\middle|\,\forall t,\ \ell_t^{(d_t)}=u_t \right]\\
&\leqslant \sum_{\substack{\vec{u}\\\mathbb Q_i[\vec{u}]>\mathbb Q[\vec{u}]}}^{}\left( \mathbb Q_i[\vec{u}]-\mathbb Q[\vec{u}] \right) \mathbb{E}_i\left[ \tau_i(T)\,\middle|\, \forall t,\ \ell_t^{(d_t)}=u_t \right]\\
&\leqslant T\sum_{\substack{\vec{u}\\\mathbb Q_i[\vec{u}]>\mathbb Q[\vec{u}]}}^{}\left( \mathbb Q_i[\vec{u}]-\mathbb Q[\vec{u}] \right)\\
&\leqslant T\sqrt{\frac{1}{2}D_{}\left( \mathbb Q[\vec{u}] \,\middle|\!\middle|\, \mathbb Q_i[\vec{u}] \right)}\\
&=T\sqrt{\frac{\mathbb B(s/2d,\varepsilon)}{2}}\sqrt{\sum_{t=1}^T\frac{\left| A_t^{(i)} \right| }{2^{t-1}}},\\
\end{align*}
where we used Pinsker's inequality in the fifth line. Moreover, we have:
\[ \frac{1}{d}\sum_{i=1}^d\mathbb{E}\left[ \tau'_i(T) \right]=\frac{1}{d}\mathbb{E}\left[ \sum_{t=1}^T\sum_{i=1}^d\mathbbm{1}_{\left\{ d'_t=i \right\} } \right]=\frac{1}{d}\mathbb{E}\left[ \sum_{t=1}^T1 \right]=\frac{T}{d}. \]
Combining this with the previous inequality gives:
\begin{align*}
\frac{1}{d}\sum_{i=1}^d\mathbb{E}_i\left[ \tau_i(T) \right]&\leqslant \frac{1}{d}\sum_{i=1}^d\mathbb{E}\left[ \tau_i'(T) \right]+T\sqrt{\frac{\mathbb B(s/2d,\varepsilon)}{2}}\frac{1}{d}\sum_{i=1}^d\sqrt{\sum_{t=1}^T\frac{\left| A_t^{(i)} \right| }{2^{t-1}}}\\
&\leqslant \frac{T}{d}+T\sqrt{\frac{\mathbb B(s/2d,\varepsilon)}{2}}\sqrt{\frac{1}{d}\sum_{t=1}^{T}\sum_{i=1}^d\frac{\left| A_t^{(i)} \right| }{2^{t-1}}}\\
&=\frac{T}{d}+T\sqrt{\frac{\mathbb B(s/2d,\varepsilon)}{2}}\sqrt{\frac{1}{d}\sum_{t=1}^T\frac{\left| \left\{ 0,1 \right\}^{t-1} \right| }{2^{t-1}}}\\
&=\frac{T}{d}+T\sqrt{\frac{\mathbb B(s/2d,\varepsilon)}{2}}\sqrt{\frac{T}{d}}\\
&=\frac{T}{d}+T^{3/2}\sqrt{\frac{\mathbb B(s/2d,\varepsilon)}{2d}}.
\end{align*}
where we used Jensen for the second inequality, and for the third
line, we remembered that $(A_t^{(i)})_{i\in [d]}$ is a partition of $\left\{ 0,1 \right\}^{t-1}$.
\subsubsection{An upper bound on $\mathbb{B}(s/2d,\varepsilon)$ for small
enough $\varepsilon$}
\label{sec:an-upper-bound}
We first write $\mathbb{B}(s/2d,\varepsilon)$ explicitely.
\begin{align*}
\mathbb{B}\left( \frac{s}{2d},\varepsilon \right)&=D_{}\left(B(1,s/2d) \,\middle|\!\middle|\, B(1,s/2d-\varepsilon) \right)\\
&=\frac{s}{2d}\log \frac{s/2d}{s/2d-\varepsilon}+\left( 1-\frac{s}{2d}
\right)\log \frac{1-s/2d}{1-s/2d+\varepsilon}\\
&=-\frac{s}{2d}\log \left( 1-\frac{2d\varepsilon}{s} \right)+\left( \frac{s}{2d}-1 \right)\log \left( 1+\frac{\varepsilon}{1-m/2d} \right).
\end{align*}
We now bound the two logarithms from above using respectively the two
following easy inequalities:
\begin{align*}
-\log (1-x)&\leqslant x+x^2,\quad \text{for $x\in [0,1/2]$}\\
-\log (1+x)&\leqslant -x+x^2,\quad \text{for $x\geqslant 0$}.
\end{align*}
This gives:
\begin{align*}
\mathbb{B}\left( \frac{s}{2d},\varepsilon \right)&\leqslant
\frac{s}{2d}\left(
\frac{2d\varepsilon}{s}+\frac{4d^2\varepsilon^2}{s^2}
\right)+\left(
1-\frac{s}{2d}
\right)\left(
-\frac{\varepsilon}{1-s/2d}+\frac{\varepsilon^2}{(1-s/2d)^2}
\right)\\
&=\frac{4d^2\varepsilon^2}{s(2d-s)},
\end{align*}
which holds for $2d\varepsilon/s\leqslant 1/2$, in other words, for $\varepsilon\leqslant s/4d$.
\subsubsection{Lower bound on the expectation of the regret of $\sigma$
against $\ell_t$}
\label{sec:lower-bound-expect}
We can now bound from below the expected regret incurred when playing
$\sigma$ against loss vectors $(\ell_t)_{t\geqslant 1}$. For $\varepsilon\leqslant s/4d$,
\begin{align*}
R_T&= \mathbb{E}\left[ \sum_{t=1}^T\ell_t^{(d_t)}-\min_{j\in
[d]}\sum_{t=1}^T\ell_t^{(j)} \right]\\
&=
\frac{1}{d}\sum_{i=1}^d\mathbb{E}_i\left[
\sum_{t=1}^{T}\ell_t^{(d_t)}-\min_{j\in
[d]}\sum_{t=1}^{T}\ell_t^{(j)}
\right]\\
&\geqslant \frac{1}{d}\sum_{i=1}^d\left( \mathbb{E}_i\left[ \sum_{t=1}^T\ell_t^{(d_t)} \right]-\min_{j\in
[d]}\sum_{t=1}^T\mathbb{E}_i\left[
\ell_t^{(j)} \right] \right)\\
&=\frac{1}{d}\sum_{i=1}^d\left( \mathbb{E}_i\left[ \sum_{t=1}^T\mathbb{E}_i\left[ \ell_t^{(d_t)}\,\middle|\,d_t \right] \right]-T\min_{j\in
[d]}\left(
\frac{s}{2d}-\varepsilon \delta_{ij} \right) \right)\\
&=\frac{1}{d}\sum_{i=1}^d\left( \mathbb{E}_i\left[
\sum_{t=1}^T\left( \frac{s}{2d}-\varepsilon \delta_{id_t} \right)
\right]-T\left( \frac{s}{2d}-\varepsilon \right) \right)\\
&=\frac{1}{d}\sum_{i=1}^d\varepsilon\left( T-\mathbb{E}_i\left[
\tau_i(T) \right] \right)\\
&=\varepsilon\left( T-\frac{1}{d}\sum_i^{}\mathbb{E}_i\left[ \tau_i(T) \right] \right).
\end{align*}
We now use the upper bound derived in Section~\ref{sec:upper-bound-frac1ds}.
\begin{align*}
R_T&\geqslant \varepsilon\left( T-\frac{T}{d}-T^{3/2}\sqrt{\frac{\mathbb
B(s/2d,\varepsilon)}{2d}} \right)\\
&\geqslant \varepsilon\left(
T-\frac{T}{d}-T^{3/2}\varepsilon\sqrt{\frac{2d}{s(2d-s)}} \right)\\
&\geqslant \varepsilon \left( T-\frac{T}{d}-2T^{3/2}\varepsilon\frac{1}{\sqrt{s}}. \right) ,
\end{align*}
where in the penultimate, we used the upper bound on
$\mathbb{B}(s/2d,\varepsilon)$ that we established above, and in the
last line, the fact that $s\leqslant d$. Let $C>0$
and we choose $\varepsilon=C\sqrt{s/T}$. Then, for
$\varepsilon\leqslant s/4d$,
\begin{align*}
R_T&\geqslant \varepsilon T\left( 1-\frac{1}{d}-2\varepsilon\sqrt{\frac{T}{s}}
\right)\\
&=C\sqrt{sT}\left( 1-\frac{1}{d} \right)-2\sqrt{sT}C^2\\
&\geqslant \sqrt{sT}\left( \frac{C}{2}-2C^2 \right),
\end{align*}
where in the last line, we used the assumption $d\geqslant 2$. The
choice $C=1/8$ give:
\[ R_T\geqslant \frac{1}{32}\sqrt{sT}, \]
which holds for $\varepsilon=C\sqrt{s/T}\leqslant s/4d$ i.e. for $T\geqslant d^2/4s$.
The above inequality does not depend on $\sigma$.
As it is a classic that a randomized strategy is equivalent to some random choice of deterministic
strategies, this lower bound holds for any strategy of
the player. In other words, for $T\geqslant d^2/4s$,
\[ \hat{v}_T^{\ell,s,d}\geqslant \frac{1}{32}\sqrt{sT}. \]
\qed
\subsection{Discussion}
\label{sec:discussion}
If the outcomes are not losses but gains, then there is an important
discrepancy between the upper and lower bounds we obtain. Indeed,
obtaining small losses regret bound as in the first displayed equation
of the proof of Theorem \ref{thm:bandit-upper bound-losses} is still
open. An idea for circumventing this issue would be to enforce exploration
by perturbing $x_t$ into $(1-\gamma)x_t + \gamma \mathcal{U}$ where
$\mathcal{U}$ is the uniform distribution over $[d]$, but usual
computations show that the only obtainable upper bounds are of
order of $\sqrt{dT}$. The aforementioned techniques used to bound the
regret from below with losses would also work with gains, which would
give a lower bound of order $\sqrt{sT}$. Therefore, finding the optimal
dependency in the dimension and/or the sparsity level is still an open question in that specific case.
\footnotesize
\bibliographystyle{siam}
| {
"timestamp": "2015-11-30T02:10:45",
"yymm": "1511",
"arxiv_id": "1511.08405",
"language": "en",
"url": "https://arxiv.org/abs/1511.08405",
"abstract": "We demonstrate that, in the classical non-stochastic regret minimization problem with $d$ decisions, gains and losses to be respectively maximized or minimized are fundamentally different. Indeed, by considering the additional sparsity assumption (at each stage, at most $s$ decisions incur a nonzero outcome), we derive optimal regret bounds of different orders. Specifically, with gains, we obtain an optimal regret guarantee after $T$ stages of order $\\sqrt{T\\log s}$, so the classical dependency in the dimension is replaced by the sparsity size. With losses, we provide matching upper and lower bounds of order $\\sqrt{Ts\\log(d)/d}$, which is decreasing in $d$. Eventually, we also study the bandit setting, and obtain an upper bound of order $\\sqrt{Ts\\log (d/s)}$ when outcomes are losses. This bound is proven to be optimal up to the logarithmic factor $\\sqrt{\\log(d/s)}$.",
"subjects": "Machine Learning (cs.LG); Machine Learning (stat.ML)",
"title": "Gains and Losses are Fundamentally Different in Regret Minimization: The Sparse Case",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9888419703960399,
"lm_q2_score": 0.7154239957834733,
"lm_q1q2_score": 0.7074412736591379
} |
https://arxiv.org/abs/2001.10650 | Connection coefficients for ultraspherical polynomials with argument doubling and generalized bispectrality | We start by presenting a generalization of a discrete wave equation that is particularly satisfied by the entries of the matrix coefficients of the refinement equation corresponding to the multiresolution analysis of Alpert. The entries are in fact functions of two discrete variables and they can be expressed in terms of the Legendre polynomials. Next, we generalize these functions to the case of the ultraspherical polynomials and show that these new functions obey two generalized eigenvalue problems in each of the two discrete variables, which constitute a generalized bispectral problem. At the end, we make some connections to other problems. | \section{Introduction}
Let $\{P_n\}_{n=0}^{\infty}$ and $\{Q_n\}_{n=0}^{\infty}$ be two families of orthonormal polynomials whose orthogonality measures are $d\mu$ and $d\nu$, respectively. Then one can see that
\[
P_i(t)=\sum_{j=0}^{i}c_{i,j}Q_j(t),
\]
where the coefficients $c_{i,j}$ can be found in the following way
\[
c_{i,j}=\int P_i(t)Q_j(t)\,d\nu(t).
\]
These coefficients are called connection coefficients and their nonnegativity for some particular cases of the ultraspherical polynomials is useful in the proof of the positivity of a certain $_3F_2$ function, which in turn, based on the work of Gasper and Askey and Gasper, played a significant role in the first proof of the Bieberbach conjecture \cite{AAR}. Also there has been much work proving the nonnegativity of integrals of products of orthogonal polynomials times certain functions which was initiated by Askey in the late 1960's. These studies have been stimulated by the fact that some of those integrals have combinatorial interpretations (see \cite{IKZ13}).
Another instance that we would like to mention is that in some early work leading to the theory of bispectral problems a matrix $S_1$, whose entries are
\[
(S_1)_{i,j}=\int_a^{\Omega} P_i(t)P_j(t)\,d\mu(t)
\]
for some real $a$ and $\Omega$, was considered (for instance, see \cite{Grunbaum83}). The question was to find eigenvectors of $S_1$. Since
$S_1$ is a full matrix, this is not an easy task. However, it was proposed to find a tridiagonal matrix commuting with $S_1$ in order to reduce the original problem to a problem of finding eigenvectors of the tridiagonal matrix, which is an easier and well understood problem. It was shown to be possible to construct such tridiagonal matrices for some families of orthogonal polynomials and this is one of the fundamental ideas in the theory of bispectral problems.
The last instance to bring up here is that in \cite{GM15} the Alpert multiresolution analysis was studied in detail and important in this study was the integral
$$
f_{i,j}=\int_0^1 \hat p_i(t)\hat p_j(2t-1)dt,
$$
where $\hat p_i$ is the orthonormal Legendre polynomial, i.e. $\hat p_j(t)=k_j t^j +\text{lower degree terms}$ with $k_j>0$ and for any two nonnegative integers $k$ and $l$ we have
\[
\int_{-1}^1 \hat p_k(t)\hat p_l(t)dt=\begin{cases}
0, \,\,k\ne l;\\
1,\,\, k=l.
\end{cases}
\]
These coefficients are entries in the refinement equation associated with this multiresolution analysis. The fact that the Legendre polynomials are involved in the above integral allowed the authors in \cite{GM15} to obtain many types of recurrence formulas in $i$ and $j$ including a generalized eigenvalue problem in each of the indices. These two equations together give rise to a bispectral generalized eigenvalue problem.
We begin by discussing a common property of the coefficients in all the above-mentioned cases: they satisfy a generalized 2D discrete wave equation. Then we observe numerically that a damped oscillatory behavior takes place in the case of the ultraspherical generalization of the coefficients $f_{i,j}$. In particular with
$$
f^{(\lambda)}_{i,j}=\int_0^1 \hat p^{(\lambda)}_i(t)\hat p^{(\lambda)}_j(2t-1)(t(1-t))^{\lambda-1/2}dt,
$$
where ${\hat p^{(\lambda)}_i}$ are the orthonormal ultraspherical polynomials and $\lambda>\ -1/2$ we find the asymptotic formula
\begin{equation*}
f^{(\lambda)}_{i,j}=k_j\frac{\cos\left(\pi\left(j + \frac{\lambda}{2} - \frac{i}{2} + \frac{1}{4}\right)\right)}{\sqrt{\pi}i^{\lambda + 1/2}}+O\left(\frac{1}{i^{\lambda+3/2}}\right),
\end{equation*}
where
\begin{equation*}
k_j=\frac{1}{2^{j+1-2\lambda}}\sqrt{\frac{(2\lambda)_j}{j!(\lambda)_j(\lambda+1)_j\lambda\Gamma(2\lambda)}}\Gamma(2j+2\lambda+1)(\lambda+\frac{1}{2})_j
\end{equation*}
which confirms the damped oscillatory behavior. We also derive some related properties and show that $f^{(\lambda)}_{i,j}$ satisfy a bispectral generalized eigenvalue problem of the form
\[
\begin{split}
\tilde A_if^{(\lambda)}_{i,j}=(j+\lambda-\frac{1}{2})(j+\lambda+\frac{1}{2})B_if^{(\lambda)}_{i,j},\\
\hat A_jf^{(\lambda)}_{i,j}=(i+\lambda+\frac{1}{2})(i+\lambda-\frac{1}{2})\hat B_jf^{(\lambda)}_{i,j},
\end{split}
\]
where $\tilde A_i$, $B_i$ are tridiagonal operators or second order linear difference operators acting on $i$ and $\hat A_j$, $\hat B_j$ are tridiagonal operators acting on $j$. Each of the two above-given relations is a generalized eigenvalue problem and the theory of such problems is intimately related to biorthogonal rational functions (for instance, see \cite{GM98}, \cite{IM95}, \cite{SZh00}).
The paper is organized as follows. In Section 2 a vast generalization of the above integral is shown to give rise to a 2D wave equation and solutions to the special case of the above integral are plotted to show the oscillations. In Section 3 the Legendre case above is analyzed and various properties of the coefficients $f_{i,j}$ are derived. One point of this section is to derive the orthogonality property of these coefficients using that they come from special functions. In Section 4 the Legendre polynomials are replaced by the ultraspherical polynomials and their scaled weight.
Here it is shown that the coefficients $f^{(\lambda)}_{i,j}$ satisfy a wave equation and also a bispectral generalized eigenvalue problem. Two proofs are given developing the generalized eigenvalue problem. One is based on the fact that the polynomials satisfy a differential equation and has the flavor of the proof given in \cite{Grunbaum83} and the second follows from the formula for $f_{i,j}^{(\lambda)}$ in terms of a ${}_2 F_1$ hypergeometric function. The two proofs emphasize different aspects of the problem that maybe useful when viewing other orthogonal polynomial systems. In Section 5 connections are made to various other problems.
\section{The 2D discrete wave equation}
Let $\{P_n\}_{n=0}^{\infty}$ and $\{Q_n\}_{n=0}^{\infty}$ be two families of orthonormal polynomials with respect to two probability measures or, equivalently, two families that obey the three-term recurrence relations
\[
a_{n+1}P_{n+1}(t)+b_nP_n(t)+a_nP_{n-1}(t)=tP_n(t), \quad n=0,1,2,\dots
\]
and
\[
c_{n+1}Q_{n+1}(t)+d_nQ_n(t)+c_nQ_{n-1}(t)=tQ_n(t), \quad n=0,1,2,\dots,
\]
where the coefficients $a_n$ and $c_n$ are positive and the coefficients $b_n$ and $d_n$ are real. In particular, the first relations are
\[
a_{1}P_{1}(t)+b_0P_0(t)=tP_0(t), \quad c_{1}Q_{1}(t)+d_0Q_0(t)=tQ_0(t).
\]
Therefore we can set $a_0=c_0=0$ for the coefficients to be defined for $n=0,1,2,\dots$. Since the families are orthonormal with respect to probability measures we know that
\[
P_0=1, \quad Q_0=1,
\]
which are the initial conditions that allow to reconstruct each of the systems from the corresponding recurrence relation. It should be stressed here that by imposing these particular initial conditions we implicitly assume that the corresponding orthogonality measures are probability measures.
In addition, suppose we are given a measure $\sigma$ on $\dR$ with finite moments.
Then, let us consider the coefficients
\begin{equation}\label{Cijg}
u_{i,j}=\int_{\dR} P_i(t)Q_j(\alpha t+\beta) d\sigma(t),
\end{equation}
where $\alpha\ne 0$ and $\beta$ are complex numbers. It turns out that these coefficients constitute a solution of a generalized wave equation on the two dimensional lattice.
\begin{theorem}[cf. Theorem 2.1 from \cite{IKZ13}]\label{GdWaveTH} We have that
\begin{equation}\label{dWaveEqG}
a_{i+1}u_{i+1,j}+b_iu_{i,j}+a_iu_{i-1,j}
=\frac{c_{j+1}}{\alpha}u_{i,j+1}+\frac{d_{j}-\beta}{\alpha}u_{i,j}+\frac{c_j}{\alpha}u_{i,j-1}
\end{equation}
for $i,j=0$, $1$, $2$, \dots.
\end{theorem}
\begin{proof}
From \eqref{Cijg} and the three-term recurrence relations we get that
\begin{align*}
&a_{i+1}u_{i+1,j}+b_iu_{i,j}+a_iu_{i-1,j}\\&=
\int_{\dR}(a_{i+1}P_{i+1}(t)+b_iP_i(t)+a_iP_{i-1}(t))Q_j(\alpha t+\beta) d\sigma(t)&
\\&=
\int_{\dR} tP_i(t)Q_j(\alpha t+\beta) d\sigma(t)\\&
=\frac{1}{\alpha}\int_{\dR}P_i(t)(\alpha t+\beta)Q_j(\alpha t+\beta) d\sigma(t)-\frac{\beta}{\alpha}\int_{\dR}P_i(t)Q_j(\alpha t+\beta) d\sigma(t)\\&=
\frac{1}{\alpha}
\int_{\dR}P_i(t)(c_{j+1}Q_{j+1}(\alpha t+\beta)+d_jQ_j(\alpha t+\beta)+c_jQ_{j-1}(\alpha t+\beta))d\sigma(t)\\&-\frac{\beta}{\alpha}u_{i,j}\\&
=\frac{c_{j+1}}{\alpha}u_{i,j+1}+\frac{d_{j}}{\alpha}u_{i,j}+\frac{c_j}{\alpha}u_{i,j-1}-\frac{\beta}{\alpha}u_{i,j}
\end{align*}
and thus \eqref{dWaveEqG} holds.
\end{proof}
\begin{remark} Given an equation of the form \eqref{dWaveEqG} then due to the Favard theorem the coefficients will uniquely determine the families $\{P_n\}_{n=0}^{\infty}$ and $\{Q_n\}_{n=0}^{\infty}$ of orthonormal polynomials. The measure $\sigma$ is responsible for the initial state when $j=0$ and $j$ can be thought of as a discrete time. Namely, for a solution of the form \eqref{Cijg} to exist they need to satisfy the initial condition
\[
u_{i,0}=\int_{\dR} P_i(t)d\sigma(t),
\]
which means that given initial function $u_{i,0}$ of the discrete space variable $i$, $\sigma$ needs to be found. The latter problem is a generalized moment problem and in this particular case it is equivalent to a Hamburger moment problem.
It is also worth mentioning here that another type of cross-difference equations on $\dZ^2_+$ was recently discussed in \cite{ADvA} and the construction was based on multiple orthogonal polynomials. Type I Legendre-Angelesco multiple orthogonal polynomials also arise in the wavelet construction proposed by Alpert \cite{GIVA}.
\end{remark}
Next, consider a particular case of the above scheme where $P_n$ and $Q_n$ are both orthonormal Legendre polynomials $\hat p_{n}$ and so verify the three-term recurrence relation
\[
\frac{(n+1)}{\sqrt{(2n+1)(2n+3)}}\hat p_{n+1}(t)+\frac{n}{\sqrt{(2n-1)(2n+1)}}\hat p_{n-1}(t)=t\hat p_n(t),
\]
for $n=0, 1, 2, \dots$. Set $\sigma$ to be the Lebesgue measure on the interval $[0,1]$. As a result, the coefficients \eqref{Cijg} take the form
\begin{equation}\label{Cij}
f_{i,j}=\int_{0}^1 \hat p_i(t)\hat p_j(2t-1) dt.
\end{equation}
It is not so hard to see that the polynomials $\hat p_j(2t-1)$ are orthogonal on the interval $[0,1]$ with respect to the Lebesgue measure, consequently
\begin{equation}\label{HalfZ}
f_{i,j}=0, \quad j>i=0, 1, 2, \dots.
\end{equation}
Since the coefficients of the three-term recurrence relation for the Legendre polynomials are explicitly known, the coefficients of equation \eqref{dWaveEqG} become explicit as well. The following Corollary can be found in \cite{GM15}.
\begin{corollary} The function $f_{i,j}$ satisfies,
\begin{equation}\label{dWaveEq}
\begin{split}
\frac{j+1}{\sqrt{(2j+1)(2j+3)}}f_{i,j+1}+f_{i,j}+\frac{j}{\sqrt{(2j-1)(2j+1)}}f_{i,j-1}=\\
=\frac{2(i+1)}{\sqrt{(2i+1)(2i+3)}}f_{i+1,j}+\frac{2i}{\sqrt{(2i-1)(2i+1)}}f_{i-1,j}
\end{split}
\end{equation}
for $i,j=0$, $1$, $2$, \dots.
\end{corollary}
Below is the MATLAB generated graphical representation of some behavior of the solution $f_{i,j}$ to equation \eqref{dWaveEq}, which is a generalization of the discretized wave equation.
\begin{figure}[h!]
\includegraphics[width=\linewidth]{dWaves.jpg}
\caption{This picture demonstrates the moving wave. Here, one can see two graphs of the function $f=f(i)=f_{i,j}$ of the discrete space variable $i$ at the two different discrete times $j=15$ and $j=20$.}
\label{Fig1}
\end{figure}
To sum up, we would like to point out here that the form \eqref{Cijg} of solutions of the discrete wave equations is very useful for understanding the behavior of solutions because there are many asymptotic results for a variety of families of orthogonal polynomials.
\section{Some further analysis of the coefficients $f_{i,j}$}
In this section, we will obtain some properties of the coefficients $f_{i,j}$ based on the intuition and observations developed in \cite{GM15}. In particular, we will rederive and expand upon some orthogonality properties of the coefficients $f_{i,j}$.
We begin with the following statement, which is based on formula \eqref{Cij} and some known properties of the Legendre polynomials.
\begin{theorem}
Let $k$ and $l$ be two nonnegative integer numbers. Then one has
\begin{equation}\label{CijOrth}
\sum_{j=0}^{\infty}f_{k,j}f_{l,j}=\begin{cases}
0, \,\,\text{if $k$ and $l$ are of the same parity but not equal};\\
1,\,\,\text{if} \,\, k=l;\\
(-1)^{\frac{k+l+1}{2}}\frac{k!l!\sqrt{2k+1}\sqrt{2l+1}}{2^{k+l-1}(k-l)(k+l+1)((\frac{k}{2})!)^2((\frac{l-1}{2})!)^2}, \,\,\text{if $k$ and $l$ are of opossite parity}.
\end{cases}
\end{equation}
\end{theorem}
\begin{proof}
Without loss of generality, we can assume that $k\le l$. Next observe that due to \eqref{HalfZ} the left-hand side of formula \eqref{CijOrth} is truncated to
\begin{equation*}
\sum_{j=0}^{\infty}f_{k,j}f_{l,j}=\sum_{j=0}^{k}f_{k,j}f_{l,j},
\end{equation*}
which can be written as
\begin{equation*}
\sum_{j=0}^{k}f_{k,j}f_{l,j}=\sum_{j=0}^{k}\int_{0}^1 \hat p_k(x)\hat p_j(2x-1) dx\int_{0}^1 \hat p_{l}(y)\hat p_j(2y-1) dy.
\end{equation*}
One can rewrite the expression in the following manner
\begin{equation*}
\sum_{j=0}^{k}f_{k,j}f_{l,j}=\int_{0}^1 \hat p_{l}(y) \left(\int_{0}^1 \hat p_k(x)\sum_{j=0}^{k} \hat p_j(2x-1)\hat p_j(2y-1) dx \right) dy.
\end{equation*}
Since the Christoffel-Darboux kernel $\displaystyle{2\sum_{j=0}^{k} \hat p_j(2x-1)\hat p_j(2y-1)}$
is a reproducing kernel, we get
\begin{equation*}
\sum_{j=0}^{k}f_{k,j}f_{l,j}=2\int_{0}^1 \hat p_{k}(y) \hat p_l(y) dy.
\end{equation*}
Next recall that one can explicitly compute the quantity
\[
\int_{0}^1 \hat p_{k}(y) \hat p_l(y) dy
\]
for any nonnegative integers $k$ and $l$. If $k$ and $l$ have the same parity the symmetry properties of the Legendre polynomials allow the above integral to be extended to the full orthonality interval $[-1,1]$ which gives the first two parts of the Theorem. The third case of formula \eqref{CijOrth} is a consequence of \cite[p.173, Art. 91, Example 2]{B59}.
\end{proof}
One can also compute the inner product of vectors $f_{i,j}$ taken the other way.
\begin{theorem}\label{OrthOfF}
Let $k$ and $l$ be two nonnegative integer numbers. Then one has
\begin{equation}\label{CijOrth2}
\sum_{i=0}^{\infty}f_{i,k}f_{i,l}=\begin{cases}
0, \,\,k\ne l;\\
1/2,\,\,\text{if} \,\, k=l.
\end{cases}
\end{equation}
\end{theorem}
\begin{proof}
Let $n$ be a nonnegative integer. Then we can write
\begin{equation*}
\sum_{i=0}^{n}f_{i,k}f_{i,l}=\sum_{i=0}^{n}\int_{0}^1 \hat p_i(x)\hat p_k(2x-1) dx\int_{0}^1 \hat p_{i}(y)\hat p_l(2y-1) dy,
\end{equation*}
which can be rewritten as follows
\begin{equation*}
\sum_{i=0}^{n}f_{i,k}f_{i,l}=\int_{-1}^1 \hat p_{k}(2x-1)\chi_{[0,1]}(x) \left(\sum_{i=0}^{n}\left(\int_{-1}^1 \hat p_l(2y-1)\chi_{[0,1]}(y) \hat p_i(y) dy \right)\hat p_i(x) \right)dx.
\end{equation*}
Since the polynomials $\hat p_i$ form an orthonormal basis in $L_2([-1,1],dt)$ we know that
\[
\sum_{i=0}^{n}\left(\int_{-1}^1 \hat p_l(2y-1)\chi_{[0,1]}(y) \hat p_i(y) dy \right)\hat p_i(x)\xrightarrow{L_2([-1,1],dt)} \hat p_{l}(2x-1)\chi_{[0,1]}(x)
\]
as $n\to\infty$. As a result we arrive at the following relation
\[
\begin{split}
\sum_{i=0}^{\infty}f_{i,k}f_{i,l}=&\int_{-1}^1 \hat p_{k}(2x-1)\chi_{[0,1]}(x)\hat p_{l}(2x-1)\chi_{[0,1]}(x)dx\\
=&\int_{0}^1 \hat p_{k}(2x-1)\hat p_{l}(2x-1)dx=\frac{1}{2}\int_{-1}^1 \hat p_{k}(t)\hat p_{l}(t)dt,
\end{split}
\]
which finally gives \eqref{CijOrth2}.
\end{proof}
As a consequence we can say a bit more about the asymptotic behavior of the coefficients $f_{i,j}$.
\begin{corollary} Let $k$ be a fixed nonnegative integer number. Then
\[
f_{i,k}\longrightarrow 0
\]
as $i\to\infty$.
\end{corollary}
\begin{proof}
The statement immediately follows from the fact that the series
\[
\sum_{i=0}^{\infty}f_{i,k}^2
\]
converges.
\end{proof}
\begin{remark} From \eqref{CijOrth2} one gets that
\[
\sum_{i=0}^{\infty}f_{i,j}^2=1/2
\]
for any nonnegative $j$. This means that the energy of the wave represented by $f=f(i)=f_{i,j}$ is conserved over the discrete time $j$.
\end{remark}
\begin{remark} The fact that $f_{i,k}$ can be represented as a hypergeometric function allows a more precise asymptotic estimate; see formula \eqref{asymi}.
\end{remark}
\section{The case of ultraspherical polynomials}
In this section we will carry over our findings from the case of Legendre polynomials to the case of the family of ultraspherical polynomials which include the Legendre polynomials as a special case.
Recall that for $\lambda>-1/2$ an ultraspherical polynomial $\hat p_n^{(\lambda)}(t)$ is a polynomial of degree $n$ that is the orthonormal polynomial with respect to the measure
\[
(1-t^2)^{\lambda-1/2}dt.
\]
In an analogous way to $f_{i,j}$, let us consider the function of the discrete variables $i$ and $j$
\begin{equation}\label{cijlam}
f^{(\lambda)}_{i,j}=\int_0^1 \hat{p}_i^{(\lambda)}(t)\hat{p}_j^{(\lambda)}(2t-1)(t(1-t))^{\lambda-1/2}dt
\end{equation}
and notice that
\[
f_{i,j}=f_{i,j}^{(1/2)}.
\]
While this generalization allows us to consider a more general case, the connection to multiresolution analysis seems to be lost due to the weight and there is no evident relation to multiresolution analysis for arbitrary $\lambda>-1/2$. Still, such a deformation of the coefficients $f_{i,j}$ gives an insight on how all these objects are connected to various problems some of which were mentioned in the introduction.
Also, it is worth mentioning that the polynomials $\hat{p}_j^{(\lambda)}(2t-1)$ are orthogonal with respect to the measure
\[
(t(1-t))^{\lambda-1/2}dt.
\]
Next since the ultraspherical polynomials satisfy the three-term recurrence relation \cite{Szego}
\[
\frac{1}{2}\sqrt{\frac{(n+1)(n+2\lambda)}{(n+\lambda)(n+\lambda+1)}}\hat{p}_{n+1}^{(\lambda)}(t)+
\frac{1}{2}\sqrt{\frac{n(n+2\lambda-1)}{(n+\lambda-1)(n+\lambda)}}\hat{p}_{n-1}^{(\lambda)}(t)=t\hat{p}_{n}^{(\lambda)}(t)
\]
the following corollary of Theorem \ref{GdWaveTH} is immediate.
\begin{corollary} The function $f^{(\lambda)}_{i,j}$ satisfies
\begin{equation}\label{dWaveUP}
\begin{split}
\frac{1}{2}\sqrt{\frac{(j+1)(j+2\lambda)}{(j+\lambda)(j+\lambda+1)}}f_{i,j+1}^{(\lambda)}+f_{i,j}^{(\lambda)}+
\frac{1}{2}\sqrt{\frac{j(j+2\lambda-1)}{(j+\lambda-1)(j+\lambda)}}f_{i,j-1}^{(\lambda)}=\\
=\sqrt{\frac{(i+1)(i+2\lambda)}{(i+\lambda)(i+\lambda+1)}}f_{i+1,j}^{(\lambda)}+
\sqrt{\frac{i(i+2\lambda-1)}{(i+\lambda-1)(i+\lambda)}}f_{i-1,j}^{(\lambda)}
\end{split}
\end{equation}
for $i,j=0$, $1$, $2$, \dots.
\end{corollary}
As one can see from the above statement, the function $f^{(\lambda)}_{i,j}$ is a solution of a discrete wave equation and Figure \ref{Fig2} demonstrates how the function changes with $\lambda$ when $j$ is fixed.
\begin{figure}[h!]
\includegraphics[width=\linewidth]{LambdaEvol.jpg}
\caption{This picture shows the $\lambda$-evolution of the function $f^{(\lambda)}=f^{(\lambda)}(i)=f_{i,j}^{(\lambda)}$ of the discrete space variable $i$ when the discrete time $j$ is fixed and $j=15$.}
\label{Fig2}
\end{figure}
It is not so hard to see that it is possible to generalize \eqref{CijOrth} and \eqref{CijOrth2} to the case of the ultraspherical polynomials.
\begin{theorem}
Let $k$ and $l$ be two nonnegative integer numbers. Then one has
\begin{equation}\label{CijOrthU}
\sum_{j=0}^{\infty}f_{k,j}^{(\lambda)}f_{l,j}^{(\lambda)}=\frac{1}{2^{2\lambda}}\int_{0}^1 \hat p_{k}^{(\lambda)}(y) \hat p_l^{(\lambda)}(y) (y(1-y))^{\lambda-\frac{1}{2}}dy
\end{equation}
for any $\lambda>-1/2$ and
\begin{equation}\label{CijOrth2U}
\sum_{i=0}^{\infty}f_{i,k}^{(\lambda)}f_{i,l}^{(\lambda)}=\int_{0}^1 \hat p_{k}^{(\lambda)}(2x-1)\hat p_{l}^{(\lambda)}(2x-1)x^{2\lambda-1}\left(\frac{1-x}{1+x}\right)^{\lambda-\frac{1}{2}}dx
\end{equation}
provided that $\lambda>0$.
\end{theorem}
\begin{proof}
As before we can assume that $k\le l$ therefore,
\small
\begin{multline*}
\sum_{j=0}^{\infty}f_{k,j}^{(\lambda)}f_{l,j}^{(\lambda)}=\sum_{j=0}^{k}f_{k,j}^{(\lambda)}f_{l,j}^{(\lambda)}=\\
\int_{0}^1 \hat p_{l}^{(\lambda)}(y) \left(\int_{0}^1 \hat p_k^{(\lambda)}(x)\sum_{j=0}^{k} \hat p_j^{(\lambda)}(2x-1)\hat p_j^{(\lambda)}(2y-1) (x(1-x))^{\lambda-\frac{1}{2}}dx \right) (y(1-y))^{\lambda-\frac{1}{2}}dy.
\end{multline*}
\normalsize
Since the Christoffel-Darboux kernel
\[
2^{2\lambda}\sum_{j=0}^{k} \hat p_j^{(\lambda)}(2x-1)\hat p_j^{(\lambda)}(2y-1)
\]
is a reproducing kernel in the corresponding $L_2$-space, we get
\begin{equation*}
\sum_{j=0}^{\infty}f_{k,j}^{(\lambda)}f_{l,j}^{(\lambda)}=\frac{1}{2^{2\lambda}}\int_{0}^1 \hat p_{k}^{(\lambda)}(y) \hat p_l^{(\lambda)}(y) (y(1-y))^{\lambda-\frac{1}{2}}dy.
\end{equation*}
To prove the second equality, consider the following representation of the finite sum
\[
\sum_{i=0}^{n}f_{i,k}^{(\lambda)}f_{i,l}^{(\lambda)}=
\int_{-1}^{1}\hat p_{k}^{(\lambda)}(2x-1)\chi_{[0,1]}(x)\frac{x^{\lambda-1/2}}{(1+x)^{\lambda-1/2}}P_n(x)(1-x^2)^{\lambda-1/2}dx,
\]
where
\[
P_n(x)=\sum_{i=0}^{n}\int_{-1}^{1}\left(\hat p_{l}^{(\lambda)}(2y-1)\chi_{[0,1]}(y))\frac{y^{\lambda-1/2}}{(1+y)^{\lambda-1/2}}p_{i}^{(\lambda)}(y)(1-y^2)^{\lambda-1/2}dy\right) \hat p_{i}^{(\lambda)}(x).
\]
If $\lambda>0$ then
\[
P_n(x)\xrightarrow{L_2([-1,1],(1-x^2)^{\lambda-1/2}dx)}\hat p_{l}^{(\lambda)}(2x-1)\chi_{[0,1]}(x)\frac{x^{\lambda-1/2}}{(1+x)^{\lambda-1/2}}
\]
as $n\to\infty$. Next since the functional
\[
F(g)=\int_{-1}^{1}\hat p_{k}^{(\lambda)}(2x-1)\chi_{[0,1]}(x)\frac{x^{\lambda-1/2}}{(1+x)^{\lambda-1/2}}g(x)(1-x^2)^{\lambda-1/2}dx,
\]
is continuous for $\lambda>0$ we arrive at the following
\[
\sum_{i=0}^{\infty}f_{i,k}^{(\lambda)}f_{i,l}^{(\lambda)}=\int_{0}^1 \hat p_{k}^{(\lambda)}(2x-1))\hat p_{l}^{(\lambda)}(2x-1)x^{2\lambda-1}\left(\frac{1-x}{1+x}\right)^{\lambda-\frac{1}{2}}dx
\]
which completes the proof.
\end{proof}
\begin{remark} The first integral in the above Theorem can be evaluated with the use of the equations~(4.7.30) in \cite{Szego}. With
$$
I^1_{k,l}=\frac{1}{2^{\lambda}}k_k k_l I^2_{k,l},
$$
where
$$
k_l=2^l\sqrt{\frac{(\lambda)_l(\lambda+1)_l}{l!(2\lambda)_l}},
$$
and
$$
I^2_{k,l}=\int_0^1 p_k^{\lambda}(y) p_i^{\lambda}(y)(y(1-y))^{\lambda-1/2}dy.
$$
With the use of the formulas alluded to above in \cite{Szego} we find
\begin{align*}
I^2_{2k,2l}&=(-1)^{k+l}\frac{(1/2)_k(1/2)_l\Gamma(\lambda+\frac{1}{2})^2}{(k+\lambda)_k(l+\lambda)_l\Gamma(2\lambda+1)}\\&\sum_{j=0}^k{\frac{(-k)_j(k+\lambda)_j(\lambda+1/2)_{2j}}{(1)_j(1/2)_j(2\lambda+1)_{2j}}\hypergeom43{-l,l+\lambda,j+\frac{\lambda}{2}+\frac{1}{4},j+\frac{\lambda}{2}+\frac{3}{4}}{\frac{1}{2},j+\lambda+1,j+\lambda+\frac{1}{2}}{1}},
\end{align*}
\begin{align*}
I^2_{2k,2l+1}&=(-1)^{k+l}\frac{(1/2)_k(3/2)_l\Gamma(\lambda+\frac{1}{2})\Gamma(\lambda+\frac{3}{2})}{(k+\lambda)_k(l+\lambda+1)_l\Gamma(2\lambda+2)}\\&\sum_{j=0}^k{\frac{(-k)_j(k+\lambda)_j(\lambda+3/2)_{2j}}{(1)_j(1/2)_j(2\lambda+2)_{2j}}\hypergeom43{-l,l+\lambda+1,j+\frac{\lambda}{2}+\frac{3}{4},j+\frac{\lambda}{2}+\frac{5}{4}}{\frac{3}{2},j+\lambda+1,j+\lambda+\frac{3}{2}}{1}},
\end{align*}
and
\small
\begin{align*}
I^2_{2k+1,2l+1}&=(-1)^{k+l}\frac{(3/2)_k(3/2)_l\Gamma(\lambda+\frac{1}{2})\Gamma(\lambda+\frac{5}{2})}{(k+\lambda+1)_k(l+\lambda+1)_l\Gamma(2\lambda+3)}\\&\sum_{j=0}^k{\frac{(-k)_j(k+\lambda+1)_j(\lambda+5/2)_{2j}}{(1)_j(3/2)_j(2\lambda+3)_{2j}}\hypergeom43{-l,l+\lambda+1,j+\frac{\lambda}{2}+\frac{5}{4},j+\frac{\lambda}{2}+\frac{7}{4}}{\frac{3}{2},j+\lambda+2,j+\lambda+\frac{3}{2}}{1}}.
\end{align*}
\normalsize
Note that all of the above hypergeometric functions are balanced. Furthermore for $\lambda=1/2$ one of the terms in the numerator cancels a denoninator term so they all become balanced ${}_3F_2$'s and can be summed using the Pfaff-Saalschiitz formula. The remaining sums in turn reduce to the Legendre case discussed earlier.
At this point we are unable to determine whether for certain values of $\lambda$ the above sums simplify or there is any orthogonality as in the Legendre case. Another interesting problem is the asymptotics of the above sums.
\end{remark}
A formula for the second integral in the above Theorem maybe obtained using equation~(4.7.6) (first formula) in \cite{Szego} and is
\begin{align*}
&\int_{0}^1 \hat p_{k}^{(\lambda)}(2x-1)\hat p_{l}^{(\lambda)}(2x-1)x^{2\lambda-1}\left(\frac{1-x}{1+x}\right)^{\lambda-\frac{1}{2}}dx\\&=(-1)^{k+l}k^{\lambda}_kk^{\lambda}_l\frac{(\lambda+\frac{1}{2})_k(\lambda+\frac{1}{2})_l}{(k+2\lambda)_k(l+2\lambda)_l}\Gamma(\lambda+1/2)\\&\times\sum_{j=0}^i\sum_{n=0}^l\frac{(-k)_j(k+2\lambda)_j(-i)_n(i+2\lambda)_n\Gamma(j+n+2\lambda)}{(1)_j(\lambda+1/2)_j(1)_n(\lambda+1/2)_n\Gamma(j+n+3\lambda+1/2)}\\&\times\hypergeom21{\lambda-1/2,j+n+2\lambda}{j+n+3\lambda+1/2}{-1}.
\end{align*}
The next step is to obtain a generalized eigenvalue problem which will be a 1D-relation for the function $f_{i,j}^{(\lambda)}$ unlike \eqref{dWaveUP}. Our first approach uses the fact that the ultraspherical polynomials satisfy second order differential equations and apparently the approach can be generalized to the case of polynomials satisfying differential equations such as Krall polynomials, Koornwinder's generalized Jacobi polynomials and some Sobolev orthogonal polynomials.
\begin{theorem}\label{UGEPth}
Let $j$ be a fixed nonnegative integer number. Then the function $f=f(i)=f^{(\lambda)}_{i,j}$ of the discrete variable $i$ satisfies the generalized eigenvalue problem
\begin{multline}\label{recurge}
2((i+\lambda)^2-1/4)(i+\lambda+\frac{3}{2})\sqrt{\frac{i+2\lambda}{(i+1)(i+\lambda+1)(\lambda+i)}}f_{i+1,j}^{(\lambda)}+\\
+2((i+\lambda)^2-1/4)(i+\lambda-\frac{3}{2})\sqrt{\frac{i}{(i-1+\lambda)(i-1+2\lambda)(\lambda+i)}}f_{i-1,j}^{(\lambda)}=\\
(j+\lambda-\frac{1}{2})(j+\lambda+\frac{1}{2})\Big[2(i+\lambda-1/2)\sqrt{\frac{i+2\lambda}{(i+1)(i+\lambda+1)(\lambda+i)}}f_{i+1,j}^{(\lambda)}+4f_{i,j}^{(\lambda)}+\\
2(i+\lambda+1/2)\sqrt{\frac{i}{(i-1+\lambda)(i-1+2\lambda)(\lambda+i)}}f_{i-1,j}^{(\lambda)}\Big],
\end{multline}
for $i=0$, $1$, $2$, \dots and, here, the number $(j+\lambda-\frac{1}{2})(j+\lambda+\frac{1}{2})$ is the corresponding generalized eigenvalue.
\end{theorem}
\begin{remark} For the case $\lambda=1/2$, formula \eqref{recurge} was obtained in \cite{GM15}.
\end{remark}
\begin{proof}
To make all the formulas shorter and, more importantly transparent, let us introduce the following operators
\begin{align}\label{tai}
A_i&=2(i+\lambda+\frac{3}{2})\sqrt{\frac{i+2\lambda}{(i+1)(i+\lambda+1)(\lambda+i)}}E_+\nonumber\\&+2(i+\lambda-\frac{3}{2})\sqrt{\frac{i}{(i-1+\lambda)(i-1+2\lambda)(\lambda+i)}}E_-\nonumber\\&=a_{i+1}E_++a_{i-1}E_-
\end{align}
and
\begin{align}\label{bi}
B_i&=4I+2(i+\lambda-1/2)\sqrt{\frac{i+2\lambda}{(i+1)(i+\lambda+1)(\lambda+i)}}E_+\nonumber\\&+2(i+\lambda+1/2)\sqrt{\frac{i}{(i-1+\lambda)(i-1+2\lambda)(\lambda+i)}} E_-\nonumber\\&=4I+b_{i+1}E_++b_{i-1}E_-,
\end{align}
where $I$ is the identity operator and $E_+$, $E_-$ are the forward and backward shift operators on $i$, respectively. With these notations, equation~\eqref{recurge} can be rewritten as
\begin{equation}\label{recurge1}
(i(i+2\lambda)+\lambda^2-1/4)A_if_{i,j}^{(\lambda)}=(j(j+2\lambda)+\lambda^2-1/4)B_if_{i,j}^{(\lambda)}
\end{equation}
or
\begin{equation}\label{recurge2}
i(i+2\lambda)A_if_{i,j}^{(\lambda)}+(\lambda^2-1/4)(A_i-B_i)f_{i,j}^{(\lambda)}=j(j+2\lambda)B_if_{i,j}^{(\lambda)}.
\end{equation}
Notice that
\begin{align}\label{aimbi}
A_i-B_i&=-4I+4\sqrt{\frac{i+2\lambda}{(i+1)(i+\lambda+1)(\lambda+i)}}E_+\nonumber\\&-4\sqrt{\frac{i}{(i-1+\lambda)(i-1+2\lambda)(\lambda+i)}}E_-
\end{align}
As is known \cite{Szego}, the ultraspherical polynomials satisfy the differential equation
\begin{equation}\label{diffe}
\frac{d}{dt}((t(1-t))^{\lambda+1/2}\frac{d}{dt}\hat p_j^{(\lambda)}(2t-1))+j(j+2\lambda)(t(1-t))^{\lambda-1/2}\hat p_j^{(\lambda)}(2t-1)=0.
\end{equation}
Thus after two integration by parts we have
\begin{equation*
\begin{split}
j(j+2\lambda)B_i f_{i,j}^{(\lambda)}=-\int_0^1\frac{d}{dt}((t(1-t))^{\lambda+1/2}\frac{d}{dt}B_i \hat p^{(\lambda)}_i(t)) \hat p^{(\lambda)}_j(2t-1)dt\nonumber\\=-\int_0^1((t(1-t)\frac{d^2}{dt^2}+(\lambda+1/2)(1-2t)\frac{d}{dt})B_i \hat p^{(\lambda)}_i(t))\hat p^{(\lambda)}_j(2t-1)(t(1-t))^{\lambda-1/2}dt\nonumber\\=-\int_0^1((1-t^2)\frac{d^2}{dt^2}-(2\lambda+1)t\frac{d}{dt})B_i \hat p^{(\lambda)}_i(t))\hat p^{\lambda}_j(2t-1)(t(1-t))^{\lambda-1/2}dt\nonumber\\-\int_0^1((t-1)\frac{d^2}{dt^2}+(\lambda+1/2)\frac{d}{dt})B_i \hat p^{(\lambda)}_i(t))\hat p^{\lambda}_j(2t-1)(t(1-t))^{\lambda-1/2}dt.
\end{split}
\end{equation*}
Now
\[
\begin{split}
-((1-t^2)\frac{d^2}{dt^2}-(2\lambda+1)t\frac{d}{dt})B_i \hat p^{(\lambda)}_i(t))=(i+1)(i+1+2\lambda)b_{i+1} \hat p^{(\lambda)}_{i+1}(t)\\+i(i+2\lambda)b_{i} \hat p^{(\lambda)}_i(t)+(i-1)(i-1+2\lambda)b_{i-1} \hat p^{(\lambda)}_{i-1}(t).
\end{split}
\]
Since
$$
(i\pm 1)(i\pm1+2\lambda)2(i+\lambda\mp \frac{1}{2})-i(i+2\lambda)2(i+\lambda\pm \frac{3}{2})
\mp 4(\lambda^2-1/4)=0,
$$
it follows that
\begin{align}\label{recurge4}
&(j(j+2\lambda)B_i-i(i+2\lambda)A_i-(\lambda^2-1/4)(A_i-B_i))f^{(\lambda)}_{i,j}\nonumber\\&=-\int_0^1((t-1)\frac{d^2}{dt^2}+(\lambda+1/2)\frac{d}{dt})B_i \hat p^{(\lambda)}_i(t))\hat p^{(\lambda)}_j(2t-1)(t(1-t))^{\lambda-1/2}dt\nonumber\\&+4(i(i+2\lambda)+\lambda^2-1/4)\int_0^1\hat p^{(\lambda)}_i(t)\hat p^{(\lambda)}_j(2t-1)(t(1-t))^{\lambda-1/2}dt.
\end{align}
We note that
$$
b_{i+1}=4\frac{i+\lambda-1/2}{i+1}a_{i+1}=4(1+\frac{\lambda-3/2}{i+1})a_{i+1}
$$
and
$$
b_{i-1}=4\frac{i+\lambda+1/2}{i+2\lambda-1}a_{i}=4(1-\frac{\lambda-3/2}{i+2\lambda-1})a_{i}.
$$
The substitution of these relations in \eqref{recurge4} leads to the following
\begin{align*}
B_i\hat p^{(\lambda)}_i(t)&=4(1+\frac{\lambda-3/2}{i+1})a_{i+1}\hat p_{i+1}^{(\lambda)}(t)+4(1-\frac{\lambda-3/2}{i+2\lambda-1})a_{i} \hat p_{i-1}^{(\lambda)}(t)+4\hat p_i^{(\lambda)}(t)\\&=
4(1+t+\frac{\lambda-3/2}{i+1}t)\hat p_i^{(\lambda)}(t)-8a_i\frac{(\lambda-3/2)(\lambda+i)}{(i+1)(i+2\lambda-1)}\hat p_{i-1}^{(\lambda)}(t)).
\end{align*}
Using the first equation in \cite[equation~(4.7.28)]{Szego} gives
$$
\frac{d}{dt}\hat p_{i-1}^{(\lambda)}(t)=2\frac{(i+\lambda-1)a_{i}}{i}(t\frac{d}{dt}\hat p_{i}^{(\lambda)}(t)-i \hat p_{i}^{(\lambda)}(t))
$$
so we find
\begin{align*}
\frac{d}{dt}B_i\hat p^{(\lambda)}_i(t)&=
4\frac{d}{dt}(1+t+\frac{\lambda-3/2}{i+1}t)\hat p_i^{(\lambda)}(t)-8a_i\frac{(\lambda-3/2)(\lambda+i)}{(i+1)(i+2\lambda-1)}\frac{d}{dt}\hat p_{i-1}^{(\lambda)}(t))\\&=4(\lambda-1/2)\hat p^{(\lambda)}_i(t)+4(1+t)\frac{d}{dt}\hat p^{(\lambda)}_i(t).
\end{align*}
Thus we have
\begin{align*}
&((1-t)\frac{d}{dt}-(\lambda+1/2))\frac{d}{dt} B_i\hat p^{(\lambda)}_i(t)\\&=
4((1-t^2)\frac{d^2}{dt^2}-(2\lambda+1)t\frac{d}{dt}-(\lambda^2-1/4))\hat p^{(\lambda)}_i(t)
\end{align*}
and the result follows.
\end{proof}
\begin{remark} At first, we can see that equation \eqref{recurge} has the form
\begin{align*}
\tilde A_if^{(\lambda)}_{i,j}=(j+\lambda-\frac{1}{2})(j+\lambda+\frac{1}{2})B_if^{(\lambda)}_{i,j},
\end{align*}
where
\begin{equation*}
\tilde A_i=(i+\lambda-1/2)(i+\lambda+1/2)A_i,
\end{equation*}
the operators $A_i$ and $B_i$ are given by \eqref{tai} and \eqref{bi}, respectively. At second, the above-given proof shows that the three-term recurrence relation \eqref{recurge} is a consequence of the fact that ultraspherical polynomials are eigenfunctions of a second order differential operator of a specific form. However, there is another way to see the validity of equation \eqref{recurge}.
\end{remark}
We first prove the following statement.
\begin{proposition}\label{ultracoeff} The following representation holds
\begin{equation}\label{Ione}
f^{(\lambda)}_{i,j}=\begin{cases}
0,\,\, i< j;\\
\frac{1}{2^{3j+1}}\sqrt{\frac{i!(\lambda+1)_i(2\lambda)_i(2\lambda)_j}{j!(\lambda)_i(\lambda)_j(\lambda+1)_j}}\frac{(i+2\lambda)_j}{(\lambda+\frac{1}{2})_j(i-j)!}\hypergeom21{-i+j,\ i+j+2\lambda}{2j+2\lambda+1}{\frac{1}{2}},\, i\ge j.
\end{cases}
\end{equation}
\end{proposition}
\begin{proof} Write
\begin{equation}\label{cijlama}
f^{(\lambda)}_{i,j}=k_{i,j,\lambda}\int_0^1 p_i^{(\lambda)}(t)p_j^{(\lambda)}(2t-1)(t(1-t))^{\lambda-1/2}dt,
\end{equation}
where $p_n^{(\lambda)}$ is the monic orthogonal polynomial and
\begin{equation}\label{norm}
k_{i,j,\lambda}=\frac{\Gamma(\lambda+1)}{\Gamma(\lambda+\frac{1}{2})\sqrt{\pi}}2^{i+j+2\lambda+1}\sqrt{\frac{(\lambda)_i(\lambda+1)_i}{i!(2\lambda)_i}}\sqrt{\frac{(\lambda)_j(\lambda+1)_j}{j!(2\lambda)_j}}.
\end{equation}
If we denote the integral in equation~\eqref{cijlama} as $I^{(1)}$ we find using the representation
\[
p^{(\lambda)}_i(t)=2^i\frac{(\lambda+\frac{1}{2})_i}{(i+2\lambda)_i}\hypergeom21{-i,\ i+2\lambda}{\lambda+\frac{1}{2}}{\frac{1-t}{2}},
\]
and set
\begin{equation}\label{ione}
I^{(1)}=2^{i+j}(-1)^j\frac{(\lambda+\frac{1}{2})_i}{((i+2\lambda)_i)}\frac{(\lambda+\frac{1}{2})_j}{((j+2\lambda)_j)}I^{(2)},
\end{equation}
with
\begin{align*}\label{itwo}
I^{(2)}&=\int_0^1\hypergeom21{-i,\ i+2\lambda}{\lambda+\frac{1}{2}}{\frac{1-t}{2}}\hypergeom21{-j,\ j+2\lambda}{\lambda+\frac{1}{2}}{t}(t(1-t))^{\lambda-1/2}dt\\&=\sum_{k=0}^i\frac{(-i)_k(i+2\lambda)_k}{(1)_k(\lambda+\frac{1}{2})_k 2^k}\sum_{n=0}^j\frac{(-j)_n(j+2\lambda)_n}{(1)_n(\lambda+\frac{1}{2})_n}\int_0^1(1-t)^{k+\lambda-1/2}t^{n+\lambda-1/2}dt.
\end{align*}
The integral can be evaluated as $\frac{\Gamma(k+\lambda+\frac{1}{2})\Gamma(n+\lambda+\frac{1}{2})}{\Gamma(k+n+2\lambda+1)}=\frac{(\lambda+\frac{1}{2})_k(\lambda+\frac{1}{2})_n\Gamma(\lambda+\frac{1}{2})^2}{(2\lambda+1)_k(k+2\lambda+1)_n\Gamma(2\lambda+1)}$. From the Chu-Vandermonde formula the sum on $n$ yields
$$
\sum_{n=0}^j\frac{(-j)_n(j+2\lambda)_n}{(1)_n(k+2\lambda+1)_n}=\frac{(k-j+1)_j}{(k+2\lambda+1)_j},
$$
and the sum on $k$ now becomes
$$
\sum_{k=j}^i\frac{(-i)_k(i+2\lambda)_k(k-j+1)_j}{(1)_k(2\lambda+1)_k(k+2\lambda+1)_j2^k}=\sum_{k=0}^{i-j}\frac{(-i)_{k+j}(i+2\lambda)_{k+j}(k+1)_j}{(1)_{k+j}(2\lambda+1)_{k+j}(k+j+2\lambda+1)_j2^{k+j}}.
$$
With the identities
$$
(k+b)_j=\frac{(j+b)_k (b)_j}{(b)_k},\ (a)_{k+j}=(a+j)_k (a)_j,
$$
the above sum becomes
\begin{align*}
&\sum_{k=0}^{i-j}\frac{(-i)_{k+j}(i+2\lambda)_{k+j}(k+1)_j}{(1)_{k+j}(2\lambda+1)_{k+j}(k+j+2\lambda+1)_j2^{k+j}}\\=&\frac{(-i)_j(i+2\lambda)_j(1)_j}{(2\lambda+1)_j(j+2\lambda+1)_j}\sum_{k=0}^{i-j}\frac{(-i+j)_k(i+j+2\lambda)_j}{(1)_j(2j+2\lambda+1)_k}\frac{1}{2^k}\\=&\frac{(-i)_j(i+2\lambda)_j(1)_j}{(2\lambda+1)_j(j+2\lambda+1)_j}\hypergeom21{-i+j,\ i+j+2\lambda}{2j+2\lambda+1}{\frac{1}{2}}.
\end{align*}
Combining all this together gives the result.
\end{proof}
The above hypergeometric representation \eqref{Ione} for $f^{(\lambda)}_{i,j}$ gives a recurrence relation among them.
\begin{proof}[Another Proof of Theorem \ref{UGEPth}]
To see this use the contiguous relation (see \cite[equation (2.5.15)]{AAR})
\begin{align*}
&2b(c-b)(b-a-1)\hypergeom21{a-1,\ b+1}{c}{\frac{1}{2}}\\&-(b-a)(b+a-1)(2c-b-a-1)\hypergeom21{a,\ b}{c}{\frac{1}{2}}\\&-2a(b-c)(b-a+1)\hypergeom21{a+1,\ b-1}{c}{\frac{1}{2}}=0,
\end{align*}
which with $a=-i+j,\ b=i+j+2,$ and $c=2j+2\lambda+1$ yields the relation
\begin{align
&(2i+2\lambda-1)(j+2\lambda+1)\sqrt{\frac{i+2\lambda}{(i+1)(i+\lambda+1)(\lambda+i)}}(i+1-j)f^{(\lambda)}_{i+1,j}\nonumber\\&-(2j+2\lambda-1)(2j+2\lambda+1)f^{(\lambda)}_{i,j}\nonumber\\&+(2i+2\lambda+1)(i-j-1)\sqrt{\frac{i}{(i-1+\lambda)(i-1+2\lambda)(\lambda+i)}}(i+j-2\lambda-1)f^{(\lambda)}_{i-1,j}=0.
\end{align}
The latter relation leads to \eqref{recurge}.
\end{proof}
A generalized eigenvalue problem can also be found for $i$ fixed. To this end we need to use the relation
\[
\hypergeom21{-n,\ b}{c}{x}=\frac{(b)_n}{(c)_n}(-x)^n\hypergeom21{-n,\ -c-n+1}{-b-n+1}{1/x}.
\]
Therefore we find
\begin{align}\label{althyp}
&\hypergeom21{-i+j,\ i+j+2\lambda}{2j+2\lambda+1}{1/2}\nonumber\\&=\frac{(i+j+2\lambda)_{i-j}}{(2j+2\lambda+1)_{i-j}}(-2)^{j-i}\hypergeom21{-i+j,\ -i-j-2\lambda}{-2i-2\lambda+1}{2}.
\end{align}
Following the steps used to obtain the recurrence formula for $j$ fixed in the second proof we find that
\begin{align*}
c_j f^{(\lambda)}_{i,j-1}+d_jf^{(\lambda)}_{i,j+1}+e_jf^{(\lambda)}_{i,j}=0,
\end{align*}
where
$$
c_j=-2(i+j+2\lambda-1)(i-j+1)(2j+2\lambda+1)(j+\lambda+1),
$$
$$
d_j=-4(i-j-1)(i+j+2\lambda+1)(j+\lambda-\frac{1}{2})\sqrt{\frac{j(j+1)(j+\lambda-1)(j+\lambda+1)}{(j+2\lambda-1)(j+2\lambda)}},
$$
and
\begin{align*}
e_j&=-2(2i+2\lambda+1)(2i+2\lambda-1)(j+\lambda+1)\sqrt{\frac{j(j+\lambda)(j+\lambda-1)}{(j+2\lambda-1)}}\\&+6(2j+2\lambda-1)(2j+2\lambda+1)(j+\lambda+1)\sqrt{\frac{j(j+\lambda-1)(j+\lambda)}{(j+2\lambda-1)}}.
\end{align*}
Since
$$
(i+j+2\lambda\mp1)(i-j\pm1)=(i+\lambda+\frac{1}{2})(i+\lambda-\frac{1}{2})-(j+\lambda\mp\frac{1}{2})(j+\lambda\mp\frac{3}{2})
$$
the above recurrence can be recast as the generalized eigenvalue equation
\begin{equation*}
\hat A_jf^{(\lambda)}_{i,j}=(i+\lambda+\frac{1}{2})(i+\lambda-\frac{1}{2})\hat B_jf^{(\lambda)}_{i,j},
\end{equation*}
where the operator $\hat A_j$ is the second order difference operator
\begin{align}\label{hai}
\hat A_j&=(2j+2\lambda+1)(2j+2\lambda-1))\bigg(3(j+\lambda+1)I\nonumber\\&+(j+\lambda+\frac{3}{2})\sqrt{\frac{(j+1)(j+\lambda+1)}{(j+\lambda)(j+2\lambda)}}
\hat E_+\\&+(j+\lambda-\frac{3}{2})(j+\lambda+1)\sqrt{\frac{(j+2\lambda-1)}{j(j+\lambda)(j+\lambda-1)}}\hat E_-\bigg)\nonumber,
\end{align}
the operator $\hat B_j$ is another second order difference operator given by the formula
\begin{align}\label{hbi}
\hat B_j&=4(j+\lambda+1)I\nonumber\\&+(2j+2\lambda-1)\sqrt{\frac{(j+1)(j+\lambda+1)}{(j+\lambda)(j+2\lambda)}}\hat E_+\\&+(2j+2\lambda+1)(j+\lambda+1)\sqrt{\frac{(j+2\lambda-1)}{j(j+\lambda)(j+\lambda-1)}}\hat E_-\bigg)\nonumber,
\end{align}
the operator $I$ is the identity operator, and $\hat E_+$, $\hat E_-$ are the forward and backward shift operators on $j$, respectively. Thus we have just proved the following statement.
\begin{theorem}\label{ifixedTH}
Let $i$ be a fixed nonnegative integer number. Then the function $f=f(j)=f^{(\lambda)}_{i,j}$ of the discrete variable $j$ satisfies the generalized eigenvalue problem
\[
\hat A_jf^{(\lambda)}_{i,j}=(i+\lambda+\frac{1}{2})(i+\lambda-\frac{1}{2})\hat B_jf^{(\lambda)}_{i,j}
\]
for $i=0$, $1$, $2$, \dots and where the operators $\hat A_j$ and $\hat B_j$ are given by \eqref{hai} and \eqref{hbi}, respectively.
Also, here, $(i+\lambda+\frac{1}{2})(i+\lambda-\frac{1}{2})$ is the corresponding generalized eigenvalue.
\end{theorem}
\begin{remark} For the case $\lambda=1/2$, Theorem \ref{ifixedTH} was obtained in \cite{GM15}.
\end{remark}
Using the asymptotic results for the Gauss hypergeometric function from \cite{J01} and \cite{W18} (see also \cite{P1}, \cite{T03}) one can easily get asymptotic behavior of the solution $f^{(\lambda)}_{i,j}$ for $j$ fixed and when $i$ tends to infinity.
\begin{theorem}
For sufficiently large $i$ the following formula holds
\begin{equation}\label{asymi}
f^{(\lambda)}_{i,j}=k_j\frac{\cos\left(\pi\left(j + \frac{\lambda}{2} - \frac{i}{2} + \frac{1}{4}\right)\right)}{\sqrt{\pi}i^{\lambda + 1/2}}+O\left(\frac{1}{i^{\lambda+3/2}}\right),
\end{equation}
where
\begin{equation}\label{ksubj}
k_j=\frac{1}{2^{j+1-2\lambda}}\sqrt{\frac{(2\lambda)_j}{j!(\lambda)_j(\lambda+1)_j\lambda\Gamma(2\lambda)}}\Gamma(2j+2\lambda+1)(\lambda+\frac{1}{2})_j .
\end{equation}
\end{theorem}
\begin{proof}
According to Proposition \ref{ultracoeff} for $i\ge j$ we have
\begin{equation*}
f^{(\lambda)}_{i,j}=\frac{1}{2^{3j+1}}\sqrt{\frac{i!(\lambda+1)_i(2\lambda)_i(2\lambda)_j}{j!(\lambda)_i(\lambda)_j(\lambda+1)_j}}\frac{(i+2\lambda)_j}{(\lambda+\frac{1}{2})_j(i-j)!}\hypergeom21{-i+j,\ i+j+2\lambda}{2j+2\lambda+1}{\frac{1}{2}}.
\end{equation*}
Then, since
$$
\sqrt{\frac{i!(\lambda+1)_i(2\lambda)_i}{(\lambda)_i}}\frac{(i+2\lambda)_j}{(i-j)!}=\sqrt{\frac{1}{\lambda \Gamma(2\lambda)}} i^{(2j+\lambda)}(1+O(1/i)),
$$
formula \eqref{asymi} follows from \cite[formula (36)]{J01}.
\end{proof}
\begin{remark} Formula \eqref{asymi} along with the fact that $f^{(\lambda)}_{i,j}=0$ for $i<j$ show that the moving wave behavior of the solution demonstrated in Figure \ref{Fig1} is also characteristic for the solution $f^{(\lambda)}_{i,j}$ of the discrete wave equation \eqref{dWaveUP} for any $\lambda>-1/2$.
\end{remark}
Another useful asymptotic is when $i=k_1 t$ and $j=k_2 t$ where $k_1>k_2$ are fixed and $t$ is large.
\begin{theorem}
For $k_1 t$ and $k_2 t$ integers with $k_1>k_2>0$, and $\frac{\sqrt{2}k_2}{k_1}>1$
\begin{equation}\label{fasyf}
f^{(\lambda)}_{k_1 t,k_2 t}=\frac{c(\epsilon,\lambda)}{2^{k_1 t+1} (k_1 t)^{\frac{1}{2}}}\left(\frac{1+\hat b(\epsilon)}{\epsilon-\hat b(\epsilon)}\right)^{(k_1-k_2)t}\left(\frac{1+2\epsilon-\hat b(\epsilon)}{1+\epsilon}\right)^{(k_1+k_2)t+2\lambda}(1+O(1/t)),
\end{equation}
where
\begin{equation}\label{ck1k2}
c(\epsilon,\lambda)=\epsilon^{\lambda}\frac{1}{\sqrt{\pi(1-\epsilon^2)(2\epsilon^2-1)^{\frac{1}{2}}}},
\end{equation}
$\epsilon=\frac{k_2}{k_1}$, and $\hat b(\epsilon)=\sqrt{2\epsilon^2-1}$.
\end{theorem}
\begin{proof}
In this case the representation given by equation~\eqref{althyp} is most convenient. An application of the transformation T3 in \cite{P1} yields
\begin{align*}
&\hypergeom21{-i+j,\ -i-j-2\lambda}{-2i-2\lambda+1}{2}\nonumber\\&=\frac{2^{i-j-1}(i-j)!}{(i+j+2\lambda+1)_{i-j-1}}\hypergeom21{-i+j+1,\ i+j+2\lambda+1}{2}{1/2}\nonumber\\&=-\frac{2^{i+j+2\lambda-1}(i-j)!}{(i+j+2\lambda+1)_{i-j-1}}\hypergeom21{i-j+1,\ -i-j-2\lambda+1}{2}{1/2},
\end{align*}
where Euler's transformation has been used to obtain the last equality.
Thus with the use of the duplication formula for the $\Gamma $ function it follows
\begin{equation}\label{asyf}
f^{(\lambda)}_{i,j}=d_{i,j}\hypergeom21{i-j+1,\ -i-j-2\lambda+1}{2}{1/2},
\end{equation}
where
$$
d_{i,j}=(-1)^{i-j+1}2^{j+2\lambda-1}\sqrt{\frac{(i+\lambda)(j+\lambda)i!\Gamma(2\lambda+j)}{j!\Gamma(2\lambda+i)}}.
$$
This becomes
\begin{align}\label{dijtl}
d_{k_1 t,k_2 t}&=(-1)^{(i-j+1}2^{j+2\lambda-1}\left(\frac{j^{2\lambda}}{i^{2\lambda-2}}\right)^{1/2}(1+O(1/i)\nonumber\\&=(-1)^{(k_1-k_2)t+1}2^{k_2 t+2\lambda-1}\left(\frac{k_2}{k_1}\right)^{\lambda} (k_1 t) (1+O(1/t)).
\end{align}
The hypergeometric function on the right hand side of equation~\eqref{asyf} is in the form to use the type B formulas in \cite{P1} and leads to considering the hypergeometric function $\hypergeom21{\epsilon_1 w+1,\ -w-2\lambda+1}{2}{1/2}$ where $\epsilon_1 w$ is an integer. Equation~(4.4) in \cite{P1} shows that the saddle points occur at $\frac{1+\epsilon_1}{2}\pm\sqrt{(\frac{1+\epsilon_1}{2})^2-2\epsilon_1}$. If the discriminant is positive both saddles are real and equation (4.9) in \cite{P1} yields
\begin{align*}
&\hypergeom21{\epsilon_1 w+1,\ -w-2\lambda+1}{2}{1/2}\\&=\frac{(-1)^{\epsilon_1 w+1}}{w^{\frac{3}{2}}\sqrt{\pi\epsilon_1 b(\epsilon_1)}}(\frac{1+\epsilon_1+b(\epsilon_1)}{1-\epsilon_1-b(\epsilon_1)})^{\epsilon_1 w}\frac{(3-\epsilon_1-b(\epsilon_1))^{w+2\lambda}}{2^{w+4\lambda+\frac{1}{2}}}(1+O(1/w)),
\end{align*}
where
\begin{equation}\label{bep1}
b(\epsilon_1)=\sqrt{(1+\epsilon_1)^2-8\epsilon_1}.
\end{equation}
With $\epsilon_1=\frac{k_1-k_2}{k_1+k_2}$ and $w=(k_1+k_2)t$ the above equations yield \eqref{fasyf}.
\end{proof}
\begin{remark}
When the discriminant is negative, the two saddle points are conjugates of each other and so in this case equation~(4.7) in \cite{P1} is used to obtain the asymptotics for $\hypergeom21{\epsilon_1 t+1,\ -t-2\lambda+1}{2}{1/2}$ which then are used to obtain the asymptotics of $f^{(\lambda)}_{k_1 t,k_2 t}$.
\end{remark}
We finish this section with a couple of statements where we start with the recurrence formulas. Write the recurrence formula in equation~\eqref{recurge} as
\begin{equation}\label{iirecur}
a_{i,j}f^{(\lambda)}_{i+1,j}+b_{i,j}f^{(\lambda)}_{i,j}+c_{i,j}f^{(\lambda)}_{i-1,j}=0,
\end{equation}
and the recurrence formula in $j$ as
\begin{equation}\label{jjrecur}
\hat a_{i,j}f^{(\lambda)}_{i,j+1}+\hat b_{i,j}f^{(\lambda)}_{i,j}+\hat c_{i,j}f^{(\lambda)}_{i,j-1}=0,
\end{equation} with $i\ge j\ge0$.
We can now prove the following simple statement.
\begin{proposition}\label{iiirecur}
Given $a_{i,j}$, $b_{i,j}$, $c_{i,j}$ and $\lambda>-1/2$. For each $j>0$ the unique solution of equation~\eqref{iirecur} with initial conditions
\[
f_{j-1,j}=0, \quad f_{j,j}=\int_0^1 \hat p^{(\lambda)}_j(t)\hat p^{(\lambda)}_j(2t-1)(t(1-t))^{\lambda-1/2}dt
\]
is the function
\[
f_{i,j}=I^{(\lambda)}_{i,j}:=\int_0^1 \hat p^{(\lambda)}_i(t)\hat p^{(\lambda)}_j(2t-1)(t(1-t))^{\lambda-1/2}dt.
\]
If $j=0$,
$\lambda>-1/2$, and $\lambda\ne1/2$ then $f_{0,0}=I_{0,0}^{(\lambda)} $ gives the unique solution $f_{i,0}=I^{(\lambda)}_{i,0}$. If $\lambda=1/2$ then the initial conditions $f_{0,0}= I^{(1/2)}_{0,0}$ and $f_{1,0}= I^{(1/2)}_{1,0}$ are needed to give $f_{i,j}=I^{(1/2)}_{i,j}$.
\end{proposition}
\begin{proof}
For $j>0,\ a_{i,j}\ne 0$ for $i\ge j$ so the result follows from equation~\eqref{iirecur}. For $j=0$ and $\lambda\ne1/2,\ c_{0,0}=0\ne a_{0,0}$ so that only $f_{0,0}$ is needed to compute $f_{1,0}$. The remaining $f_{i,j}$ are computed in the standard fashion from equation~\eqref{iirecur}. For the last case when $\lambda=1/2,\ a_{0,0}=0=b_{i,0}$ so $f_{2,0}=\frac{c_{1,0}}{a_{1,0}} f_{0,0}$ and $f_{3,0}=\frac{c_{2,0}}{a_{2,0}} f_{1,0}$. The remaining $f_{i,0}$ are computed in the same way using the fact that $a_{i,0}\ne0$ for $i>0$.
\end{proof}
Similarly, for the recurrence in $j$ we have the following.
\begin{proposition}\label{jjjrecur}
Given $a_{i,j}$, $b_{i,j}$, $c_{i,j}$ and $\lambda>-1/2$, for each $i>0$ the unique solution of equation~\eqref{jjrecur} with initial conditions
$f_{j,j+1}=0$ and $f_{j,j}=I_{j,j}^{(\lambda)}$ is $f_{i,j}=I^{(\lambda)}_{i,j}$.
\end{proposition}
Since $\hat c_{i,j},\ \hat b_{i,j}$, and $\hat a_{i,j} $ are not equal to zero for $i\ge j$ the result follows from equation~\eqref{jjrecur}.
\section{Connections to other problems}
Recall that it is said that a function $\Psi(x,y)$ is a solution of a bispectral problem if it satisfies the following
\[
\begin{split}
A\Psi(x,y)&=g(y)\Psi(x,y)\\
B\Psi(x,y)&=f(x)\Psi(x,y),
\end{split}
\]
where $A$, $B$ are some operators, with $A$ acting only on $x$ and $B$ acting only on $y$, and $f$, $g$ are some functions \cite{DG}. It is shown in \cite{LM98} that if $A$ and $B$ are tridiagonal operators then the solutions of the corresponding discrete bispectral problem are related to the Askey-Wilson polynomials.
The problem we are dealing with in this paper is the following generalization of a bispectral problem:
\begin{equation}\label{gBP}
\begin{split}
A\Psi(i,j)&=g(j)B\Psi(i,j)\\
C\Psi(i,j)&=f(i)D\Psi(i,j),
\end{split}
\end{equation}
where $i$, $j$ are discrete variables, the operators $A$ and $B$ are tridiagonal operators acting on the index $i$, and $C$, $D$ are tridiagonal operators acting on the index $j$. Note that each equation in \eqref{gBP} is a generalized eigenvalue problem and, hence, the problem \eqref{gBP} includes a bispectral problem as a particular case (for instance when $B$ and $D$ are the identity operators).
Setting $\Psi(i,j)=f^{(\lambda)}_{i,j}$ we see that Theorems \ref{UGEPth} and \ref{ifixedTH} tell us that $f^{(\lambda)}_{i,j}$ is a solution of a generalized bispectral problem of the form \eqref{gBP}. Actually, it would be nice to find a characterization of such generalized bispectral problems similar to what was done in \cite{LM98} for discrete bispectral problems. It would also be interesting to study the consistency relations for the system \eqref{jjjrecur}, \eqref{iiirecur} and those relations will constitute a nonlinear system of difference equations on the coefficients of \eqref{jjjrecur}, \eqref{iiirecur}.
Another link that is worth discussing here is the relation to linear spectral transformations. To see this in its simplest form, let us consider two families $\hat{p}_j^{(1/2)}(t)$ and $\hat{p}_j^{(3/2)}(t)$ of the ultraspherical polynomials. That is, we consider the two measures on $[-1,1]$
\[
d\mu_{1/2}(t)=dt, \quad d\mu_{3/2}(t)=(1-t^2)dt,
\]
which are clearly related in the following manner
\begin{equation}\label{GeronimusNM}
d\mu_{1/2}(t)=\frac{d\mu_{3/2}(t)}{(1-t^2)}.
\end{equation}
In such a case, one usually says that $d\mu_{1/2}(t)$ is a Geronimus transformation of $d\mu_{3/2}(t)$ of the second order or $d\mu_{1/2}(t)$ is the inverse quadratic spectral transform of $d\mu_{3/2}(t)$ (for instance, see \cite{MandCo11}). As a matter of fact, the Geronimus transformation of $d\mu_{3/2}(t)$ is more general than just \eqref{GeronimusNM} and it has the form
\[
d\mu^{(G)}(t)=\frac{d\mu_{3/2}(t)}{(1-t^2)}+M\delta_{-1}+N\delta_{1},
\]
where $\delta_{a}$ denotes the Dirac delta function supported at $a$ and $M$, $N$ are some nonegative real numbers. For the corresponding orthogonal polynomials we have that
\[
\hat{p}_i^{(G)}(t)=\alpha(1,i)\hat{p}_i^{(3/2)}(t)+\alpha(2,i)\hat{p}_{i-1}^{(3/2)}(t)+\alpha(3,i)\hat{p}_{i-2}^{(3/2)}(t),
\]
where $\alpha(1,i)$, $\alpha(2,i)$, and $\alpha(3,i)$ are some coefficients and they are of the form \eqref{Cijg}. For instance, in the simplest case \eqref{GeronimusNM}, introducing the coefficients
\[
f_{i,j}^{(1/2,3/2)}=\int_{-1}^1\hat p_i^{(1/2)}(t)\hat p_j^{(3/2)}(t)(1-t^2)dt
\]
leads to the relation
\begin{equation}\label{Geronimus}
\hat{p}_i^{(1/2)}(t)=f_{i,i}^{(1/2,3/2)}\hat{p}_i^{(3/2)}(t)+f_{i,i-1}^{(1/2,3/2)}\hat{p}_{i-1}^{(3/2)}(t)+f_{i,i-2}^{(1/2,3/2)}\hat{p}_{i-2}^{(3/2)}(t),
\end{equation}
where
\[
f_{i,i-2}^{(1/2,3/2)}=-\sqrt{\frac{k_{i-2,i-2,3/2}}{k_{i,i,1/2}}},\quad
f_{i,i-1}^{(1/2,3/2)}=0 \quad
f_{i,i}^{(1/2,3/2)}=\sqrt{\frac{k_{i,i,1/2}}{k_{i,i,3/2}}}
\]
and $k_{i,j,\lambda}$ is defined by formula \eqref{norm}. In fact, this can be generalized to the case of arbitrary Geronimus transformation but the formulas will get messier.
Due to Theorem \ref{GdWaveTH}, the coefficients $f_{i,j}^{(1/2,3/2)}$ satisfy the discrete wave equation in question. Besides, formula \eqref{Geronimus} shows that in the sequence $f_{i,j}^{(1/2,3/2)}$ when $j$ is fixed there are at most three nonzero coefficients and we know how to find them explicitly. Moreover, returning to the moving wave interpretations we did before we see that in this case we have a localized wave and below is the simulation.
\begin{figure}[h!]
\includegraphics[width=\linewidth]{SingleWave.png}
\caption{This picture shows three graphs of the function $f=f^{(1/2,3/2)}(i)=f^{(1/2,3/2)}_{i,j}$ of the discrete space variable $i$ at the three different discrete times $j=15$, $j=30$, and $j=45$.}
\label{Fig3}
\end{figure}
The phenomenon of localized waves is related to the fact that the measures are related to one another through spectral transformations. Still, one can define even more general coefficients
\[
f_{i,j}^{(\lambda,\mu)}=\int_{-1}^1\hat p_i^{(\lambda)}(t)\hat p_j^{(\mu)}(t)(1-t^2)^{\mu-1/2}dt
\]
and, as before, they form a solution to a wave equation. Moreover, these coefficients are known explicitly \cite[Section 7.1]{AAR} and are called the connection coefficients. It will be shown in a forthcoming paper that the family $f_{i,j}^{(\lambda,\mu)}$ is also a solution of a bispectral problem of the form \eqref{gBP}. In addition, the coefficients
\[
f_{i,j,k}^{(1,1,1)}=\frac{2}{\pi}\int_{-1}^1 p_i^{(1)}(t) p_j^{(1)}(t)p_k^{(1)}(t)(1-t^2)^{1/2}dt,
\]
where the polynomials $p_i^{(1)}(t)$ are the monic Chebyshev polynomials of second kind, count Dyck paths \cite{deSMV85}. Thus, it would be interesting to find out if the coefficients $f_{i,j,k}^{(1,1,1)}$ still satisfy some generalized eigenvalue problems and in which case if such generalized eigenvalue problems admit a combinatorial interpretation.
\medskip
\noindent{\bf Acknowledgments.} M.D. was supported in part by the NSF DMS grant 2008844. The authors are grateful to Erik Koelink for interesting and helpful remarks. They are also indebted to the anonymous referees for suggestions that helped to improve the presentation of the results. J.G. would like to thank J.G. for the support.
| {
"timestamp": "2021-02-01T02:04:26",
"yymm": "2001",
"arxiv_id": "2001.10650",
"language": "en",
"url": "https://arxiv.org/abs/2001.10650",
"abstract": "We start by presenting a generalization of a discrete wave equation that is particularly satisfied by the entries of the matrix coefficients of the refinement equation corresponding to the multiresolution analysis of Alpert. The entries are in fact functions of two discrete variables and they can be expressed in terms of the Legendre polynomials. Next, we generalize these functions to the case of the ultraspherical polynomials and show that these new functions obey two generalized eigenvalue problems in each of the two discrete variables, which constitute a generalized bispectral problem. At the end, we make some connections to other problems.",
"subjects": "Mathematical Physics (math-ph); Classical Analysis and ODEs (math.CA); Spectral Theory (math.SP)",
"title": "Connection coefficients for ultraspherical polynomials with argument doubling and generalized bispectrality",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.988841968751916,
"lm_q2_score": 0.7154239957834733,
"lm_q1q2_score": 0.7074412724828921
} |
https://arxiv.org/abs/1605.07229 | Fibre Products of Supersingular Curves and the Enumeration of Irreducible Polynomials with Prescribed Coefficients | For any positive integers $n\geq 3, r\geq 1$ we present formulae for the number of irreducible polynomials of degree $n$ over the finite field $\mathbb{F}_{2^r}$ where the coefficients of $x^{n-1}$, $x^{n-2}$ and $x^{n-3}$ are zero. Our proofs involve counting the number of points on certain algebraic curves over finite fields, a technique which arose from Fourier-analysing the known formulae for the $\mathbb{F}_2$ base field cases, reverse-engineering an economical new proof and then extending it. This approach gives rise to fibre products of supersingular curves and makes explicit why the formulae have period $24$ in $n$. | \section{Introduction}\label{sec:intro}
For $p$ a prime and $r \ge 1$ let ${\mathbb F}_q$ denote the finite field of $q = p^r$ elements.
The number of monic irreducible polynomials in ${\mathbb F}_q[x]$ of degree $n$ is
usually denoted $I_q(n)$, and the formula for $I_q(n)$ is a classical result due to Gauss~\cite[pp. 602-629]{gauss}.
A natural extension problem is to determine the
number of monic irreducible polynomials in ${\mathbb F}_q[x]$ of degree $n$ for which certain coefficients are prescribed.
An interesting set of subproblems here consists of counting the number of monic irreducible polynomials of degree $n$ for which the first $l$ coefficients
have the prescribed values $t_1,\ldots,t_l$, while the remaining coefficients are arbitrary, i.e.,
considering polynomials of the form
\[
x^n + t_1 x^{n-1} + \cdots + t_l x^{n-l} + a_{l+1} x^{n-l-1} + \cdots + a_{n-1}x + a_n,
\]
whose number we denote by $I_q(n,t_1,\ldots,t_l)$.
While the asymptotics for such subproblems have been obtained by Cohen~\cite{Cohen}, several exact results are known.
Carlitz gave formulae for $I_q(n,t_1)$ in 1952~\cite{Carlitz}, and
Kuz'min gave formulae for $I_q(n,t_1,t_2)$ in 1990~\cite{kuzmin1,kuzmin2}.
For the base field ${\mathbb F}_2$, Cattell {\em et al. } reproduced Kuz'min's results in
1999~\cite{cattell}, while in 2001 Yucas and Mullen computed $I_2(n,t_1,t_2,t_3)$ for $n$ even~\cite{yucasmullen} and Fitzgerald and Yucas
computed $I_2(n,t_1,t_2,t_3)$ for $n$ odd~\cite{fitzyucas}. In 2013 Ri {\em et al. } gave formulae for
$I_{2^r}(n,t_1,t_2)$ for all $r \ge 1$~\cite{RMR}.
In this paper we give a formula for $I_{2^r}(n,0,0,0)$ for all $r \ge 1$.
While our aim is to count the number of monic irreducible polynomials with certain coefficients prescribed, it turns out to be easier
(and arguably more interesting) to count the number of elements of the relevant fields with correspondingly prescribed traces; a standard M\"obius inversion-type argument allows one to switch between the two. Indeed,~\cite{cattell,yucasmullen,fitzyucas,RMR} all take this approach.
With this in mind, for $r \ge 1$ let $q = 2^r$. For $a \in {\mathbb F}_{q^n}$, the characteristic polynomial of $a$ w.r.t. the extension ${\mathbb F}_{q^n}/{\mathbb F}_q$ is
defined to be:
\begin{equation}\label{characteristicpoly}
\prod_{i = 0}^{n-1} (x + a^{q^i}) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0.
\end{equation}
The coefficient $a_{n-1}$ is known as the trace of $a$, while $a_{n-2}$ is known as the subtrace. We are concerned with the first three traces,
which we henceforth denote by $T_1,T_2$ and $T_3$.
By~(\ref{characteristicpoly}) they are given by the following expressions:
\begin{eqnarray}
\nonumber T_1(a) &=& \sum_{i = 0}^{n-1} a^{q^i},\\
\nonumber T_2(a) &=& \sum_{0 \le i < j \le n-1} a^{q^i + q^j},\\
\nonumber T_3(a) &=& \sum_{0 \le i < j <k \le n-1} a^{q^i + q^j + q^k}.
\end{eqnarray}
For $t_1 \in {\mathbb F}_q$, let $F_q(n,t_1)$ be the number of elements $a \in {\mathbb F}_{q^n}$ for which $T_1(a) = t_1$.
Similarly, for $t_1,t_2 \in {\mathbb F}_q$ let $F_q(n,t_1,t_2)$ be the number of elements $a \in {\mathbb F}_{q^n}$ for which
$T_1(a) = t_1$ and $T_2(a) = t_2$. Lastly, for $t_1,t_2,t_3 \in {\mathbb F}_q$ let $F_q(n,t_1,t_2,t_3)$ be the number
of elements $a \in {\mathbb F}_{q^n}$ for which $T_1(a) = t_1, T_2(a) = t_2$ and $T_3(a) = t_3$.
The exact counts of elements with prescribed traces found in~\cite{cattell,yucasmullen,fitzyucas,RMR} are intriguing since they depend on $n \bmod 8$ in the two trace ($l=2$) cases and on $n \bmod 24$ in the three trace ($l=3$) cases. In this work we give a new explanation for this phenomenon. It happens because the formulae are derived from the number of
rational points on certain curves -- and the important point is that these curves are supersingular. The supersingularity implies that the
roots of the Weil polynomials are roots of unity. The presence of these roots of unity explains the periodicity.
The algebraic curves that arise in this paper have a certain form, namely
$S(y)=T(x)$ where $T(x)=xR_1(x)+(xR_2(x))^2+\cdots +(xR_m(x))^{2^{m-1}}$
and each $S, R_i$ is a 2-linearised polynomial.
Such curves have been studied by van der Geer-van der Vlugt~\cite{Geer1995}.
Following them, we show that our curves can be viewed as fibre products of simpler curves,
and we compute the trace of Frobenius for the simpler curves.
Finding exact formulae for $F_q(n,t_1,\ldots,t_l)$ when $l > 3$ is an open and apparently much harder problem. We note that Koma has given
approximations for these counts in the case $q = 2$ and $l = 4$, when $n \equiv 0,2 \pmod{4}$~\cite{Koma}. In forthcoming work by the third
author, by using a similar curve-based approach it is shown that the formulae for when more than three coefficients are prescribed
are in general not periodic, as they do not arise from supersingular curves alone. This fact explains why previous approaches to this problem -- which did not use our curve-based approach -- have failed to make any progress in the past $15$ years. Thus, in the present work we explain the periodicity of known formulae, extend this to larger base fields, and lay the foundations for further developments in this area.
The paper is organised as follows. In Section~\ref{sec:irreduciblecounts} we present the formula for $I_q(n,0,0,0)$ in
terms of $F_q(n,0,0,0)$, then in Section~\ref{backg} we provide some background on curves and their Jacobians.
In Section~\ref{sec:motivation} we motivate our approach by considering the ${\mathbb F}_2$ base field case with
the first two traces specified. We then extend this and consider the same base field but with
the first three traces specified, in Section~\ref{sec:3coeffsF2}. In Sections~\ref{sec:2r}, \ref{sec:rationalpoints} and~\ref{sec:explicit},
we compute $F_q(n,0,0,0)$ for all finite binary fields, and make some concluding remarks in Section~\ref{sec:conclusion}.
\section{Computing $I_q(n,0,0,0)$ from $F_q(n,0,0,0)$}\label{sec:irreduciblecounts}
In this section we express $I_q(n,0,0,0)$ in terms of $F_q(n,0,0,0)$, using simple extensions of results from~\cite{cattell} and~\cite{yucasmullen}.
Recall that $I_q(n)$ is the number of irreducible polynomials of degree $n$ in ${\mathbb F}_q[x]$ and $I_q(n,t_1)$ is the number of irreducible polynomials
of degree $n$ in ${\mathbb F}_q[x]$ with first coefficient $t_1$. Furthermore, let $\mu(\cdot)$ be the M\"obius function, and for a proposition $P$ let
$[P]$ denote its truth value, i.e., $[P] = 1$ if $P$ is true and $0$ if $P$ is false, with $1$ and $0$ interpreted as integers. The
formula is given by the following theorem.
\begin{theorem}\label{thm:irreducibles}
Let $n \ge 3$. Then
\begin{eqnarray}
\label{thm91} I_q(n,0,0,0) &=& \frac{1}{n} \sum_{\substack{d \mid n \\ d \ \emph{odd}}} \mu(d) \big( F_q(n/d,0,0,0) - [n \ \emph{even}]q^{n/2d - 1} \big)\\
\label{thm92} &=& \frac{1}{n} \sum_{\substack{d \mid n \\ d \ \emph{odd}}} \mu(d) F_q(n/d,0,0,0) - [n \ \emph{even}]I_{\sqrt{q}}(n,1).
\end{eqnarray}
\end{theorem}
For $\beta \in {\mathbb F}_{q^n}$ let $p = \text{Min}(\beta)$ denote the minimum polynomial of $\beta$ over ${\mathbb F}_q$, which has degree $n/d$ for some $d \mid n$.
Note that $T_i(\beta)$ is the coefficient of $x^{n-i}$ in $p^d$~\cite[Lemma 2]{cattell}, and so abusing notation slightly we also write $T_i(\beta)$ as
$T_i(p^d)$.
We use the following easy lemma, which was stated for the $q = 2$ case in~\cite[Prop. 1]{yucasmullen}.
\begin{lemma}
For each integer $d \ge 1$ and $p(x) \in {\mathbb F}_q[x]$,
\begin{enumerate}
\item $T_1(p^d) = d T_1(p)$
\item $T_2(p^d) = \binom{d}{2} T_1(p) + d T_2(p)$
\item $T_3(p^d) = \binom{d}{3} T_1(p) + d T_3(p)$
\end{enumerate}
\end{lemma}
Note that $\binom{d}{i} \bmod{2}$ for $i = 1,2,3$ depends on $d \bmod{4}$. In particular, $\binom{d}{2}$ is even if and only if
$d \equiv 0,1 \pmod{4}$ and $\binom{d}{3}$ is even if and only if $d \equiv 0,1,2 \pmod{4}$.
Throughout we therefore write $d \equiv a$ to denote $d \equiv a \pmod{4}$.
As in~\cite{cattell} let ${\bf Irr}(n)$ denote the set of all irreducible polynomials of degree $n$ over ${\mathbb F}_q$,
and let $a \cdot {\bf Irr}(n)$ denote the multiset consisting of $a$ copies of ${\bf Irr}(n)$.
As in~\cite{cattell} and~\cite{yucasmullen} we have:
\begin{eqnarray}
\nonumber F_q(n,t_1,t_2,t_3) &=& \Big\vert \bigcup_{\beta \in {\mathbb F}_{q^n}, T_1(\beta) = t_1, T_2(\beta) = t_2, T_3(\beta) = t_3} Min(\beta) \Big\vert\\
\nonumber &=& \Big\vert \bigcup_{d \mid n} \frac{n}{d} \big\{ p \in {\bf Irr}(\frac{n}{d}): dT_1(p) = t_1,
\binom{d}{2}T_1(p) + d T_2(p) = t_2, \binom{d}{3}T_1(p) + d T_3(p) = t_3 \big\} \Big\vert \\
\nonumber &=& \Big\vert \bigcup_{d \mid n, \ d \equiv 0} \frac{n}{d} \big\{ p \in {\bf Irr}(\frac{n}{d}):
0 = t_1, 0 = t_2, 0 = t_3 \big\} \Big\vert \\
\nonumber &+& \Big\vert \bigcup_{d \mid n, \ d \equiv 1} \frac{n}{d} \big\{ p \in {\bf Irr}(\frac{n}{d}):
T_1(p) = t_1, T_2(p) = t_2, T_3(p) = t_3 \big\} \Big\vert \\
\nonumber &+& \Big\vert \bigcup_{d \mid n, \ d \equiv 2} \frac{n}{d} \big\{ p \in {\bf Irr}(\frac{n}{d}):
0 = t_1, T_1(p) = t_2, 0 = t_3 \big\} \Big\vert \\
\nonumber &+& \Big\vert \bigcup_{d \mid n, \ d \equiv 3} \frac{n}{d} \big\{ p \in {\bf Irr}(\frac{n}{d}):
T_1(p) = t_1, T_1(p) + T_2(p) = t_2, T_1(p) + T_3(p) = t_3 \big\} \Big\vert .
\end{eqnarray}
For $t_1 = t_2 = t_3 = 0$ we have
\begin{equation}\label{eq:F000}
F_q(n,0,0,0) = \sum_{\substack{d \mid n \\ d \equiv 0}} \frac{n}{d} I_q(\frac{n}{d})
+ \sum_{\substack{d \mid n \\ d \equiv 2}} \frac{n}{d} I_q(\frac{n}{d},0)
+ \sum_{\substack{d \mid n \\ d \ \text{odd}}} \frac{n}{d} I_q(\frac{n}{d},0,0,0)
\end{equation}
To evaluate the sum of the first two terms of~(\ref{eq:F000}) we employ the following analogue of~\cite[Lemma 5]{cattell}.
\begin{lemma}
\[
\sum_{\substack{d \mid n \\ d \equiv 0}} \frac{n}{d} I_q(\frac{n}{d})
+ \sum_{\substack{d \mid n \\ d \equiv 2}} \frac{n}{d} I_q(\frac{n}{d},0) = [n \ \emph{even}]\, q^{n/2 - 1}.
\]
\end{lemma}
\begin{proof}
Recall the classical result of Gauss~\cite[pp. 602-629]{gauss}, and that of Carlitz~\cite{Carlitz}, for $t_1 \ne 0$, respectively:
\begin{equation}
\label{carlitzformula} I_q(n) = \frac{1}{n} \sum_{d \mid n} \mu(d) q^{n/d}, \hspace{3mm}
I_q(n,t_1) = \frac{1}{qn} \sum_{\substack{d \mid n \\ d \ \text{odd}}} \mu(d) q^{n/d},
\end{equation}
and consequently $I_q(n,0) = I_q(n) - (q-1)I_q(n,1)$.
Let
\[
A(n) = (q-1) \sum_{\substack{d \mid n \\ d \equiv 2}} \frac{n}{d} I_q(\frac{n}{d},1)
\]
and let
\[
B(n) = \sum_{\substack{d \mid n \\ d \equiv 0}} \frac{n}{d} I_q(\frac{n}{d})
+ \sum_{\substack{d \mid n \\ d \equiv 2}} \frac{n}{d} I_q(\frac{n}{d},0).
\]
If $n$ is odd then the sums in $B(n)$ are empty and hence the Lemma is true. Therefore let $n$ be even.
By essentially the same argument given in the proof of~\cite[Lemma 5]{cattell}, we have $A(n) = (q-1)q^{n/2-1}$.
Similarly, we have
\begin{eqnarray}
\nonumber B(n) + A(n) &=& \sum_{\substack{d \mid n \\ d \equiv 0}} \frac{n}{d} I_q(\frac{n}{d})
+ \sum_{\substack{d \mid n \\ d \equiv 2}} \frac{n}{d} I_q(\frac{n}{d},0)
+ (q-1) \sum_{\substack{d \mid n \\ d \equiv 2}} \frac{n}{d} I_q(\frac{n}{d},1) \\
\nonumber &=& \sum_{\substack{d \mid n \\ d \ \text{even}}} \frac{n}{d} I_q(\frac{n}{d})
= \sum_{d \mid \frac{n}{2}} \frac{n}{2d} I_q(\frac{n}{2d}) = q^{n/2}.
\end{eqnarray}
Hence $B(n) = q^{n/2} - A(n) = q^{n/2} - (q-1)q^{n/2 - 1} = q^{n/2-1}$. \qed
\end{proof}
Using this result,~(\ref{eq:F000}) becomes
\[
F_q(n,0,0,0) - [n \ \text{even}]\, q^{n/2 - 1} = \sum_{\substack{d \mid n \\ d \ \text{odd}}} \frac{n}{d} I_q(\frac{n}{d},0,0,0).
\]
Applying~\cite[Corollary 1]{cattell} with $\alpha(n) = n I_q(n,0,0,0)$ finally proves expression~(\ref{thm91}) of Theorem~\ref{thm:irreducibles}, and
applying Carlitz's result from~(\ref{carlitzformula}) gives~(\ref{thm92}).
\section{Background on curves and Jacobians}\label{backg}
Let $p$ be a prime number and let $q=p^r$.
Let $C$ be a curve of genus $g$ defined over $\mathbb{F}_q$.
All curves in this paper are projective plane curves, lying in $\mathbb{P}^2(k)$
where $k$ is an algebraic closure of $\mathbb{F}_q$.
As is common, we will work with affine versions of our curves and include the point at infinity in our point counts.
Note that we do not include the point at infinity when counting the relevant $F_q(n,t_1,t_2,t_3)$.
We denote by $J_C$ the Jacobian
of $C$, an abelian variety which as a group is isomorphic to
the divisor class group $\textrm{Pic}^0 (C)$
(the degree 0 divisors modulo the principal divisors).
The map $(x,y)\mapsto (x^q,y^q)$ on $C$ (defined over an
algebraic closure of $\mathbb{F}_q$) induces an endomorphism on $J_C$
called the Frobenius endomorphism.
The characteristic polynomial of the Frobenius endomorphism plays a key role.
The roots of this characteristic polynomial are called
Frobenius eigenvalues.
\subsection{Abelian varieties}
Suppose that $A$ and $B$ are Abelian varieties over the same field $K$.
A homomorphism $f : A\longrightarrow B$ is
a morphism that is also a group homomorphism. A homomorphism
is an isogeny over $K$ if $f$ is surjective and defined over $K$
and $\dim(A) = \dim(B)$. If $A$ is an Abelian variety over $K$,
$A$ is called simple (over $K$) if it
is not isogenous over $K$ to a product of lower-dimensional Abelian varieties.
Because $J_C$ is an abelian variety
we continue with some facts from the theory of abelian varieties.
Let $A$ be any abelian variety of dimension $g$ over $\mathbb{F}_q$.
It can be shown that $A$ has a Frobenius endomorphism, and that
the characteristic polynomial $P_A(t)\in \mathbb{Z}[t]$
of the Frobenius endomorphism has the form
\begin{displaymath}
P_A(t) = t^{2g} + a_{1}t^{2g-1} + \cdots+ a_{g-1}t^{g+1}+
a_gt^g +qa_{g-1}t^{g-1}+ \cdots + a_1 q^{g-1} t+q^g.
\end{displaymath}
The Weil polynomial of an abelian variety over a finite field ${\mathbb F}_q$ is the
characteristic polynomial of its Frobenius endomorphism.
The Weil polynomial of a curve over ${\mathbb F}_q$ is the Weil polynomial of its Jacobian.
The isogeny classes of abelian varieties
are completely classified by their Weil polynomials,
as the following theorem of Tate shows.
\begin{theorem}(Tate) \label{Tate}
Let $A$ and $B$ be abelian varieties defined over $\mathbb{F}_q$. Then an
abelian variety $A$ is $\mathbb{F}_q$-isogenous to an abelian subvariety of
$B$ if and only if $P_A(t)$ divides $P_B(t)$ over $\mathbb{Q}[t]$. In
particular, $P_A(t) = P_B(t)$ if and only if $A$ and $B$ are
$\mathbb{F}_q$-isogenous.
\end{theorem}
\subsection{L-polynomials}
For a curve $C$ over ${\mathbb F}_q$ we
define for any $n\geq 1$
the number $N_n(C)$ of ${\mathbb F}_{q^n}$ rational points of $C$,
and we encode these numbers
into the zeta function of $C$
\[
Z_C(t ) = exp \biggl( \sum_{n\geq 1} N_n (C) \frac{t^n}{n} \biggr).
\]
It can be shown that the zeta function is a rational function of $t$
with denominator $(1-t)(1-qt)$.
The numerator of the zeta function of a curve $C$
is called the L-polynomial of $C$.
It can be shown that
the L-polynomial of $C$ is the reverse polynomial of the Weil polynomial of $J_C$,
which we denote $P_C(t)$, i.e.,
\[
L_C (t)=t^{2g} P_{C} (1/t).
\]
Let $P_{C} (t)=\prod_{i=1}^{2g}(t-\eta_i)$, so
$L_{C} (t)=\prod_{i=1}^{2g}(1-\eta_i t)$.
It can be shown that if $L_n(t)$ denotes the L-polynomial of
C over the extension ${\mathbb F}_{q^n}$,
then $L_n(t)=\prod_{i=1}^{2g} (1-\eta_i^n t)$.
We also have that
$N_n:=|C({\mathbb F}_{q^n})|=q^n+1-\sum_{i=1}^{2g} \eta_i^n$ for all $n\geq 1$,
and the Riemann Hypothesis $|\eta_i|=\sqrt{q}$,
which together yield
$|N_n -(q^n+1)| \leq 2g\sqrt{q^n}$ (the Hasse-Weil bound).
In this paper, all L-polynomials are for curves defined over ${\mathbb F}_2$ unless otherwise stated.
\subsection{Supersingularity}
An elliptic curve $E$ over ${\mathbb F}_q$ is called supersingular if its Weil polynomial
$t^2+at+q$ has the property that $a$ is divisible by $p$.
Equivalently, $E$ is supersingular if
$E(\overline{\mathbb{F}_q})$ has no points of order $p$.
A curve $C$ of genus $g$ over a finite field $\mathbb{F}_q$ is called supersingular if its
Jacobian is a supersingular abelian variety, i.e.,
its Jacobian is isogenous (over $\overline{\mathbb{F}_q}$) to $E^g$, where $E$ is
a supersingular elliptic curve.
For a supersingular curve $C$ defined over a finite field $\mathbb{F}_q$,
it was shown by Oort that $\eta_i = \sqrt{q}\cdot \zeta_i$
where $\zeta_i$ is a (not necessarily $i$-th) root of unity, for all $i$.
\subsection{Maximal and minimal curves}
Let $g$ be genus of the curve $C$ defined over $\mathbb F_{2^r}$ and let $\eta_1,\cdots,\eta_{2g}$ be the roots of reciprocal of the $L$- polynomial of $C$ over $\mathbb F_{2^r}$. Then the number of rational points of $C$ on $\mathbb F_{2^{rn}}$ is $$\#C(\mathbb F_{2^{rn}})=(2^{rn}+1)- \sum\limits_{i=1}^{2g}\eta_i^n.$$
As stated earlier $|\eta_i|=\sqrt{2^r}$ for all $i=1,\cdots,2g$. We call $C(\mathbb F_{2^{rn}})$ maximal if $\eta_i^n=-\sqrt{2^{rn}}$ for all $i=1,\cdots,2g$, so the Hasse-Weil
upper bound is met.
We call $C(\mathbb F_{2^{rn}})$ minimal if $\eta_i^n=\sqrt{2^{rn}}$ for all $i=1,\cdots,2g$.
A supersingular curve defined over ${\mathbb F}_q$ becomes maximal over some
finite extension of ${\mathbb F}_q$, because $\eta_i = \sqrt{q}\cdot \zeta_i$
where $\zeta_i$ is a (not necessarily $i$-th) root of unity, for all $i$.
\begin{proposition} \label{minimal-prop}
If $C(\mathbb F_{2^{rn}})$ is maximal or minimal for some $n \geq 1$ then $C(\mathbb F_{2^{r(2n)}})$ is minimal.
\end{proposition}
\begin{proof}
Since $\eta_i^n=\pm \sqrt{2^{rn}}$ for all $i=1,\cdots,2g$, we have $\eta_i^{2n}=(\pm \sqrt{2^{rn}})^2=\sqrt{2^{r(2n)}}$ for all $i=1,\cdots,2g$. The proof follows by definition of a minimal curve. \qed
\end{proof}
We will need the following simple observations later.
\begin{lemma}\label{lemma-unity}
The real numbers $1$ and $\sqrt 2$ cannot be written as a $\mathbb Q$-linear combination of primitive $16$-th roots of unity. Moreover, $1$ cannot be written as a $\mathbb Q$-linear combination of primitive $8$-th roots of unity.
\end{lemma}
\begin{proof}
Let $\eta_{2^n}=\exp(\pi/2^{n-1}) \in \mathbb C$ be a primitive $2^n$-th root of unity. Then $\eta_{2^n}$ is a root of $X^{2^{n-1}}+1$ which is the $2^n$-th cyclotomic polynomial and therefore irreducible over $\mathbb Q$. Let $K_n := \mathbb Q(\eta_{2^n})$. Then $1,\eta_{2^n},\cdots,\eta_{2^n}^{2^{n-1}-1}$ are linearly independent over $\mathbb Q$. Note that $\eta_{2^n}^i$ is a primitive $2^n$-th root of unity if and only if $i$ is odd.\\
Since $1=1$ and $\sqrt{2}=\eta_{16}^2-\eta_{16}^6$, $1$ and $\sqrt 2$ cannot be written as $\mathbb Q$-linear combination of primitive $16$-th roots of unity. Since $1=1$, $1$ cannot be written as $\mathbb Q$-linear combination of primitive $8$-th roots of unity. \qed
\end{proof}
\subsection{Quadratic forms}\label{QF}
We now recall the basic theory of quadratic forms over $\mathbb{F}_{2}$,
which we will use later.
Let $K=\mathbb{F}_{2^m}$, and
let $Q:K\longrightarrow \mathbb{F}_{2}$ be a quadratic form.
The polarization of $Q$ is the symplectic bilinear form $B$ defined by
\[
B(x,y)=Q(x+y)-Q(x)-Q(y).
\]
By definition the radical of $B$ (denoted $W$) is
\[
W =\{ x\in K : B(x,y)=0 \text{ for all $y\in K$}\}.
\]
The rank of $B$ is defined to be $m-\dim(W)$, and a well known theorem
states that the rank must be even.
Next let $Q|_W$ denote the restriction of $Q$ to $ W$, and let
\[
W_0=\{ x\in W : Q(x)=0\}
\]
(sometimes $W_0$ is called the singular radical of $Q$).
Note that $Q|_W$ is a linear map $W\longrightarrow \mathbb{F}_2$
with kernel $W_0$. Therefore
\begin{equation*}
\dim W_0 =
\begin{cases}
\dim(W) -1 & \text{if $Q|_W$ is onto} \\
\dim(W) & \text{if $Q|_W =0$ (i.e. $W=W_0$)}.%
\end{cases}%
\end{equation*}
The rank of $Q$ is defined to be $m-\dim(W_0)$.
The following theorem is well known, see Chapter $6$ of \cite{lidl} for example.
\bigskip
\begin{proposition}\label{counts}
Continue the above notation.
Let $N=|\{x\in K : Q(x)=0\}|$, and let $w=\dim(W)$.
If $Q$ has odd rank then $N=2^{m-1}$.
In this case, $\sum_{x\in K} (-1)^{Q(x)} =0$.
If $Q$ has even rank then $N=2^{m-1}\pm 2^{(m-2+w)/2}$.
\end{proposition}
\section{Motivating example: $F_2(n,t_1,t_2)$}\label{sec:motivation}
Since precisely half of the elements of ${\mathbb F}_{2^n}$ have trace zero and half of the elements have trace one, we have $F_2(n,0) =
F_2(n,1) = 2^{n-1}$ for all $n \ge 1$. Hence the four $F_2(n,t_1,t_2)$ are the
first interesting cases.
Cattell {\em et al. } \ have expressed $F_2(n,t_1,t_2)$ as sums of binomial coefficients~\cite[Thm. 4]{cattell}.
For $n \ge 2$, we instead write $F_2(n,t_1,t_2) = 2^{n-2} + f_2(n,t_1,t_2)$ and deduce
the formulae for $f_2(n,t_1,t_2)$ from~\cite[Thm. 3.4]{fitzyucas} for odd $n$, and~\cite[Thm. 3]{yucasmullen} for even $n$,
which are presented in Table~\ref{table:2coeffs}, in which the rightmost four columns cover the four
$(t_1,t_2)$ pairs.
\begin{table}[t]
\caption{$f_2(n,t_1,t_2)$}
\begin{center}\label{table:2coeffs}
\begin{tabular}{c|cccc}
\hline
$n \pmod{8}$ & $(0,0)$ & $(0,1)$ & $(1,0)$ & $(1,1)$ \\
\hline
$0$ & $-2^{n/2 - 1}$ & $2^{n/2 - 1}$ & $0$ & $0$ \\
$1$ & $2^{n/2 - 3/2}$ & $-2^{n/2 - 3/2}$ & $2^{n/2 - 3/2}$ & $-2^{n/2 - 3/2}$\\
$2$ & $0$ & $0$ & $-2^{n/2-1}$ & $2^{n/2-1}$ \\
$3$ & $-2^{n/2 - 3/2}$ & $2^{n/2 - 3/2}$ & $2^{n/2 - 3/2}$ & $-2^{n/2 - 3/2}$ \\
$4$ & $2^{n/2 - 1}$ & $-2^{n/2 - 1}$ & $0$ & $0$\\
$5$ & $-2^{n/2-3/2}$ & $2^{n/2-3/2}$ & $-2^{n/2-3/2}$ & $2^{n/2-3/2}$ \\
$6$ & $0$ & $0$ & $2^{n/2-1}$ & $-2^{n/2-1}$ \\
$7$ & $2^{n/2 - 3/2}$ & $-2^{n/2 - 3/2}$ & $-2^{n/2 - 3/2}$ & $2^{n/2 - 3/2}$ \\
\hline
\end{tabular}
\end{center}
\end{table}
As one can see, for each $(t_1,t_2)$ the expression for $f_2(n,t_1,t_2)$ depends on $n \bmod 8$.
This observation naturally gives rise to two related questions: why are the
formulae periodic, and why is the period $8$?
To answer these questions we Fourier-analyse the formulae, using the complex $8$-th roots of unity.
In particular, observe that dividing each $f_2(n,t_1,t_2)$ by $2^{n/2}$ gives a constant. We can thus
normalise by this factor and expand each as follows:
\begin{equation}\label{fourier}
f_2(n,t_1,t_2) = \sum_{k=0}^{7} g(t_1,t_2)_k \, (\sqrt{2}\omega_{8}^{k})^n = 2^{n/2}
\, \sum_{k=0}^{7} g(t_1,t_2)_k \, \omega_{8}^{kn},
\end{equation}
where the $g(t_1,t_2)_k$ are unknown constants and $\omega_8 = e^{i \pi/4} = (1 + i)/\sqrt{2}$.
For each $(t_1,t_2)$, let $\overline{v}_{t_1,t_2}$ be the length-$8$ vector
$f_2(n,t_1,t_2)/2^{n/2}$, i.e.,
\begin{eqnarray}
\nonumber \overline{v}_{0,0} &=& (-2^{-1},2^{-3/2},0,-2^{-3/2},2^{-1},-2^{-3/2},0,2^{-3/2}),\\
\nonumber \overline{v}_{0,1} &=& (2^{-1},-2^{-3/2},0,2^{-3/2},-2^{-1},2^{-3/2},0,-2^{-3/2}),\\
\nonumber \overline{v}_{1,0} &=& (0,2^{-3/2},-2^{-1},2^{-3/2},0,-2^{-3/2},2^{-1},-2^{-3/2}),\\
\nonumber \overline{v}_{1,1} &=& (0,-2^{-3/2},2^{-1},-2^{-3/2},0,2^{-3/2},-2^{-1},2^{-3/2}).
\end{eqnarray}
Let $M$ be the discrete Fourier transform matrix for $\omega_8$ with entries $M_{j,k} =
\omega_{8}^{(j-1)\cdot(k-1)}$. Then $\overline{g}(t_1,t_2) = [g(t_1,t_2)_0,\ldots,g(t_1,t_2)_{7}]$
satisfies:
\[
M \cdot \overline{g}(t_1,t_2)^{T} = \overline{v}_{t_1,t_2}^T.
\]
Since $M_{j,k}^{-1} = \frac{1}{8} \cdot \omega_{8}^{-(j-1)\cdot(k-1)}$, we easily compute
\[
\overline{g}(t_1,t_2)^T = M^{-1} \cdot \overline{v}_{t_1,t_2}^T.
\]
Then substituting each $\overline{g}(t_1,t_2)$ into the Fourier expansion~(\ref{fourier}), we obtain
\begin{proposition}
For $n \ge 2$ we have
\begin{eqnarray}
\nonumber f_2(n,0,0) &=& -\frac{1}{4}((\sqrt{2}\omega_{8}^3)^n +
(\sqrt{2}\omega_{8}^5)^n) = -\frac{1}{4}((-1+i)^n + (-1-i)^n),\\
\nonumber f_2(n,0,1) &=& \frac{1}{4}((\sqrt{2}\omega_{8}^3)^n +
(\sqrt{2}\omega_{8}^5)^n) = \frac{1}{4}((-1+i)^n + (-1-i)^n),\\
\nonumber f_2(n,1,0) &=& -\frac{i}{4}((\sqrt{2}\omega_{8}^3)^{n} -
(\sqrt{2}\omega_{8}^5)^{n}) = -\frac{i}{4}((-1+i)^n - (-1-i)^n),\\
\nonumber f_2(n,1,1) &=& \frac{i}{4}((\sqrt{2}\omega_{8}^3)^{n} -
(\sqrt{2}\omega_{8}^5)^{n}) = \frac{i}{4}((-1+i)^n - (-1-i)^n).
\end{eqnarray}
\end{proposition}
By a theorem due to G\"olo\u{g}lu, McGuire and Moloney, the expressions for
$F_2(n,0,0)$, $F_2(n,1,1)$, $F_2(n,0,1)$ and $F_2(n,1,0)$ count the number of
elements $a \in {\mathbb F}_{2^n}$ for which the corresponding binary Kloosterman sum is
congruent to $0,4,8$ and $12 \pmod{16}$, respectively~\cite[Thm. 1.2]{faruk1}.
Furthermore, Lison\v{e}k and Moisio have explicitly computed the number of elements
$a \in {\mathbb F}_{2^n}$ for which the Kloosterman sum is divisible by $16$~\cite[Thm 3.6]{lisonekmoisio},
relating it to the number of ${\mathbb F}_{2^n}$-rational points on the supersingular elliptic curve
\[
E_1/{\mathbb F}_2 : y^2 + y = x^3 + x.
\]
By G\"olo\u{g}lu {\em et al. }'s result, this number is therefore related to $F_2(n,0,0)$.
We will make repeated use of the following easy lemma, which is an immediate generalisation of~\cite[Lemma 1.1]{fitzyucas},
which was proven for the base field ${\mathbb F}_2$ case.
\begin{lemma}\label{lem:T2T3}
For all $\alpha, \beta \in {\mathbb F}_{q^n}$ we have:
\begin{itemize}
\item[(i)] $T_2(\alpha + \beta) + T_2(\alpha) + T_2(\beta) = T_1(\alpha)T_1(\beta)
+ T_1(\alpha\beta)$
\item[(ii)] $T_3(\alpha + \beta) + T_3(\alpha) + T_3(\beta) =
T_2(\alpha)T_1(\beta) + T_1(\alpha)T_2(\beta)
+ T_1(\alpha^2\beta + \alpha\beta^2) + T_1(\alpha\beta)T_1(\alpha+\beta)$.
\end{itemize}
\end{lemma}
We can now reprove the connection between $F_2(n,0,0)$ and $\#E_1({\mathbb F}_{2^n})$ directly as follows.
If $T_1(a) = 0$ then $a = x^2 + x$ for two $x \in {\mathbb F}_{2^n}$.
By Lemma~\ref{lem:T2T3}(i), we have $T_2(x^2 + x) = T_1(x^3 + x)$, and so for $n \ge 2$ we have:
\begin{eqnarray}
\nonumber F_2(n,0,0) &=& \frac{1}{2} \, \#\{ x \in {\mathbb F}_{2^n} \mid T_1(x^3 + x) = 0\}\\
\nonumber &=& \frac{1}{4} \, \#\{ (x,y) \in {\mathbb F}_{2^n} \times
{\mathbb F}_{2^n} \mid y^2 + y = x^3 + x\}\\
\nonumber &=& \frac{1}{4} \, (\#E_1({\mathbb F}_{2^n}) - 1)\\
\nonumber &=& 2^{n-2} - \frac{(-1+i)^n + (-1 - i)^n}{4},
\end{eqnarray}
where in the final line we have used the fact that the characteristic polynomial of Frobenius of $E_1$
is $P_{E_1}(X) = X^2 + 2X + 2 = (X - \sqrt{2}\omega_{8}^3)(X - \sqrt{2}\omega_{8}^5)$, since the trace of $E_1$ is $-2$.
As $F_2(n,0,0) + F_2(n,0,1) = F_2(n,0) = 2^{n-1}$, we immediately have
$f_2(n,0,1) = -f_2(n,0,0)$. The periodicity for the trace zero cases thus arises from the connection
with the supsersingular elliptic curve $E_1$, whose Weil $q$-numbers are $\sqrt{2}\omega_{8}^{\pm 3}$.
For the trace one cases, note that the form of $f_2(n,1,0)$ and $f_2(n,1,1)$ implies that they do not arise from the
zeta function of an abelian variety, but are related to
$f_2(n,0,0)$ and $f_2(n,0,1)$, as derived in Lemma 2.6 of~\cite{fitzyucas} for odd $n$,
and~\cite{yucasmullen} (deduced from~\cite{cattell}) for even $n$. Since our focus is on the
application of supersingular curves to the counting problem, we omit these cases for now, but indicate how supersingular
curves may also be used for the trace one cases, for odd $n$, in the next section.
\section{Recomputing $F_2(n,0,t_2,t_3)$}\label{sec:3coeffsF2}
We now apply the approach of the previous section to compute $F_2(n,0,t_2,t_3)$.
For the general case where $t_1$ may be 0 or $1$, there are two complementary results
already known.
Fitzgerald and Yucas gave formulae for $F_2(n,t_1,t_2,t_3)$ when $n$ is odd~\cite{fitzyucas}, while Yucas and Mullen gave
formulae for $F_2(n,t_1,t_2,t_3)$ when $n$ is even~\cite{yucasmullen}. As in Section~\ref{sec:motivation} we Fourier-analyse
these formulae in order to determine which supersingular curves are relevant and then prove the formulae directly.
As before, for $n \ge 3$ we write $F_2(n,0,t_2,t_3) = 2^{n-3} + f_2(n,0,t_2,t_3)$. Table~\ref{table:3coeffs} gives the
set of formulae for $f_2(n,0,t_1,t_2)$, in which the rightmost four columns cover the possible $(0,t_2,t_3)$ triples.
\begin{table}[t]
\caption{$f_2(n,0,t_2,t_3)$}
\begin{center}\label{table:3coeffs}
\begin{tabular}{c|cccc}
\hline
$n \pmod{24}$ & $(0,0,0)$ & $(0,0,1)$ & $(0,1,0)$ & $(0,1,1)$\\
\hline
$0$ & $-5 \cdot 2^{n/2 - 2}$ & $3 \cdot 2^{n/2-2}$ & $2^{n/2-2}$ & $2^{n/2-2}$ \\
$1$ & $3 \cdot 2^{n/2 - 5/2}$ & $-2^{n/2-5/2}$ & $-2^{n/2-5/2}$ & $-2^{n/2-5/2}$ \\
$2$ & $2^{n/2-2}$ & $-2^{n/2-2}$ & $2^{n/2-2}$ & $-2^{n/2-2}$ \\
$3$ & $0$ & $-2^{n/2-3/2}$ & $-2^{n/2-3/2}$ & $2^{n/2 - 1/2}$ \\
$4$ & $0$ & $2^{n/2-1}$ & $0$ & $-2^{n/2-1}$ \\
$5$ & $-3 \cdot 2^{n/2-5/2}$ & $2^{n/2-5/2}$ & $2^{n/2-5/2}$ & $2^{n/2-5/2}$ \\
$6$ & $2^{n/2-2}$ & $-2^{n/2-2}$ & $2^{n/2-2}$ & $-2^{n/2-2}$ \\
$7$ & $3 \cdot 2^{n/2 - 5/2}$ & $-2^{n/2-5/2}$ & $-2^{n/2-5/2}$ & $-2^{n/2-5/2}$ \\
$8$ & $-2^{n/2 - 1}$ & $0$ & $-2^{n/2-1}$ & $2^{n/2}$ \\
$9$ & $0$ & $2^{n/2-3/2}$ & $2^{n/2-3/2}$ & $-2^{n/2 - 1/2}$ \\
$10$ & $2^{n/2-2}$ & $-2^{n/2-2}$ & $2^{n/2-2}$ & $-2^{n/2-2}$ \\
$11$ & $-3 \cdot 2^{n/2-5/2}$ & $2^{n/2-5/2}$ & $2^{n/2-5/2}$ & $2^{n/2-5/2}$ \\
$12$ & $3 \cdot 2^{n/2 - 2}$ & $-2^{n/2-2}$ & $-3 \cdot 2^{n/2-2}$& $2^{n/2-2}$ \\
$13$ & $-3 \cdot 2^{n/2-5/2}$ & $2^{n/2-5/2}$ & $2^{n/2-5/2}$ & $2^{n/2-5/2}$ \\
$14$ & $2^{n/2-2}$ & $-2^{n/2-2}$ & $2^{n/2-2}$ & $-2^{n/2-2}$ \\
$15$ & $0$ & $2^{n/2-3/2}$ & $2^{n/2-3/2}$ & $-2^{n/2 - 1/2}$ \\
$16$ & $-2^{n/2 - 1}$ & $0$ & $-2^{n/2-1}$ & $2^{n/2}$ \\
$17$ & $3 \cdot 2^{n/2 - 5/2}$ & $-2^{n/2-5/2}$ & $-2^{n/2-5/2}$ & $-2^{n/2-5/2}$ \\
$18$ & $2^{n/2-2}$ & $-2^{n/2-2}$ & $2^{n/2-2}$ & $-2^{n/2-2}$ \\
$19$ & $-3 \cdot 2^{n/2 - 5/2}$ & $2^{n/2-5/2}$ & $2^{n/2-5/2}$ & $2^{n/2-5/2}$ \\
$20$ & $0$ & $2^{n/2-1}$ & $0$ & $-2^{n/2-1}$\\
$21$ & $0$ & $-2^{n/2-3/2}$ & $-2^{n/2-3/2}$ & $2^{n/2 - 1/2}$ \\
$22$ & $2^{n/2-2}$ & $-2^{n/2-2}$ & $2^{n/2-2}$ & $-2^{n/2-2}$ \\
$23$ & $3 \cdot 2^{n/2 - 5/2}$ & $-2^{n/2-5/2}$ & $-2^{n/2-5/2}$ & $-2^{n/2-5/2}$ \\
\hline
\end{tabular}
\end{center}
\end{table}
As in Section~\ref{sec:motivation}, since the set of formulae have period $24$, one can express
each in terms of the complex $24$-th roots of unity. We thus expand each as follows:
\begin{equation}\label{fourier2}
f_2(n,0,t_2,t_3) = \sum_{k=0}^{23} g(0,t_2,t_3)_k \, (\sqrt{2} \omega_{24}^{k})^n =
2^{n/2} \, \sum_{k=0}^{23} g_k \, \omega_{24}^{kn},
\end{equation}
with $\omega_{24} = e^{i \pi /12} = ((1+\sqrt{3}) + (-1+\sqrt{3})i)/2\sqrt{2}$.
For each $(t_2,t_3)$, let $\overline{v}_{0,t_2,t_3}$ be the vector of constants
$f_2(n,0,t_2,t_3)/2^{n/2}$.
Let $M$ be the discrete Fourier transform matrix with $M_{j,k} =
\omega_{24}^{(j-1)\cdot(k-1)}$. Then $\overline{g}(0,t_2,t_3) = [g(0,t_2,t_3)_k]_{0 \le k \le 23}$
satisfies:
\[
M \cdot \overline{g}(0,t_2,t_3)^{T} = \overline{v}_{0,t_2,t_3}^T.
\]
Since $M_{j,k}^{-1} = \frac{1}{24} \omega_{24}^{-(j-1)\cdot(k-1)}$ we easily compute
\[
\overline{g}(0,t_2,t_3)^T = M^{-1} \cdot \overline{v}_{0,t_2,t_3}^{T}.
\]
Then substituting each $\overline{g}(t_1,t_2)$ into the Fourier expansion~(\ref{fourier2}), we obtain
\begin{proposition}\label{prop:F2formulae}
For $n \ge 3$ we have
\begin{eqnarray}
\nonumber f_2(n,0,0,0) &=& -\frac{1}{4}((\sqrt{2}\omega_{8}^3)^n + (\sqrt{2}\omega_{8}^5)^n)
-\frac{1}{8}((\sqrt{2}i)^n + (-\sqrt{2}i)^n)\\
\nonumber & & -\frac{1}{8}((\sqrt{2}\omega_{24}^{5})^n + (\sqrt{2}\omega_{24}^{11})^n
+ (\sqrt{2}\omega_{24}^{13})^n + (\sqrt{2}\omega_{24}^{19})^n)\\
\nonumber f_2(n,0,0,1) &=& \frac{1}{8}((\sqrt{2}i)^n + (-\sqrt{2}i)^n)\\
\nonumber & & +\frac{1}{8}((\sqrt{2}\omega_{24}^{5})^n + (\sqrt{2}\omega_{24}^{11})^n
+ (\sqrt{2}\omega_{24}^{13})^n + (\sqrt{2}\omega_{24}^{19})^n)\\
\nonumber f_2(n,0,1,0) &=& -\frac{1}{8}((\sqrt{2}i)^n + (-\sqrt{2}i)^n)\\
\nonumber & & +\frac{1}{8}((\sqrt{2}\omega_{24}^{5})^n + (\sqrt{2}\omega_{24}^{11})^n
+ (\sqrt{2}\omega_{24}^{13})^n + (\sqrt{2}\omega_{24}^{19})^n)\\
\nonumber f_2(n,0,1,1) &=& \frac{1}{4}((\sqrt{2}\omega_{8}^3)^n + (\sqrt{2}\omega_{8}^5)^n)
+ \frac{1}{8}((\sqrt{2}i)^n + (-\sqrt{2}i)^n)\\
\nonumber & & -\frac{1}{8}((\sqrt{2}\omega_{24}^{5})^n + (\sqrt{2}\omega_{24}^{11})^n + (\sqrt{2}\omega_{24}^{13})^n + (\sqrt{2}\omega_{24}^{19})^n).
\end{eqnarray}
\end{proposition}
As before the $(\sqrt{2}\omega_{8}^3)^n + (\sqrt{2}\omega_{8}^5)^n$ term
arises from the roots of the characteristic polynomial of Frobenius $P_{E_1}(X) = X^2 + 2X + 2$. The term
$(\sqrt{2}i)^n + (-\sqrt{2}i)^n$ arises from the roots of $P_{E_2}(X) = X^2 + 2$ corresponding to the elliptic curve
\[
E_2/{\mathbb F}_2: y^2 + y = x^3 + 1.
\]
Finally the term $(\sqrt{2}\omega_{24}^{5})^n + (\sqrt{2}\omega_{24}^{11})^n + (\sqrt{2}\omega_{24}^{13})^n + (\sqrt{2}\omega_{24}^{19})^n$
arises from the roots of $P_{H_1}(X) = X^4 + 2X^3 + 2X^2+ 4X + 4$ corresponding to the genus $2$ supersingular curve (see~\cite[\S4]{xing})
\[
H_1/{\mathbb F}_2: y^2 + y = x^5 + x^3.
\]
In order to prove Proposition~\ref{prop:F2formulae} starting from the relevant supersingular curves, we use the following easy lemma
due to Fitzgerald. For a function $q_i: {\mathbb F}_{2^n} \rightarrow {\mathbb F}_2$ let $Z(q_i)$ denote the number of zeros of $q_i$, and for $w_i \in {\mathbb F}_2$
let $N(w_1,w_2)$ be the number of $x \in {\mathbb F}_{2^n}$ such that $q_i(x) = w_i$, for $i=1,2$.
\begin{lemma}~\cite[Prop. 5.3]{fitz}\label{lem:fitz}
Let $q_1,q_2: {\mathbb F}_{2^n} \rightarrow {\mathbb F}_2$ be functions. Then
\[
N(0,0) = \frac{1}{2} \big( Z(q_1) + Z(q_2) + Z(q_1+q_2) - 2^n \big).
\]
\end{lemma}
Note that Fitzgerald stipulates that $q_1,q_2$ be quadratic forms, since his application requires only this;
the proof however only requires that they be functions. We prove a generalisation of
Lemma~\ref{lem:fitz} in Section~\ref{sec:2r}. We now prove Proposition~\ref{prop:F2formulae}.
\begin{proof}
By Lemma~\ref{lem:T2T3}(i) we have $T_2(x^2 + x) = T_1(x^3 + x)$ and by Lemma~\ref{lem:T2T3}(ii), we
have $T_3(x^2 + x) = T_1(x^5 + x)$. Therefore, let
\begin{eqnarray}
\nonumber q_1 &:& {\mathbb F}_{2^n} \rightarrow {\mathbb F}_2: x \mapsto T_1(x^3 + x),\\
\nonumber q_2 &:& {\mathbb F}_{2^n} \rightarrow {\mathbb F}_2: x \mapsto T_1(x^5 + x).
\end{eqnarray}
For $n \ge 3$ we have:
\begin{eqnarray}
\nonumber F_2(n,0,0,0) &=& \frac{1}{2} \, \#\{ x \in {\mathbb F}_{2^n} \mid T_1(x^3 + x) = T_1(x^5 + x) = 0\}\\
\nonumber &=& \frac{1}{2} \, \#\{ x \in {\mathbb F}_{2^n} \mid q_1(x) = q_2(x) = 0\}\\
\nonumber &=& \frac{1}{4} \, (Z(q_1) + Z(q_2) + Z(q_1 + q_2) - 2^n),
\end{eqnarray}
by Lemma~\ref{lem:fitz}. Treating these zero-counts in turn, we have
\begin{eqnarray}
\nonumber Z(q_1) &=& \frac{1}{2} \, \#\{ (x,y) \in {\mathbb F}_{2^n} \times {\mathbb F}_{2^n} \mid y^2 + y = x^3 + x\}\\
\nonumber &=& \frac{1}{2} \, (\#E_1({\mathbb F}_{2^n}) - 1)\\
\nonumber &=& \frac{1}{2} \, (2^n - (\sqrt{2}\omega_{8}^3)^n - (\sqrt{2}\omega_{8}^5)^n).
\end{eqnarray}
Secondly, we have
\begin{eqnarray}
\nonumber Z(q_2) &=& \frac{1}{2} \, \#\{ (x,y) \in {\mathbb F}_{2^n} \times {\mathbb F}_{2^n} \mid y^2 + y = x^5 + x\}\\
\nonumber &=& \frac{1}{2} \, (\#H_2({\mathbb F}_{2^n}) - 1)\\
\nonumber &=& \frac{1}{2} \, (2^n - (\sqrt{2}i)^n - (-\sqrt{2}i)^n - (\sqrt{2}\omega_{8}^3)^n - (\sqrt{2}\omega_{8}^5)^n ),
\end{eqnarray}
since the Jacobian of $H_2/{\mathbb F}_2:y^2 + y = x^5 + x$ is ${\mathbb F}_2$-isogenous to the product of the two supersingular elliptic curves $E_1$
and $E_2$~\cite[\S4]{xing}.
Finally, we have
\begin{eqnarray}
\nonumber Z(q_1 + q_2) &=& \frac{1}{2} \, \#\{ (x,y) \in {\mathbb F}_{2^n} \times {\mathbb F}_{2^n} \mid y^2 + y = x^5 + x^3\}\\
\nonumber &=& \frac{1}{2} \, (\#H_1({\mathbb F}_{2^n}) - 1)\\
\nonumber &=& \frac{1}{2} \, (2^n - (\sqrt{2}\omega_{24}^{5})^n -
(\sqrt{2}\omega_{24}^{11})^n - (\sqrt{2}\omega_{24}^{13})^n - (\sqrt{2}\omega_{24}^{19})^n ),
\end{eqnarray}
Combining these as per Lemma~\ref{lem:fitz} gives $2^{n-3} + f_2(n,0,0,0)$ as required.
Furthermore, since $F_2(n,0,0,0) + F_2(n,0,0,1) = F_2(n,0,0)$, the formula for $f_2(n,0,0,1)$ follows.
For the remaining two, let $F_2(n,0,*,0)$ be the count of those $a \in {\mathbb F}_{2^n}$ for which $T_1(a)= T_3(a) = 0$ and we do not mind
what $T_2(a)$ is. Using the count for $Z(q_2)$ we have:
\begin{eqnarray}
\nonumber F_2(n,0,*,0) &=& \frac{1}{2} \, \#\{ x \in {\mathbb F}_{2^n} \mid T_1(x^5 + x) = 0\}\\
\nonumber &=& \frac{1}{2} \, Z(q_2)\\
\nonumber &=& \frac{1}{4} \, (2^n - (\sqrt{2}i)^n - (-\sqrt{2}i)^n - (\sqrt{2}\omega_{8}^3)^n - (\sqrt{2}\omega_{8}^5)^n ).
\end{eqnarray}
Since $F_2(n,0,*,0) = F_2(n,0,1,0) + F_2(n,0,0,0)$, we have
\begin{eqnarray}
\nonumber & & 2^{n-2} - \frac{1}{4}((\sqrt{2}i)^n + (-\sqrt{2}i)^n + (\sqrt{2}\omega_{8}^3)^n + (\sqrt{2}\omega_{8}^5)^n)\\
\nonumber &=& F_2(n,0,1,0) + 2^{n-3} -\frac{1}{4}((\sqrt{2}\omega_{8}^3)^n + (\sqrt{2}\omega_{8}^5)^n)
-\frac{1}{8}((\sqrt{2}i)^n + (-\sqrt{2}i)^n)\\
\nonumber &-& \frac{1}{8}((\sqrt{2}\omega_{24}^{5})^n + (\sqrt{2}\omega_{24}^{11})^n + (\sqrt{2}\omega_{24}^{13})^n + (\sqrt{2}\omega_{24}^{19})^n).
\end{eqnarray}
Rearranging gives the formula for $f_2(n,0,1,0)$.
Finally, by $F_2(n,0,1,0) + F_2(n,0,1,1) = F_2(n,0,1)$, the formula for $f_2(n,0,1,1)$ follows.\qed
\end{proof}
Our proof for all $n$ is more direct than the amalgamation of the proofs given in~\cite{fitzyucas} ($n$ odd) and~\cite{yucasmullen} ($n$ even), and
also explains in a simple way why the periodicity arises. Furthermore, for $n$ odd one can derive the trace one formulae from the trace zero formulae
using~\cite[Lemma 2.6]{fitzyucas}, and thus all of the results of that paper.
One can also derive the trace one formulae for $n$ odd directly, using the same curve based approach, noting that $T_1(1) = 1$ and
expanding $T_2(x^2 + x + 1)$ and $T_3(x^2 + x +1)$ and adjusting the definitions of $q_1$ and $q_2$ accordingly.
For completeness, we also include the Fourier-analysed formulae for the trace one cases, derived from~\cite{fitzyucas} and~\cite{yucasmullen}.
\begin{proposition}
For $n \ge 3$ we have
\begin{eqnarray}
\nonumber f_2(n,1,0,0) &=& \frac{1}{8}((\sqrt{2}i)^n + (-\sqrt{2}i)^n)
-\frac{1+i}{8}(\sqrt{2}\omega_{8}^3)^n - \frac{1-i}{8}(\sqrt{2}\omega_{8}^5)^n\\
\nonumber & & -\frac{i}{8}((\sqrt{2}\omega_{24}^{5})^n - (\sqrt{2}\omega_{24}^{11})^n
+ (\sqrt{2}\omega_{24}^{13})^n - (\sqrt{2}\omega_{24}^{19})^n)\\
\nonumber f_2(n,1,0,1) &=& -\frac{1}{8}((\sqrt{2}i)^n + (-\sqrt{2}i)^n)
+\frac{1-i}{8}(\sqrt{2}\omega_{8}^3)^n + \frac{1+i}{8}(\sqrt{2}\omega_{8}^5)^n\\
\nonumber & & +\frac{i}{8}((\sqrt{2}\omega_{24}^{5})^n - (\sqrt{2}\omega_{24}^{11})^n
+ (\sqrt{2}\omega_{24}^{13})^n - (\sqrt{2}\omega_{24}^{19})^n)\\
\nonumber f_2(n,1,1,0) &=& -\frac{1}{8}((\sqrt{2}i)^n + (-\sqrt{2}i)^n)
+\frac{1+i}{8}(\sqrt{2}\omega_{8}^3)^n + \frac{1-i}{8}(\sqrt{2}\omega_{8}^5)^n\\
\nonumber & & -\frac{i}{8}((\sqrt{2}\omega_{24}^{5})^n - (\sqrt{2}\omega_{24}^{11})^n
+ (\sqrt{2}\omega_{24}^{13})^n - (\sqrt{2}\omega_{24}^{19})^n)\\
\nonumber f_2(n,1,1,1) &=& \frac{1}{8}((\sqrt{2}i)^n + (-\sqrt{2}i)^n)
-\frac{1-i}{8}(\sqrt{2}\omega_{8}^3)^n - \frac{1+i}{8}(\sqrt{2}\omega_{8}^5)^n\\
\nonumber & & +\frac{i}{8}((\sqrt{2}\omega_{24}^{5})^n - (\sqrt{2}\omega_{24}^{11})^n
+ (\sqrt{2}\omega_{24}^{13})^n - (\sqrt{2}\omega_{24}^{19})^n).
\end{eqnarray}
\end{proposition}
\section{Computing $F_{q}(n,0,0,0)$ using supersingular curves}\label{sec:2r}
In this and the following two sections we shall compute $F_{q}(n,0,0,0)$ and Fourier-analyse the formulae.
Recall that $q = 2^r$ with $r \ge 1$, which we use interchangeably as appropriate for the remainder of the paper.
As in Sections~\ref{sec:motivation} and~\ref{sec:3coeffsF2}, for $a \in {\mathbb F}_{q^n}$, if $T_1(a) = 0$ then
$a = x^q + x$ for $q$ $x \in {\mathbb F}_{q^n}$. By Lemma~\ref{lem:T2T3}(i) and (ii), we have $T_2(x^q + x) = T_1(x^{q+1} + x^2)$ and
$T_3(x^q + x) = T_1(x^{2q+1} + x^{q+2})$ respectively. Therefore, let
\begin{eqnarray}
\nonumber q_1 &:& {\mathbb F}_{q^n} \rightarrow {\mathbb F}_q: x \mapsto T_1(x^{q+1} + x^2),\\
\nonumber q_2 &:& {\mathbb F}_{q^n} \rightarrow {\mathbb F}_q: x \mapsto T_1(x^{2q+1} + x^{q+2}).
\end{eqnarray}
We have the following generalisation of Lemma~\ref{lem:fitz}.
As before, for a function $q_i: {\mathbb F}_{q^n} \rightarrow {\mathbb F}_{q}$ let $Z(q_i)$ denote the number of zeros of $q_i$, and
for $w_i \in {\mathbb F}_q$ let $N(w_1,w_2)$ be the number of $x \in {\mathbb F}_{q^n}$ such that $q_i(x) = w_i$, for $i=1,2$.
\begin{lemma}\label{lem:fitzgeneral}
Let $q_1,q_2: {\mathbb F}_{q^n} \rightarrow {\mathbb F}_q$ be any functions. Then
\[
N(0,0) = \frac{1}{q} \big( Z(q_1) + \sum_{\alpha \in {\mathbb F}_q} Z(\alpha q_1 + q_2) - q^n \big)
\]
\end{lemma}
\begin{proof}
We have the following identities:
\[
q^n = \sum_{\alpha,\beta \in {\mathbb F}_q} N(\alpha,\beta), \ Z(q_1) =
\sum_{\beta \in {\mathbb F}_q} N(0,\beta), \ \text{and} \
\sum_{\alpha \in {\mathbb F}_q} Z(\alpha q_1 + q_2) = \sum_{\alpha,\beta \in {\mathbb F}_q} N(\beta, \alpha\beta).
\]
We have
\begin{eqnarray}
\label{gen1_1} \sum_{\alpha,\beta \in {\mathbb F}_q} N(\alpha,\beta) &=& \sum_{\beta \in
{\mathbb F}_q} N(0,\beta) + \sum_{\beta \in {\mathbb F}_q} \sum_{\alpha \in
{\mathbb F}_{q}^{\times}} N(\alpha, \beta)\\
&=&
\label{gen1_2} \sum_{\beta \in {\mathbb F}_{q}} N(0,\beta) + \sum_{\beta \in {\mathbb F}_{q}} \sum_{\alpha \in
{\mathbb F}_{q}^{\times}} N(\alpha, \alpha \beta)\\
\nonumber &=& \sum_{\beta \in {\mathbb F}_{q}} N(0,\beta) + \sum_{\alpha,\beta \in {\mathbb F}_{q}}
N(\beta, \alpha\beta) - q N(0,0),
\end{eqnarray}
where the equality between the second terms on the r.h.s. of lines~(\ref{gen1_1}) and~(\ref{gen1_2}) follow from the
fact that for each $\alpha \in {\mathbb F}_{q}^{\times}$ we have
$\{N(\alpha,\beta)\}_{\beta \in {\mathbb F}_q} = \{N(\alpha,\alpha
\beta)\}_{\beta \in {\mathbb F}_q}$. Rearranging for $N(0,0)$ gives the stated result.\qed
\end{proof}
Informed by Lemma~\ref{lem:fitzgeneral} we employ the following curves, for $\alpha \in {\mathbb F}_q$:
\begin{eqnarray}
\nonumber C &:& y^q + y = x^{q+1} + x^2,\\
\nonumber C_{\alpha} &:& y^q + y = x^{2q+1} + x^{q+2} + \alpha (x^{q+1} + x^2).
\end{eqnarray}
Observe that for $\alpha \in {\mathbb F}_{q}^{\times}$ we have the ${\mathbb F}_q$-isomorphism $\sigma: C_{\alpha} \longrightarrow C_1 :
(x,y) \mapsto (\alpha x, \alpha^3 y)$ and thus their number of rational points over any extension of ${\mathbb F}_q$ are equal.
Hence the formula in Lemma~\ref{lem:fitzgeneral} simplifies to:
\begin{equation}\label{simple}
N(0,0) = \frac{1}{q} \big( Z(q_1) + Z(q_2) + (q-1)Z(q_1 + q_2) - q^n \big).
\end{equation}
In the following we therefore focus on these three curves:
\begin{eqnarray}
\nonumber C_1/{\mathbb F}_q &:& y^q + y = x^{q+1} + x^2,\\
\nonumber C_2/{\mathbb F}_q &:& y^q + y = x^{2q+1} + x^{q+2},\\
\nonumber C_3/{\mathbb F}_q &:& y^q + y = x^{2q+1} + x^{q+2} + x^{q+1} + x^2.
\end{eqnarray}
By~(\ref{simple}) one can express $F_{q}(n,0,0,0)$ in terms of the number of points of $C_1,C_2$ and $C_3$ over ${\mathbb F}_{q^n}$.
We shall compute these numbers, which incidentally proves the supersingularity of the three curves (there are many other ways to do so too).
In the rest of this section we detail some preliminary notions and results.
\subsection{Simplifying the point counting on $C_1$, $C_2$ and $C_3$}
In this subsection we show that counting the number of points on the curves $C_1$, $C_2$ and $C_3$ over ${\mathbb F}_{q^n}$ can be simplified by obtaining a partial decomposition of the Jacobian of Artin-Schreier curves.
Let $\overline{{\mathbb F}_2}(x)$ denote the rational function field over $\overline{{\mathbb F}_2}$ which is the algebraic closure of the binary field ${\mathbb F}_2$. An Artin-Schreier curve is
\[C:y^q+y=f(x)\]
where $q=2^r$ and $f(x)\in\overline{{\mathbb F}_2}(x)$ is not a constant and has no pole of an even order. When $f(x)$ is a polynomial this is equivalent to assuming that $f(x)$ is of an odd degree. If $P_1,P_2,\ldots,P_m$ are the poles of $f(x)$ of order $e_1,e_2,\ldots,e_m$ respectively, then the genus of $C$ is
\begin{equation}\label{genus}
g_C=\frac{q-1}{2}(\sum_{i=1}^m(e_i+1)-2) .
\end{equation}
If $f(x)$ is a polynomial of an odd degree $d$, then the genus of $C$ simply is
\begin{equation}
g_C=\frac{(q-1)(d-1)}{2}.
\end{equation}
We obtain a partial decomposition of the Jacobian of the Artin-Schreier curve $C$ by studying the set of automorphisms of the l.h.s. $y^q + y$ only. Computing the relevant quotient curves and applying a theorem due to Kani and Rosen~\cite{kanirosen} allows one to then infer a relation between the number of points on the original curve and on the quotient curves.
For any positive integer $m$, let $\Tr_{m}: {\mathbb F}_{2^m} \rightarrow {\mathbb F}_{2}$ be the absolute trace function $a \mapsto a + a^{2} + a^{2^2} + \cdots + a^{2^{m-1}}$. We need the following lemma to establish the main result of this subsection.
\begin{lemma}\label{subgroup}
Let $\alpha$ be a nonzero element of ${\mathbb F}_{2^r}$, and let $H_\alpha=\{x\in {\mathbb F}_{2^r}|\Tr_{r}(\alpha x)=0\}$. Then
\begin{itemize}
\item[(i)] $H_\alpha$ is an additive subgroup of ${\mathbb F}_{2^r}$ when ${\mathbb F}_{2^r}$ is viewed as an additive group,
\item[(ii)] half the elements of ${\mathbb F}_{2^r}$ are in $H_\alpha$, i.e., $\#H_\alpha=2^{r-1}$, and
\item[(iii)]$\prod_{\beta\in {\mathbb F}_{2^r}\setminus H_\alpha}\beta=\alpha^{2^{r-1}-1}$.
\end{itemize}
\end{lemma}
\begin{proof}
It is easy to check the first part of the claim. To show (ii), notice that the traces of half the elements of ${\mathbb F}_{2^r}$ are zero, and since $\alpha$ is an element of ${\mathbb F}_{2^r}$, the map $\psi:H_\alpha\longrightarrow {\mathbb F}_{2^r}$ taking $x$ to $\alpha x$ induces a bijection between elements of $H_\alpha$ and elements $z$ of ${\mathbb F}_{2^r}$ for which $\Tr_{r}(z)=0$. To prove (iii), we first notice that
\[\Tr_{r}(\alpha x)=\alpha x+(\alpha x)^2+\cdots+(\alpha x)^{2^{r-1}},\]
and hence $\Tr_{r}(\alpha x)$ is a polynomial of degree $2^{r-1}$ which has $2^{r-1}$ roots in $H_\alpha$. Thus we have
\[\Tr_{r}(\alpha x)=\alpha^{2^{r-1}}\prod_{\gamma\in H_\alpha}(x+\gamma).\]
This implies that
\[\prod_{\stackrel{\gamma\in H_\alpha}{\gamma\neq 0}}\gamma=\alpha^{-2^{r-1}+1}.\]
Now the claim follows from the fact that the product of all non-zero elements of any binary finite field is $1$. \qed
\end{proof}
Let $\alpha\in{\mathbb F}_{2^r}^{\times}$. It is easy to check that $\alpha$ induces the involution $\phi_\alpha$ of $C$ where $\phi_\alpha$ is given by
\[\phi_\alpha:(x,y)\longrightarrow(x,y+\alpha),\]
and hence the group of automorphisms of $C$, $\mbox{Aut}(C)$, has a subgroup isomorphic to the additive group of ${\mathbb F}_{2^r}$. Thus it follows from Lemma~\ref{subgroup} that $H_\alpha$ can be viewed as a subgroup of $\mbox{Aut}(C)$.
In the following for every $\alpha\in{\mathbb F}_{2^r}$, we compute a plane model for $C/H_\alpha$.
\begin{lemma}
With the notations as above we have
\[C/H_\alpha: y_\alpha^2+\alpha^{2^{r-1}-1}y_\alpha=f(x).
\]
\end{lemma}
\begin{proof}
In order to compute a plane model for $C/H_\alpha$, we need to find two elements in the function field $\overline{{\mathbb F}_2}(x,y)$ which are invariant under the action of $H_\alpha$, and then find an algebraic relation between the two elements. Since $H_\alpha$ is an additive subgroup of ${\mathbb F}_{2^r}$, it follows that $$y_\alpha=\prod_{\gamma\in H_\alpha} (y+\gamma)$$
is invariant under the action of $H_\alpha$. So we choose $y_\alpha$ to be one of the elements that we are looking for and we take the other element to be simply $x$.
Now we claim that
\[y^{q}
+y=y_\alpha^2+\alpha^{2^{r-1}-1}y_\alpha.\]
In order to prove this claim, we need to prove that any element $\delta\in{\mathbb F}_{2^r}$ is a root of the right hand side of the above equation.
If $\delta\in H_\alpha$, then the claim is trivial. Now suppose that $\delta\in {\mathbb F}_{2^r}\setminus H_\alpha$. First notice that
\[y_\alpha(\delta)+\alpha^{2^{r-1}-1}=\prod_{\gamma\in H_\alpha} (\delta+\gamma) + \alpha^{2^{r-1}-1}.\]
Furthermore since $H_\alpha$ is an additive subgroup of ${\mathbb F}_{2^r}$ of index two, we have
\[\prod_{\gamma\in H_\alpha} (\delta+\gamma)+\alpha^{2^{r-1}-1}=\Big(\prod_{\beta\in {\mathbb F}_{2^r}\setminus H_\alpha}\beta \Big) + \alpha^{2^{r-1}-1},\]
and hence the claim follows from Lemma~\ref{subgroup} part (iii).
\end{proof}
We shall use the following theorem due to Kani and Rosen, which is from Theorem C in~\cite{kanirosen}.
\begin{theorem}\label{Decompose:kanirosen}
Let $C$ be a smooth, projective, absolutely irreducible curve defined over an arbitrary field $K$, and let $H_1,\ldots, H_t < Aut(C)$ be (finite) subgroups with $H_i \cdot H_j = H_j \cdot H_i$ for all $i, j$. Furthermore let $g_{ij}$ denote the genus of the
quotient curve $C/(H_i \cdot H_j)$. Then if $g_{ij} = 0$ for $2 \le i<j \le t$ and if
\[
g_C = g_{C/H_2} + \cdots + g_{C/H_t},
\]
then we have (by taking $H_1 = \{1\}$ above) an isogeny of Jacobians:
\[
J_C \sim J_{C/H_2} \times \cdots \times J_{C/H_t}.
\]
\end{theorem}
Applying the above theorem we get the following result.
\begin{theorem}\label{JacobianDecomposition}
Let $C$ denote the Artin-Schrier curve $y^{2^r}+y=f(x)$, and for every $\alpha\in{\mathbb F}_{2^r}^{\times}$ let $C_\alpha$ denote the curve $y_\alpha^2+y_\alpha=\alpha f(x)$. Furthermore let $H_\alpha$ be as in Lemma~\ref{subgroup}. Then we have
\[
J_C \sim \prod_{\alpha\in{\mathbb F}_{2^r}^{\times}}J_{C/H_\alpha}\sim \prod_{\alpha\in{\mathbb F}_{2^r}^{\times}}J_{C_\alpha}.
\]
\end{theorem}
\begin{proof}
For every pair of distinct elements $\alpha$ and $\beta$ in ${\mathbb F}_{2^r}^{\times}$, we trivially have $H_\alpha\cdot H_\beta=H_\beta\cdot H_\alpha$ and furthermore $(H_\alpha\cdot H_\beta)={\mathbb F}_{2^r}$ where we are considering ${\mathbb F}_{2^r}$ as an additive group. Notice that for every nonzero $\alpha$, $H_\alpha$ is a subgroup of index two of ${\mathbb F}_{2^r}$. It follows that the genus $g_{\alpha\beta}$ of the curve $C/(H_\alpha\cdot H_\beta)$ is zero. The fact that
\[g_C=\sum_{\alpha\in{\mathbb F}_{2^r}^{\times}}g_{C/H_\alpha}\]
follows from~\eqref{genus}. Thus we have
\[
J_C \sim \prod_{\alpha\in{\mathbb F}_{2^r}^{\times}}J_{C/H_\alpha}.
\]
The rest of the claim follows from the fact that by the change of variable $y_{\alpha} \mapsto \alpha^{2^{r-1} - 1}y_\alpha$, the curve equations for $C/H_{\alpha}$ becomes $C_\alpha:y_\alpha^2 + y_\alpha = \alpha f(x)$.
\end{proof}
The following is an immediate corollary of the above theorem as the above theorem implies that the L-polynomial of the curve $C$ is equal to product of the L-polynomials of the curves $C_\alpha$.
\begin{corollary}\label{pointcounting-thm}
Let $C$ and $C_\alpha$ be as defined above. Then
\[
\#C({\mathbb F}_{q^{n}})-\sum_{\alpha\in{\mathbb F}_{q}^{\times}}\#C_\alpha({\mathbb F}_{q^{n}}) = (q^{n} + 1)(q-1) - 1.
\]
\end{corollary}
\section{Computing the number of rational points on $C_1$, $C_2$ and $C_3$}\label{sec:rationalpoints}
With a view to applying Lemma~\ref{lem:fitzgeneral} to compute $F_q(n,0,0,0)$, define
\begin{align*}
q_1(x)&:=x^{2^r+1}+x^2, \\q_2(x)&:=x^{2^{r+1}+1}+x^{2^r+2}, \\q_3(x) &:= q_1(x)+q_2(x),
\end{align*}
where $r \geq 1$ is an integer. Let $n \geq 1$ be an integer. In this section, our aim is to find the number of rational points on the curves $$C_{i} : y^{2^r}+y=q_i(x)$$ on $\mathbb F_{2^{rn}}$ for all $i=1,2,3$.
In order to find this we will first try to find the number of rational points on the curves $$C_{i,\alpha} \: : \: y^2+y=\alpha q_i(x)$$ on $\mathbb F_{2^{rn}}$ for all $\alpha \in \mathbb F_{2^r}^\times$ and for all $i=1,2,3$. We will then apply Corollary~\ref{pointcounting-thm} to find the number of points on $C_i$.
\begin{proposition} \label{prop-mod-p}
Let $p$ be an odd prime divisor of $n$. Then $\#C_{i,\alpha}(\mathbb F_{2^{rn}})-\#C_{i,\alpha}(\mathbb F_{2^{r(n/p)}}) \equiv 0 \mod p$ for all $\alpha \in \mathbb F_{2^r}^\times$ and for all $i=1,2,3$.
\end{proposition}
\begin{proof}
We have $\Tr_{rn}(\alpha q_i(x))=p\cdot \Tr_{r(n/p)}(\alpha q_i(x))=\Tr_{r(n/p)}(\alpha q_i(x))$ for all $x \in \mathbb F_{2^{r(n/p)}}$, for all $\alpha \in \mathbb F_{2^r}^\times$ and for all $i=1,2,3$. Let $\beta \in \mathbb F_{2^{rn}}-\mathbb F_{2^{r(n/p)}}$. Then $\Tr_{rn}(\alpha q_i(x^{2^r}))=\Tr_{rn}(\alpha q_i(x))^{2^r}=\Tr_{rn}(\alpha q_i(x))$ for all $\alpha \in \mathbb F_{2^r}^\times$ and for all $i=1,2,3$. Since order of $\text{Gal}(\mathbb F_{2^{rn}}/\mathbb F_{2^{r{n/p}}})$ conjugates of $\beta$, which is $\{\beta,\beta^{2^r},\cdots, \beta^{2^{r(p-1)}}\}$, is $p$, we have $\#C_{i,\alpha}(\mathbb F_{2^{rn}})-\#C_{i,\alpha}(\mathbb F_{2^{r(n/p)}}) \equiv 0 \mod p$. \qed
\end{proof}
The number $\#C_{i,\alpha}(\mathbb F_{2^{rn}})$ can be written as
\begin{equation}\label{equation-Lambda}
\#C_{i,\alpha}(\mathbb F_{2^{rn}})=(2^{rn}+1)+\Lambda_{i,\alpha}(n)2^{(rn+w_{i,\alpha}(n))/2}
\end{equation} where $w_{i,\alpha}(n)$ is the dimension of of radical of $\Tr_{rn}(\alpha q_i(x))$ by Proposition \ref{counts} . We call $\Lambda_{i,\alpha}(n) \in \{-1,0,1\}$ as sign of $\#C_{i,\alpha}(\mathbb F_{2^{rn}})$.
\begin{corollary} \label{cor-mod-p}
Let $n \geq 1$ be an integer such that $n=2^um$ where $u \geq 1$ be an integer and $m$ is odd. Assume $\#C_{i,\alpha}(\mathbb F_{2^{rn}})$ and dimension of radical of $\Tr_{rn'}(\alpha q_i(x))$ are known where $n'$ equals to $2^um'$ and $m'$ is odd. Then $\#C_{i,\alpha}({\mathbb F}_{2^{rn'}})$ is known.
\end{corollary}
\begin{proof}
We can assume $m=1$ and $m'=p>2$. Since dimension of radical of $\Tr_{r2^up}(\alpha q_i(x))$ is known, It is enough to find sign of $C_{i,\alpha}(\mathbb F_{r2^up})$ by Equation \ref{equation-Lambda}. Since $\#C_{i,\alpha}(\mathbb F_{r2^u})$ is known and $\#C_{i,\alpha}(\mathbb F_{r2^up}) \equiv \#C_{i,\alpha}(\mathbb F_{r2^u}) \mod p$, the sign of $C_{i,\alpha}(\mathbb F_{r2^up})$ is also known. \qed
\end{proof}
Note that Proposition \ref{minimal-prop} and Corollary \ref{cor-mod-p} imply that in order to find the number of rational points of $C_{i,\alpha}$ on
$\mathbb F_{2^{rn}}$ for all $n \geq 1$, it is enough to find the number of rational points of $C_{i,\alpha}$ on $\mathbb F_{2^{r2^u}}$ where
$0 \leq u \leq v$ and $v$ is the minimal integer for which $C(\mathbb F_{2^{r2^v}})$ is minimal, and also the dimension of the radical of
$\Tr_{rn}(\alpha q_i(x))$ for all $n \geq 1$.
\subsection{Number of $\mathbb F_{2^{rn}}$-rational points on the curve $C_1 : y^{2^r}+y= q_1(x)$}
In this section we will find the number of rational points on the curves $$C_1 : y^{2^r}+y= x^{2^r+1}+x^2$$ on $\mathbb F_{2^{rn}}$ for all $r,n \in \mathbb Z^+$. In order to find this we will first try to find the number of rational points on the curves $$C_{1,\alpha} \: : \: y^2+y=\alpha (x^{2^r+1}+x^2)$$ on $\mathbb F_{2^{rn}}$ for all $\alpha \in \mathbb F_{2^r}^\times$, and we will then apply Corollary~\ref{pointcounting-thm}. We will see that the number of points is the same for each $\alpha$.
\begin{lemma} \label{C1-dimension}
Let $B_1(x,y)=\Tr_{rn}(q_1(x+y)+q_1(x)+q_1(y))$ and let the radical of $B_1(x,y)$ be $W_1(n)=\{x \in \mathbb F_{2^{rn}} \: | \: B_1(x,y)=0 \: \text{ for all } \: y \in \mathbb F_{2^{rn}} \}$. Denote the dimension of radical of $B_1(x,y)$ over $\mathbb F_{2^{rn}}$ as $w_r(n)$, then we have
$$w_r(n)=
\begin{cases}
r & \text{if $n$ is odd}\\
2r & \text{if $n$ is even.}\\
\end{cases}
$$
\end{lemma}
\begin{proof}
Let $B_1(x,y)=\Tr_{rn}(q_1(x+y)+q_1(x)+q_1(y))$. Then we have $$B_1(x,y)=\Tr_{rn}(y^{2^r}(x^{2^{2r}}+x))$$ and the radical of $B_1(x,y)$ is $$W_1(n)=\{x \in \mathbb F_{2^{rn}} \: | \: B_1(x,y)=0 \: \text{ for all } \: y \in \mathbb F_{2^{rn}} \}= \mathbb F_{2^{rn}} \cap \mathbb F_{2^{2r}}=\mathbb F_{2^{r(2,n)}}.$$ \qed
\end{proof}
\begin{lemma}\label{lemma-W2n-C1}
Let the notation be above. $\Tr_{2r}(q_1(W_1(2)))=\mathbb F_{2}$ and $\Tr_{2r}(q_1(W_1(n)))=0$ for $4 \mid n$.
\end{lemma}
\begin{proof}
Suppose $n=2$. Since $x^{2^r+1} \in \mathbb F_{2^r}$ for all $x \in \mathbb F_{2^{2r}}$ we have $\Tr_{2r}(x^{2^r+1})=0$ for all $x \in \mathbb F_{2^{2r}}$. Therefore $$\Tr_{2r}(x^{2^r+1}+x^2)=\Tr_{2r}(x).$$
Now suppose $4|n$. Since $W_1(n)=\mathbb F_{2^{2r}}$, for all $x \in W_1(n)$ we have $$\Tr_{rn}(x^{2^r+1}+x^2)=0$$ for all $x \in W_1(n)$.\qed
\end{proof}
\begin{theorem}\label{thm-points on curve 1}
If $r$ is odd, then we have $$ \#C_{1,\alpha}(\mathbb F_{2^{rn}})=
\begin{cases}
(2^{rn}+1)+2^{r(n+1)/2} &\text{if $n \equiv 1,7 \mod 8$,}\\
(2^{rn}+1)-2^{r(n+1)/2} &\text{if $n \equiv 3,5 \mod 8$,}\\
(2^{rn}+1) &\text{if $n \equiv 2,6 \mod 8$,}\\
(2^{rn}+1)+2^{r(n+2)/2} &\text{if $n \equiv 4 \mod 8$,}\\
(2^{rn}+1)-2^{r(n+2)/2} &\text{if $n \equiv 0 \mod 8$}
\end{cases}
$$
if $r$ is even, then we have $$ \#C_{1,\alpha}(\mathbb F_{2^{rn}})=
\begin{cases}
(2^{rn}+1)+2^{r(n+1)/2} &\text{if $n \equiv 1,3 \mod 4$,}\\
(2^{rn}+1) &\text{if $n \equiv 2 \mod 4$,}\\
(2^{rn}+1)-2^{r(n+2)/2} &\text{if $n \equiv 0\mod 4$.}\\
\end{cases}
$$
\end{theorem}
\begin{proof}
In order to prove this theorem, we will find the number of rational points of $C_{1,1}$ on $\mathbb F_{2^{r}}$, $\mathbb F_{2^{2r}}$, $\mathbb F_{2^{4r}}$ and $\mathbb F_{2^{8r}}$. Then we will apply Proposition \ref{minimal-prop} and Proposition \ref{prop-mod-p} for finding the number of rational points of $C_{1,1}$ on $\mathbb F_{2^{rn}}$ for any $n \geq 1$. \\
Since $x^{2^r+1}+x^2=0$ for all $x \in \mathbb F_{2^r}$, the number of rational points of $C_{1,1}$ on $\mathbb F_{2^{r}}$ is $(2^r+1)+2^r$. Therefore, the number of rational points of $C_{1,1}$ on $\mathbb F_{2^{rn}}$ is $(2^{rn}+1)+2^{r(n+1)/2}$ if $n \equiv 1,7 \mod 8$ and the number of rational points of $C_{1,1}$ on $\mathbb F_{2^{r}}$ is $(2^{rn}+1)+2^{r(n+1)/2}$ if $n \equiv 3,5 \mod 8$ by Proposition \ref{prop-mod-p} and Lemma \ref{C1-dimension}. \\
Since $\Tr_{2r}(q_1(W_1(2)))=\mathbb F_2$ by Lemma \ref{lemma-W2n-C1}, we have the number of rational points of $C_{1,1}$ on $\mathbb F_{2^{2r}}$ is $2^{2r}+1$. Therefore, the number of rational points of $C_{1,1}$ on $\mathbb F_{2^{rn}}$ is $2^{rn}+1$ if $n \equiv 2,6 \mod 8$ by Proposition \ref{prop-mod-p} and Lemma \ref{C1-dimension}. \\
Since $\Tr_{4r}(q_1(W_1(4)))= 0$ by Lemma \ref{lemma-W2n-C1}, we have the number of rational points of $C_{1,1}$ on $\mathbb F_{2^{2r}}$ is $(2^{2r}+1) \pm 2^{r(n+2)/2}$. Since genus of $C_{1,1}$ is $2^{r-1}$ we have $C_1(\mathbb F_{2^{4r}})$ is maximal or minimal by Proposition \ref{minimal-prop}. Since we cannot write $1$ as a $\mathbb Q$ linear combination of $\{w \in \mathbb C \: | \: w^4=-1\}$ by Lemma \ref{lemma-unity}, and since we cannot write $\sqrt 2$ as a $\mathbb Q$ linear combination of $\{w \in \mathbb C \: | \: w^4=1\}$, we have $C_{1,1}(\mathbb F_{2^{4r}})$ is maximal and $C_{1,1}(\mathbb F_{2^{8r}})$ is minimal by Proposition \ref{minimal-prop} if $r$ is odd, and $C_{1,1}(\mathbb F_{2^{4r}})$ is minimal if $r$ is even. Therefore if $r$ is odd, the number of rational points of $C_{1,1}$ on $\mathbb F_{2^{rn}}$ is $(2^{rn}+1)+2^{r(n+2)/2}$ if $n \equiv 4 \mod 8$ and $(2^{rn}+1)-2^{r(n+2)/2}$ if $n \equiv 0 \mod 8$; if $r$ is even, the number of rational points of $C_{1,1}$ on $\mathbb F_{2^{rn}}$ is $(2^{rn}+1)-2^{r(n+2)/2}$ if $n \equiv 0,4 \mod 8$ by Proposition \ref{prop-mod-p} and Lemma \ref{C1-dimension}.\\
Note that since the map $(x,y) \to (\alpha^{-1/2}x,y)$ gives an isomorphism between $C_{1,\alpha} \to C_{1,1}$ where $\alpha \in \mathbb F_{2^r}^\times$, their number of rational points equal in any extension of $\mathbb F_{2^r}$. \qed
\end{proof}
Using Corollary~\ref{pointcounting-thm} and Theorem~\ref{thm-points on curve 1}, Table~\ref{table:C1points} in the appendix presents the number of
${\mathbb F}_{2^{rn}}$-rational points on $C_1:y^{2^r}+y=q_1(x)$.
\subsection{Number of $\mathbb F_{2^{rn}}$-rational points on the curve $C_2 : y^{2^r}+y= q_2(x)$}
In this section we will find the number of $\mathbb F_{2^{rn}}$-rational points on the curve $$C_2:y^{2^r}+y= x^{2^{r+1}+1}+x^{2^r+2}$$ for all $r,n \in \mathbb Z^+$. In order to find this we will first try to find the number of rational points on the curves $$C_{2,\alpha} \: : \: y^2+y=\alpha (x^{2^{r+1}+1}+x^{2^r+2})$$ on $\mathbb F_{2^{rn}}$ for all $\alpha \in \mathbb F_{2^r}^\times$,
and we will then apply Corollary~\ref{pointcounting-thm}.
We will see that the number of points depends on whether $\alpha$ is a cube, for odd $r$.
\begin{lemma} \label{lemma-C2-dimensions}
Let $B_{2,\alpha}(x,y)=\Tr_{rn}(\alpha(q_2(x+y)+q_2(x)+q_2(y)))$ where $\alpha \in \mathbb F_{2^r}$ and let the radical of $B_{2,\alpha}(x,y)$ be $W_1(n)=\{x \in \mathbb F_{2^{rn}} \: | \: B_{2,\alpha}(x,y)=0 \: \text{ for all } \: y \in \mathbb F_{2^{rn}} \}$. Denote the dimension of radical of $B_1(x,y)$ over $\mathbb F_{2^{rn}}$ as $w_r(n)$. If $\alpha \in \{x^3 \: | \: x \in \mathbb F_{2^r} \}$, then we have $$w_{r,\alpha}(n)=
\begin{cases}
r & \text{if $n$ is odd},\\
2r & \text{if $n \equiv 2 \mod 4$,}\\
2r+2 & \text{if $n \equiv 0 \mod 4$.}\\
\end{cases}
$$ If $\alpha \not \in \{x^3 \: | \: x \in \mathbb F_{2^r} \}$, then we have $$w_{r,\alpha}(n)=
\begin{cases}
r & \text{if $(n,6)=1$},\\
2r & \text{if $(n,6)=2$},\\
r+2 & \text{if $(n,6)=3$},\\%FARUK:{this should be (n,6) = 3 I guess}
2r+2 & \text{if $(n,6)=6$}.\\
\end{cases}
$$
\end{lemma}
\begin{proof}
Let $B_{2,\alpha}(x,y)=\Tr_{rn}(\alpha(q_2(x+y)+q_2(x)+q_2(y)))$ where $\alpha \in \mathbb F_{2^r}$. Then we have $$B_{2,\alpha}(x,y)=\Tr_{rn}(\alpha y^{2^{r+1}}[(x+\alpha x^4)+(x+\alpha x^4)^{2^{2r}}])$$ and the radical of $B_{2,\alpha}(x,y)$ is $$W_{2,\alpha}(n)=\{ x \in \mathbb F_{2^{rn}} \: | \: (x+\alpha x^4)+(x+\alpha x^4)^{2^{2r}}=0 \}.$$ This can be rewritten as (multiplying by $\alpha^3$ and transforming $\alpha x \to x$) $$W_{2,\alpha}(n)=\{ x \in \mathbb F_{2^{rn}} \: | \: (x^4+\alpha^2 x)+(x^4+\alpha^2 x)^{2^{2r}}=0 \in \mathbb F_{2^{2r}} \}.$$
Let $x^4+\alpha^2 x=x(x+a)(x+b)(x+c)$ where $a,b,c \in \overline{\mathbb F_{2^r}}$. Then $$(x^4+\alpha^2 x)+(x^4+\alpha^2 x)^{2^{2r}}=(x^{2^{2r}}+x)(x^{2^{2r}}+x+a)(x^{2^{2r}}+x+b)(x^{2^{2r}}+x+c).$$ Therefore $w_{r,\alpha}(n)=\frac12\#\{u \: : \: u \in \{0,a,b,c\} \text{ and } u \in \mathbb F_{2^{rn}} \}+r(n,2).$ Moreover if $\alpha^2$ (or $\alpha$) is not a cube in $\mathbb F_{2^r}$ then $x^3+\alpha^2$ is irreducible over $\mathbb F_{2^r}$ and hence all roots of $x^3+\alpha^2$ are in $\mathbb F_{2^{3r}}$, if $\alpha^2$ (or $\alpha$) is a cube in $\mathbb F_{2^r}$ then $x^{4r}+x=0$ for all $x \in W_{2,\alpha}(n)$ and hence all roots of $x^3+\alpha^2$ are in $\mathbb F_{2^{4r}}$. \qed
\end{proof}
\begin{lemma} \label{C2-W2}
Let notation be as above. Then we have $\Tr_{2r}(\alpha q_2(W_{2,\alpha}(n)))=0$ and therefore $\#C_{2,\alpha}(\mathbb F_{2^{2r}})-(2^{2r}+1) \ne 0$.
\end{lemma}
\begin{proof}
Since $\alpha(x^{2^{r+1}+1}+x^{2^{r+2}}) \in \mathbb F_{2^r}$ for all $x \in \mathbb F_{2^{2r}}$ and for all $\alpha \in \mathbb F_{2^r}$, we have $$\Tr_{2r}(\alpha(x^{2^{r+1}+1}+x^{2^{r+2}}))=0$$ for all $x \in \mathbb F_{2^{2r}}$ and for all $\alpha \in \mathbb F_{2^r}$. Therefore $\#C_{2,\alpha}(\mathbb F_{2^{2r}})-(2^{2r}+1) \ne 0$. \qed
\end{proof}
\begin{theorem}\label{thm-points on curve 2}
If $r$ is odd, then we have
$$ \#C_{2,\alpha}(\mathbb F_{2^{rn}})=
\begin{cases}
(2^{rn}+1)+2^{r(n+1)/2} &\text{if $n \equiv 1,7 \mod 8$},\\
(2^{rn}+1)-2^{r(n+1)/2} &\text{if $n \equiv 3,5 \mod 8$},\\
(2^{rn}+1)+2^{r(n+2)/2} &\text{if $n \equiv 2,6 \mod 8$},\\
(2^{rn}+1) &\text{if $n \equiv 4 \mod 8$}\\
(2^{rn}+1)-2^{r(n+2)/2+1} &\text{if $n \equiv 0 \mod 8$}.\\
\end{cases}
$$
If $r$ is even and $\alpha \in \{x^3 \: | \: x\in \mathbb F_{2^r} \}$, , then we have
$$ \#C_{2,\alpha}(\mathbb F_{2^{rn}})=
\begin{cases}
(2^{rn}+1)+2^{r(n+1)/2} &\text{if $n \equiv 1,3 \mod 4$},\\
(2^{rn}+1)+2^{r(n+2)/2} &\text{if $n \equiv 2 \mod 4$},\\
(2^{rn}+1)-2^{r(n+2)/2+1} &\text{if $n \equiv 0 \mod 4$}.\\
\end{cases}
$$
If $r$ is even and $\alpha \not \in \{x^3 \: | \: x\in \mathbb F_{2^r} \}$, then we have $$ \#C_{2,\alpha}(\mathbb F_{2^{rn}})=
\begin{cases}
(2^{rn}+1)+2^{r(n+1)/2} & \text{if $(n,6)=1$},\\
(2^{rn}+1)+2^{r(n+2)/2} & \text{if $(n,6)=2$},\\
(2^{rn}+1)-2^{r(n+1)/2+1} & \text{if $(n,3)=3$},\\
(2^{rn}+1)-2^{r(n+2)/2+1} & \text{if $(n,6)=6$}.\\
\end{cases}
$$
\end{theorem}
\begin{proof}
We will prove this theorem in same manner of Theorem \ref{thm-points on curve 1}. Note that the genus of $C_{2,\alpha}$ is $2^{r}$ for all $\alpha \in \mathbb F_{2^r}^\times$.\\
Assume $r$ is odd. Suppose $C_{2,\alpha}(\mathbb F_{2^{4r}})$ is maximal. Then it requires $\#C_{2,\alpha}(\mathbb F_{2^{2r}})-(2^{2r}+1) = 0$, but it is not possible by Lemma \ref{C2-W2}. Suppose $C_{2,\alpha}(\mathbb F_{2^{4r}})$ is minimal curve. Then we cannot write $\sqrt 2$ as a $\mathbb Q$ linear combination of $\{\eta \in \mathbb C \: | \: \eta^4=1\}$. Therefore $\#C_{2,\alpha}(\mathbb F_{2^{4r}})-(2^{4r}+1) = 0$. Furthermore, since $W_{1,\alpha}(n) \subset \mathbb F_{2^{4r}}$, we have $C_{2,\alpha}(\mathbb F_{2^{8r}})$ is maximal or minimal. Since $\sqrt 2$ cannot be written as a $\mathbb Q$ linear combination of $\{\eta \in \mathbb C \: | \: \eta^8=-1\}$ by Lemma \ref{lemma-unity}, the curve $C_{2,\alpha}(\mathbb F_{2^{8r}})$ must be minimal.\\
Assume $r$ is even. Then $\alpha q_2(wx)=\alpha q_2(w^2x)=\alpha q^2(x)$ for all $x \in \mathbb F_{2^{rn}}$ and for all $\alpha \in \mathbb F_{2^r}^\times$ where $w^2+w+1=0$ and therefore $\#C_{2,\alpha}(\mathbb F_{2^{rn}}) \equiv 0 \mod 3$. Since we know the dimensions $w_{r,\alpha}(n)$ for all $\alpha \in \mathbb F_{2^r}^\times$ and $n \geq 1$, one can easily find the signs of $C_{2,\alpha}(\mathbb F_{2^{rn}})$ for all $\alpha \in \mathbb F_{2^r}^\times$ and $n \geq 1$. \qed
\end{proof}
Using Corollary~\ref{pointcounting-thm} and Theorem~\ref{thm-points on curve 2}, Table~\ref{table:C2points} in the appendix presents the number of ${\mathbb F}_{2^{rn}}$-rational points on $C_2:y^{2^r}+y=q_2(x)$.
We remark that Moisio and Ranto~\cite{moisioranto} have counted $F_q(n,0,*,t_3)$ for $t_3 \in {\mathbb F}_{q} = {\mathbb F}_{2^r}$.
Their methods are different, and do not yield the same type of general formula as here.
\subsection{Number of $\mathbb F_{2^{rn}}$-rational points on the curve $C_3 : y^{2^r}+y= q_3(x)$}
In this section we will find the number of rational points on the curve $$C_3 : y^{2^r}+y= x^{2^{r+1}+1}+x^{2^r+2}+x^{2^r+1}+x^2$$ on $\mathbb F_{2^{rn}}$ for all $r,n \in \mathbb Z^+$. In order to find this we will first try to find the number of rational points on the curves $$C_{3,\alpha} \: : \: y^2+y=\alpha (x^{2^{r+1}+1}+x^{2^r+2}+x^{2^r+1}+x^2)$$ on $\mathbb F_{2^{rn}}$ for all $\alpha \in \mathbb F_{2^r}^\times$.
\begin{lemma}
Let $B_{3,\alpha}(x,y)=\Tr_{rn}(\alpha(q_3(x+y)+q_3(x)+q_3(y)))$ where $\alpha \in \mathbb F_{2^r}$ and let the radical of $B_{3,\alpha}(x,y)$ be $W_{3,\alpha}(n)=\{x \in \mathbb F_{2^{rn}} \: | \: B_{3,\alpha}(x,y)=0 \: \text{ for all } \: y \in \mathbb F_{2^{rn}} \}$. Denote the dimension of radical of $B_{3,\alpha}(x,y)$ over $\mathbb F_{2^{rn}}$ as $w_r(n)$. If $x^3+x+\alpha^{-1}$ has one root or three roots in $\mathbb F_{2^r}$, then we have $$w_{r,\alpha}(n)=
\begin{cases}
r & \text{if $n$ is odd},\\
2r & \text{if $n \equiv 2 \mod 4$},\\
2r+2 & \text{if $n \equiv 0 \mod 4$}.\\
\end{cases}
$$
If $x^3+x+\alpha^{-1}$ has no root in $\mathbb F_{2^r}$ $$w_{r,\alpha}(n)=
\begin{cases}
r & \text{if $(n,6)=1$},\\
2r & \text{if $(n,6)=2$},\\
r+2 & \text{if $(n,6)=3$},\\
2r+2 & \text{if $(n,6)=6$}.\\
\end{cases}
$$
\end{lemma}
\begin{proof}
Let $B_{3,\alpha}(x,y)=\Tr_{rn}(\alpha(q_3(x+y)+q_3(x)+q_3(y)))$ where $\alpha \in \mathbb F_{2^r}$. Then we have $$B_{3,\alpha}(x,y)=\Tr_{rn}(\alpha^2 y^{2^{r+1}}[(\alpha^{-1}x+ x^2+ x^4)+(\alpha^{-1}x+ x^2+x^4)^{2^{2r}}])$$ and the radical of $B_{3,\alpha}(x,y)$ is $$W_{3,\alpha}(n)=\{ x \in \mathbb F_{2^{rn}} \: | \: (\alpha^{-1}x+ x^2+ x^4)+(\alpha^{-1}x+ x^2+x^4)^{2^{2r}}=0 \}.$$ Let $x^4+x^2+\alpha^{-1} x=x(x+a)(x+b)(x+c)$ where $a,b,c \in \overline{\mathbb F_{2^r}}$. Then $$(x^4+x^2+\alpha^{-1} x)+(x^4+x^2+\alpha^{-1} x)^{2^{2r}}=(x^{2^{2r}}+x)(x^{2^{2r}}+x+a)(x^{2^{2r}}+x+b)(x^{2^{2r}}+x+c).$$ Therefore $w_{r,\alpha}(n)=\frac12\#\{u \: : \: u \in \{0,a,b,c\} \text{ and } u \in \mathbb F_{2^{rn}} \}+r(n,2).$ Moreover if $x^3+x+\alpha^{-1}$ is irreducible over $\mathbb F_{2^r}$, then all roots of $x^3+x+\alpha^{-1}$ are in $\mathbb F_{2^{3r}}$, if $x^3+x+\alpha^{-1}$ has one root in $\mathbb F_{2^r}$ and two roots in $\mathbb F_{2^{2r}}$; or three roots in $\mathbb F_{2^r}$ then $x^{4r}+x=0$ for all $x \in W_{2,\alpha}(n)$ and hence all roots of $x^3+x+\alpha^{-1}$ are in $\mathbb F_{2^{4r}}$. \qed
\end{proof}
\begin{lemma}\label{onto-radical}
Let $x^{2^{2r}}+x+b=0$ where $b \in \mathbb F_{2^{2r}}$. Then $$\Tr_{4r}(\alpha(x^{2^{r+1}+1}+x^{2^r+2}+x^{2^r+1}+x^2))=\begin{cases}0 \text { if } b \in \mathbb F_{2^{2r}} \\ 1 \text{ if } b \in \mathbb F_{2^{2r}} \backslash \mathbb F_{2^r} \end{cases}$$ where $\alpha \in \mathbb F_{2^r}^\times$. Therefore, if $x^3+x+\alpha^{-1}$ has zero root or three roots in $\mathbb F_{2^r}$, then $\Tr_{4r}(\alpha q_3(W_{3,\alpha}))=0$ and if $x^3+x+\alpha^{-1}$ has one root in $\mathbb F_{2^r}$, then $\Tr_{4r}(\alpha q_3(W_{3,\alpha}))=\mathbb F_2$.
\end{lemma}
\begin{proof}
We expand the trace
\begin{align*}
&\Tr_{4r}(\alpha(x^{2^{r+1}+1}+x^{2^r+2}+x^{2^r+1}+x^2))\\
=&\Tr_{2r}(\alpha(x^{2^{r+1}+1}+x^{2^r+2}+x^{2^r+1}+x^2)+(\alpha(x^{2^{r+1}+1}+x^{2^r+2}+x^{2^r+1}+x^2)^{2^{2r}})\\
=&\Tr_{2r}(\alpha((x^{2^{r+1}+1}+(x+b)^{2^{r+1}+1})+(x^{2^{r}+2}+(x+b)^{2^{r}+2})+(x^{2^{r}+1}+(x+b)^{2^{r}+1})+b^2))\\
=&\Tr_{2r}(\alpha b^2)+\Tr_{2r}(\alpha((b^{2^{r+1}+1}+b^{2^r+2}))+\Tr_{2r}(\alpha b^{2^r+1})\\
+&\Tr_{2r}(bx^{2^{r+1}}+b^{2^r}x^2)+\Tr_{2r}(bx^{2^{r}}+b^{2^r}x)+\Tr_{2r}(b^2x^{2^{r}}+b^{2^{r+1}}x)\\
=&\Tr_{2r}(\alpha b^2)\\
=&\begin{cases}0 \text { if } b \in \mathbb F_{2^{2r}} \\ 1 \text{ if } b \in \mathbb F_{2^{2r}} \backslash \mathbb F_{2^r}. \end{cases}
\end{align*}
If $f_{\alpha}$ has three roots in $\mathbb F_{2^r}$ (resp. one root in $\mathbb F_{2^r}$ and two roots in $\mathbb F_{2^{2r}}$), then $$(\alpha^{-1}x+ x^2+ x^4)+(\alpha^{-1}x+ x^2+x^4)^{2^{2r}}=(x^{2^{2r}}+x)(x^{2^{2r}}+x+a)(x^{2^{2r}}+x+b)(x^{2^{2r}}+x+c)$$ where $a,b,c \in {\mathbb F}_{2^r}$ (resp. $a \in {\mathbb F}_{2^r}$ and $b,c \in \mathbb F_{2^{2r}}$). If $f_{\alpha}$ has no root in $\mathbb F_{2^r}$, then $W_{3,\alpha}(4)=\mathbb F_{2^{2r}}$. \qed
\end{proof}
\begin{theorem}\label{thm:C3 points}
If $r$ is odd and $x^3+x+\alpha^{-1}$ has three roots in $\mathbb F_{2^r}$, then we have
$$ \#C_{3,\alpha}(\mathbb F_{2^{rn}})=
\begin{cases}
(2^{rn}+1)+2^{r(n+1)/2} &\text{if $n \equiv 1,7 \mod 8$},\\
(2^{rn}+1)-2^{r(n+1)/2} &\text{if $n \equiv 3,5 \mod 8$},\\
(2^{rn}+1) &\text{if $n \equiv 2 \mod 8$},\\
(2^{rn}+1)+2^{r(n+2)/2+1} &\text{if $n \equiv 4 \mod 8$},\\
(2^{rn}+1)-2^{r(n+2)/2+1} &\text{if $n \equiv 0 \mod 8$}.\\
\end{cases}
$$
If $r$ is even and $x^3+x+\alpha^{-1}$ has three roots in $\mathbb F_{2^r}$, then we have
$$ \#C_{3,\alpha}(\mathbb F_{2^{rn}})=
\begin{cases}
(2^{rn}+1)+2^{r(n+1)/2} &\text{if $n \equiv 1,3,5,7 \mod 8$},\\
(2^{rn}+1) &\text{if $n \equiv 2 \mod 8$},\\
(2^{rn}+1)-2^{r(n+2)/2+1} &\text{if $n \equiv 4 \mod 8$},\\
(2^{rn}+1)-2^{r(n+2)/2+1} &\text{if $n \equiv 0 \mod 8$}.\\
\end{cases}
$$
If $r$ is odd and $x^3+x+\alpha^{-1}$ has one root in $\mathbb F_{2^r}$, then we have
$$ \#C_{3,\alpha}(\mathbb F_{2^{rn}})=
\begin{cases}
(2^{rn}+1)+2^{r(n+1)/2} &\text{if $n \equiv 1,7 \mod 8$},\\
(2^{rn}+1)-2^{r(n+1)/2} &\text{if $n \equiv 3,5 \mod 8$},\\
(2^{rn}+1) &\text{if $n \equiv 2,4 \mod 8$},\\
(2^{rn}+1)-2^{r(n+2)/2+1} &\text{if $n \equiv 0 \mod 8$}.\\
\end{cases}
$$
If $r$ is even and $x^3+x+\alpha^{-1}$ has one root in $\mathbb F_{2^r}$, then we have
$$ \#C_{3,\alpha}(\mathbb F_{2^{rn}})=
\begin{cases}
(2^{rn}+1)+2^{r(n+1)/2} &\text{if $n \equiv 1,3,5,7 \mod 8$},\\
(2^{rn}+1) &\text{if $n \equiv 2,4 \mod 8$},\\
(2^{rn}+1)-2^{r(n+2)/2+1} &\text{if $n \equiv 0 \mod 8$}.\\
\end{cases}
$$
If $x^3+x+\alpha^{-1}$ has no root in $\mathbb F_{2^r}$, the possibilities for $\#C_{3,\alpha}(\mathbb F_{2^{rn}})$ are given by
Table~\ref{table:C3pointsnoroots} in the appendix.
\end{theorem}
\begin{proof}
We will prove this theorem in same manner of Theorem \ref{thm-points on curve 1} and we will also use Corollary \ref{cor-mod-p} for the case $x^3+x+\alpha^{-1}$ has no root in $\mathbb F_{2^r}$. Note that the genus of $C_{3,\alpha}$ is $2^{r}$ for all $\alpha \in \mathbb F_{2^r}^\times$.\\
If $x^3+x+\alpha^{-1}$ has one root or three roots in $\mathbb F_{2^r}$ then $C_{3,\alpha}(\mathbb F_{2^{8r}})$ is maximal or minimal and if $x^3+x+\alpha^{-1}$ has no root in $\mathbb F_{2^r}$ then $C_{3,\alpha}(\mathbb F_{2^{24r}})$ is maximal or minimal. Since $\sqrt {2^r}$ (or $\sqrt {2^{3r}}$) cannot be written as a $\mathbb Z$-linear combination of $\{\eta \: | \: \eta^8=-1\}$ by Lemma \ref{lemma-unity}, if $x^3+x+\alpha^{-1}$ has one root or three roots in $\mathbb F_{2^r}$ then $C_{3,\alpha}(\mathbb F_{2^{8r}})$ is minimal and if $x^3+x+\alpha^{-1}$ has zero root in $\mathbb F_{2^r}$ then $C_{3,\alpha}(\mathbb F_{2^{24r}})$ is minimal.\\
If $x^3+x+\alpha^{-1}$ has one root in $\mathbb F_{2^r}$ then $\#C_{3,\alpha}(\mathbb F_{2^{4r}})-(2^{4r}+1)=0$, if $x^3+x+\alpha^{-1}$ has three roots in $\mathbb F_{2^r}$ and $r$ is odd (even) then $C_{3,\alpha}(\mathbb F_{2^{4r}})$ is maximal (minimal), if $x^3+x+\alpha^{-1}$ has zero root in $\mathbb F_{2^r}$ and $r$ is odd (even) then $C_{3,\alpha}(\mathbb F_{2^{12r}})$ is maximal (minimal) by Lemma \ref{onto-radical}. \qed
\end{proof}
\begin{lemma}\label{quadratic-solutions}
A quadratic equation $x^2+bx+a= 0$, $a \in \mathbb F_{2^r}$ and $b \in \mathbb F_{2^r}^\times$, has two distinct roots in $\mathbb F_{2^r}$
if and only if $\Tr_{r}(a/b^2)= 0$.
\end{lemma}
\begin{proof}
Since $b \ne 0$, we can divide the equation by $b^2$. So we get $(x/b)^2+(x/b)=a/b^2$. Hence $\Tr_{r}(a/b^2)= 0$ if and only if $x/b \in \mathbb F_{2^r}$ if and only if $x \in \mathbb F_{2^r}$. \qed
\end{proof}
\begin{lemma}\label{M013}
Denote $f_{\beta}(x) = x^3 + x + \beta$, where $\beta \in \mathbb F_{2^r}^\times$. Let
$$M_i = \#\{f_{\beta} \: | \: f_{\beta}(x) \text{ has precisely $i$ roots in } \mathbb F_{2^r}\}.$$ Then we have $(M_3,M_1,M_0)=\big(\frac{2^{r-1}-1}{3},2^{r-1}-1,\frac{2^r+1}{3}\big)$ if $r$ is odd, $(M_3,M_1,M_0)=\big(\frac{2^{r-1}-2}{3},2^{r-1},\frac{2^r-1}{3}\big)$ if $r$ is even.
\end{lemma}
\begin{proof}
The derivative of $f_{\beta}(x)$ is $x^2+1$ for all $\beta \in \mathbb F_{2^r}^\times$. Thus it has multiple roots if and only if $1$ is a root of $f_\beta$. Moreover $x=1$ is root of $f_\beta$ if $\beta=0$ (which is not the case) and the other root of $f_0$ is $0$.\\
Let $F^{\times\times} := \mathbb F_{2^r}\backslash\{0,1\}$. Now assume $f_\beta(x)$ has two different roots $a,b \in F^{\times\times}$. This means $f_\beta(a)=f_\beta(b)$. It follows that $(a+b)(b^2+ab+(a^2+1))=0$; that is, $b^2+ab+(a^2+1)=0$. By Lemma \ref{quadratic-solutions}, $\Tr_{r}(\frac{a^2+1}{a^2})=\Tr_{r}(\frac1a)+\Tr_{r}(1)$ must to be $0$.\\
Note that $1/a$ is a permutation on the set $F^{\times\times}$ and half of $\mathbb F_{2^r}$ have trace $0$ and the other half of $\mathbb F_{2^r}$ have trace $1$. Therefore we have $M_3=[2^{r-1}-\#\{x \in \{0,1\} \: |\Tr_{r}(x) = \Tr_{r}(1)\}]/3$ and $M_1=(2^r-2)-3M_3$ and $M_0=(2^r-1)-M_3-M_1$.
\qed
\end{proof}
Using Corollary~\ref{pointcounting-thm} and Theorem~\ref{thm:C3 points}, and Lemma~\ref{M013}, Table~\ref{table:C3points} in the appendix presents the number of ${\mathbb F}_{2^{rn}}$-rational points on $C_3:y^{2^r}+y=q_3(x)$.
\section{Explicit formulae for $F_{q}(n,0,0,0)$}\label{sec:explicit}
In this section we give an explicit formula for $F_{q}(n,0,0,0)$. In addition we present
Fourier-analysed formula for the number of points on the curves $C_1$, $C_2$ and $C_3$, and thus for $F_{q}(n,0,0,0)$.
We use the former to infer the characteristic polynomial of Frobenius of these curves and hence the decomposition of their Jacobians.
Using Lemma~\ref{lem:fitzgeneral} and Eq.~(\ref{simple}), we now combine the results contained in Tables~\ref{table:C1points},~\ref{table:C2points}
and~\ref{table:C3points} to give the formula for $F_{q}(n,0,0,0)$, presented in Table~\ref{table:Fqn000} in the appendix.
\subsection{$C_{1}/{\mathbb F}_q: y^{q}+y = x^{q+1} + x^2$}
By Fourier-analysing the formulae in Table~\ref{table:C1points}, for $r$ odd we have:
\[
\#C_1({\mathbb F}_{q^n}) = q^n + 1 -\frac{(q-1)(q-\sqrt{2q})}{4}((\sqrt{q}\omega_8)^n + (\sqrt{q}\omega_{8}^7)^n)
-\frac{(q-1)(q + \sqrt{2q})}{4}((\sqrt{q} \omega_{8}^3)^n + (\sqrt{q}\omega_{8}^5)^n),
\]
while for $r$ even we have:
\[
\#C_1({\mathbb F}_{q^n}) = q^n + 1 -\frac{(q-1)(q-2\sqrt{q})}{4}(\sqrt{q})^n -\frac{(q-1)(q+2\sqrt{q})}{4}(-\sqrt{q})^n
-\frac{(q-1)q}{4}((\sqrt{q} \, i)^n + (-\sqrt{q} \, i)^n).
\]
The characteristic polynomial of Frobenius is therefore
\[
P_1(X) = (X^2 -\sqrt{2q}X + q)^{\frac{(q-1)(q -\sqrt{2q})}{4}} (X^2+\sqrt{2q}X + q)^{\frac{(q-1)(q+\sqrt{2q})}{4}}
\]
if $r$ is odd, and
\[
P_1(X) = (X-\sqrt{q})^{\frac{(q-1)(q - 2\sqrt{q})}{4}} (X +\sqrt{q})^{\frac{(q-1)(q + 2\sqrt{q})}{4}} (X^2 + q)^{\frac{(q-1)q}{4}}
\]
if $r$ is even. These factors all arise from dimension one supersingular abelian varieties~\cite[Theorems 12.1 \& 12.2]{VJGaryAlexey}.
\subsection{$C_2/{\mathbb F}_q: y^{q}+y = x^{2q+1} + x^{q+2}$}
By Fourier-analysing the formulae in Table~\ref{table:C2points}, for $r$ odd we have:
\begin{eqnarray*}
\#C_2({\mathbb F}_{q^n}) = q^n + 1 &-&\frac{(q-1)(q-\sqrt{2q})}{4}((\sqrt{q}\omega_8)^n + (\sqrt{q}\omega_{8}^7)^n)\\
&-&\frac{(q-1)(q + \sqrt{2q})}{4}((\sqrt{q}\omega_{8}^3)^n + (\sqrt{q}\omega_{8}^5)^n)\\
&-&\frac{(q-1)q}{2}((\sqrt{q} \, i)^n + (-\sqrt{q} \, i)^n).
\end{eqnarray*}
Let $\omega_{6} = (1 + \sqrt{3}i)/2$. Then for $r$ even we have:
\begin{eqnarray*}
\#C_2({\mathbb F}_{q^n}) = q^n + 1 &-&\frac{(q-1)(q-2\sqrt{q})}{12}(\sqrt{q})^n\\
&-&\frac{(q-1)(q+2\sqrt{q})}{12}(-\sqrt{q})^n\\
&-&\frac{(q-1)(q - \sqrt{q})}{3}((\sqrt{q}\omega_{6})^n + (\sqrt{q}\omega_{6}^{5})^n)\\
&-&\frac{(q-1)(q + \sqrt{q})}{3}((\sqrt{q}\omega_{6}^2)^n + (\sqrt{q}\omega_{6}^{4})^n)\\
&-&\frac{(q-1)q}{4}((\sqrt{q} \, i)^n + (-\sqrt{q} \, i)^n).
\end{eqnarray*}
The characteristic polynomial of Frobenius is therefore
\[
P_2(X) = (X^2-\sqrt{2q}X + q)^{\frac{(q-1)(q-\sqrt{2q})}{4}} (X^2+\sqrt{2q}X + q)^{\frac{(q-1)(q+\sqrt{2q})}{4}}
(X^2 + q)^{\frac{(q-1)q}{2}}
\]
if $r$ is odd, and
\begin{eqnarray*}
P_2(X)=
&& (X-\sqrt{q})^{\frac{(q-1)(q - 2\sqrt{q})}{12}} (X +\sqrt{q})^{\frac{(q-1)(q + 2\sqrt{q})}{12}} \cdot \\
&& (X^2 - \sqrt{q}X + q)^{\frac{(q-1)(q - \sqrt{q})}{3}} (X^2 + \sqrt{q}X + q)^{\frac{(q-1)(q + \sqrt{q})}{3}}
(X^2 + q)^{\frac{(q-1)q}{4}}
\end{eqnarray*}
if $r$ is even. Again, these factors all arise from dimension one supersingular abelian varieties~\cite[Theorems 12.1 \& 12.2]{VJGaryAlexey}.
\subsection{$C_3/{\mathbb F}_q: y^{q}+y = x^{2q+1} + x^{q+2} + x^{q+1} + x^2$}
By Fourier-analysing the formulae in Table~\ref{table:C3points}, for $r$ odd we have:
\begin{eqnarray*}
\#C_3({\mathbb F}_{q^n}) = q^n + 1 &-&\frac{(q-2)q}{8}((\sqrt{q})^n + (-\sqrt{q})^n) \\
&-&\frac{(q-2)q}{8}((\sqrt{q} \, i)^n + (-\sqrt{q} \, i)^n)\\
&-&\frac{(q+1)(q - \sqrt{2q})}{12}( (\sqrt{q}\omega_{24})^n + (\sqrt{q}\omega_{24}^{7})^n +
(\sqrt{q}\omega_{24}^{17})^n + (\sqrt{q}\omega_{24}^{23})^n)\\
&-&\frac{(q+1)(q + \sqrt{2q})}{12}( (\sqrt{q}\omega_{24}^5)^n + (\sqrt{q}\omega_{24}^{11})^n +
(\sqrt{q}\omega_{24}^{13})^n + (\sqrt{q}\omega_{24}^{19})^n)\\
&-&\frac{(q-2)(5q - 4\sqrt{2q})}{24}( (\sqrt{q}\omega_{8})^n + (\sqrt{q}\omega_{8}^{7})^n)\\
&-&\frac{(q-2)(5q + 4\sqrt{2q})}{24}( (\sqrt{q}\omega_{8}^3)^n + (\sqrt{q}\omega_{8}^{5})^n)\\
\end{eqnarray*}
For $r$ even we have:
\begin{eqnarray*}
\#C_3({\mathbb F}_{q^n}) = q^n + 1 &-&\frac{(q-2\sqrt{q})(5q + 2\sqrt{q} - 4)}{24}(\sqrt{q})^n
-\frac{(q+2\sqrt{q})(5q - 2\sqrt{q} - 4)}{24}(-\sqrt{q})^n\\
&-&\frac{q(5q - 8)}{24}((\sqrt{q} \, i)^n + (-\sqrt{q} \, i)^n)\\
&-&\frac{q^2}{8}( (\sqrt{q}\omega_{8})^n + (\sqrt{q}\omega_{8}^{7})^n)\\
&-&\frac{q^2}{8}( (\sqrt{q}\omega_{8}^{3})^n + (\sqrt{q}\omega_{8}^{5})^n)\\
&-&\frac{(q-1)q}{12}( (\sqrt{q}\omega_{12})^n + (\sqrt{q}\omega_{12}^{5})^n +
(\sqrt{q}\omega_{12}^{7})^n + (\sqrt{q}\omega_{12}^{11})^n)\\
&-&\frac{(q-1)(q - 2\sqrt{q})}{12}( (\sqrt{q}\omega_{6})^n + (\sqrt{q}\omega_{6}^{5})^n) \\
&-&\frac{(q-1)(q + 2\sqrt{q})}{12}( (\sqrt{q}\omega_{6}^2)^n + (\sqrt{q}\omega_{6}^{4})^n)
\end{eqnarray*}
The characteristic polynomial of Frobenius is therefore
\begin{eqnarray*}
P_3(X) =
&& (X^2 - q)^{\frac{(q-2)q}{8}} (X^2 + q)^{\frac{(q-2)q}{8}} (X^2-\sqrt{2q}X + q)^{\frac{(q-2)(5q-4\sqrt{2q})}{24}} (X^2+\sqrt{2q}X + q)^{\frac{(q-2)(5q+4\sqrt{2q})}{24}} \\
&& (X^4 - \sqrt{2q}X^3 + qX^2 -q\sqrt{2q}X + q^2)^{\frac{(q+1)(q-\sqrt{2q})}{12}} \\
&& (X^4 + \sqrt{2q}X^3 + qX^2 + q\sqrt{2q}X + q^2)^{\frac{(q+1)(q+\sqrt{2q})}{12}}
\end{eqnarray*}
if $r$ is odd, and
\begin{eqnarray*}
P_3 = && (X-\sqrt{q})^{\frac{(q-2\sqrt{q})(5q + 2\sqrt{q} - 4)}{24}} (X + \sqrt{q})^{\frac{(q+2\sqrt{q})(5q - 2\sqrt{q} - 4)}{24}} \\
&& (X^2 + q)^{\frac{q(5q-8)}{24}} (X^2 - \sqrt{q}X + q)^{\frac{q^2}{8}} (X^2 + \sqrt{q}X + q)^{\frac{q^2}{8}} \\
&& (X^4 - qX^2 + q^2)^{\frac{(q-1)q}{12}}
(X^2 - \sqrt{q}X + q)^{\frac{(q-1)(q - 2\sqrt{q})}{12}} (X^2 + \sqrt{q}X + q)^{\frac{(q-1)(q + 2\sqrt{q})}{12}}
\end{eqnarray*}
if $r$ is even.
These factors all arise from dimension one and two supersingular abelian varieties~\cite[Theorems 12.1 \& 12.2]{VJGaryAlexey}.
We remark that the three curves $C_1$, $C_2$, and $C_3$ have fairly large automorphism groups, and it may well be feasible to compute their
characteristic polynomials of Frobenius by taking quotients and computing the decomposition of their Jacobian's via the Kani-Rosen
theorem~\cite{kanirosen}, rather than use the techniques adopted here.
Finally, combining the expressions for $\#C_1({\mathbb F}_{q^n})$, $\#C_2({\mathbb F}_{q^n})$ and $\#C_3({\mathbb F}_{q^n})$ as per Lemma~\ref{lem:fitzgeneral} and Eq.~(\ref{simple}) gives the following theorem.
\begin{theorem}
For $r$ odd we have:
\begin{eqnarray*}
F_q(n,0,0,0) = q^{n-3} - q^{n/2 - 3}\Big(&&\frac{q(q-1)(q-2)}{8}( 1^n + (-1)^n )\\
&+&\frac{q(q-1)(q+2)}{8}( i^n + (-i)^n )\\
&+&\frac{(q-1)(q+1)(q - \sqrt{2q})}{12}( \omega_{24}^n + \omega_{24}^{7n} +
\omega_{24}^{17n} + \omega_{24}^{23n})\\
&+&\frac{(q-1)(q+1)(q + \sqrt{2q})}{12}( \omega_{24}^{5n} + \omega_{24}^{11n} +
\omega_{24}^{13n} + \omega_{24}^{19n})\\
&+&\frac{(q-1)(q - \sqrt{2q})(5q + \sqrt{2q} + 4)}{24}( \omega_{8}^n + \omega_{8}^{7n})\\
&+&\frac{(q-1)(q + \sqrt{2q})(5q - \sqrt{2q} + 4)}{24}( \omega_{8}^{3n} + \omega_{8}^{5n})
\Big).
\end{eqnarray*}
For $r$ even we have:
\begin{eqnarray*}
F_q(n,0,0,0) = q^{n-3} - q^{n/2 - 3}\Big(&&\frac{(q-1)(q - 2\sqrt{q})(5q + 2\sqrt{q} + 4)}{24} \, 1^n\\
&+&\frac{(q-1)(q + 2\sqrt{q})(5q - 2\sqrt{q} + 4)}{24}(-1)^n \\
&+&\frac{q(q-1)(5q+4)}{24}( i^n + (-i)^n )\\
&+&\frac{q^2(q-1)}{8}( \omega_{8}^n + \omega_{8}^{7n} +
\omega_{8}^{3n} + \omega_{8}^{5n})\\
&+&\frac{q(q-1)^2}{12}( \omega_{12}^n + \omega_{12}^{5n} +
\omega_{12}^{7n} + \omega_{12}^{11n})\\
&+&\frac{(q-1)(q - \sqrt{q})(q - \sqrt{q} + 2)}{12}( \omega_{6}^n + \omega_{6}^{5n}) \\
&+&\frac{(q-1)(q + \sqrt{q})(q + \sqrt{q} + 2)}{12}( \omega_{6}^{2n} + \omega_{6}^{4n}) \Big).
\end{eqnarray*}
\end{theorem}
\section{Concluding remarks}\label{sec:conclusion}
By Fourier-analysing known formulae which count the number of elements of ${\mathbb F}_{2^n}$ for which the first three coefficients of the characteristic polynomial with respect to ${\mathbb F}_2$ are prescribed, we developed a new much simpler curve-based approach to deriving them, in the trace zero cases
for all $n$ and the trace one cases for odd $n$. This approach was
used to count the number of irreducible polynomials in ${\mathbb F}_{2^r}[x]$ for which the first three coefficients are zero.
Based on the ${\mathbb F}_2$ base field case and our result for $f_q(n,0,0,0)$, we conjecture that for $q = 2^r$, $n \ge 3$ and $t_1,t_2,t_3 \in {\mathbb F}_q$,
the formulae for $f_q(n,t_1,t_2,t_3)$ all have period $24$.
For odd $n$ it is possible to compute all of the trace zero cases with our approach, but for reasons of space we only sketch the method here.
First note that although one can express $N(0,0)$ in terms of $Z(q_1)$, $Z(q_2)$ and $Z(q_1 + q_2)$ as per Lemma~\ref{lem:fitzgeneral},
it is not possible to express all $N(t_2,t_3)$ in terms of these. For example, for any $t_2 \in {\mathbb F}_{q}^{\times}$ one has
$Z(q_1) = Z(t_2 q_1) = \sum_{t_3 \in {\mathbb F}_q} N(0,t_3)$, so one does not get a linear system of full rank. However, this issue can be obviated
by changing the definition of $q_1$ and $q_2$ to:
\begin{eqnarray}
\nonumber q_1 &:& {\mathbb F}_{q^n} \rightarrow {\mathbb F}_q: x \mapsto T_1(x^{q+1} + x^2) + t_2,\\
\nonumber q_2 &:& {\mathbb F}_{q^n} \rightarrow {\mathbb F}_q: x \mapsto T_1(x^{2q+1} + x^{q+2}) + t_3.
\end{eqnarray}
Then, for odd $n$ one has $T_1(t_2) = t_2$ and $T_1(t_3) = t_3$, and so
computing the number of points on the supersingular curves $C': y^q + y = x^{q+1} + x^2 + t_2$,
$C_{\alpha}': y^q + y = x^{2q+1} + x^{q+2} + t_3 + \alpha(x^{q+1} + x^2 + t_2)$ for $\alpha \in {\mathbb F}_q$ will
give expressions for $F_{q}(n,0,t_2,t_3)$.
\bibliographystyle{plain}
| {
"timestamp": "2016-08-10T02:07:58",
"yymm": "1605",
"arxiv_id": "1605.07229",
"language": "en",
"url": "https://arxiv.org/abs/1605.07229",
"abstract": "For any positive integers $n\\geq 3, r\\geq 1$ we present formulae for the number of irreducible polynomials of degree $n$ over the finite field $\\mathbb{F}_{2^r}$ where the coefficients of $x^{n-1}$, $x^{n-2}$ and $x^{n-3}$ are zero. Our proofs involve counting the number of points on certain algebraic curves over finite fields, a technique which arose from Fourier-analysing the known formulae for the $\\mathbb{F}_2$ base field cases, reverse-engineering an economical new proof and then extending it. This approach gives rise to fibre products of supersingular curves and makes explicit why the formulae have period $24$ in $n$.",
"subjects": "Number Theory (math.NT)",
"title": "Fibre Products of Supersingular Curves and the Enumeration of Irreducible Polynomials with Prescribed Coefficients",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9888419674366166,
"lm_q2_score": 0.7154239957834733,
"lm_q1q2_score": 0.7074412715418954
} |
https://arxiv.org/abs/2109.05728 | Best proximity pairs in ultrametric spaces | In the present paper, we study the existence of best proximity pairs in ultrametric spaces. We show, under suitable assumptions, that the proximinal pair $(A,B)$ has a best proximity pair. As a consequence we generalize a well known best approximation result and we derive some fixed point theorems. Moreover, we provide examples to illustrate the obtained results. | \section{Introduction and preliminaries}
Let $F:A\rightarrow B$ be a map where $A$ and $B$ are two nonempty subsets of a metric space $M$. Recall that a point $x\in M$ is called a fixed point of a $F$ if $F(x)=x$. It is known that such an equation does not always have a solution. However, in the absence of the fixed point (for example if $A\cap B=\varnothing$), it is possible to consider the problem of finding a point $x\in A$ that is as close as possible to $F(x)$ in $B$; i.e., to minimize the quantity $d(x,F(x))$ over $A$. This type of problem is considered as part of approximation theory, more specifically best approximation point results.
\begin{definition}\cite{ref12}\label{def11}
Let $(M, d)$ be a metric space. A subset $A\subset M$ is said to be proximinal if given any $x\in M$ there exists $a_0\in A$ such that
$$d(x,a_0)=dist(x,A)=\inf\{d(x,z):\; z\in A\}.$$
Such an $a_0$, if it exists, is called a best approximation to $x$ in $A$.
\end{definition}
In the literature, some positive results concerning the existence of the best approximation points were given whether in the archimedean or non-archimedean spaces, for more details see \cite{ref9,ref10,ref11}.
\medskip
Recall that, for subsets $A$ and $B$ of a metric space $(M,d)$, we set
$$dist(A,B)=\inf\{d(a,b):\;a\in A\;\text{and}\;b\in B\},$$
and write $\delta(A)$ for the diameter of $A$, i.e., $\delta(A):=\sup\{d(x,y)\; : \;x,y\in A \}.$
\medskip
It should be noted here that if $A$ and $B$ are proximinal subsets of a metric space $(M, d)$, then, in general, there is no reason to have $d(x, y) = dist(A, B)$ for $x \in A$ and $y \in B$ (see Example \ref{CEx} below). This led the authors in \cite{ref12} to introduce the notion of the best proximity pair.
\begin{definition}\cite[Definition 1.1]{ref12}\label{def12}
Let $M$ be a metric space and let $A$ and $B$ be nonempty subsets of $M$. Let
$$A_0=\{x\in A:\; d(x, y)=dist(A, B)\; \text{for some}\; y\in B\};$$
$$B_0=\{y\in B:\; d(x, y)=dist(A, B)\; \text{for some}\; x\in A\}.$$
A pair $(x, y) \in A_0 \times B_0$ for which $d(x, y)=dist(A, B)$ is called a best proximity pair for the sets $A$ and $B$.
\end{definition}
Let $A$ and $B$ be subsets of a metric space $M$. We will say that the pair $(A, B)$ is proximinal if $A$ and $B$ are proximinal. The following question naturally arises.
\begin{question}\label{Q1}
Let $(A,B)$ be a proximinal pair of $(M,d)$. Does there exists a best proximity pair $(a,b)\in A_0\times B_0$ ? If so, is the pair $(A_0,B_0)$ also proximinal ?
\end{question}
If this question has an affirmative answer, this inspires to formulate the following:
\begin{question}\label{Q2}
Given a mapping $F:A\cup B \rightarrow A\cup B$ with $F(A)\subset A$ and $F(B)\subset B$ (also called noncyclic mapping), does there exists an ordered pair $(a,b)\in A\times B$ such that
$$
F(a)=a, \quad F(b)=b\quad \text{and}\quad d(a,b)=dist(A,B).
$$
\end{question}
There is an extensive literature contains partial affirmative answers to these two questions in the context of metric spaces and linear spaces (see \cite{ref13,ref14,ref15}). To the best knowledge of the authors, this is the first time these questions are considered in the case of ultrametric spaces.
\medskip
Recall that an ultrametric space is a metric space $M$ in which strong triangle inequality $d(x,y) \leq \max\{d(x,z),d(z, y)\}$ is satisfied for all $x,y,z \in M$.
\begin{remark}
It is interesting to note that an axiomatic characterization of proximity spaces generated by ultrametrics was obtained by A. Yu. Lemin in 1984 (see \cite{ref200}).
\end{remark}
\begin{definition}\cite{ref22}
An ultrametric space $(M,d)$ is called spherically complete if each nested sequence of closed balls $B_1 \supset B_2 \supset\cdots$ has a nonempty intersection.
\end{definition}
The spherical complete ultrametric spaces were first introduced by Ingleton \cite{ref23} in order to obtain an analog of the Hanh-Banach theorem for non-Archimedean valued fields. This notion has numerous applications in studies of fixed point results for ultrametric spaces \cite{ref24,ref3,ref17}. It was shown by Bayod and Mart\`inez-Maurica \cite{ref25} that an ultrametric space is spherically complete if and only if this space is ultrametrically injective. Recall that an ultrametric space $(Y,\rho)$ is ultrametrically injective if for each $F : A \rightarrow X$, where $A \subseteq Y$ and $X$ is a space with an ultrametric $d$, the condition
$$d(F (x), F (y)) \leq\rho(x, y),\qquad \forall x, y \in A$$
implies the existence of an extension $\tilde{F}: Y \rightarrow X$ of the mapping $F$ such that
$$d(\tilde{F} (x), \tilde{F} (y)) \leq\rho(x, y),\qquad \forall x, y \in Y.$$
Thus, an ultrametric space is ultrametrically injective if every contractive mapping from this space to arbitrary ultrametric space has a contractive extension. Some interesting results related to spherical completeness of ultrametric spaces can also be found in \cite{ref26} and \cite{ref27}.
\medskip
In \cite{ref8} the authors prove the following result.
\begin{theorem}\label{KS1}
Let $A$ be a nonempty spherically complete subspace of an ultrametric space $M$. Then $A$ is proximinal in $M$.
\end{theorem}
The next definition slightly narrows the corresponding definition from \cite{ref3}.
\begin{definition}\rm\label{Def18}
Let $(M,d)$ be an ultrametric space. Assume that $F:M\rightarrow M$ is a map and $B=B(x,r)$, $r>0$, is a closed ball in $(M,d)$. We say that $B$ is \textit{a minimal $F$-invariant ball} if :
\begin{itemize}
\item[(i)] $F(B)\subseteq B$, and
\item[(ii)] $d(y,F(y))=r$ for each $y\in B$.
\end{itemize}
\end{definition}
\begin{remark}
Definition \ref{Def18} implies, in particular, that any minimal $F$-invariant ball $B$ cannot have the form $\{p\}$, where $p$ is a fixed point of $F$ and $\{p\}$ is the one-point set consisting the only element $p$.
\end{remark}
The following definition is well known.
\begin{definition}
Let $A\subseteq M$. $F:A\rightarrow A$ is said to be nonexpansive if
\[
d(F(x),F(y))\leq d(x,y)\quad \text{for all}\; x,y\in A
\]
\end{definition}
\begin{remark}\label{Rem19}
Let $(M, d)$ be a spherically complete ultrametric space, let $F : M \rightarrow M$ be non-expensive, and let $\mathbf{B}_T$ be the set of all closed balls $B$ which are $F$-invariant,
$$F(B) \subseteq B.$$
Then $B\in\mathbf{B}_T$ is minimal $F$-invariant if and only if $B_1 \subseteq B$ implies $B_1 = B$ for every $B_1 \in \mathbf{B}_T$.
\end{remark}
The next theorem is one of the most significant results of the fixed point theory in ultrametric spaces.
\begin{theorem}\label{KS}\cite{ref3,ref2}
Suppose $M$ is a spherically complete ultrametric space and $F:M\rightarrow M$ is a nonexpansive map. Then every closed ball
$$B(x, d(x, T (x))) = \{y \in M : d(x, y) \leq d(x, T (x))\}$$
contains either a fixed point of $F$ or a minimal $F$-invariant ball.
\end{theorem}
Therefore, to guarantee the existence of a fixed point of $F : M \rightarrow M$, it is necessary to add some additional restrictions. For example, we can consider strictly contractive mappings instead of contractive one.
\begin{definition}
Let $(M,d)$ be an ultrametric space and $F : M \rightarrow M$ a mapping. We say that :
\begin{enumerate}
\item $F$ is strictly contractive if $d(F(x), F(y)) < d ( x , y )$ whenever $x\neq y$.
\item $F$ is strictly contractive on orbit if $F(x)\neq x$ implies $d( F^2x, F(x)) < d ( F(x), x )$ for each $x \in M$.
\end{enumerate}
\end{definition}
Using the following theorem, we can prove the existence of fixed points for every strictly contractive on orbit mapping $F : M \rightarrow M$ of each spherically complete ultrametric space $(M, d)$.
\begin{theorem}\label{w-regular}
Let $(M,d)$ be a spherically complete ultrametric space. Let $F:M\rightarrow M$ be a nonexpansive map satisfying
\begin{equation}\label{eq1}
\liminf_{n\rightarrow \infty}\ d(F^n(x),F^{n+1}(x))< d(x,F(x))
\end{equation}
whenever $x$ in $M$ and $x \neq F(x)$. Then $F$ has a fixed point in any $F$-invariant closed ball.
\end{theorem}
\begin{proof}
Suppose that $B$ is a $F$-invariant closed ball which does not contain a fixed point of $F$, then, by Theorem \ref{KS}, there are $r > 0$ and $x \in B$ such that
$$0<r=d(x,F(x))=d(F(x),F^2(x))=\ldots=d(F^n(x),F^{n+1}(x))< d(x,F(x)) $$
contrary to (\ref{eq1}).
\end{proof}
It should be noted here that introduced in \cite{ref18} the mappings $F : M \rightarrow M$ satisfying the weak regular property are also satisfy conditions of Theorem \ref{w-regular}.
\begin{conjecture}
Let $(M, d)$ be a spherically complete ultrametric space and let a mapping $F:M\rightarrow M$ be nonexpansive. Then the following conditions are equivalent:
\begin{itemize}
\item[(i)] $F$ has a fixed point in every nonempty spherically complete subspace of $M$.
\item[(ii)] Inequality (\ref{eq1}) holds whenever $x\in M$ and $x\neq F (x)$.
\end{itemize}
\end{conjecture}
\medskip
In this paper, we will see that the answer to the above questions is positive under natural assumptions. In the lack of this assumptions, we show by an example that $(A_0,B_0)$ may be an empty pair. As a consequence we generalize a best approximation result due to Kirk and Shahzad (see \cite[Theorem 11]{ref3}) and we derive some fixed point theorems. Throughout this paper, we provide some examples to illustrate the obtained results. Our main theorems continue and strenghten the corresponding results from \cite{CL}.
\section{Main results}
We first give a useful lemma.
\begin{lemma}\label{Lemma1}
Let $(A,B)$ be a proximinal pair of a nonempty ultrametric space $(M, d)$ and let the inequality $\delta(B)\leq dist(A,B)$ hold. Then $A_0$ is nonempty and the equality $B_0= B$ holds.
\end{lemma}
\begin{proof}
Let us consider arbitrary $a \in A$ and $b, b^\prime \in B$. We claim that the equality
\begin{equation}\label{eqq1}
d(a,b)=d(a,b^\prime)
\end{equation}
holds. Indeed, the strong triangle inequality and the inequality $\delta(B)\leq dist(A,B)$ imply that
$$
\begin{array}{ccl}
d(a,b^\prime) & \leq & \max\{d(a,b);d(b,b^\prime) \}\\
& \leq & \max\{d(a,b);\delta(B) \}\\
& \leq & \max\{d(a,b);dist(A,B) \}\\
& \leq & d(a,b).
\end{array}
$$
Thus, $d(a,b^\prime)\leq d(a, b)$ holds. Similarly, we obtain $d(a, b) \leq d(a,b^\prime)$. Equality (\ref{eqq1}) follows.
\medskip
Now we can easily prove that $B_0 = B$ and $A_0 \neq\varnothing$. Indeed, let $b^\prime$ be an arbitrary point of $B$. Then, using equality (\ref{eqq1}), we obtain
\begin{equation}\label{eq2}
dist(A, B) =\inf_{a\in A,b\in B} d(a, b) = \inf_{a\in A} d(a, b^\prime) = dist(b^\prime, A).
\end{equation}
Since $A$ is proximinal, there is $a^\prime\in A$ such that $dist(b^\prime , A) = d(a^\prime, b^\prime)$. The last equality
and (\ref{eq2}) imply
$$d(a^\prime, b^\prime)=dist(A,B).$$
Thus, for every $b^\prime\in B$ there is $a^\prime \in A$ such that (\ref{eqq1}) holds. It implies $A_0\neq\varnothing$ and $B_0 = B$.
\end{proof}
Next, we give an example to show that if $\delta(B)>dist(A,B)$, the pair $(A_0,B_0)$ may be an empty pair.
\begin{example}\label{CEx}
Let $M=\mathbb{N}_0$ be the set of positive integers and define the ultrametric distance $d$ on $M$ as follows:
\[
d(n,m)=\begin{cases}
\begin{array}{ll}
0 & \text{if}\; n=m\\
\max\{\frac{1}{n},\frac{1}{m}\} & \text{otherwise.}
\end{array}
\end{cases}
\]
Then $(M,d)$ is an ultrametric space. Set $A=2\mathbb{N}_0=\{2,4,\ldots\}$ and $B=2\mathbb{N}_0-1=\{1,3,\ldots\}$.
\medskip
It is clear that $A$ and $B$ are proximinal and the equalities $\delta(B)=1$, $\delta(A)=\frac{1}{2}$ and $dist(A,B)=0$ hold. Now note that
$$dist(A,B)\leq\min\{\delta(B),\delta(A)\}$$
and $A_0=B_0=\varnothing$.
\end{example}
\begin{remark}
The ultrametric that we use in Example \ref{CEx} was apparently first constructed by Delhommé, Laflamme, Pouzet and Sauer \cite[Proposition 2]{ref201}. This construction is very often useful in the study of various topological and geometrical properties
of ultrametric spaces \cite{ref202,ref203,ref204,ref205} and has a natural generalization to Priess-Crampe and Ribenboim Ultrametrics with totally ordered range sets (see \cite[Proposition 4.10]{ref206}).
\end{remark}
\begin{lemma}\label{Lemma23}
Let $A$ and $B$ satisfy the conditions of Lemma \ref{Lemma1} and let $b_0$ be an arbitrary point of $B$. Write $r := dist(A, B)$. Then we have
\begin{equation}\label{eq3}
A_0=A\cap S(b_0,r),
\end{equation}
where
\begin{equation}\label{eq4}
S(b_0 , r) := \{x \in M : d(b_0 , x) = r\}
\end{equation}
is the sphere with the radius $r$ and the center $b_0$.
\end{lemma}
\begin{proof}
Suppose that $r = 0$. Then the inequality $\delta(B)\leq dist(A,B)$ implies $B = \{b_0\}$, where $\{b_0\}$ is the set containing the point $b_0$ only. Hence, by Definition \ref{def12},
\begin{equation}\label{eq5}
A_0 = \{x \in A : d(x, b_0 ) = 0\}
\end{equation}
holds. Since $A_0$ is nonempty, from (\ref{eq5}) it follows that $A_0 =\{b_0\}$. The last equality and equality (\ref{eq4}) imply (\ref{eq3}) for $r = 0$.
\medskip
For $r > 0$ we consider the open ball
$$B^\prime(b_0 , r) := \{x \in M : d(b_0 , x) < r\}$$
and the "exterior" $E(b_0 , r)$ of this ball,
$$E(b_0 , r) := \{x \in M : d(b_0 , x) > r\}.$$
It is clear that $M = B^\prime(b_0 , r) \cup S(b_0 , r) \cup E(b_0 , r)$. Consequently, the set $A_0$ can be represented as
$$A_0=A_0\cap M= [A_0\cap B^\prime(b_0 , r)] \cup [A_0\cap S(b_0 , r)] \cup [A_0\cap E(b_0 , r)].$$
Thus, equality (\ref{eq3}) holds if and only if
\begin{equation}\label{eq6}
A_0\cap B^\prime(b_0 , r)=\varnothing=A_0\cap E(b_0 , r).
\end{equation}
To prove (\ref{eq6}), we first note that
$$d(a,b_0)\geq dist(a,B)\geq dist(A,B)=r$$
holds for every $a \in A$. Consequently, the intersection $A\cap B^\prime(b_0 , r)$ is empty,
\begin{equation}\label{eq7}
A\cap B^\prime(b_0 , r)=\varnothing
\end{equation}
It implies $A_0\cap B^\prime(b_0 , r)=\varnothing$ because $A_0 \subseteq A$.
\medskip
If $A_0\cap E(b_0 , r)\neq\varnothing$, then there is $a_0\in A_0$ such that
\begin{equation}\label{eq8}
d(a_0 , b_0 ) = r_1 > r = dist(A, B).
\end{equation}
It follows from the proof of Lemma \ref{Lemma1} (see (\ref{eqq1})) that
\begin{equation}\label{eq9}
d(a_0 , b) = d(a_0 , b_0 )
\end{equation}
holds for every $b \in B$. Since $a_0 \in A_0$, there is $b^\prime\in B$ such that $d(b^\prime , a_0 ) = dist(A, B)$ by
Definition \ref{def12}. Using (\ref{eq8}), the last equality and equality (\ref{eq9}) with $b^\prime= b$, we obtain the
contradiction,
$$d(a_0,b_0)<dist(A,B)=d(b^\prime,a_0)=d(a_0,b_0),$$
that implies the second equality in (\ref{eq6}).
\end{proof}
The next lemma follows from \cite[Proposition 18.5]{ref26}.
\begin{lemma}\label{Lemma24}
Let $(M, d)$ be an ultrametric space and let $B$ be a ball in $(M, d)$. Then
\begin{equation}\label{eq10}
d(x, b) = d(x, a)
\end{equation}
holds for every $x \in M \setminus B$ and all $a, b \in B$.
\end{lemma}
The following theorem provides a partial answer to Question \ref{Q1}.
\begin{theorem}\label{Thm25}
Let $(A, B)$ be a proximinal pair in a nonempty ultrametric space $(M, d)$. Then the following statements are equivalent:
\begin{itemize}
\item[(i)] There is a point $a^\prime\in A$ such that
\begin{equation}\label{eq11}
d(a^\prime, b) = dist(A, B)
\end{equation}
for every $b \in B$.
\item[(ii)] The inequality $\delta(B)\leq dist(A, B)$ holds.
\item[(iii)] The sets $A_0$ and $B_0$ are proximinal subsets of $(M, d)$, and the equalities $B_0 = B$ and $dist(A_0 , B_0 ) = dist(A, B)$ hold, and every $(a, b) \in A_0 \times B_0$ is a best proximity pair for both $(A, B)$ and $(A_0 , B_0 )$.
\end{itemize}
\end{theorem}
\begin{proof}
(i)$\Rightarrow$(ii). Let (\ref{eq11}) hold with a fixed $a^\prime\in A$ and all $b \in B$. Then, for arbitrary $b_1 , b_2 \in B$, the strong triangle inequality and (\ref{eq11}) imply
$$d(b_1,b_2)\leq dist(A,B).$$
Statement (ii) follows.
\medskip
(ii)$\Rightarrow$(iii). Let $\delta(B)\leq dist(A, B)$ hold. Let us prove that $(A_0 , B_0 )$ is proximinal. By Lemma \ref{Lemma1}, the equality $B_0 = B$ holds. Thus, $B_0$ is proximinal. Hence, it suffices to show that $A_0$ is proximinal.
Let $b_0$ be an arbitrary point of $B_0$. By Lemma \ref{Lemma23}, we have
\begin{equation}\label{eq12}
A_0 = A \cap S(b_0 , r)
\end{equation}
with $r = dist(A, B)$. Let us consider the closed ball $B(b_0 , r)$ with the center $b_0$ and the radius $r$,
\begin{equation}\label{eq13}
B(b_0 , r) = \{x\in M : d(b_0 , x) \leq r\} = B^\prime(b_0 , r) \cup S(b_0 , r)
\end{equation}
Using (\ref{eq12}), (\ref{eq13}) and equality (\ref{eq7}), we can represent the set $A_0$ as
\begin{equation}\label{eq14}
A_0 = A \cap B(b_0 , r)
\end{equation}
By Definition \ref{def11}, $A_0$ is proximinal if for every $x^\prime\in M$ there is $a_0 \in A_0$ such that $d(x^\prime , a_0 ) = dist(x^\prime , A_0 )$. Let $x^\prime$ be an arbitrary point of $M$. If $x^\prime\notin B(b_0 , r)$, then
\begin{equation}\label{eq15}
d(b_0,x^\prime)>r
\end{equation}
holds. Using (\ref{eq14}), (\ref{eq15}) and Lemma \ref{Lemma24}, we obtain that
\begin{equation}\label{eq16}
d(a_0,x^\prime)=d(b_0,x^\prime)>r
\end{equation}
holds for every $a_0\in A_0$. Consequently, the equality
\begin{equation}\label{eq17}
d(x^\prime,a_0)=dist(x^\prime,A_0)
\end{equation}
holds for every $a_0\in A_0$.
Suppose now that $x^\prime \in B(b_0 , r)$. Since $A$ is proximinal, there exists $a^\prime\in A$ such that
\begin{equation}\label{eq18}
d(x^\prime,a^\prime)=dist(x^\prime,A)
\end{equation}
It is easy to see that $a^\prime \in B(b_0 , r)$. Indeed, if $a^\prime\notin B(b_0 , r)$, then, similarly to (\ref{eq16}), we
obtain $d(x^\prime , a^\prime ) > r$.
Now if $a_0$ is an arbitrary point of $A_0$, then from (\ref{eq14}) and $x^\prime\in B(b_0 , r)$ it follows that $d(x^\prime , a_0 ) \leq r$. (The diameter of any ultrametric ball is less than or equal to its radius see \cite[Proposition 1.2]{ref16}.) Thus, we have the contradiction,
$$r < d(x^\prime, a^\prime ) = dist(x^\prime , A) \leq dist(x^\prime , A_0 ) \leq dist(x^\prime , a_0 ) \leq r$$
and, consequently, the equality $d(x^\prime , a^\prime ) = dist(x^\prime, A_0 )$ holds. Thus, $A_0$ is proximinal.
To complete the proof of validity of the implication (ii)$\Rightarrow$(iii), it suffices to note that
$$d(a_0 , b_0 ) = dist(A, B)$$
holds for every pair $(a_0 , b_0 )\in A_0 \times B_0$ by Lemma \ref{Lemma23}.
\medskip
(iii)$\Rightarrow$(i). If (iii) holds, then (\ref{eq11}) yields with any $a^\prime\in A_0$ for each $b\in B$ because $B_0 = B$.
\end{proof}
\medskip
Theorem \ref{Thm25} and Lemma \ref{Lemma23} completely describe the structure of the set of all best proximity pairs for $A$ and $B$ if $\delta(B) \leq dist(A, B)$. It seems to be interesting to find a generalization of these results for the case when the proximinal pair $(A, B)$ is arbitrary.
\medskip
Since every spherically complete set is proximinal, and every point of each ultrametric ball is a center of this ball, and every closed ball in spherically complete space is spherically complete, formula (\ref{eq14}) and Theorem \ref{Thm25} imply the following
\begin{corollary}\label{Cor27}
Let $(A, B)$ be a nonempty spherically complete pair in an ultrametric space $(M, d)$. If $\delta(B)\leq dist(A, B)$ holds, then $(A_0 , B_0 )$ is also a nonempty spherically complete pair in $(M, d)$.
\end{corollary}
The next theorem partially answers Question \ref{Q2} for nonexpansive mappings.
\begin{theorem}\label{Thm1}
Let $A$ and $B$ be nonempty spherically complete sets in an ultrametric space $(M, d)$ and let $\delta(B) \leq dist(A, B)$ hold. Suppose $F: A \cup B \rightarrow A \cup B$ is a noncyclic nonexpansive mapping. Then there exists a best proximity pair $(a^*, b^* ) \in A\times B$ satisfying exactly one of the following statements:
\begin{itemize}
\item[(i)] $a^*$ and $b^*$ are fixed points of $F$.
\item[(ii)] $a^*$ is a fixed point of $F$, and $B(b^*,d(b^*,F(b^*)))$ is a minimal $F$-invariant ball in $B$, each point of which is a nearest point to $a^*$.
\item[(iii)] $b^*$ is a fixed point of $F$ in $B$, and $B(a^*,d(a^*,F(a^*)))$ is a minimal $F$-invariant ball in $A$, and each point of which is a nearest point to $b^*$.
\item[(iv)] $B(a^*,d(a^*,F(a^*)))$ and $B(b^*,d(b^*,F(b^*)))$ is a minimal $F$-invariant balls in $A$ and, respectively, in $B$, and in addition, every pair
$$(x,y)\in B(a^*,d(a^*,F(a^*)))\times B(b^*,d(b^*,F(b^*)))$$
is a a best proximity pair for $B(a^*,d(a^*,F(a^*)))$ and $B(b^*,d(b^*,F(b^*)))$
\end{itemize}
\end{theorem}
\begin{proof}
It follows from Definition \ref{Def18} that Statements (i)--(iv) are pairwise inconsistent (see Remark \ref{Rem19}). Therefore, it is enough to find a best proximity pair $(a^*, b^* ) \in A\times B$ for which at least one of statements (i)--(iv) is fulfilled.
\medskip
By Corollary \ref{Cor27}, $A_0$ and $B_0$ are nonempty spherically complete subsets of $(M, d)$. Let $x\in A_0$ and $y \in B_0$ be arbitrary. Then, using the implication (ii)$\Rightarrow$(iii) from Theorem \ref{Thm25},
we obtain the equality
\begin{equation}\label{eqv420}
dist(A, B) = d(x, y).
\end{equation}
Since $F$ is nonexpansive, equality (\ref{eqv420}) implies the inequality
\begin{equation}\label{eqv421}
d(F(x), F (y)) \leq dist(A, B).
\end{equation}
The mapping $F$ is noncyclic, thus, we have $F(x) \in A$ and $F(y) \in B$. Consequently, the inequality $dist(A, B) \leq d(F(x), F (y))$ holds. The last inequality and (\ref{eqv421}) imply
$$d(F(x), F(y)) = dist(A, B).$$
Thus, we have $(F(x), F(y)) \in A_0 \times B_0$ for every $(x, y) \in A_0 \times B_0$. It implies the inclusions $F(A_0 ) \subseteq A_0$ and $F(B_0 ) \subseteq B_0$.
\medskip
If $dist(A, B) = 0$ holds, then from $\delta(B) \leq dist(A, B)$ it follows that $B$ is a single-point set. Hence, there is a unique $b^*\in M$ such that $B = B_0 = \{b^*\}$. Now the inclusion $F(B_0 ) \subseteq B_0$ gives us $F(b^* ) = b^*$. Similarly, by Lemma \ref{Lemma23}, for the case $dist(A, B) = 0$, we can find a unique $a^*\in A$ such that $A_0 = \{a^*\}$. The last inequality and the inclusion $F(A_0 ) \subseteq A_0$ imply $F(a^*) = a^*$. Thus, Statement (i) holds if $dist(A, B) = 0$.
\medskip
Let us consider the case when $dist(A, B) > 0$. Using Statement (iii) of Theorem \ref{Thm25} and applying Theorem \ref{KS} to the mappings $F_{|A_0}$ and $F_{|B_0}$, where $F_{|A_0}$ and $F_{|B_0}$ are the restrictions of $F$ on $A_0$ and, respectively, on $B_0$, we see that the theorem is true when $A_0 = A$.
\medskip
To complete the proof, it suffices to show that every closed ball in $A_0$ is a closed ball in $A$. To see it, we note that
\begin{equation}\label{eqv422}
A_0 = A \cap S(b_0 , r)
\end{equation}
holds by Lemma \ref{Lemma23}, when $S(b_0 , r) = \{x \in M : d(x_0 , x) = r\}$, $b_0 \in B$ and $r = dist(A, B)$.
Moreover, by (\ref{eq6}), we have the equality
\begin{equation}\label{eqv423}
A_0 \cap B^\prime(b_0 , r)=\varnothing
\end{equation}
for $B^\prime(b_0 , r) = \{x\in M : d(b_0 , x) < r\}$, where $b_0$ and $r$ are the same as in (\ref{eqv422}). Now
equalities (\ref{eqv422}) and (\ref{eqv423}) imply
$$A_0 = A \cap B(b_0 , r),$$
i.e., $A_0$ is a closed ball in $A$. Since $A$ is a subspace of ultrametric space $(M, d)$ and $A_0 \subseteq A$ holds, every closed ball in the closed ball $A_0$ is also a closed ball in $A$.
\end{proof}
The following example shows that Theorem \ref{Thm1} cannot be strengthen by removing any of Statements (i)--(iv).
\begin{example}\label{ex29}
Let $A = \{a_1 , a_2\}$ and $B = \{b_1 , b_2\}$ be disjoint sets and let $d$ be an ultrametric on $A \cup B$ such that $d(a_1 , a_2 ) = d(b_1 , b_2 ) = 1$ and $d(a, b) = 2$ whenever $a \in A$ and $b \in B$. Let us consider the permutations
\begin{equation*}
F_1=\begin{pmatrix}
a_1 & a_2 & b_1 & b_2 \\
a_1 & a_2 & b_1 & b_2
\end{pmatrix}, \qquad F_2=\begin{pmatrix}
a_1 & a_2 & b_1 & b_2 \\
a_1 & a_2 & b_2 & b_1
\end{pmatrix},
\end{equation*}
\begin{equation*}
F_3=\begin{pmatrix}
a_1 & a_2 & b_1 & b_2 \\
a_2 & a_1 & b_1 & b_2
\end{pmatrix}, \qquad F_4=\begin{pmatrix}
a_1 & a_2 & b_1 & b_2 \\
a_2 & a_1 & b_2 & b_1
\end{pmatrix}.
\end{equation*}
Then every $F_i,$ $i = 1,\ldots, 4$, is a noncyclic nonexpansive mapping. In addition, $F_1$ satisfies Statement (i) but does not satisfy any of statements (ii), (iii) and (iv). $F_2$ satisfies Statement (ii) but does not satisfy any of statements (i), (iii) and (iv), and so on.
\end{example}
\begin{conjecture}
Statements (i)--(iv) from Theorem \ref{Thm1} remain valid for all nonempty spherically complete sets $A$ and $B$ even if the inequality $\delta(B) \leq dist(A, B)$ does not hold.
\end{conjecture}
\begin{corollary}\cite[Theorem 11]{ref3}
Let $A$ be a nonempty spherically complete subspace of an ultrametric space $M$, and let $b^* \in M \setminus A$. Suppose $F : M \rightarrow M$ is a mapping for which $F(b^*) = b^*$. Also assume that $F$ is nonexpansive on $A \cup \{ b^* \}$ and that $A$ is $F$-invariant. Then $F$ has a fixed point in $A$ which is a nearest point of $b^*$ in $A$, or $A$ contains a minimal $F$-invariant set, each point of which is a nearest point to $b^*$ in $A$.
\end{corollary}
Next, we derive some future fixed point results partially answering Question \ref{Q2}.
\begin{theorem}\label{Thm2}
Let $A$ and $B$ be nonempty spherically complete subsets of an ultrametric space $(M, d)$ and let $\delta(B) \leq dist(A, B)$ hold. Suppose $F: A \cup B \rightarrow A \cup B$ is a noncyclic nonexpansive mapping satisfying
$$\liminf_{n\rightarrow \infty}\ d(F^n(x),F^{n+1}(x))< d(x,F(x))$$
whenever $x$ in $M$ and $x \neq F(x)$. Then, there exist $a\in A$ and $b\in B$ such that
$$F(a)=a, \quad F(b)=b\quad \text{and}\quad d(a,b)=dist(A,B).$$
\end{theorem}
\begin{proof}
By Corollary \ref{Cor27} $A_0$ and $B_0$ also are nonempty spherically complete subsets of $(M, d)$. As in the proof of Theorem \ref{Thm1}, we obtain that $F(A_0 ) \subseteq A_0$ and $F(B_0 ) \subseteq B_0$.
\medskip
Now the existence of $a \in A_0$ and $b \in B_0$ which satisfy $F(a) = a$ and $F(b) = b$ follows from Theorem \ref{w-regular}. The pair $(A, B)$ is proximinal by Theorem \ref{KS1}. Using Theorem \ref{Thm25}, we obtain $d(a, b) = dist(A, B)$ since $a \in A_0$ and $b \in B_0$.
\end{proof}
Since every strictly contractive mapping $F: M \rightarrow M$ is nonexpansive and has a unique fixed point if $M$ is spherically complete (see \cite{ref19}), we obtain the following.
\begin{theorem}\label{Thm3}
Let $A$ and $B$ be nonempty spherically complete subspaces of an ultrametric space $(M, d)$ and let $\delta(B)\leq dist(A,B)$ hold. If there is a noncyclic strictly
contractive $F: A \cup B \rightarrow A \cup B$, then this $F$ has a unique fixed point $p$ and the equalities
\begin{equation}\label{eqv424}
B = B_0 = A_0 = \{p\}
\end{equation}
hold.
\end{theorem}
\begin{proof}
Suppose that there is a noncyclic strictly contractive mapping $F : A \cup B \rightarrow A \cup B$. Then $F_{|A}$ and $F_{|B}$, the restrictions of $F$ on $A$ and, respectively, on $B$, are also strictly contractive. Since $A$ and $B$ are nonempty and spherically complete, we can find a unique $a^*\in A$ and a unique $b^*\in B$ such that
\begin{equation}\label{eqv425}
F_{|A}(a^*)=a^*\quad\text{and}\quad F_{|B}(b^*)=b^*.
\end{equation}
As noted in the proof of Theorem \ref{Thm1}, the inclusions $F(A_0 ) \subseteq A_0$ and $F(B_0 ) \subseteq B_0$ hold. Moreover, $A_0$ and $B_0$ are also nonempty and spherically complete by Corollary \ref{Cor27}. Hence, we can find the unique $a_0 \in A_0$ and $b_0 \in B_0$ such that
\begin{equation}\label{eqv426}
F_{|A_0}(a_0)=a_0\quad\text{and}\quad F_{|B_0}(b_0)=b_0.
\end{equation}
Now the inclusions $A_0 \subseteq A$, $B_0 \subseteq B$, equalities (\ref{eqv425}), (\ref{eqv426}), and the uniqueness of $a^*$ and $b^*$ satisfying (\ref{eqv425}) imply the equalities $a^*= a_0$ and $b^*= b_0$. If $a^*$ and $b^*$ are distinct, then
\begin{equation}\label{eqv427}
d(F(a_0),F(b_0))<d(a_0,b_0).
\end{equation}
because $F$ is strictly contractive. Using Theorems \ref{KS1} and \ref{Thm25}, and inequality (\ref{eqv427}), we obtain the contradiction,
$$dist(A, B) = d(a_0 , b_0 ) > d(F(a_0 ), F(b_0 )) > dist(A, B).$$
Thus, the point $b_0$ is a unique fixed point of $F : A \cup B \rightarrow A \cup B$ and $dist(A, B) = 0$ holds. To complete the proof, it suffices to note that (\ref{eqv424}) with $p = b_0$ now follows from $a_0 \in A_0$ and $b_0 \in B_0$ by Theorems \ref{KS1} and \ref{Thm25}.
\end{proof}
As a consequence of Theorem \ref{Thm3}, we obtain the following corollary that strengths Corollary 12 from \cite{ref3}.
\begin{corollary}
Let $A$ be a nonempty spherically complete subspace of an ultrametric space $M$ and let $F : M \rightarrow M$ be a mapping having a fixed point $b^* \in M$. Assume that $F$ is strictly contractive on $A \cup \{ b^*\}$ and $A$ is $F$-invariant. Then $b^*$ is a point in $A$.
\end{corollary}
\bibliographystyle{plain}
| {
"timestamp": "2021-09-14T02:32:13",
"yymm": "2109",
"arxiv_id": "2109.05728",
"language": "en",
"url": "https://arxiv.org/abs/2109.05728",
"abstract": "In the present paper, we study the existence of best proximity pairs in ultrametric spaces. We show, under suitable assumptions, that the proximinal pair $(A,B)$ has a best proximity pair. As a consequence we generalize a well known best approximation result and we derive some fixed point theorems. Moreover, we provide examples to illustrate the obtained results.",
"subjects": "General Topology (math.GN)",
"title": "Best proximity pairs in ultrametric spaces",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9888419671077918,
"lm_q2_score": 0.7154239957834733,
"lm_q1q2_score": 0.7074412713066462
} |
https://arxiv.org/abs/2101.09243 | Hamiltonian paths, unit-interval complexes, and determinantal facet ideals | We study d-dimensional generalizations of three mutually related topics in graph theory: Hamiltonian paths, (unit) interval graphs, and binomial edge ideals. We provide partial high-dimensional generalizations of Ore and Posa's sufficient conditions for a graph to be Hamiltonian. We introduce a hierarchy of combinatorial properties for simplicial complexes that generalize unit-interval, interval, and co-comparability graphs. We connect these properties to the already existing notions of determinantal facet ideals and Hamiltonian paths in simplicial complexes. Some important consequences of our work are:(1) Every almost-closed strongly-connected d-dimensional simplicial complex is traceable. (This extends the well-known result "unit-interval connected graphs are traceable".)(2) Every almost-closed d-complex that remains strongly connected after the deletion of d or less vertices, is Hamiltonian. (This extends the fact that "unit-interval 2-connected graphs are Hamiltonian".)(3) Unit-interval complexes are characterized, among traceable complexes, by the property that the minors defining their determinantal facet ideal form a Groebner basis for a diagonal term order which is compatible with the traceability of the complex. (This corrects a recent theorem by Ene et al., extends a result by Herzog and others, and partially answers a question by Almousa-Vandebogert.)(4) Only the d-skeleton of the simplex has a determinantal facet ideal with linear resolution. (This extends the result by Kiani and Saeedi-Madani that "only the complete graph has a binomial edge ideal with linear resolution".)(5) The determinantal facet ideals of all under-closed and semi-closed complexes have a square-free initial ideal with respect to lex. In characteristic p, they are even F-pure. | \section*{Introduction}
The first Combinatorics paper in History is apparently Leonhard Euler's 1735 solution of the K\"onigsberg bridge problem. In that article, Euler introduced the notion of graph, and studied cycles (now called `Eulerian') that touch all edges exactly once. Euler proved that the graphs admitting them, are exactly those graphs with all vertices of even degree. \emph{Hamiltonian} cycles are instead cycles that touch all \emph{vertices} exactly once; they are named after sir William Rowan Hamilton, who in 1857 invented a puzzle game which asked to find one such cycle in the icosahedron.
Unlike for the Eulerian case, figuring out if a graph admits a Hamiltonian cycle or not is a hard problem, now known to be NP-complete \cite{Karp}.
Even if simple characterizations are off the table, in the 1950s and 1960s Dirac, Ore, P\'osa and others were able to obtain simple conditions on the vertex degrees (in the spirit of Euler's work) that are \emph{sufficient} for a graph to admit Hamiltonian cycles \cite{Dirac, Ore, Posa}. Ore's theorem, for example, says, ``{\it Any graph with $n$ vertices such that
$\deg u + \deg v \ge n$ for all non-adjacent vertices $u,v$, admits a Hamiltonian cycle}''. Ore's condition is far from being necessary: In any cycle, no matter how large, one has $\deg u + \deg v =4$ for all $u,v$.
In the same years, the two papers \cite{LekBol} and \cite{GilmoreHoffman} initiated the study of \emph{unit-interval graphs}. This very famous class consists, as the name suggests, of all intersection graphs of a bunch of length-one open intervals on the real line. (That is, we place a node in the middle of each interval, and we connect two nodes with an arc if and only if the corresponding intervals overlap). Bertossi's theorem says that if they are connected, such graphs always admit \emph{Hamiltonian paths}, i.e. paths that touch all vertices once \cite{Bertossi}. Chen--Chang--Chang's theorem states that $2$-connected unit-interval graphs admit Hamiltonian cycles \cite{ChChCh}. For these results, the length-one request can be weakened to ``pairwise not-nested'', but it cannot be dismissed: Within the larger world of \emph{interval graphs}, one encounters connected graphs such as $K_{1,3}$ that do not admit Hamiltonian paths, and also $2$-connected graphs like the $G_5$ of Remark~\ref{rem:G5} that do not admit Hamiltonian cycles.
In the 1970s, the work of Stanley and Reisner established a fundamental bridge between Combinatorics and Commutative Algebra, namely, a natural bijection between labeled simplicial complexes on $n$ vertices and radical monomial ideals in a polynomial ring with $n$ variables. This correspondence lead Stanley to prove the famous Upper Bound Theorem for triangulated spheres~\cite{StanleyUBT}. After this success, many authors have investigated ways to encode graphs into monomial ideals. In 2010, Herzog et al.~\cite{HerzogEtAl} first considered a natural way to encode graphs into binomial ideals, the so-called \emph{binomial edge ideals}. The catch is that all such binomial edge ideals are radical \cite{HerzogEtAl}. In the process, Herzog et al. re-discovered unit-interval graphs, characterizing them as the graphs whose binomial edge ideals have quadratic Gr\"obner bases with respect to a diagonal term order \cite[Theorem 1.1]{HerzogEtAl}.
So far, we sketched three graph-theoretic topics from three different centuries: Hamiltonian paths, (unit) interval graphs, binomial edge ideals. In the last years, there has been an increasing interest in expanding these three notions to higher dimensions. Specifically:
\begin{compactitem}
\item Katona--Kierstead \cite{KatKie} and many others \cite{HS, Ketal, RSR} have studied ``tight Hamiltonian paths'' and ``loose Hamiltonian paths'' in $d$-dimensional simplicial complexes; both notions for $d=1$ boil down to ordinary Hamiltonian paths. The good news is that extremal combinatorics provides a non-trivial way to extend Dirac's theorem for $d$-complexes with a very large number of vertices that satisfy certain ridge-degree conditions. The bad news is that already Ore and P\'osa's theorems seem very hard to extend.
\item Ene et al. \cite{EneEtAl} introduced ``determinantal facet ideals'', which directly generalize binomial edge ideals, and ``closed $d$-complexes'', which generalize `unit-interval graphs'. The good news is that the definitions are rather natural. The bad news is that determinantal facet ideals are not radical in general (see Example \ref{ex:notradical}), and they are hard to manipulate; alas, the two main results of the paper \cite{EneEtAl} are incorrect, cf.~Remark \ref{rem:correction}.
\end{compactitem}
\medskip
In the present paper we take a new, unified look at these approaches. In Chapter 1, we introduce a notion of `weakly-Hamiltonian paths' for $d$-dimensional simplicial complexes that for $d=1$ also boils down to ordinary Hamiltonian paths. This weaker notion enables us to obtain a first, partial extension of Dirac, Ore and P\'osa's theorem to higher dimensions:
\begin{thmnonumber}[Higher-dimensional Ore and Dirac, cf.~Proposition~\ref{prop:Ore1} and Corollary \ref{cor:Dirac}] Let $\Delta$ be any traceable $d$-complex on $n>2d$ vertices. If in some labeling that makes $\Delta$ traceable the two $(d-1)$-faces $\sigma$ and $\tau$ formed by the first $d$ and the last $d$ vertices, respectively, have facet degrees summing up to at least $n$, then $\Delta$ admits a weakly-Hamiltonian cycle.\\
In particular, if in a traceable pure $d$-complex with $n$ vertices, every $(d-1)$-face belongs to at least $\frac{n}{2}$ facets, then the complex admits a weakly-Hamiltonian cycle.
\end{thmnonumber}
\begin{thmnonumber}[Higher-dimensional P\'osa, cf.~Proposition~\ref{prop:PosaD}] \label{prop:PosaD} Let $\Delta$ be any traceable pure $d$-complex on $n$ vertices, $n > 2d$. Suppose that with any labeling in which $\Delta$ has a weakly-Hamiltonian path, $\Delta$ is traceable.
Let $\sigma_1, \sigma_2, \ldots, \sigma_{s}$ be the $(d-1)$-faces of $\Delta^*$, ordered so that $d_1 \le d_2 \le \ldots \le d_{s}$, where $d_i \eqdef d_{\sigma_i}$ is the number of $d$-faces containing $\sigma_i$. If for every $d \le k < \frac{n}{2}$ one has $ d_{k-d+1} >k$,
then $\Delta$ admits a weakly-Hamiltonian cycle.
\end{thmnonumber}
As you can see these results are conditional: `Traceability', i.e. the existence of a \emph{tight} Hamiltonian path, must be known a priori, in order to infer the existence of a \emph{weakly-}Hamiltonian cycle. This sounds like a bad deal, but in the one-dimensional case our results above still immediately imply the original theorems by Ore and P\'osa for graphs. Moreover, since no extremal combinatorics is used in the proof, there is an advantage: Main Theorems I and II do not require the number of vertices to be extremely large. On the contrary: In the two-dimensional case, they already apply to complexes with five vertices.
\medskip
In Chapter 2, we introduce a hierarchy of four natural properties that progressively weaken (for strongly-connected complexes) the notion of ``closed $d$-complexes'', as originally proposed in \cite{EneEtAl}. We introduce ``unit-interval'', ``under-closed'', and ``weakly-closed'' complexes, as natural combinatorial higher-dimensional generalizations of unit-interval graphs, of interval graphs, and of co-comparability graphs, respectively. The forth property, called ``semi-closed'', is intermediate between ``under-closed'' and ``weakly-closed''; it is also defined very naturally, but it seems to be new already for graphs. We will see its algebraic consequence in Main Theorem VI below.
The main goal of Chapter 2 is to connect this hierarchy to the notions of Chapter 1:
\begin{thmnonumber}[Higher-dimensional Bertossi, Theorem \ref{thm:CTSC}] Every unit-interval strongly-connected $d$-dimensional simplicial complex is traceable.
\end{thmnonumber}
\begin{thmnonumber}[Higher-dimensional Chen--Chan--Chang, Theorem \ref{thm:Hi0}] Every unit-interval $d$-dimensional simplicial complex that remains strongly connected after the deletion of~$d$ or less vertices, however chosen, is Hamiltonian.
\end{thmnonumber}
Finally, Chapter 3 is dedicated to the connection with commutative algebra. For a homogeneous ideal of polynomials, having a square-free Gr\"obner degeneration is a strong and desirable property. In 2020, Conca and the third author proved Herzog's conjecture that if a homogeneous ideal $I$ has a square-free initial ideal $\operatorname{in}(I)$, then the extremal Betti numbers of $I$ and $\operatorname{in}(I)$ are the same \cite{CV}. This allows us to infer the depth, the Castelnuovo--Mumford regularity, and many other invariants of the ideals $I$ with squarefree initial-ideal, simply by computing these invariants on the initial ideal --- which is a much simpler task, because the aforementioned Stanley--Reisner correspondence activates techniques from combinatorial topology.
Building on the very recent work of the second author \cite{Se2}, we are able to revise one of the results claimed in Ene et al \cite{EneEtAl} as follows:
\begin{thmnonumber}[Theorem \ref{t:a-c-gb} and \ref{t:a-c-gb1}]
A strongly-connected $d$-dimensional simplicial complex $\Delta$ is unit-interval if and only if the complex is traceable and with respect to the same labeling, the minors defining the determinantal facet ideal of $\Delta$ form a Gr\"obner basis with respect to any diagonal term order.
\end{thmnonumber}
We conclude our work with a result that provides a broad class of determinantal facet ideals that are radical, and even $F$-pure (if the characteristic is positive):
\begin{thmnonumber}[Theorem \ref{t:s-c-f}] The determinantal facet ideals of all semi-closed complexes are radical. Indeed, they have a square-free initial ideal with respect to any diagonal term order. Moreover, in characteristic $p>0$, the quotients by these ideals are all $F$-pure.
\end{thmnonumber}
The proof relies once again on the recent work by the second author \cite{Se2}. Since all shifted complexes are under-closed, and in particular semi-closed, Theorem \ref{t:s-c-f} immediately implies that the determinantal facet ideals of shifted complexes admit a square-free Gr\"obner degeneration and, in positive characteristic, define $F$-pure rings.
As a consequence of Main Theorem VI, we can extend to all dimensions the result by Kiani and Saeedi-Madani that ``among all graphs, only complete graphs have a binomial edge ideal with a linear resolution'' \cite{SMK}.
Namely, we prove that among all $d$-dimensional simplicial complexes with $n$ vertices, only the $d$-skeleta of simplices have a determinantal facet ideal with a linear resolution (Corollary \ref{cor:SMK}).
\section*{Notation}
Throughout $d, n$ are positive integers, with $d < n$. We denote by $\Sigma^d$ the $d$-simplex, and by $\Sigma^d_n$ the $d$-skeleton of $\Sigma^{n-1}$.
We write each face of $\Sigma^d_n$ by listing its vertices in increasing order. We describe simplicial complexes by listing their facets in any order, e.g. $ \Delta = 123, 124, 235$. For any $d$-face $F=a_0a_1\cdots a_d$ of $\Sigma_n^d$, we call \emph{gap of $F$} the integer $\operatorname{gap}(F) \eqdef a_d - a_0 - d$, which counts the integers $i$ strictly between $a_0$ and $a_d$ that are not present in $F$.
For each $i$ in $\{1, 2, \ldots, n-d\}$, we call $H_i$ the $d$-face of $\Sigma_n^d$ with vertices $i, i+1, \ldots, i+d$. Clearly, $H_1, H_2, \ldots, H_{n-d}$ are exactly those faces of $\Sigma_n^d$ that have gap zero. With abuse of notation, we extend the definition of $H_i$ also to $i \in \{n-d+1, \ldots, n\}$ using ``congruence modulo $n$''. Namely, by ``$n+1$'' we mean vertex $1$, by ``$n+2$'' we mean vertex $2$, and so on. So $H_n$ will be the $d$-face adjacent to $H_1$ and of vertices $\{n, 1, 2, 3, \cdots, d\}$, which we write down in increasing order, so $H_n =123\cdots d \, n$. Note that $\operatorname{gap}(H_i)>0$ when $i>n-d$.
\begin{definition}[traceable, Hamiltonian] \label{def:1}
A complex $\Delta$ is \emph{(tight-) traceable} if it has a labeling such that $H_1, \ldots, H_{n-d}$ are in $\Delta$.
It is \emph{(tight-) Hamiltonian} if it has a labeling such that all of $H_1, \ldots, H_n$ are in $\Delta$.
\end{definition}
Clearly, Hamiltonian implies traceable. For $d=1$, Definition~\ref{def:1} boils down to the classical notions of traceable and Hamiltonian graphs, that is, graphs that admits a Hamiltonian path and a Hamiltonian cycle, respectively. In fact, nobody prevents us from relabeling the vertices in the order in which we encounter them along such path (or cycle).
Recall that two facets of a pure simplicial $d$-complex are \emph{adjacent} if their intersection has cardinality $d$, or equivalently, dimension $d-1$. For example, each $H_i$ is adjacent to $H_{i+1}$. The \emph{dual graph} of a pure simplicial $d$-complex $\Delta$ has nodes corresponding to the facets of $\Delta$; two nodes are connected by an arc if and only if the corresponding facets of $\Delta$ are adjacent.
A pure simplicial $d$-complex $\Delta$ is \emph{strongly-connected} if its dual graph is connected. For $d \ge 1$, every strongly-connected $d$-complex is connected, and when $d=1$ the two notions coincide. According to our convention, all strongly-connected simplicial complexes are pure.
\begin{remark} \label{rem:HnotSC} The statement ``the dual graph of any Hamiltonian $d$-complex is Hamiltonian" holds true only for $d=1$: For example, the Hamiltonian simplicial complex
\[\Delta_1= 123, 234, 345, 456, 567, 678, 789, 189, 129, 147 \]
is not even strongly connected, because the facet $147$ is isolated in the dual graph. The deletion of vertex $1$ from $\Delta_1$ yields a simplicial complex that is not even pure.
\end{remark}
\newpage
\section{Weakly-traceable/Hamiltonian complexes and ridge degrees}
In this section, we introduce two weaker notions of traceability and Hamiltonicity that first appeared in \cite{Ketal}, and we study their nontrivial relationship with the ``ridge degree'', i.e. how many $d$-faces contain any given $(d-1)$-face. This relationship has a long history, beginning in 1952 with one of the most classical results in graph theory, due to Gabriel Dirac \cite{Dirac}, the son of Nobel Prize physicist Paul Dirac:
\begin{theorem}[Dirac \cite{Dirac}] Let $G$ be a graph with $n$ vertices.
If $\deg v \ge \frac{n}{2}$ for every vertex~$v$, then $G$ is Hamiltonian.
\end{theorem}
Later {\O}ystein Ore \cite{Ore} improved Dirac's result and extended it to traceable graphs:
\begin{theorem}[Ore \cite{Ore}] \label{thm:OreGraphs} Let $G$ be a graph with $n$ vertices.
\begin{compactenum}[ \rm (A) ]
\item If $\deg u + \deg v \ge n$ for all non-adjacent vertices $u,v$, the graph $G$ is Hamiltonian.
\item If $\deg u + \deg v \ge n-1$ for all non-adjacent vertices $u,v$, the graph $G$ is traceable.
\end{compactenum}
\end{theorem}
Two years later P\'osa extended Ore's condition (A) much further:
\begin{theorem}[P\'osa \cite{Posa}] \label{thm:PosaGraphs} Let $G$ be a graph with $n$ vertices. Order the vertices $v_1, \ldots, v_n$ so that the respective degrees are weakly increasing, $d_1 \le d_2 \le \ldots \le d_n$.
\begin{compactenum}[ \rm (C) ]
\item If for every $k < \frac{n}{2}$ one has $d_k > k$, the graph $G$ is Hamiltonian.
\end{compactenum}
\end{theorem}
These theorems have been generalized in five main directions, over the course of more than a hundred papers (see also Li \cite{Li} for a survey with a different perspective than ours):
\begin{compactenum}
\item
Bondy and Chv\'atal \cite{BondyA, Bondy, Chvatal, BondyChvatal} weakened the antecedent in the implication (C) of P\'osa's theorem (see \cite{Farrugia} for an application to self-complementary graphs);
\item Bondy \cite{BondyB} strengthened the conclusion of Ore's theorem, from Hamiltonian to \emph{pancyclic} (=containing cycles of length $\ell$ for any $3 \le \ell \le n$); later Schmeichel--Hakimi \cite{SchmeichelHakimi} showed that P\'osa and Chv\'atal's theorems can be strengthened in the same direction;
\item Fan \cite{Fan} showed that for $2$-connected graphs, it suffices to check Ore's condition for vertices $u$ and $v$ at distance $2$; and even more generally, it suffices to check that for any two vertices at distance two, at least one of them has degree $\ge \frac{n}{2}$. With these weaker assumptions he was still able to achieve a pancyclicity conclusion. See \cite{BedrossianChenSchelp}, \cite{LiLiFeng}, \cite{ChaoSongZhang} for recent extensions of Fan's work.
\item A forth line of generalizations of Ore's theorem involved requiring certain vertex sets to have large neighborhood unions, rather than large degrees: Compare Broersma--van den Heuvel--Veldman \cite{BHV} and Chen--Schelp \cite{ChenSchelp}.
\end{compactenum}
Here we are interested in the \emph{fifth} main direction, namely, the generalization to higher dimensions. This is historically a rather difficult task: As of today, no straightforward extension of Ore's theorem or of P\'osa's theorem is known. However, some elegant positive results were obtained in 1999 by Katona and Kierstead \cite{KatKie}, who applied extremal graph theory to generalize Dirac's theorem to simplicial complexes with a huge number of vertices. Building on the work by Katona and Kierstead \cite{KatKie}, R\"odl, Szemer\'edi, and Ruci\`nski \cite{RSR} were able in 2008 to prove the following `extremal' version of Dirac's theorem:
\begin{theorem}[{R\"odl--Szemer\'edi--Ruci\'nski \cite{RSR}}] \label{thm:RSR}
For all integers $d \ge 2$ and for every $\varepsilon > 0$ there exists a (very large) integer $n_0$ such that every
$d$-dimensional simplicial complex $\Delta$ with more than $n_0$ vertices, and such that every $(d-1)$-face of $\Delta$ is in at least
$n ( \frac{1}{2} + \varepsilon )$
facets, is Hamiltonian.
\end{theorem}
Now we are ready to introduce the main definition of the present section. Recall that two facets of a pure simplicial $d$-complex are \emph{incident} if their intersection is nonempty.
\begin{definition}[weakly-traceable, weakly-Hamiltonian] \label{def:WH}
A $d$-dimensional simplicial complex $\Delta$ is \emph{weakly-traceable} if if it has a labeling such that $\Delta$ contains faces $H_{i_1}, \ldots, H_{i_k}$ from $\{H_1, \ldots, H_{n-d}\}$ that altogether cover all vertices, and such that $H_{i_j}$ is incident to $H_{i_{j+1}}$ for each $j \in \{1, \ldots, k-1\}$. In this case, we call the list $H_{i_1}, \ldots, H_{i_k}$ a \emph{weakly-Hamiltonian path}.\\
A $d$-dimensional simplicial complex $\Delta$ is \emph{weakly-Hamiltonian} if it has a labeling such that $\Delta$ contains faces $H_{i_1}, \ldots, H_{i_k}$ from $H_1, \ldots, H_n$ that altogether cover all vertices, such that $H_{i_j}$ is incident to $H_{i_{j+1}}$ for each $j \in \{1, \ldots, k-1\}$, and in addition $H_{i_k}$ is incident to $H_{i_1}$. In this case, we call the list $H_{i_1}, \ldots, H_{i_k}$ a \emph{weakly-Hamiltonian cycle}.
\end{definition}
\begin{remark} These notions are not new. For what we called ``weakly-Hamiltonian'', Keevash et al. \cite{Ketal} use the term ``generic Hamiltonian''. Their paper \cite{Ketal} focuses however on the stronger notion of ``loose-Hamiltonian'' complexes, which are weakly-Hamiltonian complexes where all of the intersections $H_{i_j} \cap H_{i_{j+1}}$ consist of a single point (with possibly one exception). By definition, all Hamiltonian complexes are loose-Hamiltonian, and all loose-Hamiltonian complexes are weakly-Hamiltonian. For $d=1$ all these different notions converge: ``Weakly-Hamiltonian $1$-complexes'' are simply ``graphs with a Hamiltonian cycle'', and ``weakly-traceable $1$-complexes'' are ``graphs with a Hamiltonian path''. In 2010 Han--Schacht \cite{HS} and independently Keevash et al. \cite{Ketal} proved the following extension of Theorem \ref{thm:RSR} above:
\begin{theorem}[{Han-Schacht \cite{HS}, Keevash et al. \cite{Ketal}}]
For all integers $d \ge 2$ and for every $\varepsilon > 0$ there exists a (very large) integer $n_0$ such that every
$d$-dimensional simplicial complex $\Delta$ with more than $n_0$ vertices, and such that every $(d-1)$-face of $\Delta$ is in at least
$n (\frac{1}{2d} + \varepsilon)$
facets, is loose-Hamiltonian, and in particular weakly-Hamiltonian.
\end{theorem}
\end{remark}
\begin{remark} In Definition \ref{def:WH}, note that if $\Delta$ is weakly-traceable, necessarily $i_1 = 1$ and $i_k = n-d$, because otherwise $1$ and $n$ would not be covered. So equivalently, in Def.~\ref{def:WH} we could demand \[ \{i_2, \ldots, i_{k-1}\} \subset \{2, \ldots, n-d-1\}.\] Note also that if a labeling $v_1, \ldots, v_n$ makes $\Delta$ (weakly-) traceable, so does the ``reverse labeling'' $v_{n}, \ldots, v_{1}$.
As for Hamiltonian complexes: If a labeling $v_1, \ldots, v_n$ makes $\Delta$ weakly-Hamiltonian, so does its reverse, and also $v_{i_1}, \ldots, v_{i_n}$, where $(i_1, \ldots, i_n)$ is any cyclic permutation of $(1, \ldots, n)$. So we may assume that $i_1=1$. Or we may assume that $i_k=n-d$. But as the next remark shows, we cannot assume both.
\end{remark}
\begin{remark} \label{rem:surprise}
When $d>1$, not all weakly-Hamiltonian $d$-complexes are weakly-traceable. For $d=2$, a simple counterexample is given by
\[ \Delta_0 = 123, 156, 345.\]
The weakly-Hamiltonian cycle is of course $H_1, H_3, H_5$. Any labeling that makes $\Delta_0$ weakly-Hamiltonian is either the reverse or a cyclic shift (or both) of the labeling above. For parity reasons, in any labeling that makes $\Delta_0$ weakly-Hamiltonian, only one of $H_1$ and $H_4$ is in $\Delta_0$. \end{remark}
\begin{remark} \label{rem:An}
Weakly-traceable complexes are obviously connected. Weakly-Hamiltonian complexes are even $2$-connected, in the sense that the deletion of any vertex leaves them connected. The converses are well-known to be false already for $d=1$. In fact, let $n \ge 4$. Let $A_{n-2}$ be the edge-less graph on $n- 2$ vertices. Let $x, y$ be two new vertices. The ``suspension''
\[ \operatorname{susp}(A_{n-2}) \ \eqdef \ A_{n-2} \: \cup \: \{ x \ast v \, : \: v \in A_{n-2}\} \: \cup \: \{ y \ast v \, : \: v \in A_{n-2}\} \]
is a $2$-connected graph on $n$ vertices that is not Hamiltonian for $n \ge 5$, and not even traceable for $n \ge 6$. In higher dimensions, the $\Delta^d_3$ of Lemma \ref{lem:WheelComplex} is $d$-connected, but neither weakly-traceable nor weakly-Hamiltonian.
\end{remark}
We start with a few Lemmas that are easy, and possibly already known; we include nonetheless a proof for the sake of completeness. For the following lemma, a \emph{subword} of a word is a subsequence formed by \emph{consecutive} letters of a word: So for us ``word'' is a subword of ``subword'', whereas ``sword'' is not.
\begin{lemma} Let $d \ge 2$. If a $d$-complex $\Delta$ is weakly-Hamiltonian (resp. weakly traceable), then for any $k \in \{1, \ldots, d\}$ the $k$-skeleton of $\Delta$ is weakly-Hamiltonian (resp. weakly-traceable).
\end{lemma}
\begin{proof}
Given a weakly-Hamiltonian path/cycle, replace any $d$-face $H_1$ with its $(k+1)$-letter subwords, ordered lexicographically. The result, up to canceling possible redundancies, will be a weakly-Hamiltonian path/cycle for the $k$-skeleton.
\end{proof}
For example: if $d=3$ and $k=1$, suppose that a $3$-complex on $8$ vertices admits the Hamiltonian path \[1234, \quad 2345, \quad 5678.\]
Then the $1$-skeleton admits the Hamiltonian path
\[ 12, \, 23, \, 34, \quad \cancel{ 23, } \, \, \cancel{ 34, } \, 45, \quad \, 56, \, 67, \, 78.\]
The next Lemma is an analog to the fact that Hamiltonian complexes are traceable.
\begin{lemma} \label{lem:deletion} Let $\Delta$ be a $d$-dimensional complex that has a weakly-Hamiltonian cycle $H_{i_1}, \ldots, H_{i_k}$, with $k \ge 3$. For any $j$ in $\{1, \ldots, k\}$, let $m_j$ be the number of vertices of $H_{i_j}$ that are neither contained in $H_{i_{j-1}}$ nor in $H_{i_{j+1}}$ (where by convention $i_{k+1} \eqdef i_1$).
\begin{compactitem}
\item If $m_j >0$, the deletion of those $m_j$ vertices from $\Delta$ yields a weakly-traceable complex.
\item If $m_j=0$, and in addition $H_{i_{j-1}}$ and $H_{i_{j+1}}$ are disjoint, then $\Delta$ itself is weakly-traceable.
\end{compactitem}
\end{lemma}
\begin{proof} Fix $j$ in $\{1, \ldots, k\}$. If $m_j >0$, the $m_j$ vertices that belong to $H_{i_j}$ and to no other facet of the cycle are labeled consecutively. So up to relabeling the vertices cyclically, we can assume that they are the vertices $n-m_j + 1, \, n- m_j + 2, \ldots, n-1, n.$ Thus the facet in the cycle they all belong to is the last one, $H_{i_k}$. Now let $D$ be the complex obtained from $\Delta$ by deleting these $m_j$ vertices. It is easy to see that
\[ H_{1}=H_{i_1}, H_{i_2}, \ldots, H_{i_{k-1}}\]
is a weakly-Hamiltonian path for $D$. \\The case $m_j=0$ is similar: Up to relabeling the vertices cyclically, $i_{j+1}=1$ and thus $j=k$. By assumption $H_{i_{k-1}}$ and $H_{1}$ are disjoint. But since $m_k =0$, and vertex $n$ does not belong to $H_1$, it must belong to $H_{i_{k-1}}$. Therefore $H_{i_{k-1}}=H_{n-d}$. So
\[ H_{1}=H_{i_1}, H_{i_2}, \ldots, H_{i_{k-1}}\]
is a weakly-Hamiltonian path for $\Delta$ itself.
\end{proof}
The next Lemma can be viewed as a $d$-dimensional extension of the fact that the cone over the vertex set of a graph $G$ is a Hamiltonian graph if and only if the starting graph $G$ is traceable.
\begin{lemma} \label{lem:buciodiculo2} Let $\Delta$ be any $d$-complex on $n$ vertices. Let $\Sigma^{d-1}$ be the $(d-1)$-simplex.
Let $\Gamma$ be the $d$-complex obtained by adding to $\Delta$ a $d$-face $v \ast \Sigma^{d-1}$ for every vertex $v$ in $\Delta$. Then
\[ \Delta \textrm{ is weakly-traceable } \Longleftrightarrow \Gamma \textrm{ is weakly-Hamiltonian. }\]
\end{lemma}
\begin{proof}
``$\Rightarrow$'': If $H_{i_1}, \ldots, H_{i_k}$ is a list of facets proving that $\Delta$ is weakly-traceable, then the list $H_{i_1}, \ldots, H_{i_k}, H_n, H_{n+1}$ shows that $\Gamma$ is weakly-Hamiltonian.\\
``$\Leftarrow$'': Pick a labeling that makes $\Gamma$ weakly-Hamiltonian. By how the complex $\Gamma$ is constructed, the vertices of $\Sigma^{d-1}$ must be labeled consecutively; so without loss, we may assume that they are $n+1, \ldots, n+d$. Take a weakly-Hamiltonian cycle for $\Gamma$ and delete from the list all the $d$-faces containing any vertex whose label exceeds $n$.
\end{proof}
\begin{remark} \label{rem:trouble} The following statements are valid \textbf{only for $\mathbf{d=1}$}.
\begin{compactenum}[(i)]
\item ``$\Delta$ is weakly-traceable $ \Longleftrightarrow$ $\Delta \: \cup \: \: \: w \ast \operatorname{(d-1)-skel}(\Delta)$ is weakly-Hamiltonian.''
\item ``Deleting a single vertex from a weakly-Hamiltonian $d$-complex yields a weakly-traceable complex.''
\item ``Deleting (the interior of) any of the $H_i$'s from a weakly-Hamiltonian $d$-complex yields a weakly-traceable complex.''
\end{compactenum}
Simple counterexamples in higher dimensions are:
\begin{compactenum}[(i)]
\item $\Delta_1=126, 234, 456, 489, 678$ is not weakly-traceable, yet
$\Delta_2 \eqdef \Delta_1 \, \cup \, \left( 10 \ast \, \operatorname{2-skel}(\Delta_1) \right)$ admits the weakly\--Hamiltonian cycle $234, \; 456, \; 678, \; 89 \, 10, \; 12\, 10$.
This is a counterexample to ``$\Leftarrow$''. $\, $ In contrast, the direction ``$\Rightarrow$'' holds in all dimensions.
\item If from the $\Delta_2$ above we delete vertex $10$, we get back to $\Delta_1$, not weakly-traceable.
\item $\Delta_3=1234, \; 2345, \; 5678, \; 167 \,10, \; 189\, 10$ is weakly-Hamiltonian, but the deletion of (the interior) of $5678$ yields a complex that is not weakly-traceable.
\end{compactenum}
\end{remark}
Our first non-trivial result is an ``Ore-type result'': We shall see later that in some sense it extends `most' of the proof of Ore's theorem \ref{thm:OreGraphs}, part (A), to all dimensions.
\begin{definition} Let $\Delta$ be a pure $d$-dimensional simplicial complex, and let $\sigma$ be any $(d-1)$-face of $\Delta$. The \emph{degree} $d_{\sigma}$ of $\sigma$ is the number of $d$-faces of $\Delta$ containing $\sigma$.
\end{definition}
\begin{proposition}\label{prop:Ore1}
Let $\Delta$ be a traceable $d$-dimensional simplicial complex on $n$ vertices, $n > 2d$.
If in some labeling that makes $\Delta$ traceable the two $(d-1)$-faces $\sigma$ and $\tau$ formed by the first $d$ and the last $d$ vertices, respectively, satisfy
$d_{\sigma} + d_{\tau} \ge n$, then $\Delta$ is weakly-Hamiltonian.
\end{proposition}
\begin{proof}Since $n > 2d$, the two faces $\sigma$ and $\tau$ are disjoint. Let $J \eqdef \{d+2, d+3, \ldots, n-d\}$. For every $i$ in $J$, which has cardinality $n-2d-1$, consider the two $d$-faces of $\Sigma^d_n$
\[ S_i \eqdef \sigma \ast i \quad \textrm{ and } \quad T_i \eqdef (i-1) \ast \tau.\]
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.64]{OreA1.png}
\includegraphics[scale=0.64]{OreA2.png}
\label{fig:OreA}
\caption{\textsc{left}: The dashed triangles are $S_5$ and $T_5$. Were they both in $\Delta$, then one could relabel the vertices and create a weakly-Hamiltonian cycle (\textsc{right}).}
\end{center}
\end{figure}
Now there are two cases, both of which will result in a weakly-Hamiltonian cycle:
\smallskip
\textsc{Case 1: } \emph{For some $i$, both $S_i$, $T_i$ are in $\Delta$.}
We are going to introduce a new vertex labeling $\ell_1, \ldots, \ell_n$. The ``consecutive facets of the new labeling'' will be called $L_1=\ell_1 \cdots \ell_d \ell_{d+1}$, $L_2=\ell_2 \cdots \ell_{d+2}$, and so on. The following describes a weakly-Hamiltonian cycle:
\begin{compactitem}
\item Start with the first $i-1$ vertices in the same order: That is, set $\ell_1 \eqdef 1$, $\ldots$, $\ell_{i-1} \eqdef i-1$. Hence $L_1=H_1, L_2=H_2, \ldots, $, up until $L_{i-d-1}=H_{i-d-1}$, which (since $\Delta$ is traceable) is the first of the $H_i$'s that contains the vertex $i-1$.
\item Then set $L_{i} \eqdef T_i$. The vertices of $\tau$ are to be relabeled by $\ell_{i}, \ell_{i+1}, \ldots, \ell_{i+d}$: Specifically, label by $\ell_i$ the vertex that is in $H_{n-d}$ but not in $H_{n-d-1}$, by $\ell_{i+2}$ the vertex in in $H_{n-d-1}$ but not in in $H_{n-d-2}$, and so on. Facet-wise, we are traveling in reverse order across the \emph{last} facets of the original labeling. Stop until you get to relabel vertex $i$ by $\ell_n$. (Or equivalently, if you prefer to think about facets, stop once you reach facet $H_i$.)
\item The weakly-Hamiltonian cycle gets then concluded with $S_i$, which is adjacent to $L_1=H_1$ via $\sigma$. The facets previously called $H_{i-d}$, $H_{i-d+1}$, $\ldots$, $H_{i-1}$ are not part of the new weakly-Hamiltonian cycle.
\end{compactitem}
\smallskip
\textsc{Case 2: } \emph{For all $i$, at most one of $S_i$, $T_i$ is in $\Delta$.} Since the two sets
$\{ i \in J : \sigma * i \in \Delta\}$ and $\{ i \in J : (i-1) * \tau \in \Delta\}$ are disjoint, the sum of their cardinalities is the cardinality of their union, which is contained in $J$. So
\begin{equation} \label{eq:Ore1} |\{ i \in J : \sigma * i \in S\}| + |\{ i \in J : (i-1) * \tau \in \Delta\}|
\le \ | J | \; = \; n-2d+1.\end{equation}
Now, we claim that either $\sigma \ast n$ or $1\ast \tau$ is a face of $\Delta$. From the claim the conclusion follows immediately, as such face creates a weakly-Hamiltonian cycle. We prove the claim by contradiction. Suppose $\Delta$ contains neither $\sigma \ast n$ nor $1\ast \tau$. Every $d$-face containing $\sigma$ is of the form $\sigma \ast v$, where $v$ is either in $J$ or in the set $\{d, n-d+1, n-d+2, \ldots, n-1\}$ (which has size $d$). So
\begin{equation} \label{eq:Ore2} d_{\sigma} \le |\{ i \in J : \sigma * i \in \Delta\}| \; + d.\end{equation}
Symmetrically, the $d$-faces containing $\tau$ are of the form $w \ast \tau$, with $w$ either in $J$ or in the size-$d$ set $\{2, 3, \ldots, d, n-d+1\}$.
So
\begin{equation} \label{eq:Ore3} d_{\tau} \le |\{ i \in J : (i-1) * \tau \in \Delta\}| \; + d.\end{equation}
Putting together inequalities \ref{eq:Ore1}, \ref{eq:Ore2} and \ref{eq:Ore3}, we reach a contradiction:
\[d_{\sigma} + d_{\tau} \le (n - 2d - 1) + d + d = n-1. \qedhere \]
\end{proof}
\begin{corollary}\label{cor:Ore1}
Let $\Delta$ be a traceable $d$-dimensional simplicial complex on $n$ vertices, $n > 2d$.
If for any two disjoint
$(d-1)$-faces $\sigma$ and $\tau$ one has
$d_{\sigma} + d_{\tau} \ge n$, then $\Delta$ is weakly-Hamiltonian.
\end{corollary}
\begin{corollary} \label{cor:Dirac}
Let $\Delta$ be a traceable $d$-dimensional simplicial complex on $n$ vertices, $n > 2d$.
If every $(d-1)$-face of $\Delta$ belongs to at least $\frac{n}{2}$
facets of $\Delta$, then $\Delta$ is weakly-Hamiltonian.
\end{corollary}
\begin{example} Let $n > 2d$. Let $\Delta_4$ be the simplicial complex on $n$ vertices obtained from $\Sigma^d_n$ by removing the interior of the
$d$-faces $H_{n-d+1}$, $H_{n-d+2}$, $\ldots$, $H_{n}$. By construction $\Delta_4$ is traceable, but the given labeling (as well as any labeling obtained from it by reversing or cyclic shifting) fails to prove that $\Delta_4$ is weakly-Hamiltonian. Now, in $\Delta_4$, the
$(d-1)$-faces $ \mu_i \eqdef H_i \cap H_{i+1}$, with $i \in \{ n-d+1, n-d+2, \ldots, n-1\}$, have degree $n-d-2$. All other $(d-1)$-faces $\nu_j$ contained in one of $H_{n-d+1}, H_{n-d+2}, \ldots, H_n$ have degree $n-d-1$. Finally, all $(d-1)$-faces \emph{not} contained in any of $H_{n-d+1}, H_{n-d+2}, \ldots, H_n$ have degree $n-d$. Therefore:
\begin{compactitem}
\item If $n \ge 2d +4$, Corollary \ref{cor:Dirac} tells us that $\Delta_4$ is weakly-Hamiltonian, because $n - d - 2 \ge \frac{n}{2} $.
\item If $n=2d+3$ or $n=2d+2$,
any two of the $\mu_i$'s are incident, and any $\nu_j$ is incident to all of the $\mu_i$'s. Hence, for any two \emph{disjoint} $(d-1)$-faces $\sigma$ and $\tau$, we do have
$d_{\sigma} + d_{\tau} \ge 2n - 2d - 2 \ge n$. So we can still conclude that $\Delta_4$ is weakly-Hamiltonian via Corollary \ref{cor:Ore1}.
\item If $n=2d+1$, then the assumptions of Corollaries \ref{cor:Dirac} and \ref{cor:Ore1} are not met, but Proposition~\ref{prop:Ore1} is still applicable. In fact, for the facets $\sigma$ resp. $\tau$ formed by the first resp. the last vertices of the given labeling, one has
$d_{\sigma} + d_{\tau} = (n - d ) + (n-d-1) = 2n - (2d + 1) = n$.
\end{compactitem}
So in all cases, $\Delta_4$ is weakly-Hamiltonian. The proof of Proposition \ref{prop:Ore1} also suggests a relabeling that works:
$\ell_1\eqdef 1,\; \ell_2 \eqdef 2, \, \ldots \,, \; \ell_{d+1}\eqdef d+1, \; \ell_{d+2}=n, \; \ell_{d+3}\eqdef n-1, \, \ldots \,, \; \ell_{n}\eqdef d+2$.
\end{example}
To see in what sense Proposition \ref{prop:Ore1} is a higher-dimensional version of Ore's theorem \ref{thm:OreGraphs}, part (A), the best is to give a proof of the latter using the former:
\begin{proof}[\bf Proof of Ore's theorem \ref{thm:OreGraphs}, part (A)] By contradiction, let $G$ be a non-Hamiltonian graph satisfying $\deg u + \deg v \ge n$ for all non-adjacent vertices $u,v$. Add edges to it until you reach a \emph{maximal} non-Hamiltonian graph $G^*$. Since any further edge between the existing vertices would create a Hamiltonian cycle, $G^*$ is traceable, and obviously it still satisfies $\deg u + \deg v \ge n$.
By Proposition \ref{prop:Ore1} $G^*$ is (weakly-)Hamiltonian, a contradiction.
\end{proof}
It is possible that the bound of Proposition \ref{prop:Ore1} can be improved. But in any case, the possible improvement could only be small, as the following construction shows.
\begin{nonexample}
Let $d < m$ be positive integers. Take the disjoint union of two copies $A', A''$ of $\Sigma^d_{m}$. Let $\mu$ be any facet of $\Sigma^d_{m}$ and let $\mu', \mu''$ be its copies in $A'$ and $A''$, respectively. Glue to $A' \cup A''$ a triangulation without interior vertices of the prism $\mu \times [0,1]$, so that the lower face $\mu \times \{0\}$ is identified with $\mu'$, and the upper face $\mu \times \{1\}$ is identified with $\mu''$. Let $\Delta_5$ be the resulting $d$-complex on $n = 2m$ vertices. This $\Delta_5$ is traceable: the added prism, triangulated as a path of $d$-faces, serves as ``bridge'' to move between the two copies of $\Sigma^d_m$. However, this bridge can only be traveled once, so $\Delta_5$ is not weakly-Hamiltonian. For the labeling that makes it traceable, $d_{\sigma}+d_{\tau}=(m-d)+(m -d) = n - 2d$.
\end{nonexample}
Our next result is a ``P\'osa--type'' result, in the sense that it extends most of Nash--Williams' proof \cite{NashWilliams} of P\'osa's theorem \cite{Posa} to all dimensions. We focus on complexes $\Delta$ with the property that any labeling that makes them weakly-traceable, makes them also traceable. Such class is nonempty: for example, it contains all $1$-dimensional complexes and all trees of $d$-simplices (i.e. all triangulations of the $d$-ball whose dual graph is a tree).
\begin{proposition} \label{prop:PosaD} Let $\Delta$ be any traceable pure $d$-complex on $n$ vertices, $n > 2d$. Suppose that any labeling that makes $\Delta$ weakly-traceable makes it also traceable.
Let $\sigma_1, \sigma_2, \ldots, \sigma_{s}$ be an ordering of the $(d-1)$-faces of $\Delta^*$,
such that the respective degrees $d_i \eqdef d_{\sigma_i}$ are weakly-increasing, $d_1 \le d_2 \le \ldots \le d_{s}$.
If for every $d \le k < \frac{n}{2}$ one has $ d_{k-d+1} >k$,
then $\Delta$ is weakly-Hamiltonian.
\end{proposition}
\begin{proof} Among all possible labelings that make $\Delta$ weakly-traceable (and thus traceable, by assumption), choose one that maximizes $d_{\sigma} + d_{\tau}$, where $\sigma$ is the $(d-1)$-face of $H_1$ spanned by the first $d$ vertices (that is, $1,2,\cdots d$) and $\tau$ is the $(d-1)$-face of $H_{n-d}$ spanned by the last $d$ vertices (that is, $n-d+1, \ldots, n$). Since $n>2d$, the faces $\sigma$ and $\tau$ are disjoint.
If $d_{\sigma} + d_{\tau} \ge n$, using the proof of Proposition \ref{prop:Ore1} we get that $\Delta$ is weakly-Hamiltonian, and we are done. If not, then one of $\sigma$, $\tau$ has degree $< \frac{n}{2}$. Up to reversing the labeling, which would swap $\sigma$ and $\tau$, we can assume that $d_{\sigma} < \frac{n}{2}$.
Now let $J \eqdef \{d+2, d+3, \ldots, n-d\}.$
For every $i$ in $J$, which has cardinality $n-2d-1$, consider the two $d$-faces of $\Sigma^d_n$
\[ S_i \eqdef \sigma \ast i \quad \textrm{ and } \quad T_i \eqdef (i-1) \ast \tau.\]
We may assume that at most one of these two faces is in $\Delta$, otherwise a weakly-Hamiltonian cycle arises, exactly as in the proof of Proposition \ref{prop:Ore1}. Now for each $i$ in $J_1 \eqdef \{ i \in J \: : \: \sigma \ast i \in \Delta\}$, consider the $(d-1)$-face $\rho_i$ with vertices $\{ i-d, i-d+1, \ldots, i-1\}$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.63]{Posa1.png}
\caption{A higher-dimensional ``P\'osa flip'': Since $125$ is in $\Delta$, the vertex $5$ is in $J_1$. Now the red triangle $\rho_5 \ast w = 348$ cannot be in $\Delta$, or else we would have a weakly-Hamiltonian cycle with the blue labeling. The blue labeling makes $\Delta$ weakly-traceable, with $\rho_5$ playing the role of the ``first'' $(d-1)$-face; the ``last'' $(d-1)$-face is the same as before. By how our original labeling was chosen, $d_{\rho_5} \le d_{\sigma} < \frac{n}{2}$.}
\label{fig:Posa}
\end{center}
\end{figure}
If for some $i$ in $J_1$ the $d$-face $\rho_i \ast w$ is in $\Delta$, then there is a new relabeling $\ell_1, \ldots, \ell_n$ of the vertices for which we have a weakly-Hamiltonian cycle: see Figure \ref{fig:Posa} above. (The proof is essentially identical to that of Proposition \ref{prop:Ore1}, up to replacing $T_i$ with $T'_i \eqdef \rho_i \ast w$, reversing the order, and permuting it cyclically, so that $\rho_i$ is the first face.) So also in this case, we are done.
It remains to discuss the case in which for all $i \in J_1 \eqdef \{ i \in J \: : \: \sigma \ast i \in \Delta\}$, the $d$-face $\rho_i \ast w$ is not in $\Delta$. In this case the relabeling $\ell_1, \ldots, \ell_n$ introduced above makes $\Delta$ weakly-traceable, and thus traceable by assumption. For such relabeling, the $(d-1)$-faces spanned by the first and the last $d$ vertices are $\rho_i$ and $\tau$, respectively. So by the way our original labeling was chosen, $d_{\rho_i} + d_{\tau} \le d_{\sigma} + d_{\tau}$, and in particular
\[d_{\rho_i} \le d_{\sigma} < \frac{n}{2}.\]
Now, any $d$-face containing $\sigma$ is of the form $\sigma \ast v$, where $v$ is either in the set $J_1$ or in the set $Z \eqdef \{ d+1, n-d+1, \ldots, n-1\}$, which has cardinality $d$. So $d_{\sigma} \le |J_1 \cup Z |$. Since $J$ and $Z$ are disjoint, and $J_1 \subset J$, the sets $J_1$ and $Z$ are also disjoint and we have
\[d_{\sigma}-d = d_{\sigma}- |Z| \le |J_1 \cup Z | - |Z| = |J_1| + |Z | - |Z| = | J_1 |.\]
So the set $\{\rho_i \: : \: i \in J_1\}$ contains at least $d_{\sigma} - d\,$ faces of dimension $d-1$ and degree $\le d_{\sigma}$. If we count also $\sigma$, we have in $\Delta$ at least $d_{\sigma} - d +1\,$ faces of dimension $d-1$ and degree $\le d_{\sigma}$. But then, setting $k \eqdef d_{\sigma}$, we obtain
\[ d_{k-d+1} \le k < \frac{n}{2},\]
which contradicts the assumption.
\end{proof}
Again, to see in what sense Proposition \ref{prop:PosaD} is a higher-dimensional version of P\'osa's Theorem~\ref{thm:PosaGraphs}, perhaps the best is to see how easily the latter follows from the former:
\begin{proof}[\bf Proof of P\'osa's theorem \ref{thm:PosaGraphs}] By contradiction, if $G$ is not Hamiltonian, we can add edges to it until we reach a maximal non-Hamiltonian graph $G^*$, which still satisfies the degree conditions and is traceable. By Proposition \ref{prop:PosaD}, $G^*$ is (weakly-)Hamiltonian, a contradiction.
\end{proof}
A natural question is whether one can generalize to higher dimensions also part (B) of Ore's theorem \ref{thm:OreGraphs}. The answer is positive, although some extra work is required. In fact, for graphs part (B) of Ore's theorem can be quickly derived from part (A) by means of a coning trick.
This trick however does not extend to higher dimensions, as we explained in Remark \ref{rem:trouble}, so we'll have to take a long detour, which makes the proof three times as long. The bored reader may skip directly to the next section.
\begin{definition} \label{def:quasi-traceable}
A $d$-dimensional complex $\Delta$ is \emph{quasi-traceable} if there exists a vertex labeling for which $\Delta \cup H_j$ is weakly-traceable, and moreover, with respect to the same labeling,
\begin{compactenum}[ (a) ]
\item if $j=1$, then $\Delta$ contains all of $H_{2}, \ldots, H_{n-d}$ (i.e., $\Delta \cup H_1$ is traceable);
\item if $j \in \{2, \ldots, n-2d\}$, then $\Delta$ already contains all of $H_1, \ldots, H_{j-1}$ and $H_{j+d}, \ldots, H_{n-d}$ (i.e., $\Delta \cup H_j \cup \ldots \cup H_{j+d-1}$ is traceable);
\item if $j \in \{n-2d+1, \ldots, n-d-1\}$, then $\Delta$ contains all of $H_1, \ldots, H_{j-1}$ and also $H_{n-d}$ (i.e., $\Delta \cup H_j \cup \ldots \cup H_{n-d-1}$ is traceable);
\item if $j=n-d$, then $\Delta$ already contains all of $H_1, \ldots, H_{n-d-1}$ (i.e., $\Delta \cup H_{n-d}$ is traceable).
\end{compactenum}
\end{definition}
\begin{example} The complex $\Delta_6=123, 234, 567, 678, 789$ is quasi-traceable, although not weakly-traceable. In fact, $\Delta_6$ becomes weakly-traceable if we add one of the facets $345$ and $456$, and it becomes even traceable if we add both.
\end{example}
Definition \ref{def:quasi-traceable} allows the ``added faces'' to be already present in $\Delta$. In particular, all traceable complexes are quasi-traceable.
Here comes our high-dimensional version of Theorem \ref{thm:OreGraphs}, part (B):
\begin{proposition}\label{prop:Ore2}
Let $\Delta$ be a quasi-traceable $d$-dimensional simplicial complex on $n$ vertices, $n > 2d$.
If in some labeling that makes $\Delta$ quasi-traceable the two $(d-1)$-faces $\sigma$ and $\tau$ formed by the first $d$ and the last $d$ vertices satisfy
$d_{\sigma} + d_{\tau} \ge n-1$, then $\Delta$ is weakly-traceable.
\end{proposition}
\begin{proof} By contradiction, suppose $\Delta$ is not weakly-traceable; we treat the four cases of Definition \ref{def:quasi-traceable} separately.
\textbf{Case (a)} is symmetric to Case (d), so we will leave it to the reader.
\smallskip
\textbf{Case (b)} is the main case. Since $j \in \{2, \ldots, n-2d\}$, by definition $\Delta$ contains all of $H_1, \ldots, H_{j-1}$ and also $H_{j+d}, \ldots, H_{n-d}$. Since $\Delta$ is not weakly-traceable, it does not contain $H_j$. Moreover, $\sigma \ast (d+j)$ cannot be a facet of $\Delta$, otherwise the two ``halfpaths'' above would be connected into a weakly-Hamiltonian path. For the same reason, since $(d+j -1) \in H_{j-1}$, the $d$-face $(d+j-1) \ast \tau$ cannot be in $\Delta$. So let
$J' \eqdef \{d+2, d+3, \ldots, n-d\} \setminus \{ d+j\}$. For every $i$ in $J'$, which has cardinality $n-2d-2$, consider the two $d$-faces of $\Sigma^d_n$
\[ S_i \eqdef \sigma \ast i \quad \textrm{ and } \quad T_i \eqdef (i-1) \ast \tau.\]
Now there are two subcases: Either there exists an $i$ such that $S_i, T_i$ are both in $\Delta$, or not.
\smallskip
\textsc{Case} (b.1): \emph{For some $i$, both $S_i$ and $T_i$ are in $\Delta$.}
There are two subsubcases, according to whether $i$ is ``before the gap'' or ``after the gap''.
\begin{itemize}
\item[ -- ] \textsc{Case} (b.1.1): $i<d+j$.
A weakly-Hamiltonian path arises from a relabeling as follows: We start at the beginning of the second halfpath, with the facets previously called $H_{j+d}, H_{j+d+1}, $ etc., until we reach $H_{n-d}$. Then we use $T_i$ to get back to the vertex previously labeled by $i-1$. Next, we use in reverse order the facets previously called $H_{i-d-1}, H_{i-d-2}, \ldots, H_2, H_1$. Finally use $S_i$ to jump forward to the vertex previously called $i$, and conclude the path with the facets previously called $H_i, H_{i+1}, \ldots, H_{j-1}$.
\item[ -- ] \textsc{Case} (b.1.2): $i > d+j$.
A weakly-Hamiltonian path arises from a relabeling as follows: We start at the beginning of the second halfpath, with the facets previously called $H_{j+d}, H_{j+d+1}, $ etc., until $H_{i-d-1}$. Then we use $T_i$ to jump forward. As next faces, we use in reverse order the facets previously called $H_{n-d}, H_{n-d-1}, \ldots, H_2, H_i$. Finally, we use $S_i$ to jump back to $H_1$, and conclude the path with the facets previously called $H_1, H_{2}, \ldots, H_{j-1}$. So also in this case $\Delta$ is weakly-traceable, a contradiction.
\end{itemize}
\textsc{Case} (b.2): \emph{For all $i$, at most one of $S_i$ and $T_i$ is in $\Delta^*$.} Since the two sets
$\{ i \in J' : \sigma * i \in \Delta\}$ and $\{ i \in J' : (i-1) * \tau \in \Delta\}$ are disjoint, we obtain a numerical contradiction:
\[
\begin{array}{ll}
d_{\sigma} + d_{\tau} &\le d + \; |\{ i \in J' : \sigma * i \in \Delta\}| \: + d + \: | \{ i \in J' : (i-1) * \tau \in \Delta\}| =\\
&= 2d + \; |\{ i \in J' : \sigma * i \in \Delta\} \cup \{ i \in J' : (i-1) * \tau \in \Delta\}| \le \\
&\le 2d + | J' | \; = \; 2d + n - 2d - 2 \; = \; n-2. \end{array}
\]
\textbf{Case (c)} is the easiest. If $j \in \{n-2d+1, \ldots, n-d-1\}$, then $H_{j-1}$ intersects $H_{n-d}$. Since $\Delta$ contains $H_1, \ldots, H_{j-1}$ and also $H_{n-d}$, it is weakly-traceable, a contradiction.
\smallskip
\textbf{Case (d)} is the last one. So, assume $j=1$ and set
$J'' \eqdef \{d+2, d+3, \ldots, n-d\}$. For every $i$ in $J''$, which has cardinality $n-2d-1$, consider the two $d$-faces of $\Sigma^d_n$
\[ S_i \eqdef \sigma \ast i \quad \textrm{ and } \quad T_i \eqdef (i-1) \ast \tau.\]
Now there are two subcases: Either there exists an $i$ such that $S_i, T_i$ are both in $\Delta$, or not.
\bigskip
\textsc{Case} (d.1): \emph{For some $i$, both $S_i$ and $T_i$ are in $\Delta$.} Then we obtain a weakly-Hamiltonian path as follows: Starting with $\ell_1=1$, first we use the face $\sigma \ast i$, then $H_2, \ldots, H_{i-d-1}$ in their order, then we use $(i-1)\ast n$ to jump forward, and then we come back with $H_{n-d}, \ldots, H_{i}$.
\smallskip
\textsc{Case} (d.2): \emph{For all $i$, at most one of $S_i$ and $T_i$ is in $\Delta$.} We know by that $\sigma \ast d=H_1$ is not in $\Delta$ because we are treating the case $j=1$, and we know that $\sigma \ast n$ is not in $\Delta$ otherwise we would have a weakly-Hamiltonian path. Thus any $d$-face containing $\sigma$ is of the form $\sigma \ast v$, where $v$ is either in $J''$ or in the disjoint set $\{n-d+1, \ldots, n-1\}$, which has cardinality $d-1$. In contrast, any $d$-face containing $\tau$ is of the form $(i-1) \ast \tau$, where $i$ is either in $J''$ or in the set $\{2, \ldots, d+1\}$, which has cardinality $d$. Since the two sets
$\{ i \in J'' : \sigma * i \in \Delta\}$ and $\{ i \in J'' : (i-1) * \tau \in \Delta\}$ are disjoint, the sum of their cardinality is equal to the cardinality of their union, which is a subset of $J''$. So also in this case we obtain a contradiction
\[
\begin{array}{ll}
d_{\sigma} + d_{\tau} &\le d - 1 + \; |\{ i \in J'' : \sigma * i \in \Delta\}| \: + d + \: | \{ i \in J'' : (i-1) * \tau \in \Delta\}| =\\
&= 2d - 1 + \; |\{ i \in J'' : \sigma * i \in \Delta\} \cup \{ i \in J'' : (i-1) * \tau \in \Delta\}| \le \\
&\le 2d - 1 + | J'' | \; = \; 2d -1 + n - 2d - 1 \; = \; n-2. \end{array} \qedhere
\]
\end{proof}
\begin{example}
Let $\Delta_7$ be the simplicial complex on $5$ vertices obtained from $\Sigma^2_5$ by removing the interior of the two triangles $123$ and $124$. Clearly $\Delta_7$ is quasi-traceable with $j=1$, because $\Delta_7 \cup H_1$ is traceable.
Since $d_{12} + d_{45} = 4 = n-1$,
by Proposition \ref{prop:Ore2} $\Delta_7$ is weakly-traceable. In fact, the reader may verify that $\Delta_7$ is even Hamiltonian with the relabeling $\ell_1=1$, $\ell_2=2$, $\ell_3=5$, $\ell_4=3$, $\ell_5=4$.
\end{example}
For completeness, we conclude this section by showing how Proposition \ref{prop:Ore2} implies part (B) of Ore's theorem \ref{thm:OreGraphs}:
\begin{proof}[\bf Proof of Ore's theorem \ref{thm:OreGraphs}, part (B)] By contradiction, let $G$ be a non-traceable graph satisfying $\deg u + \deg v \ge n-1$ for all non-adjacent vertices $u,v$. Add edges to it until we reach a maximal non-traceable graph $G^*$. This $G^*$ is quasi-traceable and still satisfies $\deg u + \deg v \ge n-1$. By Proposition \ref{prop:Ore2} $G^*$ is (weakly-)traceable, a contradiction.
\end{proof}
\newpage
\section{Interval graphs and semiclosed complexes}
In the present section,
\begin{compactenum}[ (1) ]
\item we introduce ``weakly-closed $d$-complexes'', generalizing co-comparability graphs;
\item we create a hierarchy of properties between closed and weakly-closed complexes, among which a $d$-dimensional generalization of interval graphs; and
\item we connect such hierarchy to traceability and chordality.
\end{compactenum}
\subsection{A foreword on interval graphs and related graph classes}
\textsc{Interval graphs} are the intersection graphs of intervals of $\mathbb{R}$. They have long been studied in combinatorics, since the pioneering papers by Lekkerkerker--Boland \cite{LekBol} and Gilmore--Hoffman \cite{GilmoreHoffman}, and have a tremendous amount of applications; see e.g.~\cite[Ch.~8, Sec.~4]{Golumbic} for a survey. \textsc{Unit-interval graphs}, also known as ``indifference graphs'' \cite{Roberts} or ``proper interval graphs'',
are the intersection graphs of \emph{unit} intervals, or equivalently, the intersection graphs of sets of intervals no two of which are nested.
The claw $K_{1,3}$ is the classical example of a graph that can be realized as intersection of four intervals, three of which contained in the forth; but it cannot be realized as intersection of unit intervals.
Bertossi noticed in 1983 that connected unit-interval graphs are traceable \cite{Bertossi}, whereas connected interval graphs in general are not: The claw strikes. All $2$-connected unit-interval graphs are Hamiltonian \cite{ChChCh}\cite{PandaDas}; again, this does not extend to $2$-connected interval graphs.
That said, for interval graphs (and even co-comparability graphs, see below for the definition) the Hamiltonian Path Problem and the Longest Path Problem can be solved in polynomial time \cite{DeogunSteiner} \cite{MerCor}, whereas for arbitrary graphs both problems are well known to be NP-complete, cf.~\cite{Karp}.
Given a finite set of intervals in the horizontal real line, we can swipe them ``left-to-right'', and thus order them by increasing left endpoint. This so-called ``canonical labeling'' of the vertices of an interval graph obviously satisfies
the following property: for all $a<b<c$,
\begin{equation} \label{eq:Interval} ac \in G \Longrightarrow ab \in G.\end{equation}
This ``under-closure'' is a characterization: It is easy to prove by induction that any graph with $n$ vertices labeled so that (\ref{eq:Interval}) holds can be realized as the intersection graph of $n$ intervals. This result was first discovered by Olario, cf.~\cite[Proposition 4]{LO}.
There is a ``geometrically dual argument'' to the one above: Given a finite set of intervals in $\mathbb{R}$, we could also swipe them right-to-left, thereby ordering the intervals by decreasing right endpoint. This yields a vertex labeling that again satisfies (\ref{eq:Interval}), for the same geometric reasons. In general, since some of the intervals may be nested, this ``dual labeling'' bears no relation with the canonical one. But if we start with a finite set of \emph{unit} intervals, then the dual labeling is simply the reverse of the canonical labeling. Thus in \emph{unit-interval} graphs, not only the canonical labeling is under-closed, but also its reverse is. Or equivalently, in unit-interval graphs, the canonical labeling is closed `both below and above': in mathematical terms, for all $a<b<c$,
\begin{equation} \label{eq:UnitInterval} ac \in G \Longrightarrow ab, bc \in G.\end{equation}
Again, it is not difficult to prove by induction that any graph with $n$ vertices, labeled so that (\ref{eq:UnitInterval}) holds, can be realized as the intersection graph of $n$ unit intervals \cite[Theorem 1]{LO}; see Gardi \cite{Gardi} for a computationally-efficient construction.
Recently Herzog et al.~\cite{HerzogEtAl, EneEtAl} rediscovered unit-interval graphs from an algebraic perspective, which will be discussed in the next chapter. They called them \emph{closed graphs} and expanded the notion to higher dimensions as well (\emph{``closed $d$-complexes''}).
Later Matsuda \cite{Matsuda} extended this algebraic approach to the broader class of ``co-comparability graphs'' (or ``weakly-closed graphs''), that we shall now describe in terms of their complement.
Any graph can be given an acyclic orientation by choosing a vertex labeling and then by directing all edges from the smaller to the larger endpoint. Every acyclic orientation can be induced this way. (This is not a bijection: different labelings may induce the same orientation). The \textsc{drawings of posets}, also called \emph{comparability graphs}, admit also \emph{transitive} orientations, namely, orientations such that if $\vec{ab}$ and $\vec{bc}$ are present, so is $\vec{ac}$. Let us rephrase this in terms of a vertex labeling, which happens to be the same as a choice of a linear extension of the poset: Comparability graphs are those graphs~$\overline{G}$ that admit a labeling such that, for all $a<b<c$,
\[ ab \in \overline{G} \textrm{ and } bc \in \overline{G} \Longrightarrow ac \in \overline{G}.\]
Not all graphs admit transitive orientations: The pentagon, for example, does not.
\textsc{Co-comparability graphs}, also called \textsc{weakly-closed graphs} in \cite{Matsuda}, are by definition the complements of comparability graphs. So they have a labeling that satisfies the contrapositive of the property above: Namely, for all $a<b<c$,
\begin{equation}\label{eq:Cocomparability} ac \in G \Longrightarrow ab \in G \textrm{ or } bc \in G.\end{equation}
By comparing (\ref{eq:Interval}) and (\ref{eq:Cocomparability}), it is clear that all interval graphs are co-comparability.
We should mention other two famous properties that all interval graphs enjoy. A graph is \textsc{perfect} if its chromatic number equals the size of the maximum clique. For example, even cycles are perfect, but odd cycles are not, because they have chromatic number $3$ and maximal cliques of size $2$. Note that in poset drawings, a clique (resp. an independent set) is just a chain (resp. an antichain) in the poset, whereas a coloring represents a partition of the poset into antichains.
Thus Dilworth's theorem (``for every partially ordered set, the maximum size of an antichain equals the minimum number of chains into which the poset can be partitioned'' \cite{Dilworth} -- see Fulkerson \cite{Fulkerson} for an easy proof) can be equivalently stated as ``every co-comparability graph is perfect''.
Not all perfect graphs are co-comparability, as shown by large even cycles.
\enlargethispage{3mm}
Last property: A graph is \textsc{chordal} if it has no induced subcycles of length $\ge 4$.
One can characterize chordality in the same spirit of (\ref{eq:Interval}), (\ref{eq:UnitInterval}) and (\ref{eq:Cocomparability}): Namely, a graph is chordal if and only if it admits a labeling such that, for all $a<b<c$,
\begin{equation} \label{eq:Chordal}
ac, bc \in G \Longrightarrow ab \in G.
\end{equation}
In fact, if a graph $G$ has a labeling that satisfies (\ref{eq:Chordal}), then $G$ is obviously chordal, because if $c$ is the highest-labeled vertex in any induced cycle, then its neighbors $a$ and $b$ in the cycle
must be connected by a chord by (\ref{eq:Chordal}). The converse, first noticed by Fulkerson--Gross \cite{FulkersonGross}, follows recursively from Dirac's Lemma that every chordal graph has a ``simplicial vertex'', i.e. a vertex whose neighbors form a clique (cf.~\cite[p.~83]{Golumbic} for a proof). In fact, let us pick any simplicial vertex and label it by $n$. Then, in the (chordal!) subgraph induced on the unlabeled vertices, let us pick another simplicial vertex and label it by $n-1$; and so on. The result is a labeling that satisfies (\ref{eq:Chordal}). See \cite[pp.~84--87]{Golumbic} for two algorithmic implementations.
Now, if the \emph{same} labeling satisfies (\ref{eq:Cocomparability}) \& (\ref{eq:Chordal}), then it trivially satisfies (\ref{eq:Interval}); and conversely, if (\ref{eq:Interval}) holds, then also (\ref{eq:Cocomparability}) \& (\ref{eq:Chordal}) trivially hold. Thus it is natural to conjecture that interval graphs are the same as the \emph{co-comparability chordal} graphs. The conjecture is true, although the `obvious' proof does not work: Some labelings on chordal graphs satisfy (\ref{eq:Cocomparability}) but not (\ref{eq:Interval}), like $13, 23, 24$ on the three-edge path.
However, Gilmore--Hoffman proved that any labeling that satisfies (\ref{eq:Cocomparability}) on a chordal graph (or more generally, on a graph that lacks induced $4$-cycles) can be modified in a way that `linearly orders' all maximal cliques \cite[Theorem 8.1]{Golumbic} and thus satisfies (\ref{eq:Interval}).
For more characterizations,
and a proof that all chordal graphs are perfect, see Golumbic \cite[Chapter 4]{Golumbic}.
\subsection{Higher-dimensional analogs and a hierarchy}
A $d$-dimensional extension\footnote{Several different $d$-dimensional generalizations of chordality exist in the literature, e.g. toric chordality \cite{ANS} or ridge-chordality, cf.~e.~g.~\cite{Bolo}. Emtander chose the name ``$d$-chordal'' for what here we call ``chordal''.} of Characterization (\ref{eq:Chordal}) of chordality was provided in 2010 by Emtander \cite{Emtander}, and is equivalent to the following:
\begin{definition}[chordal] Let $\Delta$ be a pure $d$-dimensional simplicial complex with $n$ vertices. $\Delta$ is called \emph{chordal} if there exists a labeling $1, \ldots, n$ of its vertices (called a \emph{``PEO''} or ``\emph{Perfect Elimination Ordering}'') such that for any two facets $F=a_0 a_1\cdots a_d$ and $G= b_0\cdots b_d$ of $\Delta$ with $a_d=b_d$, the complex $\Delta$ contains the full $d$-skeleton of the simplex on the vertex set $F \cup G$.
\end{definition}
In 2013, Characterization (\ref{eq:UnitInterval}) of unit-interval graphs was generalized as well:
\begin{definition}[closed \cite{EneEtAl}]
Let $\Delta$ be a pure $d$-dimensional simplicial complex with $n$ vertices. $\Delta$ is called \emph{closed} if there exists a labeling $1, \ldots, n$ of its vertices such that for any two facets $F=a_0 a_1\cdots a_d$ and $G= b_0\cdots b_d$ of $\Delta$ with $a_i=b_i$ for some $i$, the complex $\Delta$ contains the full $d$-skeleton of the simplex on the vertex set $F \cup G$.
\end{definition}
Obviously, closed implies chordal. We now present four notions that in the strongly connected case are progressive weakenings of the closed property (see Theorem \ref{thm:Hierarchy} and Proposition \ref{prop:closed2} for the proofs); the first property still implies chordality, whereas the last three do not. In Section \ref{sec:ClosedTraceable}, we connect all these notions to traceability (Theorem \ref{thm:Hi}). One of these properties is ``new'' even for $d=1$: We will see its importance in Chapter 3.
\begin{definition}[unit-interval]
Let $\Delta$ be a pure $d$-dimensional simplicial complex with $n$ vertices. The complex $\Delta$ is called \emph{unit-interval} if there exists a labeling $1, \ldots, n$ of its vertices such that for any $d$-face $F=a_0 a_1 \cdots a_d$ of $\Delta$, the complex $\Delta$ contains the whole $d$-skeleton of the simplex with vertex set
$\{a_0, a_0 +1, a_0 + 2, \ldots, a_d\}$.
\end{definition}
\begin{definition}[under-closed]
Let $\Delta$ be a pure $d$-dimensional simplicial complex with $n$ vertices. The complex $\Delta$ is called \emph{under-closed} if there exists a labeling $1, \ldots, n$ of its vertices such that for any $d$-face $F=a_0 a_1 \cdots a_d$ of $\Delta$ the following condition holds:
\begin{compactitem}
\item all faces $a_0i_1i_2\ldots i_d$ of $\Sigma^d_n$ with $i_1\leq a_1, i_2\leq a_2,\ldots , i_d\leq a_d$, are in $\Delta$.
\end{compactitem}
\end{definition}
\begin{definition}[semi-closed]
Let $\Delta$ be a pure $d$-dimensional simplicial complex with $n$ vertices. The complex $\Delta$ is called {\it semi-closed} if there exists a labeling of its vertices such that for any $d$-face $F=a_0a_1\ldots a_d$ of $\Delta$, at least one of the two following conditions holds:
\begin{compactenum} [ (i) ]
\item either all faces $a_0i_1i_2\ldots i_d$ of $\Sigma^d_n$ with $i_1\leq a_1, i_2\leq a_2,\ldots , i_d\leq a_d$, are in $\Delta$,
\item or all faces $i_0i_1\ldots i_{d-1}a_d$ of $\Sigma^d_n$ with $i_0\geq a_0, i_1\geq a_1,\ldots ,i_{d-1}\geq a_{d-1}$ are in $\Delta$.
\end{compactenum}
\end{definition}
\begin{definition}[weakly-closed]
Let $\Delta$ be a pure $d$-dimensional simplicial complex with $n$ vertices. $\Delta$ is called \emph{weakly-closed} if there exists a labeling $1, \ldots, n$ of its vertices such that for each $d$-face $F=a_0 a_1 \cdots a_d \in \Delta$, for every integer $g \notin F$ with $a_0 < g < a_d$, there exists a $d$-face $G=b_0 b_1 \cdots b_d$ in $\Delta$ such that $G$ contains $g$, $G$ is adjacent to $F$, and at least one of the following two conditions hold:
\item \begin{compactenum}[ (i)]
\item either $b_d \ne a_d$,
\item or $b_0 \ne a_0$.
\end{compactenum}
\end{definition}
\begin{remark}\label{rem:matsuda} For $d=1$, and assuming connectedness:
\begin{compactitem}
\item ``closed $1$-complexes'' and ``unit-interval $1$-complexes'' are the same as the \emph{unit interval graphs}; compare Looges--Olario \cite[Theorem 1]{LO} and Matsuda \cite[Prop.~1.3]{Matsuda}.
\item ``under-closed $1$-complexes'' are the same as the \emph{interval graphs}, cf.~\cite[Proposition 4]{LO}.
\item ``weakly-closed $1$-complexes'' are the same as the \emph{co-comparability graphs}; this is clear from the definition we gave, but a proof is also in Matsuda \cite[Theorem 1.9]{Matsuda}.
\end{compactitem}
We will see that ``semi-closed $1$-complexes'' are an intermediate class between the previous two. For example, such class contains the $4$-cycle but not the complement of long even cycles, as we will prove in Theorem \ref{thm:Hierarchy}.
\end{remark}
\begin{remark}[``unit-interval'' vs. ``chordal''] \label{rem:almost}
Suppose $F$ and $G$ are two faces of a complex $\Delta$ with $\min F = \min G$. Then any of the two conditions ``$\Delta$ is closed'', ``$\Delta$ is unit-interval'' forces $\Delta$ to contain the full $d$-skeleton of the simplex on the vertex set $F \cup G$. (Instead, the condition ``$\Delta$ is under-closed'' does not suffice: See Remark \ref{rem:UCdC} below).
Symmetrically, if $F$ and $G$ are $d$-faces of $\Delta$ with $\max F = \max G$, and $\Delta$ is either closed or unit-interval, then $\Delta$ must contains the full $d$-skeleton of the simplex on the vertex set $F \cup G$. For this reason, all unit-interval $d$-dimensional complexes are chordal.
\end{remark}
\begin{remark}[``Under-closed'' vs. ``chordal''] \label{rem:UCdC}
Not all chordal complexes are under-closed: Alread for $d=1$, the chordal graph $G=12, 13, 14, 23, 25, 36$, known as ``$3$-sun'' or ``net graph'', is neither interval nor co-comparability. However, while all interval graphs are chordal (and co-comparability), the statement ``all under-closed $d$-complexes are chordal'' is false for $d > 1$. In fact, we leave it to the reader to verify that the smallest counterexample is the $2$-complex
\[ \Delta \eqdef 123, 124, 234, 235.\]
The other direction in Gillmore--Hoffman's theorem (namely, ``all chordal co-comparability graphs are interval graphs'') does not extend to $d>1$ either, as the next Proposition shows.
\end{remark}
\begin{proposition} \label{prop:GilmoreHoffman}
\begin{compactenum}[\rm (i) ]
\item Some chordal simplicial complexes are semi-closed, but not under-closed.
\item If a simplicial complex is chordal and semi-closed with respect to the \emph{same} labeling, then with respect to that labeling the complex is also under-closed.
\end{compactenum}
\end{proposition}
\begin{proof}
\begin{compactenum}[\rm (i) ]
\item The example we found is the complex
\[ \Sigma = 123, 124, 134, 135, 167, 234, 246.\]
The labeling above is a PEO, so $\Sigma$ is chordal. A convenient relabeling (we leave it to the reader to figure out the bijection from the vertex degrees) allows us to rewrite it as
\[ \Sigma= 123, 256, 345, 346, 347, 356, 456.\]
With this new labeling we see that $\Sigma$ is weakly- and semi-closed. However, with the help of a software designed by Pavelka \cite{Pavelka}, we verified that $\Sigma$ is not under-closed.
\item Let $\Delta$ be a simplicial complex with a labeling that is a PEO and makes $\Delta$ semi-closed. Let $F=a_0 \cdots a_d$ be a face of $\Delta$ with $\operatorname{gap} F >0$.
Let $G=a_0 b_1 \cdots b_d$ be a different $d$-face of $\Sigma^d_n$ such that $G \le F$ (componentwise) and $\min G = \min F$.
We claim that for any $b_i$ not in $F$, there exists a $d$-face $A_i$ of $\Sigma^d_n$ that contains $b_i$, such that $A_i \ge F$ (componentwise) and $\max A_i = \max F$. In fact, by construction $a_0 < b_i \le b_d \le a_d$. Since $b_i$ is not in $F$,
there exists a unique $j \in \{0, \ldots, d-1\}$ such that $a_{j} < b_i < a_{j+1}$. Thus if we set
\[ A_i \eqdef a_0 \cdots a_{j-1} b_i a_{j+1} \cdots a_{d}\]
the claim is proven. Now, either $F$ satisfies condition (i) of the semi-closed definition, and then $G \in \Delta$; or $F$ satisfies condition (ii), in which case all $A_i$'s are in $\Delta$. But by construction, the maximum of all these $A_i$'s is $a_d$, the same maximum of $F$. So by chordality, $\Delta$ must contain all the $d$-faces of $\Sigma^d_n$ with vertex set contained in
\[F \: \cup \: \bigcup_{i \textrm{ s.t. } b_i \notin F} A_i \: \: = \: \{a_0, a_1, \ldots, a_d\} \cup \{b_1, \ldots, b_d\} \: = \: F \cup G.\] So also in this case $G \in \Delta$. \qedhere
\end{compactenum}
\end{proof}
\begin{remark}
Part (ii) of Proposition \ref{prop:GilmoreHoffman} is false if one replaces the assumption ``semi-closed'' with ``weakly-closed'': The subcomplex $\Sigma' = 123, 124, 134, 135, 234$ of $\Sigma$ is weakly-closed and chordal with respect to this labeling, but to prove it under-closed, we need to change labeling.
\end{remark}
\begin{remark}[``Under-closed'' vs.~``Shifted'']\label{rem:kalai}
Recall that a simplicial complex $\Delta$ on $n$ vertices is called \emph{shifted} if for every face $F$ of $\Delta$, and for every face $G$ of the simplex on $n$ vertices, if $\dim F = \dim G$ and $F \le G$ componentwise, then also $G \in \Delta$. Shifted complexes are obviously under-closed. The converse is false, as shown by the graph $12, 23, 34$.
\end{remark}
\begin{remark}\label{rem:cones}
Being shifted is maintained under taking cones, by assigning label $1$ to the new vertex. In contrast, $G=12,13,23$ is closed and chordal, but the cone over it is neither closed nor chordal. In fact, \emph{none} of the five properties (closed, unit-interval, under-closed, semi-closed, weakly-closed) is maintained under taking cones. A counterexample for all is the unit-interval graph $G= 12, 34, 56, 78$. The cone over $G$ is the $U^2_4$ of Lemma \ref{lem:bouquet} below.
\end{remark}
Let us start exploring the relations between all the new properties with some Lemmas.
\begin{lemma} \label{lem:skeleton}
Let $d \ge k \ge 1$ be integers. If a pure $d$-dimensional simplicial complex is unit-interval (resp. under-closed, resp. semi-closed, resp. weakly-closed), then its $k$-skeleton is also unit-interval (resp. under-closed, resp. semi-closed, resp. weakly-closed).
\end{lemma}
\begin{proof} It suffices to prove the claim for $k=d-1$; the general claim follows then by iterating. We prove only the weakly-closed case; the others are easier. Let $\Delta$ be a pure weakly-closed $d$-complex. Let $\sigma=a_0 \cdots a_{d-1}$ be a $(d-1)$-face of $\Delta$. Let $g \notin \sigma$ be an integer such that $a_0 < g < a_{d-1}$. Since $\Delta$ is pure, there exists a $d$-face $F$ of $\Delta$ that contains $\sigma$. Let $v$ be the vertex of $F$ not in $\sigma$. If $v=g$, i.e. if $F = \{g\} \cup \sigma$, then all the $d$ facets of $\Delta$ different than $\sigma$ are adjacent to $\sigma$ and contain $g$; if we choose one of these $d$ facets that has either different minimum or different maximum than $\sigma$, we are done. So let us assume that $ v \ne g$, or equivalently, that $F$ does not contain $g$. By the weakly-closed assumption, there exists a $d$-face $G$ in $\Delta$ such that $G$ contains $g$, $G$ is adjacent to $F$, and $G$ and $F$ do not have same minimum and maximum. If $G$ contains the entire face $\sigma$, i.e. $G = \sigma \cup g$, then again we could conclude as above, choosing some facet of $G$ different than $\sigma$. So we can assume that $G$ does not contain the whole of $\sigma$, or in other words, that the vertex $v$ is present in $G$. Let $\tau$ be the unique face of $G$ that does not contain $v$. By construction, $\sigma$ and $\tau$ are adjacent, and $g \in \tau$. If $\sigma$ and $\tau$ had same minimum and maximum, then also $F$ and $G$ would, because $F$ and $G$ are obtained by adding to $\sigma$ and $\tau$, respectively, the same element $v$. Hence, the $(d-1)$-skeleton of $\Delta$ is weakly-closed.
\end{proof}
\begin{lemma} \label{lem:Bd} Let $d \ge 2$. Let $C^{d+1}$ be the $(d+1)$-dimensional simplicial complex with facets $H_1$ and $H_2$. The boundary $S^d$ of $C^{d+1}$ is strongly-connected, semi-closed, but not under-closed. The $d$-skeleton $B^d$ of $C^{d+1}$ is traceable, strongly-connected, unit-interval, but not closed. \\In particular, the $k$-skeleton of a closed complex need not be closed.
\end{lemma}
\begin{proof}
Note that $S^d$ is $B^d$ minus a $d$-face, so since $d \ge 2$ the $1$-skeleta of $B^d$ and of $S^d$ coincide. The vertices of $B^d$ (respectively, of $S^d$) can be partitioned with respect to the number of edges containing them, as follows: exactly two vertices have degree $d+1$, and we shall call them ``apices''; the remaining $d+1$ have degree $d+2$, and we shall call them ``basepoints''. The crucial remark is that in $B^d$ (resp. $S^d$) the two apices are not connected by any edge. We claim that any labeling that makes $B^d$ or $S^d$ closed \emph{must} assign labels $1$ and $d+3$ to the two apices. In fact:
\begin{compactitem}
\item If the label $1$ is assigned to a basepoint, let $b_1, \ldots, b_d$ be the other $d$ basepoints and let $v,w$ be the apices, with $v<w$. Then $B^d$ (resp.~$S^d$) contains a $d$-face $F$ of vertices $\{1,b_1, \ldots, b_{d-1}, v\}$ and a $d$-face $G$ of vertices $\{1,b_1, \ldots, b_{d-1}, w\}$.
Note that $1$ is in the same position in $F$ and $G$,
yet $B^d$ (or $S^d$) does not contain the whole $d$-skeleton of the simplex on $F \cup G$, because $vw$ is missing. So the closed condition is not satisfied.
\item Symmetrically, if $d+3$ is assigned to a basepoint, call $b_1, \ldots, b_d$ the other basepoints and $v,w$ the apices, with $v<w$. Then $B^d$ (resp.~$S^d$) contains a $d$-face $F$ of vertices $\{v, b_1, \ldots, b_{d-1}, d+3\}$ and a $d$-face $G$ of vertices $\{v, b_1, \ldots, b_{d-1}, d+3\}$. So $d+3$ is the maximum of both faces, and again $B^d$ (resp.~$S^d$) does not contain the edge $vw$, so the closed condition is not met.
\end{compactitem}
Next, we claim that any labeling that makes $S^d$ under-closed \emph{must} assign labels $1$ and $d+3$ to the two apices. (Caveat: This claim is valid only for $S^d$, since already $B^2$ is under-closed with the labeling $123, 124, 134, 234, 125, 135, 235$, where the apices are $4$ and $5$.)
In fact:
\begin{compactitem}
\item If the label $1$ is assigned to a basepoint, then any other vertex is contained in a facet that contains also $1$. The same is true if $d+3$ is assigned to a basepoint. So either way, there is a face $H$ containing both $1$ and $d+3$. Thus $\operatorname{gap}H =2$. But then if the labeling is under-closed, the complex must contain all three facets
$12 \cdots d \, j $, with $j \in \{d+1, d+2, d+3\}$. So we found in $S^d$ three different facets containing the $(d-1)$-face $\sigma \eqdef 12 \cdots d$. This is a contradiction because $S^d$ is topologically a sphere: Every $(d-1)$-face in it lies in exactly two $d$-faces.
\end{compactitem}
Thus the two claims are proven. So up to a rotation that does not affect the list of facets, both for $B^d$ and $S^d$ we may focus on the labeling that we introduced from the start. With respect to that labeling, $S^d$ is clearly semi-closed, but it is not under-closed, because the $d$-face with vertices $2, 3, \ldots, d+1, d+2$ is missing. Similarly, with respect to that labeling, $B^d$ is traceable and unit-interval, but it is not closed for the following reason. Let $F$ (resp. $G$) be the face of vertices $1, 3, 4, \ldots, d+1, d+2$ (resp. $2, 3, 4, \ldots, d+1, d+3$). Since $F$ (resp. $G$) is contained in the facet $H_1$ (resp. $H_2$) of $C^{d+1}$, it is in $B^d$. Yet vertex $3$ appears in second position in both $F$ and $G$. However, the face $H_3$ of vertices $1, 3, 4, \ldots, d+1, d+3$ contains the edge connecting the two apices, so $H_3$ is not in $B^d$.
\end{proof}
\begin{figure}[htbp]
\begin{center}
\hskip-4mm \includegraphics[scale=0.37]{ToyExamples3.png}
\caption{(i) A $2$-complex $B^2=123, 124, 134, 234, 235, 245, 345$ that is unit-interval, but not closed; if we remove the triangle $234$, we get a $2$-complex $S^2$ that is semi-closed, not under-closed, cf.~Lemma~\ref{lem:Bd}.
(ii) A $2$-complex $U^2_3 = 124, 345, 467$ that is closed, but not weakly-closed, cf.~Lemma~\ref{lem:bouquet}. \\
(iii) A~$2$-complex $\Delta^2_3 = 123, 124, 125$ that is under-closed, but not unit-interval, cf.~Lemma~\ref{lem:WheelComplex}.\\
(iv) A $2$-complex $Q^2=123, 125, 234, 245$ that is weakly-closed, but not semi-closed, cf.~Lemma~\ref{lem:StarredSquare}.}
\label{fig:ToyExamples1}
\end{center}
\end{figure}
\begin{lemma} \label{lem:bouquet}
Let $d$ and $k$ be positive integers. Let $U^{d}_k$ be a one-point union of $k$ copies of $\Sigma^d$. Then $U^{d}_{k}$ is closed if and only if $k \le d+1$, and it is weakly-closed if and only if $k \le 2$. In particular, for all $d \ge 2$, the $d$-complex $U^d_{d+1}$ is closed, but not weakly-closed.
\end{lemma}
\begin{proof} Let $v$ be the vertex common to all facets. When $k>d+1$, by the pigeonhole principle there are two facets in which $v$ appears in the same position; were $U^{d}_{k}$ closed, its dual graph would have to contain a clique, which is not the case. When $k \le d+1$, we force the closed property by giving $v$ a label so that $v$ appears in a different position in all facets. We show an algorithm to do this in case $k=d+1$, leaving the case $k < d+1$ to the reader. We label $v$ by $f_d \eqdef \binom{d+1}{2} + 1$. We label the vertices of the first facet by $123\cdots d\, f_d$: so in the first facet, $v$ comes last.
Then for all $i=2, 3, \ldots, k=d+1$, we label the $i$-th facet by using the next available $d-i$ integers below $f_d$, then $f_d$, then the first $i-1$ available integers after $f_d$. This way in the $i$-th facet, $v$ comes ``$i$-th last''.
For example, the labeling we construct for $U^3_4$, since $f_3 \eqdef \binom{4}{2} + 1 = 7$, $\, $ is $\: U^{3}_{4} \, = \, 1237, \; 4578, \; 679\,10, \; 7 \, 11 \, 12 \, 13$. \\
Finally, suppose that $U^{d}_{k}$ is weakly-closed. No face of $U^{d}_{k}$ has an adjacent facet. Hence, the labeling satisfying the weakly-closed condition must consist only of gap-$0$ faces. But labeling all facets with consecutive vertices is possible if and only if $k \le 2$.
\end{proof}
\begin{lemma} \label{lem:WheelComplex}
Let $k\ge 1$ and $d\ge 2$ be integers.
Let $\Delta^{d}_k$ be the $d$-dimensional complex on $d+k$ vertices obtained by joining the $(d-1)$-simplex $\Sigma^{d-1}$ to a $0$-complex consisting of $k$ points.
Then
\begin{compactenum}[ \rm (a)]
\item $\Delta^d_k$ is under-closed for all $k$.
\item $\Delta^d_k$ is closed, if and only if it is unit-interval, if and only if it is (weakly) traceable, if and only if $k \le 2$.
\end{compactenum}
\end{lemma}
\begin{proof}
Let us label the vertices of $\Sigma^{d-1}$ by $1, 2, \ldots, d$. This labeling immediately shows that $\Delta^d_k$ is under-closed.
Moreover, the $d$-complex $\Delta^d_k$ is strongly-connected. It has exactly $d+k$ vertices and $k$ facets. When $k \le 2$ its dual graph is a path, so clearly the obvious, consecutive labeling makes $\Delta^d_k$ a closed, unit-interval, and traceable complex. But when $k \ge 3$, the ``path of $k$ $d$-simplices'' is not a subcomplex of $\Delta_d$. Hence, for $k \ge 3$ the complex $\Delta^d_k$ is not traceable, not weakly-traceable, and not weakly-Hamiltonian. The fact that $\Delta^d_k$ is neither unit-interval nor closed can be verified either directly, or using Proposition \ref{prop:closed2} and Theorem \ref{thm:CTSC} below.
\end{proof}
\begin{remark} \label{rem:G5} The $1$-skeleton of $\Delta^2_3 = 123, 124, 125$ (cf.~Figure \ref{fig:ToyExamples1}) is the graph
\[ G_5 = 12, 13, 14, 15, 23, 24, 25 \]
which is under-closed by Lemma \ref{lem:skeleton}. It is not difficult
to see that $G_5$ is the smallest $2$-connected interval graph that is not Hamiltonian.
\end{remark}
\begin{lemma} \label{lem:StarredSquare}
Let $d\ge 2$ be an integer. Let $Q^{d}$ be the $d$-dimensional complex on $d+3$ vertices obtained by taking $d-1$ consecutive cones over the square.
Then $Q^{d}$ is weakly-closed, but not semi-closed.
\end{lemma}
\begin{proof}
Both $Q^2 = 123, 125, 234, 245$ and $Q^3=1236, 1256, 2346, 2456$ are weakly-closed.
If we label further coning vertices using consecutive labels after $6$, we claim that the weakly-closed property is maintained. (This is not obvious, as the weakly-closed property is not maintained under arbitrary cones, cf.~Remark \ref{rem:cones}.)
In fact, since every face $F$ of $Q^3$ contains $6$, the gap of $F$ equals the gap of $F \cup \{7\}$, and the missing integers are the same, so the calculations proving weakly-closedness end up being the same for $Q^3$ and $Q^4$.
For the same reasons, one can show that if some $Q^{d}$ is semi-closed with a labeling that assigns consecutive labels to two apices, then $Q^{d-1}$ is semi-closed too. But if $d \ge 7$, $Q^d$ has $\ge 10$ vertices, and only $4$ of them are not apices; so necessarily two apices are assigned consecutive labels. So to complete the proof we only need to show that $Q^2, Q^3, Q^4, Q^5$ and $Q^6$ are not semi-closed, which can be verified with \cite{Pavelka}.
\end{proof}
\vskip3mm
\begin{lemma}\label{lem:SemiClosedGraph}
Let $\Delta$ be a pure $d$-complex where every vertex is in at most $k$ facets.
\begin{compactenum}[\rm (1)]
\item In any labeling that makes $\Delta$ weakly-closed, every facet has gap $\le 2k-2$.
\item In any labeling that makes $\Delta$ semi-closed, every facet has gap $\le k-1$.
\\ If in addition $d=1$ and $\Delta$ is a $k$-regular graph, then in any labeling that makes $\Delta$ semi-closed, the $k$ edges of the type 1j , with $2 \le j \le k+1$, are all in $\Delta$; and so are all the $k$ edges of the type $\,in$, with $ n-k \le i \le n-1$.
\item In any labeling that makes $\Delta$ unit-interval, every facet has gap $\le g$, where $g$ is the largest integer such that $\binom{g+d}{d} \le k$; in particular, every facet has gap $ \le \sqrt[d]{k d!} - 1$.
\end{compactenum}
\end{lemma}
\begin{proof} For any vertex $v$ of $\Delta$, let $\deg v$ be the number of facets of $\Delta$ containing it. For any facet $F$ of $\Delta$, let $S_F$ be the set of integers $i \notin F$ such that $\min F < i < \max F$. By definition, $S_F$ has cardinality equal to $\operatorname{gap} F$. For brevity, set $a \eqdef \min F$ and $b \eqdef \max F$.
\begin{compactenum}[(1)]
\item For every $i$ in $S_F$, there is a face $G_i$ adjacent to $F$ that contains the vertex $i$ and exactly $d$ vertices of $F$, among which exactly one of $a, b$. Clearly as $i$ ranges over $S_F$, the $G_i$'s are all different. So $\deg a + \deg b \ge \operatorname{gap} F + 2$. (The summand $2$ is due to the fact that we should count also $F$ itself, once contributing to $\deg a$ and once to $\deg b$). Since $k \ge \deg a$ and $k \ge \deg b$, we conclude that $\operatorname{gap} F \le 2k - 2$.
\item For every $i$ in $S_F$, either $\Delta$ contains the $n_a \ge \operatorname{gap} F+1$ facets (including $F$ itself) with minimum $a$ that are componentwise $\le F$, or $\Delta$ contains the $n_b\ge \operatorname{gap} F + 1$ facets (including $F$ itself) with maximum $b$ that are componentwise $\ge F$. Either way, there is a vertex $v$ (either $a$ or $b$) with $\deg v \ge \operatorname{gap} F+1$. Since $\deg v \le k$ by assumption, we conclude that $\operatorname{gap} F \le k -1$.
So the first claim is settled. From this applied to $d=1$, it follows that
\[ \{ \textrm{ edges of $\Delta$ containing $1$ } \} \ \subseteq \ \{ \; 1j \textrm{ such that } 2 \le j \le k+1 \;\}.\]
The two sets above have size $\deg 1$ and $k$, respectively. If $\Delta$ is $k$-regular, the two quantities are equal, hence the sets coincide.
The same argument applies to the edges containing $n$.
\item For every $i$ in $S_F$, by definition of unit-interval, $\Delta$ contains the $\binom{\operatorname{gap} F + d}{d}$ $d$-faces that contain vertex $i$ and have vertices in $\{a, a+1, \ldots, b\}$. So we must have $\binom{\operatorname{gap} F + d}{d} \le k$. In particular, since $\binom{g+d}{d} \ge \frac{(g+1)^d}{d!}$ for all positive integers $g,d$, we cannot have $\frac{(\operatorname{gap} F+1)^d}{d!} > k$. \qedhere
\end{compactenum}
\end{proof}
Our next Lemma is a $d$-dimensional version of the well-known fact that cycles of length $5$ or more are not co-comparability, cf.~Matsuda \cite{Matsuda}.
\begin{lemma} \label{lem:dMatsuda}
For $n \ge 2d + 3$, the $d$-dimensional annulus $A^d_n \eqdef H_1, H_2, \ldots, H_n$ and
any $k$-skeleton of it are not weakly-closed.
\end{lemma}
\begin{proof}
By Lemma \ref{lem:skeleton}, it suffices to prove that the $1$-skeleton $G$ of $ A^d_n$ is not weakly-closed. By contradiction, let $a_1, \ldots, a_n$
be a re-labeling of the vertices $1, \ldots, n$ (respectively) that proves $G$ weakly-closed. Up to rotating the labeling cyclically, we can assume that $a_1$ is the smallest of the $a_i$'s. Since $n \ge 2d + 3$, in particular $n-d> d+2$, so the labels $a_{n-d}, a_{n-d+1}, \ldots, a_{n}, a_1, a_2, \ldots, a_{d+2}$ are all distinct. Were $a_{d+2} < a_n$, we would have a contradiction with the weakly-closed assumption: $a_1a_n$ is in $G$, but neither $a_1a_{d+2}$ nor $a_{d+2}a_n$ is. So $a_n < a_{d+2}$. Symmetrically, were
$a_{d+1}>a_{n-d}$, we would have a contradiction: $a_1a_{d+1}$ is in $G$, but neither $a_1a_{n-d}$ nor $a_{n-d}a_{d+1}$ is. So $a_{d+1} < a_{n-d}$. Now let us compare $a_{d+1}$ and $a_n$:
\begin{compactitem}[--]
\item If $a_{d+1} > a_n$, then $a_n < a_{d+1} < a_{n-d}$ by what we said above; so we get a contradiction, because the edge $a_n a_{n-d}$ is in $G$, but neither $a_na_{d+1}$ nor $a_{d+1} a_{n-d}$ is.
\item If $a_{d+1} < a_n$, then $a_{d+1} < a_n < a_{d+2}$ by what we said above; so symmetrically we get another contradiction, because $a_{d+1}a_{d+2}$ is in $G$, but neither $a_{d+1}a_n$ nor $a_n a_{d+2}$ is. \endproof
\end{compactitem}
\end{proof}
\begin{remark}
$A^2_6$ is not weakly-closed, even if its $1$-skeleton is semi-closed \cite{Pavelka}. ($A^2_5$ instead is weakly-closed.) So the bound $n \ge 2d+3$ of Lemma \ref{lem:dMatsuda} is best possible in general, but if one only cares about $A^d_n$ and not about its skeleta, then it can be improved.
\end{remark}
\vskip3mm
\begin{theorem} \label{thm:Hierarchy}
For each $d \ge 1$, for (pure) simplicial $d$-complexes, one has the hierarchy
\[ \{ \textrm{ \rm unit-interval } \} \subsetneq \{ \textrm{ \rm under-closed } \} \subsetneq \{ \textrm{ \rm semi-closed } \}
\subsetneq \{ \textrm{ \rm weakly-closed } \} \subsetneq \{ \textrm{ \rm all } \} .\]
\end{theorem}
\begin{proof} All inclusions are obvious except perhaps the third one. Let $F=a_0a_1\ldots a_d$ be a face of~$\Delta$.
If $F$ satisfies condition (i) in the definition of semi-closed, and there is a $g$ such that $a_i < g < a_{i+1}$, then $G' \eqdef a_0 a_1 \cdots a_i \: g \: a_{i+1} \cdots a_{d-1}$
is componentwise $\le F$ and thus belongs to $\Delta$; moreover, since $\max G' < \max F$, the face $G'$ satisfies condition (i) in the definition of weakly-closed. If instead $F$ satisfies condition (ii) in the definition of semi-closed, and $a_i < g < a_{i+1}$ for some $g$, then
$G'' \eqdef a_1 \cdots a_i \: g \: a_{i+1} \cdots a_d$
is componentwise $\ge F$, so $G''$ is in $\Delta$; and since $\min G'' > \min F$, this $G''$
satisfies condition (ii) in the definition of weakly-closed.
\vskip2mm
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.43]{Hierarchy.png}
\caption{One-dimensional simplicial complexes that are: (i) Not unit-interval, but under-closed. (ii) Not under-closed, but semi-closed. (iii) Not semi-closed, but weakly-closed. (iv) Not even weakly-closed.}
\label{fig:Hierarchy}
\end{center}
\end{figure}
Next, we discuss the strictness of the inclusions, which is the interesting part of the theorem.
\begin{compactenum}[ (i) ]
\item For $d=1$, the claw graph $12, 13, 14$ is under-closed only with this labeling, which is not unit-interval because for example $23$ is missing. \\For $d \ge 2$, strictness follows by Lemma~\ref{lem:WheelComplex}.
\item For $d=1$, the $4$-cycle is semi-closed with the labeling $12, 13, 24, 34$. By Lemma \ref{lem:SemiClosedGraph}, part (2), \emph{only} this labeling makes the $4$-cycle semi-closed. This labeling is not under-closed, because $24$ is an edge, but $23$ is not. More generally, for any $n \ge 4$, one can show that the graph $\operatorname{susp}(A_{n-2})$ of Remark \ref{rem:An} is semi-closed (with the suspension apices labeled by $1$ and $n$), but not under-closed. \\For $d \ge 2$, the strictness of the inclusion follows by Lemma \ref{lem:Bd}.
\item
For $d=1$: Since $C_{2k}$ is a comparability graph (it is the nonempty-face poset of the $k$-gon), $\overline{C_{2k}}$ is co-comparability.
We claim that $\overline{C_{2k}}$ is not semi-closed for any $k \ge 3$. For notational simplicity, we give the proof for $k=3$; the case of arbitrary $k$ has a completely analogous proof. Suppose by contradiction that $\overline{C_{6}}$ has a semi-closed labeling. Since $C_6$ is $2$-regular, its complement is $(6-1-2)$-regular, i.e. $3$-regular. By Lemma \ref{lem:SemiClosedGraph}, part (2), all of $12, 13, 14$ and $36, 46, 56$ are edges. In contrast, $15$, $16$ and $26$ are not edges, again by Lemma~\ref{lem:SemiClosedGraph}. But then $25$ must be an edge of $\overline{C_{6}}$, for otherwise $15$, $16$, $26$ and $25$ would form a $4$-cycle inside the complement, which is $C_{6}$. We claim that this edge $25$ cannot satisfy the semi-closed condition. In fact, if all of $23, 24, 25$ were edges, together with $12$ we would have $4$ edges containing vertex $2$, contradicting $3$-regularity; and similarly, if all of $25, 35, 45$ were edges, counting also $56$ we would have $4$ edges containing vertex $5$.\\
This shows strictness of the inclusion for $d=1$; the case $d \ge 2$ is settled by Lemma \ref{lem:StarredSquare}.
\item
For any $d \ge 1$, this is settled by Lemma \ref{lem:dMatsuda}. \qedhere
\end{compactenum}
\end{proof}
\vskip4mm
\subsection{Shortest dual paths and relation with traceability} \label{sec:ClosedTraceable}
As we saw in Lemma \ref{lem:bouquet}, there exist complexes like $U^2_3= 124, 345, 467$ that are closed but not weakly-closed. So at this point we owe the reader some explanation: Why did we (and before us, Matsuda \cite{Matsuda} and others, in the $1$-dimensional case) choose to call ``weakly-closed'' a property not implied by ``closed''? Here is the reason.
We are going to show that \emph{all strongly-connected closed complexes are unit-interval} (Proposition \ref{prop:closed2}), so in particular under-closed, semi-closed, and weakly-closed. We will then prove that all such complexes are traceable (Theorem \ref{thm:CTSC}), which can be viewed as a higher-dimensional generalization of the graph-theoretical results by Bertossi \cite{Bertossi} and Herzog et al's \cite[Proposition 1.4]{HerzogEtAl}. The key to our generalization is to focus on shortest paths in the \emph{dual} graph.
\begin{definition}
Let $F$ be a facet a pure $d$-dimensional simplicial complex $\Delta$. Let $v$ be a vertex of $\Delta$. A \emph{shortest path between $F$ and $v$} is a path in the dual graph of $\Delta$ of minimal length from $F$ to some facet containing $v$. The \emph{distance} between $F$ and $v$ is the length of a shortest path, if any exists, or $+ \infty$, otherwise.
\end{definition}
\begin{definition} Let $\Delta$ be a pure $d$-dimensional simplicial complex, with vertices labeled from $1$ to $n$.
A path $ F_0, \; F_1, \: \ldots, \; F_\ell$ in the dual graph of $\Delta$ is called \emph{ascending}, if each $F_i$ is obtained from $F_{i-1}$ by replacing the smallest vertex of $F_{i-1}$, with a vertex greater than all remaining vertices of $F_{i-1}$. A path is called \emph{descending}, if the reverse path is ascending.
\end{definition}
For example, suppose that a $2$-complex $\Delta$ contains the facets $124, 245, 456$, and $356$. The dual path they form is not ascending -- or better, it is ascending, except for the last step. Such dual path demonstrates that the vertex $v=3$ is at distance $\le 3$ from $124$. Now suppose that we know in advance that $\Delta$ is closed: Then from $356, 456 \in \Delta$, we immediately derive that $\Delta$ must contain the whole $2$-skeleton of the simplex $3456$. Note that the same conclusion could be reached also if we knew in advance that $\Delta$ is \emph{unit-interval}, rather than closed. Either way: $\Delta$ contains the facet $G=345$ which contains $3$ and is adjacent to $245$. So $124, 245, 345$ yields a ``shortcut'' to the original path, thereby proving that $v=3$ is actually at distance $\le 2$ from $124$. And it gets even better: Since $245$ and $345$ are in $\Delta$, by the closed assumption (or the unit-interval assumption) on $\Delta$, we may conclude that $\Delta$ contains the whole $2$-skeleton of the simplex $2345$. So also $234$ is in $\Delta$, which means that $v=3$ is at distance $1$ from $124$.
This example generalizes as follows, in what can be viewed as a higher-dimensional version of Cox--Erskine's narrowness property \cite{CoxErskine}:
\begin{lemma} \label{lem:closed}
Let $\Delta$ be a pure $d$-dimensional simplicial complex, with a labeling that makes it either closed or unit-interval. Let $F=a_0a_1\cdots a_d$ be a facet of $\Delta$. Let $v$ be a vertex. If the distance between $F$ and $v$ is a finite number $\ell \ge 2$, then
\begin{compactitem}
\item either there is a shortest path from $F$ to $v$ that is ascending (and thus $v > a_d$),
\item or there is a shortest path from $F$ to $v$ that is descending (and thus $v < a_0$).
\end{compactitem}
If instead $a_0 < v < a_d$, and some facet containing $v$ is in the same strongly-connected component of $F$, then the distance between $F$ and $v$ is at most one, and $\Delta$ contains the whole $d$-skeleton of the simplex on the vertex set $F \cup \{ v \}$.
\end{lemma}
\begin{proof} Let \[F= F_0, \ldots, F_{i-1}, \; F_i, \; F_{i+1}\] be a shortest path from $F$ to a vertex $v \in F_{i+1}$. Suppose the path is ascending until $F_i$, but it stops being ascending when passing from $F_i$ to $F_{i+1}$. This means that $\max F_i = \max F_{i+1}$. By Remark~\ref{rem:almost}, $\Delta$ contains the whole $d$-skeleton of the simplex with vertex set $F_i \cup F_{i+1}$. In particular, if we set $\gamma \eqdef F_{i-1} \cap F_i$, the complex $\Delta$ contains $G \eqdef \gamma \cup v$. But since $G$ is a $d$-face that contains $v$ and is already adjacent to $F_{i-1}$, \[F= F_0, \ldots, \; F_{i-1}, \; G\]
is a shorter path from $F$ to $v$ than the one we started with, a contradiction.
The same argument applies to descending paths. If instead $a_0 < v < a_d$, clearly there cannot be any ascending or descending path from $F$ to $v$. So either $v \in F$, in which case the distance from $F$ to $v$ is $0$ and there is nothing to prove, or $v \notin F$, in which case the distance is $1$. In the latter case, $F$ and the adjacent face $G$ containing $v$ have same maximum, so again by Remark \ref{rem:almost} the complex $\Delta$ contains the $d$-skeleton of the simplex on $F \cup G = F \cup \{v \}$.
\end{proof}
\begin{proposition} \label{prop:closed2} All strongly-connected closed simplicial complexes are unit-interval.
\end{proposition}
\begin{proof} Let $\Delta$ be a strongly-connected $d$-dimensional simplicial complex that is closed with respect to some-labeling. Let $F=a_0 a_1 \cdots a_d \in \Delta$. We claim the following:
\begin{compactitem}[ (*) ]
\item If there exist $m \in \{1, \ldots, d\}$ and $g_1, \ldots, g_m$ not in $F$, with $a_0 < g_1 < g_2 < \ldots < g_m < a_d$, then $\Delta$ contains the $d$-skeleton of the simplex with vertex set $\{a_0, \ldots, a_d, g_1, \ldots, g_m\}$.
\end{compactitem}
If $\operatorname{gap}(F) = 0$, then the implication is trivially true, because the antecedent is never verified. So suppose $\operatorname{gap}(F) > 0$, and let us proceed by induction on $m$.
For $m=1$: Pick a vertex $g$ of $\Delta$ not in $F$, with $a_0 < g < a_d$. Since $\Delta$ is strongly connected, by the second part of Lemma~\ref{lem:closed} the complex $\Delta$ has a facet $G$ that contains $g$ and is adjacent to $F$. Had $G$ neither same minimum nor same maximum of $F$, then either $G=a_1a_2 \cdots a_d g$ or $G=ga_0 a_1 \cdots a_{d-1}$. But both cases contradict the assumption $a_0 < g < a_d$. Hence, $F$ and $G$ have either same minimum or same maximum (or both), so they share at least one vertex in the same position. Since $\Delta$ is closed, $\Delta$ contains the $d$-skeleton of the simplex on $F \cup G = F \cup \{g \}$.
For $m>1$: let $H$ be a subset of $\{a_0, \ldots, a_d, g_1, \ldots, g_m\}$ of cardinality $d+1$. If $H$ contains at most $m-1$ elements of $\{g_1 \ldots, g_m\}$, then we know that $H \in \Delta$ by the inductive assumption. If $g_1, \ldots, g_m$ are all vertices of $H$, let us consider a new face $H'$ with exactly the same vertices of $H$, except for one replacement, to be decided as follows:
\begin{compactitem}
\item If $\min H = a_0$ and $\max H = a_d$, we shall replace $g_1$ with any vertex $v$ of $F$ that is not in~$H$. This way, since $a_0 \le v \le a_d$, we have that as real intervals \[(\min H , \max H) = (a_0, a_d) = (\min H' , \max H').\]
\item If $\min H = g_1$, or if $\min H = a_i$ for some $i >0$, we shall replace $g_1$ with $a_0$. This way \[(\min H , \max H) \subsetneq (a_0, \max H) = (\min H', \max H').\]
\item If $\max H = g_m$, or $\max H = a_i$ for some $i<d$, we shall replace $g_m$ with $a_d$. This way \[(\min H , \max H) \subsetneq (\min H, a_d) = (\min H', \max H').\]
\end{compactitem}
In all three cases, if $w$ is the only element that belongs to $H$ but not to $H'$, then $w$ is either $g_1$ or $g_m$, and we have \[\min H' < w < \max H'.\] Moreover, $H'$ contains at most $m-1$ elements of $\{g_1 \ldots, g_m\}$,
so by the inductive assumption $H'$ is in $\Delta$. But since $\min H' < w < \max H'$, by the second part of Lemma \ref{lem:closed} we conclude that also $H$ is in $\Delta$. By the genericity of $H$, this proves Claim (*). From the Claim the conclusion follows immediately, by choosing $m$ maximal.
\end{proof}
\begin{remark}
The converse is false: The complex with $k$ disjoint $d$-simplices is obviously not strongly-connected, yet it is unit-interval with the natural labeling below:
\[\Delta = H_1, \: H_{d+2}, \: H_{2d+3}, \ldots, H_{(k-1)d+k}.\]
For connected graphs, it is obvious that ``closed'' and ``unit-interval'' are the same: This is noticed also in Matsuda~\cite[Proposition 1.3]{Matsuda} and Crupi--Rinaldo \cite{Rinaldo}. However, as we saw in Lemma \ref{lem:Bd}, higher-dimensional complexes that are both strongly-connected and unit-interval might not be closed.
\end{remark}
We have arrived to the main result of this section, the generalization of Bertossi's theorem:
\begin{theorem}[Higher-dimensional Bertossi] \label{thm:CTSC}
Let $\Delta$ be a pure $d$-dimensional simplicial complex that is either closed or unit-interval. Then
\[ \Delta \textrm{ is strongly-connected }\: \ \Longleftrightarrow \ \: \Delta \textrm{ is traceable.} \]
\end{theorem}
\begin{proof}\hspace*{\fill}
\begin{compactdesc}
\item{$\Leftarrow$:} Let $F$ be a $d$-face of $\Delta$. We want to find a walk from $\Delta$ to $H_1$ in the dual graph. If $\operatorname{gap} F= 0$, then $F=H_j$ for some $j$, and $H_1, H_2, \ldots, H_j$ is the desired path. If $\operatorname{gap} F> 0$, let $i \eqdef \min F$. Since $F$ and $H_i$ have same minimum, by Remark \ref{rem:almost} $\Delta$ contains the whole $d$-skeleton of the simplex on $F \cup H_i$. But the $d$-skeleton of a higher-dimensional simplex is strongly-connected, which means that in the dual graph of $\Delta$ we can walk from $F$ to $H_i$. And since $H_i$ has gap $0$, we can walk from it to $H_1$.
\item{$\Rightarrow$:} Fix a labeling for which $\Delta$ is (almost-)closed. We are going to show by induction on $j$ that with the \emph{same} labeling, every $H_j$ is in $\Delta$. For $j=1$, since $\Delta$ is pure, it contains a face $F=a_0 a_1 \cdots a_d$ with $a_0=1$, and then it is easy to derive (either directly, or using that the labeling satisfies the under-closed condition by Theorem \ref{thm:Hierarchy}) that $H_1$ is in $\Delta$.
Now suppose that $\Delta$ contains $H_j$ and let us show that $\Delta$ contains $H_{j+1}$. By Lemma \ref{lem:closed}, $\Delta$ has a $d$-face $H'$ that contains $d+j+1$ and is adjacent to $H_j$. Such $H'$ has the same vertices of $H_j$, with the exception of a single vertex $i$ that was replaced by $d+j+1$. Now either $i=j$, in which case $H' = H_{j+1}$ and we are done; or $i > j$. If $i> j$, then $j$ was not replaced, so it is still present in $H'$. Hence $H'$ and $H_j$ are adjacent faces with the same minimum, namely, $j$. By Remark \ref{rem:almost}, this implies that $H_{j+1}$ is in $\Delta$. \qedhere
\end{compactdesc}
\end{proof}
\begin{remark} \label{rem:WCareSC} If the ``unit-interval'' assumption is weakened to ``under-closed'', then the direction ``$\Rightarrow$'' of Theorem \ref{thm:CTSC} no longer holds, with $K_{1,3}$ playing the usual role of the counterexample. The direction ``$\Leftarrow$'' instead is still valid. We claim in fact that \textit{all weakly-closed traceable complexes are strongly-connected}. To see this, it suffices to show that from any $d$-face $F$ of positive gap we can walk in the dual graph to some gap-$0$ face. But the weakly-closed definition tells us how to move in the dual graph from $F$ to a face $F'$ of smaller gap than $F$. So if we iterate this, eventually we get from $F$ to a gap-0 face. (The same type of argument is carried out in details in the proof of Theorem \ref{thm:Hi}, item (5), below.) That said, the ``weakly-closed'' assumption is needed for ``$\Leftarrow$''. In fact, for any $d \ge 2$, if $G_d \eqdef \{1, d+2, 2d+3, \ldots, (k-1)d+k, kd+(k+1), \ldots, d^2+d+1\}$, then the traceable $d$-complex with $d^2+d+1$ vertices $\Delta = H_1, H_2, \ldots, H_{d^2 }\,, H_{d^2+1 }\,, G_d\; $ is not strongly-connected. Its dual graph is a path of length $d^2+1$ plus an isolated vertex.
\end{remark}
Generalizing a result by Chen, Chang, and Chang \cite[Theorem 2]{ChChCh}, we can push Theorem \ref{thm:CTSC} a bit further. If $D$ is a simplicial complex obtained from $\Delta$ by deleting some vertices $v_1, \ldots, v_k$, then any labeling of $\Delta$ naturally induces a \emph{compressed labeling} for $D$, just by ordering the vertices of $D$ in the same way as they are ordered inside $\Delta$. For example, if $\Delta=123, 134, 345$, the compressed labeling for $D=\operatorname{del}(2, \Delta)$ is $123, 234$. A priori, this $D$ need not be pure.
\begin{lemma} \label{lem:compressed}
Let $\Delta'$ be a $d$-dimensional simplicial complex obtained by deleting some vertices from a $d$-dimensional simplicial complex $\Delta$.
If $\Delta$ is unit-interval (resp. under-closed, resp. semi-closed), then so is $\Delta'$.
\end{lemma}
\begin{proof}
If the original labeling satisfied the unit-interval (resp. under-closed, resp. semi-closed) condition, so does the compressed labeling.
\end{proof}
\begin{lemma} \label{lem:ChChCh} Let $\Delta$ be a $d$-dimensional strongly-connected simplicial complex, with a labeling that makes it unit-interval. The following are equivalent:
\begin{compactenum}[ \rm (a)]
\item The deletion of $d$ or less vertices, however chosen, yields a $d$-complex that is strongly connected.
\item The deletion of $d$ or less vertices, however chosen, yields a pure $d$-complex that with the compressed labeling is traceable.
\item $\Delta$ contains all faces of gap $\le d$.
\end{compactenum}
\end{lemma}
\enlargethispage{4mm}
\begin{proof}
\begin{compactdesc}
\item{(a) $\Leftrightarrow$ (b):} By Lemma \ref{lem:compressed} the compressed labeling satisfies the unit-interval condition. Via Theorem \ref{thm:CTSC}, we conclude.
\item{(b) $\Rightarrow$ (c):} By deleting zero vertices we notice that $\Delta$ is itself traceable. Let $F=a_0 \cdots a_d $ be any $d$-face of $\Sigma^d_n$ that has gap $\le d$. If $\operatorname{gap}(F)=0$, then $F$ is one of $H_1, \ldots, H_{n-d}$, so $F$ is in $\Delta$ by definition of traceable. Otherwise, set $S_F \eqdef \{ j \notin F \textrm{ such that } a_0 < j < a_d\}$. Let $\Delta'$ be the complex obtained from $\Delta$ by deleting the vertices in $S_F$, which are at most $d$. By assumption, $\Delta'$ is traceable with the ``compressed labeling''. So $\Delta'$ contains a gap-0 face of minimum $a_0$. But by how the compressed labeling is defined, this face has exactly the vertices that in the original labeling for $\Delta$ were called $a_0, a_1, \ldots, a_d$. So $F$ is in $\Delta$.
\item{(c) $\Rightarrow$ (b):} Let $\Delta'$ be the $d$-complex resulting from the deletion.
With the compressed labeling, $\Delta'$ is traceable, because any gap-$0$ $d$-face of $\Delta'$ with the compressed labeling, is a $d$-face of $\Delta$ that had gap $\le d$ in the original labeling. It remains to see that $\Delta'$ is pure. We prove that $\Delta'$ has no facets of dimension $d-1$, leaving the case of facets of even lower dimensions to the reader. We claim that every $(d-1)$-face $\sigma$ of $\Delta$ lies in at least $d+1$ distinct $d$-faces of $\Delta$. From the claim the conclusion follows via the pigeonhole principle: If we delete $d$ vertices, however chosen, then at least one of the $d$-faces containing $\sigma$ will survive the deletion, which implies that $\sigma$ is not a facet in $\Delta'$. \\
So let us prove the claim. Let $\sigma= b_0 \cdots b_{d-1}$. If $b_{d-1} - b_0 -d +1 \eqdef \operatorname{gap} (\sigma) \le d$, then $b_{d-1} + 1 \le b_0 + 2d$. So for each $i$ in the $(d+1)$-element set
\[ T_{\sigma} \eqdef \{ b_0, b_0 + 1, \ldots, b_{d-1}, b_{d-1} + 1, \ldots, b_0 + 2d \} \: \setminus \: \{ b_0, b_1, \ldots, b_{d-1}\}\]
the $d$-face $\sigma \cup \{i\}$ has gap $\le d$, and thus is in $\Delta$ by assumption. If instead $\operatorname{gap} (\sigma) \ge d+1$, we use the unit-interval assumption:
for every $i$ in $S_{\sigma} \eqdef \{ i \notin \sigma \textrm{ such that } b_0 < i < b_{d-1}\}$, the $d$-face $\sigma \cup \{i\}$ is in $\Delta$. So either way the claim is proven. \qedhere
\end{compactdesc}
\end{proof}
\begin{theorem}[Higher-dimensional Chen--Chang--Chang] \label{thm:Hi0}
Let $\Delta$ be a pure $d$-dimensional simplicial complex.
\begin{compactitem}
\item If $\Delta$ is unit-interval and the deletion of $\le d$ vertices, however chosen, yields a strongly-connected $d$-complex, then $\Delta$ is Hamiltonian.
\item If $\Delta$ is weakly-closed and Hamiltonian, the deletion of $\le 1$ vertices, however chosen, yields a strongly-connected $d$-complex.
\end{compactitem}
\end{theorem}
\begin{proof} For the second claim: Up to a cyclic reshuffling, the vertex we wish to delete is $n$. The argument of Remark \ref{rem:WCareSC} yields a dual path in $\Delta$ from each $d$-face $F$ to $H_1$. If $F$ does not contain $n$, none of the $d$-faces in such dual path does, so the path belongs to the dual graph of the deletion of $n$ from $\Delta$.
Now we prove the first claim. By Lemma \ref{lem:ChChCh}, $\Delta$ contains all $d$-faces of gap $\le d$. In particular:
\begin{compactitem}
\item for any odd $i$ such that $1 \le i \le n-2d$, $\Delta$ contains the gap-$d$ face $O_i$ formed by $i$ and by the first $d$ consecutive odd integers after~$i$;
\item for any even $j$ such that $2 \le j \le n-2d$, $\Delta$ contains the gap-$d$ face $E_j$ formed by $j$ and by the first $d$ consecutive even integers after $j$;
\item $\Delta$ contains the gap-$(d-1)$ face $F=1,2,4, \ldots, 2d$
formed by $1$ and by the $d$ smallest even natural numbers;
\item $\Delta$ contains the gap-$(d-1)$ face $G$ formed by the largest even integer $\le n$ and by the $d$ largest odd integers $\le n$.
\end{compactitem}
Now consider the following sequence $\mathfrak{C}$ of $d$-faces in $\Delta$: First all $O_i$'s in increasing order, then $G$, then all $E_j$'s in decreasing order, then $F$. Note that any two $O_i$'s are adjacent, and the last of them is adjacent to $G$; symmetrically, any two $E_j$'s are adjacent, and $F$ is adjacent to $E_2$. We claim that this sequence would form a weakly-Hamiltonian cycle if we relabeled the vertices of $\Delta$ first by listing the odd ones increasingly, and then the even ones decreasingly.
Formally, if $n$ is odd, we introduce the new labeling
\[ \ell_1 \eqdef 1, \, \ell_2 \eqdef 3, \, \ell_3 \eqdef 5, \ldots, \, \ell_{\frac{n+1}{2}} \eqdef n, \, \ell_{\frac{n+1}{2} + 1} \eqdef n-1, \, \ell_{\frac{n+1}{2} + 2}\eqdef n-3, \ldots, \, \ell_{n-1} \eqdef 4, \, \ell_{n} \eqdef 2.\]
And if instead $n$ is even, we introduce the new labeling
\[ \ell_1 \eqdef 1, \, \ell_2 \eqdef 3, \, \ell_3 \eqdef 5, \ldots, \, \ell_{\frac{n}{2}} \eqdef n-1, \, \, \ell_{\frac{n}{2} + 1} \eqdef n, \,\, \ell_{\frac{n+1}{2} + 2} \eqdef n-2, \ldots, \, \ell_{n-1} \eqdef 4, \, \ell_{n} \eqdef 2.\]
Let us set $L_1 \eqdef \ell_1 \ell_2 \cdots \ell_{d+1}$, $L_2 \eqdef \ell_2 \ell_3 \cdots \ell_{d+2}$, and so on.
Then the sequence $\mathfrak{C}$ described above is equal (whether $n$ is even or odd) to
\[ L_1, \, L_2, \, \ldots, \, L_{\lfloor \frac{n+1}{2} \rfloor- (d-1)}, \, L_{\lfloor \frac{n+1}{2} \rfloor + 1}, \, L_{\lfloor \frac{n+1}{2}\rfloor + 2}, \ldots, L_{n - d}, L_{n - (d-1)}.\]
This shows that with the new labeling $\Delta$ is \emph{weakly-}Hamiltonian. It remains to show for $d \ge 2$ that our weakly-Hamiltonian cycle can indeed be `completed' to a Hamiltonian cycle, in the sense that the $L_i$'s that were not mentioned in $\mathfrak{C}$ are anyway contained in $\Delta$. First of all, note that $\Delta$ with the original labeling contained all the $d$-faces of gap $\le d$, so in particular it contained all $d$-faces containing $1$ and with vertex set contained in $F \cup O_1$. This shows that with the new labeling, $L_{n-(d-2)}$, $\ldots$, $L_{n}$ are all in $\Delta$. So it remains to consider the missing $L_i$'s from the `center' of the sequence $\mathfrak{C}$. For the ``$n$ odd'' case (the case for $n$ even is analogous), we have to see whether $\Delta$ contains also the $d-1$ facets
\[ L_{\frac{n+1}{2} - d + 2}, \: L_{\frac{n+1}{2} - d+ 3}, \: \ldots, \: L_{\frac{n+1}{2}}.\]
When we translate these $d$-faces back into the \emph{old} labeling, it is easy to see that the face with the largest gap is the last one, which has gap $d-1$. So all these faces are in $\Delta$ by assumption.
\end{proof}
\begin{example}
Let $\Delta$ be an unit-interval $3$-complex on $n=9$ vertices that contains all tetrahedra with gap $\le 3$. With the notation of Theorem \ref{thm:Hi0} the complex $\Delta$ contains the sequence $\mathfrak{C}$ below:
\[O_1=1357, \: O_2=3579, \:G = 5789, \:E_2=2468, \:F=1246.\] If we relabel the vertices as in the proof of Theorem \ref{thm:Hi0}, the list above becomes
\[L_1, \:L_2, \:L_3, \:L_6, \:L_7.\]
Thus $\Delta$ is weakly-Hamiltonian. To prove that it is Hamiltonian, we need to check that $L_4, L_5$ and $L_8, L_9$ are in $\Delta$. Translated into the original labeling, this means checking that $6789, 4689$ and $1234, 1235$ are in $\Delta$, which is clearly the case because they all have gap $\le 2$.
\end{example}
\begin{remark}
For $d=1$, Theorem~\ref{thm:Hi0} boils down to Chen--Chang--Chang's result that ``unit interval graphs are Hamiltonian if and only if they are $2$-connected'' \cite[Theorem 2]{ChChCh}. The $G_5$ of Remark \ref{rem:G5} is $2$-connected and not Hamiltonian; hence the ``unit-interval'' assumption in the first claim of Theorem~\ref{thm:Hi0} is necessary. As for the second claim, the ``weakly-closed'' assumption is necessary for $d>1$, because we saw in Remark \ref{rem:HnotSC} that some Hamiltonian $d$-complexes are not strongly-connected.
\end{remark}
We may condense most of the results of this chapter in the following summary:
\begin{theorem} \label{thm:Hi}
Let $\Delta$ be a $d$-dimensional simplicial complex.
\begin{compactenum}[\rm (1) ]
\item If $\Delta$ is closed (or unit-interval) and strongly connected, then $\Delta$ is traceable.
\item If $\Delta$ is closed (or unit-interval), and the deletion of $d$ or less vertices, however chosen, yields a strongly connected complex, then $\Delta$ is Hamiltonian.
\item If $\Delta$ is under-closed, it contains $H_1$. If in addition $\Delta$ has a face of minimum $i$ for each $i\in \{2, \ldots, n-d\}$, then $\Delta$ is traceable.
\item If $\Delta$ is semi-closed, then for every face $F=a_0 \cdots a_d$ of $\Delta$ either $H_{a_0}$ or $H_{a_d-d}$ is in $\Delta$.
\item If $\Delta$ is weakly-closed, then $\Delta$ contains at least one of the $H_i$'s. \\If in addition $\Delta$ contains $H_1$, plus a face with minimum $i$ and of gap smaller than $d$ for each $i$ in $\{2, \ldots, n-d\}$, then $\Delta$ is weakly-traceable.
\end{compactenum}
\end{theorem}
\begin{proof}
\begin{compactenum}[\rm (1) ]
\item This is given by Proposition \ref{prop:closed2} and Theorem \ref{thm:CTSC} above.
\item This is given by Proposition \ref{prop:closed2} and Theorem \ref{thm:Hi0} above.
\item By definition of under-closed, if $\Delta$ has a face of minimum $i$, then $\Delta$ contains $H_i$. The fact that $\Delta$ has a face of minimum $1$ follows from the assumption that $\Delta$ is pure.
\item This is straightforward from the definition of semi-closed.
\item Let $F=a_0 a_1 \cdots a_d$ be any facet of $\Delta$ with $\operatorname{gap}(F)>0$. Let $g \notin F$ such that $a_0 < g < a_d$. By definition of ``weakly-closed'', some face $G=b_0 b_1 \cdots b_d$ of $\Delta$ contains $g$, is adjacent to $F$, and has either $b_0 \ne a_0$ or $b_d \ne a_d$. Thus $\operatorname{gap} G < \operatorname{gap} F$. Iterating the process, eventually we find in $\Delta$ a gap-$0$ face, which has to be one of
\[H_{a_0}, \; H_{a_0+1}, \ldots, H_{a_d - d}. \]
As for the second claim: By assumption, $\Delta$ contains $H_1$. Also, $\Delta$ contains $H_{n-d}$, because no other face has minimum $n-d$. Now let $H'=2a_1\cdots a_d$ be a face of $\Delta$ with minimum $2$ and gap $\le d-1$. By the argument above, we know that $\Delta$ must contain at least one of
\[H_2, H_3, \ldots, H_{a_d-d}.\]
Let us call this face $H_{i_2}$. By how $H'$ was chosen,
\[ 2 \le i_2 \le a_d - d = \operatorname{gap}(H')+2 \le d+1.\]
But since $H_1$ contains all vertices from $1$ to $d+1$, in particular it contains $i_2$. So $H_{i_2}$ is incident with $H_1$. Now let $H''=a_0a_1\cdots a_d$ be a face of $\Delta$ with gap smaller than $d$, and minimum $a_0=i_2+1$. Repeating the argument above, $\Delta$ contains one of
\[H_{i_2+1}, H_{i_2+2}, \ldots, H_{a_d-d}.\]
Call this facet $H_{i_3}$; as above, it must intersect $H_{i_2}$. And so on. Eventually, we obtain a list $H_1 = H_{i_1}, H_{i_2} \ldots, H_{i_{k-1}}, H_{i_k} = H_{n-d}$ of facets of $\Delta$ that makes it weakly-traceable.
\qedhere
\end{compactenum}
\end{proof}
\begin{remark}
In the previous theorem, a relabeling was necessary only to prove item (2). For all other items, the \emph{original} labeling was already suitable for the desired conclusion. So for item (1) we proved a slightly stronger statement: ``If $\Delta$ is strongly-connected, then any labeling that makes $\Delta$ unit-interval automatically makes $\Delta$ traceable''. Same for items (3), (4), (5).
\end{remark}
\newpage
\section{Algebraic motivation}
In this section, we review Ene et al's definition of determinantal facet ideals \cite{EneEtAl}. We find out a large class of them that are radical. In fact, we prove the following:
\begin{compactitem}
\item If a simplicial complex is semi-closed, then its determinantal facet ideal has a square-free Gr\"obner degeneration (and in particular is radical), and the quotient by such ideal in positive characteristic is $F$-pure (Theorem \ref{t:s-c-f}).
\item If the simplicial complex is unit-interval, then the natural generators of its determinantal facet ideal form a Gr\"obner basis with respect to a diagonal term order (Theorem \ref{t:a-c-gb}). Moreover, the converse is true if with respect to the same labeling, the simplicial complex is traceable (Theorem \ref{t:a-c-gb1}).
\end{compactitem}
\subsection{A foreword on $F$-pure rings, $F$-split rings, and Knutson ideals}
Let $p$ be a prime number. Let $R$ be a ring of characteristic $p$. Recall that the \emph{Frobenius map} is the ring homomorphism from $R$ to itself that maps an element $r \in R$ to $r^p$.
We denote by $F_*R$ the $R$-module defined as follows: $F_*R \eqdef R$ as additive group, and
$r\cdot x \eqdef r^px$ for all $r\in R$ and $x\in F_*R$. This allows us to view the Frobenius map as a map of $R$-modules,
\begin{eqnarray*}
F: & R\longrightarrow F_*R \\
& r\mapsto r^p.
\end{eqnarray*}
The ring $R$ is reduced if and only if $F$ is injective. So the following definitions are natural:
\begin{definition}
$R$ is \emph{$F$-pure} if $F\otimes 1_M:M\rightarrow F_*R\otimes_R M$ is injective for any $R$-module $M$.
\end{definition}
\begin{definition}
$R$ is \emph{$F$-split} if there exists a homomorphism $\theta:F_*R\rightarrow R$ of $R$-modules such that $\theta\circ F=1_R$. Such a $\theta$ is called an $F$-splitting of $R$.
\end{definition}
If a ring is $F$-split, it is clearly $F$-pure. The converse does not hold in general. However, the two concepts are equivalent in a number of cases, for example:
\enlargethispage{4mm}
\begin{lemma}\label{l:split=pure}
Let $R=\bigoplus_{i\in\mathbb{Z}}R_i$ be a Noetherian graded ring of characteristic $p$ having a unique homogeneous ideal $\mathfrak{m}$ that is maximal with respect to inclusion. Furthermore, assume that the Noetherian local ring $R_0$ is complete. Then the following are equivalent:
\begin{compactenum}[ \rm (a)]
\item $R$ is $F$-split.
\item $R$ is $F$-pure.
\item $F\otimes 1_E: \: E \, \longrightarrow \, F_*R\otimes_RE$ is injective, where $E$ is the injective hull of $R/\mathfrak{m}$.
\end{compactenum}
\end{lemma}
\begin{proof}
$(a)\implies (b)\implies (c)$ are obvious implications. To see $(c)\implies (a)$: the map
\[F\otimes 1_E: \: E \,\longrightarrow \: F_*R\otimes_RE\] is injective if and only if the corresponding map
\[\mathrm{Hom}_R(F_*R,\mathrm{Hom}_R(E,E))\cong \mathrm{Hom}_R(F_*R\otimes_RE,E)\, \longrightarrow \, \mathrm{Hom}_R(E,E)\]
is surjective. Hence, by \cite[Corollary 3.6.7, Proposition 3.6.16, Theorem 3.6.17]{BH93}, the corresponding map $\alpha:\mathrm{Hom}_R(F_*R,R)\rightarrow R$ is surjective. So there exists $\theta\in \mathrm{Hom}_R(F_*R,R)$ such that $\alpha(\theta)=1$. On the other hand, by construction $\alpha(\theta)=\theta(F(1))$, so $\theta \circ F=1_R$.
\end{proof}
Since we want to study homogeneous quotients of a polynomial ring over a field, by Lemma~\ref{l:split=pure} we may as well regard the $F$-split notion and the $F$-pure notion as equivalent.
\medskip
In the following the concept of {\it Knutson ideal} will be fundamental. The name arises from the work of Knutson \cite{Knutson}, later systematically investigated by the second author \cite{Se1}, who extended several properties from $\mathbb{Z}/p\mathbb{Z}$ to any field. The result from \cite{Se1} that we shall need is the following:
\begin{theorem}[{Seccia \cite{Se1}}]\label{t:lisa}
Let $K$ be a field. Let $g\in S=K[x_1,\ldots ,x_n]$ be a polynomial with $\mathrm{in}_<(g)$ square-free for some term order on $S$. Let $\mathcal{C}_g$ be the smallest set of ideals of $S$ containing $(g)$ and such that:
\begin{compactenum}
\item $I\in \mathcal{C}_g \implies I:h\in \mathcal{C}_g$ whenever $h\in S$,
\item $I,J\in \mathcal{C}_g \implies I+J\in \mathcal{C}_g, \ I\cap J\in \mathcal{C}_g$.
\end{compactenum}
If $I\in \mathcal{C}_g$, then $\mathrm{in}_<(I)$, and therefore $I$, is radical. Furthermore, if $I,J\in \mathcal{C}_g$, then $\mathrm{in}_<(I+J)=\mathrm{in}_<(I)+\mathrm{in}_<(J)$ and $\mathrm{in}_<(I\cap J)=\mathrm{in}_<(I)\cap\mathrm{in}_<(J)$. Finally, if $K$ has positive characteristic, $S/I$ is $F$-pure whenever $I\in\mathcal{C}_g$.
\end{theorem}
\begin{example}
It can be shown that, if $g=x_1x_2\cdots x_n$, then $\mathcal{C}_g$ is the set of squarefree monomial ideals.
\end{example}
\subsection{Determinantal facet ideals: basic properties}
Let $d,n$ be positive integers with $d+1\leq n$. Let $S \eqdef K[x_{ij}:i=1,\ldots ,n, j=0,\ldots ,d]$ be a polynomial ring in $(d+1)n$ variables over some field $K$. Set
\[X=\begin{bmatrix}
x_{01}&x_{02}&\ldots &x_{0n}\\
x_{11}&x_{12}&\ldots &x_{1n}\\
\vdots&\vdots&\ldots&\vdots\\
x_{d1}&x_{d2}&\ldots &x_{dn}
\end{bmatrix}.\]
Given $1\leq r\leq d$, and integers $0\leq a_0<a_1<\ldots <a_r\leq d$ and $1\leq b_0<\ldots <b_r\leq n$, an \emph{$(r+1)$-minor of $X$} is any element of the form \[ [a_0a_1\ldots a_r|b_0b_1\ldots b_r] \ \eqdef \ \det \;
\begin{bmatrix}
x_{a_0b_0}&x_{a_0b_1}&\ldots & x_{a_0b_r}\\
x_{a_1b_0}&x_{a_1b_1}&\ldots & x_{a_1b_r}\\
\vdots&\vdots&\ldots&\vdots\\
x_{a_rb_0}&x_{a_rb_1}&\ldots & x_{a_rb_r}\\
\end{bmatrix}.\]
If $r=d$, the row indices are forced to be $a_0=0,a_1=1,\ldots ,a_d=d$. For this reason we denote $[01\ldots d|b_0b_1\ldots b_d]$ simply by $[b_0b_1\ldots b_d]$. The ideal of $S$ generated by the $r+1$-minors of $X$ is denoted by $I_{r+1}(X)$. This ideal defines the variety of $(d+1)n$ matrices with entries in $K$ and with rank at most $r$.
The set $\Pi$ of all the minors of $X$ can be partially ordered by the relation
\[[a_0a_1\ldots a_r|b_0b_1\ldots b_r]\leq [c_0c_1\ldots c_s|d_0d_1\ldots d_s] \iffdef r\geq s, \ \ a_i\leq c_i \mbox{ and } b_i\leq d_i \ \forall \ i=0,\ldots ,s.\]
In particular, for maximal minors the previous definition restricts to
\[[a_0a_1\ldots a_d]\leq [b_0b_1\ldots b_d] \iff a_0\leq b_0, a_1\leq b_1,\ldots ,a_d\leq b_d.\]
It is not our intent to review the theory of {\it Algebras with Straightening Law} here, as the interested reader can learn it directly from the standard source \cite{BrunsVetter}. However, we wish to introduce a few concepts for the sake of clarity. The starting observation is that the polynomial ring $S$ is generated by $\Pi$ as a $K$-algebra. In fact, a basis of $S$ as $K$-vector space is given by
\[\{\pi_1\cdots \pi_m:m\in\mathbb{N}, \ \pi_i\in\Pi, \ \pi_1\leq \pi_2\leq \ldots \leq \pi_m\}.\]
The elements of this $K$-basis are called {\it standard monomials}. It may happen that the product of two standard monomials is not a standard monomial. However, such product will be uniquely writable as $K$-linear combination of standard monomials, which is in some sense compatible with the poset structure on $\Pi$. This is what is known as `Straightening Law'; compare \cite[Theorem 4.11]{BrunsVetter}. What we wish to outline is that the ideals of $S$ generated by poset ideals of $\Pi$ (i.e. subsets $\Omega\subset \Pi$ such that for all $\omega\in \Omega$, $\pi\in\Pi$, $\pi\leq \omega \implies \pi\in\Omega$) are particularly nice.
\begin{example}
For any $r\leq d$, the ideal $I_{r+1}(X)$ is generated by the poset ideal $\Omega_{\geq r+1}$ of all $t$-minors of $X$ with $t\geq r+1$. This $\Omega_{\geq r+1}$ has a unique maximal element, $[d-r\ldots d|n-r\ldots n]$.
\end{example}
Some new notation: if $1\leq i<j\leq n$, by $X_{[i,j]}$ we mean the matrix
\[X_{[i,j]}=\begin{bmatrix}
x_{0i}&x_{0,i+1}&\ldots &x_{0j}\\
x_{1i}&x_{1,i+1}&\ldots &x_{1j}\\
\vdots&\vdots&\ldots&\vdots\\
x_{di}&x_{d,i+1}&\ldots &x_{dj}
\end{bmatrix},\]
so $I_{r+1}(X_{[i,j]})$ is the ideal of $S$ generated by the $r+1$-minors of $X_{[i,j]}$, whenever $r\leq \min\{d,j-i\}$.
Eventually, we say that a term order $<$ on $S$ is a {\it diagonal term order} if, for all $1\leq r\leq d$ and integers $0\leq a_0<a_1<\ldots <a_r\leq d$ and $1\leq b_0<\ldots <b_r\leq n$, $\mathrm{in}_<([a_0a_1\ldots a_r|b_0b_1\ldots b_r])=x_{a_0b_0}x_{a_1b_1}\cdots x_{a_rb_r}$. For example, the lexicographic term order on $S$ extending the linear order of the variables given by $x_{ij}>x_{hk}$ if and only if $i<h$ or $i=h$ and $j<k$ is a diagonal term order. We will use the following result from \cite{St}:
\begin{theorem}[Sturmfels \cite{St}]\label{t:sturmfels}
If $<$ is a diagonal term order, $1\leq i<j\leq n$ and $r\leq\min\{d,j-i\}$, then $\{[a_0a_1\ldots a_r|b_0b_1\ldots b_r]: 0\leq a_0<a_1<\ldots <a_r\leq d \mbox{ and } i\leq b_0<\ldots <b_r\leq j\}$ is a Gr\"obner basis of the $I_{r+1}(X_{[i,j]})$.
\end{theorem}
So far, by a ``simplicial complex on $n$ vertices'' we have always implicitly assumed that each vertex $i=1,\ldots ,n$ appears in the complex. From now on, we will drop this convention, i.e. henceforth a simplicial complex on a set $A$ is also a simplicial complex on any finite set $B\supset A$.
\begin{definition} Let $\Delta$ be a $d$-dimensional simplicial complex on $n$ vertices. Let $K$ be any field. Let $S = K[x_{ij}:i=1,\ldots ,n, j=0,\ldots ,d]$. The \emph{determinantal facet ideal} of $\Delta$ is the ideal
\[J_{\Delta}:=([a_0a_1\ldots a_d]:a_0a_1\ldots a_d \in \Delta)\subset S.\]
\end{definition}
When $d=1$, then $\Delta$ is a graph, and $J_{\Delta}$ is the {\it binomial edge ideal} of $\Delta$. Binomial edge ideals have been intensively studied in the recent literature: Among the many papers on this topic, see for example \cite{HerzogEtAl}, \cite{Ohtani0}, \cite{MaMu}, \cite{Matsuda}. Unlike binomial edge ideals, determinantal facet ideals are not always radical -- not even if the complex is weakly-closed:
\begin{example}\label{ex:notradical}
Consider the weakly-closed $2$-dimensional simplicial complex on five vertices
\[ \Delta = 124, 145, 234, 345.\]
Thus in the polynomial ring with $15$ variables $x_{i,j}$, for $i \in \{0,1,2\}$ and $j \in \{1, \ldots, 5\}$, the ideal $J_{\Delta}$ is generated by the four degree-3 polynomials
\[ -x_{0,4}x_{1,2}x_{2,1} + x_{0,2}x_{1,4}x_{2,1}+x_{0,4}x_{1,1}x_{2,2}-x_{0,1}x_{1,4}x_{2,2}-x_{0,2}x_{1,1}x_{2,4}+x_{0,1}x_{1,2}x_{2,4} \:, \]
\[ -x_{0,5}x_{1,4}x_{2,1}+x_{0,4}x_{1,5}x_{2,1}+x_{0,5}x_{1,1}x_{2,4}-x_{0,1}x_{1,5}x_{2,4}-x_{0,4}x_{1,1}x_{2,5}+x_{0,1}x_{1,4}x_{2,5} \:, \]
\[ -x_{0,4}x_{1,3}x_{2,2}+x_{0,3}x_{1,4}x_{2,2}+x_{0,4}x_{1,2}x_{2,3}-x_{0,2}x_{1,4}x_{2,3}-x_{0,3}x_{1,2}x_{2,4}+x_{0,2}x_{1,3}x_{2,4} \: , \]
\[ -x_{0,5}x_{1,4}x_{2,3}+x_{0,4}x_{1,5}x_{2,3}+x_{0,5}x_{1,3}x_{2,4}-x_{0,3}x_{1,5}x_{2,4}-x_{0,4}x_{1,3}x_{2,5}+x_{0,3}x_{1,4}x_{2,5} \: . \]
It can be checked using the software Macaulay 2 \cite{m2} that $J_{\Delta}$ is not radical.
\end{example}
Determinantal facet ideals are multi-graded. To see this, we endow $S$ with the multi-grading defined by $\deg(x_{ij})={\bf e_j}\in\mathbb{N}^n$ for all $i=0,\ldots ,d, \ j=1,\ldots ,n$. Here ${\bf e_j}$ is the vector with a one in position $j$, and zeroes everywhere else.
With such grading $J_{\Delta}$ is homogeneous, and $S/J_{\Delta}$ admits a multi-graded minimal free resolution
\[0\rightarrow \bigoplus_{{\bf v}\in\mathbb{N}^n}S(-{\bf v})^{\beta_{p,{\bf v}}}\rightarrow \ldots \rightarrow \bigoplus_{{\bf v}\in\mathbb{N}}S(-{\bf v})^{\beta_{1,{\bf v}}}\rightarrow S\rightarrow S/J_{\Delta}\rightarrow 0,\]
where $p$ is the projective dimension of $S/J_{\Delta}$. We set $|{\bf v}|=v_1+\ldots +v_n$ for each ${\bf v}=(v_1,\ldots ,v_n)\in \mathbb{N}^n$; this way the {\it graded Betti numbers} with respect to the standard grading are
\[\beta_{i,j}=\sum_{\substack{{\bf v}\in\mathbb{N}^n \\ |{\bf v}|=j}}\beta_{i,{\bf v}}.\]
In particular, $\mathrm{reg}(S/J_{\Delta})=\max\{|{\bf v}|-i:\beta_{i,{\bf v}}\neq 0\}$. In the next result, inspired by \cite[Lemma 2.1]{MaMu} $\mathrm{supp}({\bf v})=\{i:v_i\neq 0\}\subset [n]$ for each ${\bf v}=(v_1,\ldots ,v_n)\in \mathbb{N}^n$. For each subset $W\subset
\{1,\ldots ,n\}$, by $\Delta_W$ we denote the subcomplex of $\Delta$ induced on $W$.
\begin{proposition}\label{p:betti}
Let $\Delta$ be a $d$-dimensional simplicial complex on $n$ vertices and $W\subset [n]$. Whenever ${\bf v}\in\mathbb{N}^n$ is such that $\mathrm{supp}({\bf v})\subset W$,
\[\beta_{i,{\bf v}}(S/J_{\Delta})=\beta_{i,{\bf v}}(S/J_{\Delta_W}) \ \ \ \forall \ i\in\mathbb{N}.\]
In particular, $\mathrm{reg}(S/J_{\Delta})\geq \mathrm{reg}(S/J_{\Delta_W})$.
\end{proposition}
\begin{proof}
Let $\mathbb{F}$ be the multi-graded minimal free resolution of $S/J_{\Delta}$:
\[\mathbb{F}: 0\rightarrow \bigoplus_{{\bf v}\in\mathbb{N}^n}S(-{\bf v})^{\beta_{p,{\bf v}}}\rightarrow \ldots \rightarrow \bigoplus_{{\bf v}\in\mathbb{N}^n}S(-{\bf v})^{\beta_{1,{\bf v}}}\rightarrow S\rightarrow 0.\]
Consider the complex of multi-graded $S$-modules
\[\mathbb{F}':0\rightarrow \bigoplus_{\substack{{\bf v}\in\mathbb{N}^n \\ \mathrm{supp}({\bf v})\subset W}}S(-{\bf v})^{\beta_{p,{\bf v}}}\rightarrow \ldots \rightarrow \bigoplus_{\substack{{\bf v}\in\mathbb{N}^n \\ \mathrm{supp}({\bf v})\subset W}}S(-{\bf v})^{\beta_{1,{\bf v}}}\xrightarrow{\phi} S\rightarrow 0.\]
The cokernel of $\phi$ is $S/J_{\Delta_W}$, hence all we need to show is that $\mathbb{F}'$ is acyclic. But since the minimal generators of the free $S$-modules in $\mathbb{F}'$ involve only the variables $x_{ij}$ with $j\in W$, to show that $\mathbb{F}'$ is acyclic is enough to show that $\mathbb{F}'_{{\bf u}}$ is acyclic for any ${\bf u}\in\mathbb{N}^n$ with $\mathrm{supp}({\bf u})\subset W$.
On the other hand, for any ${\bf v}\in\mathbb{N}^n$, $S(-{\bf v})_{{\bf u}}$ is nonzero if and only if ${\bf u}-{\bf v}\in\mathbb{N}^n$: in particular $S(-{\bf v})_{{\bf u}}\neq 0$ implies $\mathrm{supp}({\bf v})\subset\mathrm{supp}({\bf u})\subset W$, hence $\mathbb{F}'_{{\bf u}}=\mathbb{F}_{{\bf u}}$ whenever $\mathrm{supp}({\bf u})\subset W$. We conclude since $\mathbb{F}_{{\bf u}}$ is acyclic for any ${\bf u}\in\mathbb{N}^n$.
\end{proof}
\subsection{Many radical and many F-pure determinantal facet ideals}
Let us warm up by studying the algebraic counterpart of the traceability of $\Delta$:
\begin{proposition}
Let $\Delta$ be a traceable $d$-dimensional simplicial complex on $n$ vertices. Then $\mathrm{height}(J_{\Delta})=n-d$. Furthermore, if $J_{\Delta}$ is radical and unmixed, then it admits a square-free initial ideal. If in addition $K$ has positive characteristic, then $S/J_{\Delta}$ is even $F$-pure.
\end{proposition}
\begin{proof}
Let us fix a labeling for which $\Delta$ is traceable. Set \[C \eqdef ([1\ldots d+1],[2\ldots d+2],\ldots ,[n-d\ldots n])\subset J_{\Delta}.\] Let us fix a diagonal term order $<$ on $S$. Note that
\[ \mathrm{in}_<([i\ldots i+d])=x_{0i}x_{1(1+i)}\cdots x_{d(d+i)} \quad \textrm{ and } \quad \mathrm{in}_<([j\ldots j+d])=x_{0j}x_{1(1+j)}\cdots x_{d(d+j)}\]
are coprime if $i\neq j$. So $\{[1\ldots d+1],[2\ldots d+2],\ldots ,[n-d\ldots n]\}$ is a Gr\"obner basis of $C$ and
\[\mathrm{in}_<(C)=(x_{01}x_{12}\cdots x_{d (d+1)}, \: x_{02}x_{13}\cdots x_{d (d+2)}, \; \ldots \; , \: x_{0(n-d)}x_{1(1+n-d)}\cdots x_{dn})\]
is a complete intersection of height $n-d$. Hence $C$ is a complete intersection of height $n-d$ inside $J_{\Delta}$, which implies $\mathrm{height}(J_{\Delta})\geq n-d$. On the other hand $\mathrm{height}(J_{\Delta})\leq n-d$ because $J_{\Delta}$ is contained in $I_{d+1}(X)$, which has height equal to $n-d$. As for the final claim, set $g=[1\ldots d+1]\cdots [n-d\ldots n]$. Notice that $\mathrm{in}_<(g)$ is square-free. Obviously, we also have $C\in \mathcal{C}_g$. But if $J_{\Delta}$ is radical and unmixed, since $\mathrm{height}(J_{\Delta})=\mathrm{height}(C)$ by the previous part, then $J_{\Delta}$ must be of the form $C:h$ for some $h\in S$. Thus $J_{\Delta}\in \mathcal{C}_g$ and we conclude via Theorem \ref{t:lisa}.
\end{proof}
The next lemma will help us identify a large class of complexes whose determinantal facet ideal is indeed radical.
\begin{lemma}\label{l:poset}
Let $1\leq a_0<a_1<\ldots <a_d\leq n$, and $\Gamma_{{\bf a}}$ the simplicial complex generated by the facets $a_0i_1\ldots i_d$ with $i_j\leq a_j$ for all $j=1,\ldots ,d$. Then
\[J_{\Gamma_{{\bf a}}}=I_{d+1}(X_{[a_0,a_d]})\cap I_{d}(X_{[a_0,a_{d-1}]}) \cap I_{d-1}(X_{[a_0,a_{d-2}]})\cap \ldots \cap I_1(X_{[a_0,a_0]}).\]
Analogously, if $\Gamma^{{\bf a}}$ is the simplicial complex generated by the facets $i_0i_1\ldots a_d$ with $i_j\geq a_j$ for all $j=0,\ldots ,d-1$, then
\[J_{\Gamma^{{\bf a}}}=I_{d+1}(X_{[a_0,a_d]})\cap I_{d}(X_{[a_1,a_{d}]}) \cap I_{d-1}(X_{[a_2,a_{d}]})\cap \ldots \cap I_1(X_{[a_{d-1},a_d]}).\]
\end{lemma}
\begin{proof}
Since the two identities are symmetric, we will only prove the first one. The containment `$\subseteq$' is obvious; so let us show `$\supseteq$'.
To make the notation lighter, we make the harmless assumption that $a_0=1$. Note that $J_{\Gamma_{{\bf a}}}$ is generated by a poset ideal, namely by \[\Omega=\{\pi \in \Pi: \pi \leq [a_0\ldots a_d]\}.\] Similarly, for all $j=0,\ldots ,d$, the ideal $I_{j+1}(X_{[1,a_{j}]})$ is generated by the poset ideal
\[\Omega_j=\{\pi \in \Pi: \pi \leq [d-j\ldots d|a_j-j\ldots a_j]\}.\]
Since it is easy to check that $\Omega=\cap_{j=0}^d\Omega_j$, via \cite[Proposition (5.2)]{BrunsVetter} we obtain
\[J_{\Gamma_{{\bf a}}}=I_{d+1}(X_{[1,a_d]})\cap I_{d}(X_{[1,a_{d-1}]}) \cap I_{d-1}(X_{[1,a_{d-2}]})\cap \ldots \cap I_1(X_{[1,1]}). \qedhere\]
\end{proof}
Now, let ${\bm \mu} \in S$ be the product of the minors whose main diagonals are illustrated in the $7\times 13$ matrix below.
\centerline{\includegraphics[height=5.5cm]{fundamentalproduct.jpg}}
More precisely,
\begin{align*}
{\bm \mu} \eqdef & [d|1][d-1,d|1,2]\cdots [1,2,\ldots ,d-1,d|1,2,\ldots ,d-1,d]\cdot \\
& [1,2,\ldots ,d,d+1]\cdots [n-d,n-d+1,\ldots ,n-1,n]\cdot \\
& [n-d+1,n-d+2,\ldots ,n-1,n|0,1,\ldots ,d-2,d-1]\cdots [n-1,n|0,1][n|0].
\end{align*}
The reason we defined ${\bm \mu}$ this way is that if $<$ is a diagonal term order, we have
\[\mathrm{in}_{<}({\bm \mu})=\prod_{i=0}^d\prod_{j=1}^nx_{ij}.\]
Using this ${\bm \mu}$, we are now ready to prove the first main result of this Chapter.
\begin{theorem}\label{t:s-c-f}
Let $\Delta$ be a $d$-dimensional semi-closed simplicial complex on $n$ vertices. Then $J_{\Delta}$ is a radical ideal. Moreover:
\begin{compactenum}[ \rm (1)]
\item For any diagonal term order (compatible with the labeling which makes $\Delta$ semi-closed), $\mathrm{in}(J_{\Delta})$ is a squarefree term ideal.
\item If the field $K$ has positive characteristic, $S/J_{\Delta}$ is $F$-pure.
\end{compactenum}
\end{theorem}
\begin{proof}
We will prove that if $\Delta$ is semi-closed with respect to the given labeling then $J(\Delta)\in \mathcal{C}_{\bm \mu}$, whence both claims follow by Theorem \ref{t:lisa}.
Let $1\leq a_0<a_1<\ldots <a_d\leq n$. Using the notation of Lemma \ref{l:poset}, since $\Delta$ is semi-closed, either $\Gamma_{{\bf a}}$ or $\Gamma^{{\bf a}}$ is contained in $\Delta$ whenever $a_0a_1\cdots a_d\in \Delta$. For any $a_0a_1\cdots a_d\in \Delta$, set
$\Delta_{{\bf a}}=\Gamma_{{\bf a}}$ if $\Gamma_{{\bf a}}\subset \Delta$, and $\Delta_{{\bf a}}=\Gamma^{{\bf a}}$ otherwise. Then
\[\Delta \ = \ \bigcup_{a_0a_1\cdots a_d\in \Delta}\Delta_{{\bf a}}.\]
In particular,
\[ J(\Delta) \ = \ \displaystyle\sum_{a_0a_1\cdots a_d\in \Delta}J(\Delta_{{\bf a}}).\]
Since $\mathcal{C}_{\bm \mu}$ is closed under sums, in order to show that $J(\Delta)\in \mathcal{C}_{\bm \mu}$ we only need to check that each $J(\Delta_{\bf a})\in \mathcal{C}_{\bm \mu}$. To verify this, we use a result in \cite{Se2}: The ideal $I_{r+1}(X_{[ij]})\in \mathcal{C}_{\bm \mu}$ whenever $1\leq i<j\leq n$ and $0\leq r\leq \min\{d,j-i\}$. Since $\mathcal{C}_{\bm \mu}$ is closed under intersections, Lemma \ref{l:poset} guarantees that $J(\Delta_{{\bf a}})\in \mathcal{C}_{\bm \mu}$, as desired.
\end{proof}
\begin{remark}
The assumption ``semi-closed'' is best possible: if we replace it with ``weakly-closed'', the theorem no longer holds, cf.~Example \ref{ex:notradical}. That said, the converse of Theorem \ref{t:s-c-f} is false. To see this, consider the non-weakly-closed complex $U^2_3=124,345,467$ of Figure \ref{fig:ToyExamples1}. If $g=[124][345][467]$ then for a diagonal term order $\mathrm{in}(g)=x_{01}x_{12}x_{24}x_{03}x_{14}x_{25}x_{04}x_{16}x_{27}$, which is squarefree. Obviously $[124],[345],[467]\in \mathcal{C}_g$, hence $J_{\Delta}\in \mathcal{C}_g$. So $\mathrm{in}(J_{\Delta})$ is squarefree, and, in the positive characteristic case, $S/J_{\Delta}$ is $F$-pure by Theorem \ref{t:lisa}. On the other hand, when $d=1$ Theorem \ref{t:s-c-f} is true for all weakly closed graphs, via the main result of Matsuda \cite{Matsuda}. This shows that the techniques used in \cite{Matsuda} do not generalize to higher dimensions.
\end{remark}
\begin{remark}
Suppose that $K$ has positive characteristic. Theorem \ref{t:s-c-f} implies that, whenever $\Omega$ is a poset ideal of $\Pi$ consisting only of maximal minors, then the corresponding ASL is $F$-pure. On the other hand, some ASLs are not $F$-pure, as explained in \cite[Remark 5.2]{KoVa}. We do not know whether all the ASLs on a poset ideal of $\Pi$ are $F$-pure.
\end{remark}
\begin{remark}
In positive characteristic, having a square-free initial ideal or an $F$-pure quotient are unrelated properties. Many ideals, like $I=(x^2+xy+y^2)\subset S \eqdef \mathbb{Z}/p\mathbb{Z}[x,y]$, for $p$ prime, have the property that $S/I$ is $F$-pure even if $\mathrm{in}(I)$ is not square-free for any term order. On the other hand, the binomial edge ideal of a 5-cycle is not $F$-pure in characteristic 2 \cite[Example 2.7]{Matsuda}, even if it admits a squarefree initial ideal.
See \cite{KoVa} for a discussion on the relationship between the two properties of being $F$-pure and having a squarefree initial ideal.
\end{remark}
Theorem \ref{t:s-c-f} allows us to characterize the determinantal facet ideals having a linear resolution: It turns out that there is only one. This extends to all dimensions the result for graphs by Saeedi-Madani and Kiani \cite{SMK}.
\begin{corollary} \label{cor:SMK}
Let $\Delta$ be a pure $d$-dimensional simplicial complex on $n$ vertices.
\[ J_{\Delta} \textrm{ has a linear resolution } \ \Longleftrightarrow \ \Delta = \Sigma^d_n. \]
\end{corollary}
\begin{proof} ``$\Leftarrow$'': If $\Delta$ is the $d$-skeleton of the $(n-1)$-simplex, $J_{\Delta}$ is the ideal of maximal minors of the matrix $X$. This ideal is resolved by the Eagon-Northcott complex \cite{EagonNorthcott}, which is linear. \\
``$\Rightarrow$'': By contradiction, suppose there is a subset $W\subset [n]$ of cardinality $d+2$ such that $\Delta_W$ is not the $d$-skeleton of the $(d+1)$-simplex on $W$. We can re-label the vertices so that $W=\{1,2,\ldots ,d+2\}$ and
\[\Delta_W=12\ldots (d+1), \ 12\ldots d(d+2), \ \ \ \ldots \ \ \ ,\ 1\ldots i(i+2)(i+3)\ldots (d+2)\]
where $2\leq i\leq d$. With respect to such a labeling $\Delta_W$ is semi-closed. So by Theorem \ref{t:s-c-f}, $\mathrm{in}(J_{\Delta_W})$ is a squarefree monomial ideal for any diagonal term order. Hence, by the work of Conca--Varbaro \cite{CV}, $\mathrm{reg}(S/J_{\Delta_W})=\mathrm{reg}(S/\mathrm{in}(J_{\Delta_W}))$. But by Lemma \ref{l:poset}
\[J_{\Delta_W}=I_{i}(X_{[1,i]})\cap I_{d+1}(X_{[1,d+2]}),\]
so by Theorem \ref{t:lisa} $\mathrm{in}(J_{\Delta_W})=\mathrm{in}(I_{i}(X_{[1,i]}))\cap \mathrm{in}(I_{d+1}(X_{[1,d+2]}))$. Via Theorem \ref{t:sturmfels}, it is easy to check that the monomial $(x_{d-i+1,1}x_{d-i+2,2}\cdots x_{d,i})(x_{0,2}x_{1,3}\cdots x_{d,d+2})$ is a minimal generator of $\mathrm{in}(I_{i}(X_{[1,i]}))\cap \mathrm{in}(I_{d+1}(X))$. Hence $\mathrm{in}(J_{\Delta_W})$ has a minimal generator of degree $i+d+1$. In particular,
\[ \mathrm{reg}(S/J_{\Delta})\ge \mathrm{reg}(S/J_{\Delta_W})=\mathrm{reg}(\mathrm{in}(S/J_{\Delta_W}))\geq i+d>d.\]
So by Proposition \ref{p:betti}, $\mathrm{reg}(S/J_{\Delta})\geq \mathrm{reg}(S/J_{\Delta_W})>d$. So $J_{\Delta}$ cannot have a linear resolution.
\end{proof}
\subsection{Determinantal facet ideals defined by a Gr\"obner basis}
If $\Delta$ is a closed simplicial complex, it is easy to see that the minors generating $J_{\Delta}$ form a Gr\"obner basis with respect to a diagonal monomial order, corresponding to the labeling that makes $\Delta$ closed: See \cite{EneEtAl}. In \cite{EneEtAl} it has been incorrectly claimed that the converse of the above statement holds true. The following result, which is a consequence of \cite[Corollary 2.4]{Se2}, shows that there are many other complexes $\Delta$ for which the minors generating $J_{\Delta}$ form a Gr\"obner basis:
\begin{theorem}\label{t:a-c-gb}
Let $\Delta$ be a $d$-dimen\-sional simplicial complex, with a labeling that makes it unit-interval.
The set $\{[a_0\ldots a_d]:a_0\ldots a_d]\}$ is a Gr\"obner basis of $J_{\Delta}$ with respect to any diagonal term order. If in addition the field $K$ has positive characteristic, then $S/J_{\Delta}$ is $F$-pure.
\end{theorem}
\begin{proof}
By definition, $\Delta$ is the union of $d$-skeleta of simplices on consecutive vertices. We can choose these $d$-skeleta to be maximal with respect to inclusion. This yields a decomposition
\[\Delta \ = \ \Sigma_{[i_1,j_1]}^{d} \: \cup \: \Sigma_{[i_2,j_2]}^{d} \: \cup \: \ldots \: \cup \: \Sigma_{[i_l,j_l]}^{d},\]
where $\Sigma_{[i_k,j_k]}^{d}$ denotes the $d$-skeleton of the simplex on vertices $i_k,i_k +1, i_k+2, \ldots, j_k$. Therefore
\[J_{\Delta} \ = \ I_{d+1}(X_{[i_1,j_1]}) \: + \: I_{d+1}(X_{[i_2,j_2]}) \: + \: \ldots \: + \: I_{d+1}(X_{[i_l,j_l]}).\]
So by \cite[Corollary 2.4]{Se2}
\[\mathrm{in}_<(J_{\Delta}) \ = \ \mathrm{in}_< (I_{d+1}(X_{[i_1,j_1]})) \: + \: \mathrm{in}_< (I_{d+1}(X_{[i_2,j_2]})) \: +\: \ldots \: + \: \mathrm{in}_< (I_{d+1}(X_{[i_l,j_l]})).\]
By Theorem \ref{t:sturmfels}, $\{ [a_0, \ldots, a_d] \mid a_0\ldots a_d \in \Delta\}$ is a Gr\"obner basis for $J_\Delta$. Finally, the $F$-purity claim in the case of positive characteristic follows again from \cite[Corollary 2.4]{Se2}.
\end{proof}
\begin{remark}
That the set $\{[a_0\ldots a_d]:a_0\ldots a_d \in \Delta]\}$ is a Gr\"obner basis when $\Delta$ is unit-interval has been independently proved, using a completely different method, in Almousa--Vandebogert \cite[Theorem 2.16]{AV21}. They also obtained the analogous result for $r$-determinantal facet ideals (a more general concept than determinantal facet ideals) of unit-interval simplicial complexes. We were not aware of the paper \cite{AV21} of Almousa and Vandebogert before posting the first version of the present work on the arXiv. (We coordinated efforts to adopt the same name ``unit-interval complexes'' in the two papers.)
For the sake of completeness, we point out that \cite[Corollary 2.4]{Se2} implies that also $r$-determinantal facet ideals of unit-interval simplicial complexes define $F$-pure quotients in positive characteristic. We do not know, however, whether the ($r$-)determinantal facet ideals of ``lcm-closed'' complexes, as defined in \cite{AV21}, or whether those of ``closed complexes'', as defined here, are all $F$-pure.
\end{remark}
\begin{remark} \label{rem:closedNotAlmost}
The converse of Theorem \ref{t:a-c-gb} is false: as explained above, any closed but not unit-interval complex is a counterexample. For a more interesting example, consider
\[ W = 123, \; 124, \; 134, \; 234, \; 235, \; 245, \; 345, \; 568, \; 789, \; 8 \, 10 \, 11\]
corresponding to a one-point union of the $B^2$ and the $U^2_3$ of Figure \ref{fig:ToyExamples1}. This complex $W$ is not unit-interval, not closed, and not even weakly-closed \cite{Pavelka}.
However, one can verify with Macaulay2 \cite{m2} or via \cite[Theorem 2.15]{AV21} that $\{[a_0, a_1, a_2]: a_0\ldots a_d \in \Delta]\}$ form a Gr\"obner basis of $J_{W}$ for any diagonal term order.
\end{remark}
\begin{remark}\label{rem:correction}
Two of the results of \cite{EneEtAl} are incorrect because of the following counterexamples. As we already mentioned, the complex $B_d$
of Lemma \ref{lem:Bd} (cf.~Figure \ref{fig:ToyExamples1}) is not closed, but the set of all the minors $[abc]$, where $abc$ ranges over all facets of $B^d$, is a Gr\"obner basis of $J_{B^d}$ for any diagonal term order by Theorem \ref{t:a-c-gb}. Thus one direction of \cite[Theorem 1.1]{EneEtAl} is incorrect for all $d>1$. Moreover, the graph $G_0 = 12, 13, 23, 24, 34$
is closed, but one can verify that $S/J_{G_0}$ is not Cohen-Macaulay. Thus \cite[Corollary 1.3]{EneEtAl} is incorrect already for $d=1$.
\end{remark}
The final part of our work is dedicated to the delicate quest for some partial converse for Theorem \ref{t:a-c-gb}.
To increase the chances of success, we restrict ourselves to \emph{traceable} complexes. The traceable assumption is rather natural in this case,
as we have anyway seen in Theorem \ref{thm:CTSC} that all strongly-connected unit-interval complexes are traceable.
We start off with a Lemma:
\begin{lemma}\label{l:gb-ac}Let $\Delta$ be a simplicial complex such that $\mathcal{GB} \eqdef \left\lbrace [a_0,\ldots,a_d] \mid a_0\ldots a_d \in \Delta \right\rbrace$ is a Gr\"obner basis of $J_{\Delta}$ for some diagonal term order. Let $F= a_0\ldots a_d$ and $G=b_0 \ldots b_d$ be two facets of $\Delta$. If for some integer $l \in \{0,\ldots d-1\}$
\begin{compactitem}
\item[\rm (i)] $a_i=b_i$ for all $i \in \{0,\ldots, l\}$,
\item[\rm (ii)]$a_{l+1}> a_l +1$,
\item[\rm (iii)] $b_{l+k}= b_l +k$ for all $k\geq 1$,
\end{compactitem}
then the facet $a_0 \ldots a_{l-1}(a_l +1) \ a_{l+1} \ldots a_d$ is also in $\Delta$.
Symmetrically, if for some $l \in \{1,\ldots d\}$
\begin{compactitem}
\item[\rm (iv)] $a_i=b_i$ for all $i \in \{l,\ldots, d\}$,
\item[\rm (v)]$a_{l-1}< a_l -1$,
\item[\rm (vi)] $b_{l-k}= b_l -k$ for all $k \in \{1, \ldots, l\}$,
\end{compactitem}
then the facet $a_0 \ldots a_{l-1} (a_l -1) \ a_{l+1} \ldots a_d$ is also in $\Delta$.
\end{lemma}
\begin{proof}
It is harmless to assume that the term order $<$ is the lexicographic term order defined before, cf.~Theorem \ref{t:sturmfels}.
Let $F$ and $G$ be two facets of $\Delta$ satisfying (i), (ii) and (iii). Let us compute the initial term of the polynomial \[f\eqdef [l+1 \ldots d \mid a_{l+1} \ldots a_d][b_0 \ldots b_d]- [l+1 \ldots d \mid b_{l+1} \ldots b_d][a_0 \ldots a_d].\]
If we set
\begin{equation*}
\begin{split}
p&\eqdef [a_0 \ldots a_d], \ \ p'\eqdef [l+1 \ldots d \mid a_{l+1} \ldots a_d]\\
q&\eqdef [b_0 \ldots b_d], \ \ \ q' \eqdef [l+1 \ldots d \mid b_{l+1} \ldots b_d]
\end{split}
\end{equation*}
then $f=p'q-pq'$, and by Laplace expansion we have
\begin{equation*}
\begin{split}
p'q&=\underbrace{(x_{0 b_0}\cdots x_{l-1 b_{l-1} } x_{l b_{l}} p'q')}_{h}+ g_1, \quad \mu_1 < \alpha \ \forall \ \mu_1 \in \mathrm{supp} (g_1), \ \forall \ \alpha \in \mathrm{supp} (h), \\
pq'&=\underbrace{(x_{0 a_0}\cdots x_{l-1 a_{l-1} } x_{l a_{l}} p'q')}_{h}+ g_2,\quad \mu_2 < \alpha \ \forall \ \mu_2 \in \mathrm{supp} (g_2), \ \forall \ \alpha \in \mathrm{supp} (h).\\
\end{split}
\end{equation*}
Furthermore
\begin{equation*}
\begin{split}
\mathrm{in}_< (g_1)=( x_{l+1 a_{l+1}} \cdots x_{d a_d})(x_{0 b_0}\cdots x_{l-1 b_{l-1} } x_{l b_{l+1}} x_{l+1 b_l} x_{l+2 b_{l+2}} \ldots x_{d b_d}),\\
\mathrm{in}_<(g_2)=( x_{l+1 b_{l+1}} \cdots x_{d b_d})(x_{0 a_0}\cdots x_{l-1 a_{l-1} } x_{l a_{l+1}} x_{l+1 a_l} x_{l+2 a_{l+2}} \ldots x_{d a_d}).
\end{split}
\end{equation*}
Since $\mathrm{in}_<(g_2)$ is smaller than $\mathrm{in}_<(g_1)$, we conclude that\[\mathrm{in}_< (f)=\mathrm{in}_< (g_1-g_2)=( x_{l+1 a_{l+1}} \cdots x_{d a_d})(x_{0 b_0}\cdots x_{l-1 b_{l-1} } x_{l b_{l+1}} x_{l+1 b_l} x_{l+2 b_{l+2}} \ldots x_{d b_d}).\]
In addition $f \in J_\Delta$ because $F, G \in \Delta $. Thus, there must be a minor $g=[c_0 \ldots c_d]$ in $\mathcal{GB}$ such that $\mathrm{in}_<(g)$ divides $\mathrm{in}_<(f)$. Note that for $c_0, \ldots,c_l$ we only have one option, namely,
\begin{equation*}
\left\{
\begin{array}{ll}
c_0 &= b_0=a_0\\
& \ \vdots \\
c_{l-1}&=b_{l-1}=a_{l-1}\\
c_l&=b_{l+1}=b_l+1=a_l+1.
\end{array} \right.
\end{equation*}
For $c_{l+1}$ we have a priori two possibilities: either $c_{l+1}= b_l$ or $c_{l+1}=a_{l+1}$. But $b_l < b_{l+1}=c_l$, so it must be $c_{l+1}=a_{l+1}$. Similarly, for $c_{l+2}$ we have a priori two options: Either $c_{l+2}= b_{l+2}$, or $c_{l+2}=a_{l+2}$. But by the assumptions, we have that $b_{l+2}\leq a_{l+1}=c_{l+1}$, so since $c_{l+2}>c_{l+1}$ it must be $c_{l+2}=a_{l+2}$.
In general, for any $k\geq 2$ we have $b_{l+k}\leq a_{l+k-1}=c_{l+k-1}$. Since $c_{i}>c_{i-1}$, arguing recursively we obtain that the only possible option is $c_{l+k}=a_{l+k}$ for all $k\geq 2$.
Hence we have proved that
\[g=[c_0,\ldots,c_d]=[a_0\ldots a_{l-1} (a_l+1) a_{l+1} \ldots a_d].\]
Since $g$ is an element of $\mathcal{GB}$, we conclude that $a_0\ldots a_{l-1} (a_l+1) a_{l+1} \ldots a_d \in \Delta$.
The proof of the second part of the lemma is symmetric; namely, one considers the polynomial
\[f' \eqdef [0 \ldots l-1 \mid a_{0} \ldots a_{l-1}][b_0 \ldots b_d]- [0 \ldots l-1 \mid b_{0} \ldots b_{l-1}][a_0 \ldots a_d] \in J_{\Delta}\] whose leading term is
\[\mathrm{in}_< (f')=( x_{0 a_{0}} \cdots x_{l-1 a_{l-1}})(x_{0 b_0}\cdots x_{l-2 b_{l-2} } x_{l-1 b_{l}} x_{l b_{l-1}} x_{l+1 b_{l+1}} \ldots x_{d b_d}),\]
and one proceeds analogously to the argument above.
\end{proof}
\begin{theorem}\label{t:a-c-gb1}
Let $\Delta$ be a $d$-dimensional simplicial complex. If with respect to the same labeling $\Delta$ is traceable and the set $\{[a_0\ldots a_d]:a_0\ldots a_d]\}$ is a Gr\"obner basis of $J_{\Delta}$ with respect to some diagonal term order, then such labeling makes $\Delta$ unit-interval.
\end{theorem}
\begin{proof}
Let $F=a_0 \ldots a_d$ be a facet of $\Delta$ with $\operatorname{gap} (F)=k$. We proceed by induction on $k$. For $k=0$ there is nothing to prove, so we assume $k>0$.
Let $g_1, \ldots, g_k$ be the vertices not in $F$, and such that $a_0<g_1< \ldots <g_k<a_d$. We want to show that $\Delta$ contains the $d$-skeleton of $\{a_0, \ldots,a_d,g_1,\ldots,g_k\}$. The strategy is to first show that $\Sigma_{[a_0+1,a_d]}^{d}, \Sigma_{[a_0,a_d-1]}^{d} \subseteq \Delta$ by inductive assumption, and then to prove that $\Delta$ contains also the facets of the form $a_0 c_1 \ldots c_{d-1} a_d$. So let us proceed. Let $l$ be the greatest integer such that $a_l <g_1$, so that $g_1=a_l+1$. Consider the two facets $F$ and $H_{a_0}$ of $\Delta$.
They satisfy the assumptions of Lemma \ref{l:gb-ac}, so
\[F'=a_0 \ldots a_{l-1} \ g_1 \ a_{l+1} \ldots a_d \in \Delta.\]
If $l=0$, then $\operatorname{gap}(F')=k-1$, so by the inductive assumption $\Sigma_{[g_1,a_d]}^{d}=\Sigma_{[a_0+1,a_d]}^{d}\subset \Delta$. Otherwise, since $\operatorname{gap} (F')=k$, we cannot apply the inductive assumption yet. However, we have \lq \lq shifted" the first gap to the left and now the first missing vertex is $a_l=a_{l-1}+1$. We can apply again Lemma \ref{l:gb-ac} to the facets $F'$ and $H_{a_0}$ and we get
\[F''=a_0 \ldots a_{l-2} \ a_l \ g_1 \ a_{l+1} \ldots a_d \in \Delta.\]
If $l=1$, then $\operatorname{gap}(F'')=k-1$, so by the inductive assumption $\Sigma_{[a_1,a_d]}^{d}=\Sigma_{[a_0+1,a_d]}^{d}\subset \Delta$. Otherwise, once again $\operatorname{gap} (F'')=k$ and the first missing vertex $a_{l-1}=a_{l-2}+1$ has been shifted by one to the left.
Iterating this procedure, we eventually get that \[(a_{0}+1)\ldots a_{l}\ g_1 \ a_{l+1} \ldots a_d \in \Delta.\]
This face has gap equal to $k-1$ and we can finally apply induction: We get $\Sigma_{[a_0+1,a_d]}^{d} \subseteq \Delta$.\par
To prove that $\Sigma_{[a_0,a_d-1]}^{d} \subseteq \Delta$ we use a similar argument. Let $l$ be the smaller integer such that $g_k<a_l$, so that $g_k=a_l-1$, and consider the two facets of $\Delta$
\begin{equation*}
\begin{split}
F&=\ \ \ a_0 \ldots \ a_{l-1}\ a_l\ldots a_d\\
H^{a_d} \eqdef H_{a_d-d}&=(a_d-d)(a_d-d+1) \ldots g_k \ a_l\ldots \ a_d.
\end{split}
\end{equation*}
Iteratively applying the second part of Lemma \ref{l:gb-ac}, we can shift the last missing vertex to the right until we reach the facet
\[a_0 \ldots a_{l-1} g_k a_l \ldots a_d -1 \in \Delta,\]
which has gap $k-1$. So by induction $\Sigma_{[a_0,a_d-1]}^{d} \subseteq \Delta$.\par
It remains to prove that all the facets of the form $G=a_0 c_1 \ldots c_{d-1} a_d $ are in $\Delta$. To do so, we start from $F=a_0 a_1 \ldots a_d$ and we replace one by one each $a_i$ with the corresponding $c_i$. In detail: For $i=1$, we have three possibilities:
\begin{compactitem}
\item $c_1=a_1$, or
\item $a_0< c_1 < a_1$, or
\item $c_1 > a_1$.
\end{compactitem}
If $c_1=a_1$ there is nothing to do. If $a_0< c_1 < a_1$, consider the two facets
\begin{equation*}
\begin{split}
F&= \ \ \ a_0 \ \ \ a_1 \ldots a_d\\
\widetilde{F}&=(a_1-1) \ a_1 \ldots a_d.
\end{split}
\end{equation*}
Since $a_0< c_1 < a_1$, we have that $a_1-1 > a_0$. Hence $\widetilde{F} \in \Sigma_{[a_0+1 ,a_d]}^{d} \subseteq \Delta$. So by Lemma \ref{l:gb-ac} $a_0 (a_1-1 ) a_2 \ldots a_d \in \Delta$. If $c_1=a_1-1$ we stop, otherwise we repeat the same argument. At each iteration of Lemma \ref{l:gb-ac}, the second vertex in the facet decreases by one unit; eventually, we obtain that $a_0 c_1 a_2 \ldots a_d \in \Delta$.
As for the third possibility ($c_1 > a_1$), we claim that we can simply dismiss it without loss of generality.
In fact, for every $i\in \{1,\ldots,d\}$, we can always ``flatten all the vertices after $a_{i-1}$ to the right'': that is, we can always replace $F$ with another face in $\Delta$ of the form
\[F_i=a_0 a_1\ldots a_{i-1} (a_d-d-i) \ldots (a_d-1) a_d.\]
To see it, let $0\leq l\leq d-1$ be the largest index for which $a_l+1<a_{l+1}$ (such an $l$ must exist because $\operatorname{gap} (F)>0$). Applying Lemma \ref{l:gb-ac} to the facets $F$ and $\widetilde{F}=a_0\ldots a_{l}(a_{l}+1)(a_l+2)\ldots a_l+(d-l)$ in $\Sigma_{[a_0,a_d-1]}^{d}\subseteq \Delta$, we get that the facet $a_0 \ldots a_{l-1}(a_l+1)a_{l+1}\ldots a_d$ is in $\Delta$. Proceeding this way we end up with the face
\[F_l=a_0 \ldots a_{l-1}(a_{l+1}-1)a_{l+1}\ldots a_d\in \Delta.\]
Replacing $F$ with $F_{l}$, and arguing the same way, we infer that $F_i\in\Delta$ for all $i=0,\ldots ,d-1$.
In particular, for $i=1$, we could replace $F$ with a face with same minimum and maximum \[ F_1=a_0 \ (a_d-d+1) \ (a_d-d+2) \ldots \ a_d\in \Delta.\] Note that $c_1 \le a_d-d+1$. So our claim is proven: Up to replacing $F$ with $F_1$, we can assume that $c_1 \le a_1$.
So the case $i=1$ is settled. Consider now $i=2$. If $c_2=a_2$, there is nothing to do. Otherwise, flattening the vertices after $c_1$ of $a_0c_1a_2\ldots a_d$ to the right, we may assume that $c_2 < a_2$. Consider the two facets
\begin{equation*}
\begin{split}
F &= \ \ \ a_0 \ \ \ \ \ c_1 \ \ \ \ a_2 \ldots a_d \in \Delta\\
\widetilde{F} &=(a_2-2) \ (a_2-1) \ a_2\ldots a_d \in \Delta.
\end{split}
\end{equation*}
Since $c_2 < a_2$, we have that $c_1 < a_2-1$, so applying Lemma \ref{l:gb-ac} we obtain that \[a_0 \, c_1 \, (a_2-1) \, a_3 \ldots a_d \in \Delta.\]
If $c_2=a_2-1$ we stop, otherwise we repeat the same argument. At every iteration of Lemma \ref{l:gb-ac}, the third vertex in the facet decreases by one unit; eventually, we obtain that $a_0 c_1 c_2 a_3 \ldots a_d \in \Delta.$
Iterating this procedure for all $i$'s, we conclude that
\[G=a_0 c_1 c_2 \ldots c_{d-1} a_d \in \Delta. \qedhere\]
\end{proof}
\begin{remark} \label{rem:lcm} Very recently Almousa and Vandebogert \cite{AV21} introduced a technical property of simplicial complexes, called ``lcm-closed'', that simultaneously generalizes the two properties of being ``closed'' and being ``unit-interval''. They asked \cite[Question 2.19]{AV21} whether such property for simplicial complexes would characterize the fact that the minors of the determinantal facet ideal form a Gr\"obner basis with respect to any diagonal term order. With a little ingenuity, one can see that for traceable complexes, ``lcm-closed'' is simply equivalent to ``unit-interval''. Thus Theorem \ref{t:a-c-gb1} answers Almousa--Vandebogert's question positively, for complexes that with respect to the same labeling are traceable.
\end{remark}
| {
"timestamp": "2021-04-13T02:34:29",
"yymm": "2101",
"arxiv_id": "2101.09243",
"language": "en",
"url": "https://arxiv.org/abs/2101.09243",
"abstract": "We study d-dimensional generalizations of three mutually related topics in graph theory: Hamiltonian paths, (unit) interval graphs, and binomial edge ideals. We provide partial high-dimensional generalizations of Ore and Posa's sufficient conditions for a graph to be Hamiltonian. We introduce a hierarchy of combinatorial properties for simplicial complexes that generalize unit-interval, interval, and co-comparability graphs. We connect these properties to the already existing notions of determinantal facet ideals and Hamiltonian paths in simplicial complexes. Some important consequences of our work are:(1) Every almost-closed strongly-connected d-dimensional simplicial complex is traceable. (This extends the well-known result \"unit-interval connected graphs are traceable\".)(2) Every almost-closed d-complex that remains strongly connected after the deletion of d or less vertices, is Hamiltonian. (This extends the fact that \"unit-interval 2-connected graphs are Hamiltonian\".)(3) Unit-interval complexes are characterized, among traceable complexes, by the property that the minors defining their determinantal facet ideal form a Groebner basis for a diagonal term order which is compatible with the traceability of the complex. (This corrects a recent theorem by Ene et al., extends a result by Herzog and others, and partially answers a question by Almousa-Vandebogert.)(4) Only the d-skeleton of the simplex has a determinantal facet ideal with linear resolution. (This extends the result by Kiani and Saeedi-Madani that \"only the complete graph has a binomial edge ideal with linear resolution\".)(5) The determinantal facet ideals of all under-closed and semi-closed complexes have a square-free initial ideal with respect to lex. In characteristic p, they are even F-pure.",
"subjects": "Combinatorics (math.CO); Commutative Algebra (math.AC)",
"title": "Hamiltonian paths, unit-interval complexes, and determinantal facet ideals",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9888419664501421,
"lm_q2_score": 0.7154239957834733,
"lm_q1q2_score": 0.7074412708361478
} |
https://arxiv.org/abs/1711.05354 | On the Numerical Solution of Fourth-Order Linear Two-Point Boundary Value Problems | This paper introduces a fast and numerically stable algorithm for the solution of fourth-order linear boundary value problems on an interval. This type of equation arises in a variety of settings in physics and signal processing. Our method reformulates the equation as a collection of second-kind integral equations defined on local subdomains. Each such equation can be stably discretized and solved. The boundary values of these local solutions are matched by solving a banded linear system. The method of deferred corrections is then used to increase the accuracy of the scheme. Deferred corrections requires applying the integral operator to a function on the entire domain, for which we provide an algorithm with linear cost. We illustrate the performance of our method on several numerical examples. | \section{Introduction}
This paper describes the numerical solution to differential equations of the form
\begin{align} \label{eq:main_prob}
\sum_{j=0}^{4}a_j(x) \frac{d^j \phi}{d x^j}(x) = f(x)
\end{align}
for $x$ in an interval $[a,b]$, with specified boundary conditions
\begin{align}
\phi(a) &= \alpha_{l,0} \label{bdry1}\\
\phi(b) &= \alpha_{r,0} \label{bdry2}\\
\phi^\prime(a) &= \alpha_{l,1} \label{bdry3}\\
\phi^\prime(b) &= \alpha_{r,1} \label{bdry4}
\end{align}
and given coefficients $a_j(x)$ and right hand side $f(x)$.
Fourth-order equations of this kind arise in a variety of physical problems. In the small-bending regime, the shape of a beam under an external force is described as the solution to an equation of the form
\begin{align}
\frac{d^2}{d x^2} \left( c(x) \frac{d^2 \phi}{d x^2}(x) \right) = f(x)
\end{align}
where $c(x)$ is the stiffness of the beam, and $f(x)$ is an external force applied to the beam \cite{gere-mechanics-1984}. One-dimensional equations also arise from separation-of-variables for higher-dimensional problems, such as vibration of plates \cite{courant-methods-1937}. A notable fourth-order operator on the unbounded domain $[0,\infty)$ is defined by:
\begin{align}
-\frac{d^2}{dx^2}\left(x^2 \frac{d^2 \phi}{dx^2}(x)\right)
+ (a^2 + b^2)\frac{d}{dx} \left(x^2 \frac{d \phi}{dx}(x)\right)
- (a^2 b^2 x^2 - 2a^2)\phi(x).
\end{align}
This operator arises in many applications because it commutes with the truncated Laplace transform composed with its adjoint \cite{grunbaum1, grunbaum2}. It is useful when working with the family of decaying exponential functions, and has been the subject of recent investigation \cite{roy-thesis,lederman-laplace-i,lederman-laplace-ii}.
Existing methods for solving fourth-order boundary value problems of the form \eqref{eq:main_prob}--\eqref{bdry4} employ finite difference or finite element schemes. To obtain the solution on $m$ nodes, these methods involve solving an $O(m)$-by-$O(m)$ banded system of linear equations. While the solution can be obtained with asymptotic cost $O(m)$, the discretizations used introduce a condition number of size $O(m^4)$ \cite{strang-fem-2008}. The resulting loss of accuracy in the solution is entirely due to the choice of discretization, and is not a result of the conditioning inherent to the problem \eqref{eq:main_prob}--\eqref{bdry4}.
This paper proposes an algorithm for the solution of \eqref{eq:main_prob}--\eqref{bdry4} that maintains the $O(m)$ running time of finite element and finite difference schemes, but is as accurate as the problem allows. Our method centers on expressing the solution $\phi$ in the form
\begin{align}
\phi(x) = \int_{a}^{b} G_0(x,t) \sigma(t) dt + \psi_\alpha(x),
\end{align}
where $G_0(x,t)$ is the Green's function for the biharmonic equation $\phi^{(4)} = f$ with zero boundary values; $\psi_\alpha$ is a third degree polynomial with the desired boundary values; and $\sigma = \phi^{(4)}$ is the new function to be solved for.
The key observation is that the function $\sigma$ can be expressed as the solution to a second-kind integral equation. While direct discretizations of the differential equation \eqref{eq:main_prob}--\eqref{bdry4} are ill-conditioned, second-kind integral equations can be stably discretized. More precisely, the values of $\sigma$ on a grid of points is expressible as the solution to a linear system whose condition number does not markedly exceed the condition number of the original continuous problem.
The challenge with using second-kind integral equations is that while their discretized linear systems are well-conditioned, they are dense; consequently, a naive solver will have cubic cost $O(m^3)$. It has been observed that for many physical problems, these dense linear systems can nevertheless be solved in linear or nearly linear time. This is the observation underpinning a variety of methods devised for the solution of second-order two-point boundary value problems \cite{starr-bvp,greengard-two-point,lee-stiff}.
This paper employs a more direct approach to solving for $\sigma$. Briefly, our method contains three steps. First, we use the integral equation form of the problem to produce $m$ local solutions of the equation \eqref{eq:main_prob} with homogeneous boundary values. Next, we solve a banded linear system to match the boundary values of the local solutions. While this step introduces extraneous loss of accuracy similar to a finite element scheme, it is corrected by the method of deferred corrections, in which we recursively solve for the residual solutions on the entire interval $[a,b]$. Computing the right hand side of the residual equation requires applying the integral operators defined by the biharmonic Green's function and its derivatives to an arbitrary function on the entire interval $[a,b]$, for which we provide a linear time algorithm. This linear time algorithm is modeled after the fast multipole method \cite{greengard-fast} for applying certain dense matrices to vectors.
The asymptotic CPU time of our algorithm is $O(m n^3 \log(1/\epsilon))$, where $\epsilon$ is machine precision, $n$ is a user-selected integer (the number of points for each local solution), and $m$ is the number of local solutions, or discretization nodes. The complexity of the algorithm is linear in the number of discretization nodes, but nevertheless achieves full machine accuracy. In this sense, our algorithm realizes the advantages of finite element methods' small CPU time while maintaining the numerical accuracy afforded by second-kind integral equations.
The rest of the paper is structured as follows. In Section \ref{sec:prelim}, we review the mathematical and numerical tools we will be using in our algorithm. In Section \ref{sec:algorithm}, we describe in detail the algorithm for solving \eqref{eq:main_prob}--\eqref{bdry4}. In Section \ref{sec:experiments}, we provide the results of numerical experiments.
\section{Mathematical and numerical preliminaries} \label{sec:prelim}
In this section, we review the mathematical and numerical tools that we will be using throughout the paper. In particular, we will show how to express the function $\phi$ which solves \eqref{eq:main_prob}--\eqref{bdry4} in terms of the solution to a second-kind integral equation; review the properties of Gaussian quadrature, which we will use to discretize this integral equation; and review the method of deferred corrections.
\subsection{Rescaling the problem domain} \label{sec:rescale}
It will be convenient to rescale the problem \eqref{eq:main_prob}--\eqref{bdry4} to convert the interval $[a,b]$ into $[-1,1]$. Define:
\begin{align}
\tilde{\phi}(y) &= \phi\left(\frac{(b-a)(y+1)}{2}+a\right) \\
\tilde{a}_j(y) &= \left( \frac{2}{b-a} \bigg)^j
a_j \bigg( \frac{(b-a)(y+1)}{2} + a \right) \\
\tilde{f}(y) &=f \left( \frac{ (b-a) (y+1) }{2 } + a \right).
\end{align}
The equations \eqref{eq:main_prob}--\eqref{bdry4} for $\phi$ are then equivalent to the equation
\begin{align} \label{eq:rescale}
\sum_{i=0}^{4} \tilde{a}_j(y) \frac{d^j \tilde{\phi}}{dy^j}(y)
= \tilde{f}(y)
\end{align}
with modified boundary values
\begin{align}
\tilde{\phi}(- 1) &= \alpha_{l,0} \\
\tilde{\phi}(1) &= \alpha_{r,0} \\
\tilde{\phi}^\prime(-1) &= \left(\frac{b-a}{2}\right)\alpha_{l,1} \\
\tilde{\phi}^\prime(1) &= \left(\frac{b-a}{2}\right)\alpha_{r,1} .
\end{align}
After solving for $\tilde{\phi}$ and its derivatives, we can perform the inverse change of variables to arrive at the solution $\phi$ and its derivatives, as follows:
\begin{align} \label{eq:inverse_rescale}
\phi^{(j)}(x) = \left(\frac{2}{b-a}\right)^j
\tilde{\phi}^{(j)} \left( \frac{2(x-a)}{b-a} - 1 \right).
\end{align}
\subsection{The biharmonic Green's function on $[-1,1]$}
We will make central use of the Green's function $G_0(x,t)$ for the biharmonic equation
$\phi^{(4)} = f$ with homogeneous boundary conditions ($\alpha_{l,0} = \alpha_{r,0} = \alpha_{l,1} = \alpha_{r,1} = 0$ in \eqref{bdry1}--\eqref{bdry4}), on the interval $[-1,1]$. The biharmonic Green's function is given by the formula:
\begin{align} \label{eq:Green}
G_0(x,t) =
\begin{cases}
(1-t)^2(1+x)^2(1+2t-2x-tx) / 24, \,\,\, \text{ if } t>x\\
(1-x)^2(1+t)^2(1+2x-2t-tx) / 24, \,\,\,\text{ if } t<x\\
\end{cases}
\end{align}
It can be checked by direct calculation that $G_0(x,t)$ satisfies the defining properties of the Green's function (see \cite{courant-methods-1937}). Specifically, the following properties hold:
\begin{align}
\frac{\partial^4 G}{\partial x^4}(x,t) = 0, \quad t \in [-1,1],
\end{align}
\begin{align}
G_0(-1,t) = G_0(1,t) = \frac{\partial G_0}{\partial x} (-1,t)
= \frac{\partial G_0 }{\partial x}(1,t) = 0,
\quad t \in [-1,1],
\end{align}
\begin{align}
\lim_{t \to x^+}\frac{\partial^j G}{\partial x^j} (x,t)
= \lim_{t \to x^-}\frac{\partial^j G}{\partial x^j} (x,t),
\quad x \in [-1,1], \quad 0 \le j \le 2,
\end{align}
and
\begin{align}
\lim_{h \to 0} \left[\frac{\partial^3 G}{\partial x^3}(x+h,t)
- \frac{\partial^3 G}{\partial x^3}(x-h,t)\right] = -1,
\quad t \in [-1,1].
\end{align}
We will use the notation $G_j$ to denote the $j^{th}$ partial derivative of the Green's function, $0 \le j \le 3$; that is, we define:
\begin{align}
G_j(x,t) = \frac{\partial^j G_0}{ \partial x^j} (x,t) .
\end{align}
We will use the boldface letter $\mathbf{G}_j$ to denote the corresponding integral operator. In this notation, for a function $f$ on $[-1,1]$ we will write:
\begin{align}
(\mathbf{G}_j f)(x) = \int_{-1}^{1} G_j(x,t)f(t) d t
= \int_{-1}^{1} \frac{\partial^j G_0}{\partial t^j}(x,t) f(t) d t.
\end{align}
The significance of the Green's function and its derivatives is that the solution to the equation $\phi^{(4)} = f$ with zero boundary conditions is the function $\phi = \mathbf{G}_0 f$, with derivatives $\phi^{(j)} = \mathbf{G}_j f$, $1 \le j \le 3$. In Section \ref{sec:fredholm}, we will use the biharmonic Green's function to reformulate the problem \eqref{eq:main_prob}--\eqref{bdry4} as a second-kind integral equation.
\subsection{Integral form of the boundary value problem \eqref{eq:main_prob}--\eqref{bdry4}} \label{sec:fredholm}
In this section, we will use the biharmonic Green's function to reformulate the differential equation \eqref{eq:main_prob}--\eqref{bdry4} as a second-kind integral equation. For comprehensive background on second-kind integrals equations, see \cite{riesz-functional, courant-methods-1937, byron-mathematics-2012}. We let $\L$ denote the differential operator on the left side of \eqref{eq:main_prob}; that is,
\begin{align}
(\L \phi)(x)
= \sum_{j=0}^{4} a_j(x) \frac{d^j \phi}{dx^j}(x)
\end{align}
The equation \eqref{eq:main_prob} can then be written in the more compact form
\begin{math}
\L \phi = f.
\end{math}
Any four-times differentiable function with vanishing values and first derivatives on $[-1,1]$ can be written in the form $\mathbf{G}_0 \sigma$, for some function $\sigma$. Consequently, we can express the solution $\phi$ to \eqref{eq:main_prob}--\eqref{bdry4} as being of the form $\phi = \mathbf{G}_0 \sigma + \psi_{\alpha}$, where $\sigma = \phi^{(4)}$, and where $\psi_\alpha$ is a function satisfying the boundary conditions \eqref{bdry1}--\eqref{bdry4} and $\psi_\alpha^{(4)} = 0$ on $[-1,1]$.
It can be directly checked that the unique function $\psi_\alpha$ on $[-1,1]$ satisfying $\psi^{(4)} = 0$ and the boundary conditions \eqref{bdry1}--\eqref{bdry4} (with $a=-1$ and $b=1$) is given by the following formula:
\begin{align} \label{eq:psi_alpha}
\psi_\alpha(x) = \alpha_{l,0} \psi_{l,0}(x)
+ \alpha_{r,0} \psi_r(x) + \alpha_{l,1} \psi_{l,1}(x)
+ \alpha_{r,1}\psi_{r,1}(x)
\end{align}
where the functions $\psi_{l,0}$, $\psi_{r,0}$, $\psi_{l,1}$ and $\psi_{r,1}$ are defined by the formulas:
\begin{align}
\psi_{l,0}(x) &= (1-x)^2 (2+x) /4 \label{eq:psi_l0}\\
\psi_{r,0}(x) &= (1+x)^2 (2-x) /4 \label{eq:psi_r0}\\
\psi_{l,1}(x) &= (1-x)^2 (x+1) /4 \label{eq:psi_l1} \\
\psi_{r,1}(x) &= (1+x)^2 (x-1) /4 \label{eq:psi_r1}.
\end{align}
The function $\sigma = \phi^{(4)}$ is expressible as the solution to a second-kind integral equation, as we now show. The equation $\L (\mathbf{G}_0 \sigma + \psi_\alpha) = f$
can be written equivalently as
\begin{align} \label{eq:int_eq}
\L \mathbf{G}_0 \sigma = f - \L \psi_\alpha.
\end{align}
Because $\frac{\partial^4 G}{\partial x^4} (x,t) = \delta(x-t)$, the operator $\L \mathbf{G}_0$ is of the form
\begin{align}
\label{LG0}
(\L\mathbf{G}_0\sigma)(x)
= a_{4}(x) \sigma(x)
+ \sum_{j=0}^{3} a_j(x) \int_{-1}^{1} G_j(x,t) \sigma(t) dt.
\end{align}
After solving \eqref{eq:int_eq} for $\sigma$, the solution $\phi$ to \eqref{eq:main_prob}--\eqref{bdry4} and its derivatives are given by $\phi^{(j)} = \mathbf{G}_j \sigma + \psi_\alpha^{(j)}$, for $0 \le j \le 3$, and $\phi^{(4)} = \sigma$. The reformulation of the differential equation as a second-kind integral equation is beneficial, as the latter can be stably discretized.
\begin{rmk}
The integral equation formulation of \eqref{eq:main_prob}--\eqref{bdry4} makes use of the biharmonic Green's function $G_0$. However, in principle this can be replaced by the Green's function $G_0$ for any equation $\L_0 \phi = f$ with a fourth-order differential operator $\L_0$.
In this general setting, $G_0$ is known as the \emph{background} Green's function \cite{greengard-two-point,lee-stiff}. We choose to work with the biharmonic equation because it is so analytically tractable. We also suspect that the numerical performance of our method will not depend substantially on the choice of background Green's function, as was observed for second-order equations in the numerical experiments of \cite{greengard-two-point}.
\end{rmk}
\subsection{Legendre polynomials and Gaussian quadrature} \label{sec:legendre}
The $n^{th}$ Legendre polynomial $P_n$ is defined for $x \in [-1,1]$ by
\begin{align}
P_0(x) = 1; \quad
P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2-1)^n, \quad
n \ge 1.
\end{align}
As is well known \cite{abramowitz}, the $n$ roots $y_1,\dots,y_n$ of $P_n$ lie in
$[-1,1]$. Together with a suitable choice of weights $w_i > 0$, they can be used to
evaluate the integral on $[-1,1]$ of any polynomial $f$ of degree less than or equal
to $2n+1$ via the formula
\begin{align}
\int_{-1}^{1} f(x) dx = \sum_{i=1}^{n} f(y_i) w_i.
\end{align}
By rescaling the interval $[-1,1]$ to $[a,b]$, we obtain the quadrature
\begin{align}
\label{eq:gauss_quadr}
\int_{a}^{b} f(x) dx = \sum_{i=1}^{n} f\bigg( \frac{b-a}{2} (y_i + 1) + a \bigg)
\frac{b-a}{2} w_i.
\end{align}
The rescaled nodes $ \frac{b-a}{2} (y_i + 1) + a $ are the roots of the polynomials
$P_n(2(x-a)/(b-a) - 1)$ on $[a,b]$, which form an orthogonal basis for $L^2([a,b])$.
The nodes $y_i$ and weights $w_i$ on $[-1,1]$ can be computed at cost $O(n)$
\cite{rokhlin1}.
For an arbitrary $(2n+2)$-times continuously differentiable function $f$ on $[a,b]$ (not
necessarily a degree $2n+1$ polynomial), the same quadrature formula
\eqref{eq:gauss_quadr} has error
\begin{align}
\bigg|\int_{a}^{b} f(x) dx
- \sum_{i=1}^{n} f\bigg( \frac{b-a}{2} (y_i + 1) + a \bigg)
& \frac{b-a}{2} w_i \bigg| \nonumber \\
&\le \frac{|f^{(2n+2)}(\xi)|}{(2n+2)!} \int_{a}^{b} \prod_{i=0}^{n} (x-x_i)^2 dx
\end{align}
for some $\xi \in [a,b]$; for a proof, see Theorem 7.4.5 in \cite{dahlquist}. It follows
immediately that the error can be bounded above by
\begin{align} \label{leg_err}
\frac{(b-a)^{2n+3}}{(2n+2)!} \sup_{x \in [a,b]} |f^{(2n+2)}(x)|.
\end{align}
We also note that the Legendre polynomials $P_n$ satisfy the three-term recurrence
\cite{dahlquist,gradshteyn}
\begin{align}
P_{n+1}(x) = \frac{2n+1}{n+1}x P_n(x) - \frac{n}{n+1}P_{n-1}(x), \quad n\ge1
\end{align}
Using this recurrence, for any value of $x \in [-1,1]$ all values $P_0(x),\dots,P_n(x)$
can be computed in $O(n)$ floating-point operations.
There is a one-to-one correspondence between the values of $f$ on the $n$ nodes $y_1,\dots y_n$ and the first $n$ Legendre coefficients of $f$. More precisely, given the values of $f$ on $n$ Gaussian nodes $y_i$ of $[-1,1]$, the first $n$ coefficients of the Legendre expansion of $f$ can be computed at cost $O(n^2)$, by applying the matrix $[P_i(y_j)w_j]_{1\le i,j \le n}$ to the vector $(f(y_1),\dots,f(y_n))^\top$. Conversely, from the first $n$ Legendre coefficients of $f$ we can compute the values of $f$ on the $n$ Gaussian nodes at cost $O(n^2)$, by applying the inverse matrix to the vector of coefficients.
\subsection{Deferred corrections}
\label{sec:corrections}
In this section, we review the technique of deferred corrections for improving the accuracy of the solution to a linear system $\mathbf{A} \phi = f$. Informally, the method proceeds by solving a sequence of equations for the residuals of the previous solution. The setting where this method may be applied is when there already exists an inaccurate method for inverting $\mathbf{A}$ but a highly accurate method for applying $\mathbf{A}$ to an arbitrary vector. The algorithm is as follows:
\begin{enumerate}
\item
Produce an initial solution $\hat{\phi}$.
\item
Apply $\mathbf{A}$ to $\hat{\phi}$ and compute the residual right hand side $\Delta = f - \mathbf{A} \hat{\phi}$.
\item
Compute a vector $\hat{r}$ that solves the residual system $\mathbf{A} r = \Delta$ for the residual $r = \phi - \hat{\phi}$.
Update the solution: $\hat{\phi} \leftarrow \hat{\phi} + \hat{r}$.
\item
Repeat steps 2 and 3 until convergence.
\end{enumerate}
Algorithms based on this method have been employed for discretized second-kind integral operators $\mathbf{A}$ in, for example, \cite{dutt2000spectral,glaser2009solvers,kushnir2012stiff}. Its convergence properties for certain differential equations have been studied \cite{hansen2011deferred}. When employed for general linear systems, the method also goes by the name ``iterative refinement'', where its numerical properites have been well-studied \cite{higham,wilkinson}.
At each step of the algorithm, the current solution $\hat{\phi}$ has a residual vector $r = \phi - \hat{\phi}$ that satisfies the residual linear system:
\begin{align}
\mathbf{A} r = \mathbf{A} \phi - \mathbf{A} \hat{\phi} = f - \mathbf{A} \hat{\phi}.
\end{align}
Since $\mathbf{A} \hat{\phi}$ can be computed accurately, the right hand side $f - \mathbf{A} \hat{\phi}$ can be computed accurately as well (though see Remark \ref{rmk:itref} below). Consequently, the residual can be obtained to some relative accuracy $\epsilon$, or in other words, we can produce a vector $\hat{r}$ with $\|\hat{r} - r\| \le \epsilon\|r\|$. Then the updated solution $\hat{\phi} + \hat{r}$ satisfies:
\begin{align}
\|(\hat{\phi} + \hat{r}) - \phi\| = \|\hat{r} - (\phi - \hat{\phi}) \|
= \| \hat{r} - r\| \le \epsilon \|r\| = \epsilon \|\phi - \hat{\phi}\|.
\end{align}
In other words, the error of $\hat{\phi} + \hat{r}$ is smaller than the error of $\hat{\phi}$ by a factor of $\epsilon$. If this factor is gained after every iteration, then only $\lceil \log(\ep^*) / \log(\ep) \rceil$ iterations are required to achieve machine precision $\ep^*$.
\begin{rmk} \label{rmk:itref}
We explain one subtlety with the method of deferred corrections. In order to solve for the residual $r$ to relative error $\epsilon$, the right hand side $\Delta = f - \mathbf{A} \hat{\phi}$ must be computed to relative error at most $\epsilon$. Even if $\mathbf{A} \hat{\phi}$ is computed to full machine precision, if $\phi \approx \phi$ then $f \approx \mathbf{A} \hat{\phi}$, and the difference $f - \mathbf{A} \hat{\phi}$ may have large relative error. More precisely, when $\hat{\phi}$ is close enough to $\phi$ so that $\| f - \mathbf{A} \hat{\phi}\| \le \delta \| f\|$, then the difference $f - \mathbf{A} \hat{\phi}$ will be computed to relative error $\ep^* / \delta$, due to loss of digits. So long as $\ep^*/\delta \le \epsilon$, the next iteration of the algorithm will decrease the error by a factor of $\ep$. However, when $\hat{\phi}$ is close enough to $\phi$ that $\delta < \ep^*/\ep$, the next iteration will only increase the accuracy by a factor of $\ep^* / \delta$.
\end{rmk}
\section{The algorithm for solving \eqref{eq:main_prob}--\eqref{bdry4}} \label{sec:algorithm}
In this section, we provide a detailed description of the algorithm for solving \eqref{eq:main_prob}--\eqref{bdry4}. The algorithm has three main components. First, the interval $[a,b]$ is broken into $m$ equal-length subintervals $[x_{i}, x_{i+1}], 1 \le i \le m$, where $x_i = -1 + 2(i-1)/m$. A solution with zero boundary values is produced on each subinterval, employing the second-kind integral equation described in Section \ref{sec:fredholm}. In addition, four linearly independent solutions to the homogeneous equation are also produced on each subinterval.
Second, we form a linear combination of the solutions on each subinterval. The coefficients of this linear combination are chosen so that the resulting functions on adjacent intervals have matching boundary values. Solving for these coefficients is inexpensive but ill-conditioned, resulting in a solution that does not achieve machine precision.
Third, we remedy the ill-conditioning introduced in the second step by deferred corrections. To do so we must accurately apply the second-kind integral operator $\L \mathbf{G}_0$ to the right hand side of the equation; concretely, this entails integrating the right hand side against the functions $G_j$. We describe an algorithm for doing so with asymptotic cost $O(m)$. This step is then iterated until full machine accuracy is achieved.
The algorithm can be summarized as follows:
\begin{enumerate}
\item \label{step:solve}
Solve the problem with zero boundary conditions on each subinterval $[x_i,x_{i+1}]$.
\item \label{step:adjust}
Adjust the boundary values on each subinterval $[x_i,x_{i+1}]$ so that the solutions match at the endpoints.
\item \label{step:defer}
Compute the residual right hand side on the full domain $[a,b]$ and apply deferred corrections.
\end{enumerate}
Steps \ref{step:solve}--\ref{step:defer} will produce the solution $\phi$ sampled at $n$ Gaussian nodes of each subinterval $[x_i,x_{i+1}]$. These $n$ nodes can be converted into the first $n$ Legendre coefficients of the solution $\phi$ on each subinterval $[x_{i},x_{i+1}]$. The solution can then be evaluated at any point in $[a,b]$ by evaluating this expansion on the subinterval $[x_i, x_{i+1}]$ containing that point, as described in Section \ref{sec:legendre}.
Finally, we note that by applying the rescaling described in Section \ref{sec:rescale}, we may assume that the problem domain is the interval $[-1,1]$. After producing the solution, we can apply the inverse rescaling \eqref{eq:inverse_rescale} to return to the original domain $[a,b]$.
\subsection{The solution on $n$ nodes in a single subinterval} \label{sec:alg_nnodes}
In this section, we describe how to obtain a solution $\phi$ on $n$ Gaussian nodes of a subinterval $[x_{i},x_{i+1}]$. We will rescale the subinterval $[x_i,x_{i+1}]$ to be $[-1,1]$, as described in Section \ref{sec:rescale}; when the algorithm is completed, we will rescale back to the original domain $[x_i,x_{i+1}]$, as in equation \eqref{eq:inverse_rescale}. We explain how to produce the solution $\phi$, with specified boundary values, to the equation
\begin{math}
\L \phi = f
\end{math}
on $n$ Gaussian nodes $y_1,\dots, y_n$ in $[-1,1]$. We will also assume that we have divided the entire equation \eqref{eq:main_prob} by the leading coefficient $a_4(x)$, so that without any loss in generality we assume that the leading coefficient $a_4(x) = 1$.
As described in Section \ref{sec:fredholm}, we can write the solution $\phi$ as $\phi = \mathbf{G}_0 \sigma + \psi_\alpha$, where $\psi_\alpha$ is defined by formulas \eqref{eq:psi_alpha}--\eqref{eq:psi_r1}, and the function $\sigma$ satisfies the following second-kind integral equation:
\begin{align} \label{eq:int_eq000}
\sigma(x) + \int_{-1}^{1} \sum_{j=0}^{3} a_j(x) G_j(x,t) \sigma(t) dt
= f - \L \psi_\alpha \equiv f_{\alpha}.
\end{align}
We discretize the equation \eqref{eq:int_eq000} by sampling the coefficients $a_j(x)$, the right hand side $f_\alpha(x)$, and the functions $G_j(x,t)$ in each variable on the $n$ Gaussian nodes $y_1,\dots,y_n$. We will produce the solution $\sigma$ evaluated on the same nodes. We will write the integrals in \eqref{eq:int_eq000} using the quadrature formula for Gaussian nodes and weights $w_k$ described in Section \ref{sec:legendre}. The discrete system of equations resulting from discretizing \eqref{eq:int_eq000} is given by:
\begin{align} \label{eq:syst000}
\sigma(y_i) + \sum_{k=1}^{n} \left[ \sum_{j=0}^{3}
a_j(y_i) G_j(y_i,y_k) \right] \sigma(y_k) w_k = f_\alpha(y_i),
\quad 1 \le i \le n.
\end{align}
We can compactly write this as a linear system $A \hat{\sigma} = \hat{f}_\alpha$, where $\hat{f}_\alpha = (f_\alpha(y_1), \dots, f_\alpha(y_n))^\top$, $\hat{\sigma} = (\sigma(y_1), \dots, \sigma(y_n))^\top$ and $A$ is the $n$-by-$n$ matrix with entries
\begin{align}
A(i,k) = \delta_{i,k} + \sum_{j=0}^{3} a_j(y_i) G_j(y_i,y_k) w_k.
\end{align}
We invert the $n$-by-$n$ matrix $A$ using the $QR$ factorization at a cost of $O(n^3)$ floating point operations \cite{dahlquist}; other algorithms for solving linear systems may be used as well.
The solution $\phi$ to \eqref{eq:main_prob}--\eqref{bdry4} and its first three derivatives on the $n$ nodes $y_1,\dots,y_n$ are now given by:
\begin{align}
\phi^{(j)}(y_i) = \sum_{k=1}^n G_j(y_i,y_k) \hat{\sigma}_k w_k
+ \psi_\alpha^{(j)}(y_i),
\quad 0 \le j \le 3
\end{align}
and $\phi^{(4)}(y_i) = \hat{\sigma}_i$. We then perform the inverse rescaling in equation \eqref{eq:inverse_rescale} to complete the computation of the solution on the subinterval $[x_i,x_{i+1}]$. Since the cost of solving \eqref{eq:syst000} on each of the $m$ subintervals $[x_i,x_{i+1}]$ is $O(n^3)$, the total cost is $O(m n^3)$.
\subsection{Matching the boundary values} \label{sec:alg_match}
In Section \ref{sec:alg_nnodes} we detailed how to obtain a solution to \eqref{eq:main_prob}--\eqref{bdry4} on $n$ Gaussian nodes in each subinterval $[x_i,x_{i+1}]$ with specified boundary values at $x_{i}$ and $x_{i+1}$. We will denote by $\tilde{\phi}_i$ the solution on $[x_i,x_{i+1}]$. In this section we explain how to adjust the boundary values of these functions so that the resulting functions $\phi_i$ and $\phi_{i+1}$ have matching values and three derivatives at the interface $x_{i+1}$, and so that $\phi_{1}$ satisfies \eqref{bdry1} and \eqref{bdry3} at $-1$, and $\phi_{m}$ satisfies \eqref{bdry2} and \eqref{bdry4} at $1$. We will assume that the $\tilde{\phi}_i$ were chosen to have zero boundary values at $x_{i}$ and $x_{i+1}$.
In addition to constructing $\tilde{\phi}_i$, on each subinterval $[x_i,x_{i+1}]$, we use the method of Section \ref{sec:alg_nnodes} to solve the equation $\L g = 0$ four times to obtain functions $g_{i,j}, 1 \le j \le 4$, with a single non-zero boundary value; concretely, $g_{i,1}(x_i) = 1, g_{i,2}(x_{i+1}) = 1, g_{i,3}^{\prime}(x_i)=1$, and $g_{i,4}^{\prime}(x_{i+1})=1$, and the other values and first derivatives at $x_i$ and $x_{i+1}$ are zero. We will find coefficients $\beta_{i,j}, 1 \le i \le m, 1 \le j \le 4$, such that the functions
\begin{align}
\label{adjust111}
\phi_i(x) = \tilde{\phi}_i(x) + \sum_{j=1}^{4} \beta_{i,j} g_{i,j}(x)
\end{align}
have matching boundary values, and the desired values at $\pm 1$ given by \eqref{bdry1}--\eqref{bdry4}. The derivatives of $\phi_i$ are found by adding the same linear combination of the derivatives of the $g_{i,j}$, namely:
\begin{align}
\label{adjust222}
\phi_i^{(k)}(x) = \tilde{\phi}_i^{(k)}(x) + \sum_{j=1}^{4} \beta_{i,j} g_{i,j}^{(k)}(x),
\quad 1 \le k \le 4.
\end{align}
The function $\phi^{(k)}(x)$, $0 \le k \le 4$, is then defined by $\phi_i^{(k)}(x)$ when $x \in [x_i,x_{i+1}]$. In particular, $\sigma = \phi^{(4)}$ solves the integral equation \eqref{eq:int_eq000} on the entire interval $[-1,1]$.
The coefficients $\beta_{i,j}$ can be found as the solution to a banded linear system of
size $4m$-by-$4m$. For every pair of adjacent intervals $[x_{i}, x_{i+1}]$ and
$[x_{i+1},x_{i+2}]$, we require that
\begin{align}
\label{eq:phii_val}
\tilde{\phi}_{i}(x_{i+1}) + \beta_{i,2}
&= \tilde{\phi}_{i+1}(x_{i+1}) + \beta_{i,1} \\
\label{eq:phii_der1}
\tilde{\phi}_{i}^{\prime}(x_{i+1}) + \beta_{i,4}
&= \tilde{\phi}_{i+1}^{\prime}(x_{i+1}) + \beta_{i,3} \\
\label{eq:phii_der2}
\tilde{\phi}_{i}^{\prime \prime} (x_{i+1})+ \sum_{j=1}^{4} \beta_{i,j} g_{i,j}^{\prime
\prime}(x_{i+1})
&= \tilde{\phi}_{i+1}^{\prime \prime} (x_{i+1})+ \sum_{j=1}^{4} \beta_{i+1,j}
g_{i+1,j}^{\prime \prime}(x_{i+1}) \\
\label{eq:phii_der3}
\tilde{\phi}_{i}^{(3)} (x_{i+1})+ \sum_{j=1}^{4} \beta_{i,j} g_{i,j}^{(3)}(x_{i+1})
&= \tilde{\phi}_{i+1}^{(3)} (x_{i+1})+ \sum_{j=1}^{4} \beta_{i+1,j} g_{i+1,j}^{(3)}
(x_{i+1})
\end{align}
These conditions ensure that the functions $\phi_i$ have four matching derivatives at
the interfaces $x_{i+1}, 1 \le i \le m-1$. To ensure the boundary conditions \eqref{bdry1}--\eqref{bdry4} on $[a,b]$, we also require the following equations:
\begin{align}
\label{eq:phi1_leftval}
\tilde{\phi}_{1}(-1) + \beta_{1,1} &= \alpha_{l,0} \\
\label{eq:phi1_rightval}
\tilde{\phi}_{1}^{\prime}(-1) + \beta_{1,3} &= \alpha_{l,1} \\
\label{eq:phim_leftder}
\tilde{\phi}_{m}(1) + \beta_{m,2} &= \alpha_{r,0} \\
\label{eq:phim_rightder}
\tilde{\phi}_{m}^{\prime}(1) + \beta_{m,4} &= \alpha_{r,1}.
\end{align}
Ordering the variables $\beta_{1,1},\beta_{1,2},\beta_{1,3},\beta_{1,4},\dots \dots,\beta_{m,1},\beta_{m,2},\beta_{m,3},\beta_{m,4}$, and ordering the equations as above and by the intervals they involve, the linear system described by equations \eqref{eq:phii_val}--\eqref{eq:phim_rightder} is $9$-diagonal and so can be solved in $O(m)$ floating-point operations. In Figure \ref{fig:band_matrix}, we plot this matrix for $m=8$ subintervals. After solving for the coefficients $\beta_{i,j}$, the solutions $\phi_i$ on each subinterval can then be obtained by equations \eqref{adjust111} and \eqref{adjust222}, at an additional cost of $O(n m)$.
\begin{figure}[h]
\centerline{
\includegraphics[width=0.5\linewidth]{pic1}
}
\caption{The banded matrix used to match the boundary values, shown here for $m=8$ subintervals; non-zero values are marked with dots.}
\label{fig:band_matrix}
\end{figure}
\begin{rmk} \label{rmk:inaccurate}
Because the initial local solutions $\tilde{\phi}_i$ on each subinterval $[x_i,x_{i+1}]$ are obtained via the second-kind integral formulation described in Section \ref{sec:fredholm}, they can be computed with high accuracy. However, the method we have described in this section for matching their boundary values departs from this integral equation formulation. In practice, we have observed that the banded linear system defined by equations \eqref{eq:phii_der1}--\eqref{eq:phim_rightder} behaves like the matrices encountered in finite element or finite difference schemes, in that it has a condition number of size $O(m^4)$. To achieve greater accuracy, in Section \ref{sec:alg_corrections} we show how to apply deferred corrections to this problem.
\end{rmk}
\subsection{Applying deferred corrections} \label{sec:alg_corrections}
The solution we obtain from Sections \ref{sec:alg_nnodes} and \ref{sec:alg_match} will not be accurate to the precision permitted by the problem, as explained in Remark \ref{rmk:inaccurate}. To improve the accuracy, we apply the method of deferred corrections, as described in Section \ref{sec:corrections}. If $\hat{\sigma}$ is the vector of length $m n$ with the estimated solution to the integral equation \eqref{eq:int_eq000} evaluated on the $m n$ nodes, then to apply deferred corrections we perform the following steps:
\begin{enumerate}
\item \label{apply_L}
Apply the operator $\L \mathbf{G}_0$ to $\hat{\sigma}$, evaluated on the $m n$ nodes.
\item \label{solve_new}
Solve the new system $\L \mathbf{G}_0 r = f_\alpha - \L \mathbf{G}_0 \hat{\sigma}$, obtaining an estimated residual $\hat{r}$.
\item \label{update}
Update the solution: $\hat{\sigma} \leftarrow \hat{\sigma} + \hat{r}$.
\end{enumerate}
Step \ref{solve_new} is performed by following Sections \ref{sec:alg_nnodes} and \ref{sec:alg_match} at a cost of $O(m n^3)$ floating point operations, and Step \ref{update} requires $m n$ additions. We will now exhibit an algorithm for performing Step \ref{apply_L} with asymptotic cost $O(m n^2)$.
Because $a_4 \equiv 1$, from equation \eqref{LG0} we have
\begin{align}
(\L\mathbf{G}_0 \hat\sigma)(x)
= \hat \sigma(x)
+ \sum_{j=0}^{3}a_j(x) \int_{-1}^{1} G_j(x,t) \hat \sigma(t) dt.
\end{align}
Therefore, we must integrate $\hat \sigma(x)$ against the functions $G_j(x,t)$. We now explain how to compute the integrals $\int_{-1}^{1} G_j(x,t) \hat \sigma(t) dt$ with $O(m n^2 )$ floating-point operations. The function $G_j(x,t)$ can be written in the form
\begin{align} \label{eq:green_poly000}
G_j(x,t) =
\begin{cases}
\sum_{i=0}^{3-j} x^i p_i(t) , \text{ if } x < t \\
\sum_{i=0}^{3-j} x^i q_i(t), \text{ if } t < x
\end{cases}
\end{align}
where $p_i$ and $q_i$ are polynomials of degree $3-j$ that can be explicitly computed from
\eqref{eq:Green}. (Of course, $p_i$ and $q_i$ depend on $j$, though we suppress this for notational simplicity.) Using \eqref{eq:green_poly000}, we obtain:
\begin{align} \label{green_expand000}
\int_{-1}^{1} G_j(x,t) \hat\sigma(t) dt
= \sum_{i=0}^{3-j} x^{i} \bigg\{ \int_{-1}^{x} q_i(t) \hat\sigma(t) dt
+ \int_{x}^{1} p_i(t) \hat\sigma(t) dt \bigg\}
\end{align}
We must compute $\int_{-1}^{1} G_j(x,t) \hat\sigma(t) dt$ for each of the $m n $ values of $x$ on which we have discretized the problem -- the $n$ Gaussian nodes on each of the $m$ subintervals $[x_{i},x_{i+1}]$. For each such $x$, let us suppose $x \in [x_{i^*},x_{i^* + 1}]$. We can then write:
\begin{align}
\int_{-1}^{x} q_i(t) \hat \sigma(t) dt
= \sum_{k=1}^{i^*-1} \int_{x_{k}}^{x_{k+1}} q_i(t) \hat \sigma(t) dt
+ \int_{x_{i^*}}^{x} q_i(t) \hat \sigma(t) dt
\end{align}
and similarly
\begin{align}
\int_{x}^{1} p_i(t) \hat \sigma(t) dt
= \sum_{k=i^*+1}^{m} \int_{x_{k}}^{x_{k+1}} p_i(t)\hat \sigma(t) dt
+ \int_{x}^{x_{i^*+1}} p_i(t)\hat \sigma(t) dt .
\end{align}
Each of the $2m$ integrals $\int_{x_{k}}^{x_{k+1}} q_i(t) \hat\sigma(t) dt $ and $\int_{x_{k}}^{x_{k+1}} p_i(t) \hat\sigma(t) dt $ can be computed using an $n$-point Gaussian quadrature, at a total cost of $O(m n)$ floating-point operations. For each Gaussian node $x$ in the interval $[x_{i^*}, x_{i^*+1}]$, the integrals $\int_{x_{i^*}}^{x} q_i(t) \hat\sigma(t) dt$ and $\int_{x}^{x_{i^*+1}} p_i(t) \hat\sigma(t) dt$ can also be computed using an $n$-point Gaussian quadrature, at a total cost of $O(m n^2)$ floating-point operations.
From \eqref{green_expand000}, the integrals $\int_{-1}^1 G_j(x,t) \hat{\sigma}(t) dt$, and hence the evaluation of $(\L \mathbf{G}_0 \hat{\sigma})(x)$, can be computed at cost $O(m n^2)$. Consequently, each iteration of deferred corrections requires $O(mn^2)$ operations. As explained in Section \ref{sec:corrections}, the number of iterations is $O(\log(1/\ep))$, where $\ep$ is the machine precision. This brings the asymptotic running time of the entire algorithm to $O(m n^3 \log(1/\ep))$.
\section{Numerical results} \label{sec:experiments}
The algorithm of Section \ref{sec:algorithm} has been implemented in Fortran. In this section, we use the algorithm to obtain the numerical solution $\phi$ to several specific problems of the form \eqref{eq:main_prob}--\eqref{bdry4}. All experiments were performed on a Dell XPS-L521X laptop computer with an Intel Core i7-3632QM CPU at 2.20GHz with 15.6 GB RAM, running the 64-bit Ubuntu 14.04 LTS operating system. In our experiments, the code was compiled with the gfortran compiler using extended (16-bit) precision; this is called with the compilation flag \texttt{-freal-8-real-16}.
In the presentation of the CPU times of the experiments, we distinguish between the preprocessing steps -- which depend only on the differential operator $\L$ and which we refer to as ``factorizing'' $\L$ -- and the actual solution of \eqref{eq:main_prob}--\eqref{bdry4} for a specified right side $f$ and boundary values $\alpha_{l,0}$, $\alpha_{r,0}$, $\alpha_{l,1}$, $\alpha_{r,1}$. Factorizing $\L$ consists of constructing the $m n$ Gaussian nodes and weights on the interval $[a,b]$; computing the QR factorization of the discretized operators $\mathbf{G}_j$, $0 \le j \le 3$; and computing the values of the functions \eqref{eq:psi_l0}--\eqref{eq:psi_r1}, and their derivatives, on the Gaussian nodes.
Five experiments are presented below, and their results are contained in Tables \ref{errors:sin5}--\ref{residuals:bessel}. In each experiment, we solve a specific boundary value problem on a specified interval $[a,b]$, and measure the relative error of the solution and its first four derivatives on a grid of $N=10,000$ equispaced points $x_1,\dots,x_N$ on $[a,b]$, including the endpoints $a$ and $b$. If $\hat{\phi}^{(j)}$ is the estimated $j^{th}$ derivative, then the relative error is defined as follows:
\begin{align}
R(\phi^{(j)}) = \sqrt{
\frac{\sum_{i=1}^{N} | \hat{\phi}^{(j)}(x_i) - \phi^{(j)}(x_i) |^2}
{\sum_{i=1}^{N} |\phi^{(j)}(x_i) |^2} }
\end{align}
For each experiment, we display three tables. In each table, the first column contains the number $m$ of subintervals of $[a,b]$ used for that experiment. In the first table, the remaining five columns display the errors of the solution and its first four derivatives. In the second table, the remaining three columns show the factorization time, the solution time, and the total CPU time. In the third table, the remaining columns display the relative size of the residual norm after each iteration of the algorithm, to demonstrate the convergence of deferred corrections.
\begin{rmk}
For each example, the relative errors $R(\phi^{(j)})$ decrease by approximately a fixed number of digits whenever $m$ is doubled. This behavior is expected from the error of the
Gaussian quadrature, as described in Section \ref{sec:legendre}, equation \eqref{leg_err}, since the quadrature error should scale like $m^{-(2n+2)}$ (the exponent is not $-(2n+3)$ because we multiply by $m$, the number of subintervals).
\end{rmk}
\begin{rmk}
For each example, the CPU times -- both for the factorization of $\L$ and the solution of the equation -- scale approximately linearly with $m$. This behavior is expected from the $O(m)$ asymptotic running time of the algorithm.
\end{rmk}
\begin{rmk}
For each example and for each value of $m$, the residual size decreases by an approximately constant number of digits after approximately two iterations of deferred corrections. This is likely because two linear systems are solved each step -- the local linear systems on each subinterval, and the global linear system that matches the local solutions' boundary values. Up to taking two steps instead of one, this rate of decrease in the residual size is what we expect from Section \ref{sec:corrections}. In particular, very few iterations of the algorithm are required until convergence is achieved.
\end{rmk}
\subsection{The function $\phi(x) = \sin(5x)$}
The first example is the following simple, well-conditioned equation:
\begin{align} \label{eq_sin5x}
\sum_{j=0}^4 (1 + x^{4-j}) \phi^{(j)}(x)
= \sum_{j=0}^4 (1 + x^{4-j}) \frac{d^j}{dx^j} \sin(5x)
\end{align}
on the interval $[0,2\pi]$, subject to the boundary conditions $\alpha_{l,0}=0$, $\alpha_{r,0}=0$, $\alpha_{l,1}=5$ and $\alpha_{r,1}=5$. The solution is apparently $\phi(x) = \sin(5x)$. This example is included to illustrate that for well-conditioned problems, our algorithm obtains the solution to full machine accuracy. The algorithm was run using $n=10$ Gaussian nodes per subinterval. The results are displayed in Tables \ref{errors:sin5}--\ref{residuals:sin5}.
\subsection{The function $\phi(x) = \sin(150x)$}
For the second example, we solve the following variation on equation \eqref{eq_sin5x}:
\begin{align}
\sum_{j=0}^4 (1 + x^{4-j}) \phi^{(j)}(x)
= \sum_{j=0}^4 (1 + x^{4-j}) \frac{d^j}{dx^j} \sin(150x)
\end{align}
on the interval $[0,2\pi]$, subject to the boundary conditions $\alpha_{l,0}=0$, $\alpha_{r,0}=0$, $\alpha_{l,1}=150$ and $\alpha_{r,1}=150$. The solution is apparently $\phi(x) = \sin(150x)$. However, since the frequency is much higher, this is a worse-conditioned problem than \eqref{eq_sin5x}, and so cannot be solved to full machine precision independent of the choice of algorithm. However, our algorithm successfully recovers the solution to within the error permitted by the problem's condition number. The algorithm was run using $n=15$ Gaussian nodes per subinterval. The results are displayed in Tables \ref{errors:sin150}--\ref{residuals:sin150}.
\subsection{Beam with fixed ends} \label{sec:fixed_ends}
In this example, we consider the problem of determining the shape of a beam with non-uniform stiffness subjected to an external force. We parametrize the $x$-axis by the interval $[0,1]$. We take the stiffness of the beam to be the function $c(x)$ given by:
\begin{align}
c(x) = (x-1/2)^2 + 1
\end{align}
and the external force to be the function $f(x)$ given by:
\begin{align}
f(x) = \sin(2 \pi x) + 1.
\end{align}
If $\phi(x)$ is the shape of the beam, then $\phi$ satisfies the following differential equation \cite{gere-mechanics-1984}:
\begin{align}
\frac{d^2}{dx^2} \left( c(x) \frac{d^2 \phi}{dx^2} \right) = f(x)
\end{align}
or equivalently
\begin{align} \label{spelled_out}
((x-1/2)^2 + 1) \frac{d^4 \phi}{dx^4}(x) + 4(x-1/2)\frac{d^3 \phi}{dx^3}(x)
+ 2 \frac{d^2 \phi}{dx^2}(x) = \sin(2 \pi x) + 1.
\end{align}
We impose fixed endpoints on the beam, meaning that $\alpha_{l,0}=0$, $\alpha_{r,0}=0$, $\alpha_{l,1}=0$ and $\alpha_{r,1}=0$. The algorithm was run using $n=10$ Gaussian nodes per subinterval. The results are displayed in Tables \ref{errors:fixed}--\ref{residuals:fixed}. The solution is plotted on the left side of Figure \ref{fig:beams}.
\begin{figure}[h]
\centerline{
\includegraphics[width=.8\linewidth]{beams}
}
\caption{Left: the beam with fixed ends. Right: the beam with simply-supported ends.}
\label{fig:beams}
\end{figure}
\subsection{Beam with simply-supported ends}
In this example, we consider the same beam-bending problem \eqref{spelled_out} as in Section \ref{sec:fixed_ends}, except we require that the ends of the beam are simply-supported, or $\phi(0) = \phi(1) = 0$ and $\phi^{\prime\prime}(0) = \phi^{\prime\prime}(1) = 0$. Because the boundary vales involve the second derivatives, this problem does not immediately fit into the form of \eqref{eq:main_prob}--\eqref{bdry4}, and we include it as an example for that reason.
To introduce the correct boundary conditions, we let $\L$ denote the differential operator on the left side of \eqref{spelled_out}, and we solve the equation $\L \phi = f$ with boundary conditions $\alpha_{l,0} = \alpha_{r,0} = \alpha_{l,1} = \alpha_{r,1} = 0$; we will call this function $\tilde{\phi}$. We also produce four linearly independent solutions to the equation $\L \phi = 0$ by solving it with linearly independent boundary conditions; we will call these solutions $\phi_{1}$, $\phi_2$, $\phi_3$ and $\phi_4$.
We will find coefficients $b_i$, $1 \le i \le 4$, so that the function
\begin{align} \label{phi_plus_bphi}
\phi(x) = \tilde \phi(x) + \sum_{i=1}^4 b_i \phi_i(x)
\end{align}
satisfies the sought-after boundary conditions $\phi(0) = \phi(1) = 0$ and $\phi^{\prime\prime}(0) = \phi^{\prime\prime}(1) = 0$. The $b_i$ satisfy the linear equations:
\begin{align}
0 &= \sum_{i=1}^4 b_i \phi_i(0) \\
0 &= \sum_{i=1}^4 b_i \phi_i(1) \\
0 &= \tilde{\phi}^{\prime\prime}(0) + \sum_{i=1}^4 b_i \phi_i^{\prime\prime}(0) \\
0 &= \tilde{\phi}^{\prime\prime}(1) + \sum_{i=1}^4 b_i \phi_i^{\prime\prime}(1) .
\end{align}
This system is non-singular because the $\phi_i$ are linearly independent. This permits us to solve for the $b_i$, and then define the solution $\phi$ by \eqref{phi_plus_bphi}. The algorithm was run using $n=10$ Gaussian nodes per subinterval. The results are shown in Tables \ref{errors:simply}--\ref{residuals:simply}. The solution is plotted on the right side of Figure \ref{fig:beams}.
\subsection{The Bessel function $J_{10}(x)$}
In our final example, we solve for the Bessel function $J_{10}(x)$ on the interval $[0,100]$, which is plotted in Figure \ref{fig:bessel}. $J_{10}$ satisfies the second-order equation:
\begin{align}
x^2 \frac{d^2}{dx^2} \phi(x) + x \frac{d}{dx} \phi(x) + (x^2-100) \phi(x) = 0.
\end{align}
We differentiate this equation twice to arrive at the fourth-order equation:
\begin{align}
\label{eq:bessel}
\left(x^2 \frac{d^4}{dx^4} + 5x \frac{d^3}{dx^3} + (x^2-96) \frac{d^2}{dx^2}
+ 4x \frac{d}{dx} + 2 \right) \phi(x)= 0.
\end{align}
We solve this equation subject to the boundary values matching the known values of $J_{10}(x)$ and $J_{10}^\prime(x)$ at 0 and 100. Because of the singularity at $x=0$, we represent the 0 endpoint by the square root of machine zero. We ran the algorithm with $n=20$ nodes per subinterval. We include this example to demonstrate the algorithm's performance when the leading coefficients has a root, as does the leading coefficient $x^2$ at the left endpoint $x=0$. Because of this singularity, the solution cannot be obtained to full machine precision even on the $m n$ Gaussian nodes themselves. Nevertheless, our algorithm obtains the solution up to this necessary loss of accuracy. The results are shown in Tables \ref{errors:bessel}--\ref{residuals:bessel}.
\begin{figure}[h]
\centerline{
\includegraphics[width=0.5\linewidth]{bessel}
}
\caption{The Bessel function $J_{10}$ on the interval $[0,100]$.}
\label{fig:bessel}
\end{figure}
\section*{Acknowledgements}
William Leeb was supported by a postdoctoral fellowship from the Simons Collaboration on Algorithms and Geometry. Vladimir Rokhlin was supported in part by the Office of Naval Research under the grant N00014-16-1-2123 and the Air Force Office of Scientific Research under the grant FA9550-16-1-0175.
\markright{Bibliography}
\bibliographystyle{plain}
| {
"timestamp": "2018-03-01T02:04:23",
"yymm": "1711",
"arxiv_id": "1711.05354",
"language": "en",
"url": "https://arxiv.org/abs/1711.05354",
"abstract": "This paper introduces a fast and numerically stable algorithm for the solution of fourth-order linear boundary value problems on an interval. This type of equation arises in a variety of settings in physics and signal processing. Our method reformulates the equation as a collection of second-kind integral equations defined on local subdomains. Each such equation can be stably discretized and solved. The boundary values of these local solutions are matched by solving a banded linear system. The method of deferred corrections is then used to increase the accuracy of the scheme. Deferred corrections requires applying the integral operator to a function on the entire domain, for which we provide an algorithm with linear cost. We illustrate the performance of our method on several numerical examples.",
"subjects": "Numerical Analysis (math.NA)",
"title": "On the Numerical Solution of Fourth-Order Linear Two-Point Boundary Value Problems",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9888419680942663,
"lm_q2_score": 0.7154239897159439,
"lm_q1q2_score": 0.7074412660125661
} |
https://arxiv.org/abs/1812.04911 | The Crossing Tverberg Theorem | Tverberg's theorem is one of the cornerstones of discrete geometry. It states that, given a set $X$ of at least $(d+1)(r-1)+1$ points in $\mathbb R^d$, one can find a partition $X=X_1\cup \ldots \cup X_r$ of $X$, such that the convex hulls of the $X_i$, $i=1,\ldots,r$, all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed.As a concrete application, we show that any $n$ points in the plane in general position span $\lfloor n/3\rfloor$ vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Rebollar et al.\ guarantees $\lfloor n/6\rfloor$ pairwise crossing triangles. Our result generalizes to a result about simplices in $\mathbb R^d,d\ge2$. | \section{Introduction}
The following theorem was published by Johann Radon in 1921 \cite{Radon1921}: any set of $d+2$ points in $\mathbb R^d$ can be partitioned into two (disjoint) subsets, whose convex hulls intersect. In 1966, Helge Tverberg \cite{T1966_tverberg} proved the following important generalization of Radon's result.
\begin{theorem}[Tverberg~\cite{T1966_tverberg}]
\label{thm:Tverberg}
Let $X$ be a set of at least $(d+1)(r-1)+1$ points in $d$-space. Then $X$ can be partitioned into $r$ sets whose convex hulls all have a point $o$ in common.
(In the literature, the point $o$ is referred to as a \emph{Tverberg point} and the partition as a \emph{Tverberg partition}.)
\end{theorem}
Radon's theorem covers the case $r=2$.
Figure~\ref{fig:tverberg_partition} illustrates Tverberg's theorem for $d=2$ and $r=3$.
\begin{figure}[htb]
\includegraphics[width=\textwidth]{tverberg_partition}
\caption{Tverberg's theorem for $d=2$ and $r=3$: any set of at least $7$ points can be partitioned into three sets whose convex hulls all have a point in common. Our example uses $9$ points and shows two Tverberg partitions as well as corresponding Tverberg points. Many other Tverberg partitions exist in this example.\label{fig:tverberg_partition}}
\end{figure}
This theorem largely influenced the course of discrete geometry and spurred a lot of research in the area. We do not go into more details in this paper and refer the reader to a recent survey by B{\'{a}}r{\'{a}}ny and Sober{\'{o}}n~\cite{Barany2018}.
Another question, motivating our work, comes from the field of geometric graphs. In \cite{Aronov1994}, Aronov et al.\ conjectured that there exists an absolute constant $c>0$, such that, given any set of $n$ points in general position in the plane, one can find at least $cn$ disjoint pairs among them such that their connecting segments pairwise cross. Such a collection of segments is called a \emph{crossing family}. Despite considerable interest in this problem, the best published bound still comes from the original paper \cite{Aronov1994}, stating that one can always find a crossing family of size at least $c\sqrt n$ for some absolute $c>0$.
In an attempt to approach this problem, Rebollar et al.\ asked whether one can find at least $cn$ disjoint \emph{triples} whose connecting \emph{triangles} cross pairwise~\cite{RLU2016_crossingFam}. Throughout this paper, we say that two convex bodies in $\mathbb{R}^d$ \emph{cross} if their boundaries have a non-empty intersection. We remark that if a convex body in $\mathbb{R}^d$ is not full-dimensional then it has no interior and therefore coincides with its boundary. Our main results below would be false in general if relative boundaries were considered in the above definition of crossing of two convex bodies (an easy counterexample in this situation is a set of points lying on a line).
Rebollar et al.\ showed the following: For every finite point set of size $n$ in the plane in general position, i.e., no three points on a line, there exist $\left\lfloor\frac{n}{6}\right\rfloor$ vertex-disjoint and pairwise crossing triangles with vertices in $P$. As at most $\left\lfloor\frac{n}{3}\right\rfloor$ disjoint triangles can be found, this leaves a factor-2 gap.
If we only want triangles that have a common point, this gap can be closed using a simple strenghtening of Tverberg's theorem for point sets of size at most $(d+1)r$ that we present next. However, these triangles might not be pairwise crossing, since triangles can be nested; see Figure~\ref{fig:tverberg_partition} (right).
\begin{theorem}\label{thm:Tverberg2}
Let $X$ be a set of at least $(d+1)(r-1)+1$ and at most $(d+1)r$ points in $d$-space. Then $X$ can be partitioned into $r$ disjoint sets $X_1,\ldots, X_r$ of size at most $d+1$, whose convex hulls all have a point in common.
\end{theorem}
To prove this, we apply Theorem~\ref{thm:Tverberg} to $X$. We get sets $X'_1,\ldots, X'_r$ whose convex hulls contain a common point, say, the origin. Using Carath\' eodory's theorem, from every $X'_i$ of size larger than $d+1$ we can select $d+1$ points $X_i\subseteq X'_i$, whose convex hull still contains the origin. Finally, some of the sets $X'_i$ of size smaller than $d+1$ are filled up to size $d+1$ with the points removed from other $X'_i$'s. The origin is still a common point for all $\mathop{\mathrm{conv}}(X_i)$'s.
The main result of this paper provides a crossing version of Tverberg's Theorem~\ref{thm:Tverberg2}.
\begin{theorem} \label{thm:main} Let $X$ be a set of at least $(d+1)(r-1)+1$ and at most $(d+1)r$ points in $d$-space. Then $X$ can be partitioned into $r$ disjoint sets $X_1,\ldots, X_r$ of size at most $d+1$, whose convex hulls all have a point in common. Moreover, for any $X_i,X_j$ of size $d+1$, $\mathop{\mathrm{conv}} (X_i)$ and $\mathop{\mathrm{conv}} (X_j)$ cross, meaning that their boundaries have a non-empty intersection.
\end{theorem}
We call such a partition a \emph{crossing Tverberg partition}. An easy calculation shows that the number of sets with exactly $d+1$ elements is at least $|X|-dr\in\{r-d,r-d+1,\ldots,r\}$. In particular, for sets $X$ of size exactly $(d+1)r$, we immediately deduce the following simpler-looking corollary.
\begin{corollary}\label{cor1} Let $X$ be a set of $(d+1)r$ points in $d$-space. Then $X$ can be partitioned into $r$ sets $X_1,\ldots, X_r$ of size $d+1$, whose convex hulls all have a point in common and such that for any $i,j\in [n]$, $\mathop{\mathrm{conv}} (X_i)$ and $\mathop{\mathrm{conv}} (X_j)$ cross, meaning that their boundaries have a non-empty intersection.
\end{corollary}
We also obtain an optimal strenghtening of the result by Rebollar et al.~\cite{RLU2016_crossingFam} that moreover generalizes to all dimensions $d\geq 2$.
\begin{corollary}
For every finite point set $X$ of size $n$ in the plane in general position, i.e., no three points on a line, there exist $\left\lfloor\frac{n}{3}\right\rfloor$ vertex-disjoint and pairwise crossing triangles with vertices in $X$.
More generally, for every finite point set $X$ of size $n$ in $\mathbb R^d$ in general position, i.e., no $d$ points lying on a hyperplane, there exist $\left\lfloor\frac{n}{d+1}\right\rfloor$ vertex-disjoint and pairwise crossing simplices with vertices in $X$.
\end{corollary}
To derive this from Theorem~\ref{thm:main}, we remove $n\mod (d+1)$ points from $X$ and then apply Theorem~\ref{thm:main} on the remaining set of $(d+1)\left\lfloor\frac{n}{d+1}\right\rfloor$ points.
Finally, we get a crossing version of the actual Tverberg's theorem~\ref{thm:Tverberg}, for point sets of arbitrarily large size.
\begin{theorem}[Crossing Tverberg theorem]\label{thm:crossing_tverberg}
Let $X$ be a set of at least $(d+1)(r-1)+1$ points in $d$-space. Then $X$ can be partitioned into $r$ sets whose convex hulls all have a point in common. Moreover, for any $X_i,X_j$ of size at least $d+1$, $\mathop{\mathrm{conv}} (X_i)$ and $\mathop{\mathrm{conv}} (X_j)$ cross, meaning that their boundaries have a non-empty intersection.
\end{theorem}
This is also easy to prove, using Theorem~\ref{thm:main} and Corollary~\ref{cor1}. If $n:=|X|\leq (d+1)r$, we apply Theorem~\ref{thm:main}. Otherwise, we apply Corollary~\ref{cor1} to an arbitrary subset $Y\subset X$ of size $(d+1)r$, resulting in a crossing Tverberg partition into $r$ sets $Y_1,\ldots,Y_r$ of size $d+1$ each. Now we consecutively add the remaining points to suitable sets in such a way that the crossings between the convex hulls are maintained (the Tverberg point automatically remains valid). Suppose that some points have already been added, resulting in sets $X'_1,\ldots,X'_r$ whose convex hulls still cross pairwise. If the next point $p$ is contained in one of these convex hulls, we simply add it to the corresponding set. Otherwise, we select an inclusion-minimal set $\mathop{\mathrm{conv}}(X'_k\cup\{p\})$ among the (different) sets $\mathop{\mathrm{conv}}(X'_i\cup\{p\}), i=1,\ldots,r$, and add $p$ to $X'_k$. We claim that this cannot result in nested convex hulls. For this, we only have to rule out that $\mathop{\mathrm{conv}}(X'_k\cup\{p\})$ ``swallows'' some other $\mathop{\mathrm{conv}}(X'_i)$. But this cannot happen, as otherwise $\mathop{\mathrm{conv}}(X'_i\cup\{p\})\subset \mathop{\mathrm{conv}}(X'_k\cup\{p\})$, contradicting the choice of $k$.
It remains to prove Theorem~\ref{thm:main} which we will do in the next section. The main idea is the following. We show that we can assume the points to be in general position. We start with a Tverberg partition according to Theorem~\ref{thm:Tverberg2}, i.e.\ a partition into sets of size at most $d+1$ such that their convex hulls have a point in common. Such a partition might look like in Figure~\ref{fig:tverberg_partition} (right). If the full-dimensional convex hulls (which are simplices) cross pairwise, we are done. In general, however, we still have pairs of nested simplices. As long as this is the case, we \emph{fix} one pair of nested simplices at a time until no pair of nested simplices exists anymore, and our desired partition is obtained.
By fixing, we mean that we repartition the $2(d+1)$ points involved in the two simplices in such a way that the resulting simplices are not nested anymore but still contain the common point. In the example of Figure~\ref{fig:tverberg_partition}, there is one pair of nested simplices (red edges), and after fixing it (blue edges), we are actually done in this case; see Figure~\ref{fig:tverberg_partition2}.
\begin{figure}[htb]
\includegraphics[width=\textwidth]{tverberg_partition2}
\caption{Fixing a pair of nested simplices spanned by the 6 black points: two nested simplices (red) are transformed into two simplices that cross (blue).\label{fig:tverberg_partition2}}
\end{figure}
There are of course some things to prove here. First of all, we need to show that we can actually perform the desired repartitioning in all cases. Then, by fixing one pair, new pairs might arise that need fixing, so it has to be proved that fixing terminates.
The key for repartitioning the points is the following lemma, which may be of independent interest.
\begin{lemma}\label{lem:unnesting}
Let $T,T'$ be two disjoint $(d+1)$-element sets in $\mathbb R^d$ such that $\mathbf{0} \in \mathop{\mathrm{conv}}(T)\cap\mathop{\mathrm{conv}}(T')$. Then there exist two disjoint $(d+1)$-element sets $S,S'$ such that $S\cup S'=T\cup T'$, $\mathbf{0} \in \mathop{\mathrm{conv}}(S)\cap\mathop{\mathrm{conv}}(S')$, and moreover, $\mathop{\mathrm{conv}}(S)$ and $\mathop{\mathrm{conv}}(S')$ cross.
\end{lemma}
Since applications of Lemma~\ref{lem:unnesting} allow to keep the Tverberg point and the sizes of the parts in the (Tverberg) partition, we actually have the following strengthening of Theorem~\ref{thm:main}.
\begin{theorem}\label{thm:main-stronger}
Let $X$ be a set of points in $d$-space. Suppose that there is a Tverberg partition of $X$ into $r$ parts of sizes $s_1\dots,s_r$, for which $o$ is a Tverberg point, and $s_i\leq d+1, i=1,\ldots r$. Then there is also a \emph{crossing} Tverberg partition of $X$ into $r$ parts of sizes $s_1\dots,s_r$, for which a Tverberg point is $o$.
\end{theorem}
In the next section, we prove Theorem~\ref{thm:main} and (on the way) Lemma~\ref{lem:unnesting}. We remark that for each of the other theorems and corollaries in this section, we have already explained how to derive them from Tverberg's theorem, or from our main Theorem~\ref{thm:main}. In Section~\ref{sec3}, we discuss possible generalizations as well as some algorithmic aspects of the problem.
\section{Proof of Theorem~\ref{thm:main}}
First of all, using a standard topological argument, we may assume that the points of $X$ are in general position, meaning that no $d+1$ points lie in a common hyperplane. To justify this assumption, we observe that the set of point sets of some fixed size $n$ that allow a crossing Tverberg partition is closed, since its complement is open. Indeed, if we take a point set that does not allow a crossing Tverberg partition, then all (finitely many) partitions have the property that there is no common point in all the convex hulls, or that some convex hulls are (properly) nested. This property is maintained under any sufficiently small perturbation. Since every point set is the limit of a sequence of point sets in general position, it follows from closedness that if all points sets in general position allow a crossing Tverberg partition, then all point sets do.
Under this general position assumption, we may also assume that the union of the $X_i$'s of size $d+1$ and the common point of all convex hulls are in general position. Henceforth, we assume general position without explicitly mentioning it.
We start with a Tverberg partition of $X$ into sets $X_1,\ldots,X_r$ of size at most $d+1$; such a partition is guaranteed by Tverberg's Theorem~\ref{thm:Tverberg2}. W.l.o.g.\ we assume that the Tverberg point is the origin.
\subsection{\emph{Fixing Pairs}}
The sets $X_1,\ldots,X_{r}$ may not yet satisfy the requirements of Theorem~\ref{thm:main}, since there may be pairs $\{X_i, X_j\}$, $X_i\not= X_j$ such that $\mathop{\mathrm{conv}}(X_i)$ and $\mathop{\mathrm{conv}}(X_j)$ do not cross. As the simplices $\mathop{\mathrm{conv}}(X_i)$ and $\mathop{\mathrm{conv}}(X_j)$ themselves intersect in the origin, the only remaining possibility is that one of them contains the other one, i.e., they are nested; see Figure~\ref{fig:tverberg_partition} (right). Lemma~\ref{lem:unnesting} whose proof we approach next is a key step, allowing us to ``unnest'' simplices.
In fact, it is enough to show that for any pair $\{T,T'\}$ as in the lemma, there exists a \emph{different} pair
$\{S,S'\}$ such that $S\cup S'=T\cup T'$, $\mathbf{0} \in \mathop{\mathrm{conv}}(S)\cap\mathop{\mathrm{conv}}(S')$. The reason is that at most one of these pairs can be nested: the outer simplex of a nested pair coincides with $\mathop{\mathrm{conv}}(T\cup T')$ and is therefore uniquely determined. Hence, Lemma~\ref{lem:unnesting} is implied by the following.
\begin{lemma}\label{lemparity} Let $V\subset \mathbb R^d$, $|V|=2(d+1)$, such that $V\cup\{\mathbf{0}\}$ is in general position. Then
$$\Big|\Big\{\{F,G\}:F,G\in {V\choose d+1}, F\cap G=\emptyset, \mathbf{0}\in \mathop{\mathrm{conv}} F\cap \mathop{\mathrm{conv}} G\Big\}\Big| \ \ \ \text{is even.}$$
\end{lemma}
In particular, if there is one such pair $\{F,G\}$, then there is another.
\vskip+0.1cm
\textbf{Remark. } We note that this statement and its application is quite unusual. Typically, one proves that the number of objects of a certain type is always odd and thus at least one object of the type exists.
\subsubsection{Geometric proof of Lemma~\ref{lemparity} for $d=2$}
The purely combinatorial proof that we give in the next section for general $d$ is not difficult, but does not provide any geometric intuition. We therefore start with a simple proof in the plane.
Consider a set $V$ of $2(d+1)=6$ points in the plane. We remark that the statement is invariant under scaling points, and thus we may assume that all points lie on a circle with the center in the origin. Due to our general position assumption, no two points from $V$ lie on a line passing through the origin; see Figure~\ref{fig:geomproof} (left).
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.6\textwidth]{geomproof}
\caption{6 points in the plane (black) and their mirror images (white); there must be two consecutive black points $p,p'$.\label{fig:geomproof}}
\end{center}
\end{figure}
Now we mirror each point at the origin and obtain another $6$ points, drawn in white in Figure~\ref{fig:geomproof} (right). We observe that in the circular order of points, there must be two consective ones of the same color. Indeed, an alternating pattern (black, white, black, white,\ldots) would lead to pairs of mirrored points having the \emph{same} color. Let $p,p'$ be two consecutive points of the same color; by going to the mirror points if necessary, we may assume that they are black and hence belong to $V$.
We make two observations (actually, just one). (i) $p$ and $p'$ cannot belong to a triple $F\subset V$ such that $\mathbf{0}\in\mathop{\mathrm{conv}}(F)$, as otherwise, the third point would get mirrored to a white point between $p$ and $p'$; see Figure~\ref{fig:geomproof2} (left). (ii) For any $q\in V\setminus\{p,p'\}$, the two segments $\mathop{\mathrm{conv}}(\{p,q\})$ and $\mathop{\mathrm{conv}}(\{p',q\})$ have the origin on the same side; see Figure~\ref{fig:geomproof2} (right).
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.6\textwidth]{geomproof2}
\caption{Consecutive black points are combinatorially indistinguishable.\label{fig:geomproof2}}
\end{center}
\end{figure}
This implies the following: if $\{F,G\}$ is a partition of $V$ that we count in Lemma~\ref{lemparity}, then (i) $p$ and $p'$ are in different parts, and (ii) swapping $p$ and $p'$ between the parts leads to a different partition $\{F',G'\}$ that we also count. In other words, $p$ and $p'$ are ``combinatorially indistinguishable'' with respect to the relevant properties, and the operation of swapping them between parts establishes a matching between the partitions that we want to count. Hence, their number is even.
\subsubsection{Combinatorial proof of Lemma~\ref{lemparity}}
Recall that for $|V|=2(d+1)$, we need to show that there is an even number of partitions $\{F,G\}$ of $V$ into parts of equal size $d+1$ such that $0\in\mathop{\mathrm{conv}}(F)\cap\mathop{\mathrm{conv}}(G)$.
We show that this follows from the fact that the $(d+1)$-element subsets $F$ with $\mathbf{0}\in\mathop{\mathrm{conv}}(F)$ form a \emph{cocycle}, a concept borrowed from topology. The proof itself does not use any topology, though.
\begin{definition} Let $n\ge k\ge 1$ be integers, and let $V$ be a set with $n$ elements. A family $\mathcal C\subset {V\choose k}$ of $k$-element subsets of $V$ is a {\rm cocycle} if $$|\{F\in \mathcal C: F\subset M\}| \ \ \text{is even for every } M\subset V\ \ \text{with } \ |M|=k+1. $$\end{definition}
\textbf{Example. } Fix a set $D\in {V\choose k-1}$. Then $\delta D:=\{F\in {V\choose k}: D\subset F\}$ is a cocycle. Indeed, a $(k+1)$-element subset $M\subset V$ either contains zero sets in $\delta D$ (if $D\not \subset M$) or exactly two sets in $\delta D$ (If $D=M\setminus\{p,q\}$).
\begin{lemma}\label{lem1}
Let $V\subset \mathbb R^d$ be such that $V\cup\{\mathbf{0}\}$ is in general position. Then
$$\mathcal C(V):=\big\{F\in {V\choose d+1}: \mathbf{0} \in \mathop{\mathrm{conv}} F\big\}$$
is a cocycle.
\end{lemma}
\begin{proof}
Let $M:=\{v_1,\ldots,v_{d+2}\}\subset V$. Lift the points to dimension $d+1$ such that the convex hull of the lifted point set $\hat{M}$ is a full-dimensional simplex $\Delta$; see Figure~\ref{fig:lifting}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.6\textwidth]{lifting}
\caption{Geometric proof of Lemma~\ref{lem1}\label{fig:lifting}}
\end{center}
\end{figure}
Any set $\hat{F}\subset \hat{M}$ of size $d+1$ spans a facet of $\Delta$, and we have $\mathbf{0}\in\mathop{\mathrm{conv}}(F)$ if and only if the vertical line through $\mathbf{0}$ intersects that facet. As $V\cup\{\mathbf{0}\}$ is in general position, this line intersects zero or two facets of $\Delta$, hence $|\{F\in \mathcal C(V): F\subset M\}|\in\{0,2\}$.
We remark that there is also an elementary linear algebra version of this ``proof by picture'' (omitted in this extended abstract).
\end{proof}
By Lemma~\ref{lem1}, Lemma~\ref{lemparity} is now simply a special case of the following main result of this section.
\begin{theorem}\label{prop2} Let $k\ge 1$, $|V|=2k$ and let $\mathcal C\subset {V\choose k}$ be a cocycle. Set $$\mathcal P_{\mathcal C}:=\big\{\{F,G\}: F,G\in \mathcal C, F\cap G=\emptyset \big\}.$$
Then $|\mathcal P_{\mathcal C}|$ is even.
\end{theorem}
\begin{proof}
We double count the edges of a suitable bipartite graph. Let $v\in V$ be an arbitrary but fixed element, and let one part be
\[
\mathcal D := \big\{D\in \binom{V\setminus\{v\}}{k-1}: D\cup\{v\}\in\mathcal C\big\}.
\]
The other part is
\[
\mathcal P := \big\{\{F,G\}: F\in {V\choose k}, G\in \mathcal C, F\cap G=\emptyset \big\}.
\]
In particular, $\mathcal P\supset \mathcal P_{\mathcal C}$. We connect $D\in\mathcal D$ and $\{F,G\}\in\mathcal P$ by an edge if and only if $D\subset F$.
Now, we make the following two claims about this bipartite graph.
\begin{itemize}
\item[(i)] Every set $D\in\mathcal D$ has even degree.
\item[(ii)] A pair $\{F,G\}\in\mathcal P$ has odd degree if and only if $\{F,G\}\in\mathcal P_{\mathcal C}$.
\end{itemize}
The statement of the lemma immediately follows from~(i) and (ii) due to the fact that the number of odd-degree vertices in every graph is even. It remains to prove the two claims.\smallskip
To see (i), fix $D\in\mathcal D$. The edge condition $D\subset F$ is equivalent to $G\subset V\setminus D$. As $\mathcal C$ is a cocycle and the size of $M:=V\setminus D$ is $k+1$, the number of such $G\in\mathcal C$ is even, and each one determines a unique pair $\{V\setminus G,G\}\in\mathcal P$ that is connected to $D$.
For (ii), fix a pair $\{F,G\}\in\mathcal P$. If $v\in F$, there is only one candidate for a neighbor, namely $D=F\setminus\{v\}$. By definition of $\mathcal D$, this candidate is actually a neighbor if and only if
$F=D\cup\{v\}\in\mathcal C$. Hence, $\{F,G\}$ has degree 1 if $F\in\mathcal C$ and degree 0 otherwise.
If $v\notin F$, there are $|F|$ candidates for neighbors, namely the sets $D_w=F\setminus\{w\},w\in F$.
As before, $D_w$ is actually a neighbor if and only if $D_w\cup\{v\}= F\cup\{v\}\setminus\{w\}\in\mathcal C$.
Consider the set $M:=F\cup\{v\},|M|=k+1$. Its $k$-elements subsets are $F$ as well as all the $D_w\cup\{v\}$.
As $\mathcal C$ is a cocycle, an even number of them is contained in $\mathcal C$. Hence, if $F\in\mathcal C$, then $\{F,G\}$ has an odd number of neighbors $D_w$, otherwise an even number.
\end{proof}
\subsection{Fixing terminates}
Now we are prepared to finish the proof of Theorem~\ref{thm:main}, according to the outline already given in the introduction. We start with an arbitrary Tverberg partition $\mathcal{X}=\{X_1,\ldots, X_n\}$. As long as there exists a pair of nested simplices $\mathop{\mathrm{conv}}(X_i)\subset\mathop{\mathrm{conv}}(X_j)$, we apply the fixing operation (Lemma~\ref{lem:unnesting}) to replace $T=X_i$ and $T'=X_j$ with $S,S'$ such that $\mathop{\mathrm{conv}}(S)$ and $\mathop{\mathrm{conv}}(S')$ cross. We need to show that after finitely many fixes, there are no nested pairs anymore, in which case we have a crossing Tverberg partition.
To see termination, we observe that in any fixing operation that replaces $X_i$ and $X_j$ such that $\mathop{\mathrm{conv}}(X_i)\subset \mathop{\mathrm{conv}}(X_j)$, the simplex $\mathop{\mathrm{conv}}(X_j)$ is volume-wise the unique largest $d$-dimensional simplex that can be formed from the $2(d+1)$ points $X_i\cup X_j$ involved in the operation. Hence, the two simplices $\mathop{\mathrm{conv}}(S)$ and $\mathop{\mathrm{conv}}(S')$ replacing $\mathop{\mathrm{conv}}(X_i)$ and $\mathop{\mathrm{conv}}(X_j)$ are volume-wise both strictly smaller than $\mathop{\mathrm{conv}}(X_j)$. Therefore, if we order all full-dimensional simplices by decreasing volume, the sequence of these volumes goes down lexicographically in every fixing operation.
Formally, let ${\cal V}=(V_1,V_2\ldots V_s), s\leq r$ be the sequence of volumes in decreasing order before the fix, and ${\cal V}'=(V'_1,V'_2\ldots V'_s)$ the decreasing order after the fix. Moreover, suppose that $k$ is the largest index at which the volume of $\mathop{\mathrm{conv}}(X_j)$ appears in ${\cal V}$ (by even more general position, we could assume that there is a unique such index, but this does not really help here). The volume of $\mathop{\mathrm{conv}}(X_i)$ appears at a position $\ell>k$. As the fixing operation removes a volume equal to $V_k$ and inserts two volumes smaller than $V_k$, we have that ${\cal V}$ and ${\cal V}'$ agree in the first $k-1$ positions, but $V'_{k}<V_k$. This exactly defines the relation ``${\cal V'}<{\cal V}$'' in decreasing lexicographical order, and as this is a total order, fixing must eventually terminate.
\vskip+0.1cm
\textbf{Remark. } Instead of volume, we may use the number of points from $X$ inside $\mathop{\mathrm{conv}}(X_j)$ as a measure.
\section{Discussion}\label{sec3}
\subsection{A Topological version of Theorem~\ref{thm:main}}
If $r$ is a prime power, then Tverberg's theorem admits a topological generalization, known as the \emph{Topological Tverberg Theorem}:
\begin{theorem}
\label{thm:topTverberg}
If $d\in \mathbb{N}$ and if $r$ is a prime power, then for every continuous map from the $(d+1)(r-1)$-dimensional simplex $\Delta_{(d+1)(r-1)}$ to $\mathbb{R}^d$, there exist $r$ pairwise disjoint faces of $\Delta_{(d+1)(r-1)}$ whose images intersect in a common point.
\end{theorem}
This result was first proved in the case when $r$ is a prime by {B{\'a}r{\'a}ny, Shlosman and Sz{\"u}cs~\cite{BSS1981_topTverbergPrime} and later extended to prime powers by \"{O}zaydin~\cite{O1987_topTverberg}. On the other hand, Theorem~\ref{thm:topTverberg} is false if $r$ is not a prime power: By work of Mabillard and Wagner \cite{MabillardWagner2015} and a result of \"{O}zaydin~\cite{O1987_topTverberg}, a closely related result (the \emph{generalized Van Kampen--Flores Theorem}) is false whenever $r$ is not a prime power and, as observed by Frick~\cite{F15_approx}, the failure of Theorem~\ref{thm:topTverberg} for $r$ not a prime power follows from this by a reduction due to Gromov \cite{Gromov} and to Blagojevi\'{c}, Frick, and Ziegler \cite{BFZ_Constraints_2014}. The lowest dimension in which counterexamples are known to exist is $d=2r$ \cite{AMSW15_codim2,MabillardWagner2015}.
We refer to the recent surveys~\cite{Barany2018,BZ2017_story,S18_user,Z18} for more background on the Topological Tverberg Theorem and its history.
In the same vein, it is natural to wonder if our Theorem~\ref{thm:main} also extends to the topological setting. The straightforward approach to generalize our result would be to use the Topological Tverberg Theorem instead of Tverberg's theorem and then keep fixing the pairs of simplices whose boundaries do not mutually intersect. To this end we need an adaptation of Lemma~\ref{lem1} and the fixing procedure
to the topological setting. The rest of our argument is free of any geometry except for the use of Carath\'{e}odory's theorem which is not really crucial. While extending Lemma~\ref{lem1} is easy, showing the termination of the fixing procedure appears to be quite difficult except under the scenario which we discuss below.
First, we discuss planar extensions of Theorem~\ref{thm:main}.
An \emph{arrangement of pseudolines} $\mathbb{P}$ is a finite set of not self-intersecting open arcs, called \emph{pseudolines}, in $\mathbb{R}^2$ such that (i) For every pair $P_1,P_2\in \mathbb{P}$ of two distinct pseudolines, $P_1$ and $P_2$ intersect transversely in a single point, and (ii) $\mathbb{R}^2\setminus P$ is not connected for every $P\in \mathbb{P}$.
A drawing of a complete graph on $n$ vertices $K_n$ in the plane is \emph{pseudolinear} if the edges can be extended to an {arrangement of pseudolines}.
In the plane, it is not hard to see that Theorem~\ref{thm:main} and its proof almost extends to the setting of pseudolinear drawings of complete graphs. Since the Topological Tverberg Theorem is only valid for prime powers $r$, the number of pairwise crossing triangles is slightly smaller then the number of vertices divided by 3.
\begin{theorem} \label{thm:mainTopPlane}
In a pseudolinear drawing of a complete graph $K_{3m}$ we can find $n=(1-o(1))m$ vertex-disjoint and pairwise crossing triangles. Moreover, the topological discs bounded by these triangles intersect in a common point.
\end{theorem}
\begin{proof}
Let $\mathcal{D}$ be a pseudolinear drawing of $K_{3m}$.
We put $n$ to be the largest prime power not larger than $m+1$.
By the asymptotic law of distribution of prime numbers $n=(1-o(1))m$.
Next, apply the Topological Tverberg theorem with $r=n$ and $d=2$ to a map $\mu:\Delta_{3n-3}\rightarrow \mathbb{R}^2$
which extends $\mathcal{D}$ as follows.
We define $\mu$ on the 1-dimensional skeleton of $\Delta_{3n-3}$ as a restriction of $\mathcal{D}$ to some $K_{3n-2}$.
Note that every triangle of $K_{3n-2}$ is drawn by $\mathcal{D}$ as a closed arc without self intersections. The map $\mu$ extends to the 2-dimensional skeleton of $\Delta_{3n-3}$ so that every 2-dimensional face is mapped homeomorphically in $\mathbb{R}^2$. We define the map $\mu$ on the rest of $\Delta_{3n-3}$ arbitrarily while maintaining continuity.
Analogously to the proof of Theorem~\ref{thm:main}, an application of the Topological Tverberg Theorem gives us $n-1$ disjoint $2$-dimensional faces $F_1,\ldots, F_{n-1}$ of $\Delta_{3n-3}$ whose images under $\mu$ intersect in a common point.
To this end we apply Carath\'eodory's theorem for drawings of complete graphs~\cite[Lemma 4.7]{BFK15_monotone} instead of the original version of Carath\'eodory's theorem.
Let $T_i$ denote the boundary of $F_i$, for $i=1,\ldots, n-1$.
Note that each $T_i$ is a triangle in $K_{3n-2}$.
If all pairs $T_i$ and $T_j$ are crossing then we are done. Otherwise, we perform the fixing operations, which can be done since Lemma~\ref{lem1} easily extends to the setting in which we replace simplices by images of 2-dimensional faces of $\Delta_{3n-3}$ under $\mu$.
The procedure of applying successively the fixing operation terminates.
To this end it is enough to show that if $\mu(T_i) \cap \mu(T_j)=\emptyset$, and let's say $\mu(F_i)\subset \mu(F_j)$, then the restriction of $\mu$ to the subgraph of $K_{3n-2}$ induced by $V(T_i)\cup V(T_j)$ is contained in $\mu(F_j)$.
Indeed, then the volume argument goes through if we consider volumes of $\mu(F_i)$'s.
The previous claim follows since
every 4 vertices in a pseudolinear drawing of a complete graph induce
either a crossing free drawing of $K_4$ or a drawing of $K_4$ with exactly one pair of crossing edges in the interior of a disc bounded by a crossing free 4-cycle, see Figure~\ref{fig:pseudo} for an illustration.
\end{proof}
\begin{figure}
\centering
\includegraphics[scale=1]{pseudolinear}
\caption{The allowed drawings of $K_4$ in a pseudolinear drawing $K_n$ (left). The forbidden drawing of $K_4$ in a pseudolinear drawing $K_n$ (right).\label{fig:pseudo}}
\end{figure}
A drawing of a graph in the plane is \emph{simple} if every pair of edges intersect at most once either at a common end point or in a proper crossing.
Clearly, all pseudolinear drawings of complete graphs are also simple, but not vice-versa. Hence, it might be worthwhile to extend Theorem~\ref{thm:mainTopPlane} to simple drawings of complete graphs.
If we want the interiors of the triangles to be pairwise intersecting, we only known that we can take $m=O(\log n^{1/6})$ which is easily derived from the following result of Pach, Solymosi and T\'oth~\cite{PST2003_unavoidable}. Every simple drawing of $K_n$ contains a drawing of $K_m$ that is weakly isomorphic to a so-called convex complete graph or a twisted complete graph, see Figure~\ref{fig:twisted}, for which Theorem~\ref{thm:mainTopPlane} holds. We omit the proof of the latter which is rather straightforward. For example, if in a twisted drawing of $K_{3n}$ the vertices are labeled as indicated in the figure, $\{\{0+i,n+i,2n+i\}| \ i=0,\ldots, n-1\}$ is a crossing Tverberg partition.
If we do not insist on the interiors of the triangles to be pairwise intersecting, we know that $m$ can be taken to be at least $O(n^{\varepsilon})$ for some small $\varepsilon>0$ by the following result of Pach and T\'oth~\cite{FP2012_coloring}.
Every simple drawing of $K_n$ contains $O(n^{\varepsilon})$ pairwise crossing edges for some $\varepsilon>0$.
\begin{figure}
\centering
\includegraphics[scale=0.7]{twisted}
\caption{The twisted drawing of $K_6$. The bold grey and black edges form boundaries of two crossing triangles. The symbol x marks a point contained in the interior of these two triangles.\label{fig:twisted}}
\end{figure}
Theorem~\ref{thm:mainTopPlane} could be generalized to hold for an appropriate high dimensional analog of pseudolinear drawings. Since this would require introducing many technical terms and would not offer substantially interesting content we refrain from doing so.
\subsection{Stronger conditions on crossings}
A pair of vertex disjoint $(\lceil d/2 \rceil-1)$-dimensional simplices in general position in $d$-space does not intersect.
Hence, the pairwise intersection of the boundaries of $\mathop{\mathrm{conv}}(X_i)$'s in the conclusion of Theorem~\ref{thm:main}, cannot be strengthened to the pairwise intersection of lower than
$\lceil d/2 \rceil$-dimensional skeleta of the boundaries.
Nevertheless, for $d=3$ one may ask if in the setting of Lemma~\ref{lem:unnesting}, we get a stronger property along the following lines.
Can we guarantee the existence of a pair of vertex-disjoint tetrahedra $\{S,S'\}$ that both contain the origin and such that the boundary of a 2-dimensional face of $S$ is linked with the boundary of a 2-dimensional face of $S'$?
Again, the answer to this question is negative. Stefan Felsner and Manfred Scheucher found the following set of 8 points:
$$(3, -2, 2),\ \ \ \
(2, -5, 3),\ \ \ \
(-3, 0, -4),\ \ \ \
(-1, 2, 0),$$
$$(1, -5, -4),\ \ \ \
(4, 1, -2),\ \ \ \
(-2, -5, -4),\ \ \ \
(-3, 1, 3).$$
This set contains a pair of disjoint tetrahedra both containing the origin $(0,0,0)$, but no two disjoint linked tetrahedra
both containing the origin.
The example was found using a SAT solver who found an abstract order type with the require property. A realization of the order type with actual points was obtained with a randomized procedure.
\subsection{Computational complexity of finding a crossing Tverberg partition}
A natural question is whether we can find the partition of the point set given by Theorem~\ref{thm:main} efficiently, i.e., in polynomial time in the size of $X$.
A straightforward way to construct an algorithm is to make the proof of Theorem~\ref{thm:main} algorithmic.
To this end we first need an algorithm for finding a Tverberg partition and also a Tverberg point. Unfortunately, several results suggest that
an efficient algorithm for this problem is rather unlikely to exist. Since a Tverberg partition is guaranteed to exist, NP-complexity theory does not apply to the algorithm problem of finding it, and for example, PPAD completeness theory~\cite{P94_complexity} appears to be more suitable for this problem.
Nevertheless, we are not aware of any result in this direction.
The only closely related hardness result we are aware of is the one by Teng~\cite[Theorem 8.14]{T92_points}, who proved that checking whether a given point is a Tverberg point of a given point set is NP-complete.
A line of research on finding an approximate Tverberg partition efficiently was initiated by Miller and Sheehy~\cite{MS10_approx} and further developed in~\cite{MW13_approx,RS16_approx}.
In particular, Mulzer and Werner~\cite{MW13_approx} showed that it is possible to find in time $d^{O(\log d)|X|}$ an approximate Tverberg partition of size $\left\lceil \frac{|X|}{4(d+1)^3}\right\rceil$, whereas Theorem~\ref{thm:Tverberg} guarantees the partition of size $\left\lceil \frac{|X|}{d+1}\right\rceil$.
If we aim only at an approximate algorithmic version of our Theorem~\ref{thm:main} along the lines of the result of Mulzer and Werner, we face the problem of efficiently fixing an (approximate) Tverberg partition to make it crossing. Due to the fact that our termination argument for the iterated \emph{Fixing Pairs} procedure relies on progress in the lexicographical ordering of the simplex volumes, we may potentially need exponentially many (in the size of~$X$) iterations before we arrive at a crossing partition.
We leave it as an interesting open problem to prove or disprove that there is always a way to invoke the \emph{Fixing Pairs} operation only polynomially (or least subexponentially) many times in order to arrive at a partition required by Theorem~\ref{thm:main}.
| {
"timestamp": "2018-12-13T02:10:43",
"yymm": "1812",
"arxiv_id": "1812.04911",
"language": "en",
"url": "https://arxiv.org/abs/1812.04911",
"abstract": "Tverberg's theorem is one of the cornerstones of discrete geometry. It states that, given a set $X$ of at least $(d+1)(r-1)+1$ points in $\\mathbb R^d$, one can find a partition $X=X_1\\cup \\ldots \\cup X_r$ of $X$, such that the convex hulls of the $X_i$, $i=1,\\ldots,r$, all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed.As a concrete application, we show that any $n$ points in the plane in general position span $\\lfloor n/3\\rfloor$ vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Rebollar et al.\\ guarantees $\\lfloor n/6\\rfloor$ pairwise crossing triangles. Our result generalizes to a result about simplices in $\\mathbb R^d,d\\ge2$.",
"subjects": "Computational Geometry (cs.CG); Combinatorics (math.CO)",
"title": "The Crossing Tverberg Theorem",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.988841966778967,
"lm_q2_score": 0.7154239897159439,
"lm_q1q2_score": 0.7074412650715693
} |
https://arxiv.org/abs/0707.3895 | Knot colouring polynomials | This article introduces a natural extension of colouring numbers of knots, called colouring polynomials, and studies their relationship to Yang-Baxter invariants and quandle 2-cocycle invariants.For a knot K in the 3-sphere let \pi_K be the fundamental group of the knot complement, and let (m_K,l_K) be a meridian-longitude pair in \pi_K. Given a finite group G and an element x in G, we consider the set of representations \rho from \pi_K to G that map the meridian m_K to x, and define the colouring polynomial P(K) as the sum over all longitude images \rho(l_K). The resulting invariant maps knots to the group ring Z[G]. It is multiplicative with respect to connected sum and equivariant with respect to symmetry operations of knots. Examples are given to show that colouring polynomials distinguish knots for which other invariants fail, in particular they can distinguish knots from their mutants, obverses, inverses, or reverses.We prove that every quandle 2-cocycle state-sum invariant of knots is a specialization of some knot colouring polynomial. This provides a complete topological interpretation of these invariants in terms of the knot group and its peripheral system. Furthermore, we show that P can be presented as a Yang-Baxter invariant, i.e. as the trace of some linear braid group representation. This entails in particular that Yang-Baxter invariants can detect non-inversible and non-reversible knots. | \section{Introduction and statement of results} \label{sec:Introduction}
To each knot $K$ in the $3$-sphere $\S^3$ we can associate its knot group,
that is, the fundamental group of the knot complement,
denoted by $\pi_{K} := \pi_1(\S^3\minus K)$.
This group is already a very strong invariant: it classifies
unoriented prime knots \cite{Whitten:1987,GordonLuecke:1989}.
In order to capture the complete information,
we consider a meridian-longitude pair $m_K,l_K \in \pi_{K}$:
the group system $(\pi_{K},m_K,l_K)$ classifies
oriented knots in the $3$-sphere \cite{Waldhausen:1968}.
In particular, the group system allows us to tackle
the problem of detecting asymmetries of a given knot
(see \textsection\ref{sub:KnotGroupSymmetries}).
Using this ansatz, M.\,Dehn \cite{Dehn:1914} proved in 1914 that
the two trefoil knots are chiral, and, half a century later,
H.F.\,Trotter \cite{Trotter:1963} proved
that bretzel knots are non-reversible.
We will recover these results using knot colouring
polynomials (see \textsection\ref{sub:Examples}).
Given a knot $K$, say represented by some planar diagram $D$,
we can easily read off the Wirtinger presentation of $\pi_{K}$
in terms of generators and relations (see \textsection\ref{sub:WirtingerQuandles}).
In general, however, such presentations are very difficult to analyze.
As R.H.\,Crowell and R.H.\,Fox \cite[\textsection VI.5]{CrowellFox:1963} put it:
\begin{quote}
``What is needed are some standard procedures
for deriving from a group presentation some
easily calculable algebraic quantities
which are the same for isomorphic groups
and hence are so-called group invariants.''
\end{quote}
The classical approach is, of course, to consider
abelian invariants, most notably the Alexander polynomial.
In order to effectively extract non-abelian information,
we consider the set of knot group homomorphisms
$\Hom(\pi_{K};G)$ to some finite group $G$.
The aim of this article is to organize this information
and to generalize colouring numbers to colouring polynomials.
In doing so, we will highlight the close relationship
to Yang-Baxter invariants and their deformations on the one hand,
and to quandle cohomology and associated state-sum invariants on the other hand.
\subsection{From colouring numbers to colouring polynomials}
A first and rather crude invariant is given by
the total number of $G$-representations, denoted by
\[
\TF{G}(K) := |\Hom(\pi_{K};G)|.
\]
This defines a map $\TF{G} \colon \knots \to \Z$
on the set $\knots$ of isotopy classes of knots in $\S^3$.
This invariant can be refined by further specifying
the image of the meridian $m_K$, that is,
we choose an element $x\in G$ and consider only
those homomorphisms $\rho\colon \pi_{K} \to G$ satisfying $\rho(m_K) = x$.
Their total number defines the knot invariant
\[
\F{x}{G}(K) := |\Hom(\pi_{K},m_K;G,x)|.
\]
\begin{example}
Let $G$ be the dihedral group of order $2p$, where $p\ge3$ is odd,
and let $x\in G$ be a reflection. Then $\F{x}{G}$ is the number of $p$-colourings
as introduced by R.H.\,Fox \cite{Fox:1962}, here divided by $p$
for normalization such that $\F{x}{G}(\uk) = 1$.
\end{example}
We will call $\F{x}{G}$ the \emph{colouring number} associated with $(G,x)$,
in the dihedral case just as well as in the general case of an arbitrary group.
Obviously $\TF{G}$ can be recovered from $\F{x}{G}$ by summation over all $x\in G$.
In order to exploit the information of meridian \emph{and}
longitude, we introduce knot colouring polynomials as follows:
\begin{definition} \label{def:ColPoly}
Suppose that $G$ is a finite group and $x$ is one of its elements.
The \emph{colouring polynomial} $\P{x}{G}\colon \knots \to \Z{G}$
is defined as
\[
\P{x}{G}(K) := \sum_\rho \; \rho(l_K) ,
\]
where the sum is taken over all homomorphisms
$\rho\colon \pi_{K} \to G$ with $\rho(m_K) = x$.
\end{definition}
By definition $\P{x}{G}$ takes its values in the semiring $\N{G}$,
but we prefer the more familiar group ring $\Z{G} \supset \N{G}$.
Note that we recover the colouring number $\F{x}{G} = \varepsilon \P{x}{G}$
by composing with the augmentation map $\varepsilon\colon \Z{G} \to \Z$.
As it turns out, colouring polynomials allow us in a simple
and direct manner to distinguish knots from their mirror images,
as well as from their reverse or inverse knots.
We will highlight some examples below.
\subsection{Elementary properties}
The invariant $\P{x}{G}$ behaves very much like classical knot polynomials.
Most notably, it nicely reflects the natural operations on knots:
$\P{x}{G}$ is multiplicative under connected sum and equivariant
under symmetry operations (\textsection\ref{sub:KnotGroupSymmetries}).
Strictly speaking, $\P{x}{G}(K)$ is, of course,
not a polynomial but an element in the group ring $\Z{G}$.
Since $l_K$ lies in the commutator subgroup $\pi_{K}'$ and commutes with $m_K$,
possible longitude images lie in the subgroup $\Lambda = C(x) \cap G'$.
Very often this subgroup will be cyclic, $\Lambda = \gen{t}$ say, in which case $\P{x}{G}$
takes values in the truncated polynomial ring $\Z\Lambda = \Z[t]/(t^n)$.
Here is a first and very simple example:
\begin{example} \label{exm:TrefoilAlt5}
We choose the alternating group $G = \Alt{5}$ with basepoint $x = (12345)$.
Here the longitude subgroup $\Lambda = \gen{x}$ is cyclic of order $5$.
The colouring polynomials of the left- and right-handed trefoil knots
are $1+5x$ and $1+5x^{-1}$ respectively, hence the trefoil knots are chiral.
(A typical colouring is shown in \textsection\ref{sec:Quandles},
Figure \ref{fig:TrefoilAlt5}.)
Starting from scratch, i.e.\ knot diagrams and Reidemeister moves,
one usually appreciates Fox' notion of $3$-colourability \cite{Fox:1962}
as the simplest proof of knottedness. In this vein, the preceding example
is arguably one of the most elementary proofs of chirality,
only rivalled by Kauffman's bracket leading to
the Jones polynomial \cite{Kauffman:1987}.
\end{example}
Section \ref{sub:Examples} displays some further examples
to show that colouring polynomials distinguish knots
for which other invariants fail:
\begin{itemize}
\item
They distinguish the Kinoshita-Terasaka knot
from the Conway knot and show that none of them is
inversible nor reversible nor obversible.
\item
They detect asymmetries of bretzel knots;
they distinguish, for example, $B(3,5,7)$
from its inverse, reverse and obverse knot.
\item
They distinguish the (inversible)
knot $8_{17}$ from its reverse.
\end{itemize}
We also mention two natural questions that will not be pursued here:
\begin{question}
Can knot colouring polynomials detect other geometric properties of knots?
Applications to periodic knots and ribbon knots would be most interesting.
\end{question}
\begin{question}
Do colouring polynomials distinguish all knots?
Since the knot group system $(\pi_{K},m_K,l_K)$ charaterizes
the knot $K$ \cite[Cor.\,6.5]{Waldhausen:1968}, and knot groups
are residually finite \cite[Thm.\,3.3]{Thurston:1982},
this question is not completely hopeless.
\end{question}
\subsection{Colouring polynomials are Yang-Baxter invariants}
Moving from empirical evidence to a more theoretical level,
this article compares knot colouring polynomials with
two other classes of knot invariants:
Yang-Baxter invariants, i.e.\ knot invariants
obtained from Yang-Baxter representations of the braid group,
and quandle colouring state-sum invariants
derived from quandle cohomology.
The result can be summarized as follows:
\[
\Biggl\{ \begin{matrix} \text{Yang-Baxter} \\
\text{invariants}
\end{matrix} \Biggr\}
\supset
\Biggl\{ \begin{matrix} \text{colouring} \\
\text{polynomials}
\end{matrix} \Biggr\}
\supset
\Biggl\{ \begin{matrix} \text{quandle $2$-cocycle} \\
\text{state-sum invariants}
\end{matrix} \Biggr\}
\supset
\Biggl\{ \begin{matrix} \text{colouring} \\
\text{polynomials} \\
\text{with $\Lambda$ abelian}
\end{matrix} \Biggr\}
\]
P.J.\,Freyd and D.N.\,Yetter \cite[Prop.\,4.2.5]{FreydYetter:1989}
have shown that every colouring number $\F{x}{G}\colon \knots \to\Z$
can be obtained from a certain Yang-Baxter operator $c$ over $\Z$.
We generalize this result to colouring polynomials:
\begin{theorem}[\textsection\ref{sub:YBclosed}] \label{Thm:YBclosed}
Suppose that $G$ is a group with basepoint $x$
such that the subgroup $\Lambda = C(x)\cap G'$ is abelian.
Then the colouring polynomial $\P{x}{G}\colon \knots \to \Z\Lambda$
is a Yang-Baxter invariant of closed knots: there exists
a Yang-Baxter operator $\tilde{c}$ over the ring $\Z\Lambda$,
such that the associated knot invariant
coincides with (a constant multiple) of $\P{x}{G}$.
\end{theorem}
In the general case, where $\Lambda$ is not necessarily abelian,
Section \ref{sub:YBlong} gives an analogous presentation of $\P{x}{G}$
as a Yang-Baxter invariant of long knots (also called $1$-tangles).
\begin{corollary}
Since $\Lambda$ is abelian in all our examples of \textsection\ref{sub:Examples},
it follows in particular that Yang-Baxter invariants
\emph{can} detect non-inversible and non-reversible knots.
\end{corollary}
\begin{remark}
It follows from our construction that $\tilde{c}$
is a deformation of $c$ over the ring $\Z\Lambda$.
Conversely, the deformation ansatz leads to quandle
cohomology (see \textsection\ref{sub:Conclusion}).
Elaborating this approach, M.\,Gra\~na \cite{Grana:2002}
has shown that quandle $2$-cocycle state-sum invariants
are Yang-Baxter invariants. The general theory
of Yang-Baxter deformations of $c_Q$ over
the power series ring $\K\mathopen{[\![} h \mathclose{]\!]}$
has been developed in \cite{Eisermann:2005}.
\end{remark}
\begin{remark}
The celebrated Jones polynomial and, more generally,
all quantum invariants of knots, can be obtained
from Yang-Baxter operators that are formal
power series deformations of the trivial operator.
This implies that the coefficients in this expansion
are of finite type \cite[\textsection2.1]{BarNatan:1995}.
Part of their success lies in the fact that these invariants distinguish
many knots, and in particular they easily distinguish mirror images.
It is still unknown, however, whether finite type invariants
can detect non-inversible or non-reversible knots.
For colouring polynomials the construction is similar in that
$\P{x}{G}$ arises from a deformation of a certain operator $c$.
There are, however, two crucial differences:
\begin{itemize}
\item
The initial operator $c$ models conjugation (and is not the trivial operator),
\item
Its deformation $\tilde{c}$ is defined over $\Z\Lambda$ (and not over a power series ring).
\end{itemize}
As a consequence, the colouring polynomial $\P{x}{G}$ is not of finite type,
nor are its coefficients, nor any other rational-valued invariant
computed from it \cite{Eisermann:2000}.
\end{remark}
\subsection{Quandle invariants are specialized colouring polynomials}
A quandle, as introduced by D.\,Joyce \cite{Joyce:1982},
is a set $Q$ with a binary operation whose axioms model conjugation in a group,
or equivalently, the Reidemeister moves of knot diagrams.
Quandles have been intensively studied
by different authors and under various names;
we review the relevant definitions in \textsection\ref{sec:Quandles}.
The Lifting Lemma proved in \textsection\ref{sub:LiftingLemma}
tells us how to pass from quandle to group colourings
and back without any loss of information.
On the level of knot invariants this implies the following result:
\begin{theorem}[\textsection\ref{sub:LiftingLemma}]
Every quandle colouring number $\F{q}{Q}$ is the specialization
of some knot colouring polynomial $\P{x}{G}$.
\end{theorem}
Quandle cohomology was initially studied in order
to construct invariants in low-dimensional topology:
in \cite{CarterEtAl:1999,CarterEtAl:2003} it was shown
how a $2$-cocycle $\lambda\in Z^2(Q,\Lambda)$
gives rise to a state-sum invariant of knots,
$\CSSI{Q}{\lambda}\colon \knots \to \Z\Lambda$,
which refines the quandle colouring number $\TF{Q}$.
We prove the following result:
\begin{theorem}[\textsection\ref{sub:SS2CP}] \label{Thm:SS2CP}
Every quandle $2$-cocycle state-sum invariant of knots
is the specialization of some knot colouring polynomial.
More precisely, suppose that $Q$ is a connected quandle,
$\Lambda$ is an abelian group, and $\lambda \in Z^2(Q,\Lambda)$
is a $2$-cocycle with associated invariant
$\CSSI{Q}{\lambda}\colon \knots \to \Z{\Lambda}$.
Then there exists a group $G$ with basepoint $x$ and
a $\Z$-linear map $\varphi\colon \Z{G} \to \Z{\Lambda}$
such that $\CSSI{Q}{\lambda} = \varphi\P{x}{G} \cdot |Q|$.
\end{theorem}
This result provides a complete topological interpretation
of quandle $2$-cocycle state-sum invariants in terms of
the knot group and its peripheral system.
Conversely, we prove that state-sum invariants
contain those colouring polynomials $\P{x}{G}$ for which
the longitude group $\Lambda = C(x) \cap G'$ is abelian:
\begin{theorem}[\textsection\ref{sub:CP2SS}] \label{Thm:CP2SS}
Suppose that $G$ is a colouring group with basepoint $x$
such that the subgroup $\Lambda = C(x)\cap G'$ is abelian.
Then the colouring polynomial $\P{x}{G}$ can be presented
as a quandle $2$-cocycle state-sum invariant.
More precisely, the quandle $Q = x^G$ admits
a $2$-cocycle $\lambda\in Z^2(Q,\Lambda)$
such that $\CSSI{Q}{\lambda} = \P{x}{G} \cdot |Q|$.
\end{theorem}
\subsection{How this article is organized}
Section \ref{sec:KnotGroups} recalls the necessary facts
about the knot group and its peripheral system.
It then discusses connected sum and symmetry operations with
respect to knot colouring polynomials and displays some applications.
The main purpose is to give some evidence as to
the scope and the usefulness of these invariants.
Section \ref{sec:Quandles} examines quandle colourings
and
explains how to replace quandle colourings
by group colourings without any loss of information.
The correspondence between quandle extensions and quandle cohomology
is then used to show how quandle $2$-cocycle state-sum invariants
can be seen as specializations of colouring polynomials.
Section \ref{sec:YangBaxter} relates colouring polynomials with
Yang-Baxter invariants. After recalling the framework of linear
braid group representations, we show how colouring polynomials
can be seen as Yang-Baxter deformations of colouring numbers.
\subsection{Acknowledgements}
The author would like to thank the anonymous referee
for his careful reading and numerous helpful comments.
The results of Section \ref{sec:KnotGroups} were part of
the author's Ph.D.\ thesis \cite{Eisermann:Thesis},
which was financially supported by the Deutsche Forschungs\-gemeinschaft
through the Gra\-du\-ierten\-kolleg Mathe\-matik at the University of Bonn.
Sections \ref{sec:Quandles} and \ref{sec:YangBaxter} were elaborated while
the author held a post-doc position at the \'Ecole Normale Sup\'erieure de Lyon,
whose hospitality is gratefully acknowledged.
\section{Knot groups and colouring polynomials} \label{sec:KnotGroups}
This section collects some basic facts about the knot group
and its peripheral system (\textsection\ref{sub:PeripheralSystem})
and their homomorphic images (\textsection\ref{sub:ColouringGroups}).
We explain how connected sum and symmetry operations affect
the knot group system and how this translates to colouring polynomials
(\textsection\ref{sub:KnotGroupSymmetries}).
We then display some examples showing that colouring polynomials
are a useful tool in distinguishing knots where other invariants fail
(\textsection\ref{sub:Examples}).
\subsection{Peripheral system} \label{sub:PeripheralSystem}
We use fairly standard notation, which we recall
from \cite{Eisermann:2003} for convenience.
A \emph{knot} is a smooth embedding $k \colon \S^1\hookrightarrow\S^3$,
considered up to isotopy. This is equivalent to considering
the oriented image $K = k(\S^1)$ in $\S^3$, again up to isotopy.
A \emph{framing} of $k$ is an embedding
$f \colon \S^1 \times \D^2 \hookrightarrow \S^3$ such that $f|_{\S^1{\times}0} = k$.
As basepoint of the space $\S^3 \minus K$ we choose $p = f(1,1)$.
In the fundamental group $\pi_{K} := \pi_1(\S^3\minus K,p)$
we define the \emph{meridian} $m_K = [f|_{1{\times}\S^1}]$
and the \emph{longitude} $l_K = [f|_{\S^1{\times}1}]$.
Up to isotopy the framing is characterized
by the linking numbers $\lk(K,m_K) \in \{\pm1\}$
and $\lk(K,l_K) \in \Z$, and all combinations are realized.
We will exclusively work with the \emph{standard framing},
characterized by the linking numbers $\lk(K,m_K) = +1$ and $\lk(K,l_K)=0$.
Up to isomorphism, the triple $(\pi_{K},m_K,l_K)$ is a knot invariant,
and even a complete invariant: two knots $K$ and $K'$ are isotopic
if and only if there is a group isomorphism $\phi \colon \pi_{K} \to \pi_{K'}$
with $\phi(m_K) = m_{K'}$ and $\phi(l_K) = l_{K'}$.
This is a special case of Waldhausen's theorem on sufficiently large
$3$-manifolds; see \cite[Cor.\,6.5]{Waldhausen:1968} as well as
\cite[\textsection{3C}]{BurdeZieschang:1985}.
Besides closed knots $k\colon \S^1 \hookrightarrow \S^3$
it will be useful to consider long knots (also called $1$-tangles),
i.e.\ smooth embeddings $\ell\colon \R \hookrightarrow \R^3$
such that $\ell(t) = (t,0,0)$ for all parameters $t$
outside of some compact interval. We refer to
\cite{Eisermann:2003} for a detailed discussion
with respect to knot groups and quandles.
\begin{figure}[hbtp]
\centering
\includegraphics[width=0.9\linewidth]{merilong}
\caption{Meridian and longitude of a long knot}
\label{fig:MeriLong}
\end{figure}
\subsection{Colouring groups} \label{sub:ColouringGroups}
Since knot groups are residually finite \cite[Thm.\,3.3]{Thurston:1982},
there are plenty of finite knot group representations.
But which groups do actually occur as homomorphic images of knot groups?
This question was raised by L.P.\,Neuwirth \cite{Neuwirth:1965},
and first solved by F.\,Gonzalez-Acu\~na \cite{Gonzalez:1975}:
\begin{theorem}[\cite{Gonzalez:1975,Johnson:1980}]
A pointed group $(G,x)$ is the homomorphic image of
some knot group $(\pi_{K},m_K)$ if and only if $G$
is finitely generated and $G = \gen{x^G}$.
\qed
\end{theorem}
The condition is necessary, because every knot group $\pi_{K}$
is finitely generated by conjugates of the meridian $m_K$.
(See the Wirtinger presentation, recalled in \textsection\ref{sub:WirtingerQuandles}.)
For a proof of sufficiency we refer to the article of D.\,Johnson \cite{Johnson:1980},
who has found an elegant and ingeniously simple way to construct
a knot $K$ together with an epimorphism $(\pi_{K},m_K) \to (G,x)$.
Here we restrict attention to \emph{finite} groups:
\begin{definition} \label{def:ColouringGroup}
Let $G$ be a finite group and $x\in G$.
The pair $(G,x)$ is called a \emph{colouring group}
if the conjugacy class $x^G$ generates the whole group $G$.
For example, every finite simple group $G$ is a colouring group
with respect to any of its non-trivial elements $x \ne 1$.
\end{definition}
\begin{remark} \label{rem:ConnectedGroups}
Given a finite group $G_0$ and $x \in G_0$,
every homomorphism $(\pi_{K},m_K) \to (G_0,x)$
maps to the subgroup $G_1 := \gen{x^{G_0}}$.
If $G_1$ is strictly smaller than $G_0$,
then we can replace $G_0$ by $G_1$.
Continuing like this, we obtain a descending chain
$G_0 \supset G_1 \supset G_2 \supset \cdots$,
recursively defined by $G_{i+1} = \gen{x^{G_i}}$.
Since $G_0$ is finite, this chain must stabilize,
and we end up with a colouring group $G_n = \gen{x^{G_n}}$.
Hence, we can assume without loss of generality
that $(G,x)$ is a colouring group.
\end{remark}
Given $(G,x)$ let $\Lambda^*$ be the set of longitude images
$\rho(l_K)$, where $\rho$ ranges over all knot group homomorphisms
$\rho \colon (\pi_{K},m_K) \to (G,x)$ and all knots $K$.
Then $\Lambda^*$ is a subgroup of $G$ \cite{JohnsonLivingston:1989}.
Since meridian $m_K \in \pi_{K}$ and longitude $l_K \in \pi_{K}'$ commute,
$\Lambda^*$ is contained in the subgroup $\Lambda = C(x)\cap G'$,
which will play an important r\^ole in subsequent arguments.
D.\,Johnson and C.\,Livingston \cite{JohnsonLivingston:1989}
have worked out a complete characterization of
the subgroup $\Lambda^*$ in terms of homological obstructions.
As an application, consider a colouring group $(G,x)$
that is perfect, i.e.\ $G'=G$, and has cyclic centralizer,
say $C(x) = \gen{x}$. Then \cite{JohnsonLivingston:1989}
affirms that $\Lambda^* = \Lambda = C(x)$. All of our examples
in \textsection\ref{sub:Examples} are of this type.
\subsection{Knot and group symmetries} \label{sub:KnotGroupSymmetries}
The knot group $\pi_{K}$ is obviously independent of orientations.
In order to define the longitude, however,
we have to specify the orientation of $K$,
and the definition of the meridian additionally
depends on the orientation of $\S^3$.
Changing these orientations defines
the following symmetry operations:
\begin{definition}
Let $K\subset\S^3$ be an oriented knot.
The same knot with the opposite orientation of $\S^3$
is the \emph{mirror image} or the \emph{obverse} of $K$,
denoted $K^\obv$. (We can represent this as $K^\obv = \sigma K$,
where $\sigma\colon \S^3 \to \S^3$ is a reflection.)
Reversing the orientation of the knot $K$
yields the \emph{reverse} knot $K^\rev$.
Inverting both orientations yields
the \emph{inverse} knot $K^\inv$.
\end{definition}
Please note that different authors use different terminology,
in particular reversion and inversion are occasionally interchanged.
Here we adopt the notation of J.H.\,Conway \cite{Conway:1969}.
\begin{proposition}
\newcommand{\GS}[1]{\check\pi(#1)}
Let $K$ be an oriented knot with group system $\GS{K} = (\pi_{K},m_K,l_K)$.
Obversion, reversion and inversion affect the group system as follows:
\begin{alignat*}{3}
& \text{obversion:} \quad && \GS{K^\obv} && = (\pi_{K},m_K^{-1},l_K) \\
& \text{reversion:} \quad && \GS{K^\rev} && = (\pi_{K},m_K^{-1},l_K^{-1}) \\
& \text{inversion:} \quad && \GS{K^\inv} && = (\pi_{K},m_K,l_K^{-1})
\end{alignat*}
The fundamental group of the connected sum $K \cs L$
is the amalgamated product $\pi_{K} \ast \pi_{L}$ modulo $m_K=m_L$.
Its meridian is $m_K$ and its longitude is the product $l_K l_L$.
\qed
\end{proposition}
\begin{corollary} \label{cor:Multiplicative}
Every colouring polynomial $\P{x}{G}\colon \knots \to \Z{G}$ is multiplicative, that is,
we have $\P{x}{G}(K\cs L) = \P{x}{G}(K) \cdot \P{x}{G}(L)$ for any two knots $K$ and $L$.
\qed
\end{corollary}
In order to formulate the effect of inversion,
let ${\;}^\inv\colon \Z{G} \to \Z{G}$ be the linear extension
of the inversion map $G \to G$, $g \mapsto g^{-1}$.
\begin{corollary} \label{cor:Inversion}
Every colouring polynomial $\P{x}{G}\colon \knots \to \Z{G}$ is equivariant
under inversion, i.e.\ $\P{x}{G}(K^\inv) = \P{x}{G}(K)^\inv$ for every knot $K$.
In particular, the colouring number $\F{x}{G}(K)$ is invariant under inversion of $K$.
\qed
\end{corollary}
Obversion and reversion of knots can similarly
be translated into symmetries of colouring polynomials,
but to do so we need a specific automorphism of $G$:
\begin{definition}
An automorphism ${\;}^\obv\colon G \to G$ with $x^\obv = x^{-1}$
is called an \emph{obversion} of $(G,x)$.
An anti-automorphism ${\;}^\rev\colon G \to G$ with $x^\rev = x$
is called a \emph{reversion} of $(G,x)$.
\end{definition}
Obviously a group $(G,x)$ possesses a reversion
if and only if it possesses an obversion.
They are in general not unique, because they can
be composed with any automorphism $\alpha \in \Aut(G,x)$,
for example conjugation by an element in $C(x)$.
\begin{remark}
The braid group $\Br{n}$, recalled in \textsection\ref{sub:YangBaxter}
below, has a unique anti-automorphism ${\;}^\rev\colon \Br{n} \to \Br{n}$
fixing the standard generators $\sigma_1,\dots,\sigma_{n-1}$.
Analogously there exists a unique automorphism
${\;}^\obv\colon \Br{n} \to \Br{n}$ mapping each standard generator
$\sigma_i$ to its inverse $\sigma_i^{-1}$.
The exponent sum $\Br{n} \to \Z$ shows that
this cannot be an inner automorphism.
These symmetry operations on braids correspond
to the above symmetry operations on knots:
if a knot $K$ is represented as the closure
of the braid $\beta$ (see \textsection\ref{sub:YangBaxter}),
then the inverse braid $\beta^{-1}$ represents the inverse knot $K^\inv$,
the reverse braid $\beta^\rev$ represents the reverse knot $K^\rev$,
and the obverse braid $\beta^\obv$ represents the obverse knot $K^\obv$.
\end{remark}
Given an obversion and a reversion of $(G,x)$,
their linear extensions to the group ring $\Z{G}$
will also be denoted by ${\;}^\obv\colon \Z{G}\to\Z{G}$
and ${\;}^\rev\colon \Z{G}\to\Z{G}$, respectively.
We can now formulate the equivariance of
the corresponding colouring polynomials:
\begin{corollary} \label{cor:CPEquivariance}
Suppose that $(G,x)$ possesses an obversion
${\;}^\obv$ and a reversion ${\;}^\rev$.
Then the colouring polynomial $\P{x}{G}$ is equivariant
with respect to obversion and reversion,
that is, we have $\P{x}{G}(K^\obv) = \P{x}{G}(K)^\obv$
and $\P{x}{G}(K^\rev) = \P{x}{G}(K)^\rev$ for every knot $K$.
In this case the colouring numbers of
$K$, $K^\inv$, $K^\obv$, and $K^\rev$ are the same.
\qed
\end{corollary}
\begin{example}
Every element $x$ in the symmetric group $\Sym{n}$ is conjugated
to its inverse $x^{-1}$, because both have the same cycle structure.
Any such conjugation defines an obversion $(\Sym{n},x) \to (\Sym{n},x^{-1})$.
This argument also applies to alternating groups: given $x \in \Alt{n}$
we know that $x$ is conjugated to $x^{-1}$ in $\Sym{n}$.
Since $\Alt{n}$ is normal in $\Sym{n}$, this conjugation
restricts to an obversion $(\Alt{n},x) \to (\Alt{n},x^{-1})$.
This need not be an inner automorphism.
\end{example}
On the other hand, some groups do not permit any obversion at all:
\begin{example}
Let $\GF$ be a finite field and let
$G = \GF \rtimes \GF^\times$ be its affine group.
We have $\Aut(G) = \Inn(G) \rtimes \Gal(\GF)$,
where $\Gal(\GF)$ is the Galois group
of $\GF$ over its prime field $\GF[p]$.
If $\GF = \GF[p]$, then every automorphism of $G$ is inner and thus
induces the identity on the abelian quotient $\GF^\times$.
If $p\ge5$, we can choose an element $x=(a,b) \in G$
whose projection to $\GF^\times$ satisfies $b \ne b^{-1}$.
Hence there is no automorphism of $G$ that maps $x$ to $x^{-1}$.
Indeed, searching all groups of small order with GAP \cite{GAP},
we find that the smallest group having this property
is $\GF[5] \rtimes \GF[5]^\times$ of order $20$.
\end{example}
For the sake of completeness we expound the following elementary result:
\begin{proposition}
The affine group $G = \GF \rtimes \GF^\times$
satisfies $\Aut(G) = \Inn(G) \rtimes \Gal(\GF)$.
\end{proposition}
\begin{proof}
The product in $G$ is given by $(a,b)(c,d) = (a+bc,bd)$,
and so $\Gal(\GF)$ can be seen as a subgroup of $\Aut(G)$,
where $\phi \in \Gal(\GF)$ acts as $(a,b) \mapsto (\phi(a),\phi(b))$.
Since $\Inn(G)$ is a normal subgroup of $\Aut(G)$
with $\Inn(G) \cap \Gal(\GF) = \{\id_G\}$, we see that $\Aut(G)$
contains the semi-direct product $\Inn(G) \rtimes \Gal(\GF)$.
It remains to show that every $\alpha \in \Aut(G)$
belongs to $\Inn(G) \rtimes \Gal(\GF)$.
This is trivially true for $\GF = \GF[2]$,
so we will assume that $\GF$ has more than two elements.
It is then easily verified that $G' = \GF \times \{1\}$.
Let $\zeta$ be a generator of the multiplicative group $\GF^\times$.
We have $\alpha(1,1) = (u,1)$ with $u \in \GF^\times$, and
$\alpha(0,\zeta) = (v,\xi)$ with $v \in \GF$, $\xi \in \GF^\times$, $\xi \ne 1$.
Conjugating by $w = ( v (1-\xi)^{-1}, u )$,
we obtain $(u,1)^w = (1,1)$ and $(v,\xi)^w = (0,\xi)$.
In the sequel we can thus assume $u = 1$ and $v = 0$.
This implies $\alpha(0,b) = (0,\phi(b))$
with $\phi \colon \GF^\times \to \GF^\times$,
$\zeta^n \mapsto \xi^n$ for all $n \in \Z$.
Extending this by $\phi(0)=0$ we obtain a bijection $\phi \colon \GF \to \GF$
satisfying $\phi(ab) = \phi(a)\phi(b)$ for all $a,b \in \GF$.
Moreover, we find $\alpha(a,1) = (\phi(a),1)$:
this is clear for $a = 0$, and for $a \ne 0$
we have $(a,1) = (0,a)(1,1)$ and thus
$\alpha(a,1) = (0,\phi(a)) (1,1) = (\phi(a),1)$.
This proves that $\phi(a+b) = \phi(a) + \phi(b)$
for all $a,b \in \GF$, whence $\phi \in \Gal(\GF)$.
We conclude that $\alpha(a,b) = (\phi(a),\phi(b))$, as claimed.
\end{proof}
\subsection{Examples and applications} \label{sub:Examples}
The preceding discussion indicates
that symmetries of the group $(G,x)$
affect the colouring polynomial $\P{x}{G}(K)$
just as well as symmetries of the knot $K$.
We point out several examples:
\begin{example} \label{exm:PSLObv}
Let $p$ be a prime and let $G=\PSL{2}{\GF[p]}$ be equipped with basepoint
$z = \left[\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right]$
of order $p$. Inversion, obversion, and reversion are realized by
\[
\left[\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right]^\inv =
\left[\begin{smallmatrix} d & -b \\ -c & a \end{smallmatrix}\right], \qquad
\left[\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right]^\obv =
\left[\begin{smallmatrix} a & -b \\ -c & d \end{smallmatrix}\right], \qquad
\left[\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right]^\rev =
\left[\begin{smallmatrix} d & b \\ c & a \end{smallmatrix}\right].
\]
We have $C(z) = \gen{z}$. For $p=2$ and $p=3$ one finds
that the longitude group $\Lambda = C(z)\cap G'$ is trivial.
For $p \ge 5$ the group $G$ is perfect (even simple), hence $\Lambda = \gen{z}$.
We conclude that the colouring polynomial $\P{z}{G}$
is insensitive to reversion: we have $\P{z}{G}(K) \in \Z\gen{z}$
and reversion fixes $z$ and therefore all elements in $\Z\gen{z}$.
\end{example}
\begin{example} \label{exm:AltObv}
Consider an alternating group $G = \Alt{n}$ with $n \ge 3$,
and a cycle $x = (123\dots l)$ of maximal length,
that is, $l=n$ for $n$ odd and $l=n-1$ for $n$ even.
As we have pointed out above, a suitable conjugation
in $\Sym{n}$ produces an obversion $(G,x) \to (G,x^{-1})$.
We have $C(x) = \gen{x}$. For $n = 3$ and $n = 4$ one finds that
the longitude group $\Lambda = C(x) \cap G'$ is trivial. For $n\ge5$ the group $G$
is perfect (even simple), hence the longitude group is $\Lambda = \gen{x}$.
Again we conclude that the colouring polynomial $\P{x}{G}$ is insensitive to reversion.
We observe that for $n=3,4,7,8,11,12,\dots$ an obversion
of $(G,x)$ cannot be realized by an inner automorphism:
consider for example $G = \Alt{11}$ and $x = (\mathtt{abcdefghijk})$:
in $\Sym{11}$ the centralizer is $C(x) = \gen{x}$ and consequently every permutation
$\sigma \in \Sym{11}$ with $x^\sigma = x^{-1}$ is of the form
$\sigma = x^k (\mathtt{ak})(\mathtt{bj})(\mathtt{ci})(\mathtt{dh})(\mathtt{eg})$
and thus odd. The same argument shows that for $n=5,6,9,10,\dots$
an obversion of $(G,x)$ can be realized by an inner automorphism.
\end{example}
\begin{example} \label{exm:MathieuObv}
As a more exotic example, let us finally consider the Mathieu group $M_{11}$,
i.e.\ the unique simple group of order $7920 = 2^4{\cdot}3^2{\cdot}5{\cdot}11$,
and the smallest of the sporadic simple groups \cite{ATLAS}.
It can be presented as a subgroup of $\Alt{11}$,
for example as
\[
G = \gen{x,y} \quad\text{with}\quad
x = (\mathtt{abcdefghijk}),\; y=(\mathtt{abcejikdghf}).
\]
This presentation has been obtained from GAP \cite{GAP}
and can easily be verified with any group-theory software
by checking that $G$ is simple of order $7920$.
The Mathieu group $M_{11}$ is particularly interesting for us,
because it does \emph{not} allow an obversion. To see this it suffices
to know that its group of outer automorphisms is trivial \cite{ATLAS},
in other words, every automorphism of $M_{11}$ is realized by conjugation.
In $M_{11}$ the element $x$ is not conjugated to its inverse ---
this is not even possible in $\Alt{11}$ according to the preceding example.
Hence there is no automorphism of $M_{11}$ that maps $x$ to $x^{-1}$.
\end{example}
Applied to colouring polynomials, this means that there is a priori
no restriction on the invariants of a knot and its mirror image.
As a concrete example we consider the Kinoshita-Terasaka knot $K$
and the Conway knot $C$ displayed in Figure \ref{fig:ConwayKnot}.
\begin{figure}[hbtp]
\centering
\includegraphics[height=30mm]{conway}
\caption{The Kinoshita-Terasaka knot and the Conway knot}
\label{fig:ConwayKnot}
\end{figure}
Both knots have trivial Alexander polynomial.
They differ only by rotation of a $2$-tangle,
in other words they are mutants
in the sense of Conway \cite{Conway:1969}.
Therefore neither the Jones, \textsc{Homflypt} nor Kauffman polynomial
can distinguish between $K$ and $C$, see \cite{Lickorish:1997}.
With the help of a suitable colouring polynomial
the distinction is straightforward:
\begin{example} \label{exm:ConwaySL}
R.\,Riley \cite{Riley:1971} has studied knot group homomorphisms
to the simple group $G = \PSL{2}{\GF[7]}$ of order $168$.
Let $z$ be an element of order $7$, say
$z = \left[\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right]$.
Then the associated colouring polynomials are
\begin{alignat*}{3}
& \P{z}{G}(K) & \;=\; & \P{z}{G}(C) & \;=\; & 1 + 7z^5 + 7z^6 , \\
& \P{z}{G}(K^\inv) & \;=\; & \P{z}{G}(C^\inv) & \;=\; & 1 + 7z + 7z^2 .
\end{alignat*}
This shows that both knots are chiral.
By a more detailed analysis of their coverings,
Riley could even show that $K$ and $C$ are distinct.
\end{example}
\begin{example}
To distinguish $K$ and $C$ we give a simple and direct argument
using colouring polynomials. For every element $x\in\PSL{2}{\GF[7]}$ of order $3$,
say $x = \left[\begin{smallmatrix} 0 & 1 \\ -1 & 1 \end{smallmatrix}\right]$,
the associated colouring polynomial distinguishes $K$ and $C$:
\begin{alignat*}{4}
& \P{x}{G}(K) & \;=\; & 1 + 6x \hspace{2cm} &
& \P{x}{G}(C) & \;=\; & 1 + 12x \\
& \P{x}{G}(K^\inv) & \;=\; & 1 + 6x^2 \hspace{2cm} &
& \P{x}{G}(C^\inv) & \;=\; & 1 + 12x^2
\end{alignat*}
Both invariants, $\P{z}{G}$ and $\P{x}{G}$,
show chirality but are insensitive to reversion.
\end{example}
These and the following colouring polynomials were calculated with
the help of an early prototype of the computer program \textsl{KnotGRep},
an ongoing programming project to efficiently construct
the set of knot group homomorphisms to a finite group.
Even though general-purpose software may be less comfortable,
our results can also be obtained from the Wirtinger
presentation (\textsection\ref{sub:WirtingerQuandles})
using GAP \cite{GAP} or similar group-theoretic software.
\begin{example} \label{exm:ConwayAlt}
The alternating group $G=\Alt{7}$ with basepoint $x=(1234567)$ yields
\begin{alignat*}{4}
& \P{x}{G}(K) & \;=\; & 1 + 7x^2 + 28x^5 + 28x^6 \hspace{1cm} &
& \P{x}{G}(C) & \;=\; & 1 + 7x^2 + 7x^3 +21x^5 + 14x^6 \\
& \P{x}{G}(K^\inv) & \;=\; & 1 + 28x + 28x^2 + 7x^5 \hspace{1cm} &
& \P{x}{G}(C^\inv) & \;=\; & 1 + 14x + 21x^2 + 7x^4 + 7x^5
\end{alignat*}
Again this invariant distinguishes $K$ et $C$
and shows their chirality, but is insensitive to reversion,
as explained in Example \ref{exm:AltObv} above.
\end{example}
\begin{example} \label{exm:ConwayMathieu}
More precise information can be obtained using the Mathieu group $M_{11}$,
presented as the permutation group $(G,x)$ in Example \ref{exm:MathieuObv} above.
For the Kinoshita-Terasaka knot $K$ and the Conway knot $C$ one finds:
\begin{alignat*}{4}
& \P{x}{G}(K) & \;=\; & 1 + 11x^3 + 11x^7 \hspace{2cm} &
& \P{x}{G}(C) & \;=\; & 1 + 11x^3 + 11x^7 \\
& \P{x}{G}(K^\inv) & \;=\; & 1 + 11x^4 + 11x^8 \hspace{2cm} &
& \P{x}{G}(C^\inv) & \;=\; & 1 + 11x^4 + 11x^8 \\
& \P{x}{G}(K^\obv) & \;=\; & 1 + 11x^4 + 22x^8 \hspace{2cm} &
& \P{x}{G}(C^\obv) & \;=\; & 1 + 11x^4 + 11x^6 + 11x^8 \\
& \P{x}{G}(K^\rev) & \;=\; & 1 + 22x^3 + 11x^7 \hspace{2cm} &
& \P{x}{G}(C^\rev) & \;=\; & 1 + 11x^3 + 11x^5 + 11x^7
\end{alignat*}
Consequently all eight knots are distinct;
$K$ and $C$ are neither inversible nor obversible nor reversible.
(This example was inspired by G.\,Kuperberg \cite{Kuperberg:1996},
who used the colouring number $\F{x}{G}$ to distinguish
the knot $C$ from its reverse $C^\rev$.)
\end{example}
Usually it is very difficult to detect non-reversibility of knots.
Most invariants fail to do so, including the usual knot polynomials.
In view of the simplicity of our approach, the success
of knot colouring polynomials is remarkable.
We give two further examples:
\begin{example} \label{exm:BretzelMathieu}
The family of bretzel knots $B(p_1,p_2,p_3)$,
parametrized by odd integers $p_1,p_2,p_3$,
is depicted in Figure \ref{fig:Bretzel}a.
According to the classification of bretzel knots
(see \cite{BurdeZieschang:1985}, \textsection 12),
the bretzel knot $B=B(3,5,7)$ is neither
reversible nor obversible nor inversible.
For the Mathieu group $G=M_{11}$ with basepoint $x$
as in Example \ref{exm:ConwayMathieu} we obtain:
\begin{alignat*}{4}
& \P{x}{G}(B) & \;=\; & 1 + 11x \hspace{2cm} &
& \P{x}{G}(B^\obv) & \;=\; & 1 + 11x^7 \\
& \P{x}{G}(B^\inv) & \;=\; & 1 + 11x^{10} \hspace{2cm} &
& \P{x}{G}(B^\rev) & \;=\; & 1 + 11x^4
\end{alignat*}
Again the colouring polynomial shows that the knot $B$
possesses none of the three symmetries.
Historically, bretzel knots were
the first examples of non-reversible knots.
Their non-reversibility was first proven
by H.F.\,Trotter \cite{Trotter:1963} in 1963
by representing the knot group system on a suitable
triangle group acting on the hyperbolic plane.
\end{example}
\begin{figure}[hbtp]
\centering
\includegraphics[height=30mm]{bretzel}
\caption{(a) the bretzel knot $B(p_1,p_2,p_3)$, (b) the knot $8_{17}$}
\label{fig:Bretzel}
\end{figure}
\begin{example} \label{exm:817Mathieu}
Figure \ref{fig:Bretzel}b shows the knot $8_{17}$,
which is the smallest non-reversible knot.
It is a $3$-bridge knot but not a bretzel knot,
and there is no general classification theorem available.
To analyze this example we choose once more
the Mathieu group $M_{11}$ with basepoint $x$ as above.
The knot $8_{17}$ then has colouring polynomial $1+11x^5+11x^6$
whereas the reverse knot has trivial colouring polynomial $1$.
(Here even the colouring number $\F{x}{G}$
suffices to prove that this knot is non-reversible.)
We remark that $8_{17}$ is inversible and that this symmetry
is reflected in the symmetry of its colouring polynomials.
\end{example}
The colouring polynomial $\P{x}{G}(K)$ is,
by definition, an element in the group ring $\Z{G}$,
and it actually lies in the much smaller ring $\Z\Lambda$.
The following symmetry consideration
further narrows down the possible values.
It is included here to explain one of the observations
that come to light in the previous examples,
but it will not be used in the sequel.
\begin{proposition} \label{prop:PrimeMultiple}
Let $(G,x)$ be a colouring group.
If conjugation by $x$ has order $p^k$ for some prime $p$, then
the colouring polynomial satisfies $\P{x}{G}(K) \equiv 1 \pmod{p}$.
\end{proposition}
\begin{proof}
The cyclic subgroup $\gen{x}$ acts on the set
$\Hom(\pi_{K},m_K \,;\, G,x)$ by conjugation.
The only fixed point is the trivial representation
$(\pi(K),m_K) \to (\Z,1) \to (G,x)$. This can be most easily
seen by interpreting group homomorphisms $\rho \colon (\pi_{K},m_K) \to (G,x)$
as colourings $f \colon (D,0) \to (G,x)$ of a knot diagram $D$,
see \textsection\ref{sub:WirtingerQuandles} below.
If $f^x = f$ then all colours of $f$ commute with $x$:
following the diagram from the first to the last arc
we see by induction that all colours are in fact equal to $x$.
Since there is only one component, we conclude
that $f$ is the trivial colouring, corresponding
to the trivial representation.
Every non-trivial representation $\rho$ appears
in an orbit of length $p^\ell$ for some $\ell\ge1$.
Since $\rho(l_K)$ commutes with $x$, all representations
in such an orbit have the same longitude image in $G$.
The sum $\P{x}{G}(K)$ thus begins with $1$ for the trivial representation,
and all other summands can be grouped to multiples of $p$.
\end{proof}
\section{Quandle invariants are specialized colouring polynomials} \label{sec:Quandles}
The Wirtinger presentation allows us to interpret
knot group homomorphisms as colourings of knot diagrams.
Since such colourings involve only conjugation,
they are most naturally treated in the category of quandles,
as introduced by D.\,Joyce \cite{Joyce:1982}.
We recall the basic definitions concerning quandles
and quandle colourings in \textsection\ref{sub:WirtingerQuandles},
and explain in \textsection\ref{sub:LiftingLemma}
how to pass from quandles to groups and back
without any loss of information.
Quandle cohomology was studied in \cite{CarterEtAl:1999,CarterEtAl:2003},
where it was shown how a $2$-cocycle gives rise to
a state-sum invariant of knots in $\S^3$.
We recall this construction in \textsection\ref{sub:CP2SS}
and show that every colouring polynomial $\P{x}{G}$
can be presented as a quandle $2$-cocycle state-sum invariant,
provided that the subgroup $\Lambda=C(x)\cap G'$ is abelian
(Theorem \ref{thm:CP2SS}).
In order to prove the converse, we employ the
cohomological classification of central quandle extensions
established in \cite{Eisermann:2003,CarterEtAl:2003b},
recalled in \textsection\ref{sub:QuandleCoverings} below.
This allows us to prove in \textsection\ref{sub:SS2CP}
that every quandle $2$-cocycle state-sum invariant is the specialization
of a suitable knot colouring polynomial (Theorem \ref{thm:SS2CP}).
\subsection{Wirtinger presentation, quandles, and colourings} \label{sub:WirtingerQuandles}
Our exposition follows \cite{Eisermann:2003},
to which we refer for further details.
We consider a long knot diagram as in Figure \ref{fig:MeriLong}
and number the arcs consecutively from $0$ to $n$.
At the end of arc number $i-1$, we undercross
arc number $\kappa{i}=\kappa(i)$ and continue on arc number $i$.
We denote by $\varepsilon{i}=\varepsilon(i)$ the sign of
this crossing, as depicted in Figure \ref{fig:ColouredCrossing}.
The maps $\kappa\colon \{1,\dots,n\}\to\{0,\dots,n\}$
and $\varepsilon\colon \{1,\dots,n\}\to\{\pm1\}$
are the \emph{Wirtinger code} of the diagram.
\begin{theorem} \label{thm:Wirtinger}
Suppose that a knot $L$ is represented by a long knot diagram
with Wirtinger code $(\kappa,\varepsilon)$ as above.
Then the knot group allows the presentation
\[
\pi_{L} = \langle x_0,x_1,\dots,x_n \,|\, r_1,\dots,r_n \rangle
\quad\text{with relation $r_i$ being}\quad
x_i = x_{\kappa{i}}^{-\varepsilon{i}} \, x_{i-1} \, x_{\kappa{i}}^{\varepsilon{i}} .
\]
As peripheral system we can choose $m_L = x_0$ and
$l_L = \prod_{i=1}^{i=n} \; x_{i-1}^{-\varepsilon{i}} \, x_{\kappa{i}}^{\varepsilon{i}}$.
\qed
\end{theorem}
For a proof see Crowell-Fox \cite[\textsection{VI.3}]{CrowellFox:1963}
or Burde-Zieschang \cite[\textsection{3B}]{BurdeZieschang:1985}.
The Wirtinger presentation works just as well for a closed knot diagram.
Since arcs $0$ and $n$ are then identified, this amounts to adding
the (redundant) relation $x_0=x_n$ to the above presentation.
The group is, of course, the same.
The Wirtinger presentation allows us to interpret
knot group homomorphisms $\pi_{L} \to G$ as colourings.
More precisely, a \emph{$G$-colouring} of the diagram $D$
is a map $f \colon \{0,\dots,n\} \to G$ such that
$f(i) = f(\kappa{i})^{-\varepsilon{i}} f(i-1) f(\kappa{i})^{\varepsilon{i}}$.
In other words, at each coloured crossing as in
Figure \ref{fig:ColouredCrossing} the colours $a$ and $c$
are conjugated via $a^b=c$. Such a colouring is denoted by $f \colon D \to G$.
We denote by $\Col(D;G)$ the set of colourings of $D$ with colours in $G$.
For a long knot diagram $D$, we denote by $\Col(D,0;G,x)$
the subset of colourings that colour arc number $0$ with colour $x$.
The Wirtinger presentation establishes natural bijections
$\Hom(\pi_{K};G) \cong \Col(D;G)$ and $\Hom(\pi_{K},m_K;G,x) \cong \Col(D,0;G,x)$.
\begin{example}
Figure \ref{fig:TrefoilAlt5} shows a colouring of the left-handed trefoil knot
(represented as a long knot) with elements in the alternating group $\Alt{5}$.
Note that all definitions readily extend to closed knot diagrams.
\end{example}
\begin{figure}[hbtp]
\centering
\includegraphics[width=0.8\textwidth]{trefoil}
\caption{$\Alt{5}$-colouring of the left-handed trefoil knot}
\label{fig:TrefoilAlt5}
\end{figure}
The Wirtinger presentation of $\pi_{K}$ involves
only conjugation but not the group multiplication itself.
The underlying algebraic structure can be described as follows:
\begin{definition}
A \emph{quandle} is a set $Q$ with two binary operations
$\ast,\tsa\colon Q\times Q\to Q$ satisfying the following axioms
for all $a,b,c\in Q$:
\begin{enumerate} \addtolength{\itemindent}{\parindent}
\item[(Q1)]
$a\ast a = a$ \hfill (idempotency)
\item[(Q2)]
$(a \ast b) \tsa b = (a \tsa b) \ast b = a$ \hfill (right invertibility)
\item[(Q3)]
$(a\ast b)\ast c = (a\ast c)\ast (b\ast c)$ \hfill (self-distributivity)
\end{enumerate}
\end{definition}
The name ``quandle'' was introduced by D.\,Joyce \cite{Joyce:1982}.
The same notion was studied by S.V.\,Matveev \cite{Matveev:1982}
under the name ``distributive groupoid'', and by L.H.\,Kauffman
\cite{Kauffman:2001} who called it ``crystal''.
Quandle axioms (Q2) and (Q3) are equivalent to saying that for every $b \in Q$
the right translation $\varrho_b \colon a \mapsto a \ast b$ is an automorphism of $Q$.
Such structures were called ``automorphic sets'' by E.\,Brieskorn \cite{Brieskorn:1988}.
The somewhat shorter term \emph{rack} was preferred by R.\,Fenn and
C.P.\,Rourke \cite{FennRourke:1992}. The notion has been generalized
to ``crossed $G$-sets'' by P.J.\,Freyd and D.N.\,Yetter \cite{FreydYetter:1989}.
\begin{definition}
As before, let $D$ be a long knot diagram,
its arcs being numbered by $0,\dots,n$.
A \emph{$Q$-colouring}, denoted $f\colon D\to Q$,
is a map $f\colon \{0,\dots,n\}\to Q$ such that
at each crossing as in Figure \ref{fig:ColouredCrossing}
the three colours $a,b,c$ satisfy the relation $a \ast b=c$.
We denote by $\Col(D;Q)$ the set of $Q$-colourings,
and by $\Col(D,0;Q,q)$ the subset of colourings satisfying $f(0)=q$.
\end{definition}
\begin{figure}[hbtp]
\centering
\includegraphics[height=20mm]{wirting}
\caption{Wirtinger rules for colouring a knot diagram}
\label{fig:ColouredCrossing}
\end{figure}
\begin{proposition}[Joyce \cite{Joyce:1982}]
The quandle axioms ensure that each Reidemeister move $D \leftrightharpoons D'$
induces bijections $\Col(D;Q) \leftrightharpoons \Col(D';Q)$ and
$\Col(D,0;Q,q) \leftrightharpoons \Col(D',0;Q,q)$, respectively.
In particular, if $Q$ is finite, then the colouring numbers $\TF{Q}(D) = |\Col(D;Q)|$
and $\F{q}{Q}(D) = |\Col(D,0;Q,q)|$ are knot invariants.
\qed
\end{proposition}
\subsection{From quandle colourings to group colourings and back} \label{sub:LiftingLemma}
In many respects quandles are close to groups.
For colourings we will now explain how to pass
from quandles to groups and back without any loss of information.
\begin{definition}
A \emph{quandle homomorphism} is a map $\phi\colon Q\to Q'$
that satisfies $\phi(a\ast b) = \phi(a) \ast \phi(b)$,
and hence $\phi(a\tsa b) = \phi(a) \tsa \phi(b)$, for all $a,b\in Q$.
\end{definition}
\begin{definition}
The automorphism group $\Aut(Q)$ consists of
all bijective homomorphisms $\phi\colon Q\to Q$.
We adopt the convention that automorphisms of $Q$ act on the right,
written $a^\phi$, which means that their composition $\phi\psi$
is defined by $a^{(\phi\psi)} = (a^\phi)^\psi$ for all $a\in Q$.
\end{definition}
\begin{definition}
The group $\Inn(Q) = \gen{ \varrho_b \mid b \in Q}$
of \emph{inner automorphisms} is the subgroup of $\Aut(Q)$
generated by all right translations $\varrho_b \colon a \mapsto a \ast b$.
The quandle $Q$ is called \emph{connected}
if the action of $\Inn(Q)$ on $Q$ is transitive.
\end{definition}
In view of the map $\varrho \colon Q \to \Inn(Q)$, $b \mapsto \varrho_b$,
we also write $a^b = a \ast b$ for the operation in a quandle.
Conversely, it will sometimes be convenient to write
$a \ast b = b^{-1} a b$ for the conjugation in a group.
In neither case will there be any danger of confusion.
\begin{definition}
A \emph{representation} of a quandle $Q$
on a group $G$ is a map $\phi\colon Q\to G$ such that
$\phi(a\ast b) = \phi(a) \ast \phi(b)$ for all $a,b\in Q$.
In other words, the following diagram commutes:
\[
\begin{CD}
Q \times Q @>{\phi \times \phi}>> G \times G \\
@V{*}VV @VV{\conj}V \\
Q @>{\phi}>> G
\end{CD}
\]
\end{definition}
For example, the natural map $\varrho\colon Q \to \Aut(Q)$
satisfies $\varrho(a \ast b) = \varrho(a) \ast \varrho(b)$.
We call $\varrho$ the \emph{inner} representation of $Q$.
Moreover it satisfies $\varrho(a^g) = \varrho(a)^g$
for all $a \in Q$ and $g \in \Aut(Q)$.
This is the prototype of an augmentation:
\begin{definition}
Let $\phi \colon Q \to G$ be a representation and let
$\alpha \colon Q \times G \to Q$, $(a,g) \mapsto a^g$,
be a group action. We call the pair $(\phi,\alpha)$
an \emph{augmentation} if $a \ast b = a^{\phi(b)}$ and
$\phi(a^g) = \phi(a)^g$ for all $a,b \in Q$ and $g \in G$.
In other words, the following diagram commutes:
\begin{equation} \label{eq:Augmentation}
\begin{CD}
Q \times Q @>{\id \times \phi}>> Q \times G @>{\phi \times \id}>> G \times G \\
@V{*}VV @VV{\alpha}V @VV{\conj}V \\
Q @>{\id}>> Q @>{\phi}>> G
\end{CD}
\end{equation}
\end{definition}
\begin{remark}
We will usually reinterpret the group action $\alpha$
as a group homomorphism $\bar\alpha \colon G \to \Aut(Q)$,
and denote the augmentation by $Q \lto[\phi] G \lto[\bar\alpha] \Aut(Q)$.
If $G$ is generated by the image $\phi(Q)$, then $\phi$ is equivariant and
the action of $G$ on $Q$ is uniquely determined by the representation $\phi$.
In this case we simply say that $\phi \colon Q \to G$ is an augmentation.
For example, every quandle $Q$ comes equipped with
the inner augmentation $\varrho \colon Q \to \Inn(Q)$.
\end{remark}
Suppose that $Q$ is a quandle and $\phi\colon Q \to G$
is a representation on some group $G$. Obviously every
quandle colouring $\tilde{f}\colon D \to Q$ maps to a
group colouring $f = \phi\tilde{f} \colon D \to G$.
If $\phi$ is an augmentation, then this process can be reversed,
and we can replace quandle colourings by group colourings
without any loss of information:
\begin{lemma} \label{lem:LiftingLemma}
Let $\phi \colon (Q,q) \to (G,x)$ be an augmentation
of the quandle $Q$ with basepoint $q \in Q$
on the group $G$ with basepoint $x \in G$.
If $D$ is a long knot diagram, then every group colouring $f \colon (D,0) \to (G,x)$
can be lifted to a unique quandle colouring $\tilde{f} \colon (D,0) \to (Q,q)$.
In other words, $\phi$ induces a bijection
\[
\phi_* \colon \Col(D,0;Q,q) \lto[\sim] \Col(D,0;G,x) ,
\quad \tilde{f} \mapsto f = \phi\tilde{f} .
\]
Moreover, let $\rho \colon (\pi_{K},m_K) \to (G,x)$ be the knot group representation
associated with $f$. Then the lifted colouring $\tilde{f}$ begins
with $\tilde{f}(0)=q$ and ends with $\tilde{f}(n) = q^{\rho(l_K)}$.
\end{lemma}
\begin{proof}
Every representation $\phi \colon (Q,q) \to (G,x)$ induces a map $\phi_*$
sending each quandle colouring $\tilde{f} \colon (D,0) \to (Q,q)$ to
the associated group colouring $\phi\tilde{f} \colon (D,0) \to (G,x)$.
In general $\phi_*$ is neither injective nor surjective,
lest $\phi$ is an augmentation. In order to define the inverse map
$\psi_*\colon \Col(D,0;G,x) \to \Col(D,0;Q,q)$, we use the action
$\alpha \colon Q \times G \to Q$, which we temporarily denote
by $(a,g) \mapsto a \bullet g$ for better readability.
The crucial ingredient in the proof is
the commutativity of the diagram \eqref{eq:Augmentation}.
Let us first show how the condition $a \ast b = a \bullet \phi(b)$
ensures injectivity of $\phi_*$. Let $D$ be a long knot
diagram with Wirtinger code $(\kappa,\varepsilon)$.
Assume that $\tilde{f},\hat{f} \colon (D,0) \to (Q,q)$
are colourings with $\phi\tilde{f} = \phi\hat{f}$.
By hypothesis we have $\tilde{f}(0) = \hat{f}(0) = q$.
By induction suppose that $\tilde{f}(i-1) = \hat{f}(i-1)$
for some $i \ge 1$. In the case of a positive crossing
($\varepsilon i = +1$) we then obtain
\begin{align*}
\tilde{f}(i)
& = \tilde{f}(i-1) \ast \tilde{f}(\kappa i)
= \tilde{f}(i-1) \bullet \phi\tilde{f}(\kappa i)
\\
& = \hat{f}(i-1) \bullet \phi\hat{f}(\kappa i)
= \hat{f}(i-1) \ast \hat{f}(\kappa i)
= \hat{f}(i) .
\end{align*}
The case of a negative crossing ($\varepsilon i = -1$) is analogous.
We now show how the equivariance condition $\phi(a \bullet g) = \phi(a) \ast g$
of diagram \eqref{eq:Augmentation} ensures surjectivity.
For every colouring $f \colon (D,0) \to (G,x)$, denoted by $i \mapsto x_i$,
the colours $x_0,\dots,x_n$ satisfy $x_i = x_{i-1} \ast x_{\kappa{i}}^{\varepsilon{i}}$.
We define partial longitudes $\ell_0,\dots,\ell_n$ by setting
$
\ell_i := \prod_{j=1}^{i} x_{j-1}^{-\varepsilon{j}} x_{\kappa{j}}^{\varepsilon{j}}.
$
In particular we have $x_0 = x_n = x$ and $x_i = x_0 \ast \ell_i$ for all $i=0,\dots,n$.
By definition, $\ell_n = \rho(l_K)$ is the (total) longitude of the colouring $f$.
We define $\tilde{f} \colon (D,0) \to (Q,q)$ by assigning
the colour $q_i = q \bullet \ell_i$ to arc number $i=0,\dots,n$.
By hypothesis, $\phi \colon Q \to G$ is an equivariant map, whence
\begin{equation}
\label{eq:ProjectionProperty}
\phi(q_i) = \phi(q \bullet \ell_i) = \phi(q) \ast \ell_i = x \ast \ell_i = x_i.
\end{equation}
At each positive crossing we find the
following identity, using axiom (Q1):
\begin{equation}
\label{eq:PositiveCrossing}
q_{i-1} \ast q_{\kappa{i}}
= (q_{i-1} \tsa q_{i-1}) \ast q_{\kappa{i}}
= (((q \bullet \ell_{i-1}^{\phantom{-1}}) \bullet x_{i-1}^{-1}) \bullet x_{\kappa{i}}^{\phantom{-1}}
= q \bullet \ell_i = q_i.
\end{equation}
Analogously at each negative crossing:
\begin{equation}
\label{eq:NegativeCrossing}
q_{i-1} \tsa q_{\kappa{i}}
= (q_{i-1} \ast q_{i-1}) \tsa q_{\kappa{i}}
= (((q \bullet \ell_{i-1}^{\phantom{-1}}) \bullet x_{i-1}^{\phantom{-1}}) \bullet x_{\kappa{i}}^{-1})
= q \bullet \ell_i = q_i.
\end{equation}
We can thus define $\psi_* \colon \Col(D,0;G,x) \to \Col(D,0;Q,q)$
by $f \mapsto \tilde{f}$.
Equation \eqref{eq:ProjectionProperty} shows that $\phi_*\psi_* = \id$,
and Equations \eqref{eq:PositiveCrossing} and \eqref{eq:NegativeCrossing}
imply that $\psi_*\phi_* = \id$.
\end{proof}
\begin{remark}
Obviously, the condition $a \ast b = a^{\phi(b)}$
cannot be dropped because it connects the quandle
operation $\ast$ with the group action $\alpha$.
Likewise, the equivariance condition $\phi(a^g) = \phi(a)^g$
cannot be dropped: as an extreme counter-example, consider
a trivial quandle $Q = \{q\}$ and an arbitrary group $(G,x)$.
We have a unique representation $\phi \colon (Q,q) \to (G,x)$
and a unique group action $\alpha \colon Q \times G \to Q$.
The map $\phi$ is equivariant if and only if $x \in Z(G)$.
In general $\phi_*$ cannot be a bijection, because
the only $(Q,q)$-colouring is the trivial one,
while there may be non-trivial $(G,x)$-colourings.
\end{remark}
The Lifting Lemma has the following analogue for closed knots:
\begin{lemma} \label{lem:ClosedLifting}
Let $\phi \colon (Q,q) \to (G,x)$ be an augmentation
of the quandle $Q$ on the group $G$.
If $D$ is a closed knot diagram,
then $\phi$ induces a bijection between
$\Col(D,0;Q,q)$ and those homomorphisms
$\rho \colon (\pi_{K},m_K) \to (G,x)$
satisfying $q^{\rho(l_K)} = q$.
\qed
\end{lemma}
As an immediate consequence we obtain the following result:
\begin{theorem}
Every quandle colouring number $\F{q}{Q}$ is the specialization
of some knot colouring polynomial $\P{x}{G}$.
\end{theorem}
\begin{proof}
We consider an augmentation $\phi \colon (Q,q) \to (G,x)$
with $G = \gen{\phi(Q)}$, for example the inner augmentation
on $\phi \colon Q \to G = \Inn(Q)$ with basepoint $x = \phi(q)$.
For long knots, Lemma \ref{lem:LiftingLemma} implies $\F{q}{Q} = \F{x}{G}$.
Hence $\F{q}{Q} = \varepsilon\P{x}{G}$, where $\varepsilon\colon \Z{G}\to\Z$
is the augmentation map of the group ring, with $\varepsilon(g) = 1$ for all $g \in G$.
For closed knots we define the linear map
$\varepsilon \colon \Z{G}\to\Z$ by setting $\varepsilon(g) = 1$
if $q^g = q$, and $\varepsilon(g) = 0$ if $q^g \ne q$.
Then Lemma \ref{lem:ClosedLifting} implies that
$\F{q}{Q} = \varepsilon \P{x}{G}$.
\end{proof}
This argument will be generalized in \textsection\ref{sub:SS2CP},
where we show that every quandle $2$-cocycle state-sum invariant
is the specialization of some colouring polynomial.
\subsection{Quandle coverings, extensions, and cohomology} \label{sub:QuandleCoverings}
Following \cite{Eisermann:2003} we recall how quandle colourings
can be used to encode longitudinal information.
To this end we consider a long knot diagram with
meridians $x_0,\dots,x_n$ and partial longitudes $l_0,\dots,l_n$
as defined in the above proof of the lifting lemma.
In particular we have $x_0 = x_n = m_K$ and $x_i = x_0 \ast l_i$
with $l_0=1$ and $l_n=l_K$.
If we colour each arc not only with its meridian $x_i$
but with the pair $(x_i,l_i)$, then at each crossing we find that
\[
x_i = x_{i-1} \ast x_{\kappa{i}}^{\varepsilon{i}} \quad\text{and}\quad
l_i = l_{i-1} x_{i-1}^{-\varepsilon{i}} x_{\kappa{i}}^{\varepsilon{i}} .
\]
This crossing relation can be encoded in a quandle as follows.
\begin{lemma}[\cite{Eisermann:2003}] \label{lem:CoveringQuandle}
Let $G$ be a group that is generated by a conjugacy class $Q=x^G$.
Then $Q$ is a connected quandle with respect to conjugation
$a \ast b = b^{-1} a b$ and its inverse $a \tsa b = b a b^{-1}$.
Let $G'$ be the commutator subgroup and define
\[
\tilde{Q} = \tilde{Q}(G,x) := \{\; (a,g)\in G\times G' \;|\; a=x^g \;\}.
\]
The set $\tilde{Q}$ becomes a connected quandle when equipped with the operations
\[
(a,g) \ast (b,h) = ( a \ast b, ga^{-1}b ) \quad\text{and}\quad
(a,g) \tsa (b,h) = ( a \tsa b, gab^{-1} ).
\]
Moreover, the projection $p\colon \tilde{Q}\to Q$ given by
$p(a,g)=a$ is a surjective quandle homomorphism.
It becomes an equivariant map when we let $G'$ act on $Q$
by conjugation and on $\tilde{Q}$ by $(a,g)^b = (a^b,gb)$.
In both cases $G'$ acts transitively and
as a group of inner automorphisms.
\qed
\end{lemma}
The construction of the quandle $\tilde{Q}(G,x)$
has been tailor-made to capture longitude information.
Considered purely algebraically, it is a covering
in the following sense:
\begin{definition} \label{def:Covering}
A surjective quandle homomorphism $p\colon \tilde{Q}\to Q$
is called a \emph{covering} if $p(\tilde{x}) = p(\tilde{y})$
implies $\tilde{a}\ast\tilde{x} = \tilde{a}\ast\tilde{y}$
for all $\tilde{a},\tilde{x},\tilde{y}\in\tilde{Q}$.
In other words, the inner representation
$\tilde{Q}\to\Inn(\tilde{Q})$ factors through $p$.
This property allows us to define an action of $Q$ on $\tilde{Q}$ by setting
$\tilde{a}\ast x := \tilde{a}\ast\tilde{x}$ with $\tilde{x}\in p^{-1}(x)$.
\end{definition}
In the construction of Lemma \ref{lem:CoveringQuandle},
the projection $p\colon \tilde{Q}\to Q$ is a covering map.
Moreover, covering transformations are given by the left action of
$\Lambda = C(x)\cap G'$ defined by $\lambda\cdot(a,g) = (a,\lambda g)$.
This action satisfies the following axioms:
\begin{enumerate} \addtolength{\itemindent}{\parindent}
\item[(E1)]
$(\lambda\tilde{x}) \ast \tilde{y} = \lambda (\tilde{x}\ast\tilde{y})$
and $\tilde{x} \ast (\lambda\tilde{y}) = \tilde{x}\ast\tilde{y}$
for all $\tilde{x},\tilde{y}\in\tilde{Q}$ and $\lambda\in\Lambda$.
\item[(E2)]
$\Lambda$ acts freely and transitively on each fibre $p^{-1}(x)$.
\end{enumerate}
Axiom (E1) is equivalent to saying that $\Lambda$ acts by automorphisms
and the left action of $\Lambda$ commutes with the right action of $\Inn(\tilde{Q})$.
We denote such an action by $\Lambda \curvearrowright \tilde{Q}$.
In this situation the quotient $Q := \Lambda\backslash\tilde{Q}$
carries a unique quandle structure that turns the projection
$p \colon \tilde{Q} \to Q$ into a quandle covering.
\begin{definition} \label{def:CentrExt}
An \emph{extension} $E\colon \Lambda \curvearrowright \tilde{Q} \to Q$
consists of a surjective quandle homomorphism $\tilde{Q}\to Q$
and a group action $\Lambda \curvearrowright \tilde{Q}$
satisfying axioms (E1) and (E2).
We call $E$ a \emph{central extension} if $\Lambda$ is abelian.
\end{definition}
Quandle extensions are an analogue of group extensions,
and central quandle extensions come as close
as possible to imitating central group extensions.
Analogous to the case of groups,
central quandle extensions are classified by
the second cohomology group $H^2(Q,\Lambda)$,
see \cite{Eisermann:2003,CarterEtAl:2003b}. More precisely:
\begin{theorem}[\cite{Eisermann:2003}] \label{thm:CentrExtClass}
Let $Q$ be a quandle, let $\Lambda$ be an abelian group,
and let $\Ext(Q,\Lambda)$ be the set of equivalence classes
of central extensions of $Q$ by $\Lambda$.
Given a central extension $E\colon \Lambda \curvearrowright \tilde{Q} \to Q$,
each section $s\colon Q\to\tilde{Q}$ defines a $2$-cocycle $\lambda\colon Q \times Q \to \Lambda$.
If $s'$ is another section, then the associated $2$-cocycle
$\lambda'$ differs from $\lambda$ by a $2$-coboundary.
The map $E \mapsto [\lambda]$ so constructed induces
a natural bijection $\Ext(Q,\Lambda) \cong H^2(Q,\Lambda)$.
\qed
\end{theorem}
The relevant portion of the cochain complex
$C^1 \lto[\delta^1] C^2 \lto[\delta^2] C^3$
is formed by $n$-cochains $\lambda\colon Q^n \to \Lambda$ satisfying
$\lambda(a_1,\dots,a_n)=0$ whenever $a_i=a_{i+1}$ for some index $i$,
and the first two coboundary operators
$\delta^1(\mu)(a,b) = \mu(a) - \mu(a^b)$ and
$\delta^2(\lambda)(a,b,c) = \lambda(a,c) - \lambda(a,b) + \lambda(a^c,b^c) - \lambda(a^b,c)$.
For details, see \cite{CarterEtAl:1999,CarterEtAl:2003,Eisermann:2003}
\subsection{From colouring polynomials to state-sum invariants} \label{sub:CP2SS}
Let $D$ be a knot diagram and let $f$ be a colouring of $D$ with colours in $Q$.
Suppose that $\Lambda$ is an abelian group, written multiplicatively,
and that $\lambda \colon Q^2\to \Lambda$ is a $2$-cocycle.
For each coloured crossing $p$ as in Figure \ref{fig:ColouredCrossing},
we define its \emph{weight} by $\weight{\lambda}{p} := \lambda(a,b)^\varepsilon$.
The total weight of the colouring $f$ is the product
$\weight{\lambda}{f} := \prod_p \weight{\lambda}{p}$ over all crossings $p$.
The \emph{state-sum} of the diagram $D$ is defined to be
$\CSSI{Q}{\lambda}(D) := \sum_f \; \weight{\lambda}{f}$,
where the sum in $\Z\Lambda$ is taken over all colourings $f\colon D \to Q$.
We recall the following results:
\begin{lemma}[\cite{CarterEtAl:1999,CarterEtAl:2003}]
The state-sum $\CSSI{Q}{\lambda}$ is invariant under Reidemeister moves
and thus defines a knot invariant $\CSSI{Q}{\lambda}\colon \knots \to \Z\Lambda$.
\qed
\end{lemma}
\begin{lemma}[{\cite[Prop.\,4.5]{CarterEtAl:2003}}] \label{lem:GaugeInvariance}
If the colouring $f\colon D\to Q$ is closed, that is $f(0) = f(n)$,
then the weight $\weight{\lambda}{f}$ is invariant under addition of coboundaries.
As a consequence, the state sum $\CSSI{Q}{\lambda}$ of a closed knot
depends only on the cohomology class $[\lambda]$.
\qed
\end{lemma}
\begin{lemma}[{cf.\ \cite[Lem.\,32]{Eisermann:2005}}] \label{lem:ConjugationInvariance}
The diagonal action of $\Inn(Q)$ on $Q^n$
induces the trivial action on $H^*(Q,\Lambda)$.
As a consequence, for each closed colouring $f \colon D \to Q$
and every inner automorphism $g\in\Inn(Q)$ we have
$\weight{\lambda}{f^g} = \weight{^{g\!}\lambda}{f} = \weight{\lambda}{f}$.
\qed
\end{lemma}
This last result is well-known in group cohomology,
cf.\ Brown \cite[Prop.\,II.6.2]{Brown:1994}.
It seems to be folklore in quandle cohomology,
but I could not find a written account of it.
The necessary argument is provided by \cite[Lem.\,32]{Eisermann:2005}
in the more general setting of Yang-Baxter cohomology,
which immediately translates to Lemma \ref{lem:ConjugationInvariance}
\begin{lemma}[{\cite[Lem.\,50]{Eisermann:2003}}] \label{lem:LiftingMonodromy}
Let $p \colon (\tilde{Q},\tilde{q}) \to (Q,q)$ be a central quandle extension.
Given a long knot diagram $D$, every colouring $f \colon (D,0) \to (Q,q)$
uniquely lifts to a colouring $\tilde{f} \colon (D,0) \to (\tilde{Q},\tilde{q})$
such that $f = p \tilde{f}$.
If $f$ is closed then $\tilde{f}(n) = \weight{\lambda}{f} \cdot \tilde{q}$,
where $[\lambda] \in H^2(Q,\Lambda)$ is the cohomology class
associated with the extension $p$.
\qed
\end{lemma}
These preliminaries being in place, we can now
prove that every colouring polynomial $\P{x}{G}$
can be presented as a $2$-cocycle state-sum invariant,
provided that the subgroup $\Lambda=C(x)\cap G'$ is abelian.
\begin{theorem} \label{thm:CP2SS}
Suppose that $G$ is a colouring group with basepoint $x$
such that the subgroup $\Lambda = C(x)\cap G'$ is abelian.
Then the colouring polynomial $\P{x}{G}$ can be presented
as a quandle $2$-cocycle state-sum invariant.
More precisely, the quandle $Q = x^G$ admits
a $2$-cocycle $\lambda\in Z^2(Q,\Lambda)$
such that $\CSSI{Q}{\lambda} = \P{x}{G}\cdot|Q|$.
\end{theorem}
\begin{proof}
Let $Q=x^G$ be the conjugacy class of $x$ in the group $G$,
and let $\tilde{Q} = \tilde{Q}(G,x)$ be the covering quandle
constructed in Lemma \ref{lem:CoveringQuandle}. Since $\Lambda$ is abelian,
we obtain a central extension $\Lambda\curvearrowright\tilde{Q}\to Q$.
Let $[\lambda]\in H^2(Q,\Lambda)$ be the associated cohomology class.
As basepoints we choose $q=x$ in $Q$ and $\tilde{q} = (x,1)$ in $\tilde{Q}$.
Let $D$ be a long diagram of some knot $K$,
let $f \colon (D,0) \to (Q,q)$ be a colouring,
let $\rho \colon (\pi_{K},m_K) \to (G,x)$ be
the corresponding knot group homomorphism, and let
$\tilde{f} \colon (D,0) \to (\tilde{Q},\tilde{q})$ be the lifting of $f$.
On the one hand we have $\tilde{f}(n) = (x,\weight{\lambda}{f})$
from Lemma \ref{lem:LiftingMonodromy}.
On the other hand we have $\tilde{f}(n) = (x,\rho(l_K))$
from the Wirtinger presentation.
Thus $\rho(l_K) = \weight{\lambda}{f}$, and summing over all
colourings $f \colon (D,0) \to (Q,q)$ yields $\P{x}{G}(K)$.
To obtain the state-sum $\CSSI{Q}{\lambda}$
we have to sum over all colourings $f \colon D\to Q$.
We have $\Col(D,Q) = \bigcup_{q'\in Q} \Col(D,0;Q,q')$.
Since $Q$ is connected, for each $q'\in Q$ there exists $g\in G$
such that $q^g = q'$. Hence $f \mapsto f^g$ establishes
a bijection between $\Col(D,0;Q,q)$ and $\Col(D,0;Q,q')$.
By Lemma \ref{lem:ConjugationInvariance} we have
$\weight{\lambda}{f} = \weight{\lambda}{f^g}$.
Thus the state-sum over all colourings $f \colon (D,0) \to (Q,q')$
again yields $\P{x}{G}$. We conclude that
$\CSSI{Q}{\lambda}(K) = \P{x}{G}(K) \cdot |Q|$.
\end{proof}
\subsection{From state-sum invariants to colouring polynomials} \label{sub:SS2CP}
Theorem \ref{thm:CP2SS} has the following converse,
which allows us to express quandle $2$-cocycle state-sum
invariants by knot colouring polynomials.
\begin{theorem} \label{thm:SS2CP}
Every quandle $2$-cocycle state-sum invariant of knots
is the specialization of some knot colouring polynomial.
More precisely, suppose that $Q$ is a connected quandle,
$\Lambda$ is an abelian group, and $\lambda \in Z^2(Q,\Lambda)$
is a $2$-cocycle with associated invariant
$\CSSI{Q}{\lambda}\colon \knots\to\Z{\Lambda}$.
Then there exists a group $G$ with basepoint $x$ and
a linear map $\varphi\colon \Z{G}\to\Z{\Lambda}$ such that
the colouring polynomial $\P{x}{G}\colon \knots\to\Z{G}$
satisfies $\CSSI{Q}{\lambda} = \varphi\P{x}{G} \cdot |Q|$.
\end{theorem}
\begin{proof}
We first construct a suitable group $(G,x)$ together with
a linear map $\varphi\colon \Z{G}\to\Z{\Lambda}$.
Let $\Lambda \curvearrowright \tilde{Q} \lto[p] Q$ be
the central extension associated with the $2$-cocycle $\lambda$,
as explained in Theorem \ref{thm:CentrExtClass}.
We put $G:= \Inn(\tilde{Q})$. The inner representation
$\tilde\varrho\colon \tilde{Q} \to G$ defines an augmented quandle
in the sense of \textsection\ref{sub:LiftingLemma}.
We choose a basepoint $\tilde{q}\in\tilde{Q}$
and set $x:=\tilde{\varrho}(\tilde{q})$.
We choose $q = p(\tilde{q})$ as basepoint of $Q$.
Let $s\colon Q\to\tilde{Q}$ be a section that realizes the $2$-cocycle $\lambda$.
Since $p$ is a covering, we obtain a representation
$\varrho\colon Q \to G$ by $\varrho = \tilde\varrho\compose s$.
Conversely, we can define an action of $G$ on $Q$
by setting $a^g = p( s(a)^g )$. This turns
the representation $\varrho \colon Q \to G$ into an augmentation
and $p \colon \tilde{Q} \to Q$ into an equivariant map.
Our notation being in place, we can now define the linear map
\[
\varphi\colon \Z{G}\to\Z\Lambda \quad\text{by setting}\quad
\varphi(g) = \begin{cases} 0 & \text{if } q^g \ne q , \\
\ell & \text{if $q^g = q$ and $\ell \in \Lambda$
such that $\tilde{q}^g = \ell\cdot\tilde{q}$.}
\end{cases}
\]
It remains to prove that $\CSSI{Q}{\lambda} = \varphi\P{x}{G}\cdot|Q|$.
Let $K$ be a knot represented by a long knot diagram $D$.
The Lifting Lemma \ref{lem:ClosedLifting} grants us a bijection
between closed colourings $f\colon (D,0) \to (Q,q)$ and those homomorphisms
$\rho \colon (\pi_{K},m_K) \to (G,x)$ that satisfy $q^{\rho(l_K)} = q$.
Regarding the covering $\tilde{Q}$, we claim that
$\tilde{q}^{\rho(l_K)} = \weight{\lambda}{f} \cdot \tilde{q}$.
To see this, let $\tilde{f} \colon (D,0) \to (\tilde{Q},\tilde{q})$ be the lifting of $f$.
On the one hand we can apply the Lifting Lemma \ref{lem:ClosedLifting}
to the augmentation $\tilde{Q}\to G$,
which yields $\tilde{f}(n) = \tilde{q}^{\rho(l_K)}$.
On the other hand we can apply Lemma \ref{lem:LiftingMonodromy},
which yields $\tilde{f}(n) = \weight{\lambda}{f} \cdot \tilde{q}$.
The map $\varphi$ thus specializes
the knot colouring polynomial $\P{x}{G}(K)$
to the state-sum $\sum_f \weight{\lambda}{f}$, at least
if we restrict the summation to colourings $f \colon (D,0) \to (Q,q)$.
Since $Q$ is connected, any other basepoint $q'$ yields
the same state-sum by Lemma \ref{lem:ConjugationInvariance}.
Summing over all $q'\in Q$, we thus obtain
$\CSSI{Q}{\lambda} = \varphi\P{x}{G}\cdot|Q|$, as claimed.
\end{proof}
\section{Colouring polynomials are Yang-Baxter invariants} \label{sec:YangBaxter}
P.J.\,Freyd and D.N.\,Yetter \cite{FreydYetter:1989} have shown that
the colouring number $\F{x}{G}\colon \knots\to\Z$ is a Yang-Baxter invariant.
This means that $\F{x}{G}$ can be obtained as the trace of a linear
braid group representation arising from a suitable Yang-Baxter operator $c$.
In this section we will show that the colouring polynomial
$\P{x}{G}\colon \knots\to\Z\Lambda$ is also a Yang-Baxter invariant,
obtained from a certain Yang-Baxter operator $\tilde{c}$ defined below.
It will follow from our construction that $\tilde{c}$
is a deformation of $c$ over $\Z\Lambda$.
\subsection{Braid group representations and Yang-Baxter invariants} \label{sub:YangBaxter}
The notion of Yang-Baxter invariants rests on two
classical theorems: Artin's presentation of the braid groups
and the Alexander-Markov theorem, which we will now recall.
Our exposition closely follows \cite{Eisermann:2005}
and is included here for convenience.
\begin{theorem}[E.\,Artin \cite{Artin:1947}]
The braid group on $n$ strands can be presented as
\[
\Br{n} = \left\langle\; \sigma_1,\dots,\sigma_{n{-}1} \;\Big|\;
\begin{matrix}
\hfill{} \sigma_i \sigma_j = \sigma_j \sigma_i \hfill{} \quad \text{for } |i-j|\ge2 \\
\sigma_i \sigma_j \sigma_i = \sigma_j \sigma_i \sigma_j \quad \text{for } |i-j|=1
\end{matrix} \;\right\rangle,
\]
where the braid $\sigma_i$ performs a positive half-twist of the strands $i$ and $i+1$.
\end{theorem}
\begin{definition}
Let $\K$ be a commutative ring and $V$ a $\K$-module.
A \emph{Yang-Baxter operator} (or \emph{R-matrix})
is an automorphism $c\colon V \tensor V \to V \tensor V$
that satisfies the \emph{Yang-Baxter equation},
also called \emph{braid relation}:
\[
(c \tensor\id_V)(\id_V\tensor c)(c \tensor\id_V) =
(\id_V\tensor c)(c \tensor\id_V)(\id_V\tensor c)
\qquad\text{in}\quad \Aut_\K(V^{\tensor3}).
\]
Here and in the sequel tensor products are
taken over $\K$ if no other ring is indicated.
\end{definition}
\begin{corollary}
Given a Yang-Baxter operator $c$ and some integer $n\ge2$, we can define
automorphisms $c_i\colon V^{\tensor n} \to V^{\tensor n}$ by setting
\[
c_i = \id_V^{\tensor(i{-}1)} \,\tensor\, c \,\tensor\, \id_V^{\tensor(n{-}i{-}1)}
\quad\text{for } i=1,\dots,n-1.
\]
The Artin presentation implies that there exists,
for each $n$, a unique braid group representation
$\rho_c^n\colon \Br{n} \to \Aut_\K(V^{\tensor n})$
defined by $\rho_c^n(\sigma_i)=c_i$.
\qed
\end{corollary}
We orient braids from right to left as in Figure \ref{fig:BraidClosure}.
Braid groups will act on the left, so that composition of braids
corresponds to the usual composition of maps.
The passage from braids to links is granted by the closure map
$[\,]\colon \bigcup_n \Br{n}\to\links$ defined as follows:
for each braid $\beta$ we define its \emph{closure} $[\beta]$ to be
the link in $\S^3$ obtained by identifying opposite endpoints,
as indicated in Figure \ref{fig:BraidClosure}.
\begin{figure}[hbtp]
\centering
\includegraphics[width=100mm]{closure}
\caption{A braid $\beta$ and its closure $[\beta]$}
\label{fig:BraidClosure}
\end{figure}
\begin{theorem}[Alexander-Markov, see \cite{Birman:1974}]
Every link can be represented as the closure of some braid.
Two braids represent the same link if and only if
one can be transformed into the other by a finite sequence
of the following \emph{Markov moves}:
\begin{enumerate} \addtolength{\itemindent}{\parindent}
\item[(M1)]
Pass from $\beta\in\Br{n}$ to
$\beta\sigma_n^{\pm1}\in\Br{n+1}$, or vice versa.
\hfill \emph{(Stabilization)}
\item[(M2)]
Pass from $\beta\in\Br{n}$ to
$\alpha^{-1}\beta\alpha$ with $\alpha\in\Br{n}$.
\hfill \emph{(Conjugation)}
\end{enumerate}
\end{theorem}
Constructing a link invariant $F\colon \links\to\K$ is thus equivalent
to constructing a map $F\colon \bigcup_n \Br{n}\to\K$ that is
invariant under Markov moves. The most natural approach is
to consider traces of linear braid group representations:
invariance under conjugation is automatic, so we only have
to require invariance under stabilization:
\begin{definition}
Suppose that $V$ is a free $\K$-module with finite basis.
Let $c \colon V \tensor V \to V \tensor V$ be a Yang-Baxter
operator. An automorphism $m \colon V \to V$ is called
\emph{Markov operator} for $c$ if it satisfies
\begin{enumerate} \addtolength{\itemindent}{\parindent}
\item[(m1)]
\makebox[35mm][l]{the trace condition}
$ \tr_2 ( \; (m \tensor m) \compose c^{\pm1} \; ) = m$ \qquad and
\item[(m2)]
\makebox[35mm][l]{commutativity}
$c \compose (m \tensor m) = (m \tensor m) \compose c$.
\end{enumerate}
\end{definition}
Here the partial trace $\tr_2 \colon \End(V \tensor V) \to \End(V)$
is defined as follows. Let $(v_1,\dots,v_n)$ be a basis of $V$ over $\K$.
Every $f \in \End(V \tensor V)$ uniquely corresponds to a matrix $\smash{f_{ij}^{k\ell}}$
such that $f( v_i \tensor v_j ) = \sum_{k,\ell} \smash{f_{ij}^{k\ell}} \; v_k \tensor v_\ell$.
We can then define $g = \tr_2(f) \in \End(V)$, $g(v_i) = \sum_k \smash{g_i^k} \, v_k$,
by the matrix $\smash{g_i^k} = \sum_j \smash{f_{ij}^{kj}}$.
(See Kassel \cite[\textsection II.3]{Kassel:1995}.)
\begin{corollary}
Given a Yang-Baxter operator $c$ with Markov operator $m$,
we define a family of maps $F_n\colon \Br{n}\to\K$ by
$F_n(\beta) = \tr( m^{\tensor n} \compose \rho_c^n(\beta) )$.
Then the induced map $F\colon \bigcup_n \Br{n}\to\K$
is invariant under both Markov moves and
thus defines a link invariant $F\colon \links\to\K$.
\qed
\end{corollary}
The proof of this corollary is straight-forward:
the trace condition (m1) implies invariance under stabilization (M1),
and commutativity (m2) implies invariance under conjugation (M2).
Much more intricate is the question how to actually
\emph{find} such a Yang-Baxter-Markov operator $(c,m)$.
Attempts to construct solutions in a systematic way have led
to the theory of quantum groups \cite{Drinfeld:1987}.
For details we refer to the concise introduction
\cite{KasselRossoTuraev:1997} or the textbook \cite{Kassel:1995}.
\begin{remark}
For some Yang-Baxter operators $c$ there
does not exist any Markov operator $m$ at all.
If it exists, $m$ is in general not the identity,
as in the case of the Jones polynomial
or other quantum invariants.
The Yang-Baxter operators derived
from knot diagram colourings below
are very special in that they allow
the Markov operator $m=\id$, which is
equivalent to saying that $\tr_2(c^{\pm1}) = \id$.
\end{remark}
\subsection{Colouring polynomials of long knots} \label{sub:YBlong}
Before we consider colouring polynomials,
let us first recall how colouring numbers
can be obtained from a suitable Yang-Baxter operator.
The following result is due to Freyd and Yetter,
see \cite{FreydYetter:1989}, Prop.\ 4.2.5 and
the remark following its proof.
\begin{theorem}[\cite{FreydYetter:1989}] \label{thm:CN2YB}
Let $Q$ be a quandle and let $\K{Q}$ be the free $\K$-module with basis $Q$.
The quandle structure of $Q$ can be linearly extended to a Yang-Baxter operator
\[
c_Q\colon \K{Q}\tensor\K{Q} \to \K{Q}\tensor\K{Q} \quad\text{with}\quad
a\tensor b \mapsto b \tensor (a\ast b) \quad\text{for all}\quad a,b\in Q.
\]
Axiom (Q2) ensures that $c_Q$ is an automorphism,
while Axiom (Q3) implies the Yang-Baxter equation.
If $Q$ is finite, then (Q1) ensures that $\tr_2(c_Q^{\pm1})=\id$.
In this case the corresponding Yang-Baxter invariant
$\TF{Q} = \tr\compose\rho_Q$ coincides with the number
of $Q$-colourings (defined in \textsection\ref{sub:WirtingerQuandles})
followed by the ring homomorphism $\Z \to \K$.
\qed
\end{theorem}
As an example consider a finite group $G$ with basepoint $x$.
The Yang-Baxter operator constructed from the quandle $Q=x^G$
then leads to the colouring number $\TF{Q} = \F{x}{G}\cdot|Q|$.
We will now move from colouring numbers to colouring polynomials:
consider the quandle extension $\Lambda\curvearrowright\tilde{Q}\to Q$
as defined in \textsection\ref{sub:QuandleCoverings},
where the quandle $Q=Q(G,x)$ is covered by $\tilde{Q}=\tilde{Q}(G,x)$,
and the deck transformation group is $\Lambda = C(x) \cap G'$.
As before, we linearly extend the quandle structure of $\tilde{Q}$
to a Yang-Baxter operator $c_{\tilde{Q}}$, and denote the associated
linear braid group representation by $\rho_{\tilde{Q}}$.
We will, however, not take the total trace
as before, but rather use the partial trace
$\tr'\colon \End_\K(\K\tilde{Q}^{\tensor n}) \to \End_\K(\K\tilde{Q})$,
contracting the tensor factors $2,\dots,n$.
\begin{theorem} \label{thm:YBlong}
Let $(G,x)$ be a finite group such that
the conjugacy class $Q=x^G$ generates $G$.
Let $\tilde{Q} = \tilde{Q}(G,x)$ be the covering quandle
and let $\rho_{\tilde{Q}}$ be the associated braid group representation.
Suppose that the knot $K$ is represented by a braid $\beta$.
Then the partial trace $\tr'( \rho_{\tilde{Q}}(\beta) )\colon \K\tilde{Q} \to \K\tilde{Q}$
is given by multiplication with $\P{x}{G}(K)$.
\end{theorem}
Note that the free left action of $\Lambda$ on $\tilde{Q}$
turns $\K\tilde{Q}$ into a free left module over $\K\Lambda$.
In particular, multiplication by $\P{x}{G}(K)$ is a $\K$-linear endomorphism.
If $\K$ is of characteristic $0$, then
the endomorphism $\tr'(\rho_{\tilde{Q}}(\beta))$
uniquely determines $\P{x}{G}(K)$.
\begin{proof}
We use the obvious bases $\tilde{Q}$ for $\K\tilde{Q}$
and $\tilde{Q}^n$ for $\K\tilde{Q}^{\tensor{n}}$.
Each endomorphism $f\colon \K\tilde{Q}^{\tensor n} \to \K\tilde{Q}^{\tensor n}$
is then represented by a matrix $M^{p_1 p_2\dots p_n}_{q_1 q_2 \dots q_n}$,
indexed by elements $p_i$ and $q_j$ in the basis $\tilde{Q}$.
The partial trace $\tr'(f)\colon \K\tilde{Q} \to \K\tilde{Q}$
is given by the matrix $T^{p_1}_{q_1} = \sum M^{p_1 p_2\dots p_n}_{q_1 p_2 \dots p_n}$,
where the sum is taken over all repeated indices $p_2,\dots,p_n$.
By construction, each elementary braid $\sigma_i$
acts as a permutation on the basis $\tilde{Q}^n$,
thus each braid $\beta\in\Br{n}$ is represented
by a permutation matrix with respect to this basis.
We interpret this action as colouring
the braid $\beta$ with elements of $\tilde{Q}$:
we colour the right ends of the braid
with $v = p_1\tensor\cdots\tensor p_n$.
Moving from right to left, at each crossing the new arc
is coloured according to the Wirtinger rule
as depicted in Figure \ref{fig:ColouredCrossing}.
We thus arrive at the left ends of the braid being coloured
with $\rho(\beta) v = q_1\tensor\cdots\tensor q_n$.
We conclude that colourings of the braid $\beta$ that satisfy
the trace conditions $p_2 = q_2, \dots, p_n = q_n$
are in natural bijection with colourings
of the corresponding long knot $K$.
We now turn to the remaining indices $p_1$ and $q_1$.
Let us first consider the special case $p_1 = (x,1)$
and $q_1=(y,\lambda)$. From the preceding argument we see that
$T^{p_1}_{q_1}$
equals the number of $\tilde{Q}$-colourings of the long knot $K$
that start with $(x,1)$ and end with $(y,\lambda)$.
According to Lemma \ref{lem:LiftingLemma},
such colourings exist only for $y=x$ and $\lambda\in\Lambda$,
hence we have $q_1 = \lambda \cdot p_1$.
We conclude that $T^{p_1}_{q_1}$ equals the number
of representations $(\pi_{K},m_K,l_K) \to (G,x,\lambda)$.
In total we get $\tr'(\rho(\beta))\,(p_1) = \P{x}{G}(K) \cdot p_1$.
The preceding construction is equivariant under the right-action
of the group $G'$ on the covering quandle $\tilde{Q}$.
According to Lemma \ref{lem:CoveringQuandle} this action
is transitive: for every $p \in \tilde{Q}$ there exists
$g \in G'$ and $p = p_1^g$, so we conclude that
$\tr'(\rho(\beta))\,(p) = \P{x}{G}(K) \cdot p$.
This means that the endomorphism
$\tr'(\rho(\beta))\colon \K\tilde{Q} \to \K\tilde{Q}$
is given by multiplication with $\P{x}{G}(K)$.
\end{proof}
\begin{remark}
The partial trace $\tr' \colon \End_\K(\K\tilde{Q}^{\tensor n})
\to \End_\K(\K\tilde{Q})$ corresponds to closing the strands $2,\dots,n$
of the braid $\beta$, but leaving the first strand open:
the object thus represented is a long knot.
The natural setting for such constructions is the category of
tangles and its linear representations \cite{Kassel:1995}.
The previous theorem then says that the long knot $K$
is represented by the endomorphism $\K\tilde{Q} \to \K\tilde{Q}$
that is given by multiplication with $\P{x}{G}(K)$.
If we used the complete trace $\tr \colon \End_\K(\K\tilde{Q}^{\tensor n})
\to \K$ instead, then we would obtain a different invariant
$\TF{\tilde{Q}} = \tr\compose\rho_{\tilde{Q}}$.
By the preceding arguments, $\TF{\tilde{Q}}(K)$ equals $|\tilde{Q}|$
times the number of representations $(\pi_{K},m_K,l_K) \to (G,x,1)$,
which corresponds to the coefficient of the unit element
in the colouring polynomial $\P{x}{G}(K)$.
\end{remark}
\subsection{Colouring polynomials of closed knots} \label{sub:YBclosed}
We will now show how the colouring polynomial $\P{x}{G}$ of closed knots
can be obtained as the trace of a suitable Yang-Baxter representation.
To this end we will modify the construction of the preceding paragraph
in order to replace the partial trace $\tr'$ by the complete trace $\tr$.
We proceed as follows: the quandle $Q=x^G$ admits
an extension $\Lambda \curvearrowright \tilde{Q} \to Q$
as defined in \textsection\ref{sub:QuandleCoverings}.
The quandle structure of $\tilde{Q}$ linearly extends
to a Yang-Baxter operator $c_{\tilde{Q}}$ on $\K\tilde{Q}$.
The free $\Lambda$-action on $\tilde{Q}$ turns $\K\tilde{Q}$
into a free module over $\A = \K\Lambda$.
If $\Lambda$ is abelian, we can pass to an $\A$-linear operator
\[
\tilde{c}_Q\colon \K\tilde{Q}\tensor[\A]\K\tilde{Q} \to \K\tilde{Q}\tensor[\A]\K\tilde{Q}
\quad\text{with}\quad \tilde{a} \tensor \tilde{b}
\mapsto \tilde{b} \tensor ( \tilde{a}\ast\tilde{b} )
\quad\text{for all}\quad \tilde{a}, \tilde{b} \in \tilde{Q}.
\]
The difference between $c_{\tilde{Q}}$ and $\tilde{c}_Q$
is that the tensor product is now taken over $\A$,
which means that everything is bilinear
with respect to multiplication by $\lambda\in\Lambda$.
In the following theorem and its proof
all tensor products are to be taken over the ring $\A$,
but for notational simplicity we will write
$\tensor$ for $\tensor[\A]$.
\begin{theorem} \label{thm:YBclosed}
If $(G,x)$ is a colouring group such that $\Lambda = C(x)\cap G'$ is abelian,
then the colouring polynomial $\P{x}{G}\colon \knots\to\Z\Lambda$ is a Yang-Baxter invariant.
More precisely, the preceding construction yields a Yang-Baxter-Markov operator
$(\tilde{c}_Q,\id)$ over the ring $\A = \K\Lambda$, and the associated
knot invariant satisfies $\tilde{F}_Q = \varphi\P{x}{G}\cdot|Q|$
where $\varphi \colon \Z\Lambda \to \K\Lambda$ is the natural
ring homomorphism defined by $\varphi(\lambda)=\lambda$
for all $\lambda \in \Lambda$.
\end{theorem}
If $\K$ is of characteristic $0$, then $\tilde{F}_Q$ is
equivalent to the knot colouring polynomial $\P{x}{G}$.
If $\K$ is of finite characteristic, then we may lose
some information and $\tilde{F}_Q$ is usually weaker than $\P{x}{G}$.
In the worst case $|Q|$ vanishes in $\K$ and $\tilde{F}_Q$ becomes trivial.
\begin{proof}
It is a routine calculation to prove that
$\tilde{c}_Q$ is a Yang-Baxter operator over $\A$:
as before, axiom (Q2) implies that $\tilde{c}_Q$ is an automorphism,
while axiom (Q3) ensures that $\tilde{c}_Q$ satisfies the Yang-Baxter equation.
Axiom (Q1) implies the trace condition $\tr_2(\tilde{c}_Q^{\pm1})=\id$,
hence $(\tilde{c}_Q,\id)$ is a Yang-Baxter-Markov operator.
We thus obtain a linear braid group representation
$\tilde\rho^n_Q\colon \Br{n} \to \Aut_\A(\K\tilde{Q}^{\tensor n})$,
whose character $\smash{\tilde{F}_Q} = \tr\compose\tilde{\rho}_Q$
is Markov invariant and induces a link invariant $\tilde{F}_Q\colon \links\to\A$.
Restricted to knots we claim that $\tilde{F}_Q = \P{x}{G}\cdot|Q|$.
The proof of the theorem parallels the proof of Theorem \ref{thm:YBlong},
but requires some extra care.
To represent $\tilde{c}_Q$ by a matrix, we have to choose a basis
of $\K\tilde{Q}$ over $\A$. Let $s\colon Q\to\tilde{Q}$ be a section
to the central extension $\Lambda\curvearrowright\tilde{Q}\to Q$.
Then $B=s(Q)$ is a basis of $\K\tilde{Q}$ as an $\A$-module.
For the basepoint $x$ we can assume $s(x) = (x,1)$,
but otherwise there are no canonical choices.
In general, $s$ will not (and cannot) be a homomorphism of quandles,
but we have $s(a) \ast s(b) = \lambda(a,b) \cdot s(a\ast b)$
with a certain $2$-cocycle $\lambda\colon Q\times Q\to \Lambda$
that measures the deviation of $s$ from being a homomorphism.
Just as $c_Q$ is represented by a permutation matrix, we see that
$\tilde{c}_Q$ is represented by the same matrix except that the $1$'s
are replaced with the elements $\lambda(a,b) \in \Lambda$.
This is usually called a \emph{monomial matrix}
or \emph{generalized permutation matrix}.
Since $\K\tilde{Q}$ is a free $\A$-module with finite basis $B = s(Q)$,
the tensor product $\K\tilde{Q}^{\tensor n}$ is also free and has finite basis $B^n$.
The trace $\tr\compose\tilde\rho(\beta)$ is calculated
as the sum $\sum_{v\in B^n} \langle \tilde\rho(\beta)v | v \rangle$.
Note that $\tilde\rho(\beta)$ is again a monomial matrix in the sense
that each row and each column has exactly one non-zero entry.
Hence a vector $v\in B^n$ contributes to the trace sum if and only if
$\tilde\rho(\beta) v = \lambda(v) v$ with some $\lambda(v)\in\Lambda$.
It remains to characterize eigenvectors and identify their eigenvalues.
Given a braid $\beta\in\Br{n}$ we can interpret the action
of $\tilde\rho(\beta)$ as colouring the braid $\beta$:
we colour the right ends of the braid with a basis vector $v\in B^n$,
\[
v = (a_1,g_1) \tensor (a_2,g_2) \tensor \dots \tensor (a_n,g_n).
\]
Moving from right to left, at each crossing
the new arc is coloured according to the Wirtinger rule
as depicted in Figure \ref{fig:ColouredCrossing}.
We thus arrive at the left ends of the braid, being coloured with
\[
\tilde\rho(\beta) v = (b_1,h_1) \tensor (b_2,h_2) \tensor \cdots \tensor (b_n,h_n).
\]
Since the tensor product is defined over $\A$,
we have $\tilde\rho(\beta) v = \lambda(v) v$
if and only if $a_1=b_1, a_2=b_2, \dots, a_n=b_n$.
Hence each eigenvector $v\in B^n$ naturally corresponds
to a $Q$-colouring of the closed braid $K=[\beta]$.
In order to identify the eigenvalue $\lambda(v)$,
we will further assume that $(a_1,g_1) = (x,1)$,
where $x$ is the basepoint of $G$.
Such an eigenvector will be called \emph{normalized}.
Using the tensor product-structure over $\A = \K\Lambda$, we obtain
\[
\tilde\rho(\beta) v
= (x,\lambda) \tensor (a_2,g_2) \tensor \cdots \tensor (a_n,g_n)
= \lambda(v) v
\]
as in the proof of Theorem \ref{thm:YBlong}.
We conclude that each normalized eigenvector $v \in B^n$
with $\tilde\rho(\beta) v = \lambda(v) v$ corresponds to
a $\tilde{Q}$-colouring of the long knot,
where the first arc is coloured by $(x,1)$
and the last arc is coloured by $(x,\lambda)$.
This means that the eigenvalue $\lambda(v)$
is the associated colouring longitude.
We finally show that $\tilde{F}_Q = \P{x}{G}\cdot|Q|$ by calculating
the trace $\sum_{v\in B^n} \langle \tilde\rho(\beta)v | v \rangle$.
Normalized eigenvectors $v\in \{(x,1)\} \times B^{n-1}$ with
$\tilde\rho(\beta) v = \lambda(v) v$ correspond to colourings
$\rho \colon (\pi_{K},m_k) \to (G,x)$ with $\rho(l_K) = \lambda(v)$.
Summing over these vectors only, we thus obtain the
colouring polynomial $\P{x}{G}(K)$.
To calculate the total sum we use again the fact that
the right-action of $G'$ on $\tilde{Q}$ is transitive.
Hence for every $q\in Q$ there exists $g\in G'$ such that $s(q)^g = (x,1)$.
The action of $g$ induces a bijection between the set of basis vectors
$\{s(q)\} \times B^{n-1}$ and $\{(x,1)\} \times B^{n-1}$.
Since the preceding trace calculation is $G'$-invariant, each vector
$v \in \{s(q)\} \times B^{n-1}$ contributes $\P{x}{G}(K)$ to the trace.
In total we obtain $\tilde{F}_Q = \P{x}{G}\cdot|Q|$, as claimed.
\end{proof}
\subsection{Concluding remarks} \label{sub:Conclusion}
It follows from our construction that $\tilde{c}_Q$
is a deformation of the Yang-Baxter operator $c_Q$.
More precisely we have $\tilde{c}_Q(a\tensor b) = \lambda(a,b) \cdot c_Q(a,b)$
for all $a,b\in Q$ with a suitable map $\lambda \colon Q \times Q \to \Lambda$.
Our construction via quandle coverings and central extensions
provides a geometric interpretation in terms of meridian-longitude information.
This interpretation carries through all steps of our construction,
which finally allows us to interpret the resulting Yang-Baxter
invariant as a colouring polynomial.
Conversely, it is natural to consider the ansatz
$\tilde{c}_Q(a\tensor b) = \lambda(a,b) \cdot c_Q(a,b)$ and to ask
which $\lambda$ turn $\tilde{c}_Q$ into a Yang-Baxter operator.
This idea can, though in a restricted form, already
be found in \cite[Thm.\,4.2.6]{FreydYetter:1989}.
A direct calculation shows that $\tilde{c}_Q$ is
a Yang-Baxter operator if and only if $\lambda$
is a $2$-cocycle in the sense of quandle cohomology.
Moreover, two such deformations will be equivalent
if the cocycles differ by a coboundary.
This observation has been worked out by M.\,Gra\~na \cite{Grana:2002},
who independently proved that quandle $2$-cocycle
state-sum invariants are Yang-Baxter invariants.
\bibliographystyle{amsplain}
| {
"timestamp": "2007-07-26T13:05:51",
"yymm": "0707",
"arxiv_id": "0707.3895",
"language": "en",
"url": "https://arxiv.org/abs/0707.3895",
"abstract": "This article introduces a natural extension of colouring numbers of knots, called colouring polynomials, and studies their relationship to Yang-Baxter invariants and quandle 2-cocycle invariants.For a knot K in the 3-sphere let \\pi_K be the fundamental group of the knot complement, and let (m_K,l_K) be a meridian-longitude pair in \\pi_K. Given a finite group G and an element x in G, we consider the set of representations \\rho from \\pi_K to G that map the meridian m_K to x, and define the colouring polynomial P(K) as the sum over all longitude images \\rho(l_K). The resulting invariant maps knots to the group ring Z[G]. It is multiplicative with respect to connected sum and equivariant with respect to symmetry operations of knots. Examples are given to show that colouring polynomials distinguish knots for which other invariants fail, in particular they can distinguish knots from their mutants, obverses, inverses, or reverses.We prove that every quandle 2-cocycle state-sum invariant of knots is a specialization of some knot colouring polynomial. This provides a complete topological interpretation of these invariants in terms of the knot group and its peripheral system. Furthermore, we show that P can be presented as a Yang-Baxter invariant, i.e. as the trace of some linear braid group representation. This entails in particular that Yang-Baxter invariants can detect non-inversible and non-reversible knots.",
"subjects": "Geometric Topology (math.GT)",
"title": "Knot colouring polynomials",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9888419664501421,
"lm_q2_score": 0.7154239897159439,
"lm_q1q2_score": 0.7074412648363202
} |
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